Towards a Mathematical Understanding of Ventilator-Induced Lung Injury in Preterm Rat Pups

Abstract

Approximately 1% of infants are born extremely preterm and underweight and are prone to respiratory distress and subsequent morbidity. Typical treatments for respiratory distress in late preterm and term infants, such as non-invasive pressure support, are less effective in preterm infants. Invasive mechanical ventilation applied as a last resort causes trauma, leading to ventilator-induced lung injury (VILI). Maternal infection, such as chorioamnionitis, can cause prenatal and neonatal lung infection, inflammation, and often very preterm birth. Inflammation is expected to stiffen the lungs with increased resistance and lowered compliance, but exceptions occur. A complete picture of the mechanisms of stiffening remains unknown. In an attempt to elucidate this information, we applied custom parameter inference and image analysis procedures to a neonatal rat model of chorioamnionitis and VILI, incorporating subject-specific pressure-volume measurements and histology. Numerical optimizations on a nonlinear compartmental model identified key parameter differences between healthy and unhealthy groups that may suggest mechanisms of VILI in infected respiratory systems. Combined analyses of the two strategies identified new correlations between model parameters, imaging metrics, and inflammatory markers from the data, suggesting that mathematical approaches provide an important path towards understanding VILI and infection.

1 Introduction

Infants born at less than 28 weeks gestation or less than 1000 g in weight are considered extremely preterm and are prone to a multitude of breathing issues associated with bronchopulmonary dysplasia (BPD) and lifelong co-morbidities [1]. BPD may also be linked to maternal infection during pregnancy [2], such as chorioamnionitis, which can lead to preterm birth and lung distress [3]. Respiratory therapies, such as non-invasive pressure support and surfactant replacement, are less effective in the extremely preterm demographic, resulting in the use of invasive mechanical ventilation applied as a last resort. However, such treatment can cause various forms of trauma, such as hyperoxia or endotracheal tube injury, leading to ventilator-induced lung injury (VILI) that can be exacerbated by infection. Inflammation from VILI and co-infections changes lung structure in ways that are hypothesized to stiffen them by increasing their resistance (opposition to movement) or decreasing compliance (ability to change the volume with applied pressure). However, it is not clear which particular histological changes are the proximate agents of inflammation-induced stiffening. The challenges associated with clinical studies in this and other fragile demographics necessitate computational and animal experiments to understand these mechanisms.

Mandell et al. [2] addressed this question using a neonatal rat model of chorioamnionitis to investigate what altered lung mechanics underlie the pressure-volume (PV) responses to mechanical ventilation. Maternal infection was simulated by prenatal exposure to an endotoxin (ETX). Their data included rat pup pressures, volumes, and histology images acquired at birth (D0) and day 7 (D7) of life, with and without ETX, and under either protective (SAFE) ventilation or two levels of injurious ventilation. Metrics of respiratory mechanics were determined from a forced oscillation technique and by applying standard image analysis techniques to the histology. Their results suggested that infection-related inflammation was correlated with a stronger VILI response, presenting in part as progressive airspace enlargement and increased compliance with increased inspiratory pressure. Interestingly, this is counter to what happens in adults. In the preterm pup demographic, this was speculated to result from insufficient collagen due to incomplete development of lung tissue. However, the group receiving ETX and the highest level of injurious ventilation exhibited decreased compliance and increased stiffness, possibly resulting from alveolar flooding, reducing lung capacity. This counter-intuitive result (see Fig. 2 of [2] for visualization) suggests additional analyses are needed to help elucidate mechanisms.

The histology was evaluated in Mandell et al. [2] with radial alveolar counts (RAC) and mean linear intercept (MLI). These techniques, while well-established, are subject to sampling bias [4, 5], can be labor-intensive, and may quantify only some aspects of the lung structure. Hence, there may be some benefit to the development of additional histological metrics. In particular, we expect that septal thinness and crimp (tortuosity) would be significantly correlated with parenchymal compliance; thickened and/or straight septa would be expected to be stiffer. Localized pockets of inflammation, in the form of locally thickened tissue, would be expected to pre-stress the parenchyma, rendering it stiffer. Bespoke image processing techniques could quantify such features.

Adverse respiratory system mechanics have traditionally been assessed with the classical single-compartment hydraulic model, dating back at least as far as the 1950s [6, 7]. This model consists of a pressure drop across the lung represented as a combination of a resistive component proportional to the rate of change of volume and an elastic component proportional to volume. Such models have grown in complexity over the years with the addition of new features such as lung heterogeneity, chest wall and other compartments, nonlinear components, respiratory muscle driving functions, ventilation, and even gas composition and cardiovascular dynamics [8,9,10,11,12] but have been overlooked in favor of simpler models in the context of data fitting [13,14,15]. The newer constant phase model used by Mandell et al. [2, 16] fits impedance measurements from a forced oscillation technique to determine a different set of mechanics metrics describing airways resistance, energy dissipation, and energy storage in the lungs [17]. However, it is also only a single-compartment model and a linear characterization of respiratory dynamics. Despite their growing prevalence and complexity, respiratory models are still primarily applied to adult physiology, in part due to the scarcity of relevant data and associated parameterization for children and infants. Applying parameter inference to a compartmental model specifically designed and evaluated against typical PV data obtained in a clinical setting may help assess the mechanics of a challenging demographic such as the extremely preterm infant.

In the current study, we explored new alternate parameter inference techniques via nonlinear modeling and image analysis applied to respiratory mechanics data from a neonatal demographic that is difficult to obtain and analyze. The experimental data described above obtained from ventilated rat pups in Mandell et al. [2] was used because it serves as an accepted surrogate of human data, is current, is amenable to our techniques, and generates an interesting open question. The primary objective was to determine if the proposed parameter inference techniques can uncover new information about breathing mechanics in the context of additional model complexity and for preterm infants. Secondary to the main objective was the opportunity to address the particular open questions from Mandell et al. [2] regarding mechanisms underlying changes in breathing mechanics related to inflammation and VILI.

We proposed a reduced compartmental model of lung pressure and volume that can capture the observed PV dynamics in the data for simulated experimental mechanical ventilation (Sect. 2.2). We then applied global and local sensitivity analyses to remove non-influential model components and obtain a minimal model (Sect. 2.3). Next, we performed parameter estimation on this minimal model (Sect. 2.4) with a gradient-based optimization algorithm and compared key parameter values for different groups. Concurrently, we developed and applied novel image analysis procedures for quantifying injury-related parameters extracted from histological images (Sect. 2.5). Metrics extracted from the images were statistically analyzed and their connections with the model parameters from this study and previously obtained biomechanical and biochemical data [2] were investigated. Results for all analyses are given in Sect. 3. Finally, we discussed our results in Sect. 4, including correlations that inspire new hypotheses that may improve understanding of the driving mechanisms of VILI response.

2 Methods

2.1 Experimental Data

The experimental groups from Table 1 of [2] were divided into pups from mothers injected with ETX and pups from healthy controls injected with saline (SAL). These were further divided into a group ventilated immediately upon birth at D0 and a group ventilated at D7. A protective level and two harmful levels (maximum pressures of 20 cm H${ }_2$O and 24 cm H${ }_2$O, denoted P20 and P24, respectively) of ventilation were administered. See Fig. 1 for a flowchart describing the experimental ventilation procedure.

Fig. 1

Flowchart of the experimental procedure of ventilation. The experiment includes several recruitment maneuvers within the measurement block, which would add complexity that may be challenging to model if we chose to extend our model to simulate the entire experimental sequence

2.2 Reduced Compartmental Model of Pressure-Volume Dynamics

We adapted a “reduced model” from a previous compartmental model developed by Ellwein Fix et al. [18] to simulate breathing mechanics in a preterm human infant. We retained compartments describing the airways with nonlinear resistance and the lungs with a nonlinear compliance in series with a parallel viscoelastic resistance and compliance. The diaphragm driving pressure from the previous model was replaced by mechanical ventilation. The original model [18] included components that could differentiate between types of airways and levels of chest wall compliance; given the limited experimental pressure and volume data characterizing the lungs directly, it was expected that the impact of these components would be unobservable. The extremely high chest wall compliance observed in preterm rat pups and human neonates was expected to contribute a nearly negligible amount to total compliance, further supporting the choice to remove the related compartment from the model [2, 19]. A schematic of the reduced model is given in Fig. 2. Tables 1, 2 give definitions for all variables and parameters, respectively.

Fig. 2

Schematic of compartmental model shown as (a) an electrical circuit analog and (b) a pressure-volume analog. Simulated mechanical ventilation pressure drives the model at the airway opening ($P_{ao}$). Fixed pressures include alveolar ($P_A$) and lung tissue ($P_T$). Pressures across compliant boundaries include lung elastic ($P_{el}$) and lung viscoelastic ($P_{ve}$), such that transmural pressure across the lung $P_l=P_{el}+P_{ve}$. Volumetric air flow $\dot {V}$ is positive in the direction of the arrows across the small airways with resistance $R_s$ and across the lungs with viscoelastic resistance $R_{ve}$, thereby increasing lung volume with positive ventilation. Compliances are denoted by $C_i$, where i matches the subscript for the corresponding zone. $V_A$ denotes the volume of the airway lung compartment. Pleural pressure ($P_{pl}$) is modeled as equal to atmospheric pressure ($P_{atm}$) since chest wall effects and diaphragm breathing are considered negligible

Table 1 Variables (states and constitutive quantities) used in the reduced and minimal models. (-) denotes a dimensionless quantity

Table 2 Parameters used in the reduced and minimal models. Nominal values are those used for simulation (Sect. 3.1) and as initial guesses in optimization (Sect. 2.4); ranges are the bounds used in Morris screening (Sect. 2.3.1). $R_s$ is included here as a parameter due to results from sensitivity analyses. The fixed parameters are not varied via sensitivity analyses (Sect. 2.3). (-) denotes a dimensionless quantity

2.2.1 State Equations

State variables include pressures $P(t)$ and volumes $V(t)$ within each compartment. Airflow through the cumulative airways is equal to the change in volume across lung compartments and is represented by both $\dot {V}(t)$ and $dV/dt$. The basic mass balance, Ohm’s law, and Kirchoff’s law are used with the circuit/hydraulic analog of flow such that resistances $R=\Delta P / \dot {V}$ and compliances $C = \partial V/ \partial P$. Pressures denoted by $P_{ao}$, $P_A$, and $P_T$ represent airway opening, alveolar, and (lung) tissue pressures, respectively. The dynamic transmural pressure across the lung $P_{l,\text{dyn}}$ is the sum of elastic recoil pressure and the viscoelastic pressure:

$$\displaystyle \begin{aligned} P_{l,\text{dyn}}=P_{el}+ P_{ve}, {} \end{aligned} $$

(1)

which are described respectively by

$$\displaystyle \begin{aligned} P_{el} = P_A - P_T, \end{aligned} $$

(2)

$$\displaystyle \begin{aligned} P_{ve} = P_T - P_{atm}. \end{aligned} $$

(3)

Differentiating $P_{el}$ with respect to time and applying the chain rule gives

$$\displaystyle \begin{aligned} \frac{d P_{el}}{dt} = \frac{\partial P_{el}}{\partial V_A} \frac{dV}{dt} = \frac{1}{C_A(V_A)} \frac{dV}{dt}, {} \end{aligned} $$

(4)

where lung elastic compliance is calculated via symbolic computation as ${C_A(V_A)}=\frac {dV_A}{dP_{el}}$. Likewise, for the viscoelastic component, we can state

$$\displaystyle \begin{aligned} \frac{d P_{ve}}{dt} = \frac{\partial P_{ve}}{\partial V_{Cve}} \frac{dV_{Cve}}{dt} = \frac{1}{C_{ve}} \frac{dV_{Cve}}{dt}, {} \end{aligned} $$

(5)

where $C_{ve}$ is a constant viscoelastic compliance. Since $\frac {dV}{dt}=\frac {dV_{Rve}}{dt}+\frac {dV_{Cve}}{dt}$, i.e., the sum of the volume changes (flows) over $R_{ve}$ and $C_{ve}$, Eq. (5) can be rewritten as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{d P_{ve}}{dt} & = &\displaystyle \frac{1}{C_{ve}} \left(\frac{dV}{dt} - \frac{dV_{Rve}}{dt} \right) \end{array} \end{aligned} $$

(6a)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & = &\displaystyle \frac{1}{C_{ve}} \left(\frac{dV}{dt} - \frac{P_{ve}}{R_{ve}} \right). {} \end{array} \end{aligned} $$

(6b)

2.2.2 Constitutive Equations

The previous representation of upper airway dependence on airflow from [18] does not apply in our minimal model due to the quasi-static ventilation scheme and resulting data. We restrict the airway model to the resistance of the small airways such as bronchioles, which are presumed to decrease resistance with increased alveolar volume $V_A$. The equation for the resistance of the small airways $R_s$ reflects this inverse relationship:

$$\displaystyle \begin{aligned} R_s = R_{s,d} \exp \left[ \frac{-K_s V_A}{\text{IC}} \right] + R_{s,m}, {} \end{aligned} $$

(7)

where $R_{s,d}$ is the change in small airway resistance, $K_s$ is the small airway resistance low pressure coefficient, IC is the inspiratory capacity volume, and $R_{s,m}$ is the minimum small airway resistance. Parameter $K_s > 0$. As $V_A\rightarrow 0$, $R_s\rightarrow R_{s,m}+R_{d,m}$; likewise, as $V_A\rightarrow \text{IC}$, $R_s\rightarrow R_{s,m}$.

The volume of the lung compartment $V_A$ is given by the product of the volume due to aggregate elasticity of the lung unit structure $V_{el}$ and the fraction of recruited alveoli $F_{\text{rec}}$. Both quantities are functions of lung elastic recoil pressure:

$$\displaystyle \begin{aligned} V_A = V_{el}(P_{el}) F_{\text{rec}}(P_{el}). {} \end{aligned} $$

(8)

We model $V_{el}$ as in prior studies [18, 20, 21] with a saturated exponential function

$$\displaystyle \begin{aligned} V_{el} =\text{IC}[1 - \exp(-k P_{el})], {} \end{aligned} $$

(9)

where k is a parameter that characterizes slope and aggregate lung elasticity. The alveolar (de)recruitment function $F_{\text{rec}}$ changes for recruitment and derecruitment of alveoli. Following [21], we assume that the form of the models for recruitment and derecruitment are identical except that the closing alveolar pressures are lower than the opening pressures. Thus, we model (de)recruitment by the piecewise function

$$\displaystyle \begin{aligned} F_{\text{rec}} = \begin{cases} \beta + (\gamma - \beta) \left\{ \frac{1 - \exp(-P_{el}/d_{F_R})}{1 + \exp \left[-(P_{el} - c_{F_R})/d_{F_R} \right]} \right\}, & 0 \leq t < \tau/2 \\ \beta + (\gamma - \beta) \left\{ \frac{1 - \exp(-P_{el}/d_{F_D})}{1 + \exp \left[-(P_{el} - c_{F_D})/d_{F_D} \right]} \right\}, & \tau/2 \leq t \leq \tau \end{cases}, {} \end{aligned} $$

(10)

where $\beta$ is the baseline fraction of lung recruited at $P_{el} = 0$, $\gamma$ is the maximum recruitable fraction of lung, $d_{F_R}$ is the variance in opening pressure, $c_{F_R}$ is the mean opening pressure, $d_{F_D}$ is the variance in closing pressure, $c_{F_D}$ is the mean closing pressure, and $\tau$ is the total time of the ventilation procedure. Here, the parameters $c_F$ and $d_F$ are allowed to vary between inspiration and expiration for one breathing loop for a single rat. The subscripts R and D denote recruitment and derecruitment, respectively.

The data in this study come from a quasi-static stepwise ventilation [2], thus the driving pressure is a simulated protocol applied at the airway opening $P_{ao}$. The stepwise ventilation is a 16-second procedure, whereby the airway opening pressure $P_{ao}$ is increased from zero (or nearly zero) to 30 cm H${ }_2$O in a stair-step fashion: the maneuver comprises seven increases in pressure (7 “steps up”) with roughly equal duration and comparable decreases from maximum pressure back down to zero. We approximate the stair-step function with the sum of narrow $\tanh$ functions and include the possibility for a slight delay $t_d$ in order to better match the experiments:

$$\displaystyle \begin{aligned} P_{ao}(t) = \frac{1}{2}\frac{P_{\max}}{n_{\text{steps}}} \text{switch}(\tau/2 - t) \sum_{j = 0}^N \left[ 1 + \tanh \left( \frac{t - [j \tau/(2n_{\text{steps}}) + t_d]}{t_w} \right) \right], {} \end{aligned} $$

(11)

where $P_{\max }$ is the maximum recruitment maneuver pressure, $n_{\text{steps}}$ is the number of stair-step pressure increases, $\tau$ is the total time of the ventilation procedure, $t_d$ is the time delay in the recruitment maneuver, and $t_w$ is the stair-step width. The narrowness of the stair-step is enforced by using $t_w = 0.05$ s; the value is selected to best match experimental ventilation pressure curves [2]. Here, $N = \lfloor t/(\tau /2 n_{\text{steps}}) \rfloor$, $P_{\max } = 30$ cm H${ }_2$O, $n_{\text{steps}} = 7$, $\tau = 16$ s, $t_d = 0.25$ s, and switch$(x)$ is essentially the sign function that returns 1 or $-1$ depending on the sign of the argument, but takes the value 1 at 0:

$$\displaystyle \begin{aligned} \text{switch}(x) = \begin{cases} \text{sgn}(x), & x \neq 0 \\ 1, & x = 0 \end{cases}. \end{aligned} $$

(12)

The switch function facilitates the step direction: “up” until $t = 8$ s and then “down” until $t = 16$ s.

2.2.3 Model Formulation

Recalling the relationship between resistance, change in pressure, and time derivative of volume, we may write

$$\displaystyle \begin{aligned} \frac{dV}{dt} = \frac{P_{ao} - P_{l,\text{dyn}}}{R_s}. \end{aligned} $$

(13)

We also recall that $P_{l,\text{dyn}}$ is calculated via Eq. (1). Substitution into Eqs. (4) and (6b) yields the system for viscoelastic and elastic pressure, $P_{ve}$ and $P_{el}$, which is given by

$$\displaystyle \begin{aligned} \frac{d P_{el}}{dt} = \frac{1}{C_A(V_A)\;R_s }(P_{ao} - P_{el} - P_{ve}), {} \end{aligned} $$

(14)

$$\displaystyle \begin{aligned} \frac{d P_{ve}}{dt} = \frac{1}{C_{ve}} \left(\frac{P_{ao} - P_{el} - P_{ve}}{R_s} - \frac{P_{ve}}{R_{ve}} \right). {} \end{aligned} $$

(15)

Equations (14)–(15) are solved on $t \in [0, 16]$ s because this time frame represents one mechanical PV ventilation maneuver. The initial conditions used are $P_{el}(0)=0.954$ cm H${ }_2$O and $P_{ve}(0) = 0$ cm H${ }_2$O following [18]. Then, $V_{el}$ and $V_A$ are calculated via Eqs. (8) and (9).

The system in Eqs. (14)–(15) is solved using ode15s in MATLAB. We use default ODE solver and optimization tolerances of $10^{-3}$ for the relative error tolerance and $10^{-6}$ for the absolute error tolerance.

2.3 Sensitivity Analysis

The reduced model includes components that we hypothesize could reflect the physiology of breathing mechanics as evident in PV data, especially nonlinearities. Determination of the impact of model parameters on model output and estimation of reasonable parameter values requires a mathematical understanding of the practical identifiability of the underlying parameters in the context of available data [22]. Parameters that are identified as insensitive or correlated with other sensitive parameters are commonly handled either by keeping them at a reasonable nominal value for all data sets or by removing them from the model altogether. We perform two analyses for this purpose: (1) the Morris global screening method, a one-at-a-time method that calculates a set of randomized but structured finite difference derivatives over the full parameter space versus a set of key scalar outputs; and (2) a coarse univariate local sensitivity analysis, whereby each parameter is perturbed by a factor of two starting with nominal parameters describing an ‘average’ animal subject and the magnitude of change in scalar outputs is categorized by effect size. Sensitivity analyses are applied to the parameter set:

$$\displaystyle \begin{aligned} \boldsymbol{p} = \{ K_s, \text{IC}, C_{ve}, R_{ve}, R_{s,d}, R_{s,m}, k, c_{F_R}, c_{F_D}, d_{F_R}, d_{F_D}, \beta,\gamma\}. {} \end{aligned} $$

(16)

Note that the ventilation setting parameters are not included, as they are taken directly from the experimental settings.

2.3.1 Morris Effects Analysis

As a global sensitivity analysis, we use the technique of Morris elementary effects, or Morris screening analysis [23]. In contrast to local sensitivity analysis, which studies the effect of individual parameter perturbations, global sensitivity analysis explores the effect of combinations of parameters that are perturbed across the feasible parameter space [24]. The method aims to compute “elementary effects” that approximate derivatives of model output changes with respect to parameter perturbations [25, 26]. These are then combined in order to rank parameter sensitivities.

Let $f(t; p_1, \ldots , p_n)$ be the model output with respect to parameters $\boldsymbol {p} = (p_1, \ldots , p_n)$. Let $\boldsymbol {e}_i$ denote the ith unit vector. Then the ith Morris elementary effect is given by

$$\displaystyle \begin{aligned} \text{EE}_i = \frac{f(t; \boldsymbol{p} + \boldsymbol{e}_i \delta) - f(t; \boldsymbol{p})}{\delta}. \end{aligned} $$

(17)

Here, $\delta = \ell /2/(\ell - 1)$ describes a step in the parameter space such that $\boldsymbol {p} + \boldsymbol {e}_i \delta$ is still within the allowable bounds for the parameter. We choose $\ell = 60$, which gives $\delta \approx 0.51$. This choice allows for a symmetric sampling distribution [23]. We also normalize the parameters to the interval $[0,1]$ for the sampling following Colebank and Chesler [24].

The summary statistics are computed after conducting N random initializations for each $\text{EE}_i$. The arithmetic mean associated with each parameter is given by

$$\displaystyle \begin{aligned} \mu_i = \frac{1}{N} \sum_{j = 1}^N \text{EE}_i^j, \end{aligned} $$

(18)

and the corresponding sample variance is found by

$$\displaystyle \begin{aligned} \sigma_i^2 = \frac{1}{N-1} \sum_{j = 1}^N (\text{EE}_i^j - \mu_i)^2. \end{aligned} $$

(19)

To study the magnitude of the elementary effects while avoiding potential cancellation issues, we use the mean absolute elementary effect $\mu _i^*$ in our analysis following [27]:

$$\displaystyle \begin{aligned} \mu_i^* = \frac{1}{N} \sum_{j = 1}^N \left|\text{EE}_i^j \right|. {} \end{aligned} $$

(20)

To compare and rank the relative effects of the parameters on the model output, we use the combined statistic termed the Morris ranking:

$$\displaystyle \begin{aligned} M_i = \sqrt{\mu_i^{*2} + \sigma_i^2}. {} \end{aligned} $$

(21)

We vary thirteen free parameters (the set listed in Eq. (16)) for our Morris sensitivity analysis; descriptions and their nominal values are given in Table 2. We choose six scalar metrics as key characteristics of the PV curves that were observed in the data to differ across treatment and ventilation groups. These are work of breathing (WOB) [mL $\cdot$ cm H${ }_2$O], maximum lung volume $v_{\max }$ [mL], and four slopes of the PV curve [mL/(cm H${ }_2$O)]: (1) recruitment and (2) derecruitment curves between pressures of 0 and 10 cm H${ }_2$O and (3) recruitment and (4) derecruitment curves between pressures of 10 and 30 cm H${ }_2$O. These four slopes are denoted by $sl_R$ 0–10, $sl_D$ 0–10, $sl_R$ 10–30, and $sl_D$ 10–30, respectively. WOB is defined as

$$\displaystyle \begin{aligned} \text{WOB} = \int_{P_{l,\text{dyn}}} \left[ V_{A,D}(P_{l,\text{dyn}}) - V_{A,R}(P_{l,\text{dyn}}) \right] \ d P_{l,\text{dyn}}, {} \end{aligned} $$

(22)

where the D and R subscripts on $V_A$ denote the derecruitment and recruitment portions of the lung volume, respectively. This integration is calculated numerically with a standard trapezoid rule in MATLAB. The second metric $v_{\max }$ is the maximum of $V_A$, the lung volume, over the PV loop. The slope $sl_R$ between pressures of $P_1$ and $P_2$ cm H${ }_2$O is calculated by

$$\displaystyle \begin{aligned} sl_R = \frac{V_{A,R}(P_2) - V_{A,R }(P_1) }{P_2 - P_1}, {} \end{aligned} $$

(23)

and the slope $sl_D$ between pressures of $P_1$ and $P_2$ cm H${ }_2$O is calculated by

$$\displaystyle \begin{aligned} sl_D = \frac{V_{A,D}(P_2) - V_{A,D}(P_1) }{P_2 - P_1}. {} \end{aligned} $$

(24)

The metrics are summarized in Table 3.

Table 3 Scalar metrics used by the Morris screening analysis

2.3.2 Local Sensitivity Analysis

We perform a coarse univariate sensitivity analysis in which we change one parameter at a time by a fixed percent of its nominal value and note the percent change in the scalar output, while holding all other parameters constant at their nominal values. In particular we look at the effect of doubling and halving each of the following parameters $R_{s,m}$, $R_{s,d}$, $K_s$, $R_{ve}$, $c_{F_R}$, $c_{F_D}$, $d_{F_R}$, $d_{F_D}$, and k on the six scalar effects defined in Sect. 2.3.1. Parameters $\gamma$ and $\beta$ nominally represent fractions of recruitable lung such that $0<\gamma ,\beta \le 1$, so their low and high values are chosen explicitly to be $[0.85,0.95]$ and $[0.1,0.9]$, respectively. These also represent reasonable possible values for these fractions with respect to the physiology. Finally, for $C_{ve}$ we use 70% as opposed to 50% to ensure simulation output remains physiologically viable. The nominal values are listed in Table 2.

2.4 Optimization

For the parameters identified as sensitive for at least one metric (Table 3), we determined optimal values by minimizing the sum of the squared differences between the quasi-static points of the PV curves given in the data and the quasi-static points of our simulated PV curves. Parameters that were identified as non-sensitive across all six metrics used in the Morris screening were kept constant at the nominal values reported in Table 2. Optimization was performed using the constrained optimization algorithm lsqnonlin in MATLAB using the Levenberg-Marquardt option and an ODE solver relative tolerance of 10${ }^{-12}$. The step tolerance for the optimization algorithm was set equal to the square root of the relative tolerance of the ODE solver. The nominal values from our sensitivity analyses were used as the initial guesses for k, $c_{F_R}$, $c_{F_D}$, $\beta$, and $\gamma$.

We determined optimal parameters for each rat in the data by minimizing the least-squares objective function J:

$$\displaystyle \begin{aligned} J=\frac{1}{2}\sum [V_A(P_{l,\text{dyn}_{\text{data}}})-V_{A_{\text{data}}}(P_{l,\text{dyn}_{\text{data}}})]^2. \end{aligned} $$

(25)

Since pulmonary system elastance and tissue damping coefficient for individual rats were reported in the data as obtained from the forced oscillation impedance fit to the constant phase model ([2, 16], denoted as H and G), we based our initial iterates for $C_{ve}$ and $R_{ve}$ on these data. Constant upper and lower bounds were imposed on the algorithm. Initially, we used the same bounds that were imposed in the Morris screening for $C_{ve}$, which were estimated from pulmonary system elastance in [2]. However, we were unable to achieve acceptable fits without decreasing the lower bound for $C_{ve}$ (equivalent to increasing the upper bound on $1/C_{ve}$, which is how we later present our results). Since the viscoelastic compliance, $C_{ve}$, in our model does not actually encapsulate the whole system compliance/elastance, it is unsurprising that we had to adjust the bounds.

For some rat data sets, the optimization appeared to stagnate or converge to a local minimum that creates a non-physiological simulated output. In these cases, the initial iterate was modified so the optimization algorithm converged appropriately. We calculated the mean optimal values for each rat group along with the variance. The average values and the variances are reported in Table 5.

2.5 Image Analysis

In order to quantify morphological differences relevant to biomechanical differences among treatment and age groups, we develop a procedure to analyze lung histology images using known techniques. The implementation uses the MATLAB Image Processing Toolbox. RGB images from a previous study [2], of uniform pixel dimensions and magnification, are passed through an entropy filter, using a filter neighborhood equivalent to a disk of radius 5 microns, which minimizes the impact of irrelevant small-scale details such as individual cells. Filtered images are converted to grayscale and binarized with Otsu thresholding. Binarized images are then analyzed by skeletonization, erosion, connected region identification, and other methods to yield image metrics. The basic procedure is illustrated in Fig. 3. We do not identify or segment specific non-alveolar structures in the images, such as blood vessels.

Fig. 3

Schematic of image processing procedure. Raw RGB image is entropy filtered and converted to grayscale, before thresholding (Otsu method) to create a binary image where tissue is white and lumens are black. The binary image is quantitatively analyzed by a variety of additional methods, including skeletonization. Here, the skeletonized image is dilated for visibility

2.5.1 Metrics of Lumens

2.5.1.1 Lumen Count

A raw count c of lumens in each image will include both lumens falling completely within the image and also lumens clipped by the edge of the image (Fig. 4). Considering them together would skew the lumen metrics (count, area, and others), as well as other tissue metrics, such as tortuosity, that are derived from lumen counts. Therefore, we use a corrected lumen count, on the assumption that, on average, each edge lumen is missing half its true area.

Fig. 4

The images include lumens cropped by the image boundaries. The lumen count is used as an independent metric and in calculating other image analysis metrics. (a) When segmenting lumens to count them, it is crucial to make a distinction between complete lumens (black) and those clipped by the edge of the image (green). (b) A simple estimate of the corrected image analysis metrics in the presence of the clipped lumens begins with assuming a regular grid of square lumens and deriving the correction terms for the lumen count

Consider a grid of equal square lumens, each of side length l, in a square image of side length L (Fig. 4). On average, the image length and width each hold $n = L/l$ lumens, each of area $l^2$, so the whole image holds $N=n^2=(L/l)^2$ lumens, each of area $l^2$, for a total area of $L^2$. The image has a perimeter of 4L, and the image edge cuts 4n lumens. The edge lumens are assumed to be, on average, cut in half. Thus, if N is the true lumen count, and c is the raw lumen count, on average, $c=N+2n=N+2\sqrt {N}$. Solving for N gives the corrected lumen count, N, based on the raw lumen count c:

$$\displaystyle \begin{aligned} N=c+2-2\sqrt{c+1}. {} \end{aligned} $$

(26)

2.5.1.2 Lumen Areas

For each lumen found, we measure its area $A_i$. Again, the image contains a significant fraction of lumens that are cut by the image edge, so aggregate area metrics have to account for the estimated true areas of clipped lumens. Again considering a regular grid of squares, the total lumen area of the image is $L^2=Nl^2=c\bar {A}_{\text{meas}}$ where $\bar {A}_{\text{meas}}=[(N-2n)l^2+4nl^2/2)]/c$. Thus, the true mean area per lumen $\bar {A}$, for the square grid model, would be $\bar {A}=l^2=\bar {A}_{\text{meas}}c/N = \bar {A}_{\text{meas}} c/(c+2-2\sqrt {c+1})$.

For comparison with the existing lumen metric mean linear intercept (MLI, a length), we define a length-based area metric for image lumens, by normalizing the lumen areas $A_i$ to an equivalent disk of radius $R_i$

$$\displaystyle \begin{aligned} R_{\text{equiv},i} = \sqrt{A_i/\pi}. \end{aligned} $$

(27)

Since the set of lumens with measured areas includes clipped lumens, when finding the average $R_{\text{equiv}}$, we adjust the calculation of the mean by the expected edge counts:

$$\displaystyle \begin{aligned} \bar{R}_{\text{equiv}} = \sqrt{c/N} \frac{1}{N} \sum_{i = 1}^N \sqrt{A_i/\pi}. \end{aligned} $$

(28)

All analyses of $R_{\text{equiv}}$ use this corrected quantity, $\bar {R}_{\text{equiv}}$, but we drop the bar for convenience.

We define another area metric using the cumulative distribution of areas $A_i$:

$$\displaystyle \begin{aligned} A_{\text{cumul}}(n) = \sum_{i = 1}^n A_i. {} \end{aligned} $$

(29)

Then the total area of lumens $A_{\text{lum}} = A_{\text{cumul}}(c)$. We define $R_{\text{mid}}$ as the equivalent radius $R_{{\text{equiv}},j}$ such that $A_{\text{cumul}}(j) \approx \frac {1}{2} A_{\text{cumul}}(c)$ as closely as possible.

2.5.2 Metrics of Tissue

2.5.2.1 Tissue Area Fraction

We calculate the total tissue area $A_{\text{tiss}}$ by counting tissue pixels in the binarized image. Tissue area fraction $\phi$ is then (tissue pixel count)/(total pixel count).

2.5.2.2 Tissue Length

For each image, tissue length or total perimeter $\Lambda$ is calculated from the skeletonized image as the total length of tissue centerlines. We expect an allometric relation between $\Lambda$ and the number of lumens N. Following the extremely simplified square grid models of Sect. 2.5.1 and Fig. 4, an image of a tissue grid has total area $A=L^2=Nl^2$ and the total length (perimeter) is $\Lambda = 2Nl = 2(L/l)L = 2nL = 2L\sqrt {N}$. Thus, for a general image of lung parenchyma, we expect that the total tissue length $\Lambda$ is proportional to $\sqrt {N}$.

2.5.2.3 Tortuosity

At the length scale of whole alveoli, septal tortuosity results from adjacent alveoli pushing into the septum. Collagen tortuosity is at a much smaller length scale. We quantify tortuosity at an intermediate length scale, smaller than a single alveolus, but substantially larger than the polymer length scale. For a single geometric object of area A and perimeter $\Lambda$, we define its dimensionless tortuosity ratio T as

$$\displaystyle \begin{aligned} T = \frac{\Lambda^2}{4 \pi A}. \end{aligned} $$

(30)

A circle has $T = \frac {(2\pi r)^2}{4 \pi (\pi r^2)} = 1$, the minimum possible value. For a square, $T = 4/\pi \approx 1.3$.

To extend this metric to a non-simply-connected object or a partitioning of the space, we propose the following. Suppose an image contains N lumens with total perimeter length $\Lambda$ and total of lumen areas $A_{\text{lum}}$, which in our images may be close to the total area of the image. Because of the shared boundaries between adjacent lumens, we need to count the total perimeter twice. We generalize the tortuosity ratio T for a single object to an image tortuosity metric

$$\displaystyle \begin{aligned} \tilde{T} = \frac{\bar{\Lambda}^2}{4 \pi \bar{A}} = \frac{(2 \Lambda_{\text{tot}}/N)^2}{4\pi A_{\text{lum}}/N} =\frac{\Lambda^2}{N \pi A_{\text{lum}}}, {} \end{aligned} $$

(31)

where N is the estimated true lumen count from Eq. (26). Note that $\Lambda$ does not need an adjustment for intersecting the edge.

For a grid of squares, it is easy to confirm that $\tilde {T} = \frac {4}{\pi } \approx 1.3$, and for regular hexagonal packing $\tilde {T} = \frac {2 \sqrt {3}}{\pi } \approx 1.1$, which is the lower bound, approaching the minimum of 1 for a circle. We expect a greater tortuosity ratio $\tilde {T}$ to correspond to higher tissue compliance since the lumens would expand by straightening their crimps.

2.5.2.4 Septal Width and Its Distribution

Our most basic width metric is the mean width:

$$\displaystyle \begin{aligned} \bar{w} = A_{\text{tiss}}/\Lambda, \end{aligned} $$

(32)

where $A_{\text{tiss}}$ is the total area of septa and $\Lambda$ is the total length of septa. We can infer more about the histology, and potentially the biomechanics, if we go beyond the mean of width to look at its distribution.

Consider a 2D image containing a network of septa with a fixed total tissue area $A_{\text{tiss}}$ or area fraction $\phi$ and total tissue length $\Lambda$. If the tissue is uniformly thick, i.e., the tissue area is spread evenly along its length, we expect that network to have a certain stiffness. If that same area is distributed unevenly over the same total length, i.e., in a mixture of thick and thin septa, we would expect that network to be more compliant, as the thin portions would stretch more than the thick portions under the same load. We expect uneven width from a variety of pathological conditions, including fibrosis and inflammation.

To obtain the distribution of septa of width w, we follow a previous study [28] quantifying vascular networks. We define a pruning algorithm as follows:

• 1. Starting with the binarized image, erode the image using a disk of radius $w/2$.

• 2. Dilate the eroded image by the same disk.

• 3. Measure the area of tissue remaining in the image.

• 4. Repeat 1–3 until no tissue remains.

This serves to remove any image details finer than the gauge w (Fig. 5). For a large enough w, no pixels remain after the pruning. We can consider the functions $A_{\text{tiss}}(w)$ or $\phi (w)$, which show the distribution of structures of width greater than w.

Fig. 5

Example of applying the pruning algorithm with a disk of radius r to a representative lung tissue image, where r increases from left to right. Eventually using a large enough disk, no tissue remains. (a) The binarized tissue image before pruning. The structuring element used in (b) is a disk with radius 5 $\mu$m, in (c) 10 $\mu$m, in (d) 15 $\mu$m. Hence (b), (c), and (d) show all tissue in (a) thicker than 10, 20, and 30 $\mu$m, respectively

2.6 Statistical Analysis

Statistical analyses were performed in MATLAB and JMP (SAS Institute, Cary, NC) and using the FactoMineR package [29] in R [30]. Analyses included distributions of image metrics and their fits to multivariate models, as well as correlations with previously reported metrics of biomechanics and cytokine concentrations [2] and with fitted model parameters. Multivariate statistical models for metric distributions were evaluated by significance (strictly $p<0.05$, but generally much lower) and by small-sample corrected Akaike information criterion (AICc). Optimized parameter values are compiled by mean and variance.

3 Results

3.1 Model Solutions

A representative dimensional solution of the reduced model output airway lung compartment volume $V_A$ against dynamic pressure $P_{l, \text{dyn}}$ is shown in Fig. 6. This solution uses nominal values for a D0 control (SAL) group rat under SAFE ventilation, also used for the later sensitivity analyses (Sect. 2.3), and is plotted against the corresponding data for a single rat from this group. The effect of the stair-step ventilation pressure applied at the airway opening $P_{ao}$ on model solutions can be seen in Fig. 6b, as the curve exhibits cusps at quasi-static points corresponding to the edge of a stair. The expiration curve is higher than the inspiration curve, which is consistent with the lung air sacs requiring a higher pressure on average to open than to close for a given volume.

Fig. 6

(a) Simulated stair-step ventilation pressure applied at the airway opening. (b) Simulated pressure-volume loop for nominal parameter values (see Tables 1, 2). The experimental quasi-static points from a pressure-volume loop [2] are shown for comparison

3.2 Morris Screening

We implement Morris screening using 100 randomized initializations of model parameters. A small number of outlier results are excluded: any model implementation with an integration tolerance failure, an infeasible metric output (e.g., a negative WOB), or a drastic outlier $EE_i^j$ (defined as more than 1000 times the median magnitude) that suggests an infeasible combination of parameter values. These excluded results comprise less than 1% of the original simulations.

The absolute value of the mean $\mu _i^*$ from Eq. (20) is plotted on a log-log scale against the sample variance $\sigma _i^{2}$ for the thirteen parameters studied over the six metrics analyzed (Fig. 7). Parameters with both large $\mu _i^*$ and $\sigma _i^{2}$ are considered influential because they have both nonlinear and interacting effects with other parameters. Conversely, parameters with both small $\mu _i^*$ and $\sigma _i^{2}$ are considered non-influential. For example, inspiratory capacity IC is consistently the most influential parameter, and change in small airway resistance $R_{s,d}$ is consistently a non-influential parameter across all six metrics. This categorization follows [23].

Fig. 7

Absolute value of the mean elementary effect $\mu _i^*$ of signal (Eq. 20) is plotted against sample variance of signal for the results of the Morris effects analysis. The six metrics analyzed are work of breathing (WOB), maximum lung compartment volume ($v_{ \max }$), and recruitment or inspiratory (R) and derecruitment or expiratory $(D$) slopes between pressures 0 and 10 cm H${ }_2$O and 10 and 30 cm H${ }_2$O

The Morris statistic rankings per elementary effect by Eq. (21) are shown in Fig. 8 on a log scale. Note that the Morris ranking plot for the recruitment slope between pressures of 0 and 10 cm H${ }_2$O does not show values for $c_{F_D}$ and $d_{F_D}$ because the analysis determined those rankings to be zero. Parameter rankings falling above the mean ranking, shown by the dashed horizontal line, are deemed sensitive and are indicated in red [31, 32]. Eight out of the thirteen reduced model parameters form the union of the sensitive parameters over all six effects (Table 4), which we call the “Morris sensitive set” and summarize as

$$\displaystyle \begin{aligned} \boldsymbol{p}_{\text{Mor}}=\{\text{IC},C_{ve},R_{ve},k,c_{F_R},c_{F_D},\beta,\gamma\}. {} \end{aligned} $$

(33)

Fig. 8

Ranking from the Morris effects analysis. The black dashed line shows the mean ranking. Parameters with rankings above the mean are colored in red, and those with ranking below the mean are colored in blue. The six metrics analyzed are work of breathing (WOB), maximum lung compartment volume ($v_{ \max }$), and recruitment or inspiratory (R) and derecruitment or expiratory $(D$) slopes between pressures 0 and 10 cm H${ }_2$O and 10 and 30 cm H${ }_2$O

Table 4 Sensitive parameters as determined by the Morris screening. The middle six columns indicate the parameters that had ranking larger than the mean for each of the metrics. The far right column indicates the union of the rankings above the mean to form the Morris sensitive set

Table 5 Optimal values were determined for parameters identified as sensitive by at least one metric (see Fig. 8 and Table 3). Optimal values were determined for each rat and then averaged across each group (n = number of rats in the group). Here, we report the mean $\mu$ optimal parameter values for each rat group as well as the variances $\sigma ^2$

Inspiratory capacity was evaluated as the most sensitive parameter for all six metrics. This is not surprising since IC has a direct effect on $V_A$ through Eqs. (8)–(9), which comprise one axis of a PV curve. The parameter k, representing lung elasticity and contributing to the saturation of the function $V_{el}$, is sensitive across all six metrics. For the four slope metrics, the Morris rankings of the mean and variance in recruitment or decruitment pressure seem to correspond to the part of the PV curve studied. For example, $c_{F_R}$ and $d_{F_R}$ have higher Morris rankings than the same for derecruitment ($c_{F_D}$ and $d_{F_D}$) for $sl_R$ from 0 to 10 cm H${ }_2$O. This holds for all $c_{F_R}$ and $c_{F_D}$ pairs and for all but one pair of $d_{F_R}$ and $d_{F_D}$; in this case, the values are nearly identical.

Certain parameter groupings were found to be either all part of $\boldsymbol {p}_{\text{Mor}}$ (Eq. 33) or all non-influential. The pressure range parameters $d_{F_R}$ and $d_{F_D}$ describing the variance of the opening or closing pressures were found to be non-influential; however, the mean opening and closing pressures, $c_{F_R}$ and $c_{F_D}$, and baseline and maximum recruitment fractions, $\beta$ and $\gamma$, respectively, were found to be influential. We note that $d_{F_R}$ and $d_{F_D}$ characterize the heterogeneity of the lung by allowing for a transition to full (de)recruitment [18, 21]; since these are not sensitive, this suggests that our model cannot identify this alveolar variation in the opening or closing of lung units. In contrast, $c_{F_R}$ and $c_{F_D}$ represent average information. This suggests that a limitation of our model is that it must treat alveolar recruitment as a homogeneous action across the lung, although we recognize that in actual lungs, recruitment occurs in specific locations. The parameters $K_s$, $R_{s,d}$, and $R_{s,m}$ govern the nonlinear small airway resistance, $R_s$, and are not part of $\boldsymbol {p}_{\text{Mor}}$ (Eq. 33). Note that the function for $R_s$ is also dependent upon inspiratory capacity IC, but IC governs the variable volume $V_A$ as well. This suggests that the reduced model can be minimized further to eliminate the nonlinearity of our small airway resistance so that $R_s$ is set to a constant value: $R_s = R_{s,\text{const}}.$ For the optimization that follows in Sect. 3.4, we use this “minimal” version of our reduced model. We also create our vector of free parameters to vary in the optimization as $\boldsymbol {p}_{\text{Mor}}$ (Eq. 33) without IC, which is given by

$$\displaystyle \begin{aligned} \boldsymbol{p}_{\text{free}}=\{C_{ve},R_{ve},k,c_{F_R},c_{F_D},\beta,\gamma\}. {} \end{aligned} $$

(34)

While IC was determined to be sensitive by the Morris screening, subject-specific metrics were directly measured by Mandell et al. [2], so we use each animal’s IC data for our analysis.

3.3 Local Sensitivity Analysis

Results from the univariate local sensitivity analysis are shown in the tornado plots in Fig. 9. The parameters in a given plot are organized from the largest increase to the largest decrease in metric upon an increase in parameter, with parameters with the least effect on the metric in the center of the plot. For most parameters, there is either a positive relationship across all metrics or a negative relationship across all metrics.

Fig. 9

Local sensitivity analysis of the twelve parameters discussed in Sect. 2.3.2 and their impact on six scalar metrics defined in Table 3. The blue bar for each parameter represents the percent change in the scalar metric given an increase (typically 200%) in the parameter, while the orange bar represents the percent change in the scalar output given a decrease (typically 50%) in the parameter. Parameters are organized from the largest increase to the largest decrease in metric upon an increase in parameter

Of note are the parameters that do not have a strictly positive or negative relationship. WOB increases when viscoelastic resistance $R_{ve}$ is both increased and decreased. While WOB increases when change in airway resistance $R_{sd}$ increases, there is no change to the metric when $R_{sd}$ is decreased. Both positive and negative changes to k cause a decrease in $sl_R$ 10–30. When decreased, mean closing pressure $c_{F_D}$ causes no change in $sl_D$ 0–10 and $sl_D$ 10–30; however, increasing $c_{F_D}$ increases $sl_D$ 10–30 and decreases $sl_D$ 0–10.

We see that k has the largest impact on $sl_{D}$ 0–10, $sl_D$ 10–30, and $v_{\max }$, while $\beta$ has the largest impact on $sl_R$ 0–10 and $sl_R$ 10–30. Multiple parameters affect WOB; $c_{F_R}$, $R_{ve}$, and $\beta$ have the largest impacts. Encouragingly, the findings of local sensitivity analysis (Fig. 9) confirm the Morris screening conclusions of the least influential parameters.

3.4 Optimization Results

Optimized PV curves are plotted against the data in Fig. 10 for one rat in the D0 SAL group under SAFE ventilation and one rat in the D7 ETX group under high-pressure ventilation (P24). Using the nominal parameters, the initial guesses for the optimization are shown as dashed lines to indicate reasonable convergence to the optimal solution representing observed rat lung dynamics.

Fig. 10

Optimized pressure-volume curves (red solid curves), experimental quasi-static points (black circles), model simulated quasi-static points (red circles), and initial optimization guess (blue dashed curves) are plotted for a representative rat in (a) the D0 SAL-SAFE group and (b) the D7 ETX-P24 group. The optimized curves are close fits to the experimental data [2]

3.4.1 Mean Values

After fitting parameters to the data for each rat and then calculating summary statistics per group, we obtained the mean optimized parameter values for each group shown in Fig. 11 (blue circles) with normalized maximum and minimum values (black squares) and one standard deviation above and below the mean (solid red line). Normalizing by the parameter constraint upper bound allows for easier comparison between groups and shows the degree of variability of optimized parameter values. Mean and variance for each optimized parameter per rat group are also reported in Table 5. The largest variation is observed for the optimal values for viscoelastic elastance $1/C_{ve}$ across all groups, with non-normalized variances on the order of $10^4$. Large variation is also seen in viscoelastic resistance $R_{ve}$ in the D0 ETX group. The mean optimized value for k for the D7 group decreases with increased ventilation for both the SAL and ETX groups. In contrast, $c_{F_D}$ decreases from P20 to P24 in the D7 SAL group but increases in the D7 ETX group.

Fig. 11

Normalized optimal parameter values by rat group. We show mean values (blue circles), standard deviation range (red lines), and maximum and minimum values (black squares) for each group

3.4.2 Biomechanical Metrics

Figure 12 shows comparisons to tissue damping coefficient and pulmonary system elastance, two biomechanical metrics obtained from Mandell et al. [2] who fit the constant phase model to forced oscillation technique data. We adopt their notation as G for tissue damping coefficient and H for pulmonary system elastance. Total pulmonary system elastance $H = 1/C_{\text{total}}$ was calculated in this study from lung compliance $C_A$ and viscoelastic compliance $C_{ve}$ in series by

$$\displaystyle \begin{aligned} {} \frac{1}{C_{\text{total}}} = \frac{1}{C_A}+\frac{1}{C_{ve}}, \end{aligned} $$

(35)

(see also Fig. 2). Since $C_A$ is a dynamic variable (cf. Sect. 2.2), the time-averaged lung compliance for each rat is computed via

$$\displaystyle \begin{aligned} C_{A_{\text{avg}}} = \frac{1}{T-t_0}\int_{t_0}^T C_A \,dt \end{aligned} $$

(36)

using optimal parameter values. Pulmonary system compliance/elastance is calculated from mean compliances for each group via Eq. (35). Viscoelastic resistance $R_{ve}$ serves as a proxy for tissue damping, G. We observe in Fig. 12a,b that compared to the experimentally estimated H, average $1/C_{\text{total}}$ values are about 50% larger for D0 rats and approximately one order of magnitude larger for D7 rats. Figure 12c,d show closer agreement between $R_{ve}$ and G, especially for the D7 rats.

Fig. 12

Comparisons of optimized parameter values to estimated quantities from Mandell et al. [2]. (a)–(b) Comparing the averaged pulmonary system elastance from Eq. (35) in the current model ($1/C_{\text{total}}$) to the pulmonary system elastance from Mandell (H) for (a) day 0 and (b) day 7 rats. (c)–(d) Comparing the averaged viscoelastic resistance from the current model ($R_{ve}$) to the tissue damping from Mandell (G) for (c) day 0 and (d) day 7 rats. Simulated values are plotted as blue triangles, and experimental values are plotted as red circles

3.4.3 Sensitivity Analyses

The sensitivity analysis (Sect. 3.2) addresses how model parameters affect the slopes of the PV curve. We considered recruitment and derecruitment slopes between pressures of 0 and 10 cm H${ }_2$O and between 10 and 30 cm H${ }_2$O ($sl_R$ 0–10, $sl_D$ 0–10, $sl_D$ 10–30, and $sl_D$ 10–30). PV curves simulated using averaged optimal parameter values for each D7 rat group are shown in Fig. 13. Each curve is divided into four regions bounded by the four slopes; divisions are shown by black dashed lines. The parameters that were determined to be sensitive by the Morris screening analysis for each slope of the curve are stated in red.

Fig. 13

Pressure-volume curves computed using averaged optimized parameter values for each group at day 7

Lung elasticity constant k highly influences all four slopes, suggesting that small changes in k can significantly change the shape of the PV curve. The airway opening pressure $c_{F_R}$ was not found to be highly influential for any of the PV slopes and thus is not represented in Fig. 13. In contrast, the airway closing pressure $c_{F_D}$ was determined to be highly influential for the derecruitment slope between 10 and 30 cm H${ }_2$O and slightly less influential for the derecruitment slope between 0 and 10 cm H${ }_2$O. We find $c_{F_D}$ to be higher in the D0 rat groups. For the D7 control group, $c_{F_D}$ decreases as the ventilation pressure increases. However, for the D7 endotoxin group, it increases. The parameter $\beta$ is the baseline fraction of lung recruited and was determined to be influential for all four PV slopes. In every rat group, we notice an increase in $\beta$ from safe ventilation to P20 and P24 ventilation, except for the D7 endotoxin group where it decreases by 10% from safe to P24 ventilation. The viscoelastic elastance $1/C_{ve}$ was only found to be highly influential for the derecruitment slope between 10 and 30 cm H${ }_2$O. A lower viscoelastic elastance flattens the dynamics between the static points, while a higher value makes them more pronounced. Viscoelastic resistance, $R_{ve}$, is highly influential for the derecruitment slopes. The viscoelastic resistance is higher for the D0 rat pups and for the endotoxin rats in both age groups.

3.4.4 Comparisons of Saline Versus Endotoxin

The relative change, as a percent, of the D7 optimal parameter values for the P20 and P24 groups as compared to the SAFE ventilation groups are reported in Table 6. The two levels of injurious ventilation produce relative decreases in k with a larger change for P24 than P20. Similarly, $\gamma$ increases with increasing ventilation for both the SAL and ETX groups, but the increases are small ($\sim 3\%$) and may be considered negligible. The P24 SAL group average $c_{F_D}$ is roughly half the SAFE average; in contrast, the corresponding ETX value is a 50% increase. Although the changes are relatively small overall (up to 10%), the P24 SAL average $\beta$ increases from the SAFE average whereas the same for ETX decreases. The parameter $R_{ve}$ also shows differing responses under SAL versus ETX for P24 ventilation, as $R_{ve}$ is decreased further from that of P20 for the SAL whereas the value increases beyond the SAFE average for ETX.

Table 6 Relative change (%) from SAFE ventilation to VILI for each parameter in the D7 control group (SAL) and D7 endotoxin group (ETX). A red down arrow indicates a decrease from the baseline value, and a blue up arrow indicates an increase. Relative changes are calculated from average parameter values from optimizations taken over each group

3.5 Image Analysis

We developed a customized image analysis procedure to analyze lung histology data in novel ways. Calculations of metrics for 424 histology images (207 for D0 and 217 for D7) of 47 individual rats (23 for D0 and 24 for D7) from [2] are presented below.

3.5.1 Distributions of Image Metrics

We performed model screening and, based on AICc, found the best statistical model. Lumen count per image depends significantly on the day and ventilation strategy ($p<0.01$) but was not found to depend significantly on SAL versus ETX treatment ($p>0.07$) (Fig. 14). The best model (by AICc) for lumen count N per standard-area image is

$$\displaystyle \begin{aligned} N = 48.8 + \left\{ \begin{array}{l} \text{D0} \Rightarrow 0\\ \text{D7} \Rightarrow 22.3 \end{array} \right\} + \left\{ \begin{array}{l} \text{NV} \Rightarrow 6.0\\ \text{SAFE} \Rightarrow 1.7 \\ \text{P20} \Rightarrow -6.0 \\ \text{P24} \Rightarrow -1.7 \end{array} \right\}. {} \end{aligned} $$

(37)

Fig. 14

Distribution of lumen counts categorized by experimental groups

The lumen equivalent radii $R_{\text{equiv}}$ had a strong dependence on postnatal age, with lumens decreasing in size as the lungs developed (Fig. 15). We performed model screening and, based on AICc, found the best statistical model for lumen radius $R_{\text{equiv}}$ to be (in microns)

$$\displaystyle \begin{aligned} R_{\text{equiv}} = 25.4 + \left\{ \begin{array}{l} \text{D0} \Rightarrow 0\\ \text{D7} \Rightarrow -5.5 \end{array} \right\} + \left\{ \begin{array}{l} \text{NV} \Rightarrow 1.2\\ \text{SAFE} \Rightarrow 0.4 \\ \text{P20} \Rightarrow 1.7 \\ \text{P24} \Rightarrow 0 \end{array} \right\}. {} \end{aligned} $$

(38)

Fig. 15

Lumen equivalent radii $R_{\text{equiv}}$ by experimental groups

For the area fraction, we calculated the proportion of each image $\phi$ that is tissue, and for each pruning width w, the proportion $\phi (t)$ of the w-pruned image that is tissue. For each image, tissue area fraction at each pruning width w is shown in Fig. 16 grouped by experimental day and treatment. Most of the tissue is thinner than 40 microns; only a few experimental categories show any amount of tissue thicker than 80 microns, notably D0 ETX P20. At D0, ETX-treated lungs show about 10% greater tissue fraction $\phi (0)$, corresponding to about a 30% increase in tissue area, when compared with the controls (SAL). At D7, the differences between SAL and ETX are minimal, but there is a substantial difference between the most aggressively ventilated lungs (P24) and the others.

Fig. 16

Comparison of the tissue area fraction $\phi (w)$ across the 15 experimental groups for which histology was collected [2]. Here, w is width in microns. At D0, the ETX subgroup shows about 10% greater fraction of tissue. At D7, the differences between SAL and ETX are minimal, but there is a substantial difference between the most aggressively ventilated lungs (P24) and the others

Tissue length $\Lambda$ (per standard image size) was confirmed to have the expected allometric relationship with lumen count N, with total length $\Lambda$ proportional to $\sqrt {N}$ (Fig. 17). Comparison among all experimental groups revealed a significant dependence of tissue mean width w on all categories (Fig. 18). We performed model screening and, based on AICc, found the best model of mean tissue width (in microns) to be

$$\displaystyle \begin{aligned} \bar{w} = 19.3 + \left\{ \begin{array}{l} \text{D0} \Rightarrow 0\\ \text{D7} \Rightarrow -1.7 \end{array} \right\} + \left\{ \begin{array}{l} \text{NV} \Rightarrow -1.3\\ \text{SAFE} \Rightarrow -0.3 \\ \text{P20} \Rightarrow 0.4 \\ \text{P24} \Rightarrow 1.2 \end{array} \right\} + \left\{ \begin{array}{l} \text{SAL} \Rightarrow -0.2\\ \text{ETX} \Rightarrow 0.2 \end{array} \right\} {} \end{aligned} $$

(39)

with the SEM for each parameter ranging from 0.07–0.15 microns. Tissue is seen to thin by a mean of 1.7 microns from D0 to D7. Each pressure increase in the ventilation strategy correspondingly increases tissue width. Interestingly, the least significant variable was SAL versus ETX, which only made a difference at P20. Day and ventilation strategy were more significant.

Fig. 17

Total tissue length, as calculated by the length of centerlines per standard image, goes as $\sqrt {N}$, where N is the adjusted lumen count per image

Fig. 18

Distribution of mean tissue width $\bar {w}$, categorized by experimental groups

The tortuosity metric from Eq. (31) had substantial variation between images, so even the best predictive models had a very small $R^2$. However, there were statistically significant trends (Fig. 19; $p < 0.01$ for SAL/ETX and P24 vs. other ventilation). Notably, there was an increase in tortuosity at the highest ventilation pressures. Model screening determined the best fit to include linear terms for ventilation and SAL versus ETX, as well as an interaction term between those two variables. Postnatal day was not found to be significant.

Fig. 19

Tortuosity ratio by experimental groups ($p < 0.01$ for SAL/ETX and P24 vs. other ventilation). Dependence on postnatal day was negligible. The reference line indicates the tortuosity ratio for standard hex packing

3.6 Correlations Between Optimization Metrics, Image Metrics, and Biomechanical and Inflammatory Markers

The results from our customized modeling and analysis procedures were examined via exploratory data analysis techniques to search for connections between the optimized model parameters, the imaging metrics, and biomarkers determined in [2]. Principal component analysis (PCA) provided a measure of broad relationships between variables, and cluster analysis indicated natural groupings of variables.

3.6.1 Optimized Parameters vs Biomechanical and Inflammatory Markers

PCA was applied to the mean optimized parameter values in the current study (Table 5) together with biomarkers reported by [2] for 10 available groups based on age (D0 or D7), exposure (SAL or ETX), and ventilation type (SAFE, P20, or P24). As seen in both the correlation heat map (Fig. 20) and factor map (Fig. 21), the parameter k, which characterizes the aggregate lung elasticity in the model, is negatively correlated with $\gamma$, the maximum recruitable lung fraction, and the inflammatory metrics IL-6, TNF-a, and CXCL2. Interestingly, k is also included in the sensitive Morris set $\boldsymbol {p}_{\text{Mor}}$ (Eq. 33, Table 4). These generally represent the second principal component PC2 (23%). The parameter $\beta$ describing baseline lung recruitment is strongly correlated with IC and biomechanical compliance biomarkers, Cst. Negatively correlated with $\beta$ are the lung opening pressures $c_F$, viscoelastic parameters $R_{ve}$ and $1/C_{ve}$, and biomechanical markers G and H. These generally comprise the first principal component PC1 (62%). Cst15 and TNF-a strongly correlate with each other but not as strongly with the other variables. Additionally, a cluster analysis revealed that k, $\gamma$, and Cst15 grouped together with inflammatory metrics IL-6, TNF-a, and CXCL2 in the same cluster, while the rest of the variables grouped into a second cluster. The cluster analysis results are not explicitly shown, as they are a subset of the analyses in Sect. 3.6.3.

Fig. 20

Correlations of optimized parameter values with biomechanical and inflammatory markers from [2]. The parameter k, which characterizes aggregate lung elasticity in the model, is negatively correlated with inflammatory markers

Fig. 21

Principal component analysis of optimized parameter values as seen in Table 5 with biomechanical and inflammatory markers from [2]

3.6.2 Image Metrics vs Biomechanical and Inflammatory Markers

We calculated the mean values of image metrics, biomechanical variables, and chemical measures of inflammation for the 14 different groups for which there was data of all three types. The 14 groups are formed by dividing the available data according to the age of the rat (D0 or D7), its exposure to endotoxin (SAL or ETX), and the ventilation pressure that it received (NV, SAFE, P20, or P24) [2]. PCA was applied to the mean values calculated for each category, as seen schematically in the correlation heat map (Fig. 22) and factor map (Fig. 23).

Fig. 22

Heat map of correlations for image, biomechanical, and inflammatory variables

Fig. 23

Principal component analysis of image, biomechanical, and inflammatory metrics: first two principal components

The biomechanical measures of compliance, such as Cst, are strongly correlated among themselves and with inspiratory capacity IC. This is consistent with the findings in [2]. The measures of tissue stiffness, pulmonary system elastance H, tissue damping coefficient G, and central airway resistance $R_N$ are all strongly correlated with each other and inversely correlated with measures of compliance and IC. Among the image metrics, we observe a strong correlation between $R_{\text{equiv}}$ and MLI measurements. Between the image metrics and biomechanical variables, biomechanical measures of tissue stiffness show a strong positive correlation with $R_{\text{equiv}}$ and a strong negative correlation with the number of lumens. Therefore, the first principal component PC1, accounting for almost half (45%) of the variation (Fig. 23), can be described loosely as the compliance versus resistance axis. In terms of image metrics, this axis can also be described as many small lumens versus few large lumens.

In addition to these findings regarding the biomechanical data, we note a strong correlation between all the inflammatory markers. The biomechanical measures of compliance and IC are strongly correlated with TNF-a and less correlated with the other inflammatory markers. Each of the inflammatory markers has its highest correlation, among the measures of compliance, with $\mathrm {Cst}_{15}$. Tortuosity $\tilde {T}$ is strongly correlated with tissue area fraction $\phi (0)$ and not strongly correlated with most other variables. The inflammatory variables are generally weakly correlated with the image metrics, with the following exceptions. IL-6 is moderately correlated with $\tilde {T}$, $\phi (0)$, and $\phi (20)$. $\tilde {T}$ is the best predictor for IL-6. IL-6 is moderately negatively correlated with MLI. TNF-a is moderately correlated with $\tilde {T}$, $\phi (0)$, and $\phi (20)$. It is also moderately negatively correlated with $R_{\text{equiv}}$ and MLI. CXCL2 is moderately correlated with tissue width $\bar {w}$, $\phi (20)$, and $\phi (40)$. Therefore, the second principal component, PC2 (29%), is to a great extent a broad representation of tissue width, which is a key identifier of inflammation.

The cluster analysis on this data set produced five clusters to explain 82% of the variation among the 14 experimental groups. The clusters are, in order of proportion of variation explained,

• $C_1$: [$\mathrm {Cst}_{3}$, IC, $\mathrm {Cst}_{6}$, $-G$, $-H$]

• $C_2$: [$R_{\text{equiv}}$, $-N$, $R_N$, $- \phi (0)$, $R_{\text{midpoint}}$, MLI]

• $C_3$: [$\phi (60)$, $\phi (80)$, $\phi (100)$, $\bar {w}$]

• $C_4$: [TNF-a, $\mathrm {Cst}_{15}$, IL-6, CXCL2]

• $C_5$: [$\phi (20)$, $\phi (40)$, $\tilde {T}$]

Here, the minus signs signify negative cluster coefficients. Within each cluster, variables are ordered by their component size.

Cluster $C_1$ is a measure of low-pressure compliance, or inversely, elastance H and damping G. Notably, airway resistance $R_N$ appears in cluster $C_2$, which is otherwise a measure of lumen size. Cluster $C_3$ represents tissue width. Cluster $C_4$ represents inflammatory markers and high-pressure compliance. Cluster $C_5$ represents tissue thinness and tortuosity.

3.6.3 All Variables

There were 10 groups for which we had image metrics, biomechanical markers, inflammatory markers, and P-V data for parameter optimization. We analyzed means for each of these 10 groups for metrics obtained by the four different approaches and applied PCA and cluster analysis. The strongest correlations ($\pm 0.7$) were between 1/$C_{ve}$ and both $R_{\text{equiv}}$ and $R_{\text{midpoint}}$ (positive correlation), as well as with N (inverse correlation), which are strongly correlated (positively and negatively) with each other. The other strong correlation was between $R_{ve}$ and $\phi (60)$.

We also performed a cluster analysis on this data set, spanning the four major approaches. Four clusters serve to explain 82% of the variation among these 10 experimental groups. The clusters are, in order of proportion of variation explained,

• $K_1$: [H, G, $-$$\mathrm {Cst}_{3}$, $-$$\mathrm {Cst}_{6}$, $R_{ve}$, IC, $-\beta$, $c_{F_D}$, $1/C_{ve}$, $c_{F_R}$, $R_N$]

• $K_2$: [$\phi (0)$, $-$MLI, $\phi (20)$, $-R_{\text{midpoint}}$, $-R_{\text{equiv}}$, $\tilde {T}$, N]

• $K_3$: [$-k$, $\gamma$, CXCL2, $\mathrm {Cst}_{15}$, IL-6]

• $K_4$: [$\phi (60)$, $\phi (80)$, $\phi (100)$, $\bar {w}$, $\phi (40)$]

Here, the minus signs signify negative cluster coefficients.

Notably, cluster $K_1$ includes only biomechanical variables, but these span both the Mandell et al. [2] data and the optimized model parameters. Thus $K_1$ represents the measured and fitted biomechanics. Clusters $K_2$ and $K_4$ contain only metrics of histology. Of these, cluster $K_4$ is a measure of thickened regions of tissue, and cluster $K_2$ is a measure of thin, tortuous, and numerous septa between small lumens. Cluster $K_3$ contains inflammatory markers, the two optimized parameters k and $\gamma$, and $C_{st6}$.

4 Discussion

Mandell et al. [2] found that the rat lungs subjected to the most injurious ventilation pressure (P24) exhibited decreased compliance and increased stiffness compared to lower ventilation pressure (P20). Our model is able to replicate their findings and show the counter-intuitive pressure-volume curve trend for the highest ventilation (Fig. 13). The large variance in $C_{ve}$ as seen in Table 5 and Fig. 11 suggests that the optimizer was not able to uniquely identify this parameter despite the sensitivity analysis findings. The structure of Eq. (6b) suggests that the viscoelastic parameters $C_{ve}$ and $R_{ve}$ have similar effects on the dynamics of $P_{ve}$, and the correlation heat map (Fig. 20) indicates a strong correlation with each other. While the PV curves are adequately replicated, this observation suggests that only one of the two viscoelastic parameters is observable.

4.1 Compartmental Model Analyses

The sensitivity analyses (Sects. 3.2–3.3) identified the key parameters of the fractional recruitment function k, $\gamma$, and $\beta$ as sensitive, though they are correlated with different variables from the compartmental model and the data. The parameter k, which characterizes the aggregate lung elasticity, is a highly sensitive model parameter. A lower k value models a less compliant or stiffer lung, meaning higher pressure must be applied to expand and fill the lung for a given volume of air. While k does not strongly depend on the age of the rat, we observe k trending down as more aggressive ventilation is applied. We note that k and $\gamma$ have no direct analog in the biomechanical or inflammatory markers from Mandell et al. [2]; however, they were anti-correlated (k) and correlated ($\gamma$) with inflammatory markers CXCL2 and IL-6. This suggests a strong connection between the level of inflammation and the ability of the lungs to inflate, as expected. The parameter $\beta$, the recruitment fraction at $P = 0$ cm H${ }_2$O, is anti-correlated with biomechanical markers H, G, and $R_n$ from [2], suggesting that increases in these markers might oppose a healthy lung status. Of the Morris sensitive set $\boldsymbol {p}_{\text{Mor}}$ (Eq. 33), the mean opening airway pressures $c_{F_R}$ and $c_{F_D}$ were the least sensitive across the six metrics but were strongly correlated with optimized parameters and data markers representing biomechanics.

The relative changes in average optimized parameter values from SAFE ventilation to P20 and P24 for D7 rats (Table 6) are of particular interest because these may connect to the counter-intuitive nature of the PV curves from the observed data [2]. The direction of change differs noticeably between SAL and ETX groups for $c_{F_D}$, $\beta$, and $R_{ve}$. Generally, a lower mean opening pressure is a sign of a healthier respiratory system. Increased mechanical ventilation should aid in opening the lungs and reduce $c_{F_D}$. However, for the ETX group, when P24 ventilation has been applied, there is subsequently a much greater pressure required to open the lungs. This was speculated to occur due to alveolar flooding and increased elastance [2], which is consistent with our parameterization. The contrasting relative changes in $\beta$ may not be surprising, as an unhealthy lung further stressed by a high level of injurious ventilation may not be able to recruit as large a fraction of alveoli at zero pressure. The inverse relationship between the relative changes in $R_{ve}$ for P24 SAL and ETX may suggest that a stretching limit has been reached in the latter case. For the ETX group, the unhealthy and injured lung may be more resistant to expansion, and therefore a higher resistance is observed. Interestingly, the relative changes for the P20 SAL and ETX groups are all of the same sign, although they can differ significantly in magnitude. This may reinforce the findings of Mandell et al. [2], who observed such dynamics in their experiments.

4.2 Image Analysis

The customized image metrics from this study, in some cases, confirm other related measurements. The lumen metrics $R_{\text{equiv}}$ and $R_{\text{mid}}$ are seen to be highly correlated with MLI (Sect. 3.6.2), which was used in [2]. This makes intuitive sense and provides alternate methods for quantifying lumen size. Moreover, the image processing via segmentation that is summarized in $R_{\text{equiv}}$ and $R_{\text{mid}}$ provides a route to a more detailed analysis of lumen sizes and shapes, and their distribution, for future histological analysis.

Other image metrics in our analysis quantitatively reveal patterns in the experimental groups that may have only been reported qualitatively. For example, from our model fitting, we see a clear developmental signature in the increase in the lumen count N from D0 to D7 (Fig. 14 and Eq. 37) and the decrease in lumen radius $R_{\text{equiv}}$ from D0 to D7 (Fig. 15, Eq. 38), which correspond to the process of secondary septation. We can also see quantitatively, from the model fit for N, that secondary septation is inhibited by aggressive ventilation; our statistical fit from Eq. (37) shows that P20 ventilation results in approximately 3 days’ developmental delay relative to nonventilated pups. Our measurements of the distribution of tissue width for each treatment group (Fig. 18) show small but statistically significant differences in the quantitative histology by Eq. (39).

Similarly, we found that tortuosity shows modest increases with endotoxin treatment and/or higher-pressure ventilation, though, surprisingly, it is not seen to be dependent on the developmental day (Fig. 19). That our tortuosity metric does not correlate with developmental day, despite our expectation of increasing parenchymal surface area, suggests that either the differences are not significant at these specific stages, or that they are not visible at the length scale of the imaging or with the sample preparation techniques used. It may also suggest that changes in crimp (tortuosity) are histologically insignificant relative to the much larger changes observed in lumen count and size (Figs. 14 and 15).

The correlations (Sect. 3.6.2) between the image metrics and each other are somewhat expected, but the correlations (Figs. 22 and 23) between image metrics and biomechanical and inflammatory metrics from Mandell et al. [2] are to some extent surprising. The first principal component PC1 and the first two variable clusters $C_1$ and $C_2$ show a relationship between the biomechanical variables from [2] and lumen size (or number, which is inversely related to lumen size). Larger lumens, which might be expected to provide less fluid drag, and hence less resistance, were instead associated with greater airway resistance, damping, and elastance and lower compliance. Tortuosity, which would be expected to reflect the nonlinearity of the typical stress-strain curve, was not here found to be correlated with mechanical parameters. It is only one of several factors in tissue compliance, along with tissue width and material elasticity. It was, however, highly correlated with inflammatory marker IL-6.

4.3 Relationship Between Approaches

Examining the full set of metrics using PCA and cluster analysis was the first step toward generating hypotheses about possible relationships between compartmental model parameters and lung tissue histology metrics. Viscoelastic elastance (reciprocal of compliance) was strongly correlated with lumen metrics; however, the variance was large enough to suspect that the values for compliance were not uniquely identified. The other strong correlation was between viscoelastic resistance and tissue area fraction with a pruning diameter of 60 microns. This significance is unclear; however, it is notable that the largest correlations were with the viscoelastic parameters.

The cluster analysis identified groupings that may hold significance. Cluster $K_1$ grouped all the biomechanical metrics together regardless if derived from the data or from the compartmental model. Some of the related correlations as discussed earlier suggest that the model parameters could act as surrogates for the experimentally derived biomechanical markers. An additional biomechanical marker $C_{st15}$, the compliance of the PV curve during derecruitment at a pressure of 15 cm H${ }_2$O, was grouped with the inflammatory markers in a separate cluster $K_3$ along with k and $\gamma$. The compliances $C_{st3}$ and $C_{st6}$ represent compliance at low pressures of 3 and 6 cm H${ }_2$O, whereas $\gamma$ impacts the curve at high pressures and k impacts the entire pressure range. This may indicate that inflammation has a greater effect on PV dynamics at higher pressures.

The separate correlation and cluster analyses were done because of different numbers of experimental groups for the P-V fitting than for the images. The results of the separate analyses are consistent, except that cluster $C_2$ reveals a relationship between the image metrics and one of the biomechanical variables, $R_N$.

4.4 Extensions and Clinical Implications

A goal of this work was to apply mathematical techniques to a neonatal rat model of chorioamnionitis and VILI to better understand the mechanisms of breathing and quantify differences between healthy and diseased groups in a challenging population. To this end, our approaches focus on what is feasible given the available experimental data [2]. This allows for several extensions given additional measurement types. Here, we detail potential future steps and subsequent clinical implications.

The image analysis metrics could be inputs into an augmentation of our compartment model. However, since these metrics cannot be obtained from human subjects except postmortem, the clinical applications of such a pipeline remain unclear. At present, we envision that our model optimization could be applied to recorded pressure-volume data from human patients, and then lung histology relationships could be inferred based on our identified correlations with model parameter values. For example, if an optimal parameter value that is positively correlated with inflammatory markers is high, this suggests that a scan of the patient’s lung might show inflammation.

Our results suggest trends in safe versus injurious ventilation between healthy and unhealthy lungs that could be of clinical interest, but additional work is needed to verify these hypotheses, including a validation with a significantly larger data set. Analysis of our optimal parameters identified that for the ETX group with P24 ventilation, a much greater pressure is needed to open the lungs than for a healthy rat and that a stretching limit may be reached. Together these confirm the need for caution during ventilation of neonates that have mothers with histories of chorioamnionitis or other infections during pregnancy in order to prevent BPD and other respiratory conditions.

4.5 Limitations

The Morris screening and coarse local sensitivity analysis are both conducted using scalar model outputs, whereas a classical gradient-based local sensitivity analysis would calculate a sensitivity index across the full time course of data. Given the sizeable number of rats and associated data sets, the latter analysis was out of the scope of this study to perform on each rat pup. Thus, it is possible that parameters affected the scalar outputs differently than the quasi-static points of the pressure-volume loop, and the related optimizations in Sect. 3.4 might be based on an incomplete understanding of parameter sensitivities. In future work, an optimization algorithm could be formulated in which the objective function is weighted based on the WOB or $v_{\max }$. Further, the Morris screening uses the mean ranking in Eq. (21) as a sensitivity threshold [31, 32]; we note that other options are available, such as using 5% of the maximum rank. Using this alternative method on our Morris rankings results in eleven out of thirteen parameters deemed sensitive, rather than the eight that we report in Sect. 3.2. The subjectivity of this choice allows for other interpretations of relative parameter importance; Colebank and Chesler [24] state the need for a consistent selection method.

It is expected that inflammation from VILI or infections stiffens lung tissue by increasing resistance and decreasing compliance. Indeed, we saw increases in the viscoelastic resistance parameter $R_{ve}$ between most saline and endotoxin groups, but only one difference was statistically significant (Day 0, SAFE ventilation, $p < 0.04$, two-sample t-test with unequal variances). As previously mentioned, $C_{ve}$ and $R_{ve}$ may not be uniquely identifiable by our optimization; therefore, an important next step is to independently measure or estimate one quantity and re-run the optimizer.

We developed a new correction method for counting objects (lumens) that are clipped by the image edge. Our correction was, for simplicity, based on an assumption of monodispersity (equal sizes). It improves the accuracy of lumen count and other metrics based on lumen count, even though the lung lumens are quite polydisperse. A more detailed correction method might use a kind of bootstrapping to estimate lumen size distribution and, therefore, the size distribution of clipped lumens. Our image analysis protocols did not make a distinction between alveoli, alveolar ducts, bronchioli, and blood vessels, on the assumption that these other structures comprise a relatively small proportion of each image and can be neglected. Our quantitative analysis could potentially be improved by performing additional segmentation on the images to identify these structures and consider them separately. For example, by not separately segmenting the blood vessels, they contribute to the quantification of the non-vascular tissue and may skew the results.

The biggest challenges with quantitative image analysis reside in the image acquisition. Our image metrics were defined in a relatively straightforward fashion but can be thwarted by fields of view that encompass too few alveoli. Our estimates of the corrections for lumen counts, areas, etc. assumed regular, convex, and uniform lumen shapes and sizes. The actual lumens in lung slices are far from regular, convex, and uniform. If lumens are of even moderate size relative to the image size (as in Fig. 4a), most—possibly even all—will be clipped by the edge of the image. Lumens clipped by the edge may appear small and distinct but may actually be fingers of the same larger lumen. These considerations complicate the estimation of true lumen counts and sizes. An alternative approach would ignore all lumens clipped by the image edge and only measure lumens with complete edges. However, this approach is again complicated by the presence of lumens that are moderate in size relative to the image, which will skew the statistics.

Ideally, each image would be large relative to the lumens it contains, but that is not always possible, either due to imaging constraints or due to the particular slice or lung itself. An additional factor out of our control is sample preparation. The tissue in this study was fixed at 20 cm H${ }_2$O and, therefore, shows a tortuosity characteristic of a specific portion of the breathing cycle. Different inflation states at fixation would certainly be expected to alter most of our metrics, including lumen count, tissue width, and tortuosity.

4.6 Conclusions

We applied parameter estimation to a compartment model of pressure-volume lung dynamics and created novel image analysis metrics in an attempt to better understand the mechanisms of stiffening and inflammation and affected locations within the pulmonary structure. Importantly, our optimizations identified key parameter differences between healthy and unhealthy groups in data from a neonatal rat model from Mandell et al. [2] that may suggest the mechanisms of VILI in infected respiratory systems. Further, combined analyses of the two strategies identified correlations between inflammatory markers and model parameters with no analog in the data, suggesting that mathematical approaches provide an important path toward understanding VILI and infection.