New kinetic model of drug release from swollen gels under non-sink conditions

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1 Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) New kinetic model of drug release from swollen gels under non-sink conditions Delia L. Bernik, Diana Zubiri, María Eugenia Monge, R. Martín Negri Instituto de Química Física de Materiales, Ambiente y Energía (INQUIMAE), Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 2, 3 er Piso, C1428EHA Buenos Aires, Argentina Received 16 February 2005; received in revised form 22 July 2005; accepted 15 August 2005 Available online 22 September 2005 Abstract A kinetic model of drug release from slabs of swollen gels under non-sink conditions is presented. The model is based on the mass transfer of drug from the interfacial region to an external solution of finite volume. Mass transfer is coupled with drug diffusion from the bulk gel towards the interfacial region together with a non-equilibrium adsorption desorption of drug to the polymer chains. The coupled kinetic equations obtained for the whole mentioned process allowed us to describe the time evolution of drug concentration released to the external volume, without the needs of keeping perfect sink conditions. The remaining bulk drug concentration can also be recovered. The model is tested with experimental data of teophylline release from xanthan gum gels placed in a steady-state Franz cell. It is observed that a pseudo steady-state concentration of free (non-adsorbed) drug is reached at the interfacial region after a short transitory time, t 0. Besides, the kinetic features of the model have been simulated for several different parameters like slab dimensions, drug diffusion coefficient in the bulk gel, the equilibrium constant for the drug partition between gel and solution and the initial drug loading in the gel Elsevier B.V. All rights reserved. Keywords: Drug diffusion; Drug release; Kinetic model; Mass transfer; Swollen gels 1. Introduction Most of the release kinetic models for drugs encapsulated in matrices have been developed under three particular conditions. Two of these are related to (i) drug diffusion from the matrix bulk considered as the main or even the only process accounted in the kinetics; (ii) imposed perfect sink conditions in the external media. The third condition refers to the encapsulation matrix: they are usually modelled as swellable and/or erodible, so a time-dependent matrix change is considered [1 7]. In our experimental studies of drug release, we deal with systems under opposite conditions: the encapsulation media are swollen gels, the release is towards a closed recipient so non-sink conditions must be considered and the release kinetics does not follow a diffusional pattern. We study the release from swollen gels because are becoming object of increasing interest due to the advantages found in transdermic delivery as an alternative path- Corresponding author. Tel.: ; fax: address: dbernik@qi.fcen.uba.ar (D.L. Bernik). way for the administration of drugs with undesired secondary effects [8,9]. Therefore, the aim of the present work is to introduce a theoretical model that can be applied for fitting the kinetic of drug release from swollen gels towards a finite external volume under non-sink conditions. It is worth noting that sink conditions are frequently not achieved in laboratory experiments, particularly when using steady state Franz cells or dissolution tests (depending on drug loading) [10]. Non-sink conditions are often found in real practical cases. It has been indicated that drug release in vivo can be some times better correlated with an in vitro experiment with loss of sink conditions. For example, Ikegami et al. showed that non-sink conditions would reflect drug release in the lower region of the gastrointestinal tract because of poor drug solubility and local lack of fluid. They found that the in vitro release profiles obtained combining sink and non-sink conditions were similar to the in vivo profiles [11]. In the kinetic model depicted here drug diffusion is considered just one element within the total transport phenomena. Diffusion process is coupled with mass transfer across the gel-solution interface and with a drug to polymer /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.colsurfa

2 166 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) Nomenclature b Langmuir adsorption desorption parameter (= k 1 /k 2 ) C(t) molar concentration of drug in the external solution at time t. C 0 = C(t = 0); C = C(t = ) C b (x, t) molar concentration of bound drug in the gel at time t and the position x C f (x, t) molar concentration of free drug in the gel at time t and the position x Cb max (t) maximum allowed concentration of bound drug in the gel Cb s(t) molar concentration of bound-drug at the contact surface at time t Cb 0(t) initial molar concentration of bound drug in the gel (at t =0) Cb (t) final molar concentration of bound drug in the gel (at t = ) Cf s(t) molar concentration of free-drug at the contact surface at time t Cf 0(t) initial molar concentration of free drug in the gel (at t =0) Cf (t) final molar concentration of free drug in the gel (at t = ) Cgel 0 (t) total molar concentration of drug loaded in the gel D diffusion coefficient of the drug in the gel (in cm 2 s 1 ) k 1 rate constant of adsorption (in M 1 s 1 ) k 2 rate constant of desorption (in s 1 ) k r forward release rate constant from the gel to the external solution (in s 1 ) k r back rate constant from the external solution to the gel (in s 1 ) K eq gel-solution partition equilibrium constant L total length or thickness of the gel S area of the gel in contact with the external solution volume of the external solution V ext adsorption desorption processes. The integration of basic analytical concepts such as total mass balance and the appropriate boundary conditions brings about equations able to predict an exponential growth of the released concentration with a characteristic rate constant independent of drug concentration. 2. Model justification in connection with experimental data Fig. 1 shows experimental data of teophylline release from xanthan gum gel towards an external closed volume in a Franz diffusion cell. A 3% (w/w) xanthan gum containing 0.06% (w/w) teophylline was placed in the donor compartment and the release was followed during 60,000 s (more than 16 h), being the percentage of drug released in that period about 80%. The time dependent concentration of drug in the receptor compartment, C(t), was plotted together with the fits obtained using two well Fig. 1. Fits of experimental data in the whole release time range for the case of teophylline released from 3% xanthan gum swollen gels towards an external solution in a closed volume under non-sink condition. The dark and wide line always represents the experimental data for C(t). The thinner line corresponds to model fits. The experimental data were fitted in the whole release time range from 300 to 60,000 s (about 16 h), (a) fit by C(t) = A t; (b) fit by C(t)=kt n ; (c) fit by Eq. (13) C(t)=C (1 exp( k r t)), recovered k r = s 1. known equations, and the third equation is the one proposed in this work. The kinetic expressions used to fit the data are: a diffusional Fickian s dependence, C(t) = A t, the so-called Peppas model C(t)=kt n [12] and a growing exponential equation, C(t)=B(1 e kt ). As shown in this figure, only the growing

3 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) exponential expression gives acceptable fits in the time range assayed. The departure observed at the end of the fit is discussed later (see Section 7.4). The first two equations in Fig. 1 are based on drug diffusion inside the gel. Therefore, we developed an alternative model where other processes but diffusion can rule out the release kinetic. Although drug diffusion is the only mechanism by which the drug moves from the bulk towards the contact surface [13,14 and references therein], a mass transfer barrier between the matrix and the external medium coupled to a desorption mechanism can rule the release. The present model is expected to be realistic as far as the mass transfer at the contact surface is the rate limiting kinetic process. These characteristics are schematised in Fig. 2. At the beginning of the release the concentration gradient and the difference of drug chemical potential between the gel and the external solution ( µ) is relatively high, thus the limiting process is the crossing over the mass-transfer barrier at the contact surface. As far as the release progresses the difference of chemical potential decreases and the diffusion from the bulk towards the surface may then become the limiting kinetic process. Finally, dynamic partition equilibrium is reached ( µ = 0) at the end of the release (t = ). From now on the terms gel or matrix involve the sum of components present in the gel: the polymer, the aqueous media and the drug content. The drug in the gel is distributed between free drug and bound drug (adsorbed to the polymer chains) with molar concentrations referred as C f and C b, respectively, while the molar drug concentration in the external solution is referred as C(t) (see Fig. 3). It is assumed that C(t) is always homogeneous in the volume occupied by the external solution (V ext ). The initial concentration of the drug in the external solution is a given value named C 0, while the total analytical molar concentration of drug initially loaded in the gel is called C 0 gel.for clarity, all the symbols used in the equations are listed separately. The model considers that only the free molecules can be released towards the external solution and that a mass transfer barrier at the contact surface hinders the release (Fig. 3). The cross over the barrier has a rate constant k r (in s 1 ). The concentration C f at the contact surface is restored by the diffusion of C f molecules from the bulk but also by the back mass transfer of C-type molecules from the external solution to the surface with a rate constant k r (in s 1 ). The dynamics of the adsorption desorption in the gel is also taken into account in the present model. C f, C b and C are functions of the release time t, but C f and C b are also functions of the distance x, from x =0to x = L (see Fig. 3) after the release starts. 3. The adsorption desorption process The adsorption process is considered as a formal bimolecular process between C f molecules and the unoccupied adsorption sites. Assuming that there is a finite number of available adsorption sites and that one site can host only one drug molecule, the concentration of adsorbed drug has a maximum value Cb max. Therefore, the number of free sites at any point of the gel at any time is proportional to the difference (Cb max C b ) and the Fig. 2. Scheme indicating the presence of a mass-transfer barrier and a concentration gradient. In (a), the concentration gradient and the difference of chemical potential ( µ) between the gel and the external solution are high. These differences are lower as the release progresses, as shown in (b). Equilibrium between the gel and the external solution is reached at the end of release ( µ = 0) (c). The mass-transfer barrier remains constant.

4 168 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) Fig. 3. A swollen gel in contact with an adjacent solution, showing the bulk, the contact gel-solution surface and the external solution. The C f -type molecules may release towards the external solution, while the C b -type molecules remain adsorbed to the polymer chains. The C f -type molecules also diffuse from the bulk towards the surface interfacial region while the C b -type remain adsorbed in the bulk. The molar concentration of drug in the external solution, C, increases with time. The constants k 2, k r and k r are first order rate constants (in s 1 ) for the desorption, release and back-release processes respectively, while k 1 is the second order rate constant (in M 1 s 1 ) for the adsorption process. rate of adsorption is given by k 1 (Cb max C b )C f, where k 1 is a second-order rate constant for the adsorption process (expressed in M 1 s 1 ). Desorption is a first-order process with a rate equal to k 2 C b, where k 2 is a first-order rate constant expressed in s 1. The initial values Cf 0 and C0 b can be now calculated with this model, as shown in Appendix A. 4. Kinetic description Our model is based on the mass transfer of free molecules from the gel at the liquid contact surface, coupled with diffusion from the bulk and considering the non-equilibrium adsorption desorption process. The system evolves from the initial situation (t = 0), referred as the time when the gel is placed in contact with the external solution and the drug release starts, to the final equilibrium situation at t =. The following kinetic equations describe the time evolution of the concentration of free and bound molecules at the surface, Cf s(t) and Cs b (t), respectively, and at the external volume, C(t): dc s f dt dcb s dt dc dt = k 1 (C max b Cb s )Cs f + k 2Cb s + J diff k r Cf s + k rc(t) (1) = k 2 C s b + k 1(C max b C s b )Cs f (2) = k r C s f k rc(t) (3) where C s f (t) C f(x = 0,t) and C s b (t) C b(x = 0,t). The first two terms in Eq. (1) are due to the adsorption desorption process. The third term, J diff, is the diffusional flux of free-like molecules from the gel towards the surface, evaluated at the surface (x = 0) as a function of time. This term must be considered in Eq. (1) regarding that the freelike molecules are those which are not adsorbed to the polymer chains and can move inside the water cavities of the gel. Its concentration, C f (x, t) is dependent not only on the release time, t, but also on the position inside the matrix, x, decreasing from the bulk towards the surface. Therefore, a time and position dependent diffusional flux, J diff (x, t), of free-like molecules is established. Its particular value at the surface, x = 0, is referred as J diff in Eq. (1) and should be proportional to the gradient concentration D( C f / x) evaluated at x = 0, where D is the diffusion coefficient of the drug in the gel, according to the first Fick s Law. The two last terms in Eq. (1) are due to the mass transfer process with rate constants k r and k r for the direct and reverse transfer, respectively. These rate constants depend on the nature of the encapsulated drug, the matrix, the drug matrix interactions and the temperature, but are independent of the drug concentration. 5. Solutions for the steady state situation compatible with the non-sink constraints It is possible to obtain expressions for C(t) and Cb s (t) for a situation where Cf s is kept constant and some specific conditions for the non-sink situation are applied. We assume that the system rapidly evolves towards a situation at which the concentration of free molecules at the surface may be regarded as constant. This situation is reached at a time named t 0, so the so-called steady sate condition: dcf s /dt = 0 is valid since t = t 0. The time interval between t = 0 and t = t 0 is a transient time, which depends not only on the physical properties of the system but also on the instrumental and experimental factors (surface area, porous membrane used, temperature, etc.). When using Franz cells t 0 has been found to be between 1 and 5 min [14,15], which is a very short time in comparison with the time scale of drug release (several hours). Therefore, we assume that: C(t 0 ) = C 0 (4) C s b (t 0) = C s b (0) C0 b (5) (given by Eq. (AI.4) in Appendix A) That is, from now on it is assumed that the steady-state regime is maintained in the whole time interval t 0 t and the validity of this assumption will be discussed later. Now, the specific requirements for the case of drug release under non-sink conditions are described here below Condition 1: mass conservation The total mass of drug must remain constant because the system is closed, thus the mass-conservation law relates all concentrations at any time during the release: C 0 gel LS + C 0V ext = S 0 + S L 0 C f (x, t)dx L C b (x, t)dx + C(t)V ext (6)

5 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) Condition 2: no drug gradient inside the gel at the final equilibrium That is, the concentration gradient of free molecules inside the gel at t = must be zero: C f x 0 when t. Or equivalently : C f (x, t = ) = C s f C f (7) Otherwise, there should exist a concentration gradient inside the gel, which would keep on the release. This condition must hold also for sink-conditions, although is not always explicitly mentioned. In our case, the condition must be explicitly indicated in order to use it together with the mass conservation relationship Condition 3: drug partition equilibrium between gel and external solution at the end of the release The mass-transfer finishes when the chemical potential of C- type molecules in a differential volume placed in contact with the surface of the gel at the external solution equals to the chemical potential of C f -type molecules in the same differential volume in the gel. The final equilibrium concentrations, C f and C, are connected by the partition equilibrium constant, referred as K eq : K eq = C Cf (8) Note that the ratio of rate constants, k r /k r, is equal to the equilibrium constant, K eq : K eq = k r (9) k r Applying the mass-balance relationship (Eq. (6)) for t = together with the other conditions, we obtained: ( ) Cf V ext 1 + K eq + Cb LS = C0 gel + C V ext 0 (10) LS Up to here the developed equations are completely general, in the sense that the adsorption desorption process at t = 0 and t = were considered. However, in the present work, we will focus on the case of a neutral drug, soluble in water and with low binding affinity with the polymer. We show in Appendix B, the conditions where the adsorption desorption effects can be neglected and it is also shown that under those conditions it can be obtained a simple and useful expression for C : C = K eq 1 + K eq (V ext /LS) C0 gel (negligible adsorption and C 0 = 0) (11) In that expression it was also used C 0 = 0, i.e. no drug in the receptor compartment, which is the most common case. Eq. (11) (if C 0 = 0) predicts the maximum concentration of drug that can be reached in the external volume, C. Eq. Fig. 4. Simulations of C as function of K eq for different values of Cgel 0. Eq. (11) is used with C 0 =0,V ext =20cm 3, S =4cm 2 and L = 0.5 cm. (11) is used for fitting experimental data in the case of weak drug polymer interactions, like when using neutral drugs and hydrophilic polymers in high ionic strength media. Simulations of C as a function of K eq for different values of C 0 gel using Eq. (11) are shown in Fig. 4, where it is observed that C increases with K eq. This is expected because K eq represents the partition constant that increases when the drug partition is favored towards the external solution. It is also noted that if K eq (V ext /LS) 1 then C becomes independent of K eq and the highest possible value for C is reached (which is equal to C = (LS/Vext )C 0 gel ). Note that the condition K eq(v ext /LS) 1 is reached in the particular case of sink-conditions, where the release is independent of K eq and continues until all drug is released. Plots of C simulated as function of L are shown in Fig. 5. The parameters values used for all the simulations of the present work are shown in Table 1. Fig. 5 shows that C increases with L for a given initial loaded drug concentration, because the total number of drug molecules available for the release is larger. Now it is easy to integrate the kinetic equations with the steady-state condition using the constraints of the initial conditions and mass conservation. We found the following solutions Fig. 5. Simulations of C as function of L for different values of C 0 gel. Eq. (11) is used with C 0 =0,V ext =20cm 3, S =4cm 2 and K eq = 0.1.

6 170 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) Table 1 Typical values used for the simulations Parameter Typical value or range Units K eq 0.01 K eq 10 k r s 1 D < D <10 l0 6 cm 2 s 1 V ext 20 cm 3 S 4 cm 2 t 0 0 s L 0.05 L 1 cm Cgel 0 0 Cgel M C 0 0 C M for the time dependence of C(t)and Cb s (t) under those conditions: C s b (t) = C b + (C0 b C b ) exp( kt) (12) C(t) = C 0 + (C C 0 )(1 exp( k r t)) (13) where: k k 2 + k 1 C K eq C b = Cmax b b(c /K eq ) 1 + b(c /K eq ) (14a) (14b) Cb s(t) exponentially decreases from the initial value C0 b to its final equilibrium value Cb, with a rate constant k, which depends linearly on C. That is, the adsorbed drug is desorbed through the release from its initial value to the lower value, Cb. Note that a Langmuir-like expression for Cb is exactly compatible with the solutions of the kinetic scheme (see Appendix A and Eq. (14b)). C(t) exponentially grows from C 0 up to C, with a rate constant k r, which is independent of any concentration. C is dependent of Cgel 0, according to Eq. (11). Simulations of C(t)as function of the release time for different values of Cgel 0 with C 0 fixed equal to zero are shown in Fig. 6. Fig. 6. Simulations of C(t) as function of the release time, t for different values of Cgel 0. Eq. (13) is used with C 0 = 0 and k r = s 1. Eq. (11) is used to obtain C (C 0 =0,V ext =20cm 3, S =4cm 2, L = 0.5 cm and K eq = 0.1). 6. Time interval for the validity of the model in real situations In every process where diffusion is involved the rate constants determine characteristic distances associated to them, in our case given by D/k r and D/k, where D is the diffusion constant of the drug molecules in the gel. These characteristic distances define a region of the space close to the boundaries, where the molecules can diffuse fast enough to keep the stationary condition in the interfacial region. The shorter of these two distances ( D/k r and D/k) will be referred as L diff from now on. In our model, the mass transfer at the surface and the adsorption desorption from the polymer are coupled with diffusion from the bulk. Under the steady-state condition all the process are interconnected in such a way to keep Cf s constant. Briefly, in the steady-state regime the diffusion is fast enough to provide sufficient amount of material to the interfacial region for the release. This kind of situation can be found in the initial times of impurity evaporation with high-vapor pressures [17]. This is possible if the typical diffusion length of the problem (L diff ) is larger than the characteristic geometrical distance of the system (given here by L). In a real release system the concentration gradient decreases with the release time, thus the velocity of diffusion decreases with time and at a certain time the diffusion can not provide material to the interface at an enough velocity to keep the steady-state condition. The above considerations indicate that it is possible to determine the time interval where this model has physical meaning in a real situation. After that time interval, the diffusion becomes slow, the above-mentioned connection between the velocities is lost and the release progressively becomes limited by diffusion: a change of regime is expected. The steady-state condition can be hold only until the time at which almost all drug molecules initially located at a distance equal to L diff from the interfacial region are released to the external volume. This allows for obtaining the time at which the steadystate situation is lost, referred as t 1 from now on. Assuming that at t = t 1 a number of moles equal to (Cgel 0 SL)(L diff/l) has been released towards the external solution, then C(t 1 ) = C 0 + Cgel 0 SL(L diff/l)/v ext. Thus, according to Eq. (13): ( ) t 1 = t C C 0 ln k r C C 0 (Cgel 0 SL (15) diff/v ext ) The time interval for the validity of the results predicted assuming the steady-state condition is (t 0, t 1 ) given by Eq. (15). The model assumes that the diffusion constant of drug molecules in the gel, D, does not change over the release. This is found for weak drug polymer interactions in swollen hydrophilic gels for which the mesh size is larger than the van der Waal volume of the drug. In these cases, the diffusion coefficient does not change considerably with drug concentrations at the low drug loadings usually used in release experiments. For example, it has been reported D = cm 2 s 1 for teophylline in water and D = cm 2 s 1 for

7 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) teophylline in scleroglucan gels, both values at 25 C [18]. 7. Predictions of the model The principal predictions of the model are summarized: 7.1. Prediction 1 The molar concentration of drug in the external volume, C(t), will exponentially increase with time with a rate constant k r that is independent of any concentration. The rate constant k r can be experimentally obtained by fitting the experimental release data with Eq. (13). We have verified the invariance of k r with concentration (to be published in a fully experimental work). The central equation to be used for data fitting is Eq. (13): C(t)=C 0 +(C C 0 )(1 exp( k r t)). Note that this expression is reduced to the one shown in Fig. 1 for the case C 0 = 0 (no drug in the receptor compartment at t = 0): C(t)=C (1 exp( k r t)) Prediction 2 According to Section 7.1, the released concentration is expected to reach a plateau value, C, that can be determined from Eq. (AII.2) (or Eq. (11) if C 0 = 0), in the case of weak drug polymer interactions. The model predicts that the plateau value should increase with both the loaded drug concentration in the gel, Cgel 0, and the initial drug concentration in the external volume, C 0 (see Eqs. (11) and (AII.1) in Appendix II). This increase is expected to be linear with Cgel 0 or C 0, except in the cases where the drug adsorption at the interfacial region plays a very significant role Prediction 3 The model predicts that the amount of drug released at any time t, equal to (C(t) C 0 ), decreases with the initial concentration of drug in the external volume, C 0, because the difference between the chemical potentials is reduced. In fact, using Eq. (11) (or (AII.2) for the case C 0 > 0) and Eq. (13), then: (C(t) C 0 ) 1 (16) C (K eq V ext /LS) That is, the released concentration (C(t) C 0 ) must always decrease with C 0, at any time t, because the derivative in Eq. (16) is always negative. It is worth noting that the dependence of C(t) with C 0 at a fixed Cgel 0, or with C0 gel atafixedc 0, can be used for determining K eq. Fig. 7. Simulations of t 1 as function of L, using Eq. (15). L diff is taken equal to D/k r with D = cm 2 s 1 and k r = s 1. Eq. (11) is used to obtain C /Cgel 0 (C 0 =0,V ext =20cm 3, S =4cm 2, t 0 = 0 and K eq = 0.1, 0.2, 1 and 10, respectively). contact with the external solution. Therefore, for t > t 1, C(t) must be fitted by a different expression than the exponential growth predicted by the present model. This would be the probable reason of the fit departure observed at about 50,000 s in Fig. 1c. At this time the drug released reached 79% of the total drug loaded in the gel. More experimental results must be evaluated to confirm this t 1. The model also allows making some simple predictions about t 1 : - If k r is fixed, then t 1 should increase with D and L. - For fixed k r, D and L, then t 1 should decrease with increasing K eq. - In the case of C 0 =0,t 1 becomes independent of C 0 gel. These dependencies are shown in Figs. 7 and 8. Fig. 7 shows t 1 dependence on the device width for different K eq values. t 1 will be longer if the device is thinner. On the other hand, at constant L, a smaller K eq will yield larger t 1, because diffusion 7.4. Prediction 4 The model is valid within a time interval (t 0, t 1 ). That is, C(t) would fit a growing exponential function described by Eq. (13) in the interval (t 0, t 1 ) where t 1 is predicted from Eq. (15). The release after the time t 1 will turn to be controlled by the diffusion of drug molecules from the bulk of the gel to the surface in Fig. 8. Simulations of t 1 as function of D, using Eq. (15). L diff is taken equal to D/k r with k r = s 1. Eq. (11) is used to obtain C /Cgel 0 (C 0 =0, V ext =20cm 3, S =4cm 2 and K eq = 0.1, 0.2, 1 and 10, respectively).

8 172 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) (at constant D value) will be able to restore the drug released at the interface. In Fig. 8, we keep the device size constant and we see the influence of K eq varying the diffusion coefficient. Again, a very low K eq will favor longer t 1 values, whereas for larger K eq it is necessary to increase D in one order to have significant influence on t 1 value. 8. Final remarks The most popular mathematical equation used to fit general drug release data as a function of time is perhaps the expression kt n proposed by Ritger and Peppas, usually called power law [12]. It was inspired in the mathematical solution of the Fick differential equation for drug diffusion in a slab assuming sink conditions in the external solution. The model is very easy to handle and gives a first glance about release kinetic behaviour using different types of devices. However, we have found some drawbacks when using the power law equation: the recovered n value changes with the time interval used to fit the data. For example, in the case of teophylline release from scleroglucan swollen gels n changes from values around 0.7 when fitting the first release hour to values around 0.5 when using data collected during 4 h or more [16]. At long time intervals, the recovered fits using Peppas s equation showed systematic deviations with respect to the experimental curves (see Fig. 1). On the other hand, the power law equation does not allow the prediction of release kinetics behaviour upon variables like drug concentration in the matrix, dependence with the drug diffusion coefficient or the slab geometry. Another proposed model developed by Singh et al. [19] accounted for diffusion coupled with a desorption mechanism. Nevertheless, this model assumes again sink boundary conditions and considers that the drug release has a time dependence proportional to t, which is not the observed time dependence in many cases [15,20]. The lost of sink conditions has been rarely studied and modelled. As an example, the release of drug into a limited volume has been studied in the work of Zhou and Wu (2003, and references therein) for permeable polymeric matrixes with relatively high drug loadings [21]. In this case, the authors assume the presence of a moving front of dispersed drug into the external volume and that drug release is controlled by diffusion through that moving front. Analytical expressions were found for some limit cases, e.g. initial drug concentration much larger than drug solubility. Nevertheless, the hypothesis of a moving front cannot be particularly applied in the case of swollen gel matrixes in which the drug is completely dissolved prior to the release start. When an encapsulated molecule goes from one phase (the gel) to another (the external solution), the transport process is not only influenced by diffusion from the bulk to the surface, but the partition of the molecule between both phases has to be also considered. So, the release mechanism from the gel to the water depends on the drug affinity to the external phase. As described by Cussler [22], a mass transfer approach must be used when diffusion occurs across an interface. Cussler considered a mass transfer coefficient as type of reversible rate constant (in our notation k r = k r ). According to all these statements, we have considered in this work that diffusion across the interface is just one element within the total transport phenomena. The diffusion process is coupled with the mass transfer across the gel-solution interface and with the drug to polymer adsorption desorption processes. The integration of basic analytical concepts such as total mass balance and the appropriate boundary conditions brings about the present model. With it, it is possible to predict, under non-sink conditions, an exponential growth of the released concentration with a characteristic rate constant independent of concentration. Finally, we remember our statement that the model and their predictions are valid when diffusion is not controlling the release, but is only one ingredient of the process. Diffusion becomes more important and the model fails after the time referred here as t 1, which is very much dependent on the slab length, L, and the drug diffusion coefficient, D (see Eq. (15) and comments before). In summary, herein we intend to help filling the vacancy of new models based on kinetic equations, which go beyond the semi-empirical expressions. These models must be able to predict the dependence of the release kinetics with parameters such us drug concentration, diffusion coefficients, slab geometry, etc., accounting for the putative processes involved: diffusion from the bulk to the surface in contact with the external solution, adsorption of the drug to the polymer chains, desorption and mass transfer to the solution without assuming sink-conditions. Acknowledgements RMN and DLB are members of the Consejo Nacional de Investigaciones Cientfficas y Técnicas (CONICET, Argentina). DZ has a fellowship of Fundación YPF (Argentina). MEM has a doctoral fellowship of CONICET (Argentina). This work was supported by the University of Buenos Aires (UBACyT , Projects X171 and I007 and UBACyT , Project X267), Fundación Antorchas and CONICET (PEI 6069 and PEI 6323). Appendix A. Initial conditions and the Langmuir relationships We present in this Appendix, the relationships that allow calculating the initial values of the concentrations for the bound and free drug inside the gel, as well as the final value for C b (at the end of the release). Remember that t = 0 is referred as the time when the gel is placed in contact with the external solution and the drug release starts, the initial concentration of the drug in the external solution is C 0, while the total analytical molar concentration of drug initially loaded in the gel is Cgel 0. It is assumed that drug is initially homogeneously and randomly distributed in the gel before the release starts. Therefore: C 0 f + C0 b = C0 gel where C 0 f C f(t = 0) and C 0 b C b(t = 0). (AI.1)

9 D.L. Bernik et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 273 (2006) The initial molar concentration in the external volume is named C 0 : C 0 C(t = 0). In real practical applications of drug release, the condition C 0 = 0 usually holds. Nevertheless, we are not going to assume C 0 = 0 in the formalism, even when that condition is used for the simulations performed. Since t = 0, when the release starts, the above concentrations change as function of the release time, t, up to a final equilibrium situation, reached at long times, indicated here as t =. From now on the symbol always indicates the concentrations time infinite (the release stops). At time zero, previous to the release, the rate of adsorption must be equal to the rate of desorption (a Langmuir-like relationship): k 1 (Cb max Cb 0 )C0 f = k 2Cb 0 (AI.2) Therefore: C 0 b = Cmax b bc 0 f 1 + bc 0 f (AI.3) with b k 1 k 2 This set of relationships and Eq. (AI.1) allows obtaining Cf 0 and Cb 0, as functions of C0 gel. For example, a quadratic equation is obtained for Cb 0 as function of C0 gel, whose solution is: (1 + C 0 C 0 b = gel b + bcmax b ) (1 + C 0 gel b + bcmax b 2b ) 2 4C 0 gel Cmax b b 2 (AI.4) Were the minus sign holds before the square root in order to obtain solutions with Cb 0 <Cmax b. Then Cf 0 can be calculated from Eq. (AI.1). The adsorption desorption equilibrium condition is lost after t = 0, but must be restored at t =. This means that at t = the concentrations Cf and Cb must be also related by a Langmuirlike relationship, analogue to Eq. (AI.3): C b bc f = Cmax b 1 + bcf (AI.5) In summary, the concentration of adsorbed drug, C b, evolves from the value Cb 0, given by Eq. (AI.4), to a value C b ; related to Cf by Eq. (AI.5). Appendix B. The relative influence of the adsorption effects We deduce Eq. (11) in this Appendix, by observing that the relative weight of the adsorption effect can be evaluated by the product (bcb max ). In fact, if this term is much lower than (1 + K eq (V ext /LS)), then Cb is negligible in Eq. (10): Condition for negligible absorption effects : ( ) bcb 1 + V ext K eq (AII.1) LS If this condition holds then a simple expression is found for C from Eq. (10): ( ) C K eq = Cgel K eq (V ext /LS) + C V ext 0 LS (case of negligible adsorption effects) which reduces to Eq. (11) in the case of C 0 =0. References (AII.2) [1] P. Colombo, R. Bettini, G. Massimo, P.L. Catellani, N.A. Peppas, J. Pharm. Sci. 84 (1995) 991. [2] J. Siepman, H. Kranz, R. Bodmeier, N.A. Peppas, Pharm. Res. 16 (1999) [3] B. Narasimhan, Adv. Drug Deliv. Rev. 48 (2001) 195. [4] J. Siepmann, A. Göpferich, Adv. Drug Deliv. Rev. 48 (2001) 229. [5] J. Siepmann, A. Steubel, N.A. Peppas, Pharm. Res. 19 (2002) 306. [6] K.G. Papadokostaki, J.K. Petrou, J. Appl. Polym. Sci. 92 (2004) [7] K.G. Papadokostaki, J. Appl. Polym. Sci. 92 (2004) [8] E. Escribano, A.C. Calpena, J. Queralt, R. Obach, J. Domenech, Eur. J. Pharm. Sci. 19 (2003) 203. [9] J.P. Venter, D.G. Muller, J. du Plessis, C. Goosen, Eur. J. Pharm. Sci. 13 (2001) 169. [10] A.G. Thombre, L.E. Appel, M.B. Chidlaw, P.D. Daugherity, F. Dumont, L.A.F. Evans, S.C. Sutton, J. Control. Rel. 94 (2004) 75. [11] K. Ikegami, K. Tagawa, M. Kobayashi, T. Osawa, Int. J. Pharm. 258 (2003) 31. [12] P.L. Ritger, N.A. Peppas, J. Control. Rel. 5 (1987) 37. [13] B. Amsden, Macromolecules 31 (1998) [14] L. Masaro, X.X. Zhu, P.M. Macdonald, J. Polym. B Polym. Phys. 37 (1999) [15] M. Grassi, R. Lapasin, S. Pricl, I. Colombo, Chem. Eng. Commun. 155 (1996) 89. [16] N.J. François, A.M. Rojas, M.E. Daraio, D.L. Bernik, J. Control. Rel. 90 (2003) 355. [17] B.S. Bokshtein, Diffusion in Metals, Mir, Moscow, [18] I. Colombo, M. Grassi, R. Lapasin, S. Pricl, J. Control. Rel. 47 (1997) 305. [19] M. Singh, J.A. Lumpkin, J. Rosenblatt, J. Control. Rel. 32 (1994) 17. [20] N.A. Peppas, J.J. Sahlin, Int. J. Pharm. 57 (1989) 169. [21] Y. Zhou, X.Y. Wu, J. Control. Rel. 90 (2003) 23. [22] E.L. Cussler, Diffusion. Mass Transfer in Fluid Systems, 2nd ed., Cambridge University Press, 1999.