^{1,a)}, Sicong Li

^{2}, Slobodan Devic

^{3}, Bruce R. Whiting

^{4}and Fritz A. Lerma

^{5,b)}

### Abstract

The goal of this study is to evaluate the theoretically achievable accuracy in estimating photon cross sections at low energies from idealized dual-energy x-ray computed tomography(CT)images. Cross-section estimation from dual-energy measurements requires a model that can accurately represent photon cross sections of any biological material as a function of energy by specifying only two characteristic parameters of the underlying material, e.g., effective atomic number and density. This paper evaluates the accuracy of two commonly used two-parameter cross-section models for postprocessing idealized measurements derived from dual-energy CTimages. The parametric fit model (PFM) accounts for electron-binding effects and photoelectric absorption by power functions in atomic number and energy and scattering by the Klein–Nishina cross section. The basis-vector model (BVM) assumes that attenuation coefficients of any biological substance can be approximated by a linear combination of mass attenuation coefficients of two dissimilar basis substances. Both PFM and BVM were fit to a modern cross-section library for a range of elements and mixtures representative of naturally occurring biological materials. The PFM model, in conjunction with the effective atomic number approximation, yields estimated the total linear cross-section estimates with mean absolute and maximum error ranges of 0.6%–2.2% and 1%–6%, respectively. The corresponding error ranges for BVM estimates were 0.02%–0.15% and 0.1%–0.5%. However, for photoelectric absorption frequency, the PFM absolute mean and maximum errors were 10.8%–22.4% and 29%–50%, compared with corresponding BVM errors of 0.4%–11.3% and 0.5%–17.0%, respectively. Both models were found to exhibit similar sensitivities to image-intensity measurement uncertainties. Of the two models, BVM is the most promising approach for realizing dual-energy CT cross-section measurement.

This work was supported in part by Grants No. CA R01-46640 and CA R01-75371 awarded by the National Institutes of Health. The authors are indebted to two Virginia Commonwealth University summer work/study students, Suresh Joel (Department of Biomedical Engineering) and James Liang (School of Medicine), who contributed significantly to the MATLAB codes used for data analysis in this project.

I. INTRODUCTION

II. METHODS AND MATERIALS

II.A. Parametric fit model

II.B. Basis-vector model

II.C. Evaluation of two-parameter fit model accuracy

II.C.1. Parametric fit model predictive accuracy

II.C.2. Accuracy of idealized dual-energy CT cross-section estimation

II.C.3. Accuracy endpoints

II.C.4. Sensitivity to CT measurement uncertainty

III. RESULTS

III.A. Accuracy of parametric and basis-vector model fits to elemental cross-section data

III.B. Accuracy of cross-section estimation from idealized dual-energy imaging

III.C. QDECT sensitivity to measurement uncertainties

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Atomic force microscopy
- 72.0
- Computed tomography
- 43.0
- Medical imaging
- 40.0
- Photons
- 22.0
- Error analysis
- 18.0

## Figures

Ratio of the PFM predictions to the DLC-146 reference cross-section data for selected elements. (a) Total linear attenuation coefficient (single fit); (b) total linear attenuation coefficient (double fit); (c) frequency of photoelectric absorption collisions (double fit); and (d) mass-energy absorption coefficients (double fit).

Ratio of the PFM predictions to the DLC-146 reference cross-section data for selected elements. (a) Total linear attenuation coefficient (single fit); (b) total linear attenuation coefficient (double fit); (c) frequency of photoelectric absorption collisions (double fit); and (d) mass-energy absorption coefficients (double fit).

Ratio of the BVM predictions to the DLC-146 reference cross-section data for the same elements shown in Fig. 1. (a) Total linear attenuation coefficient (single-basis pair); (b) total linear attenuation coefficient (double-basis pair); (c) frequency of photoelectric absorption collisions (double-basis pair); and (d) mass-energy absorption coefficients (double-basis pair). Note that (a) and (b) have a more magnified -axis scale compared with Figs. 1(a) and 1(b), while panels (c) and (d) have the same -axis scale in both Figs. 1 and 2.

Ratio of the BVM predictions to the DLC-146 reference cross-section data for the same elements shown in Fig. 1. (a) Total linear attenuation coefficient (single-basis pair); (b) total linear attenuation coefficient (double-basis pair); (c) frequency of photoelectric absorption collisions (double-basis pair); and (d) mass-energy absorption coefficients (double-basis pair). Note that (a) and (b) have a more magnified -axis scale compared with Figs. 1(a) and 1(b), while panels (c) and (d) have the same -axis scale in both Figs. 1 and 2.

The MPAE (mean percent absolute error over the to energy range) of idealized dual-energy scanning of elemental materials as a function of atomic number for (a) the single- and double-basis pair BVM model and (b) the double- or single-fit PFM QDECT analysis. Total attenuation (solid line) and energy absorption (broken line) linear coefficients are shown along with frequency of photoelectric absorption for the BVM QDECT analysis only. Red lines denote double fit or basis pair and blue lines denote single fits or basis pairs.

The MPAE (mean percent absolute error over the to energy range) of idealized dual-energy scanning of elemental materials as a function of atomic number for (a) the single- and double-basis pair BVM model and (b) the double- or single-fit PFM QDECT analysis. Total attenuation (solid line) and energy absorption (broken line) linear coefficients are shown along with frequency of photoelectric absorption for the BVM QDECT analysis only. Red lines denote double fit or basis pair and blue lines denote single fits or basis pairs.

Ratios of effective to actual atomic numbers and electron densities for elements from the PFM QDECT analysis.

Ratios of effective to actual atomic numbers and electron densities for elements from the PFM QDECT analysis.

Comparison of simulated QDECT cross-section predictions derived from the double-pair BVM [left column graphs (a), (c), and (e)] and the double-fit PFM [right column, graphs (b), (d), and (f)]. Each graph compares the QDECT prescription with the DLC-146 reference cross section for selected mixtures as a function of energy from . The top, middle, and bottom rows show, respectively, the total linear attenuation coefficient [graphs (a) and (b)], the frequency of photoelectric collisions [graphs (c) and (d)], and the mass-energy absorption coefficient [graphs (e) and (f)].

Comparison of simulated QDECT cross-section predictions derived from the double-pair BVM [left column graphs (a), (c), and (e)] and the double-fit PFM [right column, graphs (b), (d), and (f)]. Each graph compares the QDECT prescription with the DLC-146 reference cross section for selected mixtures as a function of energy from . The top, middle, and bottom rows show, respectively, the total linear attenuation coefficient [graphs (a) and (b)], the frequency of photoelectric collisions [graphs (c) and (d)], and the mass-energy absorption coefficient [graphs (e) and (f)].

Percent unexpanded uncertainty of linear attenuation coefficients at the indicated energy for PMMA (blue lines) and a sodium chlorate solution (red lines) for idealized QDECT measurements based on (a) the double basis-pair BVM and (b) the double-fit PFM. The number pairs in parentheses, shown in the legends, denote the standard deviations of the 45- and image intensities, respectively, as percentages of the attenuation coefficient relative to water.

Percent unexpanded uncertainty of linear attenuation coefficients at the indicated energy for PMMA (blue lines) and a sodium chlorate solution (red lines) for idealized QDECT measurements based on (a) the double basis-pair BVM and (b) the double-fit PFM. The number pairs in parentheses, shown in the legends, denote the standard deviations of the 45- and image intensities, respectively, as percentages of the attenuation coefficient relative to water.

## Tables

Composition and physical properties of basis substances and selected mixtures used to test the BVM. Aqueous solutions are specified as percent by weight.

Composition and physical properties of basis substances and selected mixtures used to test the BVM. Aqueous solutions are specified as percent by weight.

Best fit parameters for the PFM model fit to DLC-146 cross-section data for energies of and .

Best fit parameters for the PFM model fit to DLC-146 cross-section data for energies of and .

MPAE in , , and values predicted by double-pair BVM and double-fit PFM dual-energy CT analysis over the energy range for selected mixtures. The BVM partial mass densities are also tabulated.

MPAE in , , and values predicted by double-pair BVM and double-fit PFM dual-energy CT analysis over the energy range for selected mixtures. The BVM partial mass densities are also tabulated.

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