Optimization design of configurable standard-type coils based on singular value decomposition for transcranial magnetic stimulation

Objective: This study aims to introduce a configurable standard-type (CST) coil design methodology that can accommodate different stimulation needs through the overlapping of standard coil components, while ensuring the performance of CST coil approaches the existing optimal physical limit as closely as possible. Approach: The study utilizes planar, spherical, and hemispherical stream function to represent the spatial geometry of the coils. Singular value decomposition is employed to design the standard coil components while minimizing the number of coil components. Initially, this study delineates the relationship between singular vectors and coil geometry, defining a multi-objective optimization problem for the research. By contrasting the decay characteristics of singular values in the inductance and electric field matrices of the stream functions, the singular vectors of the electric field matrix are utilized as the foundation for CST coil optimization. Subsequently, the relationships between the number of singular vectors and the coil’s Pareto fronts are determined through the optimization algorithms, and the geometric patterns of the singular vectors, i.e., the standard coil components, are analyzed. Main results: By constructing the CST coil components using seven singular vectors and comparing with the Pareto front of the performance of over 50 existing coil types, the stimulation depth range of the CST coil based on the planar stream function decreased by only 7.7%; the focality performance of the CST coil based on the spherical stream function improved by 22.4%; the focality performance of the CST coil based on the hemispherical stream function decreased by only 1.6%. Significance: The construction of CST coils with a minimal number of components is explored to approximate the performance at any discrete point on the physical limit curve. This approach overcomes the limitation of one coil associating with one performance point in traditional designs, presenting a novel perspective on coil design.