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 given electric current limits for safety, how much current should be delivered by each electrode for optimal targeting? Several solutions, apparently unrelated, have been independently proposed depending on how "optimality" is defined and on how this optimization problem is stated mathematically. Among them, there are closed-formula solutions such as ones provided by the least squares (LS) or weighted LS (WLS) methods, that attempt to fit a desired stimulation pattern at ROI and non-ROI, or reciprocity-based solutions, that maximize the directional dose at ROI under safety constraints. A more complete optimization problem can be stated as follows: maximize directional dose at ROI, limit dose at non-ROI, and constrain total injected current and current per electrode (safety constraints). To consider all these constraints (or some of them) altogether, numerical convex or linear optimization solvers are required. We theoretically demonstrate in this work that LS, WLS and reciprocity-based closed-form solutions are particular solutions to the complete optimization problem stated above, and we validate these findings with simulations on an atlas head model. Moreover, the LS and reciprocity solutions are the two opposite cases emerging under variation of one parameter of the optimization problem, the dose limit at non-ROI. LS solutions belong to one extreme case, when the non-ROI dose limit is strictly imposed, and reciprocity-based solutions belong to the opposite side, i.e., when this limit is loose. As we couple together most optimization approaches published so far, these findings will allow a better understanding of the nature of the TES optimization problem and help in the development of advanced and more effective targeting strategies.

Optimization of interferential stimulation of the human brain with electrode arrays

“…In that case we are left with criterion (3) subject to the power constraint (5). Fernandez-Corazza et al (2019) show that this is equivalent to the least-squares criterion, which was introduced by Dmochowski et al…”

“…When adjusting the power constraint P max at off-target area, we first calculate a default value of P max as e T A(A T Γ 2 A) −1 A T e, which is the value that makes the criterion equivalent to leastsquares criterion for maximum-focality (Fernandez-Corazza et al, 2019). We then vary this value across different orders of magnitude: for superficial targets (Figure 3, A1&A3), we vary the power constraint from P max ×10 −3 to P max ×10 8 ; for deep targets (Figure 3, A2&A4), we relax the power constraint furthermore to P max × 10 12 as the optimization is numerically unstable for deep targets when the power constraint is very stringent.…”

“…We also provide a mathematical proof that the solution to maximum-intensity stimulation can be obtained directly from the forward model for TES, i.e. the "lead field" as first proposed by Fernandez-Corazza et al (2019). Motivated by Guler et al (2016) andFernandez-Corazza et al (2019), we convert the computationally intractable maximum-focality optimization for IFS into a maximum-intensity problem with constraint on the energy in the off-target area.…”

“…the "lead field" as first proposed by Fernandez-Corazza et al (2019). Motivated by Guler et al (2016) andFernandez-Corazza et al (2019), we convert the computationally intractable maximum-focality optimization for IFS into a maximum-intensity problem with constraint on the energy in the off-target area. This non-convex optimization is treated as a goal attainment problem (Gembicki and Haimes, 1975) and then solved with sequential quadratic programming (Brayton et al, 1979).…”

“…Therefore, the optimal solution can be found directly from the forward model A. This has been first noted by Fernandez-Corazza et al (2016) and is discussed in Fernandez-Corazza et al (2019) and Saturnino et al (2019). We could have also relaxed the total current constraint by allowing twice as much current compared to the limit on each electrode, |s i | ≤ I max .…”