Estimating Variability in Functional Images Using a Synthetic Resampling Approach
Ranjan Maitra and Finbarr O’Sullivan
Statistics
and Data Analysis Research Group, Bellcore, Morristown, NJ 07960-6438, USA
Department of Statistics, University of Washington, Seattle WA 98195, USA
Abstract
Functional imaging of biologic parameters like in vivo
tissue metabolism is made possible by Positron Emission
Tomography (PET). Many techniques, such as mixture
analysis, have been suggested for extracting such images from
dynamic sequences of reconstructed PET scans. Methods
for assessing the variability in these functional images are
of scientific interest. The nonlinearity of the methods used
in the mixture analysis approach makes analytic formulae
for estimating variability intractable. The usual resampling
approach is infeasible because of the prohibitive computational
effort in simulating a number of sinogram datasets, applying
image reconstruction, and generating parametric images
for each replication. Here we introduce an approach that
approximates the distribution of the reconstructed PET images
by a Gaussian random field and generates synthetic realizations
in the imaging domain. This eliminates the reconstruction steps
in generating each simulated functional image and is therefore
practical. Results of experiments done to evaluate the approach
on a model one-dimensional problem are very encouraging.
Post-processing of the estimated variances is seen to improve
the accuracy of the estimation method. Mixture analysis is
used to estimate functional images; however, the suggested
approach is general enough to extend to other parametric
imaging methods.
I. I NTRODUCTION
The potential to quantitate tissue metabolism from a
sequence of PET scans is one of the most powerful features of
this imaging modality. In this context, the protocol typically
consists of injecting a patient with a radio-tracer and recording
the emissions at discrete time-points. From emissions recorded
at each time-point, the tissue isotope concentration or source
distribution is estimated, giving us a time-course sequence
of reconstructed PET scans. These scans form the input
for algorithms that output pixel-wise estimates of biologic
parameters like metabolic rate, phosphorylation ratio, etc. A
number of techniques have been developed to generate these
images; we have been using a mixture analysis technique
(O’Sullivan [12]) in our experiments.
Assessing variability in these estimated functional images
is of scientific importance because these can potentially be
used to develop inference tools like calculating significance
levels of tests of hypothesis on biologic activities in different
regions in single-patient studies. The problem of developing
practical variability measures for reconstructed PET scans at
fixed time-points has been studied extensively ([3],[7],[8]).
Blomqvist et al. [2] noted the desirability of extending these
results to functional images. Unfortunately, the nonlinear
formulations used in constructing the biologic parameter
estimates make analytic variance formulae intractable.
Extending the resampling approach of Haynor and Woods
[7] would involve simulating a number of sinogram data sets,
applying image reconstruction, mixture analysis and generating
parametric images. This approach is impractical because of the
excessive computational effort required to replicate dynamic
reconstructed PET sequences.
In this paper, we suggest a simulation approach via the
parametric bootstrap [5] executed in the imaging domain.
We use the result in Maitra [11] that with increased count
rate, each reconstructed PET scan has an approximate
multivariate Gaussian distribution. The mean is estimated
by the reconstructed image. Computationally feasible and
accurate dispersion estimates are suggested. This model is
used to simulate dynamic PET sequences, from each of which
biologic parameters are extracted. This yields a bootstrap
sample of the functional images, which can be used to assess
variability. The advantage of this synthetic approach over the
usual one is that it eliminates the computationally expensive
step of reconstructing time-course sequences after simulating
from the observation process.
In the sequel, Section 2 formulates the problem and
outlines the theory and develops the methodology behind our
approach. Section 3 details the experimental evaluations that
were carried out to examine the performance of our suggested
approach. Since it is not possible to validate our methods in
a two-dimensional PET setup, the suggestions are evaluated
on experiments performed in a model one-dimensional
deconvolution problem with reconstruction characteristics
similar to PET. The results are presented in Section 4. Finally,
Section 5 summarizes the contributions of this paper and poses
questions for future research.
II. THEORY
AND
M ETHODS
A. Problem Formulation
1) Image Reconstruction
The standard reconstruction methodology for PET is
an algorithm known as filtered backprojection (FBP). In
convolution form, this method involves filtering of the data
from each projection angle followed by back-projection. The
equation for the ’th reconstructed pixel value
!"$#&%('
is,
(1)
&#*),+-%/.10 32
65
Here, % denotes angle, 0 distance, 42
is the corrected
sinogram data and 78 is the convolution filter with resolution
size (FWHM) 9 .
In matrix notation, the reconstruction equation can be
written in terms of the expression,
: <;>=?;@BA
(2)
;>=C2
where 2 is the vector of corrected projection data, ; is the
discretized version of the Radon transform, and : represents
the smoothing operation of FWHM 9 that is applied to the raw
reconstructions in order to obtain acceptable solutions [13].
2) Functional Imaging via Mixture Analysis
Local tissue metabolism has usually been assessed from
dynamic PET scans by modeling locally averaged time-course
measurements [15]. Functional imaging techniques, like
mixture analysis [12], generate more comprehensive pixel-wise
representations.
Let 3D represent the true source distribution in the D ’th
time-bin
at the ’th pixel in the PET5 imaging domain. The
vector 78 4 EDF*D HG IJKKLK*M is called the true timeactivity curve (TAC) at the ’th pixel. A ; -component mixture
model represents the ’th pixel TAC as a weighted average of ;
underlying curves (sub-TACs), NLOJP QG BI
LKKLK*; .
R
3D (3)
3ONLO3D
OTS $U
R
3O
O S [U
6
W X OTZ
W
(4)
W X O6Z
3 O
O S U
T
3) Assessing Variability
(6)
W
Analytic expressions for Var( ) are intractable because of
the nonlinear methods used in the extraction. The prohibitive
cost of generating time-course sequences from realizations in
the observation domain makes the usual resampling approach
impractical. This suggests the need for development of variance
estimation strategies.
B. A Synthetic Variability Estimation Strategy
1) Approximate Distribution of
Maitra [11] shows that under idealized projection conditions
of no detector effects such as scatter, attenuation,
etc., the
at any fixed
distribution of the reconstructed PET scan
time-point can be approximately and adequately specified
by a multivariate Gaussian distribution. The mean of this
distribution
is : while From (2), the dispersion matrix b of
is given by,
5
where the mixing proportions 4 3O FP lie
G IJKKLK*;
U
in the ; -dimensional simplex. The
physical basis for such
a representation is that the sub-TACs (N ’s) correspond to
the different tissue types represented in the image and the
underlying <V ’s indicate the anatomic tissue composition of
U
the underlying
pixel.
Functional
imaging
maps a metabolic parameter of interest,
W
, at each pixel in the image.
The mixture analysis approach
WYX
fits the metabolic parameter O6Z to each tissue sub-TAC NLOJ78
and following (3) regards each pixel biologic parameter as a
composition of the component tissue parameters,
W
The problem of estimating N ’s, given the ’s, is a lowdimensional problem and usually robust to theU choice of the
estimation method. On the other hand, the dimensionality of
the 3O ’s is high and so the estimation problem is delicate.
U
Many
methods have been proposed: among them is a quadratic
(weighted) least-squares algorithm which constrains 3O ’s to
U
belong to the ; -dimensional simplex.
WYX O6Z
The tissue metabolic parameters
’s are estimated from
the NLO78 ’s and the pixel metabolic parameters are estimated
following (4),
R
b
c: <;d=,;eA
;>=
Var ?2f];g<;>=?;@BA
: (7)
Reconstruction in PET is practical because Fast Fourier
Transforms are used in the implementation. This is not readily
possible in the case of (7). So, we need approximation methods
whereby reconstruction-type convolution procedures can be
used in calculating dispersion.
Theoretically, using the Poisson nature of the observed
counts, one can develop exact formulae for the variances of
the reconstructed pixel values directly from (1) and obtain
unbiased estimates by repeating the reconstruction
procedure
after replacing the kernel 78 with its square &*78 .
h
jVar
i
< k !"$#f%l'nm!#)?+l%/.e0 32
(8)
W
In estimating the ’s, the data
are the
time-course sequence
5
of reconstructed PET scans
.
4
3DF*D \G BI
LKKLK]M
The number of tissue types, ; , the sub-TACs NLO&*7^ , and the
mixing proportions 3O ’s have to be determined. ; is obtained
U
from anatomic considerations
or through clustering or other
sophisticated algorithms ([12],[14]). Estimation of N ’s and ’s
U
are usually done alternately to fit the model,
3D`_
R
3ONLO3DBF\D aG IJKLKKM-K
O S [U
6
(5)
This was suggested by Alpert, et. al. [1]. In practice,
(1) is implemented via discrete convolution through Fast
Fourier Transforms.
Interpolation steps are required
during backprojection. Studies show that ignoring these
over-approximates variances and can be corrected with
negligible added computational effort ([9],[10]).
To approximate correlations, notice that if the variances of 2
are assumed uniform, (7) reduces to,
Var K
ab o
q
Hp : <;
= ;@ A
: (9)
where q p is the assumed common variance of the observables.
r
The approximately
Toeplitz/Fourier form of ; = ; (and hence,
of : <; = ;@ A : ) means that the correlation between the ’th
and s ’th reconstructed pixel values is,
u
where
y$
G KLKK
sxw y$
|{!
(10)
5
= ;@ A
4
u
: .
b
|;
h :
u
= ;@ A
: h
]y[
15
is the first row of
u
u
(This correlation
structure
is equivalent to that
developed by Carson, et al. [3] and may also be regarded as a
quick and ready implementation for his approach.) Writing
}
h
h h h~
LKKK
, the estimated dispersion matrix is,
diag : |;
z
(11)
The approximately Toeplitz form of the correlation matrix also
means that discrete convolution via Fast Fourier Transforms
may be used in simulating from the multi-Gaussian distribution.
C. Synthetic Bootstrap for Estimating Variances
Since the reconstructed PET images are independent over
time, the distribution of a time-course PET sequence can be
approximated by a Gaussian random field. This suggests
development of a practical variance estimation strategy via
resampling in the imaging domain. The exact implementation
is as follows:
1. Obtain time-course reconstructed images of radio-tracer
uptake from the PET study. Also, obtain the variance
estimates of the reconstructed pixel values for each
of these scans and the approximate spatially invariant
Fourier correlation structure. From this reconstructed
PET sequence, obtain a functional image of the estimated
pixel-wise tissue biologic parameters.
2. Simulate from a Gaussian random field with mean
estimated by the above reconstructed time-course
sequence. The spatially invariant correlation structure
means that Fourier methods can be used in the simulation
of correlated multivariate Gaussian realizations. From
each simulated PET sequence, obtain pixel-wise
simulated images of the desired biologic parameters.
3. Estimate variability of the estimated functional image
from this synthetic bootstrap sample.
The suggested variance estimation strategy is practical because
it resamples in the imaging domain and thus eliminates the
cumulative computational burden of the many reconstruction
steps that would be needed in an extension of the resampling
techniques in Haynor and Woods [7].
III. EXPERIMENTS
A. One-dimensional Convolution Model
Experiments were conducted to assess the performance of
the suggested approach in estimating the pixel-wise variances
Activity
10
5
Bw .
0
&
&
t vu
Corr W
of the ’s. Since it was not possible to estimate the true
variances in a two-dimensional PET setup under existing
computer resources, evaluations were done in a simplified
one-dimensional deconvolution setting [13] with characteristics
matching PET reconstruction. A 6-component mixture
model
WYX O6Z
was specified. In this set-up, N ’s (and hence
’s) were
assumed known. The source distribution 78 (Figure 1) was
specified using (3) with mixing proportions ( 3O ’s) that are
U
blurred step functions [4].
Time activity curves over 60 time-points were reconstructed
at 216 bins (pixels) from realizations of a inhomogeneous
6ˆ 0
5‡ 0
4† 0
t‰ im3… 0
e b „2
in 0

€
150
€ 0
er
b
0
m
1 nu

l
50 ‚ pixe
1ƒ 0
Fig. 1 Perspective plot of the source distribution
experiments.
Š&‹]ŒC
200
used in the
Poisson process in the observation domain [4].
The
reconstructions were smoothed by a Gaussian kernel with
bandwidth preset to correspond to smoothing parameters
that are reasonable for the given total expected number of
emissions. Since as explained earlier, most of the variability
is in the estimation of the mixing proportions, the component
WYX O6Z
’s) were known.
sub-time activity curves N ’s (and hence
WYX O6Z
The relationship between N O ’s and
’s was specified by the
equation
NLO&ED W X O6Z6Ž&
WYX OTZ
5
4 .
9O6D
F
P
QG BI
LKKK‘JK
(12)
N O y$ . This is called the “amplitude
This implies that
parameter”. 9 O is another functional parameter (the “half-life”)
but this parameter was
not of interest in this experiment. The
source distribution *7^ (Figure 1) was specified using (3) with
mixing proportions ( 3O ’s) that were blurred step functions [4].
TheWYtarget
functionalU parameter was defined using
the ’s and
X O6Z
U
the
’s in (4). The 3O ’s were estimated from 78 and
used
W
U
to obtain ’s. Figure 2 is a plot of the functional parameter —
the “amplitude” — along with a sample estimate obtained using
the mixture analysis approach.
1000 simulated reconstructions of the
TAC were obtained
W
by simulating
the observed process and ’s were extracted from
W
each 78 . Sample pixel-wise standard deviations of these ’s
are assumed to be the truth in our performance evaluations.
Realizations were simulated from the
approximate
multi-Gaussian
model for the estimated TACs 78 . Bootstrap
W
samples of ’s were obtained as outlined in Section 3.1.2 and
standard deviations calculated. The experiment was done
15
10
0.4
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•••
••
•
• •••
•
••
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•
•
•
•
0
0.1
5
Standard Deviation
0.2
0.3
amplitude
•
0
50
100
pixel number
150
200
›
Fig. 2 True amplitude (broken line) and a sample estimate (bold line)
obtained using mixture analysis.
with bootstrap sample sizes ’ =10, 30 and replicated 500
times in order to study the distributional properties of these
bootstrapped standard deviations.
The above experiments were performed for low ( G K8y[I”“ G y[• ),
medium
( IJK8y$–`“ G y[• ) and high (—&K G “ G y$• ) expected total counts,
˜™
3D . Corresponding bandwidths for the smoothing kernel
were set at 9.7, 8.4 and 7.8 pixels. (Different expected total
counts can be interpreted as different dosage levels of the radiotracer.)
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200
Fig. 3 True standard deviation (broken line) and its unsmoothed
(points), and smoothed (bold line) bootstrap estimates (10 bootstrap
replications).
absolute biases are not altered appreciably as a result of the
smoothing; however, the variability measures are considerably
improved. It is observed that the bias and error rates do not
differ appreciably for different total expected counts. However,
as expected, the error rates decrease with increasing bootstrap
sample size.
IV. R ESULTS
V. D ISCUSSION
The percent relative absolute bias, averaged over pixels
was about 4-5% for all count rates and bootstrap sample sizes.
Figure 3 shows a set of pixel-wise bootstrapped standard
deviation estimates (points). Here ’ =10. The estimates were
post-processed by smoothing with the variable-span smoother
of Friedman [6] which uses a local cross-validation scheme
to obtain the smoothing parameter. The smoothed estimate
(Figure 3, bold line) is shown to give a better fit. Variability of
the estimates was measured by the average, over pixels, of the
mean percent absolute error in estimating standard deviation.
Table 1 summarizes the bias and the variability measures of the
estimated bootstrap standard deviations. The percent relative
This paper suggests a practical approach towards variability
assessment in functional images. Preliminary results reported
here are very encouraging. In our evaluations, we have
concentrated on the problem of estimating pixel-wise variances
of functional images obtained using the mixture analysis
approach; the technique can be easily extended to assess other
variability measures like correlations. Also, the method can be
applied to functional imaging techniques other than mixture
analysis.
Table 1
Bias and variability measures for smoothed bootstrap standard
deviation estimates over different total expected counts and bootstrap
sample sizes. The bias measure is the percent relative bias averaged
over pixels and the variability measure is the mean relative percent
absolute error in estimating standard deviations averaged over pixels.
Corresponding measures for unsmoothed estimates are in parenthesis.
Counts
( “ G y$• )
1.02
Bias
10 rep
30 rep
4.6 (5.6) 4.8 (3.7)
Variability
10 rep
30 rep
17.3 (27.7) 14.0 (21.6)
2.05
5.1 (3.8)
5.1 (5.4)
16.1 (27.6)
12.9 (21.3)
4.10
5.1 (5.3)
4.9 (3.7)
15.5 (27.3)
12.3 (21.0)
A number of issues remain to be addressed. In smoothing
our synthetic bootstrap standard deviation estimates, the
variable-span fitting algorithm of Friedman [6] was used to
obtain the resolution size (FWHM) of the smoothing filter.
This strategy chooses the smoothing parameter adaptively
by local cross-validation. As a result, the estimates may
be under-smoothed in the presence of positively correlated
variates. Hence, the obtained error rates may potentially
be decreased by a more sophisticated choice of smoothing
parameter. Another question of interest is determining the
number of bootstrap samples. We also need to evaluate
this scheme in the context of two-dimensional PET images.
While it does not seem possible to do a complete evaluation
given existing computational resources, it may be possible to
evaluate the performance in a limited frame-work. Hence,
while this seems a promising new technique towards variability
estimation in functional images, a number of issues remain to
be investigated.
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