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Current Directions in Biomedical Engineering

Joint Journal of the German Society for Biomedical Engineering in VDE and the Austrian and Swiss Societies for Biomedical Engineering

Editor-in-Chief: Dössel, Olaf

Editorial Board: Augat, Peter / Buzug, Thorsten M. / Haueisen, Jens / Jockenhoevel, Stefan / Knaup-Gregori, Petra / Kraft, Marc / Lenarz, Thomas / Leonhardt, Steffen / Malberg, Hagen / Penzel, Thomas / Plank, Gernot / Radermacher, Klaus M. / Schkommodau, Erik / Stieglitz, Thomas / Urban, Gerald A.

CiteScore 2018: 0.47

Source Normalized Impact per Paper (SNIP) 2018: 0.377

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Inspiratory and expiratory elastance in a non-linear autoregressive model of pulmonary mechanics

Ruby Langdon / Paul D. Docherty / Knut Möller
Published Online: 2016-09-30 | DOI: https://doi.org/10.1515/cdbme-2016-0138


For patients with acute respiratory distress syndrome (ARDS), the use of mathematical models to determine patient-specific ventilator settings can reduce ventilator induced lung injury and improve patient outcomes. A non-linear autoregressive model of pulmonary mechanics was used to identify inspiratory and expiratory pressure-dependent elastance (Ei and Ee) as independent variables. The analysis was implemented on 19 data sets of recruitment manoeuvres (RMs) that were performed on 10 mechanically ventilated patients. At pressures p = 15–20 cmH2O the agreement between Ei and Ee was low. However, Ei was a well-matched predictor of Ee for p = 25–40 cmH2O, with R2 ≥ 0.78, and there was no significant bias in the difference between Ei and Ee. Since many other models cannot uniquely identify Ei and Ee, the outcome may provide further insight into the characteristics of ARDS lungs in sedated patients.

Keywords: autoregressive models; parameter identification; pulmonary modelling

1 Introduction

Acute respiratory distress syndrome (ARDS) is a condition treated via mechanical ventilation in the intensive care unit (ICU). ARDS generally involves inflammation in the lungs and excess fluid in the airspaces, which increases elastance. While ventilation is necessary, it can sometimes cause ventilator induced lung injury (VILI) [1]. Excessively high airway pressure or high tidal volumes can cause distension and sometimes rupture of alveoli. Low pressures cause the cyclical opening and closing of alveoli with each breath that can damage healthy alveoli, known as atelectrauma [2].

The use of patient specific ventilator settings can help to avoid VILI [3]. As each patient and their disease state are unique, the optimal ventilator settings are patient specific. Mathematical models that describe the physiology of ARDS lungs can allow these optimal patient specific ventilator settings to be found. VILI is associated with a high mortality rate [4], thus a model that effectively describes lung behaviour and reduces the chances of VILI could significantly improve patient outcomes and reduce morbidity and mortality.

While there are a wide range of physiologically and clinically relevant models [5], [6], [7], in general these models are not able to uniquely identify inspiratory elastance Ei, and expiratory elastance Ee as independent variables. This is because the flow and volume both follow exponential decays during relaxed expiration of the sedated lung. As flow and volume are not linearly independent, single elastance and resistance terms are non-identifiable.

A nonlinear autoregressive (NARX) model of pulmonary mechanics has been described by Langdon et al. [8]. The NARX model uses basis functions to describe a pressure-dependent elastance, allowing it to describe recruitment and distension effects across recruitment manoeuvres (RMs). The model also uses time-dependent, flow-dependent terms to fit the passive lung relaxation during an inspiratory pause.

The aim of this paper is to determine the ability of the NARX model to identify independent inspiratory and expiratory elastances, and show the relationship between Ei and Ee. This will help to further validate the NARX model and its usefulness in accurately describing patient physiology.

2 Material and methods

2.1 Data

Data from a pilot clinical utilisation of respiratory elastance (CURE) software trial was used in this analysis. Airway pressure and flow data were collected from ten fully sedated ARDS patients, seven of which were ventilated in pressure controlled mode, and three of which were ventilated in volume controlled mode. Patient age ranged from 18 to 88 with a mean of 50.3 years. The breathing rate was approximately 18 breaths per minute. Pressure and flow were recorded from a Puritan Bennett 840 ventilator at a sampling rate of 50 Hz. Volume was calculated from continuous integration of the flow, with compensation for volume drift to maintain a volume of zero at PEEP.

As patients often underwent multiple RMs during the trial, 19 sets of data were obtained. PEEP at the beginning of the RM varied between 8 cmH2O and 16 cmH2O for different patients. During each RM, PEEP was increased in steps of 2 cmH2O. The RMs of different patients contained between four and nine PEEP step increases. The maximum pressure reached for each patient ranged between 36 cmH2O and 52 cmH2O.

Ethics approval for the study and use of collected data was granted by the New Zealand South Regional Ethics Committee. Informed consent was obatined from all individuals.

2.2 Respiratory models

The NARX model contains first order b-spline basis functions that describe a pressure dependent elastance, and time dependent terms that capture the pressure responses that occur due to changes in flow:


where: Pawi and Pawe are the measured inspiratory and expiratory airway pressure (cmH2O), V˙ is the airway flow rate (l/s), Vi and Ve are the inspiratory and expiratory volume (l), and P0 is the end-expiratory pressure (cmH2O). There are four first order basis-functions to be used, k is the index of a particular basis function, ak is the coefficient for a given basis function (cmH2O/l), and k(Paw(t)) is the basis function value for a given pressure measurement. The sum of the basis functions multiplied by their ak coefficients defines elastance. There are 170 bj coefficients (cmH2Os/l) that capture airway resistance, viscoelastic effects, and expiratory relaxation. The subscript –j in the third term refers to the previous time samples. Thus, each Paw(t) is calculated from information from the previous 170 data points.

The optimal number of basis functions and bj terms was determined in previous analyses of the NARX model fit for this data set [9].

To identify the NARX model coefficients, a linear system of equations must be generated and inverted to find:


where: ai1ai4 are the inspiratory elastance coefficients, and ae1ae4 are the expiratory elastance coefficients.

The inspiratory elastance coefficients are identified based on inspiratory data only, and the expiratory elastance coefficients are identified on expiratory data only. The b coefficients are identified on both inspiratory and expiratory data.

2.3 Analysis

The ai, ae, and b coefficients were identified for each data set. The basis functions multiplied by the a terms gives a continuous elastance across pressure between the minimum and maximum pressures present in each data set. The coefficient of determination (R2) was calculated for the linear relationship between Ee and Ei at different pressures. Bland-Altman analysis was performed to determine any bias in the difference between Ee and Ei as pressure increased. To further quantify intra-patient versus inter-patient variability, the variance of Ei, Ee, and (EiEe) were calculated. All analysis was undertaken on an i7 quad core PC with 16GB RAM using MATLAB 2014a 64 bit functions and the statistical toolbox (Mathworks, Natick, MA, USA).

3 Results

The inspiratory and expiratory elastances were determined for each of the 19 data sets. The agreement between Ei and Ee was generally poor at low pressure, and good at pressures greater than the second basis function knot. The behaviour of a typical patient is shown in Figure 1.

Inspiratory and expiratory elastance across pressure for one patient.
Figure 1

Inspiratory and expiratory elastance across pressure for one patient.

The relationship between Ei and Ee at various pressures is shown in Figure 2. The plot at p = 15 cmH2O contains 18 data points because one data set did not contain pressures as low as 15 cmH2O. Similarly, two data sets did not contain pressures at 40 cmH2O or greater.

Expiratory elastance vs. inspiratory elastance for p = [15 20 25 30 35 40] cmH2O. R2 values are given for the linear relationship between Ee and Ei, plotted in red. The 1:1 line is plotted in black.
Figure 2

Expiratory elastance vs. inspiratory elastance for p = [15 20 25 30 35 40] cmH2O. R2 values are given for the linear relationship between Ee and Ei, plotted in red. The 1:1 line is plotted in black.

The R2 value for the Ei to Ee linear relationship describes how well the Ei value predicts the Ee value. Figure 2 shows the strength of prediction is weak at p = 15 cmH2O, but strong for p ≥ 25 cmH2O, with R2 ≥ 0.78. The 1:1 lines plotted plotted on Figure 2 show that there is a tendency for Ei to be > Ee at low pressures, and for Ee to be slightly higher than Ei at high pressures.

Bland-Altman plots allow any fixed bias between Ei and Ee to be more easily observed. Figure 3 shows the Bland-Altman plots for pressures 15–40 cmH2O. The mean and corresponding p value are specified on each plot. A p value > 0.05 indicates that the mean difference is not significantly different from zero, based on a one sample t-test, thus there is no significant difference between Ei and Ee for this pressure. The analysis found that there is a significant difference between Ei and Ee for p = 15 cmH2O and p = 20 cmH2O only. The bias is towards Ei being larger than Ee.

Bland-Altman plots for p = [15 20 25 30 35 40] cmH2O. Solid line = the mean of the difference. Dotted lines = standard error of the mean difference.
Figure 3

Bland-Altman plots for p = [15 20 25 30 35 40] cmH2O. Solid line = the mean of the difference. Dotted lines = standard error of the mean difference.

Variances of Ei, Ee, and (EiEe) were calcualted to determine intra-patient versus inter-patient variability. For pressures of 20 cmH2O and above, the variance of (EeEi) was smaller than the variance of both Ei and Ee at each measured pressure. Based on the t-test, the variance of Ei and Ee were not significantly different. The variance of (EeEi) was significantly smaller than the variance of Ei (t-test, p = 0.0007) and significantly smaller than the variance of Ee (t-test, p = 0.03).

4 Discussion

This analysis shows that the NARX model is capable of identifying unique inspiratory and expiratory elastance profiles. Ei was a well-matched predictor of Ee for p = 25–40 cmH2O (Figure 2). There was no significant bias in the difference between Ei and Ee for p = 25–40 cmH2O (Figure 3). The intra-patient variability was significantly lower than the inter-patient variablility for p = 20–40 cmH2O. Overall, this indicates that for this cohort, Ei and Ee were comparable for 25 ≤ p ≤ 40 cmH2O, and thus may be equally valuable as an indicator of patient condition.

There was low agreement between Ei and Ee at low pressures (Figure 1). Ei was a bad predictor of Ee at p = 15 cmH2O. There was a significant positive bias in (EiEe) at p = 15 cmH2O and p = 20 cmH2O. In addition, the variance of (EiEe) was larger than the variance of Ee at p = 15 cmH2O. The cause of this behaviour relates to the use of distinct basis functions used to define elastance. With M = 4, the first basis function (∅1) is non-zero for only the lowest third of the pressures present in the data set. Therefore ∅1 is identified using only the volume data that exists when these low pressures are present in inspiration.

At the beginning of inspiration, the pressure rises very rapidly. There are relatively few data points here, as the gradient of the pressure increase is so steep. Thus there are relatively few data points used to identify ∅1, compared to ∅2, ∅3, and ∅4. There is not enough useful information for determining the elastance in this small portion of inspiratory data. Since the gradient of the pressure drop during expiration is shallower at lower pressures, there is more data available to identify expiratory elastance, and the issue does not occur. Therefore Ee is likely to be more reliable than Ei at low pressure using the method presented in this paper.

A similar problem with inspiratory elastance would be likely to occur if the NARX model was identified using this method on any single PEEP level, where a recruitment manoeuvre was not carried out. In this scenario, either the Ei and Ee should not be identified separately, or the Ei should not be relied on for diagnostic use.

While a strong agreement between Ei and Ee was found for 25 ≤ p ≤ 40 cmH2O, the sample size of 19 data sets is relatively small. Also, due to the limited size of the RMs for some patients, the p = 15 cmH2O analysis was based on 18 data sets, and the analysis at p = 40 cmH2O was based on only 17 data sets. The methods should be tested on a larger patient cohort to verify the results. A larger number of patients with RMs that reached pressures of greater than 40 cmH2O would also allow an assessment of any possible variability in Ei and Ee at very high pressures.

Separate inspiratory and expiratory elastances may not be currently used as a diagnostic aid by clinicians. However, this analysis has shown that unique Ei and Ee values can be obtained using the NARX model. The outcomes of this type of analysis may provide further insight into sedated ARDS patient conditions that other models cannot accomplish.

Author’s Statement

Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: Ethics approval for the study and use of collected data was granted by the New Zealand South Regional Ethics Committee.


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About the article

Published Online: 2016-09-30

Published in Print: 2016-09-01

Citation Information: Current Directions in Biomedical Engineering, Volume 2, Issue 1, Pages 629–632, ISSN (Online) 2364-5504, DOI: https://doi.org/10.1515/cdbme-2016-0138.

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©2016 Ruby Langdon et al., licensee De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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