Mathematical Modelling of Solute Transport Through Stratum Corneum

Mollee, Thomas (2005). Mathematical Modelling of Solute Transport Through Stratum Corneum PhD Thesis, School of Physical Sciences, The University of Queensland.

       
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Author Mollee, Thomas
Thesis Title Mathematical Modelling of Solute Transport Through Stratum Corneum
School, Centre or Institute School of Physical Sciences
Institution The University of Queensland
Publication date 2005
Thesis type PhD Thesis
Supervisor Tony Bracken
Yuri Anissimov
Total pages 119
Collection year 2005
Language eng
Subjects L
230101 Mathematical Logic, Set Theory, Lattices And Combinatorics
780101 Mathematical sciences
Formatted abstract

The principle barrier to the penetration of chemicals through human skin is the outermost layer, termed the stratum corneum. The stratum corneum consists of approximately 20 layers of essentially dead, flattened cells called corneocytes that are embedded in a lipid bilayer domain. This lipid domain occupies only a small fraction of the volume of the stratum corneum but for small, neutral solutes is generally regarded as providing the primary transport pathway through it. A theoretical understanding of the relative impermeability of the stratum corneum is of practical importance to the application of transdermal drug delivery. 

Existing mathematical models of this transport process are discussed in Chapter 2, following a summary of the physiological background in Chapter 1. The majority of these models use the relatively simple one-dimensional diffusion equation, treating the diffusional path length and diffusion coefficient of the solute molecules in the lipid bilayer medium as unknowns. This is a useful approach for the description of experimental data, but it does not describe the transport process in terms of fundamental physical parameters of the membrane. Models that do try to incorporate such physical parameters are also discussed in Chapter 2. 

Chapter 3 discusses existing one-dimensional diffusion models in which capture and release of solute molecules throughout the lipid domain are incorporated. In these models the solute molecules are divided into two species; one in which solute molecules are bound to fixed sites in the lipid medium, and one for molecules that diffuse freely. The capture and release mechanism appears in the equation expressing conservation of mass through a source/sink term. Two cases are considered in the literature. The first involves instantaneous equilibrium between "bound'' and "free" solute molecules [21] and in the second this process is time-dependent [9]. In the former case the relation between bound and free solute is nonlinear, resulting in the equation for conservation of mass being nonlinear. Despite this, exact expressions for the steady-state flux and time lag of the system have been obtained. In the latter case the capture (sink) component of the source/sink term, at a given position in the lipid medium, is assumed to be related linearly to the concentration of free solute while the release (source) of previously captured solute is proportional to the concentration of bound solute. Conservation of mass is then expressed as a system of coupled linear partial differential equations. These equations have been analyzed in the Laplace domain, and expressions for the steady-state flux and time lag of the system have been given. We complete these earlier analyses by giving analytic expressions for the concentration and flux of free solute along the diffusional pathway, as well as the cumulative amount of solute having left the membrane. 

The capture and release process discussed by Anissimov & Roberts is related to a more general linear capture and release process discussed by Bass Bracken & Hilden. The model of Bass et al. treats the release of previously captured solute by introducing a probability density function that governs release times. As shown by Bass et al., if the probability density function is a decaying exponential function of time the equation representing conservation of mass reduces to a system of coupled, linear partial differential equations that are equivalent to those studied by Anissimov & Roberts for solute transport through skin and which have been treated in a general context by Aifantis & Hill, and later by Hill & McNabb. For the system discussed by Bass et al. we obtain the Laplace transform of concentration, flux and cumulative amount of solute having passed through the membrane for the case when the rate constants for capture and release are independent of position. From these, expressions for the steady-state flux and time lag are given for the case of a general probability density function governing captured solute release times. We then consider as an example a density function of the form tne-kt /kn+1. When n is a positive integer the equation representing conservation of mass is expressible as a system of n + 1 coupled, linear partial differential equations. 

Chapters 4 and 5 contain the main new results of this thesis. 

Chapter 4 presents a new model of solute transport through the intercellular domain of the stratum corneum that is time dependent and that accounts for some structural features of the membrane. Here transport is viewed as one-dimensional in the "vertical" space between "horizontally') adjacent corneocytes. Once solute molecules reach the thin lipid layers separating vertically adjacent corneocytes (these regions we refer to as trapping layers) they diffuse horizontally until they reach a subsequent vertical channel, after which they continue to diffuse vertically. This process is repeated until the boundary of the membrane in contact with the receptor is reached. The model consists of section-wise one-dimensional diffusion with concentration and flux matching conditions on either side of infinitesimally thin trapping layers. Capture and release of solute molecules is used to obtain the matching relation for solute flux on opposite sides of trapping layers. Two explicit probability density functions governing solute molecule residence times are found by considering the "horizontal" diffusion of a solute molecule in the trapping layers, assuming simple geometrical shapes of corneocytes. 

However, we were unable to establish on physical grounds the appropriate relation for matching solute concentration on opposite sides of a trapping layer. Instead a linear relation identical to the matching condition for flux is assumed, although it is recognised that this may not always be appropriate. Under the assumed concentration and flux matching conditions, expressions for the Laplace transform of solute concentration, flux and cumulative amount having passed through the membrane are found. From these, estimates of steady-state flux and time lag of the system are found in terms of the structural parameters of the system and the results are related to those given in Section 2.2. It is found that the estimated steady-state flux is approximately one order of magnitude greater than those given in Section 2.2, while the time lag is at worst a factor of six less than those of Section 2.2. Such discrepancies are not surprising, especially given the assumed form of the matching condition for solute concentration. Unfortunately, existing experimental data are not sufficient to discriminate between this and earlier models. Obviously further investigations are needed to clarify the effects that different forms of concentration matching conditions have on steady-state flux and time lag, and also investigations into more appropriate concentration-matching conditions. 

Transdermal iontophoresis is the transport of ions across the skin due to the application of an external electric field. Constant direct currents or voltages are most commonly used for this purpose and can be regulated to enable control of the rate and duration of the drug delivery. However, a number of side effects are associated with its use such as erythema and electrode burns of the skin. To overcome these problems, low frequency AC iontophoresis has been proposed as an alternative means to transport ions across the skin. Chapter 5 of this thesis examines the mean flux of a charged tracer across a homogeneous membrane subject to alternating, symmetric voltage waveforms. The analysis is based on the Nernst-Planck flux equation, with electric field varying with time only, and is integrated numerically for four different voltage waveforms. Approximations for small and large frequencies are obtained and an approximation formula for all frequencies, due to Anissimov, is discussed. 

A brief discussion in Chapter 6 of possible future research concludes the thesis. 

Keyword Skin -- Permeability -- Mathematical models
Skin absorption -- Mathematical models

 
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