Associating dose–volume characteristics with theoretical radiobiological metrics for rapid Gamma Knife stereotactic radiosurgery plan evaluation

Abstract Purpose To examine general dose–volume characteristics in Gamma Knife (GK) plans which may be associated with higher tumor control probability (TCP) and equivalent uniform dose (EUD) using characteristic curve sets. Methods Two sets of dose–volume histograms (DVHs) were exported alongside an analytical purpose‐generated DVH: (a) single‐shot large collimator (8 or 16 mm) emulated with multiple shots of 4 mm collimator. (b) shot‐within‐shot (SWS) technique with isodose lines (IDLs) of 40–75%. TCP, average dose, EUD in single‐fraction (EUDT) and 2 Gy fractionated regimens (EUDR) were examined for trends with cumulative DVH (cDVH) shape as calculated using a linear‐quadratic cell survival model (α/β = 10.0 Gy, N0 = 1 × 106) with both α = 0.20 Gy−1 and α = 0.23 Gy−1. Results Using α = 0.20 Gy−1 (α = 0.23 Gy−1), plans in the analytical set with higher shoulder regions had TCP, EUDT, EUDR increased by 180%, 5.9%, 10.7% (11.2%, 6.3%, 10.0%), respectively. With α = 0.20 Gy−1 (α = 0.23 Gy−1), plans with higher heels had TCP, EUDT, EUDR increased by 4.0%, <1%, <1% (0.6%, <1%, <1%), respectively. In emulating a 16 (8) mm collimator, 64 (12) shots of the small collimators were used. Plans based on small collimators had higher shoulder regions and, with α = 0.20 Gy−1 (α = 0.23 Gy−1), TCP, EUDT, EUDR was increased up to 351.4%, 5.0%, 8.8% (270.4%, 5.0%, 6.8%) compared with the single‐shot large collimator. Delivery times ranged from 10.2 to 130.3 min. The SWS technique used 16:8 mm collimator weightings ranging from 1:2 to 9.2:1 for 40–75% IDL. With α = 0.20 Gy−1 (α = 0.23 Gy−1), the 40% IDL plan had the highest shoulder with increased TCP, EUDT, EUDR by 130.7%, 9.6%, 17.1% (12.9%, 9.1%, 16.4%) over the 75% IDL plan. Delivery times ranged 6.9–13.8 min. Conclusions The magnitude of the shoulder region characteristic to GK cDVHs may be used to rapidly identify superior plan among candidates. Practical issues such as delivery time may require further consideration.


| INTRODUCTION
Gamma Knife (GK) radiosurgery treatment plans are unique in their low (typically~50%) prescription isodose lines, 1,2 which leads to a highly heterogeneous dose distribution within the target. Given that portions of the target will receive doses up to twice as high as the prescription dose, along with the versatility of the GK collimator delivery system, vastly different dose distributions can be generated depending on the planning team. This, in addition to dose ranges which are typically at least 15-18 Gy in a single fraction, 1-3 results in a persistent quest in GK treatment planning on identifying favorable dosimetric characteristics of GK plans that would produce the best tumor control is acheived. 4 Common plan quality indices such as the conformity index 1,5 or the Paddick index 6 are focused upon the shape and coverage of the prescription isodose. These indices are convenient, as simple ratios of relevant volumes, and can be calculated with minimal additional training for the treatment planning team. In exchange for its simplicity, however, these indices do not account for the shape and/or coverage of other isodose lines enclosed within the prescription isodose volume (PIV), which could also affect the clinical effectiveness of the GK plan. Dose-volume traits within the target are a source of raw data which could be used to compute a more clinically relevant objective measure of the plan quality.
Two metrics which can capture the complete dose-volume characteristics of the target are the tumor control probability (TCP) and equivalent uniform dose (EUD). Both metrics provide an objective quantification of plan quality through comprehensive dose-volume characteristics. Furthermore, these metrics also correlate with an intuitive interpretation -TCP represents a straightforward biological endpoint and EUD aids in reframing a dose distribution as a more familiar uniform dose distribution. However, these metrics require additional calculations not readily available within the current GK treatment planning software.
Since ideal dose-volume histograms (DVHs) should inherently possess some general shapes and/or visual features that maximize the plan quality, the aim of this study was to examine and identify favorable dose-volume histogram characteristics or features of GK plans that would result in the highest TCP and EUD. 4 If it is indeed possible to uncover favorable shapes and/or visual features within a dose-volume histogram, this could be incorporated by the treatment planning team to rapidly identify the most robust plan from a cohort of candidate plans.

2.A | Radiobiological analysis
Our investigation is based on the linear-quadratic (LQ) cell survival model. In general, the utility of the LQ model is predicated on its simplicity in only using a handful of parameters. Thus, the resultant TCP is especially sensitive to values chosen for its radiobiologic parameters: α, α/β, and N 0 . It is well-known that, due to the interpatient heterogeneity, no single set of parameters is feasible for any patient population. 7 With this in mind, we selected the two sets of radiobiological parameters. The first set was designed to exaggerate differences in cell survival fraction by assigning a lower α value -α = 0.20 Gy −1 , α/β = 10.0 Gy. Parameters were selected within the sharp gradient region of the sigmoidal TCP curve to increase sensitivity to uncover trends.
The second set was a more clinically realistic parameter set which was calibrated to provide more familiar TCP results by assigning a higher α value: α = 0.23 Gy −1 , α/β = 10.0 Gy. These parameters lie in the flatter plateau region of the TCP sigmoidal distribution.
The LQ cell survival model calculates the tumor cell survival fraction, S, for each plan, assuming a homogeneous initial tumor cell population as where α and β are radiobiological parameters of the LQ model, dDVH(D j ) is the differential DVH value associated with dose D j , and D is the dose increment in the DVH. Tumor control probability (TCP) was calculated assuming a Poisson distribution as where N 0 was the initial number of tumor cells before irradiation, and was assigned a value of N 0 = 1x10 6 for both datasets. The equivalent uniform dose (EUD) for a single-fraction (EUD T ) was calculated as 8 where SF 2 was the survival fraction applied with a single 2 Gy irradiation and D ref was the reference dose per fraction (2 Gy). The corresponding EUD for a fractionated regimen in 2 Gy fractions was calculated as 8 2.B | Analytical purpose-generated dose-volume-  The equations used to generate the dDVH curves are shown below, where each curve was defined by Prescription dose was set as 18 Gy at the 50% IDL for all plans, which is our institution's most common value. Thus, D min and D max were set at 18 and 36 Gy, respectively, in Eqs.  For the single-shot distribution, the 50% IDL surface of a singleshot large (8 or 16 mm) collimator was converted into a target contour ("large-8" or "large-16," respectively). The complementing multiple-shot distribution was created using exclusively 4 mm collimator shots with multiple isocenters to mimic the reference contour (large-8 or large-16) created using single-shot of the large collimator. Each plan was prescribed 18 Gy at the 50% IDL.

2.D | Nested shot-within-shot (SWS) technique
The third set of DVH curves was generated using the shot-within- Prescription isodose lines of 40%, 45%, 50%, 55%, 60%, 65%, 70%, and 75% were each assigned to different plans using a 16 to 8 mm SWS pair. For each isodose line, the weighting was adjusted to match the absolute prescription (18 Gy) to the prescription IDL such that the PIV was identical to the reference planthe 50% IDL of an evenly weighted 1:1 plan. For the dose-volume analysis, the target volume GTV was defined as the PIV of the reference 1:1 plan minus a 1 mm margin.

3.A | Purpose-generated analytical histograms
A set of five dDVH curves was created with accompanying cDVH curves, which are shown together in Fig Table A1) and used to derive remaining parameters: C 1 , C 2 , C mid . For these curves, values were chosen to create distinct, yet paired variations in each curve for the shoulder and fall-off regions.
The last curve, E is a Dirac step function in the cDVH, created by setting the dDVH to 100.0 at 18.0 Gy. This was performed manually rather than using the analytical equations to avoid dividing by zero for curve.
The "shoulder" and "heel" of the cDVH regions are shown in  Using a different radiobiological parameter set with α = 0.23 Gy −1 , the same overall trends were observed to a milder degree: for curves with similar heels, increasing the shoulder region yielded considerably higher (11.2%) TCP than the heel region (0.6%).
With the larger α value, the overall TCP values were much larger.
While the increase in TCP were milder, the EUD T and EUD R were slightly higher, with increases of 6.3% and 10.0%, respectively.    As summarized in Table 2

3.C | Nested shot-within-shot(SWS) technique
Isodose lines of 40%, 50%, 55%, 60%, 65%, 70%, and 75% were prescribed 18 Gy to various weightings of 16 and 8 mm collimators. An evenly weighted 1:1 distribution prescribed to the 50% IDL was used as a reference PIV, which had a volume of 1.443 cm 3 . For cDVH values shown in Fig. 4, the target volume was defined as the reference PIV minus a 1 mm margin.
As summarized in Table 3 T A B L E 2 Average dose, equivalent uniform dose (EUD), and tumor control probability (TCP) for analytically generated cumulative dosevolume histogram (cDVH) curves with α = 0.20 Gy −1 and α = 0.23 Gy −1 . Each with α/β = 10.0 Gy, and N 0 = 1 × 10 6 for curves created by using 64 or 12  Each value of the cDVH was uniformly larger for lower IDL prescriptions. Unlike the previous section, higher shoulders were also observed with higher heels. The lower IDL plans had a higher global maximum dose, which contributed to the larger shoulders observed in the cDVH. Additionally, the high global maximum dose in the lower IDL plans contributed to a heel region which was both higher and more extensive than the higher IDL plans.
For the curves in this portion of the investigation, the highest D avg yielded the highest values of EUD T , EUD R , and TCP; there was a monotonic direct relationship observed between D avg and these dosimetric parameters. For α = 0.20 Gy −1 , the 40% IDL plan yielded a TCP of 91.8%, which is more than double (230%) the TCP observed with the 75% IDL plan. The EUD T and EUD R increased by 9.6% and 17.1%, respectively.
With the higher value of α = 0.23 Gy −1 , the same trends were observed, but the differences were less mild; only a 12.9% increase in TCP changing from the 75% IDL plan to the 40% IDL plan. Note that the TCP values appear to be well within the plateau region of the TCP sigmoidal distribution. The EUD T and EUD R increased by 9.2% and 16.4%, respectively.
As with prior sections, the results of this portion of the investigation suggested that the shoulder value and shape were significant.
However, this set of curves also demonstrated that the absolute value of the heel does carry significance.

4.A | Radiobiological model limitations
It is well-known that there is no single unique value for each radiobiological parameterrather, it is realistically a range which is dependent on the population examined. Therefore, two values of α were  Within the expanded dose axis of the cDVH, GK plans tended to follow a general characteristic shape with the following three general regions: a shoulder, a fall-off region, and a heel. The last set of curves introduced a variable maximum dose.
These curves revealed that given two plans with otherwise similar minimum doses and shoulder shapes, the plan with the higher maximum dose would produce the superior TCP. In the curves created using the SWS technique, a higher maximum dose resulted in a smaller heel region on the cDVH, but this was counteracted by larger values throughout the remainder of each curve.
As the prescription isodose line was increased, the maximum dose was closer to prescription dose and the overall dose heterogeneity was eliminated. As a theoretical exercise, if an isodose could actually be increased further to 100%, the result would be a completely uniform delivery (i.e. EUD = prescription dose) and the cDVH would be a step-function curve as in curve E of Fig. 1. This was earlier already shown to have zero TCP assuming an 18 Gy prescription as in Table 1. Again, this would be a purely theoretical result in GK, as a flat beam profile is unachievable without some type of flattening filter. However, these results imply that there may be an upper limit on the utility of higher isodose lines in GK. This is currently being investigated by our group. In other delivery sites, such as the prostate, the effects of intrafraction repair and source decay have been shown to reduce the BED by as much as 36%. 20

4.E | Clinical impact and notes
By associating TCP and EUD with simple features within the cDVH, the general trends from this study may be integrated into GK treatment planning without relying upon the separate third-party analysis software normally required for radiobiological calculations. This visual method expedited treatment planning by allowing prompt evaluation which also circumvented repeated data transfer between GK and third-party analysis software. The efficient usage of time can be especially sensitive for GK clinics which perform frame-placement, imaging, treatment planning, and delivery all within a timespan of a few hours.
Our investigation was performed exclusively for the large dose ranges ubiquitous with GK. Therefore, while the visual method can be utilized in GK, the reader is reminded that the same trends among dose-volume characteristics, TCP, and EUD may not necessarily be observed with different dose ranges. In general, as the overall dose range becomes narrower, the heel region is expected to have a larger impact. Users should be especially vigilant in applying our observed trends in GK to conventional fractionation, where the dose range is much smaller

| CONCLUSIONS
This investigation used two separate radiobiological parameter sets to identify general trends among TCP, EUD, and dose-volume characteristics by examining basic features represented with a cDVH.
Our investigation revealed general trends in the cDVH shape that can be used to quickly and reliably identify the superior plan among candidates. However, practical issues such as delivery time may require consideration by the planning team to choose the most clinically viable plan.
ACKNOWLEDGMENTS N/A.

CONFLICT(S) OF IN TEREST
No conflict of interest.