Magnetic Resonance Imaging: Equations and Relations
Magnetic Resonance Imaging: Equations and Relations
Magnetic Resonance Imaging: Equations and Relations
Characteristic hydrogen factor, Hγ: 2.79 Speed of light: c = 3.0 x 10⁸ m/s
Magnetic resonance imaging, MRI, has become an essential diagnostic tool worldwide due to its
ability to non-invasively depict and distinguish soft tissues within the body. At the most basic
level, it utilizes an induced magnetic field and a pulsed radio frequency wave to create detailed
images of a patient. Several key discoveries paved the way for its creation including the
theories of electromagnetism, nuclear physics, and quantum mechanics, which led to the
discovery of characteristic particle spin.
History:
Over the course of the 19th century Michael Faraday, James Clerk Maxwell, and others laid the
foundational theories of electromagnetism. There was found to be an inherent duality in
nature between magnetic and electric energy fields, which could be manipulated and
quantified in concert with one another.
A classical MRI device contains a solenoid that carries high currents to generate a magnetic field
of a few Tesla. The figure above shows a cross section of solenoid where the crosses represent
a current into the paper and the dots a current out of the paper. From the right hand rule you
can find the direction of the resulting magnetic field depicted in the figure.
1) On the image of the scanner, predict and draw the direction of propagation of the induced
magnetic field. The arrows denote the direction of current through the solenoid
The discovery of nuclear spin states has had a great impact on how we understand the
quantum nature of particles. Imaging technologies such as nuclear magnetic resonance (NMR)
spectroscopy and MRI are the result of our growing knowledge in this field. It has been shown
that a nucleus will occupy either one of two orientations. This intrinsic physical property is
what is manipulated by an MRI machine.
2) Both protons and neutrons have a magnetic moment. Unless the nucleus contains an even
number of protons and an even number of neutrons, the atomic nucleus has a net magnetic
moment. A nucleus must have a net magnetization for it to be discernable using magnetic
resonance spectroscopy. From the list below circle possible candidates for magnetic resonance
spectroscopy (you may need to look at a periodic table, unless specified otherwise choose the
most abundant isotop for each element ):
First, open the MRI java simulation (found at: http://phet.colorado.edu/en/simulation/mri )and
click on the simplified MRI tab.
3) Now adjust the frequency bar to different levels and describe what you notice. Find the
frequencies that stimulate the nuclei. Are there a wide range of frequencies that promote a
resultant photon? What happens as we decrease the power?
The value that you just recorded for your resonance frequency is known as the Larmor
frequency. This is the frequency at which the Hydrogen atoms will flip between spin states and
emit resultant photons. In order to calculate the Larmor frequency we must first find the
difference in energy states for the hydrogen nucleus.
b) Using the equations from the front page, create a single equation that relates the energy
difference and the associated frequency.
c) Calculate the theoretical Larmor frequency for a magnetic field of 1 T. How close was your
value to the predicted value above from the applet?
5) Calculate the Larmor frequency for 2 T. Check your answers using the java program. This
time, set the power to 100 % and change your frequency to the result you calculated and drag
your magnetic field accordingly until you reach the max output. Were the values close?
6) On the MRI images below, give the name of the specific plane that each image was taken
from and suggest which region of the body is shown.
a) b) c)
In order to generate a slice the MRI needs to excite hydrogen nuclei in one plane. This is done
by adding a gradient magnetic field.
7) Adjust the slider of the vertical field first to 0.04 T and then to 0.08 T. What do you observe?
8) Press Add tumor and try to selectively target the hydrogen nuclei at the location of the
tumor. You can achieve this by changing the frequency of the radio wave or by changing the
strength of the main magnetic field. Note: in an actual MRI neither is done and the gradient
magnetic field is changed. Explain why changing the strength of the gradient field allows the
selective stimulation of the hydrogen nuclei in one slice.