The investigator and student propose to develop computational methods and software to solve systems of partial differential equations that arise in multiscale models of tumor invasion. 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 infl ating the number of nodes in the physiological variables. Collaboration with cancer 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 cancer biologists at Vanderbilt University, with the expectation that they will have a significant and near-term impact on understanding the mechanisms of 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 and his student propose the development of computer software and algorithms -- the mathematical rules that determine how the software works -- to study how cancer tumors, in middle and later stages of their development, invade nearby tissue. In particular, the software will handle the complicated situation where the different genetic profiles of the multitudes of individual cells within a tumor, and the different stages in the cell-division cycle of each of these 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 their genetic type or 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, resulting in software that can be used effectively with the computers of today and the near future. An important part of this research proposal is the collaboration between the investigators and colleagues in mathematics and cancer biology at Vanderbilt University in Nashville, TN, and at the University of Dundee in Scotland.
