310
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
An Efficient 3-D Eddy-Current Solver Using an Independent Impedance Method for Transcranial Magnetic Stimulation Nele De Geeter*, Guillaume Crevecoeur, and Luc Dupr´e
Abstract—In many important bioelectromagnetic problem settings, eddy-current simulations are required. Examples are the reduction of eddy-current artifacts in magnetic resonance imaging and techniques, whereby the eddy currents interact with the biological system, like the alteration of the neurophysiology due to transcranial magnetic stimulation (TMS). TMS has become an important tool for the diagnosis and treatment of neurological diseases and psychiatric disorders. A widely applied method for simulating the eddy currents is the impedance method (IM). However, this method has to contend with an ill conditioned problem and consequently a long convergence time. When dealing with optimal design problems and sensitivity control, the convergence rate becomes even more crucial since the eddy-current solver needs to be evaluated in an iterative loop. Therefore, we introduce an independent IM (IIM), which improves the conditionality and speeds up the numerical convergence. This paper shows how IIM is based on IM and what are the advantages. Moreover, the method is applied to the efficient simulation of TMS. The proposed IIM achieves superior convergence properties with high time efficiency, compared to the traditional IM and is therefore a useful tool for accurate and fast TMS simulations. Index Terms—Eddy currents, impedance method (IM), transcranial magnetic stimulation (TMS), volume conductor model.
I. INTRODUCTION IOELECTROMAGNETIC field computations are needed for a wide spectrum of biomedical applications, and eddycurrent simulations comprise an essential class thereof. This is mainly because of the high importance and significance of eddy-current-based applications. Examples are the electrical impedance tomography for conductivity and permittivity estimation [1], [2] and the design of biomedical devices. The latter comprises either imaging devices such as high-resolution magnetic resonance scanners (MRIs), where the eddy currents need to be limited so as to reduce artifacts, e.g. [3], or devices for
B
Manuscript received June 15, 2010; revised August 20, 2010; accepted September 26, 2010. Date of publication October 18, 2010; date of current version January 21, 2011. Asterisk indicates corresponding author. ∗ N. De Geeter is with the Department of Electrical Energy, Systems and Automation, Ghent University, Ghent 9000, Belgium (e-mail:
[email protected]). G. Crevecoeur and L. Dupr´e are with the Department of Electrical Energy, Systems and Automation, Ghent University, Ghent 9000, Belgium (e-mail:
[email protected]; Luc.Dupr´
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2010.2087758
therapy that employ the interaction of eddy currents with the biological system [4], [5]. Transcranial magnetic stimulation (TMS) is a widely applied technique for activating or suppressing regions of the nervous system through exposure to electromagnetic fields. In 1980, Merton and Morton showed that it was possible to stimulate the motor areas of the human brain in a noninvasive way using induced eddy currents [6]. The precise interaction mechanism of electric currents with the neurological system is still an open question and needs to be answered by neurophysiologists [7], [8]. TMS has been used in a variety of clinical applications for the diagnosis and treatment of psychiatric disorders, such as depression, obsessive-compulsive behavior, etc [9], [10], and for therapy of neurological disorders such as Parkinson, tinnitus, epilepsy, etc [11]–[13]. One or multiple external coils are positioned above the head and create rapidly changing magnetic fields. These time-varying applied fields induce eddy currents in the brain tissue through Faraday’s induction mechanism. Literature dealing with the design of TMS systems reports mostly numerical solutions carried out on the basis of spatial and time-varying magnetic induction calculations and not on the induced eddy currents themselves [14], [15]. Moreover, these simulations are not patient specific since they are not using MR segmented images for the volume conductor head model. A better quality in TMS design could be obtained by the introduction of an accurate and time-efficient eddy-current solver. Such a numerical scheme is also of high importance to determine the effect of the biomedical devices on the human body, because in practice, in vivo measurements of the induced fields and currents are difficult, expensive, and in some cases impossible. This paper aims to propose an efficient eddy-current solver. The 3-D impedance method (IM) was introduced in [16] and [17], and has been frequently used for simulating the induced eddy currents in human bodies [18]–[20]. The method discretizes the geometry into regular mesh elements (voxels) and assigns material properties, represented as impedances, to each voxel. Based on this volume conductor model, a 3-D network of impedances is built with the varying magnetic induction in each voxel as source model. IM is a simple method in the sense that the 3-D network can be built in a straightforward way, starting from MR segmented images, and that a linear system of equations needs to be solved by state of the art solvers. Moreover, the eddy currents can be simulated for various excitation sources. Although these advantages make this method widely applied, IM struggles with some problems. The linear system of equations can be significantly ill conditioned, leading to a long
0018-9294/$26.00 © 2010 IEEE
DE GEETER et al.: EFFICIENT 3-D EDDY-CURRENT SOLVER USING INDEPENDENT IM FOR TMS
convergence time or in some situations to no solution at all [21]. In this paper, we introduce an independent IM (IIM) that improves the performance of the traditional IM. IIM enables us to speed up the computation of 3-D eddy currents and is therefore useful for the simulation and design of TMS. The remainder of the paper is organized as follows. In Section II, the methodology of the traditional IM and the IIM used to calculate the induced eddy currents are extensively described. IIM is applied to TMS. The numerical results and the performance results such as the computational time and the memory consumption are discussed in Section III. The paper is concluded in Section IV.
311
TABLE I TISSUE PROPERTIES USED FOR THE CALCULATIONS (3.6 KHZ)
II. METHODOLOGY Electromagnetic phenomena are modeled by the electric and magnetic fields E and H, and by the electric and magnetic inductions D and B. These fields and inductions are position r and time t dependent, and their relationship is expressed by Maxwell’s equations and constitutive laws. In order to avoid time stepping, both IM and IIM are described in the frequency domain. Through the used 4-Cole–Cole model [22], [23], the permittivity ε and conductivity σ are position and frequency dependent. In this model, the spectrum of each biological tissue is described in terms of multiple Cole–Cole dispersions εr (ω) = ε∞ +
4 n =1
εn σi + . 1−α n 1 + (jωτn ) jωε0
(1)
The complex relative permittivity is a function of the angular frequency ω with j the imaginary unit and the parameters εn , τn , and αn are chosen appropriate to each tissue. ε∞ is the permittivity in the high frequency limit and σi is the static ionic conductivity. Having calculated εr , the conductivity of each tissue can be computed as follows: σ(ω) = −ωε0 [εr (ω)]
(2)
with being the imaginary part, ε0 = 1/μ0 c2 F/m the permittivity and μ0 = 4π · 10−7 H/m the permeability in vacuum, and c = 3 · 108 m/s the speed of light. In this paper, we assume isotropic material properties. Extending IM and the proposed IIM toward space-dependent anisotropic conductivity is complicated, needs further research, and is out of the scope of this paper. Also note that we omitted the position dependence of the properties, since they are piecewise constant. Indeed, each material has its own permittivity and conductivity. The tissue properties thus obtained and used in the present calculations are shown in Table I. Remark that the permeability of each tissue is equal to the permeability in vacuum μ = μ0 . When simulations need to be carried out in a complex geometry, i.e., a realistic head model, the Maxwell’s equations cannot be analytically solved and numerical techniques become unavoidable. In the literature, spatially distributed eddy currents are typically simulated on the basis of the finite-difference method (FDM) [24], the finite-element method (FEM) [25], the finite-integration method (FIM) [26], the IM [18], etc. In our application of TMS, we prefer impedance-based methods
Fig. 1. Voxel (i, j, k) with its corresponding branch currents Ix , Iy , and Iz , voltages V x , V y , and V z , and impedances Z x , Z y , and Z z .
for computing the eddy currents due to several reasons. For the simulation of complex coil configurations, FDM, FIM, and FEM are difficult to apply because a large number of unknowns (coil configuration included) need to be solved. In practice, the head model is often discretized to cubic voxels with dimension 1 mm × 1 mm × 1 mm, leading to the problem that the coils cannot be modeled with a resolution less than 1 mm. Moreover, it is very difficult to convert the 3-D voxel grid, obtained from the MR scanner, to the finite-element discretization of FEM. A. Traditional Impedance Method As mentioned previously, the human head model is discretized using a uniform 3-D Cartesian grid in small cubic voxels. Each voxel is labeled by its node in the lower left hand rear corner. We assign branch currents and impedances to each cell edge and loop voltages to each cell face. The sign of branch currents is positive in the direction of the corresponding axes and the sign of loop voltages is chosen using the right-hand rule, with the defined (x, y, z)-axes as reference; examining the loop from the positive direction of the loop axes, the voltage is positive if it travels counterclockwise. Fig. 1 illustrates the branch currents Ix , Iy , and Iz , the voltages Vx , Vy , and Vz , and impedances Zx , Zy , and Zz belonging to the voxel (i, j, k). The IM is derived from the integral form of Faraday’s law in the frequency domain E(r, ω) · dl = −jω B(r, ω) · ds. (3) lo op
surface
This law describes how an electric field can be induced by a changing magnetic field. The electric and magnetic fields E and H are connected with the electric and magnetic induction, D and B, respectively, by the so-called constitutive equations
with ε = εr ε0 .
D(r, ω) = ε(ω)E(r, ω)
(4)
B(r, ω) = μ0 H(r, ω)
(5)
312
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
Since Amp`ere’s law states that magnetic fields can be generated by an electrical current density and by changing electric induction, Jtot in the head can be defined as the total current density comprising the electric current density J and the displacement current density jωD. Moreover, J can be written as a function of the electric field by Ohm’s law J(r, ω) = σ(ω)E(r, ω).
(6)
Based on (4), (5), and (6), (3) and the total current density become E(r, ω) · dl = −jωμ0 H(r, ω) · ds (7) lo op
surface
Jtot (r, ω) = jωε(ω)E(r, ω) + σ(ω)E(r, ω) leading to lo op
Jtot (r, ω) · dl = −jωμ0 jωε(ω) + σ(ω)
(8)
H(r, ω) · ds surface
(9) in which the right-hand side can be written as VTM S , the voltage in a closed loop due to a time-varying magnetic field. This equation can also be seen as the Kirchhoff’s voltage equation around each loop in the system, stating that the sum of all the voltages around the loop is equal to zero Z(ω) · Itot (r, ω) − VTM S (r, ω) = 0 (10)
Fig. 2. x-directed loop in voxel (i, j, k) and corresponding voltage V x , branch currents I, and impedances Z .
lo op
in which the frequency-dependent lumped impedance Z can be calculated for each subcell, with voxel size r as follows: Z(ω) =
r . (jωε(ω) + σ(ω)) · r2
(11)
In this paper, we assume cubic cells, but in general, one can replace in the last formula r by the length of the edge and r2 by the cross-sectional area of a cell in the direction perpendicular to the edge. Note that in our case, the impedance is a scalar, due to the assumption of used isotropic materials. The magnetic induction B, originating from the current density through the coils Jcoil , can be calculated by Biot-Savart’s law as follows: Jcoil (r, ω) × (r − s) μ0 · dv (12) B(r, ω) = 4π r − s3 volum e resulting for VTM S in
Fig. 3.
x-directed loop in voxel (i, j, k) and corresponding loop currents.
Figs. 2 and 3, with 1 ≤ i ≤ Nx , 1 ≤ j ≤ Ny and 1 ≤ k ≤ Nz
VTM S (r, ω) = −jω
B(r, ω) · ds.
(13)
surface
In the aforementioned equations, we neglect the contribution of the eddy currents to the magnetic induction, as in [18] and [21]. Kirchhoff’s voltage law is in the traditional IM considered on each cell face of the voxels, by which the loop passes through the four impedances of the four edges. Because of this, each branch current can be written as a sum of the four loop currents surrounding each edge. For instance, the equation for the x-directed loop in the voxel (i, j, k) becomes, according to
Vx (i, j, k) = Zy (i, j, k) · Iy (i, j, k) + Zz (i, j + 1, k) · Iz (i, j + 1, k) − Zy (i, j, k + 1) · Iy (i, j, k + 1) − Zz (i, j, k) · Iz (i, j, k)
(14)
Vx (i, j, k) = Zy (i, j, k) · [Lx (i, j, k) − Lx (i, j, k − 1) + Lz (i − 1, j, k) − Lz (i, j, k)] + Zz (i, j + 1, k) · [Lx (i, j, k) − Lx (i, j + 1, k)
DE GEETER et al.: EFFICIENT 3-D EDDY-CURRENT SOLVER USING INDEPENDENT IM FOR TMS
+ Ly (i, j + 1, k) − Ly (i − 1, j + 1, k)]
Z˜(r x,3(iN y N z +j N z +(k +1))+1) = −Zy (i, j, k + 1)
− Zy (i, j, k + 1) · [−Lx (i, j, k) + Lx (i, j, k + 1)
Z˜(r x,3(iN y N z +j N z +(k +1))+3) = Zy (i, j, k + 1)
− Lz (i, j, k + 1) + Lz (i − 1, j, k + 1)]
Z˜(r x,3(iN y N z +(j +1)N z +k )+1) = −Zz (i, j + 1, k)
− Zz (i, j, k) · [−Lx (i, j, k) + Lx (i, j − 1, k) − Ly (i − 1, j, k) + Ly (i, j, k)]
(15)
where instead of branch currents, loop currents Lx , Ly , and Lz are introduced. For simplicity of notation, the frequency dependence as well as the index tot are omitted. Similar sets of equations can be derived for y- and z-directed loops. Practical implementation: Assuming a Nx × Ny × Nz discretized head model, the IM generates 3Nx Ny Nz equations (10). In matrix notation, this becomes Z ·I =V
(16)
with I being the unknown 3Nx Ny Nz × 1 vector of branch currents, and V the 3Nx Ny Nz × 1 vector of voltages. The branch currents can be written as a function of loop currents L, with the transformation matrix T I = T · L.
(17)
By defining Z˜ as the 3Nx Ny Nz × 3Nx Ny Nz coefficient matrix of the impedance network, the aforementioned equations lead to Z˜ = Z · T
(18)
Z˜ · L = V
313
Z˜(r x,3(iN y N z +(j +1)N z +k )+2) = Zz (i, j + 1, k). (20) One can conclude that every row of the coefficient matrix Z˜ contains at the most 13 nonzero elements, namely 13 complex impedances or sum of impedances. It is wasteful to reserve storage for zero elements. This is the reason why we use a compressed column storage mode, sometimes called the Harwell–Boeing format [27], for the sparse matrix Z˜ while saving computation time and memory space. To store one complex number, there are 16 bytes needed. Therefore, Z˜ has a storage area smaller than 13 · (3Nx Ny Nz ) · 16 bytes. This system of equations is solved numerically using a standard iterative method. Once we have obtained the loop currents L, we can calculate the branch currents I along the edges of each cell by adding the values of their neighboring four loop currents, as in (17). The system of equations seems to be however ill conditioned. The iterative method has difficulties solving it and results in poor convergence, i.e., an increased number of iterations required and a limited accuracy to which a solution can be obtained, and therefore, a long computing time or possibly no solution. See also the similar results in [21].
(19)
where Z and T are both determined by the choice of the loops. Here, each voxel with coordinates (i, j, k) is indexed as i · Ny Nz + j · Nz + k, hence, the x, y, and z components of a value belonging to voxel (i, j, k) are at row 3(iNy Nz + jNz + k) + c with c = 1, 2, 3, respectively. For example, similar as in (15), the elements in matrix Z˜ on row rx = 3(iNy Nz + jNz + k) + 1 for the x-directed loop in the voxel (i, j, k) become Z˜(r x,3((i−1)N y N z +j N z +k )+2) = Zz (i, j, k) Z˜(r x,3((i−1)N y N z +j N z +k )+3) = Zy (i, j, k) Z˜(r x,3((i−1)N y N z +j N z +(k +1))+3) = −Zy (i, j, k + 1) Z˜(r x,3((i−1)N y N z +(j +1)N z +k )+2) = −Zz (i, j + 1, k) Z˜(r x,3(iN y N z +(j −1)N z +k )+1) = −Zz (i, j, k) Z˜(r x,3(iN y N z +j N z +(k −1))+1) = −Zy (i, j, k) Z˜(r x,3(iN y N z +j N z +k )+1) = Zy (i, j, k) + Zz (i, j + 1, k) + Zy (i, j, k + 1) + Zz (i, j, k) Z˜(r x,3(iN y N z +j N z +k )+2) = −Zz (i, j, k) Z˜(r x,3(iN y N z +j N z +k )+3) = −Zy (i, j, k)
B. Independent Impedance Method The conventional IM is suboptimal and the main reason for the ill conditionality is the presence of dependent equations. Appendix A verifies the existence of dependent loops in a simple 2 × 2 × 2 circuit using IM. For a general case with problem scale Nx × Ny × Nz , we introduce graph theory to demonstrate these statements. One voxel is represented by one node and the equations in (19) belonging to the voxel are constructed, considering three branches and three faces, as one can see in Fig. 1. There are N = Nx Ny Nz nodes, B = 3Nx Ny Nz branches, and F = 3Nx Ny Nz faces considered for the equations in total. In the traditional IM, Kirchhoff’s voltage law is considered on the three cell faces of each voxel, resulting in 3Nx Ny Nz loops and therefore 3Nx Ny Nz equations. However, by considering the voxel network as a connected graph with N nodes and B branches, a maximum tree can be defined, i.e., a connected subset of the graph that contains all nodes, but not any loop. There exist more than one maximum tree for the given graph, but they all contain the same number of edges, namely N − 1 edges. The graph theory now states that the number of independent loops is equal to B − (N − 1). For our problem, this means 3Nx Ny Nz − (Nx Ny Nz − 1) independent loops, which is approximately a factor 1/3 less than the considered total loops in the traditional method. This explains the ill conditionality of IM, namely, the existence of dependent loops and consequently dependent equations.
314
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
Fig. 4. Maximum tree for a 2 × 2 × 2 cube, containing all edges in the xdirection, for x = u, all edges in the y-direction, and for x = u and y = v, all edges in the z-direction. The loop L y (u − 1, v, w) is obtained by adding the y-directed branch (u − 1, v, w), and the loop L z (u + 1, v + 1, w − 1) is obtained by adding the z-directed branch (u + 1, v + 1, w − 1).
Based on graph theory, we introduce the IIM by determining a maximum tree and consequently obtaining a new set of independent equations. The chosen maximum tree meets some predefined criteria. The obtained loops have to contain as few branches as possible so as to reduce the memory storage, while the constructed tree has to preserve some regularity so that an easy implementation remains possible. Fig. 4 illustrates the choice of the tree, used in this paper. It consists of all x-directed edges, for x = u all y-directed edges, and for x = u and y = v, all z-directed edges. Hereby, u and v are chosen as centralized as possible to limit the dimension of the loops. We remark that alternative graph trees are possible. For adding every edge, which is not in the tree, we obtain a loop. We name the loop after the replenished branch and take its sign positive in the direction of that branch, see Fig. 4. All these loops form a maximal set of independent loops, for which we have to express Kirchhoff’s voltage law. The number of equations in IIM that have to be solved is significantly reduced in comparison with IM, to the benefit of the computation time. IIM has the additional advantage that the coefficient matrix Z˜ is now well conditioned. Appendix B illustrates the independence of IIM by a simple example. Practical implementation: We solve the linear system (19) Z˜ · L = V for the loop currents L. In this paper, we use the iterative preconditioned biconjugate gradient stabilized method (PBiCGSTAB) [28], [30]. Other iterative solvers such as multigrid, etc., are possible. The BiCG methods are very attractive for solving large sparse systems of equations, because they reference Z˜ only through its multiplication by a vector or the multiplication of its Hermitian transpose and a vector [27], [29]. For coefficient matrices that are close to the identity matrix, the ordinary BiCG converges in a fast way. This suggests a technique called preconditioning (PBiCG), in which (19) can be solved indirectly through the use of the preconditioned form of it ˜ ˜ −1 (Z˜ −1 p · Z) · L = Z p · V.
(21)
The preconditioner matrix Z˜ p will be a matrix that approxi˜ but is easier to invert. Here, we choose it equal to the mates Z, ˜ Because Z˜ −1 · Z˜ ≈ 1, the preconditioner diagonal part of Z. p allows the algorithm to converge in fewer steps. The stabilized PBiCGSTAB is a faster and more stable method than its ancestor PBiCG. From L, the induced branch currents I can be calculated using (17). As aforementioned in (18), matrix Z˜ is the multiplication of the impedance matrix Z and the transformation matrix T . Z and T are constructed similarly, because they are both determined by the choice of the loops (a simple example is illustrated in Appendix B). The smallest loop consists of only four edges, and the biggest one of Nx + Ny + 2 edges. Every time a loop, indexed as m, passes through a c-directed edge (i, j, k), with c = 1, 2, 3, it corresponds with a nonzero value on row m and column 3(iNy Nz + jNz + k) + c in matrix Z, and a nonzero value on row 3(iNy Nz + jNz + k) + c and column m in matrix T . We will use this property in combination with PBiCGSTAB as the advantage that Z˜ does not have to be stored in the memory explicitly. The pth diagonal element of the preconditioner matrix becomes Z˜p(p,p) = diag(Z · T )(p,p) (Z(p,q ) · T(q ,p) ). =
(22) (23)
q
Also for the used multiplications with vector v in PBiCGSTAB, it suffices to store the sparse matrices Z and T Z˜ · v = Z · (T · v) ˜H
Z
·v =T
H
· (Z
H
(24) · v).
(25)
Moreover, the human head model is situated in air, leading to some boundary conditions. The branch currents corresponding to border voxels, i.e., the branch currents at the upper, right, and front side of the domain, can be assumed zero due to the very small conductivity of air. Kirchhoff’s current law states that at any node in an electrical network, the sum of currents flowing into that node is equal to the sum of currents flowing out of the node. Considering this, the boundary conditions are Ix (Nx , j, k) = 0
(26)
Iy (i, Ny , k) = 0
(27)
Iz (i, j, Nz ) = 0
(28)
for every value of i, j, and k. The number of branch currents that have to be found decreases consequently to RB = 3Nx Ny Nz − Nx Ny − Nx Nz − Ny Nz . Analogously, the number of loop currents that is necessary to solve (17) reduces with Nx Ny + Nx Nz + Ny Nz . As previously verified, there are 3Nx Ny Nz − (Nx Ny Nz − 1) independent loops, leading to a RL × 1 = (2Nx Ny Nz − Nx Ny − Nx Nz − Ny Nz + 1) × 1 vector L. Z becomes the RL × RB impedance matrix and T the RB × RL transformation matrix. A flowchart of the 3-D eddy-current solver IIM for the specific problem of TMS is shown in Fig. 5. Starting from the geometry and the material properties of the human head and
DE GEETER et al.: EFFICIENT 3-D EDDY-CURRENT SOLVER USING INDEPENDENT IM FOR TMS
Fig. 5.
315
Flowchart of the 3-D eddy-current solver using IIM.
the TMS coil configuration, the impedance, transformation, and voltage matrices Z, T , and V, respectively, can be obtained. The unknown eddy currents I can be calculated by solving (17) and (19).
Fig. 6. Induced current density along three z-directed lines. Corresponding with the line located at x = 2.0 cm and y = 2.0 cm, the interfaces of the different layers are indicated by vertical dotted lines.
III. RESULTS AND DISCUSSION In this section, we discuss the efficiency of IIM with respect to memory consumption and time efficiency. For this, we simulate the eddy currents of TMS devices. Moreover, we make a thorough comparison of IIM with the traditional IM. We validate the IIM against analytical and other numerical (IM) solutions. A. 3-D Eddy-Current Simulations of TMS For the field calculations of TMS, we use two different head models: a simple concentric three-layered spherical head model and a more realistic head model. The spherical model contains the layers brain, skull, and scalp, with radii 8.0, 8.6, and 9.2 cm, respectively [31], [32]. It is centralized in a cubic box of 20 cm × 20 cm × 20 cm and surrounded by air. The second model contains, besides the layers grey matter, cortical bone and wet skin, also the tissues white matter and cerebrospinal fluid, and has a realistic geometry based on the ZUBAL phantom [33]. A TMS coil with inner radius of 2.0 cm and outer radius of 2.5 cm is positioned ≈1.8 cm above the head model. The current through the coil is a sine wave with amplitude 7.66 kA and working frequency 3.6 kHz, which is typical for clinical applications [34]. A list of the assigned conductivities according to the 4-Cole–Cole model (1, 2) for the applied frequency, is provided in Table I. The first simulations are done on the spherical head model, with a discretization of Nx = Ny = Nz = 100 corresponding with a resolution of 2 mm. The PBiCGSTAB-based IIM needed 8.4 GB and 2.3 h for calculating the eddy currents 1 . Total 966 iterations were needed for reaching a normalized error less than a tolerance EPS of 2.0% Z˜ · L − V < EPS. V
(29)
1 Two dual core Intel Xuon of 2.0 GHz with 16 GB RAM memory on a 64 bit platform
Fig. 7. Induced current density for a realistic head model on a logarithmic scale. The model is composed by five layers representing scalp (orange), skull (green), CSF (dark blue), white matter (light blue), and grey matter (blue). The arrows show the local current direction.
Fig. 6 illustrates the induced current density distribution along three z-directed lines that were calculated using IIM, the first line located at x = 0.0 cm and y = 0.0 cm, the second line at x = 1.0 cm and y = 1.0, cm and the third line at x = 2.0 cm and y = 2.0 cm. One can see the effect of the different conductivities on the eddy-current amplitude within the three layers, i.e., relatively lower amplitude within layers with lower conductivity, and the decrease of the current amplitude with increasing distance from the coil. In the second stage, we use the 86 × 111 × 60 five-layered realistic-geometry head model, whereby the ZUBAL phantom [33] was mapped onto a 2.2-mm grid with volume-averaged conductivities. Fig. 7 shows 2-D cross sections of the model with the corresponding induced current density distributions. Table II illustrates the convergence performance, i.e., the computing time (time) and the number of iterations (iter) of
316
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
TABLE II CONVERGENCE PERFORMANCE FOR DIFFERENT TOLERANCES
Fig. 9. Number of nonzero elements in the stored matrices, namely, the ˜ and the matrix Z in traditional impedance matrix Z, the coefficient matrix Z, IM, for different problem scales.
Fig. 8. Error of the simulation results for a specific tolerance Jto l relative to Jre f for a resolution of 5 mm.
the IIM simulations required for reaching an error less than a desired tolerance EPS. When decreasing the tolerance, the accuracy of the results increases as well as the computational burden. A compromise has to be made. We can consider the numerical results with a normalized error, as defined in (29), less than a tolerance EPS of 0.1% as the correct solution Jref . The average error of the eddy-current density distribution for a specific tolerance Jtol can be described by 1 |Jref c (i, j, k) − Jtolc (i, j, k)| . (30) 3N c=x,y ,z Jref c (i, j, k) ∀(i,j,k )
Fig. 8 shows this error together with the maximum and minimum errors for a resolution of 5 mm. A tolerance of 2.0% is a good compromise. B. Comparison IM—IIM While comparing the IIM with the traditional method, one can notice important differences. The number of equations and the number of unknowns significantly reduce for IIM. These advantages will lead to a smaller dimension of the matrix of the system that have to be solved and consequently to a reduced memory allocation. The number of rows of both matrices Z corresponds with the number of defined loops and decreases from 3Nx Ny Nz for IM to 2Nx Ny Nz − Nx Ny − Nx Nz − Ny Nz + 1 independent loops for IIM. The number of columns of the matrices Z corresponds with the number of branch currents that have to be found and decreases from 3Nx Ny Nz to 3Nx Ny Nz − Nx Ny − Nx Nz − Ny Nz due to the boundary conditions (26), (27), and (28). However, with increasing problem scale, the obtained loops will contain more branches, resulting in an increasing number of nonzero values for each equation in matrices Z and T and consequently more memory storage. Fig. 9 illustrates the two
opposite effects on the total number of nonzero elements in the matrices in function of the problem scale. For small 3-D circuits, the effect of the reduced number of equations dominates. Take for example a 10 × 10 × 10 discretized model. The orig˜ used in IM, is a 3000 × 3000 sparse inal coefficient matrix Z, matrix, containing 37 668 nonzero elements. To solve the same system of equations in IIM, we use a 1701 × 2700 sparse matrix Z, containing 16 902 nonzero elements. Besides the fact that the memory occupation for this problem scale is significantly reduced, the new impedance matrix becomes well conditioned. This means that the linear equation solution will be more accurate and less sensitive to errors in the data. Fig. 9 also shows the benefit of storing the impedance ma˜ as explained in trix Z instead of the coefficient matrix Z, Section II-B. Conventionally, the traditional IM relies on the successive overrelaxation (SOR) algorithm for solving the linear system of equations (19) Z˜ · L = V. SOR is a variant of the Gauss– Seidel method. An overcorrection to the value of Ln(g) at the nth stage of Gauss–Seidel iteration is made, anticipating future corrections and consequently resulting in faster convergence. When solving the system, each solution element can be written as [35] follows: −1 n −1 (31) + ω LGS Ln(g) = Ln(g) (g) − L(g) with ω the overrelaxation parameter. The method is convergent only for 0 < ω < 2. LGS (g) is the Gauss–Seidel iterate LGS (g) = V(g) −
g −1 h=1
Z˜(g,h) · Ln(h) −
3N
−1 Z˜(g,h) · Ln(h) . (32)
h=g +1
However, when using the traditional method, the equation system can become significantly ill conditioned, as in [21]. This results in a long convergence time and even in some situations, where the solution will not converge at all. This is the main reason why we have introduced the IIM. IIM eliminates the dependent components by the use of graphs. We have determined a well-chosen tree and confirmed the independence of the
DE GEETER et al.: EFFICIENT 3-D EDDY-CURRENT SOLVER USING INDEPENDENT IM FOR TMS
317
TABLE III COMPARISON OF THE PERFORMANCE RESULTS FOR IM AND IIM
Fig. 11. Conventional IM calculations on the same cross sections and for the same setup, as in Fig.10, with a resolution of 5 mm.
C. Validation of IIM We validate the method by comparing the analytical solutions with those obtained by IIM for the simulation of TMS. The mathematical method for computing the current densities induced by a current flowing in a coil is based on [36] and extended for computations inside a three-layered spherical head model. Fig. 10 shows the obtained IIM calculations and analytical results for 2-D cross sections of the head model. Both results correspond well with each other qualitatively as well as quantitatively. The IIM is also validated against traditional method solutions. As illustrated in Fig. 11, a good agreement is obtained between IM and IIM, with an average difference of 6%. IV. CONCLUSION
Fig. 10. Validation involving a three-layered spherical model excited by a current carrying loop coil centralized in x = 0.0 cm, y = 0.0 cm, and z = 11.0 cm. We compare the induced current density results obtained with the proposed IIM (above), and the analytical solutions (middle) on cross section x = 0.0 cm (left) and on cross section z = 5.0 cm (right). The lowest figures represent the difference between IIM and mathematical results.
obtained loops using IIM. This independence improves the condition of the coefficient matrix Z˜ and the convergence rate of the method. The linear system of equations can easily be solved now. Table III illustrates the performance results for IM and IIM applied to the simulation of TMS in the implemented spherical head model. According to the computing time (time) and the number of iterations (iter) required for reaching an error less than a tolerance of 10%, IIM solved with the PBiCGSTAB algorithm clearly outperforms the SOR-based solutions of IM. However, the memory consumption (mem) was higher than for the conventional method, as previously explained in Section III-B. Compared to the traditional IM, the proposed IIM achieves superior time efficiency and is therefore a useful tool for accurate, fast, and numerical converging TMS simulations.
The 3-D IM has been widely applied for simulating the eddy currents. The method discretizes the geometry into voxels and assigns material properties, represented as impedances, to each of them. This leads to a 3-D network of impedances with the time-varying magnetic induction in each voxel as source model. However, the linear system of equations that have to be solved in IM, can be significantly ill conditioned, leading to a high number of iterations and an inaccurate solution, or even in some situations to no solution at all. In this paper, we have developed an efficient 3-D eddy-current solver IIM, based on IM, whereby a set of independent equations is identified by defining independent loops in the 3-D circuit. The novelty of IIM lies in the use of graphs. This results in a better conditioned problem and consequently a better time efficiency. For the application of TMS, the IM can start from MR segmented images. In this way, the solver can be performed in a patient-specific way. Much better convergence properties are observed during the simulations of IIM in comparison with IM. By using the efficient IIM, optimal design of TMS becomes feasible and IIM offers the possibility of solving systems with a huge number of unknowns in a realistic period of time. APPENDIX PROOF OF THE DEPENDENCE OF IM AND THE INDEPENDENCE OF IIM Consider a 2 × 2 × 2 3-D network, as in Fig. 12. For simplicity of comparison between IM and IIM, we apply equations (26), (27), and (28), resulting in the unknown matrix of branch
318
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
A. Dependence of IM In the traditional IM, Kirchhoff’s voltage law Z · I = V is considered on each cell face of the voxels, see Fig. 13(a). Therefore, each loop passes though the four impedances of the four edges of the face. The voltage matrix V and the impedance matrix Z become ⎛
Fig. 12.
2 × 2 × 2 3-D network with the corresponding branch currents.
⎞ Vx (1, 1, 1) ⎜ Vy (1, 1, 1) ⎟ ⎜ ⎟ ⎜ V (1, 1, 1) ⎟ V =⎜ z ⎟ ⎜ Vz (1, 1, 2) ⎟ ⎝ ⎠ Vy (1, 2, 1) Vx (2, 1, 1) ⎛ 0 −Zx Zx 0 −Zy ⎜ Zy ⎜ Zz 0 ⎜ −Zz ⎜ 0 Zx ⎜ 0 ⎜ 0 0 ⎜ −Zy ⎜ 0 −Zx ⎜ 0 T Z =⎜ 0 0 ⎜ Zz ⎜ 0 0 ⎜ 0 ⎜ 0 Zy ⎜ 0 ⎜ −Zz 0 ⎜ 0 ⎝ 0 0 0 0 0 0
(34)
0 0 0 Zx −Zy 0 0 −Zx 0 0 Zy 0
0 0 0 0 0 −Zx Zz Zx 0 0 0 −Zz
0 0 0 0 0 0 0 0 Zy −Zz −Zy Zz
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . (35) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
The six rows of the impedance matrix correspond respectively to the loops A (Lx (1, 1, 1)), B (Ly (1, 1, 1)), C (Lz (1, 1, 1)), D (Lz (1, 1, 2)), E (Ly (1, 2, 1)), and F (Lx (2, 1, 1)), as illustrated in Fig. 13(a). It is important to notice that one of these loops can be written as a linear combination of the other loops. For instance, one can write D = A + B + C − E − F.
Fig. 13. (a) Definition of the loops for the conventional IM. (b) Definition of the loops for IIM.
(36)
From this simple example, we can conclude that the traditional IM contains a set of dependent equations, causing the problem to be ill conditioned.
B. Independence of IIM
currents I ⎛
⎞ Ix (1, 1, 1) ⎜ Iy (1, 1, 1) ⎟ ⎜ ⎟ ⎜ Iz (1, 1, 1) ⎟ ⎜ ⎟ ⎜ Ix (1, 1, 2) ⎟ ⎜ ⎟ ⎜ Iy (1, 1, 2) ⎟ ⎜ ⎟ ⎜ I (1, 2, 1) ⎟ I=⎜ x ⎟. ⎜ Iz (1, 2, 1) ⎟ ⎜ ⎟ ⎜ Ix (1, 2, 2) ⎟ ⎜ ⎟ ⎜ Iy (2, 1, 1) ⎟ ⎜ ⎟ ⎜ Iz (2, 1, 1) ⎟ ⎝ ⎠ Iy (2, 1, 2) Iz (2, 2, 1)
(33)
For the IIM developed in this paper, we firstly have to determine a well-chosen maximum tree. In Fig. 13(b), this tree is indicated by the brown branches for a 2 × 2 × 2 problem. The loops are obtained by adding the remaining branches, indicated by the dotted yellow lines. We then consider Kirchhoff& rsquo;s voltage law Z · I = V over this set of loops. The voltage matrix V, the impedance matrix Z, and the transformation matrix T become ⎛
⎞ −Vz (1, 1, 1) ⎜ Vy (1, 1, 1) − Vx (2, 1, 1) ⎟ ⎜ ⎟ −Vz (1, 1, 2) V =⎜ ⎟ ⎝ ⎠ Vy (1, 2, 1) −Vx (2, 1, 1)
(37)
DE GEETER et al.: EFFICIENT 3-D EDDY-CURRENT SOLVER USING INDEPENDENT IM FOR TMS
⎛
−Zx ⎜ Zy ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ Z ZT = ⎜ x ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ −Zy ⎜ ⎜ 0 ⎝ 0 0 ⎛ −1 ⎜ 1 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 1 T =⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ −1 ⎜ ⎜ 0 ⎝ 0 0
−Zx 0 Zz Zx 0 0 0 0 −Zy 0 Zy −Zz −1 0 1 1 0 0 0 0 −1 0 1 −1
0 0 0 −Zx Zy 0 0 Zx 0 0 −Zy 0
0 0 0 0 0 0 0 0 0 0 −Zx 0 0 Zz Zx 0 0 −Zy 0 Zz 0 Zy −Zz −Zz ⎞ 0 0 0 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎟ −1 0 0 ⎟ ⎟ 1 0 0 ⎟ ⎟ 0 −1 0 ⎟ ⎟. 0 1 0 ⎟ ⎟ 1 1 0 ⎟ ⎟ 0 0 −1 ⎟ ⎟ 0 0 1 ⎟ ⎠ −1 0 1 0 −1 −1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(38)
(39)
Remark that the number of loops are decreased from six (IM) to five (IIM), and that they are, according to the graph theory, constructed in such a way that they are independent. Moreover, one can verify that the voltage matrix V and the coefficient matrix of the impedance network Z˜ = Z · T have both full rank and that this rank equals the number of variables, namely the five loop currents. Consequently, the solution of the system of linear equations Z˜ · L = V is unique. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of GOA07/GOA/006 and IUAP P6/21. G. Crevecoeur is a postdoctoral researcher for the FWO. REFERENCES [1] P. Metheral, D. C. Barber, R. H. Smallwood, and H. B. Brown, “Three dimensional electrical impedance tomography,” Nature, vol. 380, pp. 509– 512, 1996. [2] A. P. Bagshaw, A. D. Liston, R. H. Bayford, A. Tizzard, A. P. Gibson, A. T. Tidswell, M. K. Sparkes, H. Dehghani, C. D. Binnie, and D. S. Holder, “Electrical impedance tomography of human brain function using reconstruction algorithms based on the finite element method,” Neuroimage, vol. 20, pp. 752–764, 2003. [3] F. Liu and S. Crozier, “An FDTD model for calculation of gradientinduced eddy currents in MRI system,” IEEE Trans. Appl. Supercond., vol. 14, no. 3, pp. 1983–1989, Sep. 2004. [4] T. A. Wagner, M. Zahn, A. J. Grodzinsky, and A. Pascual-Leone, “Threedimensional head model simulation of transcranial magnetic stimulation,” IEEE Trans. Biomed. Eng., vol. 51, no. 9, pp. 1586–1594, Sep. 2004. [5] T. Wagner, A. Valero-Cabre, and A. Pascual-Leone, “Noninvasive human brain stimulation,” Annu. Rev. Biomed. Eng., vol. 9, pp. 527–565, 2007. [6] P. Merton and H. Morton, “Stimulation of the cerebral cortex in the intact human subject,” Nature, vol. 285, p. 227, 1980. [7] A. Post and M. E. Keck, “Transcranial magnetic stimulation as a therapeutic tool in psychiatry: What do we know about the neurobiological mechanisms?,” J. Psychiat. Res., vol. 35, pp. 193–215, 2001.
319
[8] W. F. Boron and E. L. Boulpaep, Medical Physiology—A Cellular and Molecular Approach. New York: Elsevier, 2004. [9] A. Pascual-Leone, B. Rubio, F. Pallardo, and M. D. Catala, “Rapid-rate transcranial magnetic stimulation of left dorsolateral prefrontal cortex in drug-resistant depression,” Lancet, vol. 348, pp. 233–237, 1996. [10] R. H. Belmaker, H. Einat, Y. Levkovitz, M. Segal, and N. Grisaru, “TMS effects in animal models of depression and mania implications of hippocampal electrophysiology,” in Transcranial Magnetic Stimulation in Neuropsychiatry, M. S. George and R. H. Belmaker, Eds. Washington, DC, American Psychiatric Press, 2000, pp. 99–114. [11] J. Mally and T. W. Stone, “Improvement in Parkinsonian symptoms after repetitive transcranial magnetic stimulation,” J. Neurol. Sci., vol. 62, pp. 179–184, 1999. [12] M. Kobayashi and A. Pascual-Leone, “Transcranial magnetic stimulation in neurology,” Lancet Neurol., vol. 2, pp. 145–156, 2003. [13] E. H. Bae, L. M. Schrader, K. Machii, M. Alonso-Alonso, L. J. Riviello, Jr, A. Pascual-Leone, and A. Rotenberg, “Safety and tolerability of repetitive transcranial magnetic stimulation in patients with epilepsy: A review of the literature,” Epilepsy Behav., vol. 10, pp. 521–528, 2007. [14] K. Hsu, S. S. Nagarajan, and D. M. Duran, “Analysis of efficiency of magnetic stimulation,” IEEE Trans. Biomed. Eng., vol. 50, no. 11, pp. 1276– 1285, Nov. 2003. [15] S. L. Ho, G. Xu, W. N. Fu, Q. Yang, H. Hou, and W. Yan, “Optimization of array magnetic coil design for functional magnetic stimulation based on improved genetic algorithm,” IEEE Trans. Magn., vol. 45, no. 10, pp. 4849–4852, 2009. [16] O. P. Gandhi, J. F. Deford, and H. Kanai, “Impedance method for calculation of power deposition patterns in magnetically induced hyperthermia,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 10, pp. 644–651, Oct. 1984. [17] N. Orcutt and O. P. Gandhi, “A 3-D impedance method to calculate power deposition in biological bodies subjected to time varying magnetic fields,” IEEE Trans. Biomed. Eng., vol. 35, no. 8, pp. 577–583, Aug. 1988. [18] M. Nadeem, T. Thorlin, O. P. Gandhi, and M. Persson, “Computation of electric and magnetic stimulation in human head using the 3D impedance method,” IEEE Trans. Biomed. Eng., vol. 50, no. 7, pp. 900–907, Jul. 2003. [19] Y. Han, Y. Lu, J. Zhang, and H. Zhang, “A parallel 3-D impedance method to compute the induce current in human head by TMS,” in Proc. IEEE/ICME Int. Conf. Complex Med. Eng., Beijing, China, 2007, pp.1430–1434. [20] M. Lu, S. Ueno, T. Thorlin, and M. Persson, “Calculating the activating function in the human brain by transcranial magnetic stimulation,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1438–1441, Jun. 2008. [21] H. Wang, F. Liu, L. Xia, and S. Crozier, “An efficient impedance method for induced field evaluation based on a stabilized bi-conjugate gradient algorithm,” Phys. Med. Biol., vol. 53, pp. 6363–6375, 2008. [22] K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics: Alternating current characteristics,” J. Chem. Phys., vol. 9, pp. 341–351, 1941. [23] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues—Part III: Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, pp. 2271–2293, 1996. [24] T. Dawson, J. De Moerloose, and M. Stuchly, “Comparison of magnetically induced ELF fields in humans computed by FDTD and scalar potential FD codes,” Appl. Comp. Electro. Soc. J., vol. 11, pp. 63–71, 1996. [25] T. Wagner, F. Fregni, U. Eden, C. Ramos-Estebanez, A. Grodzinsky, M. Zahn, and A. Pascual-Leone, “Transcranial magnetic stimulation and stroke: A computer-based human model study,” Neuroimage, vol. 30, pp. 857–870, 2006. [26] A. Barchanski, T. Steiner, H. De Gersem, M. Clemens, and T. Weiland, “Local grid refinement for low-frequency current computations in 3-D human anatomy models,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1371– 1374, Apr. 2006. [27] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing. New York: Cambridge Univ. Press, 2007. [28] H. Van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput., vol. 13, pp. 631–644, 1992. [29] J. R. Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Pittsburgh, PA: Carnegie Mellon Univ., 1994. [30] R. W. Freund, G. H. Golub, and N. M. Nachtigal, Iterative Solution of Linear Systems. Cambridge, U.K.: Cambridge Univ. Press, 1992.
320
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 2, FEBRUARY 2011
[31] D. A. Brody, F. H. Terry, and R. E. Ideker, “Eccentric dipole in a spherical medium: Generalized expression for surface potentials,” IEEE Trans. Biomed. Eng., vol. BME-20, no. 2, pp. 141–143, Mar. 1973. [32] S. Homma, T. Musha, Y. Nakajima, Y. Okamoto, S. Blom, R. Flink, and K. Hagbarth, “Conductivity ratios of the scalp-skull-brain head model in estimating equivalent dipole sources in human brain,” Neurosci. Res., vol. 22, pp. 51–55, 1995. [33] I. G. Zubal, C. R. Harrell, E. O. Smith, Z. Rattner, G. Gindi, and P. B. Hoffer, “Computerized three-dimensional segmented human anatomy,” Med. Phys., vol. 21, no. 2, pp. 299–302, 1994. [34] M. Lu, T. Thorlin, S. Ueno, and M. Persson, “Comparison of maximum induced current and electric field from transcranial direct current and magnetic stimulations of a human head model,” PIERS Online, vol. 3, no. 2, pp. 178–183, 2007. [35] F. Liu, S. Crozier, and H. Zhao, “On the induced electric field gradients in the human body for magnetic stimulation by gradient coils in MRI,” IEEE Trans. Biomed. Eng., vol. 50, no. 7, pp. 804–815, Jul. 2003. [36] H. Eaton, “Electric field induced in a spherical volume conductor from arbitrary coils: Application to magnetic stimulation and MEG,” Med. Biol. Eng. Comput., vol. 30, no. 4, pp. 433–440, 1992.
Nele De Geeter was born in 1986. She received the electromechanical engineering degree from Ghent University, Ghent, Belgium, in 2009, where she is currently working toward the Ph.D. degree in engineering at Department of Electrical Energy, Systems and Automation. Her research interest includes numerical methods for bioelectromagnetics.
Guillaume Crevecoeur was born in 1981. He received the physical engineering degree and the Ph.D. degree in engineering sciences from Ghent University, Ghent, Belgium, in 2004 and 2009, respectively. Since 2009, he has been a Postdoctoral Researcher for the Fund of Scientific Research Flanders at Ghent University. His main research interests include numerical methods in bioelectromagnetics, the solution of inverse problems in (bio-) electromagnetics for magnetic material characterization, geometrical optimization, and source localization.
Luc Dupr´e was born in 1966. He received the electrical and mechanical engineering degree and the Ph.D. degree in applied sciences from the Ghent University, Belgium, in 1989 and 1995, respectively. In 1989, he joined the Department of Electrical Power Engineering, Ghent University, as a Research Assistant. In 1996, he was a Postdoctoral Researcher for the Fund of Scientific Research-Flanders. Since 2002, he has been a Professor at the Engineering Faculty, Ghent University. His research interests mainly include numerical methods for electromagnetics, especially in electrical machines, modeling, and characterization of magnetic materials.
