Lorenzi Tommaso, Painter Kevin J, Villa Chiara
Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy.
Dipartimento Interateneo di Scienze, Progetto e Politiche del Territorio, Politecnico di Torino, Viale Pier Andrea Mattioli, 39, 10125, Torino, Italy.
J Math Biol. 2025 May 16;90(6):61. doi: 10.1007/s00285-025-02223-y.
Populations are heterogeneous, deviating in numerous ways. Phenotypic diversity refers to the range of traits or characteristics across a population, where for cells this could be the levels of signalling, movement and growth activity, etc. Clearly, the phenotypic distribution - and how this changes over time and space - could be a major determinant of population-level dynamics. For instance, across a cancerous population, variations in movement, growth, and ability to evade death may determine its growth trajectory and response to therapy. In this review, we discuss how classical partial differential equation (PDE) approaches for modelling cellular systems and collective cell migration can be extended to include phenotypic structuring. The resulting non-local models - which we refer to as phenotype-structured partial differential equations (PS-PDEs) - form a sophisticated class of models with rich dynamics. We set the scene through a brief history of structured population modelling, and then review the extension of several classic movement models - including the Fisher-KPP and Keller-Segel equations - into a PS-PDE form. We proceed with a tutorial-style section on derivation, analysis, and simulation techniques. First, we show a method to formally derive these models from underlying agent-based models. Second, we recount travelling waves in PDE models of spatial spread dynamics and concentration phenomena in non-local PDE models of evolutionary dynamics, and combine the two to deduce phenotypic structuring across travelling waves in PS-PDE models. Third, we discuss numerical methods to simulate PS-PDEs, illustrating with a simple scheme based on the method of lines and noting the finer points of consideration. We conclude with a discussion of future modelling and mathematical challenges.
群体是异质的，在许多方面存在差异。表型多样性是指一个群体中各种性状或特征的范围，对于细胞而言，这可能是信号传导、运动和生长活性等水平。显然，表型分布以及它如何随时间和空间变化，可能是群体水平动态的一个主要决定因素。例如，在癌性群体中，运动、生长和逃避死亡能力的差异可能决定其生长轨迹和对治疗的反应。在本综述中，我们讨论了如何扩展用于模拟细胞系统和集体细胞迁移的经典偏微分方程（PDE）方法，以纳入表型结构。由此产生的非局部模型——我们称之为表型结构偏微分方程（PS - PDEs）——形成了一类具有丰富动态的复杂模型。我们通过结构化群体建模的简要历史来引入主题，然后回顾将几个经典运动模型——包括Fisher - KPP方程和Keller - Segel方程——扩展为PS - PDE形式。我们接着进行一个教程式的章节，介绍推导、分析和模拟技术。首先，我们展示一种从底层基于主体的模型正式推导这些模型的方法。其次，我们讲述空间扩散动力学的PDE模型中的行波以及进化动力学的非局部PDE模型中的浓度现象，并将两者结合起来推导PS - PDE模型中行波的表型结构。第三，我们讨论模拟PS - PDEs的数值方法，用一个基于线方法的简单方案进行说明，并指出需要考虑的细节。我们最后讨论未来建模和数学方面的挑战。
