A Novel Approach for Estimating the Effective Atomic Number Using Dual Energy

Jeong Heon Kim; So Hyun Ahn; Kwang Woo Park; Jin Sung Kim

doi:10.14316/pmp.2025.36.1.1

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Kim, Ahn, Park, and Kim: A Novel Approach for Estimating the Effective Atomic Number Using Dual Energy

Original Article

Progress in Medical Physics 2025; 36(1): 1-7.

Published online: 31 March 2025

DOI: https://doi.org/10.14316/pmp.2025.36.1.1

A Novel Approach for Estimating the Effective Atomic Number Using Dual Energy

Jeong Heon Kim^1,^2,³

, So Hyun Ahn^4,^5,

, Kwang Woo Park^6,

, Jin Sung Kim^1,^2,³

¹Department of Medicine, Yonsei University College of Medicine, Seoul, Korea

²Medical Physics and Biomedical Engineering Lab (MPBEL), Yonsei University College of Medicine, Seoul, Korea

³Department of Radiation Oncology, Yonsei Cancer Center, Heavy Ion Therapy Research Institute, Yonsei University College of Medicine, Seoul, Korea

⁴Ewha Medicine Research Institute, School of Medicine, Ewha Womans University, Seoul, Korea

⁵Ewha Medical Artificial Intelligence Research Institute, Ewha Womans University College of Medicine, Seoul, Korea

⁶Department of Radiation Oncology, Yongin Severance Hospital, Yonsei University College of Medicine, Yongin, Korea

Corresponding author So Hyun Ahn, (mpsohyun@gmail.com), Tel: 82-2-6986-6305, Fax: 82-0504-158-4052, Corresponding author, Kwang Woo Park, (kpark02@yuhs.ac), Tel: 82-2-2258-4362, Fax: 82-2-2227-7823

Received 31 October 2024 Revised 10 December 2024 Accepted 27 December 2024

(open-access):

This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Purpose

This study aimed to present a novel method for estimating the effective atomic number (Z_eff) using dual-energy computed tomography (DECT) designed to improve accuracy and streamline clinical workflows by reducing computational complexity.

Methods

The proposed model leverages the DECT-derived mass attenuation coefficients without detailed compositional analysis. By incorporating additional parameters into the conventional Rutherford model, such as exponential and trigonometric functions, the model effectively captures complex variations in attenuation, enabling precise Z_eff estimation. Model fitting was performed using dual-energy data and evaluated using the percentage difference in error rates.

Results

Compared with the Rutherford model, which recorded a maximum error rate of 0.55%, the proposed model demonstrated a significantly lower maximum error rate of 0.15%, highlighting its precision. Z_eff estimates for various materials closely matched the reference values, confirming the improved accuracy of the model.

Conclusion

The proposed DECT-based model provides a practical and efficient approach to Z_eff estimation, with potential applications in radiation oncology, particularly for accurate stopping power ratio calculations in proton and heavy ion therapies.

Keywords: Effective atomic number, Dual-energy computed tomography, Stopping power ratio, Radiation oncology, Proton therapy

Introduction

In radiation oncology, the accurate characterization of patient tissues is essential for precise dose calculation and effective treatment planning [1,2]. The effective atomic number (Z_eff) is a crucial parameter that influences the interaction of ionizing radiation with matter [3,4]. This parameter significantly affects photon attenuation and energy absorption, playing a vital role in both diagnostic imaging and delivery of therapeutic radiation [5,6]. Traditional methods for determining Z_eff involve complex calculations and often require detailed knowledge of the elemental composition of a material, frequently relying on approximation models [7,8].

Originally introduced by Murty [9], the conventional method for calculating Z_eff in compounds and mixtures has been widely adopted to date. Although this approach provides reliable results, its computational complexity—stemming from the need for compositional analysis—can make it impractical in clinical settings that require rapid assessment [10,11]. With advancements in imaging technologies, more efficient methods are increasingly needed to determine Z_eff that can be seamlessly integrated into clinical workflows [12,13].

Dual-energy computed tomography (DECT) is emerging as a powerful medical imaging tool that can provide energy-dependent attenuation information by scanning at two different photon energies [14,15]. DECT facilitates better tissue differentiation and characterization without invasive procedures [16,17]. Leveraging the data obtained from DECT, new methodologies can be developed to calculate Z_eff more accurately and practical clinically [18].

In this study, we present a novel method for defining the Z_eff of compounds and mixtures using the dual-energy mass attenuation coefficients. This approach simplifies the calculation by using readily available attenuation data without complex compositional analysis.

Materials and Methods

1. Relationship between Z_eff and the ratio of the linear attenuation coefficients

This proposed model is designed to explain the relationship between the linear attenuation coefficient and the atomic number. By incorporating additional parameters into the existing Rutherford model, the linear attenuation coefficient can be more accurately expressed as a function of the atomic number to estimate the Z_eff [19]. Linear attenuation coefficients, atomic number, and energy were sourced from the NIST XCOM database.

The Rutherford model describes the relationship among the linear attenuation coefficient, atomic number, and energy and is expressed using the following equation:

(1)

μ = ρ N_{A} \frac{1}{A} [Z K^{K N} (E) + Z^{n} K^{S C A} (E) + Z^{m} K^{P E} (E)],

where m represents the linear attenuation coefficient of the material; ρ, material density; N_A, Avogadro’s number (6.022×10²³ mol⁻¹); A, atomic mass of the material; Z, atomic number; and E, radiation energy. The Rutherford model expresses the linear attenuation coefficient of a material based on the contributions from Compton scattering (K^KN(E)), Rayleigh scattering (K^SCA(E)), and photoelectric effect (K^PE(E)). This model explains the variations in the attenuation coefficients across different energy ranges and is used to estimate the Z_eff.

In this study, a simplified version of the Rutherford model was used, where the linear attenuation coefficient is expressed as a function related to the atomic number. The simplified model is as follows:

(2)

\hat{μ} (Z) \approx \frac{A μ (z)}{Z} = a_{1} + a_{2} Z^{n - 1} + a_{3} Z^{m - 1},

where a₁, a₂, and a₃ are constants determined through experimental fitting, and Z represents the atomic number.

The proposed model is designed to achieve more precise attenuation coefficient predictions by incorporating new parameters to the simplified Rutherford model. The proposed model is as follows:

(3)

\hat{μ} (Z) \approx \frac{A μ (z)}{Z} = a_{1} e^{a_{2} Z} + a_{3} Z + a_{4} + a_{5} \sin (a_{6} Z + a_{7}),

where a₁–a₇ represent constants determined through experimental fitting, and Z represents the atomic number. By incorporating the exponential and trigonometric functions, the model can more accurately present the variations in the attenuation coefficients based on the atomic number.

The model’s parameter fitting was performed using dual-energy data through the nonlinear least squares method. The parameters were determined based on the attenuation coefficient data from the two energy levels.

2. Z_eff estimation

Z_eff was estimated using a fitting model based on the relationship between the linear attenuation coefficient and the atomic number. This process employed the following equation:

(4)

\begin{array}{l} μ_{x, H} {a_{1 L} Z e^{a 2 L^{Z}} + a_{3 L} Z^{2} + a_{4 L} Z + a_{5 L} Z \sin (a_{6 L} Z + a_{7 L})} \\ = μ_{x, L} {a_{1 H} Z e^{a 2 H^{Z}} + a_{3 H} Z^{2} + a_{4 H} Z + a_{5 H} Z \sin (a_{6 H} Z + a_{7 H})}, \end{array}

This equation is based on a fitting model that explains the linear attenuation coefficients at the two energy levels (H and L). To estimate Z_eff, a one-dimensional root-finding algorithm, such as the Newton method, was applied.

3. Evaluation

To evaluate the performance of the proposed model, it was compared with the existing Rutherford model and models by Landry et al. [16], Hünemohr et al. [20], and Bourque et al. [21] for human tissues. The fitting accuracy of each model was assessed using the percentage difference to analyze the differences in fitting between the proposed model and the Rutherford model. By comparing the results of these models, a more accurate method that can explain the attenuation coefficient of the materials was identified.

In addition, to evaluate Z_eff estimation performance, the Z_eff values for materials such as C₅O₂, C₅H₈O₂, H₂O, adipose, and inner bone were calculated and compared with the results from the Rutherford model and models by Landry et al. [16], Hünemohr et al. [20], and Bourque et al. [21]. This comprehensive comparison allowed us to verify the accuracy of the proposed model in estimating Z_eff and validate its performance for human tissue characterization.

Results

1. Fitting results

The parameters were determined at each energy level using the nonlinear least squares method to explain the attenuation coefficients. The fitting parameters for the Rutherford model were estimated at 50 and 100 keV (Table 1). For the Rutherford model, the percentage difference was found to be a maximum of 0.55%. The percentage difference and the fitting model graph are presented in Fig. 1.

The fitting parameters for the proposed model are summarized in Table 2. For the proposed model, the percentage difference was found to be a maximum of 0.15%. The percentage difference and the fitting model graph for the proposed model are also shown in Fig. 2.

The comparison of the fitting results of both models revealed that the proposed model demonstrated a higher fitting accuracy.

2. Z_eff estimation

The proposed model and the Rutherford model were compared to predict the Z_eff of the materials. Table 3 presents the comparison of the reference Z_eff with the Z_eff predicted by both models for H₂O, C₅O₂, and C₅H₈O₂.

For C₅O₂, the Rutherford model predicted 6.882, and the proposed model estimated 6.854, both close to the reference value of 6.831. For C₅H₈O₂, the Rutherford model predicted 6.533, and the proposed model estimated 6.518. Both models provided values close to the reference value of 6.475; however, the proposed model yielded slightly more accurate results. For H₂O, the Rutherford and proposed models predicted 7.840 and 7.632, respectively, which were close to the reference Z_eff of 7.742. Other comparative models, such as models by Landry et al. [16] (7.8 ± 0.4), Hünemohr et al. [20] (7.40 ± 0.14), and Bourque et al. [21] (7.66 ± 0.03), also provided values close to the reference. The proposed model demonstrated similar accuracy to these models for low-density materials such as H₂O. For adipose, the reference Z_eff was 6.213, with the Rutherford model predicting 6.261 and the proposed model estimating 6.259, closely matching the reference value. Other models showed comparable accuracy, with predictions such as 6.2 ± 0.6 (Landry et al. [16]), 6.30 ± 0.20 (Hünemohr et al. [20]), and 6.17 ± 0.02 (Bourque et al. [21]).

However, for the inner bone, a high-density material, the proposed model demonstrated superior performance. The reference Z_eff for inner bone was 10.424, with the Rutherford and proposed models predicting 10.464 and 10.441, respectively. Other models yielded 10.4 ± 0.3 (Landry et al. [16]), 10.11 ± 0.09 (Hünemohr et al. [20]), and 10.28 ± 0.03 (Bourque et al. [21]). Although all models were relatively accurate, the proposed model was marginally closer to the reference value.

Discussion

The results of this study demonstrate that the proposed model demonstrates higher accuracy in estimating Z_eff than the conventional Rutherford model. Specifically, a comparison of the error rates in Z_eff estimation for both low (50 keV) and high (100 keV) energy levels revealed that the proposed model consistently achieves lower error rates, confirming improved precision over traditional approaches. Specifically, the maximum error rates for the Rutherford and proposed models were 0.55% and 0.15%, respectively.

For soft tissues such as H₂O and adipose, the proposed model shows comparable performance to other models, such as the models by Landry et al. [16] and Bourque et al. [21]. The error rates for these materials are close across the models, with the proposed model achieving low errors: 1.42% for H₂O and 0.74% for adipose. For high-density materials such as inner bone, the proposed model outperformed all other models, achieving the lowest error rate of 0.16% compared with 0.38%, 0.23%, 3.01%, and 1.38% for Rutherford, Landry, Hünemohr, and Bourque models, respectively. These results demonstrate the exceptional accuracy of the proposed model, particularly for high-density materials, while maintaining competitive performance for soft tissues.

Traditional Z_eff estimation methods require prior knowledge of the elemental composition of a material, resulting in complex and time-intensive analysis, which poses challenges for routine clinical use. This limitation can limit their applicability in fast-paced clinical settings that require rapid assessments. In contrast, the proposed approach leverages DECT data to estimate Z_eff using only mass attenuation coefficients, thus eliminating the need for compositional information. This feature places the proposed method as a promising tool for clinical environments that demand quick and accurate Z_eff calculations.

The proposed model introduces a new calculation method for estimating Z_eff based on the linear attenuation coefficients by improving the Rutherford model. By incorporating additional parameters such as exponential and trigonometric functions, this model effectively reflects the complexities of attenuation variations across different atomic numbers and energy levels. Our findings indicate that the proposed model provides accuracy comparable with those of existing models for low-density materials and soft tissues. However, for high-density materials, the proposed model yields results that are either comparable with or slightly better than those of other models, showcasing its effectiveness in specific applications. Consequently, the proposed model shows potential for applications not only in diagnostic imaging but also in therapeutic environments such as proton and heavy ion therapies, where high Z_eff-based precision is essential.

However, this study has some limitations. The proposed model relies on data from only two energy levels, meaning that Z_eff estimation accuracy may be affected by the resolution and quality of the DECT. In addition, the predictive performance of the model may be limited to specific tissues and materials, requiring further validations in diverse clinical scenarios. Therefore, the broader clinical application of the model will require additional validation across various data and conditions.

The proposed model offers substantial benefits for clinical applications, particularly in radiation oncology, where Z_eff-based SPR calculations are widely adopted for particle therapy. The proposed model simplifies the SPR estimation process by enabling fast and accessible Z_eff calculations, indicating greater integration potential in clinical workflows. Ultimately, the proposed model enhances the dose calculation precision in particle therapy, supporting more accurate and effective treatments for patients. Moreover, this simplified calculation method could be extended to other clinical contexts that rely on Z_eff, such as the early detection of tissue abnormalities, broadening its applicability in medical imaging and beyond.

Conclusions

This study presents a novel method for estimating the Z_eff using DECT, achieving superior accuracy and efficiency compared with the traditional Rutherford model. By leveraging the DECT-derived mass attenuation coefficients, the model achieves a low error rate of 0.15% without compositional analysis, making it highly suitable for clinical workflows. Its precision supports accurate SPR calculations in proton and heavy ion therapies, with further potential applications in diagnostic imaging and treatment planning.

Notes

FUNDING

This research supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant was funded by the Korea government (MOTIE) (20227410100050) and supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number NRF-2022R1H1A2092091) and this work was supported by the Technology development Program (RS-2023-00257618) funded by the Ministry of SMEs and Startups (MSS, Korea). This Study was supported by a grant of the SME R&D project for the Start-up & Grow stage company, Ministry of SMEs and Startups (RS-2024-00426787).

Conflicts of Interest

The authors have nothing to disclose.

Availability of Data and Materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Conceptualization: Jeong Heon Kim, Jin Sung Kim, So Hyun Ahn. Formal analysis: Kwang Woo Park. Resources: Jin Sung Kim, So Hyun Ahn. Writing – original draft: Jeong Heon Kim. Writing – review & editing: Jeong Heon Kim, Jin Sung Kim, Kwang Woo Park.

References

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15. Goodsitt MM, Christodoulou EG, Larson SC. 2011; Accuracies of the synthesized monochromatic CT numbers and effective atomic numbers obtained with a rapid kVp switching dual energy CT scanner. Med Phys. 38:2222–2232. DOI: 10.1118/1.3567509. PMID: 21626956.

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Fig. 1

Fitting result of the Rutherford model. (a) Percentage difference between NIST data and model fitting at 50 keV and 100 keV as a function of atomic number (Z); (b) Mass attenuation coefficient per Z/A (μ) as a function of atomic number (Z) at 50 keV and 100 keV, with NIST data and fitting results.

Fig. 2

Fitting result of the proposed model. (a) Percentage difference between NIST data and model fitting at 50 keV and 100 keV as a function of atomic number (Z); (b) Mass attenuation coefficient per Z/A (μ) as a function of atomic number (Z) at 50 keV and 100 keV, with NIST data and fitting results.

Table 1

Parameters of the Rutherford model

Parameter	50 keV	100 keV
a1	−0.463656	−1.45272
a2	0.000111037	1.74864
a3	0.803159	1.43277E-05
m	0.999333	4.20913
n	4.21807	1.00088

Table 2

Parameters of the proposed model

Parameter	50 keV	100 keV
a1	1.15516	2.84926
a2	−0.113633	−6.79687
a3	0.385816	0.0509277
a4	−2.62348	0.139045
a5	3.378	−0.685833
a6	−0.0891485	0.0720003
a7	0.564329	−6.50793

Table 3

Comparison of reference Z_eff with other models and proposed model predictions

Material	Reference Z_eff	Rutherford Z_eff (error %)	Landry Z_eff (error %)	Hünemohr Z_eff (error %)	Bourque Z_eff (error %)	Proposed model Z_eff (error %)
C₅O₂	6.831	6.882 (0.75)	-	-	-	6.854 (0.34)
C₅H₈O₂	6.475	6.533 (0.90)	-	-	-	6.518 (0.66)
H₂O	7.742	7.840 (1.27)	7.8 (0.75)	7.40 (4.42)	7.66 (1.06)	7.632 (1.42)
Adipose	6.213	6.261 (0.77)	6.2 (0.21)	6.30 (1.40)	6.17 (0.69)	6.259 (0.74)
Inner bone	10.424	10.464 (0.38)	10.4 (0.23)	10.11 (3.01)	10.28 (1.38)	10.441 (0.15)

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A Novel Approach for Estimating the Effective Atomic Number Using Dual Energy

Abstract

Purpose

Methods

Results

Conclusion

Introduction

Materials and Methods

1. Relationship between Zeff and the ratio of the linear attenuation coefficients

(1)

(2)

(3)

2. Zeff estimation

(4)

3. Evaluation

Results

1. Fitting results

2. Zeff estimation

Discussion

Conclusions

Notes

References

Fig. 1

Fig. 2

Table 1

Table 2

Table 3

1. Relationship between Z_eff and the ratio of the linear attenuation coefficients

2. Z_eff estimation

2. Z_eff estimation