Predictive models of metabolite concentration for organoid expansion

This thesis is motivated by the culture of organoids, specifically developing mathematical models for metabolite transport and organoid growth within a fed-plate bioreactor. The bioreactor of interest is the proprietary CXP1 bioreactor, invented by biotechnology company Cellesce, which consists of a layer of hydrogel over which culture media is flowed. Organoids are embedded within the hydrogel and flow is thought to enhance nutrient delivery to, and facilitate waste removal from, the organoids. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on bioreactor design and operating conditions. We show how mathematical modelling can be used to improve the yield of organoids grown within CXP1, by predicting metabolite concentrations during culture for different operating conditions.

In this thesis, we develop a series of models focused on different spatial scales and aspects of organoid growth and bioreactor operation, including metabolite transport and fluid flow. We exploit the slender bioreactor geometry and use an asymptotic approach to derive a longwave approximation of the transport problem within a 2D representation of CXP1, where the organoids are modelled via volumetric (bulk) reaction terms. We show these reduced models are excellent approximations to the full 2D model and explore the behaviour of the longwave approximation using both analytical and numerical approaches. We then develop a model for growth of individual organoids embedded within the hydrogel, modelling the organoids as spheres with temporally and spatially varying radii within a cubic lattice. Hence, we derive, via a homogenisation approach, the effective macroscale behaviour across the organoid-hydrogel region while also retaining the relevant organoid-scale information. Next, we consider the corresponding 3D flow problem. We exploit the slow nature of the flow and the geometry of the media domain, and systematically reduce the problem using a lubrication scaling to a 2D stream function problem of a circle with a point source and point sink. Finally, we combine the 3D flow model into the model for metabolite transport and organoid growth. We perform an asymptotic analysis on this model to reduce its spatial dimension --- this allows us to efficiently predict the metabolic environment and associated organoid size across the CXP1 bioreactor.