Il picco di Bragg G. Battistoni INFN Milano 08/06/015 G. Battistoni 1
Φ(z) The physics of Bragg Peak 180 MeV proton in water Longitudinal profile: Transversel profile: Φ(z,x) dominated by interaction with electrons MCS, Energy loss fluctuations and nuclear interactions do affect the shape Elastic nuclear scattering Non-elastic nuclear reactions move dose from the peak upstream. Pencil Beam (cm)
Anatomy of the Bragg Peak (Proton Therapy) overall shape from increase of de/dx as proton slows 1/r and transverse size set peak to entrance ratio nuclear buildup or low energy contamination width from range straggling and beam energy spread nuclear reactions take away from the peak and add to this region depth from beam energy
Nuclear Reaction in Proton Therapy about 1% cm -1 H O of the protons undergo nuclear interactions - about 0% in a typical treatment plan - 60% of the energy is deposited locally by charged fragments - 40% in n and γ out of the field F. Tommasino & M. Durante Cancers, 015, 6, 353-381 Does target fragmentation play a role which has been neglected so far?
in Words 1. The increase of de/dx as the proton slows down causes the overall upwards sweep.. The depth of penetration increases with beam energy. 3. The width of the peak is the quadratic sum of range straggling and beam energy spread. 4. The overall shape depends on the beam s transverse size. 5. Non-elastic nuclear reactions move dose from the peak upstream. 6. A short effective source distance reduces the peak/entrance ratio. 7. Low energy beam contamination (as from collimator scatter) may affect the entrance region.
Effect of Nonelastic Nuclear Reactions This figure is a Monte Carlo calculation by Martin Berger (NISTIR 56 (1993)). Dashed line: nuclear reactions switched off; solid line: actual BP. Buildup, not usually a problem because of the likely presence of buildup material near the patient, is ignored. Dose from the EM peak shifts upstream, lowering the peak and flattening the entrance region, especially at high proton energies.
Accurate MC calculations: playing with a proton beam Dose vs depth energy deposition in water for a 00 MeV p beam with various approximations for the physical processes taken into account 08/06/015 G. Battistoni 7
Dose vs depth energy deposition in water for a 14 MeV real p beam under various conditions. Exp. Data from PSI Playing with a proton beam II part 08/06/015 G. Battistoni 8
Close-up of the dose vs depth distribution for 70 MeV/n 1 C ions on a water phantom. The green line is the FLUKA prediction with the nominal 0.15% energy spread The dotted light blue line is the prediction for no spread, and the dashed blue one the prediction for I increased by 1 ev Bragg peaks vs exp. data: 1 C @ 70 MeV/n Exp. Data Jpn.J.Med.Phys. 18, 1,1998 08/06/015 G. Battistoni 9
Straggling dependence on mass 08/06/015 G. Battistoni 10
Charged track scatter also on nucleus!! The charged track is also scattered by the field of the nucleus. The process is described by the Rutherford cross section: dσ R dω = zze pv 1 4sin 4 ϑ ze, m v Ze, M The small angle deflections are more likely è 1/sin 4 (θ/) and the scattering is elastic. Crossing a finite Dx of material single scattering are not likely: the tracks undergoes multiple little scatterings. The total effect can be derived statistically (Moliere theory) v θ
Coulomb collisions: Molìere cross section The Rutheford cross section can be written as: dσ dω Ruth = z Z 4β p re me c sin 4 = θ z Z 4 4β E 4 re me c sin 4 = θ 4 z Z re me c 4 β E (1 cosθ ) Molìere (in the 30 s) has derived an approximate expression for the screened cross section: dσ Mol dω = dσ Ruth dω K scr(θ, β) = dσ Ruth dω (1 cosθ) (1 cosθ + 1 χ a ) = = z Z r e m e c 4 4β 4 E sin 4 θ (1 cosθ) (1 cosθ + 1 χ a ) = z Z r e m e c 4 β 4 E (1 cosθ) (1 cosθ) (1 cosθ + 1 χ a ) 1
Coulomb collisions: Molìere theory In the small angle limit: dσ dσ Mol Mol dω dω small small 4z Z r = 4 β E me c 4 θ 4 θ 4 ( θ + χ ) dω The expression for the screening angle Χ a as computed by Moliere is: e 4 a small = πθ dθ where Χ cc, b c are the only material dependent quantities, given by: Where n is the number of atoms per unit volume, z and Z are the atomic numbers of projectile and target, ξ e is a parameter taking into account the scattering on atomic electrons 1 3 αz χa = 0.855 1 β β α z Z 1.13 + 3.76 β 13 1
Coulomb collisions: Molìere cross section The Molìere cross section can be integrated exactly, giving: σ Mol = r e z Z( Z + ξe) me c 4 β E 4 4π χ 1+ a 1 1 χ 4 a T(MeV) Χ a (mrad) σ Mol (kb) R (g/cm ) Σ -1 (g/cm ) 0.1 38 900 0.0187 4.98x10-5 Al 1.0 8.4 346 0.555 1.30x10-4 10.0 1.14 308 5.86 1.46x10-4 0.1 317 517 0.0311 6.66x10-4 Pb 1.0 35 784 0.784 4.39x10-4 10.0 4.5 793 6.13 4.34x10-4 14
Molìere: example Transmitted (forward) and backscattered (backward) electron angular distributions for 1.75 MeV electrons on a 0.364 g/cm thick Copper foil Measured (dots) and simulated (histos, Molìere like algorithm) data 15
Multiple scattering: gaussian approximation Due to MS the charged particles wiggles along its path Δx through the material. The deflection angle is approximately gaussian f (ϑ ) = 1 exp ϑ πϑ 0 ϑ 0 ϑ 0 = 13.6MeV βcp z Δx X 0 1+ 0.038ln Δx X 0 X 0 is the same of photons: 7/9 of the mean free path for pair production by a high-energy photon. θ 0 is smaller for heavy and/or energetic particles. Electrons of MeV kinetic energy suffers a lot of multiple scattering
Courtesy by M.Durante
Courtesy by M.Durante
Ion Therapy: the lateral scattering 1 C @ 99.94 MeV/u K. Parodi et al Journal of Radiation Research, 013, 54, i91 i96 Entrance channel Near to Bragg peak Measured lateral distributions with corresponding MC simulations (normalized to the data) for carbon ion 99.94 MeV/u beams in water, sampled at a depth of ~1.5 cm in the entrance channel (left, c) and of ~16.5 cm shortly before the Bragg peak (right, d). The double Gauss fit of the experimental data is also shown in comparison to the
New ion beams for therapy Beam size at the Isocenter MC simulation of the CNAO beamline