Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

Personalized blood flow computations: A hierarchical parameter estimation framework for tuning boundary conditions

We propose a hierarchical parameter estimation framework for performing patient-specific hemodynamic computations in arterial models, which use structured tree boundary conditions. A calibration problem is formulated at each stage of the hierarchical framework, which seeks the fixed point solution of a nonlinear system of equations. Common hemodynamic properties, like resistance and compliance, are estimated at the first stage in order to match the objectives given by clinical measurements of pressure and/or flow rate. The second stage estimates the parameters of the structured trees so as to match the values of the hemodynamic properties determined at the first stage. A key feature of the proposed method is that to ensure a large range of variation, two different structured tree parameters are personalized for each hemodynamic property. First, the second stage of the parameter estimation framework is evaluated based on the properties of the outlet boundary conditions in a full body arterial model: the calibration method converges for all structured trees in less than 10 iterations. Next, the proposed framework is successfully evaluated on a patient-specific aortic model with coarctation: only six iterations are required for the computational model to be in close agreement with the clinical measurements used as objectives, and overall, there is a good agreement between the measured and computed quantities. Copyright © 2016 John Wiley & Sons, Ltd.

Uncertainty Quantification in a Patient-Specific One-Dimensional Arterial Network Model: EnKF-Based Inflow Estimator

Successful clinical use of patient-specific models for cardiovascular dynamics depends on the reliability of the model output in the presence of input uncertainties. For 1D fluid dynamics models of arterial networks, input uncertainties associated with the model output are related to the specification of vessel and network geometry, parameters within the fluid and wall equations, and parameters used to specify inlet and outlet boundary conditions. This study investigates how uncertainty in the flow profile applied at the inlet boundary of a 1D model affects area and pressure predictions at the center of a single vessel. More specifically, this study develops an iterative scheme based on the ensemble Kalman filter (EnKF) to estimate the temporal inflow profile from a prior distribution of curves. The EnKF-based inflow estimator provides a measure of uncertainty in the size and shape of the estimated inflow, which is propagated through the model to determine the corresponding uncertainty in model predictions of area and pressure. Model predictions are compared to ex vivo area and blood pressure measurements in the ascending aorta, the carotid artery, and the femoral artery of a healthy male Merino sheep. Results discuss dynamics obtained using a linear and a nonlinear viscoelastic wall model.

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure-radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.

Comparison of the Windkessel model and structured-tree model applied to prescribe outflow boundary conditions for a one-dimensional arterial tree model

One-dimensional (1D) modeling is a widely adopted approach for studying wave propagation phenomena in the arterial system. Despite the frequent use of the Windkessel (WK) model to prescribe outflow boundary conditions for 1D arterial tree models, it remains unclear to what extent the inherent limitation of the WK model in describing wave propagation in distal vasculatures affect hemodynamic variables simulated at the arterial level. In the present study, a 1D model of the arterial tree was coupled respectively with a WK boundary model and a structured-tree (ST) boundary model, yielding two types of arterial tree models. The effective resistances, compliances and inductances of the WK and ST boundary models were matched to facilitate quantitative comparisons. Obtained results showed that pressure/flow waves simulated by the two models were comparable in the aorta, whereas, their discrepancies increased towards the periphery. Wave analysis revealed that the differences in reflected waves generated by the boundary models were the major sources of pressure wave discrepancies observed in large arteries. Additional simulations performed under aging conditions demonstrated that arterial stiffening with age enlarged the discrepancies, but with the effects being partly counteracted by physiological aortic dilatation with age. These findings suggest that the method adopted for modeling the outflow boundary conditions has considerable influence on the performance of a 1D arterial tree model, with the extent of influence varying with the properties of the arterial system.

Study of cardiovascular function using a coupled left ventricle and systemic circulation model

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV–SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i.e. the SA model feeds back the pressure as a boundary condition to the LV model, and the aortic flow rate from the LV model is used as the input for the SA model. The governing equations of the coupled system are solved using a combined immersed-boundary finite-element (IB/FE) method and a Lax–Wendroff scheme. A baseline case using physiological measurements of healthy subjects, and four exemplar cases based on different physiological and pathological scenarios are studied using the LV–SA model. The results of the baseline case agree well with published experimental data. The four exemplar cases predict varied pathological responses of the cardiovascular system, which are also supported by clinical observations. The new model can be used to gain insight into cardio-arterial interactions across a range of clinical applications.

Computational fluid dynamics modelling in cardiovascular medicine

This paper reviews the methods, benefits and challenges associated with the adoption and translation of computational fluid dynamics (CFD) modelling within cardiovascular medicine. CFD, a specialist area of mathematics and a branch of fluid mechanics, is used routinely in a diverse range of safety-critical engineering systems, which increasingly is being applied to the cardiovascular system. By facilitating rapid, economical, low-risk prototyping, CFD modelling has already revolutionised research and development of devices such as stents, valve prostheses, and ventricular assist devices. Combined with cardiovascular imaging, CFD simulation enables detailed characterisation of complex physiological pressure and flow fields and the computation of metrics which cannot be directly measured, for example, wall shear stress. CFD models are now being translated into clinical tools for physicians to use across the spectrum of coronary, valvular, congenital, myocardial and peripheral vascular diseases. CFD modelling is apposite for minimally-invasive patient assessment. Patient-specific (incorporating data unique to the individual) and multi-scale (combining models of different length- and time-scales) modelling enables individualised risk prediction and virtual treatment planning. This represents a significant departure from traditional dependence upon registry-based, population-averaged data. Model integration is progressively moving towards ‘digital patient’ or ‘virtual physiological human’ representations. When combined with population-scale numerical models, these models have the potential to reduce the cost, time and risk associated with clinical trials. The adoption of CFD modelling signals a new era in cardiovascular medicine. While potentially highly beneficial, a number of academic and commercial groups are addressing the associated methodological, regulatory, education- and service-related challenges.

A unified method for estimating pressure losses at vascular junctions

In reduced-order (0D/1D) blood or respiratory flow models, pressure losses at junctions are usually neglected. However, these may become important where velocities are high and significant flow redirection occurs. Current methods for estimating losses rely on relatively complex empirical equations that are only valid for specific junction geometries and flow regimes. In pulsatile multi-directional flows, switching between empirical equations upon reversing flow may introduce unrealistic discontinuities in simulated haemodynamic waveforms. Drawing from work by Bassett et al. (SAE Trans 112:565-583, 2003), we therefore developed a unified method (Unified0D) for estimating loss coefficients that can be applied to any junction (i.e. any number of branches at any angle) and any flow regime. Discontinuities in simulated waveforms were avoided by extending Bassett et al.'s control volume-based method to incorporate a 'pseudodatum' supplier branch, an imaginary effective vessel containing all inflow to the junction. Energy exchange between diverging flow streams was also accounted for empirically. The formulation was validated using high resolution computational fluid dynamics in a wide range flow conditions and junction configurations. In a pulsatile 1D simulation exhibiting transitions between four different flow regimes, the new formulation produced smooth transitions in calculated pressure losses.

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.

A computational study of pressure wave reflections in the pulmonary arteries

Experiments using wave intensity analysis suggest that the pulmonary circulation in sheep and dogs is characterized by negative or open-end type wave reflections, that reduce the systolic pressure. Since the pulmonary physiology is similar in most mammals, including humans, we test and verify this hypothesis by using a subject specific one-dimensional model of the human pulmonary circulation and a conventional wave intensity analysis. Using the simulated pressure and velocity, we also analyse the performance of the P-U loop and sum of squares techniques for estimating the local pulse wave velocity in the pulmonary arteries, and then analyse the effects of these methods on linear wave separation in the main pulmonary artery. P-U loops are found to provide much better estimates than the sum of squares technique at proximal locations, but both techniques accumulate progressive error at distal locations away from heart, particularly near junctions. The pulse wave velocity estimated using the sum of squares method also gives rise to an artificial early systolic backward compression wave. Finally, we study the influence of three types of pulmonary hypertension viz. pulmonary arterial hypertension, chronic thromboembolic pulmonary hypertension and pulmonary hypertension associated with hypoxic lung disease. Simulating these conditions by changing the relevant parameters in the model and then applying the wave intensity analysis, we observe that for each group the early systolic backward decompression wave reflected from proximal junctions is maintained, whilst the initial forward compression and the late systolic backward compression waves amplify with increasing pathology and contribute significantly to increases in systolic pressure.

Bioengineering Strategies for Polymeric Scaffold for Tissue Engineering an Aortic Heart Valve: An Update

The occurrence of dysfunctional aortic valves is increasing every year, and current replacement heart valves, although having been shown to be clinically successful, are only short-term solutions and suffer from many agonizing long-term drawbacks. The tissue engineering of heart valves is recognized as one of the most promising answers for aortic valve disease therapy, but overcoming current shortcomings will require multidisciplinary efforts. The use of a polymeric scaffold to guide the growth of the tissue is the most common approach to generate a new tissue for an aortic heart valve. However, optimizing the design of the scaffold, in terms of biocompatibility, surface morphology for cell attachments and the correct rate of degradation is critical in creating a viable tissue-engineered aortic heart valve. This paper highlights the bioengineering strategies that need to be followed to construct a polymeric scaffold of sufficient mechanical integrity, with superior surface morphologies, that is capable of mimicking the valve dynamics in vivo. The current challenges and future directions of research for creating tissue-engineered aortic heart valves are also discussed.