Mathematical model of a hybrid dispersed network-membrane-based controlled release system

Journal of Controlled Release 70 (2001) 51–61www.elsevier.com/ locate / jconrel

Mathematical model of a hybrid dispersednetwork-membrane-based controlled release system

1 *Stephanie Farrell , Kamalesh K. SirkarDepartment of Chemical Engineering, Chemistry and Environmental Science, New Jersey Institute of Technology, Newark, NJ 07102,

USA

Received 18 April 2000; accepted 4 September 2000

Abstract

A mathematical model with an exact solution is presented for the controlled release of a drug from a hybrid dispersednetwork–membrane based system. Both hollow fiber and flat membrane device geometries are considered. The reservoir isloaded with a drug dispersed in a liquid phase. This reservoir is bounded by a microporous membrane, the pores of which arefilled with liquid immiscible with the reservoir phase liquid. The drug dissolves from the solid network into the reservoirliquid and migrates through the reservoir toward the microporous membrane. At the interface between the reservoir and thepore, the solute partitions between the reservoir and the pore liquid phases, before diffusing outward through the membranepore. Experimental results are in close agreement with the release profiles predicted by the mathematical model. Parametricstudies reveal the interaction between system parameters and the controlled release behavior. The presence of a disperseddrug phase in the reservoir results in the release of drug for an extended time. The release rate of the drug may be controlledby its rate of diffusion through the membrane pores or by its rate of dissolution into the reservoir liquid. 2001 ElsevierScience B.V. All rights reserved.

Keywords: Membrane-controlled release; Dispersed drug; Aqueous–organic partitioning; Diffusional release kinetics; Mathematical model

1. Introduction action, and solvent activation. Most controlled re-lease systems used today employ one or more of

Several types of polymeric controlled release these basic mechanisms for rate control. There is asystems have been developed for commercial appli- wide range of formulation of systems which usecations. These can be categorized according to the these mechanisms. This paper considers a hybridthree basic types of rate-controlling mechanisms as controlled release system that employs a disperseddescribed by Langer [1]: diffusion, chemical re- drug in the reservoir of a microporous membrane-

based device with aqueous–organic partitioning ofthe drug from the reservoir liquid to the pore liquid.

*Corresponding author. Tel.: 11-973-596-8447; fax: 11-973- Membrane-based controlled release systems have642-4854. been extensively investigated for a wide range of

E-mail address: [email protected] (K.K. Sirkar).1 therapeutic applications, from implants to oralPresent address: Department of Chemical Engineering, Rowan

formulations and transdermal patches [2,3,4]. SeveralUniversity, 331 Henry Rowan Hall, Glassboro, NJ 08028-1701,USA. models have been presented that describe the release

0168-3659/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S0168-3659( 00 )00336-9

52 S. Farrell, K.K. Sirkar / Journal of Controlled Release 70 (2001) 51 –61

of a drug from a membrane-based reservoir system The organic solvent was introduced into the pores of[5,6]. Farrell and Sirkar [7] developed a reservoir- a hollow fiber by injection using a syringe. Flattype controlled release device that exploits aqueous– membranes, cut to size, were wetted with a feworganic partitioning as a rate-controlling mechanism drops of organic solvent and blotted with a Kimwipe.of drug release, and also presented a mathematical The solution or suspension of the drug was nextmodel of this system[8]. The dispersed phase matrix introduced into the reservoir of the device. Forcontrolled release system has also been extensively hollow fibers, the drug was loaded using a syringeinvestigated. A model for the dispersed phase matrix connected to the end of the fiber. For flat membranesystem was introduced by Higuchi [9], and variations systems, a measured amount of drug in solution orhave been developed by Valeras et al. [10] and Paul suspension was introduced into the reservoir using aand McSpadden [11]. micropipette.

We have developed a mathematical model which After preparation of the hollow fiber or flatpredicts the release profile of an agent from a hybrid membrane system, the device was submerged indispersed network–membrane system. The model water for in vitro release study. The volume of thedescribes the release of small molecules such as water bath was large enough to maintain a negligiblecaffeine and benzoic acid which are initially dis- concentration relative to the saturation concentrationpersed in the reservoir. The reservoir is bounded by a throughout the experiment. Samples were withdrawnmicroporous membrane. The pores of the membrane periodically from the water bath and analyzed byare filled with a pore liquid immiscible with the HPLC.reservoir phase liquid, so that the agent partitions For experiments using nylon hollow fibers, thebetween the reservoir and the pore at the interface total length of fiber used was approximately 11 cm.between the two phases. The reservoir of the flat membrane device had a

We have investigated two geometries using both volume of 0.22 ml with a diameter of 1.27 cm. Otherflat microporous membranes and hollow fibers. The membrane dimensions and properties are provided indispersed agent may dissolve quickly into the reser- Table 1.voir solution to replace the agent that has beenreleased by diffusion, or it may dissolve very slowly;both cases have been considered. We have also 3. Model formulationconsidered reservoir-pore solvent configurations in-volving aqueous and organic phases. The mathemati- The problem treated here is an extension of thecal model is used to describe the experimentally uncoated membrane device considered by Farrell andobtained release data from this type of controlled Sirkar [7,8], but with dispersed drug having beenrelease device, and to explore the relative importance added to the reservoir liquid containing the dissolvedof different parameters on the mechanisms of rate drug. A scheme of the device with undissolved agentcontrol within the system. in the reservoir is shown in Fig. 1. This diagram

represents the device in either flat or cylindricalgeometry, with the distance variable designated x.

2. Materials and methods Region 1, (0#x#a), represents the reservoir wherethe agent is in suspension in an aqueous or organic

The procedure used to obtain the experimental carrier, or is in pure liquid form. Also within therelease profiles is outlined below. A detailed descrip- reservoir is the dispersed phase, designated region s.tion of the materials, methods, and analysis are Region 2 represents the membrane, and extends fromprovided elsewhere [7,12]. x5a to x5b. The surrounding water bath, x $ b, is

The pores of a microporous membrane are first designated region w and has volume V . As de-w

filled with the desired pore phase liquid, either water scribed previously, the agent partitions between theor an organic solvent. The wetting procedures for reservoir and the pore liquid phases at their interface,hydrophobic membranes in flat and hollow fiber and diffuses out through the membrane pore. As theforms are discussed by Bhave and Sirkar [13,14]. agent is depleted by diffusion through the membrane,

S. Farrell, K.K. Sirkar / Journal of Controlled Release 70 (2001) 51 –61 53

Table 1Membrane dimensions and properties

aMembrane Material O.D. I.D. Membrane Pore size Porosity Tortousitya a(mm) (mm) thickness (mm) (mm)

cNylon 6 Polyamide 6 1000 600 200 0.2 0.75 1.0(hollow fiber)

bCelgard 2400 (flat) Polypropylene – – 25 0.03 0.38 5.0a Supplied by the manufacturer.b Prasad and Sirkar [16].c Assumed to be 1.0 due to large pore size.

it is replaced by agent dissolving into the reservoir since the volume fraction is typically well undersolution. If dissolution is fast relative to diffusion, 10%. Equilibrium between the dispersed region andthe rate of release will be controlled primarily by the reservoir liquid exists at the interface between theaqueous–organic partitioning. If however, dissolu- two regions, and the concentrations are related bytion is slow, additional rate control will be provided. m 5 C /C u , which is equivalent to the solubility1,s 1 s eq

The physical situation that is assumed for the in the reservoir liquid relative to the solids con-reservoir is similar to that described by Varelas et al. centration in region s. Here C is the drug con-1

[10]. The dispersed agent occupies a constant vol- centration in region 1, the reservoir of the device,ume fraction f ; neglecting any change in volume and C is the drug concentration in region s, thes

fraction over time should be of little consequence dispersed phase. The solid agent is assumed tooccupy small spheres (it will become apparent thattheir geometry is unimportant), which are assumed tobe dispersed evenly throughout the reservoir (thesettling that may occur is neglected). The volume ofeach sphere is so small that diffusional resistancewithin the sphere may be neglected. The concen-tration of solid within each sphere changes with timeas the solid dissolves into the reservoir phase. Theconcentration of the agent within the reservoirsolution is a function of both distance and time.

Other simplifying assumptions that were made indeveloping the model are

• The diffusion of the agent molecules is Fickian.• There are no interfacial boundary layers.• Diffusivities of the agent in the reservoir and the

pore liquid phase are independent of concen-tration.

• The aqueous–organic partition coefficient for theagent is independent of concentration.

• The surrounding water bath is perfectly stirred.• The aqueous and organic phases are completely

immiscible.• The organic solvent and water are completely

insoluble in each other.• Reservoir and bath solutions are ideal.Fig. 1. Controlled release system with dispersed solid in the

reservoir. • Temperature is uniform.


3.1. Dimensional equations ratio of equilibrium concentrations of agent in regioni to region j)

Assuming that mass transfer across the interfaceCibetween the dispersed and dissolved phases is first ]m 5 (9)i,j Cjorder, the governing equation for the dispersed

region is The third boundary condition represents continuity ofagent flux across the interface between the reservoir≠C ks

] ]f 5 2 m C 2 C (1) and the pore. The last boundary condition is a simples d1,s s 1≠t V1 unsteady state material balance on the agent at thewhere k represents the product of the overall interfa- outer wall of the membrane; m is the equilibrium2,w

cial mass transfer coefficient at the interface between partition coefficient between the pore phase and thethe dispersed and dissolved reservoir phases and the water bath (equal to 1 if both are water and the poreinterfacial mass transfer area. does not impose any steric restrictions to solute

The governing equation for the concentration of partitioning, etc.) and a is the membrane area at the2

agent in the reservoir (region 1) is outer wall.The initial conditions indicate that all of the agent≠C ≠C1 s2]] ]1 2 f 5 1 2 f D = C 2 f (2) is initially present in the reservoir, the concentrations d s d 1 1≠t ≠t

of the dispersed phase is known, and the agentconcentration in the pore is zero:The governing equation for the agent in the

membrane (region 2) is 0C (0) 5 C (10)s s

≠C22 0]]D = C 5 (3)2 2 C x # a, 0 5 C (11)s d≠t 1 1

Here D is the effective diffusivity of the agent in the2 C a # x # b, 0 5 0 (12)s d2membrane, and is expressed as its free diffusivity,D , in the pore liquid medium times the ratio offree 3.2. Dimensionless equationsmembrane porosity, e, and tortuosity, t

The equations may be written in nondimensionalD efree]]D 5 (4)2 form by introducing the following dimensionlesst

variablesThe boundary conditions are

D t Cx 1 nC 0, t , ` (5) ] ] ]s d j 5 u 5 U 5 (n 5 1, 2, s) (13)1 2 n 0b a C 1

C a, t 5 m C a, t (6)s d s d The dimensionless governing equation for the solid1 1,2 2

region isD =C a, t 5 D =C a, t (7)s d s d1 1 2 2 V D f ≠U1 1 s

]]]5 2 k m U 2 U (14)s d2 1,s s 1≠C ≠u2 b]]V b, t 5 2 D a m =C b, t (8)s d s dw 2 2 2,w 2≠tThe governing equations for the reservoir phase and

The first boundary condition reflects the condition the pore phase, written in dimensionless form, arethat the concentration at x 5 0 (the center of the

≠U ≠Uf1 s2hollow fiber or the bottom of the reservoir) must be ]] ]]]5= U 2 (15)1≠u 1 2 f ≠ufinite. The second boundary condition representsequilibrium partitioning at the interface between the D ≠U1 22 ]]]reservoir and the pore liquid phases; m is the = U 5 (16)1,2 2 D ≠u2equilibrium partition coefficient, assumed to beindependent of concentration (m is defined as the The boundary conditions, in dimensionless form arei,j


U 0, t , ` (17) and the second term in square brackets equal to d,s d1

this equation is written very simply asa a] ]U S , uD5 m U S , uD (18) ] ]21 1,2 2b b = U 5 g 1 dU (26)1 1

a aFor the membrane pore (region 2), Eq. (16) can be] ]D =U S , uD5 D =U S , uD (19)1 1 2 2b b written in the Laplace domain

V D ≠Uw 1 2 D]]]]] ] ]1, u 5 2=U 1, u (20) 1s d s d 22 ]m D a ≠u = U 5 sU (27)2,w 2 2 2 2D2

Finally, the initial conditions are written in dimen- and transformed into the Laplace domain, the bound-sionless form ary conditions become

0C ]s0 ] U 0; s , ` (28)U 0 5 U 5 (21) s ds d 1s s 0C 1a a] ]] ]a U S ; sD5 m U S ; sD (29)1 1,2 2b b]U Sj # , 0D5 1 (22)1 b

a a] ]a ] ]D =U S ; sD5 D =U S ; sD (30)1 1 2 2b b]U S # j # 1, 0D5 0 (23)2 bV D ] ]w 1

]]]sU 1, u 5 2=U 1, u (31)s d s d3.3. Laplace transforms 2 2m D a2,w 2 2

Next the dimensionless governing equations for 3.4. Solution: flat membrane (Cartesianregions s, 1 and 2, and the boundary conditions are coordinates)transformed with respect to u into the Laplacedomain. Eq. (14) for the solids region becomes The solutions to the Laplace domain differential

equations in Cartesian coordinates for regions 1 and0C] s 2, expressed by Eqs. (26) and (27) are]bU 11 0C] 1 g] ] ]]]]U 5 (24) Œ Œs ]U 5 A exp fj d g 1 A exp f 2 j d g 2 (32)s 1 bm 1 1 21,s d

]2kb D] 1]]where b 5 has been introduced to simplify ]U 5 A exp j sF G2 3V D f D1 1 œ 2the expression. ]

D1Now Eq. (15) for the reservoir solution (region 1) ]1 A exp 2 j s (33)F G4 Dcan be written in the Laplace domain. Upon substitu- œ 2]

tion of Eq. (24) for U into the Laplace domains Expressions for the constants A , A , A and A in1 2 3 4governing equation for the reservoir solution, theEqs. (32) and (33) can be found by application of the

following expression is obtained for region 1boundary conditions in Eqs. (28)–(31). This results

0 in a system of algebraic equations that were solvedCf s ]]]s simultaneously using Mathematica (Wolfram Re-0 01 2 fC Cf] s 12 search, 1991) to obtain expressions for A , A , A]]] ]]]]= U 5 2 1 2 1 1 2 33 41 01 2 f s 1 bmC 1,s and A . These expressions are then substituted into1 4

the solutions for regions 1 and 2 in Eqs. (32) andf]]b (33) which are inverted numerically by the IMSL1 2 f ]

]]]1 s 1 1 U (25)3 1 24 (Visual Numerics, Houston, TX, USA) subroutine1s 1 bm1,sDINLAP which uses the method of De Hoog [15].

Setting the first term in square brackets equal to g The concentration of the agent in the well stirred


water bath is related to the concentration of the agent 4. Resultsat the outer wall of the membrane (j 51) by

By having a solution of the agent in the reservoirU 1, us d2]]] of the membrane-based controlled release device, theU 5 (34)w m2,w loading of the agent in the reservoir is limited by its

solubility in the reservoir solvent. If, however, the3.5. Hollow fiber (cylindrical coordinates)saturation concentration of the agent in the reservoirsolvent is exceeded, a suspension of the agent willThe solutions to the Laplace domain governingresult. Results are presented for experiments using aequations in cylindrical coordinates for regions 1 andreservoir initially containing a suspension of the2 expressed by Eqs. (26) and (27) areagent. These experiments were performed using both

1 g] ] ] hollow fibers and flat films, and for both an organicŒ Œ] ]U 5 1 A I j s 1 A K j s 2 (35)s d s d1 5 0 6 0s d reservoir and an aqueous reservoir. The agents] ] studied were benzoic acid (organic reservoir andD D] 1 1] ]U 5 A I j s 1 A K j s (36) water-filled pores) and caffeine (aqueous reservoirS D S D2 7 0 8 0D Dœ œ2 2 and organic-filled pores).

The boundary condition at j 5 0 can immediately be Fig. 2 shows a comparison between release pro-used to determine that A must be zero. The files obtained by using a suspension of benzoic acid6

remaining three boundary conditions are now written in decanol, and a solution of benzoic acid in decanolin terms of the solutions to obtain three algebraic for a hollow fiber configuration. While the releaseequations which are solved simultaneously for A , profile obtained by using the solution is approximate-5

A , A and A using Mathematica . The solutions ly zero order for a period of 30 min, this constant6 7 8

are then inverted as described previously using release is extended to about 180 min using aDINLAP, using the FORTRAN code listed in Farrell [12]. suspension. In addition, more benzoic acid is ulti-Eq. (34) can be used to determine the surrounding mately released from the suspension system, due tobath concentration as a function of time. its higher initial loading. The data indicate that the

Fig. 2. Extended release of benzoic acid using a suspension. System used: nylon porous hollow fibers with water-filled pores; benzoic acid27in decanol suspension in the reservoir (170 mg/ml). Predicted results are shown with the solid curves. k 5 2 3 10 . j, Solution; d,

suspension.


dissolution of benzoic acid is slightly rate limiting, solution is rapid relative to the depletion of benzoicsince the release rate from the suspension system is acid from the reservoir solution due to diffusion.slightly slower than the release rate from the system Benzoic acid has a very limited solubility inwith no suspension. The results are described by the mineral oil. In order to achieve a high loading in amodel by determining a best-fit value of the parame- system with a reservoir containing benzoic acid inter k. The data show a more pronounced zero-order mineral oil, a suspension of benzoic acid must betrend than is exhibited by the model. This is proba- used. Fig. 4 shows the release profile from such ably due to a nonuniform distribution of dispersed system, with a suspension of benzoic acid in mineraldrug in the hollow fiber lumen, which could occur oil in the reservoir of a flat membrane system, using

upon loading or subsequently as some settling oc- a Celgard 2400 flat membrane with water-filledcurs. Additional concentration gradients within the pores. The partition coefficient of benzoic acidreservoir region would reduce the release rate of the between mineral oil and water is 1.03, and no benefitdrug. is derived from partitioning. In this system the rate

A similar system was studied using the flat of release of benzoic acid from the device is severelymembrane device. In this experiment, a suspension dissolution limited. The release rate is constant for aof benzoic acid in octanol was used for release period of about 45 days. The model describes thethrough water-filled pores of a Celgard 2400 mem- data well.brane. Fig. 3 shows the results of this experiment, as The next system that was investigated was awell as the results of an experiment which used a caffeine suspension in an aqueous reservoir, diffus-solution of benzoic acid in the reservoir. When the ing through octanol-filled pores of a Celgard 2400suspension of benzoic acid is used in the reservoir, flat membrane. This system was investigated usingthe period of constant release is extended to about 40 both the 0.22 ml reservoir cell and the 0.95 mlh. In this case the model describes the data well. The reservoir cell. Fig. 5 shows the release profilerate of release (for the constant release portion of the obtained by using an aqueous caffeine suspension incurve) is the same for the solution and suspension the reservoir of the smaller (0.22 ml) cell. Theexperiments, and from the data it can be concluded release profile is constant for a period of 24 h. Thethat the dissolution of the solids into the reservoir model describes the data well.

Fig. 3. Extended release of benzoic acid using a suspension. System used: benzoic acid in octanol suspension (40.7 mg/ml), Celgard flat24.membrane with water-filled pores. Predicted results are shown with the solid curves, k 5 1 3 10 .


Fig. 4. Effect of rate limiting dissolution on the release profile. Benzoic acid initially contained in a mineral oil suspension (106 mg/ml), 26Celgard flat membrane with water-filled pores. Predicted results are shown by the solid curve, k 5 6 3 10 .

4.1. Dimensional analysis of each region within the system can be found bywriting the original governing differential equations

The time constants for the change in concentration in semidimensionless form as described by Farrell

Fig. 5. Extended release of caffeine using a suspension. System was caffeine in an aqueous 0.22 ml reservoir (146 mg/ml), Celgard 240025membrane with octanol-filled pores. Predicted results are shown by the solid curve, k 5 5 3 10 .


and Sirkar [8]. The time constants for dissolution, t , as z decreases, there is much more dispersed agentd

diffusion through the reservoir region, t , and and the dissolved drug molecules have to travel ar

through the membrane t , are given by longer distance around many dispersed regions;m

therefore the flux of agent from the device is lower2 2 and apparently constant for a much longer time. Allk a b 2 as d

(37)] ] ]]]t 5 ; t 5 ; t 5d r m the curves show the same initial release rate for theV D D1 2period of time for which the saturation concentration

When the ratio of t to t is large, diffusion through is maintained in the reservoir. The results were basedr d

the reservoir is rate controlling; when this ratio is on the assumption of constant f. For high drugsmall, dissolution is rate controlling. In Fig. 6 the loadings, the assumption of constant f would not beeffect of this ratio on the release profile is examined. valid; values of (1 2 f) /fm will change with time1,s

As t /t decreases, dissolution becomes more rate due to decreasing f. Observed release rates will ber d

controlling. And the flux is lower and apparently faster than that calculated from the model.constant for a longer period of time. At very smalltimes, the initial release rates are likely to be similarfor different cases of t /t and determined primarily 5. Conclusionsr d

by the rate of diffusion through the membrane. Thisis generally true for reservoir systems. Using microporous membranes as controlled re-

Next the effect of the group z 5 (1 2 f) /fm lease devices in aqueous–organic partition-based1,s

on the release profile is examined. This group is systems, the release of a drug can be extendedrelated to the ratio of the time constant for change in considerably by introducing a dispersed phase intodispersed solid concentration and the time constant the reservoir. This results in a hybrid system whichfor the change in reservoir solution concentration. As may exhibit diffusion-limited or dissolution-limitedthis group is the ratio of the amount of agent present rate control. A basic analysis of the system as well asin the reservoir solution to the amount present in the information regarding the agent release rates weresolid, the smaller its value, the greater the fraction of presented. Various solvent–agent systems were in-agent present in the solid region. As shown in Fig. 7, vestigated, different membranes were studied, and

Fig. 6. Effect of dissolution rate control on the release profile from a flat membrane device; t /t 5 1.r m


Fig. 7. Effect of z on the release profile; t /t 5 100.d r

[3] S. Borodkin, F.E. Tucker, Linear drug release from lami-the significance of various factors in achievingnated hydroxypropyl cellulose–polyvinyl acetate films, J.extended release of the agent were evaluated.Pharm Sci. 64 (1975) 1289.Drug present in the reservoir at concentrations

[4] J.E. Shaw, Pharmacokinetics of nitroglycerine and clonidineabove its solubility will exist as a dispersion. The delivered by the transdermal route, Am. Heart J. 108 (1984)presence of a dispersed drug phase in the reservoir 217.

[5] F. Theeuwes, R. Gale, R. Baker, Transference: A com-results in the release of drug for an extended time. Ifprehensive parameter concerning permeation of solutesthe dissolution is fast relative to diffusion, thethrough membranes, J. Membr. Sci. 1 (1976) 3–16.dispersed drug is immediately available to replace

[6] G.L. Flynn, S.H. Yalkowsky, T.J. Roseman, Mass transportdrug that has been released by diffusion. The con-

phenomena and models: theoretical concepts, J. Pharm Sci.centration in the reservoir phase is maintained at the 63 (1974) 479–510.solubility level until the dispersion has been de- [7] S. Farrell, K.K. Sirkar, A reservoir-type controlled release

device using aqueous–organic partitioning and a porouspleted. If, however, dissolution is slow relative tomembrane, J. Membr. Sci. 130 (1997) 265–274.diffusion, the drug in the reservoir solution is

[8] S. Farrell, K.K. Sirkar, A mathematical model of an aque-maintained at a very low concentration because it isous–organic partition-based controlled release system using

not immediately replaced by drug from the dispersed microporous membranes, J. Control. Rel. 61 (1999) 345–phase. This results in a slower rate of release for a 360.longer period of time, due to the small concentration [9] T. Higuchi, Rate of release of medicaments from ointment

bases containing drugs in suspension, J. Pharm. Sci. 50gradient of the dissolved drug within the system.(1961) 874.

[10] C.G. Valeras, D. Dixon, C. Steiner, Mathematical model ofmass transport through dispersed phase polymer networks,AIChE. J. 41 (4) (1995) 805–811.References [11] D. Paul, S.K. McSpadden, Diffusional release of a solutefrom a polymer matrix, J. Membr. Sci. 1 (1976) 33–48.

[1] R. Langer, New methods of drug delivery, Science 249 [12] S. Farrell, Ph.D. Thesis, A Controlled Release Technique(1990) 1527. Using Microporous Membranes, New Jersey Institute of

[2] Y.W. Chien, in: Novel Drug Delivery Systems: Fundamen- Technology, 1996.tals, Developmental Concepts, and Biomedical Assessments, [13] R. Bhave, K. Sirkar, Gas permeation and separation byMarcel Dekker, New York, 1982, Chapter 2. aqueous membranes immobilized across the whole thickness


or in a thin section of hydrophobic microporous Celgard [15] F. DeHoog, J. Knight, A. Stokes, An improved method forfilms, J. Memb. Sci. 27 (1986) 225. numerical inversion of Laplace transforms, Siam J. Sci. Stat.

[14] R. Bhave, K. Sirkar, Gas permeation and separation with Comput. 3 (1982) 357.aqueous membranes immobilized in microporous hydropho- [16] R. Prasad, K. Sirkar, Dispersion-free solvent extraction withbic hollow fibers, ACS Symp. Ser. 347 (1987) 138. microporous hollow-fiber modules, AIChE J. 34 (1988) 177.

Mathematical model of a hybrid dispersed network-membrane-based controlled release system

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