# Local invariants identify topology in metals and gapless systems

Alexander Cerjan<EMAIL_ADDRESS>Center for Integrated Nanotechnologies,
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA  Terry A.
Loring<EMAIL_ADDRESS>Department of Mathematics and Statistics,
University of New Mexico, Albuquerque, New Mexico 87131, USA

###### Abstract

Although topological band theory has been used to discover and classify a wide
array of novel topological phases in insulating and semi-metal systems, it is
not well-suited to identifying topological phenomena in metallic or gapless
systems. Here, we develop a theory of topological metals based on the system’s
spectral localizer and associated Clifford pseudospectrum, which can both
determine whether a system exhibits boundary-localized states despite the
presence of degenerate bulk bands and provide a measure of these states’
topological protection even in the absence of a bulk band gap. We demonstrate
the generality of this method across symmetry classes in two lattice systems,
a Chern metal and a higher-order topological metal, and prove the topology of
these systems is robust to relatively strong perturbations. The ability to
define invariants for metallic and gapless systems allows for the possibility
of finding topological phenomena in a broad range of natural, photonic, and
other artificial materials that could not be previously explored.

##  I Introduction

Topological band theory has enabled enormous progress in the discovery and
classification of novel states of matter. The preponderance of these
developments have been predicted and realized in insulators [1, 2, 3, 4, 5, 6,
7, 8, 9, 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] and semi-metals [35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50], where these systems’ topological features
are easily identified due to their isolation in energy and wavevector
$(E,\mathbf{k})$-space. However, in metals and other materials lacking a bulk
band gap, any states of topological origin are degenerate with bulk states,
generally resulting in hybridization between the two sets of states. This
hybridization makes it difficult to say whether a particular set of states
remains localized to the system’s boundaries, or retains any of the other
topological properties that they would possess in an insulating system.
Moreover, even if boundary-localized states could be identified, the absence
of a bulk band gap means that traditional topological band theories would be
unable to predict whether these states would be robust to perturbations, or
quantify the strength of that protection. Although detailed studies in
particular metallic and gapless systems have demonstrated the existence of
some topological behaviors [51, 52, 53, 54, 55, 56, 57, 58, 59], a general
theory for predicting topological phenomena in any metallic or gapless system
has remained elusive.

Theories of topological materials predicated upon diagonalizing a system’s
Hamiltonian to determine its topology possess an inherent challenge when
considering metals or other gapless materials. In general, the Hamiltonians,
$H$, of topologically non-trivial systems do not commute with position
operators, $X$, i.e., $[H,X]\neq 0$. Thus, it is impossible to find
eigenstates of both operators simultaneously. In insulators and semi-metals,
where any energy eigenstates of topological origin can be spectrally isolated,
position expectation values provide a measure of the state’s location and
localization. However, in metals or gapless systems, any potential topological
energy eigenstate is a member of a large degenerate subspace consisting
primarily of bulk states, which renders position expectation values
meaningless without some other discriminant between possible choices of basis
within this subspace. This argument suggests that a theory of topological
metals should be pursued using real-space definitions of topology that do not
require diagonalizing the Hamiltonian [60, 61, 62, 63, 64, 65, 66]. Such real-
space topological theories have recently been used to identify distinct phases
in aperiodic systems, such as quasicrystals [67, 68, 69], amorphous structures
[70, 71, 72, 64, 73, 74, 75, 76], and fractal lattices [77].

Here, we develop a general theory of topological metals and other gapless
materials defined using local invariants derived from the system’s “spectral
localizer” [78]. This theory has three inherent features that allow it to
directly solve the difficulties facing any topological theory of gapless
systems: First, as the spectral localizer treats the system’s Hamiltonian on
equal footing with its position operators, it is able to simultaneously
identify the approximate energy and position of the system’s states. Second,
in the absence of a state, the spectral localizer returns a measure of the
strength of the perturbation required to move a state to that position and
energy — in particular, one can calculate the Clifford pseudospectrum (a set
defined by the spectral localizer) in the immediate vicinity of a boundary-
localized state to determine the strength of its protection against disorder.
Third, the spectral localizer is mathematically proven to be connected to the
system’s $K$-theory and thus can be used to define local invariants that
classify the system’s topological phase at a given energy and position [78,
79, 80]. To demonstrate the generality of this method, we explicitly determine
the topological character and quantify the strength of its protection in two
disparate models, a Chern metal and a higher-order topological metal. As part
of this study, we also provide a definition of a local, real-space invariant
for higher-order topological phases. Altogether, the theory we present here
provides the groundwork for classifying the topology of metals and other
gapless systems of any dimension and in any symmetry class [78, 81] across a
broad range of physical platforms.

The remainder of this paper is organized as follows. First, in Sec. II we
provide a brief, physically motivated review of the spectral localizer and
Clifford pseudospectrum, and discuss how they can be used to determine a
system’s topology and the strength of its topological protection. In Sec. III
we provide a demonstration of how the spectral localizer can be used to
identify both the topology and associated boundary-localized states of a
metallic Chern lattice. In Sec. IV we demonstrate the generality of this
method by identifying the topology of a higher-order topological metal, and
derive a local invariant for classifying such systems. Finally, in Sec. V we
offer some concluding remarks.

##  II Review of the spectral localizer

From a broad perspective, the spectral localizer takes a view of a material’s
topology that is similar to that of topological quantum chemistry [82, 83,
84]: a material is topologically non-trivial if it cannot be continued to an
atomic limit without either closing a gap or breaking a symmetry. But, whereas
topological quantum chemistry determines whether given material can be
continued to an atomic limit by analyzing its band representations, the
spectral localizer seeks to make the same determination by instead using the
material’s real-space description. This real-space picture of topology is
predicated on:

Definition 1: A material is in an atomic limit if and only if its Hamiltonian,
$H$, commutes with all of its position operators, $X_{j}$,
$[H,X_{j}]=0\;\;\forall j$.

Using this definition, the question of whether a material is topologically
non-trivial becomes synonymous with whether there is an obstruction to
continuing a material’s Hamiltonian and position operators to be commuting
without closing a gap or breaking a symmetry, enabling one to leverage
developments from the study of $C^{*}$-algebras to make this determination
[78, 85] (in particular, see Fig. 1.1 from Ref. [85]). We note that this
definition of the atomic limit is consistent with previous statements about
the real-space behavior of this limit [86].

Over the last decade, the spectral localizer has emerged as a versatile tool
for identifying whether given set of matrices can be continued to commuting
matrices [78, 79, 80]. For a physical material in $d$ dimensions, the spectral
localizer is

$L_{\bm{\lambda}=(x_{1},\cdots,x_{d},E)}(X_{1},\cdots,X_{d},H)=\\\
\sum_{j=1}^{d}\kappa(X_{j}-x_{j}I)\otimes\Gamma_{j}+(H-EI)\otimes\Gamma_{d+1},$
(1)

where $I$ is the identity matrix and the matrices $\Gamma_{j}$ form a non-
trivial Clifford representation, $\Gamma_{j}^{\dagger}=\Gamma_{j}$,
$\Gamma_{j}^{2}=I$, and $\Gamma_{j}\Gamma_{l}=-\Gamma_{l}\Gamma_{j}$ for
$j\neq l$. Here, $\kappa>0$ is a scaling coefficient that ensures $X_{j}$ and
$H$ have compatible units, and
$\bm{\lambda}=(x_{1},\cdots,x_{d},E)\in\mathbb{R}^{d+1}$ is a choice of
position and energy where the spectral localizer is evaluated. There are no
restrictions on the choices of $\mathbf{x}=(x_{1},\cdots,x_{d})$ and $E$ in
$\bm{\lambda}$, these quantities can be chosen to be anywhere inside or
outside of the material’s spatial and spectral extent. Also, note that the
underlying theorems that prove the utility of the spectral localizer currently
assume that the system’s operators $X_{1},\cdots,X_{d},H$ are Hermitian,
operate on a finite-dimensional Hilbert space, and represent a system with
open boundaries.

Intuitively, the spectral localizer can be viewed as a composite of the
eigenvalue equations (which have the form $(M-\lambda)\mathbf{v}=0$) of
multiple not-necessarily-commuting operators using a Clifford representation.
However, unlike in typical eigenvalue problems where the eigenvalues,
$\lambda$, are quantities that are solved for, the spectral localizer takes
$\bm{\lambda}$ as an input and determines whether the system possesses a state
with approximate energy $E$ that is approximately at $\mathbf{x}$. If the
spectral localizer possesses an eigenvalue that is sufficiently close to zero,

$\min(|\sigma(L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H))|)\leq\sum_{j=1}^{d}\|[H,\kappa
X_{j}]\|$ (2)

where $\sigma(L_{\bm{\lambda}})$ denotes the spectrum of $L_{\bm{\lambda}}$,
$\|\cdot\|$ is the $L_{2}$ matrix norm, and it is assumed here that
$[X_{i},X_{j}]=0$, then the physical system supports a state in the vicinity
of $(\mathbf{x},E)$. If $L_{\bm{\lambda}}$ does not possess such an
eigenvalue, the system exhibits a local gap at $(\mathbf{x},E)$, i.e., a
region in position-energy space that cannot support a state (see [87, §II] for
what are currently the best known estimates on how the spectral localizer
predicts state localization). Thus, it is convenient to define the “localizer
gap” as $\min(|\sigma(L_{\bm{\lambda}})|)$, which, heuristically, can be
viewed as a spatially-resolved band gap.

The ability for the spectral localizer to calculate a quantity similar to a
band gap without determining the system’s band structure plays a crucial role
in using $L_{\bm{\lambda}}$ to categorize a system’s topology. Formally, a set
of matrices $\\{M_{j}^{(0)}\\}$ can be continued to some other set
$\\{M_{j}^{(1)}\\}$ if a continuous path of matrices can be defined between
the two sets, $\\{M_{j}^{(\tau)}\\}$ for $0\leq\tau\leq 1$. Assessing a
material’s topology via continuation is typically done for infinite systems,
and requires that every set along the path
$\\{X_{1}^{(\tau)},\cdots,X_{d}^{(\tau)},H^{(\tau)}\\}$ must both preserve the
system’s symmetries and maintain the bulk gap (i.e., the system’s band gap if
it is periodic and infinite). This process also assumes some locality criteria
on $H^{(\tau)}$ and $X_{j}^{(\tau)}$, such that two sites that are
sufficiently far apart cannot be coupled [86]. Instead, if $H^{(\tau)}$ and
$X_{j}^{(\tau)}$ represent a finite system (as is necessary to use the
spectral localizer), the concept of a bulk gap in the continuation process
needs to be replaced in someway. One option is to impose periodic boundary
conditions and insist that $H^{(\tau)}$ stays gapped, c.f. [86]. In the case
of open boundary conditions, a concept of a local gap is necessary, as the
system may possess boundary-localized states (of trivial or topological
origin) that would otherwise obscure the spectrum of a system with an
insulating interior.

In particular, the criteria of preserving a local gap in
$\\{X_{1}^{(\tau)},\cdots,X_{d}^{(\tau)},H^{(\tau)}\\}$ can be guaranteed by
monitoring its localizer gap along this path—a system’s topology at
$\bm{\lambda}$ cannot change so long as the localizer gap at that
$\bm{\lambda}$ remains open. Specifically, it has been proven that a symmetry
preserving perturbation to the Hamiltonian, $\delta H$, is unable to close the
localizer gap at $\bm{\lambda}$ so long as $\|\delta
H\|<\min(|\sigma(L_{\bm{\lambda}})|)$, [78, Lemma 7.2], and a similar
statement can be made about perturbations to the position operators, $\delta
X_{j}$. Moreover, this measure of the strength of the topological protection
inherently includes the possibility of correlated disorder that is “designed”
to defeat the system’s topology. As such, in most systems, the localizer gap
will underestimate the strength of the system’s topological robustness for
uncorrelated disorder. There has also been recent progress in understanding
the effects of perturbations that only approximately respect the system’s
symmetry class [88].

With these definitions in place, we can present a complete picture of how the
spectral localizer determines a material’s topology. Overall, there is a
constellation of theorems that dictate how the spectral localizer can be used
to assess whether a given set of matrices can be continued to commuting
matrices for systems of any physical dimension and in any symmetry class [78,
80, 79]. In general, there will be some property of $L_{\bm{\lambda}}$ that
identifies whether such a continuation is possible at a given $\bm{\lambda}$
while preserving the system’s symmetries and without closing the localizer gap
at that $\bm{\lambda}$; this same property also defines a local topological
invariant. If there is an obstruction to finding a continuation to commuting
operators, the non-trivial topology at that $\bm{\lambda}$ is protected
against perturbations that do not close the localizer gap. Moreover, as the
localizer gap is a continuous (but not smooth) function of $\bm{\lambda}$,
neighboring choices of $\bm{\lambda}$ in position-energy space possess
similarly sized localizer gaps (and thus, topology). Finally, bulk-boundary
correspondence is built right into this picture, the localizer gap associated
with a topologically non-trivial region of position-energy space must close
around the perimeter of the system where the topological boundary-localized
states have strong support, as far away from the finite material the spectral
localizer must exhibit trivial topology.

To provide an example for a specific class of topology, consider 2D systems
with broken time-reversal symmetry that may possess a non-zero Chern number.
For this symmetry class, the system’s operators are $H$, $X$, and $Y$, and the
system possesses non-trivial topology for some region in position-energy space
if $H-EI$, $X-xI$, and $Y-yI$ cannot be continued to commuting while
preserving their Hermiticity. In this case, the property of $L_{\bm{\lambda}}$
that identifies the possible obstruction is its signature,
$\textrm{sig}(L_{\bm{\lambda}})$, which is the number of its positive
eigenvalues minus the number of its negative eigenvalues. In particular:

  1. 1)

If $H$, $X$, and $Y$ commute, then $\textrm{sig}(L_{\bm{\lambda}}(X,Y,H)=0$
for any choice of $\bm{\lambda}$ [89, Lemma 4].

  2. 2)

The signature of $L_{\bm{\lambda}}(X,Y,H)$ cannot change through continuation
of $H-EI$, $X-xI$, and $Y-yI$ without closing the localizer gap at
$\bm{\lambda}$ [90].

Thus, even if $H$, $X$, and $Y$ do not commute, if
$\textrm{sig}(L_{\bm{\lambda}}(X,Y,H)=0$ for every choice of $\bm{\lambda}$,
then the system can be continued to an atomic limit without closing the band
gap / localizer gap and thus the system (assuming it is sufficiently large) is
topologically trivial everywhere [91]. Conversely, if there is a choice of
$\bm{\lambda}$ for which $\textrm{sig}(L_{\bm{\lambda}}(X,Y,H)\neq 0$, then
the system possesses non-trivial topology within the localizer gap surrounding
this choice of $\bm{\lambda}$.

In comparison with traditional theories of topology, the local nature of the
topology predicted by the spectral localizer can seem unusual, but this is
simply the language required to describe widely appreciated properties of
topological systems within a real-space picture. Consider a two band
topological insulator. Filling only the lower band results in a system that is
not Wannierizable (and cannot be continued to the atomic limit), but if both
bands are filled the system becomes Wannierizable again. For the spectral
localizer, the first case is represented by choosing $\bm{\lambda}$ for $E$
within the band gap (and $\mathbf{x}$ in the system’s bulk) and finding non-
trivial local topology, while the second case corresponds to choosing
$\bm{\lambda}$ with an $E$ greater than the maximum energy of the upper band
and finding trivial local topology. In the latter case ($E$ outside both
bands), the process of continuing the system’s operators to commuting will
maintain the localizer gap at that $E$, but will close the localizer gap for
energies between the bands. Finally, we note that the process of continuing a
system’s operators to commuting will generally involve changes to both $H$ and
$X_{j}$.

###  II.1 The Clifford pseudospectrum

For completeness and to aid a reader in understanding previous works on the
spectral localizer, we provide a brief discussion of how the spectral
localizer can be used to calculate a system’s Clifford pseudospectrum.
Mathematically, the localizer gap defines the system’s Clifford
$\epsilon$-pseudospectrum,

$\Lambda_{\epsilon}(X_{1},\cdots,X_{d},H)=\\\
\\{\bm{\lambda}\;|\;\min(|\sigma(L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H))|)\leq\epsilon\\}.$
(3)

A system’s Clifford spectrum is given by $\Lambda_{0}(X_{1},\cdots,X_{d},H)$,
i.e., the set of $\bm{\lambda}$ for which the localizer gap vanishes. Thus, a
system’s Clifford pseudospectrum is a useful tool for finding surfaces in
position-energy space with constant localizer gap.

Intuitively, a system’s Clifford pseudospectrum (regardless of the system’s
topology) can be viewed as a method for constructing an approximate joint
spectrum of non-commuting operators. In other words, if
$\min(|\sigma(L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H))|)$ is small (relative to
the norms of the commutators), the system exhibits an approximate eigenstate
that almost diagonalizes all of the operators simultaneously with approximate
eigenvalues given by $\bm{\lambda}$ [87, 78]. But, note that even if
$\bm{\lambda}$ is a member of the system’s Clifford spectrum (i.e.,
$\min(|\sigma(L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H))|)=0$), that does not
imply that there is an exact eigenstate that exactly diagonalizes all of the
constituent operators. Finally, the Clifford pseudospectra is not the only
tool that can be used to understand the approximate joint spectra of non-
commuting matrices, which can be tackled using both traditional two-operator
pseudospectra [92, 93, 94] (in the present context, these could only be used
for 1D lattices), or other constructions of multi-operator pseudospectra [87]
(i.e., different ways of combining eigenvalue equations that need not use a
Clifford representation). However, as the Clifford pseudospectra is computed
using $L_{\bm{\lambda}}$, it is the only (currently known) tool for finding
approximate joint spectra that is also related to the system’s topology.

###  II.2 A note on numerical computation

We conclude this brief review of the spectral localizer with a few comments
about the numerical calculation of its properties. Although the spectral
localizer is agnostic to the choice of basis used for its constituent
operators, if a material’s Hamiltonian is written in a tight-binding basis,
the position operators, $X_{j}$, are simply diagonal matrices that index the
$j$th coordinate of each of the lattice sites (different orbitals at the same
site have the same position). Thus, in the tight-binding basis,
$L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H)$, is a (usually large) sparse
Hermitian matrix and the localizer gap,
$\min(|\sigma(L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H))|)$, can be efficiently
calculated using standard sparse eigenvalue solvers as only a single
eigenvalue is needed. The properties of the spectral localizer that reveal the
system’s $K$-theory only need to be calculated a few times, and only once per
region in position-energy space where the localizer gap is large, as these
properties cannot change without the localizer gap closing.

Moreover, we note that for at least some of the relevant properties of
$L_{\bm{\lambda}}$, there are significant numerical speedups available. For
example, to find a matrix’s signature, it is not necessary to find its full
set of eigenvalues. Instead, as $L_{\bm{\lambda}}(X_{1},\cdots,X_{d},H)$ is
Hermitian, one can make use of Sylvester’s law of inertia [95, 96], which
states that

$\textrm{sig}(L_{\bm{\lambda}})=\textrm{sig}(D),$ (4)

where $L_{\bm{\lambda}}=PDP^{\dagger}$ is the LDLT decomposition of the
spectral localizer. Thus, as $D$ is diagonal (or block diagonal in some
numerical implementations with 1-by-1 and 2-by-2 blocks), the computational
cost of finding $\textrm{sig}(L_{\bm{\lambda}})$ is entirely dictated by the
speed of the sparse LDLT decomposition algorithm, which is, in general, more
efficient than finding the full spectrum of $L_{\bm{\lambda}}$.

##  III Topological Chern metal

To illuminate how the spectral localizer can be used to classify topological
metals, we first consider a 2D Chern insulator with an added intervening band
that is degenerate with the Chern insulator’s chiral edge states. A minimal
tight-binding model for this system can be constructed from a Haldane lattice
coupled to a single-band triangular lattice whose vertices are located in the
center of each honeycomb [51, 55], and whose Hamiltonian is schematically
illustrated in Fig. 1a. The Haldane lattice is parameterized by the nearest
neighbor coupling, $t_{1}$, next-nearest neighbor couplings with amplitude and
phase $t_{C}$ and $\phi$, and the on-site sublattice energy difference $2m$.
The triangular lattice has nearest neighbor coupling $t_{2}$ and on-site
energy $m_{2}$. The coupling strength between the two lattices is $t_{3}$. In
the absence of coupling between the two lattices, $t_{3}=0$, the Haldane model
exhibits topological and trivial insulating phases separated by semi-metal
phases and protected by the Chern number, $C$, see Fig. 1b, while the
triangular lattice exhibits a single band centered around $E=m_{2}$. When the
coupling between the two lattices is turned on, $|t_{3}|>0$, the intervening
band from the triangular lattice prohibits the unique identification of chiral
edge states within its extent, as the chiral edge states will generally
hybridize with the degenerate states of the interstitial triangular lattice.

Figure 1:  (a) Schematic of the tight-binding model for a Haldane lattice,
green and orange circles, coupled to a trivial lattice, blue triangles. Some
couplings are only shown in a portion of the system for clarity. (b) Haldane
model phase diagram, with the topological (red) and trivial (blue) systems
considered indicated. (c-f) Simulations of a metallized Haldane lattice, with
$m/t_{1}=0$, $t_{C}/t_{1}=0.5$, $\phi=\pi/2$, $m_{2}/t_{1}=-0.35$,
$t_{2}/t_{1}=0.2$ and $t_{3}/t_{1}=0.3$. (c) Ribbon band structure with two
zig-zag edges. Chiral edge states can be identified outside of the intervening
band. $\Delta E$ is the bulk gap between the top and bottom bands. (d) Local
density of states at $E=0$. Each lattice site is represented as a 2D Gaussian
with radial width $r_{0}=0.5a$. (e) 2D localizer gap,
$\textrm{min}(|\sigma(L_{\bm{\lambda}}(X,Y,H))|)/\Delta E$ at
$\bm{\lambda}=(x,y,E=0)$ with $\kappa=1$. Overlay shows the local Chern
number, $C_{\textrm{L}}(x,y,0)=1$ (red) or $=0$ (clear). (f) Localizer
spectrum along the green line in (e). The eigenvalue which yields a change in
topology is highlighted in magenta. (g-j) Same as (c-f), except for a trivial
metal, with $m/t_{C}=4\sqrt{3}$.

Previous numerical studies have shown that this metallized Haldane model
exhibits unusual transport properties that are robust against disorder, even
for choices of the Fermi energy that are within the range of the middle band
across the entire Brillouin zone (e.g., $E=0$ in Fig. 1c) [51, 55]. However,
for choices of the Fermi energy within the middle band, the states responsible
for these transport properties cannot be uniquely identified using
conventional analysis methods, as any boundary-localized states are degenerate
with the bulk states of the triangular lattice. For example, the local density
of states (LDOS) at $E=0$ cannot distinguish between topological (Fig. 1d) and
trivial (Fig. 1h) metals as the contributions from the degenerate bulk in the
system’s LDOS outweigh the contributions of any topological boundary-localized
phenomena. Moreover, the absence of a bulk band gap at $E=0$ not only poses
problems for defining an invariant for this system using band theory, but also
inhibits the use of other real-space definitions of topology such as
topological markers [61] as a system’s ground-state projection operators are
only exponentially localized in gapped systems [97, 98, 99].

Here, we use the spectral localizer to show that boundary-localized resonances
exist in the metallized Haldane model, even for energies within the extent of
the middle band, quantify the topological protection of these boundary-
localized phenomena, and identify a topological invariant that classifies this
behavior. For a 2D system, the spectral localizer can be explicitly written as

$L_{\bm{\lambda}=(x,y,E)}(X,Y,H)=\\\
\left(\begin{array}[]{cc}H-EI&\kappa(X-xI)-i\kappa(Y-yI)\\\
\kappa(X-xI)+i\kappa(Y-yI)&-(H-EI)\end{array}\right).$ (5)

As the spectral localizer directly incorporates information about the system’s
spatial and spectral properties on equal footing, it is able to identify the
approximate presence (or absence) of a state at a given position and energy
regardless of other degenerate states elsewhere in the system. Thus, the
localizer gap, $\textrm{min}(|\sigma(L_{\bm{\lambda}}(X,Y,H))|)$, immediately
reveals the difference between the topological and trivial phases of the
metallized Haldane model. In its topological phase (Fig. 1e), the presence of
the chiral edge states causes the localizer gap to close around the entire
perimeter of the system regardless of whether the boundary-localized states
hybridize with the bulk states of the interstitial triangular lattice. This
behavior is qualitatively distinct from the metallized Haldane model’s trivial
phase (Fig. 1i), where the lack of any boundary-localized states means that
the localizer gap remains open around the system’s boundary. Moreover, the
sizeable localizer gap just inside the boundary closing of the topological
system indicates that these boundary-localized states (or resonances) are
robust against disorder despite the absence of a bulk band gap. Since the
localizer gap is not, in this case, a local manifestation of any sort of a
bulk band gap, it is a mathematical mystery why it occurs, related to the
discovery in [50] of how the spectral localizer can exhibit larger gaps than
the underlying system in semi-metals.

The topological invariant at any $\bm{\lambda}=(x,y,E)$ with non-zero
localizer gap for 2D systems in symmetry class A is given by

$C_{\textrm{L}}(x,y,E)=\tfrac{1}{2}\textrm{sig}\left(L_{(x,y,E)}(X,Y,H)\right)\in\mathbb{Z},$
(6)

Thus, this formulation of the local Chern number is necessarily an integer,
even for finite systems. Calculating this invariant for the metallized Haldane
model in Figs. 1c-f proves it is topological (i.e., $X-xI$, $Y-yI$, $H-EI$
cannot be continued to be commuting for some $\bm{\lambda}$), as it exhibits a
non-trivial local Chern number in its bulk even for energies residing within
the extent of the middle band. Furthermore, this non-trivial bulk topology can
be viewed as forcing the localizer gap to close around the Chern metal’s
entire edge, as the local Chern number must be trivial far away from the
system and, thus, the localizer gap along any path connecting the system’s
interior and exterior must close for one of the localizer’s eigenvalues to
switch signs, see Figs. 1f,j.

As the spectral localizer yields a set of local, real-space definitions for
finding boundary-localized states and determining topological invariants, its
entire mathematical machinery is immediately applicable in the presence of
disorder, without any alteration. Thus, we can show that the topology of the
metallized Haldane lattice is robust against perturbations that do not close
the gap between the system’s first and third bands, $\Delta E$ in Fig. 1c,
i.e., those bands which originate from the insulating Haldane lattice. To
demonstrate this, we add on-site disorder to the system with strength $W$,
such that each vertex has an independent, uniformly distributed random on-site
energy in the range $[-W/2,W/2]$. For $W<\Delta E$, the topological character
of the system remains unchanged, and the entire bulk still possesses a non-
trivial local Chern number (Fig. 2a shows one disorder realization). As the
strength of the disorder is further increased, $W>\Delta E$, the system begins
to revert to a trivial phase (Fig. 2b). Nevertheless, even at this strength of
disorder, regions within the system can still be in a topological phase, and
these topological islands can be identified using the local Chern number, Eq.
(6).

Figure 2: 2D localizer gap,
$\textrm{min}(|\sigma(L_{\bm{\lambda}}(X,Y,H))|)/\Delta E$ at
$\bm{\lambda}=(x,y,E=0)$ with $\kappa=1$, for a metallized Haldane lattice
(same as Fig. 1) with added on-site disorder with strength $W/\Delta E=0.89$
(a) and $W/\Delta E=1.77$ (b). The colored overlay shows the local Chern
number, $\textrm{sig}(L_{\bm{\lambda}}(X,Y,H))/2=1$ (red) or $=0$ (clear).
Here, $\Delta E$ is the gap between the first and third bands of the system,
see Fig. 1c.

##  IV Higher-order topological metal

To demonstrate the generality of using the spectral localizer to identify
topological metals in any symmetry class, we show the existence of robust
higher-order topological metallic phases with in-band corner-localized states
and find the associated local topological invariant. Moreover, one may argue
that in the metallized Haldane model, it is possible to continue the system to
a Haldane insulator with an intervening decoupled flat band from the
interstitial triangular lattice, enabling the topology of the Haldane
insulator to perhaps be inferred from traditional methods. Instead, in this
section, we consider a system where the intervening metallic bands are
intrinsic to the underlying lattice.

Here, we consider a 2D chiral and $C_{4v}$ symmetric lattice with four sites
per unit cell, whose tight-binding model is schematically shown in Fig. 3a,
and in which $v$ and $w$ are the intra- and inter-unit cell couplings,
respectively. When a magnetic flux is uniformly threaded through this system,
it becomes an insulator at $E=0$, and when the system is in its topological
phase, $w>v$, zero-energy corner-localized states appear [28, 29, 30, 31].
Without this flux, the middle two bulk bands of this system are degenerate and
centered at $E=0$, Fig. 3b. Previous studies of this flux-less system with
$w>v$ have shown that so long as $C_{4v}$ (and chiral) symmetry are preserved,
corner-localized states exist that are prohibited from hybridizing with the
degenerate bulk states [57, 58], and are associated with a non-trivial
fractional corner charge invariant [59]. However, these arguments do not hold
in the absence of $C_{4v}$ symmetry, nor do they readily generalize to other
gapless systems suspected of exhibiting higher-order topological behaviors.

Figure 3: (a) Schematic of the tight-binding model for a higher-order
topological metal with intra-unit cell couplings $v$, and inter-unit cell
couplings $w$. (b) Bulk band structure with $w/v=3$. $\Delta E/v=2$ is the
bulk gap between the bottom and middle bands. (c) Local density of states at
$E=0$. Each lattice site is represented as a 2D Gaussian with radial width
$r_{0}=0.5a$. (d) 2D localizer gap,
$\textrm{min}(|\sigma(L_{\bm{\lambda}}(X,Y,H))|)/\Delta E$ at
$\bm{\lambda}=(x,y,E=0)$ with $\kappa=0.1$.

We first use the 2D spectral localizer, Eq. (5), to see that the topological
states must remain approximately localized to the corners until any added
disorder is strong enough to close the gap between the system’s bottom and
middle bands (or, equivalently, the gap between its middle and top bands),
information which is not available in the system’s local density of states,
Figs. 3c,d. However, as the metallic system in Fig. 3a is in symmetry class
AIII, the 2D spectral localizer is not connected to a non-trivial topological
invariant [100, 86, 81]. Nevertheless, strong topological invariants in 1D
protect 0D edge states, and the 0D corner states of higher-order topological
phases are, in the absence of crystalline symmetries, boundary obstructed,
rather than bulk obstructed [29]. Thus, it seems reasonable to try to treat
this system as if it were 1D and borrow the mathematics of the 1D class AIII
invariant [78, §4.1] by projecting the lattice into a lower dimension.

To isolate a corner of the system in the reduced dimension, we use the
diagonal position operator $D=(X+Y)/2$. As a Gedankenexperiment, we are
tilting the system and looking in from a corner, and this choice of diagonal
position successfully isolates two of the corners. By symmetry, similar
behavior will be assured at the remaining two corners. The 1D spectral
localizer can be explicitly written as

$L_{\bm{\lambda}=(d,E)}(D,H)=\\\
\left(\begin{array}[]{cc}0&\kappa(D-dI)-i(H-EI)\\\
\kappa(D-dI)+i(H-EI)&0\end{array}\right),$ (7)

which allows for the local topological index (assuming non-zero localizer gap)
along the diagonal coordinate $d=(x+y)/2$ to be defined as

$\displaystyle\nu_{\textrm{L}}\left(d,0\right)$
$\displaystyle=\tfrac{1}{2}\text{sig}\left[\left(\begin{array}[]{cc}0&I\end{array}\right)L_{\left(d,0\right)}(D,H)\left(\begin{array}[]{c}\Pi\\\
0\end{array}\right)\right]$ (11)
$\displaystyle=\tfrac{1}{2}\text{sig}\left[(\kappa(D-dI)+iH)\Pi\right]\in\mathbb{Z},$
(12)

where $\Pi$ is the system’s chiral operator, $\Pi H\Pi=-H$ (and $\Pi D\Pi=D$)
[78, 79]. Note, $\nu_{\textrm{L}}$ is only well-defined for $E=0$, which is a
mathematical consequence of the fact that the topological states that these
chiral-symmetric systems exhibit (if they exist) are guaranteed to be at zero
energy. For a true 1D system, $\nu_{\textrm{L}}$ is a local version of the 1D
winding number, while for higher-order topological phases in higher
dimensional systems we conjecture that it is a local version of the
corresponding multipole chiral number [101] and is related to the infinite-
volume invariants in Ref. [102]. In all cases, as $\nu_{\textrm{L}}$ depends
on a matrix’s signature, it is guaranteed to be an integer, and it is
identifying whether $D$ and $H$ can be continued to be commuting while
preserving both their Hermiticity and chiral symmetry (and local localizer
gap).

Figure 4: (a) 1D localizer gap,
$\textrm{min}(|\sigma(L_{\bm{\lambda}}(D,H))|)/\Delta E$, for a disordered
higher-order topological metal projected along the diagonal position
$\bm{\lambda}=((x+y)/2,0)$, with $\kappa=2$. Added disorder has strength
$W/\Delta E=[0,0.5,1,1.5,2]$, increasing from cyan to magenta. Solid lines
show the average of an ensemble of 100 different disorder realizations, while
filled regions show the average $\pm 1$ standard deviation. Data is offset
vertically for clarity, each horizontal gray line corresponds to a change in
$\Delta E=1$, and in the bulk of the system $(x+y)/2>2$ the ensemble average
of the localizer gap is nearly zero for each strength of disorder. (b) Similar
to (a) for the local topological invariant $\nu_{\textrm{L}}$ given by Eq.
(12).

Calculating Eq. (12) along the diagonal of the metallic system in Fig. 3a
reveals its higher-order topology: the system acquires a non-trivial invariant
after the localizer gap first closes at a corner of the system, indicating the
presence of a corner-localized state (cyan curves in Fig. 4). Moreover, the
nearby localizer gap prevents this state from moving into the system’s bulk
for disorder strengths $W<0.5\Delta E$. ($\Delta E$ is the bulk band gap
between the top or bottom band and the middle bands.) Thus, even though the
corner-localized states will hybridize with the bulk for any strength of
disorder, the system’s spectral localizer identifies that these states must
maintain support on the system’s corners until the disorder is strong enough
to close the localizer gap. We can explicitly confirm this topological
protection by adding disorder to the system which breaks all crystalline
symmetries and time-reversal symmetry, where we numerically observe the
ensemble averaged localizer gap to remain open and the topological index to
remain pinned to $\nu_{\textrm{L}}=1$ with little variance even for
$W=1.5\Delta E$, see Fig. 4. This provides numerical evidence for the notion
that for uncorrelated disorder the localizer gap is usually an underestimate
of the strength of the topological protection in a system. Note that the
traces of $\nu_{\textrm{L}}$ in Fig. 4b are ensemble averages, and thus will
generally deviate from having an integer value, but each constituent curve in
the average is always an integer for any choice of $d$.

##  V Discussion

In conclusion, we have developed a general theory for assessing a metallic or
gapless system’s topology using its spectral localizer, even in the presence
of disorder. This theory is able to both demonstrate the existence of
boundary-localized modes despite a degenerate background continuum, and yields
a measure of the strength of these systems’ topological protection. To our
knowledge, other methods of defining a local or global index all rely on some
notion of a gap in the bulk spectrum, perhaps a mobility gap, and are not
designed to work in a gapless setting. Indeed, the localizer index was
initially designed to work in the presence of a bulk gap, since a bulk gap
causes a localizer gap [78, 79, 80]. Nevertheless, we have found that useful
localizer gaps can still appear even in the absence of a bulk gap, due to the
spatial separation between degenerate states that can be revealed using
pseudospectral methods. Although we have only explicitly demonstrated this
theory for Chern and higher-order topological metals, this theory should
extend without difficulty to all symmetry classes and for systems in any
dimension, as the necessary local invariants based on the spectral localizer
have already been derived [78, 79, 80], enabling the discovery of novel
topological phases of matter in natural and artificial metals and other
gapless materials.

###### Acknowledgements.

T.L. acknowledges support from the National Science Foundation, grant
DMS-2110398. A.C. and T.L. acknowledge support from the Center for Integrated
Nanotechnologies, an Office of Science User Facility operated for the U.S.
Department of Energy (DOE) Office of Science, and the Laboratory Directed
Research and Development program at Sandia National Laboratories. Sandia
National Laboratories is a multimission laboratory managed and operated by
National Technology & Engineering Solutions of Sandia, LLC, a wholly owned
subsidiary of Honeywell International, Inc., for the U.S. DOE’s National
Nuclear Security Administration under contract DE-NA-0003525. The views
expressed in the article do not necessarily represent the views of the U.S.
DOE or the United States Government.

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#  Application of Time Separation Technique to Enhance C-arm CT Dynamic Liver
Perfusion Imaging

Hana Haseljić  Institute for Medical Engineering and Research Campus
STIMULATE, University of Magdeburg, Magdeburg, Germany  Vojtěch Kulvait
Institute for Medical Engineering and Research Campus STIMULATE, University of
Magdeburg, Magdeburg, Germany  Robert Frysch  Institute for Medical
Engineering and Research Campus STIMULATE, University of Magdeburg, Magdeburg,
Germany  Bennet Hensen  Institute of Diagnostic and Interventional Radiology,
Hannover Medical School, Hannover, Germany  Frank Wacker  Institute of
Diagnostic and Interventional Radiology, Hannover Medical School, Hannover,
Germany  Georg Rose  Institute for Medical Engineering and Research Campus
STIMULATE, University of Magdeburg, Magdeburg, Germany  Thomas Werncke
Institute of Diagnostic and Interventional Radiology, Hannover Medical School,
Hannover, Germany

Abstract Perfusion imaging is an interesting new modality for evaluation and
assessment of the liver cancer treatment. C-Arm CT provides a possibility to
perform perfusion imaging scans intra-operatively for even faster evaluation.
The slow speed of the C-Arm CT rotation and the presence of the noise,
however, have an impact on the reconstruction and therefore model based
approaches have to be applied. In this work we apply the Time separation
technique (TST), to denoise data, speed up reconstruction and improve
resulting perfusion images. We show on animal experiment data that Dynamic
C-Arm CT Liver Perfusion Imaging together with the processing of the data
based on the TST provides comparable results to standard CT liver perfusion
imaging.
---

##  1 Introduction

Perfusion CT imaging is an important modality for the treatment of the liver
cancer, see [1]. It would be the additional benefit to have C-arm CT available
as part of the interventional suite, see [2, 3]. The ability of C-arm systems
to measure parenchymal blood volume (PBV) has been the subject of research
during the last few years, see [2, 4, 5, 6, 7]. The study in [8] has used
animal model to evaluate the dynamic reconstruction algorithm [9] and measure
Arterial Liver Perfusion using C-arm CT data.

Using C-Arm CT perfusion imaging in the liver cancer management could provide
an option to evaluate the performed embolization intraoperatively. The data
moreover could be used to plan ablation. The slow rotation time, limited
number of projections and time gap between rotations causes undersampling and
artifacts in the contrast agent dynamics reconstruction. Neglecting these
limitations by static reconstruction i.e. reconstruction of each rotation
individually as if it was a native CT scan, will cause loss of accuracy and
could result in incorrect perfusion measurements, see [10]. To overcome these
problems we use the model-based approach to describe time attenuation curves
(TAC) as a weighted sum of temporal basis functions, so called Time separation
technique (TST), see [11, 12].

In this paper we use the data from the C-Arm CT perfusion reconstruction of
the swine liver. We are solving the aforementioned problems by using an
analytical basis derived from Fourier analysis and applying TST. Then we use
the deconvolution based algorithms together with Tikhonov stabilization to
compute perfusion parameters, see [13]. Using this approach we reduce noise in
the data and show that produced perfusion maps are comparable to the CT
perfusion data.

##  2 Materials and Methods

### Procedure

We used 2 anaesthetized domestic pigs to perform C-Arm CT perfusion scans of
the liver using iodinated contrast agent after the embolization that induced
the area of decreased perfusion. To induce areas of hypoperfusion in the swine
liver model we embolized branches of the right hepatic artery with tantalum-
based embolization material (Onyx) and coils. Two matching C-arm and CT
perfusion scans were acquired using Siemens ARTIS pheno C-arm and SOMATOM
Force CT. A 15ml of contrast material Imeron 300 was injected with the
duration of 5s. The tube voltage was set to $90\text{\,}\mathrm{kVp}$. Each
C-arm scan consisted of five forward-backward sweep pairs. Each sweep covered
rotation of $200\text{\,}\mathrm{\SIUnitSymbolDegree}$ and with angular step
of $0.8\text{\,}\mathrm{\SIUnitSymbolDegree}$ acquired 248 projections. The
scans with the two scanners were performed ten minutes apart to insure the
contrast material has washed out.

### Time separation technique

Let the interval $\mathcal{I}=[0,T]$ represent the duration of the scan. The
TACs are modeled as a linear combination of a defined set of $N$ orthogonal
functions

$\mathcal{B}=\\{\Psi_{1},\ldots,\Psi_{N}\\},$ (1)

where for each $i\in\\{1,\ldots,N\\}$ $\Psi_{i}=\Psi_{i}(t),t\in I$ are scalar
functions of the time. We call the set $\mathcal{B}$ basis and refer to
functions $\Psi_{i}$ as to the basis functions. In practice these functions
might be analytical functions and therefore $I=\mathcal{I}$ or $\Psi_{i}$ can
be represented as a vector of its values in the $M$ time points
$I=\\{0,T/(M-1),\ldots,T\\}$. According the TST, the time attenuation curve in
a particular volume point $x_{v}$ is given by the linear combination of the
basis functions

$x_{v}(t)=\sum_{i=1}^{N}w_{v,i}\psi_{i}(t).$ (2)

Under the assumption of the orthogonality of the basis functions, we can
transform the contrast agent dynamic reconstruction problem to the $N$
standard CT reconstruction problems to reconstruct weight coefficients
$w_{v,i}$, see [12] for the details.

To do so, we assume that the projection data for any C-Arm spatial
configuration, namely angle and pixel position, encoded by index $k$ satisfies

$p_{k}(t)=\sum_{i=1}^{N}\omega_{k,i}\psi_{i}(t).$ (3)

Note that by means of equations (2) and (3) we separate time development,
encoded by basis functions $\Psi_{i}(t)$, from the spatial configuration
encoded by weighting coefficients $w$ and $\omega$ respectively, thus the
method is called time separation technique.

In order to reduce noise in the data and extract important information about
the contrast agent dynamics, we use trigonometric functions as a basis. Based
on the numerical experiments we figured out that $N=5$ provides a good tradeof
between the number of reconstructions and the image quality. Therefore we have
chosen $N=5$ and $\Psi_{i}$ to be

$\begin{split}\Psi_{0}=1,\quad\Psi_{1}&=\sin(\frac{2\pi
t}{T}),\quad\Psi_{2}=\cos(\frac{2\pi t}{T}),\\\ \Psi_{3}&=\sin(\frac{4\pi
t}{T}),\quad\Psi_{4}=\cos(\frac{4\pi t}{T}).\end{split}$ (4)

These functions are orthogonal with respect to the scalar product
$<\Psi_{i},\Psi_{j}>=\int_{0}^{T}\Psi_{i}(t)\Psi_{j}(t)\,\mathrm{d}t$. To find
the weighting coefficients of the projection data $w^{p}$, we performed a
least squares fitting. This was followed by the 40 iterations of the algebraic
reconstruction, see [14] 111Source code of the reconstruction technique,
namely CGLS, which has been used can be found at
https://github.com/kulvait/KCT_cbct. and the TAC data was obtained using (2).

In case of CT perfusion data, the acquisition on SIEMENS SOMATOM Force was
followed by the analytical reconstruction using Br36 kernel in syngo CT VA50A
software. The reconstructed volume data was interpolated by means of cubic
splines to obtain TAC data. The estimation of the perfusion maps by our
software is identical for both modalities by the method that follows.

### Perfusion Parameters Estimation

The artery input function (AIF) is needed to compute the blood flow through
the organ. It describes the contrast agent flow over time. To generate
comparable perfusion maps artery was detected as suggested in [13].

We estimate TAC in every voxel as a convolution of AIF with residual function

$\mathrm{tac(t)}=\mathrm{aif(t)}*f_{r}\mathrm{(t)}.$ (5)

We discretize the function $aif(t)$ by its values in $k=100$ time points. Then
we apply the pseudoinverse with Tikhonov regularization to (5) in order to
recover function $f_{r}\mathrm{(t)}$.

We compute four perfusion parameters, blood flow (BF), blood volume (BV), mean
transit time (MTT) and time to peak (TTP) using the following formulas from
[13]:

$\begin{split}BF=\max{f_{r}(t)},\quad BV=\sum_{i=1}^{n}f_{r}\mathrm{(i)},\\\
MTT=\frac{BV}{BF},\quad TTP=\arg\max_{t}f_{r}(t).\end{split}$ (6)

##  3 Results

We processed the data from the swine liver perfusion using methods described
in M&M section. To compute perfusion parameters, first we have to locate the
arterial inlet in order to derive AIF function. In Figure 1 the detected
location of AIF is shown together with the time attenuation profile of this
voxel from both modalities. Due to the injection duration, an undersampling of
the contrast material peak in C-arm CT compared to CT can occur.

(a) Location of CT AIF

(b) TAC for CT AIF

(c) Location of C-arm AIF

(d) TAC for C-arm AIF

Figure 1: C-arm and CT artery input function

Using the model (5) and (6) we have computed perfusion maps. The results for a
selected C-arm slice and corresponding CT slice are shown in Figure 2. The
perfusion maps generated by the means of described TST technique are given in
the first row, in the second row are the results from the TAC obtained by
spline interpolation of the static reconstructions and the third row contains
the CT perfusion maps.

Figure 2: Perfusion maps, BF in
$\mathrm{m}\mathrm{L}\mathrm{/}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{g}\mathrm{/}\min$,
BV in
$\mathrm{m}\mathrm{L}\mathrm{/}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{g}$, MTT
in $\mathrm{s}$ and TTP in $\mathrm{s}$, white arrow is pointing to the area
of reduced perfusion induced by embolization.

To compare the results we mainly look for the hypoperfused areas. They are
easily distinguishable in the reconstruction image, see Figure 3. We can
clearly observe these areas in the perfusion maps as they have different
color, see Figure 2.

The unhealthy tissue induced by embolization is very well distinguishable
namely in the BF and BV maps of the TST reconstruction, where they have dark
blue color code.

Figure 3: Hypoperfused area in C-arm (left) and CT (right).

We observe that in the TST C-arm reconstruction perfusion maps the
hypoperfusion area is much more pronounced than in the static C-arm
reconstructions.

To better evaluate the differences between hypoperfusion area and healthy
tissue, we selected regions of interest in both CT and C-Arm CT images, see
Figure 4. We then computed mean value and standard deviation of the respective
perfusion coefficient.

Figure 4: C-arm (left) and CT (right) regions of interest.

We have found that not only the means are different, but when performed T-test
to find out the significance of that difference, we found a p-value of <
$10^{-10}$. The mean value and standard deviation of BF and BV for selected
regions are given in Tables 1 and 2.

Reconstruction | Hypoperfused area | Healthy tissue
---|---|---
TST | $20.7\pm 20.4$ |  $95.6$ $\pm$ | $24.3$
Static | $38.8\pm 39.6$ |  $133.4$ $\pm$ | $41.4$
CT | $91.0\pm 32.9$ |  $1162$ $\pm$ | $122$
Table 1: BF measurements in units of $\mathrm{m}\mathrm{L}\mathrm{/}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{g}\mathrm{/}\min$. Mean and the standard deviation taken over the selected regions of interest. Reconstruction | Hypoperfused area | Healthy tissue
---|---|---
TST |  $2.3$ $\pm$ | $3.0$ |  $13.4$ $\pm$ | $3.1$
Static |  $1.9$ $\pm$ | $2.4$ |  $19.1$ $\pm$ | $5.0$
CT |  $18.8$ $\pm$ | $8.6$ |  $58.9$ $\pm$ | $14.1$
Table 2: BV measurements in units of
$\mathrm{m}\mathrm{L}\mathrm{/}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{g}$. Mean
and the standard deviation taken over the selected regions of interest.

##  4 Discussion

In this paper we have shown that C-Arm CT liver perfusion imaging can provide
similar results as CT perfusion imaging when we use an adequate model based
approach.

It can be seen that proposed basis set of the TST provides a perfusion maps
that reduce the noise and clearly separates the hypoperfusion areas from
healthy tissues, see Figure 2 and Tables 1 and 2. Thus, this method enables
detection of hypoperfusion regions and has the potential to be introduced into
clinical practice.

C-arm perfusion maps show also some differences in well perfused areas when
compared to CT perfusion maps. The difference in values are mostly noticeable
on BF and BV. This can be mainly attributed to the fact that the positioning
of the animal in two different modalities is different but the different setup
of the reconstruction for the two modalities can also play its role, see [6].
C-Arm and CT devices also were not calibrated to provide equal attenuation
values.

##  5 Conclusion

From the results it can be seen that model based reconstruction of the C-Arm
CT perfusion scans outperforms methods based on individual static
reconstructions. The data are less noisy and the area with reduced perfusion
is more visible on perfusion maps. Therefore it provides the results
comparable to the CT perfusion imaging. We plan to perform clinical evaluation
of these data to assess whether this approach should be part of the clinical
setup.

Additional improvement of the TST results is expected by including dedicated
perfusion basis functions based on CT data as prior knowledge, as studied for
brain data in [15, 12].

## Acknowledgments

This work was partly funded by the European Structural and Investment Funds
(International Graduate School MEMoRIAL, project no. ZS/2016/08/80646) and the
German Ministry of Education and Research (Research Campus STIMULATE, grant
no. 13GW0473A and 13GW0473B).

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