11institutetext: University of California, Los Angeles, USA 22institutetext:
University of Kansas Medical Center, USA 33institutetext: UCLA David Geffen
School of Medicine, USA 44institutetext: Baylor College of Medicine, USA
44email<EMAIL_ADDRESS>
# Detecting Mitoses with a Convolutional Neural Network for MIDOG 2022
Challenge

Hongyan Gu 11 0000-0001-8962-9152 Mohammad Haeri 22 0000-0001-6055-9779 Shuo
Ni 11 0000-0002-4125-2373 Christopher Kazu Williams 33 0000-0003-3253-7630
Neda Zarrin-Khameh 44 0000-0001-7506-156X Shino Magaki 33 0000-0003-0433-5759
Xiang ‘Anthony’ Chen 11 0000-0002-8527-1744

###### Abstract

This work presents a mitosis detection method with only one vanilla
Convolutional Neural Network (CNN). Our method consists of two steps: given an
image, we first apply a CNN using a sliding window technique to extract
patches that have mitoses; we then calculate each extracted patch’s class
activation map to obtain the mitosis’s precise location. To increase the model
performance on high-domain-variance pathology images, we train the CNN with a
data augmentation pipeline, a noise-tolerant loss that copes with unlabeled
images, and a multi-rounded active learning strategy. In the MIDOG 2022
challenge, our approach, with an EfficientNet-b3 CNN model, achieved an
overall F1 score of 0.7323 in the preliminary test phase, and 0.6847 in the
final test phase (task 1). Our approach sheds light on the broader
applicability of class activation maps for object detections in pathology
images.

###### Keywords:

Mitosis detection Domain shift Convolutional neural network Class activation
map.

##  1 Introduction

Mitotic activity is a crucial histopathological indicator related to cancer
malignancy and patients’ prognosis [7]. Because of its importance, a
considerable amount of literature has proposed datasets [2, 3] and deep
learning models [11, 12] for mitosis detection. To date, a number of state-of-
the-art deep learning approaches detect mitoses with the segmentation task
[10, 16]. However, these methods usually require generating pixel-level
segmentation maps as ground truth. Another school of techniques utilizes
object detection with a two-stage setup — a localization model (e.g.,
RetinaNet) is employed for extracting interest locations, followed by a
classification model to tell whether these locations have mitoses [2, 11, 12].
Such a two-stage setup was reported to improve the performance of mitosis
detection compared to that with the localization model only [2].

Since adding a classification model can improve the performance, we argue that
using only one CNN model for mitosis detection is also viable. Because CNNs
cannot directly report the location of mitosis, previous works either modified
the structure of CNNs [6], or used CNNs with a small input size to reduce the
localization errors [14]. Instead, our approach extracts the location of
mitoses with the class activation map (CAM) [17], which allows CNNs to accept
a larger input size for more efficient training. Also, our approach is model-
agnostic, and can work with vanilla CNNs because calculating CAMs does not
require changing the network structure.

We validated our proposed method in MItosis DOmain Generalization (MIDOG) 2022
Challenge [1]. The challenge training set consists of 403 Hematoxylin & Eosin
(H&E) regions of interest (ROIs, average size=$5143\times 6860$ pixels),
covering six tumor types scanned from multiple scanners. 354/403 ROIs have
been labeled and have 9,501 mitotic figures. The preliminary test set includes
20 cases from four tumor types, and the final test set has 100 independent
tumor cases from ten tumor types. Given the dataset’s high variance, we
employed three techniques to improve the CNN’s performance:

  1. 1.

An augmentation pipeline with balanced-mixup [8] and stain augmentation [14];

  2. 2.

The utilization of unlabeled images for training (treated as negative), and
training with the Online Uncertainty Sample Mining (OUSM) [15] to gain
robustness with noisy labels;

  3. 3.

A multi-rounded, active learning training strategy that adds false-positive,
false-negative, and hard-negative patches after each round of training.

##  2 Methods

###  2.1 Extracting Patches for Initial Training

We randomly used $\sim 90\%$ of the image instances in the MIDOG 2022
Challenge to generate the training set and $\sim 10\%$ for the validation set.
To maximally utilize the dataset, we included unlabelled images in the
training set and treated them as negative images (i.e., no mitoses inside).
For each image, we extracted patches with the size of $240\times 240\times 3$
pixels surrounding the location of each annotation (provided by the challenge)
and placed them into the train/validation set.

Figure 1: (a) Illustration of the active learning training strategy used in
this work; (b) Examples of augmented patches according to our augmentation
pipeline; (c) Overall data processing pipeline our approach: detecting mitosis
using a convolution neural network and the class activation map.

###  2.2 Model Training

We trained an EfficientNet-b3 [13] model (input size: $240\times 240\times 3$)
with pre-trained ImageNet weights. Here, we tried to improve model performance
by constructing an online data augmentation pipeline. The pipeline includes
general image augmentation techniques, including random rotation, flip,
elastic transform, grid distortion, affine, color jitter, Gaussian blur, and
Gaussian noise. Besides, we added two augmentation methods – stain
augmentation [14] and balance-mixup [8] – to deal with the domain shift in
pathology images. Examples of augmented patches are shown in Figure 1(b). The
model was trained with an SGD optimizer with momentum 0.9, a Cosine Annealing
learning rate scheduler with warm restart (max LR=$6\times 10^{-4}$). Since we
treated all unlabeled images as negative, we further used an OUSM[15] \+
COnsistent RAnk Logits (CORAL) loss [4] to deal with noisy labels. Each round
of training had 100 epochs, and we selected the model with the highest F1
score on the validation set for inferencing.

###  2.3 Inferencing

We slid the trained EfficientNet on train and validation images with window
size $240\times 240$ and step-size 30. We then cross-referenced the CNN
predictions with the ground truth. Here, we define a positive window
classification as a true-positive if mitoses were inside the window and a
false-positive otherwise. We further define false-negative if no positive
windows surround a mitosis annotation.

###  2.4 Incrementing the Patch Dataset with Active Learning

We employed a multi-round active learning process to boost the performance of
the EfficientNet (Figure 1(a)). Each round starts with the model training on
the current train/validation set (Section 2.2). Then, the best model is
selected and applied to the images (Section 2.3). After that, false-positive,
false-negative, and hard-negative patches are added to the train/validation
set. The procedure was repeated six times until the model’s F1 score on the
validation set does not increase. Eventually, there are 103,816 patches in the
final training set, and 23,638 in the validation set.

###  2.5 Extracting Mitosis Locations with CAMs

We used the best model from the final round in Section 2.4 for the test
images. A window with a CNN probability $>0.84$ was considered positive, and
non-maximum suppression with a threshold of 0.22 was used to mitigate the
overlapping windows. For each positive window, we calculated the CAM with
GradCAM++ [5], and extracted the hotspot’s centroid as the mitosis location
(Figure 1(c)). Examples of CAMs are shown in Figure 2(a). Specific numbers and
thresholds were selected according to the best F1 performance from the
validation images.

##  3 Results

On the preliminary test phase of the MIDOG 2022 Challenge, our approach
achieved an overall F1 score of 0.7323, with 0.7313 precision and 0.7333
recall. In the final test phase of the challenge (task 1), our approach
achieved the F1 score of 0.6847 (precision:0.7559, recall: 0.6258). In sum,
our approach is $2.34\%$ higher than the baseline RetinaNet approach regarding
the overall F1 score in the preliminary test phase, but $4.21\%$ lower in the
final test phase. Please refer to the Grand-Challenge Leader-
board111https://midog2022.grand-challenge.org/evaluation/final-test-phase-
task-1-without-additional-data/leaderboard/ for more details of the test
result.

Figure 2: Examples of positive patches detected by CNN and corresponding CAMs
generated by GradCAM++ from image 240.tiff. (a) Patch samples and CAMs if
there is one mitosis inside (pointed by the green arrow); (b) Patch samples
and CAMs if there are multiple mitoses inside. Note that generated CAMs might
not report strong signals for some mitoses.

##  4 Discussion & Conclusion

Although CAMs are primarily used for explaining CNN classifications, we
demonstrate their potential usage to help detect mitoses in H&E images. We
believe our approach has the potential to work with mobile/edge devices, which
have limited computational power and are not optimized for special-structured
deep-learning models.

It is noteworthy that CAMs might fail to highlight all mitoses when there are
multiple in an image. As shown in Figure 2(b), if there were multiple mitoses
inside one positive patch, CAMs might report a strong signal for one mitosis,
while giving weak signals for others. However, we argue that our approach as-
is can still be helpful in diseases where mitosis is not a high-prevalent
histological pattern (e.g., meningiomas [9]).

On the other hand, we believe that the limitation is partly caused by using
classification models for localization tasks: given a patch with multiple
mitoses, a CNN only needs to find at least one to predict it as positive. As
such, other mitoses are more or less ignored. To compensate for the
limitation, we suggest future works to improve the quality of CAMs by aligning
them with ground-truth mitosis location maps. For example, a “CAM-loss” might
be designed to penalize the misalignment between the CAM and mitosis location
heatmaps [18]. Different from the segmentation task, generating mitosis
location maps does not require pixel-level segmentation masks: simply applying
a Gaussian kernel over the locations of mitoses can generate a mitosis
location map similar to Figure 2.

##  5 Acknowledgement

This work was funded in part by the Office of Naval Research under grant
N000142212188.

## References

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Further author information: (Send correspondence to L. Ferro, E. Virgilli)
Lisa Ferro: E-mail<EMAIL_ADDRESS>
Enrico Virgilli<EMAIL_ADDRESS>
# The TRILL project: increasing the technological readiness of Laue lenses

Lisa Ferro  Department of Physics and Earth Science, University of Ferrara,
Via Giuseppe Saragat 1/C, Ferrara (FE), 44122, Italy  Enrico Virgilli
INAF/OAS of Bologna, Via Piero Gobetti 93/3, Bologna (BO), 40129, Italy
Miguel Fernandes Moita  Department of Physics and Earth Science, University of
Ferrara, Via Giuseppe Saragat 1/C, Ferrara (FE), 44122, Italy  Filippo
Frontera  Department of Physics and Earth Science, University of Ferrara, Via
Giuseppe Saragat 1/C, Ferrara (FE), 44122, Italy  INAF/OAS of Bologna, Via
Piero Gobetti 93/3, Bologna (BO), 40129, Italy  Piero Rosati  Department of
Physics and Earth Science, University of Ferrara, Via Giuseppe Saragat 1/C,
Ferrara (FE), 44122, Italy  INAF/OAS of Bologna, Via Piero Gobetti 93/3,
Bologna (BO), 40129, Italy  INFN of Ferrara, Via Giuseppe Saragat 1/C, Ferrara
(FE), 44122, Italy  Cristiano Guidorzi  Department of Physics and Earth
Science, University of Ferrara, Via Giuseppe Saragat 1/C, Ferrara (FE), 44122,
Italy  INAF/OAS of Bologna, Via Piero Gobetti 93/3, Bologna (BO), 40129, Italy
INFN of Ferrara, Via Giuseppe Saragat 1/C, Ferrara (FE), 44122, Italy  Claudio
Ferrari  Institute of Materials for Electronics and Magnetism (CNR-IMEM),
Parco Area delle Scienze 37/A, Parma (PR), 43124, Italy  Riccardo Lolli
Department of Physics and Earth Science, University of Ferrara, Via Giuseppe
Saragat 1/C, Ferrara (FE), 44122, Italy  Ezio Caroli  INAF/OAS of Bologna, Via
Piero Gobetti 93/3, Bologna (BO), 40129, Italy  Natalia Auricchio  INAF/OAS of
Bologna, Via Piero Gobetti 93/3, Bologna (BO), 40129, Italy  John B. Stephen
INAF/OAS of Bologna, Via Piero Gobetti 93/3, Bologna (BO), 40129, Italy
Stefano Del Sordo  INAF/IASF of Palermo, Via Ugo La Malfa 153, Palermo (PA),
90146, Italy  Carmelo Gargano  INAF/IASF of Palermo, Via Ugo La Malfa 153,
Palermo (PA), 90146, Italy  Stefano Squerzanti  INFN of Ferrara, Via Giuseppe
Saragat 1/C, Ferrara (FE), 44122, Italy  Mauro Pucci  National Institute of
Optics (CNR-INO),
Largo Enrico Fermi 6, Florence (FI), 50125, Italy  Olivier Limousin
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France  Aline Meuris  Université Paris-Saclay, Université Paris
Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Philippe Laurent
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France  Hugo Allaire  Université Paris-Saclay, Université Paris
Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France

###### Abstract

Hard X-/soft Gamma-ray astronomy ($\mathrm{>}$100 keV) is a crucial field for
the study of important astrophysical phenomena such as the 511 keV positron
annihilation line in the Galactic center region and its origin, gamma-ray
bursts, soft gamma-ray repeaters, nuclear lines from SN explosions and more.
However, several key questions in this field require sensitivity and angular
resolution that are hardly achievable with present technology. A new
generation of instruments suitable to focus hard X-/soft Gamma-rays is
necessary to overcome the technological limitations of current direct-viewing
telescopes. One solution is using Laue lenses based on Bragg’s diffraction in
a transmission configuration. To date, this technology is in an advanced stage
of development and further efforts are being made in order to significantly
increase its technology readiness level (TRL). To this end, massive production
of suitable crystals is required, as well as an improvement of the capability
of their alignment. Such a technological improvement could be exploited in
stratospheric balloon experiments and, ultimately, in space missions with a
telescope of about 20 m focal length, capable of focusing over a broad energy
pass-band. We present the latest technological developments of the TRILL
(Technological Readiness Increase for Laue Lenses) project, supported by ASI,
devoted to the advancement of the technological readiness of Laue lenses. We
show the method we developed for preparing suitable bent Germanium and Silicon
crystals and the latest advancements in crystals alignment technology.

###### keywords:

Laue lenses, focussing instruments, Gamma-ray astronomy

##  1 Laue Lenses: concentrators for Hard X-/Soft Gamma-rays

Laue lenses are a type of hard X-/Gamma-ray focusing optics based on Bragg’s
law of diffraction in crystals. In this section we will briefly describe the
basic concept of a Laue lens.

###  1.1 Laue lenses basics

Bragg’s law of diffraction in a crystal lattice can be written as[1]:

$2d_{hkl}\sin\theta_{B}=n\frac{hc}{E}$ (1)

where $\theta_{B}$ is the Bragg’s angle, i.e. the angle between the chosen
diffraction planes of the crystal and the incident beam, $d_{hkl}$ is the
inter-planar spacing of the diffracting planes, $(hkl)$ are the Miller indexes
of the planes, $E$ is the energy in keV of the gamma-ray photons and $n$ is
the order of diffraction. If we consider a simple, flat crystal, and we
imagine that the crystal divides the space in two regions, we can define two
possible configurations:

  1. 1.

Bragg configuration, or ”reflection”, geometry, in which the diffracted beam
is in the same region of space as the beam incoming on the crystals (Fig. 1,
left).

  2. 2.

Laue configuration, or ”transmission”, geometry, in which the diffracted beam
crosses the crystal and is transmitted in the region of space opposite to the
incoming beam (Fig. 1, right).

Figure 1: Left: Bragg’s diffraction in reflection (Bragg) configuration. The
diffracted beam is reflected by the crystal in the same region of space as the
incoming beam. Right: Bragg’s diffraction in transmission (Laue)
configuration, in which the beam crosses the crystals and is diffracted in the
opposite region of space with respect to the incoming beam.

Bragg’s diffraction in transmission geometry is effective up to energies of a
few MeV, meaning that it is possible to build hard X-/Gamma ray optics based
on the diffraction process. Specifically, the transmission configuration can
be advantageous in an astrophysical context since it allows us to obtain a
larger effective area with respect to focusing techniques based on multilayer
optics.

A hard X-/soft gamma-ray optics based on Bragg’s law in transmission
configuration is called a Laue lens. A Laue lens can be visualized as a
spherical cap covered by crystal tiles oriented in such a way that the
radiation coming from the sky is transmitted through the crystals and focused
towards a point which is the lens focus, similarly to optical lenses. At the
focus, a suitable position sensitive detector is placed. It can be shown that
the focal distance of the lens $f$ is equal to half of the curvature radius of
the spherical cap [2] (Fig. 2) .

Figure 2: Side and top view of a Laue lens. The crystals are positioned on a
spherical support in such a way that the radiation coming from the sky,
parallel to the optical axis of the lens, interacts with the crystals and is
focused. The diffraction planes of the crystals are oriented in such a way
that the angle between them and the incoming X-ray beam is equal to Bragg’s
angle $\theta_{B}$, so the angle between the diffracted beam and the incoming
beam is $2\theta_{B}$. The focal length is equal to half the curvature radius
of the spherical cap.

The simplest possible distribution for the crystals on the cap is to place the
crystals in concentric rings at a distance $r$ from the axis of the lens.
Using the Bragg’s law, we can express the centroid of the energy spectrum of
the photons diffracted by each ring as[2]:

$E=\frac{hc}{2d_{hkl}\sin\Big{[}\frac{1}{2}\arctan\Big{(}\frac{r}{f}\Big{)}\Big{]}}\sim\frac{hc}{d_{hkl}}\frac{f}{r}$
(2)

which means that hard photons are diffracted by the crystals in the innermost
rings of the lens, while soft photons are diffracted by the outermost rings of
the Laue lens. The approximation in Eq.2 holds true for small Bragg’s angles,
which is true for hard x-/soft gamma-rays, being of the order of 1 degree or
less. Thanks to the technology of Laue lenses, we will be able to build
lightweight ($<$150 kg), stable, broadband optics which will enable real
concentration of hard X-/Gamma-rays up to 700 keV and with a Point Spread
Function (PSF) with a Half Power Diameter (HPD) of the order of 30 arcsec or
less, at 20 m focal distance.

A comparison with one of today’s most advanced high energy instruments, the
IBIS aboard INTEGRAL, whose angular resolution is about 12 arcmin, shows the
quantum leap in imaging capabilities that would be gained by an instrument of
this type, which will allow high sensitive spectroscopy and will enable
polarimetry in the high energy domain based on Compton kinematics. Such an
improvement in terms of sensitivity and new physical information would allow
us to answer some of the most debated questions in high-energy astrophysics,
such as the origin of the annihilation line from the Galactic center, the
study of GRB afterglow emission, emission spectra of blazars and AGN galaxies,
studies on the nuclear lines from supernova (SN) explosions and many more. We
refer the reader to the two white papers submitted to the European Space
Agency for its long term program ”Voyage 2050” for more details on the science
goals. [3, 4].

###  1.2 Crystals for Laue lenses

The fundamental requirements for a Laue lens are its energy pass-band, its
effective area in that energy range, its focusing capability and, ultimately,
the instrument sensitivity. These requirements can be optimized with a correct
selection of crystals having the appropriate material, thickness and
diffraction planes.

Several types of crystals can be used to build a Laue lens, however for our
applications we considered two main families: perfect crystals and mosaic
crystals. Ideally, a perfect crystal is a crystal in which the lattice shows
negligible imperfections of any kind: no dislocations and high chemical
purity.

The rocking curve of a perfect crystal, i.e. the measurement of the intensity
of the diffracted beam with respect to the angle at which the crystal is
oriented, is a Gaussian centered on the Bragg’s angle and its Full Width at
Half Maximum (FWHM) is called Darwin width [1]. The energy passband of a
perfect crystal is very sharp, centered on the values that satisfy Bragg’s
condition. This means that perfect crystals are very good monochromators, but
they are not the best solution if we want to build, as in our case, a
broadband instrument with a smooth effective area.

Instead, a mosaic crystal, is a type of crystal which can be described as a
mosaic of microscopic perfect crystals, called crystallites. In a mosaic
crystal, all the crystallites are slightly misaligned around an average
direction. The distribution of the crystallites misalignment can be
approximated by a Gaussian:

$W(\delta)=\frac{1}{\sqrt{2\pi\eta}}\exp{\bigg{(}-\frac{\delta^{2}}{2\eta}\bigg{)}}\;,$
(3)

where $\delta$ is the angular deviation of the crystallites with respect to
their average direction. The FWHM of the distribution $\beta=2.355\,\eta$ is
called the crystal mosaicity. The presence of a non-zero mosaicity has the
effect of increasing the energy passband of the crystals, since it increases
the range of angles satisfying the Bragg’s condition. The downside is that
this wide energy passband impacts on the size of the diffracted image produced
by a mosaic crystal. Due to the mosaic spread, the diffracted photons are
focused over an enlarged area, whose extension also depends on the focal
length. This effect is called mosaic defocusing and it slightly enlarges the
Point Spread Function (PSF) of a mosaic crystal with respect to one of a
perfect crystal.

Figure 3: Explanation of the concentration effect given by cylindrical bent
crystal: due to the fact that the crystal is bent, the average direction of
the diffraction planes (red arrows) varies continuously inside the crystals,
which means that the Bragg’s angle of the incoming radiation (blue arrows)
changes accordingly. The result is that the X-ray beam is concentrated in a
point at a distance $f=R/2$ and that the crystal’s energy pass-band is
enlarged of a quantity which depends on the length of the focusing direction
$d$ of the crystal, its curvature radius $R$ and its average diffracted
energy.

With our past HAXTEL project [5], flat mosaic crystals of copper were
successfully used to build a prototype of a Laue lens with a low ($<$6 m)
focal length. However, flat crystals (perfect or mosaic) show important
limits: first, they do not possess intrinsic concentration capability: the
size of the resulting image is comparable to the cross section of the crystal
itself. Second, the diffraction efficiency for a flat crystal cannot be higher
than 50 % [1]. Instead, with bent crystals we can overcome these limitations.
While in bent mosaic crystals the average direction of the planes varies
according to a curvature radius, in bent perfect crystals the diffraction
planes themselves are bent. Indeed, for some specific crystallographic
configurations a secondary curvature is induced by impressing a primary
curvature to a perfect crystal. This secondary or internal curvature results
in an increased angular spread of the planes called quasi-mosaicity [6, 7].
The easiest type of curvature which can be obtained is a cylindrical
curvature. A crystal, either perfect or mosaic, bent in this way acts as a
concentrator along the bent direction, focusing the radiation coming from an
astrophysical source to a focal distance equal to half the curvature radius of
the crystal itself[8] (Fig. 3). Thus, through bent crystals it is then
possible to decrease the size of the PSF and to enlarge the energy passband,
since a wider range of Bragg’s angle can be accepted by a bent crystal. In
addition, depending on the arranged internal structure of the atoms within the
crystal, their diffraction efficiency can be higher than 50%.

Figure 4: Right: Schematics of the concentration effect of one cylindrical
bent, 30$\times$10 mm2 crystals such as the one used in our tests. The
radiation (in red) is concentrated along the focusing direction and the image
is produced at a distance equal to half the curvature radius of the crystal.
Left: Experimental image of a concentrated 150 keV X-ray beam obtained at the
LARIX Facility. The area of the crystal is of 30$\times$10 mm2, while the area
of the image is of 0.1$\times$10 mm2, which means that the beam was
concentrated in an area 30 times smaller than the size of the crystal itself.

Indeed, according to the dynamical theory of diffraction in bent perfect
crystals, the presence of secondary curvature of the planes increases the
reflectivity of the crystal above the limit of 50%[6]. Intuitively, we can
imagine that, since the planes are bent, the probability that a beam crossing
the crystal undergoes a reflection is highly reduced, thus increasing the
diffraction efficiency.

For the purpose of building a broadband Laue lens with focusing capabilities
and an optimized effective area, we aim to use, for our lens, bent perfect
crystals of Silicon and Germanium with (111) diffracting planes. Tests done
with mosaic Gallium Arsenide (220) crystals bent to a curvature radius of 40 m
with a surface lapping technique[9, 10] allowed to experimentally verify that
bent crystals are able to focus high energy X-rays[11] (Fig. 4). Further tests
are being made with Ge(220) crystals and we plan to switch to perfect Si(111)
and Ge(111), since the (111) planes should shows an increase of efficiency as
described by Malgrange’s theory of bent perfect crystal[12]

##  2 The TRILL project and its results

The TRILL project (Technological Readiness Level Increase for Laue Lenses) was
funded by the Italian Space Agency (ASI) through the National Institute for
Astrophysics (INAF). Its aim is to increase the technological readiness level
(TRL) of Laue lenses. For our prototype of the Laue lens, we opted for a
modular approach: a full lens is divided in spherical sectors (petals) and
each sector is then divided into smaller modules which can contain up to 30
crystals each (Fig. 5).

The crystals would be perfect crystal of Si and Ge with diffracting planes
(111) and would be bent with the same curvature radius of $40$ m, which means
that the focal length of the Laue lens for a source at infinite distance is
$20$ m.

Aim of the TRILL project is to build four modules and join them together in a
first section of a petal. The desired PSF size of the whole lens must be of 30
arcsec, which translates in requiring that the Bragg’s angle and the radial
position of the crystal tiles are set within an accuracy better than 10
arcsec.

Figure 5: Left: Schematic CAD model of a full Laue lens. The lens is divided
in spherical sectors, called petals (red), and each petal is divided in a
series of modules (cyan). The crystals are fixed on each module. Top Right:
CAD model of the four modules we are building joined together. Center Right:
Front view of a single module with 20 crystals on top. Bottom Right: Rear view
of a single module. The substrate on which the crystals are bounded is
visible.

To reach those goals, the project is structured with the following main tasks:

  1. 1.

To define a reliable and repeatable way to bend the crystals with no
deterioration of their properties. The curvature radius of the bent Si and Ge
crystals is required to be 40 m with an accuracy of $\pm~{}2$ m, and must be
as uniform as possible to avoid distortion on the lens PSF.

  2. 2.

To find the best materials and the bonding method to build a module of Laue
lens with the desired $<$10 arcsec orientation accuracy for the crystals, and
to find a way to assemble together multiple modules.

  3. 3.

To test the prototype with a suitable focal plane detector. In order to reach
an efficiency of the order of 80% at 600 keV, a solid solid state, highly
segmented, spectral-imager detector has been chosen. This choice will allow to
fully exploit the focusing capabilities of the lens both for imaging, for
spectroscopy and polarimetry.

In this section, we illustrate the status of advancement of each of those 3
tasks.

###  2.1 Crystals’ manufacturing

The manufacturing of the bent perfect crystals of Silicon and Germanium is
done at the CNR/IMEM institute in Parma (Italy). Dislocation-free wafers of Si
and Ge of diameter of 4 inches are cut to obtain about 15 crystals of size 30
x 10 x 2 mm3. The crystals are cut in such a way that the chosen diffracting
planes are (220). These planes are parallel to the 10 x 2 mm2 edge, with a
miscut error of about 0.2 degrees. Each tile is bent to the nominal curvature
radius of 40 m through a surface lapping method. This technique consists in
lapping one side of the crystal inducing a controlled damage which generates
an internal strain able to bend the crystal in a self-standing way[9, 10]. The
first tests of this technique, done with GaAs(220) mosaic crystals, showed
that there is a range of values of curvature radii (see Fig. 6) for which the
enlargement of the PSF of the crystal, induced by a not nominal curvature
radius, is hidden by the defocusing effect due to the dispersion of the
diffracting planes. This allows us to reduce the constraints on the
distribution of the curvature radius of the crystals depending on the value of
the angular dispersion of the planes. The smaller is the angular spread of a
crystal, i.e. the mosaicity or the quasi-mosaicity, the more strict is the
requirement on the curvature radius compared with the nominal one.

Figure 6: Black points: measured size of PSF of a sample of GaAs crystals
versus their curvature radius. It can be immediately seen that between 38 m
and 44 m of curvature radius, the deformation of the crystals’ curvature
radius from the optimal value has no effect on the size of the PSF. Black
solid line: fit of the data points. Black dashed line: Theoretical PSF FWHM vs
curvature radius curve for a Ge(111) crystal with quasi-mosaicity of 5 arcsec.
The region in which the defocusing prevails on the radial deformation is
reduced to about 2 m around the optimal value of 40 m.

For perfect Si(111) and Ge(111) crystals, which we aim to use for the
definitive version of the lens, we expect that the bending of the crystals
will induce a dispersion of the diffraction planes (quasi-mosaicity) of the
crystals of $\sim$5 arcsec, which means that if the curvature radius of the
crystals is within $\pm 2$ m from the nominal value of $40$ m, then the
distortion of the PSF given by crystals with a slightly wrong curvature radius
is negligible. This means that the technique to bend the crystals that we are
developing must guarantee that all the produced bent crystals have a curvature
radius in the range $\mathrm{40\pm 2}$ m.

In the context of the TRILL project, the surface lapping technique was tested
with Ge(220) crystals. The average value of the curvature radii distribution
is of 39.7 $\pm$ 0.2 m, with an estimate of the standard deviation of 1 m
(Fig. 7), meaning that all the crystals are suitable for the TRILL goals. The
curvatures of the crystals were measured at CNR/IMEM with a Phillips X’Pert
PRO high resolution X-ray diffractometer, equipped with a Cu
K$\mathrm{\alpha}$ source.

Figure 7: Curvature radius of the sample of 82 bent crystals of Ge(220) vs
their thickness after the surface lapping procedure. Each point represents a
crystal. All the curvature radii are within the limit $\pm$2 m from the
nominal curvature radius of 40 m.

The lapping procedure shows two major downsides. The first is that it
necessarily removes part of the material, therefore the thickness of the
crystals significantly decreased with respect to the initial thickness. The
average value of the final thickness of the crystals after the lapping
procedure is of (1.656 $\pm$ 0.002) mm (Fig. 7). The second downside of this
technique is that, to date, crystals with thickness higher than 2 mm are not
bendable at the nominal 40 m radius. This represents an important drawback
given that, especially for the higher energies, thicker crystals are required
for optimizing the crystals diffraction efficiency. Even with those
limitations, the surface lapping technique results to be a very accurate,
reliable and repeatable procedure to prepare crystals for a Laue lens.

###  2.2 Module and petal assembly

The second goal of the TRILL project is to define a method to build a Laue
lens module by bonding the crystals on an adequate substrate with an accuracy
sufficient to obtain a full lens PSF with 30 arcsec HPD. If the crystals are
placed in concentric rings, two angles define the position of each tile: the
Bragg’s angle at which the crystals are set and the azimuth angle which
indicates their position on the ring. The presence of a misalignment angle has
the effect to shift the PSF from the desired focal point, meaning that for a
full lens, the combined effect of the shifts of all the crystals’ PSFs will
result in a broadening of the PSF of the lens. It is then crucial to keep the
misalignment errors as small as possible. The error on the Bragg’s angle is
the more critical, since the shift it induces is proportional to the focal
length of the lens (Fig. 8). In this paper we will use the name
$\theta$-misalignment to indicate the misalignment on the Bragg’s angle and
the $\phi$-misalignment to indicate the misalignment on the azimuth angle.

Figure 8: Schematics of the two main angles that need to be set to orient a
crystal: the Bragg’s angle (left) and the azimuth angle of the crystal on the
ring (right). If a misalignment error on the angles is present (red), the
image of the crystal will be shifted with respect to the expected position
(black). In this drawing, $F$ is the focal length of the lens and $r$ is the
distance from the centre of the lens at which the crystal is positioned, i.e.
the radius of the ring on which the crystal is positioned.

Ray-tracing simulations of a full Laue lens made of bent perfect crystals of
Ge(111) established that it is necessary that all the crystals of a module are
oriented with an accuracy, on both angles, of $\mathrm{<}$10 arcsec to obtain
the desired PSF of 30 arcsec (FWHM) for the full lens, which results in a
positional accuracy of the crystals of the order of a few microns. High
precision bonding techniques need to be developed and applied to fulfill such
a technologically challenging task.

We decided to bond the crystals to a suitable substrate by means of a low-
shrinkage adhesive. The low-shrinkage requirement comes from the fact that it
is crucial that the natural shrinkage of the glue during the curing process is
as small as possible to reduce and control the misalignment on the crystal
positions induced by the bonding procedure. Furthermore, to continuously test
the positions of the crystals during the process, we use a hard X-ray
horizontal beam, so the crystals must stay in vertical position during the
bonding process. This means that the glue must be viscous enough to not drip
down after its deposition.

Figure 9: Left: The eleven Ge(220) crystals fixed on the quartz substrate.
Right: Combined, real-time image of the 11 crystals. The 4 outlier crystal and
their misalignment are highlighted. Figure 10: Left: $\theta$-Misalignment of
the crystals vs time from bonding. Right:$\phi$-Misalignment of the crystals
vs time from bonding.

The used adhesive is a single component UV-curable adhesive with a linear
shrinkage of 0.03% and a viscosity of 60,000 cP. The substrate was chosen in
order to satisfy the following requirements: 1) the hard X-rays must pass
through the material without being heavily absorbed; 2) it should be
transparent enough to UV-light to allow it to pass through the substrate to
guarantee the complete cure of the adhesive; 3) it should have a low
coefficient of thermal expansion (CTE). To satisfy all three requirements we
chose a substrate of quartz 5 mm thick, since the quartz is transparent to UV,
has an X-ray transmission sufficiently high ($\sim$84% at 150 keV) and a low
CTE ($\mathrm{5.5\times 10^{-7}~{}K^{-1}}$). Other materials, such as Zerodur
and Ultra Low Expansion Titanium Silicate glass, are under consideration. Each
substrate is shaped as an isosceles trapezium with bases of 68 mm and 56 mm
long, respectively, while the height is 183 mm. Given that the cross section
of the crystal tiles is 30$\times$10 mm2 and that the spacing between the
crystals glued on the substrate will be of about 1 mm, each substrate can hold
about 30 crystals. The surface of the substrate, on which the tiles are glued,
was worked at the CNR/INO Institute in Arcetri (Florence, Italy), to impress a
curvature radius of 40 m, which is the cylindrical curvature of the crystal
tiles. To glue the tiles to the substrate, this is hold in vertical position
though an INVAR steel frame. The crystals are positioned on the glass
substrate using a motorized hexapod with six degrees of freedom: three
translations and three rotations. The hexapod has an accuracy of the order of
1 $\mu$m for translations and of 2$\times$10-5 radians for rotations.

Figure 11: Exploded view of a Laue lens petal assembly. The crystals are
bounded on a quartz substrate, then the substrate is fixed to a steel main
frame with an intermediate frame of flexible material.

To test the correct alignment of the crystals, we hit them with a collimated,
polychromatic hard X-ray beam (beam cross section of 10 x 10 mm2) and collect
the diffracted image with a Perkin Elmer hard X-ray imager (sensitive area 20
x 20 cm2 with spatial resolution of 200 $\mathrm{\mu}$m). The shift of the
image from a target position, usually the center of the detector, allows us to
evaluate how much the crystal is misaligned from the desired position.

The bonding procedure is the following:

  1. 1.

the crystal is positioned on the hexapod and oriented to the proper Bragg’s
angle under the X-ray beam;

  2. 2.

a small drop of glue is deposited on the substrate with a glue dispenser;

  3. 3.

the crystal is placed in contact with the drop of glue. We check that the
crystal is still in the correct position. An angular correction can still be
applied at this stage;

  4. 4.

the glue is cured with UV light;

  5. 5.

the crystal is released from the hexapod and its final position is evaluated
immediately after the cure.

To test the time stability of the bonding, we checked the position of the
crystals each day. With this method we have built one module made of GaAs(220)
crystals already available at our lab and one made of Ge(220) crystals. For
the first prototype module, we tested only our capabilities to reach the
desired accuracy on the Bragg’s angle position and we found that the
misalignment of the crystals after 50 days from bonding is within the interval
+15/-10 arcsec. For the second prototype, we tested our capabilities to
correctly set both the Bragg’s angle and the azimuth angle of the crystals.
Eleven crystals of Ge(220) were fixed on the substrate following the procedure
described above and the first real-time image of the PSF of all the crystals
combined together was obtained (Fig.9). For this second test we found that,
after about 25 days from the cure, the average value of the
$\theta$-misalignment is of 101 arcsec, with a standard deviation of 161
arcsec (see left panel of Fig. 10). Four crystals (28B, 26C, 30A, 31A) results
to be strongly misaligned. We suspect that this may be due to a deformation of
the region of the substrate on which they are glued. The average value of the
$\phi$-misalignment is found to be 2 arcsec, with a standard deviation of 9
arcsec. The $\phi$-misalignment versus time from the cure is shown in the
right panel of Fig. 10.

These results state that the required accuracy on the $\phi$-misalignment has
been achieved, while we are still one order of magnitude far from reaching the
same results on the $\theta$-misalignment. One positive aspect is that, once
cured, the adhesive guarantees a good stability of the assembly on long time
scale. We argue that the $\theta$-misalignment is due to the technical
difficulty of distributing the layer of adhesive uniformly and with a
controlled thickness. The next tests will be dedicated to control the
distribution of the adhesive with an accuracy of a tenth of microns.

Finally, the development of a way to assemble the modules together is
currently in a study phase. At the moment, we plan to join the modules
together by fixing them on an intermediate frame. The frame will be oriented
with respect to a main rigid frame through high precision micrometric screws
(Fig. 11).

###  2.3 The focal plane detector

Figure 12: The WIX-HD setup, which contains the Caliste-HD detector and its
mechanical structures, enclosed in a tight thermal controlled atmosphere. The
box contains the read-out electronics.

To exploit at best the focusing capabilities of a Laue lens, the best focal
plane detector is a highly-segmented spectral-imager detector with 3-D spatial
resolution. In this way it is possible to perform spectroscopic measurements
and imaging of a source at the same time. Both the segmentation and the 3-D
resolution, together with a very good spectral resolution, allow us to use the
detector also as a Compton polarimeter. Solid state, segmented detectors based
on Cadmium Telluride (CdTe) or Cadmiun Zinc Telluried (CZT) are the candidate
solid state materials for our scope, since their high atomic number makes them
very efficient for high energy photons. Furthermore, they can be used at room
temperature or with a minimal cooling made by a Peltier cell, thus they are
easy to handle and with a low power consumption and weight, making them great
candidate for a space mission detector.

Different spectral-imager detectors were taken into consideration for a
combined use with Laue lenses. In the context of the TRILL project, we decided
that the best detector for testing our Laue lens prototype is the Caliste-HD
device[13, 14], developed at CEA (Fig. 12). Caliste-HD is made by an
Al/CdTe/Pt pixel detector (leakage current $<$10 pA) and is fully compliant
with space operations (radiation resistant, and resistant to thermal and
mechanical instabilities). The sensitive area of the detector is 1 x 1 cm2,
with a crystal thickness of 1 mm and a pixel size of 0.625 x 0.625 mm2. The
pass-band of the detector is from 2 keV to 1 MeV and it has an energy
resolution of 1.1% at 60 keV.

At the moment, the spectroscopic and imaging capabilities of the Caliste-HD
detector are being studied at INAF-OAS Bologna (Italy). In order to evaluate
the spectral-imaging capability of the detector, it will be used in
combination with the Ge(220) Laue lens module.

##  3 The future: space missions based on Laue lenses

Figure 13: Artistic representation of ASTENA’s in flight configuration. The
WFM-I/S units surrounds the body of the spacecraft, on which the Laue lens
(orange) will be put on top. The focal plane detector (cyan) will be kept at a
distance of 20 m from the lens by means of an extensible mast.

Our studies on Laue lenses, including the TRILL project, are aimed to develop
the technology for building the Narrow Field Telescope (NFT) on-board ASTENA,
the Advanced Surveyor for Transient Events and Nuclear Astrophysics, a concept
mission that we proposed to achieve the goals discussed in two white papers we
submitted to ESA for its long term program ”Voyage 2050” [4, 3] (Fig. 13).
ASTENA’s payload consists of two main instruments: the Wide Field Monitor -
Imager/Spectrometer (WFM-I/S) and the Narrow Field Telescope (NFT). The
WFM-I/S consists of twelve Position Sensitive Detector (PSD) units surmounted
by a double-scale coded mask, one of which (for low energies) is one-
dimensional. The PSD is based on the same principles of the X/Gamma-ray
Imaging Spectrometer (XGIS) proposed for the THESEUS space mission [15]. The
NFT will be a revolutionary hard X/soft Gamma-ray focusing telescope working
in the energy band 50-700 keV based on the technology of Laue lenses. At the
moment, a possible pathfinder of the ASTENA mission concept, with only the
NFT, is also under study, for a satellite or balloon experiment. The NFT is a
Laue lens of 3 m in diameter with a 20 m focal length. The lens will be made
of about 20 thousand crystal tiles of perfect Si(111) and Ge(111) of cross
section 30$\times$10 mm2 and thickness optimized for the diffraction
efficiency vs. energy. The crystals will be radially bent with a curvature
radius of 40 m. In the context of the TRILL project, we are developing and
testing some possible techniques that will be used for manufacturing the
crystals and build the whole lens. The focal plane detector will be a
pixelated CdZnTe spectral-imager detector, with a size of 8$\times$8$\times$8
cm3, a pitch of 300 $\mu$m and an efficiency $>$80% in the whole energy band
of NFT. In this configuration, the NFT is expected to achieve an unprecedented
angular resolution in the sub-MeV energy range of 30 arcsec and a point source
localization accuracy $<$10 arcsec, with a Field of View of 4 arcmin. The NFT
will bring a leap in sensitivity of two order of magnitude with respect to the
best current instruments operating in the same energy bands, opening a new
range of possibilities for high energy astronomical observations.

###### Acknowledgements.

This work is partly supported by the AHEAD-2020 Project grant agreement 871158
of the European Union’s Horizon 2020 Programme and by the ASI-INAF agreement
no. 2017-14-H.O ”Studies for future scientific missions”.

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