Mathematical model and computational scheme for multi-phase modeling of cellular population and microenvironmental dynamics in soft tissue

In this paper we introduce a system of partial differential equations that is capable of modeling a variety of dynamic processes in soft tissue cellular populations and their microenvironments. The model is designed to be general enough to simulate such processes as tissue regeneration, tumor growth, immune response, and many more. It also has built-in flexibility to include multiple chemical fields and/or sub-populations of cells, interstitial fluid and/or extracellular matrix. The model is derived from the conservation laws for mass and linear momentum and therefore can be classified as a continuum multi-phase model. A careful choice of state variables provides stability in solving the system of discretized equations defining advective flux terms. A concept of deviation from normal allows us to use simplified constitutive relations for stresses. We also present an algorithm for computing numerical approximations to the solutions of the system and discuss properties of these approximations. We demonstrate several examples of applications of the model. Numerical simulations show a significant potential of the model for simulating a variety of processes in soft tissues.