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Computational models in cardiac electrophysiology are notorious for longruntimes, restricting the numbers of nodes and mesh elements in the numericaldiscretisations used for their solution. This makes it particularly challengingto incorporate structural heterogeneities on small spatial scales, preventing afull understanding of the critical arrhythmogenic effects of conditions such ascardiac fibrosis. In this work, we explore the technique of homogenisation byvolume averaging for the inclusion of non-conductive micro-structures intolarger-scale cardiac meshes with minor computational overhead. Importantly, ourapproach is not restricted to periodic patterns, enabling homogenised models torepresent, for example, the intricate patterns of collagen deposition presentin different types of fibrosis. We first highlight the importance ofappropriate boundary condition choice for the closure problems that define theparameters of homogenised models. Then, we demonstrate the technique's abilityto correctly upscale the effects of fibrotic patterns with a spatial resolutionof 10 $\mu$m into much larger numerical mesh sizes of 100-250 $\mu$m. Thehomogenised models using these coarser meshes correctly predict criticalpro-arrhythmic effects of fibrosis, including slowed conduction, source/sinkmismatch, and stabilisation of re-entrant activation patterns. As such, thisapproach to homogenisation represents a significant step towards whole organsimulations that unravel the effects of microscopic cardiac tissueheterogeneities.

Homogenisation for the monodomain model in the presence of microscopic fibrotic structures