A poroelastic immersed finite element framework for modelling cardiac perfusion and fluid–structure interaction

Abstract

Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid–structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre‐reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three‐phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid. We benchmark our approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry considered in the prior work by Chapelle et al (2010). This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton. With this framework, we also simulate the poroelastic dynamics of a three‐dimensional left ventricle, in which the myocardium is described by the Holzapfel–Ogden law. Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously. We find that the poroelastic LV behaves differently from the hyperelastic LV model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well‐known wall‐stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.

A stabilized linear finite element method for anisotropic poroelastodynamics with application to cardiac perfusion

“…20. Like many previous studies [11,13,22], we only consider a portion of the LV. As illustrated in Fig.…”

“…( 38)) is used for the hyperelastic strain energy function with solid and fluid densities ρ s 0 = ρ f 0 = 1.0 g/cm 3 , material constants a = 2244.87 dyne/cm 2 , b = 1.6215, a f = 24267 dyne/cm 2 , b f = 1.8268, a s = 5562.38 dyne/cm 2 , b s = 0.7746, a fs = 3905.16 dyne/cm 2 , and b fs = 1.695, along with initial porosity φ 0 = 0.1, permeability K = 10 −8 I cm 4 dyne −1 s −1 , and pore pressure constants q 1 = 220 dyne/cm 2 , q 2 = 10090 dyne/cm 2 , and q 3 = 75. The anisotropic properties in the myocardial skeleton are modelled using myofibres and collagen sheets as specified in Richardson et al [13]. The myocardium is considered porous, with source/sink terms dependent on the pore pressure [11],…”

“…In such models, the solid part is often called the skeleton [11,12]. For cardiac perfusion, the myocardium is considered a homogeneous mixture of a solid skeleton compartment consisting of cardiac myocytes and collagen and a fluid compartment composed of coronary vessels to transport blood to and from the myocardium [11][12][13]. In homogenization, the length scale of the pores is characterized by capillary permeability and porosity [14].…”

“…Chapelle et al [11] studied the poroelastodynamics in an elliptical LV with near incompressibility of the skeleton using FEM with first-order elements for the displacement and piecewise constant elements for the pressure. Richardson et al [13] studied the poroelastodynamics in a realistic human LV using an Immersed Boundary/Finite Element method [20] that used first-order elements for both the displacement and velocity of the skeleton and the pore pressure together with stabilization using a volumetric energy term. Cookson et al [21] used a mixed method with second-order elements for the displacement and first-order elements for the pressure for a comparably realistic LV.…”

Review of cardiac–coronary interaction and insights from mathematical modeling

“…Existing cardiac–coronary mathematical models are either geometrically simplified (Fan, Namani, Choy, Kassab, et al, 2021; Fan et al, 2022; Munneke et al, 2022) in terms of myocardium and coronary vasculature or homogenized based on the poroelastic modeling framework (Chapelle et al, 2010; Lee, Nordsletten, et al, 2016; Richardson et al, 2021). These models do not capture complete anatomical, morphological, and structural details of the coronary vasculature, especially the microcirculation where blood transport and flow regulation are significant.…”