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

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418 Nuclear Medicine Physics

F =

∞



(t)dt

(7.205)

Thus, one can calculate the blood ﬂow from the total quantity injected and

from the area under the tracer time concentration curve, no matter over what

period of time the tracer was injected. However, we cannot calculate the vol-

ume of the system without knowing more about the input function. Unless

the injection is truly instantaneous, the mean transit time of the concentration

function at the output is not the same as the frequency function of transit

times through the system.

The true mean transit time through the system is the difference between the

mean transit time of the concentration function at output and the mean time

of the tracer input

t =

−

(7.206)

7.2.7 Regeneration of the Frequency Function of Transit Times by

Deconvolution

We are now going to consider some applications of convolution and decon-

volution methods.

These are important operations concerning the applicability of the input–

output problems of systems to any biological system (not only to dilution sys-

tems), provided that certain conditions are satisﬁed by the system (linearity

as well as time invariance, that is, constant behavior over time).

Figure 7.45 is a schematic representation of the functions involved in a

process in which a system responds to stimuli.

Function h(t) is the response of the system to a unit impulse, that is, a delta

function of unit amplitude at zero time, or in the case of NM, for example, a

very fast injection of unit activity at the beginning of the experiment.

The convolution of functions h(t) and I(t) is represented by the integral

R(t) =



I(t −τ)h(τ)dτ = h(t)∗I(t)

where h(t) is an operator that converts an object or input function I(t) into

an output function R(t). In the above equation, t is the sole variable, and τ is

used as an auxiliary variable.

There can be any variables and they may be applied in convolution in 2D

or multidimensional spaces.

Through convolution, a response function can be calculated if the input

function and the unit response function are known (Figure 7.46).

Systems in Nuclear Medicine 419

δ(t)

I(t) R(t)

h(t)

FIGURE 7.45

h(t) is the response of the system to a unit impulse, transit time distribution (or retention function

or residence times function). I(t) is the input function and R(t) is the response function (or a

residue function).

In most cases, however,the important problemis the inverse of convolution,

thatis, I(t) iswanted when R(t) andh(t) areknown, or h(t) is wanted whenR(t)

and I(t) are known. In such cases, the unknown integrand function is said to

be deconvoluted from the known functions, and the necessary mathematical

procedures are called deconvolution. Deconvolution is, then, an operation

deﬁned as the inverse of convolution.

The shape of the activity dilution curve of an organ depends on the organ,

the blood ﬂow, and also the shape of the tracer bolus applied. In principle, the

deconvolution of these curves will remove the effects of the bolus shape, but

in many practical situations the results have been shown to be too unstable

for reliable use. Continuous deconvolution after ﬁtting with known curves

often leads to integral equations with general solutions only in constrained

(t) h(t)

R(t)

R(t) = h(t–τ) I(τ) dτ

R(t) = h(t)

I(t) = I(t)

h(t)

FIGURE 7.46

Input I(t) and response R(t) functions of a system portrayed by a unit response function h(t).

420 Nuclear Medicine Physics

conditions or somewhat special cases. In metabolic processes, the organ is

often regarded as the system, the input as the time-activity function, and the

response(or retentionfunction) asthe time-activityfunctionin theorganitself.

Using deconvolution, the residue function of the organ can be recovered.

Biomedical uses of 1D and 2D deconvolution in dynamic and metabolic

studies have been reported since the 1950s, particularly in cardiovascu-

lar, renographic, brain, kidney, and gastroenterology applications; in image

processing; and in reconstruction and restoration algorithms.

Discrete deconvolution is carried out through polynomial division, matrix

division, fast Fourier transform algorithms, and numerical function mini-

mization methods. When noise affectsthe data, the deconvolution procedures

are strongly perturbed. Deconvolution is an ill-conditioned procedure; this

means that small errors in data obtained for the two measured functions can

give rise to large deviations in the solution. The perturbation due to noise in

these techniques can be studied by adding known noise content to the func-

tions under deconvolution and using these data to develop efﬁcient methods

of noise ﬁltering.

Assuming an output concentration C

(t) in response to an arterial concen-

tration at input C

(t) and that these functions are known, then

(t) =



(t −τ)h(τ)dτ (7.207)

Now we want to calculate h(t).

The Laplace transform of Equation 7.207 is

(s) = C

(s)H(s) (7.208)

and the inverse transform of H(s) is

−1

{H(s)}=L

−1

(s)

= h(t). (7.209)

In general, C

(s) and C

(s) are only empirically determined and have no

recognizable, simple, analytical form for formal inverse transformation.

There are two ways of overcoming this difﬁculty. In the ﬁrst one, the convo-

lution integral is considered a simple summation and h(t) is calculated either

numerically or on a digital computer. In the second, the curves C

(t) and C

(t)

are empirically ﬁtted to arbitrary formal expressions that can be more easily

deconvoluted.

The convolution integral in summation form becomes (Figure 7.47)

(t) =



k=1

h(t +1 −k)C

(k) (7.210)

Systems in Nuclear Medicine 421

(k)

(a)

(b)

(t)

k–1

h[t–(k–1)]

(k)

(t)

(t) =

∑

k = 1

h(t+1–k)C

(k)

Δk = 1

FIGURE 7.47

(a) Input function is decomposed in a sequence of time shifted delta functions represented by

small rectangles whose area is equal to the mean value of the ordinate. To each of these functions

corresponds an output numbered as 1, 2, and 3. (b) The summation of all input delta functions

is C

(t) and of all the output curves is C

(t).

where k takes only integer values for integer values of t.

Theconvolution processcan bevisualized if theinput functionI(t) isdecom-

posed in an inﬁnite number of time shifted delta functions with amplitude

equal to the value of the function in each time and considering as ﬁnal result

the summation of the responses to all these delta functions.

Most of the situations in practical NM need a discrete approximation of con-

volution. When the functions involved in convolution are sampled functions,

the convolution integral becomes

R(k) =



n=0

h(k −n)I(n) (7.211)

where R(k), h(k − n), and I(n) are the sampled values of the functions.

Equation 7.211 can be written in matrix form as

R = H × I (7.212)

where I is a triangular matrix with elements (k + 1) (k +1) with n +1 line

consisting of terms I(n), I(n −1), I(n −2),…, I(0) followed by (k − n) zeros.

H and R are column matrixes with k + 1 elements.

In Figure 7.47a, the input function is decomposed into a sequence of delta

functions represented by small rectangles whose area is equal to the mean

422 Nuclear Medicine Physics

value of the ordinate. To each of these functions corresponds an output func-

tion numbered as 1, 2, and 3. The summation of all input delta functions is

(t) and the summation of all the output curves is C

(t).

For example,

(t) = h(1)C

(1) (7.213)

(2) = h(2)C

(1) +h(1)C

(2) (7.214)

(t) = h(t)C

(1) +h(t −1)C

(2) +...+ h(1)C

(t) (7.215)

This gives a system of t simultaneous linear equations, the solution of which

yields t values for h.

The ﬁtting procedures have included several functions (sum of exponen-

tials, Fourier series, etc.) to ﬁt the frequency functions to formal expressions.

7.2.8 Data Analysis and Models in PET Studies

Immediately after reconstruction, SPECT and PET images are static maps of

values proportional to activity in units of counts per second per pixel.

These values are generally converted to units of counts per second per

milliliter of tissue by calibration with a known volume of tracer activity.

NM images can be used as static, dynamic, or parametric entities [30–32].

When the biological processes under analysis evolve slowly over time, static

images may be good enough to supply the wanted information (e.g., tumor

detection). A color look-up table is often used to assign a color to each pixel

value in static images.

When functional information is wanted, dynamic imaging acquisition

is required. For example, it may be necessary to estimate the parameters

of a compartmental model that supposedly ﬁts data from PET or SPECT.

These kinetic parameters are important, because they quantify physiological

processes.

In dynamic imaging, series of scans are acquired by continuous scanning,

often for long times; and after the reconstruction of sequences of PET images,

some kind of spatiotemporal quantiﬁcation is associated with the activity

data.

One common approach, for a sequence of PET or SPECT images, is to get

the variation over time of the mean value of pixels in ROIs that have been

previously rescaled and aligned. To help with the deﬁnition of anatomical

ROIs in the brain, the higher-resolution MRI or CT is commonly used. The

TACs are generated with data points that correspond to the mean pixel value

in a speciﬁc ROI at a given time.

Systems in Nuclear Medicine 423

These TACs can be further normalized by plotting, from point-to-point, the

ratio of the values of one ROI TAC to the values of another ROI TAC that is

taken as a reference.

For quantiﬁcation some reference is needed, and this can be the tracer con-

centration in arterial plasma (blood sampling required) or some other tissue

(a different area of the same image).

In PET studies, the tracers are introduced into the body by intravenous

injection or inhalation. The tracers are mixed with blood in the heart chambers

to give a nearly constant concentration in arterial blood (model input). The

concentration of the tracer delivered to the tissue capillaries can be obtained

from any peripheral artery, and the amount of tracer delivered to the tissue

is proportional to the blood ﬂow (perfusion).

Some of the possibilities for ﬁnding the input function to an organ are by

external measurement in the left ventricle, aorta, or large artery as a function

of time (i.e., the respective TAC). This method, which requires fast response

equipment to cope with the temporal sampling requirements of the particular

input function, is also limited to those cases in which the left ventricle or a

great vessel is located within the limits of the image [33,34].

Statistical methods that will be discussed later can also be used to effectively

obtain these input functions.

Depending on their biochemical properties, the radiotracers used in PET

studies are grouped in two broad categories.

The ﬁrst category comprises the nonspeciﬁc radiotracers that trace a bio-

chemical path and lead to the measurement of the extraction parameters

of a tissue which describe its uptake and metabolism. Examples of these

radiotracers are

, which is an inert, freely diffusible tracer used to mea-

sure cerebral blood ﬂow;

FDG, which traces the initial phases of glucose

metabolism but does not follow Kreb’s cycle after phosphorylation and is,

therefore, effectively retained in cells, permitting the evaluation of glucose

metabolism in tissue; and

F-ﬂuoromisonidazole, which is a bioreductor

drug that follows an intracellular path of reduction and can be used to localize

viablehypoxic tissue.The kineticsof theradiopharmaceuticals ofthis kindcan

be evaluated through simple systems consisting of one or two compartments

apart from plasma.

The second category includes speciﬁc radiotracers involved in interactions

with a receptor, a carrier, or a speciﬁc interaction site. Examples of these radio-

tracers are

C-ﬂumazenil, which is an antagonist with high afﬁnity and selec-

tivity for central benzodiazepine receptors, and

C-SCH23390, an antagonist

withhigh afﬁnityand selectivityfor dopamine D1receptors. Both radiotracers

are used to study alterations in the density and afﬁnity of central recep-

tors. The behavior of the radiopharmaceuticals in this category is generally

assessed by means of three-compartment models: free, nonmetabolized lig-

and in plasma; free ligand in tissues; and ligand speciﬁcally bound to tissues.

In tracer studies, the data acquisition methodology and the recommended

data analysis method depend on the aim and type of study being performed.

424 Nuclear Medicine Physics

Different approaches have been used for the kinetic modeling of images

from SPECT and PET tracer studies, with the purpose of estimating quanti-

tative biological parameters.

Despite their many differences, these methods—mostly only used for brain

PET studies—can be roughly placed into general categories; for example, in

those based on data (data-driven), where no models or compartments are

assumed, and those that use models (model-driven) which tend to assume a

compartmental system.

The data-driven methods require no previous decision about a model

associated with the system (Figure 7.48).

After reconstruction, the images are used to draw ROIs and obtain TACs

that will be subjected to one of several graphical procedure sequences. These

correspond to the path 3-4-5-6 in Figure 7.48.

Tracer ROI curves can also be used for modeling. This corresponds to the

path 3-4a in Figure 7.48 and 4a-5a-6a in Figure 7.49.

Data driven

Raw data

MTGA

compart. models

spect. Analysis

SUV

Model

calculations

(blood data)

Reconst.

12 3

PET image

Drawing ROIs

Regional

TACs

To model driven

Results

FIGURE 7.48

Data-driven methods for data analysis in NM.

Model

calculations

(blood data)

1a 2a 3a 4a

4a 5a

Paramatric

sinogram

Paramatric

images

Reconstruction

Drawing ROIs

SPM

Model driven

From data driven

Sinogram

Results

FIGURE 7.49

Model-driven methods of analyzing NM data.

Systems in Nuclear Medicine 425

The ROI TACs are ﬁtted to mathematical models to estimate parameters

that are biochemically or physiologically meaningful (path 4-5). These, when

the input function is known, can be rate constants, receptor densities, or blood

ﬂows. The input function is usually the arterial plasma activity that is sampled

during the PET study and counted in the well counter. This approach allows

comparison of the model parameters in PET studies.

Multiple time graphical analysis methods and spectral analysis are gener-

ally used to quantify the dynamics of the processes.

The MTGA is a series of data-driven graphical procedures used to quan-

tify PET data that involve the study of the distribution and accumulation

of radioactive tracers in organs and tissues over time. The technique is an

alternative to compartmental models, as it does not assume an imposed

chain of predetermined events and it is suitable for both homogeneous and

nonhomogeneous tissues.

The data are usually acquired as a series of dynamic frames, beginning with

the tracer injection and continuing until either the statistics of the images is

compromised by prolonged decay or there is a limitation in memory storage

capacity.

The MTGA methods (Patlak and Logan plots) employ a transformation of

the data such that a linear regression of the transformed data yields macrosys-

tem parameters which are computed with better reproducibility from fewer

data than the separate rate constants [35,36]. By plotting these macroparam-

eters, the functional parameters are directly estimated from the slope of the

linear phase of the curves obtained.

Some negative points about these methods are the uncertainty about the

choice of the point in the plot when linearity begins, the possibility of bias

introduced by statistical noise [29], and the failure to return any information

about the underlying compartmental structure.

Spectral analysis [35] characterizes the system’s impulse response function

(IRF) as a positive sum of exponentials and uses nonnegative least squares to

ﬁt a set of these basic functions to the data. The macro system parameters of

interest are then calculated as functions of the IRF [34,35]. Spectral analysis

also returns information on the number of tissue compartments evident in

the data and is deﬁned as a transparent technique.

Schmidt [30] showed that for the majority of plasma input models the obser-

vation of all compartments led to only positive coefﬁcients; and, as such, the

spectral analysis [35] solution using nonnegative least squares is valid.

The ROI-based methods may be further classiﬁed into linear and nonlinear

methods. The former transform the data so that the parameters of interest

can be estimated by linear regression methods, whereas nonlinear techniques

generally estimate the kinetic parameters by iterative minimization.

The two different types of analysis (Patlak and Logan) are used according

to the type of situation.

The Patlak-plot applies to irreversibly binding radiotracers and involves

creating an ordinate and an abscissa using combinations of the input function

426 Nuclear Medicine Physics

and image data at each of the observed time points for every PET frame. A

TAC is obtained for a particular ROI.

For each time point, the ROI values represent the overall concentration of

radioactivity in the tissue, which can differ from the tracer concentration.

A TAC is created for the input function Cp(t), the concentration of tracer

in the plasma, by interpolating the measured plasma curve to the middle of

each frame.

For every image frame, the ratio of the ROI value in that frame divided by

the plasma concentration is plotted as a function of the integral of the plasma

curve (from t = 0 to the current time) divided by the plasma concentration at

that time point, that is,

ROI(t)/C

(t) versus



(t)dt/C

(t) (7.216)

where ROI(t) is the concentration of a tracer in a tissue ROI, t is the time

elapsed from injection to a particular frame, and Cp(t) is the concentration of

tracer in plasma. In these instances, the net inﬂux rate K

(min

−1

) equals the

slope of the Patlak plot, that is, k

×k

/(k

This method can be thought of as a way of scaling the tracer concentration

in a region of tissue by the capacity of the tissue to absorb the tracer over time.

However, it was noticed that this plot becomes linear at those time instants

when the transport of tracer trapped into the compartment is essentially

unidirectional.

Patlak showed that the slope of a straight line ﬁtted to this linear region

is equivalent to the inﬂux constant, K

, for the system. K

can be deﬁned as

the ratio of the total amount of tracer accumulated in a tissue region after an

inﬁnite amount of time, divided by the integral of the plasma TAC from t = 0

to inﬁnity.

Graphical methods are easy to perform and are generally considered more

robust than kinetic analysis with (full) compartmental models, especially for

noisy data sets.

The traditional linear regression model takes into account only the errors

in the Y variable.

The bias in the (reference input) Gjedde–Patlak slope may be diminished

by using a linear regression model that accounts for errors in both variables

[32]. This method can also take into account separate weights for both

variables.

In another work (1985), Patlak has shown that the technique still applies

when the tracer is simply “nearly” irreversibly bound and that a tissue ref-

erence region, with no speciﬁc tracer uptake, can be used as input function

instead of the measured blood TAC.

Systems in Nuclear Medicine 427

This analysis is used in neurotransmitters ([

F]-F-DOPA, [

F]-FMT, [

C]-

raclopride, etc.), metabolism markers ([

F]-FDG), and even for cardiac

applications.

The Logan plot is a graphical linearization method that applies to reversibly

bound tracers which have a constant net efﬂux that is nonnegligible when

compared with the constant inﬂux. The approach involves creating a plot

with the following coordinate axis:



ROI(t)dt/ROI(t) versus



(t)dt/ROI(t) (7.217)

where ROI(t) is the concentration of a tracer in a tissue ROI, t is the time

elapsed from injection to a particular frame, and Cp(t) is the concentration of

tracer in plasma [33].

Similar to the Patlak plot, the slope and intercept of a Logan plot have

distinct interpretations depending on the actual model one chooses to ascribe

to the underlying system.

The straight line in a Logan plot should be ﬁtted through those frames that

are linear; these frames usually occur somewhere in the middle of the plot.

This is another way of saying that data should be collected over a long enough

period to see a net efﬂux of tracer from the ROI.

A formal analysis of the Logan plot shows that it is valid for an arbitrary

number of compartments for both plasma and reference input models when

the data are free from noise.

7.2.9 Parametric Analysis

Parametric analysis is required when the studies involve complex analy-

sis and advanced statistical methods to localize the regional distribution of

functions, particularly in situations in which input function is affected by

noise. Parametric images incorporate the highest degree of quantiﬁcation and

require highly developed statistical methods that are mostly used in brain

studies.

In parametric analysis, information is obtained up to the image resolution

level,that is, the parameters arecalculated for each image voxel and the results

are represented as maps of the parameter.

The values bound to the image voxels are the physiological parameters

studied (perfusion, glucose consumption, receptor density, etc.).

Existing methods for computing kinetic parametric images work by ﬁrst

reconstructing a sequence of PET images and then estimating the kinetic

parameters for each voxel in the imaged volume. Special computer programs

convert the dynamic information to parametric functional information.

Parameter estimates are obtained from a priori speciﬁed compartmental

structures using oneof a variety of least-squaresﬁttingprocedures:linearleast