A mathematical model for the human menstrual cycle

A simple mathematical model framework is developed to describe the hormonal interactions of the human menstrual cycle along the hypothalamus–pituitary–ovaries axis. The framework is designed so that it can be readily extended to model processes that disrupt the normal functioning cycle. The model in its most basic formulation exhibits multiple periodic solutions, one of which shows the key characteristics of a menstrual cycle, while the others indicate possible abnormalities sometimes observed in women of reproductive age. The basic model is extended to encompass receptor down-regulation as a mechanism to describe the desensitization of the pituitary to continuous stimulation of hypothalamic hormone, a hormonal therapy that is commonly prescribed prior to the surgical procedure for the removal of uterine myomas. Though the mechanisms for desensitization are likely to be more complex, the model results are in good qualitative agreement with physiological observations

A mathematical model of the growth of uterine myomas

Uterine myomas or fibroids are common, benign smooth muscle tumours that can grow to 10 cm or more in diameter and are routinely removed surgically. They are typically slow- growing, well-vascularised, spherical tumours that, on a macro-scale, are a structurally uniform, hard elastic material. We present a multi-phase mathematical model of a fully vascularised myoma growing within a surrounding elastic tissue. Adopting a continuum approach, the model assumes the conservation of mass and momentum of four phases, namely cells/collagen, extracellular fluid, arterial and venous phases. The cell/collagen phase is treated as a poro-elastic material, based on a linear stress–strain relationship, and Darcy’s law is applied to describe flow in the extracellular fluid and the two vascular phases. The supply of extracellular fluid is dependent on the capillary flow rate and mean capillary pressure expressed in terms of the arterial and venous pressures. Cell growth and division is limited to the myoma domain and dependent on the local stress in the material. The resulting model consists of a system of nonlinear partial differential equations with two moving boundaries. Numerical solutions of the model successfully reproduce qualitatively the clinically observed three-phase “fast–slow–fast” growth profile that is typical for myomas. The results suggest that this growth profile requires stress-induced resistance to growth by the surrounding tissue and a switch-like cell growth response to stress. Analysis of large-time solutions reveal that while there is a functioning vasculature throughout the myoma, exponential growth results, otherwise power-law growth is predicted. An extensive survey of the effect of parameters on model solutions is also presented, and in particular, the enhanced growth caused by factors such as oestrogen is predicted by the model

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human–human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established

Numerical modelling of effects of biphasic layers of corrosion products to the degradation of magnesium metal in vitro

© 2017 by the authors. Magnesium (Mg) is becoming increasingly popular for orthopaedic implant materials. Its mechanical properties are closer to bone than other implant materials, allowing for more natural healing under stresses experienced during recovery. Being biodegradable, it also eliminates the requirement of further surgery to remove the hardware. However, Mg rapidly corrodes in clinically relevant aqueous environments, compromising its use. This problem can be addressed by alloying the Mg, but challenges remain at optimising the properties of the material for clinical use. In this paper, we present a mathematical model to provide a systematic means of quantitatively predicting Mg corrosion in aqueous environments, providing a means of informing standardisation of in vitro investigation of Mg alloy corrosion to determine implant design parameters. The model describes corrosion through reactions with water, to produce magnesium hydroxide Mg(OH) 2 , and subsequently with carbon dioxide to form magnesium carbonate MgCO 3 . The corrosion products produce distinct protective layers around the magnesium block that are modelled as porous media. The resulting model of advection-diffusion equations with multiple moving boundaries was solved numerically using asymptotic expansions to deal with singular cases. The model has few free parameters, and it is shown that these can be tuned to predict a full range of corrosion rates, reflecting differences between pure magnesium or magnesium alloys. Data from practicable in vitro experiments can be used to calibrate the model's free parameters, from which model simulations using in vivo relevant geometries provide a cheap first step in optimising Mg-based implant materials

Modelling foot-and-mouth disease virus dynamics in oral epithelium to help identify the determinants of lysis

Foot-and-mouth disease virus (FMDV) causes an economically important disease of cloven- hoofed livestock; of interest here is the dfference in lytic behaviour that is observed in bovine epithelium. On the skin around the feet and tongue the virus rapidly replicates, killing cells and resulting in growing lesions, before eventually being cleared by the immune response. In contrast there is usually minimal lysis in the soft palate, but virus may persist in tissue long after the animal has recovered from the disease. Persistence of virus has important implications for disease control, while identifying the determinant of lysis in epithelium is potentially important for the development of prophylactics. To help identify which of the differences between oral and pharyngeal epithelium are responsible for such dramatically divergent FMDV dynamics a simple model has been developed, in which virus concentration is made explicit to allow the lytic behaviour of cells to be fully considered. Results suggest that localised structuring of what are fundamentally similar cells can induce a bifurcation in the behaviour of the system, explicitly whether infection can be sustained or results mutual extinction, although parameter estimates indicate that more complex factors may be involved in maintaining viral persistence, or that there are as yet unquantified differences between the intrinsic properties of cells in these regions

Dynamical density functional theory based modelling of tissue dynamics: application to tumour growth

We present a theoretical framework based on an extension of dynamical density functional theory (DDFT) for describing the structure and dynamics of cells in living tissues and tumours. DDFT is a microscopic statistical mechanical theory for the time evolution of the density distribution of interacting many-particle systems. The theory accounts for cell pair-interactions, different cell types, phenotypes and cell birth and death processes (including cell division), in order to provide a biophysically consistent description of processes bridging across the scales, including describing the tissue structure down to the level of the individual cells. Analysis of the model is presented for a single species and a two-species cases, the latter aimed at describing competition between tumour and healthy cells. In suitable parameter regimes, model results are consistent with biological observations. Of particular note, divergent tumour growth behaviour, mirroring metastatic and benign growth characteristics, are shown to be dependent on the cell pair-interaction parameters

A simulation model of Rhizome networks for Fallopia japonica (Japanese Knotweed) in the United Kingdom

Fallopia japonica (Japanese knotweed) is an aggressively invasive herbaceous perennial that causes substantial economic and environmental damage in the United Kingdom (UK). As such, it is of considerable concern to councils, environmental groups, private landowners and property developers. We construct a 3D correlated random walk model of the development of the subterranean rhizome network for a single stand of F. japonica. The formulation of this model uses detailed knowledge of the morphology and physiology of the plant, both of which differ in the UK to that of its native habitat due to factors including a lack of predation and competition, longer growth seasons and favourable environmental conditions in the UK. Field data obtained as a part of this study are discussed and used in the model for parameterisation and validation. The simulation captures the field data well and predicts, for example, quadratic growth in time for the stand area. Furthermore, the role of a selection of parameters on long-term stand development are discussed, highlighting some key factors affecting vegetative spread rates

Timescale analysis of a mathematical model of acetaminophen metabolism and toxicity

Acetaminophen is a widespread and commonly used painkiller all over the world. However, it can cause liver damage when taken in large doses or at repeated chronic doses. Current models of acetaminophen metabolism are complex, and limited to numerical investigation though provide results that represent clinical investigation well. We derive a mathematical model based on mass action laws aimed at capturing the main dynamics of acetaminophen metabolism, in particular the contrast between normal and overdose cases, whilst remaining simple enough for detailed mathematical analysis that can identify key parameters and quantify their role in liver toxicity. We use singular perturbation analysis to separate the different timescales describing the sequence of events in acetaminophen metabolism, systematically identifying which parameters dominate during each of the successive stages. Using this approach we determined, in terms of the model parameters, the critical dose between safe and overdose cases, timescales for exhaustion and regeneration of important cofactors for acetaminophen metabolism and total toxin accumulation as a fraction of initial dose

Predicting tyrosinaemia: a mathematical model of 4-hydroxyphenylpyruvate dioxygenase inhibition by nitisinone in rats

Nitisinone or 2-(2-nitro-4-trifluoromethylbenzoyl)cyclohexane-1,3-dione, is a reversible inhibitor of 4- hydroxyphenylpyruvate dioxygenase (HPPD), an enzyme important in tyrosine catabolism. Today, nitisinone is successfully used to treat Hereditary Tyrosinaemia type 1, although its original expected role was as a herbicide. In laboratory animals, treatment with nitisinone leads to the elevation of plasma tyrosine (tyrosinaemia). In rats and Beagle dogs, repeat low-dose exposure to nitisinone leads to corneal opacities whilst similar studies in the mouse and Rhesus monkey showed no comparable toxicities or other treatment related findings. The differences in toxicological sensitivities have been related to the upper limit of the concentration of tyrosine that accumulates in plasma, which is driven by the amount/activity of tyrosine aminotransferase. A physiologically based, pharmacodynamics ordinary differential equation model of HPPD inhibition to bolus exposure of nitisinone in vivo is presented. Going beyond traditional approaches, asymptotic analysis is used to separate the different timescales of events involved in HPPD inhibition and tyrosinaemia. This analysis elucidates, in terms of the model parameters, a critical inhibitor concentration (at which tyrosine concentration starts to rise) and highlights the contribution of in vitro measured parameters to events in an in vivo system. Furthermore, using parameter-fitting methods, a systematically derived reduced model is shown to fit well to rat data, making explicit how the parameters are informed by such data. This model in combination with in vitro descriptors has potential as a surrogate for animal experimentation to predict tyrosinaemia, and further development can extend its application to other related medical scenarios

Mathematical modelling of contact dermatitis from nickel and chromium

Dermal exposure to metal allergens can lead to irritant (ICD) 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