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Mechanistic model for transdermal transport including iontophoresis
journal of
ELSEVIER
controlled release
Journal of Controlled Release 41 (1996) 177-185
Mechanistic model for transdermal transport including iontophoresis Kyrsti Kontturi*, Lasse Murtom~iki Laboratory of Physical Chemistry and Electrochemistry, Helsinki University of Technology, Kemistintie IA, FIN-02150 Espoo, Finland
Received 28 June 1995; accepted 8 December 1995
Abstract
A mechanistic model is presented to describe the transdermal transfer of drugs. In the model, flux is divided into the contributions through the lipid matrix and through aqueous pores. During iontophoresis, only flux through aqueous pores is enhanced because the relative permittivity of the lipid matrix is too low to allow the existence of free ions which could work as current carriers. The transfer through the lipid matrix is assumed to take place in three steps: (i) partition at the skin/reservoir interface; (ii) diffusion through the lipid matrix; (iii) desorption at the stratum corneum/epidermis interface. Simulations to the present model agree with experimental results in the literature. Keywords: Transdermal transfer; Iontophoresis; Aqueous pores; Permeability vs. partition; Desorption
1. I n t r o d u c t i o n The ability of drugs to penetrate human skin depends on several properties: skin structure, lipophilicity, molecular size or configuration of the drug molecule etc. [1]. The quantitative estimation of these properties is quite difficult, and the number of unknown parameters makes it difficult to predict whether a particular molecule can be delivered transdermally or how its permeation can be enhanced. Several models for the transdermal transport have been proposed (see [2-6]. Although some of them [2-4] are, in principle, rigorous enough to describe transport with sufficient accuracy, they are not feasible to rapid estimation in the practice due to their complexity, but demand lengthy computations. Another class of the models is given by fitting measured permeabilities with some suitably chosen Correspondmg author.
parameters. Such models can have certain practical value in the prediction of the permeabilities of like drugs, but without a physical significance of the fitted parameters such a model remains only practice of statistical calculus. Still, an excellent exception to these models is given by Potts and Guy [5] who were able to give a sound physical basis for the parameters. In their model, the overall permeability, Kp, is given by l o g Ke =
+ 0.71 l o g P - 0.0061M r
(1)
where P is the water/n-octanol partition coefficient and M r the molecular weight of the molecule. The number of molecules fitted to Eq. (1) was 93, but the correlation coefficient was only 0.67, although the authors find it high. Moreover, the partition coefficient and molecular weight are not independent parameters but depend on each other, as the authors themselves point out. The basic idea of the modelling of transport
0168-3659/96/$15.00 © 1996 Elsevier Science Ireland Ltd. All rights reserved P I I S01 68-3659(96)01323-5
-6.3
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K. Kontturi, L. Murtomdki / Journal of Controlled Release 41 (1996) 177-185
processes is not to know the exact mechanism of transport in great detail but to know what is its rate determining step. Furthermore, the model has to take account all the relevant factors, but not to be too complicated; the accuracy of any model is in any case bound to be limited because of the inherent variance of the permeabilities of different skin samples. Here, we try to compromise between these two requirements and create an alternative macroscopic and mechanistic model for transdermal transport, emphasizing specially iontophoresis of partially ionized drugs, and collect the properties affecting it into coefficients which can be experimentally determined. To avoid unnecessary mathematical complications originating from the solution of the partial differential equation (Fick's Second Law) [6], the model considers only steady-state fluxes. The model is justified in the view of the existing permeability vs. partition data, e.g. [7,8] and to explain our data about transdermal iontophoresis of sotalol and salicylate [9].
2. Model As in the previous paper [9], flux is divided into the two contributions (Fig. 1): flux through aqueous pathways, jw, and flux through the lipid matrix of stratum corneum, jo, without any closer consideration of the routes of penetration:
jp = jw + jo
The subscript 'p' denotes passive without enhancement. Furthermore, that iontophoresis enhances only the way, and, therefore, at the constant tophoretic flux is given by
J(/ = J ' . E
transport, i.e. it is assumed aqueous pathpotential ion-
+ jo
(3)
where E is the enhancement factor which can be theoretically calculated by, e.g. the Goldman's constant field approximation
p E=
~; u -
1-e
or the [9,10].
Planck's
E = (1 +
zFd cp RT
(4)
electroneutrality
~.__e__'~(._)(lnx,_-]\Xe_~j,X_ 1 "~.
approximation
i - ~,c/3
(5)
~...d i i
z is the charge number of the drug, and Aqb is the potential drop over the membrane; g is the ratio of the total ion concentrations on the both sides of the membrane; F, R and T have their usual significance. Discrepancies between the calculated and observed enhancement factors are discussed later on. Eq. (3) is justified by the very low conductivity of the lipid matrix. From Eq. (2) and Eq. (3), the contributions of the two mechanisms can be evaluated as
Jp
(2)
jw _
E-
1
(6)
Jp'E-Jif jo
jo_
jo
The flux through aqueous pathway is given by Fick's Law in perfect sink conditions as usual:
jw
E-
1
Cb jw
jw jo jw
jo v Fig. 1. Penetration routes through skin assumed in the model.
= ~D w_
h
(7)
(8)
where e is the fraction of the area of the aqueous pathway, D w is the aqueous diffusion coefficient and C b the bulk concentration of the drug; h is the length of the diffusion pathway which can be different from the membrane thickness due to tortuosity of the pathway. Because of the very low relative permittivity of
179
K. Kontturi, L. Murtomiiki / Journal o f Controlled Release 41 (1996) 1 7 7 - 1 8 5
the lipid matrix only nonionic solutes can exist inside it: the fraction of drug in a nonionized form (often depending on pH) is denoted with a. The more lipophilic drug is, the greater is its partition at the reservoir/stratum corneum interface, and thus the greater is its concentration inside the lipid matrix. If the transport were entirely due to Fickian diffusion, and perfect sink conditions applied below stratum corneum, the flux would increase without any limit with an increasing partition coefficient (see Discussion). This is not the case, but even convex shapes of log(Kp) vs. log(P) curves have been found [8,11]; Kp is the permeability and P partition coefficient of the non-ionized drug. Therefore, another mechanism has to be involved, as addressed later on. It is known that drugs are bound into stratum corneum, and long after iontophoretic treatment flux remains on a higher level than during passive diffusion [12]. This may seem to conflict with our assumption that current passes only through aqueous pores, but the law of mass action must be realized: the more current forces drug into the pores, the more it also brings the neutral form which is readily adsorbed onto the pore walls, after which more drug is dissociated to the neutral form etc. Thus, we postulate that flux through the lipid matrix consists of three steps: (a) partitioning at the reservoir/ stratum corneum interface; (b) Fickian diffusion through the lipid matrix; (c) desorption with the first order kinetics at the stratum corneum/viable epidermis interface (Fig. 2). As seen in Fig. 2, the concentration of drug in a nonionized form at the reservoir side of stratum corneum is o~pcb; C and F* are the volume and surface concentrations at the stratum corneum/epidermis interface. At the steady-state, the rate of diffusion through stratum comeum equals to the rate of desorption:
Interracial layer with F* stratum corneum
jo = D °
h
(~ -
kF*
(9)
epidermis
partition P
aC b ,jo v
Fig. 2. Transport through the lipid matrix in three steps: partition, diffusion and desorption. Fz ~
/.z *
_o C tz + R T I n - ~
= Iz
.o
F* + RTln Fo
o
_
,~ * o
=(#°-
/.twO) + (/zwo _
.o)
F*C o = RT l n - F° Cbar
D ° is the diffusion coefficient in the lipid matrix and
k is a homogeneous desorption rate constant (s -1). The two unknown quantities C and F * can be eliminated using the equality of their chemical potentials if it is assumed that the thermodynamic equilibrium prevails between them:
(10)
The interfacial layer does not mean that the drug is cumulated but it has a different chemical environment from that in the bulk phase. That is why chemical potentials need to be considered instead of simply converting volume concentrations to surface concentrations. Solving for the difference of the standard chemical potential at the interface, /.~,o, and in the lipid matrix, /z°, with adding and subtracting of the standard chemical potential in the aqueous phase, w o we obtain -
apc b -
I
/zwO _ * o )
F * = F ° ~ o exp ( / 2 ° ~ T -~w°) exp \
R-T
(11) The standard concentrations corresponding to each
K. Kontturi, L. MurtomiT"ki / Journal of Controlled Release 41 (1996) 177-185
180
other are C ° = 1.0 mol dm -3 and F ° = 1.18-10 -1° mol cm 2 [13]. The former exponent term clearly is P-~ because from the thermodynamic equilibrium between the lipid and aqueous phase it is obtained
wo_ o)
P -
a w - exp \-
._:6.5
..........................!ii . .... j - / / J
~-7
-R'T /
-7.5
= exp
" RT
where A G ° p is the free energy of partition. Assuming that viable epidermis can be considered as an aqueous phase, the latter difference between the standard chemical potentials is the free energy of desorption, A G ° p , and Eq. (11) can be written as
/
I.'ivo
(12)
/
/
-2
-1
0
1 log P
0,v _:-°-v-I 2
3
4
Fig. 3. S i m u l a t i o n to the m o d e l with the f o l l o w i n g p a r a m e t e r v a l u e s : e = 10 4, D w = 1 0 - 6 c m 2 s - ] , h = 1 0 0 / x m , a = 0.01, k o = 10 -4 c m s - I , D ° = 10 8 c m 2 s 1. T h e e n h a n c e m e n t factor E has b e e n c a l c u l a t e d using Eq. (4).
roe F* -
cO p exp \ - - ~ /
(13)
3. Simulations
Defining a heterogeneous desorption rate constant k o (cm S - l )
as
r 0
kD = k~
exp \ ~ - - /
(14)
we find
d
(15)
k F * = ko--f i
Inserting Eq. (15) into Eq. (9) and eliminating, we finally obtain jo
-
acbkD 1 + A'A-
kDh DOp
(16)
where all the constants have been collected into k o and a. Two limiting cases of Eq. (16) are readily seen: when A > > 1, i.e. P is very small (hydrophilic drug) otPC b jo = D o _
h
(17)
which is the familiar Fick's law; when A < < 1, i.e. P is large (lipophilic drug) j o = olCbkD
(18)
according to which desorption is the rate determining step. In the Appendix A, an example calculation is given, using our data of sotalol transfer [9], to show how different quantities can be evaluated.
Combining Eq. (8) and Eq. (16), a simulation in Fig. 3 has been carried out, and it indeed reminds of the behaviour presented in the literature; the permeability Kp is flux divided by the bulk concentration. The parameter values has been estimated from the experimental data [7]. As can be seen, the flux of hydrophilic drugs is enhanced quite a lot due to iontophoresis while the permeability of lipophilic drugs remains practically constant. Hence, the present model helps to predict if iontophoresis is a relevant method to enhance the flux of a particular drug. In Fig. 3, D ° and k o have constant values at the whole range of P which may not be realistic. Taking that the diffusion coefficient decays exponentially as a function of the molecular size [5], and assuming that log(P) increases as a linear function of the molecular size, it follows that D ° = a P b (a and b are appropriate positive constants). Similarly, assuming that also k D obeys this kind of a relationship, simulations in Fig. 4 have been carried out with different values of b indicated in the figure. This assumption for k D produces a convex shape for the permeability curve while for D ° it gives only a slowly increasing curve. It is remarkable that varying only one parameter, the partition coefficient of noionized drug, the sigmoidal curve is obtained. It is true that our model does not explicitly explain the decrease of the
K. Kontturi, L. Murtomiiki / Journal of Controlled Release 41 (1996) 177-185
-6
Fig. 3 :
]
:~
Second, it is not always quite clear how voltage drops across the membrane have been measured. If the resistance of the skin sample is R a and the intrinsic resistance of the voltage meter is R m, the following error analysis applies: Let the actual voltage drop across the membrane be Aqb and the measured value AqS'. Now:
-
-7.5
:12
A~' A~
-8 -2
-
I1
I
0
I
1 logP
I
2
3
181
1
-
1 + Ra/R,,
(19)
I
4
Fig. 4. Simulation s h o w i n g the effect o f partition coefficientd e p e n d e n t diffusion coefficient, D ° a n d desorption coefficient, k o. The other quantities are the same as in Fig. 3.
permeability at very high values of P [8,11] but it is obviously related either to strong binding to stratum corneum [14] or to a large molecular size [1,5,1517]. The former effect can be accomplished by a decreased desorption rate constant, k D, and the latter by the diffusion coefficients, D w [18] or D °.
The measured value thus is always lower than the actual one. We have measured in accordance with [21] resistances R a > 100 k~. To keep the error in Aq~ lower than 1%, R m has to be at least 10 M~. When using a potentiometric circuit as in [19,21], no errors should occur, but the comment of 'correction for solution iR drop' is enigmatic [21]. Third, enhancement due to electroosmosis has not been included in the model. In our own experience its effect is rather insignificant, although the phenomenon exists without any doubt. From Pikal and Shah's paper [22], we have estimated that the transport number of water, rw, defined as
FJw
rw-
4.1. Problems with the enhancement factor E
takes values 5 - 1 0 in hairless mouse skin (F is Faraday constant, Jw water flux and I electric current density). In human skin we have observed quite low charge densities [23] which means even lower values of r w. From Eq. (20) it is possible to calculate the order of magnitude of water flux, and inserting this value into the Nernst-Planck equation as a convective term, it is clear that electroosmotic contribution should be insignificant. Still, in the paper by Sims et al. [24] the value of an electroosmotic enhancement factor for electrically neutral mannitol was found to be as high as 7 when Aqb was 1.0 V. (From Eq. (4): E ~ 40, z = 1). Their value apparently corresponds to the combined effect of skin damage and electroosmosis because of the large deviation from Pikal and Shah's results where water flux had been measured. They also conclude that 'solvent flow considerations will probably impact in only a secondary manner in the iontophoretic transport of ionic species'. Because the quantitative incorporation of the
There are several examples in the literature, e.g. [19-21], where the observed enhancement factor is higher than the predicted one, which seems to make our model suspect. We can only suggest a few possible explanations to this dilemma, yet we have not observed enhancement factors higher than the theory predicts. First, instead of real potentiostatic control, controlled current has been administered, i.e. regardless of the resistance of the skin sample, certain current has been forced through, probably causing irreversible changes in the permeability. This has been documented as a lowered membrane resistance [21], and also in a potentiostatic mode as a 'damage factor, DF' [20]. Thus, the theoretical enhancement factor can be multiplied by DF, if so wanted, but it is always better to restrict oneself to potentiostatic control and not to overcome a voltage drop of 1.0 V. It also seems obvious that long hydration times are facilitating skin damage [21].
I
(20)
4. D i s c u s s i o n
182
K. Kontturi, L. Murtomiiki /Journal of Controlled Release 41 (1996) 177-185
effects of skin damage and electro-osmosis into our model is not straightforward, we want to suggest to avoid them fulfilling three requirements: (i) potentiostatic control: (ii) no higher voltage drops than 1.0 V; (iii) hydration time should not exceed 24 h. With these criteria fulfilled we have not observed anomalously high flux enhancements.
4.2. E x i s t e n c e o f a q u e o u s p o r e s
Potts and Guy have remarked in their rigorous paper [5] that the assumption of aqueous pores is not needed to explain the low permeability of small polar solutes, but 'the stratum corneum lipids alone can fully characterize the barrier properties of mammalian skin'. Furthermore, 'there is no physical evidence supporting the existence of aqueous pores'. Still, this cannot be true because if there were no aqueous pores, there would be no electroosmosis or streaming potential in human skin. The relative permittivity of lipid matrix certainly is low enough to prevent the existence of free ions; any attempt to measure the relative permittivity leads to an average value for skin, including the contribution of aqueous domains. If n-octanol is accepted as a model for lipid matrix the arguments above are easy to prove. The solubility of electrolytes into n-octanol is negligible, no values are even reported in the literature. Although if there were some dissolving electrolytes, say drugs, the degree of ion association would be very high because the relative permittivity of n-octanol is 10.34. Ion association is traditionally estimated by Bjerrum's or Fuoss' theories [25]. The former is valid in the cases where the thermal energy of ions ( k s T ) is comparable to the coulombic attraction which means, e.g. loose ion pairs of aqueous electrolytes. In the lipid matrix of skin, however, the situation is very much different: there are practically no free solvent molecules but organized layers of alkyl chains with very low relative permittivity (~-2) and polar groups with higher local permittivity (-~30). In such an environment contact ion pairs are more likely to exist, which can be described by the Fuoss' theory. According to this theory, the association constant, K , is given by
K
= 4. lO-247rNa(a/nm) 3
exp (b) 3
, b
Izizjle 2 -
47r~osrksT a
(21)
In Eq. (21), Nz is Avogadro number, e elementary charge, eo vacuum permittivity, Er relative permittivity of the solvent, k B Boltzmann constant, a contact distance between the ions, and zi,j are their charge numbers. For NaC1 in n-octanol a = 0.28 nm, b = 19.4 and K a ~ 1.4-10 7 M , i.e. the concentration of free ions is negligible. For a drug molecule with a = 0.7 nm, K a ~ 2000 M according to which ca. 50% would be dissociated if the total concentration were 0.1 M. But in reality, the concentration of the drug in a salt form is much less than that. If the partition coefficient of a drug is, say, 105 the free energy of partition is - 23 kJ m o l - l (see Eq. (12)). The free energy of transferring HC1 from water to n-octanol could be 4 5 - 5 0 kJ mol-1 [26]; then the free energy of partition of drug in HC1 form could be at least + 22 kJ mol-1 and its partition coefficient accordingly c a . 1 0 - 4 . Hence, even if the drug patch had 1 M solution the total concentration of the salt form drug would be ca. 0.1 mM only. Furthermore, in the above calculations, the distance between the ion centers is taken as the contact distance, which means that the net charges of the ions are assumed to be spread uniformly. In many cases, however, the charge is localized into specific sites of the ion, and the distance between the 'charge centers' should be taken as the contact distance of the Fuoss' theory. This would decrease the value of a and increase the association constant by orders of the magnitude (see e.g. NaC1 above). As a further evidence for aqueous pores, we were recently able to measure streaming potential in human skin [23] and to interpret the results with the classical theory of irreversible thermodynamics, i.e. with equations for transport in a charged capillary. The average capillary radius was found to be ca. 20 nm. The ohmic resistance of a skin sample, R a, with area A is Ra -
1 h
KA
(22)
where K is the conductivity of the solution depend-
K. Kontturi, L. MurtomiT"ki / Journal of Controlled Release 41 (1996) 177-185
ing on its concentration. We measured Rs~ with AC impedance techniques [27] in several NaCI concentrations (A = 1.0 cm 2) and found a typical value of the membrane coefficient A / h of 0.03 cm. Combining this information to the pore radius and taking h ~ 100 /xm, the average number of pores/cm 2 2.4.10 7, and their average separation distance --~2 /zm. It is therefore clear that these pores cannot be hair follicles or sweat glands. The chance that we are creating such pores when perturbing or freezing skin is still not overruled. 4.3. L i m i t a t i o n o f f l u x b y k i n e t i c c o n t r o l
The specific feature in our model is the first-order kinetic control of desorption of drug from stratum corneum to viable epidermis. Why is it needed? Potts and Guy [5] could account the flux limitation of highly lipophilic drugs solely for their large molecular weight, i.e. their low diffusivity. They could not even find limitation among small molecular but highly lipophilic n-alkenes, but the dispersion of their data (r 2 = 0.67) probably is too wide to recognize it. Moreover, specially in this case the partition coefficient and molecular weight were mutually dependent. It is also worth noticing that, in Fig. 4, we did take into account the molecular weight dependence of diffusivity, but did not find any saturation of log(Kp). In the model of Cleek and Bunge [6], it is assumed that the stratum corneum/viable epidermis interface is characterized with another partition coefficient, let it be here P ' and let the thickness of viable epidermis be h' and the drug diffusivity D ' . They solved the transient behaviour of the composite stratum corneum-viable epidermis system, but the following ideas can be compiled from their paper at the steady-state, although the boundary conditions are not explicitly given. Flux through this composite system is given by jo
~
D ° apc b h 1 + B; B
D°h'p ' D'h
183
drugs is limited by partition at the stratum corneum/ viable epidermis boundary [6]. This is true if P ' is substantially greater than P. But first, it is expected that P increases along with P ' , and second, as discussed above, the partition coefficient depends mainly on the polarity of the media. Because the polarity of viable epidermis must be close to that of water, P ~ P ' , and for highly lipophilic drugs Eq. (23) becomes j o ~ D , / h , . a C b. The effect of the partition coefficient thus vanishes but instead, the rate determining step would be diffusion through viable epidermis. The only limitation comes then from D', agreeing with Potts and Guy [5]. Therefore, to make the flux limitation of highly lipophilic drugs explicitly visible, kinetic control is needed. In the model, kinetic control is assigned to desorption but some another process is possible, too. Instead of desorption, there might be consumption of drug below stratum corneum-call it metabolism or clearance, for instance-described in chemical terms as a homogeneous first-order reaction with an appropriate rate constant, k n. Now, replacing k D with kn, the whole formalism, Eq. (16) Eq. (17) Eq. (18)), remains the same. If the homogeneous reaction were of the zeroth order the situation would become even more simple: in the case the reaction rate is slower than diffusion through stratum corneum, flux is limited by the reaction; in the case of fast reaction, any molecule passed through stratum corneum in readily consumed and perfect sink conditions prevail all the time, and once again there would be no explicit limitation of the flux of lipophilic drugs.
Acknowledgments The financial support of the Technology Development Centre of Finland (TEKES) is gratefully acknowledged.
Appendix A (23)
Note the similarity with Eq. (16). Now they deduce that for highly lipophilic drugs P ' takes high values (consequently B > > 1), and the flux of these
Considering our data for sotalol in [9] the following calculations are easily carried out: the theoretical enhancement factor, E, calculated from the Planck's electroneutrality approach is 14. l, while the observed
184
K. Kontturi, L. Murtomiiki / Journal of Controlled Release 41 (1996) 177-185
e n h a n c e m e n t f a c t o r w a s 7.2. T h e p a s s i v e p e r m e a b i l i ty (no i o n t o p h o r e s i s ) in T a b l e 1 in [9], d e f i n e d as Kp = J p / C b, w a s 1 3 . 3 . 1 0 -9 c m s 1. A p p l y i n g Eq. (6) a n d Eq. (7) it is f o u n d that jw
:
(7.2 -
=
0.53-
l).Jp/13.1
F r o m Eq. (8) J W / C
~ 0.47.Jp
--> j o
= 0 . 4 7 . 1 3 . 3 - 1 0 -9 c m s 1 =
e D W / h . In [26] w e f o u n d that for sotalol D w = 3 . 7 4 . 1 0 -6 c m 2 s -~. T a k i n g h --~ 100 / z m --> • 1 . 7 - 1 0 -5 w h i c h is a l o w p o r o s i t y indeed. S i m i l a r l y , J°[Cb
= 0.53 • 13.3 • 10 -9 c m s - l 7 . 0 0 . 1 0 -9 c m s - 1
F r o m Eq. (9), Eq. ( 1 5 ) a n d Eq. (16): jo aC b -
D ° -~ (P -
C / a C b) -
kD ~ P o~C b
kD 1 + A
U n f o r t u n a t e l y , w e d i d n o t m e a s u r e the flux o f sotalol in d i f f e r e n t b u l k c o n c e n t r a t i o n s , so that t h e separate evaluation of D ° and A would be possible, b u t a s s u m i n g t h a t D ° ~ 10 -8 c m 2 s -1 ( d i f f u s i v i t y c a n b e t w o or e v e n t h r e e o r d e r s o f m a g n i t u d e s m a l l e r in s t r a t u m c o r n e u m t h a n in a q u e o u s s o l u t i o n s [7]) a n d n o t i n g that P = 1.7 [28], p K , = 9.05 [28] a n d p H = 7.0 [9] ---> ~ ~ 0.0088, a n d it is finally f o u n d (h = I 0 0 /zm): C[olC
b =
0.909; k o =
1.5" 10 -6 c m s -1", ,~
= 0.875 P a r a m e t e r A takes the v a l u e 0.875 w h i c h is an i n t e r m e d i a t e one, a n d the t r a n s f e r o f n o n i o n i z e d sotalol t h r o u g h the lipid m a t r i x is u n d e r m i x e d d i f f u s i o n a n d d e s o r p t i o n c o n t r o l . It m u s t b e n o t i c e d t h a t ce ~ 0 . 0 0 8 8 only. A t h i g h e r pH, ce w o u l d take h i g h e r values, a n d it w o u l d b e i n f o r m a t i v e to r e p e a t o u r e x p e r i m e n t s at d i f f e r e n t pH.
References [1] N.H. Yoshida and M.S. Roberts, Structure-transport relationships in transdermal iontophoresis. Adv. Drug Deliv. Rev. 9 (1992) 239.
[2] R.H. Guy and J. Hadgraft, in: R.H. Guy and J. Hadgrafl (Eds.), Transdermal Drug Delivery, Marcel Dekker, New York, 1989, pp. 59-81. [3] K. Tojo, Mathematical model of iontophoretic transdermal drug delivery. J. Chem. Eng. Jpn. 22 (1989) 512. [4] J.C. Keister and G.B. Kasting, A kinetic model for ion transport across skin. J. Membr. Sci. 71 (1992) 257. [5] R.O. Potts and R.H. Guy, Predicting skin permeability. Pharm. Res. 9 (1992) 663. [6] R.L. Cleek and A.L. Bunge, A new method for estimating dermal absorption from chemical exposure. 1. General approach. Pharm. Res. 10 (1993) 497. [7] A.S. Michaels, S.K. Chandrasekaran and J.E. Shaw, Drug permeation through human skin: theory and in vitro experimental measurements. AIChE J. 21 (1975) 985. [8] J. Hadgrafl, Structure-activity relationships and percutaneous absorption. J. Control. Release 15 (1991) 221. [9] J. Hirvonen, K. Kontturi, L. Murtom~ki, P. Paronen and A. Urtti, Transdermal iontophoresis of sotalol and salicylate; the effect of skin charge and penetration enhancers. J. Control. Release 26 (1993) 109. [10] G.B. Kasting and J.C. Keister, Application of electrodiffusion theory for a homogeneous membrane to iontophoretic transport through skin. J. Control. Release 8 (1989) 195. [11] N. El Tayar, R.-S. Tsai, B. Testa, P.-A. Carrupt, C. Hansch and A. Leo, Percutaneous penetration of drugs: a quantitative structure-permeability relationship study. J. Pharm. Sci. 80 (1991) 744. [12] L.L. Wearley and Y.W. Chien, Iontophoretic transdermal permeation of verapamil. III. Effect of binding and concentration gradient on reversibility of skin permeation rate. Int. J. Pharm. 59 (1990) 87. [13] Y. Cheng, V.J. Cunnane, D.J. Schiffrin, L. Murtom~iki and K. Kontturi, Interfacial capacitance and ionic association at electrified liquid/liquid interfaces. J. Chem. Soc. Faraday Trans. 87 (1991) 107. [14] P.G. Green, J. Hadgraft and G. Ridout, Enhanced in vitro skin permeation of cationic drugs. Pharm. Res. 6 (1989) 628. [15] W.M. Deen, Hindered transport of large molecules in liquidfilled pores (Journal Review). AIChE J. 33 (1987) 1409. [16] S.B. Ruddy and B.W. Hadzija, Iontophoretic permeability of polyethylene glycol through hairless rat skin: application of hydrodynamic theory for hindered transport through liquidfilled pores. Drug Des. Discovery 8 (1992) 207. [17] N.H. Yoshida and M.S. Roberts, Solute molecular size and transdermal iontophoresis across excised human skin. J. Control. Release 25 (1993) 177. [18] K.A. Johnson, G.B. Westermann-Clark and D.O. Shah, Diffusion of charged micelles through charged microporous membranes. Langmuir 5 (1989) 932. [19] G.B. Kasting, E.W. Merrit and J.C. Keister, An in vitro method for studying the iontophoretic enhancement of drug transport through skin. J. Membr. Sci. 35 (1988) 137. [20] V. Srinivasan, W.I. Higuchi, S.M. Sims, A.H. Ghanem and R. Behl, Transdermal iontophoretic drug delivery: mechanistic analysis and application to polypeptide delivery. J. Pharm. Sci. 78 (1989) 370.
K. Kontturi, L. Murtomiiki / Journal of Controlled Release 41 (1996) 177-185 [21] G.B. Kasting and L.A. Bowman, DC electrical properties of frozen, excised human skin. Pharm. Res. 7 (1990) 134. [22] M.J. Pikal and S. Shah, Transport mechanisms in iontophoresis. II. Electroosmotic flow and transference number measurement for hairless mouse skin. Pharm. Res. 7 (1990) 213. [23] V. Aguilella, K. Kontturi, L. Murtomiiki and P. Ramfrez, Estimation of the pore size and charge density in human cadaver skin. J. Control. Release 32 (1994) 249. [24] S.M. Sims, W.I. Higuchi and V. Srinivasan, Skin alteration and convective solvent flow effects during iontophoresis. I. Neutral solute transport across human skin. Int. J. Pharm. 69 (1991) 109.
185
[25] J.O.M. Bockris and A.K.N. Reddy, Modern Electrochemistry, Chpt. 19, Elsevier, Amsterdam, 1985. [26] K. Kontturi and L. Murtom~iki, Electrochemical determination of partition coefficients of drugs. J. Pharm. Sci. 81 (1992) 970. [27] K. Kontturi, L. Murtomaki, J. Hirvonen, P. Paronen and A. Urtti, Electrochemical chararacterization of human skin by impedance spectroscopy: the effect of penetration enhancers. Pharm. Res. 10 (1993) 381. [28] C. Hansch and C.J. Drayton (Eds.) Comprehensive Medicinal Chemistry, Vol. 6, Pergamon Press, Oxford, 1990.
ELSEVIER
controlled release
Journal of Controlled Release 41 (1996) 177-185
Mechanistic model for transdermal transport including iontophoresis Kyrsti Kontturi*, Lasse Murtom~iki Laboratory of Physical Chemistry and Electrochemistry, Helsinki University of Technology, Kemistintie IA, FIN-02150 Espoo, Finland
Received 28 June 1995; accepted 8 December 1995
Abstract
A mechanistic model is presented to describe the transdermal transfer of drugs. In the model, flux is divided into the contributions through the lipid matrix and through aqueous pores. During iontophoresis, only flux through aqueous pores is enhanced because the relative permittivity of the lipid matrix is too low to allow the existence of free ions which could work as current carriers. The transfer through the lipid matrix is assumed to take place in three steps: (i) partition at the skin/reservoir interface; (ii) diffusion through the lipid matrix; (iii) desorption at the stratum corneum/epidermis interface. Simulations to the present model agree with experimental results in the literature. Keywords: Transdermal transfer; Iontophoresis; Aqueous pores; Permeability vs. partition; Desorption
1. I n t r o d u c t i o n The ability of drugs to penetrate human skin depends on several properties: skin structure, lipophilicity, molecular size or configuration of the drug molecule etc. [1]. The quantitative estimation of these properties is quite difficult, and the number of unknown parameters makes it difficult to predict whether a particular molecule can be delivered transdermally or how its permeation can be enhanced. Several models for the transdermal transport have been proposed (see [2-6]. Although some of them [2-4] are, in principle, rigorous enough to describe transport with sufficient accuracy, they are not feasible to rapid estimation in the practice due to their complexity, but demand lengthy computations. Another class of the models is given by fitting measured permeabilities with some suitably chosen Correspondmg author.
parameters. Such models can have certain practical value in the prediction of the permeabilities of like drugs, but without a physical significance of the fitted parameters such a model remains only practice of statistical calculus. Still, an excellent exception to these models is given by Potts and Guy [5] who were able to give a sound physical basis for the parameters. In their model, the overall permeability, Kp, is given by l o g Ke =
+ 0.71 l o g P - 0.0061M r
(1)
where P is the water/n-octanol partition coefficient and M r the molecular weight of the molecule. The number of molecules fitted to Eq. (1) was 93, but the correlation coefficient was only 0.67, although the authors find it high. Moreover, the partition coefficient and molecular weight are not independent parameters but depend on each other, as the authors themselves point out. The basic idea of the modelling of transport
0168-3659/96/$15.00 © 1996 Elsevier Science Ireland Ltd. All rights reserved P I I S01 68-3659(96)01323-5
-6.3
178
K. Kontturi, L. Murtomdki / Journal of Controlled Release 41 (1996) 177-185
processes is not to know the exact mechanism of transport in great detail but to know what is its rate determining step. Furthermore, the model has to take account all the relevant factors, but not to be too complicated; the accuracy of any model is in any case bound to be limited because of the inherent variance of the permeabilities of different skin samples. Here, we try to compromise between these two requirements and create an alternative macroscopic and mechanistic model for transdermal transport, emphasizing specially iontophoresis of partially ionized drugs, and collect the properties affecting it into coefficients which can be experimentally determined. To avoid unnecessary mathematical complications originating from the solution of the partial differential equation (Fick's Second Law) [6], the model considers only steady-state fluxes. The model is justified in the view of the existing permeability vs. partition data, e.g. [7,8] and to explain our data about transdermal iontophoresis of sotalol and salicylate [9].
2. Model As in the previous paper [9], flux is divided into the two contributions (Fig. 1): flux through aqueous pathways, jw, and flux through the lipid matrix of stratum corneum, jo, without any closer consideration of the routes of penetration:
jp = jw + jo
The subscript 'p' denotes passive without enhancement. Furthermore, that iontophoresis enhances only the way, and, therefore, at the constant tophoretic flux is given by
J(/ = J ' . E
transport, i.e. it is assumed aqueous pathpotential ion-
+ jo
(3)
where E is the enhancement factor which can be theoretically calculated by, e.g. the Goldman's constant field approximation
p E=
~; u -
1-e
or the [9,10].
Planck's
E = (1 +
zFd cp RT
(4)
electroneutrality
~.__e__'~(._)(lnx,_-]\Xe_~j,X_ 1 "~.
approximation
i - ~,c/3
(5)
~...d i i
z is the charge number of the drug, and Aqb is the potential drop over the membrane; g is the ratio of the total ion concentrations on the both sides of the membrane; F, R and T have their usual significance. Discrepancies between the calculated and observed enhancement factors are discussed later on. Eq. (3) is justified by the very low conductivity of the lipid matrix. From Eq. (2) and Eq. (3), the contributions of the two mechanisms can be evaluated as
Jp
(2)
jw _
E-
1
(6)
Jp'E-Jif jo
jo_
jo
The flux through aqueous pathway is given by Fick's Law in perfect sink conditions as usual:
jw
E-
1
Cb jw
jw jo jw
jo v Fig. 1. Penetration routes through skin assumed in the model.
= ~D w_
h
(7)
(8)
where e is the fraction of the area of the aqueous pathway, D w is the aqueous diffusion coefficient and C b the bulk concentration of the drug; h is the length of the diffusion pathway which can be different from the membrane thickness due to tortuosity of the pathway. Because of the very low relative permittivity of
179
K. Kontturi, L. Murtomiiki / Journal o f Controlled Release 41 (1996) 1 7 7 - 1 8 5
the lipid matrix only nonionic solutes can exist inside it: the fraction of drug in a nonionized form (often depending on pH) is denoted with a. The more lipophilic drug is, the greater is its partition at the reservoir/stratum corneum interface, and thus the greater is its concentration inside the lipid matrix. If the transport were entirely due to Fickian diffusion, and perfect sink conditions applied below stratum corneum, the flux would increase without any limit with an increasing partition coefficient (see Discussion). This is not the case, but even convex shapes of log(Kp) vs. log(P) curves have been found [8,11]; Kp is the permeability and P partition coefficient of the non-ionized drug. Therefore, another mechanism has to be involved, as addressed later on. It is known that drugs are bound into stratum corneum, and long after iontophoretic treatment flux remains on a higher level than during passive diffusion [12]. This may seem to conflict with our assumption that current passes only through aqueous pores, but the law of mass action must be realized: the more current forces drug into the pores, the more it also brings the neutral form which is readily adsorbed onto the pore walls, after which more drug is dissociated to the neutral form etc. Thus, we postulate that flux through the lipid matrix consists of three steps: (a) partitioning at the reservoir/ stratum corneum interface; (b) Fickian diffusion through the lipid matrix; (c) desorption with the first order kinetics at the stratum corneum/viable epidermis interface (Fig. 2). As seen in Fig. 2, the concentration of drug in a nonionized form at the reservoir side of stratum corneum is o~pcb; C and F* are the volume and surface concentrations at the stratum corneum/epidermis interface. At the steady-state, the rate of diffusion through stratum comeum equals to the rate of desorption:
Interracial layer with F* stratum corneum
jo = D °
h
(~ -
kF*
(9)
epidermis
partition P
aC b ,jo v
Fig. 2. Transport through the lipid matrix in three steps: partition, diffusion and desorption. Fz ~
/.z *
_o C tz + R T I n - ~
= Iz
.o
F* + RTln Fo
o
_
,~ * o
=(#°-
/.twO) + (/zwo _
.o)
F*C o = RT l n - F° Cbar
D ° is the diffusion coefficient in the lipid matrix and
k is a homogeneous desorption rate constant (s -1). The two unknown quantities C and F * can be eliminated using the equality of their chemical potentials if it is assumed that the thermodynamic equilibrium prevails between them:
(10)
The interfacial layer does not mean that the drug is cumulated but it has a different chemical environment from that in the bulk phase. That is why chemical potentials need to be considered instead of simply converting volume concentrations to surface concentrations. Solving for the difference of the standard chemical potential at the interface, /.~,o, and in the lipid matrix, /z°, with adding and subtracting of the standard chemical potential in the aqueous phase, w o we obtain -
apc b -
I
/zwO _ * o )
F * = F ° ~ o exp ( / 2 ° ~ T -~w°) exp \
R-T
(11) The standard concentrations corresponding to each
K. Kontturi, L. MurtomiT"ki / Journal of Controlled Release 41 (1996) 177-185
180
other are C ° = 1.0 mol dm -3 and F ° = 1.18-10 -1° mol cm 2 [13]. The former exponent term clearly is P-~ because from the thermodynamic equilibrium between the lipid and aqueous phase it is obtained
wo_ o)
P -
a w - exp \-
._:6.5
..........................!ii . .... j - / / J
~-7
-R'T /
-7.5
= exp
" RT
where A G ° p is the free energy of partition. Assuming that viable epidermis can be considered as an aqueous phase, the latter difference between the standard chemical potentials is the free energy of desorption, A G ° p , and Eq. (11) can be written as
/
I.'ivo
(12)
/
/
-2
-1
0
1 log P
0,v _:-°-v-I 2
3
4
Fig. 3. S i m u l a t i o n to the m o d e l with the f o l l o w i n g p a r a m e t e r v a l u e s : e = 10 4, D w = 1 0 - 6 c m 2 s - ] , h = 1 0 0 / x m , a = 0.01, k o = 10 -4 c m s - I , D ° = 10 8 c m 2 s 1. T h e e n h a n c e m e n t factor E has b e e n c a l c u l a t e d using Eq. (4).
roe F* -
cO p exp \ - - ~ /
(13)
3. Simulations
Defining a heterogeneous desorption rate constant k o (cm S - l )
as
r 0
kD = k~
exp \ ~ - - /
(14)
we find
d
(15)
k F * = ko--f i
Inserting Eq. (15) into Eq. (9) and eliminating, we finally obtain jo
-
acbkD 1 + A'A-
kDh DOp
(16)
where all the constants have been collected into k o and a. Two limiting cases of Eq. (16) are readily seen: when A > > 1, i.e. P is very small (hydrophilic drug) otPC b jo = D o _
h
(17)
which is the familiar Fick's law; when A < < 1, i.e. P is large (lipophilic drug) j o = olCbkD
(18)
according to which desorption is the rate determining step. In the Appendix A, an example calculation is given, using our data of sotalol transfer [9], to show how different quantities can be evaluated.
Combining Eq. (8) and Eq. (16), a simulation in Fig. 3 has been carried out, and it indeed reminds of the behaviour presented in the literature; the permeability Kp is flux divided by the bulk concentration. The parameter values has been estimated from the experimental data [7]. As can be seen, the flux of hydrophilic drugs is enhanced quite a lot due to iontophoresis while the permeability of lipophilic drugs remains practically constant. Hence, the present model helps to predict if iontophoresis is a relevant method to enhance the flux of a particular drug. In Fig. 3, D ° and k o have constant values at the whole range of P which may not be realistic. Taking that the diffusion coefficient decays exponentially as a function of the molecular size [5], and assuming that log(P) increases as a linear function of the molecular size, it follows that D ° = a P b (a and b are appropriate positive constants). Similarly, assuming that also k D obeys this kind of a relationship, simulations in Fig. 4 have been carried out with different values of b indicated in the figure. This assumption for k D produces a convex shape for the permeability curve while for D ° it gives only a slowly increasing curve. It is remarkable that varying only one parameter, the partition coefficient of noionized drug, the sigmoidal curve is obtained. It is true that our model does not explicitly explain the decrease of the
K. Kontturi, L. Murtomiiki / Journal of Controlled Release 41 (1996) 177-185
-6
Fig. 3 :
]
:~
Second, it is not always quite clear how voltage drops across the membrane have been measured. If the resistance of the skin sample is R a and the intrinsic resistance of the voltage meter is R m, the following error analysis applies: Let the actual voltage drop across the membrane be Aqb and the measured value AqS'. Now:
-
-7.5
:12
A~' A~
-8 -2
-
I1
I
0
I
1 logP
I
2
3
181
1
-
1 + Ra/R,,
(19)
I
4
Fig. 4. Simulation s h o w i n g the effect o f partition coefficientd e p e n d e n t diffusion coefficient, D ° a n d desorption coefficient, k o. The other quantities are the same as in Fig. 3.
permeability at very high values of P [8,11] but it is obviously related either to strong binding to stratum corneum [14] or to a large molecular size [1,5,1517]. The former effect can be accomplished by a decreased desorption rate constant, k D, and the latter by the diffusion coefficients, D w [18] or D °.
The measured value thus is always lower than the actual one. We have measured in accordance with [21] resistances R a > 100 k~. To keep the error in Aq~ lower than 1%, R m has to be at least 10 M~. When using a potentiometric circuit as in [19,21], no errors should occur, but the comment of 'correction for solution iR drop' is enigmatic [21]. Third, enhancement due to electroosmosis has not been included in the model. In our own experience its effect is rather insignificant, although the phenomenon exists without any doubt. From Pikal and Shah's paper [22], we have estimated that the transport number of water, rw, defined as
FJw
rw-
4.1. Problems with the enhancement factor E
takes values 5 - 1 0 in hairless mouse skin (F is Faraday constant, Jw water flux and I electric current density). In human skin we have observed quite low charge densities [23] which means even lower values of r w. From Eq. (20) it is possible to calculate the order of magnitude of water flux, and inserting this value into the Nernst-Planck equation as a convective term, it is clear that electroosmotic contribution should be insignificant. Still, in the paper by Sims et al. [24] the value of an electroosmotic enhancement factor for electrically neutral mannitol was found to be as high as 7 when Aqb was 1.0 V. (From Eq. (4): E ~ 40, z = 1). Their value apparently corresponds to the combined effect of skin damage and electroosmosis because of the large deviation from Pikal and Shah's results where water flux had been measured. They also conclude that 'solvent flow considerations will probably impact in only a secondary manner in the iontophoretic transport of ionic species'. Because the quantitative incorporation of the
There are several examples in the literature, e.g. [19-21], where the observed enhancement factor is higher than the predicted one, which seems to make our model suspect. We can only suggest a few possible explanations to this dilemma, yet we have not observed enhancement factors higher than the theory predicts. First, instead of real potentiostatic control, controlled current has been administered, i.e. regardless of the resistance of the skin sample, certain current has been forced through, probably causing irreversible changes in the permeability. This has been documented as a lowered membrane resistance [21], and also in a potentiostatic mode as a 'damage factor, DF' [20]. Thus, the theoretical enhancement factor can be multiplied by DF, if so wanted, but it is always better to restrict oneself to potentiostatic control and not to overcome a voltage drop of 1.0 V. It also seems obvious that long hydration times are facilitating skin damage [21].
I
(20)
4. D i s c u s s i o n
182
K. Kontturi, L. Murtomiiki /Journal of Controlled Release 41 (1996) 177-185
effects of skin damage and electro-osmosis into our model is not straightforward, we want to suggest to avoid them fulfilling three requirements: (i) potentiostatic control: (ii) no higher voltage drops than 1.0 V; (iii) hydration time should not exceed 24 h. With these criteria fulfilled we have not observed anomalously high flux enhancements.
4.2. E x i s t e n c e o f a q u e o u s p o r e s
Potts and Guy have remarked in their rigorous paper [5] that the assumption of aqueous pores is not needed to explain the low permeability of small polar solutes, but 'the stratum corneum lipids alone can fully characterize the barrier properties of mammalian skin'. Furthermore, 'there is no physical evidence supporting the existence of aqueous pores'. Still, this cannot be true because if there were no aqueous pores, there would be no electroosmosis or streaming potential in human skin. The relative permittivity of lipid matrix certainly is low enough to prevent the existence of free ions; any attempt to measure the relative permittivity leads to an average value for skin, including the contribution of aqueous domains. If n-octanol is accepted as a model for lipid matrix the arguments above are easy to prove. The solubility of electrolytes into n-octanol is negligible, no values are even reported in the literature. Although if there were some dissolving electrolytes, say drugs, the degree of ion association would be very high because the relative permittivity of n-octanol is 10.34. Ion association is traditionally estimated by Bjerrum's or Fuoss' theories [25]. The former is valid in the cases where the thermal energy of ions ( k s T ) is comparable to the coulombic attraction which means, e.g. loose ion pairs of aqueous electrolytes. In the lipid matrix of skin, however, the situation is very much different: there are practically no free solvent molecules but organized layers of alkyl chains with very low relative permittivity (~-2) and polar groups with higher local permittivity (-~30). In such an environment contact ion pairs are more likely to exist, which can be described by the Fuoss' theory. According to this theory, the association constant, K , is given by
K
= 4. lO-247rNa(a/nm) 3
exp (b) 3
, b
Izizjle 2 -
47r~osrksT a
(21)
In Eq. (21), Nz is Avogadro number, e elementary charge, eo vacuum permittivity, Er relative permittivity of the solvent, k B Boltzmann constant, a contact distance between the ions, and zi,j are their charge numbers. For NaC1 in n-octanol a = 0.28 nm, b = 19.4 and K a ~ 1.4-10 7 M , i.e. the concentration of free ions is negligible. For a drug molecule with a = 0.7 nm, K a ~ 2000 M according to which ca. 50% would be dissociated if the total concentration were 0.1 M. But in reality, the concentration of the drug in a salt form is much less than that. If the partition coefficient of a drug is, say, 105 the free energy of partition is - 23 kJ m o l - l (see Eq. (12)). The free energy of transferring HC1 from water to n-octanol could be 4 5 - 5 0 kJ mol-1 [26]; then the free energy of partition of drug in HC1 form could be at least + 22 kJ mol-1 and its partition coefficient accordingly c a . 1 0 - 4 . Hence, even if the drug patch had 1 M solution the total concentration of the salt form drug would be ca. 0.1 mM only. Furthermore, in the above calculations, the distance between the ion centers is taken as the contact distance, which means that the net charges of the ions are assumed to be spread uniformly. In many cases, however, the charge is localized into specific sites of the ion, and the distance between the 'charge centers' should be taken as the contact distance of the Fuoss' theory. This would decrease the value of a and increase the association constant by orders of the magnitude (see e.g. NaC1 above). As a further evidence for aqueous pores, we were recently able to measure streaming potential in human skin [23] and to interpret the results with the classical theory of irreversible thermodynamics, i.e. with equations for transport in a charged capillary. The average capillary radius was found to be ca. 20 nm. The ohmic resistance of a skin sample, R a, with area A is Ra -
1 h
KA
(22)
where K is the conductivity of the solution depend-
K. Kontturi, L. MurtomiT"ki / Journal of Controlled Release 41 (1996) 177-185
ing on its concentration. We measured Rs~ with AC impedance techniques [27] in several NaCI concentrations (A = 1.0 cm 2) and found a typical value of the membrane coefficient A / h of 0.03 cm. Combining this information to the pore radius and taking h ~ 100 /xm, the average number of pores/cm 2 2.4.10 7, and their average separation distance --~2 /zm. It is therefore clear that these pores cannot be hair follicles or sweat glands. The chance that we are creating such pores when perturbing or freezing skin is still not overruled. 4.3. L i m i t a t i o n o f f l u x b y k i n e t i c c o n t r o l
The specific feature in our model is the first-order kinetic control of desorption of drug from stratum corneum to viable epidermis. Why is it needed? Potts and Guy [5] could account the flux limitation of highly lipophilic drugs solely for their large molecular weight, i.e. their low diffusivity. They could not even find limitation among small molecular but highly lipophilic n-alkenes, but the dispersion of their data (r 2 = 0.67) probably is too wide to recognize it. Moreover, specially in this case the partition coefficient and molecular weight were mutually dependent. It is also worth noticing that, in Fig. 4, we did take into account the molecular weight dependence of diffusivity, but did not find any saturation of log(Kp). In the model of Cleek and Bunge [6], it is assumed that the stratum corneum/viable epidermis interface is characterized with another partition coefficient, let it be here P ' and let the thickness of viable epidermis be h' and the drug diffusivity D ' . They solved the transient behaviour of the composite stratum corneum-viable epidermis system, but the following ideas can be compiled from their paper at the steady-state, although the boundary conditions are not explicitly given. Flux through this composite system is given by jo
~
D ° apc b h 1 + B; B
D°h'p ' D'h
183
drugs is limited by partition at the stratum corneum/ viable epidermis boundary [6]. This is true if P ' is substantially greater than P. But first, it is expected that P increases along with P ' , and second, as discussed above, the partition coefficient depends mainly on the polarity of the media. Because the polarity of viable epidermis must be close to that of water, P ~ P ' , and for highly lipophilic drugs Eq. (23) becomes j o ~ D , / h , . a C b. The effect of the partition coefficient thus vanishes but instead, the rate determining step would be diffusion through viable epidermis. The only limitation comes then from D', agreeing with Potts and Guy [5]. Therefore, to make the flux limitation of highly lipophilic drugs explicitly visible, kinetic control is needed. In the model, kinetic control is assigned to desorption but some another process is possible, too. Instead of desorption, there might be consumption of drug below stratum corneum-call it metabolism or clearance, for instance-described in chemical terms as a homogeneous first-order reaction with an appropriate rate constant, k n. Now, replacing k D with kn, the whole formalism, Eq. (16) Eq. (17) Eq. (18)), remains the same. If the homogeneous reaction were of the zeroth order the situation would become even more simple: in the case the reaction rate is slower than diffusion through stratum corneum, flux is limited by the reaction; in the case of fast reaction, any molecule passed through stratum corneum in readily consumed and perfect sink conditions prevail all the time, and once again there would be no explicit limitation of the flux of lipophilic drugs.
Acknowledgments The financial support of the Technology Development Centre of Finland (TEKES) is gratefully acknowledged.
Appendix A (23)
Note the similarity with Eq. (16). Now they deduce that for highly lipophilic drugs P ' takes high values (consequently B > > 1), and the flux of these
Considering our data for sotalol in [9] the following calculations are easily carried out: the theoretical enhancement factor, E, calculated from the Planck's electroneutrality approach is 14. l, while the observed
184
K. Kontturi, L. Murtomiiki / Journal of Controlled Release 41 (1996) 177-185
e n h a n c e m e n t f a c t o r w a s 7.2. T h e p a s s i v e p e r m e a b i l i ty (no i o n t o p h o r e s i s ) in T a b l e 1 in [9], d e f i n e d as Kp = J p / C b, w a s 1 3 . 3 . 1 0 -9 c m s 1. A p p l y i n g Eq. (6) a n d Eq. (7) it is f o u n d that jw
:
(7.2 -
=
0.53-
l).Jp/13.1
F r o m Eq. (8) J W / C
~ 0.47.Jp
--> j o
= 0 . 4 7 . 1 3 . 3 - 1 0 -9 c m s 1 =
e D W / h . In [26] w e f o u n d that for sotalol D w = 3 . 7 4 . 1 0 -6 c m 2 s -~. T a k i n g h --~ 100 / z m --> • 1 . 7 - 1 0 -5 w h i c h is a l o w p o r o s i t y indeed. S i m i l a r l y , J°[Cb
= 0.53 • 13.3 • 10 -9 c m s - l 7 . 0 0 . 1 0 -9 c m s - 1
F r o m Eq. (9), Eq. ( 1 5 ) a n d Eq. (16): jo aC b -
D ° -~ (P -
C / a C b) -
kD ~ P o~C b
kD 1 + A
U n f o r t u n a t e l y , w e d i d n o t m e a s u r e the flux o f sotalol in d i f f e r e n t b u l k c o n c e n t r a t i o n s , so that t h e separate evaluation of D ° and A would be possible, b u t a s s u m i n g t h a t D ° ~ 10 -8 c m 2 s -1 ( d i f f u s i v i t y c a n b e t w o or e v e n t h r e e o r d e r s o f m a g n i t u d e s m a l l e r in s t r a t u m c o r n e u m t h a n in a q u e o u s s o l u t i o n s [7]) a n d n o t i n g that P = 1.7 [28], p K , = 9.05 [28] a n d p H = 7.0 [9] ---> ~ ~ 0.0088, a n d it is finally f o u n d (h = I 0 0 /zm): C[olC
b =
0.909; k o =
1.5" 10 -6 c m s -1", ,~
= 0.875 P a r a m e t e r A takes the v a l u e 0.875 w h i c h is an i n t e r m e d i a t e one, a n d the t r a n s f e r o f n o n i o n i z e d sotalol t h r o u g h the lipid m a t r i x is u n d e r m i x e d d i f f u s i o n a n d d e s o r p t i o n c o n t r o l . It m u s t b e n o t i c e d t h a t ce ~ 0 . 0 0 8 8 only. A t h i g h e r pH, ce w o u l d take h i g h e r values, a n d it w o u l d b e i n f o r m a t i v e to r e p e a t o u r e x p e r i m e n t s at d i f f e r e n t pH.
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