- Email: [email protected]
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrović1*, S. Ibrić1, J. Jocković2, J. Parojčić1, Z. Đurić1 1
Institute of Pharmaceutical Technology, 2Institute of Physics and Mathematics, Faculty of Pharmacy, Belgrade, Serbia *Correspondence: [email protected]
The purpose of this study was to implement the concepts of percolation theory in the characterization of drug release from hydrophilic matrix tablets. Percolation theory is a powerful statistical tool that enables mathematical insight into geometrically complex and disordered systems. Matrix tablets are effective substrate for the implementation of percolation theory because of their inherent disordered structure. The objective was to predict percolation thresholds of polyethylene oxide and polyacrylic polymers in diclofenac sodium hydrophilic matrices. Matrix tablets were prepared using polyethylene oxide or polyacrylic acid as matrix forming materials and diclofenac sodium was used as a model drug substance. Ten formulations with different drug/excipient ratios were prepared using the direct compression method. Dissolution studies were performed using the paddle apparatus method. For estimating percolation threshold the change of the kinetic parameters in aspect to the volumetric fraction of excipient plus initial porosity of the tablets was studied. Observed critical points with sudden changes in behavior of kinetic parameters can be attributed to the percolation thresholds. Percolation threshold is found to be 60.22% v/v polyethylene oxide + initial porosity and 39.94% v/v polyacrylic acid + initial porosity. The results obtained demonstrate that percolation theory can be used to design and develop matrix tablet formulations. Determination of percolation threshold is a useful tool for preparing robust formulations. Key words: Percolation threshold – Controlled release – Polyethylene oxide (PEO) matrix – Polyacrylic acid matrix – Tablets.
Treatment of chronic diseases often implies taking medicines several times a day. This is especially the problem for drugs with short half-life, which is one of the main reasons for designing and developing sustained-release dosage forms that are more therapeutically efficient and more suitable for patients. Novel excipients for sustained-release oral dosage forms, Polyox – polyethylene oxide (PEO) polymers and Carbopol – polyacrylic acid polymers, have been successfully used in the formulation of matrix tablets demonstrating sustained-release action over a period of 8 h. In order to improve their application, drug release mechanisms from Polyox and Carbopol matrix systems have been investigated. One of the most important characteristics about both polyethylene oxide and polyacrylic acid polymers is their hydrophilic nature which makes them highly water – soluble and swellable. Formation of the hydrophilic gel layer on the tablet surface is responsible for controlled release of the drug [1]. Water penetrates the polymer matrix leading to swelling by decreasing the glass transition temperature of polymer to the experimental temperature thus leading to transformation of glassy polymer into a rubbery phase [2]. In vitro release of water-soluble drugs from hydrophilic matrix systems is mainly controlled by diffusion out of the gel layer, whereas release of poorly soluble drugs is usually controlled by polymer relaxation – dissolution [3]. Diffusion, swelling and erosion are the most important rate-controlling mechanisms of commercially available controlled-release products [4]. Drug release from hydrophilic matrices shows a typical time-dependent profile (i.e., decreased drug release with time because of increased diffusion path length). For a description of kinetics of drug release from hydrophilic matrices the following models are most commonly used: zero-order, first-order, Higuchi’s, Hixon-Crowell’s and Peppas-Sahlin’s model [5]. Concentration of a matrix forming polymer in the matrix tablet formulation is crucial for adequate drug release mechanism and also for other tablet characteristics (mechanical properties, drug dissolution rate, disintegration time, water uptake, etc.) [6]. Percolation theory has been used for better understanding and calculation of the optimal concentration of the polymer in the tablet formulation. It was first introduced by Leuenberger and coworkers [7]. Percolation theory
has been applied for studying inert matrix tablets [8], sustained drug release in diffusion controlled matrices [9] and hydrophilic matrix tablets [10]. In the present study binary systems of hydrophilic polymer and drug have been studied with diclofenac sodium being chosen as a model drug substance. Diclofenac sodium acts as a non-steroidal anti-inflammatory agent. The aim of this study was to characterize the drug release profiles from matrix tablets incorporating these profiles in the concepts of percolation theory. Drug release mechanisms from hydrophilic matrix tablets are, as mentioned, complex including diffusion, relaxation and dissolution processes. A connection will be made between the changes in the structure of the matrix tablet and drug-release mechanisms.
I. PERCOLATION THEORY
Percolation theory is a general mathematical theory of connectivity and transport in geometrically complex systems [11]. It is, together with the theory of fractals, an approach for characterizing disordered systems [12]. An inner matrix system of the tablet may be considered a lattice of percolation clusters. A percolation cluster is a group of occupied nearest neighbor sites in the disordered matrix structure. Occupied sites are elements of the matrix (polymer or drug substance) and each site is occupied independently of its neighbors with probability p. The empty sites are the pores through which solute particles diffuse. There is a certain value of occupancy probability p, called percolation threshold at which there exists a connected cluster of sites or pores (called the infinite cluster). Percolation threshold pc is attributed to the volumetric ratio (% v/v) of the percolating component. So, for p: - p < pc, no spanning cluster exists and all clusters are finite, - p ≥ pc, one spanning cluster exists. Two types of percolation may be considered: site percolation, where an infinite cluster is described by occupied sites, and bond percolation where the infinite cluster is formed by bonds between sites [13]. Also, both polymer and drug in the binary matrix system can percolate but only the percolation of a polymer is considered in this study. 359
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
The probability of forming an infinite cluster follows the Kolmogorov’s zero-one law: for any given concentration p of occupied sites or bonds, the probability of forming an infinite cluster is either zero or one [14]. Pc values are independent of the physical or chemical nature of lattice forming material, they are only influenced by the geometrical organization of the lattice. Lattices can be two- or three-dimensional, hexagonal, square, triangular, diamond, simple cubic, face-centered cubic, body-centered cubic and random close packed. Pc values are determined by measuring electrical resistance of composites [14]. An issue that has to be discussed is that previously given characteristics consider an infinite percolation lattice. The size of the system influences the probability of percolation. It means that in an infinite lattice the system percolates at given value pc; whereas for a finite lattice (e.g. pharmaceutical compact - tablet) pc is not a discrete value and depends on the nature of the scaling function Φ, the lattice size L and the correlation length exponent ν [11]. Therefore, for a finite system, percolating probability Π is: Π = Φ [(p – pc) L1/ν]
Table I - Composition of the matrices prepared with diclofenac sodium/Polyox WSR 1105 or Carbopol 71G and % v/v polymer + initial porosity. % w/w diclofenac sodium
% w/w Polyox WSR 1105
% v/v Polyox + initial porosity
% w/w Carbopol 71G
% v/v Carbopol + initial porosity
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10
95 90 80 70 60 95 90 85 80 70
5 10 20 30 40 -
45.78 48.53 54.22 59.92 65.82 -
5 10 15 20 30
29.33 33.05 36.77 40.49 47.92
% w/w is the weight ratio and % v/v is the volumetric ratio of the component.
Eq. 1
The weight of tablets was kept constant at 450 mg and a diameter of cylindrical tablets was 12 mm and 11 mm for Polyox and Carbopol tablets, respectively. The initial porosity was calculated using the following equation:
The transition from a state with no spanning cluster to a state with one spanning cluster is a type of phase transition [10]. Although phase transition is not widely recognized and emphasized, it has to be considered, due to changes in the structure and physicochemical properties. The percolation threshold is a critical point where some tablet properties (percentage of drug released, release rate, mechanical properties, etc.) may undergo sudden changes [10]. According to percolation theory, a system property X follows a power law at the percolation threshold pc: X = S |p – pc|q
Batch
ε = VT - [(w·%drug)/rd] - [(w·%excipient)/re]
Eq. 3
where ε is the initial porosity, VT is the total volume, w is the tablet weight, rd is the drug density and re is the excipient density. The data for polyethylene oxide density were found in the literature [15]. Densities of Carbopol 71G and diclofenac sodium were measured by Stereopycnometer (Quantachrome Instruments, FL, United States). Drug release was studied in dissolution apparatus (Erweka DT6, Hausenstamm, Germany) using the rotating paddle method (50 rpm). Six tablets of each batch were used for each dissolution profile. Dissolution tests were conducted for 8 hours in 900 mL of USP30 phosphate buffer pH 6.8 and samples were taken every hour. The amount of diclofenac sodium was determined using a UV spectrophotometer (λ = 275 nm). In order to further explain and predict release characteristics of the drug substance mathematical modeling has been used. Being hydrophilic matrices with noticeable swelling of polymer, Polyox and Carbopol matrix tablets demonstrate complex drug release mechanisms. Drug release is diffusion, swelling and polymer dissolution controlled process. The mathematical models given in Table II were used to fit the experimental data. Values for exponent m were taken from literature [10] since they only depend on the geometrical shape (i.e. aspect ratios of height and thickness) of the tablet. Therefore, for Polyox tablets m value was 0.44 and for Carbopol tablets 0.42.
Eq. 2
where S is the scaling factor and q is the critical exponent. A system property can be a percentage of drug released, disintegration time of the tablet, etc. This equation is theoretically only valid close to the percolation threshold. But, in practical cases it showed a much larger validity than anticipated [7]. The defining characteristic of percolation is connectedness. At percolation threshold interior pathways of matrix system allow water to diffuse within, dissolve drug substance and interact with polymer chains. Since the polymers studied are hydrophilic their interaction with water leads to swelling of the polymer on the tablet surface creating a gel layer responsible for controlled release of the drug. Initial porous volume, entrapped air in the matrix structure, is considered to be the difference between the total volume of the tablet and sum of volumes of both drug and polymer. Interior pathways are made of initial porous network and hydrophilic polymer particles. Percolating components in the study are hydrophilic pathways (made up of porous network and hydrophilic polymer particles). Percolation threshold assumes the existence of clusters of connected polymer chains and empty pore spaces.
Table II - Study of drug release mechanism.
II. MATERIALS AND METHODS 1. Materials
The following chemicals were obtained from commercial suppliers: diclofenac sodium (Galenika, Belgrade, Serbia), Sentry Polyox WSR 1105 – Leo NF Grade (Dow Chemical Company, Charleston, United States) and Carbopol 71G (Noveon, Gattefossé, Switzerland).
2. Methods
Model - equation
Drug release mechanism
Higuchi equation Q = kht1/2 Peppas – Sahlin equation Q = kdtm + krt2m Zero-order drug delivery Q = k 0t
Fickian diffusion Coupled effects of Fickian diffusion and Case II transport
Q is the amount of the drug remaining at time t; kh is the Higuchi rate constant, kd and kr represent kinetic constants associated with diffusional and relaxational release, respectively, and m is the Fickian diffusion exponent for a device of any geometrical shape which exhibits controlled release; k0 is the zero-order kinetic constant.
Drug-containing matrices were prepared by compressing a homogeneous mixture of the drug and polymer powders with an excenter tablet press (EK0 Korsch, Germany). Table I shows the composition of the studied batches. 360
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
Correlation coefficient r2 was calculated for every model in order to evaluate its merits. The percolation threshold was estimated by studying the behavior of kinetic parameters (calculated from mathematical models) with changes in the sum of excipient volumetric fraction plus initial porosity in different batches [10]. The kinetic parameters used were: relaxational and diffusional constants of Peppas-Sahlin’s equation kr and kd, Higuchi’s slope kh and zero order kinetic constant k0.
% diclofenac sodium release
100
III. RESULTS AND DISCUSSION 1. Release profiles and kinetics
Figures 1 and 2 demonstrate release profiles obtained from matrices with diclofenac sodium and polymers (Polyox and Carbopol, respectively). From the release profiles and their fitting in the mathematical models it was concluded that Peppas-Sahlin’s equation is the most appropriate for describing drug release from hydrophilic matrices (the highest values for correlation coefficient r2). The results obtained by fitting the experimental data in mathematical models are given in Table III. Peppas-Sahlin’s equation takes into account both diffusion of the drug and relaxation of the polymer swelling chains, whereas Higuchi’s equation considers only the diffusion process. Zero order kinetics is always desirable in the controlled-release formulations and it was achieved with certain concentrations of polymers (batch F5). But, as discussed below, it is sometimes best avoided because it may be in the region of the polymer percolation threshold. Polyethylene oxide matrices are an excellent example of hydrophilic polymer matrix tablets with polymeric gel layer on the surface governing the drug-release mechanism. When the concentration of the polymer is low, the drug can dissolve and be released at faster rates. As the % w/w of the polymer increases release becomes controlled by the gel layer formed on the surface. When there is enough polymer to form the gel layer, the release becomes slower and even gets to zero order kinetics (batch F5). The sudden change in the release pattern of drug from different matrices can be explained on the basis of percolation theory. There has to be a certain concentration of polymer in matrices in order to have an interior network of pathways which allow water to diffuse within and a consequent polymeric gel layer to form on the surface. This concentration of polymer is obtained as a polymer percolation threshold. This is because at the percolation threshold there exists an infinite, connected cluster which is made up of empty sites (initial porosity of the matrix tablet) and hydrophilic polymer particles. When hydrophilic particles come into contact with the medium (water) they dissolve, thus forming an infinite cluster together with initial porosity of the matrix tablet. The findings are in agreement with previously reported studies on the effect of polyethylene oxide amount on the release of hydrophilic drug [16].
80
60
40
20
0 0
2
4
6
8
time (hours) F1 - 95% diclofenac sodium
F2 - 90% diclofenac sodium
F3 - 80% diclofenac sodium
F4 - 70% diclofenac sodium
F5 - 60% diclofenac sodium
Figure 1 - Diclofenac sodium release from matrix tablets with a drug content of 95, 90, 80, 70 and 60% prepared with diclofenac sodium and Polyox WSR 1105
% diclofenac sodium release
100
80
60
40
20
0 0
2
4
6
8
time (hours) F6 - 95% diclofenac sodium
F7 - 90% diclofenac sodium
F8 - 85% diclofenac sodium
F9 - 80% diclofenac sodium
F10 - 70% diclofenac sodium
Figure 2 - Diclofenac sodium release from matrix tablets with a drug content of 95, 90, 85, 80 and 70% prepared with diclofenac sodium and Carbopol 71G.
Polyacrylic acid matrices have shown slightly different behavior from PEO matrices. In polyacrylic acid matrices the amount of drug released varied with different % w/w of the polymer. This phenomenon
Table III - Kinetic parameters for all batches obtained by fitting the experimental data in the mathematical models. Batch
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
Higuchi equation
kh ss total ss resid r2
3.69 1777.9 923.4 0.481
3.93 2495.7 817.6 0.672
4.31 8537.3 694.3 0.919
3.12 8027.9 1742.1 0.783
2.61 5673.8 1168.4 0.794
8.01 6738.1 724.6 0.892
4.25 9206.8 2023.9 0.780
1.36 2218.7 600.4 0.729
0.20 22.1 6.7 0.697
0.24 11.3 10.9
PeppasSahlin equation
kd kr ss total ss resid r2
8.29 -0.25 1777.9 105.7 0.940
8.21 -0.21 2495.7 163.4 0.934
3.59 0.24 8537.3 603.1 0.929
-1.65 0.55 8027.9 95.4 0.988
-1.38 0.46 5673.8 19.4 0.997
-2.13 1.77 6738.1 94.49 0.986
-3.3925 0.9297 9206.8 40.33 0.996
-1.2928 0.2642 2218.7 53.33 0.976
-0.2091 0.0486 22.1 1.08 0.9512
-0.1905 0.0524 11.3 3.97 0.65
Zeroorder equation
k0 ss total ss resid r2
0.19 1777.9 5147.2
0.21 2495.7 5233.6
0.24 8537.3 2202.9 0.742
0.18 8027.9 126.4 0.984
0.15 5673.8 32.4 0.994
0.8581 6738.1 101.2 0.985
0.3164 9206.8 168.4 0.982
0.081 2218.7 90.04 0.959
0.0149 22.1 1.54 0.93
0.0179 11.3 4.03 0.64
kh (% min-1/2) is Higuchi’s slope; kd (% min-m) is diffusional constant of Peppas-Sahlin model; kr (% min-2m) is relaxational constant of Peppas-Sahlin model; m is the diffusional exponent that depends on geometric shape of the releasing device.
361
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
can also be attributed to the formation of the gel layer on the tablet surface. In this case, as the concentration of polymer increases the gel layer becomes so thick and compact that the drug cannot diffuse through it which explains the rapid decrease in the amount of released drug as the concentration of the polymer increases. The same finding was observed by El-Malah and Nazzal [17]. The results indicate that there is a change in the release mechanism and in the amount of drug released at a certain point (i.e. at a certain concentration of the polymer). This point is actually the polymer percolation threshold.
Relaxational and diffusional constants kr and kd were chosen since it was shown (Table III) that Peppas-Sahlin’s equation is the most adequate for describing drug release mechanism. The results for percolation threshold obtained with these two constants are given in Tables IV and V. A critical point has been found as the point of intersection of two regression lines. Percolation threshold is found to be 2
1,5
2. Estimation of the polymer percolation thresholds
Kr (%min-m)
In order to estimate the polymer percolation threshold changes in the behavior of kinetic parameters have been studied. Kinetic parameters changed because the release mechanism or the amount of drug released from the matrix tablets changed, all of which was influenced by differing volumetric fractions of polymer in the matrix tablet. Percolation theory assumes the existence of the critical point where a system property suddenly changes (Equation 2). Kinetic parameters chosen were following: relaxational and diffusional constants kr and kd for determining the percolation threshold for Polyox and in addition, Higuchi’s slope kh for Carbopol tablets. The results are shown in Figures 3 and 4 for Polyox matrices, and in Figures 5, 6 and 7 for Carbopol matrices. Double regression has been employed and the point of intersection of two lines represents the percolation threshold.
1
0,5
0 25
35
45
% v/v excipient plus initial porosity
Figure 5 - Relaxational constant of Peppas-Sahlin model, kr, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10). 0 25
35
45
0,6 -1
Kd (%min-m)
Kr (%min-2m)
0,4
0,2
0
-2
-3
40
50
60
70
-0,2 -4 % v/v excipient plus initial porosity
-0,4
Figure 6 - Diffusional constant of Peppas-Sahlin model, kd, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10).
% v/v excipient plus initial porosity
Figure 3 - Relaxational constant of Peppas-Sahlin model, kr, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Polyox (F1-F5).
0
10
8
8
-1 Kd (% min-m)
Kd (%min-m)
4
-2
4
Kh (% min-1/2)
6
6
2 -3
2
0
-4
0 40
50
60
70
-2 20
30
40
50
% v/v excipient plus initial porosity
-2
Figure 7 - Comparison of changes in diffusional constants of PeppasSahlin model, kd, and Higuchi’s slope, kh, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10). Full lines are connecting kd values while dashed lines are connecting kh values.
% v/v excipient plus initial porosity
Figure 4 - Diffusional constant of Peppas-Sahlin model, kd, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Polyox (F1-F5).
362
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
hydrophilic matrices optimal polymer ratio for achieving sustainedrelease of diclofenac sodium is in the range of % w/w 10-15 but, since there is a significant difference between diclofenac sodium release rate within increase of Carbopol ratio from % w/w 10-15, authors suggest that Carbopol should be avoided as matrix forming material, i.e. it is better used in combination with certain other polymers.
Table IV - The values of percolation thresholds for Polyox (batches F1-F5) according to studied kinetic parameters. Kinetic parameters
Equations
r2
Point of the intersection
Diffusion rate constant kd Relaxation rate constant kr
y = -0.7374x + 43.034 y = 0.0473x - 4.4875 y = 0.0602x - 3.0546 y = -0.0156x + 1.4843
0.963 1 0.978 1
x = 60.56 x = 59.87
REFERENCES
Table V - The values of percolation thresholds for Carbopol (batches F6-F10) according to studied kinetic parameters. Kinetic parameters
Equations
r2
Point of the intersection
Diffusion rate constant kd Relaxation rate constant kr Higuchi’s slope kh
y = 0.2792x - 11.504 y = 0.0025x - 0.3105 y = -0.1567x + 6.224 y = 0.0005x + 0.0279 y = -0.7078x + 28.167 y = 0.0054x - 0.0207
0.9944 1 0.9437 1 0.9525 1
x = 40.45
1. 2.
x = 39.42
3.
x = 39.52 4.
at 60.22% v/v Polyox + initial porosity (mean value of two obtained results) which is % w/w 34.4. It is important to notice that almost identical thresholds were obtained for Polyox matrices when changes in diffusional and relaxational constants were studied. These values are close to ratio of the polymer in the tablets that leads to zero order rate of drug release. Therefore, as much as the zero order release is favored when sustained-release formulations are in development, in this case it is best avoided. For polyacrylic acid matrices the percolation threshold is found to be at 39.94% v/v Carbopol + initial porosity (mean value of two obtained results) which is % w/w 17.8. Once again, almost identical values for thresholds were obtained when diffusional and relaxational constants were studied. Authors suggest that the gel layer formed by polyacrylic acid on the surface of the tablet is too compact to control the drug release. The release is therefore mainly governed by diffusion of the drug prior to formation of the gel layer which is why approximately the same percolation threshold is observed if Higuchi’s constant kh is studied instead of diffusional constant kd. In this case the percolation threshold obtained is 39.52% v/v Carbopol + initial porosity, which is % w/w 17.2. These findings are demonstrated in Figure 5 and Table V. It is noticeable that percolation threshold is at a lower % v/v for Carbopol matrices compared to Polyox matrices. Less Carbopol is needed to achieve sustained-release with increase of % v/v of Carbopol leading to incomplete release of diclofenac sodium.
5. 6.
7. 8.
9. 10. 11. 12. 13. 14.
* The results approved the existence of a percolation threshold – the critical point at which tablet properties suddenly change. The formulations with excipients concentrations close to their percolation thresholds should be avoided because the robustness of the formulation cannot be maintained – even small changes in the concentration close to percolation threshold sometimes lead to dramatic changes in the formulation characteristics. Characteristics that can change are: the amount of drug released, the drug release mechanism, mechanical properties of the tablet, etc. Therefore, in the case of Polyox hydrophilic matrices the optimal ratio of the polymer for achieving sustained-release of diclofenac sodium is in the range of % w/w 20-30. For matrices with % w/w 20-30 of Polyox, sustained-release is achieved with invariable release mechanism and other tablets properties. In the case of Carbopol
15.
16.
17.
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Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
Acknowledgment
Manuscript
This study has been done under the project No. TR 23015 supported by the Ministry of Science and Technological Development, Republic of Serbia.
Received 23 February 2009, accepted for publication 2 June 2009.
364
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrović1*, S. Ibrić1, J. Jocković2, J. Parojčić1, Z. Đurić1 1
Institute of Pharmaceutical Technology, 2Institute of Physics and Mathematics, Faculty of Pharmacy, Belgrade, Serbia *Correspondence: [email protected]
The purpose of this study was to implement the concepts of percolation theory in the characterization of drug release from hydrophilic matrix tablets. Percolation theory is a powerful statistical tool that enables mathematical insight into geometrically complex and disordered systems. Matrix tablets are effective substrate for the implementation of percolation theory because of their inherent disordered structure. The objective was to predict percolation thresholds of polyethylene oxide and polyacrylic polymers in diclofenac sodium hydrophilic matrices. Matrix tablets were prepared using polyethylene oxide or polyacrylic acid as matrix forming materials and diclofenac sodium was used as a model drug substance. Ten formulations with different drug/excipient ratios were prepared using the direct compression method. Dissolution studies were performed using the paddle apparatus method. For estimating percolation threshold the change of the kinetic parameters in aspect to the volumetric fraction of excipient plus initial porosity of the tablets was studied. Observed critical points with sudden changes in behavior of kinetic parameters can be attributed to the percolation thresholds. Percolation threshold is found to be 60.22% v/v polyethylene oxide + initial porosity and 39.94% v/v polyacrylic acid + initial porosity. The results obtained demonstrate that percolation theory can be used to design and develop matrix tablet formulations. Determination of percolation threshold is a useful tool for preparing robust formulations. Key words: Percolation threshold – Controlled release – Polyethylene oxide (PEO) matrix – Polyacrylic acid matrix – Tablets.
Treatment of chronic diseases often implies taking medicines several times a day. This is especially the problem for drugs with short half-life, which is one of the main reasons for designing and developing sustained-release dosage forms that are more therapeutically efficient and more suitable for patients. Novel excipients for sustained-release oral dosage forms, Polyox – polyethylene oxide (PEO) polymers and Carbopol – polyacrylic acid polymers, have been successfully used in the formulation of matrix tablets demonstrating sustained-release action over a period of 8 h. In order to improve their application, drug release mechanisms from Polyox and Carbopol matrix systems have been investigated. One of the most important characteristics about both polyethylene oxide and polyacrylic acid polymers is their hydrophilic nature which makes them highly water – soluble and swellable. Formation of the hydrophilic gel layer on the tablet surface is responsible for controlled release of the drug [1]. Water penetrates the polymer matrix leading to swelling by decreasing the glass transition temperature of polymer to the experimental temperature thus leading to transformation of glassy polymer into a rubbery phase [2]. In vitro release of water-soluble drugs from hydrophilic matrix systems is mainly controlled by diffusion out of the gel layer, whereas release of poorly soluble drugs is usually controlled by polymer relaxation – dissolution [3]. Diffusion, swelling and erosion are the most important rate-controlling mechanisms of commercially available controlled-release products [4]. Drug release from hydrophilic matrices shows a typical time-dependent profile (i.e., decreased drug release with time because of increased diffusion path length). For a description of kinetics of drug release from hydrophilic matrices the following models are most commonly used: zero-order, first-order, Higuchi’s, Hixon-Crowell’s and Peppas-Sahlin’s model [5]. Concentration of a matrix forming polymer in the matrix tablet formulation is crucial for adequate drug release mechanism and also for other tablet characteristics (mechanical properties, drug dissolution rate, disintegration time, water uptake, etc.) [6]. Percolation theory has been used for better understanding and calculation of the optimal concentration of the polymer in the tablet formulation. It was first introduced by Leuenberger and coworkers [7]. Percolation theory
has been applied for studying inert matrix tablets [8], sustained drug release in diffusion controlled matrices [9] and hydrophilic matrix tablets [10]. In the present study binary systems of hydrophilic polymer and drug have been studied with diclofenac sodium being chosen as a model drug substance. Diclofenac sodium acts as a non-steroidal anti-inflammatory agent. The aim of this study was to characterize the drug release profiles from matrix tablets incorporating these profiles in the concepts of percolation theory. Drug release mechanisms from hydrophilic matrix tablets are, as mentioned, complex including diffusion, relaxation and dissolution processes. A connection will be made between the changes in the structure of the matrix tablet and drug-release mechanisms.
I. PERCOLATION THEORY
Percolation theory is a general mathematical theory of connectivity and transport in geometrically complex systems [11]. It is, together with the theory of fractals, an approach for characterizing disordered systems [12]. An inner matrix system of the tablet may be considered a lattice of percolation clusters. A percolation cluster is a group of occupied nearest neighbor sites in the disordered matrix structure. Occupied sites are elements of the matrix (polymer or drug substance) and each site is occupied independently of its neighbors with probability p. The empty sites are the pores through which solute particles diffuse. There is a certain value of occupancy probability p, called percolation threshold at which there exists a connected cluster of sites or pores (called the infinite cluster). Percolation threshold pc is attributed to the volumetric ratio (% v/v) of the percolating component. So, for p: - p < pc, no spanning cluster exists and all clusters are finite, - p ≥ pc, one spanning cluster exists. Two types of percolation may be considered: site percolation, where an infinite cluster is described by occupied sites, and bond percolation where the infinite cluster is formed by bonds between sites [13]. Also, both polymer and drug in the binary matrix system can percolate but only the percolation of a polymer is considered in this study. 359
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
The probability of forming an infinite cluster follows the Kolmogorov’s zero-one law: for any given concentration p of occupied sites or bonds, the probability of forming an infinite cluster is either zero or one [14]. Pc values are independent of the physical or chemical nature of lattice forming material, they are only influenced by the geometrical organization of the lattice. Lattices can be two- or three-dimensional, hexagonal, square, triangular, diamond, simple cubic, face-centered cubic, body-centered cubic and random close packed. Pc values are determined by measuring electrical resistance of composites [14]. An issue that has to be discussed is that previously given characteristics consider an infinite percolation lattice. The size of the system influences the probability of percolation. It means that in an infinite lattice the system percolates at given value pc; whereas for a finite lattice (e.g. pharmaceutical compact - tablet) pc is not a discrete value and depends on the nature of the scaling function Φ, the lattice size L and the correlation length exponent ν [11]. Therefore, for a finite system, percolating probability Π is: Π = Φ [(p – pc) L1/ν]
Table I - Composition of the matrices prepared with diclofenac sodium/Polyox WSR 1105 or Carbopol 71G and % v/v polymer + initial porosity. % w/w diclofenac sodium
% w/w Polyox WSR 1105
% v/v Polyox + initial porosity
% w/w Carbopol 71G
% v/v Carbopol + initial porosity
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10
95 90 80 70 60 95 90 85 80 70
5 10 20 30 40 -
45.78 48.53 54.22 59.92 65.82 -
5 10 15 20 30
29.33 33.05 36.77 40.49 47.92
% w/w is the weight ratio and % v/v is the volumetric ratio of the component.
Eq. 1
The weight of tablets was kept constant at 450 mg and a diameter of cylindrical tablets was 12 mm and 11 mm for Polyox and Carbopol tablets, respectively. The initial porosity was calculated using the following equation:
The transition from a state with no spanning cluster to a state with one spanning cluster is a type of phase transition [10]. Although phase transition is not widely recognized and emphasized, it has to be considered, due to changes in the structure and physicochemical properties. The percolation threshold is a critical point where some tablet properties (percentage of drug released, release rate, mechanical properties, etc.) may undergo sudden changes [10]. According to percolation theory, a system property X follows a power law at the percolation threshold pc: X = S |p – pc|q
Batch
ε = VT - [(w·%drug)/rd] - [(w·%excipient)/re]
Eq. 3
where ε is the initial porosity, VT is the total volume, w is the tablet weight, rd is the drug density and re is the excipient density. The data for polyethylene oxide density were found in the literature [15]. Densities of Carbopol 71G and diclofenac sodium were measured by Stereopycnometer (Quantachrome Instruments, FL, United States). Drug release was studied in dissolution apparatus (Erweka DT6, Hausenstamm, Germany) using the rotating paddle method (50 rpm). Six tablets of each batch were used for each dissolution profile. Dissolution tests were conducted for 8 hours in 900 mL of USP30 phosphate buffer pH 6.8 and samples were taken every hour. The amount of diclofenac sodium was determined using a UV spectrophotometer (λ = 275 nm). In order to further explain and predict release characteristics of the drug substance mathematical modeling has been used. Being hydrophilic matrices with noticeable swelling of polymer, Polyox and Carbopol matrix tablets demonstrate complex drug release mechanisms. Drug release is diffusion, swelling and polymer dissolution controlled process. The mathematical models given in Table II were used to fit the experimental data. Values for exponent m were taken from literature [10] since they only depend on the geometrical shape (i.e. aspect ratios of height and thickness) of the tablet. Therefore, for Polyox tablets m value was 0.44 and for Carbopol tablets 0.42.
Eq. 2
where S is the scaling factor and q is the critical exponent. A system property can be a percentage of drug released, disintegration time of the tablet, etc. This equation is theoretically only valid close to the percolation threshold. But, in practical cases it showed a much larger validity than anticipated [7]. The defining characteristic of percolation is connectedness. At percolation threshold interior pathways of matrix system allow water to diffuse within, dissolve drug substance and interact with polymer chains. Since the polymers studied are hydrophilic their interaction with water leads to swelling of the polymer on the tablet surface creating a gel layer responsible for controlled release of the drug. Initial porous volume, entrapped air in the matrix structure, is considered to be the difference between the total volume of the tablet and sum of volumes of both drug and polymer. Interior pathways are made of initial porous network and hydrophilic polymer particles. Percolating components in the study are hydrophilic pathways (made up of porous network and hydrophilic polymer particles). Percolation threshold assumes the existence of clusters of connected polymer chains and empty pore spaces.
Table II - Study of drug release mechanism.
II. MATERIALS AND METHODS 1. Materials
The following chemicals were obtained from commercial suppliers: diclofenac sodium (Galenika, Belgrade, Serbia), Sentry Polyox WSR 1105 – Leo NF Grade (Dow Chemical Company, Charleston, United States) and Carbopol 71G (Noveon, Gattefossé, Switzerland).
2. Methods
Model - equation
Drug release mechanism
Higuchi equation Q = kht1/2 Peppas – Sahlin equation Q = kdtm + krt2m Zero-order drug delivery Q = k 0t
Fickian diffusion Coupled effects of Fickian diffusion and Case II transport
Q is the amount of the drug remaining at time t; kh is the Higuchi rate constant, kd and kr represent kinetic constants associated with diffusional and relaxational release, respectively, and m is the Fickian diffusion exponent for a device of any geometrical shape which exhibits controlled release; k0 is the zero-order kinetic constant.
Drug-containing matrices were prepared by compressing a homogeneous mixture of the drug and polymer powders with an excenter tablet press (EK0 Korsch, Germany). Table I shows the composition of the studied batches. 360
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
Correlation coefficient r2 was calculated for every model in order to evaluate its merits. The percolation threshold was estimated by studying the behavior of kinetic parameters (calculated from mathematical models) with changes in the sum of excipient volumetric fraction plus initial porosity in different batches [10]. The kinetic parameters used were: relaxational and diffusional constants of Peppas-Sahlin’s equation kr and kd, Higuchi’s slope kh and zero order kinetic constant k0.
% diclofenac sodium release
100
III. RESULTS AND DISCUSSION 1. Release profiles and kinetics
Figures 1 and 2 demonstrate release profiles obtained from matrices with diclofenac sodium and polymers (Polyox and Carbopol, respectively). From the release profiles and their fitting in the mathematical models it was concluded that Peppas-Sahlin’s equation is the most appropriate for describing drug release from hydrophilic matrices (the highest values for correlation coefficient r2). The results obtained by fitting the experimental data in mathematical models are given in Table III. Peppas-Sahlin’s equation takes into account both diffusion of the drug and relaxation of the polymer swelling chains, whereas Higuchi’s equation considers only the diffusion process. Zero order kinetics is always desirable in the controlled-release formulations and it was achieved with certain concentrations of polymers (batch F5). But, as discussed below, it is sometimes best avoided because it may be in the region of the polymer percolation threshold. Polyethylene oxide matrices are an excellent example of hydrophilic polymer matrix tablets with polymeric gel layer on the surface governing the drug-release mechanism. When the concentration of the polymer is low, the drug can dissolve and be released at faster rates. As the % w/w of the polymer increases release becomes controlled by the gel layer formed on the surface. When there is enough polymer to form the gel layer, the release becomes slower and even gets to zero order kinetics (batch F5). The sudden change in the release pattern of drug from different matrices can be explained on the basis of percolation theory. There has to be a certain concentration of polymer in matrices in order to have an interior network of pathways which allow water to diffuse within and a consequent polymeric gel layer to form on the surface. This concentration of polymer is obtained as a polymer percolation threshold. This is because at the percolation threshold there exists an infinite, connected cluster which is made up of empty sites (initial porosity of the matrix tablet) and hydrophilic polymer particles. When hydrophilic particles come into contact with the medium (water) they dissolve, thus forming an infinite cluster together with initial porosity of the matrix tablet. The findings are in agreement with previously reported studies on the effect of polyethylene oxide amount on the release of hydrophilic drug [16].
80
60
40
20
0 0
2
4
6
8
time (hours) F1 - 95% diclofenac sodium
F2 - 90% diclofenac sodium
F3 - 80% diclofenac sodium
F4 - 70% diclofenac sodium
F5 - 60% diclofenac sodium
Figure 1 - Diclofenac sodium release from matrix tablets with a drug content of 95, 90, 80, 70 and 60% prepared with diclofenac sodium and Polyox WSR 1105
% diclofenac sodium release
100
80
60
40
20
0 0
2
4
6
8
time (hours) F6 - 95% diclofenac sodium
F7 - 90% diclofenac sodium
F8 - 85% diclofenac sodium
F9 - 80% diclofenac sodium
F10 - 70% diclofenac sodium
Figure 2 - Diclofenac sodium release from matrix tablets with a drug content of 95, 90, 85, 80 and 70% prepared with diclofenac sodium and Carbopol 71G.
Polyacrylic acid matrices have shown slightly different behavior from PEO matrices. In polyacrylic acid matrices the amount of drug released varied with different % w/w of the polymer. This phenomenon
Table III - Kinetic parameters for all batches obtained by fitting the experimental data in the mathematical models. Batch
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
Higuchi equation
kh ss total ss resid r2
3.69 1777.9 923.4 0.481
3.93 2495.7 817.6 0.672
4.31 8537.3 694.3 0.919
3.12 8027.9 1742.1 0.783
2.61 5673.8 1168.4 0.794
8.01 6738.1 724.6 0.892
4.25 9206.8 2023.9 0.780
1.36 2218.7 600.4 0.729
0.20 22.1 6.7 0.697
0.24 11.3 10.9
PeppasSahlin equation
kd kr ss total ss resid r2
8.29 -0.25 1777.9 105.7 0.940
8.21 -0.21 2495.7 163.4 0.934
3.59 0.24 8537.3 603.1 0.929
-1.65 0.55 8027.9 95.4 0.988
-1.38 0.46 5673.8 19.4 0.997
-2.13 1.77 6738.1 94.49 0.986
-3.3925 0.9297 9206.8 40.33 0.996
-1.2928 0.2642 2218.7 53.33 0.976
-0.2091 0.0486 22.1 1.08 0.9512
-0.1905 0.0524 11.3 3.97 0.65
Zeroorder equation
k0 ss total ss resid r2
0.19 1777.9 5147.2
0.21 2495.7 5233.6
0.24 8537.3 2202.9 0.742
0.18 8027.9 126.4 0.984
0.15 5673.8 32.4 0.994
0.8581 6738.1 101.2 0.985
0.3164 9206.8 168.4 0.982
0.081 2218.7 90.04 0.959
0.0149 22.1 1.54 0.93
0.0179 11.3 4.03 0.64
kh (% min-1/2) is Higuchi’s slope; kd (% min-m) is diffusional constant of Peppas-Sahlin model; kr (% min-2m) is relaxational constant of Peppas-Sahlin model; m is the diffusional exponent that depends on geometric shape of the releasing device.
361
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
can also be attributed to the formation of the gel layer on the tablet surface. In this case, as the concentration of polymer increases the gel layer becomes so thick and compact that the drug cannot diffuse through it which explains the rapid decrease in the amount of released drug as the concentration of the polymer increases. The same finding was observed by El-Malah and Nazzal [17]. The results indicate that there is a change in the release mechanism and in the amount of drug released at a certain point (i.e. at a certain concentration of the polymer). This point is actually the polymer percolation threshold.
Relaxational and diffusional constants kr and kd were chosen since it was shown (Table III) that Peppas-Sahlin’s equation is the most adequate for describing drug release mechanism. The results for percolation threshold obtained with these two constants are given in Tables IV and V. A critical point has been found as the point of intersection of two regression lines. Percolation threshold is found to be 2
1,5
2. Estimation of the polymer percolation thresholds
Kr (%min-m)
In order to estimate the polymer percolation threshold changes in the behavior of kinetic parameters have been studied. Kinetic parameters changed because the release mechanism or the amount of drug released from the matrix tablets changed, all of which was influenced by differing volumetric fractions of polymer in the matrix tablet. Percolation theory assumes the existence of the critical point where a system property suddenly changes (Equation 2). Kinetic parameters chosen were following: relaxational and diffusional constants kr and kd for determining the percolation threshold for Polyox and in addition, Higuchi’s slope kh for Carbopol tablets. The results are shown in Figures 3 and 4 for Polyox matrices, and in Figures 5, 6 and 7 for Carbopol matrices. Double regression has been employed and the point of intersection of two lines represents the percolation threshold.
1
0,5
0 25
35
45
% v/v excipient plus initial porosity
Figure 5 - Relaxational constant of Peppas-Sahlin model, kr, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10). 0 25
35
45
0,6 -1
Kd (%min-m)
Kr (%min-2m)
0,4
0,2
0
-2
-3
40
50
60
70
-0,2 -4 % v/v excipient plus initial porosity
-0,4
Figure 6 - Diffusional constant of Peppas-Sahlin model, kd, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10).
% v/v excipient plus initial porosity
Figure 3 - Relaxational constant of Peppas-Sahlin model, kr, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Polyox (F1-F5).
0
10
8
8
-1 Kd (% min-m)
Kd (%min-m)
4
-2
4
Kh (% min-1/2)
6
6
2 -3
2
0
-4
0 40
50
60
70
-2 20
30
40
50
% v/v excipient plus initial porosity
-2
Figure 7 - Comparison of changes in diffusional constants of PeppasSahlin model, kd, and Higuchi’s slope, kh, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Carbopol (F6-F10). Full lines are connecting kd values while dashed lines are connecting kh values.
% v/v excipient plus initial porosity
Figure 4 - Diffusional constant of Peppas-Sahlin model, kd, in aspect to volumetric fraction of the excipient plus initial porosity for batches with Polyox (F1-F5).
362
Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
hydrophilic matrices optimal polymer ratio for achieving sustainedrelease of diclofenac sodium is in the range of % w/w 10-15 but, since there is a significant difference between diclofenac sodium release rate within increase of Carbopol ratio from % w/w 10-15, authors suggest that Carbopol should be avoided as matrix forming material, i.e. it is better used in combination with certain other polymers.
Table IV - The values of percolation thresholds for Polyox (batches F1-F5) according to studied kinetic parameters. Kinetic parameters
Equations
r2
Point of the intersection
Diffusion rate constant kd Relaxation rate constant kr
y = -0.7374x + 43.034 y = 0.0473x - 4.4875 y = 0.0602x - 3.0546 y = -0.0156x + 1.4843
0.963 1 0.978 1
x = 60.56 x = 59.87
REFERENCES
Table V - The values of percolation thresholds for Carbopol (batches F6-F10) according to studied kinetic parameters. Kinetic parameters
Equations
r2
Point of the intersection
Diffusion rate constant kd Relaxation rate constant kr Higuchi’s slope kh
y = 0.2792x - 11.504 y = 0.0025x - 0.3105 y = -0.1567x + 6.224 y = 0.0005x + 0.0279 y = -0.7078x + 28.167 y = 0.0054x - 0.0207
0.9944 1 0.9437 1 0.9525 1
x = 40.45
1. 2.
x = 39.42
3.
x = 39.52 4.
at 60.22% v/v Polyox + initial porosity (mean value of two obtained results) which is % w/w 34.4. It is important to notice that almost identical thresholds were obtained for Polyox matrices when changes in diffusional and relaxational constants were studied. These values are close to ratio of the polymer in the tablets that leads to zero order rate of drug release. Therefore, as much as the zero order release is favored when sustained-release formulations are in development, in this case it is best avoided. For polyacrylic acid matrices the percolation threshold is found to be at 39.94% v/v Carbopol + initial porosity (mean value of two obtained results) which is % w/w 17.8. Once again, almost identical values for thresholds were obtained when diffusional and relaxational constants were studied. Authors suggest that the gel layer formed by polyacrylic acid on the surface of the tablet is too compact to control the drug release. The release is therefore mainly governed by diffusion of the drug prior to formation of the gel layer which is why approximately the same percolation threshold is observed if Higuchi’s constant kh is studied instead of diffusional constant kd. In this case the percolation threshold obtained is 39.52% v/v Carbopol + initial porosity, which is % w/w 17.2. These findings are demonstrated in Figure 5 and Table V. It is noticeable that percolation threshold is at a lower % v/v for Carbopol matrices compared to Polyox matrices. Less Carbopol is needed to achieve sustained-release with increase of % v/v of Carbopol leading to incomplete release of diclofenac sodium.
5. 6.
7. 8.
9. 10. 11. 12. 13. 14.
* The results approved the existence of a percolation threshold – the critical point at which tablet properties suddenly change. The formulations with excipients concentrations close to their percolation thresholds should be avoided because the robustness of the formulation cannot be maintained – even small changes in the concentration close to percolation threshold sometimes lead to dramatic changes in the formulation characteristics. Characteristics that can change are: the amount of drug released, the drug release mechanism, mechanical properties of the tablet, etc. Therefore, in the case of Polyox hydrophilic matrices the optimal ratio of the polymer for achieving sustained-release of diclofenac sodium is in the range of % w/w 20-30. For matrices with % w/w 20-30 of Polyox, sustained-release is achieved with invariable release mechanism and other tablets properties. In the case of Carbopol
15.
16.
17.
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Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets J. Petrovic, S. Ibric, J. Jockovic, J. Parojcic, Z. Duric
J. DRUG DEL. SCI. TECH., 19 (5) 359-364 2009
Acknowledgment
Manuscript
This study has been done under the project No. TR 23015 supported by the Ministry of Science and Technological Development, Republic of Serbia.
Received 23 February 2009, accepted for publication 2 June 2009.
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