Mathematical Medicine and Biology-A Journal of the Ima最新文献

Hepatitis is the term used to describe inflammation in the liver. It is associated with a high rate of mortality, but the underlying disease mechanisms are not completely understood and treatment options are limited. We present a mathematical model of hepatitis that captures the complex interactions between hepatocytes (liver cells), hepatic stellate cells (cells in the liver that produce hepatitis-associated fibrosis) and the immune components that mediate inflammation. The model is in the form of a system of ordinary differential equations. We use numerical techniques and bifurcation analysis to characterize and elucidate the physiological mechanisms that dominate liver injury and its outcome to a healthy or unhealthy, chronic state. This study reveals the complex interactions between the multiple cell types and mediators involved in this complex disease and highlights potential problems in targeting inflammation in the liver therapeutically.

This paper describes computer models of three interventions used for treating refractory pulmonary hypertension (RPH). These procedures create either an atrial septal defect, a ventricular septal defect or, in the case of a Potts shunt, a patent ductus arteriosus. The aim in all three cases is to generate a right-to-left shunt, allowing for either pressure or volume unloading of the right side of the heart in the setting of right ventricular failure, while maintaining cardiac output. These shunts are created, however, at the expense of introducing de-oxygenated blood into the systemic circulation, thereby lowering the systemic arterial oxygen saturation. The models developed in this paper are based on compartmental descriptions of human hemodynamics and oxygen transport. An important parameter included in our models is the cross-sectional area of the surgically created defect. Numerical simulations are performed to compare different interventions and various shunt sizes and to assess their impact on hemodynamic variables and oxygen saturations. We also create a model for exercise and use it to study exercise tolerance in simulated pre-intervention and post-intervention RPH patients.

As the SARS-CoV-2 virus spreads around the world new variants are appearing regularly. Although some countries have achieved very swift and successful vaccination campaigns, on a global scale the vast majority of the population is unvaccinated and new variants are proving more resistant to the current set of vaccines. We present a simple model of disease spread, which includes the evolution of new variants of a novel virus and varying vaccine effectiveness to these new strains. We show that rapid vaccine updates to target new strains are more effective than slow updates and containing spread through non-pharmaceutical interventions is vital while these vaccines are delivered. Finally, when measuring the key model inputs, e.g. the rate at which new mutations and variants of concern emerge, is difficult we show how an observable model output, the number of new variants that have been seen, is strongly correlated with the probability the virus is eliminated.

Estimating the longevity of an individual's immune response to the SARS-Cov-2 virus is vital for future planning, particularly of vaccine requirements. Neutralizing antibodies (Nabs) are increasingly being recognized as a correlate of protection and while there are many studies that follow the response of a cohort of people, each study alone is not enough to predict the long-term response. Studies use different assays to measure Nabs, making them hard to combine. We present a modelling method that can combine multiple datasets and can be updated as more detailed data becomes available. Combining data from seven published datasets we predict that the NAb decay has two phases, an initial fast but short-lived decay period followed by a longer term and slower decay period.

Chronic wounds, such as venous leg ulcers, are difficult to treat and can reduce the quality of life for patients. Clinical trials have been conducted to identify the most effective venous leg ulcer treatments and the clinical factors that may indicate whether a wound will successfully heal. More recently, mathematical modelling has been used to gain insight into biological factors that may affect treatment success but are difficult to measure clinically, such as the rate of oxygen flow into wounded tissue. In this work, we calibrate an existing mathematical model using a Bayesian approach with clinical data for individual patients to explore which clinical factors may impact the rate of wound healing for individuals. Although the model describes group-level behaviour well, it is not able to capture individual-level responses in all cases. From the individual-level analysis, we propose distributions for coefficients of clinical factors in a linear regression model, but ultimately find that it is difficult to draw conclusions about which factors lead to faster wound healing based on the existing model and data. This work highlights the challenges of using Bayesian methods to calibrate partial differential equation models to individual patient clinical data. However, the methods used in this work may be modified and extended to calibrate spatiotemporal mathematical models to multiple data sets, such as clinical trials with several patients, to extract additional information from the model and answer outstanding biological questions.

We focus on modelling of cancer hyperthermia driven by the application of the magnetic field to iron oxide nanoparticles. We assume that the particles are interacting with the tumour environment by extravasating from the vessels into the interstitial space. We start from Darcy's and Stokes' problems in the interstitial and fluid vessels compartments. Advection-diffusion of nanoparticles takes place in both compartments (as well as uptake in the tumour interstitium), and a heat source proportional to the concentration of nanoparticles drives heat diffusion and convection in the system. The system under consideration is intrinsically multi-scale. The distance between adjacent vessels (the micro-scale) is much smaller than the average tumour size (the macro-scale). We then apply the asymptotic homogenisation technique to retain the influence of the micro-structure on the tissue scale distribution of heat and particles. We derive a new system of homogenised partial differential equations (PDEs) describing blood transport, delivery of nanoparticles and heat transport. The new model comprises a double Darcy's law, coupled with two double advection-diffusion-reaction systems of PDEs describing fluid, particles and heat transport and mass, drug and heat exchange. The role of the micro-structure is encoded in the coefficients of the model, which are to be computed solving appropriate periodic problems. We show that the heat distribution is impaired by increasing vessels' tortuosity and that regularization of the micro-vessels can produce a significant increase (1-2 degrees) in the maximum temperature. We quantify the impact of modifying the properties of the magnetic field depending on the vessels' tortuosity.

Our aim in this paper is to study a mathematical model for high grade gliomas, taking into account lactates kinetics, as well as chemotherapy and antiangiogenic treatment. In particular, we prove the existence and uniqueness of biologically relevant solutions. We also perform numerical simulations based on different therapeutical situations that can be found in the literature. These simulations are consistent with what is expected in these situations.

The Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, and generalizations thereof, involves simple reaction-diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $lambda $. For the Fisher-KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2sqrt {lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed $c=2sqrt {lambda D}> 0$. This means that, for biologically relevant initial data, the Fisher-KPP model cannot be used to study invasion with $c ne 2sqrt {lambda D}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, $-infty < c < infty $. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.

We propose a realistic model for the evolution of the COVID-19 pandemic subject to the lockdown and quarantine measures, which takes into account the timedelay for recovery or death processes. The dynamic equations for the entire process are derived by adopting a kinetic-type reactions approach. More specifically, the lockdown and the quarantine measures are modelled by some kind of inhibitor reactions where susceptible and infected individuals can be trapped into inactive states. The dynamics for the recovered people is obtained by accounting people who are only traced back to hospitalized infected people. To get the evolution equation we take inspiration from the Michaelis Menten's enzyme-substrate reaction model (the so-called MM reaction) where the enzyme is associated to the available hospital beds, the substrate to the infected people, and the product to the recovered people, respectively. In other words, everything happens as if the hospitals beds act as a catalyzer in the hospital recovery process. Of course, in our case, the reverse MM reaction has no sense in our case and, consequently, the kinetic constant is equal to zero. Finally, the ordinary differential equations (ODEs) for people tested positive to COVID-19 is simply modelled by the following kinetic scheme $S+IRightarrow 2I$ with $IRightarrow R$ or $IRightarrow D$, with $S$, $I$, $R$ and $D$ denoting the compartments susceptible, infected, recovered and deceased people, respectively. The resulting kinetic-type equations provide the ODEs, for elementary reaction steps, describing the number of the infected people, the total number of the recovered people previously hospitalized, subject to the lockdown and the quarantine measure and the total number of deaths. The model foresees also the second wave of infection by coronavirus. The tests carried out on real data for Belgium, France and Germany confirmed the correctness of our model.