Drug Release From Hydrophilic Matrices. 2. A Mathematical Model Based on the Polymer Disentanglement Concentration and the Diffusion Layer

Drug Release From Hydrophilic Matrices. 2. A Mathematical Model Based on the Polymer Disentanglement Concentration and the Diffusion Layer ROBERTT. C.Jutx, PHILLIP R.

NIXOd,

MAHESHV. PAT EL^, AND DONALDM. TONG*

Received December 22, 1994, from the +DrugDelivery R&D and ‘Research Support Biostatistics, The Upjohn Company, Accepted for publication August 22, 1995? Kalamazoo, M149001. Abstract 0 A comprehensive model is developed to describe the swelling/ dissolution behaviors and drug release from hydrophilic matrices. The major thrust of this model is to employ an important physical property of the polymer, the polymer disentanglementconcentration, &,&,the polymer concentration below which polymer chains detach off the gelled matrix. For (hydroxypropyl)methylcellulose (HPMC) in water, we estimate that pp,dls scales with HPMC molecular weight, M, as pp,dls = W o e Further, . matrix dissolution is considered similar to the dissolution of an object immersed in a fluid. As a result, a diffusion layer separating the matrix from the bulk solution is incorporated into the transport regime. An anisotropic expansion model is also introduced to account for the anisotropic expansion of the matrix where surface area in the radial direction dominates over the axial surface area. The model predicts that the overall tablet size and the characteristic swelling time correlate with pp,dis qualitatively. Two scaling laws are established for fractional polymer (mp(t)/mp(-)) and drug (md(t)/m(-))released as mp(t)/mp(-) 125’ O5 and md(f)/md(-) = Wo24, consistent with the limiting polymer molecular weight effect on drug release. Model predictions for polymer and drug release agree well with observations, within 15% error. Evolution of water concentration profiles and the detailed structure of a swollen matrix are discussed. 0~

1. Introduction Numerous works have modeled the swelling of cross-linked where the swelling reached an equilibrium state at which the elastic interaction of the network balances the osmotic pressure created by the penetrating ~ o l v e n t .For ~ uncross-linked p ~ l y m e r , ~however, -~ swelling becomes more complicated. For example, polymer dissolution occurs simultaneously with polymer relaxation. In addition, anisotropic expansion has been observed.8 In general, modeling the swelling of un-cross-linkedpolymer has not been a focus in the literature. Lee,g assuming constant drug diffusion coefficients and a quadratic drug concentration form, employed a refined integral method to solve the moving boundary problem involved. Further, approximate solutions on the basis of a linear solvent concentration profile yielded a square-root increase of the gel layer thickness with time.l0 Diffusion and dissolution terms were later incorporated to gauge qualitatively the contributions of swelling and dissolution, respectively.ll Moreover, a diffusional model on the basis of the concentration-dependent diffusion coefficient of solvent was developed for lithography purposes.lZ Recently, a one-dimensional model on the basis of the “disentanglement time” was introduced to describe the dissolution of polystyrene films.I3 Agreements between model predictions and experimental results were achieved in various systems. Despite the progress made in modeling the swelling of uncross-linked polymer, several topics remain to be addressed: @

Abstract published in Advance ACS Abstracts, October 1, 1995.

1464 / Journal of Pharmaceutical Sciences Vol. 84,No. 12, December 1995

the anisotropic expansion, the boundary condition, the moving boundary, the strong concentration dependence of the drug diffusion coefficient, swelling inhomogeneity, and polymer stress relaxation. Each of these is discussed below. Anisotropic expansion has been observed for (hydroxypropy1)methylcellulose (HPMC)I4-l6 tablets, polystyrene sheets,I7 and cellulose nitrate matrices.18 Specifically, axial expansion greatly exceeded radial expansion, leading to a predominant surface area in the radial direction. A simple anisotropic expansion model was proposed in this work to establish a constrained relationship between the extent of expansion in both directions. Most models assumed a glassy-rubbery interfacegJOas the boundary beyond which solvent molecules did not penetrate further into the inner part of the matrix. After reaching the matrix center, however, the glassy-rubbery interface disappeared, and a zero-flux boundary condition a t the center of the matrix was used instead. Nevertheless, Ueberreiterlg pointed out that solvent concentrations did not decrease to zero beyond the glassy-rubbery interface. A glassy polymer contained a great number of channels and holes of molecular dimensions and the first penetrating solvent molecules filled the voids so that the subsequent glassy-rubbery transition was achieved. Ueberreiter’s proposal was consistent with the porosity of 0.1 found for tablets made of HPMC.zOA swollen glassy layer was thus suggested by Ueberreiter as part of the structure. In addition, on the basis of scanning electron microscopy, a partially hydrated layer interposed between the dry core and the gel layer was found for HPMC matrices,21 confirming the presence of the swollen glassy layer. Further, anisotropic expansion produced a highly expanded core in the axial direction while the axial gel layer was similar to the radial one, suggesting that water molecules diffused beyond the glassy-rubbery interface. A moving boundary a t the gelled matrix-bulk solution interface results from simultaneous polymer swelling and dissolution, and this variation of the boundary position with time makes modeling more complex. We adopted the approach of Murray and Landis,22who scaled the axis with the instantaneous overall dimension to transform the moving boundary problem into a fixed boundary one. Lack of reliable data for the diffusion coefficient of drug, Dd, also hampered past modeling efforts.23 A common approach to address this problem has been to assume a constant d i f f u s i ~ i t y , ~exponential J~ or other forms.z5 For two-component systems, free-volume theoryz6predicted that the diffusion coefficient of a solute decayed exponentially with the polymer concentration, up to some critical polymer content, above which a more rapid decay in Dd was expected. This exponential relationship, up to 17%HPMC content, was confirmed by Gao and FagernessF7 who used NMR techniques to measure the self-diffusion coefficientof water, D,,,, and the diffusion coefficient of drug, Dd, in HPMC solutions. We use the diffusion coefficients data generated by Gao and Fagerness. Swelling i n h o m ~ g e n e i t y and ~ ~polymer ~ ~ ~ stress relaxation30 are important topics for polymer gels. In the early stages of

0022-3549/95/3184-1464$09.00/0

0 1995, American Chemical Society and American Pharmaceutical Association

swelling, less hydrated polymer surrounded by more extensively hydrated patches was found in the outer gel region of a gelled HPMC matrix.21 In the later stages, non-uniform hydration was observed around the central part of the gel layer.21 As a result of nonuniform hydration, only part of the polymer contributed to diffisional hindrance. However, since only part of the gel layer exhibited inhomogeneity, the effect of inhomogeneous hydration was not accounted for in the model. The contribution of inhomogeneous hydration to the disparity between model predictions and experiments is discussed in this work. Gehrke and c o - ~ o r k e r sproposed ~~ that polymer stress relaxation was not important for small solvent molecules such as water, whereas it became pronounced for large solvent molecules. This is consistent with the finding of two group^^^^^^ who reported that the self-diffusion coefficient of water in polyelectrolyte gels was independent of the polymer cross-link density. These authors attributed their finding to the small size of water molecules. That is, as water molecules penetrated through the free volume of the polymer network, they felt no effect of polymer chain relaxation. This agrees with the work of Gao and Fagerness2I who found that the diffusion coefficients of water and drug were independent of the HPMC molecular weight. They concluded that obstruction of a swollen polymer was the primary diffusional hindrance for the water/HPMC system. The objectives of this work are to consider the topics discussed above by identifylng the key parameters and, consequently, to develop a new model on the basis of these parameters for polymer and drug release from un-cross-linked polymer. Two novel approaches are introduced. First, we incorporate a diffusion layer, the transitional regime from a gelled matrix to the bulk solution, into the transport regime. As a result, polymer dissolution could be characterized. Second, an important parameter, the polymer disentanglement concentration, was introduced in the model. The polymer disentanglement concentration, pp,dis (g/mL), is defined as the polymer concentration at the gelled tabletdiffusion layer interface. On the basis of the measured Dd, the polymer diffusion coefficient, Dp, the diffusion layer, and pp,dis, this paper discusses model calculations for the evolution of solvent concentration profiles and for the detailed structure of a swollen matrix. The effects of HPMC molecular weight on polymer and drug release are also addressed. To demonstrate the appropriateness of the model, model predictions are compared with experimental data for HPMC and drug release from formulations consisting of HPMC, filler (lactose), and drug (adinazolam mesylate). It is shown that the model predicts polymer and drug release well within 15%deviation from observations.

x=O

r=R(O) I

x=O

x=R(tJ

x=R(tJ+A

Scheme 1-Evolution of the swollen matrix dimension. At an early time, 6 , solvent uptake dominates over polymer dissolution, leading to matrix expansion, R(t,) > R(0). At a later time, t2, polymer dissolution governs over solvent uptake, giving rise to matrix shrinkage, R(t2) < R(0).Note that layer thickness for the diffusion layer, A, remains unchanged. Relative thickness of A to R(r) is not to scale. The space defined by the double lines represents the undissolved matrix, which includes the dry glassy core, the swollen glassy layer, and the gel layer.

2.2. Model Descriptions-For clarity, assumptions made and the associated limitations are summarized in Table 1. On the basis of the above discussion regarding the swollen glassy layer, the model used the zero flux condition at the matrix center as one of the boundary conditions. The predominance of axial expansion over radial expansion (discussed later) leads to relatively high surface area in the radial direction. As such, the majority of polymer and drug is released in the radial direction. Therefore, it is justified to construct a model on the basis of radial diffusion only. On the basis of the assumptions in Table 1,changes in the mass concentration, pi (g/mL), and weight fraction, wi,for species i as a functions of radial position, r, and time, t, are due to three terms: convection, diffision, and source terms. Mathematically, this is stated as

2. Theory 2.1. Evolution of Matrix Dimension-Qualitative dimensional changes of hydrophilic matrices with time are illustrated in Scheme 1 (only half of the matrix is shown). Positions corresponding to the boundaries used in the model, the center, the gelled matrix-diffusion layer interface, and the diffusion layer-bulk solution interface, are at 0, R(t) and R(t) A, respectively. As described above, the commonly used glassy-rubbery interface is not included because small water molecules31 are able to diffuse across this interface. The diffusion layer is where the dissolved polymer chains diffuse toward the bulk solution. At early times, say tl, solvent uptake dominates over polymer dissolution, leading to matrix expansion in both axial and radial directions. At a later time, t2, polymer dissolution becomes governing, and the overall matrix dimension decreases.

+

The first term on the right hand side of eq 1 is the convection term, arising from the moving boundary. One can also consider the first term as the interpolation term for calculating the concentrations in each new time step “t 6t”. Murray and Land# assumed a grid point at r to move with a velocity drldt, or

+

Substitution of (2) into (1)produces

Journal of Pharmaceutical Sciences / 1465 Vol. 84, No. 12, December 1995

Table 1-Assumptions Made in the Model and the Associated Limitations

Limitations Dissolution of drug is much faster than diffusion. Only diffusion is considered. Drug does not interfere with matrix hydration. Diffusion coefficients of drug, &, decay exponentially with polymer concentration, pp Homogeneous swelling. No poiymer chains detach off a matrix before reaching pp,ds. Diffusion layer as a transition regime from the gelled tablet to the bulk solution. Relationship between pp,d$sand Mis similar to that between

Valid for fairly soluble drugs. Not valid for poorly soluble drugs. Valid for low drug load or no drug-polymer interactions. Valid if there is no drug-excipient interaction. Some deviation is expected in high polymer concentration regimes where a more rapid reduction in Dd than

exponential is expected. Not valid as (1)the sizes of solvent molecules are large or (2) solvency becomes poor for polymer, such as for HPMC in high ionic strength media or for hydrophobic polymer, or (3) pp is high. Deviation is expected,as discussed in the text. Same as the homogeneous swelling,except (3). The prefactor 5 varies with experimental setting and Dpdepends on polymer molecular weight and concentration. Sensitive to polymerlsolvent pair. Some deviation is expected.

pp’ and M.

Polymer stress relaxation is not important.

Valid only for small solvent molecules, such as water. The semiempirical relation developed in the anisotropic expansion model is good for HPMC in water only.

The second term accounts for the Fickian diffusion for species i, with diffusion coefficients Di and the local overall mass concentration, pt. The presence of the second term also implies that the time scale for drug dissolution is much faster than that for drug diffusion, i.e., diffusion, instead of dissolution, of drug molecules is the controlling mechanism of drug release. This assumption is valid only for soluble drugs. For poorly soluble drugs, a dissolution term needs to be incorporated into the governing equation. The second term also considers variable matrix density. This differs from the case for a trace amount of the diffusion species, where matrix density is assumed to be constant and pt(awi/ar)replaces the commonly used (api/ar). Matrix density variation is significant for swollen matrices because water uptake could amount t o more than 100%of the initial matrix mass. Moreover, the density of HPMC (-1.3 g/mLSS)is 30% more than that of water (1g/mL). Thus, we assume pt is linear with matrix composition

et = z e i

(4)

where the summation is operated on all the species. The last term in eq 3 is the source term, which considers concentration changes that result from matrix volume changes. For example, water uptake causes matrix expansion and an increase in the matrix volume. This in turn reduces the concentration of drug in the more expanded matrix, leading to a negative sign being applicable for this term. The NMR measurements by Gao and FagernessZ7produced the self-diffusion coefficients for water, D,,,, and mutual diffusion coefficients for adinazolam mesylate (ADM), Dd, as

D,,JD,,, = k,’ exp(k,w,) DdlDd,o

= eXP(-kd,pwp - kd,lwl - k d , d w d )

(5) (6)

where k,’ is the prefactor, k , and k d , i are the weighting factors, and Di,o is the diffusion coefficient for species i in a dilute solution. For water and polymer, the mutual diffusion coefficient, D,, is related to the self-diffusion coefficient of water, D,,,, and polymer, D,,,,, a d 9

(7) where @i is the volume fraction of species i . The approximate form of eq 7 was obtained on the basis of D,,+,>> D,,p. Once the concentrations of water are calculated, the corresponding polymer concentrations are obtained by invoking the mass conservation of eq 4. Table 2 lists the values of k d , i , k,, and 1466 / Journal of Pharmaceutical Sciences Vol. 8.4, No. 12, December 1995

Table 2-Values of the Parameters Used in Equations 5 and 6 for the Diffusion Coefficients of Water, Lactose, and Drug Molecules

Effective Component Water Lactose

Weighting Factor

Prefactor k,’

= 0.00466

Effective Component

Prefactor

k, = 5.29 Polymer k d ~= 3.48 Drug (adinazolam mesylate)

Weighting Factor lqp= 7.85 /Gl,d

= 5.92

kw‘, obtained by linear regression of the NMR data. As stated above, the exponential relationship may not hold a t high HPMC concentrations. Given the limited data available, however, we assume that the exponential relationships are valid across the entire gelled matrix. 2.3. Boundary Conditions-The gelled matrix-diffusion layer interface is located at the tablet surface. Our calculation indicated that the thickness of the diffusion layer was several orders of magnitude thinner than the tablet dimension, thus steady-state diffusion within the diffusion layer is assumed. The polymer concentration a t the interface is kept at pp,&, while the drug concentration on the tablet surface is maintained a t zero, on the basis of the fact that drug molecules diffuse much faster than polymer. The associated boundary conditions for eq 3 are gp = e p , d i s @d

=

aeilar = 0

a t r = R(t)

(tablet surface)

(3-1)

a t r =R(t)

(tablet surface)

(3-2)

at r = 0

(tablet center)

(3-3)

Initial conditions for all the diffusion species throughout the matrix are Qi

=ei(0)

at t = 0

(3-4)

Equations 3 can be made dimensionless through the following normalizations. z = tD,,J(R(t=O))2

R = R/R(t=O)

r’ = rlR(t)

ei’

DL = DiIDw,,

The normalized governing equation becomes

=

eilei,o

with boundary conditions

rre-"3

JO

@p(r)

=0

at r' = 1

(tablet surface)

(8-2)

a@,'lar' = 0

at r' = 0

(tablet center)

(8-3)

@d'

and the initial condition

Weight fraction w , for species i is related to its mass concentration pi as wi = pi/(&). Through the normalization of r ' = r/R(t),the moving boundary associated with eqs 3 at r = R(t) is now converted into a fixed boundary a t r ' = 1. 2.4. Diffusion Layer and Polymer Dissolution FluxWe envision matrix dissolution as consisting of polymer dissolution from the matrix, followed by mutual diffusion with respect to water through a diffusion layer toward the bulk solution. The diffusion layer originates from the boundary layer associated with a fluid flowing past an object,34and significant concentration gradients are expected to develop. The importance of the diffusion layer was confirmed for polystyrene matrix dissolution in toluene,lg where faster stirring produced faster polymer dissolution. We assume that polymer dissolution can be modeled as mass-transfer fluxes for forced convection around an object.35 Thus, the polymer dissolution flux, J,, is expected take the form of Jp

=f

'

-

213 -1/6 112 ~ ( ~ p ) O a p p @p,dis

(9)

where the value of the prefactor f, changes with experimental settings and the fluid dynamics, (D,) is an averaged D, within the diffusion layer and v is the kinematic viscosity of the solvent. The effect of stirring is reflected through the apparent rotating rate, Gap,. In this study, we used a magnetic stir bar and Gap, is the rotating rate of the stir bar. Unlike small molecules, the diffusion coefficient of a polymer is a strong function of its molecular weight and concent r a t i ~ n a, ~result ~ of the significant entanglement among polymer chains. One piece of evidence for chain entanglement is the abrupt increase in solution viscosity with polymer concentration, leading to the corresponding abrupt decreases in the diffusion coefficient of the polymer. The correlation between D, and the polymer concentration, along with the significant polymer concentration gradient across the diffusion layer, justified the need to determine (D,). Recently, a relationship between D, for HPMC of molecular weight M in a solution with HPMC concentration pp has been developed as37

du

= @p,dis hAe-u3du

Substitution of eq 11, along with eqs 10 and eq 12 into eq 9, produces J,.

2.5. Polymer Disentanglement Concentration, &,disThe polymer disentanglement concentration is defined as the polymer concentration below which polymer chains detach off the matrix to approach the bulk phase. The concept of polymer disentanglement concentration has been introduced p r e v i o u ~ l y ,where ~ ~ , ~ &,,dis ~ is related to the polymer concentration in the bulk, pp,b, as @p,dis = k p , b

MP0%

+ 700(M/96000)o~7~,,/81-2(10)

The averaged diffusion coefficient, (D,), is approximated as

l D p r dr

(Dp)=

sr

(11)

dr

where eq 11 integrates across the diffusion layer, from r = R(t) to r = R(t) A. The concentration profiles of the diffusion species within the diffusion layer is assumed be similar to that for a rotating disk. Thus, we employ the concentration profiles, pp(r),given by L e v i ~ h ~ ~

+

(13)

The proportionality k is equal to unity in the case of high transfer rates while k is greater than unity as the transfer rate turns low. Equation 13 connects flow patterns to the disentanglement concentration, yet no information regarding the molecular weight effect is provided. The value of pp,dis is specific t o a given polymerlsolvent pair and molecular weight.12 For polystyrene in methyl ethyl ketone of molecular weight M , it was estimated that40,41 @,,dis

= 27OOOlM

(14)

A similar result was recently obtained by Peppas and cow o r k e r ~ .Unfortunately, ~~ no literature work assessed the relationship between pp,dis and M for HPMC. However, using well-established polymer solution concepts, we were able to estimate pp,dis for HPMC in water, resulting in better predictions of drug release. The procedure for the determination of pp,dls for HPMC in water is given in a separate publication by this The result is summarized below. The relationship between pp,dis and M is in general given as

where the value of the exponent /3 depends upon the solvent quality. For example, p = 0.6 for polymers in a good solvent, while p = 0.5 for polymers in a I!? solvent.36 The e condition is defined as the second virial coefficient, A2, being zero. On the other hand, A2 is positive for polymers in a good solvent. The task now is to determine the exponent p for the HPMC/ water pair. From end-to-end distance measurements for HPMC,43,i? was estimated to be 0.6, indicating that water is a good solvent for HPMC. Thus, eq 15 becomes M-0.8 @p,dis OC

Dp = (7.24 x

(12)

(16)

Equation 16 implies that higher M has lower pp,dis. Stated differently, polymers of high molecular weight induce greater chain entanglement than polymers of lower molecular weight, such that it is harder for the former to dissolve in the solvent. On the basis of our preliminary flow property experiments, the reference pp,dis for the HPMCIwater pair is estimated to be 0.05 g/mL for HPMC K4M (M = 96 kDa), consistent with literature values.44 Thus, pp,dls for HPMC of any molecular weight M can now be calculated as ep,dis

= 0.05(M/96000)-0~s

(17)

Equation 1 7 is used for the boundary condition in eq 8-1 and for the calculation of matrix dissolution of eq 9. Journal of Pharmaceutical Sciences / 1467 Vol. 84, No. 12, December 1995

2.6. Anisotropic Expansion-It is intriguing that onedimensional swelling would produced two-dimensional matrix expansion. For the case of HPMC tablets sealed on two faces,14 the one-dimensional (radial direction) swelling was supposed to induce expansion in the radial direction only. However, expansion was observed in both the radial and the axial directions, even though the possibility of axial diffusion was ruled out because of the uniform concentration profile in the axial direction. It was also observed that the apparent radii measured became smaller than those corresponding to otherwise purely radial expansion. More importantly, the magnitude of the axial expansion, in terms of the normalized thickness L(t)/L(O),was much higher than the normalized radius, R(t)/R(O). On the basis of the results of Papadimitriou et al.14 for HPMC-containing tablets with an aspect ratio of about unity, we found that L(t)/L(O)was connected to R(t)/R(O)as

Equation 18 imposes a constraint between the axial thickness and radius. We emphasize that the relationship between L(t)/L(O)and R(t)/R(O)varies from one system to the other, and eq 18 is only valid for tablets with unity aspect ratios. Variations of tablet size with time were then characterized with the constraint of eq 18. At any time t , the onedimensional diffusion equation calculates the remaining polymer and water uptake by the matrix, which in turn gives the normalized volume, V(t)/V(O)as

Equation 22 can be rewritten as

L(O)LR(O’ ep(t’=O)r2dr =

The instantaneous polymer concentration profile, pp(r),is from eq 8, while L(t) is related to R(t) by eq 18. Thus, the calculation of R(t)becomes straight forward because R(t) is the only unknown in eq 24. Calculation of drug release is similar to polymer release. Once R(t)is known, drug released can be determined as

%(t)R(t)[h1ed(r’)2r’dr‘ + drug released (25) where pd(r) is from eq 8 directly. Equation 8 is solved numerically using the IMSL Math/ Library package MOLCH.45 Grid sizes were unequal in order to reduce the stiffness in the proximity of the tablet surface where grid sizes are smaller than those close to the tablet center. Concentration discontinuity at the tablet surface at t = 0 created another problem. For polymer, the boundary condition a t the tablet surface underwent a smooth transition from the value pp a t t = 0 to the value of pp,dis at z = t h . The transition period t d used in the numerical program is 0.1.

3. Results and Discussion

By substituting eq 19 into eq 18, we obtain

(20) For the case of one-dimensional expansion where L(t)is kept constant, V(t)is connected to R(t)as

(21) If one compares (20) with (211, it becomes evident that the correct R(t),on the basis of eq 20, is far below the incorrect R(t) restricted by eq 21. We use eq 20, along with mass balance of polymer (to be discussed below), t o calculate the instantaneous tablet radius; tablet thickness can then be determined through eq 18. 2.7. Calculationsof the InstantaneousMatrix Radius, R ( t ) , and Polymer and Drug Release-The model calculated the instantaneous tablet radius R(t)on the basis of the mass balance of HPMC, i.e.,

The right-most term of eq 22, HPMC dissolved in the bulk solution, is obtained by integrating Jpfrom t = 0 to t

3.1. Variation of pp,aswith HPMC Molecular WeightVariation of computed HPMC disentanglement concentrations with molecular weight is plotted in Figure 1. The reciprocal form of eq 14 results in a tight correlation between pp,dis and M at low M , while the correlation becomes null a t high M . For example, pp,& decreases from 0.13 to 0.05 g/mL as M increases from 30 to 100 kDa. However, a further increase in M from 100 to 300 kDa causes a decrease in pp,dis by less than 0.03 g/mL. This suggests that the extent of polymer chain entanglement gradually reaches a limit at high molecular weight. This plateau behavior of pp,dis with M may well explain the limiting phenomena observed for tablets made of M = 96, 134, and 267 kDa, where the drug release rate is independent of HPMC molecular weight, whereas the rate for M = 29 kDa is significantly h i g h e ~ - . ~ Correlations ,~ between polymer/drug release and pp,& will be discussed below. 3.2. Evolution of Water Concentration ProfilesScheme 2 illustrates qualitative profiles for chain entanglement and for water concentrations across a swollen matrix. The water concentration in the glassy core is equal to the initial concentration. From the swollen glassy layer to the diffusion layer, water concentrations increase monotonically, as a result of diffusion; concentration profiles are in general non-linear. Evolutions of water concentration profiles for matrices made of two molecular weights, 250 and 10 m a , are given in parts a and b of Figure 2, respectively. For 250 kDa, a t 2 h, water molecules begin to penetrate into the matrix, producing a steep, nearly linear profile and matrix expansion of -10%. The water front, indicated by P1, moves into the matrix a t a normalized position of -0.7; i.e., <70% of the initial matrix remains unswollen. At 5 h, the water concentration profile becomes less steep as the water front, indicated by P2, moves further into the matrix a t r ‘ 0.3, leaving less than 30% of the dry glassy core unhydrated. The corresponding tablet radius increases by 20% from its initial value. At 8 h, the water front reaches the matrix center, indicating the disap-

-

where A(t ’1 is the surface area for dissolution. 1468 / Joufnal of Pharmaceutical Sciences Vol. 84, No. 12, December 7995

0.6

-

c.

-E

0.8

A

\

Y

0.5

5

0

v

c

8

s

0.4

\

e

0.3

-E

P

e f

0.6

f

0

5 50!

1

E \

0.2

;ii

f

5 0.1 n 0 0

s

0.4

0.2

0

0.5

1

1.5

NormalizedPosition

I

KlOOLV

1

I

I

I

, I

I

50

103

150

2m

250

300

HPMC M.W. (KDa) Figure 1-Computed polymer disentanglement concentration, pp,dis,for HPMC in water. The solid circle corresponds to pp,d$ for HPMC KlOOLV, while the solid squares (from left to right) correspond to pp,d. for HPMC K4M, K15M, and KlOOM, respectively.

pearance of the dry glassy core. Continued water di&sion afterward promotes the water concentration a t the matrix center to be above the initial concentration; e.g., a t 11h, the normalized water concentration a t the center becomes 0.07, twice the initial concentration. Also, the corresponding matrix radius grows by 35%. We note that the normalized concentration at the matrix surface is kept at -0.98, the corresponding water concentration for a pp,dis of 250 kDa. At early times (tI5 h), these profiles appear to be relatively linear. Concentration gradients on the basis of the slopes of the fitted lines decrease with time, a result of the expanded swollen regime; however, the overall concentration difference remains unchanged. At later times ( t = 11h), a second-order polynomial better fits the profile. Further, the water concentration at the center exceeds the initial concentration, resulting in a lower overall concentration difference and a decrease in the rate of diffusion. At early stages, say t < 5 h, water concentration profiles for M = 10 kDa are similar to those of 250 kDa. Yet, the corresponding overall dimensions decrease with time. Conversely, tablet dimensions increase with time for 250 kDa. At 8 h, the water concentration profile for 10 kDa is much more elevated than for 250 kDa. Moreover, the tablet dimension becomes 0.6, compared t o 1.25, respectively. We are currently employing optical techniques to measure the water concentration profiles across the matrix so that results from the model predictions may be tested. The rapid matrix dissolution for M W = 10 kDa is attributed to its high &,dis and large (Dp). we estimate pp,dis for 10 kDa to be 0.37 g/mL, 18 times greater than that for 250 kDa, 0.02 g/mL. Equation 11 also indicates that (D,) for 10 kDa is higher than for 250 kDa. Therefore, on the basis of eq 9, the polymer dissolution flux for 10 kDa is expected to be greater than that for 250 kDa, leading to a smaller matrix size for 10

t 0.0

0.5

1 .o

1.5

Normalized Position

Figure 2-Water concentration profiles as a function of time for HPMC of molecular weights = 250 kDa (a) and 10 kDa (b). Concentrations are computed at times = 2, 5, 8, 11, and 14 h. Each tablet contains 500 mg of HPMC and a small amount of drug. The initial tablet radius and thickness are 3.85 and 7.7 mrn, respectively. The prefactor 6 used in eq 23 is 0.41 and the apparent rotating rate Oappis assigned to be 270 rpm.

kDa than for 250 kDa. Recall that the diffusion time scales with the square of matrix dimension; it is estimated that, at t = 8 h, the time for water molecules to diffuse across the gelled matrix for 10 kDa is only (0.6/1.2512= 0.23 of that for 250 kDa. For soluble drugs, diffusion is the primary release mechanism, and the difference in diffusion paths dictates drug release. Thus, we expect the drug release rates for 250 kDa matrices to be much slower than those for 10 kDa matrices. 3.3. Evolution of Matrix Structure: Gel Layer, Swollen Glassy Layer, Dry Glassy Layer, and "Core"Information regarding the detailed and time-dependent structure of a gelled matrix is a key to understanding swelling kinetics. As shown in Scheme 2, a swollen matrix is identified as consisting of three regimes: the gel layer, the swollen glassy layer, and the dry glassy layer. The dry glassy coreswollen glassy layer interface is the water front, while the interface between the gel layer and the swollen glassy layer is the glassy-rubbery interface. Determination of the concentration at the glassy-rubbery interface, ps, remains a task. Our preliminary optical image experiments suggested that ppr is -0.5 g/mL, close to published data for the HPMC K series.44 On the basis of the calculated water concentration profile, we use interpolation t o determine the position of the glassyrubbery interface. Thicknesses for the dry glassy core, the swollen glassy layer, and the gel layer are defined by the water front, the glassy-rubbery interface, and the instantaneous matrix radius. Structural evolution, in terms of the normalized dimensions associated with the gelled matrix, for 250 kDa matrices is depicted in Figure 3a. The normalized overall matrix dimenJournal of Pharmaceutical Sciences / 1469 Vol. 84, No. 12, December 1995

1.6

I

2

1.4

1.2

Y

E

u

3

Unhydrated Regime Center

z

Very Strong

Strong

WeaL

Entanglement

Entanglement

Entaaplement

Dry Glassy

core

Swollen GIaasy

Gel

Diffusion

Layer

I.ayer

Layer

l

0.8 0.6 0.4 0.2

BulL

0

5

10

15

20

25

lime (hr)

3 '

t $j0.8 ~

Scheme 2-(top) The profile of polymer chain entanglement for a swollen matrix. Within the dry glassy layer, polymers take up most of the space, producing an unhydrated regime. In the swollen glassy layer, solvent diffusion promotes water concentrations to a slight degree, leading to a more mobile (compared to the dry glassy layer) network and very strong chain entanglement. As a result of significant swelling, less polymers are present in the gel layer, inducing less (compared to the swollen glassy layer), yet strong, entanglement. Finally, in the water-rich diffusion layer, the polymer concentration is low and chain entanglement becomes weak. At the gel layer-diffusion layer interface, chain entanglement becomes so weak that it can no longer hold polymers together; thus, polymer dissolution lakes place at this interface. (bottom) A typical water concentration profile corresponding to the top portion of the scheme. The space defined by the double lines represents the undissolved matrix.

sion, RIR,, grows monotonically with time over 20 h; its low dissolution rate leads to the domination of solvent uptake over polymer dissolution. Similarly, a monotonic increase in the gel layer thickness, ldR,, is predicted during the same time period. The swollen glassy layer thickness, ZsdR,, is greater than the gel layer thickness up to 8 h and disappears at 16 h. Interestingly, the &IR, profile is approximately symmetrical. The solvent front penetration, being faster than polymer dissolution, contributes to the ascending component of lsJRo up to 5 h when the solvent front reaches the center. Thereafter, the descending component is controlled by polymer dissolution only. At 16 h, when the swollen glassy layer is diminishing rapidly, the matrix is completely gelled and the gel layer thickness undergoes a steep increase. There are two important time scales associated with the swelling processes, the time at which the dry glassy core disappears (or ZsdRoreaches maximum) and the time when the swollen glassy core diminishes. On the basis of the symmetry profile of 18JR,, the latter is roughly twice the former. The relationship of these two times with HPMC molecular weight will be discussed below. Finally, the normalized dry glassy layer thickness, zd#?,, decays with time and disappears at time = 5 h when ZsdRois a maximum. At early time intervals (I2 h), solvent swelling takes place only in the outer regimes of the matrix and the dry glassy core is the major constituent. As swelling proceeds, the gel and swollen glassy layers increase, resulting in the reduction of the dry glassy core. Figure 3b demonstrates the corresponding structural evolution for the case of 10 kDa. Except for RIR,, profiles for l,/Ro, 1470 / Journai of Pharmaceuticaf Sciences Voi. 84, No. 12, December 1995

-Overall Sze __-

E

--3

Gel Layer

. . . . . Swollen Glassy Layer

U 0.6 0.4

-.-

Dry Glassy Layer

0

= 0.2 0

5

10

15

20

25

Time (hr) Figure 3-Structural dimensions associated with the gelled matrix versus time for HPMC of molecular weight 250 kDa (a) and 10 kDa (b), in terms of thicknesses of overall matrix radius, gel layer, swollen glassy layer, and dry glassy layer.

lsJR,, and Zdp/R, are qualitatively similar to those for 250 kDa. After one hour, the RIR, profile for 10 kDa declines monotonically with time, due to the domination of polymer dissolution over solvent uptake. The normalized gel layer thickness increases with time up to 8 h, after which time the matrix becomes fully gelled and ZdR, exhibits a steep decline. The swollen glassy layer thickness, being symmetrical in shape, reaches its peak a t 4 h and diminishes a t about 9 h. The dry glassy layer thickness decreases with time and disappears at 4 h, 1 h earlier than for 250 kDa. Again, fast polymer dissolution for 10 kDa contributes to the fast disappearance of the swollen glassy and dry glassy layers. We now assess the molecular weight effect and demonstrate that pp,& manifests itself as the key parameter. 3.3.1. Matrix Radius, RIR,-Figure 4a shows RIR, for a series of molecular weights, ranging from 10 to 250 kDa. The durations of the growing and reduction periods vary systematically with molecular weight. Low &,,& and (D,) for high molecular weight HPMC give rise to slow dissolution such that solvent uptake governs over matrix dissolution, and the growing period duration increases. For example, RIR, for a 250 kDa matrix grows from unity to 1.5 monotonically up to 20 h, whereas RIR, for a 10 kDa matrix shows a monotonic decrease with time to -0.2 in 10 h. Molecular weight dependence of RIR, is better shown in Figure 4b a t five elapsed times. In general, RIR, increases with M sharply a t low M and approaches a plateau at high M . This feature is not fully developed a t early times (e.g., I2 h) when solvent uptake dominates over matrix dissolution for all matrices. At later times, matrix dissolution becomes controlling for matrices of low molecular weight, while solvent

(a) 1.6

1.6

-

1.4

--

1.2

--

T

3

0)

c

.

3 0

5

-i4 a

s

1--

0.8 --

'CI

-.-03 0.6 -E

8 z

0.4 --

\ I

10KDo

0.2

0

t = l l hr

t=5hr

-t = 2 h r

50

100

150

200

15

20

25

approaches a plateau at high M.

t=8hr

0

10

15KDa

lime (hi) Figure 5-Normalized gel layer thicknesses (Id&) from the five HPMC molecular weights of Figure 4. Similar to the radius, at low M, &increases I& with M and

----. --I i

5

\

250

HPMC Molecular Weight (KDa) Figure 4-(a) Evolution of normalized matrix radius for HPMC of five molecular weights = 250, 60, 30, 15, and 10 kDa. (b) Effect of HPMC molecular weight on matrix dimension at times = 2, 5, 8, 11, and 14 h. The assumed conditions are the same as those in Figure 2. At low M, calculated radii increase with M and approach a plateau at high M.

uptake is still dominating for high molecular weight matrices. This is consistent with the relationships among M , and pp,dis, and (Dp). 3.3.2. Gel Layer Thickness, l,/R,--The effect of molecular weight on the gel layer thickness is shown in Figure 5, where lJR, shows consistent growth with time up to a critical time at which lglRoincreases sharply. For M < 30 kDa, lglRo undergoes a decline with time after the critical period, whereas lJR, increases slowly thereafter for M > 60 kDa. The molecular weight dependence of lJR, after the initial time is similar to RIR,, i.e., lglRois relatively independent of M for high M (>30kDa) and decreases with decreasing M for low M (<30kDa). 3.3.3. Swollen Glassy Layer Thickness, lsglR,-The swollen glassy layer thicknesses are shown in Figure 6a, in which ZJ R, increases with time, reaches a maximum, and declines to zero afterward. Interestingly, tsg,zero is approximately twice tsg,max, a result of the symmetrical lsR.lRo profiles. Rank-order increases in tsg,eero and tsg,maxwith increasing M are also predicted. We plot the times corresponding to the maximum &IR0 (tsg,max) and the zero ls&Ro(tsg,zero) as a function of vary with and molecular weight in Figure 6b; both tsg,max M in a similar fashion to RIR,. The plateau characteristics associated with tsg,maxand tsg,zero are less evident than that for RIR,, possibly because the characteristic diffusion times scale with the squares of the tablet dimensions. If one

considers and tsg,zero in Figure 6b as the squares of RIR, from Figure 4b,the square root operation dilutes the plateau feature. 3.3.4. Dry Glassy Layer Thickness, ld,IR,-As shown in Figure 7, the corresponding ldglR,profiles do not vary to a significant degree with M . At a given time, the dry core dimension increases slightly with increasing M . The insensitivity of IdJR, to M suggests that the penetration of the water front is a weak function of M , consistent with the finding that the diffusion coefficients of water were independent of HPMC molecular weight.27 Water front penetration occurs at a region where the polymer concentration is high, -50% wlw, and polymer chain entanglement is very strong. The extent of chain entanglement is so high that any chain segment larger than a critical size36(smaller than the overall chain dimension) loses its own identity. As such, segments larger than the critical size do not recognize which chain they belong to and water-front penetration becomes less molecular weight dependent. Moreover, the & f R ,profiles are highly nonlinear and can be roughly divided into two linear periods, with a transition period in between. Slopes for the early linear period are greater those in the later period, indicating that the rate of water front movement at later times decreases as the water concentration gradient becomes less steep. The distance of the water front advance, (not shown) can be calculated through the relation (1 - &/R,). 3.3.5. "Core"Thickness, I, lR,-Experimentally, limitations in resolution prevent one from measuring lsglR0and ldp/Ro separately. Instead, measurement of the sum of lsglRoand &/Rot or the "core" thickness, lJR,, is feasible by optical microscopy6 or NMR imaging.47 Calculated core thicknesses are shown in Figure 8, where lJR, shows monotonic decreases with time and, eventually, diminishes to zero. This is consistent with the result by NMR imagine7 where the measured lJR, in the radial direction for HPMC matrices decreased with time monotonically. In general, a three-stage decline in lJR, is predicted. In the early stage, which lasts up to 2 h, the decline in lJR, is fast. In the intermediate stage, the rate of decline slows down. In the final stage, lJRo Journal of Pharmaceutical Sciences / 1471 Vol. 84, No. 12, December 1995

I

0.9

0.8 rn

t

C s 0

.

0.7

_

I

.

30 KDa

- - -15 KDa

E

$ 0.6 -0E 0.5 L.

-

-

I

10 KDa

A

5

0

15

10

c3

lime (hr)

2 0

(b)

0.4

.-2 0

0.3

0

z 0.2

t

0.1

Maximum Isg I d

t 0

50

lo3

150

200

250

HPMC Molecular Weight (KDa) Figure 6-(a) Normalized swollen glassy layer thicknesses for the the five HPMC molecular weights in Figure 4 and (b) elapsed times corresponding to the maximum swollen glassy layer thickness and zero swollen glassy layer thickness. The assumed conditions are the same as those in Figure 2.

undergoes a rapid decay. At t < 2 h, ldR, appears to be independent of M , followed by a monotonic increase with M afterward. Finally, in contrast to IdglRo,the molecular effect appears to be pronounced for lJRo. This finding is attributed to the inclusion of the molecular weight sensitive lsglR,in Id R,, rendering l&R,a strong function of molecular weight. 3.4. Polymer Release-Matrix dissolution is a critical determinant of drug release. For example, fast matrix dissolution leads to small matrix dimensions, which, in turn, produces fast drug release. It is in this regard that the RIR, can be used to gauge matrix dissolution qualitatively. However, if quantitative matrix dissolution information is desired, rigorous measurements of polymer concentrations should be employed. In this model, the polymer dissolution rate is calculated through eq 9. Polymer dissolution kinetics (on the basis off, = 0.41 and wapp = 4.5 s-l) are shown in Figure 9a, where the fractional polymer released, m,(t)lm,(.o),versus time is plotted for the above five molecular weights. As expected, m,(t)lm,(-) increases with decreasing molecular weight. For example, a t t = 10 h, 98% polymer is released for a 10 kDa matrix, compared with only 5% for a 250 kDa matrix. The effect of molecular weight on polymer release is better assessed in Figure 9b, where m,(t)/m,(.o)is plotted versus M at five elapsed times. In general, m,(t)/m,(m) undergoes a sharp decline with M at low M and gradually approaches a low limit at high M ,consistent with the characteristic variation of pp,dis with M . Conceptually, the molecular weight effect can also be explained from the perspective of polymer chain entangle1472 / Journal of Pharmaceutical Sciences Vol. 84, No. 12, December 1995

0 0

1

2

3

4

5

6

Time (hr) Figure 7-Normalized dry glassy layer thicknesses. The layer thicknessesdecrease with time as water molecules diffuse toward the center, leading to the eventual disappearance of the dry glassy layer. The assumed conditions are the same as those in Figure 2. The model predicts that the dry glassy layer dimensions do not vary significantly with M.

ment. At the same polymer concentration, a polymer of higher molecular weight induces greater chain entanglement than a polymer of lower molecular weight. Therefore, it is harder for longer chains to dissolve, because of the higher energy required for pulling them off the matrix. The result is slower dissolution rates for matrices made of higher molecular weight polymer. The power-law relationships of eqs 10 and 16 indicate that a power law between m,(t)lm,(.o)and M may be obtained. Our calculation showed that, on the basis of eq 11, (D,) scales with M as

(D,) oc

(26)

Substitution of eqs 16 and 26 into eq 9 yields

Upon conversion of the linear scales in Figure 9b to semilog scales, the straight lines in Figure 10 imply

Values of a and b are shown in parts a and b of Figure 11, respectively. A decrease in a from 1.1 to 1.0 over 14 h is obtained, close to the expected value of 1.15. On the other hand, b increases from 2 to 14 during the same time period.

(a) 4

- 250 KDa --

60 KDa

--..

30 KDa

--.

15 KDa

-..-

10 KDa

15 KDo

10 KDo/

30 KDo

60 KDa

,

.’

250 KDo _y_1

5

0

15

10

20

25

- hr - - 1 1 hr 14

- - - . a hr n 0.501 I

5 hr

- 0.401

g

E

0.301

- I .

2 hr

0 0.m

e 0

5

10

15

20

Time (hr) Figure 8-Time-dependence of the “core thickness”, defined as the sum of the glassy layer and the swollen glassy layer thicknesses. The assumed conditions are the same as those in Figure 2.

The slight change (10%)in a leads to

m,(t)/m,(-) = K 1 . 0 5

0.101

0.001

0

50

150

100

200

250

HPMC Molecular Weight (KDa) Figure 9-(a) Fractional polymer release for the five HPMC molecular weights in Figure 4. The assumed conditions are the same as those in Figure 2. (b) Effect of molecular weight on polymer release at elapsed times = 2, 5, 8, 11, and 14 h. As expected, fractional polymer release decreases with increasing M and approaches a plateau at high M.

(29)

Equation 29 is valuable in that, if the polymer release kinetics for one molecular weight is known, the corresponding profiles for any other molecular weight can be predicted. Accordingly, eq 29 can be adapted to speed up formulation development. We believe that the same approach can be applied to other polymer/solvent systems. Fortunately, polymer solution properties such as solvent quality can now be routinely measured through well-developed light-scattering techniques. What is intriguing is that light scattering probes the solution behavior of single polymer chains, yet it appears that information gathered in the solution state experiment can then be used to predict the swelling behavior of highly entangled matrices. This is due to the fact that matrix dissolution is dictated by polymer dissolution. On a molecular basis, dissolution is in turn dictated by the process of how single polymer chains detach off the matrix. Therefore, the resultant dissolution data can be interpreted on the basis of the solution behavior of single polymer chains. 3.5. Drug Release-Fractional drug release, mdft)/mdt=), is depicted in Figure 12a. As expected, lower M produces faster drug release. For example, a t t = 11h, 100% drug is release for 10 kDa, compared with 45% for 250 kDa. The molecular weight dependence of drug release is illustrated in Figure 12b, where md(t)/?nd(m) is plotted versus M a t five elapsed times. The model predicts that md(t)/md(m) decreases strongly with increasing M at low M and approaches a plateau at high M. This plateau characteristic is consistent with the observed

limiting molecular weight e f f e ~ t . ~The ? ~ similarity in the molecular weight dependence characteristics for polymer and drug release suggests that a power law exists between m&)/ md(-) and M. The fitted straight lines of Figure 13 suggest

m&t)/m,(-)= b‘K”’

(30)

As shown in Figure 14, the values of a ‘ decrease from 0.27 to 0.21 over 11 h. A corresponding increase in b ‘ from 0.2 to 1.5 is predicted during the same time period. Similar to polymer release, the slight change in a ’ leads to

md(t)/m,J(-)

OC

~ 0 . 2 4

(31)

Equation 31 provides a means for predicting drug release. If the drug release kinetics for one molecular weight are attained, the corresponding release profiles for other molecular weights can be predicted with accuracy. Accordingly, with eq 31, the time for designing an optimal formulation should be significantly reduced. 3.6. Comparisons of Predicted Polymer and Drug Release-The magnitudes of the exponents in eqs 29 and 31 reflect different degrees of dependence on molecular weight for polymer and drug release, respectively. The larger exponent for polymer (1.05)than for drug (0.24)indicates that polymer release is more sensitive to molecular weight than drug release, agreeing with our experimental finding.46 It may be attributed to the difference in release routes for Journal of Pharmaceutical Sciences / 1473 Vol. 84, No. 12, December 1995

(0)

.-

10 KDO

.-16KDa

. .-

30KDa

14 hrs 1 1 hrs 8 hrs

I

5

0

10

5 hrs

20

15

25

Time (hr) 1

--

-8 0.8

2 hrs

14 hr

6

8o, 0.6

-n2g

I

0.001

I

0 z

1MX)

t

1

10

1

100

HPMC Molecular Weight (KDa)

I

0.8 5

10

15

10

15

lime (hr)

(b)

0

'

0

.

.

'

: 5

4

Time (hr) Figure 11-Values of the exponent a and the prefactors b of eq 28 in the text. The averaged a (1.05) is used in eq 29 for the power-law relationship between fractional polymer release and M.

polymer and drug. The controlling step for polymer release is polymer dissolution at the matrix surface. On the other 1474 / Journal of Pharmaceutical Sciences Vol. 84, No. 12, December 1995

-

1

.

8 hr

- - 5-

0.4

hr

- I

8 0.2

- 2hr

0

50

loo

150

200

250

HPMC Molecular Weight (KDa)

Figure 12-(a) Fractional drug released for the five HPMC molecular weights in Figure 4. The assumed conditions are the same as those in Figure 2. (b) Effect of molecular weight on drug release at elapsed times = 2, 5, 8, 11, and 14 h. As expected, fractional drug release decreases with increasing M and approaches a plateau at high M.

(a)

0

1

0

Figure 10-Power-law plots for fractional polymer release at five elapsed times. Symbols are from model predictions and lines are from the fitted power-law relationships.

1'2

1 1 hr

hand, drug molecules traverse a gelled matrix in order to be released from the matrix surface. The controlling step for drug release is diffusion across the matrix. Part of the diffusion process within the matrix involves the region of high polymer concentrations, where diffusion coefficients are independent of molecular weight. This contrasts with polymer, the entire release process of which takes place a t the matrix surface where the primary parameters, &,din and (&), are molecular weight sensitive. Thus, polymer release is more sensitive to molecular weight than drug release. To illustrate the above concept, polymer release is compared to drug release in Figure 15 a-c for three molecular weights. Drug release is in general faster than polymer release; this feature is most manifested for the highest molecular weight 250 kDa. For 10 kDa, however, both polymer and drug are released at comparable rates. We note that the predicted drug release profiles in Figure 15 were obtained from HPMC-drug matrices. In reality, fillers such as lactose are incorporated into the matrix for commercial formulations. As indicted by eqs 5 and 6 , diffusion coefficients for drugs are strong functions of the matrix composition. Addition of lactose would increase Dd and will in turn alter the drug release characteristics. As a test to the model, comparison of model predictions with observation for three molecular weights is shown below. 3.7. Comparisons of Model Predictions w i t h Experimental D a t a f o r Polymer and Drug Release-Polymer and ADM release data were collected and compared to model predictions for the three formulations in Table 3. AS shown in Figure 16,fractional polymer release (symbols) decreases with increasing molecular weight, a result of increased chain entanglement. The calculated polymer release profiles (curves)

,

d)

K

.- - 'Drug

1 1 hrs 8 hrs

HPMC

5 hn

n 0.1

0

15

10

5

20

Time (hr)

4

'1

2 hn

Drug

5

0

10

15

-.-

20

Time (hr)

0

5

10

Time (hr) Figure 15-Comparisons of polymer (solid curves) and drug release (dashed curves) for (a) 250 kDa, (b) 60 kDa, and (c) 10 kDa.

Table 3-Molecular Weight of HPMC Used in the Formulationsa

0 0

5

10

15

Time (hr)

*I

0

5

10

15

Time (hr) Figure 14-Values of the exponent a' and the prefactors b' of eq 30 in the text. The averaged a ' (0.24) is used in eq 31 for the power-law relationship between fractional drug release and M.

were obtained on the basis off, = 0.41, happ = 270 rpm, and i? = 0.01 cm2/sfor water at 20 "C. The model predictions agree well with the experimental data, within 15% error. The agreement implies that the model is able to describe polymer dissolution with accuracy.

Code

HPMC Molecular Weight (kDa)

Code

HPMC Molecular Weight (kDa)

KlOOLV K4M

29 96

K15M K100M

134 267

a Each tablet weighed 500 mg, with 35% HPMC, 62% lactose, 2.5% adinazolam mesylate, and 0.5% magnesium stearate.

The corresponding ADM release profiles are illustrated in Figure 17. Both data and predictions show decreasing drug release with increasing molecular weight, as a consequence of decreasing polymer dissolution. Again, model predictions agree well with the data, within 15% error, indicating the credibility of the model. The above agreement suggests that, for the system studied, the polymer disentanglement concentration and the diffusion coefficient of polymer are the primary determinants of the molecular weight dependence of drug release. However, other factors such as the molecular weight dependence of the diffusional hindrance, such as D, and Dd, within the tablets are not considered, and they may contribute to the disparity. There are two other possible causes for the deviations observed. First, at early times, drug close to the matrix surface may be released before the surrounding polymer reaches pp,dis,because the diffusion coefficients for drug molecules are higher than polymer. This is confirmed by the faster drug release than that predicted by the model in the first hour, Secondly, the model assumed homogeneous swelling and no polymer disentanglement before reaching pp,dis, while in reality, a heterogeneous structure may be presentF1 and polymer may detach before reaching Cp&. Journal of Pharmaceutical Sciences / 1475 Vol, 84, No. 12, December 1995

suggests that these two causes, though possibly contributing to the deviation, play secondary roles.

1 -

4. Summary

0.8 --

Q

A comprehensive model has been developed to predict swellingldissolutionbehaviors for HPMC matrices. The theme of the work is to introduce the polymer disentanglement concentration, pp,dls, and a diffusion layer into the model. The diffusion layer is a transitional regime from a gelled matrix to the bulk solution, while pp,dls is the polymer concentration below which polymer chains detach from the matrix and, subsequently, traverse the diffusion layer toward the bulk solution. We estimate pp,& to scale with molecular weight as pp,& M-O 6. Further, on the basis of the domination of axial expansion relative t o radial expansion, the model considers swelling in the radial direction only. A simple “anisotropic expansion” model is thus introduced to account for the anisotropic expansion. Evolution of water concentration profiles are calculated for HPMC matrices of various molecular weights. A swollen matrix is divided into three regimes: the gel layer, the swollen glassy layer, and the dry glassy core. The dry glassy core is the dominant structure in the early stage, the swollen glassy layer takes over in the intermediate stages, and in the later stages, the gel layer becomes the primary regime. Model predictions show that the effects of molecular weight on the overall matrix dimension, characteristic time scales associated with swelling/dissolution, polymer release, and drug release can be understood from the relationships between pp& and (Dp) and molecular weight. and The model predicts that fractional polymer (mp(t)/mp(c-)) drug (md(t)/md(m))released scales with M as mp(t)/mp(c-) M-’ O5 and md(t)/md(c-)=M-O 24, respectively; both decrease with an increase in M and approach limiting values as M becomes large. Finally, in comparison with experimental data for HPMC and drug release from HPMC-containing matrices, model predictions agree well with the experimental data, within 15%disparity. The agreement indicates that pp,&s, the diffusion layer, the concentration dependent D, and Dd, and the molecular weight dependent D,are the primary determinants for the water/HPMC swelling system. Deviation is attributed to ( 1 ) fast release of drug at early stages and (2) inhomogeneous swelling.

3

Notations

Q

-t! 0.6 --

Q

0

HI

8

-

0.4 --

e

Y

0.2 --

0

2

4

6

8

10

12

14

16

lime (hr)

Figure 16-Comparison of model predictions and experimental data46for fractional HPMC release from HPMC-based matrices. Each tablet weighed 500 mg, with a radius and thickness of 3.85 mm and 7.7 mm, respectively. The composition of each tablets is 35% HPMC, 62% lactose, 2.5% adinazolam mesylate, and 0.5% magnesium stearate. Model calculation was on the basis of 6 = 0.41 v = 0.01 cm2/s and Uapp= 270 rpm. KlOOLV

1

K4M K1 OOM

0.8

2

0.6

0~

C

d

a, a’

0’4

&t

A(t ’) (cm2) 0.2

b, b ’

0

I

T

0

2

4

6

,

8

/

I

10

Z

t

12

I

I

14

Di (cm2/s), Di ’

1

16

lime (hr) Figure 17-Comparison of model predictions and experimental d a t k for fractional adinazolam mesylate release from HPMC-based matrices, Each tablet weighed 500 mg, with a radius and thickness of 3.85 mm and 7.7 mm, respectively. The composition of each tablets is 35% HPMC, 62% lactose, 2.5% adinazolam mesylate, and 0.5% magnesium stearate. Model calculation was on the basis of 6 = 0.41, 1) = 0.01 cm% and Oapp= 270 rpm.

However, the pronounced disparity between our model calculations and experimental observations was expected if these two factors were significant. The good agreement observed 1476 /Journal of Pharmaceutical Sciences Vol. 84, No. 12, December 1995

D, (cm2/s) Di,O(cm2/s) Ds,i (cm2/s) (Dp)(cm2/s) fP

Jp(g/cm2s) k

exponents of the scaling laws for polymer and drug, respectively Surface area of a matrix a t a normalized time t’ Prefactors of the scaling laws for polymer and drug, respectively Diffusion coefficient and the normalized diffusion coefficient for species z mutual diffusion coefficient between water and polymer diffusion coefficient for species i in the dilute solution self-diffusion coefficient for species i averaged diffusion coefficient for polymer in the diffusion layer prefactor for the polymer dissolution flux dissolution (mass) flux of polymer proportionality constant connecting the polymer disentanglement concentration and the bulk polymer concentration

weighting factor for the normalized diffusion coefficient of water prefactor for the normalized diffusion coefficient of water weighting factor of species i to the the normalized diffusion coefficients of drug matrix thickness at time t and at time = 0, respectively dissolved drug at time t and at infinite time, respectively dissolved polymer at time t and at infinite time, respectively molecular weight of polymer radial coordinate and normalized radial coordinate matrix radius at time t and at time = 0, respectively normalized matrix radius elapsed time bulk velocity for a fluid passing a dissolving object matrix volume at time t and at time = 0, respectively normalized volume of the matrix weight fraction of species i mass of HPMC mass concentration and normalized mass concentration for species i, respectively polymer concentration in the bulk polymer disentanglement concentration overall mass concentration of the matrix diffusion layer thickness normalized time apparent rotating rate for polymer dissolution flux

k,’

References and Notes 1. Korsmeyer, R. W.; Lustig, S. R.; Peppas, N. A. J . Polym. Sci.: Polym. Phys. Ed. 1986,24, 395-408. 2. Hariharan, D.; Peppas, _ _ N. A. J. Controlled Release 1993, 23, 123-136. 3. LaDidus H.: Lordi. N. G. J. Pharm. Sci. 1968.57. 1292-1301. 4. Fl&y, P. J: Principles of Polymer Chemistry; Cornell Press: Ithaca, NY, 1953. 5. Feely, L. C.; Davis, S. S. Znt. J . Pharm. 1988, 41, 83-90. 6. Ford. J . M.: Rubinstein. M. H.: Hogan. J. E. Znt. J . Pharm. 1985. 24. 327-338. 7. Shah, N.; Zhang, G.; Apelian, V.; Zeng, F.; Infeld, M. H.; Malick, A. W. Pharm. Res. 1993.10, 1693-1695. 8. Thomas, N. L.; Windle, A. H. Polymer 1981,22, 627-639. 9. Lee, P. I. J . Membr. Sci. 1980, 7, 255-275. 10. Lee, P. I.; Peppas, N. A. J. Controlled Release 1987, 6, 207215. 11. Harland, R. S.; Gazzaniga, A.; Sangalli, M. E.; Colombo, P.; Peppas, N. A. Pharm. Res. 1988,5,488-494. Carothers, J. A. In Structure-Solubility 12. Ouano, A. C.; Tu, Y. 0.; Relationships in Polymers; Harris, F. A., Seymour, R. B., Eds., Academic Press: New York. 1977. 13. Peppas, N. A,; Wu, J . C.; von Meerwall, E. D. Macromolecules 1994,27,5626-5638. 14. Papadimitriou, E.; Buckton, G.; Efentakis, M. Znt. J . Pharm. 1993,98, 57-62. 15. Colombo, P.; Conte, U.; Gazzaniga, A,; Maggi, L.; Sangalli, M. E.; Peppas, N. A.; Manna La, A. Znt. J . Pharm. 1990, 63, 43I

I

48

16. Mitchell, K.; Ford, J . M.; Armstrong, D. J.; Elliott, P. N. C.; Hogan, J. E.; Rostron, C. Znt. J . Pharm. 1993, 100, 143-154. 17. Park, G. J. J . Polym. Sci. 1953, 11, 97.

18. Dreschel, P.; Hoard, J. L.; Long, F. A. J . Polym. Sci. 1953, 10, 241. 19. Ueberreiter, K. In Diffusion in Polymers; Crank, J., Park, G. S., Eds.; Academic Press: New York, 1968. 20. Wan, L. S. C.; Heng, P. W. S.; Wong, L. F. S . T . P Pharm Sci. 1993,3, 477-487. 21. Melia, C. D.; Rajabi-Siahboomi, A. R.; Davies, M. C. Proceed. Znt. Symp. Controlled Release Bioact. Muter. 1992, 19, 28-29. 22. Murray, W. D.; Landis, F. J . Heat Transfer 1959, 81, 106. 23. Harland, R. S.; Peppas, N. A. S. T. P. Pharm Sci. 1993,3,357361. 24. Vergnaud, J. M. Znt. J. Pharm. 1993,90,89-94. 25. Vergnaud, J . M., Ed. Liquid-Transport Processes in Polymeric Materials; Prentice Hall: Englewood Cliffs, NJ, 1991. 26. Yasuda, H.; Peterlin, A.; Colton, C. K.; Smith, K. A.; Merrill, E. W. Die Makromol. Chem. 1969,126, 177-186. 27. Gao, P.; Fagerness, P. E. Pharm. Res. 1995, in press. 28. Hsu, T.; Ma, D. S.; Cohen, C. Polymer 1983,24, 1273-1278. 29. Matsuo, E. S. et al, Macromolecules 1994, 27, 6791-6796. 30. Wu, J. C.; Peppas, N. A. J . Appl. Polym. Sci. 1993, 49, 18451856. 31. Gehrke, S. H.; Biren, D.; Hopkins, J. J. J. Biomater. Sci. Polym. Ed. 1994, 6 , 375-390. 32. Muhr, A. H.; Blanshard, J. M. V. Polymer 1982,23, 1012. 33. Handbook on Methocel-Cellulose Ether Products; Dow Chemical Co.: Midland, MI, 1982. 34. Schlichting, H. Boundary Layer Theory;McGraw-Hill, New York, 1955, Chapter 7. 35. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; Chapters 17-21. 36. Doi, M; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1989; Chapter 5. 37. Ju, R. T. C.; Nixon, P. R., Patel, M. V., Manuscript in preparation. 38. Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, N.J. 1962; pp 60-72. 39. Colombo, P. Adv. Drug Delivery Rev. 1993, 11, 37-57. 40. Berry, G. C.; Fox, T. G. Adv. Polym. Sci. 1968, 5, 261. 41. Graessley, W. W. Adv. Polym. Sci. 1974, 16, 1. 42. Ju, R. T. C.; Nixon, P. R., Patel, M. V. J . Pharm. Sci. 1995,84, 1455-1463 (preceding paper in this issue). 43. Kato, T.; Asami, H.; Takahashi, A. Kobunshi Ronbunshu 1986, 43, 399-404. 44. Melia, C. D.; Hodsdon, A. C.; Davies, M. C.; Mitchell, J. R. 1994 Proceedings of the 21st Inter. Symposium on Controlled Release of Bioactive Materials, June 27-30, 1994, Nice, France, paper #1397, pp 724-725. 45. FORTRAN Subroutines for Mathematical Applications, IMSL Matmibrary, IMSL, Inc., 1991, Chapter 5. 46. Sung, K. C.; Nixon, P. R.; Skoug, J . W.; Gao, P.; Ju, R. T. C.; Patel, M.; Topp, E. M.; 1994 AAPS annual meeting, San Diego, paper No. PDD 7490, p S297. 47. Rajabi-Siahboomi, A. R.; Bowtell, R. W.; Mansfield, P; Henderson, A.; Davies, M. C.; Melia, C. D. J . Controlled Release 1994, 31, 121-128.

Acknowledgments We appreciate the many discussions with Dr. Norman Ho, whose input was pivotal during the model set-up. Input from Drs. John W. Skoug and Ping Gao in the discussions of the diffusion coefficient measurements, gel layer characterization, and polymeddrug release were also critical. We also want to acknowledge valuable comments from Drs. Gail L. Zipp and Robert P. Zipp regarding the polymer dissolution rate. R.T.C.J. would like to thank the Postdoctoral Program of the Upjohn Co. through which this work was made possible. JS940691V

Journal of Pharmaceutical Sciences / 1477 Vol. 84, No. 12, December 1995