# Thermal effect in hot QCD matter in strong magnetic fields

Xin-Jian Wen and Jia Zhang  Institute of Theoretical Physics, State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University,
Taiyuan, Shanxi 030006, China

###### Abstract

The quasiparticle model is improved by the free magnetic contribution to
investigated the QCD matter in a strong magnetic field. The temperature-
dependent bag function is determined by the thermodynamic consistency to
represent the difference in energy density between physical vacuum and lowest
state of QCD. It is found that the positive bag function vanishes at high
temperature indicating the deconfinement. The rapid decrease of the bag
function in stronger magnetic fields reveals the so-called inverse magnetic
catalysis. The interaction measure at high temperature remains so large that
the usual Stefan-Boltzmann limit can not be reached. We suggest a limit
$|q_{i}B_{m}|T^{2}/4$ for each landau level pressure. Finally, it is
demonstrated that the positive magnetization modified by the bag function and
free magnetic contribution indicates the paramagnetic characteristic of QCD
matter.

##  I Introduction

It has been observed that the strongly interacting matter, quark-gluon plasma
(QGP), produced in the Relativistic Heavy Ion Collider (RHIC) and the Large
Hadron Collider (LHC) behaves more like a near-perfect fluid. The QCD theory
in the magnetic background may reveal a better understanding of the QGP that
have rich collective effects Lai:2000at ; Koothottil:2020riy . Many efforts
have been done in theoretical work revealing interesting properties of
strongly interacting matter in the strong magnetic field Andersen:2014xxa ;
Miransky:2015ava . With the development of relativistic heavy ion collisions,
the study of medium effect of quark-gluon plasma becomes more active. Quark
self-energy at high temperature receives the contribution of both the electric
scale and magnetic scale, which have profound impact of confinement effects on
thermal quark collective excitation Su:2014rma ; Jamal:2017dqs .

In hot dense quark matter, one of the most important medium effects is the
effective mass generated by the nonperturbative interaction of the particles
with the system. In literature, the phenomenological models overcome the
difficulty of the QCD theory at finite temperature and/or chemical potentials.
A natural mechanism for quark confinement is given by the MIT bag model
DeTar:1983rw . The bag model was proposed to explain hadrons and quark
confinement, which artificially constrains the quarks inside a finite region
in space. However, the bag model is not able to well exhibit the phase
transition of the deconfinement. The quark quasiparticle model, as an extended
bag model, has been developed in studying the bulk properties of the dense
quark matter at finite density and temperature QPM ; Thaler:2003uz ;
Schertler:1996tq ; He:2023gva  and the strangeles in finite size Wen:2009zza .
The quasiparticle description has been assumed to be valid also in the case of
sufficiently high temperature Gorenstein:1995vm ; Peshier:1995ty ;
Levai:1997yx . The advantage of the quasiparticle model is the introduction of
the medium-dependent quark mass scale to reflect the nonperturbative QCD
properties Srivastava:2010xa  and color confinement mechanism Schneider:2001nf
. The transport properties of the quark gluon plasma have been well
investigated by the noninteracting/weakly interacting particles with effective
masses Albright:2015fpa . The hard thermal loop(HTL) approximation can also be
used to calculate the effective quark mass, but these calculations are valid
only in the perturbative regime of QCD Bellac:2011kqa ; Peshier:2002ww ;
Levai:1997yx . There are also self-consistent quasiparticle models and single
parameter quasiparticle model Koothottil:2018akg ; Bannur:2010zz ;
Bannur:2007tk . The medium effects are taken into account by considering
quarks and gluons as quasiparticles. Their temperature dependent masses are
proportional to the plasma frequency. More recent developments have shown that
quasiparticles with effective fugacity have been successful in describing the
lattice QCD results Chandra:2009jjo ; Chandra:2011en , which was initially
proposed by Chandra and RavishankarChandra:2011en ; Chandra:2007ca .

It is well known that quarks are bound inside hadron through strong
interaction, and it has not yet been found that quarks can exist
independently. Based on this fact, the nature of quark confinement was
derived. According to lattice simulations, the deconfinement phase transition
is of first order Boyd:1996bx . In some phenomenological models, an order
parameter of the deconfinement transition is the Polyakov loop which is the
trace of a Wilson line along a closed loop in the time directionLi:2018ygx .
For the SU(3) pure gauge theory, the deconfined phase transition at high
temperature corresponds to the spontaneous Z(3) symmetry breaking. However,
the study of symmetry can be complicated in QCD due to the quark dynamics. In
particular, the quark confinement and vacuum energy density can be well
described by the density-dependent bag constant Goloviznin:1992ws ;
Srivastava:2010xa ; Thaler:2003uz . In the present work, the formula of QGP
has been described as a non-interacting gas of quarks at zero chemical
potential but finite temperature, taking into account the phenomenological bag
constant. As for gluon gas, the equation of state for gluons has been
excellently described using the ideal gas approximation at high temperatures
in lattice calculations Peshier:1994zf ; Peshier:1995ty .

The paper is organized as follows. In Section II, we present the
thermodynamics of the magnetized QGP in the quasiparticle model. In Section
III, the numerical results for the confinement bag function and the
thermodynamic quantities in the strong magnetic field at finite temperatures.
The last section is a short summary.

##  II Thermodynamics of quasiparticle model in strong magnetic fields

The important feature of the quasiparticle model is the medium dependence of
quark masses in describing QCD nonperturbative properties. The quasiparticle
quark mass is derived at the zero-momentum limit of the dispersion relations
from an effective quark propagator by resuming one-loop self-energy diagrams
in the hard dense loop (HDL) approximation. The dynamical information for
gluonic degrees of freedom can also be accessed through the effective gluon
mass. In this paper, the effective quark mass $m_{q}$ and gluon mass $m_{g}$
are adopted as Levai:1997yx ; Pisarski:1989cs ; Plumari:2011mk

$\displaystyle m_{g}(T)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{6}(N_{c}+\frac{1}{2}N_{f})g^{2}T^{2}}\,,$ (1)
$\displaystyle m_{q}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}m_{i0}+\sqrt{m_{i0}^{2}+\frac{N_{c}^{2}-1}{2N_{c}}g^{2}T^{2}}\big{)}\,,$
(2)

where $m_{i0}$ denotes the quark current mass of the quark flavor $i$. The
constant $g$ is the strong interaction coupling. In order to reflect the
asymptotic freedom of QCD, one can also use a running coupling constant
$g(Q/\Lambda)$ in the equations of state of strange matter Shirkov:1997wi .
The parameterization of the coupling as a function of temperature close to the
theory is adopted as Peshier:1994zf

$\displaystyle
g^{2}(T,T_{c})=\frac{48\pi^{2}}{(11N_{c}-2N_{f})\ln(\lambda^{2}(T/T_{c}-T_{s}/T_{c})^{2})}(\frac{\Lambda
T_{c}}{T})^{\eta}.$ (3)

The current mass can be neglected for up and down quarks, while the strange
quark is taken to be massive. Because the vanishing current mass is assumed
for up and down quarks, Eq. (2) is reduced to the simple form

$\displaystyle m_{u,d}=\frac{1}{\sqrt{3}}gT.$ (4)

As a typical treatment in the quasiparticle model, one implements confinement
by introducing a bag pressure, measuring the level difference between the
physical vacuum and the ground state in the colorful world of QCD
Castorina:2011ja . To account for the essential non-perturbative features, the
corresponding thermodynamics is based on the ideal gas partition function with
additional contributions,

$\displaystyle T\ln Z=T\ln(Z_{0}Z_{\mathrm{vac}}Z_{\mathrm{mag}}),$ (5)

where the vacuum partition depends on the bag function $T\ln
Z_{\mathrm{vac}}=-B(T)V$ Castorina:2011ja  and the term $Z_{\mathrm{mag}}$
stands for the free magnetic field contribution. The conventional matter
contribution is introduced by $Z_{0}$. So the total partition function can
produce the thermodynamic quantities from the self-consistent relation. Within
the framework of the temperature-dependent mass $m(T)$, the pressure of the
system is expressed as

$\displaystyle P(T,B_{m})=\frac{T}{V}\ln
Z(T,B_{m})=-\sum_{i=q,g}[\Omega_{i}(T,B_{m})+B_{i}(T,B_{m})]-B_{0}-{\cal
V}(T_{0},B_{m}).$ (6)

The first term $\Omega_{i}$ is the conventional matter contribution. The bag
constant $B_{0}$ stands for the vacuum energy density at zero temperature. The
variant term $B_{i}(T,B_{m})$ resembles the interaction term from
quasiparticles and can be interpreted as the thermal vacuum energy density. We
assume Maxwell term ${\cal V}(T_{0},B_{m})$ is independent on the quark mass
but on the magnetic field, which represent the free pressure of magnetic
contribution Menezes:2008qt ; Mizher:2010zb ; Fraga:2012fs

$\displaystyle{\cal
V}(T_{0},B_{m})=-\sum_{i=q}\frac{N_{c}|e_{i}B_{m}|^{2}}{2\pi^{2}}\left[\zeta^{\prime}(-1,x_{i})-\zeta^{\prime}(-1,0)-\frac{1}{2}(x_{i}^{2}-x_{i})\ln(x_{i})+\frac{x_{i}^{2}}{4}\right],$
(7)

where the parameter $x_{i}=m_{i}^{2}/(2|q_{i}B_{m}|)$ is defined at the
moderate temperature $T_{0}=150$ MeV in the thermal bag. The constant
$\zeta^{\prime}(-1,0)=-0.165421...$ was introduced in Ref Fraga:2012fs , which
is helpful to maintain a positive magnetic pressure. The presence of ${\cal
V}(T_{0},B_{m})$ would not change the thermodynamically self-consistent
relation $\partial P/\partial m_{i}=0$ in quasiparticle model. So we have the
following differential equation

$\displaystyle\frac{dB_{i}}{dT}\frac{dT}{dm_{i}}=-\frac{\partial\Omega_{i}}{\partial
m_{i}}.$ (8)

So the temperature dependent term $B_{i}(T,B_{m})$ is

$\displaystyle B_{i}(T,B_{m})=-\int_{0}^{T}\frac{\partial\Omega_{i}}{\partial
m_{i}}\frac{dm_{i}}{dT}dT.$ (9)

The derivative $\partial\Omega_{i}/\partial m_{i}$ is

$\displaystyle\frac{\partial\Omega_{i}}{\partial
m_{i}}=\frac{d_{i}|e_{i}B_{m}|}{\pi^{2}}\sum_{\nu=0}^{\infty}(2-\delta_{\nu
0})\int_{0}^{\infty}f(\varepsilon_{i})\frac{m_{i}}{\varepsilon_{i}}dp_{z}.$
(10)

where the fermion distribution function is
$f(\varepsilon_{i})=1/({1+\exp(\frac{\varepsilon_{i}}{T})})$ and the single
particle energy is $\varepsilon_{i}=\sqrt{p_{z}^{2}+m_{i}^{2}+2\nu
e_{i}B_{m}}$ due to the quantization of orbital motion of charged particles in
the presence of a strong magnetic field along the $z$ axis Chakrabarty:1996te
. The derivative of the mass $m(T)$ with respect to the temperature is

$\displaystyle\frac{dm_{i}}{dT}=\frac{N_{c}^{2}-1}{4N_{c}}\frac{g^{2}T}{\sqrt{m_{0}^{2}+(N_{c}^{2}-1)g^{2}T^{2}/(2N_{c})}}+\frac{\partial
m_{i}}{\partial g^{2}}\frac{\partial g^{2}}{\partial T},$ (11)

which will be simplified as $\frac{dm_{i}}{dT}=\frac{g}{\sqrt{3}}$ for zero
current mass and the constant coupling $\frac{\partial g}{\partial T}=0$. If
one takes into account the running coupling $g(T)$, the derivative of the mass
$m_{i}(T)$ with respect to the $T$ should be calculated through the full
differential relation.

The entropy density, as a measure of phase space, is unaffected by
$B_{i}(T,B_{m})$ Thaler:2003uz , which is clearly understood from the relation
(8). Similar to the number density, the entropy density is written based on
the fundamental thermodynamic relation,

$\displaystyle s_{i}=-\frac{\partial\Omega_{i}}{\partial
T}=\frac{d_{i}|e_{i}B_{m}|}{\pi^{2}}\sum_{\nu=0}^{\infty}(2-\delta_{\nu
0})\int_{0}^{\infty}f(\varepsilon_{i})\frac{p^{2}+\varepsilon_{i}^{2}}{T\varepsilon_{i}}dp_{z}.$
(12)

The net effect of the bag function is to cancel the entropy density
contribution, which would arise from the dependence of the mass $m(T)$ on the
temperature. It is well known to us that the energy density and pressure
should vanish in vacuum. So the pressure should be normalized by requiring
zero pressure at zero temperature as

$\displaystyle P^{\mathrm{eff}}(T,B_{m})=P(T,B_{m})-P(0,B_{m}).$ (13)

The magnetization is an important thermodynamic quantity in understanding the
QCD matter Ferrer:2019xlr . The development of the study on the magnetization
in various methods has been summarized in Ref. Cao:2023bmk . At zero
temperature, the magnetization is found to be positive and to be responsible
for the anisotropic pressures Felipe:2007vb ; Strickland:2012vu . We propose
the expression of the magnetization in quasiparticle model as,

$\displaystyle{\cal{M}}=\frac{\partial P^{\mathrm{eff}}}{\partial
B_{m}}=-\sum_{i=u,d,s}\left(\frac{\partial\Omega_{i}}{\partial
B_{m}}+\frac{\partial B_{i}}{\partial B_{m}}+\frac{\partial{\cal
V}_{i}}{\partial B_{m}}\right),$ (14)

where the first term is the conventional contribution from the pure
quasiparticle Menezes:2015fla . The second term demonstrates that the
effective bag function would have additional contribution to the
magnetization, which reveals the medium effect on the hot quark matter. It can
be written as,

$\displaystyle\frac{\partial B_{i}}{\partial
B_{m}}=\frac{N_{c}|e_{i}|m_{i}}{\pi^{2}}\sum_{\nu=0}^{\infty}(2-\delta_{\nu
0})\int_{0}^{\infty}f(\varepsilon_{i})\left(\frac{\nu
B_{m}}{\varepsilon^{2}}+f(\varepsilon_{i})\frac{\nu|e_{i}|B_{m}}{\varepsilon_{i}T}-1\right)\frac{dp_{z}}{\varepsilon_{i}}.$
(15)

##  III Numerical result and conclusion

In the framework of the preceding quasiparticle model, we have done the
numerical calculations with the quark current mass values $m_{u}=m_{d}$= 0,
and $m_{s}$=120 MeV. The constant term $B_{0}$ is $(145\mathrm{MeV})^{4}$. The
effective bag constant acts as an energy penalty for the deconfined phase. In
Fig. 1, the effective bag constant $B(T,B_{m})$ is shown as a function of the
temperature at different magnetic fields $B_{m}=0.2$, 0.4, 0.6 GeV2. It falls
to zero at high temperature, which means that the deconfined state has larger
pressure and is energetically preferred. Compared with the fixed coupling
constant $g=3$, the running coupling constant $g(T)$ leads to a decrease of
$B(T,B_{m})$ at higher temperature. The increase of the magnetic field will
not change the vacuum energy $B(T,B_{m})$ at $T=0$. But the decrease of the
bag constant would be more rapidly in stronger magnetic fields, which
indicates a lower value of the critical temperature for the deconfinement
transition.

Figure 1: The effective bag function $B(T,B_{m})$ is shown as functions of
temperatures denoting the deconfinement transition. The $B(T,B_{m})$ decreases
more rapidly with increasing temperatures at stronger magnetic fields for both
constant coupling and running coupling constants.

In the quasiparticle model, the decrease of the effective bag constant
$B(T,B_{m})$ denotes the deconfinement transition. The critical temperature
can be determined by the position of half-height of the bag constant. In Fig.
2, the pseudocritical temperature is plotted by the red solid curve. For the
convenience of comparison, the result from PNJL is marked by the black dashed
curve Tavares:2021fik . It can be clearly seen that the trend of the decrease
of the $T_{pc}$ with the magnetic field is close to the LQCD result marked by
the shadow band in the panel Bali:2011qj . Moreover, the pseudocritical
temperature decreases by about 10 percent of its original value at zero
magnetic field. Our result is in agreement with the so-called inverse magnetic
catalysis effect revealed by the LQCD.

Figure 2: The pseudocritical temperature for the deconfinement transition in
quasiparticle model (red curve) compared with the results of the PNJL model
(dashed curve) Tavares:2021fik  and of LQCD (shadow band )Bali:2011qj .

Fig. 3 illustrates how the pressure and the energy density (in unit of
$T^{4}$) depend on the temperature of the medium. In the region around and
just above the critical temperature, the energy density rises much more
rapidly than the pressure, which gives rise to a observed rapid increase of
the interaction measure. At stronger magnetic fields, the larger pressure and
energy density are obtained at the high temperature. However, the temperature
is not the only scale with the dimension of the energy. The pressure as well
as the energy density can not asymptotically converge to their Stefan-
Boltzmann value $P^{\mathrm{SB}}/T^{4}=$constant at $T\rightarrow\infty$. The
deviations from the usual Stefan-Boltzmann values are due to the quarks are
constrained by the landau level in strong magnetic fields.

Figure 3: The scaled pressure $P^{\mathrm{eff}}/T^{4}$ on left panel and
energy density $\epsilon/T^{4}$ on right panel are always increasing functions
of the temperature. It is apparent that the Stefan-Boltzmann (S-B) limit is
absent at high temperatures in strong magnetic fields.

Interaction measure is the trace of energy-momentum tensor. For non-
interacting massless constituents (the “conformal” limit), is zero, so that
the temperature is the only scale. In a strong magnetic field, the interaction
measure is defined as $\Delta(T)\equiv(\varepsilon-3P)/T^{4}$ for quark gluon
plasma. In Fig. 4, the trace anomaly of the energy momentum tensor is plotted
as a function of temperature. The so-called interaction measure normalized by
$T^{4}$ gives the deviation from the free gas relation between the energy
density and the pressure and is also a measure of the breakdown of conformal
symmetry. Even though the temperature dependent coupling constant is employed
to realize the asymptotic freedom at high temperature, the non-zero value
indicates that some interactions must still be present due to the landau
levels in magnetic fields. The $\Delta(T)$ at high temperature will remain
larger at stronger magnetic fields, which is in agreement with LQCD that the
interaction measure remains large even at very high temperatures, where the
Stefan-Boltzmann (S-B) limit is not yet reached.

Figure 4: The interaction measure is shown as functions of the temperatures in
different magnetic fields. It remains so larger at high temperatures that the
interaction must be still present at $eB_{m}=0.6$ GeV2 marked by the solid
curve.

In fact, the S-B limit for the the $n$-th landau level can be defined as,

$\displaystyle\frac{P^{\mathrm{(n)}}_{\mathrm{SB}}}{|q_{i}B_{m}|T^{2}}=\frac{1}{4}.$
(16)

In Fig. 5, the pressures and entropy of $i$-flavor quarks in the $n$-th Landau
level are shown as the function of temperatures. At sufficiently high
temperatures, the scaled pressure $P^{(n)}/(|q_{i}B_{m}|T^{2})$ on the left
panel can approach the limit marked by the dash-dotted horizontal line. In
particular, the pressure from the lowest landau level (n=0) marked by the
solid line is close to the S-B limit line. Moreover, much higher temperature
is required for any excited levels (n=1, 2) to reach the limit. It can be
accounted for by the fact that the higher level leads to the larger effective
mass and therefore results in a larger deviation from the S-B limit.
Correspondingly, the entropy $S^{(n)}/(|q_{i}B_{m}|T)$ in the $n$-level has
the S-B limit on right panel in the strong magnetic field.

Figure 5: The scaled pressure and entropy of $i$-flavor quarks lying in Landau
levels $n=0$ , 1 and 2. At high enough temperatures, the three lines approach
the Stefan-Boltzman limit marked by the dash-dotted horizontal line. It is
characteristic that the limit is early approached in the lowest landau level.

In Fig. 6, the magnetization of strange quark matter is shown as a function of
the temperature at fixed magnetic fields $eB_{m}=0.2$, 0.4, 0.6 GeV2 marked by
the black, red, and blue solid curves, respectively. The positive value
produces the paramagnetic characteristic for the whole temperature range
Bali:2013owa . The magnetization increases with increasing temperature at
fixed magnetic fields. It can be understood that the more Landau levels at
high temperature the stronger the magnetization. The effective bag function
marked by the red dotted line enhances the magnetization at finite
temperature. The discrepancy of the magnetization at low temperature is
sizable due to the free magnetic contribution. By comparison with the lattice
result marked by the scattering triangles at $eB_{m}=0.2$ and 0.4 GeV2
Bali:2014kia , it can be concluded that the ascending trend and the magnetic
effect are consistent. In particular, increase of the magnetic field enhances
the magnetization of the quark matter.

Figure 6: The magnetization of the strange quark matter is shown as a function
of the temperature at $eB_{m}=0.2$, 0.4, 0.6 GeV2. The LQCD result at
$eB_{m}=0.2$ and 0.4 GeV2 is marked by scattering triangles for comparison
Bali:2014kia .

##  IV Summary

In this paper, we have investigated the hot QCD matter exposed to sufficiently
high magnetic fields, which could be generated in RHIC experiments. The
quasiparticle model is extended by including the free magnetic contribution
and the effective thermomagnetic bag constant, which is self-consistent
derived to represent the confinement. The running coupling constant has been
employed to reflect the asymptotic freedom of QCD. It has been found that the
decrease of the effective bag constant at the high temperature indicates the
occurence of the deconfinement transition. Moreover, the stronger magnetic
field results in a more rapidly decrease of the effective bag constant, which
provides a novel method to account for the so-called inverse magnetic
catalysis effect. Moreover, the paramagnetic characteristic of QCD is obtained
in the quasiparticle model. The effective bag constant would have an
additional contribution in the new definition of the magnetization due to the
medium effect. It is concluded that the magnetization modified by the function
and free magnetic contributions can only account for the trend revealed by the
lattice result. It would be of some interest to improve the quasiparticle
model to quantitatively interpret the lattice results in future.

For the quark-gluon plasma in strong magnetic field, the interaction measure
remains larger even at very high temperature and indicates some interactions
are present. Therefore, the usual S-B limit is not applicable. The deviation
form the S-B limit becomes remarkable with the massive effective mass led by
the stronger magnetic field. Not only the temperature but also the magnetic
field are the scale of energy. We suggested that for single landau level, the
S-B limit of the quark pressure can be defined as $|q_{i}B_{m}|T^{2}/4$. It
has been shown that the lowest landau level is close to the S-B limit, while
higher temperature is required for the excited levels to approach the limit.

###### Acknowledgements.

The authors would like to thank support from the National Natural Science
Foundation of China under the Grant Nos. 11875181, 12047571, and 11705163.
This work was also sponsored by the Fund for Shanxi ”1331 Project” Key
Subjects Construction.

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Axially Symmetric Cosmological Mesonic Stiff Fluid Models in Lyra’s Geometry
Ragab M. Gad111Email Address<EMAIL_ADDRESS>
Mathematics Department, Faculty of Science,

Minia University, 61915 El-Minia, EGYPT.

###### Abstract

In this paper, we obtained a new class of axially symmetric cosmological
mesonic stiff fluid models in the context of Lyra’s geometry. Expressions for
the energy, pressure and the massless scalar field are derived by considering
the time dependent displacement field. We found that the mesonic scalar field
depends on only $t$ coordinate. Some physical properties of the obtained
models are discussed.

Keywords: Lyra’ geometry; axially symmetric space-time; mesonic scalar field;
stiff fluid.

##  1 Introduction

After Einstein proposed his theory of general relativity he succeeded in
geometrizing the phenomena of gravitation by expressing gravitational in terms
of metric tensor $g_{ij}$. This idea of geometrizing gravitation inspired many
physicists to generalize the theory in order to incorporate electrodynamics as
a purely geometrical construct. One of the first attempts in this direction
was made in 1918 by Weyl [1] who suggested a theory based on a generalization
of Riemannian geometry, by formulating a new kind of gauge theory involving
metric tensor, to geometrize gravitation and electromagnetism. This theory was
criticized due to non-integrability of length of vector under parallel
displacement 222In the theory of general relativity if a vector undergoes a
parallel displacement, its direction may change, but not its length. While in
Weyl’s geometry not only the direction but the length may change and depends
on the path between two points, which show that the length is not integrable..
Also as pointed out by Einstein that this theory implies that frequency of
spectral lines emitted by atoms would not remain constant but would depend on
their past histories, which is in contradiction to observed uniformity of
their properties [2]. Nevertheless, completely apart from these criticisms,
Weyl’s geometry provides an interesting example of non-Riemannian connections.
Folland [3] gave a global formulation of Weyl manifolds clarifying
considerably many of Weyl’s basic ideas thereby.

In 1951 Lyra [4] proposed a modification of Riemannian geometry by introducing
a gauge function into the structure less manifold which is in close
resemblance to Weyl’s geometry [5]. This modification was to overcome the
problems appeared in Weyl’s geometry and is more in keeping with the spirit of
Einstein’s principle of geometrization, since both the scalar and tensor
fields have more or less intrinsic geometrical significance. In this way
Riemannian geometry was given a new modification and the modified geometry was
named as Lyra’s geometry. For physical motivation of Lyra’s geometry we refer
to the literature [6]-[8]. However, in contrast to Weyl’s geometry, in Lyra’s
geometry the connection is metric preserving and length transfers are
integrable as in Riemannian geometry. Subsequently, Sen [9] and Sen and Dunn
[2] proposed a new scalar tensor theory of gravitation. They constructed an
analog of Einstein field equation based in Lyra’s geometry, see equation
(2.6). Sen [9] found that static model with finite density in Lyra’s geometry
is similar to the static Einstein model, but a signification differences was
that the model exhibited red shift. Halford [10] pointed out that the constant
displacement vector field $\phi_{i}$ in Lyra’s geometry plays the role of a
cosmological constant in the normal general relativistic treatment. Halford
[11] showed that the scalar tensor treatment based in Lyra’s geometry predicts
the same effects, within observational limits, as in Einstein theory. Several
attempts have been made to cast the scalar tensor theory of gravitation in
wider geometrical context [12].

Many Authors [13] have studied cosmological models based on Lyra’s geometry
with a constant displacement field vector in the time-direction. Singh and his
collaborators [14] have studied Bianchi type I, III, Kantowski-Sachs and new
class of models with a time dependent displacement field. They have made a
comparative study of Robertson-Walker models with a constant deceleration
parameter in Einstein’s theory with a cosmological terms and in the
cosmological theory based on Lyra’s geometry.

Recently, several authors [15]\- [21] studied cosmological models based on
Lyra’s geometry in various contexts. With these motivations, in this paper, we
obtained exact solutions of the field equations for mesonic stiff fluid models
in axially symmetric space-times within the frame work of Lyra’s geometry for
time varying displacement field vector.

Axially symmetric cosmological models have been studied in both Riemannian and
Lyra geometries. In context of general relativity theory, by adopting the
comoving coordinate system, these models with string dust cloud source are
studied by Bhattacharaya and Karade [22]. They shown that some of these models
are singular free even at an initial epoch. In the context of Lyra’s geometry
these models are studied in the presence of cosmic string source and thick
domain walls [23] and in the presence of perfect fluid distribution [24].

This paper is organized as follows: The metric and field equations are
presented in section 2. Section 3 deals with solving the field equations.
Finally, in section 4, concluding remarks are given.

##  2 Fundamental Concepts and Field Equations

Consider the axially symmetric metric [22] in the form

$ds^{2}=dt^{2}-A^{2}(t)(d\chi^{2}+f^{2}(\chi)d\phi^{2})-B^{2}(t)dz^{2},$ (2.1)

with the convention $x^{0}=t$, $x^{1}=\chi$, $x^{2}=\phi$, $x^{3}=z$ and $A$
and $B$ are functions of $t$ only while $f$ is a function of the coordinate
$\chi$ only.

The volume element of the model (2.1) is given by

$\mathcal{V}=\sqrt{-g}=A^{2}fB$ (2.2)

The four-acceleration vector, the rotation, the expansion scalar and the shear
scalar characterizing the four velocity vector field, $u^{a}$, respectively,
have the usual definitions as given by Raychaudhuri [25]

$\begin{array}[]{ccc}\dot{u}_{i}&=&u_{i;j}u^{j},\\\
\omega_{ij}&=&u_{[i;j]}+\dot{u}_{[i}u_{j]},\\\ \Theta&=&u^{i}_{;i},\\\
\sigma^{2}&=&\frac{1}{2}\sigma_{ij}\sigma^{ij},\end{array}$ (2.3)

where

$\sigma_{ij}=u_{(i;j)}+\dot{u}_{(i}u_{j)}-\frac{1}{3}\Theta(g_{ij}+u_{i}u_{j}).$

In view of the metric (2.1), the four-acceleration vector, the rotation, the
expansion scalar and the shear scalar given by (2.3)can be written in a
comoving coordinates system as

$\begin{array}[]{ccc}\dot{u}_{i}&=&0,\\\ \omega_{ij}&=&0,\\\
\Theta&=&\frac{2\dot{A}}{A}+\frac{\dot{B}}{B},\\\
\sigma^{2}&=&\frac{1}{9}\big{(}11\big{(}\frac{\dot{A}}{A}\big{)}^{2}+5\big{(}\frac{\dot{B}}{B}\big{)}^{2}+\frac{2\dot{A}\dot{B}}{AB}\big{)}.\end{array}$
(2.4)

The non vanishing components of the shear tensor$\sigma_{ij}$ are

$\begin{array}[]{ccc}\sigma_{11}&=&A(\frac{1}{3}\Theta A-\dot{A}),\\\
\sigma_{22}&=&Af^{2}(\frac{1}{3}\Theta A-\dot{A}),\\\
\sigma_{33}&=&B(\frac{1}{3}\Theta B-\dot{B}),\\\
\sigma_{44}&=&-\frac{2}{3}\Theta.\par\end{array}$ (2.5)

The field equations in normal gauge for Lyra’s geometry as obtained by Sen [9]
(in gravitational units $c=8\pi G=1)$ read as

$R_{ij}-\frac{1}{2}Rg_{ij}=-T_{ij}-\frac{3}{2}\phi_{i}\phi_{j}+\frac{3}{4}g_{ij}\phi_{\alpha}\phi^{\alpha},$
(2.6)

the left hand side is the usual Einstein tensor, whereas $\phi_{i}$ is a time-
like displacement field vector defined by

$\phi_{i}=(0,0,0,\lambda(t)),$

and $T_{ij}$ is the energy momentum tensor corresponding to perfect fluid and
massless mesonic scalar field and is given by

$T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij}+V_{,i}V_{,j}-\frac{1}{2}g_{ij}V_{,k}V^{,k}.$
(2.7)

Here $p$ is the pressure, $\rho$ the energy density and $u_{i}$ the four
velocity vector satisfying the relation in co-moving coordinate system

$g_{ij}u^{i}u^{j}=1,\qquad u^{i}=u_{i}=(1,0,0,0).$

However, $V$ is the massless scalar field and we assume it to be a function of
$t$ and $\chi$ coordinates. The Scalar field $V$ is governed by the Klein-
Gordan wave equation

$g^{ij}V_{;ij}=0.$

For the line element (2.1), the field equations (2.6) with equation (2.7) lead
to the following system of equations

$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})-\frac{V^{\prime
2}}{2A^{2}},$ (2.8)
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.9)
$\frac{2\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}-\frac{f^{\prime\prime}}{fA^{2}}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.10)
$(\frac{\dot{A}}{A})^{2}+\frac{2\dot{A}\dot{B}}{AB}-\frac{f^{\prime\prime}}{fA^{2}}-\frac{3}{4}\lambda^{2}=(\rho+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.11)
$\dot{\rho}+(\rho+p)(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})=0,$ (2.12)
$\ddot{V}-\frac{1}{A^{2}}V^{\prime\prime}+(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})\dot{V}-\frac{ff^{\prime}}{f^{2}A^{2}}V^{\prime}=0.$
(2.13)

Here the over heat dot denotes differentiation with respect to $t$ and over
head prime denotes differentiation with respect to $\chi$.
From equations (2.8) and (2.9), we get

$V^{\prime}=0.$ (2.14)

Consequently, the mesonic scalar field does not exit in the direction of
$\chi$.

The functional dependence of the metric together with equations (2.9) and
(2.10), using equation (2.14), imply that

$\frac{f^{\prime\prime}}{f}=k^{2},\qquad k^{2}=\text{constant}.$ (2.15)

If $k=0$, then the solution of this differential equation is
$f(\chi)=k_{1}\chi+k_{2}$, $k_{1}$ and $k_{2}$ are constants of integration.
Without loss of generality, we choose $k_{1}=1$ and $k_{2}=0$. Thus we shall
have

$f(\chi)=\chi.$ (2.16)

In the case $f(\chi)=\chi$ the field equations (2.8)-(2.13), using equation
(2.14), reduce to

$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2}),$
(2.17)
$\frac{2\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2}),$
(2.18)
$(\frac{\dot{A}}{A})^{2}+\frac{2\dot{A}\dot{B}}{AB}-\frac{3}{4}\lambda^{2}=(\rho+\frac{1}{2}\dot{V}^{2}),$
(2.19) $\dot{\rho}+(\rho+p)(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})=0,$ (2.20)
$\ddot{V}+(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})\dot{V}=0.$ (2.21)

We can easily find from (2.21) that

$\dot{V}=\frac{n}{A^{2}B},$ (2.22)

where $n(\neq 0)$ is a constant of integration.

From equations (2.17) and (2.18), we get

$\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}=\frac{\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}.$
(2.23)

We assume $A$ to be some arbitrary function of $B$, say

$A=\psi(B).$ (2.24)

So equation (2.23) becomes

$\big{(}\frac{\psi_{B}}{\psi}-\frac{1}{B}\big{)}\ddot{B}+\big{[}\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}\big{]}\dot{B}^{2}=0,$
(2.25)

where $\psi_{A}=\frac{d\psi}{dA}$. Equation (2.25) results in the following
possibilities.
(i-1)

$\frac{\psi_{B}}{\psi}-\frac{1}{B}=0,\quad\text{and}\qquad\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}=0,$
(2.26)

(i-2)

$\ddot{B}=0,\quad\text{and}\qquad\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}=0,$
(2.27)

(i-3)

$\dot{B}=0.$ (2.28)

##  3 Solutions of Field Equations

We have only five highly non-linear field equations (2.17)-(2.21) in sex
unknowns , $A,B,p,\rho,V$ and $\lambda$. In order to obtain its exact
solution, we assume one more physically reasonable condition amongst these
variables. We consider here the effective "stiff fluid" distribution, that is,
a perfect fluid with the equation of state:

$p=\rho.$ (3.1)

The equation of state (3.1) was apparently first proposed by Zeldovich [26].
It should have applied in the early Universe, the justification being the
observation that with (3.1) the velocity of sound equals the velocity of
light, so no material in this Universe could be more stiff.
Using the condition (3.1) in equation (2.20) and by integrating, we get

$p=\rho=\frac{m}{A^{4}B^{2}},$ (3.2)

where $m(\neq 0)$ is a constant of integration.

Case (i-1):
From equation (2.26), using the first equation in the second equation, then by
integrating, we get

$\psi=c_{1}B+c_{2},$ (3.3)

where $c_{1}(\neq 0)$ and $c_{2}$ are integration constants. Using this result
in the first equation in (2.26), we get $c_{2}=0$. So that equation (3.3)
becomes

$\psi=c_{1}B.$ (3.4)

Using this result in equation (2.25), we have

$A=c_{1}B.$ (3.5)

Now using this equation and the condition (3.1), in equations (2.17)-(2.19),
we get

$\frac{2\ddot{B}}{B}+(\frac{\dot{B}}{B})^{2}+\frac{3}{4}\lambda^{2}=-\frac{3}{2}\rho,$
(3.6) $3(\frac{\dot{B}}{B})^{2}-\frac{3}{4}\lambda^{2}=\frac{3}{2}\rho.$ (3.7)

These equations yield

$B=(at+b)^{\frac{1}{3}},$ (3.8)

where $a(\neq 0)$ and $b$ are constants of integration.
According to equations (3.5) and (3.8) the line element (2.1) can be written
in the following form

$ds^{2}=dt^{2}-(at+b)^{\frac{2}{3}}(c_{1}^{2}d\chi^{2}+c_{1}^{2}\chi^{2}d\phi^{2}+dz^{2}),$
(3.9)

Physical properties of the model

Using equations (3.5) and (3.8) in equations (2.19)-(2.21), take into account
(3.1), the expressions for density $\rho$, pressure $p$, massless scalar field
$V$ and displacement field $\lambda$ are given by

$\rho=p=\frac{c}{(at+b)^{2}},\qquad c=\frac{m}{c_{1}^{4}},$

which shows that $\rho$ and $p$ are not singular,

$V=\frac{n}{ac_{1}^{2}}\log(at+b)+c_{3},$
$\lambda^{2}=\frac{c_{4}}{(at+b)^{2}},\qquad
c_{4}=\frac{4a^{2}}{9}-\frac{2(2m+n^{2})}{3c_{1}^{4}}.$

It is observed, from equations (3.5) and (3.8) that $A(t)$ and $B(t)$ can be
singular only for $t\rightarrow\infty$. Thus the line element (3.9) is
singular free even at $t=0$.

For the line element (3.9), using equations (2.2), (2.4) and (2.5), we have
the following physical properties:
The volume element is

$\mathcal{V}=c_{1}^{2}\chi(at+b).$

This equation shows that the volume increases as the time increases, that is,
the model (3.9) is expanding with time.
The expansion scalar, which determines the volume behavior of the fluid, is
given by

$\Theta=\frac{a}{at+b}.$

The only non-vanishing component of the shear tensor, $\sigma_{ij}$, is

$\sigma_{44}=-\frac{2a}{3(at+b)}.$

Hence, the shear scalar $\sigma$ is given by

$\sigma^{2}=2\big{(}\frac{a}{3(at+b)}\big{)}^{2}.$

Since $\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, then the model
(3.9) does not approach isotropy for large value of $t$. Also the model does
not admit acceleration and rotation, since $\dot{u}_{i}=0$ and $\omega_{ij}=0$
.

Case (i-2):

The first equation in (2.27) gives

$B=b_{1}t+b_{2},$ (3.10)

where $b_{1}(\neq 0)$ and $b_{2}$ are constants of integration. Using this
result in the second equation in (2.27) and take into account (2.24), we get

$A^{2}=b_{3}(b_{1}t^{2}+2b_{2}t)+b_{4},$

where $b_{3}(\neq 0)$ and $b_{4}$ are constants of integration. This equation
can be written in the form

$A=(b_{3}B^{2}+b_{5})^{\frac{1}{2}},$ (3.11)

where the constant $b_{5}$ depends on the constants $b_{1}$ and $b_{2}$.
Using equations (3.10) and (3.11) in equations (2.17)-(2.21), take into
account the condition $\rho=p$, we get

$\rho=p=\frac{m}{(b_{1}t+b_{2})^{2}}[b_{3}(b_{1}t+b_{2})^{2}+b_{5}]^{2},$
$V=\frac{n}{b_{3}b_{5}}\log\frac{b_{1}t+b_{2}}{\sqrt{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}}+b_{6},$

where $b_{6}$ is the constant of integration,

$\lambda^{2}=\frac{2(2b_{1}^{2}b_{3}^{2}(b_{1}t+b_{2})^{3}-2m-n^{2})}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})^{2}(b_{1}t+b_{2})^{2}}-\frac{8b_{1}^{2}b_{3}}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}.$

In this case the line element (2.1) takes the following form

$ds^{2}=dt^{2}-(b_{3}(b_{1}t+b_{2})^{2}+b_{5})(d\chi^{2}+\chi^{2}d\phi^{2})-(b_{1}t+b_{2})^{2}dz^{2}.$
(3.12)

We shall now give the expression for kinematic quantities. A straightforward
calculation leads to the expressions for element volume $\mathcal{V}$,
expansion scalar $\Theta$ and shear tensor $\sigma_{ij}$ of model (3.12) are
given, respectively, by

$\mathcal{V}=\chi(b_{1}t+b_{2})(b_{3}(b_{1}t+b_{2})^{2}+b_{5}),$

which shows that the model is expanding with time,

$\Theta=\frac{2b_{1}b_{3}(b_{1}t+b_{2})}{(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}+\frac{b_{1}}{(b_{1}t+b_{2})},$
$\sigma_{11}=\frac{b_{1}(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}{3(b_{1}t+b_{2})}-\frac{b_{1}b_{3}(b_{1}t+b_{2})}{3},$
$\sigma_{22}=\chi^{2}\sigma_{11},$
$\sigma_{33}=\frac{2b_{1}b_{3}(b_{1}t+b_{2})^{3}}{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}-\frac{2b_{1}(b_{1}t+b_{2})}{3},$
$\sigma_{44}=-\frac{4b_{1}b_{3}(b_{1}t+b_{2})}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}-\frac{2b_{1}}{3(b_{1}t+b_{2})},$

and other components of the shear tensor, $\sigma_{ij}$, being zero. Hence

$\sigma^{2}=\frac{1}{9}\Big{(}11\frac{b_{1}^{2}b_{3}^{2}(b_{1}t+b_{2})^{2}}{(b_{1}t+b_{2})^{2}+b_{5})^{2}}+5\big{(}\frac{b_{1}}{b_{1}t+b_{2}}\big{)}^{2}+\frac{2b^{2}_{1}b_{3}}{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}\Big{)}.$

Moreover, this model represents non-rotating and has vanishing acceleration.
Since $\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, then the model
(3.12) do not approach isotropy for large value of $t$.

Case (i-3):

Equation (2.28) can be easily integrated to give

$B=\ell,$ (3.13)

where $\ell(\neq 0)$ is the constant of integration. Using this equation in
equations (2.17)-(2.19) and take into account the condition (3.2),we have

$A=(\ell_{1}t+\ell_{2})^{\frac{1}{2}},$ (3.14)

where $\ell_{1}(\neq 0)$ and $\ell_{2}$ are constants of integration. Using
equations (3.13) and (3.14) in equations (2.17)-(2.21), take into account the
condition $\rho=p$, we get

$\rho=p=\frac{m}{\ell^{2}(\ell_{1}t+\ell_{2})^{2}},$
$V=\frac{n}{\ell\ell_{1}}\log(\ell_{1}t+\ell_{2})+\ell_{3},$

where $\ell_{3}$ is the constant of integration,

$\lambda^{2}=\frac{\ell_{4}}{(\ell_{1}+\ell_{2})^{2}},\qquad\ell_{4}=\big{(}3\ell_{1}^{2}-\frac{2(2m+n^{2})}{3\ell^{2}}.$

In this case the line element (2.1) can be written in the form

$ds^{2}=dt^{2}-(\ell_{1}t+\ell_{2})(d\chi^{2}+\chi^{2}d\phi^{2})-\ell^{2}dz^{2}.$
(3.15)

From equation (3.14), one can observed that $A(t)$ is singular only when
$t\rightarrow\infty$. Consequently, the line element (3.15) is singular free
even $t=0$.
For the line element (3.15), the volume element $\mathcal{V}$ and the
kinematics properties (acceleration $\dot{u}_{i}$, rotation $\omega_{ij}$,
expansion scalar $\Theta$, shear tensor $\sigma_{ij}$ and shear scalar
$\sigma$) respectively found to have the following expressions:

$\mathcal{V}=\ell\chi(\ell_{1}t+\ell_{2}),$

which shows that the model is expanding with time,

$\dot{u}=0,$ $\omega_{ij}=0,$ $\Theta=\frac{\ell_{1}}{\ell_{1}t+\ell_{2}},$

$\sigma_{11}=-\frac{\ell_{1}}{6},$ $\sigma_{22}=\chi^{2}\sigma_{11},$
$\sigma_{33}=\frac{\ell\ell_{1}}{3(\ell_{1}t+\ell_{2})},$
$\sigma_{44}=-\frac{2\ell_{1}}{3(\ell_{1}t+\ell_{2})}.$
$\sigma^{2}=\frac{11\ell_{1}^{2}}{36(\ell_{1}t+\ell_{2})}.$

As in the above two cases,
$\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, the model (3.15) do
not approach isotropy for large value of $t$.

##  4 Conclusions

This paper deals with axially symmetric space-time in the presence of mesonic
stiff fluid distribution within the framework of Lyra’s geometry for time
dependent displacement field. We have presented a new class of exact solutions
of Einstein’s field equations for this space-time. The obtained models
represent shearing, non-rotating and expanding with time $t$. Moreover, these
models are singular free even at the initial epoch $t=0$ and have vanishing
accelerations. For all models, we found also that
$\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, this means that they
are not approach isotropy for large time $t$. We found also that the mesonic
scalar field in axially symmetric space-time exists only in the t-direction.

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# Objective Surgical Skills Assessment and Tool Localization: Results from the
MICCAI 2021 SimSurgSkill Challenge

Aneeq Zia  Intuitive, Inc.  Kiran Bhattacharyya  Intuitive, Inc.  Xi Liu
Intuitive, Inc.  Ziheng Wang  Intuitive, Inc.  Max Berniker  Intuitive, Inc.
Satoshi Kondo  Muroran Institute of Technology  Emanuele Colleoni  University
College London  Dimitris Psychogyios  University College London  Yueming Jin
University College London  Jinfan Zhou  Shandong University  Evangelos
Mazomenos  University College London  Lena Maier-Hein  German Cancer Research
Center (DKFZ)  Danail Stoyanov  University College London  Stefanie Speidel
National Center for Tumor Diseases (NCT)  Anthony Jarc  Intuitive, Inc.

###### Abstract

Timely and effective feedback within surgical training plays a critical role
in developing the skills required to perform safe and efficient surgery.
Feedback from expert surgeons, while especially valuable in this regard, is
challenging to acquire due to their typically busy schedules, and may be
subject to biases. Formal assessment procedures like OSATS and GEARS attempt
to provide objective measures of skill, but remain time-consuming. With
advances in machine learning there is an opportunity for fast and objective
automated feedback on technical skills. The SimSurgSkill 2021 challenge
(hosted as a sub-challenge of EndoVis at MICCAI 2021) aimed to promote and
foster work in this endeavor. Using virtual reality (VR) surgical tasks,
competitors were tasked with localizing instruments and predicting surgical
skill. Here we summarize the winning approaches and how they performed. Using
this publicly available dataset and results as a springboard, future work may
enable more efficient training of surgeons with advances in surgical data
science. The dataset can be accessed from
https://console.cloud.google.com/storage/browser/isi-simsurgskill-2021.

##  1 Introduction

Surgical training requires trainees to master many technical skills before
they can move on to performing clinical procedures. The traditional approach
for evaluating surgical skills has been through expert supervision [1].
However, this requires significant time from expert surgeons and is prone to
subjectivity. Hence there is a need for methods that are both more efficient
(i.e. faster) and more objective [2, 3, 4]. Attempts to provide automated
skill assessment in the form objective feedback, aim to meet these needs.

Studies on automated surgical skills assessment [5, 6, 7, 8] target many
different data modalities, like video, wearable sensors, robot-kinematics,
etc. Within the surgical robotics domain, much work has focused on “video
understanding,” that is, work that attempts to solve problems like surgical
tool detection [9, 10], anatomy recognition [11] and task/phase recognition
[12, 13, 14, 15]. Solving these core computer vision problems can allow for
many useful downstream applications, such as automated skills assessment [8,
16].

Recent research shows that objective performance indicators (OPIs) quantifying
tool use and movement during specific tasks within surgeries can differentiate
surgeon skill [17, 5, 18], correlate to outcomes [19, 20], and describe
workflows [21]. Paired with advances in computer vision to track surgical
tools, OPIs could also be automated. Thus there is the promise of fully
automated and objective assessment of surgical skill, though much work remains
to be done.

Technical competitions, i.e. “Challenges,” have become an integral part of
many top computer vision and machine learning conferences. Such challenges
have made significant contributions in advancing algorithms that impact
surgical assistive technologies. They also serve as a platform making datasets
available to the public. MICCAI has hosted many such challenges in the recent
past, with Endoscopic Vision (EndoVis) challenge being one example [22, 23,
24, 25, 26, 27]. The EndoVis challenge is hosted annually at MICCAI and
consists of multiple sub-challenges that target various research problems in
the surgical domain. To promote and foster work on automated surgical skills
assessment, our group hosted the SimSurgSkill 2021 challenge.

In the SimSurgSkill challenge, competitors used videos of simulated surgical
tasks to predict OPIs, which served as a proxy for skill. Specifically, a
large dataset consisting of videos of simulated surgical tasks and the
resulting OPIs were released to participants. The videos were obtained during
virtual reality (VR) robotic training sessions from surgical trainees of
varying skill levels. In addition, annotated bounding boxes (for tools and
needles) were made available. Teams were tasked with localizing the surgical
tools as a first step, and then predicting the resulting OPIs. Overall, we
found that the teams did generally well in localizing surgical tools but had
difficulty in predicting OPI values. Extensive details regarding the challenge
are given in the following sections.

##  2 SimSurgSkill 2021 Challenge Description

###  2.1 Overview

The SimSurgSkill 2021 challenge took place at the Medical Image Computing and
Computer Assisted Interventions (MICCAI) 2021 conference as a sub-challenge of
the annual Endoscopic Vision (EndoVis) challenge. The challenge was designed
to be based on BIAS [28] standards and a full design document is attached (see
Appendix A). Details regarding the dataset and catagories for this challenge
are given below.

###  2.2 Data

The dataset consisted of videos, video annotations and objective performance
indicators (OPIs). The videos were obtained from multiple surgical virtual
reality exercises being performed by participants of varying skill levels. A
total of 315 video recordings of various VR exercise were included in the
dataset, including 157 videos for model training and 158 videos for testing.
As shown in Figure 1, the majority of training and testing cases are from
running suture and needle pose matching exercises. To test model
generalizibility, posteriror needle driving and VU anatomosis were included
only in the testing set. In addition to the videos, bounding boxes of the
visible surgical tools and needle were provided, along with OPIs for the tasks
performed within each video.

####  2.2.1 Bounding box annotation

A portion of the training data is accompanied with bounding boxes around tool
center, or clevis, and the needle. The videos are first downsampled to 1 fps
and then annotated through crowd-sourcing. A few example annotations and video
frames are provided in Figure 2. The needle in the needle pose match exercise
was not considered of interest and was excluded from annotations.
Consequently, false detections of the needle were considered
misclassifications and penalized.

Figure 1: Number of video clips across different VR exercises for model
training and testing Figure 2: Examples of bounding box annotations from
training and testing dataset. A) needle pose match exercise. B) running
suture. C) anterior needle driving. D) VU anatomosis

####  2.2.2 Objective performance indicators (OPI)

For each video, values for 3 OPIs were also provided. These values were
generated by the VR engine automatically, hence no human annotation was
required. The OPIs along with their definitions are given below:

  * •

Economy of motion (EOM): Length of the total path travelled by the clevis of
all instruments in use. This distance is calculated in the 3D space and is
measured in centimeters. The final value of this metric is a sum of EOM of all
tools in use.

  * •

Instrument out-of-view (IOV): Count of the total number of times the clevis of
a tool goes out of view. Each time any of the instrument’s clevis goes out of
view, the IOV metric is incremented. The final value of IOV metric is a sum of
all IOV events for all tools in use.

  * •

Needle drop (ND): Count of the number of times the needle is dropped.

The training and testing datasets consists of VR exercises video clips. The VR
exercises were completed using the da Vinci simulator by participants of
varying skill levels. The videos were captured at 30 fps with a resolution of
720p (1280*720) from one channel of the endoscope.

The dataset has now been released publicly at
https://console.cloud.google.com/storage/browser/isi-simsurgskill-2021 but can
only be used for non-commercial purposes.

###  2.3 Challenge Categories

The challenge was divided into two categories (Figure 3). Participating teams
could participate individually in either category or both. Details regarding
each category follow below.

####  2.3.1 Category 1: Surgical tool and needle detection

In this category, the teams were required to train models that take VR video
as input to detect and localize needle drivers and needles using bounding
boxes. For this object detection task, evaluation was performed at frame-level
at 1 fps. Mean average prevision (mAP) of needle and needle drivers were
averaged over discrete Intersection over Union (IoU) values (0.5 to 0.95 in
increments of 0.05). Note that this evaluation metric is the same as the
primary metric used in MS-COCO object detection challenge [29].

####  2.3.2 Category 2: Objective performance indicators prediction

In this category, the teams were required to predict the objective performance
indicators using the videos. The three OPIs considered here include needle
drop (ND) count, instrument out-of-view (IOV) count, and economy of motion
(EOM) with more details listed in section 2.2.2. Teams could either train
models that train directly on video and predict these metrics, or they could
use the bounding boxes predicted from category 1 and generate logic to predict
the objective metric values. To evaluate the performance, mean squared error
(MSE) was used for ND and IOV counts, while Pearson correlation coefficient
was used for EOM. A ranking table was then produced to rank each team on the
three performance metrics. The final ranking of each team was generated by
multiplying the 3 rankings. For example, if a team has rankings 1(for ND), 3
(for IOV) and 4 (for EOM), the final ranking will be 12 (1 x 3 x 4). The
winner of this category was the one with the highest final rank.

Figure 3: The two categories within the challenge.

##  3 Team Submissions

There were a total of 16 teams/individuals that had requested access to this
challenge’s dataset and made an intent to participate. However, only 3 teams
were able to make a timely, complete submission to one or both of the
categories. Details regarding the methodology of these 3 teams are given
below.

###  3.1 Muroran Institute of Technology

The first participating team was from Muroran Institute of Technology, Japan
and consisted of only one member - Satoshi Kondo. For Category 1, this team
proposed to use EfficientDet [30] for tool detection. The object detector was
pretrained with MS-COCO dataset [29] and fine-tuned only the head with the
SimSurgSkill dataset. Object detections having scores higher than a threshold
of 0.15 were selected until one needle and two surgical tool clevises were
found. Since only objects having scores higher than a threshold were selected,
the object detector sometimes missed objects. In the case the object detector
is unable to find one needle and two surgical tool clevises, this method tried
to find the missed object by tracking. The Kernelized Correlation Filter (KCF)
[31] was used as the object tracker. The tracker used the most recent object
rectangle found by the object detector as its template. When the tracker found
an object successfully, the tracking result was used as the object detection
result. This tracking method was not used for needles since most of area in a
rectangle including a needle was background which made it hard to track the
needle.

Figure 4: Architecture of EfficientDet used by team from Muroran Institute of
Technology.

In Category 2, this team used different logic for predicting the three OPIs.
For economy of motion (EOM), as the 3D coordinates or camera parameters of the
VR simulation system were not known, it was assumed that the size of object is
proportional to the distance between the object and the virtual camera. The
distance (in pixels) was accumulated along the object trajectory normalized
with the object size. The accumulated distance for a whole video was then
multiplied by a constant of 1.68 to convert its unit from pixels to 3D volume
space.

For instrument out-of-view (IOV), this method utilized the output from the
object detection results. When the left or right clevis of a tool was not
detected for a duration of 0.5 second, the value of IOV was counted up. For
needle drop (ND), there were two conditions defined for counting. The first
condition was met when the needle is lower than both surgical tool clevises in
image coordinates. The second condition was met when the distance from the
needle to two surgical tool clevises was greater than twice the object size.
When any of these conditions was satisfied for a duration of 0.5 second, the
value of ND was counted up.

###  3.2 UCL $+$ Shandong University

The second team was a collaboration between University College London (UCL)
and Shandong University, China and consisted of 3 team members - Jinfan Zhou,
Evangelos Mazomenos and Danail Stoyanov. This team only participated in
category 2 of the challenge. This team proposed an architecture combining an
object detection model and an action recognition model. They first input video
frames at 1fps to the object detector to estimate bounding boxes designating
the 2D locations of needle and clevises. The sequence of bounding box
estimation was used to formulate the Sequence of Interest (SoI) feature which
is employed in the detection of ND and IOV events. Then, the SoI was fed at
30fps to both an action recognition model and the previous object detector to
produce the final prediction results. Finally, the location information of the
bounding boxes was used to estimate the EOM of needle drivers.

For Object Detection, this team trained a Faster R-CNN and Deformable DETR on
SimSurgSkill using the open-source detection toolbox MMDetection. ResNet-50
was used as the backbone network for feature extraction, Feature Pyramid
Network was used as neck network to fuse the features in different scales,
along with a Region Proposal Network to generate region proposals. Fig. 5
shows the architecture of proposed Faster R-CNN model. To train the detector,
the team used Stochastic Gradient Descent (SGD) for optimization with a
learning rate of 0.02, momentum of 0.9 and weight decay of 1e-4. The training
was run for 20 epochs with a batch size of 8.

For Instrument Out-of-View (IOV) detection, the team utilized the output of
object detector and calculated a coarse count of IOV events by simply grouping
together sequences of frames which contain less than two clevises and
calculating their total number. For Needle Drop (ND) Estimation, the team used
an Inception-based I3D model for feature extraction and replaced the original
prediction head with a two-layer network to provide a binary classification.
The selected frames with 63 (32 previous and 31 after) adjacent frames in
30fps were used as input to the I3D model. The training was performed using
SGD optimization with an initial learning rate of 0.1, momentum of 0.9, a
weight decay of 1e-7 and batch size of 8. The team used a weighted cross
entropy loss with a positive(dropped)-to-negative(non-dropped) ratio of 3:1 to
avoid over-fitting due to sample imbalance. In addition to the frame-based
classifier, the team also proposed an algorithm for ND event detection by
comparing the cosine similarity between the needle and each clevis, as shown
in Fig. 6.

Figure 5: UCL $+$ Shandong university’s team proposed architecture combining
an object detection and an action recognition model. Figure 6: UCL $+$
Shandong university’s team proposed needle drop detection algorithm. The
needle drop was detected by comparing the cosine similarity between the needle
and each clevis.

###  3.3 UCL

Figure 7: Pipeline of UCL team’s proposed method. The architecture comprises
two twin models, one trained on simple frames while the other was trained on
simple frames concatenated with segmented frames. Each model is composed by a
segmentation module followed by a bounding box prediction module.

The last participating team was from University College London and consisted
of 3 team members - Emanuele Colleoni, Dimitris Psychogyos and Yueming Jin.
This team addressed the tool and needle location tasks by exploiting motion-
based video background subtraction as a pre-processing step, along with Mask-
RCNN based approaches for object detection. The team hypothesized that by
doing so they could reduce the noise introduced by backgrounds not seen during
the training phase, thus mitigating over-fitting of their proposed network.
Once a bounding box model was trained, the team tackled EOM and IOV tasks
based on bounding box predictions. An additional binary classification model
was trained to determine whether or not the tool it grabbed was one of the two
tools for each frame. Fig. 7 shows the overall workflow of this team’s
proposed architecture.

Since the background in training videos provided in this challenge does not
move continuously, the team proposed to facilitate object detection by first
removing the background based on pixel movements over time. Two detection
models based on simplified Mask-RCNN were built for tool detection and needle
detection using raw image frames and a concatenation of raw and processed
(background removal) frames, respectively, as input. Each model consisted of a
first step where a segmentation network performs a pixel-based detection of
the tool clevis and needle. The output of the segmentation network and the
original input were concatenated and then processed into the object detection
model. The final outputs were three bounding boxes directly correlated to a
specific tool (needle, left/right clevis) for each frame.

Moreover, the team used a Best Model Selection (BMS) module to select the best
prediction from the two detection models based on the Intersection over Union
(IoU) score between the generated bounding boxes and the segmentation
prediction for each frame. The best prediction for each tool was chosen by
selecting the model with the highest IoU score, where a tool was considered
‘not present’ when the IoU fell below the threshold $\tau=0.05$ in both
models. Both segmentation and bounding box models used Imagenet pre-trained
Xception net as backbone.

For needle drop detection, the team trained an image-based binary
classification model to detect whether or not one of the two tools is grabbing
the needle. As the team experimentally observed that training such a model on
full frames produced unreliable results, they produced a new dataset by first
extracting the needle from all frames using the bounding boxes predicted at
the previous stage and then resizing the crops to 128$\times$128\. The same
Xception net was used to build the binary classification model as backbone.

##  4 Results

###  4.1 Category 1: Surgical tool and needle detection

Two teams submitted to Category 1. Example bounding box predictions on two VR
exercises from each team in comparison to ground truth labels are shown in
Figure 8. In addition to the primary evaluation metric $(AP_{0.5:0.95})$ used
to decide winning teams, additional metrics are also provided in Table 1 for
comparison.

Figure 8: Example bounding box detections on the test set. Category 1
---
Team | $AP_{0.5:0.95}$ | $AP_{0.5}$ | $AP_{0.75}$ | $AR_{0.5:0.95}$
UCL | 0.33 | 0.71 | 0.26 | 0.37
Muroran Institute of Technology | 0.35 | 0.65 | 0.34 | 0.44
Table 1:  Category 1 objection detection metrics

###  4.2 Category 2: Objective performance indicators prediction

All 3 teams prepared submissions for Category 2. Since Needle drop and
Instrument out-of-view counts are integer values, we compared ground truth
with predictions using mean-squared error. However, we used the Pearson
correlation coefficient to compare the Economy of motion ground truth and
predicted values. Figure 9 shows some scatter plots comparing ground truth
with predicted values, whereas Table 2 shows the evaluation metric values

Figure 9: Results from Category 2: Objective performance indicators prediction
Category 2
---
Team | ND (mse) | IOV (mse) | EOM (pearsonr)
UCL | 4.413 | 3.796 | 0.942
Muroran Institure of Technology | 11.036 | 12.778 | 0.964
UCL $+$ Shandong | 5.167 | 5.144 | 0.879
Table 2:  Evaluation metric values for predictions of Needle drop (ND),
Instrument out-of-view (IOV), and Economy of motion (EOM). Predicted Needle
drop and Instrument out of view counts were compared to ground truth using
mean-squared error (mse) while Economy of motion predictions were compared to
ground truth values using the Pearson correlation coefficient.

##  5 Discussions

In Category 1, teams had similar performance in tool and needle detection with
team Muroran Institure of Technology marginally out-performing Team UCL in all
metrics except for one. Though the teams used different approaches, it is
unclear whether their similarity in performance indicates a performance limit
on the data set.

In Category 2, teams predicted the EOM with high correlation to the real
values with all Pearson’s r coefficients from teams $>$0.87. However,
predicted values for EOM were quite different from the real values indicating
that models performed well at differentiating between relative motion between
videos but did not correspond directly the distance measured by the
simulation. Furthermore, the high correlation of predicted values with actual
EOM across all teams suggests that different modeling approaches are equally
capable of estimating tool motion.

The performance of teams in predicting ND and IOV was more variable. Team UCL
had the lowest mean-squared-error in the prediction of ND and IOV while team
Muroran Institute of Technology had the highest error in prediction. Team UCL
$+$ Shandong marginally under-performed team UCL. These differences in
performance between team UCL and team Muroran Institute of Technology in
predicting ND and IOV while similarity in performance in tool detection
(Category 1) suggests that certain methods of processing tool and needle
detection to arrive at measure of needle drops or instruments-out-of-view are
better than others.

Overall, from all teams, we see a step-wise approach of estimating objective
performance indicators by layering models. We also start to see how model
performances in Category 1 interact to influence the performance of the teams
in Category 2. This provides an interesting example of how complex problems
like automated objective assessment of surgical skill may require an
orchestration of multiple models which directly or indirectly influence one
another. The complex technical context of an orchestration of models for skill
assessment must also be explainable and actionable. In this challenge, we
target 3 easily understood OPIs for which higher values indicate lower skill.
However, there may be future use-cases where skill is inferred from combining
multiple OPIs. With these considerations, we identify a focus on
interpretability as an important and valuable future direction in automated
surgical skill assessment.

##  6 Conclusion

In this challenge, we asked the surgical data science community to develop
computer vision algorithms which predict 3 objective measures of surgical
skill related to instrument-use and needle-handling from videos of surgical
tasks completed in a simulated environment. The objective performance
indicators used in this challenge were quite interpretable and, generally,
have a simple relationship with skill. However, since we did not have skill
labels for the dataset, it is unclear whether and how the inference of skill
from OPIs would be influenced by the error in estimating the OPIs from the
different approaches.

Nonetheless, initial results from the teams suggest that the target of
estimating OPIs from computer vision algorithms seems reachable. Correlation
to EOM—estimation of tool motion—is quite high for all approaches. For IOV and
ND, specific approaches are able to perform quite well but other approaches
are less successful. Approaches layered models to estimate OPIs suggesting
that the performances in Category 2 were influenced by the performance in
Category 1. We hope that the dataset and approaches presented here enable and
encourage further investigations into automated surgical skill assessment—an
important innovation which could accelerate and scale surgical training.

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##  Appendix A Appendix

The initial design document for the challenge can be found below. See pages -
of Appendix.pdf