# Non-minimal coupled warm inflation with quantum-corrected self-interacting
inflaton potential

Daris Samart<EMAIL_ADDRESS>Khon Kaen Particle Physics and Cosmology Theory
Group (KKPaCT), Department of Physics, Faculty of Science, Khon Kaen
University, 123 Mitraphap Rd., Khon Kaen, 40002, Thailand  Patinya Ma-adlerd
<EMAIL_ADDRESS>Khon Kaen Particle Physics and Cosmology Theory Group
(KKPaCT), Department of Physics, Faculty of Science, Khon Kaen University, 123
Mitraphap Rd., Khon Kaen, 40002, Thailand  Peeravit Koad<EMAIL_ADDRESS>School of Informatics, Walailak University, Thasala, Nakhon Si Thammarat,
80160, Thailand  Phongpichit Channuie<EMAIL_ADDRESS>College of Graduate
Studies, Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand
School of Science, Walailak University, Thasala, Nakhon Si Thammarat, 80160,
Thailand

###### Abstract

In this work, we investigate the non-minimally coupled scenario in the context
of warm inflation with quantum-corrected self-interacting potential. We
transform the potential in the Jordan frame to the Einstein frame and consider
a dissipation parameter of the form $\Gamma=C_{T}T$ with $C_{T}$ being a
coupling parameter. We focus on the strong regime of which the interaction
between inflaton and radiation fluid has been taken into account. We compute
inflationary observables and constrain the parameters of our model using
current Planck 2018 data. With the sizeable number of e-folds and proper
choices of parameters, we discover that allowed values of $C_{T}$ lie in the
range $0.014\lesssim C_{T}\lesssim 0.020$ in which the predictions are in good
agreement with the latest Planck 2018 results at the $2\,\sigma$ confident
level.

Warm inflation, Quantum correction

##  I Introduction

Warm inflation scenario has been received attentions as an alternative
approach of the reheating phase of the universe in order to generate the
thermal bath in the standard cosmology. The warm inflation was originally
proposed to resolve some problems in the standard cold inflation picture [1,
2], for instances, providing sufficiently hot thermal bath after inflaton
decaying to other matter fields in the reheating epoch [3, 4] the large
quantum correction of the inflaton field might spoiling the flatness of the
observed universe or a so-called eta problem [5, 6], fine tuning of the
initial values of the inflation models motivated by beyond standard model
physics [7, 8, 9] and other salient features see [10, 11, 12] for reviews. In
the warm inflation stage, the inflaton decays into radiation matter during the
slow-roll period. In the meantime, the quantum fluctuations of the density
perturbation amplitudes are generated by the friction of the inflaton
propagating in the thermal bath. At the end of inflation, the universe is
automatically heated up with out requiring the preheating and reheating phases
before radiation dominated era. Moreover, the energy density of the radiation
is smoothly joined with the energy density of the inflaton field. The
dissipative coefficient, $\Gamma$ plays a crucial role in the warm
inflationary universe for describing the dynamics of warm inflation. All
information of the microscopic dynamical processes during warm inflationary
universe is contained in the dissipative coefficient and it has been
constructed and calculated by using the supersymmetric models with finite
temperature field analysis in various aspects see Refs. [13, 14, 15, 16, 17,
18, 19, 20, 21, 22] for more details and references therein.

The self-interacting inflaton potential ($V\sim\phi^{4}$) has been largely
used to study of the standard (cold) inflation dynamics in numerous
perspectives see [23, 24, 25] for reviews. According to the requirements of
the standard quantum field theory, the self-interacting potential is
renormalizable theory and it is naturally received the quantum-corrected
effect. The quantum correction of the perturbative loop expansion known as
Coleman-Weinberg potential [26] is one of the famous approach. In addition,
the phenomenological quantum-corrected self-interacting potential is proposed
and employed to study the quantum-corrected effect due to the non-vanishing
primordial tensor modes by Ref.[27]. On the other hand, there are a number of
investigations that also used the self-interacting potential to study warm
inflationary universe in both minimal and non-minimal coupling to gravity, for
examples see Refs.[28, 29, 30, 31, 32, 33, 34]. However, there is no previous
work in a study of non-minimally coupled warm inflation with the quantum-
corrected self-interacting potential according to the literature.

Therefore, in this work, we will investigate the quantum-correction of the
self-interacting potential due to the thermal effect inflation with
$\Gamma\propto T$, where $T$ is a temperature. Our study might shed some light
on the the quantum-correction of the inflaton due to the finite temperature
reaction which plays significant role in warm inflationary universe. In
particular, the results in this work might reveal to what extend the model’s
parameters deviate from cold inflation when the thermal effect is taken into
account. Moreover, we will constrain our theoretical results with Planck 2018
via the COBE normalization and the prediction in this work will be compared to
the latest observational data.

The paper is organized as follows: all relevant dynamical equations in the
non-minimal coupling warm inflation under the slow-roll approximation are
determined in section II. Next, in section III, we will compare the results in
this work with the observational data. Finally, we close this paper by
providing discussions and conclusions in section IV.

##  II Formalism

###  II.1 Non-minimal coupling gravitational action and conformal
transformation

We start with a gravitational action of the non-minimal coupling of the scalar
field to Ricci scalar (gravity) with a general form of the effective potential
$V(\phi)$, one finds,

$\displaystyle
S_{J}=\int\sqrt{-g}\left[-\frac{1}{2}\left(M_{p}^{2}+\xi\,\phi^{2}\right)R+g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]$
(1)

where the action $S_{J}$ stands for the gravitational action in the Jordan
frame. While $M_{p}^{2}\equiv 1/8\pi G$ and $\xi$ are reduced Plank mass and
the non-minimal coupling constant, respectively. It is more convenient to
study the inflation dynamics of the non-minimal coupling in the Einstein
frame, i.e., the gravitational sector of the action written in the Einstein-
Hilbert form only. The Einstein frame can be achieved by using the conformal
transformation via a re-defining metric tensor as,

$\displaystyle\widetilde{g}_{\mu\nu}=\Omega(\phi)^{2}\,g_{\mu\nu}\,,\qquad\Omega(\phi)=1+\frac{\xi\,\phi^{2}}{M_{p}^{2}}\,.$
(2)

Here all variables with tilde symbol represent the quantities in the Einstein
frame. Applying the conformal transformation to the action (1), the action in
Einstein frame is given by,

$\displaystyle
S_{E}=\int\sqrt{-\widetilde{g}}\left[-\frac{1}{2}\,M_{p}^{2}\,\widetilde{g}^{\mu\nu}\widetilde{R}_{\mu\nu}+\widetilde{g}^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-U(\chi)\right].$
(3)

We have used the re-definition of new scalar field, $\chi$ in the Einstein
frame to obtain the canonical form of the kinetic term of the scalar field,
$\chi$ as

$\displaystyle\frac{1}{2}\left(\frac{d\chi}{d\phi}\right)^{2}=\frac{1+3\,M_{p}^{2}\,\Omega_{\phi}^{2}}{\Omega^{2}}\,,$
(4)

where $\Omega_{\phi}\equiv d\Omega/d\phi$ and the new effective potential in
the Einstein frame, $U(\chi)$ is also given by,

$\displaystyle U(\chi)=\Omega^{-4}\,V\left(\phi(\chi)\right).$ (5)

In this work, we will consider the self-interacting potential with
phenomenological quantum correction in the warm inflation scenario. This
potential has been proposed by Ref.[27] in order to analyze the characters of
the quantum correction in the self-interacting scalar field
phenomenologically. The potential in the Jordan frame is written in the
following form

$\displaystyle
V(\phi)=\lambda\,\phi^{4}\left(\frac{\phi}{\Lambda}\right)^{4\gamma}\,.$ (6)

Here we introduced the quantum correction (real) parameter $\gamma$ that use
to characterize the quantum behavior of the self-interacting potential and the
$\Lambda$ parameter is the cut-off at a given energy scale. It was shown that
the range of the $\gamma$ should be $\mathcal{O}(\gamma)\sim 0.1$ according to
the constraint from observational data [27]. In the latter, we will construct
the slow-roll dynamics in warm inflation in the Einstein frame with the
potential in Eq.(6).

###  II.2 Slow-roll dynamics in warm inflation

In this subsection, we collect all relevant cosmological equations of the
slow-roll paradigm in warm inflation. Recalling the Friedmann equation from
the gravitational action in Einstein frame with the flat FRW background in the
warm inflation scenario, it reads,

$\displaystyle
H^{2}=\frac{1}{3\,M_{p}^{2}}\left(\frac{1}{2}\,\dot{\chi}^{2}+U(\chi)+\rho_{R}\right),$
(7)

where $\dot{\chi}\equiv d\chi/dt$ and $\rho_{R}$ is energy density of the
radiation fluid with the equation of state $w_{R}=1/3$. The Klein-Gordon
equation of motion for the scalar field, $\chi$ in Einstein frame with the
dissipative coefficient, $\Gamma$ is given by

$\displaystyle\ddot{\chi}+3H\,\dot{\chi}+U_{\chi}=-\Gamma\,\dot{\chi}\,,$ (8)

with $U_{\phi}\equiv dU/d\chi$. The conservation of the energy-momentum tensor
of the radiation fluid leads to the continued equation as

$\displaystyle\dot{\rho}_{R}+4H\,\rho_{R}=\Gamma\,\dot{\chi}^{2}\,.$ (9)

Based on the finite temperature field theory approaching to the supersymmetry
models, the generic form of the dissipative coefficient, $\Gamma$ for several
warm inflation models can be written in terms of the coupling between
temperature $(T)$, scalar (inflaton) field $(\chi)$ and the mass of the some
heavy field during warm inflation $(M_{X})$ with $M_{X}>T$ as the following
form [14, 15, 16, 17, 35]

$\displaystyle\Gamma=C_{(m)}\,\frac{T^{m}\,\chi^{n}}{M_{X}^{l}}\,,\qquad
m+n-l=1\,.$ (10)

The dissipative coefficient, $\Gamma$ represents the energy transfer from the
inflaton field to the thermal bath in the warm inflationary universe. The
parameter $C_{m}$ encodes the microscopic dynamics of the inflaton interacting
with other particles and the $m$, $n$ and $l$ are the integer number.
Particularly for the $m=1$ case, this corresponds to the high temperature
supersymmetric model [16] or considering inflaton as a pseudo Goldstone boson
that can be coupled to other fields in the thermal bath such as warm natural
inflation [36] and warm little inflation in analogy to the little Higgs model
[37]. In the following, we will consider the slow-roll approximation framework
with the dissipative coefficients $\Gamma$ for $m=1$ and at the strong regime.

In the standard slow-roll approximation, we can re-write the Friedmann
equation as well as the equations of motion for the inflaton and the radiation
matter as

$\displaystyle H^{2}$ $\displaystyle\approx$
$\displaystyle\frac{1}{3M_{p}^{2}}\,U(\chi)\,,$ (11) $\displaystyle\dot{\chi}$
$\displaystyle\approx$ $\displaystyle-\frac{U_{\chi}}{3H(1+Q)}\,,\qquad
Q\equiv\frac{\Gamma}{3H}\,,$ (12) $\displaystyle\rho_{R}$
$\displaystyle\approx$
$\displaystyle\frac{\Gamma\,\dot{\chi}}{4H}\,,\qquad\rho_{R}=C_{R}\,T^{4}\,,$
(13)

where the $Q$ is called dimensionless parameter that use to identify the
regime of the dissipative effects in the latter and $C_{R}=g_{R}\,\pi^{2}/30$.
In addition, the minimal supersymmetric standard model gives the number of
relativistic degrees of freedom, $g_{R}=228.76$ and leads to $C_{R}\simeq 70$
[4]. We have been used the following approximations for the slow-roll
scenario,

$\displaystyle\rho_{R}$ $\displaystyle\ll$
$\displaystyle\rho_{\chi}\,,\qquad\rho_{\chi}=\frac{1}{2}\,\dot{\chi}^{2}+U\,,$
(14) $\displaystyle\dot{\chi}^{2}$ $\displaystyle\ll$ $\displaystyle
U(\chi)\,,$ (15) $\displaystyle\ddot{\chi}$ $\displaystyle\ll$ $\displaystyle
3H\left(1+Q\right)\dot{\chi}\,,$ (16) $\displaystyle\dot{\rho}_{R}$
$\displaystyle\ll$ $\displaystyle 4H\,\rho_{R}\,,$ (17)

As mentioned earlier, it is more convenient to separate warm inflation into
two regimes by using the dimensionless $Q$ as

$\displaystyle Q$ $\displaystyle\gg$ $\displaystyle 1\,,\qquad{\rm
strong~{}regime}\,,$ (18) $\displaystyle Q$ $\displaystyle\ll$ $\displaystyle
1\,,\qquad{\rm weak~{}regime}\,.$ (19)

In addition, we can express the temperature as a function of the inflaton
field, $\chi$ by using the Eqs. (10,11,12,13) for the general $m$ integer
values. It reads,

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\frac{U_{\chi}^{2}\,\chi^{m-1}}{4H\,C_{(m)}\,C_{R}}\right)^{\frac{1}{4+m}}\,,\qquad{\rm
for}\,~{}Q\gg 1\,,$ (20) $\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\frac{C_{(m)}\,U_{\chi}^{2}\,\chi^{1-m}}{36H^{3}\,C_{R}}\right)^{\frac{1}{4-m}}\,,\qquad{\rm
for}\,~{}Q\ll 1\,.$ (21)

In the following, we will concentrate and investigate warm inflation in the
strong dissipative regime only since this regime might show better thermal
effect in the inflationary universe.

Before calculating the slow-roll parameters, we would like to express the form
of the effective potential in the Einstein in Eq.(5) under the large field
assumption during the inflation i.e., $\phi\gg{M_{p}/\sqrt{\xi}}$ . One finds,

$\displaystyle\chi\simeq\kappa\,M_{p}\ln\Big{(}{\sqrt{\xi}\,\phi\over
M_{p}}\Big{)},~{}~{}~{}~{}\kappa\equiv\sqrt{{2\over\xi}+6}$ (22)

Then the Einstein frame potential then takes the following form

$\displaystyle U(\chi)=\Omega^{-4}V(\phi(\chi))$ $\displaystyle=$
$\displaystyle\frac{M_{p}^{4}}{\left(M_{p}^{2}+\xi\phi^{2}\right)^{2}}\lambda\,\phi^{4}\left(\frac{\phi}{\Lambda}\right)^{4\gamma}$
(23) $\displaystyle=$ $\displaystyle{\lambda
M_{p}^{4}\over\xi^{2}}\left(\exp\Bigg{[}{\frac{-2\chi}{\kappa
M_{p}}}\Bigg{]}+1\right)^{-2}\left({M_{p}\over\sqrt{\xi}\Lambda}\right)^{4\gamma}\exp\Bigg{[}{4\gamma\chi\over\kappa
M_{p}}\Bigg{]}$

Next, we provide the slow-roll parameters in warm inflation for general $m$
and they read,

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{M_{p}^{2}}{2}\left(\frac{U_{\chi}}{U}\right)^{2}\,,\quad\eta=M_{p}^{2}\,\frac{U_{\chi\chi}}{U}\,,\quad\beta=M_{p}^{2}\left(\frac{U_{\chi}\,\Gamma_{\chi}}{U\,\Gamma}\right)\,.$
(24)

The inflationary phase of the universe occurs under the following conditions

$\displaystyle\epsilon\ll 1+Q\,,\qquad\eta\ll 1+Q\,,\qquad\beta\ll 1+Q\,.$
(25)

We firstly calculate the slow-roll parameters, $\epsilon$ and $\eta$ with the
potential in Eq.(5) and they are given by

$\displaystyle\epsilon$ $\displaystyle\approx$
$\displaystyle\frac{8}{\kappa^{2}}\left(\frac{M_{p}^{2}}{\xi\,\phi^{2}}\right)^{2}\left(1+\gamma\,\frac{\xi\,\phi^{2}}{M_{p}^{2}}\right)^{2}$
(26) $\displaystyle\eta$ $\displaystyle\approx$
$\displaystyle\frac{16}{\kappa^{2}}\,\frac{M_{p}^{4}}{\xi^{2}\,\phi^{4}}\left(1+\left(2\gamma-\frac{1}{2}\right)\frac{\xi\,\phi^{2}}{M_{p}^{2}}\right)\,,$
(27)

where we have presented $\chi$ of the Einstein frame in terms of $\phi$ of the
Jordan frame with
$\phi=M_{p}\,\exp\left({{\chi}/{\kappa\,M_{p}}}\right)/\sqrt{\xi}$ . The
$\beta$ parameter depends on the dissipative coefficient, $\Gamma$ and we can
calculate the $\beta$ parameter after introducing the explicit form of the
$\Gamma$.

Next We start with the dissipative coefficient of warm inflation, the
dissipative coefficient for $m=1$ is read,

$\displaystyle\Gamma=C_{T}\,T\,.$ (28)

On the one hand, the dissipative coefficient in this form can be achieved from
high temperature approximation of the thermal supersymmetric model [13]. On
the other hand, the warm little inflation [37] might generates the disipative
coefficient in (28) where the inflaton in this scenario is considered as a
pseudo Nambu-Goldstone boson of a broken gauge symmetry in the warm little
inflation similar to ”Little Higgs” model for electroweak symmetry breaking.
Moreover, warm inflation can naturally occur for $T>H$. The coupling $C_{T}$
in Eq.(28) is given by

$\displaystyle
C_{T}\simeq\frac{3\,g^{2}}{h^{2}\big{(}1-0.34\log(h)\big{)}}\,,$ (29)

where $g$ is the Yukawa coupling of the inflaton (super scalar field) and
heavy fermions in the warm little inflation scenario while $h$ is Yukawa
coupling of the heavy fermions and light singlet scalar and fermion fields
[37].

By using the dissipative coefficient in Eq.(28), this leads to the expression
of the temperature as a function of the inflaton field by using Eq.(20) as

$\displaystyle
T=\left(\frac{U_{\chi}^{2}}{4H\,C_{T}\,C_{R}}\right)^{\frac{1}{5}}.$ (30)

Having use the results in Eqs.(23), (28) and (30), the slow-roll parameter
$\beta$ up to the first order of the $\gamma$ correction at the large field
approximation is given by

$\displaystyle\beta\approx\frac{24\,M_{p}^{4}}{5\,\kappa^{2}\,\xi^{2}\,\phi^{4}}-\frac{16\,M_{p}^{2}}{5\,\kappa^{2}\,\xi\,\phi^{2}}+\frac{8\,M_{p}^{4}}{\kappa^{2}\,\xi\,\phi^{2}}\,\gamma\,,$
(31)

where the relation
$\phi=M_{p}\,\exp\left({{\chi}/{\kappa\,M_{p}}}\right)/\sqrt{\xi}$ is implied.
In addition, the dimensionless parameter $Q$ for $\Gamma=C_{T}\,T$ can be
written by

$\displaystyle
Q=\left[\left(\frac{2}{3}\right)^{2}\mathcal{K}\,\frac{M_{p}^{4}}{\phi^{4}}\left(\frac{\Lambda}{\phi}\right)^{4\,\gamma}\left(1+\gamma\,\frac{\xi\phi^{2}}{M_{p}^{2}}\right)^{2}\,\right]^{\frac{1}{5}}\,,\qquad\mathcal{K}\equiv\frac{C_{T}^{4}}{C_{R}\,\kappa^{2}\,\lambda}\,.$
(32)

At the end of inflation requiring $\epsilon_{\rm end}=Q$, one finds

$\displaystyle\frac{8}{\kappa^{2}}\left[\frac{M_{p}^{4}}{\xi^{2}\,\phi_{\rm
end}^{4}}\left(1+\gamma\,\frac{\xi\phi_{\rm
end}^{2}}{M_{p}^{2}}\right)^{2}\,\right]^{\frac{4}{5}}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{2}{3}\right)^{2}\mathcal{K}\,\xi^{2}\left(\frac{\Lambda}{\phi_{\rm
end}}\right)^{4\,\gamma}\right]^{\frac{1}{5}}$
$\displaystyle\left(1+\gamma\,\frac{\xi\phi_{\rm end}^{2}}{M_{p}^{2}}\right)$
$\displaystyle=$
$\displaystyle\widetilde{K}\,\Lambda^{2}\,\frac{\xi}{M_{p}^{2}}\left(\frac{\phi_{\rm
end}}{\Lambda}\right)^{2-\frac{\gamma}{2}}\,,$ (33)

where the $\widetilde{K}$ parameter is defined by

$\displaystyle\widetilde{K}\equiv\left[\left(\frac{\kappa^{2}}{8}\right)^{5}\,\left(\frac{2}{3}\right)^{2}\,\mathcal{K}\,\xi^{2}\,\right]^{\frac{1}{8}}\,.$
(34)

Applying the assumption that the given order of the $\gamma$ parameter,
$\mathcal{O}(\gamma)\lesssim 0.1$ as mentioned earlier and leading to
$2-\gamma/2\approx 2$, the inflaton field at the end of warm inflation is read

$\displaystyle\phi_{\rm
end}=\frac{1}{\sqrt{\widetilde{K}-\gamma}}\,\frac{M_{p}}{\sqrt{\xi}}$ (35)

Therefore, the universal bound of the quantum correction for the self-
interacting inflaton field due to the modification of warm inflation is given
by

$\displaystyle\gamma<\widetilde{K}\,.$ (36)

The bound in Eq.(36) represents the thermal effects on the quantum-corrected
parameter, $\gamma$ in warm inflation. It is worth noting that the universal
bound in warm inflation is different from the standard (cold) inflation given
by Ref.[27] as $\gamma<\sqrt{3}/2$ which is equivalent to the weak regime of
warm inflation, i.e., $\epsilon(\phi_{\rm end})=1$. For given values
$C_{T}=0.02$, $C_{R}=70$, $\xi=10^{4}$ and $\lambda=0.5\times 10^{-4}$, we
find $\widetilde{K}=1.106$. This means the inflaton value at the end of warm
inflation is smaller than that of the inflaton in cold inflation.
Moreover, the e-folding number, $N$ in the strong regime $Q\gg 1$ is given by

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{M_{p}^{2}}\int_{\chi_{\rm
end}}^{\chi_{N}}\frac{Q\,U}{U_{\chi}}\,d\chi$ (37) $\displaystyle=$
$\displaystyle\int_{\phi_{\rm
end}}^{\phi_{N}}\frac{Q(\phi)}{\sqrt{2\,\epsilon(\phi)}}\,\frac{1}{\phi}\,d\phi\approx\frac{5\,\kappa\,\xi}{24}\left[\frac{\mathcal{K}}{9\sqrt{2}}\left(\frac{\Lambda}{M_{p}}\right)^{6}\left(\frac{\phi}{\Lambda}\right)^{6\left(1-\frac{2}{3}\,\gamma\right)}\,\right]^{\frac{1}{5}}\Bigg{|}_{\phi_{\rm
end}}^{\phi_{N}}\,,$

where we have expanded the $\gamma$ parameter up to the first order. Having
use the $\phi_{N}\gg\phi_{\rm end}$ and $\mathcal{O}(\gamma)\sim 0.1$
approximations, the inflaton field at the Hubble horizon crossing in terms of
the e-folding number, $N$ is written by

$\displaystyle\phi_{N}$ $\displaystyle\approx$
$\displaystyle\left[\frac{2187}{3125}\cdot
32768\sqrt{2}\right]^{\frac{1}{6-4\gamma}}\left[\left(\frac{N}{\kappa\,\xi}\right)^{5}\left(\frac{M_{p}^{6}}{\mathcal{K}\,\Lambda^{4\gamma}}\right)\right]^{\frac{1}{6-4\gamma}}$
(38) $\displaystyle=$ $\displaystyle\frac{12\cdot
2^{\frac{7}{12}}\,3^{\frac{1}{6}}}{5^{\frac{5}{6}}}\left(\frac{N}{\kappa\,\xi}\right)^{\frac{5}{6}}\left(\frac{M_{p}}{\mathcal{K}^{\frac{1}{6}}}\right)$
$\displaystyle\times\,\Bigg{[}1+\frac{\gamma}{18}\left(\ln\left[\frac{2^{31}\cdot
3^{14}}{5^{10}}\right]+2\ln\left[\left(\frac{N}{\kappa\,\xi}\right)^{5}\left(\frac{M_{p}^{6}}{\mathcal{K}\,\Lambda^{6}}\right)\right]\right)+\mathcal{O}\left(\gamma^{2}\right)\Bigg{]}$
$\displaystyle\simeq$ $\displaystyle
5.647\left[\frac{N^{5}}{\kappa^{5}\,\xi^{2}\,\mathcal{K}}\right]^{\frac{1}{6}}\Bigg{[}1+\frac{\gamma}{9}\left(10.387+\ln\left[\left(\frac{N}{\kappa\,\xi}\right)^{5}\left(\frac{M_{p}^{6}}{\mathcal{K}\,\Lambda^{6}}\right)\right]\right)\Bigg{]}\frac{M_{p}}{\sqrt{\xi}}\,,$

here we have considered the quantum-corrected character of the self-
interacting potential in warm inflation up to the leading order of the
$\gamma$ parameter only. In addition, one might re-write the dimensionless
parameter $Q$ in Eq.(32) in terms of the e-folding number, $N$ by using
Eq.(38) as

$\displaystyle Q(N)$ $\displaystyle\approx$
$\displaystyle\left[\left(\frac{2}{3}\right)^{2}\frac{\mathcal{K}\,M_{p}^{4}}{\phi_{N}^{4}}\left(\frac{\xi^{2}\,\Lambda^{4}}{M_{p}^{4}}\right)^{\gamma}\,\right]^{\frac{1}{5}}$
(39) $\displaystyle\simeq$
$\displaystyle\left[\left(\frac{2}{3}\right)^{2}\frac{\mathcal{K}\,M_{p}^{4}}{\phi_{N}^{4}}\left(1+2\,\gamma\,\ln\left[\frac{\xi\,\Lambda^{2}}{M_{p}^{2}}\right]\right)\right]^{\frac{1}{5}}$
$\displaystyle=$
$\displaystyle\left[\frac{\left(\frac{2}{3}\right)^{2}\frac{\mathcal{K}^{\frac{5}{3}}\,\xi^{2}}{5.647^{4}}\left(1+2\,\gamma\,\ln\left[\frac{\xi\,\Lambda^{2}}{M_{p}^{2}}\right]\right)}{\left[\frac{N^{5}}{\kappa^{5}\,\xi^{2}}\right]^{\frac{2}{3}}\left[1+\frac{\gamma}{9}\left(10.387+\ln\left[\left(\frac{N}{\kappa\,\xi}\right)^{5}\left(\frac{M_{p}^{6}}{\mathcal{K}\,\Lambda^{6}}\right)\right]\right)\right]^{4}}\right]^{\frac{1}{5}}$

Taking the back reaction of the inflaton fluctuation in the thermal heat bath
into account, in addition, the power spectrum is given by [17, 38, 34, 4, 35,
11, 39, 40],

$\displaystyle\Delta_{\mathcal{R}}$ $\displaystyle=$
$\displaystyle\frac{U\,\big{(}1+Q_{N}\big{)}^{2}}{24\,\pi^{2}\,M_{p}^{4}\,\epsilon}\left(1+2\,n_{N}+\left(\frac{T_{N}}{H_{N}}\right)\frac{2\sqrt{3}\,\pi\,Q_{N}}{\sqrt{3+4\pi\,Q_{N}}}\right)G(Q_{N})\,,$
(40) $\displaystyle\simeq$
$\displaystyle\frac{5\,C_{T}^{3}}{12\,\pi^{4}\,g_{R}\,Q_{N}^{2}}\left(1+\frac{\sqrt{3}\,\pi\,Q_{N}}{\sqrt{3+4\pi\,Q_{N}}}\right)G(Q_{N})\,$

where the subscript $``N"$ is labeled for the values of all quantities in warm
inflation at the Hubble horizon crossing and $n=1/\big{(}\exp{H/T}-1\big{)}$
is the Bose-Einstein statistical function. In addition, the power spectrum in
Eq.(40) can be constrained by observational data and yields the upper bound of
the $C_{T}$ parameter as $C_{T}\lesssim 0.02$ [34]. Moreover, the information
of the coupling between the inflaton and the radiation in the heat bath
leading to a growing mode is contained in the function $G(Q_{N})$ and it reads
[29],

$\displaystyle G(Q)=1+0.335\,Q^{1.364}+0.0185\,Q^{2.315}\,.$ (41)

In addition, we have used the relation
$\rho_{r}/V(\phi)=\epsilon\,Q/2(1+Q)^{2}$ and the approximation of the
thermalized inflaton fluctuation, $1+2\,n_{N}\simeq 2\,T_{N}/H_{N}$ and
$T_{N}/H_{N}=3\,Q_{N}/C_{T}$ in order to get the last line in Eq.(40) as
Ref.[34]. By using Eqs.(40,41), furthermore, the scalar spectral index is
determined as

$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle
1+\frac{d\ln\Delta_{\mathcal{R}}}{dN}$ (42) $\displaystyle=$ $\displaystyle
1+\frac{Q_{N}}{3+5\,Q_{N}}\frac{\big{(}6\,\epsilon-2\,\eta\big{)}}{\Delta_{\mathcal{R}}}\,\frac{d\Delta_{\mathcal{R}}}{dQ_{N}}\,,$
$\displaystyle\frac{d\Delta_{\mathcal{R}}}{dQ_{N}}$ $\displaystyle=$
$\displaystyle\frac{5\,C_{T}^{3}}{12\,\pi^{4}\,g_{R}}\Bigg{[}\left(1+\frac{\sqrt{3}\,\pi\,Q_{N}}{\sqrt{3+4\pi\,Q_{N}}}\right)\frac{\left(0.457\,Q^{0.364}+0.0428\,Q^{1.315}\right)}{Q_{N}^{2}}$
$\displaystyle-$
$\displaystyle\left(2+\frac{\sqrt{3}\,\pi\,Q_{N}}{\sqrt{3+4\pi\,Q_{N}}}+\frac{2\,\sqrt{3}\,\pi^{2}\,Q_{N}^{2}}{(3+4\,\pi\,Q_{N})^{\frac{3}{2}}}\right)\frac{\big{(}1+0.335\,Q^{1.364}+0.0185\,Q^{2.315}\big{)}}{Q_{N}^{3}}\Bigg{]}.$

While the tensor-to-scalar perturbation ratio, $r$ is obtained by the
following formula

$\displaystyle
r=\frac{\Delta_{T}}{\Delta_{\mathcal{R}}}=16\,\epsilon\left[\frac{6\,Q_{N}^{3}}{C_{T}}\left(1+\frac{\sqrt{3}\,\pi\,Q_{N}}{\sqrt{3+4\pi\,Q_{N}}}\right)G(Q_{N})\right]^{-1}$
(43)

where $\Delta_{T}$ is the power spectrum of the tensor perturbation and we
have used $\Delta_{T}=2H^{2}/\pi^{2}M_{p}^{2}=2U(\chi)/3\pi^{2}M_{p}^{4}$
which is the same form as in the standard (cold) inflation result for the
primordial gravitational waves.

##  III Confrontation with the data

Figure 1: We display $\lambda$ as a function of $\gamma$ obtained from Eq.(45) for $M_{p}=10\Lambda,\,\xi=10^{4}$, $C_{R}=70,\,\kappa\approx\sqrt{6}$ and $N=60$. As $\gamma$ increases, the magnitude of $\lambda$ needed to produce the correct amount of scalar perturbations also increases. Table 1: We show a set of parameters $(C_{T},\,\gamma,\,\lambda)$ obtained from Eq.(45) in which their values are constrained by the COBE normalization condition given in Eq.(44). Here we have used various values of $C_{T}$ in order to obtain viable values of $\gamma,\,\lambda$ and applied $M_{p}=10\Lambda$, $\xi=10^{4}$, $C_{R}=70$, $\kappa\approx\sqrt{6}$ and $N=60$. $C_{T}$ |  $\gamma\times 10^{-2}$ |  $\lambda\times 10^{-5}$
---|---|---
0.014 | 7.00 | 2.80
8.50 | 3.74
10.00 | 4.68
11.50 | 5.60
0.015 | 7.00 | 2.41
8.50 | 3.80
10.00 | 4.78
11.50 | 5.73
0.020 | 7.00 | 3.04
8.50 | 4.09
10.00 | 5.19
11.50 | 6.28

In this section, we will constrain the inflation potential with the COBE
normalization condition [41] to fix the parameters in the non-minimal warm
inflation with the quantum corrected self-interacting potential. According to
the Planck 2018 data, the inflaton potential must be normalized by the slow-
roll parameter, $\epsilon$ and satisfied the following relation at the horizon
crossing $\phi=\phi_{N}$ in order to generate the observed amplitude of the
cosmological density perturbation ($A_{s}$):

$\displaystyle\frac{U(\phi_{N})}{\epsilon(\phi_{N})}\simeq(0.0276\,M_{p})^{4}\,.$
(44)

Having used the potential in Eq.(23) and the slow-roll $\epsilon$ parameter in
Eq.(26), we find

$\displaystyle\left(\frac{M_{p}}{\Lambda}\right)^{2}\left(1+\gamma\,\frac{\xi\,\phi_{N}^{2}}{M_{p}^{2}}\right)=\frac{\sqrt{3\,\lambda}}{2\,(0.0276)^{2}}\left(\frac{\phi_{N}}{\Lambda}\right)^{2(1+\gamma)}\,.$
(45)

The resulting constraint is plotted in Fig.1 by using the definition of
$\phi_{N}$ in Eq.(39). The magnitude of $\lambda$ needed to produce the
observed amplitude of scalar perturbations increases linearly for increasing
$\gamma$. For reference, we consider various values of $C_{T}$ and figure out
a pair of $(\lambda,\,\gamma)$ in which their values produce the observed
amplitude of scalar perturbations. More interestingly, the numerical values of
the self-interacting coupling, $\lambda$, shown in Fig.1 are consistent with
the results of the running coupling $\lambda$ up to two-loop corrections in
the standard model of particle physics that is very close to zero at the GUT
scale which would be a typical scale of inflation [42]. In addition, the
values of the $\lambda$ coupling is in order $\lambda\sim 10^{-5}$ and still
are much bigger than the unnaturally small of $\lambda\sim 10^{-13}$ for the
minimal coupling cold inflation [43, 44].

We present the COBE constrained results for relations between the $\lambda$
and $\gamma$ parameters with varying values of the $C_{T}$ in Table 1 and then
compare the predictions in the $(r-n_{s})$ plane of the latest Planck 2018
data by using the expressions of the $n_{s}$ and $r$ in Eqs.(42) and (43),
respectively with the growing mode in (41).

Figure 2: We compare the theoretical predictions of the strong limit $Q>1$
including the growing mode effects Eqs.(42) & (43) for $C_{T}=0.020$ (purple),
$C_{T}=0.015$ (orange) and $C_{T}=0.014$ (black) in the $(r-n_{s})$ plane for
various values of $\gamma$ and $\lambda$ given in Table (1) constrained by the
COBE renormalization condition by using $C_{R}=70$, $\xi=10^{4}$, $N=60$ and
$M_{p}=10\Lambda$ with Planck’18 results for TT, TE, EE,
+lowE+lensing+BK15+BAO. Figure 3: The allowed region of the possible values of
the $g$ and $h$ from Eq.(29) due to the range of $C_{T}$ for $0.014\lesssim
C_{T}\lesssim 0.02$ where $g$ and $h$ are the Yukawa couplings of the
inflaton-heavy fermions and the heavy fermions-light singlet scalar and
fermion fields, in the supersymmetric model respectively.

From Fig.2, we present the confidence contours in the $(n_{s},\,r)$ plane. The
value of $C_{T}$ is varied for each trajectory. The curves in this figure are
related to $C_{T}$ as: $0.014$ (black), $0.015$ (orange) and $0.020$ (purple)
from the bottom curve to the top one. With a set of input parameters when
$C_{T}$ decreases the curve is shifted upward. The proper set of the
parameters $C_{R}=70$, $\xi=10^{4}$, $N=60$ and $M_{p}=10\Lambda$ is used.
With these values of the parameters, we find that in order to fit inside the
$2\,\sigma$ confidence level of the Planck 2018 data, the range of $C_{T}$ is
in $0.014\lesssim C_{T}\lesssim 0.02$ and it dose not exceed the upper bound
$0.020$ from the constraint of the power spectrum [34]. More importantly, the
given range of the parameter $C_{T}$, $0.014\lesssim C_{T}\lesssim 0.02$ can
consequently provide the possible values of the couplings $g$ and $h$ that are
encoded in the $C_{T}$ as shown in Eq.(29). The allowed region in the
parameter space of the $g$ and $h$ is depicted in Fig.3. To make consistent
results between theory and observation, in addition, this requires that the
cut-off, $\Lambda$ of the inflaton field should be less than the Planck mass
around one order of magnitude in contrast to cold inflation that usually
imposes $\Lambda\sim M_{p}$.

##  IV Conclusion

In this work, we presented the theoretical study of the non-minimal coupling
warm inflation with the quantum-corrected self-interacting inflaton potential.
The slow-roll dynamics of warm inflation in the Einstein frame is analyzed by
using the dissipative coefficient as linear function of temperature. At the
large field approximation in warm inflation, the universal bound for the
quantum-corrected parameter, $\gamma$ is modified by the dissipative
coefficient. With the proper set of the parameters, the universal bound of
warm inflation is bigger than that of cold inflation. This indicates that the
inflaton field in warm inflation is smaller that of the cold one at the end of
inflation. Having used the COBE normalization of the observed amplitude, we
found that the relationship between the self-interacting coupling, $\lambda$
and the quantum-corrected parameter, $\gamma$ is linear and the value of the
$\lambda$ is in order $\mathcal{O}(\lambda)\sim 10^{-5}$ for
$0.06<\gamma<0.1$. The constraint of the $\lambda$ coupling from COBE is
consistent with the renormalization group result at the GUT scale. We
continuously compared the tensor to scalar ratio ($r$) and spectral index
($n_{s}$) from the theoretical results to the Planck 2018 data. As results,
the given sets of the model’s parameters provide good agreement with the
Planck 2018 observational data. To make the theoretical results locating
inside the 2$\sigma$ confidence level, it was found that the range of the
parameter from the dissipative coefficient, $C_{T}$ is in range $0.014\lesssim
C_{T}\lesssim 0.02$ and the lower bound of the $C_{T}$ parameter is
constrained in this work. Consequently, we have also used the range of $C_{T}$
to evaluate the allowed region in the parameter space $g$ and $h$ in terms of
the supersymmetric model that are used to calculate the dissipative
coefficient. In addition, the self-interacting coupling, $\lambda$ should be
very small and this is consistent with the constraint from COBE. More
importantly, in contrast to cold inflation scenario, the cut-off scale of the
inflaton, $\Lambda$ is smaller than that of the Planck scale of one order of
magnitude to obtain the results compatible with the data. Furthermore, higher
order quantum-correction and other form of the inflaton potential are worth
for extensively study. More information and accurate observational data might
provide more details about the quantum-correction of the inflaton and validity
of the warm inflationary universe, especially the observation data of the
primordial tensor modes.

###### Acknowledgements.

D. Samart is financially supported by the Mid-Career Research Grant 2021 from
National Research Council of Thailand under a contract No. N41A640145. P. Ma-
adlerd is supported by National Astronomical Research Institute of Thailand
(NARIT). P. Channuie acknowledged the Mid-Career Research Grant 2020 from
National Research Council of Thailand under a contract No. NFS6400117.

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11institutetext: 1Electrical Engineering, Yale University, New Haven, CT USA
11email<EMAIL_ADDRESS>
2Electrical and Computer Engineering, The University of Washington, WA USA
3Biomedical Engineering, Yale University, New Haven, CT USA
4Radiology & Biomedical Imaging, Yale School of Medicine, New Haven, CT USA

# Incremental Learning Meets Transfer Learning: Application to Multi-site
Prostate MRI Segmentation

Chenyu You1(✉)⋆ Jinlin Xiang2⋆ Kun Su2 Xiaoran Zhang3 Siyuan Dong1
John Onofrey4 Lawrence Staib1,3,4 James S. Duncan1,3,4

###### Abstract

Many medical datasets have recently been created for medical image
segmentation tasks, and it is natural to question whether we can use them to
sequentially train a single model that (1) performs better on all these
datasets, and (2) generalizes well and transfers better to the unknown target
site domain. Prior works have achieved this goal by jointly training one model
on multi-site datasets, which achieve competitive performance on average but
such methods rely on the assumption about the availability of all training
data, thus limiting its effectiveness in practical deployment. In this paper,
we propose a novel multi-site segmentation framework called incremental-
transfer learning (ITL), which learns a model from multi-site datasets in an
end-to-end sequential fashion. Specifically, “incremental” refers to training
sequentially constructed datasets, and “transfer” is achieved by leveraging
useful information from the linear combination of embedding features on each
dataset. In addition, we introduce our ITL framework, where we train the
network including a site-agnostic encoder with pretrained weights and at most
two segmentation decoder heads. We also design a novel site-level incremental
loss in order to generalize well on the target domain. Second, we show for the
first time that leveraging our ITL training scheme is able to alleviate
challenging catastrophic forgetting problems in incremental learning. We
conduct experiments using five challenging benchmark datasets to validate the
effectiveness of our incremental-transfer learning approach. Our approach
makes minimal assumptions on computation resources and domain-specific
expertise, and hence constitutes a strong starting point in multi-site medical
image segmentation.

###### Keywords:

Incremental Learning Transfer Learning Medical Image Segmentation.

11footnotetext: Equal contribution.

##  1 Introduction

Many medical image datasets have been created over the year, and recent
breakthrough achieved by supervised training accelerates the pace in medical
image segmentation. Despite great promise, many prior works have limited
clinical value, since they are separately trained on small datasets in terms
of scale, diversity, and heterogeneity of annotations. As a result, such
single-site methods [22, 14, 21, 10, 32, 29, 35, 31, 36, 41, 38, 40, 37, 39]
are vulnerable to unknown target domains, and linearly expand parameters since
they assume to train a new model in isolation when adding new datasets. This
jeopardizes their trustworthiness and practical deployment in real-world
clinical environments.

In this paper, we carry out the first-of-its-kind comprehensive exploration of
how to build a multi-site model to achieve strong performance on the training
domains and can also serve as a strong starting point for better
generalization on new domains in the clinical scenarios. Multi-site training
[8, 24, 25, 1, 11, 3, 7] has been proposed to consolidate the generalization
on multi-site datasets, but it has the following limitations: (1) it still
exhibits certain vulnerability to different domains (i.e., different imaging
protocols), which yields sub-optimal performance [1, 13, 34]; (2) due to
various constraints (i.e., imaging time, privacy, and copyright status), it
could become challenging or even infeasible for the requirement on the
availability of all training data in a certain time phase. For example, when a
new site’s data will be available after training, the model requires
retraining, which largely prohibits the practical deployments; and (3)
consider the relatively small size of the single medical imaging dataset,
simply training a dense network from scratch usually leads to sub-optimal
segmentation quality because the model might over-fit to those datasets.

Our key idea is to combine the benefits of incremental-learning (IL) and
transfer-learning by sequentially training a multi-dataset expert: we
continually train a model with corresponding pretrained weights as new site
data are incrementally added, which we call Incremental-Transfer Learning
(ITL). This setting is appealing as: (1) the common IL setting [23, 4, 15, 17,
27, 5, 42, 28] is to train the base-learner when different site datasets
gradually come; thus the effectiveness of this approach heavily depends on the
optimality of the base-learner. Consider each single medical image dataset is
usually of relatively small size, it is undesirable to build a strong base-
learner from scratch; (2) transfer-learning [43, 26, 30, 44, 33] typically
leads to better performance and faster convergence in medical image analysis.
Inspired by these findings above, we develop a novel training strategy for
expanding its high-quality learning abilities to our multi-site incremental
setting, considering both model-level and site-level. Specifically, our system
is built upon a site-agnostic encoder with pretrained weights from natural
image datasets such as ImageNet, and at most two segmentation decoder heads
wherein only one head is trainable, and the other is fixed associated with
specific sites - a parameter-efficient design. Our intuition is that the
shared site-agnostic encoder network with pretrained weights encodes
regularities across different medical image datasets, while the target and
source segmentation decoder heads model the sub-distribution by our proposed
site-level incremental loss, resulting in an accurate and robust model that
transfers better to new domains without sacrificing performance. We conduct a
comprehensive evaluation of ITL on five prostate MRI datasets. Our approach
can consistently achieve competitive performance and faster convergence
compared to the upper-bound baselines (i.e., isolated-site and mixed-site
training), and has a clear advantage on overall segmentation performance
compared to the lower-bound baselines (i.e., multi-site training). We also
find that our simple approach can effectively address the forgetting issues.
Our experiments demonstrate the benefits of modeling both multi-site
regularities and site-specific attributes, and thereby serve as a strong
starting point on this important practical setting.

Figure 1: Overview of (a) our proposed Incremental Transfer Learning
framework, and (b) the multi-site expert model. Note that in this study, we
only use one multi-site expert model and one source decoder network, which
will not introduce additional parameter.

##  2 Method

###  2.1 Problem Setup

In ITL, a model incrementally learns from a sequential site stream wherein new
datasets (namely, medical image segmentation tasks with new sites) are
gradually added during the training, as illustrated in Figure 1. More
formally, we denote the sequence of multi-site datasets to be trained as a
multi-domain data sequence $\mathcal{D}\\!=\\!\\{D_{1},D_{2},\cdots,D_{N}\\}$
of $N$ sites, and $i$-th site $D_{i}$ contains the training images
$X\\!=\\!\\{x_{j}\\}_{j=1}^{M}$ and segmentation labels
$Y\\!=\\!\\{y_{j}\\}_{j=1}^{M}$, where ${x}_{j}\in\mathbb{R}^{H\times W\times
3}$ is the augmented image input, and ${y}_{j}\in\\{0,1\\}^{{H\times W}}$ is
the ground-truth label. Here the augmented input setting is appealing: the
axial context naturally provided by a 3D volume can uniquely yield more robust
semantic representations to the downstream tasks. We assume access to a multi-
site expert model $F_{i}\\!=\\!\\{E_{i},G_{i}\\}$ for $i$-th (site) phase,
including a pretrained model as a site-agnostic encoder network $E_{i}$ with
the weight $\theta_{i}$, a target decoder network $G_{i}^{t}$ with the weight
$\theta^{t}_{i}$. During training, we additionally attach a source decoder
network $G_{i}^{s}$ (i.e., using $G_{i-1}^{s}$ from previous phrase) with the
weight $\theta^{s}_{i}$. In the $i$-th incremental (site) phase, the multi-
site expert model has access to two types of domain knowledge: the site-
specific knowledge from the current dataset $D_{i}$ and old exemplars $P_{i}$.
The latter refers to a set of old exemplars from all previous training
datasets $D_{1:i-1}$ in the memory protocol $\mathcal{M}$. This is highly
nontrivial to preventing the challenging “catastrophic forgetting” problem
[20] of the current dataset $i$ against previous sites in clinical practice.
Note that, in this study, we only use one multi-site expert model and one
source decoder network, which will not introduce additional parameters. Based
on the setting above, we define the ITL problem below.

Problem of ITL In the current site $i$, our goal is to continuously learn a
multi-site expert model based on the knowledge from both $(D_{i},P_{i})$ and
the pretrained weight, making the model (1) generalizes well on the unseen
data at site $i$, and (2) achieves competitive performance on the previous
sites.

###  2.2 Preliminary

Our goal is to build a strong multi-site model by learning a site-agnostic
encoder with pretrained weights as well as a segmentation decoder over multi-
site datasets. This naturally raises several interesting questions: How well
will ITL-based methods perform in multi-site medical image datasets? Will
transfer learning make the base learner stronger on the unseen site? If yes,
can they perform stably well? To answer the above questions, a prerequisite is
to define the upper bound and lower bound. Here we introduce three common
paradigms for multi-site medical image segmentation: (1) isolated-site
training, (2) mixed-site training, and (3) multi-site training. It is well-
known that the isolated-site and mixed-site training approaches can achieve
state-of-the-art performance when evaluating the same dataset, while the
performance catastrophically drops when evaluating new datasets. On the other
hand, the multi-site training approach often yields inconsistent performance
across multiple sites. For all training paradigms, we minimize Dice loss
between the predicted outputs and the ground truth label.

Table 1:  Information about five different sites from three benchmark
datasets.

Dataset | Modality | # of cases | Field strength (T) | Resolution(in/through plane)(mm) | Coil | Source
---|---|---|---|---|---|---
Site0 | MRI | 30 | 3 | 0.6-0.625 / 3.6-4 | Surface | NCI-ISBI13 [2]
Site1 | MRI | 30 | 1.5 | 0.4 / 3.0 | Endorectal | NCI-ISBI13 [2]
Site2 | MRI | 19 | 3 | 0.67-0.79 / 1.25 | - | I2CVB [12]
Site3 | MRI | 12 | 1.5 | 0.625 / 3.6 | Endorectal | PROMISE12 [16]
Site4 | MRI | 13 | 1.5 and 3 | 0.325-0.625 / 3-3.6 | - | PROMISE12 [12]

Upper Bound We consider two training paradigms (i.e., isolated-site and mixed-
site training) as our upper bound baseline. For isolated-site training, given
each site $D_{i}$, we train isolated-site models separately. The architecture
of the isolated-site model consists of a pretrained encoder $E_{i}$ and a
segmentation decoder network, same architecture as $G_{i}$. Then, we apply
different isolated-site models to predict results based on the site-specific
data at inference. However, this approach dramatically increases memory and
computational overhead, making it practically challenging at scale. For mixed-
site training, we train one full model on the full mixed-site data $D$, and
then use the well-trained model for inference. However, it requires the
simultaneous presence of all data in training and inference.

Lower Bound For multi-site training, we sequentially train only one model
coupled with the pretrained weights on all sites. This can get rid of large
parameter counts, making it appealing in practice. However, due to the
forgetting quandary, it inevitably suffers from severe performance
degradation. This naturally questions: can we improve performance on multi-
site medical image segmentation with minimal additional memory footprint? In
the following, we give an affirmative answer.

###  2.3 Proposed Incremental Transfer Learning Multi-Site Method

To address the aforementioned problems, we develop the incremental transfer
learning framework to perform well on the training distribution and generalize
well on the new site dataset with minimal additional memory. To our best
knowledge, we are the first work to apply incremental transfer learning to the
limited clinical data regimes. To control the parameter efficiency, we
decompose the model into a share site-agnostic encoder $E_{i}$ and two
segmentation decoder heads (i.e., source decoder $G_{i}^{s}$ and target
decoder $G_{i}^{t}$). In this way, we can keep the network parameters the same
when adding a new site. Specifically, $G_{i}^{s}$ is designed to transfer the
knowledge of a previously learned site, and $G_{i}^{t}$ is designed to
comprehensively train on a new site and previous datasets. During training, we
only update $G_{i}^{t}$ while $G_{i}^{s}$ is frozen. It is worth mentioning
that our proposed framework is independent of the encoder architecture, and
can be easily plugged in other pretrained vision models.

The full ITL algorithm is summarized in Algorithm 1. We describe our ITL
algorithm as follows. We first randomly initialize $G_{i}^{t}$, $G_{i}^{s}$,
and then iteratively train our full model (i.e., a pretrained encoder $E_{i}$
and two decoders $G_{i}^{t}$, $G_{i}^{s}$) with $N$-site training samples.
Bounded by the computational requirements, it is challenging or even
infeasible to retain all data for training. Inspired by recent work [23], to
maintain the knowledge of previous sites, we “store” all the old site data
exemplars in the memory protocol $\mathcal{M}_{i}$. In the $i$-th incremental
(site) phase, we first load $P_{i}$, and then use both $P_{i}$ and $D_{i}$ to
train $F_{i}$ initialized by $\theta_{i}^{s}$. This setting is appealing as
(1) it can substantially alleviate the imbalance between the old and new site
knowledge, and (2) it is efficient to train on them. Of note, we do not use
the source decoder when training on the first-site dataset. We formulate ITL
as model-level and site-level optimization.

Model-level Optimization To perform better on all these training
distributions, we propose improving generic representations by distilling
knowledge from previous data. In each incremental phase, we jointly optimize
two groups of learnable parameters in our ITL learning by minimizing the
model-level incremental loss (i.e.,
$\mathcal{L}_{\text{model}}\\!=\\!\mathcal{L}_{\text{target}}+\mathcal{L}_{\text{source}}$)
on all training samples (i.e., $D_{i}\bigcup D_{0:i-1}$): (1) a share site-
agnostic encoder $E_{i}$ and a target decoder $G_{i}^{t}$; (2) a share site-
agnostic encoder $E_{i}$ and a source decoder $G_{i}^{s}$. This helps ITL
avoid catastrophic forgetting of prior site-specific knowledge.

Algorithm 1 Incremental-Transfer Learning(ITL) Algorithm

0: Dataset: $\mathcal{D}$; Hyper-parameters: $\alpha,\delta,\gamma$

1: Initialize the $\mathcal{M}\textit{ (Memory)}$ : $\mathcal{M}$

2: Initialize the Model $F_{0}$: $\textit{Pertrained Encoder}\longrightarrow
E_{0},G_{0}$

3: for $i$ = 1,2,3,….N do

4: for All training Sample in $D_{i}$ and $\mathcal{M}_{i-1}$ do

5:
$\mathcal{L}_{\text{target}}=\sum_{j=0}^{N-1}\alpha_{j}\mathcal{L}_{\text{Dice}}^{E_{i},G_{i}^{t}}(\mathcal{M}_{j},Y_{j})$
or $0$ When $N=1$

6:
$\mathcal{L}_{\text{source}}=\sum_{j=0}^{N-1}\delta_{j}\mathcal{L}_{\text{Dice}}^{E_{i},G_{i}^{s}}(\mathcal{M}_{j},Y_{j})$
or $0$ When $N=1$

7:
$\mathcal{L}_{\text{model}}=\mathcal{L}_{\text{source}}+\mathcal{L}_{\text{target}}$

8:
$\mathcal{L}_{\text{site}}=\mathcal{L}_{\text{Dice}}^{E_{i},G_{i}^{t}}(\mathcal{D}_{i},Y_{i})$

9:
$\mathcal{L}_{\text{all}}=\mathcal{L}_{\text{site}}+\mathcal{L}_{\text{model}}$

10: $F_{i}=(E_{i},G_{i}^{t})$ by minimizing the $\mathcal{L}_{\text{all}}$

11: end for

12: Update Memory: $\mathcal{M}+\gamma\%D_{N}\longrightarrow\mathcal{M}$

13: Save Teacher Model: $G_{N}$

14: end for

Site-level Optimization The above model-level optimization is used to maintain
previously learned knowledge. In contrast, this step is design to train the
multi-site model to learn site-specific knowledge on the newly added site.
Specifically, we minimize the site-level incremental loss
$\mathcal{L}_{\text{site}}$ between the probability distribution from $F_{i}$
and the ground truth. This essentially learns the site-specific knowledge for
the downstream medical image segmentation tasks. Of note,
$\mathcal{L}_{\text{source}}$, $\mathcal{L}_{\text{target}}$, and
$\mathcal{L}_{\text{site}}$ use the Dice loss. The overall loss combines the
model-level loss and the site-level loss as follows:

$\mathcal{L}_{\text{all}}=\mathcal{L}_{\text{model}}+\mathcal{L}_{\text{site}}.$
(1)

##  3 Experiments

Datasets and Settings We evaluate our proposed incremental transfer learning
method on three prostate T2-weighted MRI datasets with different sub-
distributions: NCI-ISBI13 [2], I2CVB [12], and PROMISE12 [16]. Due to the
diverse data source distributions, they can be split into five multi-site
datasets, which is similar to [19]. Table 1 provides some dataset statistics.
For pre-processing, we follow the setting in [18] to normalize the intensity,
and resample all 2D slices and the corresponding segmentation maps to
$384\times 384$ in the axial plane. For all five site datasets, we randomly
split each original site dataset into training and testing with a ratio of 4 :
1. For each site training, we divide the data from the previous site into a
small subset with a certain portion (i.e., 1%, 3%, 5%), and combine it with
the current site data for training.

Table 2: Comparison of segmentation performance (DSC[%]/95HD[mm]) across
datasets. Note that a larger DSC ($\uparrow$) and a smaller 95HD
($\downarrow$) indicate better performing ITL models. We use four models
pretrained on ImageNet: ResNet-18, ResNet-34, ResNet-50, and ViT under
different portions (i.e., 1%, 3%, 5%) of exemplars from previous data for
every incremental phase. We consider multi-site training as the lower bound,
isolated-site, and mixed-site training as the upper bound.

HK UCL ISBI ISBI1.5 I2CVB Backbone Scheme DSC[%] 95HD[mm] DSC[%] 95HD[mm]
DSC[%] 95HD[mm] DSC[%] 95HD[mm] DSC[%] 95HD[mm] RES-18 Multi 59.38 64.17 66.26
54.19 54.38 73.40 66.89 44.49 84.54 11.70 1% 67.82 56.08 67.12 58.05 59.47
70.46 77.34 34.77 82.94 6.06 3% 71.60 18.41 82.18 23.92 72.26 20.91 81.53
19.21 84.08 13.75 5% 81.81 5.50 84.45 13.95 84.52 15.65 89.32 10.11 86.72
11.70 Isolated 93.46 2.06 88.29 6.20 93.35 2.04 90.89 7.53 88.74 13.93 Mixed
92.17 7.60 83.38 12.22 91.70 2.46 90.08 9.20 89.12 13.86 RES-34 Multi 57.75
55.13 64.87 52.50 57.47 65.38 65.61 56.83 91.46 8.83 1% 67.40 24.18 79.55
30.43 69.61 44.69 84.68 18.71 89.38 15.24 3% 80.90 28.41 82.57 22.18 75.89
26.26 84.68 10.57 90.35 13.15 5% 80.46 22.92 87.79 17.32 88.14 14.64 90.29
8.57 91.30 8.52 Isolated 93.87 1.89 89.03 4.05 92.08 2.19 92.57 7.96 91.57
7.98 Mixed 93.85 1.71 87.81 16.85 91.49 3.35 93.82 5.30 92.58 6.64 RES-50
Multi 63.24 53.98 64.79 56.59 72.95 26.63 69.41 49.89 90.40 8.21 1% 69.01
60.70 69.85 44.21 75.30 28.74 80.27 20.10 90.08 8.02 3% 78.72 16.89 83.74
12.81 84.96 8.51 86.95 6.18 92.34 5.24 5% 92.46 2.92 88.79 10.97 92.16 2.04
92.18 4.87 91.35 2.12 Isolated 93.73 2.12 89.03 7.23 93.26 4.39 93.48 5.10
93.20 2.40 Mixed 94.38 1.34 88.28 9.77 92.71 9.43 92.27 5.29 90.45 5.29 VIT
Multi 66.94 53.57 65.85 54.69 92.66 6.37 72.80 51.35 90.56 7.02 1% 71.99 48.61
85.29 11.35 75.99 17.87 84.73 12.32 90.11 7.23 3% 79.33 20.84 88.16 7.08 85.48
7.97 87.64 9.95 90.07 6.94 5% 93.25 1.37 87.62 9.23 92.22 4.82 91.62 2.82
91.87 6.59 Isolated 94.44 1.88 88.80 8.21 93.23 4.76 92.47 6.27 93.23 6.43
Mixed 93.30 1.38 87.20 9.21 92.86 9.29 86.92 12.28 92.01 6.99

Training and Evaluation In this study, we implement all models using Pytorch.
We set $H,W$ as 384, $\alpha,\delta$ as 0.5, and the batch size as $5$. To
mitigate the overfitting, we augment the data by random horizontal flipping,
random rotation, and random shift. We adopt ResNet family [9] (i.e., ResNet18,
ResNet34, ResNet50) and ViT [6] (i.e., R50+ViT-B/16 hybrid model) as our
pretrained encoder. We evaluate the model performance by Dice coefficient
(DSC) and 95% Hausdorff Distance (95HD). For a fair comparison, we adopt the
same decoder architecture design in [18] are shown in Appendix Table A1, and
do not use any post-processing techniques. All of our experiments are
conducted on two NVIDIA Titan X GPUs. All the models are trained using Adam
optimizer with $\beta_{1}=0.9$, $\beta_{2}=0.999$. For 100 epochs training, a
multi-step learning rate schedule is initialized as $0.001$ and then decayed
with a power of $0.95$ at epochs $60$ and $80$.

Main Results We conduct extensive experiments on five benchmark datasets. We
adopt four models: ResNet-18, ResNet-34, ResNet-50, and ViT. We select three
portions (i.e., 1%, 3%, 5%) of exemplars from previous data for every
incremental phase. Our results are presented in Table 2 and Appendix Figure
A1. First and foremost, we can see ITL-based methods generalize across all
datasets under two exemplar portions (i.e., 3% and 5%), yielding the
competitive segmentation quality comparable to the upper bound baselines
(i.e., isolated-site and mixed-site training), which are much higher than the
lower bound counterparts. The 1% exemplar portion seems slightly more
challenging for ITL, but its superiority over the lower bound counterparts is
still solid. A possible explanation for this finding is that using two
exemplar portions (i.e., 3% and 5%) maintains enough information of ITL, which
mitigates the catastrophic forgetting, while ITL trained in the setting of 1%
exemplar portion is not powerful enough to inherit prior knowledge and
generalize well on newly added sites. Second, we consistently observe that ITL
using larger models (i.e., ResNet-50 and ViT) generalize substantially better
than those using small models (i.e., ResNet-18 and ResNet-34), which
demonstrate competitive performance across all datasets. These results suggest
that our ITL using the large model as our pretrained encoder leads to
substantial gains in the setting of very limited data.

##  4 Analysis and Discussion

We address several research questions pertaining to our ITL approach. We use a
ResNet-18 model as our encoder in our experiments. For comparisons, all models
are trained for the same number of epochs, and all results are the average of
three independent runs of experiments. To study the effectiveness of our
proposed ITL framework, we performed experiments with $5\%$ exemplars ratio.

Does transfer learning lead to better ITL? We draw two perspectives that may
intuitively explain the effectiveness of transfer learning in our proposed ITL
framework. As a first test of whether transfer learning makes the base-learner
stronger, we plot the training loss/validation loss (i.e.,
$\mathcal{L}_{\text{all}}$) to iteration to demonstrate the convergence
improvements in Appendix Figure A2. We can see that training from pretrained
weights can converge faster than training from scratch. Another (perhaps not
so surprising) observation we can get from Appendix Figure A2 is that using
pretrained weights usually yields slightly smaller loss compared to training
from scratch. We then ask whether transfer learning produces increased
performance on multi-site datasets. Since each single medical image dataset is
usually of relatively small size, training the model from scratch tends to
overfit a particular dataset. To evaluate the impact of transferring learning,
we compare w/pretraining to w/o pretraining. As shown in Appendix Table A4,
training from scratch does not bring benefits to the ITL framework. Instead of
training from scratch, we find that simply incorporating transfer learning
significantly boots the performance of ITL while achieving faster convergence
speed, suggesting that transfer learning provides additional regularization
against overfitting.

Table 3: Comparison of segmentation performance in different phases.

HK UCL ISBI ISBI1.5 I2CVB DSC[%] 95HD[mm] DSC[%] 95HD[mm] DSC[%] 95HD[mm]
DSC[%] 95HD[mm] DSC[%] 95HD[mm] 94.06 1.96 - - - - - - - - 93.68 1.98 88.74
8.72 - - - - - - 93.20 1.83 87.38 9.30 92.87 1.82 - - - - 90.37 8.34 86.73
12.75 89.84 13.32 91.57 11.05 - - 88.88 8.91 85.14 10.97 85.98 14.23 89.74
13.11 88.46 12.15

Does ITL generalizes well on multi-site datasets? We investigate whether the
ITL framework generalizes well on multi-site datasets. We report the
segmentation results of different phases in Table 3, from which we observe
that ITL achieves good performance in different phases. This reveals that our
approach is greatly helpful in reducing forgetting issues. We evaluate the
proposed ITL methods with two random ordering (i.e., (1)
{HK$\rightarrow$UCL$\rightarrow$ISBI$\rightarrow$ISBI1.5$\rightarrow$I2CVB},
and (2)
{ISBI$\rightarrow$ISBI1.5$\rightarrow$I2CVB$\rightarrow$HK$\rightarrow$ UCL}).
The results are shown in Appendix Table A2. We perform experiments using both
ordering strategies and observe comparable performance.

Efficiency of ITL We report the network size and memory costs in Appendix
Table A3. We observe that ITL achieves competitive performance and utilizes
less network parameters compared to isolated-site training (upper bound),
which requires the new model when adding new site data. We also examine the
required memory footprint at each incremental phase. We observe that ITL is
significantly more memory-efficient than mixed-site training (upper bound),
although the latter remains the same network size when adding a new training
phase. These results further demonstrate the efficiency of our proposed ITL
framework.

##  5 Conclusion

In this paper, we present a novel incremental transfer learning framework for
incrementally tackling multi-site medical image segmentation tasks. We pose
model-level and site-level incremental training strategies for better
segmentation, generalization, and transfer performance, especially in limited
clinical resource settings. Extensive experimental results on four different
baseline architectures demonstrate the effectiveness of our approach, offering
a strong starting point to encourage future work in these important practical
clinical scenarios.

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## Appendix

Figure A1: Visualization of segmentation results on five benchmarks using
ResNet-18 as the encoder. Different site results are shown in different
colors.

Table A1: Segmentation decoder head architecture

Deocder Layer Feature size Upsample 1 48x48 Residual block 1 48x48 Upsample 2
96x96 Residual block 2 96x96 Upsample 3 192x192 Residual block 3 192x192
Upsample 4 384x384 Residual block 4 384x384 Output prediction 384x384

Table A2: Comparison of different ordering strategies using ResNet-18. We
report mean and standard deviation across three random trials. Note that a
larger DSC ($\uparrow$) and a smaller 95HD ($\downarrow$) indicate better
performing ITL models.

Training sequence DSC[%] 95HD[mm]
HK$\rightarrow$UCL$\rightarrow$ISBI$\rightarrow$ISBI1.5$\rightarrow$I2CVB
85.36$\pm$0.33 11.38$\pm$0.36
ISBI$\rightarrow$ISBI1.5$\rightarrow$I2CVB$\rightarrow$HK$\rightarrow$UCL
86.27$\pm$0.27 12.01$\pm$0.68

Table A3: Comparison of different training strategies using ResNet-18. We
report mean and standard deviation across three random trials.

Scheme | DSC[%] | 95HD[mm] | Model size(Mb) | Add new sites? | Memory Cost
---|---|---|---|---|---
Isolated | 90.95$\pm$0.27 | 6.35$\pm$0.68 | 77.9$\times$Site Num. | Linearly increase | new Data
Mixed | 89.29$\pm$0.38 | 9.06$\pm$0.84 | 77.9 | constant | old Data + new Data
ITL | 85.36$\pm$0.33 | 11.38$\pm$0.36 | 77.9 | constant |  $5\%$ old Data + new Data

Table A4: Ablation of each component in the proposed ITL when using ResNet-18
under 5% exemplar portion. We report mean and standard deviation across three
random trials. Note that a larger DSC ($\uparrow$) and a smaller 95HD
($\downarrow$) indicate better performing ITL models. The best results are in
bold.

HK UCL ISBI ISBI1.5 I2CVB Backbone Component DSC[%] 95HD[mm] DSC[%] 95HD[mm]
DSC[%] 95HD[mm] DSC[%] 95HD[mm] DSC[%] 95HD[mm] Avg. DSC Avg. 95HD RES-18
pretraining only 59.38 64.17 66.26 54.19 54.38 73.40 66.89 44.49 81.54 28.70
65.69$\pm$1.51 52.99$\pm$0.72 $\mathcal{L}_{\text{model}}$ only 74.02 18.86
73.79 31.90 51.23 51.66 80.89 21.96 80.88 58.44 72.16$\pm$0.38 36.56$\pm$0.85
pretraining + $\mathcal{L}_{\text{model}}$ 81.81 5.50 84.45 13.95 84.52 15.65
89.32 10.11 86.72 11.70 85.36$\pm$0.33 11.38$\pm$0.36

Figure A2: Comparison of training from scratch against using pretraining. We
use ResNet-18 on ImageNet as the encoder. Under 5% exemplar portion, we plot
(a) training loss (Scratch), (b) training loss (Pretraining), (c) validation
loss (Scratch), (d) validation loss (Pretraining).