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

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238 Nuclear Medicine Physics

Other auxiliary corrections include the decay and the branching ratio cor-

rections. The ﬁrst is important when very short-lived markers are used, and

it corrects for the decline in activity that occurs from the instant when it is

administered to the patient until the start of the acquisition (and, if the exam

is long, the change in activity during the acquisition) by directly applying the

radioactive decay law. The second corrects the fraction of radioactive markers

that decay by processes other than β

decay, a correction which is important

in quantitative procedures if that fraction is not negligible.

From the geometric point of view, there are two more corrections that may

proveto benecessary in PET. Oneof themstems fromthe curvatureof therings

and consists of correcting the nonuniformity of spacing between consecutive

LORs along a sinogram row. The spacing narrows as the LORs become more

peripheral, being given by

Δx

= Δd



1 −





, (6.23)

where Δd

is the width of each crystal and R

is the camera’s radius. This

correction must be taken into account during image reconstruction whenever

the region being observed in a patient lies near to the limit of the FOV. The

usual practice involves interpolating new sinograms with a constant spacing

between radial coordinates, based on the measured ones.

The second correction of a geometrical nature that may be important is

related to the DOI effect, sometimes also referred to as the parallax error. This

effect arises from the possibility that photons hitting a given crystal obliquely

might penetrate beyond that crystal and be detected in neighboring crystals,

leading to an error in determining the direction of the LOR of the photon

(Figure 6.15). The primary result of this effect is the reduction of image reso-

lution in the periphery of the camera. It is only possible to correct or minimize

this effect using detectors providing DOI information, that is, detectors capa-

ble of measuring the distance from the crystal surface at which the photons

interact [79].

6.3 Image Processing

The analysis capabilities of medical images produced by different tech-

niques have improved considerably over time. These analysis procedures

have been the object of constant optimization so that they can better reveal

structural anomalies or functional behavior. Segmentation algorithms (i.e.,

for anomaly delimitation) have been developed in this ﬁeld, along with

procedures to determine image characteristics (by deﬁning parameters that

quantify visual details), and their coping with previously obtained limits. The

Imaging Methodologies 239

FIGURE 6.15

Depth of interaction (DOI) effect. If a photon pair hits the crystals obliquely, it is possible that the

two photons penetrate neighboring crystals and produce a coincidence count in an LOR with a

wrong direction (dashed line).

limits are established by observing a very large number of normal individuals,

preferably belonging to the population one intends to study.

The practical objective of anomaly detection in medical imaging is to pro-

vide a simple yes or no answer on the existence of a given anomaly. This

answer is, in principle, substantiated by a set of elements based on certain

image features; but when the detection is doubtful or it is not possible to iden-

tifythe relevantfeatures,then we have to look for contextualinformation. This

is a procedure of image interpretation, somewhat more complex than simple

detection. Different image modalities can help to reach a conclusion and they

provide an important way of clarifying the presence of pathologies.

Image processing consists of applying a broad set of algorithms and tools

to repair, analyze, develop, and visualize images. For instance, it is possi-

ble to improve noisy or degraded images to enhance certain image details

in order to better detect lesions, to extract a range of parameters from the

image, to characterize shapes and textures, and to perform the coregistration

of images obtained with different modalities. There are basically three levels

of information processing in image techniques:

• Low-level processing is characterized by both the input and the out-

put data being images. It involves basic initial operations, such as

preprocessing for noise reduction, contrast enhancement, and image

detail improvement.

240 Nuclear Medicine Physics

• Medium-level processing: At this level, the input is an image, but the

outputs are image attributes extracted from the input image, such as

peripheral regions, contourlines, and even the identiﬁcation of image

objects. Medium-level processing involves tasks such as segmenta-

tion (partitioning an image into regions or objects), the description

of objects and corresponding data reduction for digital processing,

and the classiﬁcation (and recognition) of individual objects.

• High-level processing: This type of processing involves searching for

a set of characteristic objects based on the analysis of images for the

performance of cognitive functions usually associated with vision.

In general terms, the most important steps in image processing are the

acquisition, transformation (preprocessing), segmentation, selection of char-

acteristics,and classiﬁcation.The processingoperationsthat canbe performed

on medical images can be grouped as follows:

• Image enhancement and repair, which includes image addition

and subtraction, linear and nonlinear ﬁltering, ﬁlter projections,

deblurring, and automatic contrast enhancement.

• Image analysis, which includes texture analysis, detection of con-

tour lines, detection of morphology, segmentation (partition into

features), region-of-interest (ROI) processing, measurement of image

characteristics, and the extraction of relevant image information.

• Image recognition.

•

Image compression.

• Visualization and processing of multidimensional images (3D

and 4D).

• Colour image processing, which includes color format conversions

and the import and export of machine-independent ICC (Inter-

national Color Consortium) proﬁles; an example of this type of

processingis the color transformation fromthe RGB (red,green, blue)

format used in the visualization to the CMY (cyan, magenta, yellow)

format used for hard copy printing.

•

Image transformation to alternate versions that are more efﬁcient,

where the FFT (fast-Fourier transform), DCT (discrete cosine trans-

form), Radon, and fan-beam projection transforms are included.

•

Multimodal images processing and analysis.

• Reconstruction from projections.

•

DICOM (digital imaging and communications in medicine) import

and export.

• Interactive visualization of images and modular tools to generate

GUI (graphical user interface) images.

Imaging Methodologies 241

In digital NM imaging systems, a computer is involved in the acquisition

process, as the measured signals become image data only after signal process-

ing. In these systems, it is usual to make a distinction between the processes

that operate on the raw data and lead to the formation of an image directly

observable on a screen (preprocessing) and the computer-assisted methods

for subsequent analysis and modiﬁcation of the image content. The latter are

often called digital image processing, or postprocessing, methods, and they

target the extraction and restoring of implicit information that is not directly

accessible via visual inspection.

The preprocessing of data has a number of purposes, including attenuation

and scattered radiation correction, normalization correction, for example, in

intrinsic efﬁciency, geometric, dead time, decay, and partial volume, or the

scaling, sampling, time realignment of data, etc., all of which are intended

to produce corrected digital images optimized for input in the subsequent

processing tasks.

As a ﬁrst step, the camera or acquisition device generates an image with

supposedly appropriate statistics. This is followed by a second phase, where

the raw data are subjected to preprocessing. The next step consists of adding

context to the image, which is done by a specialist aided by computer-assisted

analysis. The image context takes into account the probable pathology of

the patient, its position, the type of study, the working parameters of the

equipment, etc., and converts the recorded data into new images, possibly

with quantitative information that will support the task of the clinician.

The nuclear imaging software should be able to examine extensive regions

of the patient (up to his whole body) in a short time and make use of sophis-

ticated correction methods (e.g., the quality control of the camera) based on

efﬁcient algorithms. As such, it must be able to handle very large sets of data

and allow the processing and analysis of complex 3D data. It is also neces-

sary to have fast CPU processing speeds for the direct acquisition of 3D or

4D images in real time. The computing industry seems to be reaching a point

where it may no longer respond to the highly demanding conditions of real-

time image processing, in which case image segmentation techniques will

prove to be an important support.

It is also usually necessary to perform several corrections for scintillation

cameras to prevent anomalies such as spectral shifts (energy correction),

wrong positioning of regions (spatial linearity correction), local counting

differences (uniformity correction), loss of counts due to pile-up (pile-up

correction), and stabilization of the photomultiplier tubes (high-voltage

adjustment).

The radiopharmaceuticals administered to the patients to image physio-

logical functions and identify diseases are becoming increasingly specialized

(i.e., organ and disease speciﬁc). Those employed in PET and SPECT are

widely used at present, but only PET is able to quantify certain important

metabolic measurements in absolute physiological conditions and offer the

most cost-effective means of staging several types of cancer. The high cost

242 Nuclear Medicine Physics

of PET studies and, in particular, of the facilities needed for those studies

have restricted its application to a limited number of centers. Nevertheless,

PET has provided much information that helped the development of radio-

pharmaceuticals marked with gamma emitters for SPECT, which, in many

cases, became a good alternative to PET, for imaging the same function. New

perspectives were introduced with coincidence detection using dual-head

gamma cameras. Comparing SPECT and PET imaging devices in terms of

sensitivity, spatial resolution, SNR, performance limits, etc., it seems unlikely

that there will be any major changes in the current state of the two techniques

in the near future.

Although there have been many advances in radiochemistry and radio-

pharmacology,

99m

Tc is the most widely used radioisotope in conventional

NM diagnostics, and it has been so for over 30 years. It will probably retain

its popularity due to its availability, low cost, low radiation dose, and highly

convenient chemical properties. Research is currently going on in many labo-

ratories all over the world to develop ligands marked with

99m

Tc and

123

I for

SPECT, and with

For

C for PET, designed to bond to chosen receptors in

neurology, oncology, and cardiology.

The radiopharmaceuticals used in NM imaging diagnostics should provide

useful clinical information and expose patients to minimal radiation doses.

For this, the radiation emitted by the tracer should not contain particles, the

half-life of the nuclide has to be of the order of the exam duration, the energy

of the gamma photons needs to be appropriate for efﬁcient detection with

gamma cameras, and the speciﬁc activity of the radiopharmaceutical has to

be high enough not to affect the biochemistry of the patient.

The accurate reconstruction of 3D and 4D images is difﬁcult, mainly owing

to noise; the lack of efﬁcient analytical reconstruction methods, especially

for certain acquisition geometries; image degradation factors; and the mis-

alignment of imaging systems, particularly in devices used for the imaging

of small animals. Further, with the increasing interest in minimal invasive

surgery and in intensity modulated radiotherapy (IMRT), the precise locating

of anatomical targets has become more and more important.

6.3.1 Image Reconstruction

As noted in Chapter 5, tomographic images obtained in PET and SPECT result

from the application of reconstruction algorithms that can be classed as ana-

lytical and iterative algorithms and which, in general, take some model with

regard to how the activity distribution of the radioactive tracer in the volume

of the patient is translated into measurements, normally in the form of parallel

projections of the distribution. The inversion of the image formation model

allows recovery of the original tracer distribution as a volume of images cor-

responding to sequential slices of the object along a chosen direction. The

model most often used to describe the formation of the parallel projections of

Imaging Methodologies 243

the object presumes that, in the absence of attenuation or Compton scattering,

the number of photons detected in a projection line which passes through the

object is proportional to the integrated activity of the distribution f (x, y, z)

along that direction, that is,



direction i

(γ counts) ∝



direction i

f (x, y, z) dr

direction i

. (6.24)

This notion corresponds to the notion of a Radon transform of an object, a

mathematical operation that is central to all tomographic techniques.

Taking the notation introduced in Chapter 5 (see Figure 5.27), this line-

integral approximation [80] can be expressed in terms of the parallel

projections of the object’s activity, in two or three dimensions, by

, φ, z) = c



−R

f (x, y, z)dy

, y

, z, θ) = c



−R

f (x, y, z)dz

. (6.25)

For image reconstruction purposes, the constant c can be ignored, although

it is usually experimentally determined in PET when we want to have an

absolute calibration of the image values. This constant is a scale factor, which

endows the imaging technique with the ability to measure the absolute

concentration of the radioactive tracer throughout the body of the patient

and whose value can be found with the aid of phantoms during camera

calibration.

The reconstruction algorithms normally assume that the data are organized

in parallel projections and follow analytical or iterative strategies, depending

on whether they invert Equations 6.25 or numerically compute the activity

distribution that better reproduces the 2D or 3D projections acquired.

6.3.1.1 2D vs. 3D Reconstruction

The term 2D reconstruction refers to the determination of the activity distri-

bution for a given plane or slice of the object, using for that purpose the

projections of the object measured along that plane. This term also means

the process of computing the 3D distribution of the activity of the object

by independently reconstructing the activity in each 2D plane, using a 2D

reconstruction algorithm. For the latter, the 3D volume obtained has the

same dimensions as those obtained by the 3D reconstruction of the object.

The 3D reconstruction in PET differs from 2D reconstruction in that it uses

244 Nuclear Medicine Physics

Acquired data

2D data

2D reconstruction

(several

reconstructions

one per plane)

2D reconstruction

(one reconstruction

for all the planes)

3D data

Rebinning

Final volume of images

Pilling of the images of

each plane

etc...

, θ

, ...

= 0°

f(x, y, z)

FIGURE 6.16

Outline of the PET image reconstruction process in 2D and 3D.

information collected in oblique planes and by the fact that it uses the infor-

mation from all the projections together (rather than independently, plane by

plane) to construct the volume of the object (Figures 6.16 and 6.17).

6.3.1.2 2D Reconstruction

6.3.1.2.1 Analytical Methods

In a 2D reconstruction algorithm, image reconstruction is carried out in planes

perpendicular to the axis of the system, which means that the algorithms han-

dle projection arrays corresponding to ﬁxed z planes, p(x

, φ, z)



z=z

. These

arrays can be looked at as the compilation of one-dimensional parallel projec-

tions along the different acquisition angles φ (Figure 6.18a), which enables the

simplest analytical reconstruction procedure to be deﬁned, the backprojection

[81,82]. In this procedure, the number of photon counts for a given pair of

, φ) coordinates is added to all the image points located along the cor-

responding projection line, as depicted in Figure 6.18b. By performing the

backprojectionof all the one-dimensional projections over all the φ angles, one

obtains an estimate f

est

(x, y, z)



z=z

of the activity distribution f (x, y, z)



z=z

which gave rise to those projections.

Imaging Methodologies 245

2D acquisition

2D reconstruction

(AW)OSEM, ...

OSEM-3D, ...

3DRP, ...

Iterative methods

Analytic methods

FBP, ...

3D reconstruction

2D data

Rebinning

FORE, ...

3D data

3D acquisition

FIGURE 6.17

Summary of the reconstructionstrategies in PET, indicating some of the most popular algorithms.

However, the ﬁnite sampling of projection angles, together with the uni-

form distribution of the photon count values resulting from backprojection

along the projection line, generates artifacts, that is, there are points in the

reconstructed image to which some counts are attributed but that in reality

have no activity (Figure 6.19). In the limit when the number of projections

used in the reconstruction becomes inﬁnite, those artifacts give rise to a blur-

ring effect [81], which is the spreading out of counts from the point where

the activity source is located; the magnitude of the spread is proportional to

1/r, with r being the distance from the source (Figure 6.18c). This effect is

directly due to the oversampling that the spatial point where the source lies is

subjected to, as it concentrates a density of projection lines, which, in the limit

when the variation of φ is continuous, tends to inﬁnity [80]. Mathematically,

this effect corresponds to a convolution in each point (x, y) of the true image

(a)

(b) (c)

FIGURE 6.18

(a) One-dimensional projections in three different Π angles generated by a point source located

in the detector axis. (b) Image of the point source shown in (a) reconstructed by the backpro-

jection of 3, 6, and a very large number of projections. When the image is reconstructed using

few projections, star-shaped artifacts are created, which become blurred when the number of

projections is very large. (c) The blurring effect affecting the backprojection.

246 Nuclear Medicine Physics

(a) (b) (c)

(d) (e)

FIGURE 6.19

Comparison between the true activity distribution (a) with images reconstructed using the

backprojection of 1, 2, 4, and 16 parallel projections of a (b–e).

with the function 1/r [82], that is,

est

(x, y, z)



z=z

= f (x, y, z)



z=z

∗

1/r, (6.26)

where ∗denotes the convolution operation, and r is the distance to the point

(x, y) for the plane z = z

The solution to the blurring effect is based on not allowing the addition

of counts along the projection lines to be uniform, but instead weighting it

so that fewer counts are added to points near the activity source. A simple

way of implementing this unequal distribution is by convoluting, or ﬁltering,

the backprojected image with a given ﬁlter [81], an operation that is per-

formed by multiplying in the Fourier space the Fourier transforms of the ﬁlter,

w(ν), and of the image data. In the method known as ﬁltered-backprojection

(FBP) [81,82], the ﬁlter is directly applied to the one-dimensional projections

p(x

, φ, z)



z=z

for each φ angle, and the result, a set of ﬁltered projections

p(x

, φ, z)



z=z

∗

w(x

), is then backprojected. If {}denotes the backprojection

operation

∗





p(x

, φ, z)



z=z



2π



p(x

, φ, z)



z=z

dφ = f

est

(x, y, z)



z=z

, (6.27)

∗

It should be noted that with PET, the upper limit of the integral in Equation 6.27 is π, because

there are two detected photons and not just one, as in SPECT.

Imaging Methodologies 247

then one has for the FBP method

estFBP

(x, y, z)



z=z

=



p(x

, φ, z)



z=z

∗

w(x

)



. (6.28)

An alternative method consists not of ﬁltering the one-dimensional projec-

tions but of ﬁrst backprojecting them and only then applying the ﬁlter; this is

the backprojection-ﬁltering method (BPF) [81,82]:

estBPF

(x, y, z)



z=z

=



p(x

, φ, z)



z=z



∗

w(x, y). (6.29)

This method is seldom used, as it produces images of a quality comparable

to those obtained using the FBP, with the aggravation that it involves the

computation of a 2D Fourier transform, which makes it slower to execute.

The ﬁltering tries to correct the effect of convolution with a function of

the type 1/r; as the Fourier transform of 1/r is 1/ν; the function that nat-

urally appears as the ﬁlter to use is the ramp function in the frequency

domain, w(v) = v. In practice, however, the ﬁnite spatial sampling of the

activity distribution f forces the ﬁlter to be truncated at the Nyquist fre-

quency, v

= 1/2Δx

, and prevents the exact shape of f from being restored

[80]. Further, the statistical noise existing in the data is ampliﬁed by the

ramp ﬁlter in the high frequencies, which diminishes the contrast of the

reconstructed image. To minimize this problem, the ramp ﬁlter is usually

apodized with decreasing functions of v, such as the Hann and Hamming

windows [83] (Figure 6.20). Apodization causes the smoothing of the image

by suppressing abrupt changes but decreases image resolution [81]; for that

reason, the apodization window parameters must be chosen while taking into

account that one is trading less noise for less resolution and that the level of

compromise between the two must be governed by each speciﬁc situation.

6.3.1.2.2 Iterative Methods

Theanalytical reconstructionmethods, based on Fourier transformsand back-

projection operations, are very important in clinical practice, as they are fast,

stable, and easy to implement. However, these methods are rather inﬂexible,

because the possibility of incorporating information from the process of image

formation in the reconstruction process is very limited, as is the imposition

of certain known physical conditions, such as the degree of smoothness of

the imaged object or the shape and volume occupied by it. In particular, the

fact that they employ ﬁlters truncated at or near the Nyquist frequency often

generates artifacts by underestimation at the frontiers between regions with

very different activities, which sometimes yields negative (i.e., nonphysical)

values for the activity in certain areas of the image [80].

Iterative reconstruction methods, on the other hand, are very ﬂexible and

allow the incorporation of image formation models that correct the effects of

image deterioration, such as Compton scattering or attenuation, during the

reconstruction process itself [80]. This is a major advantage over the analytical