Experimental determination of the theophylline diffusion coefficient in swollen sodium-alginate membranes

Journal of Controlled Release 76 (2001) 93–105 www.elsevier.com / locate / jconrel

Experimental determination of the theophylline diffusion coefficient in swollen sodium-alginate membranes M. Grassi a

a,b ,

*, I. Colombo c , R. Lapasin a

Department of Chemical, Environmental and Raw Materials Engineering, DICAMP, Piazzale Europa 1, I-34127 Trieste, Italy b Eurand International S.p. A., via del Follatoio 12, I-34148 Trieste, Italy c Eurand International S.p. A., via Martin Luther King 13, I-20060 Pessano con Bornago, Milan, Italy Received 5 March 2001; accepted 21 June 2001

Abstract In this paper attention is focused on the determination of the drug diffusion coefficient in a swollen polymeric membrane referring to a recent mathematical model (linear model). The main advantage deriving from its use is that, despite its analytical nature and its ability to account for the most important aspects characterising a permeation experiment, it can also be applied in the case of thick membranes. To check the model reliability, a comparison is made with a more complex numerical model and with a largely employed model in terms of data fitting quality. To this purpose, particular care is devoted to the experimental and theoretical tools employed to calculate the auxiliary parameters required by the three models, and with the aim of getting a drug diffusion coefficient value as accurate as possible. Theophylline was chosen as model drug owing to its wide employment in the pharmaceutical field. Membranes were prepared with sodium alginates hydrogels at three different polymer concentrations. The present analysis demonstrates the reliability of the linear model and reveals that the theophylline diffusion coefficient is not significantly affected by the polymer concentration. Indeed, such a parameter is reflected in different membrane thicknesses rather than in different mesh sizes of the polymeric network.  2001 Elsevier Science B.V. All rights reserved.

1. Introduction The increasing demand for deriving more accurate and reliable information from experimental data necessarily implies that more refined theoretical tools must be developed to interpret and describe the physical phenomena and processes occurring in experiments. So doing, the model parameters can be *Corresponding author. Department of Chemical, Environmental and Raw Materials Engineering, DICAMP, Piazzale Europa 1, I-34127 Trieste, Italy. Tel.: 139-40-6763435; fax: 139-40569823. E-mail address: [email protected] (M. Grassi).

treated more correctly as the synthetic identity card of the whole experimental data set from which they are derived, while they can be more reliably used as predictive tools, if known. Despite these significant and powerful features, the practical use of the theoretical approaches may be hindered by their complexity, as they become more and more refined but less user friendly. Since the model complexity is often connected, at least partly, with the testing conditions and procedure, the experimental set-up must be designed in order to minimise this complexity, without renouncing the model precision. In this light, the present paper focuses on the determination of the theophylline diffusion coeffi-

0168-3659 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0168-3659( 01 )00424-2

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M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

cient in swollen sodium alginate membranes characterised by increasing polymer volume fraction. Theophylline was chosen since it is commonly used as a model drug [1–5], its water concentration can be easily measured by using an UV spectrophotometer and, finally, it does not require particular handling care, not being a dangerous substance. Sodium alginate gives origin to strong hydrogels and this makes it a suitable candidate for the realisation of membranes easily mountable on the permeation apparatus. Furthermore, alginate hydrogels have been widely studied [6–15] and no chemical reaction is required for gel formation, since this process is physical in nature. The determination of the theophylline diffusion coefficient inside the gel, as well as the analysis of the whole permeation process, was performed by resorting to two recent models [16]: the linear model and the numerical model. Moreover, a comparison was made with the results obtained from the widely used steady state model [17,18]. In order to achieve an accurate determination of the diffusion coefficient, all the model parameters were previously determined by means of independent experiments. For this reason, particular care is devoted to the description of the experimental and theoretical tools employed for the parameters determination.

2. Modelling For a better understanding, we briefly recall the main features of the linear model, the numerical model and the steady state model [16]. First of all, we have to be mindful that the problem under examination consists of drug permeation through a water swollen polymeric membrane sandwiched by two outer stagnant layers in both the donor and receiver bulk volumes. This should be considered a recurrent condition since stagnant layers can be removed only through very strong stirring [19]. Moreover, solid drug dissolution has to be accounted for as the donor compartment is initially filled with saturated drug solution in presence of an excess amount of undissolved drug powder. Indeed, this experimental condition is sometimes set in the attempt of neglecting the drug concentration decrease

in the donor volume during the whole permeation time. The linear model is based on the following main assumptions: (a) the profiles of the drug concentration inside the membrane and the two stagnant layers are linear; (b) the drug diffusion coefficient does not depend on drug concentration; (c) all the possible interactions between drug and polymer can be properly accounted for by means of the partition coefficient; and (d) the drug dissolution constant is time independent [16]. The linear model analytical form is, then Crstd 5 B1 1 B2 e sm 1st 2t rdd 1 B3 e sm 2st 2t rdd

(1)

where Cr is the theophylline concentration in the receiver volume at time t, t r is an empirical parameter accounting for the lag time after which the concentration profiles become linear within the membrane and in the two stagnant layers (trilaminate system). The parameters B1 , B2 , B3 , m 1 and m 2 are defined in Appendix A and depend on the geometrical and physical characteristics of the permeation apparatus under examination besides on the membrane diffusion properties. The numerical model simply requires that Fick’s law holds inside the trilaminate system, with constant drug diffusion coefficients. Indeed, it represents nothing more that the solution of Fick’s second law inside the trilaminate, provided that the pertinent boundary and initial conditions are attained. It is the most general model considered here but, unfortunately, it does not lead to an analytical solution and that is why it may be considered not so user friendly. Nevertheless, it is able to take into account phenomena such as the time dependency of the drug dissolution constant occurring in the powder dissolution [16]. The steady state model [17] represents the analytical solution of Fick’s second law assuming the sink conditions in the receiver volume, a constant value of the drug concentration in the donor volume and neglecting the presence of the stagnant layers. Its analytical form is

H

C0 Km S Dm hm Crstd 5 ]] ]t 2 ] Vr hm 6

O

2h m ` s 2 1d n s2sn p / h md2 D m td 1 ]] ]] e p n51 n 2

J

(2)

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

where C0 is the constant donor concentration, Km is the partition coefficient, S is the diffusing surface area, Vr is the receiver volume, h m is the membrane thickness and Dm is the drug diffusion coefficient inside the membrane. For very long times, the profile of Cr (t) becomes linear and Eq. (2) can be approximated as C0 SP Crstd 5 ]] ht 2 t L j Vr

(3)

where Km Dm P 5 ]] hm

h 2m t L 5 ]] 6Dm

(4)

P being the membrane permeability and t L the time corresponding to the axis intercept given by Eq. (3). For a trilaminate system, Barrie et al. [18] demonstrated that Eq. (3) still holds, but with a different expression for P and t L D1 Dm D3 K1 Km K3 P 5 ]]]]]]]]]]]]] h 1 Dm D3 Km K3 1 h m D1 D3 K1 K3 1 h 3 D1 Dm K1 Km (5)

j1 1 j2 1 j3 1 j4 t L 5 ]]]]] j5

S S S

h 21 h1 hm h3 j 1 5 ] ]] 1 ]] 1 ]] D1 6D1 K1 2Dm Km 2D3 K3

(6)

D D D

(7)

h 2m h1 hm h3 j 2 5 ] ]] 1 ]] 1 ]] Dm 2D1 K1 6Dm Km 2D3 K3

(8)

h 23 h1 hm h3 j 3 5 ] ]] 1 ]] 1 ]] D3 2D1 K1 2Dm Km 6D3 K3

(9)

Km h 1 h m h 3 j 4 5 ]]] D1 D3 K1 K3

(10)

h1 hm h3 j 5 5 ]] 1 ]] 1 ]] D1 K1 Dm Km D3 K3

(11)

where Di , Ki and h i are the drug diffusion coefficients, the drug partition coefficients and the thicknesses of the stagnant layers, respectively (i 5 1, first stagnant layer, i 5 3, second stagnant layer).

95

3. Experimental For its extensive use in the pharmaceutical field theophylline monohydrated (C 7 H 8 N 4 O 2 ?H 2 O; Carlo Erba, Milan, Italy) is used as a model drug, while sodium alginate (Protanal LF 20 / 60, Pronova Biopolymer, Drammen, Norway) represents the polymer selected for membrane preparation. Membranes are prepared by gradually adding the dry polymer powder into a highly stirred thermostatic (408C) vessel containing demineralised water. In order to eliminate air bubbles produced by stirring, the solution undergoes a further mixing stage under vacuum for 5 min. The solution is put in a Petri disk which in turn is immersed for 30 min in an aqueous solution containing 0.05 M CaCl 2 and 0.4 M NaCl. CaCl 2 represents the Ca 21 source necessary for the gel formation (based on Ca 21 mediated egg-box junctions [20]), while NaCl is added in order to guarantee a better gel homogeneity ˚ as suggested by Skiak-Bræk et al. [21]. The gel membrane is washed for 2 min in demineralised water in order to remove salts from its surfaces and then its thickness is determined as the average of four measurements taken at different membrane points by means of an electronic calibre (Mitutojo, type IDC 112MCB, Japan). Permeation experiments are performed using a side by side apparatus cell [22,23] consisting of the donor compartment (Vd 5100 cm 3 ), initially containing a saturated theophylline solution in presence of undissolved theophylline, and the receiver com3 partment (Vr 5100 cm ) initially filled by demineralised water. Each compartment is equipped with a constant temperature jacket and a magnetic stirrer (600 rpm) housed at the bottom of the cell (see Fig. 1). The membrane (diffusing area |10 cm 2 ) is located in the PTFE adapter connecting the receiver and the donor compartments (Fig. 1). The drug concentration increase is measured and recorded by means of a personal computer managing a UV spectrophotometer (271 nm, UV–Vis spectrophotometer, Lambda 6, Perkin Elmer, USA) connected to the receiver environment as indicated in Fig. 1. In order to prevent bubble formation in the detecting system, a surge chamber is inserted between the spectrophotometer and the peristaltic pump that provides for solution recirculation. Permeation

96

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

Fig. 1. Schematic representation of the experimental set-up. The drug moves from the donor to the receiver compartment diffusing through the membrane and the receiver concentration increase is monitored and recorded by means of a personal computer managing an UV spectrophotometer.

experiments, led in duplicate, are performed at 258C and 378C, and for three different polymer concentrations (%P): 1, 2 and 4 wt.%. Lower concentrations produce very fragile membranes, while at higher concentrations the preparation of homogeneous membranes cannot be easily ensured. The theophylline diffusion coefficient inside the membrane can be evaluated more properly and accurately, if some physical parameters characterising the permeation apparatus, the diffusing drug and the swollen gel are previously determined. In particular, we need to measure the theophylline water solubility Cs , its water diffusion coefficient Dw , the theophylline powder dissolution constant Ktp , its partition coefficient Km between the membrane (sodium alginate gel) and the donor–receiver fluid (water) and, finally, the thicknesses, h 1 and h 3 , of the two stagnant layers.

3.1. Water solubility The solubility, Cs , of theophylline monohydrate in distilled water is measured by shaking a theophylline

suspension for 24 h, filtering it through a cellulose ¨ acetate membrane (0.45 mm; Sartorius, Gottingen, Germany) properly diluting the filtered solution and determining the diluted theophylline concentration via spectrophotometric analysis (271 nm, UV–Vis spectrophotometer). Cs is then easily calculated from the dilution ratio, Cs 56681642 mg / cm 3 at 258C and Cs 512 4956104 mg / cm 3 at 378C.

3.2. Water diffusion coefficient The theophylline monohydrate diffusion coefficient, Dw , in distilled water at 378C is determined with the intrinsic dissolution rate technique (IDR) [24–26]. A cylindrical theophylline disk (5 mg, 1 cm diameter, compressed at 5 tons) is attached, by means of liquefied paraffin, to a rotating stainless steel disk of the same diameter. The whole apparatus is immersed in a 150 cm 3 thermostatic water volume (378C), the motion of the disk is immediately started and the theophylline time–concentration is collected by means of a personal computer managing an UV detector (UV–Vis spectrophotometer) set at 271 nm

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

for 300 s. The experimental data are then fitted by means of the usual dissolution equation [26]

F

C 5 Cs 1 2 e

S

Sd 2K d ]t Vr

DG

(12)

where C is the theophylline time concentration, t is the time, Sd is the surface of the cylindrical disk basis and Kd is the theophylline intrinsic dissolution constant defined by Dw Kd 5 ] h

(13)

where h is the thickness of the boundary layer arising between the solid surface of the rotating disk and the bulk volume [27]. By using Eq. (12), the following experimental values of Kd corresponding to different disk rotational speeds v are obtained from data fitting: (1.4960.002)?10 23 cm / s (v 57.85 rad / s), (1.8760.001)?10 23 cm / s (v 511.52 rad / s), (2.1760.0012)?10 23 cm / s (v 515.71 rad / s), (2.3760.02)?10 23 cm / s (v 518.32 rad / s). The four Kd values are then correlated with v by means of the Levich equation [27] D 2w/ 3 0.5 Kd 5 0.6203]] v 1 A0 n 1/6

(14)

where n is the kinematic viscosity of distilled water and A 0 is an adjustable parameter accounting for the slight dissolution taking place at v 50. The Dw value obtained from data fitting at 378C is (8.2160.57)? 10 26 cm 2 / s. Such a value is then used for determining the Dw value at 258C by employing the Stokes– Einstein equation [28]. Indeed, we firstly calculate the theophylline molecular radius r according to: kT r 5 ]]] 6phDw378C

(15)

where k is the Boltzmann constant, T is the absolute temperature and h is the water viscosity. Then, the r ˚ is adopted to calculate Dw at 258C: value (3.98 A) kT Dw258C 5 ]] 5s6.1560.43d10 26 cm 2 / s 6phr

(16)

The correctness of this result is also supported by other similar values previously published [29]. Moreover, the determined r value is very close to that ˚ [30]. calculated by the DTMM software (r53.56 A)

97

3.3. Powder dissolution constant In order to determine Ktp , an amount M0 of powdered theophylline monohydrate is put in the 100 ml distilled water contained in the thermostatic donor compartment of the permeation apparatus. Mixing is guaranteed by a magnetic stirrer placed at the donor compartment bottom (Fig. 1), while the dissolved drug concentration is monitored and collected by means of a personal computer managing an UV detector (UV–Vis spectrophotometer) set at 271 nm for 100 s. Assuming that the sink conditions are attained during the whole dissolution time and the size distribution is quite narrow so that the drug powder can be considered as a monodisperse system made up by Np spherical particles of mean radius R p0 , the powder dissolution can be schematised by the following system of differential equations [26]

S

D

dCp Kdp ]] 5 4p Np ]Cs R p2 dt Vd

(17)

ds4p R p3 r / 3d ]]]] 5 2 4p Cs Kdp R 2p dt

(18)

where Cp is the drug concentration, R p is the particle radius at time t, Vd is the donor volume, r is the particle density and Kdp is the intrinsic powder dissolution constant. Eq. (17) accounts for the increase of the theophylline concentration in the donor compartment, while Eq. (18) describes the particle radius reduction with time. The solution of the above system is:

e e 2 Cp 5 ]2 sKdp Csd 3 t 3 2 ]R p0sKdp Csd 2 t 2 1 e R p0 sKdp Csdt; r 3r 4p Np e 5 ]] Vd

(19)

Kdp Cs R p 5 R p0 2 ]]t r

(20)

which directly descends from the Hixon–Crowell equation [31]. Kdp is determined by fitting Eq. (19) to the experimental data Cp vs. t, provided that Np and R p0 are known. Their determination is achieved by resorting to the following equations [16]:

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

98

3M0 3 R p0 5 ]; Np 5 ]]] Ar r 4p R 3p0

(21)

where M0 is the initial powder mass and A is its specific surface (per unit mass). The measurement of the theophylline monohydrate density r (1.49060.0093 g / cm 3 ) is performed by an helium pycnometer (Micromeritics multivolume helium picnometer 1305), while the A value (2941 cm 2 / g) is measured with a mercury porosimeter (Porosimeter 2000, Carlo Erba). Finally, Ktp is given by 4p R 2p0 Np Ktp 5 Kdp ]]] Vd

(22)

Thus, it is implicitly assumed that the particle radius does not sensibly decrease during the whole permeation test. If it were not the case, Ktp would be time dependent and proportional to R 2p . For T5258C Kdp 51.57?10 23 cm / s and Ktp 55.87?10 24 1 / s, while for T5378C Kdp 51.52?10 23 cm / s and Ktp 5 5.71?10 24 1 / s.

3.4. Partition coefficient Due to their small volume, gel membranes are not suitable for determining the theophylline partition coefficient, Km , since very small volumes of theophylline solution would be required to perform the

experimental tests. Accordingly, cylindrical shaped gels of approximately 3 cm 3 are considered. The polymer solution is prepared, as previously described and poured into an hollow cylindrical vessel sealed at the bottom by a polyethersulfone membrane (100mm pores) enabling only ion-exchange. The vessel is closed with another polyethersulfone membrane and immersed in a 0.05 M CaCl 2 and 0.4 M NaCl aqueous solution for 72 h, so that a complete and uniform crosslinking is achieved [21]. Then, in order to speed up the attainment of equilibrium conditions, the washed cylindrical gel is cut into slices and immersed, for 24 h, in an aqueous theophylline solution of known volume (V515 cm 3 ) and concentration (C0 520 mg / cm 3 ). The partition coefficient is calculated according to the following expression, accounting for the drug mass balance made up on the whole gel–solution environment VsC0 2 C`d Km 5 ]]]] C`Vgel

(23)

where C` is the measured theophylline concentration in the solution at equilibrium and Vgel is the gel volume considered. This measurement, led in triplicate, gives the results shown in Table 1.

3.5. Stagnant layer thickness The stagnant layer thickness h ss (5h 1 5h 3 ) is

Table 1 Theophylline partition coefficient (Km ), membrane thickness (h m ), surface area (A) and stagnant layers thickness (h ss ) relative to the experimental tests performed; T is the temperature and %P is the polymer percentage in the membrane T (8C) 25

37

No.

%P (–)

Km (–)

hm (cm)?10 4

A (cm 2 )

h ss (cm)?10 4

1 2

1

0.9260.02

475 425

10.8

54.8

3 4

2

0.8360.04

450 430

10.7

5 6

4

0.8860.01

870 700

10.2

7 8

1

0.8360.04

280 380

10.8

9 10

2

0.7960.001

490 480

10.7

11 12

4

0.8460.008

720 655

10.2

60.7

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

determined at both 25 and 378C, by fixing a theophylline cylindrical compress (2-cm diameter) in an appropriate PTFE adapter located in place of the polymeric membrane and thus lying in between the two cells. One cell is filled with PBS solution, pH 7.4, while the other is empty and serves only to press the PTFE adapter against the other cell. As soon as the surface of the theophylline disk is put in contact with the PBS solution, it begins to dissolve and the theophylline concentration in the release volume increases. This phenomenon is recorded for 10 min by a personal computer managing an UV spectrophotometer (271 nm, UV–Vis spectrophotometer) and the experimental data, obtained in triplicate, are fitted by the usual dissolution equation [26]

F

C 5 Cs 1 2 e

S

Dw Sc 2 ] ]t h ss Vr

DG

(129)

where h ss is the fitting parameter and Sc is the compress surface area. Finally, we get h 258C 5 ss (54.860.1) mm and h 378C 5(60.760.2) mm. ss

4. Results and discussion Due to the nature of theophylline and alginate membranes (potentially electrically charged compounds), the determination of the theophylline diffusion coefficient should be performed taking into account possible electrostatic interactions between the drug and the polymer. Indeed, these interactions could manifest through a drug adsorption on the polymeric chains during diffusion [32,33] and the consequent change in the permeation curve would result in an incorrect diffusion coefficient when using an improper mathematical model. Nevertheless, on the basis of the determined partition coefficient values (see Table 1), we can rule out that the above mentioned drug–polymer electrostatic interaction (adsorption) takes place in our case. Indeed, this would determine partition coefficient values greater than those measured as demonstrated by the following equation: Cm` Km 5 ] C`

S

D

S

D

%PrH 2 O 100rp C`Vm ]]] 1 MbVm rp ]]] 100rp 1 %PrH 2 O 100rp 1 %PrH 2 O 5 ]]]]]]]]]]] Vm C`

(24)

99

where Km is the partition coefficient, Cm` and C` are the equilibrium drug concentration in the membrane and in the surrounding medium, respectively, Vm is the membrane volume, rp (1.781 g / cm 3 [34]) and rH 2 O are the polymer (alginate) and water density, respectively and, according to Singh [32], Mb is the amount of adsorbed drug per unit mass of polymer. Eq. (24) is derived by considering the membrane as a two-phase system (polymer and swelling agent) and assuming that, at equilibrium, the drug present in the swelling agent pervading the polymeric network reaches the concentration (C` ) of the surrounding medium, while a drug excess is adsorbed on the polymeric chains. Accordingly, if no adsorption occurs (Mb 50), Km would be equal to 0.978, 0.988 and 0.994 for %P54, 2, 1, respectively. The measured Km values (Table 1) are always lower than those calculated in the hypothesis of setting Mb 50 and then we can reasonably argue that no adsorption related to drug–polymer electrostatic interactions takes place. Consequently, the experimental data analysis performed by means of the mathematical model shown in the modelling section is fully authorised. Fig. 2 shows the trend of the receiver drug concentration, Cr , vs. time, t, for the test performed at 258C at three different %P (test No. 2, 3 and 5). The difference between the curves obtained from tests 2 and 3 are mainly due to different values of Dm , since the membrane thickness is comparable (see Tables 1 and 2), while in the case of test 5 the membrane thickness is the key parameter governing the difference. Moreover, the nonlinear trend in the beginning part of the curves, regardless the %P value, indicates that the membranes under consideration cannot be defined as ‘thin’. Depending on the %P considered, after a sufficiently long time, the permeation curves assume an almost linear trend indicating that, correspondingly, a linear drug concentration profile is attained inside the trilaminate (two stagnant layers and membrane) [16]. Similar considerations can be done for Fig. 3 where the Cr 2t trend refers to T5378C. Alginates do not produce thermosensitive hydrogels and, hence, an increase in temperature does not lead to appreciable changes in the membrane structure, whereas, the theophylline diffusion coefficient increases with temperature. Accordingly, the permeation kinetics is

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

100

Fig. 2. Experimental trend of the receiver theophylline concentration Cr vs. time t relatively to three different membrane polymer concentrations: test No. 2 %P51%, test No. 3 %P52%, test No. 5 %P54% (T5258C).

improved at 378C and Cr increases more rapidly than at 258C. Before presenting the results of the data analysis, an interesting comparison between the experimental data and the best fits obtained with the numerical model and the linear model can be seen in Fig. 4. The excellent agreement between the numerical model and the experimental data proves the suitabili-

ty of the theoretical frame used to generate the model. Moreover, the coincidence of the two models in terms of best fitting (linear part of the permeation curve) and in terms of diffusion coefficient (see Table 2) guarantees the reasonablessness of the lag time value t r . Such an empirical fitting parameter has been introduced in the linear model in order to simulate the time needed to reach a linear profile

Table 2 Theophylline diffusion coefficient calculated according to the linear (DLIN ), Barrie (DBARRIE ) and numerical (DNUMERIC ) model. The Barrie (t L ) and linear (t r ) model lag time is also shown; T is the temperature and %P is the polymer percentage in the membrane T (8C) 25

37

No.

%P (–)

DLIN (cm 2 / s)?10 6

tr (s)

DBARRIE (cm 2 / s)?10 6

tL (s)

DNUMERIC (cm 2 / s)?10 6

1 2

1

5.660.34

90.6 66.2

5.560.30

92.6 62.3

5.660.35

3 4

2

4.360.43

141.7 165.1

4.360.40

161.1 196.5

4.260.46

5 6

4

5.060.70

270.0 185.9

5.0160.60

275.5 236.0

5.160.64

7 8

1

4.360.59

38.2 51.1

4.360.42

29.9 48.0

4.360.22

9 10

2

4.560.21

154.1 133.6

4.560.20

190.0 148.8

4.460.22

11 12

4

4.260.11

167.1 194.6

4.360.11

196.3 244.7

4.260.12

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

101

Fig. 3. Experimental trend of the receiver theophylline concentration Cr vs. time t relative to three different membrane polymer concentrations: test No. 7 %P51%, test No. 9 %P52%, test No. 11 %P54% (T5378C).

concentration inside the trilaminate. Consequently, the use of the much simpler linear model instead of the complex numerical model is fully authorised also in presence of ‘thick’ membranes with great advantages especially for the fitting procedure. Accordingly, the linear model can be profitably used when the Barrie model fails, namely when the drug concentration in donor compartment is not constant with time and when the sink conditions are not attained in the receiver compartment. It is important to remind

that the use of the linear model is possible only because all the model parameters, except for the diffusion coefficient, Dm , and the lag time, t r , have been previously determined by independent experiments. If it were not the case, considerable differences could arise in the estimation of the diffusion coefficient by adopting the numerical or the linear model. An inspection of Table 2 further underlines the correctness of the use of the linear model in place of

Fig. 4. Best fitting of the numerical (thick solid line) and linear (thin solid line) models on the experimental data (open circles) referring to test No. 5 (T5258C, %P54%). The two models best fitting practically coincide in the linear part of the curve.

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

102

the numerical or the Barrie models. Indeed, it is easy to verify that, for all the experimental tests performed, the agreement of the three models, in terms of calculated theophylline diffusion coefficient, is quite good. Moreover, a satisfactory agreement also exists between the calculated lag time determined by the linear model (t r ) and the Barrie model (t L ) for all the performed tests. The fact that the theophylline diffusion coefficient does not significantly vary with %P, regardless of temperature, leads to the conclusion that the %P increase is not reflected in an increase of the polymeric network crosslink density, responsible for a reduction of the network mesh size. On the contrary, the %P increase is mostly reflected in a membrane thickness increase and this is why, generally, the higher %P the lower the permeation curve (see Figs. 2 and 3). Finally, it is interesting to consider the linear model behaviour in no sink conditions when the steady model fails. For this purpose we generate, by means of the numerical model, different permeation curves (Cr vs. t) corresponding to smaller and smaller Vr values (see Table 3) and characterised by DNUMERIC 55.1?10 26 cm 2 / s, all the other parameters being equal to those in test 5. These permeation curves are then fitted by means of the linear model, DLIN and t r being the only two adjustable parameters. Table 3 shows that the linear model yields correct values for the drug diffusion coefficient inside the membrane up to Vr /Vd 51 / 32 (clearly non sink conditions) and, even for lower Vr /Vd values, the discrepancies at the DLIN /DNUMERIC level are not excessive (also in the light of the DNUMERIC standard

error determined from experimental permeation curve). Furthermore, it can be seen that t r does not change significantly with decreasing Vr /Vd .

Table 3 Linear model behaviour in non sink conditions; the ratio DLIN / DNUMERIC and t r are evaluated for decreasing Vr /Vd values in order to match the non sink conditions

Cr C`

Vr /Vd

100?DLIN /DNUMERIC

t r (s)

1 1/2 1/4 1/8 1 / 16 1 / 32 1 / 64 1 / 128

98.2 97.9 98.8 97.6 94.8 90.4 82.4 71.2

270 270 290 286 286 286.9 286.9 280.7

5. Conclusions In this paper we demonstrated the reliability and the effectiveness of a simple mathematical model (linear model) for the calculation of the drug diffusion coefficient also in the case of thick membranes. This demonstration has been made by considering the diffusion of a widely employed model drug, theophylline, through alginates hydrogels characterised by different polymer concentrations. The fundamental prerequisite for the safe use of the linear model is that all the model parameters, except for the lag time, must be determined in advance through independent experiments. Moreover, this study has also revealed that an increase of the polymer concentration in the hydrogel does not significantly affect the network mesh size but, on the contrary, it increases the membrane thickness. Indeed, the theophylline diffusion coefficient is almost independent of the hydrogel polymer concentration, regardless of temperature.

6. List of symbols A A0 C

C0 Cm` Cp Cs D1

theophylline powder surface area per unit mass fitting parameter (Eq. (14)) theophylline drug concentration (Eq. (12)) reservoir drug concentration equilibrium theophylline solution concentration (Eqs. (23) and (24)) initial theophylline solution concentration equilibrium drug concentration in the membrane (Eq. (24)) theophylline drug concentration (Eq. (17)) theophylline water solubility drug diffusion coefficient in the first stagnant layer (Eq. (5))

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

D3 Dm DBARRIE DNUMERIC DLIN Dw h h1 h3 hm h ss k K1 K2 Kd Kdp Km Ktp M0 Mb Np P %P r Rp R p0 S Sc Sd t T tL tr V Vgel Vm

drug diffusion coefficient in the second stagnant layer (Eq. (5)) drug diffusion coefficient inside the membrane (Eq. (2)) Dm value calculated according to the Barrie and Flynn model Dm value calculated according to the numeric model Dm value calculated according to the linear model theophylline water diffusion coefficient hydrodynamic boundary layer thickness first stagnant layer thickness (Eq. (5)) second stagnant layer thickness (Eq. (5)) membrane thickness (Eq. (2)) stagnant layer thickness (Eq. (129)) Boltzman constant first stagnant layer partition coefficient (Eq. (5)) second stagnant layer partition coefficient (Eq. (5)) theophylline intrinsic dissolution constant intrinsic theophylline powder dissolution constant membrane partition coefficient (Eq. (2)) theophylline powder dissolution constant theophylline amount amount of drug adsorbed per unit mass of polymer number of spherical particles constituting M0 membrane permeability (Eqs. (4) and (5)) polymer concentration in the gel theophylline molecular radius theophylline particles radius initial theophylline particles radius diffusing surface area compress surface (Eq. (129)) rotating disk surface time absolute temperature time axis intercept of Eq. (3) lag time (Eq. (1)) theophylline solution volume gel volume (Eq. (23)) membrane volume (Eq. (24))

Vr

103

receiver volume

Greek letters v rotational speed n distilled water kinematic viscosity h distilled water viscosity r theophylline particles density rp polymer (alginate) density rH 2 O water density

Appendix A The parameters involved in the linear model (Eq. (1)) are connected to the physical constants through the following relations [16] D1 SK1s 5 1d D1 S G 5 Kt Cs 1 ]]]]; T 5 ]]]; Vd h 1 Vd h 1 Km D3 S D3 S K3s 5 1d x 5 ]]; Y 5 ]]]] h 3Vr Km h 3Vr D3 h m a 5 1 1 ]]]; Dm h 3 Km

(A.1)

D3 h m K3 b 5 2 ]]]; Dm h 3

K1 1 g 5 ]]]]; d 5 ]]]] Dm h 1 D1 h m 1 ]] ]]] 1] 11 D1 h m Km Dm h 1 Km

g b a 5 ]]; b 5 ]]; a 2d a 2d db z 5 ]] a 2d

(A.2)

ag g 5 ]]; a 2d (A.3)

m 1 5 0.5 f 2sTg 2 G 1 xb 2 Yd ]]]]]]]]]]] 1œsTg 2 G 1 xb 2 Yd 2 2 4ssxb 2 YdsTg 2 Gd 2 Tzxad g

(A.4) m 2 5 0.5 f 2sTg 2 G 1 xb 2 Yd ]]]]]]]]]]]]] 2œsTg2G1xb2Yd 2 24ssxb2YdsTg2Gd 2Tzxad g (A.5) sxb 2 YdKt Cs Kt Cs 2 ]]]] 2sm 2 2sTg 2 Gdd Cs m1 A 2 5 ]]]]]]]]]]]] m1 2 m2 (A.6)

104

M. Grassi et al. / Journal of Controlled Release 76 (2001) 93 – 105

sxb 2 YdKt Cs Kt Cs 2 ]]]] 2sm 1 2sTg 2 Gdd Cs m2 A 3 5 ]]]]]]]]]]]] m2 2 m1

[12]

(A.7) [13]

Kt Cs 1sTg 2 Gd A 2 B1 5 2 ]]]]]] Tz

(A.8)

A2 B2 5 ] f m 1 2sTg 2 Gd g Tz

(A.9)

A3 B3 5 ] f m 1 2sTg 2 Gd g Tz

(A.10)

[14]

[15]

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