Finite element analysis of biological soft tissue surrounded by a deformable membrane that controls transmembrane flow

Abstract

Background Many biological soft tissues are hydrated porous hyperelastic materials, which consist of a complex solid skeleton with fine voids and fluid filling these voids. Mechanical interactions between the solid and the fluid in hydrated porous tissues have been analyzed by finite element methods (FEMs) in which the mixture theory was introduced in various ways. Although most of the tissues are surrounded by deformable membranes that control transmembrane flows, the boundaries of the tissues have been treated as rigid and/or freely permeable in these studies. The purpose of this study was to develop a method for the analysis of hydrated porous hyperelastic tissues surrounded by deformable membranes that control transmembrane flows. Results For this, we developed a new nonlinear finite element formulation of the mixture theory, where the nodal unknowns were the pore water pressure and solid displacement. This method allows the control of the fluid flow rate across the membrane using Neumann boundary condition. Using the method, we conducted a compression test of the hydrated porous hyperelastic tissue, which was surrounded by a flaccid impermeable membrane, and a part of the top surface of this tissue was pushed by a platen. The simulation results showed a stress relaxation phenomenon, resulting from the interaction between the elastic deformation of the tissue, pore water pressure gradient, and the movement of fluid. The results also showed that the fluid trapped by the impermeable membrane led to the swelling of the tissue around the platen. Conclusions These facts suggest that our new method can be effectively used for the analysis of a large deformation of hydrated porous hyperelastic material surrounded by a deformable membrane that controls transmembrane flow, and further investigations may allow more realistic analyses of the biological soft tissues, such as brain edema, brain trauma, the flow of blood and lymph in capillaries and pitting edema.

An investigation of coupled solution algorithms for finite‐strain poroviscoelasticity applied to soft biological tissues

“…Poroviscoelastic models also require an overall rule to model the fluid flow through the porous domain. In this work, considering the absence of body forces, the fluid flow is governed by the Darcy's law, which is well‐known in the literature of porous media: 30,52,53 …”

“…In addition, this method generally results in non-symmetric systems of equations, making it computationally costly. [29][30][31] On the other hand, iteratively-coupled schemes split the biphasic problem into two subproblems: one related to the mechanical equilibrium and the other regarding to the conservation of mass. At each stage, the nonlinear subproblems are solved iteratively until a certain convergence criterion is satisfied.…”

“…While this approach is unconditionally stable and convergent, it can lead to ill‐conditioning problems, in which case preconditioners and robust linear solvers may be required. In addition, this method generally results in non‐symmetric systems of equations, making it computationally costly 29‐31 . On the other hand, iteratively‐coupled schemes split the biphasic problem into two subproblems: one related to the mechanical equilibrium and the other regarding to the conservation of mass.…”

Application of finite element analysis to tissue differentiation and bone remodelling approaches and their use in design optimization of orthopaedic implants: A review

“…Multiphasic finite elements which have the capability to represent the behaviour of biological tissues and can provide better insights into biological and physiological phenomena in biological systems. [11][12][13] Furthermore, FE analysis of implants may aid clinicians in the decision-making process for selecting a personalized orthopaedic implant based on a patient's need, allowing for improved surgical planning. 3 Nonetheless, results obtained from most of these FEA based studies of the bone-implant system tend to be qualitative rather than quantitative in nature.…”

“…Nevertheless, such simulations bear inherent limitations of projecting physical reality within the ambit of certain presumptions and hence, may never replace the essence of experimental denouements. Multiphasic finite elements which have the capability to represent the behaviour of biological tissues and can provide better insights into biological and physiological phenomena in biological systems 11–13 . Furthermore, FE analysis of implants may aid clinicians in the decision‐making process for selecting a personalized orthopaedic implant based on a patient's need, allowing for improved surgical planning 3 .…”