Optimizing the Electric Field Strength in Multiple Targets for Multichannel Transcranial Electric Stimulation

Abstract

Objective:

Most approaches to optimize the electric field pattern generated by multichannel Transcranial Electric Stimulation (TES) require the definition of a preferred direction of the electric field in the target region(s). However, this requires knowledge about how the neural effects depend on the field direction, which is not always available. Thus, it can be preferential to optimize the field strength in the target(s), irrespective of the field direction. However, this results in a more complex optimization problem.

Approach:

We introduce and validate a novel optimization algorithm that maximizes focality while controlling the electric field strength in the target to maintain a defined value. It obeys the safety constraints, allows limiting the number of active electrodes and allows also for multi-target optimization.

Main Results:

The optimization algorithm outperformed naïve search approaches in both quality of the solution and computational efficiency. Using the amygdala as test case, we show that it allows for reaching a reasonable trade-off between focality and field strength in the target. In contrast, simply maximizing the field strength in the target results in far more extended fields. In addition, by maintaining the pre-defined field strengths in the targets, the new algorithm allows for a balanced stimulation of two or more regions.

Significance:

The novel algorithm can be used to automatically obtain individualized, optimal montages for targeting regions without the need to define preferential directions. It will automatically select the field direction that achieves the desired field strength in the target(s) with the most focal stimulation pattern.

1. Introduction

Transcranial electric stimulation (TES) methods inject weak direct or alternating currents via scalp electrodes in order to create an electric field in the brain that modulates neural activity [1]. In order to improve the robustness of the stimulation outcome and to be able to causally relate the behavioral stimulation effects with the modulation of specific brain areas, it is important to limit the electric field region to one or more regions of interest. However, as the electric field is shaped by the individual head anatomy [2], targeting TES electric fields is not a trivial task. Therefore, multiple optimization approaches have been proposed in order to automatically plan multichannel TES interventions [3–11].

Most TES optimization methods aim to maximize, control or approximate projections of the electric field in a specific direction in the target region [11], rather than the absolute field strength (or norm) irrespective of direction. This can be a good choice for many cortical targets, as it is thought that the physiological TES effects are direction dependent, such that an electric field pointing into and out of the cortical surface correspond to anodal and cathodal stimulation [12,13]. However, a preferential direction might not always be easily definable. In some target regions, a regular and homogeneous spatial organization of the targeted neural structures might be lacking at the microscopic level (such as in the case of some subcortical areas). Even when such an organization is present and known, sufficient knowledge on how the physiological stimulation effects depend on the direction of the currents relative to the structures might be missing. Optimizing the electric field strength in the target region(s) irrespective of the field direction might be preferred in these cases. To our knowledge, this problem was so far only tackled in two prior studies that proposed methods to maximize the field strength in a single target, but without control of the focality of the resulting field [4,5].

Here, we introduce a novel TES optimization algorithm which controls the electric field strength in one or more targets while minimizing it elsewhere to improve focality. At the same time, it accounts for safety constraints and allows limiting the number of active electrodes. In contrast to approaches that maximize the electric field strength in the target [4,5], controlling the target field strength to reach a desired value instead of maximizing it allows for leveraging the trade-off between strength and focality, and for the balanced stimulation of two or more targets. We demonstrate a computationally efficient implementation of our approach that outperforms naïve brute-force search and show that it succeeds in optimizing montages for the balanced stimulation of the bilateral amygdala.

2. Methods

2.1. Head Model

We used the example head model from SimNIBS 3.1 (www.simnibs.org) [14] Ernie with six tissues compartments, White Matter (WM), Gray Matter (GM), Cerebrospinal Fluid (CSF), Skull, Scalp, and Eyes (figure 1(a)). The standard tissue conductivities in SimNIBS were used [15]. Details of the MR image types and parameters and of the methods used to create the head model are given in [16].

Figure 1.

(a) Horizontal slice through the Ernie head model, automatically created by the SimNIBS headreco pipeline. The six tissues are white matter (dark blue), gray matter (light blue), CSF (green), skull (yellow) and scalp (red). (b) Optimized electric fields in gray and white matter while controlling the field strength in the left caudal amygdala to be 0.2 V/m, 0.4 V/m, or while maximizing it. The target is delineated in blue. Notice that the color scale changes proportionally with the electric field in the target. (c) Optimized electric fields in the central gray matter surface together with the active electrodes.

2.2. Electric Field Simulations

The optimization algorithm builds upon a leadfield matrix $\mathit{A}$ [10,17], which is constructed by selecting a fixed return electrode and then injecting a unity current sequentially through each remaining electrode. The leadfield allows for quick evaluations of the electric fields produced by any current combination by leveraging the linearity of the field with respect to the injected currents. Here, we used $n=74$ electrodes placed according to the EEG 10–10 system and performed simulations in SimNIBS 3.1 using Neumann boundary conditions in the electrode surfaces and the MKL PARDISO solver [18].

2.3. Mathematical Formulation

The mean electric field strength in a target region ${\text{Ω}}_t$ is given by

$$E_0=\frac{1}{V_{{\text{Ω}}_t}}\underset{{\text{Ω}}_t}{\int }\parallel \mathit{E}(\mathit{p})\parallel dV,$$ #(1)

where $\mathit{E}$ is the electric field, $\mathit{p}$ a position in space, and $V_{{\text{Ω}}_t}$ the volume of the target region. However, due to the square root term inside the integral, we cannot represent this quantity using linear operations. We will therefore work with an approximation

$$t^2=\frac{1}{V_{{\text{Ω}}_t}}\underset{{\text{Ω}}_t}{\int }\parallel \mathit{E}(\mathit{p}){\parallel }^{2}dV.$$ #(2)

To relate $t$ to $E_0$, we can write the quantity $\parallel \mathit{E}(\mathit{p})\parallel$ as a sum of a constant component $E_0$, and a spatially varying component $ϵ\left(\mathit{p}\right)$

$$\parallel \mathit{E}(\mathit{p})\parallel =E_0+ϵ(\mathit{p}).$$ #(3)

Substituting Equation 3 in Equation 2 and applying Equation 1, we obtain

$$t=E_0\sqrt{\frac{1}{V_{{\text{Ω}}_t}}\underset{{\text{Ω}}_t}{\int }1+\frac{ϵ{(\mathit{p})}^2}{E_0^2}dV}.$$ #(4)

Therefore, $t\ge E_0$, and the larger the variations in field norm inside the target region, the larger is the difference between the two quantities.

Discretizing the system, we can write

$$t^2={\mathit{x}}^{\top }{\mathit{Q}}_t\mathit{x},$$ #(5)

where $\mathit{x}$ is an $n\times 1$ vector of electrode currents, and ${\mathit{Q}}_t$ is a $n\times n$ symmetric positive semidefinite matrix given by

$${\mathit{Q}}_t=\frac{1}{V_{{\text{Ω}}_t}}{\mathit{A}}^{\top }{\mathit{G}}_t\mathit{A}.$$ #(6)

${\mathit{G}}_t$ is a diagonal matrix with zeros in entries outside the target region and element volume values for entries inside the target region.

2.4. Optimization Problems

We set up an optimization problem to minimize the field outside the target region, while keeping the mean field strength in the target region at a desired value $t$, in line with [10]:

Problem 1:

Minimize field energy while controlling the electric field strength in the target region.

Here, P1.1 calculates the total field energy, P1.2 controls the field strength in the target, P1.3 enforces Kirchhoff’s law, P1.4 limits the total current injected, and P1.5 limits the current injected through each electrode. Please see [10] for more details. In the following, we only consider the field energy in GM for optimization (P1.1). However, in general, any region can be used.

We can extend Problem 1 to limit the number of active electrodes to $N$, and to control the electric field strength in $n_t$ target regions:

Problem 2:

Minimize field energy while controlling the electric field strength in n_t regions and limiting the number of active electrodes to N

2.5. Implementation

In contrast to the control of a field component along a specific direction, constraint P1.2 makes Problem 1 non-convex. However, this constraint can be dealt with effectively using convex-concave programming (CCP) [19]. This class of optimization algorithms works in problems where each term can be written as a difference of a convex and a concave function, called difference of convex (DC) programming problems. It then proceeds by linearizing the concave part of the functions, thereby obtaining convex optimization problems which can be readily solved.

In order to apply the CCP algorithm, we first substitute the equality constraint P1.2 with an inequality constraint

$${\mathit{x}}^{\top }{\mathit{Q}}_t\mathit{x}\ge t^2.$$ #(7)

Any solution to the modified problem is also a solution to the original one, as for any point $\mathit{x}\prime$ in the feasible region of the modified problem there exists another point $x^{″}=x^{\prime }/\sqrt{{\mathit{x}}^{\top }{\mathit{Q}}_t{\mathit{x}}^{\top }}$, which fulfills both Constraint P1.2 and Inequality 7 and has a smaller objective value. Linearizing Inequality 7 around a point ${\mathit{x}}_k$, we obtain the term

$$2{\mathit{x}}_k^{\top }{\mathit{Q}}_t\mathit{x}-{\mathit{x}}_k^{\top }{\mathit{Q}}_t{\mathit{x}}_k\ge t^2.$$ #(8)

Because of the convexity of the quadratic term, any point that obeys Inequality 8 also obeys Inequality 7. We then solve at each step $k$ an optimization problem:

Problem 3:

Linearization of problem 1 around a point x_k

where ${\xi }_k$ is a nondecreasing penalty term and $s$ is a slack variable. The introduction of the slack variable allows for the algorithm to be initialized at a point ${\mathit{x}}_0$ which violates Inequality 7, and for it to better explore the optimization domain and find regions of lower objective value [19]. In order to obtain a feasible solution, we increase the penalty term at each iteration by a factor $\mu >1$, until a maximum value of ${\xi }_{\text{max}}$ is reached

$${\xi }_{k+1}=\min \left(\mu {\xi }_k,{\xi }_{\max }\right).$$ #(9)

For the current work, we used

$${\xi }_0=\frac{{\mathit{x}}_0^{\top }\mathit{Q}{\mathit{x}}_0}{{\mathit{x}}_0^{\top }{\mathit{Q}}_t{\mathit{x}}_0}\times {10}^{-2},$$ #(10)

$${\xi }_{\max }={\xi }_0\times {10}^4,$$ #(11)

$$\mu =2.$$ #(12)

The initial points ${\mathit{x}}_0$ are obtained by solving the constrained eigenvalue problem

$$\{\begin{array}{c}{\mathit{Q}}_t\mathit{x}=\lambda \mathit{x}\\ 1^{\top }\mathit{x}=0\end{array},$$ #(13)

and then scaling the eigenvalues so that they obey Constraints P1.4 and P1.5. In order to obtain solutions closer to the global optimum, we performed a total of 20 starts for each optimization. We considered that the optimization converged when the objective stopped decreasing and the electric field strength in the target stopped increasing

$$\{\begin{array}{c}{\mathit{x}}_{k-1}^{\top }\mathit{Q}{\mathit{x}}_{k-1}-{\mathit{x}}_k^{\top }\mathit{Q}{\mathit{x}}_k<{\delta }_0\\ {\mathit{x}}_k^{\top }{\mathit{Q}}_t{\mathit{x}}_k-{\mathit{x}}_{k-1}^{\top }{\mathit{Q}}_t{\mathit{x}}_{k-1}<{\delta }_t\end{array},$$ #(14)

where

$${\delta }_0={\mathit{x}}_{k-1}^{\top }\mathit{Q}{\mathit{x}}_{k-1}\times {10}^{-3},$$ #(15)

$${\delta }_t={\mathit{x}}_{k-1}^{\top }{\mathit{Q}}_t{\mathit{x}}_{k-1}\times {10}^{-3}.$$ #(16)

Notice that, if the target strength value $t$ is too large, it might not be reachable, which makes Problem 1 infeasible. In this case, the slack term ${\xi }_{k}s$ dominates and the optimization algorithm will naturally convert to a problem of maximizing field strength in the target region.

In order to deal with many target regions simultaneously (Constraint P2.2), we go through the same steps as with a single constraint, but modify the constrained eigenproblem used for initialization (Equation 13) to use the sum of the ${\mathit{Q}}_{t_i}$ matrices.

Constraining the number of active electrodes (Constraint P2.6) leads to a combinatorial problem. However, this problem can be efficiently solved using the branch-and-bound algorithm described in [10].

2.6. Validation

In order validate our optimization algorithm, we used it to optimize 1000 randomly chosen GM target regions, each with a 10 mm radius. To test whether the results obtained were indeed optimal, we compared the solutions to the best value obtained by optimizing the electric fields while controlling directional components (i.e. solving Problem 3 in [10]) in 25 directions equally spaced in a half-sphere.

Afterwards, we performed optimizations while controlling the field strength in two targets simultaneously. Two hundred pairs of target regions were randomly selected in GM. To validate the algorithm, we compared the results with the best values obtained from minimizing the total field energy while controlling the field component in each target independently (Problem 8 in [10]). We searched all combinations of 12 equally spaced directions in a half sphere for one of the targets and 25 directions in a full sphere for the other target, which gives a total of 300 directions. In all cases, we limited the total current injected to 2 mA and the current injected per electrode to 1mA. The target intensity was set to 0.2 V/m. Performance of the optimization was assessed using ratio of the energy (P1.1) of the solutions obtained with the optimization and the search method (ratios < 1 indicate better performance of the optimization method).

Additionally, we evaluated the error produced by the approximation introduced in Equation 3 in 100 randomly chosen targets of 20 mm radius, setting the target intensity to 0.2 V/m, maximum current per electrode to 1 mA and total current to 2 mA.

2.7. Subcortical Target

To illustrate the effect of the target field strength on the electric field, we optimized electrode montages while controlling the field strength either in the left amygdala (figure 1(b)-(c)) or the bilateral amygdala (figure 2), and also maximized the field strength in these regions. The current flow through each electrodes was limited to 1 mA, the total current injected to 4 mA and the number of active electrode was limited to 8 using the branch-and-bound algorithm described in [10]. The target intensity was set to 0.2 V/m, 0.4 V/m or maximized. In the two-target case, the later was done while keeping the electric field strength in both targets the same. Even though the images show the electric field in grey and white matter, only the electric field in gray matter was considered during optimization.

Figure 2.

(a) Optimized electric fields in gray and white matter while controlling the field strength inside the two targets delineated in blue. The electric field strength in both targets was set to 0.2 V/m, 0.4 V/m, and increased further until the achieved field strength started differing between both targets. The color scale changes proportionally with the electric field strength in the target. (b) Optimized electric fields shown in the central gray matter surface with the electrode montages overlaid.

2.8. Effect of conductivity uncertainties

To address how sensitive our optimized results are to the chosen conductivity values, we calculated the electric fields produced by the optimized montages in models with altered electric conductivity values of the skull and CSF. The methodology and results are described in the Supplementary Material. Consistent with the findings from our previous study [1], the conductivity variations lead to changes of the overall field magnitude in gray matter, while the spatial distribution of the field and the ratio between field magnitude inside versus outside the target was stable. That is, a montage optimized for a specific choice of tissue conductivities will still ensure that the currents are well focused onto the target also when the conductivities are varied.

3. Results

3.1. Validation

When controlling the field strength of one target, our optimization approach outperformed the naïve search in all the cases (range of energy ratios between optimized vs searched solutions: 0.87 to 0.99; 95% confidence interval, CI), while running ~1.7 times faster (2.2 vs. 3.7 seconds). For the two-target control, the optimization was better in 99.5% of the cases (range of energy ratios between optimized vs naïve search: 0.82 to 0.97; 95% CI), while running ~3.5 times faster (4.0 vs. 14.5 seconds).

When evaluating the error caused by the approximation introduced in Equation 3 in 100 randomly chosen targets of 20 mm radius, the approximate field norm of $t=0.2\mathrm{V}/\mathrm{m}$ was reached in all targets, while the 95% confidence interval (CI) for the actual mean field norm ($E_0$, Equation 1) was between 0.176 and 0.197 V/m. That is, the approximation deviated less than 15% from the actual value, even for relatively large target regions, making it uncritical for the intended practical applications.

3.2. Subcortical Target

The peak electric fields are not located in the subcortical target, but in more superficial structures (figure 1(b)), which is expected from earlier findings [10]. Interestingly, however, as the field strength in the target increases, the fields in superficial regions get disproportionally stronger (i.e. the focality decreases strongly). This is expected due to physical limitations in the distribution of electric fields [3,10]: Increasing the field strength in the target is achieved by increasing the distance between anodes and cathode (Figure 1C), which lowers focality. This suggests that controlling the field strength rather than simply maximizing is preferred to maintain a better focality.

Figure 2 shows the same intensity-focality trade-off for the two target case. Interestingly, the trade-off between 0.2 V/m and 0.4 V/m seems small. The electrode montages changes from being symmetric along the sagittal plane to a frontal-posterior montage as the target intensity increases beyond 0.4 V/m (Figure 2B).

4. Discussion

Our new algorithm is capable of optimizing the focality of multichannel TES montages while controlling the field strength in multiple targets. In contrast, existing algorithms either optimize a specific electric field component in the target along a direction that needs to be specified by the user [3,6–11], or maximize the electric field strength in the target without accounting for the field focality [4,5]. In addition, our algorithm leverages an efficient implementation that performs better than simple search both in terms of the optimality of the solution and time. When compared to the results obtained by maximizing the field strength in the target, the algorithm can strongly improve the focality of the stimulation with only a moderate decrease of fields at the target.

Exemplary optimization of the field strength in the uni- and bilateral amygdala reveal that relatively high field strengths can be obtained also for bilateral targeting, but also shows an expected intensity-focality trade-off and stronger fields in cortical regions. Controlling the field strengths in the bilateral amygdala rather than merely maximizing it achieves a balanced montage in which both targets are similarly stimulated.

For superficial targets, it might often be more desirable to control a specific field component in order to ensure that the field is oriented perpendicularly to the cortical surface in the target area. For subcortical targets, however, a preferential direction might not be easily defined and the control of the field strength might be preferred. The new algorithm is computationally efficient and obeys safety and practical constraints, rendering it suited for use in empirical TES studies. It will be released as open source in a future version of the transcranial brain stimulation simulation and optimization software SimNIBS [14].

The optimization results will be affected by inaccuracies in the underlying electric field estimates. This is a general limitation, which our approach shares with the other methods proposed so far. Inaccurate field estimates can arise from segmentation errors, that still occur even though automated segmentation methods have improved over the past few years [16,20]. Visually confirming the anatomical accuracy of the head models is thus important to ensure reliable optimization results. Moreover, the ohmic conductivity values of many head tissues are uncertain, which can impact the simulated fields and optimization results [21,22]. However, our control experiments suggest that that the optimization results are relatively robust to conductivity changes, in line with our prior findings [21]. Specifically, the results show that that variations of tissue conductivity mostly affect the overall magnitude of the field induced by an optimized montage, while the spatial distribution remains largely unchanged, ensuring that the fields stay focused onto the target(s).