Optimizing Metrics for the Assessment of Bioequivalence Between Topical Drug Products

Abstract

Purpose

Stratum corneum tape stripping post-application of a drug product followed by analysis of the active agent in this tissue layer is an approach being seriously considered for the comparative assessment of topical bioavailability. Key issues revolve around how best to perform this experiment and interpret the data.

Methods

Using previously published results from a comparative study of three 0.025% tretinoin gel products, alternative data analysis approaches are presented that may render the technique more accessible to the evaluation of new and generic topical dosage forms.

Results

For the tretinoin gel study, the conclusions for bioequivalence from measurements of drug levels at only one uptake and one clearance time were the same as those from the original study, which required measurements at eight different treatment times. Furthermore, comparisons of drug levels at one uptake and one clearance time discriminated differences in bioequivalence for clearance and uptake, which had previously been missed. Half-life estimates, derived from time course data of drug clearance, can be related to lag time for drug penetration through the SC.

Conclusions

This new data analysis demonstrates that comparative bioequivalence might be assessed more easily.

Appendix

Appendix A

The concentration of drug at a given location within the SC will vary with time relative to the lag time (t _lag) during the uptake and clearance periods as described by Eqs. 10 and 11, respectively:

$${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{C} = 1 - \xi - 2{\sum\limits_{n = 1}^\infty {\frac{{\sin {\left( {n\,\pi \,\xi } \right)}}}{{n\,\pi }}} }\exp {\left[ { - n^{2} \pi ^{2} \tau } \right]}}$$

(10)

$$\begin{array}{*{20}l} {{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{C} = 2{\sum\limits_{n = 0}^\infty {{\left( {\frac{1}{{\lambda ^{2}_{n} }} - 2{\sum\limits_{m = 1}^\infty {\frac{{\exp {\left[ { - m^{2} \pi ^{2} \tau _{{\text{o}}} } \right]}}}{{m^{2} \pi ^{2} - \lambda ^{2}_{n} }}} }} \right)}} }\exp {\left[ { - \lambda ^{2}_{n} {\left( {\tau - \tau _{{\text{o}}} } \right)}} \right]}\cos {\left( {\lambda _{n} \xi } \right)},\quad \lambda _{n} = \frac{{2n + 1}}{2}\pi } \hfill} & {{{\text{for}}\;{\text{t}} > {\text{t}}_{{\text{o}}} } \hfill} \\ \end{array} $$

(11)

where t _o is the time at which drug is removed from the skin surface. In these equations ${\tilde C}$is the concentration in the SC (C, with units of drug mass per volume of SC) normalized by K·C _v, where K is the SC-vehicle partition coefficient and C _v is the drug concentration in the vehicle. The relative position (ξ) is the location x within the SC measured from the skin surface normalized by the SC thickness L. The dimensionless time (τ) is defined as the time since the drug was applied to the skin normalized by 6·t _lag [i.e., τ = (t/(6·t _lag)]; τ _o is the dimensionless time at which the drug is removed from the skin surface [t _o /(6·t _lag)], initiating the clearance phase.

Equations 10 and 11 were developed assuming that: (1) drug permeation in the SC can be treated as Fickian diffusion with a constant diffusion coefficient in a pseudo-homogeneous membrane of thickness L, (2) constant drug concentration in the vehicle applied to the skin surface, and (3) sink conditions (i.e., drug concentration is zero) at the inside boundary of the SC. According to Eq. 10, at a given position within the SC, the concentration will increase in time until steady state is established, which will occur when τ is approximately 0.4 (i.e., at t ≈ 2.4 × t _lag). After this time, the concentration profile will be linear (i.e., ${\tilde C}$= 1 – x/L) and will not vary in time unless the drug concentration in the vehicle depletes or the drug is cleaned from the skin surface.

The predicted average concentration of drug in the SC at a given time can be derived by integrating the concentration expressions represented by Eqs. 10 and 11 over the SC thickness. The resulting expressions for the average normalized concentration $\left\langle {\tilde C} \right\rangle$during uptake and clearance are given by Eqs. 12 and 13, respectively.

$$\begin{array}{*{20}l} {{{\left\langle { \ifmmode\expandafter\tilde\else\expandafter\~\fi{C}} \right\rangle } = \frac{1}{2} - \frac{4}{{\pi ^{2} }}{\sum\limits_{n = 0}^\infty {\frac{{\exp {\left[ { - {\left( {2n + 1} \right)}^{2} \pi ^{2} \tau } \right]}}}{{{\left( {2n + 1} \right)}^{2} }}} }} \hfill} & {{{\text{for}}\;{\text{t}} \leqslant {\text{t}}_{{\text{o}}} } \hfill} \\ \end{array} $$

(12)

$$\begin{array}{*{20}l} {{{\left\langle { \ifmmode\expandafter\tilde\else\expandafter\~\fi{C}} \right\rangle } = 2{\sum\limits_{n = 0}^\infty {\frac{{{\left( { - 1} \right)}^{n} \exp {\left[ { - \lambda ^{2}_{n} {\left( {\tau - \tau _{{\text{o}}} } \right)}} \right]}}}{{\lambda _{n} }}{\left( {\frac{1}{{\lambda ^{2}_{n} }}2{\sum\limits_{m = 1}^\infty {\frac{{\exp {\left[ { - m^{2} \pi ^{2} \tau } \right]}}}{{m^{2} \pi ^{2} - \lambda ^{2}_{n} }}} }} \right)}\quad \lambda _{n} = \frac{{2n + 1}}{2}} }} \hfill} & {{{\text{for}}\;{\text{t}} > {\text{t}}_{{\text{o}}} } \hfill} \\ \end{array} $$

(13)

The ratio of the amount of drug in the SC normalized to the amount in the SC when the drug was cleaned from the skin surface is denoted by the symbol W. The ratio of the drug amounts is also the ratio of the average concentrations. Thus, the curves in Fig. 5 showing the variation in W with clearance time (t − t _o) were derived by dividing the average drug concentration in the SC (i.e., $\left\langle {\tilde C} \right\rangle$calculated from Eq. 13 at a given time (t) that is after drug was removed from the SC surface), by the average drug concentration in the SC at the time drug was removed (t _o) [(i.e., $\left\langle {\tilde C} \right\rangle$calculated from Eq. 12 for τ = t _o/(6 t _lag)].

Appendix B

For times at which some measurements were below the LOQ (equal to 3 ng cm⁻²), the remaining measurements, on the log scale, can be considered to come from a truncated Gaussian distribution, with a truncation point of log (3 ng cm⁻²). In order to impute values for the measurements below the LOQ, we needed to estimate the mean and standard deviation of the full distribution. We chose estimates so that the mean of the truncated distribution matched the sample mean of the measurements above the LOQ, and so that the area under the tail of the curve to the left of log 3 matched the sample proportion of subjects whose measurements were below the LOQ. For each subject whose measurement was below the LOQ, we then generated a random value from the tail of the curve to the left of log 3.

After values were imputed for all subjects whose original measurements were below the LOQ, we computed means and confidence intervals as described above. Because the imputed values are random, the means and confidence intervals will differ when the process is repeated. To determine how large this variation would be, we repeated the process several times. The results were similar across replications. For the ratios of the total amount of retinoid of product B to product A and product C to product A (Fig. 4), the results at t = 13.5 h, for which the proportion of measurements below LOQ was greatest, differed by 10–20% from the values obtained when the values below LOQ were dropped. Smaller differences were observed for other times. For the ratios of the amount of total retinoid in the SC at a specified time after the drug is removed compared with the amount in the SC when the drug is first removed (Fig. 6), the geometric mean ratios decreased by about 30% at 12 h after removal, for which the proportion of measurements below LOQ was greatest. Smaller decreases were observed for earlier times.