Numerical and experimental investigation of the diffusional release of a dispersed solute from polymeric multilaminate matrices

Journal of Controlled Release 70 (2001) 309–319 www.elsevier.com / locate / jconrel

Numerical and experimental investigation of the diffusional release of a dispersed solute from polymeric multilaminate matrices G.Ch. Charalambopoulou a,b , E.S. Kikkinides a,c , K.G. Papadokostaki a , *, A.K. Stubos a , A.Th. Papaioannou b b

a National Center for Scientific Research Demokritos, 153 10 Ag. Paraskevi Attikis, Athens, Greece Department of Chemical Engineering, National Technical University of Athens, 7 Iroon Polytechniou Str., Zografou Campus, Athens 157 73, Greece c Chemical Process Engineering Research Institute, P.O. Box 361, Thermi-Thessaloniki 570 01, Greece

Received 2 June 2000; accepted 10 November 2000

Abstract In the present work the release behavior of special, multilaminate matrix-type polymer systems, is studied both theoretically and experimentally. Two different mathematical models have been employed to describe the release of a dispersed solute from both single- and multilayer matrices. A parameter sensitivity study shows that the incorporation of supersaturated matrices in the formation of multilaminate devices, with a nonuniform initial solute loading, can provide a delivery system with optimized performance compared to monolithic ones. Finally, the findings of this theoretical analysis show good agreement with measurements of the release rates of a model disperse dye from both single- and multilayer matrices.  2001 Published by Elsevier Science B.V. Keywords: Zero-order release; Controlled release; Multilaminates; Mathematical modeling

1. Introduction Zero-order release kinetics is desired for the optimal application of bioactive chemicals, such as drugs, fertilizers, pesticides, etc. Among the most widely used controlled delivery systems are properly designed matrix devices, mainly because of their low manufacturing cost and prolonged delivery time *Corresponding author. Tel.: 130-1-650-3639; fax: 130-1651-1766. E-mail address: [email protected] (K.G. Papadokostaki).

period. When the release device incorporating the bioactive agent (solute), is in contact with a liquid solvent (usually water), the solvent diffuses into the matrix while at the same time the solute starts diffusing through the matrix to the surrounding fluid. Most matrix devices have been designed in the form of a rectangular slab, and it has been observed that the cumulative release rate of drug is inversely proportional to the square root of time [1]. Generally, in conventional diffusion controlled matrix systems, where the solute to be released is uniformly distributed in an appropriate vehicle (usually a polymer), the release of a dissolved drug inherently follows first-

0168-3659 / 01 / $ – see front matter  2001 Published by Elsevier Science B.V. PII: S0168-3659( 00 )00357-6

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order diffusion with an initially high release rate followed by a rapidly declining drug release rate. The enhanced release rate, observed at small times of the release process, is known as burst effect and is usually undesirable since it may have negative consequences (e.g., toxicity due to increase of the concentration of the delivered substance beyond the accepted limits). To overcome this, various matrix geometries have been designed over the last two decades that can achieve an almost constant release of the solute with time [2–5]. An additional element is the use of multilayer matrix devices where the matrix core containing the active solute, is covered by one or more modulating layers (barriers) that delay the interaction of the solute with the dissolution medium, by reducing the surface available for the release and at the same time controlling the solute release and solvent penetration time [6]. In an alternative approach, it has been shown [6–11], that manipulation of the spatial variation in the initial solute loading in a matrix-type device, e.g., by the formation of multilaminates based on the same matrix material and containing different solute loadings, can intimately control the burst effect and at the same time moderate the variation in time of the release rate. This behavior can be significantly improved [12,13], by the use of supersaturated matrices, i.e. slabs where the initial solute loading per unit volume, A, is higher than the corresponding solute solubility Cs , in the matrix. To understand and simulate the actual mechanisms of solute release from these complex, multilaminate matrices, mathematical modelling of the diffusion process is necessary [14]. Despite the technical importance of the matrix systems, the mathematical analysis of the corresponding release kinetics is not always straightforward. The case of multilaminate systems is even more difficult. In general, one must deal with the counter-diffusion of the solute–solvent mixture through the matrix. In most instances the interaction between solute–solvent and between each of the two species with the matrix is complex, resulting in concentration dependent expressions for diffusivities and solubilities for each of the two species. The situation is further complicated by the existence of a moving front separating the dispersed from the dissolved state of the solute. Additional effects such as matrix erosion and / or swelling can complicate the problem even further [14–16].

The majority of the theoretical models [1,15,17– 21] that have been developed up to now, describe the release kinetics from single-layer devices (mainly in one-dimension) and are based on certain, common assumptions. They neglect for example, the effect of the boundary layer resistance or the solute accumulation in the release medium, except from the work of a few groups [8,19,21]. On the other hand, they consider any amount of solute introduced into the polymer matrix beyond the saturation limit to be present in the form of discrete particles. As the solute depletion progresses, the dispersed phase is assumed to dissolve and diffuse out to the surrounding medium through the matrix surface, leading to the formation of a sharp concentration front moving through the medium. Therefore, the pertinent mathematical approximation suggested by Crank [22,23] is in most cases adopted. In addition, it must be noted that the analytical, approximate solutions of the above models cannot be applied to the whole release process, but only up to a time which corresponds to the transition from the solute dispersion to the solute solution in the polymer matrix. Wu and Zhou [24] have recently employed the finite element method to study the above problem. In their method they use Galerkin finite elements to discretize the model partial differential equations in space and then they locate the moving front as a function of time by using a trial and error technique. Their method although more powerful than previous analytical or semi-analytical approaches is expected to suffer from limitations resulting mainly from the assumption (also made in most of the previous models) of fast solvent penetration in comparison with solute transport. However, in many cases a diffusion time for the solvent penetration comparable to the one for solute release is desirable since this will further delay the release of the active agent from the matrix device. In such a case, the particular model of [24] based on fast solvent penetration, is no longer valid since the position of the front depends on the diffusivity of the solvent as well. A general model that accounts for both solvent penetration and solute transport, as well as for various important aspects of the interdependence of these two processes in single-layer systems, has been proposed by Petropoulos et al. [25]. This model considers that the driving force for diffusion is due to the difference in the chemical potential of each

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species in the matrix enabling the introduction of solubility and activity instead of concentration in the diffusion equation. The main objective of the present study is to investigate both numerically and experimentally the release process of a solute from special, multilaminate matrix-type polymer systems. Two different mathematical models are used to describe the release of a dispersed solute from both single- and multilayer matrices. The first is based on the assumption of fast solvent penetration, while the second is a simplified version of the model described in [25], to account for this special case. A parameter sensitivity study shows that the incorporation of supersaturated matrices in the formation of multilaminate matrix devices, with a spatial variation in the initial solute loading, influences significantly the obtained release pattern and can provide a delivery system with optimized performance compared to conventional, monolithic ones. In a further step, the numerical simulations of this work are found to be in very good agreement with measurements of the release rates that we have performed using a model disperse dye in both single- and multilayer matrices.

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thickness 2L and initial solute loading per unit volume equal to A, is exposed to a solute-free liquid, the dispersed solute dissolves and diffuses out from the surface to the surroundings, in accordance with Fick’s law. As the depletion progresses with time, the solute concentration in the outer layers of the device decreases to reach at some point the saturation level (Cs ). A concentration profile of the dissolved solute that forms a zone between x50 (exposed surface assuming C50) and x5d (C5Cs ), where x denotes position in the polymer matrix along the diffusion axis, can therefore be assumed. In the region x.d, the solute content is still A, resulting in an undissolved core. Between these two zones, a sharp concentration gradient exists and a moving boundary separating the dissolved from the dispersed solute occurs. The corresponding concentration difference is A2Cs . The boundary moves inward gradually (layer by layer) until all the immobilized solute is dissolved in order to take further part in the diffusion process. The whole release process is mathematically described by Fick’s diffusion equation [23]: ≠ 2 C ≠C D ]] 5] ≠t ≠x 2

(1)

2. Theoretical study

with the following initial and boundary conditions:

The present work mainly focuses on the especially important case of treating supersaturated monolithic devices and multilaminate composites with constant diffusivity and constant thickness in each layer. Two different mathematical models have been properly adopted to describe the release of the dispersed solute, as shown below.

C 5 A at t 5 0, 0 # x # L

(2)

C 5 0 at x 5 0, t . 0

(3)

2.1. Moving boundary model ( M1)

where C denotes the solute concentration, t is the release time, x is the position measured from the edge of the matrix, D is the solute diffusivity, A is the initial matrix loading and d is the position of the moving boundary.

The release of the dispersed solute from the supersaturated matrix device can be postulated to proceed with the solution of the dispersed solute, followed by the diffusion of the dissolved solute, assuming that (1) the excess solute is present in the form of a sufficiently fine dispersion so that it does not affect the transport properties of the polymer matrix and (2) equilibrium with the mobile or dissolved solute is maintained during desorption (i.e. instantaneous dissolution of the dispersed solute in the polymer matrix occurs). When the system of

C 5 Cs

at x 5 d (t)

≠C ≠d D ] 5 (A 2 CS )] ≠x ≠t

(4) at x 5 d (t)

(5)

2.2. General model ( M2) The above mathematical approach has been proved useful in practice. One should however keep in mind that its applicability is restricted by the basic assumption that the solvent does not affect the overall diffusion process, which therefore is con-

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trolled only by the solute extraction. In addition, the position of the moving boundary varies with time and this induces an additional difficulty in the solution of the problem. As an alternative, a more general model that takes proper account of the simultaneous uptake of a liquid solvent by a polymer matrix and the consequent release of the loaded solute has been developed [25]. The transport in x-direction, of each species through a slab matrix of thickness 2L which is exposed to a well-stirred solvent bath, is described by the following generally formulated diffusion equations [25]:

S

D

≠m ≠C ≠ ] m T,i Ci ]i 5 ]i ≠x ≠x ≠t

(i 5 W or N)

(6)

where m T is the ‘thermodynamic’ mobility coefficient, while W and N denote solvent and solute species, respectively. Note that the driving forces for diffusion are expressed in terms of chemical potential gradients ≠mi / ≠x, instead of the concentration gradients employed in previous attempts to describe the simultaneous influx of solvent and efflux of solute [16,26]. It is known from thermodynamics that: dm 5 RT dln a

(7)

Substituting Eq. (7) into Eq. (6) the latter becomes:

F

S DS D G

C ≠ ] m T,i RT ]i ≠x ai

≠a ]i ≠x

≠Ci 5] ≠t

(8)

where Si and Di represent the ‘thermodynamic’ partition and diffusion coefficient, respectively. The boundary conditions for the set of Eqs. (9) are:

aW 5 aW0 , aN 5 aN0 , CN 5 A at t 5 0, 0 , x # L (12a)

aW 5 1, aN 5 0, CN 5 0 at t $ 0, x 5 0

(12b)

≠aW / ≠x 5 ≠aN / ≠x 5 0 at t $ 0, x 5 L

(12c)

where aW0 , aN0 , A are constant input parameters. The objective of the present work is to provide enough evidence that the above theoretical analysis can adequately describe the diffusive transport process in simple controlled delivery devices, and can thus serve as a basis for modeling more complicated systems. To this end, an initial validation of model M2 is attempted by comparing it to with the previously described M1. For this reason we consider (both theoretically and experimentally) systems with fast penetrating solvent where DN , DW and SW are constant, while DW . .DN . For the solute we assume constant SN when CN ,Cs . According to the definition 0 aN 5 c NS /c NS

(13)

where c NS denotes concentration of the solute in the external solution and c 0NS is the corresponding saturation value [25], the activity of the solute will be equal to 1 when CN .Cs . Thus we have:

aN 5 CN /SN

if CN , Cs

(14a)

which can also be written as follows:

S

D

≠a ≠C ≠ ] Di Si ]i 5 ]i ≠x ≠x ≠t

aN 5 1 if CN $ Cs

In Eq. (8), a is the activity of the diffusing species i in the matrix and it is defined as equal to that of the same species in an external phase at equilibrium. Therefore, a is related to the concentration of each species by the equilibrium sorption (partition) isotherm [25]. Comparison of Eqs. (8) and (9) leads to the following definitions: Di 5 m T,i RT Ci Si 5 ] ai

(14b)

(9)

(10) (11)

The dispersed solute is physically characterized both by aN 51 and DN 50. From the computational point of view, the former condition is sufficient to ensure immobility. In the case of a composite matrix, the basic equations are the same and the only difference is in the initial conditions at each laminate, and if necessary on the value of the diffusivity in each layer. Specifically, for t50 Eq. (12a) becomes: C 5 A1

at x 0 , x # x 1

(15a)

C 5 A2

at x 1 , x # x 2

(15b)

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C 5 Aj

at x j 21 , x # x j

(15c)

where A j and x j is the initial loading and the thickness of the jth layer of the system.

2.3. Numerical solution of the models In order to solve the two above described models, we have employed the Galerkin finite element method to discretize the model partial differential equations in space. In both cases, 500 to 1000 quadratic elements were used for the discretization. The resulting set of ordinary differential equations is integrated in time using an implicit Euler scheme, properly modified for stiff sets of equations. It must be noticed that for the case of the model M1, the position of the moving front is not known a priori. For this reason an iterative scheme based on a shooting method is employed to determine d(t) at each time step, in accordance with [23,24].

3. Experimental In an attempt to verify the findings of the theoretical study and define the appropriate parameters that will optimize the zero-order release, we have measured the experimental release rates of a model disperse dye (4-aminoazobenzene) from both monoand multilayer, planar matrices.

3.1. Materials selection and film preparation Cellulose acetate (type CA-398-30; acetyl content 39.8% by weight) in powder form, was kindly offered from Eastman Chemicals, Switzerland. The water uptake of this type of cellulose acetate has been measured to be around 0.15 g / g, while the corresponding water diffusivity is 6310 28 cm 2 / s [27]. 4-aminoazobenzene was purchased from FLUKA. Films 30–55-mm-thick were prepared by casting a solution of the polymer in acetone on a glass plate with a doctor knife blade. The solvent was removed by evaporation to the atmosphere followed by evacuation for 2 to 3 days. The organic solute was introduced in the cellulose

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acetate matrix employing two different techniques [28]: 1. Equilibration in aqueous solutions in order to produce saturated films (A /Cs 51) and determine the corresponding saturation concentration. Specifically, blank films were immersed in saturated dye solutions in a constant temperature (258C) water bath for periods long enough to ensure equilibrium. 2. Introduction of the dye at the film preparation stage, by dissolving the appropriate amount of dye in the acetone solvent prior to casting and curing of the membrane. In this way supersaturated matrices (initial solute content A /Cs .1) were obtained.

3.2. Release kinetics The release kinetic experiments were performed at 258C under conditions of strong, constant agitation so as to avoid the formation of a boundary layer at either side of the membrane [19,28]. In particular, the dyed film(s) (ca. 10 cm 2 ) was (were) mounted on a vertical frame fixed to a stirring rod rotated at a constant rate in a known volume of water. For measuring multilaminate matrices, individual dyed films of different initial loading were held together to form a stack. The device was properly sealed to prevent leakage from the side and finally both surfaces of the system were exposed to the solvent (water). The dye concentration in the desorbing aqueous phase was kept sufficiently low, by quick replacement of the water content of the receptor bath at frequent time intervals, in order to obtain sink boundary condition. The amount of desorbed dye was monitored as a function of time. The concentration of 4-aminoazobenzene in the aqueous solutions was measured with a UV-VIS spectrophotometer at the adsorption maximum (375 nm).

4. Results and discussion One of the main objectives of the present work is to simulate the actual mechanism of the release of a solute incorporated in special multilaminate matrixtype devices, by developing an adequate numerical

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model. Once this is achieved, the release process can be optimized by properly designing these delivery systems. In this respect, the effect of some structural parameters on the rate of the active agent release with time, is examined both theoretically and experimentally.

4.1. Comparison of the theoretical models For comparison, both of the theoretical approaches described in Section 2, have been employed in the case of a supersaturated single-layer matrix device, for DW . .DN to ensure fast penetrating solvent. In particular, a hypothetical system with initial, uniform loading A /Cs 54 is considered. In Fig. 1, the calculated fractional amount, Mt /M0 , of both uptaken solvent and eluted solute, during the release process, as a function of t 1 / 2 , are plotted against the following available analytical solution for a semi-infinite medium [19,23] M 2Cs ]t 5 ]] M0 A erf z

]] DN t ]] pL 2

œ

(16)

where z is given by Cs zŒ] p exp(z 2 ) erf z 5 ]] A 2 Cs

(17)

Fig. 1. Predictions of the simple (M1) and general (M2) models. Calculated amount of solute released from a monolayer system with initial loading A /Cs 54 as a function of t 1 / 2 , is plotted against available analytical solution.

It is obvious that there is excellent agreement between the resulting curves of fractional release with time. The two numerical models are practically indistinguishable and at the same time they agree well with the analytical solution at early times (at a later stage the analytical solution is no longer applicable, while the numerical approaches continuously describe the release process). Moreover, the evolution of the solute concentration profile as this was determined from the general model M2 at various times during the delivery of the active agent is illustrated in Fig. 2. It is obvious that the general model can predict the existence of a moving boundary separating the dissolved from the dispersed solute during the release process, rather than imposing it, as is the case in model M1. Based on these observations, model M2, is chosen to be further employed to study the release patterns obtained from more complex slab matrices (multilaminates). In the rest of this study, all the systems employed both in experiments and simulations, are characterized by fast diffusing solvent (DW . .DN ). Model M2, which also applies in the general case as described in detail in Section 2.2, is then considerably simplified since all equations and boundary conditions referring to the solvent need not be implemented, while DN remains constant during the release process. This results in computational savings

Fig. 2. Concentration profiles of solute in a monolayer device with initial loading A /Cs 54, obtained by the general model M2, where t is the dimensionless time DN t /L 2 .

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which render its application even simpler than in the M1 case.

4.2. Parametric study A parametric study has been undertaken in order to determine the design parameters that will lead to multilaminate systems with optimum release performance. Composite matrices consisting of four layers of equal thickness and diffusivity have been examined. Such a system is represented in Fig. 3. Due to its symmetry about the center-line, this multilaminate can be considered to consist of two layers of thicknesses L1 and L2 (in each half of the membrane) and solute loadings A 1 and A 2 , where the subscripts (1) and (2) refer to the outer and inner layer, respectively. In order to show how these parameters can manipulate the release behavior of the composite system, three combinations were selected, distributing the initial loading of the multilaminates among the separate layers of equal thickness (L1 5L2 ), in such a way that the total solute amount is the same in both devices (A /Cs 54): (a)

Fig. 3. A symmetrical multilaminate system.

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A 1 /Cs 52 and A 2 /Cs 56, (b) A 1 /Cs 51 and A 2 /Cs 5 7 and finally (c) A 1 /Cs 50 and A 2 /Cs 58. The performance of these hypothetical systems is also compared with that of the monolithic device that has an equivalent initial loading A /Cs 54. In Fig. 4 we present the obtained release profiles which show that the number of the system’s layers affect significantly the length of the period of the sustained delivery. Indeed, the increased number of layers allows for a more controlled release of the solute, as the corresponding rate is kept constant for prolonged temporal periods. On the other hand, one can realize from the same results, how small variations of the initial concentration distribution significantly alter the obtained release pattern. The effect is definitely more pronounced during the early stages of the process. For the case of the single layer system, corresponding to uniform initial loading, a burst effect is observed at small times. By distributing the initial solute loading in an increasingly nonuniform mode (i.e. by decreasing the content of the outer layer of the symmetrical device in favor of the inner one), the burst duration steadily diminishes. It is noteworthy that for the extreme case of A 1 /Cs 5 0 and A 2 /Cs 58 (curve c in Fig. 4), a time-lag release behavior occurs instead of the burst effect. The same remarks also apply for the duration of the sustained release. Increasing the nonuniformity of

Fig. 4. Effect of the multilaminate’s number of layers upon the release pattern. Curves a, b, c correspond to four layer systems with: (a) A 1 /Cs 52 and A 2 /Cs 56, (b) A 1 /Cs 51 and A 2 /Cs 57 and (c) A 1 /Cs 50 and A 2 /Cs 58, respectively.

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the initial solute distribution enhances the time interval during which the flux attains a plateau value, without affecting substantially its magnitude. The distribution of the solute concentration in the two separate compartments of the above symmetrical, composite systems is presented, at two different instants, in Fig. 5. It is evident that at early times, the variation of the initial content in the outer layer of the device differentiates the shape of the solute profiles in this compartment. In the case of the composite with A 1 /Cs 52 and A 2 /Cs 56, the initial loading of this compartment is beyond the saturation level and as expected, a moving boundary that separates the core with loading A 1 from the regime of the dissolved solute, occurs (curve a 1 ). On the other hand, when the outer layer is saturated (composite with A 1 /Cs 51 and A 2 /Cs 57, curve b 1 ) the distribution curve has a typical smooth and convex, with respect to the x-axis, shape. Finally, in the case of the composite with A 1 /Cs 50 and A 2 /Cs 58, solute is transferred to the initially empty outer layer by desorption from the regime of loading A 2 , and as a result a concave distribution curve is formed (curve c 1 ). At a later stage of the release process and as a result of the behavior described above, the multilayer with A 1 /Cs 52 is almost empty (curve a 2 ) while substantial amount of solute is still found in the other two systems with A 1 /Cs 51 and A 1 /Cs 50 (curves b 2 and c 2 , respectively).

4.3. Comparison with experiments

Fig. 5. Profiles of solute concentration distribution in the systems of Fig. 4. Curves a 1 , b 1 , c 1 correspond to early times and curves a 2 , b 2 , c 2 correspond to late times of the release process.

Fig. 6. Experimentally determined amount of solute released from monolayer devices with different initial loading. Comparison with numerical results.

Taking into account the indications of the previous theoretical parametric study, we finally studied experimentally the release rate by fabricating real multilaminate systems. At first, in some control measurements, monolithic systems of various uniform initial loadings, equal or greater than the corresponding saturation point, were measured. The total amount of dye extracted at time t is expressed as a function of (DN t /L 2 )1 / 2 , for (a) A /Cs 51, (b) A /Cs 53.1, (c) A /Cs 54.1 and (d) A /Cs 57.9 in Fig. 6. As expected, a linear dependency of cumulative release on square root of time occurs, and at the same time the effect of increased solute content favoring a more controlled release is depicted. The data from the single-layer measurements have been used to estimate the dye diffusion coefficients by fitting the whole experimental release curve to the numerical calculations, also shown in Fig. 6. An almost perfect fit is obtained for the diffusion coefficients presented in Table 1. It is observed that the diffusion coefficient D varies with the initial loading in the membrane. Due to the glassy nature of the cellulose acetate polymeric matrix, the swelling of the film caused by the relatively large amounts of dye incorporated therein during the formation of supersaturated systems, has been found not to be fully recoverable upon release of the excess dye [28].

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Table 1 Estimated diffusion coefficients of the solute for monolithic matrices A /Cs

2L (mm)

D (cm 2 / s)

1.0 3.1 4.1 7.9

32 32 27 32

1.5310 211 2.45310 211 4.42310 211 5.42310 211

As a result, the uptake of water and therefore the release rate is enhanced, leading to a monotonic increase of the effective diffusivity with the initial loading. The diffusion coefficients of Table 1 have been employed for the prediction of the release curves in multilaminate systems, as shown in detail below. Subsequently, multilaminate devices were produced, by selecting two different cases of initial concentration profiles. In particular, the following symmetrical systems (actually consisting of three layers according to the physical system of Fig. 3), were examined: (a) multilaminate A with A 1 /Cs 51 and A 2 /Cs 57.9, L1 538 mm and L2 516 mm, (b) multilaminate B with A 1 /Cs 50 and A 2 /Cs 57.9, L1 532 mm and L2 516 mm and (c) multilaminate C with A 1 /Cs 50 and A 2 /Cs 53.1, L1 532 mm and L2 516 mm. The corresponding release kinetic patterns in terms of the fractional amount of the solute released Mt /M0 vs. dimensionless time DN2 t /L 2 , are plotted against the numerical predictions of model M2 in Fig. 7. Note that the diffusivity of the empty layers has been considered to be equal to that of the saturated layers, based on the findings of previous measurements [29]. The agreement between model and experiments is excellent. In the case of multilaminate B, the theoretical and the experimental curve are essentially indistinguishable. These results verify the predictive capability of the developed numerical model and validate its use as a basic tool in the design of proper multilaminated release devices. The same observation can also be made when comparing the rates of solute release vs. real time, illustrated in Fig. 8. When empty membranes are used as external compartments (multilaminates B and C), the burst effect is eliminated, in full agreement

Fig. 7. Fractional release of solute from multilaminates: A (A 1 / Cs 51 and A 2 /Cs 57.9), B (A 1 /Cs 50 and A 2 /Cs 57.9), and C (A 1 /Cs 50 and A 2 /Cs 53.1). Comparison with numerical predictions.

with the results obtained in the parametric study section. In that case, the external layer with zero initial solute concentration serves as an effective buffer in the solute release process. It is also shown that in such systems, the increased initial loading of the supersaturated inner layer allows for a more sustained release process. Thus we can see that the multilaminate device C, which has empty outer layers and A 2 /Cs 53.1, shows a constant release period of around 10 days, while the similar device B with A 2 /Cs 57.9 shows a constant release period of 25 days. On the other hand, when saturated mem-

Fig. 8. Comparison between theoretical and experimental rates of release for the case of multilaminates A, B and C.

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branes are used in the outer layers (multilaminate A in Fig. 8), a substantial burst effect is observed at short times. Note that from Fig. 8, system A gives more prolonged release than system B, a result which from a first look seems to contradict with the findings of the parametric study. However, this deviation can be justified if one considers the differences between the experimental multilaminate systems A and B in both: (a) the total initial solute loading and (b) the thickness of the outer layers.

5. Conclusions The main objective of the present work is to study both theoretically and experimentally the release behavior of special, multilaminate matrix-type polymer systems that are used in controlled release applications. In a first step, two different mathematical models have been employed to describe the release of a dispersed solute from single-layer, supersaturated matrices. The first model relies on the assumption of fast solvent penetration, while the second is a general diffusion model accounting for all relevant mechanisms. Excellent agreement between the results of these two approaches is observed when the fast solvent penetration assumption holds. Subsequently, the general model is further employed to study the release patterns obtained from more complex (multilaminate) systems and provide valuable information for an improved design of such controlled release devices. A parameter sensitivity study shows that the incorporation of supersaturated matrices in the formation of multilaminate devices, with a spatial variation in the initial solute loading, influences significantly the obtained release pattern and can provide a delivery system with optimized performance compared to conventional, monolithic ones. It appears that the increase of the number of layers allows for a more controlled solute release process. On the other hand, by continuously distributing the initial solute loading in an increasingly nonuniform mode, the burst effect steadily diminishes, while the duration of the sustained release increases, without affecting significantly the corresponding plateau flux value. Finally, the measurements of the release kinetics

of a model disperse dye from both single- and multilayer matrices, verify the findings of the suggested models and of the optimization procedure. Very reasonable agreement is found between experiments and predictions for the case of complex multilaminate systems. The experiments on multilaminate systems confirm that sustained release for significantly long time periods can be achieved. In addition, when empty membranes are used as external compartments, the burst effect is eliminated, in full agreement with the results obtained in the parametric study section, while the increased initial loading of the supersaturated inner layer controls more efficiently the release process. On the other hand, when saturated membranes are used in the outer layers, a noticeable burst effect is observed at short times. The theoretical approach presented above can sufficiently describe the diffusive transport process in controlled delivery devices. Although simple planar matrices have been analyzed so far, the present model can easily be extended to e.g., study the effect of: the system’s geometry, the properties of the solute, the solvent and the polymer and their interaction, etc. This work is currently underway.

6. List of symbols L A Cs C x

d D mT m a S c NS c 0NS Mt M0

half-thickness of the matrix device total initial solute loading per unit volume solute saturation concentration solute concentration position measured from the edge of the matrix, along the diffusion axis position of the moving boundary diffusion coefficient ‘thermodynamic’ mobility coefficient chemical potential activity ‘thermodynamic’ partition coefficient concentration of solute in the external solution saturation concentration value of solute in the external solution amount of solute released at time t amount of solute released at infinite time

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Subscripts W N 1 2

solvent solute outer layer of the multilaminate device inner layer of the multilaminate device

[12] [13]

[14]

Acknowledgements The authors wish to acknowledge the internal support of the National Center for Scientific Research ‘Demokritos’ (DEMOEREVNA Programme, Contract No 600).

[15]

[16]

[17]

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