a nonlinear finite element model of the electrode-electrolyte-skin system

8/2/2019 A Nonlinear Finite Element Model of the Electrode-electrolyte-skin System

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 41, NO. 7 , JULY 1994 68

A Nonlinear Finite Element Model ofthe Electrode-Electrolyte-Skin System

Dorin P a n e s c u , Member-, IEEE , J o h n G. Webster , Fellow, IEEE , and R o b e r t A , St r a t bucke r , Member, IEEE

Abstract-This study prese nts a two-dimensional finite elementmodel of the electrode-electrolyte-skin system which takes intoaccount the nonlinear behavior of the skin with respect to theamplitude of the voltage. The nonlinear modeling approach haspractical value for studies related to transcutaneous stimula-tion (e.g. maximizing the dyn amic rang e of sensory substitutionsystems, optimization of TENS, optimization of transcutaneouscardiac pacing, etc.). The model has three main regions: 1) theelectrolyte; 2) the skin; and 3) the body. The model consists of364 nodes, 690 elements and was generated on a MacIntosh I1using a version of FEHT (Finite Element for Heat Transfer)adapte d for electromagnetics. The electrodes are equipotentiallines and the electrolyte is modeled as a pure resistive regionwith constant conductivity. Although the electrode-electrolyteinterface can introduce nonlinearities, we did not take them intoaccount because the skin displays a much higher impedance. Theskin is modeled as a nonlinear material with the conductivitydependent on the applied voltage. To account for the mosaicstructure of the skin, we used ten different nonlinear subregionsof five different values of breakdown voltage. The region des-ignated body models the effects of the resistance associatedwith the dermis and the tissues underneath the skin, and has aconstant high conductivity. We studied the effects of two differentelectrolytes on the comfort of stimulation and found that therewas less potential pain delivered when high-resistivity electrolyteswere used. This was due to the larger nonuniformities in thecurrent density distribution which appeared for low-resistivityelectrolytes. Moreover, increasing the skin temperature made thecurrent density even more nonuniformly distributed for low-resistivity electrolytes. Experiments performed on the skin ofthe left arm, using 1-cm Ag-AgCI electrodes, showed that theskin broke down at spots of lowest breakdown voltage. This isconsistent with reports of previous experimental studies and haspractical value for the design of optimal electrodes.

I. INTRODUCTIONHE DESIGN of optimal electrodes for transcutaneousT lectrical stimulation usually requires the investigation

of current flow through the skin. Although it is possible tomeasure the current density undemea th the electrodes directly,this requires complex instrumentation. The direct measurementis also time consuming because it requires mounting fineprobes undemeath the electrode or skin. Usually lumpedmodels of the electrode-electrolyte-skin system are used foran initial assessment of this current flow. However, thisManuscript received May 21 , 1992; revised November 26 , 199 2, and MarchD. Panescu is with E P Technologies, Incorporated, Sunnyvale CA 94086J . G. Webster is with the Department of Electrical Engineering, UniversityR. A . Stratbucker is with the Radiation Health Center of the State ofIEEE Log Number 9402026.

1. 1994. Supported by a grant from Marquette Electronics, Incorporated.USA.of Wisconsin, Madison, WI 53706 USA.Nebraska, Omaha, NE 68 15 2 USA.

type of model has limited practical value since the currentdensity underneath the electrode is not uniformly distributed[ l] . Therefore, a lumped electrical model would not be ableto predict, for example, under which part of the electrodethe skin breakdown is more likely to occur. The lumpedmodels, as well as one-dimensional electric networks usedto model the skin, also have the disadvantage that they cannot quantify the current density vector (i.e., its magnitudeand direction). Moreover, a one-dimensional electric networkhas to include many nonlinear devices to yield resolutioncomparable with a nonlinear finite element model. The circuitsimulators employed to analyze this network (e.g. SPICE)might have convergence problems for a large number ofnonlinear devices. Also, the display of the current densityvectors during the postprocessing phase might be difficultto achieve. Although approximate analytical solutions to thisproblem have been reported [ 2 ] , he skin is usually consideredto have a constant conductivity and its mosaic structure is nottaken into account.

This study proposes a new approach of assessing the currentflow through the electrode-electrolyte-skin system based onnonlinear two-dimensional finite element analysis. The elec-trolyte and the tissues underneath the skin can be modeledwith good approxim ation as linear regions [3]. Moreover, untilit breaks down, the skin mainly determines the resistance ofthe system [ 3 ] , [4]. Therefore, we modeled only the skin asa nonlinear material, assigned constant conductivities to theother elements of the system, and did not take into accountthe nonlinearities of the electrode-electrolyte interface. Themodel also accounted for the effects of the mosaic electricalstructure of the skin.

The current flow through the skin has a nonlinear de-pendence on the voltage across it [ 5 ] , [6]. Hereafter, weconsider the voltage-current characteristic (VCC) of the skinto be: 1) quasilinear, 2) nonlinear and symmetric, or 3)nonlinear and asymmetric if there is a corresponding 1 )quasilinear, 2) nonlinear and symmetric, or 3) nonlinear andasymmetric, relationship between the biphasic current flowingthrough the skin and the symmetric biphasic voltage acrossit. For low amplitude voltages or for high frequencies, theVCC of the skin is quasilinear. As the amplitude of thestimulation increases or the frequency decreases, the VCCof the skin becomes nonlinear and symmetric. For largevoltages or for very low frequencies, the VCC becomesasymmetric. The sudden decrease of the skin impedance,causing current runaway, is known as electrical breakdownof the skin. As shown in [6], the onset of the breakdown

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occurs at the transition of the skin VCC from the nonlinearand symm etric into nonlinear and asymm etric. After thistransition, the subject experiences a stinging sensation ofpain. We used cyclic voltamm ogram s, similar in shape toLissajous figures, to describe the VCCs of the skin becausethey provide information about the imaginary part of theskin impedance, as well as its real part. A quasilinear VCCimplies a corresponding cyclic voltammogram very close toan ellipse. For a nonlinear and symm etric VCC, the cyclicvoltammogram is a distorted ellipse with identical distortionsfor the positive and negative phases. A system with a nonlinearand asymm etric VCC yields a cyclic voltammogram that is adistorted ellipse with the positive phase distortion differentfrom the negative phase distortion.The skin has a mosaic electrical structure mainly becausethe stratum comeum is crossed by sweat ducts, which arethe main current pathways and nonuniformly distributed onthe surface of the skin [l] , [7]-[ll]. To demonstrate themosaic electrical structure of the skin, [9] reported that theimpedance of the skin had very different values when itwas measured using a suction microelectrode at spots thatwere only few hundreds of micrometers apart. Webster [l]showed that, on a m icroscopic scale, the current passes throug hmany small pores. When the applied voltage became largeenough, one or more of these pores broke down generatingthe current runaway. Grimnes [101 presented evidence that themain current pathways through the skin were the sweat ducts.Based on this information, hereafter, the sweat ducts will beconsidered the regions of lowest breakdown voltage.We studied the effects of two different electrolytes oncomfort of stimulation and found that it was potentiallyless painful to stimulate when high-resistivity electrolyteswere used. When the temperature of the skin increased,for the low-resistivity electrolyte case, the current becameeven more nonunif ormly distributed, potentially causing morepain, but for the high-resistivity electrolyte its distributionwas not significantly affected. The design of electrodes fortranscutaneous stimulation should choose the optimal combi-nation of electrode size and electrolyte resistivity which yieldsthe lowest ratio between the maximal and minimal currentdensities through the skin, and, therefore, the most uniformcurrent distribution.

11. THE NONLINEARWO-DIMENSIONALFINITEE L E ME N TMO D E LTo perform our finite element computations we employedFEHT (Finite Element for Heat Transfer) [121, a programdevelop ed for solving heat transfer and electromagnetic prob-lems on Apple MacIntosh series computers configured with a

minimum of 1 MB of m ain memo ry. To build a finite elemen tmodel, the user has to manually generate a coarse mesh, thesize of which can be autom atically reduced within the regionsof interest. FEHT uses triangular elements and first-order basisfunctions. Before starting to calculate the solution of anyforward problem, FEHT checks if the mesh is free of errors(i.e., intersecting edges and nodes closer than a prespecifiedtolerance). The system of linear equations involved in the

finite eleme nt computatio ns is solved based on the Ch oleskyfactorization method.We considered the biological tissues as continuous materi-als, their conductivity being averaged over a volume contain-ing many cells. Each of the regions within the finite elementmodel (FEM) was considered to be time-invariant, isotropicand free of electric charges. We w ere interested in the valuesof the current density, given by

and because J = a E nd E = -V Vv . ( a V V ) = 0 ( 2 )

where J is the current density in [A/m2], E is the electricfield intensity in [V/m], V is the electric potential in [VI anda is the material conductivity in [S/m]. For constant a, (2 )becomes a Laplace equation. To solve (2) (i.e., to find theelectric potential), one of the following types of boundaryconditions can be used:

1) V is specified all the way around the model outline;2) the normal current density is specified all the way aroundthe model outline;3) V is specified only on some part of the model outlineand the normal current density on the rest of it.The function V which satisfies (2) and one of the conditionsl ) , 2) or 3) is uniquely determined. The boundary valueproblem s of type 1 ) are known as Dirichlet problem s, the onesof type 2) as Neumann problems, and the ones of type 3) asmixed problem s. We solved mixed problems by specifying thevoltage over the region corresp onding to the electrodes and setthe normal current density to zero on the outline between the

electrodes.The region co rrespond ing to the skin had a conductivitydependent on the potential, thus, (2) was nonlinear. EEHTemployed an iterative algorithm to solve this nonlinear equa-tion. Initially, (2) was solved for the skin conductivity equal tothe value corre sponding to 0 V across it. Then, based on thesecomputed values of the potential, the skin conductivity wasupdated. The iterations continued until the Euclid ean norm, ofthe difference between subsequent solutions of the potentialinside the model, was less than a chosen tolerance. The currentdensity was further computed applying J = - a V V insideeach element of the model. The last value in the series ofupdated skin conductivities was used to find J for the regionmodeling the skin.We checked the accuracy of FEHT for solving nonlinearproblems by comparison against the analytic solution for asimple model. For a homogeneous rectangular model, con-sisting of a single type of nonlinear material, for which thecurrent-voltage equation is

(3 )where I is the current flowing through the model expressedin microamperes, IO s a constant chosen to be 1.5 pA an dV is the voltage across the model expressed in volts, the


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~

683

TABLE IT H E M A X I M A LRRORS ERSUSHE N U M B E ROF NODESN THE SIMPLE ONLINEAR ODEL

Number of nodes 33 113 417Maximal error 3% 0.5% 0.4%

conductivity is equal to1 exp (F )g ( V )= Io--S V (4 )

where 1 and S denote the depth and the area of our rectangle.The voltage, V, was swept from 0 to 50 V. Table I lists themaximal errors, between the theoretical value of the currentand the value returned by FEHT, as a function of the numberof nodes in the model. We concluded that for more than 200nodes in the nonlinear region the accuracy of FEHT wassuitable for our purposes.Fig. 1 shows the lumped model for the skin proposed by[6]. The current flow through the skin is mainly affected byiontophoretic rather than electroosmotic mechanisms. There-fore, different ion species account for the outflow than for the

inflow through the skin. D1 an d D2 are nonlinear devices tosimulate both the nonlinear and asymmetric behavior of theskin. They are described by

where Io is the value of I for large negative V when the skintemperature, T , s at its normal value, Enit.,onsidered to be34C [13], and f l i (T ) , 2 ( T ) are functions of skin temperature[61

( 7 )where Vo is the value of f 2 (T) when the skin is at itsnormal temperature. The regions of skin with differentbreakdown voltage were assigned different VO.The parametersV~:Io,kl,kzl(klor D1 ) , k 2 2 ( k 2 or D z ) , an d k s are givenin Table 11. RcOrneums the resistance corresponding tothe leakage through the stratum corneum and has a veryhigh value. Csk is the equivalent capacitance of the stratumcorneum. Rbod y is the resistance associated with the dermisand it has a value of about 1 kR [3]. Except for Rb o d v , al lthe other elements in this model have their values dependenton the skin temperature. As [6] shows, when the VCC of theskin is asymmetric its impedance is mostly determined bydevices D1 an d D2 in series with Rbody . Therefore, in thefinite element model, the regions of skin are formed onlyby the antiparallel connected diode-like devices D1 and D2.Thus, the current flowing through the skin in our model isgiven by

Fig. 1. The electrical model of the skin. RCOrllellr.,orresponds to theresistance through the stratum corneum and it has a high initial value. Whenthe temperature of the skin increases its value decreases. D1 an d D z arenonlinear devices to account for the nonlinearities and asymmetries in theelectrical characteristic of the skin. C,k is the equivalent capacitance ofthe stratum corneum and increases with temperature of the skin. Rbodlcorresponds to the resistance of the dermis and has a low value.TABLE I1T H E PARAMETERS FOR TH E ELECTRICALODELN F IG. 1

Parameter ValueI O 1.5 p Ak21 (for the positive phase)l i 2 2 (for the negative phase)k3 0.003

Average of 10 s vk1 2

0.380.17

When the skin temperature increases, for a constant voltage V ,the current through the skin I , also increases. Therefore, theresistance of the skin decreases as its temperature increases.Fig. 2 shows the two-dimensional finite element model(FEM) we generated for this study. It consists of three mainregions: 1) the electrolyte (gel); 2) the skin; and 3) the body.The stimulating electrode was placed on top of the electrolyte,whereas the reference electrode was placed at the bottom endof the body region. The model comprised 364 nodes and690 elements and its dimensions were roughly scaled up by10, with respect to the experimen tal setup , where we useda 1-cm2 stimulating electrode. Out of these nodes, 218 werein the regions corresponding to skin. The meshes for the geland body had a higher density of nodes at the boundary withthe skin to ensure a smooth transition between the regions ofdifferent conductivities. The electrodes were equipotential. T heconductivities of the regions corresponding to the electrolyteand the body were constant. The conductivity of the skindepended nonlinearly on the applied voltage, based on (8).Table I11 shows the conductivities of the regions in our FEM.We studied the effects of two types of electrolyte (gel): a)the low-conductivity electrolyte yields a 1.5-kR equivalentlumped resistance of the gel layer; and b) the high-conductivityelectrode yields resistance of 80 R. Their effects upon thecurrent density through the system have been analyzed at twotemperatures of the skin: T = 34C and T = 40C. As Fig.2 shows, to simulate the mosaic structure of the skin, wedivided it into ten regions of conductivity, each containing anaverage of 20 nodes of the finite element mesh. The se regionshad the following values of VO (at the normal temperatureof the skin, T = T,,,it,): avg had VO= 5 V (breakdownvoltage of 15 V); 1 ha d V = 4 V (breakdown voltage of12 V); 2 had VO = 4.5 V (breakdown voltage of 13.5 V);3 had V, = 5.5 V (breakdown voltage of 16.5 V); 4 hadV, = 6 V (breakdown voltage of 18 V). For the skin, Table111 gives the conductivities at T = Tin;, , (34C). The values


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F I ~ . 2. The two-dimensional finite element model of th eelectrode-electrolyre-skin system. To account for the mosaic structureof the skin. we divided it into ten regions of f i ve different values for thehreakdown voltage.

T ABL E Il lT H E OUULICTIVITIES OF TH E REGIONSN TH E FINITE ELEMENTODELRegion ConductivityGcl (high-re\istivity) 0.003 S/ mGe l (low-re5istivity) 0.06 S/ mSkin "avg" at 34OCSkin " 1 " at 34OCSkin "2 " at 34OCSkin "3 " ar 34CSkin "4 " at 34CBody 0.7 S/m

0.7he-h/V*(exp(V/S V) + exp(-V/S V) ) S/ m0.76e-6/V*(exp(V/4 V) + exp(-V/4 V)) S/m0.76e-6/V*(exp(V/4.5 V) + exp(-V/4.5 V)) S/m0.76e-6/V*(exp(V/5.5 V ) + exp(-V/5.5 V)) S/m0.76e-6/V*(exp(V/6 V) + exp(-V/6 V )) S/ m

corresponding to T = 40C ca n be computed by making useof ( 6 ) , 7) and (S) , and they are higher than those in Table 111.We chose two regions of lowest breakdown voltage because, inthe experiment presented in Section 111, the skin broke down atfive spots underneath the 1-cm2 electrode. The results reportedby [SI and [lo] motivated us to consider these spots as thelowest breakdown voltage regions of skin. Hence, these yieldabout two regions of lowest breakdown voltage per 1-cm oflinear dimension. The "body" was modeled to have a verylow equivalent lumped resistance [ 3 ] .We assessed the effectsof the proximity of the vertical Neumann type boundaries byperforming simulations with increased distance between themand the edge of the gel. The results were not significantlydifferent from those corresponding to the configuration in Fig.2.

111. SIMULATION A N D EXPERIMENTSWe performed simulations at two temperatures of the skin,

34C and 40C, for two types of electrolyte (gel). The voltageacross the system varied from 5 V to above 50 V , in orderto cover all of the quasilinear, nonlinear and symmetric, andasymmetric ranges.A . Siniidatioris Usirig the Hig h-Re sistility Elrctrnlyte

At 5 V applied across the system the ratio R, between thehighest and lowest current densities in the skin undemeath theelectrode, was 1.79, when the skin temperature was 34"C , and1.65, at 40C. The current was rather uniformly distributed,and the whole voltage was developed across the skin.

(b)Fig. 3. (a ) The current f low through the model at 10V for the high-resistivityelectrolyte, when the skin temperature was 34OC. The ratio between themaximal and minimal current densities in the skin undemeath the electrodewas 2.7: ( b ) the equipotential contours. The entire voltage was developedacross the thickness of the skin because of its very high resistance. Theregions " I" are hatched.

At 10 V applied across the system, R was 2.7 when theskin temperature was 34"C , and 2.2, at 40C . The current was,again, rather uniformly distributed and the whole voltage wasdeveloped across the skin. Fig. 3(a) shows the current densitydistribution in the model for the skin temperature equal to34C. Fig. 3(b) shows the equipotential contours at 10 V fo rthe skin temperature equal to 34 C. Note that all the 10 V aredeveloped across the skin as we would expect (i.e., becauseof the very large resistance of the skin).At 22 V applied across the system, R was 6.23, when theskin temperature was 34"C, and 6.06, at 40C. The currentwas no longer uniformly distributed; a larger amount wasflowing through regions "1 ,"which were the lowest breakdownvoltage regions. However, most of the applied voltage wasstill developed across the skin. Above 22 V , the higherskin conductivity at 40C made the current slightly morenonuniformly distributed than at 34C.At 50 V applied across the system, the ratio R was 14.87,when the skin temperature was 34"C, and 15.7, at 40C.The current was funneled through regions "1" where the skincurrent density reached its maximum of 0.56 mA /cm2, at34"C, and 0.72 mA/cm2 at, 40C. Fig. 4(a) shows the currentdensity distribution and Fig. 4(b) shows the equipotentialcontours for this case, both at 34C. The length of the shaftsof the arrows varies linearly with the current density, from itsminimum, shown in Fig. 3(a), to the values in Fig. 4(a). Notethat the major part of the applied voltage was developed acrossthe electrolyte. Across the regions "1" only 4 V, at 3 4 C a nd


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(b )Fig. 4. (a ) The current flow through the model at 50 V for the high-resistivityelectrolyte, when the skin temperature was 34OC. The current density distribu-tion was much more nonuniform and the current flows mostly through the tworegions of lowest voltage breakdown. The ratio X wa s 14.87 and the maximalcurrent density was 0.56 mA/cm2; (b) the equipotential contours. The voltagedeveloped across the thickness of the two regions of lowest voltage breakdownwas about 4 V . The voltage developed across the thickness of the skin in themiddle of the model was about 20 V. Anomalies in arrow directions may becaused by current flowing around regions having high breakdown voltages.Shaft length representing minimal current density is short. Length increaseslinearly with current density. The regions I are hatched.0.8 V, at 40C, were developed, while across regions in themiddle of the model the voltage was about 20 V , at 34 C, and1.3 V , at 40C. T he difference was a consequence of the factthat the skin first broke down at the regions 1, which had thelowest voltage breakdown. At 40C, the skin resistance wassignificantly reduced, therefore, the voltage across it decreasedas well. The display of the current density and equipotentialcontours looked similar for 40C, therefore we did not includefigures.B . Simulations Using the Low -Resistiiity Electmlyre

At 5 V applied across the system, the ratio R was 1.78,when the skin temperature was 34C, and 1.7, at 40C. Thecurrent was rather uniformly distributed and the whole voltagewas developed across the skin.At 10 V applied across the system, R was 2.75, when theskin temperature w as 34C , and 2.25, at 40C. T he currentwas, again, rather uniformly distributed, and the whole voltagewas developed across the skin. Because the current distributionand the equipotential contours were very similar to those forthe high-resistivity electrolyte we did not include figures.At 28 V applied across the system, R was 7.67, when theskin temperature was 34C, and 7.8, at 40C. The currentwas no longer uniformly distributed, a larger amount wasflowing through regions . However, most of the applied

voltage was still developed across the skin. At 40C andabove 20 V, the higher skin conductivity made the currentmore nonuniformly distributed than at 34OC. Also, the changesdue to the temperature increase were greater than for thehigh-resistivity electrolyte.When we injected above the same amount of current intothe system as for 50 V applied across the system with thehigh-resistivity electrolyte, the ratio R was 33.7, at 34C. and40.1 V , at 40C. The voltage dro p across regions 1 was 41V. at 34 C and 6.5 V, at 40C. The voltage across the regionsin the middle of the model was 47 V, at 34C and 1 1 V, at40 C. Fig. 5(a ) shows the current density distribution and Fig.5(b) shows the equipotential contours for the skin temperatureequal to 34C. More current is funneled through the regionsof lowest breakdown voltage than for the case when the high-resistivity electrolyte was used. The current density reacheda maximum of 5.9 mA/cm2. at 34C. and 11.8 mA/cm2, at40C, inside regions I . There is no relation between thelength of the shafts of the arrows in Fig. 4(a) and in Fig. S(a).If a proportional representation were used, the shafts in Fig.5(a) would have been about ten times longer. The maps ofthe current density and equipotential contours looked similarat 40C, therefore we did not include figures.

C . Espet-inient to Check the Hypothesis that the Skin InitiallyBreaks DoMn at the Regions of Lovt9est Breakdovtw Voltuge

Both of the simulations in Sections 1II.A and B showedthat the skin initially broke down at the regions of lowestbreakdown voltage. This suggests that for a test on theskin, we should observe the breakdown initially occurringonly at several spots underneath the stimulating electrode.To test this assertion, we built two I-cm Ag-AgC 1 (becauseAg-AgC1 approaches the characteristics of perfectly nonpolar-izable electrodes (41) square electrodes from a larger tab ECGelectrode. We tested the skin on the left forearm. The referenceelectrode, a pregelled 36-cm2 Ag-AgC1 electrode, was appliedin the region of the left biceps muscle. To have reproducibleexperiments, to keep the electrode in place and to distributethe pressure uniformly, we used a small rubber band betweenthe shaft of a force gage and the electrode, and applied aconstant force of 0.5 N. First, we applied a sine wave of IO-Vamplitude and 2 0-Hz frequency. The n, after the skin recoveredfor about 15 min, we increased the amplitude to 2 0 V. Theinstrumentation acquired the VCC of each individual electrodeand then that of their parallel combination. The VCCs aredisplayed using cyclic voltammograms. To prevent changes inthe characteristic of the skin due to its recovery, the wholetest lasted about 45 s.Fig. 6( a) shows that at 10-V amplitude. the breakdown didnot occur and by connecting the electrodes in parallel. thusincreasing the area twofold. the skin admittance increasedapproximately twofold. Fig. 6(b ) shows that at 20-V amplitudethe skin broke down under electrode # 2 and increasing thearea twofold, by connecting the electrodes in parallel. theadmittance of the system did not increase twofhld and wasabout equal to the admittance corresponding to the electrodeunder which the skin broke down. At the onset of the skin


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(b )Fig. 5 . (a) The current flow through the model for the low-resistivityelectrolyte, at 34OC, when the injected current was about the same asthat in Fig. 4 (E 60 mA). The ratio R wa s 34.2 and more current wasfunneled through the regions of lowest breakdown voltage. The highest currentdensity was 5.9 mA/cm, about ten times larger than for the high-resistivityelectroly te; (b) the equipotential contours. The voltage developed across thethickness of the two regions of lowest voltage breakdown was about 41 V.The voltage developed across the thickness of the skin in the middle of themodel was about 47 V. Anomalies in arrow directions maybe caused bycurrent flowing around regions having high breakdown voltages. Shaft lengthrepresenting minimal current density is short. Length increases linearly withcurrent density. The regions 1 are hatched.

breakdown, the subject experienced a stinging pain sensation.Looking at the skin under electrode #2, immediately after theexperiment was over, we noticed five black tiny spots with areddish circumference. The skin was permitted to recover forabout 2 h. To test that these spots were the lowest breakdownvoltage sites of the tested location, we set the amplitudeto 20 V and, as presented above, observed the breakdowninitially occurring underneath electrode #2 (i.e., its cyclicvoltammogram displayed the current runaway as shown in Fig.6(b)). The voltage amplitude was then increased to 22 V andthis generated the current runaway underneath electrode # I aswell. Whereas at the onset of the skin breakdown undemeathelectrode #2 the pain sensation could be tolerated, when theskin undemeath electrode # 1 broke down the pain was toointense to be tolerated, therefore, the test was immediatelyinterrupted.IV . DISCUSSION

The experiment presented herein confirmed the hypothesisthat, initially, the breakdown of the skin occurred at the regionsof lowest breakdown voltage, which are believed to be thesweat ducts. Also, it showed that the conductance of the

--I-20 -15 -10 -5 0 5 10 15Voltage [VI(a )

700

50 0

300

cg -1000

-300

-500c7004

25 -20 -15 -10 -5 0 5 10 15 20 25Voltage [VI(b )

Fig. 6. (a ) Th e VCC of the parallel combina tion of two 1-cm2 electrodesand of each of the electrodes before the breakdown of the skin occurred; (b)th e VCCs for the same set up after the skin broke down undemeath electrode#2. Once the skin was broken down, the characteristic of the system wasdetermined mainly by the region where the breakdown occurred.

electrode-skin system was approximately proportional to thesystem area only until the breakdown occurred. After that,the relationship was dictated by that part of the system whichbroke down. We found five spots of lowest breakdown voltageunder electrode #2, which can be approximated with twosuch spots per 1 cm of linear dimension. For this reason,we included two regions of lowest breakdown voltage inour finite element model. Fig. 7 plots the dependence ofthe ratio R on the current injected into the system for eachof the electrolytes at both the normal temperature of theskin, 34C, and at 40C. Although some electrolytes havea negative temperature coefficient of resistivity, because itsvalue is of only few percentPC, its effects were not accountedfor. Note that the low-resistivity electrolyte yielded muchlarger values for the ratio R, hence, more nonuniform currentdistribution than the high-resistivity electrolyte. Also, when thetemperature increased the ratio R, hence the nonuniformity ofthe current distribution, increased more for the low-resistivityelectrolyte than for the high-resistivity one. Also in Fig. 5(b) ,note that the voltage developed across the skin was largerthan that presented in Fig. 4(b) for the high-resistivity case,


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.. . .0 b w resiotwitygel ( W C )

40

30

20

huh resistivitygel (WC)0 high resistivky gel ( W C )

b w resistivitygel(WC)0 b w resiotwitygel ( W C )

10 huh resistivitygel (WC)0 high resistivky gel ( W C )

b w resistivitygel(WC)10

0 5000 10000 15000Stimulating current, 1.5e-2 mA

Fig. 7. The ratio of Jmax/Jmin as a function of the stimulating current forthe two types of electrolytes we used, at the normal temperature of the skin,34OC, and at 4OOC. The low-resistivity electrolyte yielded much larger ratioR and the current was even more nonuniformly distributed when the skintemperature increased.

even though the current injected into the system was aboutthe same (60 mA). The local current density was also higher.This means that more heat is dissipated in the skin when low-resistivity electrolytes are used. As [6] and (6), (7) an d (8)show, the increase of the skin temperature favors the onsetof the breakdown by further reducing its electric resistance.Therefore, the low-resistivity electrolytes contribute to a fasteronset of the current runaway and, thus, of the breakdown, bytwo paths: 1) the larger nonuniformity of the current flowthrough the skin; and 2 ) larger heat dissipation in the skin,which causes the increase of the local skin temperature, which,therefore, increases the ratio R, hence, makes the current evenmore nonuniformly distributed.

The onset of the breakdown generates an uncomfortablesensation or even pain. Thus, using low-resistivity electrolytescan be more painful for the patient who undergoes the tran-scutaneous stimulation. This assertion is consistent with theresults presented by Monzon [141 who found experimentallythat the onset of pain occurs faster when low-resistivityelectrolytes are applied, and also with the results presentedby Zoll et al . [15]. Knowing how the electrolyte may affectthe current distribution pattem is useful for the design ofoptimal electrodes. Webster [ l ] shows that it is less painfulto transcutaneously stimulate using an electrode with 800-Rface-to-face impedance than with 8-0 face-to-face impedance.The finite element method presented herein could be used foroptimal electrode design as follows: 1) To specify a finiteset of practical electrode areas for the specific application tobe optimized. As shown in Section III.C, it is important tocheck the effects of the electrode size because its conductanceis no longer proportional to the area once the VCC of thesystem becomes nonlinear and asymmetric. 2 ) To accountfor the mosaic electrical structure of the skin by randomlydistributing as many regions of lowest breakdown voltage aspredicted by the average density of sweat ducts for the site ofthe skin involved in the specific application [16]. 3) To specify

a finite set of practical electrolyte resistivities for the specificapplication. 4) To perform finite element simulations for eachcombination of electrode size and electrolyte resistivity andchoose the one that yields the minimal ratio R (maximal overminimal current density in the skin). Examples of specificapplications that could benefit from the design of optimalelectrodes are: the optimization of the dynamic range of stimulifor a sensory substitution system, transcutaneous electricalnerve stimulation, and transcutaneous cardiac pacing.

REFERENCESJ. G. Webster, Minimizing cutaneous pain during electrical stimula-tion, in Proc. 9th Annu. Con$ IEEE Eng. Med. Biol. Soc., vol. 9, pp.986-987, 1987.C. M. Reddy and J. G. Webster, Uniform current density electro des fortranscutaneous electrical nerve stimulation, in Proc. IEEE Frontiers ofEngineering and Computing in Health Care Con$, pp. 187-190, 1984.E. Freiberger, The electrical resistance o the human body to commercialdirect and alternating currents. Berlin: Verlag van Julius Springer, 19 34.M. R. Neuman, Biopotential electrodes, in Medical Instrumentation:Application and Design, J. G. Webster, Ed., 2nd ed. Boston: HoughtonMifflin, 1992.T. Yamamoto and Y. Yamamoto, Nonlinear electrical properties ofskin in the low frequency range, Med. Biol. Eng. Comput., vol. 19, pp.302-310, 1981.D. Panescu, J. G. Webster and R. A. Stratbucker, A nonlinear electrical-thermal model of the skin, IEEE Trans. Biomed. Eng., vol. 41, no. 7,D. P. Burbank and J. G. Webster, Reducing skin potential motionartefact by skin abrasion, Med. Bid . Eng. Comput. , vol. 16, pp. 31-38,1978.A. van Boxtel, Skin resistance during square-wave electrical pulses of1 to 10 mA, Med. Bid. Eng. Comput. , vol. 15 , pp. 679-687, 1977.D. Panescu, K. P. Cohen, J. G. Webster and R. A . Stratbucker, Themosaic electrical structure of the skin, IEEE Trans. Biomed. Eng., vol.15 , p p . 4 3 4 4 3 9 , 1 9 9 3 .S . Grimnes, Pathways of ionic flow through human skin in vivo, ActaDerm. Venereol., vol. 64, pp. 93-98, 1984.T. Yamamoto, Y. Yamamoto and A. Yoshida, Formative mechanism ofcurrent concentration and breakdown phenomena dependent on directflow through the skin by a dry electrode, IEEE Trans. Biomed. Eng.,vol. BME-33, pp. 396-404, 1986.S. A. Klein, W. A. Beckman and G. E. Myers, FEHT a f inite elementanalysis program for heat transfer and electromagnetics, F-ChartSoftware, Middleton WI, 1989.V. C. Rideout, Mathematical and Com puter Modeling o PhysiologicalSystems, Englewood Cliffs NJ: Prentice-Hall, 1991.J. E. Monzon, Noninvasive cardiac pacing electrodes, M.S. thesis,Univ. of Wisconsin, Madison, WI, 1986.P. M. Zoll, R. M. Zoll an d A . H. Belgrad, External noninvasive electricstimulation of the heart, Crit. Care Med., vol. 9, pp . 393, 1981.J . P. Reilly, Electrical Stimulation and Electropathology. Cambridge:Cambridge Univ. Press, 1992.S . Grimnes, Dielectric breakdown of human skin in vivo, Med. Biol.Eng. Comput. , vol. 21, pp. 379-381, 1983.G . B. Kasting and L. A . Bowman, DC electrical properties of frozen,excised human skin, Pharm. Res. , vol. 7, pp. 134-143, 1990.

pp. 672-680, July 1994.

Dorin Panescu (S92-M94), for photograph and biography, see p. 679 ofthis issue.

John G. Webster (M59-SM69-F86), for photograph and biograp hy, seep. 680 of this issue.

Robert A. Stratbucker(S54-M58), for biography, see p. 680 of this issue.

a nonlinear finite element model of the electrode-electrolyte-skin system

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