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

Authors: Hisashi Johno; Masahide Saito; Hiroshi Onishi
Journal: Computational and Mathematical Methods in Medicine
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

Respiratory-gated radiotherapy (RGRT) is a widely employed means of treating tumors that move with respiration [1–3]. 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). The 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. Therefore, here, we propose a prediction-based system to address the problem.Table 1
Gate on and off latencies of some gating systems.
Monitor
Linac
Gate on delay
Gate off delay
Reference
Abches (APEX)
Elekta Synergy
336 ms
88 ms
[4]
AlignRT (VisionRT)
Varian Clinac iX
356 ms
529 ms
[5]
Calypso (Varian)
Varian Clinac iX
209 ms
60 ms
[5]
Catalyst (C-RAD)
Elekta Synergy
851 ms
215 ms
[6]Figure 1
Problem with conventional RGRT. (a) Given a respiratory signal and a gating threshold, gating windows should ideally be the time when the signal is lower than the threshold. (b) In conventional RGRT, gate on and off commands are sent just when the signal is lower and higher than the threshold, respectively. Hence, gate on and off delays cause deviations of gating windows.
(a)
(b)This paper is organized as follows. The devised framework is described in Section2, experimental results are in Section 3, and the conclusions follow in Section 4.


## 2. Methods

In this section, we describe our new approach to compensate for gate on/off latency. This consists of two steps: (1) multistep-ahead prediction and (2) prediction-based gating.


### 2.1. 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. Therefore, we have devised an algorithm for this purpose.A respiratory signal is regarded as a sequence(1)xt,t=0,1,2,…,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 xt,…,xt+m−1 of subsequent observations, given the previous n-tuple xt−n,…,xt−1. Hence, our goal here is to form a predictor mapping on χn to χm. Suppose χn is a metric space with a metric dn. Let us have a learning set ℒ=x˜i,y˜i∈χn×χm, where i ranges over some finite totally ordered set (see Section 2.3 for an example of the learning set preparation). Then, for a test tuple x∈χn, we predict the next m-tuple as(2)Ψℒx=y˜p,where p is the largest index such that dnx,x˜p≤dnx,x˜i for all i. Throughout this paper, we suppose that χ=ℝ, and χk, which equals ℝk (k=1,2,3,…), is a real k-space with the Euclidean metric, i.e.,(3)dka,b=∑j=0k−1aj−bj2a=a0,…,ak−1∈ℝkb=b0,…,bk−1∈ℝk.


### 2.2. Prediction-Based RGRT

Letxt∈ℝ1 (t≥n) be the current observation, β∈ℝ1 be a gating threshold, and m1 and m0 be the numbers of time points corresponding to gate on and off delays, respectively. Given learning sets ℒ1⊂ℝn×ℝ2m1+1 and ℒ0⊂ℝn×ℝ2m0+1 (see Section 2.3 for an example of the learning set construction), the function Gℒ1,ℒ0 defined below is used for a prediction-based gating.(1)
Casem1≥m0:(4)Gℒ1,ℒ0t,β=1,ifξ2m1+1,βΨℒ1xt<0orξ2m0+1,βΨℒ0xt<0,0,otherwise.(2)
Casem1<m0:(5)Gℒ1,ℒ0t,β=1,ifξ2m1+1,βΨℒ1xt<0andξ2m0+1,βΨℒ0xt<0,0,otherwise,where ξm,β:ℝm⟶ℤ (the set of integers) is defined by(6)ξm,βa0,…,am−1=∑k=0m−1sgnak−β,and xt=xt−n,…,xt−1∈ℝn. Note that sgn:ℝ1⟶−1,0,1 denotes the signum function, i.e.,(7)sgna=1,ifa>0,0,ifa=0,−1,ifa<0.In our prediction-based RGRT system (pRGRT), gate on command is sent ifGℒ1,ℒ0t,β=1, while gate off command is sent if Gℒ1,ℒ0t,β=0.


### 2.3. Construction of a Learning Set

To begin with, a respiratory signal tuplex0,…,xN−1∈ℝN is smoothed using the finite Fourier transform [9]. In detail, the mapping Φα,N:ℝN⟶ℝN α∈ℝ1 defined below is applied for the smoothing.(8a)Φα,Nx0,…,xN−1=x˜0,…,x˜N−1,(8b)s0,…,sN−1=ℱNWx0,…,xN−1,(8c)u0,…,uN−1=ℱN−1Fαs0,…,sN−1,(8d)x˜0,…,x˜N−1=W−1Ru0,…,uN−1,where ℱN is the finite Fourier transform on ℂN (a complex N-space) defined by(9)ℱNa0,…,aN−1=a^0,…,a^N−1,a^k=∑j=0N−1ajexp−2π−1jkNk=0,…,N−1,while its inverse is given by(10)ℱN−1a0,…,aN−1=aˇ0,…,aˇN−1,aˇk=1N∑j=0N−1ajexp2π−1jkNk=0,…,N−1,W
:
ℝ
N
⟶
ℝ
N is defined by(11)Wx0,…,xN−1=w0x0,…,wN−1xN−1,wk=0.54−0.46cos2πkN−1k=0,…,N−1,while its inverse is given by(12)W−1x0,…,xN−1=x0w0,…,xN−1wN−1,F
α
:
ℂ
N
⟶
ℂ
N is defined by(13)Fαs0,…,sN−1=s˜0,…,s˜N−1,s˜k=0,ifk−N2<N2−α,sk,otherwise,k=0,…,N−1,andR:ℂN⟶ℝN is by(14)Ru0,…,uN−1=Reu0,…,ReuN−1.Note thatW defined above is called the Hamming window [10]. The parameter α∈ℝ1 can be set freely, e.g., we set(15)α=NΔτf0≤f≤12Δτ,to filter out signal components with frequencies larger than f hertz (Hz).For a signal tuplex0,…,xN−1∈ℝN,(16)x˜0,…,x˜N−1=Φα,Nx0,…,xN−1,is called the smoothed signal tuple and used to construct a learning set x˜i,y˜i⊂ℝn×ℝm (n+m≤N) by putting(17)x˜i=x˜i,…,x˜i+n−1,y˜i=x˜i+n,…,x˜i+n+m−1,fori=0,…,N−n−m.


## 2.1. 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. Therefore, we have devised an algorithm for this purpose.A respiratory signal is regarded as a sequence(1)xt,t=0,1,2,…,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 xt,…,xt+m−1 of subsequent observations, given the previous n-tuple xt−n,…,xt−1. Hence, our goal here is to form a predictor mapping on χn to χm. Suppose χn is a metric space with a metric dn. Let us have a learning set ℒ=x˜i,y˜i∈χn×χm, where i ranges over some finite totally ordered set (see Section 2.3 for an example of the learning set preparation). Then, for a test tuple x∈χn, we predict the next m-tuple as(2)Ψℒx=y˜p,where p is the largest index such that dnx,x˜p≤dnx,x˜i for all i. Throughout this paper, we suppose that χ=ℝ, and χk, which equals ℝk (k=1,2,3,…), is a real k-space with the Euclidean metric, i.e.,(3)dka,b=∑j=0k−1aj−bj2a=a0,…,ak−1∈ℝkb=b0,…,bk−1∈ℝk.


## 2.2. Prediction-Based RGRT

Letxt∈ℝ1 (t≥n) be the current observation, β∈ℝ1 be a gating threshold, and m1 and m0 be the numbers of time points corresponding to gate on and off delays, respectively. Given learning sets ℒ1⊂ℝn×ℝ2m1+1 and ℒ0⊂ℝn×ℝ2m0+1 (see Section 2.3 for an example of the learning set construction), the function Gℒ1,ℒ0 defined below is used for a prediction-based gating.(1)
Casem1≥m0:(4)Gℒ1,ℒ0t,β=1,ifξ2m1+1,βΨℒ1xt<0orξ2m0+1,βΨℒ0xt<0,0,otherwise.(2)
Casem1<m0:(5)Gℒ1,ℒ0t,β=1,ifξ2m1+1,βΨℒ1xt<0andξ2m0+1,βΨℒ0xt<0,0,otherwise,where ξm,β:ℝm⟶ℤ (the set of integers) is defined by(6)ξm,βa0,…,am−1=∑k=0m−1sgnak−β,and xt=xt−n,…,xt−1∈ℝn. Note that sgn:ℝ1⟶−1,0,1 denotes the signum function, i.e.,(7)sgna=1,ifa>0,0,ifa=0,−1,ifa<0.In our prediction-based RGRT system (pRGRT), gate on command is sent ifGℒ1,ℒ0t,β=1, while gate off command is sent if Gℒ1,ℒ0t,β=0.


## 2.3. Construction of a Learning Set

To begin with, a respiratory signal tuplex0,…,xN−1∈ℝN is smoothed using the finite Fourier transform [9]. In detail, the mapping Φα,N:ℝN⟶ℝN α∈ℝ1 defined below is applied for the smoothing.(8a)Φα,Nx0,…,xN−1=x˜0,…,x˜N−1,(8b)s0,…,sN−1=ℱNWx0,…,xN−1,(8c)u0,…,uN−1=ℱN−1Fαs0,…,sN−1,(8d)x˜0,…,x˜N−1=W−1Ru0,…,uN−1,where ℱN is the finite Fourier transform on ℂN (a complex N-space) defined by(9)ℱNa0,…,aN−1=a^0,…,a^N−1,a^k=∑j=0N−1ajexp−2π−1jkNk=0,…,N−1,while its inverse is given by(10)ℱN−1a0,…,aN−1=aˇ0,…,aˇN−1,aˇk=1N∑j=0N−1ajexp2π−1jkNk=0,…,N−1,W
:
ℝ
N
⟶
ℝ
N is defined by(11)Wx0,…,xN−1=w0x0,…,wN−1xN−1,wk=0.54−0.46cos2πkN−1k=0,…,N−1,while its inverse is given by(12)W−1x0,…,xN−1=x0w0,…,xN−1wN−1,F
α
:
ℂ
N
⟶
ℂ
N is defined by(13)Fαs0,…,sN−1=s˜0,…,s˜N−1,s˜k=0,ifk−N2<N2−α,sk,otherwise,k=0,…,N−1,andR:ℂN⟶ℝN is by(14)Ru0,…,uN−1=Reu0,…,ReuN−1.Note thatW defined above is called the Hamming window [10]. The parameter α∈ℝ1 can be set freely, e.g., we set(15)α=NΔτf0≤f≤12Δτ,to filter out signal components with frequencies larger than f hertz (Hz).For a signal tuplex0,…,xN−1∈ℝN,(16)x˜0,…,x˜N−1=Φα,Nx0,…,xN−1,is called the smoothed signal tuple and used to construct a learning set x˜i,y˜i⊂ℝn×ℝm (n+m≤N) by putting(17)x˜i=x˜i,…,x˜i+n−1,y˜i=x˜i+n,…,x˜i+n+m−1,fori=0,…,N−n−m.


## 3. 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 respiration-monitoring 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.


### 3.1. 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 tuplex=x0,…,xN−1. Then Φα,Nx 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.Figure 2
Smoothing of a respiratory signal. The phantom’s signal was measured for 20 s, and an artificial noise was added (13.65–13.7 s), forming a signal tuplex=x0,…,xN−1. Then Φα,Nx was calculated (Equations (8a)–(8d)), setting α=NΔτ to filter out high frequency (>1 Hz) components. Note that power indicates s0,…,sN−1/2, where s0,…,sN−1=ℱNWx0,…,xN−1, s denotes the absolute value of s∈ℂ1, and a is the largest integer smaller than or equal to a∈ℝ1. The units of signal value, time, and frequency are mm, s, and Hz, respectively.


### 3.2. Prediction of a Respiratory Signal

The prediction algorithm was tested using respiratory signals of ten volunteers, measured for 300 s (10000 time points) (Figure3). 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 x0,…,xM−1 denote a signal sample, where M=10000. For each t=N+n,…,M−m, the signal tuple xt−n−N,…,xt−n−1∈ℝN was used to construct a learning set ℒt⊂ℝn×ℝm as in Section 2.3. Then, Ψℒtxt∈ℝm was calculated (Section 2.1), where xt=xt−n,…,xt−1. To evaluate the prediction accuracy, the mth coordinate of Ψℒtxt, denoted as x^t+m−1, was compared with the corresponding actual observation xt+m−1. In accordance with the previous studies of predicting respiratory motion [7], the root mean square error (RMSE) (mm)(18)∑​i=N+n+m−1M−1x^i−xi2M−N−n−m+1,was calculated as an indicator of prediction error (Figure 4). The 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.Figure 3
Respiratory signal samples of ten volunteers measured for 300 s. The units of signal value and time are mm and s, respectively.Figure 4
Prediction errors for the ten samples. For each sample, RMSE (mm) value was calculated (Section3.2) and is plotted here.Figure 5
Representative prediction result. Predicted signal corresponds tox^5000,…,x^5999 of the serial prediction trials (Section 3.2) using the sample numbered 7. The units of signal value and time are mm and s, respectively.Figure 6
Representative prediction result. Predicted signal corresponds tox^6000,…,x^6999 of the serial prediction trials (Section 3.2) using the sample numbered 9. The units of signal value and time are mm and s, respectively.


### 3.3. 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 (Table1). For each time point t≥N+n of a sample x0,…,xM−1, the signal tuple xt−n−N,…,xt−n−1∈ℝN was used to construct learning sets ℒ1,t⊂ℝn×ℝ2m1+1 and ℒ0,t⊂ℝn×ℝ2m0+1 as in Section 2.3, where M=10000 (300 s), N=4000 (120 s), n=100 (3 s), m1=12 (0.336 s), and m0=3 (0.088 s). We put gj and g^j as in Algorithm 1 and Algorithm 2, respectively, where β was fixed to the median of x0,…,xN−1.Algorithm 1:Simulation of conventional RGRT.
(1)
for t⟵N+n,…,M−minm0,m1, do
(2)
if xt−1<β, then
(3)
g
t
+
m
1
−
1
,
…
,
g
M
−
1
⟵
1,
(4)
else
(5)
g
t
+
m
0
−
1
,
…
,
g
M
−
1
⟵
0,
(6)
end if
(7)
end forAlgorithm 2:Simulation of pRGRT.
(1)
for t⟵N+n,…,M−minm0,m1, do
(2)
if Gℒ1,t,ℒ0,tt,β=1, then
(3)
g
^
t
+
m
1
−
1
,
…
,
g
^
M
−
1
⟵
1,
(4)
else
(5)
g
^
t
+
m
0
−
1
,
…
,
g
^
M
−
1
⟵
0,
(6)
end if
(7)
end forForj∈S=N+n+m1−1,…,M−1, we assumed that gate on command is executed at j,(i)
if and only ifgj=1 (in conventional RGRT).(ii)
if and only ifg^j=1 (in pRGRT).In each of the RGRT simulations, letS1 be the set of j∈S at which gate on command is executed, and put S0=S\S1. To quantify possibly inappropriate irradiation during RGRT, the value(19)∑j∈SχS1jxj++χS0jxj−M−N−n−m1+1,was calculated and denoted as nErr (normalized error), whose unit is mm. Here, χS represents the characteristic function of a set S defined as(20)χSj=1,ifj∈S,0,ifj∉S,xj+=maxxj−β,0, and xj−=−minxj−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 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].Figure 7
Conventional and prediction-based RGRT systems, denoted here as RGRT and pRGRT, respectively. (a) In RGRT, gate on and off delays cause shifts of gating windows. Stated informally, nErr corresponds to the mean absolute height of the shaded area (mm). (b) In pRGRT, gate on and off commands are expected to be sent in advance to compensate for the latencies.
(a)
(b)Figure 8
nErr values for the selected five samples in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) mimicking the Abches system. The unit of nErr is mm.Figure 9
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the Abches system, using the sample numbered 7. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 5000≤j≤5333. The units of signal value and time are mm and s, respectively.Figure 10
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the Abches system, using the sample numbered 8. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 6334≤j≤6999. The units of signal value and time are mm and s, respectively.The above are cases wherem1≥m0. To see whether pRGRT works when m1<m0, similar simulations were performed with gate on and off delays being 0.356 s (m1=12) and 0.529 s (m0=18), respectively, in accordance with the the AlignRT system (Table 1). The outcome was that nErr values for all the samples decreased in pRGRT (Figure 11) and gating window shifts in conventional RGRT were ameliorated in pRGRT (Figure 12).Figure 11
nErr values for the selected five samples in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) mimicking the AlignRT system. The unit of nErr is mm.Figure 12
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the AlignRT system, using the sample numbered 7. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 4334≤j≤4666. The units of signal value and time are mm and s, respectively.


## 3.1. 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 tuplex=x0,…,xN−1. Then Φα,Nx 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.Figure 2
Smoothing of a respiratory signal. The phantom’s signal was measured for 20 s, and an artificial noise was added (13.65–13.7 s), forming a signal tuplex=x0,…,xN−1. Then Φα,Nx was calculated (Equations (8a)–(8d)), setting α=NΔτ to filter out high frequency (>1 Hz) components. Note that power indicates s0,…,sN−1/2, where s0,…,sN−1=ℱNWx0,…,xN−1, s denotes the absolute value of s∈ℂ1, and a is the largest integer smaller than or equal to a∈ℝ1. The units of signal value, time, and frequency are mm, s, and Hz, respectively.


## 3.2. Prediction of a Respiratory Signal

The prediction algorithm was tested using respiratory signals of ten volunteers, measured for 300 s (10000 time points) (Figure3). 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 x0,…,xM−1 denote a signal sample, where M=10000. For each t=N+n,…,M−m, the signal tuple xt−n−N,…,xt−n−1∈ℝN was used to construct a learning set ℒt⊂ℝn×ℝm as in Section 2.3. Then, Ψℒtxt∈ℝm was calculated (Section 2.1), where xt=xt−n,…,xt−1. To evaluate the prediction accuracy, the mth coordinate of Ψℒtxt, denoted as x^t+m−1, was compared with the corresponding actual observation xt+m−1. In accordance with the previous studies of predicting respiratory motion [7], the root mean square error (RMSE) (mm)(18)∑​i=N+n+m−1M−1x^i−xi2M−N−n−m+1,was calculated as an indicator of prediction error (Figure 4). The 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.Figure 3
Respiratory signal samples of ten volunteers measured for 300 s. The units of signal value and time are mm and s, respectively.Figure 4
Prediction errors for the ten samples. For each sample, RMSE (mm) value was calculated (Section3.2) and is plotted here.Figure 5
Representative prediction result. Predicted signal corresponds tox^5000,…,x^5999 of the serial prediction trials (Section 3.2) using the sample numbered 7. The units of signal value and time are mm and s, respectively.Figure 6
Representative prediction result. Predicted signal corresponds tox^6000,…,x^6999 of the serial prediction trials (Section 3.2) using the sample numbered 9. The units of signal value and time are mm and s, respectively.


## 3.3. 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 (Table1). For each time point t≥N+n of a sample x0,…,xM−1, the signal tuple xt−n−N,…,xt−n−1∈ℝN was used to construct learning sets ℒ1,t⊂ℝn×ℝ2m1+1 and ℒ0,t⊂ℝn×ℝ2m0+1 as in Section 2.3, where M=10000 (300 s), N=4000 (120 s), n=100 (3 s), m1=12 (0.336 s), and m0=3 (0.088 s). We put gj and g^j as in Algorithm 1 and Algorithm 2, respectively, where β was fixed to the median of x0,…,xN−1.Algorithm 1:Simulation of conventional RGRT.
(1)
for t⟵N+n,…,M−minm0,m1, do
(2)
if xt−1<β, then
(3)
g
t
+
m
1
−
1
,
…
,
g
M
−
1
⟵
1,
(4)
else
(5)
g
t
+
m
0
−
1
,
…
,
g
M
−
1
⟵
0,
(6)
end if
(7)
end forAlgorithm 2:Simulation of pRGRT.
(1)
for t⟵N+n,…,M−minm0,m1, do
(2)
if Gℒ1,t,ℒ0,tt,β=1, then
(3)
g
^
t
+
m
1
−
1
,
…
,
g
^
M
−
1
⟵
1,
(4)
else
(5)
g
^
t
+
m
0
−
1
,
…
,
g
^
M
−
1
⟵
0,
(6)
end if
(7)
end forForj∈S=N+n+m1−1,…,M−1, we assumed that gate on command is executed at j,(i)
if and only ifgj=1 (in conventional RGRT).(ii)
if and only ifg^j=1 (in pRGRT).In each of the RGRT simulations, letS1 be the set of j∈S at which gate on command is executed, and put S0=S\S1. To quantify possibly inappropriate irradiation during RGRT, the value(19)∑j∈SχS1jxj++χS0jxj−M−N−n−m1+1,was calculated and denoted as nErr (normalized error), whose unit is mm. Here, χS represents the characteristic function of a set S defined as(20)χSj=1,ifj∈S,0,ifj∉S,xj+=maxxj−β,0, and xj−=−minxj−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 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].Figure 7
Conventional and prediction-based RGRT systems, denoted here as RGRT and pRGRT, respectively. (a) In RGRT, gate on and off delays cause shifts of gating windows. Stated informally, nErr corresponds to the mean absolute height of the shaded area (mm). (b) In pRGRT, gate on and off commands are expected to be sent in advance to compensate for the latencies.
(a)
(b)Figure 8
nErr values for the selected five samples in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) mimicking the Abches system. The unit of nErr is mm.Figure 9
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the Abches system, using the sample numbered 7. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 5000≤j≤5333. The units of signal value and time are mm and s, respectively.Figure 10
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the Abches system, using the sample numbered 8. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 6334≤j≤6999. The units of signal value and time are mm and s, respectively.The above are cases wherem1≥m0. To see whether pRGRT works when m1<m0, similar simulations were performed with gate on and off delays being 0.356 s (m1=12) and 0.529 s (m0=18), respectively, in accordance with the the AlignRT system (Table 1). The outcome was that nErr values for all the samples decreased in pRGRT (Figure 11) and gating window shifts in conventional RGRT were ameliorated in pRGRT (Figure 12).Figure 11
nErr values for the selected five samples in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) mimicking the AlignRT system. The unit of nErr is mm.Figure 12
Gating windows in the conventional and prediction-based RGRT simulations (denoted here as RGRT and pRGRT, respectively) which mimic the AlignRT system, using the sample numbered 7. The colored rectangles of RGRT and pRGRT correspond toj:gj=1 and j:g^j=1, respectively, where 4334≤j≤4666. The units of signal value and time are mm and s, respectively.


## 4. 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 prediction-based 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.The developed method has potential applicability in RGRT, but there are several issues to be addressed, e.g.,(1)
Are there better algorithms for multistep-ahead prediction?(2)
Are there better algorithms for prediction-based gating?(3)
Is it possible to deal with baseline drift?(4)
Is it possible to provide theoretical foundations to the methods?(5)
Is the method valid in a real clinical setting?Further studies on these matters would be needed for the system to be of practical use.



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*Source: 5919467-2018-11-27.xml*