Implications of regular solution theory on the release mechanism of catanionic mixtures from gels

Implications of regular solution theory on the release mechanism of catanionic mixtures from gels

Colloids and Surfaces B: Biointerfaces 71 (2009) 214–225 Contents lists available at ScienceDirect Colloids and Surfaces B: Biointerfaces journal ho...

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Colloids and Surfaces B: Biointerfaces 71 (2009) 214–225

Contents lists available at ScienceDirect

Colloids and Surfaces B: Biointerfaces journal homepage: www.elsevier.com/locate/colsurfb

Implications of regular solution theory on the release mechanism of catanionic mixtures from gels Tobias Bramer ∗ , Göran Frenning, Johan Gråsjö, Katarina Edsman 1 , Per Hansson Department of Pharmacy, Uppsala University, Box 580, SE-751 23, Uppsala, Sweden

a r t i c l e

i n f o

Article history: Received 3 December 2008 Accepted 10 February 2009 Available online 21 February 2009 Keywords: Vesicle Micelle Catanionic Regular solution theory Diffusion

a b s t r a c t The aim of this study was to apply the regular solution theory of mixed micelles to gain new insights on the drug release mechanism, when using catanionic mixtures as a method of obtaining prolonged release from gels. Synergistic effects were investigated at equilibrium and quantified in terms of regular solution theory interaction parameters. The drug release from catanionic aggregates was studied both in a polymer free environment, using dialysis membranes, and in gels, using a modified USP paddle method. The drug release kinetics was modelled theoretically by combining the regular solution theory with Fick’s diffusion laws assuming a contribution to the transport only from monomeric species (stationary aggregates). The theoretical predictions were found to be in reasonably good agreement with experiments. An analysis of the calculated distribution of species between aggregated and monomeric states was shown to provide further insights into the release mechanism. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Mixtures of two oppositely charged surfactants, or catanionic mixtures, have been studied extensively during the last decades for several kinds of surfactant [1–5]. Lately, a number of potential pharmaceutical applications for catanionic mixtures have been considered [6]. One of the ideas is to use catanionic mixtures, in which one of the surfactants is an amphiphilic drug, for obtaining prolonged release from gels, which has already been successfully applied in several studies [6–10]. Gels are popular pharmaceutical dosage forms, mainly due to their mucoadhesive [11] and rheological [12] properties, facilitating an extended contact time at the site of absorption. However, as a gel is usually constituted of about 99% water the release of water soluble drugs is rapid and, hence, a strategy for prolonging the release is a necessity to benefit from the long contact time facilitated by the gel. The idea of using catanionic mixtures as a way of prolonging the release is dependent on the drug properties: the drug has to have a net charge and show some surface activity. If these requirements are met it is possible to let it form catanionic complexes with oppositely charged surfactants, and when incorporated in and released from a gel, it has been shown that the apparent diffusion coefficient decreases 10–100 times as compared with the release of non-complexed drug compound from the gel [7,9].

∗ Corresponding author. Present address: Recipharm AB, Lagervägen 7, SE-136 50, Haninge, Sweden. Tel.: +46 8 602 4573; fax: +46 8 270 380. E-mail address: [email protected] (T. Bramer). 1 Present address: Q-med AB, Seminariegatan 21, SE-752 28, Uppsala, Sweden. 0927-7765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2009.02.008

The complete mechanism behind the prolonged release is unknown. However, it has been hypothesized that the drug and the surfactant form mixed aggregates that are too large to diffuse through the gel matrix, and that only monomers and small aggregates coexisting with the large aggregates diffuse freely out of the gel [9]. This hypothesis was supported by a study by Brohede et al. [13], although it was also concluded that further studies are needed on this subject. Vesicles or very large wormlike or branched micelles are often observed in catanionic mixtures of common surfactants [9,14–20]. In these mixtures a major factor controlling the size and shape is the cationic/anionic ratio in the aggregates, which is often very close to the overall value, due to the low concentration of monomers in these systems. This is reflected also in the phase behavior where the sequence of phases depends primarily on the cationic/anionic ratio [7,9]. However, effects from aggregate–aggregate interactions as well as pH and ionic strength are also present [8]. Since the cationic/anionic ratio has been found to be a major factor controlling the size also of catanionic drug/surfactant aggregates [7–9,21] this parameter should be important also for the possibilities to retain aggregates by the polymer network in a gel. However, it is expected to control the release rate from such systems also by its influence on the concentration of monomers in (local) equilibrium with the aggregates. In fairly dilute solutions both aspects are expected to depend chiefly on the cationic/anionic ratio in the aggregates. It is therefore of great importance to know how the latter is related to the overall ratio. This is particularly so when surfactant and drug are released from a gel at different rates so that the overall ratio changes during the release process. For very dilute mixtures it is unclear to what extent the cationic/anionic

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Table 1 The three surfactants used in this study. Surfactant

Structure

Diphenhydramine

Tetracaine

CMC* (mM)

Molar mass (g/mol)

105a

291.82

75a

300.82

1.2b

SDS

* a b

288.38

Measured in 0.9% NaCl. From Bramer et al. [9]. Measured in this study.

ratio in the mixed aggregates deviates from the overall value. In a study by Caillet et al. it was reported that the aggregates formed were very close to equimolar [22]. This is in support of a previously published study by Brasher and Kaler, but in that study it was also shown how addition of salt made the cationic/anionic ratio in the aggregates more similar to that in the bulk solution [23]. In the present paper we demonstrate how a theory describing the equilibrium distribution of the components between aggregates and water can be used to predict the release rates from formulations containing catanionic drug/surfactant mixtures when combined with Fick’s diffusion laws. Many properties of mixed surfactant systems have been accounted for quantitatively or semi-quantitatively by various molecular thermodynamic models. One such important property of catanionic mixtures is that the mixed micelles with a cation/anion ratio close to unity can form without the entropic penalty of binding counterions [24]. This explains the strong synergistic effects, i.e., why the critical micelle concentration (CMC) is reduced much more than for mixtures of two non-ionic surfactants. A drawback with many theoretical models is that they require a detailed knowledge of the molecular properties of the components, often not at hand for drugs and, furthermore, that the models are not developed for amphiphilic molecules lacking a typical head-and-tail structure. Since our purpose here is to find a functional description of the distribution of the components between micelles and water rather than a detailed understanding of the underlying interactions, we will resort to the simple regular solution theory of mixed micelles [25], requiring in return the input of experimentally determined parameters. The theory is particularly suitable for demonstrator purposes because it is transparent, thereby facilitating the interpretations of results from numerical model calculations of transport processes. The major drawback is that the theory is less accurate for charged components unless there is salt present in excess. In the first part of the paper we will use the regular solution theory to model the interaction between sodium lauryl sulfate (SDS) and two drugs, the local anesthetic tetracaine and the antihistamine diphenhydramine (Table 1), in the presence of 0.9 wt.% NaCl. The applicability of the theory to these systems has been demonstrated elsewhere [26]. In subsequent parts the theory will be combined with Ficks’s diffusion laws to model the drug release with time from gel prepara-

tions. The results from the theoretical calculations will be compared with release experiments. To test the regular solution approach in a more direct way, we will investigate also the release of monomers from a liquid catanionic solution through a dialysis membrane. As a key to better understanding the release mechanism we will, in both types of experiment, analyze the release of not only the drug compound, but also the oppositely charged surfactant. 2. Theory In this section we describe first the regular solution theory of mixed micelles, and then consider the implications of it for the release from catanionic surfactant mixtures when the micelles are retained either by a semi-permeable membrane or a polymer gel network. 2.1. Regular solution theory of micelles For a surfactant i in a solution above the CMC the relationship between the monomer concentration Ci and the mole fraction in binary mixed micelles xi can be written Ci = fi xi CMCi

i = 1, 2

(1)

where fi is an activity coefficient in micelles, and CMCi is the pure component CMC. Eq. (1) is quite general but is usually derived by neglecting the dependence of the entropy of mixing the micelles on the concentration and aggregation number of the micelles. This so called phase separation approximation is consistent with observations of an essentially constant monomer concentration above the CMC in many one-surfactant systems but is equally valid for twosurfactant mixtures where the monomer concentrations of both components can vary substantially above the CMC for the mixture, CMCmix . Ideal mixing (fi = 1) is often a good approximation for mixtures of two non-ionic surfactants [25,27]. In other cases significant deviations from ideal mixing can be observed [25]. In the regular solution approach [25] deviations are taken into account by letting the activity coefficient vary with xi in the following way: fi = eˇ(1−xi )

2

(2)

where ˇ is an interaction parameter characteristic of a given surfactant pair equal to w/kB T, where w is the pair interaction energy,

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kB is Boltzmann’s constant, and T is the absolute temperature. The form of Eq. (2) is common for all mean field theories, but derives originally from a description of nearest-neighbour interactions in the Bragg–Williams lattice theory of liquids [28]. The interaction parameter, ˇ, can be determined at the mixed CMC, which has a unique value for a given molar ratio between surfactant components. At CMCmix the amount of surfactant in micelles is considered to be negligible, and so Ci = ˛i CMCmix

i = 1, 2

(3)

where ˛i is the overall mole fraction of i (calculated on a surfactantonly basis). Eqs. (1)–(3) applied to both components allow ˇ to be calculated from experimental values of CMCmix and the pure CMC:s, either separately for different ˛i :s or from a global fit to a set {˛i ; CMCmix }. Once ˇ is determined the theory can predict the composition of the micelles and the concentration of monomers at any composition above the mixed CMC when combined with the material balance equations: xi =

Citot − Ci C tot − C1 − C2

i = 1, 2

(4)

where Citot is the total concentration of i and C tot = C1tot + C2tot . However, when at least one of the components is charged the theory and the procedure described above give reliable results only when there is excess of electrolyte added to the solutions, which will be the case for all systems considered here. At other conditions fi cannot be approximated by the simple Eq. (2), and the pure component micelles at their respective CMC:s cannot in general be used as reference states, as shown elsewhere [29]. 2.2. Release through membrane The experimental situation of interest here is a mixed surfactant solution of volume V in a diffusion cell in contact with an initially surfactant-free solution via a planar dialysis membrane of thickness L and area A. The membrane is permeable to dissolved monomers, simple ions, and water, but not to surfactant aggregates. When the concentration of monomers is Ci on the inside and <
(5)

where ni is the number of moles of i and Di is its monomer diffusion coefficient. For each component the change of the total concentration on the inside will thus be given by −

∂Citot ∂t

=

ADi C = Ki Ci , LV i

(6)

where the last equality defines the transport constant Ki . Eq. (6) is valid when steady state in the membrane is re-established much faster than the variation of Ci (quasi-steady state). For two surfactants forming mixed micelles, C1 and C2 are dependent. In this case Eq. (6) generates two coupled differential equations. In the regular solution theory these can be solved numerically for prescribed valtot using Eqs. (1), ues of the initial concentrations on the inside, Ci0 (2) and (4); see Section 3.7. 2.3. Release from gel Here the experimental system of interest is an initially uniform solution of two surfactants in a polymer gel. The polymer network is considered to prevent diffusion of the mixed aggregates (wormlike micelles or vesicles). Assuming that transport occurs through

monomer diffusion only, mass conservation of component i may be expressed by Fick’s second law, ∂Citot ∂t

= Di

∂2 Ci , ∂X 2

(7)

where X is the spatial coordinate and the remaining quantities have been defined above. Remember that Ci on the right hand side is the monomer concentration whereas Citot on the left hand side also includes surfactant in micelles. For release from a planar system with a sealed boundary at X = L, in contact with a well-mixed external medium at X = 0 that maintains the sink condition, the boundary conditions are



Ci (t, X = 0) = 0

and

∂Ci  ∂X 

= 0.

(8)

X=L

We assume that the initial total concentration of either component is uniform throughout the system, and hence the initial tot , where C tot is a prescribed constant condition is Citot (t = 0, X) = Ci0 i0 value. When combined with Eqs. (1), (2) and (4), Eq. (7) can be solved numerically for both components. 3. Materials and methods 3.1. Materials Diphenhydramine hydrochloride, tetracaine hydrochloride and sodium dodecyl sulphate (SDS) were purchased from Sigma Chemical Co. (St. Louis, MO, USA). The carbopol gel (C940) was a kind gift from Noveon Inc. (Breeksville, OH) and Agar-agar was purchased from MERCK (Darmstadt, Germany). All other chemicals were from Sigma Chemical Co. and were of analytical grade or “Ultra” quality. Ultra-pure water, prepared using a MilliQ Water Purification System (Millipore, France), was used in all preparations. 3.2. Determination of critical micelle concentration (CMC) Solutions of each surfactant by itself, as well as solutions of mixed surfactants, were prepared in 0.9% NaCl. The CMC of each solution was determined in room temperature, using the dropweight technique on a custom built instrument reproduced from Tornberg [30]. For each concentration of every solution, 9–11 measurements were made and the median values were used. All measurements were performed at the water/air interface. Details of the method are given elsewhere [31]. 3.3. Preparation of gels Two different gel forming polymers were used, carbopol 940 and Agar-agar. Carbopol 940 is a chemically cross-linked poly(acrylic acid) polymer, insoluble in water at low pH. When the pH is above 6 the polymer is highly charged and swells in water, forming a highly viscous gel. Agar-agar is uncharged and insoluble in water at room temperature, but dissolved upon heating, forming a stiff gel once it has cooled down to room temperature again. The carbopol gels were prepared in 0.9% NaCl, at concentrations twice as high as the proposed final concentrations. The polymer powder was dispersed in the saline solution and stirred with a magnetic stirrer for 2 h at room temperature. NaOH was then added to a pH of about 7 and the gel was allowed to swell over night, whereupon the pH was finally adjusted to 7.4. The drugs and/or the oppositely charged surfactant were dissolved in 0.9% NaCl, also at concentrations twice as high as the final concentrations. The carbopol gel and the drug/surfactant solutions were then mixed at a 1:1 ratio and the pH was again checked and, if needed, adjusted to 7.4.

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Fig. 1. Illustrations of the sample containers used in this study for release from gels (left) and liquid solutions (right).

When preparing the Agar gels, the drug alone or together with the oppositely charged surfactant was dissolved together with the polymer in 0.9% NaCl. To properly dissolve the Agar the solutions were then heated and allowed to boil on water baths for 20 min, while being stirred with a magnetic stirrer. While still being hot, the Agar gels were then dispensed directly into the gel containers where they rapidly solidified. 3.4. Release studies The drug release was measured using a modified USP method, using a Pharma Test PTW bath (Pharma Test Apparatebau, Germany). The gels and solutions were placed in specially built containers with a fixed volume of 6 ml and a fixed area of 21 cm2 . Two different kinds of container were used (Fig. 1). The containers used for the solutions were covered with a dialysis membrane ((CE) Membrane Sheets, MWCO 3,500, Spectra/Por® Biotech, USA) before filling, and were filled through a small hole in the bottom of the container, which was then sealed. The gels, however, were first applied into the containers, whereupon they were covered with a coarse size plastic net and a stainless steel net. The release medium was composed of 0.9% NaCl in water, and the volumes fitted between 300 and 750 ml, with regards to the detection levels and so that sink conditions were kept throughout the entire experiments. The temperature was kept at 37 ◦ C and stirring was performed at 20 rpm. The release medium was continuously pumped through a Shimadzu UV-1601 spectrophotometer (Shimadzu, Kyoto, Japan), where the concentration of the drug was analysed at 258 nm for diphenhydramine and 310 nm for tetracaine. Plots of the release of drugs and SDS where then constructed and compared with similar plots calculated applying the regular solution theory. All drug release studies were performed in triplicates. The release profiles of the measured data are shown as mean values +/− standard deviations (SD). 3.5. Detection of SDS Samples of the receiving solution were continuously collected during the release experiments. The SDS concentration in these was then determined, with regard to the sulphuric content, using an inductively coupled plasma atomic emission spectroscopy (ICP-AES) instrumentation. The instrument was a Spectroflame P ICP-AES (Spectro Analytical Instruments, Kleve, Germany), equipped with a standard Meinhard nebulizer. Sulphur was

monitored at the 180.734 nm atomic emission line. Details of the instrumental setting used are given elsewhere [32]. 3.6. Determination of transport constants The values of K were determined from release data recorded separately for each one-component system. In the case of SDS, the concentration in the cell was above CMCSDS at all times. KSDS was obtained from a fit of the following integrated form of Eq. (5) to experimental data: Cout = CMCSDS KSDS t,

(9)

where Cout is the concentration in the outer solution and  is the volume ratio of the inner and outer solutions (=6.0 × 10−3 ). For the cationic drugs tetracaine and diphenhydramine the concentration was below the CMC during the entire experiment. Kdrug was determined by fitting the function: Cout = C0 (1 − e−Kdrug t )

(10)

where C0 is the initial concentration in the release cell. Eq. (10) is the integrated form of Eq. (5) taking into account that the monomer concentration inside the cell decreases with time. The following values of the transport constants were obtained: 6.07 × 10−4 s−1 for SDS, 1.45 × 10−4 s−1 for diphenhydramine, and 1.69 × 10−4 s−1 for tetracaine. 3.7. Calculation of theoretical release profiles tot and C tot were calculated as For the membrane systems, CSDS drug tot , C tot , ˇ, CMC functions of time for prescribed values of CSDS0 drug , drug0 CMCSDS , Kdrug and KSDS . To this end, Eq. (6) was applied to SDS and drug, respectively, and the right hand sides were expressed as functions of xdrug only with the help of Eqs. (1) and (2). The two coupled differential equations were solved numerically by the Runge–Kutta method, where xdrug was determined numerically at each time step using Eq. (4). For each component the concentration in the outer solution was calculated as





tot Ci,out (t) =  Ci0 − Citot (t)

(11)

For the gels, Eq. (7) was solved numerically by using the finite element (FE) method. Following the standard FE procedure [33], Eq. (7) is first converted to a matrix equation of the form Mu˙ + Kc = 0

(12)

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Fig. 2. Phase maps for the catanionic mixtures diphenhydramine/SDS (left) and tetracaine/SDS (right) in brine. White areas represent micellar solution, grey areas represent multi-phase regions, black areas represent precipitates plus dilute solution and hatched areas represent vesicle phase.

where M and K are the capacity and diffusivity matrices, respectively, u and c are vectors that are formed from the nodal values of Citot and Ci , respectively, and the superposed dot denotes time differentiation. This step amounts to, firstly, transforming Eq. (7) to a weak form and, secondly, using FE shape functions to interpolate Wi , Ci and Citot . The first of these steps is accomplished by multiplication by a weighting function Wi (required to fulfill the essential boundary condition at X = 0), integrating over the spatial domain, and utilizing partial integration to convert the second order spatial derivative of Ci to first order derivatives of Wi and Ci . The second step produces the matrix Eq. (12). Since sharp gradients were expected in Citot , a lumped capacity matrix was used. The semi-discrete Eq. (12) was integrated in time using the secondorder accurate Crank–Nicholson scheme [33]. The equation to be solved for each time step t may be written Mun+1 +

t ˜ n+1 Kcn+1 = Mu 2

micellar solution samples of fixed drug/SDS molar ratios while monitoring the variation of the surface tension. The result is shown in Fig. 3 where the CMCmix (symbols) is presented as a function of ˛drug . The solid lines show CMCmix calculated from Eqs. (1)–(3) with ˇ equal to −9.0 and −8.4 for tetracaine/SDS and diphenhydramine/SDS, respectively. The values were obtained as the mean of individual ˇ-values determined for each ˛drug . The variation of ˇ within these sub-sets is small, as evident also by the fact that the curves drawn from single values fit quite well the experimental data in the entire range. As can be seen, in both systems CMCmix is lower than the pure component CMC:s. In particular, for all compositions investigated it is very much lower than the CMC for the drugs. The upper lines in Fig. 3 show the behaviour expected for “ideal mixing”, i.e. ˇ = 0. The fact that the measured values devi-

(13)

˜ n+1 = un + (t/2)u˙ n is a predictor value and the index where u indicates points in time. Since the monomer concentration may be considered as functions of the total concentration (and hence c as a function of u), Eq. (13) represents a nonlinear equation for un+1 , which was solved by using Newton’s method with line search. Once un+1 has been determined, u˙ n+1 may be calculated ˜ n+1 ) and the solution may advance to as u˙ n+1 = (2/t)(un+1 − u the next time step. In these computations, 500 equally large linear elements were used, and the time step was not larger than 0.02 s (sometimes a smaller time step was needed to ensure convergence of the iterative solution). 4. Results and discussion 4.1. Synergistic effects in the catanionic mixtures Phase maps for tetracaine/SDS and diphenhydramine/SDS, respectively, in 0.9 wt.% NaCl (≈0.15 M) aqueous solutions are shown in Fig. 2. In both systems the sequence of single and multi-phase regions is primarily determined by the overall drug/surfactant molar ratio. Note also the symmetry of the diagrams, particularly for tetracaine where the precipitate + dilute region (black) centred on the equimolar line is flanked on both sides by, in turn, liquid vesicle solutions (hatched), multi phase regions (grey), and micellar liquid solutions (white). The phase maps where published and discussed earlier [9]. In the present work we will investigate mixtures from these maps in order to facilitate comparisons with earlier work [6–9,13]. To investigate the relevance of the regular solution theory for mixed micelles, CMCmix was determined by diluting vesicle and

Fig. 3. Non-ideal mixing plots of the (A) tetracaine/SDS system and the (B) diphenhydramine/SDS system. The bottom curves show the actual, non-ideal, mixing modeled by the regular solution theory, whereas the top curves show the conditions would there have been ideal mixing. Symbols show experimentally determined CMC values.

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ate markedly from these lines shows that the formation of mixed micelles are favoured not only by the ideal entropy of mixing the two ions in the micelles, but also that there are synergistic effects. These are expected from the columbic attraction between the two ions and the reduced counterion binding to catanionic micelles as compared to the pure component reference systems. In salt-free systems the latter contribution can be very large for catanionic mixtures of regular surfactants [25,34], but should be less dominating at the present salt concentration. For example, for sodium decylsulfate/decyltrimethylammonium bromide, ˇ was reported to change from −18.5 in pure water to −13.2 in 0.05 M NaBr [35]. When discussing large interaction parameters it is important to identify the major factors determining their magnitude. It is instructive to compare with more detailed thermodynamic descriptions. The chemical potential (expressed here in units of kB T) of a surfactant in a micelle is often expressed as a sum of contributions [36,37]: mic = 0,mic + ln xi + surf + el , i i i i

(14)

where the first term is a standard chemical potential and the second term is the contribution from the (ideal) entropy of mixing i with the other surfactant in the micelle. The third and fourth terms are, respectively, the non-electrostatic and electrostatic contributions from the work of creating the interface between the micelle and the aqueous solution. The corresponding expression in the regular solution theory is: mic = ∗,mic + ln xi + ˇ(1 − xi )2 , i i

(15)

∗,mic i

is the chemical potential of i in a pure micelle at where CMCi . Comparison between Eqs. (14) and (15) offers the following interpretation of ˇ [29]: ˇ=

+ el surf i i (1 − xi )2

,

(16)

where  means that the difference is taken between the value in the mixed micelle (at the actual composition of the system) and the pure micelle at CMCi . This shows that ˇ is expected to be a function of xi , and not constant as assumed in the regular solution approach. However, at high salt concentration the variation with xi is small as illustrated by the following example where el i is evaluated using the Poisson–Boltzmann theory [37]. For simplicity we consider spherical micelles of fixed radius of 19 Å and with the surface area per molecule in the micelle equal to 75 Å2 = 0). In this case for both components at all compositions (surf i

we obtain at 150 mM salt and 298 K, el = 2.5 for the pure micelles i

and el + = −5.9 for a mixed micelle in the limit as x+ → 0. This means that el + , corresponding to the process of transferring a cationic surfactant molecule from a pure cationic micelle to a pure anionic micelle, is −8.4, which according to Eq. (16), is also the value of ˇ. For a net neutral catanionic micelle (x+ = 0.5) we can put el + = 0, and so el + = −2.5. Eq. (16) gives in this case ˇ = −10. The fact that the latter value is of the same order of magnitude as that in the fully charged micelle, shows that the parabolic form of the excess free energy in Eq. (2) captures the variation of the activity coefficient quite well, at least in charged systems where electrostatic effects are expected to dominate. It should be mentioned that, at the present salt concentration, the result is only slightly dependent on the curvature of the aggregate, and, within reasonable limits, also on the area per molecule. A more elaborate analysis is given in a separate paper [26]. As may have been noted, ˇ calculated here from the Poisson–Boltzmann theory is close to those obtained for the drug/SDS systems [26]. This is not surprising as electrostatic effects are expected to dominate. However, due to the large structural difference between the components one cannot preclude other types

Fig. 4. (a) Molefraction of drug in aggregates, xdrug , plotted vs. the overall molefraction of drug in the catanionic mixture ˛drug , as calculated from the regular solution theory (ˇ = −8.4) at CMCmix and at higher concentrations. Shown are also the result at CMCmix for ideal mixing (ˇ = 0) in the aggregates. (b) Monomer concentration Ci of drug and surfactant in equilibrium with mixed aggregates plotted vs. ˛drug , as calculated from the regular solution theory (ˇ = −8.4) at two different total concentrations of drug and surfactant (1.2 and 40 mM).

of interactions, the importance of which are difficult to foresee as theoretical models for amphiphilic molecules lacking a typical head-and-tail structure are still poorly developed. However, with support from the above considerations and the results in Fig. 3 it is safe to conclude that the regular solution approach gives a reasonable description of the solutions close to the CMCmix . This motivates an investigation of its capacity to predict the composition of mixed micelles and monomer concentrations in the solution also at higher concentrations. In Fig. 4 we have calculated xdrug as a function of ˛drug at the CMCmix and also at a few higher concentrations. Since the results are very similar for tetracaine/SDS and diphenhydramine/SDS, only data for the former system is shown. As can be seen, at or near CMCmix the composition of the micelles differs from the overall (bulk) composition. This is an effect not caused by non-ideal mixing. In fact ideal mixing would in the present case give even stronger deviations from the bulk composition, as evident from the lower curve in Fig. 4a. The “plateau” in the curves for concentrations close to CMCmix develops because of the electrostatic driving force for a cationic component to bind with a net negatively charged micelle, and vice versa for an anionic component. For components with equal CMC values the resulting micellar composition would have been 0.5. Here, SDS is the more hydrophobic (see Table 1), and therefore the resulting xdrug < 0.5. Caillet et al. reported that catanionic vesicles often form at equimolar ratio [22]. The results in Fig. 4 indicate that this may

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be true only for fairly symmetric systems, i.e., when the components have similar CMC:s. In general, the situation can actually be more complicated than evident from Fig. 4 since phase separation is not taken into account in the present calculations. In particular, this can lead to different molar ratios in different phases. When looking at the data in Fig. 4 it is important to point out that a successful use of the model at the mixed CMC (Fig. 3) does not imply an equally good description at finite micelle concentrations. For example, the model does not take into account aggregate–aggregate interactions. However, in coming the sections we show that release kinetics is quite well predicted by the theoretical framework in Section 2, in which the regular solution theory plays a major role. This gives some reliability also to the calculations in Fig. 4. Furthermore, in a study by Brohede et al. [13], transient current measurements were performed on a gel containing a catanionic vesicle mixture, at 14 mM diphenhydramine and 26 mM SDS. The concentration of free monomer in this gel was found to be ∼0.9 mM, which is in good agreement with the CMCmix plotted in Fig. 4. 4.2. Drug release from catanionic mixtures Drug release from gels is mediated by diffusion of monomers and micelles. However, in the catanionic mixtures the monomer concentration is very low and the large aggregates typically formed can be retained by the polymer network. The very slow release from carbopol gels containing catanionic mixtures, demonstrated in earlier works [7,9], indicates that the diffusion of aggregates (vesicles and wormlike micelles) is effectively hindered. Here we test the idea that the release kinetics is controlled by the monomer diffusion, by comparing experimental release profiles with the predictions of the theory in Section 2 assuming stationary aggregates. It is further assumed in the theory that local equilibrium between monomers and aggregates is maintained at all times and correctly described by the regular solution theory of mixed micelles. In an attempt to study the latter aspect separately we investigate first the simpler case of release from a liquid solution through a membrane retaining the aggregates. This is followed by release studies on catanionic mixtures in gels. The section ends with two purely theoretical investigations, one showing how the diffusion coefficients influence the calculations and one dealing with the release from cationic/nonionic mixtures. 4.2.1. Release through membrane Fig. 5a and b shows release from, respectively, a 14 mM diphenhydramine/26 mM SDS vesicle solution and an 8 mM diphenhydramine/32 mM SDS micellar solution through a dialysis membrane. The molecular weight cut off of the membrane used was 3500 g/mol, allowing only monomers, water and simple electrolytes to diffuse through the membrane. Shown is also the prediction from the theory in Section 2.2 derived to handle this situation. The theoretical curves were calculated using the ˇ parameters presented in the previous section and the transport constants KSDS and Kdrug determined for the unmixed systems in separate experiments; see Section 3.6. Fig. 6 shows the corresponding data for a 14 mM tetracaine/26 mM SDS vesicle solution (the abscissa shows t1/2 to facilitate comparison with gel release profiles where such a presentation can be expected to give linear plots). The quite large error bars for SDS are due to another method of analysis being used for that molecule than for the drugs, owing to that SDS cannot be detected by spectrophotometer. The best agreement is seen for the 14 mM diphenhydramine/26 mM SDS vesicle mixture (Fig. 5a), where the model places the release of both SDS and diphenhydramine within one standard deviation of the measured release for the first 7 h of the process. We emphasize that the theoretical curves are calculated

Fig. 5. The release profiles from (a) 14 mM diphenhydramine/26 mM SDS vesicle solution and (b) 8 mM diphenhydramine/32 mM SDS micellar solution through a dialysis membrane. Symbols are experimental points for diphenhydramine (black) and SDS (gray). The standard deviation is given for each measured value (n = 3). Lines are the theoretical predictions (ˇ = −8.4), colored correspondingly to the measured data. (c) Theoretical release profiles calculated for the system in (a) (dash-dotted) and in (b) (dashed) using the approximate Eq. (17) with Ci0 taken from the regular solution model (ˇ = −8.4).

directly from ˇ and K values obtained from separate experiments with no fitting to the experimental curves. For the micellar solution at 8 mM diphenhydramine and 32 mM SDS, however, the agreement is not quite as good (Fig. 5b). The model correctly predicts that SDS is released faster than diphenhydramine, but it overestimates the release rate for SDS and underestimates the release rate for diphenhydramine. For the 14 mM tetracaine/26 mM SDS system (Fig. 6) the agreement between theory and measured data is decent, although not as good as for the diphenhydramine vesicle system. The release of SDS is predicted quite well, but there is an overestimation of the speed of release for tetracaine. We conclude that, on

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in contact with the membrane would be different from xdrug for micelles in the bulk. According to Fig. 4 it can in fact be larger, which would result in lower SDS monomer concentration and higher drug monomer concentration, and a corresponding change in the release rates. However, while this is perhaps the most likely explanation to the discrepancies between theory and experimental data in Fig. 5b, it does not explain the overestimated drug release rate in Fig. 6. During initial stages of the release, when ˛drug has not had time to change much, the release of the components should be rather independent of each other. Under these conditions the monomer concentrations can be approximated as constant, and so Eq. (5) can be integrated independently for each component. By Eq. (11) the concentration in the outer solution becomes: Ci,out (t) = Ki Ci0 t

(17)

The predictions of Eq. (17), with Ci0 calculated from the regular solution model, are included in Figs. 5 and 6. When comparing with the exact solution of the theory, the approximation appears to be good during the time window of the experiment (11 h). This is explained by the fact that only a small fraction of the components are released. To summarize, although the theory used here does not perfectly agree with the experimental data in all three cases, it is quite remarkable how such a simple theory still manages to predict the experiments this well.

Fig. 6. (a) The release profiles from 14 mM tetracaine/26 mM SDS vesicle solution through a dialysis membrane. Symbols are experimental points for tetracaine (black) and SDS (gray). The standard deviation is given for each measured value (n = 3). The lines are the theoretical predictions (ˇ = −9.0), colored correspondingly to the measured data. (b) Theoretical release profiles calculated for the system in (a) as described in Fig. 5c.

a qualitative level, the theory predicts the correct release order for drug and SDS in all three cases. One reason for the poor quantitative agreement in Fig. 5b could be, of course, that the regular solution theory fails to correctly predict the monomer concentrations. However, as the quantitative agreement is much better when the same model is applied to gels (see below), at the same compositions of the mixtures, this does not seem to be the case. Errors are expected if micelles pass through the membrane, which is not allowed for in the theory. However, this cannot explain the discrepancies in Fig. 5b since if the catanionic aggregates would penetrate it would increase the release rate for both components, inconsistent with the results. Likewise, an overestimation of KSDS due to penetration of SDS micelles in the pure component experiments, would affect the calculated SDS profiles in all systems, not only that in Fig. 5b. Furthermore, an underestimation of Kdrug for diphenehydranine and overestimation for tetracaine does not seem likely. It can be mentioned here that SDS was found to diffuse faster through the membrane than the drug compounds, as indicated by the K values given in Section 3.6. The reason for this is unclear. Yet another explanation would be that a stagnant layer develops at the membrane in the cell due to poor convection. If the layer would be depleted from micelles it would act as an additional diffusion barrier for the monomers and slow down the release SDS and the drugs, in conflict with Fig. 5b. On the other hand, if there is a concentration gradient in the layer but the concentration remains above CMCmix , xdrug in the micelles

4.2.2. Drug release from gels Figs. 7 and 8 show the release from, respectively, diphenhydramine/SDS and tetracaine/SDS catanionic mixtures contained in 1% carbopol gels. The experiments were performed in a manner similar to that through membranes. All full drawn lines were calculated with the theory in Section 2.3. The SDS monomer diffusion coefficient was set equal to 4.8 × 10−6 cm2 /s as determined by others [38]. No literature values were found for diphenehydramine and tetracaine, but since they have nearly the same molecular weight as SDS we used this value for both drugs. The influence of the diffusion coefficients on the theoretical predictions are examined in Section 4.2.3. It should be stated also that a reduction of the diffusion coefficient of the positively charged drug molecules due to binding with the negatively charged gel networks can be neglected in the presence of 0.9 wt.% NaCl (0.15 M), since the concentration of sodium ions widely exceeds the drug monomer concentration. As evident the theory is in fairly good agreement with the experiments, in particular for the tetracaine/SDS systems (Fig. 8). For the diphenhydramine/SDS mixtures in Fig. 7, it is evident that the theory correctly describes the drug release profiles during the first hour (t1/2 < 60 s1/2 ), both for the 14 mM/26 mM vesicular and the 8 mM/32 mM micellar compositions. At longer times the theory underestimates the release. For SDS, the theory is within the standard deviation of the experimental data for the vesicle composition (Fig. 7a) but slightly underestimates the release for the micellar composition (Fig. 7b). However, the large error bars hamper the evaluation of the SDS release. It should be noted that for the micellar compositions in Fig. 7b the theory underestimates the release rate of both components. This may indicate a contribution from micellar diffusion in the gels. Even though large wormlike or branched micelles should become entangled with the network, they can move by the reptation mechanism. Furthermore, smaller micelles coexisting with larger micelles should be able move quite freely in the 1% carbopol gels, as indicated by a previous study [13]. Interestingly, whereas our theory to a good approximation predicts the release to be linear in t1/2 in the entire time window, the experimental data suggest two linear regimes. In the case of tetracaine two vesicle compositions were investigated, one with 14 mM drug/26 mM SDS (˛drug = 0.35) (Fig. 8a) and one with 26 mM drug/14 mM SDS (˛drug = 0.65) (Fig. 8b). The

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Fig. 7. The release profiles of a (a) 14 mM diphenhydramine and 26 mM SDS vesicle composition and a (b) 8 mM diphenhydramine and 32 mM SDS micellar composition from 1% C940 gels, where the symbols are experimentally measured levels of tetracaine (black) and SDS (gray). The standard deviation is given for each measured value (n = 3). Modelled profiles are shown as lines, colored correspondingly to the measured data. (c) Theoretical release profiles calculated for the system in (a) (solid) and in (b) (dashed) using the approximate Eq. (21) with Ci * calculated from the regular solution model (ˇ = −8.4) using the initial concentration of both components.

Fig. 8. The release profiles of a (a) 14 mM tetracaine and 26 mM SDS vesicle composition and a (b) 26 mM tetracaine and 14 mM SDS vesicle composition from 1% C940 gels, where the symbols are experimentally measured levels of tetracaine (black) and SDS (gray). The standard deviation is given for each measured value (n = 3). Modelled profiles are shown as lines, colored correspondingly to the measured data. (c) Theoretical release profiles calculated for the system in (a) (dashed) and in (b) (solid) as described in Fig. 7c.

agreement between theory and experiment is fairly good for both components in both cases, despite the considerable difference in the drug/SDS ratio. Note in particular that the release order of the components is reversed when tetracaine is in excess. Similar observations were made in a previous study of 1% Agar gels where it was found that the apparent diffusion coefficient (see definition below) for tetracaine in a 14 mM tetracaine/26 mM SDS vesicle composition was 6.04 × 10−8 cm2 /s, whereas the apparent diffusion coefficient

for tetracaine in a 26 mM tetracaine and 14 mM SDS vesicle composition was 1.89 × 10−6 cm2 /s [9]. These results were not in line with those from other systems examined, where different vesicle compositions within the same drug/surfactant system were associated with very similar diffusion coefficients among them [6,7]. It is important therefore that the regular solution theory offers a simple explanation to these observations. As can be seen in Fig. 4b, the monomer concentrations of drug and SDS vary strongly but in oppo-

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tration gradients in that region, suggest an approximate treatment similar to one used by Higuchi for solid suspensions in gels [39]. Thus we make the quasi-steady state approximation that both components have linear concentration gradients at all times and for all thicknesses x of the outer region in the gel. The release rate can then be written Di ACi∗ dni = , dt x

(18)

where Ci∗ is the monomer concentration at the boundary. The total tot to the concentration is assumed to be constant and equal to Ci,0 right of the boundary, so that the number of moles released after a given time period is: tot ni = AxCi,0 −

Fig. 9. The concentration of monomers (full symbols/solid lines) and aggregated components (open symbols/dashed lines) in a gel initially containing 14 mM diphenhydramine and 26 mM SDS calculated from theory 2500 s into a release experiment. The abscissa shows the distance from the gel/water interface for a 2 mm thick gel.

site directions as functions of ˛drug . In particular, at ˛drug = 0.35 the drug monomer concentration is slightly lower than for SDS, and at ˛drug = 0.65 the drug monomer concentration is much higher than for SDS. This explains the differences in release rate if the rate of diffusion for each component in the gel is controlled by the monomer concentration. The good quantitative agreement with experiments gives strong support to this explanation. The 26 mM tetracaine/14 mM SDS case is instructive also since the drug is so quickly released that the profile starts to level out at long times as an effect of the finite amount of drug in the gel. Importantly, the theory captures this behavior quite well. According to the phase maps in Fig. 1, dilution of a homogeneous 26 mM/14 mM tetracaine/SDS mixture is soon expected to lead to precipitation, as was in fact observed when such a mixture was used in the membrane release setup (data not shown). No phase separation was observed in the release from gels experiment, nor have they been in previous studies using this composition [9], indicating that the vesicles are fixed in the gels. Fig. 9 shows the concentration profiles for monomers and aggregated components as a function of distance from the gel surface from the theoretical calculation for 14 mM diphenhydramine/26 mM SDS. The plot shows the situation after 2500 s but is representative, on a qualitative level, for a major part of the release in Fig. 7. Two major regions can be distinguished in the gel: (1) an outer region free from micelles with apparently linear monomer concentration gradients and (2) an inner region where both the concentration of monomers and the concentration in aggregates depend on the position. For small times and/or thick gels, a third inner bulk region could also be seen, where the concentrations of both components in monomer form and in aggregates are independent of position, but this region is not apparent in Fig. 9. It can be noted first that the opposite sign of the variation of the monomer concentrations of the two components in (2) is consistent with the Gibbs–Duhem equation regulating the local equilibrium at every distance in the solution. With time the sharp border between regions (1) and (2) propagates further into the gel, but the linear concentration gradients are maintained in the outer region (not shown). As will be shown below, these observations account for the observed t1/2 dependences. 4.2.3. Small-time approximation The numerical calculations of the diffusion in the gels are both demanding and time consuming. Therefore, even approximate analytical solutions are of value. The sharp boundary of the micelle-free region seen in Fig. 9 and the apparently linear monomer concen-

AxCi∗

(19)

2

Using Eq. (19) to substitute for x in Eq. (18) gives:





tot − C ∗ /2 Di A2 Ci∗ Ci,0 dni i = , ni dt

(20)

Eq. (20) can be integrated and the result expressed as the number of moles of i released per unit area of the gel surface:

 tot  1/2   ni (t)  tot 1/2 = 2Di Ci∗ Ci,0 ≈ 2Di Ci∗ Ci,0 − Ci∗ /2 t t A

(21)

The t1/2 dependence is in agreement with initial release profiles observed experimentally and in the exact calculations. The release profiles obtained with Eq. (21) are shown in Figs. 7c and 8c (dashed lines). The boundary concentration Ci∗ was calculated from the regular solution theory using the initial concentrations of both components. Since this is the only information available in advance, this is the relevant test of the usefulness of the expression. The agreement with the exact solutions is surprisingly good, considering the fact that the monomer concentrations at the boundary are different from those in the bulk as seen in Fig. 9. (Note, however, that the agreement is poorest for the system in Fig. 9.) 4.2.4. Model sensitivity to Ddrug As discussed in Section 3.2.2, the drug diffusion coefficients used as input values in the theoretical calculations were not actually measured, but assumed to be the same as for SDS. In an attempt to evaluate its influence on the results, the diffusion coefficients of both components in all systems investigated were varied between 1 × 10−6 and 1 × 10−5 cm2 /s, and the effect on the release of each component was followed. To reduce the calculation times we investigated only the initial release process where in all cases the amount released was linear in t½ . This is the well known type of smalltime kinetics expected for a single component diffusing out of a gel, where the release is often analyzed using the relationship:



ni (t) tot Di t = 2Ci,0 A 

1/2 ,

(22)

Eq. (22) is valid as an approximation when the fractional release does not exceed 0.6. When Di is determined from a fit of Eq. (22) to data from a gel containing i in an unknown state, Di should be app interpreted as an apparent diffusion coefficient Di . Fig. 10 shows app app Ddrug and DSDS obtained from the numerical solutions with parameters relevant for 14 mM drug/26 mM SDS. For clarity we show app app only how Ddrug and DSDS are affected by variations of Ddrug with a fixed DSDS = 4.8 × 10−6 cm2 /s. The latter value is the one used for both components in the calculations presented in previous sections, taken from NMR self-diffusion in water below the CMC. There is no reason to believe that the diffusion coefficient for SDS monomers in the aqueous regions of the gels should deviate much from this app app value. As can be seen both Ddrug and DSDS increase with increasing

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Fig. 10. Dependence of the apparent diffusion coefficients for the drugs and SDS on the self diffusion coefficient for the drug. app

Ddrug . The effect on DSDS is moderate, however, again showing that the release processes are only weakly coupled. This is surprising, considering the strong interaction between the components and the large synergy effects observed at equilibrium. In a similar manapp ner, increasing DSDS gives a significant increase of DSDS but small app increase of Ddrug (not shown). app

app

In Fig. 10, Ddrug is nearly proportional to Ddrug with Ddrug /Ddrug ≈ 0.05. The proportionality is accounted for by Eq. (21) offering the following simple interpretation of this ratio: app

Di

Di

=

Ci∗ tot 2Ci,0

(23)

However, insertion of the monomer concentration calculated from regular solution theory at time zero gives a ratio of 0.1. As a check, by using instead the values of Ci∗ and Ci,tot at the border between regions (1) and (2) in Fig. 10 one obtains 0.05. 4.2.5. Cationic/non-ionic mixtures As evident from previous sections, one of the advantages with catanionic mixtures for accomplishing prolonged release is that the large aggregates formed are prohibited from diffusing in the gel by the polymer network. Alternative strategies involving mixtures of amphiphiles would be large vesicles formed by (net neutral) lipids or micelles of non-ionic surfactants. If necessary these could be anchored to the polymer network by means of hydrophobic modifications on the polymers. It is interesting to see therefore what the regular solution theory predicts regarding the possibilities of obtaining prolonged release for the types of drug molecules used in the present study. In Fig. 11 we have simulated the release from gels containing mixtures of 14 mM diphenehydramine and 26 mM non-ionic surfactant. The non-ionic components, referred to as x, y, and z, were assigned CMC’s of 1 × 10−2 M, 7 × 10−5 M and 1 × 10−9 M, respectively, corresponding roughly to surfactants (or lipids) with 8, 12, and 18 carbons in the tail group. A ˇ-parameter equal to −2 was used in all three simulations, a value typical for non-ionic/cationic mixtures, as evident from the results summarized by Holland and Rubingh [35]. All other conditions are the same as in the previous calculations. It can be mentioned that the value of ˇ is in good agreement with expectations from Poisson–Boltzmann calculations; Eq. (16). The negative sign arises because the charged drug molecule gains electrostatic free energy from interacting with the non-ionic component; the latter “dilutes” the charges on the pure drug micelles.

Fig. 11. Theoretically expected release profiles of diphenhydramine and three imagined non-ionic surfactants (denoted by x, y and z) with differing CMC’s (1 × 10−2 , 7 × 10−5 and 1 × 10−9 M, respectively).

The mixture with z is the most clear cut case since practically all of z remains in the gel during the entire release, as evident from Fig. 11. Despite this, the retention of the drug is not very significant, as can be seen when comparing with the curve for just drug in the gel, included for reference. It may be noted that the drug release quantitatively is well described by Eq. (21) for the z case. Since z, intended to model fixed vesicles of an “insoluble” lipid is expected to produce the longest release times it is not surprising that mixtures with x and y gives even faster release. The drug release profiles for mixtures with y and z are almost indistinguishable. As shown in Fig. 11, the release of diphenhydramine from a nonionic/cationic system is just barely prolonged as compared to the release of diphenhydramine alone. This is not surprising, as there is no electrostatic contribution to the interaction between diphenhydramine and surfactant. It is, however, encouraging that this is reflected also in the model calculation. Varying the lipophilicity of the non-ionic surfactant does not seem to affect the release of diphenhydramine in this scenario, whereas the release of the nonionic surfactant itself is very much affected by its CMC. 5. Conclusions In conclusion, this paper has contributed to understanding the release of both drug and SDS from gels, when using catanionic mixtures. We have shown that it is possible, by the use of regular solution theory in combination with Fick’s diffusion laws, to model the lapse of release in good conformity with measured release data. The results presented in this work support previous conclusions on the mechanism of release, i.e., that only monomers, and possibly small aggregates, diffuse through the gel, whereas large aggregates such as vesicles are unable to move. We have also, for the first time, detected the release of SDS from such aggregates, which contributes to revealing the release mechanism. However, more studies need to be made before a complete understanding of the mechanism of release is accessible, and it will be of interest to see if future studies confirm the usefulness of the regular solution theory for this purpose. Acknowledgements The excellent experimental skills of Sara Rostedt and Noel Dew are gratefully acknowledged. Göran Svensk is gratefully acknowledged for sharing his expertise on the drop-volume equipment. Financial support by the Swedish research Council is acknowledged.

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