#  $D^{0}$ MIXING AND CP VIOLATION IN $D$ DECAYS

Érica Polycarpo  Universidade Federal do Rio de Janeiro
CP 68528, 20511-010, Rio de Janeiro, RJ, Brasil
E-mail<EMAIL_ADDRESS>
###### Abstract

We present a brief review of CPV and mixing measurements in the charm sector,
with emphasys in results published or presented since the previous edition of
the Physics in Collision Symposia.

##  1 Introduction

Neutral mesons mixing occur in nature due to the fact that the mass
eigenstates are not weak interaction eigenstates. An initially pure beam of
$D^{0}$ or $\bar{D}^{0}$ mesons evolves in time to a mixture of $D^{0}$ and
$\bar{D}^{0}$ states with rates determined by the solution of the Schröedinger
equation

$i\frac{d}{dt}\begin{pmatrix}D^{0}\\\ \bar{D}^{0}\\\ \end{pmatrix}(t)={\bf
H_{eff}}\begin{pmatrix}D^{0}\\\ \bar{D}^{0}\\\ \end{pmatrix}(t),$ (1)

where the diagonal elements of the effective Hamiltonian $\bf H_{eff}\equiv
M-\frac{i}{2}\Gamma$ and of the hermitian matrices $M$ and $\Gamma$ are
related to the flavour conserving $D^{0}(\bar{D}^{0})\rightarrow
D^{0}(\bar{D}^{0})$ transitions, while the non-diagonal ones are related to
the flavour changing $D^{0}\leftrightarrow\bar{D}^{0}$ transitions. The
solution of Eq.1 are the eigenstates $|D_{1,2}\rangle=p|D^{0}\rangle\pm
q|\bar{D}^{0}\rangle$, with complex coefficients $p$ e $q$ satisfying
$\sqrt{|p|^{2}+|q|^{2}}=1$. The eigenstates have well defined masses and
widths $m_{1},\Gamma_{1}$ and $m_{2},\Gamma_{2}$. The parameters which govern
the mixing are $x\equiv\frac{m1-m2}{\Gamma}$ and
$y\equiv\frac{\Gamma_{1}-\Gamma_{2}}{2\Gamma}$, where
$\Gamma\equiv\frac{\Gamma_{1}+\Gamma_{2}}{2}$ is the mean decay width. For
charm mesons, both $x$ and $y$ are small. There are essentially two approaches
used to predict their values in the Standard Model (SM), none of them leading
to fully reliable results (see, for example, [1, 2, 3] and references
therein). The transitions leading to the $D^{0}-\bar{D}^{0}$ oscillations are
those represented in Fig. 1. Short distance contributions are illustrated by
the box diagram on the left and long distance transitions, involving
intermediate states accessible by both $D^{0}$ and $\bar{D}^{0}$, on the
right. As seen from these diagrams, charm mixing gives us access to
transitions involving intermediate states with down-type quarks. In fact, this
is the only system which offers this probe, since the top quark is too heavy
to hadronize and the $\pi^{0}$ is its own anti-particle.

While $x$ receives contributions from both the short distance and long
distance processes, $y$ only receives contributions from the long distance
transitions. Since $x$ is the only one to receive contributions from virtual
intermediate states, new physics introducing off-shell particles in the box
diagrams would only affect $x$ and therefore it is a sort of consensus that an
unambiguous sign of NP would be given by a $x$ value well above $y$. The first
evidences of mixing were obtained in 2007 by BaBar [4], Belle [5] and CDF [6].
At present, both $x$ and $y$ are constrained by several measurements at the
values $x=(0.65^{+.018}_{-0.19})$% and $y=(0.73\pm 0.12)$% and the no mixing
hypothesis is excluded with a statistical significance above 10 standard
deviations ($\sigma$) [7], under the assumption of no CP violation. Although
theoretical predictions are not precise, the experimental values of the mixing
parameters are now consistent with SM expectations [8], even if still
considered at the upper level of possible values [9].

|
---|---
Figure 1: Short and long distance contributions to $D^{0}-\bar{D}^{0}$ mixing.

The CP symmetry can be violated in the mixing, if $|q|\neq|p|$; in the decay,
if the magnitudes of the instantaneous decay amplitudes ${\cal A}_{f}=\langle
f|{\cal H}|D^{0}\rangle$ and $\bar{\cal A}_{\bar{f}}=\langle\bar{f}|{\cal
H}|\bar{D}^{0}\rangle$ for CP conjugate processes are not the same; and, for
decays reachable by both the $D^{0}$ and $\bar{D}^{0}$, in the interference
between mixing and decay. This kind of mixing induced CPV can occur even in
the absence of CPV in the mixing and in the decay, if there is a non null
phase difference $\phi$ between the mixing and the decay amplitudes.

In the SM, CP violation (CPV) is suppressed by the small values of the CKM
parameters $V_{ub}$ and $V_{cb}$. In the mixing, it is expected to be at the
level of 10-4 [10], independently of the final state. Direct CPV can only
occur in singly Cabibbo suppressed (SCS) decays, since their amplitudes can
receive contributions from processes with virtual b-quarks, involving the
complex elements of the CKM matrix. Existing predictions for SCS modes in the
SM were, at least until recently, of the order of $10^{-3}$ or less. The
interest in CPV in charm decays is, therefore, related to searches for new
physics effects in a SM suppressed environment. Experimentally, the only
existing evidence of CPV in the charm sector is the measurement of the
difference between the time integrated asymmetries of the decays $D^{0}\to
K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ [11], that will be further discussed
in this document.

A very good review on CPV and mixing on the charm sector is given in [3]. Here
we report on some recent measurements performed by BaBar, Belle, CDF and LHCb
since the previous edition of the Physics in Collision conference series.

##  2 Time dependent measurements of $D^{0}-\bar{D}^{0}$ mixing and CP
violation

There are several ways to access the mixing parameters $x$ and $y$
experimentally, using decays to various final states. Each method provide
different sensitivities to these parameters and, by combining these
measurements, we can improve our knowledge about them. The HFAG group [12]
provides world averages and best fit values to these parameters using results
provided by several experiments at different machines around the world. A good
overview of the most up to date averages are given in [7].

One of the most sensitive ways of determining $x$ and $y$ is via a time
dependent amplitude analysis of the decays $D^{0}\to K_{s}\pi^{+}\pi^{-}$ and
$D^{0}\to K_{s}K^{+}K^{-}$. The two final states are CP eigenstates reachable
by both the $D^{0}$ and $\bar{D}^{0}$. The time dependent rate for these
decays is proportional to terms arising from the interference between the
direct decay and the decay via mixing:

${\mathcal{R}}(t)\propto{|A_{1}|^{2}e^{-yt}+|A_{2}|^{2}e^{-yt}+2Re[A_{1}A^{*}_{2}]\cos(xt)+2Im[A_{1}A^{*}_{2}]\sin(xt)}$
(2)

where $A_{1,2}=\frac{{\cal A}\pm\bar{\cal A}}{2}$ and ${\cal A}$ and
$\bar{\cal A}$ are the instantaneous decay amplitudes for $D^{0}$ and
$\bar{D}^{0}$, written as a function of the 2-body invariant masses $M^{+}$
($K^{0}_{S}K^{+}$) and $M^{-}$ ($K^{0}_{s}K^{-}$). In order to determine the
flavour of the meson at production, it is required to form a vertex with a low
momentum pion $\pi_{s}$, consistent with the decay vertex of a
$D^{*+}\to\pi^{+}_{s}D^{0}$ or $D^{*-}\to\pi^{-}_{s}\bar{D}^{0}$. The charge
of the $\pi_{s}$ identifies the $D^{0}$ flavour. This analysis was first
performed by CLEO [13] and subsequently by Belle [14] and BaBar [15], with
much larger samples. An update of the $D^{0}\to K_{s}\pi^{+}\pi^{-}$ analysis
by Belle was presented this summer, using their full data set of 920 fb-1 and
an improved tracking, which provides a signal yield of 1.23 M events with a
purity of 95.6% [16]. They parametrize the amplitudes ${\cal A}$ and
$\bar{\cal A}$ as a sum of 12 quasi-two-body amplitudes describing the P and D
waves and two components to describe the $K\pi$ (LASS model) and the $\pi\pi$
(K-matrix model) s-waves. From a maximum likelihood fit to the data in the
signal region, the mixing parameters $x$ and $y$, the average $D^{0}$-meson
lifetime $\tau_{D^{0}}$ and the magnitudes and phases of the resonances are
directly extracted. The preliminary results shown are $x=(0.56\pm
0.19^{+0.03+0.06}_{0.09-0.09})$% and $y=(0.30\pm
0.15^{+0.04+0.03}_{-0.05-0.06})$%, where the first uncertainty is statistical,
the second is due to experimental systematic effects and the third
systematical due to the model used to fit the Dalitz plot. At the time of the
conference, this was the most precise single measurement of the mixing
parameters.

Another measurement which is sensitive to mixing is the deviation from unity
of the ratio of the effective lifetimes measured in decays to the CP
eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ with respect to the CP mixed
state $K^{-}\pi^{+}$. Since mixing is very slow, the decay time distributions
are, to a good approximation, simply exponential. The fits to the decay time
distributions of the different samples use a model where signal events are
described by a single exponential and provide the effective lifetimes used to
build the observable

$y_{CP}\equiv\frac{\tau(D^{0}\to h^{-}h^{+})}{\tau(D^{0}\to
K^{-}\pi^{+})}-1\approx\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right]$
(3)

where $h^{-}h^{+}$ corresponds to $\pi^{-}\pi^{+}$ or $K^{-}K^{+}$, the phase
$\phi$ is considered universal and a contribution from CPV in the decay is
neglected. In the absence of CPV, $|q/p|=1$ and $\phi=0$, hence $y_{CP}=y$.
This measurement was previously performed by several collaborations and
provided the first evidence for mixing obtained by the BELLE experiment in
2007 [5]. In April 2011, both BaBar and Belle presented new results using
their full data samples, with integrated luminosities of 468 fb-1 [17] and 976
fb-1 [18], respectively. LHCb also published a first result for $y_{CP}$ using
29 pb-1 of data collected in 2010 [19]. In addition to the $D^{*\pm}$ tagged
samples, BaBar used also untagged samples of $D^{0}(\bar{D}^{0})\to
K^{\pm}\pi^{\mp}$ and $D^{0}(\bar{D}^{0})\to K^{-}K^{+}$ decays. These results
are presented in Table 1, together with the size of the samples used in each
measurement.

The effective lifetimes obtained by fitting the decay time distributions of
the tagged samples separately for the CP conjugate decays provide a second
quantity, $A_{\Gamma}$, sensitive to CPV:

$\small A_{\Gamma}(h^{-}h^{+})\equiv\frac{\tau(\bar{D}^{0}\to
h^{-}h^{+})-\tau(D^{0}\to h^{-}h^{+})}{\tau(\bar{D}^{0}\to
h^{-}h^{+})+\tau(D^{0}\to
h^{-}h^{+})}\approx\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right],$
(4)

considering as valid the same approximations used for $y_{CP}$.

The three results for $y_{CP}$ and $A_{\Gamma}$ are consistent with each other
and with previous measurements and also compatible with the hypothesis of no
CPV. They bring down the world average for $y_{CP}$ from $(1.064\pm 0.209)$%
to $(0.866\pm 0.155)$% [7], relaxing the previously existing tendency for
$y_{CP}>y$ [3]. The new world average for $A_{\Gamma}$ is $(-0.022\pm 0.161)$%
[7].

Table 1: Recent experimental results for the quantities $y_{CP}$ and $A_{\Gamma}$. The $h^{-}h^{+}$ yield is given for $K^{-}K^{+}$ and inside the parentheses for $\pi^{-}\pi^{+}$. First uncertainty is statistical, second is systematical. BaBar measures, instead of $A_{\Gamma}$, $\Delta Y=(1+y_{CP})A_{\Gamma}$. Experiment | $K^{-}\pi^{+}$ | $h^{-}h^{+}$ |  $y_{CP}$ (%) |  $A_{\Gamma}$ ($\Delta Y$) (%)
---|---|---|---|---
|  yield ($\times 10^{6}$) |  yield ($\times 10^{3}$) |  |
LHCb | 0.225 | 30 | $5.5\pm 6.3\pm 4.1$ | $-5.9\pm 5.9\pm 2.1$
BaBar | 7.312 | 633 (65) | $0.72\pm 0.18\pm 0.12$ | $0.09\pm 0.26\pm 0.09$
Belle | 2.61 | 242 (114) | $1.11\pm 0.22\pm 0.11$ | $-0.03\pm 0.20\pm 0.08$

##  3 Time integrated CP asymmetries

###  3.1 Two-body decays

One of the most discussed results from the LHC in the past year was the first
evidence of CPV in the charm sector, provided by the difference of the time
integrated CP asymmetries measured in the decays of the $D^{0}$ meson to
$K^{-}K^{+}$ and $\pi^{-}\pi^{+}$, presented by LHCb [11] and supported by new
results by CDF [20] and Belle [21]. To first order of approximation, the raw
asymmetry can be written as a sum of small terms

$A_{raw}(f)=A_{CP}(f)+A_{det}(f)+A_{det}(\pi_{s})+A_{prod}(D^{*+}),$ (5)

where $A_{CP}(f)$ is the physical CP asymmetry of interest, $A_{det}(f)$ is
the asymmetry in the detection efficiency for the decay to final state $f$,
$A_{det}(\pi_{s})$ is the reconstruction efficiency for the slow pion from the
$D^{*+}\to\pi^{+}_{s}D^{0}$ tagging chain and $A_{prod}(D^{*+})$ is a physical
asymmetry for $D^{*+}$ production. This asymmetry vanishes in CP conserving
strong collisions, as the $p\bar{p}$ collisions at the Tevatron, if the events
are reconstructed symmetrically in the pseudo-rapidity $\eta$. However, it is
not expected to be null in the $pp$ collisions at the LHC. In $e^{+}e^{-}$
colliders, there is a forward-backward asymmetry in the production of charm
mesons arising from $\gamma Z^{0}$ interference and higher order QED effects
[22].

When dealing with two-body decays of a spin 0 particle to self-conjugate final
states $f$, the detection efficiency $A_{det}(f)$ is null. Since the remaining
terms, apart from $A_{CP}(f)$, are independent of the final state, they cancel
out in the difference of asymmetries $\Delta
A_{raw}=A_{raw}(K^{+}K^{-})-A_{raw}(\pi^{+}\pi^{-})$, which thus becomes a
robust measurement of the difference between the physical CP asymmetries:

$\Delta A_{CP}\equiv A_{CP}(K^{+}K^{-})-A_{CP}(\pi^{+}\pi^{-})=\Delta
A_{raw}.$ (6)

Experimentally, the raw asymmetries are obtained from

$A_{raw}(f)\equiv\frac{N(D^{0}\to f)-N(\bar{D}^{0}\to f)}{N(D^{0}\to
f)+N(\bar{D}^{0}\to f)}$ (7)

where $N(X)$ corresponds to the number of reconstructed decays $X$ after
background subtraction.

LHCb performed this measurement using 0.62 fb-1 of data collected in 2011.
Maximum likelihood fits to the $\delta m=m(\pi_{s}h^{-}h^{+})-m(h^{-}h^{+})$
spectra provide the number of signal events used to calculate the raw
asymmetries in different kinematic bins, in order to guarantee that the
detection and production asymmetries are cancelled. The kinematic bins are
defined in intervals of the $D^{*\pm}$ candidate transverse momentum $p_{T}$
and $\eta$ and the final result, shown in Table 2, is obtained from a weighted
average over this kinematic space. This result deviates from zero with a
significance of 3.5 $\sigma$.

CDF also performed a new $\Delta A_{CP}$ measurement, using the full data
sample collected at Tevatron Run II, which corresponds to 9.7 fb-1 [20].
Compared to the previously published measurement [23], their new result relies
on a larger data sample and an optimized selection procedure. In order to take
into account differences in the instrumental asymmetries due to phase space
variations among the two final states, a reweighting of the samples is
performed, so as to equalize the $\pi_{s}$ impact parameter, $p_{T}$ and
$\eta$ distributions. A simultaneous $\chi^{2}$ fit of the resulting
$D^{0}\pi^{+}$ and $\bar{D}^{0}\pi^{-}$ mass distributions provides the number
of signal events used in the raw asymmetries. The result, also shown in Table
2, is 2.7$\sigma$ away from zero, with a precision level very close to the
obtained by LHCb. The individual asymmetries, measured with a sample of 6.0
fb-1, are $A_{CP}(K^{-}K^{+})=-0.24\pm 0.22\pm 0.09$% and
$A_{CP}(\pi^{-}\pi^{+})=(+0.22\pm 0.24\pm 0.11)$% [23].

In July 2012, Belle presented a preliminary update of the individual
asymmetries $A_{CP}(K^{-}K^{+})$ and $A_{CP}(\pi^{-}\pi^{+})$, reporting also
a $\Delta A_{CP}$ measurement, using a data sample corresponding to 976 fb-1.
Instrumental asymmetries are obtained using control samples of CP conserving
decays and the production asymmetry is eliminated by making the measurement in
bins of $cos\theta^{*}$, where $\theta^{*}$ is the polar angle of the $D^{*+}$
production in the center of mass system. The results are
$A_{CP}(K^{-}K^{+})=-0.3\pm 0.21\pm 0.09$% and
$A_{CP}(\pi^{-}\pi^{+})=(+0.55\pm 0.36\pm 0.09)$%. The $\Delta A_{CP}$ value,
given in Table 2, deviates from zero with a 2.1$\sigma$ significance.

Table 2: Recent experimental results for $\Delta A_{CP}$. The last column shows the difference between the average proper time of the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ samples used to extract $\Delta A_{CP}$. Experiment | $K^{-}K^{+}$ | $\pi^{-}\pi^{+}$ | $\delta A_{CP}$ | $\Delta\langle t\rangle$
---|---|---|---|---
|  yield ($\times 10^{6}$) |  yield ($\times 10^{6}$) |  |
LHCb | 1.44 | 0.38 |  ($−0.82\pm 0.21\pm 0.11$)% | $\sim 0.098\tau_{D^{0}}$
CDF | 0.55 | 1.2 |  ($-0.62\pm 0.21\pm 0.10$)% | $\sim 0.26\tau_{D^{0}}$
Belle | 0.282 | 0.123 |  ($-0.87\pm 0.41\pm 0.06$)% | 0

The asymmetry difference can be written, to first order of approximation, in
terms of the direct and indirect components as [23]

$\Delta A_{CP}\approx\Delta a^{dir}_{CP}+\frac{\Delta\langle
t\rangle}{\tau_{D^{0}}}a^{ind},$ (8)

where $\langle t\rangle$ is the average reconstructed decay time,
$a^{dir}_{CP}(f)=(|{\cal A}_{f}|^{2}-|\bar{\cal A}_{f}|^{2})/(|{\cal
A}_{f}|^{2}+|\bar{\cal A}_{f}|^{2})$ and
$a^{ind}_{CP}=-\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right]$
and the phase $\phi$ is assumed to be universal. From this equation and the
$\Delta\langle t\rangle/\tau_{D^{0}}$ values given in Tab. 2 for the different
experiments, it becomes clear that $\Delta A_{CP}$ is mostly a measurement of
direct CPV.

Including higher order corrections, the first term in the right side of Eq. 8
[24] changes to $\Delta a^{dir}_{CP}[1+y_{CP}(\langle
t(K^{-}K^{+})\rangle+\langle t(K^{-}K^{+})\rangle)/(2\tau_{D^{0}})]$. Assuming
that the current precision of $A_{\Gamma}$ is such that it is not sensitive to
$\Delta a^{dir}_{CP}$, HFAG fits the contributions of direct and indirect CPV
to the most recent results of $y_{CP}$, $A_{\Gamma}$ and $\Delta A_{CP}$ by
Babar, Belle, CDF and LHCb [12], as shown in Fig. 2. This fit yields $\Delta
a^{dir}_{CP}=(-6.78\pm 1.47)\times 10^{-3}$ and $a^{ind}_{CP}=(0.27\pm
1.63)\times 10^{-3}$, again favouring a direct CPV effect over an indirect,
and is consistent with CP conservation with a confidence level of only 0.002%.

Figure 2: Combined measurements of $\Delta A_{CP}$ and $A_{\Gamma}$, where the
bands represent $\pm 1\sigma$ intervals. The point of no CP violation $(0,0)$
is shown as a filled circle, and two-dimensional 68% CL, 95% CL, and 99.7% CL
regions are plotted as ellipses with the best fit value as a cross indicating
the one-dimensional uncertainties in their center. Reproduced from [3, 12].

BaBar and Belle searched for CPV in the decays of the $D^{+}$ and $D^{+}_{s}$
mesons to two-body final states containing a $K^{0}_{S}$ using data samples
corresponding to 469 fb-1 [25, 27] and 673 fb-1 [26], respectively. The
results are presented in Table 3. The first two decays are SCS decays, while
the latter result from the coherent sum of Cabibbo favoured and doubly Cabibbo
suppressed transitions and are not expected to exhibit CPV in the charm
dynamics. For all those decays, however, the SM predicts a small CP asymmetry
due to the $K^{0}-\bar{K}^{0}$ mixing. This contribution is measured to be
$(+0.332\pm 0.006)$% for the $D^{+}_{s}\to K^{0}_{S}\pi^{+}$ mode and
$(-0.332\pm 0.006)$% for the other modes [29]. The Belle result for the
$D^{+}\to K^{0}_{s}\pi^{+}$ asymmetry was performed with a larger sample,
corresponding to the full data set of 977 fb-1, and deviates from zero with a
significance of 3.2$\sigma$. In this work, they estimate a correction to the
$K^{0}-\bar{K}^{0}$ mixing induced asymmetry following [30] and, after
subtracting this contribution, they estimate a CP asymmetry due to the charm
dynamics alone of $(-0.024\pm 0.094\pm 0.067)$%, which is the most precise
measurement of $A_{CP}$ in charm decays to date. The $A_{CP}$ values obtained
for the other channels are all consistent with zero and with the SM
expectation. In particular the Babar measurements presented in May 2011 are
the most precise [27] determination of these asymmetries.

Table 3: Most recent experimental results for CP asymmetries in the decays $D^{+}\to K^{0}_{S}K^{+}$, $D^{+}_{s}\to K^{0}_{S}\pi^{+}$, $D^{+}\to K^{0}_{S}\pi^{+}$ and $D^{+}_{s}\to K^{0}_{S}K^{+}$. Decay |  BaBar $A_{CP}$(%) |  Belle $A_{CP}$(%)
---|---|---
$D^{+}\to K^{0}_{S}K^{+}$ |  $+0.13\pm 0.36\pm 0.25$[27] |  $-0.16\pm 0.58\pm 0.25$[26]
$D^{+}_{s}\to K^{0}_{S}\pi^{+}$ |  $+0.6\pm 2.0\pm 0.3$[27] |  $+5.45\pm 2.50\pm 0.33$[26]
$D^{+}_{s}\to K^{0}_{S}K^{+}$ |  $-0.05\pm 0.23\pm 0.24$[27] |  $+0.12\pm 0.36\pm 0.22$[26]
$D^{+}\to K^{0}_{S}\pi^{+}$ |  $-0.44\pm 0.13\pm 0.10$[25] |  $-0.363\pm 0.094\pm 0.067$[28]

###  3.2 Multi-body decays

In the decays of D mesons to final states containing more than 2 hadrons, the
rich resonant structure of the intermediate state provide the strong phases
necessary to the observation of CPV, under the hypothesis of existing weak CPV
phases. The interference between the resonances can lead to CPV asymmetries
which vary across the phase space. These local asymmetries can be large
compared to the overall asymmetry or can even happen in the absence of phase
space integrated asymmetry.

LHCb used a model independent approach to look for a CPV signal in the SCS
decay $D^{+}\to K^{-}K^{+}\pi^{+}$ using a high purity sample ($\sim$90%)
containing about 370$\times 10^{3}$ decays selected from about 35 pb-1 of data
collected in 2010 [31]. Within this technique [32, 33], the Dalitz plot is
divided in bins and for each bin the statistical significance of the asymmetry

$S^{i}_{CP}=\frac{N^{i}_{+}-\alpha
N^{i}_{-}}{\sqrt{N^{i}_{+}+\alpha^{2}N^{i}_{-}}}$ (9)

is calculated, where $N^{i}_{\pm}$ is the number of $D^{\pm}$ decays in each
bin $i$ and $\alpha$ is the ratio between the total number of $D^{+}$ and
$D^{-}$ events and is used as a correction due to a global production
asymmetry. The distribution of $S^{i}_{CP}$ is normal under the hypothesis of
CP conservation. A $\chi^{2}$ test using $\chi^{2}=\sum^{N}_{i=1}S^{i}_{CP}$
provides a numerical evaluation for the degree of confidence for the
assumption that the differences between the $D^{+}$ and $D^{-}$ Dalitz plots
are driven only by statistical fluctuations. Four different binning schemes
are considered, two of them uniform and two adaptative which account for the
resonant structure of the decay. Higher statistics control samples consisting
of the CF decays $D^{+}_{s}\to K^{-}K^{+}\pi^{+}$ and $D^{+}\to
K^{-}\pi^{+}\pi^{+}$ are used to demonstrate that possible instrumental
asymmetries varying over the Dalitz plot are under the sensitivity of the
method. The method is also applied to the events in the side bands of the
$D^{+}$ mass distribution to prove that no effect can be introduced by the
background. The p-values obtained with the $\chi^{2}$ tests for the
$S^{i}_{CP}$ distributions using the four different binning choices are 12.7%,
10.6%, 82.1% and 60.5%. The widths and means obtained from the fits to the
distributions are all consistent with 1 and 0. No evidence for CPV is found in
any of the binning schemes.

BaBar has also performed a similar analysis of $D^{+}\to K^{-}K^{+}\pi^{+}$
events [34], using a sample of 476 fb-1 of data with $\sim 228\times 10^{3}$
events (92% pure). They divide the Dalitz plane into 100 equally populated
bins and find a 72% probability for the consistency with the assumption of no
CPV. The measured integrated asymmetry is $A_{CP}=(0.35\pm 0.30\pm 0.15)$%. A
full amplitude analysis provides CP-violating differences for the phases and
magnitudes of the intermediate resonances, all consistent with zero.

CDF searched for CPV in the resonant substructure of $D^{*+}$ tagged $D^{0}\to
K^{0}_{S}\pi^{-}\pi^{+}$ decays using a sample corresponding to 6 fb-1
collected at Run II [35]. The SM predictions for this channel are of the order
of 10-6, dominated by the $K^{0}_{S}$ mixing contribution. The selected sample
is about 90% pure and contains approximately $350\times 10^{3}$ events. A
simultaneous maximum likelihood fit to the $D^{0}$ and $\bar{D}^{0}$ Dalitz
plots, taking into account the effect of efficiency over the Dalitz space,
provide resonances parameters and CP-violating fractions, amplitudes and
phases. Differences between the $p_{T}$ distributions of the slow pions of
opposite charges are used to reweight the $\bar{D}^{0}$ distribution. All the
asymmetries are measured to be consistent with zero, as is the overall
integrated asymmetry, $A_{CP}=(−0.05\pm 0.57\pm 0.54)$%. A model independent
analysis similar to the one described above for the $D^{+}\to
K^{-}K^{+}\pi^{+}$ decay also provides null CPV signal.

For the decay $D^{0}\to\pi^{-}\pi^{+}\pi^{-}\pi^{+}$, the phase space becomes
5-dimensional. LHCb performed the first model independent CPV analysis of this
decay, using about $180\times 10^{3}$ $D^{*+}$ tagged candidates, selected
from a sample of 1 fb-1 collected in 2011 [36]. The decay $D^{0}\to
K^{-}\pi^{+}\pi^{-}\pi^{+}$ is used as control mode and events in the $D^{0}$
sample are rejected in a way such to ensure that the $\eta$ and $p_{T}$
distributions of the $D^{0}$ decays match those of the $\bar{D}^{0}$ decays.
The distribution of the asymmetry significances across the phase space is
shown to be compatible with the hypothesis of CP conservation.

##  4 Summary and Conclusions

Thanks for the pioneering work of the charm factories and the excellent work
done by Babar, Belle and CDF, mixing in the $D^{0}-\bar{D}^{0}$ system is well
established, though no single measurement with a significance above 5 standard
deviations was available until the beginning of the preparation of these
proceedings. In November 2012, LHCb has submitted a paper reporting the
observation of oscillations in the time dependent rate of $D^{0}\to
K^{+}\pi^{-}$ relative to $D^{0}\to K^{-}\pi^{+}$, which excludes the no
mixing hypothesis with a significance of 9.1 $\sigma$ [37].

The recent results obtained for $\Delta A_{CP}$ raised the interest about CPV
in charm decays. A lot of effort was put, from the theory side, to understand
on whether this can be, if confirmed, due to a new physics effect or to an
enhancement of SM penguin contributions [38]. Until now, no clear picture
emerged yet. More precise measurements of the difference of asymmetries and of
the individual asymmetries $A_{CP}(K^{-}K^{+})$ and $A_{CP}(\pi^{-}\pi^{+})$
are needed. Advances in the theoretical understanding of charm decays can also
help to disentangle this puzzle, as well as precise measurements in additional
neutral and charged decay modes. CDF, BaBar and Belle ended operation and are
finishing to analyse their samples. LHCb has shown the ability to reconstruct
charm decays with purities comparable to the experiments at $e^{+}e^{-}$
machines and has already collected the largest charm sample of the world,
corresponding to a luminosity of 1 fb-1 at $\sqrt{s}=7$ TeV and 2 fb-1 at
$\sqrt{s}=8$ TeV. Precisions comparable or below the SM predictions will only
be achievable at the LHCb upgrade [39] and the super B factories [40, 41].

## Acknowledgements

I am grateful to the organizers of PIC2012 and to my colleagues at the LHCb
charm working group for useful discussions. My participation at this
conference was supported by CNPq and FINEP.

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††thanks: Author to whom correspondence should be addressed.

# Examining the Influence of Digital Phantom Models in Virtual Imaging Trials
for Tomographic Breast Imaging

Amar Kavuri1 Mini Das1,2<EMAIL_ADDRESS>1Department of Biomedical Engineering,
University of Houston, Houston,TX-77204,USA  2Department of Physics,
University of Houston, Houston,TX-77204,USA

###### Abstract

Purpose: Digital phantoms are one of the key components of virtual imaging
trials (VITs) that aims to assess and optimize new medical imaging systems and
algorithms. However, these phantoms vary in their voxel resolution, appearance
and structural details. This study aims to examine whether and how variations
between digital phantoms influence system optimization with digital breast
tomosynthesis (DBT) as a chosen modality.

Methods: We selected widely used and open access digital breast phantoms
generated with different methods. For each phantom type, we created an
ensemble of DBT images to test acquisition strategies. Human observer
localization ROC (LROC) was used to assess observer performance studies for
each case. Noise power spectrum (NPS) was estimated to compare the phantom
structural components. Further, we computed several gaze metrics to quantify
the gaze pattern when viewing images generated from different phantom types.

Results: Our LROC results show that the arc samplings for peak performance
were approximately $2.5^{\circ}$ and $6^{\circ}$ in Bakic and XCAT breast
phantoms respectively for 3-mm lesion detection task and indicate that system
optimization outcomes from VITs can vary with phantom types and structural
frequency components. Additionally, a significant correlation (p¡0.01) between
gaze metrics and diagnostic performance suggests that gaze analysis can be
used to understand and evaluate task difficulty in VITs.

Conclusion: Our results point to the critical need to evaluate realism in
digital phantoms as well as ensuring sufficient structural variations at
spatial frequencies relevant to the signal size for an intended task. In
addition, standardizing phantom generation and validation tools might aid in
lower discrepancies among independently conducted VITs for system or
algorithmic optimizations.

VIT; VCT; DBT; Tomosynthesis; optimization; structure variability; digital
phantom; virtual phantom; LROC; gaze metrics

##  I INTRODUCTION

Rapid advances in medical imaging technologies and methods make it impossible
to evaluate and optimize several emerging and competing systems and algorithms
using clinical imaging trials. These trials can take extensive resources and
long duration. Virtual imaging trials (VITs) are alternative approach to
assess the potential of an imaging system, software or specific
system/software combinations.bakic2013mo ; badano2017silico ;
das2011comparison ; das2015examining ; barufaldi2018openvct ; sharma2019silico
; abadi2020virtual ; barufaldi2021virtual  VITs are based on in-silico
methods, where digital phantoms replace patients, software platform mimics
imaging process, and virtual interpreters represent the human readers.

As VITs are becoming more accurate, realistic digital phantom development has
drawn much attention from the medical imaging community. Our group recently
examined the contribution of anatomical and quantum noise in signal detection
and performance for human observers in such imaging trials kavuri2020relative
; kavuri2018interaction . Using VITs, we have also examined in the past, the
development of novel visual search observer models to match humans
gifford2016visual ; lau2013towards ; jiang2017analyzing , system and algorithm
optimization questions gifford2008optimizing ; das2015examining ;
das2011comparison ; das2008evaluation ; das2009evaluation ;
gifford2018assessment , understanding image texture features as relevant to
human observer performance nisbett2020correlation ; nisbett2018investigating ,
and radiomics variability andrade2022sources . As in other group’s work, all
of these studies were conducted with one breast phantom type.

Here we will evaluate what implications a VIT study outcome may have by
changing the phantom type. We once again take DBT as the modality of choice to
examine this critical question and use two widely accepted types of breast
phantoms - both considered anthropomorphic. For the last few years, many
computational breast phantoms have been developed using different
methods.glick2018advances  Some commonly used breast phantoms include power-
law based phantomssechopoulos2009optimization ; gang2010anatomical ;
lau2012statistically ; baneva2017evaluation , anthropomorphic
phantomsbakic2002mammogram ; bliznakova2010evaluation ;
chen2011anthropomorphic ; mahr2011three ; elangovan2017design ;
carton2014virtual , modified patient tomographic datahsu2013generation ;
erickson2016population ; sturgeon2016eigenbreasts ; chen2017high ;
sarno2021dataset , and mastectomy specimenschawla2009optimized ;
michael2013generation ; o2010development . These phantoms vary in their
resolution, model, structures, and details. Power spectrum analysis has been
used as a method to assess phantom structures and realism.
bliznakova2010evaluation ; chen2011anthropomorphic ; mahr2011three ;
elangovan2017design ; michael2013generation ; bakic2011development ;
cockmartin2013comparative  Cockmartin et al cockmartin2013comparative
observed in 2013 that none of the evaluated phantoms matched with the patient
data in terms of power-spectrum parameters. Few studies extended the realism
assessment with trained human’s (physicists and radiologists) rating.
bliznakova2010evaluation ; elangovan2017design ; hsu2013generation ;
sturgeon2016eigenbreasts  Badanobadano2017much , however, argued that realism
is subjective and simulating relevant properties for the task is sufficient.
But it is unclear which properties are relevant and what level of realism is
sufficient for the task of tomographic imaging system/algorithmic
optimizations.

Past work by various groups have used different phantom types for evaluating
DBT system configurationssechopoulos2009optimization ; chawla2009optimized ;
reiser2010task ; zeng2017optimization . Because of the differences in
phantoms, configurations, and interpreters among the studies, there is little
agreement on the optimal configuration. Specifically, the contribution of
phantom differences in these inconsistencies is not well understood. In
preliminary analysis, Park et al. park2010statistical  and Zhao et al
zhao2014noise  observed that phantom differences (uniform vs structured)
influence tomographic system optimal configurations. However, differences
resulting from various available structured phantoms (when they are each
considered realistic by the research community) has never been explored.

We explore this critical question and examine whether and how phantom
structure variability would influence the study outcome of a VIT such as for
DBT acquisition parameters. In addition, we also investigated the influence of
task difficulty due to change in the phantom structures using analysis of
observer gaze pattern and interpretation process when viewing images generated
with different phantoms in the VIT studies. We present results from our study
examining the influence of breast phantom types in VIT for DBT when all other
aspects remain the same. To the best of our knowledge, ours is the first study
to evaluate this critical aspect.

To accomplish this, we selected two widely used and open access breast phantom
types which were generated using different procedures. Further, human
observers analyzed simulated in-plane DBT images of the selected phantoms for
similar configurations. Our results show a comparison of predicted system
optimizations between VITs using these two phantom types.

##  II MATERIALS AND METHODS

The study methods are summarized below. First, the selected phantoms are
listed. Second, our simulation methodology is described to generate both
abnormal and normal cases of DBT images. Next, the experimental method to
estimate human observer performance and gaze pattern is discussed. Finally,
the procedure to characterize phantom structures using a power spectrum is
described.

###  II.1 Phantom Selection

We selected two types of digital breast phantoms for this study: three-
dimensional anthropomorphic breast models generated by Bakic et al. at the
university of Pennsylvania bakic2002mammogram  (hereafter referred to as Bakic
phantom) and XCAT breast phantoms generated at Duke university using
compressed volumes of patient breast CT data erickson2016population
(hereafter referred to as XCAT breast phantoms). Bakic phantoms are based on
mathematical models which define the breast structural variability. These
phantoms can be manipulated easily through the model configurations to
simulate changes in anatomy. XCAT breast phantoms are based on patient breast
CT images. These phantoms may appear more realistic, but changing their
anatomical variations or resolution is difficult abadi2020virtual . For our
analysis, six digital phantoms of 5-cm thickness for each type were selected.
The phantoms of each type were categorized based on volumetric glandular
fractions (VGFs) with 25% density and other three with 50% density. Bakic and
XCAT breast phantoms have a voxel resolution of $(0.2mm)^{3}$ and
$(0.25mm)^{3}$ respectively. The selected case numbers of XCAT phantoms of
approximately 25% density are CTA1608, CTA0357, and CTA1326 and of
approximately 50% density are CTB6013, CTA1284, and CTA1285.

Figure 1 shows sample slices of both types of phantoms on the top. The
transitions in XCAT breast phantoms from 100% glandular to adipose tissue have
intermediately dense (25%,50%, and 75%) voxels while these transitions are
sharp (100% dense to adipose) in Bakic phantoms. The XCAT breast phantoms lack
small structures such as cooper‘s ligaments. Erickson et al.
erickson2016population  noted that the ”lack of very fine-detail structures
like cooper‘s ligaments can negatively impact the realism of digital
phantoms”. References sturgeon2016eigenbreasts ; chen2017high ;
rajagopal2018evaluation  tried different methods to improve XCAT breast
phantoms, but these phantoms were not released publicly.

Figure 1: A sample slice of 25% VGF Bakic (top left) and XCAT breast (top
right) phantoms and the corresponding 1-mm DBT slices with 3-mm spherical
lesion on the bottom.

###  II.2 Image Generation

The DBT images used in this study were generated using a simulation platform
based on serial cascade modelsiewerdsen1997empirical ; vedula2003computer .
The simulation platform modeled generic DBT systems with both source and
detector rotating geometry, which was detailed in our previous work
kavuri2020relative ; vedula2003computer ; das2011penalized  and is described
briefly here. The x-ray spectrum modeled a 30-kVp molybdenum anode source with
0.7-mm thick Al filter and the x-ray fluence scaled to provide a 1.5 mGy mean
glandular dose (MGD) to a 5-cm-thick compressed breast. This total dose was
evenly distributed at each projection angle and the x-ray fluence per
projection was determined based on the breast dosimetry data (Dgn
coefficients) generated by a Monte Carlo simulatorboone1999glandular . Focal-
spot blurring with a 0.1-mm focal-spot size was modeled using a Gaussian
modulation transfer function. The detector was modeled as a 0.1-mm thick CsI
based a:Si flat panel detector with a 0.1-mm pixel size. The scintillator
blurring was modeled using an empirically measured pre-sampling MTF. Quantum
noise was modeled by a Poisson distribution for each keV (at the absorption of
x-rays with in the scintillator), while additive electronic noise followed a
Gaussian process with a standard deviation of 2200 electrons. Scatter was not
modeled in this simulation.

The lesion targets were homogeneous spherical masses with a 3-mm diameter.
While this is smaller than the average lesion sizes detected by the current
DBT systems, VITs are also aimed at evaluating future imaging system designs.
This signal size of 3-mm was also chosen to achieve sufficient task-challenge
without requiring to artificially alter attenuation values of the signal and
to the capture the human observer performance trends more accurately for a
range of breast densities. An abnormal case was generated by substituting the
lesion into the randomly selected location in the glandular region prior to
the projection imaging. The lesion contrast or local glandularity were not
matched between phantom types while selecting lesion locations as absolute
performance was not relevant for this study. The simulated mass was assumed to
have an energy-dependent attenuation coefficient for invasive ductal carcinoma
as reported by Johns and Yaffe.johns1987x  Eight abnormal cases and one normal
case were formed for each phantom. The projections were acquired using
Siddon’s ray tracing method siddon1985fast  to model x-ray transmission
through the breast. Two different sets of projections were generated with
different phantom types. In order to filter the random noise, an adaptive
Wiener filter based denoising algorithm was applied on each
projection.lim1990two ; vieira2013effect  The denoising algorithm was shown to
reduce the noise effectively in our prior work kavuri2020relative . Each data
set was acquired over an angular span of $60^{\circ}$ with projection number
$P\in${3, 7, 11, 15, 19, 21, 25, 31, 35, 41, 45} by keeping the total dose
steady at 1.5 mGy for each DBT acquisition. Feldkamp filtered back-projection
(FBP) algorithmfeldkamp1984practical  was used for image reconstruction. A
three-dimensional Butterworth filter with a cutoff of frequency 0.25
cycles/pixel was applied on reconstructed volumes. In-plane DBT images of 1-mm
thickness were produced by applying a boxcar averaging. Eight lesion-present
(abnormal) and eight lesion-absent (normal) images were created for each
phantom for the human observer studies as described in our earlier
workdas2015examining . A set of 96 images were produced from the six phantoms
of each phantom type for a given projection number. Figure 1 shows sample 1-mm
abnormal DBT slices of both types of phantoms on the bottom.

###  II.3 Human Observer Study

Three non-radiologists took part in the Localization ROC (LROC) experiments.
An LROC study entails both detection and localization of a 3-mm spherical
lesion. The task in our study is search and localization/detection of
spherical mass in simulated in-plane DBT images thus justifying the non-
radiologists as observers. Physicists and engineers who participated as
observers had the same level of experience in reading simulated images. The
human observer experimental method was same as that described in our prior
work das2015examining ; das2011comparison ; lau2013towards ; gifford2016visual
and summarized briefly here. In this study, in-plane DBT slices of the Bakic
phantom set correspond to five acquisition protocols with projections of 7,
11, 19, 25 and 35 and XCAT breast phantom set correspond to six acquisition
protocols with projections of 3, 7, 11, 19, 25 and 35 were evaluated. The 96
images per set (48 pairs of abnormal/normal images) were divided into 72 test
images (36 pairs) and 24 training images (12 pairs). All sets included an
initial training session followed by a test session. Each observer thus read
10 image sets. Observers were asked to select the lesion location, and a four-
point ordinal scale was used to collect the confidence rating. Localizations
were considered as correct when the observer selected the location within a
2-mm radius of lesion center (radius of spherical lesion + 0.5mm additional
radius for human selection error). The observers performance was quantified
with area under LROC curves (AUC). The estimate of AUC for a given observer
and protocol was obtained with a Wilcoxon-based non-parametric ranking method.
In order to assess the consistency of the AUC values between the observers,
the intraclass correlation coefficient (ICC) was calculated using Python
(v3.7.6) software (www.python.org).

###  II.4 Eye Gaze Collection

Numerous studies have demonstrated that gaze metrics correlate with diagnostic
performance and can reveal human observers interpretation
processbrunye2019review ; kundel1978visual ; kundel2008using ;
voisin2013investigating . In order to evaluate the differences in the
interpretation process due to the change of phantom type, we collected gaze
data using a screen based eye-tracking system (Tobii pro X3-120 with EPU ). An
additional three more non-radiologists participated for the eye tracking
study. The eye gaze data was collected for a single acquisition protocol of 35
projections for both type of phantoms. Both presentation of stimuli and eye-
tracking were controlled by a Lenovo thinkpad p52s laptop using an in-house
built Python software. A calibration and validation procedures were performed
before starting a session. The images were displayed on a standard Dell 23.8
inch LCD monitor with a resolution of 1920 $\times$ 1080\. Once the software
collects and stores the gaze logs, post processing was applied to estimate
fixation locations, fixation durations, and saccade durations using an I-VT
filter based on Tobii pro white paperolsen2012tobii . Figure 2 shows a sample
gaze pattern of an observer for the task of searching and locating a 3-mm
lesion. Each vertex indicates the fixation center with numbers indicating
fixation order and lines indicating the saccade paths and lengths.

Figure 2: Example gaze pattern with fixations and saccades on DBT slices of
25% dense (top) and 50% dense (bottom) phantoms.

We computed five gaze metrics to characterize the gaze pattern. Namely, we
found total time spent on each image, total number of fixations made on each
image, time taken to first fixate on the lesion region (first hit time),
number of fixations on lesion region, and accumulated lesion dwell time. These
gaze metrics were estimated using MATLAB2018a (The MathWorks, Inc.) with in-
house built scripts. Spearman rank correlation was estimated between the
average values of each gaze metric and the corresponding AUC values of the six
observers.

###  II.5 Noise Power Spectrum (NPS)

Noise power spectrum (NPS) based analysis has been used in literature to
quantify the similarity between phantom structures and clinical
data.cockmartin2013comparative ; chen2011anthropomorphic . Our NPS estimation
methods are briefly summarized here and can be found in detail
elsewherekavuri2020relative . We selected multiple regions of interest (ROIs)
of size 2.43cm$\times$2.43cm from each lesion-absent DBT slice from the breast
region. The mean of each ROI was subtracted from the corresponding ROI and a
Hann tapering window was applied to each ROI to reduce the edge artifacts. The
2D NPS was calculated by ensemble averaging the square of the magnitude of the
discrete Fourier transform of each tapered ROI , and radial averaging of the
2D NPS was performed, resulting in a 1D NPS. A linear regression fit (from
lower frequency ranges between 0.1 and 0.15 $mm^{-1}$ to higher frequency
ranges between 0.4 and 0.7 $mm^{-1}$) was estimated to the natural logarithm
of the 1D NPS that maximizes the coefficient of determination ($R^{2}$). The
NPS parameter $\beta$ is estimated as the slope of linear fit.

##  III RESULTS

###  III.1 Human Observer LROC Results

In our studies, each human observer was tasked with search and localization of
a 3-mm lesion within the displayed DBT image slice (similar to the ones shown
in fig 1). Figure 3 shows sample regions of DBT slices with lesion at the
center of region to illustrate the changes in the visibility of the signal for
varying number of projections in both types of phantom backgrounds. The signal
is better visible in Bakic phantom background for the acquisition
configuration of 35 projections, whereas the signal is better visible in XCAT
breast background for the acquisitions of 11 to 35 projections. Figure 4
presents LROC plots for the three observers, for a sample acquisition of 35
projections over $60^{\circ}$ arc span, in both phantom backgrounds. The
y-axis of the LROC represents the joint probability of correctly localizing a
lesion in a case reported as positive. Therefore, the curve reaches up to the
percentage of cases with correct lesion localization. The LROC AUC values
above zero are considered as better than guessing as the likelihood of
guessing the lesion’s location is zero. Figure 5 shows the average performance
of the three observers in both Bakic and XCAT breast phantom images. Error bar
lengths indicate twice the standard error of the three observers’ AUC values.
We observed a greater improvement in performance up to 11 projections in XCAT
breast phantom backgrounds and a steady performance thereafter. This
corresponds to an arc sampling of approximately $6^{\circ}$ between adjacent
projections for peak performance. In Bakic phantom backgrounds, observers’
performance improved up to 25 projections and required finer arc sampling of
approximately $2.5^{\circ}$ to achieve peak performance. We also plotted the
detection performance separately for 25% dense and 50% dense slices, where 25%
dense indicates easy level of task difficulty while 50% dense indicates higher
level of task difficulty. Regardless of phantom type, both levels of task
difficulty show similar trends and suggest that optimization may not change
with the task difficulty, which is in accordance with earlier observations by
Zeng et al. zeng2017optimization  and Mackenzie et al.mackenzie2021effect . We
also noticed in figures 4 and 5 that observers had overall slightly lower
performance in XCAT breast backgrounds than in Bakic backgrounds in particular
for 25% dense images. This difference can not be attributed to phantom
structures necessarily as the local densities in the region where lesions are
inserted could also influence these LROC AUC. The magnitude of AUC values are
less relevant in this particular study as only relative change in AUC values
with changing system parameters (such a number of projection in this study)
were used to deduce the final conclusions. The inter-observer agreement,
quantified using the ICC, ranged from 0.92 to 0.95 with the average AUC
values, suggesting a strong agreement between the observers.

Figure 3: A sample lesion present regions of DBT slices of both phantoms
acquired with different number of projections.  Figure 4: Human observer
performance plotted as localization ROC (LROC) curves for a sample acquisition
of 35 projections over $60^{\circ}$ in 3-mm mass detection study in Bakic
phantom (top) and XCAT breast phantom (bottom) backgrounds.  Figure 5: Human
observer performance plotted as area under LROC (AUC) against number of
projections in 3-mm mass detection study in Bakic phantom (top) and XCAT
breast phantom (bottom) backgrounds. The results suggest the optimal
configurations does not change with the task difficulty but changes with the
background structure type.

###  III.2 Power Spectrum Analysis

In order to characterize the structural variations of the selected phantom
types, we estimated the NPS of the simulated DBT images. We selected the NPS
of in-plane DBT images corresponding to the acquisition configurations of 35
projections over a $60^{\circ}$ arc span for comparing the phantoms’
structures. Figure 6 shows the averaged NPS of both the phantom types in log-
log scale along with linear fits. These plots show that Bakic phantom
backgrounds have higher spectral densities at higher frequencies (considered
as the anatomical region) than that of XCAT breast phantom backgrounds. We
note that this high anatomic noise is not reflected in the calculated $\beta$
values. The estimated values of $\beta$ were 2.52 and 3.32 respectively for
Bakic and XCAT breast phantom backgrounds. This result is contradictory to
popular belief that lower $\beta$ values indicate lower anatomical noise and
in agreement with observations made in our prior workkavuri2020relative .

Figure 6: The power spectra analysis of DBT slices of both type of backgrounds
for a sample acquisition protocol of $60^{\circ}$ arc span, 35 projections
suggest that lack of small and sharp structures in XCAT breast phantoms
resulted in lower spectral density at higher frequencies than that of Bakic
phantoms.

###  III.3 Gaze Analysis

Figure 7: The average amount of time spent and the average number of fixations
made on images (includes lesion absent and lesion present images) plotted for
both phantom types. Observers spent longer time and made more fixations to
make decisions on images with Bakic phantom backgrounds in comparison to those
with XCAT breast backgrounds.

Figure 8: Average value of first hit time, lesion dwell time and number of
fixations on lesion were plotted against AUC value of each observer (total 6)
for both phantom types and two densities.

From the eye-tracking data, the total time spent as well as the total number
of fixations made on images (both lesion present and lesion absent images)
were estimated. For lesion present images, we also estimated the first hit
time (on the lesion), lesion dwell time and the number of fixations on lesion.
Each of these gaze metric was averaged across all images for a given phantom
type and breast density for each observer. Figure 7 shows the influence of
phantom type on two of the gaze metrics plotted separately for lesion absent
and lesion present images. Observers spent longer time and made more fixations
to diagnose lesion absent images than lesion present images, which is in-line
with previous findings timberg2013investigation ; suwa2001analyzing . The most
striking result to emerge from this data is that observers spent longer time
and made more fixations on images with the Bakic phantom backgrounds than XCAT
breast backgrounds. This difference is significant ($p-value<0.01$) in lesion
absent images only. A possible explanation for this difference may be that the
greater anatomical noise of Bakic phantom backgrounds than the XCAT breast
backgrounds (see Fig.6) made observers less confident and hence spent longer
time to make decision.

Spearman rank correlation coefficient was computed between the average values
of each gaze metric and the AUC values of both phantom types and densities.
Table 1 shows that all of the estimated gaze metrics have good correlation
with diagnostic performance. Our observation of first hit time, lesion dwell
time, and number of fixations on lesion showed that the observers took longer
to first fixate on the lesion, spent less cumulative time on lesion, and had
fewer fixations on the lesion as the task difficulty increased due to higher
breast density. No significant difference was observed in these gaze metrics
due to change of phantom type in lesion present images. This result indicates
that lesion visibility or task difficulty is a major factor in defining the
gaze pattern in lesion present images where as anatomical complexity is the
major factor influencing the gaze pattern in lesion absent images. First hit
time showed a strong negative (-0.85) correlation with the AUC values as shown
in figure 8. This result suggest that the quicker an observer fixate on the
lesion the better the diagnostic performance which is in accordance with the
observation made by Kundel et al kundel2008using . The positive correlation of
lesion fixations and dwell time with AUC suggests that observers fixate the
lesion longer and multiple times to locate accurately. All the five gaze
metrics showed significant correlation ($p-value<0.01$) with AUC values.

Table 1: Spearman rank correlation between the AUC values of the six observers and the corresponding average values of the five gaze metrics. All five metrics show good and significant correlation with the diagnostic performance. Gaze metric | Correlation coefficient ($\rho$) | p-value
---|---|---
Total Number of Fixations | -0.52 | 0.0085
Total time | -0.6 | 0.0019
Lesion Dwell Time | 0.67 | 0.0003
Lesion Number of fixations | 0.68 | 0.0003
First Hit Time | -0.85 | 1.2e-7

##  IV DISCUSSION AND CONCLUSION

This work evaluated the influence of variations between digital breast
phantoms on DBT optimization and interpretation process for a small lesion
localization task. Our results indicate that the phantoms should have adequate
structures at spatial frequencies that are relevant for the signal size and
the intended task for sufficient realism. We observed that optimal number of
projections for peak detection performance could change with the structural
complexity of the phantoms. In addition, we observed that the number of
projections required to achieve maximum performance is smaller for XCAT breast
phantoms than for Bakic phantoms. Our power spectrum analysis revealed that
the complex structures in Bakic phantoms contribute to high frequencies while
the high-frequency content of XCAT breast phantoms showed lower amplitudes.
These high-frequency structures resulted in more aliasing artifacts in Bakic
phantom backgrounds compared to XCAT breast phantom backgrounds under similar
sparse sampling conditions. Hence, more projections were required to resolve
these high-frequency structures in Bakic phantom backgrounds.

Our gaze analysis suggest that all the five gaze metrics showed good
correlation with diagnostic performance, which is in accordance with the
observations made by Voisin at al voisin2013investigating  in mammograpy. This
result indicates that gaze metrics and interpretation process were influenced
by the task difficulty. Hence, these gaze metrics can be used to understand
the task difficulty in VIT imaging.

In this study, we evaluated the optimal number of projections in an arc span
of $60^{\circ}$, which is wider than the arc ranges used in the clinical DBT
systems. The arc samplings between the adjacent projections for peak
performance were approximately $2.5^{\circ}$ and $6^{\circ}$ in Bakic and XCAT
breast phantoms respectively for 3-mm lesion detection task. For other arc
spans, we expect similar amount of aliasing artifacts for the similar arc
sampling. Thus our conclusions should hold for other arc spans used in
clinical DBT prototypes which have not been examined here in our VIT study.

The difference in the optimal arc sampling from VIT conducted with two widely
used phantom types, raises the need to standardize and unify the frequency
contents and complexity of digital phantoms if results published from multiple
groups need to be compared. The goal of our study was not to determine if one
of the two phantoms are favorable or better than the other for use in VIT for
breast. It is likely that the perceived realism and correlation with observer
performance for VIT with actual clinical imaging trials would show results in
favor of one of these phantoms based on the chosen task and the signal type.

One of the limitations of our study is that only 3 observers participated in
estimating AUC trends. This number is smaller than the number of observers
participated in DBT optimization studies in literaturezeng2017optimization ;
hadjipanteli2019threshold ; goodsitt2014digital ; vancoillie2021impact .
However, the observers had higher agreement with each other which is backed by
our estimation of intraclass correlation coefficient value of greater than
0.92. Secondly, we also estimated correlation between the average AUC trends
of Bakic and XCAT breast phantoms. The spearman rank correlation of 0.2
suggests that the two trends are not similar.

Furthermore, the study was conducted only for the task of searching and
locating a 3-mm spherical lesion as conducting human observer studies for
different signal sizes and types is time consuming and expensive. Prior
studies suggest that smaller (high frequency) lesions dictate the optimal
number of projections because their visibility is affected more by both
aliasing artifacts (at lower number of projections) and random noise (mainly
at higher number of projections) reiser2010task ; gang2010anatomical . Many
studies also used 3-mm sechopoulos2009optimization ; gang2010anatomical ;
chawla2009optimized  and similar sizes zeng2017optimization ;
goodsitt2014digital ; vancoillie2021impact ; ikejimba2021assessment  of
lesions as a signal target in optimization studies. We believe that for larger
signals the differences in optimal arc sampling due to differences in the
selected phantom structures may be less evident than that of smaller signals.
We chose a sphere lesion as target because detecting a sphere lesion is
relatively easy task compared to complex spicule lesion study, hence requiring
minimal training. In addition, the performance of non-radiologists and
radiologists was shown to be similar for relatively easy tasks
bertram2013effect  and multiple studies have used spherical lesions in virtual
imaging trialssechopoulos2009optimization ; gang2010anatomical ;
chawla2009optimized ; reiser2010task ; zeng2017optimization . On the other
hand, for high contrast smaller signals such as micro-calcification, random
noise was shown to be the dominating factor in determining the optimal
configuration rather than the background structures.reiser2010task  Future
studies will include signals of different sizes and types.

Another limitation of our study is that only six phantoms (to generate a much
larger number of independent cases in each study set) were selected for each
phantom type, this sample may not represent the entire population of each
phantom type. Although, our goal was not to compare between Bakic and XCAT
phantoms, but to validate how optimization estimation changes with phantoms
that were generated differently. One may chose 6 phantoms in their studies in
evaluating optimal configurations as some of the studies in literature chose
around 2 to 9 phantoms gifford2016visual ; das2008evaluation ;
hadjipanteli2019threshold ; goodsitt2014digital ; vancoillie2021impact . If
the selected new phantoms have similar NPS properties, we anticipate the
similar differences in the optimal number of projections. Instead, if the
selected phantoms have different NPS properties and results in different
optimal number of projections, which strengthens our argument that phantom
structures influence the estimation of system optimal configurations. The
selected phantoms (25% and 50% dense) are in the higher range of breast
densities observed in the clinical data. The average AUC trends (see figure 5)
did not change due to change of breast density, suggesting that AUC trends may
remain same for other density phantoms. These results also point to the need
to include more phantoms with variations in VIT studies.

In conclusion, our results indicate that both the structural complexity and
the relevant spatial frequency magnitudes in digital phantom structures can
influence the estimation of optimal system configurations. Our results
highlight the importance of accurate modeling of phantoms to resemble the
patient anatomy and the importance of assessment for their realism (for chosen
tasks) before use in VITs. This results can be generalized to digital phantoms
used for multiple imaging modalities. As a final note, our goal in this
particular study was not to discuss superiority of one phantom type over the
other based on results shown here. The key aspect is for the VIT and medical
physics community to be mindful that optimization or other results shown using
one ”realistic” breast phantom may not always agree with results when using
another ”realistic” phantom unless additional standardization efforts are
pursued.

##  V ACKNOWLEDGEMENTS

This work was partially supported by funding from the NIH National Institute
of Biomedical Imaging and Bioengineering (NIBIB) grant R01 EB EB029761, the US
Department of Defense (DOD) Congressionally Directed Medical Research Program
(CDMRP) Breakthrough Award BC151607 and the National Science Foundation CAREER
Award 1652892. We would also like to thank both UPenn and Duke groups for
making the digital phantoms freely available for research.

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