imaging radiotracer model parameters in pet: a mixture
TRANSCRIPT
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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-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