Best practices in mathematical modeling.

Mathematical modeling is a vehicle that allows for explanation and prediction of natural phenomena. In this chapter we present guidelines and best practices for developing and implementing mathematical models, using cancer growth, chemotherapy, and immunotherapy modeling as examples.

Team research at the biology-mathematics interface: project management perspectives.

The success of interdisciplinary research teams depends largely upon skills related to team performance. We evaluated student and team performance for undergraduate biology and mathematics students who participated in summer research projects conducted in off-campus laboratories. The student teams were composed of a student with a mathematics background and an experimentally oriented biology student. The team mentors typically ranked the students' performance very good to excellent over a range of attributes that included creativity and ability to conduct independent research. However, the research teams experienced problems meeting prespecified deadlines due to poor time and project management skills. Because time and project management skills can be readily taught and moreover typically reflect good research practices, simple modifications should be made to undergraduate curricula so that the promise of initiatives, such as MATH-BIO 2010, can be implemented.

A validated mathematical model of cell-mediated immune response to tumor growth.

Mathematical models of tumor-immune interactions provide an analytic framework in which to address specific questions about tumor-immune dynamics. We present a new mathematical model that describes tumor-immune interactions, focusing on the role of natural killer (NK) and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune-mediated tumor rejection. The model describes tumor-immune cell interactions using a system of differential equations. The functions describing tumor-immune growth, response, and interaction rates, as well as associated variables, are developed using a least-squares method combined with a numerical differential equations solver. Parameter estimates and model validations use data from published mouse and human studies. Specifically, CD8+ T-tumor and NK-tumor lysis data from chromium release assays as well as in vivo tumor growth data are used. A variable sensitivity analysis is done on the model. The new functional forms developed show that there is a clear distinction between the dynamics of NK and CD8+ T cells. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge are able to reproduce data from the published studies. A sensitivity analysis reveals that the variable to which the model is most sensitive is patient specific, and can be measured with a chromium release assay. The variable sensitivity analysis suggests that the model can predict which patients may positively respond to treatment. Computer simulations highlight the importance of CD8+ T-cell activation in cancer therapy.