Generation of multicellular spatiotemporal models of population dynamics from ordinary differential equations, with applications in viral infection

Abstract

Background The biophysics of an organism span multiple scales from subcellular to organismal and include processes characterized by spatial properties, such as the diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales, through the generation of representative models. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent the spatial and stochastic features of a biological system, limiting their insights and applications. However, spatial models describing biological systems with spatial information are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them and highlights the need for simpler methods able to model the spatial features of biological systems. Results In this work, we develop a formal method for deriving cell-based, spatial, multicellular models from ODE models of population dynamics in biological systems, and vice versa. We provide examples of generating spatiotemporal, multicellular models from ODE models of viral infection and immune response. In these models, the determinants of agreement of spatial and non-spatial models are the degree of spatial heterogeneity in viral production and rates of extracellular viral diffusion and decay. We show how ODE model parameters can implicitly represent spatial parameters, and cell-based spatial models can generate uncertain predictions through sensitivity to stochastic cellular events, which is not a feature of ODE models. Using our method, we can test ODE models in a multicellular, spatial context and translate information to and from non-spatial and spatial models, which help to employ spatiotemporal multicellular models using calibrated ODE model parameters. We additionally investigate objects and processes implicitly represented by ODE model terms and parameters and improve the reproducibility of spatial, stochastic models. Conclusion We developed and demonstrate a method for generating spatiotemporal, multicellular models from non-spatial population dynamics models of multicellular systems. We envision employing our method to generate new ODE model terms from spatiotemporal and multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.

Choice of spatial discretisation influences the progression of viral infection within multicellular tissues

“…The values for the diffusion coefficient which we have presented here, D = 0.1 CD 2 h -1 in particular, are at the lower end of the range of biologically realistic values (given the conditions of our model, including tissue size, model components etc.) [16,23,26]. As such, the magnitude of the numerical artefacts which we have shown here are somewhat of a worst case scenario for a realistic model.…”

“…Spatially-realised models of viral infections within the host represent an opportunity to account for biological detail not possible in ODE models, however, representing the two July 27, 2022 14/21 spatial scales of cells and virus (or other small particles) represents a conceptual challenge [13,26]. One popular and "natural" approach to account for these two spatial scales is by coupling a discrete grid of cells to a virus density surface [23,26]. Simulating such a system, however, requires the discretisation of the viral density.…”

“…However, when simulating such models numerically, the three spatial scales which are present — virus, cells, and the size of the tissue or computational domain — introduce a layer of complexity when it comes to the discretisation of space. Whilst some works have provided some discussion of practical concerns in the design of spatial models of this nature, and in particular the role of the viral diffusion coefficient [16,23,26], there has been no direct analysis to our knowledge of the role of the resolution of spatial discretisation in these models. However, the choice of Δ x , the spacing between discrete nodes or elements for the approximate viral surface, relative to the spacing of the cell grid, is non-trivial.…”

A multiscale multicellular spatiotemporal model of local influenza infection and immune response

“… de Jong et al, 2006 , Lucas et al, 2020 Aug , Mitchell et al, 2011 , van den Brand et al, 2010 , van den Brand et al, 2012 , Zeng et al, 2007 , Shoemaker et al, 2015 , Simon et al, 2016 , Krummel et al, 2016 , Beauchemin, 2006 , Baccam et al, 2006 , Hancioglu et al, 2007 , Pawelek et al, 2012 , Price et al, 2015 , Howat et al, 2006 , Reperant et al, , Beauchemin et al, 2005 , Bouchnita et al, 2017 , Kadolsky and Yates, 2015 , Wessler et al, 2020 , Sego et al, 2020 Dec 21 , Sego et al, 2021 Sep 8 , Graner and Glazier, 1992 , Swat et al, 2012 , Ball et al, 2003 , Quiñones-Mateu et al, 2000 , Blinov et al, 2004 .…”

Modelling the within-host spread of SARS-CoV-2 infection, and the subsequent immune response, using a hybrid, multiscale, individual-based model. Part I: Macrophages

“…Therefore, improving the parameter estimation and fitting to experimental data, either directly or indirectly via ODE models (see e.g. [70]) is a subject of future work. The presented model has the capability to include more complex intracellular pathways.…”