[3] Achim Schweikard, Greg Glosser, Mohan Bodduluri, Martin J. Murphy, and John R. Adler. Robotic motion compensation for respiratory motion during ...
A Family of Linear Algorithms for the Prediction of Respiratory Motion for Image-Guided Radiotherapy Floris Ernst, Achim Schweikard Institute for Robotics and Cognitive Systems, University of Lübeck {ernst,schweikard}@rob.uni-luebeck.de
Purpose
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Over the last years the CyberKnife system has won a firm spot in clinical treatment of cancerous regions in the body stem without using respiratory coaching, gating or fixation. To achieve this, the excursion of the patient’s chest is recorded and correlated to landmarks obtained during stereoscopic X-ray imaging [3]. This model is subsequently used to guide the robotic arm carrying a linear accelerator to reduce or eliminate respiratory motion. It has been shown that successful prediction of the time series stemming from human respiration to compensate for system latency is indeed possible [1, 2, 4]. We propose a new family of linear prediction algorithms requiring only little computational complexity while yielding highly competitive prediction results.
Methods 10
Let us assume that y is the signal we want to predict, that k is the current position in time and that is the prediction horizon. Furthermore, let yˆ be the predicted signal. The simplest member of the new family of prediction algorithms is based on the assumption that the difference between the delayed signal and the real signal stays approximately constant over time. Based on this assumption, we define the simple linear prediction algorithm according to Equation (1). Yˆk0 yk ( y, ) k ,
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(1)
where ( y, ) D( y, ) y and D( y, ) is the signal y delayed by sampling steps. This approach fails as soon as the above assumption does not hold. This clearly is the case whenever the signal’s first derivative changes. We thus further expand the prediction error ( y, ) k , by taking higher order differences, to take this change into account. By repeating this process and introducing a step-size control parameter l and an exponential smoothing parameter , we arrive at the final form of the family of linear prediction algorithms:
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yˆ k0 Yˆk0 (1 ) Yˆk0 1 yˆ ki ,l Yˆki,l (1 ) Yˆki,l 1 , i 1,2,3,
(2)
Yˆk1,l yk 2( y, ) k ( y, ) k l , Yˆk2,l yk 4( y, ) k 4( y, ) k l ( y, ) k 2l , Yˆk3,l yk 8( y, ) k 12( y, ) k l 6( y, ) k 2l ( y, ) k 3l .
(3)
where
Results The new prediction algorithms were tested using several types of signals, using a prediction horizon of 150ms. First, we used the simulated breathing signal described in Equation (4). y 2 sin(0.25 t ) 4 , t (0,0.01,...,100) T
(4)
Second, the simulated signal was corrupted with Gaussian noise of zero mean and a standard deviation of = 25
0.025mm. In a third experiment, the algorithms were tested on a real breathing signal. This signal is sampled at a rate of 26Hz and has a length of 7,500 sampling points. In all cases, the relative nRMS error was computed and used as a measure for the quality of the prediction. The same signals were also predicted using the multi-
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frequency Extended Kalman Filter approach presented in [2] and by the LMS-based prediction algorithms outlined in [1]. The results, comparing all prediction algorithms, are shown in Table 1. It is important to note that these results were obtained by global optimization, i.e. by using a grid search to determine the optimal prediction parameters. In the case of the wLMS and the EKF predictors, there is also the need for a learning window to initialise the signal’s wavelet decomposition (wLMS) or to find the frequencies present in the signal (EKF). Predictor LMS nLMS wLMS EKF best linear alg.
simulated data no noise 0.9926 0.3174 0.0817 0.0093 0.0245
with noise 0.5116 0.3286 0.5606 0.6271 0.2312
real data 0.7170 0.6376 0.6281 0.7763 0.6459
Table 1 – Relative nRMS errors for different prediction algorithms
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We can see clearly that, on real world data, the simple linear prediction algorithm outperforms the by far more complex EKF algorithm and comes very close to the best value attainable using the LMS and wLMS prediction algorithms. In the setting of simulated data without noise the EKF algorithm clearly outperforms all the other algorithms. This is due to the fact that we have a signal which can be modelled very accurately using super positioned sinusoidals. This advantage is, however, hardly significant since the EKF’s prediction error sharply increases as soon as the signal is corrupted by (relatively moderate) Gaussian noise. To further evaluate the performance of the new family of algorithms, a database of 299 breathing signals of patients treated with the CyberKnife system at Georgetown University Hospital was used. From each signal we selected 10,000 sampling points (26Hz sampling rate) and evaluated all algorithms on those data sets. The results are shown as cumulative histograms in Figure 1.
Figure 1 – Cumulative histograms of the relative performance of the best linear predictors compared to LMS, nLMS, wLMS and EKF prediction
Conclusion
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The algorithms presented in this work are of high importance when it comes to the prediction of respiratory motion in image-guided radiosurgery. Since we expect that the sampling rate of the tracking cameras as well as their accuracy will improve and the repositioning speed of the robotic system will be reduced, the need for simple, robust, and fast algorithms is evident. Using our algorithms, the prediction of one signal point can be achieved using less than twenty arithmetical operations whereas the more sophisticated EKF and wLMS algorithms require highly time-consuming computations. The performance of the new algorithms is about as good as the performance of the by far more complex wLMS or EKF approaches.
References [1] Floris Ernst, Alexander Schlaefer, and Achim Schweikard. Prediction of respiratory motion with wavelet-based multiscale
autoregression. In N. Ayache, S. Ourselin, and A. Maeder, editors, MICCAI 2007, Part II, volume 10 of Lecture Notes in Computer Science, pages 668–675, Brisbane, Australia, November 2007. MICCAI, Springer. [2] Lukas Ramrath, Alexander Schlaefer, Floris Ernst, Sonja Dieterich, and Achim Schweikard. Prediction of respiratory motion with a multi-frequency based Extended Kalman Filter. In Proceedings of the 27th International Conference and Exhibition on Computer Assisted Radiology and Surgery (CARS'07), volume 27, Berlin, Germany, June 2007. CARS. [3] Achim Schweikard, Greg Glosser, Mohan Bodduluri, Martin J. Murphy, and John R. Adler. Robotic motion compensation for respiratory motion during radiosurgery. Journal of Computer-Aided Surgery, 5(4):263–277, September 2000. [4] Gregory C. Sharp, Steve B. Jiang, Shinichi Shimizu, and Hiroki Shirato. Prediction of respiratory tumour motion for real-time image-guided radiotherapy. Physics in Medicine and Biology, 49:425–440, 2004.
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