Basic features of the models
The variations of
α and
β parameters for proton irradiation were shown as a function of LET for the two sets of tissue parameters and for the three models (see Fig.
1). The parameter
α increases, as expected, with increasing LET in the clinically relevant range for protons, according to all the three models as shown in the left panels of Fig.
1. The LEM and the WED models show
α to approach
α
x
at very low LET, whereas the CAR model assumes
α/α
x
= 0.834 for LET → 0 keV/μm. Nevertheless, for (
α/
β)
x
= 2 Gy all models predict quite similar
α values up to approximately 8 keV/μm. Beyond this value, the LEM exhibits an enhancement in
α, as described also in [
14], that the WED and CAR models do not show. For (
α/
β)
x
= 10 Gy
α increases more slowly but, while the LEM and the WED model show again similar trends, the CAR model predicts a smaller
α, that is lower than
α
x
for LET values up to approximately 4 keV/μm. It should be noted that the CAR model has been fitted to experimental data for V79 cells ((
α/
β)
x
≈ 2.7 Gy) and, as a result, it is supposed to be more reliable for a low (
α/
β)
x
value than for a high one. No relevant differences for different
D
t
values are apparent up to 15 keV/μm, for both tissues. Moreover, as discussed in [
14], the LEM predicts a vanishing slope for the RBE-LET dependence in the limit of LET → 0 keV/μm, as shown in Fig.
2 of this work.
Trends of the
β parameter are quite different for the three models (see right panels in Fig.
1). The CAR model predicts
β to slowly increase as the LET increases, with a slope that decreases with increasing (
α/
β)
x
ratio. According to the WED model the parameter
β is constant and equal to
β
x
, whereas in the LEM framework
β decreases with increasing LET for the applied LEM implementation. An improved description of the
β term within the LEM framework can be found in [
12]. Moreover, while the LEM predicts
β to approach
β
x
for very low LET values, the CAR model assumes
β/β
x
= 1.09 for LET → 0 keV/μm. Understandably, variations of
D
t
affect much more
β than
α and are more noticeable for high LET values, since
D
t
is expected to become more important with increasing dose.
Figure
2 shows RBE predictions as a function of the LET for the two considered tissues at two different photon dose levels: 2 and 4 Gy. As expected, the RBE increases for increasing LET in the LET range analyzed, decreases for increasing dose, and increases for decreasing (
α/
β)
x
of the tissue.
Despite different trends found for
β in case of (
α/
β)
x
= 2 Gy (see upper panels in Fig.
2), the three models predict similar RBE values for LET values up to 8 keV/μm for both dose values (mainly due to compensation effects between
α and
β), whereas beyond 8–10 keV/μm the LEM shows a RBE prediction higher than the other models (due to the enhancement of
α). The RBE is equal to 1.1 approximately at 2.5 keV/μm for the LEM and between 1 and 1.5 keV/μm for the other models at a dose level of 2 Gy. However, at a dose level of 4 Gy, the RBE reaches the value of 1.1 approximately at 4 keV/μm according to the LEM, at 1 keV/μm for the CAR model, and at 2 keV/μm for the WED model. According to the CAR model, at
D
x
= 4 Gy the RBE does not approach unity for very low LET values due to the parameter
β being higher than
β
x
in the low LET region (and becoming more important at higher doses). For (
α/
β)
x
= 10 Gy, as shown in the lower panels in Fig.
2, the LEM and the WED model show a similar trend to each other, while the CAR model predicts the RBE to be smaller than one for low LET values. This is a consequence of
α being lower than
α
x
in the low LET region. The effect is reduced at
D
x
= 4 Gy because of the reduced importance of
α. The RBE is equal to 1.1 approximately at 3 keV/μm for the WED model, at 4.5 keV/μm for the LEM, and at 6 keV/μm for the CAR model at a dose level of 2 Gy. Conversely, at a dose level of 4 Gy, the RBE achieves the value of 1.1 approximately at 4 keV/μm according to the WED model, at 5 keV/μm for the LEM, and at 6 keV/μm for the CAR model. The RBE predictions by the LEM model when varying
D
t
differ beyond approximately 5 keV/μm. The difference increases as LET and dose increase.
Figure
3 shows predictions of RBE as a function of the proton dose by the three models for low LET (1 keV/μm) and high LET (6.5 keV/μm). The LEM and the WED model exhibit similar trends. The RBE increases with decreasing dose with only an exception at low LET and high (
α/
β)
x
ratio. For (
α/
β)
x
= 2 Gy (upper panels in Fig.
3), and low LET, according to the LEM, the RBE is between 1 and 1.1 at any dose level, whereas it is higher than 1.1 for dose values smaller than ≈ 1.2 Gy according to the WED model. At high LET, the RBE is higher than 1.1 in the studied dose range for all the three models, and reaches values between 2 and 2.3 at very low dose values. For (
α/
β)
x
= 10 Gy (bottom panels in Fig.
3), at low LET, the RBE increases slightly with decreasing dose according to the WED model, whereas it is almost constant in the studied dose range according to the LEM. At high LET, the RBE is higher than 1.1 in the whole dose range and reaches values between 1.2 and 1.3 at very low doses. Variations in the RBE predictions by the LEM when changing
D
t
are apparent only for high LET and at high dose. The CAR model predicts that for high LET the RBE decreases as the dose increases for (α/β)x = 2 Gy while for (α/β)x = 10 Gy it remains nearly constant. Moreover, the CAR model predicts the RBE to increase with increasing dose, at least for low LET and in particular for (
α/
β)
x
= 10 Gy. This is due to the fact that RBE
max can be lower than RBE
min under certain conditions. In fact, expressing Eq. (
4) as a function of the proton dose instead of the photon dose (see for example [
9]):
$$ \mathrm{R}\mathrm{B}\mathrm{E}\left[{\left(\alpha /\beta \right)}_x,D,{\mathrm{RBE}}_{\max },{\mathrm{RBE}}_{\min}\right]=-\frac{1}{2D}{\left(\frac{\alpha }{\beta}\right)}_x+\frac{1}{D}\sqrt{\frac{1}{4}{\left(\frac{\alpha }{\beta}\right)}_x^2+RB{E}_{\max }{\left(\frac{\alpha }{\beta}\right)}_xD+RB{E}_{\min}^2{D}^2}, $$
its derivative
1 with respect to the proton dose results to be negative only if RBE
max > RBE
min:
$$ \frac{\mathrm{d}}{\mathrm{d}\kern0.1em D}\mathrm{R}\mathrm{B}\mathrm{E}\left[{\left(\alpha /\beta \right)}_x,D,{\mathrm{RBE}}_{\max },{\mathrm{RBE}}_{\min}\right]<0\kern1em \Rightarrow \kern1em {\mathrm{RBE}}_{\max }>{\mathrm{RBE}}_{\min }, $$
namely:
$$ \Rightarrow \kern1em \frac{{\mathrm{LET}}_D}{{\left(\alpha /\beta \right)}_x}>0.62\;\left[\frac{\mathrm{keV}/\upmu \mathrm{m}}{\mathrm{Gy}}\right]\;. $$
(14)
If Eq. (
14) is not fulfilled the CAR model should be considered inapplicable. As a consequence, for the tissues studied in this work, the CAR model can only be considered applicable if:
$$ \mathrm{LET}>1.24\;\left[\mathrm{keV}/\upmu \mathrm{m}\right]\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {\left(\alpha /\beta \right)}_x=2\;\mathrm{Gy}, $$
(15)
$$ \mathrm{LET}>6.20\;\left[\mathrm{keV}/\upmu \mathrm{m}\right]\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {\left(\alpha /\beta \right)}_x=10\;\mathrm{Gy}\;. $$
(16)
For this reason, the CAR model has not been taken into account for the considered clinical investigations with (
α/
β)
x
= 10 Gy. However, the results presented in [
25] suggest that the condition RBE
max > RBE
min could, in certain cases, not be fulfilled, i.e., RBE at low doses of low LET particles (e.g., low LET carbon ions in the Karger and collaborators’ paper) is lower than the RBE for high doses of the same particles with the same LET. Additional experimental investigations are needed for further understanding this scenario and in general the capabilities and the limits of the models analyzed.
Model dependencies in a SOBP
D
RBE and RBE profiles as function of depth calculated with the three biological models for the (
α/
β)
x
= 2 Gy tissue and assuming an RBE of 1.1 are depicted in Fig.
4 (left panels) together with the dose and LET
D
values (upper-left panel).
The CAR and WED models produce similar D
RBE values always higher than the clinically assumed one (with a RBE of 1.1). The CAR/WED RBE values in the entrance and in the middle of the SOBP are, respectively, 1.15/1.18 and 1.27/1.26. The LEM D
RBE prediction assuming D
t
= 10 Gy is close to the D
RBE values with RBE = 1.1 in the entrance channel, while it increases as function of depth in the high dose region of the SOBP, eventually exceeding the CAR and the WED predictions in the last millimeter of the SOBP. Applying higher D
t
values produces an enhancement of the D
RBE at the depths with higher LET
D
values. LEM RBE values in the entrance and in the middle of the SOBP are respectively, 1.07/1.1 and 1.21/1.29 for D
t
= 10 Gy / 40 Gy.
Lateral
D
RBE and RBE profiles in the middle of the SOBP for the three biological models for the (
α/
β)
x
= 2 Gy tissue are depicted in the right panels of Fig.
4, together with the LET
D
values.
Analyzing the lateral D
RBE profiles in terms of 80 – 20 % fall-off we have found the following values: 13.3 mm with RBE 1.1, 13.9 mm for LEM-“TPS” approximation, 14.0 mm for the WED model, 13.9 mm for the CAR model and 14.1/14.7 for the LEM with D
t
= 10 Gy / 40 Gy. Hence, a widening of the field in terms of DRBE of roughly 1.0 mm in comparison to assuming a constant RBE of 1.1 has been found independent of the model used (CAR/WED/LEM). Properly taking into account the variation of the mixed radiation field (secondary charged particles produced in nuclear reactions) not only as a function of depth, but also laterally, results in an increase of the RBE in the low dose region (comparing LEM and LEM-“TPS” like predictions) and an increase in the lateral fall-off of about 0.2 mm. This region corresponds to higher LET
D
values compared to the central part of the field, due to the primary protons and the secondary higher LET particles stopping.
Model dependencies in patient cases
The aim of proton treatment planning is to deliver a dose as uniform as possible to the target, sparing healthy tissues (see Fig.
5). Uniform dose distributions do not ensure a homogeneous LET
D
distribution. Moreover, equivalent dose distributions do not necessarily correspond to equivalent LET
D
distributions. High LET
D
regions can be distributed in a complex way in a patient geometry, especially if more than one field is applied. Representative slices of LET
D
distributions and the LET
D
VH of the two patients are shown in Fig.
5.
For patient 1, there is a region of intermediate LET
D
values (2.5–5 keV/μm), covering almost the whole PTV (95 %) and in the remaining volume values up to 8.1 keV/μm are observed. In the region posterior to the target (with respect to the beam direction) the LET
D
is between 8 and 12 keV/μm.
For patient 2, LET
D
is between 3 and 4.5 keV/μm in almost the entire PTV (95 %), and it is up to 6.5 keV/μm in the remaining volume. For optic chiasma and nerves, which are partially included in the PTV, LET
D
values do not exceed 5 keV/μm. Higher LET
D
spots are located in tissues surrounding the PTV, e.g., 5 % of the brain stem volume exhibits LET
D
values beyond 5 keV/μm and a maximum LET
D
of 7.4 keV/μm. LET
D
values found in this study are consistent with the values found in [
26].
Taking into account LET
D
conditions on the LET values for the CAR model reported above, one can conclude that the CAR approach can be applied for RBE/D
RBE calculations only for (α/β)
x
= 2 Gy in our case.
In Figs.
6 and
7, RBE distributions obtained from the three (two) models for (
α/
β)
x
= 2(10) Gy are shown, respectively, for the two patients.
For patient 1, the three models predict the RBE to be between 1.1(1.05) and 1.9(1.5) in the PTV for (
α/
β)
x
= 2(10) Gy, respectively (see Table
2).
For patient 2, the three models predict the RBE to be between 1.2 and 1.6 in the PTV (see Table
2). High RBE spots correspond, as expected, to high LET
D
values. For (
α/
β)
x
= 10 Gy, the RBE varies more slowly with increasing LET. These results are in line with the values found in a previous publication [
7]. There, the authors found RBE values between 1.1 and 1.2 within the clinical target volume in a multiple field hypopharinx case ((
α/
β)
x
≈ 10 Gy) and higher values in the spinal cord ((
α/
β)
x
≈ 2 Gy). Moreover, our results are in agreement with values found by Gerweck and Kozin in [
27] for (
α/
β)
x
≈ 7–13 Gy in cell survival experiments.
Representative D
RBEVHs are shown in Fig.
8 for the two patient cases. For patient 1 in case of (α/β)
x = 2 Gy, when applying RBE = 1.1, a
D
RBE between 1.9 and 2.2 Gy (RBE) is obtained in the PTV. Conversely, applying the three models yields
D
RBE values between 2.0 and 2.7 Gy (RBE) (see Table
3). CAR, WED and LEM with
D
t
= 40 Gy give similar mean
D
RBE values, while LEM with
D
t
= 10 Gy estimates an about 6 % lower mean
D
RBE value inside the PTV.
D
RBE predictions for patient 1 for (
α/
β)
x
= 10 Gy by all the models are consistent (within about 3 % looking at mean
D
RBE) and similar to the
D
RBE obtained with a fixed RBE of 1.1. Applying the LEM with
D
t
= 10 Gy or 40 Gy seems to have a low impact on D
RBEVH PTV for (
α/
β)
x
= 10 Gy (see Table
3).
For patient 2, larger variations between fixed and variable
D
RBE have been found for (
α/
β)
x
= 2 Gy with respect to (
α/
β)
x
= 10 Gy, especially within the PTV. The PTV exhibits
D
RBE values between 1.9 and 2.2 Gy (RBE) for RBE = 1.1 and between 2.0 and 2.7 Gy (RBE) for variable RBE, in the case of (
α/
β)
x
= 2 Gy (see Table
3). For the brain stem, with the (
α/
β)
x
= 2 Gy tissue, the
D
RBE,5% values range from 0.9 Gy (RBE) applying RBE = 1.1 up to 1.2 Gy (RBE) for variable RBE.
In summary,
D
RBE predictions by the three considered models are often higher than the values used in clinics with a fixed RBE of 1.1, especially for high LET
D
areas and low (
α/
β)
x
ratios. Variations exceeding 10 % of the prescription dose were found within the PTV. Since 5 % dose variations can produce 10–20 % variations in tumor control probability and 20–30 % variations in normal tissues complication probability [
28], differences found in this study can be clinically significant. However, it should be kept in mind that lower RBE values are typically found in vivo compared to the in vitro ones, which are inherently affecting our model calculations.
It is interesting to notice that similar variations were also observed when using the more recent model of McNamara et al. [
29], which draws on very similar assumptions as the WED model for the α term, but on a more recent re-evaluation of in-vitro proton experiments reported in [
3]. In this case, agreement within about 6 % in terms of mean D
RBE in the PTV was observed with the other LET
D-based models.
In terms of range variations, Fig.
9 shows, as an example, biological range shift values reported as histograms for patient 1, when using the (
α/
β)
x
= 2 Gy tissue. Mean biological range shifts for the two patients and the two tissues are reported in Table
4. Carabe and collaborators found similar range shift values (2–3 mm) for a (
α/
β)
x
approximately equal to 2 Gy, applying the CAR model to a patient case [
4]. Moreover, increasing the
D
t
value for LEM produces on average larger biological range shifts. All presented results suggest caution in proton therapy treatment planning, especially if OARs are close to the target. Usually, safety margins are applied to take into account range uncertainties and to ensure the target dose coverage.
The observed findings suggest that, regardless of the used biological model, a biological range shift margin of few millimeters distal to the target volume (with respect to the chosen beam direction) should be taken into account when designing the treatment [
30]. Since safety margins are calculated with different procedures at each facility [
31], findings comparing the outcome of different models, as performed in this study, may help developing some general recommendations for medical physicists during treatment planning. For example, opposite beams arrangements, when clinically available, should be preferred to single field/orthogonal ones to reduce the resulting LET
D
and RBE values, as shown comparing the patient cases in this work. Moreover, active beam scanning techniques should be further exploited to push LET
D
areas away from OARs, while preserving dose coverage in the PTV. The PTV definition should take into account biological range uncertainties. In addition to obvious depth dependences of LET variations, also lateral variations due to scattering effects should not be neglected and could become important at the tumor edges, as more clearly shown by the SOBP study in water (Fig.
4 - right panel).