Modelling And Simulation of Topical Drug Diffusion in The Dermal Layer of Human Body

Abstract

We consider the problem of drug diffusion in the dermal layer of human body. Two existing mathematical models of the drug diffusion problem are recalled. We obtain that the existing models lead to inconsistent equations for the steady state condition. We also obtain that solutions to the existing models are unrealistic for some cases of the unsteady state condition, because negative drug concentrations occur due to the inappropriate assumption of the model. Therefore, in this paper, we propose a modified mathematical model, so that the model is consistent, and the solution is nonnegative for both steady and unsteady state conditions of the drug diffusion problem in the dermal layer of human body. For the steady state condition, the exact solution to the proposed model is given. For unsteady state condition, we use a finite difference method for solving the models numerically, where the discretisation is centred in space and forward in time. Simulation results confirm that our proposed model and method preserve the non-negativity of the solution to the problem, so the solution is more realistic than that of the old model.

Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

“…From these representations, we observe that the rate of diffusion of drug in the skin decreases rapidly for α = 0.9 compared to α = 1.0. The results demonstrate excellent agreement with those reported by S. Mungkasi (15) . This consistency further validates the reliability of the proposed fractional-order explicit finite difference method and reinforces its capability to accurately predict drug diffusion in the human dermal layer.…”

“…Given that U k represents the exact solution of the TFDDE described by Equations ( 3), ( 4) and ( 5), it satisfies Equations ( 13), ( 14) and (15). Therefore,…”

“…Subsequently, the drug undergoes a downward diffusion, propagating to locations situated beneath the skin's surface. The outermost layer of the dermal layer corresponds to the epidermis, while the underlying dermal layer possesses an approximate thickness of 2 mm, as referenced in the literature (15) . The domain of interest in this study encompasses the dermal layer, extending up to about 0.1 mm from the epidermis.…”

“…As a result, a length of one unit corresponds to an actual measurement of two millimeters. Considering the provided assumptions and applying scaling, we examine the temporal fractional drug diffusion equation (TFDDE), derived from the classical drug diffusion equation (15) . It is expressed as follows:…”

“…This advantage arises due to the power-law kernal of the Caputo derivative, where the memory effect become more pronounced at small time values. Drawing from this context and considering the memory effect associated with fractional derivatives, our study extends the conventional model of drug diffusion (15) to a fractional-order mathematical model with the goal of predicting drug diffusion within the human dermal region. Unfortunately, finding exact solutions to most fractional differential equations is a challenging task due to their complex nature and non-linearity.…”