Biological tissue is distinguished from materials described historically by continuum mechanical theory by its ability to grow and remodel adaptively, driven by a wide range of processes across multiple spatial and temporal scales. In this thesis we derive new mathematical descriptions that capture details from across various scales and their effect on the resulting overall behaviour. Motivated by tissue engineering, we consider tissue growth on a porous scaffold. Using the multiscale homogenisation method of O'Dea \emph{et al.}, [Mathematical Medicine and Biology, 32(3):345--366, 2014] and Penta \emph{et al.}, [The Quarterly Journal of Mechanics and Applied Mathematics, 67(1):69--91, 2014] we derive a macroscale description from one posed on the microscale. Through use of a multiphase mixture model for the tissue we extend the ideas in the above to incorporate interstitial growth and cell motility. Macroscale models are obtained via two simplifications which facilitate the homogenisation: first, by taking the limit of large interphase drag and second, by linearisation about a uniform steady state. These models consist of Darcy flow and differential equations for the cell volume fraction within the scaffold and concentration of nutrient, required for growth. Effective parameters are obtained via solution of a cell problems, hence providing explicit dependence on the microscale geometry and dynamics. Closure of the model is provided by an expression for the tissue-interstitium boundary velocity, obtained from numerical investigation of the underlying multiphase description, and solutions for a sample geometry are given. The same multiscale homogenisation technique is then employed in a different context: drug uptake by cancer cells and spheroids. Beginning with a description of drug uptake and binding for a single spheroid, two different macroscale models are derived based on different scaling assumptions. These are fitted to experimental data to provide insight into uptake behaviour, with a view to revealing underlying dynamics.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:765429 |
| Date | January 2018 |
| Creators | Holden, Elizabeth |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://eprints.nottingham.ac.uk/53213/ |
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