Bridging spatial and evolutionary game theory: Implications in mathematical oncology
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
In this research programme, we aim to develop new mathematical equations that are designed to find optimal anti-cancer treatment strategies in solid tumours. These equations will be formulated using game theory.
Classical game theory provides a mathematical framework in which to study strategic interactions amongst rational agents (or players) that aim to maximise their utility (or payoff) by choosing the best strategies. In the context of mathematical oncology, game theory has been adapted to describe cell-cell interactions amongst cancer and tumour cells, where the cells are players that do not rationally choose their strategies, but rather act according to their genotypic and phenotypic makeup. The interactions are then abstractions of biological exchanges of signalling molecules, or competition for space or nutrients, and the payoffs describe any gain or loss in reproductive ability (or fitness) that cells acquire as a consequence of interacting with other cells. Upon identification the payoffs, the reproductive rates (and by extension the evolution) of various genotypic and phenotypic subpopulations that co-exist in a tumour can be mathematically modelled. Two branches of game theory that have been applied to model interactions amongst cancer cells are spatial game theory (SGT) and evolutionary game theory (EGT).
In an SGT model, cells can be modelled as autonomous agents that evolve on a spatial grid and partake in interactions with other cells in their vicinity. Such agent-based models are naturally able to incorporate spatial heterogeneity. However, they do not lend themselves to rigorous mathematical analysis, and are often difficult to parameterise and computationally expensive.
In EGT models, the temporal evolution of phenotypic subpopulations of cells can be described by a set of mathematically tractable equations, called the replicator equations, after imposing a set of simplifying modelling assumptions. According to a mean-field assumption, the replicator equations notably assume that the cells are well-mixed, so that each cell interacts with all other cells in the system with equal probability. The replicator equations allow for rigorous mathematical analysis and feasible clinical applications. Consequently, EGT is one of the mathematical approaches that is currently being used to inform pre-clinical and clinical treatment strategies in oncology, where the general premise is that by perturbing the cell-cell interactions elucidated by EGT, tumour evolution can be pushed into a state that is better manageable by treatments.
The EGT replicator equations are, however, not spatially resolved and can therefore not faithfully capture the dynamics of heterogeneous solid tumours. In fact, previous work has shown that imposing spatial constraints on agent-agent interactions, via SGT models, often results in system dynamics that vastly contradicts that simulated by EGT!
Due to the pre-clinical and clinical applications of EGT, there exists a need to bridge spatial and evolutionary game theory. In this research program, we aim to achieve this SGT-EGT bridging by formulating spatio-temporal EGT equations that capture the heterogeneity found in solid tumours, whilst being more mathematically tractable than SGT models. The novel equations developed in this research programme will enrich the mathematical research fields of game theory and mathematical oncology, and may also have applications in pre-clinical and clinical cancer research!
This work will be led by Dr Sara Hamis (University of St Andrews, UK), who will be supported by an international and interdisciplinary team with mathematical, experimental and clinical expertise. Team members are Prof Mark A.J. Chaplain (University of St Andrews), Dr Alexander J. Stewart (University of St Andrews), Dr Tommaso Lorenzi (Politecnico di Torino, Italy), Dr Philip Gerlee (Chalmers University of Technology, Sweden) and Jacob G. Scott, MD (Cleveland Clinic, USA).
Classical game theory provides a mathematical framework in which to study strategic interactions amongst rational agents (or players) that aim to maximise their utility (or payoff) by choosing the best strategies. In the context of mathematical oncology, game theory has been adapted to describe cell-cell interactions amongst cancer and tumour cells, where the cells are players that do not rationally choose their strategies, but rather act according to their genotypic and phenotypic makeup. The interactions are then abstractions of biological exchanges of signalling molecules, or competition for space or nutrients, and the payoffs describe any gain or loss in reproductive ability (or fitness) that cells acquire as a consequence of interacting with other cells. Upon identification the payoffs, the reproductive rates (and by extension the evolution) of various genotypic and phenotypic subpopulations that co-exist in a tumour can be mathematically modelled. Two branches of game theory that have been applied to model interactions amongst cancer cells are spatial game theory (SGT) and evolutionary game theory (EGT).
In an SGT model, cells can be modelled as autonomous agents that evolve on a spatial grid and partake in interactions with other cells in their vicinity. Such agent-based models are naturally able to incorporate spatial heterogeneity. However, they do not lend themselves to rigorous mathematical analysis, and are often difficult to parameterise and computationally expensive.
In EGT models, the temporal evolution of phenotypic subpopulations of cells can be described by a set of mathematically tractable equations, called the replicator equations, after imposing a set of simplifying modelling assumptions. According to a mean-field assumption, the replicator equations notably assume that the cells are well-mixed, so that each cell interacts with all other cells in the system with equal probability. The replicator equations allow for rigorous mathematical analysis and feasible clinical applications. Consequently, EGT is one of the mathematical approaches that is currently being used to inform pre-clinical and clinical treatment strategies in oncology, where the general premise is that by perturbing the cell-cell interactions elucidated by EGT, tumour evolution can be pushed into a state that is better manageable by treatments.
The EGT replicator equations are, however, not spatially resolved and can therefore not faithfully capture the dynamics of heterogeneous solid tumours. In fact, previous work has shown that imposing spatial constraints on agent-agent interactions, via SGT models, often results in system dynamics that vastly contradicts that simulated by EGT!
Due to the pre-clinical and clinical applications of EGT, there exists a need to bridge spatial and evolutionary game theory. In this research program, we aim to achieve this SGT-EGT bridging by formulating spatio-temporal EGT equations that capture the heterogeneity found in solid tumours, whilst being more mathematically tractable than SGT models. The novel equations developed in this research programme will enrich the mathematical research fields of game theory and mathematical oncology, and may also have applications in pre-clinical and clinical cancer research!
This work will be led by Dr Sara Hamis (University of St Andrews, UK), who will be supported by an international and interdisciplinary team with mathematical, experimental and clinical expertise. Team members are Prof Mark A.J. Chaplain (University of St Andrews), Dr Alexander J. Stewart (University of St Andrews), Dr Tommaso Lorenzi (Politecnico di Torino, Italy), Dr Philip Gerlee (Chalmers University of Technology, Sweden) and Jacob G. Scott, MD (Cleveland Clinic, USA).
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