Helen Byrne


1Helen Byrne (University of Oxford)

Approaches to modelling the growth and response to treatment of solid tumours

There is a long tradition of developing mathematical and computational models that describe the growth and response to treatment of Solid tumours. Such models range from simple, phenomenological models to  detailed mathematical and computational models.  The phenomenological models typically contain small numbers of parameters and, as a result, can be Validated and parameterised against available experimental and/or clinical data. By contrast, more detailed models Are typically too complex to be fully validated against data, although they can generate valuable mechanistic insight into the processes that regulate tumour growth. In this lecture course, I will introduce a range of mathematical approaches that have been used to study different aspects of tumour growth, highlighting their relative strengths And weaknesses, while also identifying directions for future investigation.

Lecture 1, will focus on one-dimensional, continuum models of avascular tumour growth, formulated as non-standard systems of partial and integro differential equations. We will learn how the earliest models of avascular tumour growth can be derived from more complex, multiphase models PDE models and also discuss their extension to account for Vascular tumour growth.

Lecture 2, we will extend the models from Lecture 1 to investigate tumour responses to treatment with, e.g., radiotherapy, chemotherapy and immunotherapy. We will also compare the predictions of the PDE models with those generated from simpler, phenomenological ones.  Attention will then turn to PDE models of tumour angiogenesis, the process by which avascular tumours stimulate the growth of a new blood vessels from pre-existing ones, And transition from relatively harmless, localised masses to rapidly growing and potentially life-threatening vascular tumours. Accordingly, in

Lecture 3, we will study the classical, snail-trail model of angiogenesis, focussing on the assumptions on which it is based, its solutions and its relationship to alternative mesoscopic and microscopic models The course will conclude, in

Lecture 4, with an introduction to hybrid, multi scale models of avascular and vascular tumour growth. We will learn How these computationally intensive models couple sub cellular processes (such as cell cycle progress and protein production) with cell scale processes (such as cell death and proliferation) And macro scale phenomena (such as the transport of nutrients and drugs within the tumour microenvironment). We will discuss the insight that these models can generate and the challenges associated with their validation.