# sections and Chapters

Meson-$\Delta$ and meson-nucleon-$\Delta$ transition coupling constants in the
soft-wall model of holographic QCD at finite temperature

Narmin Nasibova 111Corresponding author<EMAIL_ADDRESS>

Institute of Physics, National Academy of Sciences,
AZ-1148, H.Cavid ave. 131, Baku, Azerbaijan

ABSTRACT

In this paper, minimal coupling constants of meson-$\Delta$ baryon and meson-
nucleon to $\Delta$ baryon transition have been investigated in the AdS/QCD
soft-wall model at finite temperature limits. Initially, the right and left
profile functions of $\Delta$ baryons have been calculated using the Rarita-
Schwinger field in this model. After this, coupling constants have been
written by taking into account the profile functions of thermal hadrons
according to the zero-temperature case. Then, the temperature dependence
graphs are plotted for strong couplings $g_{\rho\Delta\Delta}(T)$,
$g_{a_{1}\Delta\Delta}(T)$ and $g_{\rho N\Delta}(T)$. As a result, it has been
observed by an increase in the temperature, values of the minimal coupling
constants decreased and approached zero nearby Hawking temperature.

Key words: AdS/CFT, holographic duality, soft-wall model, meson, baryon,
coupling constant.

##  1 Introduction

One notable application of string theory is the anti-de Sitter/conformal field
theory (AdS/CFT) 1 ; 2 ; 3 ; 4 ; 5  or holographic duality, which is used to
study the coupling regime of quantum field theories. This theory is also
called gravity/gauge duality. This principle is based on closed and open
string duality according to the relationship between the open and closed
strings. As such, in this principle, the gravitational field corresponding to
the closed string is defined by the boundary and the field theory
corresponding to the open string by the bulk of the AdS space. QGP (Quark
Gluon Plasma), condensed matter systems, and other QCD problems which is
impossible to solve by helping perturbative theory have been investigated
using this duality. The duality between gravitational and gauge theories is
the strong-weak duality, which also was adapted to describe the low- and high-
energy dynamics of Quantum Chromodynamics. It is also important to investigate
hadron interaction processes in the nuclear medium in a proton-proton and
heavy ion collisions in order to study phase transition of QGP and evolution
of the early Universe.

AdS/QCD (AdS/Quantum Chromodynamics) which was created based on this duality
has two hard- and soft-wall models for modifying the metric in the AdS space-
time. These models are widely applied in particle physics for quantifying
phenomenological prediction quantities and investigating other strong
interaction problems.

In theoretical physics, many QCD problems were solved by holographic QCD
models in the vacuum and low-temperature medium in Refs. 6 ; 7 ; 8 ; 9 ; 10 ;
11 ; 12 ; 13 ; 14 . It has been investigated the derivation of analytical
formulas for the mass spectrum of mesons and $\Delta$ baryons and their
coupling constants in Ref. 15 ; 16 . In addition, it has been found reasonable
results for the transition and electromagnetic form factors of nucleons in the
soft-wall model of holographic QCD at finite temperature. Continuing these
investigations In Ref. 17  we have studied the temperature dependence of
vector meson coupling constant in the soft-wall model of AdS/QCD. It was
observed the coupling constant of $\rho$ vector meson decreases on temperature
increases and vanishes at the temperature close to the confinement-
deconfinement phase transition temperature.

It is interesting to check whether this situation takes place for other meson
sectors of the model as well. As a continuation of Ref. 17  we have decided to
study, for the simplicity, minimal coupling constant of meson-baryon and
meson-nucleon to $\Delta$ baryon transitions based on the case reviewed in Ref
18  at finite temperature. Our aim here is to study temperature dependence in
the spectrum of spin 3/2 baryons ($\Delta$ resonances) and the coupling
constants between mesons and baryons in the soft-wall model of holographic QCD
at finite temperature.

The remainder of this paper is organized as follows: Sections II and III are
about the soft-wall model and the breaking of chiral symmetry at finite
temperature. In Sec. IV., the profile functions of $\Delta$ baryons have been
obtained at finite temperature in the bulk of AdS space-time. In Sec.V, we
construct the Lagrangian for the vector-spinor interaction in the bulk and
obtain the temperature-dependent integral expression for the the minimal
coupling constants of $g_{\rho\Delta\Delta}(T)$, $g_{a_{1}\Delta\Delta}(T)$
and $g_{\rho N\Delta}(T)$ using holographic correspondence at the boundary of
QCD. In Sec. VI., the free parameters are fixed and the graphs of strong
coupling constants are plotted at the different values of the quark flavor
parameter $N_{f}$. The last section is devoted to discussion and conclusions.

##  2 Thermal SOFT-WALL MODEL of holographic QCD

The Schwarzschild metric can be written as follows 15 :

$\displaystyle
ds^{2}=e^{2A(z)}\left[f(z)dt^{2}-\left(d\vec{x}\right)^{2}-\frac{dz^{2}}{f(z)}\right],$
$\displaystyle f(z)=1-\frac{z^{4}}{z_{H}^{4}},$ (1)

The relation between tortoise coordinate $r$ and $z$ is as follows:

$r=\int\frac{dz}{f(z)}$

İn the finite temperature limit the r coordinate can be expanded as

$r\approx z\left[1+\frac{z^{4}}{5z_{H}^{4}}+\frac{z^{8}}{9z_{H}^{8}}\right].$
(2)

So, AdS-Schwarzschild space time metric in the Regge-Wheeler (RW) tortoise
coordinate $r$ 15  will be written as follows:

$ds^{2}=e^{2A(r)}f^{\frac{3}{5}}(r)\left[dt^{2}-\frac{\left(d\vec{x}\right)^{2}}{f(r)}-dr^{2}\right]$
(3)

here $A(r)=log(\frac{R}{r})$ and the thermal factor $f(r)$ has the following
form:

$f(r)=1-\frac{r^{4}}{r_{H}^{4}}.$ (4)

where, $r_{H}$ \- is the position of the event horizon. It is related to the
Hawking temperature as $T=1/(\pi r_{H})$, $x=(t,\vec{x})$ is the set of
Minkowski coordinates, $A(r)=log(\frac{R}{r})$, and $R$ is the AdS space
radius. $k$ is a scale parameter.

The AdS-Schwarzchild geometry is more suitable at high $T$, while at small $T$
this metric can be also used by making a small $T$-expansion. The limit $T=0$
corresponds to a mapping of the AdS-Schwarzchild geometry onto AdS Poincare
metric and the small $T$-behavior of hadron properties can be generated in the
formalism based on AdS Poincare metric and with the use of a thermal dilaton.
It leads to equivalent results: AdS-Schwarzchild geometry with small $T$ is
equal to AdS Poincare metric with thermal dilaton.

The dilaton field $\varphi$ has following form:

$\varphi(r,T)=K^{2}(T)r^{2},$ (5)

where

$K^{2}(T)=k^{2}\left[1+\rho(T)\right].$ (6)

So, $K^{2}(T)$ is the parameter in spontaneous breaking of chiral symmetry,
while the thermal function $\rho(T)$ up to $T^{4}$ order has a form:

$\rho(T)=\delta_{T_{1}}\frac{T^{2}}{12F^{2}}+\delta_{T_{2}}\left(\frac{T^{2}}{12F^{2}}\right)^{2}.$
(7)

Here F is the decay constant and there are relation between the coefficients
$\delta_{T_{1}}$ and $\delta_{T_{2}}$ and quark flavors $N_{f}$

$\delta_{T_{1}}=-\frac{N_{f}^{2}-1}{N_{f}},$ (8)

and

$\delta_{T_{2}}=-\frac{N_{f}^{2}-1}{2N_{f}^{2}}.$ (9)

##  3 CHIRAL symmetry breaking at finite temperature

In AdS/CFT correspondence, the currents in the boundary will correspond to the
bulk gauge fields at finite temperature. The scalar field $X$ transforms as a
bi-fundamental under the $SU(2)_{L}\times SU(2)_{R}$ group. The five-
dimensional mass $M^{2}_{5}=\Delta_{0}(\Delta_{0}-4)$ of the scalar which is
fixed with the scaling dimension $\Delta_{0}$. The solution for the bulk
scalar $X$ at finite temperature is defined as,
$\chi(r,T)\approx\frac{1}{2}am_{q}r+\frac{1}{2a}\Sigma(T)r^{3}=\vartheta(r,T)$
which the mass of quarks is $m_{q}$ and $\Sigma(T)$ is the value of thermal
chiral condensate and $a=\frac{\sqrt{N}_{c}}{2\pi}$. The conventions in that
the temperature dependence of $\Sigma(T)$ quark condensate is defined as
$\Sigma(T)=\Sigma[1+\Delta_{T}]$ and
$\Sigma(T)=<0|\bar{q}q|0>_{T}=-N_{f}B(T)F^{2}(T)$. In the chiral limit,
$N_{f}$ is the number of quark flavors, $B(T)$ is the quark condensate
parameter, and $F(T)$ is the pseudo-scalar meson decay constant in the chiral
limit at finite temperature. $F(T)$ and $B(T)$ has been studied and calculated
in 19 .

##  4 Meson Fields at finite temperature

In this section, the author derives the results for the couplings of hadrons
at low temperatures. First, the author calculates the meson profile function
at low temperatures using the universal action derived in Ref 15 . The
corresponding EOM (Equation of motion) for the Fourier transform of the bulk-
to-boundary profile function of mesons $\phi_{n}(r,T)$ in Euclidean metric is
as follows:

$\left[-\frac{d^{2}}{dr^{2}}+U(r,T)\right]\phi_{n}(r,T)=M_{n}^{2}(T)\phi_{n}(r,T).$
(10)

$U(r,T)$ is the effective potential and is consist of the temperature-
dependent and non-dependent parts:

$U(r,T)=U(r)+\Delta U(r,T).$ (11)

Explicit forms of the $U(r)$ and $\Delta U(r,T)$ terms were given as

$U(z)=k^{4}r^{2}+\frac{(4m^{2}-1)}{4r^{2}},$ (12) $\Delta
U(r,T)=2\Delta(T)k^{4}r^{2}.$ (13)

The meson mass spectrum $M_{n}^{2}$ is shown the following form by the sum of
zero and finite temperature parts:

$M_{n}^{2}(T)=\ M_{n}^{2}(0)+\Delta M_{n}^{2}(T),$ (14) $\Delta
M_{n}^{2}(T)=\rho(T)M_{n}^{2}(0)+\frac{R\pi^{4}T^{4}}{k^{2}},$ (15)
$M_{n}^{2}(0)=4k^{2}\left(n+\frac{m+1}{2}\right),$ (16)

$R=(6n-1)(m+1).$ In the low-temperature case, the hadronic mass spectrum is:

$M_{n}^{2}(T)=\ M_{n}^{2}(0)+\Delta M_{n}^{2}(T),$ (17) $\Delta
M_{n}^{2}(T)=\rho_{T}M_{n}^{2}(0)+\frac{R\pi^{4}T^{4}}{k^{2}},$ (18)
$R=(6n-1)(m+1).$ (19)

The profile function of mesons in general form is given 15  as follows:

$\phi_{n}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m+1)}}K(T)^{m+1}r^{m+\frac{1}{2}}e^{-\frac{K(T)^{2}r^{2}}{2}}L_{n}^{m}(K(T)^{2}r^{2}).$
(20)

For the meson with two quarks $m=L$. By taking m=1 for the vector and axial
vector mesons, we have replaced $\phi_{n}(r,T)$ by $M_{0}(r,T)$ as the meson
profile function in the expression of the coupling constant.

##  5 RARITA-SCHWINGER FIELDS AT FINITE TEMPERATURE

According to the principle of holographic duality, the fields set against the
$\Delta$ baryon operators and the nucleon operator with a defined spin 1/2 at
the boundary of the AdS space-time differ from each other. Thus, while the
Dirac field corresponds to nucleons with spin 1/2, the $\Delta$ baryon
operators with spin 3/2 are opposed to the Rarita-Schwinger fields $\Psi_{M}$
within this space-time. 20 ; 21 ; 22 ; 23 . The action for Rarita-Schwinger
field at finite temperature can be written correspond to zero temperature case
as follows:

$\int
d^{5}x\sqrt{G}\left(i\bar{\Psi}_{A}\Gamma^{ABC}D_{B}\Psi_{C}-m_{1}\bar{\Psi}_{A}\Psi^{A}-m_{2}\bar{\Psi}_{A}\Gamma^{AB}\Psi_{B}\right)\,,$
(21)

where $\Psi_{A}=e_{A}^{M}\Psi_{M}$ and we used notations
$\Gamma^{ABC}=\frac{1}{3!}\Sigma_{\rm{perm}}(-1)^{p}\Gamma^{A}\Gamma^{B}\Gamma^{C}=\frac{1}{2}(\Gamma^{B}\Gamma^{C}\Gamma^{A}-\Gamma^{A}\Gamma^{C}\Gamma^{B})$
and $\Gamma^{AB}=\frac{1}{2}[\Gamma^{A}\,,\Gamma^{B}]$. The Rarita-Schwinger
equations in $AdS_{5}$ are written as

$i\Gamma^{A}\Big{(}D_{A}\Psi_{B}-D_{B}\Psi_{A}\Big{)}-m_{-}\Psi_{B}+\frac{m_{+}}{3}\Gamma_{B}\Gamma^{A}\Psi_{A}=0$
(22)

where $m_{\pm}=m_{1}\pm m_{2}$. $m_{1}$ and $m_{2}$ correspond to those of
spinor harmonics on $S^{5}$ of $\text{AdS}_{5}\times S^{5}$ 24 .

As is well known, the Rarita–Schwinger field contains the states of the field
with spin 1/2 in addition to spin 3/2 components. In 4-dimensional space, the
components with spin 1/2 are eliminated by imposing the Lorentz condition on
this field:

$\gamma^{\mu}\Psi_{\mu}=0\,.$ (23)

The following Lorentz-covariant constraint will project out one of the
spin-1/2 components from the Rarita-Schwinger fields in five dimensional space
correspond to four dimensional space.

$e_{A}^{M}\Gamma^{A}\Psi_{M}=0\,,$ (24)

which then gives $\partial^{M}\Psi_{M}=0$ for a free particle if combined with
equations of motion.

This field has extra spin-1/2, $\Psi_{z}$, if decreased to four dimensional
space-time at finite temperature. By choosing $\Psi_{r}=0$, it is possible to
reduce the extra spin-1/2 degrees of freedom, because there is no extra spinor
at finite temperature.

$(iz\Gamma^{A}\Psi_{A}+2i\Gamma^{5}-m_{-})\Psi_{\mu}=0,$ (25)

$\quad(\mu=0,1,2,3)$

$\Psi_{M(R)}=\frac{1}{2}(1+\gamma^{5})\Psi_{NE}$ (26)
$\Psi_{M(L)}=\frac{1}{2}(1-\gamma^{5})\Psi_{M}$ (27)
$[\partial_{r}^{2}-\frac{2(m_{-}+K(T)^{2}r^{2})}{r}\partial_{r}+\frac{2(m_{-}-K(T)^{2}r^{2})}{r^{2}}+p^{2}]\Psi_{L}=0$
(28)
$[\partial_{r}^{2}-\frac{2(m_{-}+K(T)^{2}r^{2})}{r}\partial_{r}+p^{2}]\Psi_{R}=0$
(29)

The solution of these equations with the polynomial is similar to the fermion
field, as follows:

$\displaystyle
f_{n}^{L}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m_{L}+1)}}K^{m_{L}+1}r^{m_{L}+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m_{L}}\left(K^{2}r^{2}\right),$
$\displaystyle
f_{n}^{R}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m_{R}+1)}}K^{m_{R}+1}r^{m_{R}+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m_{R}}\left(K^{2}r^{2}\right),$
(30)

where $m_{L,R}=m\pm\frac{1}{2}$. To describe the spin 3/2 baryons ($\Delta$
resonances) in the soft-wall model at finite temperature one has to introduce
a pair of Rarita-Schwinger fields in the bulk, $\Psi_{1}^{A}$ (for the left-
handed spin-3/2) and $\Psi_{2}^{A}$ (for the right-handed spin-3/2), which
obey the above Rarita-Schwinger equations. The author has considered Ref 14
to describe the spin 1/2 baryons (nucleons) profile functions in the soft-wall
model at a finite temperature which obeys the Dirac equation in the bulk 16 .
The left-handed spin-1/2 baryon profile function is given by $F_{1}^{A}(r,T)$,
whenever the right-handed spin-1/2 baryon is given by $F_{2}^{A}(r,T)$. Note
that the profile functions of spin 1/2 and 3/2 baryon are the same in a
certain approximation. The profile functions of $\Delta$ baryon $f_{m}(r,T)$
and nucleon $F_{n}(r,T)$ obey normalization conditions as

$\int_{0}^{\infty}dre^{-\frac{3}{2}A(r)}f_{m}^{L,R}(r,T)F_{n}^{L,R}(r,T)=\delta_{mn}$
(31)

##  6 STRONG COUPLING CONSTANTS $g_{\rho\Delta\Delta}(T)$,
$g_{a_{1}\Delta\Delta}(T)$, AND $g_{\rho N\Delta}(T)$ AT FINITE TEMPERATURE

In this section the author interested in the properties of $\Delta$ resonances
in AdS/QCD at finite temperature. The resonance to nucleons of $\Delta$
baryons are very necessary in the study of nucleon potential, as the decay
channel of $\Delta$ resonance at finite temperature. Generally, the
interaction Lagrangian is constructed based on the gauge invariant of the
model and contains a term of minimal gauge interaction of the meson field with
the fermions corresponding zero temperature limit 18 . Meson-baryon transition
coupling constants have been obtained from the action by including the thermal
dilaton field in the action.

The thermal couplings have expressions of thermal profile functions of the
bulk fields. It has calculated the terms of thermal action in the momentum
space and taken the variation derivative from Lagrangian terms. This variation
gives us the following contribution of each Lagrangian term to the nucleon
current:

${\cal
L}_{\pi\Delta\Delta}=\bar{\Psi}_{1}^{\mu}\Gamma^{r}A_{r}\Psi_{1\mu}-\bar{\Psi}_{2}^{\mu}\Gamma^{r}A_{r}\Psi_{2\mu}$
(32)

here $\Gamma$ is Dirac matrices in the reference frame. $\Gamma^{r}$ matrices
are chosen as $\Gamma^{r}=(\gamma^{\mu},\ -i\gamma^{5})$. The expression for
the five dimensional spinors has following form:
$\Psi^{\sigma}_{L,R}(p,r)=\sum_{n}F^{(n)}_{L,R}(p,r)\psi^{(n)\sigma}_{L,R}(p)$,
The 4D coupling constant for pions as

$\displaystyle g^{(0)nm}_{\pi\Delta\Delta}(T)$ $\displaystyle=$
$\displaystyle\\!-\int^{\infty}_{0}\frac{dr}{2r^{2}}\left[M_{0}(r,T)\Big{(}f^{(n)*}_{1L}(r,T)f^{(m)}_{1R}(r,T)-f^{(n)*}_{2L}(r,T)f^{(m)}_{2R}(r,T)\Big{)}\right.$
(33)

The Lagrangian of $\rho$ meson is as follows 25 :

${\cal
L}_{\rho\Delta\Delta}=\bar{\Psi}_{1}^{\nu}\Gamma^{\mu}V_{\mu}\Psi_{1\nu}+\bar{\Psi}_{2}^{\nu}\Gamma^{\nu}V_{\nu}\Psi_{2\nu}$
(34)

For the 5D spinors
$\Psi^{\sigma}_{L,R}(p,r)=\sum_{n}F^{(n)}_{L,R}(p,r)\psi^{(n)\sigma}_{L,R}(p)$,
the 4D coupling constant for $\rho$ meson is as follows:

$\displaystyle g^{(0)nm}_{\rho\Delta\Delta}(T)$ $\displaystyle=$
$\displaystyle\\!-\int^{\infty}_{0}\frac{dz}{r^{2}}M_{0}(r,T)\left[\Big{(}f^{(n)*}_{1L}(r,T)f^{(m)}_{1L}(r,T)+f^{(n)*}_{2L}(r,T)f^{(m)}_{2L}(r,T)\Big{)}\right.$
(35)

$a_{1}$ meson coupling constant arise from the bulk gauge coupling constant 26
and the minimal interaction Lagrangian terms has following form:

${\cal
L}_{a_{1}\Delta\Delta}=\bar{\Psi}_{1}^{M}e_{A}^{M}\Gamma^{M}A_{M}\Psi_{1M}-\bar{\Psi}_{2}^{M}e_{A}^{M}\Gamma^{M}A_{M}\Psi_{2M},$
(36)

the magnetic gauge interaction or Pauli terms ${\cal L}_{a_{1}\Delta\Delta}$
is as follows:

${\cal
L}_{a_{1}\Delta\Delta}=k_{1}\bar{\Psi}_{1}^{M}e_{A}e_{B}^{N}\Gamma^{MN}F_{MN}\Psi_{1M}-\bar{\Psi}_{2}^{M}e_{A}^{M}e_{B}^{N}\Gamma^{MN}F_{MN}\Psi_{2M},$
(37)

Thus, after making certain simplifications, we obtain from the minimum
interaction Lagrangian terms that $a_{1}$ meson $\Delta$ baryon minimal
coupling constant $g^{(0)nm}_{a_{1}\Delta\Delta}(T)$ in the framework of the
soft-wall framework which can be written as

$g^{(0)nm}_{a_{1}\Delta\Delta}(T)=\int^{\infty}_{0}\frac{dr}{r^{2}}M_{0}(r,T)\Big{(}f^{(n)}_{1R}(r,T)^{2}-f^{(m)}_{1L}(r,T)^{2}\Big{)}.$
(38)

$M_{0}(r,T)$ is expression of the profile function of a meson in the ground
state. The additional contributions to $g^{(1)nm}_{a_{1}\Delta\Delta}(T)$
coupling constant can arise from the magnetic type of interaction in the bulk
of AdS space-time as following form:

$g^{(1)nm}_{a_{1}\Delta\Delta}(T)=\frac{k_{1}}{2}\int^{\infty}_{0}\frac{dr}{r^{2}}M_{0}^{`}(r,T)\Big{(}f^{(n)}_{1L}(r,T)^{2}+f^{(n)}_{1L}(r,T)^{2})\Big{)}.$
(39)

The $\pi$ meson transition coupling constant have been obtained from the
gauge- invariant coupling constant of gauge fields with $\Delta$ resonances
and nucleons at finite temperature. The Lagrangian for this fields is given by

$\displaystyle{\cal L}_{FN\Delta}$ $\displaystyle=$
$\displaystyle\left[\alpha_{1}\Big{(}\bar{\Psi}_{1}^{M}\Gamma^{N}(F_{L})_{MN}N_{1}-(1\leftrightarrow
2\,\,\&\,\,L\leftrightarrow R)\Big{)}\right.$

where $\alpha_{1}$ is a parameter 18 . The terms contribute to the 4D $\rho$
meson thermal coupling constant.By KK reduction of 5D spinors as
$\Psi_{iL,R}(p,z)=\sum_{n}f^{(n)}_{iL,R}(p,z)\psi^{(n)}_{L,R}(p)$ for nucleons
and $\Psi_{L,R}(p,z)=\sum_{n}F^{(n)}_{L,R}(p,z)\psi^{(n)}_{L,R}(p)$ for
$\Delta$ resonances, one can write the pion-nucleon-$\Delta$ couplings as,

$\displaystyle g^{nm}_{\pi N\Delta}(T)$ $\displaystyle=$ $\displaystyle-
f_{\pi}\int^{\infty}_{0}dz\left[\frac{M_{0}(r,T)}{r^{2}}\Big{(}\kappa\big{(}F^{(n)*}_{1L}(r,T)f^{(m)}_{1R}(r,T)+F^{(n)*}_{2L}(r,T)f^{(m)}_{2R}(r,T)\big{)}\right]$
(41)

and similarly the rho-nucleon-$\Delta$ couplings at finite temperature as,

$\displaystyle g^{nm}_{\rho N\Delta}(T)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}dr\left[\frac{M_{0}(r,T)}{r^{2}}\Big{(}\kappa\big{(}F^{(n)*}_{1L}(r,T)f^{(m)}_{1R}(r,T)-F^{(n)*}_{2L}(r,T)f^{(m)}_{2R}(r,T)\big{)}\right]$
(42)

##  7 Numerical Results

The coupling constants at finite temperature $g_{\rho\Delta\Delta}(T)$,
$g_{a_{1}\Delta\Delta}$(T), and $g_{\rho N\Delta}(T)$ consists in numerical
calculation of the integrals, and in numerically plotting their temperature
dependencies by means of the Mathematical package. The author presents
numerical results for the parameters $N_{f}=2$, $F=87$ MeV, $N_{f}=3$, $F=100$
MeV, $N_{f}=4$, $F=130$ MeV and flovour $N_{f}=5$, $F=140$ MeV. These sets of
parameters were taken from 18  There are free parameters $k$, $k_{1}$, $m_{q}$
and $\Sigma$ in this work. The value of parameter $k=383$ MeV 15  and
parameter $\overline{k}=-733$ 18 . The parameters $k_{1}$ are fixed at the
values $k_{1}=-0.78$ $GeV^{3}$ in the 26 . The $\Sigma=0.368^{3}$ $MeV^{3}$
value and the $m_{q}=0.0023$ $GeV$ value of these parameters were found from
the fitting of the $\pi$ meson mass 27 . Having an idea of the relative
contributions of different flavors of hadrons, the author present results for
the temperature dependencies of the various numbers separately.

In Fig. 1 the $\rho$ meson $\Delta$ baryon coupling constant
$g^{0}_{\rho\Delta\Delta}(T)$, In Fig 2 the $a_{1}$ meson $\Delta$ baryon
coupling constant $g^{0}_{a_{1}\Delta\Delta}(T)$ and In Fig. 4 the $\rho$
meson-nucleon-$\Delta$ -transition coupling constant $g_{\rho N\Delta}(T)$
were plotted at the parameter $N_{f}=2$, $F=87$ MEV, $N_{f}=3$, $F=100$ MEV,
$N_{f}=4$, $F=130$ MEV, and $N_{f}=5$, $F=140$ MEV. In Fig 3 the author also
has considered these dependencies for $g^{1}_{a_{1}\Delta\Delta}(T)$ of the
nucleons and drawing graphs for the different number of flavors. The purple
graph curve represents two $N_{f}=2$, $F=87$ $MeV$, green one curve shows the
three $N_{f}=3$, $F=100$ MeV, orange curve shows the four $N_{f}=4$, $F=130$
MeV , and blue one shows the five flavor $N_{f}=5$, $F=140$ MeV of thermal
minimal coupling constants in the figures. This analysis shows a weak
dependence on the parameter $N_{f}$ of the coupling constants.

##  8 SUMMARY

In the present work, the author study the temperature dependency of minimal
coupling constants of meson-$\Delta$ baryon and meson-nucleon to $\Delta$
baryon transition in the framework of the soft-wall model of holographic QCD.
It has been observed that the value of all terms become zero at the same point
near the Hawking temperature by increasing of temperature. The minimal
coupling constants of meson-$\Delta$ baryon and meson-nucleon to $\Delta$
baryon transition may be of use in the deeper study of the nucleon-delta
baryon transition and in understanding processes of the early Universe.

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Figure 1: The temperature dependence of $g^{0}_{\rho\Delta\Delta}(T)$ for
$N_{f}=2$, $F=87$ MEV, $N_{f}=3$, $F=100$ MEV, $N_{f}=4$, $F=130$ MEV,
$N_{f}=5$, $F=140$ MEV. Figure 2: The temperature dependence of
$g^{0}_{a_{1}\Delta\Delta}(T)$ for $N_{f}=2$, $F=87$ MEV, $N_{f}=3$, $F=100$
MEV, $N_{f}=4$, $F=130$ MEV, $N_{f}=5$, $F=140$ MEV. Figure 3: The temperature
dependence of $g^{1}_{a_{1}\Delta\Delta}(T)$ for $N_{f}=2$, $F=87$ MEV,
$N_{f}=3$, $F=100$ MEV, $N_{f}=4$, $F=130$ MEV, $N_{f}=5$, $F=140$ MEV. Figure
4: The temperature dependence of $g_{\rho N\Delta}(T)$ for $N_{f}=2$, $F=87$
MEV, $N_{f}=3$, $F=100$ MEV, $N_{f}=4$, $F=130$ MEV, $N_{f}=5$, $F=140$ MEV.


# VertMatch: A Semi-supervised Framework for Vertebral Structure Detection in
3D Ultrasound Volume

Hongye Zeng1,2,3, kang Zhou4, Songhan Ge1,2,3, Yuchong Gao1,2,3,
Jianhao Zhao1, Shenghua Gao1,5, Rui Zheng1,5,

1School of Information Science and Technology, ShanghaiTech University,
Shanghai 201210, China
2Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai,
China
3University of Chinese Academy of Sciences, Beijing, China
4Department of Computer Science and Engineering, The Chinese University of
Hong Kong, Hong Kong, China
5Shanghai Engineering Research Center of Intelligent Vision and Imaging,
ShanghaiTech University, Shanghai, China
<EMAIL_ADDRESS>
###### Abstract

Three-dimensional (3D) ultrasound imaging technique has been applied for
scoliosis assessment, but current assessment method only uses coronal
projection image and cannot illustrate the 3D deformity and vertebra rotation.
The vertebra detection is essential to reveal 3D spine information, but the
detection task is challenging due to complex data and limited annotations. We
propose VertMatch, a two-step framework to detect vertebral structures in 3D
ultrasound volume by utilizing unlabeled data in semi-supervised manner. The
first step is to detect the possible positions of structures on transverse
slice globally, and then the local patches are cropped based on detected
positions. The second step is to distinguish whether the patches contain real
vertebral structures and screen the predicted positions from the first step.
VertMatch develops three novel components for semi-supervised learning: for
position detection in the first step, (1) anatomical prior is used to screen
pseudo labels generated from confidence threshold method; (2) multi-slice
consistency is used to utilize more unlabeled data by inputting multiple
adjacent slices; (3) for patch identification in the second step, the
categories are rebalanced in each batch to solve imbalance problem.
Experimental results demonstrate that VertMatch can detect vertebra accurately
in ultrasound volume and outperforms state-of-the-art methods. VertMatch is
also validated in clinical application on forty ultrasound scans, and it can
be a promising approach for 3D assessment of scoliosis.

_K_ eywords Vertebral structure detection $\cdot$ Landmark detection $\cdot$
Semi-supervised learning $\cdot$ Anatomical prior $\cdot$ Multi-slice
consistency

##  1 Introduction

Adolescent idiopathic scoliosis (AIS) is a 3D deformity of spine featured with
lateral deviation and axial vertebral rotation ([1];[2]), and can greatly
influence the patient’s life and even give rise to death ([3]). Therefore, the
early diagnosis and frequent follow-up exams of AIS are very important to the
treatment. The radiography is usually used to monitor scoliosis ([4]), but it
can accumulate ionizing radiation for patients during the long-term follow-up
exams ([5]). The 3D ultrasound imaging technique has become a promising
diagnostic tool since it is radiation-free, cost-effective and portable
comparing to radiography ([6];[7]).

(a)

(b)

(c)

Figure 1: Introduction of task and annotation: (a) AIS assessment only using
coronal projection image of 3D ultrasound volume; (b) Vertebral structures on
transverse slice. Five landmarks are annotated to represent possible positions
of spinous process (SP) and laminae; (c) Challenging slices with annotation.
These slice contain real (red dot) or fake (red cross) structures.

As shown in Fig. 1, 3D ultrasound technique is usually used for AIS assessment
by projecting the 3D volume to 2D coronal spine image and measuring the spinal
curvature ([8];[9];[10]). Although many researches have proved that coronal
spine image is reliable for curvature measurement ([11];[12]), the 2-D
projection image cannot expose the 3D deformity and vertebra rotation.

Therefore, it is necessary to further detect vertebra in the ultrasound volume
and provide more 3D spine information for better AIS assessment. The main
structures of vertebra to be detected are spinous process (SP) and laminae on
transverse slice as shown in Fig. 1. The SP is usually merged into the soft
tissue and presents an obvious hollow (white curve). The laminae are usually
located in the middle region of the transverse slice and shown as the two
adjacent short line segments with high intensity. However, it is challenging
to detect SP and laminae on all transverse slices in volume due to three
obstacles as illustrated in Fig. 1: 1) Subject Diversity: slices from two
subjects especially noise (white arrow) are quite different, due to different
fat and muscle content; 2) Regular Missing: from $750^{th}$ to $810^{th}$
slices from the same subject, the laminae and SP regularly disappears and
reappears at the junction of two vertebrae; 3) Random Missing: the soft tissue
layer or unilateral lamina disappear randomly, since the ultrasound signal may
not be received by the probe or be blocked by other bones due to vertebra
rotation or scanning manipulation. Subject diversity enriches the data and
causes large demand of annotations for learning-based method, and missing
problem requires the detection method to distinguish real vertebral structures
accurately in complex ultrasound volume.

The straightforward thought is to apply existing vertebra detection method on
3D ultrasound volume. Some methods have been proposed on sagittal slices to
segment the transverse process ([13]), which is quite different from SP and
laminae on transverse slice. [14] and [15] successfully segmented the spinous
process and spine surface on transverse slice respectively using the acoustic
shadow method and U-Net. Excepting for the ultrasound image, many mature
methods are applied to detect vertebrae on Computed Tomography (CT) ([16]) and
Magnetic resonance imaging (MRI) ([17]). These methods usually uses the
relation between several vertebrae as anatomical prior to achieve better
predictions ([18]), which is informative but not entirely suitable for our
transverse slice containing information of single vertebra. Besides, above
learning-based methods need large amounts of annotated data, which is very
costly and time-consuming for medical experts especially on 3D images. The
semi-supervised leaning (SSL) shows potential for improving network
performance when labeled data is scarce by using pseudo-labeling and
consistency regularization to utilize unlabeled data ([19];[20]). Recently,
the SSL algorithm has been successfully applied for general vision tasks
([21];[22];[23]) and medical image analysis ([24];[25];[26];[27]). They
usually proposed some novelties of consistency regularization, which are
instructive for our task.

In this paper, we propose a two-step landmark detection framework for vertebra
detection in 3D ultrasound volume. We firstly consider the vertebral structure
detection as landmark detection task to avoid complex segmentation
annotations, which means that the SP and laminae are annotated as five
landmarks as illustrated in Fig. 1. Specifically, five landmarks only
represent possible positions of SP and laminae, and contain real (dots) and
fake (cross) vertebral structures as shown in Fig. 1. Secondly, we design a
two-step framework containing the detector to detect possible positions of SP
and laminae globally, and the classifier to identify possible positions
whether contain real structures by using local patch. On one hand, it is
easier to detect possible position since five landmarks usually locate in the
middle of slice and form an acute triangle. On the other hand, the fake
structures are obviously different from real structures when focusing on local
patch. In a word, two-step method can reduce difficulty of vertebral structure
detection compared to find real structures in one step.

To address the limited annotation problem, we propose to use SSL to train the
detector and classifier, and propose three novel components for our task: 1)
To generate more convinced pseudo labels, anatomical prior is leveraged to
screen the pseudo labels produced from high confidence predictions. Anatomical
prior is summarized that the SP and laminae usually form an acute triangle due
to the vertebra shape, and the bone length usually is within an approximate
range. 2) To utilize more unlabeled data, a slice and its adjacent slices are
fed into the network together. This procedure is based on multi-slice
consistency, which declares that there are only slight differences between
adjacent slices while the possible positions of SP and lamina are supposed to
be consistent. Therefore the detector can be trained by forcing to output
similar predictions of adjacent slices. As regular missing slices shown in
Fig. 1, although the SP and laminae disappear and reappear, the possible
positions of SP and laminae are approximate on consecutive slices ($750^{th}$
and $755^{th}$) and locate at the center of slice. 3) To avoid classifier
bias, the categories in pseudo labels are rebalanced in each batch, since the
slices containing SP and laminae are minority in ultrasound scans.

The main contributions are summarised as follow:

  * •

We propose VertMatch framework for vertebral structure detection in 3D
ultrasound volume. VertMatch contains two steps of position detection and
patch identification, and can tackle complicated ultrasound slices from
various subjects.

  * •

We train VertMatch in semi-supervised manner and propose three novel
components: for position detection, (1) anatomical prior is used to select
better pseudo labels; (2) multi-slice consistency is proposed to utilize more
unlabeled data; (3) for patch identification, the batch rebalance is used to
balance the categories in pseudo labels.

  * •

We further verify VertMatch for AIS assessment on 40 subject scans by
measuring spinous process angle (SPA), which is a clinical parameter to reveal
axial vertebral rotation and lateral curvatures. The results of SPA from
ultrasound data are comparable to the gold standard from radiograph, which
shows VertMatch’s potential for 3D AIS assessment.

##  2 Related work

###  2.1 Vertebra detection

The spine can be visualized using 3D imaging technique such as CT, MRI and
ultrasound, and the vertebra localization and identification are required in
many clinical applications of spine disorder diagnosis and surgery planning.

Some methods have been explored for vertebra detection on ultrasound volume.
[28], [14], [29], [13] proposed novel methods to segment transverse process on
sagittal slices, and [30], [15] focused on spine surface segmentation on
transverse slices. Most methods implemented their algorithm using the U-Net,
and [15] proposed a late fusion U-Net to fuse the B-mode and shadow-enhanced
B-mode image. All current methods for 3D ultrasound volume cannot tackle
challenges in our task.

Anatomical prior has been exploited for vertebra detection in CT and MRI
volume, and these methods inspire us to derive and utilize new anatomical
prior on ultrasound transverse slices. [31] used the Bi-RNN to incorporate the
spine anatomical knowledge into neural networks implicitly, [17] used a
hierarchical learning strategy to detect different level of spine based on
prior, and [32] explicitly considered the spatial distribution of the
vertebrae by using an information aggregation layer. More commonly, anatomical
prior was used to post-process and constrain the network outputs, by using the
Hidden Markov Model ([33]) or solving the anatomical-constraints optimization
problem ([18]).

###  2.2 Landmark detection

The vertebra detection on image volume described in 2.1 is usually implemented
as a segmentation task which is time-consuming to annotate. While the vertebra
detection on radiograph is usually formulated as a landmark detection task,
since the landmark annotation is simpler and faster to accomplish compared to
segmentation annotation. [34] used the BoostLayer to eliminate the outlier
features for robust landmark detection. [35] proposed the support vector
regression, and [36] applied the HR-Net, to predict landmarks and estimated
the spinal curvature directly. [37] proposed the multi-view correlation
network to integrate the anterior-posterior and lateral radiographs, and
achieved accurate results for landmark detection and curvature estimation. The
landmark detection is a typical task for computer vision such as human pose
estimation. The stacked hourglass network (SHN) ([38]) is a commonly used
network by encoding low-resolution representation and recovering high-
resolution representation. In contrast, the high-resolution network (HRNet)
([39]) is another popular method and maintain high-resolution representations
through the whole process.

###  2.3 Semi-supervised learning

The Semi-supervised Learning (SSL) has been extensively studied to efficiently
utilize unlabeled images, and recent SSL methods usually use the pseudo-
labeling and consistency regularization. Pseudo-labeling based method predicts
pseudo labels for unlabeled data using a trained model, and then all data with
real and pseudo labels are used to retrain the model. [40] was the first to
introduce the pseudo-labeling for neural networks, and [41] proposed MixMatch
to average the predictions of stochastic augmented inputs by employing a
sharpen function. Consistency regularization utilizes unlabeled data based on
the assumption that the network should output similar predictions when being
fed perturbed versions of the same image. [42] proposed the temporal
ensembling model to improve the quality of consistency targets by using the
exponential moving average strategy for averaging predictions over the
previous epoch, and [43] proposed the Mean Teacher framework which averaged
model weights instead of label predictions by including an additional teacher
model. FixMatch ([20]) generated pseudo label on weakly-augmented unlabeled
images, and the model was trained to predict the pseudo label when being fed a
strongly-augmented version of the same image.

The SSL methods has been widely studied in medical image fields. [44] was
first to apply the SSL for 3D medical image detection, and the SSL also were
beneficial to the segmentation of muscle ([45]) and breast structures ([46]).
Most of the methods proposed novel consistency regularization to suit their
task. [24] proposed the adaptive consistency to regularize different
components in the predictions. [25] proposed the sample relation consistency
to model the relationship information among different samples. [26] used the
attention consistency to ensure the focusing of the student and teacher
networks on the same defect regions for cartilage defect level classification.
[27] defined the dual-task which consisted of pixel-level segmentation task
and geometry-aware level set representation task, and proposed the dual task
consistency which declared that the networks should behave similarly between
two different tasks when being fed same image. In this study, we also propose
new consistency regularization adapted to our task.

##  3 Methods

We propose a semi-supervised framework, termed as VertMatch, to detect
vertebral structures in 3D ultrasound volume. To reduce task difficulty, we
consider the vertebral structures detection as landmark detection task and
annotate the possible positions of structures as five landmarks, furthermore
we decompose entire task into two steps containing global detection and local
identification. To efficiently use the data and avoid a large annotation, we
train the networks using semi-supervised manner, and three novel components
are proposed to improve SSL algorithm based on anatomical prior and
observation of ultrasound data.

The overview of VertMatch is shown in Fig. 2. The first step is to detect
possible positions of structures on transverse slices. The detection network
is trained following a semi-supervised manner by utilizing multi-slice
consistency, and anatomical prior is used to select more credible pseudo
labels during training. Afterwards, the patches are cropped based on the
detected positions. The second step is to identify whether these patches
contains real vertebral structures. In this step, the SSL algorithm is
deployed with batch rebalance to solve the imbalanced data problem.

Figure 2: Overview of the two-step VertMatch framework for vertebral
structures detection. The first step is to detect the possible positions of
vertebral structures as five landmarks, and then the patches are cropped based
on the detection results; the second step uses a classifier to distinguish
whether the patches contain real vertebral structures. Both detector and
classifier are trained via similar semi-supervised manner using the labeled
and unlabeled data. In the first step, anatomical prior is used to select
better pseudo labels, and the unlabeled slice ($u_{k}$) and its adjacent
slices ($u_{k}^{-},u_{k}^{+}$) are strong-augmented ($A(\cdot)$) and input
together to utilize more unlabeled data based on proposed multi-slice
consistency. In the second step, the batch rebalance is used to balance the
number of pseudo labels in different categories in a batch.

###  3.1 Position detection

####  3.1.1 Landmark detection

The possible position of SP and laminae are detected as five landmarks. The
detector uses the SHN ([38]) or HRNet ([39]) as backbone network to extract
feature map, and uses $3\times 3$ convolution layers to process feature map
and output predicted heatmaps.

Given an input slice $I\in\mathbbm{R}^{W\times H\times 1}$ with width $W$ and
height $H$, its annotated five landmarks are encoded to a set of heatmaps
$h\in[0,1]^{\frac{W}{d}\times\frac{H}{d}\times 5}$ by using a Gaussian kernel
centering at landmarks, where $d$ is downsampling ratio and empirically set as
4. The detector $f_{\theta}$ outputs predicted heatmaps
$\hat{h}\in[0,1]^{\frac{W}{d}\times\frac{H}{d}\times 5}$ and the mean squared
error (MSE) loss $L^{l}_{det}$ is calculated between two sets of heatmaps.
Note that the coordinates of predicted landmarks need to be decoded from the
output heatmaps and restored to original resolution. Let
$\textit{X}=\\{(x_{k},h_{k}):k\in(1,...,N)\\}$ be $N$ labeled slices with
heatmap annotation $h_{k}$ in a batch, and the supervised MSE loss can be
written as:

$L^{l}_{det}=\frac{1}{N}\sum_{k=1}^{N}\big{\|}h_{k}-\hat{h_{k}}\big{\|}^{2}$
(1)

where $\hat{h_{k}}=f_{\theta}(x_{k})$ is the predicted heatmaps when inputting
$x_{k}$.

####  3.1.2 SSL-based position detection

To avoid large annotations, we proposed a novel SSL method to train the
detector as shown in Fig. 2 Step 1. Following basic SSL training procedure,
the detector is trained by using labeled (green arrow) and unlabeled data
(orange arrow) simultaneously in each batch. For labeled slice, the supervised
MSE is calculated directly as illustrated in Eq. 1. For unlabeled slice, the
pseudo labels are generated from weakly-augmented slice, and then the detector
is trained by forcing to output similar predictions when being fed strongly-
augmented slice. Especially, we use anatomical prior to improve quality of
pseudo labels, and propose multi-slice consistency to utilize more unlabeled
data.

Figure 3: Generating standard-compliant pseudo label. Firstly, coordinates of
all five landmarks are decoded from heatmaps $v_{k}$. The left image shows
decode procedure of SP heatmap $v_{k}^{0}$ by finding the maximal activation
above threshold $\tau$, and the black dot is the maximal activation. Secondly,
as shown in the right image, all decoded landmarks (orange dots) should obey
anatomical prior: 1) forming acute triangle; 2) following specific order along
rotation direction; 3) $D$ and $d$ are within a reasonable range.

We uses both confidence threshold and anatomical prior to generate better
pseudo labels as illustrated in Fig. 3. Given a unlabeled slice $u_{k}$, the
weak augmentation $\alpha(\cdot)$ is applied and then the detector outputs
heatmaps $v_{k}=f_{\theta}(\alpha(u_{k}))$. One landmark is decoded from one
heatmap $v_{k}^{i}$ by finding the maximal activation above threshold $\tau$,
moreover the unlabeled slice $u_{k}$ will be abandoned directly if there are a
maximal activation less than threshold $\tau$. Afterwards, five decoded
landmarks will be accepted as pseudo label if they obey anatomical prior.
According to prior acknowledge, the basic shape of vertebra and the relative
position between SP and laminae are consistent, and sizes of various vertebrae
are usually within reasonable range for different subjects. Therefore,
anatomical prior can be summarised into three aspects: 1) five landmarks
should form an isosceles acute triangle approximately; 2) five landmarks
should follow the order $\\{L_{0},\ L_{1},\ SP,\ L_{2},\ L3\\}$ along rotation
direction; 3) the distance $D$ between SP and laminae and length $d$ of two
laminae should be within reasonable range. Anatomical prior can screen pseudo
labels produced by confidence threshold, and provide high quality pseudo
labels for unlabeled data.

Consistency regularization assumes that the predictions
$f_{\theta}(\alpha(u_{k}))$ and $f_{\theta}(A(u_{k}))$ should be similar when
fed perturbed versions of one slice, where $A(\cdot)$ is the strong
augmentation. We further proposed multi-slice consistency, which declares that
the possible positions of vertebral structures of adjacent slices do not
mutate, and the detector should output similar predictions when being fed
perturbed versions of the adjacent slices. Let $u_{k}^{-}$ and $u_{k}^{+}$ be
adjacent slices of $u_{k}$, the detector will output three sets of heatmaps
$\hat{v_{k,1}}=f_{\theta}(A(u_{k}^{-})),\hat{v_{k,2}}=f_{\theta}(A(u_{k})),\hat{v_{k,3}}=f_{\theta}(A(u_{k}^{+}))$,
and each set of heatmaps should be similar to pseudo label
$v_{k}=f_{\theta}(\alpha(u_{k}))$. Multi-slice consistency can provide more
unlabeled slices for training and obtain better detector.

Let $\textit{U}=\\{u_{k}:k\in(1,...,\mu N)\\}$ be $\mu N$ unlabeled slices in
a batch, where $\mu$ is a hyperparameter that determines the relative sizes of
$X$ and $U$. The unsupervised loss and final semi-supervised detection loss
can be written as:

$L^{u}_{det}=\frac{1}{\mu N}\sum_{k=1}^{\mu
N}\psi(v_{k})\sum_{j=1}^{3}\sum_{i=1}^{5}\mathbbm{1}\Big{\\{}max(v_{k}^{i})\geq\tau\Big{\\}}\
\big{\|}v_{k}^{i}-\hat{v_{k,j}^{i}}\big{\|}^{2}$ (2)
$L_{det}=L^{l}_{det}+\lambda_{u}L^{u}_{det}$ (3)

where the $\psi(\cdot)=1$ if predicted five landmarks obey anatomical prior,
else $\psi(\cdot)=0$. The $\lambda_{u}$ is a hyper-parameter used to weight
the unlabeled loss.

###  3.2 Patch generation

The local patches are obtained based on possible positions as illustrated in
Fig. 4. The possible positions can be obtain directly from annotations for
labeled data, and need to be decoded from outputted heatmaps of detector for
unlabeled data. There is no threshold for this decode procedure, and the
maximal activation on each heatmap is regarded as the predicted landmark
directly.

Normally, the middle point of two lamina landmarks are calculated, and three
patches are finally obtained on one transverse slice by cropping rectangle
areas centering at the SP position and two middle points. According to
experiment, the SP patch is set to 110$\times$140 pixels and the laminae patch
is set to 80$\times$100 pixels. Compared to fixed centers of patches, we move
the centers randomly within small range during each training. The unfixed
centers can be regarded as data augmentation, and could eliminate the
influence of imprecise detection.

###  3.3 Patch identification

After obtaining patches, the classifier is used to identify whether the
patches contain real vertebra structures as shown in Fig. 2 Step 2. We uses
ResNet18 as the backbone, and train the classifier in semi-supervised manner
by utilizing $N$ labeled patches and $\mu N$ unlabeled patches in each batch.
For labeled patches (green arrow), the standard cross-entropy loss is applied
directly:

$L_{cls}^{l}=\frac{1}{N}\sum_{k=1}^{N}H(y_{k},p_{k})$ (4)

where $y_{k}$ is category label and $p_{k}$ is the predicted probability
distributions, $H$ is cross-entropy loss.

For an unlabeled patch $I_{k}$ (orange arrow), the weak augmentation
$\alpha(\cdot)$ and the strong augmentation $A(\cdot)$ are applied and then
the classifier $g_{\theta}$ outputs two probability distributions
$q_{k}=g_{\theta}(\alpha(I_{k}))$ and $\hat{q_{k}}=g_{\theta}(A(I_{k}))$.
Normally, the pseudo label is produced using a high confidence threshold
$\tau$, and the cross-entropy loss is calculated between pseudo label and
$\hat{q_{k}}$ to train classifier according to consistency regularization.

However, the categories in unlabeled patches are imbalanced, since the slices
with vertebral structures are minority in ultrasound scan, especially the SP
only appears on small amount of slices. The imbalanced unlabeled patches will
cause the bias toward majority classes when generating pseudo labels.
Therefore, we rebalance the categories of pseudo labels in a batch, which
means the number of majority is forced to be equal to the number of minority
category. Let $M$ be a mask array with $\mu N$ elements, and each element
$m_{k}=\mathbbm{1}\Big{\\{}max(\hat{q_{k}})\geq\tau)\Big{\\}}$ is initialized
based on confidence threshold. The rebalance operation can be presented as:

$M={Reb}(M,n)$ (5)

where $n$ is the number of minority category. $Reb(\cdot)$ is the rebalance
operation putting on mask array $M$, to randomly keep $n$ elements belonging
to majority category. The unsupervised loss and classification loss can be
written as:

$L_{cls}^{u}=\frac{1}{\mu N}\sum_{k=1}^{\mu N}m_{k}\ H(z_{k},\hat{q_{k}})$ (6)
$L_{cls}=L^{l}_{cls}+\lambda_{u}L^{u}_{cls}$ (7)

where $z_{k}=\arg\max(q_{k})$ is category prediction.

Figure 4: Cropping patches using random center on labeled and unlabeled
slices. Three patches are cropped on one slice centering at SP position and
two middle points (white dots) of two lamina landmarks. The white arrows mean
that the centers of patches will move in small range in every epoch.

##  4 Experiments

###  4.1 Subjects and datasets

The ultrasound spine data are collected from 90 adolescents (17 males and 73
females; mean age:14.2±2.0; body mass index: 14.3$\sim$22.7). They fulfill the
following inclusion criteria: 1) no constraints surgical treatment; 2) receive
a posterior-anterior standing radiograph and US scan within an hour; 3) with
the major curve of 10° to 45°. Before being enrolled into the study, all
participating subjects signs a written consent, and ethics approval is granted
from the local health research ethics board. The radiographic curve
information including SPAs and Cobb angles are exported from the medical
record to be used for the following data analysis.

One Standing ultrasound spine scan contains approximate 900-2300 transverse
slices acquired along the spine curve, and is obtained using the SonixONE
system equipped with 128-element C5-2/60 array transducer and SonixGPS
(Analogic Ultrasound - BK Medical, Peabody, Massachusetts, US). The center
frequency of transducer probe is 2.5 MHz, and the imaging depth is set as
6-8cm.

Fifty ultrasound scans are used to build the training and testing datasets,
and another forty ultrasound scans are applied for clinical application. The
detailed information of our datasets is illustrated in Table.1. For the
training labeled data, we conditionally selected slices in each ultrasound
scan, and try to maintain the category balance between slices with and without
vertebral structures. For testing data, it is not necessary to annotate all
slices in subject scans. In order to cover more cases, we sample every ten
slices in 10 subjects scans, and randomly select 20 slices in another 10
subjects scan.

Table 1: Information of datasets built from fifty ultrasound scans. Dataset | Subject Number | Slice Number
---|---|---
train | labeled | 10 | 1000
unlebeled | 20 | 26510
test | 20 | 1675

###  4.2 Evaluation metrics

We use the Percentage of Correct Keypoints (PCK) to measure the performance of
position detection in first step, which reports the percentage of detected
landmarks that are within the region of the ground truth, and could expressed
as:

$PCK=\frac{\sum_{n}\sum_{5}\delta(d\leq T)}{n}$ (8)

where $n$ is number of landmarks, and $d$ is the distance between the
predicted landmark and ground-truth. The $\delta(\cdot)=1$ when distance less
than threshold $T$, which is set to 15 pixels and 30 pixels for real and fake
vertebral structures respectively. The landmarks of fake vertebral structures
are hard to match the ground-truth perfectly due to inapparent features, and
the large threshold is acceptable since they will be excluded by classifier.

We use the accuracy, precision, recall and F1-score to evaluate the entire
method, which are widely-used metrics in classification task. Note that these
metrics are redefined in our study, and the true positive (TP) and true
negative (TN) are defined as:

$\begin{split}&TP:\mathbbm{1}\\{y=1\\}\ and\ \delta(d\leq T)\\\
&TN:\mathbbm{1}\\{y=0\\}\end{split}$ (9)

where the newly defined TP requests both position prediected by detector and
category predicted by classifier are correct.

###  4.3 Implmentation details

All experiments are implemented using Pytorch on two 24 GB NVIDIA RTX 3090
GPUs. Both the detector and classifier are trained from scratch with 500
epochs. The detector uses the Adam optimizer, and initial learning rate is
1e-3 with decay schedule of 0.1 in 60, 150 epoch. We set $N=8$, $\mu=3$,
$\tau=0.8$, $\lambda_{u}=1$ for detector SSL. The classifier uses the SGD
optimizer, and the learning rate is initialized to 3e-2 and decays by 0.1
after 200 epochs. The momentum is 0.9 and the weight decay is 5e-4. We set
$N=32$, $\mu=15$, $\tau=0.97$, $\lambda_{u}=1$ for classifier SSL.

The weak and strong data augmentations are important for the semi-supervised
learning, and have been widely used in SSL method ([20][41]). In our study,
the weak augmentations are random Gaussian noise augmentation for detector,
and random horizontal flip augmentation for classifier. We use strong
augmentations based on RandAugment [20], but adjust some parameters and delete
color augmentation to fit the ultrasound data. Besides, we abandon all
augmentations about spatial transform when training detector, since it is
difficult to perform the same spatial transforms for the landmarks and the
results showed no influence without them.

###  4.4 Quantitative and qualitative analysis

Quantitative Comparison with Alternatives To evaluate our proposed method,
considering there are no existing work for this task, we compare it with some
one-step and two-step methods as shown in Table. 2. The one-step methods use
two well-known detection networks SHN ([38]) and HRNet ([39]) to directly
predict real vertebral structures. The other two-step methods combine
SHN/HRNet and ResNet18, and consist of possible position detection and patch
identification following supervised manner. VertMatch also uses the SHN and
HRNet as backbone network in detector, and the performances of two backbones
are compared. All methods are trained using labeled data, and VertMatch
utilizes unlabeled data additionally.

Table.2 lists the quantitative results of different methods. The one-step
supervised methods show unsatisfied results especially in SP detection, and
the two-step supervised methods attains significant improvements in accuracy
and f1-score. Two-step methods include global position detection and local
patch identification, and these improvements demonstrate that the
decomposition of entire task can truly reduce difficulty. VertMatch using SHN
achieves the best performance in accuracy and f1-score. Compared to
SHN+ResNet18, the accuracy and f1-score increase by 2.5%, 3.4% of SP detection
and 1.7%, 2.5% of laminae detection respectively, which illustrates that
VertMatch can utilize the unlabeled data efficiently and reach better results.
Since the SHN outperforms the HRNet in most results, we use the SHN as the
backbone in the following experiments. Besides, all methods achieve a better
performance for lamina detection, since the laminae have more notable
characters compared to SP.

Table 2: Quantitative comparison between different methods. VertMatch is semi-supervised and uses 1000 labeled and 26510 unlabeled slices, and other methods only use 1000 labeled slices. Method | Spinous Process | Laminae
---|---|---
Accuracy | Precision | Recall | F1-Score | Accuracy | Precision | Recall | F1-Score
One-step | HRNet | 71.6 | 47.5 | 64.6 | 54.7 | 81.3 | 65.2 | 92.6 | 76.5
SHN | 63.4 | 41.7 | 98.9 | 58.6 | 82.1 | 65.6 | 97.6 | 78.4
Two-step | HRNet+ResNet18 | 85.2 | 68.8 | 84.4 | 75.8 | 89.9 | 83.9 | 86.3 | 84.5
SHN+ResNet18 | 86.9 | 72.7 | 85.8 | 78.7 | 89.6 | 83.0 | 86.8 | 84.8
VertMatch(HRNet) | 87.9 | 74.2 | 85.7 | 79.2 | 89.1 | 79.1 | 90.8 | 85.1
VertMatch(SHN) | 89.4 | 77.7 | 87.0 | 82.1 | 91.3 | 83.6 | 92.2 | 87.3

Quantitative Analysis of Labeled Data Volume To explore the impact of labeled
data volume on VertMatch, the evaluation is performed with different number of
labeled data and the number of used data decreases with the number of
subjects. As illustrated in Table.3, the performance of the VertMatch is
promoted when more labeled data are available, and there is no distinctive
upgrade between results from 700 and 1000 slices, which means that the labeled
data play a limited role unless the number of data is massively scaled up.
When using 700 labeled slices, VertMatch still outperforms all supervised two-
step methods using 1000 labeled slices, and the SP detection achieves 2.8% and
3.5% improvement in accuracy and f1-score respectively compared to
SHN+ResNet18. It demonstrates that VertMatch is reliable for vertebral
structures detection although without a large amount of labeled data, and our
semi-supervised technique can learn the acknowledge from unlabeled data
efficiently.

Table 3: Quantitative analysis of different number of labeled data when using VertMatch. Labeled / Unlabeled | Spinous Process | Laminae
---|---|---
Accuracy | Precision | Recall | F1-Score | Accuracy | Precision | Recall | F1-Score
100(1) / 26510 | 77.6 | 56.5 | 67.4 | 61.5 | 86.0 | 74.5 | 87.6 | 80.5
300(3) / 26510 | 81.9 | 62.9 | 76.1 | 68.9 | 85.7 | 74.3 | 86.2 | 79.8
500(5) / 26510 | 83.5 | 65.8 | 82.9 | 73.2 | 86.9 | 76.1 | 89.0 | 82.0
700(7) / 26510 | 89.7 | 79.6 | 84.8 | 82.1 | 89.5 | 81.3 | 89.2 | 85.1
1000(10) / 26510 | 89.4 | 77.7 | 87.0 | 82.1 | 91.3 | 83.6 | 92.2 | 87.3
∗300(3) indicates 300 slice are selected from 3 ultrasound scans

Qualitative Comparison We compare the visual results of some challenging
transverse slices using SHN+ResNet18 and VertMatch with SHN, respectively. As
shown in Fig. 5, four transverse slices from different subjects are detected,
and it is observed that the VertMatch is better at detecting the real SP and
handling the complexity of ultrasound data. Both two methods can detect the
possible position of SP and laminae accurately, but the supervised method
cannot distinguish the real structures well due to limited labeled data. On
the fourth slice, the predicted landmarks of left lamina are far from
annotated possible positions but can be excluded by the classifier.

Besides on transverse slices, we also compared these two methods on whole
ultrasound scans. For entire scan, the SP and laminae are detected firstly on
all transverse slices using two detection methods. Based on the predicted
coordinates, a rectangular region around the detected laminae are segmented to
remove ribs and soft tissue layer; the coordinates of the detected SP and its
8-neighbor pixels are assigned to 255. Subsequently, the Voxel-Based Nearest
Neighbor (VNN) and Fast Dot Projection (FDP) method ([7]) are used to
reconstruct the US scan to obtain the projected coronal image. As shown in
Fig. 6, the laminae and SPs are clearly separated in pairs, and the SPs are
visualized on coronal plane to provide more information compared to original
image. At the areas marked by the dotted boxes, VertMatch represents more
detected structures which can expose the clear posterior anatomy of spine; at
the areas marked by the dotted circles, VertMatch achieves better
classification results and the pairs of SP and laminae are more obvious. Both
two methods perform worse in lumbar region, since ultrasound signals are
largely attenuated by thick and multi-layered soft tissue, and SP and laminae
are very difficult to detect on transverse slice.

Figure 5: Visual comparison on challenging transverse slices. AS shown in the
$1^{st}$-$3^{rd}$ columns (left to right), the SHN+ResNet18 method can detect
possible positions accurately, but cannot distinguish real structures (white
cross $vs$ red dot). On the fourth column, the classifier can exclude the fake
structures although the predicted landmarks (white cross) are far from labeled
landmarks (red cross).

Figure 6: Visual comparison on coronal projection images of ultrasound scans.
The red dots are detected SPs, which are visualized on coronal image. The
dotted boxes reveal missing detection of SP or lamina, the dotted circles
reveal wrong or redundant detection.

###  4.5 Ablation study

We conduct the ablation experiments to validate the effectiveness of the
proposed techniques of semi-supervised learning, and all experiments are
implemented using 700 labeled data and all unlabeled data.

Anatomical Prior The influence of anatomical constraint on position detection
is explored by calculating the PCK as shown in Table. 4. Anatomical prior is
used to select better pseudo labels, and detector benefits from high-quality
pseudo labels in SSL training and achieves 2.6% improvement of PCK.

Multi-Slice Consistency Based on assumption that possible positions of
structures are consistent on adjacent slices, multi-slice consistency can
utilize more unlabeled data by inputting three consecutive slices. As shown in
Table. 4, the position detection PCK raises 2.6% using multi-slice
consistency, since more unlabeled data can train a detector with better
generalization.

More importantly, the detector using both anatomical prior and multi-slice
consistency achieves remarkable improvement of 5.4% in PCK, indicating
complementary advantage of two techniques.

Table 4: Influence of multi-slice consistency and anatomical prior on position detection. Anatomical prior |  Multi-Slice Consistency |  PCK(%)
---|---|---
✗ |  ✗ |  79.5
✓ |  ✗ |  82.1
✗ |  ✓ |  82.1
✓ |  ✓ |  84.9

Batch Rebalance & Random Center The random center and batch rebalance are used
to crop and identify patch, and their influence is investigated as illustrated
in Table.5. If we use random center to obtain patches, the accuracy and
f1-score of laminae detection increase by 3.8% and 3.4%, but the SP detection
results only improve slightly. If the batch rebalance is contained, both SP
and laminae detection are improved and achieve 3.8% and 2.0% increase in
f1-score, which proves that more true position are predicted and the batch
rebalance helps to solve the imbalanced problem. Finally, if we add both two
techniques into basic method, the final results achieve the highest accuracy
and f1-score among all the four methods, which demonstrate the effectiveness
of the random center and batch rebalance.

Table 5: Influence of random center and batch rebalance. Random Center |  Batch Rebalance | Spinous Process | laminae
---|---|---|---
Accuracy | Precision | Recall | F1-Score | Accuracy | Precision | Recall | F1-Score
✗ | ✗ | 85.7 | 77.5 | 69.4 | 73.2 | 84.5 | 69.4 | 95.5 | 80.4
✓ | ✗ | 86.4 | 80.9 | 67.5 | 73.6 | 88.3 | 78.0 | 90.1 | 83.8
✗ | ✓ | 86.8 | 75.1 | 79.0 | 77.0 | 86.8 | 74.1 | 92.9 | 82.4
✓ | ✓ | 89.7 | 79.6 | 84.8 | 82.1 | 89.5 | 81.3 | 89.2 | 85.1

###  4.6 Clinical application

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The demonstration of SPA measurement and 3D rendering result: (a)
original ultrasound coronal image. (b) the processed coronal ultrasound image
with SP and laminae. (c) the automatic measurement on ultrasound image, and
(d) the manual SPA measurement (yellow curve) on radiograph. The SPAs are
15/20° on (c) and 14/22° on (d). The Cobb angles (indicated by red lines) on
(d) are 22/27°. (e)-(f) the sagittal views of 3D rendering result of original
and processed ultrasound volume.

(a)

(b)

Figure 8: Comparison results of spinal curvature measurements on forty
ultrasound scans: (a) The comparison of SPAs between ultrasound and
radiographic measurements of SPAs. (b) The comparison between SPAs from US
images and Cobb angles.

We further investigate the potential of VertMatch on AIS assessment. The Cobb
angles is usually used to diagnose and monitor AIS ([47]; [10]), but it only
provides the information of lateral spinal curvature. The spinous process
angle (SPA) is an alternative measurement parameter which can reveal the
information of both the axial vertebral rotation and lateral curvatures
([48]). However, the spinous process is usually merged into soft tissue and
invisible on ultrasound coronal images, hence the SPA cannot be measured on
ultrasound image.

VertMatch can visualize the SP on coronal spine image by detecting SP position
on transverse slices of ultrasound scan, and the SPA can be measured
automatically on the coronal image ([49]; [50]) as shown in Fig. 7. Forty
ultrasound scans outside the training and testing datasets are applied for the
SPA measurement, fifty-five SP curves are obtained and compared to the SPAs
measured from radiographs.

Fig. 7 & fig. 7&7 illustrated an example of SPA measurement from two
modalities. As shown in the white box, the SP curves reveal apparent vertebral
rotation and are deviated to the same direction (from the middle line of
vertebra to the left side) on both ultrasound images and radiograph. It
indicated that the SP curves obtained by ultrasound qualitatively keep the
similar trend as radiography. Fig. 8 compared the SPA measurements between two
modalities quantitatively. The automatic measurements show high agreement
($R=0.84$) to radiographic results, and the average measurement differences is
$3.63\pm 2.60^{\circ}$ and less than the clinical acceptance error
($5^{\circ}$). Fig. 7 shows the 3D rendering result of process volume, and the
SP and laminae are clearly separated for different vertebral levels compared
to Fig. 7. All the results illustrate that the VertMatch is a reliable and
promising tool for SPA measurement and AIS assessment.

Fig. 8 illustrates the comparison between SPAs from ultrasound measurements
and Cobb angles from radiographs, and the moderate correlation between two
angles indicates that there is inconsistency between these two evaluation
parameters of scoliotic curves. As show in Fig. 7, the Cobb angle is used to
reveal the spinal curvature, which is apparently different from yellow SP
curve. The SPA usually is smaller compared to Cobb angle especially for large
curvature, because the vertebrae generally rotate to the curve convexity and
thus cause the SP tips offset laterally to the direction of the curve
concavity ([51]).

##  5 Discussion and conclusion

The 3D Ultrasound imaging technique has become a promising diagnostic tool for
AIS, since it is radiation-free, cost-effective and portable comparing to
radiography. However, current 3D ultrasound volume is only applied to measure
the spinal curvature by using coronal projection image, and it is necessary to
detect vertebra in ultrasound volume to reveal 3D deformity of spine. In this
paper, we proposed VertMatch for vertebra detection, which detect real
vertebral structures on transverse slices from global to local, and utilize
unlabeled data efficiently with three novel semi-supervised learning
technique: multi consistency, anatomical prior and batch rebalance. VertMatch
outperforms state-of-the-art methods and achieves good results using limited
labeled data, furthermore it can be a promising approach for 3D assessment of
scoliosis.

Multi-slice consistency inputs adjacent slices and trains the detector by
forcing to output similar detection results, and has been proved to be an
effective manner to utilize more unlabeled data. In this paper, we only input
one slice and its two adjacent slices into detector. However, there are more
than two slices which are similar to this slice, and the number of input
slices is worth investigating. In the future work, we shall explore multi-
slice consistency and try to input more unlabeled slices. Anatomical prior is
used to generate more convinced pseudo labels. Actually, the confidence
threshold is important to landmark decode, furthermore influences the
performance of anatomical prior. For example, lower threshold may challenge
the soft constraints from anatomical prior and ignore wrong detection in
pseudo labels, higher threshold makes it difficult to decode all five
landmarks and establish relationship between landmarks. The confidence
threshold therefore is an essential parameter worth to be explored.

We validate VertMatch on forty ultrasound scans for clinical application, and
the SPA can be measured on processed coronal image. The results show a good
correlation (0.84) between US and radiographic measurements, and automatic SPA
measurement based on VertMatch can be regarded as a reliable and promising
tool for the diagnosis, monitoring and screening of spine deformities.
Furthermore, the vertebral structure detection in 3D ultrasound volume can be
used for parametric spine modeling imaging, and it could provide clearer
visualization of spine which will be comparable to CT and MRI. The individual
parametric model is now generated from pose and shape parameters manually
measured from 3D ultrasound volume [Gao2022]. The vertebra detection technique
proposed in this paper is expected to extract these parameters automatically
based on the detected landmarks, and establish intuitive 3D parametric spine
model in the future.

## Acknowledgments

The authors would like to extend sincere gratitude to the Glenrose Hospital
for providing the subjects’ US scans in this study. And the authors also are
profusely grateful to the sponsorship from Natural Science Foundation of China
(NSFC), Grant No. 12074258.

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