Efficient optimization of R50% when planning multiple cranial metastases simultaneously in single isocenter SRS/SRT

Abstract Simultaneous optimization of multiple Planning Target Volumes (PTVs) of varying size and location in the cranium is a non‐trivial task. The rate of dose falloff around PTV structures is variable and depends on PTV characteristics such as the volume. The metric R50% is one parameter that can be used to quantify dose falloff achieved in a given treatment plan. An important treatment planning question is how to construct optimization conditions that result in the efficient production of acceptable plan outcomes considering metrics such as R50%. Guidance provided in literature suggests generating multiple shell control structures around each PTV. The constraints applied to these shells can vary significantly depending on PTV volume. Additionally, there is no clear guidance on how to prospectively determine objective constraints for the optimization shells to achieve a specified goal of R50%. Based on physical principles and empirical evidence, we provide clear quantitative guidance on how to translate the desired R50% outcome into appropriately sized optimization structures around PTVs via an equation that depends on a desired goal for R50% and the volume of PTV. Optimization schema are also provided that allow the goal R50% to be approached or achieved for all PTVs individually. We demonstrate the application of the methodology using commercially available treatment planning software and radiotherapy treatment equipment.

treatment planning process. The single isocenter approach for targeting multiple sites simultaneously has increased the required treatment planning effort needed to produce acceptable dose distributions, especially when considering normal tissue objectives (NTO). Ballangrud et al. comment on the increased treatment planning effort required in the statement; "To further improve VMAT planning for multiple cranial metastases, better tools to shorten planning time[s] are needed." 5 The cranium contains many normal structures requiring consideration in planning optimization such as the brainstem, optic chiasm, and optic nerves. Minimizing normal brain tissue dose is also an important optimization objective as it is always directly adjacent to Planning Target Volume (PTV) surfaces and subject to the high doses being delivered to these PTVs. Indeed, radiation necrosis of normal brain tissue is one of the more relevant adverse effects after SRS/ SRT. 6 Various publications have evaluated the intermediate dose spill from PTV surfaces using metrics such as V12 Gy (the volume of brain receiving 12 Gy) and have identified potential complications associated with excessive volumes for these quantities. [7][8][9] These studies stress the need to minimize dose spill metrics to reduce normal brain tissue doses and the incidence of associated complications.
A widely accepted approach used to control intermediate dose spill uses multiple, contiguous shells surrounding the PTVs. 2,10-12 Typically, three shells are defined that step-down dose in a controlled manner. The size and dose constraints on these shells can be dependent on the PTV size. More importantly, initial conditions applied to these shell structures do not necessarily translate into acceptable intermediate dose spill values. When targeting multiple PTVs using a single isocenter, the multiple fixed shell approach can be susceptible to unacceptable dose falloff requiring adjustment of shell parameters and repeat optimizations adding to the treatment planning effort. In addition, no clear guidance on shell parameter modification is available.
Various metrics have been devised to quantify and potentially control this intermediate dose spill. The metric R50% is the parameter we utilize in this work to develop schema to control intermediate dose spill from the PTV surface. R50% is defined as the 50% isodose cloud volume (V IDC50% ) normalized by the PTV volume (V PTV ). 13 Thus: An alternative intermediate dose spill metric commonly used metric in SRS planning when evaluating competing plans is Gradient Index (GI). Paddick defines GI as the ratio of V IDC50% to the 100% isodose cloud volume (V IDC100% ). 14 Clearly, if the plan is perfectly conformal in the high dose region, V IDC100% is equivalent to V PTV , and GI is equivalent to R50%. However, if V IDC100% is not perfectly conformal to the PTV, plan flaws can be masked. For example, a V IDC100% larger than V PTV is possible, and, in such a case, GI would not adequately account for the normal tissue that falls within V IDC100% but outside the PTV surface. A plan with an acceptable GI could consequently be an inferior plan in terms of the normal tissue outside of the PTV being radiated to a high dose. In a study of LINAC based RapidArc SRS plans, such a phenomenon was identified where the RapdiArc plans appeared to have noticeably larger GI values compared to Gamma Knife plans. 15  Several authors have suggested such planning goals and even some strategies to achieve those goals. 2,12,18 Given an optimal treatment geometry and a well-specified optimal planning goal, what has been missing is a concise optimization approach for achieving that final goal. This work addresses that third issuethe concise optimization approach to achieve the final goal, at least as it applies to R50%.
Note: A table of abbreviations is provided in Appendix A.
2 | METHODS 2.A | "Ask For It" Approach for R50% The "Ask For It" (AFI) inverse planning approach is a two-step process. The first step is the construction of an optimization shell specifically dependent on the volume of the PTV and the R50% goal one wishes to achieve. The second step is the prospective determination of the optimizer volumetric constraint dependent on the R50% goal, the PTV volume, and the optimization shell volume. As such, one is able to explicitly ask the optimizer for the desired R50% goal final resultwe ask for R50% Goal . Below is the summary of our empirically determined AFI approach. The detailed derivation and articulation of the approach are given in the Appendix B.
Given a R50% goal (R50% Goal ), we construct a unique optimization shell and inverse planning optimization criterion customized for each PTV. The unique optimization shell scales to the characteristics of the individual PTV and the specified R50% Goal . The exact nature of the R50% Goal is independent of the AFI approach, so as our understanding of the appropriate R50% Goal changes and improves, the AFI approach does not change.
As an idealized case, consider a spherical PTV with volume V PTV surrounded by an isodose cloud of 50% of the prescription (Rx) dose (IDC50%) as illustrated in Fig. 1 where V PTV is in units of cm 3 . M is given in cm and depends on two important factors, V PTV and R50% Goal . Expanding the PTV by the margin M creates the outer surface of a structure called OptiForR50.
OptiForR50shell is the difference between OptiForR50 and the PTV and has a volume V OptiForR50shell . Next, we have to determine the fractional percentage of OptiForR50shell that should receive 50% of the prescription dose (%V Opti ) to achieve the R50% Goal . This value is given by: The above procedure is applied separately for each individual PTV. As such, the OptiForR50shell and the %V Opti is unique to each PTV and is dependent on the R50% Goal .
Finally, a single global structure that is outside all the PTVs and OptiForR50shell structures is constructed, which we call iShell. This is an insurance shell with the only purpose to insure that the 50% isodose cloud remains within each individual OptiForR50shell. The structure Boolean algebra expression for iShell is: where the summation is over the n number of PTVs within the cranium.
It is important to note that the AFI approach described above would only need to be performed one time with this one set of optimization parameters; multiple iterations are typically not required.

2.B | R50% Goal determination
The AFI approach summarized in section 2.A and derived in detail in Appendix B depends on having a goal for R50%, the R50% Goal . This R50% Goal is currently not a settled question, but some guidance has been published. 2,5 In this work, we take our R50% Goal from repro- ) for plans that have been determined to be optimal.
In addition, the mean Conformity Index (CI) is reported for the plans assessed (CI = 1.2 AE 0.1). Considering the product of GI and CI: 14 Therefore, these data of Ballangrud et al. can be used to estimate a planning goal for R50% Goal .
The R50% Goal values obtained from Eq. (6) are used in Eqs. (2) and (3) to determine the critical parameters for implementing the AFI approach.

2.C | Phantom studies
The 3D anthropomorphic patient model used in the study was obtained from a treatment planning CT of the IROC Head Phantom ® (IROC Houston QA Center, Houston, TX). We ignored the IROC PTV and created a unique set of 5 PTVs (PTV1 -PTV5) distributed throughout the cranial cavity. Three planning scenarios were constructed using different PTV shapes, sizes, and locations. The first, the shape and size of the PTV changed. The term "jack" refers to a 3D solid composed of three orthogonal ellipsoids sharing a common center. This shape resembles the six-pointed "Jacks" game piece.
Thus, the three ellipsoids of the jack are oriented with their longest axes in different directions: superior-inferior, anterior-posterior, and right-left. These jacks represent a combination of extremely concave and convex shapes (at the tips of the ellipsoids). Although the jack shape is not likely for a clinical PTV, it represents an extremely nonspherical case that tests the PTV shape limits of our AFI approach.
The PTV characteristics are summarized in Table 1.

PTV M
F I G . 1. Simplified anatomy of the optimization structure surrounding the PTV. OptiForR50shell is generated by expanding the PTV by a value of M and includes everything within its boundary (gray shaded area) except for the PTV. IDC50%shell is the patterned area within the gray shaded area (IDC50%shell = IDC50% -PTV).
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The Rx dose utilized for all optimizations and final plans in this study is 9 Gy x 3 fractions for a total dose of 27 The PTV least covered by the prescription dose was used to renormalize the entire plan and to ensure that every PTV has at least 95% of its volume covered by at least the full prescription dose (D95% Rx).

2.D | Plan assessment using OptiForR50shell
Another challenge of a multiple target SRS/SRT case is how to efficiently evaluate the plan quality for each PTV independently to ensure that the dose coverage and drop-off for each PTV is optimal.
This can be achieved with the creative use of the same structures used in the optimization. There will be a unique set of such shells and optimization parameters for each PTV in a multiple target case. OptiForR50shell is a shell expansion of the PTV with the expansion margin given by M in Eq. (2), and % V Opti is given by Eq. (3), and both parameters are dependent on the R50% Goal specific to that PTV. The iShell is a single global structure defined by Eq. (4), which is the brain minus all the PTVs and Opti-ForR50shell. The penalties listed are a defined part of the AFI approach but may need minor modifications for planning systems other than Eclipse.
and %V100% OptiForR50shelln ¼ % of OptiForR50shell n that received at least the Rx dose for target n The first term in the numerator defines the amount of volume of the 100% Rx dose cloud that falls inside the PTV; the second term defines the amount of volume of the 100% Rx dose cloud spills outside the PTV. All four quantities in Eq. (9) can be conveniently read from the DVH and the structure statistics provided by the TPS or extracted by a script.
Similarly, R50% n can be acquired for each PTV individually. Define: %V50% OptiForR50shelln ¼ % of OptiForR50shell n that received at least 50% Rx dose for target n (10) Then: The numerator is the total volume of the IDC50% cloud. This analysis was used to assess quality of the final optimized plans for each target individually.

2.E | Plan QA
It is important that the plans created with the AFI approach be clinically deliverable and meet reasonable quality assurance standards.
Thus, each of the three plans was delivered and quality was assured using an ArcCHECK ® (Sun Nuclear, Melbourne, FL), with all couch angles set to 0°. Evaluation criteria of 3%, 2 mm DTA, and a 10% dose threshold were utilized to determine passing rates.

| RESULTS
The final results for plans optimized by the AFI approach are shown visually in Fig. 3. This is the product of the first and only  The qualitative representation of the AFI approach optimized plans shown in Fig. 3 are summarized quantitatively in The results for R50% can also be plotted graphically as in Fig. 4

| DISCUSSION
In this work, we address two issues in the simultaneous treatment of multiple cranial lesions: 1. how best to optimize multiple targets independently at the same time, and 2. how to achieve a stated goal for R50% without an iterative approach. We offer an approach that translates the R50% Goal into direct optimizer inputs. Although we do not address the question of the validity of the R50% Goal , the values used are obtained from a published treatment planning study. We only claim that, given a reasonable R50% Goal , a method to produce optimization parameters that approximately achieve that goal is possible.
Our AFI approach employs a specially constructed optimization shell (OptiForR50shell) around each individual PTV, whose dimensions scale with the size of the individual PTV. One can see in Table 4 that each PTV has a unique expansion margin, M, to construct the OptiForR50shell. Since our benchmark R50% Goal is dependent on the V PTV , the expansion margin and thus the  Fig. 3(a).
The R50% values achieved in an optimization for the multiple target test plans, R50% Achieved , are listed along with the R50% Goal for each PTV individually in Table 4. Those R50% values are plotted in Fig. 4  The third column labeled "M" is the PTV expansion margin [Eq. (2)] that created the OptiForR50shell used in the optimization. The fourth column labeled "%V Opti " is an optimization parameter given by [Eq. (3)]. Plan 1 (five spherical shaped PTVs) is depicted in Fig. 3(a). Plan 2 (five irregularly shaped PTVs) is depicted in Fig. 3(b). Plan 3 (five jack-shaped PTVs) is depicted in Fig. 3(c). Notice that the R50% Achieved values are better than the R50% Goal values for all spherical and irregular PTVs. Because each OptiForR50shell surrounds its respective PTV and touches the PTV outer surface, the OptiForR50shell also provides a convenient analysis tool for the optimization of the PTV within the shell. One can assess the high dose spill from an individual PTV by taking the ratio of the volume of the OptiForR50shell that has 105% of the Rx dose to V PTV . This is the V105% metric common in Lung SBRT. One can directly measure the volume of the IDC50%shell by simply finding the volume of OptiForR50shell that is within the 50% isodose cloud, and thus, the final R50% for each individual PTV can be computed from Eq. (11). Noting that R50% = CI × GI, this one metric encapsulates both CI and GI common in SRS/SRT into one metric. To extract the CI directly for each PTV, one needs to use Eq. (9), which involves only the commonly reported dose data for a given PTV and its corresponding OptiForR50shell.
This study was conducted in a phantom with well-spaced targets, and no consideration was given to other critical structures such as the brainstem. A potential advantage of the AFI approach is the ability to accommodate the asymmetric intermediate dose shell. Another potential problematic situation may occur when two PTVs are near each other. In this situation, the OptiForR50shell structures may overlap, and the resulting performance of the AFI approach is uncertain. One possible modification for nearly coincident PTVs is to combine these into a single PTV and proceed as described previously. Further study is required to evaluate the performance of the AFI approach in these more demanding clinical situations.
The AFI method was developed within a particular set of condi-   on the volume and surface area of the PTV. Thus, using R50% Goal = R50% Analytic in the AFI strategy would allow one to incorporate the PTV surface area into the optimization criterion.
While this is not the only successful approach for achieving R50% goals, this is one quantitative method that allows the planner to prospectively customize the optimization structures and parameters based on the R50% goal for a multi-target, single isocenter plan.
The potential for reducing the treatment planning time in these cases may be important in a busy radiation therapy clinic. Additionally, the AFI strategy could be built into a knowledge-based optimization system, which could be particularly powerful if one used R50% Analytic as R50% Goal . 25 A future comparison of the AFI strategy with current NTO-driven optimizations (such as HyperArc) or knowledge-based techniques (such as RapidPlan) could also prove fruitful.

| CONCLUSION
One can prospectively determine the size of an optimization struc-

CONF LICT OF I NTEREST
No conflicts of interest.

FUN DING
There are no funders to report for this submission.

D A T A A V A I L A B I L I T Y S T A T E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

R E F E R E N C E S
From the basic definition of VIDC50%shell, we know: The substitution of Eq. (B2) for V IDC50%shell into Eq. (B1) gives: By factoring V PTV out of the numerator of Eq. (B3), the resulting equation is: The following equation is the definition of R50%: The substitution of Eq. (B5) into Eq. (B4) gives: Quantitatively, the only unknown value in Eq. (B6) is the volume of OptiForR50shell (V OptiForR50shell ). We know the optimization structure OptiForR50shell must be larger than IDC50%shell but how much larger is uncertain. Therefore, V OptiForR50shell is expressed in the following symbolic terms where N is a real number larger than 1: terms of V IDC50%shell is given: Further algebraic manipulation of Eq. (B8), yields: The substitution of Eq. (B9) into Eq. (B7) gives an expression for VOptiForR50shell in terms of R50%.

B.3 | FORM 2 OF V O P T I F O R R 5 0 S H E L L
The optimization structure OptiForR50shell is generated by uniformly expanding PTV by some margin M (Fig. 1) and subtracting PTV from the resulting structure. The volume of the generated Opti-ForR50shell structure is then expressed as: where (PTV + M) represents an operation of expanding PTV by some margin M.
For the sake of simplicity, the PTV is assumed to be of a spherical shape, and the volume of the PTV is expressed as: where r PTV is the radius of the PTV.
Thus, V OptiForR50shell in Eq. (B11) is expressed in algebraic terms as: