Simple hierarchical models of the Transcranial Magnetic Stimulation

This paper proposes simple models of high computational efficiency for Transcranial Magnetic Stimulation (TMS). Since the magnetic field is produced by currents of low frequency, the physical model is based on a unidirectional coupling between a Magneto-“Steady-state” (MG) formulation and an Electric Conduction (EC) formulation. The coupling is ensured by the Faraday's law of induction and it is unidirectional because the magnetic effect of the resultant eddy currents is neglected. The models we propose consider simplified geometries. However, the source coil in which an imposed current flows is described by a parametric curve, which can be of any shape. The human head is modeled by a conductive homogeneous sphere. The paper describes all the modeling steps: geometrical, physical, mathematical, analytical, numerical and computational models. Numerical results are compared with analytical ones and, aiming to quantify the uncertainties (UQ), the expression of the modeling error was determined. The result is a hierarchical series of approximate surrogate models, which have different levels of accuracy and complexity. The performed UQ may be used to control the modeling error, keeping the complexity at a minimal level. Three test cases are studied, for different shapes of the source coil: 1-circular, 2-circular planar 8-shaped, 3-circular arranged in the 3D space.