On the uncertainty quantification of blood flow viscosity models

A Bayesian approach to blood rheological uncertainties in aortic hemodynamics

Computational hemodynamics has received increasing attention recently. Patient-specific simulations require questionable model assumptions, for example, for geometry, boundary conditions, and material parameters. Consequently, the credibility of these simulations is much doubted, and rightly so. Yet, the matter may be addressed by a rigorous uncertainty quantification. In this contribution, we investigated the impact of blood rheological models on wall shear stress uncertainties in aortic hemodynamics obtained in numerical simulations. Based on shear-rheometric experiments, we compare the non-Newtonian Carreau model to a simple Newtonian model and a Reynolds number-equivalent Newtonian model. Bayesian Probability Theory treats uncertainties consistently and allows to include elusive assumptions such as the comparability of flow regimes. We overcome the prohibitively high computational cost for the simulation with a surrogate model, and account for the uncertainties of the surrogate model itself, too. We have two main findings: (1) The Newtonian models mostly underestimate the uncertainties as compared to the non-Newtonian model. (2) The wall shear stresses of specific persons cannot be distinguished due to largely overlapping uncertainty bands, implying that a more precise determination of person-specific blood rheological properties is necessary for person-specific simulations. While we refrain from a general recommendation for one rheological model, we have quantified the error of the uncertainty quantification associated with these modeling choices.

Computational fluid dynamics investigation on aortic hemodynamics in double aortic arch before and after ligation surgery

Double aortic arch (DAA) malformation is one of the reasons for symptomatic vascular rings, the hemodynamics of which is still poorly understood. This study aims to investigate the blood flow characteristics in patient-specific double aortic arches using computational fluid dynamics (CFD). Seven cases of infantile patients with DAA were collected and their computed tomography images were used to reconstruct 3D computational models. A modified Carreau model was used to consider the non-Newtonian effect of blood and a three-element Windkessel model taking the effect of the age of patients into account was applied to reproduce physiological pressure waveforms. Numerical results show that blood flow distribution and energy loss of DAA depends on relative sizes of the two aortic arches and their angles with the ascending aorta. Ligation of either aortic arch increases the energy loss of blood in the DAA, leading to the increase in cardiac workload. Generally, the rising rate of energy loss before and after the surgery is almost linear with the area ratio between the aortic arch without ligation and the ascending aorta.

Machine Learning for Cardiovascular Biomechanics Modeling: Challenges and Beyond

Recent progress in machine learning (ML), together with advanced computational power, have provided new research opportunities in cardiovascular modeling. While classifying patient outcomes and medical image segmentation with ML have already shown significant promising results, ML for the prediction of biomechanics such as blood flow or tissue dynamics is in its infancy. This perspective article discusses some of the challenges in using ML for replacing well-established physics-based models in cardiovascular biomechanics. Specifically, we discuss the large landscape of input features in 3D patient-specific modeling as well as the high-dimensional output space of field variables that vary in space and time. We argue that the end purpose of such ML models needs to be clearly defined and the tradeoff between the loss in accuracy and the gained speedup carefully interpreted in the context of translational modeling. We also discuss several exciting venues where ML could be strategically used to augment traditional physics-based modeling in cardiovascular biomechanics. In these applications, ML is not replacing physics-based modeling, but providing opportunities to solve ill-defined problems, improve measurement data quality, enable a solution to computationally expensive problems, and interpret complex spatiotemporal data by extracting hidden patterns. In summary, we suggest a strategic integration of ML in cardiovascular biomechanics modeling where the ML model is not the end goal but rather a tool to facilitate enhanced modeling.

Effect of non-Newtonian fluid rheology on an arterial bypass graft: A numerical investigation guided by constructal design

In post-operative scenarios of arterial graft surgeries to bypass coronary artery stenosis, fluid dynamics plays a crucial role. Problems such as intimal hyperplasia have been related to fluid dynamics and wall shear stresses near the graft junction. This study focused on the question of the use of Newtonian and non-Newtonian models to represent blood in this type of problem in order to capture important flow features, as well as an analysis of the performance of geometry from the view of Constructive Theory. The objective of this study was to investigate the effects rheology on the steady-state flow and on the performance of a system consisting of an idealized version of a partially obstructed coronary artery and bypass graft. The Constructal Design Method was employed with two degrees of freedom: the ratio between bypass and artery diameters and the junction angle at the bypass inlet. The flow problem was solved numerically using the Finite Volume Method with blood modeled employing the Carreau equation for viscosity. The Computational Fluid Dynamics model associated with the Sparse Grid method generated eighteen response surfaces, each representing a severe stenosis degree of 75% for specific combinations of rheological parameters, dimensionless viscosity ratio, Carreau number and flow index at two distinct Reynolds numbers of 150 and 250. There was a considerable dependence of the pressure drop on rheological parameters. For the two Reynolds numbers studied, the Newtonian case presented the lowest value of the dimensionless pressure drop, suggesting that the choice of applying Newtonian blood may underestimate the value of pressure drop in the system by about 12.4% (Re =150) and 7.8% (Re = 250). Even so, results demonstrated that non-Newtonian rheological parameters did not influence either the shape of the response surfaces or the optimum bypass geometry, which consisted of a diameter ratio of 1 and junction angle of 30°. However, the viscosity ratio and the flow index had the greatest impact on pressure drop, recirculation zones and wall shear stress. Rheological parameters also affected the recirculation zones downstream of stenosis, where intimal hyperplasia is more prevalent. Newtonian and most non-Newtonian results had similar wall shear stresses, except for the non-Newtonian case with high viscosity ratio. In the view of Constructal Design, the geometry of best performance was independent of the rheological model. However, rheology played an important role on pressure drop and flow dynamics, allowing the prediction of recirculation zones that were not captured by a Newtonian model.

Function of arteries and veins in conditions of simulated cardiac arrest

Introduction: The study examined the behavior of vasculature in conditions of eliminated cardiac function using mathematical modeling. In addition, we addressed the question of whether the stretch-recoil capability of veins, at least in part accounts for the slower response to simulated cardiac arrest.

Methods: In the first set of computational experiments, blood flow and pressure patterns in veins and arteries during the first few seconds after cardiac arrest were assessed via a validated multi-scale mathematical model of the whole cardiovascular system, comprising cardiac dynamics, arterial and venous blood flow dynamics, and microcirculation. In the second set of experiments, the effects of stretch-recoil zones of venous vessels with different diameters and velocities on blood velocity and dynamic pressure analyzed using computational fluid dynamics (CFD) modeling.

Results: In the first set of experiments, measurement of changes in velocity, dynamic pressure, and fluid flow revealed that the venous system responded to cardiac arrest more slowly compared to the arteries. This disparity might be due to the intrinsic characteristics of the venous system, including stretch-recoil and elastic fiber composition. In the second set of experiments, we attempted to determine the role of the stretch-recoil capability of veins in the slower response to cardiac arrest. During the second set of experiments, we found that this recoil behavior increased dynamic pressure, velocity, and blood flow. The enhancement in dynamic pressure through combining the results from both experiments yielded a 15-40% increase in maximum dynamic pressure due to stretch-recoil, depending on vein diameter under normal conditions.

Conclusion: In the situation of cardiac arrest, the vein geometry changes continue, promoting smooth responses of the venous system. Moreover, the importance of such vein behavior in blood displacement may grow as the pressure on the venous side gradually decreases with time. Our experiments suggest that the driving force for venous return is the pressure difference that remains within the venous system after the energy coming from every ventricular systole spent to overcome the resistance created by arterial and capillary systems.

A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications

As we shift from population-based medicine towards a more precise patient-specific regime guided by predictions of verified and well-established cardiovascular models, an urgent question arises: how sensitive are the model predictions to errors and uncertainties in the model inputs? To make our models suitable for clinical decision-making, precise knowledge of prediction reliability is of paramount importance. Efficient and practical methods for uncertainty quantification (UQ) and sensitivity analysis (SA) are therefore essential. In this work, we explain the concepts of global UQ and global, variance-based SA along with two often-used methods that are applicable to any model without requiring model implementation changes: Monte Carlo (MC) and polynomial chaos (PC). Furthermore, we propose a guide for UQ and SA according to a six-step procedure and demonstrate it for two clinically relevant cardiovascular models: model-based estimation of the fractional flow reserve (FFR) and model-based estimation of the total arterial compliance (CT ). Both MC and PC produce identical results and may be used interchangeably to identify most significant model inputs with respect to uncertainty in model predictions of FFR and CT . However, PC is more cost-efficient as it requires an order of magnitude fewer model evaluations than MC. Additionally, we demonstrate that targeted reduction of uncertainty in the most significant model inputs reduces the uncertainty in the model predictions efficiently. In conclusion, this article offers a practical guide to UQ and SA to help move the clinical application of mathematical models forward. Copyright © 2015 John Wiley & Sons, Ltd.