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 clearence. 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

Upper bounds on the non-random fluctuations in first passage percolation with low moment conditions

We consider first passage percolation with i.i.d. weights on edges of the d-dimensional cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in the d-dimensional cubic lattice, and has moments bigger than 1, we investigate upper bounds on the so-called non-random fluctuations of the model. In addition, we give an application of our result to a lower bound for variance of the first passage percolation in the case where the limit shape has flat edges.Comment: This is the corrected version of the paper. 11 pages, title change

Usage of air-conditioners and windows in residential areas in Johor Bahru City : planning methods of coastal residential areas in consideration of wind flow

This study discusses planning methods of residential areas in coastal cities, focusing on the effect of wind flow. This aims to determine strategies that reduce energy consumption in residential areas, targeting especially a reduction in the use of air-conditioners. As a part of the above study, this paper presents the findings of a questionnaire survey on usage of windows and air-conditioners among selected households in apartment houses in Johor Bahru City, Malaysia and compares the results with those in terraced houses surveyed previously