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Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials

Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials DE GRUYTER Current Directions in Biomedical Engineering 2020;6(3): 20203135 Zehra Ese* and Waldemar Zylka Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials https://doi.org/10.1515/cdbme-2020-3135 terial. Typically, materials are mixtures of chemical elements and 𝜌 must be determined using the stoichiometric or another Abstract: Nowadays, patients with metallic implants under- approach [7]. going radiotherapy may suffer from inaccuracy in the treat- In a clinical environment, CT images, particularly ment plan caused by the implant. To ensure a precise plan an Hounsfield numbers representing the material or tissue, are accurate relation between Hounsfield values of the computer used to obtain 𝜌 . The majority of commercial CT machines tomographic (CT) images and the electron density of the ele- implement a conventional 12–bit scale which is suitable for ments and material mixtures is indispensable. In order to ex- soft tissue and bones. For metallic materials the use of an ex- tend the stoichiometric calibration approach known for tissues tended scale, e.g. 16–bit, is highly beneficial as it maps high–𝑍 to the regime of metallic materials, the basic physical equa- materials, like metals, without an ambitious failure [1–3]. tions as well as approximations in the parametrization and fit- A stoichiometric calibration is usually based on the factor- ting are carefully reviewed. CT images of a standard calibra- ization of the cross–section in terms of functions of 𝑍 and 𝐸. tion phantom and pure metallic samples up to the atomic num- This, however, conflicts with rigorous basic physics ab initio ber 𝑍 = 29 were acquired for various energies. Hounsfield val- calculations rendering such factorization impossible, at least ues were determined on an extended Hounsfield scale which inaccurate [5]. Since the exact physics formulas are hard to allows the mapping of material having high atomic number compute, particularly for material compounds and polychro- 𝑍 . It is found that from basic physics an empirical factoriza- matic X–ray distribution, an empirical parameterization pro- tion of the cross–sections into a function of 𝑍 and a function cedure based on factorization is used for tissue like materials of photon energy 𝐸 is not allowed over a wide range of 𝑍 . in the (narrow) diagnostic energy range 𝐸 = 80 − 140 keV. Specifically, the parameterization for tissue like materials can- This approach is very convenient as the coefficients can be fit- not be prolonged to materials with high–𝑍 . Thus, the calibra- ted to measured CT numbers, thus coping with (vendor and tion is subdivided into regions of materials and its accuracy scanner dependent) spectral X–ray energy distribution. Once is quantified in each region. It depends, among others, on the fitted to a calibration–material set, one is able to predict the knowledge of the X–ray photon spectra, the segmentation of electron density or CT numbers of other materials. the material samples and the empirical parameterization of the In this work, we apply the stoichiometric calibration used linear–attenuation coefficient. for biological tissues to mixtures of 𝑍 ≤ 29 materials, thus, Keywords: computed tomography, calibration, extended potentially violating some of its assumptions. One objective is Hounsfield Units, stoichiometric calibration, electron density, to systematically review all assumptions imposed on the sto- radio therapy ichiometric calibration used in the clinical environment. The results of a standard calibration procedure for 𝐸 = 80 kV and 𝐸 = 120 kV are presented in detail, the validity and accuracy 1 Introduction are investigated for high 𝑍 materials. The accuracy of radiotherapy treatment of patients with metal- lic implants is still challenging as treatment plans rely on the accuracy of the electron density 𝜌 of the tissue or implant ma- 2 Methods and Materials Extended Hounsfield scale. In CT imaging the spatial distri- *Corresponding author: Zehra Ese, Department of Electrical bution of the linear x-ray attenuation coefficients 𝜇 represent- Engineering and Applied Natural Science, Westphalian University, Campus Gelsenkirchen, Germany, and Department of General ing the tissues is reconstructed into a matrix of voxels. The at- and Theoretical Electrical Engineering, University of tenuation coefficient 𝜇 of a particular image voxel is expressed Duisburg-Essen and CENIDE – Center of Nanointegration in (Hounsfield or) CT numbers 𝐻 using the linear function Duisburg-Essen, Bismarckstr. 81, 47048 Duisburg, Germany, e-mail: zehra.ese@stud.uni-due.de 𝐻 (𝑢) = 𝑠𝑢 + 𝑖 , (1) Waldemar Zylka, Department of Electrical Engineering and where 𝑢 = 1000𝜇/𝜇 is dimensionless with 𝜇 being the at- 𝑤 𝑤 Applied Natural Science, Westphalian University, Campus tenuation coefficient of water, 𝑠 is the slope and 𝑖 the intercept. Gelsenkirchen, Germany Open Access. © 2020 Zehra Ese and Waldemar Zylka, published by De Gruyte r. This work is licensed under the Creative Commons Attribution 4.0 License. 2 Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials We denote the unit of 𝐻, 𝑠 and 𝑖 by HU. Note, that 𝑠, 𝑖 and 𝑢 [5]. Specifically, none of the cross–section can be written as ∘ ∘ ∘ are stored in the DICOM file. Equation (1) can be used to rep- 𝜎 (𝐸, 𝑍 ) = 𝐾 (𝐸)𝐹 (𝑍 ), where∘ denotes one of the phys- 𝑖 𝑖 resent various scales, e.g. setting 𝑠 = 1 and 𝑖 = −1000 the ical processes mentioned above. Such factorization, however, traditional Hounsfield definition is found. Often CT scanners is (frequently) assumed to hold as it is very beneficial when use 12–bit when storing CT measurements. It has been found calculating 𝜇(̂︀𝑆, 𝑍 ). in [1, 2] that with this scale metallic materials provoke errors in By restricting 𝑍 to the vicinity of a reference element in a dose calculations since they are incorrectly mapped to scale’s mixture, e.g. carbon, a (potentially non-integer) Taylor expan- maximum. An extended scale, e.g. 16–bit, allows correct map- sion of 𝐹 (𝑍 ) is admissible at some level of accuracy [5, 6, 9]. ping [2]. Thus, by utilizing (3), 𝜇(̂︀𝑆, 𝑍 ) can be parameterized in 𝑍 di- ̂︀ X–ray attenuation-coefficient and electron density. rection by spectrum dependent expansion coefficients 𝐾 (𝑆). The linear attenuation coefficient of a mixture of 𝑀 chemical The primary benefit of a parameterization scheme is that its elements can be expressed, as a function of energy 𝐸, by: coefficients could be obtained by a least square regression (fit) to Hounsfield values of calibration materials measured with a ∑︁ 𝑒 𝑒 particular spectrum 𝑆(𝐸). This allows a subsequent prediction 𝜇(𝐸, 𝑍 ) = 𝜌 𝜎 (𝐸, 𝑍 ) , (2) 𝑖 𝑖 𝑖 of CT numbers and electron densities of new materials for the 𝑖=1 𝑒 particular energy spectrum. where 𝜌 = 𝜌𝑁 𝑤 𝑍 /𝐴 is the volume electron density 𝐴 𝑖 𝑖 𝑖 ∑︀ 𝑒 Simplifying the Taylor expansion in 𝑍 to one single power (electrons per unit volume) of the 𝑖–th element and 𝜌 = 𝜌 ∘ 𝑠 law, i.e. 𝐹 (𝑍 ) ∝ 𝑍 , one recovers the parameterization orig- the total electron density of the mixture (here and in the fol- inally introduced in [4] for elements near to oxygen and for lowing the 𝑖–summation runs from 1 to 𝑀 ). The mass den- 𝑒 spectra of the EMI scanner at 60 keV and 80 keV: sity of the mixture is 𝜌, 𝑁 is Avogadro’s number, 𝜎 , 𝑍 𝐴 𝑖 and 𝐴 are, respectively, the total scattering cross–section per 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑖𝑛𝑐𝑜ℎ 𝑟−1 𝜎 (𝐸, 𝑍 ) = 𝐾 𝑍 + 𝐾 𝑍 + 𝐾 𝑍 . (4) 𝑖 𝑖 𝑖 𝑖 𝑖 electron, the atomic number and atomic weight of the 𝑖–th el- ∘ ∘ ∘ The coefficients 𝐾 depend on energy only, 𝐾 = 𝐾 (𝐸), ement and 𝑤 is its proportion by weight. For pure elements the non-integer exponents 𝑝 = 4.62, 𝑞 = 2.86 were obtained 𝑀 = 𝑤 = 1. The total cross-section per atom is 𝜎 = 𝑍 𝜎 . 𝑖 𝑖 𝑖 from fitting to tabulated cross–sections and 𝑟 = 1 originates A CT system must at least be described as a polychro- from Compton scattering physics at 𝐸 ≫ 1 MeV. With the aid matic beam of photons (potentially) undergoing filtration be- of (2)–(4) it follows: fore crossing the material and being detected by an energy sen- (︁ )︁ sitive detector. As in a clinical setting, we assume the X–ray 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑖𝑛𝑐𝑜ℎ 𝑟−1 ̂︀ ̃︀ ̂︀ ̃︀ ̂︀ ̃︀ 𝜇(̂︀𝑆, 𝑍 ) = 𝜌 𝐾 𝑍 + 𝐾 𝑍 + 𝐾 𝑍 , spectrum 𝑆(𝐸) to be unknown, but normalized (to 1); for sin- (5) (︀ )︀ ∑︀ gle energy beams 𝑆(𝐸) = 𝛿 (𝐸 − 𝐸 ) holds. All monochro- 1/𝑠 𝑒 𝑒 𝑠 ̃︀ where 𝑍 = (𝜌 /𝜌 )𝑍 is a weighted (effective) 𝑖 𝑖 matic quantities in (2) must be averaged over 𝑆(𝐸) and will ̂︀ atomic number. The coefficients 𝐾 are functions of the spec- be denoted by a hat (̂︀), e.g. ∘ ∘ ̂︀ ̂︀ trum, i.e. 𝐾 = 𝐾 (𝑆). In practice, each scanning proto- 𝑚𝑎𝑥 ∫︁ col uses its own energy spectrum and algorithms, potentially 𝜇(̂︀𝑆, 𝑍 ) = 𝑑𝐸𝑆(𝐸)𝜇(𝐸, 𝑍 ) . (3) including additional corrections, e.g. during reconstruction, 𝑖 𝑖 which affect the energy spectrum and are included in 𝑆(𝐸) in this paper. While passing an object the beam of a polychromatic spectrum ̂︀ 𝑒 The parameterization coefficients 𝐾 (𝑆) must be de- is shifted to higher energies since the cross–sections 𝜎 (𝐸, 𝑍 ) termined for the particular 𝑆(𝐸) by a fitting procedure to are larger for low–energy photons. This beam–hardening ef- Hounsfield values, via (1), of calibration materials measured fect is position dependent and particularly prominent for high– with a calibration phantom (with known densities and elemen- 𝑍 materials. In other words, if beam–hardening is not ac- tal compositions). Among others, the least square regression counted for, the spectrum must be seen as dependent on po- procedure in [7] and [8] can be used. The former is a three pa- sition ⃗𝑟 in the material, i.e. 𝑆(𝐸,⃗𝑟). ̂︀ rameter polynomial fit to (5) with 𝐾 normalized to the (a pri- The total X–ray cross–section per atom 𝜎 in the energy ori known) attenuation coefficient 𝜇̂︀ of water. In the approach range 80–140 keV is a sum of cross–sections of three phys- ̂︀ 𝑝ℎ from [8], which was used in this investigation, 𝐾 of the cal- ical processes: photoelectric absorption, 𝜎 (𝐸, 𝑍 ), coherent 𝑖𝑛𝑐𝑜ℎ ̂︀ 𝑐𝑜ℎ ibration material are normalized by 𝐾 of water forming (Rayleigh) scattering, 𝜎 (𝐸, 𝑍 ), and incoherent (Compton) ∘ ∘ 𝑖𝑛𝑐𝑜ℎ ̂︀ ̂︀ ̂︀ 𝑖𝑛𝑐𝑜ℎ 𝑘 (𝑆) = 𝐾 /𝐾 to be obtained from a non–linear fit to scattering, 𝜎 (𝐸, 𝑍 ). For monochromatic beams exact for- (︂ )︂ mula for each of these cross–sections were calculated from 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑟−1 ̂︀ ̃︀ ̂︀ ̃︀ ̃︀ 𝜇̂︀ 𝜌 𝑘 𝑍 + 𝑘 𝑍 + 𝑍 𝑝ℎ 𝑐𝑜ℎ ̂︀ ̂︀ (𝑘 , 𝑘 ) = . fundamental physics and found that neither a cross–section nor 𝑒 𝑝−1 𝑞−1 𝑟−1 ̂︀𝑝ℎ ̂︀𝑐𝑜ℎ 𝜇̂︀ 𝜌 ̃︀ ̃︀ ̃︀ 𝑤 𝑤 𝑘 𝑍 + 𝑘 𝑍 + 𝑍 𝑤 𝑤 𝑤 their sum factorize into a function 𝐾 (𝐸) and a function 𝐹 (𝑍 ) (6) Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials 3 ∘ ∘ ̂︀ ̂︀ Note that this implies 𝐾 ≈ 𝐾 which might be inaccurate method, i.e. Eq. (6), are displayed. To quantify the error, mea- for elements or mixtures with atomic number or electron den- sured CT numbers were additionally set to 𝐻 ± 𝜎 . The 𝑚 𝑒𝑟𝑟 sity well away from that of water. Once the scanner–specific calculated CT number is expected to be within [𝐻 − 𝜎 , 𝑚 𝑒𝑟𝑟 ̂︀ coefficients 𝑘 are known for each energy spectrum, the values 𝐻 + 𝜎 ]. 𝑚 𝑒𝑟𝑟 𝑒 𝑒 of the relative electron density 𝜌 = 𝜌 /𝜌 of other materi- For material regions in the range 𝜌 = 3.7− 8.0 a consid- 𝑟𝑒𝑙 𝑤 𝑟𝑒𝑙 als and mixtures with known elemental composition can be erable mismatch between measured and calculated CT num- computed and look–up–tables for X–ray radiotherapy created bers is observed. This cannot be explained by the statistical using (6). errors 𝜎 and must be seen as a systematic error steaming, 𝑒𝑟𝑟 We employed a tissue characterization phantom model e.g. from CT image artefacts and an incorrect parametrisation 467 (Gammex) phantom containing known tissue equivalent of the cross–section. materials in cylindrical design (𝑑 = 28 mm, ℎ = 70 mm) Discrepancies between the calculated and the measured which was scanned using 80 keV and 120 keV spectra of the CT numbers due to the choice of the material segments can Siemens Somatom Force scanner. Additionally, four metal- be found in the region of tissue equivalent materials. This is lic coin–shaped samples provided by umicore®(𝑑 = 20 mm, shown in the inset in Fig. 1 for 𝜌 = 1 − 2. While for 𝑟𝑒𝑙 ℎ = 5 mm) from Al (𝑍 = 13), Cr (𝑍 = 24), Ti (𝑍 = 22), Cu 𝜌 ≤ 𝜌 the calculated CT numbers are comparable, dis- 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑠 𝑠 (𝑍 = 29) were scanned being in a self-constructed cubic wa- crepancies are noted for 𝜌 ≥ 𝜌 . For the 𝜌 = 1 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 80𝑘𝑉 80𝑘𝑉 ter phantom (ℎ = 350 mm) made of polystyrene material. The separation point Δ𝐻 = 42 HU, Δ𝐻 = 630 HU 𝑚𝑎𝑥 𝑚𝑖𝑛 120𝑘𝑉 120𝑘𝑉 𝑠 samples were fixed at a water equivalent plate made from RW3 and Δ𝐻 = 44 HU, Δ𝐻 = 251 HU. For 𝜌 = 𝑚𝑎𝑥 𝑚𝑖𝑛 𝑟𝑒𝑙 80𝑘𝑉 80𝑘𝑉 (PTW Freiburg GmbH) and submerged in the water phantom 1.69 we observed Δ𝐻 = 0 HU, Δ𝐻 = 1 HU and 𝑚𝑎𝑥 𝑚𝑖𝑛 120𝑘𝑉 120𝑘𝑉 to make sure, that the samples are fully covered by water. The Δ𝐻 = 10 HU, Δ𝐻 = 11 HU. Separating mate- 𝑚𝑎𝑥 𝑚𝑖𝑛 samples were positioned in a water depth of 3 cm, measured rials at 𝜌 = 1 shows an increasing improvement with in- 𝑟𝑒𝑙 from the sample surfaces to the waterline. All materials were creasing energy for the whole material area. In contrast, for subdivided into two segments separated by a relative electron 𝜌 = 1.69 we observe almost energy independence, espe- 𝑟𝑒𝑙 density 𝜌 . The least square fit was separately done in the cially for tissue equivalent materials. 𝑟𝑒𝑙 𝑠 𝑠 𝑠 𝜌 ≤ 𝜌 and in the 𝜌 > 𝜌 region. We used 𝜌 = 1.0 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 and 𝜌 = 1.69 representing water and SB3 cortical bone, 𝑟𝑒𝑙 respectively. 4 Discussion and Summary While former stoichiometric calibration for relative electron 3 Results density focused on human tissue substitutes, the present study extends the approach to metallic materials. Moving to high–𝑍 At 80 kV and 120 kV, respectively, the following mean CT materials requires review of basic physics and approximations numbers 𝐻 are determined from images: Al 2587 HU and traditionally used. Consequently, the parameterization of the 2071 HU, for Ti 8186 HU and 7273 HU, for Cr 9263 HU and cross–sections for metals apparently differs from that for ma- 9085 HU and for Cu 10916 HU and 12168 HU. The effect terials nearby human tissue. As metallic materials do not nec- of decreasing CT numbers with increasing tube voltage can essarily obey the calibration curve of tissues, the calibration be observed for almost all metal samples. The expression of should be done for various material regions. The computation cupping artifacts becomes stronger with increasing tube volt- of Hounsfield values of metallic materials from CT images is age, as well as with increasing atomic number 𝑍 , mainly due susceptible to beam hardening artifacts. Even if the statistical to beam hardening effects [1–3]. Since, we use a ROI of size error is small, the systematic error due to that artifact requires 1.8 mm x 6.6 mm at the central part of a material to determine additional quantification. the mean CT number 𝐻 and its error 𝜎 . The applicability of the stoichiometric calibration based 𝑚 𝑒𝑟𝑟 ̂︀ The 𝑘 values were calculated segmentally for two sep- on parametrization from [4] is limited in the presence of metal- aration points 𝜌 given above. Figure 1 shows the relative lic materials. Among others, accuracy and robustness relies on 𝑟𝑒𝑙 electron densities 𝜌 along all materials versus the corre- the division of materials in segments and the knowledge of the 𝑟𝑒𝑙 sponding CT numbers. Figure 1a vs. Fig. 1c and Fig. 1b X–ray photon spectra. Specific spectra can, however, be incor- vs.Fig. 1d display the comparison of the fits in two regions porated into the approach. A one power–law parameterization separated by the separation points 𝜌 and obtained at 80 kV for wide 𝑍 ranges remains in conflict with rigourous physics 𝑟𝑒𝑙 and 120 kV, respectively. In each sub–figure the measured CT and must be replaced by an improved expression. numbers and those calculated by the stoichiometric calibration 4 Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials 𝑠 𝑠 (a) 𝐸 = 80 kV, separation at 𝜌 = 1 (b) 𝐸 = 120 kV, separation at 𝜌 = 1 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑠 𝑠 (c) 𝐸 = 80 kV, separation at 𝜌 = 1.69 (d) 𝐸 = 120 kV, separation at 𝜌 = 1.69 𝑟𝑒𝑙 𝑟𝑒𝑙 Fig. 1: The relative electron density 𝜌 versus measured and calculated CT numbers at 80 kV and 120 kV for Gammex 467 phan- 𝑟𝑒𝑙 tom materials and metals. Measured CT numbers are displayed with the error of the means (bars). The least square fits are shown as dashed lines. At each energy, two different fittings are compared (a) vs (c) and (b) vs (d), they differ in separation point 𝜌 . 𝑟𝑒𝑙 Author Statement [4] Rutherford R A, Pullan B R, Isherwood I. Measurement of effective atomic number and electron density using an EMI Research funding: The author state no funding involved. Con- scanner. Neuroradiology 1976;11(1):15-21. flict of interest: Authors state no conflict of interest. [5] Jackson D F, Hawkes D J. X-ray attenuation coefficients of elements and mixtures. Elsevier - Physics Reports 1981;70(3):169-233. [6] R G Ouellet R G and Schreiner L J. A parameterization of the References mass attenuation coefficients for elements with Z=1 to Z=92 in the photon energy range from approximately 1 to 150 keV. [1] Ese Z, Kressmann M, Kreutner J, Schaefers G, Erni D, Zylka Physics in Medicine and Biology 1991;36:987-999. W. Influence of conventional and extended CT scale range [7] Schneider U, Pedroni E, Lomax A. The calibration of CT on quantification of Hounsfield units of medical implants and Hounsfield units for radiotherapy treatment planning. Physics metallic objects. technisches messen - tm. 2018; 85(5):343- in Medicine and Biology 1996;41:111-124. [8] Schneider W, Bortfeld T and Schlegel W. Correlation be- [2] Ese Z, Qamhiyeh S, Kreutner J, Schaefers G, Erni D, Zylka tween CT numbers and tissue parameters needed for Monte W. CT Extended Hounsfield Unit range in radiotherapy treat- Carlo simulations of clinical dose distributions. Physics in ment planning for patients with implantable medical devices. Medicine and Biology 2000;45:459–478. Springer Nature Singapore IFMBE Proc 2019;68(3),599-603. [9] Midgley S M. A parameterization scheme for the x-ray linear [3] Ese Z, Zylka W. Influence of 12-bit and 16-bit CT values of attenuation coefficient and energy absorption coefficient. metals on dose calculation in radiotherapy using PRIMO, a Physics in Medicine and Biology 2004; 49:307-325. Monte Carlo code for clinical linear accelerators, Biomed Eng - Biomed Tech 2019;5(1):597-600. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Current Directions in Biomedical Engineering de Gruyter

Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials

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Abstract

DE GRUYTER Current Directions in Biomedical Engineering 2020;6(3): 20203135 Zehra Ese* and Waldemar Zylka Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials https://doi.org/10.1515/cdbme-2020-3135 terial. Typically, materials are mixtures of chemical elements and 𝜌 must be determined using the stoichiometric or another Abstract: Nowadays, patients with metallic implants under- approach [7]. going radiotherapy may suffer from inaccuracy in the treat- In a clinical environment, CT images, particularly ment plan caused by the implant. To ensure a precise plan an Hounsfield numbers representing the material or tissue, are accurate relation between Hounsfield values of the computer used to obtain 𝜌 . The majority of commercial CT machines tomographic (CT) images and the electron density of the ele- implement a conventional 12–bit scale which is suitable for ments and material mixtures is indispensable. In order to ex- soft tissue and bones. For metallic materials the use of an ex- tend the stoichiometric calibration approach known for tissues tended scale, e.g. 16–bit, is highly beneficial as it maps high–𝑍 to the regime of metallic materials, the basic physical equa- materials, like metals, without an ambitious failure [1–3]. tions as well as approximations in the parametrization and fit- A stoichiometric calibration is usually based on the factor- ting are carefully reviewed. CT images of a standard calibra- ization of the cross–section in terms of functions of 𝑍 and 𝐸. tion phantom and pure metallic samples up to the atomic num- This, however, conflicts with rigorous basic physics ab initio ber 𝑍 = 29 were acquired for various energies. Hounsfield val- calculations rendering such factorization impossible, at least ues were determined on an extended Hounsfield scale which inaccurate [5]. Since the exact physics formulas are hard to allows the mapping of material having high atomic number compute, particularly for material compounds and polychro- 𝑍 . It is found that from basic physics an empirical factoriza- matic X–ray distribution, an empirical parameterization pro- tion of the cross–sections into a function of 𝑍 and a function cedure based on factorization is used for tissue like materials of photon energy 𝐸 is not allowed over a wide range of 𝑍 . in the (narrow) diagnostic energy range 𝐸 = 80 − 140 keV. Specifically, the parameterization for tissue like materials can- This approach is very convenient as the coefficients can be fit- not be prolonged to materials with high–𝑍 . Thus, the calibra- ted to measured CT numbers, thus coping with (vendor and tion is subdivided into regions of materials and its accuracy scanner dependent) spectral X–ray energy distribution. Once is quantified in each region. It depends, among others, on the fitted to a calibration–material set, one is able to predict the knowledge of the X–ray photon spectra, the segmentation of electron density or CT numbers of other materials. the material samples and the empirical parameterization of the In this work, we apply the stoichiometric calibration used linear–attenuation coefficient. for biological tissues to mixtures of 𝑍 ≤ 29 materials, thus, Keywords: computed tomography, calibration, extended potentially violating some of its assumptions. One objective is Hounsfield Units, stoichiometric calibration, electron density, to systematically review all assumptions imposed on the sto- radio therapy ichiometric calibration used in the clinical environment. The results of a standard calibration procedure for 𝐸 = 80 kV and 𝐸 = 120 kV are presented in detail, the validity and accuracy 1 Introduction are investigated for high 𝑍 materials. The accuracy of radiotherapy treatment of patients with metal- lic implants is still challenging as treatment plans rely on the accuracy of the electron density 𝜌 of the tissue or implant ma- 2 Methods and Materials Extended Hounsfield scale. In CT imaging the spatial distri- *Corresponding author: Zehra Ese, Department of Electrical bution of the linear x-ray attenuation coefficients 𝜇 represent- Engineering and Applied Natural Science, Westphalian University, Campus Gelsenkirchen, Germany, and Department of General ing the tissues is reconstructed into a matrix of voxels. The at- and Theoretical Electrical Engineering, University of tenuation coefficient 𝜇 of a particular image voxel is expressed Duisburg-Essen and CENIDE – Center of Nanointegration in (Hounsfield or) CT numbers 𝐻 using the linear function Duisburg-Essen, Bismarckstr. 81, 47048 Duisburg, Germany, e-mail: zehra.ese@stud.uni-due.de 𝐻 (𝑢) = 𝑠𝑢 + 𝑖 , (1) Waldemar Zylka, Department of Electrical Engineering and where 𝑢 = 1000𝜇/𝜇 is dimensionless with 𝜇 being the at- 𝑤 𝑤 Applied Natural Science, Westphalian University, Campus tenuation coefficient of water, 𝑠 is the slope and 𝑖 the intercept. Gelsenkirchen, Germany Open Access. © 2020 Zehra Ese and Waldemar Zylka, published by De Gruyte r. This work is licensed under the Creative Commons Attribution 4.0 License. 2 Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials We denote the unit of 𝐻, 𝑠 and 𝑖 by HU. Note, that 𝑠, 𝑖 and 𝑢 [5]. Specifically, none of the cross–section can be written as ∘ ∘ ∘ are stored in the DICOM file. Equation (1) can be used to rep- 𝜎 (𝐸, 𝑍 ) = 𝐾 (𝐸)𝐹 (𝑍 ), where∘ denotes one of the phys- 𝑖 𝑖 resent various scales, e.g. setting 𝑠 = 1 and 𝑖 = −1000 the ical processes mentioned above. Such factorization, however, traditional Hounsfield definition is found. Often CT scanners is (frequently) assumed to hold as it is very beneficial when use 12–bit when storing CT measurements. It has been found calculating 𝜇(̂︀𝑆, 𝑍 ). in [1, 2] that with this scale metallic materials provoke errors in By restricting 𝑍 to the vicinity of a reference element in a dose calculations since they are incorrectly mapped to scale’s mixture, e.g. carbon, a (potentially non-integer) Taylor expan- maximum. An extended scale, e.g. 16–bit, allows correct map- sion of 𝐹 (𝑍 ) is admissible at some level of accuracy [5, 6, 9]. ping [2]. Thus, by utilizing (3), 𝜇(̂︀𝑆, 𝑍 ) can be parameterized in 𝑍 di- ̂︀ X–ray attenuation-coefficient and electron density. rection by spectrum dependent expansion coefficients 𝐾 (𝑆). The linear attenuation coefficient of a mixture of 𝑀 chemical The primary benefit of a parameterization scheme is that its elements can be expressed, as a function of energy 𝐸, by: coefficients could be obtained by a least square regression (fit) to Hounsfield values of calibration materials measured with a ∑︁ 𝑒 𝑒 particular spectrum 𝑆(𝐸). This allows a subsequent prediction 𝜇(𝐸, 𝑍 ) = 𝜌 𝜎 (𝐸, 𝑍 ) , (2) 𝑖 𝑖 𝑖 of CT numbers and electron densities of new materials for the 𝑖=1 𝑒 particular energy spectrum. where 𝜌 = 𝜌𝑁 𝑤 𝑍 /𝐴 is the volume electron density 𝐴 𝑖 𝑖 𝑖 ∑︀ 𝑒 Simplifying the Taylor expansion in 𝑍 to one single power (electrons per unit volume) of the 𝑖–th element and 𝜌 = 𝜌 ∘ 𝑠 law, i.e. 𝐹 (𝑍 ) ∝ 𝑍 , one recovers the parameterization orig- the total electron density of the mixture (here and in the fol- inally introduced in [4] for elements near to oxygen and for lowing the 𝑖–summation runs from 1 to 𝑀 ). The mass den- 𝑒 spectra of the EMI scanner at 60 keV and 80 keV: sity of the mixture is 𝜌, 𝑁 is Avogadro’s number, 𝜎 , 𝑍 𝐴 𝑖 and 𝐴 are, respectively, the total scattering cross–section per 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑖𝑛𝑐𝑜ℎ 𝑟−1 𝜎 (𝐸, 𝑍 ) = 𝐾 𝑍 + 𝐾 𝑍 + 𝐾 𝑍 . (4) 𝑖 𝑖 𝑖 𝑖 𝑖 electron, the atomic number and atomic weight of the 𝑖–th el- ∘ ∘ ∘ The coefficients 𝐾 depend on energy only, 𝐾 = 𝐾 (𝐸), ement and 𝑤 is its proportion by weight. For pure elements the non-integer exponents 𝑝 = 4.62, 𝑞 = 2.86 were obtained 𝑀 = 𝑤 = 1. The total cross-section per atom is 𝜎 = 𝑍 𝜎 . 𝑖 𝑖 𝑖 from fitting to tabulated cross–sections and 𝑟 = 1 originates A CT system must at least be described as a polychro- from Compton scattering physics at 𝐸 ≫ 1 MeV. With the aid matic beam of photons (potentially) undergoing filtration be- of (2)–(4) it follows: fore crossing the material and being detected by an energy sen- (︁ )︁ sitive detector. As in a clinical setting, we assume the X–ray 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑖𝑛𝑐𝑜ℎ 𝑟−1 ̂︀ ̃︀ ̂︀ ̃︀ ̂︀ ̃︀ 𝜇(̂︀𝑆, 𝑍 ) = 𝜌 𝐾 𝑍 + 𝐾 𝑍 + 𝐾 𝑍 , spectrum 𝑆(𝐸) to be unknown, but normalized (to 1); for sin- (5) (︀ )︀ ∑︀ gle energy beams 𝑆(𝐸) = 𝛿 (𝐸 − 𝐸 ) holds. All monochro- 1/𝑠 𝑒 𝑒 𝑠 ̃︀ where 𝑍 = (𝜌 /𝜌 )𝑍 is a weighted (effective) 𝑖 𝑖 matic quantities in (2) must be averaged over 𝑆(𝐸) and will ̂︀ atomic number. The coefficients 𝐾 are functions of the spec- be denoted by a hat (̂︀), e.g. ∘ ∘ ̂︀ ̂︀ trum, i.e. 𝐾 = 𝐾 (𝑆). In practice, each scanning proto- 𝑚𝑎𝑥 ∫︁ col uses its own energy spectrum and algorithms, potentially 𝜇(̂︀𝑆, 𝑍 ) = 𝑑𝐸𝑆(𝐸)𝜇(𝐸, 𝑍 ) . (3) including additional corrections, e.g. during reconstruction, 𝑖 𝑖 which affect the energy spectrum and are included in 𝑆(𝐸) in this paper. While passing an object the beam of a polychromatic spectrum ̂︀ 𝑒 The parameterization coefficients 𝐾 (𝑆) must be de- is shifted to higher energies since the cross–sections 𝜎 (𝐸, 𝑍 ) termined for the particular 𝑆(𝐸) by a fitting procedure to are larger for low–energy photons. This beam–hardening ef- Hounsfield values, via (1), of calibration materials measured fect is position dependent and particularly prominent for high– with a calibration phantom (with known densities and elemen- 𝑍 materials. In other words, if beam–hardening is not ac- tal compositions). Among others, the least square regression counted for, the spectrum must be seen as dependent on po- procedure in [7] and [8] can be used. The former is a three pa- sition ⃗𝑟 in the material, i.e. 𝑆(𝐸,⃗𝑟). ̂︀ rameter polynomial fit to (5) with 𝐾 normalized to the (a pri- The total X–ray cross–section per atom 𝜎 in the energy ori known) attenuation coefficient 𝜇̂︀ of water. In the approach range 80–140 keV is a sum of cross–sections of three phys- ̂︀ 𝑝ℎ from [8], which was used in this investigation, 𝐾 of the cal- ical processes: photoelectric absorption, 𝜎 (𝐸, 𝑍 ), coherent 𝑖𝑛𝑐𝑜ℎ ̂︀ 𝑐𝑜ℎ ibration material are normalized by 𝐾 of water forming (Rayleigh) scattering, 𝜎 (𝐸, 𝑍 ), and incoherent (Compton) ∘ ∘ 𝑖𝑛𝑐𝑜ℎ ̂︀ ̂︀ ̂︀ 𝑖𝑛𝑐𝑜ℎ 𝑘 (𝑆) = 𝐾 /𝐾 to be obtained from a non–linear fit to scattering, 𝜎 (𝐸, 𝑍 ). For monochromatic beams exact for- (︂ )︂ mula for each of these cross–sections were calculated from 𝑒 𝑝ℎ 𝑝−1 𝑐𝑜ℎ 𝑞−1 𝑟−1 ̂︀ ̃︀ ̂︀ ̃︀ ̃︀ 𝜇̂︀ 𝜌 𝑘 𝑍 + 𝑘 𝑍 + 𝑍 𝑝ℎ 𝑐𝑜ℎ ̂︀ ̂︀ (𝑘 , 𝑘 ) = . fundamental physics and found that neither a cross–section nor 𝑒 𝑝−1 𝑞−1 𝑟−1 ̂︀𝑝ℎ ̂︀𝑐𝑜ℎ 𝜇̂︀ 𝜌 ̃︀ ̃︀ ̃︀ 𝑤 𝑤 𝑘 𝑍 + 𝑘 𝑍 + 𝑍 𝑤 𝑤 𝑤 their sum factorize into a function 𝐾 (𝐸) and a function 𝐹 (𝑍 ) (6) Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials 3 ∘ ∘ ̂︀ ̂︀ Note that this implies 𝐾 ≈ 𝐾 which might be inaccurate method, i.e. Eq. (6), are displayed. To quantify the error, mea- for elements or mixtures with atomic number or electron den- sured CT numbers were additionally set to 𝐻 ± 𝜎 . The 𝑚 𝑒𝑟𝑟 sity well away from that of water. Once the scanner–specific calculated CT number is expected to be within [𝐻 − 𝜎 , 𝑚 𝑒𝑟𝑟 ̂︀ coefficients 𝑘 are known for each energy spectrum, the values 𝐻 + 𝜎 ]. 𝑚 𝑒𝑟𝑟 𝑒 𝑒 of the relative electron density 𝜌 = 𝜌 /𝜌 of other materi- For material regions in the range 𝜌 = 3.7− 8.0 a consid- 𝑟𝑒𝑙 𝑤 𝑟𝑒𝑙 als and mixtures with known elemental composition can be erable mismatch between measured and calculated CT num- computed and look–up–tables for X–ray radiotherapy created bers is observed. This cannot be explained by the statistical using (6). errors 𝜎 and must be seen as a systematic error steaming, 𝑒𝑟𝑟 We employed a tissue characterization phantom model e.g. from CT image artefacts and an incorrect parametrisation 467 (Gammex) phantom containing known tissue equivalent of the cross–section. materials in cylindrical design (𝑑 = 28 mm, ℎ = 70 mm) Discrepancies between the calculated and the measured which was scanned using 80 keV and 120 keV spectra of the CT numbers due to the choice of the material segments can Siemens Somatom Force scanner. Additionally, four metal- be found in the region of tissue equivalent materials. This is lic coin–shaped samples provided by umicore®(𝑑 = 20 mm, shown in the inset in Fig. 1 for 𝜌 = 1 − 2. While for 𝑟𝑒𝑙 ℎ = 5 mm) from Al (𝑍 = 13), Cr (𝑍 = 24), Ti (𝑍 = 22), Cu 𝜌 ≤ 𝜌 the calculated CT numbers are comparable, dis- 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑠 𝑠 (𝑍 = 29) were scanned being in a self-constructed cubic wa- crepancies are noted for 𝜌 ≥ 𝜌 . For the 𝜌 = 1 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 80𝑘𝑉 80𝑘𝑉 ter phantom (ℎ = 350 mm) made of polystyrene material. The separation point Δ𝐻 = 42 HU, Δ𝐻 = 630 HU 𝑚𝑎𝑥 𝑚𝑖𝑛 120𝑘𝑉 120𝑘𝑉 𝑠 samples were fixed at a water equivalent plate made from RW3 and Δ𝐻 = 44 HU, Δ𝐻 = 251 HU. For 𝜌 = 𝑚𝑎𝑥 𝑚𝑖𝑛 𝑟𝑒𝑙 80𝑘𝑉 80𝑘𝑉 (PTW Freiburg GmbH) and submerged in the water phantom 1.69 we observed Δ𝐻 = 0 HU, Δ𝐻 = 1 HU and 𝑚𝑎𝑥 𝑚𝑖𝑛 120𝑘𝑉 120𝑘𝑉 to make sure, that the samples are fully covered by water. The Δ𝐻 = 10 HU, Δ𝐻 = 11 HU. Separating mate- 𝑚𝑎𝑥 𝑚𝑖𝑛 samples were positioned in a water depth of 3 cm, measured rials at 𝜌 = 1 shows an increasing improvement with in- 𝑟𝑒𝑙 from the sample surfaces to the waterline. All materials were creasing energy for the whole material area. In contrast, for subdivided into two segments separated by a relative electron 𝜌 = 1.69 we observe almost energy independence, espe- 𝑟𝑒𝑙 density 𝜌 . The least square fit was separately done in the cially for tissue equivalent materials. 𝑟𝑒𝑙 𝑠 𝑠 𝑠 𝜌 ≤ 𝜌 and in the 𝜌 > 𝜌 region. We used 𝜌 = 1.0 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑟𝑒𝑙 and 𝜌 = 1.69 representing water and SB3 cortical bone, 𝑟𝑒𝑙 respectively. 4 Discussion and Summary While former stoichiometric calibration for relative electron 3 Results density focused on human tissue substitutes, the present study extends the approach to metallic materials. Moving to high–𝑍 At 80 kV and 120 kV, respectively, the following mean CT materials requires review of basic physics and approximations numbers 𝐻 are determined from images: Al 2587 HU and traditionally used. Consequently, the parameterization of the 2071 HU, for Ti 8186 HU and 7273 HU, for Cr 9263 HU and cross–sections for metals apparently differs from that for ma- 9085 HU and for Cu 10916 HU and 12168 HU. The effect terials nearby human tissue. As metallic materials do not nec- of decreasing CT numbers with increasing tube voltage can essarily obey the calibration curve of tissues, the calibration be observed for almost all metal samples. The expression of should be done for various material regions. The computation cupping artifacts becomes stronger with increasing tube volt- of Hounsfield values of metallic materials from CT images is age, as well as with increasing atomic number 𝑍 , mainly due susceptible to beam hardening artifacts. Even if the statistical to beam hardening effects [1–3]. Since, we use a ROI of size error is small, the systematic error due to that artifact requires 1.8 mm x 6.6 mm at the central part of a material to determine additional quantification. the mean CT number 𝐻 and its error 𝜎 . The applicability of the stoichiometric calibration based 𝑚 𝑒𝑟𝑟 ̂︀ The 𝑘 values were calculated segmentally for two sep- on parametrization from [4] is limited in the presence of metal- aration points 𝜌 given above. Figure 1 shows the relative lic materials. Among others, accuracy and robustness relies on 𝑟𝑒𝑙 electron densities 𝜌 along all materials versus the corre- the division of materials in segments and the knowledge of the 𝑟𝑒𝑙 sponding CT numbers. Figure 1a vs. Fig. 1c and Fig. 1b X–ray photon spectra. Specific spectra can, however, be incor- vs.Fig. 1d display the comparison of the fits in two regions porated into the approach. A one power–law parameterization separated by the separation points 𝜌 and obtained at 80 kV for wide 𝑍 ranges remains in conflict with rigourous physics 𝑟𝑒𝑙 and 120 kV, respectively. In each sub–figure the measured CT and must be replaced by an improved expression. numbers and those calculated by the stoichiometric calibration 4 Z. Ese et al., Extension of the Stoichiometric Calibration of CT Hounsfield values to Metallic Materials 𝑠 𝑠 (a) 𝐸 = 80 kV, separation at 𝜌 = 1 (b) 𝐸 = 120 kV, separation at 𝜌 = 1 𝑟𝑒𝑙 𝑟𝑒𝑙 𝑠 𝑠 (c) 𝐸 = 80 kV, separation at 𝜌 = 1.69 (d) 𝐸 = 120 kV, separation at 𝜌 = 1.69 𝑟𝑒𝑙 𝑟𝑒𝑙 Fig. 1: The relative electron density 𝜌 versus measured and calculated CT numbers at 80 kV and 120 kV for Gammex 467 phan- 𝑟𝑒𝑙 tom materials and metals. Measured CT numbers are displayed with the error of the means (bars). The least square fits are shown as dashed lines. At each energy, two different fittings are compared (a) vs (c) and (b) vs (d), they differ in separation point 𝜌 . 𝑟𝑒𝑙 Author Statement [4] Rutherford R A, Pullan B R, Isherwood I. Measurement of effective atomic number and electron density using an EMI Research funding: The author state no funding involved. Con- scanner. Neuroradiology 1976;11(1):15-21. flict of interest: Authors state no conflict of interest. [5] Jackson D F, Hawkes D J. X-ray attenuation coefficients of elements and mixtures. Elsevier - Physics Reports 1981;70(3):169-233. [6] R G Ouellet R G and Schreiner L J. A parameterization of the References mass attenuation coefficients for elements with Z=1 to Z=92 in the photon energy range from approximately 1 to 150 keV. [1] Ese Z, Kressmann M, Kreutner J, Schaefers G, Erni D, Zylka Physics in Medicine and Biology 1991;36:987-999. W. Influence of conventional and extended CT scale range [7] Schneider U, Pedroni E, Lomax A. The calibration of CT on quantification of Hounsfield units of medical implants and Hounsfield units for radiotherapy treatment planning. Physics metallic objects. technisches messen - tm. 2018; 85(5):343- in Medicine and Biology 1996;41:111-124. [8] Schneider W, Bortfeld T and Schlegel W. Correlation be- [2] Ese Z, Qamhiyeh S, Kreutner J, Schaefers G, Erni D, Zylka tween CT numbers and tissue parameters needed for Monte W. CT Extended Hounsfield Unit range in radiotherapy treat- Carlo simulations of clinical dose distributions. Physics in ment planning for patients with implantable medical devices. Medicine and Biology 2000;45:459–478. Springer Nature Singapore IFMBE Proc 2019;68(3),599-603. [9] Midgley S M. A parameterization scheme for the x-ray linear [3] Ese Z, Zylka W. Influence of 12-bit and 16-bit CT values of attenuation coefficient and energy absorption coefficient. metals on dose calculation in radiotherapy using PRIMO, a Physics in Medicine and Biology 2004; 49:307-325. Monte Carlo code for clinical linear accelerators, Biomed Eng - Biomed Tech 2019;5(1):597-600.

Journal

Current Directions in Biomedical Engineering – de Gruyter

Published: Sep 1, 2020

Keywords: computed tomography; calibration; extended Hounsfield Units; stoichiometric calibration; electron density; radio therapy

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