A Complete-Electrode-Model-Based Forward Approach for Transcranial Temporal Interference Stimulation with Linearization: A Numerical Simulation Study

TL;DR

This study develops a complete electrode model (CEM) forward framework for transcranial temporal interference stimulation (tTIS) that accounts for frequency-dependent complex tissue admittivity and electrode impedance. A linearized surrogate of the resistance matrix with respect to impedance changes is derived to enable fast updates of the lead-field mapping and interference patterns, facilitating optimization of stimulation currents. Numerical simulations on a high-resolution head model demonstrate that the CEM forward can reproduce interference fields and that the linearized model matches the nonlinear reference within a defined PSNR tolerance, with the strongest discrepancies localized near altered electrodes. The work supports efficient, impedance-aware optimization of tTIS protocols and provides a basis for rapid sensitivity analyses, while acknowledging limitations from tissue capacitance omission and solver constraints, and outlining directions for further validation and methodological enhancements.

Abstract

Background and Objective: Transcranial temporal interference stimulation (tTIS) is a promising non-invasive brain stimulation technique in which interference between electrical current fields extends the possibilities of electrical brain stimulation. The objective of this study is to develop an efficient mathematical tTIS forward modelling scheme that allows for realistic and adaptable simulation and can be updated accurately when the contact resistance is modified in one or more electrodes. Such a model is vital, for example, in optimization processes that seek the best possible stimulation currents to exhibit or inhibit a given brain region. This study aims to establish and evaluate the complete electrode model (CEM), i.e., a set of boundary conditions incorporating electrode impedance and contact patch, as a forward finite-element-method-based simulation technique for tTIS and investigate linearized CEM as a surrogate. Results: The CEM-based forward simulation successfully reproduced the volumetric stimulating fields induced by tTIS. Sensitivity analysis showed that variations in electrode resistance affects the field distribution, especially in regions where the interfering currents have nearly equal amplitudes. The linearized CEM model closely matched the full nonlinear model within a predefined peak signal-to-noise ratio (PSNR) threshold for relative error. Both models exhibited the highest sensitivity near the focal region.

Figures (24)

• Figure 2.1: Expected behaviour of $\lvert \IfNoValueTF{-NoValue-}{ {Z_\ell} }{ {Z_{-NoValue-}} } \rvert$ as function of stimulation frequency electrode-modelling-2017, computed via \ref{['eq:impedance-of-f']}, with $R_{\mathrm{c}}\in\{100.0,1000.0,10000.0\}\,Ω$, $R_{\mathrm{d}}\in\{1000.0,10000.0\}\,Ω$ and $C_{\mathrm{d}}=\qty{0.1}{\micro\farad}$.
• Figure 2.2: Interference patterns of $2$ sinusoidal signals $s_1$ and $s_2$ of roughly equal source strength, in a 1-dimensional domain. Interference mainly occurs where the envelope of each signal is of roughly equal magnitude. The difference between summation and subtraction mainly introduces a phase shift in the envelope of the modulating signal, as can be seen from Subfigures \ref{['fig:sum-of-signals']} and \ref{['fig:abs-diff-of-signals']}. The frequency of the stimulating interference signal is considered as the envelope of the summed signals $s_1$ and $s_2$ , seen in red.
• Figure 2.3: The volumetric head model used in our experiments.
• Figure 2.4: The 4-electrode or 2-voltage-source tTIS configuration used in this study. The electrodes are labeled according to the standard 10-by-10 system Nuwer-2018, with electrode pairs TP9 + C3, and C4 + FT10 sharing voltage sources. The electrode contact surfaces had a diameter of $\sim\qty{1.0}{\centi\meter}$.
• Figure 2.5: Schematic of the column-wise lead field differences analyzed in this study.