Modeling Hemodynamics in Vascular Networks Using a Geometrical Multiscale Approach: Numerical Aspects

Abstract

On the one hand the heterogeneity of the circulatory system requires the use of different models in its different compartments, featuring different assumptions on the spatial degrees of freedom. On the other hand, the mutual interactions between its compartments imply that these models should preferably not be considered separately. These requirements have led to the concept of geometrical multiscale modeling, where the main idea is to couple 3D models with reduced 1D and/or 0D models. As such detailed information on the flow field in a specific region of interest can be obtained while accounting for the global circulation. However, the combination of models with different mathematical features gives rise to many difficulties such as the assignment of boundary conditions at the interface between two models and the development of robust coupling algorithms, as the subproblems are usually solved in a partitioned way. This review aims to give an overview of the most important aspects concerning 3D-1D-0D coupled models. In addition, some applications are presented in order to illustrate the potentialities of these coupled models.

Multi-scale modeling of hemodynamics in the cardiovascular system

“…However, 0D or 1D models provide only spatially averaged flow rates at each cross-section. A common approach amongst various techniques consists of prescribing an a priori selected velocity profile in terms of a flat, parabolic, or Womersley manner fitting the given flow rate, which has the drawback that the chosen velocity profile strongly affects the numerical solution in the 3D domain [4]. One well-used and relatively effective technique is to extend the computational domain at the outlets artificially, which is usually taken as sufficiently long to have the velocities developed so as to reduce the sensitivity of the numerical solution in the zone of interest to the selected velocity profile.…”

“…Arterial vessels usually have compliant walls, which deform significantly due to the pulsation of blood pressure. One of the major challenges in the field of cardiovascular mechanics is to model interactions between blood flow and deforming vascular walls [1,3,4]. The interaction between blood flow and vessel walls can be resolved by coupling two sets of equations of the incompressible Navier-Stokes equations for blood flow and the elastodynamics equations governing the motion of vascular walls:…”

“…One strategy is to handle geometrical multi-scale models of different scales in an iterative way based on an explicit approach scheme [4]. In the explicit coupling case, the 3D-based spatially averaged flow rates derived from the boundaries of the 3D model are imposed as boundary conditions to the 0D/1D models (referred to as defective boundary conditions), which in turn compute the mean (total) pressures at the boundaries and transfer the pressure data onto the 3D model in the next time step.…”

“…The difficulty in modeling such heterogeneous hemodynamics has led to the concept of geometrical multiscale modeling [4]. The basic idea of multi-scale modeling is to couple low-dimensional (0D, 1D) to high-dimensional (3D) models so that the resulting model can reasonably capture the hemodynamic phenomena of concern whilst demanding affordable computational resource.…”

“…In this review, we overview recent studies on the macro-hemodynamic modeling of the arterial tree, geometrical multi-scale hemodynamic modeling of the heterogeneous composition of the vascular network, micro-hemodynamic modeling of the micro-vascular system (MVS), and the cellular mechanics of blood. Unlike other reviews in literature, for instance, the patient-specific modeling of the cardiovascular system by Taylor and Figueroa [1], pulse-wave propagation by van de Vosse and Stergiopulos [2], fluid-structure interactions by Holzapfel and Ogden [3], geometrical multi-scale modeling by Taelman et al [4], and flow in the microcirculation by Popel and Johnson [5], the present review aims at providing an in-depth systemic description of the framework, methodology, and strategy associated with multi-scale modeling of hemodynamics in the cardiovascular system.…”