Analytical Stopping Power and Range Parameterization for Therapeutic Energy Intervals
The literature on stopping powers and ranges for proton and heavier ions is vast. It includes measurements, theory, evaluations and tabulations, and software programs and routines. Electronic stopping powers are typically calculated with either the Bethe-Bloch Equation or empirically derived functions, depending on the energy regime of the ion. Nuclear stopping powers are typically calculated using a theoretical approach. Ranges are commonly calculated numerically using path length integration, stopping power data and the continuous slowing down approximation. The Stopping and Ranges of Ions in Matter (SRIM) by Ziegler et al., which provides these and other capabilities, exemplifies the integration of highly-realistic theories, evaluated data, and radiation transport capabilities. In applications where speed and simplicity are paramount considerations, analytical calculations of range offer considerable advantages over numerical methods. Range has long been calculated analytically using a simple power law known as the range-energy relationship. This provides excellent accuracy over a surprisingly wide interval of ion species (e.g. protons, carbon), ion energies (e.g. 10-200 MeV/u), absorber compositions (e.g. plastic, water, metal), and thickness (e.g. 0.5-1 10 mm). The major limitation of formulas derived from a simple power law relation was their limited interval of applicability, 1.e., 10-200 MeV for protons in water. Thus, fora projectile in a stopping length target, such as a cancer patient, the simple power law model breaks down in the biologically important interval below 10 MeV. In current standard-of-care treatment planning, this is typically not a concern because the resulting range errors are less than 1 mm (and can be accounted for) and because biological effects are not calculated based on ion stopping power. However, there is considerable interest to perform multi-scale calculations of ion beams, e.g., at dimensions corresponding to cellular DNA, with ion energies below 10 MeV. Thus, there is an increasing need for new fast and simple formulae to calculate range and stopping power across a broader energy intervals of ion energy. It was not known if simple analytical equations would meet this need and, if so, the limits of their applicability. The aim of this work was to develop a simple, non-piecewise continuous model of total stopping power which could be analytically integrated to provide a simple formula for range. More specifically, we sought to develop a model to provide stopping powers with accuracy of better than 20% and range errors less than 1 mm for hydrogen, carbon, iron, and uranium projectile ions and target materials including water, carbon, aluminum, lead, copper, iron, gadolinium, and gold. (authors)