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

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388 Nuclear Medicine Physics

System here means an entity with broad latitude in its complexity, theoret-

ically linear, shift time invariant, and able to supply a speciﬁc response to

stimuli.

InNM, the inputfunction isbasically the radiopharmaceuticaladministered

and the detected response, that is, the output function, the result of both the

transport and the biological processing of the tracer. This methodology has

beenextensively usedto studythe biokinetics,biodistribution, and occupancy

of thousands of radioactive molecules.

NM has progressed as a result of advances at each of the stages of this

process, that is, advances in the input functions due to the introduction of

new radiotracers, advances in the technology and reliability of the detectors

to improve the output, and, ﬁnally, the increase in the capacity to extract and

process the resulting data.

Recent PET imaging science has synthesized new molecules to be used in

genetically engineered or transplanted tissues, in animal models. They have

been developed to provide molecular information in human diseases, that is,

to assess the molecular and genetic basis of normal cellular function and the

transformations associated with diseased cells.

New and powerful statistical algorithms have been developed to extract

spatiotemporal information from PET data sets in medical specialities such

as oncology, neurology, and cardiology.

The development of novel radioactive tracers, multimodality, 3D acquisi-

tion, the creation of new data and image processing tools, etc., are all part of

this promising effort to characterize the molecular basis of health and disease

through imaging.

The extraordinarily high sensitivity of PET techniques (∼ 10

−9

−10

−12

and a speciﬁcity that reaches molecular level make them unique in the

in vivo studies of internal organs, particularly with speciﬁc molecules present

in the body at very low concentrations (<10

−8

M). For example, extra-

striatal dopamine D1 receptors present in the brain at concentrations of

approximately 10

−9

M can be measured with PET using

C-NNC756 [21].

7.2.2 Concepts in Tracer Kinetics Modeling

The study of the movements of molecules within organisms, as well as the

immediate consequences of such movements, is often called biokinetics.

The methods used to study molecular kinetics, particularly in pharmaco-

logical applications, have beneﬁted from developments in a number of areas,

particularly in nuclear methods and data analysis.

The purpose of NM imaging techniques (PET, SPECT, and planar scintig-

raphy) is to map physiological processes in vivo. With these techniques,

information on local and temporal variations in functional activity of a given

labeled molecule are detected as image contrast and analyzed either quali-

tatively or using quantitative approaches of varying degrees of complexity,

based on the intrinsic or statistical properties of the system under observation.

Systems in Nuclear Medicine 389

Functional parameters can be estimated from dynamically acquired NM

data by a number of different techniques such as reconstructing a tempo-

ral sequence of images, generating regions of interest (ROIs) overlaid on

the image sequence, drawing time–activity curves (TACs), and subsequently

ﬁtting the parameters using appropriate relationships.

Theoretically, PET allows the dynamic study of any organic molecule

labeled with a positron emitter, either an organic element (

N) gen-

erating molecules chemically identical to the native ones (isotopic labeling) or

anelement not presentinnative molecules (e.g.,

Fto substitute H),leading to

tracermolecules that aredifferent fromnative ones(nonisotopic labeling); and

these will, therefore, be discriminated at a later stage of the metabolic chain.

In SPECT, almost all of the molecules used as tracers are different from

the native molecules, because they have to be labeled with nonbiological γ

emitter elements (

99m

Tc,

201

Tl,

123

I, etc). The radioisotopes of low Z elements

such as the biological C, O, H, and N do not have any usable artiﬁcial γ ray

emitter radioisotopes, because they are pure β emitters (β

or β

−

Apart from the advantage of isotopic labeling, PET detection efﬁciency and

spatial resolution are generally higher than in SPECT. Further, accurate atten-

uation correction is easy in PET, and it is difﬁcult or almost impossible in

SPECT.

Temporal constraints in SPECT measurements are also a common source of

difﬁculties in studying dynamic processes.

An additional problem in SPECT is related to the gantry movement during

data acquisition. Since the projections at different angles occur at different

times during acquisition, they correspond to different tracer distributions, an

effect that is maximized in fast dynamic studies. Images reconstructed from

these incompatible projections can contain artifacts that lead to errors in the

estimation of kinetic parameters.

Most of this chapter is dedicated to PET techniques, because they are the

most representative in tracer applications.

Models are currently used in data analysis and in research projects on the

biokinetics of radiopharmaceuticals.

Amodelusesthe magnitudes and physical dimensionsof thebiological phe-

nomena being studied and may make it possible to mathematically predict

the behavior of an endogenous molecule or drug and to evaluate its concen-

tration and its active metabolites at different stages and locations, including

nonaccessible pools of the biological system.

A model is a tool for studying and understanding phenomena and, simul-

taneously, it is a shift from the experimental ﬁeld to a phase of theoretical

analysis and interpretation. Models are used to gain a better knowledge of

biological systems; test new ideas, hypotheses, or other logical processes;

evaluate future consequences of the observed phenomena; secure control of

processes in speciﬁc situations, etc.

The aim of modeling in the particular case of NM is to apply analysis tools

that are appropriate for SPECT, PET data, or biological sample counting in

390 Nuclear Medicine Physics

order to evaluate the physiological kinetics of tracers in the body and to study

the function of its organs and tissues.

Modeling allows the prediction of the metabolic steps of an endogenous

molecule or drug and the evaluation of its concentration and its active

metabolites in nonaccessible compartments of the biological system.

First principles of chemistry and physics can be used to quantitatively

model and simulate static and dynamic physiological processes that are usu-

ally described only qualitatively. A model may help us to both understand

the phenomena qualitatively and analyze the process quantitatively, using

mathematical methods.

Modeling techniques may try to represent a phenomenon exhaustively, that

is, by being its replica, or just translate its functional performance in a sim-

pliﬁed or partial way by showing the molecular paths by which to derive

parameters of interest, such as binding potentials.

Most systems in which we use models to try to understand the involvement

of tracer molecules are in a steady or quasi-equilibrium state. This state exists

due to homeostatic control that keeps the concentration of products constant,

but it does not correspond to true chemical equilibrium. Examples of this

quasi-equilibrium are the tight control of body electrolyte concentrations,

blood glucose level, even blood cell counts, and many other populations of

molecules or cells. Metabolism and other dynamic processes have to occur

for a steady state to be maintained.

Some models are essentially deterministic. These models assume that

the system is governed by known physical–chemical laws, and the val-

ues of the parameters to be introduced are obtained from the measured

samples.

Other systems are stochastic in nature. The statistical laws of the system as

well as the functional parameters have to be assessed from collected data.

Compartmental analysis, functional mathematical models, and systems

theory have been widely used in studies with radionuclides since the

1960s [22].

In modeling, a compartment may be an anatomical, physiological, chemi-

cal, or physical subdivision of the system under investigation. A steady state

is generally assumed for the system, and it is also held that the tracer is rapidly

mixed in compartments so that the speciﬁc activity is uniform throughout the

compartment at any time. The deterministic approach may be an approxima-

tion in one or more respects. In fact, the steady state condition is frequently

not achieved; sometimes, instant dilution cannot be considered to prevail;

the reaction in some processes is not determined just by a constant but by a

statistical distribution of constants; the order of the reactions is not known;

and the pathological model may not coincide with the normal one. Com-

partmental models are the most common tracer kinetic models used in both

SPECT and PET studies. These models are formulated by using differential

equations that describe exchanges between the system compartments, and

Systems in Nuclear Medicine 391

they can provide accurate measurements of rates of exchange and binding

potentials, which are proportional to the concentration of available receptor

sites and other quantities.

In most cases, the complexity of physiological systems makes it difﬁcult

to directly associate a predeﬁned biological kinetic model with a tracer. An

analysis of the dynamic processes that are plausibly observable within the

constraints of heterogeneity of the object is necessary. This analysis has to

take into account the detector resolution, the detector overall spatiotemporal

response, and the statistical quality of the data. For a well-deﬁned kinetic

model to be accepted as a simplifying tool from which quantitative biological

information on exchange and targeting can be accessed, we need a sound

knowledge of the biological fate of the tracer. This is related to the exchange

mechanisms that take place in the tissues and the contribution of the different

exchange paths.

As an example, brain studies with PET and SPECT require radiotracers with

high afﬁnity for neurotransmitter receptors and transporters. However, the

peripheral metabolism of radiotracers often leads to lower radioactivity in

the brain. Some metabolites nonspeciﬁcally bind to receptors, whereas others

even bind preferentially to other target receptors.

Thea priori analysis ofkinetic data toderive theidentiﬁable kinetic processes

before implementing a model may involve powerful preprocessing analytical

methods, some of which are still at a developmental stage. Examples of such

methods are cluster analysis, factor analysis, spectral analysis, and fractal

analysis [23–25].

Mainly, as a result of the possibilities offered by PET in brain studies and

the large number of radiotracers and radioligands that can be used to probe

molecular binding sites, kinetic modeling is nowadays a vast and highly

specialized ﬁeld. Estimation of quantitative biological images from complex

4D spatiotemporal data sets requires the application of appropriate image

processing and tracer kinetic modeling techniques.

Extending the analysis of ROIs and time or activity curves to voxel level,

parametric images can be generated where the voxel index represents a

parameter (e.g., glucose metabolism or blood perfusion) instead of tracer

concentration as in the original PET images. Mathematical modeling plays

an important role in the conversion of the original images into local reaction

rates, that is, parametric images.

Interest has recently been increasing in the formation of parametric images

that individually model the kinetic behavior of each volume element (voxel)

in the image domain under analysis. This approach is more appropriate when

the volume cannot be effectively segmented into homogeneous regions that

would be modeled with a single kinetic parameter set.

The merging of information from different modalities or from different trac-

ers can also be very useful in identifying structure or function relationships

or assessing the normal and disease states of a particular organ.

392 Nuclear Medicine Physics

7.2.3 Transport of Tracers and Localization Mechanisms

Tracer studies are based on the principle that the mass (or activity) of tracer

delivered to an organ or tissue is proportional to perfusion (blood ﬂow/mass)

to the organ or tissue, that is,

Mass of tracer in A

Blood ﬂow in A

Mass of tracer in B

Blood ﬂow in B

= C

. (7.141)

If tissue B is taken as a reference with known blood ﬂow for a given activity

of tracer in B, then blood ﬂow to tissue A can be determined by measuring

the activity of tracer in A.

The use of

N-ammonia for imaging the relative myocardial blood ﬂow is

based on the above principle.

The uptake of a tracer by a tissue perfused by blood also depends on tissue

extraction and clearance ﬂow. Extraction can be deﬁned for steady state or for

ﬁrst-pass conditions.

The former (net extraction, E

) is a relative difference between the arterial

and venous tracer concentrations (C

and C

) of an organ or tissue in steady

state, that is,

= (C

−C

)/C

(7.142)

In the latter (unidirectional extraction, E

), only the amount of tracer

extracted from blood to tissue is taken into account. The amount of tracer

deliveredback fromtissue to blood is not included. Equation 7.141 still applies

but now the concentration C

refers to ﬁrst pass.

If there is no utilization of the tracer, that is, if the tracer transferred to the

tissue is returned to blood, the E

is zero. This situation applies to inert gases.

is generally larger than the net extraction fraction. An exception to this

general rule occurs with O

. Virtually, all oxygen extracted by tissue is metab-

olized; thus, the net and unidirectional extraction fractions are the same. For

all other substances, a major portion of what is extracted by the tissue is

transported back to the blood.

After intravenous injection, radiopharmaceuticals are transported to the

capillaries by blood ﬂow (perfusion) and move out of the vascular compart-

ment, mostly across the capillary walls by several passive or active transport

mechanisms.

From the interstitial space, the radiopharmaceuticals can move across the

cellular membrane; and, in the cell, they participate in biochemical reactions

or bind to receptors.

In passive transport, the movement of the tracer is determined by diffusion

forces, following the direction of the concentration gradient across the mem-

brane, either in the membrane phase or in tiny passages between the capillary

endothelial cells.

For a constant tracer concentration gradient, the net ﬂux in passive diffusion

depends only on the diffusion coefﬁcient and on the concentration gradient.

Systems in Nuclear Medicine 393

Another type of passive transport is facilitated transport, which is per-

formed by special carriers that exist in the membrane and permits a selective

and controlled uptake of substrates, especially amino acids and glucose.

Facilitated transport is generally considered a three-stage process: the sub-

strate bonding with carrier molecules on one side of the membrane; the

assisted movement of the substrate–carrier complex; and the release of the

substrate on the other side. For low concentrations of labeled tracers, this pro-

cess can be regarded as linear and similar to catalyzed enzyme kinetics with

competitive inhibition.

In active transport, tracers are carried against concentration gradients,

thanks to energy supplied by the adenosine triphosphate (ATP) present in

the membrane. The known pump mechanism that regulates the Na

and

concentrations on either side of the cellular membrane and the tubu-

lar reabsorption of glucose in the kidneys are examples of active transport

mechanisms.

The literature on the transport and uptake of elements and compounds after

their administration into the body is vast [25].

It is known that the localization of elements in tissues after injection is

closely related to their speciﬁc electronic conﬁguration.

On the left-hand side of the periodic table, vertical relations predominate,

and the tissue distribution of elements of a particular periodic family is simi-

lar. However, on the right-hand side of the periodic table, horizontal relations

tend to predominate, especially for the transition metals. The tissue distribu-

tion of an element is similar to neighboring elements with almost the same

atomic weight.

As a rule, anions, including halogens, oxygenated or halogenated ions

of groups IV, V, and VI and the transition metals, are rapidly eliminated

by the kidneys. In contrast, the retention time of metallic cations with the

inner orbitals completely occupied by electrons tends to be prolonged in the

skeleton and soft tissues.

The main properties of an element that determine tissue localization and

excretion are oxidation or valence state, for the pH of the blood (pH=7.4);

relative solubility in aqueous media or, if insoluble, the size of the colloidal

particle; and tendency to be incorporated into organic compounds or speciﬁc

proteins.

The biological fate of compounds is determined by many factors, and it

is much more complex than for elements. After intravenous injection, most

radiopharmaceuticals leave the vascular compartment, mainly through cap-

illaries. An average man has about 4 ×10

capillaries with an exchange area

of around 1000 m

and a volume that contains, at rest, less than 5% of the total

blood circulation.

The uptake of a drug by an organ or tissue depends on the fraction of cardiac

output received by the organ and on there being receptor sites with a speciﬁc

biochemical afﬁnity for the extraction and concentration of the labeled com-

pound. In addition, factors related to capillaries such as their number per

394 Nuclear Medicine Physics

cubic millimeter of tissue and the type of capillary endothelium within the

organ are also important. There are three main groups of capillaries in the

body: continuous, fenestrated, and discontinuous. Capillaries with continu-

ous, nonfenestrated endothelium occur in the muscles, fat, connective tissue,

and lungs.

The endothelium of fenestrated capillaries may be closed as in the brain,

gastrointestinal tract, and endocrine glands or open as in the heart or renal

glomerula where the effective pore radius is 45–50 A

◦

. Capillaries with dis-

continuous endothelium allow large molecules to leak into the perivascular

space; they are found in the sinusoids of the reticuloendothelial organs—liver,

spleen, and bone marrow.

Other factors that are important in the localization of drugs within the

body are molecular size and shape in plasma, degree and strength of protein

binding in blood and tissues, liposolubility, and speciﬁc cellular mechanisms.

Afteradministrationof aradioactive tracer, slowpassivediffusionprocesses

due to concentration gradients through permeability barriers may prevail.

When this occurs, the tracer can be used to measure parameters within its

area of localization, such as volume, or those related to the barrier, such as

permeability and dynamics.

After administration, a radiopharmaceutical passes through a series of

membranes before reaching the target tissue, and it is often important to

know the permeability of these membranes for the different tracers.

In general terms, the potential energy gradients responsible for the passive

transport of noncharged radioactive molecules in biological media result from

concentration and pressure. The latter is usually the most important factor in

dilution processes.

The distribution and localization of radiopharmaceuticals is not a simple

process, as a rule; it tends to be a complex chain of processes with multiple

interactions and probably involving several of the phenomena described next.

To reach a particular cell, after administration, a radiopharmaceutical must

penetrate a series of membranes. These are predominantly lipidic and may

include the capillary endothelial membrane, the gastrointestinal membrane,

and even membranes surrounding intracellular organelles.

The more common processes involved in the movement and distribution

of tracer molecules after the injection are brieﬂy considered next.

•

Ion exchange, which consists of removing ions from the native site in

the organ of interest by exchange with identical radiotracer ions that

are concentrated in the organ. Fluoride, phosphate, and thallium ions

are initially extracted from circulating plasma, mainly by a process

of ion exchange.

• Metabolic incorporation, in which the radioactive tracer follows

the same biochemical pathways as those of the natural, nonra-

dioactive substrate and is, thus, actively incorporated during the

Systems in Nuclear Medicine 395

metabolic processes. Labeled glucose and radioactive iodide are typi-

cal examples; the ﬁrst is used to detect local glycolysis and the second

is used to study the production and use of thyroid hormone.

• Binding to receptors, in which the radiotracer (ligand) interacts with

receptors that are macromolecules (glycoproteins), to produce char-

acteristic complexes. Ligands are often regarded as consisting of two

parts, one that contains the essential binding site and a second part

that does not intervene in the reaction and can be modiﬁed to a cer-

tain extent. When a radionuclide is introduced into a ligand molecule

to produce a radiopharmaceutical, incorporation should take place

in the modiﬁable part of the molecule. After a radioligand leaves

the vascular pool, it reversibly binds to nonspeciﬁc binding sites

in the interstitial and cellular spaces and subsequently diffuses to

the speciﬁc sites. Many receptors have been investigated, including

polypeptide hormones, acetylcholine, etc.

• Enzyme-catalyzed reactions. Most chemical reactions in living tis-

sues are catalyzed by enzymes and can be described by Michaelis–

Menten kinetics. For the tracer, the equation is linear with regardto its

concentration; linearity holds even when the reactions do not strictly

follow Michaelis–Menten kinetics, as long as the concentration of

tracer is low compared with the natural substance.

• Labeled monoclonal antibodies, derived from cultured hybridoma

cells, can recognize and interact with single antigens and can poten-

tially identify and target speciﬁc groups of cells such as blood and

tumor cells.

A major problem in applying the technique to imaging is tissue speciﬁcity,

which may differ only slightly between unaffected tissues and tumor cells

and so give rise to poorly speciﬁc or nonspeciﬁc binding ratios.

• When gaseous radiopharmaceuticals are administered by inhalation,

they pass into the pulmonary venous circulation from the lungs and

can freely exchange between the lungs and tissues. In conventional

NM studies, the gases used are principally isotopes of xenon (

133

Xe,

127

Xe) and

81m

Kr, which are chemically inert, and in PET

N and

are gases with great potential.

Other processes such as phagocytosis, pinocytosis, and capillary trapping

may be involved when radioactive particles or cells are used.

7.2.4 Compartmental Models

It is simple and often useful to assume that an organism consists of a number

of linked and interacting units called compartments, which are studied as

kinetic compartmental models.

396 Nuclear Medicine Physics

Acontinuous exchange of matter and energy occurs between such compart-

ments and between some of these and the environment. These exchanges are

the physical–chemical processes relatedto absorption, distribution, synthesis,

degradation, and excretion. The interrelationships between these compart-

ments are so close that generally the perturbation of one particular exchange

can induce modiﬁcations in several compartments of the system. Illness is an

alteration of one or moreof these exchanges or of their regulatorymechanisms

[26,27].

Compartmental models originated in the ﬁeld of pharmacokinetics, and

they are a widely used mathematical tool for analyzing data from PET and

SPECT. Acompartment is not necessarily a physical volume or space but, in a

generic sense, it is an amount of a chemical compound that can be kinetically

studied in the same way as a well-deﬁned homogeneous mass of the same

product. In living organisms, there are many compartments, either physically

well separated or theoretically deﬁned.

Radiopharmaceuticals may achieve their initial bio-distribution by simple

dilution, after a fast injection, in an accessible compartment taken as being

strategicfor the experiment. The length of time a tracer stays in a compartment

depends on the speciﬁc biochemical and biophysical processes in which it

will be involved. The evolution over time of the amount of tracer present in a

compartment after a fast injection is often the time–activity function obtained

by external detection (residue detection).

A compartment model that perfectly ﬁts the NM experimental data is the

gold standard for kinetic studies [28,29].

In compartmental models, when the metabolite-corrected arterial blood

curve and dynamic data are available from the time of injection to the time

when all important changes in tracer kinetics have occurred and the data

have good statistics, it may be possible to estimate perfusion, blood volume in

tissue vasculature, metabolism rate constants, and speciﬁc binding potentials.

In practice, this situation mostly applies to simple compartmental models,

because the mathematical complexity drastically increases as the number of

compartments grows. Moreover, in most cases, there is no need to measure

all the parameters, but only those characteristic parameters that determine

the property under study.

Common problemsin compartmental analysis arethe development of plau-

sible models for particular biological systems and of analytical theories for

each class of systems.

The metabolite-corrected arterial plasma curve is the input function to the

compartment model. Although plasma is not a compartment of the model

itself, it is usually counted as such.

A common important category of compartment models is those in which

the rates of all transfers between compartments are ﬁrst-order rate constants.

This is a deterministic constraint based on experience and it has a wide

application.

Systems in Nuclear Medicine 397

7.2.4.1 Single-Compartment Models and Multicompartment Models

7.2.4.1.1 Single-Compartment Models

Consider a simple model for the injection of a radiopharmaceutical into the

bloodstream and its disappearance. The blood pool is taken to be a single

compartment of volume V; the blood concentration of tracer is c(t) =[q(t)/V],

(in activity per volume unity); and i(t) is the tracer input rate into blood (input

activity per time unit), for t ≥ 0.

The disappearance of the radiopharmaceutical is assumed to occur accord-

ing to a ﬁrst-order reaction with effective constant λ

Between times t and t +dt, effective elimination will reduce the tracer activ-

ity in the compartment from q(t) to q(t)(1 −λ

dt), and input will increase it

by the amount i(t)dt and so, the activity at time t +dt is

q(t +dt) = q(t) − q(t)λ

dt +i(t) dt (7.143)

Taking the function q(t) as continuous,

q(t +dt) = q(t) +q



(t) dt +..... (7.144)

and, neglecting terms of second order and above,



(t) =−λ

+i(t) or (7.145)

q(t) = q

−λ



−λ

(t −s)i(s)ds (7.146)

and, dividing by V,

c(t) = c

−λ



−λ

(t−s)

i(s)ds/V (7.147)

If there is no input for t > 0, that is, i(t) = 0 and, at t = 0, c(0) = c

= q

/V,

then,

q(t) = q

−λ

and c(t) = c

−λ

. (7.148)

An example of this situation (Figure 7.28) is the instantaneous injection of

activity q

= c

V,att = 0, of a diffusible tracer, with instantaneous mixing in

the compartment.

If the concentration in the compartment for t = 0 is zero, that is, c

= 0, but

i(t) is not null, then

c(t) =



−λ

(t−s)

i(s) ds/V (7.149)