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

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.