Effect of Interparticulate Interaction on Release Kinetics of ...

Effect of Interparticulate Interaction on Release Kinetics of Microsphere Ensembles X. Y. WU,* G. ESHUN,

AND

Y. ZHOU

Contribution from the Faculty of Pharmacy, University of Toronto, Toronto, Ontario, Canada M5S 2S2. Received September 5, 1997. Final revised manuscript received February 5, 1998. Accepted for publication February 5, 1998. Abstract 0 The release kinetics of microsphere ensembles is complicated by the mutual influence of the microspheres which are entrapped in small compartments such as body cavities. This work focused on the effect of interparticulate interaction on the release kinetics of microsphere ensembles with limited spreading. Experiments and finite element modeling were conducted to investigate diffusional drug release from a single sphere, a monolayer, and multiple layers of microspheres. Poly(methyl methacrylate-co-methacrylic acid) (P(MMA/MAA)) microspheres and azidothymidine (AZT) were used in the experiments. The order of the release rate of AZT from various microsphere populations was observed to be single sphere > monolayer > multiple layers. This evidenced the importance of interparticulate interaction. The finite element simulations elucidated the influence of various factors on the release kinetics of microsphere ensembles including the separation distance, location of the spheres, and the drug accumulation in the medium. Calibration of overall release kinetics for the neighboring effect was proposed on the basis of the spreading factor. Overall release profiles of microsphere ensembles were predicted using the release profiles of individual microspheres at various locations.

Introduction Microspheres have been extensively used for both systemic and regional drug delivery.1-5 Compared with singleunit dosage forms such as tablets, microspheres are advantageous because of less risk of “dose dumping” and more flexibility in administration route and dose. However, prediction of drug release kinetics of these multipleunit pellets is much more difficult because of the heterogeneity in a population and the interparticulate interactions, though the release kinetics for a single sphere is welldescribed.6-9 The effect of heterogeneity on the overall release kinetics of microcapsules has been investigated statistically.10-15 Assumptions have been made in these studies such as that the microcapsules release their payload independently and sink condition or constant drug solution concentration is maintained during the release process. In other words, the mutual influence of neighboring particles and drug accumulation in the medium on the release kinetics have been neglected, perhaps due to the complexity of mathematical modeling. In real situations, the effect of drug accumulation and interparticulate interaction on the release kinetics can be significant because of poor mixing and close packing. It is well-understood that in the case of limited mixing, drug accumulates in the unstirred boundary layer adjacent to the surface of dosage forms, thus reducing the release * Corresponding author. Tel. (416)978-5272. Fax: (416)978-8511. E-mail: [email protected].

586 / Journal of Pharmaceutical Sciences Vol. 87, No. 5, May 1998

rate.6,16-18 In the case of well-stirred liquid of finite volume, drug accumulation can also occur in the bulk liquid resulting in a lower release rate than that in a perfect sink.6-9,18-21 This effect of drug accumulation could be more profound when the microsphere ensembles are applied in vivo because of the limited separation between the particles. It has been shown by γ-scintigraphy that, in the gastrointestinal tract, multiple-unit pellets migrate together without much spreading.22-25 Some other studies of γ-scintigraphy indicate that the pellets separate at the beginning and then come together as they reach the colon. Even less spreading may be expected when the particles are injected subcutaneously, intramuscularly, or intratumorally. In these cases, simultaneous drug release from the neighboring particles could enhance the drug accumulation and consequently reduce the release rate. Although the influence of drug accumulation on the release kinetics of single devices has been studied,6-9,19-21 this kind of neighboring effect has not drawn much attention so far. The emphasis of this work was on the effect of interparticulate interaction, i.e., the neighboring effect, on the overall release kinetics of microsphere ensembles. For such a kinetic problem, multicenter release and finite liquid volume are involved and no analytical or semianalytical solutions are available to date. The finite element and experimental models were developed to investigate the dependence of release kinetics of microsphere ensembles on the drug accumulation and interparticulate interaction. The results of this work are expected to facilitate the design of multiparticulate delivery systems based on the better understanding of release kinetics of microsphere ensembles.

Experimental Section Materials and Model DrugsMethyl methacrylate (MMA), methacrylic acid (MAA), and ethylene glycol dimethacrylate (EGDMA) were purchased from Aldrich Chemicals. Azidothymidine (Sigma) was selected as the model drug because it is one of the most effective anti-HIV drugs to date and has a short half-life (1.1 h). Preparation of P(MMA/MAA) MicrospheressPoly(methyl methacrylate-co-methacrylic acid) [P(MMA/MAA)] microspheres were selected as model microspheres because their chemical composition is similar to Eudragit L and Eudragit S resins (Ro¨hn Pharma Inc.). These resins are commercial pharmaceutical excipients from which drug products have been developed and approved by the FDA. In this study, P(MMA/MAA) microspheres with a MMA/MAA weight ratio of 30:70 were prepared by suspension free radical polymerization of MMA and MAA.18,26,27 The polymerization was carried out at 70 °C using tert-butyl peroxy-2-ethylexanoate (ATOCHEM) as an initiator and EGDMA as a cross-linking agent. After the polymerization, the microspheres were cleaned by washing and extensive extraction and then dried in a vacuum oven. The microspheres were fractionated with sieves and their diameters were measured by Wild M420 stereomicroscope equipped with a Wild MMS 235 digital optical accessory.18,27

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Measurement of Solubility and Partition CoefficientsThe solubility of AZT in pH 7.4 phosphate buffer at 37 °C was determined from the equilibrium drug concentration in the solution. Partition of the drug between the microspheres and the buffer solution was evaluated from the slope of the plot of drug concentration in the microspheres against that in the solution at equilibrium, assuming that the partition coefficient was independent of the drug concentration in the concentration range of interest. Drug Loading into the MicrospheressThe solution sorption method12 was used to load AZT into the microspheres. For a typical monolayer experiment, 50 mg of dry spheres was equilibrated in 1 mL of 15.3 mg/mL solution of AZT in pH 7.4 Sorensen buffer (ionic strength 0.15 M) and then filtered through a sieve and quickly rinsed off with deionized water right before the release tests. For the multilayer experiment, 200 mg of dry spheres was soaked in buffer solutions of AZT and treated similarly as above. Determination of the Release KineticssAll the release experiments were carried out using preswollen, AZT-loaded microspheres. Release kinetics of single spheres was determined in a UV cuvette containing pH 7.4 buffer at 37 °C by a UV-vis spectrophotometer (Hewlett-Packard HP8452A) equipped with kinetics software. Vigorous mixing of the solution was obtained with a built-in magnetic stirrer, which was believed to be the best mixing available in the cuvette setting for minimizing the boundary layer around the spheres. The accuracy of the instrument is (0.005 au (absorbance unit). The absorbance unit of the AZT solution ranged from 0.1 to 2.2 during the release process, far above the limit of detection and within the linear range of the instrument (au ) 0.0022-3.3). Release profiles of microsphere ensembles were measured at two different settings as illustrated in Figure 1: (a) unrestricted setting, the USP dissolution protocol using a basket was adapted; and (b) semirestricted setting, where microspheres were entrapped in a sample holder whose top and side were closed by Teflon, allowing the drug to diffuse out only from the screen on the bottom. The basket or sample holder was connected with a motor rotating at 50 rpm, in which the microspheres were packed to form a monolayer or multiple layers. The experiments were performed in a vessel containing a certain volume (e.g. 300 mL) of pH 7.4 buffer at 37 °C. The dissolution solution was pumped through flow cells where the drug concentration was monitored by UV spectrophotometer. To calibrate the delay in the transport of solution from the vessel to the flow cells, manual sampling was also applied. In this case, fresh buffer solution of equal volume was added to keep the total volume constant.

Finite Element Modeling

Figure 1sIllustrative schemes of experimental settings for multiparticulate release.

and Ω is the region of the liquid. The definition of these regions is illustrated in the following schematic diagram:

The corresponding boundary and initial conditions are

∂C ) 0, (x,y,z) ∈ Φ, t g 0 ∂n

(3)

Cm ) C0, (x,y,z) ∈ Γi, t ) 0

(4)

C ) 0, (x,y,z) ∈ Ω, t ) 0

(5)

where n represents the outward normal to the closed boundary Φ, i.e., the boundary of the whole system, and C0 is the initial drug loading. Equation 3 defines a system of finite volume, i.e., a closed system with no mass transfer beyond the boundary, differing from a perfect sink. Equations 4 and 5 give the initial conditions of the microspheres and the liquid, respectively. For drug concentration in an unstirred boundary layer, eqs 1 and 2 are coupled through two boundary conditions at the interface of liquid and spheres6,28,29 based on the conservation law of mass that the diffusive flux must be continuous at the interface

Cδ- ) KCmδ+ Drug diffusion in a finite domain consisting of microspheres and liquid can be described by the following governing equations,6-9

(

)

∂Cm ∂2Cm ∂2Cm ∂2Cm + + ) Dm ∂t ∂x2 ∂y2 ∂z2 in the microspheres, (x,y,z) ∈ Γi (1) (i ) 1, 2, ..., N for an ensemble consisting of N microspheres)

(

)

∂C ∂2C ∂2C ∂2C )D 2 + 2 + 2 ∂t ∂x ∂y ∂z

in the liquid, (x,y,z) ∈ Ω (2)

where Cm, C, Dm, and D denote drug concentration and diffusion coefficients of a drug in the microspheres and in the liquid, respectively; x, y, and z are the Cartesian coordinates; Γi denotes the region of the ith microsphere;

(

D

)

(

(6)

)

∂Cδ- ∂Cδ- ∂Cδ∂Cmδ+ ∂Cmδ+ ∂Cmδ+ + + + + ) Dm ∂x ∂y ∂z ∂x ∂y ∂z

(7) where δ- and δ+ denote adjacent values at the interface and K is the partition coefficient of the drug between the spheres and the liquid. As mentioned previously, the problem treated here involves multicenter release and variable boundary conditions for each sphere, that is, drug concentration in the surrounding medium is a function of time and location in the ensemble. For example, drug accumulates in the surrounding medium more significantly at the center than that at the edge of an ensemble due to a longer distance of diffusion and stronger neighboring effect. To date, analytical or semianalytical solutions are only available for 1-D diffusion from a single sphere into a perfect sink or a wellstirred medium. Therefore, three-dimensional finite element models were developed to simulate microsphere ensembles of a monolayer or multiple layers with cubic packing geometry. Figure 2 illustrates the typical models Journal of Pharmaceutical Sciences / 587 Vol. 87, No. 5, May 1998

Figure 3sFractional release of AZT from microspheres of P(MMA/MAA) in pH 7.4 buffer at 37 °C. Comparison of the release profiles of various populations: 4, single sphere; b, unrestricted monolayer, 200 spheres; 9, unrestricted multilayer; [, semirestricted multilayer.

Figure 2sIllustration of finite element models of (a) a multilayer of microspheres and (b) a monolayer of microspheres (only 1/16 of the population is shown), where the numbers identify the locations of the spheres which will be used in later presentation.

of multilayer and monolayer of microspheres used in the present study, where only 1/16 of the population is shown. The numbers in the figure identify the spheres at various locations which will be used for comparison and calculation later. Outside the microspheres is a stagnant or poorly mixed surrounding medium whose elements are not shown here for clarity. The volume of the medium varies, resulting in different values of effective volume ratio, λ, which is defined as

λ ) V/VmK

(8)

where V and Vm are the volume of liquid and microspheres, respectively, and K is the partition coefficient. The computational procedures of the finite element method used in the previous work19,20 were modified and applied in the present studies. Two different environmental conditions were simulated which often exist in vivo: (a) diffusion into a stagnant liquid and (b) diffusion through a stagnant boundary layer and then into a liquid with certain degree of mixing.

Results and Discussion Experimental ResultssThe solubility of AZT in the pH 7.4 buffer at 37 °C was 29.5 mg/mL, and its partition coefficient between P(MMA/MAA) microspheres and the buffer solution at 37 °C was determined to be K ) 1.22. The microspheres used in the release studies were of diameters d ) 0.134 cm for the single sphere tests, d ) 0.134 ( 0.036 cm for the monolayer tests, and d ) 0.125 ( 0.036 cm for the multilayer tests. The average diameters of the multiparticulate samples were determined from 30 randomly selected spheres. Release profiles of AZT from various populations of preswollen P(MMA/MAA) microspheres are shown in Figure 3. As expected, a single microsphere gives the highest release rate due to better mixing and no interaction from other particles. Next to this is the monolayer and then the multilayer of microspheres in an unrestricted basket. 588 / Journal of Pharmaceutical Sciences Vol. 87, No. 5, May 1998

Figure 4sInfluence of initial drug concentration in the medium on the release rate of AZT from a single P(MMA/MAA) microsphere in pH 7.4 buffer at 37 °C.

The drug release rate is the lowest when the microspheres were packed into multiple layers in a semirestricted sample holder, because of extra layers of diffusion barrier, greater buildup of drug concentration, and the limited surface area for drug to diffuse out. Figure 4 shows the fractional release for a single microsphere varying with the initial AZT concentration in the medium. The release rate decreases as the initial concentration increases from 0 to 38.5 µg/mL, evidencing the importance of drug accumulation in the reduction of the release rate. This result is consistent with our previous observation on the release rate of theophylline from a controlled release tablet, which also decreases with increase in drug concentration in the medium. In both cases, the total drug concentration is much lower than 1/10 of the drug solubility and thus should not affect the release rate as predicted by the rule of thumb. This raises the question whether drug concentration influences the drug diffusivity or the thickness of the diffusion boundary layer, which is presently under investigation in our laboratory. To confirm the release mechanism of AZT from the preswollen P(MMA/MAA) microspheres and evaluate the diffusion coefficient of AZT in the spheres, Crank’s analytical model for diffusional drug release from a single sphere into a well-stirred finite volume was applied6

Mt M∞

∞

)1-

∑

n)1

6R(R + 1) 9 + 9R +

exp(-Dmqn2t/a2) (10)

qn2R2

where Mt/M∞ is the fraction of drug released at time t, a is

Figure 5sModel predicted and experimentally determined fractional release of AZT from a single P(MMA/MAA) microsphere into a well-stirred finite volume. Parameters used in the model fitting: Dm ) 1.65 × 10-6 cm2/s, C0 ) 0.035 g/cm3, K ) 1.22, and λ ) 1300.

the radius of the sphere (a ) 0.067 cm in the present study), qn are the nonzero roots of

tan q )

3q 3 + Rq2

(11)

and R ) λ, which is defined in eq 8. Experimentally determined parameters K ) 1.22, C0 ) 0.035 g/cm3, and λ ) 1300 were used for the model fitting. The diffusion coefficient of AZT in the microspheres at 37 °C was then evaluated, Dm ) 1.65 ( 0.05 × 10-6 cm2/s, from the duplicate experimental data by nonlinear regression. Figure 5 plots the experimental data together with the theoretical and the finite element model predictions. Both analytical and the numerical models fit the data very well, indicating that the release of AZT was controlled by diffusion. Besides, this result has also verified the computational procedures of FEM. Results of Finite Element SimulationssVerification of Computational Procedures and ProgramssThe problem analyzed in this work involves three-dimensional (3D) diffusion. Therefore, an analytical solution for heat conduction in a 3D brick geometry30 was employed to test the finite element solutions. The notation of the heat conduction was modified for the diffusion problem correspondingly. The drug concentration at location (x,y,z) and time t is then given by

C(x,y,z,t) ) ψ(x,a)ψ(y,b)ψ(z,c) ) ∞ 64 C0 π3 l)0 (2l + 1)πx

∞

∞

(-1)l+m+n

∑ ∑ ∑ (2l + 1)(2m + 1)(2n + 1) ×

cos

m)0 n)0

(2m + 1)πy cos

2a

2b

cos

(2n + 1)πz -R t e l,m,n (12) 2c

where

Rl,m,n )

[

]

2 (2m + 1)2 (2n + 1)2 Dπ2 (2l + 1) + + 4 a2 b2 c2

(13)

Figure 6a shows the drug concentration versus time at various locations in a 3D brick under the sink condition calculated by FEM and the analytical solution using D ) 7.66 × 10-6 cm2/s, a ) b ) c ) 0.2 cm, and C0 ) 0.2 g/cm3. Figure 6b compares the drug concentration distribution along the x-direction at various time intervals calculated

Figure 6s(a) Drug concentration vs time at various locations in a 3D brick under the sink condition calculated by FEM and by the analytical solution (Carslaw and Jaeger). (b) Drug concentration distribution along the x-direction in a 3D brick at various time intervals calculated by FEM and the analytical solution (Carslaw and Jaeger).

by the two methods. Excellent agreement between the finite element and the analytical solutions was observed, confirming the validity of the computational procedures. Effect of Separation DistancesFigure 7 shows the fractional release from three microspheres of the same diameter, d ) 0.134 cm, at different locations of a monolayer (see also Figure 2): V8, at the center; V9, next to the center; and V10, at the edge. The curves were obtained using Dm ) 1 × 10-6 cm2/s, diffusion coefficient in the liquid, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, λ ) 60, and L ) 0.01, 0.05, and 0.1 cm, where L is the separation distance of two immediate microspheres defined as the shortest distance between the surfaces of the two spheres. It is seen that the drug release from V10 is the fastest, followed by V9, and then V8, indicating the smallest diffusion barrier for V10 and the largest for V8. The figure also depicts that, as L increases from 0.01 to 0.1 cm, the release rate increases and the difference in release profiles of the three microspheres is almost diminished. Nevertheless, this can only be considered as equal interaction between the two spheres since the overall release rate is still noticeably lower than the single one, as will be discussed below. In addition, it is worth noting that the curves for V8 and V9 do not show much discrepancy, suggesting the possibility of approximating the release curves of microspheres at inner locations using either V8 or V9. The effect of separation distance on the overall release kinetics is manifested by Figure 8 where the overall release curves for microspheres of diameter 0.134 and 0.07 cm and various L values are compared with those of single spheres. It is shown that the overall release kinetics of microsphere ensembles differs from the single release kinetics considerably, even if they consisted of the same spheres. This Journal of Pharmaceutical Sciences / 589 Vol. 87, No. 5, May 1998

Figure 7sInfluence of separation distance on the release kinetics of individual microspheres at different locations in the monolayer: V8, at the center; V9, next to the center; V10, at the edge (d ) 0.134 cm, Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, λ ) 60).

Figure 8sOverall release kinetics of a monolayer of microspheres as a function of separation distance and compared with the release profile of a single sphere of the same diameter (Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, λ ) 60). (a) d ) 0.134 cm, (b) d ) 0.07 cm.

deviation decreases as the interparticulate distance increases. If one defines Lcr as the distance at which the neighboring spheres start to “feel” each other, apparently, 590 / Journal of Pharmaceutical Sciences Vol. 87, No. 5, May 1998

Figure 9sDrug accumulation in the medium between two neighboring microspheres, V8 and V9 (L ) 0.05, 0.10 cm, Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, d ) 0.134 cm, C0 ) 0.03 g/cm3, λ ) 60).

Lcr ≈ 0.15 cm for the microspheres of d ) 0.134 cm in the present study, as suggested by Figure 8a. In practice, a series of simulations could be performed for a multiparticulate system of interest to find out the value of Lcr which is important for approximation of overall release kinetics using the single one. Effect of Drug AccumulationsThe influence of separation distance can be interpreted as the result of drug accumulation in the medium between the microspheres. Figure 9 depicts the drug concentration between V8 and V9 as a function of distance and time. The figure shows that when L ) 0.05 cm, the drug diffusion fronts in the medium meet at a time as early as 10 s, while when L ) 0.1 cm, the diffusion fronts do not meet until 20 s. In the case of L ) 0.01 cm (data not shown here), drug concentration has already built up markedly at t ) 10 s. The separation distance does not only affect the time at which the spheres start to “feel” each other, but also the drug concentration in the medium between the spheres before it reaches equilibrium. As indicated by the figure, the drug concentration in the center of the medium between the spheres doubles as L decreases from 0.1 to 0.05 cm. This higher drug concentration is the main cause for the reduction of the release rate. Moreover, Figure 9 illustrates that the drug concentration in the medium increases at an earlier time and then decreases after a certain time, e.g., t > 200 s. This implies that, at an earlier time, the drug release is faster than the drug removal from the medium and, at later time, the removal rate is higher than the release rate. Generally, the relative rate of drug removal and drug release determines the drug accumulation. Crank6 and Baker and Lonsdale16 have described this relationship for a single slab geometry using the parameter J/(CsDliq), where Cs is the solute solubility and J is the solute flux from the surface of the delivery system. It is demonstrated that the solute accumulation increases with the increase in J and the

Figure 10sDependence of relative time on L/d for a monolayer of microspheres to reach the same fractional release as a single sphere (Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, λ ) 60).

Figure 11sOverall release profiles predicted by finite element models of single sphere, monolayer, and multilayer of microspheres (L ) 0.01 cm, d ) 0.134 cm, Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, λ ) 60).

decrease in Dliq.6,16 Similarly, our numerical simulation using various values of Dm, Dliq, and C0 also suggested that, in the microsphere ensembles, the drug accumulation increased and consequently the drug release rate decreased as Dm and C0 increased or as Dliq decreased. The details are not shown here for the sake of concise presentation. Calibration for the Interparticulate InteractionsThough the release profile of a single microsphere is useful to provide the release mechanism, calibration for the interparticulate interaction is needed if a single release profile is used to predict the overall release kinetics of a microsphere ensemble. To correlate the overall release profile with the single one, a relative time can be defined as the ratio of time for the ensemble and the single sphere to reach the same fractional release under the same conditions,

Figure 11 together with a curve for a single sphere under the same conditions. As expected, the release rate of a monolayer is slower than the single one, followed by that of multilayer, i.e., the release rate decreases as the interparticulate interaction becomes stronger. It is worth noting that the release profile of a monolayer consisting of 49 spheres is almost the same as that of 25 spheres. This suggests that it is reasonable to use the individual release profiles in a smaller population to approximate the release profiles in a larger population, or vice versa. Assuming that the individual microspheres at different locations could be used to construct a microsphere ensemble, a monolayer of 200 spheres was assembled using the spheres at various locations in the monolayer of 49 spheres. The monolayer consisted of four different layers of spheres as illustrated below.

tr )

tensemble tsingle

(14)

Besides, a spreading factor can be defined as the dimensionless distance, L/d. The correlation between tr and L/d is displayed in Figure 10 for microspheres of diameter d ) 0.07 and 0.134 cm at fractional release Mt/ M∞ ) 0.2, 0.5, and 0.8. It is shown that a correction of tr up to ∼2.3 times is required, depending on the values of d, L/d, and Mt/M∞ if a single release profile is used to approximate the overall release kinetics. Seemingly, more correction is needed for microspheres of d ) 0.07 cm at small L/d values, probably due to the larger specific surface area for the smaller spheres. Moreover, when L/d g 1, the tr deviates from unity, representing the single spheres only by about 10%, implying that the single release kinetics could be applied to approximate the overall release kinetics if the spreading factor is sufficiently large. It should be pointed out that in the present simulation, the same values of λ, Dm, and Dliq were used for single spheres and ensembles, that is, the single sphere released the drug under the same condition as the ensembles. In practice, the same experimental conditions are difficult to attain for single spheres and ensembles. For example, the hydrodynamic condition around a single sphere may differ from that around a monolayer or multiple layers of spheres, even if the stirring rate is kept identical. Therefore, extra calibration for experimental conditions may be required. Prediction of Overall Release Kinetics of Microsphere EnsemblessOverall release profiles of microsphere ensembles were evaluated from the finite element models of monolayer and multilayer using Dm ) 1 × 10-6 cm2/s, Dliq ) 1 × 10-5 cm2/s, C0 ) 0.03 g/cm3, L ) 0.01 cm, and λ ) 60. The fractional release of a monolayer of 25 spheres, 49 spheres, and a multilayer of 75 spheres is plotted in

It was estimated that, in the basket of radius about 1.01 cm, there were 41 spheres at the first outer layer, 35 at the second outer layer, and 29 at the third outer layer. The remaining 95 spheres were all treated as the spheres at the fourth outer layer. Then the release profiles of individual spheres at different layers in the model of 49 spheres, i.e., seven spheres in a row, were applied to compute the overall release profile of the monolayer of 200 spheres. To mimic the experimental condition, the finite element models of two liquid layers were used: one was the stagnant boundary layer close to the surface of the spheres, and the other was the bulk medium with mixing. The experimentally determined parameters Dm ) 1.65 × 10-6 cm2/s, C0 ) 0.0125 g/cm3, λ ) 976, and K ) 1.22 were used in the calculation of the release profiles. The model fitted parameters were Dliq,1 ) 1 × 10-5 cm2/s, Dliq,2 ) 5 × 10-5 cm2/s, where Dliq,1 and Dliq,2 are the diffusion coefficients in the boundary layer and in the bulk liquid, respectively. The computed fractional release was compared with the experimental data in Figure 12a. The results reveal that Journal of Pharmaceutical Sciences / 591 Vol. 87, No. 5, May 1998

diffusion problem with noticeable convection. In addition, there may be some other mechanisms involved such as size distribution and packing geometry of particles. Due to insufficient experimental investigation, we cannot draw a conclusion at this moment. We are presently developing a model for the multilayer with consideration of the contribution of convection and studying the influence of particle size distribution and packing geometry.

Conclusions

Figure 12sRelease profiles of AZT for P(MMA/MAA) microsphere ensembles predicted by the finite element models and measured by experiments. (a) A monolayer of about 200 spheres (average d ) 0.134 cm, L ) 0.01 cm, K ) 1.22, λ ) 976, Dm ) 1.65 × 10-6 cm2/s, Dliq,1 ) 1 × 10-5, Dliq,2 ) 5 × 10-5 cm2/s, C0 ) 0.0125 g/cm3); (b) a multilayer consisting of about 800 spheres (average d ) 0.125 cm, L ) 0.01 cm, K ) 1.22, λ ) 325, Dm ) 1.65 × 10-6 cm2/s, Dliq,1 ) 2 × 10-5 cm2/s, Dliq,2 ) 1 × 10-2 cm2/s, C0 ) 0.0082 g/cm3).

the model prediction agrees with the experimental data reasonably well, as a 5-10% experimental error is normally expected. The experimental release kinetics of multilayer microspheres was simulated using the multilayer finite element model of 75 spheres (three layers, 25 spheres in each). For the experiment with 200 mg of dry spheres, approximately four layers and 200 microspheres in each layer were there. Assuming that all the spheres at the third outer layer and the inner layers gave the same release rate, a multilayer of 800 spheres was assembled using the similar approach for the monolayer. The parameters used in the computation were Dm ) 1.65 × 10-6 cm2/s, K ) 1.22, C0 ) 0.00825 g/cm3, and λ ) 325, which were determined from the experiments, and parameters Dliq,1 ) 2 × 10-5 cm2/s and Dliq,2 ) 1 × 10-2 cm2/s, from the model fitting. The calculated release profile is plotted together with the experimental data in Figure 12b. It may be noticed that the model-determined Dliq,2 for the multilayer is too high for a solute diffusion in a liquid. This is perhaps due to the experimental conditions differing from the modeling conditions. Even though the stirring rate was fixed to 50 rpm in all the experiments, the influence of hydrodynamics may not be negligible, especially in the case of a multilayer. First, a bed of four layers of microsphere could generate better mixing in the bulk liquid than the monolayer does. Second, while stirring, the microspheres in the multilayer might experience slight motion, causing a certain degree of mixing in the medium between the spheres, which may be less marked in a monolayer. To find out whether a more gentle stirring could eliminate the mixing effect in the multilayer, several release experiments were run at various stirring rates (15, 25 and 50 rpm). As expected, the release rate decreased with the decrease in stirring rate. Fitting of data at a stirring rate of 15 rpm resulted in smaller diffusion coefficients of the drug in the liquid, but they were still high for diffusion in liquid, probably due to the limitation of the present static model, which cannot handle the 592 / Journal of Pharmaceutical Sciences Vol. 87, No. 5, May 1998

The overall release rate of microsphere ensembles is lower than that of a single microsphere due to the influence of interparticulate interaction. The extent of the influence depends on the separation distance between the particles and the rate and amount of drug accumulation in the medium. Release kinetics of microsphere ensembles could be correlated with the single release profile by the factors indicating the interparticulate interaction, such as the dimensionless distance. The overall release profile of a population could be assembled from individual release profiles with the calibration for the neighboring effect. The finite element method is useful for studying the kinetics of drug release from various populations of microspheres and thus for the design of multiparticulate delivery systems. Although only ensembles containing microspheres of the same diameter have been treated here, microspheres with various sizes can be investigated as well with the same method. Influences of particle size distribution and hydrodynamics are important in prediction of the overall release kinetics of multiparticulate systems. Investigation on these aspects with consideration of neighboring effect is undergoing.

Acknowledgments This work was supported in part by the Star-Up Fund from the Faculty of Pharmacy at the University of Toronto.

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Journal of Pharmaceutical Sciences / 593 Vol. 87, No. 5, May 1998