A linear-elastic model of anisotropic tumour growth

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.

Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

A simulation model of the within-host dynamics of Plasmodium vivax infection

Background The benign reputation of Plasmodium vivax is at odds with the burden and severity of the disease. This reputation, combined with restricted in vitro techniques, has slowed efforts to gain an understanding of the parasite biology and interaction with its human host. Methods A simulation model of the within-host dynamics of P. vivax infection is described, incorporating distinctive characteristics of the parasite such as the preferential invasion of reticulocytes and hypnozoite production. The developed model is fitted using digitized time-series’ from historic neurosyphilis studies, and subsequently validated against summary statistics from a larger study of the same population. The Chesson relapse pattern was used to demonstrate the impact of released hypnozoites. Results The typical pattern for dynamics of the parasite population is a rapid exponential increase in the first 10 days, followed by a gradual decline. Gametocyte counts follow a similar trend, but are approximately two orders of magnitude lower. The model predicts that, on average, an infected naïve host in the absence of treatment becomes infectious 7.9 days post patency and is infectious for a mean of 34.4 days. In the absence of treatment, the effect of hypnozoite release was not apparent as newly released parasites were obscured by the existing infection. Conclusions The results from the model provides useful insights into the dynamics of P. vivax infection in human hosts, in particular the timing of host infectiousness and the role of the hypnozoite in perpetuating infection.

Mathematical models of calcium and tight junctions in normal and reconstructed epidermis

This project investigated the calcium distributions of the skin, and the growth patterns of skin substitutes grown in the laboratory, using mathematical models. The research found that the calcium distribution in the upper layer of the skin is controlled by three different mechanisms, not one as previously thought. The research also suggests that tight junctions, which are adhesions between neighbouring skin cells, cannot be solely responsible for the differences in the growth patterns of skin substitutes and normal skin.

The mathematical modelling of cheese ripening

A mathematical model is developed for the ripening of cheese. Such models may assist predicting final cheese quality using measured initial composition. The main constituent chemical reactions are described with ordinary differential equations. Numerical solutions to the model equations are found using Matlab. Unknown parameter values have been fitted using experimental data available in the literature. The results from the numerical fitting are in good agreement with the data. Statistical analysis is performed on near infrared data provided to the MISG. However, due to the inhomogeneity and limited nature of the data, not many conclusions can be drawn from the analysis. A simple model of the potential changes in acidity of cheese is also considered. The results from this model are consistent with cheese manufacturing knowledge, in that the pH of cheddar cheese does not significantly change during ripening.

Limitations of a laboratory scale model in predicting optimal pilot scale conditions for dilute acid pretreatment of sugarcane bagasse

Pilot and industrial scale dilute acid pretreatment data can be difficult to obtain due to the significant infrastructure investment required. Consequently, models of dilute acid pretreatment by necessity use laboratory scale data to determine kinetic parameters and make predictions about optimal pretreatment conditions at larger scales. In order for these recommendations to be meaningful, the ability of laboratory scale models to predict pilot and industrial scale yields must be investigated. A mathematical model of the dilute acid pretreatment of sugarcane bagasse has previously been developed by the authors. This model was able to successfully reproduce the experimental yields of xylose and short chain xylooligomers obtained at the laboratory scale. In this paper, the ability of the model to reproduce pilot scale yield and composition data is examined. It was found that in general the model over predicted the pilot scale reactor yields by a significant margin. Models that appear very promising at the laboratory scale may have limitations when predicting yields on a pilot or industrial scale. It is difficult to comment whether there are any consistent trends in optimal operating conditions between reactor scale and laboratory scale hydrolysis due to the limited reactor datasets available. Further investigation is needed to determine whether the model has some efficacy when the kinetic parameters are re-evaluated by parameter fitting to reactor scale data, however, this requires the compilation of larger datasets. Alternatively, laboratory scale mathematical models may have enhanced utility for predicting larger scale reactor performance if bulk mass transport and fluid flow considerations are incorporated into the fibre scale equations. This work reinforces the need for appropriate attention to be paid to pilot scale experimental development when moving from laboratory to pilot and industrial scales for new technologies.

Excessive bone formation in a mouse model of ankylosing spondylitis is associated with decreases in Wnt pathway inhibitors

Introduction: Ankylosing spondylitis (AS) is unique in its pathology where inflammation commences at the entheses before progressing to an osteoproliferative phenotype generating excessive bone formation that can result in joint fusion. The underlying mechanisms of this progression are poorly understood. Recent work has suggested that changes in Wnt signalling, a key bone regulatory pathway, may contribute to joint ankylosis in AS. Using the proteoglycan-induced spondylitis (PGISp) mouse model which displays spondylitis and eventual joint fusion following an initial inflammatory stimulus, we have characterised the structural and molecular changes that underlie disease progression. Methods: PGISp mice were characterised 12 weeks after initiation of inflammation using histology, immunohistochemistry (IHC) and expression profiling. Results: Inflammation initiated at the periphery of the intervertebral discs progressing to disc destruction followed by massively excessive cartilage and bone matrix formation, as demonstrated by toluidine blue staining and IHC for collagen type I and osteocalcin, leading to syndesmophyte formation. Expression levels of DKK1 and SOST, Wnt signalling inhibitors highly expressed in joints, were reduced by 49% and 63% respectively in the spine PGISp compared with control mice (P < 0.05) with SOST inhibition confirmed by IHC. Microarray profiling showed genes involved in inflammation and immune-regulation were altered. Further, a number of genes specifically involved in bone regulation including other members of the Wnt pathway were also dysregulated. Conclusions: This study implicates the Wnt pathway as a likely mediator of the mechanism by which inflammation induces bony ankylosis in spondyloarthritis, raising the potential that therapies targeting this pathway may be effective in preventing this process.

5-year Chlamydia vaccination programme could reverse disease-related koala population decline: Predictions from a mathematical model using field data

BACKGROUND Many koala populations around Australia are in serious decline, with a substantial component of this decline in some Southeast Queensland populations attributed to the impact of Chlamydia. A Chlamydia vaccine for koalas is in development and has shown promise in early trials. This study contributes to implementation preparedness by simulating vaccination strategies designed to reverse population decline and by identifying which age and sex category it would be most effective to target. METHODS We used field data to inform the development and parameterisation of an individual-based stochastic simulation model of a koala population endemic with Chlamydia. The model took into account transmission, morbidity and mortality caused by Chlamydia infections. We calibrated the model to characteristics of typical Southeast Queensland koala populations. As there is uncertainty about the effectiveness of the vaccine in real-world settings, a variety of potential vaccine efficacies, half-lives and dosing schedules were simulated. RESULTS Assuming other threats remain constant, it is expected that current population declines could be reversed in around 5-6 years if female koalas aged 1-2 years are targeted, average vaccine protective efficacy is 75%, and vaccine coverage is around 10% per year. At lower vaccine efficacies the immunological effects of boosting become important: at 45% vaccine efficacy population decline is predicted to reverse in 6 years under optimistic boosting assumptions but in 9 years under pessimistic boosting assumptions. Terminating a successful vaccination programme at 5 years would lead to a rise in Chlamydia prevalence towards pre-vaccination levels. CONCLUSION For a range of vaccine efficacy levels it is projected that population decline due to endemic Chlamydia can be reversed under realistic dosing schedules, potentially in just 5 years. However, a vaccination programme might need to continue indefinitely in order to maintain Chlamydia prevalence at a sufficiently low level for population growth to continue.

An integrated model of an open-pit coal mine: Improving energy efficiency decisions

This article contributes an original integrated model of an open-pit coal mine for supporting energy-efficient decisions. Mixed integer linear programming is used to formulate a general integrated model of the operational energy consumption of four common open-pit coal mining subsystems: excavation and haulage, stockpiles, processing plants and belt conveyors. Mines are represented as connected instances of the four subsystems, in a flow sheet manner, which are then fitted to data provided by the mine operators. Solving the integrated model ensures the subsystems’ operations are synchronised and whole-of-mine energy efficiency is encouraged. An investigation on a case study of an open-pit coal mine is conducted to validate the proposed methodology. Opportunities are presented for using the model to aid energy-efficient decision-making at various levels of a mine, and future work to improve the approach is described.

On the order of the fractional Laplacian in determining the spatio-temporal evolution of a space-fractional model of cardiac electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.

Analytical model of reactive transport processes with spatially variable coefficients

Analytical solutions of partial differential equation (PDE) models describing reactive transport phenomena in saturated porous media are often used as screening tools to provide insight into contaminant fate and transport processes. While many practical modelling scenarios involve spatially variable coefficients, such as spatially variable flow velocity, v(x), or spatially variable decay rate, k(x), most analytical models deal with constant coefficients. Here we present a framework for constructing exact solutions of PDE models of reactive transport. Our approach is relevant for advection-dominant problems, and is based on a regular perturbation technique. We present a description of the solution technique for a range of one-dimensional scenarios involving constant and variable coefficients, and we show that the solutions compare well with numerical approximations. Our general approach applies to a range of initial conditions and various forms of v(x) and k(x). Instead of simply documenting specific solutions for particular cases, we present a symbolic worksheet, as supplementary material, which enables the solution to be evaluated for different choices of the initial condition, v(x) and k(x). We also discuss how the technique generalizes to apply to models of coupled multispecies reactive transport as well as higher dimensional problems.

A couple stress model of blood flow in the microcirculation

A simple mathematical model depicting blood flow in the capillary is developed with an emphasis on the permeability property of the blood vessel based on Starling's hypothesis. In this study the effect of inertia has been neglected in comparison with the viscosity on the basis of the smallness of the Reynolds number of the flow in the capillary. The capillary blood vessel is approximated by a circular cylindrical tube with a permeable wall. The blood is represented by a couple stress fluid. With such an ideal model the velocity and pressure fields are determined. It is shown that an increase in the couple stress parameter increases the resistance to the flow and thereby decreases the volume rate flow. A comparison of the results with those of the Newtonian case has also been made.

Glass transition. A new approach based on cluster model of glasses

The structure of real glasses has been considered to be microheterogeneous, composed of clusters and connective tissue. Particles in the cluster are assumed to be highly correlated in positions. The tissue is considered to have a truly amorphous structure with its particles vibrating in highly anharmonic potentials. Glass transition is recognized as corresponding to the melting of clusters. A simple mathematical model has been developed which accounts for various known features associated with glass transition, such as range of glass transition temperature,T g, variation ofT g with pressure, etc. Expressions for configurational thermodynamic properties and transport properties of glass forming systems are derived from the model. The relevence and limitations of the model are also discussed.

Mathematical modelling of human corneal oxygenation and cell kinetics in corneal epithelial wound healing

Healthy transparent cornea depends upon the regulation of fluid, nutrient and oxygen transport through the tissue to sustain cell metabolism and other critical processes for normal functioning. This research considers the corneal geometry and investigates oxygen distribution using a two-dimensional Monod kinetic model, showing that previous studies make assumptions that lead to predictions of near-anoxic levels of oxygen tension in the limbal regions of the cornea. It also considers the comparison of experimental spatial and temporal data with the predictions of novel mathematical models with respect to distributed mitotic rates during corneal epithelial wound healing.