# Tidal Disruption Events from Stripped Stars

Brenna Mockler The Observatories of the Carnegie Institution for Science,
Pasadena, CA 91101, USA  Monica Gallegos-Garcia Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA  Center for
Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern
University, Evanston, IL 60208, USA  Ylva Götberg Institute of Science and
Technology Austria, 3400 Klosterneuburg, Austria  The Observatories of the
Carnegie Institution for Science, Pasadena, CA 91101, USA  Jon M. Miller
Department of Astronomy, University of Michigan, 1085 South University Avenue,
Ann Arbor, MI 48109, USA  Enrico Ramirez-Ruiz Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA

###### Abstract

Observations of tidal disruption events (TDEs) show signs of Nitrogen
enrichment reminiscent of other astrophysical sources such as active galactic
nuclei (AGN) and star-forming galaxies. Given that TDEs probe the gas from a
single star, it is possible to test if the observed enrichment is consistent
with expectations from the CNO cycle by looking at the observed
Nitrogen/Carbon (N/C) abundance ratios. Given that $\approx 20\%$ of solar
mass stars (and an even larger fraction of more massive stars) live in close
binaries, it is worthwhile to also consider what TDEs from stars influenced by
binary evolution would look like. We show here that TDEs from stars stripped
of their Hydrogen-rich (and Nitrogen-poor) envelopes through previous binary-
induced mass loss can produce much higher observable N/C enhancements than
even TDEs from massive stars. Additionally, we predict that the time-
dependence of the N/C abundance ratio in the mass fallback rate of stripped
stars will follow the inverse behavior of main-sequence stars, enabling a more
accurate characterization of the disrupted star.

stars: black holes — stars: tidal disruption events — galaxies: nuclei –
galaxies: active — galaxies: supermassive black holes

††journal: ApJL††software: astropy (Astropy Collaboration et al., 2013)

##  1 Introduction

With recent observations and new, more sensitive instruments, Nitrogen
enrichment has been appearing in a wide variety of astrophysical research:
from quasars and active galactic nuclei (AGN) disks (e.g. Bentz et al., 2004)
to distant star-forming galaxies (e.g. Bunker et al., 2023). Nitrogen
enrichment is typically closely linked to the CNO cycle – the nuclear reaction
that fuses Hydrogen to Helium in the deep interiors of massive stars. How
Nitrogen enriched material is then exposed from its origin in the interior of
massive stars remains a largely unsolved problem. This emphasizes the
importance of understanding links between stellar evolution and a range of
different astrophysical research fields and in a variety of different
environments.

Recent observations of tidal disruption events (TDEs) show surprisingly high
N/C abundance ratios (e.g. abundance ratios relative to solar of N/C $>10$ in
ASASSN-14li, iPTF15af, iPTF16fnl, see Cenko et al. 2016a; Blagorodnova et al.
2017, 2019; Kochanek 2016; Yang et al. 2017a)111In addition to these three
events, there are also a few newer TDEs with potential detections of broad UV
Nitrogen and/or Carbon lines that might be used to constrain N/C ratios with
further analysis. These include ASASSN-14ko (Payne et al., 2023), AT2018zr
(Hung et al., 2019), AT2019qiz (Hung et al., 2021), and AT2020vdq (Somalwar et
al., 2023).. These TDEs occur when stars are ripped apart by the supermassive
black holes (SMBHs) in the centers of galaxies, after which the gas from the
star is eventually accreted by the SMBH, producing bright transient flares in
the process (e.g. Rees, 1988; Evans & Kochanek, 1989; Guillochon & Ramirez-
Ruiz, 2013). High N/C abundance ratios suggest the disrupted star was subject
to the CNO cycle and therefore relatively massive, thus constituting a
promising channel for enriching gas surrounding the SMBH (Mockler et al.,
2022). With the increasing sample of TDEs now reaching the $\gtrsim 100$ (e.g.
Hammerstein et al., 2023), and with upcoming data from the Vera Rubin
Observatory predicted to increase this number to $\gtrsim 1000$ (e.g. Bricman
& Gomboc, 2020), we can expect a wealth of TDE data soon to help improve our
understanding not only of SMBHs, but also stellar populations in the centers
of galaxies.

Tidal disruptions of stars around supermassive black holes (SMBHs) probe
stellar populations on size scales that cannot be observed directly outside
our own galactic neighborhood. The stars that are disrupted originate from
near the sphere of influence of the black hole, corresponding to size scales
of order $\approx 0.5-10$pc for SMBHs with masses between $\approx
10^{5}-10^{8}M_{\odot}$.

The increasing number of TDE observations has motivated the development of
numerous light curve and spectral models to attempt to explain the unique
features of these events and also to use observations to constrain properties
of the black hole and the disrupted star (Roth & Kasen, 2018; Mockler et al.,
2019; Ryu et al., 2020; Wen et al., 2020; Metzger, 2022) and better understand
black hole accretion (Ramirez-Ruiz & Rosswog, 2009a; Dai et al., 2018;
Andalman et al., 2022).

Much like in observations of AGN (e.g. Bentz et al., 2004; Jiang et al.,
2008), measurements of spectral lines in TDEs hold the potential to constrain
the composition of the irradiated gas in galactic nuclei. Unlike in AGN, in
TDEs the gas producing the lines is predominantly from a single star, and so
composition constraints in TDE spectra tell us about the properties of
individual stars in the spheres of influence of the host galaxies.

The observations of TDEs with unexpectedly high Nitrogen abundances have
complicated the previous assumption that most TDEs originated from low-mass
main-sequence stars (which are generically Nitrogen-poor), as is predicted if
disrupted stars are drawn randomly from the stellar population given standard
stellar initial mass functions. Mockler et al. (2022) showed that massive
stars ($M>2\,M_{\odot}$) that fuse Hydrogen through the CNO cycle are somewhat
more consistent with these observations, since the abundance of Nitrogen rises
at expense of Carbon and Oxygen in regions that experience the CNO cycle.
However, recent re-analysis of X-ray data from the TDE ASASSN-14li by Miller
et al. (2023) find Nitrogen/Carbon abundances that push the limits of what is
expected even from the disruption of massive stars. Main-sequence stars have
Nitrogen-rich cores, but their envelopes are Nitrogen-poor. During a TDE,
dilution by in-mixing of the Nitrogen-poor envelope should make the abundance
ratio less pronounced.

Motivated by these results, in this work we explore what the disruption of a
star stripped of its Hydrogen-rich (and therefore Nitrogen-poor) envelope
would look like and how it would compare to the disruption of stars at various
stages of stellar evolution. Such “stripped stars” are expected to be created
through binary interaction over the full stellar mass range, but we explore
stripped stars originating from massive stellar evolution as they will result
in higher N/C abundance ratios. Throughout this work, we will use N/C to mean
the abundance ratio of Nitrogen to Carbon relative to the solar value222Note
that the mass abundance ratio relative to solar is equivalent to the number
abundance ratio relative to solar as the ratio of the mass of the elements
cancels out..

##  2 The compositional structure of stripped stars

Figure 1: Composition as a function of mass fraction for three representative
stars. The top plot is of a $2M_{\odot}$ MS star born at $Z_{\odot}$, the
middle plot is of a $2M_{\odot}$ MS star born at $3Z_{\odot}$, and the bottom
plot is of a stripped star born at $Z_{\odot}$ with an initial mass of
$3M_{\odot}$ and a current mass of $0.58M_{\odot}$. All three stars have
$Y_{c}=0.5$. The two MS stars are halfway through Hydrogen burning, and the
stripped star is halfway through Helium burning. The gray shaded regions
denote the convectively burning core (including convection and convective
overshoot).

Stripped stars have naturally Nitrogen-enriched and Carbon-depleted outer
layers (Götberg et al., 2018). This originates from the previous main-sequence
evolution when Hydrogen fused to Helium through the CNO cycle. The slowest
reaction in the CNO cycle is proton capture onto Nitrogen, which causes a
build-up of Nitrogen at the expense of Carbon and Oxygen in regions that have
experienced or are experiencing the CNO cycle (e.g. Burbidge et al., 1957). As
the convective main-sequence core recedes in mass coordinate, seen from
outside and inwards, it leaves a layer that is gradually enriched in Helium,
but that also has Nitrogen-enrichment from the CNO cycle throughout it. During
envelope-stripping in binaries, the thick Hydrogen-rich (and Nitrogen-poor)
envelope is removed, revealing the chemically enriched layer. Because of this,
the surfaces of stripped stars contain a small amount of Hydrogen (notably
still sufficient to produce strong Hydrogen lines in supernova, Dessart et
al., 2011, 2012), a large amount of Helium, and are enriched in Nitrogen
(Hirsch, 2009; Heber, 2016; Götberg et al., 2023, see also Drilling et al.
2013).

After envelope-stripping is complete, the stripped star starts fusing Helium
to Carbon and Oxygen in the center (Laplace et al., 2021). The central region
is convective, but it is relatively small and there is also a substantial
radiative layer that previously was the main-sequence core and now extends to
the stellar surface. The described composition structure is depicted in Figure
1, where we show a model for a $0.58M_{\odot}$ stripped star (with initial
mass $M_{i}=3M_{\odot}$, bottom panel) compared with models for $2M_{\odot}$
main-sequence stars born at two different metallicities that are undergoing
CNO cycle Hydrogen burning (top, middle). In this study, we consider the gray
shaded (convective) regions the core, and the white (radiative) regions the
envelope. The main-sequence stars are born at solar metallicity ($Z_{\odot}$,
top panel) and $3\times$ solar metallicity ($3Z_{\odot}$, middle panel). In
Figure 1, all three models are plotted when the mass fraction of Helium at the
center of the star ($Y_{c}$)$=0.5$. The main sequence (MS) models are
$\sim$halfway though Hydrogen burning, and the stripped star is $\sim$halfway
through Helium burning333This is equivalent to $0.47\times$ and $0.4\times$
the terminal age main sequence (TAMS) lifetimes respectively for the
$Z_{\odot}$ and $3Z_{\odot}$ MS models, and to $0.5\times$ the stripped star’s
Helium burning lifetime for that model.. All models were computed with the
MESA stellar evolution code (Paxton et al., 2011, 2013, 2015, 2018, 2019;
Jermyn et al., 2023) and the stripped star model was published in (Götberg et
al., 2018, see their Table 1). We have chosen these masses for analysis
because they are on the lower end of the mass range that experience CNO
burning, and are therefore representative of the most common stars with
Nitrogen enhancements from CNO burning444We choose this $3M_{\odot}$ model
because the structure was numerically robust at smaller scales. The
qualitative features are the same for a stripped star with a $2M_{\odot}$
progenitor. We will continue to use these same three stellar models throughout
the paper (at various stages of their evolution).

To summarize, the entirety of stripped stars are made up of material that has
previously experienced CNO burning, however their convective cores are
currently undergoing Helium burning. Because of this, their N/C abundance
ratios are high in their radiative outer regions (leftover from their MS core
CNO burning) but low in their convective cores where Helium burning is
ongoing. This is in contrast to MS stars sufficiently massive to experience
CNO burning, whose Nitrogen-rich cores are embedded within Nitrogen-poor outer
layers. Therefore, to first order, the abundances in the center versus the
outer layers of stripped stars exhibit the opposite trends compared to solar
type main-sequence stars (see Figure 1). As we will show, this means that the
time-dependent composition evolution in the material returning to a SMBH after
the tidal disruption of stripped stars should follow the inverse behavior of
CNO burning main-sequence stars.

##  3 Connections to observations

We begin with a very brief overview of the tidal disruption process and go on
to describe how the disruption of a stripped star differs from (and is similar
to) that of a main-sequence star.

Tidal disruption occurs when the binding energy of a star is overwhelmed by
the tidal gravity of a black hole. The radius of disruption for a star of mass
$M_{*}$ and radius $R_{*}$ by a black hole of mass $M_{h}$ can be approximated
by setting the self-gravity of the star equal to the tidal force of the black
hole, which gives the canonical tidal radius
$R_{t}=R_{*}\big{(}\frac{M_{*}}{M_{h}}\big{)}^{1/3}$. The ‘depth’ of
disruption relative to $R_{t}$ can be described using the impact parameter
$\beta=R_{t}/R_{p}$, where deeper (smaller pericenter radius, $R_{p}$)
disruptions have larger values of $\beta$. The rate of return of gas to a SMBH
after a star is tidally disrupted depends on the binding energy distribution
in the gas (e.g. Lodato et al., 2009a; Guillochon & Ramirez-Ruiz, 2013; Law-
Smith et al., 2020; Ryu et al., 2020). In a full disruption, where the core is
destroyed, material in the outer layers of the star (which is more bound to
the black hole), will return first, and material from the core of the star
(which is less bound to the black hole), will return later. Once the slope of
the binding energy distribution as a function of stellar mass becomes
relatively constant (as we approach the core of the star), the mass fallback
rate approaches the canonical value of $t^{-5/3}$:

$\frac{dm}{dt}=\frac{dm}{de}\times\frac{de}{dt}$ (1)
$\frac{dm}{dt}=\frac{dm}{de}\times\frac{(2\pi GM_{h})^{2/3}}{3}t^{-5/3}$ (2)

Here, $\frac{dm}{dt}$ is the mass return rate to the black hole,
$\frac{dm}{de}$ is the mass-energy distribution in the debris, and
$\frac{de}{dt}$ is the derivative of the orbital energy with respect to
orbital timescale and orbital energy, which can be calculated analytically
from Kepler’s laws.

###  3.1 Composition evolution

Figure 2:  We plot analytical fallback rates calculated for various MESA
stellar models. Fallback rates for different elements are plotted separately
in addition to the overall fallback rate. All fallback rates are scaled by the
peak fallback rate and peak timescale for all material (see Figure 4 for a
comparison with hydro simulations in physical units). The top two panels
feature our fiducial $2M_{\odot}$ single stars near the end of their MS
lifetime at $Y_{c}=0.9$. The bottom panel features our $0.58M_{\odot}$
stripped star halfway through He-core burning at $Y_{c}=0.5$.

If the star’s composition varies with radius (and therefore also with binding
energy), this radial stratification transforms into a time-dependence, where
the composition of the gas returning to the black hole varies with time. This
has been explored for main sequence stars analytically in Gallegos-Garcia et
al. (2018) and with hydrodynamical simulations in Law-Smith et al. (2019).

A key result in both is that the ratio of N/C in the fallback material varies
dramatically with time and is dependent on the mass and age of the star. As
discussed in Section 2, the composition profile of a stripped star is, to
first order, the inverse of the profile of a main sequence star (see also
Figure 1). In Figure 2, we plot the normalized, time-dependent composition of
the fallback rate for our two main sequence stars and compare it to the
composition of the fallback rate of our stripped star (of similar progenitor
mass). We calculate the fallback rates analytically using the frozen-in
approximation described in Lodato et al. (2009b) and Kesden (2012) and apply
this framework to MESA stellar models as outlined in Gallegos-Garcia et al.
(2018).

While we know the analytical fallback rates do not exactly match the peak
magnitude and timescales of fallback rates from hydrodynamic simulations (and
for this reason we scale out these quantities from our composition plots),
they should capture overall trends in the composition as a function of
time555see Section 3.2 and Figure 4 for a comparison with fallback curves from
simulations. Hydrodynamic simulations of solar mass main sequence stars have
shown a similar transition in the abundances of Nitrogen, Carbon and Oxygen
(and the abundance ratio of N/C), in the fallback material of a TDE compared
to analytical calculations. This transition corresponds to when material from
the core begins to return to the SMBH. Simulations do find that this
transition occurs more gradually than analytical treatments predict, and that
it begins slightly before peak instead of slightly after peak (Law-Smith et
al., 2019). Because of this, we expect mixing in the hydrodynamics of
disruption to also smear out the sharp transition we see in the relative
abundance of these elements in the stripped star (these complications are
discussed in more detail in Section 3.3).

Figure 3: The analytical N/C abundance ratio (relative to solar) in the mass
fallback rate as a function of time since disruption (scaled by $t_{\rm
peak}$, which we expect to be of order $\sim$ week timescales, see Figure 4).
We plot our $0.58M_{\odot}$ stripped star ($M_{i}=3M_{\odot}$) at the
beginning of Helium core burning ($Y_{c}=0.9$, red solid line) and halfway
through Helium core burning ($Y_{c}=0.5$, red dashed line). We plot our
$2M_{\odot}$ MS stars partway through the main sequence and nearing the end of
the main sequence. The $Z_{\odot}$ MS model is plotted in dark blue, the
$3\times Z_{\odot}$ MS model is plotted in light blue ($Y_{c}=0.5$ plotted in
solid lines, $Y_{c}=0.9$ plotted in dashed lines). Constraints from UV and
X-ray spectra from the transient ASASSN-14li plotted as dotted and dash-dotted
lines respectively.

We find that the mass fractions of Nitrogen and Carbon vary much more
dramatically in the disruption from our stripped star compared to the main
sequence stars throughout the fallback evolution (see Figure 2). This is
despite comparing to main sequence stars near the end of their MS lifetimes
with $Y_{c}=0.9$666This is equivalent to $0.94$ and $0.96\times\rm TAMS$ for
the $Z_{\odot}$ and $3\times Z_{\odot}$ stars respectively.. Naively, we might
expect the material returning at late times in a main sequence star disruption
(and therefore including material from the core) to be similar to the material
that returns at early times for the stripped star (given that the envelope of
the stripped star is made up of material that was near the core when it was on
the main sequence). However, because tidal disruption is an inherently
aspherical process, we are not probing purely radial slices at each point in
time. Instead we are probing vertical slices (compared to the orbital plane of
the black hole and star), and so, while material that returns at later times
does have more material from the core, it is mixed with material from the
outer layers as well (see Figure 1 in Gallegos-Garcia et al. 2018 for
reference). On the other hand, material that returns at early times originates
almost completely from the envelope of the star. This means that for a main
sequence star, material from the core that returns at late times and is
enhanced in N/C will be diluted by material from the outer layers, whereas for
a stripped star, material that returns at early times from the envelope and is
also enhanced in N/C will not be as diluted and will retain a higher N/C
abundance.

We also find that there is comparatively little difference between the
relative mass fractions and N/C abundance ratios in the fallback rates of a
main sequence star born at solar metallicity and one born at $3\times$ solar
metallicity (top two panels of Figure 2, light and dark blue lines in Figure
3). The curves for elements other than Hydrogen are shifted up for the higher
metallicity star compared to the solar metallicity star (as expected), but the
relative abundances are very similar. The N/C abundance ratio is actually
slightly higher at late times in the solar metallicity model than the higher
metallicity model. This is because the core of the solar metallicity model
makes up a slightly larger fraction of the total mass (the N/C ratio in the
core is not significantly larger, it is simply slightly less diluted by the
outer layers of the star).

In Figure 3, we focus on just the N/C abundance in the fallback material, and
compare this time-dependent composition ratio for stellar models at different
stages of evolution. We plot the stripped star and the main sequence stars at
$Y_{c}=0.5$ and $Y_{c}=0.9$. For the main sequence stars, this is partway
through the main sequence ($Y_{c}=0.5$) and near the end of the main sequence
($Y_{c}=0.9$). For the stripped star, this is early-on in Helium burning
($Y_{c}=0.9$) and halfway through Helium burning ($Y_{c}=0.5$)777This is
equivalent to $0.5$ and $0.12$ of the way through the stripped star’s Helium
burning lifetime..

Notably, we find that the composition of the fallback rate from the stripped
star TDE can naturally explain the very high N/C ratios observed in the X-ray
spectra of ASASSN-14li (dash-dotted lines in Figure 3). While moderately
massive ($M_{*}\gtrsim 1-2M_{\odot}$) main-sequence stars provide a good
explanation for the high N/C constraints measured in the UV (Yang et al.,
2017b; Mockler et al., 2022), the much higher constraints from X-ray
measurements are better matched by the stripped star’s fallback rate (Miller
et al., 2023). We discuss ASASSN-14li as a potential stripped star TDE
candidate further in Section 3.4.

###  3.2 Fallback rate

Figure 4: Top: Comparison of the density profile of a stripped star
($M_{*}=0.58M_{\odot}$, $M_{i}=3M_{\odot}$) with the density profiles of a
$2M_{\odot}$ star at various stages of stellar evolution. A polytropic stellar
profile ($\gamma=4/3$) is also plotted for comparison. We scale the density
profile by the central density ($\rho_{c}$), and scale the radii by the
stellar radius ($R_{*}$). Middle: The central density ($\rho_{c}$) divided by
the average density ($\bar{\rho}=M_{*}/(4/3\pi R_{*}^{3})$) for the same
stellar models. For the stripped star, we plot $\rho_{c}/\bar{\rho}$ for both
the main sequence (pre-stripping) and the He-core burning sequence (post-
stripping). Bottom: Mass fallback rates from various hydrodynamical
simulations (blue, from Law-Smith et al., 2020) compared to the analytical
fallback rate calculated for the stripped star (red). The hydro fallback rates
are from full disruptions of stars with similar density profiles and central
concentrations to the stripped star, scaled to the mass and radius of the
stripped star (and to $M_{h}=10^{6.8}M_{\odot}$, a reasonable estimate for
ASASSN-14li).

The fallback rate of gas to a supermassive black hole after disruption has
been well-studied and predictions from hydrodynamic simulations have largely
converged (e.g. Ramirez-Ruiz & Rosswog, 2009b; Guillochon & Ramirez-Ruiz,
2013; Cheng & Bogdanović, 2014; Tejeda et al., 2017; Gafton & Rosswog, 2019;
Law-Smith et al., 2020; Ryu et al., 2020). However, previous work focuses on
the disruptions of non-interacting stars that evolved through single stellar
evolution, and most previous work also focuses on main-sequence
stars888Notable exceptions being papers on red giant disruptions (e.g. MacLeod
et al., 2012; Bogdanović et al., 2014) and white dwarf disruptions (e.g.
Rosswog et al., 2009; Haas et al., 2012; Law-Smith et al., 2017; Kawana et
al., 2018)..

Stripped stars are more compact (have smaller radii) than main-sequence stars,
however we find that their scaled density profiles are actually well
approximated by 2 solar mass main sequence stars. Relatedly, their central
density divided by their average density ($\rho_{c}/\bar{\rho}$) – a proxy for
the disruptability of a star (Law-Smith et al., 2020; Ryu et al., 2020), is
also similar to the value for main-sequence stars. This is convenient, as the
density profile and central concentration of a star determines both the
critical impact parameter for full disruption (the minimum impact parameter
for full disruption, above which all disruptions are full disruptions
Guillochon & Ramirez-Ruiz, 2013; Law-Smith et al., 2020; Ryu et al., 2020;
Coughlin & Nixon, 2022), and the shape of the resulting mass fallback rate to
the black hole.

In Figure 4, we compare the density profiles of our stripped stellar models
(taken from Götberg et al. 2018) with the density profiles of various main
sequence stars as well as a $\gamma=4/3$ polytropic model. We see that the
density profile of a $2M_{\odot}$ main sequence star with $Y_{c}=0.5$ is a
very good approximation of our fiducial stripped star model (also with
$Y_{c}=0.5$). From looking at the evolution between the $Y_{c}=0.5$ and
$Y_{c}=0.9$ models for the MS star and stripped star respectively, it is also
clear that the structure of the stripped star changes less dramatically over
its Helium burning lifetime as the structure of the MS star changes over its
Hydrogen burning lifetime. We also compare the central density over the
average density ($\rho_{c}/\bar{\rho}$) for the $2M_{\odot}$ star and the
stripped star over their full evolution (main sequence for the $2M_{\odot}$
star, main sequence and Helium core burning for the stripped star). The values
for $\rho_{c}/\bar{\rho}$ during Helium core burning for the stripped star are
similar to the values for the main sequence star during the middle third of
its main sequence evolution. Using the analytical formula for impact parameter
as a function of $\rho_{c}/\bar{\rho}$ from Law-Smith et al. (2020)
($\beta_{\rm crit}\approx 0.5(\rho_{c}/\bar{\rho})^{1/3}$), we find that this
corresponds to critical impact parameters for full disruption between
$2.5<\beta_{\rm crit}<3.5$.

Using the information from the top two panels of Figure 4, in the bottom panel
we have plotted a range of hydrodynamical fallback rates from Law-Smith et al.
(2020) for stars with similar stellar structures to our stripped star999We
included full disruptions of all stars with $36<\rho_{c}/\bar{\rho}<756$ from
Law-Smith et al. (2020). These include: TAMS $0.7M_{\odot}$ star, ZAMS-TAMS
$1M_{\odot}$ star, ZAMS $1.5M_{\odot}$, ZAMS $3M_{\odot}$ star.. To compare
with our analytical fallback rate, we scale these hydrodynamical fallback
rates by the mass and radius of the stripped star using the canonical $t_{\rm
peak}\propto M_{*}^{-1}R_{*}^{3/2}$ and $\dot{M}_{\rm peak}\propto
M_{*}^{2}R_{*}^{-3/2}$ relations. Given that we have specifically chosen stars
with similar density profiles and values of $\rho_{c}/\bar{\rho}$, this should
give consistent peak timescales and fallback rates to the updated equations
for these parameters presented in Coughlin & Nixon (2022) and Bandopadhyay et
al. (2023) (which reduce to the canonical relations when holding
$\rho_{c}/\bar{\rho}$ constant).

###  3.3 Outflows

Observations of tidal disruption events provide estimates for the size scales
of gas around the black hole from the disrupted star through measurements of
blackbody photosphere radii and wind radii and velocities (e.g. Hung et al.,
2017; Alexander et al., 2016; Kara et al., 2018; Mockler et al., 2019; Wevers
et al., 2023). Additionally, broad, blue-shifted lines have been observed in
many TDEs, consistent with lines forming in dense, outflowing gas (e.g. Arcavi
et al., 2014; Holoien et al., 2016; Nicholl et al., 2020; Roth & Kasen, 2018).
Lines are expected to form somewhere between the photosphere and outer wind
radius, and therefore should be dependent on the composition of the
photosphere and outflows.

We predict that outflows from TDEs of stripped stars will have a Nitrogen
enhanced outer layer, with N/C abundance ratios up to $50\times$ higher than
is achievable with main-sequence stars (see Figure 3). Emission lines produced
by the irradiated outflow would therefore show strong N/C enhancements at
early times. Then, as the wind expands and material that returned to the black
hole at later times is illuminated, the relative strength of N/C would flip,
and Carbon would be enhanced relative to Nitrogen. The exact timing of this
behavior and the amount of gas with very high N/C abundance ratios depends on
how much mixing occurs. If the gas in the outermost layers of the star that is
most bound to the black hole is mixed with material deeper in the star either
in the disruption process or in the circularization and accretion processes
that launch the outflows, then the N/C abundance will be lower, and will
decrease more rapidly.

While hydro simulations are necessary to truly determine the extent of the
mixing, we provide some limits here. If the material in the outer layer of the
star experiences minimal mixing during the disruption process and outflows are
launched promptly after material first returns to the black hole, then it is
possible that all material launched in outflows during the rise of the light
curve is highly Nitrogen enriched. While there is debate on the length of time
between when material begins to return to the black hole and when a flare is
produced, most previous results agree that once a bright flare is produced,
outflows should also be launched. This could either be through shocks and
stream collisions, or through disk winds, or through both processes (e.g., Dai
et al., 2018; Bonnerot et al., 2021; Steinberg & Stone, 2022; Huang et al.,
2023; Price et al., 2024).

In Figure 5, we plot the range of radii for a wind launched between model
estimates of when material began to return to the black hole and when the
light curve peaked for the transient ASASSN-14li, and compare to observations
(this event is discussed in more detail in the following section). We assume
there is minimal delay between when material returns to the black hole and
when luminosity and outflows are produced. If there is a longer delay,
material will be N/C enhanced for less time/over a smaller radii range.
Regardless, if N/C enhancement is observed from the disruption of a stripped
star, the N/C abundance ratio of material surrounding the black hole should
decrease with time

###  3.4 Comparing to ASASSN-14li

Nitrogen lines are measured in optical spectra of ASASSN-14li from MJD
56993-57129 (Cenko et al., 2016b; Holoien et al., 2016; Brown et al., 2018),
however the more constraining UV and X-ray detections of high Nitrogen/Carbon
abundance ratios occurred between MJD 56999-57002 (XMM and Chandra X-ray
spectra, Miller, 2015; Miller et al., 2023) and on MJD 57032 (HST UV spectra,
Cenko et al., 2016b). The X-ray spectra constrain the N/C abundance ratio to
be $\geq 300$ relative to the sun in material close to the black hole near the
peak of the light curve, and the UV spectra constrain the N/C ratio to be
$\geq 10$ in material at much larger radii on the decline of the light curve.
Interestingly, the N/C ratio in the envelope of the stripped stars we analyze
is also $\geq 300$ (see Figure 3). As we show in this section, the
measurements from both X-ray and UV observations are consistent with coming
from material that returned to the black hole before the peak of the fallback
rate, and was then subsequently fed to the accretion disk or ejected in
outflows.

Figure 5: Radii of gas in TDE ASSASN-14li, originally estimated in Mockler et
al. (2022) using the MOSFiT transient fitting code. The x-axis is time in
rest-frame days since the first observation. We have plotted in blue the radii
outflows would reach if launched between first fallback and peak of the mass
fallback rate (assuming a velocity of 0.1c, based on radio observations of the
event). This corresponds to the time range in the mass fallback rate during
which our analytical model predicts a large N/C enhancement. We overplot the
virial radii of various Nitrogen lines observed in the event, including a UV
measurement (white diamond) that puts constraints on the N/C ratio of the
debris. We also plot the radii of the blackbody photosphere and semi-major
axis of returning debris in purple and green respectively.

As discussed in the previous section, if the flare was from the disruption of
a stripped star, material that returns to the black hole at early times and up
to the peak of the fallback rate (which we are assuming here to be
approximately coincident with the peak of the optical and UV light curve, as
is predicted by many models, e.g. Mockler et al., 2019; Ryu et al., 2020;
Steinberg & Stone, 2022) is likely to be heavily enhanced in Nitrogen with
respect to Carbon (see Figure 3). Unfortunately the peak of the transient
ASASSN-14li was not observed, however radio observations place rough estimates
on the time that outflows were first launched to be between MJD $\sim
56884-56990$ (Alexander et al., 2016), and light curve analysis place
estimates on the peak of the light curve to be between MJD $\sim 56944-56990$
(e.g., Mockler et al., 2019). Given this information, the early outflows
launched before MJD $\sim 56990$ and material accreted around the same time
would likely have high N/C abundance ratios if the material came from a
stripped star.

There is much theoretical debate on the nature of the initial luminosity
production on the rise of TDEs, however multiple theoretical models have shown
that it is reasonable for material to start to circularize into a disk by the
time of the fallback rate and light curve’s peak (e.g. Bonnerot et al., 2021;
Steinberg & Stone, 2022; Andalman et al., 2022). The observations of X-ray
spectra in Miller et al. (2023) were taken starting on MJD 56999. The X-rays
are therefore consistent with being emitted from a disk made up of material
that mostly fell back before peak but only recently circularized (and would be
Nitrogen-enriched if from a stripped star), combined with additional material
that has fallen back after peak. While it is expected that the X-ray emission
originates from close to the black hole in the forming disk, the UV emission
is consistent with originating from optically thick outflows at larger radii.
These outflows would come from gas that returned to the black hole at earlier
times, and was then ejected through, e.g. stream collisions or disk winds (Dai
et al., 2018; Bonnerot et al., 2021).

In Figure 5 we show that the virial velocities measured from the widths of
Nitrogen lines in the optical and UV spectra are consistent with the range of
radii where gas from outflows launched before peak would be at the time of the
observations. We plot an estimate of the gas geometry surrounding ASASSN-14li
based on light curve fits and the wind velocity measured in radio observations
(adapted from Mockler et al., 2022). If the material is not virialized, and
gas broadening dominates the linewidths instead (Roth & Kasen, 2018; Parkinson
et al., 2022), the gas could also originate at smaller radii where it would be
more likely to be mixed with material from the core of the star that returns
at later times. For a stripped star, this would mean material that is not
enhanced in N/C (e.g., see Figure 3). The UV spectra constrain the N/C
abundance ratio to be $\geq 10\times$ solar metallicity – a limit that is much
lower than the N/C ratio in the envelope of a stripped star ($N/C\geq 300$).
Therefore, even if the lines originate at smaller radii from a mix of gas from
the outer layers and the core, this could still be consistent with the
disruption of a stripped star.

##  4 Discussion

Here we discuss additional relevant points that can be investigated further in
the future.

Partial and Repeated TDEs: If a stripped star is partially disrupted instead
of fully disrupted, the core will remain intact and the material that is bound
to the black hole will come entirely from the envelope. In this case, all of
the material that returns to the black hole will have high N/C abundance
ratios (both before and after peak). Partial disruptions are likely
intrinsically more common than full disruptions, as two-body relaxation
processes produce significantly more partial disruptions than full disruptions
(e.g. Stone & Metzger, 2016). However, they are expected to be more difficult
to observe. First of all, less accreted material likely means less luminous
events. Additionally, if optical and UV emission is due to reprocessing, then
less accreted material also means that optical and UV emission will make up a
smaller fraction of the total emitted luminosity (e.g. Roth et al., 2016).
Given this, we expect partial TDEs from stripped stars to produce lower
luminosity, X-ray bright TDEs with consistently high N/C ratios.

In the case of repeated partial disruptions (‘repeated TDEs’), the flares
should eventually switch from Nitrogen-rich to Nitrogen-poor as subsequent
disruptions remove material from deeper within the star. Interestingly,
repeated TDEs from stripped stars are theoretically predicted to survive
longer and therefore potentially produce more flares than repeated TDEs from
main sequence stars. This is because the core fraction is larger in stripped
stars than most main sequence stars, which results in the star contracting
instead of expanding when a small amount of mass is removed (Liu et al.,
2023). It is possible a stripped star could explain the repeated partial
disruption ASASSN-14ko (Payne, 2021), as this event has survived for years
without significant evolution observed in its flares (as would be expected if
its core mass fraction was $\lesssim 0.3$ Liu et al., 2023). This event
actually might be Nitrogen-poor at later times given the appearance of strong
CIII lines in the UV without obvious NIII counterparts in the HST spectra
taken during flares 18 and 19 (Payne et al., 2023, unfortunately there is no
UV spectra from earlier flares to compare with). However, this event also
appears to require a star of at least $\sim 2M_{\odot}$ to explain the
observed luminosity with reasonable mass-to-energy efficiencies (e.g.
$\epsilon\sim 0.01$, consistent with predictions of inefficient accretion for
these events Liu et al., 2023), and so a stripped star from a larger mass
progenitor ($\gtrsim 7M_{\odot}$) would be required (Götberg et al., 2018).

Rates of stripped star TDEs: A detailed rates calculation is outside the scope
of this paper, but a brief overview of how the number of stripped stars
compares to the total stellar population is still useful. The fraction of
roughly solar mass, solar metallicity stars born in close binaries is $\approx
0.2$ (Moe et al., 2019). This fraction increases with stellar mass and
decreases with metallicity (e.g. it is $\approx 0.3$ for higher mass stars,
Geller & Mathieu, 2012; Badenes et al., 2018; Lee et al., 2020; Jadhav et al.,
2021). It also increases towards the centers of clusters due to mass-
segregation (e.g., Mathieu & Latham, 1986; Geller & Mathieu, 2012; Milone et
al., 2012). Envelope-stripping is completed after the main sequence, meaning
that stripped stars can be quite long-lived, as the Helium-core burning stage
is left. For massive stars, the stripped star stage correspond to about 10% of
the total stellar lifetime. However, for stars with initial masses $\lesssim
4M_{\odot}$, the lack of a thick Hydrogen-burning shell stops the Helium core
from substantial growth and lower mass stars can therefore live much longer as
stripped – up to half of the total stellar lifetime (e.g. Götberg et al.,
2018; Arancibia-Rojas et al., 2023). Given this, we might expect at least
$\sim 3-10\%$ of stars in galactic nuclei that are enriched from the CNO cycle
to be stripped stars ($0.2\times 0.5=0.1$ for the smallest stars, $0.3\times
0.1=0.03$ for the largest stars).

The smaller radii and greater compactness of stripped stars make them more
difficult to partially disrupt than main-sequence stars, however, their
critical disruption radii for core disruption are less than a factor of 10
different101010For example, the critical disruption radius for our fiducial
stripped star is within a factor of 5-6 of the critical disruption radius for
the $2M_{\odot}$ main-sequence star we compare it to. See Section 3.2 for a
more in depth discussion of how the critical disruption radius is determined.
than the critical radii for main-sequence stars (as might be expected given
their similar values of $\rho_{c}/\bar{\rho}$). Given that the TDE rate from
two-body relaxation scales as $\propto R_{t}^{1/4}$ for a wide range of tidal
radii and black hole masses (e.g. Wang & Merritt, 2004; Milosavljević et al.,
2006; MacLeod et al., 2012), the likelihood of full disruption for stars after
being stripped and during their Helium burning sequence is, at any given time,
within a factor of two ($\lesssim 10^{1/4}=1.78$) of the likelihood of full
disruption for the same stars during their main sequence. Therefore we might
expect at least $\sim 1-5\%$ of the disruptions of stars born at $M_{*}\gtrsim
1M_{\odot}$ to come from stripped stars (including other factors such as the
age and mass-segregation of stellar populations should increase this
percentage). We can organize this estimate as follows:

$f_{\rm TDE,\>stripped}\approx f_{\rm close\>binary}\times\frac{\tau_{\rm
stripped}}{\tau_{\rm*}}\times\Big{(}\frac{R_{t,\>\rm stripped}}{R_{t,\>\rm
MS}}\Big{)}^{1/4}$ (3)

Here $f_{\rm close\>binary}$ is the fraction of stars born in closed binaries
that are expected to end their lives as stripped stars – $f_{\rm
close\>binary}\sim 0.2(0.3)$ for low mass (high mass) stars. Then $\tau_{\rm
stripped}$ and $\tau_{*}$ are the stripped star and total stellar (MS $+$
stripped) lifetimes – $\frac{\tau_{\rm stripped}}{\tau_{\rm*}}\sim 0.5(0.1)$
for low mass (high mass) stars. And finally, $R_{t,\>\rm stripped}$ and
$R_{t,\>\rm MS}$ are the tidal radii of stripped stars and MS stars
respectively – $\frac{R_{t,\>\rm stripped}}{R_{t,\>\rm MS}}\gtrsim 0.1$ for
the majority of binary stripped stars (using stripped star parameters from
Götberg et al. 2018 throughout). These leads to $f_{\rm TDE,\>stripped}\sim
0.056(0.017)$ for low mass (high mass) stars, or $\sim 1-5\%$ of TDEs from
stars born at masses such that they will undergo CNO burning ($M_{*}\gtrsim
1M_{\odot}$).

Connections to Nitrogen-enrichment in AGN disks:

Regions associated with galactic centers have been found to be Nitrogen
enriched, for example through the spectra of AGN broad line regions (BLR)
(e.g. Bentz et al., 2004; Batra & Baldwin, 2014). TDE searches have, up to
this point, mostly excluded AGN host galaxies, and so a pre-existing Nitrogen-
rich BLR is disfavored to explain Nitrogen-rich TDEs. However, it is still
possible that AGN and TDEs share the same source of Nitrogen-enrichment.

The centers of galaxies also often have higher than solar overall
metallicities. Some studies have found that all BLR line measurements in those
galaxies (not just N/C ratios) are consistent with super-solar metallicities
(e.g. Batra & Baldwin, 2014), and therefore Nitrogen enhancements can be
explained by these higher overall metallicities. Others find that the
enhancement of Nitrogen is higher than that of other metals (Jiang et al.,
2008; Matsuoka et al., 2017), and therefore can’t be explained solely by
increasing the overall metallicity of the galactic nuclei. The most common
explanation put forward for general metallicity enhancement is high star
formation rates in the vicinity of the SMBH (consistent with observations of
star formation gradients in galaxies Nelson et al. 2016, e.g., but at size
scales not easily testable observationally). To explain anomalously high
Nitrogen abundances with respect to the overall metallicity, Matsuoka et al.
(2017) invoke mass loss from AGB winds enriching the BLR gas with Nitrogen
(which was dredged up from CNO processed material near the core). This
scenario also requires in-situ star formation to create sufficient AGB stars.
Additionally, it predicts higher AGN accretion rates as some of the winds will
end up accreted by the SMBH (Davies et al., 2007). It does appear that
Nitrogen-loud quasars have higher Eddington ratios than the general
population, in line with the prediction of AGB winds enriching BLR clouds and
also increasing SMBH accretion rates (Matsuoka et al., 2017).

High star formation rates in the centers of galaxies would also help explain
the Nitrogen enhancement found in some TDEs, as it would increase the fraction
of higher mass stars undergoing CNO cycle burning. However, high star
formation rates alone are likely not sufficient to explain the unexpectedly
high rate of Nitrogen enhanced TDEs (Mockler et al., 2022). Additionally,
higher overall metallicities in the centers of galaxies are not sufficient to
explain the particularly extreme Nitrogen enhancement found in ASASSN-14li
(see Figures 2, 3), unless these high birth metallicities are accompanied by
unusually high N/C abundance ratios, or young stars are polluted by winds from
massive stars. These two scenarios have been suggested to explain highly
Nitrogen-enhanced stars in globular clusters (although these globular cluster
stars are still less Nitrogen-enhanced than ASASSN-14li appears to be Larsen
et al., 2014; Bastian & Lardo, 2018). On the other hand, TDEs can clearly
contribute to the Nitrogen-enhancement of a SMBH’s environment, and it is
possible that this contribution is important to the population of Nitrogen-
enhanced AGN. This connection is particularly intriguing if the galaxy is in a
post-starburst phase of evolution, or if the nuclear cluster hosts a SMBH
binary given that these environments appear 1) conducive to AGN activity,
(e.g. Dodd et al., 2021), and 2) produce TDE rates that are observed or
predicted to be $\gtrsim 20-100\times$ higher than average, (e.g. French et
al., 2016; Graur et al., 2018; Law-Smith et al., 2019; French et al., 2020;
Hammerstein et al., 2021; Melchor et al., 2023; Mockler et al., 2023).

##  5 Summary of Key Points

Here we briefly summarize the main takeaways from this work:

  * •

TDEs probe stellar populations in galactic nuclei, and recent observations of
N/C abundances in TDEs point to the disruption of stars born at moderately
high masses that show chemical enrichment indicative of the CNO cycle.

  * •

X-ray spectral observations of ASASSN-14li show N/C abundance lower limits so
high ($\gtrsim 300$) it is difficult to explain with main-sequence evolution
alone (Miller et al., 2023). In a main-sequence star, the core has high N/C
ratios but the envelope is more Carbon than Nitrogen. In a TDE, when material
from the core eventually falls back to the black hole it is diluted by the
Carbon in the envelope material that falls back with it, limiting the maximum
N/C abundance in fallback material (see Figure 3).

  * •

Binary-stripped stars have the reverse compositional structure of main-
sequence stars. Their Helium-rich envelopes have high N/C abundance ratios,
and it is their cores that are Nitrogen-poor and Carbon-enhanced (see Figure
1). When these stars are tidally disrupted, Nitrogen-rich material from the
envelope will return to the black hole first, without being diluted by
Nitrogen-poor material from the core. This means the composition of fallback
material from a tidally disrupted stripped star can have much higher
enhancements of N/C than disruptions of main-sequence stars.

  * •

Stripped stars also naturally produce the opposite N/C time evolution as main-
sequence stars and should therefore be observationally distinct in time series
of their spectra (see Figure 3).

## Acknowledgments

We thank the participants and organizers of the summer Aspen 2023 workshop on
“Stellar Interactions and the Transients They Cause” for fruitful discussions.
B.M. is grateful for support from the Carnegie Theoretical Astrophysics
Center. M.G.G. is grateful for the support from Northwestern University’s
Presidential Fellowship. E.R.R. thank the Heising-Simons Foundation, NSF
(AST-2150255 and AST-2307710), Swift (80NSSC21K1409, 80NSSC19K1391) and
Chandra (22-0142) for support.

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11institutetext: Anonymous11institutetext: Department of Computer Science,
Vanderbilt University 22institutetext: Department of Electrical and Computer
Engineering, Vanderbilt University 33institutetext: Department of Biomedical
Engineering, Vanderbilt University 44institutetext: Siemens
Healthineers55institutetext: Department of Biomedical Informatics, Vanderbilt
University Medical Center66institutetext: Department of Radiology, Vanderbilt
University Medical Center

# Characterizing Renal Structures with 3D Block Aggregate Transformers

Anonymous  Xin Yu 1*1* Yucheng Tang 2*2* Yinchi Zhou 11 Riqiang Gao 11 Qi Yang
11 Ho Hin Lee 11 Thomas Li 33 Shunxing Bao 11 Yuankai Huo 1122 Zhoubing Xu 44
Thomas A. Lasko 1155 Richard G. Abramson 66 Bennett A. Landman 11223366

###### Abstract

Efficiently quantifying renal structures can provide distinct spatial context
and facilitate biomarker discovery for kidney morphology. However, the
development and evaluation of transformer model to segment the renal cortex,
medulla, and collecting system remains challenging due to data inefficiency.
Inspired by the hierarchical structures in vision transformer, we propose a
novel method using 3D block aggregation transformer for segmenting kidney
components on contrast-enhanced CT scans. We construct the first cohort of
renal substructures segmentation dataset with 116 subjects under institutional
review board (IRB) approval. Our method yields the state-of-the-art
performance (Dice of 0.8467) against the baseline approach of 0.8308 with the
data-efficient design. The Pearson R achieves 0.9891 between the proposed
method and manual standards, and indicates the strong correlation and
reproducibility for volumetric analysis. We extend the proposed method to the
public KiTS dataset, the method leads to improved accuracy compared to
transformer-based approaches. We show that the 3D block aggregation
transformer can achieve local communication between sequence representations
without modifying self-attention, and it can serve as an accurate and
efficient quantification tool for characterizing renal structures.
00footnotetext: * equal contribution

###### Keywords:

Renal Substructures Computed Tomography Transformer Model.

Figure 1: Left: visual and 3D illustration of the kidney components. Right:
Demonstration of the hierarchical transformer design, the 3D block aggregation
is conducted every two hierarchies, blocks at a factor of 8 are merged to
perform communication of sequence representations.

##  1 Introduction

Hierarchical models [5, 20, 24] are received significant interest in medical
image analysis due to their advantages of modeling heterogeneous high-
resolution radiography images. Recent works on vision transformers [8, 18]
show superior performance on visual representations compared to state-of-the-
art convolution-based networks [12]. However, ViT usually requires large-scale
training data with expensive clinical expertise [24, 32]. When trained on
smaller cohorts, transformer-based models often suffer from a lack of
inductive bias [6, 8] and lead to data inefficiency. Moreover, the self-
attention mechanism on modeling multi-scale features for high-resolution
medical volumes is computationally expensive [1, 9, 18]. These challenges
inspire designing hierarchical transformer structures in analogy to
convolution-based networks (e.g., 3D UNet [5]). Addressing the data
inefficiency of transformers is critical for its application on medical image
analysis, especially for understanding small targets on 3D high-dimensional
image volumes.

One such challenge is segmenting the small structures of kidney sub-
components. Renal structure volumes from clinical CT scans have been recently
suggested as a useful surrogate for evaluating renal function [17, 7]. These
investigations elucidate the correlations of the volumetric measurements on
the renal cortex, medulla, and pelvicalyceal system with kidney function. In
such studies, manual segmentation is performed as the gold standard for visual
and quantitative morphological assessment on CT scans [21] as shown in Figure
1. However, manual quantification by clinical experts is resource-intensive,
time-consuming, and may suffer from insufficient inter- and intra-
reproducibility.

To improve the representation learning of transformers in small datasets,
recent works envision the use of local self-attention to form hierarchical
transformers [18, 2, 9]. To leverage information across embedded sequences,
”shifted window” transformers [18] are proposed for dense predictions and
modeling multi-scale features. However, these attempts that aim to complicate
the self-attention range often yield high computation complexity and data
inefficiency. Inspired by the aggregation function in the nested ViT [31], we
propose a new design of a 3D U-shape medical segmentation model with Nested
Transformers (UNesT) hierarchically with the 3D block aggregation function,
that learn locality behaviors for small structures or small dataset. This
design retains the original global self-attention mechanism and achieves
information communication across patches by stacking transformer encoders
hierarchically.

Our contributions in this work can be summarized as:

  * •

We introduce a novel 3D medical segmentation model, named UNesT with a 3D
block aggregation function. This method achieves hierarchical modeling of
high-resolution medical images and outperforms local self-attention variants
with a simplified design, which leads to improved data efficiency.

  * •

We collect and manually delineate the first renal substructures dataset (116
patients) on characterizing multiple kidney components. We show that our
method achieves state-of-the-art performance to accurately measure the
cortical, medullary, and pelvicalyceal system volumes.

  * •

We demonstrate the clinical utility of this work by accurate volumetric
analysis, strong correlation, and reproducibility. Validation on external
public dataset KiTS shows the generalizability of the proposed method.

##  2 Related Works

3D Medical Segmentation with Transformers. Transformer-based 3D medical image
segmentation models [27, 11, 29, 30, 15, 19, 32, 25, 3] are popular and
achieve state-of-the-art performance in several benchmarks. The self-attention
mechanism [26] allows the inputs at different positions of a sequence to
interact with each other, and then compute the overall representation from the
sequences. Although transformers exhibit outstanding performance in learning
global context, their deficiency in capturing localized information remain. To
address this, TransFuse [30], TransBTS [27], CoTr [29], UNETR [11] are
proposed architectures which combine transformers and CNNs into hybrid
designs. More recently, hierarchical transformers are proposed with shifted-
window [18], it enables cross-patch self-attention connections. Based on Swin
ViT, Swin UNETR [10, 24] and SwinUNET [2] are introduced for capturing multi-
scale features in CT images. However, the modification on local self-attention
results in quadratic increase of complexity.

Hierarchical Feature Aggregation. The aggregation of multi-level features
could improve the segmentation results by merging the features extracted from
different layers. Modeling hierarchical features, such as U-Net [5] and
pyramid networks [20], multi-scale representations are leveraged. The extended
feature pyramids compound the spatial and semantic information through two
structures, iterative deep layer aggregation which fuses multi-scale
information as well as hierarchical deep aggregation which fuses
representations across channels. In addition to single network, nested UNets
[33], nnUNets [14], coarse-to-fine [34] and Random Patch [22] suggest multi-
stage pathways that enrich the different semantic levels of feature
progressively with cascaded networks. Different from the above CNN-based
methods, we explore the use of data-efficient transformers for modeling
hierarchical 3D features by the block aggregation.

##  3 Method

###  3.1 UNesT Architecture

The proposed network contains a hierarchical transformer as the encoder, which
consists of three hierarchies to perform self-attention communications among
image blocks. Following the motivation of NesT [31] for natural images, we
process the volumetric information between 3D adjacent blocks by the
aggregation layer every two hierarchies. The overall architecture, as shown in
Figure 2, also contains skip connections with convolution modules and a
decoder for better capturing localized information.

Given the input image sub-volume
$\mathcal{X}\in\mathbb{R}^{H\times{W}\times{D}}$, the volumetric embedding
token is with patch size of $S_{h}\times S_{w}\times S_{d}$. Then all
projected sequences of embeddings are partitioned to blocks with a resolution
of $\mathcal{X}\in\mathbb{R}^{b\times T\times{n}}$, where $T$ is the number of
blocks at the current hierarchy, $b$ is the batch size, $n$ is the total
length of sequences. The dimensions of the embeddings follow
$T\times{n}=\frac{H}{S_{h}}\times\frac{W}{S_{w}}\times\frac{D}{S_{d}}$. In the
subsequent transformer layers, we use the canonical multi-head self-attention
(MSA), multi-layer perceptron (MLP), and Layer normalization (LN). We add
learnable position embeddings to sequences for capturing spatial relations
before the blocked transformers. The output of encoder layers $t-1$ and $t$
are computed as follows:

$\begin{array}[]{l}\hat{{z}}^{t}=\text{MSA\textsubscript{HRCHY\textsubscript{l}}}(\text{LN}({z}^{t-1}))+{z}^{t-1}\\\
{z}^{t}=\text{MLP}(\text{LN}(\hat{{z}}^{t}))+\hat{{z}}^{t},\\\ \end{array}$
(1)

where MSAHRCHYl denotes the multi-head self-attention layer of hierarchy $l$,
$\hat{z}^{t}$ and $z^{t}$ are the output representations of MSA and MLP. In
the practice, MSAHRCHYl is applied parallel to all partitioned blocks:

$\begin{array}[]{l}\text{MSA\textsubscript{HRCHY\textsubscript{l}}(Q, K,
V)}=\text{Stack}(\text{BLK}\textsubscript{1},...,\text{BLK}\textsubscript{T})\\\
\text{BLK}=\text{Attention(Q, K,
V)}=\text{Softmax}(\frac{QK^{\top}}{\sqrt{\sigma}})V,\end{array}$ (2)

where $Q,K,V$ denote queries, keys, and values vectors in the multi-head
attention, $\sigma$ is the size of each vector. All blocks at each level of
hierarchy share the same parameters given the input $\mathcal{X}$, which leads
to hierarchical representations without increasing complexity. Finally, the
block aggregation is merged spatially by adjacent 8 blocks.

Figure 2: Overview of the proposed UNesT with the hierarchical transformer
encoder. Block aggregation and image feature down-sampling are performed
between hierarchies.

###  3.2 3D Block Aggregation

Following [31], we extend the spatial nesting operations to 3D blocks where
each volume block is modeled independently. Information across blocks is
communicated by the aggregation module. At hierarchy $l$, the spatial
operations are conducted to down-sampled feature maps at $\mathbb{R}^{b\times
H^{\prime}/2\times{W^{\prime}/2}\times{D^{\prime}/2}}$. At the bottom of each
hierarchy, the embeddings are blocked back to feature
$Z_{l+1}\in\mathbb{R}^{b\times{T/8}\times{n}}$ for hierarchy $l+1$. There are
three hierarchies in our model design, a factor of 8 is reduced in a total
number of blocks which results in [64, 8, 1] blocks. In the volumetric plane,
the encoded blocks are merged among adjacent blocks representations. The
design and use of the aggregation modules in the 3D scenario leverage local
attention, lead to a data-efficient design.

###  3.3 Decoder

To better capture localized information and further reduce the effect of
lacking inductive bias of transformer, we use a hybrid design with a
convolution-based decoder for segmentation. The features from different
hierarchies of the transformer encoder are fed into skip connections followed
by convolution layers. As shown in Figure 2, we extract the output
representations at the image level and each hierarchy to $3\times 3\times 3$
conv layers, then upsample by a factor of $2$. Next, the output of the
transposed conv is concatenated with the prior hierarchy representations. The
segmentation mask is acquired by $1\times 1\times 1$ conv layer with a softmax
activation function. Compared to some prior related works such as TransBTS
[27] and CoTr [29], our design employs the hierarchical transformer directly
on images and extract representations at multiple scales without conv layers.

##  4 Experiments

###  4.1 Dataset

Renal Substructure Dataset. The study design uses clinically collected renal
CT of 116 de-identified patients accessed under IRB approval. We use selected
ICD codes related to kidney dysfunction as exclusion criteria, that could have
a potential influence on kidney anatomies. The left and right renal structures
are outlined manually by three interpreters under the supervision of clinical
experts. The annotation for the cortex label also includes the renal columns,
the medulla is surrounded by the cortex, and the pelvicalyceal systems contain
calyces and pelvis that drain into the ureter. All manual labels are verified
and corrected independently by expert observers. For the test set of 20
subjects, we perform a second round of manual segmentation (interpreter 2) to
assess the intra-rater variability and reproducibility.

KiTS19. To validate the generalizability of the proposed method while
remaining the target of characterizing renal tissues, we apply the model to
the public KiTS19 dataset. The KiTS19 [13] task focuses on the whole kidney
and tumor segmentation. We perform five-fold cross-validation experiments and
show results of the held-out 20% as testing.

###  4.2 Implementation Details

Five-fold cross-validation is used for all experiments on 96 subjects, while
20 subjects are used for held-out testing. The five-fold models’ ensemble is
used for inferencing and evaluating test set performance. For experiment
training, we used 1) CT window range of [-175, 275] HU; 2) scaled intensities
of [0.0,1.0]; 3) training with single Nvidia RTX 2080 11GB GPU with Pytorch
and MONAI implementation at batch size of 1 (input image sub-volume size of
$96\times{96}\times{96}$); 4) AdamW optimizer with warm-up cosine scheduler of
500 steps. The learning rate is initialized to 0.001 followed by a decay of
$1e^{-5}$ for 50K iterations. For fair comparison and direct evaluation of the
effectiveness of models, no pre-training is performed for all segmentation
tasks.

Metrics. Segmentation performance is evaluated between ground truth (rater 1)
and prediction by Dice-Sorensen coefficient (DSC), and symmetric Hausdorff
Distance (HD). Volumetric analyses are evaluated under R squared error,
Pearson R, absolute deviation of volume, and the percentage difference between
the proposed method and manual label.

Table 1: Segmentation results of the renal substructure on testing cases. The
UNesT achieves state-of-the-art performance compared to prior kidney
components studies and 3D medical segmentation baselines. The number of
parameters and GFLOPS (with a single input volume of $96\times 96\times 96$)
are shown for deep learning-based approaches. * indicates statistically
significant ($p<0.01$) by Wilcoxon signed-rank test.

Method | #Param | GFLOPS | Cortex | Medulla | Pelvicalyceal System | Avg.
---|---|---|---|---|---|---
DSC | HD | DSC | HD | DSC | HD | DSC | HD
Chen et al. [4] | N/A | N/A | 0.7512 | 40.1947 | N/A | N/A | N/A | N/A | N/A | N/A
Xiang et al. [28] | N/A | N/A | 0.8196 | 27.1455 | N/A | N/A | N/A | N/A | N/A | N/A
Jin et al. [16] | N/A | N/A | 0.8041 | 34.5170 | 0.7186 | 32.1059 | 0.6473 | 39.9125 | 0.7233 | 35.5118
Tang et al. [23] | 40.9M | 423.9 | 0.8601 | 19.7508 | 0.7884 | 18.6030 | 0.7490 | 34.1723 | 0.7991 | 24.1754
nnUNet [14] | 19.1M (3DUNet) | 412.7 | 0.8915 | 17.3764 | 0.8002 | 18.3132 | 0.7309 | 31.3501 | 0.8075 | 22.3466
TransBTS [27] | 33.0M | 359.4 | 0.8901 | 17.0213 | 0.8013 | 17.3084 | 0.7305 | 30.8745 | 0.8073 | 21.7347
CoTr [29] | 46.5M | 399.2 | 0.8958 | 16.4904 | 0.8019 | 16.5934 | 0.7393 | 30.1282 | 0.8123 | 21.0707
nnFormer [32] | 158.9M | 146.5 | 0.9094 | 15.5839 | 0.8104 | 15.9412 | 0.7418 | 29.4407 | 0.8205 | 20.3219
UNETR [11] | 92.6M | 41.2 | 0.9072 | 15.9829 | 0.8221 | 14.9555 | 0.7632 | 27.4703 | 0.8308 | 19.4696
UNesT | 87.3M | 37.5 | 0.9201 | 14.5401 | 0.8356 | 13.5933 | 0.7843 | 24.5445 | 0.8467* | 17.5593

Table 2: Comparison of volumetric analysis metrics between the proposed method
and the state-of-the-art clinical study on kidney components.

Metrics | Cortex | Medulla | Pelvicalyceal System
---|---|---|---
Tang et al. [23] | UNesT |  Tang et al. [23] | UNesT |  Tang et al. [23] | UNesT
R Squared | 0.9200 | 0.9359 | 0.6652 | 0.6837 | 0.4586 | 0.5917
Pearson R | 0.9838 | 0.9891 | 0.8156 | 0.8368 | 0.6772 | 0.7148
Absolute Deviation of Volume | 3.0233 | 2.7254 | 3.5496 | 3.2958 | 0.9443 | 0.8012
Percentage Difference | 4.8280 | 3.9478 | 7.4750 | 7.0382 | 19.0716 | 13.5737

##  5 Results

###  5.1 Characterization of Renal Structures

We evaluate the UNesT performance on two groups of methods: 1) the clinical
kidney components studies such as CortexSeg [4], CorteXpert [28], AAM [16],
and 2) recent conv- [14] and transformer-based [27, 29, 32, 11] 3D medical
segmentation baselines.

Segmentation Results. Compared to canonical kidney studies using shape model
or random forests in Table 1, the deep learning-based methods improve the
performance by a large margin from 0.7233 to 0.7991. Among the nnUNet [14] and
extensive transformer models, we obtain the state-of-the-art average Dice
score of 0.8467 compared to the second-best performance of 0.8308, with a
significant improvement $p<0.01$ under Wilcoxon signed-rank test. We observe
higher improvement on smaller anatomies such as medulla and collecting
systems. We compare qualitative results in Figure 3. Our method demonstrates
the distinct improvement of detailed structures for medulla and pelvicalyceal
systems.

Figure 3: Qualitative comparisons of representative renal sub-structures
segmentation on two right (top) and two left (bottom) kidneys. The average DSC
is marked on each image. UNesT shows distinct improvement on the medulla (red)
and pelvicalyceal system (green) against baselines.

Volumetric Analysis. Table 2 lists the volume measurement with the proposed
method. The UNesT achieves an R squared error of 0.9359 on the cortex. The
correlation performance metric with Pearson R achieves 0.9891 for the UNesT
against the manual label on the cortex. Our method obtains 2.7254 with an
absolute deviation of volumes. The percent difference in the cortex is 3.9478.
Quantitative results show that our workflow can serve as the state-of-the-art
volumetric measurement compared to prior kidney characterization pipeline
[23].

###  5.2 Ablation Study

Effect of the Block Aggregation. We show the hierarchical architecture design
(with 3D block aggregation) is critical for medical image segmentation (as
shown in Figure 4 left and middle). The result shows that the hierarchy
mechanism achieves superior performance at 20% to 100% of training data. At
the low data regime, the block aggregation achieves a higher improvement
($>4\%$ of DSC) compared to the second-best method. We notice that the model
without block aggregation (canonical transformer layers) obtains lower
performance. The results show that block aggregation performs as a critical
component for representation learning for transformer-based models.

Data Efficiency. The Figure 4 shows the data efficiency of our proposed
method. First, UNesT achieves better performance when training with fewer
data. Second, UNesT with block aggregation demonstrates a faster convergence
rate (15% and 4% difference at 2K/30K iterations) compared to the backbone
model without hierarchies.

Generalizability. To validate the generalizability of the UNesT, we compare
KiTS19 results among nnUNet [14] and transformer-based methods. Our approach
achieves moderate improvement at DSC of 0.9778 and 0.8398 for kidneys and
tumors, indicating that the designed architecture can be used as a generic 3D
segmentation method.

Figure 4: Left: DSC comparison on the test set at different percentages of
training samples. Middle: Comparison of the convergence rate for the proposed
method with and without hierarchical modules, validation DSC along training
iterations are demonstrated. Right: Results on the KiTS19 dataset show the
generalizability of the proposed UNesT.

##  6 Discussion and Conclusion

In this paper, we target the critical problem that transformer-based models
are commonly data-inefficient, which leads to unsatisfied performance when
tasked with learning small structures and small datasets. In this work, we
develop the first cohort of renal sub-structures study, specifically the renal
cortex, medulla, and pelvicalyceal system. Upon the clinically acquired
subjects, we propose a novel hierarchical transformer-based 3D medical image
segmentation approach (UNesT). We show that the proposed method is data-
efficient for accurately quantifying kidney components and can be used for
volumetric analysis such as the medullary pyramids. Figure 5 in the
supplementary materials shows the proposed automatic segmentation method
achieves better agreement compared to inter-rater assessment, 0.01 against
0.29 of mean difference indicating reliable reproducibility.

Figure 5: The Bland-Atman plots compare the medulla volume agreement of inter-
rater and auto-manual assessment. We show the cross-validation on interpreter
1, interpreter 2 manual segmentation on the same test set. Interpreters
present independent observation without communication. The auto-manual
assessment shows the agreement between UNesT and interpreter 1 annotations.

Clinical Impact. Visual quantitative analysis of renal structures remains a
complex task for radiologists. Some of the histomorphometry features of
regions of the kidney (e.g. textural or graph features) are poorly adapted for
manual identifications. In this study, we show that UNesT achieves
consistently reliable performance. Compared with previous studies on cortex
segmentation, the proposed approach significantly facilitates derivation of
the visual and quantitative results.

Efficient segmentation is critical for clinical practice in deploying
individual assessment. We note that, unlike other large organs, the renal
segmentation dataset can be different in terms of imaging protocols, patient
morphology, and institutional variations. We consider the framework adaptable
to the segmentation of abnormal primitives in the future. In terms of
sensitivity, we believe that the approach can be further improved from two
perspectives. First, pre-registration of the kidney region of interest can
help to reduce the shape and size variations and thus boost the segmentation
performances. Second, incorporating dose usage in the segmentation loop can be
very helpful. It can be expected that augmented contrast can be measured to
better identify adjacent tissues among renal structures.

Acknowledgements. This research is supported by NIH Common Fund and National
Institute of Diabetes, Digestive and Kidney Diseases U54DK120058, NSF CAREER
1452485, NIH grants, 2R01EB006136, 1R01EB017230 (Landman), and R01NS09529. The
identified datasets used for the analysis described were obtained from the
Research Derivative (RD), database of clinical and related data. The imaging
dataset(s) used for the analysis described were obtained from ImageVU, a
research repository of medical imaging data and image-related metadata.
ImageVU and RD are supported by the VICTR CTSA award (ULTR000445 from
NCATS/NIH) and Vanderbilt University Medical Center institutional funding.
ImageVU pilot work was also funded by PCORI (contract CDRN-1306-04869). We
thank Ali, Vishwesh, Dong, Holger and Daguang at Nvidia for the 3D transformer
discussions on 2021 summer.

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