Robustness of anatomically guided pixelbypixel algorithms for partial volume effect correction in positron emission tomography.
ABSTRACT Several algorithms have been proposed to improve positron emission tomography quantification by combining anatomical and functional information in a pixelbypixel correction scheme. The precision of these methods when applied to real data depends on the precision of the manifold correction steps, such as fullwidth halfmaximum modeling, magnetic resonance imagingpositron emission tomography registration, tissue segmentation, or background activity estimation. A good understanding of the influence of these parameters thus is critical to the effective use of the algorithms. In the current article, the authors present a monodimensional model that allows a simple theoretical and experimental evaluation of correction imprecision. The authors then assess correction robustness in three dimensions with computer simulations, and evaluate the validity of regional SD as a correction performance criterion.
 [Show abstract] [Hide abstract]
ABSTRACT: Summary Aim: We assess the influenc eo ft he reconstruction algo  rithm s( OSEM for th ei terative on ev s. af iltered backpro jection in Fourie rs pace (DiFT)) on partial volume correction in PE Te mployin gaf ully 3D 3compartment MR based PV correction algorithm. Th eg ray matter voxel si nt he PE T imag e afte rr emoval of the whit em atter an dc erebrospi na lf luid contribution a re corrected voxelbyvoxel usin g the imag er esolution . Material, methods: Phantom measurements an do ne healthy human brain FD Gs tudy were carrie do ut .F or the OSEM reconstruction ,ac om bination of iteration step sa nd subset numbers (It/Sub )w as used, whereb yi nc ase of noconvergenc et he imag er esol ution ha dt ob ef itted. The result s from the DiF Tr econstruc tion wer ee quivalent to those obtaine df rom the OSEM re  construction with 10/32 combination fo ro bject sw it hw ide  spread activity concentration .F or the sphere phantom, the mean recovery based on the actual value sa chieve d9 9.2% ±1 .8 fo ra ll spheres an da ll reconstruction mode sa nd It / sub combinations (except fo r2 /8) .I nc ase of th eH offman 3D brain phantom th em ea nr ecovery of the cortica lr egion s wa s1 01% ±1 .2 (the increas eb ase do nt he uncorrected values: 35.5% ±1 .5), while the subcortical region s reached am ea nr ecovery of 80% with an increase of 43.9 %±2 .5. For the human data, an increas eo ft he metabolize dv alue so fs everal cortical region sr ange db e tween 42% an d4 8% independent from the reconstruction mode . Conclusions: Ou rd ata show that th e3 compart  ment fully 3 DM Rb ase dP Vcorrection is sensitive to the choice of reconstruction algorithms an dt ot he parameter choice. They indicat et ha td espite improved spatial resol ution ,t he use of the iterative reconstruction algorith mf or PVcorrection result si ns imilar recovery factors when com pared to ac orrection usin gD iF Tr econstruction ,i nsofar the imag er esolution value sa re fitted at th eI t/Sub com binations.  SourceAvailable from: ieeexplore.ieee.org[Show abstract] [Hide abstract]
ABSTRACT: The ability to correctly quantify activity concentration with single photon emission computed tomography (SPECT) is limited by its spatial resolution. Blurring of data between adjacent structures, which is known as partial volume effects, can be compensated for by utilizing high resolution structural information from other imaging modalities such as CT or MRI. Previously developed partial volume correction (PVC) methods normally assume a spatially invariant point spread function. In SPECT this is not a good approximation, since the resolution varies with the distance from the collimator. A new method, pPVC, was developed in this paper, which takes into account the distance dependent blurring. The method operates in projection space and is combined with filtered backprojection (FBP) reconstruction. Results from simulations show that similar quantitative results could be obtained with pPVC+FBP as with OSEM with resolution recovery, although with better structural definition and an order of magnitude faster.Tsinghua Science & Technology 02/2010; 15(1):5055.  SourceAvailable from: Isabella Castiglioni[Show abstract] [Hide abstract]
ABSTRACT: We have developed, optimized, and validated a method for partial volume effect (PVE) correction of oncological lesions in positron emission tomography (PET) clinical studies, based on recovery coefficients (RC) and on PET measurements of lesiontobackground ratio (L/B m ) and of lesion metabolic volume. An operatorindependent technique, based on an optimised threshold of the maximum lesion uptake, allows to define an isocontour around the lesion on PET images in order to measure both lesion radioactivity uptake and lesion metabolic volume. RC are experimentally derived from PET measurements of hot spheres in hot background, miming oncological lesions. RC were obtained as a function of PET measured spheretobackground ratio and PET measured sphere metabolic volume, both resulting from the thresholdisocontour technique. PVE correction of lesions of a diameter ranging from 10 mm to 40 mm and for measured L/B m from 2 to 30 was performed using measured RC curves tailored at answering the need to quantify a large variety of real oncological lesions by means of PET. Validation of the PVE correction method resulted to be accurate (>89%) in clinical realistic conditions for lesion diameter > 1 cm, recovering >76% of radioactivity for lesion diameter < 1 cm. Results from patient studies showed that the proposed PVE correction method is suitable and feasible and has an impact on a clinical environment.BioMed research international. 01/2013; 2013:780458.
Page 1
lournal qf Cerebral Blood Flow and Metaholism
19:547�559 © 1999 The inlemalional Society for Cerebral Blood Flow and Metabolism
Published by Lippincott Williams & Wilkins, Inc., Philadelphia
Robustness of Anatomically Guided PixelbyPixel Algorithms
for Partial Volume Effect Correction in Positron
Emission Tomography
Daniel Strul and Bernard Bendriem
Service Hospitalier Frederic foliot, CEA, Orsay, France
Summary: Several algorithms have been proposed to improve
positron emission tomography quantification by combining
anatomical and functional information in a pixelbypixel cor
rection scheme, The precision of these methods when applied
to real data depends on the precision of the manifold correction
steps, such as fullwidth halfmaximum modeling, magnetic
resonance imagingpositron emission tomography registration,
tissue segmentation, or background activity estimation, A good
understanding of the int1uence of these parameters thus is criti
A major limitation of positron emission tomography
(PET) is the finite spatial resolution of PET scanners,
Activity concentration is systematically underestimated
or overestimated in objects smaller than two to three
times the fullwidth at halfmaximum (FWHM) of the
device point spread function (PSF), Indeed, because of
image smoothing by the device PSF, the activity in a
structure partly spreads into the nearby tissues ("spill
out" effect), whereas the neighborhood activity partly
contributes to the structure measurement ("spillin" ef
fect), This socalled partial volume effect (PVE) is glo
bally characterized by the recovery coefficient, defined
as the ratio of the measured over the true activity con
centration (Hoffman et aI., 1979; Mazziotta et aI., 1981;
Kessler et aI., 1984; Bendriem et aI., 1991; Kuwert et aI.,
1993).
Several algorithms have been proposed for restoring
Received April 3, 1998; final revision received August 10, 1 998;
accepted August 1 3 , 1 998.
Address correspondence and reprint requests to Dr. Daniel Strul,
Clinical PET Centre, Lower Ground Floor, Lambeth Wing, S1. Thom
as' Hospital, Lambeth Palace Road, London SEI 7EH, U.K.
Abbreviations used: APA, anatomically guided pixe1bypixel algo
rithm; BRC, bias reduction coefficient; FWHM, fullwidth at half
maximum; GM, gray matter; MRI, magnetic resonance imaging; NT,
neighbor tissue; TT, target tissue; PET, positron emission tomography;
PSF, point spread function; PVE, partial volume effect; ROT, range of
interest; VOl, volumes of interest; WM, white matter.
547
cal to the effective use of the algorithms, In the current article,
the authors present a monodimensional model that allows a
simple theoretical and experimental evaluation of correction
imprecision. The authors then assess correction robustness in
three dimensions with computer simulations, and evaluate the
validity of regional SD as a correction performance criterion,
Key Words: Positron emission tomographyPartial volume
effect correctionConstrained image restorationData fu
sionGray matter positron emission tomography,
the images on a pixelbypixel basis. Because of the ill
conditioned nature of the restoration problem, some sort
of regularization must be used to avoid highfrequency
noise amplification (Natterer, 1988; Haber et aI., 1990;
Links et a!., 1992; Chan et aI., 1997). With the increasing
resolution of computed tomography and magnetic reso
nance imaging (MRI), a family of algorithms has been
proposed, where regularization is achieved by incorpo
rating anatomical information and tissue homogeneity
constraints in a noniterative pixelbypixel correction
scheme. These algorithms include the binary atrophy
correction (Videen et a!., 1988; Meltzer et aI., 1990), the
gray matter (GMPET) method (MullerGartner et aI.,
1992), and the fourtissue correction (Meltzer et aI.,
1996). In this report, we refer to these algorithms as the
anatomically guided pixelbypixel algorithms (APA).
Since they were first proposed, the AP A have gener
ated an increasing interest, and several improvements or
variants have been published (Kosugi et aI., 1996; Labbe
et aI., 1996; Knorr et aI., 1997). However, the impreci
sion of the AP A preprocessings, such as MRI segmen
tation or MRIPET registration, significantly affect the
corrected image and may make correction reliability
questionable (Fahey et aI., 1996; Strul and Bendriem,
1996; Yang et aI., 1996; Meltzer et aI., 1997).
This article addresses several issues of AP A correction
robustness. We first briefly formulate the overall math
Page 2
548
D. STRUL AND B. BENDRIEM
ematical frame of AP A correction and show the main
sources of instability. We then present a monodimen
sional model, which allows a simple theoretical and ex
perimental evaluation of correction imprecision. Correc
tion robustness in three dimensions then is assessed with
computer simulations. Finally, we examine the useful
ness and the limitations of regional SD as a criterion for
GMPET correction performance.
THEORY
The anatomically guided pixelbypixel algorithm
correction formulas
In APA correction, an anatomical image (from MRI or
computed tomography) is subdivided by segmentation
into several tissue classes, which do not need to be either
contiguous or homogeneous. For each tissue, a separate
distribution image is derived from the segmented image,
with values of 1 in all the pixels that belong to the tissue,
and 0 everywhere else. The AP A correction restores the
true pixelbypixel activity concentration ATT for one
tissue only, named here the target tissue (TT). The other
classes, for which no restoration is performed, are termed
neighbor tissues (NT). Each NT is assumed to be homo
geneous, with a mean activity concentration ANT'
The observed image lobs is written as the sum of all the
tissue contributions, where each tissue contributes ac
cording to its activity concentration and to its distribution
function:
lobs = ( 2: AtissueXtissue) *h
all tissues
where * signs the convolution operator and "h" is the
PET image PSF.
The AP A correction assumes that the TT is at least
approximately homogeneous (MUllerGartner et al.,
1992), so that
(2)
A corrected estimation CTT of the activity concentration
ATT is obtained by combining equations (1) and (2):
(3)
This correction can be understood as a twostep restora
tion process. First, the spillin from the NT is subtracted
from the TT measurement. Second, the l/(XTT*h) term
acts as an amplification coefficient, which compensates
for the loss resulting from the partial spillout of TT
J Cereb Blood Flow Melab, Vol. 19, No.5, 1999
activity to the NT. The correction must be limited to
areas where (XTT*h) is much greater than zero, to mini
mize noise amplification.
The first APA procedure was the binary atrophy cor
rection, which relied on only one radioactive compart
ment, comprising all of the brain tissue voxels (Videen et
al., 1988; Meltzer et al., 1990):
Cbrain tissue = IObS/ (h*Xbrain tissue)
The GMPET algorithm improves correction by separat
ing white matter (WM), GM, and CSF (MUllerGartner
et al., 1992):
COM = [lobs h*(AwM XWM + ACSFXCSF)] / (h*XoM)
(4)
(5)
The fourtissue correction is a twostep extension of
GMPET (Meltzer et al., 1996). The image is first cor
rected using GMPET equation, then a focal hypoactive
or hyperactive GM area is set apart from the rest of the
GM as a specific TT:
CTT = [lobs h*(AGM XOM
+ AWMXWM + ACSrXCSF)] / (h*XTT)
Factors of correction instability
Equation (3) shows that APA corrections strongly de
pend on the precision of all processing steps: segmenta
tion of the anatomical image, anatomicalfunctional reg
istration, sampling rate conversion, measurement of the
NT mean activity concentration, and evaluation of the
PSF. The limited precision of most of these procedures
may lead to significant correction instability. For ex
ample, the measurement of the mean activity concentra
tions in the NT often is biased by the TT spillout and
scattering, especially when NT TT contrast is important.
The precision of the tissue segmentation also is limited
by instrumental artefacts and by classification problems
affecting mixed voxels. The accuracy of the PSF model
generally is limited by the neglect of resolution nonuni
formity or scattering to shorten computation time. Last,
unless specially designed head holders and protocols are
used, the limited precision of current registration tech
niques further limits correction robustness.
The AP A corrections depend on numerous parameters,
and an exhaustive study of APA precision would lead to
complicated equations. Thus, we address this issue using
a simplified monodimensional model, valid both for
GMPET and binary corrections, which allows us to re
duce the number of free parameters. To keep generality,
we make assumptions only on the tissue maps used for
correction, with absolutely no assumption on the real
tissue map. This independence is necessary to treat bi
nary correction, where the binary tissue map used for
correction neglects the WMGM separation.
The parameters used for AP A correction are assumed
(6)
Page 3
ROBUSTNESS OF PVE CORRECTIONS
549
to follow the model presented on Fig. lAo The seg
mented TT is an active bar, of center Xo and halfwidth w,
thus extending between two locations XI
= Xo  w and x2
= Xo + w. It is inserted between two NT, NTI (activity
concentration AI' ranging from X3 to XI) and NT2 (activ
ity concentration A20 ranging from x2 to x4). The PSF
used for correction is a normalized Gaussian function,
whose width is specified by its SD (a), with FWHM =
2.35*a.
This model is approximately met in various common
situations, such as GMPET correction of a small cortex
area or an internal GM nucleus, where the NT may be
WM and CSF, respectively, or both WM. No distant NT
is taken into account in this model for reasons of sim
plicity, but the equations presented later are easily ex
tended to an arbitrary number of NT. The model may be
further simplified if the outer NT boundaries are distant
from TT (a few FWHM values are enough) by setting X3
"" oc and X4 "" +oc. This model also may apply to binary
correction by merging all of the active tissues (Fig. I B).
If we neglect the uncertainty on NT segmentation, the
correction now depends on only five parameter estima
tions: the activity concentrations A1 and A2 (if there are
any), the PSF SD a, the TT center location Xo (obtained
by registration) and the TT halfwidth w (obtained by
segmentation). Thus, by neglecting interparameter cor
relations, and by assuming the parameter imprecision to
be small, we can write:
This equation expresses the relationship between the pre
cision of the AP A parameters and the statistical precision
of the corrected image. This statistic is not directly con
nected to image noise but rather to correction reproduc
ibility: in any point where var[CTT] is high, APA cor
FIG. 1. Monodimensional
model. (A) General case (like
gray matterpositron emission
tomography [GMPETJ). The
target tissue (TT) is inserted
between two neighbor tissues
(NT). The influence of the tis
sues outside of the range X3 to
around the TT (e.g., they are
not active or they are distant
from more than 3 fullwidth at
either neglected like NT 2 or in
cluded in the target tissue like
NT1·
x4 is assumed to be negligible
halfmaximum [FWHMJ). (8)
Special case corresponding to
binary correction: the NT are
� ()
'"
�
8
�
oS
.2
IS '"
�
co
A
f�'j
Target
2w
I l�
X3
Xl
Xo
x (mm)
rection may yield different values from one correction
process to another. For noisy images, another term
should be added, corresponding to the image noise am
plification by AP A correction. Since correction FWHM
is initially fixed by the user and then used for all correc
tions, FWHM errors tend to introduce systematic rather
than random errors. Then, it was not included in equation
(7) but was investigated as a bias later in this report.
The expressions of the partial derivatives needed for
computation of the correction variance are summarized
below (see appendix for details of the derivation):
=(x) = (x) and =(x) =  (x)
aCTT
aAI
axo (x) =
(x) =
aw
P3 PI
PI P2
(AID)h3 [AID  N]hl +
[A2D  N]h2 (A2D )h4
(PI _ P2)2
[AID  N]h1 + [A2D N]h2
(P1  P2f
LlX3(AID)h3 LlxI[AID  N]hl +
Llx2 [A2D  N]h2 Llx4 (A2D)h4
a (PI _ P2)2
aCTT
aA2
P2 P4
PI P2
aCTT
(x)
aCTT
(x)
aCTT
�(x) =
(x)
where we use the following notations:
hex) == (I / ayl2;)exp( _x2 / 2(2)
Vi = 1 .. .4, Llxj == (x xJ, hJx) == hex xJ,
N(x) == Iobs(x) AI[P3(X) PI (x)]
A2[P2(X) P4(x)]
D(x) == PI (x) P2 (x)
P(x) == f�h(t)dt
PJx) == P(x xJ,
(8)
(9)
These equations are firstorder approximations of the ac
Neighbour 2
.
l (merging all active tissues) l
:
Binary la"get
2w
. Littleactive tisau ..
(neglected)
:
. :
t �
X
x'
2
4
B
.
:.
.
• l
x (mm)
J Cereb Blood Flow Metab, Vol. 19, No. 5, 1999
Page 4
550
D. STRUL AND B. BENDRIEM
tual error and variance functions and are thus limited to
small errors. Within their field of validity, these equa
tions might provide a foundation for the analysis of APA
correction errors and imprecision. In the following text,
their characteristic shapes are exemplified, and assessed
by comparison with computer simulations.
Quality criteria
The correction robustness is crucial for the effective
use of AP A algorithms. Brain PET imaging often is used
to discriminate between several groups of subjects such
as controls and patients. If the correction performances
differ greatly from one subject to another, the results of
the comparison may be affected and thus misleading. In
this regard, as is pointed out by several authors, absolute
quantification is less important than comparison signifi
cance (Di Chiro and Brooks, 1988; Strother et aI., 1991;
Kuwert et aI., 1993). Thus, the correction errors intro
duced earlier could be a major limitation to the practical
use of AP A corrections.
Given the complexity of MRIPET registration and
MRI segmentation, improving correction robustness
might be difficult. It may then be more relevant to define
correction quality criteria. Ideally, such criteria would
reveal erroneous corrections and help determine the
cause of the error. Correction parameters then could be
fixed, in a manual or automatic manner, to improve data
processing.
As shown later, correction errors can lead to large
artefacts, which severely decrease regional homogeneity
in the corrected areas with regard to errorfree correction.
This then suggests homogeneity as one possible crite
rion. In the material that follows, we assess the quality of
one criterion measuring homogeneity, the regional SD,
to bring into evidence both the utility and the limitations
of this approach.
MATERIALS AND METHODS
Monodimensional bar experiment
The APA correction artefacts and variance were explored in
the monodimensional case. A model consistent with the cor
rection of a short brain profile around the cortex was consid
  TIll e activity
GMPET
ered (Fig. 2A). The NTl, corresponding to the WM, was as
sumed to be wide enough to be approximated as semiinfinite
monly used in GMPET correction, the NT 2 (corresponding to
the CSF and bone) was assumed inactive and thus was ne
glected. The cortex was assumed to be 20mm wide, and the
contrast between cortex and WM was set to 4: I. Simulations
were performed according to the following equation (centered
on the middle of the cortex):
(x] was rejected toward (0) to simplify the analysis of the
correction artefacts. To be consistent with the hypotheses com
(10)
with AGM
= 4 nCi . mLl, Al = I nCi . mLI, w = 10 mm,
Xo
ing equations (4) and (5) (detailed in equations [15] and [21] of
the appendix). The tissue distribution used for GMPET cor
rection was the same as the one used for simulation, with the
TT being the cortex. For binary correction, the binary tissue
= 0 mm, and FWHM
Both binary and GMPET corrections were applied follow
= 9 mm.
map was obtained by merging the NT 1 and the cortex maps
(thus rejecting Xl toward _00. Various processing errors were
introduced in both corrections: inaccurate NT activity concen
tration estimates, incorrect FWHM, segmentation errors, and
registration errors, and the corrected image profiles were com
pared with those predicted by equation (8).
The pixelbypixel corrected activity expectation and SD
were computed by performing 10,000 correction tries in pres
ence of random parameter errors, extracted from Gaussian dis
tributions, with SD[Ad
and SD[w]
pared with those predicted by equation (7).
= 0.2 nCi . mLI, SD[xo]
= I mm,
= 1 mm. The experimental SD profiles were com
Implementation of anatomically guided
pixelbypixel algorithm correction
For the following experiments, we used a C implementation
of APA correction, which accepts any number of tissue classes
for noniterative processing. The correction PSF is the product
of a transaxial and an axial Gaussian functions. Sampling rate
conversion is achieved by trilinearly downsampling the tissue
images after PSF blurring. The correction mask is automati
cally created by downsampling the original TT tissue map, then
thresholding the resultimage at 0.5 to avoid large correction
amplification.
Simulation library
We used a simulation library in which any digital phantom
model is defined as a combination (addition, subtraction, or
embedding) of simple radioactive geometrical shapes, or primi
tives. The corresponding model activity function and primitive
Observed
   Binary correction
20
·10
o
10
20
A
x (mm)
B
J Cereb Blood Flow Metah. Vol. 19. No.5. 1999
,
·10
,
,
,
,
,.
10
x (mm)
20
FIG. 2. Model used for the
onedimensional experiment.
(A) True and observed activity
distributions. This model is
similar to the one presented on
NT 2 neglected. (8) Optimal
GMPET and binary correc
tions.
Fig. 1 A, with NT 1 semiinfinite
(X3 rejected toward (0) and
Page 5
ROBUSTNESS OF PVE CORRECTIONS
551
distribution functions are sampled on a highresolution image
grid to compute the activity concentration image and the primi
tive distribution images, respectively. A PET simulation is de
rived from the. activity concentration image using a convolu
tiondownsampling scheme: the activity image is first con
volved with the PSF model, then downsampled to the sampling
characteristics of typical PET images. Measurement volumes of
interest (VOl) are computed for any primitive by downsam
piing its distribution image, then thresholding the result, with
threshold
= 0.5.
Quantitative analysis
For the sphere experiments described later, all measurements
were performed inside six VOl covering the whole spheres.
Quantitative accuracy was measured by computing the VOl
regional bias, equal to the difference between the measured and
the real activity concentrations inside the VOL
To evaluate the quantitation recovery performed by APA
correction, a performance index, the bias reduction coefficient
(BRC), was defined as follows:
uncorrected image biasl
correcte Image las
_I
BRC 
d' b'
(I I)
The BRC index is used to determine whether APA correction
has improved or worsened PVE errors. Large BRC values
(» 1) show excellent quantitative recoveries. On the contrary,
a BRC lower than 1 indicates that AP A correction has in
creased the image bias.
As mentioned previously, we also measured the regional
image SD, computed for each VOl as follows:
( 2:(JIvol)/(N1)
where N is the number of voxels in the VOl, and I and lyOi are
the activity concentration observed in each voxel and in the
VOl (mean value), respectively.
SDyOJ =
( 12)
all VOl voxcls
Digital sphere phantom experiment
A digital model from a Data Spectrum Sphere phantom
(Data Spectrum Corp., Chapel Hill, NC, U.S.A.) was modeled
(Fig. 3). The model comprised six hot spheres (diameters = 9.5,
FIG. 3. The digital sphere phantom (medium slice).
12.7, 15.9, 19.1,25.4, and 31.9 mm, respectively; activity
1000 nCi/mL) embedded in a warm cylinder (diameter
200 mm, activity
produce highdefinition activity images (256*256* 128 voxels,
voxel dimensions
definition simulations from the activity images. The character
istics of these simulations were chosen to be similar to those of
ECAT 953/31 B images (128* 128*31 voxels, voxel dimensions
= 5 mm). We also derived six lowdefinition VOl
from the model, with the downsamplingthresholding scheme
described previously, each fully covering one sphere. We in
vestigated the quantitative influence on GMPET correction of
four error factors.
Influence of the neighbor tissue activity concentration mea
surement. We introduced inaccurate measurements of cylinder
activity concentration in APA correction, modulating the error
from 50% to +50% of the true value (i.e., 125 to 375 nCi/mL).
Influence of the modeling. Similarly, we used inaccurate
transaxial FWHM for correction, with error ranging from 30%
(FWHM
FWHM.
Influence of magnetic resonance imagingpositron emission
tomography registration. We first computed the PET simula
tion using the true phantom model, then displaced it (from 5
to +5 mm by steps of 0.5 mm), and used thc mispositioned
model to produce the tissue images used in the AP A algorithm
and to produce the measurement VOL We chose this procedure
to reproduce the current image analysis, in which VOl are
drawn on the anatomical images.
Influence of segmentation threshold. Since the most com
monly used segmentation approaches rely on the determination
of classification thresholds, the modification of these thresholds
leads to displacements of tissue boundaries. To assess the in
fluence of these global displacements, we added a fixed value
(from 1.25 to + 1.25 mm) to each sphere radius after simula
tion. We then derived the tissue images and the measurement
VOl from the modified model, as was done in the previous
investigation.
=
=
= 250 nCi/mL). We used this model to
= I * I * 1 mm). We then derived low
= 1.995* 1.995*3.375 mm, transaxial FWHM
FWHM
= 9 mm, axial
= 6 mm) to +30% (FWHM
= 12 mm) of the true
RESULTS
Monodimensional experiment
The corrected activity profiles are plotted on Fig. 2B.
For the GMPET correction, the correct activity concen
tration is fully recovered within the correction mask (the
GM area), whereas no correction is performed outside
this mask. For the binary correction, since the binary
tissue map merges the NT] and the GM, only the X2 edge
is corrected for PYE.
Profiles of the derivatives of CTT(X), as recovered with
GMPET method, are plotted on Fig. 4. They should be
understood as follows: when a partial derivative is posi
tive, any overestimation of the corresponding parameter
yields a positive artefact, and parameter underestimation
leads to a negative artefact. The direction of the artefacts
is inverted for negative partial derivatives. Notice that, in
the conditions that were chosen for computation,
1 nCi'mL1 corresponds to a 25% error, with regard to
the true GM activity concentration.
All correction artefacts appear as error functions
propagating from the TT edges toward the center. The
errors thus are maximum near the TT edges, except for
J Cereh Blood Flow Metah. Vo!' 19. No.5. 1999
Page 6
552
D. STRUL AND B. BENDRIEM
1.5
J
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U�
·10
·10
0.0
·0.5
·1.0
'1;
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A ·1.5
1.5
0.5
·0.5
J
.s
1.0
0.0
1.0
13
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d
""
x(mm)
·10
·5
10
FWHM=wl2
·FWHM=w
....... FWHM=2w
...FWHM = 3w
x(mm)
·10
·5
FWHM= w/2
·FWHM=w
....... FWHM=2w
._._.FWHM = 3w
10
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0.0
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o
FWHM mismatches (Fig. 4B). In the latter case, each
edgeerror function reaches its maximum (for small
FWHM) only at a distance from the edges nearly equal to
vanish at a distance (roughly 3 0' from the TT edges. The
ratio between the structure width and the FWHM thus is
critical: when the structure is overly small, the edge ar
tefacts tend to cover the whole structure, leading to a
global increase of the correction error. For FWHM, seg
mentation, and registration errors, the contrast is critical,
too, with the artefacts on Fig. 4B to D being distinctly
smaller on the Xl edge than on the opposite one.
The segmentation and registration errors share the
same basic artefact pattern, corresponding to the error
induced by the shift in the positions of the TT map
boundaries Xl and X2 (Fig. 4C and D). In qualitative
terms, there is a negative artefact wherever the correction
TT map overlaps the NT, and there is a positive artefact
wherever the correction TT map does not reach the real
TT boundary. Strikingly, the error magnitude on the TT
edges increases rapidly as the FWHM decreases. Indeed,
approximating the earlier equation for small FWHM
shows that the maximum error is inversely proportional
to the FWHM.
Figure 5 displays some typical GMPET correction
artefacts. The dashed lines represent the theoretical pre
dictions derived from equation (8). The agreement is
excellent for NT activity or FWHM errors (Fig. 5A and
B). For registration and segmentation errors (Fig. 5C and
the PSF SD 0'. For all errors, the correction artefacts
J Cereb Blood Flow Metab. Vol. 19, No.5, 1999
·5
·5

0
x (mm)
x(mm)
FWHM= w/2
·FWHM=w
.. • ... . FWHM=2w
_.FWHM = 3w
10
10
... 
·FWHM=w
...... ·FWHM=2w
_._._.FWHM = 3w
FIG. 4. Theoretical GMPET
partial derivatives profiles for
the onedimensional experi
ment model at various resolu
tions. (A) Derivative relative to
Al activity estimation. (8) De
rivative relative to errors on the
resolution SO. (C) Derivative
relative to registration errors.
(0) Derivative relative to seg
mentation errors. The deriva
tives are written in (nCi·mL 1)
of corrected activity per
(nCi·mL1) of NT activity on
(A), and in (nCi·mL1) of cor
rected activity variation per
millimeter of parameter varia
tion everywhere else. Compu
tations performed with AGM = 4
nCi·mL1, A, = 1 nCi·mL1, W
= 10 mm.
D), discrepancies appear, with the positive artefacts be
ing larger than the negative ones. Figure 5E and F ex
emplifies some combinations of segmentation and regis
tration errors: they compensate each other on one TT
edge and are added on the opposite edge, leading to
strong artefacts on this edge.
Similar artefact patterns appear for binary correction,
but only near the outer edge (Fig. 6). Since the Xl edge of
the binary map is neglected in this model, registration
and segmentation errors lead to identical artefacts (Fig.
6B and C), which may as well combine in addition (Fig.
6D) or in subtraction. Notice that binary correction al
ways neglects the activity of the tissues located outside
of the brain. If this assumption was erroneous, a small
edge artefact would appear (symmetrical to the one pre
sented on Fig. SA) and would combine with any other
correction error.
Figure 7 compares the measured pixelbypixel cor
rected activity SD with theoretical predictions. The
agreement is perfect when correction imprecision comes
from NT activity estimation imprecision (Fig. 7 A). For
identical registration and segmentation imprecisions, the
profiles are identical, and we have plotted only the reg
istration profiles. When the correction mask is ideally
located (matching the real TT), the measured imprecision
is only slightly superior to the theoretical one (Fig. 7B).
In real conditions, the correction mask is affected in size
and position by the registration and segmentation errors.
The experimental SD profile then is distinctly different
Page 7
FIG. 5. Correction artefacts in
GMPET method. (A) Inaccu
rate estimation of white matter
activity concentration (±O.2
FWHM estimation (±2 mm).
(C) Missegmentation errors
(±1 mm on each TT edge). (0)
mm). (E) Combination of mis
segmentation and misregistra
1 mm. (F) Combination of op
posite missegmentation and
misregistration errors. The dot
ted lines represent the theoret
ical predictions.
nCi·mL1). (8) Inaccurate
Misregistration errors (±1
tion errors, both + 1 mm or both
ROBUSTNESS OF PVE CORRECTIONS
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from the theoretical one (Fig. 7C), being both lower and
wider. If registration imprecision is increased to 2 mm,
or if lmm registration and segmentation imprecisions
are combined, the discrepancy is more noticeable (Fig.
7D), the experimental imprecisions being substantially
lower than the predicted values.
The influence of the correction mask results from the
asymmetry between inward TT edgeshifts, which ex
clude the edge points from correction, and outward TT
edgeshifts, which induce undercorrection errors. In both
cases, the corrected activity is lowered compared with
the true activity. Consequently, the imprecision is low
ered, but at the price of a negative statistical bias (Fig
8A). When the correction mask is fixed, there is a posi
tive statistical bias, since positive registration or segmen
tation artefacts are globally higher than negative ones
(Fig. 8B).
Digital sphere phantom
Figure 9 displays the performance of GMPET correc
tion for each sphere. When it was possible, the data have
been fitted with empirical models (first or second
degree polynomials and their inverse functions), and
missing BRC values were extrapolated.
Correction instability is greater for the smallest
spheres. In presence of registration and segmentation er
rors, this trend is amplified by their influence on the
correction masks. Segmentation is the most critical factor
contributing to the imprecision of correction, with the
maximum bias (+2350 nCi/mL) appearing for the 1.25
mm undersizing of the smallest sphere. The influence of
sphere oversizing is less dramatic (maximum bias: 480
nCi/mL), and is comparable with FWHM overestimation
(maximum bias: +420 nCiimL). Maximum biases are
lower for errors induced by incorrect WM activity con
J Cereb Blood Flow Metab, Vol. 19, No.5, 1999
Page 8
554
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x(mm)
J Cereb Blood Flow Metab. Vol. 19, No. 5. 1999
20
20
20
20
D. STRUL AND B. BENDRIEM
5
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.:.
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1
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x(mm)
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f 0.8
(.)
u 0.4
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l:
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til 0.6
0.5
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x(mm)
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f
.. 0.4
I
0.9
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l:0.5
�
Q
til 0.6
to
t: 0.2
0.3
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0
20
10
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20
D
x(mm)
FIG. 6. Correction artefacts in
rate FWHM estimation (±2
rors (± 1 mm on each TT
edge). (C) Misregistration er
rors (±1 mm). (0) Combination
of missegmentation and mis
or both 1 mm. The dotted
lines represent the theoretical
predictions.
binary correction. (A) Inaccu
mm). (8) Missegmentation er
registration errors, both + 1 mm
FIG. 7. Correction precision in
GMPET method, measured
by corrected activity SO. (A)
With SO[A11
(8) With SO[xol = 1 mm and
ideal correction mask. (C) With
SO[xol = 1 mm and correction
With SO[wl = SO[Xol = 1 mm
and correction mask in real
conditions.
= 0.2 nCi·mL1.
mask in real conditions. (0)
Page 9
FIG. 8. Mean corrected activ
ity profile (10,000 tries) in pres
ence of segmentation and reg
istration imprecision (SD[w] =
(8) With ideal correction mask.
SD[xol = 1 mm). (A) With cor
rection mask in real conditions.
FIG. 9. Influence of the pro
cessing errors on the cor
rected image regional bias
(left) and BRC (right) for each
sphere. (A) white matter activ
ity estimation. (8) Correction
FWHM. (C) Segmentation er
rors (error on sphere radii). (0)
Registration.
ROBUSTNESS OF PVE CORRECTIONS
5
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J Cereb Blood Flow Metvb, Vol. 19, No. 5, 1999
Page 10
556
D. STRUL AND B. BENDRIEM
centration values or by misregistrations (290 and 190
nCi/mL, respectively). The BRC plots show that GM
PET correction usually improves quantification (BRC
above 1). The sole exception occurred during sphere un
dersizing, where GMPET correction degraded the re
gional bias (BRC less than 1) when the segmentation
errors are larger than 0.75 mm.
Table 1 summarizes results extracted from these data.
For the smallest sphere, a moderate error of cylinder
activity or FWHM can result in a significant corrected
image bias (115 nCi/mL). The bias induced by a typical
registration error is smaller (corrected image bias = 55
nCi/mL), whereas even a moderate segmentation error
can lead to a severe quantitation error, about 250 nCil
mL when sphere is oversized and +450 nCi/mL when it
is undersized. Even with these error values, image bias is
reduced from 4 to 10 times (respectively 8 to 17 times)
for the smallest (respectively largest) sphere, compared
with the noncorrected image bias. Notice that the GMPET
performances are significantly worse when there are seg
mentation errors, with the BRC ranging from 1.1 to 2.4.
Figure 10 compares the relation between the SD and
the bias for each sphere when there are some processing
errors. These curves show that there is no direct relation
between SD and bias. Even if SD always increases with
bias, its evolution rate also depends on the origin of the
error and on the sphere diameter. The corrected image
SD is extremely sensitive to the presence of registration
errors.
DISCUSSION
Previous studies suggest the strong influence of po
tential errors on the performance of AP A correction.
Videen and coworkers demonstrated the influence of
misregistration errors on the binary correction for atro
phy of PET simulations of a brain phantom (Videen et
aI., 1988). A later study of Meltzer and associates inves
tigated the effects of misregistration and of incorrect seg
mentation thresholding on the binary correction of the
brain PET image of a patient with Alzheimer's disease
(Meltzer et aI., 1990). The binary correction also was
applied to a 16cm sphere phantom, and the corrected
image recovery coefficient was 79%, which is smaller
than what was expected. An exhaustive error analysis
was performed for the GMPET method on computer
PET simulations of a patient brain (MullerGartner et aI.,
1992). Recently, the importance of this topic was under
lined by Fahey et aI., who presented a study on the ac
curacy of GMPET (Fahey et aI., 1996). All of these
previous analyses were applied on complex brain images
in which corrected structures are irregularly shaped and
affected by interstructure spillover. On such images, it
was difficult to conclude the effect of each correction
step separately. The current work provides a comprehen
sive analysis of these issues by using simplified models,
where all correction parameter and artefacts are more
readily controlled and isolated.
The onedimensional theoretical and experimental
study demonstrated that each correction artefact exhibits
a different and specific shape pattern, at least when the
TT is larger than the system FWHM. Correction errors
thus may be identified from visual examination. For in
stance, the presence of a spike on one structure edge and
of a fall on the opposite edge is likely to indicate a
registration error. Correct registration thus may be
achieved by moving slightly the tissue images toward the
spike. Once edge homogeneity is restored, a hypoactive
or hyperactive structure outline would indicate either a
segmentation error or an erroneous NT activity estima
tion (since these two errors produce artefacts that are
more difficult to discriminate). The FWHM errors are
less critical than the other errors, since this parameter can
be optimized based on phantom studies before applying
APA correction to real data. Notice that these conclu
sions were derived from onedimensional examples
where chosen structures were regularly shaped and at
least bigger than 1 FWHM. However, they were ob
served for more complex structures in the current use of
the GMPET algorithm in our laboratory.
Correction artefacts are mainly located on the bound
aries of the active structures, and this suggests that the
robustness of the corrected measure may be improved by
lessening or suppressing the influence of the edge voxels
TABLE 1. Performance of APA correction on smallest and largest spheres for typical errors
Smallest sphere
Largest sphere
 
Uncorrected image
bias (nCilml)
Corrected image
bias (nCi/ml)
Uncorrected image
bias (nCi/ml)
Corrected image
bias (nCi/ml)
BRC
BRC
20% error on cylinder activity
I mm FWHM underestimation
512
512
512
475
541
582
115
110
126
451
253
55
4.4
4.7
4.1
l.l
2.1
10.5
179
179
179
134
234
220
21
21
22
86
98
13
8.7
8.6
8.0
1.6
2.4
17.2
I mm FWHM overestimation
1 mm spherediameter undersizing
I mm spherediameter oversizing
2 mm misregistration
Real activity in spheres is 1000 nCi/mL. APA, anatomically guided pixelbypixel algorithm; BRC, bias reduction coefficient; FWHM, fullwidth
at half maximum.
J Cereb Blood Flow Metab. Vol. 19, No. 5, 1999
Page 11
ROBUSTNESS OF PVE CORRECTIONS
300 l
100
557
200 j
100
200
Bias versus SO incase of
c.ylinder activity errors
.... .. .. ... .. . 9.5 mm
*" 31.9 mm
012.7 mm
____¥_ __ __ _ 15.9 mm
   19.1 mm
025.4 mrn
300
200
100
�
'"
�
0
.!!1
OJ
100
Bias versus SO In case of
FWHM errors
FIG. 10. Relationship be
tween SO and bias for each
sphere in presence of param
eter errors_ (A) WM activity es
timation. (8) Correction
FWHM. (C) Segmentation er
rors (errors on sphere radi
uses). (0) MRIPET registra
tion.
300 +�  ' '  r_ _ �  ,
A
200 j'
B
300 +�.= ,�___;
300 l
J
'
� O��
200 1
100
200 100
200
Standarddeviation (nCVml)
Standarddeviation (nCVml)
300
200
100
segmentation errors
100
200
200 c
100 l
100 l
E
'"
'"
iIi
i
Bias vs SD in case of
registration errors
300 +�  '"_, "  _  ,
c
300 + I_,�___;
D
100
200
200
Standard deviation (nCi/ml)
100
Standard<leviation (nCVml)
in the determination of the final measure. Several ap
proaches may be used for this purpose: limiting the mea
surement region of interest (ROI) to small areas around
the structure centers, or using suitable measurement sta
tistics (Videen et a!., 1988). However, remember that for
small structures, edge artefacts propagate through the
structure center. In that latter case, the approaches pro
posed earlier are insufficient to ensure accurate quanti
fication.
The aim of the sphere phantom experiment was to
determine which parameters were most determinant to
ensure correction robustness. It appears unequivocal that
the quality of tissue segmentation is critical in that mat
ter. For small structures, segmentation errors may lead to
dramatic biases, especially if active tissue dimensions are
underestimated. Second comes the estimation of the NT
activity and of the FWHM. Notice that the influence of
the NT is likely to be even bigger when the TT is irregu
larly shaped. Finally, only severe registration errors (",, 5
mm) may lead to significant biases, since the misregis
trationinduced artefacts have opposite directions and
thus partly compensate for each other. However, this
compensation effect requires that the measurement ROI
covers both opposite artefacts; registration errors thus
dramatically increase the influence of the ROI design
strategy.
Another objective of the sphere phantom experiment
was to determine a performance index to verify and
quantify correction performance. Our initial estimation
was that the regional SD could serve this purpose. The
rationale for this choice was that SD strongly depends on
ROI homogeneity, which is severely reduced by correc
tion edge artefacts. As Figure 10 shows, SD is extremely
sensitive to registration errors, and an increase in SD
could appear even without any noticeable bias. Obvi
ously, the design of reliable correction evaluation in
dexes requires taking into account the characteristic
shapes of the correction artefacts. This could be achieved
by independently computing statistics for the innermost
and the outermost voxels. Segmentation or NT activity
estimation errors then would be detected by comparing
the center mean value and the boundary means value.
Similarly, segmentation failure areas could be revealed
by detecting local activity peaks.
In summary, APA correction provides a powerful tool
for improving PET quantification, but its application to
real data is limited by its lack of robustness. Precise
segmentation and NT activity estimation are the most
crucial issues to ensure a stable correction. The correc
tioninduced artefacts arising from registration or seg
mentation inaccuracies exhibit characteristic shapes
when the structures are big enough, which allows direct
J Cereh Blood Flow Metab. Vol. 19, No. 5, 1999
Page 12
558
D. STRUL AND B. BENDRIEM
recognition from visual examination. Automated detec
tion and removal of these artefacts, however, are more
complex. The regional SD was assessed as a correction
performance index, and our analyses show that it is in
adequate as a global performance criterion, since it is
overly sensitive to registration errors. The design of re
liable performance indexes will require more sophisti
cated methods that should take into account the charac
teristic shape of each correction artefact.
APPENDIX
Consider the onedimensional model presented on Fig.
lA. Since all of the tissues have similar bar shapes, the
PSF blurring of their tissue distribution functions may be
written as follows:
[XI*h](x) = P3(x)  PI(x)
[X2*h](x) = P2(x)  P4(x)
where the functions PI . . 4 are primitives of the PSF,
centered on the locations XI . . . 4'
[XTT*h](x) = P1(x)  P2(x)
(13)
PJx) == P(ilxJ == P(x  xJ
I IX
By introducing these expressions into APA equation (3),
we obtain the correction equation specific to our model:
with
2 2
P(x) ==
,, � exp( t l2u )dt
u V 27T
0
(14)
The derivatives of CTT(x) relative to the NT activity es
timations are then
For the other parameters, we introduce the following
notations:
N(x) == Iobs(x) A1[P3(X) PI (x)] A2[P2(X) P 4(X)]
D(x) == P1(X) P2(x)
(17)
Then, the partial derivatives of CTT(x) relative to any
parameter p except the NT activity estimations have the
following form:
1
Jp
(X) =
Jxo

JP3
 JPI
Jp
JP
A D  ' +[A D  N] 
JPo
 [A2D  N] 
(PI P2)2
I
_
_
C
 + A2D a
Jp
J TT
P (x)
(18)
J Cereb Blood Flow Metab. Vol. 19, No. 5, 1999
where the partial derivatives of Pl . . . 4 relative to Xo and
u are, respectively,
JPi

J
Xo
JPi
Ju
(x) =  hex  xJ
= (x xJhJx  xY u
(19)
Segmentation errors are assumed to affect only the TT,
so that
. (x) = h(x x )
Jw
i)w
JPI
JP2
JP3 JP4
Jw
(x)
= +h(x  x ) 
 ' =  = 0
Jw
1
2
(20)
These equations may be simplified in several manners,
depending on the specific application. Notice in particu
lar that, when a location Xi is distant enough (more than
a few FWHM), its influence may be neglected by setting
Xi '" 00 (so that Pi(x) '" +1/2) or Xi '" +00 (so that Pi(x) '"
_1/2), Also notice the binary correction, where no NT is
acknowledged, so that equation (15) simplifies to
The derivatives of CTT(x) then are readily computed as
follows:
(22)
Acknowledgments: The authors thank Vincent Brulon and
Pascal Merceron for technical support, and Ken Maya for care
fully reviewing the manuscript.
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