A two-phase model for controlled drug release from biphasic polymer hydrogels

Journal of Controlled Release 51 (1998) 313–325

A two-phase model for controlled drug release from biphasic polymer hydrogels a, a b a E.S. Kikkinides *, G.Ch. Charalambopoulou , A.K. Stubos , N.K. Kanellopoulos , c d C.G. Varelas , C.A. Steiner b

a Institute of Physical Chemistry, NCSR Demokritos, 15310 Ag. Paraskevi Attikis, Greece Institute of Nuclear Technology and Radiation Protection, NCSR Demokritos, 15310 Ag. Paraskevi Attikis, Greece c Department of Research and Development, Interchem-Hellas S. A., 17672 Athens, Greece d Department of Chemical Engineering, The City College of CUNY, New York, NY 10031, USA

Received 27 March 1997; received in revised form 11 August 1997; accepted 17 October 1997

Abstract A comprehensive two phase model is developed to describe the sustained release of a solute or drug from a biphasic hydrogel substrate. Such a material consists of a continuous hydrophilic phase (polymer backbone in water) and a dispersion of spherical microdomains made of the hydrophobic side chains of the polymer organised in a micelle like fashion. The solute or drug is assumed to be encapsulated within the dispersed microdomains, and to diffuse from the interior to the surface of the microdomain where it exchanges following a Langmuir isotherm. Mass transfer to the bulk phase occurs by desorption of the drug from the surface through a driving force that is proportional to the difference of surface and bulk concentration. Accordingly the drug is released to the surroundings by diffusion through the bulk. Depending on the values of the Langmuir constant and assuming well stirred behaviour in the interior of the microdomain, the present model results in either of the two asymptotic models developed in previous studies. The results of a parametric study show that the desired steady state flux of a specific drug to the surroundings may be obtained given appropriate values of structural properties of the material. This conclusion is further supported when using this model to simulate earlier experimental results. The polymer structural properties can be manipulated easily during the fabrication of dispersed-phase networks, as indicated by preliminary experiments.  1998 Elsevier Science B.V. Keywords: Zero-order release; Two-phase model; Langmuir isotherm; Controlled release; Biphasic hydrogel

Abbreviations: c solute (drug) concentration; Dm diffusion coefficient in the microdomains; Db diffusion coefficient in the polymer matrix; c sm ultimate solute capacity; c* solute saturation concentration in the bulk phase; a partition coefficient between the saturation concentrations; B Langmuir parameter k distribution coefficient of the solute inside the gel; b Langmuir constant; ¨ ¨ l external Damkohler number; lm internal Damkohler number; J molar flux; R radius; t time; h viscosity; u contact angle; l mean

free path; r fluid density; s surface tension; f volume fraction of microdomains; x amount of fluid sorbed per unit of solid mass; Subscripts b bulk; m microdomain; s surface; Superscripts – averaged over the microsphere volume; ∧ dimensionless quantity. *Corresponding author.

0168-3659 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved. PII S0168-3659( 97 )00182-X

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1. Introduction The diffusional release of a dispersed solute from a polymeric matrix is a problem of special interest in areas ranging from the migration of additives and impurities into the environment [1] to the controlled release of pharmaceuticals [2–4]. Zero-order release, or the release of a solute or drug from a substrate to the surroundings at a steady rate for prolonged times, is a much sought after property for controlled-release devices. During the last decade, increasing attention has been paid to controlled drug delivery systems made from amphiphilic polymers. Because of their unique structure, potential applications of amphiphilic materials may be quite extensive. A significant increase has been experienced in research studies involving applications, especially pharmaceutical, of amphiphilic materials. Most of these studies elaborate on the preparation, structural analysis and biocompatibility of polymeric devices used as drug delivery systems [5,6]. Some of the authors have been involved in several research papers dealing with the synthesis and characterisation of a new type of material (dispersedphase hydrogels) made from surface active graft copolymers. It has been observed that in these systems the side chains aggregate into hydrophobic microregions or microdomains, resulting in the formation of a network structure. The hydrophobic aggregates of these networks remain intact, exposing only the hydrophilic moiety to the surroundings [7]. In addition the hydrogels may be loaded with lipophilic drugs by soaking or by injection. The drug partitions into the microphases, leaving a saturated aqueous solution of the drug in the gel pores. Thus, the microphases serve as reservoirs for the excess material. As the drug diffuses into the surroundings through the surface of the gel, it is instantaneously replaced in the bulk from the microphases. The solute will leach out of the microdomains at a rate proportional to the permeability of the microdomain interface, the total interfacial area of the microphases in the gel, and the concentration of solute in the surrounding aqueous medium. It is evident that such a complex system cannot be modelled by simple monolithic models that exist in the literature [8–14]. Two mathematical models have

been recently developed by Varelas et al. [15,16] that assume different rate expressions (linear and zeroorder respectively) from the microphase to the bulk phase. A zero-order kinetics model seems to describe the results of preliminary experimental studies, at least qualitatively, as indicated by comparisons with experiments [16]. In this work, a general comprehensive mathematical model is developed to describe the sustained release of a drug encapsulated in the microphase of a biphasic polymer. This model accounts for diffusion in the microphase as well as in the bulk phase. The solute diffuses from the interior of the microdomain to its surface where it rapidly exchanges following a Langmuir type of isotherm. Subsequently, it is desorbed to the bulk phase through a driving force that is proportional to the difference between the surface and bulk concentrations. The drug is then released to the surroundings by diffusion through the bulk. The present model is general enough to incorporate as limiting cases the two simplified approaches previously derived by Varelas et al. [15,16]. Its predictions will enable one to design dispersedphase polymer networks to a broad range of specifications by varying the structural parameters of the polymer.

2. Theoretical Firstly, we consider the combination of transport mechanisms that are responsible for the sustained release of the drug encapsulated within the microdomains of the porous matrix. The present approach is developed in analogy with the previous work of Varelas et al. [15,16]. Consider the physical picture shown in Fig. 1. Initially, the material is impregnated with a drug or another solute such that the drug partitions preferentially into the microdomains. On exposure of the gel to drug-free surroundings, the drug exits the gel from the bulk phase in accordance with Fick’s law. ≠c b 2 (1 2 f )] 5 (1 2 f )Db= c b 1 Js ≠t

(1)

with boundary conditions: =c b 5 0

at r 5 0

(2a)

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Fig. 1. Schematic of a two-phase controlled-release device. Inset shows a hydrophobic microdomain.

cb 5 0

at r 5 R b

(2b)

where c b is the concentration of drug in the bulk phase, f is the volume fraction of microdomains, r is the radial direction in the bulk phase and R b is the radius of the polymer matrix (assuming a cylindrical shape). Db is the diffusion coefficient of the solute in the polymer matrix. The drug quantity in the bulk phase is simultaneously restored from the microdomains (flux term Js in Eq. (1)) through a combination of mass transfer mechanisms: (a) Diffusion from the interior to the surface of the microdomain (assumed to be spherical in shape) where it exchanges following a Langmuir isotherm. The corresponding mass balance equation is ≠c m f ]] 5 f Dm= 2 c m ≠t

(3)

=c m 5 0

at x 5 0

cs Bc m ] 5 ]]] c sm 1 1 Bc m

at x 5 R m

where c m is the drug concentration in the microdomains, Dm is the corresponding diffusion coefficient of the drug in the interior of the microdomain, x is the radial direction in the microdomain and R m is the radius of the microdomain; c s is the drug concentration at the surface of the microdomain, c sm is the ultimate solute capacity, c mo 5c* /a is the corresponding saturation concentration in the microphase, c* is the solute saturation concentration in the bulk phase, a is a partition coefficient between the two saturation concentrations and B is the Langmuir parameter. Note that the product c sm B corresponds to the Henry’s law constant which represents the slope of the isotherm as c m →0. (b) Mass transfer to the bulk phase occurs by desorption of the drug from the surface through a driving force that is proportional to the difference of surface and bulk concentration: dc] m f ]] 5 k s (c b 2 c s ) 5 2 Js (5) dt Where c¯ m is the volume-averaged solute concentration in the microdomain. The initial condition for the above system is that at t50 c b (r,0)5c* and c m (x,0)5c mo 5c* /a. The above model equations can be recast in dimensionless form by introducing the following dimensionless variables and parameters: cb cˆ b 5 ] c* cs cˆ s 5 ] c* acm cˆ m 5 ]] c* r rˆ 5 ] Rb

a (1 2 f ) k 5 ]]] f

with boundary conditions: (4a) (4b)

315

Db t t 5] R2b

or by eliminating c sm

1 b 5 ]]] 1 1 Bc * /a

c c m (1 1 Bc mo ) ]s 5 ]]]] c * c mo (1 1 Bc m )

ksR 2 b l 5 ]]] Db (1 2 f )

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3. Asymptotic case ( lm →`)

f Dm R 2 b lm 5 ]]]] Db R 2 m (1 2 f )

In this case the microdomain is assumed to be a well-stirred droplet containing drug which exchanges rapidly between the interior and the surface of the microdomain following a Langmuir isotherm. In this case c m is constant everywhere inside the microsphere and thus Eq. (10) becomes:

x xˆ 5 ] Rm Then Eqs. (1) through (5) become: (i) Bulk Phase ≠cˆ 2 ]b 5 =ˆ cˆ b 1 Jˆs ≠t

(6)

with boundary conditions: (7a)

cˆ b 5 0 at rˆ 5 1

(7b)

(ii) Microdomain (8)

with boundary conditions: ˆ ˆ m 5 0 at xˆ 5 0 =c cˆ m cˆ s 5 ]]]] b 1 (1 2 b)cˆ m

(9a) at xˆ 5 1

(9b)

(iii) Microdomain–bulk interface 1 dcˆ¯ m ] ]] 5 l(cˆ b 2 cˆ s ) 5 2 Jˆs k dt

(11)

where

ˆ ˆ b 5 0 at rˆ 5 0 =c

≠cˆ m 2 ]] 5 klm =ˆ cˆ m ≠t

dcˆ m ]] 5 kl(cˆ b 2 cˆ s ) dt

(10)

The initial condition for the above system is that at t 50 cˆ b (r,0) 5 1 and cˆ m (x,0) 5 1. The parameter k is a distribution coefficient of the solute inside the gel, defined such that a low k corresponds to a high proportion of drug in the microdomains at time zero. ¨ The parameter l is the Damkohler number [17], giving the ratio of the characteristic times for diffusion through the bulk and introduction of solute in the bulk from the dispersed phase. Accordingly, lm is the ratio of the characteristic times for diffusion through the bulk and diffusion through the interior of the microdomain, which will be called the ¨ internal Damkohler number. Note that the equilibrium isotherm parameter b in Eq. (9b) is termed Langmuir constant or constant separation factor and is similar to the relative volatility in vapour liquid equilibria [18].

cˆ m cˆ s 5 ]]]] b 1 (1 2 b)cˆ m

(12)

It is well known from the theory of adsorption [19] that it is the slope of an isotherm that determines whether this isotherm is favourable during adsorption or not. Using Eq. (12) it follows: ≠cˆ b ]]s 5 ]]]]]2 ≠cˆ m (b 1 (1 2 b)cˆ m )

(13)

It is evident that when 0,b,1 the above isotherm is favourable whereas when b. 1 it becomes unfavourable. When b50 the isotherm is a straight horizontal line for c m .0 corresponding to an irreversible isotherm. In that case c s is constant everywhere except from the point of c m 50 where there is a discontinuity in the isotherm and c s drops to zero. Thus, for b50 Eq. (12) becomes dcˆ m ]] 5 kl(cˆ b 2 1) dt

(14)

This is identical to the rate expression for the zeroorder model developed previously by Varelas et al. [16]. When b51 the isotherm is linear and its slope is constant and equal to Henry’s law constant. Then Eq. (12) becomes dcˆ m ]] 5 kl(cˆ b 2 cˆ m ) dt

(15)

This expression is again identical to that of the linear-rate model developed previously by Varelas et al. [15]. Since there is good evidence in the literature that

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Fig. 2. Dimensionless Langmuir isotherms for various values of parameter b.

such drug-polymer interactions can be represented by a favourable Langmuir isotherm [10,20,21] it appears that b will be in the range (0,1) under realistic conditions. Typical isotherms are shown in Fig. 2. It is evident that values of b between 0.01 and 0.1 are of special importance because they correspond to quite steep, yet reversible isotherms.

4. Solution of the model The above set of Eqs. (6)–(10) in general cannot be solved analytically due to the nonlinear character of the Langmuir isotherm (Eq. (7a)). Thus a numerical scheme has been developed using Galerkin finite elements for discretizing the equations in the radial direction in the bulk phase, and orthogonal collocation for discretizing the diffusion equation in the radial direction in the microdomains. This numerical scheme has been successfully used in modelling adsorption–desorption processes [22,23]. Twenty to thirty quadratic elements were used for the bulk phase whereas three to five collocation points were used for the microdomains. For the solution of the model, the program was run on a IBM Pentium demanding 5–10 min CPU time. As the model is

phenomenological, the microdomains were uniformly distributed and had equal size.

5. Results and discussion

5.1. Parametric study The main objective of the present work is the development of a comprehensive mathematical model that can adequately predict the sustained release of the drug encapsulated in the microdomains. Once the correct mechanism for such a behaviour is found, then it will be easier to design those hydrogels in such a way that the sustained release will be favoured. In this respect the effect of several structural parameters on the rate of the drug release with time is examined and optimum ranges of these parameters are determined. The base case corresponds to the following set of parameter values: b50.01, lm →`, l 50.5, k 51.

5.2. Effect of Langmuir constant, b The Langmuir constant, b, determines explicitly the shape of the isotherm as can be seen from Fig. 2. In this Figure it is clear that if b,0.1 the isotherm is

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considered to be steep whereas if b.0.5 the isotherm is closer to the linear case which corresponds to b51. Note that the steeper the isotherm the longer is the interval where the isotherm is flat. In the limit of b50 (irreversible isotherm) the isotherm remains flat for every value of c m .0. On the other hand, when b51 (linear isotherm) the flat part of the isotherm disappears. The present model was solved for a series of values of b, and the results are shown in Fig. 3. It follows that the steeper the isotherm the longer the release rate is kept constant. Since the steepness of the isotherm is directly related to the extent of its flat region it is evident that the longer the interval where the isotherm is flat the longer is the plateau region in the flux of solute to the surroundings. Furthermore, it appears that if the isotherm is not steep enough, the rate of release cannot sustain a constant value with time. This means that there is a lower limit of steepness in the solute isotherm, beyond which no zero-order release can be obtained. ¨ 5.3. Effect of internal Damkohler number (lm ) ¨ In previous studies [15,16] the internal Damkohler number, lm , was assumed to be equal to infinity; that is to say that the microphases act as well-stirred

reservoirs of liquid drug uniformly distributed inside them. However, this may not necessarily be the case. To investigate the effect of internal diffusion of drug in the microdomain, we varied lm from infinity to a very low value of 0.05. The fluxes obtained are presented in Fig. 4. The results in Fig. 4(a) correspond to the case of a very steep isotherm (b50.01), while those of Fig. 4(b) correspond to the case of a less steep isotherm (b50.1). It is evident that internal diffusion reduces the period of sustained drug release and results in a more ‘dispersive’ release following the plateau region of the curve. This effect is more pronounced at high lm values for relatively less steep isotherms. It is however important to point out that as long as internal diffusion is faster than external diffusion, i.e. lm . . l, then internal mass transfer does not influence the rate of release from the bulk phase to the surroundings. In addition, there is again a lower limit for the value of lm , beyond which even very steep isotherms cannot give sustained solute release. In Fig. 5(a) and b the concentration profiles in the bulk phase at various times are plotted for lm 5` and 0.05 respectively. It is observed that in the first case the concentration profile first decreases with time, then remains unchanged for a certain period (corresponding to the plateau region in Fig. 4(a)) and subsequently de-

Fig. 3. Dimensionless drug flux to the surroundings for various values of b. The rest of the parameters correspond to the base case: lm →`, l 50.5, k 51.

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319

Fig. 4. a. Dimensionless drug flux to the surroundings for b50.01 and various values of lm . The rest of the parameters correspond to the base case: l 50.5, k 51. b. Dimensionless drug flux to the surroundings for b50.1 and various values of lm . The rest of the parameters correspond to the base case: l 50.5, k 51.

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Fig. 5. a. Radial solute concentration profile in the bulk phase at different times for lm →`. The rest of the parameters correspond to the base case: b50.01, l 50.5, k 51. b. Radial solute concentration profile in the bulk phase at different times for lm 50.05. The rest of the parameters correspond to the base case: b50.01, l 50.5, k 51.

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creases further to its zero value as time progresses. On the other hand, when lm 50.05, the concentration profile drops steadily to zero without reaching any intermediate frozen profile. This is equivalent to the result for the flux with time, which for the case where lm 50.05 it does not experience any plateau value, i.e. no sustained release is observed under these conditions.

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to the surroundings is presented in Fig. 7. A reduction of k results in an increase of the ‘sustained release’ time, in other words the time interval during which the flux reaches a plateau value increases as k decreases. However, in this case, the plateau value of the flux remains constant and does not depend on the value of k.

5.6. Asymptotic case ¨ 5.4. Effect of external Damkohler number (l) ¨ The external Damkohler number, l, expresses the effect of diffusion through the bulk phase. Based on ¨ the analysis made above for the internal Damkohler number, it follows that diffusion through the bulk is important for the release of the solute as long as lm . . l. The effect of l is shown in Fig. 6. Again we see that sustained release is favoured for the case of the steeper isotherm. However, lower values of l result in longer ‘sustained release’ periods characterised by reduced fluxes.

5.5. Effect of distribution coefficient (k) The distribution coefficient, k, is defined such that the lower its value, the higher the drug loading in the microdomains. The effect of k on the flux of solute

For the sake of completeness, we should mention at this point the physical relevance of the asymptotic cases of the present model which have been presented earlier in the literature [16]. According to the model, if the parameter b becomes equal to unity the isotherm is linear and its slope is constant (Eq. (15)). This mechanism produces a slow but steady decrease over time in the flux of drug to the surroundings due to the fact that the driving force for interfacial mass transfer, kl(cˆ b 2 cˆ m ), also varies with time. This particular model does not predict a plateau followed by a sharp drop-off in the flux. On the other hand, if the Langmuir constant becomes zero (Eq. (14)), the microdomains act as a perfect source, that is, one whose composition remains unchanged for some period of operation. The solute which is available for interfacial mass

Fig. 6. Dimensionless drug flux to the surroundings for various values of l. The rest of the parameters correspond to the base case: b50.01, lm →`, k 51.

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Fig. 7. Dimensionless drug flux to the surroundings for various values of k. The rest of the parameters correspond to the base case: b50.01, lm →`, l 50.5.

transfer resides in a saturated phase (of constant concentration c s ) at the surface of the microdomain. This phase remains saturated for some period of time because it is replenished rapidly from within the source. At the end of this period all of the available solute within the microdomains has been depleted and interfacial mass transfer ceases.

6. Comparison with experiments In an attempt to test the validity of the present work, the model was tested against previously obtained experimental results [16]. In these tests, biphasic polymers were initially saturated with a solute, either theophylline (180 Da) or tryptophan (203 Da). In particular, during the experimental procedure Varelas et al. tested the performance of hydrophobically modified hydroxyethyl cellulose (HMHEC). The HMHEC used, was composed of 1.33% (w / w) n-C 12 H 25 side chains, with a hydroxyethyl molar substitution of 3.8 ethylene oxide groups per anhydroglucose unit and a molecular weight of 10 6 . Gels were prepared by dispersing 1 g of polymer in 50 ml of water, then slowly adding 50 ml

of ethanol while stirring. The resulting mixture was stirred overnight and then allowed to stand. After 2–3 days a gel precipitated out. Subsequently semibatch release experiments were carried out, where the loaded gels were immersed in an excess of distilled water and the solute concentration was monitored with time. The external aqueous phase was removed and replaced by pure water rapidly at frequent time intervals to ensure constant boundary condition for the bulk concentration (Eq. (2b)). Further details on the experimental work can be found elsewhere [16]. In order to use the present model to simulate the experiments, information is needed on several structural parameters and properties characterizing the solute and the gel. As only rough approximations could be extracted from [16] the internal and external ¨ Damkohler number and the Langmuir parameter, were varied in a narrow range before selecting their final values for the simulations. This does not mean that they have been used as fitting parameters because, as the parametric study of the previous section indicates, each parameter has a specific and pronounced effect on the plateau value of the flux and the time period of sustained release. Thus the ability to use these parameters for fitting purposes is

E.S. Kikkinides et al. / Journal of Controlled Release 51 (1998) 313 – 325 Table 1 Model input parameters Parameter

b k l lm Db /R b2 (sec 21 ) c* (g / l)

Solute Theophylline

Tryptophan

0.01 2.0 0.1 0.6 2.7310 24 7

3310 23 1.43 0.16 0.5 2.7310 24 11.4

quite limited and the range in which they may vary is small. The distribution coefficient, k, was calculated by dividing the concentration of excess solute, c xs , to its saturated concentration, c*. From [16] the values of c xs for tryptophan and theophylline are 8 and 3.5 g / l respectively and their corresponding values of c* are 11.4 and 7 g / l. Thus the resulting values of k are 1.43 for tryptophan and 2 for theophylline. From [16] the volume of the gel was 20 ml and the volume fraction of microdomains, w, was around 10%. All the input parameters used in the model are summarised in Table 1 for each case. The results in terms of rates of solute-release to the surroundings are presented in Fig. 8. It can be seen that there exists excellent agreement between

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model and experiments. When looking at the parameter values listed in Table 1, it follows that in both cases the effect of internal diffusion cannot be ignored since this phenomenon is mainly responsible for the dispersive behaviour of the flux following the plateau region. In addition, the higher value of l for the case of tryptophan results in a higher plateau flux value and a shorter time period of sustained release, as expected. Finally both solutes have to follow a quite steep Langmuir isotherm behaviour (as implied by the b values) when exchanged at the surface of the microdomain. A further comment is due concerning the Db /R b2 entry of Table 1. In fact, this has been inferred knowing that the characteristic time of the diffusion process in the bulk is of the order of 1 h for both solutes, as obtained from the theoretical initial slope of the flux vs time curve in Fig. 7. Typical R b values during the tests range between 0.2 and 0.4 cm, i.e. the resulting bulk diffusivities (taking in account the microdomain volume fraction in the gel) from the entries of Table 1 are between 0.85310 25 and 3.4310 25 cm 2 / s. These values should be compared to the measured 0.66310 25 cm 2 / s for tryptophan in water [24] which is considered to be close to the diffusivity of the solute in the bulk polymers used for controlled release purposes.

Fig. 8. Comparison between theoretical and experimental rates of release for the case of theophylline and tryptophan.

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7. Conclusions A comprehensive two phase model is developed to describe the sustained release of a solute or drug from a biphasic hydrogel substrate. The drug is assumed to be encapsulated within the dispersed microdomains, and to diffuse from the interior to the surface of the microdomain where it exchanges following a Langmuir isotherm. Mass transfer to the bulk phase occurs by desorption of the drug from the surface through a driving force that is proportional to the difference between surface and bulk concentrations. Accordingly the drug is released to the surroundings by diffusion through the bulk. Depending on the values of the Langmuir constant and assuming well stirred behaviour in the interior of the microdomain, the present model results in either of the two asymptotic models developed in previous studies. A parametric sensitivity study shows that the desired steady state flux of a solute or drug to the surroundings may be obtained given appropriate values of structural properties of the material. Specifically, a low value of the Langmuir constant, b (corresponding to a steep isotherm) favours sustained release, whereas as the isotherm is closer to linear the ‘sustained release’ period disappears. In addition, ¨ the internal Damkohler number has a profound effect on the release of the drug. If this number is kept ¨ above the external Damkohler number, the drug is released in a controlled way. On the other hand, ¨ when the internal Damkohler number is of the same order of magnitude or drops below the external ¨ Damkohler number, the plateau value of the flux of drug released to the surroundings disappears and no sustained release can be obtained. Finally the distribution coefficient of drug within the gel affects the length of the period of sustained release but not the plateau value of the flux released. The model is employed to simulate experimental results, previously obtained on this type of polymers. It follows that the present two-phase approach is the only available model that can predict accurately the controlled release of a specific solute or drug on such a biphasic polymer. Furthermore, the effect of internal diffusion cannot be ignored since this phenomenon is mainly responsible for the dispersive behaviour of the flux following the plateau region. In addition, both solutes used in the tests, have to

follow a quite steep Langmuir isotherm behaviour when exchanged at the surface of the microdomain.

Acknowledgements This work has been partly supported by the EPETII programme of the General Secretariat for Research and Technology of the Hellenic Ministry of Industry, Energy and Technology (Subprogramme 1, Project No 410).

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