Mathematical modelling and identifiability applied to positron emission tomography data

Positron emission tomography (PET) is an in vivo tracer kinetic technique. This thesis is concerned with the analysis of data derived from PET studies in humans. There are two related themes in the thesis. Firstly, the derivation of mathematical models with particular reference to the modelling of radiolabelled metabolite formation in plasma and tissue. Secondly, the identifiability of model structures is examined, and a method for the reparameterisation of unidentifiable models is derived. Compartmental models describing the accumulation of radiolabelled metabolites in plasma following the intravenous administration of [11C]flumazenil and [11C]diprenorphine are presented. A theorem is presented which gives conditions for a unique solution to the spectral analysis approach (a kinetic modelling technique used in PET which is based on the a priori definition of a large set of basis functions). Mathematical techniques are presented for the analysis of expired 11CO2, a major labelled metabolite in many PET studies. This range of analytical and modelling techniques is then applied to the analysis of [11C]thymidine scans. [11C]Thymidine is a PET tracer being developed for the measure of tumour proliferation in cancer patients. The techniques developed in the thesis allow for the removal of the confounding labelled metabolite signals from both plasma and tissue data.