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The dynamic performances of two different controlled-release systems were analyzed. In a reservoir-type drug-delivery patch, the transdermal flux is influenced by the properties of the membrane. A constant thermodynamic drug activity is preserved in the donor compartment. Monolithic matrices are among the most inexpensive systems used to direct drug delivery. In these structures, the active pharmaceutical ingredients are encapsulated within a polymeric material. Despite the popularity of these two devices, to tailor the properties of the polymer and additives to specific transient behaviors can be challenging and time-consuming. The heuristic approaches often considered to select the vehicle formulation provide limited insight into key permeation mechanisms making it difficult to predict the device performance. In this contribution, a method to calculate the flux response time in a system consisting of a reservoir and a polymeric membrane was proposed and confirmed. Nearly 8.60 h passed before the metoprolol delivery rate reached ninety-eight percent of its final value. An expression was derived for the time it took to transport the active pharmaceutical ingredient out of the polymer. Ninety-eight percent of alpha-tocopherol acetate was released in 461.4 h following application to the skin. The effective time constant can be computed to help develop optimum design strategies.

Controlled-release devices are manufactured to deliver a specific dosage of a medication over an extended period of time. Some of the advantages of transdermal drug-delivery systems (TDDS) are improved patient compliance and a smooth plasma drug concentration profile as compared to oral administrations. In the development of TDDS, reservoir and monolithic (matrix) type devices are two main designs that have received increased attention. In the reservoir system, the drug is enclosed in a compartment located between a backing layer and a membrane that is used to control the delivery rate. The drug diffusion coefficient (D), and the thickness and compositions of the polymer can be manipulated to achieve a desired flux. Fick's second law of diffusion is implemented to describe drug transport across the rate limiting barrier. A lag effect is usually observed in these devices and equations are available in the literature to estimate physicochemical parameters necessary to simulate the process [

The analysis conducted in this work takes the skin membrane into account. Drug transport is affected by resistances in the polymer and skin membranes. Because a constant thermodynamic activity is sustained in the donor, a steady-state permeation rate is achieved when reservoir devices are employed. Governing equations are derived that help explain the concentration profiles in terms of physicochemical parameters of the reservoir-polymer-skin system (hereafter called TDDS 1). Closed-form solutions can be obtained using techniques, such as the Residue method, to simulate the effects of membrane and drug properties on the release profile. These factors are expected to influence both the steady-state release rate (_{ss}_{ss}_{eff}

When the release kinetics is influenced by both, the stratum corneous and a monolithic film, a steady-state flux is not attained in the absence of a reservoir. Instead, the fraction of drug released from the matrix is monitored. The drug transport mechanism in these matrix-skin systems (hereafter called TDDS 2) has been studied by several researchers [_{eff}

This contribution focuses on the calculation of _{eff}_{eff}

The transport equations through both devices have been applied in several publications. These expressions are repeated here for completeness and to lay the foundation for the approach proposed in the Results and Discussion section. The temporal change in drug concentrations in the membrane, _{1}, and in the skin, _{2} are:
_{1} and _{2} are the diffusion coefficients in the polymeric and dermal membranes, respectively. The following initial conditions apply in the two layers:

The boundary conditions associated with

_{a} to the donor concentration. The infinite source condition is adequate for applications in which the loading dose is well above the drug saturation limit in the donor. The equilibrium partition coefficient at _{a} is assumed to have a value of 1; _{m}

The governing equations are similar to the ones describing drug transport through TDDS 1 except for the following changes.

A first-order system can be written as:
_{p}_{p}_{p}_{ss}_{p}_{ss}_{p}_{p}

It can be shown that _{p}

Without solving for _{ss}

A similar approach can be adopted for processes in which the variable of interest can be approximated by a series:
_{n}_{n}_{n}_{n}_{n}_{m}_{n}_{m}_{n}_{ss}_{eff}_{eff}_{eff}

The experimental data used to analyze the TDDS 1 device are based on the release of metoprolol in hairless rats published in [^{3} was maintained on the external side of a polymeric membrane 0.12-cm thick (Scotch Pack 1006, 3M Company, USA) [^{−5} cm^{2}/s and 2.2 × 10^{−8} cm^{2}/s, respectively. These parameters, when incorporated in a mathematical model, were able to simulate the permeation study conducted by Ghosh

Released data from the permeation of alpha-tocopherol acetate (ATA) were used to examine the performance of TDDS 2 [^{−2} cm/h; the thickness of the donor solution and the membrane were 0.56 cm and 0.005 cm, respectively. The initial concentration in the donor compartment was 50000 μg/cm^{3}.

The variables and original model equations are converted into their dimensionless counterparts:

Similarly,

If we define the normalized flux by:

The dimensional expression is:

It can be shown that the ratio of the flux to its steady-state value in the Laplace domain is:

The model parameters were obtained from [_{0} = 0.038 g/cm^{3}, _{1} = 4.4 × 10^{−5} cm^{2}/s, _{2}^{−8} cm^{2}/s, _{a}_{b}_{m}^{−3}, _{t}

The flux profiles are shown in _{eff}_{eff}^{2} h. This prediction is confirmed by _{t}_{8.60}) is 77.62 μg/cm^{2} h, which is 98% of the ultimate value of the flux.

The time lag (_{lag}_{lag}_{lag}_{ss}

The normalized equations for TDDS 1 are also appropriate for TDDS 2 with the following modifications.

The cumulative amount of drug released is:
_{t}

Considering that:

Mahamongkol _{1} = 2.27 × 10^{−3} cm^{2}/h, _{2}^{−7} cm^{2}/h, _{m}^{−3} Additional data were obtained from [_{a}_{b}_{0} = 50000 μg/cm^{3} (_{t}_{eff}_{a}C_{1,0}_{t}_{∞}

Product designers can use the effective time constant to estimate the performances of TDDS 1 and TDDS 2. The method provides manufacturers with an analytical tool to help estimate the time it takes the medication to begin working after the application of a patch of type TDDS 1. In studies involving removal and reapplication of transdermal matrix systems [_{eff}_{eff}_{eff}

The time to reach a steady-state flux and to release the drug from the polymer matrix was derived for two types of devices TDDS 1 and TDDS 2. Although the methodology used the original partial differential equations (PDEs) and Laplace transforms, the final formulas contain user-friendly expressions that can be easily coded in spreadsheet packages. The effective time constant computed for the reservoir-polymer-skin system (TDDS 1) correctly predicted the period required to attain an equilibrium steady-state delivery rate of metoprolol in hairless rats. Analysis of data collected from the delivery of alpha-tocopherol acetate in an isopropyl myristate vehicle shows that it would take 461.4 h to release 98% of the drug. This estimation was in agreement with the numerical solution of the governing PDEs. The approach can be implemented in cases such as repeated applications of a patch and the optimal design of controlled release devices.

Cumulative amount of metoprolol released (experimental: +; predicted: −).

Metoprolol flux

Cumulative amount of ATA released (experimental: +; predicted: −) and the response time.

The authors declare no conflict of interest.