A two-dimensional mathematical model of percutaneous drug absorption.

BACKGROUND: When a drug is applied on the skin surface, the concentration of the drug accumulated in the skin and the amount of the drug eliminated into the blood vessel depend on the value of a parameter, r. The values of r depend on the amount of diffusion and the normalized skin-capillary clearance. It is defined as the ratio of the steady-state drug concentration at the skin-capillary boundary to that at the skin-surface in one-dimensional models. The present paper studies the effect of the parameter values, when the region of contact of the skin with the drug, is a line segment on the skin surface. METHODS: Though a simple one-dimensional model is often useful to describe percutaneous drug absorption, it may be better represented by multi-dimensional models. A two-dimensional mathematical model is developed for percutaneous absorption of a drug, which may be used when the diffusion of the drug in the direction parallel to the skin surface must be examined, as well as in the direction into the skin, examined in one-dimensional models. This model consists of a linear second-order parabolic equation with appropriate initial conditions and boundary conditions. These boundary conditions are of Dirichlet type, Neumann type or Robin type. A finite-difference method which maintains second-order accuracy in space along the boundary, is developed to solve the parabolic equation. Extrapolation in time is applied to improve the accuracy in time. Solution of the parabolic equation gives the concentration of the drug in the skin at a given time. RESULTS: Simulation of the numerical methods described is carried out with various values of the parameter r. The illustrations are given in the form of figures. CONCLUSION: Based on the values of r, conclusions are drawn about (1) the flow rate of the drug, (2) the flux and the cumulative amount of drug eliminated into the receptor cell, (3) the steady-state value of the flux, (4) the time to reach the steady-state value of the flux and (5) the optimal value of r, which gives the maximum absorption of the drug. The paper gives valuable information which can be obtained by this two-dimensional model, that cannot be obtained with one-dimensional models. Thus this model improves upon the much simpler one-dimensional models. Some future directions of the work based on this model and the one-dimensional non-linear models that exist in the literature, are also discussed.

A mathematical model for the burden of diabetes and its complications.

BACKGROUND: The incidence and prevalence of diabetes are increasing all over the world. Complications of diabetes constitute a burden for the individuals and the whole society. METHODS: In the present paper, ordinary differential equations and numerical approximations are used to monitor the size of populations of diabetes with and without complications. RESULTS: Different scenarios are discussed according to a set of parameters and the dynamical evolution of the population from the stage of diabetes to the stage of diabetes with complications is clearly illustrated. CONCLUSIONS: The model shows how efficient and cost-effective strategies can be obtained by acting on diabetes incidence and/or controlling the evolution to the stage of complications.

A model of dengue fever.

BACKGROUND: Dengue is a disease which is now endemic in more than 100 countries of Africa, America, Asia and the Western Pacific. It is transmitted to the man by mosquitoes (Aedes) and exists in two forms: Dengue Fever and Dengue Haemorrhagic Fever. The disease can be contracted by one of the four different viruses. Moreover, immunity is acquired only to the serotype contracted and a contact with a second serotype becomes more dangerous. METHODS: The present paper deals with a succession of two epidemics caused by two different viruses. The dynamics of the disease is studied by a compartmental model involving ordinary differential equations for the human and the mosquito populations. RESULTS: Stability of the equilibrium points is given and a simulation is carried out with different values of the parameters. The epidemic dynamics is discussed and illustration is given by figures for different values of the parameters. CONCLUSION: The proposed model allows for better understanding of the disease dynamics. Environment and vaccination strategies are discussed especially in the case of the succession of two epidemics with two different viruses.

A sequential algorithm for the non-linear dual-sorption model of percutaneous drug absorption.

A sequential algorithm is developed for the non-linear dual-sorption model developed by Chandrasekaran et al. [1,2] which monitors pharmacokinetic profiles in percutaneous drug absorption. In the experimental study of percutaneous absorption, it is often observed that the lag-time decreases with the increase in the donor concentration when two or more donor concentrations of the same compound are used. The dual-sorption model has sometimes been employed to explain such experimental results. In this paper, it is shown that another feature observed after vehicle removal may also characterize the dual-sorption model. Soon after vehicle removal, the plots of the drug flux versus time become straight lines on a semilogarithmic scale as in the linear model, but the half-life is prolonged thereafter when the dual-sorption model prevails. The initial half-life after vehicle removal with a low donor concentration is longer than that with a higher donor concentration. These features, if observed in experiments, may be used as evidence to confirm that the dual-sorption model gives an explanation to the non-linear kinetic behaviour of a permeant.

Drug release from a suspension with a finite dissolution rate: theory and its application to a betamethasone 17-valerate patch.

A random walk method for a diffusion equation is applied to the model for a suspension with a finite dissolution rate developed by Ayres and Lindstrom in 1977. In the method, the diffusion of dissolved drug and dissolution of crystal are calculated separately using a simple BASIC program. The random walk method strictly meets the principle of the conservation of mass as the drug amount in each sublayer rather than the concentration at each subinterval is concerned in the ointment. The model is used to analyze the release of betamethasone 17-valerate from a pressure-sensitive silicone adhesive into a sink. The drug release from the 1.50 mg/mL patch shows no substantial discrepancy from that predicted by the classic suspension model assuming an infinite dissolution rate. However, the classic model overestimates the release from the 3.08 and 5.88 mg/mL patches. The disagreement is lessened when the dissolution rate is assumed to be finite. However, the model does not give a perfect explanation because the drug release from the 3.08 and 5.88 mg/mL patches in the early phase is faster than the model predicts.

Lag time in the dual sorption model for percutaneous absorption with finite skin-receptor boundary clearance.

The lag time in the dual sorption model for percutaneous drug absorption with finite skin-receptor boundary clearance is examined. The effect of the change in the perfusional resistance on the lag time in the dual sorption model is found to be similar to, but a little less prominent than, that in the linear model. It is suggested that the lag time is roughly proportional to the amount in the skin at the steady state. Two numerical methods are used in predictor-corrector combination to obtain a numerical solution of the parabolic partial differential equation, together with the associated initial and boundary conditions, which arise in a nonlinear mathematical model of percutaneous drug absorption. Numerical estimates are obtained for the drug concentration profiles, the cumulative amount of drug excreted into the blood, the flux in the transient and steady states, the lag time, and the amount of drug in the skin per unit area in the steady state.

Dual sorption model for the nonlinear percutaneous permeation kinetics of timolol.

The nonlinear percutaneous permeation kinetics of timolol were studied in vitro with human cadaver skin. In the sorption isotherm (equilibrium) study, the plots of the amount of timolol per unit area of epidermis versus aqueous timolol concentration in equilibrium were curvilinear as the dual sorption model predicts. In the penetration study, several aqueous timolol concentrations were maintained in the donor cell for 25 h. The lag-time was prolonged as the timolol concentration in the donor cell was decreased. The change in the lag time was analyzed by a newly proposed method with the lag-time prolongation factor that is calculated from the parameter values obtained in the sorption isotherm study. The ratio of the lag-time to lag-time prolongation factor was of the same magnitude, indicating that the dual sorption model explains the nonlinear percutaneous permeation kinetics of timolol. Finally, the diffusion parameter for the mobile solute was estimated by fitting the data of the cumulative amount excreted into receptor cell versus time to the numerical solution of the dual sorption model. The observed data were roughly compatible with the values predicted by the dual sorption model.

A nonlinear numerical model of percutaneous drug absorption.

A nonlinear mathematical model developed by Chandrasekaran et al. is examined to monitor pharmacokinetic profiles in percutaneous drug absorption and is addressed to several associated problems that could occur in the data analysis of in vitro experiments. The formulation of the model gives rise to a nonlinear partial differential equation (PDE) of parabolic type, and a family of finite-difference methods is developed for the numerical solution of the associated initial/boundary-value problem. The value given to a parameter in this family determines the stability properties of the resulting method and whether the solution is obtained explicitly or implicitly. In the case of implicit members of the family it is seen that the solution of the nonlinear PDE is obtained by solving a linear algebraic system, the coefficient matrix of which is tridiagonal. The behaviors of two methods of the family are examined in a series of numerical experiments. Numerical differentiation and integration procedures are combined to monitor the cumulative amount of drug eliminated into the receptor cell per unit area as time increases. It is found that the use of the equation for the simple membrane model to estimate the permeability coefficient and lag time is warranted even if the system should be described by the dual-sorption model, provided cumulative amount versus time data collected for a sufficiently long time are used. However, being different from the behavior in the simple membrane model, the lag time, which can be estimated in this way, is dose-dependent and decreases with increasing donor cell concentration. On the other hand, the permeability coefficient in the dual-sorption model remains constant irrespective of the donor cell concentrations as in the simple membrane model.

The global extrapolation of numerical methods for computing concentration profiles in percutaneous drug absorption.

A family of numerical methods is developed and analyzed for the numerical solution of the parabolic partial differential equation together with the associated initial and boundary conditions, which arise in a mathematical model of the transient stage of percutaneous drug absorption. Two global extrapolation procedures are described, the first in time only, the second in both space and time, for improving the accuracy of the computed concentration profiles. The behaviours of two members of the family of methods, before and after extrapolation, are examined by repeating a number of experiments reported in the literature. Modifications to the algorithms, which are necessary in computing concentration profiles after the ointment is removed at the steady state, are outlined.

A transient two-dimensional model of thermoregulation in a human subject.

This paper describes a mathematical model of thermoregulation in a nude human subject. Heat flow within and from the body is expressed as a system of partial differential equations. The nonlinearity of the system reflects the dependence of physiological factors such as blood flow, sweating, and shivering upon tissue temperature. The model incorporates the effect of longitudinal heat flow in the torso. The resultant time-dependent nonlinear system in two dimensions is solved using high-accuracy finite-difference techniques and the simulation is validated against laboratory data and tested in two numerical experiments.

Finite element solution of steady state temperature distribution in the human torso.

A finite element model of the steady state temperature distribution in the human torso is developed. The torso is approximated by a circular cylinder of core surrounded by a layer of muscle and insulating layers of fat and skin. The model is simplified by neglecting longitudinal heat flow. The region occupied by a circular cross-section of the torso is discretized into a mesh of triangles and the boundary of the torso, that is, the skin surface, is consequently approximated by a polygon. The elliptic partial differential equation governing the steady state temperature distribution, together with the associated boundary conditions, are expressed in equivalent variational form. Linear basis functions are used and the resulting integral is minimized over the region bounded by the approximating polygon. Results for two numerical experiments are determined by solving systems of linear equations.