Numerical Simulation of Controlled Nifedipine Releasefrom Chitosan Microgels

Hua Li, Guoping Yan, Shunnian Wu, Zijie Wang, K. Y. Lam

Institute of High Performance Computing, National University of Singapore, 1 Science Park Road, #01-01,The Capricorn, Singapore Science Park II, Singapore 117528

Received 4 September 2003; accepted 3 March 2004DOI 10.1002/app.20652Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: The numerical simulating investigation ofdrug delivery can provide deeper insight into drug releasemechanisms and elucidate efficiently the influences of vari-ous physical parameters. Controlled nifedipine release fromnonswellable chitosan sphere-based microgels is investi-gated numerically with a mathematical model in this article.The mathematical model takes into account both drug dis-solution and drug diffusion through the continuous matri-ces of the spherical microgels. Meshless Hermite-cloudmethod is employed to solve the formulated partial differ-

ential equations. It is seen that the model well describes thenifedipine release process from the spherical chitosan mi-crogels. The effect of several important physical parameterson drug release is evaluated, which include microsphereradius, equivalent drug saturation concentration, drug dif-fusion coefficient, and drug dissolution rate constant. © 2004Wiley Periodicals, Inc. J Appl Polym Sci 93: 1928–1937, 2004

Key words: modeling; simulation; drug delivery systems;kinetics; microgels

INTRODUCTION

Nifedipine is a poorly water-soluble drug with solu-bility less than 10 mg/L.1 As a well-known calciumchannel blocker, nifedipine is most commonly usedfor the treatment of hypertension, a chronic diseasethat affects 10–20% of the world population and in-duces cardiovascular complications.2 However, manyserious adverse effects associated with immediate ni-fedipine release have been revealed, such as hypoten-sion, myocardial ischemia or infarction, ventricularfibrillation, and cerebral ischemia.3 Given the serious-ness of the reported adverse events and the lack of anyclinical documentation attesting to a benefit, the Foodand Drug Administration (FDA) of the USA con-cluded that the use of immediately released nifedipinefor hypertensive emergencies is neither safe nor effec-tive, and therefore, should not be used.4

Microspheric drug release system has increasinglyattracted attention recently.5 Controlled nifedipine re-lease was investigated by using various polymer-based microgels, such as spherical chitosan micro-gels,6–7 Eudragit microcapsules,2,8 and poly(D,L-lac-tide-co-glycolide acid) microspheres.9–10 Obviously,the microspheric nifedipine release system reduces thetotal administration frequency to the patient. It isprobably cycled over a long period, or triggered by

specific environment or external events. Microsphericnifedipine release can maintain nifedipine at desiredlevels over a long period and thus eliminate the po-tential for both under- and overdosing. Consequently,it decreases the possible adverse effects of immediatenifedipine release. Additional advantages of micro-spherical nifedipine release include optimal adminis-tration dosage, better patient compliance, and im-proved drug efficacy.

The underlying drug release mechanisms for poly-mer-based microgels have obtained much under-standing. When drug-loaded polymeric microgels areplaced in contact with release medium, the drug re-lease process can be divided into these four consecu-tive steps2: the imbibition of release medium into themicrospherical system driven by osmotic pressurearising from concentration gradients, drug dissolu-tion, drug diffusion through the continuous matricesof microgels due to concentration gradients, and drugdiffusional and convective transport within the releasemedium. One or more of these steps can control thedrug release process. Varshosaz and Falamarzianclaimed that the drug release process could be via thediffusion through the continuous matrices or drugdissolution mechanism.11 In the diffusion mechanism,drug diffusion through the continuous matrices ofmicrogels controls the drug release process, whereas,in the dissolution mechanism, the drug release is con-trolled by the process involving drug dissolutionwithin the microgels followed by drug diffusionthrough the continuous matrices of microgels. How-ever, the drug release process is usually modeled with

Correspondence to: H. Li (lihua@ihpc.a-star.edu.sg).

Journal of Applied Polymer Science, Vol. 93, 1928–1937 (2004)© 2004 Wiley Periodicals, Inc.

the classical Fick’s diffusion equation integrating withappropriate boundary conditions or with the simpli-fied expressions developed by Higuchi and Higu-chi.12–15 A mathematical theory with simultaneousconsideration of drug dissolution and diffusion in thecontinuous matrices of microgels was put forward byGrassi et al.16 and fitted well to the experimentallymeasured temazeoan and medroxyprogesterone ace-tate release data. Recently, Hombreiro-Perez et al.2

pointed out that an adequate description of nifedipinerelease from microgels must consider drug dissolu-tion, drug diffusion in the continuous matrices of mi-crogels, and the limited solubility of nifedipine in therelease medium. Unfortunately, no effort was made tomodel the nifedipine release process because of thecomplexity.

The objective of this work was to investigate thenifedipine (Pliva, Croatia) release from chitosan mi-crogels (Protan Laboratories, Inc., Drammen, Norway)by a mathematical model incorporating this complex-ity. The drug diffusion coefficient and drug dissolu-tion rate constants were identified numerically by us-ing the model. The effect of several physical parame-ters on drug release is simulated and discussed indetail, which include microsphere radius, drug satu-ration concentration, drug diffusion coefficient, anddrug dissolution rate constant.

MATHEMATICAL MODEL ANDIMPLEMENTATION

Model development

The initial drug loading concentration C0 in sphericalmicrogels is generally greater than the drug saturationconcentration Cs. This can be achieved either by prep-aration of a solution and total evaporation of the sol-vent or by partial evaporation or phase inversion.12

When the polymeric microgels are put into a well-stirred release medium, the following four mass trans-fer steps take place consequently2: (1) drug dissolutionwithin the microgels; (2) drug diffusion within thematrices of microgels; (3) drug diffusion through theunstirred liquid boundary layers on the surfaces of themicrogels; and (4) drug diffusion and convectionwithin the release medium. Because the convectivetransport within the medium is usually fast comparedwith that of the diffusive mass, the convective trans-port can be neglected when calculating the overall rateof drug release from the polymeric microgels.2 There-fore, it is reasonable to assume that drug dissolutionand diffusion from the continuous matrices of spher-ical nonswellable microgels control the drug release ina well-stirred release medium.

The kinetics of drug release from the microgels withradius R can be simulated mathematically by the fol-lowing partial differential governing equation2

�C�r, t��t � D��2C�r, t�

�r2 �2r

�C�r, t��r �� k��Cs � C�r, t�� (1)

and the initial and boundary conditions below for thedrug release process in a well-stirred release medium,

t � 0 0 � r � R C�r, t� � �Cs (2a)

t � 0 r � 0�C�r, t�

�r � 0 (2b)

t � 0 r � R C�r, t� � 0 (2c)

where C(r, t) (g/cm3) is the drug concentration in theradial position r (cm) of the microgel system at therelease time t (s). D (cm2/s) is the drug diffusioncoefficient, k (s�1) is the first-order drug dissolutionrate constant, � is a parameter for the polymeric net-work meshes of microgels and is directly related to thecrosslinking density of the polymeric microspheres. IfCs (g/cm3) is defined as drug saturation concentrationin the system, �Cs (g/cm3) refers to the equivalentdrug saturation concentration in microgels with a net-work mesh parameter �.

The first term of the right-hand side in eq. (1) is thewell-known Fick’s second law of diffusion for a spher-ical system,17 which describes the diffusional drugrelease process in the microgels due to the continuousdissolution of the drug. The second term on the right-hand side in eq. (1) corresponds to the potential rate-limiting drug dissolution process.12 It is observed that,when the drug loading concentration C0 is smallerthan the drug saturation concentration Cs, eq. (1) isreduced to the classic Fick’s diffusion equation. Al-though the drug diffusion coefficient D in the poly-meric microgels may be solvent-concentration-depen-dent, usually it is reasonable to assume approximatelya constant D for simplicity.

It is assumed that the drug is uniformly distributedthroughout the microgels with equivalent drug satu-ration concentration �Cs initially. Under perfect sinkconditions, the release medium can be considered tobe well stirred; thus, the drug concentration outside ofmicrogels is further assumed to be constant and equalto zero.

By defining dimensionless parameters,12 � r/R fordimensionless radius, � Dt/R2 for dimensionlessFourier time, � � kR2/D called dimensionless disso-lution/diffusion number, and dimensionless concen-tration �(, ) � 1 � C(r, t)/�Cs, which indicates thenondimensional drug concentration additionally re-quired to reach saturation dissolution, the partial dif-ferential governing equation, and initial and boundaryconditions are thus rewritten in the dimensionlessforms as,

���, �

��

�2��, �

�2 �2

���, �

�� ���, � (3)

SIMULATION OF CONTROLLED NIFEDIPINE RELEASE 1929

� 0 0 � � 1 ��, � � 0 (4)

� 0���, �

�� 0 at � 0,

and ��, � � 1 at � 1 (5)

After solving the set of above governing equationsand conditions, �(, ) is obtained and then the drugconcentration C(r, t) is computed. According to Fick’sfirst law,18 the flux J � J(r, t), the rate of drug transferper unit area of section, is considered as

J�r, t� � �D�C�r, t�

�r (6)

The rate of drug release from the microgels is thuscalculated by18

�Mt

�t � AJ�r, t��r�R (7)

where A is the area of microgels with radius R, Mt

represents the amount of drug released after time tand can be calculated by integrating eq. (7)

Mt � �0

t

AJ�r, t���r�Rdt� � 4 R2 �0

t ��D�C�r, t��

�r �r�R

�dt�

� �4 R2D �0

t �C�r, t���r �

r�R

dt� (8)

Meshless Hermite-cloud method

The numerical technique employed in this work is thetrue meshless Hermite-cloud method, which is basedon the Hermite interpolation theorem and the fixedkernel approximation.19 Compared with the finite el-ement method as traditionally principal numericalanalysis tool for the modeling and simulation of wide-range science and engineering problems, the presentmeshless Hermite-cloud method is highly efficient be-cause it does not require the construction of meshes orelements with high computational cost. Moreover,compared with generic weak-form meshless tech-niques, the present Hermite-cloud method, as a collo-cation-based true meshless strong-form procedure, isa much simpler implementation because no weak-form integration is required.

By using the Hermite-cloud method, an approxima-tion f(x) of a one-dimensional unknown real functionf(x) can be constructed by

f�x� � �n�1

NT

Nn�x�fn � �m�1

NS �x � �n�1

NT

Nn�x�xn�Mm�x�gxm

(9)

where NT and NS (�NT) are the total numbers ofscattered points, covering both the interior computa-tional domain and the surrounding edges, for thepoint values fn and gxm, respectively. fn is the pointvalue of unknown real function f(x) at the nth point,and gxm is the point value of the first-order differential,gx(x) � df(x)/dx, at the mth point. The correspondingdiscrete approximation is written as

gx�x� � �m�1

NS

Mm�x�gxm (10)

In eq. (9), the shape function Nn(x) and Mm(x) for thefunctions f(x) and gx(x), respectively, are defined as20

Nn�x� � B�un�A�1�xk�BT�x�K�xk � un��Ln

n � 1, 2, . . . , NT (11)

Mm�x� � B�um�A�1�xk�BT�x�K�xk � um��Lm

m � 1, 2, . . . , NS (12)

in which �Ln is defined as the cloud length of the nthpoint, B(u) and ˜B(u) are the linearly independentbasis-function vectors which are given by

B�u� � �b1�u�, b2�u�, b3�u� � �1, u, u2 (13)

B�u� � �b1�u�, b2�u� � �1, u (14)

and A(xk) and ˜A(xk) are symmetric constant matricescorresponding to the central point xk of the fixed cloud

Aij�xk� � �n�1

NT

bi�un�K�xk � un�bj�un��Ln

�i, j � 1, 2, 3� (15)

Aij�xk� � �m�1

NS

bi�um�K�xk � um�bj�um��Lm �i, j � 1, 2�

(16)

where K(xk � u) is a kernel function constructed bycubic spline function as

1930 LI ET AL.

K�xk � u� �1

�Lk� 0 2 � z

�2 � z�3/6 1 � z � 2�2/3� � z2�1 � z/2� 0 � z � 1

(17)

where z � �(xk � u)�/�Lk.Because the additional unknown function gx(x) is

introduced, the auxiliary condition is required to im-plement the Hermite-cloud method. Imposing thefirst-order partial differential with respect to x on theapproximation f(x) expressed by eq. (9), and noting eq.(10), the auxiliary condition is obtained as

�n�1

NT

Nn,x�x�fn � �m�1

NS � �n�1

NT

Nn,x�x�xn�Mm�x�gxm � 0 (18)

where Nn,x(x) denotes the first derivative of Nn(x) withrespect to the variable x.

The formulation of the present Hermite-cloudmethod has thus far been completed and is able tosolve generic engineering partial differential bound-ary value (PDBV) problems, such as

Lf�x� � P�x� PDEs in computational domain (19)

f�x� � Q�x� Dirichlet boundary condition on �D (20)

�f�x�/�n � R�x� Neumann boundary condition on �N

(21)

where L is differential operator and f(x) is an unknownreal function. By using the point collocation technique,the approximation of the above PDBV problem can beexpressed in discrete forms as

Lf�xj� � P�xj� �j � 1, 2, . . . , N� (22)

f�xj� � Q�xj� �j � 1, 2, . . . , ND� (23)

� f�xj�/�n � R�xj� �j � 1, 2, . . . , NN� (24)

where N, ND, and NN are the numbers of scatteredpoints in the interior computational domain and alongthe Dirichlet and Neumann edges, respectively. Thus,total number of scattered points NT � N � ND � NN.

Substituting eqs. (9) and (10) into eqs. (22)–(24) andconsidering the auxiliary condition (18), a set of dis-crete algebraic equations with respect to unknownpoint values fi and gxi is derived in matrix form as

�Hij��NT�NS� �NT�NS��Fi�NT�NS� 1 � �di�NT�NS� 1 (25)

where {di} and {Fi} are (NT � NS)-order column vectors

�Fi�NT�NS� 1 � ��fi1 NT, �gxi1 NST (26)

�di�NT�NS� 1 � ��P�xi�1 N, �Q�xi�1 ND,

�R�xi�1 NN, �01 NST (27)

and [Hij] is a (NT � NS) (NT � NS) coefficient matrix

�Hij�

� ��LNj�xi��N NT L��xi � �

n�1

NT

Nn�xi�xn�Mj�xi��N NS

�Nj�xi��ND NT �0�ND NS

�0�NN NT �Mj�xi��NN NS

�Nj,x�xi��NS NT ���n�1

NT

Nn,x�xi�xn�Mj�xi�NS NS

�(28)

By solving numerically the set of linear algebraiceqs. (25), (NT � NS) point values {Fi} are obtained.Accordingly, the approximate solutions f(x) and thefirst-order differential gx(x) can be computed througheqs. (9) and (10), respectively.

Model implementation

To simulate the kinetics of drug release, the nondi-mensional partial differential governing equation isdiscretized in spatial domain by the Hermite-cloudmethod and discretized in time domain by linear in-terpolation technique. Namely, for the drug releasegoverning eq. (3) at time ,

���, �

��

�2��, �

�2 �2

���, �

�� ���, � (29)

By using the Hermite-cloud method for spatial dis-cretization, we have

��, � � �n�1

NT

Nn���n�� � �m�1

NS � � �n�1

NT

Nn��n�� Mm���m�� (30)

��, � ����, �

�� �

m�1

NS

Mm���m�� (31)

�2��, �

�2 � �n�1

NT

Nn,���n�� (32)

SIMULATION OF CONTROLLED NIFEDIPINE RELEASE 1931

Substituting eqs. (30)–(32) into eq. (29), the govern-ing equation is discretized at i of spatial domain inthe following form,

���i, �

�� �

n�1

NT

�Nn,�i� � �Nn�i���n��

� � 2i

� ��i � �n�1

NT

Nn�i�n�� �m�1

NS

Mm�i��m��

(33)

By the linear interpolation technique, a weightedaverage of the time derivative ��/� can be approxi-mated at two consecutive time steps as follows,21

�1 � �����i, �

�� �

���i, � ��

�

���i, � �� � ��i, �

�(34)

where � is a weighted coefficient (0 � � � 1).Substituting eqs. (30) and (33) into eq. (34) and

considering the auxiliary condition (18), the governingequation with time iteration is finally discretized inboth spatial and time domains and reduced to a set ofdiscrete algebraic equations in the following matrixform,

�Gij11�NT NT �Gij

12�NT NS

�Gij21�NS NT �Gij

22�NS NS� ��i� � ��NT 1

��i� � ��NS 1�

� �G*ij11�NT NT �G*ij12�NT NS

�G*ij21�NS NT �G*ij22�NS NS� ��i��NT 1

��i��NS 1� (35)

where

�Gij11� � ��1 � ����Nj�i� � ��Nj,�i�� (36a)

�Gij12� � �

2��

i� �1 � ����

� �i � �n�1

NT

Nn�i�n�Mj�i� (36b)

�Gij21� � �G*ij21� � �Nj,�i�� (36c)

�Gij22� � �G*ij22� � �� �

n�1

NT

Nn,�i�n�Mj�i� (36d)

�G*ij11� � ��1 � ���Nj,�i�

� �1 � ��1 � ����Nj�i�� (36e)

�G*ij12� � �2�1 � ���

i� �1 � ��1 � ����

� �i � �n�1

NS

Nn�i�n��Mj�i� (36f)

NUMERICAL SIMULATION

The experimentally measured nifedipine release datafor the spherical nifedipine-loaded chitosan microgelsexposed to phosphate buffer (pH 7.4), achieved byFilipovic-Grcic et al. through the chitosan microgelpreparation and characterization and nifedipine re-lease determination,6 are simulated numerically by thepresent mathematical model. A series of B samples(B1–B5) are selected and the corresponding micro-sphere radii R are listed in Table I.6 The total loadeddrug mass M� listed in Table I is calculated based onthe mass of drug-loaded microgels m (g), total drugcontent d (%), the mean radius of dry microgels R(cm), and the volume of the dissolution medium V(cm3) obtained from the experimental data.6 Assum-ing that all the drug is released and dissolved in thedissolution medium

M� �43 � � R3 �

m � dV

One is reminded that such an assumption will bringabout unpredictable error in M�; however, because theratio of Mt/M� is concerned in this study, this inac-curacy would not affect the prediction.

TABLE IExperimental and Identified Parameters of Nifedipine Microgels

Type

Experimental data6 Identified parameters

R ( 10�4 cm) M� ( 10�13 g) D ( 10�11 cm2/s) k ( 10�7 s�1) �Cs ( 10�6 g/cm3)

B1 12.10 0.20 0.40 7.0 1.225B2 13.90 0.24 0.40 7.0 1.225B3 13.05 0.20 0.35 7.0 1.033B4 12.20 0.16 0.30 7.0 0.823B5 14.50 0.32 0.40 7.0 1.225

1932 LI ET AL.

Identification of physical parameters of model

Figure 1 shows the in vitro drug release kineticsfrom the microgels with different microsphere radiiand total initially loaded drug amounts. As the timeincreases, the initial drug release amount increasesrapidly, followed by a gradual drug release. B5 haslarger microspheric radius R and higher total loadeddrug amount M� (see Table I). It is observed thatnifedipine from B5 is released a little faster than thatfrom B1. Good agreement between fitted results andexperimental data is obtained for both B1 and B5with the present mathematical model. It can be seenthat the model successfully captures the effect ofmicrospherical radius R. The values of the diffusioncoefficient D, drug dissolution rate constant k, andequivalent drug saturation concentration �Cs areidentified by best-fitting the computed results to theexperimental data. The identified D, k, and �Cs aresummarized in Table I. The D value is found to besmaller than the reported value of nifedipine incrosslinked hydrogels of polyacrylamide-graftedguar gum.22 Generally, the D values for variousdrugs in polymeric hydrogels range from 10�6 to10�9 cm2/s. Several factors may contribute to theextremely low D value of nifedipine in the studiedmicrogels. First, the solubility of nifedipine in therelease medium is very low (11 �g/mL), whichimplies relatively a large partition coefficient of ni-fedipine between the polymeric hydrogels and therelease medium. Second, the microgels that areloaded with a high content of drug tend to absorbless water than those containing a lower content ofdrug; therefore, the diffusion processes are re-tarded. Last, it is reported that an increase in drug

content will also increase the crystallinity of thedrug and thus slow down the release of such acrystalline drug.23

Figure 2 illustrates the nifedipine release kineticsfrom chitosan microgels with a different networkmesh parameter �. These microgels are formed withthe same nifedipine amount but different glutaralde-hyde reaction times. With an increase in the glutaral-dehyde reaction time, the crosslinking degree of themicrogels increases, which brings about a decrease inthe network mesh parameter � of the microgels. Thisresults in the decrease of the equivalent saturationconcentration �Cs. The fitting well represents the ex-perimental results. The corresponding D and �Cs areidentified by best-fitting the calculating results to theexperimental data, and they are summarized in TableI. The diffusion coefficient D is observed to be depen-dent on the crosslinking density, and thus, on thenetwork mesh parameter �. An increase of �Cs from0.823 to 1.225 increased the D value from 0.30 10�11

to 0.40 10�11 cm2/s. An increased network meshparameter � increases the drug diffusion coefficient D,which is consistent with the findings by Pillay andFassihi.24

Figures 1 and 2 validate that the present mathemat-ical model is able to describe well the nifedipine re-lease from chitosan microgels with different micro-sphere conditions. It successfully captures the charac-teristics of the various important physical parametersaffecting the nifedipine release kinetics. Therefore, it isconcluded that this model provides a suitable simu-lating approach to obtain deeper insight into the ef-fects of the important physical parameters on drugrelease from the microgels.

Figure 1 Rate of nifedipine release from chitosan microgels with different radii R.

SIMULATION OF CONTROLLED NIFEDIPINE RELEASE 1933

Discussion about effect of physical parameters ofmodel

To study the effect of the physical parameters, includ-ing the microsphere radius R, drug diffusion coeffi-cient D, drug dissolution rate constant k, and equiva-lent saturation concentration �Cs, the mathematicalmodel is employed to simulate drug release kinetics.When the sensitivity analysis on one of the parametersis carried out, the other parameters remained the

same. In Figures 3-6, the experimental drug releasedata from the B5 sample are taken as a comparativeexample.

Figure 3 shows the effect of microspherical radius Ron controlled drug release, where the drug diffusioncoefficient D is set equal to 0.4 10�11 cm2/s; drugdissolution rate constant k is set equal to 7.0 10�7

s�1, and drug equivalent saturation concentration �Cs

is set equal to 1.225 10�6 g/cm3. The microsphere

Figure 2 Rate of nifedipine release from chitosan microgels with different network mesh parameter �.

Figure 3 Effect of the microsphere radius R on the rate of nifedipine release from chitosan microgels when D � 0.4 10�11

cm2/s, k � 7.0 10�7 s�1, �Cs � 1.225 10�6 g/cm3.

1934 LI ET AL.

radius R in the figure ranges from 9.5 10�4 to 17.0 10�4 cm. By increasing the microsphere radius R,the overall drug release rate becomes slower. It isnoted that a slight change of microsphere radius Rresults in remarkable alteration of nifedipine releaserate. A smaller microgel has larger specific surfacearea of contact with the release medium and wouldfacilitate the drug diffusion through the continuousmatrices of microgels into the release medium in com-parison with a larger microgel. A decrease in micro-sphere radius increases both the initial fast release rateand the following gradual release rate. This impliesthat variation of microspheric radius affects both drugdissolution and diffusion process.

The effect of equivalent drug saturation concentra-tion �Cs on drug release kinetics is shown in Figure 4,where the equivalent drug saturation concentration�Cs changes from 0.5 10�6 to 1.7 10�6 g/cm3. Thedrug release is remarkably increased with increasingporosity. An increase of the network mesh parameter� of microgels (i.e., a decrease of the crosslinking den-sity of microgels) simultaneously increases the drugequivalent saturation concentration �Cs and the drugdiffusion coefficient D, which results in an increase inboth drug dissolution and diffusion rates. These syn-ergically lead to the increased nifedipine release rate.

Figure 5 is obtained for discussion about the effectof drug diffusion coefficient D on drug release, wherethe D value varies from 0.012 10�11 to 4.0 10�11

cm2/s. It is observed that the diffusion process con-trols significantly the initial drug release stage. As thediffusion coefficient D increases, the initial drug re-lease rate increases distinctly, and the drug release

amount becomes a linear function of time at relativelyshorter times. However, after a certain period of re-lease time, the drug release arrives at a constant level.This indicates that the release of drug with low diffu-sion coefficient D is via diffusion mechanism, whereasthe diffusion process does not solely control the re-lease of drug with high diffusion coefficient D.

Figure 6 illustrates the effect of the drug dissolutionrate constant k on drug release, where the value ofdissolution rate constant k varies from 0.70 10�7 to24.0 10�7 s�1. It is manifest that alteration of drugdissolution rate constant k has insignificant effects oninitial drug release rate. However, after a period ofrelease time, the drug release rate increases with in-creasing dissolution rate constant k. This indicates thatdifferent mechanisms control different drug releasestages. At the initial stage, diffusion through continu-ous matrices of microgels predominantly affects thedrug release, but after a period of drug release, drugdissolution starts to significantly affect the drug re-lease rate.

CONCLUSION

The present mathematical model based on drug dis-solution and diffusion through the continuous matri-ces of microgels is well validated by comparison withthe experimental nifedipine release from the sphericalnonswellable chitosan microgels. It provides a betterunderstanding of the underlying mechanisms in mi-crospheric drug release system and represents a high-efficiency tool to study the roles played by importantcharacteristics of the drug and the microgels such as

Figure 4 Effect of the equivalent drug saturation concentration �Cs on the rate of nifedipine release from chitosan microgelswhen D � 0.4 10�11 cm2/s, R � 14.5 10�4 cm, k � 7.0 10�7 s�1.

SIMULATION OF CONTROLLED NIFEDIPINE RELEASE 1935

the diffusion coefficient, the microspherical radius,and the network mesh parameter. Consequently, itcan be used to analyze and optimize the design of thecontrolled drug release process.

NOTATION

A area of microgels, cm2

C concentration of solute dissolved in microgels,g/cm3

C0 initial solute loading in microgels, g/cm3

Cs drug saturation concentration in microgels, g/cm3

D drug diffusion coefficient, cm2/sJ drug diffuse flux, g/cm2/sk dissolution rate constant, s�1

Mt absolute cumulative amount of drug released attime t, g

M� absolute cumulative amount of drug released attime t � �, g

Figure 5 Effect of the diffusion coefficient D on the rate of nifedipine release from chitosan microgels when R � 14.5 10�4

cm, k � 7.0 10�7 s�1, �Cs � 1.225 10�6 g/cm3.

Figure 6 Effect of the dissolution rate constant k on the rate of nifedipine release from chitosan microgels for R � 14.5 10�4

cm, D � 0.4 10�11 cm2/s, �Cs � 1.225 10�6 g/cm3.

1936 LI ET AL.

r radial position in spherical microgels, cmt release time, s� a weighted coefficient (0 � � � 1)� dimensionless dissolution/diffusion number� network mesh parameter of microgels�Cs equivalent drug saturation concentration, g/cm3

dimensionless radius dimensionless Fourier time� dimensionless concentrationm mass of drug-loaded microgels, gd total drug content, %V volume of the dissolution medium, cm3

R mean radius of dry microgels, cm

References

1. Liu, L. X.; Ku, J.; Khang, G.; Lee, B.; Rhee, J. M.; Lee, H. B. JControlled Release 2000, 68, 145.

2. Hombreiro-Perez, M.; Siepmann, J.; Zinutti, C.; Lamprecht, A.;Ubrich, N.; Hoffman, M.; Bodmeier, R.; Maincent, P. J Con-trolled Release 2003, 88, 413.

3. Mansoor, A. F.; von Hagel Keefer, L. A. Pharmacol Ther 2002,27, 362.

4. Grossman, E.; Messerli, F. H.; Grodzicki, T.; Kowey, P. JAMA1996, 276, 1328.

5. Okada, H.; Yamamoto, M.; Heya, T.; Inoue, Y.; Kamei, S.;Ogawa, Y.; Toguchi, H. J Controlled Release 1994, 28, 121.

6. Filipovic-Grcic, J.; Becirevic-Lacan, M.; Skalko, N.; Jalsenjak, I.Int J Pharm 1996, 135, 183.

7. Chandy, T.; Sharma, C. P. Biometerials 1992, 13, 949.8. Chowdary, K. P. R.; Sankar, G. G. Drug Dev Ind Pharm 1997, 23,

325.9. Dhawan, S.; Singla, A. K. Biotechnic & Histochem 2003, 78, 243.

10. Soppimath, K. S.; Aminabhavi, T. M. J Microencapsul 2002, 19,281.

11. Varshosaz, J.; Falamarzian, M. Eur J Pharmaceutics and Bio-pharmaceutics 2001, 51, 235.

12. Harland, R. S.; Dubernet, C.; Benoit, J. P.; Peppas, N. A. JControlled Release 1988, 7, 207.

13. Higuchi, T. J Pharm Sci 1961, 50, 874.14. Roseman, T. J.; Higuchi, W. I. J Pharm Sci 1970, 59, 353.15. Higuchi, W. I. J Pharm Sci 1962, 51, 802.16. Grassi, M.; Colombo, I.; Lapasin, R. J Controlled Release 2000,

68, 97.17. Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon

Press: Oxford, 1975.18. Robert, A. A. Physical Chemistry, 2nd ed.; Wiley: New York,

1996.19. Li, H.; Ng, T. Y.; Cheng, J. Q.; Lam, K. Y. Comput Mech 2003, 33,

30.20. Ng, T. Y.; Li, H.; Cheng, J. Q.; Lam, K. Y. Eng Struct 2003, 25, 141.21. Reddy, J. N. An Introduction to the Finite Element Method, 2nd

ed.; McGraw-Hill: New York, 1993.22. Soppimath, K. S.; Kulkarni, A. R.; Aminabhavi, T. M. J Con-

trolled Release 2001, 75, 331.23. Soppimath, K. S.; Kulkarni, R. A.; Sminsbhavi, T. M. J Biomat Sci

Polym Ed 2000, 11, 27.24. Pillay, V.; Fassihi, R. Pharm Res 1999, 16, 333.

SIMULATION OF CONTROLLED NIFEDIPINE RELEASE 1937