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Transference: a comprehensive parameter governing permeation of solutes through membranes
Journal of Membrane Science, 1 (1976) 3-16 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
TRANSFERENCE: A COMPREHENSIVE PARAMETER PERMEATION OF SOLUTES THROUGH MEMBRANES
FELIX
THEEUWES’,
ROBERT
ALZA Corporation, Palo Alto, (Received May 30, 1975)
GOVERNING
M. GALE and RICHARD W. BAKER** Calif. 94304
(U.S.A.)
summary Experimental studies of the diffusion of moderate molecular weight drugs through synthetic polymeric membranes has led us to define a new parameter to characterize solute permeation. The accepted parameter which describes trans-membrane flux is permeability, P, defined as the product of the flux times the membrane thickness divided by the concentration difference across the membrane in the external solution. The parameter described here is called transference, r?; and is defined as the mass of solute transferred from a saturated solution, under perfect sink conditions, across the membrane per unit time and area, multiplied by the thickness of the membrane. Regardless of whether there is solvent-membrane interaction or not, the permeability coefficient, P, for a given solute and membrane differs from one solvent to another. In contrast, transference is independent of solvent, unless there is solvent-membrane interaction, in which case the inconsistency of transference for a particular solvent signifies that the membrane is modified by the presence of that solvent or solvent--solute combination, It is for these reasons that transference is a more comprehensive parameter than permeability, and is basic at the same time to solute-membrane permeation. When transference is introduced into Fick’s law it is immediately obvious that the flux of solute through the membrane, at equal fraction of saturation in any donor solvent, is equal and independent of the donor solvent in the absence of solvent-membrane interaction, when the solute has a constant solvent-membrane partition coefficient. Two examples of the use of the transference parameter are provided.
Introduction
The permeability coefficient, defined as the product of the flux through the membrane times the membrane thickness divided by the concentration difference across the membrane in the external solution, is commonly used to describe membrane permeation. When fluxes of the same drug through different membranes are studied from a common donor solvent, the permeability coefficients can conveniently be compared to give information as to which membrane is more permeable to the drug. Most drug permeability measurements *To whom inquiries should be directed. Present address: ALZA Research, 2631 Hanover Street, Palo Alto, Calif. 94304, U.S.A. **Present address: BEND Research, Inc., 64550 Research Road, Bend, Oreg. 97701, U.S.A.
reported in the literature are carried out from aqueous solutions as donor solvents and such data can unambiguously be interrelated. The development of controlled drug delivery systems, using the principle of solution-diffusion [ 1,2] , has stimulated permeation studies from a variety of nonaqueous solutions. In such systems, the permeability coefficient is insufficiently descriptive and a new parameter, transference, F, is introduced as defined below. The permeability coefficient is not a characteristic number for the solute (drug)-membrane system but differs depending on each donor solvent. The transference is an intensive property of the solute-membrane system which is independent of the solvent in the absence of solvent-membrane interaction. The term interaction is defined as the process whereby the membrane is modified by the presence of the solvent such that the membrane appears to the solute as a different medium when different solvents are present. Theory Fick’s law is generally written as: 1 dm -=-#, A dt where dm/dt is the mass transported per unit time through an area A caused’ by the concentration gradient dC!/dscwith D as the solute diffusion coefficient. Considering solute transport across membranes the solution-diffusion model is .usually observed as illustrated in Fig. 1. At each face of the membrane, equilibrium with the respective bulk solution is assumed, and a gradient in activity and concentration exists within the membrane. Thus for the left side of the membrane: a(O)
=
am(O),
where a(o) is the solute activity in the left solution phase and amto) is the activity in the membrane at the interface x = 0. A similar expression can be written for the right hand interface. Substituting for the activity the product of an activity coefficient (y) and the concentration (C), gives eqn. (3) 7( O)C( 0) = cm(
o)Ym(O)-
Equation (3) may also be written as:
cm(o) Y(o) C(o) Ym(0) -=-
=K
’
where K is the solute distribution coefficient between solvent and membrane. Equation (4) thus shows the link between the concentration of solute in the bulk solution and the concentration in the adjacent membrane phase. The partition coefficient K is constant for regular solutions and eqn. (4) is an analogue expression to Henry’s law. When the solute is in equilibrium at the
ACTIVITY CONCENTRATED SOLUTE SOLUTION
PROFILE
MEMBRANE
CONCENTRATION
I
DILUTE SOLUTE SOLUTION
PROFILES
C(d
C(P) I
I
x=0
x=P
Fig. 1. Solute activity and concentration membrane into the acceptor side.
*
X
profiles from the donor solvent acrw
the
solvent-membrane interface a step in concentration occurs depending on the value of K, as shown in Fig. 1, from the donor solvent into the membrane phase. At equilibrium there is never a discontinuity in the activity profile as expressed by eqn. (2). If, for simplicity, the diffusion coefficient is assumed constant, eqn. (1) can be integrated over the membrane thickness 2 to give 1 dm --= A dt
D[Crn(o)-
1
Cm(l)]_
DAcrn 1
*
(5)
If the solute is dissolved in the same solvent at both membrane surfaces, and if K is independent of concentration, eqn. (4) can be substituted into eqn. (5) and the flux is described as a function of the solute concentration in the external solvent by eqn. (6)
16) Here AC is the solute concentration difference in the solvent external to the membrane. Frequently, the product DK is termed the solute permeability, P, and hence
6
1 dm PAC -=Adt I’
(7)
However, the parameter, permeability, has limited generality, for it combines both membrane and solvent characteristics. Thus, when permeation experiments are performed in different solvent media with the same membrane, different permeabilities result. D will remain unchanged but K will vary to reflect the altered solubihty in the liquid medium. In addition, a common form of permeability experiment is to maintain a negligibly low concentration on one side of the membrane and a fixed concentration on the other side simply by maintaining the solution saturated with excess solute. This method of maintaining a constant driving force is particularly useful with solutes sparingly soluble in the external solvent [3-6]. Unfortunately, solubilities of sparingly soluble solutes are often imprecisely known and are difficult to measure. The result is that widely different values of drug permeabihties through the same membrane from the same solvent systems are reported in the literature. For these reasons, we prefer to characterize membranes by a term, F, defined by eqn. (8), as the transference
sat
=J,,l=PC”=DC&.
The transference is the product of the membrane thickness and the solute fh=, Jut, through the membrane from a saturated solution to the pure solvent. It is also given by the product of the permeability coefficient and the solubility in the donor solvent, or the product of the diffusion coefficient of the solute in the membrane and the solubility of the solute in the membrane. The transference is invariant with the selection of the solvent for solvents which do not solvate or otherwide modify the membrane. The quantity JZ has been referred to as the “normalized flux” [ 71. The transference, T, is the normalized flux at the specified boundary conditions: saturated solution and zero concentration. Transference completely describes the permeation process, and the drug solubility in the donor solvent is not required to calculate fluxes through the membrane. The transference is a reproducible intensive property of the solutemembrane system, independent of whether solvent-membrane interaction occurs or whether the solvent-membrane partition coefficient for the solute is constant and is defined for a specific crystalline state. In the absence of solvent-membrane interaction, ? is a constant of the solute-membrane system, independent of the solvent system. The finding that ?1is different for different solvent systems will signify solvent-membrane interactions and can be used as a tool for such investigations. When the solute is in equilibrium at saturation with a membrane (m) in different solvents (1, 2, . . . II), eqn. (9) holds in the absence of solvent-membrane interaction: Cfm=K,G=K&=....~,Cs,,
(9)
7
which is the reason for the constancy of ?; for different solvent-membrane systems. When, in addition, the solvent-membrane partition coefficients K1, K, are constant and independent of concentration, Fick’s law can be K,,... combined with eqns. (7) and (8) to arrive at eqn. (10) for solute i 1 dmj
Tl ACi
A dt
E q’
__=Ji=_-
(10)
It follows that when the partition coefficient Ki is independent of concentration, the solute flux through the membrane is proportional to the fraction of saturation in the donor solvent. Experimental Permeabilities were measured with a conventional two-chamber cell. The upstream chamber of the cell was maintained at a constant concentration, while the downstream chamber was maintained at a negligible concentration. Both chambers were vigorously stirred, and the rate of appearance of the drug on the downstream side was measured spectrophotometrically by recirculating through a flow-through cell by means of a pump. Membranes 50 to 150 microns thick were placed between wire screens to prevent deformation. Permeabilities were reproducible to f 15 %. Drug sorptions were measured by placing weighed samples of membrane material in a solution of known solute concentration until equilibrium had been reached. The samples were then removed, rapidly rinsed, patted free of surface moisture, and weighed. The samples were then desorbed completely in a large volume of water and the mass of solute desorbed was measured with the UV. With the solute content and the total weight gain of the film known, the water and drug sorption could be calculated. Subsequent experiments showed that the samples had reached equilibrium before removal from the solution and that only a negligible quantity of solute was lost during the rinse step. Both permeability and sorption experiments were carried out at 37°C. Membranes were prepared from three commercially available polymers. Two of these,Elvax 40 from DuPont and Levaprene 400 from Bayer, are random copolymers of ethylene/vinyl acetate containing ~40 wt.% vinyl acetate. The third polymer, Elvamide 8063, is an alcohol-soluble nylon terpolymer supplied by DuPont. Ethylene/vinyl acetate copolymer membranes were solution cast while Elvamide 8063 membranes were melt pressed. The Elvamide membranes were thoroughly washed in water prior to use to remove a small quantity of slowly released UV absorbing impurity. Hydrocortisone alcohol and progesterone are commercially available as pure compounds. Pilocarpine base was prepared from the commercially available nitrate by conventional methods. The supersaturated solutions were prepared by dissolving hydrocortisone alcohol in acetone-water solutions. Acetone is a volatile, good solvent for
8
hydrocortisone while water is a relatively poor, non-volatile solvent. The acetone was flashed off under vacuum with some heat after which the solution was cooled, leaving a supersaturated aqueous hydrocortisone alcohol solution. The concentration of drug was obtained by diluting an aliquot of solution and measuring the UV absorption. The UV also showed the absence of acetone. Solutions less than four times saturated were quite stable for several hours, although more concentrated solutions precipitated spontaneously. This method of preparing supersaturated solutions has been used [S] as a method of increasing the rate of diffusion of drugs through the skin. Higuchi has prepared supersaturated solutions in a variety of ways, and has also measured the increased permeability of microporous membranes to drugs from this type of solution [ 91. Results and discussion Diffusion up to unit activity of a solute with constant solvent-membrane partition coefficient AS shown previously, eqns. (6) and (7) are subject to the misinterpretation that the flux through a membrane is proportional only to the concentration difference between the solutions on either side of the membrane. This is untrue if the solvent in which the solute is dissolved is changed, for, with different solvents, the flux from similar concentration differences can be very different. For example, Fig. 2 shows the results of permeation experiments with the solute, progesterone, dissolved in three different solvents. The downstream concentration was always maintained at a negligibly low value, while the upstream concentration was varied up to the saturation concentration. The solubility of the drug in the solvents varied from 13.6 ppm for water, and 700 ppm for Dow 360 silicone oil, to 20,000 ppm in polyethylene glycol. The permeability coefficient _Pthus varied by more than lOOO-fold. However, in all cases, the transference T was 1.28 X lo-” g cm/cm2 sec. Figure 2 demonstrates the distinction between the true driving force, i.e. the chemical potential gradient, and the apparent driving force, i.e. the concentration gradient. At saturation the progesterone in all three solutions is at the same chemical potential (unit activity) and hence the driving force and the resulting membrane fluxes are equal despite the wide range of the concentration differences. The indication that the transference was a constant of the drug-membrane system independent of the solvent, and further that the fluxes in each case are linearly proportional to drug concentration in the donor solvent indicates that eqn. (10) describes these experiments. It therefore follows that the results shown in Fig. 2 can all be normalized by plotting the normalized flux J1 against fractional saturation AC/f?, rather than against concentration C. Figure 3 shows this plot. Figure 4 shows the results of an experiment illustrating the same idea. A solution of ~2000 pg/ml progesterone in polyethylene glycol600 (in which
2.0
H20
1.5 -
1.0 -
10 PROGESTERONE
-0
100
200
PROGESTERONE
2.0 c
CONCENTRATION
300
400
500
CONCENTRATION
12
13.5
(pg/m4?)
600
700
(pg/mQ)
PEG 600
PROGESTERONE
CONCENTRATION
Fig. 2. Progesterone flux across an ethylene-vinyl function of concentration from three solvents.
(pg/mP)
acetate copolymer
membrane as a
10
1.5
1.25 -
FRACTIONAL
SATURATION
(AC&)
Fig. 3. Progesterone flux across an ethylene-vinyl acetate copolymer membrane as a function of the fraction of saturation from three different solvents. Same symbols as in Fig. 2.
(-
POLYETHYLENEGLYCOL
WATER (SOL.
Cs = 13.5pg/mP.)
(SOL.
CS = 20,00O~g/mP)
MEMBRANE \ 1
C = 2,00Opg/mV
C = lOpg/mP
i
B
INITIAL
CONDITION
C = 1.2pg/mY.
C = 2,009pg/mP
EQUILIBRIUM
s
00
CONDITION
c
-\
I 50
I 100
I 150
I 200 TIME
I 250
I 300
A ”
I
I
950
1000
(minutes)
Fig. 4. Progesterone flux across an ethylene-vinyl acetate copolymer membrane from a water solution at 10 fig/ml to a polyethylene glycol solution at 2000 rg/ml.
II
11
progesterone has a solubility of =ZO,OOOpg/ml) was placed on one side of an Elvax 40 membrane while a solution of 10 pg/ml progesterone dissolved in water (in which progesterone has a solubility of 13.5 pg/ml) was placed on the other. Progesterone diffused from the 10 fig/ml solution to the 2000 pg/ ml solution, and the concentration of progesterone in the aqueous solution fell, finally reaching 1.2 pg/ml. At 10 pg/ml in the aqueous solution, progesterone was at 0.75 of saturation, while at 2000 pg/ml in the polyethylene glycol solution, the steroid was at only 0.1 of saturation. Diffusion thus occurred from the high to the low activity solution until the two solutions were equi-active, ie. until the aqueous progesterone solution had dropped to -0.1 saturation (1.2 pg/ml in water). Diffusion above unit activity of a solute with constant solvent-membrane partition coefficient The correlation between flux and degree of saturation, eqn. (lo), has been found to hold above the normal saturation concentration of a drug. Although a saturated solution is the highest activity driving force normally used, some supersaturated solutions are fairly stable and can be used as hyperactive solutions. Figure 5 shows the results of an experiment with those solutions. Initially a normal saturated hydrocortisone alcohol solution containing a small excess of solid solute was placed on one side of the membrane and water on the other. The flux of solute was measured by periodically measuring the concentration on the dilute side, which never contained more than a few fig/ ml of hydrocortisone. After 354 minutes, the solution was removed, the cell quickly rinsed, and a previously prepared supersaturated solution of hydrocortisone alcohol added. The flux of drug was again monitored until the 560th minute, when the supersaturated solution was nucleated with a crystal of hydrocortisone. The solution immediately turned cloudy and the flux through the membrane dropped to the normal saturated solution value. The results shown in Fig. 5 represent to our knowledge the first example in which a supersaturated solution has been used to induce supersaturation in the membrane phase. As with normal subsaturated solutions, the fluxes obtained were directly proportional to the degree of saturation, as shown in Fig. 6. This plot suggests that the activity coefficient (y) is constant even in a 4 times supersaturated solution. Permeation of a solute with concentration-dependent solvent-membrane partition coefficient In a final example, the permeation of pilocarpine through a nylon membrane from water as the solvent is discussed. Pilocarpine is a liquid at 37”C, and completely miscible with water, such that the coefficient ?; cannot be defined from an experiment at 37°C. The permeation results are plotted in Fig. 7, representing a case where the flux is not proportional to the drug concentration on the upstream side (the downstream solution was water). Deviations from linearity are well known and reflect a variable diffusion coefficient
12
3
NORMAL
SOLUTION
EXCHANGED
FOR SUPERSATURATED (2.9 X SATURATION)
f” i
a E
SOLUTION
NORMAL SATURATED SOLUTION
SUPERSATURATED NUCLEATED WITH HYDROCORTISONE
SOLUTION SOL10
100
0
TIME
(MINUTES)
Fig. 5. Accumulative amount of hydrocortisone which permeated an ethylene-vinyl acetate copolymer membrane from saturated and supersaturated solutions. 1.0I-
2.6 X SATURATION
1 X SA
2
0.25
HYDROCORTISONE
CONCENTRATION
Fig. 6. Normalized flux of hydrocortisone tration above saturation.
(pg/mP)
alcohol as a function of hydrocortisone
concen-
13
20 COMPOSITION
40
60
(W% PILOCARPINE
60
0
IN Hz01
Fig. 7. Normalized flux of pilocarpine through a nylon membrane as a function of pilocarpine concentration in water as the donor solvent.
or partition coefficient. Figure 7 is to our knowledge the first reported example of a flux versus concentration plot exhibiting a maximum. The reason for this behavior is apparent from the sorption isotherms for pilocarpine and water shown in Fig. 8. The plot shows a maximum in pilocarpine and water uptake by the membrane as a function of the external solution concentration, which can be explained either by a maximum in water and pilocarpine activity in the presence of each other, or a synergistic swelling effect of the membrane caused by the drug and water at about 40% pilocarpine 60% water concentration. Such membrane swelling behavior is well known and several examples have been given in the literature [ 10). The diffusion coefficient was calculated from the flux data presented in Fig. 7, and the membrane sorption data
16
6
)
0 100
I
I
I
I
20 80
40 60
60 40
80 20
COMPOSITION
100 W% PILO 0 W% Ii20
(W%)
Fig. 8. Equilibrium concentration of pilocarpine and water in a nylon membrane as a function of the concentration of the pilocarpine-water solution,
presented in Fig. 8. It is plotted as a function of the concentration of pilocarpine in water in Fig. 9. As can be seen the diffusion coefficient is almost independent of concentration and the maximum in flux is caused by the maximum in the sorption isotherm for pilocarpine. Acknowledgements The authors are grateful to Dr. John Urquhart for helping to bring this paper to its final form.
15
120
105
90
\
75 0 “0 ii -cc -z
\
\
60
-\__
i v IP 45
30
15
O-
0
I
I
I
I
20
40
60
80
COMPOSITION
(W% PILOCARPINE
1010
IN I-I201
Fig. 9. Pilocarpine diffusion coefficient in a nylon membrane as a function of the concentration of pilocarpine in water on the donor side.
References 1 R.W. Baker and H.K. Lonsdaie, in A.C. Tanquary and R.E. Lacey (Eds.), Controlled Release of Biologically Active Agents, Plenum Press, New York, 1974, p. 15. 2 F. Theeuwes, K. Ashida and T. Higuchi, in the press, J. Pharm. Sci. 3 F.A. Kinci, G. Benagiano and I. Angee, Steroids, 11 (1968) 673. 4 K. Sundaram and F.A. Kincl, Steroids, 12 (1968) 517. 5 R.L. Shippy, ST. Hwang and R.E. Bunge, J. Biomed. Mater. Res., 7 (1973) 96. 6 E.R. Garrett and P.B. Chemburkar, J. Pharm. Sci., 57 (1968) 949; 57 (1968) 1401.
16
7 A.S. Michaels and H.J. Bixler, in E.S. Perry (Ed.), Progress in Separation and Purification, Interscience, New York, 1968. 8 H.F. Coldman, B.J. Poulsen and T. Higuchi, J. Pharm. Sci., 58 (1969) 1098. 9 W. Higuchi, personal communication. 10 G. Gee, Trans. Inst. Rubber Ind., 18 (1943) 266.
TRANSFERENCE: A COMPREHENSIVE PARAMETER PERMEATION OF SOLUTES THROUGH MEMBRANES
FELIX
THEEUWES’,
ROBERT
ALZA Corporation, Palo Alto, (Received May 30, 1975)
GOVERNING
M. GALE and RICHARD W. BAKER** Calif. 94304
(U.S.A.)
summary Experimental studies of the diffusion of moderate molecular weight drugs through synthetic polymeric membranes has led us to define a new parameter to characterize solute permeation. The accepted parameter which describes trans-membrane flux is permeability, P, defined as the product of the flux times the membrane thickness divided by the concentration difference across the membrane in the external solution. The parameter described here is called transference, r?; and is defined as the mass of solute transferred from a saturated solution, under perfect sink conditions, across the membrane per unit time and area, multiplied by the thickness of the membrane. Regardless of whether there is solvent-membrane interaction or not, the permeability coefficient, P, for a given solute and membrane differs from one solvent to another. In contrast, transference is independent of solvent, unless there is solvent-membrane interaction, in which case the inconsistency of transference for a particular solvent signifies that the membrane is modified by the presence of that solvent or solvent--solute combination, It is for these reasons that transference is a more comprehensive parameter than permeability, and is basic at the same time to solute-membrane permeation. When transference is introduced into Fick’s law it is immediately obvious that the flux of solute through the membrane, at equal fraction of saturation in any donor solvent, is equal and independent of the donor solvent in the absence of solvent-membrane interaction, when the solute has a constant solvent-membrane partition coefficient. Two examples of the use of the transference parameter are provided.
Introduction
The permeability coefficient, defined as the product of the flux through the membrane times the membrane thickness divided by the concentration difference across the membrane in the external solution, is commonly used to describe membrane permeation. When fluxes of the same drug through different membranes are studied from a common donor solvent, the permeability coefficients can conveniently be compared to give information as to which membrane is more permeable to the drug. Most drug permeability measurements *To whom inquiries should be directed. Present address: ALZA Research, 2631 Hanover Street, Palo Alto, Calif. 94304, U.S.A. **Present address: BEND Research, Inc., 64550 Research Road, Bend, Oreg. 97701, U.S.A.
reported in the literature are carried out from aqueous solutions as donor solvents and such data can unambiguously be interrelated. The development of controlled drug delivery systems, using the principle of solution-diffusion [ 1,2] , has stimulated permeation studies from a variety of nonaqueous solutions. In such systems, the permeability coefficient is insufficiently descriptive and a new parameter, transference, F, is introduced as defined below. The permeability coefficient is not a characteristic number for the solute (drug)-membrane system but differs depending on each donor solvent. The transference is an intensive property of the solute-membrane system which is independent of the solvent in the absence of solvent-membrane interaction. The term interaction is defined as the process whereby the membrane is modified by the presence of the solvent such that the membrane appears to the solute as a different medium when different solvents are present. Theory Fick’s law is generally written as: 1 dm -=-#, A dt where dm/dt is the mass transported per unit time through an area A caused’ by the concentration gradient dC!/dscwith D as the solute diffusion coefficient. Considering solute transport across membranes the solution-diffusion model is .usually observed as illustrated in Fig. 1. At each face of the membrane, equilibrium with the respective bulk solution is assumed, and a gradient in activity and concentration exists within the membrane. Thus for the left side of the membrane: a(O)
=
am(O),
where a(o) is the solute activity in the left solution phase and amto) is the activity in the membrane at the interface x = 0. A similar expression can be written for the right hand interface. Substituting for the activity the product of an activity coefficient (y) and the concentration (C), gives eqn. (3) 7( O)C( 0) = cm(
o)Ym(O)-
Equation (3) may also be written as:
cm(o) Y(o) C(o) Ym(0) -=-
=K
’
where K is the solute distribution coefficient between solvent and membrane. Equation (4) thus shows the link between the concentration of solute in the bulk solution and the concentration in the adjacent membrane phase. The partition coefficient K is constant for regular solutions and eqn. (4) is an analogue expression to Henry’s law. When the solute is in equilibrium at the
ACTIVITY CONCENTRATED SOLUTE SOLUTION
PROFILE
MEMBRANE
CONCENTRATION
I
DILUTE SOLUTE SOLUTION
PROFILES
C(d
C(P) I
I
x=0
x=P
Fig. 1. Solute activity and concentration membrane into the acceptor side.
*
X
profiles from the donor solvent acrw
the
solvent-membrane interface a step in concentration occurs depending on the value of K, as shown in Fig. 1, from the donor solvent into the membrane phase. At equilibrium there is never a discontinuity in the activity profile as expressed by eqn. (2). If, for simplicity, the diffusion coefficient is assumed constant, eqn. (1) can be integrated over the membrane thickness 2 to give 1 dm --= A dt
D[Crn(o)-
1
Cm(l)]_
DAcrn 1
*
(5)
If the solute is dissolved in the same solvent at both membrane surfaces, and if K is independent of concentration, eqn. (4) can be substituted into eqn. (5) and the flux is described as a function of the solute concentration in the external solvent by eqn. (6)
16) Here AC is the solute concentration difference in the solvent external to the membrane. Frequently, the product DK is termed the solute permeability, P, and hence
6
1 dm PAC -=Adt I’
(7)
However, the parameter, permeability, has limited generality, for it combines both membrane and solvent characteristics. Thus, when permeation experiments are performed in different solvent media with the same membrane, different permeabilities result. D will remain unchanged but K will vary to reflect the altered solubihty in the liquid medium. In addition, a common form of permeability experiment is to maintain a negligibly low concentration on one side of the membrane and a fixed concentration on the other side simply by maintaining the solution saturated with excess solute. This method of maintaining a constant driving force is particularly useful with solutes sparingly soluble in the external solvent [3-6]. Unfortunately, solubilities of sparingly soluble solutes are often imprecisely known and are difficult to measure. The result is that widely different values of drug permeabihties through the same membrane from the same solvent systems are reported in the literature. For these reasons, we prefer to characterize membranes by a term, F, defined by eqn. (8), as the transference
sat
=J,,l=PC”=DC&.
The transference is the product of the membrane thickness and the solute fh=, Jut, through the membrane from a saturated solution to the pure solvent. It is also given by the product of the permeability coefficient and the solubility in the donor solvent, or the product of the diffusion coefficient of the solute in the membrane and the solubility of the solute in the membrane. The transference is invariant with the selection of the solvent for solvents which do not solvate or otherwide modify the membrane. The quantity JZ has been referred to as the “normalized flux” [ 71. The transference, T, is the normalized flux at the specified boundary conditions: saturated solution and zero concentration. Transference completely describes the permeation process, and the drug solubility in the donor solvent is not required to calculate fluxes through the membrane. The transference is a reproducible intensive property of the solutemembrane system, independent of whether solvent-membrane interaction occurs or whether the solvent-membrane partition coefficient for the solute is constant and is defined for a specific crystalline state. In the absence of solvent-membrane interaction, ? is a constant of the solute-membrane system, independent of the solvent system. The finding that ?1is different for different solvent systems will signify solvent-membrane interactions and can be used as a tool for such investigations. When the solute is in equilibrium at saturation with a membrane (m) in different solvents (1, 2, . . . II), eqn. (9) holds in the absence of solvent-membrane interaction: Cfm=K,G=K&=....~,Cs,,
(9)
7
which is the reason for the constancy of ?; for different solvent-membrane systems. When, in addition, the solvent-membrane partition coefficients K1, K, are constant and independent of concentration, Fick’s law can be K,,... combined with eqns. (7) and (8) to arrive at eqn. (10) for solute i 1 dmj
Tl ACi
A dt
E q’
__=Ji=_-
(10)
It follows that when the partition coefficient Ki is independent of concentration, the solute flux through the membrane is proportional to the fraction of saturation in the donor solvent. Experimental Permeabilities were measured with a conventional two-chamber cell. The upstream chamber of the cell was maintained at a constant concentration, while the downstream chamber was maintained at a negligible concentration. Both chambers were vigorously stirred, and the rate of appearance of the drug on the downstream side was measured spectrophotometrically by recirculating through a flow-through cell by means of a pump. Membranes 50 to 150 microns thick were placed between wire screens to prevent deformation. Permeabilities were reproducible to f 15 %. Drug sorptions were measured by placing weighed samples of membrane material in a solution of known solute concentration until equilibrium had been reached. The samples were then removed, rapidly rinsed, patted free of surface moisture, and weighed. The samples were then desorbed completely in a large volume of water and the mass of solute desorbed was measured with the UV. With the solute content and the total weight gain of the film known, the water and drug sorption could be calculated. Subsequent experiments showed that the samples had reached equilibrium before removal from the solution and that only a negligible quantity of solute was lost during the rinse step. Both permeability and sorption experiments were carried out at 37°C. Membranes were prepared from three commercially available polymers. Two of these,Elvax 40 from DuPont and Levaprene 400 from Bayer, are random copolymers of ethylene/vinyl acetate containing ~40 wt.% vinyl acetate. The third polymer, Elvamide 8063, is an alcohol-soluble nylon terpolymer supplied by DuPont. Ethylene/vinyl acetate copolymer membranes were solution cast while Elvamide 8063 membranes were melt pressed. The Elvamide membranes were thoroughly washed in water prior to use to remove a small quantity of slowly released UV absorbing impurity. Hydrocortisone alcohol and progesterone are commercially available as pure compounds. Pilocarpine base was prepared from the commercially available nitrate by conventional methods. The supersaturated solutions were prepared by dissolving hydrocortisone alcohol in acetone-water solutions. Acetone is a volatile, good solvent for
8
hydrocortisone while water is a relatively poor, non-volatile solvent. The acetone was flashed off under vacuum with some heat after which the solution was cooled, leaving a supersaturated aqueous hydrocortisone alcohol solution. The concentration of drug was obtained by diluting an aliquot of solution and measuring the UV absorption. The UV also showed the absence of acetone. Solutions less than four times saturated were quite stable for several hours, although more concentrated solutions precipitated spontaneously. This method of preparing supersaturated solutions has been used [S] as a method of increasing the rate of diffusion of drugs through the skin. Higuchi has prepared supersaturated solutions in a variety of ways, and has also measured the increased permeability of microporous membranes to drugs from this type of solution [ 91. Results and discussion Diffusion up to unit activity of a solute with constant solvent-membrane partition coefficient AS shown previously, eqns. (6) and (7) are subject to the misinterpretation that the flux through a membrane is proportional only to the concentration difference between the solutions on either side of the membrane. This is untrue if the solvent in which the solute is dissolved is changed, for, with different solvents, the flux from similar concentration differences can be very different. For example, Fig. 2 shows the results of permeation experiments with the solute, progesterone, dissolved in three different solvents. The downstream concentration was always maintained at a negligibly low value, while the upstream concentration was varied up to the saturation concentration. The solubility of the drug in the solvents varied from 13.6 ppm for water, and 700 ppm for Dow 360 silicone oil, to 20,000 ppm in polyethylene glycol. The permeability coefficient _Pthus varied by more than lOOO-fold. However, in all cases, the transference T was 1.28 X lo-” g cm/cm2 sec. Figure 2 demonstrates the distinction between the true driving force, i.e. the chemical potential gradient, and the apparent driving force, i.e. the concentration gradient. At saturation the progesterone in all three solutions is at the same chemical potential (unit activity) and hence the driving force and the resulting membrane fluxes are equal despite the wide range of the concentration differences. The indication that the transference was a constant of the drug-membrane system independent of the solvent, and further that the fluxes in each case are linearly proportional to drug concentration in the donor solvent indicates that eqn. (10) describes these experiments. It therefore follows that the results shown in Fig. 2 can all be normalized by plotting the normalized flux J1 against fractional saturation AC/f?, rather than against concentration C. Figure 3 shows this plot. Figure 4 shows the results of an experiment illustrating the same idea. A solution of ~2000 pg/ml progesterone in polyethylene glycol600 (in which
2.0
H20
1.5 -
1.0 -
10 PROGESTERONE
-0
100
200
PROGESTERONE
2.0 c
CONCENTRATION
300
400
500
CONCENTRATION
12
13.5
(pg/m4?)
600
700
(pg/mQ)
PEG 600
PROGESTERONE
CONCENTRATION
Fig. 2. Progesterone flux across an ethylene-vinyl function of concentration from three solvents.
(pg/mP)
acetate copolymer
membrane as a
10
1.5
1.25 -
FRACTIONAL
SATURATION
(AC&)
Fig. 3. Progesterone flux across an ethylene-vinyl acetate copolymer membrane as a function of the fraction of saturation from three different solvents. Same symbols as in Fig. 2.
(-
POLYETHYLENEGLYCOL
WATER (SOL.
Cs = 13.5pg/mP.)
(SOL.
CS = 20,00O~g/mP)
MEMBRANE \ 1
C = 2,00Opg/mV
C = lOpg/mP
i
B
INITIAL
CONDITION
C = 1.2pg/mY.
C = 2,009pg/mP
EQUILIBRIUM
s
00
CONDITION
c
-\
I 50
I 100
I 150
I 200 TIME
I 250
I 300
A ”
I
I
950
1000
(minutes)
Fig. 4. Progesterone flux across an ethylene-vinyl acetate copolymer membrane from a water solution at 10 fig/ml to a polyethylene glycol solution at 2000 rg/ml.
II
11
progesterone has a solubility of =ZO,OOOpg/ml) was placed on one side of an Elvax 40 membrane while a solution of 10 pg/ml progesterone dissolved in water (in which progesterone has a solubility of 13.5 pg/ml) was placed on the other. Progesterone diffused from the 10 fig/ml solution to the 2000 pg/ ml solution, and the concentration of progesterone in the aqueous solution fell, finally reaching 1.2 pg/ml. At 10 pg/ml in the aqueous solution, progesterone was at 0.75 of saturation, while at 2000 pg/ml in the polyethylene glycol solution, the steroid was at only 0.1 of saturation. Diffusion thus occurred from the high to the low activity solution until the two solutions were equi-active, ie. until the aqueous progesterone solution had dropped to -0.1 saturation (1.2 pg/ml in water). Diffusion above unit activity of a solute with constant solvent-membrane partition coefficient The correlation between flux and degree of saturation, eqn. (lo), has been found to hold above the normal saturation concentration of a drug. Although a saturated solution is the highest activity driving force normally used, some supersaturated solutions are fairly stable and can be used as hyperactive solutions. Figure 5 shows the results of an experiment with those solutions. Initially a normal saturated hydrocortisone alcohol solution containing a small excess of solid solute was placed on one side of the membrane and water on the other. The flux of solute was measured by periodically measuring the concentration on the dilute side, which never contained more than a few fig/ ml of hydrocortisone. After 354 minutes, the solution was removed, the cell quickly rinsed, and a previously prepared supersaturated solution of hydrocortisone alcohol added. The flux of drug was again monitored until the 560th minute, when the supersaturated solution was nucleated with a crystal of hydrocortisone. The solution immediately turned cloudy and the flux through the membrane dropped to the normal saturated solution value. The results shown in Fig. 5 represent to our knowledge the first example in which a supersaturated solution has been used to induce supersaturation in the membrane phase. As with normal subsaturated solutions, the fluxes obtained were directly proportional to the degree of saturation, as shown in Fig. 6. This plot suggests that the activity coefficient (y) is constant even in a 4 times supersaturated solution. Permeation of a solute with concentration-dependent solvent-membrane partition coefficient In a final example, the permeation of pilocarpine through a nylon membrane from water as the solvent is discussed. Pilocarpine is a liquid at 37”C, and completely miscible with water, such that the coefficient ?; cannot be defined from an experiment at 37°C. The permeation results are plotted in Fig. 7, representing a case where the flux is not proportional to the drug concentration on the upstream side (the downstream solution was water). Deviations from linearity are well known and reflect a variable diffusion coefficient
12
3
NORMAL
SOLUTION
EXCHANGED
FOR SUPERSATURATED (2.9 X SATURATION)
f” i
a E
SOLUTION
NORMAL SATURATED SOLUTION
SUPERSATURATED NUCLEATED WITH HYDROCORTISONE
SOLUTION SOL10
100
0
TIME
(MINUTES)
Fig. 5. Accumulative amount of hydrocortisone which permeated an ethylene-vinyl acetate copolymer membrane from saturated and supersaturated solutions. 1.0I-
2.6 X SATURATION
1 X SA
2
0.25
HYDROCORTISONE
CONCENTRATION
Fig. 6. Normalized flux of hydrocortisone tration above saturation.
(pg/mP)
alcohol as a function of hydrocortisone
concen-
13
20 COMPOSITION
40
60
(W% PILOCARPINE
60
0
IN Hz01
Fig. 7. Normalized flux of pilocarpine through a nylon membrane as a function of pilocarpine concentration in water as the donor solvent.
or partition coefficient. Figure 7 is to our knowledge the first reported example of a flux versus concentration plot exhibiting a maximum. The reason for this behavior is apparent from the sorption isotherms for pilocarpine and water shown in Fig. 8. The plot shows a maximum in pilocarpine and water uptake by the membrane as a function of the external solution concentration, which can be explained either by a maximum in water and pilocarpine activity in the presence of each other, or a synergistic swelling effect of the membrane caused by the drug and water at about 40% pilocarpine 60% water concentration. Such membrane swelling behavior is well known and several examples have been given in the literature [ 10). The diffusion coefficient was calculated from the flux data presented in Fig. 7, and the membrane sorption data
16
6
)
0 100
I
I
I
I
20 80
40 60
60 40
80 20
COMPOSITION
100 W% PILO 0 W% Ii20
(W%)
Fig. 8. Equilibrium concentration of pilocarpine and water in a nylon membrane as a function of the concentration of the pilocarpine-water solution,
presented in Fig. 8. It is plotted as a function of the concentration of pilocarpine in water in Fig. 9. As can be seen the diffusion coefficient is almost independent of concentration and the maximum in flux is caused by the maximum in the sorption isotherm for pilocarpine. Acknowledgements The authors are grateful to Dr. John Urquhart for helping to bring this paper to its final form.
15
120
105
90
\
75 0 “0 ii -cc -z
\
\
60
-\__
i v IP 45
30
15
O-
0
I
I
I
I
20
40
60
80
COMPOSITION
(W% PILOCARPINE
1010
IN I-I201
Fig. 9. Pilocarpine diffusion coefficient in a nylon membrane as a function of the concentration of pilocarpine in water on the donor side.
References 1 R.W. Baker and H.K. Lonsdaie, in A.C. Tanquary and R.E. Lacey (Eds.), Controlled Release of Biologically Active Agents, Plenum Press, New York, 1974, p. 15. 2 F. Theeuwes, K. Ashida and T. Higuchi, in the press, J. Pharm. Sci. 3 F.A. Kinci, G. Benagiano and I. Angee, Steroids, 11 (1968) 673. 4 K. Sundaram and F.A. Kincl, Steroids, 12 (1968) 517. 5 R.L. Shippy, ST. Hwang and R.E. Bunge, J. Biomed. Mater. Res., 7 (1973) 96. 6 E.R. Garrett and P.B. Chemburkar, J. Pharm. Sci., 57 (1968) 949; 57 (1968) 1401.
16
7 A.S. Michaels and H.J. Bixler, in E.S. Perry (Ed.), Progress in Separation and Purification, Interscience, New York, 1968. 8 H.F. Coldman, B.J. Poulsen and T. Higuchi, J. Pharm. Sci., 58 (1969) 1098. 9 W. Higuchi, personal communication. 10 G. Gee, Trans. Inst. Rubber Ind., 18 (1943) 266.