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Congenital abnormalities of the kidney and urinary tract collectively form the most common type of prenatally diagnosed malformations. Whilst many of the crucial genes that direct the kidney developmental program are known, the mechanisms by which kidney organogenesis is achieved is still largely unclear. In this paper, we propose a mathematical model for the localisation of the ureteric bud, the precursor to the ureter and collecting duct system of the kidney. The mathematical model presented fundamentally implicates Schnakenberg-like ligand-receptor Turing patterning as the mechanism by which the ureteric bud is localised on the Wolfian duct as proposed by Menshykaul and Iber (2013) . This model explores the specific roles of regulatory proteins GREM1 and BMP as well as the domain properties of GDNF production. Our model demonstrates that this proposed pattern formation mechanism is capable of naturally predicting the phenotypical outcomes of many genetic experiments from the literature. Furthermore, we conclude that whilst BMP inhibits GDNF away from the budding site and GREM1 permits GDNF to signal, GREM1 also stabilises the effect of BMP on GDNF signalling from fluctuations in BMP sensitivity but not signal strength. [ABSTRACT FROM AUTHOR]

• Novel numerical method for capturing fine-scale heterogeneities using a coarse mesh. • Competitive homogenisation technique that functions naturally on irregular grids. • Error from Graph-based method is consistent with upscaling the homogeneous problem. • Performs well on some challenging problems of interest in Cardiac electrophysiology. Fibrosis, the excess of extracellular matrix, can affect, and even block, propagation of action potential in cardiac tissue. This can result in deleterious effects on heart function, but the nature and severity of these effects depend strongly on the localisation of fibrosis and its by-products in cardiac tissue, such as collagen scar formation. Computer simulation is an important means of understanding the complex effects of fibrosis on activation patterns in the heart, but concerns of computational cost place restrictions on the spatial resolution of these simulations. In this work, we present a novel numerical homogenisation technique that uses both Eikonal and graph approaches to allow fine-scale heterogeneities in conductivity to be incorporated into a coarser mesh. Homogenisation achieves this by deriving effective conductivity tensors so that a coarser mesh can then be used for numerical simulation. By taking a graph-based approach, our homogenisation technique functions naturally on irregular grids and does not rely upon any assumptions of periodicity, even implicitly. We present results of action potential propagation through fibrotic tissue in two dimensions that show the graph-based homogenisation technique is an accurate and effective way to capture fine-scale domain information on coarser meshes in the context of sharp-fronted travelling waves of activation. As test problems, we consider excitation propagation in tissue with diffuse fibrosis and through a tunnel-like structure designed to test homogenisation, interaction of an excitation wave with a scar region, and functional re-entry. [ABSTRACT FROM AUTHOR]