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A kinetic model for ion transport across skin
257
Journal of Membrane Scuznce, 71 (1992) 257-271 Elsevrer Scrence Publishers B V , Amsterdam
A kinetic model for ion transport across skin J.C Keiste+* and Gerald B. Kastingb “7690 Floyd Hampton Road, Crowley, TX 76036 (USA) bThe Procter & Gamble Company, Mtamr Valley Laboratones, P 0 Box 398707, Cvunnat~, OH 45239-8707 (USA) (Received January 2,1992, accepted m revrsed form Apnl 1,1992)
Abstract A mathematical model for ion transport across a multrlammate barrier ISpresented whrch satlsfactorrly describes the exponentially nonlinear and slightly aaymmetrrc current-voltage properties of skm subJetted to an applied DC potential This is accomplished usmg rate equations, analogous to the ButlerVolmer theory of electrode reaction rates, m which voltage-dependent potent& energy barrrers govern ion transport In skin, the physical locatron of these bamers may be hppldlamellae m the stratum corneum and/or the walls of the eprthehal cells lmmg skm appendages Model parameters for the number of barriers, n, the symmetry of the bamers, (Y,and rate constants for sodium and chlonde Ion, KNaand Kcl, are calculated based on the current-voltage characteristrc of excised human skm immersed m saline solutrons and the sodium ion transference number for this trssue obtained durmg constant current iontophoresls The ratio K&n 1s found to be equal, wrthm expenmental error, to the sodnnn ion permeability coefficient, PNa, obtained by measurmg the passive diffusion of 22Na+ through the same skm samples, as predicted by the theory Using the parameter values developed m this paper m conJunctron with the kinetic model, rt should be possible to obtain a prediction of rontophoretrc delivery rates for other Ions based on the passive drffusron of these ions through skm Keywords blologrcal membranes, drug permeability, electrochemistry, electrodrffusron, lontophoresrs, skin, theory
Introduction Electrically enhanced delivery (iontophoresis) of drugs through the skin has received considerable attention in recent years as a potential nonparenteral means of delivering peptides and other poorly orally bloavailable drugs [ 11. The theoretical approaches which have been taken toward a quantitative description of Correspondence to G B Kastmg, The Procter & Gamble Company, Mramr Valley Laboratones, P 0 Box 398707, Cmcmnatl, OH 452396707, (USA) *Present address 3M Pharmaceutrcals, 260-3A-06, 3M Center Bmldmg, St Paul, MN 55144-1000 (USA)
this phenomenon have recently been reviewed [ 2,3]. Most of these models are based on solutions to the Nernst-Planck equations obtained under the constant field approximation [ 4-81; m several cases they have been extended to include electroosmotic effects by the inclusion of solute-solvent coupling terms [ 5-81. In addition, the authors have considered the electroneutrality approach (Planck’s approximation) to solvmg the Nernst-Planck equations, thereby obtaining solutions more appropriate for the case of a thick membrane surrounded by solutions of &fferent ionic strengths [ 91. The difficulty with all of these analyses is that
0376-7388/92/$05 00 0 1992 Elsevrer Scrence Publishers B V All rights reserved
258
J C Kester and G B KastmglJ Membrane Scl 71(1992) 257-271
none of them yields the exponentially nonlinear and slightly asymmetric current-voltage relationship observed in DC experiments with excised skin [lO,ll]. This paper presents a model which leads naturally to this behavior. Our approach is to assume that ion transport in skin 1s governed by a series of potential energy barrrers, the height of which can be modified by an applied voltage The theory is thus &rectly analogous to the Butler-Volmer theory for electrode reaction kinetics [ 121, except that multiple current carriers and multiple potential energy barriers are considered. Accordmg to Bockris and Reddy [ 131, the existence of voltage-dependent barriers is a necessary condition for a system to exhibit exponential current-voltage behavior. The physical origin of the potential energy barriers in skin is discussed in a later section of this paper. Similarities may be noted between the present theory and those reviewed by Bender [ 141 in the context of ion transfer across lipid bilayer membranes. Indeed the physics of ion transport in skin and in model lipid bilayers appears to have much in common. By fitting the model parameters to experimental currentvoltage data obtained on excised human skin, we show here that the kinetic model can quantitatively describe sodium and chloride transport in skin. Theory Our model for the skin is that of the lipidwater multilaminate membrane shown in Fig. 1. The lipid layers, represented by the solid lines, are considered to be potential energy barriers which an ion must surmount in order to pass from one aqueous layer to the next. Each aqueous layer (the regions labeled cl, cz, etc.) is considered to have a uniform composition and electrical potential. This description therefore does not include a diffusive resistance for these layers, 1.e. this resistance is considered to be
(receptorSOIll);,
Skin Barrier
Device i 1 (donorsoln)
Fig 1 Multllammate potential bamer model for skm The bamers, represented here by sohd vertical lines, are considered to have the form shown m Fig 2.
\x
f
CO __-._--_ ---_--________ _____-____ -__--C, _____ ____
I
Distance
Fig 2 Schematic diagram of a voltage-dependent potential energy bamer for ion transport This model 19 dvectly analogous to that used m the Butler-Vohner approach to electrode kmetlcs [ 121
neghgible compared to that of the lipid banrers. The key feature of the model lies in the voltage-dependent nature of the potential energy barriers. Experimentally, one notes that the resistance of the membrane drops rapidly and reversibly with increasing voltage [10,11,15]. In the model this phenomenon is treated by assuming the barriers to have the form shown in Fig. 2. In the absence of an externally applied electric field, the height of each potential energy barrier encountered by an ion passing either from left-to-right or from right-to-left is E, and the net rate of ion transfer across this barrier is:
J C Kelster and G B Kastmg/J
J=c,kexp(-E/RT)-c,,kexp(--E/RT) (I) Net ion flux = leftward ion flux - rightward ion flux Here J 1sion flux per umt area, c1 and c0 are the ion concentrations on the lefthand and righthand sides of the barrier, respectively, k ISthe frequency factor associated with barrier crossing for ion species L,R is the gas constant, and T ISthe absolute temperature. If the electrical potential on one side of the barrier (assumed to be the righthand side m Fig 2) is altered by an amount S#, and if the shape of the potential energy versus &stance curve is not altered by this change in potential, then a simple geometrical argument (Fig. 2) shows that the height of the potential energy barrier must increase for ions traveling in one direction and decrease for ions traveling m the other &rection, dependmg on then charge. For an ion of charge z, the barrier for left-to-right transport is now E+ azF&$ and for right-to-left transport is E - (1 - a )zF&b,where F is Faraday’s constant and cy is a parameter describing the symmetry of the barrier (0 < cr < 1, for a symmetric barrier, cy= 0.5 ). Therefore the rate at which the ion species crosses a single barrier is: J=c,kexp{-
[E-
259
Membrane Set 71(1992) 257-271
Once the concept of voltage-dependent transport barriers is accepted, the development of the kinetic rontophoresls model follows closely along the lines of the Butler-Volmer theory, with the additional feature that multiple charge carriers and multiple barriers are considered. We introduce a subscript, z, to refer to a particular ionic species and a second subscript, 1, to describe a particular barrier. Defining K, = k$xp ( - E,/RT) and Q, = z,F&$~/RTone has for the zth species at thejth barner
J,IK,=c,exp[(l--a!)rl,l-c,-,exp(--curl,~) (3) For a system in which m ions traverse n potential barriers, an equation of the form of eqn (3 ) may be written for each ion at each barrier The sum of the voltage drops across each barrier, A@,, must be equal to the total voltage drop across the membrane, A$. Additional constraints are imposed by the requirement of electroneutrality in the aqueous layers separatmg the potential barriers. Thus*
J&/K,= ctlexp [ (I- ff )tl,, I- czoew( - q,l) (4)
=c,,exp[(1--a)~~21-cLlexp(--crrl,z) (5) =. .
(l-a)zF&b]/RT}
-c,kexp[-(E+azF&)/RT]
=c,,ew[(l-ah,1 (2)
This model for voltage-dependent transport barriers is entirely analogous to that used to describe the kinetics of charge transfer at electrode-solution interfaces according to the Butler-Volmer theory [ 121. That theory describes the transfer of n electrons of charge - 1 across an mterfacial energy barrier, whereas eqn. (2) describes the transfer of one ion species of charge z across a potential energy barrier onginatmg within a membrane (possibly, as argued later, m association with lamellar hplds ) .
exp(-crq,,
)
-c,,-I z=l,
.,m
(6)
where rlLl+~2
+
+
vrn
=
d’4VRT
(7)
and m
Cz,c,=O
j=l,
..,n-1
(8)
1=1
One furthermore assumes that the ionic concentrations and electrical potentials external to the membrane are known to the investigator. For a system m which there are n barriers and
260
J C Kezster and G B Kastmng/J Membrane Scl 71(1992) 257-271
m ionic species, there are then m (n- 1) unknown ion concentrations, n- 1 unknown voltages and m unknown ion fluxes, for a total of m n+ n- 1 unknown quantities. These quantities may be obtained via the simultaneous solution of the m n flux equations (eqns. 4-6) m combination with n - 1 electroneutrality conditions (eqn. 8)) a total of m n + n - 1 equations. These equations cannot in general be solved analytically, but they can be treated by numerical methods. Furthermore, analytical solutions have been found for several cases of practical interest, as shown below. Analysis
Exact solutions to eqns. (4)- (8) are given for the trivial case of zero voltage drop across the membrane and for the cases of a symmetric electrolyte with either identical total ion concentrations on both sides of the membrane or a high voltage drop across the membrane. (A symmetric electrolyte is one m which the magnitude of the charge on all ions is identical, i.e. all ions are either univalent, divalent, trivalent, etc Elsewhere, we have called this a 1.1 electrolyte [3,9].) These solutions are used together with sodmm chloride transport data in skin [ 10,111 to develop parameter values for the model. After these special cases are consrdered, a numerical method for obtaining the general solution to the equations is described and an example is given of its use. Zero voltage ucross the membrane Since there is zero total potential drop A$ across the membrane, the individual S&, and, hence, the g,, in eqns (4 )- (7 ), are also zero Inspection of these equations shows that the concentration drops for each ton across each barrier must therefore be identical, i.e. J,/K, =c,l -cro
=c,2
-c,l
=
=c,,
-c,,_l
(9)
In order to satisfy eqn. (9) and the boundary conditions on the ion concentrations external to the membrane, the ionic fluxes must be given by:
J, = (K/n) (c,, - cco)
(10)
Thus the ion fluxes and distribution within the membrane are Ficklan m this limit. Since the passive permeability constant, P,, is defined as the ratio of the steady-state flux, J,, to the concentration difference, c,, - cZo,it follows that: P, = K,/n
(11)
This important relatlonshlp implies that rate constants for iontophoretlc delivery are attainable from passive diffusion measurements once an appropriate value of n is known.
Symmetru: electrolyte wtth tdentlcal total wn concentrattons on both srdes of the membrane
The details of this analysis are given m Appendix A. Considerable slmplificatlons to eqns. (4)- (8) arise because the total ion concentration is equal in all layers and equal voltage drops occur between layers. Individual ion species, however, may still have a complicated flux and concentration profile. The flux of species z, normalized as m eqn. (9)) 1sgwen by either eqn. (12) or eqn. (13). For positive ions: J&/K, zX(~-@”
& 1
-1
1 -c,OX-a’n (12) x c,,(1-x-““)+c,,(x-‘~“_x-1) [
For negative ions:
J C Kelster and G B Kastmg/J
Membrane Set 71 (1992) 257-271
x CJl-Xl/n)
1
+c,,(x”“-X)
-C,gXa’n
261
b (13)
where m both cases, X=exp(
]z]Fd@/RT).
(14)
The current-voltage relationship for the membrane does not have a simple form, but may be obtained by summing eqns. (12) and (13) for all the ions in the system, i.e I= CzLFJl The
(15)
iontophoretic
enhancement
factor,
EF= J, (A#) /J, (0)) for an ion initially present on only one side of the membrane (taken to be the right side m Fig 1) is given by:
0-1
0
10
20
30
40
zFA+VRT
Fig 3 Iontophorekc enhancement factor, EF, for steadystate flux according to eqn ( 16) Hnth (Y= 0 5 The parameter n IS the number of potential energy barriers m the model This calculation applies to a symmetnc electrolvte having equal total ion concentrations on both sides o”fa symmetric membrane The dashed line shows the predxtlon of the constant field solution to the Nernst-Planck model [4] 120
EFznX”-“‘/” (
&
>
(l-X-l/n)
a=03
piq 80
(16)
(for positive ions ) EFznX-“-“‘/”
-100
+x ( -
>
(for negative ions )
J + =cK+ (X(1-d/“-X-a/n) =&_
(X-
(l-a)/n-Xcu/n)
-50
(l-Xl/n)
50
(17)
A plot of eqn. (16) is shown m Fig. 3. For small values of the applied voltage, d#, the enhancement is identical to that found for the constant field solution to the Nernst-Planck equations [ 41, but at larger voltages the exponential nature of eqn. (16) leads to larger enhancements for the kinetic model. A special case of a system with identical total ion concentrations on both sides of the membrane is that of a binary symmetric electrolyte, e.g. NaCl In this case the flux of each ion is the sum of two exponentials,
J_
04
40-
-80 1 120
I/MFcK+
Fig 4 Normalized current-voltage relatlonshlp for a multdammate membrane nnmersed m a bmary symmetnc electrolyte havmg equal concentration on both sides of the membrane, but widely different ion moblhtles The curves are calculated from eqn (19) using the values n = 15 and K+/K_ = 10
and the total current is: I= ]z] Fc(K+ (X(l-a)‘n-X-a’n)
-K_(X(18)
100 zF&/RT
wwn_Xcr/n)) (19)
Figure 4 shows how a combination of asym-
J C KeLster and G B Kastmng/J Membrane See 71 (1992) 257-271
262
metric barriers (a! # 0.5) with ions having &fferent rate constants K+ and K_ for crossing these barriers leads to an asymmetric currentvoltage characteristic for the membrane. These curves were calculated from eqn. ( 19) using the values n = 15 and K, /K_ = 10. Higher currents are obtained when the ion species having the larger rate constant (m this case, the positive ions) is accelerated according to the larger of the two exponents, cyor 1 - cy.To dramatize the effect, a value of K+/K_ far from unity was selected for the graph in Fig. 4; however, this is not necessary to produce the smaller asymmetries seen m skin (see Figs. 6 and 7 ) . For the special case of a symmetric membrane (a = 0.5), eqns. (18) and (19) reduce to: J + = ZcK+ sinh ( 1z 1FA$/2nRT) J_ = - 2cK_ sinh ( 1z 1FA@/2nRT)
(20)
and 1=2)2lFc(K+
+K_)sinh(
lzlFd@/2nRT) (21)
The electrical transference numbers for the two ion species are in this case: t, =K+/(K+
+K_)
t- =K_/(K+
+K_)
(22)
Equation (22) shows that the transference numbers for a symmetric membrane immersed m a binary symmetric electrolyte are constants which are independent of the applied voltage. This is not true for an asymmetric membrane, as may be seen by &Cling eqn. (18) by eqn (19). As the voltage apphed to an asymmetric membrane is increased, the ion flux which is described by the larger of the two exponents, a! or 1 - cy,becomes a greater fraction of the total current. SymmetrK electrolyte wrth hgh voltage across the membrane The details of this analysis are given m Appendix B. These solutions apply when the volt-
age across the membrane is sufficiently large that the negative exponential terms in eqns. (4)- (6) may be neglected. Referring to Fig. 1, this means that for large positive d$ the solutions apply to cations traveling from right to left and to anions traveling from left to right. For large negative A@ the situation is exactly reversed. The normalized flux for an ion traveling from right to left in Fig. 1 is J,/Kz =c,,X(~-“)‘~Y~
(23)
For an ion traveling from left to right, J,/K, = - c,~X~‘~Y~
(24)
In these equations X is the quantity defined in eqn. (14) except that A$ 1s replaced by I A$ I, and Y is the ratio of the total ion concentration on the righthand side of the membrane to that on the lefthand side, i.e (25)
Y= &J&l
The quantities f and g are the following functions of n and (x* f=&-n(lYy); 1 g’l_y”-
(26) 1 n(l--7)
where y= (l-(x)/a
(27)
The current-voltage relationship for the membrane, obtained by summing all of the signed ion fluxes, is given by eqn. (28):
J C Keuter and G B Ka.sttng/J
Membrane Scr 71(1992) 257-271
The enhancement factor for an ion initially present only on the righthand side of Fig. 1 is: EF_,.&l-“)hyf -
(29)
For a symmetric membrane (cu=O.5, y= l), g= -f= (n- 1)/2n, leading to corresponding simplifications to eqns. (28) and (29). For example, for a symmetric membrane in the high voltage limit, eqn. (29 ) simplifies to: EF=nX’/2”(1/Y)‘“-“/2”
(30)
Figure 5 shows a plot of eqn. (30) for several choices of the total ion concentration ratio Y Larger enhancement factors are obtained when the donor side is dilute with respect to the receptor side, i.e. Y< 1, than when the situation is reversed. This prediction is reminiscent of 600
400 EF 300
i
100 0
0
20
40
60
80
zFAq/Rl
Fig 5 Iontophoretlc enhancement factor for steady state flux for a symmetrrc 15 barner membrane (cy = 0 5, n = 15) immersed m symmetnc electrolyte solutrons of arbrtrary concentratron The parameter Y 1s the ratio of the total ion concentratron on the donor side of the membrane (nghthand side m Fig 1) to that on the receptor side The curves were calculated numencally (sohd hnes) or accordmg to the high voltage approxrmatron, eqn (30) (dashed hnes)
263
the finding for the corresponding NernstPlanck system [ 91. In both cases, although the total flux of a particular species still increases with increasing concentration, diminishing returns are obtained at higher concentrations due to the lower degree of electrical enhancement. Also shown in Fig. 5 are results calculated according to the numerical method described below. By comparing the curves one can estimate the range of validity of the high voltage approximation. For the situations shown, the maximum deviation of eqn. (30) from the numerical solution was 7.5% for a dimensionless voltage zFA$/RT=40, decreasing to less than 0.5% for zFA#/RT=BO For a univalent electrolyte at room temperature, these voltages correspond to A$ = 1 and 2 V, respectively.
NumerEal sol&on to the general problem The model represented by eqns. (4)- (6) and (8) can be cast into the form f(x) = 0 and solved using conventional techniques for coupled nonlinear equations [ 161. We found the multidimensional Newton-Raphson method, which has a quadratic rate of convergence [ 161, to be sufficiently stable for most situations of interest. A BASIC program implementing this approach for small to moderately-sized equation systems is available from the authors. This program has been tested successfully for systems of up to about 50 coupled equations, which is sufficient to solve a problem mvolving 3 ion species and 16 barriers. As an example of its use, the enhancement factors for the systems shown in Fig. 5 were computed numerically and used to establish the range of validity of eqn. (30). Experimental support for theory The parameters for the kmetlc model were estimated by fitting the model to previously published current-voltage and sodmm ion
264
J C Keuter and G B Kastmng/J Membrane SCL 71(1992) 257-271
transport data for excised human skin immersed in pH 7.4 normal saline buffers [ 10,111. These studies employed split-thickness (250 pm) cadaver skm mounted in side-by-side diffusion cells (0.7 cm2). In one study [lo] the tissue had been stored frozen, in the other [ 111 it had not; the latter was termed “fresh skin ” The available data for each skm sample included the current-voltage characteristic, the permeability coefficient for Na+ under passive diffusion conditions, the Na+ transference number, tNa,obtained at a current of 50 ,uA (71 @/cm2), and the steady-state voltage drop associated with the latter measurement. Since the buffer ions were primarily sodium and chloride, we made the simplifying assumption that all of the electrical current was carried by either Na+ or Cl-. This gave a model having four parameters - n, a, KNaand Kc,. The parameter estimation process proceeded as follows: Using eqn. (19) and a least squares minimization routine from Bevington [ 171, the four parameters hsted above (three for fits with a fixed value of a) were estimated for each skin sample from its experimental current-voltage characteristic. Parameter values were further constrained by the requirement that the value of tNa calculated from the ratio of eqn. (18) to eqn. (19) match the experimental value of tNa at the steady state voltage determined in the study. Parameter values so obtained were crosschecked for consistency with the Na+ passive diffusion results using eqn. ( 11) . Three additional experiments were carried out in which tNain excised skin was determined as a function of current and voltage. These studies were conducted according to previously described procedures [lo], except that the 22Na+ solution was placed in either the donor or receptor chamber so that the measurements might be made for both polarities of the applied voltage. Sodmm ion transport was allowed to reach a steady state (approximately 60 min) prior to determining the transference numbers.
Current-voltage relationships were also determined for these samples. Table 1 and Figs. 6 and 7 show the results of fitting eqn. (19) to the current-voltage data in Refs. [ 10,111. Acceptable fits to both datasets were obtained, with significantly better agreement for the asymmetric model, (Y# 0 5, than for the symmetric model, LY=0.5 (F-test on squared residuals). Optimum values of a! ranged from 0.58 to 0.68 and of n ranged from 16 to 19. The combination of an exponent cx> 0.5 with a rate constant ratio KNa/KcIappreciably greater than unity gave an excellent match to the r-V data for most samples. The curves shown in Figs. 6 and 7 represent fits to the mean data, whereas the parameter values in Table 1 are the means of the mdlvidual sample values. Comparable parameter values were obtained by either method. The quality of the fits obtained was quite sensitive to the number of barriers m the model, n, even when the symmetry parameter (Ywas allowed to float freely. This was not unexpected in light of the fact that the value of n determmes the voltage at which the onset of exponential r-Vbehavior may be detected, as shown m Fig. 3. For example, the dashed curve m Fig. 6 was drawn using the optimum value of n for these data, n= 16. The 95% confidence limits on n for this dataset were 14 and 22, based on an F-test on the ratio of squared residuals. Other datasets tested gave similar results. The implications of the fact that about 16-19 barriers were required to describe the z-V data are discussed later. From eqn. (11) and the data m Table 1, the mean passive sodmm ion permeability coefficient for the 7 frozen skin samples, PNa, was estimated to be 3.5~ 10m5 cm/hr (using the values on line 1 of Table 1) or 4 0 x 10v5 cm/ hr (using the values on hne 2 ) These estimates may be compared with the mean experimental value, PNa ~4.1 x 10m5 cm/hr, determined by direct measurement of 22Na+ passive transport
265
J C Ketster and G B KastmngfJ Membrane SCL 71(1992) 257-271 TABLE 1
Kmetlc model parameters obtamed by fitting eqns (18) and (19) to human skm current-voltage and sodmm Ion transference number data Mean parameter value f SD
Dataset
log Frozen skin” (7 samples) Fresh skin” (9 samples)
KNIT
-325+027 -3 17+026 -345fO45 -339fO48
MSRb x lo3
log&P
n
(Y
-327+020 -340fO21 -362+045 -402fO41
159f19 1681!125 184+26 19Of37
10 51d 05t3+003 10 51d 068+0 12
3 63 2 60 10 51 5 30
“Base 10 loganthm of rate constant m cm/hr bMean squared residuals m fit (@/cm2)’ “Data from Ref [lo] dParameter value fixed at 0 5 ‘Data from Ref [ 111
400
1
300 -
300
200 -
200 loo-
loo-5
-2 2 -- 100
-3
-1 1
4
89, volt
--200 --300
:* I,’ ‘.
I, pa/cm2--400
Fig 6
Fit of the kmetlc model to mean current-voltage data for excised human skm which had been stored frozen [lo] The curves are calculated accordmg to eqn (19) usmg parameters determined by least squares fitting as described m the text Solid curve - symmetnc model, dashed curve - asymmetnc model
m these samples [lo]. Similarly, for the 9 fresh skin samples, mean values of PNa of 1.9 x 10m5 cm/hr and 2.1~10~~ cm/hr were calculated from the data in Table 1, whereas the directly measured value was PNa= 1.8 x 10s5 cm/hr. In both cases the agreement between the two mdependent methods of determining PNa was excellent. In addition to fitting the current-voltage and
I, pa/cm2
-
-100
-
-200
t
3
5
AI,
volt
-300 -400
Fig 7 Fit of the kinetic model to current-voltage data for fresh excised human skm [ 1 l] The curves are calculated as m Fig 6
passive sodium ion transport data, the parameters m Table 1 are also consistent with the measured sodium ion transference numbers during constant current iontophoresis. These values were tNe= 0.512 0.05 for frozen skm [lo] and tN, ~0.60 2 0.02 for fresh skin [ 111. Accordmg to eqn. (22)) the parameters m lines 1 and 3 of Table 1 yield &=0.512 and 0.595, respectively. For the asymmetric fits (lines 2 and 4), however, these values of tNa are matched only for the voltage at which they were deter-
J C Kelster and G B Kastmg/J
266
+Na
plotted versus the steady state voltage drop obtained during the constant current treatments (see Fig. 9 ) , No systematic dependence of tNa on voltage was evident, either for the mdividual samples or for the data as a whole. Two of the three samples tested had asymmetnc currentvoltage properties similar to those shown m Fig. 7. Thus the pre&ction of voltage-dependent ion transference numbers for skin samples with asymmetrrc current-voltage properties was not supported by this study.
07 t
. .
. .
.
. .
.
.
.
08
12
.
-16-12-08-04
0
04
16
BP, volt
Frg 8 Sodmm ion transference numbers obtained by measuring the unidrrectronal flux of “Na+ through excrsed human skm samples (0 7 cm2 ) durmg constant current rontophoresrs m a normal saline buffer, pH 7 4
2
0
1 1
**
--.----0-o-*--0
+++-+
-+-+-+-+
C
P -
.-.-.
0
2i
.-.
-.
-.
““,““,““,I”,,
"AA-b-5o xx-x-x ,r"
-l- (199
A-A1oo-A
150 x-x
200 Time,
Membrane Scl 71(1992) 257-271
mln
-2-
Frg 9 Voltage drop across the skm obtamed durmg constant current rontophoresls for one of the samples m Frg 8 The order m whrch the treatments were conducted was +io M(M), +50 M(+), +250 -50@(x), -250@(V) Theexpenment wasternnnated after 20 mm at - 250 ,uA
ti(o), -10,~AL
mmed (about l-2 V, depending on the sample). The asymmetric models would have tNa increasing substantially at lower voltages and decreasing at higher voltages. For example, the parameters on line 2 lead to the prediction that tNa should change from 0.75 to 0.49 over the voltage range - 1.5 to -t 1.5 V. In order to test this prediction, transference number determinations for 22Na+ were made over a wide voltage range for three excised skin samples. Figure 8 shows the results, which are
Discussion
The concept of voltage-dependent potentral energy barriers to ion transport has been applied to a variety of membrane transport problems [ 13,141 and appears from this work to be apphcable to skin. The exact form the model should take 1s to some extent at the discretion of the Investigator, as several different mathematical constructs lead to very similar predictrons for observable phenomena [ 141. Based on the evidence at hand, we have chosen a model which yields an exponential voltage dependence and a Fickian concentration dependence for ion fluxes, and has otherwise a mimmum of extra detail. The skin’s cation permselectlvity 1saccounted for in the model by the assignment of larger rate constants to positrve ions than to negative ions of comparable solutron mobrlity. This may be interpreted in terms of greater potentral energy barriers E, for negative ions than positrve ions due to the net negative charge on skin [see eqns. (1) and (2) 1. Elaborations on this basic framework would be required to explain the gradual loss (Fig. 9) and even more gradual recovery [ 10,151 of resrstance in skin subjected to an applied electric field, or to describe the coupling of solute and solvent fluxes observed in some experimental models [ 5-81. Considering the aforementioned time course for resistance changes m skin, it seems likely that the source of the voltage-dependent bar-
J C KeLster and G B KastwglJ
Membrane SCL 71(1992) 257-271
riers in this tissue could involve reorientation of the molecules comprising the transport barrier under the influence of an electric field, followed by diffusion of these molecules and their neighbors into a new minimum free energy configuration. The sites in skin having the highest current density during iontophoresis are often sweat ducts and hair follicles [l&20]; hence, reorientation of the lipids comprising the epithelial cell walls of these appendages may play a role in producing the effect. However, the fact that 16-19 barriers were required to fit the current-voltage data in excised skin (Table 1) suggests the involvement of the stratum corneum hpzd lamellae. (The base of sweat ducts and hair follicles are usually lined with only 23 epithelial cell layers, which would lead one to expect only 4-6 barriers if these structures were the primary determinates of the z-Vbehavior. ) This finding seems consistent with the concept of a diffuse current pathway through the stratum corneum which dominates the overall electrical properties of the tissue despite much higher current densities in the appendages. The mechanism of ion flux through this pathway may well involve transient pore formation in lipid bilayers, as described by Sims et al. [ 151 Their description 1sconsistent with the picture we have presented if one considers the “pores” to be simply regions of the bilayer in which the lipids have been temporarily reoriented by the electric field, forming conductzve pathways through the bzlayer. It is worth notmg, however, that the z-V curves from which the model parameters were estimated (Figs. 6 and 7) do not reflect the true steady state behavior of the tissue. Experimental constraints limited the data collection time m those studies to about 30 set per data point, whereas 20-30 min would have been required to achieve steady state voltage drops at the higher voltage levels (see Fig. 9). Had sufficient time been allotted for achievement of the steady state, smaller voltage would have been
267
associated with each of the higher current levels. This would have lead to z-V curves that could be described by models having smaller rate constants and fewer barriers than those reported here. Hence the possibihty of sigmficant appendageal currents with associated smaller values of n should not at this point be ruled out. The barrier symmetry is another feature which warrants further attention It 1s clear from Figs. 6 and 7 and the residuals m Table 1 that models incorporating asymmetric barriers ((Y# 0.5) most closely match the current-voltage properties of excised skin. However, the close mathematical coupling of the rate constant ratio KN,/Kc, with the barrier symmetry parameter a! lends uncertainty to the determination of both parameters. Chloride zon transport data m excised [21] and intact [22] human skin would suggest that its permeability coefficient Pc, (and, hence, its rate constant Kcz) is not less than one half that of sodium ion; hence the ratio &J&=4.2 calculated from the values in Table 1, line 4 may be unrealistically high This observation and the fact that ion transference numbers have m several studies been found to be independent of voltage (Fig. 8) or current [23-261 raise the question as to whether the barrier symmetry at steady state under a strong electric field is the same as that in untreated tissue. In other words, could reorientation of the barrier molecules by an electric field lead to a change m barrier symmetry? We note, however, that the two skin samples in Fig. 8 which initially had asymmetric current-voltage properties retained most of that asymmetry following the iontophoresis treatments, based on current-voltage measurements conducted at the end of the studies Conclusions A kinetic model m which ion transport through skin is governed by voltage-dependent
J C Kelster and G B Kastmng/J Membrane SCL 71(1992) 257-271
268
potentral energy barriers has the capability of describing the steady state DC electrical properties of excised human skin. The rate constants for ion transport in thus model can be determined from passive diffusion permeabrlrty coefficients once the effective number of barriers in skin is known. Model parameters determined by analyzing current-voltage curves for excused human skin samples immersed in saline solutions were consistent with passive sodium Ion transport measurements in these samples.
11
12
13
14
15
16
References 17 1
2
3 4
5
6
7
8
9
10
A K Banga and Y W Chlen, Iontophoretlc delivery of drugs Fundamentals, developments and blomedlcaI applications, J ControlledRelease, 7 (1988) l-24 R R Burnette, Iontophoresis, m. J Hadgraft and R H Guy (Eds ), Transdermal Drug Delivery Developmental Issues and Research Imtiatlves, Marcel Dekker, New York, NY, 1989, pp 247-291 G B Kastmg, Theoretical models for lontophoretic dehvery, Adv Drug Del Rev ,9 (1992)) m press J C Ken&r and G B Kastmg, Ionic mass transport through a homogeneous membrane m the presence of a uniform electric field, J Membrane SCI ,29 (1986) 155-167 V Srmlvasan and W I Hlguchl, A model for lontophoresls mcorporatmg the effect of convective solvent flow, Int J Pharmaceut ,60 (1990) 133-138 K TOJO, Mathematical model of lontophoretic transdermai drug delivery, J Chem Eng Jpn ,22 (1989) 512-518 L Wearley and Y W Chien, Enhancement of the cn mtro skm permeablhty of azldothymldme (AZT) via lontophoresis and chemical enhancer, Pharmaceut Res ,7 (1990) 34-40 M J Plkai, Transport mechamsms in rontophoresis I. A theoretical model for the effect of electroosmotlc flow on flux enhancement m transdermal lontophoresis, Pharmaceut Res ,7 (1990) 118-126 G B Kastmg and J C Ken&r, Application of electrodiffusion theory for a homogeneous membrane to iontophorekc transport through skin, J Controlled Release, 8 (1989) 195-210 G B Kastmg and L A Bowman, DC electrical properties of frozen, excised human skm, Pharmaceut Res , 7 (1990) 134-143
18
19
20
21
22 23
24
25
26
G B Kastmg and L A Bowman, Electrical analysis of fresh, excised human skm A comparison with frozen skin, Pharmaceut Res ,7 (1990) 1141-1146 A J Bard and L R Faulkner, Electrochemical Methods Fundamentals and Applications, Wiley, New York, NY, 1990, pp 86-118 J O’M Bockns and A K N Reddy, Modern Electrochemistry, Vol 2, Plenum Press, New York, NY, 1970, pp 935-937 C J Bender, Vohammetnc studies of ion transfer across model biologcal membranes, Chem Sot Rev , 17 (1988)317-346 S M Sims, W I Hlguchi and V Snmvasan, Skm alteratron and convective solvent flow effects during iontophoresls I Neutral solute transport across human skin, Int J Pharmaceut ,69 (1991) 109-121 G Dahlqulst and A BJorck, Numerical Methods, Prentice-Hall, Englewood Chffs, NJ, 1974, pp 218254 P R Bevmgton, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, NY, 1969, pp 222-231 R R Burnette and B Ongplpattanakul, Charactenzation of the pore transport properties and tissue alteration of excised human skm dunng lontophoresls, J Pharm Scl ,77 (1988) 132-137 C Cuhander and R H Guy, Sites of lontophoretlc current flow mto the skm Identification and charactenzation with the vlbratmg probe electrode, J Invest Dermatol (97 (1991) 55-64 S Grunnes, Pathways of ionic flow through human m skin m LUUO,Acta Derm Venereol (Stockh ), 64 (1984) 93-98 R R Burnette and B Ongpipattanakul, Charactenzatlon of the permselective properties of excised human skm dunng lontophoresls, J Pharm Scl, 76 (1987) 765-773 R T Tregear, The permeability of mammahan skm to ions, J Invest Dermatol ,46 (1966) 16-23 G B Kastmg, E W Merntt and J C Keister, An zn uwo method for studymg the lontophoretic enhancement of drug transport through skin, J Membrane Scl , 35 (1988)137-159 J B Phipps, R V Padmanabhan and GA Lattm, Iontophoretlc dehvery of model morgamc and drug ions, J Pharm Scl ,78 (1989) 365-369 G A Lattm, R V Radmanabhan and J B Phlpps, Electronic control of iontophoretm drug delivery, m Ann New York Acad SCI, Vol 618 (Wm J M Hrushesky, R Langer and F Theeuwes, Eds ) , Temporal Control of Drug Delivery, New York, NY, 1991, pp 450-464 J D DeNuzzlo and B Berner, Electrochemical and iontophoretlc studies of human skm, J Controlled Release, 11 (1990) 105-112
269
J C KeLster and G B KastmngjJ Membrane SCL 71(1992) 257-271
Appendix A Analysts of wntophoresu for the case of a symm.etru: electrolyte havmg Identuzal concentratwns on both sides of the membrane In accordance with eqns. (4)- (6)) the followmg equations describe the flux of charged species across the skin:
J,lK,=c,exp[(1--cw)9vl--Czr-lexp(--~v) L=l...m; J= l...n (Al) where ?I,= z,F&b,/RT.For a symmetric electrolyte, all the ionic charges z, have the same magnitude. Letting this quantity be 1z 1 and defining X, = exp ( 1z 1FGQl/RT), eqn. (Al ) can be rewritten as follows: J,/K, =c~~X;-~ -cy_lX;a(for J/K, =c;,X;-’
positive ions) (A2)
- C8, _ 1Xy (for negative ions) (A3)
where c,, is used to denote positive ions and c;l to denote negative ions. Equating the normalized current from barrier to barrier yields: c,~X:-~ -c,X,”
=c,~X~-~ -c,~X;~
=etc. (A4)
Summing eqns (A4 ) for all positive ions in the system and eqns. (A5) for all negative ions, and mvokmg the requirement of electroneutrality m each layer (eqn. 8)) one obtains: clx:-”
-c,x,Ol
C,Xy-’
-C,XF
=C,Xk-a =C2Xg-’
-C,X;” -C,Xg
=etc. (A6) =etc. (A7)
where C, is the total concentration of either positive or negative ions in layer 1. It is now assumed that the total ion concentration on both sides of the membrane is iden-
tical, i.e. C, = C, Under this circumstance it is shown below that having identical ion concentrations in each layer, C,= Cl = C, =etc., and identical voltage drops across each barrier, X1 =X,=X, = etc., is mathematically consistent. To do this, first assume that C, = C, Substituting for Cl m eqns. (A6) and (A7) and solving for C, yields: c,x;-a
=c,(x,(y
c2x;-’
=C,(XF
+x:-a +xy-’
-XT”) -XY)
(A3) (A9)
Dlvulmg eqn. (A8) by (A9 ) and crossmultiplying yields:
zX;-~(X:
+Xy-’
-Xy)
(AlO)
Now try equating X, =X,=v. This relationship is rea&ly seen to satisfy eqn. (AlO). Substituting w for Xl and X, m eqn. (A8) then leads to c, = c,. Therefore, identical concentrations in regions 0, 1, and 2 and identical voltage drops across barriers 1 and 2 are consistent with the assumption of identical concentrations in the 0 and 1 layers. This process can be extended to the nth barrier, where it will be observed that the final concentration (on the outside of the membrane) must be identical with the concentration on the other side of the membrane. Thus, conditions of identical total ion concentration in each aqueous layer and identical voltage drops across each barrier are consistent with having identical total ion concentrations on both sides of the membrane. Invoking the uniqueness principle, it is clear that these conditions represent the solution to the problem. The expression for the m&vidual flux of a given species is derived from either eqn. (A4) or (A5), depending on the sign of the charge. Note first that equal voltage drops across each barrier implies Xl =X, = .. =X1jn, where X= exp ( I z I Fd@/RT). For convenience, let
270
J C KeLster and G B Kastmg/J Membrane SCL 71(1992) 257-271
y = X1/“. Considering first the posrtlve ions, this relationship is substituted into eqn. (A4) and multiphed through by y” to obtain: CllY -Go
=%?Y-c*l
(All)
cz3y- cL2etc.
Cz2Y- Ccl =
Solving for the highest compartment concentration in each case yields: cc2=czlo_c, Y
Y
=C,*(l-Y2)
c,o(l-Y) - Y(l-Y)
Y(l-Y) =c,l
(l+y+y2)
czo(l+y)
Cl3 y2
(A=)
-
The solution to the problem with the opposite polarity proceeds similarly and leads to an equivalent result. defining The solution begms by X,=exp( IzI Fd@,/RT) as in Appendix A Equating ion currents from barrier to barrier and summing as m Appendix A yields eqns. (A6 ) and (A7 ) . For large voltages it is seen that the X, are also large and the terms containing X, raised to a negative exponent may be neglected. Consequently, eqns. (A6) and (A7) simplify mto the followmg: c1x:-“=c2x;-~=...=c,x~-~
(Bl)
-cox~=-c1x~=...=-c,_1x~
(B2)
where the C, represent the total concentratron of either positive or negative ions m layer 1. By induction, It 1s seen that eqn. (B2) lead to the following relation for the lathX
Y2
Continuing this process through to the nth barrier and solving for czl yields: Crl =
cznY"-lu-Y)+czoU-Yn-‘) l-y”
l-y”
(A13)
Substituting eqn. (A13) into eqn. (A2) and making use of the relationships y =X, = Xl”’ leads directly to eqn. (12) m the main text. A similar process can be carried out for negative ions, starting with eqn. (A5) and leading to eqn. (13) in the main text. The enhancement factor, EF, for an ion present on only one side of the membrane [ eqns. (16) and (17) ] follows directly from eqns. (lo), (12), and (13) by setting czo= 0 and dividmg eqns. (12) and (13) byeqn. (10).
033 Using both eqns. (Bl ) and (B2), mductron produces the following relation for the lathC: (l-a)/cu (B4) Letting p (1 -a) (B4),
(B5) 1+y+y2+
Appendix B
=-
(> Cl
CO
Hgh voltage kmrt of rontophorests for a symmetrrc electrolyte The solution to eqns. (4)-( 6) for the sltuatlon in which @,,- @o>> 0 is presented below.
/cu, one obtains from eqn.
+y*-1
Cl-Yk)/(l-Y)
Cl
=-
(
co
>
W) Setting k = n, one obtains a relationship for Cl in terms of the known values of C on both sides of the membrane, i.e.
J C KeLster and G B Kastuzg/J
Membrane Set 71(1992)
Cl-Y")/(l-Y)
(B7)
271
257-271
Combunngeqns. (B3), (B8),and the followmg for X,: X,=X1 y[cYk-l-l)/cl-Yn)m
(Bll)
yields
Equations (B6) and (B7) then grve: Cl-Yk)/(l-Y”)
= 038)
Note that the above relationship is completely independent of the applied voltage. This means that in the case of large voltages, the total ion concentration in each aqueous layer is independent of the applied voltage. The value X=exp( IzjF&/RT) is now defined as in eqn. (14), so that X=X,X,...X,,. As X is related to the total voltage drop across the membrane, it 1s a known quantity Writing each of the X, in terms of X, according to eqn. (B3) gives the following expression:
x=
[xl]~l($“][x,($“]. . (EN)
Using eqns. (B8) and (B9) we can write X1 in terms of known quantities. Letting Y= CJC, (equivalent to eqn. 25) one obtains: ~_~;y~Y-l~/~~l-Y”~culy~Y~-l~/~~l-Y~~~l =x;y-n/t(l-Ywy(l+Y+ =X~y[-n/cl-Y~,+l/(l-Y)lcu
... +Y”-‘)/[(l-Y”)al
@lo)
Therefore, x1 =jXY
{l/(l-Y”)-l/[n(l-Y)l}/~
011)
“XY{ r
y~-‘/(l-Y”)-ll[~(l-Y)l}/~
(B12)
To calculate the flux of a given ion we return to eqns. ( 4 ) - (6 ) in the main text, recalling that one of the two terms on the righthand side is negligible m the high voltage limit. For the polarity presently being considered, en >> &, cations will travel only from right to left in Fig. 1 and anions will travel only from left to right. Thus, for a cation traveling from right to left, eqns. (6) and (B12) yield:
J&/K, =c,,X:-~
which is equivalent to eqn. (23 ). For an amon traveling from left to right, eqns. (4) and (Bll ) yield:
J,/K, = - c,,,X: =-
C~~~~/ny~l/~l-Y”~-l/~n~l-Y~l~
(B14) which is equivalent to eqn. (24). The enhancement factor, EF, grven in eqn. (29) may be calculated from the ratio of eqn. (23) to eqn. (10) after setting c,,-,=O.The symmetric membrane formula given m eqn. (30) follows by taking the limit of eqns. (26) and (29) as a-+0.5, using two successive applications of l’Hospital’s rule to obtain the exponent.
Journal of Membrane Scuznce, 71 (1992) 257-271 Elsevrer Scrence Publishers B V , Amsterdam
A kinetic model for ion transport across skin J.C Keiste+* and Gerald B. Kastingb “7690 Floyd Hampton Road, Crowley, TX 76036 (USA) bThe Procter & Gamble Company, Mtamr Valley Laboratones, P 0 Box 398707, Cvunnat~, OH 45239-8707 (USA) (Received January 2,1992, accepted m revrsed form Apnl 1,1992)
Abstract A mathematical model for ion transport across a multrlammate barrier ISpresented whrch satlsfactorrly describes the exponentially nonlinear and slightly aaymmetrrc current-voltage properties of skm subJetted to an applied DC potential This is accomplished usmg rate equations, analogous to the ButlerVolmer theory of electrode reaction rates, m which voltage-dependent potent& energy barrrers govern ion transport In skin, the physical locatron of these bamers may be hppldlamellae m the stratum corneum and/or the walls of the eprthehal cells lmmg skm appendages Model parameters for the number of barriers, n, the symmetry of the bamers, (Y,and rate constants for sodium and chlonde Ion, KNaand Kcl, are calculated based on the current-voltage characteristrc of excised human skm immersed m saline solutrons and the sodium ion transference number for this trssue obtained durmg constant current iontophoresls The ratio K&n 1s found to be equal, wrthm expenmental error, to the sodnnn ion permeability coefficient, PNa, obtained by measurmg the passive diffusion of 22Na+ through the same skm samples, as predicted by the theory Using the parameter values developed m this paper m conJunctron with the kinetic model, rt should be possible to obtain a prediction of rontophoretrc delivery rates for other Ions based on the passive drffusron of these ions through skm Keywords blologrcal membranes, drug permeability, electrochemistry, electrodrffusron, lontophoresrs, skin, theory
Introduction Electrically enhanced delivery (iontophoresis) of drugs through the skin has received considerable attention in recent years as a potential nonparenteral means of delivering peptides and other poorly orally bloavailable drugs [ 11. The theoretical approaches which have been taken toward a quantitative description of Correspondence to G B Kastmg, The Procter & Gamble Company, Mramr Valley Laboratones, P 0 Box 398707, Cmcmnatl, OH 452396707, (USA) *Present address 3M Pharmaceutrcals, 260-3A-06, 3M Center Bmldmg, St Paul, MN 55144-1000 (USA)
this phenomenon have recently been reviewed [ 2,3]. Most of these models are based on solutions to the Nernst-Planck equations obtained under the constant field approximation [ 4-81; m several cases they have been extended to include electroosmotic effects by the inclusion of solute-solvent coupling terms [ 5-81. In addition, the authors have considered the electroneutrality approach (Planck’s approximation) to solvmg the Nernst-Planck equations, thereby obtaining solutions more appropriate for the case of a thick membrane surrounded by solutions of &fferent ionic strengths [ 91. The difficulty with all of these analyses is that
0376-7388/92/$05 00 0 1992 Elsevrer Scrence Publishers B V All rights reserved
258
J C Kester and G B KastmglJ Membrane Scl 71(1992) 257-271
none of them yields the exponentially nonlinear and slightly asymmetric current-voltage relationship observed in DC experiments with excised skin [lO,ll]. This paper presents a model which leads naturally to this behavior. Our approach is to assume that ion transport in skin 1s governed by a series of potential energy barrrers, the height of which can be modified by an applied voltage The theory is thus &rectly analogous to the Butler-Volmer theory for electrode reaction kinetics [ 121, except that multiple current carriers and multiple potential energy barriers are considered. Accordmg to Bockris and Reddy [ 131, the existence of voltage-dependent barriers is a necessary condition for a system to exhibit exponential current-voltage behavior. The physical origin of the potential energy barriers in skin is discussed in a later section of this paper. Similarities may be noted between the present theory and those reviewed by Bender [ 141 in the context of ion transfer across lipid bilayer membranes. Indeed the physics of ion transport in skin and in model lipid bilayers appears to have much in common. By fitting the model parameters to experimental currentvoltage data obtained on excised human skin, we show here that the kinetic model can quantitatively describe sodium and chloride transport in skin. Theory Our model for the skin is that of the lipidwater multilaminate membrane shown in Fig. 1. The lipid layers, represented by the solid lines, are considered to be potential energy barriers which an ion must surmount in order to pass from one aqueous layer to the next. Each aqueous layer (the regions labeled cl, cz, etc.) is considered to have a uniform composition and electrical potential. This description therefore does not include a diffusive resistance for these layers, 1.e. this resistance is considered to be
(receptorSOIll);,
Skin Barrier
Device i 1 (donorsoln)
Fig 1 Multllammate potential bamer model for skm The bamers, represented here by sohd vertical lines, are considered to have the form shown m Fig 2.
\x
f
CO __-._--_ ---_--________ _____-____ -__--C, _____ ____
I
Distance
Fig 2 Schematic diagram of a voltage-dependent potential energy bamer for ion transport This model 19 dvectly analogous to that used m the Butler-Vohner approach to electrode kmetlcs [ 121
neghgible compared to that of the lipid banrers. The key feature of the model lies in the voltage-dependent nature of the potential energy barriers. Experimentally, one notes that the resistance of the membrane drops rapidly and reversibly with increasing voltage [10,11,15]. In the model this phenomenon is treated by assuming the barriers to have the form shown in Fig. 2. In the absence of an externally applied electric field, the height of each potential energy barrier encountered by an ion passing either from left-to-right or from right-to-left is E, and the net rate of ion transfer across this barrier is:
J C Kelster and G B Kastmg/J
J=c,kexp(-E/RT)-c,,kexp(--E/RT) (I) Net ion flux = leftward ion flux - rightward ion flux Here J 1sion flux per umt area, c1 and c0 are the ion concentrations on the lefthand and righthand sides of the barrier, respectively, k ISthe frequency factor associated with barrier crossing for ion species L,R is the gas constant, and T ISthe absolute temperature. If the electrical potential on one side of the barrier (assumed to be the righthand side m Fig 2) is altered by an amount S#, and if the shape of the potential energy versus &stance curve is not altered by this change in potential, then a simple geometrical argument (Fig. 2) shows that the height of the potential energy barrier must increase for ions traveling in one direction and decrease for ions traveling m the other &rection, dependmg on then charge. For an ion of charge z, the barrier for left-to-right transport is now E+ azF&$ and for right-to-left transport is E - (1 - a )zF&b,where F is Faraday’s constant and cy is a parameter describing the symmetry of the barrier (0 < cr < 1, for a symmetric barrier, cy= 0.5 ). Therefore the rate at which the ion species crosses a single barrier is: J=c,kexp{-
[E-
259
Membrane Set 71(1992) 257-271
Once the concept of voltage-dependent transport barriers is accepted, the development of the kinetic rontophoresls model follows closely along the lines of the Butler-Volmer theory, with the additional feature that multiple charge carriers and multiple barriers are considered. We introduce a subscript, z, to refer to a particular ionic species and a second subscript, 1, to describe a particular barrier. Defining K, = k$xp ( - E,/RT) and Q, = z,F&$~/RTone has for the zth species at thejth barner
J,IK,=c,exp[(l--a!)rl,l-c,-,exp(--curl,~) (3) For a system in which m ions traverse n potential barriers, an equation of the form of eqn (3 ) may be written for each ion at each barrier The sum of the voltage drops across each barrier, A@,, must be equal to the total voltage drop across the membrane, A$. Additional constraints are imposed by the requirement of electroneutrality in the aqueous layers separatmg the potential barriers. Thus*
J&/K,= ctlexp [ (I- ff )tl,, I- czoew( - q,l) (4)
=c,,exp[(1--a)~~21-cLlexp(--crrl,z) (5) =. .
(l-a)zF&b]/RT}
-c,kexp[-(E+azF&)/RT]
=c,,ew[(l-ah,1 (2)
This model for voltage-dependent transport barriers is entirely analogous to that used to describe the kinetics of charge transfer at electrode-solution interfaces according to the Butler-Volmer theory [ 121. That theory describes the transfer of n electrons of charge - 1 across an mterfacial energy barrier, whereas eqn. (2) describes the transfer of one ion species of charge z across a potential energy barrier onginatmg within a membrane (possibly, as argued later, m association with lamellar hplds ) .
exp(-crq,,
)
-c,,-I z=l,
.,m
(6)
where rlLl+~2
+
+
vrn
=
d’4VRT
(7)
and m
Cz,c,=O
j=l,
..,n-1
(8)
1=1
One furthermore assumes that the ionic concentrations and electrical potentials external to the membrane are known to the investigator. For a system m which there are n barriers and
260
J C Kezster and G B Kastmng/J Membrane Scl 71(1992) 257-271
m ionic species, there are then m (n- 1) unknown ion concentrations, n- 1 unknown voltages and m unknown ion fluxes, for a total of m n+ n- 1 unknown quantities. These quantities may be obtained via the simultaneous solution of the m n flux equations (eqns. 4-6) m combination with n - 1 electroneutrality conditions (eqn. 8)) a total of m n + n - 1 equations. These equations cannot in general be solved analytically, but they can be treated by numerical methods. Furthermore, analytical solutions have been found for several cases of practical interest, as shown below. Analysis
Exact solutions to eqns. (4)- (8) are given for the trivial case of zero voltage drop across the membrane and for the cases of a symmetric electrolyte with either identical total ion concentrations on both sides of the membrane or a high voltage drop across the membrane. (A symmetric electrolyte is one m which the magnitude of the charge on all ions is identical, i.e. all ions are either univalent, divalent, trivalent, etc Elsewhere, we have called this a 1.1 electrolyte [3,9].) These solutions are used together with sodmm chloride transport data in skin [ 10,111 to develop parameter values for the model. After these special cases are consrdered, a numerical method for obtaining the general solution to the equations is described and an example is given of its use. Zero voltage ucross the membrane Since there is zero total potential drop A$ across the membrane, the individual S&, and, hence, the g,, in eqns (4 )- (7 ), are also zero Inspection of these equations shows that the concentration drops for each ton across each barrier must therefore be identical, i.e. J,/K, =c,l -cro
=c,2
-c,l
=
=c,,
-c,,_l
(9)
In order to satisfy eqn. (9) and the boundary conditions on the ion concentrations external to the membrane, the ionic fluxes must be given by:
J, = (K/n) (c,, - cco)
(10)
Thus the ion fluxes and distribution within the membrane are Ficklan m this limit. Since the passive permeability constant, P,, is defined as the ratio of the steady-state flux, J,, to the concentration difference, c,, - cZo,it follows that: P, = K,/n
(11)
This important relatlonshlp implies that rate constants for iontophoretlc delivery are attainable from passive diffusion measurements once an appropriate value of n is known.
Symmetru: electrolyte wtth tdentlcal total wn concentrattons on both srdes of the membrane
The details of this analysis are given m Appendix A. Considerable slmplificatlons to eqns. (4)- (8) arise because the total ion concentration is equal in all layers and equal voltage drops occur between layers. Individual ion species, however, may still have a complicated flux and concentration profile. The flux of species z, normalized as m eqn. (9)) 1sgwen by either eqn. (12) or eqn. (13). For positive ions: J&/K, zX(~-@”
& 1
-1
1 -c,OX-a’n (12) x c,,(1-x-““)+c,,(x-‘~“_x-1) [
For negative ions:
J C Kelster and G B Kastmg/J
Membrane Set 71 (1992) 257-271
x CJl-Xl/n)
1
+c,,(x”“-X)
-C,gXa’n
261
b (13)
where m both cases, X=exp(
]z]Fd@/RT).
(14)
The current-voltage relationship for the membrane does not have a simple form, but may be obtained by summing eqns. (12) and (13) for all the ions in the system, i.e I= CzLFJl The
(15)
iontophoretic
enhancement
factor,
EF= J, (A#) /J, (0)) for an ion initially present on only one side of the membrane (taken to be the right side m Fig 1) is given by:
0-1
0
10
20
30
40
zFA+VRT
Fig 3 Iontophorekc enhancement factor, EF, for steadystate flux according to eqn ( 16) Hnth (Y= 0 5 The parameter n IS the number of potential energy barriers m the model This calculation applies to a symmetnc electrolvte having equal total ion concentrations on both sides o”fa symmetric membrane The dashed line shows the predxtlon of the constant field solution to the Nernst-Planck model [4] 120
EFznX”-“‘/” (
&
>
(l-X-l/n)
a=03
piq 80
(16)
(for positive ions ) EFznX-“-“‘/”
-100
+x ( -
>
(for negative ions )
J + =cK+ (X(1-d/“-X-a/n) =&_
(X-
(l-a)/n-Xcu/n)
-50
(l-Xl/n)
50
(17)
A plot of eqn. (16) is shown m Fig. 3. For small values of the applied voltage, d#, the enhancement is identical to that found for the constant field solution to the Nernst-Planck equations [ 41, but at larger voltages the exponential nature of eqn. (16) leads to larger enhancements for the kinetic model. A special case of a system with identical total ion concentrations on both sides of the membrane is that of a binary symmetric electrolyte, e.g. NaCl In this case the flux of each ion is the sum of two exponentials,
J_
04
40-
-80 1 120
I/MFcK+
Fig 4 Normalized current-voltage relatlonshlp for a multdammate membrane nnmersed m a bmary symmetnc electrolyte havmg equal concentration on both sides of the membrane, but widely different ion moblhtles The curves are calculated from eqn (19) using the values n = 15 and K+/K_ = 10
and the total current is: I= ]z] Fc(K+ (X(l-a)‘n-X-a’n)
-K_(X(18)
100 zF&/RT
wwn_Xcr/n)) (19)
Figure 4 shows how a combination of asym-
J C KeLster and G B Kastmng/J Membrane See 71 (1992) 257-271
262
metric barriers (a! # 0.5) with ions having &fferent rate constants K+ and K_ for crossing these barriers leads to an asymmetric currentvoltage characteristic for the membrane. These curves were calculated from eqn. ( 19) using the values n = 15 and K, /K_ = 10. Higher currents are obtained when the ion species having the larger rate constant (m this case, the positive ions) is accelerated according to the larger of the two exponents, cyor 1 - cy.To dramatize the effect, a value of K+/K_ far from unity was selected for the graph in Fig. 4; however, this is not necessary to produce the smaller asymmetries seen m skin (see Figs. 6 and 7 ) . For the special case of a symmetric membrane (a = 0.5), eqns. (18) and (19) reduce to: J + = ZcK+ sinh ( 1z 1FA$/2nRT) J_ = - 2cK_ sinh ( 1z 1FA@/2nRT)
(20)
and 1=2)2lFc(K+
+K_)sinh(
lzlFd@/2nRT) (21)
The electrical transference numbers for the two ion species are in this case: t, =K+/(K+
+K_)
t- =K_/(K+
+K_)
(22)
Equation (22) shows that the transference numbers for a symmetric membrane immersed m a binary symmetric electrolyte are constants which are independent of the applied voltage. This is not true for an asymmetric membrane, as may be seen by &Cling eqn. (18) by eqn (19). As the voltage apphed to an asymmetric membrane is increased, the ion flux which is described by the larger of the two exponents, a! or 1 - cy,becomes a greater fraction of the total current. SymmetrK electrolyte wrth hgh voltage across the membrane The details of this analysis are given m Appendix B. These solutions apply when the volt-
age across the membrane is sufficiently large that the negative exponential terms in eqns. (4)- (6) may be neglected. Referring to Fig. 1, this means that for large positive d$ the solutions apply to cations traveling from right to left and to anions traveling from left to right. For large negative A@ the situation is exactly reversed. The normalized flux for an ion traveling from right to left in Fig. 1 is J,/Kz =c,,X(~-“)‘~Y~
(23)
For an ion traveling from left to right, J,/K, = - c,~X~‘~Y~
(24)
In these equations X is the quantity defined in eqn. (14) except that A$ 1s replaced by I A$ I, and Y is the ratio of the total ion concentration on the righthand side of the membrane to that on the lefthand side, i.e (25)
Y= &J&l
The quantities f and g are the following functions of n and (x* f=&-n(lYy); 1 g’l_y”-
(26) 1 n(l--7)
where y= (l-(x)/a
(27)
The current-voltage relationship for the membrane, obtained by summing all of the signed ion fluxes, is given by eqn. (28):
J C Keuter and G B Ka.sttng/J
Membrane Scr 71(1992) 257-271
The enhancement factor for an ion initially present only on the righthand side of Fig. 1 is: EF_,.&l-“)hyf -
(29)
For a symmetric membrane (cu=O.5, y= l), g= -f= (n- 1)/2n, leading to corresponding simplifications to eqns. (28) and (29). For example, for a symmetric membrane in the high voltage limit, eqn. (29 ) simplifies to: EF=nX’/2”(1/Y)‘“-“/2”
(30)
Figure 5 shows a plot of eqn. (30) for several choices of the total ion concentration ratio Y Larger enhancement factors are obtained when the donor side is dilute with respect to the receptor side, i.e. Y< 1, than when the situation is reversed. This prediction is reminiscent of 600
400 EF 300
i
100 0
0
20
40
60
80
zFAq/Rl
Fig 5 Iontophoretlc enhancement factor for steady state flux for a symmetrrc 15 barner membrane (cy = 0 5, n = 15) immersed m symmetnc electrolyte solutrons of arbrtrary concentratron The parameter Y 1s the ratio of the total ion concentratron on the donor side of the membrane (nghthand side m Fig 1) to that on the receptor side The curves were calculated numencally (sohd hnes) or accordmg to the high voltage approxrmatron, eqn (30) (dashed hnes)
263
the finding for the corresponding NernstPlanck system [ 91. In both cases, although the total flux of a particular species still increases with increasing concentration, diminishing returns are obtained at higher concentrations due to the lower degree of electrical enhancement. Also shown in Fig. 5 are results calculated according to the numerical method described below. By comparing the curves one can estimate the range of validity of the high voltage approximation. For the situations shown, the maximum deviation of eqn. (30) from the numerical solution was 7.5% for a dimensionless voltage zFA$/RT=40, decreasing to less than 0.5% for zFA#/RT=BO For a univalent electrolyte at room temperature, these voltages correspond to A$ = 1 and 2 V, respectively.
NumerEal sol&on to the general problem The model represented by eqns. (4)- (6) and (8) can be cast into the form f(x) = 0 and solved using conventional techniques for coupled nonlinear equations [ 161. We found the multidimensional Newton-Raphson method, which has a quadratic rate of convergence [ 161, to be sufficiently stable for most situations of interest. A BASIC program implementing this approach for small to moderately-sized equation systems is available from the authors. This program has been tested successfully for systems of up to about 50 coupled equations, which is sufficient to solve a problem mvolving 3 ion species and 16 barriers. As an example of its use, the enhancement factors for the systems shown in Fig. 5 were computed numerically and used to establish the range of validity of eqn. (30). Experimental support for theory The parameters for the kmetlc model were estimated by fitting the model to previously published current-voltage and sodmm ion
264
J C Keuter and G B Kastmng/J Membrane SCL 71(1992) 257-271
transport data for excised human skin immersed in pH 7.4 normal saline buffers [ 10,111. These studies employed split-thickness (250 pm) cadaver skm mounted in side-by-side diffusion cells (0.7 cm2). In one study [lo] the tissue had been stored frozen, in the other [ 111 it had not; the latter was termed “fresh skin ” The available data for each skm sample included the current-voltage characteristic, the permeability coefficient for Na+ under passive diffusion conditions, the Na+ transference number, tNa,obtained at a current of 50 ,uA (71 @/cm2), and the steady-state voltage drop associated with the latter measurement. Since the buffer ions were primarily sodium and chloride, we made the simplifying assumption that all of the electrical current was carried by either Na+ or Cl-. This gave a model having four parameters - n, a, KNaand Kc,. The parameter estimation process proceeded as follows: Using eqn. (19) and a least squares minimization routine from Bevington [ 171, the four parameters hsted above (three for fits with a fixed value of a) were estimated for each skin sample from its experimental current-voltage characteristic. Parameter values were further constrained by the requirement that the value of tNa calculated from the ratio of eqn. (18) to eqn. (19) match the experimental value of tNa at the steady state voltage determined in the study. Parameter values so obtained were crosschecked for consistency with the Na+ passive diffusion results using eqn. ( 11) . Three additional experiments were carried out in which tNain excised skin was determined as a function of current and voltage. These studies were conducted according to previously described procedures [lo], except that the 22Na+ solution was placed in either the donor or receptor chamber so that the measurements might be made for both polarities of the applied voltage. Sodmm ion transport was allowed to reach a steady state (approximately 60 min) prior to determining the transference numbers.
Current-voltage relationships were also determined for these samples. Table 1 and Figs. 6 and 7 show the results of fitting eqn. (19) to the current-voltage data in Refs. [ 10,111. Acceptable fits to both datasets were obtained, with significantly better agreement for the asymmetric model, (Y# 0 5, than for the symmetric model, LY=0.5 (F-test on squared residuals). Optimum values of a! ranged from 0.58 to 0.68 and of n ranged from 16 to 19. The combination of an exponent cx> 0.5 with a rate constant ratio KNa/KcIappreciably greater than unity gave an excellent match to the r-V data for most samples. The curves shown in Figs. 6 and 7 represent fits to the mean data, whereas the parameter values in Table 1 are the means of the mdlvidual sample values. Comparable parameter values were obtained by either method. The quality of the fits obtained was quite sensitive to the number of barriers m the model, n, even when the symmetry parameter (Ywas allowed to float freely. This was not unexpected in light of the fact that the value of n determmes the voltage at which the onset of exponential r-Vbehavior may be detected, as shown m Fig. 3. For example, the dashed curve m Fig. 6 was drawn using the optimum value of n for these data, n= 16. The 95% confidence limits on n for this dataset were 14 and 22, based on an F-test on the ratio of squared residuals. Other datasets tested gave similar results. The implications of the fact that about 16-19 barriers were required to describe the z-V data are discussed later. From eqn. (11) and the data m Table 1, the mean passive sodmm ion permeability coefficient for the 7 frozen skin samples, PNa, was estimated to be 3.5~ 10m5 cm/hr (using the values on line 1 of Table 1) or 4 0 x 10v5 cm/ hr (using the values on hne 2 ) These estimates may be compared with the mean experimental value, PNa ~4.1 x 10m5 cm/hr, determined by direct measurement of 22Na+ passive transport
265
J C Ketster and G B KastmngfJ Membrane SCL 71(1992) 257-271 TABLE 1
Kmetlc model parameters obtamed by fitting eqns (18) and (19) to human skm current-voltage and sodmm Ion transference number data Mean parameter value f SD
Dataset
log Frozen skin” (7 samples) Fresh skin” (9 samples)
KNIT
-325+027 -3 17+026 -345fO45 -339fO48
MSRb x lo3
log&P
n
(Y
-327+020 -340fO21 -362+045 -402fO41
159f19 1681!125 184+26 19Of37
10 51d 05t3+003 10 51d 068+0 12
3 63 2 60 10 51 5 30
“Base 10 loganthm of rate constant m cm/hr bMean squared residuals m fit (@/cm2)’ “Data from Ref [lo] dParameter value fixed at 0 5 ‘Data from Ref [ 111
400
1
300 -
300
200 -
200 loo-
loo-5
-2 2 -- 100
-3
-1 1
4
89, volt
--200 --300
:* I,’ ‘.
I, pa/cm2--400
Fig 6
Fit of the kmetlc model to mean current-voltage data for excised human skm which had been stored frozen [lo] The curves are calculated accordmg to eqn (19) usmg parameters determined by least squares fitting as described m the text Solid curve - symmetnc model, dashed curve - asymmetnc model
m these samples [lo]. Similarly, for the 9 fresh skin samples, mean values of PNa of 1.9 x 10m5 cm/hr and 2.1~10~~ cm/hr were calculated from the data in Table 1, whereas the directly measured value was PNa= 1.8 x 10s5 cm/hr. In both cases the agreement between the two mdependent methods of determining PNa was excellent. In addition to fitting the current-voltage and
I, pa/cm2
-
-100
-
-200
t
3
5
AI,
volt
-300 -400
Fig 7 Fit of the kinetic model to current-voltage data for fresh excised human skm [ 1 l] The curves are calculated as m Fig 6
passive sodium ion transport data, the parameters m Table 1 are also consistent with the measured sodium ion transference numbers during constant current iontophoresis. These values were tNe= 0.512 0.05 for frozen skm [lo] and tN, ~0.60 2 0.02 for fresh skin [ 111. Accordmg to eqn. (22)) the parameters m lines 1 and 3 of Table 1 yield &=0.512 and 0.595, respectively. For the asymmetric fits (lines 2 and 4), however, these values of tNa are matched only for the voltage at which they were deter-
J C Kelster and G B Kastmg/J
266
+Na
plotted versus the steady state voltage drop obtained during the constant current treatments (see Fig. 9 ) , No systematic dependence of tNa on voltage was evident, either for the mdividual samples or for the data as a whole. Two of the three samples tested had asymmetnc currentvoltage properties similar to those shown m Fig. 7. Thus the pre&ction of voltage-dependent ion transference numbers for skin samples with asymmetrrc current-voltage properties was not supported by this study.
07 t
. .
. .
.
. .
.
.
.
08
12
.
-16-12-08-04
0
04
16
BP, volt
Frg 8 Sodmm ion transference numbers obtained by measuring the unidrrectronal flux of “Na+ through excrsed human skm samples (0 7 cm2 ) durmg constant current rontophoresrs m a normal saline buffer, pH 7 4
2
0
1 1
**
--.----0-o-*--0
+++-+
-+-+-+-+
C
P -
.-.-.
0
2i
.-.
-.
-.
““,““,““,I”,,
"AA-b-5o xx-x-x ,r"
-l- (199
A-A1oo-A
150 x-x
200 Time,
Membrane Scl 71(1992) 257-271
mln
-2-
Frg 9 Voltage drop across the skm obtamed durmg constant current rontophoresls for one of the samples m Frg 8 The order m whrch the treatments were conducted was +io M(M), +50 M(+), +250 -50@(x), -250@(V) Theexpenment wasternnnated after 20 mm at - 250 ,uA
ti(o), -10,~AL
mmed (about l-2 V, depending on the sample). The asymmetric models would have tNa increasing substantially at lower voltages and decreasing at higher voltages. For example, the parameters on line 2 lead to the prediction that tNa should change from 0.75 to 0.49 over the voltage range - 1.5 to -t 1.5 V. In order to test this prediction, transference number determinations for 22Na+ were made over a wide voltage range for three excised skin samples. Figure 8 shows the results, which are
Discussion
The concept of voltage-dependent potentral energy barriers to ion transport has been applied to a variety of membrane transport problems [ 13,141 and appears from this work to be apphcable to skin. The exact form the model should take 1s to some extent at the discretion of the Investigator, as several different mathematical constructs lead to very similar predictrons for observable phenomena [ 141. Based on the evidence at hand, we have chosen a model which yields an exponential voltage dependence and a Fickian concentration dependence for ion fluxes, and has otherwise a mimmum of extra detail. The skin’s cation permselectlvity 1saccounted for in the model by the assignment of larger rate constants to positrve ions than to negative ions of comparable solutron mobrlity. This may be interpreted in terms of greater potentral energy barriers E, for negative ions than positrve ions due to the net negative charge on skin [see eqns. (1) and (2) 1. Elaborations on this basic framework would be required to explain the gradual loss (Fig. 9) and even more gradual recovery [ 10,151 of resrstance in skin subjected to an applied electric field, or to describe the coupling of solute and solvent fluxes observed in some experimental models [ 5-81. Considering the aforementioned time course for resistance changes m skin, it seems likely that the source of the voltage-dependent bar-
J C KeLster and G B KastwglJ
Membrane SCL 71(1992) 257-271
riers in this tissue could involve reorientation of the molecules comprising the transport barrier under the influence of an electric field, followed by diffusion of these molecules and their neighbors into a new minimum free energy configuration. The sites in skin having the highest current density during iontophoresis are often sweat ducts and hair follicles [l&20]; hence, reorientation of the lipids comprising the epithelial cell walls of these appendages may play a role in producing the effect. However, the fact that 16-19 barriers were required to fit the current-voltage data in excised skin (Table 1) suggests the involvement of the stratum corneum hpzd lamellae. (The base of sweat ducts and hair follicles are usually lined with only 23 epithelial cell layers, which would lead one to expect only 4-6 barriers if these structures were the primary determinates of the z-Vbehavior. ) This finding seems consistent with the concept of a diffuse current pathway through the stratum corneum which dominates the overall electrical properties of the tissue despite much higher current densities in the appendages. The mechanism of ion flux through this pathway may well involve transient pore formation in lipid bilayers, as described by Sims et al. [ 151 Their description 1sconsistent with the picture we have presented if one considers the “pores” to be simply regions of the bilayer in which the lipids have been temporarily reoriented by the electric field, forming conductzve pathways through the bzlayer. It is worth notmg, however, that the z-V curves from which the model parameters were estimated (Figs. 6 and 7) do not reflect the true steady state behavior of the tissue. Experimental constraints limited the data collection time m those studies to about 30 set per data point, whereas 20-30 min would have been required to achieve steady state voltage drops at the higher voltage levels (see Fig. 9). Had sufficient time been allotted for achievement of the steady state, smaller voltage would have been
267
associated with each of the higher current levels. This would have lead to z-V curves that could be described by models having smaller rate constants and fewer barriers than those reported here. Hence the possibihty of sigmficant appendageal currents with associated smaller values of n should not at this point be ruled out. The barrier symmetry is another feature which warrants further attention It 1s clear from Figs. 6 and 7 and the residuals m Table 1 that models incorporating asymmetric barriers ((Y# 0.5) most closely match the current-voltage properties of excised skin. However, the close mathematical coupling of the rate constant ratio KN,/Kc, with the barrier symmetry parameter a! lends uncertainty to the determination of both parameters. Chloride zon transport data m excised [21] and intact [22] human skin would suggest that its permeability coefficient Pc, (and, hence, its rate constant Kcz) is not less than one half that of sodium ion; hence the ratio &J&=4.2 calculated from the values in Table 1, line 4 may be unrealistically high This observation and the fact that ion transference numbers have m several studies been found to be independent of voltage (Fig. 8) or current [23-261 raise the question as to whether the barrier symmetry at steady state under a strong electric field is the same as that in untreated tissue. In other words, could reorientation of the barrier molecules by an electric field lead to a change m barrier symmetry? We note, however, that the two skin samples in Fig. 8 which initially had asymmetric current-voltage properties retained most of that asymmetry following the iontophoresis treatments, based on current-voltage measurements conducted at the end of the studies Conclusions A kinetic model m which ion transport through skin is governed by voltage-dependent
J C Kelster and G B Kastmng/J Membrane SCL 71(1992) 257-271
268
potentral energy barriers has the capability of describing the steady state DC electrical properties of excised human skin. The rate constants for ion transport in thus model can be determined from passive diffusion permeabrlrty coefficients once the effective number of barriers in skin is known. Model parameters determined by analyzing current-voltage curves for excused human skin samples immersed in saline solutions were consistent with passive sodium Ion transport measurements in these samples.
11
12
13
14
15
16
References 17 1
2
3 4
5
6
7
8
9
10
A K Banga and Y W Chlen, Iontophoretlc delivery of drugs Fundamentals, developments and blomedlcaI applications, J ControlledRelease, 7 (1988) l-24 R R Burnette, Iontophoresis, m. J Hadgraft and R H Guy (Eds ), Transdermal Drug Delivery Developmental Issues and Research Imtiatlves, Marcel Dekker, New York, NY, 1989, pp 247-291 G B Kastmg, Theoretical models for lontophoretic dehvery, Adv Drug Del Rev ,9 (1992)) m press J C Ken&r and G B Kastmg, Ionic mass transport through a homogeneous membrane m the presence of a uniform electric field, J Membrane SCI ,29 (1986) 155-167 V Srmlvasan and W I Hlguchl, A model for lontophoresls mcorporatmg the effect of convective solvent flow, Int J Pharmaceut ,60 (1990) 133-138 K TOJO, Mathematical model of lontophoretic transdermai drug delivery, J Chem Eng Jpn ,22 (1989) 512-518 L Wearley and Y W Chien, Enhancement of the cn mtro skm permeablhty of azldothymldme (AZT) via lontophoresis and chemical enhancer, Pharmaceut Res ,7 (1990) 34-40 M J Plkai, Transport mechamsms in rontophoresis I. A theoretical model for the effect of electroosmotlc flow on flux enhancement m transdermal lontophoresis, Pharmaceut Res ,7 (1990) 118-126 G B Kastmg and J C Ken&r, Application of electrodiffusion theory for a homogeneous membrane to iontophorekc transport through skin, J Controlled Release, 8 (1989) 195-210 G B Kastmg and L A Bowman, DC electrical properties of frozen, excised human skm, Pharmaceut Res , 7 (1990) 134-143
18
19
20
21
22 23
24
25
26
G B Kastmg and L A Bowman, Electrical analysis of fresh, excised human skm A comparison with frozen skin, Pharmaceut Res ,7 (1990) 1141-1146 A J Bard and L R Faulkner, Electrochemical Methods Fundamentals and Applications, Wiley, New York, NY, 1990, pp 86-118 J O’M Bockns and A K N Reddy, Modern Electrochemistry, Vol 2, Plenum Press, New York, NY, 1970, pp 935-937 C J Bender, Vohammetnc studies of ion transfer across model biologcal membranes, Chem Sot Rev , 17 (1988)317-346 S M Sims, W I Hlguchi and V Snmvasan, Skm alteratron and convective solvent flow effects during iontophoresls I Neutral solute transport across human skin, Int J Pharmaceut ,69 (1991) 109-121 G Dahlqulst and A BJorck, Numerical Methods, Prentice-Hall, Englewood Chffs, NJ, 1974, pp 218254 P R Bevmgton, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, NY, 1969, pp 222-231 R R Burnette and B Ongplpattanakul, Charactenzation of the pore transport properties and tissue alteration of excised human skm dunng lontophoresls, J Pharm Scl ,77 (1988) 132-137 C Cuhander and R H Guy, Sites of lontophoretlc current flow mto the skm Identification and charactenzation with the vlbratmg probe electrode, J Invest Dermatol (97 (1991) 55-64 S Grunnes, Pathways of ionic flow through human m skin m LUUO,Acta Derm Venereol (Stockh ), 64 (1984) 93-98 R R Burnette and B Ongpipattanakul, Charactenzatlon of the permselective properties of excised human skm dunng lontophoresls, J Pharm Scl, 76 (1987) 765-773 R T Tregear, The permeability of mammahan skm to ions, J Invest Dermatol ,46 (1966) 16-23 G B Kastmg, E W Merntt and J C Keister, An zn uwo method for studymg the lontophoretic enhancement of drug transport through skin, J Membrane Scl , 35 (1988)137-159 J B Phipps, R V Padmanabhan and GA Lattm, Iontophoretlc dehvery of model morgamc and drug ions, J Pharm Scl ,78 (1989) 365-369 G A Lattm, R V Radmanabhan and J B Phlpps, Electronic control of iontophoretm drug delivery, m Ann New York Acad SCI, Vol 618 (Wm J M Hrushesky, R Langer and F Theeuwes, Eds ) , Temporal Control of Drug Delivery, New York, NY, 1991, pp 450-464 J D DeNuzzlo and B Berner, Electrochemical and iontophoretlc studies of human skm, J Controlled Release, 11 (1990) 105-112
269
J C KeLster and G B KastmngjJ Membrane SCL 71(1992) 257-271
Appendix A Analysts of wntophoresu for the case of a symm.etru: electrolyte havmg Identuzal concentratwns on both sides of the membrane In accordance with eqns. (4)- (6)) the followmg equations describe the flux of charged species across the skin:
J,lK,=c,exp[(1--cw)9vl--Czr-lexp(--~v) L=l...m; J= l...n (Al) where ?I,= z,F&b,/RT.For a symmetric electrolyte, all the ionic charges z, have the same magnitude. Letting this quantity be 1z 1 and defining X, = exp ( 1z 1FGQl/RT), eqn. (Al ) can be rewritten as follows: J,/K, =c~~X;-~ -cy_lX;a(for J/K, =c;,X;-’
positive ions) (A2)
- C8, _ 1Xy (for negative ions) (A3)
where c,, is used to denote positive ions and c;l to denote negative ions. Equating the normalized current from barrier to barrier yields: c,~X:-~ -c,X,”
=c,~X~-~ -c,~X;~
=etc. (A4)
Summing eqns (A4 ) for all positive ions in the system and eqns. (A5) for all negative ions, and mvokmg the requirement of electroneutrality m each layer (eqn. 8)) one obtains: clx:-”
-c,x,Ol
C,Xy-’
-C,XF
=C,Xk-a =C2Xg-’
-C,X;” -C,Xg
=etc. (A6) =etc. (A7)
where C, is the total concentration of either positive or negative ions in layer 1. It is now assumed that the total ion concentration on both sides of the membrane is iden-
tical, i.e. C, = C, Under this circumstance it is shown below that having identical ion concentrations in each layer, C,= Cl = C, =etc., and identical voltage drops across each barrier, X1 =X,=X, = etc., is mathematically consistent. To do this, first assume that C, = C, Substituting for Cl m eqns. (A6) and (A7) and solving for C, yields: c,x;-a
=c,(x,(y
c2x;-’
=C,(XF
+x:-a +xy-’
-XT”) -XY)
(A3) (A9)
Dlvulmg eqn. (A8) by (A9 ) and crossmultiplying yields:
zX;-~(X:
+Xy-’
-Xy)
(AlO)
Now try equating X, =X,=v. This relationship is rea&ly seen to satisfy eqn. (AlO). Substituting w for Xl and X, m eqn. (A8) then leads to c, = c,. Therefore, identical concentrations in regions 0, 1, and 2 and identical voltage drops across barriers 1 and 2 are consistent with the assumption of identical concentrations in the 0 and 1 layers. This process can be extended to the nth barrier, where it will be observed that the final concentration (on the outside of the membrane) must be identical with the concentration on the other side of the membrane. Thus, conditions of identical total ion concentration in each aqueous layer and identical voltage drops across each barrier are consistent with having identical total ion concentrations on both sides of the membrane. Invoking the uniqueness principle, it is clear that these conditions represent the solution to the problem. The expression for the m&vidual flux of a given species is derived from either eqn. (A4) or (A5), depending on the sign of the charge. Note first that equal voltage drops across each barrier implies Xl =X, = .. =X1jn, where X= exp ( I z I Fd@/RT). For convenience, let
270
J C KeLster and G B Kastmg/J Membrane SCL 71(1992) 257-271
y = X1/“. Considering first the posrtlve ions, this relationship is substituted into eqn. (A4) and multiphed through by y” to obtain: CllY -Go
=%?Y-c*l
(All)
cz3y- cL2etc.
Cz2Y- Ccl =
Solving for the highest compartment concentration in each case yields: cc2=czlo_c, Y
Y
=C,*(l-Y2)
c,o(l-Y) - Y(l-Y)
Y(l-Y) =c,l
(l+y+y2)
czo(l+y)
Cl3 y2
(A=)
-
The solution to the problem with the opposite polarity proceeds similarly and leads to an equivalent result. defining The solution begms by X,=exp( IzI Fd@,/RT) as in Appendix A Equating ion currents from barrier to barrier and summing as m Appendix A yields eqns. (A6 ) and (A7 ) . For large voltages it is seen that the X, are also large and the terms containing X, raised to a negative exponent may be neglected. Consequently, eqns. (A6) and (A7) simplify mto the followmg: c1x:-“=c2x;-~=...=c,x~-~
(Bl)
-cox~=-c1x~=...=-c,_1x~
(B2)
where the C, represent the total concentratron of either positive or negative ions m layer 1. By induction, It 1s seen that eqn. (B2) lead to the following relation for the lathX
Y2
Continuing this process through to the nth barrier and solving for czl yields: Crl =
cznY"-lu-Y)+czoU-Yn-‘) l-y”
l-y”
(A13)
Substituting eqn. (A13) into eqn. (A2) and making use of the relationships y =X, = Xl”’ leads directly to eqn. (12) m the main text. A similar process can be carried out for negative ions, starting with eqn. (A5) and leading to eqn. (13) in the main text. The enhancement factor, EF, for an ion present on only one side of the membrane [ eqns. (16) and (17) ] follows directly from eqns. (lo), (12), and (13) by setting czo= 0 and dividmg eqns. (12) and (13) byeqn. (10).
033 Using both eqns. (Bl ) and (B2), mductron produces the following relation for the lathC: (l-a)/cu (B4) Letting p (1 -a) (B4),
(B5) 1+y+y2+
Appendix B
=-
(> Cl
CO
Hgh voltage kmrt of rontophorests for a symmetrrc electrolyte The solution to eqns. (4)-( 6) for the sltuatlon in which @,,- @o>> 0 is presented below.
/cu, one obtains from eqn.
+y*-1
Cl-Yk)/(l-Y)
Cl
=-
(
co
>
W) Setting k = n, one obtains a relationship for Cl in terms of the known values of C on both sides of the membrane, i.e.
J C KeLster and G B Kastuzg/J
Membrane Set 71(1992)
Cl-Y")/(l-Y)
(B7)
271
257-271
Combunngeqns. (B3), (B8),and the followmg for X,: X,=X1 y[cYk-l-l)/cl-Yn)m
(Bll)
yields
Equations (B6) and (B7) then grve: Cl-Yk)/(l-Y”)
= 038)
Note that the above relationship is completely independent of the applied voltage. This means that in the case of large voltages, the total ion concentration in each aqueous layer is independent of the applied voltage. The value X=exp( IzjF&/RT) is now defined as in eqn. (14), so that X=X,X,...X,,. As X is related to the total voltage drop across the membrane, it 1s a known quantity Writing each of the X, in terms of X, according to eqn. (B3) gives the following expression:
x=
[xl]~l($“][x,($“]. . (EN)
Using eqns. (B8) and (B9) we can write X1 in terms of known quantities. Letting Y= CJC, (equivalent to eqn. 25) one obtains: ~_~;y~Y-l~/~~l-Y”~culy~Y~-l~/~~l-Y~~~l =x;y-n/t(l-Ywy(l+Y+ =X~y[-n/cl-Y~,+l/(l-Y)lcu
... +Y”-‘)/[(l-Y”)al
@lo)
Therefore, x1 =jXY
{l/(l-Y”)-l/[n(l-Y)l}/~
011)
“XY{ r
y~-‘/(l-Y”)-ll[~(l-Y)l}/~
(B12)
To calculate the flux of a given ion we return to eqns. ( 4 ) - (6 ) in the main text, recalling that one of the two terms on the righthand side is negligible m the high voltage limit. For the polarity presently being considered, en >> &, cations will travel only from right to left in Fig. 1 and anions will travel only from left to right. Thus, for a cation traveling from right to left, eqns. (6) and (B12) yield:
J&/K, =c,,X:-~
which is equivalent to eqn. (23 ). For an amon traveling from left to right, eqns. (4) and (Bll ) yield:
J,/K, = - c,,,X: =-
C~~~~/ny~l/~l-Y”~-l/~n~l-Y~l~
(B14) which is equivalent to eqn. (24). The enhancement factor, EF, grven in eqn. (29) may be calculated from the ratio of eqn. (23) to eqn. (10) after setting c,,-,=O.The symmetric membrane formula given m eqn. (30) follows by taking the limit of eqns. (26) and (29) as a-+0.5, using two successive applications of l’Hospital’s rule to obtain the exponent.