A comparative kinetics study of isothermal drug release from poly(acrylic acid) and poly(acrylic-co-methacrylic acid) hydrogels

Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

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A comparative kinetics study of isothermal drug release from poly(acrylic acid) and poly(acrylic-co-methacrylic acid) hydrogels B. Adnadjevic, J. Jovanovic ∗ Faculty of Physical Chemistry, Studentski trg 12-16, 11001 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 11 February 2008 Received in revised form 20 October 2008 Accepted 22 October 2008 Available online 8 November 2008 Keywords: Drug delivery systems Isothermal Kinetics Hydrogel MEPBA release

a b s t r a c t A comparative study of the isothermal kinetics of the release of the drug (E)-4-(4-metoxyphenyl)-4oxo-2-butenoic acid (MEPBA) from poly(acrylic acid) (PAA) and poly(acrylic-co-methacrylic acid) (PAMA) hydrogel was performed. The isothermal kinetic curves of MEPBA release from the hydrogels in bidistilled water at different temperatures ranging from 20 to 42 ◦ C were determined. The reaction rate constants of MEPBA release were determined using the initial rate, saturation rate and empirical equation developed by Peppas et.al. The so-called “model-fitting method” for determining the kinetics models of both the drug release and absorption of external solution into the hydrogel, was applied. It was found that the kinetics of the MEPBA release both from the PAA and PAMA hydrogels can be best described with the kinetics model of first order chemical reaction. The model’s kinetics parameters of the investigated drug release process were calculated and significant differences for the values for PAA and PAMA hydrogels were found. The possibility to describe the kinetics of drug release with the model of reversible chemical reaction of first order was considered. It was found that kinetics of adsorption of the drug’s solution can be described with kinetics model of first order chemical reaction for PAMA hydrogel, while for PAA hydrogel it can be described with the kinetics model which is characteristic for the “phase boundary controlled reaction”. Based on the established dependences of the kinetic parameters (Ea and ln A) on the degree of the MEPBA released (˛) as well as on the presence of a compensation effect a new molecular mechanism of drug delivery was established. According to that mechanism, drug release is considered as drug desorption from the xerogel/hydrogel’s active desorption centers with different energies. The procedure for determining the distribution function of activation energies was developed. Different activation energy distribution function for PAA and PAMA hydrogels was established. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Hydrogel is a versatile system that can be used in either loading or release of solutes with specific therapeutic properties [1]. For further development of hydrogel’s application as a device for drug loading and controlled drug release and for optimization the pharmaceuticals action of a drug it is of particular importance to know both the mechanism and key parameters that govern kinetics of drug release from hydrogels. There are mathematical models that have been developed to describe the solute release profiles of polymer networks [2]. The majority of reports reveal that the release of solutes is strongly dependent on a variety of factors, such as polymer composition,

∗ Corresponding author. Tel.: +381 11 3336871; fax: +381 11 2187163. E-mail address: [email protected] (J. Jovanovic). 0927-7765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2008.10.018

hydrogel geometry, degree of swelling, and dissolution and diffusion of solute in the hydrogel [3–8]. Based on the rate-limiting step for controlled release, there are three known kinetics models for drug release in up-to-date literature. They are: diffusion-controlled, swelling-controlled and chemically controlled models [9]. When the diffusion of the drug through the hydrogel is a limiting kinetic factor of the drug release process, diffusion-controlled model is used for modeling the kinetics of drug release. In that case, kinetic of drug release is best described by Fick’s law of diffusion or Stefan–Maxwell equations [10] or empirical equation developed by Peppas et al. [11] When diffusion of drug is faster than hydrogel swelling, model of swelling-controlled release is appropriate to use. This model is specific for systems in which drug release occurs from the interface of rubbery and glassy phases of swollen hydrogels [12]. Kinetic’s model of chemically controlled release of drug is used for hydrogel delivery systems in which cleavage of polymer

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chains occurs either as hydrolytic or enzymatic degradation or, reversible or irreversible reaction between the polymer network and releasable drug [13–16]. Kinetic models of the drug release, which are mainly mathematical and semi-empirical, reported in up-to-date literature, can predict only first 60% of released solute [17]. Reis and co-workers [18] gave mathematical model based on an assumption that the process of drug release from hydrogel can be considered as a combination of the process of drug diffusion transport through hydrogel and the partitioning of solutes between the solvent phase and the hydrogel. The mathematic model proposed there predicts the overall release profile of vitamin B12, methylene blue and acid orange-7 from semi-interpenetrating network (semi-IPN) hydrogels composed of PNIPAAm and PAAm. Hydrogels based on poly(acrylic acid) (PAA), poly(methacrylic acid) and their co-polymers, complexes, IPNs or grafted networks have often been used as carriers in drug release systems in recent years [19,20]. For this study the (E)-4-(4-metoxyphenyl)-4-oxo-2butenoic acid (MEPBA) was chosen due to its similar structure to Cytembena (NSC 104801), which has been commercially used as an anticancer drug. Antiproliferative activity of (E)-4-aryl-4-oxo-2butenoic acids toward human cervix carcinoma HeLa cells has been reported [21]. With the previously exposed in mind, in this work the possibility of applicability of the literature known kinetic’s model suggested by Reis et al. [18] to the (E)-4-(4-metoxyphenyl)-4-oxo-2-butenoic acid (MEPBA) release from hydrogels composed of cross-linked poly(acrylic acid) (PAA) and poly(acrylic acid-co-methacrylic acid) (PAMA) was investigated. Also, a new procedure for determining the drug release kinetics which is able to predict the overall release profile of (E)-4-(4-metoxyphenyl)-4-oxo-2-butenoic acid from PAA and PAMA hydrogels was developed and presented. 2. Materials and methods 2.1. Materials Materials for hydrogel synthesis: acrylic acid (99.5%) (AA) and methacrylic acid (MA) were supplied by Merck KGaA. Daramsatdt, Germany. N,N-methylene bisacrylamide (p.a) (MBA) was purchased from Aldrich Chemical Co., Milwaukee, USA. The initiator, 2,2-azobis-[2-(2-imidazolin-2-il)-propan dihidrohlorid (VA044) (99.8%) was supplied by Wako Pure Chemicals Industries, Ltd., Osaka, Japan. (E)-4-(4-Metoxyphenyl)-4-oxo-2-butenoic acid (MEPBA) was kindly supplied by our colleagues who synthesized it according to the previously described procedure [21]. Here, the sodium salt form solution was used and Na2 CO3 was used for neutralization. 2.1.1. Synthesis Poly(acrylic acid) hydrogel (PAA) (1) and poly(acrylic acid-comethacrylic acid) hydrogel (PAMA) (2), which were used in the present investigation have been synthesized by a procedure based on simultaneous radical polymerization of (1) acrylic acid, and (2) acrylic acid and methacrylic acid (1:1 mol ratio), and cross-linking of the polymers formed, using the procedure which we described in detail earlier [22,23]. The obtained gel-type products were transformed into the Na+ form (60%) by neutralization with a 3% Na2 CO3 . The swelling degree of hydrogel and the release properties strongly depend on the degree of neutralization. Our previous experiments, which are not the subject of the present study, show that the optimal neutralization degree is about 60% [24]. With that in mind, we chose to work with partially neutralized PAA hydrogel. For this investigation the obtained xerogels were used in the form of powder.

2.1.2. Characterizations of the synthesized xerogels The xerogel’s samples were characterized by the following structural properties: xerogel density (xg ), cross-link density (c ), and the distance between macromolecular chains (d), according to the methods proposed by Gudman and Peppas [25]. Xerogel density (xg ) determination: the apparent density of the synthesized sample was determined by the picnometry method using n-hexane as the non-solvent. The cross-link density (c ) and the distance between the macromolecular chains (d) were calculated using the following equations: xg Mc

c =

(1) 1/3

d = 0.154 · v2 [0.19 · Mc ]1/2

(2)

Where Mc is the molar mass between the network cross-links and is a nominal value estimated from initial composition: Mc =

72 2Xc

(3)

Where Xc is the nominal cross-linking ratio (moles of NMBA/moles acrylic acid in the reaction mixture) 2.2. MEPBA loading MEPBA loading was carried out by immersing the known weight of the xerogel sample (0.1 g) in an excess of MEPBA solution (50 ml of 0.25% solution) at different temperatures. The mixtures were left to load for 24 h. The loaded hydrogels were removed from the MEPBA solution and dried in a thermal oven at 105 ◦ C until constant mass. Consequently, the MEPBA loaded xerogels were obtained. Specific drug loading in hydrogel (X) is defined as the amount of drug loaded [g] per 1 g of xerogel [g] at defined temperature. 2.3. MEPBA release MEPBA release was carried out by immersing the xerogel (PAA(PAMA))-MEPBA loaded sample in bidistilled water at temperatures 20(22) ◦ C, 31(32) ◦ C and 40(42) ◦ C. The released substance was monitored spectrometrically on a Cintra 10e, UV-Visible Spectrometer, Serial No. V 3163, using the absorption at  = 410 nm. Effective quantity of the drug released (C [%]) is calculated using Eq. (4): C=

p·V × 100 mx · X

(4)

where p is MEPBA concentration in the solution at time (t) [g/mL]; V is volume of the solution [mL]; mx is the weight of the MEPBAloaded xerogel sample [g] and X is specific drug loading in hydrogel [g/g]. Degree of MEPBA released (˛) is calculated as a ratio of the effective quantity of MEPBA released at time (t) (C) and equilibrium quantity of MEPBA released (Cmax ): ˛=

C Cmax

(5)

2.4. Absorption of external medium (MEPBA solution) to the xerogel/hydrogel The absorption of external medium was determined by leaving PAA(PAMA) xerogels with an average weight of 0.1 g (±5%) to absorb the MEPBA solution with a concentration that corresponds to the maximal released MEPBA at temperatures 20(22) ◦ C, 31(32) ◦ C and 40(42) ◦ C. At the beginning of each experiment, a sample of xerogel was measured by weight and then it was immersed in excess

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solution. At predetermined time intervals the gel with absorbed solution was removed from the solution and weighed. This was done until the hydrogel obtained a constant mass, i.e. until reached equilibrium. 2.4.1. Determination of the absorption degree The isothermal absorption degree (AD), defined as the difference between the weight of the hydrogel sample at time (t) (mt ) and the weight of xerogel (mo ) divided by the weight of the xerogel sample (mo ), was calculated according to Eq. (6) and determined as a function of time: mt − mo AD[%] = × 100 (6) mo For each sample at least three absorption measurements were performed and the average values were used. The equilibrium absorption degree (ADeq) is the absorption degree of the hydrogel at equilibrium. 2.4.2. Normalized absorption degree The normalized absorption degree (˛A ) was defined as the ratio between the absorption degree (AD) at time (t) and the equilibrium absorption degree (ADeq ), for certain temperature: ˛A =

AD ADeq

(7)

3. Methods used to evaluate kinetics parameters 3.1. Empirical equation suggested by Peppas To determine the kinetic parameters of MEPBA release from PAA and PAMA hydrogels, the results were analyzed by applying the well-known semi-empirical Peppas’ equation [26]: ˛ = kP t nP

(8)

where ˛ is the degree of MEPBA released at time (t), nP is an exponent indicative for the mechanism of the process (diffusion exponent), coefficient kp is the apparent release rate and t is interaction time. 3.2. Iso-conversional Friedman’s method [27] The kinetic analysis of experimental data is based on the following rate equation:

 d˛  dt

 E  a,˛

˛=const

= Af (˛) exp −

RT

ln (v)˛=const = ln A + ln f (˛) −

Ea,˛ RT

Table 1 The characteristic properties of PAA and PAMA xerogels.

PAA PAMA

(10)

where (v)˛=const is the reaction rate for defined ˛. For ˛ = const, the plot ln v˛ vs. (1/T), obtained from the conversional curve should be a straight line whose slope allows the evaluation of the apparent activation energy. 4. Results and discussion Determined basic structural properties of the synthesized and used PAA and PAMA xerogels are given in Table 1. As can be seen from the presented results, structural properties of PAA and PAMA xerogels are slightly different. Xerogel density

xg [kg/m3 ]

Mc [g/mol]

c [×104 mol/cm3 ]

d [nm]

1100 1200

720000 500000

1.4 20.0

54.8 8.5

(xg ) and cross-link density (c ) of PAA xerogel is lower than PAMA xerogel, but the molar mass between the network cross-links (Mc ) and the distance between macromolecular chains (d) are higher for PAA xerogel. Table 2. presents specific drug loading in hydrogel (X) for the MEPBA loading in PAA and PAMA hydrogels at different temperatures. The obtained and presented results clearly indicate that for the both hydrogels specific drug loading in hydrogel (X) increases with temperature increase and that X is higher for PAA hydrogel at all of the applied temperatures. The isothermal dependences of a specific amount of MEPBA released (c) vs. time of drug release (kinetic curves) for different temperatures are shown in Fig. 1a for PAA hydrogel and in Fig. 1b for PAMA hydrogel. Three distinct ranges (a linear, non-linear and saturation) of the changes of the specific amount of MEPBA release with time can be clearly observed from the presented kinetics curves both for the PAA and PAMA hydrogels. In order to determine the influence of temperature on the shape of the kinetics curves the following parameters were defined: the time of range of linearity (tl ), the initial MEPBA release rate (vin ), the saturation time (tf ) and the saturation MEPBA release rate (vf ). The time of range of linearity (tl ) is the time interval within which the MEPBA release increases linearly with the interaction time. The initial MEPBA release rate is defined as the MEPBA release rate during this linear region of the conversion curve by Eq. (11):

vin =

cl tl

(11)

where cl is the specific amount of MEPBA released at the final point of the linear region of the kinetic curve of MEPBA release and tl is time that corresponds to this linear region of the kinetic curve. The saturation time (tf ) represents the interaction time required to achieve the maximal concentration of released MEPBA in the solution (cmax ) at a certain temperature, while the saturation MEPBA release rate can be calculated from the equation: f =

(9)

where T is the temperature, A is the pre-exponential factor, Ea,˛ is the apparent activation energy, f(˛) is general expression of the kinetics model and R is the gas constant. The logarithm form of Eq. (9) leads to:

33

cmax tf

(12)

The parameters of the shape of the kinetics curve parameters at different temperatures as well as the kinetic parameters calculated for the initial and saturation phase are given in Table 3. According to the results given in Table 3 it can easily be seen that the values of tl and tf decrease while those of ␯in and ␯f increase as temperature increases for the both of the hydrogels used. Since the increases of ␯in and ␯f with temperature are exponential, the kinetic parameters of the initial and saturation phase of the MEPBA release from hydrogels (Ea,in , ln Ain , Ea,f , ln Af ) can be determined by applying the Arrhenius equation. The obtained results are also Table 2 Specific drug loading (XT ) of the MEPBA loading in PAA and PAMA hydrogels. PAA T

[◦ C]

22 31 42

PAMA X [g/g]

T [◦ C]

X [g/g]

0.187 0.216 0.2592

20 32 40

0.134 0.1416 0.1525

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Fig. 1. The isothermal kinetic curves of released MEPBA from (a) PAA hydrogel at () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C, (b) PAMA hydrogel at () 20 ◦ C, (䊉) 32 ◦ C and () 40 ◦ C.

Fig. 2. The plot of ln˛ vs. ln(t) for MEPBA release at different temperatures from (a) PAA hydrogel at () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C, (b) PAMA hydrogel at () 20 ◦ C, (䊉) 32 ◦ C and () 40 ◦ C.

given in Table 3. As it can be seen from the obtained results, the activation energy for the initial stage of the MEPBA release from PAA hydrogel (Ea,in = 26.6 kJ/mol) is dramatically higher than the activation energy for the saturation phase of the same process (Ea,f = 5.4 kJ/mol). The same pattern is followed by the values of the activation energy for the MEPBA release from PAMA hydrogel but with a smaller difference in magnitudes, as well as the values of preexponential factor (ln A) for MEPBA release from PAA hydrogel on the contrary to the values ln A for PAMA hydrogel. The significantly higher value of the Ea,in than of Ea,f implies a non-elementary character of the process of the MEPBA release from these hydrogels, i.e on its complexity from the point of view of both the mechanism and kinetics model. Also, it can be easily seen that the values of Ea,in for PAA hydrogel is somewhat lower than MEPEBA from PAMA hydro-

gel, while the values of Ea,f for MEPEBA release from PAA hydrogel is significantly lower than that from PAMA hydrogel. In order to determine the kinetics’ parameters of MEPBA release from PAA and PAMA hydrogels, the results were analyzed by applying the linearized form of the empirical equation developed by Peppas (Eq. (8)). When the plot of ln ˛ vs. ln(t) gives a straight line it is possible to model the kinetics of the investigated process with the suggested model and to determine the values of the diffusion exponent (np ) and the apparent release rate (kp ) from the slopes and intercepts of these straight lines. Fig. 2a and b presents the plot of ln ˛ vs. ln(t) for MEPBA release from PAA and PAMA hydrogels respectively, at different temperatures.

Table 3 The parameters of the conversion curve for MEPBA release from PAA and PAMA hydrogels at different temperatures (R- correlation coefficient; SD- standard deviation). T, ◦ C

tl , min

vin , %/min

tf , min

vf , %/min

cmax , %

Kinetic parameters Initial stage

Saturation stage

PAA

22 31 42

8.00 6.70 5.44

3.25 × 10−3 4.33 × 10−3 6.43 × 10−3

37.0 35.0 32.0

0.0027 0.0028 0.0031

0.046 0.055 0.066

Ea,in = 26.6 ± 0.3 kJ/mol ln Ain /min−1 = 5.11 R = 0.998 SD = 0.010

Ea,f = 5.4 ± 0.3 kJ/mol ln Af /min−1 = 1.42 R = 0.973 SD = 0.046

PAMA

20 32 40

8.45 4.99 3.07

1.49 × 10−3 2.95 × 10−3 3.68 × 10−3

45.0 31.9 45.0

0.51 × 10−3 0.68 × 10−3 1.04 × 10−3

0.0278 0.0306 0.0328

Ea,in = 35.1 ± 0.7 kJ/mol ln Ain /min−1 = 7.9 R = 0.993 SD = 0. 018

Ea,s = 26.6 ± 0.6 kJ/mol ln As /min−1 = 3.2 R = 0.987 SD = 0.025

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Table 4 The kinetic parameters (np and kp ) and range of applicability (L) at different temperatures for MEPBA release from PAA and PAMA hydrogels. T, ◦ C PAA

PAMA

nP

lnkP

kP , min−1

R

SD

L (%␣ )

Kinetic parameters

1 × 10−4

22 31 42

0.907 0.898 0.877

−2.435 −2.372 −2.194

0.088 0.930 0.111

0.999 0.998 0.999

0.001 0.001

8.6–68.1 9.0–68.5 10.6–76.3

Ea = 9.4 ± 0.4 kJ/mol ln A/min−1 = 1.38 R = 0.974 SD = 0.048

20 32 40

0.941 1.027 1.025

−2.7659 −2.346 −2.249

0.063 0.096 0.105

0.999 0.998 0.999

1 × 10−4 0.001 0.001

6–53 9.6–50.3 10.6–55.45

Ea = 20.3 ± 0.8 kJ/mol ln A/min−1 = 5.59 R = 0.977 SD = 0.042

As shown in Fig. 2a and b, the plots of ln ˛ vs. ln(t) gave straight lines in limited ranges of the investigated release process for the both hydrogels. From the plot of ln˛ vs. ln(t), the kinetic parameters, np and kp , were calculated for the range within which the data are linear, i.e. for corresponding ranges of applicability (L). In the ranges of ˛ within which the plots of ln ˛ vs. ln(t) give the straight lines, it is possible to determine the values of the diffusion exponent (nP ) and apparent release rate (kP ) from the slopes and intercepts of these straight lines. The obtained results for the determined parameters (nP and kP ) and a range of applicability (L) for the determined parameters are presented in Table 4. As it may be seen from the results presented in Table 4, it was possible to determine the kinetic parameters, np and kp , for MEPBA release from PAA hydrogel for the range of applicability (L) of ˛ ∼ 10–70% and for the PAMA hydrogel for L of ˛ ∼ 6–55%. As could also be easily seen, for MEPBA release from PAA hydrogel the rate constant kp increases with temperature increase, meanwhile the release exponent (np ), whose value is about 0.9, slightly decreases, on the contrary to the MEPBA release from PAMA hydrogel where np slightly increases and achieves value “1”. These changes might indicate the possible changes of the mechanism with temperature increase in MEPBA release from the both hydrogels. It was possible to determine the kinetic parameters (Ea and ln A) because the rate constants “kp ” show Arrhenius dependence on temperature changes. The obtained value for activation energy for MEPBA release from PAA hydrogel was Ea = 9.4 kJ/mol, which is significantly lower than the activation energy for the initial stage of the MEPBA release (Ea,in = 26.6 kJ/mol) and higher than the value of the activation energy for saturation stage of the same process

(Ea,f = 5.4 kJ/mol). The value for activation energy for MEPBA release from PAMA hydrogel was Ea = 20.3 kJ/mol, which is lower than the activation energy both for the initial and saturation stage of the MEPBA release (Ea,in = 35.1 kJ/mol; Ea,f = 26.6 kJ/mol). Because the obtained values of the kinetic parameters for the different stages of the investigated process of MEPBA release kinetics were found to be somewhat different, another method, the so-called “model-fitting method” [28] was applied. According to the model-fitting method the kinetic reaction model is classified in 5 groups depending on the reaction mechanism: (1) power law reaction (2) phase controlled reaction (3) reaction order (4) reaction described by the Avrami equation and (5) diffusion controlled reactions. Applying the procedure of model-fitting method, the experimentally determined conversion curve ˛exp = f(t)T was transformed into the so-called universal conversion curve ˛exp = f(tN )T, where tN is the so-called normalized time. The reduced time, tN , was introduced to normalize the time interval of the monitored process and was defined by the equation: tN =

t t0.9

(13)

where t0.9 is the moment in time at which ˛ = 0.9. By applying the reduced time, it was possible to calculate the universal conversional curves for the different kinetics models [29]. The kinetics model of the investigated process was determined by comparing (graphically and analytically, using the sum of squares of the residual) the experimentally determined curves with the theoretical curves. The chosen kinetics model is the one for which the sum of squares of the residual is minimal.

Table 5 The set of the kinetics models used to determine the kinetics model of MEPBA release process from PAA and PAMA hydrogels [30]. Model

Reaction mechanism

General expression of the kinetics model, f(˛)

Integral form of the kinetics model, g(˛)

P1 P2 P3 P4 R1 R2

Power law Power law Power law Power law Zero-order (Polany-Winger equation) Phase-boundary controlled reaction (contracting area, i.e. bidimensional shape) Phase-boundary controlled reaction (contracting volume, i.e. tridimensional shape) First order (Mampel) Second order Third order Avrami-Erofe’ev Avrami-Erofe’ev Avrami-Erofe’ev One-dimensional diffusion Two-dimensional diffusion (bidimensional particle shape) Three-dimensional diffusion (tridimensional particle shape) Jander equation Three-dimensional diffusion (tridimensional particle shape) Ginstling-Brounshtein

4˛3/4 3˛2/3 2˛1/2 2/3˛−1/2 1 2(1 − ˛)1/2

˛1/4 ˛1/3 ˛1/2 ˛3/2 ˛ [1 − (1 − ˛)1/2 ]

3(1 − ˛)2/3

[1 − (1 − ˛)1/3 ]

(1 − ˛) (1 − ˛)2 (1 − ˛)3 2(1 − ˛)[−ln(1 − ˛)]1/2 3(1 − ˛)[−ln(1 − ˛)]2/3 4(1 − ˛)[−ln(1 − ˛)]3/4 1/2˛ 1/[−ln(1 − ˛)]

−ln(1 − ˛) (1 − ˛)−1 − 1 0.5[(1 − ˛)−2 − 1] [−ln(1 − ˛)]1/2 [−ln(1 − ˛)]1/3 [−ln(1 − ˛)]1/4 ˛2 (1 − ˛)ln(1 − ˛) + ˛

3(1 − ˛)2/3 /2[1 − (1 − ˛)1/3 ]

[1 − (1 − ˛)1/3 ]2

3/2[(1 − ˛)−1/3 −1]

(1 − 2˛/3)–(1 − ˛)2/3

R3 F1 F2 F3 A2 A3 A4 D1 D2 D3 D4

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Fig. 3. The plot of ˛ = f(tN ) for the theoretical kinetics reaction models: F1 (solid curve: —) and R3 (dot curve: · · ·) and the experimental plots of ˛ = f(tN ) for (a) PAA hydrogel at: () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C, (b) PAMA hydrogel at () 20 ◦ C, (䊉) 32 ◦ C and () 40 ◦ C.

A set of the reaction kinetics models used to determine the model which best describes the kinetics of the MEPBA release process is shown in Table 5. Fig. 3a and b shows the plots ˛ = f(tN ) for the selected theoretical kinetics model (F1 and R3) presented in Table 5 and the experimental plots of ˛ = f(tN ) for the MEPBA release process from (a) PAA hydrogel and (b) PAMA hydrogel at the investigated temperatures. According to the results shown in Fig. 3a and b, it can be stated with great assurance that the kinetics of MEPBA release from both hydrogels at all of the investigated temperatures can best be described with the kinetic model F1 (square root of deviation  = 10−4 ), which corresponds to the first order chemical reaction: − ln(1 − ˛) = kM · t

(14)

where kM is a model’s constant for the first order chemical reaction rate. The isothermal dependences of a −ln(1 − ˛) vs. interaction time for MEPBA release from PAA hydrogel and PAMA hydrogel are shown in Fig. 4a and b, respectively. Table 6 shows the values of the model’s constant for the rate of MEPBA release from PAA hydrogel at different temperatures which was obtained from the slopes of the isothermal dependences −ln(1 − ˛) on the time of the drug release (Fig. 4a and b). Since the increase in the model’s constant with temperature is exponential, the model’s kinetics parameters (activation energy (Ea,M ) and pre-exponent factor (ln AM )) of MEPBA release from PAA and PAMA hydrogels were determined by applying the Arrhenius equation. The obtained results are also given in Table 6. Accordingly

Fig. 4. The plots of −ln(1 − ˛) vs. MEPBA release time for (a) PAA hydrogel at: () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C, (b) PAMA hydrogel at: () 20 ◦ C, (䊉) 32 ◦ C and () 40 ◦ C.

to that, it was found that the activation energy of the MEPBA release for the entire process which was determined on the basis of the model constants (kM ) for the rate of MEPBA release from PAA hydrogel had a value of 7.8 kJ/mol and the pre-exponential factor (ln A) was 1.0. Although these values for the kinetic parameters are the closest to those calculated from the saturation MEPBA release rate and the empirical equation developed by Peppas (Ea = 9.4 kJ/mol) they are significantly dissimilar from the values obtained on the basis of the initial MEPBA release rate (Ea,in = 26.6 kJ/mol). Also, it was found that the activation energy of the MEPBA release for the entire process was determined on the basis of the model constant for the rate of MEPBA release from PAMA hydrogel had a value of 27.5 kJ/mol and the value of the pre-exponential factor was 8.8. The obtained results imply some discrepancies between the values of the activation energy for the early beginning and the final part of the investigated process of MEPBA release from PAMA hydrogel, as well as the activation energy determined by applying the model of the first order chemical reaction (F1). The established existence of limited quantity of the equilibrium concentration of MEPBA in the external medium, the increase in the equilibrium concentration of the MEPBA in the external medium with temperature increase, and also the possibility to describe the kinetics of the investigated process with the kinetic model of the first order chemical reaction, indicate that the kinetics of the MEPBA release from hydrogel can be considered as reversible first order

B. Adnadjevic, J. Jovanovic / Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

37

Table 6 The values of the model’s constant for the rate of MEPBA release (kM ) at different temperatures. T (◦ C)

kM (min−1 )

R

SD

L (%␣ )

Kinetic parameters

PAA

22 31 42

0.1169 0.1265 0.1428

0.999 0.999 0.999

0.008 0.007 0.019

0–99 0–99 0–99

Ea,M = 7.8 ± 0.1kJ/mol, ln AM /min−1 = 1.0 R = 0.997 SD = 0.012

PAMA

20 32 40

0.080 0.142 0.163

0.996 0.999 0.999

0.015 0.003 0.004

0–99 0–99 0–99

Ea ,M = 27.5 ± 1.1kJ/mol ln AM /min−1 = 1.8 R = 0.979 SD = 0.040

chemical reaction, in agreement with the Reis and co-workers [30]. According to that, it can be described with the following scheme (15): ∗ kt

A ⇔A

(15)

kab

where A* is MEPBA concentration in its solution in the hydrogel; A is MEPBA concentration in the external medium; kt is constant of the MEPBA transfer rate from its solution in the hydrogel to the external medium and kab is constant of the MEPBA absorption rate from its external medium to the hydrogel. Lets denote with A∗0 and A∗∞ the concentrations of the MEPBA in its solution in the hydrogel in time t = 0 and t = ∞, respectively, and with A0 and A∞ the concentrations of the MEPBA in the external medium in time t = 0 and t = ∞. According to the law of conservation of matter, the following is valid: A∗0 + A0 = A∗ + A = A∗∞

(16)

i.e. A∗ = A∗∞ + A∞ − A

(17)

Because:

v=

dA = kt · A∗ − kab · A dt

(18)

The introduction of (17) in (18) gives: (19)

v=

dA = kt (A∗∞ − A∞ − A) − kab A dt

(19)

Bearing in mind that: kt · A∗∞ = kab A∞ ,

i.e.

A∗∞ =

kab A kt

(20)

We can get the expression for the rate of the reversible first order chemical reaction:

v=

dA = (kt + kab )(A∞ − A) dt

(21)

If we denote with ˛ = A/A∞ , expression (21) is transformed to (22):

v=

dA = (kt + kab )(1 − ˛) dt

(22)

Accordingly to that, model’s constant of the rate of the reaction of the MEPBA release (km ) is equal to the sum of constants kt and kab , i.e: km = kt + kab

(23)

Keeping in mind that:



Ea = RT

2

dv/dT

v



order chemical reaction (Ea,R ) which is kinetically predetermined with the reversible first order chemical reaction (25): Ea,R =

kt Ea,t + kap Ea,ab kt + kab

where Ea,t is activation energy of the MEPBA transfer from its solution in the hydrogel to the external medium, Ea,ab is activation energy of the MEPBA absorption from the external medium to the hydrogel and Ea,R , is effective activation energy which is independent on the degree of released drug but is predetermined with the kinetic parameters of the MEPBA transfer from its solution in the hydrogel to the external medium and with the MEPBA absorption from the external medium to the hydrogel. Experimentally determination of A∗0 , A* and A∗∞ , and consequently determination of the constant of the MEPBA transfer rate from its solution in the hydrogel to the external medium (kt ) is an especially complex procedure. Thus, we try to determine the constant of the MEPBA absorption rate from its external medium to the hydrogel (kab ) in order to approve the previously given kinetic model. For that purposes, we used MEPBA solution with concentration that corresponds to the maximal experimentally found concentration of MEPBA released and experiments were undertaken at same temperatures as MEPBA release experiments (in further text related as MEPBA external solution (* )). The dependences of the degrees of isothermal absorption of the external MEPBA solution (˛) vs. interaction time are shown in Fig. 5a and b for PAA and PAMA hydrogels, respectively. Table 7 summarizes the equilibrium adsorption’s degrees of PAA and PAMA hydrogels at different temperatures in MEPBA external solution (as previously described (* )). Based on the presented results it is clear that in the case of both hydrogels ADeq increases with temperature increase and that ADeq is somewhat higher for PAA hydrogel. By applying the previously described “model-fitting” method it was established that kinetics of absorption of the external solution of MEPBA in the PAA hydrogel, can be described in complete with the model “R2” which is characteristically for the “phase boundary controlled reaction” (see Table 3). That would imply that the following expression should be valid: 1 − (1 − ˛A )1/2 = km,ab · t

by using the expression (22), it is easy to obtain the expression for effective activation energy of MEPBA release for the reversible first

(26)

where km,ab is the model’s constant rate.

Table 7 Equilibrium adsorption degree (ADeq ) of PAA and PAMA hydrogel for the MEPBA external solution at different temperatures. PAA

(24)

(25)

T

[◦ C]

22 31 42

PAMA ADeq [%]

T [◦ C]

ADeq [%]

7480 8640 10380

20 32 40

5360 5664 6100

38

B. Adnadjevic, J. Jovanovic / Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

Fig. 6. A plot of 1−(1 − ˛A )½ vs. time at: () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C for PAA hydrogel.

Fig. 5. Absorption of external solution in the hydrogels (a) PAA hydrogel at: () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C, (b) PAMA hydrogel at: () 20 ◦ C, (䊉) 32 ◦ C and () 40 ◦ C. Fig. 7. A plot of −ln(1 − ˛) vs. time for PAMA hydrogel at: () 22 ◦ C, (䊉) 31 ◦ C and () 42 ◦ C.

[1 − (1 − ˛)1/2 ]

Fig. 6 presents the dependence on time at different investigated temperatures, for absorption of the external solution of MEPBA in the PAA hydrogel. The dependence [1 − (1 − ˛A )1/2 ] on time, at all of the investigated temperatures for almost whole range of the absorption process of external MEPBA solution gave straight lines (range of applicability L ≥ 94%). The values of the model’s absorptions rates constants (km,ab ) were determined based on these slopes. Table 8 shows the influence of temperature on the changes of degree of applicability of applied model and the model’s absorptions rates constants (km,ab ). Based on the obtained results that are presented in Table 8, it is easy to observe that as the temperature of absorption increases the model’s absorptions rate constants (km,ab ) increase as well. Once again, the previously described “model-fitting” method was applied and it was found that kinetics of absorption of the external solution of MEPBA in the PAMA hydrogel, differently than in the case of PAA hydrogel can be described in complete with the model “F1” (characteristically for the “first order chemical reaction” (see Table 5). That means that the expression previously given with Eq.

(12) should be valid: − ln(1 − ˛) = km,ab t

(27)

where km,ab is a model’s constant for the first order chemical reaction rate. Fig. 7 presents the dependence −ln(1 − ˛) on time for the absorption of the external solution of MEPBA in the PAMA hydrogel at investigated temperatures. The dependence −ln(1 − ˛) on time, at all of the investigated temperatures for almost whole range of the absorption process of external MEPBA solution in PAMA hydrogel gave straight lines (range of applicability L ≥ 95%). The values of the model’s absorptions rate constants (km,ab ) were determined based on these slopes. Table 6 presents the influence of temperature on the range of applicability (L) and the absorption rate constants (kab,1 ). As absorption temperature increases absorption rate constants also increase. The kinetic parameters (Ea,ab and ln Aab ) for the MEPBA absorption from its external medium to the PAMA hydrogel were

Table 8 The changes of absorption model rates constants (km,ab ) with temperature for the absorption of MEPBA external solution in PAA hydrogel. Temperature [◦ C]

L [%]

R

SD

km,ab [min−1 ]

Kinetic parameters

22 31 42

0–94 0–94 0–94

1.000 1.000 0.999

2.33 × 10−6 0.000 0.008

0.0039 0.0050 0.0078

Ea,m,ab = 26.6 ± 0.1kJ/mol ln Am,ab = 5.28/min−1 R = 0.992 SD = 0.018

B. Adnadjevic, J. Jovanovic / Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

39

Table 9 The changes of absorption model rates constants (km,ab ) with temperature for the absorption of MEPBA external solution in PAMA hydrogel. Temperature [◦ C]

L [%]

20 32 40

0–100 0–100 0–95

R 0.999 0.999 0.999

determined by applying the Arrhenius equation, since the increase in the absorptions rate constants with temperature was exponential. The obtained results are also given in Table 9. Based on equation (21) and on the known values of constants km and kab , the constant kt was determined and presented in Table 10, column 2. Since the increase in the constants of the MEPBA transfer rate from its solution in hydrogel to the external medium (kt ) with temperature was exponential, the kinetics parameters of the MEPBA transfer from its solution in hydrogel to the external medium were determined by applying the Arrhenius equation and the obtained results are given in Table 10. Also, the values for effective activation energy of MEPBA release for the reversible first order chemical reaction (Ea,R ) calculated by using Eq. (25) are presented in Table 10. The established data that the MEPBA absorption from the external solution can be described for PAA hydrogel only in limited range of the absorption degree with the kinetics model of the first order chemical reaction, in significant part makes suspicion to the state that the kinetic of the MEPBA release from the hydrogel can be described with the model of the reversible first order chemical reaction. Also, according to the kinetic model of reversible first order reaction the MEPBA, release process from hydrogel is independent on the degree of MEPBA released. In order to examine the dependence of the activation energy of the investigated drug release process from PAA and PAMA hydrogels on the degree of released MEPBA, the Friedman’s iso-conversional method was applied. By using that method activation energies for different degree of released MEPBA were determined. Fig. 8 presents the dependences ln v˛,Ti = f (1/T ) (T is the temperature (K)), for different degrees of released MEPBA from (a) PAA hydrogel and (b) PAMA hydrogel. As can be seen from the obtained results presented in Fig. 8, there was a linear relationship between the ln v˛,Ti and the inverse temperature (1/Ti ) for all of the degrees of released MEPBA both from PAA and PAMA hydrogel. From the slopes and intercepts of these straight lines the values of the kinetics parameters (Ea,˛ and ln A˛ ) for each value of the degree of MEPBA released (˛) have been obtained. The dependences of Ea,˛ vs. ˛, and ln A˛ vs. ˛ are shown in Fig. 9. As can be seen from the results presented in Fig. 9, both Ea,˛ and ln A˛ decreased almost linearly with an increase in the degree of MEPBA release from PAA hydrogel for ˛ ≤ 0.6, while for ˛ > 0.6 Ea,˛

SD 0.011 0.014 0.013

km,ab [min−1 ]

Kinetic parameters

0.014 0.017 0.020

Ea,ab = 17 ± 0.5 kJ/mol ln Aab = 1.48 ± 0.05 min−1 R = 0.999 SD = 0.008

is about 8 kJ/mol and ln A˛ is about 2. By comparing these values of activation energies (Ea,˛ ) and pre-exponential factors (ln A˛ ), it may be concluded that the increase in ln A˛ also coincides with the increase in Ea,˛ and this is the so-called compensation effect [31]. As can be easily seen, both Ea and ln A almost linearly decreased with the increase in the degree of MEPBA release from PAMA hydrogel. Fig. 10 presents the changes of the pre-exponential factor (ln A˛ ) with activation energy (Ea,˛ ) with changes of the degrees of MEPBA release from (a) PAA-hydrogel and (b) PAMA hydrogel. As can be seen from the results presented in Fig. 10, the linear relationships between ln A˛ and Ea,˛ are obtained both for PAA and PAMA hydrogel. The changes of the pre-exponential factor ln A˛ with activation energy at varying degrees of MEPBA release (˛), can be expressed with the following Eqs. (28) and (29) for PAA hydrogel and for PAMA hydrogel, respectively: ln A˛ = −3.9 + 0.57Ea,˛

(28)

ln A˛ = −8.8 + 0.62Ea,˛

(29)

As can be seen, the kinetic parameters of the MEPBA release kinetics are significantly dependent on the degree of the MEPBA release for the both of applied hydrogels. The established dependence of activation energy on the degree of released MEPBA as well as the presence of compensation effect obscured the possible description

Table 10 The changes of transfer-absorption model rates constants (kt ) with temperature and the kinetic parameters of the investigated MEPBA transfer-absorption process from PAA and PAMA hydrogels. T (◦ C)

kt (min−1 )

Kinetic parameters

PAA

22 31 42

0.1130 0.1215 0.1350

Ea,t = 6.82 ± 0.08kJ/mol ln A = 0.1130 min−1 R = 0.998 SD = 0.010

Ea,R = 7.5 ± 0.2 kJ/mol Ea,R = 7.6 ± 0.2 kJ/mol Ea,R = 7.9 ± 0.2 kJ/mol

PAMA

22 31 40

0. 066 0.125 0.142

Ea,t = 32 ± 0.4 kJ/mol ln At = 10.40 min−1 R = 0.998 SD = 0.013

Ea,R = 29 ± 1 kJ/mol Ea,R = 30.2 ± 1 kJ/mol Ea,R = 30 ± 1 kJ/mol

Fig. 8. The dependences of ln ˛,Ti on inverse temperature (1/T) for different degrees of MEPBA released for (a) PAA and (b) PAAMA hydrogel.

40

B. Adnadjevic, J. Jovanovic / Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

Fig. 9. The dependences of Ea,˛ and ln A˛ vs. degree of released MEPBA (˛) from () PAA hydrogel and (䊉) PAMA hydrogel.

of the MEPBA release kinetic with the model of reversible first order chemical reaction. The linear decrease in the activation energy with the degree of released MEPBA from PAA hydrogel in the range of 0 < ˛< 0.6, clearly imply that the MEPBA absorption, which is in connection with the MEPBA desorption, has the dominant influence on the kinetics of MEPBA release. Describing the MEPBA release from the both hydrogels with a first order chemical reaction model strongly supports that the investigated process of MEPBA release is a kinetically controlled reaction and that its kinetics depends highly on the rate of the MEPBA release of the MEPBA bonded to the hydrogels’ active desorption centers that are energetically distributed through the hydrogels. Because the kinetics of the MEPBA released are determined by the rate of MEPBA release from the active center, the experimentally determined dependence of the kinetic parameters from the degree of MEPBA release (˛), the compensation effect and achievement of the maximal MEPBA release rate for low degrees of MEPBA release (ie. the high values of Ea ) could be explained with following proposals:

(a) On the hydrogel structure there are active desorption centers with different specific energy and energetic distributions. (b) The degree of efficiency of the active desorption center in the specific interaction is proportional to its own energy. (c) The value of Ea for the release process is inversely proportional to the value of the specific energy of the active desorption center. Bearing that in mind, during the release process at low degrees of release and relatively low values of activation energy, active desorption centers with low specific energies must participate. Active desorption centers with low energies are more abundant than those with high energies due to their thermodynamics and the statistics of their requirements. Consequently, the ln A value is high as it is proportional to the mass concentration of the active desorption centers and this explains the presence of the compensation effect, as well as the maximal rate of MEPBA delivery at the low degrees of the MEPBA release. On the contrary, however, the active desorption centers with high specific energies must participate when the degrees of the MEPBA release are high and Ea is low. Due to their sparse distribution, the ln A value is low and the rate of the process is minimal. If it was assumed that the distribution of the active desorption centers existed, then the function of distribution of the probable values of activation energies (f(Ea )) could be defined. Because the model of first order reaction kinetics can describe the kinetics of MEPBA release, the following equation is valid:



˛=1−

∞

˚(Ea , T )f (Ea )dEa

(30)

0

where (Ea ,T) is equated to



˚(Ea , T ) = exp



−A0

 E  a

T

exp − 0

RT

 dT

(31)

By using a variable x = Ea /RT, Eq. (31) is rewritten as follows:



˚(Ea , T ) = exp

Fig. 10. A plot of ln A˛ dependence on Ea,˛ with changes of ˛ for () PAA hydrogel and (䊉) PAMA hydrogel.

= exp

−

A0 Ea R

A E 0 a −

R



e−x − x

p(x)



x

∞

e−x dx x



(32)

B. Adnadjevic, J. Jovanovic / Colloids and Surfaces B: Biointerfaces 69 (2009) 31–42

Fig. 11. The function of the density distribution f(Ea ) for MEPBA release from ( —) PAA and (· · ·) PAMA hydrogel.

41

one well-defined peak with Ea = 27.3 kJ/mol as well as two weakdefined peaks at Ea = 31.1 kJ/mol and Ea = 33.5 kJ/mol. The value of the first, dominant peak is again in good correspondence with Ea,M = 27.5 kJ/mol). This again confirms previously exposed model and the drug release mechanism from the investigated hydrogels, as well as the differences in the established kinetic’s parameters between the PAA and PAMA hydrogels. The established differences in the energies distribution curves for PAA and PAMA xerogels/hydrogels can be explained with the changes in the hydrophobicity of their interface. Actually, the introduction of methacrylic co-monomer unit in the hydrogel network, due to the presence of hydrophobic methyl-groups, leads to the increase in its overall hydrophobicity which in turns leads to the decreases of its own energy of the desorption centers, i.e. to the increase in the activation energy of the drug release as well as to the increase in the ratio of the desorption centers with the lower specific energy. 5. Conclusions

where p(x) is the so-called “p-function” that is well known in the field of thermal analysis. By employing an approximation p(x) = e−x /x2 , we can write that





A0 RT 2 Ea exp − ˚(Ea , T ) = exp − E RT



(33)

To estimate the f(Ea ) curve from the experimental data of ˛ vs. time, the possibility of an approximate representation for Eq. (33) was examined. Since the (Ea ,T)function changes rather steeply with activation energy at a given temperature, it seems to be reasonable to assume (Ea ,T)that by the step function U at an activation energy Ea = Ea,s as: ˚(Ea , T ) = U(Ea − Ea,s )

(34)

This approximation assumes that it is only single reactions whose activation energy is Es at a given temperature T. Then Eq. (28) is simplified to:



The process of MEPBA release from the both PAA and PAMA hydrogels into a water solution is a complex heterogeneous process that can be kinetically modeled with the model of the sum of parallel desorption’s reactions of first order which occurred in the energetically heterogeneous surface. The dependence of the distribution of activation energies on degree of MEPBA release is a consequence of the energetic heterogeneity of the desorption centers. The function of density distribution of the probability of activation energies, for PAA hydrogel shows a well signed maximum for Ea = 8.1 kJ/mol. The Ea value is in good correspondence with the Ea value obtained from the model constant of MEPBA release from PAA hydrogel. The function of density distribution of the probability of activation energies, for PAMA hydrogel shows a well signed maximum for Ea = 27.5 kJ/mol, which is in good correspondence with the Ea value obtained from the model constant of MEPBA release from PAMA hydrogel.

∞

˛=1−

f (Ea )dt

(35)

Acknowledgements

Ea,s

According to Eq. (34) we can write that:



Es

˛=1−

f (Ea )dEa

(36)

0

therefore, f(Ea ) is given by differentiating Eq. (34) by Ea,s as: f (Es ) =

d˛ dEs

This investigation was supported by the Ministry of Science and Environmental Protection of Serbia, through project 142025G. We acknowledge Mr. B. Drakulic for kindly preparing the synthesis of MEPBA. References

(37)

Thus, the density distribution function of activation energies could be directly obtained by differentiating the experimentally determined relationship: ˛ vs. E, ie. f(Ea ) = d˛/dEa . Fig. 11 shows the density distribution function of activation energies f(E) for MEPBA release from PAA hydrogel and PAMA hydrogel. From Fig. 11 it can be clearly seen that the both of the investigated hydrogels on the whole interacting boundary phase had active desorption centers with different specific energies. Consequently that sites/centers had also different activation energies. At the function of the density distribution curve for MEPBA release process from PAA hydrogel, it can be observed one well-defined peak with Ea = 8.1 kJ/mol, which corresponds to the most probably activation energy for this process. That Ea value is in good correspondence with the determined Ea,M = 7,8 kJ/mol from the model constant of MEPBA release from PAA hydrogel. On the contrary however, at the function of the density distribution curve for MEPBA release process from PAMA hydrogel, it can be seen

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