Theoretical study of a simplified implementation model of a dual-energy technique for computed tomography

Accepted Manuscript Theoretical study of a simplified implementation model of a dual-energy technique for computed tomography Sergei Osipov, Sergei Chakhlov, Andrey Batranin, Oleg Osipov, Van Bak Trinh, Juriy Kytmanov PII:

S0963-8695(17)30133-0

DOI:

10.1016/j.ndteint.2018.04.010

Reference:

JNDT 1978

To appear in:

NDT and E International

Received Date: 25 February 2017 Revised Date:

10 February 2018

Accepted Date: 16 April 2018

Please cite this article as: Osipov S, Chakhlov S, Batranin A, Osipov O, Trinh VB, Kytmanov J, Theoretical study of a simplified implementation model of a dual-energy technique for computed tomography, NDT and E International (2018), doi: 10.1016/j.ndteint.2018.04.010. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Theoretical Study of a Simplified Implementation Model of a Dual-Energy Technique for Computed Tomography

Sergei Osipova , Sergei Chakhlova,∗, Andrey Batranina , Oleg Osipova , Van Bak Trinha , Juriy Kytmanova Polytechnic University, Savinykh Street, 7, 634028, Russia

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Abstract

The article describes a theoretical study of the simplified implementation of dual energy technique (DET) for computed tomography (CT). The implementation is based on the X-ray pre-filter. Two sets of projections, acquired for two maximal X-ray energies, are transformed to the projections of DET parameters. The density and effective atomic number of the test object fragments are estimated after a separate recovery of the internal structure of the test object for each of

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the DET parameters. The choise of pre-filter thickness is discussed. An example of the initial projections simulation and the estimation of the internal structure of cylindrical objects are shown.

Keywords: X-ray radiation, dual energy technique, computed tomography,

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pre-filter X-ray tube

1. Introduction

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The dual energy technique (DET) was originally developed for the CT to

compensate the X-ray beam hardening in case an attenuating object thickness increases [1, 2, 3]. DET is based on a representation of mass attenuation coeffi-

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cient (MAC) by a sum of members consisted of two multiplicative components. ∗ Corresponding

author, tel. +7 913 822 2194 Email addresses: [email protected] (Sergei Osipov), [email protected] (Sergei Chakhlov), [email protected] (Andrey Batranin), [email protected] (Oleg Osipov), [email protected] (Van Bak Trinh), [email protected] (Juriy Kytmanov)

Preprint submitted to NDT & E International

April 16, 2018

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First component depends on the energy only and has the same description for

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all attenuating materials. Second component is defined by the parameters of the test object (TO) and its material. The mentioned parameters are density

and the effective atomic number. Representation of the MAC has defined the 10

second direction of DET application in digital radiography (DR) and computed tomography (CT). The integration of DR and DET has led to the development

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of discrimination methods for materials and their separate fragments [4, 5, 6].

With respect to CT DET allows not only to reduce the beam hardening artifacts, but also to evaluate the distributions of the density and effective atomic number or DET parameters for the volume of the test object [7, 8, 9]. Note

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that the physical X-ray pre-filters are used to reduce the X-ray hardening effect and the range of the dynamic of the X-ray signal [10, 11, 12, 13]. There are monochromatic DET implementations [14, 15], in which the information processing algorithms are simple and have a high speed. Work [16] 20

emphasized that the use of pre-filtering allows approximately associate the transformed X-rays with the monochromatic X-ray source. This hardened X-ray

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beam is called pseudo-monochromatic [17]. It is natural to assume that the choice of the pre-filter thickness essentially depends on the task at hand. The scientific literature does not fully discuss the choice of the pre-filter thickness for 25

the pseudo-monochromatic DET implementation in CT for the maximal X-ray

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energies up to 250 keV. In particular, the criteria to select thickness are not formulated in a closed form.

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2. Theoretical basis

To understand the theoretical basis of the pseudo-monochromatic DET im-

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plementation we give a brief description of the monochromatic and classic DET implementations and discuss the possible limits of these methods. 2.1. Monochromatic DET implementation We use one of the options to describe a dependence of measured dynamic of the X-ray signals from DET parameters by expressions similar to [1, 2, 3]. 2

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Let the test object with the mass thickness ρH and the effective atomic num-

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ber Z is penetrated by monochromatic X-rays with energies E1 and E2 . The input of DET algorithm has the digitized signals without object Id (E1 , 0, Z),

Id (E2 , 0, Z) and with object Id (E1 , ρH, Z), Id (E2 , ρH, Z). The first stage of the algorithm estimates the TO thicknesses in mean free paths (MFP) Y (E1 , ρH, Z), Y (E2 , ρH, Z). The corresponding expressions have form Id (E1 , ρH, Z) , Id (E1 , 0, Z) Id (E2 , ρH, Z) Y (E2 , ρH, Z) = − ln . Id (E2 , 0, Z)

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Y (E1 , ρH, Z) = − ln

(1)

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For the range of monochromatic X-ray energies, not exceeding 250 keV, the dependence of the thicknesses in mean free paths (MFP) Y (E1 , ρH, Z), Y (E2 , ρH, Z) from DET parameters A and B is described by formulas Y (E1 , ρH, Z) = m(E1 , Z)ρH ≈ Afphoto (E1 ) + BfCompt (E1 )

(2)

Y (E2 , ρH, Z) = m(E2 , Z)ρH ≈ Afphoto (E2 ) + BfCompt (E2 ),

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where m(E1 , Z), m(E2 , Z) is MAC of monochromatic X-ray with energies E1 and E2 for the material with effective atomic number Z; fphoto (E), fCompt (E) 35

is energy dependencies due to photoelectric and Compton effects. The sign of approximate equality ≈ is used in the formulas (2) intentionally. This sign reminds that the variants of interaction of X-ray with matter are not

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limited to the photoelectric and Compton effects. The effect of Rayleigh has a significant influence in the range of low-energy X-ray, as evidenced by data of the interaction of X-ray with matter given in the relevant databases [18]. For the analyzed range of X-ray the DET parameters are associated with

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the TO characteristics by relations [16] B = ρH, A = Z β ρH.

(3)

The theoretical value of the exponent β is 4 [2], in practice, the value β = 3.8

is used more often [19]. Strictly speaking, the β parameter value is determined by the range of the used energies and the range of effective atomic number.

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For the approximation of energy dependence fphoto (E) a power function is

fphoto (E) =

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used 1 . eα

In the scientific literature it is recommended to use different values of the 45

parameter α, for example, α = 3 [2, 20], α = 3.2 [12, 19] etc. Most likely, these

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discrepancies are related to the underestimation of the comment made above for the parameter β.

The dependence fCompt (E) is described by the expression [2], based on the

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classic Klein-Nishina formula.   1 + α 2(1 + α) 1 1 1 + 3α fCompt (α) = − ln(1 + 2α) + ln(1 + 2α) − , α2 1 + 2α α 2α (1 + 2α)2 here α = E/511, energy E is given in keV.

In some cases this formula is not accurate due to the photon scattering by 50

free electrons only.

In real conditions the values of energies E1 and E2 are fixed. The corre-

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sponding measured values of thicknesses in MFP Y1 and Y2 are related with parameters A and B a system of two linear equations of the form (2). The solution of this system has the form of two-dimensional linear regression without displacement

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A = Y1 g22 + Y2 g12 ,

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B = Y1 g21 + Y2 g11 .

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The regression coefficients gij , i = 1, 2, j = 1, 2 have a strict physical interpretation, which follows from the expressions (2), g11 = −

fphoto (E1 ) fCompt (E1 ) fphoto (E2 ) fCompt (E2 ) , g12 = − , g21 = − , g22 = − , r r r r

r =fphoto (E1 )fCompt (E2 ) − fphoto (E2 )fCompt (E1 ). In practice the theoretical expressions are meaningful only for preliminary

estimation of the choice correctness for energies E1 and E2 . The value of the auxiliary parameter r must be significantly different from zero. 4

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In experimental applications the two-dimensional linear regression coefficients gij , i = 1, 2, j = 1, 2 are on the calibration stage. The fragments of the

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relevant test object should cover the entire area of consumer interest for the ranges of mass thickness ρH and effective atomic number Z. 2.2. Classic DET implementation

In the classic DET implementation the primary information processing is

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the same as for monochromatic one, and it is reduced to the calculation of the thicknesses MFP Y (E1 , ρH, Z), Y (E2 , ρH, Z). The difference is the usage of

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X-ray sources with maximum energies E1 and E2 . In this case the system for DET parameters A and B has form [1, 16] which is more complicated than system (4), R E1 Y (E1 , ρH, Z) = − ln

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R E2 Y (E2 , ρH, Z) = − ln

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F (E, E1 )Eab (E)ε(E)e−Afphoto (E)−BfCompt (E) dE , R E1 F (E, E1 )Eab (E)ε(E)dE 0 F (E, E2 )Eab (E)ε(E)e−Afphoto (E)−BfCompt (E) dE , R E2 F (E, E )E (E)ε(E)dE 2 ab 0

(5)

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where F (E, Ei ), i = 1, 2 are the digital energy spectra of X-ray with maximum energies Ei ; Eab (E) is a mean value of the absorbed energy of registered photon with energy E; ε(E) is an energy distribution of the detector registration efficiency.

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In work [16] the main approaches to solving a system of non-linear parametric equations (5) were analyzed. The classical solution of the system requires 65

an analytical description of all necessary functions. Above it mentioned that

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there are some problems in the description of energy dependencies fphoto (E), fCompt (E). Known approaches [21, 22] to evaluate the Eab (E) dependencies in view of leakage secondary photons and electrons are in need of updating. Above all, this is due to the miniaturization of sensitive volumes of X-ray de-

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tectors. The existing approximation of the X-ray energy spectra, for example, [23, 24, 25] do not have the accuracy levels required for the analyzed problem. It should also be noted that the DET models described by systems (2) and (5) can be sufficiently far from reality. The influence of different physical and 5

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technical factors, leading to the displacement of the DET parameter estimates, is discussed in [26].

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The high-performance and high-efficiency methods to evaluate DET parameters A and B proposed in [16] required the use of special complex test objects.

The use of pre-filter allows us to consider the transformable X-rays radiation

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2.3. Pseudo-monochromatic DET implementation

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as pseudo-monochromatic one. The following briefly outlines the method basics.

The X-ray pre-filter leads to the transformation of the original power spec-

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trum. As a result, the radiation quality is improved. When the initial digital radiographs are acquired by two scans, it is advisable to use filters with different thickness h1 and h2 . For one scan h1 = h2 . Taking into account the filter, system (5) takes the form

(6)

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Y (E1 , ρH, Z) = R E1 F (E, E1 )Eab (E)ε(E)e−mf (E)ρf h1 e−Afphoto (E)−BfCompt (E) dE − ln 0 R E1 F (E, E1 )Eab (E)ε(E)e−mf (E)ρf h1 dE 0 Y (E2 , ρH, Z) = R E2 F (E, E2 )Eab (E)ε(E)e−mf (E)ρf h2 e−Afphoto (E)−BfCompt (E) dE − ln 0 R E2 F (E, E2 )Eab (E)ε(E)e−mf (E)ρf h2 dE 0

where mf (E) is the energy dependence of MAC for the filter material; ρf , hf

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are the density of filter material and its thickness. The system (6) is the starting point for the development of the mass thickness

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pre-filters h1 and h2 selection. To do this, you must define the criterion to select 85

the filter thickness. 2.4. The criterion to select the filter thickness The mass thickness of pre-filter ρf hf is selected by the simplification of

the DET implementation. The simplification is reduced to a hypothetical replacement of X-ray source with a continuous spectrum by a source of pseudo-

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monochromatic photon radiation. As a result of such replacement the DET

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parameter estimation algorithm is based on the system (4). The main problem

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is a selection of the tarnsformed X-ray energy spectrum close to a spectrum of certain monochromatic X-ray source.

When comparing the different DET implementations there are two basic 95

approaches [27] – by the degree of closeness of initial and final informative

parameters. It should be noted that the DET parameters for the analyzed

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problem are intermediate. The final parameters of CT, supplemented by DET are the density distribution ρ, and the effective atomic number Z, or a function of Z. It is evident that at the stage of preliminary calculations of pre-filter

thicknesses the most easily implemented approach is based on the estimation

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accuracy of DET parameters A and B. To study the transformation of the displacement parameters A and B into the displacement of distributions of the density ρ and the effective atomic number Z it is convenient to use an axisymmetric test object, for example, [28, 29, 30], and apply the Abel inversion 105

of the initial projections [31, 32, 33, 34].

We give appropriate expressions to illustrate how use the Abel inversion.

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Let the test object is a cylinder with radius R, see. Fig. 1. Suppose we have a radial distribution of a parameter q. This parameter may be a MAC (for monochromatic X-ray source) or the DET parameters a and b. Let P (x) is a distribution of the parameter q integrated along a ray (projec-

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tion). Then the relationship of the radial distribution q(r) and the projection

(7)

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P (x) is described by an expression (Abel inversion) [31, 32, 33, 34] Z 1 R P 0 (x) √ q(r) = − dx. π r x2 − r 2

The projection A(x) and B(x) go to the input of the reconstruction al-

gorithm, and the distributions a(r) and b(r) are formed at the output. The estimates of the radial distributions of density and effective atomic number are derived from these distributions in accordance with the expressions (3) s a(r) ρ(r) = b(r), Z(r) = β . b(r)

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Expressions (7), (8) are derived for small size of the detectors aperture. 7

(8)

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Fig. 1. Test object and the unit radiographic projection

The expressions (1) – (8) are sufficient to investigate the influence of the

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pre-filter thickness on the evaluations of distributions of the density and the effective atomic number for the pseudo-monochromatic DET implementation. An approach based on the given maximum deviations of the final parameters can be a main criterion to select the pre-filters thicknesses. Let ∆ρ and ∆Z

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are the user defined maximal values of the systematic error estimation of the distributions of density and effective atomic number. Then the generalized criterion to select the pre-filter thickness h1 = 1 and h2 is described by the

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system of inequalities

|ρ(r, h1 , h2 ) − ρt (r)| ≤ ∆ρ; |Z(r, h1 , h2 ) − Zt (r)| ≤ ∆Z;

(ρt , H, Z) ∈ V,

(9)

there ρt (r) and Zt (r) are real distributions of the density and the effective

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atomic number; V is a discrete or continuous set, defined by the ranges of the

test object mass thickness and effective atomic numbers. The X-ray pre-filter decrease the examination performance, so an additional restriction of the criterion (9) is the permissible increase of measurement time in 8

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the specified number of times klim . Since the object examination performance

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is determined by its maximum thickness in MFP ρt Hmax , it is logical to calculate the time extension factor kg , caused by the pre-filtering of X-ray, by the expression

RE 2 ti 0 i F (E, Ei )Eab (E)ε(E)e−m(E,Z)ρt Hmax dE 1 X . (10) kg ≈ R t1 + t2 i=1 Ei F (E, Ei )Eab (E)ε(E)e−m(E,Z)ρi Hmax e−mf (E)ρf hi dE 0

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here ti is a measurement time for the X-ray radiation with the maximal energy Ei .

kg (h1 , h2 ) < klim .

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A further limitation to the system of inequalities (9) is as follows

(11)

Note that the set of inequalities (9) and (11) may not have a solution. In 120

this case, the user of the developed method should decide with his preferences. The user should choose what he can sacrifice – the density estimation precision, and (or) the effective atomic number estimation precision, and (or) examination

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performance.

An additional criterion is the workability, for example, the formation of pre125

filters from the plates of the same thickness. The foregoing is defined a generalized criterion to select the thicknesses of X-ray pre-filter for the CT method, complemented by pseudo-monochromatic

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DET implementation.

3. The influence of the pre-filter thickness on the accuracies of density and effective atomic number

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We investigate the influence of pre-filtering on the accuracies of density and

effective atomic number for two classes of axisymmetric objects. The first class includes homogeneous objects, and the second one – non-uniform. The simulation of initial projections (6), the transformation of projections

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into radial distributions of the DET parameters using expression (4) and the estimation of the radial density distributions and the effective atomic number 9

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by formulas (7), (8) were carried out by MathCad software. MathCad is widely

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used for imitation modeling and image analysis in radiography and CT [35, 36]. The recommendations from papers [33, 34, 37, 38] were used to estimate the 140

radial distributions of density and the effective atomic number by use the Abel inversion.

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3.1. Homogeneous cylindrical objects

We select to study the uniform cylindrical objects from carbon with density 2.26 g/cm3 , from aluminum with density 2.7 g/cm3 , from steel with density 7.86 g/cm3 and copper with density 8.92 g/cm3 . The diameter of cylinders is

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5 g/cm2 in mass thickness units. The maximal X-ray energies E1 = 100 keV and E2 = 225 keV are chosen from the condition of significance of the competing effects of the interaction of X-rays with matter [2, 8] – photoelectric effect and Compton effect for the testing materials from carbon to copper, taking into 150

account the change in the mass thickness ρH from 0 to 5 g/cm2 . Any material with a high density and high effective atomic number can be

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used as filter materials. Pre-filters should provide the necessary hardening of energy spectra. The choice of copper is due to the manufacturability of thin plates with thickness 0.5 mm and higher. The filter thicknesses should provide 155

the necessary levels of accuracy of the spatial density distributions and the effective atomic number and an allowable level of the time extension factor.

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Fig. 2 shows the radial dependence ρ(r) and Z(r) for cylinders made of carbon, aluminum, steel, copper and with the copper pre-filters thicknesses

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h1 = 1 mm and h2 = 5 mm. For these filter thicknesses the linear regression 160

coefficients (4) and the values of approximating parameter β were obtained by

least squares method for the DET parameter Z dependence on g11 = 9.3432; g12 = −19294; g21 = −1.6463; g22 = 17034; β = 2.73. The use of pre-filters for this calculation will increase the total scan time

approximately in 2.8 times. This significantly reduces the influence of X-ray

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beam hardening, and it is possible to estimate the radial distributions of density and effective atomic numbers with sufficient accuracy. The effect, observed on 10

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Fig. 2.

Radial dependences a – ρ(r) and b – Z(r) for the copper pre-filter thicknesses

h1 = 1 mm and h2 = 5 mm: — – carbon; — – aluminum; — – steel; — – copper

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the cylinder boundary, essentially depends on the effective atomic number of

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the test object material. For aluminum and carbon the offsets of estimations of density and effective atomic number are insignificant for the thicknesses of 170

selected filters.

As the parameters that characterize the effectiveness of the pre-filtering in DET we can use: the ranges of density changes (ρmin , ρmax ) and effective

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atomic numbers (Zmin , Zmax ); and the time extension factor kg . Table 1 shows these effectiveness parameters of the copper pre-filters usage in the pseudo175

monochromatic DET implementation in conjunction with CT.

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From the Table 1 analysis, several conclusions can be considered for the selected maximum X-ray energies. Increasing the pre-filter thickness over 3 mm is inappropriate for less energy and for higher energy – over 10 mm. The inexpediency caused a significant increase of the total scan time. There are significant 180

systematic errors for estimates of density and effective atomic number for the outer regions of the test object and less significant – for the inner regions. Hypothesis. The most likely reason of the offsets of estimates of the density

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and the effective atomic number is a non-adequate description of the energy dependence of MAC by formulas (3) and (4). 185

Above it is noted that expression (4) does not take into account the effect of coherent scattering, which is significant for the considered materials and the

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energy of monochromatic X-rays less than 150 keV. To confirm or refute the hypothesis, we made some calculations to estimate the distributions of density and effective atomic number for mono-energetic DET implementation with the energies of monochromatic X-ray E1 = 9.5 keV, E2 =

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142 keV. The selected energies correspond approximately to the effective X-ray energies for the maximal energies E1 = 100 keV, E2 = 225 keV correspondingly. Above it was shown that the most significant offset of the unknown estimations is observed at the boundary of steel objects, so Fig. 3 shows the radial dependences

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of errors δρ(r) and δZ(r) for monochromatic X-rays and pseudo-monochromatic X-ray radiation, formed by the copper filters with thicknesses h1 = 4 mm and h2 = 10 mm. 12

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Fig. 3. Errors δρ(r) – a and δZ(r) – b for steel cylinder: — — – monochromatic X-rays; .... .... – pseudo-monochromatic X-ray radiation

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Analysis of the data from Fig. 3 allows to confirm the above formulated

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hypothesis. The systematic errors of density estimates for a monochromatic X-rays and pseudo-monochromatic X-ray radiation do not differ by more than 0.6% for the points of the inner portion of the object. The specified accuracy is

sufficient for the main practical applications of CT in conjunction with DET. For the systematic errors of the radial distributions of the effective atomic number

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the differences are more significant, but does not exceed 1.2% for the central

region of the cylinder. Underestimation of the effective atomic number is not more than 0.32 units, which also can be considered as a quite satisfactory result.

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The pre-filter thicknesses are chosen taking into account the changes in the TO mass thicknesses and effective atomic numbers of materials based on the specified errors of density and effective atomic number.

Note that if the level of errors ρ and Z does not satisfy the user, then a possible solution is the use of triple-energy [39, 40, 41]. In this method TO was scanned by X-rays with a three specially selected maximal energies. A stage of pretreatment of initial projections is reduced to selection of the method param-

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eters. For coherent scattering in addition to the above mentioned parameters A and B we calculate the C parameter, which is associated with the parameters of the object ρH and Z by a simple relation

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(12)

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C = ZρH.

It is natural to expect that the MAC of X-ray in the form of a sum corresponding coefficients for the photoelectric effect, incoherent and coherent scat-

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tering will be significantly more accurate than the expression (2). The same may be said about the pseudo-monochromatic implementation of the triple-energy method on the base of pre-filtration.

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3.2. Multilayer cylindrical objects A hypothetical four-layer cylindrical object was selected to confirm the applicability of the analyzed method to more complex test objects. The cylinder radius is R0 = 15 mm. The cylinder consists of the following layers: first layer 14

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Fig. 4. Multilayer cylindrical object. The radial dependences a — ρ(r), b — Z(r) for the

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copper pre-filter thicknesses h1 = 4 mm and h2 = 10 mm

with thickness 0.2R0 of carbon with density 2.26 g/cm3 ; second layer with thickness 0.2R0 of aluminum with density 2.7 g/cm3 ; third layer with thick-

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ness 0.3R0 of copper with density 8.92 g/cm3 ; internal rod with radius 0.3R0 of

steel with density 7.86 g/cm3 . The maximal X-ray energies are E1 = 100 keV, E2 = 225 keV, the corresponding copper filter thicknesses are h1 = 4 mm and

h2 = 10 mm.

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Fig. 4 shows the evaluation results of radial distributions ρ(r) and Z(r) for the test object

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Fig. 5. Material discrimination for multilayer cylindrical object by CT with the monochro— carbon;

— aluminum;

— steel;

— copper.

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matic DET:

Considering Fig. 4 one can make a conclusion about the satisfactory discrimination of the materials fragments by density and about the good discrimination by the effective atomic number. 230

Fig. 5 shows the quality of materials discrimination by the effective atomic

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number. The maximum offset of Z estimations does not exceed the absolute value of 0.8 units of effective atomic number. More significant deviations occur

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at the material boundaries.

4. Selection of the pre-filter thickness

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The criterion to select the thicknesses of the pre-filters at tube port is based

on the inequalities (9) and (11). Inequalities (9) set the limiting level of systematic errors of density and the effective atomic number, but inequality (11) is intended to verify the possibility of increasing the measurement time. It is

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noted that the priorities in the assignment ∆ρ, ∆Z and klim are placed by the consumer of the CT system.

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We examined the features of the proposed criterion in practice. For definiteness, we considered a homogeneous test object from aluminum with diameter

of 4 g/cm2 . Let the basic consumer priority for the thicknesses of the cooper pre-filters at tube port is the effective atomic number error - ∆Z.

The minimum values of the copper pre-filter at tube port thicknesses h1

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and h2 , satisfying the inequalities (9), were calculated for ∆ρ < 0.3 g/cm3 and a pair of maximum X-ray energies E1 = 100 keV, E2 = 225 keV. The time

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extension factor kg was calculated from formula (11) for the obtained values. The calculation results are summarized in Table 2. 250

The data of Table 2 allows selection of the pre-filter at tube port thickness under the conditions ∆ρ < 0.3 g/cm3 and 0.15 < ∆Z < 1.5. For example, for ∆Z = 1, it is sufficient to use copper filters with thicknesses h1 = 0.6 mm and h2 = 1.0 mm. For such a set of filters, the total time increases approximately by two times, which is most likely acceptable for the consumer. The Table 2 analysis shows that for maximum X-ray energies E1 = 100 keV,

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E2 = 225 keV, the copper pre-filters with a thickness of less than 0.5 mm are inappropriate to use due to the insufficient accuracy of the effective atomic number (over 1.5 effective atomic numbers). The use of copper pre-filters with

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a thickness of more than 1.5 mm allows to achieve high accuracy of the effective atomic number, but leads to a significant increase in the time extension factor. Note, that the above described method is currently uncontested to solve

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the problem of simultaneous estimation of the density distribution and effective atomic number in the TO volume with high accuracy by CT system without changing the X-ray source to a monochromatic X-ray source. Because of this,

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the time extension factor increase in four or more times can be acceptable if the use of pre-filtering in DET allows one to estimate, for example, the distribution of the effective atomic number with an accuracy of 0.1.

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5. Conclusion

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Analysis of the theoretical research resultss and calculations prove a possibility of accurate estimation of the spatial distributions of density and the effective atomic number by CT method in combination with a pseudo-monochromatic

DET. The selection of pre-filter thickness is based on a compromise between an increase of accuracy of spatial distributions of density and effective atomic num-

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ber and a decrease of the examination performance compared to the classical DET implementation of CT.

The proposed approach allows to theoretically substantiate the preliminary

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choice of pre-filter thicknesses in the pseudo-monochromatic implementation of DET, starting from the specified accuracy of estimating the distributions of the effective atomic number and the density by the TO volume.

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Acknowledgments

The research is carried out at Tomsk Polytechnic University within the

gram grant.

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framework of Tomsk Polytechnic University Competitiveness Enhancement Pro-

[1] F. Kelcz, P. M. Joseph, S. K. Hilal, Noise considerations in dual energy ct scanning, Medical Physics 6 (1979) 418–25. doi:10.1118/1.594520.

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Table 1. The effectiveness of the copper pre-filters in CT with DET usage

2

1

3

5

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3

ρmin 2.22 2.52 6.62 2.29 2.63 7.52 2.30 2.60 6.60 2.31 2.59 6.09 2.33 2.60 6.13 2.24 2.67 6.99 2.26 2.65 7.42 2.28 2.65 7.71 2.29 2.64 7.23 2.25 2.66 7.20 2.27 2.65 7.60 2.28 2.65 7.64

3

5

10

ρmax 2.43 3.05 13.6 2.49 2.86 8.23 2.51 2.82 7.98 2.51 2.81 8.11 2.51 2.81 8.22 2.45 2.97 11.49 2.47 2.91 9.53 2.45 2.89 8.26 2.49 2.86 8.01 2.46 2.94 10.6 2.47 2.90 9.06 2.84 2.88 8.23

Zmin 3.32 11.87 24.63 5.35 13.05 25.55 5.1 13.11 25.06 5.01 13.14 24.71 5.09 13.11 24.54 6.03 12.98 24.94 5.58 13.08 25.96 5.42 13.09 25.58 5.21 13.16 25.25 6.04 13.04 25.19 6.41 13.04 25.94 5.12 13.18 25.56

Zmax 6.42 13.68 27.14 5.45 14.17 31.26 5.21 14.33 33.62 5.15 14.38 34.86 5.26 14.34 34.61 6.13 13.27 26.8 5.66 13.55 27.1 5.50 13.66 28.81 5.32 13.78 29.60 6.15 13.28 26.61 6.51 13.42 27.07 5.19 13.63 28.09

24

kg 2.70 1.75 1.32 2.49 1.89 1.40 2.51 2.06 1.50 3.66 2.47 2.11 5.79 4.03 2.65 4.46 2.68 1.76 4.61 2.80 1.83 4.99 3.12 2.03 6.71 4.45 2.85 6.82 3.95 2.36 7.09 4.18 2.52 8.39 5.26 3.22

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Z 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26 6 13 26

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h2 , mm

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h1 , mm

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Table 2. Cooper filter thicknesses h1 and h2 , mm

Parameter

1 0.6 1.0 1.98

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h1 h2 kg

1.5 0.5 1.0 1.85

25

∆Z 0.5 0.85 1.0 2.35

0.25 1.0 1.0 2.62

0.15 1.5 1.5 3.94

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Formulas of density and effective atomic number are given for pseudomonochromatic dual energy CT.

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The pre-filter thickness was defined to examine homogeneous cylinders by CT. Closeness of monochromatic and quasi-monochromatic versions of dualenergy method is estimated.

Density and effective atomic number were calculated by dual energy CT for

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uniform and non-uniform cylinders.

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•