Modelling acute myeloid leukaemia in a continuum of differentiation states.

Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of cellular differentiation using single cell RNA sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into ℝ 2 or ℝ 2 with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of hematopoeisis parameterized with publicly available single cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.

Modeling cancer-immune responses to therapy.

Cancer therapies that harness the actions of the immune response, such as targeted monoclonal antibody treatments and therapeutic vaccines, are relatively new and promising in the landscape of cancer treatment options. Mathematical modeling and simulation of immune-modifying therapies can help to offset the costs of drug discovery and development, and encourage progress toward new immunotherapies. Despite advances in cancer immunology research, questions such as how the immune system interacts with a growing tumor, and which components of the immune system play significant roles in responding to immunotherapy are still not well understood. Mathematical modeling and simulation are powerful tools that provide an analytical framework in which to address such questions. A quantitative understanding of the kinetics of the immune response to treatment is crucial in designing treatment strategies, such as dosing, timing, and predicting the response to a specific treatment. These models can be used both descriptively and predictively. In this chapter, various mathematical models that address different cancer treatments, including cytotoxic chemotherapy, immunotherapy, and combinations of both treatments, are presented. The aim of this chapter is to highlight the importance of mathematical modeling and simulation in the design of immunotherapy protocols for cancer treatment. The results demonstrate the power of these approaches in explaining determinants that are fundamental to cancer-immune dynamics, therapeutic success, and the development of efficient therapies.

Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations.

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attraction. These system characteristics are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of combination therapies. Numerical simulations of mixed chemo-immuno and vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor.