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188 Nuclear Medicine Physics

a projection that is employed to reconstruct the image in that plane. As we

shall see later, in PET the total number of reconstructed planes in the 2D mode

is 2N −1, where N is the total number of rings in the scanner.

5.3.2.3.3 3D Sinograms

The 3D mode also uses sinograms for storing data in histogram format but

which include the counting of coincidences between crystals in distant rings.

Thus, besides all the planes deﬁned in the 2D mode parallel to the rings,

the 3D mode also includes sinograms corresponding to planes oblique to

the rings. Each of these oblique sinograms stores the coincidences in LORs

deﬁned between two different rings (Figure 5.35a); so, in a system of N rings,

a maximum of N

sinograms can be deﬁned for the 3D mode, whereas only

2N −1 exist in the 2D mode. In the 3D mode, it is, therefore, necessary to

specify for a given plane not only its position along the detector’s OZ axis but

also its inclination relative to that axis. The latter is usually given in terms of

the ring difference between the two rings in coincidence, Δr = r

−r

, where

r is an integer that positions each ring along the detector axis (Figure 5.35a).

The axial coordinate z of the plane is the average of the positions of the

two rings,

z =

Δd

, (5.47)

Δd

Δr = r

– r

(a) (b)

FIGURE 5.35

(a) Sketch of the rings involved in the deﬁnition of an oblique sinogram, and corresponding

deﬁnition of the values for Δr and z of that sinogram. (b) Deﬁnition of the OX

reference

frame obtained by the rotation of the OXYZ frame ﬁxed to the detector by the azimuthal and

co-polar angles φ and θ. It should be noted that in the 2D mode the direction of each LOR with

azimuthal coordinate φ is parallel to the axis OY

(see Figure 5.34a), whereas the presence of the

co-polar coordinate θ in the 3D mode forces the direction of an LOR with angular coordinates

(φ, θ) to be parallel to the OZ

axis.

Radiation Detectors and Image Formation 189

where Δd

is the width of each ring. As in the 2D mode, the 3D sinograms

of an object are grouped in an array s

, φ, z, Δr), now four-dimensional,

which collects all the oblique and nonoblique sinograms. The organization of

this array as a function of Δr and z will be addressed later.

In 3D sinograms, the deﬁnition of the coordinatesx

and φ is now performed

using a reference frame OX

rotated relative to the OXYZ frame by the

azimuthalangle, φ ∈[0, π], and the copolarangle, θ ∈[0, π](Figure5.35b). The

copolar angle θ is a way of representing the inclination of the plane equiva-

lent to using Δr, as the LORs located in the central radial positions for all the

azimuthal angles (which constitute the central column of the sinogram) make

an angle of π/2 −θ with the axis of the detector. For a sinogram with incli-

nation Δr and axial coordinate z, each row groups LORs that make exactly

the same angle, φ, with the vertical axis OY, measured in the OXY plane (Fig-

ure 5.35b). Those LORs are not all strictly parallel, and, except for the central

LOR, they form an angle with the detector axis that is slightly lower than

π/2 −θ (i.e., are more slanted than the central LOR).

Since each sinogram acquired in the 3D mode does not directly deﬁne a

plane, owing to this slight nonparallelism, a problem that does not arise in

the 2D mode, 3D mode data are often immediately transformed into parallel

projections of the activity of the object being observed [42]. In this mode, par-

allel projections are 2D, generated in the plane OX

deﬁned by the angular

coordinates φ and θ (Figure 5.36), and are a function of (x

, y

). The map-

ping between all the 3D sinograms s

, φ, z, Δr) and the parallel projections

Projection

(fixed θ)

(a) (b)

Sinogram

(z, Δr fixed)

Projection

(fixed θ)

FIGURE 5.36

(a) Parallel projection of an object in the plane OX

along the angles (φ, θ). (b) Relation between

parallel projections and the rows of an oblique sinogram.

190 Nuclear Medicine Physics

, y

, φ, θ) is performed by

, y

, φ, θ) = s

, φ, z, Δr), (5.48)

where the coordinates (x

, y

, φ, θ) of the projections are given in terms of the

sinogram coordinates (x

, φ, z, Δr) by

⎧

⎪

⎨

⎪

⎩

= x

=−z cos θ

φ = φ

tan θ =

Δr ×Δd



−x

, (5.49)

with R

being the radius of the detector rings.

If the radius, R

FOV

, of the FOV and the length of the detector axis are small

when compared with R

, then it is possible to ignore the nonparallelism in

an oblique sinogram and assume that the LORs grouped in a sinogram’s row

corresponding to an azimuthal angle, φ, are all parallel and make the same

angle θ = arctan(Δr × Δd

/2R

) with the OXY plane. With this approxima-

tion, there is a direct correspondence between that sinogram row and the row

with coordinate y

=−z cos θ in the projection array along the direction (φ, θ);

the projection along (φ,θ) corresponds, thus, to the rows of coordinate φ of all

the sinograms with different z coordinates that deﬁne an angle π/2 −θ with

the detector axis. If that approximation is not valid, for each value of the radial

coordinate x

it becomes necessary to interpolate the count values stored in

the corresponding columns in sinograms with different inclinations to obtain

the parallel projections. The reconstruction of the ﬁnal image for the whole

volume of the patient in the 3D mode uses all the projections p

, y

, φ, θ)

simultaneously, which means that the reconstruction algorithms in the 3D

mode operate in a four-dimensional space and involve a considerably larger

degree of numerical sophistication than the algorithms operating in the 2D

space used for the 2D mode, as will be seen later.

5.3.2.4 Preprocessing of Data

As in the case of SPECT, image reconstruction in PET assumes that the coin-

cidence counts in each LOR are proportional to the integrated activity of

the object along the direction of the LOR, and similarly there are several

physical effects which modify that proportionality relation. The perturbation

introduced by those effects is different from LOR to LOR, and obtaining a

reconstructed image that is as near as possible to the true distribution of the

activity in the patient means there must be a preprocessing step which corrects

the data from those effects. Attenuation of the annihilation photons and scat-

tered radiation (which leads to scattered coincidences) also involves physical

Radiation Detectors and Image Formation 191

effects that are found in PET, and they considerably affect the image. There are

still more effects speciﬁc to PET, such as the existence of random coincidences

or instrument-related effects which arise from the nonuniformity of the detec-

tion efﬁciency between distinct crystals, or the fact that the system has LORs

with slightly different geometrical conditions. All the physical effects which

alter a PET exam can, in principle, be corrected, allowing a quantitative use

of this technique, that is, that images showing absolute values for the activity

concentration, in, for example, Bq/cm

, are produced. In SPECT, quantitative

measurements are much harder to obtain, because it is also considerably more

difﬁcult to properly correct attenuation effects.

5.3.2.4.1 Data Reduction

Besides data correction, image reconstruction procedures in PET are usually

precededby another preprocessingtask, one thataims at reducingthe number

of LORs used in the reconstruction. This task consists of summing coincidence

counts of adjacent LORs, a process that is known as mashing [42,55]. Mashing

techniques are used in most current scanners due to the huge number of LORs

that these systems possess, a number which results in low counting statistics

in individual LORs and creates difﬁculties in data processing because of the

large size of the data structures that have to be handled. The sampling of the

radial, azimuthal, copolar, and axial coordinates and the way LOR mashing

is accomplished are described next.

5.3.2.4.2 Radial Sampling: Interleaving

The detection of coincidence in a PET system, in which each LOR samples a

prism-like volume of the FOV, implies the discretization of the whole FOV’s

volume. The discretization process determines a maximum spatial resolution

for the system, which depends on the properties of the detector; for instance,

the sampling of the radial coordinate x

depends directly on the size of the

crystals, whereas the sampling of the azimuthal angle, φ, depends on the

number of crystals per ring.

The circular geometry of the detection system in PET entails a slight spatial

oversampling of the transaxial FOV’s central region relative to its peripheral

regions. Spatial sampling is normally further improved by a strategy known

as interleaving [42], which increases radial sampling by reducing the angular

sampling in a plane. In this process, the LORs of two consecutive φ angles are

alternately combined in a single row of the sinogram, as shown in Figure 5.37.

Supposing that the number of crystals per ring is even and equal to a value

, in the 2D mode, those LORs of consecutive angles deﬁne two sets of N

LORs that are shifted apart by a half-radial position. By combining these

two sets in a single row of the sinogram, the sampling frequency of the radial

coordinateisdoubled, at the cost of halving theazimuthal sampling frequency

and introducing a minor error in the angular position, φ, of the coincidence

counts recorded in each radial position, x

192 Nuclear Medicine Physics

180° 180°

angles

crystals

With interleaving

With no interleaving

(a) (b)

FIGURE 5.37

(a) Diagram of the two sets of LORs shifted by a half radial position combined in the interleaving

process (N

even, 2D mode). (b) The way the interleaving process combines those shifted sets

to build the interleaved sinogram rows. The spatial sampling frequency doubles at the cost of

halving the angular sampling.

In addition to the interleaving, the radial sampling is also usually restricted

to an interval smaller than the transverse dimension of the detector. The FOV

is, therefore, smaller than the volume deﬁned by the rings, but it avoids

practical problems related to the nonuniform spatial sampling of peripheral

LORs and to the low count rates they exhibit. There is rarely any activity

at the periphery, because the patient is quite a distance from the surface of

the rings, and even if there were any activity the marked inclination of the

peripheral LORs relative to the crystal surfaces would reduce the solid angle

of acquisition.

5.3.2.4.3 Angular Sampling: Angular Compression

The frequency of the angular sampling in a PET system, which depends on

the total number of crystals in each ring, also inﬂuences the maximum spatial

resolutionof the system. In most of today’s PET scanners, far more crystals are

needed to ensure good spatial resolution along the radial coordinate than are

necessary to deﬁne good spatial resolution along the azimuthal coordinate,

especially in the center of the transaxial FOV. Therefore, the reduction of the

angular sampling frequency which results from the interleaving process does

not compromise spatial resolution, and a further reduction is even achieved

by what is called angular compression (or angular mashing) [42], that is, the

postinterleavingsum of coincidence counts of LORs andconsecutive φ values.

This procedure is equivalent to adding sinogram rows in groups of two-by-

two, four-by-four, etc. (sets of powers of 2), dividing the number of sampled

angles by a factor of two, four, etc. (Figure 5.38). Angular compression has

Radiation Detectors and Image Formation 193

With compression

With no compression

FIGURE 5.38

Diagram of angular compression procedure. In the sinogram, rows are added according to the

angular compression factor, and a new sinogram is constructed with a single row per added set

corresponding to the average angular position of the set.

the advantage of lowering both storage and data processing needs without

signiﬁcantly compromising the spatial resolution.

5.3.2.4.4 Axial Sampling in the 2D Mode: 2D Mashing

Inthe 2D mode, the mashing of LORs along theaxial directionis accomplished

by summing the coincidences between crystals located in the same ring k,

with those reported between the next ring (k +1) and the preceding ring

(k −1), the rings in the next succeeding position (k + 2), second preceding

position(k − 2), etc., up to amaximum evenvalue forthe ringdifferenceΔr

max

between the rings in coincidence, as shown in Figure 5.39a. The summing is

carried out for each pair of x

and φ coordinates, which is the same as adding

together the elements in the 3D sinograms of the planes deﬁned by those ring

combinations, that is,

, φ, z = k × Δd

) = s

, φ, z = k × Δd

, Δr = 0)

, φ, z = k × Δd

, Δr =±2)

, φ, z = k × Δd

, Δr =±Δr

max

) (5.50)

z = k × Δd

z = (k+1/2) × Δd

Direct planes

(a) (b)

Crossed planes

Δr = +2

Δr = 0

Δr = –2

Δr = +3

Δr = +1

Δr = –1

Δr = –3

FIGURE 5.39

Representation of the axial mashing performed in the 2D mode, which gives rise to (a) direct and

(b) crossed planes. In a system with N rings, there are N direct planes and N − 1 crossed planes.

194 Nuclear Medicine Physics

from which a single plane is created, called a direct plane, coinciding with

the axial position of ring k. Additionally, crossed planes are created by sum-

ming the coincidences between the rings k and (k +1), (k −1), (k + 2), etc.,

(Figure 5.39b), also up to a maximum ring difference, but now an odd num-

ber. The crossed plane produced with this mashing is ascribed to the middle

position between the k and (k +1) rings, which has an axial coordinate given

by z = (k +0.5) ×Δd

, φ, z = (k + 0.5) × Δd

) = s

, φ, z = (k + 0.5) × Δd

, Δr =±1)

, φ, z = (k + 0.5) × Δd

, Δr =±3)

, φ, z = (k + 0.5) × Δd

, Δr =±Δr

max

) (5.51)

In this way, it is possible to deﬁne Ndirect planes and N − 1 crossed planes

in a detector with N rings, in a total of 2N − 1 planes for the 2D mode.

5.3.2.4.5 Copolar and Axial Sampling in the 3D Mode: 3D Mashing

and the Michelogram

For the 3D mode, the axial mashing is combined with the mashing of the

parameter that determines the inclination of the sinograms, Δr. This process

involves summing 3D sinograms with the same axial coordinate z and adja-

cent Δr inclinations, yielding a total sinogram with that same z coordinate

and an inclination which is the average of the inclinations of the planes ﬁgur-

ing in the sum. This method creates sets of total sinograms that have the same

inclination but different axial coordinates z [42], called segments (Figure 5.40).

To create the summed sinogram with axial coordinate z = q ×Δd

(where

q is an integer or half-integer, depending on whether the plane is direct or

crossed) belonging to the segment with inclination Δr = m, we add all the

original 3D sinograms with that axial coordinate z = q ×Δd

, inclinations Δr

within a range centered in m, and with a width known as span:

3Dseg

, φ, z = q ×Δd

, Δr = m)

= s

, φ, z = q ×Δd

, Δr = m −(span −1)/2)

, φ, z = q ×Δd

, Δr = m −(span −1)/2 +1)

, φ, z = q ×Δd

, Δr = m +(span −1)/2)

(5.52)

The span is the maximum number of direct and crossed planes used to

obtain the total-sinogram and is necessarily an odd quantity. It is easy to

check that the sum of Equation 5.52 is conducted over all sinograms that

Radiation Detectors and Image Formation 195

Segment 0

2D mode

3D mode

Segment +1 Segment +2 Segment –1 Segment –2

FIGURE 5.40

Diagram of the 3D mashing of sinograms. In a given segment, the total sinograms obtained

from the mashing process cross all the possible integer (direct planes) and half-integer (crossed

planes) z positions. Segment zero corresponds to all the direct and crossed planes deﬁned in the

2D mode.

collect coincidences between the rings r

and r

which simultaneously satisfy

the conditions r

−r

∈[m−int(span/2), m +int(span/2)] and (r

)/2 =

q. Each segment is labeled by its inclination m, which may take the values of

0, ±span, ±2 ×span, ±3 ×span, etc., in a total of N

seg

number (always odd)

of segments. The number of segments is determined not only by the span

but also by the maximum ring difference (MRD), Δr

max

, which expresses the

maximum inclination of combined rings that it is allowed to use, according to

Δr

max

(span −1)

seg

−1)

×span. (5.53)

Itshould be notedthat onceΔr

max

ischosen, only certainspan valuesremain

possible (and vice versa) so that the relation of Equation 5.53 holds, with the

number of segments being an integer quantity.

The 3D mashing process is very aptly described by a diagram called the

Michelogram, developed by Christian Michel. The Michelogram is a 2D plot

in which all the possible combinations of ring pairs are represented by dots

along the horizontal (ﬁrst ring index r

) and vertical (second ring index r

)

axis (Figure 5.41). Each dot represents a sinogram, and the sum of sinograms

is indicated by lines joining the corresponding dots. Segments are delimited

by dashed lines, which differentiate the separate regions in the plot. This type

of plot is highly intuitive, and its careful analysis clariﬁes all the quantitative

relations shown in this section. In addition, it makes it possible to ﬁnd the

parameters of the 3D mashing very quickly, because the maximum ring dif-

ference is read directly from the number of dots in the ﬁrst row or column,

196 Nuclear Medicine Physics

Ring

Seg –2

Seg 2

Seg –1

Seg 1

Seg 0

Ring

310

FIGURE 5.41

Example of a Michelogram for a PET system with 32 rings, with mashing span parameters = 9

and Δr

max

= 22 (which deﬁnes N

seg

= 5 distinct segments). Each point represents the sinograma

for the pair of rings (r1, r2), the lines connecting points represent the sum of sinogramas for these

points.

whereasthe number of dots in a given rowor column inside the same segment

is the span.

5.3.2.5 Image Reconstruction

Image reconstruction in PET, as in SPECT, is based on the line-integral

approximation in both the 2D and 3D modes:

, φ, z) = c



−R

f (x, y, z) dy

(2D mode)

, y

, z, θ) = c



−R

f (x, y, z) dz

(3D mode)

(5.54)

For the purpose of image reconstruction, the constant c is ignored as well,

although in PET this constant can be experimentally measured. Its value is

routinely determined during PET scanner calibration procedures, and it is

a scale factor that enables PET to measure the absolute concentration of the

radioactive tracer throughout the patient’s body.

Radiation Detectors and Image Formation 197

The image reconstruction algorithms for PET normally assume that the data

are organized in the format of parallel projections and follow analytical or

iterative methods, depending on whether they invert the equations (Equation

5.54) or numerically compute the activity distribution that best reproduces

the 2D or 3D projections acquired. Some of the main image reconstruction

algorithms are discussed in Chapter 6.

5.3.3 Multimodal and Dedicated Systems

After some years of discussion 10 or 20 years ago on the relative advantages

and disadvantages of each imaging modality and after much speculation

about which modality would survive this ﬁght, the dominant perspective

changed and discussion began to focus on the possibility of merging differ-

ent modalities to optimally achieve a certain clinical or research goal. Thus,

for each speciﬁc application the objective is to develop or use a system, mul-

timodal or not, that has the best set of performance characteristics, while

optimizing the resources to achieve this purpose. In some cases, a system

based on a single modality and optimized for a given purpose can be the

simplest and cheapest system and can perform perfectly adequately. In other

cases, the synergy achieved by integrating two or more methods may lead

to the sharing of components and functions, to a better overall performance,

and to an optimization of resources that makes a system better than the sum

of the parts, although more complex. This section will give examples of sys-

tems optimized for speciﬁc applications, unimodal and multimodal, existing

or under development.

5.3.3.1 Systems for Speciﬁc Applications

With the gradual exploitation of the potential of a technique, often we see a

diversiﬁcation of the design of systems and the introduction of variants of

existing systems to meet the demands of the market and the emergence of

new applications. In the case of PET and SPECT, the development of systems

optimized for the study of a particular organ is especially attractive, because

placing the camera close to the organ increases the solid angle, and this can

signiﬁcantly improve the sensitivity, while at the same time lowering the cost

of the system, as it may require less detector volume and fewer electronic

devices. As an additional beneﬁt, the spatial resolution may increase in both

SPECT and PET: in SPECT, the resolution improves with a shorter distance

fromthe sourceto the detector (due to the effectof the collimator,as mentioned

in Section 5.3.1.1); and in PET the smaller distances between detectors reduces

the noncolinearity effect of the two annihilation photons. There may be more

gains,too, for example, with regard to noise anddetected event rates, resulting

from the speciﬁc geometries used (e.g., radiation scattered from outside the

ﬁeld of view decreasing with the reduction of the radius of the PET detector).