A mathematical model was developed for mesenchymal stromal cell (MSC) growth in a packed bed bioreactor that improves oxygen availability by allowing oxygen diffusion through a gas-permeable wall. The governing equations for oxygen, glucose and lactate, the inhibitory waste product, were developed assuming Michaelis-Menten kinetics, together with an equation for the medium flow based on Darcy's Law. The conservation law for the cells includes the effects of inhibition as the cells reach confluence, nutrient and waste product concentrations, and the assumption that the cells can migrate on the scaffold. The equations were solved using the finite element package, COMSOL. Previous experimental results collected using a packed bed bioreactor with gas permeable walls to expand MSCs produced a lower cell yield than was obtained using a traditional cell culture flask. This mathematical model suggests that the main contributors to the observed low cell yield were a non-uniform initial cell seeding profile and a potential lag phase as cells recovered from the initial seeding procedure. Lactate build-up was predicted to have only a small effect at lower flow rates. Thus, the most important parameters to optimise cell expansion in the proliferation of MSCs in a bioreactor with gas permeable wall are the initial cell seeding protocol and the handling of the cells during the seeding process. The mathematical model was then used to identify and characterise potential enhancements to the bioreactor design, including incorporating a central gas permeable capillary to further enhance oxygen availability to the cells. Finally, to evaluate the issues and limitations that might be encountered scale-up of the bioreactor, the mathematical model was used to investigate modifications to the bioreactor design geometry and packing density.