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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 open-source software implementations of centre-based models rely on different force functions for their default behaviour and it is not straightforward for a modeller 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 two-dimensional monolayer relaxing mechanically after intense cell proliferation. 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.

Center-based models are used to simulate the mechanical behavior of biological cells during embryonic development or cancer growth. To allow for the simulation of biological populations potentially growing from a few individual cells to many thousands or more, these models have to be numerically efficient, while being reasonably accurate on the level of individual cell trajectories. In this work, we increase the robustness, accuracy, and efficiency of the simulation of center-based models by choosing the time steps adaptively in the numerical method. We investigate the gain in using single rate time stepping for the forward and backward Euler methods, based on local estimates of the numerical errors and the stability of the method in the case of the explicit forward Euler method. Furthermore,we propose a multirate time stepping scheme that simulates regions with high local force gradients (e.g. as they happen after cell division) with multiple smaller time steps within a larger single time step for regions with smoother forces. These methods are compared for different model systems in numerical experiments. We conclude that the adaptive single rate forward Euler method results in significant gains in terms of reduced wall clock times for the simulation of a linearly growing tissue, while at the same time eliminating the need for manual determination of a suitable time step size.

Background: Cell-based models are becoming increasingly popular for applications in developmental biology. However, the impact of numerical choices on the accuracy and efficiency of the simulation of these models is rarely meticulously tested. Without concrete studies to differentiate between solid model conclusions and numerical artifacts, modelers are at risk of being misled by their experiments’ results. Most cell-based modeling frameworks offer a feature-rich environment, providing a wide range of biological components, but are less suitable for numerical studies. There is thus a need for software specifically targeted at this use case.

Results: We present CBMOS, a Python framework for the simulation of the center-based or cell-centered model. Contrary to other implementations, CBMOS’ focus is on facilitating numerical study of center-based models by providing access to multiple ordinary differential equation solvers and force functions through a flexible, user-friendly interface and by enabling rapid testing through graphics processing unit (GPU) acceleration. We show-case its potential by illustrating two common workflows: (1) comparison of the numerical properties of two solvers within a Jupyter notebook and (2) measuring average wall times of both solvers on a high performance computing cluster. More specifically, we confirm that although for moderate accuracy levels the backward Euler method allows for larger time step sizes than the commonly used forward Euler method, its additional computational cost due to being an implicit method prohibits its use for practical test cases.

Conclusions: CBMOS is a flexible, easy-to-use Python implementation of the center-based model, exposing both basic model assumptions and numerical components to the user. It is available on GitHub and PyPI under an MIT license. CBMOS allows for fast prototyping on a central processing unit for small systems through the use of NumPy. Using CuPy on a GPU, cell populations of up to 10,000 cells can be simulated within a few seconds. As such, it will substantially lower the time investment for any modeler to check the crucial assumption that model conclusions are independent of numerical issues.

During early development, cartilage provides shape and stability to the embryo while serving as a precursor for the skeleton. Correct formation of embryonic cartilage is hence essential for healthy development. In vertebrate cranial cartilage, it has been observed that a flat and laterally extended macroscopic geometry is linked to regular microscopic structure consisting of tightly packed, short, transversal clonar columns. However, it remains an ongoing challenge to identify how individual cells coordinate to successfully shape the tissue, and more precisely which mechanical interactions and cell behaviors contribute to the generation and maintenance of this columnar cartilage geometry during embryogenesis. Here, we apply a three-dimensional cell-based computational model to investigate mechanical principles contributing to column formation. The model accounts for clonal expansion, anisotropic proliferation and the geometrical arrangement of progenitor cells in space. We confirm that oriented cell divisions and repulsive mechanical interactions between cells are key drivers of column formation. In addition, the model suggests that column formation benefits from the spatial gaps created by the extracellular matrix in the initial configuration, and that column maintenance is facilitated by sequential proliferative phases. Our model thus correctly predicts the dependence of local order on division orientation and tissue thickness. The present study presents the first cell-based simulations of cell mechanics during cranial cartilage formation and we anticipate that it will be useful in future studies on the formation and growth of other cartilage geometries.