Impact of force function formulations on the numerical simulation of centre-based models

Abstract

Centre-based, or cell-centre models are a framework for the computational study of multicellular systems with widespread use in cancer modelling and computational developmental biology. At the core of these models are the numerical method used to update cell positions and the force functions that encode the pairwise mechanical interactions of cells. For the latter there are multiple choices that could potentially affect both the biological behaviour captured, and the robustness and efficiency of simulation. For example, available opensource software implementations of centre-based models rely on different force functions for their default behaviour and it is not straightforward for a modeler to know if these are interchangeable. Our study addresses this problem and contributes to the understanding of the potential and limitations of three popular force functions from a numerical perspective. We show empirically that choosing the force parameters such that the relaxation time for two cells after cell division is consistent between different force functions results in good agreement of the population radius of a growing monolayer. Furthermore, we report that numerical stability is not sufficient to prevent unphysical cell trajectories following cell division, and consequently, that too large time steps can cause geometrical differences at the population level.

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“…For instance, for spherical cells in a homogeneous and isotropic environment, Γ jinormalcs=γdouble-struckI, where γ is a damping coefficient and double-struckI is the identity matrix [122]. Thus, the drag force that acts on a cell in an isotropic viscous environment is proportional to the cell’s velocity [149].…”

“…Meineke et al [150] and Drasdo [122] use this model to describe cellular mono-layers in the intestinal crypt and in vitro . For mono-layers, we might assume that all cells have roughly the same velocity, which leaves us with the first term on the left-hand side and the last term on the right-hand side of equation (3.3):γnormaldxinormaldt=∑ jfalse(F jinormaladh+F jinormalrepfalse),which amounts to a set of coupled ODEs that can be solved using explicit or implicit schemes, such as a forward Euler scheme [149,150]. In general, equation (3.3) can be written as a set of linear equations and be solved using matrix manipulation, for which standard software packages are available.…”

“…Some authors simply assume that all forces involved in cell–cell interactions are well approximated by linear springs, generalized linear springs or polynomial expressions (e.g. [ 43 , 149 , 150 ] and the references therein).…”

“…which amounts to a set of coupled ODEs that can be solved using explicit or implicit schemes, such as a forward Euler scheme [149,150]. In general, equation (3.3) can be written as a set of linear equations and be solved using matrix manipulation, for which standard software packages are available.…”

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“…However, some progress has been made on aspects of this issue. Simulation studies have been used to compare different mechanical assumptions and constitutive equations for particular classes of cell-based models, such as cell-centre models (Mathias et al, 2020;Pathmanathan et al, 2009) and vertex models (Fletcher et al, 2013). Efforts have also been made to compare and contrast five competing cell-based model paradigms, including lattice-based and off-lattice models, in a consistent computational framework (Osborne et al, 2017), to help elucidate where one may expect to see qualitative differences between model behaviours.…”