Spherical-harmonic approximation to the forward problem of electrocardiology

Journal of Electrocardiology Vol. 32 No. 2 1999

Spherical-Harmonic Approximation to the Forward Problem of Electrocardiology R. M a r t i n

Arthur,

PhD

Abstract: Forward-problem solutions were approximated using sphericalharmonic series on an adult male torso model with heart and lungs. These approximations were found using only a knowledge of torso-model geometry and were not based on a prior solution for surface potentials. Because these series depend only on polar and azimuthal angles, they allow continuous estimation of the forward-problem solution over the torso without further knowledge of the torso geometry. Compared to the conventional method, potentials estimated from fifth-degree series for eight distributed double-layer sources had an average relative error of 0.036. Relative errors were similar with and without torso inhomogeneities. The fifth-degree series solution (36 terms) was found four times faster than the conventional method and provided a data reduction factor of about 20 in the 715-node torso model studied. Spherical-harmonic series transform surface potentials into an orthogonal basis set whose spatial-frequency content increases with increasing degree. Consequently, these series may provide a structure for the systematic study of the effect on forward-problem solutions of both changes in torso shape and inclusion of inhomogeneities. K e y words: cardiac sources, forward problem, spherical harmonics, torso model.

element due to a given source in or on the heart (2,3,4). Using the integral formulation, torso-surface potentials are often f o u n d iteratively (1,5,6). Typically, the initial values for the surface potentials are those of the cardiac source of interest in an infinite medium. Convergence of the conventional, iterative process yields the b o u n d e d - m e d i u m potential at each of the discrete elements of the torso model. F o r w a r d - p r o b l e m results can be used to specify transfer coefficients for use in inverse solutions, w h i c h are aimed at finding cardiac-equivalent generators. If the sites of the f o r w a r d - p r o b l e m solutions do not coincide with electrocardiographic measurem e n t sites, however, t h e n transfer coeffidents must be interpolated to find values at the electrode locations. In this study, the polar- and azimuthal-angle

Numerical m e t h o d s for solving the far-field forward problem of electrocardiology in realistic torso models are well k n o w n . These techniques are based o n Maxwell's equations, w h i c h have been simplified because propagative, capacitive, and inductive effects can be neglected in the h u m a n body at frequencies observed in cardiac sources (1). One of the most c o m m o n l y e m p l o y e d f o r w a r d - p r o b l e m techniques is based on an integral formulation in w h i c h potentials are f o u n d directly at each surface

From the Electronic Systems and Signals Research Laboratory, Department of Electrical Engineering, Washington University in St. Louis, St. Louis, Missouri. This work was supported in part by Washington University. Reprint requests: R. Martin Arthur, PhD, Department of Electrical Engineering, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130. Copyright © 1999 by Churchill Livingstone ® 0022-0736/99/3202-0003510.00/0

103

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Journal of Electrocardiology Vol. 32 No. 2 April 1999

solutions for Laplace's equation in spherical coordinates, n a m e l y spherical harmonics, were used as the basis functions for a series approximation to forward-problem solutions (7,8,9). These sphericalharmonic series can be f o u n d w i t h o u t prior knowledge of the conventional, iterative solution. Approximating forward-problem solutions with spherical harmonics allows a list of potentials at discrete torso sites to be replaced with a smaller set of coefficients from which the forward-problem solution can be calculated e v e r y w h e r e o n the torso surface. Thus, only the azimuthal and polar angles at a site of interest are n e e d e d to estimate the forward-problem solution there. Furthermore, spherical-harmonic decomposition of the forward problem m a y offer a suitable structure in which to systematically study the influence of torso shape and inhomogeneities on torso potentials. For this study, spherical-harmonic series were constructed to approximate forward-problem solutions for b o t h uniform, double-layer sources and orthogonal, dipole, and quadrupole sources in an i n h o m o g e n e o u s adult male torso. Both distributed and discrete sources were used because any set of distributed sources is arbitrary and necessarily incomplete. Discrete multipolar sources, however, can be combined to characterize any distributed source. The quality of the spherical-harmonic series was characterized by comparison to conventional, iterative solutions. Spherical-harmonic series were also e x a m i n e d for: (a) reduction in computation time, (b) data reduction in representation, (c) use in interpolation, and (d) spatial-frequency decomposition of surface potentials.

potential due to the source in an infinite medium. The surface integral excludes the point at which V(r) is determined (1,2,3).

Torso Model The torso model used in this study was a version of an aduk male model reported previously (6), with heart and lungs added. Figure i shows the node locations of the inhomogeneous torso model. The torso surface contained 715 nodes in 23 contours of 31 nodes plus two caps. The heart surface, estimated from radiograph silhouettes of the subject of the torso study, contained 58 nodes in 7 contours of 8 nodes plus two caps. Each lung surface, taken from computed tomography scans of an aduk male, contained l l 0 nodes in 9 contours of I2 nodes plus two caps. Conductivity of the torso, heart, and lungs was set to 0.2, 0.6, and 0.05 S/m, respectively (4,10).

Discrete Forward-Problem Solutions In matrix notation the discrete analog to equation 1 for V(r) in a torso model with heart and lungs is seen here in equation 2.

Vh

V~

~th

(0"1 - - O't) ~'~tl

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Forward-Problem Solution on the Surface of an Inhomogeneous Torso

[v j Vh

The integral solution to potential distribution V(r) on the surface of an i n h o m o g e n e o u s conductor in an insulating m e d i u m (air) due to an internal s o u r c e j i is given by equation 1.

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O-t

+

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~h O~h + o-t O- 1 q- O"t

where the subscripts t, h, and 1designate the torso, heart, and lung, respectively; f~ is solid angle, o-is conductivity, and • is infinite-medium potential (1,5). The infinite-medium potential m a y be due to any source. Here, we consider both distributed and discrete sources. The infinite-medium potential due to a double layer is seen here in equation 3. ~IrL(r) --

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Fig. 1. N o d e locations of the surface e l e m e n t s of the i n h o m o g e n e o u s torso model. The torso surface, w h i c h had 715 nodes, c o n t a i n e d a 5 8 - n o d e heart (upper panels) and 110-node lungs (lower panels). Conductivity of the torso, heart, and lungs was 0.2, 0.6, and 0.05 S/m, respectively. where p the solid potential order N equation

is d o u b l e - l a y e r strength a n d df~ is a n g l e it s u b t e n d s . T h e i n f i n i t e - m e d i u m due to a multipole source through (N = 2 is a d i p o l e ) is s e e n h e r e i n 4.

~ N ( r ) = xpN (r, O, qb)

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1Yl=O

+bnm sin(mq~)]

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Journal of Electrocardiology Vol. 32 No. 2 April 1999

w h e r e pm(cos 0) is an associated Legendre polynomial of degree n and order m (7,8,9). For both distributed and discrete sources, infinite-medium potentials were evaluated using a 7-point numerical integral over each triangular surface element ( 11 ). Solid angles were calculated using the m e t h o d of Barnard and coworkers (5) as i m p l e m e n t e d by van Oosterom and Strackee (12). The solid-angle matrix was deflated to r e m o v e its singularity (1,13,14). Forward-problem solutions were f o u n d in the i n h o m o g e n e o u s torso model for eight uniform, double-layer sources. A - 1 m a / m m double-layer source of radius 32 m m was located at a position corresponding to the radiograph center of the heart, which was at (20,0) in the frontal plane and ( - 5 0 , 0 ) in the sagittal plane. The double layer was aimed in five directions: (a) - Z (back to front), (b) X (right to left), (c) m i d w a y b e t w e e n - Z and Y, (d) m i d w a y b e t w e e n X and Y, and (e) toward the apex of the heart. The double-layer disk was also m o v e d toward the apex of the heart along the heart axis at distances of 10, 19, and 29 m m from the heart center, w h e r e its radius was reduced to 30, 26, and 14 mm, respectively. The forward-problem solution for each of the disks is the same as any double-layer configuration that Iorms a closed double-layer surface with the disk, so that the solution for each disk is the same as that for an infinite n u m b e r of c o m p l e m e n t a r y double-layer sources (4). Forwardproblem solutions were also f o u n d for unit dipole and quadrupole sources. Discrete unit sources were located at the heart center. Torso-surface potentials were f o u n d for the eight distributed sources and for the eight discrete sources by iterating equation 2. For both sets of sources, the m a x i m u m ratio of RMS value of the change in potential to the RMS value of the potential for the set of sources reached - 6 0 dB after 17 iterations. Torso-surface potentials for three of the distributed sources are s h o w n in Figure 2.

Spherical-Harmonic Approximation to the Forward Problem The infinite-medium potentials for the discrete primary sources in the forward problem, equation 4, are described by terms in polar angle 0 and azimuthal angle 4). These terms are u n n o r m a l i z e d spherical harmonics (7,8,9). In addition to the prim a r y sources, w h e t h e r distributed or discrete, seco n d a r y sources are established o n the torso-model

surfaces to m e e t b o u n d a r y conditions (1,4). The potentials of b o t h primary and secondary sources m a y be described in terms of spherical harmonics, which allow continuous reconstruction of potentials over the torso-model surfaces. Spherical harmonics are a suitable choice as basis functions for describing the b o u n d e d - m e d i u m potentials V(r) due to the combined effects of primary and secondary sources, if the surface is a single-value function of angle 0 and 4) and if the harmonic series turnish an efficient representation of surface potentials. A finite-series approximation to V(r) in terms of normalized spherical harmonics is seen here in equation 5. N

n

V(r, 0, 4))= E E anmyem(0, ~) n-0

m=0

+ bnn~ Y°m(0, 4~) w h e r e (equation 6a. as seen here)

Ym(0, 4)) /2n+

1 (n-m)!

(nTm)!

Pro(COS 0) cos(m4))

and (equation 6b. as seen here)

~%(0, 4)) /2n+

1 (n-m)!

pm(cos 0) sin(m4))

In matrix notation, the spherical-harmonic approximation to V(r) of equation 5 is seen here in equation 7.

°[a m1

IV] = [Yem Ynm] bnm

w h e r e the matrix containing anm and bnm is a c o l u m n vector.

Solid-Angle, Spherical-Harmonic (SASH) Series A spherical-harmonic approximation to V(r) was f o u n d by substituting the matrix form of the approximation given by equation 7, into the conventional, iterative solution to V(r) (equation 2). The resulting spherical-harmonic coefficients were designated a~n~n and bS~n. They depend only on the infinite-medium potential for the source and the solid-angle matrix for the torso surfaces. These approximations to the forward problem were designated as solid-angle, spherical-harmonic, or SASH

Forward-Problem Approximation

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Fig. 2. Conventional, iterative solution to the body-surface potentials in the inhomogeneous torso. The source was a uniform double layer of 32-mm radius located at the heart center. Its strength was 1 m a / m m . It was directed from back j to front (upper panel), toward the apex of the heart (middle panel), and from right to left (lower panel). Equipotentials are given in millivolts.

108

Journal of Electrocardiology Vol. 32 No. 2 April 1999

series. Although the torso-surface potentials were approximated, knowledge of conventional forwardproblem solutions was not used to generate the SASH series. Simplifying the matrix notation of equation 2 for convenience: (equation 8a) [V] = [ a ] [V] + [G] or (equation 8b)

The fit of the conventional, iterative solutions Is yields a set of spherical-harmonic coefficients, anm and b~m. If (equation 12 as seen here)

ra~ml

[yenm Y°m] Lbkmj = [v] t h e n (equation 13 as seen here)

[

a

m]

b~mJ = ([Y~m Y°m]t[Y~m Y°m])-~ [Y~m Y°nm]t Iv]

([I] - [A]) [V] = [G] and substituting equation 7 for [V] yields as seen here in equation 9.

r

([I] - [A]) [Y;m YI~III] Lb~L j = [C] w h e r e [I] is the identity matrix. With [H] = ( [ I ] - [A]) l y e m Ynm], o equation 9 becomes equation 10 as seen here:

In]

[b~l~j = [G]

Solving for anmSaand bnm Sa to minimize the meansquare error of the approximation gives the result of equation 11 as seen here: b~aj = ([H]~[H]) - ' I n ] t [G] Torso-surface potentials predicted by a fifth-degree SASH series, which was based only on the infinite-medium potentials and the solid-angle matrix, are s h o w n in Figure 3 for the same three double-layer sources depicted in Figure 2. The SASH-series maps of Figure 3 are difficult to distinguish from the iterative-solution maps of Figure 2 except in the region of the shoulders. As part of the quantitative evaluations of the SASH series, the SASH series were compared to series solutions which resulted from least-squares-error fits of the conventional forward-problem solutions. This comparison d e t e r m i n e d h o w close the SASH series were to spherical-harmonic coefficients f o u n d in the least-squares-error sense.

Least-Squares-Error (LSE) Series The SASH approximations to the torso-surface potentials presented in Figure 3 did not use a knowledge of the conventional forward-problem solutions. If the conventional forward-problem solutions V(r) are k n o w n , t h e n they can also be m a t c h e d on a least-squares-error (LSE) basis using a spherical-harmonic approximation.

Quantitative evaluations were p e r f o r m e d on b o t h SASH series and LSE series as described next.

Evaluation of th e S p h e r i c a l Harmonic Approximation Torso-surface potentials on the i n h o m o g e n e o u s torso model predicted by first- t h r o u g h tenth-degree SASH and LSE series were compared to the conventional forward-problem solutions for the eight distributed sources using two measures as s h o w n in Figure 4. Comparisons were evaluated using the correlation coefficient and relative error (15), whose m e a n and standard deviation are s h o w n for the eight sources. Note that the SASH series, calculated w i t h o u t knowledge of the conventional solution, b e h a v e d almost identically to the LSE series, which was a fit of the conventional solution. Most of the i m p r o v e m e n t in the performance ol either series approximation, with increasing degree of the approximation, occurred a r o u n d the fitth to the seventh degree. Potentials calculated from fifth-degree SASH series for all eight sources had an average correlation coefficient and relative error of 0.9993 and 0.036, respectively. Average values for those measures for tenth-degree SASH series were 0.9997 and 0.026, respectively. Average values for the same measures for tenth-degree LSE series were 0.9998 and 0.020, respectively. Results presented so far apply to forward-problem solutions on the torso surface. Approximations can be found, however, on all surfaces of the torso model. Furthermore, approximations can be f o u n d for potentials due to secondary sources, as well as for the total potentials. Table 1 presents the m e a n and standard deviation for fifth- and seventh-degree LSE approximations on all four surfaces in the torso model for both total and secondary-source potentials. Note that, for each surface, the overall quality of the approximation for total compared to secondary sources was similar. The degree of the approximation can vary with each surface, depending on the accuracy desired and the n u m b e r of

Forward-Problem Approximation

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Fig. 3, Fifth-degree, spherical-harmonic series approximation to the body-surface potentials in the inhomogeneous torso. Series coefficients were determined from the solid-angle matrix for the torso and the infinite-medium potentials. The conventional forward-problem solution was not used in determining the solid-angle, spherical-harmonic (SASH) series. Sources were the same as those in the previous figure, namely a 32-mm, 1 - m a / m m double layer directed from back to front (upper panel), toward the apex of the heart (middle panel), and from right to left (lower panel). Equipotentials are given in millivolts.

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Journal of Electrocardiology Vol. 32 No. 2 April 1999

LSE SERIES

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Fig. 4. Performance of the solid-angle (SASH) and least-squares-error (LSE) spherical-harmonic approximations to the torso-surface potentials for double-layer sources. The m e a n and standard deviation of correlation coefficients and relative errors are shown for eight sources. Series approximations were compared to the conventional, iterative solution in each case. The SASH series, calculated without knowledge of the conventional solution, behaved almost identically to the LSE series, which was a fit of the conventional solution. Both series approximations appear to have reached an asymptote by about the fifth harmonic.

Forward-Problem Approximation

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Table l. Performance Measures for Uniform Double-Layer Sources: Fifth- and Seventh-Degree A p p r o x i m a t i o n s in the I n h o m o g e n e o u s Torso Total Potentials Corr. Coeff. (n = 5) Corr. Coeff. (n = 7) Rel. Error (n = 5) Rel. Error (n = 7) Secondary Potentials Corr. Coeff. (n = 5) Corr. Coeff. (n = 7) Rel. Error (n = 5) Rel. Error (n = 7)

Torso 0.9993 0.9997 0.036 0.026

Heart

± 0.0002 _+ 0.0001 ± 0.006 ± 0.004

0.9987 0.9999 0.049 0.016

Torso 0.9981 0.9991 0.062 0.042

_+ 0.0001 ± 0.0002 ± 0.006 ± 0.005

Right Lung*

_+ 0.0010 ± 0.0001 ± 0.018 -+ 0.005 Heart

0.9999 1.0000 0.011 0.006

± ± ± ±

0.0002 0.0001 0.004 0.002

0.9973 0.9994 0.074 0.034

_+ 0.0006 ± 0.0001 _+ 0.009 _+ 0.004

Right Lung* 0.9972 0.9994 0.074 0,033

± 0.0006 ± 0.0001 ± 0.009 _+ 0.005

Left Lung* 0.9976 0.9990 0.069 0,043

_+ 0.0006 ± 0.0002 _+ 0,009 _+ 0.005

Left Lung* 0.9975 0.9991 0.069 0,042

_+ 0.0008 ± 0.0002 Jr 0.012 ± 0.006

* Match using infinite-medium potentials.

n o d e s used. B e c a u s e t h e l u n g s w e r e n o t singlev a l u e d f u n c t i o n s of 0 a n d &, i n f i n i t e - m e d i u m p o t e n t i a l s ( e q u a t i o n 4) w e r e u s e d i n s t e a d of s p h e r i c a l h a r m o n i c s ( e q u a t i o n 6) to d e f i n e t h e basis f u n c tions. T h e d i f f e r e n c e b e t w e e n t h e t w o basis sets is t h a t r a d i u s is r e q u i r e d to c a l c u l a t e t h e a p p r o x i m a tion from the i n f i n i t e - m e d i u m potentials. In other w o r d s , t h e g e o m e t r y of t h e l u n g s is n e e d e d to m a p t h e a p p r o x i m a t i o n v a l u e s . I n c o n tr a s t, t h e g e o m e t ry of t h e t o r s o a n d h e a r t is n o t n e e d e d to m a p t h e s p h e r i c a l - h a r m o n i c a p p r o x i m a t i o n to f o r w a r d p r o b l e m solutions. Th e r es u l t s in Table 1 s u m m a r i z e t h e p e r f o r m a n c e of e i g h t u n i f o r m l y d i s t r i b u t e d s o u r c e s . Obviously, m a n y other distributed sources could h a v e b e e n s t u d i e d . M u l t i p o l a r sources, h o w e v e r , c a n b e c o m b i n e d to c h a r a c t e r i z e a n y d i s t r i b u t e d s o u r c e . Table 2 p r e s e n t s t h e m e a n a n d s t a n d a r d d e v i a t i o n for fifth- a n d s e v e n t h - d e g r e e LSE a p p r o x i m a t i o n o n all f o u r surfaces i n t h e t o r s o m o d e l to t h e t o t a l p o t e n t i a l s of t h e t h r e e d i p o l a r a n d five q u a d r u p o l a r o r t h o g o n a l s o u r c e s . N o t e t h a t for e a c h s u r face t h e q u a l i t y of t h e a p p r o x i m a t i o n to t h e t o t a l p o t e n t i a l for t h e fixed, d i s c r e t e d i p o l a r s o u r c e s w a s c o m p a r a -

ble to t h a t for t h e d o u b l e - l a y e r s o u r c e s (Table 1). F o r e x a m p l e , t h e r e is n o s i g n i f i c a n t d i f f e r e n c e o n a n y su r f ace b e t w e e n t h e r e l a t i v e e r r o r s for t h e distributed and dipolar sources. The p e r f o r m a n c e for t h e q u a d r u p o l a r s o u r c e s is also s i m i l a r to t h a t for t h e d i p o l a r sources.

Discussion A l t h o u g h t h e m e t h o d s for g e n e r a t i n g t h e d i r e c t a n d LSE a p p r o x i m a t i o n s to t h e f o r w a r d p r o b l e m c a n be a p p l i e d w i t h o t h e r basis f u n c t i o n s , s p h e r i c a l h a r m o n i c s w e r e s u c c e s s f u l in c h a r a c t e r i z i n g forw a r d - p r o b l e m s o l u t i o n s for a v a r i e t y of d i s t r i b u t e d a n d d i s c r e t e s o u r c e s . T h e a v e r a g e r e l a t i v e e r r o r for s e v e n t h - d e g r e e , s p h e r i c a l - h a r m o n i c series o v e r t h e torso and heart in the four-surface torso m o d e l was 0 . 0 2 4 for t h e 48 p o t e n t i a l m a p s d e s c r i b e d i n Tables 1 a n d 2. Th e c h a r a c t e r i s t i c s of t h e s e s o u r c e s s u g g e s t t h a t t h e s e r esu l t s are r e p r e s e n t a t i v e of m a n y o t h e r s o u r c e c o n f i g u r a t i o n s . E a c h of t h e e i g h t u n i f o r m double-layer sources actually represents an infinite

T a b l e 2. Performance Measures for Dipole and Q u a d r u p o l e Sources: Fifth- and Seventh-Degree A p p r o x i m a t i o n s in the I n h o m o g e n e o u s Torso Dipole Potentials Corr. Coeff. (n = 5) Corr. Coeff. (n = 7) Rel. Error (n = 5) Rel. Error (n = 7) Quadrupole Potentials Corr. Coeff. (n = 5) Corr. Coeff. (n = 7) Rel. Error (n = 5) Rel. Error (n = 7)

Torso 0.9993 0.9996 0.038 0.028

_+ 0.0003 ± 0.0002 ± 0.007 -- 0.006

Torso 0.9959 0.9991 0.089 0.041

± 0.0012 _+ 0.0001 ± 0.013 _+ 0.004

* Match using infinite-medium potentials.

Heart 0.9997 0.9999 0.022 0.014

+ 0.0000 ± 0.0000 _+ 0.002 _+ 0,001

Heart 0.9995 0.9998 0.031 0.017

_+ 0,0001 ± 0.0000 _+ 0.002 ± 0.003

Right Lung* 0.9976 0.9995 0.068 0.030

_+ 0.0009 _+ 0.0001 ± 0.013 ± 0.005

Right Lung* 0.9985 0.9997 0.055 0.026

_+ 0.0005 ± 0.0000 _+ 0.010 ± 0.001

Left Lung* 0.9980 0.9992 0.063 0.040

_+ 0.0003 ± 0.0003 ± 0.005 ± 0.007

Left Lung* 0.9970 0.9990 0.077 0.044

_+ 0.0013 ± 0.0003 ± 0.016 _+ 0.007

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Journal of Electrocardiology Vol. 32 No. 2 April 1999

n u m b e r of sources because the forward-problem solution is the same for any double layer that closes the surface containing the double-layer disks that were used (4). Furthermore, the dipole and quadrupole sources are the first terms in a multipole expansion that can be used to represent any arbitrary source. Therefore, similar relative errors can be expected for a wide variety of other possible sources. Similar relative errors are also expected for additional inhomogeneities because errors were virtually u n c h a n g e d as inhomogeneities were added to the torso model. The average relative errors of the seventh-degree, spherical-harmonic series for the total potential due to the eight distributed sources on the torso surface were 0.026, 0.025, and 0.026 for the h o m o g e n e o u s torso, for the torso with heart, and for the torso with heart and lungs, respectively. Spherical-harmonic series also had comparable relative errors w h e n used to match the transfer coefficients that relate epicardial to torsosurface potentials (16). Transfer coefficients for a 58-node heart in a 715-node torso were m a t c h e d with relative errors of 0.065, 0.037, and 0.026 for fifth-, seventh-, and tenth-degree series, respectively. Spherical-harmonic approximations succeeded in matching forward-problem solutions on the torso and heart. These surfaces were single-valued functions of the polar and azimuthal angles from an origin at the x-ray center of the heart. The lungs, of course, were not. Unfortunately, moving the origin for the spherical-harmonic approximation to the center of the lungs did not provide a comparable fit to that on the torso or heart because the sphericalh a r m o n i c series was not located in the vicinity of the primary source. Infinite-medium-potential approximations at the x-ray heart center, however, did provide matches on the lungs which were comparable to the fits on the torso and heart f o u n d using spherical harmonics. Infinite-medium approximations are composed of a function of the radius in addition to spherical harmonies (compare equation 4 with equation 6). Although spherical harmonics provided a successful basis set for the torso and heart, other inhomogeneities like the lungs presumably will require basis functions that must incorporate knowledge of the geometry of the i n h o m o g e n e i t y to reconstruct forward-problem solutions. Computation of both the SASH and LSE series required inversion of an N x N matrix, w h e r e N is the n u m b e r of coefficients in the spherical-harmonic series. N is the degree of the harmonic plus one squared. Because N can be m u c h smaller t h a n the n u m b e r of elements in the surface model, the

forward problem can be approximated faster t h a n the conventional, iterative solution can be found. Forming the SASH series was slower t h a n finding the LSE because it also required multiplying the basis functions by the solid-angle matrix. The SASH series was 3.8, 2.3, and 1.2 times faster t h a n the conventional solution for fifth-, seventh-, and tenth-degree approximations. The LSE series was 76, 38, and 7 times faster t h a n the conventional solution for fifth-, seventh-, and tenth-degree approximations. Results of the forward-problem using the iterative, conventional m e t h o d are tabulated at nodes of the torso model. Representation of torso-surface potentials with a spherical-harmonic series replaces the list of potentials with a set of coefficients. Series representation of surface potentials has two advantages. First, the set of series coefficients can be smaller t h a n the list of potentials. Although the list n e e d not contain the potentials at all nodes, to include t h e m all w o u l d require 715 values in this study because the torso model contained 715 nodes. A fifth-degree series contains 36 coefficients. Thus, the n u m b e r of items required to represent torso potentials using a fifth-degree series is reduced by a factor of about 20. The reduction is by a factor of 11 and 6 for seventh- and tenth-degree spherical harmonics, respectively. Second, the series can be used to predict surface potentials everyw h e r e on the torso surface, not just at the nodes of the torso model. This advantage m a y be useful for cases in which electrocardiographic m e a s u r e m e n t s are made at sites that differ from the location of torso-model elements. Using the conventional approach, both the torso geometry and the forwardproblem solution are required to interpolate the transfer coefficients to find the value at such an electrode site. With the spherical harmonics, only the location of the electrode, as specified by its polar angle 0 and its azimuthal angle 4~ is needed; torsomodel geometry is no longer needed. The spherical-harmonic approximations are a transformation of surface potentials into an orthogonal basis set whose spatial-frequency content increases with increasing degree. Consequently, their use m a y provide a structure for the systematic study of the effects on forward-problem solutions of both changes in torso shape and inclusion of i n h o m o g e neities. Consider Figure 5, which shows the relative contribution of each degree of a tenth-degree LSE series for the torso potentials in the torso model with and w i t h o u t inhomogeneities. The changes in the contributions to torso potentials are s h o w n with respect to the values f o u n d in the h o m o g e n e o u s torso, which was used as baseline. The h o m o g e -

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DEGREE OF SPHERICAL HARMONIC F i g . 5. R e l a t i v e c o n t r i b u t i o n of e a c h d e g r e e of a t e n t h - d e g r e e LSE s e r i e s f o r t h e t o r s o p o t e n t i a l s i n t h e t o r s o m o d e l w i t h and without inhomogeneities. Changes in the contributions to torso potentials are shown with respect to the values found in the homogeneous t o r s o w h i c h w a s u s e d as b a s e l i n e (ie, s e t to z e r o ) . T h e e f f e c t of t h e i n h o m o g e n e i t i e s d e p e n d s o n t h e d i r e c t i o n of t h e s o u r c e a n d t h e s p a t i a l f r e q u e n c i e s of t h e t o r s o p o t e n t i a l s , w h i c h i n c r e a s e w i t h t h e d e g r e e of t h e spherical harmonic.

114

Journal of Electrocardiology Vol. 32 No. 2 April 1999

neous torso values were set to zero for all h a r m o n ics and the solutions with inhomogeneities adjusted accordingly. Clearly, the effect of the i n h o m o g e n e ities depends on the degree of the h a r m o n i c and the a m o u n t of change at each h a r m o n i c depends on the nature of the source. For example, for all three sources the addition of the cardiac i n h o m o g e n e i t y reduced the contribution of the fourth h a r m o n i c by about 5 dB. W h e n the lungs were added, however, the contribution for the fourth h a r m o n i c increased by 4 dB for the frontal source, decreased by 3 dB for the axial source, and was u n c h a n g e d for the sagittal source.

Conclusions Low-degree, spherical-harmonic series provide an accurate representation of forward-problem solutions for b o t h distributed and discrete cardiac sources. These series can be f o u n d more rapidly t h a n the conventional, iterative solutions with or w i t h o u t knowledge of those solutions. Series coefficients furnish a more compact representation of surlace potentials t h a n a tabulated list of potentials at the elements of the torso model. Furthermore, the series yield c o n t i n u o u s approximations of forw a r d - p r o b l e m solutions that are valid e v e r y w h e r e on the torso surface.

References 1. Gulrajani RM, Roberge FA, Mailloux GE: The forward problem of electrocardiography, p. 197. In MacFarlane PW, VeitchLawrie TD (eds): Comprehensive Electrocardiology. Vol. 1, Pergamon Press, 1989 2. Barr RC, Pilkington TC, Boineau JP, Spach MS: Determining surface potentials from current dipoles, with application to electrocardiography. IEEE Trans Biomed Eng 13:88, 1966

3. Barnard ACL, Duck IM, Lynn MS: The application of electromagnetic theory to electrocardiography I. Derivation of the integral equations. Biophys J 7:443, 1967 4. Malmivuo J, Plonsey R: Bioelectromagnetism. pp. 134-136, 140-141, 187-190. Oxford University Press, New York, 1995 5. Barnard ACL, Duck IM, Lynn MS, Timlake WP: The application of electromagnetic theory to electrocardiography II. Numerical solution to the integral equations. Biophys J 7:463, 1967 6. Arthur RM, Geselowitz DB, Briller SA, Trost RF: Quadrupole components of the human surface electrocardiogram. Am Heart J 83:663, 1972 7. Morse PM, Fesbach H: Methods of Theoretical Physics. pp. 1264-1266. McGraw-Hill, New York, 1953 8. Arfken G: Mathematical Methods for Physicists. pp. 446-448. Academic Press, New York, 1966 9. Plonsey R, Fleming DG: Bioelectric Phenomena. pp. 324-332. McGraw-Hill, New York, 1969 10. Rush S, Abildskov JA, McFee R: Resistivity of body tissues at low frequency. Circ Res XII:40, 1963 11. Abramowitz M, Stegun IA (eds): Handbook of Mathematical Functions. p. 893. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington DC, 1968 12. van Oosterom A, Strackee J: The solid angle of a plane triangle. IEEE Trans Biomed Eng 30:125, 1983 13. Lynn MS, Timlake WP: The use of multiple deflations in the numerical solution of singular systems of equations, with applications to potential theory. SIAM J Numer Anal 5:303, 1968 14. Lynn MS, Timlake WP: The numerical solution of singular integral equations of potential theory. Numerische Mathematik 11:77, 1968 15. Johnston PR, Gulrajani RM: A new method for regularization parameter determination in the inverse problem of electrocardiography. IEEE Trans Biomed Eng 44:19, 1997 16. Barr RC, Ramsey MIII, Spach MS: Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements. IEEE Trans Biomed Eng 24:1, 1977