Estimating intrafraction tumor motion during fiducial-based liver stereotactic radiotherapy via an iterative closest point (ICP) algorithm

Traditional radiotherapy can prolong survival for patients with resectable liver cancers [1] but it offers limited efficacy for the treatment of unresectable primary and metastatic liver cancers mainly due to the low whole liver tolerance to radiotherapy [2, 3]. Stereotactic body radiation therapy (SBRT), which is an accurate external beam irradiation method to deliver conformal high doses in a few fractions, has been proven to be an effective treatment modality for liver cancers with an elevated rate of local control [3‐8].

However, target motion may compromise the accuracy of liver stereotactic radiotherapy. It was reported that liver motions of up to 25 mm and 55 mm were observed under normal respiration and deep-breathing, respectively [9‐11]. The effect of organ motion on dose has also been investigated [12‐14]. According to Velec et al. [14], 70% patients involved in their study treated with liver stereotactic radiotherapy had accumulated dose deviations relative to the planned static prescription dose > 5%, ranging from − 15 to 5% in tumors and 42 to 8% in normal tissues. Management of intrafraction motion is crucial to ensure successful liver SBRT so that nearby healthy tissues and critical organs can be spared. Thus, liver tumor translation, rotation and deformation should be considered in both planning and treatment. Many studies have quantified the rigid and non-rigid motions of liver tumors using 4D computed tomography (4DCT) and/or cone beam computed tomography (CBCT) [14‐18]. Yet liver tumors are difficult to visualize on X-ray images due to their low contrast against soft tissues around.

An effective solution to this problem is the use of implanted fiducial markers as surrogates of liver tumors [19‐21]. Xu et al. [22] proposed a geometric solution to reconstruct the 3D locations of the fiducials and quantified the rigid motion of liver via Least-squares fitting algorithm [23].This is a closed form solution based on singular value decomposition (SVD) of a 3 × 3 matrix derived from two point sets. According to Murphy et al. [24], the basic SVD solution is ambiguous in differentiating reflections from rotations especially for the case of only three fiducials (which means point sets are coplanar.) A reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points. The matrix of a reflection is orthogonal with determinant − 1. In our study, the basic SVD solution yields 175 reflections in 360 trials involving three fiducials. Thus, the basic SVD method must be carefully implemented in the appropriate situation to avoid failures caused by singularities.

According to Euler’s rotation theorem, any rotation in three dimensional space can be represented as a combination of a unit vector \( \hat{\boldsymbol{e}} \) (called the Euler axis) indicating the direction of an axis of rotation, and an angle θ describing the magnitude of the rotation about the axis. The quaternions, firstly described by W.R.Hamilton [25] in 1843, give a simple way to encode this axis–angle representation in four numbers. A quaternion representation of rotation can be written as \( \hat{\boldsymbol{q}}={q}_i\boldsymbol{i}+{q}_j\boldsymbol{j}+{q}_k\boldsymbol{k}+{q}_r={\left[{q}_i\kern0.5em {q}_j\kern0.5em {q}_k\kern0.5em {q}_r\right]}^T \). In terms of the Euler axis \( \hat{\boldsymbol{e}}={\left[{e}_x\kern0.5em {e}_y\kern0.5em {e}_z\ \right]}^T \) and angle θ, the four components of this quaternion are expressed as follows:

The quaternion-based algorithm doesn’t involve with singular value decomposition, which is preferred for our purpose since reflections are not desired. The iterative closest point (ICP) algorithm is a robust and fast algorithm which has been demonstrated useful in estimating real-time prostate motion [26]. In this study, we aim to estimate the intrafractional rigid motion of liver tumor based on real-time KV X-ray images acquired by CyberKnife stereo imaging system via a quaternion-based ICP algorithm.

This study was approved by the Institutional Review Board at the Tongji Medical College of Huazhong University of Science and Technology. All methods were carried out in accordance with the relevant guidelines and regulations.

A total of 495 pairs of kV X-ray images from 12 patients were analyzed in this study. The translational and rotational motion ranges are summarized in Table 2. Translation and rotation are measured as the mean of the maximum range of motion for each case. For all patients, the translational motion ranges in left-right (LR), anterior-posterior (AP), and superior-inferior (SI) directions were 2.92 ± 1.98 mm (∆x), 5.53 ± 3.12 mm (∆y), and 16.22 ± 5.86 mm (∆z), respectively. Translational motion range in 3D space (Δd) can be computed as:

The translational motion range in 3D space was 11.89 ± 5.11 mm (∆d).

The rotational angles in LR, AP, and SI directions were 3.95° ± 3.08° (∆θ_x), 4.93° ± 2.90° (∆θ_y), and 4.09° ± 1.99° (∆θ_z), respectively. Figure 5 displays a normalized frequency histogram of intrafractional translational shift and rotational angles of all patients. Large rotation angles exceeding 8° were only observed in two cases. Figure 6 shows the registration residual error between the reference points and the transformed target points with rigid corrections (Fig. 6, red) and with translational corrections only (Fig. 6, blue). The mean ± SD of residual errors with and without rotational correction in each direction are displayed in the Table 3. The difference in each direction was statistically significant with and without rotational correction (P < 0.001). Rotational corrections decreased 3D residual error from 1.19 ± 0.35 mm to 0.68 ± 0.24 mm.

In this study, the intrafraction translations and rotations of liver tumor were estimated via the ICP algorithm during CyberKnife-based stereotactic radiotherapy. The maximum translational motion occurred in the SI direction due to breathing motion, and the smallest translation was observed along the LR direction, which agrees with previous studies [15, 18, 20, 21]. Xu et al. [22] obtained kV images from CyberKnife imaging system and reported that the means ± SD of the absolute value of intrafraction translations in liver SBRT was 2.1 ± 2.3 mm (LR), 2.9 ± 2.8 mm (AP), and 6.4 ± 5.5 mm (SI). Higher values were obtained in this study, since the maximum motion range for each patient was used to calculate the mean and standard deviation of both translation and rotation. In addition to translational movements, large rotational motion angles were observed. This suggests that the rigid motion of liver tumor should be paid special attention to for some patients. The residual error with rotational correction was 0.65 ± 0.23 mm, probably due to liver tumor deformation during treatment.

Several ways can be utilized to compute the optimal rigid transformation of two geometric data sets. Matching up the correspondent point one-to-one between the reference image and the test image is necessary to solve this problem. Distance between fiducials should exceed 20 mm to avoid overlapping or mismatching [24]. It is not difficult to associate the target point with the corresponding point in reference image in our case. Once the correspondence is known, the orthogonal transformation that minimizes the residual error in points set registration can be found with both of the closed form and iterative methods. This indicates that true least-squares minimum exists in the solution and the convergence of the iterative algorithm is guaranteed. The closest iterative point (ICP) algorithm is commonly used in medical image registration [31, 32], and it was found to be more general and robust when dealing with corrupted data which is more common in realistic scenarios.

The algorithm for fiducial segmentation in this study might introduce minor errors. Firstly, the resolution of kV images obtained by CyberKnife imaging system was 1024 × 1024. Thus, the centroid of the ‘white blobs’ calculated by the coordinates of the pixel block may deviate from the actual centroid of the gold fiducial markers. Secondly, the process of binarization could also introduce minor errors to the calculation of the fiducial centroids. Automatic solutions to segment radiopaque fiducial markers from CBCT scans have been proposed previously [27, 33, 34]. Mao et al. [33] presented a pattern matching algorithm using matched filters and template matching for detecting fiducials specifically designed to work with both kV and MV image data. However, this algorithm might be less applicable in this study because the projections of fiducials change with projection angle caused by liver deformation. Other solutions using CBCT projections with large angular separation and good marker contrast on a uniform background might also be not suitable in this study. Considering that the image data used in this study contained considerable noise caused by the bone structure, false signals are almost inevitable with these methods. Fiducial trajectory estimation is important in tumor motion management such as tumor tracking. A robust and reliable automatic segmentation method used in CyberKnife-based kV X-ray images needs to be developed.

Tumor motion management is necessary to improve the accuracy of SBRT. The rotational and translational motions of liver tumors during SBRT were estimated based on the CyberKnife system via the ICP algorithm. The residual error after registration decreased significantly with rotational correction. The results of this study can be used in motion management and planning target volume (PTV) margin determination for both liver SBRT and conventional radiotherapy.