# Role of Oxygen States in the Low Valence Nickelate La4Ni3O8

Y. Shen<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, USA  J.
Sears  Condensed Matter Physics and Materials Science Department, Brookhaven
National Laboratory, Upton, New York 11973, USA  G. Fabbris  Advanced Photon
Source, Argonne National Laboratory, Lemont, Illinois 60439, USA  J. Li  J.
Pelliciari  I. Jarrige  National Synchrotron Light Source II, Brookhaven
National Laboratory, Upton, New York 11973, USA  Xi He  I. Božović  Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA  Department of Chemistry, Yale
University, New Haven, Connecticut 06520, USA  M. Mitrano  Department of
Physics, Harvard University, Cambridge, Massachusetts 02138, USA  Junjie Zhang
Materials Science Division, Argonne National Laboratory, Lemont, Illinois
60439, USA  Institute of Crystal Materials, Shandong University, Jinan,
Shandong 250100, China  J. F. Mitchell  Materials Science Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  A. S. Botana  Department of
Physics, Arizona State University, Tempe, Arizona 85287, USA  V. Bisogni
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  M. R. Norman  Materials Science Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  S. Johnston  Department of
Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee
37966, USA  Institute of Advanced Materials and Manufacturing, The University
of Tennessee, Knoxville, Tennessee 37996, USA  M. P. M. Dean<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA

###### Abstract

The discovery of superconductivity in square-planar low valence nickelates has
ignited a vigorous debate regarding their essential electronic properties: Do
these materials have appreciable oxygen charge-transfer character akin to the
cuprates, or are they in a distinct Mott-Hubbard regime where oxygen plays a
minimal role? Here, we resolve this question using O $K$-edge resonant
inelastic x-ray scattering (RIXS) measurements of the low valence nickelate
La4Ni3O8 and a prototypical cuprate La2-xSrxCuO4 ($x=0.35$). As expected, the
cuprate lies deep in the charge-transfer regime of the Zaanen-Sawatzky-Allen
(ZSA) scheme. The nickelate, however, is not well described by either limit of
the ZSA scheme and is found to be of mixed charge-transfer/Mott-Hubbard
character with the Coulomb repulsion $U$ of similar size to the charge-
transfer energy $\Delta$. Nevertheless, the transition-metal-oxygen hopping is
larger in La4Ni3O8 than in La2-xSrxCuO4, leading to a significant
superexchange interaction and an appreciable hole occupation of the ligand O
orbitals in La4Ni3O8 despite its larger $\Delta$. Our results clarify the
essential characteristics of low valence nickelates and put strong constraints
on theoretical interpretations of superconductivity in these materials.

##  I Introduction

Creating analogs of the cuprate high-temperature superconductors has been a
target of materials research for decades [1, 2, 3]. It is widely believed that
this requires a formal $d^{9-\delta}$ electron count on the transition-metal
(TM) site (as in hole-doped cuprates), strong electronic correlations, and a
substantial TM-O hybridization. This situation is illustrated in Fig. 1(a),
where the Cu-O charge-transfer energy $\Delta$ in cuprates is much smaller
than the on-site $3d$ Coulomb repulsion $U$. Square-planar nickelate materials
of the form $R_{n+1}$NinO2n+2 (R stands for a rare earth and $n$ is the number
of neighboring NiO2 layers) realize a $d^{9-\delta}$ valence and were
identified as promising potential cuprate analogs [1, 4]. This strategy bore
fruit when infinite-layer $n=\infty$ R1-xSrxNiO2 materials, and more recently,
$n=5$ Nd6Ni5O12, were shown to superconduct, generating intense scientific
interest [5, 6, 7, 8].

Figure 1: Schematic of the O $K$-edge RIXS process. (a)–(c) RIXS processes for
different values of the Hubbard $U$ and charge-transfer energy $\Delta$. These
are defined as the energy cost of specific atomic transitions in the material:
for a $d^{9}$ ground state, $U$ being a $d_{i}^{9}d_{j}^{9}\rightarrow
d^{8}_{i}d_{j}^{10}$ and $\Delta$ being a $d_{i}^{9}\rightarrow
d^{10}_{i}\underline{L}$ transition, where $i$ and $j$ label TM sites and
$\underline{L}$ denotes an oxygen ligand hole. For a charge-transfer insulator
($\Delta\ll U$), the doped holes are mostly in the oxygen $2p$ orbitals, while
in a Mott-Hubbard insulator ($\Delta\gg U$), the doped holes mainly occupy the
TM $3d^{n}$ state. In the mixed charge-transfer/Mott-Hubbard regime
($\Delta\sim U$), the doped holes are spread among both the TM and oxygen
sites. Arrows show examples of x-ray transition pathways, which, because of
x-ray dipole selection rules, can involve either O states or TM-O hybridized
states, making this process ideal to distinguish situations (a)–(c). Note that
$W$ is the band width for oxygen $2p$ orbitals, and $E_{\textrm{F}}$ is the
Fermi energy (marked by the horizontal dashed lines). (d) Experimental setup
for RIXS measurements at the O $K$-edge and crystal structure of La4Ni3O8.
Only the nickel and oxygen ions in the trilayers are shown.

Even before this discovery, however, it was noted that Ni has a smaller
nuclear charge than Cu, which would tend to increase $\Delta$ [9]. If $\Delta$
is substantially larger, this could potentially realize a Mott-Hubbard
situation, as illustrated in Fig. 1(c), in which O states play a minimal role
in the low-energy physics. This expectation motivated extensive interest in
determining the Mott-Hubbard versus charge-transfer nature of square-planar
nickelates, with some researchers categorizing them as Mott-Hubbard systems,
and others emphasizing the nickelates as cuprate analogs [10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]. Resolving this controversy is a crucial step
for understanding these materials. The degree of TM-O hybridization is a major
determining factor of the relative Ni versus O character of the charge
carriers and the magnetic superexchange. This issue also constrains which
minimal effective models are appropriate for these materials and is therefore
central to understanding nickelate superconductivity. Part of the challenge of
this question is that $\Delta$, $U$, the hopping integrals $t_{pd}$ and
$t_{pp}$, and the crystal fields all act together to determine the role of
oxygen and the location of the material within the Zaanen-Sawatzky-Allen
(ZSA)scheme [42, 43]. Progress in this area has also been hindered by varying
interpretations based on different experimental approaches [44, 45, 46, 18,
47, 36, 48, 41] and the fact that many-body electronic structure results are
sensitive to the method used [49]. This controversy has been further amplified
by the recent discovery of relatively large superexchange interactions, about
half that of cuprates, in both the $n=3$ and $n=\infty$ materials [50, 44].
Resolving these issues requires a spectroscopic probe that can specifically
target oxygen states and their energies and an interpretative approach that
treats $\Delta$, $U$, $t_{pd}$, and $t_{pp}$ on the same footing.

Here, we use O $K$-edge RIXS to determine the Mott-Hubbard versus charge-
transfer electronic characteristics of the $n=3$ nickelate La4Ni3O8, which can
be prepared as a bulk single crystal without the need for chemical doping, and
the oxygen-containing substrates and capping layers typically required for the
stability of the infinite-layer nickelate superconductors. We benchmark our
results for this material against a prototypical cuprate La2-xSrxCuO4
($x=0.35$), which has a similar effective doping. The nickelate is found to be
of mixed charge-transfer/Mott-Hubbard electronic character and not well
described by either the Zhang-Rice singlet states found in La2-xSrxCuO4 nor a
Ni-dominated Mott-Hubbard scenario. We confirm this interpretation using exact
diagonalization (ED) calculations and further find that the TM-oxygen hopping
is enhanced in La4Ni3O8 relative to La2-xSrxCuO4, with a $\Delta$ about 2 eV
larger. These parameters lead to a sizable superexchange interaction and an
appreciable O character of the doped holes. Our results clarify the essential
electronic characteristics of nickelates and imply that both O and Ni are
necessary ingredients of minimal effective models for these materials.

##  II Methods

High-energy-resolution RIXS measurements were performed at the SIX beamline at
the NSLS-II with an energy resolution of around 22 meV. All data shown were
collected at 40 K. We fixed the spectrometer at a horizontal scattering angle
$2\Theta=150^{\circ}$ and used either $\pi$ or $\sigma$ polarized x-rays as
shown in Fig. 1(d). Thus, we put the crystalline ($H$, 0, 0) and (0, 0, $L$)
directions in the scattering plane and changed $\bm{Q}$ by rotating the sample
around the vertical (0, 1, 0) axis. Note that $\theta$ is defined as the angle
between the incident beam and sample surface, which is perpendicular to the
sample $c$ axis.

Our chosen low-valence nickelate sample is La4Ni3O8 [51, 52, 53, 54, 50]. This
material has a nominal Ni $d^{8\frac{2}{3}}$ valence and can be considered as
$1/3$ self-doped with holes. Here, “doped holes” refers to this $\delta=1/3$
additional holes with respect to $d^{9}$, and “holes” refers to the total
holes with respect to the closed Ni $d$ shell. With no alkaline earth
substitution on the rare-earth site, the La4Ni3O8 system presents a well-
defined hole doping and avoids sample inhomogeneity challenges that have
hindered studies of R1-xSrxNiO2 [36, 55, 56, 57, 58, 41].

Single crystal samples were reduced from the as-grown Ruddlesden-Popper phase
La4Ni3O10 using H2/Ar gas [51, 52, 53] (see Appendix A for details). La4Ni3O8
has a tetragonal structure with space group $I4/mmm$ and lattice constants of
$a=b=3.97$ Å, $c=26.1$ Å. Its structure features trilayers of square-lattice
NiO2 planes, which are separated by La2O2 fluorite blocks [see Fig. 1(d)]. All
members of the $R_{n+1}$NinO2n+2 family share very similar Ni-O bonding and
would therefore be expected to have similar local correlated physics, provided
they are compared at the same effective doping. This expectation is borne out
by calculations [25, 59] and the similar magnetic exchange [50, 44] and
superconducting transition temperature in different nickelates [5, 6, 7, 8].
We compare La4Ni3O8 to La2-xSrxCuO4 ($x=0.35$) samples prepared via molecular
beam epitaxy with a similar effective doping as the nickelate.

Figure 2: Comparison of RIXS spectra in La4Ni3O8 and La2-xSrxCuO4. XAS plots
and RIXS intensity maps across the pre-peak energies for (a,b) La4Ni3O8 and
(c,d) La2-xSrxCuO4. The data were collected with $\theta=15^{\circ}$ and
$\sigma$ polarization. Different contributions can be identified, including
the fluorescence (FL) and resonant signals (labeled as features A and B). The
vertical dashed lines indicate the pre-peak resonant energy. XAS data are
shown in arbitrary units (a.u.) normalized to 0 and 1 at energies 1 eV below
and 1 eV above the pre-peak [52, 50]. Figure 3: Cluster exact diagonalization
results. (a) Sketch of the clusters used in each case. (b) RIXS curves
measured at the O pre-peak resonance with $\theta=60^{\circ}$ and $\sigma$
polarization. The feature at 0.5 eV in the La2-xSrxCuO4 sample is known to
come from a plasmon, so is not expected to appear in our model calculations
[60]. (c) Calculated spectra using the parameters listed in Table 1 with the
same experimental geometries.

##  III Results and interpretation

We start by measuring x-ray absorption spectra of our nickelate and cuprate
samples in the O $K$ pre-edge region, as shown in Figs. 2(a) and 2(c). The
spectra have a pre-peak feature, which comes from holes occupying TM-O
hybridized states [61, 62, 52, 50]. As measured previously, this pre-peak is
weaker in La4Ni3O8 compared to La2-xSrxCuO4, but this in itself is
insufficient to quantify the electronic nature of these materials in detail
[52, 50].

Figures 2(b) and 2(d) show the RIXS energy maps at $\theta=15^{\circ}$ with
$\sigma$ polarization. The maps consist of strong resonant features that
appear at the O $K$ pre-peak and diagonal structures of weaker, more diffuse
intensity above the resonance. Features that appear as diagonal lines in these
maps are assigned to x-ray fluorescence (labeled FL), which occur at constant
final energy and arise from a continuum of particle-hole excitations. Apart
from the fluorescence line, Raman-like peaks are observed at the pre-peak
incident energy in both samples, which correspond to local excitations. For
La4Ni3O8, the resonant signals can be divided into a strong manifold of
excitations between 4 and 8 eV (feature A) and a low-energy weaker structure
between 1 and 2 eV (feature B) [see Figs. 2(b) and 3(b)]. For La2-xSrxCuO4,
one manifold of excitations, which we call feature A, covers 1–8 eV, making
feature B hard to distinguish [see Figs. 2(d) and 3(b)].

Table 1: Representative parameters used for the ED calculations. Full details are provided in Appendix C. Note that $U$ is the on-site intra-orbital Coulomb interaction for the $3d$ orbitals, and $\Delta$ measures the charge-transfer energy. These parameters have an estimated error bar of about 1 eV [63]. Here, $t_{p_{\sigma}d_{x^{2}-y^{2}}}$ is the hopping integral between planar $p_{\sigma}$ and $d_{x^{2}-y^{2}}$ orbitals while $t_{p_{\sigma}p_{\sigma}}$ is that among the $p_{\sigma}$ orbitals. The column labeled $J_{\textrm{calc}}$ is the singlet-triplet energy splitting of the undoped ($d^{9}$) cluster (two holes in total), and $J_{\textrm{exp}}$ is the superexchange interaction from experiment. These parameters are given in eV. The column labeled TM holes is the hole occupation of the TM orbitals per TM site of the doped 2-TM site clusters (which have three holes in total). The column labeled O holes is the hole occupation of the oxygen orbitals per TM site.  Cluster | $U$ | $\Delta$ | $t_{p_{\sigma}d_{x^{2}-y^{2}}}$ | $t_{p_{\sigma}p_{\sigma}}$ | $J_{\textrm{calc}}$ | $J_{\textrm{exp}}$ | TM holes | O holes
---|---|---|---|---|---|---|---|---
Ni2O7 | 6.5 | 5.6 | 1.36 | 0.375 | 0.084 | 0.069 [50] | 1.05 | 0.45
Cu2O11 | 9 | 3.4 | 1.17 | 0.625 | 0.151 | 0.143 [64] | 0.76 | 0.74

In the photon emission process of O $K$-edge RIXS, electrons from either the
TM $3d$ or oxygen $2p$ orbitals can potentially be deexcited to fill the core
hole; however, the former channel is expected to be weaker since it is only
possible through the TM-O hybridization. In light of this, we might expect
feature A to come from charge-transfer excitations in the O $2p$ states and
feature B to come from the weaker inter-orbital $dd$ excitations of the TM
$3d$ orbitals. This is further verified by the Ni $L$-edge RIXS data we have
taken, which show a strong peak in the 1–2 eV energy window. The fact that
feature A for La4Ni3O8 is higher in energy than that in La2-xSrxCuO4 indicates
a larger $\Delta$ by about 2 eV. At the same time, the presence of feature B
from $dd$ excitations suggests that there is still appreciable Ni-O
hybridization, which would not occur if $\Delta\gg U$. It is also clear that
feature A is narrower in the energy loss axis in La4Ni3O8, suggesting a
smaller oxygen band width, again consistent with the larger $\Delta$.
Considering these factors together allows us to conclude that La4Ni3O8 is of
mixed charge-transfer/Mott-Hubbard character.

To confirm and quantify our conclusion of mixed charge-transfer/Mott-Hubbard
character, we make use of cluster ED calculations using the EDRIXS software
[65, 66] to reveal the electronic structure and oxygen involvement in La4Ni3O8
and La2-xSrxCuO4. The calculations are performed on Ni2O7 and Cu2O11 clusters,
respectively, with open boundary conditions and three holes in total (i.e.,
1.5 holes per TM site or $x=0.5$), as shown in Fig. 3(a). The orbital basis in
each case includes all five TM $3d$ orbitals and all three O $2p$ orbitals on
their respective atoms, and parameters are defined in hole notation such that
$\Delta$ denotes the energetic splitting between the O and TM states (see
Appendixes B and C). More specifically, $\Delta$ is defined as the energy cost
for a local $d_{i}^{9}\rightarrow d^{10}_{i}\underline{L}$ transition, and $U$
reflects the energy needed for a $d_{i}^{9}d_{j}^{9}\rightarrow
d^{8}_{i}d_{j}^{10}$ transition[42, 43]. Although in undoped $d^{9}$ infinite-
layer nickelates, the rare-earth orbitals act as effective dopants, dynamical
mean field theory (DMFT) calculations suggest that their role is minimal in
heavily doped nickelates, where these states are unoccupied and only weakly
hybridized with the planar Ni and O orbitals [25]. Omitting rare-earth
orbitals allows us to maintain a tractable basis size while still achieving a
satisfactory description of the data. Starting from the atomic limit, we
explicitly include Coulomb interactions and nearest-neighbor inter-atomic
hopping, and simulate the XAS and RIXS spectra with experimental conditions
fully accounted for (see Appendix C for more details). As explained in detail
in the Supplemental Material [63], the model parameters were then adjusted to
match the experimental spectra, and they are listed in Table 1. The strong
core-hole potential in RIXS means that the low-energy excitations seen at
resonance tend to have rather local character and minimal dispersion, which
supports and motivates the widespread cluster-based interpretation of RIXS
measurements [67, 68, 69, 70, 71, 72, 73, 74].

Our approach fully incorporates many-body and multi-orbital effects, avoiding
issues with ad hoc parameters or double counting [75, 76], and it allows us to
directly compare theory and experiment by evaluating the Kramers-Heisenberg
equation for the RIXS cross section. In this way, we can extract $\Delta$ and
$U$ in the original sense defined within the ZSA scheme as outlined in Fig. 1.
Other means of extracting these parameters, such as those relying on density
functional theory, can potentially be affected by double counting of the
Coulomb interactions [75, 76, 63]. These advantages come at the cost of having
to work with a small cluster, which means that the effective doping level
($x=0.5$) is only approximately that of the real materials ($x\approx 1/3$).
Potential minor discrepancies are expected, arising from the small cluster
size and omitted interactions with other atoms such as the rare-earth ions and
adjacent layers. In addition, the cluster does not capture the FL feature
since this requires a continuum of states [77, 41]. These factors lead to an
error bar of about 1 eV on the extracted parameters, but within this error,
the extracted parameters are robust, as has been demonstrated in several prior
small cluster calculations of related oxide materials including cuprates and
infinite-layer nickelates [67, 68, 69, 70, 71, 72, 73, 74].

The calculated RIXS spectra are shown in Fig. 3(c) and capture the onset and
bandwidth of the spectral features nicely, including feature A, and the fact
that feature B is difficult to observe in La2-xSrxCuO4. The observed level of
agreement is comparable or surpasses the typical level of agreement of
theoretical interpretations of O $K$-edge RIXS spectra [74, 70, 71, 72, 19,
73].

Figure 4: Angle dependence of La4Ni3O8 RIXS intensity at the pre-edge with
$\sigma$ polarization. (a) RIXS map data after absorption correction. (b)
Simulation of a Ni2O7 cluster. (c) Orbital energy-level diagram of the
multiplet ligand field analysis using the parameters listed in Table 1. A full
description of these states can be found in Appendix E. Since this breakdown
of the different orbitals is based on one-hole basis functions, the $y$ energy
axis has been shifted to account for the doping.

Table 1 summarizes the parameters extracted from our theoretical analysis. As
expected, these values confirm our empirical conclusions regarding the charge-
transfer versus Mott-Hubbard characteristics of La4Ni3O8 and La2-xSrxCuO4. We
find that $\Delta$ is much smaller than $U$ in La2-xSrxCuO4, confirming its
charge-transfer nature, consistent with the known character of cuprates [1, 2,
3]. Our $\Delta\sim 3.4$ eV determination is very similar to the literature
values from prior ED studies and within 1 eV (slightly larger) of those in
first-principles work [78, 74]. This is consistent with the estimated 1 eV
error bar of our parameter extraction and indicates that the finite-size
cluster effect will only change a parameter by less than this error bar. In
contrast, La4Ni3O8 has a significantly larger $\Delta\sim 5.6$ eV, which is
comparable to $U\sim 6.5$ eV. This value places La4Ni3O8 in a mixed charge-
transfer/Mott-Hubbard regime. It should be noted that different combinations
of ED parameters can produce similar RIXS spectra for La4Ni3O8, but all of
them give a $\Delta\approx 6$ eV within a 1 eV error bar and a $U$ of a
similar size to $\Delta$ [63]. As a consequence, the nickelate has enhanced Ni
hole character compared to the cuprate, but it retains appreciable O hole
character and therefore is not in the pure Mott-like regime in which the holes
would have overwhelming TM $d$ character. Our fits to the data also find that
the $p-d$ hopping is enhanced in La4Ni3O8, which we attribute to the more
extended nickel $d$ orbitals due to the smaller nuclear charge
111Interestingly, this enhancement is not seen in a DFT Wannier analysis of
the infinite-layer nickelate [16] and La4Ni3O8 [92], where a value similar to
the cuprates is found.. Coupled with the fact that $\Delta$ is only about 2 eV
larger than in the cuprates, this means that a significant portion of the
doped holes still reside on the oxygen sites (Table 1). At the same time,
La4Ni3O8 has a larger fraction of $|d^{8}\rangle$ states in its ground-state
wave function compared to La2-xSrxCuO4 because of its larger $\Delta$ and
lower $U\approx\Delta$ (see Appendix D), as also found in DMFT studies [21,
25, 28]. Overall, the TM-O hybridization, which can be quantified by the
parameter $t_{pd}^{2}/\Delta$, is slightly reduced in La4Ni3O8: 0.33 for
La4Ni3O8 compared to 0.4 for La2-xSrxCuO4.

Examining the wave functions in Appendix D, we find that feature A in La4Ni3O8
is largely of a charge-transfer type where a significant fraction of a hole
(about 0.43) is excited from the double-hole-occupied Ni ion to the ligand O
orbitals (i.e., $|d^{8}\rangle\rightarrow|d^{9}\underline{L}\rangle$). In
La2-xSrxCuO4, the excitation is more mixed; it involves both a small amount of
charge transfer (about 0.14 holes) from the Cu to ligand O and a rearrangement
of the ligand hole (i.e.,
$|d^{9}\underline{L}\rangle\rightarrow|d^{9}\underline{L}^{\prime}\rangle$).
Feature B, on the other hand, corresponds to $dd$ excitations, where the
ground-state occupations of the TM $3d$ orbitals are rearranged during the
scattering process. They are also present but weaker in the La2-xSrxCuO4
calculations and obscured by feature A’s overlapping contribution. Detailed
projections of the wave functions onto different states are provided in
Appendix D.

To further verify our model for La4Ni3O8, we show that our ED calculations
also capture the angular dependence of the spectra. In this case, the
evolution of the RIXS intensity with $\theta$ can largely be accounted for by
the matrix elements of the RIXS cross section, which select different O
orbitals. They can be easily calculated in the dipole approximation and the
results are reasonably consistent with the La4Ni3O8 data (Fig. 4). The
different levels can be conceptualized using a multiplet ligand field model
[80, 81], which is a simplified version of the cluster model we use for our
full analysis, as explained in Appendix E. In this approach, the O $2p$
orbitals hybridize with each other, forming bonding, anti-bonding and non-
bonding ligand orbitals through in-plane $p-p$ hopping. Some of these have the
same symmetry as the TM $3d$ orbitals and thus hybridize with them via the
relevant $p-d$ hopping. Using the parameters determined from our ED
calculations, we produce a simplified scheme of the ligand orbital energy
levels in Fig. 4(c). Although the ligand orbital splitting is slightly
underestimated because of omitted terms in the ligand field analysis—such as
the $p_{\sigma}-p_{\pi}$ hopping—and multiplet effects are absent in the one-
hole basis functions, the overall distribution of the hybridized states is
qualitatively consistent with the data and is well captured by our ED
calculations.

##  IV Discussion

Our detailed RIXS measurements and ED analysis put La4Ni3O8 in the mixed
charge-transfer/Mott-Hubbard parameter regime of the ZSA scheme. Although the
larger $\Delta$ in La4Ni3O8 suppresses the hybridization of the TM $3d$ and
oxygen $2p$ orbitals, the fact that $\Delta$ is only about 2 eV larger, along
with the enhanced $p-d$ hopping, guarantees that the oxygen orbitals retain a
significant role in determining the physical properties of these low valence
nickelates. This also results in a substantial superexchange interaction that
is half that of the cuprates [50]. In this regard, low valence nickelates are
a good analog of cuprate superconductors. However, the larger $\Delta$
produces a small, but possibly significant Ni $|d^{8}\rangle$ weight in the
ground state.

Since multi-orbital physics has been discussed extensively in nickelates, we
compute the involvement of different Ni $d$ orbitals in the ground state.
According to our ED calculations, the majority orbital character retains
$d_{x^{2}-y^{2}}$ symmetry—less than 8% of the holes occupy the
$d_{3z^{2}-r^{2}}$ orbitals in La4Ni3O8 (see Appendix D). Hund’s physics
therefore has only a small role in determining the properties of this
material, and the Ni states are predominantly of low-spin character,
consistent with prior experiments [52, 50, 45]. This is partly due to the
strong TM-O hybridization, along with the square-planar environment of Ni,
which gives rise to a large crystal field splitting in the $e_{g}$ states.
Since both the $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ orbitals hybridize
with the O $2p_{\sigma}$ orbitals, they can mix despite their different
symmetries. If $\Delta$ increases further, the $d_{3z^{2}-r^{2}}$ orbitals
become more involved, as shown in Appendix F.

The mixed charge-transfer/Mott-Hubbard character of the low valence nickelates
reported here suggests that minimal theoretical models must explicitly include
both Ni and O states alongside strong correlations. In this regime, the
properties of many-body methods such as DMFT can become dependent on the
choice of basis functions to represent the correlations [49]. Problems with
so-called “double counting” of $U$ are also more difficult to avoid in this
case [75, 76]. Both of these challenges likely contribute to the differences
between different studies thus far [10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41]. By combining O-resonant spectroscopy with an ED treatment of the
relevant $\Delta$, $U$, and hopping parameters, we can confidently report that
La4Ni3O8 is described by $U\sim\Delta$ with both values of order 6 eV. While
our results position La4Ni3O8 in the mixed ZSA regime, other factors such as
rare-earth orbitals, electron-phonon coupling, and La2O2 layers might
contribute to the detailed properties of these materials [18, 11, 59].
However, addressing these more subtle factors would expand the basis size well
beyond what can be handled by current state-of-the-art ED calculations and
require more approximate means of handling correlations, which is contrary to
the goal of providing a rigorous minimal description of these materials.

##  V Conclusions

In summary, we combine RIXS measurements at the O $K$-edge and ED calculations
to compare the low valence nickelate La4Ni3O8 to a prototypical cuprate
La2-xSrxCuO4 with a similar nominal electron filling. Our work is unique in
directly measuring the energy of the Ni-O hybridized states and interpreting
them while treating $\Delta$, $U$, and hoppings on the same footing. The
results reveal that La4Ni3O8 has a larger $\Delta$, smaller $U$, and increased
$p-d$ hopping, which gives it a mixed charge-transfer/Mott-Hubbard electronic
character. Despite its larger $\Delta$, the nickelate retains a sizable
superexchange through its strong hopping. Moreover, for the parameters we have
determined, Hund’s physics is less relevant than might be expected (i.e.,
despite the admixture of $d^{8}$), consistent with the square-planar
environment of Ni. Given that La4Ni3O8 and NdNiO2 have similar magnetic
exchange interactions [50, 44], and since magnetic exchange is very sensitive
to the $\Delta/U$ ratio, we suggest that this mixed charge-transfer/Mott-
Hubbard picture likely applies to the entire $R_{n+1}$NinO2n+2 series. This
idea is also supported by first-principles calculations, which find that these
materials are indeed rather similar, provided they are compared at the same
effective doping [25, 59]. Overall, we conclude that both the Coulomb
interactions and charge-transfer processes need to be considered when
interpreting the properties of these nickelates. Realistic models of low
valence nickelates must therefore include both Ni and O states at a minimum,
and we suggest this as a basis for conceptualizing these fascinating
materials.

The supporting data for the plots in this article are openly available from
the Zenodo database [82].

###### Acknowledgements.

We thank Kenji Ishii for helpful discussions. Work at Brookhaven National
Laboratory (RIXS measurement and interpretation, and cuprate sample synthesis)
was supported by the U.S. Department of Energy (DOE), Office of Science,
Office of Basic Energy Sciences. Work at Argonne was supported by the U.S.
DOE, Office of Science, Basic Energy Sciences, Materials Science and
Engineering Division (nickelate sample synthesis and first-principles
calculations). A.B. acknowledges support from NSF Grant No. DMR 2045826. S.J.
acknowledges support from the U.S. DOE, Office of Science, Office of Basic
Energy Sciences, under Award No. DE-SC0022311. X.H. was supported by the
Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No.
GBMF9074. This research used resources at the SIX beamline of the National
Synchrotron Light Source II, a U.S. DOE Office of Science User Facility
operated for the DOE Office of Science by Brookhaven National Laboratory under
Contract No. DE-SC0012704. This research used resources of the Advanced Photon
Source, a U.S. DOE Office of Science User Facility at Argonne National
Laboratory, and is based on research supported by the U.S. DOE Office of
Science-Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

Table 2: Full list of parameters used for the ED calculations including the full set of TM $3d$ and oxygen $2p$ orbital energies. We set $\epsilon_{d_{x^{2}-y^{2}}}=0$, which means that, within hole notation, we need to set $\epsilon_{p_{\sigma}}=\Delta$ [63]. Note that $V_{pd\sigma}$, $V_{pd\pi}$, $V_{pp\sigma}$, and $V_{pp\pi}$ are amplitudes of Slater-Koster parameters, and we fix $V_{pd\pi}=-V_{pd\sigma}/2$ and $V_{pp\pi}=-V_{pp\sigma}/4$. Correspondingly, $t_{p_{\sigma}p_{\sigma}}=(V_{pp\sigma}-V_{pp\pi})/2$ and $t_{p_{\sigma}d_{x^{2}-y^{2}}}=\sqrt{3}V_{pd\sigma}/2$. Here, $\eta$ is the ratio of the out-of-plane TM-O distance over the in-plane one, which controls the hopping amplitudes involving the apical oxygen. We assume $V_{pp\sigma}$ scales with $d^{-2}$, where $d$ is the O-O bond length, and $V_{pd\sigma}$ scales with $d^{-3.5}$, where $d$ is the TM-O bond length [80]. Here, $F^{0}_{dd}$, $F^{2}_{dd}$, and $F^{4}_{dd}$ are Slater integrals for the TM $3d$ orbitals, and $F^{0}_{pp}$ and $F^{2}_{pp}$ are for oxygen $2p$. Correspondingly, $U=F^{0}_{dd}+4/49\times(F^{2}_{dd}+F^{4}_{dd})$, and the Hund’s coupling is $J_{H}=(F^{2}_{dd}+F^{4}_{dd})/14$. Note that $U_{dp}$ is the inter-site Coulomb interaction between TM $3d$ and oxygen $2p$ holes, and $U_{q}$ is the core-hole potential for oxygen. Here, $\Delta$ and $U$ have an estimated error bar of about 1 eV [63]. All parameters, with the exception of $\eta$, are in units of eV.  Cluster | $\epsilon_{d_{x^{2}-y^{2}}}$ | $\epsilon_{d_{3z^{2}-r^{2}}}$ | $\epsilon_{d_{xy}}$ | $\epsilon_{d_{xz/yz}}$ | $\epsilon_{p_{\sigma}}$ | $\epsilon_{p_{\pi}/p_{z}}$ | $\epsilon_{p^{a}_{\pi}}$ | $\epsilon_{p^{a}_{z}}$ | $V_{pd\sigma}$ | $V_{pp\sigma}$ | $\eta$ | $F^{0}_{dd}$ | $F^{2}_{dd}$ | $F^{4}_{dd}$ | $F^{0}_{pp}$ | $F^{2}_{pp}$ | $U_{dp}$ | $U_{q}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Cu2O11 | 0 | 0.95 | 0.7 | 0.9 | 3.4 | 4 | 4 | 3.4 | 1.35 | 1 | 1.3 | 7.86 | 8.61 | 5.38 | 3.3 | 5 | 1 | 6
Ni2O7 | 0 | 0.2 | 0.1 | 0.3 | 5.6 | 6.1 | - | - | 1.57 | 0.6 | - | 5.58 | 6.89 | 4.31 | 3.3 | 5 | 1 | 6

##  Appendix A Sample synthesis and characterization

The La4Ni3O8 single crystal used in this study was prepared as described in
Refs. [51, 52] and was the same piece as was used in Ref. [50]. The parent
Ruddlesden-Popper La4Ni3O8 was prepared using the high-pressure optical
floating zone method. During growth, oxygen gas was maintained at 20 bar
pressure with a flow rate of 0.1 l/min. To improve homogeneity, the feed and
seed rods were counter-rotated at 30 r.p.m., and rods were advanced at 4 mm/h
over the 30 hour growth time. Sample reduction was performed by heating small
crystals cleaved from the parent La4Ni3O8 boule in a flowing 4% H2/Ar gas
mixture at 350∘C for five days.

The La2-xSrxCuO4 film reported here was synthesized using the atomic-layer-by-
layer molecular beam epitaxy (ALL-MBE) technique [83], in a similar way to
samples used for prior RIXS studies [84, 85, 86]. Our ALL-MBE system contains
sixteen metal sources (thermal-effusion or Knudsen cells) and a source of pure
(distilled) ozone. The growth is monitored in real time by means of reflection
high-energy electron diffraction (RHEED), which provides information about the
time evolution of the film surface morphology and crystal structure [87]. The
growth kinetics is controlled using pneumatic linear-motion actuators that
shutter the atomic sources. The La2-xSrxCuO4 film studied here is heavily
overdoped, $x=0.35$. It was grown on a $10\times 10$ mm2 single-crystal
LaSrAlO4 substrate polished with the [001] axis perpendicular to the surface,
with a miscut of less than $0.10^{\circ}$. The LaSrAlO4 substrate lattice
constants are $a_{0}=b_{0}=3.755$ Å, $c_{0}=12.56$ Å. The La2-xSrxCuO4 film is
40 unit cells (530 Å) thick, and it is pseudomorphic with the LaSrAlO4
substrate; thus, it is under a small 0.5% compressive strain [88]. Since
LaSrAlO4 has no pre-peak, it does not contribute to the measured signal. The
substrate temperature during the growth (measured by a two-color pyrometer)
was kept at $T=611^{\circ}$C, and the ozone partial pressure was fixed at
$p=3\times 10^{-5}$ Torr. The growth rate was about 0.03 Å/sec. After the
deposition, the film was annealed for 3.5 hours at $T=611^{\circ}$C in an
ozone partial pressure $p=1\times 10^{-4}$ Torr, and then cooled down slowly
at the same pressure.

Figure 5: Wave function analysis of Ni2O7 and Cu2O11 clusters with three
holes. (a,b) Calculated unbroadened RIXS intensity (black vertical lines) and
broadened RIXS spectra (red curves) with $\theta=60^{\circ}$ and $\sigma$
polarization. Note that in this geometry, the contribution from $p_{z}$
orbitals is minimized. (c,d) Weights of different configurations of the ground
and excited states. (e,f) Hole occupations of TM $3d$ and O $2p$ orbitals.
(g,h) Hole occupations for each type of $3d$ orbital. (i,j) Hole occupations
for each type of $2p$ orbital. Only the excited states with unbroadened RIXS
intensity stronger than 1% of the maximum value are shown.

##  Appendix B Notation of transition metal and oxygen orbitals

For isolated atoms, the TM $3d$ and oxygen $2p$ orbitals are fully localized.
In a compound, however, the hybridization among atoms will mix these orbitals,
and the wave functions of the eigenstate orbitals are composed of different
atomic-character Wannier orbitals.

In the clusters used in this study, shown in Fig. 3(a) of the main text, the
TM sites are all equivalent and include $d_{x^{2}-y^{2}}$, $d_{3z^{2}-r^{2}}$,
$d_{xy}$ and $d_{xz/yz}$ orbitals. Oxygen atoms include $p_{x}$, $p_{y}$,
$p_{z}$ orbitals, and they exist in inequivalent planar and apical locations;
thus, we use $p$ and $p^{a}$ to distinguish them. The planar oxygen orbitals
can be represented using $p_{\sigma}$ and $p_{\pi}$ orbitals that are parallel
and perpendicular to the TM-O bonds, respectively, and that are composed of
different $p_{x}$, $p_{y}$ and $p_{z}$ orbitals depending on the site. To
further distinguish between the in-plane and out-of-plane $p_{\pi}$-type
orbitals, we refer to the in-plane ones as $p_{\pi}$ and the out-of-plane ones
as just $p_{z}$. For the apical oxygens, two sets of $p^{a}_{\pi}$ orbitals
are equivalent, and $p^{a}_{\sigma}$ orbitals are labeled simply as
$p^{a}_{z}$.

##  Appendix C  Cluster exact diagonalization calculations

For this work, we compute the RIXS cross section using the cluster ED
approach. This accurately treats interactions and cross-section effects and is
widely used to treat localized direct RIXS processes in correlated oxides [89,
67, 68, 69, 70, 71, 72, 73, 74]. This makes it appropriate for simulating
features A and B, although it fails to capture x-ray fluorescence. The full
Hamiltonian is composed of several terms

$\mathcal{H}=\hat{E}_{d}+\hat{E}_{p}+\hat{\Delta}+\hat{U}_{dd}+\hat{U}_{pp}+\hat{U}_{dp}+\hat{U}_{q}+\hat{T}_{pp}+\hat{T}_{dp}$
(1)

where $\hat{E}_{d}$ and $\hat{E}_{p}$ are the on-site energies of TM $3d$ and
oxygen $2p$ orbitals, respectively, and $\hat{\Delta}$ is the charge-transfer
energy. The point-charge crystal electric field (CEF) splitting is included in
these terms. Here, $\hat{U}_{dd}$ and $\hat{U}_{pp}$ describe the on-site
Coulomb interactions for the TM and oxygen, respectively. All the Coulomb and
exchange integrals are included explicitly. Note that $\hat{U}_{dp}$ is the
inter-site Coulomb interaction between TM and oxygen. For the intermediate
states, an additional term, $\hat{U}_{q}$, is included to account for the
core-hole potential. Here, $\hat{T}_{pp}$ and $\hat{T}_{dp}$ refer to O-O
$p-p$ hopping and TM-O $p-d$ hopping, respectively. All the hopping integrals
are evaluated from the Slater-Koster parameters. The hopping terms involving
apical oxygens are renormalized according to the bond lengths for the Cu2O11
cluster [80]. The hopping phases are explicitly considered in the Hamiltonian.
For simplicity, spin-orbit coupling is not considered. The full set of
parameters used for calculating the spectra presented in the main text are
listed in Table 2.

Regarding the Hilbert space, we include all the spin-resolved TM $3d$ and
oxygen $2p$ orbitals and perform ED in the two-up, one-down spin sector for
the doped clusters (three holes). For the undoped clusters (two holes), all
the spin configurations are included. The Hamiltonian is constructed in the
hole language. In this case, $\Delta$ is the energy difference between the
$d_{x^{2}-y^{2}}$ and $p_{\sigma}$ orbitals regardless of the Coulomb
interactions. In addition, by definition, $U$ equals the intra-orbital on-site
Coulomb interaction for a $d^{9}$ system [42, 43]. The resulting eigenstates
for the initial, intermediate, and final states from ED are used to calculate
the XAS and RIXS spectra using the Kramers-Heisenberg formula in the dipole
approximation with the experimental geometry explicitly considered. The
inverse core-hole lifetime is fixed to 0.25 eV according to XAS data, and the
final-state energy loss spectra are broadened using a Lorentzian function with
$\sigma=0.2$ to account for the band width that is not captured by the small
cluster. We sum up the spectra from clusters rotated by 90∘ around the $z$
axis to restore the $C_{4}$ symmetry. These calculations are compared with the
experimental intensity after a simple self-absorption correction has been
applied to the data based on the incident and emitted x-ray angles with
respect to the surface [90].

##  Appendix D Wavefunction analysis of ED calculations

To reveal the nature of the RIXS spectral features, we show in Fig. 5 the
weights of different configurations of the excited states as well as the hole
occupations. As discussed in the main text, feature A in La2-xSrxCuO4 is
dominated by the rearrangement of ligand orbitals, while for La4Ni3O8, it
involves a significant portion of charge transfer from Ni to O. In another
aspect, feature B, which is mostly $dd$-excitations, is clearly identified in
La4Ni3O8 since it is well separated from feature A. In La2-xSrxCuO4, it is
weak and obscured by feature A.

##  Appendix E Multiplet ligand field analysis

Table 3: Energy-level splitting of the one-hole basis functions. The O $p$ orbitals (on-site energies $\epsilon_{p_{\sigma}}$, $\epsilon_{p_{\pi}}$, $\epsilon_{p_{z}}$) hybridize with each other to form ligand orbitals (energies $\epsilon_{L}$), which further hybridize with TM $d$ orbitals (energies $\epsilon_{d}$). Here, $V$ are the hopping integrals, and $T_{pp}=V_{pp\sigma}-V_{pp\pi}$. Note that the anti-bonding molecular $p_{\pi}$ ligand orbital that has an energy of $\epsilon_{p_{\pi}}-T_{pp}$ (not listed) will not hybridize with the TM $d$ states. Symmetry | $\epsilon_{d}$ | $\epsilon_{L}$ | $V$
---|---|---|---
$x^{2}-y^{2}$ | $\epsilon_{d_{x^{2}-y^{2}}}$ | $\epsilon_{p_{\sigma}}-T_{pp}$ | $\sqrt{3}|V_{pd\sigma}|$
$3z^{2}-r^{2}$ | $\epsilon_{d_{3z^{2}-r^{2}}}$ | $\epsilon_{p_{\sigma}}+T_{pp}$ | $|V_{pd\sigma}|$
$xy$ | $\epsilon_{d_{xy}}$ | $\epsilon_{p_{\pi}}+T_{pp}$ | $2|V_{pd\pi}|$
$xz/yz$ | $\epsilon_{d_{xz/yz}}$ | $\epsilon_{p_{z}}$ | $\sqrt{2}|V_{pd\pi}|$

Here, we provide a qualitative explanation of the multiplet ligand field
analysis in Fig. 4(c). The analysis is the same as that derived more formally
in Ref. [80]. For simplicity, we consider the hybridization of one hole in a
NiO4 cluster. When comparing to La4Ni3O8, the sample’s doping level is
accounted for by offsetting the energy of these states when plotting the
different hybridized orbital energies in Fig. 4(c). We start by considering
the orbitals in the absence of $p_{\sigma}-p_{\pi}$ hopping, such that the
four $p_{\sigma}$ orbitals in a NiO4 plaquette will hybridize with each other
to form bonding and anti-bonding molecular orbitals. The non-bonding orbitals
are not included [80, 68]. The same thing happens for $p_{\pi}$ orbitals,
while $p_{z}$ orbitals remain non-bonding if the weak $p_{z}-p_{z}$ hopping is
ignored. The bonding $p_{\sigma}$ molecular orbital has the same symmetry as
the nickel $d_{3z^{2}-r^{2}}$ orbital, so they will hybridize with each other
through $p_{\sigma}-d_{3z^{2}-r^{2}}$ hopping. The same applies for anti-
bonding $p_{\sigma}$ molecular and $d_{x^{2}-y^{2}}$ orbitals, bonding
$p_{\pi}$ and $d_{xy}$ orbitals, and the non-bonding $p_{z}$ and $d_{xz/yz}$
orbitals. The energy levels and hopping integrals are listed in Table 3. Note
that the anti-bonding $p_{\pi}$ orbital will not hybridize with nickel $d$ due
to phase cancellation.

##  Appendix F $\Delta$ dependence of RIXS spectra and hole occupation

Figure 6: Ni2O7 cluster calculations with different $\Delta$. Other parameters
are fixed according to Table 2. (a) Intensity map presenting the calculated
$\Delta$-dependent RIXS spectra with $\theta=60^{\circ}$ and $\sigma$
polarization. (b) Orbital occupation of holes with different $\Delta$ in a
doped cluster (three holes) with $U=6.5$ eV.

Here, we present additional Ni2O7 cluster ED results to show how RIXS spectra
and hole occupations evolve with $\Delta$. When $\Delta$ is small, only
feature A can be resolved with an onset below 2 eV, resembling the cuprate
data [see Fig. 6(a)]. Concomitantly, the holes mainly occupy the oxygen sites
[see Fig. 6(b)]. With increasing $\Delta$, feature A shifts to higher energy,
and feature B starts to emerge out of feature A. This also increases the
number of holes transferred to the nickel sites. The red and black dashed
lines indicate the $\Delta$ for La4Ni3O8 where feature A and feature B are
well separated, but a significant portion of the holes remain on the oxygen
sites. As $\Delta$ increases further, the $d_{x^{2}-y^{2}}$ hole concentration
first increases, and then the $d_{3z^{2}-r^{2}}$ orbitals come into play,
leading to multi-orbital physics.

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11institutetext: Department of Medical Physics and Biomedical Engineering,
22institutetext: Wellcome/EPSRC Centre for Interventional and Surgical
Sciences, 33institutetext: Centre for Medical Image Computing,
University College London, UK

# Strategising template-guided needle placement for MR-targeted prostate
biopsy

Iani JMB Gayo  Shaheer U. Saeed  Dean C. Barratt
Matthew J. Clarkson  Yipeng Hu

###### Abstract

Clinically significant prostate cancer has a better chance to be sampled
during ultrasound-guided biopsy procedures, if suspected lesions found in pre-
operative magnetic resonance (MR) images are used as targets. However, the
diagnostic accuracy of the biopsy procedure is limited by the operator-
dependent skills and experience in sampling the targets, a sequential decision
making process that involves navigating an ultrasound probe and placing a
series of sampling needles for potentially multiple targets. This work aims to
learn a reinforcement learning (RL) policy that optimises the actions of
continuous positioning of 2D ultrasound views and biopsy needles with respect
to a guiding template, such that the MR targets can be sampled efficiently and
sufficiently. We first formulate the task as a Markov decision process (MDP)
and construct an environment that allows the targeting actions to be performed
virtually for individual patients, based on their anatomy and lesions derived
from MR images. A patient-specific policy can thus be optimised, before each
biopsy procedure, by rewarding positive sampling in the MDP environment.
Experiment results from fifty four prostate cancer patients show that the
proposed RL-learned policies obtained a mean hit rate of 93% and an average
cancer core length of 11 mm, which compared favourably to two alternative
baseline strategies designed by humans, without hand-engineered rewards that
directly maximise these clinically relevant metrics. Perhaps more
interestingly, it is found that the RL agents learned strategies that were
adaptive to the lesion size, where spread of the needles was prioritised for
smaller lesions. Such a strategy has not been previously reported or commonly
adopted in clinical practice, but led to an overall superior targeting
performance, achieving higher hit rates (93% vs 76%) and measured cancer core
lengths (11.0mm vs 9.8mm) when compared with intuitively designed strategies.

###### Keywords:

Reinforcement Learning Prostate Cancer Targeted Biopsy Planning

##  1 Introduction

Recent development in multiparametric MR imaging (mpMRI) techniques provides a
means of noninasive localisation of suspected prostate cancer [1], which
enables clinicians to target these lesions during the follow-up ultrasound-
guided biopsy for further histopathology confirmation. This MR-targeted
approach has been shown to reduce both the false positive and false negative
detection, compared to previously adopted random biopsy [1, 2], and
subsequently motivated research in developing multimodal MR-to-ultrasound
image registration [3].

Needle sampling of the MR-identified targets, with or without registration
errors, can still be a challenging and arguably overlooked task. Operator
expertise was found to be an important predictor in detecting clinically
significant prostate cancer [4]. Planning strategies is important for
navigating the ultrasound probe, to better observe the targets with respect to
imaging, and for manual needle positioning. In transperineal biopsy, the
introduction of brachytherapy templates (See Fig.1 for an example) helps the
needle deployment - a procedure that is of interest in this study, but choice
between $13\times 13$ grid positions remains a subjective decision. For
example, a common clinical practice aims at the target centre, but it has been
shown to yield an insufficient sampling of the heterogeneous cancer [5] and
possibly an inferior diagnostic accuracy in terms of disease-representative
grading [6], compared with more spread needle placement. The design of an
optimum strategy is further complicated by the need of multiple needles for
individual targets, for maximising the hit rate, and the multifoci nature of
prostate cancer, which requires repeated sampling of one or more targets.

To the best of our knowledge, there has not been any computer-assisted
sampling strategy that takes into account the previous needle deployment(s) or
quantitatively optimises patient-and-target-specific needle distribution. In
summary, improving the targeting strategy may help reduce the significant
false negative rate found in MR-targeted biopsy (reported being as high as 13%
[7]), and hence improves the chance of early cancer detection for patients.

Reinforcement learning (RL) has been proposed for medical image analysis tasks
[8], such as landmark detection [9], plane finding [10], and for surgical
planning such as hysterectomy [11] and orthopaedic operations [12]. It has
also been used for needle path planning in minimally-invasive robotic surgery
[13], [14]. It is its ability to learn intelligent policies for sequential
decision making that provides a potential solution to problems without
requiring direct supervision for each action taken, a common constraint in
developing machine learning-assisted methods for complex and skill-demanding
surgical and intervention applications. This makes RL suitable for finding an
optimal targeting strategy, which requires complex decisions for which there
is no established best method.

In this study, we investigate the feasibility of using RL to plan patient-
specific needle sampling strategies, optimised in pre-operative MR-derived RL
environments. We present experimental results based on clinical data from
prostate cancer patients and compare the proposed method, using a set of
clinically important metrics, to baseline strategies that are designed by
human intuition and an interactive targeting performed by two observers. We
conclude by reporting a set of interesting observations that demonstrate the
benefit of using the proposed RL-learned patient-specific strategies. These
indeed adapted effectively to individual procedures and varying targets, for
improved final performance of the sequential target sampling. Consistent hit
rates were achieved with less variance for both smaller and larger lesions as
a result of the learned adaptive strategies.

##  2 Method

The agent-environment interactions are modelled as a Markov decision process
(MDP), and summarised as a 4-tuple
$\langle\mathcal{S},\mathcal{A},r,p\rangle$, where $\mathcal{S}$ and
$\mathcal{A}$ are the state and action spaces consisting of all possible
observed states as input and actions as output for the agent, respectively.
$r:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$ is the reward function
which maps state-action pairs to a real value. The state transition
distribution is defined by
$p:\mathcal{S}\times\mathcal{S}\times\mathcal{A}\rightarrow[0,1]$ which
denotes probability of transitioning to the next state, given the current
state-action pair. In this section, we develop an environment for template-
guided biopsy sampling of the cancer targets, the MDP components and the
policy learning strategy.

###  2.1 Patient-specific prostate MR-derived biopsy environment

(a) Environment

(b) Template grid

(c) States and targets

Figure 1: Simulated biopsy procedure environment. (a) Placement of ultrasound
probe and template grid within the MRI volume. (b) Visualisation of ultrasound
probe rotation which is always aligned with the chosen template grid position.
(c) Examples of (top) states and (bottom) overlaid MR-identified targets

The environment is illustrated in Figure 1(a) for the targeted biopsy
procedures, where virtual biopsy needles are inserted through the perineum via
a brachytherapy template grid consisting of 13x13 holes that are 5mm apart.
Other needle-based treatments such as cryotherapy, brachytherapy and
radiofrequency ablation [15] may also be applicable but are not discussed
further in this paper. The position of the transrectal ultrasound probe is
approximated within the rectum directly underneath the prostate gland, with a
fixed distance to the template grid such that the top of the probe is aligned
with the lower side of the template, as shown in the Figure 1(a). Both
anatomical and pathological structures can be sampled, at any sagittal
ultrasound imaging plane given an arbitrary angle, as illustrated in Figure
1(b).

We summarised a number of considerations in designing and constructing the
adopted biopsy environment as follows. 1) The prostate gland from each MR
volume, the MR-identified targets (as in Figure 1(c)), and key landmarks such
as position of the rectum are all manually segmented from individual patients
to construct the biopsy environment. Automated methods for segmenting these
regions of interest have been proposed, e.g. [16], [17]. 2) Binary
segmentation are provided as observations for the RL agents, as opposed to
ultrasound image intensities, which are neither available during planning nor
straightforward to synthesize from MR images. We argue that the use of binary
representation would be more robust to train the RL agents and the resulting
methods are more likely to generalise to different procedures and planning MR
images, especially given the existing MR and ultrasound segmentation and
registration algorithms described above. 3) Uncertainty in MR-to-ultrasound
registration can and should be added to the segmented regions, together with
other potential errors in localising these regions during observation such as
observer variability in manual segmentation used in this work. We would like
to point out that, however, these localisation errors are unlikely to be
independent to each other and the dependency of RL model generalisability on
how precisely these need to be modelled remain open research questions. Sec. 3
discusses further details adopted in our experiments. 4) In the presented
experiments, we focus on targeting the index lesions, those are of largest
volumes in each case, to provide first results that show the efficacy of
modelling the dynamic biopsy sampling process. However, the described MDP
should be directly applicable for and likely to be more effective in cases
with multiple lesions.

###  2.2 The MDP components

#### State

\- At a given time point $t$ during the procedure, the agent receives
information about its current state $s_{t}\in\mathcal{S}$ : the chosen grid
point and the re-sampled 2D image plane obtained by rotating the probe to the
current template grid position, as in Figure 1(c). This current position is
determined by the previous action. This is to test the scenario with least
assumptions, where the overall 3D anatomical and pathological information may
be corrupted or unreliable due to intra-procedural uncertainties from patient
movement and outdated registration.

#### Actions

\- The agent takes actions $a_{t}\in\mathcal{A}$ which modify its position on
the template grid. These actions are relative to the current position of the
agent $(i,j)$ and are defined as $a_{t}=(\delta_{i},\delta_{j})$ such that the
new position is given by $(i+\delta_{i},j+\delta_{j})$, where
$\delta_{i},\delta_{j}\in[-15,15]$. By formulating this relative grid-moving
action, we consider the biopsy needle is positioned on the image plane, with
an insertion depth that overlaps the needle centre and the centre of the
observed 2D target, subject to small predefined positioning errors in each
direction. These are commonly adopted practice though not strictly enforced,
and we found that increasing the flexibility by independently positioning the
ultrasound probe and needle may unnecessarily make the training difficult to
converge.

#### Reward

\- The reward at the time $t$ is computed based on the reward function
$R_{t}=r(s_{t},a_{t})$, during training. The agent is rewarded positively if
the fired needle obtains samples of the lesion. A penalty is given when chosen
grid positions are outside of the prostate, to avoid hitting surrounding
critical structures. From initial experiments, it was found that a greater
reward of +5 lead to a faster convergence during training, encouraging the
agent to hit the lesions, whilst a penalty of -1 was enough to deter the agent
from firing outside the prostate gland. Reward shaping is also introduced to
guide the agent towards the lesion, thereby speeding up the learning process.
Similar to [9] and [10], a sign function $\operatorname{Sgn}$ of the
difference between $dist_{t-1}$ and $dist_{t}$ is computed, where $dist_{t}$
represents Euclidean distance between target centre and needle trajectory at
time $t$.

$r=\begin{cases}$+5$&\textit{if biopsy needle intersected with target}\\\
$-1$&\textit{if biopsy needle placed outside prostate}\\\
\operatorname{Sgn}(dist_{t-1}-dist_{t})&\textit{otherwise}\end{cases}$ (1)

###  2.3 Policy learning

The navigation and sampling strategy is parameterised by a policy neural
network $\pi_{\theta}$, with parameters $\theta$, quantifying the probability
of performing action $a_{t}$ given state $s_{t}$. Agent’s actions then can be
sampled from the policy, $a_{t}\sim\pi_{\theta}(\cdot|s_{t})$. During the
policy training, the accumulated reward
$Q^{\pi_{\theta}}(s_{t},a_{t})=\sum_{k=0}^{T}\gamma^{k}R_{t+k}$ is maximised,
where $\gamma$ is a discount factor set to 0.9, to obtain the optimal policy
$\pi_{\theta^{*}}$, $\theta^{*}=\text{arg
max}_{\theta}\mathbb{E}_{\pi_{\theta}}[Q^{\pi_{\theta}}(s_{t},a_{t})]$. With
continuous actions, policy gradient (PG) and actor-critic (AC) algorithms can
thus be adopted for the optimisation.

##  3 Experiments

#### Data set

\- The T2-weighted MR images and their segmentation were acquired from 54
prostate cancer patients. These were obtained as part of clinical trials,
PROMIS [1] and SmartTarget [7], where patients underwent ultrasound-guided
minimally invasive needle biopsies and focal therapy procedures.

#### RL algorithm implementation

\- An agent was trained for each patient individually using the Stable
Baselines implementation of PPO [18]. Each agent was trained for 120,000
episodes and a model was selected with the highest average reward after 10
episodes. Each episode was limited to a maximum of 15 time steps, but can
terminate early if any 5 needles hit the lesion. At each episode the agent is
initialised at random starting positions on the template grid. Based on
estimated registration error reported previously [3], random localisation
error was added in the observed states, equivalent to a Gaussian noise with a
standard deviation of 1.73mm in each of the x, y and z coordinates, or a mean
distance error of 3mm. The PPO algorithm [19] was used in reported results, as
it guarantees a monotonic reward improvement and stability of training.
However, we also report a lack of substantial difference in performance to
other tested algorithms, including DDPG [20] and SAC [21]. The policy network
was based on ResNet18 [22] architecture, with an additional fully-connected
layer for a linear output. An Adam optimiser was used, with a learning rate of
0.0001. It could be of future interest to compare further network
architectures and PG/AC algorithms on the proposed RL problem, but is
considered beyond the scope of current work.

#### Biopsy performance metrics

\- To quantitatively assess target sampling performance, three biopsy-specific
metrics are used in this study: hit rate (HR), cancer core length (CCL) and
needle area (NA). The HR is the number of needle samples that contains the
target, i.e. positive samples, divided by the number of needles fired. Five
needles are chosen to represent the maximum number typically used in targeted
biopsy [23]. CCL is the total length (in mm) overlapping the target, i.e. the
sampled target tissue, in individual needles. CCL $>=6mm$ often indicates
clinical significance [24]. NA estimates the coverage of all fired needles in
each episode, defined as the area of an approximating ellipse,
$NA=\pi*std_{x}\times std_{y}$, where $std_{x}$ and $std_{y}$ are standard
deviations in the needle navigating x-y plane, defined by the template grid
position.

#### Baseline strategies

\- Two strategies were compared with the proposed agent, designed to provide
an estimate of what clinicians are likely to achieve in practice. For a fair
comparison, the same observed targets, the states, and starting positions were
used. Student’s t-tests were used when comparison is made at a significance
level of $\alpha=0.05$, unless otherwise indicated. The first strategy
(Sweeping strategy) adopted a simple sweeping of the biopsy needle together
with the ultrasound probe, from left to right in a 5mm interval. The target
was sampled at the centre of the observed target, i.e. fired, once an image
plane is encountered with a lesion. The second strategy (Scouting strategy)
moves the virtual probe to scout all candidate positions that samples the
target, before 5 random ones were selected as fired positions among these
candidates. Inter- and intra-operator variance is one of the most important
factors that impact the performance of a sampling strategy. For the baseline
strategies, additional Gaussian noise was added to the chosen needle/probe
positioning with a varying bias and a varying standard deviation (SD).
Increasing the SD leads to higher uncertainty in placing the needles, while
the bias indicates a targeting strategy that does not aim for the target
centre, e.g. for avoiding empty cores or urethra. The experiments were
repeated using different combinations of the two variables (each ranging
between 0 to 10mm), to test different strategies.

#### Interactive experiments by two observers

\- Two human operators, one computer scientist and one biomedical imaging
researcher, interacted with a custom-made interface that displays the current
template grid position and image plane observed. They were asked to choose
where to sample and how far to spread the needles. This is a simple
interactive experiment to provide preliminary results that can be compared
with the other described strategies.

##  4 Results

#### Learned strategy performance

\- From Table 1, the agent outperforms both baseline strategies in HR and CCL
(both p-values$<$0.005), but not in NA, with noticeably smaller variability in
HR. Different levels of bias did not lead to significantly different targeting
results, while higher SD increased the spread of needles, but reduced CCL and
HR. In general, the results between the sweeping and scouting strategies were
not found statistically different in CCL and NA. The scouting strategy
resulted in an increased HR (p-value$<$0.05) compared to sweeping, but does
not outperform the agent which still obtains the highest HR.

Table 1: Summary of biopsy performance from the RL agent (top row) and the sweeping and scouting strategies for different bias and SD combinations | Baseline 1 (Sweeping) | Baseline 2 (Scouting)
---|---|---
Bias | SD | CCL(mm) | HR(%) | NA($mm^{2}$) | CCL(mm) | HR(%) | NA($mm^{2}$)
Agent | 11.13±3.43 | 93.40 ± 11.44 | 22.14±18.18 | 11.13±3.43 | 93.40 ± 11.44 | 22.14±18.18
0 | 0 | 7.95±3.00 | 53.36±35.59 | 23.67±16.57 | 8.40±2.65 | 61.10±30.01 | 31.31±28.53
0 | 5 | 7.45±3.19 | 49.43±34.10 | 56.81±50.25 | 8.32±3.04 | 55.19±30.46 | 39.84±20.48
0 | 10 | 4.89±3.29 | 30.94±28.71 | 108.00±86.24 | 5.67±4.32 | 43.70±31.99 | 114.35±71.20
5 | 0 | 8.90±4.00 | 53.69±28.44 | 36.53±28.91 | 8.32±3.00 | 65.19±30.04 | 36.53±28.91
5 | 5 | 7.88±3.52 | 51.32±32.94 | 60.44±47.62 | 7.28±3.17 | 54.48±31.04 | 72.08±40.89
5 | 10 | 6.06±3.95 | 41.51±31.58 | 86.15±63.15 | 5.74±4.41 | 44.82±31.02 | 113.88±79.60
10 | 0 | 7.29±3.08 | 47.80±34.16 | 27.47±18.38 | 7.65±3.05 | 54.81±31.13 | 27.59±22.95
10 | 5 | 7.14±3.38 | 42.44±28.00 | 52.78±48.02 | 6.40±3.64 | 50.74±29.00 | 55.45±42.36
10 | 10 | 4.74±3.46 | 28.70±27.22 | 106.17±115.95 | 5.98±3.42 | 47.04±31.22 | 112.68±95.85
Table 2: Obtained biopsy strategy metrics for the agent and two human observers Observers | CCL (mm) | HR(%) | NA ($mm^{2}$)
---|---|---|---
Agent | 11.13 ± 3.43 | 93.40 ± 11.44 | 22.14 ± 18.18
Observer 1 | 9.71 ± 3.78 | 66.30 ± 20.55 | 42.85 ± 23.36
Observer 2 | 9.83 ± 3.89 | 76.30 ± 19.50 | 64.90 ± 38.84

#### Interactive experiments by two observers

\- The agent outperforms both observers in CCL (p-values=0.020 & 0.040), but
its NA values are more than double that of the agents, suggesting a potential
trade-off between sampling coverage and precision. For HR, the agent
outperforms both observers (p-value $<$ 0.001), which demonstrates that the
agent could achieve an overall comparable performance as human observers, with
a significantly higher CCL.

(a) Biopsy metrics CCL, HR and NA vs lesion size

(b) Small lesion size 0.2cc

(c) Large lesion size 0.4cc

Figure 2: (a): CCL, HR and NA as a function of lesion size. (b) and (c):
Examples of different sized targets (red), corresponding to the learned
policies, represented by the needle sampling positions (red sticks, brighter
indicates later time steps) and observed ultrasound images in green. The
bounding cube and the cylinder represent the MR prostate volume and probe,
respectively.

#### Learned strategy for varying target sizes

\- From Figure 2(a) and Table 3, we observe an interesting behaviour from the
learned agent: the smaller the lesions, the larger the spread of the needles.
At a volume threshold of 0.4 cc, the mean CCL and NA are statistically
different for smaller and larger lesions (p-values=0.002 & 0.040), whilst the
difference in HR is not (p-value=0.166). This result may seem counter-
intuitive, as one would be cautious in spreading needles for a small target.
However, the agent learned to distribute needles more widely for smaller
lesions, attempting to maintain the hit rate, given the inevitable presence of
target localisation uncertainty described in Sec. 2.1. Visual examples of the
learned strategies are shown in Figures 2(b) and 2(c). This learned behaviour
is interesting because a) it has not been observed previously, either in
literature or in clinical practice. b) it improved the overall targeting
performance compared to the target-size-agnostic baseline strategies and c)
this could be suggested to urologists and interventional radiologists with or
without the proposed RL assistance.

Table 3: CCL, HR and NA for different lesion sizes using threshold size $<$ 0.4cc Lesion size | CCL $(mm)$ | HR $(\%)$ | NA $(mm^{2})$
---|---|---|---
Small lesions | 10.26 ± 4.19 | 93.16 ± 16.03 | 25.26 ± 19.45
Large lesions | 14.52 ± 1.18 | 100.00 ± 0.00 | 13.02 ± 6.25

##  5 Discussion and Conclusion

The results show that the developed RL agents are competitive in sampling MR-
derived targets, compared with intuitively devised strategies. Higher HR and
average CCL were obtained by the agents, which was achieved by reducing the
spread of the needles compared to baseline strategies. Furthermore, the
learned strategies adapted to patient-specific procedures and varying
pathology. The agents learned to achieve similar HR for different sized
lesions, by spreading the fired needles more for smaller lesions. Such
behaviour has not been observed before, and could be suggested to clinicians
for improved targeting performance. Assumptions, such as number of allowed
needles, template positioning and uncertainties in localisation/placement,
have been made to facilitate the proposed pre-procedural planning. Some of
them may be relaxed for an intra-procedural guidance tool - as a potential
extension of this work, the others may require further validation. More
importantly, the improved targeting performance provides means in mitigating
the cancer under-sampling and help timely diagnosis of a significant number of
prostate cancer patients with current MR-targeted biopsy.

## Acknowledgement

This work is supported by the EPSRC-funded UCL Centre for Doctoral Training in
Intelligent, Integrated Imaging in Healthcare (i4health) [EP/S021930/1], EPSRC
[EP/T029404/1], and the Department of Health’s NIHR funded Biomedical Research
Centre at University College Hospital. This work is also supported by the
International Alliance for Cancer Early Detection, an alliance between Cancer
Research UK [C28070/A30912; C73666/A31378], Canary Center at Stanford
University, the University of Cambridge, OHSU Knight Cancer Institute,
University College London and the University of Manchester. This work was also
supported by the Wellcome/EPSRC Centre for Interventional and Surgical
Sciences [203145Z/16/Z].

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