The influence of positioning error on the intensity of transcranial magnetic stimulation (TMS)-induced cortical electric field: Mathematical models constructed from 55 776 electric field simulations of 28 commercial TMS coils

The clinical efficacy of transcranial magnetic stimulation (TMS) critically depends on precise coil positioning; yet, a systematic quantitative analysis of how positioning errors affect the TMS-induced cortical electric field (E-field) intensity remains unavailable. To address this gap, this study developed mathematical models for 28 commercial TMS coils to characterize the impact of six-degree-of-freedom (6-DoF) positioning errors—including translational (X, Y, Z) and rotational (yaw, pitch, roll) components—on cortical E-field intensity. These models were constructed using 55 776 E-field simulations of the primary motor cortex (M1) across 24 healthy adult subjects, enabling the quantification of translational and rotational positioning error sensitivity across coils and the development of a rapid cortical electric field evaluation method (FIEM) for rapid E-field evaluation. Key results demonstrate high accuracy in predicting positioning error effects (R > 0.94 for all 28 coils), significantly broader sensitivity ranges for translational compared to rotational errors, and FIEM achieving 95% accuracy with a 14 ms computational speed. By incorporating cross-vendor, multi-size coil comparisons, our results provide clinicians and researchers with data-driven tools for selecting TMS coils and positioning approaches, while the FIEM framework supports rapid E-field evaluation to reduce uncertainty in cortical targeting during TMS interventions.