A reduced-dimensional model for near-wall transport in cardiovascular flows

Divergence of the normalized wall shear stress as an effective computational template of low-density lipoprotein polarization at the arterial blood-vessel wall interface

BACKGROUND AND OBJECTIVE

Near-wall transport of low-density lipoproteins (LDL) in arteries plays a relevant role in the initiation of atherosclerosis. Although it can be modelled in silico by coupling the Navier-Stokes equations with the 3D advection-diffusion (AD) equation, the associated computational cost is high. As wall shear stress (WSS) represents a first-order approximation of the near-wall velocity in arteries, we aimed at identifying computationally convenient WSS-based quantities to infer LDL near-wall transport based on the underlying near-wall hemodynamics in five models of three human arterial districts (aorta, carotid bifurcations, coronary arteries). The simulated LDL transport and its WSS-based surrogates were qualitatively compared with in vivo longitudinal measurements of wall thickness growth on the coronary artery models.

METHODS

Numerical simulations of blood flow coupled with AD equations for LDL transport and blood-wall transfer were performed. The co-localization of the simulated LDL concentration polarization patterns with luminal surface areas characterized by low cycle-average WSS, near-wall flow stagnation and WSS attracting patterns was quantitatively assessed by the similarity index (SI). In detail, the latter two represent features of the WSS topological skeleton, obtained respectively through the Lagrangian tracking of surface-born particles, and the Eulerian analysis of the divergence of the normalized cycle-average WSS vector field.

RESULTS

Convergence of the solution of the AD problem required the simulation of 3 (coronary artery) to 10 (aorta) additional cardiac cycles with respect to the Navier-Stokes problem. Co-localization results underlined that WSS topological skeleton features indicating near-wall flow stagnation and WSS attracting patterns identified LDL concentration polarization profiles more effectively than low WSS, as indicated by higher SI values (SI range: 0.17-0.50 for low WSS; 0.24-0.57 for WSS topological skeleton features). Moreover, the correspondence between the simulated LDL uptake and WSS-based quantities profiles with the in vivo measured wall thickness growth in coronary arteries appears promising.

CONCLUSIONS

The recently introduced Eulerian approach for identifying WSS attracting patterns from the divergence of normalized WSS provides a computationally affordable template of the LDL polarization at the arterial blood-wall interface without simulating the AD problem. It thus candidates as an effective biomechanical tool for elucidating the mechanistic link amongst LDL transfer at the arterial blood-wall interface, WSS and atherogenesis.

Model reduction of coagulation cascade based on genetic algorithm

Fibrin is an important product of the coagulation cascade, and plays an eminent role in platelet stabilization. Since coagulation cascade models typically involve the reaction kinetics of dozens of proteins, which will incur burdensome computational costs when coupled to blood flow in complex geometries, researchers often ignore this process when constructing thrombosis models. However, previous studies have shown that fundamental aspects of coagulation can be reproduced with simpler models, which motivated us to obtain a reduced-order model of fibrin generation through a systematic approach. Therefore, we introduced a semi-automatic framework to perform model-reduction of cascade reactions in this study, which consisted of two processes. Specifically, the retained protein species and cascade reactions were determined based on published studies and simulation results from the full cascade model, while the optimal reaction rates for the new cascade network were determined using a genetic algorithm. The framework has been applied to a 19-species coagulation model that triggers fibrin generation in internal fields via reactive boundaries, and a 10-species reduced-order model was obtained to reproduce the kinetics of fibrinogenesis in the full cascade model at different boundary tissue factor concentrations. This reduced-order model of fibrinogenesis would be valuable for thrombosis modeling that considers both the coagulation cascade and platelet activity. Furthermore, the framework proposed herein can also be applied to the reductions of other cascade reaction models.

Mathematical and Computational Modeling of Device-Induced Thrombosis

Given the extensive and routine use of cardiovascular devices, a major limiting factor to their success is the thrombotic rate that occurs. This both poses direct risk to the patient and requires counterbalancing with anticoagulation and other treatment strategies, contributing additional risks. Developing a better understanding of the mechanisms of device-induced thrombosis to aid in device design and medical management of patients is critical to advance the ubiquitous use and durability. Thus, mathematical and computational modelling of device-induced thrombosis has received significant attention recently, but challenges remain. Additional areas that need to be explored include microscopic/macroscopic approaches, reconciling physical and numerical timescales, immune/inflammatory responses, experimental validation, and incorporating pathologies and blood conditions. Addressing these areas will provide engineers and clinicians the tools to provide safe and effective cardiovascular devices.

A study on the transport and interaction between blood flow and low-density-lipoprotein in near-wall regions of blood vessels

Differences in the dynamics and transport of blood make certain regions of the arterial network the preferred sites for initiation and formation of arterial diseases like stenosis and aneurysms. Understanding of such arterial diseases is directly linked to critical hemodynamic parameters such as the wall shear stress (WSS). The present work generalises the influence of WSS on the concentration of LDL that was observed in an earlier study. To this end, a wide variety of simplified flow domain, inspired by the near-wall regions of aneurysms and stenosis, are constructed and analyzed. The effects of pulsatile inflow condition, rheology of blood and curvature of the wall on the correlation between WSS and LDL concentration are investigated. It is demonstrated that the time-scale of variation of lumen-surface-concentration (LSC) of LDL is larger than a single cardiac cycle. As a consequence, the time-average values of WSS are sufficient to locate the regions of higher LSC. This idea is strengthened by making use of simplified flow domain that generates moving stagnation point. Further, it was observed that the rheology of the blood and curvature of the wall does not affect the observed correlation between the WSS and LDL concentration.

Automated reduction of blood coagulation models

Mathematical modeling of thrombosis typically involves modeling the coagulation cascade. Models of coagulation generally involve the reaction kinetics for dozens of proteins. The resulting system of equations is difficult to parameterize, and its numerical solution is challenging when coupled to blood flow or other physics important to clotting. Prior research suggests that essential aspects of coagulation may be reproduced by simpler models. This evidence motivates a systematic approach to model reduction. We herein introduce an automated framework to generate reduced-order models of blood coagulation. The framework consists of nested optimizations, where an outer optimization selects the optimal species for the reduced-order model and an inner optimization selects the optimal reaction rates for the new coagulation network. The framework was tested on an established 34-species coagulation model to rigorously consider what level of model fidelity is necessary to capture essential coagulation dynamics. The results indicate that a nine-species reduced-order model is sufficient to reproduce the thrombin dynamics of the benchmark 34-species model for a range of tissue factor concentrations, including those not included in the optimization process. Further model reduction begins to compromise the ability to capture the thrombin generation process. The framework proposed herein enables automated development of reduced-order models of coagulation that maintain essential dynamics used to model thrombosis.

Near-Wall Flow in Cerebral Aneurysms

The region where the vascular lumen meets the surrounding endothelium cell layer, hence the interface region between haemodynamics and cell tissue, is of primary importance in the physiological functions of the cardiovascular system. The functions include mass transport to/from the blood and tissue, and signalling via mechanotransduction, which are primary functions of the cardiovascular system and abnormalities in these functions are known to affect disease formation and vascular remodelling. This region is denoted by the near-wall region in the present work, and we outline simple yet effective numerical recipes to analyse the near-wall flow field. Computational haemodynamics solutions are presented for six patient specific cerebral aneurysms, at three instances in the cardiac cycle: peak systole, end systole (taken as dicrotic notch) and end diastole. A sensitivity study, based on Newtonian and non-Newtonian rheological models, and different flow rate profiles, is effected for a selection of aneurysm cases. The near-wall flow field is described by the wall shear stress (WSS) and the divergence of wall shear stress (WSSdiv), as descriptors of tangential and normal velocity components, respectively, as well as the wall shear stress critical points. Relations between near-wall and free-stream flow fields are discussed.

Finite element modeling of near-wall mass transport in cardiovascular flows

Many cardiovascular processes involve mass transport between blood and the vessel wall. Finite element methods are commonly used to numerically simulate these processes. Cardiovascular mass transport problems are typically characterized by high Péclet numbers, requiring fine near-wall mesh resolution as well as the use of stabilization techniques to avoid numerical instabilities. In this work, we develop a set of guidelines for solving high-Péclet-number near-wall mass transport problems using the finite element method. We use a steady, idealized test case to investigate the required mesh resolution and finite element basis order to accurately capture near-wall concentration boundary layers, as well as the performance of several commonly used stabilization techniques. Linear tetrahedral meshes were found to outperform quadratic tetrahedral meshes of equivalent degrees of freedom, and the commonly used discontinuity-capturing stabilization technique was found to be overly diffusive for these types of problems. Best practices derived from the idealized test case were then applied to a typical patient-specific vascular blood flow modeling application, where it was found that the commonly applied technique of avoiding numerical difficulties by artificially increasing mass diffusivity provides qualitatively similar but quantitatively erroneous results.

Hemodynamics-driven deposition of intraluminal thrombus in abdominal aortic aneurysms

Accumulating evidence suggests that intraluminal thrombus plays many roles in the natural history of abdominal aortic aneurysms. There is, therefore, a pressing need for computational models that can describe and predict the initiation and progression of thrombus in aneurysms. In this paper, we introduce a phenomenological metric for thrombus deposition potential and use hemodynamic simulations based on medical images from 6 patients to identify best-fit values of the 2 key model parameters. We then introduce a shape optimization method to predict the associated radial growth of the thrombus into the lumen based on the expectation that thrombus initiation will create a thrombogenic surface, which in turn will promote growth until increasing hemodynamically induced frictional forces prevent any further cell or protein deposition. Comparisons between predicted and actual intraluminal thrombus in the 6 patient-specific aneurysms suggest that this phenomenological description provides a good first estimate of thrombus deposition. We submit further that, because the biologically active region of the thrombus appears to be confined to a thin luminal layer, predictions of morphology alone may be sufficient to inform fluid-solid-growth models of aneurysmal growth and remodeling.

SimVascular: An Open Source Pipeline for Cardiovascular Simulation

Patient-specific cardiovascular simulation has become a paradigm in cardiovascular research and is emerging as a powerful tool in basic, translational and clinical research. In this paper we discuss the recent development of a fully open-source SimVascular software package, which provides a complete pipeline from medical image data segmentation to patient-specific blood flow simulation and analysis. This package serves as a research tool for cardiovascular modeling and simulation, and has contributed to numerous advances in personalized medicine, surgical planning and medical device design. The SimVascular software has recently been refactored and expanded to enhance functionality, usability, efficiency and accuracy of image-based patient-specific modeling tools. Moreover, SimVascular previously required several licensed components that hindered new user adoption and code management and our recent developments have replaced these commercial components to create a fully open source pipeline. These developments foster advances in cardiovascular modeling research, increased collaboration, standardization of methods, and a growing developer community.

Characterizations and Correlations of Wall Shear Stress in Aneurysmal Flow

Wall shear stress (WSS) is one of the most studied hemodynamic parameters, used in correlating blood flow to various diseases. The pulsatile nature of blood flow, along with the complex geometries of diseased arteries, produces complicated temporal and spatial WSS patterns. Moreover, WSS is a vector, which further complicates its quantification and interpretation. The goal of this study is to investigate WSS magnitude, angle, and vector changes in space and time in complex blood flow. Abdominal aortic aneurysm (AAA) was chosen as a setting to explore WSS quantification. Patient-specific computational fluid dynamics (CFD) simulations were performed in six AAAs. New WSS parameters are introduced, and the pointwise correlation among these, and more traditional WSS parameters, was explored. WSS magnitude had positive correlation with spatial/temporal gradients of WSS magnitude. This motivated the definition of relative WSS gradients. WSS vectorial gradients were highly correlated with magnitude gradients. A mix WSS spatial gradient and a mix WSS temporal gradient are proposed to equally account for variations in the WSS angle and magnitude in single measures. The important role that WSS plays in regulating near wall transport, and the high correlation among some of the WSS parameters motivates further attention in revisiting the traditional approaches used in WSS characterizations.