Unification of optimal targeting methods in transcranial electrical stimulation

Abstract

One of the major questions in high-density transcranial electrical stimulation (TES) is: given a region of interest (ROI) and electric current limits for safety, how much current should be delivered by each electrode for optimal targeting of the ROI? Several solutions, apparently unrelated, have been independently proposed depending on how "optimality" is defined and on how this optimization problem is stated mathematically. The least squares (LS), weighted LS (WLS), or reciprocity-based approaches are the simplest ones and have closed-form solutions. An extended optimization problem can be stated as follows: maximize the directional intensity at the ROI, limit the electric fields at the non-ROI, and constrain total injected current and current per electrode for safety. This problem requires iterative convex or linear optimization solvers. We theoretically prove in this work that the LS, WLS and reciprocity-based closed-form solutions are specific solutions to the extended directional maximization optimization problem. Moreover, the LS/WLS and reciprocity-based solutions are the two extreme cases of the intensity-focality trade-off, emerging under variation of a unique parameter of the extended directional maximization problem, the imposed constraint to the electric fields at the non-ROI. We validate and illustrate these findings with simulations on an atlas head model. The unified approach we present here allows a better understanding of the nature of the TES optimization problem and helps in the development of advanced and more effective targeting strategies.

Keywords: Least squares; Optimal electrical stimulation; Reciprocity theorem; Transcranial direct current stimulation (tDCS); Transcranial electrical stimulation (TES).

Fig. B1.

Feasible domains $\mathcal{D}$ for the 2D (left) and 3D (right) cases, assuming two and three electrodes respectively. All vertices correspond to only two active electrodes. The 2D case is trivial, with two electrodes, there are only two possible injection patterns: $(i_{max},-i_{max})\text{and}(-i_{max},i_{max}).$ For the 3D case there are six vertices, i.e., six possible solutions.

Fig. B2.

Example of how the standard simplexes in Eq (B.3) are altered with a current limit per electrode lower in absolute value than i_max. In this example we set $c>\frac{1}{2}(\text{left}),c=\frac{1}{2}(\text{center}),c<\frac{1}{2}(\text{right}).$ Corners v₁; v₂; v₃ do not longer belong to the truncated simplex, and points v_a; v_b; v_c are new corners and thus, possible solutions.3

Fig. 1.

Iterative solutions to the constrained directional maximization problem in Eq. (3) with constraint of Eq. (3.i.a) and computed with the SDPT3 solver. (A) Mean ROI directional intensity measured as the functional to be maximized in Eq. (3) divided by Ω_ROI volume (blue line), total injected current (red line), and integral focality (black line) as a function of the Ω_non–ROI energy upper bound (α_I). (B) Some examples of the iterative solutions: optimal current injection patterns $\widehat{\mathbf{i}}$ (first row); modulus of the electric field at the brain with Ω_ROI circled in black (second row), and absolute value of the normal-to-cortex component of the electric field (third row). Color scale limits are different, increasing from left to right. The solutions in the pale pink zone of (A) are equivalent, except for a scaling constant, to the WLS closed-form solution. The iterative solutions in the pale blue zone of (B) are equivalent to the closed-form one-to-one reciprocity solution. Between critical points “a” and “b”, there is a smooth transition between both extreme solutions.

Fig. 2.

Iterative solutions to the constrained directional maximization problem in Eq. (3) with the constraint of limiting the electric field intensity at each Ω_non–ROI element (Eq. (3.i.b)). (A) Mean ROI directional intensity measured as the functional to be maximized in Eq. (3) divided by Ω_ROI volume (blue line), total injected current (red line), and elementwise focality (black line) as a function of the Ω_non–ROI maximum electric field (α_E). (B) Some examples of the optimal solutions: optimal current injection patterns $\widehat{\mathbf{i}}$ (first row); modulus of electric field at the brain with Ω_ROI circled in black (second row), and absolute value of the normal component of the electric field (third row). Color scale limits are different, increasing from left to right. The solutions in the pale pink zone of (A) have the same pattern, except for a scaling constant. The iterative solutions in the pale blue zone of (B) are equivalent to the closed-form one-to-one reciprocity solution. Between critical points “a” and “b”, there is a smooth transition between both extreme solutions. We marked an additional point “c”, where the focality starts to decrease more sharply.

Fig. 3.

Focality values as a function of the mean electric field intensity in Ω_ROI for the solutions obtained with the Ω_non–ROI integral constraint (solid line) and with the Ω_non–ROI elementwise constraint (dotted line). Subfigure A shows the integral focality plots and subfigure B shows the elementwise focality plots for both optimal solution approaches. The red circles indicate the corresponding critical points “a” of Figs. 1A and 2A, i.e. the points where the optimal solutions reach the maximum available budget.