Contribution of serial and parallel microperfusion to spatial variability in pulmonary inter- and intra-acinar blood flow

Physics-based in silico modelling of microvascular pulmonary perfusion in COVID-19

Due to its ability to induce heterogenous, patient-specific damage in pulmonary alveoli and capillaries, COVID-19 poses challenges in defining a uniform profile to elucidate infection across all patients. Computational models that integrate changes in ventilation and perfusion with heterogeneous damage profiles offer valuable insights into the impact of COVID-19 on pulmonary health. This study aims to develop an in silico hypothesis-testing platform specifically focused on studying microvascular pulmonary perfusion in COVID-19-infected lungs. Through this platform, we explore the effects of various acinar-level pulmonary perfusion abnormalities on global lung function. Our modelling approach simulates changes in pulmonary perfusion and the resulting mismatch of ventilation and perfusion in COVID-19-afflicted lungs. Using this coupled modelling platform, we conducted multiple simulations to assess different scenarios of perfusion abnormalities in COVID-19-infected lungs. The simulation results showed an overall decrease in ventilation-perfusion (V/Q) ratio with inclusion of various types of perfusion abnormalities such as hypoperfusion with and without microangiopathy. This model serves as a foundation for comprehending and comparing the spectrum of findings associated with COVID-19 in the lung, paving the way for patient-specific modelling of microscale lung damage in emerging pulmonary pathologies like COVID-19.

Predicting Patient Status in Chronic Thromboembolic Pulmonary Hypertension Using a Biophysical Model

Chronic thromboembolic pulmonary hypertension (CTEPH) involves abnormally high blood pressure in the pulmonary vessels and is associated with small vessel vasculopathy and pre-capillary proximal occlusions. Management of CTEPH disease is challenging, therefore accurate diagnosis is crucial in ensuring effective treatment and improved patient outcomes. The treatment of choice for CTEPH is pulmonary endarterectomy, which is an invasive surgical intervention to remove thrombi. Following PEA, a number of patients experience poor outcomes or worse-than-expected improvements, which may indicate that they have significant small vessel disease. A method that can predict the extent of distal remodelling may provide useful clinical information to plan appropriate CTEPH patient treatment. Here, a novel biophysical modelling approach has been developed to estimate and quantify the extent of distal remodelling. This method includes a combination of mathematical modelling and computed tomography pulmonary angiography to first model the geometry of the pulmonary arteries and to identify the under-perfused regions in CTEPH. The geometric model is then used alongside haemodynamic measurements from right heart catheterisation to predict distal remodelling. In this study, the method is tested and validated using synthetically generated remodelling data. Then, a preliminary application of this technique to patient data is shown to demonstrate the potential of the approach for use in the clinical setting.Clinical relevance- Patient-specific modelling can help provide useful information regarding the extent of distal vasculopathy on a per-patient basis, which remains challenging. Physicians can be unsure of outcomes following pulmonary endarterectomy. Therefore, the predictive aspect of the patient's response to surgery can help with clinical decision-making.

An in silico approach to understanding the interaction between cardiovascular and pulmonary lymphatic dysfunction

The lung is extremely sensitive to interstitial fluid balance, yet the role of pulmonary lymphatics in lung fluid homeostasis and its interaction with cardiovascular pressures is poorly understood. In health, there is a fine balance between fluid extravasated from the pulmonary capillaries into the interstitium and the return of fluid to the circulation via the lymphatic vessels. This balance is maintained by an extremely interdependent system governed by pressures in the fluids (air and blood) and tissue (interstitium), lung motion during breathing, and the permeability of the tissues. Chronic elevation in left atrial pressure (LAP) due to left heart disease increases the capillary blood pressure. The consequent fluid accumulation in the delicate lung tissue increases its weight, decreases its compliance, and impairs gas exchange. This interdependent system is difficult, if not impossible, to study experimentally. Computational modeling provides a unique perspective to analyze fluid movement in the cardiopulmonary vasculature in health and disease. We have developed an initial in silico model of pulmonary lymphatic function using an anatomically structured model to represent ventilation and perfusion and underlying biophysical laws governing fluid transfer at the interstitium. This novel model was tested against increased LAP and noncardiogenic effects (increased permeability). The model returned physiologically reasonable values for all applications, predicting pulmonary edema when LAP reached 25 mmHg and with increased permeability.NEW & NOTEWORTHY This model presents a novel approach to understanding the interaction between cardiac dysfunction and pulmonary lymphatic function, using anatomically structured models and biophysical equations to estimate regional variation in fluid transport from blood to interstitial and lymphatic flux. This fluid transport model brings together advanced models of ventilation, perfusion, and lung mechanics to produce a detailed model of fluid transport in health and various altered pathological conditions.

Aerosol Transport Modeling: The Key Link Between Lung Infections of Individuals and Populations

The recent COVID-19 pandemic has propelled the field of aerosol science to the forefront, particularly the central role of virus-laden respiratory droplets and aerosols. The pandemic has also highlighted the critical need, and value for, an information bridge between epidemiological models (that inform policymakers to develop public health responses) and within-host models (that inform the public and health care providers how individuals develop respiratory infections). Here, we review existing data and models of generation of respiratory droplets and aerosols, their exhalation and inhalation, and the fate of infectious droplet transport and deposition throughout the respiratory tract. We then articulate how aerosol transport modeling can serve as a bridge between and guide calibration of within-host and epidemiological models, forming a comprehensive tool to formulate and test hypotheses about respiratory tract exposure and infection within and between individuals.

A model of the pulmonary acinar circulatory system with gas exchange function to explore the influence of alveolar diameter

Lung diseases such as acute respiratory distress syndrome affect the patient's lung compliance, which in turn affects the ability of gas exchange. Changes in alveolar diameter relate to local lung compliance. How alveolar diameter affects gas exchange, particularly oxygen concentrations in alveolar capillaries, is a topic of concern for researchers, and can be studied using mathematical models. The level of small-scale mathematical models of the pulmonary circulatory system was the alveolar capillaries, but existing models do not consider the gas-exchange function and fail to reflect the influence of alveolar diameter. Therefore, we proposed a pulmonary acinar capillary model with gas exchange function, and most importantly, introduced alveolar diameter into the model, to analyze the effect of alveolar diameter on the gas exchange function of the pulmonary acini. The model was tested by three respiratory function simulation experiments. According to the simulation results of changing diameter, we found that the alveolar diameter mainly affects the alveolar gas exchange function of lung acinar inlets and the middle section compared with the peripheral section.

Simulating Multi-Scale Pulmonary Vascular Function by Coupling Computational Fluid Dynamics With an Anatomic Network Model

The function of the pulmonary circulation is truly multi-scale, with blood transported through vessels from centimeter to micron scale. There are scale-dependent mechanisms that govern the flow in the pulmonary vascular system. However, very few computational models of pulmonary hemodynamics capture the physics of pulmonary perfusion across the spatial scales of functional importance in the lung. Here we present a multi-scale model that incorporates the 3-dimensional (3D) complexities of pulmonary blood flow in the major vessels, coupled to an anatomically-based vascular network model incorporating the multiple contributing factors to capillary perfusion, including gravity. Using the model we demonstrate how we can predict the impact of vascular remodeling and occlusion on both macro-scale functional drivers (flow distribution between lungs, and wall shear stress) and micro-scale contributors to gas exchange. The model predicts interactions between 3D and 1D models that lead to a redistribution of blood between postures, both on a macro- and a micro-scale. This allows us to estimate the effect of posture on left and right pulmonary artery wall shear stress, with predictions varying by 0.75-1.35 dyne/cm² between postures.

A computational model of contributors to pulmonary hypertensive disease: impacts of whole lung and focal disease distributions

Pulmonary hypertension has multiple etiologies and so can be difficult to diagnose, prognose, and treat. Diagnosis is typically made via invasive hemodynamic measurements in the main pulmonary artery and is based on observed elevation of mean pulmonary artery pressure. This static mean pressure enables diagnosis, but does not easily allow assessment of the severity of pulmonary hypertension, nor the etiology of the disease, which may impact treatment. Assessment of the dynamic properties of pressure and flow data obtained from catheterization potentially allows more meaningful assessment of the strain on the right heart and may help to distinguish between disease phenotypes. However, mechanistic understanding of how the distribution of disease in the lung leading to pulmonary hypertension impacts the dynamics of blood flow in the main pulmonary artery and/or the pulmonary capillaries is lacking. We present a computational model of the pulmonary vasculature, parameterized to characteristic features of pulmonary arterial hypertension and chronic thromboembolic pulmonary hypertension to help understand how the two conditions differ in terms of pulmonary vascular response to disease. Our model incorporates key features known to contribute to pulmonary vascular function in health and disease, including anatomical structure and multiple contributions from gravity. The model suggests that dynamic measurements obtained from catheterization potentially distinguish between distal and proximal vasculopathy typical of pulmonary arterial hypertension and chronic thromboembolic pulmonary hypertension. However, the model suggests a non-linear relationship between these data and vascular structural changes typical of pulmonary arterial hypertension and chronic thromboembolic pulmonary hypertension which may impede analysis of these metrics to distinguish between cohorts.

Hypoxic pulmonary vasoconstriction as a regulator of alveolar-capillary oxygen flux: A computational model of ventilation-perfusion matching

The relationship between regional variabilities in airflow (ventilation) and blood flow (perfusion) is a critical determinant of gas exchange efficiency in the lungs. Hypoxic pulmonary vasoconstriction is understood to be the primary active regulator of ventilation-perfusion matching, where upstream arterioles constrict to direct blood flow away from areas that have low oxygen supply. However, it is not understood how the integrated action of hypoxic pulmonary vasoconstriction affects oxygen transport at the system level. In this study we develop, and make functional predictions with a multi-scale multi-physics model of ventilation-perfusion matching governed by the mechanism of hypoxic pulmonary vasoconstriction. Our model consists of (a) morphometrically realistic 2D pulmonary vascular networks to the level of large arterioles and venules; (b) a tileable lumped-parameter model of vascular fluid and wall mechanics that accounts for the influence of alveolar pressure; (c) oxygen transport accounting for oxygen bound to hemoglobin and dissolved in plasma; and (d) a novel empirical model of hypoxic pulmonary vasoconstriction. Our model simulations predict that under the artificial test condition of a uniform ventilation distribution (1) hypoxic pulmonary vasoconstriction matches perfusion to ventilation; (2) hypoxic pulmonary vasoconstriction homogenizes regional alveolar-capillary oxygen flux; and (3) hypoxic pulmonary vasoconstriction increases whole-lobe oxygen uptake by improving ventilation-perfusion matching.

The relationship between regional ventilation (airflow) and perfusion (blood flow) is a major determinant of gas exchange efficiency. Atelactasis and pulmonary vascular occlusive diseases, such as acute pulmonary embolism, are characterized by ventilation-perfusion mismatching and decreased oxygen in the bloodstream. Despite the physiological and medical importance of ventilation-perfusion matching, there are gaps in our knowledge of the regulatory mechanisms that maintain adequate gas exchange under pathological and normal conditions. Hypoxic pulmonary vasoconstriction is understood to be the primary regulator of ventilation-perfusion matching, where upstream arterioles constrict to direct blood flow away from areas that have low oxygen supply, yet it is not understood how this mechanism affects oxygen transport at the system level. In this study we present a computational model of the ventilation-perfusion matching and hypoxic pulmonary vasoconstriction to better understand how physiological regulation at the regional level scales to affect oxygen transport at the system level. Our model simulations predict that this regulatory mechanism improves the spatial overlap of airflow and blood flow, which serves to increase the uptake of oxygen into the bloodstream. This improved understanding of ventilation-perfusion matching may offer insights into the etiology of, and therapeutic interventions for diseases characterized by ventilation-perfusion mismatching.

Integrative Computational Models of Lung Structure-Function Interactions

Anatomically based integrative models of the lung and their interaction with other key components of the respiratory system provide unique capabilities for investigating both normal and abnormal lung function. There is substantial regional variability in both structure and function within the normal lung, yet it remains capable of relatively efficient gas exchange by providing close matching of air delivery (ventilation) and blood delivery (perfusion) to regions of gas exchange tissue from the scale of the whole organ to the smallest continuous gas exchange units. This is despite remarkably different mechanisms of air and blood delivery, different fluid properties, and unique scale-dependent anatomical structures through which the blood and air are transported. This inherent heterogeneity can be exacerbated in the presence of disease or when the body is under stress. Current computational power and data availability allow for the construction of sophisticated data-driven integrative models that can mimic respiratory system structure, function, and response to intervention. Computational models do not have the same technical and ethical issues that can limit experimental studies and biomedical imaging, and if they are solidly grounded in physiology and physics they facilitate investigation of the underlying interaction between mechanisms that determine respiratory function and dysfunction, and to estimate otherwise difficult-to-access measures. © 2021 American Physiological Society. Compr Physiol 11:1501-1530, 2021.

The Lung Physiome and virtual patient models: From morphometry to clinical translation

The understanding or prediction of specific functions of the lung can be made using compact models that have identifiable parameters and that are custom designed to the problem of interest. However, when structure contributes to function - as is the case with most lung pathologies - structure-based, biophysical models become essential. Here we describe the application of structure-based models within the lung Physiome framework to identifying and explaining patient risk in 12patients diagnosed with acute pulmonary embolism. The model integrates perfusion, ventilation, and gas exchange to predict arterial blood gases and pulmonary artery pressure in individual patient models in response to patient-specific blood clot distribution, with full or partial arterial occlusion. The necessity for a patient-specific approach with biophysical models that account for scale-specific structure and function is demonstrated.

Ventilation/Perfusion Matching: Of Myths, Mice, and Men

Despite a huge range in lung size between species, there is little measured difference in the ability of the lung to provide a well-matched air flow (ventilation) to blood flow (perfusion) at the gas exchange tissue. Here, we consider the remarkable similarities in ventilation/perfusion matching between species through a biophysical lens and consider evidence that matching in large animals is dominated by gravity but in small animals by structure.

Pulmonary Vascular Dynamics

The pulmonary circulation carries deoxygenated blood from the systemic veins through the pulmonary arteries to be oxygenated in the capillaries that line the walls of the pulmonary alveoli. The pulmonary circulation carries the cardiac output with a relatively low driving pressure, and so differs considerably in structure and function from the systemic circulation to maintain a low-resistance vascular system. The pulmonary circulation is often considered to be a quasi-static system in both experimental and computational studies of pulmonary perfusion and its matching to ventilation (air flow) for exchange. However, the system is highly dynamic, with cardiac output and regional perfusion changing with posture, exercise, and over time. Here we review this dynamic system, with a focus on understanding the physiology of pulmonary vascular dynamics across spatial and temporal scales, and the changes to these dynamics that are reflective of disease. © 2019 American Physiological Society. Compr Physiol 9:1081-1100, 2019.

Optimizing human pulmonary perfusion measurement using an in silico model of arterial spin labeling magnetic resonance imaging

Arterial spin labeling (ASL) magnetic resonance imaging (MRI) is an imaging methodology that uses blood as an endogenous contrast agent to quantify flow. One limitation of this method of capillary blood quantification when applied in the lung is the contribution of signals from non-capillary blood. Intensity thresholding is one approach that has been proposed for minimizing the non-capillary blood signal. This method has been tested in previous in silico modeling studies; however, it has only been tested under a restricted set of physiological conditions (supine posture and a cardiac output of 5 L/min). This study presents an in silico approach that extends previous intensity thresholding analysis to estimate the optimal "per-slice" intensity threshold value using the individual components of the simulated ASL signal (signal arising independently from capillary blood as well as pulmonary arterial and pulmonary venous blood). The aim of this study was to assess whether the threshold value should vary with slice location, posture, or cardiac output. We applied an in silico modeling approach to predict the blood flow distribution and the corresponding ASL quantification of pulmonary perfusion in multiple sagittal imaging slices. There was a significant increase in ASL signal and heterogeneity (COV = 0.90 to COV = 1.65) of ASL signals when slice location changed from lateral to medial. Heterogeneity of the ASL signal within a slice was significantly lower (P = 0.03) in prone (COV = 1.08) compared to in the supine posture (COV = 1.17). Increasing stroke volume resulted in an increase in ASL signal and conversely an increase in heart rate resulted in a decrease in ASL signal. However, when cardiac output was increased via an increase in both stroke volume and heart rate, ASL signal remained relatively constant. Despite these differences, we conclude that a threshold value of 35% provides optimal removal of large vessel signal independent of slice location, posture, and cardiac output.

Analytical model of the feto-placental vascular system: consideration of placental oxygen transport

The placenta is a transient vascular organ that enables nutrients and blood gases to be exchanged between fetal and maternal circulations. Herein, the structure and oxygen diffusion across the trophoblast membrane between the fetal and maternal red blood cells in the feto-placental vasculature system in both human and mouse placentas are presented together as a functional unit. Previous models have claimed that the most efficient fetal blood flow relies upon structures containing a number of 'conductive' symmetrical branches, offering a path of minimal resistance that maximizes blood flow to the terminal villi, where oxygen diffusion occurs. However, most of these models have disregarded the actual descriptions of the exchange at the level of the intermediate and terminal villi. We are proposing a 'mixed model' whereby both 'conductive' and 'terminal' villi are presumed to be present at the end of single (in human) or multiple (in mouse) pregnancies. We predict an optimal number of 18 and 22 bifurcation levels in the human and the mouse placentas, respectively. Wherever possible, we have compared our model's predictions with experimental results reported in the literature and found close agreement between them.

Negative pressure ventilation enhances acinar perfusion in isolated rat lungs

We compared acinar perfusion in isolated rat lungs ventilated using positive or negative pressures. The lungs were ventilated with air at transpulmomary pressures of 15/5 cm H₂O, at 25 breaths/min, and perfused with a hetastarch solution at P_{pulm art}/P_LA pressures of 10/0 cm H₂O. We evaluated overall perfusability from perfusate flows, and from the venous concentrations of 4-µm diameter fluorescent latex particles infused into the pulmonary circulation during perfusion. We measured perfusion distribution from the trapping patterns of those particles within the lung. We infused approximately 9 million red fluorescent particles into each lung, followed 20 min later by an infusion of an equal number of green particles. In positive pressure lungs, 94.7 ± 2.4% of the infused particles remained trapped within the lungs, compared to 86.8 ± 5.6% in negative pressure lungs ( P ≤ 0.05). Perfusate flows averaged 2.5 ± 0.1 mL/min in lungs ventilated with positive pressures, compared to 5.6 ± 01 mL/min in lungs ventilated with negative pressures ( P ≤ 0.05). Particle infusions had little effect on perfusate flows. In confocal images of dried sections of each lung, red and green particles were co-localized in clusters in positive pressure lungs, suggesting that acinar vessels that lacked particles were collapsed by these pressures thereby preventing perfusion through them. Particles were more broadly and uniformly distributed in negative pressure lungs, suggesting that perfusion in these lungs was also more uniformly distributed. Our results suggest that the acinar circulation is organized as a web, and further suggest that portions of this web are collapsed by positive pressure ventilation.

Strain-induced inflammation in pulmonary alveolar tissue due to mechanical ventilation

Inflammation is the body's attempt at self-protection to remove harmful stimuli, including damaged cells, irritants, or pathogens and begin the healing process. In this study, strain-induced inflammation in pulmonary alveolar tissue under high tidal volume is investigated through a combination of an inflammation model and fluid structure interaction (FSI) analysis. A realistic three-dimensional organ model for alveolar sacs is built, and FSI is employed to evaluate strain distribution in alveolar tissue for different tidal volume (TV) values under the mechanical ventilation (MV) condition. The alveolar tissue is treated as a hyperelastic solid and provides the environment for the tissue constituents. The influence of different strain distributions resulting from different tidal volumes is investigated. It is observed that strain is highly distributed in the inlet area. In addition, strain versus time curves in different locations through the alveolar model reveals that middle layers in the alveolar region would undergo higher levels of strain during breathing under the MV condition. Three different types of strain distributions in the alveolar region from the FSI simulation are transferred to the CA model to study population dynamics of cell constituents under MV for different TVs; 200, 500 and 1000 mL, respectively. The CA model results suggests that strain distribution plays a significant role in population dynamics. An interplay between strain magnitude and distribution appears to influence healing capability. Results suggest that increasing TV leads to an exponential rise in tissue damage by inflammation.

Arterio-venous anastomoses in isolated, perfused rat lungs

Several studies have suggested that large-diameter (>25 μm) arterio-venous shunt pathways exist in the lungs of rats, dogs, and humans. We investigated the nature of these pathways by infusing specific-diameter fluorescent latex particles (4, 7, 15, 30, or 50 μm) into isolated, ventilated rat lungs perfused at constant pressure. All lungs received the same mass of latex (5 mg), which resulted in infused particle numbers that ranged from 1.7 × 10⁷ 4 μm particles to 7.5 × 10⁴ 50 μm particles. Particles were infused over 2 min. We used a flow cytometer to count particle appearances in venous effluent samples collected every 0.5 min for 12 min from the start of particle infusion. Cumulative percentages of infused particles that appeared in the samples averaged 3.17 ± 2.46% for 4 μm diameter particles, but ranged from 0.01% to 0.17% for larger particles. Appearances of 4 μm particles followed a rapid upslope beginning at 30 sec followed by a more gradual downslope that lasted for up to 12 min. All other particle diameters also began to appear at 30 sec, but followed highly irregular time courses. Infusion of 7 and 15 μm particles caused transient but significant perfusate flow reductions, while infusion of all other diameters caused insignificant reductions in flow. We conclude that small numbers of bypass vessels exist that can accommodate particle diameters of 7-to-50 μm. We further conclude that our 4 μm particle data are consistent with a well-developed network of serial and parallel perfusion pathways at the acinar level.

Fast blood-flow simulation for large arterial trees containing thousands of vessels

Blood flow modelling has previously been successfully carried out in arterial trees to study pulse wave propagation using nonlinear or linear flow solvers. However, the number of vessels used in the simulations seldom grows over a few hundred. The aim of this work is to present a computationally efficient solver coupled with highly detailed arterial trees containing thousands of vessels. The core of the solver is based on a modified transmission line method, which exploits the analogy between electrical current in finite-length conductors and blood flow in vessels. The viscoelastic behaviour of the arterial-wall is taken into account using a complex elastic modulus. The flow is solved vessel by vessel in the frequency domain and the calculated output pressure is then used as an input boundary condition for daughter vessels. The computational results yield pulsatile blood pressure and flow rate for every segment in the tree. This solver is coupled with large arterial trees generated from a three-dimensional constrained constructive optimisation algorithm. The tree contains thousands of blood vessels with radii spanning ~1 mm in the root artery to ~30 μm in leaf vessels. The computation takes seconds to complete for a vasculature of 2048 vessels and less than 2 min for a vasculature of 4096 vessels on a desktop computer.

Multiscale modelling of the feto-placental vasculature

The placenta provides all the nutrients required for the fetus through pregnancy. It develops dynamically, and, to avoid rejection of the fetus, there is no mixing of fetal and maternal blood; rather, the branched placental villi 'bathe' in blood supplied from the uterine arteries. Within the villi, the feto-placental vasculature also develops a complex branching structure in order to maximize exchange between the placental and maternal circulations. To understand the development of the placenta, we must translate functional information across spatial scales including the interaction between macro- and micro-scale haemodynamics and account for the effects of a dynamically and rapidly changing structure through the time course of pregnancy. Here, we present steps towards an anatomically based and multiscale approach to modelling the feto-placental circulation. We assess the effect of the location of cord insertion on feto-placental blood flow resistance and flow heterogeneity and show that, although cord insertion does not appear to directly influence feto-placental resistance, the heterogeneity of flow in the placenta is predicted to increase from a 19.4% coefficient of variation with central cord insertion to 23.3% when the cord is inserted 2 cm from the edge of the placenta. Model geometries with spheroidal and ellipsoidal shapes, but the same volume, showed no significant differences in flow resistance or heterogeneity, implying that normal asymmetry in shape does not affect placental efficiency. However, the size and number of small capillary vessels is predicted to have a large effect on feto-placental resistance and flow heterogeneity. Using this new model as an example, we highlight the importance of taking an integrated multi-disciplinary and multiscale approach to understand development of the placenta.

Comparison of generic and subject-specific models for simulation of pulmonary perfusion and forced expiration

The goal of translating multiscale model analysis of pulmonary function into population studies is challenging because of the need to derive a geometric model for each subject. This could be addressed by using a generic model with appropriate customization to subject-specific data. Here, we present a quantitative comparison of simulating two fundamental behaviours of the lung-its haemodynamic response to vascular occlusion, and the forced expiration in 1 s (FEV1) following bronchoconstriction-in subject-specific and generic models. When the subjects are considered as a group, there is no significant difference between predictions of mean pulmonary artery pressure (mPAP), pulmonary vascular resistance or forced expiration; however, significant differences are apparent in the prediction of arterial oxygen, for both baseline and post-occlusion. Despite the apparent consistency of the generic and subject-specific models, a third of subjects had generic model under-prediction of the increase in mPAP following occlusion, and half had the decrease in arterial oxygen over-predicted; two subjects had considerable differences in the percentage reduction of FEV1 following bronchoconstriction. The generic model approach can be useful for physiologically directed studies but is not appropriate for simulating pathophysiological function that is strongly dependent on interaction with lung structure.

Hypoxic pulmonary vasoconstriction as a contributor to response in acute pulmonary embolism

Hypoxic pulmonary vasoconstriction (HPV) is an adaptive response unique to the lung whereby blood flow is diverted away from areas of low alveolar oxygen to improve ventilation-perfusion matching and resultant gas exchange. Some previous experimental studies have suggested that the HPV response to hypoxia is blunted in acute pulmonary embolism (APE), while others have concluded that HPV contributes to elevated pulmonary blood pressures in APE. To understand these contradictory observations, we have used a structure-based computational model of integrated lung function in 10 subjects to study the impact of HPV on pulmonary hemodynamics and gas exchange in the presence of regional arterial occlusion. The integrated model includes an experimentally-derived model for HPV. Its function is validated against measurements of pulmonary vascular resistance in normal subjects at four levels of inspired oxygen. Our results show that the apparently disparate observations of previous studies can be explained within a single model: the model predicts that HPV increases mean pulmonary artery pressure in APE (by 8.2 ± 7.0% in these subjects), and concurrently shows a reduction in response to hypoxia in the subjects who have high levels of occlusion and therefore maximal HPV in normoxia.

Dynamic blood flow and wall shear stress in pulmonary hypertensive disease

This study provides new model of pulsatile flow in the pulmonary circulation in health and pulmonary hypertensive disease. Structural vascular remodeling typical of pulmonary hypertensive disease was implemented in the model by progressively altering the mechanical properties of the arterial geometry and progressively increasing the inlet pulse pressure (PP). The transmission of PP throughout the tree was shown to increase in advanced stages of disease, creating the potential for a `vicious-cycle' of damage to vasculature. Wall shear stress (WSS) was shown to be highest in the terminal arteries of the model and increased significantly with disease. A further trend observed in WSS results was that high WSS values began to `climb' the arterial tree towards the proximal vessels as disease progressed. This suggests a link between WSS and distal remodeling in pulmonary hypertensive disease, which initiates in the small muscular arteries and arterioles and spreads into larger arteries as the disease progresses.

Integrative approaches for modeling regulation and function of the respiratory system

Mathematical models have been central to understanding the interaction between neural control and breathing. Models of the entire respiratory system-which comprises the lungs and the neural circuitry that controls their ventilation-have been derived using simplifying assumptions to compartmentalize each component of the system and to define the interactions between components. These full system models often rely-through necessity-on empirically derived relationships or parameters, in addition to physiological values. In parallel with the development of whole respiratory system models are mathematical models that focus on furthering a detailed understanding of the neural control network, or of the several functions that contribute to gas exchange within the lung. These models are biophysically based, and rely on physiological parameters. They include single-unit models for a breathing lung or neural circuit, through to spatially distributed models of ventilation and perfusion, or multicircuit models for neural control. The challenge is to bring together these more recent advances in models of neural control with models of lung function, into a full simulation for the respiratory system that builds upon the more detailed models but remains computationally tractable. This requires first understanding the mathematical models that have been developed for the respiratory system at different levels, and which could be used to study how physiological levels of O2 and CO2 in the blood are maintained.

Blood Flow in Capillaries of the Human Lung

A novel model for the blood system is postulated focusing on the flow rate and pressure distribution inside the arterioles and venules of the pulmonary acinus. Based upon physiological data it is devoid of any ad hoc constants. The model comprises nine generations of arterioles, venules, and capillaries in the acinus, the gas exchange unit of the lung. Blood is assumed incompressible and Newtonian and the blood vessels are assumed inextensible. Unlike previous models of the blood system, the venules and arterioles open up to the capillary network in numerous locations along each generation. The large number of interconnected capillaries is perceived as a porous medium in which the flow is macroscopically unidirectional from arterioles to venules openings. In addition, the large number of capillaries extending from each arteriole and venule allows introduction of a continuum theory and formulation of a novel system of ordinary, nonlinear differential equations which governs the blood flow and pressure fields along the arterioles, venules, and capillaries. The solution of the differential equations is semianalytical and requires the inversion of three diagonal, 9 × 9 matrices only. The results for the total flow rate of blood through the acinus are within the ballpark of physiological observations despite the simplifying assumptions used in our model. The results also manifest that the contribution of the nonlinear convection term of the Navier-Stokes equations has little effect (less than 2%) on the total blood flow entering/leaving the acinus despite the fact that the Reynolds number is not much smaller than unity at the proximal generations. The model makes it possible to examine some pathological cases. Here, centri-acinar and distal emphysema were investigated yielding a reduction in inlet blood flow rate.

Blood flow redistribution and ventilation-perfusion mismatch during embolic pulmonary arterial occlusion

Acute pulmonary embolism causes redistribution of blood in the lung, which impairs ventilation/perfusion matching and gas exchange and can elevate pulmonary arterial pressure (PAP) by increasing pulmonary vascular resistance (PVR). An anatomically-based multi-scale model of the human pulmonary circulation was used to simulate pre- and post-occlusion flow, to study blood flow redistribution in the presence of an embolus, and to evaluate whether reduction in perfused vascular bed is sufficient to increase PAP to hypertensive levels, or whether other vasoconstrictive mechanisms are necessary. A model of oxygen transfer from air to blood was included to assess the impact of vascular occlusion on oxygen exchange. Emboli of 5, 7, and 10 mm radius were introduced to occlude increasing proportions of the vasculature. Blood flow redistribution was calculated after arterial occlusion, giving predictions of PAP, PVR, flow redistribution, and micro-circulatory flow dynamics. Because of the large flow reserve capacity (via both capillary recruitment and distension), approximately 55% of the vasculature was occluded before PAP reached clinically significant levels indicative of hypertension. In contrast, model predictions showed that even relatively low levels of occlusion could cause localized oxygen deficit. Flow preferentially redistributed to gravitationally non-dependent regions regardless of occlusion location, due to the greater potential for capillary recruitment in this region. Red blood cell transit times decreased below the minimum time for oxygen saturation (<0.25 s) and capillary pressures became high enough to initiate cell damage (which may result in edema) only after ~80% of the lung was occluded.

Assessing potential errors of MRI-based measurements of pulmonary blood flow using a detailed network flow model

MRI images of pulmonary blood flow using arterial spin labeling (ASL) measure the delivery of magnetically tagged blood to an image plane during one systolic ejection period. However, the method potentially suffers from two problems, each of which may depend on the imaging plane location: 1) the inversion plane is thicker than the imaging plane, resulting in a gap that blood must cross to be detected in the image; and 2) ASL includes signal contributions from tagged blood in conduit vessels (arterial and venous). By using an in silico model of the pulmonary circulation we found the gap reduced the ASL signal to 64-74% of that in the absence of a gap in the sagittal plane and 53-84% in the coronal. The contribution of the conduit vessels varied markedly as a function of image plane ranging from ∼90% of the overall signal in image planes that encompass the central hilar vessels to <20% in peripheral image planes. A threshold cutoff removing voxels with intensities >35% of maximum reduced the conduit vessel contribution to the total ASL signal to ∼20% on average; however, planes with large contributions from conduit vessels underestimate acinar flow due to a high proportion of in-plane flow, making ASL measurements of perfusion impractical. In other image planes, perfusion dominated the resulting ASL images with good agreement between ASL and acinar flow. Similarly, heterogeneity of the ASL signal as measured by relative dispersion is a reliable measure of heterogeneity of the acinar flow distribution in the same image planes.

Computational modeling of airway and pulmonary vascular structure and function: development of a "lung physiome"

Computational models of lung structure and function necessarily span multiple spatial and temporal scales, i.e., dynamic molecular interactions give rise to whole organ function, and the link between these scales cannot be fully understood if only molecular or organ-level function is considered. Here, we review progress in constructing multiscale finite element models of lung structure and function that are aimed at providing a computational framework for bridging the spatial scales from molecular to whole organ. These include structural models of the intact lung, embedded models of the pulmonary airways that couple to model lung tissue, and models of the pulmonary vasculature that account for distinct structural differences at the extra- and intra-acinar levels. Biophysically based functional models for tissue deformation, pulmonary blood flow, and airway bronchoconstriction are also described. The development of these advanced multiscale models has led to a better understanding of complex physiological mechanisms that govern regional lung perfusion and emergent heterogeneity during bronchoconstriction.

Pulmonary embolism: predicting disease severity

Pulmonary embolism (PE) is the most common cause of acute pulmonary hypertension, yet it is commonly undiagnosed, with risk of death if not recognized promptly and managed accordingly. Patients typically present with hypoxemia and hypomania, although the presentation varies greatly, being confounded by co-morbidities such as pre-existing cardio-respiratory disease. Previous studies have demonstrated variable patient outcomes in spite of similar extent and distribution of pulmonary vascular occlusion, but the path physiological determinants of outcome remain unclear. Computational models enable exact control over many of the compounding factors leading to functional outcomes and therefore provide a useful tool to understand and assess these mechanisms. We review the current state of pulmonary blood flow models. We present a pilot study within 10 patients presenting with acute PE, where patient-derived vascular occlusions are imposed onto an existing model of the pulmonary circulation enabling predictions of resultant haemodynamic after embolus occlusion. Results show that mechanical obstruction alone is not sufficient to cause pulmonary arterial hypertension, even when up to 65 per cent of lung tissue is occluded. Blood flow is found to preferentially redistribute to the gravitationally non-dependent regions. The presence of an additional downstream occlusion is found to significantly increase pressures.

Computational models of the pulmonary circulation: Insights and the move towards clinically directed studies

Biophysically-based computational models provide a tool for integrating and explaining experimental data, observations, and hypotheses. Computational models of the pulmonary circulation have evolved from minimal and efficient constructs that have been used to study individual mechanisms that contribute to lung perfusion, to sophisticated multi-scale and -physics structure-based models that predict integrated structure-function relationships within a heterogeneous organ. This review considers the utility of computational models in providing new insights into the function of the pulmonary circulation, and their application in clinically motivated studies. We review mathematical and computational models of the pulmonary circulation based on their application; we begin with models that seek to answer questions in basic science and physiology and progress to models that aim to have clinical application. In looking forward, we discuss the relative merits and clinical relevance of computational models: what important features are still lacking; and how these models may ultimately be applied to further increasing our understanding of the mechanisms occurring in disease of the pulmonary circulation.

The interdependent contributions of gravitational and structural features to perfusion distribution in a multiscale model of the pulmonary circulation

Recent experimental and imaging studies suggest that the influence of gravity on the measured distribution of blood flow in the lung is largely through deformation of the parenchymal tissue. To study the contribution of hydrostatic effects to regional perfusion in the presence of tissue deformation, we have developed an anatomically structured computational model of the pulmonary circulation (arteries, capillaries, veins), coupled to a continuum model of tissue deformation, and including scale-appropriate fluid dynamics for blood flow in each vessel type. The model demonstrates that both structural and the multiple effects of gravity on the pulmonary circulation make a distinct contribution to the distribution of blood. It shows that postural differences in perfusion gradients can be explained by the combined effect of tissue deformation and extra-acinar blood vessel resistance to flow in the dependent tissue. However, gravitational perfusion gradients persist when the effect of tissue deformation is eliminated, highlighting the importance of the hydrostatic effects of gravity on blood distribution in the pulmonary circulation. Coupling of large- and small-scale models reveals variation in microcirculatory driving pressures within isogravitational planes due to extra-acinar vessel resistance. Variation in driving pressures is due to heterogeneous large-vessel resistance as a consequence of geometric asymmetry in the vascular trees and is amplified by the complex balance of pressures, distension, and flow at the microcirculatory level.

Multi-scale lung modeling

Multi-scale modeling of biological systems has recently become fashionable due to the growing power of digital computers as well as to the growing realization that integrative systems behavior is as important to life as is the genome. While it is true that the behavior of a living organism must ultimately be traceable to all its components and their myriad interactions, attempting to codify this in its entirety in a model misses the insights gained from understanding how collections of system components at one level of scale conspire to produce qualitatively different behavior at higher levels. The essence of multi-scale modeling thus lies not in the inclusion of every conceivable biological detail, but rather in the judicious selection of emergent phenomena appropriate to the level of scale being modeled. These principles are exemplified in recent computational models of the lung. Airways responsiveness, for example, is an organ-level manifestation of events that begin at the molecular level within airway smooth muscle cells, yet it is not necessary to invoke all these molecular events to accurately describe the contraction dynamics of a cell, nor is it necessary to invoke all phenomena observable at the level of the cell to account for the changes in overall lung function that occur following methacholine challenge. Similarly, the regulation of pulmonary vascular tone has complex origins within the individual smooth muscle cells that line the blood vessels but, again, many of the fine details of cell behavior average out at the level of the organ to produce an effect on pulmonary vascular pressure that can be described in much simpler terms. The art of multi-scale lung modeling thus reduces not to being limitlessly inclusive, but rather to knowing what biological details to leave out.

The impact of micro-embolism size on haemodynamic changes in the pulmonary micro-circulation

Embolus occlusion of pulmonary arteries can result in elevated pulmonary blood pressures, often resulting in pulmonary hypertension (PH). Experimental observations have shown that small emboli (diameter <170 μm) can have a disproportionate effect on pulmonary vascular resistance (PVR) compared with larger emboli for the same tissue occlusion. We present an anatomically based theoretical model of perfusion in the acinar blood vessels designed to investigate changes in PVR following occlusion of arteries <500 μm in diameter. The model predicts that emboli lodged near proximal capillary beds have a greater effect on PVR--regardless of their size--than emboli occluding 200 μm diameter arterioles, with PH occurring for 10% less tissue occlusion. Capillary blood pressures are predicted to exceed 24 mmHg (levels initiating capillary wall damage) in regions of the capillary bed at approximately the onset of PH. This study focuses on the effect of mechanical obstruction alone; however, we present simple models of vasoconstriction illustrating an increased impact on PVR.

Computational Simulation of the Pulmonary Arteries and its Role in the Study of Pediatric Pulmonary Hypertension

The hemodynamic state of the pulmonary arteries is challenging to routinely measure in children due to the vascular circuit's position in the lungs. The resulting relative scarcity of quantitative clinical diagnostic and prognostic information impairs management of diseases such as pulmonary hypertension, or high blood pressure of the pulmonary circuit, and invites new techniques of measurement. Here we examine recent applications of macro-scale computational mechanics methods for fluids and solids - traditionally used by engineers in the design and virtual testing of complex metal and composite structures - applied to study the pulmonary vasculature, both in healthy and diseased states. In four subject areas, we briefly outline advances in computational methodology and provide examples of clinical relevance.

Evidence for minimal oxygen heterogeneity in the healthy human pulmonary acinus

It has been suggested that the human pulmonary acinus operates at submaximal efficiency at rest due to substantial spatial heterogeneity in the oxygen partial pressure (Po(2)) in alveolar air within the acinus. Indirect measurements of alveolar air Po(2) could theoretically mask significant heterogeneity if intra-acinar perfusion is well matched to Po(2). To investigate the extent of intra-acinar heterogeneity, we developed a computational model with anatomically based structure and biophysically based equations for gas exchange. This model yields a quantitative prediction of the intra-acinar O(2) distribution that cannot be measured directly. Temporal and spatial variations in Po(2) in the intra-acinar air and blood are predicted with the model. The model, representative of a single average acinus, has an asymmetric multibranching respiratory airways geometry coupled to a symmetric branching conducting airways geometry. Advective and diffusive O(2) transport through the airways and gas exchange into the capillary blood are incorporated. The gas exchange component of the model includes diffusion across the alveolar air-blood membrane and O(2)-hemoglobin binding. Contrary to previous modeling studies, simulations show that the acinus functions extremely effectively at rest, with only a small degree of intra-acinar Po(2) heterogeneity. All regions of the model acinus, including the peripheral generations, maintain a Po(2) >100 mmHg. Heterogeneity increases slightly when the acinus is stressed by exercise. However, even during exercise the acinus retains a reasonably homogeneous gas phase.