Understanding and characterising the diffusive transport of capillary oxygen and nutrients in striated muscles is key to assessing angiogenesis and investigating the efficacy of experimental and therapeutic interventions for numerous pathological conditions, such as chronic ischaemia. In articular, the influence of both muscle tissue and microvascular heterogeneities on capillary oxygen supply is poorly understood. The objective of this thesis is to develop mathematical and computational modelling frameworks for the purpose of extending and generalising the current use of histology in estimating the regions of tissue supplied by individual capillaries to facilitate the exploration of functional capillary oxygen supply in striated muscles. In particular, we aim to investigate the balance between local capillary supply of oxygen and oxygen demand in the presence of various anatomical and functional heterogeneities, by capturing tissue details from histological imaging and estimating or predicting regions of capillary supply. Our computational method throughout is based on a finite element framework that captures the anatomical details of tissue cross sections. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe oxygen transport from capillaries to uniform muscle tissues (e.g. cardiac muscle). Transport is then explored in terms of oxygen levels and capillary supply regions. In Chapter 3 we extend this modelling framework to explore the influence of the surrounding tissue by accounting for the spatial anisotropies of fibre oxygen demand and diffusivity and the heterogeneity in fibre size and shape, as exemplified by mixed muscle tissues (e.g. skeletal muscle). We additionally explore the effects of diffusion through the interstitium, facilitated--diffusion by myoglobin, and Michaelis--Menten kinetics of tissue oxygen consumption. In Chapter 4, a further extension is pursued to account for intracellular heterogeneities in mitochondrial distribution and diffusive parameters. As a demonstration of the potential of the models derived in Chapters 2--4, in Chapter 5 we simulate oxygen transport in myocardial tissue biopsies from rats with either impaired angiogenesis or impaired arteriolar perfusion. Quantitative predictions are made to help explain and support experimental measurements of cardiac performance and metabolism. In the final chapter we summarize the main results and indicate directions for further work.