title: An empirical model of diagnostic x-ray attenuation under narrow-beam geometry.
abstract: OBJECTIVE
  The purpose of this study was to develop and validate a mathematical model to describe narrow-beam attenuation of kilovoltage x-ray beams for the intended applications of half-value layer (HVL) and quarter-value layer (QVL) estimations, patient organ shielding, and computer modeling.
  
  
  METHODS
  An empirical model, which uses the Lambert W function and represents a generalized Lambert-Beer law, was developed. To validate this model, transmission of diagnostic energy x-ray beams was measured over a wide range of attenuator thicknesses [0.49-33.03 mm Al on a computed tomography (CT) scanner, 0.09-1.93 mm Al on two mammography systems, and 0.1-0.45 mm Cu and 0.49-14.87 mm Al using general radiography]. Exposure measurements were acquired under narrow-beam geometry using standard methods, including the appropriate ionization chamber, for each radiographic system. Nonlinear regression was used to find the best-fit curve of the proposed Lambert W model to each measured transmission versus attenuator thickness data set. In addition to validating the Lambert W model, we also assessed the performance of two-point Lambert W interpolation compared to traditional methods for estimating the HVL and QVL [i.e., semi-logarithmic (exponential) and linear interpolation].
  
  
  RESULTS
  The Lambert W model was validated for modeling attenuation versus attenuator thickness with respect to the data collected in this study (R2 > 0.99). Furthermore, Lambert W interpolation was more accurate and less sensitive to the choice of interpolation points used to estimate the HVL and/or QVL than the traditional methods of semilogarithmic and linear interpolation.
  
  
  CONCLUSIONS
  The proposed Lambert W model accurately describes attenuation of both monoenergetic radiation and (kilovoltage) polyenergetic beams (under narrow-beam geometry).
authors.affiliation: Department of Bioengineering, Rice University, Houston, Texas 77005, USA. kbmathieu@alumni.rice.edu
authors.forename: Kelsey B
authors.initials: KB
authors.lastname: Mathieu
authors.forename: S Cheenu
authors.initials: SC
authors.lastname: Kappadath
authors.forename: R Allen
authors.initials: RA
authors.lastname: White
authors.forename: E Neely
authors.initials: EN
authors.lastname: Atkinson
authors.forename: Dianna D
authors.initials: DD
authors.lastname: Cody
mesh_terms: D055592:Biophysical Phenomena; D003198:Computer Simulation; D005260:Female; D006801:Humans; D008327:Mammography / Q000706:statistics & numerical data; D008962:Models, Theoretical; D011859:Radiography / Q000706:statistics & numerical data*; D011874:Radiometry; D012542:Scattering, Radiation; D014057:Tomography, X-Ray Computed / Q000706:statistics & numerical data