Applied Numerical Mathematics of Structured Multiscale Models of Biological Systems

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The investigator proposes to develop computational methods and software to solve systems of partial differential equations that arise in multiscale modelsof various biological systems. The behavior of these systems depends not only on time and space, but also on physiological traits such as size or age. The methods decouple time from the physiological variables. In doing so they, unlike previous methods, prevent the often fine-resolution time discretization from inflating thenumber of nodes in the physiological variables. This family of methods remains the one with the highest-order accuracy due to the fact that approximation error is the only meaningful source of error in the computations. Collaboration with biologists is a critical component of this research plan. The software will be developed and used in the context of models developed and parameterized with biofilm researchers at Montana State University and cancer biologists at Vanderbilt University, with the expectation that they will have a significant and near-term impact on understanding the mechanisms of biofilm growth and tumor invasion. This collaboration is also vital to ensure the relevance and usefulness of the methods and software; methods developed in an application vacuum tend to lack utility.The investigator proposes the development of computer software and algorithms -- the mathematical rules that determine how the software works -- to study howbiofilms develop and how cancer tumors, in middle and later stages of their development, invade nearby tissue. Biofilms are bacteria that live in a material casing of their own creation. They are involved in many of the most difficult to treat infections and in industrial fouling, and are used in wastewater remediation. The software will handle complicated situations where individual cells within the biofilm or tumor differ in some important respect. For example, different bacteria in a biofilm will have different abilities to grow and divide, based on the number of divisions already undertaken. The different stages in the cell-division cycle of cancer cells are linked to the physically larger complete tumor. This is important, for example, in studying the overall effects of chemotherapy when using drugs that affect cells differently depending on what part of the cell-division cycle they are in. The mathematics behind the algorithms in this proposal are significantly more advanced than what came before and remain the state of the art, resulting in software that can be used effectively with the computers of today and the near future. They can also be expected to be useful for problems other than biofilms and tumors. An important part of this research proposal is the collaboration between the investigator and colleagues at Montana State University in Bozeman, MT and Vanderbilt University in Nashville, TN.