My current research is in cancer. And I'm involved in several projects involving cancer progression and evolution and response to treatment. I have worked on breast, lung, brain, and prostate cancer, at different stages of disease, and considered scales ranging from molecular to population dynamics. Each modeling effort is unique to the particular question at hand.
Many cancers, especially at early stages, are treatable. For later stage disease, cancer cells are able to employ a wide variety of resistance mechanisms. New targeted and immunological agents have benefitted cancer treatment, but recurrence is a major problem. This occurs for a variety of reasons. Tumor cells are often very heterogeneous and do not respond to treatment in the same way. The cells in a tumor grow too fast for the vasculature to keep up, so cells must compete for space, oxygen, and resources in an environment that also changes and interacts to the presence of the tumor, which in turn affects the tumor cells. This results in an ecosystem in which evolution occurs, selection happens, and the composition of the tumor is not simply monoclonal or monophenotypic or static. And there is more to the tumor microenvironment than vasculature. Stromal interactions can affect tumor growth and response to treatment, and the patient's immune system is complicated enough before considering how it interacts with the tumor.
I use a variety of mathematical modeling tools to try to understand the mechanisms underlying treatment resistance. With better knowledge of why treatment fails, we can start to employ smarter, evolutionary-informed interventions.
I'm very interested in the spatial and temporal heterogeneity of tumors. I like to use agent-based models to understand these problems. Agent-based models (ABMs) are simulations in which the cells act as individual agents, following an algorithm for how they proliferate, move, and interact with each other and the environment. ABMs are great tools for understanding how a heterogeneous population of cells evolve in a variable environment and in response to different treatments. They can also be very computationally-expensive, time-consuming, and require knowledge of many unknown parameters and interactions.
I also use partial differential equation (PDE) models to study tumor/environment interactions. These models assume that each tumor component (e.g. specific cells, vasculature, oxygen, drug) is a continuous field in space instead of individual elements. With PDEs, we can study spatial gradients of interacting tumor components and shape and temporal dynamics of tumors. These models may be more useful for larger scale spatial interactions. PDEs can't capture heterogeneity, evolution and interactions like an ABM, but they are faster and easier to analyze.
Ordinary differential equation models (ODEs) are used when the the population size and temporal dynamics of system are most important. They assume that populations are well mixed, carrying no spatial information. ODEs are easier to analyze, but focus mostly on the temporal dynamics of a system.
The challenging part is abstracting a biological system to fit a particular question by deciding what are the most essential components that contribute to the outcome.