Quasi-Static Approximation Error of Electric Field Analysis for Transcranial Current Stimulation

TL;DR

This study evaluates the accuracy of the quasi-static approximation in electric field modeling for transcranial current stimulation, finding it valid up to 1.43 MHz but highlighting errors from neglecting tissue permittivity.

Contribution

It provides a comprehensive analysis of the quasi-static approximation's validity across a broad frequency range, emphasizing the importance of tissue permittivity in accurate modeling.

Findings

Quasi-static approximation has less than 1% error up to 1.43 MHz.

Neglecting tissue permittivity causes errors over 20% at 10 Hz.

Capacitive tissue effects are crucial for pulsed waveforms.

Abstract

Objective: Numerical modeling of electric fields induced by transcranial alternating current stimulation (tACS) is currently a part of the standard procedure to predict and understand neural response. Quasi-static approximation for electric field calculations is generally applied to reduce the computational cost. Here, we aimed to analyze and quantify the validity of the approximation over a broad frequency range. Approach: We performed electromagnetic modeling studies using an anatomical head models and considered approximations assuming either a purely ohmic medium (i.e., static formulation) or a lossy dielectric medium (quasi-static formulation). The results were compared with the solution of Maxwell's equations in the cases of harmonic and pulsed signals. Finally, we analyzed the effect of electrode positioning on these errors. Main Results: Our findings demonstrate that the quasi-static approximation is valid and produces a relative error below 1% up to 1.43 MHz. The largest error is introduced in the static case, where the error is over 1% across the entire considered spectrum and as high as 20% in the brain at 10 Hz. We also highlight the special importance of considering the capacitive effect of tissues for pulsed waveforms, which prevents signal distortion induced by the purely ohmic approximation. At the neuron level, the results point a difference of sense electric field as high as 22% at focusing point, impacting pyramidal cells firing times. Significance: Quasi-static approximation remains valid in the frequency range currently used for tACS. However, neglecting permittivity (static formulation) introduces significant error for both harmonic and non-harmonic signals. It points out that reliable low frequency dielectric data are needed for accurate tCS numerical modeling.