Geometric multiscale modeling of the cardiovascular system, between theory and practice

Abstract

This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.Our review starts with the introduction of the stand-alone problems, namely the 3D fluidstructure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works -from different research groups -which addressed the geometric multiscale approach in the past years.A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled.

Inverse problems in reduced order models of cardiovascular haemodynamics: aspects of data assimilation and heart rate variability

“…The lumped parameter models, in particular, find application either individually or as boundary conditions (open or closed loop) to higher order models (1D or 3D) in the geometrical multi-scale method [1,2]. For a patient-specific analysis, however, these models must be adapted to each patient individually.…”

Integrated Heart—Coupling multiscale and multiphysics models for the simulation of the cardiac function

“…While advancements in constructing fully 3-D flow simulations in the arterial tree have recently been made [278], nevertheless considerable difficulties arise in the personalization and calibration of such highly detailed models of patient-specific vasculature (see for instance the works on collagen fiber remodelling in cardiovascular tissues [74,16]). The three models (3-D, 1-D and lumped 0-D), however, can very effectively interplay to form the so-called geometrical multiscale models (see [90,91,209]). …”

“…Since this is insufficient to provide a complete set of Dirichlet data for the fluid equations inside the LV, (3.27) is in fact a defective boundary condition [89]. As such, condition (3.27) is imposed using Lagrange multipliers, which has the benefit that no explicit velocity profile needs to be imposed at the inflow, see [209]. In order to stabilize the velocity at the mitral valve due to flow reversal effects, the tangential component of the velocity field at the inlet, Γ in has to be further constrained to zero (see [172] and the discussion therein), leading to the inflow boundary condition:…”

“…3.4), it is necessary to complete the previously described mechano-fluidic models by prescribing physiological preload and afterload conditions to the heart. In the simplest case this can be achieved with lumped parameter models or "0-D models" that account for the characteristic arterial impedance and consist of coupled sub-systems of resistive, inductive and capacitive elements in accordance to the analogy between fluid networks and electrical circuits, see [209]. The standard approximation for the afterload produced by the systemic circulation is the three-element windkessel model [274] dQ dt 35) where the two resistive components and the capacitive element correspond to different parts of the systemic circulation: R a is the peripheral resistance mainly due to small arteries and arterioles, C a the elastic compliance mainly due to the aorta and other large arteries, and R a the characteristic resistance of the aorta.…”

Uncertainty propagation of phase contrast-MRI derived inlet boundary conditions in computational hemodynamics models of thoracic aorta

“…Many sources of uncertainty can affect the accuracy of image-based CFD results: image acquisition and processing, mathematical modelling assumptions, physical parameter values and boundary condition (BC) settings. In particular, the latter are known to be relevant in the simulation of hemodynamic scenarios (Veneziani & Vergara 2005;Grinberg & Karniadakis 2008;Spilker & Taylor 2010;Gallo et al 2012;D'Elia & Veneziani 2013;Morbiducci et al 2013;Quarteroni et al 2016).…”