A comparative study of effective atomic number calculations for dual-energy CT

Abstract

Purpose: Several new formalisms of Effective Atomic Number ( $Z_{\text{eff}}$ ) have emerged recently, deviating from the widely accepted Mayneord's definition. This comparative study aims to reexamine their theories, reveal their connections, and apply them to material differentiation on dual-energy computed tomography (DECT).

Methods: The first part of this paper is an in-depth review of several highly cited $Z_{\text{eff}}$ formalisms. This part includes (1) refuting the claim in Taylor's study that the classic Mayneord's formalism was inaccurate, (2) showing that Mayneord's, Rutherford's, and Bourque's formalisms were equivalent, and (3) explaining the fundamental difference between Taylor's and Bourque's formalisms. The second part of this paper explains how we translated the theories into software implementation and added an open-source $Z_{\text{eff}}$ calculation engine to our free research software 3D Quantitative Imaging (3DQI). The work includes developing an interpolation method based on radial basis function to make Taylor's formalism applicable to DECT, and devising a table lookup method to generate $Z_{\text{eff}}$ map with high efficiency for all appropriate formalisms.

Results: Comparing Bourque's and Taylor's formalisms for six common materials over 40 $\sim$ 100 keV energy range, it was found that Bourque's $Z_{\text{eff}}$ values had a weak energy dependence by 0.18% $\sim$ 3.10%, but for Taylor's results this variation increased by a factor of 10. Further comparison showed that at 61 keV, different formalisms fall into two categories-Bourque, Mayneord, Van Abbema (a derivative of Rutherford) for the first category, and Taylor and Manohara for the second. Formalisms within each category produced similar $Z_{\text{eff}}$ values. For a material consisting of two elements, the two categories of formalisms tended to show a greater discrepancy if the constituent elements had larger difference in $Z$ . The developed $Z_{\text{eff}}$ calculation engine was successfully applied to kidney stone classification and colon electronic cleansing.

Conclusions: We renewed the understanding of several popular $Z_{\text{eff}}$ formalisms: Contrary to the conclusion of Taylor's study, Mayneord's power-law formula is well grounded in theory; Bourque's formalism (based on the average electron microscopic cross-section) is considered numerically equivalent to Rutherford's, but with the advantage of being mathematically rigorous and physically meaningful; Taylor's formalism (based on the average atomic microscopic cross-section) is theoretically not suitable for DECT but a workaround still exists; Manohara's formalism should be used with caution due to a problem in its definition of electron cross-sections. The developed $Z_{\text{eff}}$ engine in the 3DQI software facilitated accurate and efficient $Z_{\text{eff}}$ estimate for various DECT applications.

Keywords: dual energy CT; effective atomic number; electronic cleansing.