A Bayesian approach to parameter reconstruction from surface electromyographic signals

Abstract

Applying a Bayesian approach to infer the electrical conductivity of a body or body part from surface electromyographic (EMG) signals yields a non-invasive and radiation-free imaging technique. Further, measuring the surface EMG signals that stem from voluntary muscle contractions, there is no need to apply external electrical stimuli to the body. The electrical conductivity provides structural information of the corresponding tissue that is used to estimate whether the tissue has isotropic or anisotropic properties and which is the preferred conducting direction, if applicable. Additionally, changes in the magnitude of the electrical conductivity indicate changes in the tissue material. Together, these properties of the electrical conductivity provide medical images of the examined body part. This imaging process results in an inverse and mathematically ill-posed problem. Including a stochastic model of the inevitable measurement error into the mathematical problem description, the whole system is embedded into a probabilistic framework. Thus, instead of estimating the structure of the examined body part, the probability distribution of the parameters describing the tissue structure given surface EMG measurements, the so-called posterior distribution, is estimated. This Bayesian approach to inverse problems not only yields more information about the quantities of interest than classical regularization approaches, but also has a regularizing effect on the ill-posed problem. Indeed, the Bayesian inverse problem of inferring the tissue structure from surface EMG measurements is proven to be well-posed. This yields the convergence of the inversion algorithm and allows establishing error bounds and thus quantifying the uncertainties in the solution of the inverse EMG problem. Numerically, Markov chain Monte Carlo methods are used to explore the posterior distribution. Accelerations of these sampling methods are achieved by deriving a data-sparse representation of the discretized forward model for all conceivable discretizations of the parameters describing the tissue structure. The resulting approach is not only mathematically well-founded, but also faster by orders of magnitude. Finally, the proposed sampling algorithms are applied to several use cases that are related to clinical applications.