A Novel Method To Calculate Compliance and Airway Resistance in Ventilated Patients

Gutierrez Intensive Care Medicine
Intensive Care Medicine Experimental (2022) 10:55

https://doi.org/10.1186/s40635-022-00483-2 Experimental
RESEARCH ARTICLES Open Access
A novel method to calculate compliance

and airway resistance in ventilated patients
Guillermo Gutierrez*
*Correspondence:
gutier@gwu.edu Abstract
Professor Emeritus Medicine, Background: The respiratory system’s static compliance (Crs) and airway resistance
Anesthesiology and Engineering, (Rrs) are measured during an end-inspiratory hold on volume-controlled ventilation
The George Washington (static method). A numerical algorithm is presented to calculate Crs and Rrs during
University, 700 New Hampshire
Ave, NW, Suite 510, Washington, volume-controlled ventilation on a breath-by-breath basis not requiring an end-inspir-
DC 20037, USA atory hold (dynamic method).
Methods: The dynamic method combines a numerical solution of the equation of
motion of the respiratory system with frequency analysis of airway signals. The method
was validated experimentally with a one-liter test lung using 300 mL and 400 mL
tidal volumes. It also was validated clinically using airway signals sampled at 32.25 Hz
stored in a historical database as 131.1-s-long epochs. There were 15 patients in the
database having epochs on volume-controlled ventilation with breaths displaying
end-inspiratory holds. This allowed for the reliable calculation of paired Crs and Rrs
values using both static and dynamic methods. Epoch mean values for Crs and Rrs were
assessed by both methods and compared in aggregate form and individually for each
patient in the study with Pearson’s R2 and Bland–Altman analysis. Figures are shown as
median[IQR].
Results: Experimental method differences in 880 simulated breaths were 0.3[0.2,0.4]
mL·cmH2O−1 for Crs and 0[− 0.2,0.2] cmH2O·s· L−1 for Rrs. Clinical testing included
78,371 breaths found in 3174 epochs meeting criteria with 24[21,30] breaths per
epoch. For the aggregate data, Pearson’s R2 were 0.99 and 0.94 for Crs and Rrs, respec-
tively. Bias ± 95% limits of agreement (LOA) were 0.2 ± 1.6 mL·cmH2O−1 for Crs and
− 0.2 ± 1.5 cmH2O·s· L−1 for Rrs. Bias ± LOA median values for individual patients were
0.6[− 0.2, 1.4] ± 0.9[0.8, 1.2] mL·cmH2O−1 for Crs and − 0.1[− 0.3, 0.2] ± 0.8[0.5, 1.2]
cmH2O·s· L−1 for Rrs.
Discussion: Experimental and clinical testing produced equivalent paired measure-
ments of Crs and Rrs by the dynamic and static methods under the conditions tested.
Conclusions: These findings support to the possibility of using the dynamic method
in continuously monitoring respiratory system mechanics in patients on ventilatory
support with volume-controlled ventilation.
Keywords: Mechanical ventilation, Acute respiratory failure, Static compliance, Airway
resistance, Numerical analysis, Frequency analysis
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Gutierrez Intensive Care Medicine Experimental (2022) 10:55 Page 2 of 14
The respiratory system (rs) static compliance (Crs) and airway resistance (Rrs) are cal-
culated during volume-controlled (VC) mechanical ventilation with a breath-hold
maneuver at the end of quiet inspiration (static method) [1]. Under these conditions
Crs = Vtidal /(Pplateau – PEEPa), where Vtidal = tidal volume, Pplateau = breath-hold Paw, and
PEEPa = applied positive end expiratory pressure. Similarly, Rrs = (Ppeak – Pplateau)/Faw,
where Ppeak = peak inspiratory pressure and F aw is airway flow measured just prior to
breath-holding [2].
A reliable method to calculate Crs and Rrs automatically, without the need of an
inspiratory hold, would have great utility in monitoring the adequacy of ventilatory
support. One approach previously tried is the multiple least squares fit (LSF) tech-
nique [3, 4], where measures of Paw, Faw, and lung volume change (ΔV) are fitted to the
equation of motion of the respiratory system. Another is the expiratory time constant
τe method [5] where equations for Crs and Rrs are developed assuming mono-expo-
nential lung volume release [6]. Both methods require the use of complex computa-
tional techniques and absent respiratory muscle effort.
Described is a method to calculate Crs and Rrs during insufflation in the presence
of airflow (dynamic method) that combines frequency analysis of the airway signals
with a novel numerical solution of the equation of motion. The method was validated
experimentally with a one-liter test lung. It was also validated clinically using pre-
viously acquired Faw and Paw signal data from patients on VC ventilation displaying
end-inspiratory holds. This allowed for the reliable calculation of paired Crs and Rrs
values using both static and dynamic methods.
Theoretical development. The time-dependent equation of motion of the respiratory
system is:
�V (t) d2 V (t)
Paw (t) = P mus (t) + P vent (t) = + Rrs Faw (t) + I + PEEPa + PEEPi
Crs dt 2
(1)
This equation, based on the one-compartment model of Otis et al. [7], assumes con-
stant values for Crs and Rrs. The measured airway pressure Paw(t) represents the sum
of the ventilator and respiratory muscles applied pressures Pvent(t) and Pmus(t), respec-
tively. Opposing them are the elastic, resistive, and inertial components of the respira-
tory system. V(t) represents the time-dependent lung volume; ΔV(t) is the insufflation
t
lung volume at time t, equal to 0 Faw (t)dt ; I is the respiratory system inertia; and
PEEPi the intrinsic PEEP [8].
Assuming passive insufflation (Pmus = 0), negligible P EEPI, and ignoring the effect of
the inertia term [9], Eq. (1) becomes:
�V (t)
Paw (t) = Pvent (t) = + Rrs Faw (t) + PEEPa . (2)
Crs
It is possible to solve numerically this indeterminate equation with two unknowns,

Crs and Rrs, by first developing a solution matrix for each set of Paw(tk), ΔV(tk), Faw(tk),
and PEEPa values measured at successive times tk during insufflation. The elements
of the solution matrix are calculated by substituting the measured values for ΔV(tk),
Faw(tk), and P
EEPa into Eq. 2 and alternately applying a range of physiologically plausi-
ble values for Crs (C1 … Cn) and Rrs (R1 … Rn).
R1 R2 Rn
 
Paw (R1 , C1 ) Paw (R2 , C1 ) · · · Paw (Rn , C 1 ) C1
.. .. ..  .
Solution matrix =   ..

. . .
Paw (R1 , Cn ) Paw (R2 , Cn ) · · · Paw (Rn , Cn ) Cn
For example, applying a range of (C1 … Cn) values from 10 to 100 mL·cmH2O −1 and
1.0 to 50.0 c mH2O·s·L−1 for (R1 … Rn), at intervals of 0.1 each, produces a 900 × 490
solution matrix containing all possible Paw values capable of satisfying Eq. 2 for given a
set of (tk), Faw(tk), and PEEPa measurements made at time tk during insufflation.
Figure 1 shows a schematic of the proposed numerical method of solution. In this
example, a solution matrix was generated for ΔV(tk) = 300 mL, Faw(tk) = 32 L·min−1, and
PEEPa = 5 cmH2O and plotted as a three-dimensional surface in a Cartesian (Crs, Rrs,
Paw) system. According to the above reasoning, the solution of Eq. 2, in terms of Crs and
Rrs, resides on a point on that surface. Further insight is gained by noting that the solu-
tion must lie along a surface path traced by the measured Paw(tk) at time tk. This is shown
in Fig. 1 as path A, where Paw(tk) = 27 cmH2O and point ’a’ symbolizes the yet unknown
solution of Eq. 2.
Fig. 1 Schematic of the numerical method used to solve the respiratory system equation of motion for
static compliance (Crs) and airway resistance (Rrs). In this example, the solution matrix was developed for
ΔV(tk) = 300 mL, Faw(tk) = 32 L·min−1, and P
EEPa = 5 cmH2O and shown graphically as a three-dimensional
surface bounded by Crs values ranging from 10 to 50 mL·cmH2O−1 and Rrs from 0 to 20 c mH2O·s·L−1. This
surface encompasses all possible combinations of Paw, Crs and Rrs capable of satisfying Eq. 2 for a given set of
ΔV, Faw, and P
EEPa measurements made at time tk during insufflation. Paw, also measured at tk and equal in
this example to 27 cmH2O, further restricts the solution of Eq. 2 to lie along path (A). This path is defined by
surface values coinciding with the measured Paw, with point ‘a’ referring to the still unknown solution of Eq. 2.
Projecting path A onto the Crs – Rrs plane results in a two-dimensional function (B) relating Crs to Rrs. Here ‘b’
represents the unique solution of Eq. 2 defining the values for Crs and Rrs for the breath under consideration
Projecting path (A) onto the Crs–Rrs plane generates a two-dimensional function,
shown as path B, that restricts all possible combinations of Crs and Rrs able to satisfy
Eq. 2 for the set of measurements taken at time tk. The Crs–Rrs function is developed
numerically by noting the values Rx, Cy associated with those matrix elements where
Paw(Rx, Cy) = measured Paw(tk).
It only remains to identify the location of solution point ‘b’ on the Crs–Rrs plane. This
is accomplished by generating a family of Crs–Rrs functions, one for each set of Paw(tk),
ΔV(tk), Faw(tk), and P EEPa values measured at sequential times tk during insufflation.
Since the one-compartment model of Eq. 2 assumes constant Crs and Rrs, it follows that
all generated Crs–Rrs functions must pass through, and therefore intersect, at a point that
defines Crs and Rrs for the breath in question.
It is known that Crs and Rrs vary early in inspiration as unstable alveoli open and con-
ducting airways distend [10]. However, as lung volume increases past a lower inflection
point (LIP) Crs achieves steady state until reaching an upper inflection point (UIP) where
over-distention might occur. Defining the LIP and UIP by their respective lung volumes
as ΔVLIP and ΔVUIP, it is reasonable to expect all Crs–Rrs functions generated for insuf-
flation lung volumes ΔVLIP < ΔV(t) < ΔVUIP to intersect at the solution point ‘b’ uniquely
defining Crs and Rrs for that breath.
Figure 2 shows a family of Crs–Rrs functions (n = 14) generated during a single breath’s
insufflation at sequential 32-ms intervals, past ΔVLIP > 200 mL. The intersection of these
functions defines the values for Crs = 32.8 mL·cmH2O −1 and Rrs = 23.8 cmH2O·s·L−1.
The inset graph illustrates the slight uncertainty associated in determining the intersec-
tion of the Crs–Rrs functions, likely the result of random variations in measurement or
small changes in Crs and Rrs occurring during the insufflation. Accordingly, the point of
intersection is best defined by the smallest standard deviation (σ) of all Crs values meas-
ured at each Rrs increment along the Rrs axis.
Fig. 2 A family of Crs and Rrs functions (n = 14). Each function was generated at different times (tk)
measured sequentially at 32 ms during a single insufflation. The intersection of these functions defines
Crs = 32.8 mL·cmH2O −1 and Rrs = 23.8 cmH2O·s· L−1 for the breath. Shown in the inset graph is the
uncertainty associated with the intersection point, likely the result of measurement limitations or minute
alterations in Crs and Rrs during insufflation. Accordingly, the point of intersection is best defined by the
smallest standard deviation (σ) of all Crs values measured at each Rrs increment along the Rrs axis
Methods
The accuracy of the dynamic method was tested by comparing paired Crs and Rrs val-
ues predicted by the dynamic method and the static method (used here as the ‘gold
standard’) for the same breath.
Experimental validation
Validation was performed experimentally with a Maquet 190 one-liter test lung (Get-
inge, Solna, Sweden) using VC ventilation with a 0.5-s inspiratory hold. The test lung
was attached to a Servo s ventilator (Getinge, Solna, Sweden) and ventilated at a res-
piratory rate of 15 bpm with Vtidal of 300 mL or 400 mL. PEEP levels of 0, 5 and 10
cmH2O were applied sequentially at each Vtidal.
An in-house built data acquisition monitor was used to sample Faw and Paw sig-
nals from the ventilator data-port at 32.25 Hz and compile successive epochs of 4096
points, each lasting 131.1 s. Five epochs were obtained at each Vtidal–PEEP combina-
tion. Data were analyzed in situ with the monitor’s Raspberry Pi 3B processor pro-
grammed (Python 3.8) to calculate Crs and Rrs for each breath by the dynamic method.
Crs and Rrs were also determined manually by the static method for 10 breaths in each
epoch using data from the Paw and Faw signals. Average epoch values for Crs and Rrs
computed with either method were compared at each Vtidal–PEEP combination.
Clinical validation
The dynamic method also was validated with clinical data using Faw and Paw signals
obtained in a prior study of mechanically ventilated patients performed in 2011–2012
at The George Washington University Hospital Intensive Care Unit (IRB No. 110910)
[11]. The database (Additional file 1: Section S1) contains information from 176
patients with acute respiratory failure enrolled within 24 h of intubation and moni-
tored during their entire time on ventilatory support. It contains deidentified demo-
graphic information and Faw and Paw signals sampled at 32.25 Hz by the ventilator
(Servo I or Servo S ventilators, Getinge, Solna, Sweden). The signals were saved as
contiguous time-windows or epochs, each lasting 131.1 s and containing 4096 sam-
ples of each signal.
Epoch selection
Software was written (Python 3.8) to search the database for epochs on VC venti-
lation. The respiratory rate variability (RRV) for each identified epoch was used to
determine the degree of active respiratory muscle activity. RRV was determined from
the frequency spectrum of the expiratory flow signal as previously described [12]
using the fast Fourier transform (FFT) algorithm [13]. RRV was defined as 100 – H1/
DC %, where H1 is the amplitude of the spectrum’s first harmonic and DC that of the
zero-frequency component. Epochs with RRV < 55% were assumed to have negligible
respiratory muscle activity (Pmus = 0) and were chosen for the study. This RRV value
corresponds to those noted in normal individuals during quiet breathing in stages N2
and N3 of sleep [14].
Breath selection
Within each selected epoch, the software further identified breaths displaying an end-
inspiratory hold and absent voluntary respiratory effort. These breaths allowed for
the reliable measurements of static compliance and airway resistance using standard
calculations for comparison with those predicted by the dynamic method. The fol-
lowing criteria was used to choose breaths for analysis: (1) a discernible end-inspira-
tory hold > 0.25 s with mean plateau airway flow < 1 L·min−1; (2) ventilator-triggered
(PEEPa – minimal Paw < 0.3 cmH2O); (3) full volume breaths (ΔV(t) ≥ 300 mL with
EEPi (end-exhalation (EE) Faw < 3 L·min−1
insufflation time (Ti) > 0.8 s); (4) absent P
[15] and Paw(t0)—EE Paw < 2 cmH2O) [16]; and (5) no air circuit leak (inspired –
expired Vtidal <|30 mL|). Excluded were breaths with ΔV(t) ≥ 740 mL to avoid exceed-
ing the UIP [17] (see Additional file 1: Section S2 for breath exclusion example). Once
a breath was deemed adequate for analysis, the software calculated Crs and Rrs by both
the dynamic and static methods.
Results from the dynamic and static methods were compared with Pearson’s linear
regression R2 and Bland–Altman analysis [18] for bias ± 95% limits of agreement (LOA).
Since some patients had substantially more epochs meeting study criteria than others,
the methods were compared in aggregate by combining data from all epochs and indi-
vidually for each study patient. Unless otherwise specified, data are shown as median
and interquartile range. The Mann–Whitney test was used to determine significant dif-
ferences between independent samples. All reported p values are two-sided with p < 0.05
considered significant.
Results
Experimental validation
Analysis of 880 breaths from 30 epochs resulted in nearly identical values for Crs and Rrs
calculated by the static and dynamic methods for all tested combinations of Vtidal and
PEEP (Table 1; p = NS). Overall method differences were 0.3[0.2,0.4] mL·cmH2O −1 for
Crs and 0[− 0.2,0.2] cmH2O·s·L−1 for Rrs.
Table 1 Respiratory system’s static compliance and resistance values calculated by the static and
dynamic methods using a test lung
Vtidal PEEP* Compliance mL·cmH2O −1 Resistance cmH2O·s·L−1
mL cmH2O Static Dynamic Static Dynamic
0 21.0[21.0,21.0] 21.5[21.4,21.5] 7.9[7.9,8.0] 7.7[7.6,7.8]

300 5 22.8[22.8,22.9] 23.2[23.1,23.3] 8.2[5.2,8.3] 8.3[8.2,8.3]
10 26.1[26.0,26.1] 26.4[26.4,26.3] 8.2[7.9,8.2] 8.1[8.1,8.3]
0 22.9[22.9,23.0] 23.0[23.0,23.1] 10.5[10.4,10.5] 10.1[10.0,10.1]
400 5 24.4[24.4,24.5] 24.7[24.6,24.8] 10.7[10.7,10.8] 11.0[11.0,11.0]
10 27.7[27.7,27.8] 28.1[28.0,28.1] 10.7[10.7,11.1] 11.0[10.9,11.0]
Total (n = 880) 23.6[22.8,26.1] 23.9[23.0,26.4] 9.3[8.2,10.7] 9.2[8.1,10.9]
Difference† 0.3 [0.2,0.4] 0 [− 0.2,0.2]
There were no significant differences between methods in any of the variables measured
Vtidal tidal volume, PEEP positive end expiratory pressure; values shown as median [IQR]
*
Five epochs per PEEP level, each containing approximately 30 breaths
†
Difference = dynamic – static methods
Clinical validation
Of the 176 patients in the database, 15 (8.5%) were identified as meeting study criteria.
The 15 patients had a combined total of 33,371 epochs on VC ventilation and RRV < 55%.
The study patients were evenly split according to gender, but ranged widely in age, pre-
dicted body weight (PBW) and body mass index (BMI). Disease acuity was high (SAPS
II 36[32,44]), five were non-cardiac post-operative, two were trauma and the remainder
medical patients. The P/F ratio was relatively high at 337[272,429] m mH2O, reflecting
the lack of lung pathology noted in half of the patients’ chest radiographs (Additional
file 1: Table S1, Additional file 2, Additional file 3, Additional file 4).
Of the 33,371 identified epochs, 3174 (9.5%) contained breaths displaying end-inspir-
atory holds. The ventilatory parameters associated with these epochs were compatible
with those of quiet, passive ventilation with a low RR = 11[11,14] bpm and RRV = 45
[40,46] % (Additional file 1: Table S2). The 3174 chosen epochs encompassed 87,021
individual breaths with 78,371 (90.1%) considered adequate for analysis of static com-
pliance and airway resistance using standard calculations for comparison with those
predicted by the dynamic method. The median number of breaths in these epochs was
24[21,30].
Aggregate data analysis

There was an excellent correlation between the static and dynamic methods
(Fig. 3) with (Crs)stat = 1.06 (Crs)dyn – 2.26; R2 = 0.99; p < 0.001 and (Rrs)stat = 0.93
(Rrs)dyn + 1.02; R2 = 0.94; p < 0.001. Bland–Altman analysis (Fig. 4) showed bias ± LOA of
0.2 ± 1.6 mL·cmH2O −1 for Crs and – .2 ± 1.5 cmH2O·s· L−1 for Rrs.
Individual patient analysis. Table 2 shows Bland–Altman analyses for individual
study patients. There were 119 [34, 339] (range 15 to 881) epochs per patient contain-
ing 2926 [624, 7702] (range 426 to 17,953) breaths. Individual patient bias ± LOA for Crs
was 0.6 [− 0.2, 1.4] (range − 0.8 to 1.6) ± 0.9 [0.8, 1.2] (range 0.7 to 2.3) mL·cmH2O −1.
Bias ± LOA for Rrs was − 0.1[− 0.3, 0.2] (range − 1.6 to 2.1) ± 0.8 [0.5, 1.2] (range 0.2 to
2.2) cmH2O·s· L−1.
Discussion
Increases in computing power [19] allow for the application of powerful analytical tech-
niques to monitor patients on ventilatory support. The present study describes an algo-
rithm capable of providing breath-by-breath measures of Crs and Rrs without the need
for an end-inspiratory pause in patients on VC ventilation. This technique may in turn
allow for the continuous monitoring of other parameters, such as the driving pressure, a
proven indicator of ventilator associated lung injury [20].
The dynamic method used to determine static Crs and Rrs is based on a novel numeri-
cal solution of the equation of motion of the respiratory system. This equation depicts
the behavior of respiratory mechanics in normal individuals and has been applied suc-
cessfully to ventilated patients with respiratory failure [21]. In the form used here, the
equation of motion ignores the inspired gas inertia and the resistance to energy trans-
fer by visco-elastic lung tissue, whereas both terms may be quantitatively significant
under extreme ventilatory conditions, they are likely inconsequential under the studied
Fig. 3 Pearson’s linear regression using data generated by the 15 patients in the study. Compared were
average epoch measurements of Crs and of Rrs by the static and dynamic methods (n = 3174). (Crs)stat = 1.06
(Crs)dyn – 2.26; R2 = 0.99; p < 0.001 and (Rrs)stat = 0.93 (Rrs)dyn + 1.02; R2 = 0.94; p < 0.001
conditions [22]. It should be noted that the one-compartment model of Otis et al. [7]
does not allow for the partitioning of respiratory system mechanics. On the other hand,
a similar numerical approach may be considered when solving a more complex model of
the respiratory system, one that accounts for both lung and chest wall compliances.
Method validation was done with matching pairs of Crs and Rrs calculated by the
static and the dynamic methods. Experimental method validation yielded nearly iden-
tical Crs and Rrs values when tested with a test lung ventilated using different Vtidal
and applied PEEP levels. Since Crs and Rrs were more or less fixed for the test lung,
the dynamic method also was validated using airway signals from a previous study
on mechanically ventilated patients. The use of clinical data yielded a more realis-
tic assessment of the dynamic method, allowing for method comparison at Crs values
Fig. 4 Bland–Altman analysis of average epoch measurements of Crs and Rrs (n = 3174) from the data
generated by the 15 patients in the study. Bias ± 95% LOA was 0.2 ± 1.6 mL·cmH2O −1 for Crs and − 0.2 ± 1.5
cmH2O·s· L−1 for Rrs
ranging from 20 to 70 mL·cmH2O−1 and from 10 to 32 c mH2O·s· L−1 for Rrs. These
are ranges similar to those encountered in clinical practice.
Software was written to identify breaths meeting strict morphologic criteria that
included a discernible plateau pressure and negligible Pmus or P EEPi. This resulted in
the evaluation of a massive number of individual breaths (78,371) contained in the
3174 identified epochs. The software calculated paired Crs and Rrs values by the static
and dynamic methods in all identified breaths enclosed within each 131.1-s-long
epoch, reporting the epoch’s average for comparison. The use of epochs was dictated
both by the format initially used to store the data and by the ability to assess respira-
tory muscle activity indirectly by spectral analysis.
Table 2 Bland–Altman analysis of respiratory system’s static compliance and resistance values

calculated by the static and dynamic methods for individual patients
Patient Epochs Breaths Compliance mL·cmH2O −1 Resistance cmH2O·s·L−1
(3174) (78,371)
Mean value Bias LOA Mean value Bias LOA
1 293 5924 35.8 0.4 1.1 21.9 0.4 1.4

2 119 4917 27.5 − 0.6 0.9 15.2 0.2 2.2
3 127 2926 58.6 1.4 0.9 14.3 0.2 0.3
4 385 8632 32.8 − 0.3 1.2 20.5 − 0.3 0.9
5 15 459 59.7 1.6 0.9 16.7 − 0.1 0.6
6 38 660 46.7 1.6 2.3 13.6 − 0.4 1.9
7 399 11,714 39.7 0.2 0.9 15.4 0.1 1.1
8 881 17,953 53.5 0.8 0.8 16.8 − 0.3 0.6
9 72 1718 49.7 1.3 0.8 17.7 -0.1 0.5
10 27 426 41.2 1.4 1.3 17.4 2.1 1.1
11 38 1123 47.5 0.7 1.1 12.2 0.8 0.4
12 26 494 52.5 0.0 0.7 12.0 − 0.1 0.2
13 248 6771 34.8 − 0.8 0.7 18.9 − 1.6 0.8
14 476 14,067 33.8 − 0.5 1.2 15.6 − 0.5 1.6
15 30 587 64.7 0.6 1.3 12.9 0.2 0.3
Median 119 2926 46.7 0.6 0.9 15.6 − 0.1 0.8
IQR [34,339] [624,7702] [35.3, 53.0] [− 0.2, 1.4] [0.8, 1.2] [13.9, 17.5] [− 0.3, 0.2] [0.5, 1.2]
Mean value = average value of all measurements used in the Bland–Altman analysis; LOA = 95% limits of agreement;
IQR = interquartile range]
The cohort was composed mainly of highly sedated patients transferred from the
Emergency Department and ventilated with end-inspiratory holds that were not imme-
diately detected by the ICU team. Although the data were collected several years ago,
neither the passage of time nor changes in ICU care should have influenced the results
presented nor adversely altered the fidelity of the stored airway signals.
To provide for a balanced assessment of the data, analysis was performed in aggre-
gate form and also individually for each patient in the study. Whereas aggregate analysis
biased the results in favor of patients with many analyzed epochs, individual analysis
amplified the effect of patients with fewer epochs. Regardless of comparison strategy,
however, both methods produced nearly identical Crs and Rrs values with negligible bias
and exceedingly small LOA.
Although method bias was minimal for both Crs and Rrs, the possibility should be
acknowledged of introducing a systematic error by the software when calculating the
“gold standards” Crs and Rrs by the static method. The cessation of gas flow during the
end-inspiratory hold produces a rapid decline in Paw from Ppeak to P1, followed by a
slow decay to a plateau P2 [23]. The timing of the end-inspiratory hold (thold) could
be an important source of measurement error since a short thold may affect P1 by the
persistence of airflow during inspiratory valve closure or P2 by prematurely shorten-
ing the decay of Paw. Conversely, a long thold may allow voluntary respiratory muscle
activity to occur, also distorting P2. All breaths in the study were ventilator triggered
with no evidence of spontaneous respiratory muscle activity throughout the length of
the breath, including the end-inspiratory hold portion. For the cohort, thold was 0.4
[0.4,0.4] (range 0.3 to 0.7) seconds, allowing ample time for inspiratory valve closing
[24] and placing P2 firmly on the flat portion of the plateau, as evidenced by the small
decline in Paw (< 1.0 c mH2O) predicted by decreasing exponentials fitted to the data
(R2 = 0.96) and extrapolated from 0.4 to 1.0 s (Additional file 1: Section S4).
Several assumptions were made in the development of the dynamic method, among
them the constancy of Crs and Rrs during insufflation. This basic tenet of the one-
dimensional model of Otis et al. [7] is unlikely to hold true during the early stages of
inspiration where the volume signal is curvilinear [23]. Past a certain inflation vol-
ume, defined here as ΔVLIP, Crs becomes constant and remains so over the rest of
the tidal range [25]. The dynamic method was therefore applied to insufflation lung
volumes > 200 mL, a ΔVLIP chosen to match those reported in ARDS patients [26, 27].
This is probably a conservative estimate since no patient in the study met the Berlin
definition for ARDS [28] with half the cohort having normal chest radiographs. More-
over, all patients were ventilated with PEEPa = 5 cmH2O, likely resulting in initial lung
volumes in the region of constant Crs. It is possible, however, that small variations in
Crs and Rrs during the studied insufflation volumes resulted in the slight uncertainty
noted in determining the intersection of the Crs – Rrs functions.
The assumption of absent patient inspiratory effort during insufflation (Pmus = 0)
cannot be independently verified since esophageal balloon catheters were not used in
the original study. The validity of this assumption rests on: (1) the use of RRV < 55%
as an inclusion criterion, a value noted in heavily sedated ventilated patients [29] and
normal individuals during stages of deep sleep; (2) all analyzed breaths were ventila-
tor-initiated; and 3) a cohort of 50 epochs selected randomly from the sample popula-
tion was characterized by a regular breathing pattern, low respiratory rate (11 [11, 14]
bpm) and no signal distortion (see Additional file 1: Section S7 and Table 2e).
The assumption of absent P EEPi also could not be independently verified, but care
was taken to include in the analysis only breaths displaying minimal differences
between its onset and the prior breath’s end-exhalation Faw and Paw. In addition, (1)
no patient in the study was diagnosed with obstructive lung disease; (2) the exhalation
time for the cohort allowed ample time for expiration (3.2 ± 0.7 s); and (3) tachypnea
(RR > 20 bpm) was absent in all chosen epochs.
The dynamic method is unlikely to perform well under conditions of persistent
asynchronous breathing or in the presence of significant respiratory muscle effort. It
is also not amenable for bedside use or with ventilators lacking airway signal sam-
pling. Conversely, when used in conjunction with a computer connected to the venti-
lator’s data-port, the dynamic method may provide accurate ongoing measurements
of Crs and Rrs under most clinical conditions encountered during the provision of vol-
ume-controlled mechanical ventilation.
Although the present study was not intended as a methodological comparison, the
dynamic method appears to perform as well or better than either the LSF or the τe meth-
ods (Additional file 1: Table S3). Unlike these empirical models, the dynamic method rep-
resents a deterministic approach to the solution of the equation of motion. As such, it
may be applicable to ventilatory modes other than VC and provide insight into the rela-
tionship of respiratory system mechanics to other ventilatory variables, such as plateau
pressure, respiratory muscle effort and intrinsic PEEP. These, and other issues related to
the application of the dynamic method await further confirmation by prospective studies.
Take‑home message
A novel numerical method to calculate static compliance and airway resistance of the
respiratory system during ventilatory support is developed and validated.
Abbreviations
Crs Respiratory system static compliance
ΔV Lung volume change during insufflation
ΔVLIP Lung volume at the lower inflection point
ΔVUIP Lung volume at the upper inflection point
DC Flow spectrum zero frequency component amplitude
Faw Airway flow
FFT Fast Fourier transform algorithm
H1 First harmonic amplitude of the flow or pressure signal spectrum
I Inertia of the respiratory system
LIP Lower inflection point
LOA 95% Limits of agreement
LSF Multiple least squares fit
Paw Airway pressure
PEEPa Applied positive end expiratory pressure
PEEPi Intrinsic PEEP present at end expiration
Pmus Respiratory muscles applied pressure
Ppeak Peak inspiratory pressure
Pplateau Plateau pressure during the end-inspiratory hold
Pvent Ventilator applied pressure
rs Respiratory system
Rrs Respiratory system airway resistance
RRV Respiratory rate variability
τe Expiratory time constant
Thold Timing of the expiratory hold
tk A point in time during insufflation
UIP Upper inflection point
VC Volume controlled
V(t) Lung volume as a function of time
Vtidal Tidal volume
Supplementary Information
The online version contains supplementary material available at https://doi.org/10.1186/s40635-022-00483-2.
Additional file 1: Table S1. Demographic and ICU Admission Data, Diagnoses, and ICU Admission Chest X Rays for
Study Patients. Table S2. Average Ventilatory Parameters Computed from All Epochs Used in Method Comparison.
Table S3. Comparison of bias ± Limits of Agreement (LOA) for Crs and Rrs calculated From Individual Patient Data by
the Dynamic, Least Square Fitting (LSF) and Expiratory Time Constant (τE) Methods.
Additional file 2. Fitting double exponential.
Additional file 3. Raw data individual and aggregate.
Additional file 4. Experimental data results.
Acknowledgements
The author thanks the Commission for Educational Exchange between the United States, Belgium and Luxembourg and
the Fulbright Scholarship Board for their generous support as a Fulbright Research Scholar at the Erasme Hospital of the
Université Libre de Bruxelles.
Author contributions
The author read and approved the final manuscript.
Funding
Not applicable.
Availability of data and materials

The datasets used and analyzed during the current study can be found in the Electronic Data Repository. The database
storing the raw data is available from the author upon reasonable request.
Declarations
Ethics approval and consent to participate
The database used in the present study was collected in 2011 as part of research on respiratory rate variability (Gutierrez
et al., Intensive Care Med. 2013; 39:1359–1367). The study was approved by The George Washington University IRB (IRB
No. 110910) that allowed use of the deidentified data in further studies.
Consent for publication

Not applicable.
Competing interests
The author has applied for a U.S. patent based on the information presented in the paper.
Received: 17 August 2022 Accepted: 17 December 2022
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