Analytical considerations of beam hardening in medical accelerator photon spectra.

Beam hardening is a well-known phenomenon for therapeutic accelerator beams passing through matter in narrow beam geometry. This study assesses quantitatively the magnitude of beam hardening of therapeutic beams in water. A formal concept of beam hardening is proposed which is based on the decrease of the mean attenuation coefficient with depth. On the basis of this concept calculations of beam hardening effects are easily performed by means of a commercial spreadsheet program. Published accelerator spectra and the tabulated values of attenuation coefficients serve as input for these calculations. It is shown that the mean attenuation coefficient starts at depth zero with an almost linear decrease and then slowly levels off to a limit value. A similar behavior is found for the beam hardening coefficient. A physically reasonable, semianalytical model is given which fits the data better than previously published functions. The energy dependence of the initial attenuation coefficient is evaluated and shown. It fits well to published experimental data. The initial beam hardening coefficient, however, shows no energy dependence. Its mean value (eta0) approximately 0.006 cm(-1)) is also in close agreement to the measured data.