Multivariate regression approaches for surrogate-based diffeomorphic estimation of respiratory motion in radiation therapy

Breathing-induced location uncertainties of internal structures are still a relevant issue in the radiation therapy of thoracic and abdominal tumours. Motion compensation approaches like gating or tumour tracking are usually driven by low-dimensional breathing signals, which are acquired in real-time during the treatment. These signals are only surrogates of the internal motion of target structures and organs at risk, and, consequently, appropriate models are needed to establish correspondence between the acquired signals and the sought internal motion patterns. In this work, we present a diffeomorphic framework for correspondence modelling based on the Log-Euclidean framework and multivariate regression. Within the framework, we systematically compare standard and subspace regression approaches (principal component regression, partial least squares, canonical correlation analysis) for different types of common breathing signals (1D: spirometry, abdominal belt, diaphragm tracking; multi-dimensional: skin surface tracking). Experiments are based on 4D CT and 4D MRI data sets and cover intra- and inter-cycle as well as intra- and inter-session motion variations. Only small differences in internal motion estimation accuracy are observed between the 1D surrogates. Increasing the surrogate dimensionality, however, improved the accuracy significantly; this is shown for both 2D signals, which consist of a common 1D signal and its time derivative, and high-dimensional signals containing the motion of many skin surface points. Eventually, comparing the standard and subspace regression variants when applied to the high-dimensional breathing signals, only small differences in terms of motion estimation accuracy are found.