Introduction
Dose-volume histograms (DVHs) are used to evaluate treatment planning. Dose-volume constraints indicate that organs should not receive doses exceeding certain limits. Radiobiological studies have exhibited the same biological effect by providing a biologically effective dose (BED) through different fractionation schemes or dose per fraction [1–3]. Fowler showed that BED modeling is useful for understanding normal tissue and tumor responses across different fractionation schemes and treatment methods [4].
The relative biological effectiveness depends on the dose level, which is the dose per fraction (DPF) and the number of dose fractions. Our previous study proposed a dosimetric margin (DM) that considers dose perturbation based on the setup uncertainties. Moreover, biological effectiveness was incorporated in the DM and the biological conversion factor (BCF) between the physical DM (PDM), and biological DM (BDM) was proposed. We evaluated the PDM and BDM with regard to lung stereotactic body radiotherapy (SBRT). Our previous study showed that the size of the planning target volume (PTV) margin negligibly affected the BDM and PDM. However, the effect of the PTV margin was not included in this study. Moreover, the dose perturbation is different for different density materials, energies, dose calculation algorithms, and interplay effects. The penumbra width increases for high-energy photon beams [5]. Lateral scattered electrons generated by the beam path increase, and the penumbra width increases when low-density materials are present [6–8]. Dose build-up and rebuild-down effects occur in the peripheral tumor region around the low-density materials. It decreases with the small field size [5]. For the intensity-modulated radiotherapy (IMRT) treatment, the interplay effect occurs at hot or cold dose spots [9, 10]. Consequently, these dose perturbations may affect the BDM distribution and BCF model.
In the current study, we evaluated the PDM and BDM involving dose perturbation, which focused on the setup uncertainty in homogeneous and nonhomogeneous regions. An improved BCF model corresponding to the PTV margin was proposed.
Materials and methods
An anthropomorphic phantom, RANDO (The Phantom Laboratory, Salem, NY), was scanned using computed tomography (CT) (LightSpeed RT16, GE Healthcare, Little Chalfont, UK). The tube voltage was 120 kVp, and the slice interval and thickness were 2.5 mm. The scanned CT images were transferred to the treatment planning system — RayStation ver. 6.0 (RaySearch Medical Laboratories AB, Stockholm, Sweden). In treatment planning, a linear accelerator was used — TrueBeam (Varian Medical Systems, Palo Alta, CA, USA) — with 6-MV flattening filter-free beams. Based on the RANDO phantom, virtual lung and liver phantoms were created for treatment planning (Figs. 1 and 2). The electron density of the virtual sphere tumor was 1.0 g/cm3. The radius of the virtual sphere tumor was 1.0 cm in the homogeneous and nonhomogeneous phantoms. The electron densities of the virtual lung and liver were 0.3 and 1.0 g/cm3, respectively. The virtual tumor region was defined as the gross tumor volume (GTV) = CTV. A PTV margin was added 5–20 mm around the CTV. A volumetric modulated arc therapy (VMAT) plan was created using noncoplanar angles. The gantry angle was set counterclockwise at 0°–180°. The collimator and couch angles were fixed at 10° and 0°, respectively. A dose of 24–66 Gy was prescribed for D95% of the PTV.
Figure 3 shows the scheme for calculating the BDM and PDM which were obtained from our previous study [11]. The BED was converted from the physical dose using the linear quadratic (LQ) model (Step 1).
(1)
where n is the fractionation and d is the DPF. The LQ model fits the cell-surviving fraction through a second-order polynomial on the DPF with coefficients α/β, which shows the repair capacity of the cells. The DPF and α/βmainly affected the BDM. In this study, α/β was set to 3 Gy for the normal tissue and 10 Gy for the tumor. Dose and fraction schemes differed for each clinical trial. In the current study, the DPF was in the range of 3–20 Gy, referring to the previous study of lung SBRT [12–19] and liver SBRT [20].
The biological and physical dose distributions considering the setup uncertainty were calculated in the RayStation treatment planning system. The isocenter was shifted from –30 to 30 mm along the left–right (LR) direction (DL and DR), –30 to 30 mm along the anterior–posterior (AP) direction (DA and DP), and –30 to 30 mm along the cranio-caudal (CC) direction (DCr and DCa) (Step 2). The treated volume (TV) and DM proposed in the previous study are shown in Fig. 4. In the current study, TV is defined as the volume that satisfies
(2)
where DXROI is the dose to the region of interest (ROI) and X is 95% of the prescribed physical and biological doses by shifting the isocenter. DRx denotes the prescribed dose. Subsequently, the maximum shifts toward the LR, AP, and CC directions (DL, DR, DA, DP, DCr , and DCa) that followed the passed criteria.
(3)
The DM by shifting the isocenter was calculated as
(4)
(5)
(6)
The anisotropic physical DM (PDM) and biological DM (BDM) were calculated from TV and CTV (Step 3), respectively.
The BCF model was calculated as follows (Step 4):
(7)
The BCF was fitted using d/(α/β). The fitting equation was modified from that of our previous fitting model. This was because the PTV margin had a constant value in the previous study. The fitting model was expanded to fit the various sizes of the PTV margins in the homogeneous and nonhomogeneous tumor regions.
(8)
A, B, C, and D are the optimized fitting parameters obtained using the least squares method. M shows the PTV margin from 5 to 20 mm. The fitting equation was determined by calculating the BDM.
Results
Physical and biological DMs in a virtual nonhomogeneous phantom
Figure 5 shows the PDM and BDM at a DPF of 3–20 Gy along the LR, AP, and CC directions with an isotropic PTV margin of 5–20 mm in the nonhomogeneous phantom. The maximum difference between the directions was 0.6, 1.9, 3.1, and 5.3 mm with PTV margins of 5, 10, 15, and 20 mm, respectively. Figure 6 shows the BCF at a DPF of 3–20 Gy along the LR, AP, and CC directions. The difference in the BCF was within 0.01 with a PTV margin of 5 mm, 0.02 with a PTV margin of 10–15 mm, and 0.03 with a PTV margin of 20 mm. Figure 7A shows the average BCF along the LR, AP, and CC directions with a PTV margin of 5–20 mm. There was no significant difference in the data with a PTV margin of 10–20 mm. Thus, the BCF was fitted after combining the data with a PTV margin of 10–20 mm. Thus, M was 5 for a PTV margin of 5 mm and 10 for a PTV margin of 10–20 mm. The resulting fitting parameters of the BCF are shown in Figure 7B (the parameters are listed in Tab. 1).
|
Parameter |
Value |
SD |
|
A |
–4.1e–03 |
2.1e–04 |
|
B |
–3.5e–03 |
1.1e–04 |
|
C |
2.5e–03 |
1.6e–03 |
|
D |
8.0e–02 |
1.0e–03 |
Physical and biological DMs in a virtual homogeneous phantom
Figure 8 shows the PDM and BDM at a DPF of 3–20 Gy along the LR, AP, and CC directions with an isotropic PTV margin of 5–20 mm in a homogeneous phantom. The maximum difference between the directions was 1.15, 1.8, 4.5, and 5.7 mm with PTV margins of 5, 10, 15, and 20 mm, respectively. Fig. 9 shows the BCF at a DPF of 3–20 Gy along the LR, AP, and CC directions. The difference in the BCF was within 0.03 with a PTV margin of 5 mm, and 0.05 with a PTV margin of 10–20 mm. Figure 10 shows the average BCF along the LR, AP, and CC directions and the fitted BCF curve with a PTV margin of 5–20 mm. The fitting parameters for the BCF are listed in Table 2.
|
Parameter |
Value |
SD |
|
A |
8.7e–03 |
8.1e–04 |
|
B |
–1.3e–02 |
1.1e–03 |
|
C |
–0.1 |
1.0e–02 |
|
D |
2.8e–03 |
1.3e–04 |
Discussion
The PTV margin is a geometrical concept that incorporates geometrical variations and inaccuracies [21]. Gordon proposed a DM that considers the relation between the PTV and dose distribution. In the previous study, the BED was used to compare the biological effects of the prescribed dose [4, 12–19]. Biological damage depends on the DPF. The current study evaluated the BDM and BCF, which are the conversion factors of PDM to BDM in homogeneous and nonhomogeneous tumor regions.
In the current study, the BDM was evaluated in the virtual homogeneous and nonhomogeneous phantoms. Moreover, the BCF fitting model expanded with the size of the PTV margin in the homogeneous and nonhomogeneous regions. The BDM was smaller than the PDM in the LR, AP, and CC directions for the DPFs in the homogeneous and nonhomogeneous tumor regions. In the nonhomogeneous region, the difference in the BCF with a PTV margin of 10–20 mm was smaller, and these exhibited a larger difference with the BCF with a PTV margin of 5 mm. The tumor scattering decreased with an increase in the distance from the tumor. For a PTV margin of 5 mm, the relation between the tumor and scattering changed owing to perturbation. For a PTV margin of 10–20 mm, the ratio of the tumor to the PTV was small, and it had a negligible effect on the tumor scattering in the stationary phase without perturbation.
The PDM can be calculated using the geometric PTV margin obtained through commercial treatment planning in the clinical process. Then, the optimal BDM can be derived considering the size of the PTV margin for the BED in the homogeneous and nonhomogeneous regions when the DPF is changed.
The BED calculation using the LQ model did not correspond to the repair and proliferation of the tumors. A recent study showed that the LQ model underestimates tumor control performed by SBRT and stereotactic radiosurgery (SRS) because it does not consider indirect cell death [22–24]. Our study incorporated the BED into the margin. This BED model can be applied to other tumor types and biological models [25, 26] using the proposed procedure.
The radiobiological model has been improved to fit hypofractionation or other radiotherapy irradiation techniques. In the future, the BED should be used to compare different dose fractionation schemes and improve the margin model by evaluating the BED distribution.
Conclusion
The current study helped improve the estimation of the BDM based on the physical dose distribution and size of the PTV margin. Moreover, the improved BCF model can be useful to calculate the BED coverage of the target tumor volume in homogeneous and nonhomogeneous regions.
Conflict of interest
None declared.
Funding
None declared.