The Hallmarks of Mathematical Oncology

Abstract

Over the past twenty-five years there has been an unparalleled increase in understanding of cancer biology. This transformation is exemplified by Hanahan and Weinberg's decision in 2011 to expand their original Hallmarks of Cancer from six traits to ten! At the same time, mathematical modelling has emerged as a natural tool for unravelling the complex processes that contribute to the initiation and progression of tumours, for testing hypotheses about experimental and clinical observations, and assisting with the development of new approaches for improving its treatment. This article starts by reviewing some of the earliest models of tumour growth and tumour responses to radiotherapy. Following Hanahan and Weinberg's lead, attention then focuses on how closer collaboration with cancer scientists and access to experimental data is stimulating the development of new and increasingly detailed models which account, for example, for tumour-immune interactions and immunotherapy. The article concludes by discussing the ways in which mathematical modelling is being integrated with experimental and clinical work and outlining how this could improve disease diagnosis and the delivery of effective personalised treatments to cancer patients. As such, the article serves as an introduction to mathematical modelling of cancer and its treatments, suitable for researchers seeking to enter the field.

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability

“…Both of which we aim to avoid. Many mathematical models have been developed with additional mechanisms, but they have not been quantitatively tested with experimental data [25,[31][32][33][34]. The experimental data and framework that we provide here are suitable to test such models.…”

“…Throughout this study we quantitatively analyse experimental data using mathematical modelling and statistical uncertainty quantification. We start with the seminal Greenspan mathematical model [10,15,16,31]. Greenspan's model describes the three phases of growth and is relatively simple in comparison to other models [31][32][33][34].…”

“…We start with the seminal Greenspan mathematical model [10,15,16,31]. Greenspan's model describes the three phases of growth and is relatively simple in comparison to other models [31][32][33][34]. This simplicity is a great advantage.…”

Quantification of spatial and phenotypic heterogeneity in an agent-based model of tumour-macrophage interactions

“…Choices of w which simplify the wPCF. By appropriately choosing w u and w v , Equation (37) reduces to the wPCF presented in Equation ( 17), to the cross-PCF in Equation (15), or to the original definition of the PCF. Function name Notation u: Discrete?…”

“…ABMs describing tumour growth have been developed within a range of software frameworks, including PhysiCell [26], HAL (Hybrid Automata Library) [27], CompuCell3D [28,29] and Chaste (Cancer, Heart and Soft Tissue Environment) [30–32]. These models have been used to study avascular and vascular tumour growth, tumour angiogenesis, tumour-immune interactions and also to simulate tumour responses to treatments that include chemotherapy, radiotherapy, immunotherapy and their combinations [36, 37]. In this paper, we use the Chaste framework to investigate the impact of microenvironmental cues on the spatial and phenotypic distribution of macrophages and their ability to control tumour growth.…”