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Purpose: The isocenter of a medical linac system is a frequently used concept in clinical practice. However, so far not all the isocenters are rigorously defined. This work is intended as an attempt of deriving consistent and operable isocenter definitions. Methods: The isocenter definition is based on a fundamental concept, the axis of rotation of a rigid body. The axis of rotation is determined using the trajectory of any point on a plane that intersects the rigid body. A point on the axis of rotation is found through the minimal bounding sphere of the trajectory when the rigid body makes a full rotation. The essential mathematical tool of the isocenter definition system is three-dimensional coordinate transformation. Results: The axes of rotation of the linac collimator, gantry, and couch are established first. The linac mechanical isocenter (linac isocenter) is defined as the center of a circle that best fits the trajectory of a select linac X-ray source position. The axis of rotation and the minimal bounding sphere are cornerstones for the rotation isocenters of the collimator, gantry and couch. The definition of radiation isocenter incorporates a surrogate of the useful beam axis. Conclusions: A framework of isocenter definitions for medical linacs is presented in this manuscript. Consistent meanings of the mechanical and radiation isocenters can be achieved using this approach.

This work is motivated by our experience as clinical medical physicists in that we constantly encounter a particular term known as the “isocenter”. Being a unique concept attributed to a medical linear accelerator (linac), isocenter is used on countless occasions, and in combinations such as mechanical isocenter, radiation isocenter, rotation isocenter, imaging isocenter, treatment isocenter, and so on. These expressions seem to be self-evident ―medical physicists intuitively understand their meaning and use them with few obstacles. The aforementioned isocenters are expected to coincide [

Medical physicists are familiar with the following scenario. Prior to frame-based stereotactic radiosurgery, a ball bearing phantom representing the treatment target is mounted on the couch and aligned to the room lasers, which have been brought to a mechanical isocenter determined by means such as a frontpointer. The target alignment as a function of couch and gantry angles is verified with multiple collimated beams [

In a popular radiotherapy physics textbook, the definition of linac mechanical isocenter initially was, “Mechanical isocenter is the intersection point of the axis of rotation of the collimator and the axis of rotation of the gantry [^{1}”.

The American Association of Physicists in Medicine (AAPM) has published Reports of Task Group No. 40 (TG-40) [

tolerance. We must admit that in this approach, a few assumptions are implicitly made and quantification is rather difficult. The validity of the isocenter determined in step 2) is an open question. The QA results depend in large on visual observations and subjective judgment. The above QA approach is an extremely simplified version of the acceptance testing of medical linacs described in the literature. Although the linac installers have increasingly precise equipment, they still follow trial-and-error approaches. In general, they initially install and adjust mechanical components, and secondly turn on the radiation to calibrate beam collimation. The linac installation and acceptance test help us establish the grounds and logic of the isocenter definitions.

Numerous methods have been developed for the verification of linac isocenters [

The proposed isocenter definitions are based on the concept of axis of rotation. We shall make the isocenter definitions self-consistent, operable, and in addition, independent of measurement techniques. Our definition system observes three criteria: 1) It avoids circular definitions that have occurred in the literature but not clearly identified; 2) the definitions are compatible with the terminologies in the AAPM TG-40 and TG-142 reports; and 3) the definitions are mathematically operable and their physical measurements are feasible. Though our proposal does not include a specific approach of isocenter determination, we believe that once rigorous isocenter definitions are agreed upon, equivalent methods can be developed for isocenter measurement and verification. Accompanying the isocenter definitions, a mathematical model will be presented in the Methods section. This mathematical model can be used to deduce or calculate isocenter locations from the fundamental definition, or to transform the coordinates of a given point among different systems. The isocenter definitions will be presented in the Results, and the clinical applications of the isocenter concepts are addressed in the Discussion.

In this section, we present the fundamental concepts and mathematical methods that are essential to the deduction of isocenters. These are considered the minimally sufficient tools for the proposed definition system.

The linac subsystems―collimator, gantry, and couch―are considered rigid bodies. When a rigid body rotates, every point in the rigid body rotates relative to other points. To determine the axis of rotation (AOR) of a subsystem, its rotation should be observed in a coordinate system that is stationary relative to the linac vault. Separate coordinate systems can be established on the collimator, gantry, and couch that rotate with these subsystems. The AOR definition is modified from the concept proposed by previous authors [

As illustrated by

This subsection presents a mathematical tool, the transformation between two coordinate systems. The coordinate systems of the collimator, gantry, and couch rotate with these subsystems. Because of the imperfection of the mechanical systems, the rotation of these coordinate systems may not follow the nominal rotation angle and can only be measured as a function of the nominal rotation angle. For simplicity, the x, y, and z axes of these coordinate systems at the 0˚ nominal rotation angle are set to be parallel to that of the reference system. The coordinates of a point in the collimator, gantry, and couch systems are denoted by subscripts C, G and T, respectively. For instance, assuming that at a nominal gantry rotation angle

For a point with coordinates (

The elements in the transformation matrix

Equations (1) and (2) differ from the regular coordinate transformations in that the subsystems do not necessarily rotate about fixed origins relative to the stationary reference system. The linac gantry and couch have independent AOR; however, the collimator AOR is a function of the gantry angle. To simplify the mathematical treatment, we use the first-order approximation by assuming that the collimator AOR trajectory is not a function of gantry angle. The transformation matrix M_{C} and coordinate offsets of the collimator can be initially measured at an arbitrary gantry angle (for example 0˚) in the reference system, and then applied to any gantry angles in the same manner.

The isocenters of a linac are defined in this section. The rotation isocenters of the collimator, gantry, and couch will be purely mechanical concepts and they are established in this order, but the linac isocenter and the radiation isocenter definitions include the linac X-ray source position.

At some point during linac installation, the installer has to turn on the radiation beams. In this work, only the photon beams are of consequence. The methods of finding X-ray source position have been reported [

The AOR’s of the collimator, gantry, and couch should be established prior to the isocenter definition. An assumption is made that the mechanical motions are accurately reproducible in a short period of time. We propose choosing the source position at 0˚ collimator and gantry angles (source position zero), and measuring the trajectory of source position zero when the gantry makes a full rotation. The linac mechanical isocenter or “linac isocenter” is defined as the center of the circle that best fits the trajectory of the source position zero. By this definition, the linac isocenter is a measureable point in space, and it is stationary in the reference frame.

In this work, the collimator rotation isocenter might not be an indispensable term. To be consistent with the AAPM reports, we define the “collimator rotation isocenter” as follows. When the collimator rotates, if the center of the minimal bounding sphere of the trajectory of a point on the collimator AOR is the closest to the gantry AOR, this sphere center is the collimator rotation center. By this definition, the collimator isocenter is not a stationary point, and it is subject to the misalignment shown in

So far in our approach, the radiation field has not utilized collimation. In order to define the radiation isocenter, it is now appropriate to consider photon beam collimation. In Nebraska, the above linac isocenter definition does not comply with its regulations to a full extent. Even with the regulations in place, however, medical physicists pay little attention to such kind of definitions. The obvious reason is that collimated radiation fields are routinely used in radiation therapy. From this point of view, Nebraska’s isocenter definition is close to clinical reality and can be considered as the radiation isocenter. Therefore, we believe that it is necessary to propose a definition of radiation isocenter that incorporates a surrogate of the “useful beam axis”. When the gantry and collimator rotate, the instantaneous X-ray source determined by beam divergence might not remain at a fixed position relative to the gantry. It has been reported that the linac X-ray focal spots of different energies might not be coincident, and that they may shift over time [

The connections among the concepts and the definitions presented above are illustrated in

In radiotherapy, the treatment target is expected to be located at a predetermined position. In this section we will consider any point in the target volume, and find the formulae that can calculate its position with respect to a radiation field, which corresponds to arbitrary rotation angles of the three subsystems. The goal of this section is

to demonstrate an example of our mathematical model, i.e., to derive the coordinate transformation to the collimator system, because the clinical radiation fields are shaped by the jaws, MLC, or cones. This derivation can directly be used to calculate the position of a target point in a radiation field. Similar treatments should be useful in the isocenter determination and QA processes. For simplicity, the coordinate transformation from the gantry system to the reference system Equation (1) is written as

The transformation from the collimator and couch has the same form except that the symbol G is replaced by C or T, respectively. Now we examine the situation in that the collimator, gantry and couch have rotation angles. In reality, the target is placed on the couch. From Equation (3) the transformation from couch system to gantry system is

Rewriting Equation (3) derives the transformation from collimator system to gantry system

The superscript G reflects the fact that collimator system rotates with the gantry. From Equations (4) and (5), it is easy to derive the coordinate transformation from the couch system to the collimator system

In the above sections, we have established the definitions of mechanical isocenters and radiation isocenter. Efforts are made to ensure that these definitions are compatible with the prevailing understanding of the isocenters of a conventional medical linac [

Our work provides medical physicists and linac installers with a rigorous system without ambiguity and circuitous definitions. Among the five isocenters defined above, the radiation isocenter is most relevant to treatment delivery. Because the radiation isocenter is associated with gantry rotation, the position of any point in the target volume relative to the useful beam axis is predictable. This will eliminate some current concerns such as gantry sag which has been integrated into the isocenter definitions in our system. In order to implement these definitions in clinical practice, physicists and engineers have to develop methods that can track the trajectory of any point with sufficient accuracy. Therefore, after the preliminary installation, the linac installer will be able to make mechanical adjustments and line up the isocenters based on accurate measurements that replace the previous trial-and-error techniques.

According to our principle of operability, ancillary devices for beam shaping and target localization are aligned with the isocenters that have been previously determined. From this point of view, those ancillary devices are not appropriate for the measurement of a mechanical or radiation isocenter. In this manuscript, we have developed an isocenter definition system that is independent of measurement techniques. There should be multiple approaches that can utilize this system to determine the isocenters of a medical linac. The pursuit of a rigorous isocenter definition system will result in the expense of complicated measurements. Because the subsystem rotations are involved, we anticipate the measurements to be tedious and not suitable for periodic QA tests. Another limitation we have realized is that, in this system the isocenters are defined in the linac coordinate system. When utilizing our system, a physicist might need a surrogate to locate an isocenter in the reference system. It is unlikely that the resources necessary for the isocenter measurements are accessible by all the clinical physicists. The majority of physicists have to rely on commercial products or some simplified approaches. In reality, a subsample of full measurements might provide acceptable accuracy and be used in periodic QA. Currently we are working on an approach that utilizes the definitions and methods proposed in this manuscript to measure the isocenters in three-dimensional space. It is expected that in the future, medical physicists will be able to quantify the rotation isocenters and the coincidence of radiation and mechanical isocenters during annual QA [

Rigorous definitions help physicists understand clinical procedures where an isocenter plays a role. Now we exemplify using analysis of the traditional frame-based SRS on a medical linac, in which the treatment target must be precisely positioned at the radiation isocenter. Because an isocenter is invisible, lasers have to be used as surrogates. Conventionally, the lasers are aligned to the linac isocenter through a mechanical approach, thus they approximately represent the linac isocenter. Before the SRS procedure, the Winston-Lutz test is performed using a ball bearing phantom [

One may ask, is the Winston-Lutz test an acceptable method for radiation isocenter measurement? In our opinion, this is doubtful. Firstly, the Winston-Lutz test takes subsamples of the measurements that are necessary to determine the optimal treatment target position; Secondly and more importantly, the collimated radiation fields that the Winston-Lutz test employs do not necessarily provide the useful beam axis for any angle combination. If a Winston-Lutz type approach is used to determine an optimal treatment target location, it should not be considered the exact radiation isocenter. We can use the Elekta “flexmap” [

Currently, image guided frameless SRS procedures are replacing frame-based SRS. We would like to point out that relative to the subsystems of a linac, image guidance systems are secondary in nature, whether or not they are integrated into the mechanical structure of the linac. Although image guidance is increasingly used in radiotherapy, this fact does not compromise the importance of appropriate isocenter definitions. Nowadays some image guidance systems still rely on the lasers for calibration―their calibration phantoms are aligned to the lasers. Even if the lasers are no longer essential for patient setup when image guidance is in place, the discrepancy between the radiation isocenter and the imaging isocenter becomes crucial [

In this manuscript, a complete system of isocenter definitions is proposed for medical linacs. Minimal bounding sphere and coordinate transformation are the fundamental mathematical tools in the system. The following terms have been defined: linac isocenter, collimator rotation isocenter, gantry rotation isocenter, couch rotation isocenter, and radiation isocenter. These definitions are derived from the axes of rotation of the linac subsystems― collimator, gantry, and couch through operable procedures. This work provides the clinical physicists and linac engineers with a framework to perform measurements and quality assurance tests pertinent to the isocenters and radiation fields of a medical linear accelerator.

The authors are grateful to the medical physics colleagues and linac engineers who shared their insights on this topic. Mr. Chuck Rhode acquired the images shown in

MutianZhang,Su-MinZhou,TanxiaQu, (2015) What Do We Mean When We Talk about the Linac Isocenter?. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology,04,233-242. doi: 10.4236/ijmpcero.2015.43028