Students’ approaches to learning a new mathematical model

In this article, we report on the findings of an exploratory study into the experience of undergraduate students as they learn new mathematical models. Qualitative and quanti- tative data based around the students’ approaches to learning new mathematical models were collected. The data revealed that students actively adopt three approaches to under- standing a new mathematical model: gathering information for the task of understanding the model, practising with and using the model, and finding interrelationships between elements of the model. We found that the students appreciate mathematical models that have a real world application and that this can be used to engage students in higher level learning approaches.

Re-use of domain knowledge to provide confidence for adoption of off-site manufacturing for construction in Australia

Many construction industry decision-makers believe there is a lack of off-site manufacture (OSM) adoption for non-residential construction in Australia. Identification of construction business process was considered imperative in order to assist decision-makers to increase OSM utilisation. The premise that domain knowledge can be re-used to provide an intervention point in the construction process led a team of researchers to construct simple base-line process models for the complete construction process, segmented into six phases. Sixteen domain knowledge industry experts were asked to review the construction phase base-line models to answer the question “Where in the process illustrated by this base-line model phase is an OSM task?”. Through an iterative and generative process a number of off-site manufacture intervention points were identified and integrated into the process models. The re-use of industry expert domain knowledge provided suggestions for new ways to do basic tasks thus facilitating changes to current practice. It is expected that implementation of the new processes will lead to systemic industry change and thus a growth in productivity due to increased adoption of OSM.

Travelling waves for a velocity–jump model of cell migration and proliferation

Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher–Kolmogorov equation. These traditional parabolic models can not be used to represent experimental measurements of individual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity–jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left–moving cells, $L(x,t)$, and a subpopulation of right–moving cells, $R(x,t)$. This leads to a system of hyperbolic partial differential equations that includes a turning rate, $\Lambda \ge 0$, describing the rate at which individuals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where $\Lambda = 0$ and in the limit that $\Lambda \to \infty$. For intermediate turning rates, $0 < \Lambda < \infty$, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as $\Lambda$ decreases through a critical value $\Lambda_{crit}$. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small $\Lambda$ limit produces results that are consistent with experimental observations.

A tensor encoding model of word meaning : theory and application to information retrieval

This project was a step forward in developing and evaluating a novel, mathematical model that can deduce the meaning of words based on their use in language. This model can be applied to a wide range of natural language applications, including the information seeking process most of us undertake on a daily basis.

Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

A new optimal control model of the interactions between a growing tumour and the host immune system along with an immunotherapy treatment strategy is presented. The model is based on an ordinary differential equation model of interactions between the growing tu- mour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treat- ment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum al- lowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

Towards a model of spray-canopy interactions : interception, shatter, bounce and retention of droplets on horizontal leaves

Pesticides used in agricultural systems must be applied in economically viable and environmentally sensitive ways, and this often requires expensive field trials on spray deposition and retention by plant foliage. Computational models to describe whether a spray droplet sticks (adheres), bounces or shatters on impact, and if any rebounding parent or shatter daughter droplets are recaptured, would provide an estimate of spray retention and thereby act as a useful guide prior to any field trials. Parameter-driven interactive software has been implemented to enable the end-user to study and visualise droplet interception and impaction on a single, horizontal leaf. Living chenopodium, wheat and cotton leaves have been scanned to capture the surface topography and realistic virtual leaf surface models have been generated. Individual leaf models have then been subjected to virtual spray droplets and predictions made of droplet interception with the virtual plant leaf. Thereafter, the impaction behaviour of the droplets and the subsequent behaviour of any daughter droplets, up until re-capture, are simulated to give the predicted total spray retention by the leaf. A series of critical thresholds for the stick, bounce, and shatter elements in the impaction process have been developed for different combinations of formulation, droplet size and velocity, and leaf surface characteristics to provide this output. The results show that droplet properties, spray formulations and leaf surface characteristics all influence the predicted amount of spray retained on a horizontal leaf surface. Overall the predicted spray retention increases as formulation surface tension, static contact angle, droplet size and velocity decreases. Predicted retention on cotton is much higher than on chenopodium. The average predicted retention on a single horizontal leaf across all droplet size, velocity and formulations scenarios tested, is 18, 30 and 85% for chenopodium, wheat and cotton, respectively.

An androgen receptor mutation in the MDA-MB-453 cell line model of molecular apocrine breast cancer compromises receptor activity

Recent evidence indicates that the estrogen receptor-a-negative, androgen receptor (AR)- positive molecular apocrine subtype of breast cancer is driven by AR signaling. The MDA-MB-453 cell line is the prototypical model of this breast cancer subtype; its proliferation is stimulated by androgens such as 5a-dihydrotestosterone (DHT) but inhibited by the progestin medroxyprogesterone acetate (MPA) via AR-mediated mechanisms. We report here that the AR gene in MDAMB- 453 cells contains a G-T transversion in exon 7, resulting in a receptor variant with a glutamine to histidine substitution at amino acid 865 (Q865H) in the ligand binding domain. Compared with wild-type AR, the Q865H variant exhibited reduced sensitivity to DHT and MPA in transactivation assays in MDA-MB-453 and PC-3 cells but did not respond to non-androgenic ligands or receptor antagonists. Ligand binding, molecular modeling, mammalian two-hybrid and immunoblot assays revealed effects of the Q865H mutation on ligand dissociation, AR intramolecular interactions, and receptor stability. Microarray expression profiling demonstrated that DHT and MPA regulate distinct transcriptional programs in MDA-MB-453 cells. Gene Set Enrichment Analysis revealed that DHT- but not MPA-regulated genes were associated with estrogen-responsive transcriptomes from MCF-7 cells and the Wnt signaling pathway. These findings suggest that the divergent proliferative responses of MDA-MB-453 cells to DHT and MPA result from the different genetic programs elicited by these two ligands through the AR-Q865H variant. This work highlights the necessity to characterize additional models of molecular apocrine breast cancer to determine the precise role of AR signaling in this breast cancer subtype. Endocrine-Related Cancer (2012) 19 599–613

Mathematical modelling of soft callus formation in early murine bone repair

Fracture healing is a complicated coupling of many processes. Yet despite the apparent complexity, fracture repair is usually effective. There is, however, no comprehensive mathematical model addressing the multiple interactions of cells, cytokines and oxygen that includes extra-cellular matrix production and that results in the formation of the early stage soft callus. This thesis develops a one dimensional continuum transport model in the context of early fracture healing. Although fracture healing is a complex interplay of many local factors, critical components are identified and used to construct an hypothesis about regulation of the evolution of early callus formation. Multiple cell lines, cellular differentiation, oxygen levels and cytokine concentrations are examined as factors affecting this model of early bone repair. The model presumes diffusive and chemotactic cell migration mechanisms. It is proposed that the initial signalling regime and oxygen availability arising as consequences of bone fracture, are sufficient to determine the quantity and quality of early soft callus formation. Readily available software and purpose written algorithms have been used to obtain numerical solutions representative of various initial conditions. These numerical distributions of cellular populations reflect available histology obtained from murine osteotomies. The behaviour of the numerical system in response to differing initial conditions can be described by alternative in vivo healing pathways. An experimental basis, as illustrated in murine fracture histology, has been utilised to validate the mathematical model outcomes. The model developed in this thesis has potential for future extension, to incorporate processes leading to woven bone deposition, while maintaining the characteristics that regulate early callus formation.

Mathematical modelling of counter flow v-grove solar air collector

Double-pass counter flow v-grove collector is considered one of the most efficient solar air-collectors. In this design of the collector, the inlet air initially flows at the top part of the collector and changes direction once it reaches the end of the collector and flows below the collector to the outlet. A mathematical model is developed for this type of collector and simulation is carried out using MATLAB programme. The simulation results were verified with three distinguished research results and it was found that the simulation has the ability to predict the performance of the air collector accurately as proven by the comparison of experimental data with simulation. The difference between the predicted and experimental results is, at maximum, approximately 7% which is within the acceptable limit considering some uncertainties in the input parameter values to allow comparison. A parametric study was performed and it was found that solar radiation, inlet air temperature, flow rate and length have a significant effect on the efficiency of the air collector. Additionally, the results are compared with single flow V-groove collector.

Existence of travelling wave solutions for a model of tumour invasion

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.

Mathematical investigation of the interactions between the inflammatory response and mechanical aspects of dermal wound repair

Dermal wound repair involves complex interactions between cells, cytokines and mechanics to close injuries to the skin. In particular, we investigate the contribution of fibroblasts, myofibroblasts, TGFβ, collagen and local tissue mechanics to wound repair in the human dermis. We develop a morphoelastic model where a realistic representation of tissue mechanics is key, and a fibrocontractive model that involves a reasonable approximation to the true kinetics of the important bioactive species. We use each of these descriptions to elucidate the mechanisms that generate pathologies such as hypertrophic scars, contractures and keloids. We find that for hypertrophic scar and contracture development, factors regulating the myofibroblast phenotype are critical, with heightened myofibroblast activation, reduced myofibroblast apoptosis or prolonged inflammation all predicted as mediators for scar hypertrophy and contractures. Prevention of these pathologies is predicted when myofibroblast apoptosis is induced, myofibroblast activation is blocked or TGFβ is neutralised. To investigate keloid invasion, we develop a caricature representation of the fibrocontractive model and find that TGFβ spread is the driving factor behind keloid growth. Blocking activation of TGFβ is found to cause keloid regression. Thus, we recommend myofibroblasts and TGFβ as targets for clinicians when developing intervention strategies for prevention and cure of fibrotic scars.

A model of workplace safety incorporating worker interactions and simple interventions

Although there was substantial research into the occupational health and safety sector over the past forty years, this generally focused on statistical analyses of data related to costs and/or fatalities and injuries. There is a lack of mathematical modelling of the interactions between workers and the resulting safety dynamics of the workplace. There is also little work investigating the potential impact of different safety intervention programs prior to their implementation. In this article, we present a fundamental, differential equation-based model of workplace safety that treats worker safety habits similarly to an infectious disease in an epidemic model. Analytical results for the model, derived via phase plane and stability analysis, are discussed. The model is coupled with a model of a generic safety strategy aimed at minimising unsafe work habits, to produce an optimal control problem. The optimal control model is solved using the forward-backward sweep numerical scheme implemented in Matlab.

Development and application of a predictive model of freshwater fish assemblage composition to evaluate river health in eastern Australia

Multivariate predictive models are widely used tools for assessment of aquatic ecosystem health and models have been successfully developed for the prediction and assessment of aquatic macroinvertebrates, diatoms, local stream habitat features and fish. We evaluated the ability of a modelling method based on the River InVertebrate Prediction and Classification System (RIVPACS) to accurately predict freshwater fish assemblage composition and assess aquatic ecosystem health in rivers and streams of south-eastern Queensland, Australia. The predictive model was developed, validated and tested in a region of comparatively high environmental variability due to the unpredictable nature of rainfall and river discharge. The model was concluded to provide sufficiently accurate and precise predictions of species composition and was sensitive enough to distinguish test sites impacted by several common types of human disturbance (particularly impacts associated with catchment land use and associated local riparian, in-stream habitat and water quality degradation). The total number of fish species available for prediction was low in comparison to similar applications of multivariate predictive models based on other indicator groups, yet the accuracy and precision of our model was comparable to outcomes from such studies. In addition, our model developed for sites sampled on one occasion and in one season only (winter), was able to accurately predict fish assemblage composition at sites sampled during other seasons and years, provided that they were not subject to unusually extreme environmental conditions (e.g. extended periods of low flow that restricted fish movement or resulted in habitat desiccation and local fish extinctions).

Cohabitation, marriage, and divorce in a model of match quality

The objective of this research is to further our understanding of how and why individuals enter and leave coresidential relationships. We develop and estimate an economic model of nonmarital cohabitation, marriage, and divorce that is consistent with current data on the formation and dissolution of relationships. Jovanovic's (Journal of Political Economy 87 (1979), 972-90) theoretical matching model is extended to help explain household formation and dissolution behavior. Implications of the model reveal what factors influence the decision to start a relationship, what form this relationship will take, and the relative stability of the various types of unions. The structural parameters of the model are estimated using longitudinal data from a sample of female high school seniors from the United States. New numerical methods are developed to reduce computational costs associated with estimation. The empirical results have interesting interpretations given the structural model. They show that a significant cause of cohabitation is the need to learn about potential partners and to hedge against future bad shocks. The estimated parameters are used to conduct several comparative dynamic experiments. For example, we show that policy experiments changing the cost of divorce have little effect on relationship choices.

Mathematical modeling of the cancer cell’s control circuitry : paving the way to individualized therapeutic strategies

Cancer is a disease of signal transduction in which the dysregulation of the network of intracellular and extracellular signaling cascades is sufficient to thwart the cells finely-tuned biochemical control mechanisms. A keen interest in the mathematical modeling of cell signaling networks and the regulation of signal transduction has emerged in recent years, and has produced a glimmer of insight into the sophisticated feedback control and network regulation operating within cells. In this review, we present an overview of published theoretical studies on the control aspects of signal transduction, emphasizing the role and importance of mechanisms such as ‘ultrasensitivity’ and feedback loops. We emphasize that these exquisite and often subtle control strategies represent the key to orchestrating ‘simple’ signaling behaviors within the complex intracellular network, while regulating the trade-off between sensitivity and robustness to internal and external perturbations. Through a consideration of these apparent paradoxes, we explore how the basic homeostasis of the intracellular signaling network, in the face of carcinogenesis, can lead to neoplastic progression rather than cell death. A simple mathematical model is presented, furnishing a vivid illustration of how ‘control-oriented’ models of the deranged signaling networks in cancer cells may enucleate improved treatment strategies, including patient-tailored combination therapies, with the potential for reduced toxicity and more robust and potent antitumor activity.