

ORIGINAL ARTICLE 

Ahead of print publication 


Calculation of setup margin in frameless stereotactic radiotherapy accounting for translational and rotational patient positing error
Arjunan Manikandan^{1}, Biplab Sarkar^{2}, Anusheel Munshi^{3}, Saneg Krishnankutty^{4}, Tharmarnadar Ganesh^{3}, Bidhu Kalyan Mohanti^{3}, Sujatha Manikandan^{5}, Pradhan Anirudh^{6}, Surekha Chandrasekharan^{7}
^{1} Department of Medical Physics, Apollo Proton Cancer Centre, Chennai; Department of Medical Physics, Bharathiar University, Coimbatore, Tamil Nadu, India ^{2} Department of Radiation Oncology, Apollo Multispeciality Hospital, Kolkata, West Bengal; Department of Physics, GLA University, Mathura, UP, India ^{3} Department of Radiation Oncology, Manipal Hospital, Delhi, India ^{4} Department of Radiation Oncology, Fortis Memorial Research Institute, Delhi, India ^{5} Department of Radiotherapy, Government General Hospital, Guntur, Andhra Pradesh, India ^{6} Director, Centre of Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura, UP, India ^{7} Department of Medical Physics, Bharathiar University, Coimbatore, Tamil Nadu, India
Date of Submission  03Mar2021 
Date of Acceptance  30May2021 
Date of Web Publication  09May2022 
Correspondence Address: Biplab Sarkar, Apollo Multispeciality Hospitals, Kolkata, West Bengal India
Source of Support: None, Conflict of Interest: None DOI: 10.4103/jcrt.jcrt_359_21
Context: Rotation corrected setup margins in stereotactic radiotherapy (SRT). Aims: This study aimed to calculate the rotational positional error corrected setup margin in frameless SRT. Settings and Design: 6D setup errors for the steriotactic radiotherapy patients were converted to 3D translational only error mathematically. Setup margins were calculated with and without considering the rotational error and compared. Materials and Methods: A total of 79 patients of SRT each received >1 fraction (3–6 fractions) incorporated in this study. Two conebeam computed tomography (CBCT) scans were acquired for each session of treatment, before and after the robotic couchaided patient position correction using a CBCT. The postpositional correction setup margin was calculated using the van Herk formula. Further, a planning target volume_R (PTV_R) (with rotational correction) and PTV_NR (without rotational correction) were calculated by applying the rotation corrected and uncorrected setup margins on the gross tumor volumes (GTVs). Statistical Analysis Used: General. Results: A total of 380 sessions of pre (190) and post (190) table positional correction CBCT was analyzed. Posttable position correction mean positional error for lateral, longitudinal, and vertical translational and rotational shifts was (x)0.01 ± 0.05 cm, (y)0.02 ± 0.05 cm, (z) 0.00 ± 0.05 cm, and (θ) 0.04° ± 0.3°, (Φ) 0.1° ± 0.4°, (Ψ) 0.0° ± 0.4°, respectively. The GTV volumes show a range of 0.13 cc–39.56 cc, with a mean volume of 6.35 ± 8.65 cc. Rotational correction incorporated postpositional correction setup margin the in lateral (x), longitudinal (y) and vertical (z) directions were 0.05 cm, 0.12 cm, and 0.1 cm, respectively. PTV_R ranges from 0.27 cc to 44.7 cc, with a mean volume of 7.7 ± 9.8 cc. PTV_NR ranges from 0.32 cc to 46.0 cc, with a mean volume of 8.1 ± 10.1 cc. Conclusions: The postcorrection linear setup margin matches well with the conventional setup margin of 1 mm. Beyond a GTV radius of 2 cm, the difference between PTV_NR and PTV_R is ≤2.5%, hence not significant.
Keywords: Frameless stereotactic radiosurgery/radiotherapy, robotic couch, rotational setup error, sixdimensional motion, stereotactic radiosurgerystereotactic radiotherapy, translational setup error
How to cite this URL: Manikandan A, Sarkar B, Munshi A, Krishnankutty S, Ganesh T, Mohanti BK, Manikandan S, Anirudh P, Chandrasekharan S. Calculation of setup margin in frameless stereotactic radiotherapy accounting for translational and rotational patient positing error. J Can Res Ther [Epub ahead of print] [cited 2022 Nov 29]. Available from: https://www.cancerjournal.net/preprintarticle.asp?id=345002 
> Introduction   
In threedimensional (3D) Euclidian space, rigid body motion exhibits six dimensions of freedom; three translational and three rotational. Patients being treated through radiotherapy are analogous to a rigid body when immobilized properly, exhibiting a similar sixdimensional (6D) degree of freedom. Therefore, positional errors are also distributed over three translational and three rotational coordinates. Advanced linear accelerators are equipped with a robotic couch, and advance onboard imaging facilities like the cone beam computed tomography (CBCT) or Exac track (BrainLAB AG, Feldkirchen, Germany) can identify and correct the 6D positional error for accurate patient positioning.
Numerous researchers have investigated the influence of rotational and translational motions for different treatment sites, including prostate, spine stereotactic body radiotherapy, and brain tumors including stereotactic radiotherapy (SRT).^{[1],[2],[3],[4],[5],[6],[7],[8]} However, the translational and rotational geometrical uncertainties are typically reported separately, except in the case of Onimaru et al.^{[4]} The clinical implication of patient positional inaccuracy involves the calculation of the setup margin to obtain the planning target volume (PTV) from a clinical target volume (CTV) or a gross tumor volume (GTV) in the case of stereotaxy. The total geometrical uncertainty used for calculating the setup margin can be divided into two parts, as the systematic component (preparation) and the random component (execution) variation.^{[9]} The systematic component can be defined as a patientdependent mean dislocation between the planning scan anatomy and the anatomy during treatment delivery. The random component is defined as the fluctuation over the time around the systematic displacement.^{[10]} A safe margin is defined to account for all geometrical uncertainties. The most commonly used margin recipes were defined by van Herk and Stroom et al.^{[10],[11]} These margin recipes were derived for homogeneous, 3D conformal radiotherapy and accounted for only the translational deviations during treatment.^{[12]} Furthermore, it is not obvious how to use these recipes for single fraction therapy since they do not fit in the calculation demonstrated by van Herk.^{[10]}
The robotic couch is a very useful tool for frameless radiosurgery patient positioning, which requires precise patient positioning and treatment delivery. It is already established that the Elekta robotic (HexaPOD) couch and CBCT system can achieve an accuracy of 0.2 mm and 0.2°, respectively.^{[13]} Other phantom studies have also demonstrated that robotic couches can achieve submillimeter and subdegree accuracy.^{[14],[15],[16],[17]} However, it must be considered that a phantom does not qualitatively represent a patient in terms of complexity and weight. Several researchers have investigated the efficacy of the robotic couch in frameless radiosurgery or other intracranial lesions using a BrainLab (BrainLAB AG, Feldkirchen, Germany) mask along with Varian (Varian Medical System, Polo Alto, CA, USA) or Elekta (Elekta, Stockholm, Sweden) linear accelerators and determined that it was easy and comparable with invasive frame radiosurgery.^{[18],[19],[20],[21],[22],[23]} In an ElektaBrainlab frameless Stereotactic combination a BrainLAB couch extension is used, an additional part that is attached to the Elekta linear accelerator couch (iBEAM evo couch). This extension does not firmly attach with the Elekta iBEAM evo couch, hence when loaded with patient there is a zag in the cranial end due to weight of the patient head and fulcrum effect. This phenomenon usually introduces a large shift in the vertical direction, which is subsequently corrected using a 6D motion enable robotic couch. Because of a design fault, both BrainLabElekta and BrainaLabVarian frameless stereotactic system shows an unusual cranial depression, which was previously studied by Sarkar et al., and therefore, robotic couchassisted positional correction is essential.^{[20]}
Another study by same group, derived and validated the mathematical formulation for converting the 6D (translational + rotation) positional correction to a 3D translational correction.^{[21]} This formulation allows the clinical end user to correct frameless stereotactic patients equally accurate without a 6D motion enabled couch, or in the case that it is not functioning in certain treatment session(s). The formula is based on the MATLAB programming which converts the translational + rotational motion in terms of the translational shifts only.^{[21]} The formula was theoretically and experimentally validated for the actual shift obtained during the patient positional correction using a 6D motion enable robotic couch and phantom measurements. Further, the formulation was used in several other clinical studies for different cranial and extracranial stereotactic techniques.^{[21],[22],[23],[24]}
Several researchers have investigated different aspects of 6D shifts; however, to the best of our knowledge, no formulation is available to incorporate the rotational shifts in the margin recipe. For example, Hyde et al. presented the shift data in sixdimension coordinates in millimeters and degrees, and Wilbert et al. presented them as systematic and random errors for 6D coordinates, while Infusino et al. presented a 6D margin.^{[2],[8],[19]} However, this information is not very useful to a radiation oncologist/physicist when expanding a CTV (GTV: In case of Stereotactic radiosurgery [SRS]/SRT) to a PTV margin in three dimensions.
The aim of this investigation was to calculate the GTV to PTV margin for frameless stereotaxy in an ElektaBrainlab setup, by converting 6D patient positional uncertainty to 3D translational only uncertainty, using the formula described earlier.^{[21]}
> Materials and Methods   
The choice of the coordinate system in a Euclidian space for a headfirst supine patient where leftright (LR) is the xaxis, the superiorinferior (SI) is the yaxis, and the vertical (AP) is the zaxis. Further, rotational motions are defined as LR (pitch), SI (roll), and AP (yaw).
Correction of the rotational error of a rigid body
A detailed overview of the theory of the rotational correction of a rigid body in a Euclidian space is presented for an inertial frame of reference. Two different scenarios are considered in this analysis. In the first case, top panel the origin of the coordinate system or origin (isocenter) is off of the target center, while the isocenter is at the target center in the second case. [Figure 1] represents the situation when the target center is away from the center of the coordinate system (origin) or isocenter. In the coordinate system shown in the top left panel, a black arrow pierces the black star at a certain angle. A rotational error which corresponds to the patient's positional setup error is introduced to the black star and represented in blue. As shown in the top panel, the angle between the black stararrow and blue star arrow is not simply defined by only a rotation. In such a situation, a linear coordinate transformation or Galilean transformation is required to obtain a new origin and to apply a rotational correction. The right top panel of [Figure 1] shows that it is possible to determine the appropriate puncture angle of the black arrowstar relative to the blue arrowstar prior to the introduction of a rotational error.  Figure 1: Theory of rotational error correction: Top panel: When target centre away from isocentre. In such scenario a Galilean/linear transformation is required along with the rotation to obtain the desired angle. Bottom panel: When the target centre at the isocentre, the rotational correction of the target can be done only with the rotation of the coordinate system no additional translational motion is required
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In the second scenario represented in lower panel, the isocenter is at the target center. This situation presents a much simpler geometrical framework since a linear coordinate transformation is not required to obtain the desired angle. In this case, the blue arrow can pierce the red star at the same angle as the black stararrow combination by simply rotating the coordinate system. On basis of the abovementioned theory, a formula has been deduced by Sarkar et al. which is used for the calculation of the rotation corrected setup margin in this article.^{[21]}
Patient selection
A total of 79 patients treated using SRT (3–6 fractions) were considered for this study. Their ages range between 15.5 years and 71 years with a mean age 45.6 ± 18.7 years, and a median age of 43.6 years. The delivered dose ranged between 18 and 36 Gy (mean 20.6 ± 7.7 Gy, median 21 Gy) and the median and mean number of the fraction was 3 and 4.7 ± 3.4, respectively. The delivered dose per fraction varies between 5 and 7 Gy with a median of 6.0 Gy and a mean of 5.7 ± 2.4 Gy. Tumor volume varied between 14.4 ± 23.8 cc (range 1.3 cc–131.8 cc). Tumor characteristics of all patients are presented in [Table 1].
Patient setup, imaging, positional correction, and treatment planning
Patients were immobilized with a doublelayered thermoplastic mask, and 1.5 mm thick computed tomography (CT) slices were acquired along with a head and neck localizer box target positioner, to generate the stereotactic coordinates. These values were generated in the iPlan (BrainLab AG, Feldkirchen, Germany) treatment planning system by localization of the CT localizer box and the target positioner. Radiotherapy treatment planning was performed using an Xray Voxel Monet Carlo based Monaco (CMS Elekta, Sunnyvale, CA) planning system without altering the stereotactic coordinates. Setup errors were acquired based on the CBCT results and performed using a 6D motion enabled HexaPOD couch (iBEAM evo, Medical Intelligence, Schwabmunchen, Germany). Acquired CBCT images were matched with the reference images on the basis of mutual information fusion algorithm by the therapist. Image fusion was verified and electronically approved by a radiation oncologist and medical physicist for every session. Acquired 6D shifts were transferred to the robotic couch control system where simultaneous translational and rotational motion was carried out keeping plan isocenter as the origin of the coordinate system.^{[25]} All patients received treatment using the Elekta Axesse (Elekta AB, Stockholm, Sweden) linear accelerator equipped with forty pairs of multileaf collimators with a uniform width of 4 mm and interdigitization facility.
Single isocentric plans were primarily considered in this study based on our clinical preference. Only 3 out of 79 patients had more than one isocenter. If multiple isolated target volumes were placed in close proximity, a single isocentric treatment plan was created by utilizing the interdigitation facility of the MLC leafs. In the case of more than one isocenter plan, each isocenter was treated as an independent entity with complete repetition of pre and posttable position correction CBCT. Further detail of treatment planning strategy can be found elsewhere.^{[20],[21],[22],[23],[24],[26]}
Setup margin calculation
Our imaging protocol consists of two sets of CBCT images acquired during setup verification in [Figure 2]. A total of 308 sessions of pre (190) and post (190) patient positional correction was considered. For prepositional correction, first CBCT imaging (correction CBCT) was performed, and complete positional corrections were done using the 6D motion enable robotic couch. Further, a postpositional correction image (verification CBCT) was acquired to ensure the correctness of the couch movement. There is a finite shift associated with the patient's positional correction, although this value is small, which is determined by postcorrection CBCT and termed as residual setup error. 6D data was converted to three dimensional (3D) (translation only) data using the formula (1a3a).^{[21]} Further, the GTV to PTV margin was calculated for both the pre and postpositional correction conditions using the van Herk formula for correction and verification CBCT, and the results are presented in [Table 2] and [Table 3], respectively. Margin defined as 2.5∑ +0.7 σ where systematic error (∑) defined as standard deviation of mean and random Error (σ): Root mean square of standard deviation.^{[10]}  Figure 2: Left panel : Pre table position correction cone beam CT images and shifts Right panel : Post 6D table correction cone beam CT matching
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 Table 2: Pre positional correction setup margins in lateral longitudinal and vertical direction with incorporating the rotational angle and without incorporating the rotational angle
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 Table 3: Post positional correction setup margins in lateral longitudinal and vertical direction with incorporating the rotational angle and without incorporating the rotational angle. Post positional correction margin attributed to the residual setup error
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Patients with 1 fraction treatment were not considered in this study since these data are not readily implemented in the current formula.^{[10]}
> Results   
Pre table positional correction error
The pretable position correction mean positional error for lateral, longitudinal, and vertical translational and rotational shifts is (x) 0.08 ± 0.09 cm, (y) 0.04 ± 0.21 cm, (z)0.12 ± 0.2 cm and (θ) 0.2° ± 1.1°, (Φ) 0.4° ± 0.9°, and (Ψ)0.1° ± 1.1° respectively.
Post table positional correction error
[Figure 3] presents the lateral, longitudinal, and vertical translational and rotational posttable positional correction CBCT imaging data. The postpatient positional correction positional errors or residual errors (in the same sequence) are 0.1 ± 0.5 cm,0.2 ± 0.5 cm, 0.0 ± 0.5 cm, and 0.0° ± 0.3°, 0.1° ± 0.4°, 0.0° ± 0.4°, respectively.  Figure 3: Post table position correction transitional (cm ) and rotational error (degree) for 79 steriotactic patients
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Pre table positional correction setup margin calculation
The prepositional correction setup margin in the lateral (x), longitudinal (y), and vertical (z) directions is 0.19 cm, 0.42 cm, and 0.34 cm, respectively, with the rotation correction. Without rotational correction, the margin in the same directions was 0.24 cm, 0.46 cm, and 0.36 cm, respectively. [Table 2] summarizes other essential parameters for uncorrected or pretable position correction condition.
Post table positional correction setup margin calculation
The postcorrection setup margins along the same axes in the lateral (x), longitudinal (y), and vertical (z) directions are 0.05 cm, 0.12 cm, and 0.1 cm, respectively.
The influence of the rotational angle in the setup margins was tested using the following methodology. GTV volumes ranged from 0.13 cc to 39.56 cc with a mean volume of 6.35 ± 8.65 cc for the tested patients. Treated PTV volumes ranged from 0.33 cc to 57.55 cc, with a mean volume of 9.8 ± 12.4 cc. Postcorrection margins (with rotational correction and without rotational correction) were calculated on the basis of the cone beam presented in [Table 3]. Postpositional correction margin attributed to the residual setup error.
Margins were applied over the GTV radius to generate two PTVs, one with a rotational correction (PTV_R) and another PTV which did not incorporate any rotational correction (PTV_NR). PTV_R values ranged from 0.27 cc to 44.7 cc, with a mean volume of 7.7 ± 9.8 cc. PTV_NR values ranged from 0.32 cc to 46.0 cc with a mean volume of 8.1 ± 10.1 cc. [Figure 4] depicts the difference between the postcorrection volume for PTV_R and PTV_NR. The mean absolute difference of the pre and postcorrectional volume for PTV_R and PTV_NR was 4.57 ± 3.39 cc and ranged from 0.77 cc (equivalent radius = 0.57 cm) to 15.6 cc (equivalent radius = 1.55 cm) and 5.0 ± 3.63 cc in the range from 0.9 cc to 16.8 cc, respectively. The mean variation of the absolute volume between the PTV_R and PTV_NR was 0.4 ± 0.3 cc and ranged from 0.05 cc to 1.35 cc. A relative variation in the range of 3%–18% was obtained with a mean variation of 7.5% ± 3.3%.  Figure 4: Absolute and relative difference of post positional correction planning target volume for planning target volume_R and planning target volume_NR as a function of gross tumour volume radius
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> Discussion   
This article presents the setup margins, both including and excluding the rotational setup errors and as a corollary the explanation of the rotation theory as well. The mean difference of the pre and postcorrectional volume for both PTV_R and PTV_NR was in the range of 4–5 cc, which accounts for a radius of GTV of 1 cm. These differences are very high and preclude frameless stereotaxy in the absence of a 6D motion enable couch. As the GTV radius increases, the absolute difference between the PTV_R and the PTV_NR increases. Moreover, the relative difference between the PTV_R and the PTV_NR is reduced. The variation exhibits a characteristic conic section like a parabola and varies as a function of the square of the GTV radius.
While analyzing the result in [Table 3], we found that the longitudinal (Y) and vertical (Z) shifts are substantially higher than the lateral (X) margins in a clinical setting. Even postpositional correction of the longitudinal and vertical margins does not exhibit submillimeter accuracy, except in the Xdirection. A similar result has been reported by another group.^{[19]} Numerous investigators reported submillimeter and subdegree accuracy with regard to the reproducibility of the stereotactic coordinate for the phantom measurements like Winston Lutz test or flexmap.^{[12],[13],[14],[15],[16]} However, during actual therapy delivery, a higher setup margin in the y and z directions was observed, which is causally related to the cranial end fall of the frameless couch extension due to patient weight. The dosimetric and mechanical characteristic of the couch extension for cranial stereotaxy was reported in our earlier studies.^{[20]}
It has also been reported by the other investigators that a substantial intrafraction and interfraction positional error is associated with the Brainlab mask and couch extension.^{[7],[17],[20],[21]} Therefore, frameless stereotaxy requires strict setup verification imaging and correction. Thus, a robotic couch is inevitable for frameless stereotaxy. The theory of rotational error correction presented in material and method section is valid for a rigid body in all inertial frames of reference where a Galilean transformation is valid.^{[21]} In the present context of the positional error and the robotic couch, only inertial frames of reference are encountered. Thus, the validity of the theory is assured in this respect. Furthermore, a wellimmobilized structure of the human anatomy behaves as a rigid body. The brain, which is welldemarcated by a bony envelope, can be considered as a rigid body. Hence, the validity of the theory is also assured in this regard. In the generalized situation where the tumor center is away from the origin/isocenter, it is not possible to obtain the required angle using only a rotational transformation and additional linear translation(s) are necessary. This is because rotational vectors are dependent functions of each other as well as translational vectors; whereas the translational vectors are independent of each other and the rotational vectors.^{[23]} Therefore, it is essential to express the rotational errors as translational errors, otherwise they cannot be utilized for any mathematical operation. Furthermore, although a robotic couch can perform a 6D movement in terms of three translations and three rotations; however, only 3D contouring margins are practically affordable to account for the 6D setup uncertainties.^{[10]} This is because no contouring station permits a user the flexibility of a 6D margin while expanding CTV (GTV: For SRS/SRT) to PTV. Other than this technical inefficiency of the contouring station, another important reason might be the difficulty in imagining the margin in the rotational directions. Therefore, it is essential to reduce the 6D positional error to a 3D translational dimensional margin for performing any spatial dosimetric evaluation. A recent study by Sarkar et al. mathematically and dosimetrically validated the principle of a 6D to 3D shift.^{[21]} A previously described matrix method by Onimaru et al. was mathematically incorrect.^{[23]}
A more pertinent question is whether it is allowed to do a partial correction of the rotational error. According to our understanding and all published literature one may ignore the rotational correction completely, which is usually the practice with 3D translational only couches. However, we strongly discourage half correction of the rotational errors, either it needs to be a full rotational correction using a 6D couch or no correction.
Translational correction is independent in each axis and also independent from the rotational errors and can be corrected in any x, y, or z axis by 3D couch; nevertheless, for 6D motion enabled robotic couch, translational and rotational motions performed simultaneously and cannot be segregated from each other.^{[23]} Rotational positional errors cannot be segregated or performed without performing the translational error correction. It may be questioned whether the van Herk formula is applicable to the stereotactic technique since it ensures that only 90% of patients receive 95% of the prescription dose. A literature review reveals the use of the formulation in case of SRT with or without modification.^{[25],[26],[27]} We included ≥3 fraction treatment only as van Herk formula is not applicable for 1 fraction treatment and 2 fractions SRS treatment is clinically not applicable therefore minimum of 3 fraction treatment was considered for this study.
van Herk demonstrated the application of this formula for a single fraction brain metastasis during the 2011 AAPM summer school.^{[25]} However, it is not possible to accurately calculate the delineation error and the setup error independently. Therefore, we restricted the use of this formula to cases involving more than 1 fraction treatment, which prevents the inaccuracies due to inefficient calculation of single fraction setup and delineation error. Our study is valid as the formula is applicable for both single and multiple fractions.
> Conclusions   
This study is the first to incorporate the rotational patient positional error in the setup margin calculation. The setup margins considering and without considering the rotational correction is around 1 mm which is in line with margins classically used in the invasive frame based steriotaxy.^{[28],[29]} Further this study also find if the tumour having diameter ≥4 cm the difference between rotational corrected and uncorrected PTV volume is <5%, hence can be considered insignificant.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4]
[Table 1], [Table 2], [Table 3]
