A physically meaningful relationship between R50% and PTV surface area in lung SBRT

Abstract Purpose We propose a novel understanding of two characteristics of the planning target volume (PTV) that affect the intermediate‐dose spill in lung stereotactic body radiation therapy (SBRT) as measured by R50%. This phantom model research investigates two characteristics of the PTV that have a marked effect on the value of R50%: the mean dose deposited within the PTV (Dav) and the surface area of the PTV (SAPTV). Methods Using a phantom model provided by a CT of the IROC Thorax‐Lung Phantom® (IROC Houston QA Center, Houston, TX) and Eclipse® Treatment Planning System (Varian Medical Systems, Palo Alto, CA), we investigate the two characteristics for spherical and cylindrical PTVs. A total of 135 plans with tightly controlled PTV characteristics are employed. A lower bound for R50% (R50%min∆r) is derived and clearly establishes a relationship between R50% and SAPTV that has not been fully appreciated previously. Results The study of PTV Dav revealed a local minimum for R50% as a function of the PTV Dav at Dav ≈ 110% of Rx dose. As PTV Dav increases above this local minimum, R50% increases; while for PTV Dav less than this local minimum, the R50% value also increases. The study of PTV surface area (SAPTV) demonstrated that as the SAPTV increases, the R50% increases if the PTV volume stays the same. The SAPTV result is predicted by the theoretical investigation that yields the R50% lower bound, R50%min∆r. Conclusions This research has identified two characteristics of the PTV that have a marked influence on R50%: PTV Dav and SAPTV. These characteristics have not been clearly articulated in the vast body of previous research in SBRT. These results could help explain plans that cannot meet the RTOG criteria for R50%. With further development, these concepts could be extended to provide additional guidance for creating acceptable SBRT plans.


| INTRODUCTION
Stereotactic body radiation therapy (SBRT) was defined in 2014 as "a method of external beam radiotherapy (EBRT) that accurately delivers a high dose of irradiation in one or few treatment fractions to an extracranial target." 1 This technique of high-precision, ultra-hypofractionated EBRT is gaining widespread clinical use facilitated by convenient, commercially available equipment that makes SBRT practical even in community-based radiation therapy facilities.
Stereotactic body radiation therapy has been rapidly adopted as the standard of care for nonsurgical early-stage lung cancer. [2][3][4][5][6] The current clinical practice of SBRT for early-stage nonsurgical lung cancer treatment is based on the clinical protocols provided by three RTOG studies: RTOG 0236, 7 RTOG 0813, 8 and RTOG 0915. 9 A principle metric for plan acceptability is R50%. R50% is a derived unitless quantity obtained from the volume of the 50% prescription isodose cloud (IDC50%) and the planning target volume (PTV) volume as follows: The IDC50% is a surrogate for damage to irradiated normal tissues as discussed in the work of Yang et al. 10 The goal of SBRT treatment planning is to deliver a high dose to the target (for local tumor control) that meets or exceeds the minimum dose specified by the prescription while minimizing the IDC50% (to minimize normal tissue damage) and avoiding unacceptable dose to organs at risk (OAR).
The RTOG protocol 9 provides guidance for acceptable values for R50% as function of the PTV volume measurement. Other RTOGdefined quantities used to assess SBRT plan quality are D2cm 7 (maximum Dose 2 cm from the PTV in any direction), V105% 9 (105% prescription isodose volume outside of the PTV), and CI 9 (Conformity Index = [100% prescription Isodose Volume]/[PTV Volume]). In addition, these protocols contain limits on various OARs, such as the spinal cord and ribs.
When considering the PTV dose heterogeneity there are three metrics available: ICRU HI, 9 RTOG HI, 11 and PTV mean dose (PTV D av ). The ICRU HI, RTOG HI, and PTV D av are defined, respectively, as follows: where D x% = the minimum dose to x% of the PTV.

RTOG HI
where D max = "maximum dose in the PTV" and D Rx = "prescription dose." And PTV D av ¼ mean dose in PTV Many authors have reported investigations on Lung SBRT treatment plan quality by analyzing adherence to the various goals set forth in the RTOG protocols. 12 The arc geometry was limited to include only the PTV and right lung and used typical clinical arc limits and collimator angles.
All PTVs were created as simple geometric shapes of uniform physical density to simplify the problem and control as many variables as possible, thus isolating the PTV characteristics under study.
The simple, well-defined geometric shapes also allowed an analytic calculation of the PTV surface area.
The geometric configuration of gantry, collimator, and phantom is shown in Fig. 1 phantom study with no respiratory motion and well-defined target properties satisfies these conditions. All plans meet the requirements of RTOG protocols 7-9 for D2cm, V105%, and CI.

2.A | R50% dependence on PTV Dose
Heterogeneity To study the relationship between R50% and PTV dose heterogeneity, we generated four spherical volumes (9, 33, 53, and 81 cc) and four cylindrical shapes of the same volumes but varying elongation.
We employed two different PTV shapes to control for PTV shape so the results for dose heterogeneity study can be clearly attributed to the dose heterogeneity independent of the PTV shape. All eight PTVs are located in the center of the right lung. The density of the PTVs was overridden to a typical 0.9 g/cc for all cases.
A series of ten plans for each PTV were created that control the PTV dose heterogeneity using only a PTV maximum dose constraint (PTV D max ) and the prescription D95% condition. The PTV D max constraint included an unlimited maximum dose and maximums ranging from 140% to 100% of the Rx dose decreasing in increments of 5%.
This resulted in plans with progressively more dose homogeneity in the PTV. We use PTV D max to contain the PTV dose heterogeneity within the limits of the prescription D95% and PTV D max . We examine these results in light of two more standard metrics for dose heterogeneity: ICRU HI and RTOG HI [defined by Eqs. (2) and (3), respectively]. We also study these results with a third metric PTV D av defined by Eq. (4).
We report the results relative to PTV D av because this is a reasonable measure of the actual global dose delivered to the target.
D av is also a readily available parameter in all commercially available RTPS and allows comparison with previous studies.
To further understand the dose distribution, we examined the average dose in the shell of the IDC50% that is outside the PTV.
For notational simplicity, we define IDC50%shell as that part of IDC50% that is not part of the PTV.
This evaluation of PTV dose heterogeneity generated a total of 80 plans (4 volumes × 2 shapes × 10 D max constraints).

2.B | Dependence on PTV surface area
To study the relationship between R50% and SA PTV , we generated a large set of PTVs of both spherical and cylindrical shape and having varying surface area. All PTVs were defined to have a uniform density of 0.9 g/cc. The optimizer PTV D max constraint was held constant at 120% of prescription yielding a PTV D av of approximately 110% of Rx.
F I G . 1. Geometry of the experimental plans run on the IROC Thorax-Lung Phantom ® . The PTV shown in the right lung is a density override volume and indicates the position of isocenter and the arc geometry used.
DESAI ET AL.

| 49
A set of cylindrical PTVs were generated having volumes equal to the 11 spherical PTVs and a maximum length equivalent to the maximum dimension given in RTOG 0236 thus fixing diameter for a given maximum length. These cylinders were placed in 2 distinct orientations: cylindrical axis in the superior/inferior direction (denoted as "RTOG Horizontal Cylinders" or "RTOG Horz Cyl"); and a second set with the cylindrical axis in the anterior/posterior direction (denoted as "RTOG Vertical Cylinders" or "RTOG Vert Cyl"). To provide additional variation in the surface area, we created an additional set of cylindrical PTVs with the same volume as the spherical PTVs but very long cylindrical axes and thus smaller radial dimensions than the cylindrical PTVs mentioned above. The maximum cylinder height was capped at 8 cm, however, to avoid clinical irrelevance. These new cylinders are also oriented in both the "horizontal" and "vertical" directions previously described. The dual cylinder orientations were used to more directly test the PTV surface area dependence of R50%, by controling for PTV orientation relative to the delivery geometry of the beams. This section of the study generated a total of 55 PTVs and associated plans.

2.C | R50% Limit Derivation
To derive the R50% limit in highly conformal treatment like SBRT, begin by considering Fig. 2.
The volume of the IDC50% can be determined by a differential shell expansion of the PTV. If the expansion is vanishingly small, Δr becomes dr. In this differential shell, the volume is given by This is strictly correct only if the differential shell is truly an infinitesimal expansion. This expansion generates a new volume and that volume, V 1 , has a surface area, SA 1 . Now that volume, V 1 , can be expanded by another dr into volume V 2.
V 2 has a surface area SA 2 , and This process continues through the entire infinitesimal differential shell series of nested shells such that Thus, SA is a function of r, SA(r), where we are concerned with r in the range from the PTV surface outward. For notational simplicity, adopt the notation that r = 0 at the surface of the PTV, and r = Δr at the surface of IDC50%.
Continue adding differential shells until the surface reaches IDC50% (at distance Δr). Thus, strictly speaking, Since each differential shell surface area is larger than the one nested inside it, one can see that Divide both sides of the inequality by V PTV to obtain the following: We can now define a lower bound for R50%. This lower limit we name R50%min Δr . , respectively). Thus, By comparison, for a perfectly conformal plan with Conformality Index = 1, the volume of the 100% isodose volume is identical to and spatially coincident with the PTV volume. Thus, for the equivalent sphere, R 100% eq = radius of the equivalent sphere PTV.
In such case, the GM is equivalent to the Δr as defined in Fig. 2 2. An arbitrary shape planning target volume of surface area SA is expanded by Δr to achieve the IDC50% volume.
where the subscript GM designates Δr min = GM from Hoffman et al.
In principle, the limit in Eq. (13)  conceptualize. In addition, D av is directly correlated with tumor control. 17 Therefore, we choose PTV D av as our metric of choice for the evaluation of PTV dose heterogeneity. Results for the dependence of R50% on PTV D av are summarized in Fig. 4.
The three regions of interest seen in all four panels of Fig. 4 are PTV D av < 105% Rx, PTV D av > 115% Rx, and the interval between 105% and 115% Rx. There appears to be a relatively broad local minimum in R50% in the interval between 105% and 115% with increases outside this range. The R50% local minimum region as a function of PTV D av (between 105% and 115% of the Rx dose) appears to vary slightly with PTV volume. The R50% local minimum as a function of PTV D av occurs at D av ≈ 110% Rx.
We also examined the mean dose in the IDC50%shell. Figure 5 shows that as PTV D av decreases below the local minimum (D av ≈ 110%), the mean dose in the IDC50%shell increases for all PTV volumes, ie, the dose spill outside the PTV increases as the PTV dose becomes more homogeneous.

3.B | Dependence on PTV surface area
The results of the PTV surface area study are summarized in Figs. 6 and 7. Figure 6 shows R50% for all 55 plans plotted as a function of the SA PTV /V PTV for all volumes and PTV shapes (spheres and various cylinders). Also displayed is the numerical value of R50%min GM [Eq. (15)], which is the limit R50%min Δr that uses Δr min = GM values given by Hoffman, et al. 20 Notice that the lower bound R50%min GM is always less than the planned R50% and thus is indeed a lower bound. Figure 7 provides the same data as Fig. 6 but alternatively plotted as a function of PTV volume. As seen in Fig. 7(a), R50% rises sharply as the PTV Volume approaches zero. Again, it is evident that the R50%min GM appears to be a true lower bound for R50% regardless of PTV volume. Figure 7(

4.A | R50% dependence on PTV D av
The observed R50% local minimum in PTV D av seen in Fig. 4 can also be seen in the results of previous work. 12,21 However, these and other studies made little comment about the phenomenon. This R50% local minimum in PTV D av is an interesting result of this study.
The R50% local minimum region as a function of PTV D av occurs between 105% and 115% of the Rx dose and appears to vary slightly with PTV volume. We are not certain that this minimum behavior will hold when other treatment planning constraints are applied, but it is a possible characteristic in a general treatment planning scenario.
The R50% local minimum as a function of PTV D av as shown in is less PTV material to absorb the beam. This would increase the exit fluence from the PTV periphery. The higher exit fluence would contribute to a larger R50% and, thus, more dose would be deposited in the IDC50%shell. This effect is illustrated in Fig. 5 where the dose in the IDC50%shell is seen to increase as the dose in the PTV becomes more homogeneous as indicated by a reduction in the PTV D av . We acknowledge that there is little clinical justification for seeking a homogeneous PTV dose in SBRT. One could even argue that this is contrary to the standard clinical practice of SBRT. Previous recommendations 5 have encouraged doses as high as 130% of Rx in the PTV. The point of this study, however, is to understand the effect that PTV dose heterogeneity, quantified by PTV D av , has on R50%.
Many previous studies have shown how a heterogeneous dose within the PTV is desirable for a steep dose falloff surrounding PTV. [13][14][15][16]22,23 Our study clearly shows that this forced dose heterogeneity may only be useful to a point, that being the R50% local minimum in PTV D av . Beyond this point, increasing the PTV dose heterogeneity will increase the R50%. Many of the steep dose falloff studies are for cranial radiosurgery targets. [14][15][16]22,23 Most of the radiosurgery target volumes are <10 cc and many involve SRS cones or a GammaKnife delivery platform. It is conceivable that the exact location of the R50% local minimum in PTV D av could be optimizer and delivery technique dependent.   20 Notice that the lower bound R50%min GM is always <R50% for all plans and thus is indeed a lower bound.
It should be noted that the difference between the spherical and cylindrical PTVs in Fig. 4 shows roughly the same local minimum in R50%. This implies that the local minimum observed is consistent regardless of PTV shape. The differences between the spherical and cylindrical PTVs are exposed in the study of PTV surface area.
The immediate clinical effect of this study is to alert the treatment planner that both homogenous PTV doses and extreme heterogeneous PTV dose distributions are undesirable if the goal is to obtain optimal values of R50%. Dosimetrists, Physicists, and Physicians need to be aware of this phenomenon.

4.B | Dependence on PTV surface area
Dependence of R50% on SA PTV appears to be underreported in the literature. The results of the PTV surface area study are illustrated in Fig. 6. R50% is seen to be highly correlated with SA PTV /V PTV and very nearly linear regardless of PTV shape. One would expect this result from the R50%min Δr expression given by  Fig. 6(b). There is also a second-order effect of cylinder orientation on R50%. Note that a larger SA PTV correlates with a larger R50% for the same volume. We also note similarity of the PTV surface area study results to the recent publication of Goldbaum et al. 25 working in cranial SRS.
In that work, the authors hypothesized that "an increase in TV12 [intermediate dose spill] could be related to an increase in the surface area of the target, and the surface area of the target could be quantified by analogy with an ellipsoid." 25 Their attempt to use effective ellipsoids to quantify the PTV surface area did not prove productive.
However, as we see from this work, the PTV surface area is a decisive factor in predicting the R50% limit given by R50%min Δr .
The results shown in Fig. 7 Fig. 1(b)]. These other publications make no specific comment about the origin of this overall trend. This work, however, provides a clinically useful and physically reasonable explanation of the trend in R50% as a function of V PTV . The smaller surface area PTV (sphere) always has a smaller R50% for any given PTV volume. Furthermore, this general PTV surface area trend is maintained even when cylinders are oriented differently as shown in Table 1. Thus, the PTV orientation is only a minor perturbation on the overall PTV surface area effect.
Considering what the PTV surface area represents, it should not be surprising that PTV surface area has a marked effect on R50% because the PTV surface area is a direct measure of how much nontarget tissue is exposed to the high dose of the PTV: PTV surface area is the interface between target and nontarget tissue. Given the same degree of conformality, a larger PTV surface area in a lung SBRT plan means more lung tissue is exposed to the highest dose region. That exposure would propagate out to all isodose lines outside the target including the 50% isodose line defining the R50%.
Since PTV surface area markedly affects R50%, it would be advantageous to know this value in all clinical situations. At present we are aware of no commercially available RTPS that can report the surface area of a segmented structure. This may be why the importance of surface area has not been previously recognized. Having this statistic available in the treatment planning systems would make the study of PTV surface area possible in clinical studies.
Other authors have observed PTV shape dependence 27 and have noted that a concave PTV has a higher R50% than a comparable volume fully convex PTV. Naturally, a concave PTV will have a higher surface area than a convex PTV of same volume. PTV surface area may prove a useful metric in future studies involving PTV shape.
The immediate clinical impact of this PTV surface area study is the understanding that for PTV shapes that are distinctly nonspheroidal, the R50% will be larger for the same volume and thus it will be more difficult to meet the R50% criterion given in the RTOG protocols 7-9 for such plans. The derived R50%min GM [Eq. (15)] is clinically useful in its own right. But it also clearly establishes the relationship of R50% with the PTV surface area (SA PTV ).
The two PTV characteristics, PTV D av and SA PTV , have a marked impact on the value of R50% and can be summarized as follows: PTV Mean Dose (D av ): Heterogeneity in the PTV dose distribution, as indicated by PTV D av , affects the value of R50%. There is a local minimum in R50% for PTV D av ≈ 110% prescription dose.
PTV Surface Area: As the PTV surface area increases, the R50% naturally increases if the PTV volume stays the same.
The principle goal of this work is to illuminate the effects of the PTV characteristics, PTV D av , and SA PTV , on the plan metric R50% and to develop a conceptual understanding of these effects. This will allow future investigations to be more detailed and discuss how the entanglement of other characteristics is affecting the in-depth study of a single PTV characteristic.

| CONCLUSION S
This research has identified two critical characteristics of the PTV that have not been clearly articulated in the vast body of previous research in lung SBRT: PTV D av and SA PTV . Equation (13) clearly establishes a relationship between R50% and SA PTV that has not been fully appreciated previously. This research provides new insight on a large body of previously published work related to lung SBRT.
A better understanding of the relationship of these PTV characteristics to the ultimate R50% could provide improved guidance for determining when an optimal SBRT plan is achieved. Future work will provide more detailed descriptions of how these PTV characteristics affect R50%.

ACKNOWLEDG MENTS
The authors would like to thank Debra Olson Desai, MS for her assistance with the editing and organization of the References, her work formatting the graphs, and formatting the equations. Ulrich