title: Discrete-to-continuum modelling of cells to tissues
Creators: name: Barry, Roxanna G.
Publisher: name: University of Glasgow
Year: 2020
Subjects: subject: upscaling, cell modelling, discrete-to-continuum, continuum mechanics.

description: Constitutive models for the mechanics of soft tissues are typically constructed by fitting phenomenological models to in vitro experimental measurements. However, a significant challenge is to construct macroscale soft tissue models which directly encode the properties of the constituent cells and their extracellular matrix in a rational manner. In this work we present a general framework to derive multiscale soft tissue models which incorporate the properties of individual cells without necessarily assuming homogeneity or periodicity at the cell level. The aim of this thesis is to derive a new model for cardiac soft tissue which we approach by forming an individual based model. First, we consider a reduced viscoelastic model for each individual cell and couple this to a network description of a one-dimensional line of cells. We utilise a discrete-to-continuum approach to upscale this array to form new (nonlinear) continuum partial- differential equation (PDE) models for the tissue which allows for gradients in the cell properties along the line. This system is implemented for a test problem inducing a prescribed displacement at one end of the array (while remaining fixed at the other) for both uniform and non-uniform stiffness of cells. A cluster of stiffer cells in the centre of the domain (mimicking a cluster of dead cells in myocardium after an infarction) is investigated and results show that the majority of the deformation is taken on by the more flexible cells while the stiff cells undergo a minimal deformation. We extend this model to include the effects of active contraction, to simulate myocardium behaviour in a periodic domain and we observe a travelling wave of contraction moving through the domain. For all formulations, the discrete and continuum results agree well. For the test problem, these systems also agree well with analytical results of the linearised continuum PDE. We further extend this model to incorporate cell growth and proliferation to consider the dy- namics of a proliferating array, examining how assumptions about cell dissipation translate into different global behaviour. Utilising the theory of morphoelasticity, we introduce cell growth into the system by multiplicative decomposition of the deformation tensor for each cell into an unstressed growth phase and an elastic deformation phase. We investigate stress-driven growth, where a cell grows fastest when it is unstressed and the growth rate reduces under compression (the set up does not allow the cells to be in tension). In order to assess the effect of cell dis- sipation on the system, we compare two cases: first, that the dissipation is independent of cell surface area; and second, that the dissipation coefficient is linearly proportional to the current cell surface area. We observe that in the latter case, cells pay an extra penalty for enlarging and overall growth of the array is decreased. We further consider cell proliferation in this system, with cells dividing when they reach double their initial size. In this case we can predict changes in the number of cells with time showing that the growth eventually attains a constant rate. Sub- strate dissipation results in division events becoming localised to the free end of the domain, replicating the behaviour of a proliferating rim. We also observe that cell proliferation generally leads to slower growth of the array (except in cases with very small substrate dissipation). We then extend the approach to a two-dimensional rectangular array of cells atop a fixed substrate and the upper boundary of cells parallel to this is subject to zero stress, again utilis- ing a discrete-to-continuum approach to form new (nonlinear) two-dimensional continuum PDE models. We specify the general formulation where each cell’s deformation must (in general) be solved numerically, and then focus on two simpler cases where the cell deformation is ap- proximated as either a uniaxial deformation or a simple shear. For cells undergoing uniaxial deformation, we consider a time-dependent prescribed deformation along one edge of the rect- angular domain (while keeping the edge parallel to this fixed) with two different cases for the boundaries normal to the moving edge. First, we consider zero external stress where the re- sulting deformation is in all three dimensions and the cell area in contact with the substrate decreases. Second, we consider the two boundaries normal to the moving edge to be periodic. In this case, there is no deformation normal to the periodic boundaries, and the prescribed com- pression on the array is in the out-of-plane direction alone. For a simple shear deformation, we apply a constant shearing force on one edge of the rectangular array (with the opposite edge held fixed) and periodic boundary conditions on the remaining two edges. In this case, we prohibit motion normal to the periodic boundaries, allowing motion only in the direction of the shearing force. Dissipation in the system results in a transient delay in the transmission of the shearing force to all the cells in the array. Cells closer to the sheared boundary move ahead of those closer to the fixed boundary. In this case we show that this deformation can be solved analytically. We conclude this thesis with an overview of how the approaches developed within can be extended to produce new models of soft tissue mechanics.