Bridging spatial and evolutionary game theory: Implications in mathematical oncology

在这个研究项目中，我们的目标是开发新的数学方程，旨在找到实体肿瘤的最佳抗癌治疗策略。这些方程将用博弈论来表述。经典博弈论提供了一个数学框架，用于研究理性代理人（或玩家）之间的战略互动，目的是通过选择最佳策略来最大化他们的效用（或收益）。在数学肿瘤学的背景下，博弈论已经被用来描述癌症和肿瘤细胞之间的细胞-细胞相互作用，其中细胞是玩家，它们不会理性地选择策略，而是根据它们的基因型和表型组成行事。这种相互作用是信号分子的生物交换的抽象，或者是对空间或营养物质的竞争，而回报则描述了细胞在与其他细胞相互作用中获得的生殖能力（或适应性）的任何增益或损失。在确定收益之后，可以对肿瘤中共存的各种基因型和表型亚群的繁殖率（以及由此延伸的进化）进行数学建模。博弈论的两个分支已经应用于模拟癌细胞之间的相互作用是空间博弈论（SGT）和进化博弈论（EGT）。在SGT模型中，细胞可以建模为在空间网格上进化的自主代理，并参与与附近其他细胞的相互作用。这种基于主体的模型自然能够纳入空间异质性。然而，它们不适合进行严格的数学分析，而且通常难以参数化，计算成本也很高。在EGT模型中，在强加了一组简化的建模假设之后，细胞表型亚群的时间进化可以用一组数学上易于处理的方程（称为复制因子方程）来描述。根据平均场假设，复制因子方程明显假设细胞混合良好，因此每个细胞与系统中所有其他细胞以相同的概率相互作用。复制因子方程允许严格的数学分析和可行的临床应用。因此，EGT是目前用于肿瘤学临床前和临床治疗策略的数学方法之一，其一般前提是，通过干扰EGT阐明的细胞-细胞相互作用，肿瘤进化可以被推进到一种更容易被治疗控制的状态。然而，EGT复制因子方程不是空间分解的，因此不能忠实地捕捉异质性实体肿瘤的动力学。事实上，先前的研究表明，通过SGT模型对代理-代理相互作用施加空间约束，通常会导致与EGT模拟的系统动力学大相径庭的结果！由于EGT的临床前和临床应用，有必要将空间和进化博弈论联系起来。在本研究项目中，我们的目标是通过制定时空EGT方程来实现SGT-EGT桥接，该方程捕获实体肿瘤中发现的异质性，同时比SGT模型在数学上更易于处理。在本研究项目中开发的新方程将丰富博弈论和数学肿瘤学的数学研究领域，并可能在临床前和临床癌症研究中有应用！这项工作将由Sara Hamis博士（英国圣安德鲁斯大学）领导，他将得到一个具有数学、实验和临床专业知识的国际跨学科团队的支持。团队成员包括Mark A.J. Chaplain教授（圣安德鲁斯大学）、Alexander J. Stewart博士（圣安德鲁斯大学）、Tommaso Lorenzi博士（意大利都灵理工大学）、Philip Gerlee博士（瑞典查尔姆斯理工大学）和Jacob G. Scott医学博士（美国克利夫兰诊所）。

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).