Parameterized algorithms for quantitative differentials in spectrally equivalent medical diagnostic x-ray beams

Qualitative and quantitative equivalence of spectra transmitted by two different elemental filters require a good match in terms of shape and size over the entire energy range of 0-150 keV used in medical diagnostic radiology. However, the photoelectric absorptions and Compton scattering involved in the interaction of x rays with matter at these relatively low photon energies differ in a nonuniform manner with energy and atomic number. By careful choice of thicknesses for filter materials with an atomic number between 12 and 39, when compared with aluminum, it is possible to obtain transmitted beams of the same shape (quality) but not of the same size (quantity). In this paper, calculations have been carried out for the matching of the shapes and sizes of beams transmitted through specified thicknesses of aluminium filter and spectrally equivalent thicknesses of other filter materials (different from aluminium) using FORTRAN source codes traceable to the American Association of Physics in Medicine (AAPM), College Park, MD, USA. Parametrized algorithms for the evaluation of quantitative differentials (deficit or surplus) in radiation output (namely, photon fluence, exposure, kerma, energy imparted, absorbed dose, and effective dose) from these transmitted spectrally equivalent beams were developed. These differentials range between 1%, and 4% at 1 mm Al filtration and between 8%, and 25% for filtration of 6 mm Al for different filter materials in comparison with aluminum. Also developed were models for factors for converting measures of photon fluence, exposure-area product, (EAP), and kerma-area product (KAP) to risk related quantities such as energy imparted, absorbed dose, and effective dose from the spectrally equivalent beams. The thicknesses of other filter materials that are spectrally equivalent to given thicknesses of aluminum filter were characterized using polynomial functions. The fact that the use of equivalent spectra in radiological practice can provide means of ranking the differentials in radiographic image quality and stochastic risk is discussed.