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1 Introduction

Deep Brain Stimulation (DBS) of Subthalamic Nucleus (STN) has become in the best treatment for Parkinson’s disease [1]. DBS consists of the implantation of a neurostimulator in one of the movement control centers (thalamus, subthalamic nucleus, globus palidus), which transmits an electric current to the surrounding neurons. The fundamental purpose of DBS is to modulate neural activity with electric fields. But, despite the clinical success of this procedure, there is not a full knowledge about the therapeutic mechanisms of DBS [2], limiting opportunities to improve treatment efficacy and simplify selection of stimulation parameters [3]. In addition, there are many questions related to the consequences that DBS generates in the nervous system, difficulting to determine the amount of brain tissue which presents excitation or electrical response to stimulation [2].

Recently the scientific community has developed methodological tools that link scientific analysis and both electrical and anatomical models of DBS applied in humans [4]. Models based on finite element method to determine the voltage distribution generated by DBS, show that therapeutic benefit can be achieved by direct stimulation in a wide range of the subthalamic region [2]. It is known that several factors influence the behavior of the electric field, and therefore the neural response of patients to therapeutic DBS. In addition to stimulation parameters, the electric potential distribution depends on size and shape of the electrode [5], and the postoperative cell growth around the implant resulting in an encapsulation layer [6].

In most studies, researchers have employed conductor homogeneous domains with square, rectangular and spherical shapes. The main problem with these models is that the optimal geometry and dimensions for the volume conductor are not defined. For example in [7] was quantified the role that geometry dimensions and boundary conditions play in the model. They found that for any variation of lengths in a box or ground positioning, the results are different for fundamental electrical quantities such as potential, field and neural activation function. Similarly, electrical properties of brain tissue affect the simulation results [8]. In [9] has been proved the influence of anisotropic and heterogeneous brain tissue in results of DBS applied in a human head model with finite elements. This brain model includes the gray matter, white matter and cerebrospinal fluid with their corresponding electrical properties and anisotropy values, through the inclusion of diffusion tensors. The authors of [10] showed that the tissue layers in the head (white matter, gray matter, cerebrospinal fluid, skull, muscle, stimulating device) and electrical properties of brain tissue are critical in results. These studies show a gap in the modeling of the DBS and most of them have been performed in North American and European populations. Therefore, little is known about the electrical behavior during DBS in patients from South America. For this reason, it is necessary to consider all the critical aspects of the DBS model applied to South American population. In this regard must be considered an optimal geometry, a heterogeneous brain tissue, the encapsulation tissue around the electrode, some layers of tissue inside the head and the anisotropic electrical behavior of neurons.

The aim of this paper is to show an objective comparison among three kind of volumes which represent the geometry, layers and structure of the head, likewise, the influence of electrical properties of anisotropic tissue and ground positioning. Another worth contribution of this work is the analysis and evaluation of the anisotropy level in relevant brain tissue (i.e. Thalamus-Thal, Subthalamic Nucleus-STN and Substantia Nigra reticulata-SNr) in Colombian patients. For this purpose, we use diffusion Magnetic Resonance Imaging (dMRI) obtained from five patients with Parkinson’s Disease located in the west-central region of Colombia.

2 Materials and Methods

2.1 Simulation Models for DBS and Geometries

The phenomenon of electrical brain activity is described by the Laplace equation [7, 8]. Here, the voltage propagation is a spatial function depending of tensorial tissue parameters:

$$\begin{aligned} \nabla \cdot \sigma \nabla V_e = 0, \end{aligned}$$
(1)

where \(V_e\) is the electrical potential, \(\sigma \) is a conductivity tensor calculated from dMRI and \(\nabla \) is the gradient operator given by \(\frac{\partial }{\partial x},\frac{\partial }{\partial y}, \frac{\partial }{\partial z}\). Equation 1 is solved by finite element approximation, because it is not possible to find the analytical solution. We employ the software Comsol Multiphysics 4.3.

We propose three solution domains that represent the skull cavity. First, a \(10\times 10\times 10\) \(cm^3\) cubic domain (see Fig. 1a), second a spherical 3D domain with 10 cm of radius (see Fig. 1b), and third a ellipsoid domain (see Fig. 1c). Also, we establish Dirichlet boundary conditions for all models and we consider the encapsulation tissue (\(\sigma _t=0.042\) S / m) of 0.1 mm thickness generated by the DBS electrode. The medium is anisotropic and heterogeneous. To address this formulation, we apply conductivity tensors estimated from dMRI obtained from 2 patients with Parkinson’s disease located in the west-central region of Colombia.

The stimulation electrode is the Medtronic DBS lead model 3389 (radius 0.635 mm)Footnote 1. This device have four platinum electrode contacts (length 1.5 mm, spaced 0.5 mm), with constant conductivity \(\sigma _e=8.6\times 10^6\) S / m. We consider an unipolar stimulus located in the Subthlamic Nucleus (STN) (see Fig. 1d). The waveform is a train of squared pulses with amplitude \(-1\) V and frequencies between 100–150 Hz.

Fig. 1.
figure 1

Geometric domains and DBS electrode.

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2.2 Brain Conductivity Through Diffusion Magnetic Resonance Imaging (dMRI)

The diffusion Magnetic Resonance Imaging (dMRI) studies the diffusion of water particles in the human brain. The direction of the diffusion is related to local micro-structure of the brain tissue and it is usually aligned through oriented pathways like the connective fibers. Diffusion can be described by a \(3\times 3\) tensor matrix proportional to the covariance of a Gaussian distribution.

$$\begin{aligned} \mathbf {D}= \begin{bmatrix} D_{xx}&D_{xy}&D_{xz} \\D_{yx}&D_{yy}&D_{yz} \\ D_{zx}&D_{zy}&D_{zz}\end{bmatrix} \end{aligned}$$
(2)

For water, the diffusion tensor (DT) is symmetric, so that \( D_ {ij} = D_ {ji} \) to \( i, j = x, y, z \). The water DT D will be completely defined by six elements: \(D_{xx},D_{yy},D_{zz},D_{xy},D_{xz},D_{yz}\). The diffusion tensor for each voxel of the dMRI is calculated using the Stejskal-Tanner formulation [11]:

$$\begin{aligned} S_k=S_0e^{-b\hat{\mathbf {g_k}}D\hat{\mathbf {g_k}}^\top }, \end{aligned}$$
(3)

where, \(S_k\) is the diffusion weighted image (DWI), \(S_0\) is the reference image, is the gradient vector and b is the diffusion coefficient. It is necessary at least 7 DWIs for each slice (\(K=0,1,...,7\)). Usually, DTs are estimated using re-weighted least squares. However, there are robust methods for DT estimation. In this work, we use the RESTORE algorithm [12] for solving the DTs.

Diffusion tensor D can be transformed to a conductivity tensor \(\sigma \) using the scalar transformation proposed in [13]:

$$\begin{aligned} \sigma =k\frac{S\times sec}{mm^3}D, \end{aligned}$$
(4)

where \(k=0.844\pm 0.0545\), S: Siemens, sec: seconds, \(mm^3\): cubic millimeters.

For evaluating anisotropy level in brain tissue, first we locate the Thal, STN and SNr through Atlas registration. Then, we calculate the fractional anisotropy (FA) from the diffusion tensors using the following relation:

$$\begin{aligned} FA=\frac{\sqrt{(\lambda _1-\lambda _2)^2+(\lambda _2-\lambda _3)^2+(\lambda _3-\lambda _1)^2}}{\sqrt{2(\lambda _1^2+\lambda _2^2+\lambda _3^2)}}, \end{aligned}$$
(5)

where, \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) are the eigenvalues of the each diffusion tensor D. The FA measures the fraction of the total magnitude of the tensor that is attributable to the anisotropic diffusion. For example a value of FA equal to 0 represents a perfect sphere, while a FA of 1 is an ideal linear diffusion. It is known that well-defined tracts have a value of FA greater than 0.20 and that very few regions have a higher FA to 0.9 [14].

3 Experimental Results

3.1 Anisotropy Level of Brain Structures for Each Patient

We evaluate the anisotropy level of relevant brain structures for each patient. The Table 1 shows the average fractional anisotropy (FA) for the Thalamus (Thal), Subthalamic Nucleus (STN) and Substantia Nigra reticulata (SNr).

Table 1. Average results of fractional anisotropy (FA) for Thal, STN and SNr.
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3.2 Simulations of DBS Models

We evaluate simulations for two patients (A and B). We test the cubic, spherical and ellipsoidal geometries and we analyze the electrical propagation for each domain. For example, we observe if there are significant modifications in potential distribution respect to a type of geometry, ground positioning and tissue properties.

Fig. 2.
figure 2

Simulation of a STN-DBS for a cubic domain (patient A)

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Fig. 3.
figure 3

Simulation of a STN-DBS for a spherical domain (patient A)

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Results for Patient A: Figures 2, 3 and 4 show the electric potential generated by DBS in cubic, spherical and ellipsoidal domains for patient A.

Fig. 4.
figure 4

Simulation of a STN-DBS for a ellipsoidal domain (patient A)

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Fig. 5.
figure 5

Simulation of a STN-DBS for a cubic domain (patient B)

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Fig. 6.
figure 6

Simulation of a STN-DBS for a spherical domain (patient B)

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Fig. 7.
figure 7

Simulation of a STN-DBS for a ellipsoidal domain (patient B)

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Results for Patient B: Figures 5, 6 and 7 show results for patient B.

3.3 Discussion

We synthesize the highlights of results in the following points:

  • The FA values are very important because they allow us to establish if the diffusion in Thalamus (Thal), Subthalamic Nucleus (STN) and Substantia Nigra reticulata (SNr) can be considered as anisotropic or not. The threshold value established for considering anisotropy is \({\ge }0.3\). FA results in Table 1 show that Thal, STN and SNr are highly anisotropic in five Colombian patients. This is a clear trend of anisotropy and heterogeneity in the tissues of relevant structures in the DBS. According to this, we can infer that a simulation model that considers the medium as isotropic and homogeneous has low accuracy and it is not consistent with the real behavior of the brain tissue from a patient with Parkinson’s disease.

  • Diffusion tensors provide quantitative information about water diffusion in brain tissue, allowing more accurate conductivity models in comparison to previous approaches of brain electrical conductivity. They also offer information of directionality and anisotropy. In the other hand, homogeneous and isotropic models consider the brain tissue as a single conductor volume with a scalar value (i.e. 0.3 S/m). This is a wrong approximation because the brain tissue is divided in several classes: gray matter, white matter, cerebrospinal fluid, skull, among others. Each tissue type has unique electrical properties.

  • All simulations performed in our experiments show that voltage propagation generated by DBS does not depend of geometry. We can observe from all figures that shape, quantity and direction of propagation did not change when we use a cube, a sphere or an ellipsoid in the same patient. In contrast, in previous approaches [7, 8] reported for DBS models with homogeneous and isotropic domains, was found a strong dependence with the selected geometry. According to this analysis, we can establish that geometry is not a key factor when the medium is anisotropic and heterogeneous.

  • We also analyze the ground positioning in all DBS models. We tested ground poles in different sides of each geometry and we found a strong dependence with the ground placement in our models. We obtained different voltage propagation for several ground locations. This demonstrates that DBS simulation models are highly sensible to the ground placement.

  • If we compare results for patients A and B, we can see a different voltage propagation. In heterogeneous models the tissue properties have a fundamental role in electrical magnitudes generated by DBS. Following this notion, the propagation of the electric potential generated by a DBS electrode is related to the stimulation parameters (amplitude, frequency, wide pulse, etc) and the electrical properties of the brain tissue. However, we found that another factors inside the simulation models as ground position are very critical. The inclusion of anisotropy with diffusion tensors allow the development of detailed models for specific patients analysis. Similarly, it is possible to realize more reliable simulations according to the real electrical behavior of neurons affected by DBS. This point has a remarkable relevance, because the therapeutic outcomes of DBS in Parkinson’s disease patients depends strongly of an accurate stimulation in the STN and surrounding structures.

4 Conclusions and Future Work

In this paper, we presented an analysis of the role of geometry and electric properties of brain tissue in simulation models for DBS. We found that type of geometry is not relevant when the medium is anisotropic and heterogeneous. However, the tissue properties and ground location are key in voltage propagation. The electric properties of brain tissue are different in all analyzed patient. For this reason, a homogeneous and isotropic model might be a wrong approach to simulate the DBS. As future work, we propose models with real brain shapes reconstructed from MRI. For example: 3D brain reconstructions to perform patient specific analysis of voltage propagation and to correlate it with therapeutic outcomes.