# Tailored Ion Beam for Precise Color Center Creation

A. Tobalina  Department of Physical Chemistry, University of the Basque
Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  C. Munuera-Javaloy
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, 48080 Bilbao, Spain  E. Torrontegui  Departamento de Física,
Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés,
Spain  Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006
Madrid, Spain  J. G. Muga  Department of Physical Chemistry, University of the
Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  J. Casanova
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, 48080 Bilbao, Spain  IKERBASQUE, Basque Foundation for Science,
Plaza Euskadi 5, 48009 Bilbao, Spain

###### Abstract

We present a unitary quantum control scheme that produces a highly
monochromatic ion beam from a Paul trap. Our protocol is implementable by
supplying the segmented electrodes with voltages of the order of Volts, which
mitigates the impact of fluctuating voltages in previous designs and leads to
a low-dispersion beam of ions. Moreover, our proposal does not rely on
sympathetically cooling the ions, which bypasses the need of loading different
species in the trap –namely, the propelled ion and, e.g., a 40Ca+ atom able to
exert sympathetic cooling– incrementing the repetition rate of the launching
procedure. Our scheme is based on an invariant operator linear in position and
momentum, which enables us to control the average extraction energy and the
outgoing momentum spread. In addition, we propose a sequential operation to
tailor the transversal properties of the beam before the ejection to minimize
the impact spot and to increase the lateral resolution of the implantation.

_Introduction.–_ The creation of structural defects on crystalline structures
is a relevant task on the road towards building solid-state quantum devices
for groundbreaking applications such as nanoscale nuclear magnetic resonance
(NMR) Wu16 ; Degen17 . For instance, nitrogen-vacancy (NV) color centres in
diamond Doherty13  have attracted considerable attention in the last years for
their potential to serve as a quantum platform for room temperature
applications Lovchinsky16 . A possible fabrication technique of such
structures consists on blasting a highly-pure diamond piece with nitrogen
atoms such that, as they penetrate the diamond, they kick a carbon atom out of
its location leading to a useful NV defect that behaves as a nanoscale
magnetic compass Smith19 . Creating shallow defects Romach15  is specially
convenient as it eases their control and enables relevant applications, such
as using the NV as a nanoscale NMR probe of molecular samples in the diamond
surface Shi15 ; Aslam17 ; Kehayias17 ; Glenn18 ; Bucher20 ; Arunkumar21 . In
this scenario, the depth at which the defect is created is directly related
with the energy of the impacting atom, thus, achieving a flying atom with a
narrow momentum distribution around a predetermined value would enable the
creation of defects near the surface of the crystalline host.

Many potential uses of those defects require high resolution placement, which
can be achieved using masks Schukraft16  or pierced atomic force microscopes
Pezzagna10 ; Riedrich-Moller15 . Creation of single defect centres with
nanometer resolution has also been demonstrated using focused ion beams
Schroder17 . A possible approach to produce an ion beam for high-precision
color centre creation is to extract the ions from a linear Paul trap and
direct them to the host though an electromagnetic lens Schnitzler09 . In these
experiments the spread of the beam is significantly reduced by cooling down
the ions. Unfortunately, nitrogen –along with many other interesting dopant
species such as Germanium Bhaskar17 ; Siyushev17 – lacks the internal
structure that allows for laser cooling Jayich16 . Thus, dopants are typically
cooled down through their interaction with different ion species that present
a closed optical transition. In this manner, besides the dopant ion, this
sympathetic cooling procedure Kielpinski00 ; Wubbena12  implies the ionization
and loading of another ion such as 40Ca+, necessarily decreasing the
repetition rate of the implantation procedure. Moreover, the necessity of
reaching an ion crystal for sympathetic cooling introduces additional issues
that may add to the duration of the operation. For example, if more than two
N${}_{2}^{\,\,+}$ ions are trapped together with a 40Ca+ the crystal will
melt, in which case the operation has to be started from scratch JacobThesis16
. Even when a two-ion crystal is formed, the heavier cooling ion that
accelerates slower than the lighter dopant ion, may be located ahead in the
shooting line. Consequently, an additional voltage sequence has to be applied
to reorientate the crystal Groot-Berning21 .

Furthermore, once the dopant ion is loaded into the trap and cooled down below
the Doppler limit, it is subjected to an electric field that propels it
through the pierced endcap of the trap towards the crystalline target. In the
latest reported experiments Jacob16 ; Groot-Berning19 ; Groot-Berning21 , the
driving field is generated by applying a constant voltage of the order of kV
to the endcap electrodes. Such large voltages generate a significant
uncertainty in the extraction energy of the ion and lower the control over the
depth of the newborn defect.

In this Letter we propose a unitary quantum control scheme with a significant
twofold benefit. On the one hand, our protocol decreases the ion-extraction-
energy-uncertainty as it uses small voltages bearable by the segmented
electrodes of a Paul trap. This leads to an ion beam with precise ejection
velocity. On the other hand, our method requires no previous cooling of the
dopant –thus, no cooling ions are needed– leading to a significant increase on
the ejection repetition rate. Our protocol relies on the existence of an
invariant operator linear in momentum and position Guasti03 ; Urzua19 , which
is exploited here as a control tool to engineer the properties of the outgoing
beam.

_Driving in the shooting direction.–_ The axial motion of the trapped nitrogen
ion is driven by a movable harmonic potential with controllable frequency.
Therefore its dynamics along the coordinate $z$ is governed by

$H_{z}(t)=\frac{p_{z}^{2}}{2m}+\frac{1}{2}m\omega_{z}(t)^{2}\left[z-z_{0}(t)\right]^{2},$
(1)

where $m$ is the mass of the ion and the center of the trap $z_{0}(t)$ and its
frequency $\omega_{z}(t)$ constitute the control parameters of the operation
(the use of the subscript $z$ is motivated by a later treatment of the radial
motion of the ion). We find that such evolution keeps constant the expectation
value of the linear operator

$I(t)=u(t)p_{z}-m\dot{u}(t)z+f(t),$ (2)

provided that the auxiliary functions $u(t)$ and $f(t)$ and the control
parameters $\omega_{z}(t)$ and $z_{0}(t)$ obey the following equations,

$\displaystyle\ddot{u}(t)+\omega_{z}(t)^{2}u(t)$ $\displaystyle=$
$\displaystyle 0,$ (3)
$\displaystyle\dot{f}(t)+m\omega_{z}(t)^{2}u(t)z_{0}(t)$ $\displaystyle=$
$\displaystyle 0.$ (4)

The novel invariant presented here generalizes the operators used in previous
works Muga20 ; Tobalina20  to introduce the displacement of the trap in the
engineering of the driving. The dynamical state of the ion can be conveniently
written in terms of the eigenvectors of the invariant $I(t)$, while the
corresponding evolution of $\langle z\rangle$ and $\langle p_{z}\rangle$ (also
of $\Delta z$ and $\Delta p_{z}$) can be computed in terms of their initial
values $\langle z\rangle_{0}$ and $\langle p_{z}\rangle_{0}$ (the explicit
expressions are in the Supplemental Material SM ). In fact, since our proposal
does not rely on sympathetically cooling the ion near the absolute zero, we
consider that the ion starts the operation at a thermal state characterised by
$\langle z\rangle_{0}=0$, $\langle p_{z}\rangle_{0}=0$, $\langle
zp_{z}+p_{z}z\rangle_{0}=0$, as well as

$\displaystyle\langle
z^{2}\rangle_{0}=\frac{\hbar}{2m\omega_{z,0}}\coth{\left(\frac{\hbar\omega_{z,0}}{2k_{B}T_{0}}\right)},$
$\displaystyle\langle
p_{z}^{2}\rangle_{0}=\frac{m\hbar\omega_{z,0}}{2}\coth{\left(\frac{\hbar\omega_{z,0}}{2k_{B}T_{0}}\right)},$
(5)

where $k_{B}$ is the Boltzmann constant and $T_{0}$ the initial temperature.

The invariance of $I(t)$ allows to establish useful relations between the
expectation values of position and momentum operators at different times of
the evolution. In particular, if $\dot{u}_{0,f}=0$ (we use subscripts $f$ and
$0$ for final and initial times), the motion of the ion yields

$\displaystyle\langle p_{z}\rangle_{f}$ $\displaystyle=$ $\displaystyle
R\langle p_{z}\rangle_{0}+P,$ (6) $\displaystyle\Delta p_{z,f}^{\,\,2}$
$\displaystyle=$ $\displaystyle R\Delta p_{z,0}^{\,\,2}-2RP\langle
p_{z,0}\rangle,$ (7)

where the coefficients $R=u_{0}/u_{f}$ and $P=(f_{0}-f_{f})/u_{f}$ are
tunable. Hence, by suitably fixing the boundary values of the auxiliary
functions we can determine the average launching momentum and its outgoing
spread, leading to the generation of a tailored ion beam.

Along with $\dot{u}_{0,f}=0$, a possible choice of boundary conditions (BC)
leading to Eqs. (6) and (7) is

$\displaystyle u_{0}=1,\hskip 17.07182ptu_{f}=\frac{1}{R},\hskip
17.07182ptf_{0}=0,\hskip 17.07182ptf_{f}=-\frac{P}{R}.$ (8)

Further BC may determine other properties of the system. For example, using
Eq. (3), we select the initial and final width of the trapping potential by
imposing

$\ddot{u}_{0}=-u_{0}\omega_{0}^{\,\,2},\hskip
22.76228pt\ddot{u}_{f}=-u_{f}\omega_{f}^{\,\,2},\hskip
22.76228pt\dddot{u}_{0,f}=0.$ (9)

Likewise, from Eq. (4) the location $z_{0}(t_{f})=d$ and the velocity
$\dot{z}_{0}(t_{f})=v$ of the minimum of the potential at the end of the
operation are specified through

$\dot{f}_{0}=\ddot{f}_{0}=0,\hskip
9.95863pt\dot{f}_{f}=-mu_{f}\omega_{f}^{\,\,2}d,\hskip
9.95863pt\ddot{f}_{f}=-mu_{f}\omega_{f}^{\,\,2}v.$ (10)

Once $u(t)$ and $f(t)$ have been suitably designed, the control parameters
leading to the tailored ion beam are found from Eqs. (3) and (4). This inverse
approach based on invariants is widely use in the context of shortcuts to
adiabaticity Guery19 .

Here we use polynomial auxiliary functions $u(t)$ and $f(t)$ (explicit
expressions in SM ) but note that any pair of auxiliary functions that
satisfies the BC in Eqs. (8), (9) and (10) yields the results predicted by
Eqs. (6) and (7) and would lead to the target ion beam. Thus, our method
provides infinite degrees of freedom to address additional goals or
constraints. Here, we exploit this flexibility to reduce the maximum potential
energy gained by the ion during its manipulation. In fact, the evolution of
the ion is accurately described by $H(t)$ provided that it never exceeds the
depth of the trap. Note that the harmonic potential in Eq. (1) is a local
approximation of the total potential generated within the Paul trap. Thus,
reducing the energy peak avoids stages where the potential felt by the ion is
not harmonic, thus ensuring the viability of the method. A detailed
explanation on the design process can be found in the Supplemental Material SM
.

Avoiding transient departures from the harmonic regime poses a bound on the
capacity of the method. Nevertheless, Fig. 1 (a) demonstrates that our
controls can be optimized to reduce transient energy peaks. In particular, in
Fig. 1 (a) we display some example energy peaks and show that, for trap depths
achievable with current technology (around few eV Leibfried03 ; Schnitzler10
), we obtain controls that boost ions starting with temperatures as high as
thousands of Kelvins to velocities of the order of km/s. Moreover, the results
shown in Fig. 1 (a) admit further optimization tools that may lead to a beam
with greater extraction energy for the same potential depth.

Figure 1:  The maximum value of the average potential energy
$E_{p}=\frac{1}{2}m\omega_{z}(t)^{2}\left(z-z_{0}(t)\right)^{2}$ for different
initial temperatures $T_{0}$ during a protocol that boosts the
N${}_{2}^{\,\,+}$ ion to launching velocities of $1$ km/s (solid blue), $3$
km/s (dashed green) and $5$ km/s (dotted brown) while reducing the width of
the momentum spread by a factor of 2. (b) Average momentum and scaling
parameter (inset) resulting from different relative uncertainties in the
control voltages for an initial ion at $T_{0}=1000K$ and a target velocity of
5 km/s. All the operations are carried out in $t_{f}=0.94$ $\mu$s and the trap
begins and ends with $\omega_{0}=\omega_{f}=(2\pi)\times 0.85$ MHz.

In previous reports on the creation of deterministic ion beams from Paul
traps, imprecise voltages lead to fluctuating extraction energies that exceed
by far the intrinsic uncertainty of the quantum system Jacob16 . Our proposal
admits an arbitrarily small scaling parameter $R$ –see Eqs. (6, 7)– so,
formally, the momentum of the ejected ion may be as well defined as desired.
However, uncertainties in the electrode voltages could spoil the outcome of
the method presented here, so, next we evaluate the impact of imprecise
electrodes in the outgoing beam. To this end, we consider a model Furst14 ;
Tobalina18  where the ion is driven by two adjacent electrodes that generate a
potential $V(z,t)=e\left[\phi_{1}(z)U_{1}(t)+\phi_{2}(z)U_{2}(t)\right]$ with
time-dependent voltages

$U_{1,2}(t)=\frac{-\phi^{\prime}_{2,1}[z_{0}(t)]m\omega(t)^{2}}{(\phi^{\prime\prime}_{2}[z_{0}(t)]\phi^{\prime}_{1}[z_{0}(t)]-\phi^{\prime}_{2}[z_{0}(t)]\phi^{\prime\prime}_{1}[z_{0}(t)])e},$
(11)

where $\phi_{1,2}(z)$ are dimensionless functions determined by the geometry
of the electrodes and $e$ the fundamental electric charge.

Figure 2:  a) Voltages to eject a single N${}_{2}^{\,\,+}$ ion, $U_{1}(t)$ in
solid blue and $U_{2}(t)$ in dashed brown, with average velocity of $5$ km/s
and $R=1/5$, see Eq. (7). The operation takes $0.94$ $\mu$s. The generated
potential at boundary times has $\omega_{0}=\omega_{f}=(2\pi)\times 0.85$ MHz,
ending at $d=250$ $\mu$m and with $v=10$ km/s. The geometric functions are
modeled as Gaussians,
$\phi_{1,2}(z)=0.2\exp{\left[-(z-\beta_{1,2})^{2}/(2\sigma^{2})\right]}$ with
$\beta_{1}=0$, $\beta_{2}=250$ $\mu$m and $\sigma^{2}=200$ $\mu$m. b) Control
functions produced by the ideal voltages, solid black, and by voltages with
relative uncertainty of $\Delta U/U=10^{-5}$, dashed red, and $\Delta
U/U=10^{-1}$, dotted green. c) Initial and final Wigner functions for an ion
starting with $T=1000$ K and driven by electrodes with a relative voltage
uncertainty of $\Delta U/U=10^{-3}$. The reference length is
$l=\sqrt{\hbar/(m\omega_{z,0})}$ and the reference momentum $p_{ref}=\hbar/l$.

Figure 1 (b) displays the average extraction momentum and scaling parameter
for simulations with different voltage uncertainties relative to the ideal
scheme in Fig. 2 (a), see Supplemental Material SM  for a detailed explanation
of the simulation. Figure 2 (b) shows two examples of the control functions
produced by inaccurate electrodes, together with the ideal driving, and Fig 2
(c) displays the result of driving the ion with fluctuating electrodes through
its Wigner function before (first panel) and after the manipulation (second
panel). From Figure 1 (b) one can learn that a relative uncertainty below
$\Delta U/U=10^{-3}$ (we consider the same fluctuation in both electrodes, so
$\Delta U/U\equiv\Delta U_{1,2}/U_{1,2}$) hardly alters the intended results.
Note that in Schnitzler09  they reported a value of $\Delta U/U=10^{-5}$,
while bigger errors would cause notable changes in the extraction velocity and
the outgoing momentum spread.

Another relevant feature of our protocol is its low voltage requirement. In
particular, supplying the two electrodes with few Volts suffices to boost the
ion to velocities of the order of km/s. Notice also that the launching
protocols are carried out in few trap oscillation periods, i.e., in a time
scale of the order of $\mu$s. Furthermore, and compared with procedures that
require to cool down the ion to near the absolute cero, our method provides a
huge improvement in the repetition rate. More specifically, since our scheme
does not need to trap and cool down a second/cooling ion, the bottleneck of
the total operation is the loading of the molecular nitrogen which typically
takes around 200 ms JacobThesis16 . On the contrary, in standard methods, ion-
crystal formation and arrangement issues limit the loading rate to 1
N${}_{2}^{\,\,+}$ ion per minute when a 40Ca+ is already present in the trap
JacobThesis16 .

_Transversal spread.–_ A deterministic creation technique of single defects in
diamond controls not only the depth, but also the transversal location of the
created defect. Here we tailor the potential acting on the radial directions
to reduce the impact spot and thus increase the resolution of the
implantation. We propose a sequential approach in which the transversal
properties of the beam are addressed first, by suitably driving the radial
motion of the ion, and then the launching operation designed in the previous
section is carried out. The evolution of the ion in one of the radial degrees
of freedom (both radial directions are symmetrically equivalent) is governed
by

$H_{r}(t)=\frac{p_{r}^{2}}{2m}+\frac{1}{2}m\omega_{r}(t)r^{2}.$ (12)

In contrast to the longitudinal motion, the radial harmonic oscillators have a
static center. In this case the design of the angular frequency $\omega(t)$
boils down to the case introduced in Muga20  where the control protocol stems
from a single auxiliary function, Eq. (3), as Eq. (4) is trivially satisfied
for $f(t)=0$. The function $u(t)$ may be designed to tailor either the
outgoing transversal position or momentum distributions. We consider two
possibilities, narrowing down the final momentum spread and focusing the
outgoing wave-packet, and compute the effect of each case in the the impact
width of the beam. To determine the worth of the operation, we compare the
results with the trivial evolution in which no potential manipulation takes
place in the radial degrees of freedom. A detailed explanation of the design
of the control function is given in SM .

The spatial dispersion of the wave-packet during the free flight after some
average time of flight $\bar{t}=md_{ff}/\langle p_{z}\rangle$ (where $d_{ff}$
is the distance of the free flight and $m$ is the mass of the ion) reads
$\Delta r=\sqrt{\langle r^{2}\rangle_{0}+\bar{t}\,^{2}\langle
p_{r}^{\,\,2}\rangle_{0}/m}$ SM . As explained before, the initial state of
the ion is thermally distributed and its statistical moments have the form
written in Eq. (Tailored Ion Beam for Precise Color Center Creation) but with
the corresponding frequency $\omega_{r,0}$. Intuitively, focusing the wave-
packet may seem the right choice, inasmuch as the goal is to reduce the
spatial width at the impact point. However, it turns out that the scattered
radial momenta blow the state up during the free flight, actually
deteriorating the result obtained from the trivial evolution. On the other
hand, appropriately scaling down the radial momenta, even at the expense of
starting with a wider state, improves considerably the impact spot one would
get form the initial thermal state as it is shown by Fig. 3 (a).

Figure 3:  (a) Radial dispersion after a free flight distance of $d_{ff}=300$
mm for invariant based drivings with different momentum scaling parameters,
see Eq. (7). Benchmark dispersion for a thermal state at $T=1000$ K (dashed
orange), which is also the starting point for the invariant based protocol.
(b) Dispersion at the impact spot for ions that have gone through an einzel
lens with focal length $F=13$mm with velocities of 5 km/s (solid blue), 10km/s
(dashes green) and 50 km/s(dotted brown). Other parameters are
$\omega_{r}=(2\pi)\times 1.4$ MHz and $t_{f}=\omega_{r}^{\,\,-1}$,
$t_{f}=0.94$ $\mu$s and $\langle p_{z}\rangle=5\times 10^{4}$ m/s.

Additionally, if we consider that the ion goes through an electromagnetic lens
before the impact, which is a typical setting in state-of-the-art experiments,
the cross area of the beam may be reduced even further. Assuming that the
target is placed at the focal point of an einzel lens Schnitzler10 , where the
outcome has minimum width, and provided that the position and momentum of the
incoming beam are Gauss distributed (which holds for the thermal state) the
impact spread can be computed as JacobThesis16

$\Delta r_{impact}=\frac{F\Delta
r\Delta\alpha}{\sqrt{\left(F-d_{ff}\right)^{2}\Delta\alpha^{2}+\Delta
r^{2}}},$ (13)

where $F$ is the focal length, the free flight distance $d_{ff}$ is the
separation between the launching point and the principal plane of the lens and
the beam divergence $\Delta\alpha$ can be computed from the ingoing radial
width as $\Delta\alpha=\Delta r/d_{ff}$. Figure 3 (b) displays the impact spot
radius in terms for various initial temperatures and launching velocities. It
exposes the bigger advantage of cooling the ion to sub-Doopler temperatures,
which is the capacity to reach nanoscale lateral resolution Groot-Berning19 ;
Groot-Berning21 . In our case, the radial driving allows to reduce the spot
below one $\mu$m for reasonable initial temperatures. This result, however,
could be further improved without invoking the time consuming sympathetic
cooling, for example through voltage sequences devoted to reduce the energy of
the ion during the loading process, via more elaborate focusing setups (for
example concatenating more lenses) or exploiting the flexibility of our method
to find optimal trajectories Stefanatos10 ; Chen11 .

_Conclusions.–_ We have developed a theoretical proposal to produce, from a
Paul trap, a highly monochromatic beam of N${}_{2}^{\,\,+}$ ions that could be
accurately implanted into a diamond host to generate shallow NV color centers.
Even for drivings produced by realistic/fluctuating electrodes, our invariant
based protocol limits the uncertainty of the extraction energy to the value
determined by the thermal distribution and even reduces it. It also provides a
substantial improvement on the repetition rate over implantation processes
that rely on cooling down the ions, so it would facilitate the deterministic
production of large amounts of defects and its expansion to industrial use. A
sequential use of our method can also tailor the radial properties of the
outgoing beam to lower its dispersion.

###### Acknowledgements.

Acknowledgements.– Authors acknowledge insightful discussions with Karin
Groot-Berning. We acknowledge financial support from Spanish Government via
PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE), PGC2018-101355-B-100
(MCIU/AEI/FEDER, UE) and EUR2020-112117, from Basque Government via IT986-16,
as well as from QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on
Quantum Technologies, as well as from the EU FET Open Grant Quromorphic
(828826). C.M.-J. acknowledges the predoctoral MICINN grant PRE2019-088519.
J.C. acknowledges the Ramón y Cajal program (RYC2018-025197-I) and support
from the UPV/EHU through the grant EHUrOPE. E.T. acknowledges financial
support from the Spanish Government through PGC2018-094792-B-I00
(MCIU/AEI/FEDER,UE), CSIC Research Platform PTI-001, CAM/FEDER Project No.
S2018/TCS-4342 (QUITEMAD-CM), and by Comunidad de Madrid-EPUC3M14.

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Tailored ion beam for deterministic color center creation.
SUPPLEMENTAL MATERIAL

##  I Dynamical state of the ion and its statistical moments

The state of the N${}_{2}^{\,\,+}$ ion during the launching operation may be
written in terms of the eigenvectors of the invariant,
$I(t)|\phi_{p_{z,0}}(t)\rangle=u_{0}p_{z,0}|\phi_{p_{z,0}}(t)\rangle$, with
corresponding wave-functions

$\displaystyle\phi_{p_{z,0}}(z,t)$ $\displaystyle=$ $\displaystyle\langle
z|\phi_{p_{z,0}}(t)\rangle$ (14) $\displaystyle=$
$\displaystyle\frac{e^{i\varphi_{p_{z,0}}(t)}}{\sqrt{\hbar}}e^{i\left[(u_{0}p_{z,0}-f)q+m\dot{u}z^{2}/2\right]/\hbar
u}.$

The phase

$e^{i\varphi_{p_{z,0}}(t)}=\sqrt{\frac{u_{0}}{u}}e^{-\frac{i}{2\hbar
m}(\mathcal{I}_{1}p_{z,0}^{2}-2\mathcal{I}_{2}p_{z,0}-\mathcal{I}_{3})}$ (15)

given here in terms of the integrals $\mathcal{I}_{1}=\int dt/u^{2}$,
$\mathcal{I}_{2}=\int dtf/u^{2}$ and $\mathcal{I}_{3}=\int
dt(\dot{f}/\ddot{u}-f^{2}/u)/u$, is chosen so that the function in Eq. (14)
satisfies $i\hbar\partial_{t}\phi_{p_{z,0}}(z,t)=H(t)\phi_{p_{z,0}}(z,t)$,
i.e., it becomes a solution of the time-dependent Schrödinger equation. Thus,
the state of our system at any given time can be expressed as

$\displaystyle\psi(z,t)$ $\displaystyle=$ $\displaystyle\int
dp_{z,0}\left(\frac{u_{0}}{\hbar
u}\right)^{\frac{1}{2}}\exp\bigg{[}\frac{i}{\hbar
u}\bigg{(}\frac{m\dot{u}}{2}z^{2}+(u_{0}p_{z,0}-f)z$ (16) $\displaystyle-$
$\displaystyle\frac{u}{2m}(\mathcal{I}_{1}p_{z,0}^{2}-2\mathcal{I}_{2}p_{z,0}-\mathcal{I}_{3})\bigg{)}\bigg{]}\langle
p_{z,0}|\psi(0)\rangle,$

where the initial momentum $p_{z,0}$ has been used as the integration variable
to expand the wave-function. The corresponding expectation values for the
position and momentum operators, given in terms of their initial values, read

$\displaystyle\langle z\rangle$ $\displaystyle=$ $\displaystyle\int
dz\psi^{*}(z,t)z\psi(z,t)$ (17) $\displaystyle=$
$\displaystyle\frac{u}{u_{0}}\langle
z\rangle_{0}+\frac{u_{0}u}{m}\mathcal{I}_{1}\langle
p_{z}\rangle_{0}-\frac{u}{m}\mathcal{I}_{2},$ $\displaystyle\langle
p_{z}\rangle$ $\displaystyle=$ $\displaystyle\int
dz\psi^{*}(z,t)(-i\hbar\frac{\partial}{\partial
z})\psi(z,t)=\frac{m\dot{u}}{u_{0}}\langle z\rangle_{0}$ (18) $\displaystyle+$
$\displaystyle\left(\frac{u_{0}}{u}+u_{0}\dot{u}\mathcal{I}_{1}\right)\langle
p_{z}\rangle_{0}-\dot{u}\mathcal{I}_{2}+\frac{(f_{0}-f_{f})}{u}.$

Notice that the solution for $\langle z\rangle$ is, according to Ehrenfest’s
theorem, a solution for the classical equation of motion

$\ddot{z}+\omega_{z}(t)^{2}z=\omega_{z}(t)^{2}z_{0}(t),$ (19)

with control parameters that obey Eqs. (3) and (4) of the main text. Thus, by
computing the expectation value of the position for a state described by (16)
we have found an analytical solution (in terms of some integral forms) for the
time-dependent forced harmonic oscillator:

$z(t)=\frac{u(t)}{u_{0}}z_{0}+\frac{u_{0}u(t)}{m}\mathcal{I}_{1}p_{z,0}-\frac{u(t)}{m}\mathcal{I}_{2}.$
(20)

From here the second order moments may be computed using the Liouville
theorem, which states that the phase-space distribution function is constant
along the trajectories of the system. The theorem regards classical dynamics,
however, the Wigner function $W(z,t)$, i.e., the quasiprobability distribution
that describes our quadratic system in phase-space, evolves classically and
thus fulfills Liouville’s theorem, leading to
$W_{0}(z_{0},p_{z,0})=W_{t}(z,p_{z}),$ allowing us to easily obtain higher
order moments for $q$ and $p$ from their expectation values:

$\displaystyle\langle z^{2}\rangle$ $\displaystyle=$ $\displaystyle\iint
dzdp_{z}W_{t}(z,p_{z})z^{2}=\left(\frac{u}{m}\mathcal{I}_{2}\right)^{2}+\left(\frac{u}{u_{0}}\right)^{2}\langle
z^{2}\rangle_{0}$ (21) $\displaystyle-$ $\displaystyle
2\frac{u^{2}}{mu_{0}}\mathcal{I}_{2}\langle
z\rangle_{0}-2\frac{u_{0}u}{m}\mathcal{I}_{1}\mathcal{I}_{2}\langle
p_{z}\rangle_{0}+\left(\frac{u_{0}u}{m}\mathcal{I}_{1}\right)^{2}\langle
p_{z}^{2}\rangle_{0}$ $\displaystyle+$
$\displaystyle\frac{u^{2}}{m}\mathcal{I}_{1}\langle zp_{z}+p_{z}z\rangle_{0},$
$\displaystyle\langle p_{z}^{2}\rangle$ $\displaystyle=$ $\displaystyle\iint
dzdp_{z}W_{t}(z,p_{z})p_{z}^{2}=\dot{u}^{2}\mathcal{I}_{2}^{2}+2\frac{\dot{u}}{u}\mathcal{I}_{2}\Delta
f+\left(\frac{\Delta f}{u}\right)^{2}$ (22) $\displaystyle-$
$\displaystyle\left(2m\frac{\dot{u}^{2}}{u_{0}}\mathcal{I}_{2}+2m\frac{\dot{u}}{u_{0}u}\Delta
f\right)\langle z\rangle_{0}+\left(\frac{m\dot{u}}{u_{0}}\right)^{2}\langle
z^{2}\rangle_{0}$ $\displaystyle-$
$\displaystyle\left(2\dot{u}^{2}u_{0}\mathcal{I}_{1}\mathcal{I}_{2}+2\frac{\dot{u}}{u}\mathcal{I}_{2}+2\frac{u_{0}}{u^{2}}\Delta
f+2\frac{u_{0}\dot{u}}{u}\Delta f\mathcal{I}_{1}\right)\langle
p_{z}\rangle_{0}$ $\displaystyle+$
$\displaystyle\left(u_{0}^{2}\dot{u}^{2}\mathcal{I}_{1}^{2}+\frac{u_{0}^{2}}{u^{2}}+2\frac{u_{0}^{2}\dot{u}}{u}\mathcal{I}_{1}\right)\langle
p_{z}^{2}\rangle_{0}$ $\displaystyle+$
$\displaystyle\left(m\dot{u}^{2}\mathcal{I}_{1}+\frac{m\dot{u}}{u}\right)\langle
zp_{z}+p_{z}z\rangle_{0}.$

##  II Radial dispersion during the free flight

Liouville’s theorem also allows us to easily compute the result of the free
motion of the flying atom, as no force is acting on the quantum system. The
average radial position mimics classical free motion,

$\langle r\rangle=\langle r\rangle_{0}+\frac{\langle p_{r}\rangle_{0}}{m}t,$
(23)

and its dispersion reads

$\displaystyle\langle r^{2}\rangle$ $\displaystyle=$ $\displaystyle\iint
drdp_{r}W_{t}(r,p_{r})r^{2}$ (24) $\displaystyle=$ $\displaystyle\langle
r^{2}\rangle_{0}+\frac{\langle p_{r}^{2}\rangle_{0}}{m}t+\frac{\langle
rp_{r}+p_{r}r\rangle_{0}}{m}t.$

From here, and considering the statistical moments of the initial thermal
state, the radial dispersion of the flying ion is

$\Delta r^{2}=\langle r^{2}\rangle-\langle r\rangle^{2}=\langle
r^{2}\rangle=\langle r^{2}\rangle_{0}+\frac{\langle
p_{r}^{2}\rangle_{0}}{m}t.$ (25)

##  III Invariant based design of the control parameters

Here we take an inverse engineering approach to deduce the driving that
produces the desired ion beam. In general, such design method requires three
ingredients: (i) the operator that describes the evolution of the system,
i.e., the Hamiltonian, (ii) a property of the system that remains constant
under that evolution, i.e., the expectation value of the invariant operator,
and (iii) some connection between the time-dependent functions that define
both of them, which in our case are the equations that relate the control
parameters $\omega_{z}(t)$ and $z_{0}(t)$ with the auxiliary parameters $u(t)$
and $f(t)$, Eqs. (3) and (4) in the main text.

Once we have identified the necessary elements, we must translate the desired
outcome into boundary conditions (BC) for the auxiliary functions. The BC
leading to the tailored ion beam are given in Eqs. (8), (9) and (10) of the
main text. Finally, all that remains is to propose functions for the auxiliary
parameters that satisfy the deduced BC and compute the control parameters from
Eqs. (3) and (4), which read

$\omega_{z}(t)=\sqrt{-\frac{\ddot{u}(t)}{u(t)^{2}}},\hskip
17.07182ptz_{0}(t)=\frac{-\dot{f}(t)}{mu(t)\omega_{z}(t)^{2}}.$ (26)

Notice that, given these definitions, we might encounter discontinuities that
render unfeasible control parameters. In particular, both $\omega_{z}(t)$ and
$z_{0}(t)$ diverge whenever $u(t)=0$, and, additionally, $z_{0}(t)$ diverges
if $\omega_{z}(t)$ becomes cero. To avoid discontinuities caused by the zeros
of $u(t)$, we will design a function that is positive at all times. On the
other hand, the zeros of $\omega_{z}(t)$ are harder to avoid, so instead of
seeking a design of the frequency that never changes sign, we will force
$\dot{f}(t)$ to match the zeros of $\omega_{z}(t)$, so that the resulting
trajectory $z_{0}(t)$ is free of divergences.

We use polynomial functions for the auxiliary parameters,

$u(t)=\sum_{j=0}^{10}a_{j}(t/t_{f})^{j},\hskip
22.76228ptf(t)=\sum_{k=0}^{11}b_{k}(t/t_{f})^{k}.$ (27)

The coefficients will be fixed in three different steps. First $a_{0-7}$ and
$b_{0-5}$ are determined by the BC that the functions must meet. Then $a_{8}$
is used to maintain $u(t)$ positive at all times (we force $u(t_{f}/2)$ to
have a certain value that hinders the change), while $b_{6-9}$ are used to
impose the necessary zeros in $\dot{f}(t)$ to avoid divergences in $z_{0}(t)$.
Finally, $a_{9,10}$ and $b_{10,11}$, two coefficients in each auxiliary
function, are used to minimise the maximum value of the average potential
energy of the ion using the ’fminseach’ tool of Matlab.

##  IV Simulation of the dynamics

The simulation of the evolution of the quantum system is considerably
simplified (regarding computational effort) by the fact that the Wigner
function in a quadratic potential evolves classically.

Thus, to simulate the dynamics of the ion we define a large amount of points,
distributed in phase-space according to the Wigner function, which for the
initial thermal state of the harmonic oscillator reads

$W(z,p_{z})=\frac{1}{2\pi\sqrt{\langle z^{2}\rangle_{0}\langle
p_{z}^{2}\rangle_{0}}}\exp{\left[\frac{1}{2}\left(\frac{z^{2}}{\langle
z^{2}\rangle_{0}}+\frac{p_{z}^{2}}{\langle
p_{z}^{2}\rangle_{0}}\right)\right]},$ (28)

and evolve then individually using the classical equation of motion in Eq.
(19).


# Towards A Device-Independent Deep Learning Approach for the Automated
Segmentation of Sonographic Fetal Brain Structures: A Multi-Center and Multi-
Device Validation

Abhi Lad  Origin Health India, Bengaluru, Karnataka, India  Adithya Narayan
Origin Health India, Bengaluru, Karnataka, India  Hari Shankar  Origin Health
India, Bengaluru, Karnataka, India  Shefali Jain  Bangalore Fetal Medicine
Center, Bengaluru, Karnataka, India  Pooja Punjani Vyas  Jaslok Hospital —&
Research Center, Mumbai, Maharashtra, India  Divya Singh  Prime Imaging and
Prenatal Diagnostics, Chandigarh, Punjab, India  Nivedita Hegde  Kasturba
Medical College, Manipal Academy of Higher Education, Manipal, Karnataka,
India  Jagruthi Atada  Origin Health India, Bengaluru, Karnataka, India  Jens
Thang  Origin Health, Singapore  Saw Shier Nee  Department of Artificial
Intelligence, Faculty of Computer Science and Information Technology,
Universiti Malaya, Kuala Lumpur, Malaysia  Arunkumar Govindarajan  Aarthi
Scans and Labs, Chennai, Tamil Nadu, India  Roopa PS  Kasturba Medical
College, Manipal Academy of Higher Education, Manipal, Karnataka, India
Muralidhar V Pai  Kasturba Medical College, Manipal Academy of Higher
Education, Manipal, Karnataka, India  Akhila Vasudeva  Prathima Radhakrishnan
Sripad Krishna Devalla

###### Abstract

Quality assessment of prenatal ultrasonography is essential for the screening
of fetal central nervous system (CNS) anomalies. The interpretation of fetal
brain structures is highly subjective, expertise-driven, and requires years of
training experience, limiting quality prenatal care for all pregnant mothers.
With recent advancement in Artificial Intelligence (AI), computer assisted
diagnosis has shown promising results, being able to provide expert level
diagnosis in matter of seconds and therefore has the potential to improve
access to quality and standardized care for all. Specifically, with advent of
deep learning (DL), assistance in precise anatomy identification through
semantic segmentation essential for the reliable assessment of growth and
neurodevelopment, and detection of structural abnormalities have been
proposed. However, existing works only identify certain structures (e.g.,
cavum septum pellucidum [CSP], lateral ventricles [LV], cerebellum ) from
either of the axial views (transventricular [TV], transcerebellar [TC]),
limiting the scope for a thorough anatomical assessment as per practice
guidelines necessary for the screening of CNS anomalies. Further, existing
works do not analyze the generalizability of these DL algorithms across images
from multiple ultrasound devices and centers, thus, limiting their real-world
clinical impact. In this study, we propose a deep learning (DL) based
segmentation framework for the automated segmentation of 10 key fetal brain
structures from 2 axial planes from fetal brain USG images (2D). We developed
a custom U-Net variant that uses inceptionv4 block as a feature extractor and
leverages custom domain-specific data augmentation. Quantitatively, the mean
(10 structures; test sets 1/2/3/4) Dice-coefficients were: 0.827, 0.802,
0.731, 0.783. Irrespective of the USG device/center, the DL segmentations were
qualitatively comparable to their manual segmentations. The proposed DL system
offered a promising and generalizable performance (multi-centers, multi-
device) and also presents evidence in support of device-induced variation in
image quality (a challenge to generalizibility) by using UMAP analysis. Its
clinical translation can assist a wide range of users across settings to
deliver standardized and quality prenatal examinations.

###### keywords:

fetal ultrasonography, transventricular, transcerebellar, U-Net variant,
segmentation, multi-center, multi-device

##  1 Introduction

The prenatal ultrasonography (USG) examination of the fetal brain at the level
of the transventricular (TV) and transcerebellar (TC) planes (axial views)
during the 2nd trimester has been clinically recommended for the screening of
central nervous system (CNS) anomalies [1]. It offers scope for timely
specialist referrals, necessary intervention and appropriate pregnancy
management. The central nervous system (CNS) anomalies are among the most
common types of fetal anomalies (1.4-1.6 per 1000 live births) [2]. With 3-6%
stillbirths, low 5-year survival rates, life-long physical and mental
disabilities, CNS anomalies take a very expensive toll on families, community,
and the healthcare system at large [3, 4]. Due to poor specialist-to-patient
ratio, access to quality and timely prenatal care is limited for pregnant
women; especially in low-resource, rural, and semi-urban settings. Several
existing deep learning (DL) approaches have focused on only certain structures
on any one of the axial views (TC or TV). Some applications include but are
not limited to, detecting standard fetal planes and structures, segmentation
of fetal anatomy (e.g., brain, head, face, abdomen, cardiac, lungs, etc.), and
classification (normal/abnormal/pathology) of fetal USG images [5, 6, 7, 8,
9]. Specifically, for USG images of the fetal brain, studies have attempted to
individually segment key structures such as the cranium, cavum septum
pellucidum (CSP), LV, etc. [10, 7, 11]. However, these studies have not
explored generalizability of these algorithms on images from unseen devices
and multiple imaging centers, thus limiting the clinical translation of these
approaches. In this study, we propose a custom DL framework for the automated
segmentation of 10 key fetal structures across both the TC and TV planes.
Further, we validate the generalizability of the proposed approach across 4
independent test sets comprising of images from 4 unseen USG devices across 3
imaging centers. We have also provided UMAP [12] analysis of images from
different USG machines from across the imaging centers to support the machine
induced variance in the USG image characteristics. Following are the key
contributions of this study:

  * •

Development, validation, and benchmarking of a U-Net inspired multi-plane,
multi-structure DL segmentation model.

  * •

UMAP analysis of multi-device USG images to establish image characteristic
variance among USG devices.

  * •

Significance of domain-specific data augmentation and its effect on DL model
generalizability.

  * •

Extensive (1626 images) multi-device, multi-center validation to assess the
robustness of the proposed approach.

##  2 MATERIALS AND METHODS

###  2.1 Fetal Ultrasound Imaging

A total of 4370 two-dimensional (2D) standard plane USG images of the TV and
TC planes were obtained from 3 centers (2 tertiary referral centers [TRC 1,2]
+ 1 routine imaging center [RIC]) using 6 USG devices (General Electric [GE]
Voluson: P8, P6, E8, E10, S10; Samsung: HERA W10). The images were acquired
from a cohort of 1349 pregnant women who received their targeted mid-trimester
USG examination (transabdominal). All USG examinations were reviewed and
approved by fetal medicine specialists for quality and correctness. Only live
singleton fetuses that did not exhibit any growth anomalies were included in
this study. Due to the study’s retrospective nature, all informed consent was
waived. All data was anonymized as per the tenets of the Declaration of
Helsinki after approval from the ethics committee of the respective centers.
The USG examination was performed based on internationally recommended
practice guidelines [1].

###  2.2 Dataset Preparation

From the 4370 images, a total of 2744 images (TC/TV images from GE E8:
641/1307; TC/TV images from GE S10: 221/575) from TRC 1 were used for training
and validation of the DL model. Data balancing (up-sampling) was done to
ensure equal representation from each USG device and care was taken to ensure
there were no duplicate images, or images from the same subjects between the
training and validation datasets (training and validation splits:
89.7%:10.3%).

The remaining 1626 images were used as a part of 4 test sets. Test set 1 (TRC
1; trained device: GE E8 [82.3 %], GE S10 [17.7 %]) consisting of of 719
images (TC/TV: 131/588) was used to test the performance of best DL model. The
robustness and generalizability of the DL model was assessed on independent
test sets 2,3 and 4. Test set 2 (TRC 1, unseen devices: Samsung HERA W10 [79.2
%], GE P6 [6.8 %], GE E10 [14 %]) consisted of 192 images (TC/TV: 90/102),
test set 3 (TRC 2, trained devices: GE E8 [69.5 %], GE S10 [30.5 %]) consisted
of 557 images (TC/TV: 223/334); and test set 4 (RIC; unseen device: GE P8)
consisted of 158 images (TC/TV: 100/58)

All images were resized to 160 (height) x 288 (width). To preserve the
inherent variability (i.e., gain, zoom, contrast, speckle noise, patient-
specific probe settings, pixel resolution, etc.) in the dataset, no additional
pre-processing was done.

Manual segmentations were prepared for all the 4370 images by well-trained
medical image annotators and reviewed by fetal medicine specialists. A total
of 10 unique fetal brain structures/classes from the TV (cranium, brain
parenchyma, cavum septum pellucidum [CSP], midline falx, choroid plexus,
lateral ventricles [LV]) and TC planes (cranium, brain parenchyma, CSP,
midline falx, cerebellum, cisterna magna, nuchal fold, skin) were segmented.
Note that each class common to both the planes (i.e., cranium, brain
parenchyma, midline falx and CSP) were assigned the same unique index while
preparing the segmentation masks.

---
Figure 1:  The U-Net inspired DL architecture used in this study is shown.

###  2.3 Deep Learning Network

We propose a fully-convolutional and end-to-end DL network inspired (Figure
1)by U-Net [13] to simultaneously segment 10 fetal brain structures across 2
axial views. Inception v4A [14] blocks were used as feature extractors for the
4-levels of the (feature maps: 64, 128, 256, 512) encoder and latent space,
while standard blocks (Figure 1) were used in the decoder for feature
extraction. Segmentation of fetal brain structures can often be difficult
especially in the cases of structural anomalies due to common co-existence of
multiple fetal anomalies. Specifically for finer structures like LV, CSP, and
nuchal fold where subtle changes in structure are associated with critical
anomalies like ventriculomegaly, agenesis of corpus callosum, and Down
syndrome. The inceptionv4A block with its multiple kernel sizes can take into
account the global features, like structures in vicinity e.g. choroid plexus
inside LV and local features, like boundaries and integrity of structures e.g.
case of choroid plexus with a cyst inside it. The input image (160 [height] x
288 [width]) was downsampled by a factor of 2 at each level of the encoder
(maxpooling, stride = 2), and sequentially upsampled to original scale (bi-
linear interpolation) in the decoder. The output from the decoder was passed
through a 1x1 convolutional layer (number of feature maps= [11; 10 fetal
structures + background], stride = 1) followed by softmax activation to obtain
the class-wise probability distributions. To get the final segmentation mask
(herein referred to as DL segmentation), each pixel was assigned the class
with maximum probability.

The entire network was trained end-to-end with Adam [15] optimizer, Dice loss,
fixed learning rate of 0.0001, and a mini batch of 2 images. The model with
the best validation loss was finally chosen. The proposed network consisted of
38 million trainable parameters and was trained and tested on an NVIDIA Tesla
T4 with CUDA v11.0 and cuDNN v7.6.5 using PyTorch 1.7.0.

###  2.4 Domain-Specific Data Augmentation

Fetal USG images from different USG devices and centers may appear visually
different in terms of image quality (e.g., noise, resolution, acoustic
shadowing), perception (e.g., brightness, contrast, intensity homogeneity) and
presentation (e.g., zoom, orientation; operator dependent factors). To counter
these effects, we used extensive domain-specific online data augmentation (DA;
implemented using Albumentations [16]; Figure 2 A) to improve the DL model’s
performance and generalizability across images from different USG
devices/centers.

Image quality and perceptual factors were augmented using color-jitter
(brightness: 0.8-1.2 and contrast: 0.8-1.2) - representative of various image
enhancing settings, quality reduction (downsampling and upsampling the images;
range: 0 to 0.5) - representative of poor quality hardware or image
compression loss, speckle noise (Gaussian multiplicative noise; mean: 0,
standard deviation: 0.1) - represents artifact introduced by proprietary post-
processing, and acoustic shadowing - characteristic artifact formed due to
acoustic property of cranium.

In this study, we propose a novel approach to accurately represent fetal
acoustic shadows occurring due to attenuation of sound beam at the edges of
cranium due to incorrect focal depth. As shown in Figure 2 B, the acoustic
shadows are created using the cranium segmentation mask which is used to
determine horizontal extremities (red dots) of the cranium. Then 2 patches of
width 75-100 pixels are extended from the extreme points at angle 15-20
degrees. These patches are further processed with blur to achieve more
realistic murky shadows. Then it is superimposed on original image resulting
in image with artificial acoustic cranium shadows as shown in Figure 2 B.

---
Figure 2:  (A) Domain-specific data augmentation (1: baseline; 2: motion blur;
3: acoustic shadows [highlighted in yellow box]; 4: affine transforms; 5:
brightness and contrast; 6: Gaussian multiplicative noise; and 7: resultant
image of combined augmentations) are shown. (B) The visualization of steps
used to create artificial cranium shadows as part of data augmentation from
left to right: cranium mask ground truth used to identify left and right
extremities; black patches created from extreme points; superimposing black
patches on correspoonding USG image to create cranium shadows.

We used affine transformations (zoom [range: 0.6 to 1.2], rotation [range: +20
to -20 degrees], translations [range: 40 to 60 pixels along height or width],
shear transformations [range: 0 to 0.2]) and motion blur (kernel size: 50x50
pixels) to simulate operator variability. Finally, variation in fetal anatomy
due to anomalies/gestation age were accounted by using elastic deformations
(standard deviation: 50, affine: 50).

###  2.5 Experiments and Analysis

####  2.5.1 UMAP Analysis

The visual feel/quality of images generated by USG devices vary depending on
model, manufacturer and provided operator settings. Such variations in image
quality occur due to proprietary image construction/processing algorithms,
image compression techniques or even probe technology used. In order to
demonstrate the USG device induced variance in images, we use UMAP [12]
algorithm. The UMAP is a dimensionality reduction technique similar to tSNE
which is useful in visualizing high dimensional data in low dimensional
(2D,3D) euclidean space. UMAP algorithm creates a connected graph based on
high-dimension input data, i.e. images, and projects it to low dimensional
(2D, 3D) graph by using force-directed graph layout algorithm. This low
dimensional graph is mathematically similar to the high dimensional graph,
i.e. the points appearing closer in lower dimensional space are actually
similar even in high dimensional space. The presentation of projected low-
dimensional graph depends on parameters like nearest-neighbour and minimum
distance used to create connected graph (manifold). Separated clusters
indicate that images from different USG devices are varied and dense clusters
indicate low intra-device variance. Overlapping clusters suggest similarity in
image characteristics among USG devices. The resultant plot as presented in
Figure 3 clearly depicts formation of clusters for each USG device. The plots
of TC and TV images are presented individually to prevent cluster formation
due to different planes. While some overlap is observed among the clusters,
specifically for Voluson S10 (Figure 3: pink) and P6 (Figure 3: cyan), overall
the images have enough variance to form a cluster with images from same
machine lying in vicinity. Interestingly, images from Samsung Hera W10 (Figure
3: red) have high intra-device variance thus resulting in separate clusters
for the same USG device.

---
Figure 3:  2D UMAP (low dimensional) representation of TC images (left) and TV
images (right) corresponding to 6 USG devices across all 4 test sets.
Overlapping clusters suggest similarity in image characteristics among USG
devices. Dense clusters indicate low intra-device variance. Separation of
clusters as observed above supports the variance among images from different
USG devices is substantial and qualitatively verifiable.
---
Figure 4:  The Dice-coefficients (when tested on model trained with and w/o
domain-specific DA) for individual structures across the 4 test sets are
shown.
---
Figure 5:  The effect of training dataset size on the segmentation performance
(when tested on model trained with and w/o domain-specific DA) across the test
sets is shown.
---
Figure 6:  Qualitative comparison of the proposed DL segmentation with and
without (w/o) domain-specific DA (vs. manual segmentation). 1st row represents
baseline images from 8 subjects (2 per test set), 2nd, 3rd, and 4th columns
represents their manual segmentation, and DL segmentations (with and w/o
domain specific DA), respectively.

####  2.5.2 Deep Learning Experiments and Ablations

We performed the following experiments: (1) benchmarking against multiple DL
segmentation models that have been widely used for segmentation problems
(U-Net [13], LinkNet [17], MNet [18], Deeplabv3+ [19] and HRNet [20]); (2)
effect of domain-specific DA (trained with and without (w/o) domain-specific
DA); (3) effect of dataset size (data ablation on fractions [0.2, 0.4, 0.6,
0.8, 1.0] of the training dataset; performed with and without domain-specific
DA) and model performance; and (4) 5-fold cross-validation to assess
repeatability.

For all experiments, the DL segmentations from all the 4 test sets were
qualitatively and quantitatively (structure wise Dice coefficient [DC] as per
standard definition) assessed by benchmarking against the expert-annotated
manual segmentations. Note that background class was excluded from qualitative
and quantitative comparison for all experiments.

##  3 RESULTS AND DISCUSSION

###  3.1 Qualitative Comparison with Manual Segmentation

Qualitatively, the DL segmentations were comparable to the manual
segmentations across all test sets for both the TV and TC plane structures. As
it is evident from Figure 6, there is substantial difference between DL
segmentations with DA and w/o DA. While DL segmentations with DA (3rd column)
are comparable to manual segmentations, the DL segmentations w/o DA (4th
column) even fail to correctly segment brain parenchyma, which covers more
than half of segmentable area inside the brain region (Figure 6: light blue)
and thus is essentially unable to predict finer structures such as CSP and LV.

###  3.2 Quantitative Comparison of Models

Quantitatively (Table 1; mean of 5-fold cross validation was used), when
benchmarked against widely used segmentation models such as U-Net [13],
LinkNet [17], MNet [18], Deeplabv3+, [19] and HRNet [20], we obtained a
performance improvement (mean DC) of 0.5% to 2% across 4 test sets.
Specifically, compared to the benchmarking models, our model localized the the
finer structures (LV, CSP, nuchal fold) that are crucial in the screening of
CNS anomalies and overall resulted in better performance, i.e. average
improvement over benchmarking models for CSP: 2.4 %, LV: 9.8 % and nuchal
fold: 8.3 % when averaged across test sets.

###  3.3 Effect of Domain Specific Data Augmentations

The effect of domain-specific DA on the DL segmentations were qualitatively
(Figure 6, 3rd and 4th columns) and quantitatively observed (Figure 4). As it
is evident from (Figure 6) columns 3rd and 4th, data augmentation can have a
profound impact on the segmentation performance especially for USG images from
different centers. Across all test sets (especially unseen devices/unseen
centers), in general, finer structures and structures of critical importance
like LV, CSP and cerebellum were affected the most (Figure 6, 4th column) in
the absence of domain-specific DA. Specifically, we observed performance gains
(percentage change in mean DC; Figure 4 and Table 1) of 2.1%, 7.11%, 18.4%,
and 22.7% across test sets 1, 2, 3, and 4 respectively. Besides, even when the
domain-specific DA were used with the benchmarking models, on an average, we
observed a minimum of 0.486% and a maximum of 30% performance gain for all
cases.

###  3.4 Ablation on Size of Training Set

When trained (with domain-specific DA) on increasing fractions of dataset
(0.2, 0.4, 0.6, 0.8, 1.0), across all test sets, the DL segmentation
performance continued to improve (Figure 5). When repeated the same without
domain-specific DA, for a given fraction of training data, the segmentation
performance was overall (mean of all fractions; vs. trained with domain-
specific DA) reduced by 1%, 4%, 11.4%, and 14.8%, across test sets 1, 2, 3 and
4 respectively.

Table 1: Benchmarking of the proposed DL method (with and without domain-specific DA) against U-Net [13], LinkNet [17], MNet [18], DeepLabv3+ [19], and HRNet [20] across 4 independent test sets. Mean Dice coefficient (DC; mean of all 10 classes) was used for the purposes of evaluation. Models | Test set 1 | Test set 2 | Test set 3 | Test set 4
---|---|---|---|---
DA | w/o DA | DA | w/o DA | DA | w/o DA | DA | w/o DA
Ours | 0.827 | 0.810 | 0.783 | 0.743 | 0.731 | 0.617 | 0.802 | 0.654
U-Net [13] | 0.823 | 0.817 | 0.769 | 0.737 | 0.712 | 0.639 | 0.781 | 0.670
LinkNet [17] | 0.724 | 0.720 | 0.665 | 0.632 | 0.608 | 0.451 | 0.663 | 0.491
MNet [18] | 0.818 | 0.797 | 0.764 | 0.712 | 0.706 | 0.593 | 0.782 | 0.619
Deeplabv3+ [19] | 0.814 | 0.778 | 0.737 | 0.656 | 0.681 | 0.448 | 0.731 | 0.431
HRNet [20] | 0.819 | 0.802 | 0.765 | 0.703 | 0.702 | 0.589 | 0.767 | 0.622

###  3.5 Structure-wise Quantitative Comparison

---
Figure 7:  The comparison of Dice-coefficients of individual structures (1:
Midline Falx, 2: LV, 3: CSP, 4: Skin, 5: Nuchal Fold, 6: Choroid Plexus, 7:
Cerebellum, 8: Cisterna Magna) averaged over the 4 test sets is presented.

Further structure-wise analysis of model performance is provided in Figure 7.
The results of each structure are averaged across all the 4 test sets and
presented in the figure. As it is evident from Table 1 and Figure 7, our
proposed approach generally performs better than benchmarking models. While
the overall performance of U-Net model is similar (1.4 % lower than our
approach when averaged across all test sets), our approach outperforms U-Net
specifically for finer structures like CSP, LV and nuchal fold which are
crucial for differential diagnosis (DD) of CNS anomalies.

In fetal brain examination certain structures are more important. Changes in
structures like deviation from normal CSP ratio and dilated lateral ventricles
are small but crucial for the differential diagnosis of several CNS anomalies.
Our proposed approach always outperforms the U-Net for these structures in all
the test sets. Numerical values for CSP: Test set 1 (Proposed-0.863,
U-Net-0.857), Test set 2 (Proposed-0.767, U-Net-0.739), Test set 3
(Proposed-0.836, U-Net-0.790), Test set 4 (Proposed-0.710, U-Net-0.698).
Numerical values for LV: Test set 1 (Proposed-0.720, U-Net-0.706), Test set 2
(Proposed-0.612, U-Net-0.604), Test set 3 (Proposed-0.603, U-Net-0.573), Test
set 4 (Proposed-0.693, U-Net-0.630).

##  4 CONCLUSION

In this study, we presented a custom DL framework for the automated
segmentation of 10 axial view structures (TC and TV planes) of the fetal
brain. On testing across 4 test sets, the proposed DL framework outperformed
other popular segmentation networks by 0.5% to 2% (mean DC), localizing fine
structures better. We have also presented the rationale behind selecting the
feature extraction block based on empirical evidence of higher performance on
important finer structures. Additionally, our proposed domain-specific DA
improved model generalizability and showed significant performance
improvements even across the benchmarking models. From training dataset
ablation studies, we observed large segmentation performance gains and
improved generalizability by simply using domain-specific DA, despite limited
training data.

We believe, our work opens doors for the development of device-independent and
vastly generalizable DL models, a necessity for the seamless clinical
translation and deployment. We hope the emergence of such assistive technology
in clinics can truly help democratize quality and standardized prenatal care
for every expecting mother. Our study faced limitations in terms of size of
dataset, variation in demographics and number of brain examination planes and
only considered normal fetal brain scans. Future works can focus on analyzing
the specific performance differences across USG devices and wider demographic,
use of domain adaptation, and performance evaluation on non-standard planes
and anomaly cases. The hyperparameters for data augmentations have been set
manually and future studies can also focus on automating the process and the
dataset scales up.

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