Innovative margin design and optimized isocenter to minimize the normal tissue in target volumes for single-isocenter multi-target stereotactic radiosurgery

Objective. Treating multiple brain metastases in a single plan is a popular radiosurgery technique. However, targets positioned off-isocenter are subject to rotational uncertainties. This work introduces two new planning target volumes (PTVs) that address this increased uncertainty. The volume of normal tissue included in these PTVs when paired with optimized isocenters are evaluated and compared with conventional methods. Approach. Sets of 1000 random multi-target radiosurgery patients were simulated, each patient with a random number of spherical targets (2–10). Each target had a random volume (0.1–15 cc) and was randomly positioned between 5 and 50 mm or 100 mm from isocenter. Two new PTVs (‘LensPTV’ and ‘SwipePTV’) and conventional isotropic PTVs were created using isocenters derived from the center-of-centroids, the center-of-mass, or optimized per PTV type. The total volume of normal tissue in the PTVs for each patient was calculated and compared using 1 mm translations and 0.5°, 1.0°, and 2.0° rotations. Main results. Using the new PTVs and/or using optimized isocenters decreased the total volume of normal tissue in the PTVs per patient. The SwipePTV, in particular, provided the greatest decrease. Compared to the SwipePTV, the LensPTV and the conventional isotropic PTV included an extra 0.68 and 0.73 cc of normal tissue per patient (median), respectively, when using 50 mm max distance to isocenter and 1° max rotation angle. Under these conditions, 25% of patients had extra volume of normal tissue ≥ 0.96 and 1.04 cc. When using 100 mm max distance to isocenter and 2° max rotation angle, 25% of patients had extra volume of normal tissue ≥ 4.35 and 5.75 cc. Significance. PTVs like those presented here, especially when paired with optimized isocenters, can decrease the total volume of included normal tissue and reduce the risk of toxicity without compromising target coverage.


Introduction
The number of patients with brain metastases treated with radiation is increasing, and stereotactic radiosurgery (SRS) has gained favor as a treatment technique due to its efficacy and improved neurocognitive outcomes (Aoyama et al 2006, Yamamoto et al 2014a, 2014b, Brown et al 2016, Palmer et al 2020, Suh et al 2020, Gondi et al 2022).Historically, brain metastases were treated one at a time with either a conical collimator or a multileaf collimator (MLC) on treatment units such as the Gamma Knife (Elekta AB, Stockholm, Sweden) or a linear accelerator (linac).More recently, treating multiple metastases at once using a linac's MLC to block the radiation between targets has become widespread (Clark et al 2010(Clark et al , 2012)).This technique-frequently referred to as single-isocenter multi-target (SIMT) SRS-is much more efficient than treating multiple targets individually.Clinicians and patients, alike, benefit from the decrease in time required to deliver treatment, as extra time on the treatment couch has been shown to correlate with an increase in patient discomfort and intrafractional motion (Feygelman et al 2008, Cervĩo et al 2010).
One challenge does arise, however, amidst this convenience when treating multiple targets simultaneously.As multiple targets are included in each treatment field, each target is most likely treated with an isocenter that is not located at its own geometric center.Considering that both the alignment of a patient as well as the coincidence of all imaging and treatment subsystems are described relative to the isocenter, targets that are offisocenter will be more affected by rotational uncertainties.The distance between the target and the isocenter acts like a lever arm where the physical displacement of a target that is subject to a given rotation increases with this distance.Other concerns are also introduced when treating targets off-isocenter, such as those regarding increased MLC leaf width or compromised beam modeling at steep angles of divergence.
In order to mitigate the emergent concerns of off-isocenter targets, clinicians have used a number of strategies.First, minimizing the distance between each target and the isocenter can reduce the positional uncertainty from rotations.Implementing a maximum allowed distance from isocenter can limit these rotational effects.It can also address the beam shaping and modeling concerns (Nakano et al 2020, Meeks et al 2022).In addition, larger margins can be used when generating the planning target volume (PTV), and procedures to determining the necessary margin magnitude have been described previously (Stanhope et al 2016, Chang 2017, 2018, Nakano et al 2020, Agazaryan et al 2021, Rojas-López et al 2021).
Often, however, these larger margins are added to the target isotropically.While this implementation makes practical sense-as it can be readily used with existing treatment planning systems (TPSs)-it is not the ideal solution.Increasing the PTV margin isotropically expands the volume in all directions including radially toward and away from the isocenter.However, there exists no rigid rotation about isocenter that will increase this radial displacement of the target, so incorporating these regions into the PTV will increase the volume of normal tissue included unnecessarily.Understanding the consequence of this approach is imperative to the objective of minimizing normal tissue toxicity without compromising target coverage.
The purpose of this work is to present two novel PTVs and their optimized isocenters together designed to minimize the total volume of included normal tissue while maintaining complete target coverage.After being introduced, the new PTVs, along with one using a conventional isotropic margin, are each combined with three separate methods for selecting isocenters.The joint effects of PTV type and isocenter selection method on the volume of normal tissue included in the PTVs are analyzed and presented for a simulated population of SIMT SRS patients under several different treatment conditions.

Simulated patient geometry
In-house software (MATLAB, Mathworks, Natick, MA) was used to generate two sets of 1000 random multitarget configurations, each configuration representing a simulated radiosurgery patient.For each patient, a random number of spherical targets was selected between 2 and 10 (uniformly distributed).Each target was randomly assigned a volume between 0.1 and 15 cc (uniformly distributed).These volumes corresponded to a GTV of roughly 1.9-14.3mm in radius with an additional 1 mm margin added to account for typical uncertainties such as target position translation.With an isocenter presumed to be located at the origin, the position of each target was randomly selected from 3D Cartesian space (uniformly distributed) consistent with the assumption that a brain met was equally likely to occur in any small volume of brain tissue.The range of possible distances between each target and the isocenter was limited to the range of 5 mm to D mm, where D = 50 mm or 100 mm depending on the patient set, by replacing any target position having a distance outside of this range with a new random position.Similarly, the inter-target distances within each patient were assessed to ensure at least a 5 mm buffer spacing between target surfaces to avoid overlapping PTVs.

PTV volumes
For each patient, three types of PTVs were considered.The derivations of the PTVs are described below with those of two new PTVs elaborated further in the Appendices.Each is defined in such a way so as to completely cover the motion of a spherical GTV under the allowed translations and rotations, thereby guaranteeing complete target coverage within the scope of this analysis.The three PTVs differ in the volume of normal tissue that they include beyond the GTV.The two new PTVs introduced in this work are specifically designed to address additional uncertainties due to rotations.They are built upon an intermediate PTV volume (PTV int ) generated from the GTV using an isotropic expansion.The PTV int is considered to account for all sources of uncertainty other than the additional rotational uncertainty introduced for off-isocenter targets.The PTV int therefore is considered to account for target translations and other localization or delineation uncertainties as well as finite linac precision.In this work, a GTV-to-PTV int margin of 1 mm was used since this is equivalent to a typical clinical margin when an SRS target is positioned at the isocenter.

Isotropic PTV
The isotropic PTV (IsoPTV) is defined here as the minimum isotropic expansion from the PTV int that is required to fully cover its motion when subject to rotations of a given magnitude.The margin added to the PTV int corresponds to the geometric displacement of its center (d) as a function of the distance to isocenter (D) and an angle of rotation (j) as calculated in equation (1) and depicted in figure 1.The magnitude of this margin increases continuously as a function of distance from the isocenter.The IsoPTV is depicted and compared with the other PTVs in figure 2

LensPTV
The first new type of PTV introduced in this work is the LensPTV (figure 2).It is designed to remove regions of the IsoPTV adjacent to the PTV int that cannot be occupied by the PTV int under any rigid rotation about isocenter.A full derivation of the LensPTV and calculation of the included normal tissue is described in appendix A. In short, the LensPTV is constructed as a Boolean operation between the IsoPTV and two spheres that are centered at the isocenter and are tangent to the PTV int on the near and far side.The intersection of each sphere with the IsoPTV results in an asymmetric lens shape with the smaller lens fully contained within the larger lens.The volume of any lens resulting from the intersection of two non-concentric, partially overlapping spheres can be calculated by determining the position of the plane that deconstructs the geometry into spherical caps and calculating the volume of the caps.The LensPTV is constructed as the Boolean difference between the two lens volumes, effectively removing the radially unnecessary regions from the IsoPTV.

SwipePTV
The second new type of PTV introduced here is the SwipePTV (figure 2).It is defined as the region covered by rotating the PTV int by the allowed magnitude around any axis with the isocenter as the center of rotation.For this work, the SwipePTV was considered the ground truth as it represented all possible positions of the GTV according to the allowed translations and rotations without including normal tissue beyond this domain.A full derivation of the SwipePTV and calculation of the included normal tissue is described in appendix B. In short, the SwipePTV's volume can be determined as the sum of two separate regions.The first region is that represented by solving the triple integral that corresponds to the portion of a spherical shell with thickness equal to the diameter of the PTV int that is contained within a solid angle defined by a half-angle equal to the max allowed rotation.The volume of the remaining portion of the SwipePTV is determined through another integral, this one using the equation for the surface area of a conical frustum with the radius and slant height varying between the integration limits.Figure 2. A comparison of the different types of PTVs for a GTV (red) located a distance D from the isocenter and subject to a rotation j in any possible direction.Immediately surrounding the GTV is the smaller isotropic expansion representing the PTV int .The IsoPTV includes normal tissue regions colored in blue, yellow, and white.Meanwhile, the LensPTV includes normal tissue colored in blue and yellow, and the SwipePTV includes normal tissue colored in blue only.

Isocenter position selection methods
The volume of normal tissue included in each type of PTV was calculated when those PTVs were created relative to isocenters positioned using the following three isocenter selection methods.

Center-of-centroid (CoC)
The first isocenter selection method was to simply place the isocenter at the center of the GTVs' geometric centroids.In this work, this is referred to as the center-of-centroid (CoC) method and it is equivalent to that used in Brainlab's Elements Multiple Brain Mets treatment planning module (Brainlab, Munich, Germany).The GTVs are the target volumes used for this step because the PTVs are determined subsequently and will depend on the position of the isocenter.

Center-of-mass (CoM)
The second method is similar to the first but places the isocenter at the center of the combined volume of all GTVs, rather than at the center of their centroids.As the density of the targets were considered uniform, this corresponded to the center of mass of the union of all GTVs and is referred to here as the center-of-mass (CoM) method.This method is equivalent to that used in Varian's HyperArc SRS treatment planning module (Varian, Palo Alto, USA).

Optimization per PTV type
The final method was to optimize the isocenter per the PTV type.The objective of the optimization was to minimize the total volume of normal tissue included within the PTVs of all targets of a given patient.As the value of the objective function was different for each PTV type, the corresponding optimal isocenter could vary, unlike for the CoC and CoM methods where the isocenter position was consistent across PTV type.This was achieved individually for each type of PTV by using MATLAB's GlobalSearch function to repeatedly perform the fmincon optimization with multiple starting points (Mathworks, Natick, MA).

Simulation and analysis
Using in-house software (MATLAB, Mathworks, Natick, MA), two independent populations of 1000 patients were created with a max target distance to isocenter of either 50 mm or 100 mm.For each patient in the two populations, the volume of normal tissue included in each type of PTV when combined with an isocenter from each selection method and subject to a max rotation angle of 0.5°, 1.0°, or 2.0°was calculated.Rotation angles of approximately 1.0°are consistent with recommendations from the American Association of Physicists in Medicine (Klein et al 2009).Clinically, our institution typically uses 1.0°and 0.5°angles as action limits for SRS treatments featuring a single target or multiple targets per isocenter, respectively.The ideal combination of PTV type and isocenter selection method was considered to be the SwipePTV paired with its optimized isocenter.For a given patient, the total volume of normal tissue included in the PTVs of all targets that resulted from this combination was considered the minimum achievable.It was therefore subtracted from those of the other combinations to calculate their excess volume of included normal tissue.

Results
For different combinations of max distance to isocenter and max rotation angle, figure 3 depicts the excess volume of included normal tissue resulting from combinations of the three PTV types and the three isocenter selection methods.As expected, the IsoPTV included the greatest amount of normal tissue, followed by the LensPTV.Also as expected, the amount of normal tissue included in the PTVs increased with max distance to isocenter and max rotation angle.
When compared to the SwipePTV paired with its optimized isocenter using a 50 mm max distance to isocenter and 1°max rotation angle, the IsoPTV and LensPTV (each with their optimized isocenter) included an extra 0.73 and 0.68 cc more normal tissue (median).Under these conditions, 25% of patients had extra volume of normal tissue that was at least 1.04 and 0.96 cc.Increasing the max distance to isocenter to 100 mm increased the median extra volume of normal tissue in the IsoPTV and LensPTV to 1.85 cc and 1.58 cc, respectively, with 25% of patients exceeding 2.51 and 2.17 cc.If, in addition, a 2°max rotation angle were allowed, the median extra volume of normal tissue was 4.22 cc and 3.15 cc, respectively, with 25% of patients exceeding 5.75 and 4.35 cc.
The PTV type can be observed to have a greater effect than the isocenter selection method.For example, while the IsoPTV and Lens PTV added 0.73 and 0.68 cc more normal tissue compared to the SwipePTV (each with their optimized isocenter and at 50 mm/1°), using the CoC and CoM isocenter selection methods with a SwipePTV only added 0.07 and 0.04 cc of normal tissue (median).These patterns held for different max distances to isocenter and max rotation angles.Even when using a 100 mm max distance to isocenter and 2°max rotation angle, using the CoC and CoM isocenter selection methods with a SwipePTV only added 0.26 and 0.12 cc of normal tissue (median).

Discussion
This work compared the amount of normal tissue included in various types of PTVs in conjunction with various methods for selecting the isocenter in SRS plans treating multiple targets simultaneously.While all PTV types were, by definition, constructed to provide complete coverage of the GTV under translations and rotations of a given magnitude, the SwipePTV did so with considerably less normal tissue than the IsoPTV or LensPTV.Optimizing the position of the isocenter per the PTV type decreased the volume of included normal tissue compared to isocenters selected using the CoC or CoM methods.However, this effect was not as dramatic as the reduction achieved using the SwipePTV.
Numerous works have described how the PTV margin must increase to account for the increased rotational uncertainty of off-isocenter targets.However, many of these still use an isotropic margin (Chang 2017, 2018, Nakano et  Although incorporating a single isotropic margin into existing treatment planning systems is straight forward, we have demonstrated here that a non-isotropic PTV specifically designed to address the rotational effects of off-isocenter targets can decrease the volume of normal tissue included. Miao et al present a more complex use of isotropic margins by applying margins of varying magnitude to small target subregions (Miao et al 2022).This method accounts for the varying effects of rotations across the target that increase with the increasing distance of target subregions from the isocenter.However, regardless of their magnitude, isotropic margins will not produce the optimal PTV shape to address rotational uncertainties that are not, at any scale, isotropic.The PTV resulting from isotropic margins applied to target subregions will still unnecessarily incorporate normal tissue adjacent to the target similar to that included in the IsoPTV but excluded from the LensPTV and SwipePTV.
Some works have taken approaches that model the probability of possible GTV positions when subject to translations and rotations (Chang 2017, 2018, Slagowski and Wen 2020, Cui et al 2021, Miao et al 2022).Although this approach is more computationally sophisticated than the work presented here, it requires additional information or assumptions to fully characterize the motion.Describing this motion precisely remains a critical challenge, particularly when considering the appropriateness of the model for an individual patient.The PTVs described here are designed to completely cover a given range of translations and rotations.Reducing them further compromises target coverage and becomes increasingly dependent on the accuracy of the presumed motion model.It is worth noting, however, that the methods described here can be also used in conjunction with probabilistic models of GTV motion.A motion model that describes the probability of possible angles of rotation can be used to determine the maximum angle that corresponds to the desired level of target coverage.The appropriate IsoPTV, LensPTV, or SwipePTV can then be generated using this as the max rotation angle parameter.
Another point of comparison between this work and prior investigations is the focus on minimizing the normal tissue included in the PTV rather than maximizing the target coverage.When dealing with targets of varying sizes, striving to maximize target coverage requires an additional assumption as to whether an absolute unit or relative unit of target volume is to be considered equivalent across multiple targets.While other works evaluate targets and even optimize the position of isocenters based on maximizing target coverage (Nakano et al 2020, Cui et al 2021, Nakano et al 2021), this work constructed PTVs that provided complete target coverage, and then within that constraint, sought to minimize the normal tissue included.This is similar to Slagowski and Wen (2020) and Miao et al (2022) but features the use of non-isotropic PTVs.
A related limitation of our method is that all normal tissue is considered equivalent, and the proximity of organs-at-risk are not considered explicitly.We feel the objective of minimizing normal tissue remains closely aligned with that of minimizing dose to nearby organs-at-risk.In scenarios where the two goals may disassociate, plan parameters and the appropriateness of treating all targets simultaneously, exacerbating the sensitivity to rotational uncertainties, should be reconsidered.Some other works describe or optimize these PTVs from a dosimetric perspective (Nakano et al 2020, Cui et al 2021, Nakano et al 2021, Rojas-López et al 2021).Such metrics would be preferred because treatment plans will ultimately be evaluated on dosimetric characteristics rather than geometric ones.However, as true dose values would require the computationally intensive steps of treatment plan optimization and dose calculation, these works often use presumed dose values derived from geometric considerations.The results of such strategies are therefore less likely to depart significantly from those of purely geometric analyses.
Although our geometric analysis does not provide dosimetric information, it does provide several meaningful advantages.The spherical approximation of the GTV used in this work is a clinically reasonable representation of the typical geometry of brain metastases.In addition, assuming a spherical target allowed us to analytically calculate the exact volume of each PTV, which also facilitated the rapid determination of the optimized isocenter position.
It is worth noting that while the specific calculations used to determine the volume of normal tissue included in the LensPTV and SwipePTV for spherical targets may not be exact for non-spherical targets, the concepts and methodology behind these PTVs do still extend to non-spherical targets.These PTVs could still be generated as long as systems provide the functionality to conduct the necessary geometric operations.A lingering challenge of our method, therefore, is the incorporation of these techniques into existing treatment planning systems.Generation of the SwipePTV is beyond the functionality currently available in most commercial TPSs except when extended with scripting capabilities.Recognizing this, we designed the LensPTV specifically to be able to be constructed using conventional contouring tools and operations.However, our work shows that while the LensPTV did include less normal tissue compared to the IsoPTV, it still included considerably more than the SwipePTV.Incorporating the LensPTV and SwipePTV into a clinical TPS is also limited by the finite precision and resolution of various TPS features and data structures.The ability to accurately delineate targets is limited by the inherent resolution of the planning image and targets are often ultimately represented digitally using relatively simple polygons.In addition, the finite width of MLC leaves limits the MLC's ability to conform dose around a geometric target, which is of particular note because the LensPTV and SwipePTV differ from the IsoPTV by featuring sharper curves and a concavity facing the isocenter.
While geometric improvements may not perfectly correlate with dosimetric improvements, at the point during treatment planning when the PTV geometry is established, many dosimetric parameters are not yet determined.It remains strategic to initiate dose optimization with the most geometrically advantageous (e.g. the smallest) PTV.The geometric analysis could also be used to determine the PTV for a spherical target that is a close approximation of a non-spherical target, leveraging the computational advantage of the analytical calculations.This PTV could then be evaluated per patient anatomy and plan dosimetry to confirm its appropriateness or else be revised.A final advantage of our geometric treatment is its generalizability.The fact that our targets were spherical and that the volumes of the PTVs were calculated analytically (as functions of GTV size, translational margin, distance to isocenter, and maximum allowed rotation angle) means our analysis can be applied beyond the analysis presented here.Future work is underway to incorporate these techniques into a conventional TPS in order to assess their impact on the treatment planning process and on treatment plan dosimetry.

Conclusion
In this work, we introduced two new types of PTVs designed to more explicitly address the effects of rotational uncertainties when treating multiple targets simultaneously with radiosurgery.By being more directly tied to the possible displacement of the GTV undergoing translations and rotations, the volume of normal tissue included in these PTVs was less than in those created using conventionally isotropic margins.Optimizing the position of the isocenter had an additional benefit but it was less than that of the PTV type.By minimizing the volume of normal tissue included in the PTV through strategic target design and isocenter placement, it may be possible to decrease the dose delivered to healthy tissue and decrease the risk of normal tissue toxicities.
• The overlap of each sphere with the IsoPTV results in an asymmetric lens-shaped volume (A and B).
• The volume of a lens-shaped intersection of any two spheres of radii r 1 and r 2 centered a distance t apart can be calculated by locating the position (x) of the plane that deconstructs the geometry into spherical caps (e.g.L 1 and L 2 ).The volume of the lens can then be calculated from the volumes of the generated caps.
• Any limiting case where the distance between the centers of the spheres is greater than the sum of the radii or less than their difference trivializes to a lens with volume 0 or equal to that of the smaller sphere, respectively.
• The position (x) of the intersection plane relative to the center of the sphere of radius r 1 can be calculated with: • For a sphere with radius r, the volume of a cap with height h can be calculated with: • The LensPTV is defined as the Boolean difference between lens-shaped volumes generated from the two spheres centered at the isocenter, and its volume can be calculated from those of the two lenses.
• The volume of normal tissue (yellow) included in the LensPTV is calculated by subtracting the volume of the GTV (red).
LensPTV Large Lens Small Lens ò ò ò • The volume of the outer region can be calculated by integrating a nested set of thin hollow conical frustums.
• The lateral surface area of a conical frustum (SA lateral ) can be calculated from the length of the slant height (l) and the average radius (r ¯)  • The volume of the SwipePTV can be calculated as the sum of the volume of the inner and outer regions.
• The volume of normal tissue included in the SwipePTV (blue) is calculated by subtracting the volume of the GTV (red).

Figure 1 .
Figure 1.Depiction of target displacement (d) due to rotation of angle j.

Figure 3 .
Figure3.Volume of excess normal tissue included in various PTVs combined with various isocenter selection methods.Values represent the amount of normal tissue included in the PTV beyond the minimum amount included in the SwipePTV with its optimized isocenter.Each PTV was designed to provide adequate target coverage for 1 mm translations and 0.5°(top), 1.0°(middle), and 2.0°(bottom) rotations for targets with a max distance to isocenter of 50 mm (left) and 100 mm (right).The median value is marked in the center of each box.The box limits represent the 25th and 75th percentile, and the whiskers represent the full range of data.
This equation can be used to calculate the volume of the outer region (V Outer ) by considering a conical frustum with a lateral surface of length l parallel to the boundary between the inner and outer regions and then integrating along its radius in the direction of y as illustrated below.