Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics

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228 Nuclear Medicine Physics

but there are three methods that can perform this conversion: segmentation,

scaling, and CT scans with two different x-ray energies.

6.2.2.2.2.2 Segmentation Segmentation methods can be used to separate the

CT image in regions that correspond to different types of organic tissue (e.g.,

soft tissue, lung, and bone). The HUs obtained with the CT scan are replaced

by the attenuation coefﬁcients for a photon energy of 511 keV. A signiﬁcant

problem arises from the fact that the density of some regions of tissue con-

tinuously varies, and so it cannot be represented by a discrete ensemble of

segmented values. For example, some lung regions exhibit density variations

amounting to a 30% difference [41].

6.2.2.2.2.3 Scaling The image values produced by the CT scan are generally

linearlycorrelatedwiththe attenuationcoefﬁcientof the correspondingtissue.

This makes it possible to estimate the patient attenuation map for 511 keV

radiation. All that is needed is to multiply the entire CT image by the ratio

of the attenuation coefﬁcients of water (representing the soft tissues) for the

energies of the CT and PET scans. Since the spectrum of energy of the CT

scan is a continuum, just one value is usually estimated (typically 70 keV),

which then represents the whole spectrum. La Croix et al. [42] carried out

simulation studies to investigate different scaling techniques for SPECT scan

attenuation coefﬁcients for an energy of 140 keV. The results show that the

mentioned rescaling technique yields correct attenuation coefﬁcients only for

low atomic number (Z) materials (e.g., air, water, and soft tissue). For bone,

with a higher atomic number, the study reveals that the results of this linear

scaling method are not satisfactory, because the photoelectric contribution

100.00

10.00

1.00

0.10

0.01

10 100

Bone–Total

Bone–Photoel. eﬀect

Bone–Compton eﬀect

Muscle–Total

Muscle–Photoel. eﬀect

Muscle–Compton eﬀect

Energy (keV)

μ (1/cm)

1000

FIGURE 6.8

Linear attenuation coefﬁcients for bone and muscle between 10 and 1000 keV. The components

corresponding to photoelectric absorption and Compton scattering are also shown. The total

absorption coefﬁcient shown considers all types of interactions possible.

Imaging Methodologies 229

is dominant for the lower energies in CT scans (Figure 6.8). In other words,

different factors have to be used in bone and soft tissue scaling to transform

CT images acquired at an effective energy of 70 keV into an attenuation map

calibrated for 511 keV.

One method to compensate for the presence of high Z material arises

from noting that HU values such that −1000 < HU < 0 correspond mostly

to regions that contain mixtures of lung and soft tissue; whereas regions with

HU > 0 contain soft tissue and bone. Blankespoor et al. [43] have suggested

that a bilinear scale could be used to convert CT images into attenuation

maps for 140-keV SPECT. In the method suggested, two factors are used for

the rescaling: One factor is used for water and air (−1000 < HU < 0), and

the other is used for water and bone (HU > 0). The attenuation coefﬁcients

resulting from this method, but corresponding to PET radiation at 511 keV,

are displayed in Figure 6.9. This method was proposed for PET by Burger

et al. [44] and Bai et al. [45].

An alternative method that could be used to convert CT images into atten-

uation maps is the so-called hybrid method. This combines the segmentation

and scaling methods. The attenuation map for 511-keV radiation is estimated

by using an initial pedestal that separates the bone component in CT, followed

bytwo otherfactors that rescalebone fromnonbonecomponents. This method

is motivated by the information displayed in Figure 6.10, which shows that the

ratio of the mass attenuation coefﬁcientsfor radiation with 70 keV and 511 keV

is quite similar for all organic materials except bone. This is due to the larger

photoelectric fraction in bone caused by the presence of calcium (amounting

to about 22.5% in cortical bone). The behavior of the hybrid method when

converting HU to linear attenuation coefﬁcients at 511 keV contrasts with the

–1000

0.05

0.10

0.15

0.20

μ (1/cm)

CT number (Hounsfield units)

–5000 5000

15001000

Hybrid

Bilinear

FIGURE 6.9

Conversion of HU into linear attenuation coefﬁcients at 511keV. Note the change in slope at

HU = 0 for the bilinear method and the discontinuity at HU =300 in the hybrid method.

230 Nuclear Medicine Physics

μ/ρ (g/cm

)

10 1000100

0.10

0.01

1.00

10.00

100.00

Energy (keV)

Muscle

Bone

Fat

FIGURE 6.10

Mass attenuation coefﬁcients for three biological materials.

bilinear rescalingmethod, as shown in Figure 6.9, where,for heuristic reasons,

the pedestal chosen to separate bone from nonbone regions was HU = 300

[46]. This means that the hybrid method assumes regions of −1000 < HU < 0

to be mixtures of air or water and regions of HU > 300 to be mixtures of air

or bone (Figure 6.10).

It has been shown that both the bilinear scaling method and the hybrid

methodyield reasonableresultsfor biological material [45,47,48].Recent stud-

ies compare patient transmission images based on both positron sources and

x-ray sources [49,50]. The impact of these transmission images on the recon-

structedimagesin studieswith thebiotracer FDGshow slight, evennegligible,

effects. This is no longer true if the patient is injected with a contrast agent or

when the effects contain metal objects.

6.2.2.2.2.4 Dual-Energy X-ray Imaging A precise solution of the problem of

converting HU into linear attenuation coefﬁcients for 511 keV radiation can

be obtained if two x-ray images with two different spectra are obtained. The

attenuation coefﬁcient is now a weighted combination of the photoelectric

absorption and Compton scattering, that is, in essence it is a system with two

components. If the individual photoelectric and Compton components could

be determined, they could be separately converted into any desired energy

and then added together to give the total attenuation coefﬁcient. Alvarez

and Macovski [51] developed one such quantitative CT imaging method.

It was used by Hagesawa et al. [52] to create a monoenergetic attenuation

map at 140 keV for a prototype SPECT or CT detector. One potential dis-

advantage of the method arises from the fact that it obtains the attenuation

map by computing the difference between the two CT scans obtained with

different energies, which means that the noise of both images is added to

the quadrature. Consequently, even though double-energy techniques poten-

tially offer the highest degree of precision, they also result in degraded noise

Imaging Methodologies 231

characteristics compared with methods obtained either from a single CT scan

or from transmission scans with positron emitting radionuclides.

6.2.2.2.3 Random Coincidences Correction

As mentioned in Chapter 5, coincidences can be true (T), random (R),or

scattered (S). The number of coincidence events, C, detected per unit time in

an LOR is the sum of these three types of coincidence,

C = T +R +S, (6.14)

but only the true coincidences contain valid information about the location of

the source that generated the photon pair detected. The correction of random

and scattered coincidences consists of obtaining an independent estimate of

the random coincidences rate

R and scattered coincidences rate

S for each

LOR in order to estimate the true coincidences rate by

T = C −

R −

S. (6.15)

The random coincidences rate can be estimated in two ways: by using

Equation 5.43 directly, if the singles count rates R

and R

are known for

each LOR and the coincidence window of the system Δt is also known [53];

or by using a hardware-implemented delay line, which delays the time sig-

nals of each detector by a time lag considerably larger than Δt [54,55]. Since

the large time lag prevents there being any correlation between the delayed

line of one detector and the nondelayed, or prompt, line of another, the coin-

cidences existing between the delayed and prompt lines of any two crystals

are random and provide a way to estimate the random coincidences rate

for the corresponding LOR (Figure 6.11). Using a delay line is the commonest

way of obtaining

R, because it can be done online during the acquisition, and

it does not need any type of input from the operator of the system.

6.2.2.2.4 Scattered Coincidences Correction

Thecorrectionofscatteredcoincidencesinvolves determiningwhetherin each

recorded coincidence any of the two annihilation photons detected interacted

with an electron inside the patient’s body via Compton scattering. Just as in

SPECT, the simplest way of reducing the effect of Compton scattering consists

of measuring the energy of the photons hitting the detectors and rejecting all

the coincidences in which one of the photons exhibits energy lower than a

preset value [56]. This discriminating value is typically around 350 keV; but it

must be chosen carefully, because if it is too low then the rejection of scattered

coincidences will not be efﬁcient, whereas if it is too high then there is the risk

of rejecting true coincidences for which one of the annihilation photons did

not deposit all its energy in the scintillator that detected it.

To supplement the simple energy window discrimination, a more precise

way of correcting the data from scattered coincidences is often used. One

232 Nuclear Medicine Physics

Detector 1

Detector 2

Coincidences

(prompt)

Detector 2

+ delay

Coincidences

(delayed)

Delay

Time

FIGURE 6.11

Diagram of the random coincidence correction using a hardware delay line. Besides the prompt

line of timing signals, each crystal has a delay line whose coincidences with the prompt lines of

other crystals enable the estimation of the random coincidence rates of the corresponding LORs.

method involves using the attenuation map determined with the transmis-

sion measurement together with a model of the geometry for the scanner or

patient set to compute the fraction of scattered photons hitting each crystal.

The computation takes into account the probability of each annihilation pho-

ton interacting with the patient (given by the attenuation map); and, should

it interact, the probability of the scattered photon being emitted along a direc-

tion θ is given by the Klein–Nishina equation for the differential cross-section

dσ/dΩ as a function of θ,

dσ

dΩ

(θ) =

1 +cos



1 +(E

)(1 −cos θ)





1 +



)(1 −cos θ)



(1 +cos

θ)



1 +(E

)(1 −cos θ)





, (6.16)

where Z is the effective atomic number of the medium, m

is the electron’s

mass, c is the speed of light, and r

is a quantity known as the classical elec-

tron radius. One of the most precise and efﬁcient methods uses a simpliﬁed

simulation, assuming that there is only one Compton scattering event in the

path of the photon pair giving rise to a scattered coincidence [57,58], a sit-

uation that corresponds to the large majority of Compton coincidences in

PET [59].

There are many other techniques for the correction of scattered coinci-

dences, as described in detail in [38]. They are all based, in one way or another,

onthe physics of Compton scattering and in the geometric and technical speci-

ﬁcities of PET. Since scatter in PET can be counted in projection lines external

to the object (contrary to SPECT where it is unlikely that this could happen),

Imaging Methodologies 233

thereareseveral methodsthat exploit this factand estimate its spatial distribu-

tion by ﬁtting the activity tail observed on the outside of the object’s frontiers

with, for example, a Gaussian function in one [60] or two dimensions [61]. An

assumption often used as a starting point is that the distribution of Comp-

ton scattering events over space slowly changes and mainly comprises low

spatial frequencies, which allow modeling of the distribution inside the object

with Gaussian or polynomial functions, or even by curves obtained by convo-

lution [62] or deconvolution [63] from the total distribution of coincidences.

Another approach uses methods based on data acquisition with one [64,65]

or more [66,67] additional energy windows covering energy ranges below

[64] or above [65] the usual energy window centered around 511 keV. Finally,

there are yet other methods that compute the scattered radiation distribution

with sophisticated Monte Carlo simulation techniques [68] and methods that

explore the complementarity of all these techniques in an effort to optimize

correction [69,70].

Scattered radiation correction is usually ignored in data acquired with the

2D mode, because the physical collimation provided by the septa drastically

reduces the amount of scattered coincidences. In the 3D mode, however, cor-

recting the data from scattered radiation is mandatory, as the sensitivity to

scattered coincidences is much greater. Figure 6.12 illustrates that require-

ment, as the impact of the scattered coincidences correction is clearly seen in

3D mode data, but it is almost imperceptible in the 2D mode.

Emission

Attenuation

Without correction

mode

With correction

FIGURE 6.12

Impact of the scattered coincidence correction in the 2D and 3D modes. On the right, the atten-

uation map of the object is shown, with the lighter regions indicating areas of higher linear

attenuation coefﬁcient. Comparing this map with the noncorrected 3D mode image, the existence

of more scattering in the more attenuating media is apparent, as expected.

234 Nuclear Medicine Physics

6.2.2.2.5 Normalization Correction

The coincidence detection efﬁciency of a PET system is not uniform for all

LORs, that is, the probability of a coincidence being detected varies from LOR

to LOR. This is due both to variations in the intrinsic efﬁciency of each crystal

and its coupled photomultiplier tube, variations that may vary over time, and

to systematic effects related to the geometrical arrangement of crystals. The

correction of the nonuniformity of response for the LORs in a PET system is

called normalization correction, in which dead-time effects are also included.

The normalization correction commences by carrying out a measurement,

the normalization measurement, in which all LORs are illuminated equally

by a uniform activity source [71]. In these conditions, the number of counts

obtained in a given LOR, properly corrected from the existence of attenuation

and random and scattered coincidences, is directly proportional to the prob-

ability of detecting a coincidence in that LOR. The inverse of that number,

1/C

uniform

, called the LOR’s normalization coefﬁcient, NC, can then be used

to correct the coincidences, C, of regular measurements using the equation

norm

= NC × C. (6.17)

This normalization method, known as the direct method [38,71], is often

used to normalize data acquired in the 2D mode, but it is frequently impracti-

cal for the normalization of 3D mode data due to the overwhelming number

of 3D LORs that are found in modern PET systems (several tens of millions),

which implies the acquisition of a massive number of coincidence events

in the normalization measurement in order to obtain a reasonable statistical

precision for the NC factors and renders the normalization procedures incom-

patible with everyday clinical use. Further, the larger solid angle covered by

the 3D mode subjects the coincidence processing system to an event rate con-

siderably higher than in the 2D mode, forcing the use of low activity sources

in the normalization procedure, which makes it even more time consuming.

Additionally, storing and handling such a large number of NC coefﬁcients

can also be a problem.

The normalization correction of 3D data, therefore, employs other methods

called indirect or component methods to model the individual response of

each detector instead of directly measuring the response of each LOR to coin-

cidence events [72–74]. The normalization coefﬁcients are separated into the

product of several components, one for each effect modeled:

NC = NC(intrinsic efﬁciency)

×NC(geometric efﬁciency)

×NC(dead time)

. (6.18)

Imaging Methodologies 235

The components relative to the intrinsic and geometric efﬁciencies of the

system are normally broken down into several other factors, depending on

the speciﬁc features of the tomograph, especially its geometry [75,76].

With this method, it is possible to reduce the number of parameters needed

for the 3D normalization from several million (the number of LORs in a PET

scanner) to just a few thousand (of the order of the total number of crystals),

making it practical to implement [71]. In addition, the number of coincidence

events that has to be accumulated during the normalization measurement

is substantially smaller than that needed for the direct method, although it

is still quite substantial, partly because the methodologies used to estimate

the different components from the normalization measurement are not fully

independent in the presence of noise.

6.2.2.2.5.1 Intrinsic Efﬁciency The correction of intrinsic efﬁciency aims at

offsetting the variations in the intrinsic efﬁciency of the crystals in the PET

scanner. The simplest way of doing this is by estimating the individual

efﬁciency ε

of each crystal from the normalization measurement using the

fan-sum method [72,73]. This method estimates the relative efﬁciency of each

crystal i with regard to the average efﬁciency of all crystals in the ring where

it is located, by dividing the sum C

fan

of all counts recorded in the fan facing

that crystal, that is, all the LORs it establishes with the other crystals in the ring

(Figure 6.13) by the average value of the fan sums of all the crystals of the ring:

fan



fan



ring

. (6.19)

Since the measurement of the efﬁciency for a crystal i is not totally inde-

pendent of the efﬁciencies of the other crystals in its fan, this method may

FIGURE 6.13

Diagram of all the LORs deﬁned between a crystal i and the other crystals of its ring; this set of

LORs is called the fan of crystal i.

236 Nuclear Medicine Physics

produce inaccurate values if, for any reason, one of the crystals in the ring

has a very different efﬁciency from the others [77].

The normalization coefﬁcient expressing the intrinsic efﬁciency of the LOR

deﬁned by the crystals i and j reads

NC(intrinsec effeciency) =

×ε

, (6.20)

assuming that all the rings exhibit identical overall efﬁciencies. Should that

not be true, the relative efﬁciency of each crystal given by Equation 6.19 needs

to be multiplied by the efﬁciency of the ring where it is located.

6.2.2.2.5.2 Geometric Efﬁciency Geometric efﬁciency correction consists of

compensating the variations arising from the geometric speciﬁcities of the

camera’s construction that affect the detection efﬁciency of each LOR. Since

these variations are purely geometrical and do not change over time, they are

typically determined only once for each PET system.

The main geometrical factor responsible for the differences in the response

of LORs belonging to the same parallel projection is the decrease of the inci-

dence angle of the photons on the crystal surface as LORs become more

peripheral (Figure 6.14). The change of angle reduces the solid angle of detec-

tion, which leads to poorer LOR efﬁciency. The systematic variation of the

detection efﬁciency with the radial coordinate is called the geometrical pro-

ﬁle [75]; its inverse is the normalization coefﬁcient for the geometric efﬁciency

of the camera.

In addition to the geometrical proﬁle, it is also usual to carry out a second-

order geometric correction in systems where the crystals are grouped in

detector blocks. The systematic variation of the crystal positions inside a

block, which is reﬂected in slightly different acquisition geometries due to

(a) (b)

FIGURE 6.14

(a) Variation of the incidence angle of the LORs on the crystal surfaces in a ring; since the inci-

dence is further from perpendicularity, the cross-section of the LORs diminishes, decreasing also

their detection efﬁciency. (b) Geometrical interference between crystals in a system assembled

from detection blocks. Each distinct position inside a block corresponds to a slightly different

acquisition geometry of the crystal.

Imaging Methodologies 237

features such as the depth of the cuts between crystals or the coupling to the

photomultiplier tube, modulates the geometrical proﬁle; this modulation is

known as crystal interference [76].

6.2.2.2.5.3 Dead Time As mentioned in Chapter 5, the sensitivity of PET is

higher than that of SPECT, because it dispenses with the use of physical colli-

mators to build parallel projections of the activity. This leads to a count rate in

PET that is much higher than in SPECT; and, therefore, it becomes important

to consider the dead time that exists after a crystal detects a photon, time dur-

ing which the crystal is incapable of detecting new incoming photons. This

dead time decreases the number of coincidences actually recorded relative

to the number of coincidences whose photons hit the detector and interacted

with the crystals. In the simplest model to describe the dead time effect in a

PET scanner, which ignores crossed interference between crystals in a block,

the number of singles counts in a crystal i, n

single

, is given as a function of the

number of incident photons n

inc

that interact with the crystal and of the dead

time parameter τ

of that crystal by

single

= (1 −τ

) ×n

inc

, (6.21)

where τ

is the dead time of the crystal divided by the total acquisition time.

Since each coincidence involves the detection of two photons in different

crystals, the normalization coefﬁcient for the dead time component in an LOR

deﬁned between the crystals i and j that corrects the incident coincidences rate

from the detected coincidences rate can be given by

NC(dead time) =

(1 −τ

) ×(1 −τ

)

. (6.22)

6.2.2.2.6 Other Corrections

In addition to the corrections just mentioned, there are others that, in certain

circumstances, should be performed when quantitative PET measurements

are wanted.

Partial volume correction can also take place in PET; it is based on the same

principles and uses correction strategies similar to those applied in SPECT (cf.

Section 6.2.2.1.5) [78]. This correction is mostly performed in PET, especially

in research studies that need a precise absolute quantiﬁcation in small struc-

tures, with dimensions up to 3 times the camera’s spatial resolution. Still, with

the increasing use of PET systems for animals, where the need for imaging

small structures is obvious and multimodal hybrid systems PET or CT and

PET or MRI, where the existence of high quality anatomical information and

good coregistration techniques means this correction can be performed easily,

partial volume correction is being employed more often, although it is not yet

routinely included in commercial cameras.