Abstract |
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Introduction In order to improve reference dosimetry of radiotherapy beams, Monte Carlo (MC) simulation codes are widely used to compute ionization chamber dose response accurately. To achieve this, the accuracy of condensed history (CH) algorithms in MC must be properly assessed. As claimed by Rogers, the most stringest test particle transport algorithm is the so-called “Fano test”, which is based on Fano’s theorem stating that the electron fluence spectrum under the condition of charged particle equilibrium (CPE) is independent of the local variations of mass density. While there are validation basis of electron transport algorithms in literature, it is not the case for proton algorithms. With the advent of delivery technique like pencil-beam scanning and the use of small fields, there is an obvious interest to evaluate dose response accuracy of ionization chamber to proton beams calculated with MC methods. Here we present a hybrid Fano test which is independent of the cross sections of incident particles and implies regeneration technique to ensure CPE condition. The approach consists by simulating virtual particles that have the property to trigger proton motion. We demonstrate that it allows achieving equivalent test as the well-known Fano test. Materials and methods We make use of an extension of the formalism described by Poon et al (Phys. Med. Biol. 50, 681–694 2005). The dose deposited into a cavity of an ion chamber exposed to a parallel photon beam may be determined as follows: Dgas=Kcoll , wall( ́Lρ )wall gas Awall Afl (1) where Dgas is the dose to the gas in the cavity, Kcoll , wall is the collision kerma in the wall material, (́Lρ )wall gas is the (spectrum averaged) restricted mass collision stopping power ratio of the gas cavity to the wall, Awall is the correction for wall attenuation and scattering and Afl corrects for electron fluence perturbation in the cavity. If the wall material is the same as the gas material, Afl is unity. Assuming identical materials for the wall and the gas and assuming that the density effect is not taken into account when computing the stopping powers, (́Lρ )wall gas becomes unity as well. Awall is also unity if photons are regenerated. Under strict Fano conditions (CPE in an infinite medium), the Fano theorem stated in the introduction section can be applied and: Kcoll , wall=(μ́tr ρ )wall ❑ ́E ϕ(1−́g wall ) (2) where ́E is the average photon energy, μ́tr ρ is the spectrum averaged mass energy transfer coefficient in the medium, ϕ is the fluence, and ́gwall is the bremsstrahlung fraction in the wall material. The methodology proposed here uses virtual particles that generate protons. The virtual particle interacts with a rate given by the total mass attenuation coefficient μρ . For each interaction, all the energy of the virtual particle is transferred to a proton, hence μ́tr ρ = ́μρ . Also, monoenergetic virtual particles with energy E are considered. Hence, combining equation (1) and (2) with (́Lρ )wall gas Awall Afl=1 leads to Dgas ϕ = μρ E . It is noteworthy to mention that the average mass stopping power (denoted above by ́L for electrons) is not restricted for protons. The geometry considered is a 10x10 cm² parallel virtual field and a cavity (2x2x0.2mm³) in a water phantom with dimensions large enough to ensure CPE. The transport of protons is being simulated as follows: 1) virtual particles with kinetic energy E and mass attenuation coefficient in water μ; 2) water-like cavity with density divided by 1000; 3) interacting virtual particles trigger protons with kinetic energy E and are regenerated after each interaction point; 4) all secondaries (i.e electrons, positrons) are locally absorbed; 5) Bremsstrahlung interactions with nucleus and shell electrons are ignored as well as nuclear interaction; 6) the kinetic energy of protons is locally absorbed once it falls below a threshold. Under these conditions, the computed cavity dose divided by virtual particle fluence equals μE. The simulations are implemented in GEANT4 and PENELOPE-proton codes. Results The left figure illustrates integral depth-dose distributions achieved with μ=0.02cm²/g. For PENELOPE-proton, agreement between computed and expected values was within 0.1% for all simulated C1 and C2 which both control step-length. The cut-off for hard electromagnetic inelastic interactions (Wcc) was 10 keV. Other values of user-defined inputs that control transport were tested, e.g Wcc=100 keV and C1=C2=0.2. For this configuration, difference from expected value of more than 0.7% was observed. This was attributed to the random-hinge method used to simulate the transport across interfaces. If the step-length is too large (i.e of the same order of magnitude or larger that the cavity size), which is the case for high values of Wcc, an artificial range straggling of the protons is introduced. This result demonstrates the ability of our Fano test to detect potential inconsistencies in simulation mechanics. Discussion and conclusion The method can be implemented in any existing code MC algorithm for proton transport, disregarding the simulation of neutrons. The interaction rate (defined by the attenuation coefficient by analogy with photons) can be completely controlled by the user. Tests showed ability to detect inconsistencies in the simulation mechanics of the algorithm. The MC simulations can be affected by the choice of physics-related user inputs. It is not the objective of the present study to test extensively the dependence of the accuracy with user inputs. Nevertheless, a comprehensive set of tests should be performed before starting to evaluate dose response accuracy of ionization chamber to proton beams computed by MC simulations. Moreover the method described may be used to verify the transport algorithms (CH) of any charged particle. |