Respiratory Motion Prediction with Empirical Mode Decomposition-Based Random Vector Functional Link

Abstract

The precise prediction of tumor motion for radiotherapy has proven challenging due to the non-stationary nature of respiration-induced motion, frequently accompanied by unpredictable irregularities. Despite the availability of numerous prediction methods for respiratory motion prediction, the prediction errors they generate often suffer from large prediction horizons, intra-trace variabilities, and irregularities. To overcome these challenges, we have employed a hybrid method, which combines empirical mode decomposition (EMD) and random vector functional link (RVFL), referred to as EMD-RVFL. In the initial stage, EMD is used to decompose respiratory motion into interpretable intrinsic mode functions (IMFs) and residue. Subsequently, the RVFL network is trained for each obtained IMF and residue. Finally, the prediction results of all the IMFs and residue are summed up to obtain the final predicted output. We validated this proposed method on the benchmark datasets of 304 respiratory motion traces obtained from 31 patients for various prediction lengths, which are equivalent to the latencies of radiotherapy systems. In direct comparison with existing prediction techniques, our hybrid architecture consistently delivers a robust and highly accurate prediction performance. This proof-of-concept study indicates that the proposed approach is feasible and has the potential to improve the accuracy and effectiveness of radiotherapy treatment.

1. Introduction

Stereo-tactic body radiation therapy (SBRT) is a widely-used treatment approach for cancer patients that involves using ionizing radiation to kill cancer cells [1]. The goal of radiotherapy treatment is to provide a conformal dosage to the clinical target volume (CTV) while minimizing the risk of damage to nearby organs and tissues. However, the precision of radiotherapy may be undermined by involuntary respiratory motion, resulting in inadvertent displacement within the abdominal and thoracic regions, including the lungs, liver, and pancreas, with deviations ranging from 20 to 50 mm. Such unintended movement can significantly impact the effectiveness of radiotherapy, potentially resulting in deviations from the prescribed dosage [1,2]. Hence, the main goal of radiotherapy is to predict the motion of abdominal and thoracic tumors, thereby enhancing the precision of treatment in the presence of motion induced by respiration. One of the main challenges in predicting tumor motion in real time is the time lag or latency that exists in existing radiotherapy systems for commercial use, such as CyberKnife™ [3]. Due to intrinsic data processing and mechanical limitations, the positional latency can range from 50 ms to 500 ms [1,4] depending on the specific system. To mitigate this delay, it is recommended to predict tumor motion with a prediction length that is equal to the delay of the system [2]. However, accurately predicting tumor motion is challenging due to the variations in respiratory motion patterns between patients, as well as the irregularities and temporal fluctuations that can occur [5]. These factors make it difficult to predict the exact motion of the tumor, even with advanced predictive models. Despite these challenges, significant progress has been made in recent years to improve the accuracy of radiotherapy in the presence of respiratory-induced motion.

Several respiratory motion prediction algorithms have been developed, which can be divided into two types: model-based and model-free approaches. In model-based approaches, functions that are either linear or nonlinear are developed to mimic the properties of respiratory motion. Existing model-based approaches, such as multi-scale regression (wLMS) [6], normalized least mean square (nLMS) [7], MULIN algorithm [8], and local circular motion-based extended Kalman filter (LCM-EKF) [9] are included. On the other hand, model-free methods, alternatively, do not formulate any specific model for motion characteristics, as an alternative to forecast the location by taking into account the preceding observations. Existing model-free approaches include support vector regression (SVR) [10,11], relevance vector machines (RVM) [12,13], extreme learning machine (ELM) [14,15,16,17,18], random vector functional link (RVFL) [19], convolution neural network (CNN) [20], long short-term memory (LSTM) [21,22,23], and forecasting random convolution node (fRCN) [24]. A recent comprehensive evaluation of existing approaches [7] reveals that model-free approaches predict respiratory motion with greater robustness and less prediction error than model-based counterparts, as a result, yielding improved prediction performance, especially at larger prediction lengths.

One of the main limitations of the existing algorithms is that they rely on modeling the raw respiratory motion trace, which includes various non-stationary components and time-varying irregularities. This makes it difficult to accurately capture the essential features necessary for estimating a generalized model for respiratory motion variations. However, detecting these irregularities early on can be highly advantageous since this allows medical physicists and doctors to halt the treatment beam at the precise moment during radiotherapy treatment, thus minimizing the risk of harm to the patient. Therefore, there is a need for more advanced algorithms that can effectively handle the complex and dynamic nature of respiratory motion and provide more accurate predictions of tumor motion during radiotherapy. Such algorithms could greatly enhance the safety and effectiveness of radiotherapy treatment and improve patient outcomes.

To deal with the irregularities and intra-trace variabilities in respiratory motion, the hybrid approach can be strategically built up by combining multiple algorithms [25]. The advantages of hybrid approaches are based on three primary reasons: statistical, computational, and representational [25]. Hybrid techniques are divided into two groups based on their combination: parallel and sequential [26]. In a parallel combined hybrid technique, the training signal is first divided into a number of sub-datasets, each of which is then examined separately and merged to obtain the final prediction result; on the other hand, in a sequentially combined hybrid approach, the outputs from one model are treated as the inputs for another prediction model. Empirical mode decomposition (EMD) is one of the examples of the parallel combined hybrid approach. This approach decomposes time-series data by examining time-domain attributes such as local maxima, local minima, and zero-crossings. EMD has been widely employed to estimate the respiratory motion from other biomedical signals [27,28]. Furthermore, it has found extensive application in various prediction tasks, including short-term load forecasting (STLF) [29], forecasting financial time-series [30], predicting wind speed [31], etc. The authors in [32] used EMD to extract IMFs from the respiratory signal, and then employed deep learning to construct an effective prediction model, enabling accurate predictions.

Rizal et al. [33] used EMD to analyze lung sounds, obtaining an impressive 98.8% overall accuracy for the first 10 intrinsic mode functions (IMFs) with multilayer perceptron (MLP) validation. Similarly, Karan et al. [34] proposed an IMF-based cepstral feature extraction for neurological disorder detection, achieving a significant 10–20% improvement in accuracy compared to mel-frequency cepstral coefficient (MFCC) features using support vector machine (SVM). While other EMD-derived algorithms like ensemble EMD (EEMD) and complete EEMD (CEEMD) excel in imaging applications, this study avoids EEMD due to its noise-assisted nature, potentially altering signal characteristics [35]. To improve the efficacy of methods such as CEEMD, CEEMDAN, and enhanced CEEMDAN, it is essential to uphold a correlation between low and high instantaneous frequencies below 0.75. This necessitates meticulous filtering operations to prevent substantial calculation errors and uphold a high frequency resolution. In this investigation, EMD takes precedence over the alternative algorithms derived from EMD to safeguard the original signal characteristics, mitigating potential influences on outcomes and ensuring accurate respiratory motion prediction.

Artificial neural network (ANN) is a model-free architecture and inspired by the structure of the human brain and nervous system [36,37]. Single-layer feedforward neural network (SLFN) is the simplest form of ANN. Due to its universal approximation capability, it has been widely used in classification and prediction problems [38]. Back propagation (BP) is employed to train the SLFN that suffers from slow convergence and is sensitive to learning rate/noisy data and getting gradient descent trapped in local minimum. A randomized neural network was described in [39], and this version is known as random vector functional link (RVFL) networks. The RVFL network, initially proposed by Pao [40], presents a distinctive neural network architecture that sets it apart from traditional feedforward networks. It achieves this distinction by solely relying on randomly generated weights connecting the input and hidden layers. The weights between the hidden layer and the output layer, however, are computed analytically. In contrast to ELM, RVFL integrates direct connections between the input and output layers [41]. This architectural choice enhances its overall generalization capabilities and significantly accelerates the learning speed. The effectiveness of these direct links within the RVFL network has been showcased across various domains, including time-series prediction [37,42], function approximation [43], as well as classification and regression tasks [19,44,45,46]. In our prior studies [19], RVFL was employed on respiratory motion for multi-step ahead prediction. The significance of direct links in RVFL was discussed by comparing the prediction performance with and without direct links. RVFL with direct links achieved a higher prediction accuracy compared to RVFL without direct links [19]. Furthermore, it is evident in [41,47] that direct links in RVFL significantly improve the performance of prediction models as they regularize the randomization. However, RVFL did not accurately model the intra-trace variabilities and irregularities of respiratory motion, resulting in worse performance at larger prediction lengths. Therefore, there is still a need to improve the prediction performance of RVFL.

To overcome the above-mentioned problems, we employed a combined structure of EMD [48] and RVFL [41,49] in this work, and named it hybrid EMD-RVFL [47]. In the hybrid EMD-RVFL approach, EMD is initially utilized to decompose the respiratory motion into multiple interpretable intrinsic mode functions (IMFs) and residue. Using this decomposition strategy, the intra-trace variabilities and irregularities of the respiratory motion could be treated effectively. Following that, RVFL networks are trained for each extracted IMF and residue. The RVFL prediction results of all the IMFs and residue are then summed up to obtain the final predicted output. We evaluated the performance of EMD-RVFL for various prediction lengths equivalent to the delays exhibited by commercially available radiotherapy systems using 304 respiratory traces acquired from 31 patients. The hybrid EMD-RVFL approach developed in our study was the first ever to be applied to the prediction of respiratory motion as a generalized model. A performance analysis conducted at various prediction lengths showed that the EMD-RVFL produced robust and accurate prediction results when compared with existing methods, namely SVR [10], wLMS [6], RVM [12,13], ELM [14], fRCN [24], EMD-SVR [50], EMD-wLMS [6], EMD-RVM [51], EMD-ELM [52], and EMD-fRCN [24].

This article is organized into sections as follows: In Section 2, we present a concise overview of EMD and RVFL, while also introducing our proposed approach, EMD-RVFL. Moving on to Section 3, we detail the step-by-step implementation process of our proposed method and provide a comprehensive performance comparison against existing techniques. In Section 4, discussion will be included and also provides the future work. Section 5 concludes our study and highlights key findings.

2. Methods and Materials

The primary objective of this study is to introduce a comprehensive framework that can effectively reduce the delay and improve the accuracy of predicting the tumor position. This section presents a detailed discussion on the prediction algorithm developed for respiratory motion prediction in this study.

2.1. Empirical Mode Decomposition

EMD [48] is a technique for decomposing time series data into a number of interpretable intrinsic mode functions (IMFs) as well as a residue that stands for trend. It is an empirically based technique for obtaining instantaneous frequency information from natural signals, which are frequently nonlinear and non-stationary in nature. Respiratory motion consists of many of intra-trace variabilities and irregularities. These variations in the respiratory motion causes erroneous prediction. Thus, EMD algorithm will be very effective in dealing with these intra-trace variabilities and the irregularities of respiratory motion [32].

An IMF is a function that consists of one extreme between zero crossings as well as a mean value of zero. The shifting process that EMD employs to decompose the respiratory motion into IMFs is explained as follows.

1.

Let $z_1$ be the mean of the upper and lower envelopes of a time series signal $x\left(t\right)$ obtained by interpolating the local maxima and minima.

2.

Subtracting $z_1$ from the original time series $x\left(t\right)$ will provide the first component $p_1$ as $p_1=x\left(t\right)-z_1$.

3.

During the second shifting process, consider $p_1$ as the data with the mean $z_{11}$ of its upper and lower envelopes: $p_{11}=p_1-z_{11}$.

4.

The stopping criteria in which the shifting process is carried out k times are as follows: (a) $z_{1k}$ is approaching towards zero; (b) there is maximum one difference between the numbers of extrema and zero-crossing of $p_{1k}$; or (c) it reaches the maximum number of iterations. Here, $p_{1k}$ can be considered as IMF and can be calculated as $p_{1k}=p_{1(k-1)}-z_{1k}$.

5.

Then, it can be represented as $a_1=p_{1k}$, which has the first IMF (the shortest component of the data), and subtracts it from the data as $x\left(t\right)-a_1=y_1$ and the process is iterated on $y_j$ as: $y_2=y_1-a_2$,..., $y_n=y_{n-1}-a_n$.

Consequently, the original time series $x\left(t\right)$ is decomposed into a set of IMF functions as:

$$\begin{array}{c}\hfill \mathit{x}\left(\mathit{t}\right)=(\sum _{i=1}^n{a}_i+y_n)\end{array}$$

where n is the number of functions in the set that is dependent on the original signal.

2.2. Artificial Neural Network Model

As a machine learning model that was motivated by the central nervous system and the human brain [53], ANN will be divided into two groups based on the network structure architecture: the feedforward neural network (FNN) and recurrent neural network (RNN). The simplest type of FNN is the single-layer feedforward neural network (SLFN) which consists of an input layer with an equal number of neurons as the dimension of input features, a hidden layer with a nonlinear activation function, and the output layer that collects the outputs from the hidden layer. The direct connections between neurons in adjacent layers have varying weights which indicate how strong these connections are. The SLFN structure, on the other hand, avoids interconnecting neurons within the same layer. The following equation is used to calculate the outputs $v_j$ from the hidden layer neurons:

$$\begin{array}{c}\hfill v_j=f(\sum _{i=1}^l{w}_{ij}{x}_i+b_i)\end{array}$$

and the final output y of SLFN is computed as follows:

$$\begin{array}{c}\hfill y=g(\sum _{j=1}^m{w}_{jo}{v}_j+b_j)\end{array}$$

where f(.) and g(.) denote the nonlinear activation functions; l and m represent a number of input features and hidden layer neurons; $x_i$ serves as the neuron’s input; $w_{ij}$ is the weight of the connection between the input variable i and the hidden neuron j; $w_{jo}$ shows the weight connected between the hidden neuron j and the output; and $b_i$ and $b_j$ represent the biases.

Traditionally, the back propagation (BP) technique is employed to update the weights of SLFN. Because of the iteration of BP, BP-based approaches are typically quite time-consuming. Furthermore, gradient descent is frequently trapped at a local minimum and these drawbacks restrict the SLFN performance.

2.3. Random Vector Functional Link

RVFL [40,41], a randomized version of FNN, is structured such that the direct links are established between the input and output layer neurons as shown in Figure 1. In the implementation of RVFL architecture, fixed random weights ${\mathbf{W}}_{\mathit{h}}$ are employed between input and hidden layers. Thus, we compute hidden features H from hidden neurons using the activation function sigmoid given as follows:

$$\begin{array}{c}\hfill \mathbf{H}=sigmoid\left({\mathbf{W}}_{\mathit{h}}{\mathbf{X}}_t\right)\end{array}$$

According to the definition of RVFL, the training matrix ${\mathbf{X}}_t$ is concatenated to the hidden features $\mathbf{H}$ as:

$$\begin{array}{c}\hfill \mathbf{D}=\left[\mathbf{H}{\mathbf{X}}_t\right]\end{array}$$

Instead of BP, a closed form solution such as ridge regression is employed due to its fast computation in training the output layer weights ${\mathbf{W}}_{\mathit{o}}$ [40]. Thus, the only weights that need to be optimized are the output layer weights ${\mathbf{W}}_{\mathit{o}}$ calculated as:

$$\begin{array}{c}\hfill \mathit{Primal}\mathit{Space}:{\mathbf{W}}_{\mathit{o}}={\left(D^{T}D+CI\right)}^{-1}{D}^T{\mathbf{Y}}_t\end{array}$$

or

$$\begin{array}{c}\hfill \mathit{Dual}\mathit{Space}:{\mathbf{W}}_{\mathit{o}}=D^T{\left(D{D}^T+CI\right)}^{-1}{\mathbf{Y}}_t\end{array}$$

where C and ${\mathbf{Y}}_t$ show the regularized constant and true labels of the training matrix ${\mathbf{X}}_{\mathit{t}}$, respectively. Applying ${\mathbf{W}}_{\mathit{o}}$ and ${\mathbf{W}}_{\mathit{h}}$ on the testing matrix ${\mathbf{X}}_{\mathit{s}}$ to obtain the multistep ahead predicted signal $\widehat{Y}$ as:

$$\begin{array}{c}\hfill \widehat{Y}={\mathbf{W}}_{\mathit{o}}\left(f\left({\mathbf{W}}_{\mathit{h}}{\mathbf{X}}_{\mathit{s}}\right)\right)\end{array}$$

2.4. Respiratory Motion Prediction with Hybrid EMD-RVFL Approach

In this paper, we utilize an ensemble method termed “divide and conquer”. This approach involves decomposing the original time series into a sequence of sub-datasets until they are sufficiently simplified for analysis. Specifically, for the proposed RVFL network based on EMD, denoted by EMD-RVFL, the respiratory motion time-series undergoes decomposition into IMFs and one residue through the EMD method. Subsequently, individual RVFL networks are trained for each IMF, encompassing the residue. The ultimate prediction results are obtained by aggregating the outputs from all sub-series through a simple summation. The procedural steps can be summarized as follows:

1.

In the first step, we implement EMD to decompose respiratory motion into IMFs ($IM{F}_1$, $IM{F}_2$, $IM{F}_3$, ..., $IM{F}_n$) and a residue (R), as depicted in the decomposition module of Figure 2. In this step, the intra-trace variabilities and irregularities of the respiratory motion are addressed with the decomposition strategy.

2.

Secondly, construct a training data ${\mathbf{X}}_t$ to be used as the input to each RVFL network ($RVF{L}_1$, $RVF{L}_2$, ..., $RVF{L}_n$, $RVF{L}_r$) for each extracted IMF and residue. ${\mathbf{X}}_t$ can be expressed as the sequence of the inputs as ${\mathbf{X}}_t=[x_1,x_2,...,x_t,x_{t+1},...]$ where $x_t$ is the magnitude of respiratory motion acquired at time instant t. Predicting the respiratory motion for a known horizon $\tau$ can be considered as a classical learning problem for estimating an unknown relation between elements in a given input space ${\mathbf{X}}_t$$\in {\mathbb{R}}^m$ to the elements in a target space $y\in \mathbb{R}$. Elements in the input space are formulated by considering the recent history of the respiratory motion trace ${\mathbf{X}}_t=[x_t,x_{t-1},...,x_{t-m}]$, with m being the dimension of the input feature vector while the target space elements corresponding to ${\mathbf{X}}_t$ are formulated as $y_t=x_{t+\tau }$. ${\widehat{x}}_{t+\tau }$ denotes the $\tau$ samples ahead predicted value at tth sample.

3.

Then, the input vectors and the corresponding target vectors formulated with the training data of $IM{F}_1$, $IM{F}_2$, $IM{F}_3$, ..., $IM{F}_n$, R, are then provided to the $RVF{L}_1$, $RVF{L}_2$, ..., $RVF{L}_n$, $RVF{L}_r$ models for learning a nonlinear mapping that represents the underlying relationship between the input feature space and the target space. This stage can visualized in the prediction module of Figure 2.

4.

In this stage, the nonlinear mapping achieved during the training phases of $RVF{L}_1$, $RVF{L}_2$, ..., $RVF{L}_n$, $RVF{L}_r$ will be utilized for predicting the unseen data of all $IM{F}_1$, $IM{F}_2$, $IM{F}_3$, ..., $IM{F}_n$, R.

5.

Lastly, sum up the predicted outputs of all RVFL networks ($RVF{L}_1$, $RVF{L}_2$, ..., $RVF{L}_n$, $RVF{L}_r$) to formulate the predicted output ${\widehat{x}}_{t+\tau }$, as it is illustrated in the summation unit of Figure 2.

3. Results

The respiratory motion dataset, performance measures, and optimization of hyperparameters used for the proposed method are all described in this part. Later, a full comparative analysis is performed for various prediction lengths by employing all the approaches, and the results are explained.

3.1. Data Description

The evaluation of different prediction algorithms was conducted using an extensive database encompassing 304 respiratory motion traces. These traces were meticulously recorded during CyberKnife treatments administered to 31 patients at Georgetown University Hospital. Notably, patients underwent treatment across multiple sessions, with up to seven fractions administered per patient, culminating in a total of 102 fractions delivered throughout the study period. The motion traces were meticulously recorded throughout patient treatment sessions, employing the optical tracking system commonly utilized alongside CyberKnife technology. This system, known as the Synchrony Respiratory Motion Tracking System developed by Accuray, Inc. (Sunnyvale, CA, USA), utilizes a customized Flashpoint FP 5500 platform produced by Boulder Innovators, Inc. (Louisville, CO, USA) to accurately capture the movement of optical fibers transmitting red light. To thoroughly monitor the respiratory motion and tumor movement, the fibers are positioned on the patient’s chest and abdomen so that their open ends align with areas of maximum respiratory motion amplitude. Throughout each treatment fraction, a comprehensive data collection process was undertaken, wherein three distinct markers were meticulously recorded in three-dimensional space, resulting in three motion traces per patient. In a single fraction, markers two and three were defective, impacting the accuracy of motion tracking.

The database encompasses the corresponding principal component traces (one- dimensional), derived via principal component analysis on the traces of each fraction. For comprehensive details on the recording process and principal component analysis methodology, please refer to [7,54]. Prior research [7,55,56] underscores the significance of assessing the prediction algorithm performance using PCA-processed traces, providing a reliable measure of their predictive efficacy for three-dimensional traces. Consequently, in this study, we assess the prediction performance of all methods using the PCA-processed traces of a fixed length of 40 min. The database organizes the PCA traces in ascending alphanumeric order. To enhance the clarity in our presentation, we have assigned numeric IDs from 1 to 304 to the traces, corresponding to their order in the list. A few examples of the respiratory motion traces are given in Figure 3 for better visualization. In traces # 286 and # 212, it is evident that there are significant amounts of intra-trace variabilities and irregularities.

3.2. Performance Measures

To measure the performance of prediction algorithms while doing comparative analysis, we employed two performance indices: the root mean square error $\left(RMSE\right)$ and the relative root mean square error $\left(rRMSE\right)$ for various prediction lengths.

Definition 1. 

Let $y_i\left(k\right)$ represent the true value for a trace i and $\widehat{y_i}\left(k\right)$ denote its corresponding predicted value at a chosen prediction length. Hence, the prediction error $\widehat{e_i}\left(k\right)$ can be computed as:

$$\begin{array}{c}\hfill \widehat{e_i}\left(k\right)=\widehat{y_i}\left(k\right)-y_i\left(k\right)\end{array}$$

Therefore, the RMS prediction error ${(RMSE}_i)$ for a trace i can be computed as:

$$\begin{array}{c}\hfill RMS{E}_i=\sqrt{\frac{1}{N_i}{\Sigma }_k\widehat{e_i}{\left(k\right)}^2}\end{array}$$

(1)

where $N_i$ denotes the total number of samples in the testing data of a trace i.

We can obtain the RMS prediction error averaged across all traces for all the prediction lengths as:

$$\begin{array}{c}\hfill RMSE=\sqrt{\frac{1}{N_I}{\Sigma }_{i}RMS{E}_i^2}\end{array}$$

(2)

where $N_I$ denotes the total number of traces.

Definition 2. 

The second performance measure that we employed is $rRMSE$. For this measure, we used no prediction method (baseline method) where the predicted position measurement will be calculated using the current position measurement at a chosen prediction length as ${\widehat{y}}_i(k+h)=y_i\left(k\right)$, and thus the prediction error with no prediction method can be calculated as:

$$\begin{array}{c}\hfill {\widehat{e_i}\left(k\right)}_{NoPrediction}=\widehat{y_i}\left(k\right)-y_i(k-h)\end{array}$$

We obtain $RMS{E}_{iNoPrediction}$ for each trace i as follows:

$$\begin{array}{c}\hfill {RMS{E}_i}_{NoPrediction}=\sqrt{\frac{1}{N_i}{\Sigma }_k{\widehat{e_i}\left(k\right)}_{NoPrediction}^2}\end{array}$$

(3)

Then, $RMS{E}_{NoPrediction}$ is obtained by averaging across all the traces for all the prediction lengths as:

$$\begin{array}{c}\hfill RMS{E}_{NoPrediction}=\sqrt{\frac{1}{N_I}{\Sigma }_{i}RMS{E}_{iNoPrediction}^2}\end{array}$$

(4)

Finally, we compute the $rRMSE$ with the following equation as:

$$rRMSE=\frac{RMSE}{RMS{E}_{NoPrediction}}$$

(5)

For respiratory motion prediction, $rRMSE$ is used to highlight the relative improvement in the performance of the prediction algorithm in comparison to the prediction results produced with the help of no prediction method. For example, if $rRMSE$ = 0.70, then it means that the prediction algorithm is able to minimize the prediction error by 30% compared to the error achieved when there is a system delay [24]. This analysis with $RMSE$ and $rRMSE$ is in line with the existing studies in [7,24,57].

3.3. Optimization of Hyperparameters

During the hyperparameter optimization process, we initially utilized RVFL to optimize the number of neurons (L) and feature vector (p). Subsequently, with the optimized parameters obtained from this step, we applied EMD-RVFL to determine the optimal values for the regularization constant (C) and IMFs, denoted by (k), as illustrated in Figure 4a. For parameters L and p, a grid search was performed on the training data, as depicted in Figure 4b, treating each trace independently. The search spanned a broad range of values, $10\le L\le 150$ and $10\le p\le 120$, with a step size of 5 over 50 random traces. Subsequently, the testing data were modeled using the nonlinear mapping generated by each combination. The RMSE according to Equation (1) was calculated for each combination. We selected the optimal initialization pair, denoted by $(L,p)$, based on the combination resulting in the lowest RMS prediction error. There are minimal differences in the ideal parameters identified among the traces. We computed the mean of all hyperparameters across the traces to obtain a single initialization value. This mean was then treated as the optimal hyperparameters. Our analysis revealed that L = 70 and p = 33 are the optimized parameters of RVFL, as illustrated in Figure 4b. These hyperparameters are employed to optimize the C and k with EMD-RVFL over the same 50 random traces to yield the least $RMSE$ [58]. As depicted in Figure 4c, it is evident that EMD-RVFL achieved the lowest RMSE at C = $10\times {10}^9$ and k = 7. Beyond these specific values, a few subsequent parameter combinations led to a significant and abrupt increase in RMSE. Hence, for all traces, we selected the optimal parameters as (p, L, C, k) = (33, 70, $10\times {10}^9$, 7). The other parameters of RVFL including the seed, scale, and scalemode were set to zero and no bias was included in the RVFL’s output layer. In addition, both ELM and RVFL employed the sigmoid as the activation function, employed the Moore–Penrose Pseudoinverse for calculating output weight, and fixed random weights ${\mathbf{W}}_{\mathit{h}}$ from uniform distribution. In this paper, we first compared the prediction performance of RVFL against SVR, wLMS, RVM, ELM, and fRCN. Subsequently, to ensure a fair comparison, we also incorporated EMD with wLMS, RVM, ELM, and fRCN, in order to assess their performance against our proposed EMD-RVFL approach.

3.4. Performance Comparison

In this performance analysis, predictions were conducted for four distinct prediction horizon lengths: 77 ms (equivalent to 2 samples); 154 ms (equivalent to 4 samples); 308 ms (equivalent to 8 samples); and 576 ms (equivalent to 15 samples). These horizon lengths were carefully selected, taking into account the typical latencies observed in commercial robotic radiotherapy systems. Specifically, the latency is approximately 115 ms in CyberKnife [3], 175 ms in Tomotherapy [59], and motion compensation devices with multi-lead collimators may exhibit latencies of up to several hundred milliseconds [1].

3.4.1. Respiratory Motion Tracking Using Various Methods

The prediction performance of the proposed EMD-RVFL method at a prediction length of 308 ms on trace-14 is illustrated in Figure 5. This trace was randomly selected from a pool of 304 traces, each containing irregularities as depicted in Figure 5A. The prediction errors of EMD-RVM, EMD-ELM, EMD-fRCN, and EMD-RVFL are plotted in Figure 5B. To demonstrate the effectiveness of all methods, the dotted portion of Figure 5 highlights their performance during irregularities in respiratory motion. It can be observed from Figure 5B that EMD-RVFL significantly reduces undershoots and overshoots, resulting in lower prediction errors. The tracking performance of the EMD-RVFL approach indicates its robust response to irregularities, exhibiting minimal phase lag in tracking or prediction error compared to existing approaches. This superior performance of EMD-RVFL is attributed to the inclusion of direct links, which serve to regularize randomization. Consequently, the RVFL component in EMD-RVFL effectively learns the local patterns and irregularities of respiratory motion at all time points [41].

3.4.2. Performance of Proposed Method during Irregularities of Respiratory Motion

In order to evaluate the prediction performance of EMD-RVFL in the presence of irregularities, the performance of the EMD-RVFL model at three prediction lengths (154 ms, 307 ms, and 576 ms) is shown in Figure 6. Any prediction method’s tracking performance is prone to substantial inaccuracies once an irregularity emerges. To illustrate the performance of EMD-RVFL during the irregularities, in Figure 6, two types of irregularities are investigated, first case; consistent variations in trace (deep breath) and second case; sudden variations in trace (cough). The performance of EMD-RVFL for these cases are enclosed in two rectangular boxes as shown in Figure 6 for better demonstration. When the tracking performance of the EMD-RVFL is carefully examined at both irregularities, it was observed that EMD-RVFL was not precise at both the instants of irregularities in terms of tracking performance, but it quickly tracks the original trace accurately. Furthermore, it proved that, whenever there are abrupt changes in the respiratory motion, EMD-RVFL quickly learns that the auto-correlation properties of respiratory motion yields less of a prediction error [60].

3.4.3. Comparative Analysis Based on Average RMSE

We computed the RMSE following the methodology outlined in [24] to ensure a fair comparison among several algorithms, including SVR, wLMS, RVM, ELM, fRCN, RVFL, EMD-SVR, EMD-wLMS, EMD-RVM, EMD-ELM, EMD-fRCN, and EMD-RVFL. Initially, we trained the models using 7800 samples (equivalent to 280 min) and subsequently evaluated each model’s performance using 1000 samples for each trace, resulting in the calculation of RMSE for each model, as depicted in Figure 7. Regarding single-structure models, RVFL consistently outperformed all existing algorithms across all prediction lengths, as illustrated in Figure 7A. The prediction performance of fRCN surpassed that of ELM except for a prediction length of 576 ms. While RVM demonstrated the minimum RMSE at smaller prediction lengths, its performance deteriorated for longer prediction lengths. SVR exhibited a superior prediction performance compared to wLMS across all prediction lengths. Figure 7B highlights a significant reduction in RMSE following the incorporation of EMD with SVR, RVM, ELM, fRCN, and RVFL, with the exception of wLMS. Among these, EMD-RVFL demonstrated the lowest RMSE compared to EMD-SVR, EMD-wLMS, EMD-RVM, EMD-ELM, and EMD-fRCN. EMD-fRCN emerged as the second-best method in terms of RMSE reduction compared to its counterparts across all prediction lengths. Consequently, RVFL and EMD-RVFL models exhibited a superior performance compared to existing algorithms.

3.4.4. Comparative Analysis Based on Individual RMSE of 304 Traces

We utilized 304 respiratory motion traces to compute the $RMS$ prediction errors with RVM, ELM, fRCN, and RVFL and their hybrid variants such as EMD-RVM, EMD-ELM, EMD-fRCN, and EMD-RVFL across all prediction lengths, as shown in Figure 8. When we compared the prediction performance of all prediction methods at 77 ms, we observed that RVFL and EMD-RVFL outperformed the existing methods by minimizing the majority of 304 RMSE values, as shown in Figure 8A,B. A similar trend was also observed for the prediction lengths of 115 ms, 308 ms, and 570 ms in Figure 8A,B with RVFL and EMD-RVFL. The fRCN method was not accurate only at the larger prediction length of 576 ms compared to ELM, but we observed that, in hybrid form, the EMD-fRCN surpassed the EMD-ELM performance at each prediction length. The RVM showed the comparable prediction performance to ELM and fRCN at small prediction lengths of 77 ms and 154 ms. Similar results were also seen among EMD-RVM, EMD-ELM, and EMD-fRCN at these prediction lengths. However, RVM-based models failed to minimize the 304 RMS prediction errors at large prediction lengths of 308 ms and 576 ms as compared to ELM- and fRCN-based models. This statistical analysis proves that RVFL and EMD-RVFL surpassed the prediction performance of existing methods for most of the 304 traces across all prediction lengths.

3.4.5. Robustness of RVFL and EMD-RVFL Models with Scatter Plots

Scatter plots were drawn to show how the $RMSE$ of RVFL and EMD-RVFL changes with respect to SVR, wLMS, RVM, ELM, and fRCN and their hybrid variants, EMD-SVR, EMD-wLMS, EMD-RVM, EMD-ELM, and EMD-fRCN, as illustrated in Figure 9A,B. The objective was to demonstrate the robustness of the RVFL and EMD-RVFL models in the presence of the inter-trace variabilities of respiratory motion over existing methods. The position of the blue circle marker will indicate the robustness of the RVFL and EMD-RVFL. For example, if the blue circle is above the diagonal, which is red in color, then RVFL or EMD-RVFL correctly offsets the existing approaches. On the other hand, if the blue circle is below the diagonal line (which is black in color), it suggests that the existing approach works better than RVFL or EMD-RVFL. The majority of the blue circles were above or very near the red-colored diagonal line, as can be seen in Figure 9A,B at all prediction lengths. Hence, this investigation demonstrates that RVFL and EMD-RVFL outperform the existing approaches in terms of prediction accuracy and robustness.

3.4.6. Effect of Prediction Lengths on the Performance of Proposed Approach

An analysis of the four chosen prediction lengths of 77 ms, 154 ms, 308 ms, and 576 ms was performed to assess the influence of the prediction length on the prediction performance of EMD-RVFL. Figure 10 shows the $RMSE$ of the prediction error of all 304 traces at all prediction lengths. The $RMSE$ medians change linearly with the prediction length, as can be seen in the scatter plot. We also noted that the $RMSE$ of the prediction errors of all the traces is grouped at all selected prediction lengths, with a few outliers. As a result, the proposed hybrid EMD-RVFL approach is robust towards the database’s irregularities and inter-trace variabilities. We also note that the $RMSE$ of prediction errors is more distributed at large prediction lengths than at others. It is mostly due to the scarcity of information regarding future occurrences across larger prediction lengths.

3.4.7. Comparative Analysis in Terms of Relative Improvement (in %)

Table 1. Relative improvement shown in % using all the methods, SVR, wLMS, RVM, ELM, fRCN, RVFL, EMD-SVR, EMD-wLMS, EMD-RVM, EMD-ELM, EMD-fRCN, and EMD-RVFL.

Method	Comparison Method	77 ms (in %)	115 ms (in %)	308 ms (in %)	576 ms (in %)
SVR	wLMS	−19.10	−13.12	−9.14	−4.20
	RVM	6.30	0.80	1.12	2.68
	ELM	−22.37	−6.90	2.16	6.69
	fRCN	4.21	2.20	3.24	3.39
	RVFL	9.78	4.27	5.27	8.05
wLMS	RVM	25.40	13.92	10.26	6.88
	ELM	−3.27	−6.22	11.30	10.89
	fRCN	23.31	15.32	12.38	7.59
	RVFL	28.88	17.39	14.41	12.00
RVM	ELM	−28.67	−7.70	1.04	4.01
	fRCN	−2.09	1.40	2.12	0.71
	RVFL	3.48	3.47	3.11	5.37
ELM	fRCN	26.58	9.1	1.08	−3.30
	RVFL	32.15	11.17	3.11	0.2
fRCN	RVFL	5.28	2.07	2.03	4.60
EMD-SVR	EMD-wLMS	−18.36	−19.37	−20.94	−25.12
	EMD-RVM	21.16	7.09	3.05	−0.25
	EMD-ELM	−12.10	−3.45	5.66	5.38
	EMD-fRCN	−1.27	3.28	9.09	6.43
	EMD-RVFL	28.94	25.30	13.10	8.39
EMD-wLMS	EMD-RVM	39.52	26.46	24.02	24.85
	EMD-ELM	6.26	15.92	26.63	30.50
	EMD-fRCN	17.09	22.65	30.06	31.55
	EMD-RVFL	47.30	34.67	34.07	33.51
EMD-RVM	EMD-ELM	−33.26	−10.54	2.04	5.65
	EMD-fRCN	−22.43	−3.8	6.04	6.00
	EMD-RVFL	7.78	8.21	10.05	8.66
EMD-ELM	EMD-fRCN	10.38	6.73	3.43	1.05
	EMD-RVFL	41.04	18.75	7.44	3.01
EMD-fRCN	EMD-RVFL	30.21	12.02	4.01	1.96

Note: negative sign shows the superiority of method in first column over the method in second column.

compares the relative performance improvements (in terms of % rRMSE) of RVFL and EMD-RVFL vs. other prediction techniques for all prediction lengths. SVR and EMD-SVR outperformed wLMS and EMD-wLMS at all prediction lengths, but the improvement in SVR declined over wLMS when the prediction length was increased, while it was the opposite in EMD-SVR compared to EMD-wLMS. Furthermore, EMD-SVR was outperformed by EMD-RVM at all prediction lengths except 576 ms. SVR and EMD-SVR showed a better % improvement compared to ELM and EMD-ELM at small prediction lengths (77 ms and 154 ms). However, ELM and EMD-ELM outperformed SVR and EMD-SVR at large prediction lengths (308 ms and 576 ms). It is observed that wLMS outperformed ELM at prediction lengths of 77 ms and 115 ms, and thereafter, the ELM prediction performance was better than wLMS with a higher accuracy at large prediction lengths. Among the EMD techniques, EMD-wLMS exhibited the lowest prediction accuracy. RVM and EMD-RVM showed a better prediction performance as compared to ELM and EMD-ELM at prediction lengths of 77 ms and 154 ms. However, when the prediction lengths were increased, ELM-based methods performed better than RVM-based models. ELM obtained the better prediction accuracy compared to fRCN only at a large prediction length (576 ms). Nevertheless, despite its EMD-ELM variant, the method failed to achieve a superior prediction performance compared to EMD-fRCN and EMD-RVFL across all prediction lengths. fRCN and EMD-fRCN yielded more erroneous prediction results as compared to RVFL and EMD-RVFL. Table 1 indicates that the RVFL and EMD-RVFL show the highest prediction performance independent of the prediction length.

4. Discussion

To mitigate the tumor positioning error in radiotherapy, respiratory motion prediction in equivalence to the system delay is discussed in this paper. In this paper, we employed the hybrid approach to improve the performance of respiratory motion prediction. Although RVFL computes the future position of respiratory motion with less computation time, its prediction error gradually increases with the larger prediction length, as discussed in Section 3 [19]. To circumvent this problem, we take advantage of the hybrid approach, where we combine the EMD with RVFL and termed it as a hybrid EMD-RVFL to further decrease the prediction error, especially at larger prediction lengths. As a result, the hybrid model-free approach outperformed the single model-free approach in reducing the prediction error. This supports the idea that using a hybrid model-free approach can improve the respiratory motion prediction [47,61].

In the result section, we compared the prediction performances of all the prediction methods using 304 respiratory motion traces from large databases at 77 ms, 154 ms, 308 ms, and 576 ms. Firstly, in the case of single-structure models, RVFL outperformed SVR, wLMS, RVM, ELM, and fRCN at all prediction lengths based on two performance measures (RMSE and rRMSE). Secondly, for hybrid models, EMD was appended with every method to compare the prediction performances. It was found that EMD-RVFL showed superior prediction performances over EMD-SVR, EMD-wLMS, EMD-RVM, EMD-ELM, and EMD-fRCN at all prediction lengths using RMSE and rRMSE. The inclusion of the direct link significantly enhanced the prediction accuracy of both RVFL and EMD-RVFL models by regularizing randomization [41]. Additionally, it proved highly effective in terms of computational efficiency. Notably, our observations indicated that EMD-RVFL outperformed EMD-ELM with a lower number of hidden neurons [62]. For example, while EMD-ELM required 200 hidden neurons to minimize the prediction error, EMD-RVFL achieved a comparable accuracy with just 70 hidden neurons for the same feature vector. Consequently, EMD-RVFL exhibited a reduced computational time compared to EMD-ELM due to the direct link while maintaining superior accuracy. Similar comparisons can also be found in [63]. As for future research, our aim is to implement EMD-RVFL in real-time applications. The detailed online implementation of EMD-RVFL is discussed in [64,65,66]. This future study aims to advance the real-time tracking of tumor motion for motion-adaptive radiotherapy. Furthermore, we will also focus on reducing the mode mixing problem of the proposed method by developing a more robust and accurate paradigm.

5. Conclusions

To accurately predict the respiratory motion, a hybrid model EMD-RVFL is developed in this paper. As part of this study, we investigate the issues related to tracking respiratory motion in the presence of irregularities and intra-trace variabilities using a hybrid EMD-RVFL approach. We employed respiratory motion traces (304 in total) to validate the feasibility of this approach at different prediction lengths in accordance with the latencies of devices that are commercially available. With the employed two error measures (RMSE and rRMSE), the results showed that EMD-RVFL outperformed the existing methods across all traces and at all prediction lengths. The proposed EMD-RVFL approach significantly reduces the prediction error, especially at large prediction lengths as compared to the existing techniques. This enhancement can be attributed to the inclusion of the direct links, which effectively facilitates RVFL in the EMD-RVFL approach by introducing the regularization to the randomization process and subsequently reducing the model complexity.