Spatial cumulant models enable spatially informed treatment strategies and analysis of local interactions in cancer systems

Abstract

Theoretical and applied cancer studies that use individual-based models (IBMs) have been limited by the lack of a mathematical formulation that enables rigorous analysis of these models. However, spatial cumulant models (SCMs), which have arisen from recent advances in theoretical ecology, can be used to describe population dynamics generated by a specific family of IBMs, namely spatio-temporal point processes (STPPs). SCMs are spatially resolved population models formulated by a system of ordinary differential equations that approximate the dynamics of two STPP-generated summary statistics: first order spatial cumulants (densities), and second order spatial cumulants (spatial covariances).Here, we exemplify how SCMs can be used in mathematical oncology by modelling a theoretical cancer cell population comprising interacting growth factor producing and non-producing cells. To formulate our models, we use the Unified Framework Software which was recently developed by Cornell et al. (2019) and enables the generation of STPPs, SCMs and mean-field population models (MFPMs) from user-defined model descriptions. Our results demonstrate that SCMs can capture STPP-generated population density dynamics, even when MFPMs fail to do so. From both MFPM and SCM equations, we derive treatment-induced cell death rates required to achieve non-growing cell populations. When testing these treatment strategies in STPP-generated cell populations, our results demonstrate that SCM-informed strategies outperform MFPM-informed strategies in terms of inhibiting population growths. We argue that SCMs provide a new framework in which to study cell-cell interactions, and can be used to deepen the mathematical analysis of IBMs and thereby increase IBMs’ applicability in cancer research.

Simulations probe the role of space in the interplay between drug-sensitive and drug-resistant cancer cells

“…A growing body of mathematical modelling work highlight that spatio-temporal factors may impact cell-to-cell interactions and the ensuing dynamics in cancer cell populations. Recent contributions to this body of work includes studies on interactions between drug-sensitive and drug-resistant cells (Strobl et al, 2022), growth-factor producing and non-producing cells (Hamis et al, 2023), and cancer cells and immune response cells (Retzlaff et al, 2023;Ruiz-Martinez et al, 2022;van Genderen et al, 2024). In line with these studies we, here, use an on-lattice agent-based modelling approach to simulate spatially structured cell populations comprising drug-sensitive and drug-resistant cells that compete for space on the lattice.…”

Spatial interactions modulate tumor growth and immune infiltration

“…Lenia demonstrates the capability to recapitulate important features required for modeling the evolution and ecology of cancer: growth dynamics (deterministic/stochastic), cell-cell interactions, and cell migration. While here we focus here on modeling tumor-immune spatial interactions, the role of spatial structure more broadly has important implications for optimizing cancer treatment 54, 55 , modulating evolutionary dynamics of tumor heterogeneity 17, 19, 56 , and altering cell-cell competitive dynamics 57 even when using the same interaction rules 58, 59 . Lenia is flexible enough to mimic non-spatial (well-mixed) ordinary differential equations (with sufficiently large kernel sizes) as well as small-scale spatial neighborhoods commonly used in agent-based methods (e.g.…”

“…a For example, one might define E as the Lennard-Jones potential function, which describes the balance of repulsive and attractive forces as a function of the distance between two interacting particles 42 Lenia demonstrates the capability to recapitulate important features required for modeling the evolution and ecology of cancer. While here we focus here on modeling tumor-immune spatial interactions, the role of spatial structure more broadly has important implications for optimizing cancer treatment 44,45 , modulating evolutionary dynamics of tumor heterogeneity 15,17,46 , and altering cell-cell competitive dynamics 47 even when using the same interaction rules 48,49 . Lenia is flexible enough to mimic non-spatial (well-mixed) ordinary differential equations (with sufficiently large kernel sizes) as well as small-scale spatial neighborhoods commonly used in agent-based methods, such as Moore or von Neumann neighborhoods.…”

Spatial interactions modulate tumor growth and immune infiltration

“…While here we focus here on modeling tumor-immune spatial interactions, the role of spatial structure more broadly has important implications for optimizing cancer treatment 51,52 , modulating evolutionary dynamics of tumor heterogeneity 17,19,53 , and altering cell-cell competitive dynamics 54 even when using the same interaction rules 55,56 . Lenia is flexible enough to mimic non-spatial (well-mixed) ordinary differential equations (with sufficiently large kernel sizes) as well as small-scale spatial neighborhoods commonly used in agent-based methods (e.g.…”