Title of the Paper (18Pt Times New Roman, Bold) s6

On the Weak and Numerical Solutions of the Forward Problem in EEG

DIANA RUBIO1 and MARÍA INÉS TROPAREVSKY2 1Instituto de Ciencias, Universidad Nacional de General Sarmiento José M. Gutiérrez entre José L. Suárez y Verdi 1613 - Los Polvorines - Prov. de Bs. As. 1Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín Calle 91 Nº 3391, V. Ballester – Prov. de Bs. As. 2 Departamento de Matemática, Facultad de Ingeniería Universidad de Buenos Aires ARGENTINA

Abstract: - The Forward Problem in EEG consists on finding the scalp potential due to current sources within the brain. Electrical activity in the brain is governed by the laws of electromagnetics. Thus, to solve the problem, Maxwell Equations must be solved in a volume representing the head. In this work we consider a simplified model of the human head and a static approximation of the dynamical model. We use BEM techniques to solve the resulting equations. We illustrate some properties of the solution by typical numerical examples.

Key-Words: - Bioelectric Potentials, Boundary Element Method, Forward Model, Spherical Head Model

1 Introduction speed of propagation of waves in the head (see [6], The potential distribution on the scalp produced by [9]). This approximation uncouples the magnetic and current sources within the brain can be measured by electric equations, in consequence the model for the an Electroencephalograph (EEG). For several electrical activity consists of the Poisson equation decades, neurologists have been interested in solving with boundary conditions (see [12]). The second the problem of determining the location of these simplification of the model is about the domain current sources from the measured potential on the where we solve the equations: the human head is a scalp. In the case of epilepsy there are a few dipoles complicated anisotropic media with tissues of that give major contribution in the generation of the different conductivity values. We simplify the electric field and we are interested in finding their geometry of the head representing it as three volumes location based on the superficially measured potential with different conductivity values. Each volume is values. In order to solve this problem, called the surrounded by the next one. They represent the brain, INVERSE PROBLEM in EEG, we must be able to the skull and the scalp. In this case where the domain calculate the superficial potential for any possible is described by concentric spheres it is possible to dipole configuration which is called the FORWARD solve the differential boundary value problem PROBLEM (FP) in EEG. In this work we present exactly by a series of functions (see [3], [4], [11] some results concerning Numerical Solutions of the among others). For general domains it is not possible Forward Problem in EEG. to obtain the solution in this way. Therefore it is Electrical activity in the brain is governed by the necessary to use numerical methods. As we are laws of electromagnetics. Thus, to solve the Forward interested in finding a solution on a realistic head Problem, Maxwell's equations must be solved in a model, at a first step we propose an approximated volume representing the head. In order to solve solution although we are working with spherical numerically the problem of finding the location of the models. We intend to adapt this scheme to the source of an epileptic seizure we need to propose a realistic head model. To solve numerically the mathematical model, i.e., we need to choose the differential problem we use the boundary element equations that will represent the process and the technique which allows us to reduce a problem in a domain where these equations will be solved. At this volume to a problem over surfaces. In this way the stage of our work we simplified the model in two numerical burden is alleviated. Other approaches with ways. The first one is about the differential system of FEM techniques [15], the reciprocity theorem [13] equations arising from Maxwell's equations: we and the cubic Hermite boundary element procedure consider the static approximation to the equations. [1] were also developed. The paper is organized as The static approximation of the dynamical model is follows: in Section 2 we present the differential standard in neurology and it is justified by the high system and establish some properties of its weak solution. In Section 3 we present the integral with boundary condition formulation of the problem. Section 4 introduces the u( x )  0,x  G, (5) Discrete System. Numerical examples are shown in n Section 5. Finally we present the conclusions in where G denotes the volume representing the head Section 6. and n represents the outward normal. This condition reflects the fact that the normal derivative to the head at the boundary must be zero since air is an insulating 2 The Differential Equations material that does not support current flow. As mentioned in the Introduction, we assume that G can be described as three concentric spheres where the 2.1 Maxwell Equations radii and conductivity values are given (see [8]). We The electrical activity of the brain consists of currents denote them from the inner one to the outer: G1 the generated by biochemical sources at the cellular G level. The electric and magnetic fields that they brain, G2 the skull and 3 the scalp. The surfaces produce can be measured and predicted because they between them are denoted by: S1 , S2 and S3 (the obey physical laws: Maxwell's Equations [7] and the scalp) respectively (Fig.1). equation of conservation of the charge .J   / t,where E is the electric field, B the magnetic field,  the density of charge and J is the total current density. The density J can be decomposed in the current density due to the macroscopic electric field, and the current density caused by synaptic activity as follows

J  E  J i (1) where  is the conductivity of the media. If we accept Fig.1 The Three Sphere Model the properties of the tissues involved (conductivity , In neuromagetics a current dipole is widely used magnetic permeability values and dielectric values to to approximate a primary current. In the case of be constant, see [9]), the velocity of propagation of epilepsy a dipole can be thought as a concentration the electromagnetic waves caused by potential of J i to a single point rq : changes within the brain is about 105 m / s . This J  M  ( r  r ), means that the effect of the potential changes may be i1 q q (6) detected simultaneously at any point in the brain or in where  (.) is the Dirac delta distribution and M q is the surrounding tissues. In other words, the currents the dipole strength. In this work we considered that caused by sources in the brain behave in a stationary there is an unique dipole responsible for the seizure way (see [6],[9]). The use of the static approximation process. A current dipole is completely described by of Maxwell's Equations to describe the process is six parameters: three to determine its location and then justified. We can simplify (1) to obtain: three to define its strength. There are physical .(E )  4 xE  0 considerations that must be taken into account. They 4 can be described by the equations: .B  0 x( B )  J . c u  u  0  0 These equations together with the law of   S   S i n  i conservation of charge, that in this case can be written as where . represents the difference between the values .J  0 (2) of the functions inside the brackets through the give rise to indicated surface. These equations represent the E  u. (3) continuity of the potential and of its normal Substituting (3) in (1), and taking the divergence of derivative across the different regions. both sides we arrive to the Second Order Partial The scalp potential u(x) is measured as a Differential Equation that describes the relationship difference between the potential value at each point between the measured potential u and the current x  S3 and its value at a reference point x0  S3 , J density i (the Poisson equation): thus u( x0 )  0 .

.(( x ).u( x ))  .J i (4) Therefore, the differential equation to be solved is values chosen for solving the equations are not .(( x ).u( x ))  .J i with boundary condition accurate [17]. We state that the weak solutions of the equations u ( x )  0, x  G are continuous with respect to . That is, if the values n of two different  are close, the corresponding scalp and continuity condition for the normal derivatives at potential solutions will be close too. Suppose that u is the transition surfaces the generalized solution obtained for a given , and   that u is the generalized solution for  . Considering u  u  0  0, the weak form of the equation (4) it can be proved   S   S   i n  i u  u  C   that 2  where C is a constant that and it must verify u( x0 )  0 . depends on G, the electric field and the conductivities. We will illustrate this relationship with several 2.2 Existence and Uniqueness of Solutions examples in Section 5. Equation (4) relates the impressed current J with In order to calculate a solution of (4), in the next i Subsection we present Integral Equations for u. the electric potential u. This equation is a second order elliptic one and it can be written in the divergence form as 3 Integral Equations for the Potential Lu  .(( x ).u( x ))  .J . i Integral equations can be derived from the differential The function (x) that contains the conductivity of boundary value problem using Green's integral the different tissues at each point x, may be theorems (see [6], [14]). The following formulas are considered discontinuous and piecewise constant obtained

 1 , x  G1 ( r )u( r )   v( r )   1 3 ( x )   2 , x  G2 .  j  j 1 r  r'   u( r' ) dS'    3  3 , x  G3 j1 4  S j r  r' In order to assure the existence of a solutions of for r  S , where (4) with boundary condition (5) we need to work with j a generalized solution [5]. It can be proved that there 1 .J v( r )   i dr'. exist generalized solutions u of (4) if Ji verifies  (7) 4 1 G r  r' .J  0  i and that these solutions differs only on a r  r  S G Let j j to obtain another equation for constant. In the case we are considering, the above points on the surfaces of conductivity transition S j : J  0 condition can be written as  i and it is  k   k 1 G u( r )   1v( r )  automatically fulfilled because, under our hypothesis, 2 3 (8) J  j  j1 r  r' i has finite support inside G1 (dipole).  u( r' )n( r' ) dS'   3 The uniqueness of solution is also justified j1 4 S j r  r' because at the reference point, x0 , on the scalp the for r  Sk . potential u( x0 )  0 . Finally if u is a solution of the The above formulas provide the solution u at problem, it is also a generalized solution. points on the surfaces of conductivity transition Sj. In the next Section we present a technique for the discretization of the integral equations (8) and 2.2.1 Continuity of the Solution with respect to describe the way we solved them. the Conductivities When solving the equation (4) we need to know the conductivity of the brain, skull and scalp. There exist 4 The Discrete Problem typical values for these constant [9]. However, it may In this section we propose a discretized version differ from the real ones, thus it is important to of the integral equations for the potential. estimate the error that appears if the conductivity 4.1 The BEM Technique at the nodal points, that is, at the vertices of the

The idea behind the Boundary Element Method elements E j ,k of the grid. More precisely, if r is a (BEM) is to transform a differential equation given r  r' on a domain into an integral equation on the nodal point, U(r,r')  u(r')n(r') 3 and boundary of that domain. Thus, the dimension of the r  r' problem is reduced yielding to a reduced order r  r' numerical system (see [2]). H( r,r' )  n( r' ) 3 , we approximate each In order to apply the BEM technique to our r  r' problem we must discretize the surface integrals (8), I ( r )  U( r,r' )dS' and approximate the volume integral over G for v(r), surface integral S j  by given by equation (7). S j Nk u j ,k ,i I ( r )  H( r,r' )dS' , S j   (9) E i1 N k 4.2 The Discretization of the Equations j ,k E j ,k The discretization error depends on the grid and on where uj,k,i are the value of the potential at vertex i of the function and integral approximations. the element Ej,k and Nk is the number of vertices of the element Ej,k. As we are working on spheres, we can 4.2.1 The Grid calculate the actual normal at each nodal point

If we divide the surfaces into elements Ej,k such that exactly. For simplicity we take the same amount of points, say N, on each surface that give rise to M verifies S j   E j ,k then the grid coincides exactly k elements per surface (see Section 4.2.1 for a with each surface Sj j=1,2,3, and there is no error due description of the chosen grid). to the surface discretization. We divide the surface There exist other ways to discretize the potential. into a set of spherical elements and choose the nodal In [16] there are several comparisons of the results points to be the vertices of these elements. We obtain for different approximations. construct spherical rectangles as follows. We consider the spherical coordinates >0,  0,2 ) 4.2.3 The Integral Approximation and  0,  . On each sphere is fixed then the Finally, to calculate the integrals involved in (7) and parameters are and . We divide [0,2) into n (9) we need to select a numerical rule. intervals and [0,] into n=n/2 intervals. This choice The integral on (7) is easy to solve since we give rise to a grid of N=n(n nodal points and model the primary current Ji as a dipole (see equation

M=nn spherical elements. These elements are (6)). The integral on (9), that is, r - r' rectangles except for   0,h  and    - h , , H(r,r' )dr'  dS , 1   r - r' v  H where they become triangles as shown E j,k E j,k in Fig. 2. is the solid angle subtended by the element Ej,k at the point r of the grid. We apply the Simpson’s rule for

r  E j ,k and geometric properties of the solid angle

in the improper case (see [10]), that is when r  E j,k .

4.3 The Linear System The implementation of all these approximation schemes lead to a Linear System of Equations (D -A) u=C (10) where D is the block diagonal matrix arising from the Fig.2 The Spherical Grid right hand side of (8), that is, D=diag(D1,D2,D3)

 i  i1 where Di  for i=1,2,3. A is the matrix 4.2.2 The Potential on the Grid 2 The next step is to decide the way in which the resulting from the discretization of the surface function u will be approximated over the elements of integrals ISj (9), C is the discretization of v(r) and u is the grid. We consider u to be an average of its values the vector containing the values of the potential at the 10 10 10 M q  (1.2 10 ,0.6 10 ,0.6 10 ) , for different nodes uj,k,i. Once the grid is chosen, the matrices of the conductivities is shown. In the first column system, D and A, depends only on the conductivities   ( 0.33,0.0065,0.33), in the second one of the brain, skull and scalp.   ( 0.33,0.0042,0.33) and in the third one Note that all the values of uj,k,i are involved in the   ( 0.33,0.0020,0.33). calculation of the whole vector u. After solving (10) we can reconstruct the scalp potential interpolating the values obtained at the nodal points of the external surface (representing the scalp).

5 Numerical Results We solve the Forward Problem for a volume conductor G composed by three concentric spheres of radii .071, .078 and .085 mm. The conductivity values commonly used are (see [9])

  ( 1 , 2 , 3 )  ( 0.33,0.0042,0.33) Am/m for the brain, skull and scalp respectively. We considered dipoles oriented in different ways. In Fig.3 the scalp potentials due to a transversal and radial dipoles are Fig.4 Surface Potentials for Different Skull Conductivities.

To quantify the differences between the

potential distribution for different  i i  1,2,3 we compare the 2-norm of the vector u containing the values of the potential over the three surfaces with the shown. 2-norm of the vector  . We present some results in the following table

  i  j ui  u j

2,3 4.50000e-003 9.95365e-008

1, 3 2.20000e-003 5.86464e-008

1, 2 2.30000e-003 4.16362e-008

Fig.3 Scalp Potential for a Tangential Dipole (on left) 4, 5 4.03150e-003 4.58856e-009 and Radial Dipole (on right). 5, 6 2.75879e-003 3.09301e-009

4, 6 1.27304e-003 1.50926e-009   2.23663e-004 1.46143e-010 5.1 Numerical Results for Different 4,Brain 1 Cond. Skull Cond. Scalp Cond. 0.330000 0.004200 0.330000 Conductivity Values 1 0.330000 0.006500 0.330000 As mentioned in Section 2 the solutions of the FP are 2 continuous with respect to the values of  . To 3 0.330000 0.002000 0.330000 illustrate this relationship we numerically solve the 4 0.329900 0.004205 0.330200

FP and found the potential distribution for different 5 0.327000 0.004150 0.333000 0.329000 0.004180 0.331100 values of  i i  1,2,3 around the references values 6 mentioned above. We observe that for different values of  i the patterns of the potential distribution It can be observed that as the order of  i  j are very similar, they only differ in their magnitude, u  u not in their position. This fact can be observed in decreases, the order of i j decreases too. Fig.4 where the scalp potential for a dipole located at r  (0,0.0062,0.04) and with q 6 Conclusions In this work we established the continuity of the weak Clinical Applications, and Related Fields, Chap.5, solution to the FP of EEG with respect to the vector 1999, pp 93-109. parameter  (conductivity) and shown how this fact [10] Meijs, J. W. H., Weier, O. W.,, Peters, M. J., is reflected in a numerical solution. From the and Van Oosterom, A., On the Numerical examples presented in Section 5, it can be seen that Accuracy of the Boundary Element Method, the changes in the conductivity produces variations IEEE Trans. Biomedical Engineering, Vol.36, only in the magnitude of the potential as we pointed No.10, 1989, pp.1038-1049. out. [11] Mosher, J. C., Leahy, R. M., Ewis, P.S., EEG We proposed a spherical grid that exactly matches and MEG: Forward Solutions for Inverse the surfaces of transition of the head model which Methods, IEEE Trans. Biomedical Engineering, reduces the discretization error. Vol.46, No.3, 1999, pp. 245-259. With respect to the potential, we discretized it at [12] Plonsey, R., The Biomedical Engineering each grid element and used geometric properties to Handbook, Boca Raton, FL.CRC Press, 1995, pp calculate the resulting integrals. 119-125. These results were illustrated by some typical [13] Riera J., Fuentes M., Electric Lead Field for a numerical examples. Piecewise Homogeneous Conductor Model of the Head, IEEE Trans. Biomedical Engineering, Vol.45, No.6, 1998, pp.746-753. References: [14] Sarvas, J., Basic Mathematical and [1] Bradley, C. P., Harris, G. M. and Pullan, A. J., Electromagnetic Concepts of the Biomagnetic The Computational Performance of a High-Order Inverse Problem, Phys. Med Biol., Vol.32, No.1 , Coupled FEM/BEM Procedure in Electropotential 1987, pp.11-22. Problems, IEEE Trans. Biomedical Engineering, [15] Schimpf, P. H., Ramon, C., and Haneisen, J., Vol.48, No.11, 2001, pp.1238-1250. Dipole Models for EEG and MEG, IEEE Trans. [2] Brebbia, C.A., Telles, J.C.F.,and Wrobel, L.C., Biomedical Engineering, Vol.49, No.5, 2002, Boundary Element Techniques: Theory and pp.409-418. Applications in Engineeering, Springer Verlag, [16] Schlitt, H., Heller, L., Aaron, R., Best, N.Y. 1984. E.,Ranken, M., Evaluation of Boundary Element [3] de Munck, J.C., The Potential Distribution in a Methods for the EEG Forward Problem: Effect of Layered Anisotropic Spheroidal Volume Linear Interpolation, IEEE Trans. Biomedical Conductor, J. Appl. Phys., Vol.64, No.2, 1988, pp. Engineering, Vol.42, No.1, 1995, pp.52-58. 464-470. [17] Tsai, Y, Will, J.A., ScottHubbard-Vsan Stelle, [4] de Munck, J.C. and Peters, M.J., A Fast Method Cao, H., Tungjitkusolmun., Error Analysis of to Compute the Potential in the Multisphere Tissue Resistivity Measurements, IEEE Trans. Model, , IEEE Trans. Biomedical Engineering, Biomedical Engineering, Vol.49, No.5, 2002, Vol.40, No.11, 1993, pp. 1166-1174. pp.484-494. [5] Evans, L.C., Partial Differential Equations, AMS Vol.19, 1999. [6] Hamalainen M., Hari R., Ilmoniemi R. J., Knuutila, J. and Lounasmaa O., Magnetoencephalography, Theory, Instrumentation and Applications to Noninvasive Studies of the Working Human Brain, Reviews of modern Physics, Vol.65, No.2, 1993, pp.414-487. [7] Jackson, J.D. , Classical Electrodynamics, J.Wiley, New York. [8] Lagerlund, T.D., EEG Source Localization (Model-Dependent and Model-Independent Methods), Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, Chap. 46, 1999, pp 809-822. [9] Lopes da Silva, F. and Van Rotterdam, A., Biophysical Aspects of EEG and Magnetoencephalogram Generation, Electroencephalography: Basic Principles,