A semi-empirical model for the therapeutic range shift estimation caused by inhomogeneities in proton beam therapy

Abstract

The purpose of this study was to devise a simple semi-empirical model to estimate the range shift in clinical practices with high-Z inhomogeneity in proton beam. A semi-empirical model utilizing the logarithmic dependence on Z in stopping power from Bohr's classical approach has been developed to calculate the range shift due to the presence of inhomogeneity. Range shift from metallic plates of atomic number Z of various thicknesses were measured in water using a parallel plate ionization chamber and calculated with the FLUKA Monte Carlo code. The proton range shifts for bone and polymethyl methacrylate (PMMA) were estimated using the semi-empirical model and compared with Monte Carlo calculation. The semi-empirical equation to determine range shift and water equivalent thickness is presented. The model predicts a shift of the therapeutic range to within 2.5% accuracy for initial proton energies of 50 to 250 MeV and atomic numbers from 3.3 (effective Z for water) to 82. This equation is independent of beam energy, and thus provides range shift from high-Z materials without the knowledge of proton energy. The proposed method of calculating the therapeutic range shift accurately requires only knowledge of the effective or actual atomic number of the inhomogeneity and the thickness of the inhomogeneity along the beam direction. The model generalizes the range shift calculation for any material based on its effective atomic number, and permits reliable prediction of the range shift for material combinations where no data is currently available. The proposed model can be readily implemented in routine clinical practice for proton range shift estimation and quality assurance on the treatment planning.

Figure 1. Geometry used to define: (a) the range of the protons in water phantom Rw; (b) water equivalent thickness WET, range shift in the presence of inhomogeneity on the surface of water phantom (typical experimental setup to determine WET); and (c) observed range shift Δx in the case of the inhomogeneity inside the water phantom.

Figure 2. Range shift in the presence of the bone material in a water phantom. Monte Carlo simulated dose in the water phantom in the presence of the 4 cm diameter cylinder, imitating the bone. Isodose line corresponding to 90% dose defines the range of protons along the normal to the surface of the beam incidence.

Figure 3. Dependence of the fitting function α(Z), representing the ratio (Sρ)WM of mass stopping power in material to water for various initial energies of protons. The symbols are the calculated values using the tabulated data on mass stopping power. ⁽¹⁴⁾ The curve is the fitted function for α(Z).

Figure 4. Comparison of the measurements (filled symbols), Monte Carlo simulations (open symbols) and results predicted by the model as described by Eq. (4) (line). Physical densities of Al, Ti, Cu, Pb used in the calculations are 2.7, 4.5, 8.96, and 11.4 g/cm³, respectively. Published values for polymethyl methacrylate (PMMA) (mass density 1.19 g/cm³, effective atomic number Zeff=3.3 ⁽¹⁹⁾ ) and bone (mass density 1.85gm/cm3,Zeff=4.9 ⁽¹⁹⁾ ) are used. Proton beam is defined by range of R90=15.84cm in water.

Figure 5. Relative percent difference δ between the semi‐empirical fitting by Eq. (4) and calculations using the data from PSTAR ⁽¹⁴⁾ in Eq. (1).