††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Chiral Dynamics of Ultracold Atoms under a Tunable SU(2) Synthetic Gauge
Field

Qian Liang  Zhaoli Dong  Zhejiang Province Key Laboratory of Quantum
Technology and Device, School of Physics, and State Key Laboratory for Extreme
Photonics and Instrumentation, Zhejiang University, Hangzhou 310027, China
College of Optical Science and Engineering, Zhejiang University, Hangzhou
310027, China  Hongru Wang  Zhejiang Province Key Laboratory of Quantum
Technology and Device, School of Physics, and State Key Laboratory for Extreme
Photonics and Instrumentation, Zhejiang University, Hangzhou 310027, China
College of Optical Science and Engineering, Zhejiang University, Hangzhou
310027, China  Hang Li  Zhaoju Yang  Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, and State Key Laboratory for
Extreme Photonics and Instrumentation, Zhejiang University, Hangzhou 310027,
China  Jian-Song Pan  College of Physics and Key Laboratory of High Energy
Density Physics and Technology of Ministry of Education, Sichuan University,
Chengdu 610065, China  Wei Yi<EMAIL_ADDRESS>CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
China  CAS Center For Excellence in Quantum Information and Quantum Physics,
Hefei 230026, China  Bo Yan<EMAIL_ADDRESS>Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, and State Key
Laboratory for Extreme Photonics and Instrumentation, Zhejiang University,
Hangzhou 310027, China  College of Optical Science and Engineering, Zhejiang
University, Hangzhou 310027, China

###### Abstract

Surface currents emerge in superconductors exposed to magnetic fields, and are
a key signature of the Meissner effect. Analogously, chiral dynamics were
observed in quantum simulators under synthetic Abelian gauge fields. The
flexible control of these simulators also facilitates the engineering of non-
Abelian gauge fields, but their impact on the chiral dynamics remains elusive.
Here, by employing the cutting-edge momentum-lattice technique, we implement a
synthetic SU(2) gauge field in a spinful 1D ladder and study the rich chiral
dynamics therein. We confirm the non-Abelian nature of the synthetic potential
by observing the non-Abelian Aharonov-Bohm effect on a single plaquette. More
importantly, the chiral current along the two legs of the ladder is observed
to be spin-dependent and highly tunable through the parameters of the gauge
potential. We experimentally map out different dynamic regimes of the chiral
current, and reveal the underlying competition between overlaying flux ladders
with distinct spin compositions. Our experiment demonstrates the dramatic
impact of non-Abelian gauge fields on the system dynamics, paving the way for
future studies of exotic synthetic gauge fields on the versatile platform of
momentum lattices.

Gauge theory underlies our modern understanding of the quantum world: from
interactions between elementary particles [1, 2] to emergent gauge symmetries
in strongly correlated matter [3, 4, 5], gauge fields play a central role in a
wealth of physical contexts. In recent years, the ever-growing ability to
engineer synthetic gauge fields in quantum simulators offers fresh
opportunities. On one hand, it is possible to emulate gauge fields that
already exist in nature, now in the highly controllable synthetic settings
such as ultracold atoms [6, 7, 8, 9, 10, 11], superconducting qubits [12], as
well as photons [13, 14, 15, 16, 17] and phonons [18, 19, 20, 21]. For
instance, the experimental synthesis of the simplest Abelian gauge field, the
scalar and vector potentials of electromagnetism, has led to the observation
of chiral edge current of charge-neutral atoms in a flux ladder [11], a
phenomenon analogous to the surface current in superconductors under the
Meissner effect [22, 23]. On the other hand, a rich variety of exotic gauge
potentials become experimentally accessible [24, 25, 26, 27, 28, 29, 30]. In
particular, the realization of synthetic non-Abelian gauge fields underlies
the recent surge of interest in simulating topological phenomena such as the
non-Abelian geometric phase and topology [31, 32, 33, 34], the (anomalous)
quantum Hall effect [35, 36], and the Yang’s monopole [37, 38]. Therein, the
non-Abelian gauge fields couple the internal and external degrees of freedom
of the system, leading to intriguing geometric structures in the Hilbert
space. Through these couplings, the non-Abelian gauge fields are also bound to
affect the dynamic or transport properties of the system, but their impact
remains largely unexplored.

Figure 1: Synthesis of the SU(2) gauge fields in the RML. (a) Energy
dispersions of the relevant ground-state hyperfine manifolds of 87Rb atoms.
Some typical two-photon couplings are shown here: the red arrows indicate the
two-photon transitions with the recoil momentum $2\hbar k_{1}$, and the blue
arrows indicate those with $2\hbar k_{2}$. (b) Schematic illustration of the
momentum lattice formed by the couplings in (a). The two internal hyperfine
states are encoded as the spin-down ($F=1$) and spin-up ($F=2$) states. The
color codes of the bonds and sites and consistent with those of the coupling
processes and hyperfine states in (a). (c) A spin-1/2 ladder with an arbitrary
SU(2) gauge field. Each site has spin-up and spin-down internal degrees of
freedom. Here $n$ and $m$ respectively represent the index in the synthetic
$x$ and $y$ direction.

In this work, we experimentally study the effects of a highly tunable SU(2)
gauge field on the chiral dynamics along a spin-$1/2$ 1D ladder. As
illustrated in Fig. 1, we encode the spin and lattice-site degrees of freedom
into the combined synthetic dimensions of hyperfine and momentum states of a
Bose-Einstein condensate of 87Rb atoms. Adopting a Raman momentum-lattice
(RML) construction, we enforce the ladder geometry and gauge couplings through
a series of appropriately designed Raman and Bragg couplings. The form and
strength of the synthetic gauge field are conveniently tuned by manipulating
the amplitude and phase of the coupling lasers. We first demonstrate the non-
Abelian nature of the gauge couplings through the non-Abelian Aharonov-Bohm
(AB) effect [39, 24, 40, 41, 42]. Specifically, focusing on a single plaquette
in the ladder, we sequentially switch on couplings to drive the atoms hopping
around the plaquette in either the clockwise or counterclockwise direction.
When the atoms come back to the starting position, we find that their final
states are path-dependent in general, which can be understood as the non-
Abelian counterpart of the well-known AB effect. We then systematically
characterize the chiral dynamics by probing the chiral current, whose
amplitude and direction are sensitive to the parameters of the gauge field. By
decomposing the synthetic lattice into two sets of overlaying flux ladders, we
reveal that the observed rich chiral dynamics derives from the competition
between the flux ladders, as each set hosts a unique spin-dependent chiral
current. Our experiment constitutes a natural starting point for studying the
interplay of non-Abelian gauge fields, lattice geometry, disorder, and many-
body interactions, all of which are independently tunable in the RML.

Figure 2: Schematic illustrations of the loop operators for the clockwise (a)
and counterclockwise (b) rotations. Both operators start and end at the
bottom-left corner of the plaquette. (c) and (d) show the numerically
simulated results for the final states following the actions of
$W_{\text{cw}}$ and $W_{\text{ccw}}$, respectively for the Abelian
($\theta=0,\varphi=0$) and non-Abelian cases
($\theta=\pi/4,\phi=-\pi/6,\varphi=0$). The experimental parameters are taken
along the black dashed lines. (e) and (f) show the experimental results of the
polarization ratio $\rho$ for different initial states characterized by
various $\beta$ and a fixed $\alpha=\pi/2$. Blue (green) dots and lines are
the corresponding experimental and numerical results for the clockwise
(counter-clockwise) rotations. The error bars represent the standard deviation
of the mean.  Figure 3: Spin-dependent chiral dynamics. (a) The spin-1/2
ladder in our experiment. The condensate is initially prepared in a
superposition of sites $\\{0,0\\}$ and $\\{0,1\\}$ and in the spin-down states
(shaded in red). (b) Time evolution of the condensate population with
$\theta=\pi/4,\phi=\pi/6,\varphi=\pi/2$, and $J=h\times 0.50(2)~{}\text{kHz}$.
(c) The average displacements $\langle n_{0,1}\rangle$ along the $m=0$ leg
(red) and $m=1$ leg (blue), extracted from (b). The black dashed line
indicates $t=0.35~{}\text{ms}$, as used in (d)(e). (d)(e) The chirality
$\langle n_{c}\rangle$ versus $\varphi$ at $t=0.35~{}\text{ms}$ for
$\theta=\pi/3,\phi=0$ (d) and $\theta=\pi/3,\phi=\pi/2$ (e), respectively. The
blue (red) dots represent the chiral displacement of the spin-down (spin-up)
states, while the green dots are the overall chiral displacement. In (c-e),
diamonds (or dots) are the experiment results and the solid lines are from
numerical simulations. Error bars stand for the standard deviation of the
mean.

Experimental implementation and model Hamiltonian. We experimentally implement
the RML following the schematics illustrated in Fig. 1. Two sets of Bragg
lasers, with wavelengths $\lambda_{1}=794.7~{}\text{nm}$ and
$\lambda_{2}=1064~{}\text{nm}$ respectively, couple discrete momentum states
in the $F=1$ and $F=2$ ground-state hyperfine manifolds. The coupled discrete
momenta are $2n\hbar k_{1}$ along the synthetic $x$ axis, and $2m\hbar k_{2}$
along the $y$ axis, respectively, where $k_{1,2}=2\pi/\lambda_{1,2}$, and
$n,~{}m\in\mathbb{Z}$. To couple states in different hyperfine manifolds, we
introduce Raman-Bragg processes, as illustrated in the inset of Fig. 1(a).
Mapping different momentum states to lattice sites in the two-leg ladder
(labeled by $\\{n,m\\}$), and the two internal hyperfine degrees of freedom to
the spins, we thus realize a spin-$1/2$ ladder in the synthetic dimensions of
the atoms. By adjusting the parameters of the coupling lasers, a tunable SU(2)
lattice gauge field is achieved, as depicted in Fig. 1(c) [43].

The system can be described by the following effective Hamiltonian

$\displaystyle
H=\sum_{n,m}J(\hat{a}^{\dagger}_{n+1,m}U_{m}^{(x)}\hat{a}_{n,m}+\hat{a}^{\dagger}_{n,m+1}U_{n}^{(y)}\hat{a}_{n,m})+\rm{H.c},$
(1)

where $J$ is the laser-induced tunneling amplitude,
$\hat{a}_{n,m}=(\hat{a}_{n,m,\uparrow},\hat{a}_{n,m,\downarrow})^{T}$ and
$\hat{a}_{n,m}^{\dagger}=(\hat{a}^{\dagger}_{n,m,\uparrow},\hat{a}^{\dagger}_{n,m,\downarrow})^{T}$
are the spinor annihilation and creation operators on the $\\{n,m\\}$ site,
respectively. The hopping between two adjacent sites along the $x$ axis is
accompanied by a spin-state rotation given by $U^{(x)}$, with

$\displaystyle U_{x}(\theta,\phi)=\begin{pmatrix}\cos\theta&i\sin\theta
e^{i\phi}\\\ i\sin\theta e^{-i\phi}&\cos\theta\end{pmatrix}.$ (2)

In our experiment, we take $U^{(x)}_{0}=U_{x}(\theta,\phi)$ and
$U^{(x)}_{1}=U_{x}(\theta,-\phi)$ by setting the coupling parameters. Along
the $y$ axis, we impose $U^{(y)}_{n}=U_{y}(n\varphi)=e^{in\varphi}\mathbb{I}$,
where $\mathbb{I}$ is an identity $2\times 2$ matrix. While $U_{x}$ and
$U_{y}$ conspire to give a synthetic SU(2) gauge field, both the form and
strength of the gauge field can be easily controlled by adjusting the
parameters of $\theta,\phi$, and $\varphi$. Under the condition
$\\{\theta=n\pi\\}$, or $\\{\phi=m\pi\\}$, or
$\\{\theta=(2n+1)\pi/2\,\,\text{and}\,\,\phi=(2m+1)\pi/2\\}~{}(n,m\in\mathbb{N})$,
the gauge field becomes Abelian. Hence, by controlling the coupling-laser
parameters, the Abelian or non-Abelian nature of the gauge field can be tuned
conveniently [43].

Figure 4: Dynamic regimes of the chiral current. (a-c) The SU(2) ladder in (b)
can be decomposed into two sets of overlaying flux ladders. When $\theta=0$,
the SU(2) ladder is reduced to a pair of decoupled flux ladders with different
spin species. We illustrate the spin-down ladder in (a), corresponding to the
blue-shaded horizontal plane in (b). When $\theta=\pi/2$, the SU(2) ladder is
reduced to a decoupled pair of zig-zag ladders with alternating spins on
adjacent sites. We illustrate one of them in (c), which is shaded in red in
(b). (d-g) show the chiral dynamic regimes in the plane of $\phi$ and
$\varphi$, under different values of $\theta$. For each panel, the upper-left
subfigure displays the numerical result of $\langle n_{c}\rangle$, and the
upper-right one shows the corresponding experimental result. The lower
subfigures are experimental data for $\langle n^{\downarrow}_{c}\rangle$ and
$\langle n^{W}_{c}\rangle$, respectively. The tunneling strength is set at
$J=h\times 0.50(2)~{}\text{kHz}$, and the time of measurement is fixed at
$t=0.35~{}\text{ms}$ for all cases.

Non-Abelian AB effect. Under the AB effect, a charged particle moving around a
closed loop acquires a geometric phase as a result of the U(1) gauge potential
along the path, which is responsible for the magnetic flux penetrating the
loop. However, the situation is quite different when the particles are
subjected to a non-Abelian gauge field. To see this, we focus on a single
plaquette, and define the loop operators of the clockwise (cw) or
counterclockwise (ccw) paths, following the effective Hamiltonian (1)

$\displaystyle W_{\text{cw}}$
$\displaystyle=U_{x}^{\dagger}(\theta,\phi)U_{y}^{\dagger}((n+1)\varphi)U_{x}(\theta,-\phi)U_{y}(n\varphi),$
(3) $\displaystyle W_{\text{ccw}}$
$\displaystyle=U_{y}^{\dagger}(n\varphi)U_{x}^{\dagger}(\theta,-\phi)U_{y}((n+1)\varphi)U_{x}(\theta,\phi).$
(4)

Under an Abelian gauge potential, $W_{\text{cw}}=e^{-i\varphi}\mathbb{I}$ and
$W_{\text{ccw}}=e^{i\varphi}\mathbb{I}$. Atoms going around a closed loop via
different paths would accumulate geometric phases with opposite signs. In the
non-Abelian scenario, however, the two loop operators are different: atoms
going around the plaquette undergo an internal-state rotation that is path-
dependent.

For our experiment, we initialize the atoms in site $|n=0,m=0\rangle$
(referred to as $|0,0\rangle$ in the following) with variable spin states

$\displaystyle|\psi_{\text{ini}}\rangle=\left(i\sin\frac{\alpha}{2}e^{i\beta}\left|\uparrow\right\rangle+\cos\frac{\alpha}{2}\left|\downarrow\right\rangle\right)\otimes|0,0\rangle.$
(5)

Here $\alpha$ determines the spin composition and $\beta$ is the relative
phase. We employ a sequence of four $\pi$-pulses to implement the loop
operators. The final states are denoted as
$|\psi^{\text{cw}}_{\text{fin}}\rangle=W_{\text{cw}}|\psi_{\text{ini}}\rangle$
and
$|\psi^{\text{ccw}}_{\text{fin}}\rangle=W_{\text{ccw}}|\psi_{\text{ini}}\rangle$
for the clockwise and counter-clockwise loops, respectively. To characterize
the internal-state response, we define the polarization ratio
$\rho=|\langle\uparrow|\psi_{\text{fin}}\rangle|^{2}-|\langle\downarrow|\psi_{\text{fin}}\rangle|^{2}$.
Under an Abelian gauge field, $\rho^{\text{cw}}=\rho^{\text{ccw}}$, regardless
of the initial state [as shown in Fig. 2(c)]. Whereas under a non-Abelian
gauge potential, $\rho^{\text{cw}}\neq\rho^{\text{ccw}}$ for most initial
states [see Fig. 2(d)]. These results are confirmed by the experimental
observation for $\alpha=\pi/2$. In Fig. 2(e), we set $\theta=0,\varphi=0$, and
observe $\rho^{\text{cw}}=\rho^{\text{ccw}}=0$ for both the clockwise and
counter-clockwise loops. Conversely, under a non-Abelian gauge field, the
polarization ratio $\rho$ exhibits an apparent dependence on the path
direction, as well as on the relative phase $\beta$ of the initial state [see
Fig. 2(f)].

Chiral dynamics under the Non-Abelian gauge field. We are now in a position to
discuss the dynamic consequence of the SU(2) gauge field. In previous studies
of flux ladders, an outstanding feature is the chiral current: particles flow
toward opposite directions along the two legs, with the overall chirality
dependent on the sign of the flux. Here we show that, when the form of the
gauge potential is continuously tuned, both the amplitude and direction of the
chiral current are significantly modified, as the system exhibits multiple
dynamic regimes.

We experimentally study the chiral dynamics on a 1D ladder of $2\times 5$
sites. As illustrated in Fig. 3(a), this corresponds to $m\in\\{0,1\\}$ and
$n\in\\{-2,-1,0,1,2\\}$. The condensate is initialized in the spin-down state,
with
$|\psi(t=0)\rangle=\left|\downarrow\right\rangle\otimes\frac{\sqrt{2}}{2}(|0,0\rangle+|0,1\rangle)$.
Under a typical non-Abelian gauge field with
$(\theta=\pi/4,\phi=\pi/6,\varphi=\pi/2)$, the atomic flow shows a shearing
behavior, as shown in Fig. 3(b). As time evolves, the atom population displays
a directional propagation in both legs, where atoms in the $m=0$ leg flow to
the left and those in the $m=1$ leg to the right.

To better visualize the directional flow, we define the average displacement
for $m$-th leg

$\displaystyle\langle
n_{m}(t)\rangle=\sum_{n}n(|\psi_{n,m,\uparrow}(t)|^{2}+|\psi_{n,m,\downarrow}(t)|^{2}),$
(6)

where $\psi_{n,m,\sigma}(t)=\langle\psi(t)|\sigma\rangle\otimes|n,m\rangle$
($\sigma\in\\{\uparrow,\downarrow\\}$). The corresponding experiment results
are shown in Fig. 3(c), where atomic currents are opposite in direction on the
two legs. Note that for $t\gtrsim 0.6$ ms, the atoms reach the lattice
boundary, and the directional flow diminishes. Based on this observation, we
characterize the chirality of the flow using the chiral displacement $\langle
n^{\sigma}_{c}\rangle=\langle n^{\sigma}_{1}\rangle-\langle
n^{\sigma}_{0}\rangle$ at $t=0.35$ ms, when the atoms are still in the bulk.
The presence of chirality in spin component $\sigma$ is reflected in a non-
vanishing $n^{\sigma}_{c}$, with the chirality given by its sign. Likewise,
the overall chirality in the atomic density flow is given by $\langle
n_{c}\rangle=\langle n^{\uparrow}_{c}\rangle+\langle
n_{c}^{\downarrow}\rangle$.

In Fig. 3(d), we show the measured $n_{c}$ and $n_{c}^{\sigma}$ for
$\theta=\pi/3,\phi=0$, where the gauge field is reduced to an Abelian form.
The synthetic flux through each plaquette is given by $\varphi$. Consistent
with previous results of a flux ladder [11, 9], we observe $\langle
n_{c}\rangle>0$ ($\langle n_{c}\rangle<0$) for $\varphi>0$ ($\varphi<0$),
indicating the change of chirality as the flux is reversed. However, when
$\theta=\pi/3,\phi=\pi/2$, as shown in Fig. 3(e), $\langle n_{c}\rangle$ is
reversed compared to that in Fig. 3(d), and the chiral current is dominated by
that of the spin-down species. This suggests that, under the non-Abelian gauge
field, chiral dynamics becomes spin-dependent, and the overall chirality can
be flipped.

To understand the rich chiral dynamics of the system, let us examine two
limiting cases. First, when $\theta=0$, the gauge field becomes Abelian, and
the two spin components are decoupled, as the system is reduced to two
independent flux ladders [see Fig. 4(a)(b)]. In this case, the measured chiral
dynamics depends only on $\varphi$, consistent with that of a simple flux
ladder [see Fig. 4(d)]. Second, when $\theta=\pi/2$, the system is again
reduced to two decoupled flux ladders, each consisting of alternating spin
species along the ladder [see Fig. 4(b)(c)]. For a condensate initially
prepared in the spin-down state, the ensuing dynamics only involves a zig-zag
flux ladder composed of the red-shaded plaquettes in Fig. 4(b). To
characterize the chirality of the dynamics on such a zig-zag ladder, we define

$\displaystyle\langle n_{c}^{\text{w}}\rangle$
$\displaystyle=\sum_{n}[2n|\psi_{2n,1,\downarrow}|^{2}+(2n+1)|\psi_{2n+1,1,\uparrow}|^{2}$
$\displaystyle-2n|\psi_{2n,0,\downarrow}|^{2}-(2n+1)|\psi_{2n+1,0,\uparrow}|^{2}],$
(7)

As shown in Fig. 4(g), multiple distinct dynamic regimes emerge in the
parameter space of $\varphi$ and $\phi$. There are two kinds of boundaries
here. One is given by $(\varphi=0,\pi)$, where the flux in Fig. 4(c) becomes
staggered. In this case, the chiralities of adjacent plaquettes cancel out
[44, 45], leading to the corresponding boundaries observed in Fig. 4(g). The
rest of the boundaries are pinned by the points $(\phi=0,\varphi=\pm\pi)$ and
$(\phi=\pm\pi/2,\varphi=0)$, where the chirality flips as the fluxes cross
$\pi$ or $-\pi$.

Finally, for $\theta$ taking values in between $0$ and $\pi/2$, the chiral
dynamics can be understood as the result of the competition between the two
sets of flux ladders. Indeed, Fig. 4(e)(f) show the transition from one limit
to the other. The boundary of $\varphi=n\pi$ persists for all values of
$\theta$, while $\phi$ plays an increasingly important role as $\theta$
changes from $0$ to $\pi/2$, giving rise to the rich chiral dynamics under a
general SU(2) gauge field.

Discussion. To summarize, we demonstrate the spin-dependent chiral dynamics in
a 1D ladder under an SU(2) synthetic gauge field. The flexibility of our Raman
momentum-lattice construction provides opportunities to study synthetic non-
Abelian gauge fields in higher dimensions, or topological models and the
associated topological phenomena where various forms of the non-Abelian gauge
fields play a key role. These include the non-Abelian Hofstadter model [24,
46, 47, 40, 48], the higher-order topological insulators [25, 49, 32], and the
quantum (spin) Hall effect [50]. It also builds a foundation for studying the
interplay of many-body interaction and non-Abelian gauge fields, with the
exciting possibility of probing intrinsic topological order and fractional
excitations on the highly tunable platform of momentum lattices.

###### Acknowledgements.

Acknowledgement: We acknowledge the support from the National Key Research and
Development Program of China under Grant No.2022YFA1404203 and No.
2023YFA1406703, The National Natural Science Foundation of China under Grant
No. U21A20437, No. 12074337, No. 11974331, and No. 12374479, Natural Science
Foundation of Zhejiang Province under Grant No. LR21A040002, Zhejiang Province
Plan for Science and Technology Grant No. 2020C01019, and the Fundamental
Research Funds for the Central Universities under Grant No. 2021FZZX001-02 and
226-2023-00131, the China Postdoctoral Science Foundation under Grant No.
2023M733122.

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MIDLMedical Imaging with Deep Learning 2019 MIDL 2019 – Extended Abstract
Track Pedro M. Gordaliza1<EMAIL_ADDRESS>
Juan José Vaquero1<EMAIL_ADDRESS>
Sally Sharpe2<EMAIL_ADDRESS>
Fergus Gleeson3<EMAIL_ADDRESS>
Arrate Muñoz-Barrutia1<EMAIL_ADDRESS>
1 Dpto.Bioingenieria e Ing. Aeroespacial, Universidad Carlos III de Madrid,
Leganés, Spain

2 Microbiology Services Division, Public Health England, Porton Down, England

3 Oncology Department, The Churchill Hospital, Headington, Oxford, UK

# A Multi-Task Self-Normalizing 3D-CNN to Infer Tuberculosis Radiological
Manifestations

###### Abstract

We propose a learning method well-suited to infer the presence of Tuberculosis
(TB) manifestations on Computer Tomography (CT) scans mimicking the
radiologist reports. Latent features are extracted from the CT volumes
employing the V-Net encoder and those are the input to a Feed-Forward Neural
Network (FNN) for multi-class classification. To overtake the issues (e.g.,
exploding/vanishing gradients, lack of sensibility) that normally appear when
training deep 3D models with datasets of limited size and composed of large
volumes, our proposal employs: 1) At the network architecture level, the
scaled exponential linear unit (SELU) activation which allows the network
self-normalization, and 2) at the learning phase, multi-task learning with a
loss function weighted by the task homoscedastic uncertainty. The results
achieve $F_{1}$-scores close to or above $0.9$ for the detection of TB lesions
and a Root Mean Square Error of $1.16$ for the number of nodules.

###### keywords:

Self-Normalizing Neural Networks, multi-task, homoscedastic uncertainty

††editors: Under Review for MIDL 2019

##  1 Introduction

Tuberculosis (TB) is an infectious disease which generally affects the lungs
and has a high incidence and mortality [World Health Organization and
others(2018)]. Due to the severity of the pandemic, the World Health
Organization (WHO) has launched an ambitious plan to eradicate TB by 2030, for
which, the extraction of sensitive radiological biomarkers [Nachiappan et
al.(2017)Nachiappan, Rahbar, Shi, Guy, Mortani Barbosa Jr, Shroff, Ocazionez,
Schlesinger, Katz, and Hammer] is a clear need. Traditionally, radiologists
through visual inspection of x-ray Computed Tomography (CT) volumes generate
reports summarizing the presence of TB-related manifestations. However, using
this approach for the extraction of robust TB radiological biomarkers, it is
unfeasible and automation required. In recent years, the introduction of deep
learning techniques has drastically contributed to this task [Wang et
al.(2017)Wang, Peng, Lu, Lu, Bagheri, and Summers, Litjens et
al.(2017)Litjens, Kooi, Bejnordi, Setio, Ciompi, Ghafoorian, van der Laak, van
Ginneken, and Sánchez, Hinton(2018)]. For deep learning, knowledge is usually
injected into the model in the form of manually segmented masks of the lung
lesions. Instead, our proposal directly employs the expertise acquired by the
radiologist through years of clinical practice as synthesized in tabular
reports.

##  2 Material and Methods

Chest CT scans ($56$) acquired from $14$ male Cynomolgus macaques at $3$, $7$,
$11$ and $16$ weeks after TB aerosol exposure were employed. First, the CT
volumes were cropped [Gordaliza et al.(2018)Gordaliza, Muñoz-Barrutia, Abella,
Desco, Sharpe, and Vaquero] and resampled to $1\mbox{mm}\times
1\mbox{mm}\times 2\mbox{mm}$. During the training, data augmentation is
performed (elastic transformation, addition of Gaussian noise). As labelled
data, we employ the tabular reports elaborated by a radiologist ($20$ years
experienced) that contain the number of detected nodules ($0-15$) and binary
annotations indicating the presence per lung lobe of the most common TB
manifestations (e.g., cavitations, conglomerations, consolidations and trees
in bud) [Nachiappan et al.(2017)Nachiappan, Rahbar, Shi, Guy, Mortani Barbosa
Jr, Shroff, Ocazionez, Schlesinger, Katz, and Hammer].

###  2.1 Model Architecture

Our implementation (fig:architecture) extracts latent features from the
volumes employing the V-Net [Milletari et al.(2016)Milletari, Navab, and
Ahmadi] encoder. The extracted features are used as input of the Feed-Forward
Neural Networks (FNNs) for multi-class classification. The encoder generates
$1,376.256$ features, which feed FNN1 (task-shared). The outputs of FNN1 are
employed as the input of two independent FNNs, FNNR and FNNB, corresponding
with regression (nodule counting) and binary tasks. Dropout or Batch
Normalization (BN) [Ioffe and Szegedy(2015)] layers are included where is
needed. When employing SNN, BN is unnecessary.

fig:architecture

Figure 1: 3D-CNN $+$ three FNNs (Feed Forward neural networks): $FNN_{1}$
(tasks-shared parameters) and, $FNN_{R}$ and $FNN_{B}$ for prediction of
regression and binary tasks. BN is not present with SELU.

###  2.2 Self-Normalizing Neural Networks

For regularization and normalization purposes, we apply the Self-Normalizing
Neural Networks (SNNs) strategy to our model [Klambauer et al.(2017)Klambauer,
Unterthiner, Mayr, and Hochreiter] that preserves the activation magnitude
close to zero mean and unit variance. As activation function, we use the
Scaled Exponential Linear Unit (SELU) and as dropout, the $\alpha$-dropout
[Klambauer et al.(2017)Klambauer, Unterthiner, Mayr, and Hochreiter].

###  2.3 Learning Principle: Uncertainty Weighted Multi-task Loss

When working in multi-task classification, the decision on the influence of
each task in the final loss is non-trivial. Traditionally, the loss is
computed as $\mathbf{\mathcal{L}}=\sum_{i}w_{i}\mathcal{L}_{i}$, where the
weights $w_{i}$ are selected either manually or after an exhaustive grid
search. This approach is highly influenced by the units and the scale of each
task and extremely time-consuming. Recently, [Kendall et al.(2018)Kendall,
Gal, and Cipolla] et al. proposed a method to compute the weights guiding each
specific task loss by homoscedastic uncertainty of the predictions. In this
work, we adapt this approach to our multi-label classification problem and
derive a loss function for our model (fig:architecture).

##  3 Experiments and Results

For the evaluation, we employed $5$-fold Cross-Validation (CV): Validation set
($4$ CT volumes of one subject); training set (remaining $13$ subjects). We
compared the proposed model (SELU) and a modified version which employs a
Parametric Rectified Linear Unit (PReLU), Batch Normalization and standard
dropout (BN+PReLU). The models are trained with $10.000$ iterations of the
ADAM optimizer [Kingma and Ba(2014)] (learning rate$=10^{-5}$, mini-batch
size$=15$). For BN+PReLU, standard parameters were employed for the ADAM
optimizer and a dropout rate of $0.5$. For SELU: $\beta_{2}=0.9$,
$\epsilon=0.01$ and alpha-dropout$=0.1$ [Klambauer et al.(2017)Klambauer,
Unterthiner, Mayr, and Hochreiter]. fig:loss_chart shows for the validation
data, the loss at each fold for the BN+PReLU (in red) and the proposed model
(in blue). It can be observed that SELU presents improved convergence. The
inference error is estimated by the Root Mean Square Error (RMSE) for the
nodules count tasks and the $F_{1}$-score for the twenty binary tasks.
tab:Fold_results presents the results per fold. No significant statistical
differences were found for paired t-test ($p\nleq 0.05$). Nevertheless, the
proposed model presents better RMSE and $F_{1}$-score results.

2gray!6white tab:Fold_results

Table 1: Performance measure results the for BN+PReLU and the proposed model
(SELU).

[HTML]9B9B9B [HTML]343434[HTML]FFFFFF Manifestation/ |  [HTML]9B9B9BNodules [RMSE] |  [HTML]9B9B9B[HTML]343434 Cavitations [$F_{1}$] |  [HTML]9B9B9B[HTML]343434 Conglomeration [$F_{1}$] |  [HTML]9B9B9B[HTML]343434 Consolidation [$F_{1}$] |  [HTML]9B9B9BTree in bud [$F_{1}$]
---|---|---|---|---|---
[HTML]343434[HTML]FFFFFF Fold |  [HTML]C0C0C0BN+PReLU |  [HTML]EFEFEFSELU |  [HTML]C0C0C0BN+PReLU |  [HTML]EFEFEFSELU |  [HTML]C0C0C0BN+PReLU |  [HTML]EFEFEFSELU |  [HTML]C0C0C0BN+PReLU |  [HTML]EFEFEFSELU |  [HTML]C0C0C0BN+PReLU |  [HTML]EFEFEFSELU
[HTML]9B9B9B[HTML]343434 1 | $0.73_{0.84}$ | $0.85_{0.35}$ | $0.88_{0.11}$ | $0.88_{0.12}$ | $0.90_{0.13}$ | $0.92_{0.12}$ | $0.88_{0.18}$ | $0.83_{0.22}$ | $0.83_{0.22}$ | $0.79_{0.23}$
[HTML]EFEFEF [HTML]9B9B9B[HTML]343434 2 | $1.15_{0.89}$ |  [HTML]EFEFEF$1.09_{0.83}$ | $0.86_{0.23}$ |  [HTML]EFEFEF$0.88_{0.22}$ | $0.94_{0.17}$ |  [HTML]EFEFEF$0.93_{0.18}$ | $0.93_{0.15}$ |  [HTML]EFEFEF$0.93_{0.18}$ | $0.97_{0.08}$ | $0.97_{0.08}$
[HTML]9B9B9B[HTML]343434 3 | $0.41_{0.34}$ | $0.23_{0.39}$ | $0.85_{0.12}$ | $0.87_{0.11}$ | $0.89_{0.19}$ | $0.97_{0.11}$ | $0.96_{0.11}$ | $0.98_{0.03}$ | $0.95_{0.14}$ | $0.96_{0.12}$
[HTML]EFEFEF [HTML]9B9B9B[HTML]343434 4 | $1.22_{0.6}$ |  [HTML]EFEFEF$0.78_{0.74}$ | $0.94_{0.15}$ |  [HTML]EFEFEF$0.9_{0.19}$ | $0.9_{0.19}$ |  [HTML]EFEFEF$0.92_{0.15}$ | $0.88_{0.18}$ |  [HTML]EFEFEF$0.87_{0.18}$ | $0.87_{0.21}$ | $0.94_{0.15}$
[HTML]9B9B9B[HTML]343434 5 | $0.41_{0.8}$ | $0.45_{0.8}$ | $0.93_{0.17}$ | $0.94_{0.17}$ | $0.94_{0.14}$ | $0.96_{0.12}$ | $0.90_{0.18}$ | $0.94_{0.14}$ | $0.91_{0.18}$ | $0.92_{0.17}$
[HTML]EFEFEF [HTML]9B9B9B[HTML]343434 Total | $0.78_{0.69}$ |  [HTML]EFEFEF$\mathbf{0.68_{0.62}}$ | $0.89_{0.16}$ |  [HTML]EFEFEF$\mathbf{0.90_{0.17}}$ | $0.91_{0.16}$ |  [HTML]EFEFEF$\mathbf{0.94_{0.16}}$ | $0.91_{0.16}$ |  [HTML]EFEFEF$0.91_{0.15}$ | $0.91_{0.17}$ | $\mathbf{0.92_{0.15}}$

fig:loss_chart

Figure 2: Evolution of the validation loss for each fold of the Cross-
Validation.

##  4 Conclusions

Although further validation on large datasets is needed, the work presents a
promising inference of the radiologist reports. This is achieved with a
reduced computational complexity by avoiding normalization layers and
hyperparameter tuning of the loss weights.

The research leading to these results received funding from the Innovative
Medicines Initiative (www.imi.europa.eu) Joint Undertaking under grant
agreement no. 115337, whose resources comprise funding from EU FP7/2007-2013
and EFPIA companies in kind contribution. This work was partially funded by
projects RTC-2015-3772-1, TEC2015-73064-EXP and TEC2016-78052-R from the
Spanish Ministry of Economy, Industry and Competitiveness (MEIC), TOPUS
S2013/MIT-3024 project from the regional government of Madrid and by the
Department of Health, UK. This material is based upon work supported by Google
Cloud Platform.

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