the use of electrical impedance tomography with the inverse problem of eeg and meg

7/27/2019 The Use of Electrical Impedance Tomography With the Inverse Problem of EEG and MEG

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Proceedings of the 22"dAnnual EMBS International Conference, July 23-28, ; !OOO, Chicago IL.

THE USE OFELECTRICAL IMPEDANCE TOMOGRAPHYWITH THE INVERSE PROBLEM OF EEG AND M[EGs.GonGalves***' nd J.C. de Munck*

'MEG Center KNAW Nniversity Hospital Vrije Universiteit, Amsterdam, The NetherlandsInstitute of Biophysics and Biomedical Engineerin gFaculty of Sciences, Lisbon., Portugal8

1. IntroductionThe Inverse Problem (IF') of EEG [ I ]and MEG [2] aims to estimate the sourcesinside the brain based on measurements of theelectrical potential on the scalp surface (in thecase of the EEG) or on measurements of themagnetic field outside the head (in the case ofthe MEG). In the solution of the IP it isassumed that the head is divided into a numberof compartments, and that the conductivities ofthese compartments are known. The head canbe modelled as a set of concentric spheres [3],each sphere representing each of the headcompartments, or by using a realistic model,which consists of a set of nested triangulatedsurfaces. In both types of models, theconductivities of the different headcompartments are included as knownconstants. In the EEG case it appears that the

solution of the IP [4]-[7] is highly dependenton the values of these constants. Thereforelarge systematic errors in the localisation ofthe sources may arise due to errors in thevalues of the elec trical conductivities.In practice, the values of the electricalconductivities are taken from literature andthey result from very few conductivitymeasurements which, most of the times, wereperformed on isolated samples of tissue. Thesevalues vary over a wide range and there mightbe a factor of 7 between the maximum andminimum conductivity values reported for acertaiin tissue [ 8 ] , [ 9 ] . his wide variation of theconductivity values is related to the variationof tissue properties between individuals, to thefact that the tissues are neither homogeneousnor isotropic and also to the variation inmeasuring techniques.Electrical Impedance Tomography(EIT) r lO ] is a technique used to determine thedistribution of electrical conductivities inside

an object from the measurement of theelectrical potential distribution at the surface,which results from the injection of currents

0-7803-6465-1 00/$10.00 02000 IEEE

into the object. In practice, the current isinjected into a pair of electrodes and measuredon the remaining. In the case of EEG IP nodetailed distribution of the conductivities isneeded. On the contrary, the knowledge of theequivalent conductivities of the headcompartments is sufficient to srolve the sourcelocalization problem. In this paper theprinciples of EIT are applied to determinethese equivalent conductivities and it is shownthat their use in solving the EEG IP decreasethe systematic errors of the solution.2. Materials and Methods

In this study, the model represented infigure 1was used to model thle head. In thissituation the solution of the fclrward problem(i.e. given a grid of electrodes placed at theouter sphere, the conductivies values and theinjected electrical current, determine thepotential distribution at the outer surface) wasobtained by using a spherical harmonicsexpansion of the potential.Electrodes of finite dimensions wereconsidered and the distribution of the currentinjected at a given electrode was assumed to begiven by:

In (1) 0 is defined in figure 2.a, 0, is theangular dimension of the injecting electrode asdefined in figure 2.b and k is a constant whichis adjusted such that:j j @) R sin(8)tle d q =0 (2)dF

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Proceedings of t he 22"dAnnual EMBS International Conference, Ju ly 23-28, 2000, Chicago IL.

Figure 1. Schematic representation of a3 layer sphere of radius RI, R2, R3 andconductivities ol,2, 03.Solving the problem in the situationconsidered in figure 2.b by imposing theappropriate boundary conditions one arrives ata linear relation between the injected currentand the potential generated at the su rface:

Y

Z - ar i s

Injected currentpattem

Figure 2. a) Schematic representationof 8 and 9 ngle. The angle 8 s theangular distance of a point P on thesurface of a sphere of radius R to the Z-axis, and 9 is the angular distance tothe X-axis. b) The angle named 0, is theangular dimension of the e lectrodes. Inthis case, there is only one injection

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axis

electrode (located on the z-axis) and thecurrent is extracted throughout the restof the sphere. The thin line representsthe current density distribution that isequal to kl=Jo on the su$ace of theelectrode and equal tol - c o s e e

I +cos e e, = J, on the rest of thesu$ace. The angle nam ed 0 (as de$nedin figu re 2.a) is the angular distance ofa measuring point on the surface of thesphere to the injection electrode.In practice, there are two currentinjection electrodes at arbitrary positions onthe surface of the sphere and two potentialmeasurement electrodes. Because of the linearrelation between cu rrent and potential (eq. (3)),the overall potential distribution, which resultsfrom an injecting and extracting electrode

(respectively located on I , a n d $ * ) , a n dmeasured on ? with respect to a reference(located on Imrcan be defined as:

In this method the Ip,which consists inthe determination of the electricalconductivities given the generated potentialdistribution originated by a known injectedelectrical current, was solved through theminimisation of the cost function:

Where K k are the measured (or simulatedpotentials) and Kk((3) the potentials predictedby the model on the i-th potential measu rementelectro de pair, caused by the k-th cu rrentinjection electro de pair.2347


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Proceedings of the 22"dAnnual EMBS International Conference,Ju ly 23-28, :2000,Chicago IL.

SNR1020305070100t ru e values

3.Results

01 U2Ed%) EX%)11.67 7.871.21 0.020.22 0.380.01 0.070.02 0.070.09 0.152.600E-04 5.69E-05

The efficiency of the proposed EITmethod was studied in several simulations.The results regarding the robustness of theimplemented algorithm to noise proved to bevery encouraging, as can be observed in table1. In fact, even for the lowest signal-to-noiseratio (SNR). onductivity values within 12%of the true values were obtained. Also, it isvisible the efficiency of the cost function inimproving the estimations with the increase ofthe SNR alues.

One of the ideas supporting this methodis to use the same head model in both the EITestimation of the conductivities and in theEEG IP. It was observed that if the head modelwas affected by some geometrical error thenthe estimated electrical conductivities wouldalso be affected by an error which w ould tendto compensate for the geometrical error. Forinstance, if for the skull thickness a value wasused which was twice as large as the truevalue, its conductivity estimated by E IT w ouldalso be twice as large as the true value, andvice versa. Therefore, when using both thehead geometry with errors and the EITestimated conductivies in the solution of theEEG IP, a decrease in the systematic errorscould be expected. To test this hypothesisseveral simulations were performed in which acomparison was made between the case that adipole was estimated using the wrong skullthickness but the correct skull conductivity andthe case where the erroneous skull thicknesswas compensated using the EIT determinedskull conductivity. This comparison was madefor various skull thickness errors (expressed infactors of the true thickness) and various ratiosof o1 nd oz.The results are plotted in figure 3.

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The ability of the presented EIT to correct forconductivity parameter errors was also studiedin a series of simulations in which the EEGinverse problem was solved with systematicconductivity errors, but the correct geometry(figure 4). In all cases the true values for c1and c3were assumed to be differeat.The EEG inverse problem was solvedand a comparison was made between thedipole position and strength errors obtained intwo ways:1. The conductivities used to find the inversesolution were set equal to the true values(ol ,0 2,0 3) , a ffected by a 50% error;2. Th e conductivities, used to find the EEGinverse solution, were given by the EIT

method, in the approximation 01=(~3.In the plot of figure 4t js representedthe error in the dipole position when using:EIT given conductivities in the a.pproximation

0 1 = 0 3 (EIT), (q,2+50%, cr3) (Case l), (cl,02-50%. 03)Case 2), (a1+50%, az-50%,03+50%) (Case 3) and (01-50%,2+50%, 03-50%)(Case 4).

The results are presented in figure 4.Effect of geometric errors on diipole positio n

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ct r l g m l b l @ = 7 78 0.20 -j 0.10 -

0.00 70.00 0.50 1.00 1.50 2.00

Factor affectlng ru e "sku l l thickness"

Figure 3. See text.

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Proceedings of the 22 Annual EMBS International Conference, July 23-28,2000, Chicago IL.

I . Effect of conductivity errors on dipole positionI 1.4 IF 2g 10E

0.8X 0.6Bf 0.4(D 0.2

0.0

0-I

0 sigma%igmal=1.121W si~ma3sigmal=2.000W sigmaWsigmal=3.000

Er C a s e 1 C a s e 2 C a s e 3 C a s e 4slhratlon

Figure 4. See text.4. Conclusions

This study dem onstrates the theoreticalfeasibility of the proposed BIT method toestimate the equivalent electricalconductivities of brain ( G ~ ) , kull (q) ndscalp (0,).n the other hand, it clearly showsthat the presented method has the ability tocompute equivalent electrical conductivitieswhich compensate for errors committed on thegeometry of the head. The use of the EITestimated conductivities in the solution of theEEG IP with a wrong head geometry proved tobe effective in the decrease of the systematicerrors of the dipole position. Also the use ofEIT es timated conductivities in the solution ofthe EEG IP instead of con ductiv ities affectedwith errors also improves, in general, thedipole position error. It was concluded (datanot shown) that the method is not efficient inimproving the dipole strength error.It is therefore demonstrated that thecombination of EIT and EEG has the potentialto reduce s ystem atic errors in estimating theunderlying generators of the EEG. Moredetails supporting this conclusion arepresented in [111.

AcknowledgementsThe work of S . Gonqalves on thisprojec t was financially supp orted by a PhD.Scholarship (Praxis XXI/BD/15502/96)awarded by the Portuguese Foundation forScience and Technology.

References[ l l Z.J Koles, Trends in EEG sourcelocalization Electroenceph. Clin.Neurophysiol, vol. 106, pp. 127-37, 1998.[21 M. Hihalainen, R. Hari, J. Risto, J.Knuutila,O.Lounasmaa,Magnetoencephalogphy - heory, instmmentation, and applicationsto noninvasive studies of the working hum anbrain, Reviews of Modern Physics, vol. 65,[3] J.C. de Munck, M.J. Peters, A fast methodto compute the potential in the multi spheremodel IEEE, Trans. Biomed. Eng., vol. BME-[4] R.M. Leahy, J.C. Mosher, M.E. Spencer,M.X. Huang, J.D. Lewine, A study of dipolelocalization accuracy for MEG and EEG usinga human skull phantom, Electroenceph. Clin.Neurophysiol., vol. 107, pp. 159-73, 1998.[5] J.C. Mosher, M.E. Spencer, R.M. Leahy,P.S. Lewis, Error bounds for EEG and MEGdipole source localization, Electroenceph.Clin. Neurophysiol.,vol. 86, pp. 303-21, 1993.[6] R. Pohlmeier, H. Buchner, A. Knoll, R.Beckmann, J. Pesh, The Influenc e of Skull -Conductivity Misspecification on InverseSource Localization in Realistically ShapedFinite Element Models Brain Topography,[7] C.J. Stok, The influence of modelparameters on EEG/ME G single dipole sourceestimation, IEEE Trans. Biomed. Eng., vol.[8] T J C Faes, H A van der Meij, J C deMunck and R M Heethaar, The electricresistivity of human tissues (100 Hz-10 MHz):a meta-analysis of review studiesPhysiological Measurement, vol. 20(4), pp.[9] L.A. Geddes, L.E. Baker, The specificresistance of biological material - Acompendium of data for the biomedicalengin eer and physiologist, Med. &Biol. Eng.,[lo] D.C. Barber and B.H. Brown, Appliedpotential tomography, J. Phys. E.: SciInstrum.,vol. 37, pp. 723-732, 1984.[ l l ] S . Gonqalves, J.C. de Munck, R.M.Heethaar, F.H. Lopes da Silva, B.W. van Dijk,The Application of Electric ImpedanceTomography to reduce sy stematic errors in theEEG Inverse Problem - A Simulation Study,Physiol. Meas., 1999, submitted.

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the use of electrical impedance tomography with the inverse problem of eeg and meg

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