THE FINITE VOLUME HEAD MODEL FOR EEG SOURCE LOCALIZATION

L. A. Neilson, Z. J. Koles
Department of Electrical and Computer Engineering, University of Alberta,
2^nd Floor ECERF, Edmonton, Alberta, CANADA T6G 2V4

Abstract: An accurate computer model for EEG source localization was implemented based on the finite volume method, using lead field analysis to facilitate solution of the inverse problem in EEG.  A new integrated algorithm that exploits polynomial preconditioning and the Conjugate Gradient Method makes it possible to solve the forward problem faster than ever before so that a realistic head model of very high spatial resolution can be processed in a reasonable amount of time.  Single dipole localization experiments were performed on an inhomogeneous finite volume spherical conductor model.  The less-refined model (1.84 mm resolution) provided average displacement error of 2.4 (0.0–4.1) mm, orientation error of 0.7 (0.0–4.6) degrees, and magnitude error of 11.1 (4.8–25.7)%.  The highly discretized model (0.92 mm resolution) rendered average displacement error of 1.4 (0.0–5.9) mm, orientation error of 0.2 (0.0–0.9) degrees, and magnitude error of 4.2 (1.4–7.3)%.

INTRODUCTION

The finite volume method is a numerical technique used to model current flow in an inhomogeneous volume by representing the conductor as a complex assemblage of many equally sized cubes [1].  Specifying current sources anywhere in the model, the resulting resistive network can be translated into a large, sparse system of finite difference equations and solved to determine potentials at each node.  Using this procedure, the position of a dipole source in the brain can be determined from a set of measured EEG scalp potentials.

The Finite Volume Head Model (FVHM), unlike analytical methods, can precisely account for actual head shape and tissue discontinuities.  Furthermore, this approach can easily accommodate anisotropic tissue in the conductivity model of the head volume.

The concept of lead field analysis is central to this localization technique.  Similar implementations are discussed in [2] and [3], but in this study, the application of polynomial preconditioning has made it possible to solve much larger systems of finite difference equations, allowing for head models of very high spatial resolution (< 1mm).

METHODS

For the purpose of simulating the flow of current from a source in the brain to electrodes on the scalp (the forward problem), the assemblage of discrete conductive elements are interpreted as 3-D mesh circuit.  To allow for anisotropic tissues, the head volume can be represented by three distinct conductivity models, one for each orientation.  This does not add any computational complexity to the FVHM algorithm.

A relationship between any node in the 3-D circuit and its six neighbours can be expressed using Kirchoff’s Current Law (KCL).  The finite difference equation that describes the relationship amongst any set of adjacent nodes is

      (1)

where V₀ is the potential of the central node, V_n are the potentials of the adjacent node, s_effn are the effective combination of the conductivities between the central node and each adjacent node (conductivity per unit length), I₀ is the total current flow into the central node, and h is the side length of every voxel.  Equation (1) is applied to each of the millions of conductive voxels to create a system of equations governed by Ohm’s Law.  Solving the system and extracting the potential values from those nodes that represent scalp electrodes constitutes calculation of the forward problem.  This task presents significant computational demands. 

Conjugate Gradient Method and Polynomial Preconditioning

The Conjugate Gradient Method (CGM) is an iterative algorithm for solving sparse, linear systems of equations.  Integration of polynomial preconditioning with the CGM provides the ability to find a solution to the large system of equations in a relatively short time frame.

The condition (proximity of the spectral bounds) of the system is an important factor in how quickly the CGM will converge toward the correct solution vector. Polynomial preconditioning manipulates the equation matrix to yield an equivalent system whose eigenvalues are as tightly clustered around unity as possible.  A special algorithm for polynomial preconditioning is developed in [1].  Through this unique method, preconditioning is applied “on the fly” during the CGM, which offers important computational savings.

Collection Of Lead Field Data

EEG source localization requires calculation of the lead field matrix – the scalp potentials induced by dipoles at every possible location and orientation in the head model.  From the traditional perspective, acquisition of this data means solving the forward problem thrice for each voxel in the solution space.  Obviously, these computational requirements are prohibitively time consuming.  An alternative means of compiling the required data is presented here.

The 3-D mesh circuit that is used to model current flow through the head is a linear (thus reciprocal) network. This special property can be exploited to calculate the lead field matrix by computing the forward solution just once for each electrode lead rather than hundreds of thousands of times.

Method of Localization

For the purposes of this model, surface potentials are interpreted as a superposition of the voltages produced by three single dipoles in one location with orthogonal orientations.  One need only discover the most likely dipole location in the brain and the dipole moment of each orientation which result in scalp potentials that are best fit (in a least squares sense) to the measured data.

A desirable feature of this approach is that the lead field data for a head model can be used repeatedly to localize any number of equivalent dipole sources based on various EEG recording samples.  The lead field matrix depends only on the head model and the electrode montage [3].  Also, localization results from many EEG samples can be superimposed to identify distributed cortical sources, or dipoles can be localized in time series to describe a moving dipole trace.

Testing Source Localization Accuracy

The 3-shelled sphere is an analytic model that serves as a “gold standard” to which forward solutions from any other conductivity model can be compared.  To test the accuracy of the FVHM, simulated EEG recordings were obtained from the analytic model at 25 electrodes.  These data were used to localize single dipoles within a finite volume version of the same 3-shelled sphere (outer radius = 92 mm).  The sphere was decomposed in a cubic grid of 100³ voxels (h=1.84 mm). Experiments were performed for dipoles at eccentricities = 0.5, 0.6, 0.7, and 0.8 along three axes, in both radial and tangential orientations.  The complete set of localization experiments was repeated for the same sphere with double the spatial resolution – 200³ cubic grid (h=0.92 mm).

Accuracy of a calculated dipole location is quantified by displacement, orientation, and magnitude errors.  The importance of achieving small displacement error is apparent, but orientation also hints at source location since real EEG sources (pyramidal cells) tend to be aligned normal to the surface of the cortex.  The perceived magnitude of a localized dipole gives an indication of the surface area of coherently active cortical cells – the affected region of brain tissue.

RESULTS

TABLE I
Localization Errors

Table I shows the error calculations for all localization tests.  Shaded cells indicate the maximum and minimum values over all dipoles in each category of error.  Although there was less variation in displacement error for the 100³ grid model, the averages of all three errors were significantly less for the higher resolution (200³ grid) sphere (2.4 mm versus 1.4 mm average displacement errors). Localization error on this scale is an improvement on the results of many previous EEG source localization systems that make use of idealized or low-resolution head models.

DISCUSSION

The results obtained from testing FVHM source localization on an inhomogeneous spherical volume indicate that it is a valid approach to the inverse problem.  It can accommodate realistic head models of very high spatial resolution and deliver patient-specific source localization for multiple EEG samples a few days following head data acquisition.  Further testing of the method should be carried out using genuine EEG recordings and realistic head models derived from patient imaging.

In [4], EEG sources were induced in patient heads by exciting implanted depth electrodes.  Scalp recordings were used to localize 176 dipoles in realistically shaped boundary element (BEM) head models of 13 subjects.  The average localization error of 10.5mm indicates that the BEM technique offers very little advantage over the spherical model (10.6mm error).  However, as discussed in [4], the BEM yields a conductivity model that is only vaguely representative of the head volume.  It is therefore expected that the FVHM will show superior performance in source localization with EEG data recorded from patients.

A complete physical and mathematical justification for the FVHM development, as well as a detailed description of the decomposition algorithm is available in [1].

Acknowledgments:  This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

REFERENCES

[1]  V.E. Agapov. “Numerical solution of the forward problem in electroencephalography.” Ph.D. thesis. Departement of Mathematical Sciences, University of Alberta – Edmonton, 2001.

[2]  P. Kauppinen, et al.. “A software implementation for detailed volume conductor modeling in electrophysiology using finite difference method.” Comput Methods Programs Biomed, vol. 58, pp. 191-203, 1999.

[3] P. Laarne, et al.. “Accuracy of two dipolar inverse algorithms applying reciprocity for forward calculation.” Comput Biomed Res, vol. 33, pp. 172-185, 2000.

[4] N. Cuffin, et al..  “Experimental tests of EEG source localization accuracy in realistically shaped head models.”  Clin Neurophysiol, vol. 112, pp. 2288-2292, 2001.