Collaborative Research: Analysis and Control in Multi-Scale Interface Coupling between Deformable Porous Media and Lumped Hydraulic Circuits

This project will develop theoretical and computational techniques to advance understanding of the circulation of blood in the cardiovascular system, from large arteries to small vessels embedded within the tissues. This is particularly relevant for the millions of people who suffer from glaucoma, a progressive degenerative disease of the eye that leads to irreversible vision loss; understanding the role of hemodynamic factors could bring new hope to the more than 25% of glaucoma patients who continue to progress to blindness despite medical treatment. The project will cultivate science literacy among a broad audience and diversify the next generation of STEM students. Undergraduate and graduate students will be mentored by the investigators on both the theoretical and applied aspects of this project. The investigators will run Virtual Summits on “The Eye: A Window into Our Body” to build interest in mathematics research and mathematics-related careers among women and underrepresented minorities at the high school level in North Carolina and Missouri. The research program is focused on analysis and control in models of the perfusion of a tissue, a local phenomenon studied in correlation with the global features of the surrounding blood circulation. The models involve multi-scale interface coupling between partial differential equations (quasi-linear Biot model with incompressible constituents and permeability nonlinearly depending on dilation) and ordinary differential equations (nonlinear system describing fluid flow through a vascular network). The research will bridge applied and computational mathematics and apply methods from analysis and control theory, numerical analysis, non-convex optimization, and sensitivity analysis to study how local factors and global features affect the solution of the system. In the application to glaucoma, the study will elucidate the roles played by intraocular pressure (local factor) and blood pressure (global, or systemic, feature) in securing adequate blood flow to the eye, thereby facilitating the development of novel therapeutic approaches coregulating intraocular pressure and blood pressure that could potentially help many glaucoma patients. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.