Differential equation techniques for modeling a cycle-specific oncolytic virotherapeutic

Abstract

The development of a mathematical model of oncolytic virotherapeutic vesticular stomatitis virus (VSV) is presented in stages. Standard mathematical tools are discussed along with the development and analysis of the model. A defining property of VSV is that it only affects tumor cells when they are in the active phases of the cell cycle. To model this characteristic, we first model tumor growth and separate cells into active and resting, which takes the form of a linear system of differential equations. We then take into account the minimum time needed for cells to travel through the active phases of the cell cycle, first using delay-differential equations and then later age-structured partial differential equations. Our basic tumor growth model allows us to investigate linear systems analysis (eigenvalue analysis). We then study similar techniques for delay differential equations, after adding the minimum time necessary to travel through the active phases of the cell cycle to the model. After tumor growth alone has been modeled, we include viral dynamics, which takes the form of a nonlinear system of ordinary differential equations. We investigate how linearization helps us understand how to properly develop the model. Finally we add the minimum biological time to the viral model. With the model fully developed, we arrive at a system of differential equations, one of which is an age-structured partial differential equation, which provides a nice example for discussing the method of characteristics. Finally, we show how our model can be used to investigate the dynamics of the tumor-virus system. As we travel through the development of our model, we discuss various techniques to analyze ordinary, delay, and partial differential equations.