Mathematical modelling of contact dermatitis from nickel and chromium.

Dermal exposure to metal allergens can lead to irritant and allergic contact dermatitis (ACD). In this paper we present a mathematical model of the absorption of metal ions, hexavalent chromium and nickel, into the viable epidermis and compare the localised irritant and T-lymphocyte (T-cell) mediated immune responses. The model accounts for the spatial-temporal variation of skin health, extra and intracellular allergen concentrations, innate immune cells, T-cells, cytokine signalling and lymph node activity up to about 6 days after contact with these metals; repair processes associated with withdrawal of exposure to both metals is not considered in the current model, being assumed secondary during the initial phases of exposure. Simulations of the resulting system of PDEs are studied in one-dimension, i.e. across skin depth, and three-dimensional scenarios with the aim of comparing the responses to the two ions in the cases of first contact (no T-cells initially present) and second contact (T-cells initially present). The results show that on continuous contact, chromium ions elicit stronger skin inflammation, but for nickel, subsequent re-exposure stimulates stronger responses due to an accumulation of cytotoxic T-cell mediated responses which characterise ACD. Furthermore, the surface area of contact to these metals has little effect on the speed of response, whilst sensitivity is predicted to increase with the thickness of skin. The modelling approach is generic and should be applicable to describe contact dermatitis from a wide range of allergens.

Modelling the influence of vitamin D and probiotic supplementation on the microbiome and immune response.

The intestinal microbiota play a critical role in human health and disease, maintaining metabolic and immune/inflammatory health, synthesising essential vitamins and amino acids and maintaining intestinal barrier integrity. The aim of this paper is to develop a mathematical model to describe the complex interactions between the microbiota, vitamin D/vitamin D receptor (VDR) pathway, epithelial barrier and immune response in order to understand better the effects of supplementation with probiotics and vitamin D. This is motivated by emerging data indicating the beneficial effects of vitamin D and probiotics individually and when combined. We propose a system of ordinary differential equations determining the time evolution of intestinal bacterial populations, concentration of the VDR:1,25(OH)2D complex in epithelial and immune cells, the epithelial barrier and the immune response. The model shows that administration of probiotics and/or vitamin D upregulates the VDR complex, which enhances barrier function and protects against intestinal inflammation. The model also suggests co-supplementation to be superior to individual supplements. We explore the effects of inflammation on the populations of commensal and pathogenic bacteria and the vitamin D/VDR pathway and discuss the value of gathering additional experimental data motivated by the modelling insights.

Modelling the role of enzymatic pathways in the metabolism of docosahexaenoic acid by monocytes and its association with osteoarthritic pain.

Chronic pain is a major cause of disability and suffering in osteoarthritis (OA) patients. Endogenous specialised pro-resolving molecules (SPMs) curtail pro-inflammatory responses. One of the SPM intermediate oxylipins, 17-hydroxydocasahexaenoic acid (17-HDHA, a metabolite of docosahexaenoic acid (DHA)), is significantly associated with OA pain. The aim of this multidisciplinary work is to develop a mathematical model to describe the contributions of enzymatic pathways (and the genes that encode them) to the metabolism of DHA by monocytes and to the levels of the down-stream metabolites, 17-HDHA and 14-hydroxydocasahexaenoic acid (14-HDHA), motivated by novel clinical data from a study involving 30 participants with OA. The data include measurements of oxylipin levels, mRNA levels, measures of OA severity and self-reported pain scores. We propose a system of ordinary differential equations to characterise associations between the different datasets, in order to determine the homeostatic concentrations of DHA, 17-HDHA and 14-HDHA, dependent upon the gene expression of the associated metabolic enzymes. Using parameter-fitting methods, local sensitivity and uncertainty analysis, the model is shown to fit well qualitatively to experimental data. The model suggests that up-regulation of some ALOX genes may lead to the down-regulation of 17-HDHA and that dosing with 17-HDHA increases the production of resolvins, which helps to down-regulate the inflammatory response. More generally, we explore the challenges and limitations of modelling real data, in particular individual variability, and also discuss the value of gathering additional experimental data motivated by the modelling insights.

Fiber reinforced hydrated networks recapitulate the poroelastic mechanics of articular cartilage.

The role of poroelasticity on the functional performance of articular cartilage has been established in the scientific literature since the 1960s. Despite the extensive knowledge on this topic there remain few attempts to design for poroelasticity and to our knowledge no demonstration of an engineered poroelastic material that approaches the physiological performance. In this paper, we report on the development of an engineered material that begins to approach physiological poroelasticity. We quantify poroelasticity using the fluid load fraction, apply mixture theory to model the material system, and determine cytocompatibility using primary human mesenchymal stem cells. The design approach is based on a fiber reinforced hydrated network and uses routine fabrication methods (electrohydrodynamic deposition) and materials (poly[ɛ-caprolactone] and gelatin) to develop the engineered poroelastic material. This composite material achieved a mean peak fluid load fraction of 68%, displayed consistency with mixture theory, and demonstrated cytocompatibility. This work creates a foundation for designing poroelastic cartilage implants and developing scaffold systems to study chondrocyte mechanobiology and tissue engineering. STATEMENT OF SIGNIFICANCE: Poroelasticity drives the functional mechanics of articular cartilage (load bearing and lubrication). In this work we develop the design rationale and approach to produce a poroelastic material, known as a fiber reinforced hydrated network (FiHy™), that begins to approach the native performance of articular cartilage. This is the first engineered material system capable of exceeding isotropic linear poroelastic theory. The framework developed here enables fundamental studies of poroelasticity and the development of translational materials for cartilage repair.

A change in climate causes rapid evolution of multiple life-history traits and their interactions in an annual plant.

Climate change is likely to spur rapid evolution, potentially altering integrated suites of life-history traits. We examined evolutionary change in multiple life-history traits of the annual plant Brassica rapa collected before and after a recent 5-year drought in southern California. We used a direct approach to examining evolutionary change by comparing ancestors and descendants. Collections were made from two populations varying in average soil moisture levels, and lines propagated from the collected seeds were grown in a greenhouse and experimentally subjected to conditions simulating either drought (short growing season) or high precipitation (long growing season) years. Comparing ancestors and descendants, we found that the drought caused many changes in life-history traits, including a shift to earlier flowering, longer duration of flowering, reduced peak flowering and greater skew of the flowering schedule. Descendants had thinner stems and fewer leaf nodes at the time of flowering than ancestors, indicating that the drought selected for plants that flowered at a smaller size and earlier ontogenetic stage rather than selecting for plants to develop more rapidly. Thus, there was not evidence for absolute developmental constraints to flowering time evolution. Common principal component analyses showed substantial differences in the matrix of trait covariances both between short and long growing season treatments and between populations. Although the covariances matrices were generally similar between ancestors and descendants, there was evidence for complex evolutionary changes in the relationships among the traits, and these changes depended on the population and treatment. These results show that a full appreciation of the impacts of global change on phenotypic evolution will entail an understanding of how changes in climatic conditions affect trait values and the structure of relationships among traits.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure.

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.

The role of phenotypic plasticity and pollination environment in the cleistogamous, mixed mating breeding system of Triodanis perfoliata.

The role of variable pollination environments in maintaining mixed mating systems is an active area of research. Dimorphic cleistogamy, in which a plant reproduces by both open, facultative outcrossing chasmogamous (CH) flowers and closed, cleistogamous (CL) flowers presents an excellent opportunity to study mixed mating. For example, plastic responses in allocation to an optimal floral type could serve as an adaptive strategy that maintains mixed mating under variable pollination environments. We tested for pollen limitation and plastic responses in allocation to different floral types under manipulated pollination conditions in the dimorphic cleistogamous, mixed mating annual, Triodanis perfoliata. Using a field population, we quantified pollen limitation, auto-fertility and plastic responses in the breeding system by measuring allocation to flower number and seed set of floral types. We found no evidence for pollen limitation for CH flowers, and CH flowers had low efficacy of autonomous selfing. Importantly, we found that T. perfoliata alters floral number following changes in pollination conditions, with pollen-supplemented plants having lower relative CH flower number than non-supplemented plants. Breeding system plasticity may allow for benefits from outcrossing through CH flowers, but also increased overall fitness through relatively cheap CL reproduction. After CH flowers receive pollen, subsequent production of CH flowers was reduced, which may be due to resource limitation. Our findings did not support a theoretical model predicting increased CH flowers with high pollination levels. These results increase our understanding of the role of pollination services and resource allocation in the maintenance of mixed mating systems, which also warrants further investigation.

Physiologically based pharmacokinetic modelling of human exposure to 2-butoxyethanol.

A physiologically based pharmacokinetic (PBPK) model describing the disposition of 2-butoxyethanol (2-BE) was developed in order to predict the urinary concentration of its major metabolite, butoxyacetic acid (BAA) under a range of exposure scenarios. Based on Corley et al. [Corley, R.A., Bormett, G.A., Ghanayem, B.I., 1994. Physiologically based pharmacokinetics of 2-butoxyethanol and its major metabolite, 2-butoxyacetic acid, in rats and humans. Toxicol. Appl. Pharmacol. 129, 61-79], the model included such features as multiple entry routes into the body, varying workload conditions, metabolism in the liver and elimination of free BAA in urine by glomerular filtration and acid transport. A bladder compartment simulating the fluctuations in metabolite concentration in urine caused by micturition formed a novel aspect of the model. Good agreement between model predictions and existing experimental data of total BAA levels in the blood and urine over various exposure conditions were observed. The mechanistically based PBPK model allowed comparison of disparate studies and also enabled the prediction of urinary concentrations of BAA post-shift. By calculating the total amount of BAA, any inter-individual variability in conjugation is taken into account. This led us to conclude that a biological monitoring guidance value should be proposed for total rather than free BAA with a value of 250 mmol/mol of creatinine (post-shift), based on an 8h exposure to 25 ppm 2-BE at resting working conditions.

Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties.

The stability of a planar tumour growing into neighbouring tissue is examined and, in particular, its dependence on the properties of the tumour and of the surrounding material studied. An abundant supply of nutrient is assumed, so the proliferation of cells is uninhibited (resulting in exponential growth). We consider two possible constitutive relations. Darcy's law and Stokes flow, in describing the deformation of the tissue and the resulting model takes the form of a coupled system comprising a nonlinear reaction-diffusion-convection equation for the tumour cell concentration and an elliptic system for the deformation and stress fields. Using a combination of linear-stability analysis, numerical methods and thin-film approximations. the evolution of the advancing tumour boundary is determined. It is shown that when the tumour and surrounding material properties are the same, a planar interface is always linearly unstable, with the Stokes flow problem being reducible to the Darcy one. We treat the subsequent (nonlinear) evolution and suggest possible extensions to this work.

Mathematical modelling of comedo ductal carcinoma in situ of the breast.

The growth of a tumour in a cylindrical duct with compliant walls is examined in order to model the early stages of ductal carcinoma in situ (DCIS) of the breast, the earliest known stage of breast cancer. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant basement membrane are studied using asymptotic and numerical methods. In this way we show how the duct wall deforms as the tumour grows and also how the progression of the tumour along the duct depends upon the stiffness of the wall. By varying key parameters we determine how treatment, protease production and the inclusion of the surrounding stroma affect the growth. Finally, we discuss the biological relevance of our results and suggest possible directions for future work.

A mathematical model for the absorption and metabolism of formaldehyde vapour by humans.

Epidemiological studies of occupational exposure to formaldehyde gas (HCHO) have suggested possible links between concentration and duration of exposure, and elevated risks of leukaemia and other cancers at sites distant from the site of contact. Formaldehyde is a highly water soluble gas which, when inhaled, reacts rapidly at the site of contact and is quickly metabolised by enzymes in the respiratory tissue. Inhaled formaldehyde is almost entirely absorbed in the respiratory tract and, for formaldehyde induced toxicity to occur at distant sites, HCHO must enter the blood and be transported to systemic tissues via the circulatory system. A mathematical model describing the absorption and removal of inhaled formaldehyde in the nasal tissue is therefore formulated to predict the proportion of formaldehyde entering into the blood. Accounting for the spatial distribution of the formaldehyde concentration and the metabolic activity within the mucosa, the concentration of formaldehyde in the mucus, the epithelium and the blood has been determined and was found to attain a steady-state profile within a few seconds of exposure. The increase of the formaldehyde concentration in the blood was predicted to be insignificant compared with the existing pre-exposure levels in the body, indicating that formaldehyde is rapidly removed in the nasal tissue. The results of the model thus suggest that it is highly unlikely that following inhalation by the nose, formaldehyde itself will cause toxicity at sites other than the initial site of contact in the respiratory tract.

Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast.

The growth of a tumour in a duct is examined in order to model ductal carcinoma in situ (DCIS) of the breast, the earliest known stage of breast cancer. Interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant basement membrane are studied using numerical and analytical techniques. Particular attention focuses on the impact that proteolytic enzymes have on the tumour's progression. As the tumour expands and the duct wall deforms, the tumour cells are subjected to mechanical and nutritional stresses caused by high pressures and oxygen deprivation. Such stresses may stimulate the cells to produce proteolytic enzymes that degrade the duct wall, making it more compliant and prone to penetration by the tumour cells. We use our model to compare these two hypotheses for enzyme production and find that mechanical stress is likely the dominant mechanism, with the wall deforming most at the centre of the duct. We then discuss the biological implications of our theoretical results and suggest possible directions for future work.

Modelling the early growth of ductal carcinoma in situ of the breast.

The growth of a tumour in a rigid walled cylindrical duct is examined in order to model the initial stages of tumour cell expansion in ductal carcinoma in situ (DCIS) of the breast. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The effects on the shape of the tumour boundary of the material properties (i.e. the viscosity) and the extent to which the cells adhere to the duct wall are studied using numerical and asymptotic methods. It is shown how stable, non-planar, interface configurations result and that, during these initial stages, before the duct wall has been breached, few cells die and a nutrient-rich model is usually sufficient to capture the behaviour. Finally, we discuss the relevance of this approach to DCIS and suggest possible avenues for further work.