Grant: Discovery Grants Program - Individual

The long-term objective of my research is to develop mathematical models and numerical methods for problems from various application areas. As the source of these problems evolves, my research program maintains a balanced approach between continuing existing research topics and developing new research directions, which is reflected in the current proposal. In this proposal we will focus on several related problems in electrophysiology and complex fluids. From a mathematical point of view, this proposed research focuses on the development and analysis of methods for evolution equations involved in two classes: (i) multi-scale reaction and electro-diffusion equations, (ii) mass transfer equation coupled with Navier-Stokes equations. 1. Blood flow and mass transport in tissue consists both extension of previous work and new directions.  Potassium clearance and spreading depression are two problems of practical importance but never been investigated in the context of coupled ion, water, and blood flow.  Facial blood flow modelling and vasculature reconstruction using facial blood flow have important applications but also mathematically challenging.   2. Ion transport at sub-cellular, cellular and multi-cellular scales consists both extension of previous work and new directions. Modelling transport through permeable membrane and Ion channels are extensions of our previous work and current literature but ion pump is completely new. 3. Computational issues is where we explore new methods for solving the models.  The proposed research will have significant impact in several areas including mathematics, health and other societal disciplines in Canada and globally. The coupling of vasculature and tissue models will lead to a deeper understanding of the mechanism of mass transport and the regulation of hemostasis (maintaining local equilibria) under normal conditions. The outcome of this particular aspect of the research also has clinical implications that will allow for a better understanding of potassium clearance and neurovascular coupling under normal and pathological conditions. The proposed facial blood flow modeling and analysis of facial video images is a major step forward from our previous proof of concept study. Methods for solving coupled mass transfer and fluid flow problems with moving interfaces have many applications (e.g., filtration and purification) beyond the biomedical focus in this proposal. All of these proposed projects will provide training opportunities for Canadian and international graduate students to develop and apply advanced analytical and numerical techniques which can serve to further research in the various areas and benefit Canada in raising its profile in the field of mathematical physiology.