mathematical models describing polymer dissolution: consequences for drug delivery

Advanced Drug Delivery Reviews 48 (2001) 195–210www.elsevier.com/ locate /drugdeliv

Mathematical models describing polymer dissolution:consequences for drug delivery

*Balaji Narasimhan

Department of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road Piscataway, NJ 08854-8058, USA

Received 13 October 2000; received in revised form 21 December 2000; accepted 22 December 2000

Abstract

Polymer dissolution is an important phenomenon in polymer science and engineering that has found applications in areaslike microlithography, controlled drug delivery, and plastics recycling. This review focuses on the modeling efforts tounderstand the physics of the drug release process from dissolving polymers. A brief review of the experimentally observeddissolution behavior is presented, thus motivating the modeling of the mechanism of dissolution. The main modelingcontributions have been classified into two broad approaches — phenomenological models and Fickian equations, andanomalous transport models and scaling law-based approaches. The underlying principles and the important features of eachapproach are discussed. Details of the important models and their corresponding predictions are provided. Experimentalresults seem to be qualitatively consistent with the present picture. 2001 Elsevier Science B.V. All rights reserved.

Keywords: Modeling; Dissolution-controlled systems; Drug release; Phenomenology; Molecular models

Contents

1. Introduction ............................................................................................................................................................................ 1952. Mechanisms and models .......................................................................................................................................................... 197

2.1. Amorphous polymers ....................................................................................................................................................... 1972.2. Semicrystalline polymers .................................................................................................................................................. 198

3. Current models for dissolution-controlled drug delivery systems................................................................................................. 1994. Phenomenological models........................................................................................................................................................ 2005. Molecular models.................................................................................................................................................................... 2016. Conclusions and future outlook ................................................................................................................................................ 207References .................................................................................................................................................................................. 208

1. Introduction

Dissolution of polymers in solvents is an im-portant phenomenon in polymer science and en-gineering. It has found significant applications in a*Tel.: 11-732-445-0315; fax: 11-732-445-2581.

E-mail address: [email protected] (B. Narasimhan). variety of areas. For example, in microlithography, a

0169-409X/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S0169-409X( 01 )00117-X

196 B. Narasimhan / Advanced Drug Delivery Reviews 48 (2001) 195 –210

process used in fabricating integrated circuits, a A single solvent such as xylene is used to dissolvephotosensitive polymer, called a photoresist, is five polymers — poly(vinylchloride), polystyrene,coated onto a substrate to form a thin film [1–5]. low density polyethylene, high density polyethyleneIrradiation through a glass plate or ‘mask’ bearing an and polypropylene — at five different temperatures,array of circuit patterns allows selected areas of the ranging from room temperature to 1388C. The re-photoresist to be exposed. The modified or exposed maining poly(ethyleneterephthalate) can be dissolvedregions of the polymer exhibit an altered rate of using a separate solvent. In this process, xylene isdissolution in certain solvents, resulting in the forma- placed in a chip-filled vat first at room temperature.tion of a polymeric image of the mask pattern. Polystyrene dissolves while the other five do not.

In controlled release applications of polymers, a The xylene solution is then drained to a separate partsolute is molecularly dispersed in a polymer phase, of the system where it is heated under pressure towhich is usually in the glassy state. In the presence about 2508C. The solution is then sent to a vacuumof a thermodynamically compatible solvent, swelling chamber to undergo flash devolatilization, causingoccurs and the polymer begins to release its contents xylene to vaporize instantaneously, leaving behindto the surrounding fluid. This release process can be pure polystyrene. The same xylene is then sent backcontrolled either by solute diffusion or by polymer to dissolve another polymer at a different tempera-dissolution. In this case, the presence of permanent ture and the process continues with the other poly-entanglements in the polymer becomes significant. mers.These systems can give constant release rates under The dissolution of novolak resins in solvents is anspecial conditions [6–12]. important process in many semiconductor applica-

Polymer dissolution finds applications in mem- tions [20–22]. Hence, it becomes necessary tobrane science. In phase inversion, a technique to understand the dissolution characteristics ofform asymmetric membranes, a thin film of polymer novolaks. Novolak resins, as the world’s oldestsolution is cast onto a suitable substrate followed by synthetic materials, play an important role in today’simmersion in a coagulation bath (quench step) [13– semiconductor industry. Their advantages are in their15]. During the quench period, solvent /nonsolvent non-swelling nature, aqueous –base developabilityexchange and eventual polymer precipitation occur. and etching resistance [23,24].The extent of dissolution of the polymer is in- Some of the potential applications of the dissolu-strumental in determining the ultimate structure of tion of polymers are in scaffolding for tissue regene-the membrane. Many crystallizable polymer films ration [25]. Polymers are shaped into scaffolds thatsuch as polycarbonates can be made porous by resemble the structure of tissues or organs. They areexposing a uniform film to a beam of alpha particles. treated with compounds that help cells adhere ontoThe crystalline structure is disrupted and the film their surface and multiply. They are then ‘seeded’then chemically treated with an etchant such as with the cells that grow and multiply as the polymercaustic potash. The pores produced are nearly cylin- gradually dissolves. The new permanent tissue or thedrical and of uniform radius. Such membranes are organ is implanted in the patient. So far, onlyused for microfiltration. Another technique is to cast biodegradable polymers have been used for suchfilms from pairs of compatible, non-complexing purposes, thereby considerably narrowing down thepolymers. Treatment of such films with a solvent that number of polymers that can be used. Moreover,dissolves only one of the polymers leaves behind a most degradable polymers are too weak to be used inpolymer with interconnected microvoids. This acts as load bearing implants [26]. By using semicrystallinea microfiltration membrane. Knowledge of the disso- polymers, which combine strength with ease oflution behavior of the polymers enables us to select control of the degree of crystallinity, it is possible tosuitable solvents. The membranes produced by the obtain desired dissolution rates that correspond to theabove methods are of uniform texture and are not growth rates of different tissues.skinned or asymmetric [16]. In addition to the above, polymer dissolution rate

Polymer dissolution has also been applied in the data have been used to determine glass transitiontreatment of unsorted plastics for recycling [17–19]. temperature and other thermodynamic parameters

B. Narasimhan / Advanced Drug Delivery Reviews 48 (2001) 195 –210 197

associated with polymorphic changes [27]. Dissolu- In general, polymer dissolution differs from disso-tion has also found a variety of uses in the pharma- lution of a non-polymeric material in two aspects.ceutical sciences. In the development of microcap- Polymers require an induction time before starting tosules for sustained release dosage forms [28], the dissolve, while non-polymeric materials dissolvemechanism of drug transport is governed by the instantaneously. Also, polymer dissolution can bedissolution of the polymer. Cooney [29] studied the controlled either by the disentanglement of thedissolution of pharmaceutical tablets in the design of polymer chains or by the diffusion of the chainssustained release forms. Ozturk and co-workers through a boundary layer adjacent to the solvent–[30,31] showed that the dissolution of the polyacid, polymer interface. However, the dissolution of non-which is used in enteric-coated tablets, was the polymeric materials is generally controlled by thecontrolling step in the release kinetics mechanism. external mass transfer resistance through a liquid

The dissolution of a polymer into a solvent layer adjacent to the solid–liquid interface.involves two transport processes, namely solvent This review focuses on understanding the mathe-diffusion and chain disentanglement. When an un- matical modeling of the mechanism of polymercrosslinked, amorphous, glassy polymer is in contact dissolution in drug delivery applications. An appro-with a thermodynamically compatible solvent, the priate understanding of the various controlling stepslatter diffuses into the polymer. A gel-like layer is in the dissolution process greatly enhances theformed adjacent to the solvent–polymer interface tailoring of the polymer to achieve not only desireddue to plasticization of the polymer by the solvent. rates of drug release, but also the desired profiles.After an induction time, the polymer is dissolved.However, there are also cases where a polymercrazes when placed in a solvent. When semicrystal- 2. Mechanisms and modelsline polymers are exposed to thermodynamicallycompatible solvents, the unfolding of the crystalline This section details the current understanding inregions is an additional step accompanying solvent the field with respect to mechanisms and models fordiffusion and chain disentanglement. Hence, dissolution of amorphous as well as semicrystallinesemicrystalline polymer dissolution becomes equiva- polymers. The various experimental techniques usedlent to amorphous polymer dissolution after the to characterize dissolution behavior are summarizedcrystals in the polymer unfold. A schematic of in Table 1.polymer dissolution is shown in Fig. 1.

2.1. Amorphous polymers

The main features of the dissolution mechanism ofan amorphous polymer are depicted as shown in Fig.2. During the initial stage of the dissolution process,a glassy polymer of thickness 2l starts swelling dueto the penetration of the solvent into it and thesimultaneous transition from the glassy to the rub-bery state. Thus two distinct fronts are observed — aswelling interface at position R and a polymer–solvent interface at position S. Front R movesinwards while front S moves outwards. When theconcentration of the penetrant in the polymer ex-ceeds a critical value, macromolecular disentangle-ment begins. This is when the true dissolutioncommences. After an induction time, the polymer isdissolved. During this time, front R continues toFig. 1. Schematic of one-dimensional solvent diffusion and

polymer dissolution (reproduced with permission from [46]). move towards the center of the slab, while front S


Table 1Experimental techniques to characterize polymer dissolution behavior

Technique Reference

Laser interferometry Papanu et al. [32]; Rodriguez et al. [33]Differential refractometry Ueberreiter [34]Optical microscopy Ouano and Carothers [35]Gravimetry Mallapragada and Peppas [36];

Rao et al. [37,38]; Blackadder and Le Poidevin [39]Ellipsometry Papanu et al. [40]Magnetic resonance imaging Narasimhan et al. [41]Spin echo NMR Peppas et al. [42]Front measurements Harland et al. [43]; Bettini et al. [45]

Narasimhan and Peppas [44]

to model the dissolution mechanism and to explainthe experimentally observed dissolution behavior. Weshall now describe the various modeling approachesto study polymer dissolution.

The approaches to model dissolution of amor-phous polymers can be broadly classified [46,47]into

1. use of phenomenology and models with Fickianequations [47–52]

2. models with external mass transfer as the control-ling resistance to dissolution [53,54]

3. models based on stress relaxation [55,56]4. analysis using anomalous transport models for

solvent transport and scaling laws for chaindisentanglement [42,57–59].

The various approaches to model dissolution ofamorphous polymers have accounted for phenomenalike stress relaxation, viscoelasticity of the polymer,disentanglement of polymer chains, anomalous trans-Fig. 2. Schematic representation of a one-dimensional solventport of solvent, chain reptation and external massdiffusion and polymer dissolution process. (a) Initial slab oftransfer limitations.thickness 2l; (b) initial swelling step showing the increasing

position of the rubbery–solvent interface (S) and the decreasingposition of the glassy–rubbery interface (R); (c) onset of the 2.2. Semicrystalline polymersdissolution step showing the decreasing position of the interface Salong with the decreasing position of the interface R; and (d) final

Although there has been considerable experimen-dissolution step where the slab has been transformed into atal work on the dissolution of semicrystalline poly-rubbery material (disappearance of interface R) and the position of

interface S still decreases (reproduced with permission from [59]). mers, few efforts have been made to predict theirdissolution kinetics. One approach assumed that

moves inwards as well. After the disappearance of semicrystalline polymers dissolve by unfolding ofthe glassy core, only front S exists and it continues the crystal chains [60] to join the amorphous portion,to move inwards towards the center of the slab till followed by subsequent disentanglement (Fig. 3). Inthe entire polymer is dissolved. this work, both the crystal unfolding and the disen-

Various mathematical models have been proposed tanglement process during semicrystalline polymer


Fig. 3. Schematic representation of crystal unfolding and subsequent disentanglement in the presence of a solvent (reproduced withpermission from [60]).

dissolution were modeled in order to predict the them either in dissolved or in dispersed form inkinetics of the process. The semicrystalline polymer polymers [61]. In the development of controlledwas assumed to have crystallites formed by folded release systems, mathematical modeling of the re-chains, as shown in Fig. 3. The free energy changes lease process plays a significant role as it establishesduring chain unfolding were assumed to depend on the mechanism(s) of drug (solute) release and pro-the crystal dimensions and on the surface energies vides more general guidelines for the development ofalong different crystal planes. The model considered other systems. It is accepted that numerous success-the influence of factors such as polymer degree of ful controlled delivery systems have been developedcrystallinity, polymer molecular weight, temperature, as a result of an almost arbitrary selection ofand polymer–solvent interactions on chain unfold- components, configurations and geometries. Yet,ing. development of advanced controlled release systems

All the above approaches have addressed various is increasingly dependent on judicious application ofissues that are important in the polymer dissolution the fundamentals of solute diffusion through poly-phenomenon. These include stress relaxation, vis- mers.coelasticity, disentanglement of polymer chains, From a mathematical modeling point of view,anomalous transport of solvent, chain reptation, and controlled release systems may be classified accord-external mass transfer limitations. These phenomena ing to the controlling physical mechanism(s) ofcan be quantified by accurate measurement of prop- release of the incorporated drug. A convenienterties such as the polymer molecular weight, its method [62] is based on the mechanism of transportpolydispersity, polymer–solvent interactions, radius in these systems as diffusion-controlled, swelling-of gyration, viscosity and plateau modulus. The controlled, and chemically controlled systems.comparisons with experiments are at best qualitative. Accurate mathematical models that take into accountFuture work in the area should focus on the design of the mechanistic aspects of the transport processes inexperiments that will quantitatively validate the drug delivery systems and the structural characteris-models; in particular, experiments that can provide tics of the polymer can help fulfill the abovespatial variations in molecular properties such as objectives. The model equations can be used todiffusion coefficients and radii of gyration will be design new systems by selecting the optimal geome-very valuable. try, method of formulation and size [63–69].

Modeling of such systems is complicated andrelies on careful representation of the physical

3. Current models for dissolution-controlled situation. For example, proposed models must inter-drug delivery systems pret the numerous problems that arise in testing of

the controlled release systems, e.g. in vitro or inControlled delivery of drugs, proteins and other vivo. The large drug loading and/or the solvent

bioactive agents can be achieved by incorporating penetration into the polymer carrier could result in


non-constant diffusion coefficients. The polymer M cd b]] ]]carrier could swell /deswell due to solvent transport. 5M acd,` cdThe occurrence of multicomponent transport instead

]]]]]]]]]]]2of single drug diffusion and the (possible) presence 2[2D (c* 1 c 2 c 2 c ) 1 D (c 2 c ) 1 D /c ] (1 2 c* 2 c )ts s d b d s b d b s

]]]]]]]]]]]3 1 kc tH JbD (2 2 c* 2 c )(c* 1 c 2 c 2 c ) 1 D (c* 1 c )(c 2 c )of macromolecular relaxational phenomena compli- œ s s s d b d s s b

cate the analysis. Accurate mathematical models can (1)overcome many of the above shortcomings and

Here, D is the diffusion coefficient of the solvent inhence mathematical models play a pivotal role in the s

the polymer and D is the diffusion coefficient of thedesign of systems for controlled drug delivery. d

drug in the polymer. The parameter c is the volumeHere we present a critical review and evaluation of b

fraction of the drug in the bulk. This model canimportant mathematical models for the description ofpredict both Fickian as well as non-Fickian behavior.drug release from dissolving polymeric systems.It is to be noted that front synchronization (i.e.Mechanistic aspects of the diffusion phenomenavelocity of glassy–rubbery interface5velocity ofobserved in drug delivery systems are related to anrubbery–solvent interface) leads to zero-order releaseaccurate mathematical model and to the structuralin dissolution-controlled systems. Comparisons withcharacteristics of the polymer under consideration.experiments for release of sodium diclofenac fromDissolution-controlled systems have the charac-poly(vinyl alcohol) (PVA)–mannitol tablets yieldedteristic that in addition to polymer swelling and druggood results (Fig. 4). This approach, while valuable,diffusion through the continuously changing phase,does not discuss the molecular (or physical) origin ofthey are accompanied by slow disentanglement ofsome of the important parameters used (such as cthe polymer chains, leading to complete dissolution d

and k).of the carrier. This mechanism will occur only inRecently, Siepmann and co-workers [71–73] pub-uncross-linked polymer carriers. This review clas-

lished a mathematical model describing drug releasesifies the approaches for dissolution models for drugfrom dissolving hydroxypropyl methyl celluloserelease systems into(HPMC) matrices by considering cylindrical devicesand accounting for both radial as well as axial

1. use of phenomenology and models with Fickiantransport. Diffusion coefficients for water and drug

equations [70–73]were taken to be concentration dependent following

2. analysis using anomalous transport models fora generalized free volume theory. Dissolution was

solvent transport and scaling laws for chaindescribed by a reptation model similar to the ap-

disentanglement [10,79–82].proach considered by Narasimhan and Peppas [10].A critical polymer concentration is considered,below which disentanglement processes begin todominate, resulting in convection-controlled trans-4. Phenomenological modelsport of the chains. A dissolution constant, k , wasdiss

defined to describe the velocity of dissolution perHarland et al. [70] formulated for the first mathe-unit area and could be controlled by either rate ofmatical model for drug release in a dissolvingdisentanglement or by diffusion through a boundarypolymer–solvent system. The transport was assumedlayer, adjacent to the HPMC–water interface. Ato be Fickian and mass balances were written for themass balance for the polymer chains is written asdrug and the solvent at the glassy–rubbery interface

and at the rubbery–solvent interface. The important M 5 M 2 k A t (2)t 0 diss tparameters identified in the phenomenon were thepolymer volume fraction c* at the glassy–rubbery Here M is the mass of the dry matrix at any time t,t

transition, the polymer volume fraction c for disen- M is the mass of the dry matrix at t 5 0, and A isd 0 t

tanglement of the chains and the dissolution /mass the surface area at time t. In this approach, thetransfer coefficient, k. The expression for drug dissolution rate, k , is treated as an adjustablediss

release as a function of time was obtained as parameter. While some physical meaning is attribu-


Fig. 5. Comparison of the Siepmann model [71] predictions withexperimental data on propanolol–HCl release from HPMC ma-trices.

and an incorporation of that effect into this modelmight lead to more meaningful predictions.

5. Molecular models

The above model by Harland et al. [70] wasmodified by Narasimhan and Peppas [10] by ac-counting for macromolecular chain disentanglement(Fig. 6). This enabled a molecular understanding ofthe dissolution mechanism of the polymer. Thisinformation is important to design tailor-made drugdelivery systems for specific applications. A one-dimensional water and drug diffusion is followed bychain disentanglement in amorphous, uncrosslinked,

Fig. 4. Variation of the normalized gel layer thickness (a) and the and linear polymers. This model describes transportfraction of sodium diclofenac released (b) as a function of in a film, slab, disk or tablet in the x direction. Thedissolution time. The lines indicate the prediction of the dissolu-

assumption of one-dimensional transport may betion model proposed by Harland et al. [70].relaxed without loss of generality. A three-com-ponent system is considered, with the water indicated

ted to the dissolution rate, a molecular interpretation as component 1, the polymer as component 2 and theis missing. The model has been shown to possess drug as component d. The model considers twopredictive capabilities by comparison with ex- moving boundaries, R and S, (R is the glassy–perimental data on release of propanolol–HCl from rubbery interface and S is the rubbery–solventHPMC matrices (Fig. 5). However, since k has interface) and defines (S–R) as the gel layer thick-diss

been treated as a constant, the model predictions can ness. Quasi-equilibrium conditions at the rubbery–at best be treated as qualitative, as evidenced from solvent interface enable use of the Flory–Rehnerthe over-prediction of the release rate at short times theory [59] to calculate the water (and drug) volumeand under-prediction of the release rate at long times. fractions at this interface. The disentanglement rateIt has been shown [44] that k is a function of time of the polymer was taken as the ratio between thediss


A quasi-steady state solution to the model equa-tions was presented in this work. The variation of thegel-layer thickness, (S–R), with time was obtainedas

(S 2 R) A B]]] ] ]2 2 ln h1 2 (S 2 R)j 5 t (3)2B AB

here A and B are given as

y 11,eq]]] ]]]*A 5 D (y 2 y ) ? 1S D1,eq 1 * *y 1 y y 1 y1,eq d,eq 1 d

y 1d,eq]]] ]]]*1 D (y 2 y ) ? 1S Dd d d,eq * *y 1 y y 1 y1,eq d,eq 1 d

(4)

kd]]]B 5 (5)y 1 y1,eq d,eq

An expression for the fraction of the drug releasedwas derived as

*y 1 yM d,eq d ]d Œ]] ]]]5 ( 2At 1 Bt) (6)M 2 ld,`

Here, l is the half-thickness of the polymer, D is thediffusion coefficient of the solvent and D is thed

*diffusion coefficient of the drug. y * and y are1 d

characteristic concentrations of solvent and drug,respectively, while y and y are equilibrium1,eq d,eq

concentrations of solvent and drug, respectively. kd

is the disentanglement rate of the polymer chains andis calculated using reptation theory [74–76].

To investigate the influence of various parameterson the drug release behavior, the normalized drugreleased as a function of time was simulated for

25different size drugs with values of D from 1310d2 29 2cm /s to 1310 cm /s. As expected, the amount

of drug released was higher for higher drug diffusioncoefficients (Fig. 7). Also, as a approached zeroFig. 6. Macromolecular disentanglement indicating the successive

processes of (a) system of entangled chains; (b) a reptating chain (a 5 A /B), Case II behavior (linear release profile)‘disentangling’ from the system; and (c) a completely disentangled was observed and for higher values of a, the drugchain. release was Fickian. Small values of a are indicative

of small D values, which in turn means large sized

polymer radius of gyration and its reptation time. An drugs. This could happen in the instance of proteinimportant contribution of this work was the presence delivery, where unfolding of the protein beforeof a diffusion boundary layer adjacent to the rubber– release could be the rate limiting step, leading tosolvent interface, through which the disentangled Case II behavior. For higher values of a, the drugchains (and the drug) have to diffuse. diffusion coefficient is high, indicating small size


Fig. 7. Predicted normalized drug released, M /M , as a function of normalized time, t, for different values of a (a 5 A /B). The values ofd d,`26 2 25* *the parameters used in the simulation were: 2l51 mm; y 50.10; y 50.05; y 50.85; y 50.15; D51.5310 cm /s; k 5 2 3 10d,eq d 1,eq 1 d

cm/s (reproduced with permission from [10]).

drugs. Such drugs diffuse in a Fickian manner as higher values of a, the gel layer thickness increasesindicated by the model. with time (diffusion-controlled). This is consistent

In addition, the gel layer thickness was simulated with the trends shown by the normalized drug releaseas a function of normalized time, t, defined as t 5 B profile.

2t /l. The values of the various parameters used were For the drug release rate to be zero-order, B /the same as in Fig. 7. It is observed (Fig. 8) that as a A 4 1. Hence choosing a polymer–drug–solventapproaches zero, the gel layer thickness profile system such that the above inequality is satisfiedbecomes flat (disentanglement-controlled) while for would result in a zero-order drug release. The model

Fig. 8. Predicted normalized gel layer thickness, d /l, as a function of normalized time, t, for different values of a. The values of theparameters are the same as in Fig. 7 (reproduced with permission from [10]).


also captures the transition between Fickian andCase-II type behavior (Fig. 8). Experimental verifi-cation of the approximate solution of the dissolution-controlled zero-order release model is presented.Three separate polymer–solvent–drug systems wereconsidered and the model predictions have beencompared with the experimental data. Comparisonsof the gel-layer thickness as well as the normalizedamount of drug released as a function of time weremade.

An example of tablet dissolution [8] that presentsthe variation of the gel layer thickness with time iscompared with the model predictions. The systemunder consideration is a 50% by weight solution of

Fig. 10. Normalized mass of drug released vs. time for dissolutioncimetidine hydrochloride, a hydrophilic drug (molec- of a tablet containing 40 wt.% mannitol, 10 wt.% poly(vinylal-

26 2ular weight5288.79, D 5 1.3 3 10 cm /s) from a cohol) and 50 wt.% of diprophylline at 378C; data of Conte et al.d

[8] (reproduced with permission from [10]).tablet (20% by weight of mannitol and 30% byweight of PVA of molecular weight 130 000). Man-nitol was used as filler. The dissolution was per-formed at 378C in deionized water. Fig. 9 shows the weight of mannitol and 40% by weight of PVA ofgel layer thickness, normalized with respect to the molecular weight 130 000) in deionized water isinitial half thickness of the tablet, plotted as a considered. Fig. 10 shows the fraction of drugfunction of time. There is good agreement between released as a function of time. The agreementthe model predictions and the experimental data over between the experimental data and the model predic-a range of about 100 min. tions is very good.

The second system used for comparison is from The final system considered is from the group ofthe same work [8]. Here, the release of dip- Peppas [10]. Here, the release of sodium diclofenac

26 2rophylline, a hydrophilic drug (molecular weight5 (molecular weight5318.13, D 51.1310 cm /s)d26 2254, D 5 1.5 3 10 cm /s) from a tablet (20% by from a PVA tablet (50% drug, 30% polymer) wasd

investigated. The release studies were performed inintestinal stimulating fluid at 378C. Fig. 11 presentsthe comparison between the experimental data andthe model prediction of the fraction of drug releasedas a function of time. Once again, the agreement isvery good, thus verifying the validity of the model.

A novel class of phase-erosion, controlled releasedevices based on semicrystalline polymers was de-veloped by Mallapragada and Peppas [77]. Forpreparation of these systems, a drug was dissolved ina dilute polymer solution, the system was cast ormolded and exposed to an annealing techniqueleading to significant crystallization of the polymercarrier. Drug release from such systems is controlledby the rate of crystal dissolution in water or bio-logical fluids. The degree of crystallinity of the

Fig. 9. Normalized gel layer thickness vs. time for dissolution of apolymer carrier can be varied by heat treatment andtablet containing 50 wt.% cimetidine hydrochloride, 10 wt.% PVAis the controlling factor of the drug release rate from¯(M 5 130 000) and 40 wt.% mannitol at 378C; data of Conte etn

al. [8] (reproduced with permission from [10]). such systems. A mathematical model was proposed


The diffusion coefficient of the drug through thesemicrystalline polymer, D , was assumed to dependd

not only on the volume fraction of the crystals [60]but also on the polymer tortuosity, t

1 2 y2cS]]DD 5 D (10)d a t

Here D represents the drug diffusion coefficient ofa

the drug through the purely amorphous polymer. Itwas shown by Harland and Peppas [43] that thevalue of t is 3.0 for diffusion of small moleculesthrough semicrystalline polymers, unless the volumefraction of the crystals is very low. The present

Fig. 11. Normalized mass of drug released vs. time for dissolution model assumed, t 51.0 when y #0.05, and t 53.02cof a tablet containing 20 wt.% mannitol, 30 wt.% poly(vinylal- when y .0.05. The model predictions indicated that2ccohol) and 50 wt.% of sodium diclofenac in intestinal stimulating

higher values of initial degrees of crystallinity, (i.e.fluid at 378C (reproduced with permission from [10]).larger crystal sizes) led to slower drug release rates(Fig. 12). As the unfolding rate was increased, the

[78] to predict drug release rates from semicrystal- fraction of drug released increases due to fasterline polymer systems, while taking into account the dissolution of smaller crystals. The release processdissolution of the crystals when brought into contact was found to be non-Fickian, a result attributed towith water. the process of crystal dissolution that accompanies

For crystal dissolution, a first order dependence on drug release.the concentration of the solvent was assumed. There- PVA devices loaded with metronidazole, an antit-fore, the expression for change in volume fraction of richomonal drug, were exposed to temperaturesthe crystalline portion of the polymer as a function of ranging from 90 to 1208C for times of 10–90 min intime was written as order to obtain samples with different degrees of

crystallinity. In vitro release of metronidazole from≠y2c]] 5 2 k y H(y ) (7) such systems into deionized water at 378C was1 1 2c≠t

monitored. The drug release rate was found to beThe Heavyside function, H(y ) is used to prevent the dependent on the crystallization conditions of the2c

polymer crystal volume fraction from becoming samples. The influence of annealing conditions onnegative as the crystal dissolution occurs for an the release rate of the drug is shown in Fig. 13. Theextended period. The term k the rate of unfolding sample crystallized at 1208C for 1 h released met-1,

of the crystals, was calculated using free energy ronidazole at a much slower rate than the sampleconsiderations [60]. The rate of change of the annealed at 1108C for 10 min. As the crystallizationamorphous fraction is given by temperature and/or time was increased, the degree of

crystallinity of PVA increased [36]. As a result, the≠y ≠y≠2a 2a]] ] ]]S D5 D 1 k y H(y ) (8) hindrance to drug diffusion was greater, leading to12 1 1 2c≠t ≠x ≠x

slower drug release rates. Therefore, it is evident thathere D is the mutual diffusion coefficient and the controlling annealing conditions (i.e. temperature12

source term accounts for the transformation of the and/or time) implies controlling the degree of crys-crystalline phase to the amorphous phase during the tallinity of the polymer and hence the drug releaseunfolding process. The drug diffusion through the rate.semicrystalline polymer is given by Ju and co-workers [79–81] proposed a model to

predict swelling /dissolution behavior of HPMC ma-≠y ≠y≠d d] ] ]S D5 D (9) trices. They introduced a polymer disentanglementd≠t ≠x ≠x


Fig. 13. Influence of annealing conditions on metronidazole¯release at 378C from PVA controlled release systems (PVA M 5n

17 600; metronidazole loading-2 wt.%); (d) annealed at 1108Cfor 20 min and (j) annealed at 1208C for 1 h (reproduced withpermission from [78]).

The mathematical description of the transportaccounts for swelling /dissolution in the radial direc-tion and goes on to derive ‘universal’ scaling rela-tionships for the fractional release of both polymerand drug with respect to the polymer molecular

21.05weight M [80]. A stronger dependence (|M ) is20.24predicted for the polymer than the drug (|M ). It

is postulated that both fractional release profilesapproach limiting values for very large values of M.These results agree well with experimental data onfractional release of HPMC and adinazolam mesylateFig. 12. Effect of crystal unfolding rate, k , on drug release from1

from HPMC matrices of various molecular weightsdissolving PVA matrix. The initial degree of crystallinity is X525 21 24 2140%. (a) k 510 s , (b) k 510 s (reproduced with permis- (Figs. 14 and 15). Another assumption made in this1 1

sion from [78]). contribution is that the mutual diffusion coefficientbetween HPMC and water is independent of the

concentration, r , and a diffusion layer into the self-diffusion coefficient of HPMC. Ju and co-work-p,dis

model [78]. The disentanglement concentration, ers also developed a model for the effective diffusionr , was defined as the concentration below which coefficient of disentangled HPMC chains in thep,dis

the chains detach from the matrix and diffuse diffusion layer adjacent to the matrix [81]. Theythrough the diffusion layer into the bulk solution. show that this effective diffusion coefficient scales

20.53Scaling laws with polymer molecular weight M were with M as D |M . Since the thickness of theeff20.8derived for r as r |M . This scaling rela- diffusion layer is negligible compared to the half-p,dis p,dis

tionship appears to be valid for HPMC systems. The thickness (or the radius) of the matrix, the need toscaling law is based on the premise that the disen- calculate an effective D seems less meaningful.tanglement concentration scales similarly with M as Additionally, since the chains are disentangled in thisthe crossover concentration, c* [74]. This approach, regime, it is reasonable to assume that they undergowhile reasonable, fails to account for the phenom- Zimm dynamics [76], which provides a scaling

20.5enon of reptation, which has been shown to occur in relationship of D | M , which is very close toZimm

entangled polymer melts [59]. the result derived by Ju et al. [81]


Fig. 14. Fractional release of HPMC from HPMC-based matricesFig. 15. Fractional release of adinazolam mesylate from HPMC-of various molecular weights (K100LV: M 530 000; K4M: M 5w w

based matrices of various molecular weights. The values of the96 000; K100M: M 5267 000). The dotted lines indicate modelw

parameters are the same as in Fig. 14. The dotted lines indicatepredictions (reproduced with permission from [80]).model predictions (reproduced with permission from [80]).

6. Conclusions and future outlookneeds attention is the case when the drug is poorly

The phenomenological models identified a poly- soluble in the release medium [82].mer ‘dissolution rate’ as a key parameter. However, Polymer dissolution in solvents has been longthey failed to quantitatively predict this rate from the recognized as an important phenomenon in drugmolecular properties of the polymer and the solvent delivery systems. It is only recently that a molecularas treated this rate as a model parameter. Additional- understanding of the mechanism of dissolution ofly, there are parameters used in these approaches that amorphous as well as semicrystalline polymers hascannot be obtained from experiments. This is a key been obtained, both due to experimental advances asarea where experimentalists and modelers need to well as predictive mathematical models. The rele-work together since there are numerous models vance of a number of physical phenomena as well ascontaining parameters that cannot be obtained ex- molecular properties of solvent and polymer haveperimentally and numerous experimental results that been established and characterized. However, quan-cannot be rationalized by current theories. In this titative comparisons between model predictions andcontext, the approach of using molecular theories in experimental data have not been adequately ad-a continuum framework appears to yield useful dressed. Both experimentalists as well as theoreti-insight into the understanding of the physics of the cians need to assume responsibility to bridge thisproblem and comparisons with experimental data gap; the experimentalists by developing techniques(both molecular and macroscopic) appear promising. that probe the dissolution phenomenon at a molecu-The effect of polydispersity on the dissolution mech- lar level (MRI, scattering) and the theoreticians byanism is an aspect that has not been addressed by providing fundamental explanations for phenomenaany of the above contributions. Another aspect that such as constraint removal, chain disentanglement,


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