Mathematics and Statistics' Assistant Professor Mohamed Sulman, Ph.D. and his doctoral student, Truong Nguyen are investigating mathematical models that describe cancer cells invasion of a biological tissue.
Sulman examines mathematical models simulating cancer cells invasion of the extracellular matrix (ECM). The invasion is driven by the cancer cells movement as a result of their interactions with the proteins in the ECM, their diffusion, the gradients of the chemicals (chemotaxis) within the ECM and the gradient in the structure of the ECM.
The models describing the dynamics of cancer cell invasion of a biological tissue are typically systems of nonlinear reaction-diffusion-taxis partial differential equations(PDEs). The solutions of these PDEs are unstable and exhibit very rapid variations at the boundary of the healthy and cancer cells. This poses significant theoretical and computational challenges to the development of efficient numerical methods for these models, according to Dr. Sulman.
Sulman and his Ph.D. student Nguyen are currently implementing efficient adaptive moving mesh methods for the numerical solution of a model that consists of a system of five nonlinear reaction-diffusion-taxis partial differential equations (PDEs) describing the dynamics of the tumor cells due to proteolytic enzyme activity, such as urokinase-type plasminogen activator (uPA). The goal of the adaptive grid methods is to accurately capture the regions of large solution variations where the cancer cells are expected to be clustering. This effort aims to give the better understanding of the spread of cancer in a human body which can play an important role in the design of strategies for cancer diagnosis and treatment.
