Uncertainty analysis in transcranial magnetic stimulation using non-intrusive polynomial chaos expansion

We propose a framework of nonintrusive polynomial chaos methods for transcranial magnetic stimulation (TMS) to investigate the influence of the uncertainty in the electrical conductivity of biological tissues on the induced electric field. The conductivities of three different tissues, namely, cerebrospinal fluid, gray matter (GM), and white matter, are modeled as uniformly distributed random variables. The investigations are performed on a simplified model of a cortical gyrus/sulcus structure. The statistical moments are calculated by means of a generalized polynomial chaos expansion using a regression and cubature approach. Furthermore, the results are compared with the solutions obtained by stochastic collocation. The accuracy of the methods to predict random field distributions was compared by applying different grids and orders of expansion. An investigation on the convergence of the expansion showed that in the present framework, an order 4 expansion is sufficient to determine results with an error of <1%. The results indicate a major influence of the uncertainty in electrical conductivity on the induced electric field. The standard deviation exceeds values of 20%-40% of the mean induced electric field in the GM. A sensitivity analysis revealed that the uncertainty in electrical conductivity of the GM affects the solution the most. This paper outlines the importance of exact knowledge of the electrical conductivities in TMS in order to provide reliable numerical predictions of the induced electric field. Furthermore, it outlines the performance and the applicability of spectral methods in the framework of TMS for future studies.