Electroencephalography and clinical Neurophysiology, 88 (1993) 1- I 1 1 © 1993 Elsevier Scientific Publishers Ireland, Ltd. 0168-5597/93/$(/6.00

E V O P O T 92061

Insidious errors in dipole localization parameters at a single time-point due to model misspecification of

number of shells

Zhi Zhang and Don L. Jewett Research Division, Abratech Corporation, Mill Valley, CA 94941-6610 (USA)

(Accepted for publication: 24 August 1992)

Summary Insidious errors (unexpected and difficult-to-detect under usual conditions) were found using a single time-point dipole localization method, when two dipoles were simultaneously active and there was misspecification of the number of shells (usually intended to mimic the conductivity differences of the brain, skull, and scalp). The errors involved all dipole parameters (i.e., location, orientation, and magnitude). Potentials at 65 "electrode" locations on the surface of a 3-shell sphere were computed for dipoles of known location, orientation, and magnitude. These "maps" were then used to compute the best-least-squares-fit of the surface potentials based upon dipole parameters in a 1-shell sphere when either one or two dipoles were active. The dipole paramet - rs were often significantly different when computed with two equal-magnitude dipoles active, compared with only one dipole, with location errors of 0-36 mm, orientation errors of 0-63 °, and magnitude errors of 2-98%. When the two dipole magni tudes were not the same, the errors in the computed dipole parameters were even larger. All these errors occurred when the LSE (least-square-error) was small and at or near minimum. Moreover, location errors increased as LSE decreased over iterations.

These errors generally occur because the fitted dipole parameters under different potential maps do not obey the superposition law when there is shell model misspecification, which is also the reason that presently used "correction" methods cannot satisfactorily remove these errors from the analyses. This problem must be dealt with when analyzing evoked response "maps" from simultaneously active generatorS, if correspondence to anatomy and physiology is desired.

Key words: Dipole source localization; EP generator; Evoked response; Model misspecification; Dipole parameters; Ary correction

While evoked responses have been remarkably suc- cessful in experimental and clinical work, the vast majority of the wave forms (surface potential maps) are analyzed using single-channel "peak-picking," i.e., measuring latency and amplitude of maxima or min- ima. A promise of significant advance has come from DLM (dipole localization methods) which utilize multi- ple channels of data to predict the location, orienta- tion, and magnitude of dipole equivalents of neural generators, even when more than one generator is simultaneously active. Such methods, if accurate, should markedly improve interpretation of recorded poten- tials, permitting better measures of physiological or pathological processes of interest.

The success of a given modeling effort is often evaluated by comparing the predicted surface-recorded potential map from the model with the recorded po- tential map that has been analyzed; a close correspon-

Correspondence to: Don L. Jewett, M.D., D. Phil., Research Division, Abratech Corporation, 150 Shoreline Highway, Building E, Mill Valley, CA 94941-6610 (USA). Tel.: (415) 476-5588; Fax: (415) 331-4537.

dence between the two maps is taken as evidence that the analysis is "correct," i.e., the analysis accurately describes the location, orientation, and magnitude of the equivalent dipole generators whose summed poten- tials created the recorded surface map. However, we here present evidence that close correspondence be- tween predicted and observed maps can occur with significantly large errors between fitted (computed) dipole parameters and those used to generate the surface potentials even when the correct number of dipoles is used in the fitting (a close correspondence can be guaranteed by analyzing for more dipoles than used to generate the map). Thus, other means must be used to determine whether an inverse calculation of the surface potential map provides an accurate analy- sis.

A sphere is one of the commonly used head models, as its shape is close to that of the superior cranium. To compensate for the non-homogeneity of brain, skull and scalp, a 3-shell sphere model is sometimes used. However, to simplify the computation, the 3-shell sphere is itself sometimes approximated by computing the inverse solution (dipole parameters from surface

2 Z. ZHANG, D.L. JEWETT

" ':

Lr~ I

Fig. 1. The overall procedure used in this paper. The dipoles shown in solid lines (lower-case letters) are the least-square fitted results in the 1-shell sphere (solid circle) to the potential maps generated by dipoles shown in dotted lines (denoted by upper-case letters) in the 3-shell sphere (dotted circles). Each letter (Ds and ds) represents 6 components (3 for location, 2 for orientation, 1 for magnitude), d I is the fitted dipole to the potential map generated by D1; similarly is d 2 to D 2. d'~ and d~ are the fitted results when two dipoles were used to fit the potential map generated by D~ and D 2 together. Also shown in the figure is the superposition error in location (location error) for

D2 (LE2).

potentials) in a 1-shell sphere and then applying an "Ary correction" to correct the 1-shell results to pre- dict those likely in a 3-shell model (Ary et al. 1981). However, we show that even with such corrections there are significant errors when two dipoles are simul- taneously active, which calls into question results ob- tained by this method. Errors in computations such as this can be expected on theoretical grounds, but the results presented here are needed to show the magni- tude of the errors that can occur.

Fig. 1 outlines the procedures. The potentials at the electrodes on a 3-shell sphere, due to a dipole with parameters D1 (unit magnitude at a given location and orientation) were calculated yielding a surface poten- tial map equivalent to potentials that might be ob- served from scalp recordings. This map was then ana- lyzed on a 1-shell sphere, so as to derive the best-fit dipole parameters d~. Similarly, d 2 was derived from the map computed from D 2. To study simultaneously active dipoles, the surface potential map due to two spatially distinct dipoles D~ and D 2 was computed by summing the maps from each separate dipole. This combined map was then fitted by two simultaneously active dipoles d ' 1 and d~.

We report that d't 4: d~ and d~ 4= d 2 , though they should be equal under an assumption of linear super-

position, and that the errors can be significantly large. It should be emphasized that the errors we have found occur only with model misspecification, by which we mean that the model used to create the surface poten- tial map is not the same as the model used to analyze the map. All of this work is based upon analyzing a single simultaneous set of surface potentials, i.e., is equivalent to the analysis of a single " instantaneous" time-point in a surface potential map (e.g., Henderson et al. 1977; Kavanagh et al. 1978). Neither noise nor multiple time-points were considered.

M e t h o d s

All results were created by computer simulations on an IBM RS 6000, model 530. Programs were written in Fortran 77, and computations were done using double precision floating point (8 bytes).

A 3-shell model was used to create the potential maps (Vob S) using an N-shell concentric sphere compu- tation with the dipole within the inner most shell (see Appendix). The conductivities for the 3-shell sphere were (from inside out): 1.0, 0.0125, 1.0, and the shell radii were 8.7, 9.2, 10.0 cm. These values are the same as those used by Ary et al. (1981). In the 1-shell sphere used for analysis, the conductivity was 1.0 and the radius 10 cm.

A given set of dipole parameters, D1, were used to calculate the potentials at the electrodes on a 3-shell sphere, creating the map Vob S. The dipole parameters d t were then derived by iteratively minimizing the LSE of the differences between Vob ~ and Vnt maps; where Vfi t was calculated in a 1-shell sphere:

N

LSE = ~ [V,~,(i)- voh~0)] 2 (1) i I

where N is the total number of electrodes, Vobs(i) the observed (forward computed) potential on the 3-shell sphere surface and Vnt(i) the fitted potential on the 1-shell model surface, at electrode i. The minimization in the inverse calculation was done using the Simplex algorithm (Amoeba in Press et al. 1989) for the non- linear part (e.g., dipole locations) and within each iteration singular value decomposition (SVDFIT in Press et al. 1989) for the linear part (e.g., dipole orientations and magnitudes). No constraints were added to the fitting parameters, except in some sym- metrical constraint tests (see Results). d~ was found by starting the inverse calculation program at a given location, and then iteratively computing until the LSE satisfied {LSEma × - LSEmi n [ / ILSEma x + LSEmi n I < •, where LSEma x and LSEma n are the largest and the smallest LSEs among all the vertices of the Simplex in the current iteration, and • is the tolerance, set at 10 - 7 . The inverse program reports the vertex with the minimum LSE as the final result dr.

ERRORS IN DIPOLE LOCALIZATION PARAMETERS 3

Note that , in this pape r , uppe r - ca se le t te rs indicate p a r a m e t e r s used to c rea t e the surface po ten t i a l m a p Vob,~, lower-case le t te r s indica te p a r a m e t e r s der ived f rom Vri t (see Fig. 1). W h e n more than one d ipo le is used, subscr ip ts I and 2 are used. A pr ime symbol is used when d i f fe ren t p a r a m e t e r s a re de r ived for the same d ipo le u n d e r a s imul taneous ly active condi t ion (e.g., d ' 1 and d~ in Fig. 1). Similarly, 0 ' , ~b' and m ' a re used. A d ipo le d has 6 p a r a m e t e r s : locat ion r = (x, y, z), o r i en ta t ion (0, ~h) and magn i tude m (for D, R = (X, Y, Z), (9, qb and M). Ang les a re de f ined in the Fig. 2 legend.

" S u p e r p o s i t i o n e r ro r s" ( the d i f fe rences be tween d n and d], and d 2 and d~ in Fig. 1) were found by compu t ing a d ipo le as one of a pa i r of s imul taneous ly active dipoles , as de f ined in equa t ions (2) to (4). E r ro r s a re c o m p u t e d be tween the p a r a m e t e r s of co r r e spond- ing dipoles , e.g., be tween d~ and d'l, bu t not be tween d 1 and d~:

m' - m ME (magnitude error) (%) -= x 100% (2)

m

A posi t ive M E means tha t the f i t ted m a g n i t u d e unde r a pa i r is l a rger than the f i t ted m a g n i t u d e u n d e r a single d ipole , for the same d ipo le in the forward ( " o b s e r v e d " ) ca lcula t ion.

OE (orientation error) = Cos t[sin((h) sin((/>') cos(0 - 0')

+cos(#,) cos(6')] (3)

O E is the inc luded angle be tween the two d ipo le or ien ta t ions , (0, ~b) and (0 ' , d / ) , in degrees .

tr'-rl LE (location error) (%) - x 100% (4)

R 0

L E (Fig. 1) is the Euc l i dean d i s tance be tween r ' and r no rma l i zed by the sphe re rad ius R0, whose uni ts can be expressed e i the r as % of the radius , or as the case here when the sphe re rad ius is 10 cm, as d i s tance in mm.

To see how much the f i t ted Vfit m a p due to the f i t ted d ipole(s ) d i f fe red f rom the f o r w a r d - c o m p u t e d Vob S m a p be tween models , a no rma l i zed LSE, NLSE, was c o m p u t e d as follows (where the symbols have the same mean ing as in equa t ion (1)):

[Vrit(i) - Vohs(i)] 2

NLSE(%)-= i=I N x100% (5)

[VoMi)] 2 i = l

Twenty-e igh t d ipo les were p l aced in several loca- t ions and at several o r ien ta t ions , as i l lus t ra ted in Fig. 2. The d ipo le loca t ions were chosen to a p p r o x i m a t e bo th cor t ical and b ra in - s t em locat ions. A l so inc luded are d ipo les with d i f fe ren t eccentr ic i t ies .

For ty-seven d ipo le pai rs as l is ted in Tab le I were s tudied. Pai rs were d iv ided into 10 ca tegor ies , fo rming

Z

.' / , ] / 9 / ~ ] 7 / / "_'~',~4[2:4

. . i . . ~ . ~ , t , 7" , | • i •

. ; I 7 ~ . ~ . j / / ; ; : : i / : : "..'.. ',~-- / I ,28 , / . , .%

,'."."7

)

Y

Fig. 2. Dipoles used in the computer simulation. The + z-axis points to the "vertex." The +x-axis is towards the reader. The solid circle and two dotted circles indicate the 3 shells, while the dashed lines and circle are to facilitate visualization. Since a concentric 3-shell sphere was used, the directions of x- and y-axis are not important to the results, so most of the dipole locations were first estimated from anatomical considerations and then rotated onto the y-z plane. The dipole numbers are shown around their locations (marked with *) separated by commas if more than one is at the same location. Dipoles nos. 1-10 are radial with orientation outward, nos. 11-18 are tangential with orientation pointing to + x-axis. The orientation of tangential dipole no. 19 is shown in the figure, nos. 20-28 are neither radial nor tangential, their orientations are:

Dipole 20 21 22 23 24 25 26 27 28 n o .

O - 45 45 0 0 0 0 0 65.5 0 @ 91) 90 30 45 80 80 20 47.4 20

where (9 is the azimuth angle, between + X-axis and the projection of the dipole moment onto the XY plane, a positive angle being toward the +Y-axis; q~ is the elevation angle between the dipole moment and the +Z-axis; in degrees. Dipoles 7, 26 -28 are not on the YZ plane. Dipole no. 7 can be obtained by rotating no. 6 60 ° around the z-axis. The locations (x,y, z) in cm are (1.42, 2.224.

- 3.684) for nos. 26 and 27 and ( - 1.034, 2.224, - 2.627) for no. 28.

the fol lowing groups: cor t ica l -cor t ica l ; d i f ferent eccen- tr ici t ies; ' b r a i n - s t e m - b r a i n - s t e m ; and cor t i ca l -bra in- s tem. In the 10 ca tegor ies (Table I) the re are d ipo les with var ious kinds of o r ien ta t ions . T h e r e are also sym- met r ica l pairs . Since this is a c o m p u t e r s imula t ion , we inc luded d ipo le pai rs with a large var ie ty of separa - t ions, f rom 1 to 15 cm (Table I).

Sixty-five e l ec t rodes were p l aced on the super io r he mi sphe re of the sphere (above the x-y plane) . One e l ec t rode far f rom all the d ipo les was chosen as the r e fe rence e lec t rode , making a total n u m b e r of 64 inde- p e n d e n t channels . The 64 channels exceed the gener-

4 Z. ZHANG, D.L. JEWETT

ally accepted minimum number of 6 per computed dipole (Fender 1987; Snyder 1991).

Since the surface potential expression for a dipole within the N-shell sphere model is an infinite series expansion, the potentials can be calculated only to finite accuracy. To determine how many terms were needed in the potential calculation, the 3-shell model was tested. The potentials at all the 65 electrodes for a few dipoles were calculated with different numbers of terms in the potential expansion. The results showed that with 50 terms in use, adding a further 50 terms changes the computed numerical potentials very little (maximum difference among all the 65 electrodes less than 0.0026%). However, the question remains how accurate is the computation using only the first 50 terms for the 3-shell sphere. To obtain a rough mea- sure, surface potentials for the 1-shell sphere were computed, with only the first 50 terms, and with the closed form (13rody et al. 1973) (equivalent to infinite number of terms in the expansion), and the computed

difference is still very small (maximum difference no more than 0.022%). So in all the 3-shell calculations only the first 50 terms were used, while the closed form was used for the 1-shell sphere.

When' the model generating the Vob ~ map was the same as that used in the inverse fitting (either both 3-shell or both 1-shell) and one dipole was active, there were no detectable differences between the dipole parameters fitted and the parameters generating the gobs map (0.00000). The LSEs were all less than 10-14 for such test runs. This indicates that the forward and inverse programs were consistent. With the 1-shell sphere, another test was also done, i.e., in the forward calculation, only the first 50 terms were used, while in the inverse calculation, the closed form was used. Zero difference (0.00000) in the dipole parameters indicates that the approximation in the potential expansion was accurate enough for computing the dipole parameters, at least for the homogeneous case. The LSEs for these test cases were all less than 6 x 10 -12

TABLE I

Dipole pairs studied. Pairs are grouped into 10 categories according to their locations and orientations. Pair numbers are given under each category, starting from the smallest to the largest. The two dipole numbers formed the pair are shown together with an ampersand sign. Within parentheses, the location separation of the two dipoles in either degrees or centimeters is shown. In addition, ± means the two dipole orientations are perpendicular to each other, while II indicates parallel. In forming the pairs, the following was considered: (1) in the same category, distance between the two dipoles is the same or increases when pair number increases; (2) in a category, if several pairs have the same dipole separation, then the non-parallel ones come first; (3) if there is at least one radial dipole in a pair, then it is the first one in the pair; (4) if the two dipoles in a pair have different eccentricities, then the one with larger eccentricity is the first one; (5) if a same dipole is paired with different dipoles in a category, keep the same dipole in the same sequence in the pairs whenever possible.

Category Pair number Dipoles forming the pair

Cortical-cortical I Symmetrical 20&21 (20 °, somatosensory); 6&7 (60 °, visual); 382 (60°);

(01-05) 12&l 1 (60°); 6&l (160 °, auditory)

II Radial-radial 3& 4 (15°); 6&5 (15°); 3&5 (35°); (06-11) 6& 4 (35°); 6&3 (50°); 6&2 (110 °)

III Tangential-tangential 15&14 (15°); 15&19 (35 °, ± ); 15&13 (35°); (12-16) 15&12 (50°); 15&ll (110 °)

IV Radial-tangential 6&14 (15°); 6&13 (35°); 6&12 (50°); (17-20) 6&ll (110 °)

V Others 4&22 (15°); 3&23 (15°); 6&24 (15°); (21-30) 5&25 (15°); 23&22 (15°); 25&24 (15°);

3&24 (35°); 5&22 (35°); 22&24 (35°); 2&25 (110 o)

Different eccentricities VI

VII

VIII

Brain -stem -brain -stem

IX

Cortical-brain-stem X

Radial-radial (31-33)

Tangential-tangential (34-36)

Radial-tangential (37-39)

(40-41 )

(42-47)

3& 8 (2 cm); 8& 9 (2 cm); 9&10 (2 cm)

12&16 (2 cm); 16&17 (2 cm); 17&18 (2 cm)

3&16 (2 cm); 8&17 (2 cm); 9&18 (2 cm)

27&28 (1.1 cm); 26&28 (1.1 cm, II)

3&28 (9 cm); 22&28 (9 cm); 3&27 (10 cm); 3&26 (10 cm); 22&27 (10 cm); 22&26 (10 cm)

ERRORS IN DIPOLE LOCALIZATION PARAMETERS 5

Results

In what follows, the conclusions we have reached are numbered, each followed by a description of the results upon which the conclusion was based.

(1) If the model and the number of dipoles were correctly specified (i.e., forward and inverse calcula- tions in the same model, and two dipoles were used to analyze the map generated by two dipoles), the errors in the fitted dipole parameters were very small, if the computation was started at or close to the correct location. This computation serves as a control, verify- ing that errors only occur when there is model misspec- ification. Tests were done for 25 of the 47 dipole pairs using the 3-shell sphere for both forward and inverse calculations. In the tests, if the starting location of the dipole pairs was correct, then the computed dipole parameters were the same as in the forward calcula- tion. However, starting locations affected the results in the following ways: (1) If the starting location of the dipole pair was within 0.5 mm from the correct loca- tion, the computed parameters for all the 25 pairs tested were either correct or had only very small errors (LE < 0.001%, IMEI < 0.004%). (2) If the starting lo- cation for the two dipoles was about 2 -4 cm away from their correct location, the pairs with spatially separated dipoles (distance greater than 2 cm) gave dipole pa- rameters without error, but large errors in dipole pa- rameters did occur for 7 closely spaced pairs (LE 4-33%, I MEI from a few tens to a few thousand percent). (3) When the starting locations were more than 5 cm away from their correct locations, the pa- rameters for 16 pairs were markedly wrong.

The tests were repeated with the 1-shell model. For any starting location the program gave the right param- eters for most of the spatially separated dipole pairs, but the closely spaced pairs required good starting locations.

These results showed that when multiple dipoles are simultaneously active, the LSE may have "local min- ima" over the multi-dimensional space, as argued by many investigators, such as Fender (1987) and Achim et al. (199l). Because of these results, all 2-dipole runs reported below were done starting at the "correct" location, determined from the 1-dipole case (see no. 2, below). It is unlikely that the 0.01 mm error in the "correct" locations used as the starting locations (see no. 2 below) will have affected our results, because the tests showed that when the two dipoles were put at 0.5 mm (> 0.01 mm) away from their correct location in the inverse calculation, the program still moved them back to the correct location.

(2) When only one dipole was used in the 3-shell forward calculation and one dipole was fitted in the 1-shell sphere, the differences (e.g., between D 1 and d 1 in Fig. 1) were large, but could theoretically be com-

pensated for by an "Ary-type" correction. Even when different starting locations were used for the inverse calculation, the fitted parameters for the same single dipole differed no more than 0.01 mm for location, < 0.001% for magnitude and < 1 ° for orientation. The computed differences (between d and D, computed in similar way as ME, OE and LE) were 34-41% for absolute magnitude, 6-28 mm for location (only a few gave a location difference less than 20 mm), and 0-4 ° for orientation. All these differences can be compen- sated for by an "Ary-type" correction (see Discussion). The parameters obtained in this way (e.g., dl in Fig. 1) formed the basis for determining the "superposition errors" occurring when maps from simultaneously ac- tive dipoles were analyzed. Since these parameters are the results that should be obtained if linear superposi- tion occurs when more than one dipole is active, it is these dipole parameters which, though different from those shown in Fig. 2, were taken as the "correct" parameters for a best fit in a pair of simultaneously active dipoles.

(3) When two dipoles in the 1-shell sphere were used to fit the potential map generated by two dipoles within the 3-shell sphere, the superposition errors in the fitted results were large and not "Ary-correctable" (see Discussion).

ME, OE and LE for all the 47 dipole pairs (magni- tude ratio 1:1 in the pair) are shown in Fig. 3. The errors could be significantly large in all parameters (location, orientation and magnitude) or just in some of the parameters. Fig. 3 shows that there is no clear systematic relationship between the magnitude of the errors and the dipole locations used in the forward calculation. One thing these groups have in common (except category X) is that the errors (ME, OE and LE) tend to be smaller as the pair number increases in each group, meaning that the errors generally decrease as the distance between the two dipoles increases (see Table I legend). Also pairs 31-39 show that pairs with the same dipole separation but less eccentricity had less errors.

The dipoles of the first 5 pairs are symmetrical in both location and orientation with respect to some plane across the origin of the sphere (Fig. 2). Applying symmetrical constraints (on both location and orienta- tion) in the fitting did not significantly improve the results (category I in Fig. 3). Closely spaced dipoles still showed large errors, and the errors were small for widely spaced dipoles whether constraints were applied or not. For example, pair 01 was fitted together on the z-axis by the symmetrical constraint, resulting in a large ME. These were the only cases where symmetrical constraints were applied.

(4) If the ratio of the magnitudes of two simultane- ously active dipoles was varied in the 3-shell forward calculation, then both the fitted parameters in the

z. ZHANG, D.L. JEWETT

1-shell and the errors varied as compared with the dipole parameters computed from a 1:1 magnitude ratio. In generating the Vob S maps, magnitude ratios of 1:4 , 1:2, 2 :1 and 4:1 were used for each of the 47 dipole pairs, and the ME, O E and LE were plotted. The results were similar to the 1 : 1 ratio results shown in Fig. 3 (i.e., no obvious pat tern shown in the data) and therefore only the errors at ratio l : 4 are shown in Fig. 4. Close inspection of the results led to the follow- ing observations: (1) when the magnitude ratio moves away from 1 : 1, the OE and ME tended to increase; (2) for some closely spaced pairs, the program could no longer separate the two dipoles, thus resulting in large ME (>> 100%) and OE (75-180°); (3) the dipole with smaller magnitude in the pair had larger errors; and (4) in category X (cortical-brain-stem pairs, cortical dipole is the first in the pair), the ME, OE and LE for the brain-stem dipole increased as the magnitude ratio increases (from 1:4 to 4:1), while the errors for the cortical dipole were not significantly changed. Thus, as

90 [zc

6O

~ 3 0

-5©

- 6 0

-9@ 60 .o

co • @

/-5

©

15 ©

0 I

5@

b~

O0

• O O • 0

o ° • • °

©

o I o •

. ~ o,O ~

O

I

O O

0 t l I ) ,~ r ~ f ~ T I I

o i 0 20 50 4o P a i r N u m b e r

I ' I ',~ ;,~i j Jv I v lWlW~!Vl~lux[ x I Category

• --1 s t d;po e in pa i r O - 2 q d d ipole in po i r A ? - l s t , 2 n d d ipo les in pa i r u n d e r s y r n m e t r i c a c o r s t r a i r ; t

Fig. 3. ME, OE and LE for all the 47 dipole pairs. The Vo~,s map was generated by two dipoles with magnitude ratio 1 : 1 within the 3-shell sphere. Different categories (see Table I) are separated by vertical lines. For pairs 01-05, also shown are the errors when symmetrical

constraints in both dipole locations and orientations were applied.

i 00

5O

_u 0

- 5 0

- ! 00

1 8 0

t o • o • I Ioo

,% (

0 0

Q • •

! I • 0 •

i; O O

1 3 5

~ ' 9 0 o c~ ©

rm v L~ ©

4 5

0

60

• • O o

o

0"T2 ° M

~ , 4 0 v L,I _J

QO O • • • 20 • • • ,•o

O O 0 O

o o o c.~o. " • ii~

0 0 1 0 2 0 5 0 4 0

Pair N u m b e r

I I I u r ill I rV l v I vllvulvlutlXl x I Category

• -1 st dipoteFn poFr o - 2 n d dipole in pair

Fig. 4. ME, OE and LE for the 47 dipole pairs as in Fig. 3, except the magnitude ratio being 1:4,

the brain-stem dipole magnitude lessened, the errors increased.

(5) When a given dipole was paired with different dipoles in the forward calculation~ the fitted parame- ters in the misspecified model for the given dipole also differed. This can be seen in Fig. 3 and Fig. 4 (e.g., in category III, the first dipole in the pairs is dipole no.

E R R O R S IN D I P O L E L O C A L I Z A T I O N P A R A M E T E R S 7

15, while the errors for no. 15 were different; in cate- gory IV, dipole no. 6 is always the first, but the errors for it were different). Since the result for a single dipole differs depending upon the other dipole, any "Ary-type" correction would have to allow for different pairs, and the magnitude ratios as well (no. 4 above).

(6) When there was model misspecification, an ad- ditional (third) dipole added to the fitting decreased the LSE and affected the dipole parameters of the two original dipoles. In a DLM of evoked response data, the number of active dipoles may be unknown and hence possibly misspecified in the analysis. To simulate this, 3 simultaneously active dipoles were used in the 1-shell sphere to fit the potential map due to two unit-magnitude dipoles within the 3-shell sphere. Two of the dipoles were put at the "correct" locations, while the third was put at the center of the sphere. All parameters (including location) were then computed for all 3 dipoles. Although the third dipole was put at the center of the sphere, it sometimes ended at a location closer to one of the two "correct" locations than the first two. The errors shown in Fig. 5 were computed from the two dipoles whose fitted locations were closest to the "correct" locations. In order to have a direct comparison with the errors shown in Fig. 3, Fig. 5 shows dME, dOE and dLE, the differences between the errors computed here and those shown in Fig. 3. Mathematically,

dM E (()~) = I M E 3 ] - I M E 2 ]; d O E ( d e g r e e s ) - O E 3 - O E 2 ;

d L E (%) = L E ~ - L E 2

where the subscript indicates the number of dipoles used in the fitting. If dME, dOE, dLE are less than zero, then the third dipole reduced the errors in the fitted parameters for the first two dipoles; otherwise, the third dipole increased the errors. Fig. 5 shows that the third dipole did not decrease the errors in all the parameters for any dipole pair, as there did not exist a pair where dME, dOE and dLE for both dipoles were less than zero.

The results also revealed (not shown) that the third dipole magnitude in some cases was large and the LSE was always made smaller by the presence of the super- fluous dipole. Since the superfluous dipole might theo- retically have computed to have zero magnitude (with a result then identical to that if only two dipoles were used in the fitting), it is clear that each such superflu- ous dipole makes the "fi t" bet ter (i.e., the LSE be- comes smaller) with some alteration in the parameters of the other two dipoles. This problem occurs mainly when there is a residual error, such as occurs with model misspecification, but is less likely to occur if there is no model misspecification (see Discussion).

(7) There is a general inverse relationship between LE and LSE (or NLSE). To verify that the parameters found under a single-dipole case (the "correct" param-

8o

N 40' L ~

0

- 4 0

- 8 0

9 0

60

~'~30

c~

~ ' 0 o K3

---50

- - 6 0

10

~" o v W

~ - 1 0

- 2 0

- 3 0 L i

o

I i

o c ; • I

co I

0

0

0

coo

© o

0

0

I 0 o

,i .L

I 10

II I III

,____

! o i i o o

o o o

, I r !

20 30 40 Pair N u m b e r

q Lv [ v !vl lv iv,,,IX Category

x ]

• - 1 s t d ipo le in pa i r O - 2 P d d ipo le n paT,

Fig. 5. dME, d O E and d L E when 3 d ipoles were used in tile 1-shell

sphe re to fit the po ten t ia l map due to two un i t -magn i tude d ipoles

wi thin the 3-shell sphere. S ta r t ing locat ions for the first two dipoles

were at the ( "co r r ec t " ) loca t ions li~und unde r s ingle-d ipole fit t ings,

and for the third was at the sphere center . Er ro r d i f ferences were c o m p u t e d from the two d ipoles closest to the " 'correct" locations. No

er ror is shown for the third dipole.

eters) would not also result in a minimum LSE for two simultaneously active dipoles, all of the parameters (location, orientation, magnitude) for each of the 47 pairs (with magnitude ratio 1:1) were fixed at those "correct" parameters (so that ME, OE, and LE are all zero), and the least square error LSEcM o was then computed. Similarly, to verify that the locations found under the single-dipole case would not yield a mini- mum LSE for two simultaneously active dipoles even if dipole magnitudes and orientations were allowed to

8 Z. ZHANG, D . L J E W E T T

vary, for each of the pairs, only the dipole location was fixed and the least square error LSE L was computed. The results showed, as expected, that for each pair, LSE < LSE L < LSELMo, where LSE is the least square error corresponding to the data in Fig. 3. This means that, if the two dipoles were fixed to the "correct" locations and only the magnitudes and orientations were computed, then the dipoles still have different orientations and magnitudes from the "correct" ones. The errors (shown in Fig. 6) in some pairs were not small (e.g., largest ME = 18%), while at the same time giving LSE k < LSELM o. When all the dipole parame- ters were fitted, the errors were large (Fig. 3), but the LSEs were smaller than both LSELM o and LSE L. Thus, in these models, the dipole parameters found under the single dipole calculations do not give a minimum least square fit for two simultaneously active dipoles. This explains why the dipole parameters moved away from their starting values, because with their starting values, the LSE was not at a minimum.

Fig. 7 shows the relationship between LE and NLSE for dipole pair 45. The data points were obtained by letting the inverse program report the best dipole pa- rameters found after each iteration, so the plot is actually a trace of the location error at each point along the Simplex searching path. The program stopped after 333 iterations, giving the final results leading to the errors shown in Fig. 3. It can be seen from the figure that although LE varied with NLSE in a compli- cated way, generally LE increased as NLSE decreased. In Fig. 7 the location error of the brain-stem dipole (dipole no. 26) was much more sensitive to the NLSE change than the cortical dipole (dipole no. 3). Within two regions (NLSE from 4.75% to 2.8%, and from 2.5% to 1.4%) LE of dipole no. 26 changed rapidly with the decreasing of NLSE. An inverse relationship between LSE and LE can be inferred for all dipole pairs by the finding that LSE < L S E L < LSELM o (showing that the final least square error was less than it was at the "correct" location), and by Fig. 3 in which

L , A

o 0 0 ! C 20 30 dO

P a l ~ t'q d r ' n b e r

[ I iJ ! ~i I ! v j v i\/ijvl;iV,:i,x[ x i Category

• - l st dlpole in pair o -2nddipo)einpc'r Fig. 6. ME and OE in each of the dipole pairs when the two dipoles were fixed at the "correct" location in the 1-shell sphere to fit the potential map due to two unit-magnitude dipoles within the 3-shell

sphere (i.e., LE = 0 for all pairs).

8 / ,

l

r !

~ o ~ p o l e z s !

O

0 ~-J

. ~ ~ ~ , \ , o , i

Fig. 7. LE as a function of NLSE for dipole pair 45 (dipoles 03 and 26). The data were generated by letting the inverse program report the best dipole parameters found so far after each iteration. Thus, the figure plots the results during the Simplex search. The first point (at the bottom right of the figure for both dipoles) corresponds to the

dipoles being at the starting ("correct") locations, at which LE = 0.

LEs were non-zero (indicating that the final location was not at the starting "correct" location). Fig. 7 shows that reducing LSE iteratively actually leads to increas- ing errors in the dipole location. This calls into ques- tion whether iterative methods which reduce the LSE are appropriate for dipole localization when there is model misspecification.

D i s c u s s i o n

One may argue that bet ter parameters (i.e., less superposition errors) for a simultaneously active dipole pair may be obtained by starting the inverse computa- tion at different locations and orientations (e.g., Fender 1987). While it might be possible to find dipole param- eters with still smaller LSEs than we did, such results would not change the conclusion that superposition errors are present since any dipole parameters found by specifying different starting locations would still be different from the parameters found when computing the dipoles individually. So the errors we report here are not because the program is t rapped in a local minimum of the LSE. Since we chose the starting parameters to be those found under the 1-dipole cases (the "correct" ones), if the parameters found under the 1-dipole cases were the minimum LSE under the 2-di- pole case, the program would have gtven the starting parameters as its final result. Instead we found smaller LSEs with dipole parameters different from those tound in the 1-dipole cases. Thus, the parameters corresponding to the global minimum for a single ac- tive-dipole are not at the global minimum for a simul- taneously active dipole pair.

Although we referred to "Ary-type" corrections, we

ERRORS IN DIPOLE LOCALIZATION PARAMETERS 9

did not use them. Rather than interpolate any such correction from the published data (Ary et al. 1981), an exact correction could be obtained from the known position of the dipole in the forward calculation and the position determined from the inverse calculation of a single dipole. After such a correction, if exact, there will be no errors in the parameters only when a single dipole is used in the forward and inverse calculations. But, as our results clearly show, this model misspecifi- cation "correct ion" cannot be applied to computations of two simultaneously active dipoles. The correction would work if and only if the superposition errors (ME, OE and LE) were all zero. Since the "Ary-type" cor- rection multiplies both the dipole location and dipole magnitude computed in the 1-shell sphere by a factor of approximately 1.5 (the fitted parameters showed this), after the correction, LE will be about 1.5 times larger. At the same time, OE would still be about the same as we have reported (because the orientation difference between the fitted single dipole in the 1-shell sphere and the single dipole used in the 3-shell for- ward calculation was very small), and ME would also be about as we have reported (because of the normal- ization of magnitude in the ME calculation (see eq. 2)). So even after applying an "Ary-type" correction, the errors in most of the dipole parameters will still be too large to be acceptable when more than one dipole is active.

If an "Ary-type" single-dipole-derived correction is used in a DLM computing simultaneously active dipoles, there is an implied assumption that linear superposition for the fitted dipole parameters holds. i.e., the errors in Figs. 3 and 4 will all be zero. Gener- ally, this is likely to be true only under the ideal conditions of a perfect match between the data source and the computational model.

Snyder (1991) fitted a gobs map generated by a single dipole by one or more dipoles, using the same model in both the forward and inverse calculations. The dipole magnitudes fitted were all non-zero, except when one of the fitting dipole locations was coincident with the dipole location in the forward calculation, in which the fit was perfect (i.e., NLSE = 0%, no errors in fitted dipole parameters). His results showed that the fit at the worst case became better and better (i.e., LSE become smaller) when the number of dipoles used in the fitting increased. We always used the same number of dipoles in both the forward and inverse calculations, except in Results under the numbered sentence 6, in which the same conclusion as Snyder (1991) was ob- tained, except that in our case, because of the model misspecification, the fitted results were still not "cor- rect," although two dipoles were put at the "correct" locations.

Model misspecification occurs when the real model can be precisely described but is too complicated, such

as from a 3-shell (or 4-shell) sphere to a 1-shell sphere. Or it occurs when the model cannot be precisely de- scribed, such as from a real head to a 3-shell sphere or to a 1-shell sphere. In this case, investigators are aware that there are errors made in choosing the model, but there is not much that can be done to correct the situation. In the models used by some presently used DLM's, there are misspecifications. The misspecifica- tions may include shape (e.g., head shape to sphere) and conductivity (e.g., shell conductivities, number of shells, shell thicknesses, local conductivities). When any of these occurs in the model and more than one dipole is active, there may be errors in the fitted dipole parameters, despite any "Ary-type" correction derived from single-dipole data. The magnitude of the errors depends upon many factors including the degree of model misspecification, the dipole locations, the mag- nitude ratios, etc., in complex ways we have not tried to categorize.

The errors of this report are insidious in that they occur even when the LSE is very small. Thus, a low LSE cannot and should not be used as evidence for the accuracy of a given analysis. Indeed, the computer search for a smaller LSE may increase errors (Fig. 7). In our view, the sizes of the errors in these simulations are sufficient to call into question data analyzed in this way. We have not analyzed any real evoked response data because we would not know the correct location of the dipole generators for such data. However, we have used the homogeneous sphere for the inverse calculation, just as is done by some DLMs (e.g., Hen- derson et al. 1977; Kavanagh et al. 1978; Lehmann et al. 1982). Thus, if there is model misspecification as great or greater than in our computations, then our work has shown that errors of this magnitude are possible. It is still unknown how much the superposi- tion errors under conditions of model misspecification will be affected by fitting data across time-points (Scherg 1984).

Indeed, the interaction of variables is sufficiently complex and sensitive to values of the many parame- ters, that we warn readers not to assume that specific results here can be applied to any other cases, e.g., different locations, or different conductivity ratios. Ide- ally, each user of a DLM should demonstrate that linear superposition errors have not occurred in any analyses of evoked response data asserting detection of simultaneously active generators. The promise of DLMs in accurately separating the activity of anatomically or physiologically distinct generators that are contributing simultaneously to surface potential maps will not be fully fulfilled until this problem has been adequately addressed. Alternatively, DLMs containing insidious superposition errors should be viewed as purely an empirical means of data reduction lacking accuracy with respect to underlying dipole parameters.

10 Z. ZHANG, D.L. JEWETT

We thank Avner Amir, Valerie Cardenas, George Fein, Dan Fletcher, and Jack Gerson for discussion of the issues.

This research was supported by Grant DC 00328 from the Na- tional Institution on Deafness and Other Communication Disorders.

A p p e n d i x

Derivation of equations for surface potential of a dipole within an N-shell concentrical sphere

We present here the method used to compute the surface potentials in the 3-shell model. The details of this me thod are not available in the literature, nor is it readily derived. A more general form of the potential expression due to dipole within an N-shell spherical model was given by De Munck (1988), f rom which the equat ions shown in this Appendix can be derived.

Let the radii and conductivities of the N shells be R i and ~r t (i = 1,2 . . . . . N) f rom inside to outside. Assume a dipole momen t D (magni tude D = ]D l) be located at R 0 (bold capital letters denote vectors), [R 0 ] < Ri. The observation point is an electrode at R e on the surface of the sphere.

A current dipole can be t reated as the combinat ion of one current source and one current sink, with the same magnitude, close to each other. The potential due to a current dipole can be derived from the poten- tial of ei ther a current sink or source. Let a current source with unit magni tude be located at R 0, and V(R0; R) denote the potential at R due to this current source. Then the potential U(Ro; R) at R due to a current dipole D at R 0 is given by

U(Ro; R) = D.VR,y(R,): R) (A1)

where Vrq ' indicates g opera t ing on R 0 ra ther than on R. V(R0; R) is governed by Poisson's equat ion

v. [,rVV(R(); R)] = - 6(R- Ro) (A2)

The potentials V(R0; R) within each shell f rom in- ner to outer are given by

1 ~ ( ~BnJ V,(R,); R)= 4rr~ ,~ , .A~'R~ + J(2n+ 1)P,,(cos 7)

(R i I~<RVRj); j=0, 1,2 . . . . . N;R t=-0;o-0=o- I (A3)

where 3' is the angle between vectors R 0 and R (i.e., between source location and observation location). P,s are the Legendre polynomials. A . j s and B.js are to be de termined by the boundary conditions.

The boundary condit ions are: (i)~ Potentials are cont inuous across each shell bound-

ary:

VitRo; R) = Vj+ i(R0; R ) for all /R[ = Rj (j= 0, 1,2 . . . . . N - I ) ,

(ii) Current is cont inuous cross each shell boundary:

a~ eV,+l for all JR[ = R i 0 = 1 , 2 . . . . . N - I ) .

¢',7R -=%+I OR

(iii) On the sphere surface no currents cross the boundary:

av N ~rN~- ~- =0 for all JR] = R N.

(iv) At the current source location, eq. A2 must be satisfied, too:

Let V i = ~ Fj.(R)(2n + 1/47r)Pn(cos y), then the n = l

condit ion at R = R o is given by

d Rtl ÷ 3R I

dR [RFm(R)] R. -m °'oR. (aR--+0)

(V) A t R = 0, the potential is finite:

Bn0=0 ( n = l , 2 , . . . )

Considering all the boundary conditions, the follow- ing results can be obtained:

1 ~ (2n4 l) N [R~I V N ( R 0 ; R ) = ~ E (n+l)M~ t+GR~T'M,,

N n = ]

x (n+ 1)R~ + R-~v~jp.(cos y)

When R = R~, the potential on the surface observation point is

1 ~ (2n + 1 )NR'~Rr~ VN(R[); R~)= 47ro-~- - - 5 1 (n+ l)M~-~t + nR~I~M~ P,(cos y)

(A4)

where M2~ and M = are functions of n, ~r i and R i (i = 1, 2 . . . . . N), and are given by

( MII M I 2 ] - l - I 1 n+ak( l+n) fik (A5)

M21 M 2 2 ] - - k = l n~k(rr k 1) ncek+(l+n )

crk /jk = Rkn+l (~k = - - - - ' O'k~ 1

Substituting A5 into A1, one obtains the surface potential UN(R0; R~) due to dipole D at R within the N-shell sphere:

D ~ (2n + 1)NR'~j IR~

U N ( R 0: Re ) = 4-~NN nEl (n + l~)-l~21 + nR~'G '~M22

X[n cos aP,(cos },)+cos/3 sin aP~(cos y)] (A6)

where P~s are the associated Legendre polynomials, c~ is the angle between dipole location R 0 and dipole momen t D. 3' is the angle between R o and observation R N. R 0 and D define a plane P1, R 0 and R N define another plane P2, /3 is the angle between P1 and P2.

When N = 1, the shell model becomes a homoge- neous sphere, Mlj = M22 = 1, Mi2 = M21 = 0. This can

ERRORS IN DIPOLE LOCALIZATION PARAMETERS

be seen by setting a k = 1 (k = 1, 2 . . . . . N - 1). And eq. A6 reduces to the following expression

D ~ ( 2 n + l ) R ' ~ - I UN(R(); R~) = 4W~N E nR,~+ I

n ~ l

X[n cos ¢ee,,(cos - / )+cos /3 sin c~P,~(cos y ) ] (A7)

Note that the net accumulation of charge implied by eq. A4 is not a problem for eq. A7, which is for a simultaneous current source and sink dipole. Let a current sink with unit magnitude be located at R() 1 (close to R0), V(RtH; R) be its potential at R,

V(Rol ; R) = - [V(Ro; R) + (R m - Ro)'VR,y(Ro; R) + . . . ]

The potential due to both the source and sink is just the sum of the two potentials:

V(Ro,; R) +V(Ro; R) = [(Rol - R()).r'k,y(Ro; R) + ... ] (A8)

Mathematically a current dipole is obtained by re- ducing the distance between R 0 and R0~ to zero and increasing the source and sink magnitude m to infinity, while keeping the product m(R 0 - R o t ) = D constant, and this constant is the dipole moment. This leads to eq. A1, which is exact, as the rest of the terms in eq. A8 all vanish.

Since the current dipole is no more than the sum of a current source; and a current sink, its potential obeys the same boundary conditions as the current source or sink does. Although the V operator occurs in the dipole potential expression A1, it only operates on R 0, while the boundary conditions all refer to R. Thus, the boundary conditions can be applied first to V and then the 17 operator can be applied.

From eq. A6 it can be seen that, using a different number of shells will not change the structure of the potential expression, but will change M2~ and M22 only. The difference between the potential expression for a tangential dipole and that for a radial dipole is only the part shown in the squared bracket in eq. A6. If the potential expression V for a radial dipole with a magnitude D can be expressed as

o o

V~ = Y'~ f~DrnPn(COS 7), then without solving Poisson's n~-I

equation, the potential expression V t for the tangential

11

dipole of magnitude D t can be obtained by simply replacing DrnPn(COS "/) with Dtcos/3P,l(cos y) in this equation.

References

Achim, A., Richer, F. and Saint-Hilaire, J.-M. Methodological con- siderations for the evaluation of spatio-temporal source models. Electroenceph. clin. Neurophysiol., 1991, 79: 227-240.

Ary, J.P., Klein, S.A. and Fender, D.H. Location of sources of evoked scalp potentials: corrections for skull and scalp thick- nesses. IEEE Trans. Bio-med. Eng., 1981, BME-28: 447-452.

Brody, D.A., Terry, F.H. and Ideker, R.E. Eccentric dipole in a spherical medium: generalized expression for surface potentials. IEEE Trans. Bio-med. Eng., 1973, BME-20: 141-143.

De Munck, J.C. The potential distribution in a layered anisotropic spheroidal volume conductor, J. Appl. Phys., 1988, 64: 464-470.

Fender, D.H. Source localization of brain electrical activity. In: A. Gevins and A. Rdmond (Eds.), Methods of Analysis of Brain Elec|rical and Magnetic Signals. Elsevier Science PUN., Amster- dam, 1987: 355-403.

Henderson, C.J., Butler, S.R. and Glass, A. Analysis of EEG poten- tial fields in terms of multiple equivalent dipoles. Electroenceph. clin. Neurophysiol., 1977, 43: 770.

Kavanagh, R.N., Darcey, T.M., Lehmann, D. and Fender, D.H. Evaluation of methods for 3-dimensional localization of electrical sources in the human brain. IEEE Trans. Bio-med. Eng., 1978, BME-25: 421-429.

Lehmann, D., Darcey, T.M. and Skrandies, W. lntracerebral and scalp fields evoked by hemiretinal checkerboard reversal, and modeling of their dipole generators. In: J. Courjon, F. Maugui~re and M. Revol (Eds.), Clinical Applications of Evoked Potentials in Neurology. Raven Press, New York, 1982: 41-48.

Press, W., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (Eds.). Numerical Recipes (Fortran Version). Cambridge Univ. Press, Cambridge, 1989.

Scherg, M. Spatio-temporal modelling of early auditory evoked po- tentials. Rev. Laryngol. (Bordeaux), 1984, 105: 163-170.

Snyder, A.Z. Dipole source localization in the study of EP genera- tors: a critique. Electroenceph. clin. Neurophysiol., 1991, 80: 321-325.