Using injury cost functions from a predictive single-compartment model to assess the severity of mechanical ventilator-induced lung injuries

Abstract

Identifying safe ventilation patterns for patients with acute respiratory distress syndrome remains challenging because of the delicate balance between gas exchange and selection of ventilator settings to prevent further ventilator-induced lung injury (VILI). Accordingly, this work seeks to link ventilator settings to graded levels of VILI to identify injury cost functions that predict injury by using a computational model to process pressures and flows measured at the airway opening. Pressure-volume loops were acquired over the course of ~2 h of mechanical ventilation in four different groups of BALB/c mice. A cohort of these animals were subjected to an injurious bronchoalveolar lavage before ventilation. The data were analyzed with a single-compartment model that predicts recruitment/derecruitment and tissue distension at each time step in measured pressure-volume loops. We compared several injury cost functions to markers of VILI-induced blood-gas barrier disruption. Of the cost functions considered, we conclude that mechanical power dissipation and strain heterogeneity are the best at distinguishing between graded levels of injury and are good candidates for forecasting the development of VILI.

NEW & NOTEWORTHY This work uses a predictive single-compartment model and injury cost functions to assess graded levels of mechanical ventilator-induced lung injury. The most promising measures include strain heterogeneity and mechanical power dissipation.

INTRODUCTION

Clinical care of mechanically ventilated patients suffering from acute respiratory distress syndrome (ARDS) is a compromise between providing sufficient gas exchange and avoiding ventilator-induced lung injury (VILI) (48). This task is challenging since it requires balancing the conflicting requirements of avoiding damage due to overdistension at high inspiratory pressures (volutrauma) and injury caused by cyclic recruitment and derecruitment (atelectrauma) that results from low expiratory pressure (20, 22, 37, 42, 48). Current clinical guidelines recommend the use of lower tidal volumes (4 ≤ Vt ≤ 8 ml/kg predicted body wt) and higher positive end-expiratory pressures (PEEPs) (15) to avoid overdistension and PEEP to reduce atelectrauma (1). However, the mortality rate of ARDS remains high (10), and alternative ventilation strategies such as PEEP titration to maximize lung compliance have not proven successful (52). One of the difficulties in designing ventilation strategies is that the ARDS patient population is highly heterogeneous in both the nature of injury and its severity. Practically, this means that a mode of ventilation that is ideal for one subject may be highly unsafe in another. Identifying safe ventilator settings is additionally challenging because the safe settings for a patient may change with time as the functional state of the lung evolves with injury or recovery. Indeed, our studies in mice indicate that ventilation modes that are safe in healthy animals may produce injury once VILI has been initiated (43).

Because safe ventilator parameters vary among patients and change as injury progresses, we have proposed that mechanical ventilation can be guided by continually monitoring the functional state of the lung and adjusting ventilation to minimize injury (20). However, for an injury predictor (cost function) to successfully guide safe ventilator setting selection, it must be able to assess how damaging a particular mode of ventilation is before injury occurs. Measures of mechanical power (17, 35) and driving pressure (7, 19) are examples of cost functions that can be determined at the bedside. In the case of clinical low-Vt ventilation, driving pressure, plateau pressure, and compliance have been linked to mortality rates in ARDS patients (19). However, these cost functions may best describe the combined effects of ventilation mode and functional state of the lung. For example, elevated driving pressure may be observed in patients with more severe lung injury (decreased compliance) as well as with changes in PEEP or Vt. As such, to separate ventilation mode effects from the level of lung injury, new metrics should directly reflect the underlying mechanisms of injury. Such an injury cost function should be able to differentiate between the graded levels of VILI.

To address these shortcomings, we analyze pressures and flows in mechanically ventilated mice by using a biomechanical single-compartment model and developed predictive injury cost functions (20) to assess the synergistic contributions of volutrauma and atelectrauma in VILI. This single-compartment model expands in two directions to incorporate the effects of viscoelastic lung tissue distension and recruitment on total lung measurements of pressure and volume. These metrics are compared to experimental assessments of VILI severity in order to determine their predictive efficacy. Ultimately, we seek to establish clear links between ventilator settings, the severity of injury, and the relative contributions of atelectrauma and/or volutrauma in an effort to identify better predictors of VILI to guide patient-specific ventilation.

MATERIALS AND METHODS

See glossary for definitions of parameters.¹

Animal Experiments

Four groups of 12 BALB/c mice (Jackson Laboratories, Bar Harbor, ME) were mechanically ventilated with different combinations of delivered Vt and PEEP for ~2 h with a flexiVent small-animal ventilator (SCIREQ, Montreal, QC, Canada) in a study approved by the University of Vermont Institutional Animal Care and Use Committee (IACUC no. 14-056). Table 1 summarizes the differences among the four ventilation groups. The P30-PEEP0 and P35-PEEP0 animals received pressure-controlled ventilation with large Vt and zero end-expiratory pressure. The Low Vt-PEEP1.5 and Low Vt-PEEP8 groups were subjected to volume-controlled ventilation at Vt ≈ 9 ml/kg with PEEP = 1.5 and 8 cmH₂O, respectively. Additional differences relating to the sequence of prescribed ventilation maneuvers are explained below. Within each ventilation group, 6 of the 12 mice received a single injurious bronchoalveolar lavage (27) with 0.15 ml of saline instilled and ~0.08 ml recovered (LAV group). Nonlavaged animals are referred to as the NoLAV subgroup. The saline lavage injury we used causes surfactant depletion and leads to a significant injury phenotype similar to ARDS that includes impaired gas exchange and reduced pulmonary compliance (31). In addition to the ventilated groups in Table 1, 12 mice (6 NoLAV and 6 LAV) served as minimally ventilated controls without VILI as described below.

Table 1.

Description of experimental groups

	P30-PEEP0	P35-PEEP0	LowVt-PEEP1.5	LowVt-PEEP8
PEEP, cmH2O	0	0	1.5	8
PIP, cmH2O	30	35	13.6 ± 7.6	22.8 ± 3.0
RR, breaths/min	60	50	200	200
Ventilation mode	Pressure limited	Pressure limited	Quasi-sinusoidal volume	Quasi-sinusoidal volume
Minute ventilation, (ml·kg−1)·breaths·min−1	1,914	1,780	1,740	1,740
Tidal volume, ml/kg	32.4 ± 9.5	37.8 ± 6.3	9.2 ± 2.4	8.7 ± 0.83
IE	0.5	0.5	0.67	0.67
Duration of prescribed ventilation in each epoch, min	5.2	5.0	5.0	5.0
Total epoch time, min	5.7	6.0	5.8	5.8

Description of experimental groups of BALB/c mice, each containing 6 control (nonlavaged) mice and 6 mice receiving an injurious lavage, is provided to highlight differences in positive end-expiratory pressure (PEEP), peak inspiratory pressure (PIP), respiratory rate (RR), the mode of ventilation, minute ventilation, mean ± SD tidal volume (Vt), inspiratory-to-expiratory volume ratio (IE), and duration of prescribed ventilation during each ventilation epoch.

Eight- to ten-week-old mice with an average weight of 18.7 g were anesthetized with a 90 mg/kg intraperitoneal (ip) injection of pentobarbital sodium, tracheostomized with a blunted 18-g cannula, and mechanically ventilated. During the experiment, doses of 5 mg/kg ip pentobarbital sodium were administered with 150 μl ip of 5% dextrose lactated Ringer solution every 30 min. Pancuronium bromide (0.8 ml/kg ip) was administered at the onset of ventilation to suppress respiratory drive in order to prevent spontaneous breathing efforts from interfering with lung mechanics measurements. Heart rate was continually monitored via electrocardiogram to ensure adequate depth of anesthesia.

Pulmonary system mechanics were measured with four different ventilator maneuvers. Two different types of pressure-volume (P-V) loop measurements were selected to match the type of ventilation prescribed to each of the four groups. For mice subjected to volume-controlled ventilation (Low Vt-PEEP1.5 and Low Vt-PEEP8; Table 1), the P-V relationship was determined by measuring lung volume, V(t), relative to functional residual capacity and airway pressure, P(t), during three sinusoidal breaths (PV_V). The breaths measured during the PV_V maneuver were delivered at a respiratory rate (RR) = 200 breaths/min and a ventilator cylinder displacement of 0.25 ml, which delivered a Vt ≈ 0.2 ml due to gas compression in the ventilator circuit. To determine the P-V relationship in mice subjected to pressure-controlled ventilation (P30-PEEP0 and P35-PEEP0; Table 1), a PV_P data set was acquired consisting of a 3-s ramp to peak inspiratory pressure (PIP; Table 1), a 3-s pressure hold, and then a ramp back to 0 cmH₂O. Each of the four ventilation groups also received recruitment maneuvers (RMs) and forced oscillation technique (FOT) measurements. The 2-s FOT measurements consisted of a multifrequency (0.5–20.5 Hz) volume perturbation from which mechanical impedance (Z) is calculated. The constant-phase model is then fit to Z to obtain pulmonary system elastance, tissue damping, and airway resistance (Raw) (4, 5, 21, 39, 42). Measured RMs consisted of a 3-s inflation followed by a 3-s inspiratory hold.

Mice were stabilized on the ventilator for ~5 min with volume-controlled ventilation at PEEP = 3 cmH₂O, RR = 200 breaths/min, and Vt = 0.2 ml. A comprehensive lung function assessment was performed to determine the baseline mechanical properties for each mouse. The assessment began with a RM to provide a uniform volume history as well as to measure PV_P (PIP = 30 cmH₂O) in order to document the P-V relationship. Then, derecruitment tests (DTs) were conducted at PEEP = 0, 3, and 6 cmH₂O (41). Each DT consisted of a RM followed by 3 min of Vt = 0.2 ml ventilation interspersed with FOTs at 18-s intervals at the prescribed PEEP (9 FOT measurements per DT). The DT concluded with another RM (PIP = 30 cmH₂O) at the same PEEP.

After the comprehensive lung function assessment, mice in the LAV subgroup were subjected to an injurious bronchoalveolar lavage and six additional RMs were performed over the course of 3 min to allow the animals to stabilize. This was followed by a second comprehensive lung function assessment to quantify the effects of the lavage. The 12 minimally ventilated control animals received only this first lung function assessment, a lavage (depending on subgroup assignment), the stabilization period, and the second function assessment. In the remaining groups, each mouse was subsequently ventilated for ~108 min consisting of 18 repeated epochs according to group assignments shown in Table 1.

At the start of each epoch, the ventilation specified by the PIP, PEEP, RR, mode, and duration in Table 1 was applied. Next, a P-V curve was obtained with either PV_V or PV_P as appropriate for the ventilation group assignment. Then, to provide a consistent volume history for comparison between groups, the P35-PEEP0, Low Vt-PEEP1.5, and Low Vt-PEEP8 groups received 15 s of pressure-controlled ventilation (RR = 50 breaths/min, PIP = 30 cmH₂O) and a PV_P (PIP = 30 cmH₂O). All groups then received a 2-s FOT followed by quasi-sinusoidal volume-controlled ventilation for 10 s with prescribed PEEP by group, RR = 200 breaths/min, and Vt ≈ 0.2 ml. A second FOT measurement was repeated to conclude the epoch.

At the conclusion of the ventilation period, the comprehensive lung function assessment was repeated. The 12 minimally ventilated control animals without VILI received only the first function assessment, a lavage (if prescribed), and a final function assessment. After ventilation, 1.0 ml of bronchoalveolar lavage fluid (BAL) was collected from all groups (including the minimally ventilated control animals). The BAL was centrifuged for 10 min at 1,600 rpm, and supernatant protein concentration was determined with a BCA protein assay kit (Pierce, Rockford, IL) as we have previously described (6). Bio-Safe Coomassie Stain (Bio-Rad, Hercules, CA) was used for visualization of proteins separated by polyacrylamide gel electrophoresis (PAGE) according to the manufacturer’s protocol. After PAGE, the gels were washed three times for 5 min in ddH₂O to remove SDS. Subsequently, Coomassie stain was added to completely cover the gels. They were shaken for 1 h and rinsed in ddH₂O overnight. Protein species appeared as blue bands on a clear background in the gels, which were scanned at gray setting. Densitometry was performed on the equal amount-loaded samples.

Computational Model

A subject-specific predictive single-compartment model (Fig. 1) that allows expansion in two directions is used to simulate the mechanical properties of the lung. Distension of the respiratory system tissues is represented by the vertical expansion, y(t), of the compartment. Recruitment dynamics are incorporated through horizontal expansion, x(t), of the compartment that is a continuous function of time (t). The open fraction, x(t), describes the ventilated fraction of the lung such that when x(t) = 1, all alveoli and small airways are recruited and the lung is fully patent. The volume, V_m(t), of the model compartment is defined by the product of the open fraction and the distension so that

Fig. 1.

Horizontal expansion of the single-compartment model [recruitment, x(t)] and vertical expansion [tissue distension, y(t)] were determined such that the model predictions of compartment pressure, airway opening pressure, and volume mimicked the experimentally measured pressure (P) and volume (V) signals at a forced oscillation technique-determined airway resistance (Raw). Nonlinear springs E₁ and E₂ account for tissue stiffness, and the nonlinear dashpot R₁ represents tissue resistance. The length of elements E₂ and R₁ is described by y_k.

$${\text{V}}_{\text{m}}\left(t\right)=x\left(t\right)y\left(t\right)$$ (1)

The flow of air into the compartment, dV_m(t)/dt, can also be computed directly from x and y,

$${\dot{V}}_{\text{m}}\left(t\right)=\dot{x}\left(t\right)y\left(t\right)+x\left(t\right)\dot{y}\left(t\right)$$ (2)

The distension of viscoelastic pulmonary tissue is modeled with a nonlinear Kelvin body with strain-dependent elastic springs E₁(y) and E₂(y) and resistive dashpot R₁(y). We describe this nonlinear behavior in E₁ by

$$E_1\left(y\right)=\begin{array}{cc}{E}_b& \text{if\hspace{0.17em}}y-y_k<y_c\\ E_b+E_F{\left(y-y_k-y_c\right)}^3& \text{if\hspace{0.17em}}y-y_k\ge y_c\end{array}$$ (3)

In Eq. 3, the cubic dependence was selected to provide a consistent fit across the mice analyzed in this study. The nonlinear formulation was chosen to represent the characteristic stiffening of the lung at high volumes that arises from the network of connective tissue that provides mechanical structure to the lung (26). At low volumes, the wavy collagen fibers are distensible and the elastin fibers provide the elastic recoil analogous to E_b. As alveolar volume is increased, the collagen fibers are progressively straightened and lung elastance increases, corresponding to the upper corner point on the P-V curve (Fig. 2). Parameter y_c describes this transition volume, and the length of the elements R₁ and E₂ is given by y_k (Fig. 1).

Fig. 2.

Predicted open fraction for the pressure-volume loop (PV_P) maneuver taken during the first lung function assessment, before any mouse received an injurious lavage (if prescribed). The lower limb of the hysteresis loop corresponds to inspiration, and the upper limb describes expiration.

The distension of nonlinear spring E₂ and dashpot R₁ is denoted by the variable y_k(t). The elements E₁, E₂, and R₁ are governed by the same form of nonlinear function (Eq. 3) so that the tissue time constant is well defined and consistent across volumes. Since there is a consistent ratio of energy storage (elastance) and energy dissipation (resistance) across a range of strains in most biological tissues (16), we assume that the nonlinearity of E₁ is reflected in the other elements. Therefore, E₂ and R₁ are expressed as factors (E_F2 and R_F) of E₁ so that

$$E_2\left(y\right)=E_{F2}{E}_1\left(y\right)$$ (4)

and

$$R_1\left(y\right)=R_F{E}_2\left(y\right)$$ (5)

To implement this model with measured data, discrete time points are defined as T = bΔt, where time step size Δt = 0.01 s and b is the vector index. The differential equations describing the behavior of the three model variables are implemented discretely with the following difference form:

$$\dot{x}\left(T\right)=\frac{x\left(b\text{Δ}t\right)-x\left[\left(b-1\right)\text{Δ}t\right]}{\text{Δ}t}$$ (6)

$$\dot{y}\left(T\right)=\frac{\dot{V}\left(b\text{Δ}t\right)-y\left[\left(b-1\right)\text{Δ}t\right]\dot{x}\left(T\right)}{x\left(b\text{Δ}t\right)}$$ (7)

$${\dot{y}}_k\left(T\right)=\frac{E_1\left(b\text{Δ}t\right)\left\{y\left(b\text{Δ}t\right)-y_k\left[\left(b-1\right)\text{Δ}t\right]\right\}-E_2\left(b\text{Δ}t\right)y_k\left[\left(b-1\right)\text{Δ}t\right]}{R_1\left(b\text{Δ}t\right)}$$ (8)

The variables x, y, and y_k are computed at each time step by integrating Eqs. 6–8 by the forward Euler method. The force F exerted by nonlinear spring E₁ is then

$$\text{F}\left(y,t\right)=E_1\left(y\right)\left[y\left(t\right)-y_k\left(t\right)\right]$$ (9)

so that we can ultimately determine the compartment pressure (Pa = F/x). The model pressure at the opening [P_m(t)] is found by summing the airway pressure and the compartment pressure so that

$${\text{P}}_{\text{m}}\left(t\right)=[\text{Raw}\times {\dot{V}}_{\text{m}}\left(t\right)]+\frac{\text{F}\left(y,t\right)}{x\left(t\right)}$$ (10)

Equations 1–10 were computed at each time step so that x could be determined such that it minimized the root-mean-square difference between P_m(t) and experimentally measured pressure [P(t)] with a MATLAB one-dimensional minimization function, as in Smith et al. (42). The value of Raw used in the model was that determined experimentally from a temporally adjacent FOT measurement.

Model Fitting

The model parameters E_b, E_F, y_c, E_F2, and R_F were determined independently for each mouse. In this process, we fit the single-compartment simulations to measured P(t) and V(t) signals (acquired at 256 Hz and downsampled to 100 Hz) by driving the model with V(t) and minimizing the objective function

$$\text{χ}=w_1\text{λ}+w_2\text{κ}+w_3{\text{RMSE}}_x+{\displaystyle \sum }_{m=1}^2\left(w_4{\text{RMSE}}_{\text{P}}+w_5\text{ξ}\right)$$ (11)

where RMSE is root mean square error, for each mouse with the MATLAB particle swarm optimization algorithm. Terms in χ, explained in detail in the subsequent paragraphs, were defined to enforce physiological constraints and ensure that the measurements and model predictions matched. The fit was performed with an RM and PV_P measurement recorded at PEEP = 0 cmH₂O in the baseline assessment maneuver (m = 1 and m = 2, respectively) in the healthy mouse before lavage or VILI. Each of the five terms in χ are weighted (w) as follows: w₁ = 1,000, w₂ = 1, w₃ = 10, w₄ = 100, w₅ = 10. Many permutations of weights were iteratively evaluated to determine the combination that provided a low standard deviation of model parameters and low variance in x for the PV_P maneuver used in the fit (Fig. 2). All time-dependent parameters are described here in terms of their discrete implementation where time T = bΔt.

RMSE_P requires the model to match measurements using the RMSE of P_m(T) to P(T). RMSE_x steers the fit toward full recruitment and consists of the RMSE between x(T) and 1 for the PEEP = 0 cmH₂O RM maneuver. To normalize open fraction between mice, λ constrains x = 1 (approximately full recruitment) at end-inspiration (T = iΔt) in the PEEP = 0 cmH₂O RM maneuver since the healthy lung is fully recruited at an airway pressure of 30 cmH₂O.

$$\lambda =\left|1-x\left(i\text{Δ}t\right)\right|$$ (12)

Component κ penalizes predicted x that is greater on inspiration than on expiration for the full range of pressures in the PV_P maneuver. T = eΔt is the time at end-expiration, so that x(eΔt – bΔt) is the expiratory open fraction predicted at the same pressure as x(bΔt) during inspiration. κ is calculated from the start of inspiration (b = 0) to the end of inspiration (b = i) by

$$\text{κ}={\displaystyle \sum }_{b=0}^{i}r\left[x\left(b\text{Δ}t\right),x\left(e\text{Δ}t-b\text{Δ}t\right)\right]$$ (13)

where the recruitment penalty function r (used in both κ and ξ) is given by

$$r\left(a,c\right)=\left(c-a\right)H\left(c-a\right)$$ (14)

where H is the Heaviside step function. In the case of κ, c and a are predicted open fractions at identical pressures on inspiration and expiration, respectively. Therefore, at each b the penalty κ is equivalent to the difference in x when the predicted open fraction is greater on inspiration than on expiration.

ξ penalizes parameter combinations that permit derecruitment on inspiration and recruitment on expiration by

$$\text{ξ}=\frac{1}{e}{\displaystyle \sum }_{b=2}^e\left(\frac{1}{w_5}{\displaystyle \sum }_{b=2}^{i}d\left\{x\left[b\left(\Delta t-1\right)\right],x\left(b\Delta t\right)\right\}+{\displaystyle \sum }_{b=i+1}^{e}r\left[x\left(b\Delta t-1\right),x\left(b\Delta t\right)\right]\right)$$ (15)

where b = e corresponds to end-expiration. The derecruitment penalty d is described by the predicted open fraction of the current time step (h) and the predicted open fraction for the previous time step (l), so that

$$d\left(h,l\right)=\left(h-l\right)H\left(h-l\right)$$ (16)

We assumed that the underlying tissue mechanics governing the values of E₁, E₂, and R₁ remained unchanged during the short period of ventilation. This assumption is based on the strong linear correlation between the number of ventilated alveoli (determined with the physical disector stereology probe) and pulmonary system elastance (from the FOT) in the acute phase of bleomycin-induced lung injury (25, 28) and during overexpression of transforming growth factor-β1 (8). Multicompartment simulations using structural and functional data (25) showed that alveolar elastance is relatively unchanged, whereas the functional changes are primarily caused by a loss of patent alveoli due to alveolar edema and surfactant inactivation. Therefore, model parameters found from fitting the initial baseline assessment are used in the present study to compute predicted x and y for all epochs.

Injury Cost Functions

Since x(T) and y(T) represent tissue recruitment and distension, respectively, we employed them to calculate a variety of injury cost functions purporting to capture the extent of atelectrauma and volutrauma, and their interactions, following a given regimen of mechanical ventilation. These injury cost functions are listed in Table 3. Since the cost functions consider parameters specific to each epoch (n), we use a superscript to denote the epoch that the parameters are from; for example, xⁿ(T) describes the open fraction at discrete time T during epoch n.

Table 3.

Summary of candidate injury cost functions

	Injury Cost Function
ΣX	Cumulative recruitment/derecruitment
ΣY	Cumulative overdistension
Σα	Strain heterogeneity
epk	Average peak strain
et	Total tidal distension
eRR	Cumulative distension
Pd	Driving pressure
W	Inspiratory work
N	Mechanical power dissipation
M	Mechanical power

Cumulative recruitment/derecruitment.

Intratidal recruitment/derecruitment (R/D) occurs at low end-expiratory lung volumes (23) or when regional surface tension forces are high (3), and the resulting fluid-mechanical forces can damage the parenchyma (11, 24). This causes atelectrauma, which is widely recognized as being involved in VILI pathogenesis (3, 20, 22, 37, 42, 48). We quantify the contribution of R/D to VILI using a cumulative R/D metric (ΣX) that integrates the aggregate effect of changes in open fraction between the beginning of inspiration (T = 0) and the end of inspiration (T = iΔt) over all of the 18 ventilation epochs. This metric is defined as

$$\text{Σ}X=\text{RR}{\displaystyle \sum }_{n=1}^{18}{x}^n\left(i\Delta t\right)-x^n\left(0\right)$$ (17)

where RR is respiratory rate in hertz. Since this metric describes the cumulative change in x over each breath, it does not reflect flooded or atelectatic regions that are recalcitrant to recruitment.

Cumulative overdistension.

The other key mechanical factor associated with VILI pathogenesis is volutrauma (20, 22, 23, 37, 42, 48), which results from overdistension of the lung tissues. We define overdistension in the model as extension of the lung tissue past the upper inflection point on its P-V curve, which occurs when it enters the nonlinear elastic regime for which y − y_k > y_c (Eq. 3). We quantify the contribution of volutrauma to VILI as the integrated amount of overdistension (ΣY) throughout the 18 ventilation epochs as

$$\text{Σ}Y={\displaystyle \sum }_{n=1}^{18}\text{RR}{Y}^n$$ (18)

where Y in a single epoch is described by

$$Y=\begin{array}{cc}0& \text{if\hspace{0.17em}}y-yk<y_c\\ y\left(i\Delta t\right)-y_c-y_k& \text{otherwise}\end{array}$$ (19)

Strain heterogeneity.

The progression of VILI is spatially and temporally heterogeneous, and computed tomography imaging (38) has provided evidence that regional ventilation heterogeneity plays a role in VILI development. This injury cost function uses the physiological conditions that define the most severe case of strain heterogeneity (Σα): high distension of the open lung and atelectasis. In this case, the interface between atelectatic and patent alveoli serves as a stress concentrator (29, 33) where ventilated alveoli must expand during inspiration against the mechanical constraints imposed by consolidated adjacent alveoli. Σα takes its largest value when open fraction (x) is small and tissue distension (y) is high, as demonstrated previously in a finite element model of the lung parenchyma (29). Accordingly, we define a strain heterogeneity cost function as

$$\text{Σα}=\text{RR}{\displaystyle \sum }_{n=1}^{18}{\displaystyle \sum }_{b=1}^e{y}^n\left(b\Delta t\right)\left[x_{\text{max}}-x^n\left(b\Delta t\right)\right]$$ (20)

where b = e is the discrete time point corresponding to end-expiration and b = 1 corresponds to the start of inspiration (t = 0). x_max ≈ 1 is the largest predicted open fraction value in all 18 epochs (after lavage, if prescribed). Typically, x_max < 1 for mice that received an injurious lavage.

Average peak distension.

Average peak distension (e_pk) is simply the average of the maximum distension [y(i)] across the 18 ventilation epochs.

Total tidal distension.

Total tidal distension (e_t) sums the maximum stretches from end-expiration (T = 0) to peak inspiration (t = iΔt) over the 18 ventilation epochs, defined as

$$e_t={\displaystyle \sum }_{n=1}^{18}{y}^n\left(i\Delta t\right)-y^n\left(0\right)$$ (21)

Cumulative distension.

Cumulative distension (e_RR) is similar to e_t but takes account of the frequency at which the tissues are distended by including RR in its definition, which is

$$e_{\text{RR}}=e_t\text{RR}$$ (22)

Driving pressure.

Driving pressure (P_d) has been proposed as a simple predictor of mortality in ARDS patients (7, 19). We calculated a driving pressure metric as the averaged increase in pressure from the start of inspiration (T = 0) to the plateau pressure (T = iΔt), defined as

$${\text{P}}_{\text{d}}=\frac{{\sum }_{n=1}^{18}{P}^n\left(i\Delta t\right)-P^n\left(0\right)}{18}$$ (23)

Inspiratory work.

Inspiratory work is the energy required to inflate the chest (30), and mechanical power is rate at which this work is performed. Both work and power are influenced by Raw, viscous tissue damping, and elastic recoil. Mechanical power has received considerable attention in the mechanical ventilation literature recently (17, 35) and has been suggested as a means of determining a threshold for safe ventilation (14). Inspiratory work is the time integral of the product of pressure and flow rate over the duration of inspiration (30). We calculate total inspiratory work computed over every breath in each epoch as

$$\text{W}={\displaystyle \sum }_{n=1}^{18}\text{RR}{\displaystyle \sum }_{b=1}^i{\text{P}}^n\left(b\Delta t\right){\dot{V}}^n\left(b\Delta t\right)$$ (24)

where b = 1 is the start of inspiration and b = i is the end of inspiration.

Mechanical power dissipation.

The mechanical power dissipation (N) injury cost function is similar to W in Eq. 24 except that the upper limit of summation is end-expiration (b = e), so that it includes both the inspiratory and expiratory limbs.

Mechanical power.

We use the formulation for mechanical power provided in References 19 and 35 to determine the mechanical power (M) at each epoch. Here M is found by the product of P_d with Vt and RR, or

$$\text{M}={\sum }_{n=1}^{18}{\text{RRP}}_{\text{d}}\text{VT}$$ (25)

All candidate injury cost functions were compared to three different quantifications of BAL protein: 1) total protein concentration, 2) medium-molecular weight (MW) (50,000–70,000) protein content, and 3) large-MW (80,000–250,000) protein content. To identify the injury cost functions that have a strong correlation to each biological marker, we calculated Spearman’s rank correlation coefficient (ρ). The closer ρ is to 1, the stronger the monotonic linear or nonlinear relationship between the injury cost function and marker. To determine whether the injury cost function values depend on the underlying injury phenotype, an analysis of covariance was performed on linear models fit to the NoLAV and LAV subgroups for all injury cost functions and BAL protein measures described above. A Tukey-Kramer multiple-comparison test was applied to determine significant (P < 0.05) differences in slope and intercept between cost functions in the LAV and NoLAV groups.

RESULTS

The stiffness of the model (E_b) below the distension threshold (y_c) as well as the threshold itself (y_c) were similar between mice, whereas the parameters accounting for the nonlinear viscoelastic properties of the tissues (E_F, R_F, E_F2) showed more variation between subjects at baseline (Table 2). A one-way ANOVA with the Tukey-Kramer multiple-comparison test was used to compare each parameter across the four prescribed ventilation groups. The only significant differences found were in E_F between the P35-PEEP0 and Low Vt-PEEP8 groups and in y_c between the P30-PEEP0 and Low Vt-PEEP8 groups. To assess model performance, we determined the sensitivity of χ (Eq. 11) to the model parameters by varying each parameter in turn by 10% either side of its best-fit value and determining the relative increase in χ. The average percent changes in χ (Δχ%) for all mice, provided in Table 2, show that the objective function is most sensitive to changes in E_b, E_F, and y_c. In contrast, changes in R_F and E_F2 resulted in less than a 5% change in χ.

Table 2.

Best-fit values of model parameters and sensitivity of cost function value to change in parameter values

	Eb	EF	yc	RF	EF2
Mean value ± SD	13.57 ± 2.30	951 ± 452	0.47 ± 0.07	28.0 ± 87.3	9,532 ± 15,710
Δχ% ± SD	313 ± 115	512 ± 83	2397 ± 525	0.76 ± 2.2	1.2 ± 3.2

Values are best-fit values of the model parameters across all groups of mice together with the sensitivity of the cost function value (Δχ%) to a ±10% change in parameter values. See text for model parameter definitions.

The average value of x as a function of inflation pressure for all mice determined by fitting the model to the baseline PV_P relationship measured at the start of the experiment is shown in Fig. 2. These predictions demonstrated relatively little variation between subjects. This result is expected since the mice used in this study are very similar both functionally and structurally and these measurements were recorded before injury.

The average x, y, and V for each ventilation group at the first and final epochs with and without an injurious lavage are plotted against pressure in Fig. 3. The average effect of the lavage injury is observed by comparing the solid lines in Fig. 3, left. Pressure-controlled ventilation demonstrates a loss of delivered volume with injury, whereas volume-controlled ventilation shows increased peak pressure at a given Vt. In the high-pressure ventilation groups, the degradation in lung function in mice subject to an injurious LAV is reflected by the increased hysteresis in the volume, open fraction, and distension loops compared with NoLAV loops. The amount of hysteresis in the P-V, open fraction, and distension loops for the LAV mice was decreased after ventilation. However, this was associated with a decrease in ventilated alveoli as indicated by the lower maximum values of x(t). Functional recovery in the LAV subgroup is shown in Fig. 3 by the increased compliance in the Low Vt-PEEP1.5 and Low Vt-PEEP8 groups. In both cases, there is an increase in predicted open fraction at the end of the experiment. In the Low Vt-PEEP1.5 there is also a decrease in tissue distension compared with the start. By contrast, none of the LAV mice in the P30-PEEP0 or P35-PEEP0 groups demonstrated an improvement in lung mechanics. Instead, these animals were characterized by a reduction in open fraction hysteresis and a loss of aerated alveoli over the course of ventilation.

Fig. 3.

The mean of volume (left), open fraction (center), and tissue distension (right) for the lavaged (red) and nonlavaged (black) animals within each of the ventilation groups. Solid lines are at the start of ventilation (t = 0 min), and dashed lines are at the conclusion of ventilation (t ≈ 90 min).

Spearman’s ρ was calculated to quantify the correlation between injury cost functions (Table 3) and BAL protein content at the conclusion of ventilation. Figure 4 contains ρ for each injury cost function compared with BAL total protein, medium-MW protein content, and large-MW protein content. Note that ρ cannot be used to determine the efficacy of a cost function in discriminating between graded levels of injury but rather just whether a monotonic relationship exists. This limitation occurs because ρ is calculated using ranks. Identical values of ρ are found for data that segregate in a bimodal fashion or have a continuous increase as long as the ranks of the dependent and independent variables remain the same. Therefore, ρ was only used to screen for promising injury cost functions.

Fig. 4.

Spearman’s rank correlation coefficient (ρ) is given for each injury cost function compared to the total protein concentration (black), medium-molecular weight protein content (light gray), and large-molecular weight protein content (dark gray). *Inverse relationship. Injury cost functions: N, mechanical power dissipation; P_d, driving pressure; e_t, total tidal distension; Σα, strain heterogeneity; M, mechanical power; e_pk, average peak strain; ΣX, cumulative recruitment and derecruitment; W, inspiratory work; e_RR, cumulative distension; ΣY, cumulative overdistension.

Through ρ it is evident that a strong monotonic relationship exists between total BAL protein content and the injury cost functions N, P_d, e_t, Σα, and ΣX. These injury cost functions are shown in Fig. 5 along with W, since that clinical cost function has demonstrated promise in recent publications (30). The different ventilation groups are shown in Figs. 5 and 6, P30-PEEP0, P35-PEEP0, Low Vt-PEEP1.5, and Low Vt-PEEP8, and the presence or absence of injurious lavage is indicated. Figure 5 is arranged so that Fig. 5, A and B, show cost functions that segregate subgroups while also allowing for clear distinction between graded injury levels. Injury cost functions in Fig. 5, C and D, also segregate subgroups well but do not differentiate between small changes in injury level. Assuming that BAL protein and the cost function value should be directly related, the values of inspiratory work and cumulative R/D are underpredicted for the P30-PEEP0 lavage subgroups. Figure 5E (P_d) and Figure 5F (e_t) illustrate cases in which high ρ represents the ability to detect the presence of injury and segregate groups but an inability to determine the degree of injury. M (not shown) exhibits the same behavior as P_d and e_t. Figure 6 contains the injury cost functions that were most capable of distinguishing graded levels of injury for the specific MW ranges. Figure 6, A and B, show Σα and ΣX compared to medium-MW proteins (50,000–70,000), whereas Fig. 6, C and D, contain N and e_t compared to large-MW proteins (80,000–250,000). Despite having a high ρ, P_d is not shown for either range because of its inability to detect graded injury.

Fig. 5.

The concentration of total protein found in the air space at the conclusion of ventilation is compared to injury cost functions. A: power dissipation (N). B: strain heterogeneity (Σα). C: inspiratory power (W). D: total recruitment and derecruitment (ΣX). E: driving pressure (P_d). F: total tidal distension (e_t). Circles, mice that received an injurious lavage (LAV); squares, nonlavaged (NoLAV) group control mice. BAL, bronchoalveolar lavage fluid; PEEP, positive end-expiratory pressure; Vt, tidal volume. See text for description of groups.

Fig. 6.

Injury cost functions with the strongest correlations to % changes in protein content found in the air space at the conclusion of ventilation compared with minimally ventilated (No Vent) control mice. A and B: medium-molecular weight (MW) proteins (50,000–70,000). A: total tidal distension (e_t). B: total recruitment/derecruitment (ΣX). C and D: large-MW proteins (80,000–250,000). C: power dissipation (N). D: strain heterogeneity (Σα). Circles, mice that received an injurious lavage (LAV); squares, nonlavaged (NoLAV) group control mice. PEEP, positive end-expiratory pressure; Vt, tidal volume. See text for description of groups.

The dependence of cost functions on injury phenotype was assessed with an analysis of covariance (Table 4). N (Fig. 5A), Σα (Fig. 5B), and W (Fig. 5C) did not show a statistically significant difference (P < 0.05) in slopes or intercepts between the LAV and NoLAV subgroups for total protein. However, for all other metrics the correlation between the cost function and total protein was dependent on the initial injury.

Table 4.

R² values for linear model fit to each cost function for each BAL protein measure

	Total Protein			MW 50,000–70,000			MW 80,000–250,000
Metric	R2	Slope	Intercept	R2	Slope	Intercept	R2	Slope	Intercept
ΣX	0.34	0.017	0.850	0.52	0.023	0.050	0.37	<0.01	0.150
ΣY	0.03	0.483	0.015	0.23	0.864	0.966	0.15	0.679	0.037
Σα	0.84	0.138	0.908	0.43*	0.734*	0.917*	0.40*	0.234*	0.389*
epk	0.33	<0.01	0.040	0.27	0.689	0.648	0.28	0.344	0.166
et	0.86	<0.01	<0.01	0.84	0.024	0.072	0.80	<0.01	<0.01
eRR	0.14	0.401	0.081	0.10	0.138	0.156	0.11	0.062	0.103
Pd	0.52	<0.01	<0.01	0.61	0.497	0.786	0.63	0.065	0.073
W	0.42	0.117	0.528	0.11	0.909	0.812	0.13	0.425	0.415
N	0.92	0.543	0.282	0.64*	0.312*	0.466*	0.68*	0.017*	0.029*
M	0.40	<0.01	<0.01	0.61	0.841	0.731	0.64	0.147	0.052

R² values for the linear model fit to each cost function for each bronchoalveolar lavage fluid (BAL) protein measure (MW, molecular weight). The dependence of each cost function on whether the mice had received an injurious lavage (LAV subgroup) or not (NoLAV subgroup) before ventilation was assessed with an analysis of covariance. P values comparing slopes and intercepts are in boldface when the difference is significant (P < 0.05). See Table 3 for cost function definitions.

^*A logarithmic transform was applied before fitting.

DISCUSSION

Ventilatory management of ARDS patients must balance the conflicting requirements of gas exchange and VILI. This balance is complicated by differences in severity of injury, spatial heterogeneity, and progression of VILI across individuals. The temporal and patient-to-patient variation dictates that ventilator settings, such as Vt and PEEP (40), are adjusted on the basis of the evolving functional state of the lung. Injury cost functions predicted by computer simulations (20) using subject-specific data provide a potential avenue to determine the relative safety of a given mode of ventilation in a particular patient before it is applied. However, because of the complex etiology of ARDS and VILI, cost functions that identify safe ventilation have yet to be identified. To find suitable injury cost functions, we compared the results of an array of subject-specific injury predictions to measures of the pulmonary blood-gas barrier disruption. Several of the injury cost functions considered were computed independently from the single-compartment model, whereas others were not. We found that the best predictors of graded levels of injury are N and Σα.

The majority of the injury cost functions presented here, such as Σα, are generated with a computational model that predicts recruitment and tissue distension based on measurements of pressure and flow at the airway opening. In a healthy mouse, rapid recruitment occurs at pressures <10–15 cmH₂O as our previous simulations have indicated (42). This behavior is demonstrated in Fig. 3, center in both the P30-PEEP0 and P35-PEEP0 groups. The mechanical interdependence of alveoli, due to their shared septa and air spaces (alveolar ducts), that are supplied by the same airways may be a reason that changes in open fraction can occur rapidly once a threshold pressure is reached (13). The range of recruitment pressures increases with injury (Fig. 3, center). Notably, most of the P30-PEEP0 and P35-PEEP0 LAV mice demonstrate ongoing recruitment at pressures up to 35 cmH₂O. Minimal recruitment occurs in the LowVt groups because the range of applied pressures does not reach the threshold necessary to reopen collapsed alveoli and small airways (44). Partial recovery of lung function following protective ventilation in the LowVt LAV mice from the start to the end of ventilation is suggested by the proximity of the postventilation measurements to the baseline NoLAV values.

This protective effect of low-Vt ventilation is one of the few ventilatory strategies that have demonstrated efficacy in clinical trials (1). As such, we expected that injury cost functions related to tissue distension would be strongly associated with one of our markers of injury. We looked for a monotonic association between injury cost functions and BAL protein with Spearman’s ρ. The stronger correlation of total tidal distension (e_t, ρ = 0.83) and peak distension (e_pk, ρ = 0.70) to total BAL protein concentration compared with cumulative distension (e_RR, ρ = −0.34) suggests that strain metrics should emphasize the magnitude of the insult rather than a combination of magnitude and frequency (or RR). This result supports the idea that modest changes to RR to maintain minute ventilation for low-Vt ventilation would not negatively affect regional strain. e_pk mimics the behavior of e_t (Fig. 5F), such that groups segregate more strongly according to prescribed ventilation than injury level. Within the high-pressure ventilation groups, e_t and e_pk do not discriminate between graded levels of injury. The poor correlation between cumulative overdistension (ΣY) and total BAL protein concentration suggests that overstretch, occurring when the stress-strain relationship is nonlinear, is not the best injury cost function to predict VILI. Note that the strain-based metrics e_pk, e_RR, and ΣY all exhibit a dependence on the existing level of lung injury and, as such, should be applied with caution.

Although cyclic recruitment and derecruitment (R/D) is recognized as one of the mechanisms of VILI pathogenesis, the total R/D injury cost function (ΣX; Fig. 5D) failed to clearly separate mice within the P30-PEEP0 group that received an injurious lavage from mice that did not. The increased BAL protein content in the lavage subgroup indicates that VILI was more severe in those mice. One-third of the uninjured mice in the P35-PEEP0 group also exhibited high ΣX but only moderate injury. This lack of fidelity for moderate to severe injury is likely due to the synergistic combination of R/D and overdistension, which cause VILI. Our findings relating to ΣX are consistent with other studies that show that injury is not primarily found in atelectatic areas (9, 49). Similarly, Tabuchi et al. show that it is possible for injury to develop in regions without cyclic R/D (46). From Fig. 5D, it is possible to conclude that some degree of R/D is safe. However, there also appears to be a threshold at which a subsequent increase in R/D is associated with alveolocapillary barrier disruption. One limitation of the single-compartment simulation approach is that it cannot reveal whether the same small airways and alveoli are recruited breath to breath. In vitro studies have shown that damage from repeated reopening is cumulative (24), so it is possible that the threshold effect with increasing R/D may be due to repeated application of this injurious stimulus to the same regions.

The threshold for injury observed in ΣX could also be due to the high pressures necessary to recruit the injured lung (Fig. 3, center, top 2 rows) and the amplification of strain caused by heterogeneous collapse (29). The strain heterogeneity cost function (Σα; Fig. 5B) describes this scenario and provided the best performance of the model-derived metrics. High BAL total protein at large Σα suggests that injury is the greatest when open fraction is small and the remaining open lung tissue is highly distended. This is consistent with the idea that heterogeneous alveolar distension contributes to VILI pathogenesis by promoting overdistension of the larger alveoli (34). Recent work by Motta-Ribeiro et al. also supports this theory. They found that mechanical ventilation caused injurious heterogeneous strain distributions, increasing pulmonary inflammation in tandem with decreased regional aeration due to the relocation of mechanically delivered air (36). Because we use a single-compartment model, heterogeneity is not simulated directly. As such, the model predictions represent the average distension and do not account for regional variations or gravitational gradients that, given the small size of the mouse lung, are relatively small. Incorporating a regional strain description could improve the cost function predictions at the expense of model identifiability. Since these simulations are intended to provide clinically applicable injury metrics, we have elected to retain a simplified model architecture. To predict the effects of nonuniform inflation, the strain heterogeneity cost function is based on conditions (low x and high y) in which heterogeneous strain occurs. Σα takes into account the recruitment pressures of collapsed acinar units by identifying ventilation regimes where high pressures (and thus high distension) are applied to closed units.

The Spearman rank test identifies P_d as the clinical metric most strongly correlated with markers of alveolocapillary barrier disruption (Fig. 4). However, despite strong correlations, P_d segregates the different groups more clearly by their prescribed ventilation than by their level of injury. This important limitation of P_d is best illustrated in Fig. 5E, where the animals in the P30-PEEP0 group have identical P_d but exhibit a wide range of BAL protein levels. In addition, the correlation between P_d and BAL protein is dependent on the lung injury level, as indicated by the significant differences in the slope and intercept of linear models fit to the LAV and NoLAV groups (Table 4). In this report, we propose other injury cost functions that could be readily be implemented in the clinical setting or clinic that better distinguish differences in the graded levels of injury and are independent of lung injury status.

Of the three different power injury cost functions summarized in Table 3, power dissipation (N) provided the best correlation to injury markers. We found that although an approximation of mechanical power (M) correlated with injury (ρ = 0.74), no trend exists where M is sensitive to the level of injury. It is possible that more complicated implementations of M that include PEEP and Raw derived from the single-compartment model (17) may improve the correlation. Similarly, inspiratory work (W) lacks the ability to discern the extent of injury (Fig. 5C), particularly in the case of moderate injury (W ≈ 4e4, total protein ≈ 1–2.5 mg/ml). However, N provides a sensitive measure of injury severity (Fig. 5A) that is independent of ventilation and lavage grouping. We postulate that N is an improvement over W and M because it includes both inspiratory and expiratory limbs of the P-V loop. This inclusion means that N is a measure of energy dissipated during the breath, a portion of which we attribute to injury. The strong correlation of N to total BAL protein (ρ = 0.85) supports the idea that it is largely measuring the energy lost to generating VILI.

An important cost function characteristic is independence from the underlying lung injury to allow predictions of VILI regardless of injury phenotype. The analysis of covariance presented in Table 4 shows that Σα, e_t, and W do not differ between the LAV and NoLAV subgroups. N was independent of the underlying injury for total and MW 50,000–70,000 BAL protein content. However, the correlation for MW 80,000–250,000 proteins was significantly different between the LAV and NoLAV subgroups for that metric. The dependence of injury cost functions such as P_d (Fig. 5E) and e_t (Fig. 5F) on the underlying injury phenotype complicates the use of those metrics for predicting VILI.

One limitation of our simulations is that the open fraction may be underestimated at low pressures on inspiration (Fig. 2 and Fig. 3, center) because of alveolar septal folding first shown by Gil et al. (18). Folds in the septa cause changes in alveolar surface volume that are not captured by this model. This, in effect, allows recruitment of additional septal wall within the alveolus. Consequently, we believe that this causes open fraction to be underpredicted at low pressures during inspiration. Others have noted the “wavy” structure of elastin fibers in alveolar mouths and alveoli (47), which may also allow for some initial extensibility that is not captured by the single-compartment model. One of the primary challenges in validating models predicting open fraction at discrete time points throughout an entire breath is that we are unable to verify the accuracy of model predictions on inspiration because most in vivo imaging studies and fixed-tissue morphometry rely on making assessments after a breath hold.

Some of our conclusions may be limited by the sequence of ventilator maneuvers used to establish a similar volume history between ventilation groups before mechanics measurements. Studies have shown that a short cyclic recruitment maneuver (or sigh) can be protective by improving ventilation homogeneity by opening up previously collapsed areas of the lung (6, 12, 32). Since the characteristics of P-V measurements are strongly influenced by the volume history of the lung, our efforts to standardize volume history for ventilation group comparison may have inadvertently obscured some of our findings. Specifically, the inclusion of a PV_P in functional assessment may have caused physiological changes similar to a sigh, such as increased recruitment and promoted surfactant release. Additionally, the short duration of ventilation does not have a direct clinical correlate since ARDS patients may be ventilated for days or weeks. We postulate that the physical mechanisms that cause direct physical injury to the blood-gas barrier (e.g., volutrauma and atelectrauma) will be similar whether they occur rapidly or slowly. As such, high Vt are used to rapidly produce VILI. However, since the inflammatory effects of VILI take several hours to manifest (31, 51) our choice of ventilation modes limits the findings here to short-term direct physical consequences and not the downstream effects of inflammation resulting from ventilation. Furthermore, it has been proposed that the pressure or volume threshold above which VILI occurs may be a consequence of ventilation duration (45, 50). Since our simulations track both the magnitude and number of injurious stimuli, we anticipate that the short ventilation duration that we employ is sufficient to identify the mathematical functions that describe injury progression. However, changes occurring over long timescales will not be reflected in the present study and will require evaluation of injury cost functions during prolonged ventilation to identify.

In summary, we propose a methodology to interpret measurements of pressures and flows recorded during mechanical ventilation. Several of the candidate injury cost functions were capable of distinguishing between the graded levels of injury generated by four different mechanical ventilation patterns in a mouse model of VILI. Of the injury cost functions derived from our single-compartment model, strain heterogeneity (Σα) provided the strongest correlation to alveolocapillary barrier disruption and was independent of underlying lung injury. However, mechanical power dissipation (N) is the most promising for clinical use because of its fidelity and ease of computation. We postulate that one of the benefits of N is that it measures energy lost to VILI generation. In contrast, the existing injury cost functions P_d and W lack the fidelity to clearly distinguish across moderate levels of injury. These findings suggest that Σα and N could facilitate patient-specific adjustment of ventilation parameters and identify damaging ventilation patterns before injury occurs. Whereas we only assess these injury cost functions individually in this work, it is possible that different combinations of them may yield even more promising results in the future.

GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grants R00 HL-128944, R01 HL-124052, T32 HL-076122, and T32 HL-072738.

DISCLOSURES

J. H. T. Bates owns shares and serves on the advisory board of Oscillavent, Inc. None of the other authors has any conflicts of interest, financial or otherwise, to disclose.

AUTHOR CONTRIBUTIONS

J.H.T.B. and B.J.S. conceived and designed research; S.A.S., G.S.R., E.B.-S., and B.J.S. performed experiments; M.M.M. analyzed data; M.M.M., K.L.H., and B.J.S. interpreted results of experiments; M.M.M. prepared figures; M.M.M. drafted manuscript; M.M.M., G.S.R., K.L.H., J.H.T.B., and B.J.S. edited and revised manuscript; B.J.S. approved final version of manuscript.

Glossary

DT

Derecruitability test: 3 min of low-Vt ventilation with FOT measurements every 18 s

E

Tissue stiffness modeled through a nonlinear spring coefficient, cmH₂O⋅cm

FOT

Forced oscillation maneuver from the flexiVent

H

Tissue elastance obtained from flexiVent FOT, cmH₂O⋅s⋅ml⁻¹

IE

Inspiratory-to-expiratory volume ratio

P

Pressure applied or measured by the ventilator at the airway opening, cmH₂O

Pa

Computational model pressure in the alveolar compartment, cmH₂O

PEEP

Positive end-expiratory pressure, cmH₂O

PIP

Peak inspiratory pressure, cmH₂O

P_m

Computational model pressure at the airway opening, cmH₂O

PV_P

Pressure-volume loop: inflation for 3 s to PIP, then 3-s hold at PIP and 3-s expiration

PV_V

4 breaths measured during volume-controlled ventilation, 200 breaths/min, Vt = 0.25 ml

R

Tissue resistance represented by a nonlinear dashpot, cmH₂O⋅cm⋅s⁻¹

RM

Recruitment maneuver: 3-s inflation followed by a 3-s inspiratory hold

RMSE_P

Root mean square error between measured and model pressures

RMSE_x

Root mean square error of x(T) and full recruitment (x = 1)

RR

Respiratory rate, breaths/min

t

Time, s

T

Discrete time determined by index b and time step Δt, s

V

Volume of air applied by the ventilator at the airway opening, ml

V_M

Computational model lung volume, ml

Vt

Tidal volume, ml

x

Open fraction, cm²

Χ

Objective function used to fit model parameters

y

Tissue distension or stretch, cm

κ

Penalizes x that is greater on inspiration than on expiration

λ

Penalty term that steers x to full recruitment at end-inspiration

ξ

Penalizes derecruitment on inspiration and recruitment on expiration