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

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 E_i, and expiratory elastance E_e 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 E_i and E_e. This will help to further validate the NARX model and its usefulness in accurately describing patient physiology.

2 Material and methods

3 Results

The inspiratory and expiratory elastances were determined for each of the 19 data sets. The agreement between E_i and E_e 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.

Figure 1

Inspiratory and expiratory elastance across pressure for one patient.

The relationship between E_i and E_e 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.

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 E_e and E_i, plotted in red. The 1:1 line is plotted in black.

The R² value for the E_i to E_e linear relationship describes how well the E_i value predicts the E_e 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 E_i to be > E_e at low pressures, and for E_e to be slightly higher than E_i at high pressures.

Bland-Altman plots allow any fixed bias between E_i and E_e 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 E_i and E_e for this pressure. The analysis found that there is a significant difference between E_i and E_e for p = 15 cmH₂O and p = 20 cmH₂O only. The bias is towards E_i being larger than E_e.

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

4 Discussion

This analysis shows that the NARX model is capable of identifying unique inspiratory and expiratory elastance profiles. E_i was a well-matched predictor of E_e for p = 25–40 cmH₂O (Figure 2). There was no significant bias in the difference between E_i and E_e 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, E_i and E_e were comparable for 25 ≤ p ≤ 40 cmH₂O, and thus may be equally valuable as an indicator of patient condition.

There was low agreement between E_i and E_e at low pressures (Figure 1). E_i was a bad predictor of E_e at p = 15 cmH₂O. There was a significant positive bias in (E_i–E_e) at p = 15 cmH₂O and p = 20 cmH₂O. In addition, the variance of (E_i–E_e) was larger than the variance of E_e 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 E_e is likely to be more reliable than E_i 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 E_i and E_e should not be identified separately, or the E_i should not be relied on for diagnostic use.

While a strong agreement between E_i and E_e 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 E_i and E_e 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 E_i and E_e 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.