Calculation of the biological effective dose for piecewise defined dose-rate fits

Abstract

An algorithmic solution to the biological effective dose (BED) calculation from the Lea-Catcheside formula for a piecewise defined function is presented. Data from patients treated for metastatic thyroid cancer were used to illustrate the solution. The Lea-Catcheside formula for the G-factor of the BED is integrated numerically using a large number of small trapezoidal fits to each integral. The algorithmically calculated BED is compatible with an analytic calculation for a similarly valued exponentially fitted dose-rate plot and is the only resolution for piecewise defined dose-rate functions.

Figures

Figure 1

Dose-rate values in the region of interest calculated from 3D-RD plotted as a function of time (triangles). The dotted line shows the exponential fit and the solid line shows the hybrid trapezoid-exponential fit.

Figure 2

G-factor shown for the hybrid fit (thin line). The dotted line shows the hybrid functional fit to the dose rate, and the crosses (the thick line) show the incremental change in the G-factor (dG∕dt).

Figure 3

G-factor plotted for the simple exponential fit (thin line). The dotted line shows the exponential fit, the thin dotted line represents the theoretical value, and the crosses (the thick line) show the incremental change in the G-factor (dG∕dt).

Figure 4

convergence of the G-factor and the absorbed dose. The thin line shows the convergence of the G-factor, the thick line represents the convergence of the absorbed dose to its final value, and the dotted curve shows the ratio of the convergence of the two quantities.