Fluid–structure interaction for highly complex, statistically defined, biological media: Homogenisation and a 3D multi-compartmental poroelastic model for brain biomechanics

Abstract

Numerous problems of relevance in physiology and biomechanics, have at their core, the presence of a deformable solid matrix which experiences flow-induced strain. Often, this fluid–structure interaction (FSI) is directed the opposite way, i.e. it is solid deformation that creates flow, with the heart being the most prominent example. In many cases, this interaction of fluid and solid is genuinely bidirectional and strongly coupled, with solid deformation inducing flow and fluid pressure deforming the solid. Although an FSI problem, numerous cases in biomechanics are not tractable via the traditional FSI methodologies: in the internal flows that are of interest to use, the number and range of fluid passages is so vast that the direct approach of a deterministically defined boundary between fluid and solid is impossible to apply. In these cases, homogenisation and statistical treatment of the material-fluid system is possibly the only way forward. Such homogenisation,quite common to flow-only systems through porous media considerations, is also possible for FSI systems, where the loading is effectively internal to the material. A prominent technique of this type is that of poroelasticity. In this paper, we discuss a class of poroelastic theory techniques that allow for the co-existence of a multitude of – always statistically treated – channels and passages of widely different properties: termed multiple-network poroelasticity (or multicompartmental poroelasticity). This paradigm is particularly suitable for the study of living tissue, that is invariably permeated – perfused – by fluids, often different in nature and across a wide range of scales. Multicompartmental poroelasticity is capable of accounting for bidirectional coupling between the fluids and the solid matrix and allows us to track transport of a multitude of substances together with the deformation of the solid material that this transport gives rise to or is caused by, or both. For the purposes of demonstration, we utilise a complex and physiologically very important system, the human brain (specifically, we target the hippocampus), to exemplify the qualities and efficacy of this methodology during the course of Alzheimer’s Disease. The methodology we present has been implemented through the Finite Element Method, in a general manner, allowing for the co-existence of an arbitrary number of compartments. For the applications used in this paper to exemplify the method, a four-compartment implementation is used. A unified pipeline is used on a cohort of 35 subjects to provide statistically meaningful insight into the underlying mechanisms of the neurovascular unit (NVU) in the hippocampus, and to ascertain whether physical activity would have an influence in both swelling and drainage by taking into account both the scaled strain field and the proportion of perfused blood injected into the brain tissue. A key result garnered from his study is the statistically significant differences in right hemisphere hippocampal NVU swelling between males in the control group and females with mild cognitive impairment during high and low activity states.