## Abstract

Cone beam computed tomography (CBCT) is an imaging modality that has been used in image-guided radiation therapy (IGRT). For applications such as lung radiation therapy, CBCT images are greatly affected by the motion artefacts. This is mainly due to low temporal resolution of CBCT. Recently, a dual modality of electrical impedance tomography (EIT) and CBCT has been proposed, in which the high temporal resolution EIT imaging system provides motion data to a motion-compensated algebraic reconstruction technique (ART)-based CBCT reconstruction software. High computational time associated with ART and indeed other variations of ART make it less practical for real applications. This paper develops a motion-compensated conjugate gradient least-squares (CGLS) algorithm for CBCT. A motion-compensated CGLS offers several advantages over ART-based methods, including possibilities for explicit regularization, rapid convergence and parallel computations. This paper for the first time demonstrates motion-compensated CBCT reconstruction using CGLS and reconstruction results are shown in limited data CBCT considering only a quarter of the full dataset. The proposed algorithm is tested using simulated motion data in generic motion-compensated CBCT as well as measured EIT data in dual EIT–CBCT imaging.

## 1. Introduction

Cone beam computed tomography (CBCT) is an imaging system combined into a linear accelerator (Linac) for verifying the treatment area in radiation therapy. Owing to slow rotating, CBCT motion artefacts often appear in CBCT images in applications such as lung imaging. The motion artefacts can result in target registration errors and radiation beam misalignment, which in turn can lead to over-dose of radiation to normal tissue and inhomogeneous distribution of the radiation dose to the tumour [1–3]. In addition, reconstruction with limited data has also been studied in order to reduce the radiation dose to patients [4–6]. Motion will further degrade the limited data imaging.

In the last decade, motion-compensated CBCT for removing motion blur artefacts in CBCT scans has attracted the interest of many research groups [7–12]. Many reconstruction methods have been proposed for compensating the motion. Iterative algorithms, e.g. the algebraic reconstruction technique (ART), simultaneous algebraic reconstruction technique (SART) and ordered-subset SART (OS-SART) are the methods of choice [13–16]. Furthermore, motion compensation has been combined with ART and SART [17,18].

Conjugate gradient least squares (CGLS) is an attractive iterative reconstruction algorithm, which has been applied to CBCT image reconstruction by Jia *et al.* [19]. CGLS has advantages over ART and SART, in terms of faster convergence rates and the possibility of parallel computation. In addition, CGLS provides the possibility of including a regularization term, which is useful for limited data. While the Feldkamp, Davis and Kress (FDK) reconstruction is one of the most commonly used in many volumetric CT, imaging based on filtered-back projection (FBP) showed inappropriate method for insufficient projection data [20] and less accurate reconstruction in comparison with iterative method reconstruction (SART), in particular with limited projection data [5]. CGLS has not been developed in motion-compensated CBCT to the best of our knowledge.

To our knowledge, the application of motion compensation to CGLS has not been reported. Therefore, it is proposed to apply fast image reconstruction CGLS with limited data, to remove motion blur artefacts on CBCT images and reduce the reconstruction time.

The ART-based motion-compensated algorithm has been tested in a previously proposed dual modality electrical impedance tomography (EIT) CBCT setting for potential lung imaging application [18]. EIT is an imaging system and recently found momentum in the area of lung imaging. EIT can provide a three-dimensional image of electrical conductivity distribution by measuring current and voltages at the boundary electrodes. EIT has low spatial resolution, but its high temporal resolution makes it as a very good candidate to provide motion information to CBCT. EIT-based motion data is real-time and instant observation based and so can be more reliable than the estimated model based motion data currently used in four-dimensional CBCT. This paper presents simulation and phantom experimental studies.

Figure 1 shows a more complete idea of how the proposed scheme can be considered in future clinical studies. In this general plan, a patient-specific model can be developed using high quality diagnostic CT images for patient-specific modelling EIT imaging. The combined EIT–CBCT imaging will be used just before the radiation session for treatment planning. In addition, one may be able use EIT imaging during radiation therapy for adaptive real-time treatment, which is outside of the scope of this paper.

The main aim of this work is to investigate the effectiveness of motion-compensated CBCT using the proposed CGLS algorithm together with the EIT system for limited projection data reconstruction as shown in figure 1. In this set-up, the EIT will provide information about the movement of the organ(s). The motion data will then be used to enhance the CBCT images. In this paper, motion-compensated CGLS was developed and tested for the improvement of CBCT image reconstruction using one-fourth projection data. The motion-compensated CGLS was applied to the dual modality EIT–CBCT [18] by using the motion estimated from EIT images. A better performance of motion-compensated CBCT can be seen here due to the possibility of regularization terms that can be included in the CGLS scheme.

## 2. Image reconstruction methods

This section presents the image reconstruction methods used for CBCT and EIT. Imaging results are displayed as reconstructed images and one-dimensional plots and are analysed in terms of the root mean square error (RMSE) of reconstructed images compared with the true image as follows:
2.1
where *N* is the number of voxles, *x*_{i} and *g*_{i} are the reconstructed and true value in the *i*th voxel, respectively.

### (a) Electrical impedance tomography image reconstruction

An EIT is an imaging technique of impedance distribution within electrically conductive objects from surface electrical measurements. The main advantage of EIT is its high temporal resolution. EIT imaging was proposed for use as the motion monitoring system in this study. To generate the EIT images, the forward problem needs to be solved, which can be performed using a finite-element method. The forward problem is a problem of estimating the measured EIT data with given conductivity distribution. The image reconstruction problem is an inverse problem and so to solve the inverse problem forward modelling is needed. Under low-frequency assumptions, all Maxwell's equations can be simplified to the complex-valued Poisson equation
2.2
where **u** is the complex-valued electric potential and *σ* is the conductivity of the medium. Appropriate boundary conditions (complete electrode model) are needed to enable a representative model for the EIT measurement process. In this work, the complete electrode model is adapted, taking into account both the shunting effect of the electrodes and the contact impedance between the electrodes and tissue. Using this boundary condition, the EIT model includes
2.3
2.4
2.5
where *z*_{l} is the effective contact impedance between the *l*th electrode and the tissue, *n* is the outward normal to the surface electrodes, *U* is the complex-valued voltage, *I* is the complex-valued current and *e*_{l} denotes the electrode *l*. Here, indicates a point on the boundary not under the electrodes. The difference imaging mode with a Tikhonov type algorithm has been used for the image reconstruction
2.6
where **R** is the regularization matrix (identity matrix here), **J** is the Jacobian matrix, **Δ****V** is the measurement vector, *δ* is the regularization parameter, which has been selected empirically with test samples in experimental data.

Our 16 electrode EIT system in a LabVIEW environment based on National Instruments (NI) cards was used in this study [18]. Figure 2 shows the imaging capability of the EIT system used in this study; *δ*=0.001 is a suitable choice for the regularization parameter here. Figure 2*b* shows the EIT-reconstructed image of four bottles in a tank phantom (figure 2*a*). The adjacent current pattern with the electric current of 5 mA in a single frequency of 10 kHz was used. The same EIT system, algorithm and EIT phantom was used for experimental study for dual modality EIT–CBCT presented in §4.

### (b) Motion-compensated conjugate gradient least-squares algorithm for cone beam computed tomography

The CGLS is an iterative method well suited for the large sparse least-squares problem of **Ax**=**b**; in CBCT reconstruction **x** is the image values for **x**∈ℜ^{N} and **b** is observed data for **b**∈ℜ^{M}. The weighting (**A**∈ℜ^{M×N}) is created using a forward projection programme. Matrix **A** is a combination of the sub-matrix for each projection. The length of intersection of the *m*th ray with the *n*th cell is **A**_{mn}. In the case where matrix **A** is positive-definite, then the matrix **A**^{T}**A** is positive-definite for any matrix **A**. The iterative method is terminated at most *i* steps for *k*=0,1,2,…,*i*. The residual (**r**^{k}) at each step is computed by [21–23]
2.7
Starting with initial approximation (**x**^{0}), then **r**^{0}=**b**−**Ax**^{0}, **p**^{0}=**A**^{T}**r**^{0}, where **p** is the mutually conjugate direction.
2.8
2.9
2.10
2.11
2.12
Normally, an initial approximation is **x**^{0}=0, then **r**^{0}=**b** and **p**^{0}=**A**^{T}**b**. If the residual **r** is zero, it implies that the problem is solved. Otherwise, if the residual **r** is non-zero, the desired solution for **r**^{k} is minimized, which can be monitored by the L2 norm of the **r**^{k} for each iteration. This can be achieved by updating the residual **r** into the problem iteratively.

For motion-compensated CGLS, a motion compensation technique based on our previous report [18] was used. The weight matrix **A** is partitioned into sub-matrices to be shifted; its column appropriately applied according to a motion model to each measured projection. This technique implements the motion compensation into a matrix **A** rather than image side **x**. The weight matrix **A**∈ℜ^{M×N} was partitioned by row into sub-matrices that correspond to each unique projection. The motion compensation was then applied separately for each sub-matrix, by shifting the columns within each sub-matrix according to the motion model for the relevant projection. For this reason, the motion must be estimated for each and every projection.

To evaluate the motion-compensated CGLS, two types of motions ‘correct motions’, assuming the motion was exactly known and ‘approximated motion’ are considered, assuming that the motion was known but with some error. In ‘correct motion compensation’, matrix **A** was compensated by applying the same amount of motion as applied to the phantom for each projection. Considering limited spatial resolution in EIT imaging the ‘approximated motion compensation’ is closer to EIT motion data. This has been investigated in previous study [18], where images with 10% error in motion data could sufficiently compensate for motion artefacts in CBCT images. For ‘approximated motion compensation’, matrix **A** was shifted by the motion signals extracted from EIT images for the relative projection.

Moreover, regularized CGLS was also studied here for limited data problems. Limited data generally require regularization terms for image reconstruction. The standard Tikhonov is one of the commonly used methods for regularization. It was used in this study to create a regularized CGLS as follows:
2.13
where λ is the regularization parameter, **R** is the regularization matrix, which is the identity matrix in this case.

#### (i) Conjugate gradient least-squares reconstruction of consistent and inconsistent phantom

These are the results of CGLS reconstruction of CBCT for a consistent phantom and an inconsistent phantom. Figure 3*a* shows the image error plot between reconstructed images and the true image (figure 3*b*) for both cases. The behaviours of consistency and inconsistency were similar to each other for the first few iterations. After the 12th iteration, the reconstruction of the consistent phantom remained steady, but, for the inconsistent phantom, the reconstruction after the 8th iteration diverged. Reconstructed images at the 1st and the 12th iteration of the consistent phantom are shown in figure 3*c*,*d*; and reconstructed images at the 1st and the 12th iteration of the inconsistent phantom are shown in figure 3*e*,*f*, respectively. These results clearly showed the sharper images for the consistent phantom.

#### (ii) Regularized conjugate gradient least squares for inconsistent phantom

Due to the divergence after a few iterations of the inconsistent phantom reconstruction, regularized CGLS was introduced in this study. Variation of the regularization parameter (λ) was also investigated. Figure 4*a* shows the RMSE of no regularized CGLS and regularized CGLS with different values of the λ. The reconstructed images, after the 12th iterations, are shown in figure 4*b*–*e*. The RMSE plot shows that the appropriate λ-value could reduce divergence effects of the CGLS reconstruction for the inconsistent case. In this study, λ=10 gave the best convergence compared with no regularization and regularization for λ={1,100}. In conclusion, the regularization with optimal regularization parameter has a feasibility to keep the convergence of CGLS reconstruction for inconsistent case steadily after the 12th iteration. However, in this study the 12th iteration was chosen for the motion-compensated CGLS in order to keep the computational time minimal.

## 3. Motion-compensated cone beam computed tomography: simulated motion data

### (a) Simulated motion data

A simulated phantom was created for investigating motion-compensated CGLS image reconstruction using EIT motion signals. To test the CGLS algorithm for CBCT imaging, a consistent phantom was first reconstructed. However, since most clinical cases were inconsistent, image reconstruction of inconsistent cases was also studied by corrupting the right-hand side of the **Ax**=**b** with errors, so-called ‘inconsistency’. Inconsistent cases were created by adding random Gaussian noise at 5% of the standard deviation for projection data at each relative projection.

For testing the motion-compensated CGLS, a moving phantom was created by shifting the simulated phantom with 20 mm peak-to-peak amplitude of sinusoidal signal (figure 2) in the up/down direction. Motion-compensated CGLS for the phantom when motion with additive motion error applied was subsequently evaluated. Three types of motion error (amplitude error, Gaussian error and phase error) with different percentages of error were added into the original sinusoidal signal (figure 5). For amplitude error, 5, 10, 15 and 20% of the 20 mm peak-to-peak amplitude of the original sinusoidal signal was applied. Next, motion signals with additive Gaussian noises were created by adding Gaussian noises with standard deviation set to 5, 10, 15 and 20% of the 20 mm peak-to-peak amplitude of the original motion signal. Finally, phase errors were generated by using 2–20% with 2% interval of a complete cycle of the original signal. These motion signals with motion error were used to shift the phantom when collecting projection data.

### (b) Motion-compensated conjugate gradient least squares

The motion-compensated CGLS algorithm (as described in Materials and methods) was tested in this part of the study. The RMSE of no motion applied, sinusoidal motion applied, and correct motion compensation when motion applied are shown in figure 6*a*. The plot of the correct motion compensation shows similar behaviour as to motion applied. When motion is applied into the data acquisition process, the image reconstruction shows motion blur artefacts, as can be easily seen in figure 6*c*. The results here suggested that the motion-compensated CGLS can be used to reduce the motion blur artefacts in CBCT image reconstruction.

### (c) Motion-compensated conjugate gradient least squares when motion error applied

This was the study of 20 mm peak-to-peak amplitude of sinusoidal motion signal, corrupted by three types of motion error: amplitude error, Gaussian error and phase error. Figure 7*a*–*c* shows the RMSE of reconstructed images when motion error applied and motion-compensated CGLS are used, and figure 7*d*–*g* shows the motion-compensated images after the 12th iteration of no motion error applied and motion error applied for 20% of motion. The RMSE plots show that an increase in percentages of the motion error results in an increase in RMSE for all types of error. Phase shifting was the most effective for motion-compensated CGLS when compared with amplitude error and Gaussian error. This can be clearly seen in the reconstructed images shown in figure 7*d*–*g*. The reconstructed image of phase error added (figure 7*g*) shows higher blurring artefacts at the same level of motion error.

## 4. Motion-compensated cone beam computed tomography: electrical impedance tomography phantom data

The motion information is extracted from a sequence of EIT images using a simple motion detection technique. This technique is an image processing thresholding technique developed and used in [18,24]. The image processing thresholding technique converts the grey-scale EIT images to binary images using the average of maximum and minimum pixel values for each image. The centre of an object within a binary image is extracted, and then is tracked for a sequence of images. This results in the difference in positions of centroids between the sequence images and the first image, the so-called motion signal extracted from EIT images.

The motion-compensated CGLS, using motion signals extracted from EIT images, was tested with a 50 mm diameter cylindrical object. This object was moved in the up/down direction for 20 mm peak-to-peak displacement and 60 mm peak-to-peak displacement and in the left/right direction for 40 mm peak-to-peak.

The results of the motion-compensated CGLS using EIT motion signals for the up/down movement are shown in figure 4 for 20 mm displacement and figure 6 for 60 mm displacement. Figure 8 shows the results of the motion-compensated CGLS using EIT motion signals for the left/right direction with 40 mm movement.

For the 20 mm up/down displacement, reconstructed images after the 12th iteration are shown in figure 8*a*–*d*, the RMSE is shown in figure 8*e*, and the one-dimensional plot is shown in figure 8*f*. It can be seen that figure 8*b* is corrupted by motion. When the correct motion compensation technique (as described above) was applied into the image reconstruction process, the resulting image is shown in figure 8*c*. Furthermore, figure 8*d* shows a motion-compensated image using motion signals extracted from EIT images. The RMSE shows that the behaviours of no motion applied and correct motion compensation were very similar. On the other hand, motion compensation using the EIT motion signal has higher image error than the correct motion compensation; however, lower image error than motion applied. The one-dimensional plot also supports the results of the reconstructed images and the RMSE plot.

The results of 60 mm up/down displacement motion compensation are shown in figure 9. The reconstructed images of no motion applied, motion applied, correct motion compensation applied and motion compensation using EIT motion signal applied are given in figure 9*a*–*d*, respectively. Figure 9*e*, *f* shows the RMSE and one-dimensional plot of reconstructed images. It can be seen that the motion-compensated image using the EIT motion signal of 60 mm shows higher image noise than the motion-compensated image using EIT motion signal of 20 mm.

The results of motion compensation using the EIT motion signal for 40 mm displacement in the left/right direction are shown in figure 10. Figure 10*a* is the reconstructed image without motion applied. Blurring of the movement in the CBCT reconstructed image can be noticed in figure 10*b*. Figure 10*c* and *d* shows motion-compensated CGLS images at the 12th iteration using correct motion signal and motion signal extracted from EIT images, respectively. The RMSE plot shows the behaviour of motion compensation using the correct motion signal is similar to the behaviour of no motion applied. In addition, motion compensation using the EIT motion signal can reduce the motion blur artefact on CGLS image reconstruction. A one-dimensional plot running through the centre of object in figure 10*a*–*d* is shown in figure 10*f*. The boundaries of the object, in motion-compensated images, are sharper than those of the motion image.

Comparing the motion-compensated images using the EIT motion signal for 20, 40 and 60 mm displacements, motion artefacts for the 20 mm image are lower than those for the 40 and 60 mm images. These results suggested that motion-compensated CGLS using the EIT motion signal can reduce motion blur artefacts, especially for small amounts of motion. This implied that the efficiency of removing motion artefacts may depend on the accuracy of the motion detection technique.

## 5. Conclusion

This paper presents motion-compensated CBCT with limited data (one-fourth data) using the CGLS reconstruction algorithm. CGLS is a superior iterative method compared with Kaczmarz type methods (ART, SIRT and SART), in terms of the speed of convergence, well suited for parallel computation, and the possibility of including the regularization and *a priori* knowledge in the image reconstruction process. The results suggested that the motion-compensated CGLS can improve image blur artefacts. Moreover, the motion-compensated CGLS was applied to our dual modality of EIT–CBCT, which uses the motion signal extracted from EIT images to compensate in CBCT image reconstruction. The results show that the motion signal extracted from EIT images can be used to compensate the motion artefact in CBCT image reconstruction by using the proposed motion-compensated CGLS algorithm. Furthermore, the performance of the motion compensation algorithm depends on the accuracy of the motion signal. Further work will be required to implement this dual modality strategy in more meaningful clinical settings. Motion compensation can reduce blurring artefacts in CBCT scans, especially the movement of body organs, e.g. heart, chest and abdomens.

## Footnotes

One contribution of 11 to a theme issue ‘X-ray tomographic reconstruction for materials science’.

- Accepted January 20, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.