title: Multiscale modelling of perfusion and mechanics in poroelastic biological tissues
Creators: name: Miller, Laura Mary
Publisher: name: University of Glasgow
Year: 2022

description: The Theory of Poroelasticity is embraced to model the effective mechanical behaviour of a porous elastic structure with fluid percolating in the pores. Key examples of the linear theory include hard hierarchical tissues, such as the bones, the interstitial matrix in healthy and tumorous biological tissues, the human eye, artificial constructs and biomaterials, as well as rocks and soil. Nonlinear poroelasticity has been applied to modelling tumour growth and in imaging to locate tumours in an incompressible medium, to the lungs and to consider the perfused myocardium. Poroelasticity has also been applied to studying the artery walls. The current modelling approaches assume simplistic microstructures for the materials which are in general unrealistic for the desired applications. This thesis will extend the current literature by proposing exciting, novel computationally feasible macroscale models that account for realistic microstructures and can help capture the true behaviour of materials. To fulfil this modelling goal we use the asymptotic homogenization technique. To provide a complete overview of the area we begin with a re-derivation of stan- dard Biot’s poroelasticity via the asymptotic homogenization technique. In the following chapters we build upon this to create appropriate models for complex, realistic biological scenarios. We begin our development by deriving the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. This is a novel model that can account for interactions between a variety of phases at the porescale which is much more realistic of biological tissues than the previously assumed matrix homogeneity. We then further extend this theory to assume that both the matrix and fibres/inclusions are hyperelastic, thus providing one of the first few works to use asymptotic homogenization in the context of nonlinear elasticity and making the theory more applicable to the heart and arteries. We continue the development by considering an approach over three microstructural scales. We derive the balance equations for a double poroelastic material which comprises a matrix with embedded subphases. Both the matrix and subphases can be described by Biot’s anisotropic, heterogeneous, compressible poroelasticity. This gives us a macroscale model that can account for the difference in a full set of poroelastic parameters and encodes structural details on three scales. We complete our analysis by investigating our novel poroelastic composite model nu- merically. We perform a study to investigate the role that the microstructure of a poroe- lastic material has on the resulting elastic parameters. We are considering how important an effect that multiple elastic and fluid phases at the same scale have on the estimation of the material’s elastic parameters when compared with a standard poroelastic approach. This work justifies the work of this thesis. That is, the introduction of novel models with detailed microstructures should be used instead of the previously known Biot’s poroelas- ticity for materials with non-homogeneous microstructures. The final part of this thesis applies the novel poroelastic composite model to investigate how physiologically observed microstructural changes induced by myocardial infarction impact the elastic parameters of the heart.