Int J CARS DOI 10.1007/s11548-008-0211-z
ORIGINAL ARTICLE
Predicting respiratory motion signals for image-guided radiotherapy using multi-step linear methods (MULIN) Floris Ernst · Achim Schweikard
Received: 9 January 2008 / Accepted: 2 May 2008 © CARS 2008
Abstract Objective Forecasting of respiration motion in image-guided radiotherapy requires algorithms that can accurately and efficiently predict target location. Improved methods for respiratory motion forecasting were developed and tested. Materials and methods MULIN, a new family of prediction algorithms based on linear expansions of the prediction error, was developed and tested. Computer-generated data with a prediction horizon of 150 ms was used for testing in simulation experiments. MULIN was compared to Least Mean Squares-based predictors (LMS; normalized LMS, nLMS; wavelet-based multiscale autoregression, wLMS) and a multi-frequency Extended Kalman Filter (EKF) approach. The in vivo performance of the algorithms was tested on data sets of patients who underwent radiotherapy. Results The new MULIN methods are highly competitive, outperforming the LMS and the EKF prediction algorithms in real-world settings and performing similarly to optimized nLMS and wLMS prediction algorithms. On simulated, periodic data the MULIN algorithms are outperformed only by the EKF approach due to its inherent advantage in predicting periodic signals. In the presence of noise, the MULIN methods significantly outperform all other algorithms. Conclusion The MULIN family of algorithms is a feasible tool for the prediction of respiratory motion, performing as well as or better than conventional algorithms while requiring significantly lower computational complexity. The MULIN algorithms are of special importance wherever high-speed prediction is required. F. Ernst (B) · A. Schweikard University of Lübeck, Institute for Robotics and Cognitive Systems, Ratzeburger Allee 160, 23538 Lübeck, Germany e-mail:
[email protected] A. Schweikard e-mail:
[email protected]
Keywords Respiration · Forecasting · Algorithms · Radiosurgery
Introduction In recent years, it has become more and more possible to irradiate tumours in the head and neck without using stereotactic fixation. In this process, the CyberKnife® [7] system has won a firm spot in clinical treatment of cancerous regions in the whole body without using respiratory coaching, gating or fixation. To achieve this, the excursion of the patient’s chest is recorded and correlated to artificial landmarks obtained during stereoscopic X-ray imaging [6,10]. This model is subsequently used to guide the robotic arm carrying a linear accelerator to reduce or eliminate respiratory motion. Since there inevitably will be a systematic delay between the motion of the tumour and the motion of the treatment beam, the need for compensation of this delay is clear. It has been shown previously that successful prediction of the time series stemming from human respiration is indeed possible [1,5,9]. We propose a new family of linear prediction algorithms requiring only little computational complexity while yielding highly competitive prediction results.
Methods Let us assume that y is the (evenly sampled) signal we want to predict, that k is the current position in time and that δ is the prediction horizon. Furthermore, let yˆ be the predicted signal. The simplest member of the new family of prediction algorithms, MULIN0 , is based on the assumption that the difference between the delayed signal and the real signal
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Int J CARS 9
position [mm]
Fig. 1 Top breathing signal of a CyberKnife® patient and the signal delayed by 150ms (sampling rate of 26 Hz), bottom the difference between these two signals
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stays approximately constant over time. That this is—at least partially—the case can be seen from Fig. 1. We thus assume that at position k + δ the difference between the sought-after value yk+δ and the currently measured value yk is roughly the same as between—the known values—yk and yk−δ . Based on this assumption, we define the simple linear prediction algorithm according to the following equation: MULIN0 = yk − (y, δ)k , Yˆk+δ
(1)
where (y, δ) = D(y, δ) − y and D(y, δ) is the signal y delayed by δ sampling steps. Of course, this approach fails quite badly as soon as the above assumption does not hold. This, obviously, is the case whenever the signal’s first derivative changes. The next step is therefore to further expand the prediction error (y, δ)k to take this change into account. This is done by taking higher order differences, i.e. by regarding the term (y, δ)k in Eq. 1 as the unknown and to be predicted quantity. Applying Eq. 1 to (y, δ)k results in the first-order prediction equation, shown in Eq. 2. Yˆk+δ
MULIN1,l
= yk − ((y, δ)k − ((y, δ), l)k ) = yk − 2(y, δ)k + (y, δ)k−l
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(2)
In this extended algorithm, a new parameter l was introduced. This parameter can be seen as something akin to the signal history length of LMS prediction algorithms: it controls how far back the algorithm should look to determine the change in the signal’s first derivative. Naturally, we would expect l to be equal to δ since we are trying to predict the change of the difference signal (y, δ). In those cases, however, where the signal we try to predict is highly irregular or instable, either due to the presence of noise or high signal variation, or the sampling rate is low, it is reasonable to assume that the correlation between yk−l and yk+δ is higher the smaller l becomes. This is highlighted in Fig. 2. Nevertheless, when both the sampling rate and the signalto-noise ratio are high, we can assume that long-term signal dependencies exist. To exploit these dependencies, we repeat the expansion process, generating third and fourth order linear prediction algorithms. This is done by applying Eq. 1 to (y, δ)k , (y, δ)k−l and (y, δ)k−2l . MULIN Yˆk+δ 2,l = yk − 4(y, δ)k + 4(y, δ)k−l − (y, δ)k−2l MULIN Yˆk+δ 3,l = yk − 8(y, δ)k + 12(y, δ)k−l − 6(y, δ)k−2l
+(y, δ)k−3l
(3)
Clearly, in those cases where the correlation between the quantities yk+δ and (y, δ)k−2l as well as between yk+δ and
Int J CARS MULIN0 − minimum at [µ,l,relRMS] = [1, 1, 0.30]
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Fig. 3 Relative nRMS error of the prediction of simulated data (150 ms horizon)
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MULIN2 − minimum at [µ,l,relRMS] = [0.57, 8, 0.04]
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MULIN3 − minimum at [µ,l,relRMS] = [0.7, 5, 0.02]
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(y, δ)k−3l is low, we do not expect improvement over the simple or first-order prediction algorithms, and, in fact, might even witness degradation of the prediction performance. Since we deal with real-world signals which are inevitably corrupted by noise (stemming, among other sources, from the recording process), we introduce an exponential smoothing parameter µ, yielding the final form of the new multi-step linear prediction algorithms. yˆk+δ
=
yˆk+δ
=
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MULINi,l
0 µ · Yˆk+δ i,l µ · Yˆk+δ
0 + (1 − µ) · Yˆk+δ−1 i,l + (1 − µ) · Yˆk+δ−1 ,
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(4)
This new parameter allows us to fine-tune the prediction algorithms’ sensitivity to abrupt changes in the measured time series. As human respiration does not change abruptly between subsequent sampling points (given a sufficiently high sampling rate), this parameter can aid in generating a predicted signal which does not exhibit rapid changes, thus effectively suppressing the measurement noise.
Results The MULIN algorithms were tested using several types of signals, using a prediction horizon of 150 ms. To achieve this, they were implemented in C++ on ubuntu Linux, using a common prediction framework currently under development
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at our laboratory. First, we used the simulated breathing signal described in Eq. 5. This signal thus has a sampling rate of 100 Hz and the required prediction horizon of 150 ms corresponds to 15 steps in time. y = 2 sin(0.25π · t)4 , t = (0, 0.01, . . . , 100)T
(5)
Second, the simulated signal was corrupted with Gaussian noise of zero mean and a standard deviation of σ = 0.025. In a third experiment, the algorithms were tested on the real breathing signal shown in Fig. 1, top. This signal is equidistantly sampled at a rate of 26 Hz and has a length of 7,500 sampling points (just under 5 min). The required prediction horizon thus corresponds to four steps in time. In all cases, the relative nRMS error (Eq. 6) was computed and used as a measure for the quality of the prediction. nRMSrel =
||y − yˆ || ||y − D(y, δ)||
(6)
Figures 3 and 4 show the results for the computer-generated data. It becomes clear that the prediction error can be reduced significantly in all cases and that in the case of a signal with no noise the prediction is extremely good and its quality increases with the order of the prediction algorithm. On the other hand, in case of presence of noise, using higher order algorithms, i.e. using a longer part of the signal’s past to predict the future, deteriorates the result after a certain point.
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Int J CARS MULIN 0 − minimum at [µ,l,relRMS] = [0.5, 1, 0.36]
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Fig. 4 Relative nRMS error of the prediction of simulated data with noise (150 ms horizon)
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MULIN 1 − minimum at [µ,l,relRMS] = [0.23, 21, 0.23] l
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MULIN 2 − minimum at [µ,l,relRMS] = [0.19, 7, 0.41] l
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MULIN 3 − minimum at [µ,l,relRMS] = [0.18, 1, 0.55] l
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Table 1 Time required to compute prediction on a signal with 30,000 samples (Pentium 4, 2.80 GHz, ubuntu Linux) LMS Initialisation (s)
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wLMS 0.037
EKF
MULIN 0 1 2
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Table 2 Relative nRMS errors for different prediction algorithms Predictor
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With noise
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LMS
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0.51
0.72
0.32
0.33
0.64
Prediction (s)
0.017
0.023
1.578
7.751
