1 Introduction
IN CLINICAL electrocardiography an empirical-statistical evaluation is made of the potentials on the torso surface generated by the electrical activity of the heart. These potentials are measured at specific sites on the torso. Although, in the various lead systems used, both the number and the location of these sites may be different, in all systems they are defined with respect to anatomical landmarks observable on the body surface (the intercostal spaces) (MACFARLANE,1989a). This means that the interindividual variations of the ECG, due to the differences of geometry between subjects with respect to lead positions, are implicitly assumed to be smaller than the ones that lead to a classification as abnormal. One of the most important features of interindividual geometry variation is probably the position and orientation of the heart with respect to the ribs. Whereas the general size of a subject has an overall scaling effect on the potentials measured, differences in heart position and orientation may lead to waveform changes and nonuniform changes in amplitude. ECG waveforms and changing amplitudes at certain leads are important discriminants used in E C G classification (MACFARLANE,1989b). A quantification of the influence of heart position and orientation on ECG waveforms in the single individual, when torso size and electrode position are fixed, is therefore of importance for the evaluation of the diagnostic accuracy of clinical electrocardiography. In theoretical electrocardiography, in particular in the Correspondence should be addressed to G. J. M. Huiskamp, Lab. of Medical Physics & Biophysics, University of Nijmegen, Geert Grooteplein noord 21, 6525 EZ Nijmegen, The Netherlands First received 18th March and in final form 14th November 1991 9 IFMBE: 1992
Medical & Biological Engineering & Computing
so-called inverse problem, the influence of heart position and orientation is of even greater importance. In this field, the generation of the electrocardiogram is treated as a physical problem in which the characteristics of the electrical generator within the ventricular walls are computed from measured body surface potentials. Solutions to the inverse problem are known to be very sensitive to measuring and modelling errors. In a previous paper, we have shown that it is indeed impossible to obtain stable solutions, that are physiologically interoretable, when interindividual geometry variation is disregarded completely (HuISKAMPand VANOOSTEROM,1989). Sufficiently accurate individual measurements of torso shape and relative electrode position in a practical, i.e. clinical, application can probably be taken relatively easily. For measurements of heart position and orientation however, which are not observable from the outside, the demands of practicality and accuracy may be conflicting. Quantification of the influence of heart position and orientation within a single thorax is therefore of great importance to the inverse problem of electrocardiography, in particular to the determination of an upper bound for the required accuracy of the geometry measurements. In this paper, we present the results of a model study on the influence of small changes in heart position and orientation on both simulated QRS waveforms and calculated ventricular activation sequences. For these studies, we used three sets of accurate geometry data, based on MR! crosssections, of three healthy subjects, as well as their recorded set of body surface potentials. In addition, for each of the subjects, we used an individual ventricular activation sequence, obtained from an inverse procedure using the measured individual geometry and the recorded ECG data. These activation sequences, of which the physiological realism has been discussed previously (HuISKAMP and
November 1992
613
VAN OOSTEROM, 1988), were used as a reference solution in the inverse study and as the generator of the QRS waveforms in the forward simulation study.
the ventricular surface) and in A(p, q). This transfer operator A contains, implicitly, all volume conduction effects included in the model: the differing conductivities of the lungs and the ventricular cavities and the geometry of these compartments and of the torso boundary. A(p, q) is calculated using the boundary element method, the validity and accuracy of which has been demonstrated in the past (MEIJS et al., 1989). In this study, the conductivities are fixed. The effects of including different compartments of differing conductivity have been described extensively in a previous paper (VAN OOSTEROM and HUISKAMP, 1989). The induced variations of geometry used in this paper that affect the operator A(p, q) are small rotations and translations of the entire heart (and consequently of the ventricular cavities).
2 Methods
The model used to describe the generation of the QRS complex is represented by the following equation (CuPPEN, 1984): V(p, t)
=
[
dsh
A(p, q)H[t -- z(q)] aS(q)
(1)
where V(p, t) = potential at time t within the QRS interval (te[O, T]) at torso point p, pEP, the total torso surface A(p, q) = transfer coefficient relating sources at point q on the ventricular surface to resulting potentials at torso point p z(q) = local depolarisation (breakthrough) time of points q, qESh, the ventricular surface H = Heaviside step function which switches on the element dS(q) from z(q) onward.
2.1 Forward In the forward simulation study, first reference potentials V ~ are calculated from a transfer matrix A ~ based on the actual, measured geometry of the subject concerned and a known activation sequence z defined on the ventricular surface Sh. Next, potentials Va are calculated from the transfer matrix A A, corresponding to induced slight changes in overall heart position and orientation, and from the same activation sequence z. To compare VA with V~ two measures of difference are defined. The first one is the (dimensionless) relative RMS
In this equation, geometry is present in p (the position of the leads on the torso), S h (the position and orientation of
i
Vl
R
V4
/l
/
=pl
o ,,L'
~ 1 7 6
50tl I,._..," ._ 1(30
0~ /
,%/
/%
0
50
100 0-'-'~-~
"'50-" ..... 100 0
~ 501
V
I
Ill
F
V3
~oo
o--
614
100
50
100
~%
5o~..---'-1oo
0~ - I
I
Fig. 1
50 "-"
V6
SI
....
/""~100
-
111
Measured Q RS complexes in standard leads for subjects A (broken line) and B (solid line). Vertical scale: 1 mV ; horizontal scale: 100 ms M e d i c a l & Biological Engineering & C o m p u t i n g
N o v e m b e r 1992
difference, defined as
re l d i f =
~
fe~[ V---~(p--, t)] 2-----d;d t
"
(2)
Furthermore, a maximum absolute difference maxdif is used, defined as
maxdif =
max
I VA(p, t) - V~
t) l
(3)
pEP, t~[O, T]
maxdifis expressed in mV.
2.2 Inverse In the inverse study, first the reference activation sequence z ~ is calculated from known (measured) body surface potentials V, using the transfer matrix A ~ This
~ . . ~ black~
a
activation sequence z ~ is the activation sequence used in the forward simulation study. The potentials V ~ simulated using this sequence, closely resemble the ones measured. As a next step, activation sequences z A are calculated from the same measured potentials V, but now using the transfer matrix A A derived from the changed geometry. To compare z* with z ~ the measures relerr and maxerr are used, the definition of which is analogous to that of relclifand m a x d i f maxerr is expressed in ms. An important item in inverse electrocardiology is the type and amount of regularisation used solving the illposed problem. In our inverse procedure, the surface Laplacian is used to stabilise solutions. The amount of regularisation is chosen such that solutions are obtained having a specific value for the integral of the square of the surface Laplacian over the heart surface. This specific value is chosen a priori, and it has been shown previously that solutions thus obtained are both physiologically
\
=nt
blackpoint
ypoint
( blackpoint black point
grey point
b Fig. 2
Thorax geometry of subject A (a) in a transversal crosssection at z = 0 cm and (b) in a longitudinal cross-section at x = Ocm. The fine lines represent the original (measured) geometry of the torso surface, the lung surface and the ventricular surface, including the ventricular cavities. The heavy lines depict ventricular geometry in the case of a rotation A~ of O'O3n around the z-axis. The grey point is a point beneath lead V2 on the original ventricular geometry. The black one is the same point on the rotated geometry. The measured positions of lead V1 and V2 on the torso are indicated as (a) black bars and (b) black circles
Medical & Biological Engineering & Computing
Fig. 3
Thorax geometry of subject B (a) in a transversal crosssection at z = 0 cm and (b) in a longitudinal cross section at x = 0 cm. Here the heavy lines represent the ventricular geometry in the case of translation Ay of - 0 . 0 5 cm in the y direction. See Fig. 2.for further explanation
November 1992
615
acceptable and stable with respect to small electrode displacements and noise (HuISKAMP and VAN OOSTEROM, 1989). The protocol for obtaining the solutions related to the different translations and rotations of the heart considered is the same as the one described in the same paper.
and three by applying separate orthogonal rotations A~b, A0 or A~ of about 5 ~ (0-03n) around the - z , + y and - x - a x i s (assuming the origin at the centre of mass of the heart). The magnitude of the rotation was chosen such that the effect of Aq~ and A0 on a point of the epicardium lying directly beneath lead V2 was a translation in the same direction and of about the same magnitude as the effect of the corresponding direct translations Ay and Az. Some of the rotated and translated geometries, with the original geometry as a reference, are shown in Figs. 2 and 3. Fig. 2 shows, for subject A, the geometry with the heart and cavities (bold lines) rotated - 0 . 0 3 n around the z-axis, which is pointed upwards, toward the head. On the top, the contours depict a transversal cross-section at z -- 0 cm (xy-plane); on the bottom, they depict a longitudinal crosssection at x = 0 cm (yz-plane). The lighter lines depict the original geometry. The two small circles/bars represent the sites of precordial leads Vx and V2 ; the grey and the black points indicate the effect of the rotation on a point on the epicardium beneath V2 (the grey point is at the original position). Fig. 3 shows the same cross-sections, for subject B, of the geometry, with the heart translated 0.5 cm in the direction of the negative y-axis (toward the right lung).
3 Material
Geometry data were obtained from MRI transverse cross-sections, made with the equipment from the research department of Philips Medical Systems at Best, The Netherlands. For three subjects, 28 cross-sections were produced at intervals of 1.5 cm. The recordings were triggered on the peak of the R-wave in lead 2 of the surface electrocardiogram, resulting in images of the heart geometry in diastole. The contours of the torso boundary, lung boundary, heart surface and enclosed surface of the ventricular cavities were read from photographs of the MRI scan using a digitising tablet. In a separate session, body surface potentials of the same subjects were recorded in 64 electrodes at 500 samples per second. The standard leads presentation of the QRS complexes for two of the subjects, A (broken line) and B (solid line), is shown in Fig. 1. The precordial standard leads' positions (the respective intercostal spaces), which are a subset of the total of 64 recording sites, produced clear reference points in the MRI scans. Consequently, an accurate localisation of the electrodes, with respect to the geometry of the heart and to that of the other interfaces considered, could be established. Based on the measured thorax geometry, six sets of transfer coefficients were computed for each subject: three by applying separate (orthogonal) translations Ax, Ay, or Az of - 0 - 5 c m to the heart and the ventricular cavities; R
I
4 Results
4.1 Forward QRS complexes in the 64 electrodes were calculated for each of the three subjects A, B and C, for each of the geometry configurations (transfer matrices) A ~ (the original, i.e. unrotated, untranslated, geometry), A ax, A Ay, A aZ and A 'x~', A A~ A'xL Individual activation sequences z ~ , z ~ and z ~ were used.
V4~ ~tf,~
vl
' . ."
l&0 0 I /
5 ~
' l&O
V5 '~~ o
so
Ill
0
Fig. 4
616
o
40
F
50
1()0 0
50
i
o
lOO o
v3
V6
100 0
100
0
16o
50
100
Simulated QRS complexes for subject A. The broken lines represent the simulation in the case of the original, measured geometry. The solid lines are the simulations in case Ay, in which the heart is translated O'05 cm in the y-direction. Vertical scale: 1 mV ; horizontal scale: 100 ms
Medical & Biological Engineering & Computing
November 1992
/ V4 j
Vl
~o ' ~
50
-'~,.y"-~
%0
"~--%0
16o
o
~oo
'/
~So o
io
r
io L/ 16o
~6o o
Woo
vluI
i%1
[I1| ~ 0
~
i i~k 50
i
100 0
~
V3~\/\
V6~
'"
50
100 0
I~
' 0 100
50
i
lO0
Fig. 5 Simulated Q R S complexes f o r subject C. T h e broken lines represent the simulation f o r the original geometry. T h e solid lines are the simulations in case AO, in which the heart is rotated 0.037r around the y-axis. Vertical scale: 1 m V ; horizontal scale: 100 ms
Table 1 Forward: reldif and m a x d i f values f o r simulated potentials on 64 electrodes. For two cases, the Q R S waveforms are depicted in the figures indicated.
A Ax Ay Az Adp AO
A7
rel max rel max rel max tel max rel max rel max
0'06 0-18 mV 0-05 0.14mV 0"05 0-16mV 0-09 0.32mV 0-08 0.18mV 0"09 0.15mV
Medical & Biological Engineering & Computing
B tel max rel max rel max rel max tel max rel max
C
0-07 (Fig. 4) rel 0.24 mV max 0-07 rel 0.28mV max 0"06 rel 0.27mV max 0"12
0-46mV 0"08 0.42mV 0'09 0"30mV
November 1992
tel max rel max rel max
0-10 0.62 mV 0-08 0.51mV O"11
0'66mV 0"11
0-85mV 0" 12 (Fig. 5) 0"87mV O"12
0"66mV
617
In Fig. 4, the solid lines were the standard lead representation of the QRS complexes for subject A, in the case of a translation Ay of - 0 . 5 c m in the y direction (moving the heart toward the right lung). The broken lines are the simulations in the case of the original geometry. This is the case in which the smallest effect on the QRS complexes (for all three subjects) was found: the value of reldifhere is 0.05, maxdifis 0.14mV. Fig. 5 shows, for subject C, the simulation in the case of a rotation A0 around the y-axis of 0.03n (turning the heart downward when facing it frontally). This represents the case in which the largest effect on QRS complexes was found: reldifhere is 0.12, maxdifis 0.87 mV. The values of reldif and maxdif, for each subject and for all rotations and translations, are presented in Table 1.
4.2 Inverse Ventricular activation sequences were calculated for each of the three subjects A, B and C, for each of the geometry configurations A ~ A ax, A ay, A A~ and A ar A a~ A a~. The individual recorded body surface potential measurements VA, Vn, and Vc were used as input. Figs. 6a-d show the activation sequences z ~ za~ za~ and zav, respectively, at the epicardium of subject A. Isochrones of activation are shown at steps of 2 ms. relerr and maxerr values, with respect to the reference activation sequence z ~ are (0.04, 8.4ms), (0.05, 12"0ms) and (0.08, 19.8ms), respectively. All views are from an anterior viewpoint with the heart in its natural, original position. iii O ~ ! ~ i i ! i ~ ! ~ i ~ i ~ ! ! ~ { { ~ i i ~ i i ~ ! ~ ! ~ i i ~ ! ~ ! i ~ i i ~ ! ! ~ i ~ i ! ~ i ~ ! ~ i ! ;:: . . . . . . . . . : : ~ : : : : : : : : : : : : : : : : : : : : : : : ~ : : : : : : : : : : : : : : : : : : : ~ : ~ : : : : : : : : : : ~ : : : : : : : : : ; : : : : : : : : : : : : : : : : : ~ : : : : : : : : : : : : : ~ : : : : : : : : : : : : : :
:::::~:::::::::::::::::::::::::::::::::::::::::::;::::::::::::::::::::::::::::
...................
In Fig. 7 for subject C, the reference activation sequence z ~ (a) and the sequence z ay (c) at the epicardium are shown. Figs. 7b and d show the same actix, ation sequences z ~ and z ay, but now at the endocardial aspect of the right and left ventricle, lying immediately beneath the epicardium, relerr and maxerr values for z ay are 0.20 and 32.5 ms, respectively. The values of relerr and maxerr are summarised in Table 2. 5 Discussion This study was not of a statistical nature, and we did not seek for any quantitative generalisation of the results obtained. Indeed, owing to the great interindividual geometry variations, any mean or generalised value for deviations of the ECG due to heart position and orientation will be of little significance in assessing the accuracy of the E C G measured in a single individual. We instead presented effects which can be expected in some individual, normal cases. If, in even one of these cases, significant effects of small changes in heart position and orientation are shown, the unknown geometry of many other individual cases cannot be ruled out as a major contributor to the generation of QRS waveforms.
5.1 Forward As can be seen in Figs. 2 and 3, the rotations and translations chosen in this study are small when compared to the interindividual variations in geometry. The views on the heart from leads V1 and V2 in subject A and subject B b . . . .
::::::::~::,:::.::::::::::,,::::::::
! ~ i ~ ! i i ~ i i i ~ i i ~ i i i i ~ ! ! ~ ! ~ i i ! i ~ i i i i ! ~ ! ~ i i ~ i ! i i i i i i ~ i i i ::::::~:~:,::::::~ ~:::::.:~:,:.::,::.:~,,,::::::::::,::;::::::::~:~ :~:::::::: ~ , : : : : : ~ : ~ . : : : : :~
:::::::::::::::,::::::::..:.:::::::::::::::::::::::::::::::::.,,.::::
.................
::::: :::::::::
84 / : ~
/:::::::
:
ii~ ~ii iiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiii ii!!iiiiiiiilii!ii!iiiiiiiiiili iiiiiiiiiiiiiiiii!iiiiiii iiiiiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiii iiiiiii? !i!!! ii
C
~ii~i~i~i~!!ii!~!~!!!~!!~!~!~i~!~!!!~i~i~!~!!~i:~!~::~!:!~i~::!:!~!~
d
~iii~iii~!~!!~!~!!iii!~!~ii!!!~i~!!~i~!~i~!~i~!!!i!~!~i!i!!:!~ii!!i:i:!
Fig. 6 Activation sequences: (a) z ~ (left upper) ; (b) z~~
upper) ; (c) z ~" (left lower) ; (d) z ~ (right lower), for subject A. Shown are four identical three-dimensional frontal views of the epicardium with the heart in its natural, measured position and orientation. Isochrones of activation are shown at steps of 2 ms
618
Medical & Biological Engineering & Computing
November 1992
o ii~!i~!i~i~i~!~7!~i~iii!i~iiiiiiii~iii~ii~ji~i~i~iii~!~!~ii~i~i~i~:~i~j~iii~ii~ii b ii!i•iiiii]7i•iiiiiiii•7!iiii7i•i7ii7iiiiiiiiijiiiiiiiiiiiiiiiii7iiiiiii;ii;i!iiiiiiiii7iii•i•i;i•ij•ii;i
:::::::::::::::::::::::::~
~/
~
::::;::::;: .......... !iiiiii!!"
/
9
::::::ill::;
:::
"
::::',i~
,"/
J
/- j'1
. . . . . . iiiiiiiii~i~iii!i~ ~ "~4..S
/
r-----
,\\\\Xxj~
'
\
---~'~/~'~///'~/-'--Q 1 ,~//I/j/l./
$6 S2
. -
/
~!!i!i:
:::::::::::::::::::::::::::
--"'-'-'~ ---.~.~---_~ii : ....................... i i i !i i i i i i i i
\
f ~ _ _ _ -z-,--z-:
4
Z: ;
.............
:ii i:ii::
5~
/~/t -.,--~:h:.i :::
7 ii! c"7111iiiiiiiii::iTiiiiiiiiiii{iiiiiiiiiiiiiiiiiiiiii::ili
Fig. 7
iiiiiiiiiiiiir
ii:iiiiiiiiiiiiiiiiiiii::
d
ii:i!
iii::i::r
ii
7:ii
: i:i:!iii::i::ii:.iiiiiii::
;
Activation sequences: (a, b) zo (upper) and (c, d) zA, (lower)for subject C. The left part, (a) and (c), shows frontal views of the epicardium, as in Fig. 6. The right part, (b) and (d), shows, in the same view, the endocardial aspects of the right and left ventricle that lie immediately beneath the epicardium Table 2 Inverse: relerr and maxerr values for inverse solutions. For four cases, the corresponding solutions are depicted in the figures indicated.
A Ax Ay Az Adp AO A7
rel max rel max tel max tel max tel max rel max
0.07 19-8 ms 0-05 9-9 ms 0.04
8-5 ms 0"05 (Fig. 6c) 12-0 ms 0-04 (Fig. 6b) 8"4 ms 0"04 (Fig. 6d) 8"9 ms
B tel max tel max tel max rel max rel max rel max
are quite different, and this is reflected very prominently in Q R S complexes presented in Fig. 1. Apart from the particular (individual) activation sequence, in the generation of these waveforms, the torso size, conductivity distribution and the heartposition and orientation play a role. O n e aim of this study has been to single out and quantify the latter effect. The values in Table 1 indicate that the effect on reldif of a rotation is larger than that of a translation, which is to be expected as, for m o r e distant electrodes, solid angle theory implies that, in contrast to that of a rotation, the relative effect of a translation decreases. The specific lead system used in this study, which is distributed inhomogeneously over the torso surface, obviously influences the Medical & Biological Engineering & Computing
C 0"03 6-1 ms 0"03 4-9 ms 0.07 15"2 ms 0-05 11"6 ms 0-05 11-4 ms 0"04
9-3 ms
tel max tel max tel max tel max tel max rel max
0"03 8-2 ms 0.02 6-0 ms 0"21 33-3 ms 0"20 (Figs. 7 c and d) 32-5 ms 0"02 8.0 ms 0"02 2.9 ms
values of reldif found, m a x d i f values, however, which occur in the precordial leads, can be assumed to be representative. It is found that they are as high as 0 . 8 7 m V in case C, in which the m a x i m u m Q R S amplitude is 3-61 mV. The influence on Q R S waveforms of these small changes i"_ heart r o s i t i o n and orientation m a y be, as expected, just a mln,,[ scaling of amplitudes, as in lead V2 in Fig. 4. However, liE in Fig. 5 shows a prominent decrease of its m i n i m u m value, whereas the m a x i m u m is hardly affected. In that same figure, the m a x i m u m and m i n i m u m in lead V3 remain a b o u t the same, whereas a second, relative m a x i m u m is shifted toward zero. The amplitude of an individual peak of the Q R S complex m a y be of little diagnostic significance. The
November 1992
619
notion, however, that such small induced changes (0.5 cm !) in geometry can reflect changes of QRS amplitude of the order of the observed standard deviation for large populations of normals (MACFARLANE,1989b) is of importance. It suggests that placing leads with reference to the real heart position and not to skeletal anatomy may substantially decrease variability. The unknown heart orientation will still be a source of error. The latter is much more difficult to measure than heart position, and it is not o b v i o u s in which way one might adapt electrode placement to it. If a further improvement of the diagnostic accuracy of the E C G is to be obtained, an inverse procedure, in which the E C G waveforms are no longer directly interpreted and which incorporates the physics and the individual geometry of the problem, will be needed. 5.2 Inverse Fig. 6 and the values in the first two columns of Table 2 indicate that regularisation is able to stablise solutions to the ill-posed inverse problem, even if, in addition to the modelling and measurement errors already present, systematic errors are introduced. The errors, or rather the differences resulting from the induced geometry changes, present in the inverse solutions of Fig. 6 are not a reflection of the ill-posed nature of the problem: they must occur as the geometry of the ventricles on which the potential patterns measured on the torso are 'projected' by the inverse procedure is changed by rotations or translations. The activation pattern z A~ of Fig. 6b, where the heart is rotated downward when facing it frontally, shows a shift in the opposite direction with respect to pattern z ~ in Fig. 6b, as is to be expected. In Fig. 6c, the corresponding situation for a rotation of the heart to the left (toward the right lung) is shown (za*). Here, the activation pattern is again shifted in the opposite direction (towards the left lung). In Fig. 6d, the heart is rotated clockwise in the plane of the paper, around the x-axis, which is pointing toward the observer. Therefore the central part of the figure remains in place; the shift of the pattern would, in this case, be better visible in a left sagittal view. The third column of Table 2 and Fig. 7 show that things are different in case C. O f the three subjects A, B and C involved in this study, C clearly was the most lean, with the heart relatively close to the precordial leads. The large errors and departures from the reference activation sequence z ~ (i.e. z~ and z A*) occurring in this case are therefore probably not due to the ill-posed nature of the inverse problem. Here, the error introduced by rotating the heart away from its original position, with the precordial leads so close, has become too large to agree with the particular activation sequence on the ventricular geometry (which, as can be seen in Fig. 7, includes the surface of the endocardium). In this case, an increase in the amount of regularisation applied will not alter the situation. When errors made in assessing heart position and orientation are of the order of the small translations and rotations used in this study, which is likely when, for example, echocardiographic methods are used, regularisation can ensure a stable solution to the inverse problem of electro-
620
cardiography. Extra care has to be taken, however, in cases in which the heart is very close to the precordial leads (lean persons). Then, a systematic error in the estimation of the complicated ventricular geometry will make it very difficult to find a reliable solution in terms of the ventricular activation sequence. Acknowledgment--This research was made possible by a grant of tl3e Dutch Heart Foundation NHS.
References
CUPPEN,J. J. M. (1984) Calculating the isochrones of ventricular depolarization. S I A M J. Scien. Stat. Comp., 5, 105-120.
HUISKAMP,G. J. M. and VAN OOSTEROM,A. (1988) The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans., BME-35, 1047-1058. HUISKAMP, G. J. M. and VAN OOSTEROM, A. (1989) Tailored versus realistic geometry in the inverse problem of electrocardiography. Ibid., BME-36, 827-835. MACFARLANE, P. W. (1989a) Lead systems. In Comprehensive electrocardiology. MACFARLANE,P. W. and LAWRIE, T. T. V. (Eds.), Pergamon Press, 315-352. MACFARLAN~, P. W. (1989b) The normal electrocardiogram and vectorcardiogram. In Comprehensive electrocardiology. MACFARLANE, P. W. and LAWRIE,T. T. V. (Eds)., Pergamon Press, 407-458. MEIJS, J. W. H., WEIER, O. W., PETERS, M. J. and VAN OOSTEROM, A. (1989) On the numerical accuracy of the boundary element method. IEEE Trans., BME-36, 1038-1049. VAN OOSTEROM,A. and HUISKAMP,G. J. m. (1989) The effect of torso inhomogeneities on body surface potentials quantified by using tailored geometry. J. Electrocardiol., 22, 53-72.
Authors' biographies Geertjan Huiskamp was born in 1958 in 'sHertogenbosch, The Netherlands. He received his Masters degree in Experimental Physics at the University of Nijmegen in 1985. From 1985 to 1988 he was a Ph.D. student at the Laboratory of Medical Physics & Biophysics at the University of Nijmegen, working on the inverse problem of electrocardiography. He received his Ph.D. degree at the University of Nijmegen in 1989. He is currently working on a project on inverse electrocardiography subsidised by the Dutch Heart Foundation. His main interests are numerical modelling and computer graphics representation of complex data. Adriaan van Oosterom was born in 1942 in Abcoude, The Netherlands. From 1961 to 1965 he trained as an electronics engineer at the Laboratory of Medical Physics, University of Amsterdam. He received the Masters degree in Physics in 1971, and the Ph.D. degree in Physics in 1978, both from the University of Amsterdam, studying cadiac potential distributions. From 1971 to 1982, he works at the Laboratory of Medical Physics and the Department of Experimental Cardiology, University of Amsterdam, where he researched the electrical activity of the heart. He is now a Professor of Medical Physics at the University of Nijmegen.
Medical & Biological Engineering & Computing
November 1992
IN CLINICAL electrocardiography an empirical-statistical evaluation is made of the potentials on the torso surface generated by the electrical activity of the heart. These potentials are measured at specific sites on the torso. Although, in the various lead systems used, both the number and the location of these sites may be different, in all systems they are defined with respect to anatomical landmarks observable on the body surface (the intercostal spaces) (MACFARLANE,1989a). This means that the interindividual variations of the ECG, due to the differences of geometry between subjects with respect to lead positions, are implicitly assumed to be smaller than the ones that lead to a classification as abnormal. One of the most important features of interindividual geometry variation is probably the position and orientation of the heart with respect to the ribs. Whereas the general size of a subject has an overall scaling effect on the potentials measured, differences in heart position and orientation may lead to waveform changes and nonuniform changes in amplitude. ECG waveforms and changing amplitudes at certain leads are important discriminants used in E C G classification (MACFARLANE,1989b). A quantification of the influence of heart position and orientation on ECG waveforms in the single individual, when torso size and electrode position are fixed, is therefore of importance for the evaluation of the diagnostic accuracy of clinical electrocardiography. In theoretical electrocardiography, in particular in the Correspondence should be addressed to G. J. M. Huiskamp, Lab. of Medical Physics & Biophysics, University of Nijmegen, Geert Grooteplein noord 21, 6525 EZ Nijmegen, The Netherlands First received 18th March and in final form 14th November 1991 9 IFMBE: 1992
Medical & Biological Engineering & Computing
so-called inverse problem, the influence of heart position and orientation is of even greater importance. In this field, the generation of the electrocardiogram is treated as a physical problem in which the characteristics of the electrical generator within the ventricular walls are computed from measured body surface potentials. Solutions to the inverse problem are known to be very sensitive to measuring and modelling errors. In a previous paper, we have shown that it is indeed impossible to obtain stable solutions, that are physiologically interoretable, when interindividual geometry variation is disregarded completely (HuISKAMPand VANOOSTEROM,1989). Sufficiently accurate individual measurements of torso shape and relative electrode position in a practical, i.e. clinical, application can probably be taken relatively easily. For measurements of heart position and orientation however, which are not observable from the outside, the demands of practicality and accuracy may be conflicting. Quantification of the influence of heart position and orientation within a single thorax is therefore of great importance to the inverse problem of electrocardiography, in particular to the determination of an upper bound for the required accuracy of the geometry measurements. In this paper, we present the results of a model study on the influence of small changes in heart position and orientation on both simulated QRS waveforms and calculated ventricular activation sequences. For these studies, we used three sets of accurate geometry data, based on MR! crosssections, of three healthy subjects, as well as their recorded set of body surface potentials. In addition, for each of the subjects, we used an individual ventricular activation sequence, obtained from an inverse procedure using the measured individual geometry and the recorded ECG data. These activation sequences, of which the physiological realism has been discussed previously (HuISKAMP and
November 1992
613
VAN OOSTEROM, 1988), were used as a reference solution in the inverse study and as the generator of the QRS waveforms in the forward simulation study.
the ventricular surface) and in A(p, q). This transfer operator A contains, implicitly, all volume conduction effects included in the model: the differing conductivities of the lungs and the ventricular cavities and the geometry of these compartments and of the torso boundary. A(p, q) is calculated using the boundary element method, the validity and accuracy of which has been demonstrated in the past (MEIJS et al., 1989). In this study, the conductivities are fixed. The effects of including different compartments of differing conductivity have been described extensively in a previous paper (VAN OOSTEROM and HUISKAMP, 1989). The induced variations of geometry used in this paper that affect the operator A(p, q) are small rotations and translations of the entire heart (and consequently of the ventricular cavities).
2 Methods
The model used to describe the generation of the QRS complex is represented by the following equation (CuPPEN, 1984): V(p, t)
=
[
dsh
A(p, q)H[t -- z(q)] aS(q)
(1)
where V(p, t) = potential at time t within the QRS interval (te[O, T]) at torso point p, pEP, the total torso surface A(p, q) = transfer coefficient relating sources at point q on the ventricular surface to resulting potentials at torso point p z(q) = local depolarisation (breakthrough) time of points q, qESh, the ventricular surface H = Heaviside step function which switches on the element dS(q) from z(q) onward.
2.1 Forward In the forward simulation study, first reference potentials V ~ are calculated from a transfer matrix A ~ based on the actual, measured geometry of the subject concerned and a known activation sequence z defined on the ventricular surface Sh. Next, potentials Va are calculated from the transfer matrix A A, corresponding to induced slight changes in overall heart position and orientation, and from the same activation sequence z. To compare VA with V~ two measures of difference are defined. The first one is the (dimensionless) relative RMS
In this equation, geometry is present in p (the position of the leads on the torso), S h (the position and orientation of
i
Vl
R
V4
/l
/
=pl
o ,,L'
~ 1 7 6
50tl I,._..," ._ 1(30
0~ /
,%/
/%
0
50
100 0-'-'~-~
"'50-" ..... 100 0
~ 501
V
I
Ill
F
V3
~oo
o--
614
100
50
100
~%
5o~..---'-1oo
0~ - I
I
Fig. 1
50 "-"
V6
SI
....
/""~100
-
111
Measured Q RS complexes in standard leads for subjects A (broken line) and B (solid line). Vertical scale: 1 mV ; horizontal scale: 100 ms M e d i c a l & Biological Engineering & C o m p u t i n g
N o v e m b e r 1992
difference, defined as
re l d i f =
~
fe~[ V---~(p--, t)] 2-----d;d t
"
(2)
Furthermore, a maximum absolute difference maxdif is used, defined as
maxdif =
max
I VA(p, t) - V~
t) l
(3)
pEP, t~[O, T]
maxdifis expressed in mV.
2.2 Inverse In the inverse study, first the reference activation sequence z ~ is calculated from known (measured) body surface potentials V, using the transfer matrix A ~ This
~ . . ~ black~
a
activation sequence z ~ is the activation sequence used in the forward simulation study. The potentials V ~ simulated using this sequence, closely resemble the ones measured. As a next step, activation sequences z A are calculated from the same measured potentials V, but now using the transfer matrix A A derived from the changed geometry. To compare z* with z ~ the measures relerr and maxerr are used, the definition of which is analogous to that of relclifand m a x d i f maxerr is expressed in ms. An important item in inverse electrocardiology is the type and amount of regularisation used solving the illposed problem. In our inverse procedure, the surface Laplacian is used to stabilise solutions. The amount of regularisation is chosen such that solutions are obtained having a specific value for the integral of the square of the surface Laplacian over the heart surface. This specific value is chosen a priori, and it has been shown previously that solutions thus obtained are both physiologically
\
=nt
blackpoint
ypoint
( blackpoint black point
grey point
b Fig. 2
Thorax geometry of subject A (a) in a transversal crosssection at z = 0 cm and (b) in a longitudinal cross-section at x = Ocm. The fine lines represent the original (measured) geometry of the torso surface, the lung surface and the ventricular surface, including the ventricular cavities. The heavy lines depict ventricular geometry in the case of a rotation A~ of O'O3n around the z-axis. The grey point is a point beneath lead V2 on the original ventricular geometry. The black one is the same point on the rotated geometry. The measured positions of lead V1 and V2 on the torso are indicated as (a) black bars and (b) black circles
Medical & Biological Engineering & Computing
Fig. 3
Thorax geometry of subject B (a) in a transversal crosssection at z = 0 cm and (b) in a longitudinal cross section at x = 0 cm. Here the heavy lines represent the ventricular geometry in the case of translation Ay of - 0 . 0 5 cm in the y direction. See Fig. 2.for further explanation
November 1992
615
acceptable and stable with respect to small electrode displacements and noise (HuISKAMP and VAN OOSTEROM, 1989). The protocol for obtaining the solutions related to the different translations and rotations of the heart considered is the same as the one described in the same paper.
and three by applying separate orthogonal rotations A~b, A0 or A~ of about 5 ~ (0-03n) around the - z , + y and - x - a x i s (assuming the origin at the centre of mass of the heart). The magnitude of the rotation was chosen such that the effect of Aq~ and A0 on a point of the epicardium lying directly beneath lead V2 was a translation in the same direction and of about the same magnitude as the effect of the corresponding direct translations Ay and Az. Some of the rotated and translated geometries, with the original geometry as a reference, are shown in Figs. 2 and 3. Fig. 2 shows, for subject A, the geometry with the heart and cavities (bold lines) rotated - 0 . 0 3 n around the z-axis, which is pointed upwards, toward the head. On the top, the contours depict a transversal cross-section at z -- 0 cm (xy-plane); on the bottom, they depict a longitudinal crosssection at x = 0 cm (yz-plane). The lighter lines depict the original geometry. The two small circles/bars represent the sites of precordial leads Vx and V2 ; the grey and the black points indicate the effect of the rotation on a point on the epicardium beneath V2 (the grey point is at the original position). Fig. 3 shows the same cross-sections, for subject B, of the geometry, with the heart translated 0.5 cm in the direction of the negative y-axis (toward the right lung).
3 Material
Geometry data were obtained from MRI transverse cross-sections, made with the equipment from the research department of Philips Medical Systems at Best, The Netherlands. For three subjects, 28 cross-sections were produced at intervals of 1.5 cm. The recordings were triggered on the peak of the R-wave in lead 2 of the surface electrocardiogram, resulting in images of the heart geometry in diastole. The contours of the torso boundary, lung boundary, heart surface and enclosed surface of the ventricular cavities were read from photographs of the MRI scan using a digitising tablet. In a separate session, body surface potentials of the same subjects were recorded in 64 electrodes at 500 samples per second. The standard leads presentation of the QRS complexes for two of the subjects, A (broken line) and B (solid line), is shown in Fig. 1. The precordial standard leads' positions (the respective intercostal spaces), which are a subset of the total of 64 recording sites, produced clear reference points in the MRI scans. Consequently, an accurate localisation of the electrodes, with respect to the geometry of the heart and to that of the other interfaces considered, could be established. Based on the measured thorax geometry, six sets of transfer coefficients were computed for each subject: three by applying separate (orthogonal) translations Ax, Ay, or Az of - 0 - 5 c m to the heart and the ventricular cavities; R
I
4 Results
4.1 Forward QRS complexes in the 64 electrodes were calculated for each of the three subjects A, B and C, for each of the geometry configurations (transfer matrices) A ~ (the original, i.e. unrotated, untranslated, geometry), A ax, A Ay, A aZ and A 'x~', A A~ A'xL Individual activation sequences z ~ , z ~ and z ~ were used.
V4~ ~tf,~
vl
' . ."
l&0 0 I /
5 ~
' l&O
V5 '~~ o
so
Ill
0
Fig. 4
616
o
40
F
50
1()0 0
50
i
o
lOO o
v3
V6
100 0
100
0
16o
50
100
Simulated QRS complexes for subject A. The broken lines represent the simulation in the case of the original, measured geometry. The solid lines are the simulations in case Ay, in which the heart is translated O'05 cm in the y-direction. Vertical scale: 1 mV ; horizontal scale: 100 ms
Medical & Biological Engineering & Computing
November 1992
/ V4 j
Vl
~o ' ~
50
-'~,.y"-~
%0
"~--%0
16o
o
~oo
'/
~So o
io
r
io L/ 16o
~6o o
Woo
vluI
i%1
[I1| ~ 0
~
i i~k 50
i
100 0
~
V3~\/\
V6~
'"
50
100 0
I~
' 0 100
50
i
lO0
Fig. 5 Simulated Q R S complexes f o r subject C. T h e broken lines represent the simulation f o r the original geometry. T h e solid lines are the simulations in case AO, in which the heart is rotated 0.037r around the y-axis. Vertical scale: 1 m V ; horizontal scale: 100 ms
Table 1 Forward: reldif and m a x d i f values f o r simulated potentials on 64 electrodes. For two cases, the Q R S waveforms are depicted in the figures indicated.
A Ax Ay Az Adp AO
A7
rel max rel max rel max tel max rel max rel max
0'06 0-18 mV 0-05 0.14mV 0"05 0-16mV 0-09 0.32mV 0-08 0.18mV 0"09 0.15mV
Medical & Biological Engineering & Computing
B tel max rel max rel max rel max tel max rel max
C
0-07 (Fig. 4) rel 0.24 mV max 0-07 rel 0.28mV max 0"06 rel 0.27mV max 0"12
0-46mV 0"08 0.42mV 0'09 0"30mV
November 1992
tel max rel max rel max
0-10 0.62 mV 0-08 0.51mV O"11
0'66mV 0"11
0-85mV 0" 12 (Fig. 5) 0"87mV O"12
0"66mV
617
In Fig. 4, the solid lines were the standard lead representation of the QRS complexes for subject A, in the case of a translation Ay of - 0 . 5 c m in the y direction (moving the heart toward the right lung). The broken lines are the simulations in the case of the original geometry. This is the case in which the smallest effect on the QRS complexes (for all three subjects) was found: the value of reldifhere is 0.05, maxdifis 0.14mV. Fig. 5 shows, for subject C, the simulation in the case of a rotation A0 around the y-axis of 0.03n (turning the heart downward when facing it frontally). This represents the case in which the largest effect on QRS complexes was found: reldifhere is 0.12, maxdifis 0.87 mV. The values of reldif and maxdif, for each subject and for all rotations and translations, are presented in Table 1.
4.2 Inverse Ventricular activation sequences were calculated for each of the three subjects A, B and C, for each of the geometry configurations A ~ A ax, A ay, A A~ and A ar A a~ A a~. The individual recorded body surface potential measurements VA, Vn, and Vc were used as input. Figs. 6a-d show the activation sequences z ~ za~ za~ and zav, respectively, at the epicardium of subject A. Isochrones of activation are shown at steps of 2 ms. relerr and maxerr values, with respect to the reference activation sequence z ~ are (0.04, 8.4ms), (0.05, 12"0ms) and (0.08, 19.8ms), respectively. All views are from an anterior viewpoint with the heart in its natural, original position. iii O ~ ! ~ i i ! i ~ ! ~ i ~ i ~ ! ! ~ { { ~ i i ~ i i ~ ! ~ ! ~ i i ~ ! ~ ! i ~ i i ~ ! ! ~ i ~ i ! ~ i ~ ! ~ i ! ;:: . . . . . . . . . : : ~ : : : : : : : : : : : : : : : : : : : : : : : ~ : : : : : : : : : : : : : : : : : : : ~ : ~ : : : : : : : : : : ~ : : : : : : : : : ; : : : : : : : : : : : : : : : : : ~ : : : : : : : : : : : : : ~ : : : : : : : : : : : : : :
:::::~:::::::::::::::::::::::::::::::::::::::::::;::::::::::::::::::::::::::::
...................
In Fig. 7 for subject C, the reference activation sequence z ~ (a) and the sequence z ay (c) at the epicardium are shown. Figs. 7b and d show the same actix, ation sequences z ~ and z ay, but now at the endocardial aspect of the right and left ventricle, lying immediately beneath the epicardium, relerr and maxerr values for z ay are 0.20 and 32.5 ms, respectively. The values of relerr and maxerr are summarised in Table 2. 5 Discussion This study was not of a statistical nature, and we did not seek for any quantitative generalisation of the results obtained. Indeed, owing to the great interindividual geometry variations, any mean or generalised value for deviations of the ECG due to heart position and orientation will be of little significance in assessing the accuracy of the E C G measured in a single individual. We instead presented effects which can be expected in some individual, normal cases. If, in even one of these cases, significant effects of small changes in heart position and orientation are shown, the unknown geometry of many other individual cases cannot be ruled out as a major contributor to the generation of QRS waveforms.
5.1 Forward As can be seen in Figs. 2 and 3, the rotations and translations chosen in this study are small when compared to the interindividual variations in geometry. The views on the heart from leads V1 and V2 in subject A and subject B b . . . .
::::::::~::,:::.::::::::::,,::::::::
! ~ i ~ ! i i ~ i i i ~ i i ~ i i i i ~ ! ! ~ ! ~ i i ! i ~ i i i i ! ~ ! ~ i i ~ i ! i i i i i i ~ i i i ::::::~:~:,::::::~ ~:::::.:~:,:.::,::.:~,,,::::::::::,::;::::::::~:~ :~:::::::: ~ , : : : : : ~ : ~ . : : : : :~
:::::::::::::::,::::::::..:.:::::::::::::::::::::::::::::::::.,,.::::
.................
::::: :::::::::
84 / : ~
/:::::::
:
ii~ ~ii iiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiii ii!!iiiiiiiilii!ii!iiiiiiiiiili iiiiiiiiiiiiiiiii!iiiiiii iiiiiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiii iiiiiii? !i!!! ii
C
~ii~i~i~i~!!ii!~!~!!!~!!~!~!~i~!~!!!~i~i~!~!!~i:~!~::~!:!~i~::!:!~!~
d
~iii~iii~!~!!~!~!!iii!~!~ii!!!~i~!!~i~!~i~!~i~!!!i!~!~i!i!!:!~ii!!i:i:!
Fig. 6 Activation sequences: (a) z ~ (left upper) ; (b) z~~
upper) ; (c) z ~" (left lower) ; (d) z ~ (right lower), for subject A. Shown are four identical three-dimensional frontal views of the epicardium with the heart in its natural, measured position and orientation. Isochrones of activation are shown at steps of 2 ms
618
Medical & Biological Engineering & Computing
November 1992
o ii~!i~!i~i~i~!~7!~i~iii!i~iiiiiiii~iii~ii~ji~i~i~iii~!~!~ii~i~i~i~:~i~j~iii~ii~ii b ii!i•iiiii]7i•iiiiiiii•7!iiii7i•i7ii7iiiiiiiiijiiiiiiiiiiiiiiiii7iiiiiii;ii;i!iiiiiiiii7iii•i•i;i•ij•ii;i
:::::::::::::::::::::::::~
~/
~
::::;::::;: .......... !iiiiii!!"
/
9
::::::ill::;
:::
"
::::',i~
,"/
J
/- j'1
. . . . . . iiiiiiiii~i~iii!i~ ~ "~4..S
/
r-----
,\\\\Xxj~
'
\
---~'~/~'~///'~/-'--Q 1 ,~//I/j/l./
$6 S2
. -
/
~!!i!i:
:::::::::::::::::::::::::::
--"'-'-'~ ---.~.~---_~ii : ....................... i i i !i i i i i i i i
\
f ~ _ _ _ -z-,--z-:
4
Z: ;
.............
:ii i:ii::
5~
/~/t -.,--~:h:.i :::
7 ii! c"7111iiiiiiiii::iTiiiiiiiiiii{iiiiiiiiiiiiiiiiiiiiii::ili
Fig. 7
iiiiiiiiiiiiir
ii:iiiiiiiiiiiiiiiiiiii::
d
ii:i!
iii::i::r
ii
7:ii
: i:i:!iii::i::ii:.iiiiiii::
;
Activation sequences: (a, b) zo (upper) and (c, d) zA, (lower)for subject C. The left part, (a) and (c), shows frontal views of the epicardium, as in Fig. 6. The right part, (b) and (d), shows, in the same view, the endocardial aspects of the right and left ventricle that lie immediately beneath the epicardium Table 2 Inverse: relerr and maxerr values for inverse solutions. For four cases, the corresponding solutions are depicted in the figures indicated.
A Ax Ay Az Adp AO A7
rel max rel max tel max tel max tel max rel max
0.07 19-8 ms 0-05 9-9 ms 0.04
8-5 ms 0"05 (Fig. 6c) 12-0 ms 0-04 (Fig. 6b) 8"4 ms 0"04 (Fig. 6d) 8"9 ms
B tel max tel max tel max rel max rel max rel max
are quite different, and this is reflected very prominently in Q R S complexes presented in Fig. 1. Apart from the particular (individual) activation sequence, in the generation of these waveforms, the torso size, conductivity distribution and the heartposition and orientation play a role. O n e aim of this study has been to single out and quantify the latter effect. The values in Table 1 indicate that the effect on reldif of a rotation is larger than that of a translation, which is to be expected as, for m o r e distant electrodes, solid angle theory implies that, in contrast to that of a rotation, the relative effect of a translation decreases. The specific lead system used in this study, which is distributed inhomogeneously over the torso surface, obviously influences the Medical & Biological Engineering & Computing
C 0"03 6-1 ms 0"03 4-9 ms 0.07 15"2 ms 0-05 11"6 ms 0-05 11-4 ms 0"04
9-3 ms
tel max tel max tel max tel max tel max rel max
0"03 8-2 ms 0.02 6-0 ms 0"21 33-3 ms 0"20 (Figs. 7 c and d) 32-5 ms 0"02 8.0 ms 0"02 2.9 ms
values of reldif found, m a x d i f values, however, which occur in the precordial leads, can be assumed to be representative. It is found that they are as high as 0 . 8 7 m V in case C, in which the m a x i m u m Q R S amplitude is 3-61 mV. The influence on Q R S waveforms of these small changes i"_ heart r o s i t i o n and orientation m a y be, as expected, just a mln,,[ scaling of amplitudes, as in lead V2 in Fig. 4. However, liE in Fig. 5 shows a prominent decrease of its m i n i m u m value, whereas the m a x i m u m is hardly affected. In that same figure, the m a x i m u m and m i n i m u m in lead V3 remain a b o u t the same, whereas a second, relative m a x i m u m is shifted toward zero. The amplitude of an individual peak of the Q R S complex m a y be of little diagnostic significance. The
November 1992
619
notion, however, that such small induced changes (0.5 cm !) in geometry can reflect changes of QRS amplitude of the order of the observed standard deviation for large populations of normals (MACFARLANE,1989b) is of importance. It suggests that placing leads with reference to the real heart position and not to skeletal anatomy may substantially decrease variability. The unknown heart orientation will still be a source of error. The latter is much more difficult to measure than heart position, and it is not o b v i o u s in which way one might adapt electrode placement to it. If a further improvement of the diagnostic accuracy of the E C G is to be obtained, an inverse procedure, in which the E C G waveforms are no longer directly interpreted and which incorporates the physics and the individual geometry of the problem, will be needed. 5.2 Inverse Fig. 6 and the values in the first two columns of Table 2 indicate that regularisation is able to stablise solutions to the ill-posed inverse problem, even if, in addition to the modelling and measurement errors already present, systematic errors are introduced. The errors, or rather the differences resulting from the induced geometry changes, present in the inverse solutions of Fig. 6 are not a reflection of the ill-posed nature of the problem: they must occur as the geometry of the ventricles on which the potential patterns measured on the torso are 'projected' by the inverse procedure is changed by rotations or translations. The activation pattern z A~ of Fig. 6b, where the heart is rotated downward when facing it frontally, shows a shift in the opposite direction with respect to pattern z ~ in Fig. 6b, as is to be expected. In Fig. 6c, the corresponding situation for a rotation of the heart to the left (toward the right lung) is shown (za*). Here, the activation pattern is again shifted in the opposite direction (towards the left lung). In Fig. 6d, the heart is rotated clockwise in the plane of the paper, around the x-axis, which is pointing toward the observer. Therefore the central part of the figure remains in place; the shift of the pattern would, in this case, be better visible in a left sagittal view. The third column of Table 2 and Fig. 7 show that things are different in case C. O f the three subjects A, B and C involved in this study, C clearly was the most lean, with the heart relatively close to the precordial leads. The large errors and departures from the reference activation sequence z ~ (i.e. z~ and z A*) occurring in this case are therefore probably not due to the ill-posed nature of the inverse problem. Here, the error introduced by rotating the heart away from its original position, with the precordial leads so close, has become too large to agree with the particular activation sequence on the ventricular geometry (which, as can be seen in Fig. 7, includes the surface of the endocardium). In this case, an increase in the amount of regularisation applied will not alter the situation. When errors made in assessing heart position and orientation are of the order of the small translations and rotations used in this study, which is likely when, for example, echocardiographic methods are used, regularisation can ensure a stable solution to the inverse problem of electro-
620
cardiography. Extra care has to be taken, however, in cases in which the heart is very close to the precordial leads (lean persons). Then, a systematic error in the estimation of the complicated ventricular geometry will make it very difficult to find a reliable solution in terms of the ventricular activation sequence. Acknowledgment--This research was made possible by a grant of tl3e Dutch Heart Foundation NHS.
References
CUPPEN,J. J. M. (1984) Calculating the isochrones of ventricular depolarization. S I A M J. Scien. Stat. Comp., 5, 105-120.
HUISKAMP,G. J. M. and VAN OOSTEROM,A. (1988) The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans., BME-35, 1047-1058. HUISKAMP, G. J. M. and VAN OOSTEROM, A. (1989) Tailored versus realistic geometry in the inverse problem of electrocardiography. Ibid., BME-36, 827-835. MACFARLANE, P. W. (1989a) Lead systems. In Comprehensive electrocardiology. MACFARLANE,P. W. and LAWRIE, T. T. V. (Eds.), Pergamon Press, 315-352. MACFARLAN~, P. W. (1989b) The normal electrocardiogram and vectorcardiogram. In Comprehensive electrocardiology. MACFARLANE, P. W. and LAWRIE,T. T. V. (Eds)., Pergamon Press, 407-458. MEIJS, J. W. H., WEIER, O. W., PETERS, M. J. and VAN OOSTEROM, A. (1989) On the numerical accuracy of the boundary element method. IEEE Trans., BME-36, 1038-1049. VAN OOSTEROM,A. and HUISKAMP,G. J. m. (1989) The effect of torso inhomogeneities on body surface potentials quantified by using tailored geometry. J. Electrocardiol., 22, 53-72.
Authors' biographies Geertjan Huiskamp was born in 1958 in 'sHertogenbosch, The Netherlands. He received his Masters degree in Experimental Physics at the University of Nijmegen in 1985. From 1985 to 1988 he was a Ph.D. student at the Laboratory of Medical Physics & Biophysics at the University of Nijmegen, working on the inverse problem of electrocardiography. He received his Ph.D. degree at the University of Nijmegen in 1989. He is currently working on a project on inverse electrocardiography subsidised by the Dutch Heart Foundation. His main interests are numerical modelling and computer graphics representation of complex data. Adriaan van Oosterom was born in 1942 in Abcoude, The Netherlands. From 1961 to 1965 he trained as an electronics engineer at the Laboratory of Medical Physics, University of Amsterdam. He received the Masters degree in Physics in 1971, and the Ph.D. degree in Physics in 1978, both from the University of Amsterdam, studying cadiac potential distributions. From 1971 to 1982, he works at the Laboratory of Medical Physics and the Department of Experimental Cardiology, University of Amsterdam, where he researched the electrical activity of the heart. He is now a Professor of Medical Physics at the University of Nijmegen.
Medical & Biological Engineering & Computing
November 1992
