A dissolution-diffusion model and quantitative analysis of drug controlled release from biodegradable polymer microspheres.

Zhang, Lijuan ; Long, Chunxia ; Pan, Jizheng 等

Controlled release behaviours of nifedipine loaded poly (D,L-lactide) (PLA) and poly(D,L-lactide-co-glycolide) (PLGA) microspheres are investigated and modelled in this paper. Based on the integrated consideration of diffusion, finite dissolution rate, moving front of dissolution and size distribution of microspheres, a mathematic model is presented to quantitatively describe the drug release kinetics. The coupled partial differential equations are numerically solved. Dynamic concentration profiles of both dissolved and undissolved drug in the microspheres are analyzed. In comparison with the diffusion model and Higuchi model, the proposed dissolution-diffusion model is characteristic of describing the whole release process without limitation of different dissolution rate or dissolubility. The diffusion coefficient and the dissolution rate constants are evaluated from measured release profiles. The effects of microstructures of polymer microspheres on release behaviours are related to parameters of the model. Based on the mathematical model and in vitro release data, intrinsic mass transfer mechanism is further investigated.

Dans cet article, on a etudie et modelise les comportements de liberation controlee de microspheres de poly (D,L-lactide) (PLA) et poly (D,L-lactide-co-glycolide) (PLGA) chargees en nifedipine. Un modele mathematique est presente pour decrire quantitativement la cinetique de liberation du medicament, en considerant de facon integree la diffusion, la vitesse de dissolution finie, le front de dissolution en deplacement et la distribution de tailles des microspheres. Les equations differentielles partielles couplees sont resolues numeriquement. En comparaison au modele de diffusion et au modele de Higuchi, le modele de dissolution-diffusion propose caracterise bien la description du procede de liberation entier sans limitation de vitesse de dissolution ou de dissolubilite differente. Le coefficient de diffusion et les constantes de vitesse de dissolution sont evalues a partir des profils de liberation. Les effets de la microstructure des microspheres de polymere sur le comportement de liberation sont relies aux parametres du modele. D'apres le modele mathematique et des donnees de liberation in vitro, le mecanisme de transfert de matiere intrinseque est etudie plus en profondeur.

Keywords: model, polymer microspheres, drug delivery system, controlled release

INTRODUCTION

Biodegradable polymer microspheres, as a new drug delivery system, have been studied in the last 10 years (Brannon-Peppas, 1995; Anderson and Shive, 1997; O'Donnell and McGinity, 1997; Soppimath et al., 2001). One of its most important applications is controlling drug release at a proper rate over prolonged time. For many hydrophobic drugs, the absorption is dissolution dependent and their absorption rate is low when orally administered in solid dosage forms. Some of the drugs are known to have a short elimination half life and will cause significant fluctuations in plasma drug concentrations in conventional formulations. To enhance the therapeutic efficiency of these drugs, many researchers have developed controlled delivery by encapsulating them in polymer microspheres, which improve their physical stability and dissolution properties (Verger et al., 1998; Six et al., 2004). Drugs with dissolution-limited absorption might benefit from reduced particle size and highly dispersed amorphous state in proper supplementary polymers.

Nifedipine is a calcium antagonist used widely in the treatment of angina and hypertension. It is highly crystalline and poorly soluble in water. To improve its dissolution properties, nifedipine loaded PLA and PLGA microspheres were produced in this work. Nifedipine releasing from polymer microspheres was investigated and a release model was presented for quantitative description of drug release behaviours based on diffusion and dissolution mechanism analysis. In combination with theoretical analysis, experimental characterization and modelling drug release kinetics was studied in details.

The main mechanisms of drug release from microparticles include solid drug dissolution, diffusion through polymer matrix, swelling of polymers and surface or bulk erosion of polymer matrix. For the system in this work, the effect of swelling and erosion of polymer can be negligible since the water uptake by PLA is less than 10% (Quellec et al., 1998). In addition, some researches verified that PLA and PLGA micro- or nano-particles don't experience a significant degradation during in vitro incubation after several weeks (Lemoine et al., 1996; Gorner et al., 1999; Wong et al., 2001). In cases of drug concentration lower than its solubility, drug release is only dependant on diffusion and the corresponding mathematical model was described in detail by Crank (1975) based on Fick's second law of diffusion. For the system with drug concentration higher than its solubility, the assumption of instantaneous dissolution is not suitable any more. A well-known mathematical model, Higuchi model (Higuchi, 1961) can be used to illuminate such situation. The principles of pseudo steady state, and linear drug concentration gradient were applied in the derivation of the equation. Based on Higuchi's work, Koizumi and Panomsuk (1995) derived an approximate solution for the drug release from a spherical delivery. However, Higuchi model is not applicable to the release process after solid drug dissolves completely. Afterwards, much effort has been devoted to the mathematical description of the drug release process, in which drug loading exceeds its solubility. However, most of the models are only adequate for soluble drugs, and the dissolution rate limitation of poorly soluble drugs is not taken into account. Harland proposed a diffusion-dissolution model to describe drug release process of such system, in which a linear dissolution term was added in Fick's second law of diffusion (Harland et al., 1988). Later, Wong used this model to investigate the in vitro controlled release kinetics of human immunoglobulin G (IgG) of biodegradable polymer microspheres (Wong et al., 2001). Chang and Himmelstein also presented a similar model with the assumption of a linear diffusivity (Chang and Himmelstein, 1990). These models suggested that mechanisms of drug release were mainly diffusion and dissolution controlled. However, the influence of moving boundary of dissolution was not considered.

THEORETICAL

Mathematical Model

On the basis of the diffusion-dissolution model developed by Harland et al. (1988), moving dissolution boundary and particle size distribution are further taken into account in the present model. The dissolution boundary moves from the surface towards the spherical centre, and the undissolved drug nucleus reduces until drug is completely dissolved. A function sgn(x) is added in the model to count the influence of the moving front of drug dissolution. In the liquid phase, accumulation of the drug is equal to the amount of drug diffused and dissolved:

[partial derivative][C.sub.L](r, t)/[[partial derivative].sub.t] = D([[partial derivative].sub.2][C.sub.L](r, t)/ [partial derivative][r.sup.2] + 2 [partial derivative][C.sub.L](r, t)/[partial derivative]r) + k sgn ([C.sub.S](r, t)) x ([C.sub.sat] - [C.sub.L](r, t)) (1)

where k is the dissolution constant, and [C.sub.sat] represents the solubility of drug in the medium solution. In the solid phase, the change of drug concentration with time is represented by drug dissolution rate:

[partial derivative][C.sub.S](r, t)/[partial derivative]t) + k sgn ([C.sub.S](r, t)) x ([C.sub.sat] - [C.sub.L](r, t)) (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

When the drug concentration in solid phase at position r approaches to zero, the dissolution term is in fact eliminated from the equation.

The initial and boundary conditions are as follows:

t = 0 0 <r < R, [C.sub.L] (r, 0) = 0; [C.sub.S] (r, 0) = [C.sub.in] (4)

t > 0 r = 0, [partial derivative][C.sub.L]/[partial derivative]t[|.sub.r=0] = 0; [partial derivative][C.sub.S]/[partial derivative]t[|.sub.r=0] = 0 (5)

t > 0 r = R, [C.sub.L] (R, t) = 0 (6)

By solving the coupled partial differential equations, the concentration distributions in both solid and liquid phases in the microspheres are obtained. The drug remaining in the microspheres is calculated by integrating concentration along the radial coordinate. Ultimately the cumulative released drug from a spherical particle is calculated by the following equation:

[M.sub.t]/[M.sub.[infinity]] = 1 - 3 [[integral].sup.R.sub.0] [r.sup.2] ([C.sub.S] + [C.sub.L])/[R.sup.3][C.sub.in] dr (7)

Considering the contribution of particle size distribution, the drug release from microsphere population can be calculated by:

[([M.sub.t]/[M.sub.[infinity]]).sub.population] = [[integral].sup.R.sub.0] [M.sub.t]/[M.sub.[infinity]] (r)f(r)dr (8)

Computation of the Model

Since there are symbolic functions in Equations (1) and (2), no analytical solution can be obtained for the coupled non-linear partial differential equations (PDEs). In this work, the PDEs are numerically solved with the method of lines. The main process of computation is described as follows:

1. With finite difference method, the variables discretization is made.

2. Cubic Hermite polynomials are applied in the r variable approximation so that the trial solution is expanded in serials.

[C.sub.m](r, t) = [S.summation over (i=1)] ([a.sub.i,m](t)[[phi].sub.i](r) + [b.sub.i,m](t)[[theta].sub.i](r)); m = 1, ..., M (9)

Where [phi](r) and [theta](r) are the standard basis functions for the cubic Hermite polynomials with the knots [r.sub.1] < [r.sub.2] < ... < [r.sub.N]. M represents the number of equations in the PDEs. These are piecewise cubic polynomials with continuous first derivatives. At the breakpoints, they satisfy the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

3. According to the collocation method, coefficients of the approximation are obtained so that the trial solution satisfies the differential equation at the two Gaussian points in each subinterval.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

4. By collocation approximation, the differential equations are transferred to a system of 2M(N-1) ordinary differential equations (ODEs) with 2MN unknown coefficient functions, [a.sub.i,k] and [b.sub.i,k]. The basic form is shown as Equation (12):

[da.sub.i,m]/dt[[phi].sub.i]([p.sub.j]) + [db.sub.i,m]/dt[theta]([p.sub.j]) = [f.sub.m] ([p.sub.j], t, [C.sub.1]([p.sub.j]), ..., [C.sub.N]([p.sub.j]), ..., [([C.sub.1]).sub.rr]([p.sub.j]), ... [([C.sub.N]).sub.rr]([p.sub.j])) m = 1, ..., M; j = 1, ..., 2(N - 1) (12)

Similarly, the other 2M equations are obtained by differentiating the boundary conditions:

[[alpha].sub.m] [da.sub.m]/dt + [[beta].sub.m] [db.sub.m]/dt = d[[gamma].sub.m]/dt, m = 1, ..., M (13)

5. Combined with the initial conditions, the one order initial value problem of ODEs is formed. Since the system is typically stiff, it is solved with Gear's backward differentiation formulas.

Finally, discrete drug concentrations in both solid and liquid phase are obtained. Then, the cubic spine interpolation of discrete values are computed, and substituted to Equation (7) to calculate integral quantities, which is the accumulative drug release percentage. A diagram of the numerical computation of the dissolution-diffusion model is shown in Figure 1.

[FIGURE 1 OMITTED]

EXPERIMENTAL

Materials

The following components are used. Nifedipine (98%) and polyvinyl alcohol (PVA, 87-89% hydrolyzed, WM=13 000-23 000) are products of Sigma. Poly(D,L-lactic acid) (PLA, WM = 100 000), Poly(D,L-lactide-co-glycolide) 85:15 (PLGA85:15, WM = 40 000) and Poly(D,L-lactide-co-glycolide) 50:50 (PLGA50:50, WM = 40 000) are bought from Shangdong Medial Aapparatus Research Institute (China). Other solvents are analytical pure.

Preparation Method

Nifedipine loaded microspheres are prepared using O/W emulsion solvent evaporation method as described in detail elsewhere (O'Donnell and McGinity, 1997). Briefly, a given amount of PLA and nifedipine are dissolved in 10 ml dichloromethane and then the prepared organic solution is emulsified in 100 ml aqueous solution containing 0.5% (w/V) PVA. The mixture is homogenized at 10 000 rpm for 5 min with a high shear emulsion machine (FA25, FLUKO) at room temperature. Then the O/W emulsion is further stirred at 35[degrees]C to remove solvent. During the solvent evaporation, the polymer is deposited as solid microspheres in which the drug is encapsulated. The solidified microspheres are separated by filtration (Millipore, pore size 0.45 [micro]m), and dried in a vacuum desiccator at 25[degrees]C for 24 h.

Differential Scanning Calorimetry (DSC)

Glass transition temperature ([T.sub.g]) and melting temperature ([T.sub.m]) are analyzed by differential scanning calorimetry (DSC, DSC-204, NETZSCH, Germany). Approximately 5 mg sample is weighted in a sealed aluminum pan. The DSC scan is recorded at a heating rate of 2 K/min from -50[degrees]C to 200[degrees]C under nitrogen gas purge. Analysis is performed on pure substances and drug-loaded microspheres.

In Vitro Release Test

10 mg drug-loaded microspheres are suspended in 500 ml phosphate buffered saline. The experiment is carried out at 37[degrees]C with a stirring rate of 100 rpm for 24 h. 5 ml supernatant is withdrawn at predetermined time intervals and filtered through a 0.22 [micro]m millipore filter. Drug concentration of solution is analyzed spectrophotometrically at 236 nm. At the same time, the medium solution is maintained at a constant volume by replacing the samples with fresh medium solution.

RESULTS

Shown in Figure 2 are pictures of scanning electron microscopy (SEM) of 5% drug loaded PLA microspheres. The microparticles are in good sphericity and have a narrow size distribution.

[FIGURE 2 OMITTED]

The DSC thermogram of nifedipine is shown in Figure 3 and presents a melting temperature of 173.3[degrees]C. The DSC therograms of pure PLA and PLGA 50:50 materials and microspheres with different drug loading are shown in Figure 4. PLA and PLGA 50:50 show glass transition temperatures of 43.9[degrees]C and 45.2[degrees]C, respectively. For the DSC thermograms of drug loaded microspheres, no melting peaks of nifedipine are observed, which indicates that nifedipine is in amorphous state. The DSC thermograms between 100-200[degrees]C are not shown in Figure 4 for no peaks appear. Note that the peak melting temperatures of drug loaded PLGA 50:50 microspheres are changed compared to its pure material. It suggests that a strong interaction exists between nifedipine and PLGA 50:50. Meanwhile, the peak melting temperatures remain unchanged for drug loaded PLA microspheres and it indicates that no interaction between nifedipine and PLA or the interaction is weak (Okhamafe and York, 1989; David et al., 1999).

[FIGURES 3-4 OMITTED]

In vitro release profiles of nifedipine crystal and nifedipine loaded microspheres prepared with different polymers are plotted in Figure 5. Drug release rate is much improved after nifedipine being encapsulated in polymers. Experimental data shows the drug release rate of PLGA microspheres is higher than that of PLA. A stronger interaction exists between nifedipine and PLGA than PLA. A better dispersion is achieved when nifedipine is loaded in PLGA microspheres. It accelerates the drug dissolution rate and so improves the drug release rate. The dissolution rate possesses a great effect on nifedipine release.

[FIGURE 5 OMITTED]

RELEASE MODELS ANALYSIS

Transfer Mechanism of Drug Release

Using the proposed dissolution-diffusion model, the amount of dissolved and undissolved drug within microspheres is calculated. From the results, mass transfer mechanism and the effect of dissolution rate and solubility on release kinetics are examined. First, the microsphere diameter is assumed to be 1 [micro]m and the diffusion coefficient is [10.sup.-13][cm.sup.2]/s. The initial drug loading is 10 times of solubility.

Shown in Figure 6 is the calculated drug concentration in both the solid and liquid phases in microspheres. The concentration is plotted along radial positions. Drug is not dissolved at the initial time. Drug concentration is the same everywhere in the microspheres. As time goes by, undissolved spherical drug nucleus lapses to the centre of the microspheres. As shown in Figure 6(b), the drug concentration in liquid phase is very low at the beginning. The drug concentration increases rapidly as drug began to dissolve. It reached the maximal value after 10 h and holds constant in about 30 h. During this period, drug dissolution and diffusion rate reach balance and the drug release profile shows a constant rate. As the solid drug exhausts from the surface, dissolution rate is lower than diffusion rate and drug release reduces gradually.

[FIGURE 6 OMITTED]

In comparison, if drug dissolves rapidly and the dissolution rate is high, as seen in Figure 6(d), the dissolved drug near the centre of the microspheres reaches saturation concentration in a short time. In this situation, the drug release depends entirely on drug diffusion.

Consider one critical situation when dissolution constant k=1 [s.sup.-1] and drug loading is 10 times its solubility. The dissolution rate is large enough and drug release is only limited by the solubility. In this circumstance, the drug in liquid phase reaches saturation concentration at the very beginning. Drug dissolution process in solid phase coincides with Higuchi model. After dissolution for a short time at beginning, the undissolved drug concentration at the moving front holds constant since the liquid phase is saturated, whereas the one at the front decreases rapidly. The dissolution front moves inward while drug dissolution continues. Concentration gradient of dissolved drug from the front to the surface of microspheres is developed. The drug concentrations in liquid phase between the moving front and the spherical centre maintain saturation because of the high dissolution rate of drug. The calculated concentration profiles are shown in Figure 7.

[FIGURE 7 OMITTED]

Consider another critical situation when k=1 [s.sup.-1] and [C.sub.sat]/ [C.sub.in]=1. Since drug dissolution rate is very high and the drug loading is smaller than its solubility the limitation of solubility and dissolution rate can be neglected. At the beginning, drug dissolves completely in aqueous phase and the undissolved drug concentration is zero. Thus, only the concentration profiles in liquid phase are given, as shown in Figure 8. In this case, drug release is entirely governed by drug diffusion in polymer matrixes.

[FIGURE 8 OMITTED]

Comparison of the Release Models

The calculated release profiles based on three models are shown in Figure 9. The parameters are given using the assumption in foregoing section.

[FIGURE 9 OMITTED]

The curve (a) presents the case that finite dissolution rate and solubility are neglected. The broken line is from the basic diffusion model (Crank, 1975). The solid lines are calculated numerically with the proposed dissolution-diffusion model. The broken line is not shown in Figure 9 because it coincides completely with the solid line. It indicates that the basic diffusion model can be regarded as a special case of the dissolution-diffusion model without considering the limitation of finite dissolution rate and solubility. In this case, drug release only depends on diffusion, which does not match the postulated conditions of Higuchi model which is invalid in such situation.

The case that drug loading is much higher than its solubility is plotted as curve (b). Since the dissolution rate is very large, the drug release is only restricted by solubility. The profile calculated with the proposed dissolution-diffusion model is similar to that with Higuchi model (Higuchi, 1961). However, diffusion model is not applicable here.

With respect to the other two cases (k=[10.sup.-4][s.sup.-1] and k=5 x [10.sup.-4][s.sup.-1]), the limitation of drug dissolution rate must be considered. The basic diffusion model and Higuchi model can not describe the influence of dissolution constant. Whereas, the dissolution-diffusion model proposed in our work can satisfactorily describes the release characteristic in these circumstances. As shown in curves (c) and (d), the release process is composed of two stages. In the first stage, the effect of dissolution rate is dominant, and the profiles are approximately linear, corresponding to the stage of constant concentration in microspheres. As dissolution processes, the solid drug starts to exhaust, the release process shifts to diffusion controlled. The release rate decreases gradually.

SIMULATION OF IN VITRO DRUG RELEASE

Based on the proposed dissolution-diffusion model, experimental release data are analyzed. The effects of physicochemical and structural properties on release kinetics are revealed from the examination of model parameters, which can provide insight into the drug controlled release process. Parameters are calculated from experimentally measured drug release data fitting to the proposed model. The fitting procedure is to minimize the differences between experimental and computational values. Such a non-linear least square optimization problem is solved using a modified Levenberg-Marquardt algorithm with finite-difference Jacobian matrix. First, the initial parameters are given, and then the release model is numerically solved as described above. Next, the obtained cumulative drug release percentages at various times are compared with experimental data and the optimization object minimized their difference. The iterative computation is processed until the convergence criterion is met and the parameters are finally determined. Otherwise, the parameters are optimized using Levenberg-Marquardt method, among which the path searching is performed by Jacobian matrix which is estimated by the finite difference method.

The Effect of Different Polymer on Drug Release Characteristics

During the sample preparation, the amount of nifedipine and polymer used are 20 mg and 100 mg, respectively. The release profiles calculated from the proposed diffusion-dissolution model are plotted in Figure 10 together with the experimental release data. Particle size distribution (PSD) of microspheres is considered in computation (the measurements of PSD are not listed). Good agreements are achieved between modelling results (solid curves) and experimental value (symbols) in all cases. The release process can be described by the proposed dissolution-diffusion model.

[FIGURE 10 OMITTED]

The characteristic parameters of different polymeric microspheres obtained from the modelling studies are given in Table 1. The effective diffusion efficient D is greater for PLA than PLGA microspheres. This may ascribe to smaller PLGA microspheres having a denser structure. The morphology of microspheres is affected by the phase separation kinetics during solvent evaporation. PLGA is more hydrophilic and it solidifies at a lower rate. Thereby, it shrinks more in prolonged solidification process and results in smaller microspheres (Wong et al., 2001). However, the dissolution constant k is smaller for PLA than PLGA microspheres. This is due to the different compatibility of polymers and drug. Based on the analysis of DSC thermograms before, nifedipine has stronger interaction with PLGA than PLA and has a better compatibility with PLGA. The more compatible polymer provides better drug dispersion and facilitates dissolution of drug (Francesco et al., 1998).

The Effect of Polymer Contents on Drug Release Characteristics

Consider microspheres prepared with different PLA content, the parameters fitted and the measured microsphere characters are listed in Table 2. When the polymer concentration increases, it is more difficult to homogenize the emulsion. Besides, emulsion drops are easier to collide together and get bigger. Therefore, as the content of polymer increases, the size of microspheres increases. At the same time, higher polymer concentration in emulsion conduces to forming a denser microsphere structure which retards drug diffuse in polymer matrixes. Therefore, the diffusion coefficient D decreases with polymer content increases. On the other hand, for the microspheres prepared with higher polymer content, the drug loading is less comparative. The drug is better dispersed in the polymer matrixes. Dispersibility may be the essential reason to influence the dissolution rate k. Figure 11 is the calculated and measured release profiles with different polymer contents.

[FIGURE 11 OMITTED]

From the above analysis, it is clear that drug release depends essentially on the mechanism of diffusion and dissolution. Diffusion and dissolution rates are affected by comprehensive factors, such as the compatibility of drug and polymer, the density of microspheres and particle size. During the preparation, material constitutes and operation conditions control the structural characteristics by specific action mechanism, thereby governing the drug release behaviours. The proposed dissolution-diffusion model predicts the drug release behaviours successfully. The calculated parameters of the model can quantitatively describe the influence of microsphere formulation on drug release mechanism.

CONCLUSIONS

In this paper, a new dissolution-diffusion release model is developed based on integrated consideration of diffusion, finite dissolution rate, moving front of dissolution and microsphere size distribution. Numerical solutions of the partial differential equations are achieved. Compared to Fick's diffusion model and Higuchi model, the proposed dissolution-diffusion release model is more suitable in describing the release behaviour of dissolution-limited drug. In vitro drug release data is analyzed based on the proposed model. Density of microspheres, compatibility of drug and carrier as well as microsphere size imposes influence on drug release. Increasing microsphere density will decrease drug diffusion rate and a better compatibility of drug and carriers contributes to drug dissolving faster. The release kinetics of microspheres is modelled from multi-levels, revealing the effects of structure, diffusion and dissolution on release profiles. The model is helpful in controlling the microsphere structure to achieve desirable release performance.

ACKNOWLEDGMENTS

Financial supports from the Natural Science Fund of China (NSFC) (Nos. 20376025, 20476033, 20536020), the NFSC Excellent Young Scientist Fund (No. 20225620), and the Science Fund of Guangdong (No. 04020121) are gratefully acknowledged.

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Manuscript received September 1, 2005; revised manuscript received May 26, 2006; accepted for publication June 26, 2006.

Lijuan Zhang, Chunxia Long, Jizheng Pan and Yu Qian *

School of Chemical Engineering, South China University of Technology, Guangzhou 510640, China

* Author to whom correspondence may be addressed. E-mail address: ceyuqian@scut.edu.cn

Table 1. Microsphere characters and calculated parameters by the
proposed dissolution-different model

Polymer Mean diameter Encapsulation
 [micro]m efficiency, %

PLA 3.80 [+ or -] 0.20 87.22 [+ or -] 0.51

PLGA85:15 2.55 [+ or -] 0.12 86.95 [+ or -] 0.44

PLGA50:50 2.74 [+ or -] 0.15 92.18 [+ or -] 0.55

Polymer D, [cm.sup.2] k, [s.sup.-1]
 [s.sup.-1]

PLA 5.08e-13 3.70e-3

PLGA85:15 4.11e-13 6.39e-3

PLGA50:50 3.96e-13 7.26e-3

Experimental data represent the average of three independent
experiments

Table 2. Microsphere characters of different PLA quantities and
calculated parameters by the proposed dissolution-diffusion model

Polymer Mean diameter Encapsulation
mg [micro]m efficiency, %

50 1.99 [+ or -]0.11 79.61 [+ or -] 0.41

80 2.65 [+ or -] 0.13 84.88 [+ or -] 0.35

100 3.80 [+ or -] 0.20 87.22 [+ or -] 0.51

120 4.30 [+ or -] 0.23 88.47 [+ or -] 0.47

Polymer D, [cm.sup.2] k, [s.sup.-1]
mg [s.sup.-1]

50 6.74e-13 1.37e-3

80 5.83e-13 3.49e-3

100 5.08e-13 3.70e-3

120 4.93e-13 5.14e-3

Experimental data represent the average of three independent
experiments