and the Magnetic Fleld at the Apex of the Heart

31 Pages·2001·1.87 MB·English

‘The Etfects of Splial Anisotropy on ihe Electric Potential
and the Magneto Fatd al the Apex of the Haawt®

DEADLEY 1. ROTH, WaLQUANG CO,

i Sate Pier Grow Deemed fa mt une,

austese

sorte ig re prmers Ana apcrs Bre i he ptr a
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{The rode qradte he eens of hematin th ere Ad a a
pron ltd psi

INTRODUCTION

‘A central question in biomagne‘sm is whether a blonugneic ld can
‘ontain formation not present in the bicelorc tential, We wl adress
this question hy ceulating the cece and magnetic Hels produced atthe
pos ofthe heart Fguce | shows ace ofthe wages Eeatars of hi soe:
tle ausual genet af the bers, which pial aromad 9 ceca yore st
fhe apes of the left venrle. Our goal to cvaite the eer and
_magnsic Bids produced by a propagating dine ation pont osing +
‘model ofthe apex of the lar tha snmmportes iy paling srucore, I>
antcler. we wil abow that this fhe gneery leads to bagel Bll
{ht contain information sbaut the tue that cathe hai fm the
bioeletis potential Such "ceculcaTy alent magnetic els” am be pr
doce in sues wih ether spiraling oc boil hoe geowetcs [I]. At Be
apex of the heut,thex electrically sor magnoui Gels are captions

comp. Vans es Boe I. Sa Nad T 2
MATHEMATICAT BIOSCIPNCPS MAM-2H19) 2h

#2 Vanderb Ave, Now Yor. NY LOD17 mS 5864/80 8
ie (BL ROTH. W. GLO, ANT A.B, WIKSTD 1

Piet. A detugot the peso ie at Taba Sonat tly
ana

large and shold be cay detctale ia a2 eaperiment axing So¥ID
magnetometer

INFORMATION CONTENT OF THE MOG
the Kast meacrement of dhe agaeocanngsam by Rake sk

possble 10 posture simple corel snrcey presenting the cxzdiac
letra acti that have dential lcetsie polenta at the boxy surice
boat have different mapoee eds (9). Aay ippresd caren desis inthe
Acar cam be sopheented by a Helos decompoaiien, anim 12
Flonioy used Git fact to show tht the BOG aud MCG aighe enna
Indepcndoot lforomia (4), Rush subaeqnenty agued that gysologel
conaraina prevent the BCG and the MOG from having dive infoate
on content [5], Wikswo ad Borah sypgted a coonteraample io Rass
scgumente in Ghich reo imptesed curem dirbotioas ia te heat Bate
‘desialexernal cei elds bu ditteing enteral magooi fee. Te
Phyriobogiod| movaton for Wikies and Bara

Sevaton by Corbin and Scber thar the assinpy
‘educe cura detains, associate with saal spiel or hemiaph
al xavetrons, that were not consign wih Whe-then widelY aomptat
‘worm doabislayer mode of cardiac ac vation (1). The work hy Costin
and Seber has snes been eoainsed and extoded Wy jcveal authors 1
‘Using a multipole expmsion of the impressed currat souee, Katia and
arp [2] and Wika [13 base shown thatthe anbeypomcte pat of the
limprestedeurrent quadrapole tensor produces new infomation i the MCG.
However. in 1982 Plonsey suggested that mcamrzments of the ces
bielecui field do in fact, compleety determine dhe exttmal biomgnctie
fied bona th impreaedcaneat canbe mac Ista the hicensgatic
fields than do senondarysouret duz 10 the prscace of the resistive cll
membranes [14} The graeral lationship betwen the BCG: and MCC and
‘he posable that the magnetic eld of the heart might contain information
‘ot present in the elec potential, were reviewed hy Wikswo [13]

"There have been seer eee Teoetial venir ima the tela
Information convent of ticeleeic aml bioowgneie field, Thome and
‘Kneppo[13, 18: baw shown how.mulpole expansions of the cari cine
and maguetie Teds can be wed w detest new infil, Using 4
tidomain ode of sanzactssoe, Sepubvede aad Wikswo 17 emnsrated
that measurement of ube etic action potential alone are inslficees 19
eomplowly deteraine te eeeiat propenie of earac mule but thas
‘oubined eneaswrenens af the magnetic a lactis cam Be wsed
Selene thee popenice, Roth and Wika [1] nave shown that hele
‘potentials produced in tienes with spiraling or helical Mee goemetics eam
Iced to ecru eot magnetic cds, abd dey derived anahyc equations
forthese feds inthe special case of a eyindtiea) rand of bers. The
‘umgsual ber geometry was incorpraed into thir model by inducing
f-diagonel temas in the condos fear

Only reat hus expericonel eidence been obtain 1 uddess te
questo of inforaution content of the MCG. Burstin sa Coben [180479
‘hut one-dimensional stands of fog eutiac muscle generate rougnetc and
lose Cede tm cont the sae aforiaen when mesure froma 1 $
fxg wna from a wavefront 1 ao ess Jong propagating long the stra
Hloweve>r Ue experiemis do aol eaclde dhe posit hat eect
‘lent mga fis wight be produced im ties vd more compe. Te
sonnet On Te cher hand, recordings of the mgnetocardiogann by
MAulay esl 9,20), Nowinnen ot al [2Ve Yarpoa eta [22 and
‘lone and Ayla [3] pwd evidence Wa there ay be mes fer
‘i inthe MCG.

Pasiinsry coupatsom of experincnil MCG map wth caleylatons ot
‘ho heart's nagar Hild that ke ina aecoaal De aisohoy nf cadiae
tisene are jum besianing to appear. Campos eta [24 have celeoated the
MCG sing a moliple dipole model and Ducrer et al’s [25] date for
196 2. ROTH, GUO, ANDI.P. WIKSWO, 18

repagstion of the ceriae wavefronts the normal heart, They aceminied
[oe the Gas anietropy by orienting the dipoles alan angle wiih wspect
the dteton of propagation, nd fon tha her ede] could edie many
of the features Ober in ixperimaaly mcauncd MEGS. Gonacl aod
‘Aznelo [26 have sil soodled the cardia aciaton wetctiont &s a
bemisphere, ong Suet’ [27] daa for toe hoe excntation in the heart
Their ealclalad MCG ap sls showed a suing ressnblane to comed
at,

METHODS.

‘Many of the previ teottcl and experimental sais ofthe MCG
toler from the ase of acirte knowledge of teeta shape ofthe hear
the fher yom, and the pric path of the depletion wavefront.
scans Dhl a bes ay to answer the Fndacoal question of information
content is lo study a8 ample a system as posable, bot one thot is ctl
fommplex cough Io produce ctrl seat magoct Felés, A all
‘Grallor wavlfoat inated af ths oper ofthe heats sock u tea

‘Oar calelation dons aot predict MCG maps, us"dors the work of
Cacia [24 and Gonna and Aanello (25), lowever ou moll des
‘avo aveal advantages over dee other works, Fit whe syst ead
‘aough that fey siaple anagticsoltion aan he fund Cor ech ths
‘Seve potential and the magnets ld, allowing wz Wo stoner unusbian
fculy questions boucdhe rater iofoto coment of these Fld,
Sesond the mae cam br feed by arin ne. expernent, bowing dhe
‘xperimenter 1 bane mech pure contol over the experimental paracees
{End eonaitionsAdealzed mes yo in sro experimen ace abvays opea
toe orm tha they ane are) and yr aceuately vepteseat he
properties of he mole heir ion. However, these cine mt be
‘weighed against the extreme Hilly of developing oiodels of the whisk
bear, with al es comple, and expecially the oocensinGe ia validating
Aor made eaperimenaly. Tis calauation is bor encanto quamttasly
modal the whole Hest fa ven, I does ode a poasble tote expexchok,
Sud eeay provide sono inight ino the quaiteée interpretation of the
MOG.

‘We wl gcse vera curvature ofthe bert and ass tat the apes
is exendlly 4 plane da of tieae of thicknow # (oe Figure 2. Tas wil be
f goed apponiation a foog a8 the ditance Iron the apex tthe depoler-
Eaton wavefront is smell soraperal to te sudis of curvature ofthe ew
‘The uaae is anrmed to Ueno wsbounded saline bath wth conducts
fq. We conider only suion potuaials tat exgiate at dhe apoe ind
‘ronagate radially gusord Ths i 2 good apprcaimtion to nia vt
xpeciment fu woh thea is malate atthe apex Weeuse a endecl
APEX OF THE HEART 495.

SALINE

SALINE dy

sxrdaty tt py bn is. 40d ova etl
eel ls ofthe oe a epi. Th asp #
‘linge! Cours, dv cONCE Ln EN Her AED

sectinte sytem co deste the tive, wilh dhe axa ping ibeough de
pox comer, erented perpendicular 1 the pane ef lhe thou The vavables
fad @ are the ndial and acimuthlcoonfinals sero in Fans 2 We
Sssume that the tinue enindouly smite, which cual igaring,
‘he presence of the send vertex in Mure 1 due to the ight wee, Ts
in valid it we assume thatthe waveloat lus not moved bspood de Ret
vente,

Cardiac tase Ss 2 yest, and tbercore-ean Be tpteated DY 2
tademia ave! (26 32) wih Iniaellar and feet) condos
Temors Sr, eos, The donwpasen Decency ceo
fare deteradned by the deli! properties of the Usswe andthe ber
ometry. On a large ae folnetend the er uicetinn cages ith
Dori vs De bers spiel around he apes. We seus however, Hea on 8
nile wale nis of eines) cach pate of Gas hata wedi
Teel Rr diestion (Figure 3-Ta pine his diction of bghest conde
ity wuld be ttsent ia the insole aad ies dem ale
“how ik pace hist unlikely, Movever tbe ansowopy eal ia he to
space arc test Ukely dierent (3) Let oa aseuse that for the introeealar
V6 BLT ROTIL,W. GUO, AND 1. F WIKIO IR,

to 3th ae et ton pa nd ppd
cade sytem Te sgh Ld p 3 °

main the dicins parle wa the cal ies satin, teed the Tena
final direction, has conde andthe dircion erpendicalr tthe
Joval Boer dieton, tenoed the uansvess diction, has conducts»,
sad that 6) aud 0, ue comet troughoat the tissue. Fethermors, we
sum thatthe logitinal dizecon mes everywhere a constant engl X,
ds the prdivetion (Pq, 3) Thea for ay small cepion of Gat the
intceluar condacity tomce can be seen wih rept Wo 2 lead
coordinate system exited alaig the Journal drctlon as

a 0 8
8 om w
0 0 el

“To represcat this teazoe i ends! enordngas se mise ete our Ene
of reference fron. one algaad with the fer dinction fo oe erento lens
the py & ad 2 dizesions To rotate a teas, we aust gully i 0 each
Sideby « rottion mates

ah oo
oh ot 0
ES:
coex, tiny, 0 fap 0 O}femz, —sinx, 0!
=|-snx, ox, o|[o «, o|fanx, cox, |, @
oo alla o afl oe 0 yf
(APEX GP THE HEART It

hich Ia
0
°
nyse ay smF x, (yay eon,
fay ondeosxisingy —mysie'xt apem” x, 0 |. (3)

° ° a!

‘he conductivity Lenser bas four independent nonvanishing componeats: the
thee diagonal companeots oP, of, and 0, and one off diggs) compe-
vent oP The offingonal ene rele the sping tise zeomety. The
‘assompton tht the unl x, is Constant shoughout te Hise ips that
ibe oondotity tea iv adependent of postion

I x,=U or 9%, the olf dhgonsl components van, and if x. 45°.
they have ther enximu value When the angle x, is 45%, the condactity
sense as tbe sine fort

ate
re

0 oa

Furthemore. if the Jogitdina) conducive: Ik much qhestr tha te
‘manvene condutviy, then the condocivy tensor has the even simple
fem

oe 0) f q/2 ~w/2 0}
0 fm) -m/2 a2 9 6
of | ot wx

1m this case, tb of lanl ns are a aye a te agonal ters.

a similar way the intertal condncty tnsor @ an be evalusied
{rom the interdal Iiagtadinal (0, and cransvere (9, conducts,
‘and Ge angle x, berneea p and the lntrxidal Bher diveion, For the
reminder of dis paper, we wil only comsder the case (a Which he
isc ao ner ema ee ae gid en,
ie a =4,=:

ive ie cadiaiy tenors esting te the ae none, He
slate dee soln eoedusted etic potential and carent The fund
fenlal eqvations ysverning Die electric putental are ee equation of come
1 1B.ROTH W. GUO. ANDI # WIKSWO. IK

Ansty of eument

wibtho, -1eesia 6
vano, Oy

‘and Oba lw
@
“
cc)

suze E represents he else ld, 3 caeat dons. and we abseps
2. oy and sigly he inane, incense. and ereeal ba volomes,
‘These equations can he soabibed ilk he define’ of de ects Bld in
etna ofthe potential

r--96, an
the asinpion of eine mney, ai he samgton tt fhe coe
‘Seto tensors ae ndcpendene of poi o eld to paral ferential
‘esmation desing ae fase, and exter potas,
ouand
19], 8 foe, og
th lodtore,-er0,)]
F (gra, 080) =
+ Bere, a0,)-0

vie a 0,

“The boundary vonions a he surace of dh deus, |e|= 1/2, are [32]
BABAR, 09

and
a9, 09

‘Equation (12) is quite wroplitel, bu ca bade eastable ung few
sdgemuie manipula. Following the locoigur given la 2], we ake &
shimge of variable

sas, 19
hate

(7)
APEX OF THE HEART. we

and iroduce two new poten ¥ and Sy Which are Linear combinations
of 6, and 6,

Ane) = 2.0) O

wanton Oo) o»

o), an

Egnation (16) the eal definition of th: transmembrane oven, whch
swe asmume ia known. Note cht &, essai to he 4 felon ly
‘onsitent wth our asampton of dial pragation of he ction penta
Equation (9) defies &. We ean iver the aos fan eyuaticns lo ba

SG) gil ¥eny-eto)l. oo
Cee veo Banco ee

‘Making the substitions into Equation (2), and using te fact har is
independent of, we Fd that

are -ynercty es

‘whic is Lapa’ aguation in Oh (2%, coordinate system, The boundary
conditions of Euuaios (14) and (18) hase

AY gm 88,

ao @

es

(ue problem is now roduc 19 sclting Laplace's equation [Equations (133
snd (2) wth the appropsite boundary conditions Equarians (1) and
(29). Therefore, if we ue lea the teu paramctas and the tate
brane pont uu fometion of p, we cua drive snasticexpeeione fr the
poten de introlller ni ndextral ohne. This dei
‘Son isoained ia Appeadin

"Toe anes igesGcnot teva of th euloltion ita 8 and af a not
2gpear in ump af the equations gmering- the. ihraePnar, sai, oP
‘exiemal poten. Ths dace penal ieindependent of the of-dugonal
Canduciey tems. 90 isnot alfectad by Ge spring gvomesy of the
™ BJ. ROTT, W G10, ANTE). WIKSWO, 1

tisme. Haweven, af" and al do appear in Be pressions Jor se cerent
sent

“Ths, here arc eurens cing around he apen in the Baimetion, lthobgh
there are no pete) sfaions im thas dizection. This eel esa erly
fo the spiel gromeuy of the hors, expres mathewabtlly fy the
‘off-diagonal tenue in the conduotviey teasns, and leads dscly tn the
‘producti of eectsicaly alent magaeti cds

“The Law of Biot and Savarlgrnerns the bnbivi of the augacte fel.
‘This equation canbe wba analytically er the migneic 4B prodasad at
the apex of he hearin tre af the mnmembrane potential, i dope is
“Appendix. The ees shows hat there are componens of the age
Feld ina thee diecion, and thot the p and x eamponente bot depend
an of and of. Ths, dor ofiiagoal componente ofthe condi tense
appear ithe ecresines for tee mgt fl, ta 40 mo eepsae inthe
fexpresdont forthe elena! pon. We tem the p and components of
{he magnetic Bold the eiearnaly set components they conrin ner ier
‘matin that is wot presen Inthe clactric penta!

RESULTS

‘To cakulate che poll and magnets Geld from the expresions ia
Appeatices A aad B, se must axign specs value to the parameter:
‘escting the cardiac sue and action polenta. We wwe 4 prramenizntion
‘eaved om Ploncy and Bai? sepiasentation of the anmmembmane potent
Bar

eco) 7ifisial Bea-olf | as

shown in Figue 4, vhee My is tho wniplvade of the ation porate
(110 mV), Foye 5 1 asian cal of ie (20) 7S) und a is Hs
roped vloier (020 m/s) [3]. Ths Vanioewbrane potenti pre
sens a eirular sefiaion waveroatpropugting atally ar th the
apes of the er, stinlating lettode had ean placa he wore
‘The wevetrom e-have a ioe oy abd a Wit (10 a0 908. of wou)