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Developing PDE-compartment hybrid frameworks for modelling cell migration

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Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, due to recent advances in computational power, the simulation, and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist.

The specific stochastic models with which we shall be concerned in this talk are referred to as `compartment based’ or `on-lattice’. These models are characterised by a discretisation of the computational domain into a grid/lattice of `compartments’. Within each compartment particles are assumed to be well-mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments.

Individual-based stochastic models provide microscopic/mesoscopic accuracy but at the cost of significant computational resources. Models which have regions of both low and high concentrations often necessitate coupled macroscale and microscale modelling paradigms. This is because microscale models are not feasible to simulate at large concentrations and macroscale models are often inappropriate at small concentrations.

In this work we develop hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: `how are individual particles transported between the vastly different model descriptions?’ First, we present an algorithm derived by carefully re-defining the continuous PDE concentration as a probability distribution. Whilst this first algorithm shows very strong convergence to analytic solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations to analytic solutions of PDEs for mean concentrations.

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