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Dynamics of controlled release from bioerodible matrices
Vol.15 No.2 199104 p.95-195 95
Jo~rn~ofCo~t~ol~ed~eIe~e, 15 (1991) 95-104 0 199 1 Elsevier Science Publishers B.V. 016%3659/91/$03.50 ~DO~rSOl68365991~13F
COREL 00558
Dynamics of controlled release from bioerodible matrices Abhay Joshi* and Kenneth J. ~immelstein** Aflergan Pharmaceuticals, 2525 Dupont Drive, Irvine, c4 927151599 (U.S.A.) (Received April 30, 1990; accepted in revised form August 23, 1990)
A mathematical model for the analysis of the basic physicochemical determinants that yields experimentally verifiable predictions of controlled release of bioactive agents from eroding polymeric matrices is presented. This model is applied to the erosion characteristics of acid catalyzed erosion of poly (ortho ester)s since there is considerable info~ation in the literature on the detailed physical and chemical performance of this system. The analysis shows that the dynamic changes in polymer matrix properties (namely, simultaneous reaction-diffusion-transport of matrix constituents, moving diffusion and water fronts, water-polymer partition coefficients, solubility of water and diffusivity of matrix components as a function of the extent of acid catalyzed matrix hydrolysis) play a significant role in regulating the release kinetics of bioactive agents. The analysis of poly (ortho ester) erosion as a test system predicts experimentally verifiable measurable quantities: release characteristics of the incorporated bioactive agent, water penetration into the matrix, catalysis by the anhydride and catalytic degradation of the polymer matrix. Further, an estimate of the concentration of unbroken polymer backbone linkages and hence the molecular weight of the polymer disc with time is obtained using random scission kinetics. The Thiele modulus is a good indicator of surface versus bulk erosion and has been successfully applied to characterize the erosion characteristics of the poly (ortho ester) system. Finally, an analysis for the basic design, interpretation, and prediction of overall release modes (bulk versus surface erosion) from devices that rely on simultaneous reaction and diffusion controlled erosion phenomenon is presented. Keywords: Biodegradable polymers; Erodible matrices; Release kinetics; Mathematical model
1. In~odu~tion Many recent advances in controlled release technology demonstrate that there are many uses for drug delivery systems whose governing drug release mechanism is either physical or chemical erosion, or a combination of erosion and diffusion. Such systems have inherent advantages over *To whom correspondence about this paper should be addressed. **Present address: 2 17 Gilbert Ave., Suite B, Pearle River, NY 10965, U.S.A.
other systems in that the self eroding process of the device obviates the need for retrieval or removal, drug release is more nearly drug-property independent, and the erodible device can be implanted at or near the target site of action. It is also apparent that the design and construction of these systems is not a trivial exercise. In most systems, erosion and release characteristics depend on a complex, interrelated set of mechanisms including, but not limited to, chemical reaction (often catalyzed or retarded by excipients and the physical milieu), multicomponent dif-
96
fusion, and physical changes in the system. In the effort to develop workable systems, it is often difficult to assess (and hence exploit) how these interrelated mechanisms combine to yield overall performance. A mathematical model which incorporates all of the important mechanisms can be useful to aid in the design of erodible drug delivery systems. Unsteady state mass balances which lead to parabolic differential equations have been successfully used to describe reaction/diffusion/ dissolution systems [ 11. Others have considered erosion to be a moving boundary problem, with the rate of surface disappearance being given by a rate constant. However, this approach does not predict the erosion rate [ 21. A recent attempt to develop a mechanistically based model [ 31 was able to reasonably predict overall release characteristics of an eroding system. However, this model was never applied to the simulation of experimentally derived results. Therefore, it was not possible to confirm the predictive capabilities of the approach using independently derived parameters. A further complication was that the principal dependent variable was the concentration of the unbroken polymer backbone linkages, a difficult to obtain piece of information. It is the purpose of this paper to develop a generalized mathematical model of a chemically erodible drug delivery system. Based on the previously published model [ 3 1, the model is extended to calculate an experimentally determinable quantity, the average polymer molecular weight, using random chain scission theory. Other drug release characteristics such as release rates and overall performance characteristics such as bulk versus surface erosion are also included. Because of a lack of detailed information on many erodible systems, the model is applied to the erosion characteristics of the acid catalyzed erosion of poly (ortho ester )s. There is considerable information in the literature [ 4-9 ] on the detailed performance of this system covering a wide range of system characteristics, and therefore it makes a good test case for the extended mathematical model. The effort to accomplish the above goal is ap-
proached in four steps. First, a model previously proposed [ 31 is used as a starting point for the model formulation. Second, a more nearly complete accounting of physicochemical and kinetic properties of the polymer, otherwise omitted in the previous model formulation, is done. Third, the model is applied to the description of results published related to release and erosion of a polymer matrix containing acid anhydrides. A feature of this work is the estimation of the molecular weight of the degrading polymer disc with time using random chain scission theory. The significance of the Thiele modulus is discussed and conditions, consistent with experimental findings, are obtained for diffusion and reaction limiting acid anhydride concentrations. Finally, an analysis for the device design, interpretation, and prediction of overall release modes (bulk versus surface erosion) from devices that rely on simultaneous reaction and diffusion controlled erosion phenomena is presented. 2. Model system As noted above, a general mathematical formulation is desired. However, the specific chemical reactions are system dependent. Therefore, this development is aimed at a specific polymer system, namely, poly (ortho ester )s, but the general approach will be useful for other systems as well. Poly (orthoester )s are random copolymers containing ortho ester linkages in the backbone [ lo]. These linkages are subject to hydrolysis that is acid catalyzed [ 8,113. As a result, incorporation of an acid or an acid-producing species can accelerate the hydrolysis of the matrix when placed in an aqueous medium [ 9 1. A second important property of poly (ortho ester )s is that they are relatively hydrophobic materials [ 5 1. Thus, very little water diffuses into the matrix. It has been hypothesized that this can lead to surface erosion if the reaction rate is fast enough to essentially consume the water as it enters the matrix [ 6 1. Surface erosion will, of course, lead to constant-rate drug release if the surface area of the device is relatively constant during the ero-
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WATER + ACID GENERATOR @I ACID + POLYMER (D)
ACID (Cl
POLYMER (D’)
+
ER’ + WATER (A)
+
l
-
DRUG + ERODING POLYMER (El
x.= 0
x’= a
Fig. I. Scheme depicting erosion process of polymer disc containing acid labile linkages.
sion process and drug diffusion is slow compared to the chemical erosion of the matrix. The general outline of a model system that exerts control on drug release rates by chemical erosion using an acid-producing excipient for catalysis is shown in Fig. 1. For ease of comparison, the physical eroding system and its environment were taken to be similar to the system studied by Nguyen et al. [4]. The system is a poly(ortho ester) disc about 0.8 mm thick and 10 mm in diameter with an average weight of 57 mg. The disc contains a variable amount of phthalic anhydride (denoted by B) physically dispersed into the polymer matrix (ortho ester linkages denoted by D) along with the drug to be released. In the experimental work of Nguyen et al. [ 41, a marker, Amaranth red dye (denoted by E), was used to simulate the drug release behavior (in the following text, words drug and dye are used interchangeably). The method of disc manufacture, dissolution testing and analysis procedure are detailed in Nguyen et al. [ 8 1. When such a matrix is placed in an aqueous environment (water denoted by A), water partitions into the disc at the surface of the matrix and slowly diffuses into the matrix. The slowly intruding water hydrolyzes the anhydride to activate the catalytic acid species (C). The acidic species catalyzes the polymer’s degradation in two steps: the formation of an unstable intermediate ester (D* ) followed by its reaction with water to form degradation products. These small
degradation products are water soluble and are released from the eroding matrix along with the catalyst and the bioactive ingredient into the surrounding medium. A close look at the drug release mechanism from the eroding matrix reveals that the rate of drug release is influenced by: ( 1) the chemical properties of the system - the hydrolytic and neutralization processes confined to the outer surface of the device, the catalyzed degradation of the polymer and the intrinsic backbone reactivity, and (2) the combination of physical properties and processes such as water diffusivity, solubility in the eroding matrix, drug and excipient loading, matrix dimensions, and the dissolution environment. A mathematical description that accounts for these factors is now discussed. 2.1 Scope and model assumptions A few assumptions are required for the formulation of the model. These assumptions relate to kinetic simplifications, boundary conditions, erosion milieu, hydrodynamics and transport consideration. They are: ( 1) a slab (extending infinitely in the plane parallel to the slab surface) geometry of the polymer matrix to simplify mathematical procedures, (2 ) a homogeneous distribution of drug and excipients in the polymer matrix,
98
( 3 ) low drug loading so that no phase separation occurs, (4) no volume change of polymer matrix due to water imbibement [ 5 1, and (5) perfect sink conditions, Fickian diffusion and external mass transfer resistances. 2.2 Generalized transport equations The equations that represent the dynamics of the active species of poly (ortho ester) based drug delivery systems are simply one dimensional unsteady state mass balances. The elementary reaction steps that comprise the degradation of this system, using the notation described above, are: A+&
r,=k[AIIBl
c++P
r~=k*[c] [Ill-k_,[Pl
ks D*+A- C+ products r3 =b[D*l [Al +b[Dl [Al To be consistent with previous work [ 3 J, the same alphanumeric notations are used to designate the matrix constituents and the reaction rates. The hydrolysis rate of the anhydride is given by k,; k2 and k_-2 are the rate constants for the formation and degradation of unstable ortho ester species; and k3 denotes the rate constant for ortho ester degradation and product generating reaction. The rate of uncatalyzed reaction k4 is taken to be zero. The reaction rate laws, denoted by r,, r, and r3, describing the hydrolytic activity of the model system are elementary. The equations that describe the unsteady state mass balance for all the reacting-effusing species take the following form:
where Cj is the concentration of species i; Dj is the corresponding diffusion coefficient of that species; x is the distance from the center of the matrix of totai thickness 2~; u, is the net sum of synthesis and degradation rate of species i; and t is the time. The diffusion coefficient of all the
species is related to the local extent of polymer hydrolysis and is given by the expression [ 3 ] : Di =DPexp[ ‘(‘:i
cD’],
i=A,B,C,E
(2)
where Dp is the diffusion coefficient of the species i when the polymer is not hydrolyzed and ,u is a constant. It should be noted that this functional form is chosen to yield initial and final values of known diffusivities in the unreacted matrix and water as the system is converted from pure to completely reacted polymer, respectively. No mechanistic meaning should be attributed to this form. The above equations are subjected to normal initial and boundary conditions. Initial conditions: C,(x,O)=O
forO
Cj(x,O)=Cp
for O
i=A,C,P
where Cp is the initial concentration of species i in the matrix. Boundary conditions: Di(O,t)!$(O,t)=O,
O
(3)
. ,
O
is the concentration of if” species at Ci,bulk time zero in the bulk of the aqueous phase. As the backbone cleavage of the polymer occurs, there is a transformation of the polymer causing the solubility of water in the matrix to be a function of polymer hydrolysis. This phenomenon causes a change in the solubility of water with the polymer at the boundary where hydrolysis creates a more hydrophilic region. This can be accounted for by making the partition coeffticient of water be a function of extent of hydrolysis. This boundary constraint for water is given by C,(W)=K(G)Cj: where Ci is the concentration of water in the aqueous phase. It is desirable to formulate the model so that it calculates a measurable dependent variable where
99
rather than being based on ortho ester linkages. This is especially important since the drug release rate is a complex function of many processes, so assessment of system performance needs at least a second measurement. Nguyen et al. [ 4,571 have shown that the reaction rate is relatively independent of comonomer and molecular weight, therefore random degradation statistics can be applied to the degradation process. The random hydrolysis of the ortho ester linkages in homogeneous solution can be used to determine the molecular weight of the degraded polymer with time. An analysis can be conveniently made by calculating the probability of finding a polymer molecule x-mer units long with p as the probability of finding a reacting group and d as the probability of finding an unreacted group [ 12 1. The probability of finding a molecule x units long is given by px-’ ( 1-p) or ( 1-d)“- ‘d; and the probability of finding number of molecules x units long, N,, in a polymer solution containing N chains of different lengths is N( 1 -d)“- ‘d. The number average degree of polymerization CxhT,/CN, when multiplied by the monomer unit weight gives an average molecular weight of the polymer. By similar analysis, the weight and number average molecular weights, and therefore the polydispersity in the polymer matrix, can be calculated. Thus, it is evident that given the extent of hydrolysis or polymer degradation, the average molecular weight of the polymer can be readily calculated at each point in the matrix with time. The average molecular weight for the entire disc is then obtained by summing the molecular weight at each point. The predictions generated using simple kinetics of random scission provide a means for system performance assessment and identifying the degradation mechanism of polymer matrix, namely surface erosion versus bulk erosion. As evident from the eqns. ( 1) , ( 3 ), and (4)) the issue of polymer erosion and drug release is an interaction of reaction and diffusion of multiple species. A priori, it is difficult to analyze several concurrent processes. One way to analyze and observe any physicochemical system within a proper frame of reference is to scale the
operating variables with respect to intrinsic reference scales; all the parameters would then aggregate into dimensionless property ratios which, if properly interpreted, can have clear physical significance. For the current problem, being of diffusion-reaction in nature, an important dimensionless group, the Thiele modulus, is worthy of discussion here. This dimensionless number, denoted by 9 is a ratio of time constant of diffusion and time constant for reaction [ 13 1. A particular Thiele modulus term that results from this analysis is: a21&, 92= l,k,C,,
(5)
Equation ( 5 ) can also be expressed as
where rd is the characteristic time COnStant for the diffusing species and 7, is the characteristic time constant for the reacting species. When the magnitude of the Thiele modulus (which is the square root of eqn. [ 5 ] ) is greater than Unity, rd is greater than 7, and the diffusion rate limits the overall rate of reaction; when Thiele modulus is less than unity, rd is smaller than 7, and the chemical reaction is rate limiting. In particular, the Thiele modulus & relating to the reaction and diffusion of acid anhydride and water is of most importance for the model analysis, since it determines the regimes of diffusion versus reaction controlled kinetics. The reaction terms in Thiele moduli #2 and & correspond to the chemical degradation of the polymer and its unstable intermediate; these are the reactions that occur after the release of acidic species and thus are not the primary rate limiting steps in acid catalyzed degradation of erodible matrices. Therefore, all the discussion in the following text is based on the significance of &. 2.3 Solution method The numerical method of solution of these coupled, unsteady state partial differential equations is taken from Ref. [ 141. Briefly, finite differences are used to represent the various terms
100
of the equations. The reaction terms and the nonlinear terms are written on the previous (known) time step to both linearize the equations and decouple them. At the next time step the calculated values are compared to the assumed, previous time values. If they do not agree to a predetermined value, the calculated values are used to update the nonlinear and reaction terms and the solution is iterated. The resulting linear algebraic equations are solved using the Thomas technique [ 141. 3. Results and discussion The simultaneous diffusion-reaction model as discussed above is applied to model the relationship between drug release and polymer degradation, using the experimental data of Nguyen et al. [ 4,8 ] as a test system to evaluate the practical utility of this model for bioerodible matrices. The experimental studies on the events leading to controlled drug release from such a fabricated device were conducted by Nguyen et al. [ 41. ErTABLE
1
Device dimensions
and properties
[ 21
property
dimension
Diameter of disc Average thickness Device volume Device weight Dye loading Device composition
1 cm 0.14cm 0.1099 cm3 51 mg 0.5% refer to Table 2
odible discs containing 0.25%, 0.5% and 1.O% phthalic anhydride (by weight ) were placed in a water bath (resembling sink conditions); periodically the discs and the bathing medium were analyzed to determine the water content, amount of anhydride in the disc, average polymer molecular weight, and the amount of color marker released in the aqueous medium. Computer simulations were conducted with the model to mimic the conditions of the experimental situation. The simulation results are presented for a disc device whose dimensions and physical properties are given in Table 1. The molecular weight of each monomeric unit of the polymer is about 350 with two sites of hydrolysis. The concentration of various components in the polymer matrix and the dimensionless property ratios corresponding to each disc are also given in Table 2. The results obtained from these simulations are discussed below. 3.1 Erosion and release events of discs with 0.25% anhydride
Figure 2 shows the experimental results and the computer simulations for a device initially containing 0.25% phthalic anhydride. The model simulation for the dye release compares favorably with the experimental data except during the initial release phase. This can be attributed to the premature dye release during the initial release 7
TABLE 2 Device composition and dimensionless ratios [ 3-61 Device 1 Phthalic anhydride C, (mol/cm’) C, (mol/cm’) c,, (mol/cm3) B=G& ff =C,/C, 91 A @3
0.25% 8.75X10-’ 2.94x10-3 0.0555 0.00297 6339.59 0.756 7.592 75.92
Device 2 0.5% 1.75x1o-s 2.93x10-3 0.0555 0.00596 3171.40 1.069 7.583 75.83
Device 3 1.0% 3.50x10-~ 2.91x10-3 0.0555 0.01199 1585.70 1.512 7.563 75.63
0
20
40
60
60
100
Time, h
Fig. 2. Cumulative release of active species from the disc containing 0.25% of anhydride.
101
phase due to the physical properties of the dye used to simulate the drug. Amaranth red dye is hydrophilic in nature and is insoluble in the polymeric matrix. When the matrix is exposed to the aqueous environment, both diffusion of the dye particles at or near the surface and water uptake into the matrix near the surface cause the hydrophilic dye particles to dissolve and a burst of the dye is experienced [ 15, i 6 1. As shown in Fig. 2, about 90% of the dye is released in about 90 hours. The experimental dye release time frame almost overlaps with the model time frame. An evaluation of the Thiele modulus for this system gives a #I value for anhydride of about 0.7652. Since the Thiele modulus is less than unity, it can be expected that the governing kinetic release mechanism is reaction controlled (i.e., by the rate of hydrolysis). A large resistance is offered by the hydrolysis of the catalyst; consequently, the water diffuses in the interior of the matrix much faster than the chemical reaction rate can consume it. Under these conditions, the water zone advances towards the interior of the matrix faster than the reaction zone, resulting in rapid saturation of the matrix bulk. Taking the diffusion coefficient of water to be 3.0~ lo-* cm*/s [ 5 1 and the characteristic length the water has to travel to be 0.07 cm, a time constant of about 45 hours is obtained. One time constant (45 h) elapsed corresponds to 63% of matrix saturation, and two time constants (90 h) corresponds to about 87% matrix saturation. 1.0 ..-_..* i;
I
a
f 0 G e u.
_...----
,.’
0.8 -
_S
E B E
. ..-
These numbers compare favorably with the fractional amount of water in the matrix as calculated from the simulations and the experimental results [ 41, see Fig. 3. It should be noted that these are the upper limits established by the analysis. However, the actual saturation of the matrix may be effected by several events such as inhomogeneity of the matrix, nonuniform dispersion of the catalyst, and presence of residual water. Under these circumstances, bulk degradation is expected. Figure 2 also shows the fractional amount of acid anhydride and acid remaining in the matrix as a function of time and the profiles are similar. A fair comparison could not be made with the experimental data, since the fabrication of the device and assay technique for anhydride resulted in an initial loss of anhydride [ 5 1. However, the time span over which the anhydride level drops to below 0.05% is about 50 h in both the experimental and the theoretical studies. The decrease in the molecular weight of the eroding disc containing 0.25% phthalic anhydride as a function of time is illustrated in Fig. 4. The molecular weight decreases linearly with time for about 45 to 50 h and then remains invariant for another 40 h. Based on the time constant analysis for water diffusion, the linear decrease in molecular weight corresponds to the elapsed time of one time constant when approx-
_i p
0.6 -
_.--
.*’ I’
0.4 /’
_..-
_...--
..*-
0.2 - P ! E
0.04 0
, 20
,
.
40
, 60
) 80
0
-I
1IO0
Time, h
Fig. 3. Amount of water imbi~ment in matrix containing 0.25% anhydride.
20
40 Time,
60
80
h
Fig. 4. Molecular weight of the degrading polymer with 0.25% anhydride with time. (-O-) Th~reti~Ily predicted, and ( f. 0.. ) ex~~mentaliy found values.
102
imately 63% of the matrix is saturated with water. The time elapsed after one time constant corresponds to further saturation of the matrix with water. Once the matrix is saturated with water, bulk degradation occurs; this enhances dye release and the dye release continues until the matrix collapses. These results are clearly illustrated in Figs. 2 and 3. 3.2 Erosion and release events of discs with 1.0% anhydride The events during the erosion of the polymer matrix containing 1% anhydride are completely different from the matrix containing 0.25% anhydride. The model predictions of dye release, acid anhydride concentration and acid compare favorably with the experimental data [ 41 and are shown in Fig. 5. Two distinct differences are observed when the erosion and release profiles of matrices containing 1% and 0.25% of anhydrides are compared: ( 1) About 90% of the dye is released in 30 h from the 1% anhydride-containing device as compared to 90 h from the 0.25% anhydridecontaining device. (2) Water diffusion into the center of matrix is not as rapid for the device containing 1% anhydride as for 0.25% anhydride-containing device [ 41. The Thiele modulus for this system is about
1.5 12, significantly greater than unity. Therefore, at the anhydride level of 1% it can be expected that the controlling degradation mechanism is diffusion and not reaction. The degradation front of the matrix also corresponds to the water front, and therefore the polymer breakdown happens in only those areas where the slowly diffusing water molecules are capable of hydrolyzing anhydride molecules to initiate the acid catalytic degradation of the polymer; the anhydride reaction uses water as fast as it is supplied via diffusion. The complete erosion process is controlled by only one moving zone -the water front, moving towards the center of the matrix. The surface of the device undergoes chemical degradation and the dye is released into the surrounding medium first through dissolution in the hydrophilic layer of the degraded matrix followed by diffusion. As shown in Fig. 5, the anhydride level in the disc falls rapidly; similar experimental observations were made [ 41. It can be concluded that the dye release mechanism is primarily by surface erosion and is further supported by electron micrographs in the work of Nguyen et al. [ 41. The degradation kinetics of such a device was analyzed as a function of time. In Fig. 6, the experimental results of the average molecular weight of the disc during erosion at several time intervals are compared with the model prediction and a striking qualitative similarity is ob-
1.0
3 I 8
0.8
= E a
0.8
s
0.4
a .s 0 &
0.2
-
Acid Anhydride
-
Acid
..........
Dye Released
20
10
0.0 20
30
40
50
Time, h
Fig. 5. Cumulative release of active species from the disc containing 1.O% of anhydride.
Time,
0
h
Fig. 6. Molecular weight of the degrading polymer with 1.O% anhydride with time. (. .O.. ) Theoretically predicted, and (- -O- -) experimentally found.
103
served. With the erosion process being predominantly at the surface, the availability of water in the polymeric matrix is significantly reduced. It is interesting to note that up to 20 h, the molecular weight of the disc decreases linearly with time. It can be surmised that at a later stage of erosion, the degradation front and hence the dye release front would be left behind the water diffusion front; at this point bulk erosion occurs. It can be concluded that the concentration of anhydride levels affects the erosion characteristics of the device in two ways: ( I ) it reduces the life span of the device, and (2) the governing drug release mechanism is surface erosion for higher concentration of anhydrides and bulk erosion for lower concentration of anhydrides.
diffusion and the reaction mechanism are competing against each other; there is no dominant degradation mechanism. The time constant for the transport of catalyst is about the same as that for the catalytic reaction rate, thus it is difficult to categorize whether the degradation mechanism is controlled by surface erosion or bulk erosion. The concentration of the catalyst, 0.50%, gives the borderline concentration of the catalyst that would govern the degradation mechanism. Thus, for anhydride concentrations less than 0.5%, bulk erosion is expected, and for anhydride concentrations greater than 0.5%, surface erosion is expected. Similar observations were made both theoretically and experimentally and were discussed in the preceeding sections.
3.3 Erosion and release events of discs with 0.50% anhydride
4. Conclusion
In Fig. 7, the results of erosion with 0.5% phthalic anhydride are compared with the theoretical simulations. The events during the erosion of this matrix are significantly different from the erosion events of matrix that contains 1.0% and 0.25% of anhydride. The model prediction of the dye release, as shown in Fig. 7, compares favorably with the experimental data of Nguyen etal. [4]. The Thiele modulus for this system is around 1.06. For the Thiele modulus of unity, both the Anhydride
I
A mathematical model has been developed to describe the erosion characteristics of polymeric matrices. The poly (ortho ester) system was chosen as a test system for this analysis. The model predicts measurable quantities, namely, release characteristics of the incorporated dye, water penetration into the matrix, catalysis by anhydride, and catalytic degradation of the polymer matrix. Random degradation kinetics give a good estimate of molecular weight in the polymer disc with time. The Thiele modulus is a good indicator of surface versus bulk erosion and has been successfully applied to determine the erosion characteristics of the test system, the poly (ortho ester)s. Acknowledgement We thank Dr. A.G. Thombre and Dr. T.H. Nguyen for the informative discussions during the course of this work.
References
Time, h
Fig. 7. Cumulative release of active species from the disc containing 0.5% of anhydride. (-_) Anhydride, (...) acid, and (--- ) dye released.
1
R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, NY, 1960.
104 2
3
4
5
6
7
8
P.I. Lee, Controlled release from polymeric materials involving moving boundaries, in: D.H. Lewis (Ed. ), Controlled Release of Pesticides and Pharmaceuticals, Plenum, New York, NY, 1981, pp. 39-48. A.G. Thombre and K.J. Himmelstein, A simultaneous transport reaction model for controlled drug delivery from catalyzed bioerodible polymer matrices, AIChE J., 35(1985)759-766. T.H. Nguyen, T. Higuchi and K.J. Himmelstein, Erosion characteristics of catalyzed poly (ortho ester) matrices, J. Controlled Release, 5 (1987) I-12. T.H. Nguyen, K.J. Himmelstein and T. Higuchi, Some equilibrium and kinetic aspects of water sorption in poly(ortho ester)s, Int. J. Pharm., 25 (1985) l-12. R.V. Sparer, C. Shih, C.D. Ringeisen and K.J. Himmelstein, Controlled release from erodible poly(ortho ester) drug delivery systems, J. Controlled Release, 1 ( 1984) 23-32. T.H. Nguyen, C. Shih, K.J. Himmelstein and T. Higuchi, Hydrolysis of some poly( ortho ester)s in homogeneous solutions, J. Pharm. Sci., 73 ( 1984) 1563-l 568. T.H. Nguyen, K.J. Himmelstein and T. Higuchi, Erosion of poly(ortho ester) matrices in buffered aqueous solutions, J. Controlled Release., 4 ( 1986 ) 9- 16.
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12 13 14
15
16
C. Shih, T. Higuchi and K.J. Himmelstein, Drug delivery from catalyzed erodible polymeric matrices of poly(ortho ester)s, Biomaterials, 5 (1984) 237-240. J. Heller, Synthesis of biodegradable polymers for biomedical utilization, Am. Chem. Sot. Symp. Ser., 212 (1983) 373-392. E.H. Cordes and H.G. Bull, Mechanism and catalysis for hydrolysis of acetals, ketals, and ortho esters, Chem. Rev., 94 (1974) 581-603. P.J. Flory, Principles of Polymer Chemistry, Cornell Press, Ithaca, NY, 1953. G. Astarita, Mass Transfer with Chemical Reactions, Elsevier, Amsterdam, 1967. B. Camahan, H.A. Luther and J.O. Wilkes, Approximation of the Solution of Partial Differential Equations, Applied Numerical Methods, Wiley, New York, NY, 1969. J. Heller, D.W.H. Penhale, R.F. Helwing, B.K. Fritzinger and R.W. Baker, Release of norethindrone from poly(ortho ester)s, in: T.J. Roseman and Z. Mansdorf (Eds.), Controlled Release Delivery Systems, Marcel Dekker, New York, NY, 1983, pp. 91-106. R.F. Fedors, Osmotic effects in water absorption for polymers, Polymers, 2 1 ( 1980) 207-2 10.
Jo~rn~ofCo~t~ol~ed~eIe~e, 15 (1991) 95-104 0 199 1 Elsevier Science Publishers B.V. 016%3659/91/$03.50 ~DO~rSOl68365991~13F
COREL 00558
Dynamics of controlled release from bioerodible matrices Abhay Joshi* and Kenneth J. ~immelstein** Aflergan Pharmaceuticals, 2525 Dupont Drive, Irvine, c4 927151599 (U.S.A.) (Received April 30, 1990; accepted in revised form August 23, 1990)
A mathematical model for the analysis of the basic physicochemical determinants that yields experimentally verifiable predictions of controlled release of bioactive agents from eroding polymeric matrices is presented. This model is applied to the erosion characteristics of acid catalyzed erosion of poly (ortho ester)s since there is considerable info~ation in the literature on the detailed physical and chemical performance of this system. The analysis shows that the dynamic changes in polymer matrix properties (namely, simultaneous reaction-diffusion-transport of matrix constituents, moving diffusion and water fronts, water-polymer partition coefficients, solubility of water and diffusivity of matrix components as a function of the extent of acid catalyzed matrix hydrolysis) play a significant role in regulating the release kinetics of bioactive agents. The analysis of poly (ortho ester) erosion as a test system predicts experimentally verifiable measurable quantities: release characteristics of the incorporated bioactive agent, water penetration into the matrix, catalysis by the anhydride and catalytic degradation of the polymer matrix. Further, an estimate of the concentration of unbroken polymer backbone linkages and hence the molecular weight of the polymer disc with time is obtained using random scission kinetics. The Thiele modulus is a good indicator of surface versus bulk erosion and has been successfully applied to characterize the erosion characteristics of the poly (ortho ester) system. Finally, an analysis for the basic design, interpretation, and prediction of overall release modes (bulk versus surface erosion) from devices that rely on simultaneous reaction and diffusion controlled erosion phenomenon is presented. Keywords: Biodegradable polymers; Erodible matrices; Release kinetics; Mathematical model
1. In~odu~tion Many recent advances in controlled release technology demonstrate that there are many uses for drug delivery systems whose governing drug release mechanism is either physical or chemical erosion, or a combination of erosion and diffusion. Such systems have inherent advantages over *To whom correspondence about this paper should be addressed. **Present address: 2 17 Gilbert Ave., Suite B, Pearle River, NY 10965, U.S.A.
other systems in that the self eroding process of the device obviates the need for retrieval or removal, drug release is more nearly drug-property independent, and the erodible device can be implanted at or near the target site of action. It is also apparent that the design and construction of these systems is not a trivial exercise. In most systems, erosion and release characteristics depend on a complex, interrelated set of mechanisms including, but not limited to, chemical reaction (often catalyzed or retarded by excipients and the physical milieu), multicomponent dif-
96
fusion, and physical changes in the system. In the effort to develop workable systems, it is often difficult to assess (and hence exploit) how these interrelated mechanisms combine to yield overall performance. A mathematical model which incorporates all of the important mechanisms can be useful to aid in the design of erodible drug delivery systems. Unsteady state mass balances which lead to parabolic differential equations have been successfully used to describe reaction/diffusion/ dissolution systems [ 11. Others have considered erosion to be a moving boundary problem, with the rate of surface disappearance being given by a rate constant. However, this approach does not predict the erosion rate [ 21. A recent attempt to develop a mechanistically based model [ 31 was able to reasonably predict overall release characteristics of an eroding system. However, this model was never applied to the simulation of experimentally derived results. Therefore, it was not possible to confirm the predictive capabilities of the approach using independently derived parameters. A further complication was that the principal dependent variable was the concentration of the unbroken polymer backbone linkages, a difficult to obtain piece of information. It is the purpose of this paper to develop a generalized mathematical model of a chemically erodible drug delivery system. Based on the previously published model [ 3 1, the model is extended to calculate an experimentally determinable quantity, the average polymer molecular weight, using random chain scission theory. Other drug release characteristics such as release rates and overall performance characteristics such as bulk versus surface erosion are also included. Because of a lack of detailed information on many erodible systems, the model is applied to the erosion characteristics of the acid catalyzed erosion of poly (ortho ester )s. There is considerable information in the literature [ 4-9 ] on the detailed performance of this system covering a wide range of system characteristics, and therefore it makes a good test case for the extended mathematical model. The effort to accomplish the above goal is ap-
proached in four steps. First, a model previously proposed [ 31 is used as a starting point for the model formulation. Second, a more nearly complete accounting of physicochemical and kinetic properties of the polymer, otherwise omitted in the previous model formulation, is done. Third, the model is applied to the description of results published related to release and erosion of a polymer matrix containing acid anhydrides. A feature of this work is the estimation of the molecular weight of the degrading polymer disc with time using random chain scission theory. The significance of the Thiele modulus is discussed and conditions, consistent with experimental findings, are obtained for diffusion and reaction limiting acid anhydride concentrations. Finally, an analysis for the device design, interpretation, and prediction of overall release modes (bulk versus surface erosion) from devices that rely on simultaneous reaction and diffusion controlled erosion phenomena is presented. 2. Model system As noted above, a general mathematical formulation is desired. However, the specific chemical reactions are system dependent. Therefore, this development is aimed at a specific polymer system, namely, poly (ortho ester )s, but the general approach will be useful for other systems as well. Poly (orthoester )s are random copolymers containing ortho ester linkages in the backbone [ lo]. These linkages are subject to hydrolysis that is acid catalyzed [ 8,113. As a result, incorporation of an acid or an acid-producing species can accelerate the hydrolysis of the matrix when placed in an aqueous medium [ 9 1. A second important property of poly (ortho ester )s is that they are relatively hydrophobic materials [ 5 1. Thus, very little water diffuses into the matrix. It has been hypothesized that this can lead to surface erosion if the reaction rate is fast enough to essentially consume the water as it enters the matrix [ 6 1. Surface erosion will, of course, lead to constant-rate drug release if the surface area of the device is relatively constant during the ero-
97
WATER + ACID GENERATOR @I ACID + POLYMER (D)
ACID (Cl
POLYMER (D’)
+
ER’ + WATER (A)
+
l
-
DRUG + ERODING POLYMER (El
x.= 0
x’= a
Fig. I. Scheme depicting erosion process of polymer disc containing acid labile linkages.
sion process and drug diffusion is slow compared to the chemical erosion of the matrix. The general outline of a model system that exerts control on drug release rates by chemical erosion using an acid-producing excipient for catalysis is shown in Fig. 1. For ease of comparison, the physical eroding system and its environment were taken to be similar to the system studied by Nguyen et al. [4]. The system is a poly(ortho ester) disc about 0.8 mm thick and 10 mm in diameter with an average weight of 57 mg. The disc contains a variable amount of phthalic anhydride (denoted by B) physically dispersed into the polymer matrix (ortho ester linkages denoted by D) along with the drug to be released. In the experimental work of Nguyen et al. [ 41, a marker, Amaranth red dye (denoted by E), was used to simulate the drug release behavior (in the following text, words drug and dye are used interchangeably). The method of disc manufacture, dissolution testing and analysis procedure are detailed in Nguyen et al. [ 8 1. When such a matrix is placed in an aqueous environment (water denoted by A), water partitions into the disc at the surface of the matrix and slowly diffuses into the matrix. The slowly intruding water hydrolyzes the anhydride to activate the catalytic acid species (C). The acidic species catalyzes the polymer’s degradation in two steps: the formation of an unstable intermediate ester (D* ) followed by its reaction with water to form degradation products. These small
degradation products are water soluble and are released from the eroding matrix along with the catalyst and the bioactive ingredient into the surrounding medium. A close look at the drug release mechanism from the eroding matrix reveals that the rate of drug release is influenced by: ( 1) the chemical properties of the system - the hydrolytic and neutralization processes confined to the outer surface of the device, the catalyzed degradation of the polymer and the intrinsic backbone reactivity, and (2) the combination of physical properties and processes such as water diffusivity, solubility in the eroding matrix, drug and excipient loading, matrix dimensions, and the dissolution environment. A mathematical description that accounts for these factors is now discussed. 2.1 Scope and model assumptions A few assumptions are required for the formulation of the model. These assumptions relate to kinetic simplifications, boundary conditions, erosion milieu, hydrodynamics and transport consideration. They are: ( 1) a slab (extending infinitely in the plane parallel to the slab surface) geometry of the polymer matrix to simplify mathematical procedures, (2 ) a homogeneous distribution of drug and excipients in the polymer matrix,
98
( 3 ) low drug loading so that no phase separation occurs, (4) no volume change of polymer matrix due to water imbibement [ 5 1, and (5) perfect sink conditions, Fickian diffusion and external mass transfer resistances. 2.2 Generalized transport equations The equations that represent the dynamics of the active species of poly (ortho ester) based drug delivery systems are simply one dimensional unsteady state mass balances. The elementary reaction steps that comprise the degradation of this system, using the notation described above, are: A+&
r,=k[AIIBl
c++P
r~=k*[c] [Ill-k_,[Pl
ks D*+A- C+ products r3 =b[D*l [Al +b[Dl [Al To be consistent with previous work [ 3 J, the same alphanumeric notations are used to designate the matrix constituents and the reaction rates. The hydrolysis rate of the anhydride is given by k,; k2 and k_-2 are the rate constants for the formation and degradation of unstable ortho ester species; and k3 denotes the rate constant for ortho ester degradation and product generating reaction. The rate of uncatalyzed reaction k4 is taken to be zero. The reaction rate laws, denoted by r,, r, and r3, describing the hydrolytic activity of the model system are elementary. The equations that describe the unsteady state mass balance for all the reacting-effusing species take the following form:
where Cj is the concentration of species i; Dj is the corresponding diffusion coefficient of that species; x is the distance from the center of the matrix of totai thickness 2~; u, is the net sum of synthesis and degradation rate of species i; and t is the time. The diffusion coefficient of all the
species is related to the local extent of polymer hydrolysis and is given by the expression [ 3 ] : Di =DPexp[ ‘(‘:i
cD’],
i=A,B,C,E
(2)
where Dp is the diffusion coefficient of the species i when the polymer is not hydrolyzed and ,u is a constant. It should be noted that this functional form is chosen to yield initial and final values of known diffusivities in the unreacted matrix and water as the system is converted from pure to completely reacted polymer, respectively. No mechanistic meaning should be attributed to this form. The above equations are subjected to normal initial and boundary conditions. Initial conditions: C,(x,O)=O
forO
Cj(x,O)=Cp
for O
i=A,C,P
where Cp is the initial concentration of species i in the matrix. Boundary conditions: Di(O,t)!$(O,t)=O,
O
(3)
. ,
O
is the concentration of if” species at Ci,bulk time zero in the bulk of the aqueous phase. As the backbone cleavage of the polymer occurs, there is a transformation of the polymer causing the solubility of water in the matrix to be a function of polymer hydrolysis. This phenomenon causes a change in the solubility of water with the polymer at the boundary where hydrolysis creates a more hydrophilic region. This can be accounted for by making the partition coeffticient of water be a function of extent of hydrolysis. This boundary constraint for water is given by C,(W)=K(G)Cj: where Ci is the concentration of water in the aqueous phase. It is desirable to formulate the model so that it calculates a measurable dependent variable where
99
rather than being based on ortho ester linkages. This is especially important since the drug release rate is a complex function of many processes, so assessment of system performance needs at least a second measurement. Nguyen et al. [ 4,571 have shown that the reaction rate is relatively independent of comonomer and molecular weight, therefore random degradation statistics can be applied to the degradation process. The random hydrolysis of the ortho ester linkages in homogeneous solution can be used to determine the molecular weight of the degraded polymer with time. An analysis can be conveniently made by calculating the probability of finding a polymer molecule x-mer units long with p as the probability of finding a reacting group and d as the probability of finding an unreacted group [ 12 1. The probability of finding a molecule x units long is given by px-’ ( 1-p) or ( 1-d)“- ‘d; and the probability of finding number of molecules x units long, N,, in a polymer solution containing N chains of different lengths is N( 1 -d)“- ‘d. The number average degree of polymerization CxhT,/CN, when multiplied by the monomer unit weight gives an average molecular weight of the polymer. By similar analysis, the weight and number average molecular weights, and therefore the polydispersity in the polymer matrix, can be calculated. Thus, it is evident that given the extent of hydrolysis or polymer degradation, the average molecular weight of the polymer can be readily calculated at each point in the matrix with time. The average molecular weight for the entire disc is then obtained by summing the molecular weight at each point. The predictions generated using simple kinetics of random scission provide a means for system performance assessment and identifying the degradation mechanism of polymer matrix, namely surface erosion versus bulk erosion. As evident from the eqns. ( 1) , ( 3 ), and (4)) the issue of polymer erosion and drug release is an interaction of reaction and diffusion of multiple species. A priori, it is difficult to analyze several concurrent processes. One way to analyze and observe any physicochemical system within a proper frame of reference is to scale the
operating variables with respect to intrinsic reference scales; all the parameters would then aggregate into dimensionless property ratios which, if properly interpreted, can have clear physical significance. For the current problem, being of diffusion-reaction in nature, an important dimensionless group, the Thiele modulus, is worthy of discussion here. This dimensionless number, denoted by 9 is a ratio of time constant of diffusion and time constant for reaction [ 13 1. A particular Thiele modulus term that results from this analysis is: a21&, 92= l,k,C,,
(5)
Equation ( 5 ) can also be expressed as
where rd is the characteristic time COnStant for the diffusing species and 7, is the characteristic time constant for the reacting species. When the magnitude of the Thiele modulus (which is the square root of eqn. [ 5 ] ) is greater than Unity, rd is greater than 7, and the diffusion rate limits the overall rate of reaction; when Thiele modulus is less than unity, rd is smaller than 7, and the chemical reaction is rate limiting. In particular, the Thiele modulus & relating to the reaction and diffusion of acid anhydride and water is of most importance for the model analysis, since it determines the regimes of diffusion versus reaction controlled kinetics. The reaction terms in Thiele moduli #2 and & correspond to the chemical degradation of the polymer and its unstable intermediate; these are the reactions that occur after the release of acidic species and thus are not the primary rate limiting steps in acid catalyzed degradation of erodible matrices. Therefore, all the discussion in the following text is based on the significance of &. 2.3 Solution method The numerical method of solution of these coupled, unsteady state partial differential equations is taken from Ref. [ 141. Briefly, finite differences are used to represent the various terms
100
of the equations. The reaction terms and the nonlinear terms are written on the previous (known) time step to both linearize the equations and decouple them. At the next time step the calculated values are compared to the assumed, previous time values. If they do not agree to a predetermined value, the calculated values are used to update the nonlinear and reaction terms and the solution is iterated. The resulting linear algebraic equations are solved using the Thomas technique [ 141. 3. Results and discussion The simultaneous diffusion-reaction model as discussed above is applied to model the relationship between drug release and polymer degradation, using the experimental data of Nguyen et al. [ 4,8 ] as a test system to evaluate the practical utility of this model for bioerodible matrices. The experimental studies on the events leading to controlled drug release from such a fabricated device were conducted by Nguyen et al. [ 41. ErTABLE
1
Device dimensions
and properties
[ 21
property
dimension
Diameter of disc Average thickness Device volume Device weight Dye loading Device composition
1 cm 0.14cm 0.1099 cm3 51 mg 0.5% refer to Table 2
odible discs containing 0.25%, 0.5% and 1.O% phthalic anhydride (by weight ) were placed in a water bath (resembling sink conditions); periodically the discs and the bathing medium were analyzed to determine the water content, amount of anhydride in the disc, average polymer molecular weight, and the amount of color marker released in the aqueous medium. Computer simulations were conducted with the model to mimic the conditions of the experimental situation. The simulation results are presented for a disc device whose dimensions and physical properties are given in Table 1. The molecular weight of each monomeric unit of the polymer is about 350 with two sites of hydrolysis. The concentration of various components in the polymer matrix and the dimensionless property ratios corresponding to each disc are also given in Table 2. The results obtained from these simulations are discussed below. 3.1 Erosion and release events of discs with 0.25% anhydride
Figure 2 shows the experimental results and the computer simulations for a device initially containing 0.25% phthalic anhydride. The model simulation for the dye release compares favorably with the experimental data except during the initial release phase. This can be attributed to the premature dye release during the initial release 7
TABLE 2 Device composition and dimensionless ratios [ 3-61 Device 1 Phthalic anhydride C, (mol/cm’) C, (mol/cm’) c,, (mol/cm3) B=G& ff =C,/C, 91 A @3
0.25% 8.75X10-’ 2.94x10-3 0.0555 0.00297 6339.59 0.756 7.592 75.92
Device 2 0.5% 1.75x1o-s 2.93x10-3 0.0555 0.00596 3171.40 1.069 7.583 75.83
Device 3 1.0% 3.50x10-~ 2.91x10-3 0.0555 0.01199 1585.70 1.512 7.563 75.63
0
20
40
60
60
100
Time, h
Fig. 2. Cumulative release of active species from the disc containing 0.25% of anhydride.
101
phase due to the physical properties of the dye used to simulate the drug. Amaranth red dye is hydrophilic in nature and is insoluble in the polymeric matrix. When the matrix is exposed to the aqueous environment, both diffusion of the dye particles at or near the surface and water uptake into the matrix near the surface cause the hydrophilic dye particles to dissolve and a burst of the dye is experienced [ 15, i 6 1. As shown in Fig. 2, about 90% of the dye is released in about 90 hours. The experimental dye release time frame almost overlaps with the model time frame. An evaluation of the Thiele modulus for this system gives a #I value for anhydride of about 0.7652. Since the Thiele modulus is less than unity, it can be expected that the governing kinetic release mechanism is reaction controlled (i.e., by the rate of hydrolysis). A large resistance is offered by the hydrolysis of the catalyst; consequently, the water diffuses in the interior of the matrix much faster than the chemical reaction rate can consume it. Under these conditions, the water zone advances towards the interior of the matrix faster than the reaction zone, resulting in rapid saturation of the matrix bulk. Taking the diffusion coefficient of water to be 3.0~ lo-* cm*/s [ 5 1 and the characteristic length the water has to travel to be 0.07 cm, a time constant of about 45 hours is obtained. One time constant (45 h) elapsed corresponds to 63% of matrix saturation, and two time constants (90 h) corresponds to about 87% matrix saturation. 1.0 ..-_..* i;
I
a
f 0 G e u.
_...----
,.’
0.8 -
_S
E B E
. ..-
These numbers compare favorably with the fractional amount of water in the matrix as calculated from the simulations and the experimental results [ 41, see Fig. 3. It should be noted that these are the upper limits established by the analysis. However, the actual saturation of the matrix may be effected by several events such as inhomogeneity of the matrix, nonuniform dispersion of the catalyst, and presence of residual water. Under these circumstances, bulk degradation is expected. Figure 2 also shows the fractional amount of acid anhydride and acid remaining in the matrix as a function of time and the profiles are similar. A fair comparison could not be made with the experimental data, since the fabrication of the device and assay technique for anhydride resulted in an initial loss of anhydride [ 5 1. However, the time span over which the anhydride level drops to below 0.05% is about 50 h in both the experimental and the theoretical studies. The decrease in the molecular weight of the eroding disc containing 0.25% phthalic anhydride as a function of time is illustrated in Fig. 4. The molecular weight decreases linearly with time for about 45 to 50 h and then remains invariant for another 40 h. Based on the time constant analysis for water diffusion, the linear decrease in molecular weight corresponds to the elapsed time of one time constant when approx-
_i p
0.6 -
_.--
.*’ I’
0.4 /’
_..-
_...--
..*-
0.2 - P ! E
0.04 0
, 20
,
.
40
, 60
) 80
0
-I
1IO0
Time, h
Fig. 3. Amount of water imbi~ment in matrix containing 0.25% anhydride.
20
40 Time,
60
80
h
Fig. 4. Molecular weight of the degrading polymer with 0.25% anhydride with time. (-O-) Th~reti~Ily predicted, and ( f. 0.. ) ex~~mentaliy found values.
102
imately 63% of the matrix is saturated with water. The time elapsed after one time constant corresponds to further saturation of the matrix with water. Once the matrix is saturated with water, bulk degradation occurs; this enhances dye release and the dye release continues until the matrix collapses. These results are clearly illustrated in Figs. 2 and 3. 3.2 Erosion and release events of discs with 1.0% anhydride The events during the erosion of the polymer matrix containing 1% anhydride are completely different from the matrix containing 0.25% anhydride. The model predictions of dye release, acid anhydride concentration and acid compare favorably with the experimental data [ 41 and are shown in Fig. 5. Two distinct differences are observed when the erosion and release profiles of matrices containing 1% and 0.25% of anhydrides are compared: ( 1) About 90% of the dye is released in 30 h from the 1% anhydride-containing device as compared to 90 h from the 0.25% anhydridecontaining device. (2) Water diffusion into the center of matrix is not as rapid for the device containing 1% anhydride as for 0.25% anhydride-containing device [ 41. The Thiele modulus for this system is about
1.5 12, significantly greater than unity. Therefore, at the anhydride level of 1% it can be expected that the controlling degradation mechanism is diffusion and not reaction. The degradation front of the matrix also corresponds to the water front, and therefore the polymer breakdown happens in only those areas where the slowly diffusing water molecules are capable of hydrolyzing anhydride molecules to initiate the acid catalytic degradation of the polymer; the anhydride reaction uses water as fast as it is supplied via diffusion. The complete erosion process is controlled by only one moving zone -the water front, moving towards the center of the matrix. The surface of the device undergoes chemical degradation and the dye is released into the surrounding medium first through dissolution in the hydrophilic layer of the degraded matrix followed by diffusion. As shown in Fig. 5, the anhydride level in the disc falls rapidly; similar experimental observations were made [ 41. It can be concluded that the dye release mechanism is primarily by surface erosion and is further supported by electron micrographs in the work of Nguyen et al. [ 41. The degradation kinetics of such a device was analyzed as a function of time. In Fig. 6, the experimental results of the average molecular weight of the disc during erosion at several time intervals are compared with the model prediction and a striking qualitative similarity is ob-
1.0
3 I 8
0.8
= E a
0.8
s
0.4
a .s 0 &
0.2
-
Acid Anhydride
-
Acid
..........
Dye Released
20
10
0.0 20
30
40
50
Time, h
Fig. 5. Cumulative release of active species from the disc containing 1.O% of anhydride.
Time,
0
h
Fig. 6. Molecular weight of the degrading polymer with 1.O% anhydride with time. (. .O.. ) Theoretically predicted, and (- -O- -) experimentally found.
103
served. With the erosion process being predominantly at the surface, the availability of water in the polymeric matrix is significantly reduced. It is interesting to note that up to 20 h, the molecular weight of the disc decreases linearly with time. It can be surmised that at a later stage of erosion, the degradation front and hence the dye release front would be left behind the water diffusion front; at this point bulk erosion occurs. It can be concluded that the concentration of anhydride levels affects the erosion characteristics of the device in two ways: ( I ) it reduces the life span of the device, and (2) the governing drug release mechanism is surface erosion for higher concentration of anhydrides and bulk erosion for lower concentration of anhydrides.
diffusion and the reaction mechanism are competing against each other; there is no dominant degradation mechanism. The time constant for the transport of catalyst is about the same as that for the catalytic reaction rate, thus it is difficult to categorize whether the degradation mechanism is controlled by surface erosion or bulk erosion. The concentration of the catalyst, 0.50%, gives the borderline concentration of the catalyst that would govern the degradation mechanism. Thus, for anhydride concentrations less than 0.5%, bulk erosion is expected, and for anhydride concentrations greater than 0.5%, surface erosion is expected. Similar observations were made both theoretically and experimentally and were discussed in the preceeding sections.
3.3 Erosion and release events of discs with 0.50% anhydride
4. Conclusion
In Fig. 7, the results of erosion with 0.5% phthalic anhydride are compared with the theoretical simulations. The events during the erosion of this matrix are significantly different from the erosion events of matrix that contains 1.0% and 0.25% of anhydride. The model prediction of the dye release, as shown in Fig. 7, compares favorably with the experimental data of Nguyen etal. [4]. The Thiele modulus for this system is around 1.06. For the Thiele modulus of unity, both the Anhydride
I
A mathematical model has been developed to describe the erosion characteristics of polymeric matrices. The poly (ortho ester) system was chosen as a test system for this analysis. The model predicts measurable quantities, namely, release characteristics of the incorporated dye, water penetration into the matrix, catalysis by anhydride, and catalytic degradation of the polymer matrix. Random degradation kinetics give a good estimate of molecular weight in the polymer disc with time. The Thiele modulus is a good indicator of surface versus bulk erosion and has been successfully applied to determine the erosion characteristics of the test system, the poly (ortho ester)s. Acknowledgement We thank Dr. A.G. Thombre and Dr. T.H. Nguyen for the informative discussions during the course of this work.
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Time, h
Fig. 7. Cumulative release of active species from the disc containing 0.5% of anhydride. (-_) Anhydride, (...) acid, and (--- ) dye released.
1
R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, NY, 1960.
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C. Shih, T. Higuchi and K.J. Himmelstein, Drug delivery from catalyzed erodible polymeric matrices of poly(ortho ester)s, Biomaterials, 5 (1984) 237-240. J. Heller, Synthesis of biodegradable polymers for biomedical utilization, Am. Chem. Sot. Symp. Ser., 212 (1983) 373-392. E.H. Cordes and H.G. Bull, Mechanism and catalysis for hydrolysis of acetals, ketals, and ortho esters, Chem. Rev., 94 (1974) 581-603. P.J. Flory, Principles of Polymer Chemistry, Cornell Press, Ithaca, NY, 1953. G. Astarita, Mass Transfer with Chemical Reactions, Elsevier, Amsterdam, 1967. B. Camahan, H.A. Luther and J.O. Wilkes, Approximation of the Solution of Partial Differential Equations, Applied Numerical Methods, Wiley, New York, NY, 1969. J. Heller, D.W.H. Penhale, R.F. Helwing, B.K. Fritzinger and R.W. Baker, Release of norethindrone from poly(ortho ester)s, in: T.J. Roseman and Z. Mansdorf (Eds.), Controlled Release Delivery Systems, Marcel Dekker, New York, NY, 1983, pp. 91-106. R.F. Fedors, Osmotic effects in water absorption for polymers, Polymers, 2 1 ( 1980) 207-2 10.