Evaluation of a Tube Method for Determining Diffusion Coefficients for Sparingly Soluble Drugs

PAULA Jo MEYER STOUT*, NAHLA KHOURY' JOHN MAUGER", AND STEPHEN HOWARD** Received February 1 1, 1985, from West Virginia Universiiy, School of Pharmacy, Morganfown, WV 26506. September 4, 1985.

Accepted for publication Present address: *Ortho Pharmaceuflcel Corp., Raritan, NJ 08869,

___-- ________ Abstract Evaluation of a non-steady state method using glass tubes for the determination of diffusion coefficients is the purpose of this study. Unlike capillaries, glass tubes accommodate a larger volume of solution, facilitating assay procedures. Tubes are more susceptible to convection than are capillaries, but this effect is anticipated and accounted for in experimental design and data treatment. Glass tubes, 66 or 90 mm in length and 2 mm outer diameter, were siliconized and then filled with aqueous drug solution and placed in a jacketed flask containing gently stirred solvent at 25°C. Diffusion experiments were run from 140 to 168 hours. At the end of this time period, the tubes were removed from the flask, placed in an ultrasonic vibrator for one minute, and their contents assayed spectrophotometrically. Data collected using potassium chlo- ride as the diffusant showed little tube-to-tube variability, demonstrating the precision of the tube method, while diffusion coefficients determined for benzoic acid and paminobenzoic acid using the tube method tested the accuracy of the method by comparing reasonably well with values obtained using standard methods such as the rotating disk, free boundary, and membrane cell. Experiments done with either hydrocorti- sone or sulfisoxazole as the diffusant demonstrated the appropriateness of the tube method for the study of the diffusion of sparingly soluble pharmaceutical solutes.

Diffusion coefficients are important data for assessment of mass transfer studies involving drug solutes. Several experi- mental methods for determining diffusion coefficients have been reported,'-' with each method having particular attri- butes and limitations. Pharmaceutical scientists have com- monly used quasi-steady state methods, as exemplified by the diaphragm cell method reported by Goldberg and Higu- chi.' An alternative non-steady state method using capillary tubes was introduced by Anderson and Saddington4 and used for systems of pharmaceutical interest by Farng and Nelson.6 Limitations of the capillary method have been discussed by Fawcett and Caton6 who extended the original capillary method to the use of glass tubes which accommodate a larger volume and facilitate assay procedures. Fawcett and Caton studied and verified the tube method using either electro- chemical solutes in aqueous salt solutions or organic solutes in organic solvents, such as acetonitrile. The purpose of this report is to extend and verify the use of the tube method for solute/solvent systems of pharmaceutical interest.

Theoretical Section Goldberg and Higuchi' described a diaphragm cell method

employing a silver membrane filter, which proved to be useful for determining the diffusion coefficient of pharmaceu- tical solutes dissolved in aqueous systems. This method requires stable hydrodynamic conditions via experimentally controlled stirring conditions. Following a calibration step with a solute having a known diffusion coefficient, which provides a cell constant for the device, it was demonstrated that this method provides both accurate and precise diffusion coefficient values. A potential limitation of this method is related to the dilution process which occurs as the solute transfers from the donor compartment to the receptor com- partment. If samples are taken from the dilute receptor

compartment, assay difficulties can be anticipated for poorly soluble drug solutes.

The capillary method for determining diffusion coefficients was originally presented by Anderson and Saddington4 and has been used extensively.- This experimental technique involves the use of uniform glass capillaries, with the lower end sealed and filled with a solution of radioactive tracer. The filled capillaries are placed vertically in a holder which is then placed in a large stirred tracer sink having the same chemical composition as the solution inside the capillary. Radioactive tracer is allowed to diffuse outwardly and the concentration of the tracer is maintained a t zero at the end of the capillary by convection currents in the stirred bath. Each capillary of known length, L, can be removed from the sink solution a t time, t, and the average concentration, C A ~ ~ , can be determined by radioassay.

Fawcett and Caton6 noted limitations of the capillary method. First, each capillary holds a small volume and because of the small amount of solute diffused, a sensitive analytical method such as radioassay is required. Another limitation is connected with the loss of diffusant from the capillary by mechanical convection, which was discussed by Mills and Adamson.9

Fawcett and Caton6 proposed using glass tubes instead of capillaries in order to overcome these limitations. The vol- ume of solution with the capillary method is limited (< 0.02 mL) when compared with the tube method which yields volumes of 0.1 to 0.2 mL. For poorly soluble drug solutes, assay problems related to volume limitations are alleviated if samples from several tubes are pooled. Loss of diffusant from the tube due to convection is anticipated and accounted for via experimental technique and associated data treatment. Since the solute concentration remains high within the tubes, the use of the tube method offers an assay advantage over the diaphragm cell method.

A solution of Ficks law applicable to the tube method has been given by Fawcett and Caton6 for a tube having length L (cm) and unit cross sectional area. The initial drug concen- tration in the tube is Co (g/mL), and the tube is submerged in a solvent system large enough to maintain sink conditions.

Ficks second law, assuming that the diffusion coefficient, D (cm2/s), is independent of concentration, is given by:

- ac = D - a2C at ax2

and may be solved with the following boundary and initial conditions. (Boundary Conditions at x = L, aClax = 0 and at x = 0, C = 0; Initial Conditions at t = 0, C = Co a t 0 < x 5 L)

The solution in terms of the average concentration, CAVE (g/mW, in the tube a t time, t (s), is:

CAVE = 8/7? 2 1/(2n + 1)2 x CO n=o

OO22-3549/86/0 1 OO-O065$0 1.00/0 Q 1986, American Pharmaceutical Association

Journal of Pharmaceutical Sciences / 65 Vol. 75, No. 1, January 1986

For values of (DtlL') less than 0.2, eq. 2 simplifies to: n

(3)

Since CAVE can be experimentally determined and Co, L, and t are known, the diffusion coefficient, D, can be directly determined from:

(4)

Since no calibration with a solute of known diffusion coefficient is necessary, this method provides for direct calcu- lation of the diffusion coefficient from experimental data.

Experimental Section Diffusion coefficients were determined using a modification of the

non-steady state method described by Fawcett and Caton.6 Glass tubes (capillary tubes, 1.6-1.8 x 66 or 90 mm, Kimox-51, Kimble Products) of known length and 2 mm outer diameter were first siliconized using the following procedure. The tubes were inverted in a beaker containing a 1% solution of methyltrichlorosilane in tolu- ene. The beaker was placed in an ultrasonic vibrator (Bransonic 220 Ultrasound, Branson Cleaning Equipment Co., Shelton, CT) for five min, then the tubes were removed, drained, and allowed to dry for 24 h. Siliconized tubes were found to improve reproducibility of the data, possibly by preventing leaching from or adsorption to the glass.10

The tubes were filled with an aqueous drug solution of concentra- tion Co using a 2 mL syringe and a needle (B-D Yale 2 mL Glass Insulin Syringe with 25-gauge, 8.75-cm Needle, Becton, Dickinson, and Co.). Care was taken to fill the tubes without introducing air bubbles. Depending on .the diffusant's degree of solubility, the number of tubes used varied. The tubes were secured in a vertical position within the channels of a standard USP disintegration basket (Standard USP Disintegration Basket with Plunger Accesso- ries, Van-Kel Industries, Inc., Edison, NJ) by placement in the holes of the Plexiglas discs of the basket's plungers. With the mouth of each tube held above the top of the disintegration basket, the tubes were gently lowered into a mildly stirred jacketed 1500 mL flask (Virtis 1500 mL Glass Jacketed Flask, The Virtis Co., Gardiner, NY). The flask contained 1400 mL of distilled water held constant a t 25°C through the use of a circulatory water bath (MGW Lauda Thermostat RM3, Brinkmann, West Germany). Prior to lowering the tubes into the solvent, a drop of drug solution was added to the top of each tube to ensure complete filling of the tube. The tubes were lowered to the same level, approximately 5 cm below the surface of the eolvent sink, for each run. Constant stirring at 240 rpm was achieved throughout the study by the use of a Teflon stir bar (Mono- Mold Magnetic Stir Bar with Spinning Ring, Cole-Parmer Instru- ment Co., Chicago, Illinois) and a magnetic stirrer (Nuova I1 Stirrer, Sybron-Thermolyne, Cole-Parmer Instrument Co., Chicago, 1L). Since the study ran for an extended period of time, a Plexiglas cover was placed on top of the flask to prevent evaporation.

At a predetermined time (140 to 168 h), the diffusion run was terminated. The disintegration basket was carefully removed from the stirred solvent, was placed in the ultrasonic vibrator for one min, and the contents of each tube were subsequently removed using a syringe and needle. In most cases, several tubes were pooled to give one sample prior to assay. Each sample was immediately assayed spectrophotometrically (Perkin-Elmer, Double Beam Spectropho- tometer-Coleman 124, Hitachi, Ltd., Tokyo, Japan) for the concen- tration of drug remaining, CAVE.

It was expected that some drug solution would be lost initially during the submersion process due to convection. This loss must be accounted for since it is a physical, not a diffusional, process. The modification of two parameters, Co and L, in eq. 4 allows for this convective effect.

A submersion test was carried out for the tube set using the exact procedure just described for the diffusion run. In this case, however,

the submersion time was 30 s. Previous visualization studies using a dye solution established that a stable boundary was formed within this time frame. After submersion, the tubes were removed, and their contents were thoroughly mixed and assayed spectrophotomet- rically. With the assumption of no mixing during the formation of the stable boundary, it can be shown that after boundary formation the concentration of the drug solution in the tube remains the same (Co). Upon mixing, the drug solution is diluted by the fresh solvent present in the tube above the stable boundary. This fresh solvent replaces the volume of drug solution carried away by convection. Thus, the assay value CsUR (g/mL), the concentration of the solute following submersion, accounts for the diluting of the drug solution and replaces Co in eq. 4.

Because of convection, an effective tube length, LEFF, is a better measure of the distance traveled by the diffusing molecule than is the geometric length, L. These two lengths are related by:

(5) LEFF = L - AL where AL = length lost due to the convective low of drug solution during boundary formation (cm).

While AL can physically be measured, the relationship of length to concentration allows the effective tube length to be calculated directly:

l A

Since the geometric length of the tubes was known and the ratio of the solute concentrations, Csus/Co, could be calculated following assay, an effective tube length was determined for each diffusion study. The value of Csus/Co experimentally ranged from 0.980 to 0.985. That is, 2% of the solution in the tube was lost to convection. Diffusion coefficients were then calculated based on eq. 4 which took the final form.

(7)

Results Accuracy of the Tube Method-Average values (n = 5 or

6 tubes) and their associated standard deviations are given in Table I for experimentally determined diffusion coefficients. With benzoic acid, the diffusion coefficient was calculated using tubes 6.6 cm in length. An average experimental value of CAVE/CSUB of 0.6191 was determined after a diffusion time of 141.5 h. The data for benzoic acid show that the tube method yields a value somewhat lower than that from the membrane cell and consistent with that from the rotating disk.

The diffusant p-aminobenzoic acid provides an opportunity to test the sensitivity of the tube method for a larger hydrodynamic particle where a smaller diffusion coefficient is expected. The experimental data when compared with the

Table 1-Aqueous Diff usion Coeff icients Determined at 25°C by the Tube Method Compared with Similar Data Obtained by Other Methods

D x lo6, D x lo6, D x lo8, cm2/s cm2/s cm2/s Diffusant

~~~~~ ~ ~

Benzoic Acid 9.75 t 0.643' l l . l O b 9.40'

pAminobenzoic Acid 7.34 2 0.520a 8.43d 8.47' (0.01 M)

(0.01 66 M)

' Tube method. Ref. 1 1, membrane cell method. ' Ref. 12, rotating disk method. dRef. 2, free boundary method. "Ref. 3, rotating disk method at 30°C.

06 / Journal of Pharmaceutical Sciences Vol. 75, No. 1, January 1986

Table 11-Data Indicating the Preclslon of the lube Method Uslng 1.31 M KCL as the Diffusant at 25°C

Tube CAvE/Csue as Determined from Number Refractive Indices

1 0.4800 2 0.4800 3 0.4688 4 0.4688 5 0.4800

ave. = 0.4755 * 0.006134 D (Tube Method) = 1.85 x D (Membrane CeNa = 1.87 x

* 4.38 X lo-'

'Ref. 13

Table Ill-Data lndlcatlng the Usefulness of the lube Method for Determlnlng Dlffuslon Coefficients of Pharmaceutlcal Solutes

CAVE D x lo6' D x losb Subset Csm crn2/s cm2/s

Sulfisoxazole I 0.649 8.27 11 0.663 7.63 111 0.670 7.31 IV 0.654 8.04

Average = 0.65ge 7.81 ' 5.75-8.62

Hydrocortisoned I 0.792 4.37 II 0.798 4.12

Average = 0.795 4.24 5.05-7.58

'Tube method. bRef. 14, partial molal volume. '1.87 x M, 3 M, 17 tubesfsubset. eSD = k0.00934. ' SD tubesfsubset. *4.96 x

= 20.427.

value for benzoic acid is in keeping with this expectation. The experimental conditions and tube lengths were identical with those used for benzoic acid, with an average CAVEICSUB value of 0.6697. For p-aminobenzoic acid the tube method yields a smaller value for the diffusion coefficient when compared with either a free boundary method or the rotating disk.

These data show reasonable agreement with previously determined values, and indicate that the tube method pro- vides accurate data for typical pharmaceutical solutes.

Precision of the Tube Method-An error analysis of eq. 7 will show that the diffusion coefficient is very sensitive to small changes in the ratio CAVE/CSUB. For example, with LEFF = 7 cm, t = 6 x lo5 s, and D = 1 x cm2/s, a 2% change in CAa/CSUB will result in a 6% change in D. Therefore, the precision of the method is of interest.

Aqueous solutions of 1.31 M KCl were allowed to diffuse from 6.6-cm tubes into a stirred aqueous sink for 141.5 h, and the contents were assayed using a refractometer (Bausch & Lamb Refractometer, Rochester, NY). This assay procedure is direct, not requiring diluting or other handling of the sample prior to assay nor the pooling of tube contents since

the contents of each tube are sufficient to allow for quantita- tive assessment. Thus, with these intermediate steps elimi- nated, less experimental error is introduced via this assay method.

The values of CAVE/CSuB listed in Table I1 show small variation from tube to tube, and the standard deviations associated with the average values for CAVEiCsUB and the diffusion coefficient are relatively small, indicating that the intrinsic precision of the tube method is high. The accuracy of the potassium chloride diffusion coefficient is excellent when compared with previously published data.13 Due to the sensi- tivity of the diffusion coefficient to small changes in the ratio CAVEICSUB, however, a high level of analytical precision is needed so as not to decrease both the precision and accuracy of the experimentally determined diffusion coefficient.

Use of the Tube Method for Pharmaceutical Solutes- Hydrocortisone and sulfisoxazole, sparingly soluble drugs, were chosen as prototype solutes. Aqueous solutions of each were allowed to diffuse from 9.0 cm tubes for 168 h into a stirred aqueous sink. Due to the drugs' low solubility, tubes were pooled at the termination of each study for assay purposes, as detailed in Table 111.

For sulfisoxazole, an average ratio of CAVEICSUB of 0.6590 was obtained along with a small standard deviation, indica- tive of tube subset to subset consistency. The experimentally determined diffusion coefficient for this diffusant falls well within the range of values calculated using the relationship of diffusivity to drug molecule partial molal volume given by Flynn et al.14 At 25"C, the partial molal volume of sulfisoxa- zole is 189.09 mL/rn01.~~

Since hydrocortisone is highly insoluble, tube contents were pooled into two subsets of 17 tubes each for assay purposes. The tube method, in this case, yields a value below the range of diffusion coefficients determined via the partial molal volume relationship, but of the same order of magni- tude. Hydrocortisone's partial molal volume was calculated as 278.82 mL/mol. These data suggest that the extension of the use of the tube method for pharmaceutical solute/solvent systems is appropriate.

References and Notes 1. Goldberg, A. H.; Higuchi, W. I. J . Pharm. Sci. 1968, 57, 1583-

2. Longsworth, L. J . Am. Chem. Soc. 1953, 75, 57055709. 3. Nogami, H.; Nagai, T.; Ito, K. Chem. Pharm. Bull. 1966,14,351-

4. Anderson, J. S.; Saddington, K. J . Chem. SOC. 1949, S3814386. 5. Farng, K. F.; Nelson, K. G. J . Phurm. Sci. 1973,62, 1435-1438. 6. Fawcett, N. C.; Caton, R. D. Anal. Chem. 1976,48,600-604. 7. Bacon, J.; Adams, R. N. Anal. Chem. 1970,42,524-525. 8. Miller, T. A.; Lamb, B.; Prater, K.; Lee, J. K.; Adams R. N. Anal.

9. Mills, R.; Adamson, A. W. J . Am. Chem. SOC. 1955, 77, 3454- 10. Thakker, K. D.; Higuchi, T.; Stemson, L. A. J . Pharm. Sci. 1979,

11. King, C.; Brodie, S. J . Am. Chem. SOC. 1937, 59, 1375-1379. 12. Mooney, K., personal communication. 13. Stakes, R. J . Am. Chem. SOC. 1950, 72, 2234-2247. 14. Flynn, G. L.; Yalkowsky, S. H.; Roseman, T. J. J . Pharm. Sci.

15. Sunwoo, C.; Eisen, H. J . Pharm. Sci. 1971,60,238-244.

1585.

354.

Chem. 1964,36,418-420.

3458.

68, 93-95.

1974,63, 479-510.

Journal of Pharmaceutical Sciences / 67 Vol. 75, No. 1, January 1986