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.

A high-density immunoblotting methodology for quantification of total protein levels and phosphorylation modifications.

The components of many signaling pathways have been identified and there is now a need to conduct quantitative data-rich temporal experiments for systems biology and modeling approaches to better understand pathway dynamics and regulation. Here we present a modified Western blotting method that allows the rapid and reproducible quantification and analysis of hundreds of data points per day on proteins and their phosphorylation state at individual sites. The approach is of particular use where samples show a high degree of sample-to-sample variability such as primary cells from multiple donors. We present a case study on the analysis of >800 phosphorylation data points from three phosphorylation sites in three signaling proteins over multiple time points from platelets isolated from ten donors, demonstrating the technique's potential to determine kinetic and regulatory information from limited cell numbers and to investigate signaling variation within a population. We envisage the approach being of use in the analysis of many cellular processes such as signaling pathway dynamics to identify regulatory feedback loops and the investigation of potential drug/inhibitor responses, using primary cells and tissues, to generate information about how a cell's physiological state changes over time.

Spatiotemporal modelling of CheY complexes in Escherichia coli chemotaxis.

The chemotaxis pathway of Escherichia coli is one of the best studied and modelled biological signalling pathways. Here we extend existing modelling approaches by explicitly including a description of the formation and subcellular localization of intermediary complexes in the phosphotransfer pathway. The inclusion of these complexes shows that only about 60% of the total output response regulator (CheY) is uncomplexed at any moment and hence free to interact with its target, the flagellar motor. A clear strength of this model is its ability to predict the experimentally observable subcellular localization of CheY throughout a chemotactic response. We have found good agreement between the model output and experimentally determined CheY localization patterns.

A continuum receptor model of hepatic lipoprotein metabolism.

A mathematical model describing the uptake of low density lipoprotein (LDL) and very low density lipoprotein (VLDL) particles by a single hepatocyte cell is formulated and solved. The model includes a description of the dynamic change in receptor density on the surface of the cell due to the binding and dissociation of the lipoprotein particles, the subsequent internalisation of bound particles, receptors and unbound receptors, the recycling of receptors to the cell surface, cholesterol dependent de novo receptor formation by the cell and the effect that particle uptake has on the cell's overall cholesterol content. The effect that blocking access to LDL receptors by VLDL, or internalisation of VLDL particles containing different amounts of apolipoprotein E (we will refer to these particles as VLDL-2 and VLDL-3) has on LDL uptake is explored. By comparison with experimental data we find that measures of cell cholesterol content are important in differentiating between the mechanisms by which VLDL is thought to inhibit LDL uptake. We extend our work to show that in the presence of both types of VLDL particle (VLDL-2 and VLDL-3), measuring relative LDL uptake does not allow differentiation between the results of blocking and internalisation of each VLDL particle to be made. Instead by considering the intracellular cholesterol content it is found that internalisation of VLDL-2 and VLDL-3 leads to the highest intracellular cholesterol concentration. A sensitivity analysis of the model reveals that binding, unbinding and internalisation rates, the fraction of receptors recycled and the rate at which the cholesterol dependent free receptors are created by the cell have important implications for the overall uptake dynamics of either VLDL or LDL particles and subsequent intracellular cholesterol concentration.

Understanding post-operative temperature drop in cardiac surgery: a mathematical model.

A mathematical model is presented to understand heat transfer processes during the cooling and re-warming of patients during cardiac surgery. Our compartmental model is able to account for many of the qualitative features observed in the cooling of various regions of the body including the central core containing the majority of organs, the rectal region containing the intestines and the outer peripheral region of skin and muscle. In particular, we focus on the issue of afterdrop: a drop in core temperature following patient re-warming, which can lead to serious post-operative complications. Model results for a typical cooling and re-warming procedure during surgery are in qualitative agreement with experimental data in producing the afterdrop effect and the observed dynamical variation in temperature between the core, rectal and peripheral regions. The influence of heat transfer processes and the volume of each compartmental region on the afterdrop effect is discussed. We find that excess fat on the peripheral and rectal regions leads to an increase in the afterdrop effect. Our model predicts that, by allowing constant re-warming after the core temperature has been raised, the afterdrop effect will be reduced.

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations.

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller-Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated.

Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell.

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area.

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.

Modelling the formation of necrotic regions in avascular tumours.

The mechanisms underlying the formation of necrotic regions within avascular tumours are not well understood. In this paper, we examine the relative roles of nutrient deprivation and of cell death, from both the proliferating phase of the cell cycle via apoptosis and from the quiescent phase via necrosis, in changing the structure within multicellular tumour spheroids and particularly the accumulation of dead cell material in the centre. A mathematical model is presented and studied that accounts for nutrient diffusion, changes in cell cycling rates, the two different routes to cell death as well as active motion of cells and passive motion of the dead cell material. In studying the accumulation of dead cell matter we do not distinguish between the route by which each was formed. The resulting mathematical model is examined for a number of scenarios. Results show that in many cases the size of the necrotic core is closely correlated with low levels in nutrient concentration. However, in certain cases, particularly where the rate of necrosis is large, the resulting necrotic core can lead to regions of non-negligible nutrient concentration-dependent upon the mode of cell death.

Modelling the cell cycle and cell movement in multicellular tumour spheroids.

This paper analyses a recent mathematical model of avascular tumour spheroid growth which accounts for both cell cycle dynamics and chemotactic driven cell movement. The model considers cells to exist in one of two compartments: proliferating and quiescent, as well as accounting for necrosis and apoptosis. One particular focus of this paper is the behaviour created when proliferating and quiescent cells have different chemotactic responses to an extracellular nutrient supply. Two very different steady-state behaviours are identified corresponding to those cases where proliferating cells move either more quickly or more slowly than quiescent cells in response to a gradient in the extracellular nutrient supply. The case where proliferating cells move more rapidly leads to the commonly accepted spheroid structure of a thin layer of proliferating cells surrounding an inner quiescent core. In the case where proliferating cells move more slowly than quiescent cells the model predicts an interesting structure of a thin layer of quiescent cells surrounding an inner core of proliferating and quiescent cells. The sensitivity of this tumour structure to the cell cycle model parameters is also discussed. In particular variations in the steady-state size of the tumour and the types of transient behaviour are explored. The model reveals interesting transient behaviour with sharply delineated regions of proliferating and quiescent cells.

Modelling cell growth and its modulation of the G1/S transition.

We present a model for the regulation of the G(1)/S transition by cell growth in budding yeast. The model includes a description of cell size, the extracellular nutrient concentration and a simplified model of the G(1)/S transition as originally reported by Chen et al. [Mol. Biol. Cell 11:369-391, 2000]. By considering cell growth proportional to cell size we show that the cell grows exponentially. In the case where cell growth is considered proportional to the concentration of a sizer protein within the cell, our model exhibits both exponential and linear cell growth for varying parameter values. The effects of varying nutrient concentration and initial cell size are considered in the context of whether progression through the cell-size checkpoint occurs. We consider our results in relation to recent experimental evidence and discuss possible experiments for testing our theoretical predictions.

Male and female rural probationers: HIV risk behaviors and knowledge.

Individuals involved in the criminal justice system are at substantial risk for HIV infection and have elevated rates of AIDS. Offenders under community supervision, such as probationers, have substantially more opportunities to engage in high-risk behaviors than prisoners. Furthermore, probationers in rural areas are at risk because rural areas may be slower to adopt HIV risk-reduction approaches. Consequently, the primary goal of this study is to describe the HIV risk behaviors and level of HIV knowledge of 800 rural felony probationers. Bivariate results indicate that males have substantially greater criminal histories and engage in more substance use risk behaviors than females. Overall, there was minimal and inconsistent use of condoms, but there were no significant differences by gender. Gender differences prevailed in perceived HIV knowledge, with females reporting high levels of perceived HIV knowledge. Multivariate models did not support the hypothesis that perceived knowledge would be a more robust correlate of scores on the HIV Risk Behavior Knowledge Test for males than females. Results suggest that rural residents are not protected from engaging in HIV risk behaviors and future studies should examine gender discrepancies between perceived and actual HIV knowledge among offenders under community supervision.

Variation in intubation decisions for patients with chronic obstructive pulmonary disease in one critical care network.

BACKGROUND: Anecdotal evidence suggests variation in intubation decisions for chronic obstructive pulmonary disease (COPD) patients with respiratory failure, but little is known about the extent of or reasons for this variability. AIM: To describe clinician decision-making for patients with exacerbations of COPD considered for intubation. DESIGN: Telephone simulation study. METHODS: Consultants responsible for COPD admissions in the Heart of England Critical Care network were asked to decide whether or not to admit three patients with COPD to ICU on the basis of information conveyed over the telephone. Consultants were also asked to predict patients survival in ICU hospital and at 180 days on the assumption that the patient did receive ICU care. RESULTS: Of the 120 consultants, 98 (82%) took part; 89% would admit patient 1, 64% patient 2, and 40% patient 3. The prediction of survival if ICU admission had occurred differed significantly between admitters and non-admitters. Mean predicted post-ICU hospital survival for patient 1 was 46% (95%CI 43-49) for admitters, and 13% (95%CI 6-19) for non-admitters (p < 0.001). The respective figures for patient 2 were 38% (95%CI 34-42) vs. 12% (95%CI 8-15) (p < 0.001), and for patient 3, 28% (95%CI 24-33) vs. 13% (95%CI 10-16) (p < 0.001). For a housebound COPD patient in their mid 70s, the mean (SD) threshold of predicted hospital survival below which consultants would recommend not admitting to ICU was 22% (13.2%). CONCLUSIONS: Consultants differed markedly in their admitting decisions about identical patients. Objective outcome prediction models might improve equity in ICU bed use for patients with COPD.

The migration of cells in multicell tumor spheroids.

A mathematical model is proposed to explain the observed internalization of microspheres and 3H-thymidine labelled cells in steady-state multicellular spheroids. The model uses the conventional ideas of nutrient diffusion and consumption by the cells. In addition, a very simple model of the progress of the cells through the cell cycle is considered. Cells are divided into two classes, those proliferating (being in G1, S, G2 or M phases) and those that are quiescent (being in G0). Furthermore, the two categories are presumed to have different chemotactic responses to the nutrient gradient. The model accounts for the spatial and temporal variations in the cell categories together with mitosis, conversion between categories and cell death. Numerical solutions demonstrate that the model predicts the behavior similar to existing models but has some novel effects. It allows for spheroids to approach a steady-state size in a non-monotonic manner, it predicts self-sorting of the cell classes to produce a thin layer of rapidly proliferating cells near the outer surface and significant numbers of cells within the spheroid stalled in a proliferating state. The model predicts that overall tumor growth is not only determined by proliferation rates but also by the ability of cells to convert readily between the classes. Moreover, the steady-state structure of the spheroid indicates that if the outer layers are removed then the tumor grows quickly by recruiting cells stalled in a proliferating state. Questions are raised about the chemotactic response of cells in differing phases and to the dependency of cell cycle rates to nutrient levels.

A training exercise in subjectively estimating inhalation exposures.

OBJECTIVES: This study examined whether it is possible to train occupational hygienists to estimate inhalation exposures reliably from limited occupational information using a new method and assessed improvements in the quality of the estimate using the aggregate from multiple assessors. METHODS: Five occupational hygienists estimated inhalation exposure for 40 tasks covering a range of chemical hazards using a recently developed subjective modeling technique supplemented by detailed guidance notes. The measured exposure levels were used to determine the validity of the method. The correlation coefficients of the log-transformed data were used to assess the discriminative power of the method, and the ratio of the mean estimate to measured values was used to measure accuracy. RESULTS: There was good-to-excellent agreement between the assessors' estimates and the measured data, the correlation coefficients ranging from 0.73 to 0.85. There was a tendency for assessors to overestimate the exposure levels by, on the average, two- to fourfold. Aggregating the assessors' estimates helped to improve the correlation coefficient to 0.88, the overestimation being 2.6-fold. Using more than three assessors for aggregate estimates did not improve the reliability of the method. CONCLUSIONS: Overall, the assessors found the method to be useful in generating exposure estimates that correlate well with measured levels. The provision of high-quality guidance information is likely to be important in the generation of reliable exposure estimates. The method is likely to be of use in epidemiologic studies in which limited exposure data are available.

Mitochondrial DNA deletions in human skin reflect photo- rather than chronologic aging.

We have examined the use of mitochondrial DNA (mtDNA) as a molecular marker to study the relation between chronologic aging and photoageing in human skin. Using a 3-primer quantitative polymerase chain reaction method we have studied changes in the ratio of the 4977 bp deleted to wild-type mtDNA in relation to sun exposure and chronologic age of human skin. Based on previous studies, samples showing greater than 1% deleted mtDNA were classed as abnormal. There was a significant increase in the incidence of high levels (i.e., >1%) of the 4977 bp deleted mtDNA in sun-exposed sites (27%, 27 of 100) compared with sun-protected sites (1.1%, one of 90) (Fisher's exact test, p < 0.0001). There appeared to be no relation between the frequency of the mtDNA deletion and age. Analysis of split skin samples showed that most deletions (93%, n = 27) were confined to the dermal rather than the epidermal component, and in keeping with this deletions were found in three of six primary cultures of fibroblasts from sun-exposed sites. Deletions were not seen in the epidermal component of several epidermal tumors nor were deletions seen in fibroblasts cultured from an individual with Werner's syndrome. We propose that deletions or mutations of mitochondrial DNA may be useful as a marker of cumulative ultraviolet radiation exposure.

Empirical generality of data from recognition memory receiver-operating characteristic functions and implications for the global memory models.

The experiments presented in this article examined the slope of the zeta-ROC (receiver-operating characteristic) function for recognition memory. The slope was examined as a function of strength and the variables study time, list length, word frequency, and category membership. For normal distributions of familiarity, the slope of the zeta-ROC is the ratio of the new-item to old-item standard deviations. R. Ratcliff, C.-F. Sheu, and S. D. Gronlund (1992) found that the slope was constant within standard error as a function of strength of encoding, which is inconsistent with the predictions of the global memory models. The results presented here extend this finding: The slope was constant as a function of strength of encoding, list length, and the number of related items from a category in the study list. Word frequency did affect the slope, but within a frequency class the slope was constant as a function of strength. The implications of these data for the global memory models, the attention likelihood model, and variants of these models are discussed.

Relational and item-specific information in the determination of "blocking effects".

Items organized ("blocked") by membership in common categories are generally better remembered than unorganized items. In experiments conducted by Gollin and Sharps (1988), however, this blocking effect was largely confined to verbal stimuli. In the present manuscript, we show that this result may reflect differences in the processing of item-specific and relational information, as first described by Einstein and Hunt (1980). This hypothesis requires that the blocking effect for verbal stimuli diminish over longer retention intervals and that this diminution be attenuated or arrested by instructional manipulations at encoding. The present results verified these conjectures and replicated the result of Gollin and Sharps (1988). The findings suggest that the results of Gollin and Sharps were predictable from the application of item-specific/relational information theory and that the theory may appropriately be applied to paradigms involving both verbal and nonverbal stimuli.