research papers

J. Appl. Cryst. (2015). 48, 125–137 doi:10.1107/S1600576714026235 125

Journal of

AppliedCrystallography

ISSN 1600-5767

Received 8 August 2014

Accepted 29 November 2014

# 2015 International Union of Crystallography

Design study of a Laue lens for nuclear medicine

Gianfranco Paterno, Valerio Bellucci, Riccardo Camattari and Vincenzo Guidi*

Department of Physics and Earth Sciences, University of Ferrara, Via Saragat 1/c, 44122 Ferrara and

INFN Section of Ferrara, Italy. Correspondence e-mail: guidi@fe.infn.it

A Laue lens is an ensemble of crystals capable of focusing, through diffraction in

transmission geometry, a fraction of the photons emitted by an X- or �-ray

source onto a small area of a detector. The present study facilitates a thorough

understanding of the effect of each system parameter on the efficiency, the

resolution and the field of view of the lens. In this way, the structure and the size

of the crystals can be set to achieve a compact lens capable of providing a high-

resolution image of the radioactivity distribution lying inside a restricted region

of a patient’s body. As an application, a Laue lens optimized at 140.5 keV, the �-

line emitted by 99mTc, has been designed. The lens is composed of ten rings of

Ge crystals with curved diffracting planes and focuses the photons onto a

detector 50 cm apart from the source with 1.16 � 10�5 efficiency and 0.2 mm

resolution. The combination of these two important figures of merit makes the

proposed device better performing than pinhole single photon emission

computed tomography, which is the technique employed for top-resolution

images in nuclear medicine. Finally, the imaging capability of the designed lens

has been tested through simulations performed with a custom-made Monte

Carlo code.

1. IntroductionThe techniques of diagnostic nuclear medicine, namely scin-

tigraphy, single photon emission computed tomography

(SPECT) and positron emission tomography (PET), represent

some of the best methods for medical imaging (Bushberg et al.,

2002). While X-ray-based computed tomography (CT) and

magnetic resonance imaging (MRI) provide accurate images

of anatomical districts, diagnostic nuclear medicine permits

the analysis of some metabolic processes and an early recog-

nition of tumour masses. More specifically, in a CT examina-

tion, the patient’s body is irradiated through an external X-ray

beam emitted by an X-ray tube capable of rotating around the

bed. Since the absorbtion of X-rays by tissues basically

depends on their density, morphological imaging can be

performed by detecting the unabsorbed radiation through a

detector positioned behind the patient. In nuclear medicine, a

molecule directly involved in a specific metabolic process is

marked with a short-lived radioactive atom. Such compounds,

called radiopharmaceuticals or radiotracers, are given,

generally by injection, to the patient and accumulate in a

specific organ or anatomical district. The most used radio-

tracers are shown in Table 1. They are � emitters for scinti-

graphy and SPECT and �þ emitters for PET. In the PET case,

positrons annihilate with electrons producing two 511 keV

back-to-back photons. Therefore, with both kinds of radio-

tracers, a concentration map of the metabolic activity, i.e. a

functional imaging, can be obtained by detecting the emitted

photons. The performances of scintigraphy and SPECT are

heavily influenced by the collimator positioned before the

detector and used to discriminate the direction of the photons.

This element leads to a trade-off between efficiency and

spatial resolution. Typical values for resolution of conven-

tional SPECT and PET are 5–15 and 3–5 mm, respectively. In

the PET case there is an intrinsic limitation due to the path of

the positrons within the tissue before their annihilation

(Rahmim & Zaidi, 2008). If a resolution close to 1 mm is

required, pinhole SPECT is typically employed (Beekman &

van der Have, 2007). Nevertheless, this technique allows the

investigation of a limited region and shows a resolution–effi-

ciency trade-off. Therefore, the conventional techniques suffer

from low spatial resolution and low signal-to-noise ratio for a

small radioactive source, e.g. a tumour in an initial stage of

development.

The image resolution and quality can be increased by using

an efficient focusing device. Since the real part of the index of

refraction of all materials is approximately equal to one for

high-energy photons, the use of common optical elements is

prevented (Authier, 2001).

The desired effect can be obtained by using a Laue lens,

namely a device that exploits diffraction in crystals to

concentrate a large number of photons onto a small area of a

detector. These optics were initially studied for the realization

of a high-energy telescope (Smither, 1982) and subsequently

have also been proposed for use in nuclear medicine (Smither

& Roa, 2000). Fig. 1 schematically shows the configuration of a

diagnostic system exploiting a Laue lens.

Some prototypes of Laue lens for nuclear medicine have

already been realized (Roa et al., 2005). However, they

provide at the most the same image resolution achievable with

conventional PET. Thus, there is the need to further improve

the lens performance in such a way that its clinical use can

actually become convenient.

In this paper we present a detailed study and the design

principles for a Laue lens when, like in nuclear medicine, a

monochromatic �-ray source is employed. The proposed

method allows high-resolution images of small radioactive

sources to be obtained. The resolution can be one order of

magnitude better than the level attainable in conventional

nuclear medicine examinations.

2. High-energy radiation diffraction in crystals

2.1. Basic concepts

A Laue lens is based on X- or �-ray diffraction, which is a

coherent effect carried out by parallel atomic planes within a

crystalline material. Incident photons interact elastically with

the electrons of the lattice atoms and deviate from their

trajectory. Reflected waves interfere constructively, giving rise

to a diffracted beam, provided that their paths through the

crystal lead to a phase shift that is a multiple of the wave-

length. This condition occurs if Bragg’s law is satisfied:

2dhkl sin �B ¼ �; ð1Þ

where dhkl is the spacing between atomic planes, �B the angle

subtended by the incoming �-ray trajectory and the diffracting

lattice planes, and � the wavelength of the radiation. The

Bragg angle, �B, depends on the orientation of the lattice

planes. For a cubic crystal, such as Cu, GaAs, Si or Ge, the

spacing between the planes can be expressed as

dhkl ¼a

ðh2 þ k2 þ l2Þ1=2; ð2Þ

where a is the lattice constant of the crystal and h, k, l are the

Miller indices of the planes. Since � ¼ hpc=E, where hp is

Planck’s constant, c the speed of light in vacuum and E the

energy of the radiation, by combining equations (1) and (2), it

follows that

sin �B ¼hpcðh2 þ k2 þ l2Þ

1=2

2aE: ð3Þ

Two diffraction geometries are possible. In the first case, called

Bragg (reflection) geometry and depicted in Fig. 2(a), the

diffracted beam emerges from the same crystal surface on

which the incident beam impinges. Conversely, in the Laue

(transmission) geometry, depicted in Fig. 2(b), the diffracted

beam emerges from the surface opposite to that onto which

the incident beam impinges. For high-energy photons, such as

those emitted by a radiotracer, the Bragg angle is very small

and the crystal has to be large to diffract even a small-size

research papers

126 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 1Schematic representation of a nuclear diagnostic system equipped with aLaue lens. A fraction of the �-rays emitted by the radiotracer localized ina specific area of the patient’s body are focused onto the detector throughdiffraction in the crystals of the lens.

Figure 2Diffraction geometries. (a) Bragg geometry: the diffracted beam emergesfrom the same crystal surface on which the incident beam impinges. (b)Laue geometry: the diffracted beam emerges from the surface opposite tothat on which the incident beam impinges.

Table 1Typical radioisotopes used in diagnostic nuclear medicine.

Nuclide Radiotracer Decay mode Decay product T1=2 E� (keV) SPECT/PET Usage

99mTc NaTcO4 IT 99Tc 6.02 h 140.5 SPECT General purpose67Ga Ga citrate, Ga nitrate EC 67Zn 78.3 h 93, 185, 300 SPECT Tumour imaging111In I salts EC 111Cd 67.8 h 171, 245 SPECT Brain study, intestinal disturbances123I NaI EC 67Te 13.2 h 159 SPECT Thyroid study201Tl Tl salts EC 201Hg 73.1 h 135, 167 SPECT Diagnosis of coronary artery disease18F FDG, F-DOPA �þ 18O 109.8 min 511 PET Oncology, neurology15O O2, CO2, CO �þ 15N 2.03 min 511 PET Neurology11C CO2, CO, HCN, CH3I �þ 11B 20.38 min 511 PET Cardiology13N NH3 �þ 13C 9.96 min 511 PET Cardiology

beam. For this reason, the Laue geometry represents a more

convenient choice.

The diffracted beam intensity depends on the crystal

features and can be obtained from the dynamical theory or

from the kinematic theory of diffraction (Zachariasen, 1945).

The dynamical theory, as developed by Darwin, takes into

account the interaction of X-rays with matter by solving

recurrence equations that describe the balance of partially

transmitted and partially reflected waves at each lattice plane

(Darwin, 1914a,b). On the other hand, the kinematic theory

assumes that each photon is scattered only once. The total

diffracted amplitude is simply obtained by adding the indivi-

dual amplitudes diffracted by each diffracting centre, taking

into account only the geometrical phase differences between

them and neglecting the interaction of the radiation with

matter. Even if the kinematic theory is less rigorous than the

dynamical theory, it gives correct results when a thin perfect

crystal or a highly curved crystal is considered. An exhaustive

treatment of the subject, in both perfect and distorted crystals,

can be found in specialized books (Authier, 2001) or in review

articles (Authier & Malgrange, 1998; Authier, 2006; Batterman

& Cole, 1964). Here we recall only the concepts that are

relevant to the study of the lens.

The reflectivity of a crystal is defined as the ratio of the

diffracted beam intensity over the incident beam intensity.

Instead, diffraction efficiency is defined as the ratio of the

diffracted beam intensity over the transmitted beam intensity

when no diffraction occurs.

For radiation of energy E and a crystal with lattice planes

ðhklÞ, equation (3) provides the incidence angle � at which

diffraction occurs. Actually, both kinematic and dynamical

theories predict a range around the Bragg angle �B for which

the intensity of the diffracted beam is different from zero. If

we consider a perfect crystal under Laue symmetrical

geometry,1 the rocking curve, i.e. the reflectivity (or the

diffraction efficiency) plotted as a function of �� ¼ � � �B,

shows a narrow peak. Its width at half-maximum (FWHM) is

called the Darwin width �,

� ¼ 2dhkl=�0: ð4Þ

Here �0 is defined as the extinction length

�0 ¼�Vc cos �B

re�jCjjFhklj; ð5Þ

Vc being the volume of the crystal elementary cell (Vc ¼ a3 for

a cubic cell), re the classical electron radius, C the polarization

factor and Fhkl the structure factor. For an unpolarized beam,

the polarization factor is C ¼ ð1þ cos2 �BÞ=2. The structure

factor quantifies the scattering efficiency of an elementary cell

of the crystal, by taking into account the repartition of elec-

trons in space and the vibration of lattice ions via the so-called

Debye–Waller factor (Halloin & Bastie, 2005).

For the context of this article, typical values for � are of the

order of 100. Furthermore, because of the re-diffraction of the

beam, the reflectivity of a thick flat crystal is fixed at 1/2. A

crystal can be regarded as thick if T0 � �0, T0 being the

thickness of the crystal traversed by radiation. Since in our

case �0 is small, this condition is almost always fulfilled. The

integrated reflectivity is the integral of a rocking curve over

the range of incidence angles. Since the integrated reflectivity

is very poor for a flat perfect crystal, different types of crystals

have been considered by the scientific community for the

realization of a Laue lens. Their features are summarized in

the next subsection.

2.2. Mosaic crystals

Unlike an ideal crystal, a real crystal presents imperfections

due to its growth conditions, and it can be better modelled

through the Darwin model (Halloin & Bastie, 2005). This

model, known also as the mosaic model, regards the crystal as

an ensemble of microscopic ideal crystals. The crystallites are

slightly misaligned with respect to each other according to an

angular distribution, which is usually a Gaussian function:

Wð��Þ ¼1

ð2�Þ1=2�exp���2

2�2: ð6Þ

The FWHM of this distribution m ¼ 2ð2 log 2Þ1=2� is the

mosaicity of the crystal. The reflectivity, under symmetric

Laue conditions, is expressed by

R ¼ 12 ½1� expð2T0Þ�

1=2 expð�T0= cos �BÞ; ð7Þ

T0 being the thickness traversed by the beam, the linear

absorption coefficient of the crystal and

¼ Wð��ÞQ: ð8Þ

Q is the integrated intensity diffracted by an individual crys-

tallite per unit of thickness. From the dynamical theory of

diffraction, Q can be written as

Q ¼�2ddhk

�20 cos �B

f ðAÞ: ð9Þ

Under the small-angle approximation, which is valid above

100 keV, the function f ðAÞ can be written as

f ðAÞ ¼2I0ð2AÞ

2A: ð10Þ

I0 is the integral, from 0 to 2A, of the zero-order Bessel

function. A is defined as

A ¼�t0

�0 cos �B

; ð11Þ

t0 being the thickness of the crystallites. f ðAÞ is approximately

1, which is its maximum value, when t0 � �0, namely when

the kinematic theory tends to the dynamical theory.

The reflectivity of a mosaic crystal is the product of two

terms. The first one is the diffraction efficiency of the crystal;

the second one takes into account the absorption of the beam.

As can be seen by equation (7), the reflectivity peaks at

�� ¼ 0. The peak height is at most 1/2 as in the case of a

perfect crystal.

research papers

J. Appl. Cryst. (2015). 48, 125–137 Gianfranco Paterno et al. � A Laue lens for nuclear medicine 127

1 In Laue symmetrical geometry, the angle between the lattice plane and thecrystal surface is exactly 90�.

The FWHM of the rocking curve, �, is proportional to the

mosaicity of the crystal and can be written as

� ¼ m� ln � 1

� ln 12 1þ expð��Þ½ �� �� �

ln 2

� �; ð12Þ

where � is a dimensionless coefficient given by

� ¼ 4�2 ln 2

�

� 1=2dhklT0

�20m

: ð13Þ

Even though the peak reflectivity is at most 1/2, a mosaic

crystal with large mosaicity (it can be several tens of arcse-

conds) may exhibit a large integrated reflectivity. For this

reason, such crystals were chosen for the realization of the first

prototypes of Laue lens for nuclear medicine (Smither & Roa,

2000; Roa et al., 2005).

2.3. Curved diffracting planes crystals

Another type of crystal is the so-called curved diffracting

planes (CDP) crystal. In a CDP crystal, a stress induces a

curvature in the whole lattice structure according to the elastic

properties of the material. Owing to the curvature, the crystal

possesses an angular dispersion of the lattice planes. The

continuous change in the orientation of the lattice planes

prevents re-diffraction inside the crystal and the reflectivity

limit of 50% disappears (Fig. 3). Thus, CDP crystals have the

potential to achieve better performance with respect to both

perfect and mosaic crystals.

There are many ways to fabricate a curved diffracting plane.

The easiest one is by means of an external device (holder) that

applies a bending moment to the crystal (Carassiti et al., 2010).

This method has been in use for decades for the realization of

high-efficiency monochromators employed in synchrotron

high-energy X-ray beamlines (Schulze et al., 1998; Suortti et

al., 1997). However, the use of a holder implies additional

weight and space occupation. These problems represent a

severe limitation to the use of such a crystal as a component of

a Laue lens for both satellite-borne and medical applications.

Thus, the crystal curvature is required to be self-standing. For

this purpose, various methods have been proposed. One is by

applying a thermal gradient perpendicular to the considered

diffracting planes of a perfect crystal (Smither et al., 2005).

Another is by growing a two-component crystal whose

composition varies along the crystal growth axis (Keitel et al.,

1999). A self-standing curvature can be also obtained by

depositing a coating or by grinding (Ferrari et al., 2013) or

grooving a face of the crystal (Bellucci et al., 2011; Camattari,

Guidi et al., 2013).

In the grooving method, the grooves, manufactured on the

surface of the crystal by means of a diamond saw, produce

dislocations and partial amorphizations in the surrounding

regions, which are hence compressed and prevented from

relaxation. Such strain deforms the whole crystal and induces

a net curvature. This method has been proved to be simple,

economical and highly reproducible (Camattari, Battelli et al.,

2013; Camattari, Paterno, Battelli et al., 2014). All of these

features are very important when one has to deal with several

samples as in the case of a Laue lens. The drawback using the

grooving method is the decrease in integrated reflectivity due

to the removed material and to the mosaicization of a portion

of the crystal (Camattari, Paterno, Bellucci & Guidi, 2014).

A new method to produce a self-standing CDP crystal

without losing efficiency is carbon fiber deposition onto a

mono-crystal. Some promising results have been already

achieved and further studies are under development

(Camattari, Dolcini et al., 2014).

Diffraction in curved crystals can be studied through the

Takagi–Taupin equations (Takagi, 1969; Taupin, 1964). They

are hyperbolic partial derivative equations obtained from

Maxwell’s equations in a deformed periodic medium. In the

general case, these equations cannot be solved explicitly and a

numerical approach has to be used. Alternatively, for a slightly

curved crystal, the so-called PPK theory can be adopted. This

theory, based on geometrical optics principles, has been

developed independently by Penning & Polder (1961) and by

Kato (1963). In the PPK theory the distortion of the

diffracting planes is described by the strain gradient �,

� ¼�0

cos2 �B

@2ðh uÞ

@s0@sh

; ð14Þ

where s0 and sh are unit vectors parallel to the incident and to

the diffracted beams, respectively. h is the reciprocal lattice

vector of the reflection hkl and u the displacement vector. If

the curvature of a crystal is uniform, the strain gradient

assumes the simpler form

� ¼�

T0�=2; ð15Þ

� being the angular distribution of the lattice planes, T0 the

thickness of the crystal and � the Darwin width given by

research papers

128 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 3Diffraction in CDP and perfect crystals. (a) In a CDP crystal, because ofthe continuous change of the incident angle, re-diffraction is prevented.(b) In a perfect crystal the beam is reflected many times, limiting thereflectivity to 50%.

equation (4). The angular distribution of the lattice plane is

directly proportional to the crystal curvature. Indeed, it is

� ¼ T0=Rcurv; ð16Þ

where Rcurv is the radius of curvature of the crystal.

The rocking curve of a curved crystal follows a rectangular

distribution with width �. The height of the plateau, which

corresponds to the peak reflectivity, depends on the curvature

of the diffracting planes as well. In the general case, reflectivity

cannot be expressed in a closed form. However, for a highly

curved crystal there exists an extension of the PPK theory that

provides the reflectivity under Laue symmetric condition for

diffraction (Malgrange, 2002). It holds that

R ¼ 1� exp��2dhkl

�20

T0

�

� � �exp �

T0

cos �B

� : ð17Þ

A crystal can be regarded as highly curved when the following

condition for the strain gradient is met:

�>�c ¼ �=ð2�0Þ: ð18Þ

Such a condition is fulfilled if the radius of curvature of the

diffracting planes Rcurv is smaller than the critical value

Rcurvc¼ 4�0=��.

If Rcurv >Rcurvc, a multi-lamellar model can be used for the

calculation of the crystal diffraction efficiency (Boeuf et al.,

1978). A multi-lamellar model that takes into account the re-

diffraction of the beam is reported by Bellucci, Guidi et al.

(2013). Such a model merges the results provided by equation

(17) for highly curved crystals with the results provided by the

dynamical theory for flat thick crystals.

In Fig. 4 the rocking curves for a mosaic and a CDP crystal

with the same angular acceptance are compared. The peak

reflectivity of a CDP crystal, as calculated by equation (17),

does not suffer from the 50% limitation. Hence, CDP crystals

can have a very high integrated reflectivity, making them very

good candidates for the realization of a Laue lens.

3. Design principles of the lens

The design principles of a Laue lens for astrophysics have

been well defined in the literature (see for instance Lund,

1992; Frontera & Von Ballmoos, 2010; Bellucci, Camattari &

Guidi, 2013). Even though the requirements of a lens for

nuclear medicine and a lens for astrophysics are the same, the

design approach differs significantly. In the astrophysical case,

the polychromatic hard X- or �-rays come from very distant

sources and can be considered to be parallel when they

impinge on the crystals of the lens. Conversely, in the medical

case one has to deal with a monochromatic and divergent

beam, since � photons are emitted in any direction by the

radioactive source lying inside the patient’s body.

3.1. Geometry of the system

A lens composed of concentric rings of crystals is shown in

Fig. 5(a). It lies at a distance LS from a �-ray point source and

LD from a detector. LS and LD are called the object distance

and image distance, respectively. With an appropriate design,

the lens can focus the photons toward the image point OD,

providing a point-to-point imaging.

First, the energy E of the monochromatic source and the

object distance LS have to be set. Then, the lens geometry can

be determined. The lens is formed by N rings, each of them

exploiting a different set of crystalline planes and/or a

different crystalline material. The crystals of each ring are

assumed to be identical. To obtain the diffraction of the

photons, the radius of the jth ring, rj, has to be set according to

LS ¼rj

tanð�Bj� �jÞ

; ð19Þ

where �j is the angle between the crystal axis and the lens axis

(see Fig. 5b). The photons diffracted by each crystal are

focused toward the point OD if the image distance is set to

LD ¼rj

tanð�Bjþ �jÞ

: ð20Þ

Under the small-angle approximation, tan �Bj’ sin �Bj

’ �Bj,

it is possible to obtain the same equation as for thin lenses in

visible optics (Smither et al., 1995):

1

LS

þ1

LD

¼1

f; ð21Þ

where f is the focal length of the lens, namely the distance from

the ring to the focus for a distant source,

f ¼rj

tanð2�BjÞ: ð22Þ

The source-to-detector distance, LS þ LD, is minimum when

�j ¼ 0, namely when the crystals are perfectly aligned with the

lens axis. Since the system should be as compact as possible,

hereinafter we will assume that this condition is fulfilled.

Moreover, we will always use the small-angle approximation.

Therefore, it follows that

rj=�Bj¼ LD ¼ LS: ð23Þ

research papers

J. Appl. Cryst. (2015). 48, 125–137 Gianfranco Paterno et al. � A Laue lens for nuclear medicine 129

Figure 4Rocking curves for a mosaic and a CDP Ge crystal with the samethickness T0 = 1 mm and acceptance � = 200 0 and using (111) latticeplanes to diffract 140.5 keV photons. (a) CDP crystal with radius ofcurvature Rcurv = 10.3 m. (b) Mosaic crystal with mosaicity m = 13.60 0.

By combining equations (23) and (3), the radius of the jth ring

can be written as

rj ¼hpc

2a

LS

Eðh2þ k2þ l2Þ

1=2: ð24Þ

Therefore, once the material and the lattice plane ðhklÞ used

for a ring have been set, the ring radius is determined. The

higher the Miller indices, the larger the ring radius. For the

arrangement of the crystals in each ring, we assume the

polygonal arrangement that best approximates the ring, i.e.

the polygon inscribed in the circle with radius ðrj � Lrj=2Þ=

cosð�=njÞ. To determine the number of crystals in each ring,

the tangential length Ltjand the radial length Lrj

have to be

set. The latter has been already shown in Fig. 5(b), while Ltjis

the size of the side orthogonal to the plane of the figure. The

number of crystals in each ring, nj, turns out to be

nj ¼ �

arctanLtj

2rj � Lrj

!: ð25Þ

Not all the lattice planes ðhklÞ can be selected for a given

material and size because different crystals would occupy

approximately the same position in the lens. This limitation

becomes more severe as LS decreases.

3.2. Efficiency calculation

The type and thickness of the crystals have to be set to

optimize the lens performance. An important figure of merit is

the lens efficiency. It is defined as the ratio between the

number of photons with energy E diffracted per second by the

lens and the number of photons with energy E emitted per

second by the source. The latter quantity is given by the

product of the source activity and the probability of emission

of the considered �-line. Even though the lens efficiency is a

dimensionless quantity, it can be expressed in counts per

second per becquerel, which is the unit of measurement used

in nuclear medicine. Indeed, the number of counts per second

of the signal on the detector is equal to the number of photons

diffracted per second by the lens, provided that the air

absorption is neglected, an ideal detector is considered, and

the counts due to the background and to the not coherently

scattered photons are subtracted. Furthermore, in nuclear

medicine the ratio described above is often called sensitivity.

Thus, for our purposes, the two terms are considered as

synonyms.

The efficiency of the lens is strictly related to the integrated

reflectivity of the crystals that compose the lens. Let us

consider Fig. 6, which sketches a crystal in a generic ring. The

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130 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 5Schematic representation of the geometry of the system. (a) A pointsource is positioned at OS, the lens is placed at a distance LS from thesource and a detector, centred at OD, is at a distance LD from the lens. Ageneric ring of the lens is highlighted; OCij ¼ rj is the radius of the ring.The crystals of each ring are assumed to be identical. (b) A detail of thecrystal i in the ring j. T0j

is the thickness of the crystal traversed bythe photons. �j is the angle between the crystal axis and the lens axis. Inour calculation, we assume that �j ¼ 0.

Figure 6Schematic representation of a crystal in a generic ring (not to scale).OC ¼ r is the radius of the ring. The crystal diffracts the photons emittedwith a polar angle � within the range ½�B ��=2; �B þ�=2�, where � isthe crystal acceptance.

crystal diffracts only the photons emitted with a polar angle �belonging to the interval ½�B ��=2; �B þ�=2�. � is the

angular acceptance of the crystal and is considered to be equal

to the FWHM of its rocking curve. This is exact if a CDP

crystal is considered but is only an approximation for a mosaic

crystal. Likewise, the integrated reflectivity of a mosaic crystal

is considered to be simply the product of the peak reflectivity

and the angular acceptance, as in the case of a CDP crystal.

These approximations allow us to develop our analytical

calculation in the same way for both types of crystal.

It should be noted that not all the crystal volume diffracts

the photons emitted by a point source. Equivalently, not all the

surface on which the photons impinge acts as a collecting area.

Since OSQ ¼ OSO�QO ¼ LS � T0=2 ’ LS, by considering

the two triangles dOSQEOSQE and dOSQBOSQB, it can be calculated that

AE ¼ EB ¼ �=2LS. Therefore, the collecting area of the

crystal is LS�Lt. Since �LS <Lr, the collecting area is lower

than the geometric area LrLt. The collecting area of each ring

is equal to the number of crystals nj times their collecting area:

Acj¼ njLS�jLtj

: ð26Þ

The effective area of each ring is defined as the collecting area

of the ring times the reflectivity of the crystals. The effective

area of the lens is the sum of the effective area of the N rings:

Aeff ¼PNj¼1

Aeffj¼PNj¼1

AcjRj; ð27Þ

Rj being the peak reflectivity of the crystals in the jth ring.

If _NSNS is the number of � photons per second emitted by the

point source, the photon flux on the lens is �L ¼_NSNS=4�L2

S.

Hence, the number of photons per second diffracted by the

lens is _NDND ¼ �LAeff. Therefore, the efficiency of the lens can

be written as

" ¼_NDND

_NSNS

¼Aeff

4�L2S

¼1

4�L2S

XN

j¼1

AcjRj: ð28Þ

It follows that

" ¼1

2

XN

j¼1

njLtj

2�LS

�jRj ¼1

2

XN

j¼1

njLtj

2�rj

�Bj�jRj: ð29Þ

Since

njLtj

2�rj

’ 1; ð30Þ

we obtain

" ¼PNj¼1

"j ¼12

PNj¼1

�Bj�jRj: ð31Þ

The efficiency of a ring is approximately given by

"j ¼12 �Bj

�jRj: ð32Þ

In a first approximation, "j does not depend on LS or on nj or

on the surface on which the photons impinge. Indeed, by

raising the object distance, a larger number of crystals can be

arranged in each ring. Thus, an increase in Aeff can be

obtained. However, this gain is approximately compensated

for by the decrease in the photon flux hitting the lens.

Conversely, "j depends on the Bragg angle, on the acceptance

and on the thickness of the crystals. For a ring of crystals of a

given material and a given set of lattice planes, �j and T0jare

two free parameters that can be chosen to maximize the effi-

ciency of the ring.

To proceed further in the analysis, highly curved crystals are

considered. In this case, the efficiency of a generic ring

assumes the form (neglecting the subscript j)

" ¼1

2�B� 1� exp

��2dhkl

�20

T0

�

� � �exp�T0

cos �B

� : ð33Þ

The efficiency depends linearly on �B. However, by equation

(3) it follows that a higher value of the Bragg angle implies

higher indices and hence a higher value of �0, leading to a

decrease in the ring efficiency. For this reason, the outermost

rings in the lens give a small contribution to the overall effi-

ciency. As a result, it is not convenient to increase the number

of rings over a certain value, which depends on the specific

case. Otherwise, there would be only an increase in the lens

complexity without an appreciable increase in the lens effi-

ciency.

Fig. 7 plots the efficiency at 140.5 keV of a ring exploiting

the (111) planes of Ge crystals.

The acceptance has to be as high as possible to maximize

the efficiency of the ring. Moreover, for each value of �, there

is a relatively narrow range of thickness T0 for which the

efficiency attains its maximum value.

Another parameter that is generally used to quantify the

diffraction capability in the whole lens, especially those for

astrophysics, is the lens diffraction efficiency, which has not to

be confused with the crystals’ diffraction efficiency. The lens

diffraction efficiency is defined as the ratio between the

number of photons per second diffracted by the lens _NDND and

number of photons per second incident on the lens _NLNL. It turns

out that

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J. Appl. Cryst. (2015). 48, 125–137 Gianfranco Paterno et al. � A Laue lens for nuclear medicine 131

Figure 7Normalized efficiency at 140.5 keV of a ring exploiting the (111) planes ofGe crystals as a function of the thickness and the acceptance of thecrystals. The solid black lines plot the optimum thickness as a function ofthe acceptance �.

"D ¼_NDND

_NLNL

¼Aeff

AL

; ð34Þ

where AL ¼PN

j¼1 njLtjLrj

is the geometric area of the lens.

3.3. Point spread function calculation

The point spread function (PSF), i.e. the system response to

a point-like source, is the physical quantity that fully char-

acterizes an imaging system. Indeed, the image of an extended

source is obtained by the convolution of its spatial distribution

and the PSF of the system (Webb, 1988). The PSF of the

system is given by the convolution of the PSF of each indivi-

dual element that contributes to the formation of the image. In

our case, only the lens and the detector contribute to the

formation of the image. Thus, for an ideal detector, the PSF of

the system coincides with the PSF of the lens. The spatial

resolution of the system is defined as the FWHM of its PSF.

This quantity affects the capability to distinguish small details

of the image and hence has to be as small as possible.

An ideal situation would be if the crystals, other than

possessing the curvature of the planes of diffraction, were bent

to perfectly fit in the ring in which they are positioned. In this

case, a perfect focusing onto a single point could be achieved.

Unfortunately, owing to the mechanical stiffness of the crys-

tals, this condition cannot be met and each crystal focuses only

in its scattering plane.2 Therefore, on the image plane, where

the detector is positioned, the distribution function of the

incoming photons is spread out around the centre OD. This

spread, and hence the lens resolution, depends on several

parameters.

If the lens is composed of highly curved crystals disposed

under Laue symmetric geometry, the photon trajectories and

their distribution on the detector can be calculated. Because of

the variation of the lattice plane orientation due to the

curvature, a photon penetrates into the crystal until the zone

where the Bragg condition is reached. Such a zone is narrow

enough that the photon trajectory undergoes a kink

(Malgrange, 2002). Therefore, each photon is diffracted only

once and then proceeds along its trajectory undergoing

normal absorption. These are the conditions provided by the

kinematic theory.

Let us consider the situation sketched in Fig. 8. A photon

impinges on the surface of a crystal with azimuthal angle equal

to zero. The coordinates of the diffraction point can be

calculated as

yD ¼ 0; ð35Þ

zD ¼ �T0

���; ð36Þ

xD ¼ ���T0

�� � LS

� : ð37Þ

The distance of the arrival point of the photons on the

detector, with respect to OD, turns out to be

xDET ¼ rþ xD � ðLD � zDÞð�B ���Þ; ð38Þ

where r is the radius of the ring to which the crystal belongs

and �� ¼ � � �B. Since LD ¼ LS, by considering all the

possible values for the polar angle �, the image obtained on

the detector is a line of width j2T0�B � 2�LSj. Owing to the

focusing effect provided by curved diffracting planes, this

value is lower than the radial size of the crystal Lr. Further-

more, including in our analysis the photon azimuthal angle, it

is possible to see that a point source produces a rectangle with

sides j2T0�B � 2�LSj and 2Lt as an image of each crystal. The

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132 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 8Schematic representation of diffraction in a highly curved crystal underLaue symmetric geometry. The kink of the trajectory is due to the largevariation of the lattice plane orientation over a distance equal to theextinction length.

2 The scattering plane is defined by the normal vector of the diffracting latticeplanes and the incoming beam direction. In the perpendicular direction theradiation propagates in a straight line (Stockmeier & Magerl, 2008).

PSF of the lens can be estimated by taking into account such

contributions for all crystals in each ring. In the next section a

specific lens will be considered and its PSF shown. The FWHM

of the PSF depends on the object distance, on the Bragg angle

of each ring, and on the thickness and the acceptance of the

crystals. Conversely, the tails of the PSF are basically deter-

mined by the tangential size of the crystals.

The PSF can also be calculated if the lens is composed of

mosaic crystals (Halloin, 2005). In this case, the width of the

image spot is at least equal to the radial size of the crystals Lr

and becomes larger, owing to the so-called mosaic defocusing,

as LS increases over the value 2 lnð2Þ½ �1=2

Lr=m, m being the

mosaicity of the crystals (Halloin & Bastie, 2005). Therefore,

the image resolution attainable with a lens composed of CDP

crystals is much better than for a lens composed of mosaic

crystals.

A parameter that combines the efficiency and focusing

capability of a lens is the focusing factor G. It is defined as

(Frontera & Von Ballmoos, 2010)

G ¼ fphAeff=AD; ð39Þ

Aeff being the effective area of the lens and AD the area of the

spot on the detector that contains a fraction fph of photons

reflected by the lens.

3.4. Field of view estimation

The field of view (FOV) of the lens is the portion of space

within which a source has to lie so that it can be correctly

reproduced on the image plane. This issue can be precisely

addressed by a numerical simulation. However, an estimation

can be performed as follows. Let us consider a Laue lens

composed of crystals of the same type and size. Let us also

assume that Lr Lt. If a point source is positioned at OS, the

lens provides a point-to-point imaging with a blur given by the

lens resolution. If the point source is moved away from OS, the

lens diffracts the photons with no efficiency loss or increase in

blurring only if the transaxial displacement is lower than

approximately Lr=2. Beyond this distance, diffraction may not

occur for some polar angles within the nominal acceptance

range. Indeed, through a modified version of equation (37)

which includes the source shift and the azimuthal angle

contribution (not reported here), it is possible to calculate that

the diffraction point lies outside the crystal. Since the lens

transaxial FOV is approximately given by the radial size of the

crystals, there is a trade-off between the need for small Lr (to

enable diffraction from all the desired planes) and the need

for a large value of Lr (to magnify the transaxial FOV). The

axial FOV of the lens is far broader than the transaxial one.

Indeed, it is approximately equal to �1LS=�B1, �B1

being very

small.

3.5. Response of the lens versus energy

A lens designed for a specific monochromatic source can be

used with different sources as well. Equation (24) settles a

proportionality relationship between E and LS. Thus, if the

source energy changes, the object and the image distances

have to be set accordingly. However, the lens efficiency and

resolution change as well.

The efficiency worsens at high energy. Indeed, the crystal

thickness should be increased to maintain a high value of

reflectivity. Furthermore, even if one considers the optimum

thickness, the lower Bragg angle leads to an increase in �0 and

hence to a decrease in diffraction efficiency at high energy.

Such a decrease is stronger than the increase in the absorption

factor and, as a result, the reflectivity decreases.

The shape of the lens PSF also changes with energy. Indeed,

it depends on LS and on the Bragg angles of each ring and, in

turn, these physical quantities vary with energy. As will be

shown in the next section, there is an energy range within

which the resolution assumes its best value. However, this

condition leads to a smaller fraction of photons enclosed by a

circle with diameter equal to the FWHM of the PSF.

4. High-resolution lens

In order to estimate the performance of a Laue lens designed

with the method described above, we will now work out a

practical case. In particular, we will design a lens to focus the

140.5 keV photons emitted by a point source of 99mTc located

0.25 m from the lens.

4.1. Lens features

The selection of the crystals that compose the lens heavily

affects the properties of the lens itself. We consider a lens

composed of ten rings of CDP Ge crystals whose transaxial

size is 1.0 � 0.5 mm. Furthermore, we take under considera-

tion crystals with a radius of curvature of 11.5 m. This value is

achievable through the grooving method, creating 16 grooves,

320 mm deep and 150 mm wide, on the largest face of the

crystal. The removed material would be about 30%. However,

taking into account the mosaicization that could occur in some

portions of the crystals and that not all the volume of the

crystals is required to diffract the photons emitted by a point

source, the decrease in lens efficiency can be estimated to be

20%. Since the crystals do not necessarily have to be fabri-

cated through the grooving method, the reduction factor is not

taken into account and a general treatment is kept.

In the previous section we showed that each ring should be

composed of crystals with different thicknesses. However, in

this case, the optimum thickness does not change very much

from one ring to another. Thus, for the sake of simplicity, we

set the thickness of all the crystals to be 5.0 mm. Their angular

acceptance turns out to be 9000. Finally, we consider a

symmetric Laue geometry and a perfect crystal alignment. The

features of the lens are summarized in Table 2.

For each ring, the lattice planes, the radius and the number

of crystals exploited are listed in Table 3.

By using such diffracting planes, the lens assumes the

geometry whose front view is depicted in Fig. 9. The lens is

very compact, having a diameter of 27 mm. Thus, it would be

easy to move it to perform the scan of a restricted area.

Through equation (29), the lens efficiency turns out to be

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J. Appl. Cryst. (2015). 48, 125–137 Gianfranco Paterno et al. � A Laue lens for nuclear medicine 133

1:16� 10�5, which is lower than the levels for SPECT and

PET and is indeed comparable to that of pinhole SPECT. The

efficiency of the proposed lens is also of the same order of

magnitude if compared with the already realized prototypes

(Smither & Roa, 2000; Roa et al., 2005).

By application of the method expounded in the previous

section, the PSF of the lens can be calculated. It is a function of

the two coordinates on the image (detector) planes. However,

owing to the nearly perfect cylindrical symmetry of the lens, a

cross section along any axis can be considered. Fig. 10 shows

the cross section of the PSF along the X axis.

The majority of the photons are focused near to the image

point OD. Indeed, the FWHM of the curve is 0.20 mm and the

number of diffracted photons enclosed by a circle whose

diameter is 0.20 mm turns out to be 42%. Furthermore, a spot

of 0.92 mm contains 80% of the diffracted photons. Fig. 11

plots the fraction of diffracted photons, fph, enclosed by a

circle of increasing diameter dfph.

Owing to the focusing power of the lens, a very low blur in

the image of a point source is guaranteed. A resolution of

0.20 mm would be an outstanding performance for a tech-

nique in nuclear medicine. This value is one order of magni-

tude better than that obtainable with PET and it can be only

approached by some pinhole SPECTs, but in such cases, the

efficiency would be far lower than for the lens described here.

The lens performance at 140.5 keV is summarized in Table 4.

Finally, the efficiency and the resolution of the lens as a

function of energy are shown in Figs. 12 and 13 respectively.

As explained in the previous section, the efficiency decreases

monotonically at high energy owing to the decrease in

reflectivity of the crystals.

The PSF of the lens varies versus energy because of the

variation of LS and of the Bragg angle relative to each ring.

When the condition for best resolution occurs, a smaller

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134 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 9Front view of the designed lens. The largest side of the crystals is thetangential side Lt. The radial side is Lr, while the thickness T0 traversedby the photons is in the direction perpendicular to the plane of the figure.

Figure 10Cross section of the lens PSF along the X axis of the image plane.

Table 3Rings of the lens.

Ring Lattice planes Number of crystals Radius (mm)

1 (111) 19 3.382 (220) 33 5.523 (311) 39 6.474 (400) 47 7.805 (331) 51 8.506 (422) 58 9.567 (511) 62 10.148 (440) 67 11.049 (620) 76 12.3510 (444) 83 13.53

Figure 11Fraction of the diffracted photons enclosed by a circle of increasingdiameter.

Table 2Lens features.

E (keV) 140.5LS (m) 0.25LD (m) 0.25N 10Crystals Ge (CDP)T0 (mm) 5.0Lt (mm) 1.0Lr (mm) 0.5Rcurv (m) 11.5� (0 0) 90

fraction of diffracted photons is enclosed by a circle with

diameter equal to the FWHM of the PSF. Furthermore, the

diameter of a circle enclosing 80% of diffracted photons is

larger.

It can be argued that the lens is still usable when other

monochromatic sources are considered. Naturally, the best

performance is achieved at the energy for which the lens has

been designed.

4.2. Simulations

In order to obtain in-depth knowledge about the imaging

properties of the lens, several Monte Carlo simulations have

been performed through a custom-made code named

LAUENM. The code, which is implemented in the MATLAB

language (The MathWorks Inc., Natick, MA, USA), enables

one to design the lens and to simulate the overall diffraction

process. The lens can be composed of crystals of different

material and different structure (CDP or mosaic). The code is

modular and each module deals with a particular task. A

specific module manages the photon source, which can take

various shapes and activity distributions. The photons are

emitted randomly in direction. A module works out the lens

geometry starting from the input parameters set by the user.

The main processing module manages the interaction of the

photons with the crystals and calculates the arrival point of the

photons on the detector. Finally, a post-processing module,

taking into account the detector features, computes the

obtained image and all the figures of merit regarding the lens.

Fig. 14 depicts the simulated lens PSF and a comparison

with the PSF calculated through the

analytical method described in the

previous section. An ideal detector with

100% efficiency and 40 � 40 mm pixel

size is considered.

The PSF is a narrow distribution

peaked at OD. It is also possible to

notice a very good agreement between

simulation and analytical calculation.

In order to investigate the FOV of the

lens, the lens efficiency is recorded as a

function of the position of the point

source. In particular, the source is

moved along a transaxial direction and

along the axial direction, namely along

the axes xS and zS of a frame centred at

OS. Fig. 15 plots the normalized lens

efficiency as a function of xS and zS. It is possible to recognize

that within a transaxial distance equal to 0.25 mm from OS

there is no efficiency decrease. This distance corresponds to

Lr=2. The axial distance within which there is no efficiency

decrease is approximately equal to 4.0 mm, which corresponds

to �1LS=2�B1. It is worth noting that over these distances the

response does not vanish but undergoes a progressive loss

accompanied by an increase in the photon spread. This affects

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J. Appl. Cryst. (2015). 48, 125–137 Gianfranco Paterno et al. � A Laue lens for nuclear medicine 135

Figure 12Efficiency of the lens as a function of source energy.

Figure 13Characteristic parameters of the lens PSF as a function of the sourceenergy. Red circles joined by a red solid line plot the resolution of the lens(mm). Green crosses joined by a green dot–dashed line plot the fractionof photons (%) enclosed by a circle whose diameter is equal to theFWHM of the lens PSF. Blue squares joined by a blue dotted line plot thediameter of the circle (mm) that encloses 80% of diffracted photons.

Table 4Lens performance at 140.5 keV.

Number of crystals 535Lens diameter (cm) 2.7Filling factor (%) 46Diffraction efficiency (%) 3.4Efficiency 1:16� 10�5

Resolution (mm) 0.20fph (%) 42d80% (mm) 0.92G80% 9

Figure 14(a) Simulated lens PSF. (b) Comparison of the simulated and calculated PSF cross section along thedetector X axis.

the ability to obtain a correct image of the source. Hence, the

FOV of the lens has to be considered limited to the value

within which the normalized efficiency is 1, to be capable of

reconstructing the image of an extended source.

Fig. 16 summarizes the images obtained when various

source configurations are considered, provided that the

background and the not coherently scattered photons are

neglected. Fig. 16(a) plots the image obtained in the nominal

configuration, i.e. with the point source

positioned at OS. Fig. 16(b) plots the

image related to a point source posi-

tioned at yS =�0.2 mm, which is a point

inside the FOV of the lens. It can be

seen that an exact image of the source is

reproduced on the detector. Fig. 16(c)

plots the image related to a point source

positioned at xS = +0.4 mm, a point

outside the FOV of the lens as defined

above. In this case it is still possible to

recognize that the peak of the photon

distribution is centred at X = +0.4 mm.

However, there is a broadening of the

peak and an asymmetry in the tails,

which are much more extended in the

opposite direction with respect to the

source shift and are almost absent in the

perpendicular direction. This asymmetry makes it very diffi-

cult to reconstruct the image when an extended source is

considered. In Fig. 16(d) the point source is moved 10 mm

axially with respect to OS; thus it is outside the FOV. Because

the part of the crystal closer to the lens axis diffracts hardly

any photons, a central hole appears, making image recon-

struction impossible. In Fig. 16(e) two point sources with

different activity are considered. One is centred at OS and the

other, having an activity equal to one-half that of the former, is

moved 0.25 mm along xS. Both the sources are perfectly

resolved and the difference in activity is perceptible. Indeed,

both the sources are inside the FOV and are separated by a

distance larger than the spatial resolution. Finally, Fig. 16( f)

shows the image of a circular source with a radius of 0.32 mm,

which is correctly reproduced.

5. Conclusions

The design principles of a diagnostic imaging system based on

a Laue lens have been presented. CDP crystals represent the

best choice for the composition of the lens because of their

high integrated reflectivity and the ability to fabricate them in

a reproducible manner. By properly selecting the size of the

crystals, a lens with good efficiency and high resolution can be

obtained. In particular, such a system attains the best resolu-

tion among diagnostic techniques used in nuclear medicine.

The transaxial FOV of the lens is limited by the radial size of

the crystals and ranges within 0.3–5 mm. However, by moving

the lens, it would be possible to perform a high-resolution scan

over a restricted region of interest (alternatively, the lens

could be maintained in a fixed position and the bed moved).

The obtained image may help specialists to gain a better

understanding of some metabolic processes or to recognize

more accurately the extension and the position of a tumour

mass in an early stage of evolution. Generally speaking, the

lens could be used in those cases where the results of a

conventional diagnostic examination are doubtful. Owing to

the compactness of the lens, it would be possible to make use

of a high number of identical lenses to observe the same point,

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136 Gianfranco Paterno et al. � A Laue lens for nuclear medicine J. Appl. Cryst. (2015). 48, 125–137

Figure 15(a) Normalized lens efficiency as a function of xS and zS. (b) Contour of the normalized lensefficiency.

Figure 16Image on the detector with various source configurations. (a) Point sourcepositioned at OS. (b) Point source moved 0.2 mm transversely (along yS)with respect to OS. (c) Point source moved 0.4 mm transversely (along xS)with respect to OS. (d) Point source moved 10 mm axially (along zS) withrespect to OS. (e) Two point sources of different activity are considered,one at OS and the second, having an activity equal to one-half that of theformer, moved 0.25 mm along xS. ( f ) Circular source with a radius of0.32 mm.

obtaining an amplification of the lens efficiency. In this way,

the high-resolution scan could be performed immediately

after a conventional SPECT examination without the need for

a new injection of the radiotracer.

We acknowledge partial support by the LOGOS project of

INFN and the SPINNER project of Emilia-Romagna region.

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