Department of Computational and Applied Mathematics
A Numerical Model For Edema Formation In Layered, Poroelastic Tissue
Edema is generalized clinical condition referring to the accumulation of excess fluid between cells, or in cavities; in the intestine the result is a loss of contractility by disrupting peristaltic activity. The canonical clinical model of edema used in the medical literature is an ordinary differential equation representing a simple balance of lymphatic and vascular fluid exchange based on the Drake Lane and Starling Landis equations. These ODE models rely on an tissue bed structure hypothesis that is too simplistic to represent the layered, heterogeneous physiology of the intestine; furthermore, they neglect the mechanical response of the tissue itself.
Recently, an approach for modeling the combined fluid / mechanical interaction based on a simplified, mixed form of Biot's equations of poroelasticity has been put forth (Young & Riviere); the model was discretized using the symmetric interior penalty discontinuous Galerkin method with piecewise linear elements. Clinical experiments conducted at the UT Health Sciences center in 2008 provided a means for verification; no error analysis was conducted.
In this talk I will present the full mixed form of the Biot model, discretization with the (non-conforming) discontinuous Galerkin SIPG / NIPG approaches, a-priori error estimates, implementation in 2D and 3D, numerical convergence results, and robustness to locking. I will then introduce a perturbed model, based on the generalization, suited for use in the context of the heterogeneous physiology of the intestine. I will share preliminary physiological computations, and ongoing work of both a clinical and mathematical nature.