Modelling of some biological materials using continuum mechanics

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 phenomena. Unfortunately, classical continuum mechanics is of limited use in biomechanical problems. As cells refashion the �bres that make up a soft tissue, they sometimes alter the tissue's fundamental mechanical structure. Advanced mathematical techniques are needed in order to accurately describe this sort of biological `plasticity'. A number of such techniques have been proposed by previous researchers. However, models that incorporate biological plasticity tend to be very complicated. Furthermore, these models are often di�cult to apply and/or interpret, making them of limited practical use. One alternative approach is to ignore biological plasticity and use classical continuum 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. 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 de�nition 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 morphoelasticity can be used to obtain a constitutive law for a Maxwell viscoelastic uid that is valid at large deformation gradients. Similarly, we analyse a morphoelastic 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.

Corrected mean-field models for spatially-dependent advection-diffusion-reaction phenomena

In the exclusion-process literature, mean-field models are often derived by assuming that the occupancy status of lattice sites is independent. Although this assumption is questionable, it is the foundation of many mean-field models. In this work we develop methods to relax the independence assumption for a range of discrete exclusion process-based mechanisms motivated by applications from cell biology. Previous investigations that focussed on relaxing the independence assumption have been limited to studying initially-uniform populations and ignored any spatial variations. By ignoring spatial variations these previous studies were greatly simplified due to translational invariance of the lattice. These previous corrected mean-field models could not be applied to many important problems in cell biology such as invasion waves of cells that are characterised by moving fronts. Here we propose generalised methods that relax the independence assumption for spatially inhomogeneous problems, leading to corrected mean-field descriptions of a range of exclusion process-based models that incorporate (i) unbiased motility, (ii) biased motility, and (iii) unbiased motility with agent birth and death processes. The corrected mean-field models derived here are applicable to spatially variable processes including invasion wave type problems. We show that there can be large deviations between simulation data and traditional mean-field models based on invoking the independence assumption. Furthermore, we show that the corrected mean-field models give an improved match to the simulation data in all cases considered.

Depth-dependent biomechanical and biochemical properties of fetal, newborn, and tissue-engineered articular cartilage

Adult articular cartilage has depth-dependent mechanical and biochemical properties which contribute to zone-specific functions. The compressive moduli of immature cartilage and tissue-engineered cartilage are known to be lower than those of adult cartilage. The objective of this study was to determine if such tissues exhibit depth-dependent compressive properties, and how these depth-varying properties were correlated with cell and matrix composition of the tissue. The compressive moduli of fetal and newborn bovine articular cartilage increased with depth (p < 0.05) by a factor of 4-5 from the top 0.1 mm (28 +/- 13 kPa, 141 +/- 10 kPa, respectively) to 1 mm deep into the tissue. Likewise, the glycosaminoglycan and collagen content increased with depth (both p < 0.001), and correlated with the modulus (both p < 0.01). In contrast, tissue-engineered cartilage formed by either layering or mixing cells from the superficial and middle zone of articular cartilage exhibited similarly soft regions at both construct surfaces, as exemplified by large equilibrium strains. The properties of immature cartilage may provide a template for developing tissue-engineered cartilage which aims to repair cartilage defects by recapitulating the natural development and growth processes. These results suggest that while depth-dependent properties may be important to engineer into cartilage constructs, issues other than cell heterogeneity must be addressed to generate such tissues. (c) 2005 Elsevier Ltd. All rights reserved.

A biomechanical investigation of vertebral staples for fusionless scoliosis correction

Background: Fusionless scoliosis surgery is an early-stage treatment for idiopathic scoliosis which claims potential advantages over current fusion-based surgical procedures. Anterior vertebral stapling using a shape memory alloy staple is one such approach. Despite increasing interest in this technique, little is known about the effects on the spine following insertion, or the mechanism of action of the staple. The purpose of this study was to investigate the biomechanical consequences of staple insertion in the anterior thoracic spine, using in vitro experiments on an immature bovine model. Methods: Individual calf spine thoracic motion segments were tested in flexion, extension, lateral bending and axial rotation. Changes in motion segment rotational stiffness following staple insertion were measured on a series of 14 specimens. Strain gauges were attached to three of the staples in the series to measure forces transmitted through the staple during loading. A micro-CT scan of a single specimen was performed after loading to qualitatively examine damage to the vertebral bone caused by the staple. Findings: Small but statistically significant decreases in bending stiffness occurred in flexion,extension, lateral bending away from the staple, and axial rotation away from the staple. Each strain-gauged staple showed a baseline compressive loading following insertion which was seen to gradually decrease during testing. Post-test micro-CT showed substantial bone and growth plate damage near the staple. Interpretation: Based on our findings it is possible that growth modulation following staple insertion is due to tissue damage rather than sustained mechanical compression of the motion segment.