  Computational models in cardiac electrophysiology are notorious for long
runtimes, restricting the numbers of nodes and mesh elements in the numerical
discretisations used for their solution. This makes it particularly challenging
to incorporate structural heterogeneities on small spatial scales, preventing a
full understanding of the critical arrhythmogenic effects of conditions such as
cardiac fibrosis. In this work, we explore the technique of homogenisation by
volume averaging for the inclusion of non-conductive micro-structures into
larger-scale cardiac meshes with minor computational overhead. Importantly, our
approach is not restricted to periodic patterns, enabling homogenised models to
represent, for example, the intricate patterns of collagen deposition present
in different types of fibrosis. We first highlight the importance of
appropriate boundary condition choice for the closure problems that define the
parameters of homogenised models. Then, we demonstrate the technique's ability
to correctly upscale the effects of fibrotic patterns with a spatial resolution
of 10 $\mu$m into much larger numerical mesh sizes of 100-250 $\mu$m. The
homogenised models using these coarser meshes correctly predict critical
pro-arrhythmic effects of fibrosis, including slowed conduction, source/sink
mismatch, and stabilisation of re-entrant activation patterns. As such, this
approach to homogenisation represents a significant step towards whole organ
simulations that unravel the effects of microscopic cardiac tissue
heterogeneities.
