for the NUMEN collaboration

# Study of one-proton transfer reaction for the 18O + 48Ti system at 275 MeV

O. Sgouros<EMAIL_ADDRESS>INFN - Laboratori Nazionali del Sud,
Catania, Italy  M. Cavallaro  INFN - Laboratori Nazionali del Sud, Catania,
Italy  F. Cappuzzello  INFN - Laboratori Nazionali del Sud, Catania, Italy
Dipartimento di Fisica e Astronomia ”Ettore Majorana”, Università di Catania,
Catania, Italy  D. Carbone  INFN - Laboratori Nazionali del Sud, Catania,
Italy  C. Agodi  INFN - Laboratori Nazionali del Sud, Catania, Italy  A.
Gargano  INFN - Sezione di Napoli, Napoli, Italy  G. De Gregorio  INFN -
Sezione di Napoli, Napoli, Italy  Dipartimento di Matematica e Fisica,
Università della Campania ”Luigi Vanvitelli”, Caserta, Italy  C. Altana  INFN
- Laboratori Nazionali del Sud, Catania, Italy  G. A. Brischetto  INFN -
Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy  S.
Burrello  Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
Technische Universität Darmstadt, Institut für Kernphysik, Darmstadt, Germany
S. Calabrese  INFN - Laboratori Nazionali del Sud, Catania, Italy
Dipartimento di Fisica e Astronomia ”Ettore Majorana”, Università di Catania,
Catania, Italy  D. Calvo  INFN - Sezione di Torino, Torino, Italy  V.
Capirossi  INFN - Sezione di Torino, Torino, Italy  DISAT - Politecnico di
Torino, Torino, Italy  E. R. Chávez Lomelí  Instituto de Física, Universidad
Nacional Autónoma de México, Mexico City, Mexico  I. Ciraldo  INFN -
Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy  M. Cutuli
INFN - Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy  F.
Delaunay  INFN - Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di
Fisica e Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy
LPC Caen, Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, Caen, France
H. Djapo  Ancara University, Institute of Accelerator Technologies, Turkey  C.
Eke  Department of Mathematics and Science Education, Faculty of Education,
Akdeniz University, Antalya, Turkey  P. Finocchiaro  INFN - Laboratori
Nazionali del Sud, Catania, Italy  M. Fisichella  INFN - Laboratori Nazionali
del Sud, Catania, Italy  A. Foti  INFN - Sezione di Catania, Catania, Italy
A. Hacisalihoglu  Institute of Natural Sciences, Karadeniz Teknik
Universitesi, Trabzon, Turkey  F. Iazzi  INFN - Sezione di Torino, Torino,
Italy  DISAT - Politecnico di Torino, Torino, Italy  L. La Fauci  INFN -
Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy  R.
Linares  Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil
J. Lubian  Instituto de Física, Universidade Federal Fluminense, Niterói,
Brazil  N. H. Medina  Instituto de Física, Universidade de São Paulo, São
Paulo, Brazil  M. Moralles  Instituto de Pesquisas Energeticas e Nucleares
IPEN/CNEN, São Paulo, Brazil  J. R. B. Oliveira  Instituto de Física,
Universidade de São Paulo, São Paulo, Brazil  A. Pakou  Department of Physics,
University of Ioannina and Hellenic Institute of Nuclear Physics, Ioannina,
Greece  L. Pandola  INFN - Laboratori Nazionali del Sud, Catania, Italy  F.
Pinna  INFN - Sezione di Torino, Torino, Italy  DISAT - Politecnico di Torino,
Torino, Italy  G. Russo  Dipartimento di Fisica e Astronomia ”Ettore
Majorana”, Università di Catania, Catania, Italy  INFN - Sezione di Catania,
Catania, Italy  M. A. Guazzelli  Centro Universitario FEI, São Bernardo do
Campo, Brazil  V. Soukeras  INFN - Laboratori Nazionali del Sud, Catania,
Italy  G. Souliotis  Department of Chemistry, University of Athens and
Hellenic Institute of Nuclear Physics, Athens, Greece  A. Spatafora  INFN -
Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia ”Ettore Majorana”, Università di Catania, Catania, Italy  D.
Torresi  INFN - Laboratori Nazionali del Sud, Catania, Italy  A. Yildirim
Department of Physics, Akdeniz Universitesi, Antalya, Turkey  V. A. B. Zagatto
Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil

###### Abstract

Single-nucleon transfer reactions are processes that selectively probe single-
particle components of the populated many-body nuclear states. In this
context, recent efforts are made to build a unified description of the rich
nuclear spectroscopy accessible in heavy-ion collisions. An example of this
multi-channel approach is the study of the competition between successive
nucleon transfer and charge exchange reactions, the latter being of particular
interest in the context of single and double beta decay studies. To this
extent, the one-proton pickup reaction 48Ti(18O,19F)47Sc at 275 MeV was
measured for the first time, under the NUMEN experimental campaign.
Differential cross-section angular distribution measurements for the 19F
ejectiles were performed at INFN-LNS in Catania by using the MAGNEX large
acceptance magnetic spectrometer. The data were analyzed within the Distorted-
Wave and Coupled-Channels Born Approximation frameworks. The initial and
final-state interactions were described adopting the São Paulo potential,
whereas the spectroscopic amplitudes for the projectile and target overlaps
were derived from shell-model calculations. The theoretical cross-sections are
found to be in very good agreement with the experimental data suggesting the
validity of the optical potentials and the shell-model description of the
involved nuclear states within the adopted model space.

###### pacs:

25.70.Hi, 24.10.Eq, 21.10.Jx

##  I Introduction

In recent years, the interest of the physics community to study the neutrino-
less double beta decay (0$\nu$$\beta$$\beta$ decay) [1, 2, 3, 4, 5, 6] has
been intensified although the aforementioned process has not been
experimentally observed yet. Several experimental campaigns [7, 8, 9, 10, 11,
12] have provided only lower limits on the 0$\nu$$\beta$$\beta$ decay half-
life for selected $\beta$$\beta$ decay isotopes. The 0$\nu$$\beta$$\beta$
decay rate is governed by three factors namely, the phase-space factor G0ν
related to the motion of the electrons [13], the Nuclear Matrix Elements
(NMEs) and a term containing the effective neutrino mass [14]. Thus, the
precise knowledge of the NMEs in conjunction with the experimental constraints
for the decay half-life may provide the route towards the determination of the
neutrino mass scale [15, 4].

The NUMEN (NUclear Matrix Elements for Neutrinoless double beta decay) project
[16] has been recently conceived at the Istituto Nazionale di Fisica Nucleare
- Laboratori Nazionali del Sud (INFN-LNS) in Catania, Italy, aiming at
accessing information about the NMEs of 0$\nu$$\beta$$\beta$ decay through the
study of the heavy-ion induced double charge exchange (DCE) reactions on
various $\beta$$\beta$ decay candidate targets. Among these, the 48Ti nucleus
is under investigation since it is the daughter nucleus of 48Ca in the
0$\nu$$\beta$$\beta$ decay process [17, 18]. The choice of DCE as surrogate
reactions to study 0$\nu$$\beta$$\beta$ decay stems from the fact that both
processes have several features in common. Among these, the two processes
probe the same initial and final state nuclear wave functions [16, 19, 20],
while Short-range Fermi, Gamow-Teller and rank-2 tensor components are present
in both transition operators [16]. However, in order to extract meaningful
information on the NMEs, contributions in the DCE channel from reaction
mechanisms populating the same final nuclear states (based on direct meson-
exchange or by sequential multi-nucleon transfer) should be quantitatively
determined.

The degree of competition between DCE and multi-nucleon transfer is an
important aspect for the description of the DCE reaction mechanism [19, 20,
21]. Early studies on the 40Ca(14C,14O)40Ar reaction at 51 MeV by Dasso and
Vitturi showed an important contribution to the 40Arg.s., at not very forward
angles, from the sequential transfer of proton and neutron pairs [22].
Instead, the theoretical study of the DCE mechanism based on a recent
measurement for the 40Ca(18O,18Ne)40Ar reaction at 275 MeV revealed that the
direct-meson exchange mechanism may play a leading role, at least around zero
degrees scattering angle [23, 24]. However, the theoretical analysis in
Ref.[24] suggested the combination of single charge exchange (SCE) with one-
proton and one-neutron transfer reactions as the second process in the leading
order. Therefore, the contribution of the competitive processes to the DCE
channel should not be taken for granted and measurements of all reaction
channels under the same experimental conditions as the DCE reaction are
necessary [25, 26, 27].

Heavy-ion transfer reactions may provide valuable information on the nuclear
structure and the reaction mechanism [28, 29, 30, 31, 32, 33, 34, 35]. Due to
their high selectivity in populating specific degrees of freedom in the
residual nuclei, single-nucleon transfer are well established tools for
probing single-particle configurations, while two-nucleon transfer offers an
insight of pairing correlations [36, 37, 38, 39]. One-proton transfer
reactions have been extensively used for decades aiming at determining the
spectroscopic factors. Experimentally, these quantities were traditionally
determined by re-normalizing the calculated cross-sections, provided by a
reaction model, to the measured ones (e.g. [40, 41, 42, 43]). However,
nowadays a substantial progress on reaction theory and in the computational
and numerical sciences has been undertaken. The distortion of the incoming and
outgoing scattering waves is under control by adopting double-folding optical
potentials (OP) like the São Paulo Potential (SPP) [44, 45, 46, 47]. The
available computational resources allow to perform exact finite-range
calculations rather easily. In addition, with the unprecedented growth in
computing power detailed nuclear structure calculations like large scale
shell-model ones are increasingly adopted for the determination of
spectroscopic factors. Into this context, it has been demonstrated that it is
possible to obtain reliable spectroscopic information from heavy-ion transfer
reactions without the need for any arbitrary scaling factor [48, 49, 50, 51,
52, 53, 54, 55, 56].

One of the most useful reaction models for the analysis of experimental data
on transfer reactions is the Distorted-Wave Born Approximation (DWBA)
formalism [57, 58]. The main ingredients of the distorted-wave theory are the
OP, which describe the elastic scattering at the entrance and exit channels,
and the overlap functions which contain information on the nuclear structure
and angular momentum of the involved nuclei. At energies above the Coulomb
barrier, the DWBA calculation is rather sensitive on the choice of the OP
parameters [41, 59]. On the other hand, the overlap functions are usually
determined as single-particle solutions of a Woods-Saxon potential weighted by
the corresponding spectroscopic amplitudes provided by many-body shell model
calculations. In a recent publication of our group, the importance of these
model dependencies for the description of two-nucleon transfer for 20Ne+116Cd
system was addressed [25].

The success of the DWBA to describe one-nucleon transfer data is well-
established in the literature [60, 61] but in some cases, when coupling
effects among the various reaction channels are strong, it fails to reproduce
the experimental data. In particular, projectile and/or target excitation
could play a deep role in the reaction mechanism and thus should be included
in the theoretical description by means of the Coupled-Channels Born
Approximation (CCBA) method [62, 63, 64].

The 18O has been frequently used as projectile in several studies, but
primarily involving two-neutron transfer reactions [65, 66, 67, 48, 49, 50,
51, 52, 53]. This stems from the fact that the neutron pair out of the 16O
core can be transferred to another nucleus during a nuclear collision thus
giving the opportunity to study neutron pairing correlations in the final
nuclear states. On the other hand, proton transfer reactions has not been
thoroughly investigated. The (18O,19F) reaction on 40Ca and 54Fe has been
studied in Ref.[68] yielding poor results in the description of the angular
distributions. In a more recent publication [55] the same reaction was
considered in a systematic study together with one-neutron, two-neutron and
$\alpha$-particle transfer for the system 18O+65Cu, in order to investigate
the effect of transfer reactions on the quasi-elastic barrier distribution.

In the present work, the one-proton pickup reaction for the system 18O+48Ti at
275 MeV incident energy is investigated for the first time, under the NUMEN
experimental campaign. The measured differential cross-section angular
distribution data are analyzed within the DWBA framework in order to validate
the adopted optical potentials for the description of the initial and final-
state interactions as well as to access information on the single-particle
components of nuclear wave functions of the involved nuclei. Further on, the
coupling influence of projectile and target excitations are investigated
adopting the CCBA formalism. The data analysis is also accompanied by the
study of the same reaction on 16O and 27Al targets, measured under the same
experimental conditions, in order to estimate the background arising from the
different target components (TiO2+27Al) as well as to strengthen the
conclusions of our analysis. As a result of this work we highlight the
importance of studying transfer reactions, since they provide the ground for
testing the validity of the initial and final-state interactions and the
spectroscopic information provided by the nuclear structure models, which
should be under control for the proper description of the SCE and DCE
mechanisms [19].

This paper is organized as follows: the experimental setup and data reduction
procedure are reported in Sections II and III, respectively; a brief
description of the theoretical formalism used for the calculation of the one-
proton transfer cross-sections is given in Section IV; our results are
discussed in Section V and the conclusions are presented in Section VI.

##  II Experimental setup

The experiment was carried out at INFN-LNS in Catania. The 18O8+ ion beam,
delivered by the K800 Superconducting Cyclotron at the energy of 275 MeV, was
directed onto a 510 $\mu$g/cm2 thick TiO2 target which was evaporated on a 216
$\mu$g/cm2 thick aluminum foil. Measurements using a WO3 target (284
$\mu$g/cm2) evaporated on an aluminum foil and a 226 $\mu$g/cm2 thick 27Al one
were performed, to estimate the background arising from the different target
components. The use of a collimation system composed of a 2 mm diameter
collimator followed by a 4 mm diameter anti-scattering, mounted 125 mm and 20
mm upstream the target respectively, limited the beam spot size to $\approx$3
mm at the target position and the beam angular divergence to 3 mrad. The beam
charge was collected by a Faraday cup located 150 mm downstream the target
position. The error in the measurement of the beam charge was reduced by
placing at the entrance of the Faraday cup an electron suppression ring biased
at -200V, minimizing the escape of the secondary produced electrons.

The various reaction products were momentum analyzed by the MAGNEX large
acceptance magnetic spectrometer [69], whose optical axis was set at
$\theta$opt= $None$ with respect to the beam direction. MAGNEX was operated in
full horizontal acceptance covering an angular range between $None$ and $None$
in the laboratory reference frame but with a reduced vertical one
($\pm$$None$), in order to avoid the high ion counting rate at the Focal Plane
Detector (FPD) [70, 71, 72]. The latter was used in order to identify the ions
of interest among the various reaction products, by means of the conventional
$\Delta$E-E method for the Z separation and a technique for the determination
of the mass number based on the correlation between the ions kinetic energy
and the measured horizontal position at the focal plane. An example of
identification spectra is presented in Fig.1, while a detailed description of
the identification technique is given in Ref.[73].

Figure 1: Typical identification spectra for 48Ti(18O,19F)47Sc reaction at the
energy of 275 MeV. a) $\Delta$E-Eresid correlation plot for the ions energy
loss inside the gas tracker, $\Delta$E, as a function the of residual energy,
Eresid, measured by one silicon detector of the FPD. A graphical selection in
the contour of the fluorine ion events is illustrated by the solid red line.
b) The horizontal position at the MAGNEX focal plane as a function of the
residual energy for the selected fluorine ions of panel (a). The different
loci correspond to ions with different ratio $\sqrt{m}$/q. A graphical
selection on the 19F9+ events is shown by the solid red line.

##  III Data reduction

Once the 19F ejectiles were identified in the spectra, a high-order software
ray reconstruction technique was applied to the data by solving the equation
of motion for the ejectile particles, based on a fully differential algebraic
method. Hence, it was possible to reconstruct the ion momentum vector at the
target position [74]. The reconstruction procedure was performed separately
for each of the reactions under study namely, the 48Ti(18O,19F)47Sc,
16O(18O,19F)15N and the 27Al(18O,19F)26Mg reaction corresponding to the
measurement with the TiO2, WO3 and 27Al targets, respectively.

###  III.1 The 27Al(18O,19F)26Mg and 16O(18O,19F)15N reactions

The first part of the data reduction is referred to the analysis of the one-
proton transfer events obtained with the aluminum target, since the aluminum
backing was present in both measurements performed with the TiO2 and WO3
targets. The absolute cross-sections were determined after correcting the
experimental yields for the overall efficiency of MAGNEX spectrometer [75]. In
Fig.2a the absolute differential cross-section as a function of the excitation
energy for the 27Al(18O,19F)26Mg reaction corresponding to the angular range
between $None$ and $None$ in the laboratory reference frame is shown. The
excitation energy Ex was obtained from the missing mass method [69] as:

$E_{x}=Q_{0}-Q,$ (1)

where Q0 is the ground state (g.s.) to g.s. Q-value for the 27Al(18O,19F)26Mg
reaction and Q is a term containing the masses of the nuclei at the entrance
and exit channels, and the reconstructed kinetic energy and angle of the 19F
ions. The obtained energy resolution in the current measurement was
approximately 500 keV in full width at half maximum (FWHM). Looking at Fig.2a,
various states are populated up to $\alpha$-particle emission threshold of
26Mg (Sα$\approx$ 10.6 MeV), while a rather continuous shape is observed
beyond such threshold. In order to describe the excitation energy spectrum in
terms of the well-known states of the 19F and 26Mg nuclei a fit procedure was
performed, where the shape of each state was described by a Gaussian function.
The results of the fit procedure are presented in Fig.2b. It should be pointed
out that the output of the fit is the overall contribution of the Gaussian
peaks, rather than the individual contribution of each transition. Therefore,
the experimental yields of the unresolved states corresponding to three
different excitation energy regions namely, -1.0 $<$ Ex $<$ 1.0 MeV, 1.0 $<$
Ex $<$ 3.0 MeV and 3.5 $<$ Ex $<$ 5.5 MeV were deduced from the fits
considering an angular step between $None$ and $None$ depending on the
statistics. The differential cross-section angular distributions were
extracted and are presented in Fig.3. The error bars include the statistical
uncertainty and to a lesser extent a contribution due to the uncertainty in
the determination of the solid angle. A systematic error of about 10 % due to
the uncertainty in the target thickness and the integrated value of the beam
charge, common to all the data points, is not included in the error bars.

Figure 2: a) Reconstructed excitation energy spectrum for the
27Al(18O,19F)26Mg reaction at 275 MeV. The alpha emission threshold of 26Mg
nucleus is indicated by the vertical red dashed line. b) Same as in panel (a),
but with a zoom of the first 5 MeV of the excitation energy spectrum. The
experimental data are compared to the result of a fit procedure where for the
description of each known state a Gaussian function was used. The dashed
magenta curve corresponds to the ground state to ground state transition, the
dotted-dashed blue lines are referred to the excited states of the 19F
nucleus, the dotted brown lines correspond to the excited states of 26Mg
keeping the 19F in its ground state and the long-dashed green lines correspond
to transitions where both ejectile and residual nuclei are excited. The red
line corresponds to sum of the Gaussian curves. Figure 3: Comparison between
experimental and theoretical angular distribution data for the
27Al(18O,19F)26Mg reaction measured at 275 MeV. The experimental data, denoted
with the black circles, were obtained by integrating the yields of the
unresolved states referred as a) first peak, b) second peak and c) third peak
in Fig.2. Theoretical cross-sections for the transitions to the involved
states of the ejectile and the residual nuclei were calculated within the DWBA
framework and are presented with the colored curves. In the legend, each curve
is labeled by the corresponding excitation energy of 19F for transitions to
the 26Mgg.s. and a symbol in case where 26Mg is excited. The curves marked
with an asterisk correspond to transitions to the $2$1+ (1.809 MeV) state of
26Mg, while those marked with an open circle, a filled circle and a triangle
correspond to transitions to the $4$1+ (4.319 MeV), $2$3+ (4.333 MeV) and
$3$3+ (4.350 MeV) states of 26Mg, respectively. The sum of all transitions is
illustrated by the red solid line. The sum of a CCBA calculation considering
the same final states as the DWBA one is indicated with the dashed black
curve.

As stated above, to study the 16O(18O,19F)15N reaction, a compound target of
WO3 evaporated onto a thin aluminum foil was used. Therefore, the background
due to the presence of the two contaminants should be identified in order to
isolate the 19F and 15N spectra. The reconstructed excitation energy spectrum
measured in the angular range between $None$ and $None$ in the laboratory
reference frame is shown in Fig.4. The excitation energy was determined using
Eq.1 with Q0 being the g.s. to g.s Q-value of the 16O(18O,19F)15N reaction.
The pronounced peaks at Ex $\approx$ -5 MeV and -3 MeV were identified through
the reaction kinematics with the one-proton transfer reaction on 27Al. Thus,
by using the data obtained with the pure aluminum target it was possible to
subtract the aluminum contribution, appropriately normalized, from the
spectrum. A small remnant of events located at Ex$<$ -8 MeV was attributed to
the tungsten component of the target. In the absence of a measurement with a
pure tungsten target, a uniform distribution for the tungsten contamination
was assumed throughout the whole energy range (see Fig.4). Subsequently, the
tungsten contribution was subtracted and the excitation energy spectrum for
the 18O + 16O reaction was deduced. Following the same procedure as the one
adopted for the 18O + 27Al reaction, energy and angle differential cross-
sections were deduced and are presented in Figs. 5 and 6, respectively. The
error in both spectra includes the contribution from the statistical and the
background subtraction uncertainties. Having an inspection at the energy
spectrum of Fig.5, it is seen that the first excited state of the 15N nucleus
is found at Ex= 5.27 MeV. So, the observed structure at the first $\approx$ 5
MeV of the spectrum in Fig.5 is the fingerprint of the population of 19F low-
lying states. This offers the unique possibility for validating the
spectroscopic information for the 19F states provided by our shell-model
calculations and thus to optimize the overall data interpretation. Further
details are given in the following section.

Figure 4: Decomposition of the excitation energy spectrum obtained with the
WO3+27Al target. The total spectrum is presented with the solid black line.
The red-hatched area corresponds to the normalized background originating from
the 27Al backing material, while the dotted blue area corresponds to the
background arising from tungsten component of the target. For reasons of
clarity, the latter is multiplied by a factor of 3. The solid green area is
the obtained excitation energy spectrum for the 16O(18O,19F)15N reaction,
after subtracting from the total spectrum the background contributions. Figure
5: Excitation energy spectrum for the 16O(18O,19F)15N reaction at the energy
of 275 MeV compared to the results of the fit procedure, where for the
description of each state, a Gaussian function was used. The adopted colors
and line styles are the same as those presented in Fig.2. In addition to
Fig.2, in case of several nearby states a single Gaussian function, depicted
with the double-dotted-dashed curve, was considered in the fit procedure with
the centroid fixed to the mean value of the excited states. Moreover, the
continuum background above the proton emission threshold of 15N is indicated
with the purple line. Figure 6: Comparison between experimental and
theoretical angular distribution data for the 16O(18O,19F)15N reaction
measured at 275 MeV. The experimental data, denoted with the black circles,
were obtained by integrating the yields of the unresolved states referred as
a) first peak, b) second peak and c) third peak in Fig.5. Theoretical angular
distributions for the transitions to the involved states of the ejectile and
the residual nuclei were calculated within the DWBA framework and are
presented with the colored curves. In the legend, each curve is labeled by the
corresponding excitation energy of 19F for transitions to the 15Ng.s. and a
symbol in case where 15N is excited. The curves marked with an asterisk, an
open circle and a triangle correspond to transitions to the $\frac{5}{2}$1+
(5.27 MeV), $\frac{1}{2}$1+ (5.299 MeV) and $\frac{3}{2}$1- (6.324 MeV) states
of 15N, respectively. The sum of all transitions is illustrated by the red
solid line. The sum of a CCBA calculation considering the same final states as
the DWBA one is indicated with the dashed black curve.

###  III.2  The 48Ti(18O,19F)47Sc reaction

The reconstructed excitation energy spectrum obtained with the TiO2+27Al
target is shown in Fig.7. The excitation energy was obtained through Eq.1 with
Q0 being the g.s. to g.s Q-value of the 48Ti(18O,19F)47Sc reaction. In the
same figure, the contributions from the 27Al backing and 16O component of the
target, appropriately normalized, are also shown. The background spectra were
subtracted from the total one and the excitation energy spectrum for the
48Ti(18O,19F)47Sc reaction was deduced. Unlike spectra in Figs. 2 and 5, this
one has a rather continuum shape, reflecting the high level density of the
populated 47Sc nucleus and the limited energy resolution. Therefore, angular
distributions were determined by integrating the events in a wide energy range
corresponding to 0 $<$ Ex $<$ 3.5 MeV including thus the contribution of
various excited states of 19F and 47Sc nuclei. The obtained angular
distribution data are presented in Fig.8.

Figure 7: a) Decomposition of the excitation energy spectrum obtained with the
TiO2+27Al target. The total spectrum is presented with the solid black line.
The red-hatched area corresponds to the normalized background originating from
the 27Al backing material, while the dotted green area corresponds to the
background arising from oxygen. The blue-vertically-hatched area is the
obtained excitation energy spectrum for the 48Ti(18O,19F)47Sc reaction, after
subtracting from the total spectrum the background contributions. b)
Reconstructed excitation energy spectrum for the 48Ti(18O,19F)47Sc reaction at
275 MeV. The proton emission threshold of 47Sc nucleus is indicated with the
vertical red dashed line. Figure 8: Comparison between experimental and
theoretical cross-sections for the 48Ti(18O,19F)47Sc reaction measured at 275
MeV. The experimental data, denoted with the black circles, were obtained by
integrating the spectrum of Fig.7 between 0 $<$ Ex $<$ 3.5 MeV. Theoretical
angular distributions for the transitions to the involved states of the
ejectile and the residual nuclei were calculated within the DWBA framework and
are presented with the colored curves. Due to the large number of states
introduced in the calculation, for reasons of clarity only the five stronger
transitions are shown in the graph. In the legend, each curve is labeled by
the corresponding excitation energy of 19F for transitions to the 47Scg.s. and
a symbol in case where 47Sc is excited. The curves marked with an asterisk, an
open circle and a triangle correspond to excitation in the $\frac{3}{2}$1+
(0.767 MeV), $\frac{3}{2}$1- (0.808 MeV) and $\frac{1}{2}$1+ (1.391 MeV)
states of 47Sc respectively. The contribution of all other transitions is
indicated with the blue dotted-dashed line under the notation ”sum rest”. The
sum of all transitions is illustrated by the red solid line. The sum of a CCBA
calculation considering the same final states as the DWBA one is indicated
with the dashed black curve.

##  IV Theoretical analysis

The experimental angular distribution data were analyzed within the DWBA and
CCBA frameworks using FRESCO code [76]. Considering a reaction of the form: a
+ A $\rightarrow$ b + B, the transition amplitude in the DWBA model is given
by the following expression:

$T=\int
d\vec{r_{\alpha}}d\vec{r_{\beta}}\chi_{\beta}^{(-)*}\braket{\Psi_{b}\Psi_{B}}{V}{\Psi_{a}\Psi_{A}}\chi_{\alpha}^{(+)},$
(2)

where $\chi$α and $\chi$β are the distorted waves describing the relative
motion of the involved nuclei at the entrance ($\alpha$) and exit ($\beta$)
channels, respectively and $\braket{\Psi_{b}\Psi_{B}}{V}{\Psi_{a}\Psi_{A}}$ is
the matrix element describing the interaction between the internal states of
the colliding pairs. The transfer operator V was calculated in the post
representation including full complex remnant terms. In the present analysis,
for all reactions under study the distorted waves at the entrance and exit
channels were generated adopting the double-folding São Paulo Potential for
the description of both the real and imaginary parts of the optical potential,
but with two different normalization factors NR and NI, respectively. For the
DWBA calculations, the adopted values for the normalization factors were
NR=1.0 and NI= 0.78. Based on a systematic study, it was found that with this
prescription the SPP is able to describe adequately-well the elastic
scattering data for heavy-ion reactions involving light [77, 78], medium [79,
80] and heavy [81, 82] mass targets over a wide energy range. In the CCBA
approach where couplings to the relevant projectile and target excitations are
explicitly taken into account, the normalization factor of the imaginary part
of the OP at the entrance channel was reduced to NI= 0.60. The same
prescription has been also adopted in previous studies for the description of
one and two-neutron transfer reactions [47, 54, 48, 50, 51, 52]. It should be
noted that for the 27Al(18O,19F)26Mg reaction we considered only couplings to
the $2$1+ state of 18O and thus an intermediate value for the normalization
factor NI= 0.70 was chosen.

The single-particle wave functions were calculated assuming that the
transferred proton is bound to the core by an effective potential of a Woods-
Saxon form factor with a radius R= r0Ai1/3, where Ai is the mass number of the
core nucleus. For the 18O core the reduced radius (r0) and diffuseness was set
to 1.26 and 0.70 fm, respectively, whereas the reduced radii and the
diffusenesses were set to 1.20 and 0.60 fm, respectively, for the heavier
nuclei [25, 26]. For the potential that binds the valence proton in the 15N
core, since 15N has an excess of one neutron with respect to protons, the
adopted geometry of the binding potential was the same as for the 18O neutron
rich core. Tests adopting different values for the reduced radius (1.20-1.26)
fm and diffuseness (0.60-0.70) fm of this potential were performed showing
that the theoretical cross-sections for one-proton transfer do not vary
significantly. In all cases, the depth of the potential was adjusted such as
to reproduce the separation energy of the transferred particle.

In the CCBA calculation, inelastic excitations to the low-lying states of the
projectile and target nuclei were taken into account adopting the rotational
model. Coulomb deformations were introduced in terms of the reduced transition
probabilities B(Eλ), where $\lambda$ is the multipolarity of the excitation.
In more detail, for the 18O projectile the excitation to the 21+ state was
considered in the coupling scheme using B(E2; 0+$\rightarrow$2+)= 0.0043 e2b2
as reported in [83]. For the 16O and 48Ti targets, inelastic excitations to
the 31- and 21+ states, respectively, were introduced in the calculation
adopting the values of B(E3; 0+$\rightarrow$3-)= 0.0015 e2b3 from Ref.[84] and
B(E2; 0+$\rightarrow$2+)= 0.072 e2b2 from Ref.[85]. The nuclear coupling
potentials were derived following the same procedure as described in Ref.
[80].

The spectroscopic amplitudes for the projectile and target overlaps were
computed within the framework of shell-model [86, 87] using the code KSHELL
[88]. In particular, we have adopted two different effective Hamiltonians,
namely the p-sd-mod [89] and the SDPF-MU [90] interactions, both defined in a
model space encompassing two major shells. The first one is a modified version
of the PSDWBT interaction [91] and is defined in a model space including the
1p and 2s-1d orbitals for both protons and neutrons with 4He as closed core.
It has been already adopted in many of ours previous studies (e.g. [35, 51,
54]) and is now used to compute the one-proton spectroscopic amplitudes
involved in reactions with the 16O and 27Al targets. The second interaction,
employed to calculate the spectroscopic amplitudes for the
$\braket{{}^{47}Sc}{{}^{48}Ti}$ overlaps, is constructed in a model space
spanned by the proton and neutron 2s-1d and 1f-2p orbitals on top of the
doubly magic 16O core. It is based on the VMu potential [92] and has been
widely adopted to study the effects of excitations across the sd-pf major
shells as well as to compute the spectroscopic factors of nuclei in Ca region
[93, 94, 95, 96]. However, in this case complete calculations are unfeasible
due to the very large capacity of the adopted model space and, therefore, a
truncation of the shell-model basis is required. In particular, we have
considered only one particle-one hole cross-shell excitations, while all
possible configurations within the 2s-1d and 1f-2p shells are taken into
account. A list with the spectroscopic amplitudes, used in the reaction
calculations, is presented in Tables I-III.

Table 1: List of the one-proton spectroscopic amplitudes for the projectile overlaps used in the DWBA and CCBA calculations. The symbols n, l and j correspond to the principal quantum number, the orbital and the total angular momentum of the transferred proton orbitals, respectively. Initial state |  nlj |  Final state |  Spectroscopic amplitude
---|---|---|---
| 2s1/2 |  19Fg.s. (1/2+) | -0.554
| 1p1/2 |  19F0.110 (1/2-) | -0.244
18Og.s. (0+) | 1d5/2 |  19F0.197 (5/2+) | 0.664
| 1p3/2 |  19F1.459 (3/2-) | -0.011
| 1d3/2 |  19F1.554 (3/2+) | -0.424
| 1d5/2 |  19Fg.s. (1/2+) | -0.586
| 1d3/2 |  19Fg.s. (1/2+) | 0.281
| 1p3/2 |  19F0.110 (1/2-) | 0.030
| 1d5/2 |  19F0.197 (5/2+) | 0.427
| 1d3/2 |  19F0.197 (5/2+) | -0.156
18O1.982 (2+) | 2s1/2 |  19F0.197 (5/2+) | 0.311
| 1p3/2 |  19F1.459 (3/2-) | 0.002
| 1p1/2 |  19F1.459 (3/2-) | -0.164
| 1d5/2 |  19F1.554 (3/2+) | -0.315
| 1d3/2 |  19F1.554 (3/2+) | -0.319
| 2s1/2 |  19F1.554 (3/2+) | -0.354
Table 2: List of the one-proton spectroscopic amplitudes for the target overlaps used in the DWBA and CCBA calculations for the reactions with the 16O and 27Al targets. The symbols n, l and j correspond to the principal quantum number, the orbital and the total angular momentum of the transferred proton orbitals, respectively. Initial state |  nlj |  Final state |  Spectroscopic amplitude
---|---|---|---
| 1d5/2 |  26Mgg.s. (0+) | 0.525
| 1d5/2 |  26Mg1.809 (2+) | 0.915
| 2s1/2 |  26Mg1.809 (2+) | -0.140
| 1d3/2 |  26Mg1.809 (2+) | 0.028
| 1d5/2 |  26Mg4.318 (4+) | 1.069
27Alg.s. (5/2+) | 1d3/2 |  26Mg4.318 (4+) | 0.120
| 1d5/2 |  26Mg4.332 (2+) | 0.209
| 1d3/2 |  26Mg4.332 (2+) | 0.056
| 2s1/2 |  26Mg4.332 (2+) | 0.129
| 1d5/2 |  26Mg4.350 (3+) | 0.062
| 1d3/2 |  26Mg4.350 (3+) | 0.138
| 2s1/2 |  26Mg4.350 (3+) | 0.195
| 1p1/2 |  15Ng.s. (1/2-) | -1.253
16Og.s. (0+) | 1d5/2 |  15N5.270 (5/2+) | 0.493
| 2s1/2 |  15N5.299 (1/2+) | 0.141
| 1p3/2 |  15N6.324 (3/2-) | 1.753
| 1d5/2 |  15Ng.s. (1/2-) | 0.617
| 1p3/2 |  15N5.270 (5/2+) | 0.062
16O6.130 (3-) | 1p1/2 |  15N5.270 (5/2+) | -0.834
| 1d5/2 |  15N6.324 (3/2-) | 0.160
| 1d3/2 |  15N6.324 (3/2-) | 0.268
Table 3: List of the one-proton spectroscopic amplitudes for the target overlaps used in the DWBA and CCBA calculations for the 48Ti(18O,19F)47Sc reaction. The symbols n, l and j correspond to the principal quantum number, the orbital and the total angular momentum of the transferred proton orbitals, respectively. Initial state |  nlj |  Final state |  Spectroscopic amplitude
---|---|---|---
| 1f7/2 |  47Scg.s. (7/2-) | -1.325
| 1d3/2 |  47Sc0.767 (3/2+) | -1.536
| 2p3/2 |  47Sc0.808 (3/2-) | -0.306
| 1f5/2 |  47Sc1.297 (5/2-) | 0.043
48Tig.s. (0+) | 2s1/2 |  47Sc1.391 (1/2+) | -1.094
| 1d5/2 |  47Sc1.404 (5/2+) | 0.122
| 2s1/2 |  47Sc1.798 (1/2+) | 0.332
| 1d3/2 |  47Sc2.002 (3/2+) | -0.039
| 1d5/2 |  47Sc2.381 (5/2+) | 0.422
| 2s1/2 |  47Sc2.529 (1/2+) | -0.050
| 1f7/2 |  47Scg.s. (7/2-) | -0.793
| 1f5/2 |  47Scg.s. (7/2-) | -0.003
| 2p3/2 |  47Scg.s. (7/2-) | -0.253
| 1d3/2 |  47Sc0.767 (3/2+) | 0.377
| 1d5/2 |  47Sc0.767 (3/2+) | -0.083
| 2s1/2 |  47Sc0.767 (3/2+) | -0.247
| 2p3/2 |  47Sc0.808 (3/2-) | -0.113
| 1f7/2 |  47Sc0.808 (3/2-) | -0.612
| 1f5/2 |  47Sc0.808 (3/2-) | -0.025
| 2p1/2 |  47Sc0.808 (3/2-) | 0.004
| 1f7/2 |  47Sc1.297 (5/2-) | -0.281
| 1f5/2 |  47Sc1.297 (5/2-) | 0.008
| 2p3/2 |  47Sc1.297 (5/2-) | -0.129
| 2p1/2 |  47Sc1.297 (5/2-) | -0.014
48Ti0.984 (2+) | 1d5/2 |  47Sc1.391 (1/2+) | 0.108
| 1d3/2 |  47Sc1.391 (1/2+) | 0.297
| 1d5/2 |  47Sc1.404 (5/2+) | -0.031
| 1d3/2 |  47Sc1.404 (5/2+) | -0.913
| 2s1/2 |  47Sc1.404 (5/2+) | 0.316
| 1d5/2 |  47Sc1.798 (1/2+) | -0.067
| 1d3/2 |  47Sc1.798 (1/2+) | 0.072
| 1d3/2 |  47Sc2.002 (3/2+) | -0.473
| 1d5/2 |  47Sc2.002 (3/2+) | 0.114
| 2s1/2 |  47Sc2.002 (3/2+) | -0.600
| 1d5/2 |  47Sc2.381 (5/2+) | 0.072
| 1d3/2 |  47Sc2.381 (5/2+) | -0.248
| 2s1/2 |  47Sc2.381 (5/2+) | -0.786
| 1d5/2 |  47Sc2.529 (1/2+) | -0.031
| 1d3/2 |  47Sc2.529 (1/2+) | -0.014

##  V Results and discussion

Considering the case of the 16O(18O,19F)15N reaction, as it was stated above,
the excitation energy spectrum up to $\approx$ 5 MeV underlines the
fingerprint of the 19F nucleus. The ground state of 15N is mainly described as
a 1p1/2 hole coupled to the 16O ground state and thus, the 16Og.s.
$\rightarrow$ 15Ng.s. transition proceeds via a proton removal from the 1p1/2
shell [97, 98, 99]. The 1p1/2 shell of 16O is fully occupied and the one-
proton pickup proceeds via the removal of one of the two indistinguishable
valence protons. Thus, it is expected that the absolute value of the
spectroscopic amplitude for the $\braket{{}^{15}N_{g.s.}}{{}^{16}O_{g.s.}}$
overlap should be closed to $\sqrt{2}$. Indeed, the absolute value for the
spectroscopic amplitude predicted by our shell-model calculation is equal to
1.2532 in agreement with those reported in Ref.[99] from (d,3He) experiments.
As a result, a comparison between experimental and theoretical angular
distribution data corresponding to the first two peaks of Fig.5, provides the
ground for testing the accuracy of the calculated spectroscopic amplitudes for
the projectile overlaps. To this direction, DWBA and CCBA calculations for the
one-proton transfer reaction populating 15N in its ground state and the low-
lying states of 19F(see Table I) were performed and the results are presented
in Fig.6. In all cases, the agreement between experimental and theoretical
angular distribution data is very good. The inclusion of inelastic excitations
of projectile and target in the coupling scheme improves in most of the cases
the agreement with the experimental data but the case of the ground state
region where the CCBA prediction slightly overestimates the experimental
cross-sections. The striking example of the importance of projectile and
target excitations is demonstrated in Fig.6b. Although both DWBA and CCBA
calculations are in quantitative agreement with the experimental data, it is
evident that only the latter is able to reproduce the oscillatory pattern of
the angular distribution. This result points also to a small, yet significant,
contribution from core excitation configurations in the population of the
$\frac{3}{2}$- and $\frac{3}{2}$+ states of 19F. At this point, it should be
mentioned that additional calculations by introducing the $\frac{5}{2}$- 1.346
MeV state of 19F in the coupling scheme were performed. However, due to the
small values in the spectroscopic amplitudes for this transition the
theoretical cross-sections presented deviations of about 1%. Therefore, this
state was no longer considered in the calculations. In general, both
calculations reproduce adequately-well the experimental data without need for
any additional normalization factor [43, 100, 101, 102, 103]. The good
agreement between experimental and theoretical cross-section gives further
support to the validity of the calculated spectroscopic amplitudes as well as
the adopted optical potentials for the description of the initial and final-
state interactions.

Once the spectroscopic amplitudes for the $\braket{{}^{19}F}{{}^{18}O}$
overlaps were established, similar calculations for the 27Al(18O,19F)26Mg and
48Ti(18O,19F)47Sc reactions were performed and the results are compared to the
experimental data in Figs. 3 and 8, respectively. Starting from the case of
27Al, the theoretical angular distributions are in very good agreement with
the experimental data but with a tendency to slightly underestimate the
magnitude of the experimental cross-sections at the high excited states. The
coupling to the $2$1+ state of 18O is found to be weak but its inclusion
improves the description of the angular distribution data. The shape of the
measured angular distribution is rather structure-less reflecting the strong
contribution from the 1d5/2 transfer to the $\frac{5}{2}$+ state of 19F. This
is consistent with the findings from ($\alpha$,t) experiments [104]. The
theoretical calculation for the 48Ti(18O,19F)47Sc reaction was more demanding,
since a large number of nuclear states of 47Sc was introduced, thus presenting
a good test for adopted spectroscopic amplitudes. Despite that, once more the
predicted angular distribution compares very well with the experimental data,
as seen in Fig.8.

Having completed the analysis for the three reactions under study, it is
worthwhile to notice that the leading role in the projectile transitions for
the excitation energy range under investigation is always provided by the
transition to the $\frac{5}{2}$+ state at 0.197 MeV of 19F, regardless the
transition undertaken from the target.

##  VI Summary and conclusions

Cross-section measurements for the one-proton pickup reactions induced by 18O
on 16O, 27Al and 48Ti targets were performed at INFN-LNS. The 19F ejectiles
were detected by the MAGNEX magnetic spectrometer spanning an angular range
between $None$ and $None$ in the laboratory reference frame. It has been
demonstrated that adopting the DWBA and/or CCBA formalism we are able to
describe very accurately the one-proton transfer angular distribution data,
without the need for any re-normalization factor of the theoretical curves.
The couplings to the low-lying states of the projectile and target nuclei were
found to be weak but in most of the cases improve the description of the
experimental data. It should be pointed out that these couplings were not
appropriately constrained, since elastic and inelastic scattering data were
not available, but is in our intentions to perform elastic and inelastic
scattering measurements for the same system in the near future. Considering
that the transition amplitudes in the DWBA approach are strongly dependent on
the spectroscopic factors, being also rather sensitive to the spatial
distribution of the projectile-target interaction, the good agreement observed
between the experimental and theoretical cross-sections supports the validity
of the adopted optical potentials as well as the accuracy of the spectroscopic
amplitudes which were derived from shell-model calculations. This result is
very important for the NUMEN project. In the quantum mechanical description of
the DCE mechanism the wave functions which enter in the expression of the
transition amplitude are the distorted waves at the entrance and exit channels
and the transition densities. Like in the case of transfer reactions, these
distorted waves are the solution of the Schroedinger equation using the
optical potentials, while the calculation of the transition densities relies
on nuclear structure models. Therefore, transfer reactions can provide the
testing ground for the reaction and structure models, before moving on the
description of more complicated reaction mechanisms like the SCE and DCE ones.
Finally, the results of the present analysis together with those from the
(18O,17O) one in the 18O + 48Ti reaction will clarify the degree of
competition between the direct SCE mechanism (18O,18F) and the sequential
nucleon transfer, with the former being particularly important in order to
constrain the DCE sequential mechanism in the
18O$\rightarrow$18F$\rightarrow$18Ne transition.

## Acknowledgments

We warmly acknowledge the operators of INFN-LNS Accelerator for the production
and delivery of the 18O beam and their support throughout the experiment. The
research leading to these results was partially funded by the European
Research Council (ERC) under the European Union’s Horizon 2020 Research and
Innovation Programme (Grant Agreement No 714625). We also acknowledge the
CINECA award under the ISCRA initiative (code HP10B51E4M) and through the
INFN-CINECA agreement for the availability of high performance computing
resources and support. One of us (G.D.G.) acknowledges the support by the
funding program VALERE of Univerità degli Studi della Campania ”Luigi
Vanvitelli”. One of us (S.B.) acknowledges support from the Alexander von
Humboldt foundation. We also acknowledge partial financial support from CNPq,
FAPERJ, FAPESP (proc. no. 2019/07767-1) and Instituto Nacional de Ciência e
Tecnologia - Física Nuclear e Aplicações (INCT-FNA, proc. no. 464898/2014-5),
Brazil.

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# Analog In-Memory Computing with Uncertainty Quantification for Efficient
Edge-based Medical Imaging Segmentation

Imane Hamzaoui
École nationale Supérieure d’Informatique
Algiers, Algeria
<EMAIL_ADDRESS>
&Hadjer Benmeziane
IBM Research Europe
8803 Rüschlikon, Switzerland
<EMAIL_ADDRESS>
&Zayneb Cherif
Yorktown High school
Yorktown Heights, 10598, USA
<EMAIL_ADDRESS>&Kaoutar El Maghraoui
IBM T. J. Watson Research Center
Yorktown Heights, NY 10598, USA
<EMAIL_ADDRESS>
###### Abstract

This work investigates the role of the emerging Analog In-memory computing
(AIMC) paradigm in enabling Medical AI analysis and improving the certainty of
these models at the edge. It contrasts AIMC’s efficiency with traditional
digital computing’s limitations in power, speed, and scalability. Our
comprehensive evaluation focuses on brain tumor analysis, spleen segmentation,
and nuclei detection. The study highlights the superior robustness of
isotropic architectures, which exhibit a minimal accuracy drop (0.04) in
analog-aware training, compared to significant drops (up to 0.15) in pyramidal
structures. Additionally, the paper emphasizes IMC’s effective data
pipelining, reducing latency and increasing throughput as well as the
exploitation of inherent noise within AIMC, strategically harnessed to augment
model certainty.

##  1 Introduction

Analog In-memory Computing (AIMC) marks a shift from traditional digital
computing promising efficient and scalable processing for the rapidly growing
medical data. Traditional digital systems, hindered by the Von Neumann
bottleneck where data and instructions travel separately, struggle with large-
scale data tasks, resulting in inherent inefficiencies. AIMC promises better
efficiency and lower power use but faces challenges like susceptibility to
noise, which can impact computation accuracy. In a recent study (Bonnet et
al., 2023), memristor-based Bayesian neural networks (BNNs) were investigated
for heartbeats classification. In contrast, our work explores AIMC’s
application on a wider range of medical imaging tasks, analyzing if it can
effectively address healthcare needs while managing noise issues. Our primary
focus lies on the algorithmic aspects, given that AIMC accelerators are still
in the early stages of development (Gallo et al., 2023; Wan et al., 2022; Yin
et al., 2019; Khwa et al., 2022). The code is available via this link
(https://anonymous.4open.science/r/Analog_Med-B867).

##  2 Evaluating Medical Deep learning on Analog IMC

We present a comprehensive evaluation of Analog In-memory Computing (AIMC) for
medical imaging, utilizing three benchmark datasets: Brain Tumor Segmentation,
Spleen Segmentation, and Nuclei Detection. The study incorporates advanced
architectures like U-Net (Ronneberger et al., 2015), U-Net++ (Zhou et al.,
2018), and Swin Transformer (Hatamizadeh et al., 2021), trained via AIHWKIT
(Rasch et al., 2021). The benchmarks and training methodologies are described
in the appendix A.1 and appendixA.2. This analysis aims to assess AIMC’s
effectiveness in critical medical imaging tasks, highlighting its potential
and capabilities in this evolving field.

Figure 1: (a) Dissimilarity and targeted task variance. b-c-d) Noise-induced
dice score drop in different medical datasets.

Noise-resiliency in medical imaging analysis models: We depict the diversity
of the benchmarks we evaluate in Figure 1(a), highlighting various
characteristics such as the size, the number of classes, etc., inspired by
(Isensee et al., 2020).

Our study on noise resilience in medical imaging models highlights two key
insights. First, the pyramidal structure of U-Net models leads to increased
noise vulnerability, as evidenced by dice score reductions of 0.15 and 0.22
for U-Net and U-Net++ respectively in brain segmentation tasks (see Figure 1).
This vulnerability is attributed to their alternating down-sampling and up-
sampling design which can amplify noise variations. In contrast, Swin-like
transformer architectures exhibit remarkable noise resilience, with a
negligible 0.04 performance drop as shown in Figure 1. Their isotropic design,
which treats image patches consistently and lacks hierarchical convolutional
operations, contributes to their enhanced stability against noise
disturbances.

Model |  Avg tile utilization (%) |  Avg Reuse factor |  # Parameters (M)
---|---|---|---
U-Net (Ronneberger et al., 2015) |  7.42 |  6574.7 |  7.76
UNet++ (Zhou et al., 2018) |  12.53 |  8721.0 |  19.6
Swin UNET (Hatamizadeh et al., 2021) |  43.2 |  11540.6 |  62.19
Table 1: Performance metrics of state-of-the-art medical imaging models.

Model Inference on MRIs & CT images: In medical imaging, such as MRI and CT
scans, Analog In-memory Computing (AIMC) significantly enhances data
processing efficiency. Unlike traditional methods, AIMC’s pipelining ability
allows for rapid, parallel processing of sequential image slices, crucial for
three-dimensional anatomical analysis. This approach not only improves
throughput in urgent medical scenarios but also optimizes energy usage and
minimizes latency between slices, making it particularly effective for
volumetric data in tumor segmentation.

Figure 2: Uncertainty analysis on sample brain tumor segmentation images for
UNet++.

Certainty enhanced through noise: While noise in computational models is often
seen as detrimental, in hardware-aware training (HWA training), it
inadvertently leads to more resilient models. These models, trained under
controlled noise conditions, show improved noise tolerance and prediction
certainty, as visualized in Figure 2. In critical healthcare applications, the
certainty of model predictions is vital, as it minimizes the risk of
misdiagnosis and enhances decision-making in treatment planning. This is
especially evident when comparing digital and analog-trained models, such as
the U-Net++ architectures.

##  3 Conclusion

In-memory computing (IMC) holds promise in refining medical imaging, with
transformer structures surpassing their pyramidal counterparts in resilience
to noise. Rather than being detrimental, strategic noise injection fortifies
model precision, an essential aspect in healthcare. This approach mitigates
overfitting and boosts confidence in diagnostics. Future efforts will focus on
advancing transformer-based analog-aware architectures through the application
of neural architecture search, catering to a wide array of medical imaging
tasks.

## Acknowledgment

This research work was supported by Abu Dhabi National Oil Company (ADNOC),
Emirates NBD, Sharjah Electricity Water & Gas Authority (SEWA), Technology
Innovation Institute (TII) and GSK as the sponsors of the 4th Forum for Women
in Research (QUWA): Sustaining Women’s Empowerment in Research & Innovation at
University of Sharjah.

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##  Appendix A Appendix

Figure 3: Model uncertainty density analysis across different brain tumor
detection inputs.

###  A.1 Datasets

The Brain Tumor Segmentation dataset from The Cancer Imaging Archive (Simpson
et al., 2019) features MRI scans and segmentation masks for 110 lower-grade
glioma patients, enriched with FLAIR sequences and genomic cluster data. The
Spleen Segmentation dataset, also from the Medical Segmentation Decathlon
(Simpson et al., 2019), offers CT scans for detailed spleen segmentation,
focusing on organ delineation. Lastly, the Nuclei detection dataset (Caicedo
et al., 2019) provides a broad collection of segmented nuclei images,
showcasing diversity in cell types and imaging modalities, including
brightfield and fluorescence techniques.

###  A.2 Models & training

U-Net, effective in biomedical segmentation, offers a balanced ’U’ shaped
structure for localization. U-Net++ enhances U-Net by adding nested and skip
pathways for improved detail capture. Swin Transformer, unlike these, utilizes
shifted windows to handle image patches, focusing on long-range
interdependencies. Transitioning from initial training, the models were
adapted to hardware-aware training through noise injection, facilitated by
AIHWKIT (Rasch et al., 2021). This approach, supported by MONAI (Cardoso et
al., 2022) frameworks and other open-source tools, optimized our models for
in-memory computing applications. The simulation of the analog aware
training’s parameters and characteristics is available in the provided code
and defined in the function $create\\_rpu\\_config()$.

###  A.3 Extended Certainty analysis

Figure 3 provides an overall certainty analysis of U-Net++ and Swin U-Net when
trained with hardware training versus digital training. While the main paper
presents a compelling and illustrative example, we extend the density
quantification to the whole test set. The uncertainty is computed using Monte
Carlo Sampling. This involves using multiple passes of the input data and
observing the variability in the outputs. The lower the variability, the
higher the certainty of the model’s predictions.

Results suggest that the analog-trained model exhibits a lower uncertainty
density compared to the digital-trained model. This implies that the analog-
trained U-Net++ is be more reliable and consistent in its predictions, making
it a more suitable choice for medical applications. This is mainly due to the
hardware-aware training that forces the model to adapt to the inherent
variability and constraints of the analog computing environment, thereby
enhancing its ability to handle uncertain scenarios with greater precision.