Novel Volume Integral Equation Approach for Low-Frequency E-Field Dosimetry of Transcranial Magnetic Stimulation

Abstract

A new volume integral equation (VIE) approach is introduced to study transcranial magnetic stimulation (TMS) and high-contrast media at low frequencies. This new integral equation offers a simple solution to the high-contrast breakdown observed in low-frequency electric field (E-field) dosimetry of conductive media. Specifically, we employ appropriate approximations that are valid for low frequencies and stabilize the VIE by introducing a basis expansion set that removes solutions associated with high eigenvalues in the equation. The new equation is devoid of high-contrast breakdown and does not require the use of auxiliary surface variables or projectors, providing a straightforward practical solution for the VIE analysis of TMS. Our results indicate that the novel VIE formulation matches boundary element, finite element, and analytical solutions. This new VIE represents a first step towards including anisotropy in integral equation E-field dosimetry for brain stimulation.

I. Introduction

TRANSCRANIAL Magnetic Stimulation (TMS) involves the use of coils driven by low-frequency pulses and placed on the scalp to induce E-fields in the brain [1; 2]. These E-fields, in turn, modulate brain activity [3]. Consequently, TMS has widespread applications for treating brain and mental health disorders, including major depression, migraines, and obsessive-compulsive disorder [4–6]. For TMS targeting, it is important to deliver a specified E-field dose [7]. However, measuring the induced E-fields in the brain during TMS would require the implantation of probes [8]. Therefore, computational E-field dosimetry is commonly used to determine where to place the TMS coil, determine pulse waveform amplitude, and study the effects of TMS [9–13]. This work introduces a novel approach using Volume Integral Equation (VIE) for E-field dosimetry of TMS. The advantages of employing VIEs lie in their adaptability to highly heterogeneous and anisotropic media.

There are many methods of E-field dosimetry, typically falling into one of two classes: differential equation and integral equation-based solvers. Differential equation-based solvers include the finite difference method (FDM) [12; 14] and finite element method (FEM) [13; 15–17]. Integral equations involve the derivation of equivalent boundary or volume sources to derive boundary or volume integral equations, respectively [18]. Boundary integral equations have been successfully employed for analyzing TMS [12; 19–21]. However, unlike existing differential equation methods, boundary integral equations cannot account for the conductivity anisotropy of the white matter. The bottleneck is that white matter anisotropy is highly heterogeneous, requiring the use of a large number of distinct Green’s functions to approximate it [22].

VIEs can, in principle, be used to model tissue anisotropy [23–25]. However, they remain relatively unexplored in E-field dosimetry due to breakdowns observed at low frequencies when analyzing conductive media [26; 27]. Several solutions for stabilizing the VIE for analyzing TMS and general bio-electromagnetic phenomena have been proposed. In [26; 28–30], a surface integral equation is appended to the volume integral equation to reduce the contrast of the volume equivalence currents, resulting in a high-contrast stable VIE. Furthermore, in [27], projectors are used to partition the VIE spectrum into subspaces that can be rescaled, thereby resulting in well-conditioned systems of equations. Both of these methods enable the solution of the VIE; however, they require the implementation of additional surface integral equations or projectors, thereby limiting their applicability.

The high-contrast breakdown is due to the relatively large conductivity contrast at the boundary between the head’s scalp and the surrounding insulating free-space, which is responsible for attenuating normally incident E-fields as they penetrate into the head. In this paper, we introduce a simple VIE and its form where the relevant commonly used approximations for TMS E-field dosimetry are applied. We also introduce a method to alleviate its ill-conditioning by spanning the E-fields using a reduced basis set with zero normal component on the free-space and scalp interface. We present several results indicating that this formulation is accurate and can be solved using iterative solvers.

Although the solver presented in this paper underperforms compared to BEM and FEM in terms of speed, it provides similar accuracy levels to BEM. This work offers a simple phenomenological explanation for the breakdown of VIEs when modeling conductive media at low frequencies. Furthermore, a stabilization scheme is provided by applying appropriate approximations to VIEs at low frequencies, without the need for additional surface operators or projectors. This is also a preliminary step towards integral equation modeling of white matter anisotropy, and these extensions will be addressed in future research.

The paper is structured as follows. First, the low-frequency VIE is derived, which satisfies the common approximations in TMS. Then, it is discretized by expanding the electric flux density vector using loop basis functions. Finally, the resulting matrix equation is implemented and solved. Implementation results for a four-layer spherical head model and a realistic head model are presented and discussed in detail.

II. Materials And Methods

In what follows, for simplicity of notation, we assume the variation of the TMS current pulse is time-harmonic (i.e., $\left.{\mathbf{J}}_{\mathbf{c}\mathbf{o}\mathbf{i}\mathbf{l}}(\mathbf{r};t)={\mathbf{J}}_{\mathbf{c}\mathbf{o}\mathbf{i}\mathbf{l}}\left(\mathbf{r}\right)e^{-j\omega t}\right)$ and it is suppressed. Nevertheless, because TMS driving pulses typically have frequency content between 1–10 kHz, quasi-stationary assumptions are valid. As such, the temporal and spatial variation of the fields are separable, and the equations remain valid for typical TMS pulse driving current ${\mathbf{J}}_{\mathbf{c}\mathbf{o}\mathbf{i}\mathbf{l}}(\mathbf{r};t)={\mathbf{J}}_{\mathbf{c}\mathbf{o}\mathbf{i}\mathbf{l}}\left(\mathbf{r}\right)f\left(t\right)$, where $f\left(t\right)$ is the TMS pulse driving waveform, as long as $j\omega$ is replaced with $\frac{d}{dt}f\left(t\right)$.

A. Low-frequency VIE

During TMS, a coil is driven with a current ${\mathbf{J}}_{\text{coil}}\left(\mathbf{r}\right)$, where $\mathbf{r}$ represents a Cartesian location. The time-harmonic magnetic fields produced by these currents induce E-fields inside the head. These fields must satisfy the Maxwell’s equations,

$$\begin{gathered} \nabla \times H\left(r\right)=j\omega {ϵ}_0{ϵ}_r\left(r\right)E\left(r\right)+J_{\text{coil}}\left(r\right) \\ \nabla \cdot {\mu }_{0}H\left(r\right)=0 \\ \nabla \times E\left(r\right)=-j\omega {\mu }_{0}H\left(r\right) \\ \nabla \cdot {ϵ}_0{ϵ}_r\left(r\right)E\left(r\right)=\nabla \cdot D\left(r\right)=0. \end{gathered}$$ (1)

Here, ${\mu }_0$ and ${ϵ}_0$ represent the permeability and permittivity of free space, respectively. ${ϵ}_r\left(\mathbf{r}\right)=ϵ\left(\mathbf{r}\right)/{ϵ}_0-j\sigma \left(\mathbf{r}\right)/\omega {ϵ}_0$ is the relative electrical permittivity, where $ϵ\left(\mathbf{r}\right)$ and $\sigma \left(\mathbf{r}\right)$ are the electric permittivity and conductivity, respectively. The VIE utilizes volume equivalence principles to treat the inhomo-geneity as a volume equivalent current ${\mathbf{J}}_s\left(\mathbf{r}\right)=j\omega {ϵ}_0\left({ϵ}_r\left(\mathbf{r}\right)-\right.1\left)\mathbf{E}\right(\mathbf{r})$ that radiates in free space, which results in

$$\begin{gathered} \nabla \times H\left(r\right)=j\omega {ϵ}_{0}E\left(r\right)+J_s\left(r\right)+J_{\text{coil}}\left(r\right) \\ \nabla \cdot {\mu }_{0}H\left(r\right)=0 \\ \nabla \times E\left(r\right)=-j\omega {\mu }_{0}H\left(r\right) \\ \nabla \cdot {ϵ}_{0}E\left(r\right)=-\frac{1}{j\omega }\nabla \cdot J_s\left(r\right). \end{gathered}$$ (2)

The E-fields generated by the coil current ${\mathbf{J}}_{\text{coil}}\left(\mathbf{r}\right)$ are referred to in the literature as primary E-fields ${\mathbf{E}}_p\left(\mathbf{r}\right)$, and the ones generated by ${\mathbf{J}}_s\left(\mathbf{r}\right)$ are referred to as secondary E-fields ${\mathbf{E}}_s\left(\mathbf{r}\right)$. Due to the solenoidal shape of the coil, the primary E-field is purely rotational and is equal to

$$\begin{gathered} E_p\left(r\right)=-J\omega {\mu }_0{\int }_C\frac{J_{\text{coil}}\left(r^{\prime }\right)e^{-jk\left|r-r^{\prime }\right|}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime } \\ \approx -j\omega {\mu }_0{\int }_C\frac{J_{\text{coil}}\left(r^{\prime }\right)}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }, \end{gathered}$$ (3)

where $k=\omega \sqrt{{\mu }_0{ϵ}_0}$ represents the wave number and $C$ stands for the coil support. Here, propagation effects are neglected because the wavelength of the TMS-induced fields is on the order of kilometers, while the maximum distance between the coil and any point in the head is less than a meter. As a result, the factor $k\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|$ will be orders of magnitude smaller than $2\pi$. The secondary E-field must satisfy standard source field relations and is

$$\begin{gathered} E_s\left(r\right)=-J\omega {\mu }_0{\int }_H\frac{J_s\left(r^{\prime }\right)e^{-jk\left|r-r^{\prime }\right|}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }+\frac{1}{J\omega {ϵ}_0}\nabla {\int }_H\frac{{\nabla }^{\prime }\cdot J_s\left(r^{\prime }\right)e^{-jk\left|r-r^{\prime }\right|}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime } \\ \approx \frac{1}{J\omega {ϵ}_0}\nabla {\int }_H\frac{{\nabla }^{\prime }\cdot J_s\left(r^{\prime }\right)}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }. \end{gathered}$$ (4)

Here again, we neglect propagation effects. Furthermore, the first term is neglected as it represents the relatively weak mutual inductive coupling between the coil and the head [31]. In other words, it is assumed that volume equivalent currents generate a negligible magnetic field relative to that of the coil.

The resulting integral equation is

$$\begin{gathered} E_p\left(r\right)=E\left(r\right)-E_s\left(r\right) \\ =E\left(r\right)-\frac{1}{j\omega {ϵ}_0}\nabla {\int }_H\frac{{\nabla }^{\prime }\cdot J_s\left(r^{\prime }\right)}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }. \end{gathered}$$ (5)

In the head, and for the range of frequencies of TMS, $\sigma \left(\mathbf{r}\right)/j\omega {ϵ}_0$ is much larger than $ϵ\left(\mathbf{r}\right)/{ϵ}_0$. As a result, we can approximate ${ϵ}_r\left(\mathbf{r}\right)\approx \sigma \left(\mathbf{r}\right)/j\omega {ϵ}_0$. Finally, we solve for the electric flux density to arrive at the final volume integral equation,

$$\begin{gathered} E_p\left(r\right)=\frac{D\left(r\right)}{{ϵ}_0{ϵ}_r\left(r\right)}-\frac{1}{{ϵ}_0}\nabla {\int }_H\frac{{\nabla }^{\prime }\cdot \left(1-\frac{1}{{ϵ}_r\left(r^{\prime }\right)}\right)D\left(r^{\prime }\right)}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime } \\ \approx j\omega \frac{D\left(r\right)}{\sigma \left(r\right)}+j\omega \nabla {\int }_H\frac{D\left(r^{\prime }\right)\cdot {\nabla }^{\prime }\frac{1}{\sigma \left(r^{\prime }\right)}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }. \end{gathered}$$ (6)

In the above equation, all terms have a $j\omega$ factor because the E-field induced during TMS is proportional to the time derivative of the coil currents. This factor can be canceled out from both sides of the equations, and for a non-time-harmonic coil current waveform $\mathbf{E}(\mathbf{r};t)=\frac{d}{dt}f\left(t\right)\mathbf{D}\left(\mathbf{r}\right)/\sigma \left(\mathbf{r}\right)$.

The maximum conductivity in the head is between about 1 to 1.7 S/m. Consequently, the magnitude of ${ϵ}_r\left(\mathbf{r}\right)$ can range from approximately 3 × 10⁶ to 3 × 10⁷ for frequencies between 1 to 10 kHz, respectively. The approximation of the volume integral equation (6) will result in an ill-conditioned system of equations due to the high-contrast breakdown for the VIE [26; 27; 32–34]. This ill-conditioning is related to the E-field jump conditions at the boundary between two tissue interfaces. The tangential component of the E-field must be continuous, and the normal component must jump proportionally to the jump in permittivity. Excluding potential physical resonances observed at high frequencies [33], which are not applicable here, this results in the VIE operator having two subspaces of solutions: a rotational subspace that corresponds to tangential fields and has spectrum clustering at one, and an irrotational subspace that corresponds to normal fields and has spectrum equal to each relative permittivity of the object [32–34]. Thus, when the electromagnetic scenario has high ratios of maximum to minimum permittivity, the VIE equation is ill-conditioned. (Note: The above statements only exactly apply when the VIE is written in terms of the E-field as an unknown. However, when dielectric flux or volume equivalence currents are used in the formulation, the same arguments can be made up to a multiplicative factor. As a result, the condition number of the VIE will remain unchanged, independent of E-field related unknown quantity.)

To remove this ill-conditioning, we consider the boundary condition between the scalp and air. The boundary condition dictates that the normal component of the flux must be continuous. In other words, on the scalp boundary

$$j\omega {ϵ}_0{E}^{+}\left(r\right)\cdot \hat{n}=\left(j\omega {ϵ}_0{ϵ}_{\mathit{\text{scalp}}}+{\sigma }_{\mathit{\text{scalp}}}\right)E^{-}\left(r\right)\cdot \hat{n},$$ (7)

where $\stackrel{\mathbf{ˆ}}{\mathbf{n}}$ is an outward-pointing normal on the scalp, ${\mathbf{E}}^{+}$ is the E-field just outside the scalp, and ${\mathbf{E}}^{-}$ is the E-field just inside the head, and ${ϵ}_{\text{scalp}}$ and ${\sigma }_{\text{scalp}}$ are the permittivity and conductivity, respectively. Since the scalp has a conductivity of 0.456 S/m, the term $j\omega {ϵ}_0$ is five to six orders of magnitude smaller than ${\sigma }_{\text{scalp}}$. Correspondingly, to achieve equality, ${\mathbf{E}}^{-}\left(\mathbf{r}\right)\cdot \stackrel{\mathbf{ˆ}}{\mathbf{n}}$ must be five to six orders of magnitude smaller than ${\mathbf{E}}^{+}\left(\mathbf{r}\right)\cdot \stackrel{\mathbf{ˆ}}{\mathbf{n}}$. This jump in the normal component of the E-field results in the ill-conditioned VIE for solving TMS.

As a result, the ill-conditioning is alleviated by approximating ${\mathbf{E}}^{-}\left(\mathbf{r}\right)\cdot \stackrel{\mathbf{ˆ}}{\mathbf{n}}\approx 0$. This is already typically assumed in TMS modeling (e.g., the FEM, FDM, and BEM formulations commonly used in the literature [12]). In the next section, we explain our discretization and how we impose this condition by choosing an expansion set for the electric flux that results in a zero normal component of the field on the scalp.

B. Matrix Representation of the VIE

A Galerkin procedure is employed to solve the VIE shown in (6). It is assumed that the head has been approximated by a volume mesh consisting of homogeneous tetrahedrons. We choose loop basis functions ${\mathbf{f}}_j\left(\mathbf{r}\right)$ (where $j=1,\dots ,N_e$ and $N_e$ is the total number of edges) to expand $\mathbf{D}$ at the edges of the mesh. These basis functions have zero divergence (i.e., $\nabla \cdot {\mathbf{f}}_j\left(\mathbf{r}\right)=0$) and, as a result, are a good candidate for approximating $\mathbf{D}$. The expansion is written as

$$D\left(r\right)=\sum _{j=1}^{N_e}{D}_j{f}_j\left(r\right).$$ (8)

Here, $D_j$ represents the unknown expansion coefficients. The loop basis functions ${\mathbf{f}}_j\left(\mathbf{r}\right)$ are the curl of vector first-order edge elements and have support on all tetrahedrons sharing the $j$-th edge of the mesh [35]. The basis function for the $j$-th edge of the mesh is defined as

$$f_j\left(r\right)=\sum _{t\in {𝒩}_j}\frac{L_t^j}{V_t}{f}_t\left(r\right),$$ (9)

where ${𝒩}_j$ is a set containing the index of all the tetrahedrons that share the $j$-th edge, ${\mathbf{L}}_t^j$ the vector along the edge of the $t$-th tetrahedron that does not share any point with the $j$-th edge, $V_t$ is the volume of the $t$-th tetrahedron, and $f_t\left(\mathbf{r}\right)$ is an indicator function that has a value of one if the position vector $\mathbf{r}$ lies inside the $t$-th tetrahedron and zero otherwise. In other words, the basis function defined on $j$-th edge is piece-wise constant within each tetrahedron sharing the $j$-th edge and points along the vector edge connecting the two faces sharing the $j$-th edge as a common edge as shown in Fig. 1. In panel (a) of Fig. 1, a mesh is shown with tetrahedrons shaded in blue. In this panel, two edges are selected as the parent edges to visualize their basis functions within their surrounding tetrahedrons. The two parent edges are shown in thick green lines in this figure. In panels (b) and (c), the basis functions of the two parent edges are shown in black arrows. The parent edge ${\mathbf{e}}_1$ is shown in Fig. 1(b), and its connected tetrahedrons with ids ${𝒩}_1$ and faces with ids ${ℱ}_1$ are shaded with light blue and red, respectively. In this case, the parent edge is connected to tetrahedrons $T_3,T_4,T_5$, and $T_7$ (i.e., ${𝒩}_1=\{\mathrm{3,4},\mathrm{5,7}\}$) and triangles $S_3,S_5,S_7$, and $S_8$ (i.e., $\left.{ℱ}_1=\{\mathrm{3,5},\mathrm{7,8}\}\right)$. In Fig. 1(b), the parent edge ${\mathbf{e}}_2$ is connected to tetrahedrons with ids ${𝒩}_2=\{\mathrm{1,2},6\}$ and triangles with ids ${ℱ}_2=\left\{\mathrm{9,10,11}\right\}$.

Fig. 1:

A small piece of a mesh with 9 tetrahedrons, shaded in blue, is shown in (a). Two distinct edges are indicated with blue lines. The respective basis functions of the two blue edges are represented by black arrows at some sample points of the tetrahedrons in panels (b) and (c). The tetrahedral interfaces are shaded in red in these two panels.

Testing the VIE in equation (6) with $\sigma \left(\mathbf{r}\right){\mathbf{f}}_i$, where $i=\mathrm{1,2},\dots ,N_e$, we arrive at the equation

$$\begin{gathered} {\int }_H\sigma \left(r\right)f_i\left(r\right)\cdot E_p\left(r\right)dr=j\omega {\int }_H\sigma \left(r\right)f_i\left(r\right)\cdot \frac{D\left(r\right)}{\sigma \left(r\right)}dr+j\omega {\int }_H\sigma \left(r\right)f_i\left(r\right)\cdot \nabla {\int }_H\frac{D\left(r^{\prime }\right)\cdot {\nabla }^{\prime }\frac{1}{\sigma \left(r^{\prime }\right)}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }dr \\ =j\omega {\int }_H\sigma \left(r\right)f_i\left(r\right)\cdot \frac{D\left(r\right)}{\sigma \left(r\right)}dr-j\omega {\int }_H\nabla \cdot \sigma \left(r\right)f_i\left(r\right){\int }_H\frac{D\left(r^{\prime }\right)\cdot {\nabla }^{\prime }\frac{1}{\sigma \left(r^{\prime }\right)}}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }dr. \end{gathered}$$ (10)

Plugging in the expansion (8) results in a linear system of equations as

$$\begin{gathered} A\cdot \text{x}=b \\ {A}_{ij}=-\sum _{t\in {𝒩}_i}\frac{L_t^i\cdot L_t^j}{V_t}+\sum _{q\in {ℱ}_i}\sum _{p\in {ℱ}_j}[\left(\frac{1}{{\sigma }_p^{+}}-\frac{1}{{\sigma }_p^{-}}\right)\left({\sigma }_q^{+}-{\sigma }_q^{-}\right){\int }_{S_q}{\int }_{S_p}\frac{1}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }dr] \\ x=\left(D_1,D_2,\dots ,D_{N_e}\right) \\ {b}_i={\mu }_0\sum _{t\in {𝒩}_i}\frac{L_t^i}{V_t}{\sigma }_t{\int }_{T_t}dr{\int }_C\frac{J_{\text{coil}}\left(r^{\prime }\right)}{4\pi \left|r-r^{\prime }\right|}d{r}^{\prime }. \end{gathered}$$ (11)

Here, ${\sigma }_q^{+}$ and ${\sigma }_q^{-}$ are the conductivities of each of the two tetrahedrons sharing face $S_q$, and $T_t$ is the $t$-th tetrahedron. Additionally, ${\sigma }_q^{+}$ and ${\sigma }_q^{-}$ are chosen from the tetrahedron $L_t^i$ points towards and away from, respectively, and ${\sigma }_p^{+}$ and ${\sigma }_p^{-}$ are chosen from the tetrahedron $L_t^j$ points towards and away from.

When calculating $A_{ij}$, if the observation point lies within the source triangle face (i.e., $S_q=S_p$), we use the analytical method of [36] to evaluate the integrals. The method is discussed in the first section of the Supplementary Material. On the other hand, if the observation point is positioned outside the source interface, the integral is computed using a three-point Dunavant quadrature rule defined in [37]. Additionally, we solve the matrix iteratively using a TFQMR iterative solver, and we accelerate our code using FMM libraries freely available online [38].

C. Error metrics

To obtain a quantitative measure of the accuracy of each method, the relative error is defined as

$${\mathit{\text{Err}}}_{L^2}=100\sqrt{\frac{{\int }_H{\Vert E_M\left(r\right)-E_R\left(r\right)\Vert }^{2}dr}{{\int }_H{\Vert E_R\left(r\right)\Vert }^{2}dr}}\left(\text{\%}\right).$$ (12)

Here, ‖∥⋅‖∥ represents the magnitude, ${\mathbf{E}}_R\left(\mathbf{r}\right)$ refers to the reference E-field solution, and ${\mathbf{E}}_M\left(\mathbf{r}\right)$ refers to the calculated E-field. This relative error is calculated for different methods to determine the TMS-induced E-fields. To define a local error, we use the error metric

$${\mathit{\text{Err}}}_{\mathit{\text{loc}}}\left(r\right)=100\frac{\Vert E_M\left(r\right)-E_R\left(r\right)\Vert }{{\mathit{\text{max}}}_{r\in H}\Vert E_R\left(r\right)\Vert }\left(\text{\%}\right).$$ (13)

III. Results and Discussion

In the last section, we derived and discretized the VIE required for determining the E-field induced during TMS. Here, we present numerical results indicating the validity of our formulation for analyzing TMS-induced E-fields. We compare the proposed VIE method with well-established methods such as FEM, BEM, and analytical solutions available from [12]. All numerical experiments were done using an AMD Rome CPU with 2.0 GHz, 32-core processor and 250 GB of memory.

A. Spherical 4-Layer Head Model

In this section, we consider a concentric four-layer spherical head model as depicted in Fig. 2. The spherical compartments are homogeneous and have boundaries at radii of 78, 80, 86, and 92 mm, respectively. The conductivities of these four regions, from the innermost to the outermost, are 0.33, 1.79, 0.01, and 0.43 S/m. The coil is also placed 5 mm above and centered about the apex of the spherical head model. The coil is composed of two electrically connected copper loops. Each loop has 9 turns and 17 layers, with inner and outer radii of 26 mm and 44 mm, respectively. The coil is assumed to be driven with a peak current of 3.5 kA and pulse frequency of 3 kHz. This results in a peak $dI/dt=2\pi \left(3\text{kHz}\right)\left(3.5\text{kA}\right)=6.6\times {10}^7\text{A}/\text{s}$.

Fig. 2:

The predicted E-field along the z-axis derived using different methods. “ANAT” refers to the analytical solution. In the main figure, four rectangular regions, shown as panels (a), (b), (c), and (d), are selected and magnified in the subsequent panels to illustrate the algorithm’s performance at various conductivity boundaries. The regions are labeled within the panels, and the boundaries are indicated by dashed red lines.

The results for the predicted E-field along the z-axis are shown in Fig. 2. VIE, BEM, and the second-order FEM agree well with the analytical solution. Additionally, the cross-sectional plots of the E-field magnitudes for VIE, BEM, FEM, and the analytical solution are provided in Fig. 3. It is evident that the VIE matches well with the analytical solution. The setup time (the time required to populate the matrix equation) and the solving time (the time taken to solve the matrix equation and sample the E-fields) for the different methods are depicted in Table I.

Fig. 3:

The predicted E-field distribution on the X-Z plane derived using VIE, second-order FEM, BEM, and analytical method.

TABLE I:

The setup time (the time to populate the matrix equation) and solving time (the time to solve the matrix equation and sample the E-fields) for implementing VIE, BEM, and FEM for the spherical head model.

CPU Time	FEM	BEM	VIE
Setup	0.27 sec	0.23 sec	6.43 sec
Solving	9 sec	23 sec	60 sec

The VIE for this test case was approximately three times slower than the BEM. The relative $L^2$ errors, calculated using (12) (with the analytical solution as the reference), for BEM, FEM, and VIE are 2.92%, 11.54%, and 5.08%, respectively. The relative error for the VIE highlights its high accuracy. Histograms of the point-wise error at different layers of the spherical model, along with the errors in the hot-spot regions of the layers, where the amplitude of the predicted E-field exceeds 70% of the maximum E-field, are given in Fig. 4. The maximum and mean local errors across all compartments for the VIE are 10.29% and 3.8%, respectively. The VIE accurately solves the TMS equations. Furthermore, the BEM method exhibited the closest agreement with the analytical solution, particularly within the innermost layer, which models the gray matter and is of greater importance in TMS. As such, it is used as the reference solution in the next section.

Fig. 4:

The local error distribution, calculated using (13), is shown for different layers of the spherical model. Panel (a) presents the error values across the entire regions, while panel (b) shows the errors in the hot-spot regions of the layers, where the predicted E-field amplitude exceeds 70% of the maximum E-field.

B. Ernie Head Model

In this section, a realistic head model is selected to analyze the performance of the proposed VIE method. The head model is composed of five different compartments: white matter (WM), gray matter (GM), cerebrospinal fluid (CSF), skull, and skin. The conductivities for these five regions are chosen to be 0.126 S/m, 0.275 S/m, 1.654 S/m, 0.01 S/m, and 0.465, respectively. The head model is shown in Fig. 5. However, only the outer layer and the gray matter are shown in this figure.

Fig. 5:

The predicted E-field along the peak E-field axis derived using BEM and VIE methods. In the main figure, four rectangular regions shown as panels (a), (b), (c), and (d) are selected and magnified in the following panels to show the performance of the algorithm at different conductivity boundaries. Different regions are labeled in the panels and the boundaries are shown by dashed red lines.

The TMS coil is placed 5 mm above the head and approximately centered above the motor cortex as shown in Fig. 5. This figure shows the predicted E-field in the head along the peak E-field axis. The VIE results are compared to the BEM in this figure. We used both the regular mesh and the first-order refined mesh to calculate the BEM, providing two reference BEM solutions to validate the VIE. The results are magnified in panels (a) to (d) of Fig. 5, demonstrating good agreement between the VIE and the reference methods at the conductivity boundaries. The relative $L^2$ errors for the VIE are 1.21% and 1.67% when compared to the BEM and the BEM with a refined mesh, respectively. The E-field distributions in the brain, calculated using the VIE and BEM (with a refined mesh), along with the local error distribution, are shown in panels (a) to (c) of Fig. 6. Additionally, the box plot representation of the error across the entire brain region and in the hot-spot region (where the amplitudes are higher than 70% of the maximum amplitude) is shown in panels (d) and (e). The maximum and mean point-wise errors of the VIE, calculated using the local error definition in (13), are 6.94% and 0.14%, respectively. Overall, the VIE accurately determines the TMS-induced E-field in an MRI-derived head model.

Fig. 6:

The predicted E-field distribution on the brain calculated using (a) VIE and (b) BEM with the first-order refined mesh. Panel (c) shows the point-wise error plot acquired using (13) on the brain. The BEM solution with the first-order refined mesh is selected as the reference field ${\mathbf{E}}_R$, and the VIE solution is selected as ${\mathbf{E}}_M$ in the equation. Panel (d) shows the box-plot for the error in the different regions of the head model. Panel (e) shows the errors related to the hot-spot region (the region with E-fields higher than 70% of the maximum E-field).

The CPU times for the setup and solving phases (as defined in the previous section) of the simulations are given in Table. II. The main reason for the high CPU time of the VIE is that we have used loop basis functions that results in increasing condition number with increasing number of unknowns. Future works, will use standard Schaubert Wilton Glisson (SWG) basis functions to remove this increase in iterations at the cost of having to compute volume integrals for charges [36] and only numerically satisfying the divergence free condition of the flux [39].

TABLE II:

The setup time (the time to populate the matrix equation) and solving time (the time to solve the matrix equation and sample the E-fields) for implementing VIE, BEM (labeled as BEM 1), and BEM with a first-order refined mesh (labeled as BEM 2).

CPU Time	VIE	BEM 1	BEM 2
Setup	13 min	1 min 9 sec	7 min 19 sec
Solving	3 h 32 min	7 min 34 sec	53 min 2 sec

The VIE method is also compared to FEM of different orders for ten different coil locations, and the results are presented in the second section of the Supplementary Material. The results show good agreement between VIE, BEM, and third-order FEM. Moreover, the coil is considered tangential to the scalp in this paper. However, in some cases, the coil may not be tangential. When the coil is inclined with respect to the scalp’s tangential plane, an additional charge build-up occurs on the scalp. Therefore, the VIE solver should be validated for non-tangential coil configurations. This behavior is thoroughly studied in the third section of the Supplementary Material. It is observed that the proposed VIE solver behaves similarly to the BEM for different coil angles.

IV. Conclusion

In this paper, a VIE approach for TMS analysis is introduced. The results demonstrate that this method matches the analytical solution of a spherical head model and with BEM for an MRI-derived 5-compartment head model. The error is lower in inner compartments, such as gray and white matter, which are of greater importance. In addition to providing an accurate solution to the TMS problem, the new VIE does not require the addition of projectors or auxiliary surface variables, making it more practical than other VIE methods used to analyze low-frequency high-contrast phenomena. This VIE provides progress towards realizing practical inclusion of anisotropies in white matter to TMS simulations using integral equations.