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A model for osmotic pressure driven release from cylindrical rubbery polymer matrices
Journal of Controlled Release 93 (2003) 249 – 258 www.elsevier.com/locate/jconrel
A model for osmotic pressure driven release from cylindrical rubbery polymer matrices Brian Amsden Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada, K7L 3N6 Received 19 May 2003; accepted 2 August 2003
Abstract Osmotic pressure-driven drug release from rubbery polymer matrices in a cylindrical geometry has been shown to produce a period of nearly constant release. In order to explain this behavior and in an effort to produce a tool for device design, a mathematical model of the release was developed. The model was tested by application to literature data of the release of NaI from poly(dimethylsiloxane) cylinders and found to provide good agreement with the data. The model demonstrates that, although there is a decrease in solute concentration as one moves from the exterior to the center of the cylinder, a period of nearly constant release is produced, lasting until about 60% of the initial drug load has been released. D 2003 Published by Elsevier B.V. Keywords: Mathematical model; Osmotic pressure; Rubbery polymer; Constant release
1. Introduction Implantable polymer devices for local drug delivery possess a number of advantages. These include reduced systemic toxicity, improved patient compliance, and improved bioavailability. A feature often considered desirable of such an implantable device is the ability to release the drug at a constant rate. One means of achieving constant release is the use of an osmotic pressure driving force [1 –12]. In this approach, the drug is dispersed as a solid particle throughout a rubbery polymer matrix. Drug release is thought to proceed in the following manner (Fig. 1) [12 –14]. Water vapor diffuses through the polymer until it encounters a polymer-surrounded
E-mail address: [email protected] (B. Amsden). 0168-3659/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/j.conrel.2003.08.007
drug particle (hereinafter referred to as a capsule). At the particle/polymer interface, the water phase separates and dissolves a portion of the particle. As a result of the water activity gradient between the capsule and the external medium, water is drawn in and the capsule swells. This swelling is resisted by the elastic nature of the polymer. As the elastomer is strained, energy is stored by polymer chain extension, bond bending or bond stretching. This energy is dissipated if bond breakage or viscoelastic flow occurs. Bond breakage initiates crack formation. If the degree of swelling is insufficient to initiate crack formation pressure, equilibrium is reached. If equilibrium is reached, then the osmotic pressure of the solution within the capsule is equal to the resisting pressure of the rubbery polymer. The crack or rupture formed connects the contents in the capsule to a pore network that ultimately extends to the surface of the
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Fig. 1. Schematic of the release mechanism. There are three zones within the drug-loaded polymer matrix: (1) a zone of capsules that have swollen to the fracture point and have created microcracks within the polymer matrix; (2) a zone of capsules that are swelling but have not ruptured; and (3) an inner core of dry particles.
device. This process occurs in a layer-by-layer manner throughout the device. This mechanism of release has been supported by the studies of Schirrer et al. [12] and Riggs et al. [14], who prepared cylindrical samples of poly(dimethylsiloxane) rubber containing NaI and NH4F, respectively, and immersed them in water. At specific times, Schirrer et al. removed the cylinders and sectioned them and examined their core structure. Riggs et al. examined the cylinders using NMR to determine the state of water within the matrix. These researchers found that three zones were present once salt release was established: an outer layer, which was transparent, where the salt had been released; an intermediate layer wherein the polymer encased salt particle regions were swollen with water; and an inner layer, which was dry and white. Furthermore, osmotic pressure driven release only dominates if the total volumetric loading of the active agent in the polymer matrix is less than the percolation threshold [7]. The percolation threshold is defined as that volume fraction of active agent at which sufficient agent particles are connected so as to form a pore spanning the thickness of the device. For most geometries, this percolation threshold value is a volume fraction of about 0.33 [15]. As a result of the layer-by-layer release mechanism, release is approximately zero order for slab geometries [1,3,6,7,11]. Slab geometries require a more invasive procedure to implant and so spherical and cylindrical geometries have also been investigat-
ed. For spherical geometries, where the number of capsules per layer decreases upon moving into the device, release is not zero order, but rather decreases with time [16]. Conversely, it has been reported that release is zero order for a portion of the release period from cylindrical geometries where the release occurs radially [12,17]. In this situation, the number of capsules per layer also decreases upon moving into the device and so a temporally decreasing release rate would be anticipated. In order to explain this period of constant release, it is the objective of this paper to develop a model, based on the above description of release, which describes the osmotically driven release period for a cylinder. Models have been developed for both slab [6,7] and spherical geometries [16], but have yet to be extended to a cylindrical geometry. The model will be tested by application to literature experimental data.
2. Model development In the following development, these simplifying assumptions are made, based on the reported observations of the system during drug release: (1) the elastomer is an isotropic, neo-Hookean material (2) the drug particles are uniform in size, homogeneously distributed and spherical
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(3) the drug capsules swell spherically in an isotropic fashion (4) capsule swelling and rupturing proceeds in a serial manner through the device, from the surface to the centre. In other words, the capsules in the crosssectional layer closest to the surface rupture before any in the next layer rupture. Capsules in one layer rupture spontaneously. (5) the mass of drug released by one layer from the cylinder ends is negligible. In other words, release occurs almost exclusively radially. The mass of agent released by osmotic pressure induced polymer rupturing, m, with time, t, can be expressed as [6], dm ML ¼ dt tb þ tp
ð1Þ
in which ML is the mass of agent released per crosssectional layer of the device, tb is the time required to rupture a capsule and tp is the time during which solution is forced from the ruptured capsule. ML is defined as, ML ¼ nb Vc C
ð2Þ
C is solute concentration in the capsule, Vc is the volume of the capsule at the point of rupture and nb is the number of ruptured capsules per layer. To determine nb, the cylinder is pictured as being composed of concentric cannulae (Fig. 2). The thickness of each cannula, x, is assumed to be constant and equal to (h¯ + 2ro), where h¯ is the average wall thickness of polymer surrounding a solid particle and ro is the average radius of an encapsulated particle. The total number of capsules that rupture per layer can be expressed as, nb ¼ /
VL Vp
ð3Þ
in which VL is the volume of a given layer, / is the volume fraction of agent within the matrix and Vp is the volume of an encapsulated solid particle. The volume of the cannula in question at time t is thereby, VL ¼ pH½ðR ði 1ÞxÞ2 ðR ixÞ2
ð4Þ
where i represents the cannula, which is in the process of being ruptured at time t, R is the cylinder radius and
Fig. 2. Cross-section of cylinder, showing the particle layers as concentric cannuli.
H is the length of the cylinder. This cannula can be defined as t/tb. Expanding Eq. (4) and substituting into Eq. (3) yields, pH t 2 2 nb ¼ / 2Rx þ x 2 x Vp tb
ð5Þ
Previously, it has been shown that tb is much greater than tp [18]. Therefore, tp can be ignored and the mass of agent released during the osmotic phase can be solved by substituting Eqs. (2) and (5) into Eq. (1) and integrating, " 2 # pH t 2 2 t mt ¼ /Vc C ð2Rx þ x Þ x Vp tb tb
ð6Þ
The release rate can also be expressed in terms of the fraction of the initially loaded agent released, mT, by recognizing that once the final layer of capsules ruptures (i.e. when t/tb = R/x) mt = mT. Thus, mT ¼ /Vc
pH C½ð2R2 þ xRÞ R2 Vp
ð7Þ
Using Eq. (7) results in the following expression for the mass fraction of drug released, " # mt 1 x t x t 2 2þ ¼ R tb R tb mT 1 þ R=x
ð8Þ
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2.1. Determination of time required to form a crack in capsule wall The time to rupture the capsule is governed by the rate at which the capsule swells, which in turn depends on the rate at which the capsule draws in water. Water imbibition into a capsule is given by [19],
ðh þ rÞ3 r3 ¼ ðh¯ þ ro Þ3 ro3
dV kw AðP pÞ ¼ dt h
S2 1 S2
ð10Þ
where n is the moles of particles formed upon dissolution of the solute, U is the osmotic coefficient of the solute that corrects for deviations from ideality, S2 is the volume fraction of the solute in the solution and V¯2 is the molar volume of the solute. It should be noted that U is a function of solute concentration. At any moment during swelling, the amount of water in the capsule is Vo(k3 1) where Vo is the initial volume of the capsule and k is the ratio of the swollen capsule radius to its original radius. Thus, S2 = 1/k3 and the osmotic pressure is then, nRT U P¼ V¯ 2
1 3 k 1
ð11Þ
Eq. (9) can be expressed in terms of the capsule radius, r, as, dr kðP pÞ ¼ dt h
ð13Þ
ð9Þ
where V is the volume of water imbibed, h is the capsule wall thickness, kw is the hydraulic permeability of the polymer, A is the capsule surface area, P is the osmotic pressure of the drug solution in the capsule and p is the resisting pressure of the polymer. For the period of swelling in which solid material remains in the capsule, P is constant and equal to the osmotic pressure of a saturated solution. Water imbibition continues beyond this point, provided the resisting pressure of the elastomer is less than that of the pressure of the solution within the capsule. The osmotic pressure of the solution in the capsule can be expressed as [13], nRT U P¼ V¯ z
As the capsule swells, the thickness of the surrounding wall will decrease. The material is an elastomer, and so it can be safely assumed that there is no volume change upon deformation. The capsule wall thickness, h, can therefore be expressed in terms of r as follows,
ð12Þ
The average distance between encapsulated particles within the cylinder can be estimated by assuming an ideal situation in which the particles were distributed equidistantly within a cube of volume equal to the volume of the cylinder. Assuming further that none of the particles are exposed at a surface, h¯ is given by, L nd h¯ ¼ nþ1
ð14Þ
n is the number of particles in any one-dimensional direction within the cube, d is the diameter of the spherical particle and L is the length of the cubic matrix. From geometric considerations, n is given by, n¼
L 13 / d
ð15Þ
Substituting for n results in, 1
dLð1 / 3 Þ h¯ ¼ 1 L/ 3 þ d
ð16Þ
Substituting Eq. (16) into Eq. (13) and solving for h gives, 1
h ¼ ðr3 þ nro3 Þ 3 r
ð17Þ
where, ¯ o þ 1Þ3 1 n ¼ ðh=r
ð18Þ
Finally, it is necessary to calculate the resisting pressure of the polymer. For low drug volume fractions, this can be expressed as [20], E 4 1 5 4 P¼ 6 k k
ð19Þ
B. Amsden / Journal of Controlled Release 93 (2003) 249–258
in which E is the tensile modulus of the polymer. Combining Eqs. (12) and (18), substituting r = kro and integrating yields, tb ¼
ro2 kw
Z 1
kc
1 E 4 1 5 4 hV P dk 6 k k
ð20Þ
in which kc is the ultimate radial extension ratio of the swollen capsule upon crack formation and hVis given by, hV¼ ðk3 þ nÞ1=3 k
ð21Þ
Eq. (19) must be integrated numerically for each condition of osmotic pressure (saturation in which P is a constant and diluted solution represented by Eq. (11)).
3. Application of the model As has been pointed out by Schirrer et al., the analysis of the rupture of a spherical hole in an infinite rubbery medium subjected to pressure demonstrates that the critical extension ratio, kc, is a function of the initial hole diameter (i.e. initial particle size), along with the tensile modulus and fracture surface energy of the polymer. Thus, application of the model to literature data would be limited to situations wherein the physical properties of the polymer were appropriately characterized. For these reasons, the model was applied to the data published by Schirrer et al. [12] and Torres [17], who examined sodium iodide salt release from poly(dimethylsiloxane) cylinders into distilled water. These authors provided release data at 20 jC and for three different salt particle average diameters. The cylinders had a diameter of 23 mm, a length of 50 mm and a NaI volumetric loading of 0.08 (20 wt.%). The extension ratio at which cracking begins in poly(dimethylsiloxane) as measured by Schirrer et al. are listed in Table 1. An examination of the data indicates that the extent of swelling of the capsule is drug particle size dependent, decreasing as particle size increases. This result is consistent with the results of other studies [1,11,18]. The release data of Schirrer et al. and Torres is given in Fig. 3. The solid lines in the figure represent
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Table 1 Particle diameter and extension ratio at rupture (kc) for poly (dimethylsiloxane) Diameter (Am)
kc
40 150 300
3.5 2.2 1.7
Data from Schirrer et al. [12].
the least-squares fits of Eq. (8) to the data. The curve fitting was performed with tb as the adjustable parameter. The curve fitting results are listed in Table 2. The model provided a good fit to the experimental data and the results indicate that the time to form a crack increases as the particle size of the particle encapsulated increases. The time to burst model was tested by using the fitted values for tb, and calculating for the water hydraulic permeability using Eq. (20). The calculated values were then compared for consistency. To do these calculations required the tensile modulus of the polymer and the osmotic pressure coefficient of NaI as a function of concentration. The tensile modulus was given by Schirrer et al. as 2.4 MPa. In order to obtain the manner in which the osmotic coefficient varied with concentration, the data of Hamer and Wu [21] was plotted as shown in Fig. 4, and the data linearly regressed (correlation coefficient, R2 = 0.986) to yield the following expression, U ¼ 4:1S2 þ 0:77
ð22Þ
Finally, a saturated solution of NaI has a solute volume fraction of 0.35, which corresponds to an extension ratio of 1.44. For swelling beyond this point, the NaI solution is no longer saturated. Thus, Eq. (20) was integrated twice: once from k = 1 –1.44 using P = 125 MPa (saturated NaI solution osmotic pressure) and again from k = 1.44 to kc using Eq. (11) for P and Eq. (22) for U with S2 substituted as 1/k3. These two integration results were then added. In performing these calculations, the integral in Eq. (20) was solved numerically using MAPLEk software. These integration results are listed in Table 2. A comparison of the calculated kw values (Table 2) shows that they are in close agreement for each particle size, being approximately 4 (10 12) cm2/ (sMPa). This value can be compared to those of Barrie and Machin [22] who examined water uptake
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Fig. 3. Data of Schirrer et al. [12] and Torres [17] for NaI of various particle average diameters releasing from poly(dimethylsiloxane) cylinders into distilled water. The volume fraction of NaI in the cylinders was 0.08.
rates into NaCl doped silicone. Barrie and Machin found that water diffusivity decreased as the silicone becomes saturated with water, decreasing as much as 1000 times as saturation is reached. From their values, the diffusivity of water in water-saturated silicone is approximately 12 –9.2 (10 10) cm2/s at 35 jC. The water permeability, kw, can be related to diffusivity, D, through [23], kw ¼ Dr
ð23Þ
in which r is the Henry’s law constant for water/ polymer at a given temperature. At 35 jC, r is approximately 0.1 MPa1 [23] and so kw is approximately 12– 9.2 (10 12) cm2/(sMPa) at 35 jC. Thus, the value obtained in this work at 20 jC is reasonable.
The model development outlines the important factors controlling osmotically driven release, and which should be considered when designing a release system. These are polymer properties such as hydrophobicity (kw) modulus (E) and tear resistance, solute osmotic activity, the solute loading, and the choice of solute particle size and cylinder geometry. Of these properties, only radius and length is unique to the case of cylinders. Increasing polymer hydrophobicity, modulus or tear resistance results in a decrease in release rate, by decreasing tb. Increasing the osmotic activity of the solute results in an increase in release rate, for a given polymer, solute size and volumetric loading. This prediction is confirmed by release data for slabs [6,7]. Data on the influence of volumetric loading for cylinders was not available, but its effect can be
Table 2 Curve fitting results from Fig. 3 Particle diameter (Am)
tb (days)
R2
n
I a (MPa 1), k = 1 – 1.44
Ia (MPa 1) k = 1.44 – kc
kw (mm2/ (sMPa) 1010
40 150 300
4.8 F 0.1 9.5 F 0.4 11.4 F 0.4
0.988 0.975 0.984
46.9 45.9 44.7
0.008 0.008 0.008
0.504 0.048 0.008
4.4 3.6 3.7
The water permeability values were calculated using Eq. (20) and the fitted values for tb. 1 Z kc E 4 1 a 5 4 I¼ hV P dk: 6 k k 1
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Fig. 4. Variation of NaI osmotic coefficient with volume fraction NaI in aqueous solution. The data are reported at 25 jC (from Hamer and Wu [21]).
predicted using the model equations. These predictions, for NaI particle diameters of 150 Am, are shown in Fig. 5. As has been demonstrated for slabs [1,6,7], an increase in volume fraction of the drug produces a faster release. The model also predicts a decrease in release rate with incorporated solute diameter, as is observed in the data of Schirrer et al., and others [6,7].
The time to burst is highly dependent on particle size, as it is proportional to the square of the particle size and increases as kc increases. As solute size decreases, kc increases and x decreases, both of which produce a decrease in release rate. The choice of cylinder dimensions influences the ratio x/R. As x/R increases, the release rate as pre-
Fig. 5. Simulated release profiles for NaI of particle diameter 150 Am at varying volume fractions.
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Table 3 Influence of cylinder dimensions on the ratio x/R R
H
L
h¯
x/R
11.5 11.5 11.5 15 5
50 40 30 50 50
27.49 25.52 23.19 15.78 32.82
0.386 0.386 0.385 0.379 0.388
0.060 0.060 0.060 0.135 0.046
dicted by Eq. (8) increases. Using the model equations, the influence of cylinder radius and length at a fixed volume fraction and particle size were calculated. The results are listed in Table 3. This analysis shows that there is no significant affect on the release rate of changing the cylinder length. However, the radius plays a dominant role. As the radius decreases, the value of x/R increases, thus leading to a faster release rate. The release does not remain approximately zeroorder throughout, but decreases monotonically as time progresses. This decrease in release rate is a result of the reduced number of drug particles within the cylinder as release proceeds from the surface to the center. As predicted through Eq. (5), the number of particles being released decreases in
a linear fashion, with most of the particles residing in the layers nearest the surface. The decrease in the number of particles per layer is expressed in the time term in Eq. (5) ( 2(t/tb)x2). For the first few layers, this change accumulates rather slowly, but as release proceeds into the cylinder, this accumulated change increases more noticeably. The first term in the release equation (Eq. (8)) represents the constant release potential of the cylindrical device. Comparing the release behavior of this portion of the overall equation to the full equation, it can be sent that, at the point at which mt/mT = 0.5 for the linear portion, the deviation from the full equation is only 12.5% (Fig. 6). Thus, for a significant portion of the release (roughly 60%), the release rate can be approximated as being linear. The assumptions of the model bear examination. Under the swelling conditions, poly(dimethylsiloxane) can reasonably be assumed to behave as a Hookean material, as the degree of extension is relatively small. The drug particles are likely not uniform in size, homogeneously distributed and spherical. However, their size distribution could be considered roughly Gaussian, and so the use of the average particle size is justified. Furthermore, as
Fig. 6. A comparison of the contribution of the linear term of the mass fraction release equation (Eq. (8)) to the overall predicted release behavior.
B. Amsden / Journal of Controlled Release 93 (2003) 249–258
257
Fig. 7. Water flux into swelling capsule as a function of radial extension ratio.
mixing of the particles can be considered a random event, the same argument holds for the reasonableness of the assumption of homogeneous particle distribution throughout the matrix. The particles are likely not spherical; however, it has been noted that the particle capsules do swell to produce roughly spherical cavities in the polymer matrix [1,7,13,14], and so this assumption is also reasonable. The serial release mechanism has been demonstrated by Schirrer et al. [12] and Riggs et al. [14], and has been discussed above. Finally, the use of Eq. (19) to predict the resisting pressure of the rubber is strictly valid only for very low drug volume fractions, as it assumes no interaction between swelling capsules. This restriction is valid provided the distance between two capsules is greater than or equal to four times the diameter of the initial particle. This is true only for drug volume fractions of about 4%. However, the success of the model would seem to indicate that this assumption is reasonable for the loadings considered here. For higher loadings, the approach taken in a previous paper should be used [16]. Schirrer et al. also derived a model of the release behavior. However, they assumed that the volumetric flux of water into the capsule during swelling was constant. The flux of water into the capsule is given by Eq. (9). Using this equation, with the appropriate defining equations for P and p, the
volume flux into the capsule was calculated as a function of k and the results are shown in Fig. 7. The volume flux is clearly not constant, but decreases as the capsule swells and becomes zero once the capsule reaches its equilibrium swelling point.
4. Conclusions A mathematical model of the release of solid drug particles from a cylindrical rubbery polymer matrix was developed. The model was tested by application to literature data of the release of NaI from poly(dimethylsiloxane) cylinders and found to provide good agreement with the data. The model demonstrates that, although there is a decrease in solute concentration as one moves from the exterior to the center of the cylinder, a period of nearly constant release is produced, lasting until about 60% of the initial drug load has been released. This model can be used in the rational design of osmotically driven drug delivery systems.
Acknowledgements Funding for this work was provided by the Canadian Institutes for Health Research.
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References [1] V. Carelli, G. Di Colo, C. Guerrini, E. Nannipieri, Drug release from silicone elastomer through controlled polymer cracking: an extension to macromolecular drugs, Int. J. Pharm. 50 (3) (1989) 181 – 188. [2] G. Di Colo, Controlled drug release from implantable matrices based on hydrophobic polymers, Biomaterials 13 (12) (1992) 850 – 856. [3] R. Gale, S.K. Chandrasekaran, D. Swanson, J. Wright, Use of osmotically active therapeutic agents in monolithic systems, J. Membr. Sci. 7 (3) (1980) 319 – 331. [4] M. Kajihara, T. Sugie, M. Mizuno, N. Tamura, A. Sano, K. Fujioka, Y. Kashiwazaki, T. Yamaoka, S. Sugawara, Y. Urabe, Development of a new drug delivery system for protein drugs using silicone (I), J. Control. Release 66 (2000) 49 – 61. [5] M. Kajihara, T. Sugie, T. Hojo, H. Maeda, A. Sano, K. Fujioka, S. Sugawara, Y. Urabe, Development of a new drug delivery system for protein drugs using silicone (II), J. Control. Release 73 (2001) 279 – 291. [6] J. Wright, S.K. Chandrasekaran, R. Gale, D. Swanson, Model for the release of osmotically active agents from monolithic polymeric matrices, AIChE Symp. Ser. 77 (206) (1981). [7] B.G. Amsden, Y.-L. Cheng, M.F.A. Goosen, A mechanistic study of the release of osmotic agents from polymeric monoliths, J. Control. Release 30 (1) (1994) 45 – 56. [8] B. Amsden, Y.-L. Cheng, A generic protein delivery system based on osmotically rupturable monoliths, J. Control. Release 33 (1995) 99 – 105. [9] A.S. Michaels, M.S. Guillod, Osmotic bursting drug delivery device, ALZA: USA patent 4,117,256 (1979). [10] J.W. McGinity, L.A. Hunke, A.B. Combs, Effect of watersoluble carriers on morphine release from a silicone polymer, J. Pharm. Sci. 68 (5) (1979) 662 – 664. [11] G. Golomb, P. Fisher, E. Rahamim, The relationship between
[12]
[13] [14]
[15]
[16]
[17]
[18]
[19] [20] [21]
[22]
[23]
drug release rate, particle size and swelling of silicone matrices, J. Control. Release 12 (2) (1990) 121 – 132. R. Schirrer, P. Thepin, G. Torres, Water absorption, swelling, rupture and salt release in salt-silicone rubber compounds, J. Mater. Sci. 27 (1992) 3424 – 3434. R.F. Fedors, Osmotic effects in water absorption by polymers, Polymer 21 (1980) 207 – 212. P.D. Riggs, P. Kinchesh, M. Braden, M.P. Patel, Nuclear magnetic imaging of an osmotic water uptake and delivery process, Biomaterials 22 (2001) 419 – 427. R. Siegel, J. Kost, R. Langer, Mechanistic studies of macromolecular drug release from macroporous polymers. I. Experiments and preliminary theory concerning completeness of drug release, J. Control. Release 8 (3) (1989) 223 – 236. B. Amsden, Osmotically activated protein release from electrostatically generated polymer microbeads, AIChE J. 42 (1996) 3253 – 3266. G. Torres, Controlled release of iodine salts from silicone rubber: a solution for iodine disease in developed countries, Proc. Int. Symp. Control. Release Bioact. Mater. 18 (1991) 403 – 404. B.G. Amsden, Y.-L. Cheng, Enhanced fraction releasable above percolation threshold from monoliths containing osmotic excipients, J. Control. Release 31 (1994) 21 – 32. F. Theeuwes, Elementary osmotic pump, J. Pharm. Sci. 64 (12) (1975) 1987 – 1991. A.E. Green, W. Zerna, Theoretical Elasticity, Oxford Univ. Press, Toronto, 1963. W.J. Hamer, Y.-C. Wu, Osmotic coefficients and mean activity coefficients of uni-valent electrolytes in water at 25 jC, J. Phys. Chem. Ref. Data 1 (4) (1972) 1047 – 1099. J.A. Barrie, D. Machin, The sorption and diffusion of water in silicone rubbers: Part II. Filled rubbers, J. Macromol. Sci. Phys. B3 (4) (1969) 673 – 692. J.A. Barrie, D. Machin, The sorption and diffusion of water in silicone rubbers: Part 1. Unfilled rubbers, J. Macromol. Sci. Phys. B 3 (4) (1969) 645 – 672.
A model for osmotic pressure driven release from cylindrical rubbery polymer matrices Brian Amsden Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada, K7L 3N6 Received 19 May 2003; accepted 2 August 2003
Abstract Osmotic pressure-driven drug release from rubbery polymer matrices in a cylindrical geometry has been shown to produce a period of nearly constant release. In order to explain this behavior and in an effort to produce a tool for device design, a mathematical model of the release was developed. The model was tested by application to literature data of the release of NaI from poly(dimethylsiloxane) cylinders and found to provide good agreement with the data. The model demonstrates that, although there is a decrease in solute concentration as one moves from the exterior to the center of the cylinder, a period of nearly constant release is produced, lasting until about 60% of the initial drug load has been released. D 2003 Published by Elsevier B.V. Keywords: Mathematical model; Osmotic pressure; Rubbery polymer; Constant release
1. Introduction Implantable polymer devices for local drug delivery possess a number of advantages. These include reduced systemic toxicity, improved patient compliance, and improved bioavailability. A feature often considered desirable of such an implantable device is the ability to release the drug at a constant rate. One means of achieving constant release is the use of an osmotic pressure driving force [1 –12]. In this approach, the drug is dispersed as a solid particle throughout a rubbery polymer matrix. Drug release is thought to proceed in the following manner (Fig. 1) [12 –14]. Water vapor diffuses through the polymer until it encounters a polymer-surrounded
E-mail address: [email protected] (B. Amsden). 0168-3659/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/j.conrel.2003.08.007
drug particle (hereinafter referred to as a capsule). At the particle/polymer interface, the water phase separates and dissolves a portion of the particle. As a result of the water activity gradient between the capsule and the external medium, water is drawn in and the capsule swells. This swelling is resisted by the elastic nature of the polymer. As the elastomer is strained, energy is stored by polymer chain extension, bond bending or bond stretching. This energy is dissipated if bond breakage or viscoelastic flow occurs. Bond breakage initiates crack formation. If the degree of swelling is insufficient to initiate crack formation pressure, equilibrium is reached. If equilibrium is reached, then the osmotic pressure of the solution within the capsule is equal to the resisting pressure of the rubbery polymer. The crack or rupture formed connects the contents in the capsule to a pore network that ultimately extends to the surface of the
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B. Amsden / Journal of Controlled Release 93 (2003) 249–258
Fig. 1. Schematic of the release mechanism. There are three zones within the drug-loaded polymer matrix: (1) a zone of capsules that have swollen to the fracture point and have created microcracks within the polymer matrix; (2) a zone of capsules that are swelling but have not ruptured; and (3) an inner core of dry particles.
device. This process occurs in a layer-by-layer manner throughout the device. This mechanism of release has been supported by the studies of Schirrer et al. [12] and Riggs et al. [14], who prepared cylindrical samples of poly(dimethylsiloxane) rubber containing NaI and NH4F, respectively, and immersed them in water. At specific times, Schirrer et al. removed the cylinders and sectioned them and examined their core structure. Riggs et al. examined the cylinders using NMR to determine the state of water within the matrix. These researchers found that three zones were present once salt release was established: an outer layer, which was transparent, where the salt had been released; an intermediate layer wherein the polymer encased salt particle regions were swollen with water; and an inner layer, which was dry and white. Furthermore, osmotic pressure driven release only dominates if the total volumetric loading of the active agent in the polymer matrix is less than the percolation threshold [7]. The percolation threshold is defined as that volume fraction of active agent at which sufficient agent particles are connected so as to form a pore spanning the thickness of the device. For most geometries, this percolation threshold value is a volume fraction of about 0.33 [15]. As a result of the layer-by-layer release mechanism, release is approximately zero order for slab geometries [1,3,6,7,11]. Slab geometries require a more invasive procedure to implant and so spherical and cylindrical geometries have also been investigat-
ed. For spherical geometries, where the number of capsules per layer decreases upon moving into the device, release is not zero order, but rather decreases with time [16]. Conversely, it has been reported that release is zero order for a portion of the release period from cylindrical geometries where the release occurs radially [12,17]. In this situation, the number of capsules per layer also decreases upon moving into the device and so a temporally decreasing release rate would be anticipated. In order to explain this period of constant release, it is the objective of this paper to develop a model, based on the above description of release, which describes the osmotically driven release period for a cylinder. Models have been developed for both slab [6,7] and spherical geometries [16], but have yet to be extended to a cylindrical geometry. The model will be tested by application to literature experimental data.
2. Model development In the following development, these simplifying assumptions are made, based on the reported observations of the system during drug release: (1) the elastomer is an isotropic, neo-Hookean material (2) the drug particles are uniform in size, homogeneously distributed and spherical
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(3) the drug capsules swell spherically in an isotropic fashion (4) capsule swelling and rupturing proceeds in a serial manner through the device, from the surface to the centre. In other words, the capsules in the crosssectional layer closest to the surface rupture before any in the next layer rupture. Capsules in one layer rupture spontaneously. (5) the mass of drug released by one layer from the cylinder ends is negligible. In other words, release occurs almost exclusively radially. The mass of agent released by osmotic pressure induced polymer rupturing, m, with time, t, can be expressed as [6], dm ML ¼ dt tb þ tp
ð1Þ
in which ML is the mass of agent released per crosssectional layer of the device, tb is the time required to rupture a capsule and tp is the time during which solution is forced from the ruptured capsule. ML is defined as, ML ¼ nb Vc C
ð2Þ
C is solute concentration in the capsule, Vc is the volume of the capsule at the point of rupture and nb is the number of ruptured capsules per layer. To determine nb, the cylinder is pictured as being composed of concentric cannulae (Fig. 2). The thickness of each cannula, x, is assumed to be constant and equal to (h¯ + 2ro), where h¯ is the average wall thickness of polymer surrounding a solid particle and ro is the average radius of an encapsulated particle. The total number of capsules that rupture per layer can be expressed as, nb ¼ /
VL Vp
ð3Þ
in which VL is the volume of a given layer, / is the volume fraction of agent within the matrix and Vp is the volume of an encapsulated solid particle. The volume of the cannula in question at time t is thereby, VL ¼ pH½ðR ði 1ÞxÞ2 ðR ixÞ2
ð4Þ
where i represents the cannula, which is in the process of being ruptured at time t, R is the cylinder radius and
Fig. 2. Cross-section of cylinder, showing the particle layers as concentric cannuli.
H is the length of the cylinder. This cannula can be defined as t/tb. Expanding Eq. (4) and substituting into Eq. (3) yields, pH t 2 2 nb ¼ / 2Rx þ x 2 x Vp tb
ð5Þ
Previously, it has been shown that tb is much greater than tp [18]. Therefore, tp can be ignored and the mass of agent released during the osmotic phase can be solved by substituting Eqs. (2) and (5) into Eq. (1) and integrating, " 2 # pH t 2 2 t mt ¼ /Vc C ð2Rx þ x Þ x Vp tb tb
ð6Þ
The release rate can also be expressed in terms of the fraction of the initially loaded agent released, mT, by recognizing that once the final layer of capsules ruptures (i.e. when t/tb = R/x) mt = mT. Thus, mT ¼ /Vc
pH C½ð2R2 þ xRÞ R2 Vp
ð7Þ
Using Eq. (7) results in the following expression for the mass fraction of drug released, " # mt 1 x t x t 2 2þ ¼ R tb R tb mT 1 þ R=x
ð8Þ
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2.1. Determination of time required to form a crack in capsule wall The time to rupture the capsule is governed by the rate at which the capsule swells, which in turn depends on the rate at which the capsule draws in water. Water imbibition into a capsule is given by [19],
ðh þ rÞ3 r3 ¼ ðh¯ þ ro Þ3 ro3
dV kw AðP pÞ ¼ dt h
S2 1 S2
ð10Þ
where n is the moles of particles formed upon dissolution of the solute, U is the osmotic coefficient of the solute that corrects for deviations from ideality, S2 is the volume fraction of the solute in the solution and V¯2 is the molar volume of the solute. It should be noted that U is a function of solute concentration. At any moment during swelling, the amount of water in the capsule is Vo(k3 1) where Vo is the initial volume of the capsule and k is the ratio of the swollen capsule radius to its original radius. Thus, S2 = 1/k3 and the osmotic pressure is then, nRT U P¼ V¯ 2
1 3 k 1
ð11Þ
Eq. (9) can be expressed in terms of the capsule radius, r, as, dr kðP pÞ ¼ dt h
ð13Þ
ð9Þ
where V is the volume of water imbibed, h is the capsule wall thickness, kw is the hydraulic permeability of the polymer, A is the capsule surface area, P is the osmotic pressure of the drug solution in the capsule and p is the resisting pressure of the polymer. For the period of swelling in which solid material remains in the capsule, P is constant and equal to the osmotic pressure of a saturated solution. Water imbibition continues beyond this point, provided the resisting pressure of the elastomer is less than that of the pressure of the solution within the capsule. The osmotic pressure of the solution in the capsule can be expressed as [13], nRT U P¼ V¯ z
As the capsule swells, the thickness of the surrounding wall will decrease. The material is an elastomer, and so it can be safely assumed that there is no volume change upon deformation. The capsule wall thickness, h, can therefore be expressed in terms of r as follows,
ð12Þ
The average distance between encapsulated particles within the cylinder can be estimated by assuming an ideal situation in which the particles were distributed equidistantly within a cube of volume equal to the volume of the cylinder. Assuming further that none of the particles are exposed at a surface, h¯ is given by, L nd h¯ ¼ nþ1
ð14Þ
n is the number of particles in any one-dimensional direction within the cube, d is the diameter of the spherical particle and L is the length of the cubic matrix. From geometric considerations, n is given by, n¼
L 13 / d
ð15Þ
Substituting for n results in, 1
dLð1 / 3 Þ h¯ ¼ 1 L/ 3 þ d
ð16Þ
Substituting Eq. (16) into Eq. (13) and solving for h gives, 1
h ¼ ðr3 þ nro3 Þ 3 r
ð17Þ
where, ¯ o þ 1Þ3 1 n ¼ ðh=r
ð18Þ
Finally, it is necessary to calculate the resisting pressure of the polymer. For low drug volume fractions, this can be expressed as [20], E 4 1 5 4 P¼ 6 k k
ð19Þ
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in which E is the tensile modulus of the polymer. Combining Eqs. (12) and (18), substituting r = kro and integrating yields, tb ¼
ro2 kw
Z 1
kc
1 E 4 1 5 4 hV P dk 6 k k
ð20Þ
in which kc is the ultimate radial extension ratio of the swollen capsule upon crack formation and hVis given by, hV¼ ðk3 þ nÞ1=3 k
ð21Þ
Eq. (19) must be integrated numerically for each condition of osmotic pressure (saturation in which P is a constant and diluted solution represented by Eq. (11)).
3. Application of the model As has been pointed out by Schirrer et al., the analysis of the rupture of a spherical hole in an infinite rubbery medium subjected to pressure demonstrates that the critical extension ratio, kc, is a function of the initial hole diameter (i.e. initial particle size), along with the tensile modulus and fracture surface energy of the polymer. Thus, application of the model to literature data would be limited to situations wherein the physical properties of the polymer were appropriately characterized. For these reasons, the model was applied to the data published by Schirrer et al. [12] and Torres [17], who examined sodium iodide salt release from poly(dimethylsiloxane) cylinders into distilled water. These authors provided release data at 20 jC and for three different salt particle average diameters. The cylinders had a diameter of 23 mm, a length of 50 mm and a NaI volumetric loading of 0.08 (20 wt.%). The extension ratio at which cracking begins in poly(dimethylsiloxane) as measured by Schirrer et al. are listed in Table 1. An examination of the data indicates that the extent of swelling of the capsule is drug particle size dependent, decreasing as particle size increases. This result is consistent with the results of other studies [1,11,18]. The release data of Schirrer et al. and Torres is given in Fig. 3. The solid lines in the figure represent
253
Table 1 Particle diameter and extension ratio at rupture (kc) for poly (dimethylsiloxane) Diameter (Am)
kc
40 150 300
3.5 2.2 1.7
Data from Schirrer et al. [12].
the least-squares fits of Eq. (8) to the data. The curve fitting was performed with tb as the adjustable parameter. The curve fitting results are listed in Table 2. The model provided a good fit to the experimental data and the results indicate that the time to form a crack increases as the particle size of the particle encapsulated increases. The time to burst model was tested by using the fitted values for tb, and calculating for the water hydraulic permeability using Eq. (20). The calculated values were then compared for consistency. To do these calculations required the tensile modulus of the polymer and the osmotic pressure coefficient of NaI as a function of concentration. The tensile modulus was given by Schirrer et al. as 2.4 MPa. In order to obtain the manner in which the osmotic coefficient varied with concentration, the data of Hamer and Wu [21] was plotted as shown in Fig. 4, and the data linearly regressed (correlation coefficient, R2 = 0.986) to yield the following expression, U ¼ 4:1S2 þ 0:77
ð22Þ
Finally, a saturated solution of NaI has a solute volume fraction of 0.35, which corresponds to an extension ratio of 1.44. For swelling beyond this point, the NaI solution is no longer saturated. Thus, Eq. (20) was integrated twice: once from k = 1 –1.44 using P = 125 MPa (saturated NaI solution osmotic pressure) and again from k = 1.44 to kc using Eq. (11) for P and Eq. (22) for U with S2 substituted as 1/k3. These two integration results were then added. In performing these calculations, the integral in Eq. (20) was solved numerically using MAPLEk software. These integration results are listed in Table 2. A comparison of the calculated kw values (Table 2) shows that they are in close agreement for each particle size, being approximately 4 (10 12) cm2/ (sMPa). This value can be compared to those of Barrie and Machin [22] who examined water uptake
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Fig. 3. Data of Schirrer et al. [12] and Torres [17] for NaI of various particle average diameters releasing from poly(dimethylsiloxane) cylinders into distilled water. The volume fraction of NaI in the cylinders was 0.08.
rates into NaCl doped silicone. Barrie and Machin found that water diffusivity decreased as the silicone becomes saturated with water, decreasing as much as 1000 times as saturation is reached. From their values, the diffusivity of water in water-saturated silicone is approximately 12 –9.2 (10 10) cm2/s at 35 jC. The water permeability, kw, can be related to diffusivity, D, through [23], kw ¼ Dr
ð23Þ
in which r is the Henry’s law constant for water/ polymer at a given temperature. At 35 jC, r is approximately 0.1 MPa1 [23] and so kw is approximately 12– 9.2 (10 12) cm2/(sMPa) at 35 jC. Thus, the value obtained in this work at 20 jC is reasonable.
The model development outlines the important factors controlling osmotically driven release, and which should be considered when designing a release system. These are polymer properties such as hydrophobicity (kw) modulus (E) and tear resistance, solute osmotic activity, the solute loading, and the choice of solute particle size and cylinder geometry. Of these properties, only radius and length is unique to the case of cylinders. Increasing polymer hydrophobicity, modulus or tear resistance results in a decrease in release rate, by decreasing tb. Increasing the osmotic activity of the solute results in an increase in release rate, for a given polymer, solute size and volumetric loading. This prediction is confirmed by release data for slabs [6,7]. Data on the influence of volumetric loading for cylinders was not available, but its effect can be
Table 2 Curve fitting results from Fig. 3 Particle diameter (Am)
tb (days)
R2
n
I a (MPa 1), k = 1 – 1.44
Ia (MPa 1) k = 1.44 – kc
kw (mm2/ (sMPa) 1010
40 150 300
4.8 F 0.1 9.5 F 0.4 11.4 F 0.4
0.988 0.975 0.984
46.9 45.9 44.7
0.008 0.008 0.008
0.504 0.048 0.008
4.4 3.6 3.7
The water permeability values were calculated using Eq. (20) and the fitted values for tb. 1 Z kc E 4 1 a 5 4 I¼ hV P dk: 6 k k 1
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Fig. 4. Variation of NaI osmotic coefficient with volume fraction NaI in aqueous solution. The data are reported at 25 jC (from Hamer and Wu [21]).
predicted using the model equations. These predictions, for NaI particle diameters of 150 Am, are shown in Fig. 5. As has been demonstrated for slabs [1,6,7], an increase in volume fraction of the drug produces a faster release. The model also predicts a decrease in release rate with incorporated solute diameter, as is observed in the data of Schirrer et al., and others [6,7].
The time to burst is highly dependent on particle size, as it is proportional to the square of the particle size and increases as kc increases. As solute size decreases, kc increases and x decreases, both of which produce a decrease in release rate. The choice of cylinder dimensions influences the ratio x/R. As x/R increases, the release rate as pre-
Fig. 5. Simulated release profiles for NaI of particle diameter 150 Am at varying volume fractions.
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Table 3 Influence of cylinder dimensions on the ratio x/R R
H
L
h¯
x/R
11.5 11.5 11.5 15 5
50 40 30 50 50
27.49 25.52 23.19 15.78 32.82
0.386 0.386 0.385 0.379 0.388
0.060 0.060 0.060 0.135 0.046
dicted by Eq. (8) increases. Using the model equations, the influence of cylinder radius and length at a fixed volume fraction and particle size were calculated. The results are listed in Table 3. This analysis shows that there is no significant affect on the release rate of changing the cylinder length. However, the radius plays a dominant role. As the radius decreases, the value of x/R increases, thus leading to a faster release rate. The release does not remain approximately zeroorder throughout, but decreases monotonically as time progresses. This decrease in release rate is a result of the reduced number of drug particles within the cylinder as release proceeds from the surface to the center. As predicted through Eq. (5), the number of particles being released decreases in
a linear fashion, with most of the particles residing in the layers nearest the surface. The decrease in the number of particles per layer is expressed in the time term in Eq. (5) ( 2(t/tb)x2). For the first few layers, this change accumulates rather slowly, but as release proceeds into the cylinder, this accumulated change increases more noticeably. The first term in the release equation (Eq. (8)) represents the constant release potential of the cylindrical device. Comparing the release behavior of this portion of the overall equation to the full equation, it can be sent that, at the point at which mt/mT = 0.5 for the linear portion, the deviation from the full equation is only 12.5% (Fig. 6). Thus, for a significant portion of the release (roughly 60%), the release rate can be approximated as being linear. The assumptions of the model bear examination. Under the swelling conditions, poly(dimethylsiloxane) can reasonably be assumed to behave as a Hookean material, as the degree of extension is relatively small. The drug particles are likely not uniform in size, homogeneously distributed and spherical. However, their size distribution could be considered roughly Gaussian, and so the use of the average particle size is justified. Furthermore, as
Fig. 6. A comparison of the contribution of the linear term of the mass fraction release equation (Eq. (8)) to the overall predicted release behavior.
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Fig. 7. Water flux into swelling capsule as a function of radial extension ratio.
mixing of the particles can be considered a random event, the same argument holds for the reasonableness of the assumption of homogeneous particle distribution throughout the matrix. The particles are likely not spherical; however, it has been noted that the particle capsules do swell to produce roughly spherical cavities in the polymer matrix [1,7,13,14], and so this assumption is also reasonable. The serial release mechanism has been demonstrated by Schirrer et al. [12] and Riggs et al. [14], and has been discussed above. Finally, the use of Eq. (19) to predict the resisting pressure of the rubber is strictly valid only for very low drug volume fractions, as it assumes no interaction between swelling capsules. This restriction is valid provided the distance between two capsules is greater than or equal to four times the diameter of the initial particle. This is true only for drug volume fractions of about 4%. However, the success of the model would seem to indicate that this assumption is reasonable for the loadings considered here. For higher loadings, the approach taken in a previous paper should be used [16]. Schirrer et al. also derived a model of the release behavior. However, they assumed that the volumetric flux of water into the capsule during swelling was constant. The flux of water into the capsule is given by Eq. (9). Using this equation, with the appropriate defining equations for P and p, the
volume flux into the capsule was calculated as a function of k and the results are shown in Fig. 7. The volume flux is clearly not constant, but decreases as the capsule swells and becomes zero once the capsule reaches its equilibrium swelling point.
4. Conclusions A mathematical model of the release of solid drug particles from a cylindrical rubbery polymer matrix was developed. The model was tested by application to literature data of the release of NaI from poly(dimethylsiloxane) cylinders and found to provide good agreement with the data. The model demonstrates that, although there is a decrease in solute concentration as one moves from the exterior to the center of the cylinder, a period of nearly constant release is produced, lasting until about 60% of the initial drug load has been released. This model can be used in the rational design of osmotically driven drug delivery systems.
Acknowledgements Funding for this work was provided by the Canadian Institutes for Health Research.
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References [1] V. Carelli, G. Di Colo, C. Guerrini, E. Nannipieri, Drug release from silicone elastomer through controlled polymer cracking: an extension to macromolecular drugs, Int. J. Pharm. 50 (3) (1989) 181 – 188. [2] G. Di Colo, Controlled drug release from implantable matrices based on hydrophobic polymers, Biomaterials 13 (12) (1992) 850 – 856. [3] R. Gale, S.K. Chandrasekaran, D. Swanson, J. Wright, Use of osmotically active therapeutic agents in monolithic systems, J. Membr. Sci. 7 (3) (1980) 319 – 331. [4] M. Kajihara, T. Sugie, M. Mizuno, N. Tamura, A. Sano, K. Fujioka, Y. Kashiwazaki, T. Yamaoka, S. Sugawara, Y. Urabe, Development of a new drug delivery system for protein drugs using silicone (I), J. Control. Release 66 (2000) 49 – 61. [5] M. Kajihara, T. Sugie, T. Hojo, H. Maeda, A. Sano, K. Fujioka, S. Sugawara, Y. Urabe, Development of a new drug delivery system for protein drugs using silicone (II), J. Control. Release 73 (2001) 279 – 291. [6] J. Wright, S.K. Chandrasekaran, R. Gale, D. Swanson, Model for the release of osmotically active agents from monolithic polymeric matrices, AIChE Symp. Ser. 77 (206) (1981). [7] B.G. Amsden, Y.-L. Cheng, M.F.A. Goosen, A mechanistic study of the release of osmotic agents from polymeric monoliths, J. Control. Release 30 (1) (1994) 45 – 56. [8] B. Amsden, Y.-L. Cheng, A generic protein delivery system based on osmotically rupturable monoliths, J. Control. Release 33 (1995) 99 – 105. [9] A.S. Michaels, M.S. Guillod, Osmotic bursting drug delivery device, ALZA: USA patent 4,117,256 (1979). [10] J.W. McGinity, L.A. Hunke, A.B. Combs, Effect of watersoluble carriers on morphine release from a silicone polymer, J. Pharm. Sci. 68 (5) (1979) 662 – 664. [11] G. Golomb, P. Fisher, E. Rahamim, The relationship between
[12]
[13] [14]
[15]
[16]
[17]
[18]
[19] [20] [21]
[22]
[23]
drug release rate, particle size and swelling of silicone matrices, J. Control. Release 12 (2) (1990) 121 – 132. R. Schirrer, P. Thepin, G. Torres, Water absorption, swelling, rupture and salt release in salt-silicone rubber compounds, J. Mater. Sci. 27 (1992) 3424 – 3434. R.F. Fedors, Osmotic effects in water absorption by polymers, Polymer 21 (1980) 207 – 212. P.D. Riggs, P. Kinchesh, M. Braden, M.P. Patel, Nuclear magnetic imaging of an osmotic water uptake and delivery process, Biomaterials 22 (2001) 419 – 427. R. Siegel, J. Kost, R. Langer, Mechanistic studies of macromolecular drug release from macroporous polymers. I. Experiments and preliminary theory concerning completeness of drug release, J. Control. Release 8 (3) (1989) 223 – 236. B. Amsden, Osmotically activated protein release from electrostatically generated polymer microbeads, AIChE J. 42 (1996) 3253 – 3266. G. Torres, Controlled release of iodine salts from silicone rubber: a solution for iodine disease in developed countries, Proc. Int. Symp. Control. Release Bioact. Mater. 18 (1991) 403 – 404. B.G. Amsden, Y.-L. Cheng, Enhanced fraction releasable above percolation threshold from monoliths containing osmotic excipients, J. Control. Release 31 (1994) 21 – 32. F. Theeuwes, Elementary osmotic pump, J. Pharm. Sci. 64 (12) (1975) 1987 – 1991. A.E. Green, W. Zerna, Theoretical Elasticity, Oxford Univ. Press, Toronto, 1963. W.J. Hamer, Y.-C. Wu, Osmotic coefficients and mean activity coefficients of uni-valent electrolytes in water at 25 jC, J. Phys. Chem. Ref. Data 1 (4) (1972) 1047 – 1099. J.A. Barrie, D. Machin, The sorption and diffusion of water in silicone rubbers: Part II. Filled rubbers, J. Macromol. Sci. Phys. B3 (4) (1969) 673 – 692. J.A. Barrie, D. Machin, The sorption and diffusion of water in silicone rubbers: Part 1. Unfilled rubbers, J. Macromol. Sci. Phys. B 3 (4) (1969) 645 – 672.