A study of nonuniform CTV to PTV margin expansion incorporating both rotational and translational uncertainties

Abstract Purpose In this work, we implemented a method to obtain a nonuniform clinical target volume (CTV) to planning target volume (PTV) margin caused by both rotational and translational uncertainties and evaluated it in the treatment planning system (TPS). Materials and method Based on a previously published statistical model, the relationship between a target margin and the distance d (from isocenter to target point), setup uncertainties, and significance level was established. For a single CTV, it can be thought as a combination of many small volume elements or target points. The margin of each point could be obtained using the suggested statistical model. The whole nonuniform CTV–PTV margin was determined by the union of all possible margins of the CTV boundary points. This method was implemented in the Pinnacle3 treatment planning system and compared with uniform margin algorithm. Ten vertebral metastases targets and multiple brain metastases targets were chosen for evaluation. Results The combined CTV–PTV margin as a function of d for various initial translational margin and rotational uncertainties was calculated. The combined margin increases as d, rotational uncertainties and translational margin increase. For the same rotational uncertainty, a smaller initial translational margin requires a larger rotational margin to compensate for the rotational error. Compared with the uniform margin algorithm, the advantage of this method is that it could minimize the PTVs volume for given CTVs to obtain same significance level. Using vertebral metastases targets and multiple brain metastases targets, a series of volume difference was obtained for various translational margins and rotational uncertainties. The volume difference of PTV could be more than 17% when translational margin is 2 mm and rotational uncertainty is 1.4°. Conclusion Nonuniform margin algorithm could avoid excessive compensation for the CTV boundary points near isocenter. This method could be used for clinical margin determination and might be useful for the protection of risk organs.


| INTRODUCTION
It is well known that setup uncertainties would be introduced throughout the treatment delivery. These setup errors during treatment are critical to the success of radiotherapy. 1-4 Without proper attention, these setup errors can cause misalignment of the beams and lead to the radiation dose delivered outside the target area. To solve this problem, a method is provided to account for setup errors in the ICRU 50 report and its supplement. 5 According to ICRU 50, a margin is added to the clinical target volume (CTV), thus yielding the planning target volume (PTV). The entire PTV is given a prescription dose to ensure that the CTV receives desired coverage. During the past years, many recipes for CTV-PTV margin have been developed by different groups. [6][7][8][9][10][11] Usually, PTV is formed by drawing a uniform margin around CTV to account for setup errors and possible motion during the treatment. The setup errors include both translational and rotational setup uncertainties. Currently, kilovoltage cone beam computed tomography (kV-CBCT) and Optical Surface Monitoring System (OSMS) are widely used in clinical practice to provide image guidance for treatment. 9,12 Both three-dimensional translational and rotational positioning errors can be detected and measured. In current clinical practice, translational errors can be corrected online since they are easily implemented using couch shifts along three axes.
Rotational setup differences between the patient's position in the linac and the CT scanning position can be corrected using advanced couches that have six degrees of freedom. 13,14 While the use of such couches is increasingly prevalent, there are still many linacs with conventional couches that cannot correct for pitch, roll, or yaw. errors for spine and lung stereotactic body radiotherapy. They found even if the rotational setup error was ≤2°, rotational errors alone could cause an unexpected dosimetric effect. They found the rotational setup error was related to the isocenter location, and rotational setup error would be more significant if the isocenter is far from the geometric center of the target. To account for these rotation errors, an extra margin is needed for the margin between CTV and PTV. 15,16 Zhang et al. reported an analytical formula to determine the extra margin between CTV and PTV to account for both translational and rotational setup errors. 17 Remeijer et al. described a probability-based approach to generate margins for translational and rotational uncertainties. 18 Most of the previous studies usually use uniform CTV-PTV margin which may lead to a large PTV.
In addition, a new technology called single-isocenter for multiple target (SIMT) technique has been developed in recent years. In SIMT technique, one isocenter which is usually located in the geometrical center of the total CTVs is used for treating multiple lesions. 19 With SIMT technique, significant time can be saved for patient setup and radiation delivery. But this technique will also introduce additional rotational uncertainties and these uncertainties are not easy to be corrected with IGRT and six-degree couch for all targets. The additional rotational uncertainty could not be ignored when the distance between isocenter and target is large. Roper et al. reported that target coverage was significantly lower than expected when rotational errors (>1°) were introduced. 20 Chang developed a new margin recipe using a statistical model to investigate the effect of additional rotational error for the SIMT technique. 21 But the margin size for single target in one direction (x, y, or z) is also uniform.
The size of extra CTV-PTV margin introduced by rotational uncertainties depends on the treatment site, the distance from the isocenter to the target, rotational uncertainties, and the required confidence level for tumor coverage. 10 If isocenter, rotational uncertainties, and the required confidence level have already been fixed, the extra margin size of one boundary point in target is mainly related to the distance between the point and the isocenter. So the needed extra margin size caused by rotation will change when the CTV points are in different boundary location. If a uniform value is chosen as the whole target margin, which is usually close to the maximum of all points, it would cause the target volume to be very large. As a result, large target volume will increase the dose of the surrounding organs.
In this paper, we propose a method to obtain a nonuniform CTV-PTV margin caused by setup uncertainties. This method is based on a statistical model considering both the conventional translational error and the additional rotational uncertainty. The benefits of this method will be investigated by comparing it with uniform margin algorithms. For rotational uncertainties of all points at the CTV boundary are considered, the CTV-PTV margin recipe derived from this method will better reflect the expansion necessary. This method could be used for clinical margin determination and might be useful for the protection of organs nearby. The translational error vector e S as shown in Fig. 1(a) is a random vector with fixed amplitude and direction regardless the CTV location. In Fig. 1(b), the CTV rotates around the axis that is normal to the paper passing through the isocenter (ISO) for δ degrees. The rotational error e R is also a random vector. However, its amplitude is equal to sinδ × d ≈ δd (for small δ), where d is the distance between the isocenter and the CTV point. And its direction is not fixed but along the rotational direction. When considering the CTV to PTV expansion, the combined setup error e R + e S (translated & rotated) needs to be compensated. MIAO ET AL.

| 79
Chang has developed a statistical model to analyze the rotational error for the SIMT technique. 21 In this model, the translational setup error follows the three-dimensional independent normal distribution with zero mean and a standard deviation (SD) of σ s (in mm). It is assumed that the rotation happens randomly and follows a three-dimensional (3D) independent normal distribution with a zero mean and a uniform SD of σ D (in degrees). Correspondingly, rotational setup error determined by rotation also follows a 3D-independent normal distribution with a zero mean and a uniform SD of . The value of σ R is proportional to d (the distance between the isocenter and the CTV point) and rotational uncertainty σ D . Based on the published method, 22 the combined PTV margin is (1) where χ 2 α is the critical value of Chi-square distribution with three degrees of freedom for significance level α. And χ α values for several probability levels are given in Table 1. M S ¼ χ α σ S is the required PTV margin for the translational error, and M R ¼ χ α σ R is the required PTV margin for the rotational error. From Eq. (1), it is clear that the combined PTV margin M E is related with translational margin M S , rotational uncertainty σ D , and the distance d (between the isocenter and the CTV point).

2.B | Nonuniform margin
For a single target, it can be thought as a combination of many small volume elements or target points. Based on the above statistical model, the margin of every point could be obtained using Eq. (1). The whole CTV-PTV margin is determined by all possible margins of the CTV boundary points. In this paper, χ α value, translational setup error, and rotational setup angle of boundary points are assumed to be fixed value. Since the distance between isocenter and every CTV boundary point is different, the margin of each CTV boundary point will not be equal. Finally, we would obtain PTV as the union of CTV nonuniform expansion as shown in Fig. 2(a). For the SIMT technique, the isocenter is usually outside the target area, and the nonuniform margin for every target will be calculated one by one as shown in Fig. 2(b).

2.C | Implementation of the method
The method was implemented in the Pinnacle 3 treatment planning system (version 9.10, Philips). The gross tumor volume (GTV) and CTV were delineated and reviewed by experienced radiation oncologists based on the planning CT. Pinnacle 3 Scripts and custom Python code were used to export CTV contour, ISO location and other position information into DICOM files, which includes RT-Structure and RT-Plan. Then parameters of contoured structures were extracted from the DICOM files using the software package Computational Environment for Radiotherapy Research (CERR). 23 The contoured structures were converted into 3D matrix using an open-source software visualization toolkit (VTK)-based algorithm. The matrix dimension was same as CT images, which were 512 × 512 pixels in transverse plane and 3-mm slice thickness in axial direction. CTV in 3D matrix format was processed with custom Python code to obtain a nonuniform margin. At last, the PTV in DICOM format was imported into Pinnacle 3 treatment planning system by Pinnacle 3 Scripts. A flow diagram depicting this process could be seen in

2.D | Evaluation of the proposed method
This method was compared with uniform margin algorithm provided by Pinnacle, which was widely used now. We chose vertebral F I G . 1. Illustration of the (a) translational and (b) rotational errors. The translational error in (a) is a random vector. The amplitude and direction remain the same regardless of the location of the clinical target volume (CTV). The amplitude of rotational error in (b) increases with d, the distance between the isocenter and the CTV.
T A B L E 1 Chi-square distribution table with three degrees of freedom, χ α is the critical value of Chi-square distribution, and α is significance level.     Table 2 shows the results of volume difference between V uniform and V non-uniform with a series of M S , σ D values.

3.B | Vertebral metastases
From the volume comparison, we can see that the PTV volume obtained by our nonuniform margin algorithm is less than volume obtained by the uniform margin algorithm. When the rotational uncertainty is small, volume difference is almost negligible. But if this rotational uncertainty becomes big, the volume difference of the PTV will be significant. The maximum value of volume difference in Table 2 is more than 17%, and the corresponding M S , and σ D values are 2 mm, and 1.4°, respectively. It can be easily predicted that this volume difference will become bigger if the values M S and σ D further increase. | 81 minimum and maximum of the distance between the isocenter and CTV points are 3.1 and 6.4 cm respectively. Table 3 shows the margin and volume of the ten patients' target treated with SIMT technique. tional error margin in order to make a safe significance level. Figure 6 shows an example of a vertebral metastases target treated with conventional fractionation. The margins are different for target points at different location. From Table 2, we can see that the volume difference of PTV could be more than 17% when σ D value is 1.4°.

| DISCUSSION
For a modern IGRT program equipped with kV on-board imaging device and six-degree couch, the residual rotational is generally about 0.5°. Although the volume difference of PTV could be ignored (2%-5%) when translational margin is 2 mm and rotational uncertainty is 0.5°, the result is still meaningful for some special tumor (e.g., multiple metastases). Due to relative motion between various parts of the multiple metastases (vertebral) targets, the uncertainties are impossible to be corrected as expected using sixdegree couch with CBCT. Moreover, new linacs are still being installed with standard couches. The rotational uncertainties will be bigger than 1°. For these patients, the use of a nonuniform margin will make a smaller PTV which may reduce the dose of risk organs nearby. And using a more accurate margin for each part of the target instead of a uniform margin should be the direction of our future efforts for precise radiotherapy.
As shown in Figs. 6, 7 and 8, excessive compensation for the CTV points near isocenter could be avoided in nonuniform algorithm.
This method is particularly suitable for the irregular target and multiple targets with single isocenter in which there is a significant difference in the distance from the isocenter to the target boundary points. The irregular target usually has significant differences in

| CONCLUSION S
In this paper, we proposed a method to obtain a nonuniform CTV-PTV margin caused by setup uncertainties. This method is based on a statistical model considering both the conventional translational error and the additional rotational uncertainty. The method could avoided excessive compensation for the CTV points near isocenter and was implemented in the clinical treatment planning system (Philips, Pinnacle 3 9.10). Compared with the uniform margin algorithm, the advantage of this method is that it will minimize the volume of the PTVs for the given CTVs to obtain the same significance level.
This method could be used for clinical margin determination and might be useful for the protection of risk organs nearby.

CONFLI CT OF INTEREST
The authors declare no conflict of interests.