Technical Note: Exploring the limit for the conversion of energy-subtracted CT number to electron density for high-atomic-number materials Masatoshi Saito and Masayoshi Tsukihara Citation: Medical Physics 41, 071701 (2014); doi: 10.1118/1.4881327 View online: http://dx.doi.org/10.1118/1.4881327 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/41/7?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in A method to acquire CT organ dose map using OSL dosimeters and ATOM anthropomorphic phantoms Med. Phys. 40, 081918 (2013); 10.1118/1.4816299 High-dose MVCT image guidance for stereotactic body radiation therapy Med. Phys. 39, 4812 (2012); 10.1118/1.4736416 Potential of dual-energy subtraction for converting CT numbers to electron density based on a single linear relationship Med. Phys. 39, 2021 (2012); 10.1118/1.3694111 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 (2011); 10.1118/1.3567509 Calibration of megavoltage cone-beam CT for radiotherapy dose calculations: Correction of cupping artifacts and conversion of CT numbers to electron density Med. Phys. 35, 849 (2008); 10.1118/1.2836945
Technical Note: Exploring the limit for the conversion of energy-subtracted CT number to electron density for high-atomic-number materials Masatoshi Saitoa) Department of Radiological Technology, School of Health Sciences, Faculty of Medicine, Niigata University, Niigata 951-8518, Japan
Masayoshi Tsukihara Division of Radiological Technology, Graduate School of Health Sciences, Niigata University, Niigata 951-8518, Japan
(Received 16 January 2014; revised 5 May 2014; accepted for publication 16 May 2014; published 6 June 2014) Purpose: For accurate tissue inhomogeneity correction in radiotherapy treatment planning, the authors had previously proposed a novel conversion of the energy-subtracted CT number to an electron density (HU–ρ e conversion), which provides a single linear relationship between HU and ρ e over a wide ρ e range. The purpose of this study is to address the limitations of the conversion method with respect to atomic number (Z) by elucidating the role of partial photon interactions in the HU–ρ e conversion process. Methods: The authors performed numerical analyses of the HU–ρ e conversion for 105 human body tissues, as listed in ICRU Report 46, and elementary substances with Z = 1–40. Total and partial attenuation coefficients for these materials were calculated using the XCOM photon cross section database. The effective x-ray energies used to calculate the attenuation were chosen to imitate a dualsource CT scanner operated at 80–140 kV/Sn under well-calibrated and poorly calibrated conditions. Results: The accuracy of the resultant calibrated electron density, ρecal , for the ICRU-46 body tissues fully satisfied the IPEM-81 tolerance levels in radiotherapy treatment planning. If a criterion of ρecal /ρ e − 1 is assumed to be within ±2%, the predicted upper limit of Z applicable for the HU–ρ e conversion under the well-calibrated condition is Z = 27. In the case of the poorly calibrated condition, the upper limit of Z is approximately 16. The deviation from the HU–ρ e linearity for higher Z substances is mainly caused by the anomalous variation in the photoelectric-absorption component. Conclusions: Compensation among the three partial components of the photon interactions provides for sufficient linearity of the HU–ρ e conversion to be applicable for most human tissues even for poorly conditioned scans in which there exists a large variation of effective xray energies owing to beam-hardening effects arising from the mismatch between the sizes of the object and the calibration phantom. © 2014 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4881327] Key words: dual-energy CT, electron density, atomic number, CT number, energy subtraction 1. INTRODUCTION Tissue inhomogeneity correction derived by the conversion of the computed tomography (CT) number (in Hounsfield units, HU) into an electron density relative to water (ρ e ) is one of the main processes that determine the accuracy of patient dose calculations in radiotherapy treatment planning.1 The HU–ρ e conversion is usually performed using tissue substitutes with known electron densities in a calibration phantom.2 However, the elemental composition of those tissue substitutes differs from that of real tissues; consequently, different calibration curves are used for the HU–ρ e conversion. Moreover, the CT number and ρ e value cannot be interrelated via a simple oneto-one correspondence. This is because the CT numbers depend on the electron densities and effective atomic numbers, i.e., the CT numbers of materials with equal electron density can vary owing to different effective atomic numbers. As a solution to this problem, the authors had previously proposed a novel conversion of the energy-subtracted CT 071701-1
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number (HU) to an electron density (hereafter referred to as HU–ρ e conversion).3 In the previous work, we performed analytical dual-energy CT (DECT) image simulations and predicted a single linear relationship between HU and ρ e over a wide range of ρ e values from 0.00 (air) to 2.35 (aluminum). The linearity of the HU–ρ e plot was confirmed experimentally using a clinical dual-source CT (DSCT) scanner with an electron density phantom.4 Nevertheless, questions related to the HU–ρ e conversion were raised. First, why does such a single linear relationship of HU–ρ e exist? Second, what is the upper limit of the atomic number (Z) of materials to which the HU–ρ e conversion can be applied? The main objective of this study is to investigate the limit of the HU–ρ e conversion method with respect to Z by elucidating the role of partial photon interactions in realizing the linear HU–ρ e relationship. This paper presents numerical analyses of the HU–ρ e conversion for a large variety of materials with known elemental compositions and densities using an available photon cross sections database.
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2. MATERIALS AND METHODS 2.A. HU–ρ e conversion
Let us briefly review the procedure of HU–ρ e conversion.3 We introduce a dual-energy subtracted quantity, HU, into the ρ e calibration, which is defined as follows: HU ≡ (1 + α)HUH − αHUL ,
HU + b. 1000
(2)
In the ideal case, r2 , a, and b are unity. 2.B. CT number calculation
CT numbers of all the materials tested in the present study with known elemental compositions and densities were estimated from their mass attenuation coefficients calculated using WinXCom software,5, 6 which is based on a photon crosssection database XCOM provided by the National Institute of Standards and Technology (NIST).7 WinXCom can be used to calculate partial attenuation coefficients for coherent and incoherent scattering, photoelectric absorption, and pair production in addition to total attenuation coefficients, in any element, compound, or mixture. 2.C. Effective x-ray energies of DECT
To retrieve the attenuation coefficients from the XCOM database, a photon energy must be specified. In this study, we chose to input effective x-ray energies, Eeff , which were converted from measured CT numbers of aluminum, i.e., the Eeff was determined as the monochromatic value that has the same CT number as the polyenergetic beam for aluminum.8, 9 We used the CT numbers of aluminum that we previously measured using a clinical DSCT scanner (Somatom Definition Flash, Siemens Healthcare, Forchheim, Germany) at 80 and 140 kV with an additional tin (Sn) filter.4 The DECT measurements were performed for the electron density phantom of CIRS model 062 (Computerized Imaging Reference Systems Inc., Norfolk, VA) including high-purity aluminum inserts. Table I shows the estimated Eeff values of the 80-kV and 140-kV/Sn tubes, along with the corresponding measured CT numbers of the aluminum inserts. Two different pairs of Eeff were provided. The first was estimated from DECT data using an abdomen configuration (33 cm in width and 27 cm in Medical Physics, Vol. 41, No. 7, July 2014
TABLE I. Effective x-ray energies, Eeff , converted from the CT numbers (HUL and HUH ) of aluminum that were previously measured using a clinical DSCT scanner operated at 80 kV and 140 kV/Sn under the well-calibrated and poorly calibrated conditions (Ref. 4). 80 kV
140 kV/Sn
Condition
HUL
Eeff (keV)
HUH
Eeff (keV)
Well-calibrated Poorly calibrated
2528 2733
62 58
1734 1781
97 92
(1)
where HUk is the CT number in Hounsfield units for highkV (k = H) and low-kV (k = L) scans, and α is the weighting factor for the subtraction. For materials with lower effective atomic numbers in a limited range of energies, α can be regarded as a single, material-independent parameter that is specified only by the mean energies of the high-kV and lowkV spectra of a CT scanner. Consequently, we can expect the relationship between HU and ρ e to invariably be a linear relation.3 The scanner-specific α is numerically determined using a HU–ρ e data set to maximize the coefficient of determination, r2 , by the least-squares fitting to a linear function for the calibrated ρ e , i.e., ρecal , as follows: ρecal = a
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height) of the CIRS phantom with 33-cm scan field of view (SFOV) (hereafter referred to as the “well-calibrated condition”). The second was estimated from DECT data using a phantom of the head configuration with a diameter of 18 cm; the 33-cm SFOV appropriate for the abdomen phantom was intentionally also used for this smaller sized phantom to investigate the effects of object size on the HU–ρ e conversion (hereafter referred to as the “poorly calibrated condition”). The use of such a small object would induce a large variation in Eeff values owing to beam-hardening effects arising from the change in the phantom size. Thus, the “poorly calibrated condition” may represent an ill-conditioned scan in which the scanned object is highly heterogeneous and a large amount of low-density lung tissue is present. For simplicity, the Eeff values were assumed to be unchanged among all materials at all positions of the SFOV under the same scanned conditions. 2.D. Materials tested for HU–ρ e conversion
In order to determine the weighting factor α, slope a, and intercept b in the fitted linear function of Eq. (2), we first performed the HU–ρ e conversion for 105 human body tissues as listed in ICRU Report 46 (Ref. 10) using their calculated CT numbers and given ρ e s under the well-calibrated condition. The ICRU-46 body tissues have ρ e values ranging from 0.258 (lung) to 2.895 (bone mineral-hydroxyapatite). Then, the determined conversion parameters of α, a, and b were used for the poorly calibrated case of the ICRU-46 body tissues. Furthermore, the ρ e calibrations were performed with the same conversion parameters for elementary substances with atomic number Z ranging from 1 (hydrogen) to 40 (zirconium) to explore the limit of Z for the HU–ρ e conversion. Needless to say, the elementary substances have their own characteristic ρ e values at ordinary temperature and atmospheric pressure. Nevertheless, to investigate the relationship between HU and ρ e in more detail, we virtually varied their ρ e s values from 0.0 to 3.0. 3. RESULTS AND DISCUSSION 3.A. ICRU-46 human body tissues
Figures 1(a) and 1(b) show HU–ρ e plots of the ICRU-46 body tissues for 80 kV and 140 kV/Sn, respectively, under the well-calibrated condition. A magnified portion of the softtissue region between ±100 HU is shown at the right side
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F IG . 1. The HU–ρ e conversion for the 105 human body tissues as listed in ICRU Report 46, under the well-calibrated condition for 80 kV–140 kV/Sn. (a) HUL –ρ e , (b) HUH –ρ e , and (c) HU–ρ e (α = 0.455). A magnified portion of the soft-tissue region within ±100 HU is shown at the right side of each figure. The straight line shown is fitted for the HU–ρ e data points using Eq. (2) with a = 1.003 and b = 1.001. Medical Physics, Vol. 41, No. 7, July 2014
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3.B. Elementary substances with a wider Z range
The HU–ρ e conversion method with α = 0.455, a = 1.003, and b = 1.001 determined in Sec. 3.A. was applied to elementary substances with a wider Z range. Figures 3(a) and 3(b) show ρ e variations as a function of HU for
F IG . 2. Relative deviations, ρecal /ρ e − 1, as functions of Zeff for 105 ICRU46 body tissues under the well-calibrated and poorly calibrated conditions.
of each figure. Both HU–ρ e plots display certain trends of linearity in limited regions but scatter in a somewhat unpredictable manner in other regions. Therefore, they could not be exactly interrelated via a one-to-one correspondence. In contrast, as shown in Fig. 1(c), the HU–ρ e data points of all the body tissues are in close proximity to the fitted calibration line. The weighting factor α used for calculating HUs was 0.455 because this yielded the largest r2 . This α value is consistent with previous experimental results (0.456 for 80– 140 kV/Sn),4 which may illustrate the validity of the effective energies of DECT used in this study. The resulting a, b, and r2 for the fitted line were 1.003, 1.001, and 0.99989, respectively; these are very close to unity, which is the value of these parameters in the ideal case. Figure 2 shows the relative deviations between ρecal and ρ e , i.e., ρecal /ρ e − 1, as a function of effective atomic number (Zeff ) under the well-calibrated and poorly calibrated conditions. Here, we used Mayneord’s equation to calculate Zeff : Zeff = 3.5 ai Zi 3.5 , where ai is the fractional number of electrons belonging to the ith material with atomic number Zi .11 In the well-calibrated case, the relative deviations were within ±0.4% for all ICRU-46 body tissues. In the poorly calibrated case, the corresponding ρecal /ρ e − 1 plot varies similarly to the well-calibrated case for Zeff = 1–9, but for higher Zeff values, the plot deviates downward in comparison to the well-calibrated case. Nevertheless, the resultant ρecal values agreed within ±1.0% with the true ρ e s of all ICRU-46 body tissues except for bone mineral hydroxyapatite (−1.8%), breast calcifications (−1.7%), and urinary (renal) stones oxalate (−1.2%). IPEM Report 81 recommends that agreement for ρecal in radiotherapy treatment planning should be within ±1% for water and within ±2% for lung and bone compared with their true ρ e values.12 According to the IPEM-81 tolerance levels, the agreements between ρecal and ρ e of the ICRU-46 body tissues would be sufficient even under the poorly calibrated condition. Medical Physics, Vol. 41, No. 7, July 2014
F IG . 3. The HU–ρ e conversion for elementary substances with Z = 1–30 under the well-calibrated condition for 80 kV–140 kV/Sn. Variations of ρ e as a function of (a) HUL , (b) HUH , and (c) HU (α = 0.455).
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be rewritten as μH μL w − α w, μH μL
ρecal = (1 + α)
(3)
where μk and μw k (k = H, L) are the linear attenuation coefficients of the object material and water, respectively. For simplicity, a and b in Eq. (2) were set to unity (the ideal case). The total attenuation coefficient in the diagnostic energy range can be approximated with high accuracy as the sum of the contributions from the following three partial photon interactions: μ = μpe + μcoh + μincoh ,
(4)
where μpe is the photoelectric absorption component, and μcoh and μincoh are the coherent and incoherent scattering components, respectively. Thus, using Eqs. (3) and (4), ρecal can be written as the sum of the three contributions as follows: ρecal = ρepe + ρecoh + ρeincoh ,
(5)
where F IG . 4. Relative deviations, ρecal /ρ e − 1, as functions of Z for elementary substances under the well-calibrated and poorly calibrated conditions.
Z = 1–30 at 80 kV and 140 kV/Sn, respectively, under the well-calibrated condition. Because the CT numbers estimated from attenuation coefficients are proportional to the relative electron densities, an HU–ρ e plot for a given Z should be a straight line. As illustrated in these figures, the straight lines are widely spread on each HU–ρ e plane, accompanying the Z variations, and their slopes decrease with increasing Z. These graphs also clearly demonstrate that the CT numbers of materials with equal ρ e can vary owing to different Z values. On the other hand, most of the HU–ρ e lines significantly converged, as shown in Fig. 3(c). Figure 4 shows the relative deviations ρecal /ρ e − 1 as a function of Z under the well-calibrated and poorly calibrated conditions. Following the IPEM-81 tolerance levels for reliable ρ e calibration as mentioned in Sec. 3.A, we assume here that ρecal /ρ e − 1 must be within ±2%, as in the case of measuring bone using IPEM-81. It is apparent from the ρecal /ρ e − 1 curve of the well-calibrated case that the elementary substances with Z ≤ 27 satisfy this criterion. On the other hand, the corresponding curve under the poorly calibrated condition starts to deviate downward at approximately Z = 10 from the well-calibrated case for higher Z substances. As a result, the limit of Z would be lowered to around 16 for the poorly calibrated case; this is consistent with the result of the ICRU-46 body tissues, as illustrated in Fig. 2.
3.C. Contribution of partial photon interactions to the HU–ρ e conversion process
Here, we address the limitations of the HU–ρ e conversion method by elucidating the role of partial photon interactions in the HU–ρ e conversion process. According to the definition of the CT number in Hounsfield units, Eq. (2) can Medical Physics, Vol. 41, No. 7, July 2014
pe
ρepe = (1 + α)
pe
μH μL w −α w, μH μL
ρecoh = (1 + α)
μcoh μcoh H L , w −α μH μw L
ρeincoh = (1 + α)
μincoh μincoh H L . w −α μH μw L pe
(6)
(7)
(8)
coh w incoh /μw Figure 5(a) shows plots of μk /μw k , μk /μk , and μk k (k = H, L) as functions of Z for the elementary substances in the well-calibrated and poorly calibrated conditions. To negate the influence of the variation in ρ e , the ρ e values of all of the elementary substances were assumed to be unity. This figure reveals that the photoelectric absorption increases more rapidly than the coherent scattering with increasing exponentiation of Z, whereas incoherent scattering decreases gradually with increasing Z. Corresponding graphs of the partial pe components ρe , ρecoh , ρeincoh , and their sum, ρecal , are shown in Fig. 5(b). We found that ρecoh and ρeincoh varied in a manw incoh coh ner similar to μcoh /μw k /μk and μk k , respectively; i.e., ρe incoh increases with increasing Z, whereas ρe decreases gradupe ally. In contrast, ρe exhibits a downward convex curve, and its minimum appears at approximately Z = 27 under the wellcalibrated condition. In the lower-Z region (Z ≤ 26) where pe ρe decreases with increasing Z, ρecal is very close to unity, which is the value in the ideal case; this can be achieved by compensation among the three partial components of the photon interactions. In other words, we can find a single, material-independent α for any substance within this Z range. However, this is not possible in the higher Z region (28 ≤ pe Z) owing to the change of ρe from decreasing to increasing, resulting in the apparent deviation of total ρecal from unity. Consequently, the effective limit of the HU–ρ e conversion appears to be approximately Z = 27 under the wellcalibrated condition. The curves of ρecoh and ρeincoh under the
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4. CONCLUSION The present numerical analyses using the XCOM photon cross-sections database revealed that a single linear relationship between HU and ρ e can be achieved by compensation among the three partial components of photon interactions. If we assume a criterion of ρecal /ρ e − 1 for the reliably calibrated ρ e to be within ±2%, the predicted upper limit of the HU–ρ e conversion under the well-calibrated condition is Z = 27. On the other hand, the corresponding limit is approximately 16 under the poorly calibrated condition. The deviation from the HU–ρ e linearity for higher Z substances is primarily caused by the anomalous variation in the photoelectricabsorption component. Nevertheless, the values of Zeff for most human tissues fall within this Z range and are therefore applicable to the HU–ρ e conversion, even for considerably ill-conditioned DECT scans in which there exists a large variation of effective x-ray energies owing to beam-hardening effects arising from the mismatch between the sizes of the object and the calibration phantom. ACKNOWLEDGMENT This work was partially supported by a Grant-in-Aid for Scientific Research (C) (25461908) from the Japan Society for the Promotion of Science (JSPS). a) Electronic
mail: [email protected]
1 R. P. Parker, P. A. Hobday, and K. J. Cassell, “The direct use of CT numbers
pe
coh w incoh F IG . 5. (a) μk /μw /μw k , μk /μk , and μk k (k = H, L) as functions of Z = 1–40 of the elementary substances under the well-calibrated and poorly calibrated conditions for 80–140 kV/Sn. (b) Corresponding partial components pe ρe , ρecoh , ρeincoh , along with their total: ρecal . The coordinates for the ρeincoh curves were shifted down by 0.2 to avoid overlap with the ρecal curve.
poorly calibrated condition almost coincided with those under the well-calibrated condition. However, the correspondpe ing ρe curve starts to deviate downward at approximately Z = 13 in the well-calibrated case for higher Z substances. As a result, we can confirm the findings presented in Sec. 3.B, i.e., that the limit of Z for the HU–ρ e conversion method decreases to approximately 16 under the poorly calibrated condition. Medical Physics, Vol. 41, No. 7, July 2014
in radiotherapy dosage calculations for inhomogeneous media,” Phys. Med. Biol. 24(4), 802–809 (1979). 2 C. Constantinou, J. C. Harrington, and L. A. DeWerd, “An electron density calibration phantom for CT-based treatment planning computers,” Med. Phys. 19(2), 325–327 (1992). 3 M. Saito, “Potential of dual-energy subtraction for converting CT numbers to electron density based on a single linear relationship,” Med. Phys. 39(4), 2021–2030 (2012). 4 M. Tsukihara, Y. Noto, T. Hayakawa, and M. Saito, “Conversion of the energy-subtracted CT number to electron density based on a single linear relationship: An experimental verification using a clinical dual-source CT scanner,” Phys. Med. Biol. 58(9), N135–N144 (2013). 5 L. Gerward, N. Guilbert, K. B. Jensen, and H. Levring, “WinXCom— A program for calculating X-ray attenuation coefficients,” Radiat. Phys. Chem. 71(3), 653–654 (2004). 6 L. Gerward, N. Guilbert, K. B. Jensen, and H. Levring, “X-ray absorption in matter. Reengineering XCOM,” Radiat. Phys. Chem. 60(1–2), 23–24 (2001). 7 M. J. Berger and J. H. Hubbell, XCOM: Photon Cross Sections Database, Web Version 1.2, National Institute of Standards and Technology, Gaithersburg, MD, USA, 1999 (available URL: http://physics.nist.gov/xcom), originally published as NBSIR 87-3597 (1987): XCOM: Photon Cross Sections on a Personal Computer. 8 ASTM, “Standard Test Method for Calibrating and Measuring CT Density,” ASTM E1935 (ASTM International, West Conshohocken, PA, 2013). 9 B. Li, R. H. Behrman, and A. M. Norbash, “Comparison of topogram-based body size indices for CT dose consideration and scan protocol optimization,” Med. Phys. 39(6), 3456–3465 (2012). 10 International Commission on Radiation Units and Measurements, “Photon, electron, proton, and neutron interaction data for body tissues,” ICRU Report No. 46 (ICRU Publications, Bethesda, MD, 1992). 11 H. E. Johns and J. R. Cunningham, The Physics of Radiology, 4th ed. (Charles C. Thomas, Springfield, IL, 1983). 12 W. P. H. Mayles, R. Lake, A. Mckenzie, E. M. Macauley, H. M. Morgan, T. J. Jordan, and S. K. Powley, “Physics aspects of quality control in radiotherapy,” IPEM Report No. 81 (The Institute of Physics and Engineering in Medicine, York, 1999).
Technical Note: Exploring the limit for the conversion of energy-subtracted CT number to electron density for high-atomic-number materials Masatoshi Saitoa) Department of Radiological Technology, School of Health Sciences, Faculty of Medicine, Niigata University, Niigata 951-8518, Japan
Masayoshi Tsukihara Division of Radiological Technology, Graduate School of Health Sciences, Niigata University, Niigata 951-8518, Japan
(Received 16 January 2014; revised 5 May 2014; accepted for publication 16 May 2014; published 6 June 2014) Purpose: For accurate tissue inhomogeneity correction in radiotherapy treatment planning, the authors had previously proposed a novel conversion of the energy-subtracted CT number to an electron density (HU–ρ e conversion), which provides a single linear relationship between HU and ρ e over a wide ρ e range. The purpose of this study is to address the limitations of the conversion method with respect to atomic number (Z) by elucidating the role of partial photon interactions in the HU–ρ e conversion process. Methods: The authors performed numerical analyses of the HU–ρ e conversion for 105 human body tissues, as listed in ICRU Report 46, and elementary substances with Z = 1–40. Total and partial attenuation coefficients for these materials were calculated using the XCOM photon cross section database. The effective x-ray energies used to calculate the attenuation were chosen to imitate a dualsource CT scanner operated at 80–140 kV/Sn under well-calibrated and poorly calibrated conditions. Results: The accuracy of the resultant calibrated electron density, ρecal , for the ICRU-46 body tissues fully satisfied the IPEM-81 tolerance levels in radiotherapy treatment planning. If a criterion of ρecal /ρ e − 1 is assumed to be within ±2%, the predicted upper limit of Z applicable for the HU–ρ e conversion under the well-calibrated condition is Z = 27. In the case of the poorly calibrated condition, the upper limit of Z is approximately 16. The deviation from the HU–ρ e linearity for higher Z substances is mainly caused by the anomalous variation in the photoelectric-absorption component. Conclusions: Compensation among the three partial components of the photon interactions provides for sufficient linearity of the HU–ρ e conversion to be applicable for most human tissues even for poorly conditioned scans in which there exists a large variation of effective xray energies owing to beam-hardening effects arising from the mismatch between the sizes of the object and the calibration phantom. © 2014 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4881327] Key words: dual-energy CT, electron density, atomic number, CT number, energy subtraction 1. INTRODUCTION Tissue inhomogeneity correction derived by the conversion of the computed tomography (CT) number (in Hounsfield units, HU) into an electron density relative to water (ρ e ) is one of the main processes that determine the accuracy of patient dose calculations in radiotherapy treatment planning.1 The HU–ρ e conversion is usually performed using tissue substitutes with known electron densities in a calibration phantom.2 However, the elemental composition of those tissue substitutes differs from that of real tissues; consequently, different calibration curves are used for the HU–ρ e conversion. Moreover, the CT number and ρ e value cannot be interrelated via a simple oneto-one correspondence. This is because the CT numbers depend on the electron densities and effective atomic numbers, i.e., the CT numbers of materials with equal electron density can vary owing to different effective atomic numbers. As a solution to this problem, the authors had previously proposed a novel conversion of the energy-subtracted CT 071701-1
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number (HU) to an electron density (hereafter referred to as HU–ρ e conversion).3 In the previous work, we performed analytical dual-energy CT (DECT) image simulations and predicted a single linear relationship between HU and ρ e over a wide range of ρ e values from 0.00 (air) to 2.35 (aluminum). The linearity of the HU–ρ e plot was confirmed experimentally using a clinical dual-source CT (DSCT) scanner with an electron density phantom.4 Nevertheless, questions related to the HU–ρ e conversion were raised. First, why does such a single linear relationship of HU–ρ e exist? Second, what is the upper limit of the atomic number (Z) of materials to which the HU–ρ e conversion can be applied? The main objective of this study is to investigate the limit of the HU–ρ e conversion method with respect to Z by elucidating the role of partial photon interactions in realizing the linear HU–ρ e relationship. This paper presents numerical analyses of the HU–ρ e conversion for a large variety of materials with known elemental compositions and densities using an available photon cross sections database.
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© 2014 Am. Assoc. Phys. Med.
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M. Saito and M. Tsukihara: Limitation of HU–ρ e conversion for high-Z materials
2. MATERIALS AND METHODS 2.A. HU–ρ e conversion
Let us briefly review the procedure of HU–ρ e conversion.3 We introduce a dual-energy subtracted quantity, HU, into the ρ e calibration, which is defined as follows: HU ≡ (1 + α)HUH − αHUL ,
HU + b. 1000
(2)
In the ideal case, r2 , a, and b are unity. 2.B. CT number calculation
CT numbers of all the materials tested in the present study with known elemental compositions and densities were estimated from their mass attenuation coefficients calculated using WinXCom software,5, 6 which is based on a photon crosssection database XCOM provided by the National Institute of Standards and Technology (NIST).7 WinXCom can be used to calculate partial attenuation coefficients for coherent and incoherent scattering, photoelectric absorption, and pair production in addition to total attenuation coefficients, in any element, compound, or mixture. 2.C. Effective x-ray energies of DECT
To retrieve the attenuation coefficients from the XCOM database, a photon energy must be specified. In this study, we chose to input effective x-ray energies, Eeff , which were converted from measured CT numbers of aluminum, i.e., the Eeff was determined as the monochromatic value that has the same CT number as the polyenergetic beam for aluminum.8, 9 We used the CT numbers of aluminum that we previously measured using a clinical DSCT scanner (Somatom Definition Flash, Siemens Healthcare, Forchheim, Germany) at 80 and 140 kV with an additional tin (Sn) filter.4 The DECT measurements were performed for the electron density phantom of CIRS model 062 (Computerized Imaging Reference Systems Inc., Norfolk, VA) including high-purity aluminum inserts. Table I shows the estimated Eeff values of the 80-kV and 140-kV/Sn tubes, along with the corresponding measured CT numbers of the aluminum inserts. Two different pairs of Eeff were provided. The first was estimated from DECT data using an abdomen configuration (33 cm in width and 27 cm in Medical Physics, Vol. 41, No. 7, July 2014
TABLE I. Effective x-ray energies, Eeff , converted from the CT numbers (HUL and HUH ) of aluminum that were previously measured using a clinical DSCT scanner operated at 80 kV and 140 kV/Sn under the well-calibrated and poorly calibrated conditions (Ref. 4). 80 kV
140 kV/Sn
Condition
HUL
Eeff (keV)
HUH
Eeff (keV)
Well-calibrated Poorly calibrated
2528 2733
62 58
1734 1781
97 92
(1)
where HUk is the CT number in Hounsfield units for highkV (k = H) and low-kV (k = L) scans, and α is the weighting factor for the subtraction. For materials with lower effective atomic numbers in a limited range of energies, α can be regarded as a single, material-independent parameter that is specified only by the mean energies of the high-kV and lowkV spectra of a CT scanner. Consequently, we can expect the relationship between HU and ρ e to invariably be a linear relation.3 The scanner-specific α is numerically determined using a HU–ρ e data set to maximize the coefficient of determination, r2 , by the least-squares fitting to a linear function for the calibrated ρ e , i.e., ρecal , as follows: ρecal = a
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height) of the CIRS phantom with 33-cm scan field of view (SFOV) (hereafter referred to as the “well-calibrated condition”). The second was estimated from DECT data using a phantom of the head configuration with a diameter of 18 cm; the 33-cm SFOV appropriate for the abdomen phantom was intentionally also used for this smaller sized phantom to investigate the effects of object size on the HU–ρ e conversion (hereafter referred to as the “poorly calibrated condition”). The use of such a small object would induce a large variation in Eeff values owing to beam-hardening effects arising from the change in the phantom size. Thus, the “poorly calibrated condition” may represent an ill-conditioned scan in which the scanned object is highly heterogeneous and a large amount of low-density lung tissue is present. For simplicity, the Eeff values were assumed to be unchanged among all materials at all positions of the SFOV under the same scanned conditions. 2.D. Materials tested for HU–ρ e conversion
In order to determine the weighting factor α, slope a, and intercept b in the fitted linear function of Eq. (2), we first performed the HU–ρ e conversion for 105 human body tissues as listed in ICRU Report 46 (Ref. 10) using their calculated CT numbers and given ρ e s under the well-calibrated condition. The ICRU-46 body tissues have ρ e values ranging from 0.258 (lung) to 2.895 (bone mineral-hydroxyapatite). Then, the determined conversion parameters of α, a, and b were used for the poorly calibrated case of the ICRU-46 body tissues. Furthermore, the ρ e calibrations were performed with the same conversion parameters for elementary substances with atomic number Z ranging from 1 (hydrogen) to 40 (zirconium) to explore the limit of Z for the HU–ρ e conversion. Needless to say, the elementary substances have their own characteristic ρ e values at ordinary temperature and atmospheric pressure. Nevertheless, to investigate the relationship between HU and ρ e in more detail, we virtually varied their ρ e s values from 0.0 to 3.0. 3. RESULTS AND DISCUSSION 3.A. ICRU-46 human body tissues
Figures 1(a) and 1(b) show HU–ρ e plots of the ICRU-46 body tissues for 80 kV and 140 kV/Sn, respectively, under the well-calibrated condition. A magnified portion of the softtissue region between ±100 HU is shown at the right side
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F IG . 1. The HU–ρ e conversion for the 105 human body tissues as listed in ICRU Report 46, under the well-calibrated condition for 80 kV–140 kV/Sn. (a) HUL –ρ e , (b) HUH –ρ e , and (c) HU–ρ e (α = 0.455). A magnified portion of the soft-tissue region within ±100 HU is shown at the right side of each figure. The straight line shown is fitted for the HU–ρ e data points using Eq. (2) with a = 1.003 and b = 1.001. Medical Physics, Vol. 41, No. 7, July 2014
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3.B. Elementary substances with a wider Z range
The HU–ρ e conversion method with α = 0.455, a = 1.003, and b = 1.001 determined in Sec. 3.A. was applied to elementary substances with a wider Z range. Figures 3(a) and 3(b) show ρ e variations as a function of HU for
F IG . 2. Relative deviations, ρecal /ρ e − 1, as functions of Zeff for 105 ICRU46 body tissues under the well-calibrated and poorly calibrated conditions.
of each figure. Both HU–ρ e plots display certain trends of linearity in limited regions but scatter in a somewhat unpredictable manner in other regions. Therefore, they could not be exactly interrelated via a one-to-one correspondence. In contrast, as shown in Fig. 1(c), the HU–ρ e data points of all the body tissues are in close proximity to the fitted calibration line. The weighting factor α used for calculating HUs was 0.455 because this yielded the largest r2 . This α value is consistent with previous experimental results (0.456 for 80– 140 kV/Sn),4 which may illustrate the validity of the effective energies of DECT used in this study. The resulting a, b, and r2 for the fitted line were 1.003, 1.001, and 0.99989, respectively; these are very close to unity, which is the value of these parameters in the ideal case. Figure 2 shows the relative deviations between ρecal and ρ e , i.e., ρecal /ρ e − 1, as a function of effective atomic number (Zeff ) under the well-calibrated and poorly calibrated conditions. Here, we used Mayneord’s equation to calculate Zeff : Zeff = 3.5 ai Zi 3.5 , where ai is the fractional number of electrons belonging to the ith material with atomic number Zi .11 In the well-calibrated case, the relative deviations were within ±0.4% for all ICRU-46 body tissues. In the poorly calibrated case, the corresponding ρecal /ρ e − 1 plot varies similarly to the well-calibrated case for Zeff = 1–9, but for higher Zeff values, the plot deviates downward in comparison to the well-calibrated case. Nevertheless, the resultant ρecal values agreed within ±1.0% with the true ρ e s of all ICRU-46 body tissues except for bone mineral hydroxyapatite (−1.8%), breast calcifications (−1.7%), and urinary (renal) stones oxalate (−1.2%). IPEM Report 81 recommends that agreement for ρecal in radiotherapy treatment planning should be within ±1% for water and within ±2% for lung and bone compared with their true ρ e values.12 According to the IPEM-81 tolerance levels, the agreements between ρecal and ρ e of the ICRU-46 body tissues would be sufficient even under the poorly calibrated condition. Medical Physics, Vol. 41, No. 7, July 2014
F IG . 3. The HU–ρ e conversion for elementary substances with Z = 1–30 under the well-calibrated condition for 80 kV–140 kV/Sn. Variations of ρ e as a function of (a) HUL , (b) HUH , and (c) HU (α = 0.455).
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be rewritten as μH μL w − α w, μH μL
ρecal = (1 + α)
(3)
where μk and μw k (k = H, L) are the linear attenuation coefficients of the object material and water, respectively. For simplicity, a and b in Eq. (2) were set to unity (the ideal case). The total attenuation coefficient in the diagnostic energy range can be approximated with high accuracy as the sum of the contributions from the following three partial photon interactions: μ = μpe + μcoh + μincoh ,
(4)
where μpe is the photoelectric absorption component, and μcoh and μincoh are the coherent and incoherent scattering components, respectively. Thus, using Eqs. (3) and (4), ρecal can be written as the sum of the three contributions as follows: ρecal = ρepe + ρecoh + ρeincoh ,
(5)
where F IG . 4. Relative deviations, ρecal /ρ e − 1, as functions of Z for elementary substances under the well-calibrated and poorly calibrated conditions.
Z = 1–30 at 80 kV and 140 kV/Sn, respectively, under the well-calibrated condition. Because the CT numbers estimated from attenuation coefficients are proportional to the relative electron densities, an HU–ρ e plot for a given Z should be a straight line. As illustrated in these figures, the straight lines are widely spread on each HU–ρ e plane, accompanying the Z variations, and their slopes decrease with increasing Z. These graphs also clearly demonstrate that the CT numbers of materials with equal ρ e can vary owing to different Z values. On the other hand, most of the HU–ρ e lines significantly converged, as shown in Fig. 3(c). Figure 4 shows the relative deviations ρecal /ρ e − 1 as a function of Z under the well-calibrated and poorly calibrated conditions. Following the IPEM-81 tolerance levels for reliable ρ e calibration as mentioned in Sec. 3.A, we assume here that ρecal /ρ e − 1 must be within ±2%, as in the case of measuring bone using IPEM-81. It is apparent from the ρecal /ρ e − 1 curve of the well-calibrated case that the elementary substances with Z ≤ 27 satisfy this criterion. On the other hand, the corresponding curve under the poorly calibrated condition starts to deviate downward at approximately Z = 10 from the well-calibrated case for higher Z substances. As a result, the limit of Z would be lowered to around 16 for the poorly calibrated case; this is consistent with the result of the ICRU-46 body tissues, as illustrated in Fig. 2.
3.C. Contribution of partial photon interactions to the HU–ρ e conversion process
Here, we address the limitations of the HU–ρ e conversion method by elucidating the role of partial photon interactions in the HU–ρ e conversion process. According to the definition of the CT number in Hounsfield units, Eq. (2) can Medical Physics, Vol. 41, No. 7, July 2014
pe
ρepe = (1 + α)
pe
μH μL w −α w, μH μL
ρecoh = (1 + α)
μcoh μcoh H L , w −α μH μw L
ρeincoh = (1 + α)
μincoh μincoh H L . w −α μH μw L pe
(6)
(7)
(8)
coh w incoh /μw Figure 5(a) shows plots of μk /μw k , μk /μk , and μk k (k = H, L) as functions of Z for the elementary substances in the well-calibrated and poorly calibrated conditions. To negate the influence of the variation in ρ e , the ρ e values of all of the elementary substances were assumed to be unity. This figure reveals that the photoelectric absorption increases more rapidly than the coherent scattering with increasing exponentiation of Z, whereas incoherent scattering decreases gradually with increasing Z. Corresponding graphs of the partial pe components ρe , ρecoh , ρeincoh , and their sum, ρecal , are shown in Fig. 5(b). We found that ρecoh and ρeincoh varied in a manw incoh coh ner similar to μcoh /μw k /μk and μk k , respectively; i.e., ρe incoh increases with increasing Z, whereas ρe decreases gradupe ally. In contrast, ρe exhibits a downward convex curve, and its minimum appears at approximately Z = 27 under the wellcalibrated condition. In the lower-Z region (Z ≤ 26) where pe ρe decreases with increasing Z, ρecal is very close to unity, which is the value in the ideal case; this can be achieved by compensation among the three partial components of the photon interactions. In other words, we can find a single, material-independent α for any substance within this Z range. However, this is not possible in the higher Z region (28 ≤ pe Z) owing to the change of ρe from decreasing to increasing, resulting in the apparent deviation of total ρecal from unity. Consequently, the effective limit of the HU–ρ e conversion appears to be approximately Z = 27 under the wellcalibrated condition. The curves of ρecoh and ρeincoh under the
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4. CONCLUSION The present numerical analyses using the XCOM photon cross-sections database revealed that a single linear relationship between HU and ρ e can be achieved by compensation among the three partial components of photon interactions. If we assume a criterion of ρecal /ρ e − 1 for the reliably calibrated ρ e to be within ±2%, the predicted upper limit of the HU–ρ e conversion under the well-calibrated condition is Z = 27. On the other hand, the corresponding limit is approximately 16 under the poorly calibrated condition. The deviation from the HU–ρ e linearity for higher Z substances is primarily caused by the anomalous variation in the photoelectricabsorption component. Nevertheless, the values of Zeff for most human tissues fall within this Z range and are therefore applicable to the HU–ρ e conversion, even for considerably ill-conditioned DECT scans in which there exists a large variation of effective x-ray energies owing to beam-hardening effects arising from the mismatch between the sizes of the object and the calibration phantom. ACKNOWLEDGMENT This work was partially supported by a Grant-in-Aid for Scientific Research (C) (25461908) from the Japan Society for the Promotion of Science (JSPS). a) Electronic
mail: [email protected]
1 R. P. Parker, P. A. Hobday, and K. J. Cassell, “The direct use of CT numbers
pe
coh w incoh F IG . 5. (a) μk /μw /μw k , μk /μk , and μk k (k = H, L) as functions of Z = 1–40 of the elementary substances under the well-calibrated and poorly calibrated conditions for 80–140 kV/Sn. (b) Corresponding partial components pe ρe , ρecoh , ρeincoh , along with their total: ρecal . The coordinates for the ρeincoh curves were shifted down by 0.2 to avoid overlap with the ρecal curve.
poorly calibrated condition almost coincided with those under the well-calibrated condition. However, the correspondpe ing ρe curve starts to deviate downward at approximately Z = 13 in the well-calibrated case for higher Z substances. As a result, we can confirm the findings presented in Sec. 3.B, i.e., that the limit of Z for the HU–ρ e conversion method decreases to approximately 16 under the poorly calibrated condition. Medical Physics, Vol. 41, No. 7, July 2014
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