113 (2006) 216–225www.elsevier.com/locate/jconrel

Journal of Controlled Release

A model for the drug release from a polymer matrixtablet—effects of swelling and dissolution

Per Borgquist a, Anna Körner b, Lennart Piculell b, Anette Larsson c, Anders Axelsson a,⁎

a Department of Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Swedenb Department of Physical Chemistry I, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden

c Department of Materials and Surface Chemistry, Chalmers University of Technology, SE-41296 Gothenburg, Sweden

Received 20 January 2006; accepted 4 May 2006Available online 12 May 2006

Abstract

A model for simulating the drug release from a swelling and dissolving polymer tablet is presented and verified to data. The model is based ona mechanistic approach, and it can therefore be employed to study the sensitivity of true physical constants, for instance the drug diffusioncoefficient or the drug solubility. The model generates the drug and polymer release profiles and the front positions of the total tablet, the solidcore, and of the solid–drug–solubilized-drug interface. The convective contribution to mass transfer is shown to be of great importance. This ismost markedly noticed for slowly diffusing drugs. In a simulation with a low value of the drug diffusion coefficient, it is shown that the initial drugrelease rate is faster than the polymer dissolution rate, followed by a second stage with a slower drug release rate. Furthermore, it is shown thatpolymer dissolution influences the drug release profile significantly, but not the front position of saturated drug in the gel layer. The model isverified against drug release and polymer dissolution data for the slightly soluble drug Methyl paraben and the soluble drug Saligenin in a poly(ethylene oxide) tablet, resulting in good agreement between model and experiments.© 2006 Elsevier B.V. All rights reserved.

Keywords: Controlled release; Matrix; Mathematical model; Polymer dissolution; Swelling

1. Introduction

Acommonway of controlling drug delivery is by incorporatingthe drug into a polymer matrix system [1–3]. Thus, drug dis-solution and drug diffusion through the polymer are importantphenomena in controlling the release characteristics of the formu-lation. Simplemodels, such as the classicalHiguchi equation, havesuccessfully been applied for the case of drug release from non-swelling and non-dissolving polymer matrices [4,5]. However, ifsubstantial matrix swelling takes place, the drug release processwill also be influenced by the solvent penetration process in thepolymer system [2,6]. Then the mathematical description of therelease process becomes a moving boundary problem, with in-creased complexity as a result [7]. The importance of drug disso-lution has been studied by several researchers, identifying thedistance in the gel from the gel–solvent interface to the position in

⁎ Corresponding author. Tel.: +46 46 222 82 87; fax: +46 46 222 45 26.E-mail address: Anders.Axelsson@chemeng.lth.se (A. Axelsson).

0168-3659/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jconrel.2006.05.004

the gel where the drug is saturated as an important measure of therelease characteristics [8–11]. An additional phenomenon, con-tributing to solvent penetration and the drug release process, mayappear for non-cross-linked swelling polymers. These systemsmay exhibit matrix dissolution by disentanglement of the polymerchains [6,12]. Matrix dissolution reduces the diffusion path in thepolymer, thus counteracting the swelling process. Therefore, themodelling of swelling and dissolving drug-loaded polymers is avery complicated issue. At the same time, it demonstrates thepotential of modelling. If it is possible to predict these processes itwill be possible to design a delivery system with the requiredrelease characteristics without expensive and time-consuminglaboratory tests.

Many models for drug release from swelling and dissolvingpolymer matrices can be found in the literature. However, almostall of them describe the transport in polymer spheres or slabs [13–15]. Thus, these models are not applicable to the case of tablets,having a nearly cylindrical geometry. A few models describe therelease process for cylindrical geometry [16–20], however, often

217P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

with a simplified one-dimensional transport assumption [21–23].The contribution to solvent and solute transport due to the swel-ling of the matrix is often neglected in polymer transport models.This diffusion-induced convective contribution to mass transferarises from the local transport of polymer chains in the directiontowards the pure solvent phase. This transport results in a non-zero bulk velocity, a situation well accounted for in basic masstransfer theory [24,25]. The convective contribution to masstransfer in swellable polymer systems has been included in somemodels of solvent and or solute transport [21,22], but is generallyneglected. In applications of swelling gels, e.g. as enzyme carriersin bioreactors, the importance of the convective contribution hasbeen demonstrated and even utilized to improve efficiency [26].

In this study a model for the drug release from a swelling anddissolving cylindrical polymeric tablet is presented. The model isthe result of further development of a previously presented modelfor the dissolution of polymer from a swelling tablet [27]. Themodel can predict drug release from formulations containing sol-uble aswell as poorly soluble drugs. Themass transfer rate includesthe contributions from diffusion as well as swelling (diffusion-induced convection). The transformation from solid to gel phase isassumed to be limited by the rate of penetrant transport into thesolid phase, corresponding to a critical penetrant concentration. Atthe gel–solvent interface, it is assumed that equilibrium existsbetween the hydrodynamic forces and polymer entanglement. Thisimplies that the surface polymer concentration reaches a constantvalue after the initial phase, and that this equilibrium exists for therest of the dissolution process. Hence, the initial polymer dis-solution rate is zero, until the entanglement strength has been re-duced by the increased penetrant concentration. The model is usedto study the influence of the drug diffusion coefficient, the drugsolubility and the initial drug loading on the drug release profile.Furthermore, the effects of diffusion-induced convection and poly-mer dissolution are studied. The model is verified against drugrelease data and polymer dissolution data for drug-loaded poly(ethylene oxide) (PEO) tablets, for a slightly soluble drug, Metylparaben, and a soluble drug, Saligenin.

2. Experimental methods

2.1. Materials

Poly (ethylene oxide) with a molar mass of about 2.0×106 Daaccording to the supplier, (PolyoxWSRN-60K, hereafter referredto as PEO 2.0) was supplied by Dow, Austria. Methyl paraben(Fluka Chemie GmbH), with a molar mass of 152.15 g/mol, andSaligenin (Sigma-Aldrich Chemie GmbH), with a molar mass of124.14 g/mol, were used as model drugs. The water solubilities ofthe model drugs Methyl paraben and Saligenin were 2.5 kg·m−3

and 74 kg·m−3, respectively [28,29]. Other chemicals used in thiswork were sodium chloride (MERCK, USA) and sodium azide(BDH Laboratory Supplies, UK).

2.2. Tablet preparation

The model drug (2% w/w) and the polymer (both used assupplied) were mixed for at least 5 min using a pestle and mortar.

Tablets weremade using a single-punch tabletingmachine (KilianSP300, Kilian and Co., GmbH) equippedwith 12.0mm flat-facedpunches. The distance between punches was set to 2.0 mm. Thepowder mixture was poured into the die and the automatic com-pression cycle was run with a maximum compaction force ofabout 10 kN. All tablets weighed 315.5±1 mg, had a diameter of12.0 mm and a height of about 2.7 mm.

2.3. Dissolution experiments

The dissolution experiments were carried out according to amethod described in detail earlier, in a USP dissolution apparatus(Dissolutest, Prolabo) equipped with spinning disks [30]. A tabletwasmounted at the centre of a disk using water-impermeable glueand the disc was rotated at 100 rpm in the dissolution medium(deionised water, 25 °C). Aliquots of 2 ml for analysis (herereferred to as “dissolution samples”) were withdrawn from thedissolution medium at defined times. The dissolution sampleswere analysed for both polymer and model drug content.

2.4. Polymer content

The dissolution samples were analysed for polymer con-tent using size-exclusion chromatography (SEC) on a TSK-GELGMPWXL 7.8×300 mm, particle size 13 μm, linear mixed bedsize-exclusion column (TosoHaas, Montgomeryville, PA, USA)connected online to a differential refractometer (Waters 410, Wa-ters Milford,MA, USA) according to a method described in detailearlier [30]. A solution of 10 mM NaCl with 0.02% NaN3 wasused as mobile phase, the flow rate was set to 0.3 ml/min and thesample volume injected was 200 μl. Millennium 32 software(Waters, Milford, MA, USA) was used for analysis of the re-fractive index chromatograms obtained, and the polymer contentin the dissolutionmediumwas calculated using a calibration curveobtained from polymer samples of known concentration. Thepercentage polymer released at each sampling time was calculatedfrom the known tablet weight using the following relation:

% dissolved ¼cn � ðV0−Vsðn−1ÞÞ þ Vs

Pn−1n¼0

cn

mtbl

0B@

1CA�100: ð1Þ

Here cn is the polymer concentration in dissolution sample n, V0is the total volume of the dissolution medium at time t=0, i.e.500ml,Vs is the volume of the dissolution sample, i.e. 2 ml, n is thesample number in the dissolution series andmtbl is the tabletweight.Since the dissolution profiles indicated that slightly more than100% polymer had been released by the end of the experiment(partly due to evaporation of the dissolution medium during pre-heating and during the experiment) all dissolution profiles werenormalized to 100%dissolved polymer at the end of the experiment(t=∞).

2.5. Model drug content

The dissolution samples were analysed for model drug contentin aUV spectrophotometer (Lamda 40UV/VIS Spectrometer, with

218 P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

UV Winlab software, PerkinElmer Instruments) using a wave-length of 254.9 nm forMethyl paraben and 273.0 nm for Saligenin.The model drug concentration in the dissolution medium was cal-culated using a calibration curve obtained from samples of knownconcentration, and the dissolution profiles were normalized to100% drug released at the end of the experiment (t=∞).

3. Theory

The model presented in this study is the result of further deve-lopment of a previously presented model describing polymer dis-solution from a cylindrical polymer tablet [27]. The polymerdissolution model was improved to account for drug release forfreely as well as poorly water-soluble drugs. Drug transport is as-sumed to take place only in the solvent phase. Furthermore, it isassumed that the effect of drug transport on solvent transport isnegligible. This implies that the swelling and dissolution of thepolymer–drug–solvent system depends only on the properties ofthe polymer and of the solvent. This assumption is valid for lowdrug loadings.

The polymer dissolution model was derived for the case of acylindrical tablet mounted on a spinning disk [27]. This means thatthe entire axial dimension has to be simulated (as opposed to atablet dissolving freely in solution, where symmetric conditionsimply that only one quarter of the tablet has to be simulated [17–20]. The cylindrical polymer tablet is discretized in space in PQfinite volumes, see Fig. 1. The axial dimension (index i) is dis-cretized in P slices and each slice is discretized in Q annular rings(index j). The axial length of a finite volume is denoted X and theannular ring thickness is denoted δ. Larger values of i and j indicatelocations closer to the bulk phase. It is assumed that the totalvolume change upon mixing is zero [15,17–20]. Thus, the totalvolume of a finite volume,Vij, can be expressed as a function of thepolymer mass, mij

p, and the polymer volume fraction, yijp:

Vij ¼mp

ij

qp1

ypij; ð2Þ

where ρp is the density of the pure polymer.

Fig. 1. The finite volume discretization, exemplified for 9 finite volumes (3×3).The axial dimension (index i) is discretized inP slices and each slice is discretized inQ (index j) annular rings (larger indices indicate locations closer to the bulk phase).

3.1. Polymer dissolution

The matrix dissolution of the drug-loaded polymer tablet ismodelled in the same manner as in the previously presented purepolymer case [27]. The dissolution of a boundary volume (avolume in contact with the bulk phase) can be expressed by usinga dissolution coefficient, kd

p:

dmpij

dt¼ −kpdA

Bijq

pypd ; ð3Þ

where AijB is the area available for dissolution (axial and/or radial)

and ydp is the polymer volume fraction at the boundary.

3.2. Water mass balance

The unsteady-state water mass balance over a finite volume(i,j), taking into account diffusion and convection fluxes andpolymer dissolution can be written [27]:

Vij

ypij

dywijdt

−Aes;outij ywi;jþ1

Xjk¼1

ddikdt

þAes;inij ywij

Xj−1k¼1

ddikdt

−Acsij y

wiþ1;j

Xik¼1

dXkj

dtþ Acs

ij ywij

Xi−1k¼1

dXkj

dt¼ −ywij A

BijUij

þAes;outij NDw

i;jþ1Yj−Aes;inij NDw

i;jYj−1 þ Acsij N

Dwiþ1Yi;j−A

csij N

DwiYi−1;j;

ð4Þ

where the contribution due to dissolution, Φ, is:

Uij ¼ 0; ð5Þ

for the non-dissolving volume elements, i.e. the elements not incontact with the bulk phase, and:

Uij ¼ −kpdypdypij

; ð6Þ

for the elements at the dissolving front.The expressions for the expansion in the axial and radial

direction becomes [27]:

−awij Xij

ypij

dywijdt

þ dXij

dt¼ Uij; ð7Þ

−ð1−awij Þ

ypij

Vij

Aes;outij

dywijdt

þ dijRij

Xj−1k¼1

ddikdt

þ ddijdt

¼ Uij: ð8Þ

The initial conditions of Eqs. (4), (7), (8) are:

ywij ðt ¼ 0Þ ¼ 0Xijðt ¼ 0Þ ¼ Xijj0dijðt ¼ 0Þ ¼ dijj0

;

8<: ð9Þ

219P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

and the boundary conditions of Eq. (4) are:

NwT1Y0;j ¼ NwT

i;1Y0 ¼ 0

NwDPþ1YP;j ¼ −2DPþ1YP;j

ywPj−ywd

XPj

NwDi;Qþ1YQ ¼ −2Di;Qþ1YQ

ywiQ−ywddiQ

;

8>>>>><>>>>>:

ð10Þ

where ydw is the solvent volume fraction at the gel–solvent

boundary.

3.3. Drug release

The drug mass balance, assuming that the drug transport onlytakes place in the solvent phase, can be expressed:

VijqA

qA−cAsatyATij

" #2ywij

dyATij

dt

þVijqA

qA−cAsat yATij

yATij

ypij

dywijdt

−Aes;outij yA;i;jþ1

Xjk¼1

ddikdt

þ Aes;inij yAij

Xj−1k¼1

ddikdt

−Acsij y

Aiþ1;j

Xik¼1

dXkj

dt

þ Acsij y

Aij

Xi−1k¼1

dXkj

dt

¼ −qA

qA−cAsatyATij

yATij ywijUijABij

þ Aes;outij NAD

i;jþ1Yj−Aes;inij NAD

i;jYj−1

þ Acsij N

ADiþ1Yi;j−A

csij N

ADiYi−1;j; ð11Þ

where ρA is the drug density, csatA is the drug solubility, yij

AT is theoverall normalized drug concentration and yA,ij is a normalizeddrug concentration in the solvent phase. The drug concentrationis normalized to the drug solubility. Since the drug transportonly takes place in the solvent phase, yA,ij has a maximum valueof 1.

The initial and boundary conditions of Eq. (11), for the drugconcentration are:

yAij ðt ¼ 0Þ ¼ yAij j0yAd ¼ 0NATc ¼ 0

:

8<: ð12Þ

When the solid phase has vanished, the zero flux boundarycondition is located at the tablet centre (radial coordinate) and atzero axial position, analogous to the water transport boundarycondition (Eq. (10)).

3.4. Polymer and drug in the bulk phase

The amount of polymer dissolved into the bulk phase iscalculated using a total mass balance, by summing the polymer

content in all finite volumes. This is expressed as the fraction ofinitial polymer mass dissolved as a function of time:

Y pB tð Þ ¼ 1−

PPi¼1

PQj¼1

mpijðtÞ

!

mpj0: ð13Þ

The amount of drug released into the bulk is calculated in asimilarmanner, resulting in the expression for the fraction released:

YAB tð Þ ¼ 1−

PPi¼1

PQj¼1

mAij ðtÞ

!

mAj0: ð14Þ

3.5. Simulation of the model

The model derived in this work results in a set of coupledordinary differential equations. These are solved numerically, usinga 5th-order implicit ODEsolver [31,32]. Obviously, the precisionincreases with increasing number of finite volumes, as, unfortu-nately, does the simulation time. In this work the cylindrical tabletswere discretized into 400 finite volumes (20 axial shells and 20radial annular rings). These simulations were compared with si-mulations with fewer volumes (10×10), and from this comparisonit was concluded that 20×20 finite volumeswas sufficient to obtainsatisfactory results.

4. Results and discussion

The model presented can predict the polymer dissolution rateand the drug release rate from a cylindrical polymer tablet. Simu-lation gives the volume fractions of drug and solvent in the finitevolumes at predefined time intervals, aswell as the axial and radialdimensions of each volume. From this output an axial–radialconcentration profile can be generated. Therefore, the modelallows two-dimensional studying of the dynamic behaviour of thesolid–gel interface, the gel–solvent interface, and the solid–drug–solubilized-drug interface (henceforth referred to as the solid–druginterface). Furthermore, the model gives the polymer dissolutionprofile and the drug release profile. For simplicity, only the axialdimension (R=0) is shown in the front-position figures.

Themodel was tested for the case of a drug-loaded PEO tablet.In a previous study, the rate-controlling model parameters for thedissolution of pure PEO tablets were determined for two polymermolecular masses, 2.0×106 Da and 0.3×106 Da [27]. The para-meters fitted were the solid–gel mass transfer coefficient, kc, thedissolution coefficient kd

p, the solvent (water) diffusion coe-fficient, Dw, and the gel–solvent interfacial concentration, yd

w. Ifthe drug loading is low it is reasonable to assume that the drug hasnegligible effect on the swelling and dissolution characteristics.Therefore, the parameter values obtained from the pure polymerstudy can be used in the present model [27].

4.1. Sensitivity study

Obviously, the drug diffusion coefficient and the drug solubilityare important properties in the release from drug-loaded polymer

Fig. 2. Sensitivity study of the drug diffusion coefficient. Simulation parametersare given in Table 1 (base case I). (A) Axial front positions showing the totalheight of the tablet (independent of drug diffusion coefficient), the solid coreheight (independent of drug diffusion coefficient), and the solid–drug interfacepositions from the simulations. (B) Polymer dissolution profile (independent ofdrug diffusion coefficient) and drug release profiles.

220 P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

matrices. Furthermore, the initial drug loading will influence therelease process in a similar manner to the drug solubility. There-fore, the influence of these parameters on the release characteristicswere studied for the case of a drug-loaded, high-molecular-weightPEO matrix (average molecular mass 2.0×106 Da). A base casesimulation for a hypothetical drug substance was performed withthe parameter values given in Table 1 (base case I; polymer solventparameters obtained in [27]). The drug diffusion coefficient in basecase I is 2.0×10− 10 m2 s−1 and the normalized solubility (ysat

A =csatA /ρA) is 0.02. These values represent a sparingly soluble drugsubstance with a molecular mass of about 200 Da (by scaling thedrug diffusivity assuming that the diffusion coefficient is pro-portional to M−1/3 [24,25]). The initial drug loading in base case Iwas chosen to be 10% w/w. To study the influence of diffusion-induced convection and polymer dissolution on the drug releaseprocess, a second base case was defined (see Table 1, base case II).The drug diffusion coefficient in base case II is smaller (1.0×10−10

m2 s−1), in order to increase the relative significance of theconvective transport.

4.1.1. Drug diffusion coefficientThe front positions of the solid core height (lower curve) and

the total height (upper curve) of the base case I simulation areshown in Fig. 2A, which, together with the polymer dissolutionprofile in Fig. 2B, define the polymer–solvent interactions duringthe dissolution process. The solid core (axial) height decreasesalmost linearly with time (Fig. 2A), in contrast to the total (axial)height of the tablet, which first increases due to the developmentof the gel layer, and then decreases due to the dissolution processafter the solid core has vanished. The gel layer thickness, notshown in the figure, is the difference between the total height andthe core height. The position of the solid–drug interface for thebase case I simulation is also shown in Fig. 2A. In this case, theinterface is positioned between the solid–gel interface and thegel–solvent interface. In Fig. 2B the polymer dissolution profile(independent of the drug properties, but given for comparison)and the drug release profile of the base case I simulation areshown. Polymer dissolution commences after a lag time of 75min(estimated from model validation against pure polymer swellingdata in a previous study [27]). After the initial lag phase, thedissolution rate increases with time due to the increased surfacearea of the tablet (which is dependent on the total height, see Fig.2A). After the solid core has vanished the surface area decreaseswith time, and therefore so does the polymer dissolution rate. The

Table 1Values of the parameters used in the base cases

Parameter Symbol(units)

Basecase I

Basecase II

Drug diffusion coefficient DA (m2 s−1) 2.0×10−10 1.0×10−10

Normalized drug solubility ysatA 0.02 Same as

Base case IDrug loading – 10% w/w -//-Solid–gel mass transfer coefficient kc (m s−1) 4.3×10−9 -//-Gel–solvent mass transfer coefficient kd

p (m s−1) 8.5×10−8 -//-Solvent diffusion coefficient Dw (m2 s−1) 4.5×10−10 -//-Gel–solvent polymer volume fraction yd

p 0.11 -//-

lag phase of the drug release profile (Fig. 2B) is very short, as drugrelease commences as soon as a gel layer has been formed.Theoretically, the model predicts that an infinitesimal gel layerexists after an infinitesimal dissolution time, implying zero lagtime. However, this theoretical result can only be obtained if thetablet is discretized in an infinite number of volumes. Hence thevery short lag time of the drug release profile seen in Fig. 2Bindicates that the number of finite volumes (400) is sufficient toobtain good accuracy. Due to the negligible drug-release lag time,the fraction of released drug in base case I becomes larger than thefraction of dissolved polymer during the polymer lag phase, asituation that prevails throughout the entire release process.

Also in Fig. 2B, the effect of the drug diffusivity on the drugrelease profile is shown. The effect of a larger drug diffusioncoefficient, 3.0×10−10 m2 s−1, on the drug release profile is anobvious increase in the drug release rate. The front position of thesolid–drug interface is also affected (Fig. 2A). The solid–drug

221P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

layer decreases faster when the diffusion coefficient is increased.Interestingly, when decreasing the drug diffusion coefficient(1.0×10−10 m2 s−1), the characteristics of the drug release profilechange significantly. The initial drug release rate is still faster thanthe polymer dissolution rate (obviously due to the polymer dis-solution lag time). However, after a certain period of time, thedrug release rate decreases and the drug release profile intersectsthe polymer dissolution profile. The reason for this behaviour ofthe drug release profile at this lower diffusion coefficient is thecompetition between the drug transport rates of diffusion anddiffusion-induced convection. For this simulation, the drug dif-fusive transport and the swelling rate (convection) are apparentlyof the same order of magnitude. The effect of diffusion-inducedconvection will be further discussed later in this paper.

Fig. 3. Sensitivity study of the normalized drug solubility, ysatA . Simulation

parameters are given in Table 1 (base case I). (A) Drug release profiles andpolymer dissolution profile (independent of drug solubility). (B) Axial frontpositions showing the total height of the tablet, i.e. the gel–solvent interface(independent of drug solubility), the solid core height, i.e. the solid–gel interface(independent of drug solubility), and the solid–drug interface positions given bythe simulations. The front position of the solid–drug interface for ysat

A =0.2coincides with the solid–gel interface, and the front position of the solid–druginterface for ysat

A =0.002 coincides with the gel–solvent interface.

Fig. 4. Sensitivity study of the drug loading. Simulation parameters are given inTable 1 (base case I). (A) Drug release profiles and polymer dissolution profile(assumed to be independent of drug loading). (B) Axial front positions showingthe total height of the tablet (independent of drug loading), the solid core height(independent of drug loading), and the solid–drug interface positions.

4.1.2. Drug solubility and initial loadingA sensitivity study was also performed for the drug solubility

and the initial drug loading, as shown in Figs. 3 and 4, res-pectively. The normalized drug solubility (ysat

A , as defined above)was studied for a slightly soluble compound with ysat

A =0.002(base case I with ysat

A =0.02) and for a freely soluble compoundwith ysat

A =0.2. It is clear from Fig. 3A that the effect on the drugrelease profile is much more pronounced for a decrease in drugsolubility than for an increase. The drug release profile for thelowest normalized solubility (0.002) follows the polymerdissolution profile during the entire process (except for the initialpolymer lag phase, naturally). This indicates that the solid–druginterface is located close to the gel–solvent interface; a conclusionconfirmed by the front-position profile in Fig. 3B, where thesolid–drug interface coincides with the gel–solvent interface.When the normalized drug solubility is increased to 0.2, the frontposition of the solid–drug interface coincides with the solid–gelinterface. This explains the small difference in release profile

222 P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

(Fig. 3A) between the highly soluble compound (0.2) and the basecase I simulation. Therefore, no effect will be seen on either thedrug release profile or the front-position profile with a furtherincrease in the normalized drug solubility.

The effects of a change in drug loading can be seen in Fig. 4,for the case of 5% w/w, 10% w/w (base case I) and 20% w/w.Drug loading affects the drug release profile and the front positionin a similar manner to the normalized drug solubility. However,for the interval chosen in this study, the absolute effects of adecrease or an increase in drug loading are equivalent. This isclear from both the drug release profile (Fig. 4A) and the solid–drug interface position (Fig. 4B).

4.1.3. Convection and matrix erosionThe model includes the transport phenomena of diffusion,

diffusion-induced convection (swelling) and surface erosion. Therelative importance of these phenomena can be illustrated by thesimulations in Figs. 5 and 6. In Fig. 5, the effect of the diffusion-

Fig. 5. The effect of diffusion-induced drug convection, illustrated with asimulation without diffusion-induced drug convection and a base casesimulation (base case II, including convection). Simulation parameters aregiven in Table 1 (base case II). (A) Drug release profiles and polymer dissolutionprofile. (B) Axial front positions showing the total height of the tablet, the solidcore height, and the solid–drug interface positions.

Fig. 6. The effect of matrix erosion, illustrated by a simulation without matrixerosion and a base case simulation (base case II, including matrix erosion).Simulation parameters are given in Table 1 (base case II). (A) Drug releaseprofiles. (B) Axial front positions for the eroding and non-eroding tablets,showing the total height, the solid core height and the solid–drug interfacepositions.

induced convection on the drug release process is shown. Thedrug release profile for the base case II simulation in Fig. 5Aintersects the polymer dissolution profile after about 8 h of release,due to the convectivemass transfer towards the centre of the tablet.The considerable influence of the convective contribution is evi-dent when comparing the base case II simulationwith a simulationwithout the convective drug transport (Fig. 5A, dotted line). Ob-viously, the characteristics of the drug release profile changesignificantly when excluding the convective term. Also, the frontposition of the solid–drug interface is greatly affected by the con-vective transport, as shown in Fig. 5B. Without convection, thesolid–drug core has vanished after about 15 h, while for base caseII it remains at the end of the simulation (32 h). These results areimportant, as they show that neglecting the convective contribu-tion will lead to erroneous results when, for instance, fitting thedrug diffusion coefficient to experimental data.

To study the importance of matrix erosion (polymer dissolu-tion) on drug release characteristics, a simulation was performed

Fig. 7. Model validation against drug release and polymer dissolution data of (2)drug-loaded PEO tablets (molecular mass 2×106 Da) for a slightly soluble drugand a soluble drug. Initial loading: 2% w/w. (A) Release of the slightly solubledrug Methyl paraben (solubility 2.5 kg·m−3 [29]), using a drug diffusioncoefficient of 3×10−10 m2·s−1. (B) Release of the soluble drug Saligenin(solubility 74 kg·m−3 [28]), using a drug diffusion coefficient of 3×10−10

m2·s−1. (C) Predicted axial front position of the solid–drug interface of Methylparaben and Saligenin, and the front positions of the total height (gel–solventinterface) and the core height (solid–gel interface). The front position ofSaligenin coincides with the solid–gel interface.

223P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

with the base case II parameters, except for the polymer dissolutionrate (gel–solvent mass transfer coefficient), which was set to zero.This release rate with no polymer dissolution is significantlyslower than for the base case II simulation (except for the polymerlag phase), as can be seen in Fig. 6A. This depends on the increaseddiffusion path in the gel phase, as shown in Fig. 6B (the differencebetween the total height and the solid–drug interface). However,matrix erosion does not affect the solid–gel interface or the initialpart of the solid–drug interface. This is due to the flat solventconcentration profile in the external part of the gel layer (data notshown). When the solid core has vanished, the solid–druginterface position of the non-eroding tablet decreases more slowlythan the eroding tablet due to the zero-flux boundary condition atthe tablet centre.

4.2. Model validation

The model was validated against drug release and polymerdissolution data for the release from drug-loaded PEO tablets(molecular mass 2×106 Da). The drugs studied were the slightlysoluble drugMethyl paraben (solubility 2.5 kg· m−3 [29]), and thesoluble drug Saligenin (solubility 74 kg· m−3 [28]). Due to the lowdrug loading in the formulations (2%w/w), it seems reasonable toassume that the tablet will swell and dissolve as in the purepolymer case, described elsewhere [27,30,33]. Therefore, thepolymer–solvent model parameters obtained from parameterfitting to polymer dissolution data of pure PEO tablets can be usedin the present model [27].

In Fig. 7A, drug release data and polymer dissolution data fromdrug-loaded PEO tablets are shown for the case of the slightlysoluble drug, Methyl paraben. Simulation using a drug diffusioncoefficient of 3×10−10 m2·s−1 agrees very well with the data. Thesimulated polymer dissolution profile also agrees well with data.This confirms that the presence of the drug in the tablet has neg-ligible effect on the polymer swelling and dissolution behaviour ofthe tablet. The fitted value of the drug diffusion coefficient can becompared to a simple scaling of the drug diffusion coefficient withthe solvent diffusion coefficient using a Stokes–Einstein expres-sion (assuming that the diffusion coefficient is proportional toM−1/3). This results in a value of 2.2×10−10 m2·s−1. However,these predictions are limited to diffusion in dilute solutions. Fur-thermore, the simple scaling may not be valid as the solventdiffusion coefficient (fitted to pure polymer data, [27]) in themodelalso takes into account the possible resistance to swelling of thepolymer gel, due to relaxation of the network. Thus, the solventdiffusion coefficient is underestimated if relaxations contribute tothe swelling process. Therefore, the fitted value of the drug dif-fusion coefficient, 3×10−10 m2·s−1, seems reasonable.

In Fig. 7B, the result of the simulation of the release of thesoluble drug, Saligenin, is shown. The good agreement betweenthe simulation and data is evident. The simulated polymerdissolution rate is slightly overestimated, probably due to tablet-to-tablet variation. To fit also the polymer dissolution profile, thedissolution rate coefficient has to be decreased by about 5% (datanot shown). The fitted value of the Saligenin diffusion coefficientwas the same as for Methyl paraben, 3×10−10 m2·s−1. Themolecular masses of Saligenin and Methyl paraben differ by only

224 P. Borgquist et al. / Journal of Controlled Release 113 (2006) 216–225

20%. It therefore seems reasonable that the values obtained for thedrug diffusion coefficients are of the same order of magnitude.

The solubility data of Saligenin and Methyl paraben used inthis work correspond to the solubility in water. Therefore, theeffect on the solubility of high polymer concentration is not ac-counted for in this study. Furthermore, the polymer concentrationeffect on the drug solubility is likely to increase with the drugloading. Therefore, when modelling high drug loadings the con-centration dependency needs to be considered.

The front positions of the solid–drug interface of the slightlysoluble drug and the soluble drug are compared in Fig. 7C. Thesolid–drug interface of the soluble drug coincides with the solid–gel interface, due to the high solubility of the drug, the low drugloading and the fairly large solvent volume fraction near the solid–gel interface. The solid–drug interface of the slightly soluble drugis located between the solid–gel and gel–solvent interfaces.

5. Conclusions

The model presented in this paper can simulate the drug andpolymer release from swelling and dissolving polymer tablet. Asthe model generates both the front positions at the solid–gel, gel–solvent and solid–drug–solubilized-drug (solid–drug) interfacesand the drug and polymer release profiles, it is easily validated toreal data. The model was verified against drug release andpolymer dissolution data for the slightly soluble drug Methylparaben and the soluble drug Saligenin, showing good agree-ments in both cases. The drug diffusion coefficient was fitted todata, and the values obtained were considered to be accurate, thusconfirming the reliability of the model. The model sensitivitystudy revealed that when decreasing the drug diffusivity, themodel predicts that the relative importance of the convectivecontribution increases. As a result it is possible to obtain a drugrelease profile with an initial drug release rate faster than thepolymer dissolution rate, followed by a second stage with aslower drug release rate. The drug solubility and the drug loadingboth influence the drug release profile in similar manners. For aslightly soluble drug or a formulation with a high drug loading,the solid–drug interface follows the front position of the solid–gelinterface. For the other extreme case studied, a freely soluble drugor a formulation with a low drug loading, the solid–drug frontfollows the gel–solvent interface.

6. Symbols and units

Roman letters

A

Area m2

c

Concentration kg m−3

D

Diffusion coefficient m2 s−1

k

Mass transfer coefficient m s−1

m

Mass kg N Volumetric flux m s−1

R

Radius m t Time s V Volume m3

X

Finite volume length m y Volume fraction or normalized concentration – Y Fraction dissolved/released –

Greek letters

α

Axial fraction of volume change – δ Annular ring thickness m Φ Dissolution term m s−1

ρ

Density kg m−3

Subscripts

0

Initial condition (t=0) c Solid–gel boundary d Gel–solvent boundary diss Dissolution i Index of axial discretization j Index of radial discretization n Sample number P Number of finite volumes in axial direction Q Number of finite volumes in radial direction s Sample sat Saturation tbl Tablet

Superscripts

A

Drug component B Bulk C Convection cs Cross-section (axial direction) D Diffusion in Internal es Envelope surface (radial direction) out External p Polymer T Total w Solvent (water)

Acknowledgements

Bernt Nilsson, Department of Chemical Engineering, LundUniversity, is gratefully acknowledged for discussions ondetails regarding modelling of controlled-release formulations.The experimental part of this work was financed by the Centrefor Amphiphilic Polymers from Renewable Resources at LundUniversity (A.K.), by the Chalmers Bioscience Programmefunded by a strategic initiative of Chalmers University ofTechnology Foundation (A.L.), and by the Swedish ResearchCouncil (L.P.). Ms. Ann Bek is gratefully acknowledged forcarrying out the dissolution experiments.

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