A Porous Media Approach for Foot Biomechanics

Recently a new computational model, based on the thermodynamically constrained averaging theory (TCAT), has been proposed to predict tumor initiation and proliferation [1, 2]. A similar mathematical approach is being developed for diabetic ulcer prevention. The common aspects at continuum level are the macroscopic balance equations governing the flow of the fluid phase, diffusion of chemicals and tissue mechanics, and some of the constitutive equations. We show here as first step a porous media model of the foot where the ulcer model will be introduced in future. TCAT [3] is a framework recently established for the analysis of multiphase systems, which is consistent over multiple scales. It provides a rigorous yet flexible method for developing multiphase, continuum models at any scale of interest. TCAT uses averaging theorems to formally and consistently convert micro-scale equations to the larger macro-scale. The soft plantar tissue is modeled as a multiphase system consisting of two phases: one solid for the tissue cells and their extracellular matrix (i.e. ECM components and the tissue cells are treated as a single solid phase), and one fluid (the interstitial fluid). The load history and boundary conditions are consistent with experimental measurements performed at the mega level: in-vivo foot kinematics and magnetic resonance give the input data of the model [4]. The governing equations are discretized in space by the finite element method [5] and in time domain by using the θ-Wilson Method. An example of application is presented at the end of the paper where attention is focused on quantities which will be of importance in diabetic ulcer modeling.