A mathematical model of wound healing and subsequent scarring.

Wound healing is a complex process involving the delicate interaction between elements that vary widely in nature and size scales, from the nanometre level, such as molecules, to cells measured in micrometres, and fibres with width and length measured on both scales. Hybrid approaches, where each species is represented by a model on an appropriate size scale, have received attention recently. In this study, we provide a review of earlier work on such hybrid models of wound healing. General models for each of the element types involved in dermal wound healing used in this research are described: cells, modelled as discrete individuals; chemicals, modelled as continua; and fibres, modelled with a novel tensorial representation. Techniques for integrating such disparate models are outlined. A six-species model (fibrin, collagen, macrophages, fibroblasts, transforming growth factor-beta (TGF-beta) and tissue plasminogen activator) of dermal wound healing is presented. The role of the cytokine TGF-beta in the healing cascade is investigated using the model, along with its role in the degree of scarring in the healed tissue.

Elastohydrodynamics of the eyelid wiper.

This paper presents an elastohydrodynamic model of the human eyelid wiper. Standard lubrication theory is applied to the fluid layer between the eyelid wiper and ocular surface. The role of the lubrication film is to reduce the shear stresses by preventing solid to solid contact between the eyelid wiper and ocular surface. For the lubrication film to be effective, it is required that the orientation of the eyelid wiper changes between the opening and closing phases of a blink. In order to model this, the hydrodynamic model is coupled with an elastic mattress model for the soft tissue of the eyelid wiper and ocular surface. This leads to a one-dimensional non-linear partial differential equation governing the fluid pressure in the lubrication film. In order to solve the differential equation, a loading condition or constraint equation must be specified. The resulting system is then solved numerically. The model allows predictions of the tear film flux from under the upper eyelid, as well as normal and shear stresses acting on the ocular surface. These factors are important in relation to dry eye syndrome, deformation of the cornea and contact lens design. It is found that the pressure and shear stress under the eyelid act across a length of approximately 0.1 mm which is consistent with clinical observations. It order to achieve a flow of tears from under the upper eyelid during a blink, the model requires that the normal force the eyelid applies to the ocular surface during the closing phase of the blink is significantly higher than during the opening phase of the blink.

A simple mechanistic model of sprout spacing in tumour-associated angiogenesis.

This paper develops a simple mathematical model of the sitting of capillary sprouts on an existing blood vessel during the initiation of tumour-induced angiogenesis. The model represents an inceptive attempt to address the question of how unchecked sprouting of the parent vessel is avoided at the initiation of angiogenesis, based on the idea that feedback regulation processes play the dominant role. No chemical interaction between the proangiogenic and antiangiogenic factors is assumed. The model is based on corneal pocket experiments, and provides a mathematical analysis of the initial spacing of angiogenic sprouts.

Characterizing an outbreak of vancomycin-resistant enterococci using hidden Markov models.

BACKGROUND: Antibiotic-resistant nosocomial pathogens can arise in epidemic clusters or sporadically. Genotyping is commonly used to distinguish epidemic from sporadic vancomycin-resistant enterococci (VRE). We compare this to a statistical method to determine the transmission characteristics of VRE. METHODS AND FINDINGS: A structured continuous-time hidden Markov model (HMM) was developed. The hidden states were the number of VRE-colonized patients (both detected and undetected). The input for this study was weekly point-prevalence data; 157 weeks of VRE prevalence. We estimated two parameters: one to quantify the cross-transmission of VRE and the other to quantify the level of VRE colonization from sporadic sources. We compared the results to those obtained by concomitant genotyping and phenotyping. We estimated that 89% of transmissions were due to ward cross-transmission while 11% were sporadic. Genotyping found that 90% had identical glycopeptide resistance genes and 84% were identical or nearly identical on pulsed-field gel electrophoresis (PFGE). There was some evidence, based on model selection criteria, that the cross-transmission parameter changed throughout the study period. The model that allowed for a change in transmission just prior to the outbreak and again at the peak of the outbreak was superior to other models. This model estimated that cross-transmission increased at week 120 and declined after week 135, coinciding with environmental decontamination. SIGNIFICANCE: We found that HMMs can be applied to serial prevalence data to estimate the characteristics of acquisition of nosocomial pathogens and distinguish between epidemic and sporadic acquisition. This model was able to estimate transmission parameters despite imperfect detection of the organism. The results of this model were validated against PFGE and glycopeptide resistance genotype data and produced very similar results. Additionally, HMMs can provide information about unobserved events such as undetected colonization.

Modelling external bone adaptation using evolutionary structural optimisation.

External remodelling is significant in the bone healing process, and it is essential to predict the bone external shape in the design of artificial bone grafts. This paper demonstrates the effectiveness of the evolutionary structural optimisation (ESO) method for the simulation of bone morphology. A two-dimensional ESO strategy is developed which is capable of finding the modified bone topology beginning with any geometry under any loading conditions. The morphology of bone structure is described by the quantitative bone adaptation theory, which is integrated with the finite element method. The evolutionary topology optimisation process is introduced to find the bone shape. A rectangle, which occupies a larger space than the external shape of the bone structure, is specified as a design domain; the evolutionary process iteratively eliminates and redistributes material throughout the domain to obtain an optimum arrangement of bone materials. The technique has been tested on a wide range of examples. In this paper, the formation of trabecular bone architecture around an implant is studied; as another example, the growth of the coronal section of a vertebral body is predicted. The examples support the assertion that the external shape of bone structure can be successfully predicted by the proposed ESO procedure.

Comparison of two numerical approaches for bone remodelling.

This paper addresses the "checkerboard" phenomenon, which occurs in numerical simulation of bone remodelling. It attempts to answer the question: is an element-based approach suitable for bone remodelling? Two different numerical approaches, the element-based and the node-based finite element analyses, are implemented using ABAQUS. A comparison of the numerical results demonstrates that the checkerboard phenomenon occurs only in the element-based finite element analyses; the node-based approach eradicates the checkerboard phenomenon but requires much more computational time. This study shows that it is essential to enforce the continuity of bone density across the element boundaries. As the node-based approach requires much more computational time, the first-order Adams-Bashforth integration method is introduced to reduce computational cost. The comparisons with Euler's forward method demonstrate that the first-order Adams-Bashforth method indeed enhances accuracy and reduces computational cost. This study concludes that the node-based approach with the first-order Adams-Bashforth integration scheme is to be recommended for computational bone remodelling studies.

A stochastic mathematical model of methicillin resistant Staphylococcus aureus transmission in an intensive care unit: predicting the impact of interventions.

OBJECTIVES: To estimate the transmission rate of MRSA in an intensive care unit (ICU) in an 800 bed Australian teaching hospital and predict the impact of infection control interventions. METHODS: A mathematical model was developed which consisted of four compartments: colonised and uncolonised patients and contaminated and uncontaminated health-care workers (HCWs). Patient movements, MRSA acquisition and daily prevalence data were collected from an ICU over 939 days. Hand hygiene compliance and the probability of MRSA transmission from patient to HCW per discordant contact were measured during the study. Attack rate and reproduction ratio were estimated using Bayesian methods. The impact of a number of interventions on attack rate was estimated using both stochastic and deterministic versions of the model. RESULTS: The mean number of secondary cases arising from the ICU admission of colonised patients, also called the ward reproduction ratio, R(w), was estimated to be 0.50 (95% CI 0.39-0.62). The attack rate was one MRSA transmission per 160 (95% CI 130-210) uncolonised-patient days. Results were not sensitive to uncertainty in measured model parameters (hand hygiene rate and transmission probability per contact). Hand hygiene was predicted to be the most effective intervention. Decolonisation was predicted to be relatively ineffective. Increasing HCW numbers was predicted to increase MRSA transmission, in the absence of patient cohorting. The predictions of the stochastic model differed from those of the deterministic model, with lower levels of colonisation predicted by the stochastic model. CONCLUSIONS: The number of secondary cases of MRSA colonisation within the ICU in this study was below unity. Transmission of MRSA was sustained through admission of colonised patients. Stochastic model simulations give more realistic predictions in hospital ward settings than deterministic models. Increasing staff does not necessarily lead to reduced transmission of nosocomial pathogens.

The effect of the lipid layer on tear film behaviour.

This paper investigates the effect of surfactants during tear film deposition and subsequent thinning. The surfactants occur naturally on the surface of the tear film in the form of a lipid layer. A lubrication model is developed that describes lipid spreading and film height evolution. It is shown that lipids may play an important role in drawing the tear film up the cornea during the opening phase of the blink. Further, nonuniform distributions of lipids may lead to a rapid thinning of the tear film behind the advancing lipid front (shock). Experiments using a fluorescein dye technique and using a tearscope were undertaken in order to visualise the motion of the lipid layer and any associated shocks immediately after a blink. It is found that the lipid layer continues to spread upwards on the cornea after the opening phase of the blink, in agreement with the model. Using the experimental data, lipid particles were tracked in order to determine the surface velocity and these results are compared to the model predictions.

In vitro degradation of articular cartilage: does trypsin treatment produce consistent results?

It is common practice in laboratories to create models of degraded articular cartilage in vitro and use these to study the effects of degeneration on cartilage responses to external stimuli such as mechanical loading. However, there are inconsistencies in the reported action of trypsin, and there is no guide on the concentration of trypsin or the time to which a given sample can be treated so that a specific level of proteoglycan depletion is achieved. This paper argues that before any level of confidence can be established in comparative analysis it is necessary to first obtain samples with similar properties. Consequently, we examine the consistency of the outcome of the artificial modification of cartilage relative to the effects of the common enzyme, trypsin, used in the process of in vitro proteoglycan depletion. The results demonstrate that for a given time and enzyme concentration, the action of trypsin on proteoglycans is highly variable and is dependent on the initial distribution and concentration of proteoglycans at different depths, the intrinsic sample depth, the location in the joint space and the medium type, thereby sounding a note of caution to researchers attempting to model a proteoglycan-based degeneration of articular cartilage in their experimental studies.

Type III secretion, contact-dependent model for the intracellular development of chlamydia.

The medically significant genus Chlamydia is a class of obligate intracellular bacterial pathogens that replicate within vacuoles in host eukaryotic cells termed inclusions. Chlamydia's developmental cycle involves two forms; an infectious extracellular form, known as an elementary body (EB), and a non-infectious form, known as the reticulate body (RB), that replicates inside the vacuoles of the host cells. The RB surface is covered in projections that are in intimate contact with the inclusion membrane. Late in the developmental cycle, these reticulate bodies differentiate into the elementary body form. In this paper, we present a hypothesis for the modulation of these developmental events involving the contact-dependent type III secretion (TTS) system. TTS surface projections mediate intimate contact between the RB and the inclusion membrane. Below a certain number of projections, detachment of the RB provides a signal for late differentiation of RB into EB. We use data and develop a mathematical model investigating this hypothesis. If the hypothesis proves to be accurate, then we have shown that increasing the number of inclusions per host cell will increase the number of infectious progeny EB until some optimal number of inclusions. For more inclusions than this optimum, the infectious yield is reduced because of spatial restrictions. We also predict that a reduction in the number of projections on the surface of the RB (and as early as possible during development) will significantly reduce the burst size of infectious EB particles. Many of the results predicted by the model can be tested experimentally and may lead to the identification of potential targets for drug design.

Bayesian modelling of an epidemic of severe acute respiratory syndrome.

This paper analyses data arising from a SARS epidemic in Shanxi province of China involving a total of 354 people infected with SARS-CoV between late February and late May 2003. Using Bayesian inference, we have estimated critical epidemiological determinants. The estimated mean incubation period was 5.3 days (95% CI 4.2-6.8 days), mean time to hospitalisation was 3.5 days (95% CI 2.8-3.6 days), mean time from symptom onset to recovery was 26 days (95% CI 25-27 days) and mean time from symptom onset to death was 21 days (95% CI 16-26 days). The reproduction ratio was estimated to be 4.8 (95% CI 2.2-8.8) in the early part of the epidemic (February and March 2003) reducing to 0.75 (95% CI 0.65-0.85) in the later part of the epidemic (April and May 2003). The infectivity of symptomatic SARS cases in hospital and in the community was estimated. Community SARS cases caused transmission to others at an estimated rate of 0.4 per infective per day during the early part of the epidemic, reducing to 0.2 in the later part of the epidemic. For hospitalised patients, the daily infectivity was approximately 0.15 early in the epidemic, but fell to 0.0006 in the later part of the epidemic. Despite the lower daily infectivity level for hospitalised patients, the long duration of the hospitalisation led to a greater number of transmissions within hospitals compared with the community in the early part of the epidemic, as estimated by this study. This study investigated the individual infectivity profile during the symptomatic period, with an estimated peak infectivity on the ninth symptomatic day.

The role of mechanical host-tumour interactions in the collapse of tumour blood vessels and tumour growth dynamics.

A mathematical model of residual stress evolution in a growing vascular tumour is presented, in an attempt to elucidate the poorly understood phenomenon of vascular collapse. Whereas earlier studies in this area have neglected the effects of mechanical interactions between the tumour and the surrounding host tissue, the significance of these interactions for the long-term development of a tumour is now considered. The model predicts tumour stress distributions which reflect the distinctive patterns of vascular collapse reported in experimental studies. Moreover, while neglecting mechanical host/tumour interactions results in the eventual complete regression of the tumour to its avascular dormant size in the event of vascular collapse, this new model points to the possibility of oscillations in the tumour's size in the long term.

Dynamics of tear film deposition and draining.

This paper investigates the deposition of the tear film on the cornea of the human eye. The tear film is laid down by the motion of the upper eyelid and then subsequently flows and thins. Of particular interest is the stability of the tear layer and the development of dry patches on the cornea. While there has been significant research on the behaviour of tear films between blinks, this paper focuses on understanding the mechanisms which control the shape and thickness of the deposited film and how this affects the subsequent film behaviour. Numerical and analytical methods are applied to a lubrication model which includes the effects of surface tension, viscosity, gravity and evaporation. The model reveals the importance of the eyelid velocity, motion of the surface lipid layer and the storage of tear film between blinks.

An investigation of contact transmission of methicillin-resistant Staphylococcus aureus.

Hand hygiene is critical in the healthcare setting and it is believed that methicillin-resistant Staphylococcus aureus (MRSA), for example, is transmitted from patient to patient largely via the hands of health professionals. A study has been carried out at a large teaching hospital to estimate how often the gloves of a healthcare worker are contaminated with MRSA after contact with a colonized patient. The effectiveness of handwashing procedures to decontaminate the health professionals' hands was also investigated, together with how well different healthcare professional groups complied with handwashing procedures. The study showed that about 17% (9-25%) of contacts between a healthcare worker and a MRSA-colonized patient results in transmission of MRSA from a patient to the gloves of a healthcare worker. Different health professional groups have different rates of compliance with infection control procedures. Non-contact staff (cleaners, food services) had the shortest handwashing times. In this study, glove use compliance rates were 75% or above in all healthcare worker groups except doctors whose compliance was only 27%.

A history of the study of solid tumour growth: the contribution of mathematical modelling.

A miscellany of new strategies, experimental techniques and theoretical approaches are emerging in the ongoing battle against cancer. Nevertheless, as new, ground-breaking discoveries relating to many and diverse areas of cancer research are made, scientists often have recourse to mathematical modelling in order to elucidate and interpret these experimental findings. Indeed, experimentalists and clinicians alike are becoming increasingly aware of the possibilities afforded by mathematical modelling, recognising that current medical techniques and experimental approaches are often unable to distinguish between various possible mechanisms underlying important aspects of tumour development. This short treatise presents a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti-cancer treatments. Thus, a selection of mathematical publications, including the influential theoretical studies by Burton, Greenspan, Liotta et al., McElwain and co-workers, Adam and Maggelakis, and Byrne and co-workers are juxtaposed with the seminal experimental findings of Gray et al. on oxygenation and radio-sensitivity, Folkman on angiogenesis, Dorie et al. on cell migration and a wide variety of other crucial discoveries. In this way the development of this field of research through the interactions of these different approaches is illuminated, demonstrating the origins of our current understanding of the disease.

New insights into vascular collapse and growth dynamics in solid tumors.

The experimentally-observed phenomenon of vascular collapse in tumors represents a significant barrier to the delivery of blood-borne therapeutic drugs, and has been attributed to the elevated tissue stresses resulting from confined proliferation of tumor cells. This paper presents a mathematical framework which describes the evolution of growth-induced stresses in tumors and gives new insights into both vascular collapse and tumor growth dynamics. The linear-elastic description of anisotropic growth adopted here provides the mechanical model with a realistic constitutive basis, incorporating both the solid and stress-relaxation characteristics of soft biological tissues. A particular distribution of spatially non-uniform growth is proposed which is considered representative of a vascular tumor. The stress distribution associated with this growth pattern predicts the onset of vascular collapse, producing the well-defined regions observed in vascular collapse experiments: a peripheral layer with open blood vessels adjacent to a region of vascular collapse, enclosing an inner region where the vessels are open. The model also highlights the roles of various tissue properties in inducing vascular collapse. Moreover, the tumor growth rates predicted by this model reflect experimental observations, with exponential growth taking place immediately following vascularization, followed by a period of exponential retardation.

Use of a quantitative gene expression assay based on micro-array techniques and a mathematical model for the investigation of chlamydial generation time.

Chlamydia is an important pathogen which possesses a unique developmental cycle. We used real-time PCR technology to measure gene transcript levels in Chlamydia trachomatis strain L2. By measuring 16S rRNA transcript levels, and developing a mathematical model of the chlamydial developmental cycle fitting the data, we predict an average generation time of approximately 2.6 h. Additionally, potentially this modelling also provides the foundation for the application of emerging micro-array technology in which identification of the gene signals that trigger a chlamydial body to start replicating or transform to its infectious form can be made possible.

A model of neutralization of Chlamydia trachomatis based on antibody and host cell aggregation on the elementary body surface.

Humoral immunity is that aspect of specific immunity that is mediated by B lymphocytes and involves the neutralizing of pathogens by means of antibodies attaching to the pathogen's binding sites. Antibodies bind to and block ligand sites on the pathogen which prevents these sites from attaching to target cell receptors and so cell entry is inhibited. Many studies investigate the role of humoral immunity for protection against chlamydial challenge and they have shown that neutralization of the chlamydial body requires a large number of attached antibodies. Steric hindrance greatly influences the number of available sites that may be bound, reducing relative occupancy well below 100%. We model steric effects of antibody Fab fragment attachment indicating that they must be taken into consideration to accurately model valency, the number of available binding sites. We derive a partial differential equation for the number of antibody Fabs and host cell receptors that are aggregated to extracellular chlamydial elementary bodies. We consider steric effects in describing the size distribution of aggregates. Our theory is in good agreement with Monte Carlo simulations of binding. We use our theoretical prediction for the valency in a model for the in-host population dynamics of a chlamydial infection and we fit our model to experimental data.

A mathematical model for the investigation of the Th1 immune response to Chlamydia trachomatis.

Chlamydia are bacterial pathogens of humans and animals causing the important human diseases trachoma, sexually transmitted chlamydial disease and pneumonia. Of the human chlamydial diseases, sexually transmitted disease caused by Chlamydia trachomatis is a major public health concern. Chlamydia trachomatis replicates intracellularly and is characterised by a complex developmental cycle. Chlamydia is susceptible to humoral and cell-mediated immunity. Here we investigate the Th1 cell-mediated immune response against Chlamydia-infected cells as the response changes over the chlamydial developmental cycle. We suggest a form for the immune response over one developmental cycle by modelling the change in the number of intracellular chlamydial particles and assume peptides are presented in proportion to the number of replicating forms of chlamydial particles. We predict, perhaps non-intuitively, that persistent Chlamydia should be induced and forced not to return to the lytic cycle. We also suggest that extending the length of the time of the lytic cycle will effectively decrease the required efficacy of the Th1 response to eliminate the pathogen. We produce plots of active disease progression, control and clearance for varying levels of Th1 effectiveness.