Lag Time in Diffusion-Controlled Release Formulations Containing a Drug-Free Outer Layer

Abstract

Theoretical considerations along with extensive Monte Carlo simulations are used to calculate the lag time before the initiation of diffusion-controlled drug release in multilayer planar devices with an outer layer containing no drug. The presented results are also relevant in formulations coated by a drug-free membrane as well as in other reservoir systems. The diffusion of drug molecules through the outer layer towards the release medium is considered, giving rise to the observed lag time. We have determined the dependence of lag time on the thickness and the diffusion coefficient of the drug-free outer layer, as well as on the initial drug concentration and the surface area of the planar device. A simple expression, obtained through an analytical solution of diffusion equation, provides an approximate estimate for the lag time that describes the numerical results reasonably well; according to this relation, the lag time is proportional to the squared thickness of the outer layer over the corresponding diffusion coefficient and inversely proportional to the logarithm of the linear number density of the drug that is initially loaded in the inner layer.

1. Introduction

Multilayer, composite, or coated devices, either in the form of films and tablets [1,2,3,4], or of beads and microspheres [5,6,7,8,9], are frequently considered in drug release formulations in order to control the amount and the time of administration of bioactive compounds. Moreover, these systems can be used to reduce the undesired initial burst release effect [10,11,12].

A lag time, preceding the delivery of the encapsulated drug from these carriers, appears in some cases, especially when the drug is enclosed in an inner layer. This has been observed for example in the release of etanidazole trapped in the PLGA core of composite PLGA-PLLA core-shell microspheres [10], or the release of acetaminophen and proxyphylline from composite poly(vinyl alcohol) beads with a bilayer core-shell structure containing lightly cross-linked inner cores and highly cross-linked outer shells [5].

In another context, taking advantage of the concept of lag time in erosion-controlled release, multilayer devices have been designed for the sequential release of multiple drugs, deposited in different layers at appropriate depths within the formulation [13,14]. In particular, four different therapeutic agents, targeting different stages of periodontitis, have been sequentially released, while the times where the peaks of the release of each drug appear can be manipulated by the thickness and the composition of empty layers inserted in between the drug-loaded layers [13]. This, in vitro accessed, sequential release has been successfully verified in vivo, by implanting in rats such multilayer devices containing various drug doses [14].

In theoretical investigations, the lag time is usually inserted as a free parameter, representing a time shift in empirical models describing drug release profiles [15,16,17,18]. The effect of diffusion through an external membrane can be explicitly taken into account in the steady-state, zero order release kinetics of reservoir systems with constant activity source, where the initial drug concentration in the internal part of the formulation is larger than its corresponding solubility [19]. Thombre and Himmelstein have numerically demonstrated in an early study the appearance of lag times in diffusional drug release by erodible matrices coated with non-degrading barrier membranes [20]. It has been demonstrated in that work that, as expected, the thicker the barrier membrane or the smaller the drug diffusion coefficient in this membrane, the larger the observed lag time [20]; however, a quantification of this effect was not presented.

Here, the lag time is numerically studied using Monte Carlo simulations in diffusion-controlled release from multilayer slabs with an empty outer layer. There are many works simulating drug release by Monte Carlo methods, considering either diffusion or matrix erosion as the dominant release mechanisms [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. A few of them have investigated diffusional release from composite matrices consisting of an inner core and an outer shell [32,36], or even from multilayer formulations [33], but lag time effects have not been considered therein because the drug particles were initially distributed throughout the device and thus a drug-free outer layer was absent. On the other hand, lag times have been observed in Monte Carlo simulations of protein release by degradable dextran hydrogels [28]. However, in that case the lag times are due to a different physical mechanism, the hydrolysis of crosslinks, thus driven by erosion and not by drug diffusion through an empty outer layer, as it is considered in this work.

We quantitatively investigate the dependence of lag time on the characteristics of the drug-free outer layer, viz. its thickness and the drug diffusion coefficient, as well as on the initial drug loading and the surface area of the planar device. We find that, as intuitively expected, the lag time is proportional to the squared thickness of the outer layer and inversely proportional to the drug diffusivity in this layer. Further, using analytical insights from the solution of diffusion equation in a limiting case, we provide a simple approximate formula for the dependence of lag time on the initial drug concentration and the surface area of the formulation. The quantitative description presented here could be useful for the proper design of multilayer or coated formulations, releasing their bioactive substances on desirable times.

2. Numerical Methods and Theoretical Estimates

In order to calculate the lag time in release formulations containing a drug-free external layer, due to the drug diffusion through the latter layer, which is initially empty of the drug, we minimally consider a composite slab of surface area $\mathrm{S}$ containing two layers: a drug-loaded inner layer of thickness ${\mathrm{L}}_{\mathrm{I}}$ and an empty outer layer of thickness ${\mathrm{L}}_{\mathrm{O}}$, as schematically shown in Figure 1. The drug diffusion coefficients—determined by the physicochemical characteristics of the drug molecules and the corresponding matrix layer, such as their hydrophobic or hydrophilic character, the size of the drug particles, the microscopic structure of the matrix, etc., as well as on the thermodynamic properties of the environment, such as its temperature—are generally different in these two layers, with corresponding values ${\mathrm{D}}_{\mathrm{I}}$ and ${\mathrm{D}}_{\mathrm{O}}$, respectively.

The initial drug concentration (number density) in the loaded inner layer is ${\mathrm{C}}_0$. Thus, the total number of drug particles inside the formulation initially is

$${\mathrm{N}}_0={\mathrm{C}}_0·{\mathrm{L}}_{\mathrm{I}}·\mathrm{S}$$

(1)

and the corresponding linear number density of loaded drug is

$${\mathrm{C}}_{\mathrm{l}}=\frac{{\mathrm{N}}_0}{{\mathrm{L}}_{\mathrm{I}}}={\mathrm{C}}_0·\mathrm{S}$$

(2)

Drug release is unidirectional and it is only permitted through the external boundary of the outer layer, which is facing the release medium (see the rightmost vertical dashed line in Figure 1). On the external boundary of the inner layer (see the leftmost vertical thick line in Figure 1) reflecting boundary conditions are considered, thus not allowing the exit of drug molecules from there. Unidirectional release is achieved by using periodic boundary conditions in the two directions perpendicular to the desired release direction. This means that when a drug molecule tries to exit the device through a surface boundary where periodic boundary conditions are applied, then it is inserted back into the system through the opposite-lying boundary surface. The dashed lines in Figure 1 indicate surfaces that can be crossed by drug molecules, which then are leaving the corresponding layer. The middle-dashed line (at the interface between the inner and outer layer) can be crossed in both directions as the drug particles are randomly moved within the slab, while the rightmost dashed line (at the interface between the outer layer and the release medium) is crossed only in the direction of drug release due to the considered sink boundary conditions.

The double-layer structure with unidirectional drug release considered here is completely equivalent to a symmetric three-layer formulation of the form ABA, with an initially drug-loaded inner layer B of thickness 2${\mathrm{L}}_{\mathrm{I}}$ sandwiched between two identical, empty layers A of thickness ${\mathrm{L}}_{\mathrm{O}}$ each, exhibiting a bidirectional release. Such controlled delivery systems have been considered, for example in Ref. [12]. The only difference in the case of symmetric ABA slabs is that the total number of drug molecules is twice as that given by Equation (1), because of the double thickness of the initially loaded inner layer B, while the linear density is still equal to that of Equation (2). Lag times and fractional release curves are identical in these two situations.

2.1. Monte Carlo Numerical Simulations

Monte Carlo simulations describe microscopic events through random choices based on numbers randomly selected by a homogeneous distribution in the interval [0, 1]. These methods are suitable to model stochastic events such as particle diffusion or chemical reactions. For this reason, Monte Carlo simulations have been broadly used in drug release studies, especially when bond degradation [22,23,24,25,28,30,31] or diffusion [26,27,29,32,33,34,35,36,38,41,43] are the main mechanisms of release.

Monte Carlo simulations are performed to numerically investigate drug release from the considered devices, employing the three-dimensional lattice method explained in detail in Refs. [26,35,36]. The specific application of this Monte Carlo scheme in cases of slabs, as considered in this work too, is outlined in Section 2.1 of Ref. [38]. Considering the z-axis to be oriented along the direction of drug release, the reflecting external boundary of the inner layer is located at $\mathrm{z}=0$, the interface between the inner and the outer layer at $\mathrm{z}={\mathrm{L}}_{\mathrm{I}}$, while the releasing external boundary of the outer layer at $\mathrm{z}=\mathrm{L}$, where $\mathrm{L}={\mathrm{L}}_{\mathrm{I}}+{\mathrm{L}}_{\mathrm{O}}$ is the total thickness of the simulated slab (see Figure 1). Periodic boundary conditions are used along the x and y axes, as is described in detail in Ref. [38].

There are two essential differences between our simulations here and those discussed in Section 2.1 of Ref. [38]: (i) the use of a reflecting boundary, bouncing back the drug particles within the inner layer, at $\mathrm{z}=0$ (instead of a freely releasing boundary in Ref. [38]) and (ii) the composite nature of the slab here, consisting of two discrete layers. The different physicochemical properties of these layers generally result in different drug diffusion coefficients ${\mathrm{D}}_{\mathrm{I}}$ and ${\mathrm{D}}_{\mathrm{O}}$ in the inner and the outer layer, respectively. We consider that in the outer layer the drug molecules have a diffusion coefficient lower or equal, at most, to that of the inner layer, viz. ${\mathrm{D}}_{\mathrm{O}}$ ≤ ${\mathrm{D}}_{\mathrm{I}}$. This is taken into account in the numerical simulations as described in Section 2.2 of Ref. [36]. The case ${\mathrm{D}}_{\mathrm{O}}={\mathrm{D}}_{\mathrm{I}}$ corresponds to a homogeneous slab, which is initially loaded only in an internal region of thickness ${\mathrm{L}}_{\mathrm{I}}$.

The cubic lattice on which the Monte Carlo simulations are performed consists of ${\mathrm{L}}_{\mathrm{x}}\times {\mathrm{L}}_{\mathrm{y}}\times {\mathrm{L}}_{\mathrm{z}}$ sites. The lattice constant $l_u$ represents the unit length in our simulations and it is of the order of magnitude of the linear size of the drug molecules [35]. In our computations, we consider ${\mathrm{L}}_{\mathrm{x}}={\mathrm{L}}_{\mathrm{y}}$, i.e., we have equal dimensions in the directions where periodic boundary conditions are applied, which determine a squared surface area of the planar device. Different values of ${\mathrm{L}}_{\mathrm{z}}$, along the drug release direction, correspond to different thicknesses L of the composite slab. At the beginning of each simulation, ${\mathrm{N}}_0$ drug particles are randomly distributed within the inner layer according to the desired concentration ${\mathrm{C}}_0$ (for example, for a 20% initial drug concentration, i.e., ${\mathrm{C}}_0=0.2$, one over five of the total lattice sites of the inner layer are initially occupied by drug molecules), while the outer layer is completely free of the drug.

During the simulation, a randomly chosen drug molecule is diffusing to a randomly selected neighboring site at each Monte Carlo step, under the condition that this neighboring site is not already occupied by another drug molecule (excluded volume interactions are considered). The different diffusion coefficients of the corresponding layers are taken into account as explained in Ref. [36]. We consider that the two layers have the same partitioning properties. The number $\mathrm{N}$ of drug molecules within the slab is followed during the course of the simulation and the Monte Carlo time is increased by $1/\mathrm{N}$ at each Monte Carlo step [26,35,36,38].

When a drug molecule crosses the external releasing boundary of the outer layer, it exits the formulation. If we just want to compute lag times, then the simulation stops as soon as the first drug particle exits the device and the corresponding Monte Carlo time is recorded. We consider that the time needed for the first drug molecule to be released by the formulation in our simulations provides a representation of the lag time, as after that event the other molecules successively exit the slab. In order to calculate fractional release curves, the simulations are continued to run for longer times. For each result presented here, at various parameter values, the simulations described above are repeated by a number of different random realizations, of the order of hundreds, in order to obtain sufficient statistical averages.

2.2. Analytical Estimates of Lag Time Derived through the Diffusion Equation

If one considers a homogeneous initial distribution of a substance, extending in a finite region from −h to +h at a uniform concentration ${\mathrm{C}}_0$, then the diffusion equation (second Fick’s law) can be analytically solved, yielding the concentration $\mathrm{C}\left(\mathrm{x},\mathrm{t}\right)$ in any position x at time t [47]:

$$\mathrm{C}\left(\mathrm{x},\mathrm{t}\right)=\frac{{\mathrm{C}}_0}{2}\left[erf\left(\frac{\mathrm{h}-\mathrm{x}}{2\sqrt{\mathrm{Dt}}}\right)+erf\left(\frac{\mathrm{h}+\mathrm{x}}{2\sqrt{\mathrm{Dt}}}\right)\right]$$

(3)

where D is the diffusion coefficient and $erf\left(\mathrm{x}\right)=\frac{2}{\sqrt{\mathsf{\pi }}}{\displaystyle {{\displaystyle \int }}_0^{\mathrm{x}}}{e}^{-{\mathrm{t}}^2}\mathrm{dt}$ is the error function.

Considering $\mathrm{h}={\mathrm{L}}_{\mathrm{I}}$, and reminding the equivalence of our system with the symmetric ABA structure, this situation corresponds to a continuum problem that can be related to a limiting case of our problem when the diffusion constants of the two layers are equal (${\mathrm{D}}_{\mathrm{I}}={\mathrm{D}}_{\mathrm{O}}=\mathrm{D}$) and the presence of the releasing boundary is neglected. Using this analogy, we can approximately estimate in this limiting case the lag time from Equation (3), through the time needed for the concentration at the position $\mathrm{x}={\mathrm{L}}_{\mathrm{I}}+{\mathrm{L}}_{\mathrm{O}}=\mathrm{L}$ to reach a threshold value that corresponds to just one drug particle of our discrete simulation arriving at the external releasing boundary of the outer layer in our Monte Carlo scheme. Note that Equation (3) does not describe the solution of the problem considered in this work in the aforementioned limiting case (${\mathrm{D}}_{\mathrm{I}}={\mathrm{D}}_{\mathrm{O}}$) at any time, since it gives nonzero concentrations beyond the boundary L of the slab as time increases. However, since we are interested in the lag time, i.e., the time needed by the very first molecules to reach this boundary, we anticipate that up to that time the diffusing particles do not yet “feel” the boundary and thus the evolution of the system is insensitive to the precise boundary conditions. Therefore, the approximate use of Equation (3) is restricted only for these early times up to the lag time. In any case, the analytical estimate of the lag time obtained below through this approximation will be tested against the numerical Monte Carlo data.

Taking into account that in our simulation units the thickness of just one layer of the three-dimensional lattice is $l_u$, then the threshold value ${\mathrm{C}}_{\mathrm{thr}.}$ of the concentration at $\mathrm{x}=\mathrm{L}$ in Equation (3) corresponding to just one drug molecule at the external boundary satisfies the condition $1={\mathrm{C}}_{\mathrm{thr}.}{l}_u\mathrm{S}$. Dividing this relation with Equation (1) we find that $\frac{{\mathrm{C}}_{\mathrm{thr}.}}{{\mathrm{C}}_0}=\frac{\left({\mathrm{L}}_{\mathrm{I}}/l_u\right)}{{\mathrm{N}}_0}$ and using the linear concentration ${\mathrm{C}}_{\mathrm{l}}$ of Equation (2), we then end up with $\frac{{\mathrm{C}}_{\mathrm{thr}.}}{{\mathrm{C}}_0}=\frac{1}{{\mathrm{C}}_{\mathrm{l}}·l_u}$. Substituting the last relation in Equation (3), we obtain an expression for the lag time ${\mathrm{t}}_{\mathrm{lag}}$ that corresponds to $\mathrm{C}(\mathrm{x}=\mathrm{L},\mathrm{t}={\mathrm{t}}_{\mathrm{lag}})={\mathrm{C}}_{\mathrm{thr}.}$, which reads $\frac{1}{{\mathrm{C}}_{\mathrm{l}}·l_u}=\frac{1}{2}\left[-erf\left(\frac{{\mathrm{L}}_{\mathrm{O}}}{2·\sqrt{{\mathrm{Dt}}_{\mathrm{lag}}}}\right)+erf\left(\frac{{\mathrm{L}}_{\mathrm{O}}+2{\mathrm{L}}_{\mathrm{I}}}{2·\sqrt{{\mathrm{Dt}}_{\mathrm{lag}}}}\right)\right]$. This equation can be further simplified using dimensionless parameters, viz. the relative thickness of the inner to the outer layer $r_L={\mathrm{L}}_{\mathrm{I}}/{\mathrm{L}}_{\mathrm{O}}$ and the dimensionless time ${\mathsf{\tau }}_{\mathrm{lag}}=\left(\frac{\mathrm{D}}{{\mathrm{L}}_{\mathrm{O}}^2}\right){\mathrm{t}}_{\mathrm{lag}}$, yielding $\frac{1}{{\mathrm{C}}_{\mathrm{l}}·l_u}=\frac{1}{2}\left[-erf\left(\frac{1}{2·\sqrt{{\mathsf{\tau }}_{\mathrm{lag}}}}\right)+erf\left(\frac{1+2r_L}{2·\sqrt{{\mathsf{\tau }}_{\mathrm{lag}}}}\right)\right]$.

A more useful and convenient analytical expression can be derived from the last relation, considering that the dimensionless lag time obeys the condition ${\mathsf{\tau }}_{\mathrm{lag}}<<1$, an assumption that it is found to be justified in all cases examined here (see Figure 9 below), which represent a large exploration of the parameters space of the system under investigation. Then, using the complementary error function 1-erf(x) and its expansion for large values of its argument, $\frac{1}{2·\sqrt{{\mathsf{\tau }}_{\mathrm{lag}}}}$ >> 1, one obtains a simple formula in leading order on ${\mathsf{\tau }}_{\mathrm{lag}}$ resulting in

$${\mathsf{\tau }}_{\mathrm{lag}}=\frac{1}{4·\ln \left({\mathrm{C}}_{\mathrm{l}}·l_u\right)}$$

(4)

To restore units, we have to consider the relation between dimensionless and real time, ${\mathrm{t}}_{\mathrm{lag}}=\left(\frac{{\mathrm{L}}_{\mathrm{O}}^2}{\mathrm{D}}\right){\mathsf{\tau }}_{\mathrm{lag}}$. However, as mentioned above, the situation discussed so far in this Subsection corresponds to a limiting case of our problem, where ${\mathrm{D}}_{\mathrm{I}}={\mathrm{D}}_{\mathrm{O}}\equiv \mathrm{D}$. Generalizing the derived result in order to approximately describe our composite system with different diffusion coefficients, it is natural to use the corresponding value of the outer layer’s diffusion constant in the relation between dimensionless and real time, leading thus to ${\mathrm{t}}_{\mathrm{lag}}=\left(\frac{{\mathrm{L}}_{\mathrm{O}}^2}{{\mathrm{D}}_{\mathrm{O}}}\right){\mathsf{\tau }}_{\mathrm{lag}}$. Therefore, we propose the following analytical estimation for the lag time

$${\mathrm{t}}_{\mathrm{lag}}=\frac{{\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}}{4·\ln \left({\mathrm{C}}_{\mathrm{l}}·l_u\right)}=\frac{{\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}}{4·\ln \left({\mathrm{C}}_0·\mathrm{S}·l_u\right)}$$

(5)

For the last equality, see Equation (2). As we demonstrate below, this relatively simple expression describes reasonably well all the numerical Monte Carlo data presented in the next section.

3. Results and Discussion

In the computational results discussed here, the thicknesses of the drug-loaded inner layer, the drug-free outer layer, and the whole slab (${\mathrm{L}}_{\mathrm{I}}$, ${\mathrm{L}}_{\mathrm{O}}$, and L, respectively) are given in units of the lattice constant $l_u$ of the three-dimensional ${\mathrm{L}}_{\mathrm{x}}$ × ${\mathrm{L}}_{\mathrm{y}}$ × ${\mathrm{L}}_{\mathrm{z}}$ lattice used in the Monte Carlo simulations. Thus, the slab thickness is $\mathrm{L}={\mathrm{L}}_{\mathrm{z}}$, while the surface area of the simulated slab is $\mathrm{S}={\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}$ (in units of $l_u{}^2$). The initial drug concentration ${\mathrm{C}}_0$ is expressed in units $1/l_u^3$ and the corresponding linear density ${\mathrm{C}}_{\mathrm{l}}$ in units $1/l_u$. Further, the drug diffusion coefficient ${\mathrm{D}}_{\mathrm{O}}$ in the outer layer is expressed in units of ${\mathrm{D}}_{\mathrm{I}}$ (the inner layer diffusion constant), i.e., it represents the relative value of the lower drug diffusivity in the outer layer with respect to that of the inner layer, thus taking values less or equal to 1. We investigate below the dependence of lag time on the parameters of the system: ${\mathrm{L}}_{\mathrm{I}}$, ${\mathrm{L}}_{\mathrm{O}}$, ${\mathrm{D}}_{\mathrm{O}}$, ${\mathrm{C}}_0$, and $\mathrm{S}$.

The thickness of the inner layer does not seem to have a measurable effect on the lag time. This is demonstrated in Figure 2, where fractional release curves are presented for devices with fixed values of ${\mathrm{L}}_{\mathrm{O}}$ and ${\mathrm{D}}_{\mathrm{O}}$, but varying values of ${\mathrm{L}}_{\mathrm{I}}$. Though the thickness of the inner layer influences the release kinetics, as shown by the three distinct release profiles corresponding to different thicknesses (${\mathrm{L}}_{\mathrm{I}}=$ 10, 30, and 50), the observed lag time is the same in all these cases. This can be clearly seen by the magnification of the release curves at relatively small times, depicted in the inset of Figure 2, where the initial stages of the release process are shown.

In all release profiles presented here (in Figure 2 and Figure 3), the first non-zero time point of the fractional release curve corresponds to the first drug molecule exiting the formulation. Therefore, the fractional release value of this first point is equal to 1/${\mathrm{N}}_0$, and the corresponding time (representing the average time, over the different Monte Carlo realizations, for the exit of the first drug molecule) provides the lag time. As the arrow points out in the inset of Figure 2, the lag time is unaffected by the inner layer thickness ${\mathrm{L}}_{\mathrm{I}}$. We have checked that this result holds even for very small concentrations. In particular, for ${\mathrm{C}}_0=0.01$ (i.e., 1% initial drug loading in the inner layer) and different values of ${\mathrm{D}}_{\mathrm{O}}=$ 0.2 or 0.6, while the other parameters (${\mathrm{L}}_{\mathrm{O}}$ and ${\mathrm{L}}_{\mathrm{x}}$, ${\mathrm{L}}_{\mathrm{y}}$) are as in Figure 2, the lag time is not dependent on the varying thickness ${\mathrm{L}}_{\mathrm{I}}$ (10, 30, or 50) of the inner layer.

Figure 3 demonstrates the influence of the outer layer properties on lag time. Both the outer layer’s thickness ${\mathrm{L}}_{\mathrm{O}}$ and the drug diffusion coefficient ${\mathrm{D}}_{\mathrm{O}}$ affect the lag time. The initial stages of release kinetics and the corresponding lag times are presented for two slabs with different values of outer layer thickness ${\mathrm{L}}_{\mathrm{O}}$ (Figure 3a), as well as for two formulations with different values of relative drug diffusivity ${\mathrm{D}}_{\mathrm{O}}$ in the outer layer (Figure 3b). As indicated by these plots (see the arrows), the lag time is increased by either increasing ${\mathrm{L}}_{\mathrm{O}}$, or by decreasing ${\mathrm{D}}_{\mathrm{O}}$ in accordance to what is intuitively anticipated. Note that Monte Carlo calculations in systems where drug release is driven by matrix degradation have shown lag times almost not dependent on the diffusivity [28]. However, this concerns a different release mechanism, in devices without drug-free outer layers, and therefore the result obtained in that work is not expected to valid in diffusion-controlled release systems, as those examined here.

In order to quantitatively explore the dependence of lag time on ${\mathrm{L}}_{\mathrm{O}}$ and ${\mathrm{D}}_{\mathrm{O}}$, we have run a number of Monte Carlo simulations for various values of these parameters of the outer layer, in composite slabs with a total thickness $\mathrm{L}=60$, ${\mathrm{L}}_{\mathrm{x}}={\mathrm{L}}_{\mathrm{y}}=20$ and ${\mathrm{C}}_0=50\%$. We have considered six different values of the outer layer thickness, varying from ${\mathrm{L}}_{\mathrm{O}}=10$ up to ${\mathrm{L}}_{\mathrm{O}}=55$. For each one of these cases, six different values of relative drug diffusion coefficients ${\mathrm{D}}_{\mathrm{O}}$, ranging from 0.1 to 1, have been investigated.

The dependence of lag time on the outer layer thickness ${\mathrm{L}}_{\mathrm{O}}$ is depicted in Figure 4, for different values of ${\mathrm{D}}_{\mathrm{O}}$ represented by different symbols. As already mentioned, the lag time increases with ${\mathrm{L}}_{\mathrm{O}}$, and the rate of increase depends on the diffusion coefficient of the outer layer. From dimensionality considerations and in accordance to the analytical estimate, Equation (5), the lag time is expected to be proportional to the square of the thickness ${\mathrm{L}}_{\mathrm{O}}$. Indeed, the dashed lines in Figure 4 represent very accurate fittings (with correlation coefficients larger than 0.999) of the numerically obtained data by a quadratic relation ${\mathrm{t}}_{\mathrm{lag}}=\mathsf{\alpha }·{\mathrm{L}}_{\mathrm{O}}^2$, where α is a proportionality constant, depending on the diffusion coefficient ${\mathrm{D}}_{\mathrm{O}}$. Larger values of ${\mathrm{D}}_{\mathrm{O}}$ correspond to smaller α.

Figure 5 shows the dependence of lag time on the relative drug diffusion coefficient of the outer layer, in a range of ${\mathrm{D}}_{\mathrm{O}}$ from 0.1 to 1, for different thicknesses ${\mathrm{L}}_{\mathrm{O}}$ of this layer. The lag time decreases with ${\mathrm{D}}_{\mathrm{O}}$ in consistence with dimensionality arguments and the Equation (5), which imply an inverse proportionality ${\mathrm{t}}_{\mathrm{lag}}=\mathrm{b}/{\mathrm{D}}_{\mathrm{O}}$, where the coefficient $\mathrm{b}$ depends on ${\mathrm{L}}_{\mathrm{O}}$ (actually it is proportional to ${\mathrm{L}}_{\mathrm{O}}^2$, as we have already discussed above). The numerical Monte Carlo results of lag times (points in Figure 5) are accurately fitted by this expression of inverse proportionality (with a correlation coefficient larger than 0.9998 in all cases), as shown by the dashed lines in Figure 5.

Up to now, our Monte Carlo simulations have verified that for fixed values of the other parameters, the lag time ${\mathrm{t}}_{\mathrm{lag}}$ is proportional to ${\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}$, as both the intuition and Equation (5) indicate. To further explore the implications of Equation (5), we examine the influence of the initial drug loading ${\mathrm{C}}_0$ of the inner layer on the lag time. To this end, we have repeated the calculations presented in Figure 4 and Figure 5 above, i.e., lag times have been obtained for 6 × 6 different combinations of ${\mathrm{L}}_{\mathrm{O}}$ and ${\mathrm{D}}_{\mathrm{O}}$, for various initial drug concentrations ${\mathrm{C}}_0$ ranging from 0.01 to 1. In all these cases, the total slab thickness and the lateral dimensions of the examined systems were $\mathrm{L}=60$ and ${\mathrm{L}}_{\mathrm{x}}={\mathrm{L}}_{\mathrm{y}}=20$, respectively, while the lag times have been averaged over 500 different Monte Carlo realizations. For each different value of ${\mathrm{C}}_0$ we have plotted the lag times as a function of ${\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}$, and then these data have been fitted by linear expressions of the form $y=A·\mathrm{x}$, in order to derive the numerical coefficient in the relation ${\mathrm{t}}_{\mathrm{lag}} ~ {\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}$.

Two representative examples are shown in Figure 6 in log-log scale, for two different initial drug concentrations: the points with higher lag times correspond to ${\mathrm{C}}_0=0.01$, while the points with lower lag times to ${\mathrm{C}}_0=0.5$. Note that the error bars in this plot correspond to the standard deviations of the Monte Carlo calculated lag times. The continuous straight lines in Figure 6 represent linear fittings (using a weight 1/$y^2$) of these 36 data points at each case.

The analytical estimate of Equation (5) suggests that the slope of lag time ${\mathrm{t}}_{\mathrm{lag}}$ versus ${\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}$ varies with the initial drug loading as $\frac{1}{4\cdot {\mathrm{lnC}}_{\mathrm{l}}}=\frac{1}{4\cdot \ln \left({\mathrm{C}}_0\cdot \mathrm{S}\right)}$, where we have considered $l_u=1$, as it is the case in our simulations. To test this result at a quantitative level, one has to take into account that the lag times obtained numerically so far in this Section correspond to Monte Carlo times. In order to compare with normal time units, the Monte Carlo times within our numerical scheme should be divided by a factor of 6, as discussed in Section 3.3 of Ref. [35]. This consideration has also enabled a direct comparison of the times computed in our Monte Carlo methodology with normal time units in Refs. [36,38].

Therefore, the slopes of linear fittings such as those shown in the last figure, divided by the aforementioned factor of 6, should provide the corresponding values of ${\mathrm{t}}_{\mathrm{lag}}/\left({\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}\right)$ in normal time units. These results are presented in Figure 7 with filled circles for all different drug loading concentrations ${\mathrm{C}}_0$ considered in this work, along with the analytical relation $\frac{1}{4\cdot \ln \left({\mathrm{C}}_0\cdot \mathrm{S}\right)}$ of Equation (5), shown by continuous line, where the surface area of the slab is $\mathrm{S}={\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}=400$ here. A rather good quantitative agreement is found between the theoretical estimation provided by Equation (5) and the numerically obtained lag times. The analytical and numerical results almost coincide for larger values of the initial drug concentration in the inner layer.

Finally, we investigate the scaling of lag time with the surface area $\mathrm{S}$ of the system (given by the lateral dimensions ${\mathrm{L}}_{\mathrm{x}}$, ${\mathrm{L}}_{\mathrm{y}}$ of the considered slabs) or, equivalently, by the total number of drug molecules ${\mathrm{N}}_0$. Note that ${\mathrm{N}}_0$ increases with the lateral size of the device as indicated by Equation (1), where $\mathrm{S}={\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}$. For this purpose, we have performed quite extensive numerical simulations in systems with increasing lateral dimensions ${\mathrm{L}}_{\mathrm{x}}={\mathrm{L}}_{\mathrm{y}}$ up to the order of a few thousands, initially loaded by up to hundreds of millions of drug particles. Different values of the other parameters of the system (${\mathrm{L}}_{\mathrm{O}}$, ${\mathrm{D}}_{\mathrm{O}}$ and ${\mathrm{C}}_0$) have been explored in order to fully evaluate the variation of lag time with S or ${\mathrm{N}}_0$. Lag times averaged over 500 different Monte Carlo realizations have been obtained in all these cases.

In these extensive numerical calculations, we find that the lag time decreases monotonically by increasing the lateral size (surface area $\mathrm{S}$) or the number of drug particles, as can be observed in Figure 8. For a fixed lateral size, lag times increase when ${\mathrm{C}}_0$ is decreased, as has been also observed in Figure 7. If the data are plotted as a function of the total number of drug molecules ${\mathrm{N}}_0={\mathrm{C}}_0{\mathrm{L}}_{\mathrm{I}}\mathrm{S}$, instead of the lateral size $\mathrm{S}$, then the results for different values of ${\mathrm{C}}_0$ are grouped together when the other parameters are the same. This scaling of lag time with ${\mathrm{N}}_0$ is depicted in Figure 8. Different symbols correspond to different system parameters, as indicated in the figure, exploring various initial drug concentrations ${\mathrm{C}}_0$ from 0.1 to 1, outer layer relative diffusion coefficients ${\mathrm{D}}_{\mathrm{O}}$ from 0.2 to 1, and outer layer thicknesses ${\mathrm{L}}_{\mathrm{O}}$ from 10 to 55.

Guided by the theoretical result of Equation (5), we have represented the data of last figure by plotting the dimensionless lag time ${\mathrm{t}}_{\mathrm{lag}}/\left({\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}\right)$ as a function of the linear concentration of drug ${\mathrm{C}}_{\mathrm{l}}={\mathrm{N}}_0/{\mathrm{L}}_{\mathrm{I}}$ instead of ${\mathrm{N}}_0$. This is shown in Figure 9, where the Monte Carlo lag times of Figure 8 have been further divided by the factor of 6, as discussed above, in order to obtain normal time units and allow a direct comparison with the analytical formula of Equation (5). The data of Figure 7 have been also included (crosses) in Figure 9, which thus contains the dependence of all computationally derived lag times in this work. We observe that the simple analytical formula of Equation (5), shown by the solid thick line, describes the Monte Carlo results rather well, as all the numerical data are grouped around this curve.

Figure 9. Dimensionless lag time ${\mathrm{t}}_{\mathrm{lag}}/\left({\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}\right)$ as a function of the linear number density of initially loaded drug, ${\mathrm{C}}_{\mathrm{l}}={\mathrm{N}}_0/{\mathrm{L}}_{\mathrm{I}}$. Symbols correspond to all numerical results derived through Monte Carlo simulations in this work. Continuous line depicts the analytical relation of Equation (5) with $l_u=1$.

Figure 9. Dimensionless lag time ${\mathrm{t}}_{\mathrm{lag}}/\left({\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}\right)$ as a function of the linear number density of initially loaded drug, ${\mathrm{C}}_{\mathrm{l}}={\mathrm{N}}_0/{\mathrm{L}}_{\mathrm{I}}$. Symbols correspond to all numerical results derived through Monte Carlo simulations in this work. Continuous line depicts the analytical relation of Equation (5) with $l_u=1$.

Note that the average time for covering a distance ${\mathrm{L}}_{\mathrm{O}}$ by diffusion in a three-dimensional medium with a diffusion coefficient ${\mathrm{D}}_{\mathrm{O}}$ is $\left(1/6\right)·{\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}\approx 0.17·{\mathrm{L}}_{\mathrm{O}}^2/{\mathrm{D}}_{\mathrm{O}}$. If one roughly approximates the lag time of the considered problem by the latter relation, its value is overestimated by a factor more than four for linear number densities of drug particles of the order of ${10}^2/l_u$ or larger, as compared to the result of Equation (5) derived here. This analytical result in general agrees with the numerical Monte Carlo calculations, as can be observed by the data shown in Figure 9. For higher linear densities of the initially loaded drug, the discrepancy between the average diffusion time and the lag time is even larger, as the latter one decreases inversely proportional to ${\mathrm{lnC}}_{\mathrm{l}}$. Of course, the lag time is expected to be shorter than the average time, since it refers to the first molecules that start exiting the formulation, which cover the required distance ${\mathrm{L}}_{\mathrm{O}}$ faster than those considered on the average.

4. Conclusions

We have numerically and theoretically examined the lag times that appear in diffusion-controlled drug release from planar systems containing a drug-free outer layer. The dependence of lag time on the characteristics of the outer layer, i.e., its thickness and the drug diffusion coefficient, as well as on the initial drug loading, and the surface area of the formulation has been quantified by use of extensive Monte Carlo simulations in three-dimensional lattices. The numerical results are in accordance with an approximate analytical estimation of the lag time, Equation (5), obtained through the solution of the diffusion equation in a relevant limiting case. Temperature effects on lag time are included through the corresponding dependence of the drug diffusion coefficient.

It has been demonstrated that the lag time does not depend on the thickness of the drug-loaded inner layer, while it is found to be proportional to the squared thickness of the outer layer and inversely proportional to the corresponding drug diffusion coefficient, as anticipated both intuitively and by dimensional analysis. The precise value of the coefficient in the latter relation seems to be inversely proportional to the logarithm of the linear density of drug in the initially loaded inner layer (which is equal to the product of the number concentration multiplied by the surface area of the planar device), as approximately indicated by Equation (5). The resulting lag times are significantly shorter than what is obtained through simple considerations of average diffusion times.

Even though the results presented here have been obtained for bilayers with unidirectional release or for equivalent symmetric trilayer systems with bidirectional release, it is expected to also provide the dependence of lag time in diffusion-controlled release from planar multilayer systems when an empty outer layer is present. These findings are relevant to frequently used membrane-coated or reservoir-type drug delivery devices. The presented results could be useful for the proper design of such formulations.

Finally, we emphasize that our results for the lag time concern cases where the diffusion is the dominant release mechanism. The used Monte Carlo method can be applied as well when other release mechanisms prevail (see, for example, the recent studies [48,49,50,51]). Future works could quantify the lag times in formulations where the release is determined either by bond cleavage due to hydrolysis or other chemical reactions, or by hydrogel swelling.