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.
Highlights
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
We show how the directional maximization iterative solutions evolve from the weighted LS (WLS) to the reciprocity closed-form solutions when varying the imposed bound to the energy integral over Ωnon–ROI
We theoretically proved that the apparently unrelated least squares (LS), WLS, and reciprocity-based solutions all belong to the same family of the general constrained maximizing intensity problem solutions of Eq (3), constituting a unified approach (Sections 3.2 and 3.3)
Summary
Due to the low frequencies involved, the FP is governed by the quasistatic Maxwell equations It is described by the Poisson equation for the electric potential ψ( x ) in the head volume with Neumann boundary conditions (Frank, 1952; Jackson, 1975). The FEM converts the FP formulation into a linear system of equations Kv = u, where K is the stiffness matrix and accounts for the geometry, bulk conductivity or a conductivity map of each tissue, and electrode contact impedances (if using CEM); v is the vector of unknown electric potentials at each mesh node of the head and at the electrodes, and u is a vector accounting for the electric sources and sinks (in TES, the applied currents or, equivalently, the current injection pattern).
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