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Purpose: The recommended value for the relative biological effectiveness (RBE) of proton beams is currently assumed to be 1.1. However, there is increasing evidence that RBE increases towards the end of proton beam range that may increase the biological effect of proton beam in the distal regions of the dose deposition. Methods: A computational approach is presented for estimating the biological effect of the proton beam. It includes a method for calculating the dose averaged linear energy transfer (LET) along the measured Bragg peak and published LET to RBE conversion routine. To validate the proposed method, we have performed Monte Carlo simulations of the pristine Bragg peak at various beam energies and compared the analysis with the simulated results. A good agreement within 5% is observed between the LET analysis of the modeled Bragg peaks and Monte Carlo simulations. Results: Applying the method to the set of Bragg peaks measured at a proton therapy facility we have estimated LET and RBE values along each Bragg peak. Combining the individual RBE-weighted Bragg peaks with known energy modulation weights we have calculated the RBE-weighted dose in the modulated proton beam. The proposed computational method provides a tool for calculating dose averaged LET along the measured Bragg peak. Conclusions: Combined with a model to convert LET into RBE, this method enables calculation of RBE-weighted dose both in pristine Bragg peak and in modulated beam in proton therapy.

Proton beams offer clinically favorable dose distribution due to their unique energy deposition characteristics in the form of the Bragg peak and finite penetration depth with near-zero dose beyond a certain depth known as range. These unique features provide possibilities for dose escalations to improve radiation outcomes with significant reduction in radiation related toxicities and complications [

The majority of proton therapy facilities have adopted a constant value of 1.1 for the relative biological effectiveness (RBE) of proton beams based on international recommendations [

The experimentally measured RBE dependence on linear energy transfer (LET) suggests that the RBE value increases towards the end of beam range (as the energy decreases) that could increase the biological effect of proton radiation in the distal regions of the dose deposition [

The main focus of this paper is to develop a computational method for RBE based on dose averaged LET (see Equation (2)) values from a measured depth dose data for clinical beam. While for the same treatment range Bragg peak shape may differ between different treatment centers, we do expect that the method presented here is universally applicable for converting measured Bragg peaks into RBE-weighted Bragg peaks. Once the RBE values are calculated along individual Bragg peaks, one can generate RBE-weighted SOBP dose distributions for clinical usage.

Ionization track structures and microdosimetric energy spectra of protons at various depths are typically analyzed to convert LET [

The depth dose curve containing Bragg peak of the proton beam is divided into two regions: the proximal region from the surface to the dose maximum and the distal region from the maximum to near zero dose. For every depth, point x measured from the material surface in the proximal region, we can define the residual range R_{RES} as:

where R_{80} is the proton beam penetration range measured at the distal 80% dose level of the Bragg peak. Using the published energy-range tables [

It is important to note the small disagreement between various range-energy tables. For example, for the same energy of 250 MeV ICRU Report 49 [_{80}, which introduces very small discrepancy in initial beam energy at the water surface but brings the overall agreement of stopping power calculations from various range-energy tables to within 2% as illustrated in

Next we defined a method of calculating LET in the distal region of Bragg peak. Our most general assump- tion about distal region is that the dose averaged LET should be a smooth function of depth. The continuous slowing down approximation (CSDA) does not work beyond the Bragg peak since residual range becomes negative. Furthermore, the proton beam cannot be considered monoenergetic on the distal edge of the Bragg peak. Instead, the beam energy spectrum changes when it reaches energies below 3 MeV (residual range of 0.1 mm) as illustrated in

LET spectral analysis works well in Monte Carlo but is not practical for analysis of clinically measured Bragg peaks. Looking for an alternative method, we have observed in our Monte Carlo simulations that the dose averaged LET exhibits a nearly linear increase with depth beyond the Bragg peak. Therefore, as a first approximation we assume that the dose averaged LET increases linearly with depth beyond the Bragg peak position and the slope of LET growth is defined by LET slope at the maximum dose point on the Bragg peak.

Furthermore, when the physical dose drops below 1% level the biological effect becomes important only at the theoretical level. When the physical dose is negligible, the RBE value becomes irrelevant from a clinical point of view. Therefore, beyond the distal 1% - 2% depth dose point we assume constant LET and RBE values. This prevents magnification of the experimental noise which could be present in the region beyond the Bragg peak.

To validate the proposed model for LET calculation we have carried out Monte Carlo simulation of the dose distribution along depth generated by 1 cm diameter source of mono-energetic proton shitting a water target. The Monte-Carlo calculations were carried out using MCNPx code [^{3}] by the proton beam fluence F [1/cm^{2}]:

We have calculated the depth dose distribution and dose averaged LET for monoenergetic proton beams at three energies 116 MeV, 162 MeV and 201 MeV corresponding to the penetration range R_{80} in water of about 10 cm, 18 cm and 26 cm, respectively. We applied the LET calculation model outlined above to the Bragg peaks simulated with MCNPx to validate the proposed model. Thus, we could compare the LET model calculations to the MCNPx simulation results based on identical Bragg peaks. The comparison was first done for the proximal regions and then for the distal regions of the Bragg peaks.

The most straight forward way to obtain RBE is to use experimental data from radiobiological experiment for a particular cell line. However, taking the data from published experiments [

A more generic phenomenological model proposed by Wilkens and Oelfke [_{p}, the LET, and the tissue specific parameters α and β from a linear-quadratic cell survival parameterization:

where subscripts p and x refer to proton and x-ray linear quadratic parameters, respectively. Analyzing published radiobiological experimental data Wilkens and Oelfke [_{p} on LET could be assumed, while β_{p} is essentially independent of LET and equal to β_{x}:

Note that such parameterization results in α_{p} = α_{x} at the entrance region of 170 MeV proton beam where the LET = 5 MeV/cm. In other words, monoenergetic proton beam has RBE = 1 at the entrance region into the water equivalent media. To estimate RBE variation in different tissues and for different dose fractionation regimes we have compiled a short table (

Finally we computed the RBE-weighted dose along the Bragg peak as a product of measured physical dose at a given depth x, and RBE factor calculated using phenomenological model (Equation (3)) for LET value at the same point on Bragg peak assuming proton dose delivery with 2 Gy or 4 Gy per fraction. In these calculations we used head and neck tissue alpha beta parameters since it represents a typical treatment site in proton therapy. The case of 4 Gy per fraction was calculated to evaluate the effect of hypo-fractionation in proton therapy that has been observed in many publications [

The physical depth dose measurements were taken with a Markus ion chamber in a water phantom while delivering pristine energy proton beams with field sizes exceeding 5 cm in diameter. The measured depth dose curves were smoothed with Gaussian algorithm imbedded into the water phantom data acquisition software to eliminate signal noise associated with beam intensity fluctuations. These measured Bragg peaks were part of the commissioning data set taken in uniform scanning nozzle as described Farr et al. [

For proton beams modulated in depth, we calculated the RBE-weighted dose by combining RBE-weighted dose from individual Bragg peaks scaled according to monitor unit weights of individual energy layers.

Tissue Type | α_{x} (Gy^{−1}) | β_{x} (Gy^{−2}) | α_{x}/β_{x} (Gy) | Ref. Rad. | Reference |
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Chordoma | 0.1 | 0.050 | 2.0 | Co-60 | [ |

Prostate | 0.15 | 0.048 | 3.1 | Co-60 | [ |

Lung | 0.19 | 0.052 | 3.7 | Co-60 | [ |

Head & Neck | 0.25 | 0.025 | 10 | Co-60 | [ |

To validate the proposed model for LET calculation we used Monte Carlo modeling to investigate the LET behavior around the proton Bragg peak.

The Monte Carlo simulation data indicate that LET behavior is nearly identical in the proximal region of the Bragg peak regardless of the incident beam energy. This pattern also agrees qualitatively with the LET predic- tions from the water range-energy tables. However, the stopping power values computed from the water range- energy table are consistently lower by about 7% due to missing contribution from the short range secondary particles created in the nuclear interactions. Similar nuclear buildup of about 6% has been observed experimentally at the entrance region of the Bragg peak.

Including the 7% correction into the model, we applied the LET calculation method to the set of Bragg peaks simulated with MCNPx [

With the method for calculating LET values along the measured Bragg peak, the next step is to calculate RBE based on LET values. We compared the RBE behavior in the range of LET values relevant in proton therapy for different tissues as shown in

RBE variation with dose become substantial and exceeds 10% for the LET values above 40 MeV/cm as shown in

Applying the proposed method to the set of experimentally measured Bragg peaks, we obtained the RBE-weighted absorbed dose as shown in

Our LET calculation approach is based on the observation that for nearly monoenergetic proton beam, dose deposition per unit length on the proximal side of Bragg peak is defined by the proton stopping power and small contribution from secondary charged particles. While the shape of Bragg peak can vary depending on the hardware configuration and beam properties in the treatment nozzle, the LET behavior is the same on the proximal side of the Bragg peak and is defined by the residual range of the proton beam. On the distal side of the Bragg peak, LET slope does depend on beam properties (SAD, energy spread, etc.). However, we found that linear interpolation based on the LET slope calculated at the peak point of the Bragg peak gives reasonable approximation of the LET behavior on the distal side. Applying this approach to clinically measured Bragg peaks can give us dose averaged LET values at each point of the depth dose curve.

In our calculations of RBE-weighted dose, we rely on the phenomenological model proposed by Wilkens and Olfke [

Applying our LET and RBE calculation model to the typical treatment sites in proton therapy, we found that tissue dependence of RBE is negligible. This result matches well with ICRU-78 recommendation of using tissue-independent mean RBE value of 1.1, which is based upon available in-vitro and in-vivo data. However, we suggest using caution since tissue-independence of proton RBE may not be applicable to all treatment sites and certainly may not be applicable to organs at risk adjacent to the treatment volume [

Using the RBE-weighted set of Bragg peaks measured at our center, we have computed RBE-weighted dose distributions in modulated proton beam delivery. In the middle of computed SOBPs the RBE-weighted dose was found in reasonable agreement with a generic RBE value of 1.1 as recommended [

Note that, the SOBP delivery implies superposition of Bragg peaks with different weighting factors. The Bragg peaks in the middle of SOBP deliver smaller dose than distal Bragg peaks to take into account dose already delivered by all those distal Bragg peaks. This may raise a concern about dose dependence of RBE. However, because of this “prior” dose from distal Bragg peaks and since the TOTAL physical dose within the SOBP is constant, we cannot apply dose effect to RBE weighted individual Bragg peaks comprising the SOBP.

Our method also predicts higher RBE at the distal edge of the dose distribution, which can lead to an apparent extension of the physical range by about 1 mm in depth.

Range = 6 cm SOBP = 3.6 cm | Range = 16 cm SOBP = 10 cm | Range = 27 cm SOBP = 12 cm | ||||
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2 Gy | 4 Gy | 2 Gy | 4 Gy | 2 Gy | 4 Gy | |

RBE proximal | 1.104 | 1.069 | 1.045 | 1.026 | 1.037 | 1.025 |

RBE distal | 1.334 | 1.223 | 1.253 | 1.166 | 1.247 | 1.164 |

RBE average | 1.17 | 1.113 | 1.101 | 1.063 | 1.088 | 1.058 |

Range extension | 1 mm | 0.6 mm | 1.3 mm | 0.8 mm | 1.4 mm | 0.9 mm |

of maximizing the weight of the deepest Bragg peak to sharpen the distal fall off in favor of optimizing relative weights of the Bragg peaks to smooth out distal gradient and to compensate for the RBE rise on the distal edge of SOBP.

The analysis of hypo-fractionated proton dose delivery mode with 4 Gy per fraction indicates reduction of the biological effect in the SOBP region by about 7% compared with 2 Gy per fraction dose delivery mode as also noted by Carabe-Fernandez et al. [

A computational model for estimating RBE-weighted dose based upon experimentally measured set of Bragg peaks is developed. The model allows one to calculate the dose averaged LET along the measured Bragg peak so that one can apply an LET to RBE conversion function. This routine can be applied to a set of Bragg peaks measured at any proton therapy facility. Combining the RBE-weighted pristine Bragg peaks according to the known weighting factors, one can obtain the RBE-weighted depth dose distribution in a modulated proton beam. The computed RBE-weighted absorbed dose exceeds physical dose by 5% - 30% over the SOBP region, depending on SOBP modulation width, maximum beam energy and depth along the SOBP.

The biological dose calculations strongly depend on the RBE versus LET relationship. However, we found that proton beam RBE tissue dependence is very small for four typical treatment sites. We also observed that the dose dependence of proton beam RBE has strongest effect mainly at the distal end of SOBP. The proposed method of computing the RBE-weighted dose in the proton beam may be useful in clinical practice for estimating the RBE-weighted dose delivered to the target and surrounding tissues, which could lead to treatment optimization and better definitions of safety margins during the treatment planning.

The authors are grateful to Dr. D. Nichiporov for useful discussions and critique that helped developing the computational method.