Analytical and Numerical Methods in Determining the Combined Effects of Iontophoresis and Chemical Penetration Enhancers

(25.1)

where C is the drug concentration at depth x, D represents the diffusion coefficient in the membrane, and t is the time. Initially, the membrane is free of drug:

$$$ C\left( x>0,0\right)=0 $$ ” src=”http://plasticsurgerykey.com/wp-content/uploads/2017/10/A309386_1_En_25_Chapter_Equ2.gif”></DIV></DIV><br /> <DIV class=EquationNumber>(25.2)</DIV></DIV>The boundary conditions are<br /> <DIV id=Equ3 class=Equation><br /> <DIV class=EquationContent><br /> <DIV class=MediaObject><IMG alt=$ 0\right)={C}_{\mathrm{s}}; C\left( L, t>0\right)=0 $$ ” src=”http://plasticsurgerykey.com/wp-content/uploads/2017/10/A309386_1_En_25_Chapter_Equ3.gif”>

(25.3)

In Eq. 25.3, C _s is the concentration at the membrane-vehicle interface and L is the membrane thickness. The drug concentration in the vehicle C _o is related to C _s by

${C}_{\mathrm{s}}= K{C}_{\mathrm{o}}$

(25.4)

The flux is defined at L:

$J=- D\frac{\partial C}{\partial x}\Big|{}_{x= L}=\frac{dQ}{dt}$

(25.5)

where Q is the cumulative amount of medicament released and K is the vehicle/stratum corneum partition coefficient. An expression for Q is developed by solving the system formed by Eqs. 25.1, 25.2, 25.3, and 25.5:

$Q= L{C}_s\left[\frac{tD}{L^2}-\frac{1}{6}-\frac{2}{\pi^2}\left({\displaystyle \sum}_{n=1}^{\infty}\frac{{\left(-1\right)}^n \exp \left[-{n}^2{\pi}^2\frac{tD}{L^2}\right]}{n^2}\right)\right]$

(25.6)

The steady-state flux J _ss and the cumulative amount of drug released at long time Q _ss are

${J}_{\mathrm{s}\mathrm{s}}=\frac{D{ C}_{\mathrm{s}}}{L},$

(25.7)

and

${Q}_{s s}= L{C}_s\left(\frac{tD}{L^2}-\frac{1}{6}\right)$

(25.8)

respectively. The lag time t _lag is given by

${t}_{\mathrm{lag}}=\frac{L^2}{6 D}$

(25.9)

Besides t _lag, the effective time constant t _eff can also estimate the time elapsed before reaching J _ss (Collins 1980; Simon 2009):

${t}_{\mathrm{eff}}=\frac{\underset{s\to 0}{ \lim}\left(\frac{J_{\mathrm{ss}}}{s^2}+\frac{d\overline{J}(s)}{ d s}\right)}{\underset{s\to 0}{ \lim}\left(\frac{J_{\mathrm{ss}}}{s}-\overline{J}(s)\right)}$

(25.10)

${t}_{\mathrm{eff}}=\frac{7{L}^2}{60 D}$

(25.11)

in the case of passive diffusion. In Eq. 25.10, $\overline{J}(s)$ is the Laplace transform of J. The permeability coefficient P, often used in skin absorption studies, incorporates the effects of K, D, and L:

$P=\frac{KD}{L}=\frac{J_{\mathrm{ss}}}{C_{\mathrm{o}}}$

(25.12)

25.4.2 Iontophoretic Drug Transport Across a Biological Membrane

The following equation can be used to study iontophoretic drug transport across a membrane (Keister and Kasting 1986):

$\frac{\partial C}{\partial t}= D\frac{\partial^2 C}{\partial {x}^2}-\frac{\gamma D}{L}\frac{\partial C}{\partial x}$

(25.13)

where γ is a parameter which represents the effects of the electric field. Consideration of Eqs. 25.2 and 25.3 leads to the following expression for the cumulative amount of drug released (Wei et al. 2012; Simon 2009; Keister and Kasting 1986):

$Q=\frac{D{ C}_s}{L}\frac{\gamma}{1-{e}^{-\gamma}}\left\{ t+\frac{2{L}^2}{D}\frac{ \sinh \left(\frac{\gamma}{2}\right)}{\frac{\gamma}{2}}{\displaystyle \sum}_{n=1}^{\infty}\frac{n^2{\pi}^2{\left(-1\right)}^n}{{\left(\frac{\gamma^2}{4}+{n}^2{\pi}^2\right)}^2}\left[1- \exp \left(-\frac{\left(\frac{\gamma^2}{4}+{n}^2{\pi}^2\right) Dt}{L^2}\right)\right]\right\}$

(25.14)

Similar to the case of passive diffusion, the following functions were derived (Simon 2009; Keister and Kasting 1986):

${J}_{\mathrm{s}\mathrm{s}}=\frac{D{ C}_{\mathrm{s}}}{L}\frac{\gamma}{1-{e}^{-\gamma}},$

(25.15)

${Q}_{\mathrm{s}\mathrm{s}}=\frac{D{ C}_{\mathrm{s}}}{L}\frac{\gamma}{1-{e}^{-\gamma}}\left[ t+\frac{L^2}{D}\frac{2 \sinh \left(\frac{\gamma}{2}\right)-\gamma \cosh \left(\frac{\gamma}{2}\right)}{\gamma^2 \sinh \left(\frac{\gamma}{2}\right)}\right],$

(25.16)

${t}_{\mathrm{lag}}=-\frac{L^2}{D}\frac{2 \sinh \left(\frac{\gamma}{2}\right)-\gamma \cosh \left(\frac{\gamma}{2}\right)}{\gamma^2 \sinh \left(\frac{\gamma}{2}\right)}$

(25.17)

and

${t}_{\mathrm{eff}}=\frac{L^2 \csc {\mathrm{h}}^2\left(\frac{\gamma}{2}\right)\left(\begin{array}{l}3{\gamma}^2-2 \sinh \left(\gamma \right)\gamma +\left({\gamma}^2-4\right)\\ {} \cosh \left(\gamma \right)+4\end{array}\right)}{4 D{\gamma}^2\left(\gamma \coth \left(\frac{\gamma}{2}\right)-2\right)}$

(25.18)

25.5 Results

For systems using passive diffusion alone, the lag time method is adopted to calculate C _s and D (Fig. 25.1). The affinity of the skin for the drug is assessed by computing the partition coefficient K. The flux reaches 98 % of its steady-state value at 4t _eff. This result is typical of a process that can be approximated by a first order system. One of the advantages of using t _eff as a performance criterion is the possibility of estimating the time elapsed before achieving a desired therapeutic flux. For controlled release technology, the approach can also help identify process conditions that may need to be adjusted to meet a target delivery rate (Simon 2009).

Fig. 25.1

Transient flux and cumulative amount of drug released by passive diffusion. The effective time constant and lag time are represented by t _eff and t _lag, respectively

After applying a CPE to the membrane, changes in the drug diffusivity or its partitioning behavior would clarify the transport process. The direct calculations of K and D allow scientists to develop more efficient methods to design and assess chemicals that could promote drug transport through the skin. By computing partition and diffusion ratios after and before skin treatment, Khan et al. (2011) was able to hypothesize on the mechanism by which five, 9-dimethyl-2-cyclopropyl-2-decanol and tetrahydrogeraniol increased the percutaneous penetration of 5-FU and tramadol hydrochloride (Khan et al. 2011). The increased K value might be due to a change in the structure of the stratum corneum lipid bilayers, while a modification of the intercellular lipid regions might be responsible for the increased diffusion coefficient.

Three parameters need to be estimated in iontophoretic drug delivery across a polymer membrane: C _s, D, and γ. Based on previous work, the electric field is assumed to have negligible influence on the diffusion coefficient obtained from passive transport experiments (Tojo 2003; Simon et al. 2006). In addition, an increase in the surface concentration has been reported after the onset of iontophoresis. This effect was verified by a skin-stripping method in the case of verapamil (Tojo 2003). Thus, an apparent partition coefficient should be determined using Eq. 25.4. The parameter γ corresponds to the intersection of the function

$f\left(\gamma \right)=-\frac{L^2}{D}\frac{2 \sinh \left(\frac{\gamma}{2}\right)-\gamma \cosh \left(\frac{\gamma}{2}\right)}{\gamma^2 \sinh \left(\frac{\gamma}{2}\right)}-{t}_{\mathrm{lag}}$