A simulation study of diffusion-convection interaction and its effect on multiple breath washout indices

Structure-function in smokers: when a small airways test really reflects the small airways

If multiple-breath washout (MBW)-derived acinar ventilation heterogeneity (Sacin) really represents peripheral units, the N2 phase-III of the first MBW exhalation should be curvilinear. This is essentially due to the superposed effect of gas diffusion and convection resulting in an equilibration of N2 concentrations between neighboring lung units throughout exhalation. We investigated this in smokers with computed tomography (CT)-proven functional small airway disease. Instantaneous N2-slopes were computed over 40-ms intervals throughout phase-III and normalized by mean phase-III N2 concentration. N2 phase-III (concave) curvilinearity was quantified as the rate at which the instantaneous N2-slope decreases past the phase-II peak over a 1-s interval; for a linear N2 phase-III unaffected by diffusion, this rate would amount to 0 L-1/s. N2 phase-III curvilinearity was obtained on the experimental curves and on existing model simulations of N2 curves from a normal peripheral lung model and one with missing terminal bronchioles (either 50% or 30% TB left). In 46 smokers [66 (±8) yr; 49 (±26) pack·yr] with CT-based evidence of peripheral lung destruction, instantaneous N2-slope decrease was compared between those with (fSAD+fEmphys) > 20% [-0.26 ± 0.14 (SD) L-1/s; n = 24] and those with (fSAD+fEmphys) < 20% [-0.16 ± 0.12 (SD) L-1/s; n = 22] (P = 0.014). Experimental values fell in the range predicted by a realistic peripheral lung model with progressive reduction of terminal bronchioles: values of instantaneous N2-slope decrease obtained from model simulations were -0.09 L-1/s (normal lung; 100% TB left), -0.17 L-1/s (normal lung 50% TB left), and -0.29 L-1/s (30% TB left). In smokers with CT-based evidence of functional small airway alterations, it is possible to demonstrate that Sacin really does represent the most peripheral airspaces.NEW & NOTEWORTHY In smokers with computed tomography-based evidence of functional small airway alterations by parametric response mapping, it is possible to demonstrate that the multiple-breath washout-derived Sacin, an index of acinar ventilation heterogeneity, actually does represent the most peripheral airspaces. This is done by verifying on experimental N2 washout curves of the first breath, N2 phase-III concavity predicted by the diffusion-convection interdependence model.

Algebraic approximation of the distributed model for the pressure drop in the respiratory airways

The complexity of the human respiratory system causes that one of the main methods of analyzing the dynamic pulmonary phenomena and interpreting experimental results are simulations of its computational models. Among the most compound elements of these models, apart from the bronchial tree structure, is the phenomenon of flow limitation in flexible bronchi, which causes them to collapse with increasing flow, thus their properties, such as resistance, compliance and inertance, are highly nonlinear and time-varying. Commonly, this phenomenon is ignored, or a distributed model for the airway pressure drop is applied, simulated with a modified numerical solver of this differential equation (ODE). The disadvantages of this solution are the problems with taking into account the inherent singularity of the model and the long computation time due to iterative nature of the ODE procedure. The aim of the work was to derive an algebraic approximation of this distributed model, suitable for implementation in continuous dynamic models, to validate it by comparing the results of simulations with the respiratory system model including approximate and original (ODE solver) numerical procedures, as well as to evaluate possible acceleration of calculations. All simulations, including spontaneous breathing, mechanical ventilation with the optimal ventilatory waveform and forced expiration, proved that algebraic approximation yielded results negligibly differing from the ODE solution, and shortened the computation time by an order. The proposed approach is an attractive alternative in the case of computer implementations of pulmonary models, where simulations of flows and pressures in the complex respiratory system are of primary importance.

Various reduced-order surrogate models for fluid flow and mass transfer in human bronchial tree

The bronchial tree plays a main role in the human respiratory system because the air distribution throughout the lungs and gas exchange with blood occur in the airways whose dimensions vary from several centimeters to micrometers. Organization of about 60,000 conducting airways and 33 million respiratory airways in a limited space results in a complex structure. Due to this inherent complexity and a high number of airways, using target-oriented dimensional reduction is inevitable. In addition, there is no general reduced-order model for various types of problems. This necessitates coming up with an appropriate model from a variety of different reduced-order models to solve the desired problem. Lumped formulation, trumpet, or typical path model of whole or parts of bronchial tree are frequently used reduced-order models. On the other hand, using any of these models results in underestimation of flow heterogeneity leading to inaccurate prediction of the systems whose mechanisms depend on the fluid heterogeneity. In this study, a simple robust model combining mechanistic and non-mechanistic modeling approaches of the bronchial tree is proposed which overcomes the limitations of the previous reduced-order models and gives the same results of a detailed mechanistic model for the first time. This model starts from an accurate multi-branching model of conducting and respiratory airways (i.e., the base model) and suggests a proxy model of conducting airway and reduced-order model of respiratory airways based on the base model to significantly reduce computational cost while retaining the accuracy. The combination of these models suggests various reduced-order surrogate models of the human bronchial tree for different problems. The applications and limitations of each reduced-order model are also discussed. The accuracy of the proposed model in the prediction of fluid heterogeneity has been examined by the simulation of multi-breath inert gas washout because the alveolar slope is the reflection of fluid heterogeneity where the computational time decreases from 121 h (using the base model) to 4.8 s (using the reduced-order model). A parallel strategy for solving the equations is also proposed which decreases run time by 0.18 s making the model suitable for real-time applications. Furthermore, the ability of the model has been evaluated in the modeling of asthmatic lung as an instance of abnormal lungs, and in the modeling of O2-CO2 exchange as an instance of nonlinear reacting systems. The results indicate that the proposed model outperforms previous models based on accuracy, robustness, and run time.

Ventilation heterogeneity in smokers: role of unequal lung expansion and peripheral lung structure

Smoking-induced ventilation heterogeneity measured at the mouth via established washout indices [lung clearance index (LCI) and alveolar mixing efficiency (AME)] potentially results from unequal expansion, which can be quantified by computer tomography (CT), and structural changes down to the lung periphery, characterized by CT parametric response mapping indices [percentage of lung affected by functional small airway disease (PRM^fSAD) and emphysema (PRM^Emph)]. By combining CT imaging and nitrogen (N₂) washout tests in smokers, we specifically examined the roles of unequal lung expansion and peripheral structure. We first extracted three-dimensional maps of local lung expansion from registered inspiratory/expiratory CT images in 50 smokers (GOLD 0-IV) to compute for each smoker the theoretical N₂ washout concentration curve solely attributable to unequal local expansion. By a head-on comparison with washout N₂ concentrations measured at the mouth in the same smokers supine, we observed that 1) LCI increased from 4.8 ± 0.2 (SD) to 6.6 ± 0.8 (SD) due to unequal lung expansion alone and further increased to 9.0 ± 1.5 (SD) independent of local expansion and 2) AME decreased (from 100% by definition) to 95 ± 2 (SD)% due to unequal expansion alone and further decreased to 75 ± 7(SD)% independent of local expansion. In a multiple regression between the washout indices and CT-derived PRM^fSAD and PRM^Emph, LCI was related to PRM^fSAD (r = +0.58; P < 0.001), whereas AME was related to both PRM^fSAD (r_partial = -0.44; P = 0.002) and PRM^Emph (r_partial = -0.31; P = 0.033), in line with AME being dominated by alterations in peripheral structure. We conclude that smokers showing an increased LCI without corresponding AME decrease are predominantly affected by unequal lung expansion, whereas an AME decrease with a commensurate LCI increase indicates a smoking-induced alteration of peripheral structure.NEW & NOTEWORTHY A head-on comparison between imaging and multiple breath washout in supine smokers shows that computer tomography-measured unequal local lung expansion accounts for 50% or less of smoking-induced increase in ventilation heterogeneity. The contributions from unequal lung expansion and peripheral structure to the two main washout indices also explain their respective association with parametric response mapping indices.

Unraveling Disease Pathophysiology with Mathematical Modeling

Modeling has enabled fundamental advances in our understanding of the mechanisms of health and disease for centuries, since at least the time of William Harvey almost 500 years ago. Recent technological advances in molecular methods, computation, and imaging generate optimism that mathematical modeling will enable the biomedical research community to accelerate its efforts in unraveling the molecular, cellular, tissue-, and organ-level processes that maintain health, predispose to disease, and determine response to treatment. In this review, we discuss some of the roles of mathematical modeling in the study of human physiology and pathophysiology and some challenges and opportunities in general and in two specific areas: in vivo modeling of pulmonary function and in vitro modeling of blood cell populations.

A multi-scale model of gas transport in the lung to study heterogeneous lung ventilation during the multiple-breath washout test

The multiple-breath washout (MBW) is a lung function test that measures the degree of ventilation inhomogeneity (VI). The test is used to identify small airway impairment in patients with lung diseases like cystic fibrosis. However, the physical and physiological factors that influence the test outcomes and differentiate health from disease are not well understood. Computational models have been used to better understand the interaction between anatomical structure and physiological properties of the lung, but none of them has dealt in depth with the tracer gas washout test in a whole. Thus, our aim was to create a lung model that simulates the entire MBW and investigate the role of lung morphology and tissue mechanics on the tracer gas washout procedure. To this end, we developed a multi-scale lung model to simulate the inert gas transport in airways of all size. We then applied systematically different modifications to geometrical and mechanical properties of the lung model (compliance, residual airway volume and flow resistance) which have been associated with VI. The modifications were applied to distinct parts of the model, and their effects on the gas distribution within the lung and on the gas concentration profile were assessed. We found that variability in compliance and residual volume of the airways, as well as the spatial distribution of this variability in the lung had a direct influence on gas distribution among airways and on the MBW pattern (washout duration, characteristic concentration profile during each expiration), while the effects of variable flow resistance were negligible. Based on these findings, it is possible to classify different types of inhomogeneities in the lung and relate them to specific features of the MBW pattern, which builds the basis for a more detailed association of lung function and structure.

Obstructive lung diseases, like cystic fibrosis or primary ciliary dyskinesia, lead to inhomogeneous ventilation. The degree of observed inhomogeneity represents a clinical measure for the progression of the disease. The multiple-breath washout (MBW) is a lung function test that measures this inhomogeneity in the lung. However, the factors that influence the results of the test and differentiate between health and disease are not well understood. Computational models help us to understand better the relation between anatomical structure and physiological properties of the lung, but none of them has dealt in depth with the MBW test in whole. Our aim was to create a lung model that simulates the entire MBW test and study the role of lung structure and tissue mechanics on the washout procedure. We developed a multi-scale lung model to simulate the inert gas transport in all airways including the gas exchange area. Our model offers the opportunity to understand the ventilation distribution in the healthy lung. It can also mimic certain patterns of lung disease by applying modifications in mechanical properties out of the physiological limits. Thus, it can be used to study MBW characteristics in health and disease.