# Characterization of breast tissues in density and effective atomic number
basis via spectral X-ray computed tomography

Stevan Vrbaški  Department of Physics, University of Trieste, Via Valerio 2,
34127 Trieste, Italy and Elettra-Sincrotrone Trieste S.C.p.A, 34149 Basovizza
Trieste, Italy and INFN Division of Trieste, Via Valerio 2, 34127 Trieste,
Italy  Lucia Mariel Arana Peña  Department of Physics, University of Trieste,
Via Valerio 2, 34127 Trieste, Italy and Elettra-Sincrotrone Trieste S.C.p.A,
34149 Basovizza Trieste, Italy and INFN Division of Trieste, Via Valerio 2,
34127 Trieste, Italy  Luca Brombal  Department of Physics, University of
Trieste, Via Valerio 2, 34127 Trieste, Italy and INFN Division of Trieste, Via
Valerio 2, 34127 Trieste, Italy  Sandro Donato  Department of Physics,
University of Calabria, Via Pietro Bucci, 87036 Arcavacata, Italy and INFN
Division of Frascati, Via Enrico Fermi 54, 00044 Frascati, Italy  Angelo Taibi
Department of Physics and Earth Sciences, University of Ferrara, Via Saragat
1, 44122 Ferrara, Italy and INFN Division of Ferrara, Via Saragat 1, 44122
Ferrara, Italy  Adriano Contillo  Elettra-Sincrotrone Trieste S.C.p.A, 34149
Basovizza Trieste, Italy<EMAIL_ADDRESS>Renata Longo  Department
of Physics, University of Trieste, Via Valerio 2, 34127 Trieste, Italy and
INFN Division of Trieste, Via Valerio 2, 34127 Trieste, Italy

###### Abstract

_Objective._ Differentiation of breast tissues is challenging in X-ray imaging
because tissues might share similar or even the same linear attenuation
coefficients $\mu$. Spectral computed tomography (CT) allows for more
quantitative characterization in terms of tissue density and effective atomic
number by exploiting the energy dependence of $\mu$. _Approach._ In this work,
5 mastectomy samples and a phantom with inserts mimicking breast soft tissues
were evaluated in a retrospective study. The samples were imaged at three
monochromatic energy levels in the range of 24 - 38 keV at 5 mGy per scan
using a propagation-based phase-contrast setup at SYRMEP beamline at the
Italian national synchrotron Elettra. _Main results._ A custom-made algorithm
incorporating CT reconstructions of an arbitrary number of spectral energy
channels was developed to extract the density and effective atomic number of
adipose, fibro-glandular, pure glandular, tumor, and skin from regions
selected by a radiologist. _Significance._ Preliminary results suggest that,
via spectral CT, it is possible to enhance tissue differentiation. It was
found that adipose, fibro-glandular and tumorous tissues have average
effective atomic numbers (5.94 $\pm$ 0.09, 7.03 $\pm$ 0.012, and 7.40 $\pm$
0.10) and densities (0.90 $\pm$ 0.02, 0.96 $\pm$ 0.02, and 1.07 $\pm$ 0.03
g/cm3) and can be better distinguished if both quantitative values are
observed together.

††: Phys. Med. Biol.

Keywords: spectral breast computed tomography, material density, effective
atomic number, photon-counting detectors, synchrotron radiation X-ray imaging

##  1 Introduction

Tissue differentiation in mammography is a challenging task due to the overlap
of tissues with similar linear attenuation coefficients $\mu$ in the
diagnostic energy range. A comparison of the attenuation of glandular tissue
and close-by malignant tumors found that the average attenuation values are
almost identical even in the low-energy range (?, ?). Computed tomography (CT)
solves tissue overlapping issues and provides additional diagnostic
information based on tissue morphology in 3D volumes. On the other hand,
breast-dedicated CT uses higher X-ray energies compared to mammography due to
the imaging of uncompressed breasts, thus bringing a further reduction in
attenuation contrast between similar tissues. Tissue separation can be
significantly improved using spectral CT imaging (?). New generations of
whole-body spectral CT systems obtain source-generated spectral information
from two (or more) scans at different energy levels (dual-energy), or
detector-generated spectral response with layered and photon-counting (PC)
detectors. The first breast CT scanners with PC detectors are being introduced
in clinics (?, ?, ?, ?) and synchrotron facilities (?, ?), where, in the
latter case, tomography can be performed with low-energy (20-40 keV)
monochromatic beams. Spectral CT imaging can improve tissue separation by
acquiring data at multiple energy levels, allowing for spectral material
decomposition analysis and the creation of e.g., virtual monochromatic images,
iodine maps, and density-effective atomic number ($\rho$/Z${}_{\textrm{\tiny
eff}}$) maps. In synchrotron facilities, source-generated spectral separation
can be obtained using multiple X-ray beams of precisely defined energies.
Alternatively, when dealing with polychromatic spectra such as in clinical
scanners, detectors with multiple thresholds (up to 12 (?)) could be used, but
the benefits of such an approach in breast CT imaging have not been explored
yet.

Given two (or more) scans at different energies, their content can be
described as a linear combination of the attenuation of two basis materials
where the weights of the combination are related to some physical property
(e.g., volume fractions) of the basis materials. Two-basis decomposition is
the most common approach because two physical effects contribute to image
formation in the diagnostic energy range: the photoelectric effect and Compton
scattering. Photoelectric-Compton basis itself is very convenient because of
its physical relationship to density and effective atomic number (?), but
decomposing biological materials in a pair of physical materials, for example,
poly(methyl methacrylate) (PMMA) and aluminum (Al), leads to significantly
lower decomposition errors (?), thus PMMA-Al basis was used in this study.
Estimated coefficients for the linear combination of the two basis materials
and the known energy dependence of their attenuation coefficients are used to
extrapolate virtual monochromatic images (VMIs) at an arbitrary energy value
within the diagnostic energy range (?). Clinical case studies show that
virtual monochromatic images are diagnostically valuable for certain tasks
(?), and they are a standard output in the first clinical whole-body PC-CT
(Naeotom Alpha, Siemens) (?). Basis material maps are also used for the
physical description of scanned objects in terms of material density $\rho$
and effective atomic number $Z_{\textrm{\tiny eff}}$. Several works have been
published exploring this approach (?, ?, ?, ?, ?, ?, ?). In fact, while
different materials can share the same $\mu$ value for a given energy range,
the successful physical description offers its unique characterization. Recent
work on a System Independent Rho Z (SIRZ) software showed a potential for
$\rho$/$Z_{\textrm{\tiny eff}}$ decomposition with poly-chromatic beams in
low-density plastic materials with an accuracy within 2.1 $\%$ (?).

In the present paper, for the first time, a spectral study on breast CT images
is developed to characterize tissues in terms of material density $\rho$ and
effective atomic number $Z_{\textrm{\tiny eff}}$. The work investigates the
feasibility and the potential diagnostic benefit of $\rho$/Z${}_{\textrm{\tiny
eff}}$ decomposition in a dedicated phantom and on 5 mastectomy samples in a
retrospective study. We also show the convenience of such decomposition for
computing virtual monochromatic $\mu$ values at desired energy, taking into
account the known dependence of X-ray attenuation on density and atomic
number. The overarching goal required the development of methodological
procedures that have been organized into several parts. The paper contains a
theoretical approach to material and $\rho$/Z${}_{\textrm{\tiny eff}}$
decomposition in section 2.1, calibration and Z${}_{\textrm{\tiny eff}}$
definition in section 2.3, a description of breast mastectomy samples in
section 2.4, and proposed solution to the well-known problem of the
decomposition noise (?, ?, ?) in section 2.5 which also provides a detailed
explanation of how all parts are practically joined together. CT scans were
acquired using synchrotron X-ray beams at several energies. The data
collection was performed using a high-resolution PC detector, as detailed in
section 2.2.

##  2 Materials and Methods

###  2.1 Theoretical model

The attenuation of X-rays is influenced by several physical effects, which are
dependent on the energy and material properties of the attenuator. In the
range of photon energies useful for medical CT imaging, the attenuation
coefficient $\mu$ of given material of density $\rho$, the atomic number $Z$,
and atomic mass $A$ are approximated as the sum of the two contributions

$\mu(\rho,Z,A,E)=\frac{\rho\,Z}{A}\,\left[K_{1}\,Z^{n-1}\,f_{P}(E)+K_{2}\,f_{C}(E)\right]$
(1)

where the functions $f_{P}(E)$ and $f_{C}(E)$ encode the energy dependencies
of the photoelectric and Compton effects, respectively, and K1, K2 and $n$ are
constants. Using material decomposition, the attenuation coefficient $\mu$ of
a given material is expressed as a linear combination of the (known)
attenuation coefficients of a pair of basis materials, here labeled $\mu_{1}$
and $\mu_{2}$

$\mu=x_{1}\,\mu_{1}+x_{2}\,\mu_{2}$ (2)

where $x_{1}$ and $x_{2}$ are the coordinates of the material in the reference
frame identified by the selected basis. In the first step of our method, we
used equation (2) to determine the coefficients $x_{1}$ and $x_{2}$, which
represent the (energy-independent) relative concentrations of each basis
material. Because more than two spectral scans were available for each sample
(?), instead of using traditional matrix inversion (?), $x_{1}$ and $x_{2}$
coefficients were calculated by using a least-square fit of the form:

$\sum_{i}\,\left(\mu(E_{i})-(x_{1}\,\mu_{1}(E_{i})+x_{2}\,\mu_{2}(E_{i}))\right)^{2}$
(3)

The fit procedure consists of a voxel-by-voxel minimization of the sum over
the energies $E_{i}$ of the squared residuals. From such a point of view,
including more images of different energies adds further points to the plane,
which is expected to increase the accuracy of the decomposition.

Assuming $A=2Z$, as it is almost true for any chemical element with $Z\lesssim
20$ and making use of equation (1), the linear combination coefficients of
equation (2) read

$\displaystyle x_{1}$
$\displaystyle=\frac{\rho\,Z_{1}\left(Z^{n}Z_{2}-Z\,Z_{2}^{n}\right)}{\rho_{1}Z\left(Z_{1}^{n}Z_{2}-Z_{1}Z_{2}^{n}\right)}$
(4a) $\displaystyle x_{2}$
$\displaystyle=\frac{\rho\,Z_{2}\left(Z\,Z_{1}^{n}-Z^{n}Z_{1}\right)}{\rho_{2}Z\left(Z_{1}^{n}Z_{2}-Z_{1}Z_{2}^{n}\right)}$
(4b)

$\rho_{j}$ and $Z_{j}$ being the density and the atomic number of the $j$-th
basis material. It is straightforward to notice that equations (4a) and (4b)
depend on both the density and the atomic number of the material. In order to
decouple the two dependencies, it is necessary to rotate the reference frame
(by an angle $\phi=\arctan\\!\left(\frac{\rho_{2}}{\rho_{1}}\right)$) and then
rescale the second coordinate dividing it by the first one. The resulting
coordinates, which read

$\displaystyle x_{\rho}$
$\displaystyle=\frac{\rho}{\sqrt{\rho_{1}^{2}+\rho_{2}^{2}}}$ (4ea)
$\displaystyle x_{Z}$
$\displaystyle=\frac{Z^{\ell}\left(\rho_{1}^{2}+\rho_{2}^{2}\right)-\left(Z_{1}^{\ell}\rho_{1}^{2}+Z_{2}^{\ell}\rho_{2}^{2}\right)}{(Z_{2}^{\ell}-Z_{1}^{\ell})\rho_{1}\rho_{2}}$
(4eb)

(with $\ell=n-1$) are labeled $x_{\rho}$ and $x_{Z}$ because of their
exclusive dependencies on the variables mentioned in the subscripts. Equations
(4ea) and (4eb) represent the expected relationships between $x_{\rho}$ and
$\rho$, and between $x_{Z}$ and $Z$, for a given choice of basis materials.
Such expressions are used in the calibration procedure described in section
3.1, where the (known) densities and atomic numbers of reference materials
will be fitted against the corresponding (measured) values of $x_{\rho}$ and
$x_{Z}$ using the functional forms

$\displaystyle x_{\rho}(\rho)$ $\displaystyle=\kappa\,\rho$ (4efa)
$\displaystyle x_{Z}(Z)$ $\displaystyle=p\,Z^{\lambda}+q$ (4efb)

stemming directly from equations (4ea) and (4eb) with the (physically
motivated) coefficients replaced by the effective fit parameters $\kappa$,
$\lambda$, $p$, and $q$. The resulting calibrated relationships map the
measured $x_{\rho}$ and $x_{Z}$ of any imaged material onto its actual values
of density and effective atomic number.

###  2.2 Scan setup

The experimental image acquisition was carried out at the SYRMEP beamline of
Elettra, Italian synchrotron light source in Trieste, in the framework of
SYRMA-3D (SYnchrotron Radiation for MAmmography) collaboration (?). The SYRMEP
beamline utilizes a laminar beam having a cross-section at the detector equal
to 148.5 mm (horizontal) $\times$ 3.25 mm (vertical), while the energy is
selected through a Si double-crystal (1,1,1) monochromator with a resolution
around 0.1%. To perform tomographic acquisition the sample was positioned on a
rotation stage spinning at a constant speed of 4.5 degrees/s while 1200
projections were acquired over 180 degrees. The imaging detector was a large
area high-resolution CdTe photon-counting device (Pixirad8) featuring a
honeycomb matrix of 4096 $\times$ 476 pixels with a 60 $\mu$m pitch (?, ?). It
was positioned $\sim$1.6 m from the sample to employ the propagation-based
phase-contrast imaging technique. Acquired projections were pre-processed with
an ad-hoc procedure (?) and then phase-retrieved using an algorithm based on
the homogeneous transport of intensity equation (TIE-Hom) (?). Finally, CT
reconstructions were obtained via a GPU-based filtered-back-projection
algorithm with Shepp-Logan filtering (?). It is worth noting that, despite
being a product of phase retrieval, reconstructions are maps of the
attenuation coefficients $\mu$ at a given energy, as thoroughly explained in
(?, ?, ?).

###  2.3 Calibration phantom and effective atomic number

A custom-made cylindrical phantom with a 10 cm diameter dedicated to
calibration and quality control of the synchrotron breast CT system was used
(?, ?). The phantom was filled with water and contained five inserts of
polyethylene (PE), nylon (PA), PMMA, polyoxymethylene (POM), and
polytetrafluoroethylene (PTFE) mimicking soft tissues of similar attenuation
properties. The density of each material was known and the effective atomic
number was computed from the material composition. The atomic number is a
physical property of an element, but the same concept cannot be trivially
defined in compounds. Across the literature (?, ?, ?, ?, ?, ?) several
definitions have been proposed, suggesting that it depends not just on
material properties, but also experimental conditions (?). We included most of
the published definitions in open-source GUI software ”ZcompARE” (?). Part of
the software is built on top of the python library (xraylib 4.1.0 package,
Python 3.10) of the xraylib database (?, ?) and can be used to compare
approaches for computing atomic numbers of compounds from this database or
list of user-defined materials can be provided. Comparison between the methods
using this software is analyzed in Appendix B, showing that depending on the
method chosen, $Z_{\textrm{\tiny eff}}$ number can take different values for
the same compound. However, any choice of the method led to a unique material
description in chosen basis. Quantity $x_{Z}$ derived in section 2.1, was
simply calibrated to the desired definition of effective atomic number for a
compound by putting $Z=Z_{\textrm{\tiny eff}}$ in equation (4efb). In this
paper, we used the approach by Champley et al (?) which defines
$Z_{\textrm{\tiny eff}}$ of a compound as a linear combination of two
consecutive Z numbers such that the least square error between X-ray
transmission of the compound and the transmission of a combination of the two
elements is minimized. Compounds’ brute formula, density, and
$Z_{\textrm{\tiny eff}}$ are given in table 1. The phantom served two
purposes: _i)_ to obtain the $\rho$ and $Z_{\textrm{\tiny eff}}$ calibration
curve from the decoupled set in equations (4ea) and (4eb) and _ii)_ to
validate the theoretical model against the ground truth.

Table 1: List of materials composing the phantom together with chemical formulas, effective atomic numbers, and material density values(?). Material | Water | PE | PA | PMMA | POM | PTFE
---|---|---|---|---|---|---
Brute formula | $\textrm{H}_{2}\textrm{O}$ | $\textrm{C}_{2}\textrm{H}_{4}$ | $\textrm{C}_{12}\textrm{H}_{22}\textrm{N}_{2}\textrm{O}_{2}$ | $\textrm{C}_{5}\textrm{H}_{8}\textrm{O}_{2}$ | $\textrm{C}\textrm{H}_{2}\textrm{O}$ | $\textrm{C}_{2}\textrm{F}_{4}$
Effective $Z$ | 7.44 | 5.28 | 6.16 | 6.49 | 7.01 | 8.56
Density (g/cm3) | 1.0 | 0.94 | 1.14 | 1.19 | 1.425 | 2.2

###  2.4 Breast mastectomy samples

In addition to phantom scanning, post-mastectomy breast tissue images were
analyzed in a retrospective study. The analyzed surgical samples were fixed in
formalin, sealed in a vacuum bag, and conserved at 4 ∘C. The preliminary
analysis of the same data was published (?) as a feasibility study of the
synchrotron breast CT approach. All the procedures adopted in this work
followed Directive 2004/23/EC of the European Parliament and of the Council of
31 March 2004 on setting standards of quality and safety for the donation,
procurement, testing, processing, preservation, storage, and distribution of
human tissues. In the present work, we further processed the data to extract
$\rho$ and $Z_{\textrm{\tiny eff}}$ of breast tissues. Tomographic
reconstructions of selected samples are given in figure 1. They all contained
adipose, fibro-glandular, and tumorous tissue, but only in sample 4 existed a
region of glandular tissue clearly separated from the fibrous. It also
contained calcification regions which were not evaluated in this study. All
samples contained some type of malignant tissue: Samples 1, 2 and 3 contain
infiltrating ductal carcinoma, Sample 4 contains infiltrating ductal carcinoma
with a core of desmoplastic tissue, and Sample 5 contains vastly
differentiated infiltrating ductal carcinoma. Samples 1,2,4, and 5 also
contained portions of the skin.

Figure 1: Mastectomy CT reconstructions acquired at SYRMEP beamline at Elettra
Sincrotrone Trieste at the energy of 28 keV (Sample 1,2,3, and 5) and 26 keV
(Sample 4).

The mean glandular dose of 5 mGy delivered per scan was computed according to
a dedicated Geant4 Monte Carlo simulation (?, ?). The phantom scans were
acquired at 25, 28, 32, and 35 keV, and mastectomy samples at several
monochromatic beam energy levels in the range of 24 - 38 keV, from which three
scans were selected for the spectral analysis.

###  2.5 Practical implementation

The theoretical model described in section 2.1 was implemented with Python
(Python 3.10.0) and an interactive delineation tool was developed to select an
arbitrary region of interest (ROI) within a reconstructed spectral data set
(Matplotlib, Python 3.10.0) containing a single tissue type. In the first step
of the process (framed with a dashed line in figure 2), the voxel-to-voxel
material decomposition method using the PMMA-Al basis is applied to the
selected regions (figure 2(a)), resulting in PMMA and Al maps for each ROI
(figure 2(b)). In the next step, the material maps are combined in 2D
histograms, one for each tissue type (figure 2(c)). Due to the presence of
noise, the obtained histograms are blurred and elongated.

Figure 2: The step-by-step scheme of quantitative material evaluation followed
in this work. Inputs to the algorithm (dashed rectangle) are delineated
tissues. Section _a_ shows the material decomposition task which leads to
material maps in section _b_. These maps are represented in form of a 2D
histogram and centers of clusters corresponding to each delineated material
are extracted in section _c_. These values are then transformed using
equations 4ea and 4eb and an offline calibration using materials from table 1
is performed in section _d_ to obtain the $\rho$ and $Z_{\textrm{\tiny eff}}$
values as a final output.

The correct decomposition coefficients $x_{1}$ and $x_{2}$ are considered to
be the centers of obtained distributions. They are extracted using a 2D
Gaussian fit method of the form:

$G(x_{1},x_{2})=A\,e^{\left(-\frac{u^{2}}{2\,S_{u}^{2}}+\frac{v^{2}}{2\,S_{v}^{2}}\right)}$
(4efg)

$A$ is the intensity of the peak and $S_{u}$, $S_{v}$ the spreads (that is,
the standard deviations of the associated distributions) along the major and
minor axes, respectively. The peak coordinates $\bar{x}_{1}$ and $\bar{x}_{2}$
relative to $(x_{1},x_{2})$-plane of the histogram are contained in the
quantities $u$ and $v$,

$\displaystyle
u=(x_{1}-\bar{x}_{1})\,\cos\theta+(x_{2}-\bar{x}_{2})\,\sin\theta$ (4efha)
$\displaystyle
v=(x_{2}-\bar{x}_{2})\,\cos\theta-(x_{1}-\bar{x}_{1})\,\sin\theta$ (4efhb)

$\theta$ is the tilt of the major axis of the spot with respect to the
horizontal direction. After the fitting, the peak coordinates $\bar{x}_{1}$
and $\bar{x}_{2}$ found according to equations (4efha) and (4efhb) are used in
equations (4a) and (4b) to compute the final output of the procedure: the
single values for densities and effective atomic numbers of selected tissues
(figure 2(d)). The measurement uncertainty, estimated as the standard error on
the center of fitted 2D Gaussian distribution, is propagated through all
mathematical transformation steps and measurement calibration. For the sake of
conciseness, the complete analysis is given in the A.

###  2.6 Data analysis

In addition to the experimental data obtained using the calibration phantom,
the procedure described in the theoretical model (section 2.1) was also
applied to published $\mu$ (”true”) values (?) of phantom material inserts at
the energy levels used in the experiment. The true data points are used to
evaluate the accuracy of the model by calculating the percentage error between
the experimental and the ground truth data points, as given in equation
(4efhi).

$\%\,error=\frac{|\textrm{ground\,truth -
measured}|}{\textrm{ground\,truth}}\times\,100$ (4efhi)

The segmentation of the breast CT reconstructions was performed by an
experienced radiologist, who selected ROIs containing adipose, fibro-
glandular, glandular, skin, and tumorous tissue already knowing the mastectomy
content from from specimen sampled for the histological examination. For each
sample and tissue type, density and Z${}_{\textrm{\tiny eff}}$ values and
their uncertainties were estimated respectively as the mean and standard
deviation evaluated over 10 consecutive CT slices. To estimate the
discrimination power of $\rho$/Z${}_{\textrm{\tiny eff}}$ maps, hence
diagnostic potential, a mean Silhouette score using euclidean distance as a
metric was computed on mean values of all tissue types collected from the
mastectomy samples (?). Mean Silhouette score (MSS) is a tool to quantify how
similar are samples within the same cluster and how separated they are from
other clusters, defined as:

$\textrm{MSS}=\frac{1}{n}\sum_{k=0}^{n}\frac{\textrm{NC}_{k}-\textrm{IC}_{k}}{max(\textrm{NC}_{k},\,\textrm{IC}_{k})}$
(4efhj)

where IC is the mean intra-cluster distance and NC is the mean nearest-cluster
distance for each sample $k$, and $n$ equals number of selected tissue types
$\times$ number of selected mastectomy samples. Negative values of MSS
indicate cluster overlap, values around 0 signal that samples are on or close
to the boundary between clusters, and positive values up to 1 indicate
increased cluster separation.

Finally, extracted density and effective atomic numbers from the calibration
phantom and Sample 4 were fed to the mathematical relationship given in
equation (1) to extrapolate linear attenuation coefficients of tissues for
arbitrary (virtual) monochromatic energy levels. The uncertainty was
propagated from the standard deviation of obtained $\rho$ and
Z${}_{\textrm{\tiny eff}}$ values.

##  3 Results

###  3.1 Calibration phantom results

The accuracy of material decomposition in phantom materials following the
Gaussian fitting in the histogram space is shown in figure 3(a). Quantities
$x_{1}$ and $x_{2}$ in equations (4a) and (4b) are replaced with
$x_{\textrm{\tiny{PMMA}}}$ and $x_{\textrm{\tiny Al}}$ because of the
particular basis choice. Figure 3(b) corresponds to the decoupling of density
and effective atomic number in equations (4ea) and (4eb). The fitting
parameters $A$, $S_{u}$, $S_{v}$ defined in equation (4efg), and angle
$\theta$ for phantom materials are given and further discussed in B.

(a)

(b)

Figure 3: Experimentally obtained basis material concentrations (blue) against
the true data values (red) in a) the PMMA-Al basis and b) decoupled reference
frame. The error bars show standard error a) computed using equations (4efhka)
and (4efhkb) and b) propagated using equations (4efhkmn) and (4efhkmo) in A.

The decoupled values from figure 3(b) were calibrated to absolute density and
effective atomic numbers (table 1) applying the least-squares fitting method
to the functional forms given in equations (4efa) and (4efb). Both calibration
curves were based on the phantom material inserts in the range of interest for
soft tissue imaging and the mapping to correct density and effective atomic
number was obtained at $R^{2}=0.998$ and $R^{2}=0.997$, respectively.
Calibration curves are given in figure 4 and corresponding density and
$Z_{\textrm{\tiny eff}}$ values are given in table 2.

(a)

(b)

Figure 4: Theoretical data points fitted to (a) a linear calibration curve for
density $\rho$, and (b) a power-law curve for $Z_{\textrm{\tiny eff}}$ of the
same form suggested by the equations (4efa) and (4efb).

Using the equation 4efhi, $\%$ errors in calibration materials were estimated
to be below 3$\%$ and 1.5$\%$, respectively, after the density and effective
atomic number calibration.

Table 2: $Z_{\textrm{\tiny eff}}$ and $\rho$ with the corresponding standard errors $\sigma_{Z}$ and $\sigma_{\rho}$ obtained from the calibration phantom. Material | $\rho\pm\sigma_{\rho}\;\textrm{(g/cm}^{3})$ |  Z${}_{\textrm{eff}}$ $\pm\sigma_{Z_{\textrm{\tiny eff}}}$ | $\%err\,\rho$ | $\%err\,\textrm{Z}_{\textrm{eff}}$
---|---|---|---|---
PE | $0.963\pm 0.004$ | $5.36\pm 0.02$ | $2.4$ | $1.3$
Water | $0.971\pm 0.001$ | $7.51\pm 0.02$ | $2.9$ | $0.9$
PA | $1.149\pm 0.004$ | $6.10\pm 0.01$ | $0.7$ | $1.0$
PMMA | $1.198\pm 0.004$ | $6.40\pm 0.01$ | $0.7$ | $1.4$
POM | $1.411\pm 0.004$ | $7.04\pm 0.01$ | $1.0$ | $0.4$
PTFE | $2.204\pm 0.003$ | $8.55\pm 0.01$ | $0.2$ | $0.2$

###  3.2 Breast mastectomy results

Tissues inside the mastectomies delineated by a radiologist were
quantitatively analyzed in terms of their density and effective atomic number
and presented in figure 5, with average values given in table 3.

Table 3: Average $Z_{\textrm{\tiny eff}}$ and $\rho$ from 5 mastectomy samples. Tissue type | $\rho\pm SD_{\rho}\;\textrm{(g/cm}^{3})$ |  Z${}_{\textrm{eff}}$ $\pm SD_{Z_{\textrm{\tiny eff}}}$
---|---|---
Adipose |  0.90 $\pm$ 0.02 |  5.94 $\pm$ 0.09
Fibro-glandular |  0.96 $\pm$ 0.02 |  7.03 $\pm$ 0.12
Tumor |  1.07 $\pm$ 0.03 |  7.40 $\pm$ 0.10

The level of separation between adipose, fibro-glandular and tumor tissue was
found to be 0.31 (on a scale of -1 to 1) using MSS. Pairwise comparison of
adipose and fibro-glandular (MSS = 0.59), adipose and tumor (MSS = 0.74), and
fibro-glandular and tumor (MSS = 0.17) showed that adipose tissue can be well
distinguished from the other two, while fibro-glandular and tumor are closer
togerher in $\rho$/$Z_{\textrm{\tiny eff}}$ space. From figure 5 it can be
seen that, adipose tissue can be distinguished from fibro-glandular and
tumorous purely based on effective atomic number values. On the other hand,
fibro-glandular and tumorous tissues are overlapping both in their effective
atomic numbers and densities and can be well distinguished only if both
quantitative values are observed together.

Figure 5: Quantitative description of tissues in terms of density (g/cm3) and
atomic number computed as in Champley et al (?). Adipose, fibro-glandular and
tumorous tissues are distinguished with green, blue, and red color,
respectively, while marker shapes correspond to different mastectomy samples.
Error bars represent the standard deviation of 10 ROIs.

In addition to adipose, fibro-glandular, and malignant tissue, the mastectomy
labeled as Sample 4 contains well-separated pure glandular tissue and skin.
Thus, an extensive investigation was conducted on this tissue. Extracted
density and effective atomic number were 0.98 $\pm$ 0.01 and 5.42 $\pm$ 0.06
for adipose, 1.07 $\pm$ 0.02 and 6.56 $\pm$ 0.20 for fibro-glandular, 1.18
$\pm$ 0.02 and 6.88 $\pm$ 0.09 for glandular, 1.19 $\pm$ 0.01 and 7.04 $\pm$
0.07 for tumorous, and 1.19 $\pm$ 0.02 and 6.87 $\pm$ 0.04 for skin tissue,
respectively. In this particular mastectomy sample, mean density and
$Z_{\textrm{\tiny eff}}$ are not distinguishable between the skin and
glandular tissue. Glandular tissue has higher $Z_{\textrm{\tiny eff}}$ and
density than fibro-glandular tissue, but almost the same density and slightly
lower $Z_{\textrm{\tiny eff}}$ than tumorous tissue.

Virtual $\mu$ values of phantom materials and tissues in sample 4 are given in
figure 6. The obtained $\rho$ and $Z_{\textrm{\tiny eff}}$ values correctly
map to experimentally measured $\mu$ values and provide information about
tissue separation at lower energies with respect to those used for tissue
scanning.

(a)

(b)

Figure 6: Virtual linear attenuation coefficients computed from extracted
density and effective atomic numbers. Full lines represent virtual $\mu$
values extracted from experimental data at an arbitrary energy level, while
uncertainty was propagated from the standard deviation of $\rho$ and
$Z_{\textrm{\tiny eff}}$.

##  4 Discussion

Conventional CT scans offer a relatively low specificity when distinguishing
between soft tissues of slightly different compositions. On the other hand,
spectral imaging advances the ability to distinguish such tissues by probing
their attenuation properties at several energy levels. Spectral information is
used for the decomposition of data in basis materials, such as PMMA-Al. The
decomposed material maps can be used as an intermediate step to estimating the
uniquely defined physical quantities of imaged tissues. Utilizing these
quantities, virtual monochromatic $\mu$ values at arbitrary energy levels can
be further extrapolated. In this work, a new approach to material
decomposition and $\rho$/$Z_{\textrm{\tiny eff}}$ estimation was presented to
characterize breast tissues.

A least-square fitting approach to two-basis material decomposition was
adopted for an over-determined system when $\mu$ values can be measured at
several energy levels. This was the case in our retrospective study with
monochromatic beams, but the same approach can be applied to multi-threshold
photon counting detectors. The procedure for decoupling density from
$Z_{\textrm{\tiny eff}}$ depends on the quality of performed material
decomposition, but is at the same time independent of the method itself, and
can be applied to other methods published in the literature. Experimentally
computed concentrations $x_{\textrm{\tiny PMMA}}$ and $x_{\textrm{\tiny Al}}$
of basis materials and related basis $x_{\tiny\rho}$ and $x_{\textrm{\tiny
Z}}$ in the phantom were found to be in good agreement with the ground truth
values. Moreover, small associated standard errors in figure 3 show that
materials can be well separated. The theoretically derived functions in
equations (4efa) and (4efb) were fitted to 6 data points corresponding to
tissue-equivalent plastics in the phantom. The high accuracy (R${}^{2}>$ 0.99)
of the fitting procedure justifies the assumptions made in theoretical
derivation. Density and $Z_{\textrm{\tiny eff}}$ values for phantom inserts,
given in table 2, showed excellent agreement with the ground truth data at an
average $\%$ error of 1.34 $\%$ for $\rho$ and 0.89 $\%$ for $Z_{\textrm{\tiny
eff}}$. These results demonstrate that soft-tissue-equivalent plastic
materials of similar composition can be well distinguished from spectral CT
data, based on their estimated effective atomic number and density value.

The preliminary study on the breast cancer mastectomy samples was an attempt
to demonstrate the feasibility of our method to distinguish between fibro-
glandular and tumorous tissues inside the breast. Based on the available
samples, it was shown that starting from spectral data it is possible to
separate adipose, fibro-glandular and tumorous tissues based on their physical
characteristics. This might be useful in risk assessment, cancer diagnosis and
the assessment of the status of the disease. Although pure glandular and
tumorous tissue in Sample 4 had almost the same density and slightly different
effective atomic numbers, this could be due to the desmoplastic core present
inside the tumor and no conclusions could be made based on single piece of
evidence. The importance of $Z_{\textrm{\tiny eff}}$ to X-ray attenuation can
be observed when comparing clusters related to adipose and fibro-glandular
tissue. It can be seen that lower $\mu$ values for adipose tissue are driven
by lower $Z_{\textrm{\tiny eff}}$, rather than significantly lower density.
Distinguishing tumorous and pure glandular tissue is challenging because only
slight differences exist in both density and $Z_{\textrm{\tiny eff}}$. In our
study, we observed that it is not possible to distinguish between different
tissues solely on density or effective atomic number or $\mu$ value alone
while reasonable discrimination (MSS = 0.31) can be obtained considering 2D
clusters in $\rho$/$Z_{\textrm{\tiny eff}}$ space. Therefore, using both
density and effective atomic number maps in the diagnostic workflow could be
beneficial, potentially allowing the identification of the tissue type based
on quantitative measurements.

Virtual monochromatic images are usually computed directly from decomposed
basis (_e.g.,_ $\mu_{1}$ and $\mu_{2}$) using $x_{1}$ and $x_{2}$ in equation
(2). Equivalently, physically relevant $\rho$/$Z_{\textrm{\tiny eff}}$ space
can be used to compute other quantitative maps established in clinical
practice. Linear attenuation values in figure 6 were calculated using equation
(1) at energies within and outside the energy range used in the experiment.
Virtual $\mu$ values calculated using the $Z_{\textrm{\tiny eff}}$ defined by
Champley et al (?) were in agreement (within the measurement error) with
experimentally measured $\mu$ coefficients in both the phantom and Sample 4
and they enabled comparison with experimentally obtained $\mu$ values in other
studies. For Sample 4, which contained tissues examined by Fredenberg et al
(?), $\mu$ values were compared at energies of 20, 30, and 40 keV with average
% error of 3.3, 7.3, and 3.4 %, for glandular, adipose and tumor tissue,
respectively.

Our approach to tissue analysis could be directly applied to state-of-the-art
synchrotron radiation breast CT setups currently developed in Trieste at
Elettra Sincrotrone SYRMEP beamline (?), and at ANSTO Imaging and Medical
beamline in Australia (?). With these experimental settings, it should be
mentioned that multiple energy acquisitions would required, thus resulting in
an increased dose to the breast. On the other hand, due to the high contrast-
to-noise ratio of phase-contrast images, the dose per scan could be reduced
thus bringing overall acceptable radiation exposures. Considering clinical
systems in hospitals, the advent of spectral CT paves the way to material
decomposition following a single shot acquisition without a significant
increment of the dose. Photon counting breast CT systems are in clinical
practice (?) and an extension to spectral applications is expected.

Studies estimating both $\rho$ and $Z_{\textrm{\tiny eff}}$ of human tissues
using synchrotron CT systems are almost nonexistent. The study by Torikoshi et
al (?) introduced a method to compute these quantities avoiding the material
decomposition task. An average accuracy of 0.9 $\%$ for $\rho_{e}$ and 1 $\%$
for $Z_{\textrm{\tiny eff}}$ is comparable to our method, but the dose level
used was not reported. The analysis was performed on low-$Z_{\textrm{\tiny
eff}}$ plastic materials using a pair of monochromatic acquisitions.
Considerably more papers using conventional systems have been published, but
not focusing on breast tissues or breast dedicated scanners. Szczykutowicz et
al (?) performed a methodologically similar approach to the one presented in
this paper using a clinical scanner and test object without any noise
remedying approach. They successfully decompose electron density and
$Z_{\textrm{\tiny eff}}$ but at the cost of a significant reduction in signal-
to-noise ratio. Lalonde et al (?) developed a model in which materials are
decomposed in a compressed basis with principal component analysis, using the
fact that human tissues are composed of a very limited number of elements.
Then, the first principal components are virtual materials containing a
certain fraction of those elements. From there $Z_{\textrm{\tiny eff}}$ were
computed. Azevedo et al (?) implemented their System-Independent-Rho-Z (SIRZ)
method to obtain physical quantities of phantom materials independent of the
shape of the X-ray spectrum. The photoelectric-Compton decomposition is
performed in sinogram space and absolute $\rho_{e}$ and Z values are obtained
after the calibration procedure. While physical characterization was
successfully described for materials of fairly distinguishable compositions,
the noise behavior was also not described in this work. Champley et al (?)
released a follow-up paper focusing on the optimization and simplification of
spectral modeling as a new method called SIRZ-2. Most recently, Busi et al (?)
developed a physical characterization method using spectral detectors. They
claim higher robustness and increased estimation accuracy ($~{}25\%$) compared
to SIRZ methods. Extended work from the same group was published in (?) to
optimize the computation speed. Machine learning solutions to
$\rho$/$Z_{\textrm{\tiny eff}}$ extraction were also tested by Su et al (?)
using dedicated phantoms with several tissue-equivalent inserts. Good results
in computing $Z_{\textrm{\tiny eff}}$ were obtained using artificial neural
networks and the random forest method with the relative error between 1 and 2
$\%$ at clinically relevant doses. However, a low-dose scanning and evaluation
of the model on materials that were removed from the training set led to
errors up to 6 $\%$. Nonetheless, the authors showed that the machine learning
approach is robust and computationally efficient.

Considering specifically the estimation of $\rho$ and $Z_{\textrm{\tiny eff}}$
for breast tissues, only a few studies exist and have been made with
experimental setups not used in diagnostic radiology. For instance, Gobo et al
(?) used a combination of transmission and scattering measurements with 241Am
source and an X-ray tube, while Antoniassi et al (?) performed scattering
measurements at 90 degrees by using low energy beams. Given diverse
formulations of $Z_{\textrm{\tiny eff}}$ across the literature, we made a
comparison in relative change using $Z_{\textrm{\tiny eff}}$ of nylon as a
reference value in table 4.

Table 4: Literature review of experimentally obtained density and $Z_{\textrm{\tiny eff}}$ numbers for fibro-glandular, adipose and tumorous tissues. Tissue type | Density $\rho$ |  $Z_{\textrm{\tiny eff}}$ to nylon % diff
---|---|---
| This work | Gobo et. al | This work |  Gobo et al |  Antoniassi et al
Fibro-glandular | 0.96 | 1.04 | 15.38 | 16.73 | 14.07
Adipose | 0.90 | 0.95 | 2.68 | 4.40 | 5.60
Tumorous | 1.07 | 1.05 | 21.31 | 18.87 | 14.79

Our study shows potential for quantitative breast imaging by translating
spectral information into the computation of physically relevant quantities,
but it also has some limitations. The accuracy of the presented method is
mainly governed by the quality of material decomposition, which is highly
dependent on the denoising of decomposed data. We gave up the spatial
information inside the ROI to obtain quantitatively correct material
decomposition during the 2D Gaussian denoising approach. Inaccurate tissue
segmentation would result in an erroneous Gaussian fitting procedure as the
number of peaks in 2D histogram space corresponds to the number of tissue
types being evaluated at once. The appearance of a histogram containing
several plastic materials was published in our previous study (?). Thus,
correct tissue segmentation is a critical aspect of the present model. For the
proof of concept, we relied on high-quality synchrotron beam radiation, but to
make this approach broadly used, we plan to apply the method to more
accessible polychromatic sources. Finally, the conclusions drawn in this study
were based on the analysis of 5 mastectomy samples. More samples will be
evaluated to further confirm these findings. Despite the mentioned
limitations, given the general validity of the proposed decomposition model
and the foreseeable use of spectral detectors in breast CT scanners, the
present feasibility study paves the way for its application to clinical
spectral breast CT data.

##  5 Conclusions

A model incorporating CT reconstructions of an arbitrary number of spectral
energy channels was developed to compute material density and effective atomic
number. Density and effective atomic number of soft-tissue-equivalent plastic
materials were computed with high accuracy, and the same approach was applied
to the set of 5 mastectomy samples. Quantitative analysis suggests that
adipose, fibro-glandular, and tumorous tissues could be well distinguished
based on their effective atomic number and density. Density and effective
atomic number can also be used for physics-based extrapolation of virtual
monochromatic linear attenuation coefficients outside of the experimental
energy range.

##  6 Acknowledgment

The authors acknowledge the SYRMA-3D collaboration at Elettra for the support.
We also would like to thank Dr. Cristina Marrocchio, Radiologist and Research
fellow at the University of Parma, Italy supporting us in tissue
identification from CT and histology data sets.

##  Appendix A Noise analysis and uncertainties

As it was shown in section 2.5, regions of uniform composition result in
scattered values of elongated shape (_i.e._ clusters) in the
$(x_{1},x_{2})$-plane histogram. Such behavior was ascribed to the unavoidable
amount of noise carried by the tomographic reconstructions on which the whole
decomposition method is based. Detailed analysis of the propagation of
measurement uncertainties, related to the size of the cluster, was carried
out. The Gaussian fitting procedure introduced in section 2.5 individuates a
preferred direction at angle $\theta$ aligned to the major axis $u$ of
elliptical Gaussian. The length in this direction and the orthogonal one
(minor axis) are identified as the Gaussian spreads in a reference frame
$(x_{u},x_{v})$, which is parallel to the major and minor axes of the spot.
The standard errors of the centroid coordinates can be approximated by the
ratios between the Gaussian spreads $S_{u}$ and $S_{v}$ and the square root of
the volume under the Gaussian surface

$\displaystyle\sigma_{x_{u}}=\frac{S_{u}}{\sqrt{V}}$ (4efhka)
$\displaystyle\sigma_{x_{v}}=\frac{S_{v}}{\sqrt{V}}$ (4efhkb)

where the volume is given as

$V=2\pi\,A\,\frac{S_{u}\,S_{v}}{b_{1}\,b_{2}}$ (4efhkl)

that it is equal to the total number of voxels contributing to the
corresponding two-dimensional Gaussian function. Quantities $b_{1}$ and
$b_{2}$ are bin sizes in both directions and $A$ is the peak value. Calculated
errors are then translated into uncertainties of coordinates $x_{\xi}$ and
$x_{\zeta}$ in the reference frame rotated by the angle $\phi$ defined in
section (2.1). Rotation from the frame identified by angle $\theta$ to the
frame identified by angle $\phi$ is defined as

$\left[\matrix{x_{\xi}\cr
x_{\zeta}}\right]=\left[\matrix{\cos\alpha&\sin\alpha\cr-\sin\alpha&\cos\alpha}\right]\,\left[\matrix{x_{u}\cr
x_{v}}\right]$

where $\alpha=\phi-\theta$. Propagating uncertainty through the rotation of
the reference frame leads to

$\displaystyle\sigma^{2}_{x_{\xi}}=\cos^{2}\alpha\,\sigma^{2}_{x_{u}}+\sin^{2}\alpha\,\sigma^{2}_{x_{v}}$
(4efhkma)
$\displaystyle\sigma^{2}_{x_{\zeta}}=\sin^{2}\alpha\,\sigma^{2}_{x_{u}}+\cos^{2}\alpha\,\sigma^{2}_{x_{v}}$
(4efhkmb)

From the rotated frame, to the rescaled $x_{\rho},x_{Z}$ frame, the division
of the second coordinate by the first one implicates

$\sigma^{2}_{x_{Z}}=\frac{x^{2}_{\zeta}}{x^{4}_{\xi}}\sigma^{2}_{x_{\xi}}+\frac{1}{x^{2}_{\xi}}\sigma^{2}_{x_{\zeta}}$
(4efhkmn)

while propagating $x_{\rho}$ remains trivial

$\sigma^{2}_{x_{\rho}}=\sigma^{2}_{x_{1}}.$ (4efhkmo)

Finally, shifting to the pair $\rho,Z$ requires equations (LABEL:xrhoZfuncs),
leading to the uncertainties

$\displaystyle\sigma_{\rho}$ $\displaystyle=\kappa^{-1}\,\sigma_{x_{\rho}}$
(4efhkmpa) $\displaystyle\sigma_{Z}$
$\displaystyle=\frac{(x_{Z}-q)^{\frac{1-\lambda}{\lambda}}}{\lambda\,p^{1/\lambda}}\,\sigma_{x_{Z}}\,.$
(4efhkmpb)

##  Appendix B Noise behaviour in 2D histogram space

In addition to the estimated centers of the distributions in figure 3(a), the
2D Gaussian fit method outputs several other parameters relevant for the
accurate estimation of basis material concentrations summarized in table 5.
The angle $\theta$ at which the cluster is extended remains constant
($\simeq-0.12$) for all materials indicating that blurring is not dependent on
the material type. It is rather a result of combined contributions of image
acquisition, reconstruction, and material decomposition noise. The amount of
blurring in the major direction $\textrm{S}_{u}$ and the direction orthogonal
to it $\textrm{S}_{v}$ is of the same order of magnitude for all materials.
The $\textrm{S}_{u}$ values are around 2 orders of magnitude larger than
$\textrm{S}_{v}$ values. The amplitude A of the Gaussian function remains
constant for the same size of the ROI for all plastic inserts.

Table 5: List of output parameters in 2D Gaussian fitting method for the phantom materials. Material | Amplitude (A) | Major axis spread $\textrm{S}_{u}$ | Minor axis spread $\textrm{S}_{v}$ | Angle $\theta$
---|---|---|---|---
Water | 38 | 0.40 | 0.0074 | -0.121
PE | 40 | 0.37 | 0.0065 | -0.121
PA | 38 | 0.32 | 0.0062 | -0.120
PMMA | 38 | 0.41 | 0.0073 | -0.121
POM | 38 | 0.32 | 0.0062 | -0.120
PTFE | 41 | 0.46 | 0.0078 | -0.121

These properties are important for the Gaussian fitting procedure because an
initial guess for fitting parameters can be given, improving the robustness of
the method and increasing the computational speed. A more rigorous statistical
description of these features will be the subject of a forthcoming standalone
communication.

##  Appendix C Comparison of methods used to compute the atomic number of a
compound - ZcompARE

Effective atomic number as a property of a material depends on its chemical
composition, but it is not uniquely defined in the literature. Very often
different definitions are adopted by researchers depending on the experimental
setup, energy range, and type of compounds. Several definitions that have been
proposed to compute this quantity can be divided into: i) methods that compute
$Z_{\textrm{\tiny eff}}$ as a weighted average of composing elements in the
compound, ii) and methods that rely on mass attenuation coefficients of
composing elements to compute $Z_{\textrm{\tiny eff}}$ (?). The second group
of methods was developed to solve the fundamental problem: $Z_{\textrm{\tiny
eff}}$ as the quantity defined in the first category is not specifically tied
to the absorption property of a material. Thus, weighted sums of elements in a
compound used in the first category are exclusively valid for certain energy
ranges and often to a certain set of elements contained in the mixture, while
methods in the second category rely on tabulated attenuation properties of
materials to inherently define $Z_{\textrm{\tiny eff}}$ as a quantity tied to
attenuation property of the material.

In this section, we compared several available methods for the materials in
the calibration phantom. Z _comp_ ARE (?) is user-friendly software with a
graphical interface that can be used to compare several methods most often
used. The comparison of the methods for the calibration phantom materials in
the energy range 20 - 40 keV is given in 6.

Table 6: Comparison of the most often used formulations of effective atomic numbers for phantom materials in the energy range 20 - 40 keV. Material | Water | PE | PA | PMMA | POM | PTFE
---|---|---|---|---|---|---
Brute formula | $\textrm{H}_{2}\textrm{O}$ | $\textrm{C}_{2}\textrm{H}_{4}$ | $\textrm{C}_{12}\textrm{H}_{22}\textrm{N}_{2}\textrm{O}_{2}$ | $\textrm{C}_{5}\textrm{H}_{8}\textrm{O}_{2}$ | $\textrm{C}\textrm{H}_{2}\textrm{O}$ | $\textrm{C}_{2}\textrm{F}_{4}$
Spiers et al | 7.42 | 5.44 | 6.12 | 6.47 | 6.95 | 8.43
Glasser et al | 7.96 | 5.94 | 6.60 | 6.94 | 7.38 | 8.62
Hine et al | 3.34 | 2.67 | 3.27 | 3.60 | 4.00 | 7.99
Puumalainen et al | 3.33 | 2.67 | 3.27 | 3.61 | 4.00 | 8.00
Tsai and Cho | 7.44 | 5.47 | 6.15 | 6.50 | 6.98 | 8.45
Gowda et al | 7.24 | 5.05 | 5.89 | 6.32 | 6.85 | 8.53
Champley et al | 7.44 | 5.28 | 6.16 | 6.49 | 7.01 | 8.56

It can be seen that methods by Spiers et al, Tsai and Cho, Glasser et al,
Gowda et al, and Champley et al provide very similar $Z_{\textrm{\tiny eff}}$
numbers for our experimental conditions. Methods by Hine et al and Puumalainen
et al gave considerably lower $Z_{\textrm{\tiny eff}}$ numbers for all
materials. Calibration functions of the form defined in equations (4efa) and
(4efb) were applied to all definitions for $Z_{\textrm{\tiny eff}}$ and
results are given in figure 7. The mean Silhouette score defining the level of
separation in the range -1 to 1 was found to be approx. 0.31 for all methods
used. Thus, the distinguishment of tissues could be obtained irrespective of
the proposed definition.

(a)

(b)

(c)

(d)

(e)

(f)

(g) Comparison of $Z_{\textrm{\tiny eff}}$ definitions in
$\rho$/$Z_{\textrm{\tiny eff}}$ space.

Figure 7:

The method defined by Champley et al (?) showed slightly better agreement of
virtual monochromatic $\mu$ values computed using equation 1 with
experimentally measured data. Constants $n$, $K_{1}$, and $K_{2}$ used in
equation 1 were estimated to be 4.44, 9.4, and 1.6 using the least-square fit
method for elemental materials found in the human body in the Z range of 1-20
across the diagnostic energy range (20 - 200 keV).

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