CONCAVE ELECTRODES II: THEORETICAL FOUNDATIONS

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1 Physis i Mediie ad Biology, vol. 9 a, 1994 CONCAVE ELECTRODES II: THEORETICAL FOUNDATIONS Roberto Suárez-Atola Direió Naioal de Teología Nulear, Miisterio de Idustria, Eergía y Miería, Motevideo, Uruguay Itrodutio The geometry of the eletrial urret field geerated by a give eletrode i the volume odutor formed by biologial tissues ad the properties ad positio of the target regio i exitable tissues, both determie the performae of the eletrode as a stimulatig ad as a sesig devie. Both from experimetal fats ad from theoretial reasos, the geometry of the urret field of ertai oave eletrodes deserves speial attetio. Materials ad Methods For a field with a symmetry axis the distributio of urret soures at eletrode s surfae is substituted by multipolar oeffiiets assiged to the eletri etre of the eletrode. Usig Leveberg-Marquardt algorithm, the multipolar oeffiiets a be determied from suitable experimetal results. Besides this, the urret desities at ay two poits over the same field lie are related by a futioal of the mea urvatures of the poits of the equipotetial surfaes rossed by said field lie betwee these two poits. The, Prim s theorem for soleoidal ad almost gradiet fields a be applied to the aalysis of the ritial regio of the eletrode, experimetally foud as desribed i the first part of this wor. Results ad Colusios At poits ot too far away from the symmetry axis, the field a be well represeted by three or four terms of the multipolar expasio, eve i the ase of oave eletrodes. The eletri potetial, osidered as a futio of distae o the symmetry axis, shows a ifletio poit for oave eletrodes ad oe for ovex eletrodes. This ifletio poit is the etre of the ritial regio of the eletrode. The ritial regio of the eletrode a be defied as a ertai eighbourhood of the surfae formed by the poits where the equipotetial surfaes have zero mea urvature. Its positio ad extesio is the related with the multipolar oeffiiets of the field. Key words: oave eletrodes, lothoidal eletrode, multipolar expasio of a potetial field, soleoidal fields, almost gradiet fields, ritial regios, mea urvature, Prim s theorem.

by biologial tissues ad the properties ad positio of the target regio i exitable tissues, both determie the performae of the eletrode as

Both from experimetal fats ad from theoretial reasos, the geometry of the urret field of ertai oave eletrodes deserves speial attetio.

2 Physis i Mediie ad Biology, vol. 9 a, 1994 EXTENDED SUMMARY (A) Itrodutio The geometry of the eletrial urret field geerated by a give eletrode i the volume odutor formed by biologial tissues ad the properties ad positio of the target regio i exitable tissues, both determie the performae of the eletrode as a stimulatig ad as a sesig devie. Both from experimetal fats ad from theoretial reasos, the geometry of the urret field of ertai oave eletrodes deserves speial attetio. Figure 1. Seth of the equipotetial urves, lothoidal oave eletrode Figure. Seth of the equipotetial urves, hemispherial oave eletrode (B) Materials ad Methods (B. 1) Figure shows a seth of a eletrode with a axis of rotatioal symmetry. Figure. Seth of a eletrode s head with a set of polar oordiates r ad o eah symmetry plae through the axis z.

3 Physis i Mediie ad Biology, vol. 9 a, 1994 The distributio of urret soures at eletrode s surfae a be substituted by a set of multipolar oeffiiets assiged to the eletri etre of the eletrode. The for a ubouded, isotropi ad homogeeous volume odutor, the eletri potetial V ( r, ) a be developed i terms of Legedre s polyomials ad reiproal powers of the polar distae r, beig the multipolar oeffiiet (Kellog, 199): V ( r, ) P os 1 (1) r Whe the volume odutor is bouded but the symmetry of revolutio is retaied, it is eessary to add a ostat term d ad terms ivolvig ireasig powers of r. The origi of oordiates (see Figure ), whe the eletrode ijets a et urret i the volume odutor, a always be hose to oiide with the so alled eletri etre, so that the oeffiiet of the dipolar term 1. Taig this ito aout, we foud that it is possible to approximate the equipotetial surfaes measured i the upper regio of the eletrolyti ta, with fous i a eighbourhood of the z axis, by a formula that iludes oly a ostat, a moopolar, a dipolar ad a otupolar terms: V ( r, ) d P os P os () 4 r r r The uow multipolar parameters as well as the positio of the eletri etre i the symmetry axis a be estimated miimizig the followig hi-squared variable, where V ( r, ) represets a measured voltage value at poit ( r, ) i a vertial plae through the symmetry axis: N d,,, (, ) os os V r d P P 4 () 1 r r r I this ase 1,,..., N represets the poits where the measuremets were doe. The distae r betwee the eletri etre of the eletrode ad the poit where the voltage is measured is give by: r r r os (4) I formula (4): (1) is the distae betwee the poit E of itersetio of the symmetry axis z with the surfae of the eletrode s head, as show i Figure, ad the poit P where the voltage is determied; () is the agle betwee the segmet E P ad the z axis; ad r is the distae betwee ad E (see Figure ). It ad a be alled eletri radius of the eletrode. While ad a be measured diretly, ( r, ) must be determied after the eletri radius has bee estimated, so i pratie we have to wor with the operatioal oordiates, ). ( To estimate the multipolar oeffiiets ad the eletri radius r it is possible to apply the Leveberg-Marquardt algorithm (Marquardt, 196; Press ad others, 199) i two steps. First a gross estimatio is obtaied worig with the voltage values o the eletrode axis oly. I this ase we put ad r z r, so the followig formula is used to estimate the uow parameters r, d,,, :

4 Physis i Mediie ad Biology, vol. 9 a, 1994 V ( z) d (5) z r 4 z r z r The, this gross estimatio is used as seed to obtai a more aurate estimatio employig all the measured values of voltage, aordig to formulae () ad (4). If A is the area of the odutive eletrode s head, the to begi the oliear iteratios A sequetially (as doe i ompartmet aalysis) from the umerial values of V (z) r a be equalled to A seed for parameters d, ad a be estimated distae o the axis of symmetry, from the surfae of the eletrode, grows. (B. ) whe the Let us osider ow a so alled almost gradiet field, lie the eletri urret desity J i a ohmi but perhaps heterogeeous volume odutor of variable (from poit to poit) salar odutivity G : J G V (6) If the vetorial field is soleoidal, by defiitio the divergee of the field vaishes: J (7) Aordig to Prim s theorem (Prim, 1948; Suárez-Átola, 1984) for soleoidal ad almost gradiet fields, two of the followig three oditios imply the third oe: 1- The orthogoal surfaes to the field have zero mea urvatures. -The field is soleoidal. - The magitude of the field alog a field lie is ostat. Equatio (7) a be applied to almost statioary fields, that is, i oditios of slow eough time variatio to be able to wor with stati equatios. This is the ase ommoly eoutered while paig the heart or durig futioal eletri stimulatio of erve ad musle fibres (Fiadra ad others, 1985 a, Chapter 8; Reilly ad others, 199). However, the volume odutor formed by biologial tissues is ofte aisotropi. This is partiularly importat i the ase of vetriular myoardium (Zipes ad Jalife, 199, Chapter ). Now the relatio betwee the urret desity vetor field ad the gradiet of eletri potetial field is give by the equatio J G ~ V (11) The eletrial odutivity G ~ is a positive defiite symmetrial tesor. The urret desity field is ot exatly almost-gradiet as previously assumed. The bidomai model of vetriular myoardium, that is ow used to aalyse threshold ad atio potetial propagatio, taes ito aout the so alled uequal aisotropy (differet aisotropy ratios i the itaellular domai i ompariso with the extraellular domai) ad has importat pratial osequees (Suárez-Átola, 1994 a) (C) Results (C.1) Figures 4 ad 5 show the adjustig of formula () o the axis of symmetry of both, the plae ad the lothoidal oave eletrode (arrow 1,7m), respetively, miimizig the

Physis i Mediie ad Biology, vol. 9 a, 1994 orrespodig hi-squared by Leveberg-Marquardt algorithm. Formula (5) was employed with fixed at zero. Figure 4 Figure 5 We foud: Plae eletrode: r 1. 5m d 1.

For the lothoidal eletrode with arrow 1,7m, we foud, iludig ow the otupolar 4 oeffiiet (ad with a 1,881 ):. 466 m.

5 Physis i Mediie ad Biology, vol. 9 a, 1994 orrespodig hi-squared by Leveberg-Marquardt algorithm. Formula (5) was employed with fixed at zero. Figure 4 Figure 5 We foud: Plae eletrode: r 1. 5m d 1. 66V. 1V m. 9V m Clothoidal oave: r. 87m d. 84V 1. 1V m V m Sie the multipolar oeffiiets are proportioal to the ijeted urret, a more meaigful set of parameters is give by. For the lothoidal eletrode with arrow 1,7m, we foud, iludig ow the otupolar 4 oeffiiet (ad with a 1,881 ):. 466 m. 89 m For a lothoidal eletrode with arrow 1,7m, we foud adjustig parameters with the omplete set of measured voltages, iludig also the otupolar oeffiiet :. 65 m. 5 m A omplete report of the estimatio of parameters will be give elsewhere. (C.) Now, osider the mea urvatures of the equipotetial surfaes ear the axis of symmetry of a oave eletrode i a isotropi ad ohmi volume odutor. Near the oave surfae of the eletrode, mea urvatures are positive. But if we move alog the axis away from the surfae of the eletrode, the mea urvature dereases util it vaishes at a ertai poit o the symmetry axis. I this poit the urve that represets the voltage as a futio of the absissa z has a ifletio poit. This a be see i figures 4 ad 5. I the ase of a lothoidal eletrode this ifletio poit is fairly distat from the eletrode. I the ase of a plae eletrode, the ifletio poit does t exist. Now, it is possible to apply Prim s theorem to a oave eletrode taig ito due aout the otiuity properties of the urret field. We see that i the eighbourhood of the surfae with zero mea urvature, the field lies will be almost parallel, the magitude of the urret desity will be early ostat alog the field lie ad the magitude of the

Physis i Mediie ad Biology, vol. 9 a, 1994 urret desity will be higher tha its magitude i poits of the same field lie away from the surfae with zero mea urvature.

(D) Colusios At poits ot too far away from the symmetry axis, the field a be well represeted by three or four terms of the multipolar expasio, eve i the ase of oave eletrodes.

6 Physis i Mediie ad Biology, vol. 9 a, 1994 urret desity will be higher tha its magitude i poits of the same field lie away from the surfae with zero mea urvature. As osequee, a ritial regio is formed ear the axis of a oave eletrode, where the field lies are early parallel ad more desely distributed. (D) Colusios At poits ot too far away from the symmetry axis, the field a be well represeted by three or four terms of the multipolar expasio, eve i the ase of oave eletrodes. The eletri potetial, osidered as a futio of distae o the symmetry axis, shows a ifletio poit for oave eletrodes ad oe for ovex eletrodes. This ifletio poit is the etre of the ritial regio of the eletrode. The ritial regio of the eletrode a be defied as a ertai eighbourhood of the surfae formed by the poits where the equipotetial surfaes have zero mea urvature. Its positio ad extesio is the related with the multipolar oeffiiets of the field. Figure 6 shows a qualitative piture of the aforemetioed ritial regio. I the last artile of this series (Suárez-Átola ad Artuio, 1994) the positio, size ad shape of the ritial regio will be studied quatitatively. Figure 6 Figure 7 Figure 8 Figure 7 ad 8 shows a template that was used to ostrut a optimum oave eletrode ad a photograph of the lothoidal eletrode had rafted by Dr. O. Fiadra usig his jewelry s lather. Bibliography 1. Brad, L. Vetor ad tesor aalysis, Wiley, New Yor, Erise, J. Tesor Fields, Appedix to Truesdall, C. ad R. Toupi, The lassial field theories, Eylopaedia of Physis, vol. III, Spriger, Berli, Fiadra, O. ad others, Cardia Paemaers, Departmet of Cardiology (Shool of Mediie, UdelaR) ad Natioal Istitute of Cardia Surgery, Motevideo, 1985 a. 4. Fiadra O. ad others, The athode i ardia stimulatio: ifluee of its shape i hroi thresholds, VII Uruguaya Cogress of Cardiology, Motevideo, 1985 b. 5. Kellog, O. Foudatios of potetial theory, Spriger, Berli, Klie, M. ad I. Kay, Eletromageti theory ad geometrial optis, Itersiee, New Yor, 1964.

7 Physis i Mediie ad Biology, vol. 9 a, Lidemas, F. Eletrial stimulatio of heart musle, Ph.D. Thesis, Eliwij, Uthreth, Marquardt, D. A algorithm for least-squares estimatio of oliear parameters, Joural of the Soiety for Idustrial ad Applied Mathematis,, 41, Press, W. Flaery, B. Teuolsy, S. ad W. Vetterlig Numerial Reipes i C, Cambridge Uiversity Press, Cambridge, Prim, R. O doubly lamiar flow fields havig a ostat veloity magitude alog eah sream-lie, U.S. Naval Ordae Laboratory Memoir Nº 976, Reilly, J. ad others, Eletri stimulatio ad eletro-pathology, Cambridge Uiversity Press, New Yor, Suárez-Átola, R. Costat field eletrode ad hroi paig of the heart, II Iteratioal Cogress o Bio-mathematis, Bueos Aires, Suárez-Átola, R. Thresholds: Cotributios to the study of exitatio ad propagatio of the eletri ativity of biologial tissues stimulated by exteral eletrodes, D.S. Thesis, PEDECIBA, UdelaR, Motevideo, 1994 a. 14. Suárez-Átola, R. Griego, J. ad O. Fiadra, Coave Eletrodes I: Experimetal Foudatios. Physis i Mediie ad Biology, 9 a, Suárez Atola, R. ad G. Artuio, Coave Eletrodes III: Computer Assisted Desig, Physis i Mediie ad Biology, 9 a, D. Zipes y J. Jalife (Editors), Cardia eletrophysiology: from ell to bedside, Sauders, Philadelphia, 199.

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