Modelling of some biological materials using continuum mechanics Cameron Luke Hall B. App. Sc. (Hons I) Queensland University of Technology A thesis submitted for the degree of Doctor of Philosophy in the School of Mathematical Sciences, Faculty of Science, Queensland University of Technology. 2008 Principal Supervisor: Professor D. L. Sean McElwain Associate Supervisor: Associate Professor Graeme J. Pettet Associate Supervisor: Professor Zee Upton Abstract Continuum mechanics provides a mathematical framework for modelling the physical stresses experienced by a material. Recent studies show that physical stresses play an important role in a wide variety of biological processes, including dermal wound healing, soft tissue growth and morphogenesis. Thus, continuum mechanics is a useful mathematical tool for modelling a range of biological phe- nomena. Unfortunately, classical continuum mechanics is of limited use in biomechanical problems. As cells refashion the fibres that make up a soft tissue, they some- times alter the tissue's fundamental mechanical structure. Advanced mathemat- ical techniques are needed in order to accurately describe this sort of biological `plasticity'. A number of such techniques have been proposed by previous researchers. How- ever, models that incorporate biological plasticity tend to be very complicated. Furthermore, these models are often difficult to apply and/or interpret, making them of limited practical use. One alternative approach is to ignore biological plasticity and use classical con- tinuum mechanics. For example, most mechanochemical models of dermal wound healing assume that the skin behaves as a linear viscoelastic solid. Our analysis indicates that this assumption leads to physically unrealistic results. i ii Abstract In this thesis we present a novel and practical approach to modelling biological plasticity. Our principal aim is to combine the simplicity of classical linear models with the sophistication of plasticity theory. To achieve this, we perform a careful mathematical analysis of the concept of a `zero stress state'. This leads us to a formal definition of strain that is appropriate for materials that undergo internal remodelling. Next, we consider the evolution of the zero stress state over time. We develop a novel theory of `morphoelasticity' that can be used to describe how the zero stress state changes in response to growth and remodelling. Importantly, our work yields an intuitive and internally consistent way of modelling anisotropic growth. Furthermore, we are able to use our theory of morphoelasticity to develop evolution equations for elastic strain. We also present some applications of our theory. For example, we show that mor- phoelasticity can be used to obtain a constitutive law for a Maxwell viscoelastic fluid that is valid at large deformation gradients. Similarly, we analyse a mor- phoelastic model of the stress-dependent growth of a tumour spheroid. This work leads to the prediction that a tumour spheroid will always be in a state of radial compression and circumferential tension. Finally, we conclude by presenting a novel mechanochemical model of dermal wound healing that takes into account the plasticity of the healing skin. Keywords: asymptotic analysis, biomechanics, continuum mechanics, dermal wound healing, growth, mathematical biology, mathematical modelling, morphoe- lasticity, plasticity, residual stress, viscoelasticity, zero stress state Contents Abstracti List of figures viii Statement of original authorshipx Acknowledgements xi 1 Introduction1 2 Literature Review5 2.1 Introduction..............................5 2.2 Biology of dermal wound healing..................7 2.2.1 Normal dermal wound healing in humans..........8 2.2.2 Hypertrophic scars and keloids................ 16 2.2.3 In vitro studies of traction and contraction......... 19 2.3 Mechanochemical models of dermal wound healing......... 21 iii iv Contents 2.3.1 The Tranquillo-Murray model................ 23 2.3.2 Other mechanochemical models............... 29 2.3.3 Models of lattice contraction................. 32 2.4 Mathematical approaches to growth and remodelling....... 35 2.5 The present work in context..................... 43 3 Analysis of the Tranquillo-Murray wound healing model 45 3.1 Introduction.............................. 45 3.2 Outline of the Tranquillo-Murray model.............. 47 3.2.1 Description of the model................... 47 3.2.2 Preliminary mathematical analysis............. 54 3.2.3 Numeric simulations..................... 57 3.3 Asymptotic analysis......................... 74 3.3.1 Simplifying the Tranquillo-Murray model.......... 74 3.3.2 Matched asymptotics and the Wright function....... 79 3.4 Elasticity vs. viscoelasticity..................... 97 3.4.1 Experimental measurements of dermal viscoelasticity... 97 3.4.2 Analysis of an elastic-only model.............. 103 3.5 Towards a new model of dermal wound healing........... 110 4 Zero stress state theory and the definition of strain 113 Contents v 4.1 Introduction.............................. 113 4.2 The classical theory of elastic deformation............. 116 4.2.1 Motions, deformations and coordinate systems....... 116 4.2.2 The deformation gradient tensor............... 119 4.2.3 Rotations and orthogonal tensors.............. 121 4.2.4 The Cauchy-Green tensors.................. 125 4.2.5 The classical definition of strain............... 128 4.3 Stress, strain and the zero stress state................ 133 4.3.1 Stress and plasticity..................... 133 4.3.2 Residual stresses and the zero stress state......... 136 4.3.3 The improved definition of strain.............. 146 4.4 Infinitesimal strain approximations................. 150 4.4.1 The principal zero stress deformation gradient....... 151 4.4.2 Eulerian infinitesimal strain................. 155 4.4.3 Lagrangian infinitesimal strain................ 157 4.5 The challenge of defining strain................... 158 5 Morphoelasticity 161 5.1 Introduction.............................. 161 5.2 One-dimensional morphoelasticity.................. 164 5.3 Three-dimensional morphoelasticity................. 175 vi Contents 5.3.1 Volumetric growth...................... 175 5.3.2 The pre-symmetrised growth tensor............. 181 5.3.3 A unique representation of the growth tensor........ 188 5.3.4 Incorporating morphoelasticity into a model........ 198 5.4 Evolution equations for strain.................... 203 5.4.1 One-dimensional strain evolution.............. 203 5.4.2 Three-dimensional strain evolution............. 206 5.5 Comparison with other mechanical treatments of growth..... 221 5.5.1 The Lagrangian growth equation.............. 221 5.5.2 Rodriguez et al. and the rate of growth tensor....... 227 5.5.3 Goriely and Ben Amar's cumulative theory of growth... 231 5.6 Summary of zero stress state theory and morphoelasticity..... 239 6 Applications of zero stress state theory and morphoelasticity 246 6.1 Introduction.............................. 246 6.2 A one-dimensional Maxwell material................ 249 6.3 Modelling growth........................... 260 6.3.1 Dependence of the growth tensor on stress......... 260 6.3.2 A model of spherically-symmetric growth.......... 267 6.4 A mechanochemical model of dermal wound healing........ 280 6.4.1 Modelling approach...................... 280 Contents vii 6.4.2 Development of model equations............... 288 6.4.3 Initial conditions and boundary conditions......... 298 6.4.4 Nondimensionalisation.................... 306 6.4.5 Preliminary observations and further work......... 312 6.5 Further applications of morphoelasticity.............. 315 7 Conclusions and further work 318 Appendices 322 A Asymptotic analysis of a general viscoelastic wound healing model 322 B Existence and uniqueness of the zero stress state.......... 338 B.1 Existence of the zero stress Cauchy-Green tensor..... 338 B.2 Uniqueness of the zero stress Cauchy-Green tensor..... 343 C Rodriguez et al.'s definition of Lagrangian strain.......... 346 D Finite strain evolution and related topics.............. 349 D.1 Preliminary results...................... 349 D.2 Evolution of finite strain................... 352 D.3 Observer independence.................... 353 Bibliography 358 List of Figures 2.1 Illustration of the healthy skin....................8 2.2 Inflammation............................. 11 2.3 Proliferation.............................. 12 2.4 Maturation.............................. 15 3.1 A narrow rectangular wound..................... 48 3.2 Numeric results obtained from the base Tranquillo-Murray model 59 3.3 Tranquillo-Murray model showing oscillations at large values of x 61 3.4 The Tranquillo-Murray model with and without advection: cell density profiles............................ 63 3.5 The Tranquillo-Murray model with and without advection: ECM displacement profiles......................... 64 3.6 Tranquillo-Murray model: γ = 0................... 66 3.7 Tranquillo-Murray model: r0 = 0.................. 68 3.8 Viscous-only Tranquillo-Murray model: E¯ = 0........... 70 3.9 Tranquillo-Murray model: µ 1.................. 72 viii List of figures ix 3.10 Tranquillo-Murray model: s = 0................... 73 3.11 Comparison of numeric results from the simplified Tranquillo-Murray model with a short-space asymptotic approximation........ 86 3.12 Comparison of numeric results from the simplified Tranquillo-Murray model with a long-space asymptotic approximation........ 94 3.13 ECM displacement profiles from the elastic-only Tranquillo-Murray