An analytical expression for R50% dependent on PTV surface area and volume: A cranial SRS comparison

Abstract The intermediate dose spill for a stereotactic radiosurgery (SRS) plan can be quantified with the metric R50%, defined as the 50% isodose cloud volume (VIDC50%) divided by the volume of the planning target volume (PTV). By coupling sound physical principles with the basic definition of R50%, we derive an analytical expression for R50% for a spherical PTV. Our analytical expression depends on three quantities: the surface area of PTV (SAPTV), the volume of PTV (VPTV), and the distance of dose drop‐off to 50% (Δr). The value of ∆r was obtained from a simple set of cranial phantom plan calculations. We generate values from our analytical expression for R50% (R50%Analytic) and compare the values to clinical R50% values (R50%Clinical) extracted from a previously published SRS data set that spans the VPTV range from 0.15 to 50.1 cm3. R50%Analytic is smaller than R50%Clinical in all cases by an average of 15% ± 7%, and the general trend of R50%Clinical vs VPTV is reflected in the same trend of R50%Analytic. This comparison suggests that R50%Analytic could represent a theoretical lower limit for the clinical SRS data; further investigation is required to confirm this. R50%Analytic could provide useful guidance for what might be achievable in SRS planning.

planning tend to be phenomenological constructs, and limits so obtained are based on observations from large numbers of treatment plans. We have proposed a model-based approach for the metric R50% that considers the physical characteristics V PTV and PTV surface area (SA PTV ). This approach allows for the derivation of an analytical form of R50% (R50% Analytic ) that is based on physical principles. It is necessary, however, that this analytical methodology be validated against clinical data. At least one published study on cranial SRS does provide the necessary data for a meaningful comparison of R50% Analytic to clinical data. 8

2.A | R50% Analytic derivation
Consider a spherical PTV volume, V PTV , surrounded by a spherical shell that encloses the 50% isodose cloud volume (V IDC50%shell ) as illustrated in Fig. 1. The sum of V PTV and V IDC50%shell is the total volume encompassed by the 50% isodose cloud (V IDC50% ). R50% is defined as the ratio of the volume of the 50% Isodose Cloud to the volume of the PTV as follows: Furthermore, we determined an exact value of V IDC50%shell by integrating the spherical differential shell volume, 4πr 2 dr, from r =r PTV to r = r PTV + Δr.
Given that SA PTV ¼ 4πr 2 PTV and combining Eqs. (1) and (2), the resulting analytical form of R50% can be expressed as: Equation (3)  shell, middle control shell, and outer control shell) as described by Clark et al. to directly limit the dose spill outside the PTV, in accordance with standard clinical practices. 12 Alternatively, one could use other dose limiting shell techniques. 13 We sought the minimum value of Δr one could obtain clinically in ideal circumstances. The quality of these phantom plans can be seen from the parameters given in Table 1.
Since a highly noncoplanar delivery geometry coupled with a spherical PTV was chosen, the resulting dose distribution is reasonably isotropic and can be assumed spherical. This nearly spherical dose distribution can be clearly seen in Fig. 3 as the transparent yellow isodose cloud of 50% of the prescription dose (IDC50%) surrounding the solid orange PTV. This distribution bears a marked similarity to Fig. 1 used in the derivation of R50% Analytic . Thus, it becomes simple to extract a value of Δr for each phantom PTV as follows: Based on the values of Δr obtained from the phantom study, a power law fit was generated (Microsoft Excel) for Δr as a function of V PTV as shown in Fig. 4.
The resulting power law expression for Δr, in units of cm, is: where V PTV is measured in cm 3 .
As can be seen in Table 1, the GM values reported by Eclipse for these spherical volumes are nearly identical to the Δr values obtained from Eq. (4). This should not be surprising since GM is defined as the difference, in centimeters, of the equivalent sphere radii of V IDC50% and V IDC100% (r50% eq and r100% eq , respectively). 7 Thus, GM ¼ r50% eq À r100% eq (6) By comparison, for a perfectly conformal plan (CI = 1.0), V IDC100% is identical to and spatially coincident with V PTV . Thus, for The five hemi-arcs beam arrangement for determination of Δr. This three-dimensional (3D) view of the IROC head phantom shows the beam delivery geometry used for the phantom plans used to determine Δr for a series of nine spherical planning target volumes. Each red curve in the figure represents the path of an arc around the cranium using the Varian IEC scale. For couch angles 355°(A), 315°(B), and 270°(C), the arcs span 195°to 345°. For couch angles 45°(D) and 5°(E), the arcs span 15°to 165°. and R50% can be seen as the product of Eqs. (7) and (8).
Using this approach, the data of Zhao et al. will yield the equivalent R50% to be used for comparison.

| DISCUSSION
It can be readily seen that R50% Analytic values are consistently lower than the corresponding R50% Clinical data (Fig. 5). Consideration of Typical results for the phantom study to determine Δr. The diagram shows an AP DRR and a right lateral DRR that display the position and size of the PTV (solid orange shape) and IDC50% (transparent yellow shape) within the cranium. The distance from the edge of the PTV and the outer edge of IDC50% is Δr. The volume of the PTV is 3 cm 3 . Note that the IDC50% is very nearly spherical. to V PTV ratios, which is consistent with the suppositions in Goldbaum et al. In previous work, it was quantitatively shown that an increase in the SA PTV to V PTV ratio leads to an increase in R50% values. 9 For any given volume, the shape that corresponds to the smallest surface area is a sphere, 16  At lower V PTV values, a larger difference is seen between R50% Clinical and R50% Analytic (Fig. 5

| CONCLUSION
An analytical expression for R50% was derived for the special case of spherical volumes. The expression appears to provide a lower limit of R50% when compared to peer-reviewed, clinical data. We surmise that SA PTV plays an important role in the determination of the R50% value ultimately achievable in treatment planning. Further research is needed to establish the role of SA PTV for other PTV shapes in the determination of treatment planning outcomes. Research is also needed to establish methods for obtaining Δr and investigate additional determining factors beyond V PTV .

CONF LICT OF I NTEREST
No conflict of interest.