Med Biol Eng Comput DOI 10.1007/s11517-015-1301-z

ORIGINAL ARTICLE

The effect of tissue anisotropy on the radial and tangential components of the electric field in transcranial direct current stimulation Mohamed K. Metwally1 · Seung Moo Han1 · Tae‑Seong Kim1 

Received: 31 December 2013 / Accepted: 23 April 2015 © International Federation for Medical and Biological Engineering 2015

Abstract  Transcranial direct current stimulation (tDCS) is considered to be a promising technique for noninvasive brain stimulation and brain disease therapy. Recent studies have investigated the distribution of the electric field (EF) magnitude over gyri and sulci and the effect of tissue homogeneity with isotropic electrical conductivities. However, it is well known that the skull and white matter (WM) are highly anisotropic electrically, requiring investigations of their anisotropic effects on the magnitude and the directional components of the induced EF due to the high dependency between neuromodulation and the EF direction. In this study, we investigated the effects of the skull and WM anisotropy on the radial and tangential components of the EF via gyri-specific high-resolution finite element head models. For tDCS, three configurations were investigated: the conventional rectangular pad electrode, a 4(cathodes) +1(anode) ring configuration, and a bilateral configuration. The results showed that the skull anisotropy has a crucial influence on the distribution of the radial EF component. The affected cortical regions by the radial EF were reduced about 22 % when considering the skull anisotropy in comparison with the regions with the skull isotropy. On the other hand, the WM anisotropy strongly alters the EF directionality, especially within the sulci. The electric current tends to flow radially to the cortical surface with the WM anisotropy. This effect increases the affected cortical areas by the radial EF component within the sulcal regions. Our results suggest that one must examine the * Tae‑Seong Kim [email protected] 1



Department of Biomedical Engineering, College of Electronics and Information, Kyung Hee University, 1 Seocheon‑dong, Giheung‑gu, Yongin‑si, Gyeonggi‑do 446‑701, Republic of Korea

distribution of the EF components in tDCS, not just the magnitude of the EF alone. Keywords  Transcranial direct current stimulation · Neuromodulation · Finite element simulation

1 Introduction Transcranial direct current stimulation (tDCS) is a promising technique for noninvasive brain stimulation. It modulates the membrane potentials of cortical neurons by injecting weak direct or slowly varying currents (less than 2 mA with a frequency less than 10 kHz) into the brain using electrodes positioned on the scalp [23]. The polarity of the injected current affects cortical excitability, with anodal stimulation increasing cortical excitability and cathodal stimulation inhibiting it [34]. tDCS is known to relieve the symptoms of depression [24], stroke [16, 35], Parkinson’s disease [12], and epilepsy [13]. Although tDCS is considered to be a promising therapeutic technique, the induced electric field (EF) inside the brain is not yet well understood. There are many factors that have to be considered for effective brain stimulation such as the electrode size, amount of injected current, head geometry [7, 30], tissue properties [30], and electrode configuration, which control the orientation and distribution of the EF. However, it is difficult to study the distribution of EF and the influence of tissue heterogeneity or anisotropy within the brain in vivo. Instead, scientists and clinicians extensively use numerical simulations to investigate the effect of tDCS. Most tDCS studies use finite element method (FEM) as a numerical method to solve the partial differential equations that govern tDCS because of its capability to handle complex and arbitrary-shaped

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geometries and its ability to preserve the boundaries between adjacent regions of the head [18]. Several types of head models have been used in tDCS investigations. Earlier models for representing the head and brain tissues included simplified concentric spheres [21]. Later, MRIderived high-resolution head models were used that took into consideration the geometrical complexity of the head [30]. Recently, even gyri-specific head models have been created, preserving the precise boundaries of the gyri and sulci [7]. Using these gyri-specific models, several studies have investigated the distribution of the EF on the gyri and sulci, finding that the electric current concentrates on the gyri edges [5], which may lead to an inhomogeneous influence of the sulci and gyri on the EF distribution. These findings and other studies highlight the importance of considering the gyral folding patterns to gain more realistic knowledge of the effect of noninvasive brain stimulation methods such as tDCS and transcranial magnetic stimulation (TMS) [3, 17]. Most of the tDCS simulation studies have focused on the distribution of the EF magnitude and the isotropic conductivity of the skull and white matter (WM) tissues [7, 21, 22, 29]. However, research has proven that both the skull and WM tissues are highly anisotropic in their electrical conductivity, which has a large effect on the EF distribution. It was shown that the skull anisotropy could result in a shunting effect, which increases the focality of the EF distribution over the cortex and dampens the maximum EF value, while the WM anisotropy could alter the EF distribution without affecting the maximum EF value [20, 30]. On the other side, recent in vitro studies have examined the impact of the EF orientation on neuromodulation. They reported that neurons are mainly affected by the EF that is parallel to their orientations. In other words, the neuronal segments that are oriented toward the stimulating anode have been shown to hyperpolarize, while the segments toward the cathode depolarize [25, 27]. However, if the neuron is oriented perpendicular to the EF, there will be no polarization effect. Accordingly, it is particularly relevant to investigate the directional components of the EF. Lately, it was proved numerically in a TMS study that the transverse EF (with respect to the neuron orientation) has low efficiency to stimulate neurons, yet a modification to the TMS cable equation can be done to improve the impact of the transverse field for nerve stimulation [37]. However, the EF in tDCS is induced in a different way than TMS. Thus, it was important to look at the distribution of the directional EF components in tDCS. Only one tDCS study was conducted by investigating the effect of tissue heterogeneity on the distribution of the EF components [22]. The results highlighted the potential of studying the EF orientation to explain the opposite effects of the anodal and cathodal tDCS. The EF vector was divided into two components: (1)

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a radial EF in the normal direction to the cortical surface and (2) a tangential EF in the perpendicular direction to the radial EF. The study showed that the high conductivity of CSF causes a high radial EF within sulci and a weaker tangential EF over the gyri and that using small anode electrode improves the focality of the tangential EF distribution. However, this study did not consider the anisotropic properties of the skull and WM. Since the skull and WM tissues are highly anisotropic in their electrical conductivity and the directional components of the induced EF would have differential effects on tDCS, our motivation was to investigate the effect of the anisotropic conductivities in the skull and WM tissues on the distribution of the EF components to obtain better insights into the effect of tDCS. This paper aims to investigate the directional components of the tDCS-induced EF while considering the anisotropic properties of the skull and WM in three different electrode configurations via a 3-D MRI-derived high-resolution gyri-specific FE head model. In our tDCS setup, we used three tDCS configurations stimulating the motor cortex to show the impact of electrode configurations on the EF orientation: (1) the conventional rectangular pad electrode, (2) a 4(cathodes) +1(anode) ring electrode configuration, and (3) a bilateral configuration. The EF distribution was examined in terms of the maximum-induced EF magnitude and the maximum of each EF component. Our results show the significant effect of skull anisotropy on the distribution of the radial EF by improving its focality and the role of the WM anisotropy by expanding the affected cortical surface by the radial EF.

2 Methods 2.1 Generation of 3‑D gyri‑specific FE head models Our head model was generated using a set of 146 MR image slices acquired by a 3-T MRI scanner from an Asian healthy male volunteer of 38 years old with no prior medical record. Each slice had a dimension of 212 × 181 with a voxel volume of 1 mm3. The head model was segmented into five tissues: WM, gray matter (GM), cerebrospinal fluid (CSF), skull, and scalp. The WM and GM were segmented using Freesurfer [15], while the other tissues were segmented using morphological techniques including dilation and erosion with some manual refinements [10]. Meshing was carried out using ISO2MESH [11]. The head model consisted of 1,077,808 elements and 181,846 nodes in the rectangular pad montage, 1,066,222 elements and 179,872 nodes in the 4 + 1 ring montage, and 1,018,394 elements and 169,347 nodes in the bilateral montage with their element quality [2] in a range from 0.55 to 0.86 for all montages. Figure 1 shows a schematic diagram of the steps

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Fig. 1  A schematic diagram showing the steps of building the head models and running the analysis

Fig. 2  3-D rendered tDCS montages a the rectangular pad montage with an anode (in red) positioned at C3, a cathode (in blue) over the right eyebrow, b the 4 + 1 ring electrodes montage with its anode (in red) at C3 and four cathodes (in blue) at the corners of the assumed rectangular pad electrode, and c the bilateral configuration with its anode (in red) at C3 and cathode (in blue) at C4 (color figure online)

of generating the head model and the steps of the analysis that were conducted in this study. 2.2 tDCS montages In tDCS, the electrode configuration affects the EF distribution and orientation. Thus, we have investigated three different tDCS montages in this study: (1) a standard pair of rectangular pad electrodes, (2) a set of 4(cathodes) +1(anode) ring electrodes, and (3) a pair of bilateral cylindrical electrodes. The first montage is the most widely used tDCS setup [6, 35] in stimulating the motor cortex but suffers from low focality in the induced EF. Lately, the use of multiple electrodes as anodes and cathodes has been suggested to improve the stimulation focality in some specific cortical regions [7, 9, 31]: this is known as high-definition

tDCS (HD-tDCS). In this study, two HD-tDCS montages were selected: the 4(cathodes) +1(anode) ring electrodes and a pair of bilateral cylindrical electrodes. The multiple ring electrodes configuration has a potential to enhance tDCS, given that it has been clinically proven to have a large effect at decreasing the sensory threshold for heat and cold in a pain perception study [4]. The bilateral montage also demonstrated an ability to significantly enhance the motor function for stroke patients in a pilot clinical study [14]. Figure 2 shows the rendered tDCS setup for each montage. The rectangular pad electrodes were 5 × 7 cm2 in size. The anode was centered at the electrode position of C3 of the international 10–20 EEG system, while the cathode was over the right eyebrow to stimulate the motor cortex. For the 4 + 1 ring montage, the anode was placed over C3 and

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Table 1  Tissues conductivity values in each model

Med Biol Eng Comput Model

Scalp (S/m)

Skull (S/m) Radial

Tangential

GM (S/m)

WM (S/m) Longitudinal

Transverse

Isotropic

0.33

0.0132

0.0132

1.79

0.33

0.14

0.14

Aniso-I

0.33

0.00424

0.0233

1.79

0.33

0.14

0.14

Aniso-II

0.33

0.0132

0.0132

1.79

0.33

0.65

0.065

Aniso-III

0.33

0.00424

0.0233

1.79

0.33

0.65

0.065

the cathodes at the four corners of an assumed rectangular pad. In the bilateral montage, the anode was placed at C3 while the cathode at C4. In both montages, each electrode had a radius of 4 mm and a height of 2 mm. 2.3 Isotropic and anisotropic conductivities in the skull and WM Three types of conductivity settings were used for each montage. The first conductivity setting considered the isotropic conductivities for all tissues and was named as the Isotropic model. The second setting was based on the skull anisotropy, while the other tissues were kept isotropic and named as the Aniso-I model. The third setting considered the WM anisotropy, while the other tissues including the skull kept isotropic and named as the Aniso-II model. The fourth setting considered both the skull and WM anisotropy, while the other tissues were kept isotropic and named as the Aniso-III model. These four settings were selected to investigate the influence of skull and WM anisotropy on the EF components. The effect of skull anisotropy can be observed by comparing the results of Isotropic with Aniso-I, while comparing the results of Aniso-II and Isotropic depicts the effect of WM anisotropy. Comparing the results of Aniso-III with Isotropic shows the combined influence of skull and WM anisotropy on the EF components. The following isotropic conductivity values were set in the Isotropic model of each montage: WM = 0.14 S/m, GM = 0.33 S/m, CSF = 1.79 S/m, skull = 0.0132 S/m, and scalp = 0.33 S/m [19, 36]. In the case of the anisotropic models, the anisotropic conductivity tensor, σ either for the skull or WM was calculated in a quadratic form,   σ1 0 0 σ = S −1  0 σ2 0 S (1) 0 0 σ3 where S is the eigenvector matrix, and σi is the conductivity value where σ1 > σ2 > σ3. For the skull anisotropy, it is known that the skull has three layers with different conductivities: the inner compacta, spongiosa, and outer compacta. Based on the investigations of Akhtrai et al. [1] and Rampersad et al. [26] on the effect of multilayered skull models on the homogenous

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CSF (S/m)

isotropic and anisotropic skulls in tDCS using a spherical model, it was shown that a single-layer skull with the anisotropic conductivity is the closest approximation to the three-layer skull model. Therefore, in this study, the skull was simulated as a single layer with the anisotropic conductivity. We set the skull to have an anisotropic ratio of 1:5.5 (radial: tangential) and then assigned the conductivity values of 0.00424 and 0.0233 S/m to the radial and tangential directions of the skull, respectively. This ratio is equivalent to the maximum ratio between the conductivity values of the spongiosa and compacta layers as used in Rampersad et al. [26] and covers the maximum and minimum values noted by Akhtari et al. [1]. For the WM anisotropy, the diffusion tensors from DT-MRI were used to obtain the electrical conductivity tensors [19]. Based on the assumption that the diffusion and conductivity tensors share the same eigenvectors and according to the volume constraint algorithm [36], we assumed that the WM conductivities parallel to the neural fiber directions (i.e., longitudinal) were 10 times larger than those in the normal direction (i.e., transversal) [36]. This ratio is based on the review by Sadleir et al. [28] on the WM anisotropic conductivities in which the optimal ratio between the geometric means of the longitudinal to the transverse conductivity was 10:1. The anisotropic conductivity values for the WM were 0.65 and 0.065 S/m in the longitudinal and transverse directions, respectively, which have been previously used in [20, 30, 31, 36]. Table 1 summarizes the tissue conductivity properties of all models. 2.4 tDCS simulation An electrical current of 1 mA was injected into the anode and extracted from the cathode in the rectangular pad montage [7]. In the 4 + 1 ring montage, a current of 2 mA was injected by the anode and 0.5 mA was extracted by each cathode [7]. Finally, in the bilateral montage, a current of 2 mA was injected by the anode and extracted by the cathode. The conductivity of the electrodes was assigned as 5.8 × 107 S/m [7]. The EF distribution in the head was computed by solving the following quasi-static Laplace equation,

∇ · (σ ∇V ) = 0

(2)

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where V is the electric potential and σ is the electrical conductivity of the tissue. The Neumann boundary condition on the outer surface of the head is defined as follows, (3)

(σ ∇V ) · n˜ = 0

n is the unit normal vector to the outer surface of where  the head. The sparse direct equation solver based on the direct elimination of the equation was used to compute the induced EF in the whole head. To investigate the distribution of the EF components over the brain cortex, the radial direction within the GM was defined to be in the direction of the pyramidal cells, → which was defined to be in the normal direction (− nr ) to the gyri or sulci surfaces pointing toward the CSF, and it was computed according the following formula, − → → EFradial = EF · − nr

(4)

The tangential component was defined to be the second norm of the EF components within the perpendicular plane to the radial direction, and it was computed as follows, −  →  → EFtan g = EF − EFradial · − nr  (5) 2

→ where − nr is a unit vector. 2.5 Analysis criteria

In the analysis of the effect of tDCS on our models, we performed regional analysis on the brain cortex. We defined the affected regions either by the EF magnitude or by any of the individual components as regions that encompass nodal EF values higher than the half maximum of the induced EF magnitude, as reported by Deng et al. [8]. There are four regions of EF components defined based on this threshold: (1) a region affected by the radial EF only because the norms of the tangential components were less than the threshold, (2) a region affected by the tangential EF only where the values of the radial EF were less than the threshold, (3) a region affected by both of the EF components, and (4) a region where the values of the EF components were less than the threshold. The influence of tissue anisotropy was evaluated using the following criteria: (1) the maximum value of each of the EF magnitude and components, (2) the distribution of the EF magnitude and components over the brain cortex, and (3) the area of the affected region by either the EF magnitude or EF components. The relative change in the observed features (such as the maximum EF or the affected areas) after considering the tissue anisotropy was computed according to the following equation,

Change in percent =

where Fiso is the measured feature in the Isotropic model, and Fm is the measured feature when the anisotropic conductivity is considered.

Fm − Fiso × 100 Fiso

(6)

3 Results The EF magnitude distributions over the cortical surface in the four models (Isotropic, Aniso-I, Aniso-II, and AnisoIII) over the three montages are shown in Fig. 3. The distribution maps of the EF components corresponding to the results in Fig. 3 are shown in Fig. 4. The quantitative measurements from Figs. 3 and 4, including the maximum EF (magnitude, radial, and tangential component) on the cortical surface of each model and the areas of the affected regions, are summarized in Fig. 5. The main cortical regions where the skull anisotropy alters the distribution of the EF magnitude and its components can be observed in Fig. 6, where the absolute difference maps between the EF distributions on the cortex in Isotropic and Aniso-I were computed. In order to analyze the effect of WM anisotropy on the distribution of the EF magnitude and its components over the brain cortex where this effect mostly takes place, Fig. 7 shows the absolute difference maps between the EF distributions on the cortex from Aniso-II and Isotropic. In order to understand how the WM anisotropy affects the distribution of the EF components, the flow of the electric current within the coronal and sagittal slices of the Isotropic and Aniso-II models of each montage is shown in Fig. 8. At last, the absolute difference maps between Isotropic and Aniso-III are shown in Fig. 9 in order to realize how the combined effect of skull and WM anisotropy affects the distribution of the EF components. 3.1 The effect of skull anisotropy 3.1.1 The distribution of EF magnitude To examine the effect of skull anisotropy, we compared the results of Isotropic versus Aniso-I with regard to the EF distribution over the cortex in both models and the distribution map of the EF components. Comparisons were made for these two models in each of the three montages (i.e., rectangular pad, 4 + 1 ring, and bilateral electrodes) to show the effect of the montages on the distribution of the EF components. It was observed that the skull anisotropy reduced the distribution of the EF on the cortical surface. Note the reduction in the distribution of the EF magnitude in the case of the rectangular pad montage by comparing Fig. 3b of Aniso-I to Fig. 3a of Isotropic. Note the focality enhancement in the case of the ring montage after considering the skull anisotropy as shown in Fig. 3f of Aniso-I as compared

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Fig. 3  Distribution of the EF magnitude over the brain cortical surface in all models. The first row a–d is from the rectangular pad montage; the second row e–h from the ring montage; and the third row i–l

from the bilateral montage. The first column a, e, and i Isotropic; the second column b, f, and j Aniso-I; the third column c, g, and k AnisoII; and the fourth column d, h, and l Aniso-III models

to Isotropic in Fig. 3e. The enhanced focality could also be observed in the bilateral montage by observing Fig. 3i, j of Isotropic and Aniso-I, respectively. In order to quantify the distribution of the EF magnitude, the affected area in each model was calculated as shown in the plots of Fig. 5. In the case of the rectangular pad montage, the reduction in the affected region was on a degree of 16 % with respect to the affected area in the Isotropic model (i.e., the first four columns in Fig. 5a). In the case of the ring montage, the affected region by the EF magnitude was reduced about 10 % after considering the skull anisotropy (i.e., the first four columns in Fig. 5c). In the bilateral montage, the focality was enhanced 4 % in the Aniso-I model after considering the skull anisotropy (i.e., the first four columns in Fig. 5e).

These values showed that the skull anisotropy enhanced the focality on average about 10 % more than in the case of the skull isotropy. However, the maximum-induced EF was reduced as well, as shown in the plots of Fig. 5. It was observed in the case of the rectangular pad montage that the maximum-induced EF was reduced 17 % less in Aniso-I after considering the skull anisotropy (i.e., the first four columns in Fig. 5b). This reduction was about 40 % in the case of the ring montage (i.e., the first four columns in Fig. 5d), and it was about 18 % in the case of the bilateral montage (i.e., the first four columns in Fig. 5f). This observation seems to be due to the shunting effect of skull anisotropy as reported in [32]. It was observed that the maximum EF exists in the vicinity of the anodes and over the gyri in all Isotropic models. However, it seems that

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Fig. 4  Affected regions according to two EF components (i.e., the radial EF and the tangential EF) over the brain cortical surface in all models where the radial (blue) and tangential EF components exceed 50 % of their maximum EF magnitudes. The first row a–d is from the rectangular pad montage; the second row e–h from the ring montage;

and the third row i–l from the bilateral montage. The first column a, e, and i Isotropic; the second column b, f, and j Aniso-I; the third column c, g and k Aniso-II; and the fourth column d, h and l Aniso-III models (color figure online)

the skull anisotropy has an influence on the location of the maximum EF where it was shifted toward the cathode in all Aniso-I models, which has to be considered during tDCS to achieve its maximum influence on the target region. Analyzing the main cortical regions where the skull anisotropy has a high effect is possible by observing Fig. 6. Figure 6a, d, and g shows that the influence of skull anisotropy on the distribution of EF magnitude was always over the gyri surfaces.

3.1.2 The distribution of EF components In terms of the distribution of the EF components, it was observed in all Isotropic and Aniso-I models in Fig.  4 (i.e., Fig. 4a, b for the rectangular pad montage, Fig.  4e, f for the ring montage, and Fig. 4i, j for the bilateral montage) that the tangential EF component covered most of the gyri in the area between the electrodes, although the radial component had a different

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Fig. 5  Plots of the quantitative measurements of the EF distribution over the brain cortical surface. The plots a and b are from the rectangular pad montage, c and d from the 4 + 1 ring montage, and e and f from the bilateral montage. The plots a, c, and e show the area of the affected regions either by the EF magnitude, radial EF, or tangential EF. The plots b, d, and f show the maximum value of the EF magni-

tude and its components. The blue bars the Isotropic models, the red bars the Aniso-I models, the green bars the Aniso-II models, and the brown bars the Aniso-III models. In each plot, the first four columns the EF magnitude, the middle four columns the radial EF, and the last four columns the tangential EF (color figure online)

distribution based on the electrode shape. In the case of the rectangular montage, the radial EF existed obviously within the deep sulci and over the walls of the sulci in both models (i.e., Isotropic and Aniso-I), as shown in Fig.  4a, b, respectively, whereas in the case of the circular electrodes, it was obvious in the Isotropic models of the ring and bilateral montages (i.e., Figure 4e, i, respectively) that the radial EF existed on the gyri under the anode center and also within the walls of the sulci [22]. This observation was also applied in the Aniso-I models in Fig. 4e, h. It was also noticed that the skull anisotropy more critically affected the radial EF distribution than the

tangential EF. In the rectangular pad montage, note the reduction in the affected sulci regions by the radial EF in the frontal lobe after considering the skull anisotropy in Fig. 4b with respect to those regions in Fig. 4a of the Isotropic model. In the ring montage, the reduction in the affected gyri region by the radial EF in the AnisoI model was observed with respect to the distribution of the radial EF in the Isotropic model, as shown in Fig.  4f, e, respectively. The same observation could be noticed for the radial EF distribution under the anode in the bilateral montage, whereas Fig. 4j of the Aniso-I model shows vanishing of the radial EF after considering the skull anisotropy with respect to the wide area

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Med Biol Eng Comput Fig. 6  Absolute difference maps between Isotropic and Aniso-I in all models. The first row a–c is from the rectangular pad montage, the second row d–f from the ring montage, and the third row g–i from the bilateral montage. The first column a, d, and g indicates the EF magnitude difference, the second column b, e, and h the radial EF difference, and the third column c, f, and i the tangential EF difference

affected by the radial EF in the case of Fig. 4i of the Isotropic model. The regions affected by the EF components were computed, as shown in the plots of Fig. 5. The affected regions by the radial EF in the case of the rectangular pad montage were reduced with consideration of the skull anisotropy about 25 % less than the area in the Isotropic model (i.e., the middle four columns in Fig. 5a), while this percentage in the tangential component was 15 %, as shown in Fig. 5a in the last four columns. In the ring montage, the areas of the radial EF were reduced about 28 %, as shown in Fig. 5c in the middle four columns, whereas the areas of the tangential EF were 13 % less with consideration of the skull anisotropy (i.e., the last four columns in Fig. 5c). The bilateral montage showed a reduction in the affected areas by the radial and tangential EFs by 14 % and 6 %, respectively, as shown in Fig. 5e, in the middle and last four columns, respectively.

These results show that the skull anisotropy results in a reduction in the affected regions by the radial and tangential EFs by an average of 22 and 11 %, respectively. Resolving the cortical regions where the maximum change to the distribution of the EF components can be observed in Fig. 6. It was noticed that in general, the skull anisotropy altered the EF components over the gyri surface, with the exception of the radial EF component in the rectangular pad montage. Figure 6b shows that the radial EF components were altered within the deep sulci and over the sulci walls; however, Fig. 6c shows that the tangential EF components were altered over the gyri surfaces. In the ring montage, the skull anisotropy altered the distribution of the radial EF components on the surface of the gyri, especially the gyrus under the anode and the sulcal wall under the anode, as shown in Fig. 6e, while the tangential EF components were altered over the gyri surfaces that were near the cathodes, as observed in Fig. 6f. In the case of the bilateral

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Fig. 7  Absolute difference maps between Isotropic and Aniso-II in all models. The first row a–c is from the rectangular pad montage, the second row d–f from the ring montage, and the third row g–i from the bilateral montage. The first column a, d, and g the EF magnitude difference, the second column b, e, and h the radial EF difference, and the third column c, f, and i tangential EF difference

montage, the distribution of the radial EF was altered on the gyri surfaces, specifically under the anode and cathode as shown in Fig. 6h, while Fig. 6i demonstrates that the tangential EF components were altered on the surface of the gyri between the anode and cathode. These observations confirm that the shunting effect of skull anisotropy mainly alters the EF components on the gyri surfaces. 3.2 The effect of WM anisotropy 3.2.1 The distribution of EF magnitude The influence of WM anisotropy on the distribution of the EF magnitude can be observed by comparing the results of Isotropic model to the results of Aniso-II model. It was observed that the WM anisotropy alters the EF magnitude

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distribution over the cortex. Note that for the EF distribution in the case of the rectangular pad montage, the value of the EF was decreased over the precentral and superior frontal gyri in Fig. 3c of the Aniso-II model in comparison with the same region in the Isotropic model in Fig. 3a, while it was almost the same value on the middle frontal gyrus in both models. In the case of the ring montage, a slight change in the EF magnitude distribution on the gyri under the anode after considering the WM anisotropy could be observed in Fig. 3g with respect to the EF distribution in Fig. 3e. In the bilateral montage, the alteration could be observed on the EF distribution over the gyri under the anode in Fig. 3i, k of the Isotropic and Aniso-II models, respectively. Quantitatively, the plots in Fig. 5 show the area of the EF magnitude in each montage and the corresponding maximum EF. It was observed in the rectangular pad montage

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Fig. 8  Electric current flow before and after taking the WM anisotropy into account: the first row a–c shows the anatomical slices that were selected to show the electric current flow. The blue regions are GM and the yellow regions are WM. The second row d–f shows the Isotropic models, the third row g–i shows the Aniso-II models. The first column a, d, and g shows the sagittal slices in the rectangular

pad montage, the second column b, e, h shows the coronal slices in the ring montage, and the third column d, f, i shows the coronal slices in the bilateral montage. The red ellipses highlight the regions where their differences are high in the current flow orientation. The arrow length is uniform, and the color map is for the current density (color figure online)

that the EF magnitude distribution over the brain cortical surface in Aniso-II was decreased about 8 % less than the distribution in Isotropic (i.e., the first four columns in Fig.  5a). In the case of the ring montage, the EF magnitude distribution was reduced about 5 % after considering the WM anisotropy with respect to the EF distribution in Isotropic (i.e., the first four columns in Fig. 5c). The bilateral montage showed that the reduction in the EF magnitude distribution in Aniso-II was about 2 % with respect to Isotropic (i.e., the first four columns in Fig. 5e). Note that

the maximum EF value almost did not change after taking into consideration the WM anisotropy in any montage (i.e., the first four columns in Fig. 5b, d, f). These measures indicate that the WM anisotropy seems to have less significant influence on the distribution of the EF magnitude over the cortical surface, which is consistent with the literature [32]. Recognizing the cortical sites where the WM anisotropy has a maximum effect on the EF magnitude distribution is possible by observing Fig. 7a, d, g demonstrates that the

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Fig. 9  Absolute difference maps between Isotropic and Aniso-III in all models. The first row a–c is from the rectangular pad montage, the second row d– f from the ring montage, and the third row g–i from the bilateral montage. The first column a, d, and g the EF magnitude difference, the second column b, e, and h the radial EF difference, and the third column c, f, and i the tangential EF difference

WM anisotropy usually has a significant effect on the distribution of the EF magnitude within the sulci, although the WM anisotropy had an effect on the gyri under the cathode in the case of the rectangular pad. 3.2.2 The distribution of EF components Taking into consideration the effect of WM anisotropy on the EF component distribution, it was observed that the surface areas affected by the radial EF component were expanded after considering the WM anisotropy in all AnisoII models of Fig. 4. In the case of the rectangular pad montage, more sulci regions in the frontal lobe under the cathode in Fig. 4c of the Aniso-II model were affected by the radial EF than those in Fig. 4a of the Isotropic model. The same observation was noticed in the ring montage, where the affected regions (i.e., the gyri region and the wall of the

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sulci) by the radial EF under the anode in Fig. 4g of the Aniso-II model were increased in comparison with those in Fig. 4e of the Isotropic model. In the bilateral montage, it was noticed that the affected regions by the radial EF were increased in Fig. 4k of the Aniso-II model and included the gyri regions and the wall of the sulci under the anode when compared to the same region in Fig. 4i of the Isotropic model. The observed enlargement of the affected regions by the radial EF was quantified for each montage. In the rectangular pad montage, it was observed that taking into consideration the WM anisotropy resulted in an 31 % enlargement to the affected area by the radial EF with respect to that area in the Isotropic model (i.e., the middle four columns in Fig. 5a). The enlargement in the affected region by the radial EF in the case of the ring montage was about 13 % with respect to that area in the Isotropic model (i.e., the

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middle four columns in Fig. 5c). In the bilateral montage, the affected area by the radial EF was enlarged 12 % after taking into consideration the WM anisotropy with respect to that area in the Isotropic model (i.e., the middle four columns in Fig. 5e). These observations highlight the fact that taking into consideration the WM anisotropy results in an increase in the affected sulci regions by the radial EF. A reduction in the affected area by the tangential EF associated with the enlargement of the affected regions by the radial EF was observed in all Aniso-II models after taking into consideration the WM anisotropy. In the rectangular pad montage, the affected areas by the tangential EF in the Aniso-II model were reduced about 11 % with respect to those areas in the Isotropic model (i.e., the last four columns in Fig. 5a). In the ring montage, it was observed that the reduction in the affected areas by the tangential EF in the Aniso-II model was about 5 % with respect to the Isotropic model (i.e., the last four columns in Fig. 5c). The same observation was made in the bilateral montage, where the reduction was about 5 % in the Aniso-II model with respect to the Isotropic model (i.e., the last four columns in Fig. 5e). A change in the maximum value of the radial and tangential EFs after taking into consideration the WM anisotropy has been highlighted. It was observed in the rectangular pad montage that the maximum value of the radial EF was increased about 11 % after considering the WM anisotropy in comparison with that in Isotropic (i.e., the middle four columns in Fig. 5b), while the maximum value of the tangential EF was decreased in the Aniso-II model about 4 % with respect to that in Isotropic model (i.e., the last four columns in Fig. 5b). In the ring montage, there was almost no change to the maximum value of the radial EF (i.e., the middle four columns in Fig. 5d), although the maximum value of the tangential EF was reduced about 3 % after taking into consideration the WM anisotropy (i.e., the last four columns in Fig. 5d). In the bilateral montage, the maximum value of the radial EF was increased about 2 % after taking into consideration the WM anisotropy (i.e., the middle four columns in Fig. 5f), while the maximum value of the tangential EF was not changed (i.e., the last four columns in Fig. 5f). Observing the cortical sites where the WM anisotropy affects the distribution of the EF components extremely is shown in Fig. 7. In the case of the rectangular pad montage, it appeared that the WM anisotropy altered the radial EF mainly within the deep sulci, as shown in Fig. 7b where the maximum difference existed within the sulci near the cathode; however, there are also alterations over the gyri under the cathode. The WM anisotropy in the rectangular pad montage also altered the tangential EF within the sulci in addition to the gyri as shown in Fig. 7c, where the maximum difference existed within the sulci under the anode

and cathode and over the gyrus under the cathode. In the ring montage, the alterations to the radial EF distribution occurred in the wall of the sulci and the deep sulci where the maximum difference existed, as shown in Fig. 7e, while the tangential EF distribution was altered within the deep sulci as shown in Fig. 7f. The bilateral montage depicted that the radial and tangential EFs were altered within the deep sulci, as shown in Fig. 7h, i, respectively. These observations show that the WM anisotropy affects the EF components mainly in sulci regions. Observing the electric current flow in both of the Isotropic and Aniso-II models may help to elucidate how the WM anisotropy affects the distribution of the EF components. In the case of the rectangular pad montage, Fig. 8a shows that the electric current within CSF tended to accumulate at the bottom of the sulcus under the anode in the Isotropic model. However, taking into consideration the WM anisotropy in the Aniso-II model resulted in the flow of the electric current perpendicular to the gyrus wall, as shown in Fig. 8d. In the ring montage, note that the electric current in the Isotropic model (Fig. 8b) flowed in a tangential path with respect to the wall of the gyrus under the anode toward the upper corners cathodes, whereas the current flowed perpendicular to the gyrus in Aniso-II, as shown in Fig. 8e. In the case of the bilateral montage, the streaming of electric current in the WM fibers direction in the Aniso-II model led to flow of the electric current perpendicular to the apex of a gyrus, as shown in Fig. 8f, while the electric current flowed in the CSF perpendicular to the gyri walls. There was no such constraint to the current flow in the case of the Isotropic model (Fig. 8c), which led to flow of the electric current perpendicular to the gyri walls and tangential to the apex of the gyri. 3.3 The combined effect of skull and WM anisotropy 3.3.1 The distribution of EF magnitude The combined effect of skull and WM anisotropy on the distribution of the EF magnitude can be observed by comparing the results of the Isotropic model versus Aniso-III. It was observed that the combined effect of skull and WM anisotropy reduced the distribution of the EF on the cortical surface. In the case of the rectangular pad montage, the value of the EF was decreased apparently over the precentral and the superior gyri in Fig. 3d of the Aniso-III model in comparison with the same regions in Fig. 3a of the Isotropic model. On the other side, the reduction in the EF value and distribution over the gyri under the anode in the ring montage is observable in Fig. 3h of the Aniso-III model with respect to the distribution in Fig. 3e of the Isotropic model. Note in the bilateral montage that the EF distribution in Fig. 3l of the Aniso-III model reduced at the

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gyrus under the anode and the precentral gyrus at the cathode side with respect to the distribution in Fig. 3i of the Isotropic model. The combined effect of skull and WM anisotropy was quantified in terms of the area of the affected regions by the EF magnitude and the maximum value of the induced EF magnitude and shown in the plots of Fig. 5. In the case of the rectangular pad montage, the reduction in the affected region was about 31 % with respect to the affected area in the Isotropic model (i.e., the first four columns in Fig. 5a), and the maximum EF magnitude in the Aniso-III model was damped about 17 % with respect to that value in the Isotropic model (i.e., the first four columns in Fig. 5b). In the case of the ring montage, the EF magnitude distribution was reduced about 17 % less than the distribution in the Isotropic model (i.e., the first four columns in Fig. 5c), and the maximum-induced EF magnitude was reduced about 40 % (i.e., the first four columns in Fig. 5d). The bilateral montage showed that the reduction in the EF magnitude distribution in the Aniso-III model was about 8 % with respect to the Isotropic model (i.e., the first four columns in Fig. 5e) and that the decrease in the maximum value of the EF magnitude was about 18 % (i.e., the first four columns in Fig. 5f). These measures and observations indicate that although the skull anisotropy has a dominant effect on the EF magnitude distribution, the WM anisotropy assists to shrink the EF magnitude distribution more. Recognizing the cortical sites where the combined effect of skull and WM anisotropy has a maximum influence on the EF magnitude distribution is possible by observing Fig. 9. Note that most of the changes in the EF distribution happened over the gyri in all montages (Fig. 9a from the rectangular pad, Fig. 9d from the ring, and Fig. 9g from the bilateral montage). 3.3.2 The distribution of EF components In terms of the distribution of the EF components, it was observed generally that the distribution of the radial and tangential EF was reduced. However, the distribution of the radial EF was altered over the cortex. Note that in the rectangular pad montage, more sulci regions under the cathode in Fig. 4d of the Aniso-III model were affected by the radial EF than in Fig. 4a of the Isotropic model, while the precentral sulcus that was affected by the radial EF in the Isotropic model became almost unaffected by the radial EF in the Aniso-III model. In the ring montage, more gyrus surface under the anode was affected by the radial EF in Fig. 4h of the Aniso-III model than in Fig. 4e of the Isotropic model, while the distribution of the radial EF was altered over the gyri near to the cathodes. In the

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bilateral montage, it can be observed in Fig. 4l of the Aniso-III model that some of the precentral sulcus was affected by the radial EF, which was not the case in Fig. 4i of the Isotropic model. On the other side, the affected region by the radial EF under the anode in the Aniso-III model was decreased with respect to that region in the Isotropic model. The reduction in the affected regions by the EF components was computed as shown in the plots of Fig. 5. In the rectangular pad montage, the affected regions by the radial EF in the Aniso-III model were reduced about 11 % with respect to that area in the Isotropic model (i.e., the middle four columns in Fig. 5a), while the affected regions by the tangential EF were reduced about 28 % (i.e., the last four columns in Fig. 5a). In the ring montage, the reduction in the affected regions by the radial and tangential EF in the Aniso-III model was 18 and 19 %, respectively, with respect to the measures of the Isotropic model as shown in Fig.  5c in the middle and last four columns, respectively. In the bilateral montage, the affected regions by the radial EF in the Aniso-III model were decreased about 6 % (i.e., the middle four columns in Fig. 5e), while the affected regions by the tangential EF were reduced about 16 % with respect to the Isotropic model (i.e., the last four columns in Fig. 5e). These results show that the combined effect of skull and WM anisotropy leads to reducing the affected regions by the radial and tangential EF by an average of 12 and 21 %, respectively. Recognizing the cortical regions where maximum change to the distribution of the EF components occurred can be observed in Fig. 9. It was noticed that in general, the EF components in the Aniso-III models were altered over the gyri surface, with the exception of the radial EF components in the rectangular pad montage. Figure 9b shows that the radial EF components were altered within the deep sulci and over the sulci walls; however, Fig. 9c shows that the tangential EF components were altered over the gyri surfaces. In the ring montage, the combination of skull and WM anisotropy altered the distribution of the radial EF components on the surface of gyrus under the anode, as shown in Fig. 9e, while the tangential EF components were altered over the gyri surfaces near the cathodes, as shown in Fig. 9f. In the case of the bilateral montage, the distribution of the radial EF was altered on the gyri surfaces under the anode and cathode as shown in Fig. 9h, while Fig. 9i shows that the tangential EF components were altered on the surface of the gyri between the anode and cathode. These observations confirm that the shunting effect of the skull anisotropy has the dominant influence to alter the distribution of the EF components over the gyri surfaces with respect to the influence of WM anisotropy.

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4 Discussion

4.2 The effect of WM anisotropy

4.1 The effect of skull anisotropy

The study also investigated the effect of the WM anisotropy. It was observed in the literature that taking the WM anisotropy into account alters the distribution of the EF magnitude over the brain cortex [32]. It has not been shown how these alterations take place, although studying the influence of WM anisotropy on the EF components distribution may provide some insight. We observed that: (1) the maximum value of the tangential EF is reduced on average about 5 % when considering the WM anisotropy, (2) the affected region by the tangential EF is reduced on average by 11 %, and (3) the affected region by the radial EF is increased on average by 14 %. Combining these observations, it can be deduced that the WM anisotropy affects the directionality of the EF over the cortical surface. An explanation for this phenomenon can be provided by understanding the electric current flow before and after taking the WM anisotropy into account. In the case of the isotropic electrical conductivity property of the WM, the WM region will have a homogeneous conductivity that will lead the electric current to flow freely within the WM region in any direction. Therefore, the electric current can freely flow from the CSF to the GM and WM. On the other hand, the anisotropic conductivity of the WM constrains the paths of the electric current stream to be either in the WM fiber directions within the WM regions or within the CSF tangential to the cortical surface due to its high conductivity with respect to the transverse conductivity of the WM fibers. This constraint leads to flow of the electric current within the CSF more tangentially to the cortical surface (which is perpendicular way to the gyri walls) to avoid the transverse conductivity of the WM that may confront the electric current stream when taking into consideration the WM anisotropy with respect to the current flow in the case of the isotropic WM. This organization of the electric current paths after taking the WM anisotropy into account leads to better alignment to the EF with the orientation of the pyramidal cells that are in the normal direction to the cortical surface, which is useful for modulating these pyramidal cells and the brain stimulation process.

Studying the distribution of the EF components in tDCS was done before with only considering the tissue heterogeneity in [22], indicating the importance of the EF components due to the CSF and WM. However, it is clear that tissues such as the skull and WM are highly anisotropic in their electrical conductivities. The main contribution of our study is investigating the effect of these anisotropic conductivities in the skull and WM on the EF components of tDCS. Using the rectangular pad montage, Miranda et al. [22] observed that the tangential EF covers the gyri in the region between the electrodes, and the radial EF is within the deep sulci or in the walls of the sulci. Our study showed consistency with their results in terms of the distribution of the EF components in the Isotropic models (especially the Isotropic model of the rectangular pad montage in comparison with the published work of Miranda et al. [22]) and highlighted that this distribution was also applicable when taking the tissue anisotropy into account. However, although the skull anisotropy dampens the maximum value of the EF magnitude and each of its components, it has a crucial effect on the radial EF distribution, more than on the tangential EF. It was noticed that the skull anisotropy reduced on average the region affected by the radial EF by about 22 %, while the region affected by the tangential EF was reduced about 11 %. This differential effect can be explained by smearing of the induced electric current due to the skull anisotropy, which leads the electric current to flow tangentially to the brain cortical surface, therefore resulting in less electric current flowing perpendicularly into the brain cortex. As a result, the maximum value of the EF is shifted toward the cathode after considering the skull anisotropy. The role of electrode size on the distribution of the EF magnitude was noted in the literature, with smaller anode electrode sizes associated with higher EF magnitudes under the anode [7, 22]. Miranda et al. [22] have also observed that the EF under the center of the anode is mainly the radial EF. Our work showed that the EF under the center of the anode in all Isotropic models of all montages was mainly the radial EF. However, the radial EF under the anode in the rectangular pad montage was always too small to be considered, while the radial EF under the circular electrodes such as in the ring and bilateral montages was large enough to be observed. Because of this, our study showed that the influence of skull anisotropy on the radial EF was highly observable with the circular electrodes because of the smearing effect mentioned above, but it was difficult to observe with the rectangular pad electrodes.

4.3 The combined effect of skull and WM anisotropy We investigated the combined effect of skull and WM anisotropy on the distribution of the EF magnitude and components. The results demonstrated that the skull anisotropy has the dominant effect; however, the WM anisotropy can limit or emphasize the effect of skull anisotropy. As shown before, the WM anisotropy tends to increase the affected regions by the radial EF component, but the skull anisotropy tends to reduce them. Therefore, the WM anisotropy limited the crucial effect of the skull anisotropy on the

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radial EF component by reducing on average the region affected by the radial EF about 12 % instead of 22 % in the case of considering the skull anisotropy only. On the other side, both of the skull and WM anisotropy tend to decrease the affected regions by the tangential EF. Thus, the affected regions by the tangential EF in the case of considering the skull and WM anisotropy were reduced about 21 %, while the affected regions were reduced about 11 % in the case of considering the skull anisotropy only. Finally, our study used one threshold value to define the area of the affected region either by the EF magnitude, the radial EF, or the tangential EF. It was observed that the summation of the areas affected by the EF components was less than the affected area by the EF magnitude. This means that some regions were considered to be affected by tDCS (based on the distribution of the EF magnitude) but actually neither of the EF components was strong enough to affect these regions. Miranda et al. [22] used in their study three threshold values: one to define the affected region by the EF magnitude, another for the radial EF, and the last for the tangential EF. As a result, the summation of the affected areas by the EF components was larger than the area of the affected region by the EF magnitude. This meant that some regions affected by tDCS were ignored even though one of the EF components was strong enough to have an effect. Despite the influence of selecting the threshold value, this observation depicts that studying the distribution of the EF components may contribute to understanding the causes of the long-lasting effects of tDCS. Investigating the EF components distribution may be more significant in electroconvulsive therapy (ECT), where achieving electrical stimulation is more important than neuromodulation. Our investigation basically shows the effect of tissue anisotropy in their conductivity on the EF components of tDCS. Also the observed effect indicates that in practicing tDCS, one should consider the skull and WM anisotropic conductivities. The limitation of this study is how to measure the anisotropic conductivities for WM since there is no technology to measure or image the tissue anisotropic conductivities. In this study, we rely on DT-MRI and derive the anisotropic conductivity values by assuming the diffusivity and conductivity tensors share the same eigenvectors. We emphasize that our presented results are based on the head model from an individual. Although we showed the significant effect of the tissue anisotropy on tDCS, there are other factors might affect the results such as ethnic differences and brain morphology [33], pathological conditions, or population differences. These factors should be investigated in the future studies. Nevertheless, our results present the importance of individual head modeling and accurate tissue properties, which must be utilized to improve the efficiency of tDCS on neuromodulation.

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5 Conclusion This study aims to elucidate the influence of the anisotropic properties of skull and WM on the orientation of the induced EF via MRI-derived high-resolution gyri-specific FE head models. The study emphasizes that the skull anisotropy has a crucial influence on the affected cortical areas by the radial EF more than the areas affected by the tangential EF. The skull shunting that resulted from the skull anisotropy decreased the affected area by the radial EF about 22 % on average with respect to the area in the case with the isotropic skull properties. It was also observed that the skull anisotropy caused a shifting of the maximum EF toward the cathode. The study emphasizes that the WM anisotropy mainly affects the EF components within the sulci, leading to an increase in the sulci regions affected by the radial EF. The study also highlights the influence of WM anisotropy on the EF orientation, with the WM anisotropy tending to alter the direction of the EF to be aligned with the pyramidal cells trajectories. This influence on the EF direction is the main cause of altering the EF distribution on the cortical surface, especially within the sulci. As a result, the cortical surface that is affected by the radial EF can be expanded, which can help to modulate more pyramidal cells within the targeted region. Consequently, the WM anisotropy tends to limit the effect of skull anisotropy on the radial EF component; therefore, the affected regions by the radial EF will be reduced about 12 % on average when the skull and WM anisotropy is taken into consideration with respect to the area in the case of the isotropic skull and WM properties. Given the high dependency of the electric field direction on neural modulation, this study showed that investigating the distribution of the EF magnitude may not be accurate enough to predict the effect of tDCS on the neurostimulation, but studying the distribution of the EF components can elucidate more knowledge on tDCS. Acknowledgments  This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (2014R1A2A2A09052449).

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