Mathematical Medicine and Biology - Advance Access

Chronic disease projections in heterogeneous ageing populations: approximating multi-state models of joint distributions by modelling marginal distributions

Hoogenveen, R. T., van Baal, P. H. M., Boshuizen, H. C. — 2009-06-10

To quantify the effects of changes in risk factors for chronic diseases on morbidity and mortality, Markov-type multi-state models are used. However, with multiple risk factors and many diseases relating to these risk factors, these models contain a large number of states. In this paper, we present an alternative modelling methodology implemented in the National Institute for Public Health and the Environment chronic disease model. This model includes multiple states based on risk factor levels and disease stages but only keeps track of the marginal probability values. Starting from the multi-state model, differential equations are derived that describe the change of the marginal distribution for each risk factor class and disease stage, taking into account population heterogeneity and competing mortality risks. The model is illustrated by presenting results of a scenario affecting disease incidence by altering the risk factor distribution of the population. To show the strength of the approximating model, we compare its results to those of the multi-state Markov model.

The impact of uncertainty in a blood coagulation model

Danforth, C. M., Orfeo, T., Mann, K. G., Brummel-Ziedins, K. E., Everse, S. J. — 2009-05-18

Deterministic mathematical models of biochemical processes operate as if the empirically derived rate constants governing the dynamics are known with certainty. Our objective in this study was to explore the sensitivity of a deterministic model of blood coagulation to variations in the values of its 44 rate constants. This was accomplished for each rate constant at a given time by defining a normalized ensemble standard deviation (w_{k_i}^f(t)) that accounted for the sensitivity of the predicted concentration of each protein species to variation in that rate constant (from 10 to 1000% of the accepted value). A mean coefficient of variation derived from w_{k_i}^f(t) values for all protein species was defined to quantify the overall variation introduced into the model's predictive capacity at that time by the assumed uncertainty in that rate constant. A time-average value of the coefficient of variation over the 20-min simulation for each rate constant was then used to rank rate constants. The model's predictive capacity is particularly sensitive (50% of the aggregate variation) to uncertainty in five rate constants involved in the regulation of the formation and function of the factor VIIa–tissue factor complex. Therefore, our analysis has identified specific rate constants to which the predictive capability of this model is most sensitive and thus where improvements in measurement accuracy will yield the greatest increase in predictive capability.

Stability of ecosystem: global properties of a general predator-prey model

Korobeinikov, A. — 2009-04-20

Establishing the conditions for the stability of ecosystems and for stable coexistence of interacting populations is a problem of the highest priority in mathematical biology. This problem is usually considered under specific assumptions made regarding the functional forms of non-linear feedbacks. However, there is growing understanding that this approach has a number of major deficiencies. The most important of these is that the precise forms of the functional responses involved in the model are unknown in detail, and we can hardly expect that these will be known in feasible future. In this paper, we consider the dynamics of two species with interaction of consumer–supplier (prey–predator) type. This model generalizes a variety of models of population dynamics, including a range of prey–predator models, SIR and SIRS epidemic models, chemostat models, etc. We assume that the functional responses that are usually included in such models are given by unspecified functions. Using the direct Lyapunov method, we derive the conditions which ensure global asymptotic stability of this general model. It is remarkable that these conditions impose much weaker constraints on the system properties than that are usually assumed. We also identify the parameter that allows us to distinguish between existence and non-existence of the coexisting steady state.

Importance of Taylor dispersion in pharmacokinetic and multiple indicator dilution modelling

Fallon, M. S., Howell, B. A., Chauhan, A. — 2009-03-24

Mass transfer in tissues is typically studied by multiple indicator dilution (MID) studies. Several of these studies have shown that drug concentrations can be modelled by axially distributed models. In this paper, we determine the Taylor dispersion coefficient that describes the degree of axial mixing for the MID models, while accounting for the presence of red blood cells in the capillaries. The capillaries are treated as well mixed with no radial concentration gradients. The concentration in tissue is treated as position and time dependent and the partial differential equations for mass transport are averaged using the method of multiple timescales. The calculated values of the dispersion coefficient are in reasonable agreement with the values reported in literature, suggesting that Taylor dispersion is an important contributor to dispersion in tissues. We also show that the average equations for the barrier-limited drugs reduce to the commonly used Sangren–Sheppard model. In this case, Taylor dispersion is not significant in comparison to the dispersion caused by drug exchange between the capillary and the tissue. Additionally, we utilize the average equations for both flow-limited and barrier-limited drugs in pharmacokinetic models. These simulations show that neglecting the dispersion coefficient could cause significant effects in the dynamic drug concentration profiles and thus lead to incorrect estimation of parameters if the experimental data from MID studies are fitted to a model that neglects Taylor dispersion.

Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate

Korobeinikov, A. — 2009-03-18

We consider two models for the spread of an infection with a free-living infective stage, where parasite reproduction and virulence (parasite-induced mortality) depend on the parasite dose to which the host is exposed and are given by unspecified non-linear functions of the number of the free pathogen particles, and the incidence rate is non-linear. We study the impact of these non-linearities with the focus on the global properties of these models. We consider a very general form of the non-linearities: we assume that the virulence and the parasite reproduction rates are given by unspecified non-linear functions of the number of the free pathogen particles and that the incidence rate is an unspecified function of the number of susceptible hosts and free pathogen particles; all these functions are constrained by a few biologically feasible conditions. We construct Lyapunov functions that enable us to find biologically realistic conditions which are sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number, this equilibrium state can be either positive, where parasite endemically persists, or infection free.

A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials

Richardson, G. — 2009-03-09

By application of matched asymptotic expansions, a simplified partial differential equation (PDE) model for the dynamic electrochemical processes occurring in the vicinity of a membrane, as ions selectively permeate across it, is formally derived from the Poisson–Nernst–Planck equations of electrochemistry. It is demonstrated that this simplified model reduces itself, in the limit of a long thin axon, to the cable equation used by Hodgkin and Huxley to describe the propagation of action potentials in the unmyelinated squid giant axon. The asymptotic reduction from the simplified PDE model to the cable equation leads to insights that are not otherwise apparent; these include an explanation of why the squid giant axon attains a diameter in the region of 1 mm. The simplified PDE model has more general application than the Hodgkin–Huxley cable equation and can, e.g. be used to describe action potential propagation in myelinated axons and neuronal cell bodies.

Analytical thermal-optic model for laser heating of biological tissue using the hyperbolic heat transfer equation

Trujillo, M., Rivera, M. J., Lopez Molina, J. A., Berjano, E. J. — 2009-02-20

In this paper, we solve in an analytical way the thermal-optic coupled problem associated with a 1D model of non-perfused homogeneous biological tissue irradiated by a laser beam. We consider a laser pulse duration of 200 µs and study the temperatures of areas very close to the point of laser beam application. We consider that these values of the temporal and spatial variables mean that the problem has to be solved by means of the hyperbolic heat conduction model instead of the classic or parabolic model. We therefore obtain the solution using both models and apply the temperature profiles obtained to a specific biological tissue for comparison. Finally, we theoretically study the effect of the thermal relaxation time on the temperature profiles in the tissue for both heating and cooling phases (i.e. during and after laser application).