Mathematical modelling of chronic wound healing

Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.

Ecological aspects of biosecurity surveillance design for the detection of multiple invasive animal species

Complex surveillance problems are common in biosecurity, such as prioritizing detection among multiple invasive species, specifying risk over a heterogeneous landscape, combining multiple sources of surveillance data, designing for specified power to detect, resource management, and collateral effects on the environment. Moreover, when designing for multiple target species, inherent biological differences among species result in different ecological models underpinning the individual surveillance systems for each. Species are likely to have different habitat requirements, different introduction mechanisms and locations, require different methods of detection, have different levels of detectability, and vary in rates of movement and spread. Often there is a further challenge of a lack of knowledge, literature, or data, for any number of the above problems. Even so, governments and industry need to proceed with surveillance programs which aim to detect incursions in order to meet environmental, social and political requirements. We present an approach taken to meet these challenges in one comprehensive and statistically powerful surveillance design for non-indigenous terrestrial vertebrates on Barrow Island, a high conservation nature reserve off the Western Australian coast. Here, the possibility of incursions is increased due to construction and expanding industry on the island. The design, which includes mammals, amphibians and reptiles, provides a complete surveillance program for most potential terrestrial vertebrate invaders. Individual surveillance systems were developed for various potential invaders, and then integrated into an overall surveillance system which meets the above challenges using a statistical model and expert elicitation. We discuss the ecological basis for the design, the flexibility of the surveillance scheme, how it meets the above challenges, design limitations, and how it can be updated as data are collected as a basis for adaptive management.

A mathematical model of chlamydial infection incorporating movement of chlamydial particles

We present a spatiotemporal mathematical model of chlamydial infection, host immune response and spatial movement of infectious particles. The re- sulting partial differential equations model both the dynamics of the infection and changes in infection profile observed spatially along the length of the host genital tract. This model advances previous chlamydia modelling by incorporating spatial change, which we also demonstrate to be essential when the timescale for movement of infectious particles is equal to, or shorter than, the developmental cycle timescale. Numerical solutions and model analysis are carried out, and we present a hypothesis regarding the potential for treatment and prevention of infection by increasing chlamydial particle motility.

The effect of friction on indenter force and pile-up in numerical simulations of bone nanoindentation

Nanoindentation is a useful technique for probing the mechanical properties of bone, and finite element (FE) modeling of the indentation allows inverse determination of elasto-plastic constitutive properties. However, all but one FE study to date have assumed frictionless contact between indenter and bone. The aim of this study was to explore the effect of friction in simulations of bone nanoindentation. Two dimensional axisymmetric FE simulations were performed using a spheroconical indenter of tip radius 0.6 m and angle 90°. The coefficient of friction between indenter and bone was varied between 0.0 (frictionless) and 0.3. Isotropic linear elasticity was used in all simulations, with bone elastic modulus E=13.56GPa and Poisson‟s ratio f 0.3. Plasticity was incorporated using both Drucker-Prager and von Mises yield surfaces. Friction had a modest effect on the predicted force-indentation curve for both von Mises and Drucker-Prager plasticity, reducing maximum indenter displacement by 10% and 20% respectively as friction coefficient was increased from zero to 0.3 (at a maximum indenter force of 5mN). However, friction has a much greater effect on predicted pile-up after indentation, reducing predicted pile-up from 0.27 to 0.11 m with a von Mises model, and from 0.09 to 0.02 m with Drucker-Prager plasticity. We conclude that it is potentially important to include friction in nanoindentation simulations of bone if pile-up is used to compare simulation results with experiment.

What aspects of the scoliosis correction are most important to the teenager who has keyhole scoliosis correction surgery? A prospective series of 100 patients

Introduction. Surgical treatment of scoliosis is assessed in the spine clinic by the surgeon making numerous measurements on X-Rays as well as the rib hump. But it is important to understand which of these measures correlate with self-reported improvements in patients’ quality of life following surgery. The objective of this study was to examine the relationship between patient satisfaction after thoracoscopic (keyhole) anterior scoliosis surgery and standard deformity correction measures using the Scoliosis Research Society (SRS) adolescent questionnaire. Methods. A series of 100 consecutive adolescent idiopathic scoliosis patients received a single anterior rod via a keyhole approach at the Mater Children’s Hospital, Brisbane. Patients completed SRS outcomes questionnaires before surgery and again at 24 months after surgery. Multiple regression and t-tests were used to investigate the relationship between SRS scores and deformity correction achieved after surgery. Results. There were 94 females and 6 males with a mean age of 16.1 years. The mean Cobb angle improved from 52º pre-operatively to 21º for the instrumented levels post-operatively (59% correction) and the mean rib hump improved from 16º to 8º (51% correction). The mean total SRS score for the cohort was 99.4/120 which indicated a high level of satisfaction with the results of their scoliosis surgery. None of the deformity related parameters in the multiple regressions were significant. However, the twenty patients with the smallest Cobb angles after surgery reported significantly higher SRS scores than the twenty patients with the largest Cobb angles after surgery, but there was no difference on the basis of rib hump correction. Discussion. Patients undergoing thoracoscopic (keyhole) anterior scoliosis correction report good SRS scores which are comparable to those in previous studies. We suggest that the absence of any statistically significant difference in SRS scores between patients with and without rod or screw complications is because these complications are not associated with any clinically significant loss of correction in our patient group. The Cobb angle after surgery was the only significant predictor of patient satisfaction when comparing subgroups of patients with the largest and smallest Cobb angles after surgery.