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, C₁, and in the skin, C₂ are: (1) ∂ C 1 ∂ t = D 1 ∂ 2 C 1 ∂ x 2 , − l a ≤ x ≤ 0 (2) ∂ C 2 ∂ t = D 2 ∂ 2 C 2 ∂ x 2 , 0 ≤ x ≤ l b where D₁ and D₂ are the diffusion coefficients in the polymeric and dermal membranes, respectively. The following initial conditions apply in the two layers: (3) C 1 ( x , 0 ) = 0 , − l a ≤ x ≤ 0and (4) C 2 ( x , 0 ) = 0 , 0 ≤ x ≤ l b

The boundary conditions associated with Eqs. (1) and (2) take the forms: C 1 ( − l a , t ) = C 0 (6) D 1 ∂ C 1 ∂ x ( 0 , t ) = D 2 ∂ C 2 ∂ x ( 0 , t ) (7) C 1 ( 0 , t ) = k m C 2 ( 0 , t )and (8) C 2 ( l b , t ) = 0

Equation (5) relates the concentration at −l_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 x = −l_a is assumed to have a value of 1; Eq. (6) describes the continuity of flux at the membrane-skin boundary; Eq. (7) denotes an equilibrium condition at x = 0 where k_m is the partition coefficient. A perfect sink condition is expressed by (8).

The governing equations are similar to the ones describing drug transport through TDDS 1 except for the following changes. Equation (3) is replaced by: (9) C 1 ( x , 0 ) = C 0 , − l a ≤ x ≤ 0 to express that the drug is uniformly distributed in the membrane. Equation (5) becomes a zero-flux condition: (10) ∂ C 1 ( − l a , t ) ∂ x = 0indicating that no mass is exchanged with the environment.

A first-order system can be written as: (11) τ p d y d t + y = K p u ( t )where the model parameters τ_p and K_p are the time constant and steady-state gain of the process. The variables u and y denote the input and output (or response) variables. While K_p represents the ratio of the equilibrium response (y_ss) to the size of a step change in u, τ_p is a measure of how long it takes to reach y_ss. The gain K_p is a measure of the sensitivity of a system. For example, consider a process where saturated steam is supplied to heat the liquid in a vessel. A K_p value of 5.0 °F/(lb/min) implies that an increase of 1.0 lb/min in the steam mass flow rate is needed to raise the liquid temperature by 5.0 °F.

It can be shown that y is at 63.2% of its ultimate value after one time constant. At 4τ_p , the response has attained 98% of its final value (called the response time). In the tank heater case, the response time denotes the period elapsed before the temperature changes by 5.0 °F. Using the Laplace variable s, often used to analyze the dynamics of linear systems, the response becomes: (12) Y ( s ) = K p τ s + 1 U ( s )where Y and U represent the Laplace transforms u and y assuming that y(0) = u(0) = 0. For example, a unit step change in u leads to: (13) Y ( s ) = K p s ( τ s + 1 )

Without solving for y(t) directly in Eq. (11), an inspection of Eq. (12) shows the time constant. In addition, the steady-state value y_ss is easily obtained: (14) y s s = lim s → 0 s Y ( s ) = lim s → 0 s K p s ( τ s + 1 ) = K pafter applying the final value theorem.

A similar approach can be adopted for processes in which the variable of interest can be approximated by a series: (15) ψ ( x , t ) = ∑ n = 1 ∞ f n ( x ) exp ( − λ n t ) where λ_n = 1/t_n and f_n is a function of x. The numbers t_n designate characteristic time constants (t_n > t_m or λ_n < λ_m for n < m). In general, the system dynamics is represented by the first λ_n values. To employ a single time constant that estimates how fast ψ(x,t) approaches the equilibrium ψ_ss, a first-moment relaxation time constant is applied: (16) t e f f = lim s → 0 ( ψ s s ( x ) s 2 + d ψ ¯ ( x , s ) d s ) [ lim s → 0 ( ψ s s ( x ) s − ψ ¯ ( x , s ) ) ] − 1 where t_eff is a function of position x. Note that, for this investigation, t_eff is independent of x because ψ is defined as the delivery rate or the fraction of drug released. The derivation of t_eff and other applications of this approach are found in [5]. Equation 15 is obtained by solving the two-layer diffusion problem using Laplace transform methods or the technique of separation of variables. When the former procedure is implemented, Laplace transform expressions are readily available and can be used to derive steady-state conditions (i.e., final value theorem). The representation of the solution, in terms of a single-infinite series, is based on the assumption of a one-dimensional solute transport. If a rectangular geometry was considered, a double-infinite series would be obtained.

The variables and original model equations are converted into their dimensionless counterparts: (17) U 1 = C 1 C 0 U 2 = C 2 C 0 β = D 2 l a D 1 l b p = D 1 l b 2 D 2 l a 2 τ = D 2 l b 2 t χ 1 = x l a , − l a ≤ x ≤ 0 χ 2 = x l b , 0 ≤ x ≤ l b (18) ∂ U 1 ∂ τ = p ∂ 2 U 1 ∂ χ 1 2 , − 1 ≤ χ 1 ≤ 0 (19) ∂ U 2 ∂ τ = ∂ 2 U 2 ∂ χ 2 2 , 0 ≤ χ 2 ≤ 1 with the initial conditions given by: (20) U 1 ( χ 1 , 0 ) = 0and (21) U 2 ( χ 2 , 0 ) = 0

Similarly, Eqs. (5) to (8) become: (22) U 1 ( − 1 , τ ) = 1 (23) ∂ U 1 ∂ χ 1 ( 0 , τ ) = β ∂ U 2 ∂ χ 2 ( 0 , τ ) (24) U 1 ( 0 , τ ) = k m U 2 ( 0 , τ )and (25) U 2 ( 1 , τ ) = 0 ( 25 )

If we define the normalized flux by: (26) J ( τ ) = − ∂ U 2 ( 1 , τ ) ∂ χ 2

The dimensional expression is: (27) J t = C 0 D 2 l b J ( τ )

It can be shown that the ratio of the flux to its steady-state value in the Laplace domain is: (28) j ( s ) j s s = J ( s ) J s s = β + k m s [ k m sinh ( s ) cosh ( s p ) + β p cosh ( s ) sinh ( s p ) ]and after applying Eq. (16), the normalized and dimensional effective time constants are: (29) τ e f f = 14 β ( p + 3 ) ( 3 p + 1 ) k m + ( p ( 7 p + 30 ) + 75 ) k m 2 + β 2 ( 15 p ( 5 p + 2 ) + 7 ) 60 p ( β + k m ) ( β + ( p + 3 ) k m + 3 β p ) and (30) t = l b 2 D 2 τ eff

The model parameters were obtained from [11]: C₀ = 0.038 g/cm³, D ₁ = 4.4 × 10⁻⁵ cm²/s, D₂ = 2.2 × 10⁻⁸ cm²/s, l_a = 0.12 cm, l_b = 0.038 cm and k_m = 1 (i.e., β = 1.58 × 10⁻³, p = 200.5). Data published in [10] were used to validate the mathematical approach. The experimental and calculated cumulative amounts of drug released: (31) M t = ∫ 0 t ( J t ) d tare depicted in Figure 1 to show that the theoretical model was able to predict the observations. An inversion method, developed in [15], was applied to obtain J_t from Eq. (28). The discrepancy observed between the experimental and predicted data after 15 h may be due to parameter estimation errors.

The flux profiles are shown in Figure 2. The experimental delivery rate was obtained after applying a cubic spline to the data. The disagreement between the model predictions and the laboratory data, observed in Figure 1, is accentuated by the use of the derivative approximation. In this case, t_eff and the response time (i.e., 4t_eff) are 2.15 h and 8.60 h, respectively. Based on this approach, it can be estimated, from the membrane and skin properties alone, that it should take approximately 8.60 h for the flux to reach 98% of its steady-state value of 79.2 μg/cm² h. This prediction is confirmed by Figure 2. In addition, computations with J_t show that the delivery at this time (i.e., J_8.60) is 77.62 μg/cm² h, which is 98% of the ultimate value of the flux.

The time lag (t_lag) is often calculated to determine the diffusion and partition coefficients for molecular transport through a flat membrane assuming perfect sink and infinite source conditions. In such a case, the steady-state flux is achieved after 2.7 × t_lag. However, for the TDDS 1, the expression for t_lag would involve the diffusion coefficients in the two layers, among other parameters. In addition, how this measurement is correlated with the onset of j_ss is not as well established as in the case of the one-layer model. The effective time constant is adopted in this contribution.

The normalized equations for TDDS 1 are also appropriate for TDDS 2 with the following modifications. Equations (20) and (22) are changed to: (32) U 1 ( χ 1 , 0 ) = 1 ( 32 ) (33) ∂ U 1 ∂ χ 1 ( − 1 , τ ) = 0

The cumulative amount of drug released is: (34) M t = − A l b C 1 , 0 ∫ 0 τ ( ∂ U 2 ( 1 , τ ) ∂ χ 2 ) d τwhere A is the area exposed to the dosing solution. As a result, the Laplace transform of M_t is [4]: (35) M ( s ) = A l b C 1 , 0 L 1 s 3 / 2 [ k m sinh ( s ) + β p cosh ( s ) coth ( s p ) ]

Considering that: (36) M ∞ = lim t → ∞ M t ( t ) = s lim s → 0 M ( s ) = A l b C 1 , 0 p β = A l a C 1 , 0the percentage of drug released is written as: (37) M ( s ) M ∞ = β p s 3 / 2 [ k m sinh ( s ) + β p cosh ( s ) coth ( s p ) ]and Eq. (16) yields: (38) τ eff = 20 β ( 5 p + 4 ) k m + 120 k m 2 + β 2 ( 5 p ( 5 p + 4 ) + 16 ) 20 β p ( 6 k m + β ( 3 p + 2 ) )with (39) t e f f = l b 2 D 2 τ e f f

Mahamongkol et al. (2005) provided permeation data which were analyzed, via the method outlined in [16], to determine the following parameters: D ₁ = 2.27 × 10⁻³ cm²/h, D₂ = 4.30 × 10⁻⁷ cm²/h, k_m = 7.81 × 10⁻³ Additional data were obtained from [13]: l_a = 0.56 cm, l_b = 0.005 cm and C₀ = 50000 μg/cm³ (i.e., β = 3.81, p = 0.41). Good agreement was observed between predicted and experimental M_t profiles measured in the first 50 min (Figure 3). Based on a calculated t_eff of 115.3 h, it would take 461.4 h to release 98% of ATA from the device (i.e. 0.98 Al_aC_1,0 = 49,000 μg). The same fraction of drug was estimated by calculating M_t/M_∞ at t = 461.4 h.

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 [8,9,17], the method developed for TDDS 2 can be adapted to estimate how long a person has to wear the patch. In addition, calculation of t_eff is straightforward and does not involve the solution of a system of partial differential equations. This approach can be applied to help design novel drug-delivery products where both the steady-state delivery rate and the effective time constant are controlled. The methods implemented do not rely on a short-time approximation of the original equations. Instead, the complete transient behavior of the process is included in the analysis. As a result, the effective-time-constant technique offers broad practical applications and may help researchers assess both short-term and long-term dynamic performances of a drug-delivery device. The framework also provides the possibility for investigating the effects of design specifications on t_eff. Diffusion coefficients and the membrane thickness can be manipulated to produce a desired t_eff value and satisfy end-user requirements. Because closed-form expressions are provided, sensitivity analysis can be conducted to identify which parameters influence the release time of a drug the most.