Comparison of optimization parametrizations for regional lung compliance estimation using personalized pulmonary poromechanical modeling

Interstitial lung diseases, such as idiopathic pulmonary fibrosis (IPF) or post-COVID-19 pulmonary fibrosis, are progressive and severe diseases characterized by an irreversible scarring of interstitial tissues that affects lung function. Despite many efforts, these diseases remain poorly understood and poorly treated. In this paper, we propose an automated method for the estimation of personalized regional lung compliances based on a poromechanical model of the lung. The model is personalized by integrating routine clinical imaging data – namely computed tomography images taken at two breathing levels in order to reproduce the breathing kinematic—notably through an inverse problem with fully personalized boundary conditions that is solved to estimate patient-specific regional lung compliances. A new parametrization of the inverse problem is introduced in this paper, based on the combined estimation of a personalized breathing pressure in addition to material parameters, improving the robustness and consistency of estimation results. The method is applied to three IPF patients and one post-COVID-19 patient. This personalized model could help better understand the role of mechanics in pulmonary remodeling due to fibrosis; moreover, patient-specific regional lung compliances could be used as an objective and quantitative biomarker for improved diagnosis and treatment follow up for various interstitial lung diseases.

     

Airway mechanics, gas exchange, and blood flow in a nonlinear model of the normal human lung

Amodel integrating airway/lung mechanics, pulmonary blood flow, and gasexchange for a normal human subject executing the forced vital capacity(FVC) maneuver is presented. It requires as input the intrapleuralpressure measured during the maneuver. Selected model-generated outputvariables are compared against measured data (flow at the mouth, changein lung volume, and expired O₂ and CO₂concentrations at the mouth). A nonlinear parameter-estimation algorithm is employed to vary selected sensitive model parameters toobtain reasonable least squares fits to the data. This study indicatesthat 1) all three components of the respiratory model arenecessary to characterize the FVC maneuver; 2) changes in pulmonary blood flow rate are associated with changes in alveolar andintrapleural pressures and affect gas exchange and the time course ofexpired gas concentrations; and 3) a collapsible midairway segment must be included to match airflow during a forced expiration. Model simulations suggest that the resistances to airflow offered bythe collapsible segment and the small airways are significant throughout forced expiration; their combined effect is needed toadequately match the inspiratory and expiratory flow-volume loops.Despite the limitations of this lumped single-compartment model, aremarkable agreement with airflow and expired gas concentration measurements is obtained for normal subjects. Furthermore, the modelprovides insight into the important dynamic interactions betweenventilation and perfusion during the FVC maneuver.

     

Dynamic model of the bronchial tree

We present a distributed model of the bronchial tree which simulates the global dynamic characteristics of the lung. Local mechanical characteristics of each airway are represented by RCL circuits and parameters of the electrical components are determined from local physiological data. The bronchi geometry is described by Weibel's symmetric model, the flow in each airway is assumed laminar and mixing effects at the bifurcations are neglected; the transpulmonary pressure is assumed to be sinusoidal. In simulations of quiet breathing the resistance to airflow is found to be dominant, the flow amplitude decreasing as breathing frequency increases, but remaining almost constant in all the generations. Simulations of ventilation through obstructed lungs show frequency dependence of the dynamic characteristics in very compliant lungs. The global resistance to airflow and the dynamic compliance of the bronchi decrease as the forced oscillation frequency increases in a pattern similar to in vivo measurements in diseased lungs. This may be an outcome of the RCL properties of the network rather than due to uneven distribution of mechanical properties of the lung.     

Modeling of weak blast wave propagation in the lung

Blast injuries of the lung are the most life-threatening after an explosion. The choice of physical parameters responsible for trauma is important to understand its mechanism. We developed a one-dimensional linear model of an elastic wave propagation in foam-like pulmonary parenchyma to identify the possible cause of edema due to the impact load. The model demonstrates different injury localizations for free and rigid boundary conditions. The following parameters were considered: strain, velocity, pressure in the medium and stresses in structural elements, energy dissipation, parameter of viscous criterion. Maximum underpressure is the most suitable wave parameter to be the criterion for edema formation in a rabbit lung. We supposed that observed scattering of experimental data on edema severity is induced by the physiological variety of rabbit lungs. The criterion and the model explain this scattering. The model outlines the demands for experimental data to make an unambiguous choice of physical parameters responsible for lung trauma due to impact load.     

Anaerobic digestion model No. 1-based distributed parameter model of an anaerobic reactor: II. Model validation

In this study, an ADM1-based distributed parameter model was validated using experimental results obtained in a laboratory-scale 10 L UASB reactor. Sensitivity analysis of the model parameters was used to select four parameters for estimation by a numerical procedure while other parameters were accepted from ADM1 benchmark simulations. The parameter estimation procedure used measurements of liquid phase components obtained at different sampling points in the reactor and under different operating conditions. Model verification used real time fluorescence-based measurements of chemical oxygen demand and volatile fatty acids at four sampling locations in the reactor. Overall, the distributed parameter model was able to describe the distribution of liquid phase components in the reactor and adequately simulated the effect of external recirculation on degradation efficiency. The model can be used in the design, analysis and optimization of UASB reactors.     

Vascular and airway effects of endogenous cyclooxygenase products during lung inflation

The influence of lung inflation on lung elasticity and pulmonary resistance (RL) and on pulmonary and bronchial hemodynamics was examined in five anesthetized, mechanically ventilated adult sheep before and after treatment with the cyclooxygenase inhibitor indomethacin (2 mg/kg). Lung inflation was accomplished by increasing levels of positive end-expiratory pressure (PEEP). Measurements of pulmonary vascular resistance (PVR), bronchial blood flow (Qbr), and RL were obtained with a Swan-Ganz catheter, with an electromagnetic flow probe placed around the carinal artery, and by relating airflow to transpulmonary pressure (Ptp), respectively. Before indomethacin, increasing PEEP from 5 to 15 cmH2O increased mean lung volume (VL) to 135% (P less than 0.01), Ptp to 165% (P less than 0.005), and PVR to 132% (P less than 0.05) of base line and decreased mean Qbr (normalized for cardiac output) to 53% (P less than 0.05) of base line. Mean RL showed a tendency to decrease with a mean value of 67% of base line at 15 cmH2O PEEP. After indomethacin the corresponding values were 121% for VL, 155% for Ptp, 124% for PVR, 35% for Qbr, and 31% for RL. The PEEP-dependent changes were not different before and after indomethacin except for mean VL, which increased less (P less than 0.05) after indomethacin. The failure of indomethacin to modify PEEP-induced changes in RL, PVR, and Qbr was also present when these parameters were expressed as a function of Ptp. These findings suggest that the cyclooxygenase products elaborated during lung inflation reduce lung elasticity but fail to influence airflow resistance and pulmonary and bronchial hemodynamics.     

Mechanisms of gas exchange response to lung volume reduction surgery in severe emphysema

Lung volume reduction surgery (LVRS) improves lung function, respiratory symptoms, and exercise tolerance in selected patients with chronic obstructive pulmonary disease, who have heterogeneous emphysema. However, the reported effects of LVRS on gas exchange are variable, even when lung function is improved. To clarify how LVRS affects gas exchange in chronic obstructive pulmonary disease, 23 patients were studied before LVRS, 14 of whom were again studied afterwards. We performed measurements of lung mechanics, pulmonary hemodynamics, and ventilation-perfusion (Va/Q) inequality using the multiple inert-gas elimination technique. LVRS improved arterial Po? (Pa(O?)) by a mean of 6 Torr (P = 0.04), with no significant effect on arterial Pco? (Pa(CO?)), but with great variability in both. Lung mechanical properties improved considerably more than did gas exchange. Post-LVRS Pa(O?) depended mostly on its pre-LVRS value, whereas improvement in Pa(O(2)) was explained mostly by improved Va/Q inequality, with lesser contributions from both increased ventilation and higher mixed venous Po(2). However, no index of lung mechanical properties correlated with Pa(O?). Conversely, post-LVRS Pa(CO?) bore no relationship to its pre-LVRS value, whereas changes in Pa(CO?) were tightly related (r2 = 0.96) to variables, reflecting decrease in static lung hyperinflation (intrinsic positive end-expiratory pressure and residual volume/total lung capacity) and increase in airflow potential (tidal volume and maximal inspiratory pressure), but not to Va/Q distribution changes. Individual gas exchange responses to LVRS vary greatly, but can be explained by changes in combinations of determining variables that are different for oxygen and carbon dioxide.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

A model for hypoxic constriction of the pulmonary circulation

The detailed anatomic and biodynamic data provided for the cat lung by Zhuang et al. (J. Appl. Physiol. 55: 1341-1348, 1983) allowed pressure-flow curves for the normal lung to be generated. This model has been modified to permit the stimulation of the pressure and flow distribution effects of hypoxic pulmonary vasoconstriction for a two-compartment lung and generalized to allow comparison with the experimental results from dogs (and probably other species). Hypoxic pulmonary vasoconstriction is simulated by reduction of the initial diameter of the smallest six orders of pulmonary arteries. Expressions are presented that relate the alveolar and mixed-venous O2 tensions to a graded constriction of these vessels. In addition, the diameter of the capillary sheet and the six small arteries is defined with a maximum diameter at a transmural pressure of 20 cmH2O. Pressure-flow curves are derived for any combination of alveolar and mixed-venous O2 tension, alveolar and pleural pressure, left atrial pressure, and hematocrit. The two-compartment model is solved by an iterative procedure to identify the distribution of the flow and the resulting pulmonary arterial pressure when the compartments differ by size, hypoxic constriction, or other imposed conditions. The results of the model are compared with those from a variety of experimental preparations. It is concluded that the model is useful for identifying the quantitative causes of changes in the response to hypoxic pulmonary vasoconstriction and for the exploration of the functional influence of mechanical properties of the vasculature.     

Micropuncture pressure in small arteries or veins of perfused rabbit lungs

Until now, direct micropuncture measurements of vascular pressure in lung have been limited to small vessels less than 100 microns on the pleural surface. On the other hand, direct pressure measurements using small catheters (less than 1-mm OD) in pulmonary vessels have been limited to those greater than 1.2 mm. We measured pressure in intermediate-sized microvessels (300-700 microns) using the micropuncture method in isolated perfused rabbit lungs. These microvessels are located 2 or 3 mm beneath the pleura. We exposed them by microsurgery and punctured the relatively thick-walled vessels with specially configured micropipettes. We exposed one pulmonary microvessel in each rabbit lung by microsurgery on the left middle lobe. In 15 rabbit lungs we measured pressure in a total of six small arteries (275- to 470-microns diam) and nine small veins (300- to 700-microns diam) under high zone 3 conditions, near the zone 2/3 boundary. We found approximately 35% of the total pulmonary vascular pressure drop in arteries greater than 275-microns diam and 7% in veins greater than 300-microns diam. In veins greater than 500-microns diam, there was no measurable pressure drop. After the measurements, we froze the lung and confirmed that there was no detectable interstitial or alveolar edema in the cross sections of the punctured site. Our data are compatible with those of other investigators who have used isolated perfused rabbit lungs under similar experimental conditions.     

Theoretical analysis of occlusion techniques for measuring pulmonary capillary pressure.

We have developed a model including three serial compliant compartments (arterial, capillary, and venous) separated by two resistances (arterial and venous) for interpreting in vivo single pulmonary arterial or venous occlusion pressure profiles and double occlusion. We formalized and solved the corresponding system of equations. We showed that in this model 1) pulmonary capillary pressure (Pc) profile after arterial or venous occlusion has an S shape, 2) the estimation of Pc by zero time extrapolation of the slow component of the arterial occlusion profile (Pcao) always overestimates Pc, 3) symmetrically such an estimation on the venous occlusion profile (Pcvo) always underestimates Pc, 4) double occlusion pressure (Pcdo) differs from Pc. We evaluated the impact of varying parameter values in the model with parameter sets drawn either from the literature or from arbitrary arterial and venous pressures, being respectively 20 and 5 mmHg. Resulting Pcao-Pc differences ranged from 0.4 to 5.4 mmHg and resulting Pcvo-Pc differences ranged from -0.3 to -5.0 mmHg. Pcdo-Pc was positive or negative, its absolute value in general being negligible (< 1.1 mmHg).     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Partitioning of nasal and pulmonary resistance changes during noninvasive plethysmography in mice

Double-chamber plethysmography is a well established noninvasive method of assessing airflow obstruction in small lab animals. It allows measurement of the specific airway resistance (sRaw), which unlike enhanced pause (Penh), is a meaningful airway mechanics parameter. Since sRaw is measured in spontaneously breathing mice, a limitation of the method is the inability to exclude nasal resistance changes. We recently showed that mice are not truly obligate nasal breathers and that after nasal occlusion, nasally breathing mice can transition to an oral mode of breathing. We now show that it is experimentally possible to algebraically separate the average nasal and pulmonary (including laryngeal) components of total airway resistance change by a series of measurements made across groups of mice breathing nasally or orally, assuming that oral resistance remains constant. Using this approach, we show that nasal resistance change comprises one-half or more of the total resistance change during methacholine challenge. Inhibition of mucin secretion from airway goblet cells attenuates pulmonary but not nasal airway hyperresponsiveness (AHR), and nasal AHR in a murine model of rhinitis may be related to edema.     

Pressure in the pulmonary artery and transbronchial electroplethysmography in rats

A complex technique for pulmonary circulation study involving catheterization of pulmonary artery and transbronchial electroplethysmography has been tested in rats. The technique permits pulmonary artery pressure measurement and registration of electrical resistance in the lung lobe of closed-chest animals with the subsequent estimation of blood flow, blood volume and air content in the lungs expressed in adequate units per unit of organ volume. The experimental data characterizing standard values of the above parameters in rats are presented.     

Patient-specific modeling of cardiovascular and respiratory dynamics during hypercapnia

This study develops a lumped cardiovascular–respiratory system-level model that incorporates patient-specific data to predict cardiorespiratory response to hypercapnia (increased CO₂ partial pressure) for a patient with congestive heart failure (CHF). In particular, the study focuses on predicting cerebral CO₂ reactivity, which can be defined as the ability of vessels in the cerebral vasculature to expand or contract in response CO₂ induced challenges. It is difficult to characterize cerebral CO₂ reactivity directly from measurements, since no methods exist to dynamically measure vasomotion of vessels in the cerebral vasculature. In this study we show how mathematical modeling can be combined with available data to predict cerebral CO₂ reactivity via dynamic predictions of cerebral vascular resistance, which can be directly related to vasomotion of vessels in the cerebral vasculature. To this end we have developed a coupled cardiovascular and respiratory model that predicts blood pressure, flow, and concentration of gasses (CO₂ and O₂) in the systemic, cerebral, and pulmonary arteries and veins. Cerebral vascular resistance is incorporated via a model parameter separating cerebral arteries and veins. The model was adapted to a specific patient using parameter estimation combined with sensitivity analysis and subset selection. These techniques allowed estimation of cerebral vascular resistance along with other cardiovascular and respiratory parameters. Parameter estimation was carried out during eucapnia (breathing room air), first for the cardiovascular model and then for the respiratory model. Then, hypercapnia was introduced by increasing inspired CO₂ partial pressure. During eucapnia, seven cardiovascular parameters and four respiratory parameters was be identified and estimated, including cerebral and systemic resistance. During the transition from eucapnia to hypercapnia, the model predicted a drop in cerebral vascular resistance consistent with cerebral vasodilation.     

Parameter estimation in a stochastic model of the tubuloglomerular feedback mechanism in a rat nephron

A key parameter in the understanding of renal hemodynamics is the gain of the feedback function in the tubuloglomerular feedback mechanism. A dynamic model of autoregulation of renal blood flow and glomerular filtration rate has been extended to include a stochastic differential equations model of one of the main parameters that determines feedback gain. The model reproduces fluctuations and irregularities in the tubular pressure oscillations that the former deterministic models failed to describe. This approach assumes that the gain exhibits spontaneous erratic variations that can be explained by a variety of influences, which change over time (blood pressure, hormone levels, etc.). To estimate the key parameters of the model we have developed a new estimation method based on the oscillatory behavior of the data. The dynamics is characterized by the spectral density, which has been estimated for the observed time series, and numerically approximated for the model. The parameters have then been estimated by the least squares distance between data and model spectral densities. To evaluate the estimation procedure measurements of the proximal tubular pressure from 35 nephrons in 16 rat kidneys have been analyzed, and the parameters characterizing the gain and the delay have been estimated. There was good agreement between the estimated values, and the values obtained for the same parameters in independent, previously published experiments.     

Respiratory mechanics in men following a deep air dive

The mechanical properties of the lungs were measured in 10 men before and after a simulated air dive to 285 ft of seawater (87 m). The objective was to determine whether a dive likely to produce pulmonary bubble emboli would alter lung mechanics. Lung function was measured predive and at 1, 2, 3, 6, 7, and 23 h postdive. Measurements of lung function were also made at identical times on a control day when no dive was made. Each set of measurements included precordial Doppler signals, pulmonary resistance, quasistatic lung compliance, forced vital capacity (FVC), forced expired volume after 1.0 s (FEV 1.0), the ratio of FEV 1.0 to FVC (FEV 1.0/FVC%), and maximal airflow after 50 and 75% of the vital capacity had been expired (Vmax50 and Vmax75, respectively). Base-line measurements of pulmonary resistance and quasistatic compliance were normal in all subjects. FVC and FEV 1.0 were greater than predicted for most subjects and were increased proportionately so that the FEV 1.0/FVC% was normal. Following the dive, bubble signals were heard in four subjects, and two subjects had mild symptoms of decompression sickness. No subject demonstrated any alteration in lung function that could be attributed to the dive. We concluded that stressful decompressions capable of producing "silent" pulmonary bubble emboli do not alter lung mechanics.     

Parameter estimation of kinetic models from metabolic profiles: two-phase dynamic decoupling method

MOTIVATION: Time-series measurements of metabolite concentration have become increasingly more common, providing data for building kinetic models of metabolic networks using ordinary differential equations (ODEs). In practice, however, such time-course data are usually incomplete and noisy, and the estimation of kinetic parameters from these data is challenging. Practical limitations due to data and computational aspects, such as solving stiff ODEs and finding global optimal solution to the estimation problem, give motivations to develop a new estimation procedure that can circumvent some of these constraints. RESULTS: In this work, an incremental and iterative parameter estimation method is proposed that combines and iterates between two estimation phases. One phase involves a decoupling method, in which a subset of model parameters that are associated with measured metabolites, are estimated using the minimization of slope errors. Another phase follows, in which the ODE model is solved one equation at a time and the remaining model parameters are obtained by minimizing concentration errors. The performance of this two-phase method was tested on a generic branched metabolic pathway and the glycolytic pathway of Lactococcus lactis. The results showed that the method is efficient in getting accurate parameter estimates, even when some information is missing.