Mathematical Biology

This figure shows the proliferating-cell density from a tumor-invasion simulation.  Current projects are the structured multiscale modeling and simulation of tumor invasion and Proteus mirabilis swarm-colony development. We refer to these systems as "structured multiscale" because they link mechanisms at the macroscopic or population scale with mechanisms at the individual scale using so-called structured variables such as age or size. The different spatial scales induce different time scales into the problem.

A major focus of our research is the computational methodology to solve these high-dimensional multiscale systems.  We use both finite element and finite difference formulations, depending upon the variable of interest.  Our computational methodology allows us to better understand the underlying systems of equations, and to continue to develop the models to ask and perhaps answer fundamental questions about the underlying biology.  Further details can be found in the manuscripts
 
Modeling the Role of the Cell Cycle in Regulating Proteus mirabilis Swarm-Colony Development

Computational Methods and Results for Structured Multiscale Models of Tumor Invasion

A Structured-Population Model of Proteus mirabilis Swarm-Colony Development

Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion

A Variable Time Step Method for an Age-dependent Population Model with Nonlinear Diffusion