Determination of the drug diffusion coefficient in swollen hydrogel polymeric matrices by means of the inverse sectioning method

Journal of Controlled Release 47 (1997) 305–314

Determination of the drug diffusion coefficient in swollen hydrogel polymeric matrices by means of the inverse sectioning method a b b, b Italo Colombo , Mario Grassi , Romano Lapasin *, Sabrina Pricl a

b

VectorPharma International S.p. A., Via del Follatoio 12, I-34148 Trieste, Italy Department of Chemical, Environmental and Raw Materials Engineering — DICAMP, University of Trieste, Piazzale Europa 1, I-34127 Trieste, Italy Revised 2 January 1997; accepted 12 February 1997

Abstract The accurate knowledge of the diffusion coefficient Dg of an active drug within a swollen polymeric hydrogel matrix plays a determinant role in the analysis of the controlled release processes, in the prevision of their kinetics and in the design and formulation of efficient controlled release drug delivery systems. In this paper, we applied the so-called inverse sectioning method to a scleroglucan hydrogel matrix loaded with theophylline, and illustrated a procedure for a reliable estimation of Dg . This crucial parameter is calculated by correlating the experimental data with the model resulting from the analytical solution of the second Fick law in one dimension, obtained under the appropriate initial and boundary conditions. The quality of the adopted model and the uncertainty on the calculated value of Dg are discussed on a statistic basis.  1997 Elsevier Science Ireland Ltd. Keywords: Diffusion; Scleroglucan; Hydrogel; Mathematical modeling; Release kinetics

1. Introduction A fundamental tool for a detailed study and a correct prevision of the kinetics of the release process of an active drug from a polymeric hydrogel matrix is the accurate knowledge of several factors such as the diffusion coefficient Dg of the diffusing active drug, the constitutive equation for the mass flux within the polymeric matrix and the relevant initial and boundary conditions. The simplest and most commonly used expression for the mass flux is given by Fick’s law, according to which the mass flux is proportional to the concentration gradient *Corresponding author. Tel: 139 40 6763434; fax: 139 40 569823; e-mail: [email protected].

through the diffusion coefficient Dg , assumed to be independent of the concentration of the diffusing active drug. Under this hypothesis, the mass of the active principle released Mt varies with the square root of the time t for Mt /M` ,0.6, where M` is the mass released after an infinitely long time, and the kinetics of the release process is said to be Fickian [1]. Nevertheless, many situations exist in which the kinetics of the release process are shown to deviate from the Fickian behavior, and the reasons for these experimental evidences can be ascribed to the existence of both intrinsic and extrinsic matrix factors [2]. The intrinsic matrix factors can be classified as: • Presence of a concomitant kinetics of drug disso-

0168-3659 / 97 / $17.00  1997 Elsevier Science Ireland Ltd. All rights reserved PII S0168-3659( 97 )01657-X

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lution (when it is present in the matrix in a concentration beyond its solubility value) • Matrix erosion phenomena • Intrinsically non-Fickian drug diffusion due to the peculiar structure of the matrix • Matrix swelling / deswelling phenomena caused by the diffusion process. Among the extrinsic factors we can mention both the initial and the boundary conditions of the system. Quite commonly, the intrinsic factors can be considered as the responsible for a non-Fickian kinetics of the release process; nevertheless, in some cases, the extrinsic factors play a crucial role in determining the non-Fickian character of the diffusion. From the above considerations we can conclude that, in order to study and to design a system for controlled drug delivery we must know in detail the law which rules the flux of matter, the initial and boundary conditions of the system and the drug diffusion coefficient Dg . Accordingly, an accurate and reliable experimental determination of Dg is of primary importance. Aim of this paper is to test the accuracy and the reliability of a diffusion coefficient measure obtained by means of the inverse sectioning method. This method presents several advantages, such as nonexpensive experimental setup and easy execution. Theophylline monohydrate is chosen as model drug, while the polymer constituting the non-self-sustaining hydrogel matrix is scleroglucan. Particular emphasis will be placed on the statistical meaning of the relevant data processing.

2. Materials The polysaccharide scleroglucan (Actigum CS11, | 1.2310 6 , Mero-Russelot-Satia, France) used Mw 5 for hydrogel matrix formulation was kindly supplied by VectorPharma S.p.A., Trieste, Italy. Scleroglucan is the name given to a class of fungal, neutral, water-soluble polysaccharides secreted exocellularly by certain imperfect fungi of the genus Sclerotium. The structure of scleroglucan has been characterized as a linear chain of glucopyranosyl units linked b -D-(1→3), with single glucopyranosyl units linked b -D-(1→6) to every third unit of the main chain [3].

Scleroglucan dissolves in water as a rodlike triple helix (also called a ‘triplex’), which consists of three individual strands composed of six residues in the backbone per turn. The three strands of the triplex are held together by interstrand hydrogen bonds, and the (1→6)-linked b -D-glucopyranosyl side groups protrude from the outside of the triple helix [4]. Theophylline monohydrate (C 7 H 8 N 4 O 2 ,H 2 O, Mw 5198.17, water solubility at 258C56.68 mg / g (measured by authors), diffusion coefficient at 258C in water Dw 56.15310 26 cm 2 / s (measured by authors)) was selected as a typical model drug, since it readily dissolves in water giving stable solutions, is easily detected (at concentrations as low as 1 mg / cm 3 ) by means of UV adsorption tests at the corresponding wavelength l5271 nm, is not toxic and is already employed by the pharmaceutical industry as a test drug. Furthermore, any possible interference in UV adsorption by the polymer eventually present in the release environment is avoided, since scleroglucan, as a neutral polysaccharide, does not show any adsorption peak at 271 nm. For the preparation of homogeneous hydrogel systems, scleroglucan was slowly added to wellstirred theophylline aqueous solutions at room temperature. The systems were allowed to rest overnight to obtain polymer swelling and gel formation. A polymer concentration of 2% w / w was chosen to guarantee the required final characteristics of the gel.

3. Methods

3.1. General considerations Several methods are available in literature for the experimental determination of the diffusion coefficient Dg [5–11,16]. Among this plethora, we can mention the category of methods which are founded on the drug concentration gradient under stationary and non-stationary conditions, those deriving from nuclear magnetic resonance (NMR) and dynamic light scattering (DLS) experiments, and those based on holographic relaxation spectroscopy. Within the group of methods based on the concentration gradient, we can recall the so-called sectioning method, according to which Dg is determined from the knowledge of profile of the drug concentration in a gel matrix slab. The main advantages of this tech-

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nique are its low cost and easy setup, even if it requires numerous (and time consuming) tests to obtain the final value of Dg . In detail, the sectioning method consists of setting a slab of gel, initially devoid of drug, in contact with an aqueous solution containing a known and constant amount of the active principle. After a set time, the gel slab is sectioned into a number of thin slices of known volume, the concentration of the drug in each slice is determined and the profile of drug concentration within the entire slab is thus determined. A variant of this technique consists in preparing a gel slab containing a given initial amount C0 of the drug and placing it in contact with a reservoir filled with pure water. After a fixed time, the gel slab will be sectioned and the drug concentration profile determined as described above. The conceptual advantage of this inverse sectioning method lies in the possibility of a double-checking of the experimental data by comparing the quality of the release data on the gel side (i.e., the spatial drug concentration profile at any given time t, C(x, t)) with the release data quality on the release environment side (i.e., Mt vs. t). The evaluation of the diffusion coefficient Dg is then obtained by a fitting of both experimental data sets. This cross verification cannot be performed in the direct sectioning method since, in this case, the evaluation of Dg from the Mt vs. t data set is made impossible by the imposed experimental condition of constant drug concentration in the external medium. From a theoretical point of view, the inverse sectioning method requires taking into consideration the fact that an accumulation of the active drug in the release environment takes place, which hinders a further release of the drug itself. This problem, however, can be easily overcome by using a reservoir with volume large enough to preserve sink conditions. Further, the use of this inverse method is particularly advantageous when we are able to determine the accumulation of the active principle in the release reservoir by simple techniques, such as UV spectroscopy, notwithstanding the persisting sink conditions.

the polymeric powder to well-stirred theophylline aqueous solutions (theophylline concentration C0 51 mg / cm 3 ) at room temperature. The systems are then allowed to rest overnight to obtain polymer swelling and gel formation, and the resulting gels are stored for 24 h at 58C, to ensure the development of a homogeneous matrix. A portion of the gel is slowly sucked up into a graduated plastic syringe of radius r50.705 cm, whose bottom has been cut out. The quantity of sucked gel, equivalent to a mass of ¯7 g, corresponds to a syringe filling of ¯4 cm. The bottom part of the syringe is then closed with a metallic disk with regular holes, which serves as a gel sustaining net. This device is then positioned in a Teflon container and, at a given initial time t 0 , placed in contact with the release environment (degassed bidistilled water) contained in a reservoir of volume ¯100 cm 3 , thermostated at 258C. The homogeneity of the release medium is ensured by the presence of a magnetic stirrer, whose stirring speed v is kept constant and equal to 100 rev. / min (corresponding to a Reynolds number R e 55315). A scheme of the apparatus is reported in Fig. 1. After a pre-set time t, the syringe is taken away from the device and the analysis of the relevant

3.2. Experimental setup Samples of about 100 cm 3 of scleroglucan hydrogel (polymer concentration Cp 52% w / w) can be prepared by slowly adding the required amount of

307

Fig. 1. Schematic view of the experimental setup.

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experimental data is carried out as follows: first, the exact volume of the release environment employed is determined by weight, and the concentration of theophylline in it determined by UV adsorption at its peak wavelength l5271 nm. This leads to the determination of the drug mass Mt released from the gel up to the pre-set time t. In order to determine with a greater precision the initial concentration of the theophylline (C0 ) in the gel matrix, three aliquots of ¯0.5 g of the unused gel sample are taken, weighted accurately, placed into a graduated flask and diluted with an exact quantity (20 cm 3 ) of dimethylsulfoxide (DMSO). When dissolved in DMSO, the scleroglucan triple helix dissociates into three single random coils [4]; as a consequence, the gel matrix is destroyed, the theophylline is completely released and its concentration in each sample can be determined via UV spectroscopy. The average of the three drug concentrations thus determined is then taken as the effective theophylline initial concentration C0 . It has to be noted here that, since the amount of water in each DMSO sample is less than 5% of the sample total volume, its presence can be neglected and a standard baseline for pure DMSO UV adsorption can be used for instrument calibration [12]. Correspondingly, the adsorption peak of theophylline in DMSO falls at l5273.5 nm. The profile of theophylline concentration within the hydrogel matrix C(x,t) is performed as follows: the cylindrical gel sample contained in the syringe is cut into 10 tiny slices, each of which is weighted and dissolved in 20 cm 3 of DMSO into a graduated flask. The theophylline concentration in each sample is then determined by UV adsorption at 273.5 nm and the concentration profile of the active drug within the hydrogel matrix is obtained accordingly.

3.3. Operative variables The pre-set times for the evaluation of the theo-

phylline C(x,t) profiles are the following: t50, 8, 16, 32 and 72 h, respectively. Data reproducibility was assessed by running each experiment in triplicate.

4. Results and discussion The experimental tests at t50 h are necessary to establish the influence of the time elapsed between the theophylline-hydrogel system preparation and its utilization on the uniformity of the active drug distribution within the hydrogel matrix itself. Accordingly, the concentration profile in fresh matrix samples has been evaluated after 0, 24 and 96 h from their preparation. As we can see from Table 1, a rest time of at least 24 h is necessary to ensure a greater uniformity of theophylline concentration within the polymeric hydrogel matrix. Again from Table 1 we can observe that the average initial drug concentration C0 is greater than its nominal value of 1 mg / cm 3 : it varies from 1.075 mg / cm 3 to 1.172 mg / cm 3 . These fluctuations can be sensibly ascribed to the procedure followed for the gel preparation. Indeed, after a strong mechanical agitation of the polymer / drug / water system, necessary to guarantee the total polymer dissolution and swelling, a second, milder agitation phase takes place, which is performed under vacuum, in order to remove the air bubbles entrapped in the hydrogel matrix. During this second phase, a small fraction of the water solvent is sucked away from the hydrogel matrix, resulting in a slightly higher initial theophylline concentration C0 . Previous results [13] have shown that the kinetics of the controlled release from scleroglucan hydrogel matrices exhibit evident non-Fickian macroscopic features, due to the instantaneous formation of a stagnant layer of thickness h between the gel matrix and the reservoir fluid. The cause of such phenomenon has been attributed to both a limited erosion of the gel (caused by the mechanical agitation of the

Table 1 Average values of initial theophylline concentration inside the hydrogel matrix after different gel rest times Rest time (h)

Average initial theophylline concentration C0 (mg / cm 3 )

Standard deviation (mg / cm 3 )

0 24 96

1172 1075 1114

95 43 57

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fluid reservoir) and the presence of the gel sustaining net. Under such experimental conditions, the relevant release kinetics profile can be simulated fairly well by solving the second Fick law both in the gel and in the stagnant layer, assuming an equal diffusion coefficient for the theophylline in the two phases. Therefore, we can also assume that the distribution of the initial concentration of the drug in the gel follows a step-profile, that is it is equal to its nominal value C0 for 0,x,L2h and it is 0 for L2h,x,L, where L is the sum of the thickness of the gel and the stagnant layer. Since the experimental setup used in this work is the same used in [13] (except for a larger gel volume and, hence, a larger reservoir), we cannot exclude a priori an instantaneous formation of a stagnant layer between the gel and the release environment. Accordingly, the interpretation of the experimental release data can be carried out by solving the monodimensional second Fick law: ≠C ≠ 2C ] 5 Dg ]] ≠t ≠x 2

(1)

under the appropriate initial and boundary conditions, that is: Initial conditions: C(t 5 0) 5 0 L 2 h , x , L C(t 5 0) 5 C0

(2)

0 , x , L 2 h

(29)

S

2 L2h Bn 5 ] sin ln ]] ln L

309

D

(59)

Integrating Eq. (4) over L yields the dependence of the drug mass released Mt on the release time t as:

O ]Bl sinsl d

n5`

Mt 5 S(L 2 h) C0

n

n

n50

H

3 1 2 exp

F

n

l 2n tDg 2 ]] L2

GJ

(6)

in which S is the interface area of the gel. Fig. 2, Fig. 3, Fig. 4 and Fig. 5 report the experimental results (three runs each) obtained for any pre-set test time t in a normalized form. As we can see from these figures, notwithstanding a small degree of data scattering, we can individuate a net profile of drug concentration C vs. x, which appears to level off with increasing test time t. Whereas (L 2 h) is a known quantity, the two parameters, h and Dg , in the above equations should be determined from the relevant experimental data. To this target, Eq. (4) is fitted to the experimental surface C(x,t) generated by all the data collected at 8, 16, 32 and 72 h. This fitting leads to the conclusion that h is zero and the theophylline diffusion coefficient Dg inside the gel is equal to (3.6060.18)3 10 26 cm 2 / s (F 51047). The fact that h is equal to zero is in agreement with other previous results [13]: in fact, although it is highly probable that a stagnant layer may soon form between the gel and the release

Boundary conditions: ≠C ] 5 0 x 5 0 ≠x

(3)

C 5 0 x 5 L

(39)

The solution of Eq. (1), which satisfies all the conditions set by Eq. (2), Eq. (29), Eq. (3) and Eq. (39) is the following: C ] 5 C0

O

n5`

n50

S

2 l 2n tDg x Bn cos ln ] exp ]]] L L2

S D

D

(4)

in which:

p ln 5 ](1 1 2n) 2

(5)

Fig. 2. Experimental drug concentration profiles after t58 h. (s) Test 1, (h) test 2, (D) test 3.

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Fig. 3. Experimental drug concentration profiles after t516 h. (s) Test 4, (h) test 5, (D) test 6.

Fig. 5. Experimental drug concentration profiles after t572 h. (s) Test 10, (h) test 11, (D) test 12.

environment, the considerable gel thickness adopted makes h negligible. Fig. 6 shows the comparison between the experimental surface and the calculated one. As it can be seen, in all the surface domain the agreement between the experimental data and the model prevision is fairly poor. In order to check the reliability of the value found for Dg , Eq. (6) is fitted to the theophylline released data after 8, 16, 32 and 72 h (averaged over the three

repeated experiments), being h and Dg the two fitting parameters. The results of this fitting show again that h is equal to zero, while Dg is equal to (5.3260.18)310 26 cm 2 / s (F 5532). Fig. 7 shows the good agreement thus obtained between the experimental data and the model prevision. Due to the more complicated experimental procedure requested to collect the C(x,t) data with respect to that employed to get the amount of

Fig. 4. Experimental drug concentration profiles after t532 h. (s) Test 7, (h) test 8, (D) test 9.

Fig. 6. Comparison between all sets of experimental data C(x,t) (symbols) and the relevant fitting with Eq. (4) (lines).

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selected for data fitting (we can recall here that, for a data set made up of 10 experimental data points, F must be greater than 5.12 in order to be sure that the model in use is statistically correct [14]). Nevertheless, we must observe that there is a certain fluctuation in the estimated values of Dg . This leads us to conclude that the statistic criterion alone is not sufficient for a discrimination among all the possible, reliable values of Dg . We suggest then that another possible criterion for the selection of the most reliable value of Dg can be based on a simple mass balance between the active drug concentration in the gel and in the release environment, respectively. In the absence of any experimental error, the following material balance equation should be satisfied: Fig. 7. Comparison between the experimental data Mt (symbols) and the relevant fitting with Eq. (6) (line).

M0 2 Mrel 5 Mrem

theophylline released after 8, 16, 32 and 72 h, we reasonably assume that the more reliable value for Dg is that resulting from the data fitting with Eq. (6). On the basis of this consideration, we must consider carefully the C(x,t) data in order to suggest a possible criterion able to discriminate the best C(x,t) data set among the whole C(x,t) data collected. In order to match this aim, the fitting of Eq. (4) is performed on each data set collected at different time, being Dg the fitting parameter. Table 2 reports the results of the fitting procedure. From an inspection of Table 2 we can see that the corresponding values of F are rather good, and this seems to support the statistical validity of the model

where M0 is the mass of theophylline initially present in the gel, Mrel is the mass of drug released up to time t and Mrem is the mass of the drug still remaining in the gel matrix after the time t. In the mass balance given by Eq. (7), M0 can be easily calculated from C0 and the gel matrix volume V0 , Mrel can be determined from the volume of the reservoir and the relevant theophylline concentration determined via UV adsorption, and Mrem can be obtained by integrating the profile of C within the hydrogel matrix. From all the considerations developed above, it is evident that, among the three parameters in Eq. (7), the most affected by uncertainty is Mrem , since its value is obtained by integrating C over the gel

(7)

Table 2 Results of experimental data fitting with Eq. (4) Test

Calculated Dg (cm 2 / s) 310 6

Standard error 310 7

F-value

1–8 h 2–8 h 3–8 h 4–16 h 5–16 h 6–16 h 7–32 h 8–32 h 9–32 h 10–72 h 11–72 h 12–72 h

3.35 1.45 3.47 5.45 5.80 2.53 3.89 5.13 2.53 3.49 2.34 5.86

5.45 2.32 6.09 3.03 6.21 5.10 5.02 5.64 4.98 3.96 3.15 4.76

142.1 205.2 148.7 1059.8 222.1 68.8 127.6 199.0 86.2 129.7 63.5 143.5

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thickness. In some cases, the analysis of our experimental data sets leads to an overestimation of Mrem (i.e., Mrem .M0 2Mrel ); this necessarily implies an underestimation of Dg , since Mrem increases as Dg decreases. In order to evaluate which tests are not affected by these parameter underestimation problems we can observe that, by the mean value theorem, we must have: t

Mrel 5

≠C U E D U] ≠x g

x 5L

Sdt 5 SDg gav t

(8)

0

in which gav is the concentration gradient at the interface averaged over time. From Eq. (7) and Eq. (8) it follows that the diffusion coefficient can be evaluated in two different ways, on the basis of the available experimental data, that is: Mrel D g* 5 ]] Sgav t

(9)

or: M0 2 Mrem D ** 5 ]]]] g Sgav t

(99)

From Eq. (9) and Eq. (99) we can then write: D g* Mrel ]] 5 f 5 ]]]] D ** M g 0 2 Mrem

(10)

Accordingly, the best experimental tests will be those for which f is close to unity. Table 3 reports the values for f for all our experimental release data sets. We have now to define the f range inside which an experimental data set may be retained acceptable. In other words, we have to estimate the f standard error due to the uncertainty connected to the estimation of Mrel , M0 and Mrem . To this purpose, we suppose that the standard error associated to Mrel is very little compared to that associated to M0 and Mrem , so that Mrel does not contribute to the f standard error. The M0 standard error could also be calculated, but nothing can be said about the Mrem standard error. Anyway, because of the modality employed to calculate Mrem , we may be sure that its standard error will be higher (or at least equal to) than the M0 standard error sM0 . Due to this fact, we

Table 3 Values of the calculated Dg and the correction factor f for all experimental data sets Test

Calculated Dg (cm 2 / s)310 6

f (2)

1–8 h 2–8 h 3–8 h 4–16 h 5–16 h 6–16 h 7–32 h 8–32 h 9–32 h 10–72 h 11–72 h 12–72 h

3.35 1.45 3.47 5.45 5.80 2.53 3.89 5.13 2.53 3.49 2.34 5.86

1.28 2.27 1.41 1.08 0.95 1.30 1.23 0.96 1.64 1.36 1.68 1.20

assume that the standard error associated to the quantity (M0 2Mrem ) is equal to that connected to M0 . (We stress at this point the fact that all the hypotheses adopted lead to an underestimation of the f standard error). On the basis of such hypotheses and remembering the error propagation law [15], the f standard error sf may be calculated as follows: f sf 5 ]]]] sM 0 M0 2 Mrem

(11)

Observing that the mean value of sM 0 (averaged over the twelve estimations of sM 0 . ) is 213 mg, the mean value of (M0 2Mrem ) (averaged over the twelve data sets) is 1060 mg; thus, assuming f 51, we obtain sf 50.2. We may now conclude that, for our twelve experimental data sets, a given data set is acceptable if its f value ranges between 0.8,f ,1.2. From the above considerations we can conclude that the smaller sM 0 , the closer to unity the limits of the f range for acceptable experimental data sets. Therefore, knowing the mean value of (M0 2Mrem ) and sM 0 , one can estimate the sf value relative to his own experimental data sets. From Table 3 we can deduce that, for a reliable estimation of Dg , only tests 4, 5, 8 and 12 must be taken into account. The resultant value of Dg , calculated as the average value obtained from these four tests, is Dgav 5(5.5660.50)310 26 cm 2 / s; this value is statistically equal to that previously calcu-

I. Colombo et al. / Journal of Controlled Release 47 (1997) 305 – 314

Fig. 8. Comparison between three sets (4, 8, 12) of experimental data (symbols) and the relevant fitting with Eq. (4) (lines).

lated from the release data Mt vs. t (Dg 5 (5.3260.18)310 26 cm 2 / s) and is slightly lower than that of theophylline in pure water at 258C. Fig. 8 reports the comparison between the experimental data (4, 8 and 12) and the relevant fitting with Eq. (4) adopting h50 and Dg 55.32310 26 cm 2 / s. As can be seen from this figure, the agreement is fairly good.

5. Conclusions Even if the low cost apparatus needed and the easy realisation are very attractive, the inverse sectioning method seems to not yield a very reliable measure of the diffusion coefficient in a non self sustaining hydrogel matrix. This is essentially due to uncertainty in the determination of the model drug profile inside the matrix and to the non uniform distribution of the model drug inside the matrix at the beginning of the experiment. The attempt to determine with greater accuracy the drug profile by increasing the

313

number of the parts in which the matrix has to be cut failed because of the problem of the non uniform distribution of the drug inside the matrix at the beginning of the experiments. The results of the above mentioned problems reflect in a certain scattering of the diffusion coefficient values obtained from the fitting of the model to the experimental data. These considerations lead us to conclude that a calculation of Dg from Mt is more reliable than that performed by using the C(x,t) data. Anyway, when the Mt data are not available it is compulsory to resort the C(x,t) data in order to estimate Dg . This requires the definition of a criterion able to individuate the reliable experimental sets giving the correct values for the diffusion coefficient. The present work presents a criterion based on the mass balance calculated knowing the amount of released drug at the end of the experiment, the initial mass of the drug in the matrix and the remaining drug mass in the matrix at the end of the experiment. Of course, it does not require the knowledge of the drug mass released as a function of time. By means of such criterion it is possible to individuate the experimental data sets giving the correct value of the diffusion coefficient. In order to have a reliable value of Dg , for hydrogel polymeric matrices, the inverse sectioning method has to be coupled to a check criterion like that presented. The shown methodology is very important in those situations where the measurement of Mt vs. t is not feasible (for instance, when release environment alterations must be avoided) and hence, the inverse sectioning method is the only simple method at disposal. Beside that, the proposed methodology (inv. sect method and the check mass balance) allows to perform the experimental measures at only one fixed time and do not require other measures at different times. Indeed, the only requirement needed by the proposed method for the sf estimation is to have a sufficiently high number of experimental data sets (i.e. concentration profiles), regardless the fact that they were collected after different times or after one time only. In principle, one could perform all the concentration profile determinations for a given time only, provided the number of determinations is sufficiently high and the concentration profile is completely developed. The F criterion will individuate the statistically good sets.

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Accordingly, both very short and very long times are to be avoided.

6. List of symbols C C(x,t) Cp C0 Dg Dgav Dw D g* D g** gav L h Mrel Mrem M0 Mt M` Mw r Re S t t0 V0 l v x sf sM 0

Drug concentration at x position at time t Drug concentration profile in the hydrogel at time t Polymer concentration in the hydrogel Initial theophylline concentration in the hydrogel Drug diffusion coefficient in swollen hydrogel Average drug diffusion coefficient in the hydrogel Drug diffusion coefficient in water Drug diffusion coefficient (Eq. (9)) Drug diffusion coefficient (Eq. (11)) Drug concentration gradient at the interface averaged over time Sum of the thicknesses of the hydrogel and of the stagnant layer Stagnant layer thickness The same meaning as Mt Mass of drug still remaining in the hydrogel after the time t Mass of the drug initially present in the hydrogel Mass of drug released until time t Mass of drug released after an infinite time Molecular weight Syringe radius Reynolds number Interface area of the hydrogel Time Initial time Hydrogel volume Wavelength Stirring speed Abscissa f standard error M0 standard error

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