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Correlations between external marker positions and internal fiducial positions are used by the robotic CyberKnife (CK) for tracking of tumors with respiratory motion. Applied correlation models are established pre-treatment and updated during treatment. Apart from linear models, the CK can also use non-linear models, e.g. for handling phase shifts between internal and external motion. Currently, none of the implemented models has an option to explicitly include time dependency to account for baseline drifts. In this study, we extended the currently implemented non-linear models with an explicit time dependency, and we compared accuracy of predicted internal fiducial positions with and without this option for respiratory motion tracks including baseline drifts and phase shifts between external and internal motion. 

Existing quadratic, dual quadratic and constraint dual fourth order CK models were extended with an option for fitting explicit linear time dependency during model updates. For 7500 synthetic patients, simulations were performed to establish inaccuracies in predicted internal fiducial positions with and without the novel time dependency extension. In building and updating of the novel models, the time dependency was only applied in case of detected statistically significant baseline drifts. A respiratory track generator was used to create motion tracks for the synthetic patients. For each track, respiratory motion baseline, amplitude, period, shape and noise were randomly drawn from distributions found in literature. Additionally, phase shifts of 0, 0.1, 0.2, 0.3 and 0.4rad, and linear baseline drifts of 0, 0.25 and 0.5mm/min were added to respiratory motion tracks in CC direction. For each treatment simulation, the prediction error in CC was quantified by R95; for 95% of the radiation time, the distance between the predicted and real internal fiducial position is ≤ R95.

For the three investigated non-linear models Fig. 1 shows R95 differences between tracking with and without time dependency for a range of phase shifts and baseline drifts. In contrast to the current non-linear CK models, inclusion of time dependency could to a large extent avoid increase of predicted errors caused by baseline drifts. E.g. for a 0 phase shift, mean R95 for the dual quadratic model with time dependency went up from 2.0mm for 0mm/min drift to only 2.3mm for 0.5mm/min drift, while for the conventional dual quadratic model the increase was much larger, from 2.0mm to 4.9mm (Fig. 1). For larger phase shifts, time-dependent models still did better than the original models, but differences were slightly smaller (Fig. 1).

The proposed non-linear correlation models with explicit time dependency significantly enhanced the accuracy of internal fiducial position prediction in respiratory tracking, compared to currently implemented CyberKnife models, while keeping the number of acquired X-ray images fixed.