Prediction-Based Compensation for Gate On/Off Latency during Respiratory-Gated Radiotherapy

During respiratory-gated radiotherapy (RGRT), gate on and off latencies cause deviations of gating windows, possibly leading to delivery of low- and high-dose radiations to tumors and normal tissues, respectively. Currently, there are no RGRT systems that have definite tools to compensate for the delays. To address the problem, we propose a framework consisting of two steps: (1) multistep-ahead prediction and (2) prediction-based gating. For each step, we have devised a specific algorithm to accomplish the task. Numerical experiments were performed using respiratory signals of a phantom and ten volunteers, and our prediction-based RGRT system exhibited superior performance in more than a few signal samples. In some, however, signal prediction and prediction-based gating did not work well, maybe due to signal irregularity and/or baseline drift. The proposed approach has potential applicability in RGRT, and further studies are needed to verify and refine the constituent algorithms.


Introduction
Respiratory-gated radiotherapy (RGRT) is a widely employed means of treating tumors that move with respiration [1][2][3]. In RGRT, radiation is administered within particular phases of the patient's breathing cycle (called as gating windows), which are determined by monitoring respiratory motion in the form of a respiratory signal using either external or internal markers. Note that, although there are some options for RGRT (e.g., whether to choose amplitude-based or phase-based gating and whether to gate during inhalation or exhalation), this study focuses only on amplitude-based gating during exhalation, which is a common setting in clinical practice. Several RGRT systems have been developed, and some take considerable time from the detection of a signal change to the execution of a gate on/off command (Table 1). e gate on/off latency causes deviations of gating windows in conventional RGRT (Figure 1), possibly leading to delivery of low-and high-dose radiation to tumor and normal tissues, respectively. At present, there are no RGRT systems that have definite techniques to compensate for the delays. erefore, here, we propose a prediction-based system to address the problem. is paper is organized as follows. e devised framework is described in Section 2, experimental results are in Section 3, and the conclusions follow in Section 4.

Methods
In this section, we describe our new approach to compensate for gate on/off latency.

Multistep-Ahead Prediction.
Several prediction algorithms for respiratory signals have been proposed, and most of them adopt single-output strategies [7,8]. However, in our framework, multiple-output multistep-ahead prediction is required. erefore, we have devised an algorithm for this purpose.
A respiratory signal is regarded as a sequence of equally spaced time-series observations in a space χ, with a time interval of Δτ seconds (s), where Δτ > 0. Let n and m be positive integers. For each time point t ≥ n, multistep-ahead prediction aims to forecast the m-tuple (x t , . . . , x t+m−1 ) of subsequent observations, given the previous n-tuple (x t−n , . . . , x t−1 ). Hence, our goal here is to form a predictor mapping on χ n to χ m . Suppose χ n is a metric space with a metric d n . Let us have a learning set L � (x i , y i ) ∈ χ n × χ m , where i ranges over some finite totally ordered set (see Section 2.3 for an example of the learning set preparation). en, for a test tuple x ∈ χ n , we predict the next m-tuple as where p is the largest index such that d n (x, x p ) ≤ d n (x, x i ) for all i. roughout this paper, we suppose that χ � R, and χ k , which equals R k (k � 1, 2, 3, . . .), is a real k-space with the Euclidean metric, i.e.,

Prediction-Based RGRT
. Let x t ∈ R 1 (t ≥ n) be the current observation, β ∈ R 1 be a gating threshold, and m 1 and m 0 be the numbers of time points corresponding to gate on and off delays, respectively. Given learning sets L 1 ⊂ R n × R 2m 1 +1 and L 0 ⊂ R n × R 2m 0 +1 (see Section 2.3 for an example of the learning set construction), the function G L 1 ,L 0 defined below is used for a prediction-based gating.

Construction of a Learning Set.
To begin with, a respiratory signal tuple (x 0 , . . . , x N−1 ) ∈ R N is smoothed using the finite Fourier transform [9]. In detail, the mapping Φ α,N : R N ⟶ R N (α ∈ R 1 ) defined below is applied for the smoothing.  where F N is the finite Fourier transform on C N (a complex N-space) defined by while its inverse is given by while its inverse is given by and R : Note that W defined above is called the Hamming window [10]. e parameter α ∈ R 1 can be set freely, e.g., we set to filter out signal components with frequencies larger than f hertz (Hz). For a signal tuple (x 0 , . . . , is called the smoothed signal tuple and used to construct a learning set (

Numerical Results and Discussion
To validate the devised algorithms, respiratory signals of a dynamic thoracic phantom (CIRS, Virginia, USA) and ten healthy volunteers were measured with Abches (APEX Medical, Inc., Tokyo, Japan), which is a respirationmonitoring device developed by Onishi et al. [11] and routinely used in our university hospital. Note that, for simplicity, we supposed that Δτ � 0.03 although the actual time intervals were not precisely equal to 0.03 s. Signal values were given in the unit of mm.

Smoothing of a Respiratory Signal.
To test the algorithm of smoothing a respiratory signal, the phantom's signal was measured for 20 s (667 time points) and an artificial noise was added (13.65-13.7 s), forming a signal tuple x � (x 0 , . . . , x N−1 ). en Φ α,N (x) was calculated (Equations (8a)-(8d)), setting α � NΔτ to filter out high frequency (>1 Hz) components. As shown in Figure 2, we succeeded in removing noisy components of x.

Prediction of a Respiratory
Signal. e prediction algorithm was tested using respiratory signals of ten volunteers, measured for 300 s (10000 time points) (Figure 3). For each time point of a signal sample, observations during the past 120 s (4000 points) were used to construct a learning set and a predictor is formed to forecast the next 0.3 s (10 points) given the previous 3 s (100 points). In detail, let N � 4000, n � 100, m � 10, and x 0 , . . . , x M−1 denote a signal sample, where M � 10000. For each t � N + n, . . . , M − m, the signal tuple (x t−n−N , . . . , x t−n−1 ) ∈ R N was used to construct a learning set L t ⊂ R n × R m as in Section 2.3. en, Ψ L t (x t ) ∈ R m was calculated (Section 2.1), where x t � (x t−n , . . . , x t−1 ). To evaluate the prediction accuracy, the mth coordinate of Ψ L t (x t ), denoted as x t+m−1 , was compared with the corresponding actual observation x t+m−1 . In accordance with the previous studies of predicting respiratory motion [7], the root mean square error (RMSE) (mm) ����������������� was calculated as an indicator of prediction error (Figure 4). e signal samples with RMSE less than 1.5 mm appeared to be well predictable by our approach (Figure 5), while some of the others appeared not to ( Figure 6). Hence, the former samples numbered 0, 1, 2, 7, and 8 were selected for the next experiment.

Prediction-Based RGRT.
Our prediction-based gating system, pRGRT, was tested using the selected five signal samples. In the following experiment, gate on and off delays were set to be 0.336 s and 0.088 s, respectively, in accordance with the Abches system (Table 1) . , x t−n−1 ) ∈ R N was used to construct learning sets L 1,t ⊂ R n × R 2m 1 +1 and L 0,t ⊂ R n × R 2m 0 +1 as in Section 2.3, where M � 10000 (300 s), N � 4000 (120 s), n � 100 (3 s), m 1 � 12 (0.336 s), and m 0 � 3 (0.088 s). We put g j and g j as in Algorithm 1 and Algorithm 2, respectively, where β was fixed to the median of x 0 , . . . , x N−1 .
For j ∈ S � N + n + m 1 − 1, . . . , M − 1 , we assumed that gate on command is executed at j, In each of the RGRT simulations, let S 1 be the set of j ∈ S at which gate on command is executed, and put S 0 � S\S 1 . To quantify possibly inappropriate irradiation during RGRT, the value was calculated and denoted as nErr (normalized error), whose unit is mm. Here, χ S represents the characteristic function of a set S defined as x + j � max x j − β, 0 , and x − j � −min x j − β, 0 . Schematic illustrations of nErr and pRGRT are shown in Figure 7. As a result, nErr values for four out of the five samples . , x N−1 ))), |s| denotes the absolute value of s ∈ C 1 , and ⌊a⌋ is the largest integer smaller than or equal to a ∈ R 1 . e units of signal value, time, and frequency are mm, s, and Hz, respectively. decreased in pRGRT (Figure 8). Regarding the four samples, gating window shifts observed in conventional RGRT appeared to be improved in pRGRT ( Figure 9). As for the other sample (numbered 8), considerable baseline drift was observed (Figure 10), which is an undesirable feature for gating systems with fixed threshold [12].  0  50  100  150  200  250  300  Time   24  16  8  0  0  50  100  150  200  250  300  Time   24  16  8      ( end if (7) end for ALGORITHM 1: Simulation of conventional RGRT.

Conclusions
In this paper, we proposed a framework to compensate for gate on/off latency during RGRT. It consisted of two steps: (1) multistep-ahead prediction and (2) prediction-based gating.  For each step, we devised a specific algorithm to accomplish the task. Numerical experiments were performed using respiratory signals of a phantom and ten volunteers, and our predictionbased RGRT system, pRGRT, displayed superior performance in not a few of the signal samples. In some, however, signal prediction and prediction-based gating did not work well, probably because of signal irregularity and/or baseline drift.
e developed method has potential applicability in RGRT, but there are several issues to be addressed, e.g.,   Further studies on these matters would be needed for the system to be of practical use.

Conflicts of Interest
e authors declare no conflicts of interest.