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Assessment of biological dosimetric margin for stereotactic body radiation therapy

Abstract Purpose To develop a novel biological dosimetric margin (BDM) and to create a biological conversion factor (BCF) that compensates for the difference between physical dosimetric margin (PDM) and BDM, which provides a novel scheme of a direct estimation of the BDM from the physical dose (PD) distribution. Methods The offset to isocenter was applied in 1‐mm steps along left‐right (LR), anterior‐posterior (AP), and cranio‐caudal (CC) directions for 10 treatment plans of lung stereotactic body radiation therapy (SBRT) with a prescribed dose of 48 Gy. These plans were recalculated to biological equivalent dose (BED) by the linear‐quadratic model for the dose per fraction (DPF) of d = 3–20 Gy/fr and α/β=3-10. BDM and PDM were defined so that the region that satisfied that the dose covering 95% (or 98%) of the clinical target volume was greater than or equal to the 90% of the prescribed PD and BED, respectively. An empirical formula of the BCF was created as a function of the DPF. Results There was no significant difference between LR and AP directions for neither the PDM nor BDM. On the other hand, BDM and PDM in the CC direction were significantly larger than in the other directions. BCFs of D 95% and D 98% were derived for the transverse (LR and AP) and longitudinal (CC) directions. Conclusions A novel scheme to directly estimate the BDM using the BCF was developed. This technique is expected to enable the BED‐based SBRT treatment planning using PD‐based treatment planning systems.

Particularly, stereotactic body radiation therapy (SBRT) requires the calculation of BED, as it uses hypo-fractionations and results in the delivery of a high BED. In these studies, linear quadratic (LQ) model was used. The LQ model is the most commonly used tool to model the effect of fractionation in conventionally fractionated radiotherapy and to predict tumor response to altered fractionation regimens.
Technical advances in radiation therapy, including three-dimensional conformal radiotherapy (3D-CRT) and, more recently, intensity-modulated radiotherapy (IMRT) has simultaneously enabled dose escalation and enhanced normal tissue sparing. ICRU 50 and 62 Reports were widely used as an international reference for the prescribing, recording, and reporting of photon beam radiotherapy. 4,5 However, the ICRU 83 Report was released in 2010 specifically addressing IMRT and introducing some different concepts and plan evaluation parameters such as volumetric planning target volume (PTV) prescribing. 6 For SBRT, clinical trials RTOG 0236 7 and 0813 8 used the volume prescription method. 9 The dose that covered 95 % of the PTV (D95%) was conformal in the treatment plan using the volume prescription. 10,11 The variation of the peripheral dose of PTV is expected to be significantly reduced using this prescribing method.
To accommodate inter-and intra-fractional patient setup uncertainties and organ motions, the International Commission on Radiation Units and Measurements, Inc. (ICRU) recommends expanding the clinical target volume (CTV) by a margin to obtain PTV. 12 In past studies, the van Herk formula was generally used for calculating the PTV margin from the systematic and random errors of the CTV. This formula ensures that the minimum dose of the CTV is equal to or greater than 95% of the prescribed dose for 90% of the population. 13 However, the treated volume (TV) is usually larger than the PTV, resulting in a mismatch between the theory and application of the van Herk formula. Gordon and Siebers introduced a new concept, termed the dosimetric margin (DM), to explain the sensitivity of a group of prostate IMRT treatment plans to patient setup errors. 14 The TV was defined as a volume covered by the minimum dose of the PTV. The DM, which is a margin achieved between the CTV and TV for a given plan, is a generalization of the conformity index. 15 Importantly, the sensitivity of the CTV dose to setup errors is a function of the DM.
Thus, the target coverage by the isodose surface of interest (e.g., D 95% ) should be evaluated using the DM, rather than the CTV-to-PTV margin, in the presence of setup errors. However, the DM does not consider the difference in dose distribution by setup uncertainty.
Moreover, the DM proposed by Gordon was defined using only the physical dose (PD) calculation. In practice, there are many fractionation schemes for SBRT (e.g., 48 Gy/4 fr, 60 Gy/3 fr etc.). Therefore, it is considered to be essential to take the biological effect such as DPF into account to provide appropriate DM for each fractionation scheme.
In this study, we introduced a DM involving the effects of the dose perturbation due to the setup uncertainty to take into account the setup errors in the clinical practice. The DM was defined as the isocenter shift that the CTV is satisfied with a certain dose level by setup uncertainty. The DM with physical dose distribution is defined as the physical dosimetric margin (PDM). Moreover, we proposed a novel quantity, named biological dosimetric margin (BDM), which was a margin distribution considering the biological effect of the DM. The biological effect was introduced by calculating the BED using the LQ model 16 as an example biological model.
The differences between the relative dose distribution of the PD and BED were calculated. The relative BED distribution was analyzed for the dose per fraction (DPF) from 3 to 20 Gy/fr. The α=β of the tumor and normal tissue were used different values. The α=β of the PTV includes the tumor was 3, 5, and 10 Gy, and that of the normal tissue was 3 Gy. [17][18][19][20] To provide appropriate DM for each fractionation scheme in BED-based treatment planning, a biological conversion factor (BCF) between BDM and PDM was introduced by considering the DPF and α=β to create a simple model of the BDM.

| MATERIALS AND METHODS
Ten cases of patients with lung cancer, who underwent SBRT at (institution name), were analyzed. The characteristics of the patients and their tumors are presented in Table 1. The use of clinical materials in this study was approved by the Institutional Review Board of (institution name).

2.A | Treatment planning
All patients were immobilized using a Vac-Lok cushion (CIVCO, Kalona, IA, USA). Breath-holding was coordinated in the expiratory phase using Abches (APEX Medical, Tokyo, Japan)a device that allowed patients to control their chest and abdominal respiratory motion. 21 The tumor position reproducibility during several expiratory breath-hold intervals was verified to be within 5 mm using X- The treatment plans with a prescribed dose of 48 Gy for D 95% of the PTV was created using the superposition/convolution algorithm on RayStation.

2.B | Biological equivalent dose
The BED was calculated using the LQ model as an example model to create the BED. The LQ model fits the cell-surviving fraction through a second-order polynomial on the DPF, where d is the DPF. The BED is then defined by where n is the number of treatment fractions. The ratio α=β describes the repair capacity of the cells, and thus the sensitivity to the fractionation. In the calculation of relative BED, DPF, and α=β were mainly affected with a constant DPF. The BED distribution was calculated from the physical dose distribution using Eq. (2). In this study, α=β was varied along 3, 5, and 10. 23 The DPF was in the range 3-20 Gy, referring to the past clinical trials shown in Table 2.  and Siebers [ Fig. 2(a)] is defined by

2.C | Treated volume and dosimetric margin
where D ROI X denotes the dose to the region of interest (ROI) and X = min., max., 95%, etc. The DM by Gordon and Siebers 14 (DM G ) was then defined as a volume achieved between the CTV and TV. These definitions are given based on a treatment plan with no blurring of the isocenter.
In this study, the TV was defined so that the dose perturbation effects due to the setup error were taken into account [ Fig. 2(b)].
The setup error was generated by shifting the isocenter (IC) along LR, AP, and CC directions from −20 to +20 mm with a 1-mm step (δ L , δ R , δ A , δ P , δ Cr , and δ Ca , respectively). The dose distributions were calculated with the shifted isocenter. The TV is defined as a volume that satisfied where D Rx denotes the prescribed dose. The D ROI X denotes the dose to the region of interest (ROI) and X = 95%, 98% in the physical and biological dose distributions by shifting the isocenter. Next, the maximum shift toward the left, right, anterior, posterior, cranio, and caudal directions (Δ L , Δ R , Δ A , Δ P , Δ Cr , and Δ Ca , respectively) that passed criteria of Eq. (5) were determined.
Here, the scale of the DM in this study that is the distance and the DM G that is the volume are different. The DM along each direction was calculated by was defined as the ratio between the BDM and PDM: The correlation of the BCF and α=β is evaluated. After confirming there is no significant difference for the BCF due to the α=β, the BCF is fitted using the following function of d= α=β ð Þ.
The average DG in the LR, AP, and CC directions were defined as DG LR , DG AP , and DG CC , respectively.
Then, the DG in physical dose distribution (DG PD ) and the DG in BED (DG BED ) were derived by ) 3 | RESULTS

3.D | Biological conversion factor
Figures 9 and 10 show the BCF for the D CTV 95% and D CTV 98% of CTV with α=β = 3, 5, 10 Gy in the LR, AP, and CC directions. The BCF is smaller with higher DPF and lower α=β. Figures 11 and 12 show the BCF for the D CTV 95% and D CTV 98% of CTV with d= α=β ð Þ in the LR, AP, and CC directions. The differences in the BCF due to α=β for the D CTV 95% and D CTV 98% of CTV were not significantly. The data of the transverse direction (LR and AP directions) were combined for the fitting since there was no significant difference between the LR and AP directions. The fitting results of the BCF are shown in Fig. 13. Figure 13(a) shows the measurement data and the fitted curve of D 95% in the transverse and CC directions, respectively. Figure 13

| DISCUSSION
In a past study, van Herk reported that the PTV should be a geometrical concept, and van Herk's margin was defined to select the appropriate beam sizes and arrangements, taking into consideration the net effect of all the possible geometrical variations and inaccuracies to obtain a clinically acceptable and specified probability that the prescribed dose is absorbed in the CTV. 13 Gordon and Siebers reported the use of the DM, which extended the concept of the CTV-to-PTV margin. 14 The DM by Gordon was defined as the

| CONCLUSION
A novel scheme for the direct estimation of the BDM from the PD distribution was developed in this study. The setup error was taken into account for the DM used in this study. The effects of the DPF and α=β were involved into the BCF which provided the direct conversion from the PDF to BDM. This scheme is applicable for the various prescribed doses and fractionations. It is also possible to replace the BCF by replacing the LQ model by some other biological model.
The BCF model is useful for evaluating the BED coverage to the target volume, which plays an equivalent role of the BED-based treatment planning of SBRT in the current PD-based treatment planning system.

CONFLI CT OF INTEREST
None.