The Influence of Skull - Conductivity Misspecification on Inverse Source Localization in Realistically Shaped Finite Element Head Models

Transcription

1 3rain Topography, Volume 9, Number 3, The Influence of Skull - Conductivity Misspecification on Inverse Source Localization in Realistically Shaped Finite Element Head Models Robert Pohlmeier*, Helmut Buchner*, Gunter KnoU ^, Adrian Rien~Jcker ^, Rainer Beckmann*, and Jbrg Summary: The electric conductivities of different tissues are important parameters of the head model and their precise knowledge appears to be a prerequisite for the localization of electric sources within the brain. To estimate the error in source localization due to errors in assumed conductivity values, parameter variations on skull conductivities are examined. The skull conductivity was varied in a wide range and, in a second part of this paper, the effect of a nonhomogeneous skull conductivity was examined. An error in conductivity of lower than 20% appears to be acceptable for fine finite element head models with average discretization errors down to 3ram. Nonhomogeneous skull conductivities, e.g., sutures, yield important mislocalizations especially in the vincinty of electrodes and should be modeled. Key words: Source localization; Finite element model (FEM); Skull conductivity; Inhomogeneous conductivity. Introduction The electric effects of activated brain areas can be measured in the Electroencephalogram (EEG). A prominent goal in the analysis of EEG data is a stable and accurate reconstruction of the sources within the individual anatomy. Such noninvasive analysis may prove to be very useful in the localization of epileptic foci, preoperative localization, e.g., of the central sulcus, and the investigation of the functional organization of the brain. Any inverse analysis strongly depends on the quality of the forward mode] used. Within the past years, finite element and boundary element models have been developed by a number of groups (Bertrand et ai. 1991; Buchner et al. 1996, Fuchs et al. 1994; Lfitkenh6hner et al. 1995; Meijs et al. 1985; Menninghaus et al. 1994; Peters 1994; Roth et al. 1993; Wagner et al. 1995, 1996; Yan et al. 1991). Both types of models are capable of handling complex, realistic geometries-as resulting from segmented MR data. Since the head electrically acts as a volume conductor, it is necessary *Department of Neurology, RWTH Aachen, FRG. ^J, nstitute for Machine Elements and Tribology, Urdv. Gh Kassel, for Biomedical Engineering, RWTH Aachen, FRG. Accepted for publication January 2,1997. The authors wish to thank the VW - Foundation for financial support. Correspondence and reprint requests should be addressed to Helmut Buchner M.D., Department of Neurology, Pauwelsstrage 30, Aachen, Germany. Fax: Copyright Human Sciences Press, Inc. to assign conductivity values to each compartment or even element of the discrete model. Although carefully measured conductivity values have been published by Geddes and Baker (1967) and Law (1993), it is still unclear if these values hold for living tissue and are applicable for each individual. Investigations on the influence of conductivities have only been ma de for the spherical head model until now (Ary et al. 1981; Stok 1987). In this work we investigate the question on how accurate the conductivity of the skull must be specified in a realistically shaped head model, in order to yield acceptable results in source reconstructions. At the same time this work might clarify how big an effort is justified in determining correct individual conductivities for the analysis. Model description Two realistically shaped finite element head geometries, aquired from the same MR data set of an individual (who gave his informed consent), segmented and meshed with the CURRY package (Philips, Hamburg, FRG) are taken as a reference, with standard conductivities (Cohen and Cuffin 1983) assigned to four compartments within the head. The coarse mesh, with typical tetrahedra circumsphere radius h=9mm, was used for most of the computations, while some calculations were carried out with the finer mesh (typical tetrahedra circumsphere radius h=5mm), to crosscheck some of the findings. The four compartments of the head had elements and con-

2 t58 Pohtmeier et al. Table i. Conductivities used in the reference configurations and the number of finite elements in each compartment. Compartment Brain Liquor Skull Scalp Conductivity [M(Vm)] Elements h=9mm Elements h=5 mm ductivities as in table I. Figure i shows the head geometry with 65 electrode positions marked, the finite element mesh of the outer surface of the skull with some marked elements and 5 horizontal slices through the model to visualize the spatial extent of the skull. Finite Element algorithms, described elsewhere (Buchner et al. 1996), were employed to calculate potential distributions for single, normalized dipole sources, resulting in a lead field or influence operator that connects sources from all possible source locations with potentials at the electrode positions. Physiological arguments make the use of the dipole model plausible (Nunez 1990; Ilmoniemi 1994). Electrodes are modelled by bar elements with high conductivity but no net current flow due to the homogeneous Neumann boundary condition (insulation) at the end node. Investigation In a first investigation skull conductivities for both models were decreased and increased up to a factor of 100 compared to the reference configurations. It is believed that the conductivity range covered is far beyond the range that can be observed physiologically. For each noise-free set of electrode potentials from the lead field operator of the disturbed configuration, an inverse best single dipole fit was carried out, i.e., finding the single source dipole, which explains the potentials with the minimum of remaining variance RV, all FE nodes within the brain volume were tested, This fit resulted in a spatial measure, the vector from the original dipole position to the inversely determined dipole position. For each thinkable dipole position (node within the brain volume) three orthogonal dipole directions are possible and were investigated one after another. For each node three different mislocalization vectors fx,fy,fz result, each index referring to the cartesian direction of the original dipole orientation. A sensible measure for the total mislocalization error for each node is a vector f, pointing in the normalized direction of the vector sum (~fix + ~;y + fz)/llf;x + f:y + fzll with length re- suiting from the norm of the length of the individual mislocations. The length of the vector f is an upper bound of the localization error of an arbitrarily orientated dipole at the investigated location, while the vector has the direction of that part of the mislocalization that is caused by the changed electrical conductivity independently from the dipole orientation. (tx'bty'btz) ~q_ fy 2-b[d(z 2 (1) This measure, difficult to capture at first glance, eliminates the danger of compensating effects in the vectorial addition of fx, fy and fz and thus aids in visualizing the results. Figure 2 shows a sketch of the measure f and the Figure 1, Head geometry with electrode positions, the finite element mesh of the outer surface of the skull with the elements of the temporal bone and the midsaggital suture marked and on the right side five slices (distance between successive slices 15mm) of the investigated geometry giving an impression of the spatial extension of the skull (white). addition of the mislocalization of the x-, y- and z-testdipole /f at each source location: / - Z +L+f - 2 ~ --. Figure 2. Vectorial localization error due to two times increased and decreased skull conductivities.

3 The Influence of Skull-Conductivity Misspecification 159 Figure 3. Color c o d e d length of the localization error for varied skull conductmties in a sequence of 2-D maps with a distance of 20 mm b e t w e e n successive maps, mislocalization vectors for two skull conductivity variations. If the skull conductivity is reduced, a clear tendency for a mislocalization towards the head's midpoint is recorded. This especially holds for dipole positions in the upper part of the brain, close to the electrodes. Increased skull conductivities lead to mislocalizations in radial outward direction and in direction to the apex. Figure 3 shows the mislocalizations' increasing spatial extent as skull conductivity departs from its original value. The center in mislocalization is found on the left side of the brain, where individual skull thickness is larger than on the right side (figure 1). While the largest mislocalization distances are in the top part of the brain for reduced conductivities, an increased skull conductivities leads to the

4 160 Pohlmeier et ai. 20 ented dipoles. Because increased conductivities lead to EE*816 ii!!iiii!ii!iii!!!ili!!!iiiiii!i!iii!i!iiiiiiiiiii!iiiiiii!iiiiiiiiil iii!iiiiiiii mislocations in radial outward and apex direction, where the skull surface limits the magnitude of the 9 -~ ]4 mislocation, the tendency reported before is not symmetric. In the right part of the figure, the relative probability for a mislocalization in a cartesian direction i, counted "... -~'--F-',---~-:... ~---l-" -- "... l---"... ~-... "--'- ~ i ::... i... it---i...!-i--",... 1-i-!--i !- ' ' J l I J 1, ~ ~ ~ 0 l i [ 1 t t skull conductivity divided by liquor conductivity Figure 4. Averaged overall error for the coarse and the fine mesh. largest errors in the central part of the brain. Figure 4 shows the overall average in mislocalization error g=max(f,d), where d is the local discretization error, which in a rough approximation is taken to be half of the average inter-node distance for the coarse and the finer mesh. Naturally the localization error has a lower bound, given by d, independant of changes in conductivity. A 50% errror in skull conductivity appears to be acceptable in coarse mesh and leads to an additional mislocalization of about h=2mm in the finer mesh. A further evaluation of the data is shown in figure 5, where the averaged localization error is separately evaluated for each dipole direction in the left graph. While dipoles pointing in x and y direction are equally sensitive to skull conductivity changes, a decrease in conductivity yields higher mislocalizations for z-ori- over all examined nodes in the brain (1644) and dipole directions (3) zjfi r i = zjlf/i i = x,y,z j : 1, (2) is plotted versus the relative change in skull conductivity. A relative probability close to r=l indicates a high tendency towards a mislocalization in positive i-direction, while the value r=-i indicates a rnislocalization in negative/-direction. Strong tendencies can only be reported for the z-direction, which can also be seen from figure 2. The preference in this direction cannot be explained by geometrical irregularities of the head model itself, but can only be related to the electrode arrangement, which exhibits a strong preference in this direction. In the second part of this work nonhomogeneous skull conductivity is investigated by two different approaches. First the influence of a nonhomogeneous skull conductivity was examined in an exemplary study on the basis of skull conductivities, measured by Law (1993). Law found deviant conductivities for temporal bone and for pieces of bone containing a suture line, the corresponding areas used in the presented investigation are marked in figure 1. The average of all values measured by Law was taken as the base conductivity for the 25 2O = o "~ 15 9 ~ 10 ~u ,00001 I i II I 1 in I I II I [ II I u I I ii ~I~L I I II I I II I I II I I I I I I I I I I I I I II I I II I I t. I. I)~lk'l.-i- I. i I. ~ i I.". '1 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 I I I I I In I I I I I I I I I I I I I I I I Z i i I I I I l I I I I I I I I } t l i I I ] l I I ( I I I I } 1 I I I I I I i i I i I I i I I I I IZl, '~---I ---I -i,,~--i i I I I ~ i t I IXI I I II I I I I I II I II I L I I ii I I I I II I I tl I [ I I II I X I I II I Ul I u, i, i t i - L - - L - M.... '-_U L-_J- -- '~ - - -i- '~ - - -i- - -I- I I II I ( II 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 I i I I I I i I I I I I I I l l I 1 I $ i i 1 l I [ I I ~ I { i I I i I t 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 I I I i i i i i [ I I I I It I I I I I I I I I I I I I I I l I IJ. l Jl I,,, t , ,1 skull conductivity divided by liquor conductivity ~ 0.2 '~ 0,0 o -0.2 ~ o t I II I I II u I I ~ I I I! O I I I I I II,, 1',,,,, 'i I I II I I I I ~ I I I L I ii ~ 1 I i It I I Ii i I II I I II Z I..... I- 1 T - -I- - -r -1"1 - -[ - - T -i-t- - r - - T - rl ~- ' i I i I I I ~ I i i I I ii I I I I I I I I I I I I I I I I I qm'- un i u i~ ~ i r'.l~. ~,, - la_l_..... ~ II I1-" - r -Ii-It--- [ - - ] - fll - X tl n ' I II I I IF I I~11 I I It ~1 I I I I I I I I I 1 '~1 I I I I I I I II I I I I I I II ~t~ll~ll~rll II Y I I I I i I I I I 1 I I I i I I I - ~---,- ~ r--,---;-i~--~- r-y,--'---i - ~.... t.i I i. I I~..mll~dlPV I I I I I I I I U I U -I i n - n "tl I I, _,. ' _ ',, _, ', _ - ' _...,-x~7.....,--,~--,,,~,l ~,... ~,~_,_,_,... ~...,~ ~... ~,, I In [ lu u u In I I II I l._, '--,,----'------,--',----'------'--'--'----'---'--''----'----'-- u II I t II n I LI I ~ ii [ I I Ill n nl I i II n I r, ' ' O.O00l O.Ol O,l skull conductivity divided by liquor conductivity Figure 5. Mislocali7ations, examined by dipole direction, on the right side and on the left side relative probability for a mislocalization in a cartesian direction.

5 The Influence of Skull-Conductivity Misspecification 161 Table II, Conductivities used in the investigation of inhomoeneities according to the data given by Law (1993), Investigation reference configuration temporal inhomogeneity inhomogeneity at the suture combining both inhomogeneities conductivity of temoral bone [A/(Vm)] conductivity of the suture elements [A/(Vm)I conductivity of the rest of the skull [A/(Vm) l reference solution. First, the conductivities of the temporal bone were assigned differently. Second, conductivities of elements around the midsaggital suture were varied and third both effects were combined. The assigened values are given in table II, the average conductivity was held constant. The change in temporal bone conductivity effected practically no mislocalization, while mislocalizations caused by a change of conductivity around the midsaggital suture could only be found in the area directly below the region of modified conductivity. The combination of both inhomogeneities showed no further effect on mislocalizations; the results of this investigation are presented in the upper part of figure 6 using the same scale as in figure 3. It is apparent that conductivities in the neighborhood of electrodes must be precisely modeled. Finally, motivated by physiologial arguments, the skull is modelled by three layers, the middle layer with higher conductivity taken as spongiosa and the other two layers as poorly conducting compacta. In order to define these three layers the FEM-geometry is remeshed increasing the number of skull elements up to , of which were taken as spongiosa with half the liquor conductivity (0.695 A/(Vm)), the conductivity of the compacta is defined so that the resistivity (the three layers are interpreted as series connection of resistors) of the skull is held almost constant ( A/(Vm)). In the bottom part of figure 6 it is clear that the effect of this nonhomogeneous skull conductivity is asymmetric especially in the upper front part of the brain. This effect is due to the asymmetric thickness of the skull (figure 1). The effect appears to be important and the physiologially more realistic three layer model of the skull should be considered in order to improve the accuracy of source localization. Although the examination of nonhomogeneous conductivities is theoretically possible with the boundary element method, too, in practice only the finite element method is suitable because there is no increase in needed system memory with an increasing number of different conductivities contrasting to the boundary element method. Conclusion Skull conductivities, although important, are not the decisive factor in inverse EEG analysis. An error of 20% appears to be acceptable with respect to the discretization error, even for meshes finer than that used in this paper. Local variances in skull conductivity should be considered in areas where many electrodes are concentrated. Apart from measuring these conductivities which might be difficult, one should at least identify the areas (e.g., sutures) from MR data. In that respect finite element methods may have computational advantages over the commonly used boundary element methods. The nonhomogeous structure of the skull should be modeled to take of compacta and spongiosa into account. Figure 6. Color coded length of the localization error for the investigation of nonhomogeneous skull conductivity in a sequence of 2-D maps with a distance of 20 mm between successive maps,

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