IMAGING RADIOTRACER MODEL PARAMETERS

IN PET: A MIXTURE ANALYSIS APPROACH

by

Finbarr O'Sullivan

TECHNICAL REPORT No. 235

August 1992

Department of Statistics, GN-22

University of Washington

Seattle, Washington 98195 USA

Imaging Radiotracer lVIodel Parameters in PET:

A. lVIixture A.nalysis Approach. 1

Finbarr O'Sullivan

Department of Statistics, GN-22

University of vVashington, Seattle \VA 98195.

August 10, 1992

research was SUPI)Orted m

CA-42045.

the National Institutes of Health under vA-4~u<JJ and

Abstract

A variety of sophisticated radiotracer models are available for the quantitative interpre­

tation of dynamic positron emission tomography (PET) data. Parameters in these models

are used to define quantities, such as metabolic rate, blood volume and flow, etc., character­

izing the functional physiological and/or biochemical status of tissue, in vivo. We consider

two methodologies for fitting radiotracer models on a pixel-wise basis to PET data. The

first method does parameter optimization for each pixel considered as a separate region

of interest. The second method also does pixel-wise analysis but incorporates an addi­

tive mixture representation to account for heterogeneity effects induced by instrumental

and biological blurring. Several numerical and statistical techniques including cluster anal­

ysis, constrained non-linear optimization, sub-sampling, and spatial filtering are used to

implement the methods. A computer simulation experiment, modeling a standard [F-18]

deoxyglucose imaging protocol using the UW-PET scanner, is conducted to evaluate the

statistical performance of the parametric images obtained by the two methods. The results

obtained by mixture analysis are found to have substantially improved mean square error

performance characteristics. The total compute time for mixture analysis is on the order

of 0.7 seconds per pixel on a 16 MIPS workstation. This results in a total compute time of

about 1 hour for a typical FDG brain study.

Keywords: constrained optimization, kinetic modeling, mixture analysis, parametric imag­

ing, positron emission tomography

1 Introduction

There is substantial interest in the ability of dynamic positron emission tomography (PET)

to derive quantitative characterizations of the functional, physiological and biochemical,

state of tissue, in vivo [31, 43]. The question of whether PET should be used quantitatively

or qualitatively for clinical applications has not yet been resolved, for example, DiChiro

and Brooks[13] have argued that the technical aspects of quantitative PET are so complex

that one ought to focus on the qualitative interpretation of PET data in a clinical setting.

However, the high cost of PET relative to other radiological methods is a practical incentive

to recover as much information as possible from the data it provides. In order to scientifically

evaluate the hypothesis that quantitative PET is worthwhile, we need to have an efficient

technology for generating quantitative functional images.

The functional characterizations of tissue, measurable by PET and for that matter

quantitative autoradiography, are defined in terms of parameters associated with models

describing the delivery, transport and biochemical transformation of the radiotracer. For

brain imaging, [O-lS]-labeled water and [F-18]-labeled deoxyglucose (FDG) are among the

most well studied PET radiotracers[43J. With [0-1.5] water, equations have been derived

which yield estimates of cerebral blood flow or volume from a single integrated image[7, 23,

25]. Analogous techniques have been used to map rates of glucose utilization with FDG in

the brain[39]. Many of these techniques are complex because they depend on parameters

which must be spatially adapted depending on the type of tissue under consideration, e.g.

white or grey matter in normal brain[7, 39J. Even for these well studied radiotracers, there

remain concerns with current imaging methodology; misspecification of the input function

time delay and tracer extraction fraction for [0-1.5J-water[7, 23, 2.5] and non-uniformity in

the rate constants in FDG studies[29, 47J. Many of these difficulties could be substantially

diminished if there were techniques available to on a pixel-wise basis.

OHJa(ler range of poterltiaLlly

as as

which may be found to have some diagnostic significance. The pixel-wise fitting approach

would also allow the construction offunctional images with more complex radiotracers[6, 4.5]

which cannot be adequately modeled by simple refinement to the models used for [0-15]

water or FDG.

Radiotracer models are typically only rigorously applied to average time-activity-curves

(TACs) derived from user selected regions of interest (ROIs). Key exceptions are the ap­

proaches of Blomqvist[5] and Herho1z[22]. Blomqvist adapted the so-called graphical ap­

proach of Gjedde[15, 16] and Patlak et. al.[37] to obtain images of the glucose metabolic

rate from FDG data. The method estimates unidirectional uptake which is accessible by the

graphical approach but difficult to generalize to other radiotracer models[38]. The approach

of Herholz is more general. He fits radiotracer models to TAC data derived from ROIs which

are allowed to spatially vary in size and shape over the entire imaging region. A variation on

this approach is examined here. However, there are inherent noise and heterogeneity[21, 44]

problems associated with pixel-wise data which makes the approach unappealing. A second,

more powerful, approach we examine uses a mixture analysis technique in which pixel-wise

tissue time-activity curves are approximated by a sum of several underlying curves each

representing homogeneous tissue. Similar types of mixture models have been used for some

time in the remote sensing literature, see Adams et al.[l] for example, and more recently

by Choi et al.(10] in the analysis of Magnetic Resonance image data.

The definition and implementation of the above approaches for parametric imaging with

dynamic PET data are fully developed in sections 2 and 3. Section 4 presents a simulation

experiment designed to evaluate the statistical performance of the techniques in the context

of brain imaging with FDG with the UW-PET scanner. The results for the mixture analysis

approach are quite encouraging. The method has been implemented for an operational FDG

imaging protocol. An illustration with data from an FDG brain study is presented in section

5. 'VVe conclude a summary and discussion.

2

2 Parametric Imaging Methods

Blood-tissue exchange models for PET radiotracers track the transport and transformation

of tracer in physiologically defined compartments such as plasma and cells. Mathematically,

these models are typically specified as systems of (linear) differential equations whose coef­

ficients are quantities of physiologic or biochemical interest[27]. The forcing function for the

system is a plasma blood input function which is often obtained by blood sampling[23, 39].

Summing the activity from various sources defines a model function, denoted Jt(tlp), for

t = 1,2, ... ,T, which depends on an unknown p-dimensional parameter vector p. In the

analysis to follow, we seek to obtain estimates of these parameters for each pixel in the

imaging region. The parameters may be of interest in themselves or may be used to define

other quantities describing the functional state of tissue; such as the glucose utilization rate

with FDG[39].

With dynamic PET, the data available for fitting parametric models is a time series

of radioactivity distribution images acquired over the course of the study. Suppose the

imaging region consists of I pixels with spatial coordinates Xi for i = 1,2, ... ,1. Then at

each pixel, we have a vector of measurements, Zi(t) for t = 1,2, ... , T. Zi is referred to as

the observed time activity curve (TAC) at the i'th pixel. The two methods we consider for

generating pixel-wise estimates of the parameters p are the following:

Pixel-wise ROI Analysis

This approach does pixel-wise optimization in the spirit of Blomqvist[5] and Herholz[22].

The parameters mapped at the i'th pixel, Pi, minimize the weighted residual sum of squares

fit to the pixel-wise TAC, i.e.

T

= argmina Lt=l

weights Wt can the r"!.,t;;",, accuracy of measurements at ditrerient

times. use a welg;tltlIlg; obt;Ct111led a clu.stt~r ana.LYS;lS. see sec.tlO,n

tl€:rho1t;zl:t~J, it is necessary to mclucle some sp;:tt};:U filtering; into a sCtl.eIIJle sort.

Our approach is described in section 3.3.

Mixture Analysis

This approach uses a mixture model which approximates the pixel-wise TAC data as a

convex linear combination of, say K, underlying sub- TA Cs:

K

Zi(t) ~ 2:: 1rikPk(t) ,k=l

(1)

where Pk(t) is the sub-TAC corresponding to k'th component in the model. The 1rik are

the weights given to the sub-TACs; these lie in the K -dimensional simplex, i.e. they obey

positivity and unitary constraints, 1rik 2:: 0 and I::f:l 1rik = 1. In practice, it may be

reasonable to have the flexibility to impose a sparsijying constraint on the mixing vector

1ri, e.g. only a fraction of components are allowed to be active at a given pixel[10, 35]. A

numerical scheme for imposing sparsity is presented in the next section. (In the context of

PET, the use of sparsifying constraints might ultimately prove to be a way to correct for

partial volume effects[10] .) The unknown J>arameters in the mixture model are the Pk'S,

the 1r/s and K.

Mixture models are not unfamiliar in the PET literature, for example, the work of

Herscovitch et al.[23] and Schmidt et al.[44] employs two class mixture models to study

the effects of heterogeneity on parameter estimation schemes with water and deoxyglucose

radiotracers. The mixture model in equation (1) is more general of course. Fitting the

radiotracer model to the sub-TACs J.lk(t) yields a set of parameters, ,8(k) , for k = 1,2, .. . K.

Hence we can infer a distribution of parameter values at each pixel - occurs with

frequency 1rik at the i'th pixel. There are a number of features of this distribution which

could be of interest, e.g. relative dispersion, skewness, kurtosis, multi-modality. In this

paper, we focus attention on the mean of

pixel willK

=2::k=l

distribution, i.e. parameter of interest at

3 Computational and Statistical Methods

Various techniques from numerical optimization and statistics are used in the construction

of the algorithms to implement the parametric imaging methods. An overview of the steps

involved are as follows:

Pixel-wise ROI Analysis Algorithm:

A. Cluster Initialization (section 3.1)

B. Pixel-wise optimization with filtering (section 3.2.4,3.3)

Mixture Analysis Algorithm:

A. Cluster Initialization (section 3.1)

B. Estimation of mixture model without kinetic constraints

1. Estimation of 7r and J.L given J( (section 3.2.2.)

2. Estimation of K (section 3.2.3)

C. Kinetic parameter estimation with 7r'S fixed (section 3.2.4)

D. Final updating of 7r'S including filtering (section 3.3)

In order to eliminate artifacts associated with background pixels, parameters are set to

zero on any pixel for which the TAC is too samll. The pixel-wise parameters are set to zero,

if the mean square of the TAC is less than .5% of the maximum mean square of all TACs in

the image.

3.1 Clustering

set time ri.l',tivitv curves, ZiCllllst;erJlllf,?; n,,'~tit';r.T1C or seF~men1;s

clusters with SelI-SJlmJ.rar ch;:tra,ct~~n~;tlC:S.into, say

each cluster be denoted {ikb(t) for kb = 1,2 .. .Kb. Fitting the blood-tissue exchange model

for the radiotracer to each of these curves yields a set of parameter estimates !3(kb ) and

model functions p(tl!3(kb )) for kb = 1,2 .. .Kb. For the pixel-wise ROI analysis, the results

of the clustering are used to define warm starts for the pixel-wise optimizations: i.e. /j(kb )

is used as the warm start for the optimization of all pixels falling into the kb'th cluster. In

mixture analysis clustering is used as a starting point for a more sophisticated approach to

identifying the number of components, K, in the model (see section 3.2.3 below).

A hierarchical algorithm is used for clustering[19]. Hierarchical algorithms have two

important advantages over more sophisticated model-based techniques such as maximum

likelihood[19]: (i) robustness against departures from an assumed model and (i1) compu­

tational efficiency. A hierarchical algorithm can be represented as a binary tree whose

terminal nodes define the clusters. In the crudest clustering there is just one root node

which contains all the data. At the other extreme, is the finest clustering in which each ter­

minal node contains a single data point. There are agglomerative and divisive approaches

to hierarchical clustering. Agglomerative techniques are more versatile as they can be used

with qualitative and well as quantitative data, however, the drawback is that they require

the computation of a pairwise distance matrix between all terminal nodes in the tree and,

since this matrix grows as the square of the number of data points, the approach becomes

impractical for large data sets. A divisive algorithm in the style of modern tree structured

regression techniques, see, for example, Brieman et al.[8]' is used here. This procedure

recursively splits the most impure terminal node in the tree until some desired level of

purity is reached. Node impurity is taken to be the trace of the covariance matrix of the

collection of TACs in the node. The covariance is used because it is not sensitive to the

shear number of TACs in the node; without this the clustering algorithm typically pays too

much attention to background pixels which often constitute a large but uninteresting part

of PET images. Splitting a node creates a daughter nodes and the point for a

IS to sum of of As size

may be root set

points considered for splitting is restricted to quintiles of the first principal component of

the within node covariance.

If there are Kb terminal nodes, the overall fit of the clustering can be measured by the

total sums of squares between the pixel-wise TACs in the cluster and the cluster means,

K

WK L L L[Zi(t) - fik(tW·k=l iECk t

where Ck are the set of pixels in the k'th cluster. this is known as Ward[19]'s criterion.

W Kb is monotonically decreasing and equal to zero when Kb = I. There is no advantage

to excessive splitting, so a stopping rule is employed. The rationale for the stopping rule is

different in the two parametric imaging algorithms. In the pixel-wise ROI analysis approach

estimates are obtained by pixel-wise optimization using cluster defined warm starts. Here

the choice of the number of clusters is purely driven by computational considerations and

formally has no effect on the statistical performance characteristics of the resulting para­

metric images. The situation is more delicate for the mixture analysis scheme. Here the

choice of Kb restricts the range for the optimization of K in the mixture model (see section

3.2.3). Consequently, since K behaves as a regularization parameter (similar to filter size

in reconstruction algorithms[24]) the choice of K will influence the resolution and variance

characteristics of the parametric images. Our approach is to choose K b large enough so that

this influence is minimaL In the current implementation only a crude attempt has been

made to optimize the choice of Kb in the cluster initialization of either algorithm. Based on

experiences with some real and simulated data-sets, a value of Kb = min(I/2, 3.5) for the

pixel-wise ROI approach and Kb = min(I/2, 20) for the mixture analysis (here the optimal

K in FDG brain studies is typically between 4 and 12), appeared to work reasonably welL

The averaged within cluster variance is used to define the weights Wt in (1):

where IS number to

to CllJlst,ers - again is to protect against

unimportant clusters corresponding, for example, to background. More accurate weight­

ing schemes could possibly be developed from the work of Huesman[26] and Haynor and

Woods[20].

3.2 Mixture Analysis Considerations

3.2.1 Identifiability

Before presenting an estimation algorithm to solve Step B.1 in the mixture analysis algo­

rithm, two potential identifiability problems associated with the mixture model need to be

addressed. The obvious one is labeling. The k-labels on the J-lk'S and Ttik'S can be per­

muted without affecting the value of the model prediction in equation (1). Fortunately, this

source of lack of identifiability is not important because the pixel-wise parameter estimates

in equation (2) are also invariant under such re-labeling. A more serious source of non­

identifiability arises because the model represents data in terms of convex combinations of

the sub-TACs, J-lk for k = 1,2, .. .K. If the J-lk'S are surrounded by a convex polygon with

vertices at some new set of points ILk for k. = 1,2, .. .1(, then any convex combination of

the J-lk'S can equally well be represented in terms of a convex combination of the Ilk' The

problem is that the J-l'k may have quite different kinetic parameters than the J-lk'S and so the

pixel-wise parameters in equation (2) could be substantially altered in this case.

One approach to this problem would be to find the best fit of the mixture model to

the image data subject to the constraint that the volume of the convex hull of J-lk'S be

reasonably to the distribution of the observed data. The implementation of such a

volume constraint appears to be somewhat complicated, however, so an alternative strategy

is adopted in which the J-lk'S are merely forced to lie in the convex hull of a fixed set of

T -dimensional vectors for k = 1,2, ... , K contained in the range of the data. We let i.e.

Kb

J-lk = Lkb=l

where vector lk = 12k,·· . in

vectors are taken to means

3.2.2 Algorithm for K Fixed

For fixed K, the algorithm for estimating J.L'S (really ,'s, see equation 3) and Jr's is designed

to minimize the weighted residual sum of squares:

WRSS(Jr, J.L) I:wrssi(Jri,,),

where wrSSi is the weighted residual sum of squares at the i'th pixel:

wrsSi(Jri,,) = I:Wt(Zi( t) - I: JrikJ.Lk (t)f2 ,t k

and, determines J.L by equation (3). The algorithm for minimizing WRSS is as follows:

Initialize: crit;- 00; WRSS;- WRSS(Jr,fL); tol = .0001; maxit = 100; iter = a

While ( crit > tol & iter:::; maxit ) {

WRSSo ;- WRSS ; ite1' ;- iter + 1

1. Update J.L (really ~() conditional on Jr fixed

2. Update Jr conditional on J.L fixed

End

Steps 1 and 2 of above aig;orJ.th:m involve solution of quadratic optimization

problems constraints. Step 1, the constraints are soi11tiC}fi is cornpilted

1SS01 et. IS

reduced to a more convenient form: Consider the symmetric positive semi-definite matrices

V and <P defined by

vK

(2:': 1rik1rikJ] = L aveve~v=l

Kb

[Lwt4)kb(t)¢kJ~(t)]= Lbddei e=l

(4)

where (av,ev) and (be,fe) are the eigenvalue-eigenvector pairs for V and <P, respectively.

After some algebra, the objective function in Step 1 of the algorithm is reduced to

WRSS(,) = L LavbdUe is) ev)',- gev?e v

where ® is the Kronecker product[40] and gev = Ue ® ev)'Y/~ with

Ykb = L L WtZi(t)¢kb(t).t

The LSSOL code is used to minimize the above quadratic form subject to the unitary and

positivity constraints on the J( components of I' This accomplishes Step 1 of the algorithm.

The computation of Step 2 uses the LSSOL code at each pixel in turn. However, here the

implementation of a sparsity constraint requires some additional work. Sparsity is imposed

by placing a constraint on the Gini diversity[8, 35J index of 1ri. Thus, the goal in Step 2 of

the algorithm becomes to minimize the weighted residual sum of squares, wrSSi, subject to

the constraint that the mixing vector 1ri satisfies

K

L 1r[k ~ 0: ,

k=l

where 0: E [l-,IJ. The parameter 0: determines the strength of the sparsity constraint:

Setting 0: = 1/1'l! at most allows lv! components to be present in equal proportions at a

pixel.

(and hence

in the c>vj'rnrna

to 1 of 0: :::: 0.25 is In

one component of 1ri is non-zero

some more th()U~;httul

physiologic consideration. we prove to

be a useful mechanism for addressing partial volumes (limited resolution) effects.) Numer­

ica.lly, the optimization of 1i'i subject to the sparsity constraint is achieved by an iterative

algorithm which linearizes the constraint about the current value of 1i'i and then obtains an

update by solving the resulting quadratic programming problem using LSSOL[18], see Gill

et. al. [17].

In our experience over a few hundred examples, the algorithm typica.lly converges in

fewer than 15 iterations. The minimization ofWRSS for K fixed is an example of a quadratic

optimization problem with non-linear inequality constraints. It is clear that each cycle of

the above algorithm reduces (at least does not increase) the objective function, however,

while this is a very desirable property, in general, it does not imply convergence because the

step sizes in the algorithm could possibly get sma.ller without making substantial progress

towards the global minimum. There are several techniques for addressing this potential

problem, see Gill[17]. For example, if the constraints were linear, then it would be possible

to refine the algorithm by incorporating conjugate gradient techniques and appeal to a

well developed convergence theory. Unfortunately, the incorporation of conjugate gradients

substantia.lly complicates the algorithm and so for the time being the simple implementation

above, with it's potential shortcomings, is used.

3.2.3 Estimation of the Number of Mixing Components

A cross-validated backwards elimination scheme is applied to estimate K in step B.2 of the

mixture analysis algorithm. The method works as follows: Starting with a model with a

large number of components, K = Kb where K is the number of clusters obtained in the

initialization, the backwards elimination scheme successively eliminates the most redundant

sub-TAC in the model. This leads to a sequence of mixture models indexed by !{ for

K = Kb, Kb - 1, ... ,2. For each of these models, a generalized cross-validation score is

computed and best score is steps involved in process

are described now.

For most redun,dartt IS identified

sub-TAC, Ilk, as a convex linear combination of the other sub-TACs in the model, i.e. find

a k in the (K - 1)-dimensional simplex to minimize WR88(1l, il") where

Ilk = L ai'llk'k'-:j:k

and all other parameters, Ilk"S (k' f:. k) and il"'S, are held fixed (the L880L[18J code is used

again here). This yields a set of values WR88k for k = 1,2, .. ,K, and the most redundant

class, k* , is defined as the one which has the smallest WR88k value. Once Ilk. has been

dropped the algorithm in 3.2.2 is used to re-optimize the parameters il" and 11.

There is quite an extensive statistical literature on model order selection, see [2, 36,

50], for example. We base our model selection strategy on a method known as cross­

validation[8 , 50J. This approach is appealing because it requires only limited stochastic

modeling of the data. This contrasts with more model dependent methods, such as AIC[2J

for example. The idea in cross-validation is to resample the observed data so as to construct

an estimator of the predication error associated with a model. The model with the best

(smallest) prediction error is then selected. For computational reasons, we use a variant of

cross-validation known as generalized cross-validation (GCV), see Wahba[50J. In GCV the

cross-validation prediction error estimate is replaced by a surrogate ratio of the form

GCV = Model FitDFE2

Here Model Fit represents the fit of the model (WRSS) and DFE is the effective degrees of

freedom for error. Models which are too simple have large degrees of freedom for error but

they fit the data too poorly, on the other hand complex models may fit the data well but

often at the cost of a very small error degrees of freedom. In either of these extremes the

prediction error and the GCV statistic takes on large values. Typically the model with the

smallest GCV value also has the smallest prediction error. A of validation studies are

1:> UJlIllJ.J<tl JZ,C:U In book by Wahba[50]. GCV method been successfully adapted to

a range see some ex;:tmple:s.

a term to rer)reSerLt treeC1()m uSl1aJJ.y irl"r.I",,,, some heuristics.

For mixture LH,-nJ.'-L, are two cOJltribultiC)fiS to account - one

(.5)

of the f-L/s and and the other from the estimation of IT;'s. Let DFE I· T DFM where

DFM is a count of the total number of parameters being estimated. We take the number

of parameters involved in the f-Lk'S is taken to be I( . Kb (note this ignores the unitary

constraints on the 'Yk 's). In the case ofthe IT/s, due to the sparsity constraint, it is common

for many of the elements of IT; to be essentially zero, and so we need to be more careful in

measuring degrees of freedom. We measure degrees offreedom of IT; by df(IT;) - the number

of components of IT; which exceed a small positive number E. Currently, the algorithm sets

E = 0.01. (There has been no serious attempt to optimize E except that in limited simulation

studies values above 0.1 or below .001 created noticeable biases in the results.) With this,

the overall degrees of freedom of the mixture model is taken to be

DFM(K) = K· Kb +I:. df(IT;).1

The compute time associated with evaluating a sequence of mixture models required for

cross-validated backwards elimination procedure is non-trivial. Thus, to improve computa­

tional efficiency, only a subset, S, of Is pixels, randomly selected from the image, are used

for optimizing K. The GCV statistic used is

s WRSss(I()GCVI\ = [1 - DF~fS(K)/(Is . T)J2

where WRSSS(K) is the weighted residual sum of squares fit of a K class mixture model to

the Is subsampled pixels, and

DFrvrs(K) = K· Kb + I:.df(IT;).;ES

The sample of Is cases is selected in such a manner that there are roughly equal numbers of

pixels chosen from each of the K clusters obtained in the initialization. The idea here is to

protect against the undue influence oflarge unimportant areas corresponding to background.

The sample S is built up by sequentially applying the following rule: select a cluster at

random and, provided not a pixel at random without

optmlal K was not very "",r,,,itiv,,, to one eX;lmplewe found that

I = 8000 a choice of 2.50 or choice K.

3.2.4 Kinetic Parameter Estimation

Primarily due to the identifiability issue, discussed in 3.2.1, no radiotracer modeling con­

straints are placed on the JLk'S throughout mixture model estimation process described in

Step B of the mixture algorithm. After the optimal model has been determined, radiotracer

parameters, j3(k)'s, are found as follows: Let z;( t) be the set of Zi( t) predictions available

after the mixture estimation process is complete,

zi(t) = L 7f"ikJtk(t) .k

The j3(k)'s are chosen to minimize the sum of squares

WRSS(;3) = L L[zi(t) - L 7f"ikJL(tlt3(k»)]2.i t k

Some linear algebra shows that minimizing the sum of squares can be reduced 1 to the

problem of minimizing

K TWRSS*(,6) = L L Wtav[gv(t) - e~JL(t1.6W ,

v=l t=l

where (av , ev ) for 1/ = 1,2, ... ]( are the eigenvalues and eigenvectors of the symmet­

ric positive semi-definite matrix V = [Vkk'], see equation (4). Here JL(tIl3) is the vector

(Jt(tl,8(1) , JL(tl,8(2) , ... , p( tI.6(K»)', and gv( t) = e~y(t)jva;; where for k = 1,2, ... , J(

The minimization of WRSS*(l3) is carried out using the NL2S0L code of Dennis, Gay

and Welsch[12]. To accommodate interval constraints, which are relevant for many pa­

rameters, the optimization is carried out in a logistic parameterization. Thus if the r'th

component of the parameter vector must be forced to lie in the interval [a, b], then the

optimization code is forced to work with an unconstrained parameter Or which determines

according to logistic transform

=a+

A similar reduction was used in 1 of the Mixture rl.U,UY"l" Algol:ith.m in section 3.2.2.

Once the optimization for the kinetic parameters is complete, we compute a new set of

pi's by; Pj(t) = p(tliJ(k)) for j = 1,2, .. .K. The NL2S0L code is also used in the pixel-wise

optimizations involved in the ROI analysis approach. In that case the number of iterations

allowed is limited to at most 10, otherwise the computation time of the approach becomes

excessive.

3.3 Spatial Filtering by Local Likelihood-Type Procedure

Herholz[22] noted the potential need to incorporate smoothing procedures into parametric

imaging algorithms and developed an approach in which the pixel-wise estimation process

is applied to a filtered version of the input data, i.e. the images z(t) for t = 1,2, .. .T.

As a general approach this is similar to what is known as local likelihood in the statistics

literature[48]. We use a version of Herholz's method in which the filtering is accomplished

using a linear (as opposed to the more elaborate non-linear scheme proposed by Herholz)

Butterworth-type filter[34]. Here the size of the parametric imaging region is selected so

that the fast Fourier transform methods in O'Sullivan[34] can be applied. This improves the

computational efficiency of the approach. The choice of bandwidth for use in the filtering

has not been systematically investigated or optimized. Experience showed that the mixture

analysis could tolerate smaller bandwidth than the pixel-wise ROI approach. We currently

set the bandwidth for the spatial filtering used in the mixture analysis to be the same size

as the bandwidth used in the reconstruction process. Consequently the nominal pixel reso­

lution of the parametric images is 1.4 times the pixel resolution of the raw reconstructions

(two applications of a Gaussian convolution filter with bandwidth b is equivalent to one

application with a bandwidth of V2 .b). The spatial filtering bandwidth for the pixel-wise

ROI approach was set at twice that of the mixture analysis. This may not be enough be­

cause the mean square is still found to be largely dominated by variance components

next section). The is prior to last optimizations -

du:;teI'ing m Dlxel-vnse ROI m kHltetJlC parame-

ter estimation, in mixture <tU,:Ll Y :~10l.

at earlier stages in the mixture algorithm were examined, however, the problem of then

incorporating the sparsity constraint proved too difficult. More complicated regularization

methods[lO, 11, 35, 50J for incorporating smoothness were also considered, however, at

present these techniques would require too much computation time for practical use.

4 Numerical Evaluation of Performance Characteristics

A computer experiment was conducted to examine the performance characteristics of the

parametric imaging algorithms on FDG brain studies with the SP-3000jUW(UvV-PET)

tomograph, see Lewellen et. al.[30J.

4.1 Data Generation Procedures.

UW-PET is a time-of-flight tomograph with an aperture of 45cm, a transverse detector

resolution of 4.5mm and time-of-flight resolution of 9.0cm. Reconstructions are generated

on a 216 x 216 pixel domain whose pixel dimension is 2.1mm. The machine acquires events in

list-mode and later reformats the data into atime-distance-angle (TDA) array of dimension

32 x 216 x 320 for reconstruction. In order to cut down on the memory requirements of our

simulation code, we only consider single cross-sectional slice imaging in the absence of any

trans-a)dal sampling effects. The machine aperture is reduced to 27cm while maintaining

the same pixel dimension (human brain cross-sections are easily contained within these

dimensions) and the number of angles is reduced by a factor of two. Thus our simulated

tomograph is focused on a 128 x 128 pixel imaging region with a TDA array of dimension

32 x 128 x 160. The discretization in the time-of-flight (2.3cm bin-width) and distance

(2.1mm bin-width) match those used in UWPET; the angular discretization is uniform over

[0 0 ,1800].

cross-sectional Shepp-Logan brain phantom bone region removed was to

M111Ul,<L",,~ a FDG distribution: phantom was most rec:ently

1,

Vardi et. to de1nonstrai;e

phantom, shown

in context

aOI)ro;:tch to recon~;tnlction.

and shape; all but one of the regions (region 3) consists of a a single connected set of pixels.

The well known three compartment FDG model, see Phelps et al.[39], is used to define FDG

time-course signals for the six regions in the phantom: Let ql(t) and q2(t) be the amounts

(Activityjgm) of the [F-18] label in tissue at time t (minutes) in the original(FDG) and

phosphorylated (FDG-6-phosphate) forms. Let Cp(t) be the concentration (Activityjml) of

FDG in plasma. The state equations for the model are

:~l(t) K1Cp(t - r) - (k2 + k3 + A)ql(t) + k4q2(t)

dq2 (t) = k3ql(t) - (k4 + A)q2(t)dt

and the model function is given by

(6)

Note that these model equations are a little non-standard because they include, via the

parameter A, a mechanism which accounts for decay of the F-18 isotope over the duration

of the study. The half-life of F-18 is 108.36 minutes so A = 0.00636min-1. There are

six unknown parameters in the model f3 = (r, Iv, Kll k 2 , k3 , k4 ): r is a delay, in seconds,

in the arrival of the input function, Cp , to the tissue region under consideration, Iv is

fractional volume of blood in the region, K 1 is a transfer constant (~min-I) operating on

concentration and k2 , k3 , k4 are rate constants (min-I) operating on amounts. The steady

state rate of conversion of plasma FDG to the phosphorylated form is measured by the ratio

As discussed by Phelps et. al.[39], this is proportional to the Glucose metabolic rate as

determined by FDG. Parameters values used for each of the six regions in the phantom are

given in Table 1. The variation in parameters is similar to what is seen in typical brain

scans- c.f. the In next sectIon. plasma input curve In 2,

is from an

SIX ,..,0'1("'17\<: in ph,ant;olll, see a spe~Clllcation

tracer ",..'''trihr diEitrj buticm in the at any time

In actual FDG studies, a sequence of PET reconstructions of tracer activity are obtained

over the duration of the study. At our institution at set of 30 such reconstructions are

currently acquired in a typical FDG brain scan. Time-bins for these reconstructions are

shown as tick-marks on Figure 2; similar to many other protocols of this type, the time

binning is most rapid in the early part of the study. In the numerical experiment, a sequence

of PET reconstructions are generated by computer simulation. The stochastic variability

of PET count array data is well modeled as an inhomogeneous Poisson process[20]. and

there are well established techniques for generating realizations from such a processes[42].

Using these, to simulate a count from a cell in the TDA array we only need to know

what the expected count (or rate) for that cell is. The expected value of the TDA array

corresponding to a particular reconstruction time-bin is computed as follows: First, a time­

binned tracer activity image is produced by integrating the activity (using the time-courses

in Figure 2) over the duration of the given time-bin. Next, this activity image is projected

into the TDA domain using the model described by Snyder et. al.[46] (note this model

ignores scatter). Finally, cells in the TDA array are multiplied by an attenuation factor,

depending only on the line of flight of the photon pair, to obtain the expected TDA array.

Our simulation assumes a uniform attenuation of .2cmz/ gm throughout the brain region.

The blood input function may be scaled to mimic studies with more or less injected dose.

Equivalently, the scaling can be applied to the expected TDA arrays. We examined a set

of nine studies in which total expected number of positron coincidence detections ranged

(uniformly on a logz scale) from 0.5 x 106 to 0.5 X 109 counts. As a point of reference,

a brain scan with a 10.OmCi injection of FDG typically leads to a total count per slice

over a 90 minute period of between 1.0 x 107 and 5.0 x 107 coincident events. In each

of the nine studies, a sequence of 30 simulated TDA count arrays were generated and

subsequently reconstructed using the standard confidence-weighted filtered backprojection

method[24, 46]. The resolution size used in sequence of reconstructions was selected

so mt·egrat;ed reconstruction over entire tmae-coun,e was as close

as possible, in a mean square error sense, to true mt;egral;ed :>r1"lVlfv

4.2 Simulation Results.

Parametric images of the MR(metabolic rate) parameter for a total count of 1.58 X 107

coincident events are shown in Figure 3. The target or best possible image is also shown,

this is the true MR image blurred by the limitation imposed by the transverse detector

resolution - the lower resolution limit of the reconstructions is determined by the FWHM

of the transverse detector resolution, i.e. 4.5mm. The mixture analysis estimate appears to

be quite good; the image obtained by the pixel-wise ROI approach shows substantial noise

artifacts. Additional filtering would reduce these effects but at the cost of further loss of

resolution relative to mixture analysis. The behavior of the GCV statistic (eqn 5) used in

mixture analysis is shown in Figure 4. The GCV estimates of J (number of sub-TAGs as

a function of the the number of counts is shown in Table 2. Clearly there is considerable

variability in the estimate of J but there are many theoretical results which indicate that

behavior is to be expect in model order selection, see Park and Marron[36] and the references

cited therein.

Parametric images were evaluated in more quantitative terms using percent error crite­

na. The percent error in a parameter estimate 8 is defined as

8-B%Error = -- x 100

B

where B is the true value. Since percent errors are only meaningful when the target parame­

ter is non-zero, we focus attention on pixels over the brain where the underlying radiotracer

parameters are non-zero. Over the brain there are five distinct tissue regions, see Figure 1.

The mean (BIAS) and standard deviation (SD) of the pixel-wise percent errors in each re­

gion are computed and from these, the root mean squared (RMS) percent error is evaluated

as

RMS,. /BIAS; +SD;

Here subscript 7' refers to regions, 7' = 1,2, ... 5. overall per10nIlaIlce IS c V<uUQ,"C;U

terms of the average BIAS

BIAS = ~t{) ,.=1

over5

= "\'~ L...) ,.=1

BIAS and RMS performance characteristics for the 1.58 x 107 count study are shown

in Tables 3 and 4, respectively. The error rates for mixture analysis approach are typically

more than 2 times smaller. Error rates associated with the delay and blood volume are

noticeably greater but this is to be expected since these parameters are largely determined

by the early time course data which corresponds to low activity and consequently poorer

relative precision. The estimation of individual kinetic parameters is somewhat poorer in

Region 1, this is the region with the lowest uptake in the brain. The performance on Regions

3 and 4, which are both quite small, is remarkably good. The MR parameter is the most

stable - the percent error ranges between 6.0% and 19.0%.

Performance measurements obviously depend on the parameter e under consideration

as well as the expected total counts associated with the data set. In planning studies, it is

of interest to know the count dependence of the estimation error. Figure 5 plots the RMS

errors in the metabolic rate (MR) estimate as a function of the total count in the study.

The plots show that the performance measures follow a simple log-linear relation. In theory

this is not unexpected (see Cox and O'Sullivan[ll] as well as a variety of references cited

therein). To summarize the RMS dependence on total counts. we fit the log-linear model

RMS ~ A . (!!.- )-r­No

(9)

for each parameter by brain region in the phantom. Here A is the RMS error at a count

of No (we set No = 1..58 X 107) and r is the so-called rate of estimation achieved by the

algorithm. The values of A and r are given in Table 5. Note that since we are fitting a

model, the values of A sometimes differ from the numbers in Table 4, the most substantial

difference is seen for the blood volume parameter. The larger the rate of estimation, r,

the more quickly the RMS error responds to increased counts (injected dose). In finite­

dimensional estimation theory[40], the classical optimal rate of estimation is 0.5. Table

5 shows that the response of mixture analysis algorithm to more data is slower

what one sees is COIlsis:teIlt

is a very should empn;a,S12:e

differ,ent rates differl~nt phantoms.

The experiment was carried out on a SUN4/330 workstation. This processor is bench­

marked at 16 MIPS and 2.5 MFLOPS. The total compute times per pixel were 5.0 and

0.7 seconds for pixel-wise ROI and mixture analysis, respectively. This excludes the time

for generation of the PET data sets. At this rate the mixture analysis approach is very

reasonable for practical use. We predict an analysis time of about one hour for a typical

FDG PET brain study.

5 An Application to Real Data

The mixture analysis parametric imaging methodology has been implemented for an opera­

tional FDG brain imaging protocol using the UW-PET scanner. Figure 6, shows data from

a transverse brain scan of a 34 year old male patient with a mid-calossal glioblastoma mul­

tiforme. The study protocol followed the simulation except that the scan time only lasted

one-hour, this reducing the number of samples to 24 instead of 30. The tumor region is

clearly visible on the CT and PET uptake images. Applying the mixture analysis algorithm

to the data from this study yielded the parametric images shown in Figure 7. The delay

image was computed bu is not shown. The quantitation of blood volume in normal brain,

i.e. outside of tumor, is close the expected value of 0.04. The the k4 image appears to suffer

from some noise artifacts, this is probably due to the fact that the imaging time was only

one hour, which limits the ability to properly resolve k4 [39, 44J. Several parameters are

seen to show good definition for the tumor region. In normal brain, all kinetic parameters

are in general agreement with normal brain values reported in the literature(see Phelps et.

al.[39J for example). Over the tumor region most parameters assume substantially higher

values - between 1.5 and 2.0 times the normal range.

The quality of fit of the mixture analysis model can be appreciated by examining the

RMS of the data residuals at see These qu.antltll?S are aeimea as

RMS(data) ::::1 T

L:T t=l

1 T

T t=l

are mixture residual RMS

is small and exhibits no worrisome spatial pattern. Overall residuals are a little larger in

tissue, which is to be expected since the signal, measured by data RMS, is also highest there.

If the residuals at each time-point, Zi(t) - z;(t), were mean-zero Gaussian then one would

consider modeling the distribution of the mean square residual, RMS(residual)2 in terms of

a Chi-squared distribution[40J. The distribution of the residual mean square was compared

to a range of Chi-squared distributions with different degrees of freedom and it was found

a Chi-squared distribution with 6.0 degrees of freedom gave the best fit. Figure 9 shows a

histogram and a plot of the quantiles of the residual mean square and the quantiles of the

Chi-squared distribution. The latter is known as a qq-plot. The strong linearity (R 2 = .996

for the regression line), indicates the close agreement except in the right hand tail - the

residual mean square is somewhat heavier tailed than the Chi-squared approximation. Of

course one must keep in mind a standard caveat concerning residual analysis: At best

residuals point to areas where there may be substantial discrepances between the data and

the model. However, a satisfactory residual analysis never proves the scientific validity of

the model - models are always wrong but can sometimes be useful. Nevertheless, for the

data at hand, the mixture model has the desirable property that it accurately represents

the PET data while providing estimates of kinetic parameters in the physiologic range.

6 Discussion

We have developed and implemented two approaches for generating pixel-wise functional

metabolic images from dynamic PET data. The first approach treats each pixel as a separate

ROI. The second approach uses a mixture analysis model which approximates pixel-wise

TACs in terms of a linear combination of a number of underlying model sub-TACs. In a

realistic numerical simulation of an FDG protocol with the UW-PET tomograph, mix­

ture analysis approach was found to superior RMS performance characteristics.

mtxhue <LUCLlYl:>ll:> approach is also more eltlCl(:;nt COlllp,utatjion,'Lily

FDG is on the of 60 to 70 minutes on ,..",""ontl" a'va]Jlable

Several aspects of the mixture approach could be developed further. Perhaps the most

important area is the development of appropriate variance assessment tools; an estimate

without an assessment of its variance has limited inferential value. In general this prob­

lem is complicated by the need to propagate the variability in the raw tomograph count

measurements through the reconstruction and mixture analysis steps. vVe are currently ex­

ploring an adaptation of the bootstrap scheme of Raynor and Woods[20] for this problem.

The question of whether there is some other algorithm which has a better RMS response

characteristics is an open question. One possibility, motivated by the work of Carson[9],

Ruesman[26] and Ollinger and Snyder[33] for example, would be to implement the mixture

model directly in the projection (TDA) domain. We are examining this at present.

The underlying models for the radiotracers can be extended to incorporate a more accu­

rate representations of tracer transport[4] and biochemical transformation[6]. Even though

such models tend to be more computationally complex, because of the limited number

of sub-TACs, our experience is that this does not significantly impact the computational

efficiency of the mixture analysis approach.

Mixture analysis parametric imaging addresses some data analysis difficulties tradition­

ally associated with quantitation of PET data. At the moment, the method is being imple­

mented operationally for a set of radiotracers used to image cancer at our institution; [C­

11] thymidine for cell proliferation in lymphomas, lung cancer and soft tissue sarcomas[45]'

[C-ll] glucose and [F-18] deoxyglucose for glucose utilization in brain tumors[6, 47] and [F­

18]fiuoromisonidizole for tumor hypoxia[28]. This software will be made available to other

centers.

Acknowledgements

gerter()us advice and encouragement I

re!iereies were very helptlll

I am most grateful for

entire PET imaging group at

'JJ.'''''J.J.~.J.J.J., David and Mark

to substantial imDri:)Vt=m,enl;s on

ni,;'or"ihr of Wa,shing;ton..

Cornmients of

initial draft of this paper.

from

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Figure Legends

Figure 1

• Legend: 1. Modified six-region Shepp-Logan Brain Phantom. Time activity curves

for each region are shown in Figure 2.

Figure 2

• Legend: 2. Time activity curves for each region in the brain phantom as well as the

blood input function Cp(t). The tick marks on the time-axis indicate the reconstruc­

tion time bins.

Figure 3

• Legend: 2. MR estimates obtained for a count rate of 1.58 X 107• (i) Target MR

image. The image is blurred due to the limitation imposed by finite transverse detector

resolution. (ii) Mixture analysis estimate. (iii) Pixel-wise ROI analysis estimates.

Figure 4.

• Legend: 3. The GCV Function for the 1.58 x 107 count rate study. The dashed

line shows the GCV Function, the dotted line indicates the shows the numerator of

the GCV Function (WRSS) expressed as a percentge variance explained relative to a

J( 1 model. The GCV procedure chooses a value of J( = 10 in this example.

Figure 5.

• Legend: 4. Count Rate Behavior of the Percent RMS error for the MR parameter.

are the for regions 1-5 in Phantom. shows behavior of the

overallRMS. Least the model, see 9. are also shown.

Figure 6.

• Legend: 6. FDG Scan of a Glioma Patient. (i) X-ray scout image,

uptake image (7mm filter), and (iii) CT scan.

Figure 7.

Integated FDG

• Legend: 7. Images of the FDG Model Parameters obtain by Mixture Analysis. (i)

MR = k~~k(3' (ii) Blood Volume (Iv), (iii) ](1, (iv) kz, (v) k3 , and (vi) k4 . The time

delay was was also computed but is not shown. The GCV technique identified ]( = 6

components for the mixture model.

Figure 8.

• Legend: 8. Residual Analysis. (i) RMS(data) and (ii) RMS(residual).

Figure 9.

• Legend: 9. (i) Histogram of RMS(residual)2 and (ii) a QQ-Plot versus a Chi-squared

distribution with 6.0 degrees of freedom, see text.

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,- .s:::::.0

a

90'0 90'0 vO'O 80'0 GO'O 0'0

coa0

LOaa

'V ("f_

a CCIa ::::l

-0'en('I') ~a ---a (j)

:2C\I

a:a0

,-

Table 1: Regional Model Parameters for Data Simulation

,... u. ' -l DA",i~~ R",,,.inn 1< ",,,.iAn R "'''''An n _=.L 0 ~ ~

(pixels)

T (seconds) I 0.00 10.2 10.2 10.2 10.2 I 0.00

I Iv I0.00 0.35 I 0.58 0.50 I 0.31 0.44!

I

I /(1 (gm/(ml min) ) II II

0.00 0.10 0.20 I 0.24 0.14 I 0.13

k 2 (min-I) II0.00 0.23 0.30 0.32 I 0.13 I 0.10I I

k3 (min-I) 0.00 I 0.03 I 0.05 0.07 I 0.08 0.11 I

I k4 (min-I) 0.00 I 0.0098 0.0075 0.0098 0.0106 I 0.0084

MR = K)k3 I 0.00 0.0115 0.0286 0.0430 0.0.533 I 0.0681k2+ k3 I I

2: K

Table 3: PPT'C'PTlt

Pixel-wise ROI]

Performance Characteristics for 1.58 x 107VVCLtLtO Mi)ctUJ,e fiUd,llY1:H:;;

.,.... DArt':An DA~:~~ DA~;A~ RplYlrm Dl>,.,.;~nJ. 0,1 u ~ ~

T 47: 99 13 : 10 11: 14 43 : 21 26 : 35 28 : 4

fv -39 : 99 -16 : 35 3: 13 70 : 124 -41 : 42 I -5 : 23I

J(l 26 : 99 15 : 9 4: 7 -5 : 10 0: 3 I 8: 18 II

I , I IIk2 -16 : 99 I 8: 1 2: 4 14 : 6 i -11 : 20 -6 : 23I

i

IIk3, -22 : 96 -12 : 20 -21 : 27 -27 : 21 -20 : 38 -20 : 41I I I -17 : Ik4, -55 : 90 -12 : 27 25 -26 : 34 4: 52 -21 : 46

MR 17 : 92 -1 : 14 10: 27 -11 : '"7 -6 : 19 I - 2: 32I

Table 4: Percent RMS Performance Characteristics for 1.58 X 107 Counts fMixture Analvsis:, v

Pixel-wise ROI]

Table 5: Percent RMS Dependence on Counts for the Mixture Analysis( .1:1' see equation 9)

T\ RAO'iAn R AO'iAn R AO'lAn R AO'iAn R"'O'lAnr <1>1 u u u u u

I

I 36 : 0.46 , 33 : 0.42 I 36 : 0.23 IT 45 : 0.12 22 : 0.21 24 : 0.15I

Iv 30 : 0.30 31 : 0.38 I .52 : 0.28 84 : 0.33 I 78 : 0.2.5 I .59 : 0.29 II

I I

I J(I 20 : 0.35 13 : 0.28 17 : 0.04 13 : 0.32 19 : 0.18 I 17 : 0.22 I

k2I 25 : 0.16 19 : 0.24 22 : 0.12 28 : 0.29

I

38 : 0.11 28 : 0.16 II I

I k3 I 17 : 0.08 I I29 : 0.18 ! 21 : 0.18 I13 : 0.14 16 : 0.12 I 22 : 0.41,

I

II

34 : 0.1.5 I 28 : 0.18 IIk4 I 42 : 0.11 12 : 0.20 13 : 0.19 I 23 : 0040I .

20 : 0.13 I II IIMR 6 : 0.20 10 : 0.19 I 8:0.40 12 : 0.15 12 : 0.18 II

I