Mô hình hóa Liên tục-Rời rạc Tế bào đến Mô Tissues - Luận án Tiến sĩ của Roxanna G Barry

(2020) Discrete-to-continuum modelling of cells to tissues.uk/81616/ Copyright and moral rights for this work are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This work cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Enlighten: Theses https://theses.uk/ research-enlighten@glasgow.uk Discrete-to-Continuum Modelling of Cells to Tissues Roxanna G Barry Submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy School of Mathematics and Statistics College of Science and Engineering University of Glasgow August 2020 i Abstract 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 ii 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.

Contents Declaration xi 1 Introduction 1 1.1 Soft tissue modelling .3 Individual based models .4 Discrete to continuum asymptotics. 16 2 Deformation of a Single Array of Cells 18 2.8 Equations of Motion .9 Global Boundary and Initial Conditions .10 Non-dimensional variables. 24 iii CONTENTS iv 2.2 Upscaling to continuum model (Incompressible Cells) .1 Neo-Hookean Material .2 Small Displacements in a Neo-Hookean Material .3 Comparison to macroscale static model for a long thin strip .1 Case 1: Prescribed deformation .2 Case 2: Non-uniform shear modulus .4 Application to cardiac tissue: active contraction .1 Discrete active contraction model .2 Continuum active contraction model .3 Results for the active contraction model. 48 3 Growth and Proliferation of a Single Array of Cells 51 3.7 Cell growth rate .10 Global Boundary Conditions .12 Non-dimensional variables .2 Upscaling to Continuum Model (Incompressible Cells) .1 Neo-Hookean Material .3 Results: Cell growth (Incompressible Cells) .1 Case 1: Stress dependent cell growth (no proliferation) .2 Case 2: Stress dependent cell growth with proliferation.

69 CONTENTS v 4 Two- and Three-Dimensional Discrete-to-Continuum Models 70 4.5 Discrete governing equations .7 Discrete global boundary conditions .8 Non-dimensional variables .9 Upscaling to continuum .10 Small displacements in a neo-Hookean material .12 Results Case 1: Free boundary condition along y = w0 .13 Results Case 2: Periodic boundary condition .3 Simple shear deformation .4 Discrete governing equations .6 Discrete global boundary conditions .7 Angle of shear .8 Non-dimensional variables .11 Results: Simple Shear. 98 5 Discussion 100 A Listing of Matlab Codes 105 A. 105 List of Figures 1.1 Tissues are composed of large number of cells packed together. Cells comprise of a nucleus, filament networks (e.

actin), microtubules and a cellular membrane.3 The wall of the heart is constructed from predominantly 3 layers; endocardium (inner), myocardium (middle) and pericardium (outer) (left figure from http: //stevegallik.org/sites/histologyolm.org/images/ heartwall.4 Decomposition of the deformation tensor, F into growth, G, and elastic, A, com- ponents.1 Force diagram of N viscoelastic incompressible 2D cells aligned end-to-end in a single array. The imposed a fixed boundary at x = 0 (to investigate the system in a test problem) is represented as a solid wall at the left hand side of the array. The dissipation due to cells attached to the substrate (in the plane of the page) is represented as a dashpot connected to the initial position of the cell centre (displayed on a dashed line above the array of cells here for clarity). See text for more details.2 Local reference coordinate system for cell j.3 Force balance across boundary between cells j and j + 1, summing the individ- ual forces of cell j on cell j + 1 and of cell j + 1 on cell j.4 Stretch of the N-th cell, λ1 , at t = 10.

(a) Solutions for relative tolerance values rt = 10−8 to rt = 1; (b) Solutions for absolute tolerance values at = 10−8 to at = 1. 27 vi LIST OF FIGURES vii 2.5 Solutions for a system of incompressible Neo-Hookean cells aligned in a single array subject to a prescribed displacement (10% decrease in full array length) with substrate dissipation. Solutions for N = 100 cells, η = 0 and uniform shear modulus µ = 1.6 Steady state solutions for a system of incompressible Neo-Hookean cells aligned in a single array subject to a prescribed displacement (10% decrease in full array length) with substrate dissipation (K = 10−5 ). Solutions for N = 100 cells (n = 2000 continuum nodes) and uniform shear modulus µ = 1.

The profile in x (discrete, continuum), and χ0 (static macroscale) is displayed against initial position, x0 .7 Solutions for a system of incompressible Neo-Hookean cells aligned in a single array subject to a prescribed displacement (10% decrease in full array length) with substrate dissipation. Solutions for N = 100 cells (n = 101 continuum nodes) and uniform shear modulus µ = 1.8 Solutions for a system of incompressible Neo-Hookean cells aligned in a single array subject to a prescribed displacement (10% decrease in full array length) with substrate damping in the absence of viscous dissipation (η = 0). The sys- tem is considered with a non-uniform µ(x) as defined in (2.5, 1, 10 for K = 1 (inset with the case for K = 10−5 ); (b) λ (x,t) over x at t = 0, 0.5, 1, 10 for K = 1 (inset with the case for K = 10−5 ); (c) maxx (p) for t = 0 to t = 10 for K = 10−5 , 1, 10; (d) minx (λ ) for t = 0 to t = 10 for K = 10−5 , 1, 10. 44 LIST OF FIGURES viii 2.9 Solutions for a system of incompressible Neo-Hookean cells aligned in a single array subject to a prescribed displacement (10% decrease in full array length) with viscous damping in the absence of substrate damping (K = 0).

The system is considered with a non-uniform µ(x) as defined in (2.5, 1, 10 for η = 1); (b) min( λ ) for t = 0 to t = 3 for η = 10−5 , 1, 10 (inset with λ (x,t) over x at t = 0, 0.10 Solutions for a system of incompressible neo-Hookean cells aligned in a sin- gle array in a periodic domain subject to an active contraction force, defined in (i) (i) (2. The system is con- sidered with a uniform µ = 1 with substrate damping (K ∈ [10−5 , 10]), in the absence of viscous dissipation (η = 0). Both discrete (symbols) and continuum (lines) systems are displayed.