Some notes on the inverse problem in electro cardiography

Citation for published version (APA):

Damen, A. A. H. (1974). Some notes on the inverse problem in electro cardiography. (EUT report. E, Fac. of Electrical Engineering; Vol. 74-E-48). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1974

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by

Group Measurement and Control

Department of Electrical Engineering Eindhoven University of Technology Eindhoven, The Netherlands

SOME NOTES ON THE INVERSE PROBLEM IN

ELECTRO CARDIOGRAPHY

BY

Ir. A.A.H. Darnen

TH-Report 74-E-48

July 1974

Abstract

The inverse problem in cardiography cpnsists of the estimation of

electrical heart activity from body surface measurements. This implies, that many fields of technical research are involved like field theory, information theory, model building, parameter estimation etc. This report has been written in order to provide an introduction in these aspects. In order to improve the legibility some examples and

illustrations are included. They are printed in italics so that the occasional reader may omit them.

It is shown, that the information contained in the skin potentials ( as a function of two dimensions) is not sufficient to define uniquely a volume source distribution (three dimensional). Under some conditions it is possible to constitute a unique equivalent double layer (two dimensions). This simplification of the sources is not as drastic as

the discretization by way of dipoles. Furthermore the observability can better be defined for a double layer.

Contents

1. Brief field-theoretical consideration.

2. On the dipolarity of the sources.

3. Effects of inhomogeneities.

3.1 General aspects.

3.2 Source area.

3.3 Multipole model.

4. Restrictions on the number of equivalent dipoles.

5. Suggestions for future research.

6. Conclusions. 7. References. 4 12 19 22 24 27 33 40 41

I. Brief field-theoretical consideration.

It is remarkable, that bioelectric phenomena inspired men in the past to start research in electricity and magnetism. In 7792 Galvani

(7737-7798) described (in his "De viribus electricitatis in motu musculari commentarius") how he observed by chance eleven years before, that a frog paw contracted, when it was brought into contact with zinc and copper. One should remember in this context, that Volta (7745-7827) and

Ampere (7775-7836) were his contemporaries. The cardiology dates from

an even earlier time, for alreadY in 7628 Harvey cut up a frog heart

and established the separate contraction of all pieces. The frog has proved his usefUlness during later times too. In 7856 Kalliker and

MUller demonstrated the presence of action currents in hearts by let-ting frog paws contract as a response to these currents. In fact this resulted in the first electrocardiograph.

The great impulse in electrocardiography was given, however, by

Einthoven in the beginning of the 20th century. His work and consequent research untill 7950 will not be discussed here, because this is des-cribed very well in several textbooks.

In order to give some idea of the sequence of research in the last two decennia, the reference list is chronologically arranged.

The primary electrical sources in the heart muscle generate a timevarying, electrical and magnetic field in the body. The electrical field potential can be measured by the use of electrodes, situated on the skin. The mag-netic field can also be detected outside the body, but this measurement is mostly corrupted by much noise. In this stage we will leave the magnetic field out of our scope. In literature one finds practically merely

theo-retical comments, (ref. 61, 59, 66) in stead of reports of actual measurements. Especially because the bioelectric sources may supposed to be

electrical field. This quasi-stationary approximation is rigorously motivated by Plonsey (ref. 37, 48) in view of the electromagnetic pro-perties of living tissue, the dimensions of the body and the frequency contents of the sources. He shows that propagative, inductive and capa-citive effects are negligible. We will not go into details, but will rather use the results and consider the whole system of heart and body as a system of current sources in a purely resistive medium at any moment. The medium may be anisotropic and inhomogeneous. For such a

system a reciprocity theorem can be derived (ref. 3, 18, 21, 5, 19, 48).

•

-=JfI;

.!.,

•

1:2

a

•

figure 1-1 i= 1,2,3 ... N

We will illustrate this with above example, where the medium is piece-wise homogeneous inside interfaces Si and is bounded by surface ST' which is surrounded by air (0

=

0). All

primary sources are concentrated

in a limited area and there are no surfaces, which enclose the sources, except for ST'

For every homogeneous volume we Dan apply Green's Theorem:

JJJ(~V20/

-

o/v2~)dV

V

JJ

(~

1! -

0/ .E!.)dS

an

an

S (l-l )

where surfaceS eneZoses vo Zume V and n is the outwardZy directed normd onS. This constitutes the foUowing equations:

( 1-2)

ff

(q;

~

- 'I'l!...)dS = 0 for i = I, 2, 3 •••••• N an. dn. S. 1 1 ( 1-3) 1

Let ~ be the reaZ potentiaZ due to the concerned process and 'I' the potentiaZ due to sources outside ST' This means. that concerning 'I' there are assumed conductivity and sources outside ST' InsideS_T there are no sources generating'!'. so V2'1'

=

O.

No current beZonging to the fieZd ~ can pass

Sr"

consequentZy

l!

I

= O.

d~ S

T

For the surfaces

s.

hoZds: 1

(1-4)

because there are no sources in the interfaces. Furthermore the po-tentiaZs are continuous over the entire medium.

Substitution of (7-4) into (7-3) gives:

aT

If

(iP

~

-o. anT

1

s.

( 1-5) 1

ConsequentZy (7-2) is transformed into:

- f

JJ

'I'\72 wdV=

JJ

~ ;~T

dS

VT ST

( 1-6)

which is the desired form of the reciprocity theorem.

When

the

I (r), then:

v-2

- 0 V~=I

T v (1-7)

The field 0/ is caused by sources outside ST and can be fUlly des-cribed by the currents normal to ST in all points on the surface.

Define this current per surface area as I _{s -}(r) then

00/ 0" - 0 - = I • on T s So (1-6) becomes:

HI

_V T n dV v Suppose, that

n

dS s I = I

ro

(r -r) - <5 (r

-a

s os

LS

-a - s -b - j

where <5 is a two-dimensional Dirac jUnction.

s Also:

where <5 is a three-dimensional Dirac jUnction.

Substitution of (1-10) and (1-11) leads to:

[ 'l'<.EJ) - 'l'(r2

- J

)1

I ov =

r~(r

~ -a ) -

~(rb)l

-:J

I os (1-8) ( 1-9) (1-10) (I-II) (J - J 2)

This result asserts, that a difference in the potential '1' between the points ~J and ~2 due to a unity current supplied in ~a and removed in~b is equal to the difference of potential ~ between the points rand rb

-a

-due to the same amount of current supplied in ~J and removed in ~2'

As will be discussed in chapter 2 the primary sources in the heart muscle are found to be small current dipoles, Le. a positive and a negative current source of equal strength at a small distance. In order to relate

the reciprocity theorem to current dipotes we write:

If ~1 - ~2 approaches zero, Furthermore let I (r,-r₂)

oV . -These assumptions transform

( J-13)

the higher order terms are negligible.

=

m •

- 0

(1-12) into:

( J-14)

The current los in ~a and ~b are said to generate a lead field ~ be-longing to lead abo When I is a unity current (e.g.

lmA),

then it is

os

evident, that the leadvoltage V _a b

=

~(r _-a ) - ~(rb) _- equals V~.m •

- 0

When the higher order terms in (1-13) are not neglected the above concept may be extended to higher order poles using tensor description (see ref. 19). Consequently the transfer from the sources to the electrodes can be studied in two ways:

1st) Directly by postulating the sources and determining the potential at the outer surface by way of models. This method provides infor-mation about the optimal placing of the electrodes. (ref. 9, 31, 36, 41, 43, 45, 57, 23, 25).

2nd) Introduce a reciprocal field ~ and try to find the characteristics of this field in the area of the real sources. When this is done, the contribution of one source configuration to the supposed leads is easily found. (ref. 3, 5, II).

In earlier times only three leads were used, where a combination of these three leads should constitute approximately one equivalent dipole (e.g. ref. II). In fact the sources are numerous but concentrated in a limited area. The lead voltage is then given by

V ab M =

I

i V~(r.) _{- 1 - - l .} .m(r.) ( J-15)

if there are M current dipoles m(r.) in r .• This conccntration of

sour-- -~ - l .

ces, the inhomogeneities and the configuration of the sources cause the equivalent dipole to be a rather good representation (see also ref. 8, 9). We will comment on this later on.

Different kinds of models for the human body are used. They can be dis-tinguished into three categories:

1st. Physical models, which can be distinguished into I a two dimensional fluid currents model (ref. 3)

b two dimensional electrolytic tanks (ref. 25)

I c three dimensional electrolytic tanks (ref. 57, 9, 41) d two dimensional models with teledeltos paper (ref. 5, 52) I e one could think of resistance networks according to ref. 2.

2nd. Placing electrical sources in living beings (ref. 23) or measuring surface maps due to reciprocal activation (ref. II)

3rd. Mathematical models, which can be distinguished into

3 a symplified theoretical models, which can be used to explain certain phenomena by simple analytical computation. Generally used (e.g. ref. 13, 14, 64).

3 b discretization of the differential equation, where the theory of ref. 30 could be used. In this approach one needs more memory but on the other hand there is a great flexibility e.g. with respect to anisotropies (ref. 40).

3 c discretization of the integral equation. This method is mostly applied in literature (ref. 45, 36, 31, 43).

It is worthwhile to comment on the aspects of the extrapolation from two to three dimensions. Quantitatively the fields due to comparable sources in two or three dimensies are completely different. E.g. a point charge in the origin leads to a potentialln(r) in two dimensions and I/r in three dimensions. So one should be very careful to conclude for three dimen-sions on the basis of a two-dimensional model with the same proportions. Also qualitatively the behaviour of three dimensional fields differs from the two dimensional field. In ref. 58 Rush pays attention to a typical phenomenon. In a closed, conductive, two dimensional medium, which may be inhomogeneous and anisotropic, a single current dipole can cause only one maximum and one minimum in the potential on the boundary. In three

dimensions, however, more relative maxima and minima are possible, due to the extra degree of freedom for the volume currents. Also the boun-dary effects playa different part. In two dimensions a circular theo-retical boundary in a homogeneous medium provides two parts, the inner and outer areas. The impedances seen from that circular boundary is the same for both parts (see in this context the last commentary part of this chapter). This is not true for a spherical boundary in three dimen-sions, as is shown in reference 65.

As the discretisation of the integral equation is frequently used 1n literature a small comment seems to be justifiable.

Equation (1-3) may be a suitable starting-point.

n. -J figure 1-2

If

(IP

l!...

-dll. 1 S. 1 'Y 1.!...)dS dn. 1

=

0 ( 1-3)

Let ¢ be again the real potential and 'Y the potential of an infinite homogeneous medium with conductivity a

i. We can think of an infinite

number of different source distributions for 'Y outside S .• When we 1

divide the surface S. into N smaZZ surface areas, where 1

d'Y

dn.

1

are approximately constant, we can write:

N

a. ₁

L

~

IP. - 0 d'Y - LIS. - 'Y. - 0 dIP - LIS. ] = 0

Jon. J Jon. J

j _{J J}

where j denotes a small surface area. Define: a'Y a. 0 LIS. 1 on. J J l'JI. and J dIP a - LIS. = ian. J J IiI>. J dIP 1>,

-0-'

_on. 1 'Y and ( 1-16) ( 1-17)

then:

N

I

(~. I~. - ~. I~.) ~ 0

j J J J J ( 1-18)

So N different fields ~ constitute N equations in the 2N unknowns ~. and H .. Generally these fields are generated by way of current

J J

SOL~ces in the different surface areas

nS .•

J

In fact equation (7-78) can be looked at as a three dimensional Ohm's

laL) (see ref. 34). When 4> is due to sources within S. then equation 1 (7-3) changes into:

If

(~~

-

~

1!...)dS

~

HI

~1I24>dV

an.

an.

s.

1 1 V. (1-19) 1 1

For the case of current dipoles this results into:

N

I

(4).

n.

- ~. H.) ~

I

_{IIII' 'E!l'}

j J J J J K '.

( 1-20)

There are still N equations for the 2N unknowns.,

All S. are always boundaries of other areas, where similar equations 1

are valid. Furthermore there are the conditions of continuity in

~ and I~ . All these equations are at last sufficient for the solution in <1>. and I~ ..

2. On the dipolarity of the sources.

In a cell of the heart muscle at rest a charge difference exists over the cell membrane due to a difference in ion concentrations. The outside wall is positive compared to the inside. The activation of the

ventricu-lar cells manifests itself in a depoventricu-larisation during approximately I msec followed by a depolarised state of 200 msec and a repolarisation during approximately lOa msec (see figure 2-1). A thorough description of this process is given by Plonsey (r~f. 48). We will pass a discussion of the conversion of energy (chemical, mechanical, thermal, electrical) and only look at the electrical consequences. We will use therefore results obtained by Plonsey (ref. 48), which show, that this phenomenon can be represented electrically for an observer outside the cell by an equivalent source in the form of an e·lectrical double layer in the cell membrane, varying in time.

+

0

-

-

-

+

.

--

-.

- +

0 0

+ i ii

- - -

- , - - - , . 100 mY ii iii lmsec 100msec iv 200msec iii t

-figure 2-1 idealized ventricular transmembrane potential iv

A singZe ceZZ is embedded in a

unifo~

medium with conductivity

cr • o

The inside of the ceZZ is aZso homogeneous with conductivity

a •• ~

All electromotive forces are in the membrane i.e. between surfaces

S. and S (see figure 2-2).

1 0

n·

- I p.

figure 2-2

p.

Apt> lication of Green's theorem to the inside vo lwne bounded by S.

1 and the outside volwne bounded by S _o and swnmation of the resulting equations leads to:

1

~L

J

It

;m

,j

~ (p) = - - m dS - ₊ 0 411cr r 0 1

[, I'

""",. -II·

""",~

- 411 ..2 _cr

J~'

₁

an.

₀

an

(2-1 ) o

s.

1 S 0 1 0

where r is the distance between integration and observation points and J

m is the outward membrane current density. When r » md, where

md is the thickness of the membrane, the integration may be performed over a surface inbetween S _o and S., ₁ which results in:

~ (p)

o = -411 1

II

( 2. cr • cr <P. -1 S 0

(2-2)

dQ is the solid angle sub tended at P by a surface element dS. As will be shown at page 17 this formula can be interpreted as the expression for the potential in an infinite homogeneous mediwn due to a double layer on the surface S with a strength given by the integPand multi-plied by the conductivity cr • Measurements of ~., <P , m

d, J and cr

show. that the term containing J can be neglected. This term can

m

be supposed to represent the double layer activity. if the conduo-tivity of the membrane was the same as the conducconduo-tivity cr • The term.

o

containing both ~. ₁ and ~ ₀ stands for the inhomogeneity effect of the membrane. which appears to be muoh more important.

The strength of the double layer as a function of the position on the surface (cell membrane) is given in a very good approximation by:

Ia

1>

[0 0

-

cr.~J

1

:J

(2-3)

where 1> and

o cr o are respectively the potential and specific conductivity just outside the membrane and ~. and cr. analogously just inside the

mem-1 1

brane. Macroscopically this source can be regarded as a single current dipole, which can be explained by a mUltipole description of a concen-trated source distribution.

Separation of Laplace's equation

(2-4)

with respect to spherical coordinates yields general solutions of the kind:

A

=

(V+i

+ Brv).(C~(cose) + ~(COsS».

v v

r

.(E cos~~ + F sin ~~) (2-5)

where the spherical coordinates are defined by

x = r sinS cos~ y = r sinS sin~ z

=

r cose

and A, B, C, D, E, F, v, ~ are constants depending on the boundary conditions.

cp\l

and

£\1

are generalized Legendre functions of the first and second \! \!

kind (ref. 77).

Let Uil now consider the situation of an infinite homogeneous medium,

where the sources (primary 01' equivalent) are concentrated in a sphere

around the origin with radius R.

F01'

e =

0 01'

e =

~ most generalized Legendre functions becomes infinite,

which is objectionable on physicaZ grounds. Besides this it is neces-sary, that the potential remains the same for ~

=

~ + k 2~ and k

o

an arbitrary integer constant. These physical requirements involve , that

w

can be written as:

00 n A

L L

(-+1 + B _n nm _run r)P (cose) n m _n (E cosm~ + F sinm~)

IUD. IUD. (2-6)

n=o m=o r

where m and n are integer.

When the potential is restricted to the area r > R, it should be finite,

so:

~ =

L

n=o

n

L

_ I -I pm(cose)

G

cosm~

+ b sinm¢l

n+ n t:nm tun ~

m=o r

(2-7)

where the coefficients a

nm and bnm are detennined by the source.

Further study reveals, that a _{00 -}- __ _4~a I_ where I is the total current source concentrated in the origin. So

w

is the field of a point source

00

in the origin.

4~rraIO' 4~aall' 4~abll appear to be the total dipole activity along the z, x and y axes, concentrated again in the origin.

In a similar way one can interpret the higher order poles (e.g. ref. 7).

A most simple case of a source distribution is given by a current source I on the z-axis for z

=

p.

00 n \'~ L n+,1 n::::o r p (cose:) n r

where r is the observation point and ~ the source point. For n = 0 :

~

= ---41 the field of a monopole in the origin

!Tor n = Ip cose r 2 4!Tor (2-8)

the field of a dipole Ip in the origin, directed along the z-axis.

n

=

2 Ip2P2(cOSS_r) ~ = -;.-..!:...""3,--=- a quadrupole 4!Tor field. etc. I f ~ is not replaced by

on the z-axis, the factor P (cose ) in formula 2-8 should be

n r

(n-m)! m m

(n+m)! P (cosS )p (cosS )cosm(~ -$ )

n r n p r p (2-9)

if m

=

0 i f m

+

0

For a general distribution of Iv(~) ampere per unit volume as a function

of~, the potential can be obtained by integrating the modified equation

2-8 over the source area. Nevertheless one can distinguish a magnitude

• . . . d · .

P , wh~ch denotes the d~mens~ons of the source area, an r , wh~ch stands for the distance to the 2bserver. Then the field of multipole with index n is proportional to ( ~ )n, which implies, that for

r·

» p. the fields become negligible for l~rger n. As the dimensions of a single cell are very small compared to the complete body, the electric activity can very well be represented by a single current dipole, because the monopole disappears due to a zero total current. In a completely polarized cell the poten-tial ~. inside the membrane is constant. The same holds for

~ ~ o outside

the membrane. In this situation we are confrontated by a homogeneous, completely closed double layer. An observer outside

perceive any field from this source configuration.

the cell will not

The field of a dipole

~

in

~

is in the obeepvation point

~

given

by the potential

T

E.! • (.!,-.£) I'!'-.£ 13

T

=

m (2-9)

This implies. that the field of a double layer with E.!s as dipole strength per unit surface area. is given by:

If

E!~'

(2-10)

surface

Because the direction of _-s m is perpendicular to the surface. 2-70 can be written as:

(2-11 )

where ms =

IE!" I

and (l \E,.£) is the solid angle subtended by dS at

r. The sign of the solid angle depends on the fact. whether the posi-tive or negaposi-tive side of the double layer is faced. It will be evident now. that a closed homogeneous double layer will not have any effect in the observation point.

As soon as the cell membrane depolarises, an inhomogeneous double layer arises, which indeed causes an observable field. It is true, that the magnitude of the double layer is a function of the inner and outer

poten-tials ~i and ~o' but the behaviour of the double layer is described by equivalent current sources. Consequently, when ~. and ~ are given, the

1 0

activity of one cell can be represented by a single current dipole, pro-vided, that the surroundings of the cell remain the same. The current

dipole is an equivalent source and not a primary source. We will comment

on this in chapter 3.

In view of the dimensions of the cell, it was very well admissible to in-corporate only the dipole activity in our computations. This will also be possible for groups of cells or heart muscle segments, be it with a bigger error. Further study of the dimensions of these heart segments and their equivalent dipole (magnitude and position) will be necessary (see section 4). Such an approximation of the process is called a mUltiple dipole model, and it has been used by many investigators after its introduction by Selvester (ref. 26).

In the extreme the total heart may be represented by one single dipole, and even this drastic simplification appears to be very useful in prac-tice (the vectorcardiography) (see ref. 8, 9, 10).

3. Effects of inhomogeneities.

When we define an average 0 (specific conductivity) for the human body,

we can distinguish two types of inhomogeneities. On the one side there are areas with a relatively high conductivity (e.g. the well conducting blood mass in ventricles, veins, arteries), on the other hand, hardly conducting areas are existing (e.g. the air surrounding the body and in the lungs).

Areas with high conductivity are characterised by the effect of short-circuit or "amplification". Volumes of practically zero conductivity can be regarded as reflectors for currents. These phenomena can be illus-trated by the simple situations, sketched in figure 3-1, where the

sources (dipoles) are near by the inhomogeneity boundary.

a=finite primary sources

4--i

tj

~ a=in1inite

.--1

mirror sourceS figure 3-1 u=finite

..-t

primary sou rees

tf

-.!4

a=zero

---.

2><1>=0 ()n

i

mirror sources

The effect of the extreme conductivities in the right half spaces can be accomplished for an observer in the left half spaces by placing mirror sources in the right half spaces and assuming a complete homogeneous medium with conductivity o. We observe, that, at the boundary of a well conducting inhomogeneity, a normally directed dipole will be "amplified" due to an analogously directed mirror dipole, while a tangential dipole is almost short circuited by an opposing dipole. For the effect of the intracavitary bloodmass of the ventricles this implies, that in the electrodes only the fields of the normally directed dipoles will be ob-served. Furthermore it appears, that the dipolarity of the total heart activity is emphasized by this phenomenon. (see ref. 28, 50 and the following illustration in this section).

At the boundary with an isolator, the opposite occurs. This effect can be characterized as reflection. When we substitute for this boundary the skin-to air transition, where the electrodes are placed, we notice, that for these observers the sources are also amplified, because the same sources are present at the left and the right side.

Both the intracavitary bloodmass and the body boundary intensify sources. Quantitatively these effects are difficult to evaluate, the more so

as they are not additive. This can be illustrated by the following example.

I

\

\

/

I

-

-figure 3-2

\m

~

I

/

/ / ' U=finite u=zero

A current dipole is directed along the z-axis at a distance p from the ol'igin in a homogeneous medium (conductivity 0

0) , which is

boun-ded by an insulator and a super conductor as is shown in figure 3-2. The particular solution of the Poisson equation

v

2

w

= I where

v

I (r) stands for the source distribution, is defined as the potential

v

-due to the source in an infinUe homogeneous medium. This potential is given by:

00 n ~. = - -.;m::'--;;-2

L

(n+l) (2::.) p (cose) 1 4no p n=1 p n ~ u o m 2 4no r o 00 n-I n(.E) p (co~e) r n

I

n=1

which can be derived from fo~ula 2-8.

for r < p

for r > p

(3-1 )

The influence of the boundaries is incorporated by add1:ng a suitable soluUon ¢h of the corresponding LapZace equaUon

v

2

¢

= 0

~ =

h for q .~ r ~ R

The boundary conditions are given by:

1.!1

=0

ar r=R ~I _r=q

=

0

(3-2)

(3-3)

The assumption of uniformly convergent series leads to the potential on the boundary r = R. As a next simpZication we let p approach q

which means that the sources are very close to the intracavitary blood mass. The resulting potential is given by:

(cp + u where m <P ) -

--='--::-h r=R - 2 4no r p-+q 0 (3n+ 1 )R2n+ 1 00 n-l

I

n(.E) p (cose).c 1 r n n=

c

= 2n+1 n ( n+ I) p 2n+1 +nR 2n+1 (3-4)

Comparison with formula 3-1 reveals, that due to the inhomogeneities, the multipoles are multiplied by C, where

2n+1 n 2n+1 n 3n+1 n depending on the quotient p/R.

(3-5)

Conclusion: all multipoles are amplified, but the amplification decreases as the multipole index n increases, which states, that the dipolarity is enlarged by these kinds of inhomogeneities.

When we omit the blood mass, the C becomes equal to (2n+I)/n. Also neglec-tion of the body boundary results in a C of (2n+I)/n.

So it is clear that the effects of both inhomogeneities are not additive. The fields of several dipoles in this medium however may very well be added, because of the superposition theorem.

Many articles have been written about inhomogeneities in the form of con-ductivity interfaces. (e.g. ref. 39). Such an interface can always be re-presented by an equivalent double layer or single layer on the concerning boundary.

Very few articles, however, have been written about anisotropies. (e.g. ref. 35).

In chapter 2 the electrical activity of a heart muscle cell is shown to be represented by an equivalent double layer. This double layer, however, is no primary source, because it can change under influence of neighbouring cells or inhomogeneities. The conductivity of the cell membrane is very low compared to the extra cellular medium. This means, that an alteration in the surroundings will change the equivalent double layer. This effect could influence the measurements of the depolarisation wave through the heart muscle by way of intramural electrodes. Whether this is done during operations, or in an isolated heart, the natural surroundings of the heart are disturbed.

Durrer e.g. reports, that the intramural conduction velocity of an iso-lated heart is about 40-50 cm/sec compared with the 30 em/sec of the intact human heart (ref. 42). Plonsey (ref. 48) mentions the phenomenon, that a reduction in the size of the surrounding myocardium has been observed to decrease the duration of the plateau in figure 3-3.

Results of researches on the microscopic effects of the membrane depola-risation will be very important in relation to the neighbouring inhomo-geneities, as other cells are. We could think for example of a depolari-sat ion wave, which is a.o. determined by a minimum energy principle. We can illustrate this on the basis of one dimensional example. The question is, what direction (sign) the sources should have and at what time acti-vation is required in order to obtain a minimum energy dissipation.

Suppose we possess two souroes with an inner impedance, that are connected

by

a net-work of resistances.

figure 3-3

The network of figure 3-3 is of figure 3-4.

R01 R1 R2 R02

+ +

v

1 ' til R3 i2

t

v2

figure 3-4

The total energy delivered

by

the two sources amounts to:

E (3-6)

- 00

We can use the superposition theorem and define

and (3-7)

where i _l I is the part of i

l due to source vI etc. Considering the configuration and the polarities shown in figure 3-4, we get:

v v₂ - v

2 - vI

i I I =_1 _zl i22 = -_z2 i _l2 = i₂₁ = - - (3-8)

zl2 z21

~b

2 2

~

E ~

J

l

+ v2 _ v lv 2 _ vl v 2 dt zi z2 z 12 z21 -~ (3-9) 01' ~ V 2 2 ~ E

J

( I v2 )dt

-J

~+~

dt ~ + - - v 1v2 zl z2 z12 z2 -~ --~ (3-10) ~

which implies, that the dissipation is minimal, when

J

v

1v2dt is

maximal, independent of the values of 2

1, z2' z12 and-;21'

If vI and v

2 possess similar time fUnctions, we can conclude, that

they must be simultaneous.-So v ~ I v2

=

v and ~

f,

E ~

_J

_v2_dt + -I _I

~

(3-11) z2 z12 z21 -~

Further studY of the resistances reveals the trivial conclusion, that minimal dissipation occurs for large R

1, R2 and R3•

A suitable orientation of the equivalent dipoles may lead to simultaneous activation for a minimal dissipation. The anatomy of the heart restricts an optimal placing and orientation • Deviations from a normal activation path, which result mostly in a widened and enlarged QRS-complex (in

ventricular ectopic beats) sustain a minimum energy principle. Furthermore one can think of other laws, which determine the depolarisation wave. The sequence of contraction after the depolarisation may be a predominant fac-tor. Also the evolution and grow principle of the heart as well as the guarantee of maximal safety in the case of interruptions, may play an im-portant part. These aspects lie beyond our scope for the time being.

Macroscopically the electrical activity of the heart muscle can be seen as set up by some effective sources in that region with supposed homogeneous characteristics as far as conductivity is concerned. As mentioned before all conductivity interfaces can be represented by double layers too, so

that the total concept 15 a homogeneous medium containing equivalent

sour-ce" and bounded by the skin-air interface. In such a situation all dis-tribu:ed sources may be simulated by a concentrated source in the center of tIl<, body. This concentrated source is characterized then by multipoles and tile optimal center is defined by the least contribution of multipoles of a higher order than the dipole. (ref. 15, 29). The assumed multipole E'xpan:;ion should be valid for the unbounded medium. That is:

'\ u w '" I L. 0.=1 n '" \' IA ye L L.: nm mn m=o

( e , ")

+ B yO (8" )l I 't' nm mn ' '+' ~J --;tT - r whpre TIl ( ) P coss cosm¢ n m = P (cosS) sinm¢ n

hfe make use of the reciprocity theorem (1-4):

('I

- ;J

~'72~

b dV

=

II

~

b (lnd'l' dS r

Vr Sr

for the bounded medium (ill b Define 'i' by:

(2-60) (n--m)! m (n+m)!

~ bounded med !.um) •

(3-12) (1-4) (3-13) 2 Because 17 ~b

=

'72~ u Ie v

a the equivalent volume source, the left side of

equ.at ion 1-4 equals:

a

IJJ

I v (2-6o~ (n-m)! rn m (n+m)! dV (3-14) V

T

Generalisation of formula 2-8 teaches us, that 3-14 equals the multipole

coefficients~:j

multiplied by 4rr, so that combination of 1-4 and 3-13 leads to:

1

= 4rr

(l~tJ

Formula 3-15 reveals, that integration of the surface potential mUltiplied by a suitable function over the surface area provides us with the multipole coefficients for the equivalent source in an infinite medium. This is per-formed for complete bodies (ref. 44, 7) as well as for isolated hearts in artificial media. (ref. 55, 22). Some remarks should be made about above procedure:

1st. When the torso would be immersed in an infinite medium of the same conductivity as the body itself,

because the assumed sources are

the potential would not equal ~ ,

- - u

the effective sources and not only primary sources. And as we have seen before the equivalent sources of the conductivity interfaces are highly depending on the existence of the outer boundary. Therefore, it has to be stressed, that ~ is the

u theoretical, infinite potential when the sources were all primary.

2nd. All surface points should lie beyond a sphere, which encounters all equivalent sources, which is only approximately true, because there are several conductivity interfaces near the skin.

3rd. The multipole coefficients A and B contain all information, that

urn urn

is available in the surface potentials. Furthermore there is no re-dundancy, but the coefficients are not unique as they depend upon the choice of origin of the coordinate system.

4th. An infinite number of source distributions may produce the same mul-tipole series.

4. Restrictions on the number of equivalent dipoles.

It may be clear from the foregoing, that we can express the electrode sig-nals in a linear combination of several dipole strengths f. as a good

J

approximation depending on the number of the supposed dipoles. The coeffi-cients of this linear relation are determined by the potential in elec-trode i of a unity dipole in n. and with direction m. diminished by the

L_J -J

potential in a reference electrode, which we will define zero for the sake of simplicity. In practice this reference electrode will be far away from the sources, so it takes approximately the potential at infinity. The po-tentials of N electrodes are placed in a vector

y.

The dipole time functions are brought in a vector f dimension M. The linear relationship can be ex-pressed by:

y = A f (4-1 )

where A is an N x M matrix.

If the number of electrodes is larger than or equal to the number of di-poles, there is a solution for the dipole functions by way of the

pseudo-inverse of A:

(4-2)

This solution would be unbiased if:

1st. the model is completely correct with respect to the number of dipoles and the transfer matrix A is exactly known.

2nd. the possible noise on the electrode signals y. has a flat spectrum in

1

the inverse length frequency domain (we will comment on this later on). Furthermore the noise should have a zero mean.

The second condition is difficult to check, but the first condition already is certainly not satisfied.

The total number of dipoles will be tremendous if not infinite and the matrix A will never be known exactly.

Even if matrix! was accurately known and we assumed many dipoles, the

•

A • In an infinite, homogeneous, isotropic medium an element of matrix A is given by: T (r.-n.) m.

[

A:J"

,l

= - ] L] - ] 3 - 1J 41(0Ir.-n.1 1m. I - 1 LJ -J

l.Ej I « lEi I implies:

T r. m. = - 1 - ] . 3 • lEi I I!!!j I

~ij

- - + ₄₁₍₀ 6 .. 1J (4-3) (4-4)

6 .. will be small compared to the first term and it represents all mu1ti-1J

pole activity in the electrode except for the dipole. Neglect of all 6 .. 1J results in a rank 3 for matrix!, because all assumed dipoles originate in the origin of the coordinate system, and can be decomposed along three independent axes. In first approximation only three dipoles with fixed directions are to distinguish. When inhomogeneities are involved, the ef-fect seems to be enlarged according to examples of chapter 3 if, however, these inhomogeneities are so synunetric.

With the aid of the reciprocity theorem this phenomenon can be explained too. If a current is supplied to the body via two electrodes on the skin, a lead field will exist inside the body. In first approximation this lead field will be homogeneous··) in the area of the sources and will have two degrees of freedom viz. the field strength and the direction (first

derivative). The second and higher derivatives win be zero. Basically in this situation only three independent homogeneous fields can be contructed. Then in the concerning electrode pairs we will only observe the projections of all dipoles on the directions of the three lead fields. If we intend to extract more information from the system, it is necessary to place e1ec'-trodes in such a way, that the field in the area of the sources is as

in-• In chapter 5 will be shown, that there is no unique exact solution in-•

••

A homogeneous field is given by a potential: ~

=

aw + 8 where a and S are constants, and w is the distance in a constant, arbitrary direc-tion. A field is the more inhomogeneous the more the higher derivatives of w (and perpendicular directions) playa part.

homogeneous as possible. Furthermore, what is more important, the lead fields of others electrodes should be completely different. For the field may be very inhomogeneous due to e.g. the intracavitary blood mass, yet

the other lead fields will show the same characteristics (see ref. 12).

The amount of usable information about the electrical sources in the heart muscle concealed in the electrode signals is dependent on the following error sources:

1st. Simplification of the sources by a finite number of dipoles; the source area is divided into defined segments. All higher order poles of those segments are neglected as well as the secondary sources, that are caused by them in the conductivity interfaces. 2nd. The medium in between the source area and the electrodes will

never be completely known. The accuracy of matrix ~ is limited. Also the placing of the electrode will be submitted to error. 3rd. Noise on the electrode signals originating in the body

(myo-electrical sources) or outside the body (mains, amplifiers).

The number of dipoles is restricted by the errors 2 and 3. It means, that the errors due the discretization of the source, the errors in the trans-fer matrix and the noise from undesirable sourceS have to have the same level, when they are evaluated for the resultant electrode signals. Accor-ding to figure 4-1 there are four distinguishable errors, where e₃ is negligible, because it is a product of two small errors.

Above errors have to be studied carefully. As soon as on tried for example to determine the states (dipole functions) according to a least squares

criterion without constraints, the resulting states were very unrealistic

(ref. 38, 60). Because one knows from measurements (ref. 42, 53), what the rough orientations of the simulating dipoles are, negative values found were unacceptable. Lynn et al. then proceeded to create constraints by imposing nonnegativity of the states, so that more reliable estimates followed (ref. 38, 47).

Constraints on the time function of the states is also possible in order to compensate the lack of knowledge about the transfer matrix~ (ref. 54). For a systematic research of the information contents we suggest the

follo-primary transfer skin potentials _.1''\ 1. sou rees medium currents electrode signals rea Ii ty source transfer X model model ~4=!l source transfer ~1 ₄

I~

model error

J"-V

-00 sourCe transfer !.2 error

/

model source

\

Jtranster "\ !.3 error

"-

error models

+

errors figure 4-1

wing strategies. The surface potential ~ is a function of the time t and s

two surface coordinates defined by ~ and n. When ~ is measured as a func-s

tion of these three variables, this ~ can be Fourier transformed to both s

time frequency as length frequency domains·. The interesting parts of the ECG's are the QRS-complexes of a time duration of T seconds. The Fourier transforms of QRS-complexes indicates a maximal time frequency W, which

•

Similar work is done at the Imperial College in London and for a sim-plified form see reference 51.

is relevant observing the signal to noise ratio. This W will be approxi-mately the same for all i; and n values. On the basis of Shannon's theorem we can conclude, that 2WT samples will contain all information for certain

i; and n. The same procedure can be used with respect to i; and n, which results in an upper limit of N electrodes, which provide separately sig-nals y. defined by 2WT samples. Consequently there are 2WTN data, which

1

constitute a matrix, where the rows are the time samples of y .•

1 _ I

1

N I - - Y₁(lj ) - - - -I I I I Yj(ll)- - - -I I I

I

•

2WT figure 4-2

•

These N time functions y. form an N-dimensional time space. By factor

ana-1

lysis (ref. 16, 20, 24) the orthogonal components can be .determined. These orthogonal components form an orthogonal coordinate system in that

N-dimensional space. The power contribution of the orthogonal components to the y. signals, however, differs widely. A certain group of orthogonal

1

components will appear in the y. signal at a level equal to that of the

1

noise n. If there is no a priori information about the noise, this contri-bution can not be distinguished from the noise. In short because of the

noise the N-dimensional space degenerates into an M-dimensional space.

So there are only M independent y. signals, which provide altogether 2 WTM

1

data. The rest of the y. signals (N-M) can be expressed in a linear

com-1

bination of the M independent y. within the noise level. This linear

combi-1

nation needs (N-M) M coefficients.

The total number of data is now given by:

because

M~N and M ~ 2WT

The first term contains the information about the possible M states up to a scaling factor. So a maximum number M of linearly independent states (limited by T and W) can be estimated from M

(2WT-I)

data.

The rest of the data, viz. (N-M)M+M values, provide information about the transfer to the electrodes.

A related research of Barr e.a. (ref. 56) resulted in an M of about 24. In a similar way Martin and Pilkington (ref. 63) investigated the spatial independent transfer coefficients (matrix ~). They concluded, that it is impossible to compute epicardial potentials from surface potentials, if there are no constraints on those epicardial equivalent dipoles. However they modelled the body by a homogeneous, isotropic, spherical medium, and here only a limited inhomogeneous field in the source area can be set up by external sources; furthermore that field does not differ fundamen-tally from other fields generated by external sources. Their conclusion seems to be premature.

Another example of a study about possibilities of inverse cardiography is performed by Brody (ref. 60), who also inclines towards constraints on

the dipole functions. In chapter 5 we will define a direction of research

for determining these constraints.

The states, commented above, were defined by factor analysis of the elec-trode signals. Other states can be defined by the multipole series or a multiple dipole model. Theoretically the number of these states may be

larger than 24, but it is not likely. If we should decide to generate a model with 24 states, then these states can be valued arbitrarily. In that case every state represents a high information contents expressed in bits. If we put constraints on the states however, like the nonnegativity

(ref. 47, 38) or even define twovalued dipoles (ref. 46, 49) (1 bit per state), the information contents of every state can be drastically reduced so that more states are possible. In fact then the amplitude is discretised, where we did this before in time and space.

5. Suggestions for future research.

In order to obtain all information from the body surface potentials, many electrodes (> 24) will be necessary, which could be placed by the use of a vest. At the same tiIDe information should be gained about the coordi-nates of the electrodes and the measures of the body surface. Such measurements allows determination of surface integrals, which provide

the relevant multipole components as is shown in chapter 3-3. For that purpose all electrode signals have to be measured simultaneously. It is of course possible to measure potentials at different places sequentially with one reference, on which can be triggered later on, but this supposes a very stable heart function.

The choice of an origin of coordinate system is important, as the multi-pole components are dependent on that origin.

A favourable choice seems to be the center of the smallest sphere, that encounters both ventricles completely. This implies, that we should have a priori information about the position and magnitude of the heart. That information could be obtained by x-rays photographs.

Of course an alternative origin is the one, that causes a minimal contri-bution of higher multipoles to the potential (ref. 29). Such a criterion ,however,depends upon the actual electrical activity, while the former

suggestion is based only upon the position of the heart. For the sake of comparisons between healthy and defect hearts an origin, defined by the geometrical properties of the heart, seems to be more favourable.

The next step is then to relate these multipole coefficients to the trical heart action. As is illustrated before, all possible surface elec-trode information is incorporated in the multipole coefficients, but they do not define a unique source distribution. Plonsey (ref. 33) comments on the limited information contents of the multipole components. A very clear analysis of possible source distributions is given by Morse and Feshback

(ref. 6), which can be developed for our problem. We recall therefore formula (3-J2), which represents the potential in an infinite medium, due to sources within a sphere of radius b. (of course r> a).

I

n=J n ~

[A

ye (e,iP) +

B

yO (e,iP~

"!""-J

L run mn om ron ~ n+ m=o r (3-12)

Using formula (3-15) the coefficients A _run and B _run can be 'obtained from surface measurements. According to formula (3-10 the A and B are

re-run run lated to the source distribution by:

~~

211 11 b (2-00) (n-m)!

~:n(80'~0~

n+2 !

f

(

sine de

f

dr I m

=--

d<j>.,

J

_(n+m)

_!

_Y~n (eo' ~o) r 411(1 0 0 0 v 0 0 0 0 (5-1 ) where I = I (r , e , <j> ) v v 0 0 0

Any modification of I~, which does not ,change these coefficients, will produce the same potential outside r = b.

In particular we can add arbitrarily a source distribution I • without

v

•

disturbing the field, when Iv is given by:

y

(e,

<j> ) .R(r ) m.n 0 0 0 (5-2) and b

J

R(r ) r n+2 dr o 0 0 = 0 (5-3)

This is easy to verify by substituting (,5-2) in(5-1) and applying(5-3l In general we can write I in a series of spherical harmonics (ye , yO )

v mn mn as follows: "" n I (r ,

e ,

~

)

= v 0 0 0 \' \' fie (r ) ye (e ,

.p )

+

10 (r ) yO

(e ~)l L L L'mn 0 mn 0 0 mn 0 mn 0' "'0.:1 n=1 m=o where: (2_00) (2n+l) 211 m (n-m)!

f

d.p

411 (n+m) ! 0 o 0 11

f

Sine de I (r o 0 v 0

,e',~

0 0 ) (5-4) (5-5)

Comparison of(5-~with(5-~ leads to: n+2

. r dr

o 0 (5-6)

Now we can change the radial distributions I (r), without affecting the mn 0

multipole coefficients and consequently the resulting potential, in the following manner:

let there be a radial distribution n (r) given by _mn

0 -n-3 ( ) I (ro;) nmn ro = Y mn Y o < Y < 1 Then: yb

j

ne (r ) r mn 0 0 n+2 dr o -n-3 Yfb Ie (ro

I

)r n+2 Y mn Y 0 dr _o = o n+2 (ro /y ) d(ro;y)

=

0(2n+l) A nm (5-7) (5-8)

The same relation holds of course for 1_mn 0 and B _mn Thus we can compress each partial density I into

mn a smaller sphere and increase the magnitude by an appriate factor without affecting the potential outside r

=

b in the slightest. Analogously we can expand each partial density into a sphere of radius b' > b, without changing the potential outside r

=

b' •

This ambiguity is fundamentally caused by wide variety of possible radial source positions, so let us put a restriction on this. We will prove now, that the mUltipole components define a unique equivalent double layer on the sphere of radius a.

The partial density I (r) can then be written as: mn 0

(5-9)

The coefficients of the Dirac functions are determined by the fact, that the total current has to be zero.

2 A

nn,

1 Ke

0(2n+IJ ron 15 (r -a-Aa) - -=a_--::o . 2 (a-Aa)

2

a e

r:

n+1 2

1

= -0';(""2n-+'"'I""") Kron ~a 2Aa + Aa ••••••••

:J

In the limit Aa 4 0 and K 2Aa 4 J

A appears to be: nm n+1 n a A _nm = -7:~~ 0(2n+l) mn mn

Analogously for Band JO •

nm mn n+2d r r o 0 (5-10) (5-11 )

Furthermore it is trivial, that J may be chosen arbitrarily and that

00

A has to be zero.

00

It will also be clear, that no double layer can be found, that satisfies

equation(5-3~ So we can indeed conclude, that a spherical double layer is completely defined by the multipole series. It is very likely, that this applies to any convex surface inside a sphere r = b. One of the aspects of future research will be to find a closed formula, which relates the double layer on a convex surface to the multipole coefficients.

Of course the source distribution within the chest cavity is not in the form of a double layer.

Nevertheless it can be approximated in that manner.

The thickness of the heart muscle is small compared to the dimensions of the complete chest. Furthermore the depolarisation direction is rather perpendicular to the surface and deviations from this are obscured by the short circuiting effect of the intracavitary blood mass. So in first approximation we can define a representative double layer in the form of the two ventricles. This equivalent double layer then also incorporates the effect of the conductivity interface between heart muscle and blood mass. Of course there are also equivalent double layers on the conducti-vity interfaces of the lungs, ribs, sternum and spine. We will neglect these effects at first as is done in most reported researchwork,

because of a.o. insufficient knowledge about dimensions and electrical properties. The main reason at this moment, however, is to keep the theory surveyable. Vie really keep in mind, that the conductivity inter-faces will influence the estimation of primary sources in the heart. This influence will even be much more, than in the forward problem, where the electrode potentials has to be derived from a known source. This can easily be illustrated by the following example:

The field of a simple dipole in the origin can be simulated by a sphe2'ical doubte layer with radius a. But a nonhomogeneity in the

form of a spherical insulator (thinking of lungs) will distU:1'b the field. The system is sketched in figU:1'e 5-7.

a a u=o

o

!!! m

..

b c figure 5-1

For the infinite homogeneous medium holds:

, i l l cose ,) - - 2 41Tcrr A 0 nm B 0 m except for Ao 1 = 41T0 nm

Using formulae 5-77 and 5-4 shows:

J J 00 3m cose + 41Ta2 ~ ( 5-12)

The influence of the insulating spherical volume may be simulated jar

r > b+d by a mirror dipole m* in c in a complete homogeneous medium,

3<l>

and m ~

This results in a field potential:

<I> = m cose 2 41Tcrr ~ ~ + ..2!!...,.2

I

41Tcr r n= 1 n-I b n(-) p (cose) r n

which is a convergent series for r > c.

For the inverse problem difficulties arises. as: J

=

J oo + 3m cose 2 41Ta n=1

I

(2n+1 )m 2 . 41Ta d 3 b n-I (-b) (-) _a p _n (cose) (5-13) (5-14 )

mtd this series diverges. because b > a ; there exists no equivalent double layer on the sphere r = a.

Above example also illustrates the effect, that sources within a equiva-lent closed double layer are not observable or distinguishable from the double layer. Of course this phenomenon will remain, when the double layer is discretized and simulated by a finite number of dipoles. We can concentrate on an equivalent double layer shaped like the heart ventricles, which is not a convex surface however. Because of the septum

we cannot expect, that the mUltipole coefficients define a unique double

layer intensity. We are forced to put in more a priori knowledge about the expected activation path through the heart muscle. This corresponds to the nonnegativity constraint on supposed dipoles on which we reported before. The research about what restrictions are necessary and sufficient can be made with the use of the string model (ref. 63). This string model provides a simulation of the depolarisation wave by way of a time varying double layer on a spherical surface and a plane inside the sphere through the center of it. See figure 5-2.

y

figure 5-2

The string model contains the fundamental aspects of the real ventricles as far as the electrical properties are concerned. Mathematical operations are facilitated because of the symmetrical configuration. An effect

like the intracavitary blood mass can 'easily' brought into account. In short the string model may be an ideal expedient to test theoretical procedures before applying them to the real situation.

A complete procedure for the solution of the inverse problem can be deve-loped with the aid of the string model. This procedure should depart from the multipole coefficients, which is completely different from the conven-tional procedures (26, 54, 32, 47, 27). Furthermore we are trying to con-stitute an equivalent double layer, which possibly may be divided later on into equivalent dipoles.

The main goal however remains the definition of the problem with so much a priori information, that the solution is unique. We believe, that this can be accomplished with above strategy.

As soon as a unique equivalent double layer is determined further research can be directed towards the separation of the primary sources in the heart muscle from the equivalent sources, due to conductivity interface with

6. Conclusions.

Because the body boundary potentials do not define a unique source distri-bution, more a priori information has to be obtained. This information should contain data about the configurations of the inhomogeneities as well as the electrical properties of living tissues. While this is information as far as the medium between sources and electrodes is concerned, more information about the sources itself is also necessary, whereby a restriction on the source area is not sufficient. Constraints have to be derived from the activation path as measured by Durrer. Here a difficulty arises because these constraints have to be valid for anomalies too, as the whole research is finally directed towards detection and localisation of defects. Until now the constraints were obtained in a rather heuristic way and consisted of predefined positions and directions of dipoles with a nonnegative strength as a function of time. We intend to analyse this problem more fundamentally. Therefore the multipole coeffi-cients of the source for an infinite medium are assumed to be known. These multipole coefficients contain all possible information from the surface

po-tential. By modelling the heart as a double layer with constraints on the intensity per surface area, we hope to define the problem in such a way, that a unique solution is possible. The string model may be very useful for this research. One has also to take into account the noise and model errors as defined in chapter 4.

As soon as above solution is obtained, the theory can be applied to practice. This implies the determination of the multipole coefficients from the surface potential. Many electrodes and consequent amplifiers will be necessary besides a system; which measures the dimensions of the body and the positions of the electrodes. Furthermore x-rays photographs are very desirable. So it will be clear, that a thorough theoretical foundation has to be available before such