Physiology For Engineers

Biosystems & Biorobotics
Michael Chappell
Stephen Payne
Physiology
for Engineers
Applying Engineering Methods to
Physiological Systems
Volume 13
Series editor
Eugenio Guglielmelli, Campus Bio-Medico University of Rome, Rome, Italy
e-mail: e.guglielmelli@unicampus.it
Editorial Board
Dino Accoto, Campus Bio-Medico University of Rome, Rome, Italy
Sunil Agrawal, University of Delaware, Newark, DE, USA
Fabio Babiloni, Sapienza University of Rome, Rome, Italy
Jose M. Carmena, University of California, Berkeley, CA, USA
Maria Chiara Carrozza, Scuola Superiore Sant’Anna, Pisa, Italy
Paolo Dario, Scuola Superiore Sant’Anna, Pisa, Italy
Arturo Forner-Cordero, University of Sao Paolo, São Paulo, Brazil
Masakatsu G. Fujie, Waseda University, Tokyo, Japan
Nicolas Garcia, Miguel Hernández University of Elche, Elche, Spain
Neville Hogan, Massachusetts Institute of Technology, Cambridge, MA, USA
Hermano Igo Krebs, Massachusetts Institute of Technology, Cambridge, MA, USA
Dirk Lefeber, Universiteit Brussel, Brussels, Belgium
Rui Loureiro, Middlesex University, London, UK
Marko Munih, University of Ljubljana, Ljubljana, Slovenia
Paolo M. Rossini, University Cattolica del Sacro Cuore, Rome, Italy
Atsuo Takanishi, Waseda University, Tokyo, Japan
Russell H. Taylor, The Johns Hopkins University, Baltimore, MA, USA
David A. Weitz, Harvard University, Cambridge, MA, USA
Loredana Zollo, Campus Bio-Medico University of Rome, Rome, Italy
Aims & Scope
Biosystems & Biorobotics publishes the latest research developments in three main areas:
1) understanding biological systems from a bioengineering point of view, i.e. the study of
biosystems by exploiting engineering methods and tools to unveil their functioning principles
and unrivalled performance; 2) design and development of biologically inspired machines
and systems to be used for different purposes and in a variety of application contexts. The
series welcomes contributions on novel design approaches, methods and tools as well as case
studies on specific bioinspired systems; 3) design and developments of nano-, micro-,
macrodevices and systems for biomedical applications, i.e. technologies that can improve
modern healthcare and welfare by enabling novel solutions for prevention, diagnosis,
surgery, prosthetics, rehabilitation and independent living.
On one side, the series focuses on recent methods and technologies which allow multiscale,
multi-physics, high-resolution analysis and modeling of biological systems. A special
emphasis on this side is given to the use of mechatronic and robotic systems as a tool for basic
research in biology. On the other side, the series authoritatively reports on current theoretical
and experimental challenges and developments related to the “biomechatronic” design of novel
biorobotic machines. A special emphasis on this side is given to human-machine interaction
and interfacing, and also to the ethical and social implications of this emerging research area, as
key challenges for the acceptability and sustainability of biorobotics technology.
The main target of the series are engineers interested in biology and medicine, and
specifically bioengineers and bioroboticists. Volume published in the series comprise
monographs, edited volumes, lecture notes, as well as selected conference proceedings and
PhD theses. The series also publishes books purposely devoted to support education in
bioengineering, biomedical engineering, biomechatronics and biorobotics at graduate and
post-graduate levels.
About the Cover
The cover of the book series Biosystems & Biorobotics features a robotic hand prosthesis.
This looks like a natural hand and is ready to be implanted on a human amputee to help them
recover their physical capabilities. This picture was chosen to represent a variety of concepts
and disciplines: from the understanding of biological systems to biomechatronics,
bioinspiration and biomimetics; and from the concept of human-robot and human-machine
interaction to the use of robots and, more generally, of engineering techniques for biological
research and in healthcare. The picture also points to the social impact of bioengineering
research and to its potential for improving human health and the quality of life of all
individuals, including those with special needs. The picture was taken during the
LIFEHAND experimental trials run at Università Campus Bio-Medico of Rome (Italy) in
2008. The LIFEHAND project tested the ability of an amputee patient to control the
Cyberhand, a robotic prosthesis developed at Scuola Superiore Sant’Anna in Pisa (Italy),
using the tf-LIFE electrodes developed at the Fraunhofer Institute for Biomedical
Engineering (IBMT, Germany), which were implanted in the patient’s arm. The implanted
tf-LIFE electrodes were shown to enable bidirectional communication (from brain to hand
and vice versa) between the brain and the Cyberhand. As a result, the patient was able to
control complex movements of the prosthesis, while receiving sensory feedback in the form
of direct neurostimulation. For more information please visit http://www.biorobotics.it or
contact the Series Editor.
More information about this series at http://www.springer.com/series/10421

Michael Chappell Stephen Payne
•
Physiology for Engineers

Applying Engineering Methods
to Physiological Systems
123
Michael Chappell Stephen Payne
Department of Engineering Science Department of Engineering Science
University of Oxford University of Oxford
Oxford Oxford
UK UK
ISSN 2195-3562 ISSN 2195-3570 (electronic)

ISBN 978-3-319-26195-9 ISBN 978-3-319-26197-3 (eBook)
DOI 10.1007/978-3-319-26197-3
Library of Congress Control Number: 2015955884
Springer Cham Heidelberg New York Dordrecht London

© Springer International Publishing Switzerland 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this
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Printed on acid-free paper
Springer International Publishing AG Switzerland is part of Springer Science+Business Media

(www.springer.com)
To Angela Chappell BSc (Hons) aka Mum
—Michael Chappell
To Alan
—Stephen Payne
Preface
As its name indicates, this short book is intended to give students in any branch of
engineering an introduction to human physiology and how it can be approached by
those with an engineering background. It will also provide an introduction for
students in other physical science subjects to the approaches that can be adopted in
understanding and modelling human physiology. We hope that it will provide a
starting point for both students and nonstudents to explore what is fascinating about
human physiology (which simply means the study of nature).
This book was written particularly with those wanting to enter the field of
biomedical engineering in mind. As biomedical engineers ourselves, we often get
asked what biomedical engineering actually is and what its purpose is. Neither of us
set out to train as biomedical engineers (we did our final year undergraduate pro-
jects on land mines and jet engines), but we both ended up in this area because it
opened up so many interesting problems to work on. If engineering is the “appli-
ance of science”1, then biomedical engineering is the appliance of engineering to
medicine. The whole point is to help to solve problems that will one day improve
health care.
Let us give you an example, directly related to our research.
How can we best decide what treatment to give someone coming into a hospital
with a stroke? Doctors will do brain imaging, mostly likely a CT scan, or in some
hospitals, an MRI scan. How does the doctor then decide what best to do? Stroke is
essentially a plumbing problem and comes in two forms:
Blockage—an ischaemic stroke arises when a vessel feeding part of the brain
gets blocked.
Leakage—an haemorrhagic stroke arises when blood is leaking into the brain
from a broken blood vessel.
Both forms of stroke need urgent attention and the solutions seem simple:
remove a blockage or stop a leak.
1
Our thanks to Zanussi.
vii
viii Preface
However, there are lots of complications to this. For example, in an ischaemic

stroke, should the doctor operate and try to pull the clot out or should he/she try to
dissolve the clot by injecting a blood thinning drug? Often the person is very elderly
with many other illnesses and so an operation is not a good idea; likewise, the blood
thinner may result in a blood vessel bursting and a haemorrhage (leak) elsewhere in
the brain.
The doctor has to make a rapid treatment decision and does so based on a lot of
experience. What we can do as biomedical engineers in this context is to help in the
decision-making process by providing (1) the best and most useful information
from the brain imaging and (2) the best predictions of outcome for various treat-
ment decisions. By doing this and providing the doctor with the most useful
information to make a well-informed decision, biomedical engineers can help to get
the best outcomes for these patients. This will not only help to save lives, but also
help to reduce the levels of disability that stroke patients have to live with when
they leave the hospital. Even a small improvement in the outcome of a stroke
patient can make the difference between living at home and living in a home, or
between independent and dependent living.
The tools we have to do this are both physical (imaging devices) and mathe-
matical (systems of equations), things that engineers excel in. This needs to be
matched with an understanding of the problem, and this requires the knowledge of
not only how the body works, but also how our tools apply to that physiology.
Hence, this book is an attempt to guide the reader in bridging what appears to be a
divide between physical and biomedical sciences.
There are a number of things to note about the book:
1. It is introductory: deliberately, we are covering a wide range of material at a
high level. There are many other very good textbooks that will take you into
much greater detail, but the purpose here is simply to give you an introduction to
physiology from an engineering viewpoint. We also do not attempt to cover all
of physiology, rather we have chosen what we hope are a selection of illustrative
and interesting examples.
2. It is quantitative: the focus is very much on turning the underlying physiology
into mathematics so that a system, whether it is a single cell or the whole brain,
can be modelled and interpreted using engineering techniques.
3. It is concerned with clinical measurements: since we can only know about what
we can measure, throughout the book we have detailed the most common
clinical measurement techniques that can be used to gain information about the
topics presented here.
At the end of the book, we hope that you will have gained some insight into
human physiology and how we can express this mathematically and measure it,
using core engineering techniques. You will then hopefully also be better equipped
to apply the same principles and techniques to other aspects of physiology that we
have not covered here.
The book starts with the cell, which is the fundamental unit of the body. We will
examine its structure, its function and how it operates: in particular we will examine
Preface ix
the generation of the action potential and how this is transmitted between cells. We
will then gradually move to larger scales and look at various systems in the body,
primarily the cardiovascular, respiratory and nervous systems.
We have included exercises, but instead of including them all at the end of each
chapter, we have deliberately inserted them in the text at the most appropriate point.
Many of the examples require you to explore a result that has been stated or to work
through a similar example. It is not necessary to do the examples as you read
through the book (and they are clearly highlighted), but we hope that they will be
helpful in reinforcing the ideas presented, and in particular giving you experience in
applying engineering techniques to physiological problems. There are brief solu-
tions given at the end of the book.
This book does draw on a very wide range of engineering knowledge, which we
appreciate that you will not all have. For some of you, phasor analysis will be a
mystery; for others, elasticity theory will be incomprehensible. We have tried to
keep the level of knowledge required to a minimum, but since this is not a course in
basic engineering, there may be some parts that you will not understand. We hope
that at least you will get some insight into what is going on, even if you do not
follow all of the equations and mathematics in every chapter.
As academics, we have taught a similar course for nearly ten years to under-
graduate and graduate students at the University of Oxford. We have, like all
academics, learnt a great deal ourselves over this time, and we take this opportunity
to thank the students who have attended our course in its various different incar-
nations. In all of this, we have been able to draw on our experience teaching our
students, to the book’s great benefit; however, any errors that remain are naturally
entirely our fault.
Contents
1 Cell Structure and Biochemical Reactions . . . . . . . . . . . . . . . . . . 1

1.1 Cell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Cell Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 ATP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 DNA/RNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Reaction Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Mass Action Kinetics . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Enzyme Cooperativity . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Enzyme Inhibition. . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Cellular Homeostasis and Membrane Potential. . . . . . . . . . . . . . . 19
2.1 Membrane Structure and Composition . . . . . . . . . . . . . . . . . . 19
2.2 Osmotic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Conservation of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Equilibrium Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 A Simple Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Ion Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Membrane Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 The Action Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Na+/K+ Action Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Ca2+ Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Hodgkin-Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xi
xii Contents
4 Cellular Transport and Communication. . . . . . . . . . . . . . . . . . . . 43

4.1 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Passive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 Carrier-Mediated Transport. . . . . . . . . . . . . . . . . . . . 45
4.1.3 Active Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Cellular Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Synapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Electrical Synapses . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Chemical Synapses . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Action Potential Propagation . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 ADME Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Compartmental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 One Compartment Model . . . . . . . . . . . . . . . . . . . . . 56
5.2.2 Absorption Compartment . . . . . . . . . . . . . . . . . . . . . 59
5.2.3 Peripheral Compartment. . . . . . . . . . . . . . . . . . . . . . 61
5.2.4 Multi Compartment Models . . . . . . . . . . . . . . . . . . . 61
5.2.5 Non-linear Models . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Tracer Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Tissue Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Stress-Strain Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.1 Linear Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.2 Non-linear Material . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Coupled Cell-Tissue Model . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Coupled Fluid-Tissue Model . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 Coupled Blood Vessel-Tissue Model . . . . . . . . . . . . . . . . . . . 75
6.5.1 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5.2 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Cardiovascular System I: The Heart . . . . . . . . . . . . . . . . . . . . . . 81
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Structure and Operation of the Heart . . . . . . . . . . . . . . . . . . . 83
7.3 Measurement of Cardiac Output . . . . . . . . . . . . . . . . . . . . . . 85
Contents xiii
7.4 Electrical Activity of the Heart . . . . . . . . . . . . . . . . . . . . . . . 86

7.4.1 The Action Potential . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4.2 Pacemaker Potential . . . . . . . . . . . . . . . . . . . . . . . . 88
7.4.3 Cardiac Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.4.4 Introduction to Electrocardiography . . . . . . . . . . . . . . 91
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 Cardiovascular System II: The Vasculature . . . . . . . . . . . . . . . . . 97
8.1 Anatomy of the Vascular System . . . . . . . . . . . . . . . . . . . . . 97
8.2 Blood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Haemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.4 Blood Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4.1 Long-Term Measurement Techniques . . . . . . . . . . . . 108
8.4.2 Short-Term Measurement Techniques . . . . . . . . . . . . 109
8.5 Measurement of Blood Supply . . . . . . . . . . . . . . . . . . . . . . . 109
8.5.1 Blood Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.5.2 Perfusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.6 Control of Blood Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.6.1 Individual Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.6.2 Individual Body Organs . . . . . . . . . . . . . . . . . . . . . . 112
8.6.3 Whole Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 The Respiratory System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.1 The Lungs and Pulmonary Circulation . . . . . . . . . . . . . . . . . . 117
9.1.1 Breathing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.1.2 Respiration Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.2 Gas Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.1 Inert Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.2 Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.2.3 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2.4 Tissue Gas Delivery . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2.5 Blood Oxygenation . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.2.6 Control of Acid-Base Balance. . . . . . . . . . . . . . . . . . 127
9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10 The Central Nervous System . . . . . ....... . . . . . . . . . . . . . . . . 129
10.1 Neurons . . . . . . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . 129
10.2 Autonomic Nervous System . . ....... . . . . . . . . . . . . . . . . 131
10.2.1 Autonomic Control of the Heart . . . . . . . . . . . . . . . . 131
10.2.2 Cardiac Efferents . . . ....... . . . . . . . . . . . . . . . . 132
10.2.3 Cardiac Afferents . . . ....... . . . . . . . . . . . . . . . . 133
xiv Contents
10.3 Somatic Nervous System . . . . . . . . . . . . . . . . . . ......... 134

10.3.1 Temporal and Spatial Summation
of Synaptic Potentials . . . . . . . . . . . . . . . . . . . . . . . 135
10.3.2 Excitatory and Inhibitory Synapses . . . . . . . . . . . . . . 136
10.3.3 The Brain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.3.4 Function: Introduction to EEG and FMRI . . . . . . . . . 138
10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Chapter 1
Cell Structure and Biochemical Reactions
1.1 Cell Structure
We start by examining the cell, since it is the fundamental unit of living matter.
There are brain cells, heart cells, liver cells and so on, in every part of the body.
Each of these cells has a specific function and purpose. Despite this all cells have
some characteristics in common, as shown schematically in Fig. 1.1. There are four
main components. Cells have an outer layer, called the membrane, which acts as
the boundary between the inside and outside of the cell. We will examine this in
more detail in the next chapter.
Inside the cell, there is a nucleus that contains the cell’s DNA, i.e. the genetic
code that determines what the cell does, and many small structures that carry out the
operations for the cell, termed organelles. Unsurprisingly there are lots of these,
each with particular roles. They include ribosomes, lysosomes and mitochondria,
where many of the reactions that produce energy take place. The rest of the cell is
occupied by a gel-like fluid, called the cytosol, that contains ions and other sub-
stances and that surrounds the other internal elements. The cytosol and the orga-
nelles together are termed the cytoplasm. We will examine a number of these
elements, primarily the membrane and the cytoplasm, in more detail later on.
More formally, we classify a cell as the smallest independently viable unit of a
living organism. It is therefore self-contained and self-maintaining. The smallest
organism is made up of just one cell, whereas the largest are made up of billions of
cells. All cells can reproduce by cell division and metabolise, which means that
they take in and process raw materials and release the by-products of metabolism.
Cells also respond to both external and internal stimuli, for example temperature
and pH.
© Springer International Publishing Switzerland 2016 1

M. Chappell and S. Payne, Physiology for Engineers,
Biosystems & Biorobotics 13, DOI 10.1007/978-3-319-26197-3_1
2 1 Cell Structure and Biochemical Reactions
Fig. 1.1 Structure of the cell (this figure is taken, without changes, from OpenStax College under
license: http://creativecommons.org/licenses/by/3.0/)
1.2 Cell Chemicals
Before examining the cell in detail, we will look at the different types of chemicals
found in the body and their roles, such that when we consider the functions of the
cell, it will be easier to understand what it is doing. Note that we will only look at a
few of the most important substances found within a cell.
The important chemicals within the body can be divided into two categories:
inorganic and organic. The difference is very simply that organic compounds
always contain carbon, whereas most inorganic compounds do not contain carbon.
The main inorganic substances are:
1. Water, which acts as a solvent, a biochemical reactant, a regulator of body
temperature and a lubricant;
2. Electrolytes, which balance osmotic pressure and biochemical reactants;
3. Acids and bases, which act to balance pH.
The primary organic substances are:
1. Carbohydrates;
2. Lipids;
3. Proteins;
4. Nucleic acids (including DNA and RNA);
5. Adenosine triphosphate (ATP).
1.2 Cell Chemicals 3
Carbohydrates, proteins and nucleic acids are all necessary for life to function.
Cells are made up of just three substances: water; inorganic ions; and organic
molecules. Water makes up around 70 % of cell mass, with most of the rest being
organic molecules: inorganic ions make up only a few percent. The inorganic ions
include sodium (Na+), potassium (K+), calcium (Ca2+), chloride (Cl−) and bicar-
bonate (HCO3−). The plus and minus signs are used to tell us that the ions are
positively or negatively charged: this will be very important when we consider how
the cell behaves later on. We don’t always explicitly write the plus or minus signs
though, since only a handful of ions are involved and we tend to know what their
charges are.
1.2.1 Proteins
There are many types of proteins and they perform lots of roles within the body,
including:
1. Structural (coverings and support);
2. Regulatory (hormones, control of metabolism);
3. Contractile (muscles);
4. Immunological (antibodies, immune system);
5. Transport (movement of materials, haemoglobin for oxygen);
6. Catalytic (enzymes).
Proteins are large molecules made up of amino acids (of which there are around
20 in the human body): examples of proteins are insulin, haemoglobin and
myosin. Insulin plays a key role in the regulation of blood sugar levels; hae-
moglobin is vital to the transport of oxygen around the body and myosin is used in
muscle contraction.
Proteins differ between themselves in their order of amino acids, which is
determined by their gene sequence; they are folded into specific three-dimensional
shapes. The unique folded structure of a protein is often a very important deter-
minant of its function. Proteins are formed, used and recycled with a lifespan that
can vary from minutes to years. A schematic of the structure of haemoglobin, a
typical protein, is shown in Fig. 1.2. The final structure is made up of four sub-units,
each of which is a highly folded and bonded version of the original chain of amino
acids.
1.2.2 ATP
Adenosine triphosphate (ATP) is essentially the energy source for cells and acts
like a battery. It is created by the conversion of glucose and oxygen into carbon
dioxide, water and ATP:
Fig. 1.2 Protein structure of haemoglobin (this figure is taken, without changes, from OpenStax
College under license: http://creativecommons.org/licenses/by/3.0/)
C6 H12 O6 þ 6O2 ! 6CO2 þ 6H2 O þ ATP ð1:1Þ
This is why we breathe in oxygen and breathe out carbon dioxide: to convert
glucose (a sugar) into an energy store that can be used for cells to function.
Figure 1.3 shows the three parts that make ATP molecules, with the third part
comprising three phosphate groups (hence ‘triphosphate’). When the third, terminal,
phosphate is released, energy is also liberated: ATP then becomes ADP (adenosine
diphosphate), i.e. with only two phosphate groups. If the second phosphate is also
released, to yield AMP (adenosine monophosphate), more energy is liberated. This
energy is used to power many cellular processes, as we will see later.
1.2.3 DNA/RNA
Deoxyribonucleic acid (DNA for short) is the molecule that contains most of the
genetic information that controls all of the processes that occur in the cell. As can be
1.2 Cell Chemicals 5
Fig. 1.3 Structure of ATP (this figure is taken, without changes, from OpenStax College under
seen in Fig. 1.4, there are two base pairs: adenine (A) with thymine (T), and
guanine (G) with cytosine (C). These are connected to a sugar and a phosphate
molecule, together called a nucleotide, which are arranged in the famous double
helix structure shown in Fig. 1.4. The order of the bases encodes the information
contained in each strand of DNA.
DNA is then organised into chromosomes: human cells have 23 pairs of
chromosomes including the X and Y sex chromosomes. Chromosomes are
sub-divided into genes, with each gene having its own function, instructing cells to
make specific proteins. Ribonucleic acid (RNA) is the molecule that is used to
convey genetic information by translating the information in a specific gene into a
protein’s amino acid sequence: unlike DNA, RNA is single-stranded.
1.3 Reaction Equations
Now that we have discussed how a cell is constructed, it is time to consider how we
analyse its behaviour. We do this analysis using reaction equations, which describe
how one set of atoms and molecules (reactants) combine to form a different set of
atoms and molecules (products).
Fig. 1.4 Structure of DNA

(this figure is taken, without
changes, from OpenStax
College under license: http://
creativecommons.org/
licenses/by/3.0/)
However, before we start to do this, we need to consider the correct use of units.
Throughout this book, we will use the mole, which is formally defined as the
number of atoms found in 12 grams of 12C (6.022 × 1023 atoms), as the basic unit.
Chemical equations are all based on the use of the mole, since it is a much more
convenient means of describing quantities than simple weight or volume.
We said earlier that 70 % of the cell is water and so for many physiological
elements, the molecules are not found in isolation, but in solution, primarily in
water. We thus need a way to describe how much of the solute is present in the
solution. The most common definition, although there are others, is called
molarity: this is the number of moles of the solute per litre of solution. It thus has
units of mol/l, which is normally written in shorthand as M. We will use this all the
way through the remainder of this book unless stated otherwise. Since most
physiological molarities are much less than 1 M, you should get used to seeing the
abbreviation for milli-molar, which is written mM.
1.3 Reaction Equations 7
1.3.1 Mass Action Kinetics
Let’s start by writing down one of the very simplest reaction equations:
k
AþB!C ð1:2Þ
k is termed the rate constant for this reaction, which simply takes two reactants,
A and B, and converts them into one product C. The quantity of C increases
dependent upon the quantities of both A and B, thus a simple model for rate of
change of the concentration of the product with time (termed the reaction rate) is
given by:
d ½C
¼ k½ A½B ð1:3Þ
dt
This is called the law of mass action and systems that obey this style of equation
are said to be governed by mass action kinetics. The rate constant will depend
upon the sizes and shapes of A and B; it also varies strongly with temperature (as
well as with other factors such as pH).
Mass action kinetics are based on C being produced when A and B collide and
combine, so-called elementary reactions. The rate constant is thus proportional to
the number of collisions between A and B per unit time and the probability that the
collision has enough energy.
Note that concentrations are usually denoted by square brackets, i.e. the con-
centration of A is [A]. However, for simplicity we will often use lower case letters to
refer to chemical concentrations where we need to write many equations (see the
next example below, where we do this to make writing the equations easier).
We should also note that Eq. (1.3) is not always true: for very high or very low
concentrations, the rate of change is limited by other factors. Despite this, the
equation is a good first approximation and will enable us to analyse even quite
complicated systems in a straightforward way.
If there are multiple moles on the left hand side of the reaction equation, then the
reaction rate is proportional to the concentrations to the relevant powers, i.e. the
reaction rate for the reaction:
k
A þ 2B ! C ð1:4Þ
is:
dc
¼ kab2 ð1:5Þ
dt
Also, all reactions strictly speaking must be reversible to some extent, thus there
are forward and reverse rate constants, which need not be the same:
kþ
AþB C ð1:6Þ
k
For this example, we can write down three equations, one for each of the
chemical substances:
da
¼ k c k þ ab ð1:7Þ
dt
db
¼ k c k þ ab ð1:8Þ
dt
dc
¼ k þ ab k c ð1:9Þ
dt
Although this now looks rather complicated, if we add the first and third
equations together, the rate of change of a + c is equal to zero. Integrating up this
equation tells us that the sum of these two concentrations is a constant, which we
define here as: a + c = ao, i.e. ao can be thought of as the initial amount of A before
any was converted to C.
We can use the equilibrium condition in these equations. By setting the rates of
change to zero in the three equations, we find:
k ab
¼ ¼K ð1:10Þ
kþ c
where we use the over-bar to refer to the equilibrium or steady-state values of each
substance. The ratio of the two rate constants, which we call K, is termed the
equilibrium constant. This measures the relative preference for the substances to
be in the combined, C, or separate, A + B, state. Note that we can define this either
way up: we have chosen here to use the backwards rate divided by the forward rate,
but we could just as easily have defined it the other way.
This equation can be used to determine the equilibrium constant using the
equilibrium values of the concentrations. The rule for determining the constant is to
multiply all the reactant concentrations to the power of their coefficients and divide
by the product of the product concentrations, again each to the power of their
coefficients. The following two exercises will help you get some practice at this.
Exercise A
A reaction takes place according to:
kþ
A þ 2B C þ D ðA:1Þ
k
Write down the four reaction equations and show that the equilibrium
constant is given by:
2
k ab
¼ ðA:2Þ
kþ cd
Exercise B
A reaction takes place that turns reactants A and B into products D, E and
F according to the following separate reaction steps:
kþ1
AþB CþD ðB:1Þ
k1
kþ2
C EþF ðB:2Þ
k2
a. Write down the reaction equations for A using Eq. B.1 and for C using
Eq. B.2.
b. Assuming that both reactions are in equilibrium, show that the equilibrium
constant for the whole reaction is given by:
ab k1 k2
K¼ ¼ ðB:3Þ
def k þ1 kþ2
1.3.2 Enzyme Kinetics
Now that we have considered simple reaction equations and how to analyse them,
we will look at one particular category of reaction that is very important: one where
the reaction is catalysed by an enzyme. Enzymes are essentially substances that
help other molecules called substrates change into products but which themselves
are left unaffected by the reaction: they are sometimes known as catalysts.
One way in which enzymes work is by lowering the activation energy of the
reaction, i.e. they make it easier to move from one state to another, as shown in
Fig. 1.5. Activation energy is the transient energy required to proceed with a
reaction and is often larger than the energy required or released by the reaction
itself. For example, a reaction may require breaking of some atomic bonds before
new bonds are formed. Thus to proceed along the reaction, moving from left to
right in the figure, a large and temporary source of energy is required, giving the
Fig. 1.5 Enzymes and

activation energy (this figure
is taken, without changes,
from OpenStax College under
license: http://
licenses/by/3.0/)
characteristic ‘hill’ shown here. The enzyme lowers this ‘hill’, making it easier for
the reaction to proceed.
Enzymes are particularly efficient at speeding up biological reactions and are
highly specific, thus allowing very precise control of the reaction speed. Remember
that our simple model is that of an elementary reaction that depends upon the rate of
collision of the reactants and the probability of the reaction having enough energy.
A simple example of enzyme action would be a protein that ‘fits’ a particular
molecule in such a way that it causes a bond to be stressed making that bond more
easily broken and thus reducing the activation energy, making the probability of
reaction higher. Alternatively a protein might have ‘sites’ to which the species
involved in the reaction bind, so that the enzyme increases the rate at which the
species are brought together.
The first model to consider enzyme reactions was proposed by Michaelis and
Menten. The enzyme E converts the substrate S into the product P in two stages.
S and E first combine to give a complex C (sometimes written as ES), which then
breaks down into E and P:
kþ1 kþ2
EþS C ! EþP ð1:11Þ
k1
In theory this second reaction can also work backwards, but normally P is
continually removed, which prevents the reverse reaction from occurring, so we
approximate it as a one way reaction. We can represent this process in diagram-
matic form, as shown in Fig. 1.6, where the two substrates (in this case called S1
and S2) join together with the enzyme to form a product before being released from
the enzyme.
We can write down the four differential equations in the same way as before for
the four different substances, S, E, C and P:
Fig. 1.6 Stages in an enzyme reaction (this figure is taken, without changes, from OpenStax
ds
¼ k1 c k þ 1 se ð1:12Þ
dt
de
¼ k1 c k þ 1 se þ k þ 2 c ð1:13Þ
dt
dc
¼ k þ 1 se k þ 2 c k1 c ð1:14Þ
dt
dp
¼ k þ 2c ð1:15Þ
dt
Note that this set of equations is redundant: the second and third equations add to
give zero, which means that the sum of E and C is constant over time and so we
introduce a new constant, normally represented by e + c = eo.
There are two common approaches to analysing this system of equations: the
equilibrium approximation and the quasi-steady-state approximation. We will
examine them both briefly here.
1.3.2.1 Equilibrium Approximation
The first method is the original one proposed by Michaelis and Menten and assumes
that the substrate is always in equilibrium with the complex, i.e. the first stage of the
reaction is in equilibrium and hence ds=dt ¼ 0. The rate of formation of the pro-
duct, termed the velocity of the reaction, is then given by:
dp k þ 2 eo s
V¼ ¼ k þ 2c ¼ ð1:16Þ
dt Ks þ s
where Ks ¼ k1 =k þ 1 , similarly to before.

Fig. 1.7 Michaelis-Menten

reaction behaviour (for
Ks = 0.5 and Vmax = 1)
At small substrate concentrations, the reaction rate is proportional to the amount

of available enzyme and the amount of substrate: however, at large substrate
concentrations, the rate is limited by the amount of enzyme present. The second
reaction is thus termed rate limiting since the reaction velocity cannot increase
beyond a certain value.
Equation (1.16) is often re-written in the form:
Vmax s
V¼ ð1:17Þ
Ks þ s
where Vmax is so called because it is the maximum velocity with which the reaction
can proceed. This equation is known as the Michaelis-Menten equation and is
used widely in physiological modelling to mimic processes where there is a limited
rate of reaction. It is shown in Fig. 1.7, where the asymptote and the initial slope are
shown as straight lines: note that the square marks the point at which the reaction
velocity reaches half of its maximum value (this is obviously when s ¼ Ks ).
1.3.2.2 Quasi-Steady-State Approximation
The second approximation assumes that the rates of formation and the breakdown
of the complex are equal, thus dc=dt ¼ 0. Solution of the remaining equations gives
a very similar result to the previous reaction velocity:
Vmax s
V¼ ð1:18Þ
Km þ s
where Km ¼ ðk1 þ k2 Þ=k þ 1 . Clearly the two approximations give very similar
results, and they are both of Michaelis-Menten form, but they are based on very
different assumptions.
Exercise C
Derive the results shown in Eqs. (1.17) and (1.18). Explain why the
quasi-steady-state reaction velocity is always smaller than the equilibrium
reaction velocity.
Exercise D
Explain why plotting 1/V as a function of 1/s for a Michaelis-Menten reaction
gives a straight line. Sketch this function and explain how the constants in
this equation can be found from the intercepts with the two axes.
This model is very simple and is not normally an accurate reflection of the
various stages of a true enzyme assisted reaction. A more detailed model was later
proposed by Briggs and Haldane, where the complex has two phases (ES and EP).
This has the same form as Eqs. 1.17 and 1.18, but again with a different constant.
We can obviously extend the analysis to as complex a model as we wish (and we
will look at some of these below), but the analysis does become increasingly
laborious to perform. In practice, quite often a simple approximation is sufficient to
capture most of the behaviour of the system, especially if it is operating near either a
linear or a saturated regime.
1.3.3 Enzyme Cooperativity
Some enzymes can bind more than one substrate molecule, such that the binding of
one substrate molecule affects the binding of subsequent molecules. This is known
as cooperativity and is involved in one of the most important bindings found in the
human body: that of oxygen to haemoglobin in the blood. Although the analysis is
quite complicated, the final result is relatively simple: if n substrate molecules can
bind to the enzyme, the rate of reaction is given by:
Vmax sn
V¼ ð1:19Þ
K n þ sn
This is known as the Hill equation. It is frequently used as an approximation for

reactions where the intermediate steps are not well known and is then derived from
fits to experimental data. The shape of the Hill equation for different values of n is
shown in Fig. 1.8, illustrating how this can be used to fit different shapes of curves
derived from experimental data. In particular, note that the curve becomes much
sharper for larger values of n (the asymptote is at 1, just as in the previous Figure).
Fig. 1.8 Hill equation

reaction behaviour, for
different values of n
This is particularly true in the case of oxygen binding to haemoglobin, where the
reaction equation:
kþ
Hb þ 4O2 HbðO2 Þ4 ð1:20Þ
k
turns out to imply a fraction filling of available haemoglobin sites, S, of:
½O 2 n
S¼ ð1:21Þ
K n þ ½O2 n
with n = 4. In fact, a much better fit to experimental data is found with n = 2.5,

implying that the later sites prefer to fill up if the early sites are already full.
Precisely how this positive cooperativity occurs, however, is not yet completely
understood. This example, which is known as the oxygen saturation curve and
which is very important in respiration, as we will see in Chap. 9, provides a useful
early illustration of a model that provides a good understanding of how the reaction
processes occur, but where the result has to be adapted according to the real
behaviour. The oxygen saturation curve is also affected by pH, temperature and
CO2 levels, amongst other things.
Exercise E
a. Show that the Hill equation can be written as a straight line plot if the
value of Vmax is known.
b. Sketch this function and explain how the intercepts with the two axes can
be used to calculate the values of n and Km. Given the values in the table
below, calculate the values of n and Km, assuming that Vmax = 1 mM.
Is the Hill equation a good fit to this data set?
Substrate conc. (mM) 0.2 0.5 1.0 1.5 2.0 2.5 3.5 4.5
Reaction velocity (mM/s) 0.01 0.06 0.27 0.5 0.67 0.78 0.89 0.94
1.3.4 Enzyme Inhibition
In many cases, it is very important to have control over the reaction rate, so that it
can be altered under different conditions, for example speeded up when demand for
the product is greater, or slowed down when demand drops. An enzyme inhibitor
is thus often used to reduce the rate of reaction, as explained below. There are two
types that we will look at here: competitive and allosteric inhibitors.
1.3.4.1 Competitive Inhibitors
In this case, the inhibitor species combines with the enzyme to form a compound,
which essentially removes some of the enzyme from the system, preventing it from
forming the product: hence the idea of competition for the enzyme. The reactions
are thus:
kþ1 kþ2
E þ S C1 ! E þ P ð1:22Þ
k1
kþ3
E þ I C2 ð1:23Þ
k3
There are now six differential equations:
ds
¼ k1 c1 k þ 1 se ð1:24Þ
dt
de
¼ k1 c1 þ k þ 2 c1 k þ 1 se þ k3 c2 k þ 3 ie ð1:25Þ
dt
dc1
¼ k þ 1 se ðk þ 2 þ k1 Þc1 ð1:26Þ
dt
dp
¼ k þ 2 c1 ð1:27Þ
dt
di
¼ k3 c2 k þ 3 ie ð1:28Þ
dt
dc2
¼ k þ 3 ie k3 c2 ð1:29Þ
dt
Thankfully, this isn’t as complicated as it looks. The normal assumption made in

the analysis is that both the compounds are in quasi-steady-state. We can also see
that if we add the differentials for the enzyme and the two compounds that they add
up to zero (so these three variables always add up to a constant, which we will again
call eo).
The rate of formation of the product is then found to be:
dp Vmax s
V¼ ¼ k þ 2 c1 ¼ ð1:30Þ
dt Km ð1 þ i=Ki Þ þ s
where Ki ¼ k3 =k þ 3 . Note that if i is set to zero, we get back to the original
Michaelis-Menten equation, just as we would expect.
Exercise F
a. Derive the result in (1.30), given that dc1 =dt ¼ 0 and dc2 =dt ¼ 0 in
quasi-steady-state, and that c1 þ c2 þ e ¼ eo .
b. Sketch the result for different values of inhibitor. Explain how the inhi-
bitor affects the intercepts on the 1/V versus 1/s plot.
At this point, we will just note that the reaction equations can be re-written in
schematic form (where we have omitted the rate constants for simplicity), as shown
below.
This diagrammatic form can be very helpful in seeing how the reaction equations
link together and understanding how substances ‘move’ around the system.
1.3.4.2 Allosteric Inhibitors
In this case the inhibitor binds to the enzyme in such a way as to prevent the
product being formed. For example, the inhibitor might bind to a different site on
the enzyme preventing the complex converting into the product. This can be
modelled as the inhibitor binding to the complex, written in schematic form as
shown below (note how we have moved to simply sketching the diagram to explain
the system before writing down the equations).
A schematic of allosteric inhibition is shown in Fig. 1.9, alongside allosteric

activation. The latter can be seen as the complementary process, where an activator
is required to enhance the role of the enzyme: like the enzyme, it is not used up in
the reaction, but its presence affects the overall reaction rate.
The rate of formation of product is now:
Vmax s
V¼ ð1:31Þ
Km þ ð1 þ i=Ki Þs
This is quite similar to the previous equation, but you should be able to spot the
difference in its behaviour. In practice this result is only true for an allosteric
Fig. 1.9 Allosteric inhibition and activation (this figure is taken, without changes, from OpenStax
inhibitor that is uncompetitive, i.e. one that doesn’t bind to E to form EI. It is of
course possible to have inhibitors that are both allosteric and competitive in which
case the analysis becomes more complex. We won’t go into these, as there are more
detailed treatments of these elsewhere. However, you have by now at least got an
idea of how we write down reaction equations and how we analyse them. Anything
else may be more complex, but every system will be analysed in the same way.
1.4 Conclusions
In this chapter we have looked at how human cells are constructed, what substances
they are made of and considered how to analyse simple chemical processes using
reaction equations. You should now be familiar with the basic make-up of cells and
be able to understand how to write down any system of reactions, using both
diagrammatic form and reaction equations, and to do some basic analysis.
Chapter 2
Cellular Homeostasis and Membrane
Potential
2.1 Membrane Structure and Composition
The human cell can be considered to consist of a bag of fluid with a wall that
separates the internal, or intracellular, fluid (ICF) from the external, or extracellular,
fluid (ECF): this wall is termed the plasma membrane. The membrane consists of a
sheet of lipids two molecules thick: lipids being molecules that are not soluble in
water but are that soluble in oil. The cell lipids are primarily phospholipids, as
illustrated in Fig. 2.1, they have one end that is hydrophilic and one that is
hydrophobic (are attracted to and repelled by water molecules respectively). The
hydrophobic ends tend to point towards each other, and away from the aqueous
environments inside and outside the cell, hence the two molecule thickness.
Substances can cross the membrane if they can dissolve in the lipids, which will
not be true of most of the species that are in aqueous solution, such as ions.
However, some electrically charged substances, which cannot readily pass through
the lipid sheets, do cross the membrane. This is because the membrane is full of
various types of protein molecules, some of which bridge the lipid layer. Sometimes
these form pores or channels through which molecules can pass. Figure 2.2 shows a
schematic of the membrane structure.
Outside the cell, the main positively charged ion is sodium (Na+) with a small
amount of potassium (K+) and chloride (Cl−) ions, Table 2.1. The relative quantities
are reversed in the ICF. The balance of charge is provided inside the cell by a class
of molecules that include protein molecules and amino acids (likewise outside the
cell, but these will be ignored here). One of the key features of the cell is the
balance of molecules between the inside and outside, yet at first glance it doesn’t
appear that is the case for the cell in Table 2.1. It also does not seem obvious why
the individual molecules do not diffuse in and out of the cell such that the ICF and
ECF concentrations are equal. In general we are interested in how the cell maintains
its internal conditions despite what is going on outside: the property called

20 2 Cellular Homeostasis and Membrane Potential
Fig. 2.1 The structure of a phospholipid with both hydrophobic and hydrophilic ends (this figure
is taken, without changes, from OpenStax College under license: http://creativecommons.org/
licenses/by/3.0/)
Fig. 2.2 Schematic of membrane structure (this figure is taken, without changes, from OpenStax
2.1 Membrane Structure and Composition 21
Table 2.1 Compositions of intracellular and extracellular fluids for a typical cell
Internal concentration External concentration
(mM) (mM)
K+ 125 5
Na+ 12 120
−
Cl 5 125
P−* 108 0
H2O 55,000 55,000
*
P refers to non-ionic substances, largely proteins and amino acids, inside the cell that are changed.
The overall net charge is negative
homeostasis. If we want to understand how the cell maintains this imbalance we

first need to consider the balance of cell volume.
2.2 Osmotic Balance
Consider a litre of water with 1 mol of dissolved particles: this is termed a 1 molar,
or 1 M, solution, as we saw in Chap. 1. Now consider two adjacent identical
volumes with different molarities (say 100 and 200 mM) and a barrier between
them. If the barrier allows both water and the solute to pass, equilibrium will be
reached with equal levels of the solute (150 mM) and the barrier will not move, as
might be expected. However, if the barrier allows only water to cross, enough water
will have to cross the barrier for the concentrations to balance and the barrier will
thus move. Equilibrium will then be reached with the same concentrations
(150 mM) but different volumes, 2/3 and 4/3 L, as illustrated in Fig. 2.3. The
volumes are calculated by remembering that the concentrations must balance and
that the number of moles of the solute cannot change. This process can be thought
analogous to pressurised chambers with initial pressures (equivalent to the con-
centrations) and volumes: the barrier is like a piston, moving until pressure equi-
librium is reached.
Now consider a slightly different example with a cell with an intracellular
concentration of a substance P and an extracellular concentration of a substance Q:
neither P nor Q is able to cross the membrane. There are three possibilities:
1. The concentration of Q is equal to that of P (isotonic): cell volume remains
constant.
2. The concentration of Q is greater than that of P (hypertonic solution): cell
volume decreases and the cell ultimately collapses if the difference in con-
centration is sufficiently large.
3. The concentration of Q is less than that of P (hypotonic solution): cell volume
increases eventually rupturing the membrane if the concentration difference is
sufficiently large.
1l 1l
100mM 200mM
Water & solute
can cross Water only
barrier can cross barrier
1l 1l 2/3 l 4/3 l
150mM 150mM 150mM 150mM
Fig. 2.3 Example of balance of concentration with solute able to cross (left) and with solute
unable to cross (right)
Fig. 2.4 Cell behaviour in

solutions of different
concentration. Neither P nor
Q are able to cross the cell
membrane, any adjustments
in concentration can only
occur by movement of water,
i.e. osmosis
A hypotonic solution is defined as one that makes the cell increase in size and a
hypertonic solution is one that makes the cell decrease in size. An illustration of the
three types of behaviour is shown in Fig. 2.4.
2.2 Osmotic Balance 23
Exercise A
For the following case determine the final, equilibrium, cell volume given the
starting conditions and an initial cell volume V0.
So far we have only considered what happens if water can cross the membrane:
if Q (or P) is able to cross the membrane, then the cell behaves differently. The
concentration of Q must be the same inside and outside the cell: however, the total
concentration inside and outside the cell must also be the same otherwise the
membrane will move to adjust the concentrations.
Exercise B
If Q is now able to cross the membrane determine the final, equilibrium, cell
volume given the starting conditions and an initial cell volume V0. Assume
that the concentration of the species, Q, that is outside the cell is fixed.
Note that in Exercise B it doesn’t matter that the total internal and external
concentrations are equal unlike in exercise A where this led to equilibrium. It is also
worth questioning the assumption that the concentration of Q is fixed and not
altered by Q moving into the cell. This stems from an assumption that the extra-
cellular space is a lot larger than the cell itself and/or can easily be replenished from
elsewhere, i.e. that there is an infinite reserve of Q in this case.
The requirement for the total concentration to be equal inside and outside the cell is
actually a requirement that the concentration of water balances. It might seem strange
to talk about water concentration since there is so much of it compared to the other
species, but osmosis is essentially the process of balancing water concentration.
The total concentration is often referred to as the osmolarity: the higher the
osmolarity the lower the concentration of water and vice versa. A solution containing
0.1 M glucose and 0.1 M urea would have a total concentration of non-water species
of 0.2 M and thus an osmolarity of 0.2 Osm. Care needs to be taken with solutions of
substances that dissociate, for example, a 0.1 M solution of NaCl is a 0.2 Osm
solution since you get free Na+ and Cl−. In practice the osmolarity could be lower than
this if the ions in solution interacted, but this is not common in biological systems.
Exercise C
For the following cases determine the final, equilibrium, cell volume given
the starting conditions and an initial cell volume V0.
We have considered how the cell might maintain its volume despite imbalances
in the concentrations of species inside and outside the cell. In fact we have met our
first principle, the principle of concentration balance.
2.3 Conservation of Charge 25
2.3 Conservation of Charge
Now consider the slightly more complex example in Fig. 2.5, which is a very basic
model of a cell. Inside the cell are found organic molecules, P, which cannot pass
through the barrier. The internal Na+ is also trapped, whereas Cl− can pass freely
through the barrier. The concentrations of P and Na+ inside the cell are 100 and
50 mM respectively.
To analyse this model, there are two quantities that must be in balance: charge
and concentration. The fact that the positive and negative charges must balance
within any compartment is called the principle of electrical neutrality, which
states that the bulk concentration of positively charged ions must equal the bulk
concentration of negatively charged ions. Essentially this is due to the fact that
under biological conditions, so few positively and negatively charged ions have to
move to generate any membrane potential (which we will meet later) that we can
assume that they balance at all times.
From charge balance:
a = 50 ð2:1Þ
b=c ð2:2Þ
From total concentration balance:
50 þ a þ 100 ¼ b þ c ð2:3Þ
Hence:
b ¼ c ¼ 100 ð2:4Þ
Note that, unlike in the previous section, the concentrations of Cl− are not equal
inside and outside the cell: this is due to the influence of the charge balance. In fact
at this point we can more carefully define the principle of concentration balance to
only refer to the concentrations of uncharged species. For a simple cell model the
only uncharged species that can freely cross the membrane to any significant degree
is water, thus the principle of concentration balance might be called the principle of
osmotic balance.
50 mM Na+ Na+ b
a Cl- Cl- c
100 mM P
Fig. 2.5 Cell model example

2.4 Equilibrium Potential
So far we have only considered concentration equilibrium: however, there is another

important factor that drives ions across a cell membrane. In addition to the con-
centration gradient that drives ions from a region of high concentration to a region of
low concentration, there is an electrical potential difference across the membrane.
For the membrane shown in Fig. 2.6, the difference in voltage between the inside
and the outside of the cell is given by the Nernst equation:

RT ½ X out
EX ¼ Vin Vout ¼ ln ; ð2:5Þ
ZF ½ X in
where R is the gas constant, T is absolute temperature (in Kelvin), Z is the valence
of the ion and F is Faraday’s constant (96,500 C/mol_univalent_ion). The quantity
in the equation above is known as the equilibrium potential and only applies for a
single ion that can cross the barrier. At standard room temperature, the equation can
be re-written as:

58 mV ½ X out
EX ¼ log10 ; ð2:6Þ
Z ½ X in
where we have changed from a natural logarithm to a base-10 logarithm.

There can only be a single potential across the membrane, the membrane
potential, thus if there are two ions that can cross the membrane (in the real cell
these are K+ and Cl−), then the equilibrium potential must be the same for both.
Hence:
þ
½K out ½Cl out
Em ¼ 58 mV log10 ¼ 58 mV log ; ð2:7Þ
½K þ in 10
½Cl in
which on re-arranging becomes:
½K þ out ½Cl
þ ¼ in : ð2:8Þ
½K in ½Cl out
This is known as the Donnan or Gibbs-Donnan equilibrium equation.
[X]in [X]out
Em= Vin - Vout

Vin Vout
Fig. 2.6 Membrane and membrane potential

2.4 Equilibrium Potential 27
Exercise D
Suppose that two compartments, each of one litre in volume, are connected
by a membrane that is permeable to both K+ and Cl−, but not permeable to
water or the protein X. Suppose further that the compartment on the left
initially contains 300 mM K+ and 300 mM Cl−, while the compartment on the
right initially contains 200 mM protein, with valence −2, and 400 mM K+.
(a) Is the starting configuration electrically and osmotically balanced?
(b) Find the concentrations at equilibrium.
(c) Why is [K+] in the right compartment at equilibrium greater than its
starting value, even though [K+] in the right compartment was greater
than [K+] in the left compartment initially? Why does K+ not diffuse
from right to left to equalize the concentrations?
(d) What is the equilibrium potential difference?
Exercise D is an interesting example of the counter intuitive differences in ion

concentration that can arise due to the balance of electrical configuration. However,
it is not like our cell model because we had only a fixed total amount of each ion
available and the whole system was thus closed.
2.5 A Simple Cell Model
We started out by trying to understand cell homeostasis and how an imbalance in

concentrations of ions inside and outside of the cell could be maintained. We now
have three principles to apply when we analyse cell concentrations:
1. Concentration (osmotic) balance.
2. Electrical neutrality.
3. Gibbs-Donnan equilibrium.
We are now ready to try and build a simple model of a cell at equilibrium,
Fig. 2.7. Inside the cell is found Na+, K+ and Cl− as well as some negatively
charged particles, termed P, that represent an array of different molecules, including
a Na+ Na+ 120 mM

b K+ K+ 5 mM
c Cl- Cl- d
108 mM P -11/9
Fig. 2.7 A simple cell model

proteins. Outside the cell is found Na+, K+ and Cl− where K+ and Cl− are free to
cross the membrane.
Exercise E
Figure 2.7 is incomplete since some of the concentrations have not been
given.
(a) Write down and solve the appropriate equations to calculate the
unknown concentrations in the figure. Note that the charge of P is −11/9
(about −1.22),1 which means that the charge equilibrium equation must
be written down carefully.
(b) What is the value of the membrane potential in this example?
(c) If the cell membrane was permeable to sodium, calculate the equilibrium
potential for sodium. What would happen to the cell?
This exercise takes us close to a realistic model of the cell: you should find that
the values of concentration that you get are the same as Table 2.1. Note that it will
remain in this state indefinitely without the expending of any metabolic energy: a
very efficient structure. However, you will have found that if the cell is permeable
to Na+, its equilibrium potential is very different from the membrane potential you
calculated when the membrane is impermeable to Na+. Thus if the membrane were
permeable to sodium then it would be impossible to achieve equilibrium due to the
proteins etc also present in the cell: the cell would grow until rupture.
2.6 Ion Pumps
Unfortunately, the real cell actually does expend metabolic energy in order to
remain at equilibrium. The reason for this is that the cell wall is actually permeable
to Na+, which implies that our model cell will not remain in an equilibrium state.
The answer to this problem is that there is something called a sodium pump, which
we will now examine briefly.
An ion pump is a mechanism that absorbs energy to move ions against a con-
centration or electrical gradient, rather like a heat pump. The ion pump gets its
energy from ATP as we met in Chap. 1. For Na+, as fast as it leaks in due to the
concentration and electrical gradients, it is pumped out. Na+ thus effectively acts as
if it cannot cross the membrane, but the cell is now a steady state, requiring energy,
rather than an equilibrium state, which requires no energy.
1
Some other texts round down to a charge of −1.2, which appears to be very close to that here, but
will result in quite different concentrations for some of the ions if you try to use it.
2.6 Ion Pumps 29
Na+
K+
Fig. 2.8 The sodium/potassium pump
The common symbol for the pump is shown in Fig. 2.8, which also shows that
the pump needs K+ ions outside the cell to pump inside in return for Na+ ions inside.
The protein on the cell outer surface needs K+ to bind to it before the protein can
return to a state in which it can bind another ATP and sodium ions at the inner
surface. Since the K+ ions bound on the outside are then released on the inside, the
pump essentially swaps Na+ and K+ ions across the membrane and is thus more
correctly known as the Na+/K+ pump and the membrane-associated enzyme as a
Na+/K+ ATPase. We will revisit the ion pump in Chap. 4.
2.7 Membrane Potential
Now that we have reconsidered the cell as a steady state device (rather than an
equilibrium device), we need to reconsider the membrane potential. Previously in
Exercise E we had Em = EK = ECl = −81 mV. However, now we also have a
contribution from Na+ with ENa = +58 mV, the membrane potential will have to
settle somewhere between these extremes. This actually depends upon both the
ionic concentrations and the membrane permeability to the different ions. Clearly if
the permeability to a particular ion is zero, it contributes nothing to the potential,
whereas with a high permeability it contributes significantly more. The permeability
of the membrane to different ions is absolutely vital in our understanding of the
operation of the cell.
The permeability of a membrane to a particular ion is simply a measure of how
easily those ions can cross the membrane. In electrical terms, it is equivalent to the
inverse of resistance (i.e. conductance). We will consider why the permeabilities are
different for different ions in Chap. 3, but for now, we will note that the perme-
ability is related to the number of channels that allow the ions to pass through and
the ease of passage through the channels.
The relationship between membrane potential and the concentrations and per-
meabilities of the different ions in the cell is known as the Goldman equation:

pK ½K þ o þ pNa ½Na þ o þ pCl ½Cl i
Em ¼ 58 mV log10 ; ð2:9Þ
pK ½K þ i þ pNa ½Na þ i þ pCl ½Cl o
where p denotes permeability. Note that because Cl− has a negative valence the
inner and outer concentrations are the opposite way round to those for Na+ and K+.
For a membrane that is permeable to only one ion, the Goldman equation reduces
immediately to the Nernst equation.
In practice, the contribution of Cl− is negligible and hence the equation is usually
encountered in the form:

½K þ o þ b½Na þ o
Em ¼ 58 mV log10 ; ð2:10Þ
½K þ i þ b½Na þ i
where in the resting state b ¼ pNa =pK is approximately 0.02. For the typical resting
state with the concentrations given previously, the membrane potential is approx-
imately −71 mV. The membrane potential is closer to the value for K+, since the
permeability to K+ is much greater than that for Na+. However, changes in the
relative permeability can produce large changes in the membrane potential between
these two values.
Since the membrane potential is equal to neither the values for Na+ nor for K+,
there is a leakage of both K+ out of and Na+ into the cell: hence the role of the Na+/
K+ pump to maintain the membrane potential at a steady state value. A more
complete model for cell is shown in Fig. 2.9 that also includes the forces acting on
the ions. Note that the net charge on the inside of the cell is negative therefore the
electrostatic forces acting on both Na+ and K+ in inward, whereas on Cl− it is
outward. It is easy to see why at the very least a pump for Na+ is required.
Although we have ignored Cl− in the calculation of the membrane potential, it is
affected by it: the equilibirum membrane potential for Cl− is −80 mV, so either the
concentration will change (as in some cells) or a Cl− pump is used to maintain a
steady state level of Cl−. Less is known about this pump than the Na+/K+ pump.
Since a difference in membrane potential from the equilibrium value for an
individual ion causes a movement of ions across the membrane we can introduce a
new concept, that of membrane conductance, as defined by:
iK ¼ gK ðEm EK Þ; ð2:11Þ
iNa ¼ gNa ðEm ENa Þ; ð2:12Þ
iCl ¼ gCl ðEm ECl Þ: ð2:13Þ
Fig. 2.9 Steady state cell E

model, showing the electrical 12 mM Na+ Na+ 120 mM
(E) and chemical (C) forces C
acting on the ions 125 mM K+ K+ 5 mM
5 mM Cl- Cl- 125 mM
108 mM A-11/9
K+ Na+
Em = -71 mV
2.7 Membrane Potential 31
gk=1/Rk
iK
Em Ek
in out
Fig. 2.10 Modelling the cell membrane as a resistance/conductance for each ion
Since Em ¼ 71 mV; EK ¼ 80 mV and ENa ¼ 58 mV from above, the potas-
sium current is positive and the sodium current is negative. By convention, an
outward current is positive and an inward current is negative. In the steady state the
net current is zero, which is the basis of the Goldman equation. The conductance is
related to both the permeability and the number of available ions in the solution.
Note that conductance is the inverse of resistance and so in electrical terms, the
membrane can be considered as a resistor as in Fig. 2.10.
The meaning of permeability can be explored in more detail by remembering
that the membrane is full of protein channels that permit different ions to pass
through. These channels can be considered to be controlled by a gate that is either
open or closed (this mechanism is known as channel gating). Although the channels
are slightly more complicated than this, it is a valid first approximation: we will
examine this in more detail in Chap. 3. Rather like an electrical switch, each
channel is thus either ‘on’ or ‘off’ as far as current is concerned. Since there are a
very large number of channels, the permeability of the membrane can be controlled
to a high degree of accuracy by the opening of different numbers of channels. This
ability to change the membrane permeabilities is a major factor in the behaviour of
cells and this will be examined in Chap. 3 when we consider the action potential.
Exercise F
A simple model for a cardiac myocyte (heart muscle cell) can be built using
the concentrations of the ions to which the membrane is permeable given in
the table.
Ion Internal concentration (mM) External concentration (mM)

Na+ 10 145
K+ 140 4
Cl− 30 114
Ca2+ 10−4 1.2
(a) Calculate the equilibrium potentials for all the ions in the cardiac
myocyte and hence determine if the cell is in equilibrium.
(b) Using the principles of electrical neutrality and osmotic balance deter-
mine the internal concentration and overall charge of other charged
species (e.g. proteins) within the cell that are unable to cross the cell
membrane, assuming zero external concentration of any other species.
(c) Show that this cardiac myocyte cell is not in a steady state.
This final exercise explores a cell with a specific function that we will meet in
Chap. 7. Whilst a lot of the values for ionic concentrations, charges and potentials
are not wildly different from the model cell we have considered in this chapter, this
cell is neither in equilibrium nor even in steady state. We might have been wrong to
ignore any other charged species outside the cell. Otherwise, like the simple cell
model, we might worry that this cell would expand until rupture. Note that ions
pumps wouldn’t help here as the concentrations we have do not satisfy all the
principles and thus the cell cannot be in steady state; the ion pumps only helped to
maintain the steady state in the simple cell model. It turns out that cardiac myocytes
never reach a steady state and the concentrations fluctuate over a cycle, these values
(probably) just representing the ‘resting’ state. We will return to this in Chap. 7.
2.8 Conclusions
In this chapter we have considered how the cell can maintain homeostasis despite
differences in concentration of ions and other charged species both inside and
outside of the cell membrane. You should now understand the principles of con-
centration balance, electrical neutrality and Gibbs-Donnan equilibrium and be able
to apply them to a cell model. You should also be able to calculate equilibrium
potentials for individual ions as well as the membrane potential and understand why
these are not always the same value. Finally you should now appreciate why cells
use energy to maintain homeostasis.
Chapter 3
The Action Potential
3.1 Na+/K+ Action Potential
As we saw in Chap. 2, it is the relative permeability of the membrane to Na+ and K+

that determines the membrane potential.
þ
½K o þ b½Na þ o
Em ¼ 58 mV log10 : ð3:1Þ
½K þ i þ b½Na þ i
In the resting state, the ratio is 0.02, as given in Chap. 2, but if the membrane
permeability to Na+ were suddenly increased by a significant factor, this ratio would
increase and the membrane potential swing from close to EK (−80 mV) to close to
ENa (+58 mV). This is essentially all that is required to generate the action
potential, which is a transient change in the membrane potential of the cell. The
action potential arises because the gates which determine the permeability of the
membrane are controlled by the membrane potential.
Exercise A
Calculate the membrane potential if b were to increase by three orders of
magnitude to 20, assuming that the ionic concentrations are still at the steady
state values in Table 2.1. What would be implications for the ionic concen-
trations of this change?
Since Em is a logarithmic function of the relative permeability it takes orders of

magnitude changes in b to make a substantial difference. You will have noticed in
Exercise A that an increase in relative permeability either implies that the perme-
ability to K+ has reduced or to Na+ has increased, in practice it is the latter.

34 3 The Action Potential
A change in permeability will lead to a flow of ions driven by the different

membrane potential, the cell will no longer be in steady state.
A typical action potential is shown in Fig. 3.1, this would correspond to a nerve
cell: we will see examples of variants on this later. The process of the action
potential follows these steps, summarised in Fig. 3.2 and Table 3.1:
(A) Resting state: The number of Na+ channels that are open are small and
b = pNa /pK * 0.02.
(B) Depolarization: If the membrane potential becomes more positive then more
Na+ channels open, thus b will increase and Em will become even more
positive. This positive feedback loop means that there is a large and rapid
swing in Em, meaning that the action potential continues irrespective of any
further external influence. Thus the action potential is known as an ‘all or
nothing’ event.
(C) Repolarization: The Na+ channel actually has two gates: m that is normally
closed and h that is normally open. Upon depolarization m opens rapidly, but
h closes much more slowly and it is this timing difference that leaves time for
depolarization before repolarization kicks in. Additionally, there are voltage
sensitive K+ channels with n gates that are normally closed that also respond
slowly to the depolarization. Thus pK also increases on depolarization and like
the h gate this drives Em back toward the resting value.
(D) Undershoot: Due to the n gates pK is greater than usual so that b ends up being
smaller than the resting value, which causes Em to undershoot until the n gates
eventually return to normal.
(E) Refractory period: Whilst the membrane potential may have returned to the
resting state the h gates will initially still be shut blocking the Na+ channels
even if the m gates were to reopen. Thus there is a period in which a new
action potential cannot be generated. In practice the undershoot and refractory
Fig. 3.1 The membrane potential during a typical (nerve cell) action potential
3.1 Na+/K+ Action Potential 35
Fig. 3.2 Schematic of

voltage-sensitive channels
during action potential
Table 3.1 Overview of gate behaviour during an AP

Phase Na+ gates K+ gate b
m h n
A: Resting state Closed Open Closed *0.02
B: Depolarising Opens quickly Closes slowly Opens slowly 0.02 → 20
C: Repolarising Open 20 → *0.02
D: Undershoot Closing quickly Closed Open <0.02
E: Refractory Closed Opening slowly Closing slowly <0.02 → *0.02
periods overlap and the extent to which it is raised K+ permeability or inability

for an increased Na+ permeability that prevents another AP from firing
depends upon the precise timings of the gates.
In practice the action potential only occurs once a certain threshold of membrane
potential is reached, usually around 10–20 mV above its resting value. For smaller
depolarization the efflux of K+ induced by the change in Em exceeds the efflux of
Na+ , leading to a negative feedback process that suppresses further depolarization.
The actual value of this threshold depends upon a variety of factors, particularly the
packing density of the Na+ channels and the relationship between the membrane
potential and the opening pattern. Some neurons are highly sensitive, whilst others
require a very large depolarization to stimulate a action potential.
3.2 Ca2+ Contribution
Action potentials do not only occur in neurons: they can be found in muscle cells,
as will be considered later. In most neurons, however, voltage-dependent Ca2+
channels are also found, which can contribute significantly to the action potential.
This is because they often inactivate more slowly than the corresponding Na+
channels, causing a much slower repolarization with a plateau phase caused by the
Ca2+ channels and the increase in intracellular Ca2+. This increase can be important
in its own right: in Chap. 4 we will see that this is the trigger for the release of
neurotransmitters, which are important in cell-cell communication.
Intracellular Ca2+ can also activate other kinds of ion channels (often K+
channels that are activated by Ca2+), which can lead to a hyperpolarizing under-
shoot after repolarization due to the increase in pK. Since the increase in Ca2+
concentration lasts much longer than the normal timing of the action potential, the
resulting hyperpolarization is some hundreds of times longer than it. This effect is
particularly important for heart muscle cells, as we will see in Chap. 7. This is
dependent on having enough Ca2+ influx during the action potential and enough
Ca2+ activated K+ channels.
In some case the levels of Ca2+ may build up gradually over many action
potentials before it is sufficient to activate the K+ channels and to cause hyperpo-
larization (normally called the after-hyperpolarization to distinguish it from the
undershoot). This can be a way of obtaining rhythmic bursts of activity punctuated
by periods of silent behaviour, particularly in neurons that control rhythmic events,
which we will meet in Chap. 7.
3.3 Hodgkin-Huxley Model
The generation and propagation of signals have been studied for at least 100 years:
however, the most important piece of work during this time was performed by
Hodgkin and Huxley between 1949 and 1952. Hodgkin and Huxley studied the
squid giant axon and developed the first quantitative model of the propagation of
the electrical signal. This model was such an important step in the understanding of
electrical activity that it has been called “the most important model in all of the
physiological literature”, Keener and Sneyd. Their approach was to model the cell
as a simple electrical circuit, Fig. 3.3. The cell is assumed to have a capacitance Cm
that models the insulating effects of a cell membrane that is inherently impermeable
to ions. The model includes current inputs through potassium, sodium and other
channels (these last being lumped together and termed leakage). There is also an
applied current, which we will examine in more detail later.
3.3 Hodgkin-Huxley Model 37
Fig. 3.3 Schematic of Hodgkin-Huxley model
Exercise B
Starting with the simple cell model in Fig. 3.3 use current balance to derive a
differential equation for the electrical properties of the cell.
The Hodgkin-Huxley model starts from applying current balance to the model in
Fig. 3.3, as you did in Exercise B:
dv
Cm gK n4 ðv vK Þ gNa m3 hðv vNa Þ gL ðv vL Þ þ Iapp ;
¼ ð3:2Þ
dt
The potential, v, here is defined as the deviation from the steady state value,
measured in units of mV. Notice that in Eq. 3.2 gK ¼ gK n4 and gNa ¼ gNa m3 , the
conductance’s include the variables m, n and h. These refer to the m, n and h gates
respectively (actually the gates are named after the variables named by
Hodgkin-Huxley in their model): when the values are equal to 0, the gates are
closed (the first equation shows that both m and h gates are needed for Na+ to flow,
but only the n gate is required for K+). The powers of m and n in the equation were
chosen based on experimental data and are often interpreted as the number of
binding sites on the two gates.
The gate parameters have their own governing equations:
dm
¼ am ð1 mÞ bm m; ð3:3Þ
dt
dn
¼ an ð1 nÞ bn n; ð3:4Þ
dt
dh
¼ ah ð1 hÞ bh h; ð3:5Þ
dt
Equations 3.3–3.5 are simply first order kinetic models with forward and
backward rate constants like those we met in Chap. 1. These have a sigmoidal
response to a change in potential that matches that seen in experimental data, the
powers on the m and n gates result in a steeper slope in their sigmoidal response.
The parameters in Eqs. 3.3–3.5 are themselves non-linear functions of the voltage:
25 v
am ¼ 0:1 ; ð3:6Þ
exp 25v
10 1
v
bm ¼ 4 exp ; ð3:7Þ
18
10 v
an ¼ 0:1 ; ð3:8Þ
exp 10v
10 1
v
bn ¼ 0:125 exp ; ð3:9Þ
18
v
ah ¼ 0:07 exp ; ð3:10Þ
20
1
bn ¼ : ð3:11Þ
exp 30v
10 þ1
The remaining parameters all have fixed values as shown in Table 3.2.
Exercise C
A gate in a cell membrane can have one of two states: closed and open.
A simple reaction diagram can thus be written to model changes between the
two states:
a
CO
b
where C denotes the probability that the gate is closed and O the probability
that it is open.
(a) Write down the two reaction equations governing the behaviour of this
gate.
(b) What is the sum of the two probabilities? Hence write down a single
equation in terms of the probability that the gate is open. How does this
relate to the gate equations in the Hodgkin-Huxley model?
Table 3.2 Fixed parameter gK 36 mS/cm2

values for the
gNa 120 mS/cm2
Hodgkin-Huxley model
gL 0.3 mS/cm2
vK −12 mV
vNa 115 mV
vL 10.6 mV
Cm 1 μF/cm2
(c) For the equation derived in part (b), write down the steady-state prob-
ability of the gate being open and the time constant governing its
behaviour.
We will perform some simple analysis of the Hodgkin-Huxley model here to

illustrate how a simple model can be used to simulate and to understand the
behaviour of a physiological system. The first thing we examine is the steady state
behaviour of the variables m, n and h and the time constants. These values are
plotted for a range of potential in Fig. 3.4.
Exercise D
(a) Write down expressions for the steady state probability and time constant
for each of the gates in the Hodgkin-Huxley model.
(b) Using the values in Fig. 3.4, explain how the response of the different
gates gives rise to the action potential.
Fig. 3.4 a Steady state values and b time constants for m, n and h as a function of v, the difference
in membrane potential from its resting value
Fig. 3.5. Time series of action potential with applied stimulus
We now apply a stimulus to the system in the form of the applied current: this is
equivalent to a neighbouring cell providing a change in potential. If a small stim-
ulus is applied, nothing happens, but if it reaches a threshold value, the action
potential suddenly occurs. In Fig. 3.5, a stimulus is applied at 2 s and left on: the
system ‘fires’ and there is then a refractory period before it fires again. Note that the
m gate opens very rapidly, with the n and h gates responding more slowly, as in the
model schematic shown earlier. The presence of the applied current in the model
does not reflect a physiological current source, but that the original experiments
were done by applying external electrical stimulation to excised nerve cells.
Exercise E
Implement the Hodgkin-Huxely model in a numerical mathematics package
using a differential equation solver and reproduce Fig. 3.5 using an applied
current of 5 μA. Determine the smallest current needed to stimulate the
model.
It may seem slightly confusing that in Chap. 2 we considered Na+/K+ and Cl−
ions, whereas here we have considered Na+/K+ and Ca2+ in this chapter. Na+ and
K+ are the dominant ions in the cell’s behaviour, whereas the other ions play roles
in different aspects of the cell’s behaviour, so have to be considered as and when
they are relevant. More detailed models of the cell’s behaviour include all the
different ions.
Exercise F
Returning to the cardiac myocyte model from exercise 2F, the following
figure shows a schematic of the action potential for this cell.
(a) During periods between muscle contraction (phase 4) the membrane is

found to a have a high relative permeability to potassium compared to
all the other ions. Use this to explain why the membrane potential
during this phase is in the range −85 to −95 mV.
(b) The plateau phase (2) is believed to be largely due to the influx of
calcium into the cell (through calcium-gated calcium channels). What
minimum internal concentration of calcium ions could sustain this
plateau? Assume that calcium permeability is far larger than that of all
other ions during the plateau.
3.4 Conclusions
In this chapter we have met the action potential, the fundamental electrical sig-
nalling mechanism used by the body. You should now be able to understand the
cycle of ion movements associated with the generation of an action potential and
the role of different ion gates. You should also be able to analyse the system of
action potential generation in terms of a simple electrical circuit model.
Chapter 4
Cellular Transport and Communication
4.1 Transport
One of the important processes that occurs everywhere in the body is transport:
getting substances to move from one place to another. The different types of mass
transport can be thought of in terms that are analogous to heat transfer, where we
have conduction and both forced and natural convection. For example, ‘forced’
convective transport is achieved by pumping a fluid from one place to another: this
is essentially how oxygen is transported around the body, by blood being pumped
through blood vessels and we will examine this in later chapters. We will consider a
number of different processes here very briefly that apply at the cellular level.
Although there are only a handful of mechanisms that can be used and the law of
conservation of mass always holds, there are many different conditions under which
transport occurs.
4.1.1 Passive Transport
We have already considered how concentration differences drive species such as

ions in and out of the cell. For a given entity present in solution within a region we
can write down, via conservation of mass:
@c
¼ r J þ f ; ð4:1Þ
@t
which essentially says that the rate of change of the concentration c with time
equals the rate of production of the substance within the region, f, and the rate at
which it leaves across the surface of the region, which is determined by the

44 4 Cellular Transport and Communication
flux J. Intuitively we expect the flux should be related to the concentration gradient,
which is the basis of Fick’s law:
J ¼ Drc; ð4:2Þ
If we substitute that into the conservation law then we arrive at the diffusion
equation:
@c
¼ Dr2 c þ f ; ð4:3Þ
@t
where in this case we have assumed that D, the diffusion co-efficient, is constant
across space. Since no energy is involved, diffusion is a passive means of transport.
The value of D is related to the size and geometry of the chemical species as well as
things like temperature and viscosity. If the size of the solute is much greater than
that of the solvent then an estimate of D can be found from the Stokes-Einstein
equation:
kT
D¼ ; ð4:4Þ
6pla
where k is the Boltzmann constant, μ is viscosity and a is the radius of the molecule
assuming an approximately spherical shape. We can re-write this in terms of the
molecular weight:
kT q 13
D¼ ; ð4:5Þ
3l 6p2 M
where, since ρ is approximately constant for large protein molecules we arrive at the
result that D / M 1=3 , whereas for smaller molecules, for example respiratory
molecules, D / M 1=2 turns out to be more accurate.
In the simplest case passive transport of a species through a cellular membrane
between an external concentration co and internal concentration ci can be described
by the formula for flux:
DK
J¼ ðci co Þ; ð4:6Þ
L
where D is the diffusivity of the species in the membrane, L is the membrane

thickness and K is the partition coefficient. Note that the flux is simply linearly
dependent upon the concentration difference. The partition coefficient describes the
relative solubility of the species in the membrane as compared to the external (and
internal) environments. For a cell this would reflect the typically poor solubility of
many solutes in the lipid membrane region compared to the aqueous environments
either side of the membrane. It is this relatively poor solubility that means few
species pass directly through the cell membrane in any significant quantity.
4.1 Transport 45
The combination DK/L is often termed the permeability, since these terms all relate
to properties of the membrane for the species that determine how permeable the
membrane appears.
Exercise A
The figure shows a simple 1-dimensional model of a cell membrane through
which a species might permeate, with thickness L and diffusivity D. Using the
diffusion equation derive an expression for the concentration as a function of
x and thus derive an expression for the flux.
Redraw the figure if the species is twice as soluble in the membrane

compared to the external (and internal) solute.
4.1.2 Carrier-Mediated Transport
Some substances are insoluble in the cell membrane, yet pass through by a process
called carrier-mediated transport. Essentially a substance combines with a carrier
protein at the outer membrane boundary and by means of a conformational change
is released on the inside of the cell. This is still essentially a passive transport
process since no energy is involved. One of the most important examples of this is
the transport of glucose into the cell to provide a source of energy.
Although the precise mechanics of this process are not completely understood, a
simple model for this transport is that the glucose substrate binds with an enzyme
carrier protein to form a complex, which can change (the conformational change)
from an ‘internal’ carrier to an ‘external’ carrier and vice versa. The external carrier
can reduce to the glucose substrate outside the cell with the enzyme carrier protein
in its ‘external’ state, which then reduces to an ‘internal’ state. This is shown in the
Fig. 4.1. A model reaction scheme for carrier-mediated transport of a substance S across a cell
membrane by means of a carrier C
reaction scheme in Fig. 4.1. Notice the similarities with the enzyme reaction models
in Chap. 1.
Exercise B
Using the reaction scheme in Fig. 4.1, write down the differential equations
for each of the four states of the carrier.
Analysis of this is pretty complicated, but quite possible using the mathematics
we used in Chap. 1. It turns out that in the steady state we can calculate the rate of
supply and demand, J:
J ¼ k pi k þ s i c i ¼ k þ s e c e k pe ; ð4:7Þ
1 se si
J ¼ Kd Kk þ Co ; ð4:8Þ
2 ðsi þ K þ Kd Þðse þ K þ Kd Þ Kd2
where K ¼ k =k þ ; Kd ¼ k=k þ and Co ¼ pi þ pe þ ci þ ce , the total amount of the

carrier present in any form. The most important features of this are that at low levels
flux transport is proportional to concentration difference, so it looks like a diffusion
process, and that there is saturation at high levels of external glucose. Although the
system looks more complicated, the essentials of its behaviour are not that far away
from the simple Michaelis-Menten kinetics we met in Chap. 1.
4.1.3 Active Transport
The processes considered thus far are all passive processes, where ions move down
pressure or concentration gradients. There are many processes, however, that
require energy to take place, and are thus termed active transport. A case of this is
where the concentration of the species inside the cell is actively reduced thus
maintaining the concentration gradient required for the passive transport process.
4.1 Transport 47
For example, glucose is rapidly bound within the cell keeping its concentration low;
this phosphorylation of glucose requires hydrolysis of ATP.
Another case is the ion pumps we saw in Chap. 2, where a Na+/K+ pump was
used to keep the levels of Na+ high outside the cell and K+ high inside the cell. This
requires energy to overcome the gradients and this energy was supplied in the form
of ATP. Since the Na+/K+ pump is essentially a reaction, we can write down a
reaction equation for it:
ATP þ 3Naiþ þ 2Keþ ! ADP þ Pi þ 3Naeþ þ 2Kiþ : ð4:9Þ
However, this is a rather simplified version of the actual processes. A more

precise schematic is shown in Fig. 4.2. In its dephosphorylated state, Na+ binding
sites are exposed to the intracellular space; when these ions are bound, the carrier
protein is phosphorylated by the release of energy involved in ATP converting to
ADP. This exposes the Na+ binding sites to the extracellular space, reducing the
binding affinity of these sites and releasing the bound Na+. A similar process
happens, but from extracellular to intracellular, for K+.
This can be converted into a mathematical model by writing down three reaction
equations (which we won’t do here). Note that this is still a simplified version,
where the rate of exchange is one Na+ for one K+ ion, rather than the actual three for
two. The corresponding differential equations can be written down (again we won’t
do this here as it is lengthy), where the intracellular Na+ and extracellular K+ are
supplied at a constant flux J (and extracellular K+ and intracellular Na+ are
removed). In the steady state, this rate can be expressed as:
½Naiþ ½Keþ
J ¼ Co K1 K2 ; ð4:10Þ
½Keþ K2 þ ½Kiþ K2 Kn þ ½Naiþ K1 Kk
where the rate constants are functions of the individual rate constants in the detailed
model. The important features of this are that it is very similar to an enzyme
reaction, i.e. it has dynamics similar to a Michaelis-Menten reaction, being nearly
linear at small concentrations of intracellular Na+ and saturating at large
concentrations.
Fig. 4.2. Schematic of Na+/K+ pump

4.2 Cellular Communication
So far, we have considered the cell largely in isolation: we now need to think about
how information is passed between cells. Like transport there are processes hap-
pening both at the cellular and systemic level. In the latter case the bloodstream is
involved as a means to transport signalling chemicals, hormones, from one cell or
group of cells where the hormone is produced to another region where the signal is
‘received’. This is the responsibility of the endocrine system and can be seen as a
way to broadcast information throughout the body. A similar, but more local sys-
tem, of communication is achieve by the paracrine system, where local ‘mediator’
chemicals are released by the signalling cell and received by neighbouring target
cells.
In this chapter we will be more interested in direct signalling between cells. One
way of signalling is contact-dependent, where the signalling molecule is physi-
cally bound to the cell membrane of the signalling cell. This requires cells to come
into contact with the target cell and thus only really applies to mobile cells.
4.3 Synapses
An important role of cell-to-cell communication is the passing of action potentials

that we met in Chap. 3. There are two ways in which information of this form can
be passed: electrical and chemical. The point where the transfer takes place is
termed a synapse: there are thus both electrical and chemical synapses. In both
cases there are special structures at the point where the input cell, the presynaptic
cell, communicates with the output cell, the postsynaptic cell.
Exercise C
An increase of extracellular K+ has the same effect as an applied current in the
Hodgkin-Huxley model.
(a) Explain why this is the case and derive an expression that relates the
necessary change in potassium equilibrium to the size of the applied
current
(b) If a current of 2.3 μA can initiate an action potential calculate the
equivalent change in potassium concentration required.
(c) What relative change in external potassium concentration would give
rise to such a change in potassium equilibrium potential. Would a rise in
extracellular Na+ also work?
4.3 Synapses 49
4.3.1 Electrical Synapses
In Exercise C you have explored a mechanism by which one cell could theoretically
initiate an action potential in a neighbouring cell through an increase in potassium
in the space between them. A similar mechanism is used at an electrical synapse;
the action potential spreads directly to the postsynaptic cell. An electrical synapse is
a special region of the cell where postsynaptic cell membrane touches that of the
presynaptic cell and the intracellular spaces are connected through special ion
channels called gap junctions. The two cells are deliberately put into close chemical
contact making it easy for changes in ionic concentration in one to affect another.
This is a very rapid means to pass an action potential from one cell to another and is
used by cardiac muscle cells, as we will see in Chap. 6. Apart from being used to
conduct electrical signals, it is also the mechanism used to co-ordinate activity in
the liver.
4.3.2 Chemical Synapses
At a chemical synapse, an action potential results in the release of a chemical

substance (called a neurotransmitter). This moves through the extracellular space
separating the two cells, and alters the membrane potential of the postsynaptic cell.
The best understood chemical synapse is the one between a motor neuron and a
muscle cell, termed a neuromuscular junction, Fig. 4.3. The process proceeds as
follows:
• The action potential arrives at the presynaptic cell and causes depolarization.
• Depolarization causes Ca2+ channels to open.
• Ca2+ enters the presynaptic cell.
• Synaptic vesicles fuse with the membrane.
• Neurotransmitters are released into the synaptic cleft.
• Neurotransmitters bind to special channels found on the post-synaptic cell
surface.
• Channels open allowing Na+ and K+ to cross, permeability of both thus
increases.
• Depolarization of post synaptic cell.
Whilst the absolute permeabilities of both Na+ and K+ increase by roughly the
same amount, the relative permeability to Na+ increases more since it started at a
lower absolute level. This causes the membrane potential to rise by some 50–60 mV
and hence an action potential is generated, the channels closing after a short time
(approximately 1 ms).
The neurotransmitter in this case is called acetylcholine (ACh). This is stored in
units of about 10,000 molecules and so is released in packets, or quanta: a single
action potential normally results in the release of more than 100 packets. These
Fig. 4.3. Schematic of neurotransmitter release, (this figure is taken, without changes, from
OpenStax College under license: http://creativecommons.org/licenses/by/3.0/)
packets are stored in large numbers of tiny membrane-bound structures known as

synaptic vesicles. ACh is released when the vesicles fuse with the muscle cell
membrane at special sites called release sites or active zones, which are only found
on the membrane surface opposite the postsynaptic cell. The vesicle membranes are
continuously recycled, being filled, emptied and re-filled. Once outside the cell
ACh is broken down by an enzyme (choline acetylcholinesterase), allowing the
receptor mediated channels to close. The choline produced by the process is then
taken back up into the cell and more ACh is synthesized, before being packaged
into vesicles. Many other neurotransmitters exist along with a range of different
channels that respond to them, allowing for a variety of different chemical synapses
with different properties.
We have considered the neuromuscular junction in some detail: chemical
synapses between two neurons are essentially the same, although there are some
important differences. Probably the most important is that a neuron may receive
synaptic connections from thousands of different neurons, unlike a muscle cell that
receives an input from only one neuron as we will see in Chap. 10.
4.4 Action Potential Propagation 51
4.4 Action Potential Propagation
We have seen how an action potential is generated and how it can be passed from
one cell to another, but it is clearly of no use if it remains isolated in one location in
the cell. Instead it must be able to propagate through a cell. In particular we need to
ensure that an action potential can be transmitted by nerve cells from the brain to
the rest of the body (or vice versa), for example along a neuron to the
neuro-muscular junction we met earlier. We might imagine that because the action
potential is an electrical signal and is associated with the flow of ions it could travel
along a cell in the same way that an electrical signal can travel along a wire.
We can build a simple model of this process using the electrical circuit in
Fig. 4.4, which is effectively the model of an electrical cable. In this model we have
an infinitely small segment of a cell with action potential propagation in only
one-dimension ‘along’ the cell. We have modelled the interior cell as an electrically
resistive medium, with resistance per unit length Rc, and the cell wall as having both
capacitance and conductance per unit area of membrane of Cm and 1/Rm respec-
tively. Thus we are including the fact the cell wall is ‘leaky’ to ions, but we are
simplifying the problem by ignoring the contributions from different ions that we
had in the Hodgkin-Huxley model. We are also ignoring any resistance to current in
the extra cellular space, effectively assuming that there is a larger volume for
conductance there than inside the cell. Using this model we can get a general
solution for the potential as a function of both time and space (length along the cell
in this one-dimensional case), the cable equation:
@2V @V
k2m ¼ sm þ V; ð4:11Þ
@x2 @t
qffiffiffiffi
where sm ¼ Rm Cm is the membrane time constant and km ¼ Rm
Rc is the cable
space constant.
Fig. 4.4. The cable model of

passive action potential
propagation along a cell
Exercise D
(a) Starting with Fig. 4.4 derive the cable equation in Eq. 4.11. Note that by
convention Rc is a resistance per unit length of cell, but Cm and Rm are
defined per unit area of membrane.
(b) By convention properties of the membrane are quoted as specific
resistance and specific capacitance for a unit area of membrane, rm and
cm respectively. Properties of the space inside the cell are given as
specific resistance of a unit (cross sectional) area of cytoplasm, rc. For a
cell of diameter d, rewrite the membrane time and cable space constants
in terms of these properties.
(c) Calculate the membrane time and cable space constants for a cardiac cell
with d = 20 μm, rc = 150 Ω cm, rm = 7 × 103 Ω cm2 and
cm = 1.2 μF/cm2.
(d) Consider the behaviour of the system under steady state conditions. If
one end of the cell is 100 mV above resting potential at what maximum
distance could an action potential be induced, if a threshold (above
resting potential) of 20 mV is required to initiate an action potential?
The analysis from Exercise D implies that a change in potential at one point in
the cell could initiate another action potential at a distance of the order of 1 mm
along the cell. This would be acceptable for the propagation of the action potential
in a cardiac muscle cell, which is of the order of 0.1 mm in length, but clearly isn’t
suitable for nerve cells who have cell bodies metres in length. The analysis in
Exercise D was relatively simplistic, and we could be more thorough and specify
appropriate boundary conditions and solve Eq. 4.11 analytically. However, this
starts to get messy if we want more accurately to model the propagation of an action
potential. In fact if we did the analysis more carefully we would not only find that
an action potential cannot propagate far along the cell, it also wouldn’t do so fast
enough for nerve impulses.
A nerve cell cannot act like a wire and simply carry the electrical signal along it.
As we have seen, cells are generally very leaky and a lot of the (ionic) current flows
out of the cell making them poor conductors. Instead the action potential mecha-
nism itself is employed to achieve a net flow of the signal down the nerve, where an
action potential in one region causes another to fire in the neighbouring region of
the same cell, called active propagation: Fig. 4.5. An action potential at a point in
the nerve cell causes a current of ions that means that the change in membrane
potential extends along the cell. This leads to neighbouring regions of the cell
reaching a potential above the threshold and thus also undergoing an action
potential. In this case the action potential could propagate in either direction, in
practice an action potential will be triggered at one end and propagate along.
4.4 Action Potential Propagation 53
Fig. 4.5. A schematic of

active propagation
The speed of propagation varies between nerve cells and depends upon how far
the effect of the action potential extends along the cell: the larger the region the
more rapid propagation occurs. A larger region can be achieved either by reducing
the resistance within the cell to the movement of ions, for example by making the
cell diameter larger; or by increasing the resistance of the membrane, making it less
leaky, so that more of the ionic current flows along the cell. These are both methods
to make the cable space constant in Eq. 4.11 larger.
Many vertebrate nerve cells insulate their nerve cells with a myelin sheath, as is
the case for the neuron shown in Fig. 4.6. The disadvantage of this approach is that
the action potential itself relies upon permeability of ions between the inside and
outside of the cell. Thus very efficient insulation would actually prevent an action
potential from occurring and thus preventing propagation. This is avoided by
having breaks in the myelin sheath, called nodes of Ranvier. The action potential
thus effectively skips from one node to the next.
Fig. 4.6. Schematic of a nerve cell (neuron): the axon that carries the action potential is insulated
with a myelin sheath, (this figure is taken, without changes, from OpenStax College under license:
http://creativecommons.org/licenses/by/3.0/)
4.5 Conclusions
In this chapter we have looked at how substances can get in and out of the cell. You
should now be able to identify and analyse some of the main mechanisms for
transport across the cell membrane. We then looked at the related topic of how cells
can communicate with each other. We concentrated on how an action potential can
be passed from one cell to another and also how the action potential can propagate
along a cell, so that we can achieve rapid signalling over a larger scale.
Chapter 5
Pharmacokinetics
We have already looked at the kinetics of reactions in Chap. 1, here we now

consider how kinetic modelling can be applied to a whole ‘system’ such as an organ
or the body as a whole. We will see that kinetic models and in particular com-
partmental models appear in various guises. To introduce the topic we are going to
look at their use in pharmacokinetics.
Pharmacokinetics, often abbreviated to PK, is concerned with the fate of sub-
stances introduced into the body; most obviously this includes therapeutic agents, but
might also include things like toxins. Pharmacokinetics is widely used to study
substances in the whole body, for example where a drug has been delivered by
injection and subsequently blood samples have been taken to determine the plasma
concentration as a function of time. From these measurements we are interested in
inferring what happened to the drug: how rapidly it was absorbed into tissues, how
quickly it was removed from the body etc. However, pharmacokinetics also applies to
various imaging modalities where some form of contrast agent is introduced and we
want a spatially resolved map of absorption etc. We will revisit this when we look at
Tracer Kinetics. Whilst pharmacokinetics follows what happens to the substance in
the body it is also important to know what that substance is doing to the body, which is
the aim of pharmacodynamics, but something that we will not consider here.
5.1 ADME Principles
Pharmacokinetics is commonly divided into a number of separate processes,

referred to (for obvious reasons) as the ADME scheme, Fig. 5.1:
• Absorption is concerned with how the substance taken up into the blood stream
or a specific tissue where it has been introduced.
• Distribution is concerned with how the substance is distributed throughout the
body, most commonly how it goes from blood to tissue and back again.
• Metabolism is the conversion of the substance into other products usually via
enzyme reactions.
• Excretion is the removal of the substance from the body, for example via the
kidneys.

56 5 Pharmacokinetics
Fig. 5.1 ADME principles
The final two processes together represent elimination, since they both result in
the loss of the substance. In some cases we might also need to be concerned with
liberation, the release of the substance we are interested in from the formulation
that was used to introduce it. Most of these processes, particularly absorption and
distribution, will involve some form of transport process to get the substance across
membranes in the body. We have already looked at the various passive and active
transport processes through which this might be achieved in Chap. 4.
5.2 Compartmental Models
Whilst there are a number of strategies for pharmacokinetics modelling we will

follow the method of compartmental modelling. In this approach the substance is
considered to be contained by and move between different compartments. These
compartments are very broadly defined and may not represent any specific part of
the body, but rather a collection of tissues that share similar properties. The appeal
of this method is that the model remains simple and we have some hope of using it
to interpret the data; the disadvantage is that it can be harder to then interpret the
model in terms of the underlying physiology.
5.2.1 One Compartment Model
We will start with the simple model shown in Fig. 5.2; in this case we have a single
compartment into which the substance is absorbed and out of which it is eliminated.
This is most likely to represent a substance in the blood plasma that is unable to
cross into the tissues, thus we will talk about the concentration in this compartment
as the plasma concentration. If we assume that a first order output process describes
5.2 Compartmental Models 57
Fig. 5.2 One compartment model
the excretion with a constant ke then we can write a differential equation for the
system, ignoring the input for the time being, as:
dCp
¼ ke Cp ðtÞ; ð5:1Þ
dt
where Cp is the concentration in the compartment. If the substance were introduced

by intravenous injection then we can define the input to the system via the initial
conditions as the concentration at t = 0, i.e. Cp(0). Solving this equation gives a
simple exponential decay for the plasma concentration:
Cp ðtÞ ¼ Cp ð0Þeke t : ð5:2Þ
Thus it would be possible to determine ke from data of Cp measured through

blood sampling. From this we can also define the half-life of the substance as the
time taken for 50 % of the substance to be eliminated: T1/2 = ln 2/ke. We might also
be interested in the apparent volume of distribution for the compartment, which
can be determined simply by dividing the dose given, D, by the initial concentration
observed:
D
Vc ¼ : ð5:3Þ
Cp ð0Þ
This is the volume of the compartment into which the substance has distributed
itself. We might assume in this case that this volume would equal the blood volume,
since we have modelled the dug as only being in the blood. However, the volume of
distribution is typically not equal to the blood volume, partially due to the drug only
being in the blood plasma, thus not including the substantial volume occupied by
blood cells. The volume of distribution allows us to define the total body clear-
ance, the rate of elimination per unit of concentration:
Cltotal ¼ ke Vc : ð5:4Þ
In the case of an intravenous injection it will actually take a short time from
injection for the drug to have distributed throughout the whole circulation.
Additionally, excretion will not begin until the substance has reached the right area
of the body. Thus Cp(0) will often be taken shortly after injection once a steady
‘initial’ condition should have been reached.
We can generalise our model for any arbitrary input as:
dCp
¼ ke Cp ðtÞ þ Cin ðtÞ: ð5:4Þ
dt
It is also useful to write the equivalent expression for the total amount of the
substance:
dAp dCp
¼ Vc ¼ Vc ke Cp ðtÞ þ Ain ðtÞ; ð5:5Þ
dt dt
where A(t) = Vc C(t). This allows us to deal with the case of continuous intravas-
cular infusion. Letting Ain(t) = k0 and solving for Cp:
k0
C p ðt Þ ¼ 1 eke t : ð5:6Þ
Vc k e
From which we can determine the steady state concentration as t → ∞. Once

again we can define T1/2: now it tells us how long it takes for 50 % of the steady
state concentration to be reached, with the steady state being achieved after
approximately 5 half-lives.
We can also derive the more general result:
1 Zt 1
C p ðt Þ ¼ Ain ðkÞeke ðktÞ dk ¼ Ain ðtÞ eke t ; ð5:7Þ
Vc 0 Vc
which you might recognise as a convolution as indicated by the symbol on the

right-hand side and thus the need for the dummy variable λ. Our model of
instantaneous injection is thus equivalent to Ain(t) = D · δ(t), where δ(t) is the dirac
delta function. This allows us to consider more arbitrary intravenous introductions,
such as a series of injections or an infusion with a given duration.
Exercise A
(a) Starting with Eq. 5.5 derive the general result in Eq. 5.7.
(b) Show that Eq. 5.6, for the plasma concentration during continuous
infusion, can be obtained from this result when Ain(t) = k0.
(c) Determine the steady state concentration and consider the roles of k0 and
ke in reaching that steady state.
5.2.2 Absorption Compartment
So far we have considered only a first-order output. However, the substance may
have been administered orally, thus there is likely to have been at least one
membrane for it to pass through to get into the plasma, requiring a first order input,
as in Fig. 5.3. Now we have two equations:
dAa
¼ ka Aa ðtÞ þ Ain ðtÞ; ð5:8Þ
dt
dAp
¼ ka Aa ðtÞ ke Ap ðtÞ; ð5:9Þ
dt
where we have introduced a new compartment with concentration Aa. Note that our
convolution result above can be applied here. We could also write the equation for
the amount of substance being eliminated:
dAe
¼ ke Ap ðtÞ; ð5:10Þ
dt
from which we could solve for Ae(t). This might be useful if we were able to make
some direct measurements of elimination, for example urinary sampling.
Exercise B
(a) By modelling Ain as a Dirac delta function solve Eqs. 5.8 and 5.9 for
plasma concentration Cp in response to a single oral dose.
(b) Sketch the result and comment on its shape.
(c) If ka > ke comment on the shape of ln (Cp) as t → ∞. How might this be
used to estimate ke from plasma-sampled data?
Fig. 5.3 One compartment

plus absorption model
Following Exercise B for a single oral dose of substance (Ain(t) = D · δ(t)) the
plasma concentration can be written as:
BD ka ke t
C p ðt Þ ¼ e eka t ; ð5:11Þ
V c ka ke
where we have also included, B, the bioavailability of the substance. This is the
fraction of the delivered dose that actually appears in the blood, taking into account
losses in the gut through incomplete absorption and metabolism, as well as
excretion in the liver. This can be calculated by comparing the area under the
plasma concentration-time curve for oral delivery against the same dose of drug
delivered intravenously.
Exercise C
The table below shows the measured plasma concentrations of a drug that
was delivered by both intravenous injection and orally, with sufficient time in
between for the first dose to wash-out completely. In both cases 2.0 g of agent
were given.
(a) Calculate the bioavailability of the substance under oral administration.
(b) Assuming first order kinetics calculate the half-lives of elimination and
absorption.
(c) Determine the volume of distribution for this drug.
IV injection Oral administration

Time (h) Cp (mg/L) Time (h) Cp (mg/L)
0.10 190.2 0.17 47.7
0.25 176.5 0.25 61.5
0.5 155.8 0.33 71.5
0.75 137.5 0.5 83.4
1.00 121.3 0.75 87.3
1.50 94.5 1.00 83.5
2.00 73.6 1.50 69.2
3.00 44.6 2.00 54.8
5.00 16.4 4.00 20.3
7.00 6.0 7.00 4.5
5.2.3 Peripheral Compartment
Finally (at least for our purposes) we will consider a substance that also exchanges
from the plasma into the tissue: the distribution process. This requires us to add a
further compartment to the model, Fig. 5.4. We still have a ‘central’ compartment,
which includes plasma, but may also include tissues into which the substance rapidly
exchanges (such that we are unable to distinguish them from plasma). We have
gained a peripheral compartment, to where our drug is distributed, typically an organ
or a collection thereof. This combination is generally called a two-compartment
model; note that the absorption process is not counted as a compartment even though
it behaves like one. The equation for the central compartment is now:
dAp
¼ ka Aa ðtÞ ke Ap ðtÞ k12 Ap ðtÞ þ k21 Ag ðtÞ; ð5:12Þ
dt
and the peripheral compartment:
dAg
¼ k12 Ap ðtÞ k21 Ag ðtÞ: ð5:13Þ
dt
This can be solved reasonably easily for the simple cases of single dose and
infusion that we have considered so far to give solutions for Cp that are sums of
exponential terms.
Note that in pharmacological studies we generally have access only to the
plasma compartment and have to try to determine all the unknowns of the system
from that alone, which will be difficult even with this relatively simple system.
5.2.4 Multi Compartment Models
We can build increasingly complex compartmental models if we need to include

different peripheral compartments with differing rates of exchange, these will often
Fig. 5.4 Two compartment

model (with absorption)
represent different organs (or groups of organs). The model might also need to
account for the metabolism of the substance whilst in the tissue into various
products (which may or may not then exchange back into the plasma). Metabolism
is simply the result of a reaction, thus the reaction kinetics we saw in Chap. 1 can be
applied and at their simplest match those we have been using here, which should
not be a surprise since both are called kinetics. If the metabolic products do end up
in the blood stream it might be possible to sample them too and thus to quantify
more about the system.
A way to get around (or summarise) the complexity of a multi compartmental
model is to return to the convolution expression we had earlier, but now to write the
plasma concentration as:
Cp ðtÞ ¼ Cin ðtÞ RðtÞ; ð5:14Þ
where R(t) is the impulse response of the system (and gives rise to a transfer
function in the Laplace or frequency domains). This impulse response function can
be interpreted as the fraction of the substance that arrived at any point in time that is
still present at some time later. By definition this impulse response function must
start at unity at time zero and then monotonically decay.
5.2.5 Non-linear Models
So far all of the compartmental models have been linear. However, we have already
met various non-linear processes in reactions and transport. Hence, if we were
trying to build more physiologically accurate models we might need to incorporate
some non-linear terms. The most common way to model non-linearities is using the
Michaelis-Menten kinetics that we met in Chap. 1:
dCp Vmax Cp
¼ : ð5:15Þ
dt Cp þ Km
This can be regarded as pretty much first order if Cp ≪ Km or zero order

(saturated) if Cp ≫ Km.
Exercise D
A drug has been developed that is believed to be effective when the plasma
concentration is greater than 20 mg/L, but toxic if the plasma concentration
exceeds 100 mg/L.
The drug will be administered orally as a single dose D and is assumed to
have first order absorption as well as elimination kinetics.
(a) Identify the appropriate equation for the plasma concentration.
A clinical trial is about to begin and an appropriate dosing regime needs to be

established. Initial experiments have determined the kinetic parameters for the
drug in the table.
(b) Determine the maximum dose that can be delivered.
(c) If the maximum dose is delivered, at what time would the plasma con-
centration no longer be sufficient for the drug to be effective?
(d) Subsequent study of the drug indicates that the elimination process occurs
via an enzyme-mediated process that is more accurately modelled by
Michaelis-Menten kinetics with a constant Km = 2000 mg/L. Is the
analysis performed in part (c) still valid?
Parameter Value
F 0.8
Vc 10 L
ke 0.0028 min−1
ka 0.2 min−1
5.3 Tracer Kinetics
As we have seen pharmacokinetics sets out to describe the distribution of agents

introduced into the body. Any quantitative measurements typically take the form of
concentration samples taken from a specific compartment. With imaging devices it
is possible to get spatially resolved measurements from inside the body, i.e. images.
If we can introduce a contrast agent, or tracer, that will change the images, we
could in principle measure the distribution of this agent. Two very common
examples are Positron Emission Tomography (PET) and Magnetic Resonance
Imaging (MRI).
In PET, a radioactive tracer is injected into the bloodstream; in principle any
biologically compatible substance that can be radioactively labelled can be used.
The most common is a sugar called F-18 labelled flurodeoxyglucose (FDG), which
behaves similarly to glucose, in that cells take it up. However, it then does not under
go the subsequent (metabolic) reactions that glucose would and thus gradually
accumulates in the organ of interest.
In MRI, the most common tracers are based on gadolinium, a material whose
magnetic properties mean that its presence in tissues alters the image acquired using
MRI in direct proportion to its concentration. MRI can also exploit blood water as a
naturally occurring, or endogenous, tracer. This is most commonly applied in the
brain and the magnetism of blood water is inverted in the neck prior to imaging in
the brain. This process of Arterial Spin Labelling (ASL) allows the accumulation of
Fig. 5.5 Timeseries of ASL MRI tracer kinetics in the brain
labelled water to be observed in tissue, when it has had time to exchange from the
blood into cells. In all of these imaging methods it is possible simply to take an
image at the right moment, once enough of the contrast agent has accumulated, and
to visualise the distribution of the agent. Alternatively it may be possible to acquire
time series data of the agent as it is delivered and also removed, either by excretion
from the cells or through physical decay of the agent signal. For example, the
timeseries of the delivery labelled blood water into the brain using ASL MRI (along
with the decay of the label) is shown in Fig 5.5.
The quantification of delivery from imaging data is based on the principle of
tracer kinetics, which is conceptually similar to pharmacokinetics. The contrast
agent serves the same purpose as the drug in pharmacokinetics and we need a
description of the input function to the tissue in the imaging region, commonly
called the arterial input function. This can be obtained by blood sampling, more
common in PET, or extracted directly from the images by looking at the time series
in larger arteries, common in MRI.
Unlike pharmacokinetics, with imaging we have direct measures of the time
series of our tracer in the tissue. Rather than getting a single series for a whole
organ, we get measurements from each resolution element in the 3-dimensional
imaging volume, called voxels. Thus for each voxel we have a time series for the
tracer in that volume which will include both blood vessels and tissue. The form of
this time series will depend upon what the tracer does when it gets to the tissue: for
example, whether or not it all crosses the capillary wall and ends up in the cells or
extra cellular space. The simplest case that we might adopt is the one compartment
model, making the assumption that the tracer is eliminated according to a first order
process once it has arrived in the tissue. This is often a reasonable simplification for
FDG PET, ASL MRI and MRI using a Gadolinium contrast agent when imaging
outside of the brain. The actual elimination process will be different in each case
representing a combination of metabolism, exchange out of tissue back into the
venous circulation and physical decay of the tracer itself. We can write the model
for this process as:
dCt
¼ ke Ct ðtÞ þ F Cin ðtÞ; ð5:16Þ
dt
where we are measuring the concentration of tracer in the tissue, Ct, we know from
an independent measure the AIF, Cin, and we have a first order elimination process
with rate ke. Notice that the AIF is scaled by a parameter, F, which quantifies the
amount delivered. F has units of blood (carrying the agent) per volume of tissue per
5.3 Tracer Kinetics 65
unit time, this is in fact a quantify known as perfusion that we will meet again in
Chap. 8. We could solve this equation for the function of Ct with time and use this
to quantify perfusion from time series measurements. If we could provide an
infusion of tracer we could simply acquire a single image once steady state has been
reached to quantify perfusion. However, for various reasons infusion of the agent
generally isn’t possible, often because the agent is toxic in too high a dose. For
various PET tracers the elimination process is sufficiently slow that it is possible to
use a single injection of tracer: the tracer accumulates and remains in the tissue and
thus can be imaged sometime later for quantification.
Like pharmacokinetics we can extend the model to more complex cases. For
example, we can model what happens if there is a limiting rate at which the tracer
leaves the blood into the extravascular space using a two compartment model.
Tracer kinetics has a more general result that is applicable in a wide range of more
complex tracer uptake situations:
Ct ðtÞ ¼ F Cin ðtÞ RðtÞ: ð5:17Þ
This is effectively a further generalisation of the convolution result we met in

Eq. 5.14, where again we have an impulse response function R(t), in tracer kinetics
this is often called the residue function. This function describes the fate of a unit of
tracer once it has arrived in the voxel, whether it is washed out in the venous blood,
is eliminated through metabolism, decays through some physical process etc. This
function has special properties:
• Rðt ¼ 0Þ ¼ 1: this simply says that when a unit of tracer arrives it is all there;
• Rðt [ 0Þ 1: this say that tracer either stays or is removed;
• Rðt2 Þ Rðt1 Þ; t2 [ t1 : this says that once tracer has been eliminated it cannot
come back again.
Notice that R(t) only tells us about a unit of tracer, strictly it tells us what
happens to a (Dirac) delta of tracer that has arrived in the voxel. Thus Eq. 5.17 is
modelling the voxel as a linear time invariant system with the AIF as the input. The
observation that tracer accumulates in the voxel is thus a result of a delivery of
tracer, described by the AIF, and that the tracer, once it has arrived, isn’t imme-
diately removed from the voxel, as described by the residue function.
R(t) relates to the behaviour of the tracer in the voxel, and thus represents a
property (or properties) of the volume of tissue in the voxel. Thus the shape of R
(t) might be interesting in its own right in detecting tissue that is pathological, for
example when looking for cancerous tissue. Hence, once we have time series data
for each voxel we might try to extract R(t) either by writing it as a parameterised
function and trying to estimate the parameter values from the data, or performing
some form of numerical ‘deconvolution’.
5.4 Conclusions
In this chapter we have met a general method for the treatment of the body as a
whole system in which the behaviour of substances can by described mathemati-
cally. You should now be able to use compartmental models built around the
ADME principles to write down and to solve differential equations for simple
pharmacokinetics problems. We have also seen how the same principles can be
applied to measuring the delivery of a contrast agent to tissues, as is commonly
used in medical imaging.
Chapter 6
Tissue Mechanics
6.1 Introduction
The model of the cell that we have been examining in Chaps. 1 and 2 provides a
very neat link to models of tissue. At its simplest level, we can simply consider
tissue to be a collection of cells, each of which behaves according to the rules that
we set out in Chap. 2 (conservation of charge, equilibrium potential etc.). It turns
out that we can use these rules to understand how tissue behaves at a larger scale.
This then provides a good introduction to more general models of how tissue
behaves as a material with properties such as Young’s modulus. We will concen-
trate on soft tissue, rather than bones and muscle: the main reason for this is that the
deformations are larger and there is more interaction between fluids and solids in
soft tissue. Exactly the same principles can be applied to bones and muscle, but we
will leave discussion of these to the many sources of information about this that can
be found elsewhere.
6.2 Stress-Strain Relationships
Just like any material, human tissue must satisfy the conditions of equilibrium
(where forces must balance) and compatibility (where the material must occupy all
the available space). These two conditions can be written down in any suitable
co-ordinate system, dependent upon the particular problem: we will restrict our-
selves to Cartesian co-ordinates here. We will be using exactly the same equations
as for any material, but we will see that tissue behaves in quite different ways to
traditional engineering materials.

68 6 Tissue Mechanics
The equilibrium equations in three dimensions are:
@rx @sxy @sxz

þ þ þ Fx ¼ 0 ð6:1Þ
@x @y @z
@ry @syx @syz

þ þ þ Fy ¼ 0 ð6:2Þ
@y @x @z
@rz @szx @szy

þ þ þ Fz ¼ 0 ð6:3Þ
@z @x @y
where the material has a body force per unit volume in each direction, Fx, Fy, Fz.
We are using σ to denote direct stresses and τ to denote shear stresses and the
subscripts on the stresses indicate the direction of the force.
The strains are then related to the displacements as follows:
@u
ex ¼ ð6:4Þ
@x
@v
ey ¼ ð6:5Þ
@y
@w
ez ¼ ð6:6Þ
@z
@v @u
cxy ¼ þ ð6:7Þ
@x @y
@w @v
cyz ¼ þ ð6:8Þ
@y @z
@u @w
czx ¼ þ ð6:9Þ
@z @x
where we are using ε to denote direct strains and γ to denote shear strains. The
displacements are u, v and w in the x, y and z directions.
To solve these sets of equations, we need to relate stress and strain through the
material properties of the tissue: for example using Young’s modulus (the ratio of
stress to strain), E, and Poisson’s ratio (the ratio of strain perpendicular to applied
force to strain in the direction of the applied force), ν. There are other material
properties such as shear modulus, G, and bulk modulus, K, but these are all
inter-related. These material properties assume that the material is linear and iso-
tropic (i.e. the material properties are the same in every direction), which is far from
the case for many tissues. We will think about this in more detail later, but let’s start
by looking at a linear material.
6.2 Stress-Strain Relationships 69
6.2.1 Linear Material
If the material can be assumed to be linear and isotropic, then the equations are just
the same as those for any standard engineering material (such as steel). In this case
strains in each direction are the sum of the direct and indirect strains:
1
ex ¼ r x m r y þ rz ð6:10Þ
E
1
ey ¼ r y m ð r z þ rx Þ ð6:11Þ
E
1
ez ¼ rz m r x þ ry ð6:12Þ
E
Given suitable initial and boundary conditions, any stress and strain field can be
solved from this set of equations. A very simple example is given in Exercise A to
solve a one-dimensional displacement problem for a soft tissue.
Exercise A
Consider a linear tissue where gravity is the only body force and it acts in the
negative x direction.
(a) Assume: the problem is one-dimensional in the x direction; the material
is of height L; the boundary conditions are zero displacement at x = 0 and
zero stress at x = L. Show that the displacement of the tissue as a function
of x is:
qgx
u¼
ðx 2LÞ ðA:1Þ
2E
(b) If the tissue has density the same as water and a value of Young’s
modulus of 500 Pa, calculate the displacement of the top of the tissue if it
is 1 cm tall (for convenience take g = 10 m/s2).
You will find that the displacement is surprisingly large (approximately 1 mm

for a tissue of height 1 cm). This is because the value of Young’s modulus is very
small compared to the kind of values that you may be used to for more common
engineering materials (for example, steel has a Young’s modulus of around
200 GPa). Soft tissue is inherently very deformable, as you might expect.
6.2.2 Non-linear Material
Of course many tissues are neither linear nor isotropic: this introduces additional
complexity. As we showed in the previous section, human tissue is also not very
stiff, which means that the usual assumption of small displacements is no longer
valid. The theory behind tissue movement is thus more complex, because we have
to consider two frames of reference.
We won’t consider non-isotropic behaviour here, but we will look very briefly at
non-linear behaviour. A very common equation for modelling soft tissue is that
proposed by Fung, based on the idea of strain energy density, W. The strain energy
density function is just the amount of energy stored in the material per unit volume
due to strain. The advantage of writing down the strain energy density is that it
provides a single representation of the stress-strain relationship, from which indi-
vidual stress and strain terms can be calculated. For a linear isotropic material (such
as might be assumed for steel or aluminium, for example) with only direct stresses,
this would be:
1
W ¼ E e2x þ e2y þ e2z ð6:13Þ
2
Individual stress terms are calculated by differentiating Eq. (6.13) with respect to
individual strain terms, for example:
@W
rx ¼ ¼ Eex ð6:14Þ
@ex
hence Young’s modulus. This is the simplest example and anything else becomes
very complicated very quickly. However, we can extend the linear isotropic
material to a non-linear isotropic material by adapting the strain energy density
function to give:
1h 2 i
a k1 þ k22 þ k23 3 þ c ebðk1 þ k2 þ k3 3Þ 1
2 2 2
W¼ ð6:15Þ
2
The terms λi refer to the principal strains, which are simply the ratios of the new
‘lengths’ of an element to the original ‘lengths’ in each direction and thus directly
related to strain. The constants a, b and c are set to match experimental data. The
Fung expression above then models the fact that at small strains, the Young’s
modulus of tissue is very small (the first term), but that as the strain increases, the
effective Young’s modulus increases very rapidly (the second, exponential, term).
Again, we can use these results to derive relationships for stress and strain, but
these are quite complicated in comparison with the linear model, so we won’t go
any further here. We will, however, explore some examples where tissue interacts
with other substances in non-linear ways to show how we can use stress and strain
6.2 Stress-Strain Relationships 71
to model soft tissue. This will illustrate how we have to consider tissue mechanics
very differently to more traditional engineering mechanics.
6.3 Coupled Cell-Tissue Model
We start by going back to our model of the cell from earlier chapters. Consider a
simple cell surrounded by a solution. The cell has both positive and negative ions
inside, with concentrations c+ and c− respectively, and the extracellular space has a
concentration c*. The resulting pressure is proportional to the difference in internal
and external concentrations:
p ¼ RT ððc þ þ c Þ c Þ ð6:16Þ
where R and T are the ideal gas constant and absolute temperature respectively. This
is just like the ideal gas equation, with concentration replacing density.
In Chap. 2, we noted that the inside of the cell possesses a certain amount of
fixed charge, attached to proteins that cannot move across the cell membrane
(unlike the ions, which can move through channels). The intracellular concentra-
tions can be written in terms of the extracellular and fixed charge concentrations:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c þ þ c ¼ ð c 2 þ c f 2 Þ ð6:17Þ
Of course the fixed concentration, cf, varies with the size of the cell, since it is
the total charge that is fixed, and so concentration varies inversely with cell volume.
We can relate this to the expansion or contraction of the tissue:
!
/w0 f
c ¼
f
w c0 ð6:18Þ
V þ /0
dV
where /w0 is the tissue water content in some baseline state. This then gives a
scaling for the ‘new’ volume of the cell as a fraction of the ‘original’ volume: to
calculate this we need to define how the cell volume responds to pressure. This
requires a model of the tissue, as we examined earlier.
The very simplest model is to assume that displacements are (at least relatively)
small and that an applied pressure gives rise to a strain. This is based on the
definition of bulk modulus (K):
dp
K ¼ V ð6:19Þ
dV
The volume change will be:
dV
¼ ð1 þ eÞ3 1 ð6:20Þ
V
If the strain is small, then we can approximate this by:
dV
¼ 3e ð6:21Þ
V
Hence, using Eq. 6.19:
p ¼ 3Ke ð6:22Þ
The factor of 3 essentially comes from the fact that tissue is three-dimensional
and so the volume change is approximately three times the strain in each separate
dimension.
In its simplest form, we can combine all of the equations above to give:
0v
u0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
u !2 1
w f
Bu /0 c0 A c C
3Ke ¼ RT @t@c2 þ A ð6:23Þ
3e þ /w0
This gives a relationship between tissue strain and extracellular concentration

(we can calculate the pressure from the strain). Note that it is not a straightforward
expression to solve (although it can be written in simpler form, as in Exercise B): it
is also strongly non-linear. These are both common features of models derived from
physiological systems and we often require numerical methods to solve the equa-
tions and models that we derive. This does, however, give many interesting types of
behaviour.
Exercise B
(a) Show that Eq. (6.23) can be re-written in the form:
!2
RT /w0 c0f p
c ¼ ðB:1Þ
k þ /0
p w
2p 2RT
relating concentration to pressure.

(b) Comment on the form of this relationship.
6.4 Coupled Fluid-Tissue Model 73
6.4 Coupled Fluid-Tissue Model
Having considered a very simple model relating cellular behaviour to tissue

behaviour, let’s now consider the interaction between tissue and fluid more rigor-
ously. This fluid could be water or blood in very small blood vessels. It is not
possible to model individual pores or blood vessels, except on a very small scale, so
we have to consider an alternative way of describing this. The most common way of
doing this is to use the idea of a porous medium. This is a concept that is very
familiar in soil mechanics, where it is used to describe the drainage of soils with
water content.
The early work on soil consolidation (the fact that loading a soil results in a
gradual settlement of that soil when it is saturated with water) was performed by
Terzaghi in the 1920s and developed further by Biot in the 1940s. The theory builds
upon elastic continua theory with the addition of an expression for increment of
water content, θ, as a function of stress and water pressure:
1 p
h¼ rx þ ry þ rz þ ð6:24Þ
3H R
where H and R are physical constants. For a linear material, the stresses are related
to strains through Hooke’s law:
1 p
ex ¼ rx mry mrz þ ð6:25Þ
E 3H
1 p
ey ¼ ry mrx mrz þ ð6:26Þ
E 3H
1 p
ez ¼ rz mrx mry þ ð6:27Þ
E 3H
Hence:
p
h ¼ ae þ ð6:28Þ
Q
where:
e ¼ ex þ ey þ ez ð6:29Þ
is now defined as the total volumetric strain and:
2ð1 þ mÞG
a¼ ð6:30Þ
3ð1 2mÞH
1 1 a
¼ ð6:31Þ
Q R H
The water content is thus dependent upon the total volumetric strain and the
water pressure in a linear way, with the coefficients dependent upon the material
properties of the tissue.
The flow of a fluid through a porous medium is then normally governed by
Darcy’s law, which is again very commonly found in civil engineering:
j
q ¼ rp ð6:32Þ
l
In this equation, q is the vector flow rate per unit area (so it has the dimensions
of m/s), and κ and μ are the permeability of the porous medium and the viscosity of
the fluid respectively. Essentially this equation just says that flow moves down a
pressure gradient, with this being proportional to how easily it can pass through the
porous medium and inversely proportional to the fluid viscosity.
To solve this equation, we will consider the fluid to be incompressible, which is
in fact a very good approximation since blood is largely composed of water, which
is essentially incompressible. In this case, conservation of volume gives:
@h
¼ r q ð6:33Þ
@t
Equations 6.28, 6.32 and 6.33 can be combined to give:
j 2 @e 1 @p
r p¼a þ ð6:34Þ
l @t Q @t
which is the equation governing the fluid; the only thing remaining is the rela-
tionship between pressure and strain for the solid matrix. In this context these can
be written down as:
G
Gr2 u þ r e ¼ ar p ð6:35Þ
1 2m
These come from the equilibrium equations and the statement of Hooke’s law
above. Solving these equations then just requires a set of initial and boundary
conditions and a suitable numerical solver.
6.4 Coupled Fluid-Tissue Model 75
Exercise C
(a) Consider a porous medium in one dimension. Show that the governing
equation for pressure is of the form:
@ 2 p 1 @p
¼ ðC:1Þ
@x2 c @t
and derive an expression for the coefficient, c.

(b) Consider a material occupying the positive x line, such that the boundary
is at x = 0 and the material extends infinitely in the positive x direction.
Initially the pressure is zero throughout, but at time t = 0, a step change
in pressure of magnitude P is applied at the boundary. Calculate the
pressure in the material as a function of distance and time.
The result in Exercise C is very similar to the heat equation that governs the flow
of heat in a one-dimensional passive material. Solutions to this partial differential
equation are very standard and importantly are only dependent upon one parameter,
as shown in the exercise. This means that they are well understood and can be
applied relatively easily, although of course we are assuming linear and isotropic
behaviour, so it does become much more challenging if this is not the case.
6.5 Coupled Blood Vessel-Tissue Model
As well as considering tissue as a continuum, we might also want to describe how it

interacts with blood flow in individual vessels, i.e. on a larger scale. There is here a
strong interaction between the fluid and the structure due to the fact that the fluid
has a variable pressure. This is very similar to the theory of thin-walled and
thick-walled vessels, used widely in standard pressure vessel theory, for example.
Note that we will be changing from x-y-z co-ordinates to r-θ-z co-ordinates.
Let’s start by considering a thick cylinder under uniform pressure as our model
of a blood vessel. The radial and circumferential stresses are given by:
B
rr ¼ A þ ð6:36Þ
r2
B
rh ¼ A ð6:37Þ
r2
where A and B are constants, set by the boundary conditions. Note that these come
from solving the equilibrium and compatibility equations in cylindrical polar
co-ordinates (these are standard results, so we won’t derive them here).
If we assume that we have a blood vessel with blood at a uniform pressure

p inside it and pext outside it, these equations turn into:
pR2 pext ðR þ hÞ2 þ R2 ðR þ hÞ2 ðpext pÞ=r 2

rr ¼ ð6:38Þ
ðR þ hÞ2 R2
pR2 pext ðR þ hÞ2 R2 ðR þ hÞ2 ðpext pÞ=r 2

rh ¼ ð6:39Þ
ðR þ hÞ2 R2
where the wall has thickness h and inner radius R.
Exercise D
Show that the expression in Eq. 6.38 is compatible with the pressure
boundary conditions.
At this point, we need to make an assumption about the stress or strain in the
axial direction to calculate the displacements. There are two very common choices:
plane stress (where the stress in the axial direction is zero) or plane strain (where the
strain in the axial direction is zero). We will consider these briefly in turn.
6.5.1 Plane Stress
In the first case, there is no stress in the axial direction and the vessel is allowed to
expand in this direction as it wishes. The circumferential strain in cylindrical polar
co-ordinates is given by:
u 1
eh ¼ ¼ ðrh mrr Þ ð6:40Þ
r E
where the material typically has a Poisson’s ratio, ν, of 0.5 (the value for an
incompressible material). Substituting the expressions for stress into this equation at
the inner radius then gives the internal displacement of the vessel wall. The
resulting expression is long and complicated, so we won’t quote it here; however,
the main thing to note is that the displacement is linearly proportional to both the
internal and the external pressure. This should be no surprise, since we assumed a
linear material; however, it does still illustrate that even with these simple
assumptions the final expression can be highly complicated.
6.5 Coupled Blood Vessel-Tissue Model 77
6.5.2 Plane Strain
In the second case, there is no strain in the axial direction and the vessel is con-
strained to expand only in the radial direction. The circumferential strain in
cylindrical polar co-ordinates is thus:
u ð 1 m2 Þ m
eh ¼ ¼ rh rr ð6:41Þ
r E 1m
Substituting the expressions for stress into this equation again gives a linear
relationship between pressures and wall displacement, but with different coefficients
(it can be shown that the plane stress condition gives a smaller displacement than
that for plane strain for the same applied internal pressure).
Having derived this result, we can simply say that both assumptions yield the result
that displacement is proportional to pressure. Since the vessel wall inner radius is not
zero at zero pressure, we introduce an offset and turn the expression into the form:

R Ro
p po ¼ G o ð6:42Þ
Ro
where we define the wall stiffness as Go and the vessel wall has radius Ro at
pressure po. Notice that we are slightly abusing our notation to write R as the
variable inner wall radius now (i.e. equal to the original radius plus the displace-
ment). This allows us to relate the pressure inside the vessel to the wall radius.
The pressure inside the blood vessel will be determined by the flow of blood in
the vessel, so, having considered the relationship between wall displacement and
applied pressure, we turn to the governing equations for flow inside the vessel.
We start by assuming an axisymmetric straight vessel and write down the
equations for continuity and momentum as:
@A @ ðAU Þ
þ ¼0 ð6:43Þ
@t @x
@U @U 1 @p f
þU ¼ þ ð6:44Þ
@t @x q @x qA
where the vessel has cross-sectional area A, and the fluid has average velocity U,
pressure p and density ρ. f denotes the friction term caused by the viscosity of the
fluid. The first equation (continuity) balances the storage of fluid due to changes in
cross-sectional area with the rate of change of flow along the vessel. The second
equation can be thought of as an application of Bernoulli’s equation with an added
term for the friction of the fluid (this is what dissipates energy in a real fluid).
Note that there are four variables in these two equations: we therefore need two
more equations. We can calculate the frictional force if we estimate the velocity
profile of the fluid through the vessel: it is normal to guess a polynomial profile, as
you will see in Exercise E.
Exercise E
Assume that the velocity of the fluid through a blood vessel is of the form:
r n
uðr Þ ¼ Umax 1 ðE:1Þ
R
(a) Show that the mean flow velocity, averaged over the cross-sectional
area, is given by:

n
U ¼ Umax ðE:2Þ
nþ2
(b) Hence show that the frictional force per unit length of the vessel is given
by:
f ¼ 2plðn þ 2ÞU ðE:3Þ
This result means that we have just three variables remaining. The usual way to
complete this set of equations is to consider the relationship between the fluid
pressure and the vessel wall area, along the lines that we derived earlier. Re-writing
Eq. 6.42 in terms of vessel area gives:
rffiffiffiffiffi
A
p po ¼ Go 1 ð6:45Þ
Ao
The governing equations can thus be written as:
@A @Q
þ ¼0 ð6:46Þ
@t @x
rffiffiffiffiffi
@Q @ Q2 Go A @A 2plðn þ 2ÞQ
þ þ ¼ ð6:47Þ
@t @x A 2q Ao @x qA
where we are writing them in terms of area and flow rate (defined as Q = UA), rather
than area and velocity.
6.5 Coupled Blood Vessel-Tissue Model 79
Now that we have two simultaneous equations, we can examine how they
behave. You will notice that they are very similar in form and so it is common to
write them in vector/matrix form:
@U @U
þ HðUÞ ¼ BðUÞ ð6:48Þ
@t @x
where:

A
U¼ ð6:49Þ
Q
" #
0 qffiffiffiffi 1
HðUÞ ¼ 2
QA2 þ G2qo AAo 2 QA ð6:50Þ
" #
0
BðUÞ ¼ ð6:51Þ
2plðnqAþ 2ÞQ
These are called the quasi-linear matrix form of the equations and might remind
you of the wave equation. The eigenvalues of H(U) can be shown to be the mean
flow speed plus or minus the wave speed, and are given by:
sffiffiffiffiffiffi 1
Q Go A 4
k¼ ð6:52Þ
A 2q Ao
The first term is the mean speed of the fluid and the second term is the wave
speed. Any disturbance in the flow (such as a pulse of blood) will thus cause a
movement both forwards and backwards. This movement will have a speed that is
proportional to the square root of the vessel wall stiffness: hence in very stiff
vessels, the wave speed is very high. Typical values are that the vessel wall stiffness
is of order 10 kPa and the density of blood is about 1000 kg/m3, so the wave speed
is a few metres per second.
In the limit as the vessel walls become infinitely stiff, the pulse wave speed tends to
infinity and we can get shock waves forming. This is one of the reasons why car-
diovascular problems develop with age, as vessel walls become stiffer (for example as
the result of fatty deposits on the vessel wall). The wave speed will also be higher
when the vessel is dilated, i.e. when the pressure is higher, but this is a smaller effect.
6.6 Conclusions
In this Chapter, we have examined ways in which we can model both tissue and its
interaction with cells, water and blood. The key idea of the chapter is the fact that
there is this interaction between difference components of body organs: they cannot
be considered in isolation, as for many other engineering problems. This does make
the task of understanding their behaviour very complicated and we have only
examined a few very simple cases to give you the idea of what can be done in this
field. There are very many other examples that you will be able to find elsewhere.
Chapter 7
Cardiovascular System I: The Heart
7.1 Overview
The cardiovascular system, which is also known as the circulatory system, com-
prises the human heart, blood vessels and blood. It has five main functions:
1. Distribution of O2 and nutrients, for example glucose and amino acids, to all
body tissues;
2. Transport of CO2 and metabolic waste products from the tissues to the lungs
and other excretory organs;
3. Distribution of water, electrolytes and hormones throughout the body;
4. Contributing to the infrastructure of the immune system;
5. Thermoregulation.
Its overall aim is to maintain the body within well-defined limits, irrespective of
external stimuli: this equilibrium is termed homeostasis.
A schematic outline of the system is shown in Fig. 7.1 with the heart at the
middle. Although it looks complicated, it is actually relatively simple: the blood
flows in a continuous loop, from the right ventricle to the lungs (where carbon
dioxide in the blood is removed and exchanged for oxygen), to the left atrium and
left ventricle, out to the body tissues (where oxygen passes into the tissue in
exchange for carbon dioxide) and back into the right atrium and ventricle. If you get
lost, try tracing your finger around a path until you get back to where you started
from.
Note that, although the lungs are a single entity, the remainder of the body
tissues are very widely spread out. The main body tissues in descending order of
blood flow rate are shown in Table 7.1. The circulation is sometimes divided into
pulmonary and systemic components, where the pulmonary circulation comprises
the lungs and the systemic circulation all the body tissues.

82 7 Cardiovascular System I: The Heart
Fig. 7.1. Structure of cardiovascular system and heart. This figure is taken, without changes, from
OpenStax College under license: http://creativecommons.org/licenses/by/3.0/
Table 7.1 Distribution of blood between body organs, in descending order

Kidney Spleen Skeletal muscle Brain Liver Skin Bone Heart muscle Other
22 % 21 % 15 % 14 % 6% 6% 5% 3% 8%
7.2 Structure and Operation of the Heart 83
7.2 Structure and Operation of the Heart
The most important part of the system is the heart, which acts as the pump driving
blood round the body. The heart is divided into two sections, left and right: the left
side pumps oxygenated blood from the lungs to the tissues, whilst the right side
pumps de-oxygenated blood from the tissues back to the lungs. A healthy adult
heart pumps approximately 5 L of blood around the body every minute. Over the
course of a 70-year life span, a heart will beat several billion times and deliver over
100 million litres of blood.
A schematic of the structure of the heart is shown in Fig. 7.1. Although it looks
complex, again it is relatively straightforward. The main division is into the left and
right sides (confusingly pictures of the heart are normally shown as looking front
on, whereas left and right are defined as per the person’s left and right). They are
also normally coloured in red and blue respectively since de-oxygenated blood is
blue and oxygenated blood is red.
Both sides are subdivided into an atrium and a ventricle: there are thus four
chambers in total. The two atria are thin-walled and receive blood from the veins
(pulmonary veins on the left side and superior and inferior vena cava on the right
side), whereas the two ventricles are thick-walled and pump blood out through the
arteries (aorta on the left side and pulmonary arteries on the right side). Note how
the aorta is arched as the blood comes out of the top of the left ventricle upwards
but then goes downwards (the small vessels coming off the top of this arch supply
the brain and cardiac muscle).
There are also four valves: one at the exit from each chamber. The mitral and
aortic valves are found on the left side at exit from the atrium and ventricle
respectively; the tricuspid and pulmonary valves on the right side similarly. The two
valves at exit from the ventricles are semilunar valves with three cusps, i.e. three
semicircles that act to cover the cross sectional area, whereas the mitral and tri-
cuspid valves have inextensible fine chords (chordae tendineae) extending from free
margins of the cusps to the papillary muscles, which contract during systole, thus
acting somewhat like parachutes.
The heart wall is comprised of three layers, as shown in Fig. 7.2: the epicardium,
myocardium and endocardium, from outer to inner. The endocardium is a thin layer
of cells, similar to the endothelium found in blood vessels; the myocardium is the
muscle and the epicardium is the outer layer of cells. The whole heart is contained
within the pericardium, a thin fibrous sheath or sac, which prevents excessive
enlargement. The pericardial space contains interstitial fluid as a lubricant.
The cardiac cycle comprises two parts: the resting (or filling) phase, termed
diastole, and the contractile (or pumping) phase, termed systole. During the first
phase, blood enters the heart on both sides simultaneously through the veins. The
pulmonary and aortic valves are both shut, so the atria and ventricles expand.
During the second phase, the heart contracts strongly and the pulmonary and aortic
valves open: a pulse of blood thus flows out into the pulmonary arteries and aorta.
However, it is important to remember that blood is still flowing in through the
Fig. 7.2. Structure of heart wall. This figure is taken, without changes, from OpenStax College
under license: http://creativecommons.org/licenses/by/3.0/
veins: the heart must accept this blood and make sure that blood is not forced back
into the veins at any point in the cycle.
This is when the separate chambers become important: the tricuspid and mitral
valves are shut automatically by the pressure that builds up in the ventricles and so
the atria continue to receive blood whilst it is being forced out of the ventricles. The
structure of the tricuspid and mitral valves is the reason for this unidirectional
behaviour, since they are attached to the ventricular walls and thus permit flow
through in one direction only, as shown in Fig. 7.1. Similarly, when the ventricles
relax, the pulmonary and aortic valves act to prevent blood from flowing back into
the ventricles: these valves are thus also unidirectional.
Note that the heart valves behave in a very similar way to electrical diodes,
where the current can only flow one way. In fact there is always some leakage in the
heart valve, just as there is in a diode, but as long as this doesn’t get too large, it can
safely be ignored. The structure of the heart thus enables it to receive a constant
flow of blood through the veins, whilst pumping out blood at regular intervals
through the arteries. It is important to remember that the body receives a pulsatile,
rather than a steady, flow of blood, although this pulsatile flow is smoothed out
rapidly as the blood passes into the vasculature.
The myocardium is the muscle in the heart wall whereby the heart contracts and
obviously it must be supplied with blood to function properly. The cardiac muscle has
a network of blood vessels to supply it, via the left and right coronary arteries, which
branch off the ascending aorta. After passing through the myocardium these vessels
feed back to the right atrium via the coronary sinus. The operation of the coronary
system is thus vital in maintaining continuous blood supply to the heart muscle.
7.2 Structure and Operation of the Heart 85
Ischaemic (meaning a reduction in blood supply) heart disease is caused by an

imbalance between the blood flow to the myocardium and the myocardial metabolic
demand. If the coronary arteries block up (through deposits of fatty substances on
the wall), their resistance to flow increases, reducing the supply of blood.
A myocardial infarction (the technical term for a heart attack) means that some of
the heart muscle cells die due to this lack of blood supply. This death can cause
irregular rhythms, which can be fatal even if there is enough healthy muscle to
continue pumping.
After myocardial infarction, the heart can recover, although it will rarely be able
to pump as much or as efficiently as before. Nowadays, most patients survive heart
attacks, due to rapid treatment, and bypass operations, which open up the blood
supply, can reduce the chance of a future heart attack. The heart obviously plays a
crucial role in the operation of the human body: if it stops operating for even a very
short time, the build-up of waste products and starvation of nutrients leads rapidly
to irreversible cell death.
7.3 Measurement of Cardiac Output
Given the key role of the heart in the operation of the human body, it is obviously
important to be able to monitor its behaviour. We will look at a number of ways, but
the first one that we will examine is termed Cardiac Output (CO): this is simply the
rate at which the heart pumps blood out into the pulmonary and systemic circu-
lations. There are a variety of ways to measure CO, both directly and indirectly.
The first technique is based on Fick’s principle, which simply equates the
absorption of oxygen in the capillaries to the oxygen inhaled in the lungs (effec-
tively balancing the input and output of oxygen in the body). The absorption of
oxygen in the capillaries can be calculated as the difference between the oxygen
contents in the arteries and the veins: these are normally measured in units of
ml_O2/ml_blood. The oxygen consumption of the body, measured in ml_O2/min, is
the product of this difference and the CO, measured in ml_blood/min: we are
essentially using oxygen as a tracer. CO can thus be calculated by:
VO2
CO ¼ ð7:1Þ
ðO2arterial O2venous Þ
Indicator dilution techniques are based on the idea that if a known amount of a
substance is injected into an unknown volume, the final concentration allows the
volume to be calculated. In Hamilton’s dye method a quantity of non-toxic dye is
injected into a vein: it mixes with the blood as it passes through the heart and lungs.
By taking successive arterial blood samples, the mean concentration can be cal-
culated. Cardiac output is then calculated from:
ADI
CO ¼ ð7:2Þ
MDC DFP
i.e. the ratio of the Amount of Dye Injected (ADI) to the product of the Mean Dye
Concentration (MDC) with Duration of First Passage (DFP).
A variant on this is the thermodilution technique, which is the most common
method to measure CO. A modified Swan-Ganz catheter with a thermistor at its tip
and an opening a few centimetres from the tip is inserted from a peripheral vein
such that the tip is in the pulmonary artery and the opening is in the right atrium.
A small amount of cold saline is injected into the atrium and this mixes with the
blood as it passes through the ventricle and into the pulmonary artery, thus cooling
the blood. By measuring the blood temperature, the flow rate can be calculated,
since the temperature drop is inversely proportional to the blood flow. This is very
similar to the indicator dilution technique, but measuring the dilution of temperature
rather than concentration.
More modern non-invasive techniques (i.e. those that do not require the insertion
of any probes into the body) use imaging methods to estimate real-time changes in
the size of the ventricles. This can be used to give the Stroke Volume (SV): the
volume of blood pumped out in one cardiac cycle. The CO can then be calculated,
using a simultaneous measurement of the heart rate (HR), from:
CO ¼ SV HR ð7:3Þ
Imaging methods, like MRI, have the potential for great accuracy and for
beat-to-beat measures of CO, but are of course very much more expensive and
difficult to use, despite being non-invasive in nature.
Exercise A
Calculate cardiac output if oxygen consumption is calculated to be
200 ml_O2/min and arterial and venous oxygen concentrations are 0.21 and
0.16 ml_O2/ml_blood. If the heart rate is 60 beats per minute, calculate the
stroke volume. Comment on your answer.
7.4 Electrical Activity of the Heart
Thus far, we have simply considered the mechanical activity of the heart, by
looking at it as a pump and thinking in terms of blood flow. We will now look at the
electrical activity of the heart, considering what drives it, in particular what makes it
expand and contract, before looking at how we can monitor its performance. This
builds upon Chap. 3, where we looked at the generation of the action potential.
7.4 Electrical Activity of the Heart 87
7.4.1 The Action Potential
The action potential of a cardiac muscle cell, as shown in Fig. 7.3a, is similar to that
of the cells that we examined earlier, but with the addition of a sustained plateau
due to the influence of Ca2+. Calcium ions play a very important role in cardiac
cells.
Depolarization: This is the first phase of cardiac cell firing: this lasts approxi-
mately 2 ms. As in the example shown in Chap. 3, the action potential is generated
by a sudden transient rise in Na+ permeability with a subsequent increase in
potassium permeability (remember the m, n and h gates). The inward current of Na+
ions through voltage-gated Na+ channels becomes sufficiently large to overcome the
outward current through K+ channels and so the cell potential increases (becomes
less negative): the membrane thus suddenly becomes much more permeable to Na+
ions due to the channels. This process then activates more Na+ channels and the
process becomes self-perpetuating (essentially there is positive feedback and the
system is unstable).
Plateau: The role of calcium now becomes crucially important. When the
membrane potential reaches a threshold, voltage-dependent calcium channels open
and there is a large influx of calcium ions. There are also some potassium channels
that close upon depolarization (the opposite to the potassium channels discussed in
Chap. 3). These two effects contribute to the plateau, where the membrane potential
decays only very slowly over approximately 200 ms, even though the sodium
permeability returns virtually to its resting value.
Repolarization: The next stage is a more rapid drop. This is due to a gradual
decline in the calcium permeability and an increase in potassium permeability. The
precise mechanisms are still not fully understood, but the decline in calcium per-
meability may be due to the effect on the calcium channels of the accumulation of
calcium in the cells. By the end of this phase, the outward potassium current returns
to its dominant position and the membrane potential returns to its resting value
before the next action potential begins.
Since the action potential automatically causes a mechanical response, with the
tissue contracting and becoming shorter in length, it might be wondered why there
is a need for a more complicated action potential. This is because in cardiac muscle,
the duration and strength of the contraction are important parameters that will have
to be adjusted dependent upon the circumstances: more control is needed over the
action potential and this is achieved through the use of calcium ions. This is shown
in Fig. 7.3b, where the prolonged action potential results in a more powerful
contraction: calcium ions thus give a lot of control over the magnitude of the tensile
forces.
Fig. 7.3. Action potential in cardiac muscle cells: a electrical activity; b electrical and tensile
behaviour. This figure is taken, without changes, from OpenStax College under license: http://
creativecommons.org/licenses/by/3.0/
7.4.2 Pacemaker Potential
Heart cells in isolation beat rhythmically, i.e. they exhibit a spontaneous action
potential that does not require an external stimulus to start the process, thus they do
not have a true resting potential. Like skeletal muscle, the cardiac action potential
has a phase where the open potassium channels slowly close (this was the ‘under-
shoot’ for the earlier action potential). As the system returns to ‘baseline’ the
threshold is once again reached for depolarization and the whole process restarts.
This is referred to as the pacemaker potential. In reality the rate of action potentials
in cardiac muscle cells varies from cell to cell and thus there needs to be some form
of co-ordination to ensure that the whole muscle contracts simultaneously.
7.4.3 Cardiac Cycle
The stimulus for the cardiac action potential is provided by the SinoAtrial Node
(SAN): the cells found in the SAN are often termed pacemaker cells. It has an
unstable resting potential, as shown in Fig. 7.4, which decays from approximately
−60 mV to a threshold value of −40 mV, at which an action potential is initiated.
The rate of decay of this resting potential thus determines the rate of firing and
hence the heart rate. In a healthy human at rest, this will result in action potentials
being started approximately 70–80 times each minute.
As the SAN cells depolarise, they stimulate the adjacent atrial cells, causing
them to depolarise similarly. The depolarisation wave spreads over the atria in an
outward-travelling wave from the point of origin, the action potential passing from
cell to cell via electrical synapses, as described in Chap. 4. Both atria contract
nearly simultaneously, as do both ventricles shortly afterwards. Co-ordination
between atria and ventricles is achieved by specialised cardiac muscle cells that
make up the conduction system, as shown in Fig. 7.5.
The wave is prevented from spreading past the limits of the atria by a fibrous
barrier of non-excitable cells: the only excitable tissue that crosses this barrier is the
Bundle of His. At the origin of this bundle is a mass of specialised tissue about 2 cm
long and 1 cm wide called the AtrioVentricular Node (AVN). The conduction
velocity through the AVN is approximately 0.1 m/s, some 10 % of that of the atrial
cells. The delay that this causes is vital in preventing the ventricles from contracting
before the atria have completed their contraction. The impulse from the AVN
travels through the Bundle of His, which splits into the left and right bundle
Fig. 7.4. Action potential at the SAN. This figure is taken, without changes, from OpenStax
College under license: http://creativecommons.org/licenses/by/3.0/
Fig. 7.5. Conduction system in the heart. This figure is taken, without changes, from OpenStax
College under license: http://creativecommons.org/licenses/by/3.0/
branches before dividing into the multiple fibres of the Purkinje system. This
distributes the impulse over the inner walls of the ventricles causing contraction.
Although the SAN alone would produce a constant rhythmic heart rate, there are
regulating factors present, as the heart rate may need to increase, for example during
increased physical activity, or to decrease, for example when sleeping. This is
largely controlled by the AVN and we will look at this in more detail later. Most of
the changes in heart rate are mediated through the cardiac centre in the brain via the
sympathetic and parasympathetic nervous systems, which will be examined in
Chap. 10. Obviously, there are a large number of factors involved in setting the
heart rate, such as body temperature, ion concentrations, oxygenation levels, blood
pressure and even emotions (although how emotions, such as stress, drive your
heart rate is a very complicated subject).
Heart rate increases to raise Cardiac Output (CO) since stroke volume is largely
constant (Eq. 7.3). There are only three factors that can affect CO: the filling
pressure of the right heart (the preload), the resistance to outflow from the left
ventricle (the afterload) and the functional state of the heart. The last includes heart
rate and contractility: the ability of cardiac muscle to generate force for any given
fibre length. The stroke volume is known to vary with the ventricular end-diastole
volume (EDV), according to the Frank-Starling law of the heart, such that an
increase in this volume causes an increase in stroke volume.
Figure 7.6 shows the pressure and volume changes during the cardiac cycle,
firstly over time (LHS) and then against each other (RHS), for the left ventricle. The
second of these plots is used for any cycle, rather like an internal combustion
Fig. 7.6. Cardiac pressure volume cycle (LV = Left ventricle). This figure is taken, without
changes, from Wikimedia under license: https://creativecommons.org/licenses/by-sa/2.5/deed.en
engine, where the area enclosed by the loop is the work done to power the cycle (in
the human body this must come from metabolism). Increases in preload raise stroke
volume, but increases in afterload decrease stroke volume, as the loop moves
upwards and the left side to the right. Increases in contractility move the end
systolic pressure-volume relationship (ESPVR) line upwards, thus increasing stroke
volume since line D moves to the left.
7.4.4 Introduction to Electrocardiography
The pumping cycle is controlled by a conduction system to give a series of events in

a very well-defined order. The conduction system generates current densities
through the membrane activity of the heart muscle cells and this is important in how
we measure electrical activity in the heart. The ionic currents flow in the thorax,
which contains no other sources or sinks and is thus an almost entirely passive
medium.
Currents flowing through resistive loads produce voltages: they are of course
very small, but, by placing electrodes at different positions on the human body,
potential differences can be recorded. These potential differences form the basis of
the ElectroCardioGram (ECG). There are of course difficulties involved in
attempting to measure very small potential differences (of the order of a few mV) on
a living human, but we won’t look at those here. We will confine ourselves to
understanding the production and interpretation of the signal.
The ECG is based on the idea of an equilateral triangle (known as Einthoven’s
triangle) with the heart as a current source at the centre. Since the potential dif-
ference depends upon both the current magnitude and direction, the ECG is a
vector quantity. The three bipolar leads, i.e. measurements made between two
points, are known as Lead I (right arm to left arm), Lead II (right arm to left leg) and
Lead III (left arm to left leg): hence the idea of a triangle, as shown in Fig. 7.7.
There are then a further nine unipolar leads, i.e. measurements made at one point:
Fig. 7.7. Locations of leads

used in ECG. This figure is
taken, without changes,
from OpenStax College
under license:
http://creativecommons.org/
licenses/by/3.0/
aVR (right arm), aVL (left arm), aVF (left leg) and V1–V6, as shown in Fig. 7.7,
where V1–V6 run from left to right across the chest. Note that this naming system is
simply the convention that is used for historical reasons.
Although each of these 12 leads will give a different output, the characteristics of
the heart cycle are most clearly shown in lead II, since this pair aligns most closely
with the major axis of potential differences in the heart. A typical waveform is
shown in Fig. 7.8. It is labelled at various points: P, Q, R, S and T.
P wave: The depolarisation of the atria prior to atrial contraction causes a small
low-voltage deflection, followed by a delay.
QRS complex: The depolarisation of the ventricles prior to ventricular con-
traction causes a large voltage deflection: this is the largest-amplitude section of the
ECG. Although atrial repolarization occurs before ventricular depolarisation, the
resulting signal is very small and thus not seen.
T wave: Ventricular repolarization. This last section is much the most variable
and often very hard to see.
The sections between these features are close to zero potential, since there are
few other sources of potential difference.
Fig. 7.8. Schematic of ECG signal. This figure is taken, without changes, from OpenStax College
under license: http://creativecommons.org/licenses/by/3.0/
A great deal of research has been done to interpret the ECG in terms of providing
clinical diagnoses. The simplest measure that can be extracted is the heart rate, nor-
mally taken as the inverse of the time between successive R peaks (the RR interval),
measured in beats per minute. A trained clinician will also be able to infer a very large
amount of information about the state of the heart from simply looking at an ECG,
using changes in the relative timings and amplitudes of different sections of the ECG.
Measurement of the ECG is also simple and cheap, meaning that it is normally the first
monitoring device attached to a patient. Note that Fig. 7.8 shows a perfectly regular
heartbeat, which is not what is actually found: in fact the variability in your heart rate is
a good sign and this can also tell us a lot about how the heart is working.
Exercise B
(a) What is the heart rate (in beats per minute) of the patient with the ECG
shown in Fig. 7.8? Is there any variability from beat to beat?
(b) By holding the fingers of one hand against the other wrist, estimate your
heart rate in beats per minute. After a short period of light exercise,
estimate it again. Comment on why the values are different.
A brief summary of the cardiac cycle is shown in Fig. 7.9, relating the ECG to
the pressure and volume changes. Check that you can follow the process from both
a mechanical and an electrical point of view. There is ejection during systole, which
is generated by ventricular depolarisation (shown by the QRS complex), which
causes the pressure in the left ventricle to increase rapidly and the volume to
drop. When the pressure rises above the aortic pressure, the aortic valve opens and
blood leaves the ventricle: the pressure thus drops and when it drops below aortic
pressure the aortic valve closes again. Ventricular repolarization occurs during the
remainder of systole (shown by the T wave) whilst the ventricles slowly return to
their steady state conditions.
Note that there are characteristic heart sounds, which are also shown in Fig. 7.9.
The first heart sound is low frequency and associated with closure of the atri-
oventricular valves; the second is higher frequency, reflecting a longer ejection
period in the right ventricle; and the third is associated with rapid refilling. There is
a fourth sound, corresponding to atrial systole, but this is not normally audible.
Fig. 7.9. Cardiac cycle. This figure is taken, without changes, from Wikimedia under license:
https://creativecommons.org/licenses/by-sa/2.5/deed.en
7.5 Conclusions 95
7.5 Conclusions
In this chapter we have looked at the human heart: how it is constructed, how it
works as a pump and how this is controlled. We have also looked at how we
measure how it is performing, both in terms of the cardiac output and the ECG. In
the next chapter we will look at the rest of the cardiovascular system and its
interaction with the heart.
Chapter 8
Cardiovascular System II: The Vasculature
In the previous chapter, we looked at the centre of the cardiovascular system,

the heart. The heart is responsible for receiving blood and pumping it out again.
This blood passes round the body through the vasculature, which is the complete set
of blood vessels in every part of the human body. In this chapter, we will look at
these blood vessels, how we can model them mathematically and how they con-
tribute to the maintenance of homeostasis.
8.1 Anatomy of the Vascular System
Blood vessels divide into five main types: arteries, arterioles, capillaries, venules
and veins. Blood passes through them in that order and they can be distinguished by
differences in size, characteristics and function. Blood exiting the heart first flows
through the aorta, which divides into arteries. These divide in turn into arterioles
then capillaries before joining together in venules before entering the heart from
the veins via the venae cavae. A schematic is shown in Fig. 8.1, where we will
explain many of the details shown in later sections. The vessels shown in bright red
are oxygenated, whereas the vessels shown in dark red are de-oxygenated, since the
oxygen carried by the blood has been transferred to the surrounding tissues in the
capillaries.
The arteries are thick-walled vessels that expand to accept and temporarily to
store some of the blood ejected by the heart during systole and then to pass it
downstream by passive recoil during diastole. As the arteries branch off, their
diameter decreases, but in such a manner that the total cross-sectional area increases
several-fold. The arteries have a relatively low and constant resistance to the flow.
The artery walls comprise three layers: the tunica intima (or tunica interna), the
tunica media and the tunica adventitia (or tunica externa), which are simply the
inner, middle and outer layers. The middle layer is mainly smooth muscle and is

98 8 Cardiovascular System II: The Vasculature
Fig. 8.1 Schematic of different blood vessel types and structure (this figure is taken, without
changes, from OpenStax College under license: http://creativecommons.org/licenses/by/3.0/)
usually the thickest layer: it is important as it both supports the vessel and changes
diameter to regulate the blood flow and blood pressure. A schematic of the artery
and vein wall structures is also shown in Fig. 8.1.
The arterioles have much thicker walls in proportion to their size than the
arteries and have a much greater flexibility to change their diameter and hence their
resistance to flow, due to them having lots of smooth muscle. The overall
cross-sectional area is again much larger than the arteries but they have a high and
changeable resistance, which allows for regulation of blood flow.
The capillaries are the smallest vessels with very thin walls that contain no
smooth muscle: their resistance is thus largely unchanging. The cross-sectional area
is now at its largest: the flow velocity is very small and this is where most of the
exchange processes occur. The density and distribution of the capillaries depends
upon the requirements of the local body tissues: where tissue consumes lots of
oxygen, there will be a greater density than where tissue consumes less. This
consumption level is termed the metabolic rate: tissues such as skeletal muscle,
liver and kidney have a high rate and thus very many capillaries.
A reverse branching process occurs as the flow enters the venules and the veins:
the vessels branch back together again and the flow velocity increases as the
cross-sectional area decreases. The venous vessels have very thin walls in proportion
to their diameters: as a result they are very compliant and increase in volume
significantly as the pressure changes (unlike the arterial and capillary vessels).
8.1 Anatomy of the Vascular System 99
Table 8.1 Structural properties of the vascular system

Aorta Arteries Arterioles Capillaries Venules Veins Venae
cavae
Internal 25 mm 4 mm 40 μm 7 μm 40 μm 7 mm 30 mm
diameter
Wall 2 mm 1 mm 20 μm 1 μm 7 μm 0.5 mm 1.5 mm
thickness
Length 0.4 m 20 mm 2 mm 0.5 mm 0.5 mm 25 mm 0.3 m
Number 1 280 20 × 106 16 × 109 160 × 106 260 1
They normally contain approximately 70 % of the total blood volume at any one
time. Just as the arterioles are used to control the resistance, so the venous vessels are
used to control the blood volume.
There are valves in medium and large veins to prevent blood from flowing
backwards, since the veins are mostly travelling up towards the heart and there is
very little pressure to force the blood forwards. Venous return primarily depends
upon skeletal muscle action, respiratory movements and the constriction of smooth
muscle in the venous walls.
Table 8.1 summarises the structural properties of the different parts of the vas-
cular system in more detail: we will consider how these affect the behaviour of the
cardiovascular system once we have considered the mechanics of blood flow
through vessels in more detail. Note that the number of branches is very large, as
the cardiovascular network must cover the entire body.
8.2 Blood
Blood consists of two elements: plasma and blood cells. Approximately 40–45 % of
the volume of blood is occupied by blood cells that are suspended in a watery fluid
called plasma. The fraction of blood volume occupied by cells is called the
haematocrit. The plasma consists of ions in solution and many plasma proteins. By
weight it comprises about 90 % water and 7 % plasma protein, with the remainder
made up of other organic and inorganic substances. Blood cells are divided into
erythrocytes (red blood cells), leucocytes (white blood cells) and platelets, as
shown in Fig. 8.2. Although they are all produced in the red bone marrow, they
have different functions and white blood cells come in a large number of types.
The red blood cells are by far the most numerous and they contain the hae-
moglobin that is responsible for the transport of oxygen. They are biconcave discs
and deform easily to pass through the capillaries, since they are approximately 7 μm
in diameter. White blood cells defend the body against infection whilst platelets
play an important role in haemostasis (the formation of blood clots in response to
damage to the vessel wall). An adult human contains approximately 5.5 L of blood.
Fig. 8.2 Red blood cell, platelet and white blood cell, isolated from a scanning electron
micrograph (this figure is taken, without changes, from OpenStax College under license: http://
creativecommons.org/licenses/by/3.0/)
8.3 Haemodynamics
Haemodynamics is the study of the flow of blood. Since blood is a fluid, it can be
treated like any other fluid. We will start here by assuming that the fluid is
Newtonian, that the flow is laminar and steady and that the vessel in which it is
flowing is straight and rigid. Although this might seem to be a lot of assumptions to
make, it actually gives us a very important result.
In the exercise below, you can derive the following relationship between the
pressure difference along a blood vessel and the flow rate through it:
Dp 8lL
¼ 4 ð8:1Þ
q pR
where the viscosity of blood is µ and the vessel has radius R and length L. This is in
fact a fairly standard result for any fluid and turns out to be surprisingly useful in
haemodynamics. It introduces us to the concept of hydraulic resistance, which
says that when a pressure difference is applied between two ends of a blood vessel,
a flow through the vessel will result that is linearly proportional to this pressure
difference. The ratio between applied pressure difference and resulting flow is then
termed hydraulic resistance. This is given by the RHS of Eq. 8.1 and is only
dependent upon the vessel geometry and the viscosity of the fluid. Blood flowing in
large vessels has a viscosity approximately three times that of water.
8.3 Haemodynamics 101
Exercise A
The equation governing flow in a cylinder is:
@ ðrsÞ dp
¼r ðA:1Þ
@r dx
where r is radial position, p the fluid pressure (which is only a function of

axial distance, x) and shear stress for a Newtonian fluid is given by:
@u
s¼l ðA:2Þ
@r
Given that the fluid velocity must be zero at the wall, where r = R, and that
there must be zero gradient at the origin, show that the velocity profile of the
fluid can be given by:
1 2 dp
u¼ r R2 ðA:3Þ
4l dx
Hence calculate the flow rate and show the result given in Eq. 8.1.
This idea of resistance to flow, which is simply a measure of the friction caused
by viscosity in the flow, is very similar to electrical resistance. In a resistor, when a
potential difference is applied between the two ends, current flows, linearly pro-
portional to this potential difference.
We can thus use this idea to develop something called an equivalent electrical
circuit. In the same way that we can write the equation for a resistor (i.e. Ohm’s
law):
DV
¼R ð8:2Þ
I
we can also write:
Dp
¼R ð8:3Þ
q
where the equation describing electrical current flow is mathematically identical to

that describing blood flow in a blood vessel.
We can thus draw an electrical equivalent circuit for any number of blood
vessels in exactly the same way that we draw an electrical circuit for any number of
resistors. In this analogy, voltage (strictly potential difference) equates to pressure
difference, current relates to blood flow and the ratio of the two is resistance
(electrical resistance or hydraulic resistance). The fact that there is such a close
correspondence between the two is why the analogy is used so frequently in this
and other contexts.
Hydraulic resistances can be put in series and in parallel in exactly the same way
as electrical resistances (see Exercise B). The arteries, arterioles, capillaries, venules
and veins flowing in each body organ are in series, since blood flows through them
consecutively, whereas the body organs are predominantly in parallel, since blood
flows through them simultaneously.
Exercise B
Show that a network of N blood vessels, each of resistance R, has an overall
resistance R=N when placed in parallel.
Exercise D asks you to calculate the resistances of all of the different parts of the
vascular network. You will find that the resistance is dominated by the arterioles:
this is why they are predominantly used to adjust the overall vascular resistance. As
their resistance drops, flow rate increases for fixed blood pressure or blood pressure
drops for fixed blood flow. They do, of course, only adjust the resistance within
certain limits, since blood vessels can only change resistance through changes in
diameter. The blood flow through each body organ can also be altered by adjusting
the relevant arteriolar resistance: since the organs are in parallel, the blood flow can
be changed for one organ without having a significant effect on the remainder of the
vascular system (and of course an increase can also be provided by an increase in
cardiac output).
We have made a large number of assumptions to derive Eq. 8.1, none of which
are strictly true. The most important are that the flow is not steady (remember that
the heart is a pulsatile pump) and that the vessel radius is not constant. The effects
of pulsatility and an expanding wall diameter are often modelled by means of two
extra electrical components.
Inductance is used to model the inertia of the fluid whereas capacitance is used
to model the storage of blood in vessels, due to the elasticity of the vessel walls:
normally called compliance. This storage is a very important feature of the vascular
system: in fact, the arteries and veins behave more like balloons with a single
pressure, rather than resistive pipes with a continuous decrease in pressure.
The inductance of a blood vessel is found from solving the Navier–Stokes
equations (the equations that govern the behaviour of all fluids) under certain
conditions. For simplicity, we will just quote the result here, although it can be
derived from the equations in Chap. 6:
qL
I¼ ð8:4Þ
pR2
where ρ is the density of blood. Note that we will use I for inductance, rather than L,
for obvious reasons.
The compliance of a blood vessel is very simply defined as the rate of change of
volume with pressure:
dV
C¼ ð8:5Þ
dp
where we normally define the pressure here as the difference between internal and
external pressure (rather than the difference between inlet and outlet pressure).
Compliance is analogous to capacitance, which is defined as the rate of change of
charge with potential difference.
The compliance depends upon the geometry and properties of the vessel wall.
Exercise C asks you to derive the following expression for compliance:
3pR3 L
C¼ ð8:6Þ
2Eh
This assumes that the vessel wall is of thickness h and made up of a linear elastic
material with Young’s modulus E and Poisson’s ratio of 1/2. This value of Poisson’s
ratio relates to an incompressible material, which is in fact a good approximation for
the tissue that makes up a vessel wall, as we discussed in Chap. 6.
Exercise C
(a) Show that the compliance of a vessel is given by Eq. (8.6). Assume that
the material is linear and elastic with Young’s modulus E and Poisson’s
ratio of 1/2. The incremental stress components can be taken to be those
for a thin-walled vessel:
R
drh ¼ dp ðC:1Þ
h
R
drz ¼ dp ðC:2Þ
2h
drr ¼ 0 ðC:3Þ
(b) Show that a network of N capacitors, each of capacitance C, when
placed in parallel has total capacitance CN.
We can combine all three of these electrical elements to derive the equivalent circuit
shown in Fig. 8.3 for any type of vessel. Note that we have placed the capacitor in
this equivalent electrical circuit at the outlet, but other models will place it in the
middle or at the inlet: there is no unique way of doing this. We can use the values
given in Table 8.1 to calculate values of resistance, inductance and compliance for
every type of blood vessel, as in the exercise below.
Fig. 8.3 Equivalent electrical circuit for blood flow
Exercise D
Calculate the values of resistance, inductance and compliance for all the
different types of vessel listed in Table 8.1. Assume that vessels have a
Young’s modulus of 3 kPa and that blood has a density of 1050 kg/m3.
Although we have only considered a single vessel, larger networks of vessels

can be built by adding the equivalent circuits in series and parallel as necessary. It is
common to simplify the larger networks by neglecting some resistances and
capacitances and merging others.
One simple realistic equivalent circuit model of a body organ comprises seven
components, as shown in Fig. 8.4. Obviously this requires some knowledge of the
physiology and a number of assumptions. Note that we nearly always neglect
capillary inductance and compliance, as done here, since these tend to be very small
compared to the other components. The simplest model of the systemic circulation
then becomes a series of organs in parallel, where each body organ is supplied by
the aorta and drains into the venae cavae.
The compliance of the arteries is vital in converting the pulsatile nature of the
flow exiting the heart into a continuous flow. During systole, the flow of blood into
the arteries is greater than that exiting into the arterioles, so the arteries expand,
contracting during diastole. There is a storing of energy during the first stage, which
is then used to propel the blood forward during the second stage. By the time that
the flow has reached the capillaries, it is virtually steady state, as you will show in
the exercise below.
Fig. 8.4 Approximate equivalent electrical circuit for body organ

Exercise E
(a) Derive an expression for the flow through the capillaries in terms of the
pressure difference between inlet and outlet for the circuit shown in
Fig. 8.4. Use phasor notation where the impedance of an inductor is
Z ¼ ixI and the impedance of a capacitor is Z ¼ 1=ixC. Assume for
simplicity that venous (outlet) pressure is negligible.
(b) Explain how this circuit acts like a low-pass filter. Why is this
important?
We can derive more complex equivalent circuits using more detailed models of
the fluid flow and the vessel wall. In particular, the vessel wall is not a passive
elastic material and the relationship between pressure and volume is more like an
exponential rise than a straight line (so blood vessels get less compliant as they get
bigger, as you would expect from Fung’s model in Chap. 6). Although this can be
modelled using a capacitance that varies with pressure, the model then becomes
non-linear and considerably harder to analyse.
8.4 Blood Pressure
The body’s pulse can easily be felt at the wrist by pressing your fingers against the
inside of your wrist. This is simply caused by the regular beating of the heart
forcing an artery to expand and contract rhythmically as pulses of blood pass
through the artery. When we talk about blood pressure, we conventionally mean
Arterial Blood Pressure (ABP), which is the pressure found in the arteries and
their branches. This pressure varies between a high pressure, caused by ventricular
contraction, which is termed systolic, as it occurs during systole, and a corre-
sponding low pressure, termed diastolic, as it occurs during diastole. An example
waveform is shown in Fig. 8.5: note that the rise is much sharper than the subse-
quent fall. Also visible is the ‘dicrotic notch’, which is the sudden drop and rise in
ABP that occurs when the aortic valve closes (see the previous chapter for details).
Since there is a peak in blood pressure every time the heart beats, heart rate can be
calculated from the blood pressure, in a similar manner to methods used for the
ECG.
ABP is nearly always measured in units of millimetres of mercury (mmHg),
rather than in Pascals, since the resulting numbers are around 100, which makes
them easier to quote (the conversion factor is 1 mmHg = 133 Pa). Normally, only
the systolic and diastolic values of blood pressure are measured for each heart beat
and this is then recorded in the form systolic over diastolic (for example a systolic
Fig. 8.5 Arterial blood pressure waveform (this figure is taken, without changes, from OpenStax
blood pressure of 120 mmHg and a diastolic blood pressure of 80 mmHg would be
quoted as 120/80).
Mean arterial blood pressure (MAP) is also of interest, since it gives the average
effective pressure that drives blood through the systemic circulation. Although it
should strictly be calculated by integrating the blood pressure waveform over time,
often only the systemic and diastolic values are known. The most common
weighted combination of the two is:
SBP DBP
MAP ¼ DBP þ ð8:7Þ
3
where SBP and DBP represent systolic and diastolic blood pressures respectively.
Although it would seem more logical simply to average the SBP and DBP, it turns
out that for a typical waveform, this weighting is more representative of the
time-averaged value.
There are some other relationships that we use to help to understand the flow of
blood through the body. We have already come across Cardiac Output and the
concept of resistance to flow, so we introduce the idea of Total Peripheral
Resistance:
MAP
TPR ¼ ð8:8Þ
CO
which is essentially Eq. 8.3 applied to the whole body. Of course in this equation
we are assuming that the blood pressure at return to the heart is zero. Although very
simple, Eq. 8.8 does tell us that to change MAP the body must either change CO or
TPR. We will look at both of these later.
We also define the Arterial Pulse Pressure (APP) to be:
APP ¼ SBP DBP ð8:9Þ

8.4 Blood Pressure 107
i.e. the difference between the maximum and minimum value of arterial pressure
during each heartbeat. It turns out that there is an approximate relationship between
APP and heart stroke volume:
SV
APP ¼ ð8:10Þ
Ca
where Ca is arterial compliance.

Changes in APP are thus caused either by a change in heart stroke volume or
arterial compliance. It can be seen that MAP and APP are in many ways more
useful measures of how the cardiovascular system is behaving than SBP and DBP.
Blood flow through the body has one predominant frequency, which is the heart
rate (although there are actually many other frequency components to a blood
pressure signal, caused by respiration and other processes within the human body).
The equivalent circuit shown in Fig. 8.4 has several sections that can be thought of
as filters, whereby frequency components are removed.
A resistor and inductor in series have a time constant:
I qL pR4 qR2
s¼ ¼ 2 ¼ ð8:11Þ
R pR 8lL 8l
If we assume that the dominant frequency is 1 Hz (the heart beat), then for the
inductor to be important, we need:
1
x[ ð8:12Þ
s
i.e. the frequency of oscillation must be greater than the cut-off frequency. This will
be true when:
sffiffiffiffiffiffi
4l
R[ ð8:13Þ
pq
Since blood has roughly the same density as water and three times the viscosity,
this turns out to be approximately 2 mm. Inductance is thus only important in
vessels above this size (which is why we only included it in Fig. 8.4 for the arterial
and venous compartments).
A resistor and capacitor have time constant:
12lL2
s ¼ RC ¼ ð8:14Þ
ERh
This will be important in long vessels with thin walls and low values of Young’s
modulus. This makes it most important for the venous compartment. In fact arterial
compliance also turns out to be very useful in the control of blood flow (as we will
see in Sect. 8.5), so this is normally retained.
Exercise F
For the circuit shown in Fig. 8.4, derive a simplified expression for the
transfer function of the circuit, neglecting inductance. What is the time
constant and what condition must be satisfied if this circuit is to remove
oscillations in capillary flow due to pulsatile ABP?
8.4.1 Long-Term Measurement Techniques
Measurements of blood pressure are very important to doctors when assessing the
risks of a whole range of vascular diseases, for example strokes, where the risk of a
stroke goes up rapidly with increased blood pressure. Elevated blood pressure
(hypertension) will normally be treated through the use of anti-hypertensive drugs
that act to lower MAP.
Blood pressure is normally measured using a sphygmomanometer, which
consists of an inflatable cuff that is placed around the upper arm over the brachial
artery and connected to a pressure gauge. The pressure gauge is usually a column of
mercury. The cuff is inflated to a pressure well above the systolic pressure (in the
region 175–200 mmHg): since this is higher than the largest arterial pressure, the
blood vessels collapse, preventing blood flow to or from the forearm. A stethoscope
is then placed over the artery just below the cuff and the cuff pressure allowed to fall
gradually: as soon as the cuff pressure falls below the peak arterial pressure, some
blood passes through the arteries, but only intermittently. This gives the value of
SBP.
Since this flow is turbulent and intermittent, tapping sounds, known as Korotkoff
sounds, are produced. As the pressure continues to drop, the sounds increase in
volume before decreasing again. The pressure at which the sounds disappear
completely is the DBP. Since these sounds are often difficult to hear near the
diastolic pressure, accurate determination of the DBP is often a matter of experi-
ence. This technique is known as the auscultatory technique, since it is based on
listening to sounds, and is illustrated in Fig. 8.6.
It is also possible to use oscillometry to measure blood pressure. This is per-
formed using a cuff with an in-line pressure sensor. Again, the cuff is pressurised
above SBP and allowed to deflate: the measured cuff pressure is high-pass-filtered
above 1 Hz to observe the pulsatile oscillations as the cuff deflates. The point of
maximum oscillation corresponds to a cuff pressure equivalent to MAP. Both SBP
and DBP are found where the amplitude of the oscillations is a fixed percentage of
the maximum amplitude: 55 % for SBP and 85 % for DBP.
8.4 Blood Pressure 109
Fig. 8.6 Blood pressure measurement (this figure is taken, without changes, from OpenStax
8.4.2 Short-Term Measurement Techniques
We should note that both of these methods have a measurement time that is much
longer than any individual heart beat period: they are thus used for occasional
samples rather than as continuous measures. Some attempts have been made to
provide a continuous measure of blood pressure. The continuous vascular
unloading technique works on the basis of continuously adjusting the cuff pressure
to the arterial pressure: this is done using mechanical feedback. The main disad-
vantage of this technique is that it can only be applied at a peripheral vascular
location such as the finger, where it may not be an accurate representation of true
arterial blood pressure. There are a variety of other possible techniques, each with
advantages and disadvantages, which we will not consider here.
8.5 Measurement of Blood Supply
As well as measuring blood pressure, we are often interested in measuring blood

flow to different parts of the body. Blood carries nutrients to body organs and so
blood supply must be provided continuously, particularly to organs such as the
brain and heart. Cardiac output is adjusted to provide sufficient total blood flow and
the resistances of different body organs are also altered to ensure that each organ
receives enough flow.
An important function of the circulation is to support metabolism by supplying

nutrients and removing waste. Thus we are often interested at the capillary level less
in blood flow and more in terms of blood supply or perfusion. Perfusion measures
the rate of delivery of blood to a volume of tissue, rather than simply the flow rate
through the vessels; thus it has units of volume of blood per volume of tissue per
unit time, typically ml blood/ml tissue min−1. As we will see in Chap. 9, the
delivery and the removal of gases are essentially perfusion limited processes,
because these predominantly happen over the large surface area of the capillary bed,
i.e. it is perfusion that determines how much is delivered to and taken away from
tissue.
8.5.1 Blood Flow
The easiest and simplest way to measure blood flow is through measurements of
blood flow velocity. This can be done using a Doppler probe, whereby an ultra-
sound signal with frequency of order 2 MHz is sent out towards a blood vessel. The
movement of the blood in the vessel causes a phase shift, which can be measured in
the returning signal.
The most common use of Doppler is in the brain, where it is called transcranial
Doppler (TCD). This has been widely used since the 1980s to monitor cerebral
blood flow, although recordings have to be made in what are known as insonation
windows, since ultrasound signals are blocked by bone.
It is important to remember that Doppler ultrasound only measures blood flow
velocity and so calculating blood flow requires an assumption about the
cross-sectional area of the vessel to be made. This area can vary under certain
conditions, but measuring this area is very difficult and can only really be done
using high-resolution imaging.
8.5.2 Perfusion
Perfusion is a key measure of how many organs behave. Hypoperfusion (a drop in

perfusion) can have a very detrimental impact on organs such as the heart and brain.
In the brain, hypoperfusion can be the result of either a blockage in a supply vessel
or a bleed in a major blood vessel in the brain: these cause an ischaemic stroke or a
haemorrhagic stroke respectively. In the heart, hypoperfusion in the blood vessels
supplying heart muscle can cause a heart attack, as described in the previous
chapter.
Since perfusion is so important to body organs, there has been a lot of interest in
measuring perfusion, mainly using imaging techniques to generate perfusion maps.
The common factor to all of these is the use of a contrast agent that acts as a tracer:
the basic principle is that flow is measured by tracking a substance that moves and
8.5 Measurement of Blood Supply 111
the concentration of which can be followed. Two very common examples are
Positron Emission Tomography (PET) and Magnetic Resonance Imaging
(MRI).
In PET, a radioactive tracer is injected into the bloodstream: this is most com-
monly a sugar called F-18 labelled flurodeoxyglucose (FDG). This gradually
accumulates in the organ of interest before it disperses away and the radioactivity
decays below a detectable level: a time series of the concentration can then be used
to estimate perfusion (see Chap. 5). In MRI, the most common tracers are based on
gadolinium, which is a material whose magnetic properties mean that its presence in
tissue alters the MRI image in direct proportion to its concentration.
MRI can also use the water in blood as a naturally occurring, or endogenous
tracer. This is most often done in the brain, when the magnetism of blood water is
inverted in the neck before imaging in the brain. This process, known as Arterial
Spin Labelling (ASL) allows water accumulation in tissue to be seen, by which time
it has had time to exchange from the blood into the tissue, so that perfusion can be
measured. This technique avoids the need for injections, but does then give a poorer
signal to noise ratio.
8.6 Control of Blood Flow
The fact that body organs need a continuous supply of blood means that the body
has a variety of mechanisms to maintain this blood supply. We will look at the ways
in which the central control mechanisms control blood flow to an organ in the final
chapter: we consider here how individual vessels locally control perfusion. This
happens primarily within an organ at the arteriolar level (remember that these are
the vessels with the most smooth muscle and the greatest contribution to resistance),
although there is thought to be some control at the capillary level too.
8.6.1 Individual Vessels
In Exercise G you will show that, for fixed compliance, flow is proportional to
vessel radius to the power 6 and that shear stress is proportional to vessel radius to
the power 3. These mean that the vessel is extremely sensitive to even quite small
changes in vessel radius. In fact, the vessel wall is thought to respond to a large
number of stimuli, including shear stress, and these combine to give a signal that
acts to control blood flow. Precisely how this works is not well understood,
although it is known that nitric oxide and intracellular calcium play important roles
in the behaviour.
Exercise G
Consider a vessel of fixed length with driving pressure P at the inlet and zero
pressure both at the outlet and outside the vessel.
(a) If the vessel compliance is constant, show that flow through the vessel is
proportional to vessel radius to the power 6.
(b) Show also that shear stress is proportional to vessel radius to the
power 3.
In fact, vessel compliance changes as the vessel alters in size. In Exercise H you
will show that for an incompressible vessel, the vessel wall gets thinner as the
vessel expands. Thus as pressure increases and the vessel wall gets thinner, com-
pliance increases passively very rapidly (the numerator gets larger and the
denominator gets smaller).
Exercise H
(a) Consider a vessel fixed at both ends, initially with inner radius Ro and
wall thickness ho. Assuming that the volume of the vessel wall does not
change when the pressure increases, derive a relationship for the wall
thickness as a function of the inner radius and the initial inner radius and
wall thickness.
(b) Using the expression for resistance (Eq. 8.1), explain how resistance
changes as pressure (and hence inner radius) increases.
(c) Does the vessel wall become stiffer or more compliant as pressure
increases?
However, in reality vessels become less compliant with increased diameter due
to the active mechanisms involved in autoregulation of blood flow. The final
response of a blood vessel to a decrease in blood pressure is thus made up of two
components: the early passive response that leads to an increase in diameter and a
decrease in blood flow; and the subsequent active response that counterbalances this
response to maintain blood flow nearly constant.
8.6.2 Individual Body Organs
One of the most tightly regulated body organs is the brain, where perfusion (known
here as cerebral blood flow) is kept nearly constant within a range of ABP of
8.6 Control of Blood Flow 113
approximately 50–150 mmHg. It does this by tightly controlling blood flow through
the arterioles, where there are a lot of smooth muscle cells. Precisely how this is
done is still not entirely fully understood, but impaired autoregulation in the brain is
found in a wide range of brain diseases, such as stroke, dementia and brain injury.
Body organs, of course, do not just regulate their blood supply in response to
pressure, but must also match blood (and hence nutrient) supply to metabolic
demands. If the cells need greater oxygen, more blood must be supplied. In the
brain this will only be a relatively small change, but in some organs, such as
skeletal muscle, the increase in blood supply can be very large. The balance
between these two can be modelled very simply.
Let’s consider a body organ with driving pressure difference P and metabolic
rate of oxygen M. The equations governing flow and metabolism can be written as
follows:
ðCa Cv ÞQ ¼ M ð8:15Þ
P ¼ RQ ð8:16Þ
where Ca and Cv are arterial and venous oxygen concentrations. The first equation
comes from conservation of mass (remember cardiac output in the previous chapter)
and the second one is simply Eq. 8.3.
We will assume to start with that resistance responds to changes in venous
oxygen concentration linearly, so that a decrease in venous oxygen concentration
results in a decrease in resistance:
R ¼ Ro ð1 þ ACv Þ ð8:17Þ
where A is a constant (rather like the gain in a feedback circuit). The flow through
the organ is then given by:

1 P
Q¼ MA þ ð8:18Þ
1 þ ACa Ro
Obviously if A = 0, there is no regulation and flow is linearly proportional to

pressure. However, as A increases, there is greater feedback and flow becomes less
sensitive to pressure.
If we reference everything back to baseline conditions (denoted by the overbar),
then we can re-write the equation for flow as:
Q M P
¼ a þ ð1 aÞ ð8:19Þ
Q M P
where:
AMRo
a¼ ð8:20Þ
P þ AMRo
This means that flow responds to both changes in metabolism (with sensitivity a)
and to blood pressure (with sensitivity 1 a). There is a trade-off between these
two, with it being desirable to have higher sensitivity to changes in metabolism than
to changes in pressure. This is achieved by having:
P
A ð8:21Þ
MRo
i.e. a high level of feedback. However, any system with too much feedback can
become unstable, so it is not as simple as aiming for as high a value as possible.
This again is a trade-off between competing requirements.
8.6.3 Whole Body
Now that we have looked at the whole cardiovascular system, we will finish by
considering the whole body and how the heart plays its part in controlling blood
pressure. We divide the body into arterial and venous compartments, each of which
has resistance and compliance. Assuming that both compliances are constant, we
can state that total blood volume comprises arterial and venous compartments:
VT ¼ Pa Ca þ Pv Cv ð8:22Þ
In the steady state we can also say that total resistance is related to pressure and
flow by:
Pa Pv ¼ QRT ð8:23Þ
This time, in order to maintain arterial blood pressure, we assume that cardiac
output, Q, is proportional to heart rate, H, and venous pressure:
Q ¼ kHPv ð8:24Þ
This is a simple approximation, based on the ability of the heart to provide

greater cardiac output based on heart rate and preload.
From these, we can derive an expression for arterial pressure:
VT ð1 þ kHRT Þ
Pa ¼ ð8:25Þ
Cv þ Ca ð1 þ kHRT Þ
8.6 Control of Blood Flow 115
At low values of heart rate, this approximates to:
VT
Pa ¼ ð8:26Þ
Cv þ Ca
but at high values of heart rate, this approximates to:
VT
Pa ¼ ð8:27Þ
Ca
which is larger. However, this is the maximum arterial blood pressure than can be
achieved in this model and so continuing to increase heart rate to raise arterial blood
pressure becomes less effective. The model also explains why as blood vessels
become stiffer with age (and compliance decreases) baseline arterial blood pressure
tends to increase.
We can plot this result, which is most easily done by converting it to
non-dimensional form:

Pa 1 ð1 þ hðpr 1ÞÞ
¼ 1þ ð8:28Þ
Pa pr c r 1 þ hð pr 1Þ þ 1
cr
where arterial blood pressure, as a fraction of its baseline value, is only a function of
heart rate (now as a fraction of its baseline value), h, and the ratio of arterial to
venous blood pressure, pr, and the ratio of arterial to venous compliance, cr. By
writing the result with as few variables as possible in non-dimensional form, we can
see that there are only two extra parameters determining the relationship between
heart rate and blood pressure, making it much easier to interpret. The resulting
relationship between heart rate and blood pressure is then shown in Fig. 8.7, where
Fig. 8.7 Relationship

between heart rate and blood
pressure in simple model
we have taken pr = 10 and cr = 1/3 as typical values. The saturation is very clearly
seen, where increases in heart rate have progressively less impact on arterial blood
pressure.
In these two examples, we can see both how whole organ control is achieved
through very simple feedback on oxygen concentration and whole body control
through blood pressure and heart rate. Obviously the models that we are using are
very simplistic, but they do illustrate the kind of behaviour that is seen in the human
body and help to provide an insight into the underlying relationships.
8.7 Conclusions
In this chapter, we have examined the second part of the cardiovascular system: all
the vessels that make up the vasculature. These can be modelled using very simple
techniques and we have developed the equivalent electrical circuit model that is in
very widespread use. We have examined how these models can be constructed and
developed. We then investigated how blood flow and perfusion can be measured
and finished by considering how the body regulates blood flow at a local level in
addition to the global control that we looked at in the previous chapter.
Chapter 9
The Respiratory System
The respiratory system can roughly be divided into the upper airways in the head
and neck, and the lower airways including trachea and all the structures in the
lungs. The primary functions of the respiratory system are:
• Gas exchange—most importantly the movement of O2 into the body and CO2
out.
• Host defence—it provides one of the barriers between the outside world and the
body inside.
• Synthesis and metabolism of various compounds.
9.1 The Lungs and Pulmonary Circulation
The lungs are contained in space with a volume of approximately 4 L but present a
surface area of approximately 85 m2 for gas exchange. This is achieved thorough a
highly branched structure, Table 9.1, the trachea branching into the two main stem
bronchi which themselves divide into further bronchi at progressively small
diameters. Subsequently further generations of the branches are called bronchioles
before finally terminating in the alveoli. Both the smallest bronchioles and alveoli
are involved in respiration over a gas-blood barrier that is only 1–2 μm thick. The
pulmonary capillary bed is the largest in the body having a surface area of
approximately 70–80 m2. The capillary volume of the lungs is 70 ml at rest, but can
increase up to 200 ml during exercise. This is achieved through the recruitment of
closed vessels or compressed capillary segments from an increase in pulmonary
pressure when cardiac output is increased, along with an enlargement of the cap-
illaries with a rise in internal pressure when the lungs fill with blood.

118 9 The Respiratory System
Table 9.1. Branching structure of the lungs

Generation Diameter Length Number Total
(cm) (cm) cross-sectional
area (cm2)
Conducting Trachea 0 1.80 12.0 1 2.54
zone 1 1.22 4.8 2 2.33
2 0.83 1.9 4 2.13
3 0.56 0.8 8 2.00
Bronchioles 4 0.45 1.3 16 2.48
5 0.35 1.07 32 3.11
⋮ ⋮ ⋮ ⋮ ⋮
Terminal 16 0.06 0.17 5 × 104 180.0
bronchioles
Transitional Respiratory 17 ⋮ ⋮ ⋮ ⋮
and bronchioles 18
respiratory
19 0.05 0.10 5 × 105 103
zones
Alveolar 20 ⋮ ⋮ ⋮ ⋮
ducts 21
22
Alveolar 23 0.04 0.05 8 × 105 104
sacs
Following Weibel ER. Morphometry of the Human Lung. Springer Verlag and Academic Press,
Heidelberg–New York 1963
9.1.1 Breathing
Figure 9.1 shows the breathing cycle. This is mainly achieved by the action of the
diaphragm, although the external intercostal and scalene muscles in the chest also
play a role. Air is drawn into the lungs by an increase in the chest cavity, pushing
the abdominal content downwards. Exhalation is passive during normal breathing,
but may involve active effort by muscles during exercise and hyperventilation.
The total lung capacity is the total volume of air that can be contained within the
lung. Various other volumes and capacities can be defined as shown in Fig. 9.2,
where capacities are composed of two or more volumes. Under normal breathing
only a relatively small range of the available lung volume is used.
9.1.2 Respiration Rate
The measurement of respiration rate at its simplest relies on counting the number of
breaths per minute. This would typically require either a sensor at the mouth and/or
nose to detect air movement of air in and out of the lungs, or a band around the
chest. The latter is potentially able to measure changes in chest volume and thus to
9.1 The Lungs and Pulmonary Circulation 119
Fig. 9.1 The breathing cycle (this figure is taken, without changes, from OpenStax College under
Fig. 9.2 Definition of the various lung volumes (a) and capacities (b) (this figure is taken, without
changes, from OpenStax College under license: http://creativecommons.org/licenses/by/3.0/)
infer information about lung volume. Alternatively, it can be possible to extract

information about respiration from EEG, since changes in the volume of the
abdominal cavity will alter the conductivity and thus produce small fluctuations in
the ECG signal. These changes are typically very subtle and thus hard to observe in
anything other than ideal conditions.
9.2 Gas Transport
Air is a mixture of various gases, predominantly nitrogen (78 %) with a large

proportion of oxygen (21 %) and various other gasses (1 %) including carbon
dioxide. We often refer to the partial pressures of each of the components, where
the partial pressure of gas i is defined by:
pi ¼ yi p; ð9:1Þ
where yi is the mole fraction of the gas and p is the total pressure of the gas mixture.
Here the partial pressure is the pressure that this gas would exert if it alone were
present. Since in the lungs gases are (indirectly) in contact with a liquid, the blood,
we need to be able to relate partial pressure of a gas to its concentration in the
blood. We can do this via Henry’s law:
ci ¼ ri pi ; ð9:2Þ
where σ is the Ostwald solubility co-efficient. Typical values for respiratory gases in
plasma are given in Table 9.2. All of these solubilities are fairly similar and are in
fact too small to carry sufficient quantities of oxygen and carbon dioxide in the
blood.
9.2.1 Inert Gases
A simple model for the transfer of gas across the capillary-alveoli interface assumes
that the flow is linearly proportional to the difference in partial pressures, thus:

q ¼ Ds cb rpg ; ð9:3Þ
where q is the net flux per unit area, pg is the partial pressure of gas in the air space,
cb is the concentration of the gas dissolved in the capillary blood and Ds is the
surface diffusion co-efficient. Note that like a lot of the processes we have met in
this book, this is just a first order equation with time.
Table 9.2 Solubility of Substance σ (mM/mmHg)

respiratory gases in blood
plasma O2 1.4 × 10−3
CO2 3.3 × 10−2
CO 1.2 × 10−3
N2 7 × 10−4
He 4.8 × 10−4
9.2 Gas Transport 121
Exercise A
Starting with the diffusion equation in one dimension, show that the flux of
gas per unit area of membrane from the air space into the blood is given by
Eq. 9.3, where it has been assumed that the gas has the same solubility in the
membrane to that in the blood.
As you will see in Exercise A modelling the flow of gas across the
capillary-alveoli interface is very similar to Exercise 4A modelling passive transport
through a cell membrane. Note that we haven’t worried about the relative solu-
bilities of the gas in the membrane compared to the blood here. This is because the
membrane we care about is the cellular layer that separates the gas space and the
blood. These cells will have an aqueous interior and thus similar solubility to that of
blood. We have ignored the diffusion through the cell membranes (and thus relative
solubility of blood in the cell walls) because these are much thinner than the cells
themselves.
A complete pulmonary capillary can be modelled as a cylinder with gas transport
occurring over the full surface. In the limit the expression for total gas flux reduces
to:
Q ¼ f rðp0 pin Þ; ð9:4Þ
where pin is the partial pressure of the gas in the blood at the inlet to the capillary.
Thus the gas flux depends only upon the pressure difference and the blood flow rate,
f. This only holds as long as the permeability of the membrane to the gas (i.e. the Ds
value) is sufficiently high, which is primarily achieved in the lungs through having a
thin membrane. Since the gas flux depends on blood flow and not the rate of
diffusion it is called perfusion limited.
Exercise B
Here we will derive the result in Eq. 9.4 based on a cylindrical model of a
pulmonary capillary.
(a) Using continuity derive the relationship between the change in con-
centration of gas within the blood along the capillary and the flux across
the surface for a vessel of radius r and with a blood flow rate f.
(b) Find the solution to this equation for the concentration of gas within the
blood with distance along the capillary, x. Assume that the concentration
at the inlet is c0 and that the concentration in the air space is constant
along the length.
(c) The total gas flux across the capillary surface can be calculated for a
capillary of length L by integration. Using the solution from part
(b) derive an expression for the total gas flux.
(d) Calculate the flux for a single capillary, assuming parameter values of:
Cin Co ¼ 1 mM; f ¼ 1 1013 m3 =s and 2prDs L ¼ 0:51
13 3
10 m =s: Plot the variation of flux with blood flow rate, assuming that
all other parameters remain constant.
(e) Using the result in part (d), explain why increasing the flow rate has only
a small effect on the total flux across the capillary wall.
(f) What happens in the limit as L → ∞? What properties of the capillary
bed structure would ensure that this would be true?
9.2.2 Carbon Dioxide
Whilst the analysis above holds for the inert gases, where Henry’s law holds, the
metabolic gases are more complicated. CO2 is mainly transported within the red
blood cells as HCO 3 ; the reaction being catalysed by the enzyme carbonic
anhydrase:
CO2 þ H2 O H2 CO3 HCO þ

3 þH ; ð9:5Þ
where CO2 combines with water to form carbonic acid, which then produces
bicarbonate (HCO +
3 ) and H .
Exercise C
(a) Determine the rate constant for the second part of the reaction scheme
for CO2 conversion to carbonic acid.
(b) Convert this to the ‘corrected’ rate constant assuming that almost all the
available CO2 converts to carbonic acid.
In Exercise C you have derived an expression for the corrected rate constant, KA,
that relates CO2 concentration to those of hydrocarbonate ions and H+. It is possible
to evaluate what difference this makes to the removal of CO2 from the capillary
blood in the lungs and arrive at the result:
Q ¼ f ð1 þ KA ÞrCO2 ðp0 pCO2 Þ; ð9:6Þ
which is (1 + KA) larger than our result for inert gases in Eq. 9.5, with KA = 20 at a
normal pH of 7.4. The conversion of CO2 to bicarbonate thus substantially
increases the gas flow rate, because as CO2 leaves the capillary by diffusion from
the plasma it is rapidly replenished by more from the reversal of the bicarbonate
reaction. Thus this is an example of facilitated diffusion that we met in Chap. 4.
Exercise D
(a) Starting with the model in Exercise B write down separate equations for
the concentration of CO2 and HCO 3 in the capillary blood. Assume that
you can ignore H2CO3 from your reaction scheme and thus have a single
forward and backward reaction rate.
(b) Combine your equations from part (a) to give a single expression for the
total concentration of CO2 and HCO 3 , i.e. CO2 in all its ‘forms’ for the
purposes of transport.
(c) Using a quasi-steady state approximation write HCO 3 in terms of CO2
and thus derive the result in Eq. 9.6 using a similar analysis to
Exercise B and considering the limit as L → ∞.
(d) If log10Ka = −6.1 calculate Kc at normal pH and thus comment on the
effect of blood chemistry on CO2 removal from the body.
9.2.3 Oxygen
We have already seen in Chap. 1 that oxygen binds to haemoglobin the blood:
Hb þ 4O2 HbðO2 Þ4 ; ð9:7Þ
This is the primary means by which it is transported; a negligible fraction (3 %)

is carried in solution in the plasma. Figure 9.3 shows the relationship between the
partial pressure of O2 and haemoglobin saturation, which is highly non-linear. In
theory the enhancement provided by haemoglobin on oxygen transport could be as
much as 200, but in practice an enhancement of around 32 is achieved.
9.2.4 Tissue Gas Delivery
Delivery of gases to the tissues is essentially the reverse of the process in the lungs,
and again the process is typically assumed to be perfusion limited. We can thus
equate the amount of gas delivered to the difference in partial pressure between the
arterial and venous ends of the capillary bed. This process can be modelled by a
linear first order differential equation, just as in Eq. 9.4:
Fig. 9.3 Saturation curves for haemoglobin (this figure is taken, without changes, from David
Iberri under license: http://creativecommons.org/licenses/by-sa/3.0/)
dpt rb
¼ f ðpa pv Þ; ð9:8Þ
dt rt
where pa, pa, and pt are the partial pressure in arterial blood, venous blood and
tissue respectively, and we have to consider the solubility of the gas in both the
blood and tissue.
This means that gas delivery can be thought of like the one-compartmental
models we considered when we examined pharmacokinetics in Chap. 5. In this case
the compartment is the tissue into which gas is being delivered and we assume that
there is no spatial variation in the partial pressure in the tissue, thus the compart-
ment is referred to as ‘well stirred’ or ‘well mixed’. This can be represented
graphically as in Fig. 9.4, where a number of other simple models of tissue-capillary
gas exchange are shown that extend the model by accounting for some degree of
diffusion limitation.
The one compartment model means that the tissue concentration will respond
exponentially to a change in gas concentration and a time constant can be deter-
mined for that process; often the half-life is quoted. This will vary from tissue to
tissue depending upon the solubility, but more significantly upon the perfusion rate.
Given the range of tissues and accompanying perfusion rates, simple models often
divide the body into a collection of compartments with a range of representative
time constants.
As before, this description is fine for the inert gases, but is insufficient for the
metabolic gases. As we have already seen, binding in the blood gives a non-linear
(a)
(b)
(c)
(d)
(e)
Fig. 9.4 Compartmental models of tissue gas exchange (reproduced with permission,
M.A. Chappell, DPhil thesis, University of Oxford, 2006)
relationship between concentration and partial pressure. There is further binding of

oxygen in the tissues to myoglobin (Mb), whose saturation curve is much like
the standard Michaelis–Menten function rather than the sigmoidal shape of
haemoglobin. The difference between these two curves arises from the greater
affinity of myoglobin for oxygen which means oxygen is readily transferred from
haemoglobin to myoglobin, improving the transfer rate from blood into tissue. The
presence of myoglobin in muscles also provides some limited oxygen store and
accounts for the colour of red meats.
A further aspect for the behaviour of the metabolic gases is that oxygen is
consumed in the tissues and carbon dioxide is being produced, requiring an addi-
tional metabolism term in the equation above. If we are interested in the total
pressure of gas in solution in the tissue it is often a reasonable approximation
to assume zero dissolved oxygen in tissue, because oxygen is rapidly metabolised
(or bound), and an equivalent fixed fraction for the CO2 that has been produced.
9.2.5 Blood Oxygenation
The standard measurement of blood oxygenation is the fraction of the carrying

capacity of the blood that is being used: the oxygen saturation. Pulse-oximetry is a
widely used method for measuring the oxygen saturation of the blood. The sensor is
placed in a thin part of the body, for example a fingertip or earlobe. It works by
passing two distinct wavelengths of light through the body to a photodector; the
relative absorbance at the two wavelengths allows the oxygen saturation of arterial
blood to be determined.
The measurement relies upon the changes in colour of haemoglobin with oxygen
saturation, thus pulse-oximetry strictly measures the percentage of haemoglobin
that is loaded with oxygen. Since, as we have seen, haemoglobin is the main carrier
of oxygen in the blood, this in turn is a good measure of oxygen saturation of
the blood. Figure 9.5 shows the absorption spectra for both oxy- and deoxy-
haemoglobin.
Pulse oximetry works in the Near Infrared Region (NIR) of the light spectrum
and exploits the distinctive differences in absorption seen in this region. Typically
two wavelengths, 660 and 940 nm, are used and are sampled up to 30 times per
second allowing changes in ambient light and also the amount of arterial blood
present to be corrected for. Once the ratio of light transmission at the two wave-
lengths has been calculated this can be converted to oxygen saturation, which in
normal healthy individuals will be above 95 %.
Whilst pulse oximetry is widely used for the measurement of oxygen saturation,
the time course associated with the measurement of optical transmission, the
photoplethysmogram (PPG), is less widely used. This signal captures the time
varying blood volume in the skin associated with the differences between systolic
and diastolic pressure in the arteries. Thus it is possible to monitor heart rate from
the PPG. It is also, theoretically, possible to monitor respiration, since changes in
lung volume and that of the thoracic cavity affect the heart. This leads to small
Fig. 9.5 Absorption spectra

for Oxy- and
deoxy-haemoglobin, showing
the Near Infrared Region (this
figure is taken, without
changes, from Adrian Curtin
under license: http://
licenses/by-sa/3.0/)
changes in systolic and diastolic pressure during the breathing cycle that might be
detected from the PPG. Measuring such small changes is challenging, but has not
prevented attempts to measure vital signs using the light reflected from exposed
areas of skin acquired using conventional digital video cameras.
9.2.6 Control of Acid-Base Balance
One further aspect of the body’s behaviour that we will consider here is the reg-
ulation of the acid-base status. The maintenance of the pH of arterial blood between
7.35 and 7.45 is absolutely vital for the correct functioning of the human body. The
most important influence on pH is the transport of CO2 in the blood. The reaction
equation that governs buffering is the same one we considered for CO2 transport
above. The expression for the ‘corrected’ rate constant can be re-arranged into the
form:

HCO
pH ¼ pKA þ log10 3
; ð9:9Þ
½CO2
where pH ¼ log½H þ and pKA ¼ logðKA Þ ¼ 6:1: This is the Henderson–

Hasselbach equation. The concentration of CO2 is approximately proportional to
the partial pressure of CO2. The fact that the concentrations of both bicarbonate and
carbon dioxide can be independently controlled, by the kidneys and the lungs
respectively, means that it proves relatively straightforward to maintain pH at a
constant level. This is achieved by the autonomic nervous system that we will meet
in Chap. 10.
Exercise E
(a) Derive the Henderson–Hasselbach equation from the ‘corrected’ equi-
librium constant in Exercise C.
(b) Given that the concentration of CO2 can be linearly related to the partial
pressure of CO2, with an Ostwald constant of 0.03 mM/mmHg, calculate
the pH value for a partial pressure of 40 mmHg and a concentration of
bicarbonate of 24 mM.
(c) Draw lines of constant pCO2 (20, 40 and 60 mmHg) on a plot of
bicarbonate concentration versus pH. Use a range of pH of 7.1–7.7 and
bicarbonate concentration of 10–40 mM and illustrate the point calcu-
lated in part (b).
(d) If the pCO2 rises from 40 to 60 mmHg (why might this happen?), how
can the body maintain a constant pH?
9.3 Conclusions
In this chapter we have looked at the structure of the respiratory system and
especially the respiratory circulation. We have used models of transport from earlier
chapters to explain the unique structure of the lungs that enables effective gas
exchange. You should now be able to setup compartmental models for gas transport
and analyse these for inert and metabolic gases using suitable approximations.
Chapter 10
The Central Nervous System
We have so far looked at a number of the most vital systems in the body. However,
their actions and interactions all need to be coordinated and this role falls to the
nervous system, which we will briefly look at in this final chapter. The nervous
system can be divided into two parts:
• The central nervous system (CNS): the brain and spinal cord.
• The peripheral nervous system (PNS): nerves (bundles of nerve cells) con-
necting the CNS to other parts of the body. The PNS also encompasses the
enteric nervous system, a semi-independent part of the nervous system
responsible for the gastrointestinal system.
Figure 10.1 shows a conceptual model of how this all fits together. Notice that
we have both inputs (afferents) and outputs (efferents), but also a division into parts
of which we are consciously aware and have direct control: the somatic nervous
system, and the parts that we are largely unaware of and happens automatically—
the autonomic nervous system.
10.1 Neurons
We have already met the idea of a neuron or nerve cell in Chaps. 3 and 4 where we
considered the action potential and chemical synapses. Neurons are the main actors
in the CNS, providing a way to transmit both efferent and afferent signals to and
from the CNS. A number of specialized neurons exist:
• Sensory neurons: These respond to stimuli such as light and sound in the
sensory organs and send this information back to the CNS.
• Motor neurons: These receive signals from the CNS and cause muscle con-
traction or affect glands (for the release of hormones).
• Interneurons: These connect neurons to other neurons within the same region of
the CNS.
Typically a neuron will have a cell body, dendrites and an axon, as shown in
Fig. 10.2. The dendrites are thin structures arising from the cell body that typically
130 10 The Central Nervous System
Fig. 10.1 Conceptual diagram of the nervous system
Fig. 10.2 Schematic of a typical CNS neuron (this figure is taken, without changes, from
branch multiple times, forming a complex ‘dendritic tree’. In contrast the axon (and
there is only ever a single axon) is a special extension of the cell body that may
extend as far as 1 m. The axon itself may branch hundreds of times before it
terminates at the dendrite of another neuron. At the majority of synapses signals are
sent from the axon of one neuron to the dendrite of another, although there are
exceptions to this rule. It is the axon that carries signals over long distances and is
thus insulated with a myelin sheath that in turn is interrupted at various points by
the nodes of Ranvier to aid conduction of the action potential as we saw in Chap. 4.
10.2 Autonomic Nervous System 131
10.2 Autonomic Nervous System
There are a great many systems in the body that the CNS needs to regulate over
which we have no (or very little) direct conscious control. These include the reg-
ulation of digestion, maintaining glucose balance and regulating heart rate. The
autonomic system is divided into sympathetic and parasympathetic parts, as
shown in Fig. 10.3. Most organs receive signals from both and broadly the role of
the sympathetic system is to put the organ into ‘emergency mode’, whereas the
parasympathetic system has the opposite effect of placing the organ into ‘vegetative
mode’. The connection between the CNS and PNS for the autonomic nervous
system occurs primarily in the spinal cord, but the bodies of the nerve cells largely
reside in ganglia that are distributed around the body.
10.2.1 Autonomic Control of the Heart
The control of the cardiovascular system is, unsurprisingly, a highly complicated

topic and we will only look at a few parts of this system. We will focus on the
Fig. 10.3 The autonomic system showing sympathetic and parasympathetic parts separately (this
figure is taken, without changes, from OpenStax College under license: http://creativecommons.
org/licenses/by/3.0/)
baroreceptor and chemoreceptor reflexes in the context of maintaining cardiac

output, but there are many other components to the control of the cardiovascular
system that act to maintain other parameters such as blood volume. The aim here is
just to give you a brief introduction to how the body maintains homeostasis through
measuring physiological parameters, feeding back this information to the brain and
acting on it in a co-ordinated manner. This is just like any engineering control
system, where feedback based on measurements is used to control a system and
hence to maintain equilibrium.
There are three cardiovascular centres in the medulla oblongata in the brain.
These neurons respond to changes in blood pressure, blood gas concentrations and
pH. The cardioaccelerator centre acts to increase cardiac function through sym-
pathetic activation and the cardioinhibitor centre acts to reduce cardiac function
through parasympathetic activation, whilst the vasomotor centre controls vessel
wall stiffness via smooth muscle cells.
10.2.2 Cardiac Efferents
The cardiac muscle cells are the main efferents in the heart. As we have seen, their
regular contraction is self-sustaining, coordinated by the SAN and requires no
external instruction to maintain a resting heart rate. The heart is thus a good
example of the competing role of sympathetic and parasympathetic nerves and how
these are used to tune the heart rate, as illustrated in Fig. 10.4:
• Parasympathetic stimulation: Synaptic terminals release the neurotransmitter
ACh; this slows the rate of depolarization during the pacemaker potential of the
SA node, thus increasing the interval between successive action potentials, and
slowing the heart rate. ACh acts by increasing potassium permeability, keeping
the membrane potential nearer to that of potassium, retarding the growth of the
pacemaker potential toward the threshold for triggering action potentials.
• Sympathetic stimulation: Synaptic terminals release norepinephrine. This speeds
the heart rate and increases the strength of contraction, an effect which is
mediated by an increase in calcium permeability. The greater the number of
calcium channels that are open, the lower the threshold is for triggering an AP in
the SA, thus increasing the heart rate. In the cardiac muscle cells the increase in
calcium permeability increases the calcium influx during the plateau, increasing
the strength of contraction.
In both cases the effects of the neurotransmitter are indirect, unlike those we saw
when we looked at chemical synapses where the neurotransmitter acted directly on
the ion channels. Note that this means that the body can, via an indirect mechanism,
effect longer term changes without having to provide a continuous neural signal.
10.2 Autonomic Nervous System 133
Fig. 10.4 Effects of parasympathetic and sympathetic activity on heart rate (this figure is taken,
without changes, from OpenStax College under license: http://creativecommons.org/licenses/by/3.0/)
10.2.3 Cardiac Afferents
The baroreceptor reflexes are rapid mechanisms that attempt to minimise

short-term fluctuations in blood pressure. Baroreceptors are nerve endings in the
walls of the carotid sinus, where the external and internal carotid arteries split, and
in the aortic arch. They are termed mechanoreceptors as they are sensitive to
Fig. 10.5 Baroreceptor control of blood pressure
stretching of the arterial wall, which is related to arterial blood pressure. When
there is a drop in blood pressure, resulting in a reduction in wall stretch, the firing
rate of the receptors is reduced.
This feeds back to the brain and results in an increase in heart rate and stroke
volume: as a result, there is an increase in blood pressure. This increase in heart rate
is known as tachycardia (the opposite, a reduction in heart rate, is known as
bradycardia). How this works is shown in terms of a control loop (with the large
arrows used to describe complex relationships) in Fig. 10.5: a decrease in ABP
results in a decrease in baroreceptor firing rate, an increase in cardiac output, which,
coupled with vasoconstriction, results in an increase of ABP and a return to baseline
conditions. There are also many other processes that act to maintain homeostasis in
the longer term.
The carotid sinus receptors respond to pressures of approximately 60–180 mmHg,
whilst the aortic arch receptors are less sensitive as they have a higher threshold
pressure. The maximal sensitivity is around normal ABP. The receptors are also
sensitive to the rate of change of ABP, and increases in the pulse pressure make the
baroreceptors more sensitive to changes in MAP. The baroreceptors quickly adapt to
changes in the mean level of MAP: if the level drops for a sustained period, unaltered
by the action of the baroreceptors, the reflex will gradually reset itself to this
new operating point over a period of several hours. They are thus only short-term
regulators of MAP.
The chemoreceptor reflexes originate in the aortic arch and carotid bodies and
respond to changes in arterial oxygen and carbon dioxide levels. If blood pressure
drops and oxygen levels in the blood decrease, the chemoreceptor reflexes will
respond by increasing sympathetic activity. This increase acts in the same way as a
decrease in parasympathetic activity and so heart rate increases.
10.3 Somatic Nervous System
We will now focus on the somatic nervous system. Remember that this is the part of
the CNS whose inputs, neuronal afferents, we are largely conscious of and whose
outputs or actions, neuronal efferents, we generally have voluntary control of.
The familiar patellar (knee-cap) reflex provides a good example of the somatic
nervous system in action. A simplified diagram of the circuitry involved in this
reflex is shown in Fig. 10.6. Concentrating only on the quadriceps muscle to start
10.3 Somatic Nervous System 135
Fig. 10.6 Patellar reflex (this figure is taken, without changes, from Backyard Brains under
with: tapping the knee causes a stretch in the muscle fibres that stimulates an action
potential in stretch-sensitive sensory neurons. This signal is passed to the spinal
cord arriving at a synapse with the motor neuron. This then elicits an action
potential in the motor neuron, which is carried back to the muscle arriving at a
neuromuscular synapse as described in Chap. 4. The resulting AP causes con-
traction of the muscle fibre, which when repeated across the whole muscle causes
contraction of the quadriceps muscle and the resulting leg extension.
10.3.1 Temporal and Spatial Summation of Synaptic

Potentials
The neuromuscular junction is unusual in one respect, in that a single AP that

arrives at the presynaptic side leads to a sufficiently large depolarization on the post
synaptic side to trigger an AP there. Thus such a synapse is called a ‘one-for-one’
synapse. However, most synapses are not that strong: a single presynaptic AP will
cause only a small depolarization of the postsynaptic cell, called an excitatory
postsynaptic potential (epsp). The synapse between a single starch receptor and
the quadriceps motor neuron in the patellar reflex is a typical example of this.
Each epsp is of the order of 1 mV, far smaller than the 10–20 mV threshold
required. However, if several subsequent epsp arrive before the effects of the
previous ones have died away a sufficient depolarization of the postsynaptic cell can
be achieved. This is called temporal summation. An alternative mechanism by
which epsp can sum to reach a threshold is via the postsynaptic cell receiving
multiple presynaptic inputs, this is called spatial summation. This relies on the fact
that most neurons are connected to many others via their dendrites. Both temporal
and spatial summations are involved in the patellar reflex.
10.3.2 Excitatory and Inhibitory Synapses
So far we have only met excitatory synapses where the AP in the presynaptic cell
causes a depolarization in the postsynaptic cell. However, it is also common to find
inhibitory synapses, where the release of neurotransmitter due to the presynaptic AP
tends to prevent the firing of the postsynaptic AP. In this case the neurotransmitter
causes a hyperpolarization of the post synaptic cell, making the membrane
potential more negative and moving it further way from the threshold required to
elicit an AP. Thus we now also have a class of inhibitory postsynaptic potential
(ipsp). Like the epsp we have already met, the ipsp is achieved through changes in
the ionic permeability of the postsynaptic cell membrane.
One possible mechanism for an ipsp would be an increase in the permeability to
K+ similar to the undershoot we met in Chap. 3. Many inhibitory synapses, how-
ever, rely on changes in Cl− permeability. In many neurons chloride pumps
maintain the chloride equilibrium potential more negative than the membrane
potential, thus an increase in Cl− permeability leads to a hyperpolarization of the
neuron. Even when the Cl− equilibrium potential is near to that of the membrane
potential, inhibition can occur, as, although there will be no appreciable hyperpo-
larization, the increase in Cl− permeability will resist any increases in the membrane
potential brought about by an excitatory input.
Figure 10.7 shows an example of summation of epsps and ipsps that have been
received by a neuron. At time A the sum effect of a series of epsps has been
insufficient to initiate an action potential in the axon. By time B sufficient epsps
have been received within a short enough time to reach threshold and an action
potential will be generated and will propagate along the axon.
The combination of inhibitory and exhibitory synapses plays an important role in
the patellar reflex. In Fig. 10.6 we also have to consider the flexor muscles at the
back of the thigh as well as the quadriceps muscle. As the leg extends, this will
cause a stretch in these muscles that in turn will, via the excitatory pathway through
the spinal cord, cause contraction of the flexor muscle, jerking the leg back again. In
turn this would elicit a stretch and contract reaction from the quadriceps muscle and
so it would go on. However, the extra inhibitory link between quadriceps sensory
neuron and the flexor’s motor neuron prevents this occurring and only the first
extension is seen.
10.3 Somatic Nervous System 137
Fig. 10.7 Summation of postsynaptic potentials (this figure is taken, without changes, from
10.3.3 The Brain
The ‘pinnacle’ of the nervous system is the brain, where many millions of indi-
vidual neurons interact via a complex network of dendritic connections and
synapses. This relies upon the many combinations that are possible when we bring
many epsp and ipsp together with temporal and spatial summation over a densely
connected neural network.
The brain can broadly be considered to contain both ‘grey matter’ and ‘white
matter’ reflecting different visual qualities of the tissue, Fig. 10.8. These actually
primarily represent a division of the brain into neurons in the grey matter that
constitutes a thin layer with a large surface area that takes the form of a highly
Fig. 10.8 Image of the brain

collected using MRI showing
folded structure and division
into grey and white matters
(reproduced with permission,
M.A. Chappell)
folded sheet within the head, and in the white matter bundles of axons from neurons
carrying APs to different regions of the brain and out of the brain. Within the brain
there is also another sort of cell called glial cells of which there are a number of
different types, all of which provide supporting roles either structurally or
metabolically to the neurons.
10.3.4 Function: Introduction to EEG and FMRI
The complexity of the CNS and the brain in particular makes it both very difficult to
study and one of the most studied parts of the human body. It is possible to make
measurements of neural activity, however, not with a resolution that allows indi-
vidual cells to be investigated. Like the ECG of the heart that we met in Chap. 7,
electroencephalography (EEG) measures electrical signals in the brain using an
array of electrodes placed on the head. Alternatively, magnetoencephalography
(MEG) can be used to detect magnetic fields arising from electrical activity of the
brain with highly sensitive magnetic detectors. Both methods provide very highly
sampled temporal information about signals in the brain, but provide low spatial
resolution, since it is difficult to reconstruct spatial locations from an array of
measurements made outside of the head.
Very many EEG electrodes or MEG detectors can be placed across the head to
try to localise electrical activity to different areas of the brain. In the heart the main
electrical activity arose from the propagation of the AP across the heart wall, which
gave rise to a relatively strong and highly directional vector potential. In the brain
many APs are occurring that are not necessarily so simply co-ordinated, although,
due to the structure of the brain, the main direction of travel is perpendicular to the
skull. Despite the complexity of the signals in the brain, even with just a few
electrodes placed around the head, it is possible to measure signals in specific bands
of frequencies that reflect levels of co-ordination in the signalling in the brain’s
nervous network.
Magnetic Resonance Imaging (MRI) methods have also been used to study brain
activity, typically called functional MRI (fMRI). The most common method is to
exploit the Blood Oxygen Level Dependent (BOLD) effect. BOLD relies upon the
different magnetic properties of oxygenated and deoxygenated blood, allowing
increases in oxygen usage to be indirectly measured from changes in perfusion and
blood volume in the brain during a task compared to that during rest. This method
provides far better spatial resolution, of the order of mm, than EEG or MEG, but
with much poorer temporal resolution, typically one measurement every few sec-
onds. Much research using these methods has allowed the functions of various
different regions of the brain to be identified and increasingly their connections to
each other are being explored.
10.4 Conclusions 139
10.4 Conclusions
In this chapter we have examined the nervous system and seen how the concepts of
action potentials and their propagation come together to regulate the body as part of
the autonomic system and provide for conscious and voluntary action through the
somatic system. We have seen that through the summation of many action
potentials the brain is a very complex organ, requiring in turn a very specific
structural architecture and metabolic support system. Finally, despite the com-
plexity of the brain we have seen that we are still able to measure at least some of its
functionality by examining electrical and metabolic changes.
At the end of this chapter we have completed our introductory survey of
physiology from an engineering perspective. We started with the fundamental unit
of living matter, the cell, and examined behaviour associated with it. We then
started to examine larger systems where we might bring cells together into the form
of a tissue, and then how groups of tissues form systems. Finally, we examined
some of the most important systems in the body. You will hopefully have learned a
lot about how the human body works, although we have only scratched the surface.
More importantly you will have seen that a very wide range of engineering methods
can be used to understand and study physiology, bringing interest to engineering
and benefit to medicine.
Answers
1. Cell structure and biochemical reactions

A
Reaction equations:
da
¼ k cd k þ ab2
dt
db
¼ k cd k þ ab2
dt
dc
¼ k þ ab2 k cd
dt
dd
¼ k þ ab2 k cd
dt
In steady state this gives the result in A.2.

B
(a) Reaction equations
da
¼ k1 cd k þ 1 ab
dt
dc
¼ k2 ef k þ 2 c
dt
(b) In steady state this gives the result in B.3.

C
Equilibrium approximation:

Biosystems & Biorobotics 13, DOI 10.1007/978-3-319-26197-3
142 Answers
k1 c ¼ k þ 1 se
Ks c ¼ s ð e o c Þ
s
c ¼ eo
Ks þ s
dp s
V¼ ¼ k þ 2 c ¼ k þ 2 eo
dt Ks þ s
Quasi-steady-state approximation
ðk þ 2 þ k1 Þc ¼ k þ 1 se
Km c ¼ sðeo cÞ
s
c ¼ eo
Km þ s
dp s
V¼ ¼ k þ 2 c ¼ k þ 2 eo
dt Km þ s
Since Km [ Ks ; the reaction velocity is always smaller for the quasi-steady-state

approximation.
D
This is known as the double reciprocal form:
1 K 1 1
¼ þ
V Vmax s Vmax
If we plot 1/V against 1/s, we get a straight line. This intercepts the x axis when
1=s ¼ 1=K and the y axis when 1=V ¼ 1=Vmax .
E
(a) Using a logarithmic transformation, we get:

V
ln ¼ n ln s n ln K
Vmax V

V
(b) If we plot ln against ln s, we get a straight line. This intercepts
Vmax V

the x axis when ln s ¼ ln K and the y axis when ln VmaxVV ¼ n ln K.
Plotting the values given yields values of n ¼ 2:394 and K ¼ 1:5 mM/s.
Yes it is a good fit.
F
(a) Quasi-steady-state approximation:
Answers 143
k þ 1 se ¼ ðk þ 2 þ k1 Þc1
k þ 3 ie ¼ k3 c2

s i
e þ þ 1 ¼ eo
Km Ki
dp eo s Vmax s
V¼ ¼ k þ 2 c1 ¼ k þ 2 ¼
dt s
þ i
þ1 K m K m ð 1 þ i=Ki Þ þ s
Km Ki
(b) As the inhibitor increases, the curve shifts to the right. The intercepts on
the double reciprocal plot would only be affected on the x axis, where
the intercept would move to the right (i.e. towards the origin).
2. Cellular homeostasis and membrane potential
A
(a) V ¼ V0 :
The concentrations of P and Q are the same inside and outside, this is the
isotonic case and the cell is already in equilibrium for concentrations and
thus the volume stays the same as the original.
(b) V ¼ 2V0 :
We require that [P] = [Q] in equilibrium, but we have initially ½P0 ¼
2½Q0 ; i.e. a hypertonic osmotic imbalance. Given that the volume of
solution outside the cell is far larger than the cell itself (effectively
infinite) the concentration of Q is not going to change, thus water must
move into the cell to dilute P: to halve the concentration inside we need
to double the volume.
Formally, we require ½P=V ¼ 50; initially we have ½P0 =V0 ¼ 100;
solving for V ¼ 2V0 :
B
The cell will expand indefinitely: the only way that the two requirements of
equal internal and external concentration of Q and equal total concentration can
be met is for the concentration of P to be zero, the cell expands to infinite size.
Formally, we have two equilibrium equations:
½Qi ¼ ½Qe
½Pi þ ½Qi ¼ ½Qe
where the subscripts i and e refer to the internal and external environments
respectively. The only solution is ½Pi ¼ 0:
C
(a) V ¼ V0
We have two equilibrium equations:
Concentration balance for R:
144 Answers
½Ri ¼ ½Re
Total concentration (osmolarity):
½Pi þ ½Ri ¼ ½Qe þ ½Re
These equations are satisfied with the initial concentrations so the system
is in equilibrium.
(b) V ¼ 2 V0 :
We have the same two equilibrium equations as in part a), but the system
is not initially balanced. However
• the total amount of P is fixed inside the cell
• the concentrations of Q (outside the cell) is fixed
• the concentration of R both inside and outside the cell is fixed, since
the external concentration will not be altered by movement of R into
the cell (at least to the same degree as the internal concentration) and
R is free to move in and out of the cell.
Writing the balance of total concentration in terms of quantities of P and
R and the final volume:
P/V þ R/V ¼ 50 þ 100
Concentration of R is fixed:
R/V ¼ 100
Thus:
P/V ¼ 50
Since P0 ¼ 100 V0 then V ¼ 2 V0 :

D
(a) Yes
(b) This example is a bit different to the cell examples so far, in that we have
a closed system - the external concentration is not fixed. We are also
examining a system in which water isn’t permeable - so we do not need
to consider osmolarity and there are no other permeable uncharged
species.
Using conservation of mass, since the volume is fixed this is the same as
considering the total concentrations of these species:
Answers 145
½K þ l þ ½K þ r ¼ 700 mM
½Cl l þ ½Cl r ¼ 300 mN
Charge neutrality:

½K þ l ½Cl l 2 X2 l ¼ 0
½K þ r ½Cl r ¼ 0
Gibbs-Donnan:
½K þ r ½Cl r ¼ ½K þ l ½Cl l
Solving these equations gives the following equilibrium concentrations
½K þ r ¼ 210 mM
½K þ l ¼ 490 mM
½Cl r ¼ 210 mM
½Cl l ¼ 90 mM
½X2 l ¼ 200 mM
(c) To achieve the correct concentration and electrical gradients for Cl- leads
to changes in the K+ concentrations to the left and right of the mem-
brane. As the left gets more negative and the right more positive due to
Cl- ion movement, K+ has to migrate to the left to guarantee charge
neutrality on either side.
½K þ r
(d) EK ¼ 58 mV log ½K þ l ¼ 21:34 mV
E
(a) Concentration balance:
a þ b þ c þ 108 ¼ 120 þ 5 þ d
Charge balance (both inside and outside)
11
a þ b ¼ c þ 108
9
120 þ 5 ¼ d
146 Answers
Gibbs-Donnan equilibrium:
bc ¼ 5 125
(b) For potassium ions:

½K þ o
EK ¼ 58 mV log ¼ 81 mV
½K þ i
For chloride ions

½Cl o
ECl ¼ 58 mV log ¼ 81 mV
½Cl i
As expected they are equal and the membrane potential is −81 mV.
(c) For sodium ions:

½Na þ o
ENa ¼ 58 mV log ¼ þ 58 mV
½Na þ i
F
(a) Using the Nernst equation:
Ion Equilibrium potential (mV)

Na+ +67
K+ −90
Cl− −34
Ca2+ +118
This is not in equilibrium since they not equal. This means that the
membrane potential will fall somewhere in between the extremes and be
determined also by permeability.
Answers 147
(b) Total concentration (osmotic) balance:
10 þ 140 þ 30 þ 104 þ ½P ¼ 145 þ 4 þ 114 þ 1:2
½P ¼ 84:2 mM
Internal charge balance:
10 þ 40 30 þ 2 104 þ Z ½P ¼ 0
Z ¼ 1:43 (the net charge on the internal proteins etc).

(c) External charge balance:
145 þ 4 114 þ 1:2 6¼ 0
The cell cannot be in a steady state in this case as electrical neutrality

outside the cell is not met.
3. The Action Potential
A
Em ¼ þ 47:5 mV:
B
The total current flowing from inside to outside, Iapp ; is the sum of the four
branches. For sodium: the current of sodium ions arises from the difference in
potential between the sodium equilibrium and the membrane potentials
ðEm ENa Þ and this passes through the membrane subject to a conductivity
(1/resistance) of gNa ; using Ohm’s law:
iNa ¼ gNa ðEm ENa Þ:
Likewise for potassium and the ‘leakage’ current:
iK ¼ gK ðEm EK Þ;
iL ¼ gL ðEm EL Þ:
The membrane acts like a capacitor and thus current flow in response to changes
in membrane potential can be described by:
dEm
im ¼ Cm :
dt
148 Answers
Summing:
dEm
Iapp ¼ gK ðEm EK Þ þ gNa ðEm ENa Þ þ gL ðEm EL Þ þ Cm :
dt
Re-arranging gives:
dEm
Cm ¼ gK ðEm EK Þ gNa ðEm ENa Þ gL ðEm EL Þ þ Iapp :
dt
C
(a) The two equations are:
dO
¼ aC bO;
dt
dC
¼ aC þ bO:
dt
(b) The sum of the probabilities is 1. Hence:

dO
¼ að1 OÞ bO:
dt
(c) The steady state value and time constant are:

¼ a=ða þ bÞ;
O
sO ¼ 1=ða þ bÞ:
D
(a) This is just a substitution of Equations 3.3–3.5 into the expressions from
Exercise C for each of the gates in turn.
(b) In the steady-state with the potential at zero (remember that this is
relative to the reference value), the h gates are largely open and the
m and n gates largely closed, this requires bm am and bh ah :
When the voltage increases, the gates go from open to closed and vice
versa. The time constants for the n and h gates are much larger than for
the m gate, so the m gate responds much more rapidly, as expected, this
requires bm am and bh ah :
E
About 2.3 μA.E
Answers 149
F
(a) If the permeability to K+ is high compared to other ions then EK+ will
dominate the membrane potential. From exercise 2F we know that EK+ is
−90 mV so the range suggested seems reasonable.
(b) From the figure the membrane potential plateaus at +52 mV. If, as
suggested, we ignore the other ions and assume that Ca2+ permeability
dominates then we need ECa2 þ equal to or greater than +52 mV:
!
58 mV 1:2
ECa2 þ ¼ log10 2 þ þ 52 mV:
2 Ca i

This requires that Ca2 þ i 0:019 mM:
4. Cellular Transport and Communication
A
Starting with the diffusion equation in 1-dimension:
@c @2c
¼ f þD 2:
@t @x
We are looking for transport in the steady state, thus changes with respect to time
are zero. We will also assume there is no generation (or consumption) of the
species within the membrane thus f=0:
@2c
D ¼ 0:
@x2
This is a standard second order differential equation with a solution of the form
c=ax+b. The boundary conditions from the diagram are:
cð x ¼ 0Þ ¼ co ;
cðx ¼ LÞ ¼ ci :
Hence
x
c ¼ co þ ðci co Þ :
L
Using Fick’s law we can write the flux:
@c D
J¼D ¼ ðci co Þ:
@x L
150 Answers
If we were to account for relative solubility via the partition coefficient then the
boundary conditions would change - since the concentration at the boundaries
inside the membrane would be higher (or lower) compared to the same boundary
outside the membrane:
cðx ¼ 0Þ ¼ Kco :
cðx ¼ LÞ ¼ Kci :
The solution would look like:
B
The differential equations for this process are:
dpi
¼ kpe kpi þ k þ si ci k pi ;
dt
dpe
¼ kpi kpe þ k þ se ce k pe ;
dt
dci
¼ kce kci þ k pi þ k þ si ci ;
dt
dce
¼ kci kce þ k pe k þ se ce :
dt
C
(a) This constant current source can equally well be simulated in the
equations by an increase in the resting potential for potassium, since both
contribute a fixed term on the right hand side of the equation. A change
in equilibrium potential is directly related to concentration via the Nernst
equation. The necessary rise in potassium equilibrium potential is:
Iapp 2:3
mK ¼ ¼ ¼ 6:27 mV:
gK n 4 36 0:31774
(b) Relating the potassium concentration to potential by the Nernst equation,

the relative change in extracellular potassium is given by:
Answers 151

½K þ o new
¼ ½K þ o old 10ð6:27=58Þ ¼ 1:28 ½K þ o old :
A similar approach for sodium gives a factor of over 500, which is not
physiologically possible!
D
(a) The current flowing in the membrane is given by:
@i @V cdx
iþ dx i ¼ Cm cdx þ V;
@x @t Rm
where c is the circumference of the cell wall.

The potential difference within the segment of cell is:
@V
Vþ V ¼ iRc :
@x
Combining these gives Eq. 4.12.

(b) Assuming a circular cell cross section: if rm and cm are given for a unit
area of membrane then Rm ¼ rm =ðpd Þ and Cm ¼ cm pd:; if rc is given
for a unit area of cytoplasm tens Rc ¼ 4rc =ðpd 2 Þ. Therefore:
sm ¼ r m c m :
rffiffiffiffiffiffiffiffi
rm d
km ¼ :
4rc
(c) sm ¼ 8:4 ms and km ¼ 0:15 cm.

(d) This is a second order differential equation in x with solution (including
boundary condition at x = 0) of: V ¼ 100ex=k : Solving for x when V =
20 mV gives x = 2.4 mm.
5. Pharmacokinetics
A
(a) Starting with:
dCp
Vc ¼ Vc ke Cp ðtÞ þ Ain ðtÞ:
dt
Take Laplace transform:
sVc Cp ðsÞ ¼ Vc ke Cp ðsÞ þ Ain ðsÞ:

152 Answers
Rearrange
1 Ain ðsÞ
C p ðsÞ ¼ :
Vc ðs þ ke Þ
Inverse Laplace transform (using convolution property)
Zt
1
C p ðt Þ ¼ Ain ðkÞeke ðktÞ dk:
Vc
0
(b) Ain ðtÞ ¼ k0

Zt
k0 eke t
Cp ðtÞ ¼ eke k dk;
Vc
0
k0 eke t ke t
Cp ðtÞ ¼ ðe 1Þ;
V c ke
k0
Cp ðtÞ ¼ 1 eke t :
V c ke
(c) The steady state concentration can be found in the limit t ! 1

k0
Cp1 ¼ :
Vc ke
As might be expected this concentration depends on the relative rate of elimi-

nation and delivery, as well as the volume of distribution into which the sub-
stance is dissolved.
B
(a) Starting with:
dAa
¼ ka Aa ðtÞ þ Ain ðtÞ;
dt
dAp
¼ ka Aa ðtÞ ke Ap ðtÞ:
dt
Laplace transform (with zero initial conditions):
Ain ðsÞ
Aa ðsÞ ¼ ;
s þ ka
ka Aa ðsÞ
Ap ðsÞ ¼ :
s þ ke
Answers 153
Thus:
ka Ain ðsÞ
Ap ðsÞ ¼ :
ðs þ ke Þðs þ ka Þ
Inverse Laplace Transform:

ka ke t
A p ðt Þ ¼ e eka t Ain ðtÞ:
ka ke
Ain ðtÞ ¼ D dðtÞ; where D is the dose. Also Ap ¼ Cp Vc :
D ka ke t
C p ðt Þ ¼ e eka t :
V c ka ke
(b) A sum of a growth toward saturation plus a decay toward zero leads to
something that firstly increases and then decays away.
(c) The slowest time constant, which we would expect to be elimination,
will dominate at later time points and thus a straight line fit to the log
values will give an estimate for this time const.
C
(a) Using the data in the table we need to calculate the area under the curve
for both methods of administration. Trapezoidal integration will suffice
for this calculation. We do not have measurements at very long time
post-administration so we do not know when the concentration drops to
zero, we will ignore any concentration after the last time point:
AUCoral 251:07
F¼ ¼ ¼ 0:650
AUCIV 386:04
(b) We can use the IV injection to quantify the elimination kinetics, it will
obey Equation 5.6. We can estimate the parameters either from a
semi-logarithmic plot of concentration versus time, which gives a
straight line slope of −0.500 and intercept of 5.298, or we could use
non-linear model fitting of the model to the data:
ke ¼ 0:500 h1 :
ln2
T1=2 ¼ ¼ 1:386 h:
ke
For the absorption constant, and thus the half-life, we need to use
Equation 5.11, where we have already obtained ke. Again we could use
model fitting of the equation to the data, or we could employ the ‘method
of residuals’.
154 Answers
Writing Eq. 5.11 as:
Cp ¼ Aeke t Aeka t :
If ka [ ke ; then as t ! 1
Cp
Aeke t :
We apply this to the final three time points of data: a semi-log plot allows
us to calculate A and ke from a slope of −0.500 and intercept of 5.00. Note
that we get the same value for ke as above (unsurprisingly) but the
intercept value is different because of the bioavailability.
Now define the ‘residual’:
R ¼ Aeke t Cp :
We have all the information we need to calculate R at every time point

and by definition:
R ¼ Aeka t :
Thus a semi-log plot of R against time will allow ka to be estimated from

the slope. Using the first three data points (where the residual will be the
largest, since absorption dominates here) gives a slope of −3.03. Hence:
ka ¼ 3:03 h1 ;
ln2
T1=2 ¼ ¼ 0:229 h:
ka
We should check at this point that ka > ke, which it is and sufficiently so
that the assumption we made above is reasonable.
(c) To calculate the volume of distribution we need the initial plasma
concentration. We can use the data we have in part (a), for example
using the intercept from the IV administration.
Cp0 ¼ 199:94 mg/L:
Thus
D
Vc ¼ ¼ 10:0 L:
Cp0
Answers 155
Which is larger than the total plasma volume, so must include some fast
exchanging tissue.
D
(a) We need Eq. 5.11
FD ka ke t
Cp ðtÞ ¼ e eka t :
Vc ka ke
(b) We can find the maximum dose by firstly finding the maximum con-
centration, for which we will need to determine the time of maximum
concentration by finding the turning point of the function described by
Cp. This gives

1 ka
tmax ¼ ln ¼ 21:7 min:
ka ke ke
Substituting into the expression for Cp and then converting to dose
Vc Cpmax
Dmax ¼ ¼ 1330 mg:
F
It is possible to find a nice simplified form for the maximum concen-

tration with a bit of effort:

FD ka ke=ka ke
Cpmax ¼ :
Vc ke
(c) The drug is no longer effective once C p< X where X = 20 mg/L, thus we
need to solve for the time at which:
FD ka ke t
Cp ðtÞ ¼ e eka t ¼ X:
Vc ka ke
Noting that ke ≪ ka, the absorption process is very rapid in comparison to

elimination, so we might assume there is no further absorption by the
time that the concentration drops below the level for efficacy. We can
thus simplify the expression assuming that the exponential term con-
taining ka is zero:
156 Answers
FD ka
eke t ¼ X:
Vc ka ke
Giving:

1 ka ke Vc
t¼ ln X ¼ 606 min:
ke ka FD
We could check our assumption from above that absorption isn’t

important by this time by comparing 606 min to the time constant for
absorption (1/ka) which is approximately 5 min.
(d) Michaelis-Menten kinetics are of the form:
Vmax C
:
C þ Km
This is linear if Km ≫ C, first order kinetics, but is constant if Km ≪ C,

zeroth order kinetics. In this case Km is an order of magnitude greater
than C at any point in time, given that Cp,max is around 100 mg/L. So we
would conclude that the analysis still holds as the elimination would still
look like first order and has not saturated.
6. Tissue mechanics
A
(a) The one-dimensional equations are:
@rx
qg ¼ 0
@x
@u
ex ¼
@x
rx
ex ¼
E
Hence:
@ 2 u qg
¼
@x2 E
Integrate twice, and substitute in the boundary conditions given to derive

the answer.
Answers 157
(b) At the top of the tissue:
qgL2
u¼
2E
This is approximately 1 mm (10 % of the tissue height) for the values

given.
B
(a) Re-arrange and square the result:
2 !2
3Ke /w0 c0f
þ c 2
¼c þ
RT 3e þ /w0
Then multiply out the LHS and re-arrange to get the answer given.
(b) The result is obviously very non-linear, with concentration becoming
very high at low values of pressure.
C
(a) The one-dimensional equations are:
@2u G @e @p
G þ ¼a
@x 2 1 2m @x @x
@u
e¼
@x
j @2p @e 1 @p
¼a þ
l @x2 @t Q @t
Integrate the first equation wrt x:
2Gð1 mÞ
e¼p
að1 2mÞ
Substitute into the last equation:

2
j @2p a ð1 2mÞ 1 @p
¼ þ
l @x2 2Gð1 mÞ Q @t
Hence the expression given, where the coefficient is:

1 l a2 ð1 2mÞ 1
¼ þ
c j 2Gð1 mÞ Q
158 Answers
(b) Easiest way is to take Laplace transforms (with zero initial condition):
@2p s
p¼0
@x2 c
Solve, ignoring the positive exponential term (which would tend to

infinity):
pffis
p ¼ Ae cx
Use the boundary condition (Laplace transformed):
P pfficsx
p¼ e
s
This can be inverse Laplace transformed (using a standard result) to give:

x
p ¼ P 1 erf pffiffiffiffi
2 ct
D
Boundary conditions are:
rr jr¼R ¼ p
rr jr¼R þ h ¼ pext
These are obviously satisfied.

E
(a) The mean flow velocity is found from:
ZR
pR2 U ¼ uðr ÞdA
r¼0
2Umax ZR r n
U¼ 1 rdr
R2 r¼0 R

n
U ¼ Umax
nþ2
(b) The frictional force is given by:

@u
f ¼ 2pRl ¼ 2pnlUmax ¼ 2pðn þ 2ÞlU
@r r¼R
Answers 159
7. Cardiovascular system I
A
For the numbers given, cardiac output would be 200/(0.21 − 0.16) = 4000
ml_blood/minute. Stroke volume = 4000/60 = 67 ml_blood.
B
(a) The RR interval is 0.8 s, so the heart rate = 60/0.8 = 75 bpm. There is no
variability shown here, which is not realistic.
(b) You should find that it goes up quite rapidly, the additional cardiac
output providing more oxygen to your muscles.
8. Cardiovascular system II
A
Integrate up the equation wrt radius twice:
r 2 dp
lu ¼ þ A ln r þ B
4 dx
For the solution to be finite at the origin, A = 0. Also the velocity is zero at the
wall, hence:
1 2 dp
u¼ r R2
4l dx
Flow rate:
ZR ZR 1 2 dp dp pR4
q¼ uðr ÞdA ¼ r R2 2prdr ¼
r¼0 r¼0
4l dx dx 8l
Since the pressure gradient is the pressure drop divided by the vessel length, we
get:
Dp 8lL
¼ 4
q pR
B
When placed in parallel, the pressure difference is the same for each vessel, with
the total flow rate being the sum of the individual flow rates. Hence:
X
N
Dp XN
1 N
q¼ ¼ Dp ¼ Dp
i¼1
R i¼1
R R
The overall resistance is thus R=N:

160 Answers
C
(a) It is easiest to work in terms of infinitesimal strains, since compliance is
defined as the rate of change of volume with pressure:
dV dA dR 2pR2 L dR 2pR2 L
C¼ ¼L ¼ 2pRL ¼ ¼ deh
dp dp dp dp R dp
Substitute in Hooke’s law in polar co-ordinates:
2pR2 L 1
C¼ ðdrh mdrr mdrz Þ
dp E
Substitute in the stress components to give:
2pR3 L m
C¼ 1
Eh 2
This gives the quoted result, since Poisson’s ratio is equal to 1/2.
(b) When placed in parallel, capacitors essentially simply become one large
capacitor with storage capacity (capacitance) adding. Hence N capacitors
have capacitance CN. The same applies for compliance.
D
You should be able to calculate the values easily.
E
(a) Denoting the pressure at entrance to the capillary compartment by Pa,
conservation of current/flow at this node gives:
Pin Pa 0 Pa 0 Pa
þ þ ¼0
Ra þ ixIa Rc þ Rv þ ixIv 1=ixCa
The flow through the capillaries is then:
Pa
q¼
Rc þ Rv þ ixIv
Pin
¼
ðRa þ ixIa þ Rc þ Rv þ ixIv þ ixCa ðRa þ ixIa ÞðRc þ Rv þ ixIv ÞÞ
(b) This is of the form of a (third-order) low-pass filter. This is important

because it filters out the pulsatile nature of the flow in the aorta such that
flow is steady by the time that it reaches the capillaries.
Answers 161
F
Equation 1.13 shows that inductance can be neglected even in the larger vessels
that we are considering here. The transfer function thus simplifies to:
Pin
q¼
ðRa þ Rc þ Rv þ ixCa Ra ðRc þ Rv ÞÞ
This has time constant:
Ca Ra ðRc þ Rv Þ
s¼
Ra þ Rc þ Rv
Any frequencies above the cut-off frequency set by this time constant will be
filtered out, hence we need:
Ra þ Rc þ Rv
2p [
Ca Ra ðRc þ Rv Þ
G
(a) We know that flow is proportional to the product of pressure difference
(which will be P here) and radius to the power four:
q ¼ k1 PR4
If compliance is constant, then the volume is proportional to pressure:
V ¼ k2 P
Since volume is proportional to radius squared:
P ¼ k3 R 2
Putting these all together gives:
q ¼ k1 k3 R6
(b) Force balance on the fluid gives wall shear stress as:
dp 2
sw 2pR ¼ pR
dx
162 Answers
Hence:
dp R
sw ¼
dx 2
Hence:
sw ¼ k4 R3
H
(a) If the vessel is incompressible, then the cross-sectional area must be the
same at all times:
ðR þ hÞ2 R2 ¼ ðRo þ ho Þ2 R2o
Hence:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h ¼ R þ R2 þ 2Ro ho þ h2o
(b) Clearly, as R increases, h decreases. Resistance is simply inversely

proportional to vessel radius to the power four, so it drops very rapidly
with increased radius.
(c) Compliance is proportional to radius to the power three and inversely
proportional to wall thickness, so as radius increases, compliance also
goes up very rapidly.
9. The Respiratory System
A
In the steady state with no generation or consumption of the gas involved the
diffusion equation for the partial pressure of gas in the membrane is:
@2c
D ¼ 0:
@x2
The boundary conditions are:

• Concentration of the gas on the airspace side of the membrane (x=0). For this
we need to use the Ostwald co-efficient to ‘convert’ partial pressure of gas into a
concentration: cðx ¼ 0Þ ¼ rpg :
• Concentration of gas in the blood: cðx ¼ LÞ ¼ cb .
Solving and using Fick’s law the flux is thus:
D
q¼ cb rpg :
L
We can define D/L as the surface diffusion co-efficient Ds, which is a property of
the membrane.
Answers 163
(a) f dCdxðxÞ ¼ 2prDs ðC ð xÞ Co Þ

(b) The equation is a first order differential equation, so can be written in the
form:
f dC
þ C ¼ Co
2prDs dx
which has the solution (given the boundary condition):

2prDs
C ¼ Co þ ðCin Co Þexp x
f
(c) The flux is found by integrating the function above
ZL
Q¼ 2prDs ðCð xÞ Co Þdx
x¼0
to give:

2prDs L
Q ¼ ðCin Co Þf 1 exp
f
(d) The flux is 4e−14 mM/s for baseline flow, 3.2e−14 mM/s and 4.5e
−14 mM/s for halved and doubled flow. Hence it is relatively constant
with flow rate in this region (flux saturates at high flows, increasing
proportionally less than flow).
(e) Two competing effects: increasing the flow increases the oxygen flux
into the capillary, but it passes through the vessel more quickly, so has
less chance to diffuse into the surrounding tissue.
(f) This gives Eq. 10.4, the equation no longer depends upon the diffusion
properties of the capillary wall only on perfusion and concentration
(partial pressure) differences. A large surface area for gas transfer is
present in the lungs and the membrane is thin, thus diffusion doesn’t
limit the overall transfer of gas, thus L is large compared to the diffusion
‘distances’ involved.
C
(a) Using reaction kinetic principles from Chap. 1, the equilibrium constant
for the second part of the reaction scheme is given by:
164 Answers
þ
HCO 3 ½H
KA ¼ :
½H2 CO3
(b) If almost all the available Co2 converts to carbonic acid, then the con-
centration of CO2 is very similar to that of H2CO3, thus the ‘corrected’
equilibrium constant is:
þ
HCO3 ½H
KA ¼ :
½CO2
D
(a) In Exercise B the governing equation for gas concentration in the cap-
illary blood was:
dC ð xÞ
f ¼ 2prDs ðC ð xÞ Co Þ
dx
This can be applied to CO2, but we now need to account for the con-
version to bicarbonate. As suggested model this using a simplified
reaction scheme that ignores the intermediary H2CO3:
K1
CO2 þ H2 O HCO
3 þH
þ
k1
Including the ‘loss’ of CO2 to bicarbonate and the reverse process where
bicarbonate converts back to CO2:
dC
f ¼ 2prDs ðrCO2 PCO2 C Þ þ k1 ½H þ D k1 C;
dx
where we use D to signify [HCO3-] as we have used C to represent

[CO2]. We also have an equation for D, but as this exists only in solution
we do not have to worry about the gas exchange part only the reaction
kinetics:
dD
f ¼ k1 C k1 ½H þ D:
dx
(b) Combining the two equations gives:
d
f ðC þ DÞ ¼ 2prDs ðrCO2 PCO2 C Þ:
dx
Answers 165
(c) Taking a quasi-steady state approximation allows us to write D ¼ Ka C

where Kc ¼ k1 =ðk1 ½H þ Þ: Thus, if [H+] is constant with respect to x:
dC
f ð 1 þ Kc Þ ¼ 2prDs ðrCO2 PCO2 CÞ:
dx
This is practically identical to the result in Exercise B and thus the

solution for flux in the limit will be the same except for an ‘amplifica-
tion’ of (1+Kc), hence Eq. 9.6.
(d) We can find the relationship between Kc and Ka using the result in
Exercise C:
Ka ¼ Kc ½H þ
Since log10Ka = −6.1 and pH=-log10[H+] = 7.4 typically, Kc is around

20. A substantial amplification of the flux of CO2 allowing more rapid
removal from the body than would be achievable by having CO2 alone in
solution.
E
(a) This is a matter of taking logs of the expression from Exercise C part b)
and re-arranging in terms of -log10[H+] i.e. pH.
(b) For the values given, [CO2] = 1.2 mM and pH = 7.4.
(c) This is called the Davenport diagram and is shown below. Large changes
in pH can be gained by adjusting both pCO2 and bicarbonate concen-
tration, so the lungs and kidneys have a lot of control over pH.
(d) The kidneys need to excrete H+ and reabsorb HCO-3 to balance pH again.
This would happen in a person with poor respiration (not enough
removal of CO2 from the lungs).
Further Reading
Cardiovascular Physiology (8th edition): Mohrman, Heller. McGraw-Hill, 2013.

Cellular Physiology of Nerve and Muscle (4th edition): Matthews. Blackwell, 2002.
Mathematical Physiology (2nd edition): Keener, Sneyd. Springer, 2008.
Molecular Biology of the Cell (6th edition): Alberts, Johnson, Lewis, Morgan, Raff, Roberts and
Walter. Garland Science, 2014.
The Cardiovascular System at a Glance (4th edition): Aaronson, Ward and Connolly.
Wiley-Blackwell, 2012.
The Respiratory System at a Glance (3rd edition): Ward, Ward and Leach. Wiley-Blackwell, 2010.
Tissue Mechanics: Cotwin, Doty. Springer, 2007.

Biosystems & Biorobotics 13, DOI 10.1007/978-3-319-26197-3

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About the Cover

More information about this series at http://www.springer.com/series/10421

ISSN 2195-3562 ISSN 2195-3570 (electronic)

Library of Congress Control Number: 2015955884

Springer Cham Heidelberg New York Dordrecht London

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media

However, there are lots of complications to this. For example, in an ischaemic

1 Cell Structure and Biochemical Reactions . . . . . . . . . . . . . . . . . . 1

4 Cellular Transport and Communication. . . . . . . . . . . . . . . . . . . . 43

7.4 Electrical Activity of the Heart . . . . . . . . . . . . . . . . . . . . . . . 86

10.3 Somatic Nervous System . . . . . . . . . . . . . . . . . . ......... 134

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

1.1 Cell Structure

© Springer International Publishing Switzerland 2016 1

1.2 Cell Chemicals

C6 H12 O6 þ 6O2 ! 6CO2 þ 6H2 O þ ATP ð1:1Þ

1.3 Reaction Equations

Fig. 1.4 Structure of DNA

1.3.1 Mass Action Kinetics

1.3.2 Enzyme Kinetics

Fig. 1.5 Enzymes and

1.3.2.1 Equilibrium Approximation

where Ks ¼ k1 =k þ 1 , similarly to before.

Fig. 1.7 Michaelis-Menten

At small substrate concentrations, the reaction rate is proportional to the amount

1.3.2.2 Quasi-Steady-State Approximation

1.3.3 Enzyme Cooperativity

This is known as the Hill equation. It is frequently used as an approximation for

Fig. 1.8 Hill equation

turns out to imply a fraction ﬁlling of available haemoglobin sites, S, of:

with n = 4. In fact, a much better ﬁt to experimental data is found with n = 2.5,

1.3.4 Enzyme Inhibition

1.3.4.1 Competitive Inhibitors

There are now six differential equations:

Thankfully, this isn’t as complicated as it looks. The normal assumption made in

1.3.4.2 Allosteric Inhibitors

A schematic of allosteric inhibition is shown in Fig. 1.9, alongside allosteric

2.1 Membrane Structure and Composition

© Springer International Publishing Switzerland 2016 19

homeostasis. If we want to understand how the cell maintains this imbalance we

2.2 Osmotic Balance

Fig. 2.4 Cell behaviour in

2.3 Conservation of Charge

From total concentration balance:

Fig. 2.5 Cell model example

2.4 Equilibrium Potential

So far we have only considered concentration equilibrium: however, there is another

where we have changed from a natural logarithm to a base-10 logarithm.

which on re-arranging becomes:

This is known as the Donnan or Gibbs-Donnan equilibrium equation.

Em= Vin - Vout

Fig. 2.6 Membrane and membrane potential

Exercise D is an interesting example of the counter intuitive differences in ion

2.5 A Simple Cell Model

We started out by trying to understand cell homeostasis and how an imbalance in

a Na+ Na+ 120 mM