Numerical modelling and analysis of peripheral airway asymmetry and ventilation in the human adult lung

Investigation of tracer gas transport in a new numerical model of lung acini

Obstructive pulmonary diseases are associated with considerable morbidity. For an early diagnosis of these diseases, inert gas washouts can potentially be used. However, the complex interaction between lung anatomy and gas transport mechanisms complicates data analysis. In order to investigate this interaction, a numerical model, based on the finite difference method, consisting of two lung units connected in parallel, was developed to simulate the tracer gas transport within the human acinus. Firstly, the geometries of the units were varied and the diffusion coefficients (D) were kept constant. Secondly, D was changed and the geometry was kept constant. Furthermore, simple monoexponential growth functions were applied to evaluate the simulated data. In 109 of the 112 analyzed curves, monoexponential function matched simulated data with an accuracy of over 90%, potentially representing a suitable numerical tool to predict transport processes in further model extensions. For total flows greater than 5 × 10−4 ml/s, the exponential growth constants increased linearly with linear increasing flow to an accuracy of over 95%. The slopes of these linear trend lines of 1.23 µl−1 (D = 0.6 cm2/s), 1.69 µl−1 (D = 0.3 cm2/s), and 2.25 µl−1 (D = 0.1 cm2/s) indicated that gases with low D are more sensitive to changes in flows than gases with high D.

Graphical abstract

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.

Spectral graph theory efficiently characterizes ventilation heterogeneity in lung airway networks

This paper introduces a linear operator for the purposes of quantifying the spectral properties of transport within resistive trees, such as airflow in lung airway networks. The operator, which we call the Maury matrix, acts only on the terminal nodes of the tree and is equivalent to the adjacency matrix of a complete graph summarizing the relationships between all pairs of terminal nodes. We show that the eigenmodes of the Maury operator have a direct physical interpretation as the relaxation, or resistive, modes of the network. We apply these findings to both idealized and image-based models of ventilation in lung airway trees and show that the spectral properties of the Maury matrix characterize the flow asymmetry in these networks more concisely than the Laplacian modes, and that eigenvector centrality in the Maury spectrum is closely related to the phenomenon of ventilation heterogeneity caused by airway narrowing or obstruction. This method has applications in dimensionality reduction in simulations of lung mechanics, as well as for characterization of models of the airway tree derived from medical images.

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.

Modelling structural determinants of ventilation heterogeneity: A perturbative approach

We have developed a computational model of gas mixing and ventilation in the human lung represented as a bifurcating network. We have simulated multiple-breath washout (MBW), a clinical test for measuring ventilation heterogeneity (VH) in patients with obstructive lung conditions. By applying airway constrictions inter-regionally, we have predicted the response of MBW indices to obstructions and found that they detect a narrow range of severe constrictions that reduce airway radius to 10%–30% of healthy values. These results help to explain the success of the MBW test to distinguish obstructive lung conditions from healthy controls. Further, we have used a perturbative approach to account for intra-regional airway heterogeneity that avoids modelling each airway individually. We have found, for random airway heterogeneity, that the variance in MBW indices is greater when indices are already elevated due to constrictions. By quantifying this effect, we have shown that variability in lung structure and mechanical properties alone can lead to clinically significant variability in MBW indices (specifically the Lung Clearance Index—LCI, and the gradient of phase-III slopes—S_cond), but only in cases simulating obstructive lung conditions. This method is a computationally efficient way to probe the lung’s sensitivity to structural changes, and to quantify uncertainty in predictions due to random variations in lung mechanical and structural properties.

Feasibility of automated slope III and Scond analysis in children

BACKGROUND

Multiple breath washout (MBW) derived Scond is an established index of ventilation inhomogeneity. Time-consuming post hoc calculations of the expirogram's slope of alveolar phase III (SIII) and the lack of available software hampered widespread application of Scond.

METHODS

Seventy-two school-aged children (45 with cystic fibrosis; CF) performed 3 nitrogen MBW. We tested a new automated algorithm for Scond analysis (Scondauto ) which comprised breath selection for SIII detection, calculation and reporting of test quality. We compared Scondauto to (i) standard Scond analysis (Scondmanual ) with manual breath selection and to (ii) pragmatic Scond analysis including all breaths (Scondall ). Primary outcomes were success rate and agreement between different Scond protocols, and Scond fitting quality (linear regression R(2) ).

RESULTS

Average Scondauto (0.06 for CF and 0.01 for controls) was not different from Scondmanual (0.06 for CF and 0.01 for controls) and showed comparable fitting quality (R(2) 0.53 for CF and 0.13 for controls vs. R(2) 0.54 for CF and 0.13 for controls). Scondall was similar in CF and controls but with inferior fitting quality compared to Scondauto and Scondmanual .

CONCLUSIONS

Automated Scond calculation is feasible and produces robust results comparable to the standard manual way of Scond calculation. This algorithm provides a valid, fast and objective tool for regular use, even in children.