## Abstract

The inverse problem, i.e., estimating parameters in an assumed model structure representing the system of interest, is central in mathematical modelling. Structural identifiability is a prerequisite to successful parameter estimation. If a model is structurally globally identifiable then there exists a unique solution to the inverse problem. Structural indistinguishability relates to the uniqueness of the structures in a set of candidate models. These two closely related concepts are of particular importance in the modelling of biological systems where conclusions are often drawn from the parameter estimates following parameter estimation and where candidate models are used to understand the underlying mechanisms of the biological system.

In this thesis two new definitions of structural identifiability and indistinguishability are presented in which the two concepts have been generalised to now also include the mixed-effects modelling framework which is frequently used in pharmaceutical applications. Several analytical methods applicable to study these concepts in mixed-effects models are presented. These are applicable to any arbitrary mixed-effects models written in state-space form. The developed methods can be used to determine whether the distribution of the set of output functions uniquely, or otherwise, determine the parameter/model structure.

Interesting results have followed from the application of these established techniques to mixed-effects models. It is shown using examples that result from either structural identifiability or indistinguishability analyses of non-mixed-effects models no longer necessarily hold for the corresponding mixed-effects model formulation. This is due to the random effects in the statistical sub-model in three different ways i) where the random effects enter into the structural model ii) the form of the random effects iii) the structure of the covariance matrix related to the random effects. These insights are collected in a set of conjectures.

Several such examples are provided including the well-known unidentifiable one-compartment absorption model whose mixed-effects version is shown to be identifiable depending on the choice of the statistical sub-model.

The contributions from this thesis are thus theoretical, but with direct practical use in a mixed-effects modelling context.