Mathematical modelling of tissue metabolism and growth

Abstract

The work presented in this thesis is concerned with modelling the growth of tissue constructs, with particular focus on the effects the local micro environment has on the cell cycle and metabolism. We consider two cases; multicellular tumour spheroids and orthopaedic tissue constructs. This thesis is divided into two parts. In the first part we will present a multispecies model of an avascular tumour that studies how a cell’s metabolism affects the cell cycle, spheroid growth and the mechanical forces that arise during growth. The second part consists of a study of the growth of an engineered cartilaginous tissue layer. Experimental observations will be compared to a model of the distribution of cells and extracellular matrix.

The efficiency of cancer treatments such as radiotherapy and chemotherapy are sensitive to the local environment of a cell. Therefore an essential task in tumour biology is to understand the microenvironment within a tumour. Many mathematical models study the effects of nutrients and waste products, usually assuming growth is limited by the diffusion of a single nutrient. We will look in detail at the metabolic pathways from which cells obtain energy (ATP). A multispecies model is presented that considers the transition from aerobic to anaerobic respiration and includes relevant chemical and ionic buffering reactions and transport mechanisms. Results show that potential ATP production affects the cell cycle and consequently the rate of growth. This model is simplified using mathematical analysis and is integrated with a morphoelastic model to study the development of mechanical forces. The model shows that mechanical effects are particularly important during necrosis, where large tensile forces are shown to develop. A review of the equations governing nutrient conservation is given, by developing alternative macroscopic equations based on the microscopic features of a tumour using homogenization techniques.

The second part of this thesis studies the growth of cartilaginous tissue. Biomaterials are being engineered in an attempt to replace dysfunctional tissue in the human body using cells extracted from living organisms. We model the growth of a cartilaginous tissue construct that has been grown from expanded chondrocytes seeded onto collagen coated filters. A model is developed to explain the distribution of cells and the concentration and distribution of collagen and GAGs. This is achieved by studying the local environment of the cells. Model predictions are compared to a range of experimental data and show most of the growth takes place in the upper region of the construct.

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