Accurate measurement of brain changes in longitudinal MRI scans using tensor-based morphometry.

Abstract

This paper responds to Thompson and Holland (2011), who challenged our tensor-based morphometry (TBM) method for estimating rates of brain changes in serial MRI from 431 subjects scanned every 6 months, for 2 years. Thompson and Holland noted an unexplained jump in our atrophy rate estimates: an offset between 0 and 6 months that may bias clinical trial power calculations. We identified why this jump occurs and propose a solution. By enforcing inverse-consistency in our TBM method, the offset dropped from 1.4% to 0.28%, giving plausible anatomical trajectories. Transitivity error accounted for the minimal remaining offset. Drug trial sample size estimates with the revised TBM-derived metrics are highly competitive with other methods, though higher than previously reported sample size estimates by a factor of 1.6 to 2.4. Importantly, estimates are far below those given in the critique. To demonstrate a 25% slowing of atrophic rates with 80% power, 62 AD and 129 MCI subjects would be required for a 2-year trial, and 91 AD and 192 MCI subjects for a 1-year trial.

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Comments and Controversies
Accurate measurement of brain changes in longitudinal MRI scans using
tensor-based morphometry
Xue Huaa,1, Boris Gutmana,1, Christina P. Boylea, Priya Rajagopalana, Alex D. Leowb,c,d, Igor Yanovskya,
Anand R. Kumarc, Arthur W. Togaa, Clifford R. Jack Jr.e, Norbert Schufff,g, Gene E. Alexanderh,
Kewei Cheni,j, Eric M. Reimani,k,l, Michael W. Weinerf,g,m, Paul M. Thompsona,⁎
and the Alzheimer's Disease Neuroimaging Initiative2
aLaboratory of Neuro Imaging, Dept. of Neurology, UCLA School of Medicine, Los Angeles, CA, USA
bDept. of Psychiatry, University of Illinois-Chicago, Chicago, IL, USA
cDept. of Bioengineering, University of Illinois-Chicago, Chicago, IL, USA
dCommunity Psychiatry Associates, Sacramento, CA, USA
eDept. of Radiology, Mayo Clinic, Rochester, MN, USA
fDept. of Radiology and Biomedical Imaging, UCSF, San Francisco, CA, USA
gDept. of Medicine, UCSF, San Francisco, CA, USA
hDept. Psychology and Evelyn F. McKnight Brain Institute, University of Arizona, Tucson, AZ, USA
iBanner Alzheimer's Institute and Banner Good Samaritan PET Center, Phoenix, AZ, USA
jDept. of Mathematics and Statistics, Arizona State University, Tempe, AZ, USA
kDept. Psychiatry, University of Arizona, Phoenix, AZ, USA
lNeurogenomics Division, Translational Genomics Research Institute and Arizona Alzheimer's Consortium, AZ, USA
mDept. of Psychiatry, UCSF, San Francisco, CA, USA
a b s t r a c t a r t i c l e i n f o
Article history:
Received 7 December 2010
Revised 16 January 2011
Accepted 28 January 2011
Available online 23 February 2011
This paper responds to Thompson and Holland (2011), who challenged our tensor-based morphometry
(TBM) method for estimating rates of brain changes in serial MRI from 431 subjects scanned every 6 months,
for 2 years. Thompson and Holland noted an unexplained jump in our atrophy rate estimates: an offset
between 0 and 6 months that may bias clinical trial power calculations. We identified why this jump occurs
and propose a solution. By enforcing inverse-consistency in our TBM method, the offset dropped from 1.4% to
0.28%, giving plausible anatomical trajectories. Transitivity error accounted for the minimal remaining offset.
Drug trial sample size estimates with the revised TBM-derived metrics are highly competitive with other
methods, though higher than previously reported sample size estimates by a factor of 1.6 to 2.4. Importantly,
estimates are far below those given in the critique. To demonstrate a 25% slowing of atrophic rates with 80%
power, 62 AD and 129 MCI subjects would be required for a 2-year trial, and 91 AD and 192 MCI subjects for a
1-year trial.
© 2011 Elsevier Inc. All rights reserved.
Introduction
This paper responds to a recent commentary in the journal
NeuroImage (Thompson and Holland, 2011), regarding the accurate
estimation of changes in serial brain MRI scans. Thompson and Holland
(2011)3pointed out an important issue about potential image
registration bias when computing changes in brain images, which
they noticed in a re-analysis of the data we previously published in
NeuroImage (Hua et al., 2010). We carefully studied and agreed with
the main argument in Thompson and Holland's letter and have
developed a solution to the problem by using inverse-consistent
registration. The resulting updated measures from tensor-based
morphometry are informative and powerful for use in drug trials to
assess factors that affect brain change; sample size estimates remain
competitive. Measures from our inverse-consistent algorithm show
NeuroImage 57 (2011) 5–14
⁎ Corresponding author at: Laboratory of Neuro Imaging, Dept. of Neurology, UCLA
School of Medicine, Neuroscience Research Building 225E, 635 Charles Young Drive, Los
Angeles, CA 90095-1769, USA. Fax: +1 310 206 5518.
E-mail address: thompson@loni.ucla.edu (P.M. Thompson).
1These authors contributed equally.
2Data used in preparing this article were obtained from the Alzheimer's Disease
Neuroimaging Initiative (ADNI) database (www.loni.ucla.edu/ADNI). As such, the
investigators within the ADNI contributed to the design and implementation of ADNI
and/or provided data but did not participate in analysis or writing of this report. ADNI
investigators include (complete listing available at: http://www.loni.ucla.edu/ADNI/
Collaboration/ADNI_Manuscript_Citations.pdf).
3We note for clarity that the corresponding author of this paper is Paul Thompson
(UCLA School of Medicine), and we are responding to a letter by Wes Thompson (no
relation) and Dominic Holland of UC San Diego.
1053-8119/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2011.01.079
Contents lists available at ScienceDirect
NeuroImage
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verygoodpower,andaresuperiortotheadjustmentsthatshowedpoor
statistical power in the Thompson and Holland re-analysis. We would
liketothankThompsonandHollandfornotingsurprisingaspectsofour
prior data and helping us identify and correct them.
Summary of the problem
What is tensor-based morphometry?
Tensor-based morphometry (TBM) produces 3D maps of volu-
metric brain change found by deforming one brain to match another.
Individual maps of brain changes (also called Jacobian maps) are
aligned to an average group template, and group-wise comparisons
can be made using voxel-based statistics. We note, for clarity, that
although this general type of analysis is called TBM, many nonlinear
image registration methods have been developed to compute brain
changes analyzed in this way (e.g., Freeborough and Fox, 1998;
Ashburner, 2007; Yanovsky et al., 2009). Klein et al. (2009) recently
compared 14 nonlinear registration methods; these algorithms are in
a continual state of refinement, with the goal of reducing quantifica-
tion errors.
Apparent jump in the rate of atrophy, measured using TBM
In recent work, we computed rates of brain change based on 1309
ADNI MRI scans (Hua et al., 2010) and in accordance with the
recommendationsof theADNI project,we madethe resultingnumeric
summaries from our analyses available in a public database (http://
adni.loni.ucla.edu). Several ADNI analysis groups also upload numer-
ical summaries from the MRI scans to this database. Because of the
unusually large scale of this neuroimaging project, these numeric
summaries are frequently replaced or updated over time, and
corrected datasets are uploaded as errors in the analyses come to
light.
Although TBM produces an entire 3D map of brain changes, most
MRI analysis methods compute a single number from each image
(such as the volume of the hippocampus; Schuff et al., 2009; Morra
et al., 2009a,b). Given the interest in comparing different analysis
methods, we examined two different methods to obtain a represen-
tative measure of brain change from TBM. First, we computed the
average change over an anatomically defined region of interest (ROI),
the temporal lobes. As advocated in work on FDG-PET by Eric Reiman
and colleagues (Chen et al., 2010), we also used a statistically defined
region of interest (stat-ROI) that selects the regions in the image with
greatest effect sizes, based on a pilot analysis of a non-overlapping
dataset. This region is then used to compute summaries of changes in
other scans. The validity and advantages of the stat-ROI method have
previously been discussed (e.g., Chen et al., 2010) and often
outperform standard atlas-based ROIs.
In a re-analysis of our numeric summary data from a stat-ROI,
which we uploaded to the public ADNI database, Thompson and
Holland (2011) noted an unexplained and surprising jump in the
time-series of changes. We were able to replicate this effect in our
own data, andthe graphis shown in Fig. 1. Clearly there is an apparent
jump in the atrophy rate between 0 and 6 months, with more linear
changes thereafter. The jump occurs in all diagnostic groups — AD,
MCI, and controls. In Hua et al. (2010), we showed time-series of
cumulative atrophy in MCI and AD. This jump is more coherent with
the rest of the trajectory in those groups, and we interpreted it as
natural. This was plausible, given the possibility of a biological
nonlinear change in the atrophy over time due to changes in the
disease process, measurement error, drift in scanner calibration over
time and attrition or sampling effects. However, when the trajectory
of atrophy from controls is also shown, it becomes clear that there is a
systematic bias in the measures of unknown origin.
Assessment of the source of the bias
We were alerted to this effect on September 27, 2010 (W.
Thompson, personal communication) and conducted a set of experi-
ments that hypothesized, tested, identified and subsequently cor-
rected the source of this problem. We postulate that any bias in
atrophy estimates may be comprised of a constant, additive offset, and
a component whose magnitude may depend on the true level of
atrophy, an atrophy-dependent component. We report our experi-
ments below, which consider factors that might affect the linearity of
the time-series. In turn, we discuss various sources of bias as they are
relevant to ongoing investigations of brain morphometry by our
group and others.
1. Inverse-consistency. In Hua et al. (2010), we computed atrophy
rates using a registration method developed and validated in
Yanovsky et al. (2009). This approach estimates the rate of brain
change by computing a deformation field that optimizes a
matching cost functional, in this case the mutual information
(MI) between the deforming image and the target scan, while also
optimizing a measure of the regularity of the deformation, called
the regularizer. In this case the regularizer is the symmetrized
Kullback–Leibler distance, or sKL (and the algorithm is referred to
as sKL-MI). While the regularization is symmetrized (and thus
inverse-consistent), here the follow-up scans are aligned to the
baseline scan, which serves as a registration target. This results in
inherent asymmetry in the overall functional being optimized,
due to the asymmetric nature of the registration. To address this
as a potential issue, we re-implemented the registration method
to be fully inverse-consistent by further symmetrizing the MI,
which means the computed brain changes (deformations) are
forced to be the same if the chronological order of the images is
switched; i.e., regardless of which image is the target. As first
proposed by Thirion (1998) and later popularized in several
papers by Christensen et al. (e.g., Christensen, 1999; Christensen
and He, 2001), an inverse-consistent registration algorithm gives
the same correspondences between the source and target images,
even if the order of the images is switched. This does not allow
Fig. 1. TBM-derived summaries of cumulative atrophy in 91 AD, 188 MCI and 152
normal control subjects, based on a statistical region of interest in the temporal lobes
(using summary data from Hua et al., 2010; see Methods for details). If the 6–12 month
change is extrapolated back to the origin, there is an apparent non-zero offset that
appears to shift the measures upward by about 1.2–1.4%; a linear regression fitted to all
time points would give a smaller offset. The biological trajectory may not be linear over
time, but algorithmic sources contributing to this offset are explained and analyzed in
this paper. We later show that this offset is reduced to 0.28% using the proposed
implementation of inverse-consistent registration to compute the brain changes. The
residual (much smaller) offset may be attributed to transitivity errors as well as
sampling and biological nonlinearity, as there is no reason to expect the changes to be
perfectly linear and to have zero intercept.
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either time-point to have a privileged position in assessment of
change, as change is computed in both directions between the
two images. An inverse-consistent mapping is typically achieved
numerically by mapping the source to the target and the target to
the source simultaneously, and reducing the degree of inverse-
inconsistency with an explicit regularization term (Johnson and
Christensen, 2002; He and Christensen, 2003; Rogelj and Kovacic,
2006; Christensen et al., 2006). Inverse consistency may also be
achieved using more complex 4D methods that optimize the
geodesic path connecting a pair of images, treated as a full space–
time optimization that is symmetric in the inputs (Miller, 2004;
Avants et al., 2008; Vercauteren et al., 2008; Miller and Qiu, 2009;
Qiu et al., 2009). We studied inverse-consistency extensively in
our prior work (e.g., Leow et al., 2007), and found that it could be
implemented computationally without the need to use an explicit
inverse-consistency regularization, through a numerical method
known as equivalent perturbation (Leow et al., 2007). Even so, in
our published work with sKL-MI, we did not use an inverse-
consistent version of the algorithm for the matching cost
functional, which, as shown below, accounts for all but 0.28% of
the offset in atrophy rates. In our tests below, the additive bias in
atrophy estimation was largely eliminated by using an inverse-
consistent implementation of sKL-MI: it was reduced from 1.4% to
0.28%. Thus, we conclude that inverse-consistent implementation
should be preferred in future analyses. To assess the contribution
of inverse-consistency error to the mappings, we re-computed
trajectories of atrophy rates. Using inverse consistency error (ICE)
maps, we also verified that our new equivalent perturbation
version of the sKL-MI method (ic-sKL-MI) enforces zero inverse
consistency error across the brain, to within a few thousandths of
a voxel.
2. Statistical region of interest. We studied whether the bias (non-zero
intercept) in the time-series is affected by the use of a statistical
region of interest. By non-zero intercept, we mean that, if the
trajectory of atrophy over time is approximated by a straight line,
the line should hit the y-axis at a point that is not significantly
different from zero. Because the stat-ROI is defined based on
regions that change the most (strictly speaking, voxels with
greatest effect sizes for changes between the time-points in AD),
the use of a stat-ROI may pick up a biologically plausible
nonlinearity in the rate of regional atrophy, which could represent
biological truth rather than algorithmic error. For example, if the
same stat-ROI is applied to summarize maps of brain change from
all time points, the trajectory may still appear to be nonlinear if the
focus of atrophy moves to a different part of the brain over time
(e.g., if it moves out of the statistical ROI).
The region of fastest ongoing atrophy in Alzheimer's disease
moves rapidly over a period of 1.5 years from the temporal lobes
to other cortices as the disease progresses (Thompson et al.,
2003; see also Smith, 2002; Whitwell et al., 2007). Scahill et al.
(2002) also noted that, using fluid registration of serial MRI
scans, the focus of atrophy in the brain changes as the disease
advances, and this is in line with the spreading trajectory of
pathology (Braak and Braak, 1991). Given our knowledge of
atrophy in normal aging (Sowell et al., 2003), this shifting
effect is expected to be minimal over the 2-year time-span of this
study. To avoid correcting for atrophy that is truly biologically
nonlinear, such as that due to an anatomically spreading or shif-
ting disease process, we assessed the effects of atrophy in control
subjects (N=152) in addition to assessing the same issues in MCI
and AD.
3. Transitivity error. As noted by Christensen and Johnson (2001)
[see also Johnson and Christensen, 2002; Geng et al., 2005, and
the PhD thesis by Geng (2007)] one desirable property of a
registration algorithm is that it creates mappings that are
transitive. Few existing registration algorithms are transitive,
but most will give mappings that are fairly close to transitive. In
our context of mapping brain change, transitivity means that the
total amount of change found when matching baseline (0 month)
to 24-month follow-up scans should be the same as would be
found by matching 0 to 12-month follow-up scans, and then
matching 12- to 24-month scans, and concatenating the results.
Any discrepancy between the direct and indirect mappings may
be used to compute the transitivity error (as detailed below in the
Methods). In theory, this may lead to an additive or multiplicative
bias present in all mappings, which may or may not correlate with
the true amount of change, or with the estimated amount of
change. Very few registration algorithms are forced to be transi-
tive by design (see Škrinjar and Tagare, 2004; Škrinjar et al., 2008;
Tagare et al., 2006; Geng, 2007, for exceptions). To assess this, we
examined the contributions of this source of error (transitivity
error) to the mapping of serial changes. We also tested if this
error was correlated with the estimated overall amount of
change.
Methods
As in our prior work (Hua et al., 2010), we used tensor-based
morphometry (TBM) to map the 3D profile of progressive atrophy in
91 subjects with probable AD (age: 75.4±7.5 years), and 188 with
amnestic mild cognitive impairment (MCI; 74.6±7.1 years), scanned
at 0, 6, 12, 18 and 24 months (in ADNI, only the MCI subjects were
scanned at 18 month intervals). In the current analysis, we added 152
healthy controls (age: 76.0±4.8 years), scanned at 0, 6, 12, and
24 months. To avoid sampling different individuals at each time-
point, we included only those subjects who were scanned at all time-
points. At the time of writing, far fewer people had been scanned at
36 months, so we did not include this time point in our analysis. The
inclusion of only those subjects with scans at all time-points could
have under-sampled people whose atrophy was progressing more
quickly and were more likely to drop out of the study (Scahill et al.,
2002). So, a sampling bias cannot be absolutely eliminated, and there
may be sources of nonlinearity in the trajectory of atrophy that cannot
be entirely modeled or explained.
Individual maps of atrophy rates (also known as “Jacobian maps”)
were derived from a TBM analysis of MRI scans acquired over time.
These maps represent the rates of tissue shrinkage (or CSF space
expansion) at each voxel location in the brain.
We compared two registration methods to assess brain changes
over time.
sKL-MI (Yanovsky et al., 2009)
This method nonlinearly warps the follow-up scan to match the
baseline scan of the same individual, driven by a mutual information
cost function, and a regularizing term called the symmetrized
Kullback–Leibler (sKL-MI) distance (Yanovsky, et al., 2009). This
method was used to compute change over time in our earlier paper
(Hua et al., 2010). While the penalty term (the symmetric Kullback–
Leibler distance) is designed to be inverse consistent, there is no
explicit constraint in this method to ensure inverse-consistency on
the matching cost function.
ic-sKL-MI
An inverse consistent implementation of sKL-MI was implemented
(by B.G.). Distinct from other implementations of inverse consistency,
instead of reducing the inverse consistency error, we completely
eliminate it, using the equivalent perturbation method introduced in
Leow et al. (2007). In early work by Christensen and Johnson (2001),
the inverse-consistency error of a nonlinear registration algorithm
was penalized, but not reduced to zero, by defining the following
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energies (or cost functional to be minimized) on the forward and
backward mappings:
EhT;S
ð
EgT;s
ð
Þ = ∫
Þ = ∫
Ω
jS h x ð Þ
jT g x ð Þ
ð
ð
Þ−T x ð Þj2dx + λR h
Þ−S x ð Þj2dx + λR g
ð Þ + ρ∫
ð Þ + ρ∫
Ω
jjh− g
jjg− h
ð Þ−1jj2dx
ð Þ−1jj2:
ΩΩ
Here (as in Leow et al., 2007), T (target) and S (source) are the
images to be registered, both defined on a computational domain, Ω,
h is the forward transform from the source to the target, g is the
backward transform from the target to the source, R is the regularizer
and λ and ρ are weighting terms. As noted by us in Leow et al. (2007)
and by Avants et al. (2008), this will not entirely remove the inverse
consistency error. Instead we consider an infinitesimal perturbation ξ
applied to the inverse mapping, and solve for η, the perturbation in
the forward mapping that preserves the fact that the forward-
backward mapping pair h and h−1stay inverses of each other. Thus,
the composition of the two perturbations ought to approach the
identity mapping in the limit:
given perturbation
Solve
η x ð Þsuch that
h−1+ εξ
h−1x ð Þ→h−1x ð Þ + εξ x ð Þ
lim
ε→0
??
? h + εη
ε
ðÞ x ð Þ−x
= 0
As previously shown, η(x)=−D(h(x))ξ(h(x)), where Dh is the
Jacobian matrix of h with (i,j)th element ∂hi/∂xj, so we can then
compute a forward equivalent of a body force in the backward
direction, using only the forward mapping h, and not involving h−1.
This circumvents numerical errors incurred when performing
numerical inversion operations to go between h and h−1.
Mean template construction
All subjects' maps of brain change were registered to a mean
deformation template (MDT) based on 40 subjects from the study, as
in Hua et al. (2009). The MDT represented the average shape of 40
healthy elderly controls; the procedure to construct the MDT is
detailed in Hua et al. (2008a,b). The mean template does not affect the
estimates of atrophy rates in each person. Average Jacobian maps
were computed by taking the mean at each voxel of the Jacobian maps
across subjects.
Sample size estimates (n80)
A power analysis was established by the ADNI Biostatistics Core
to estimate the minimal sample size required to detect, with 80%
power, a 25% reduction in the mean annual change, using a two-
sided test and standard significance level (α=0.05) for a hypothet-
ical two-arm study (treatment versus placebo). The estimated
minimum sample size for each arm was computed with the formula
below. Briefly, β denotes the estimated change over a one-year or
two-year period (average for the group) and σD refers to the
standard deviation of this rate of atrophy across subjects.
n =
2ˆ σ2
Dz1−α=2+ zpower
?
??2
0:25ˆβ
?2
Here zαis the value of the standard normal distribution for which
P[Zbzα]=α (Rosner, 1990). The sample size required to achieve 80%
power was computed, denoted by n80. We note in passing that this is
a linearization of the exact expression for statistical power.
Statistical analyses
We performed several statistical analyses to assess factors influ-
encing the linearity of the brain changes over time, and the effect sizes
of the resulting measures of atrophy. We assessed brain changes in
anatomical and statistical regions of interest. A statistically defined
region of interest (stat-ROI) was based on voxels with significant
atrophic rates over time (pb0.001 or pb0.0001, uncorrected) within
the temporal lobes. This was established in a non-overlapping training
set of 20 AD patients (age at baseline: 74.8±6.3 years; 7 men and 13
women) scanned at baseline and 12 months. This procedure is detailed
in Chen et al. (2010) and Hua et al. (2009, 2010). The anatomical ROI
included thetemporallobe graymatter, a regiontypically providingthe
highest statistical power for tracking AD progression (Jack et al., 1998).
A numerical summary of the atrophic rate, in the temporal lobes, was
computed by taking the arithmetic mean of Jacobian values within the
corresponding stat-ROI or anatomical ROI (Hua et al., 2009, 2010),
givingasinglemeasureoftemporallobeatrophyforeachindividual.An
evidence for an offset at time zero (which may arise from the method
and/or biological nonlinearity) was estimated by fitting a linear mixed
effects model through (1) measures of cumulative atrophy at 6, 12, and
24 months but leaving out the data point at baseline, (2) measures of
cumulative atrophy at 6, 12, and 24 months and the known data point
at time zero (having a zero change at month zero is known), in the
control group (n=152). The lmer and other statistical functions
from the R statistical package (version 2.10.1: library (lme4)) were
used to estimate the intercept (offset at month zero) and 95% confi-
dence intervals.
We hypothesized that inverse-consistent registration would:
(1) yield temporal trajectories of atrophy with a greatly reduced
offset close to zero;
(2) yield sample size estimates (n80) with substantial power,
considerably better than those computed from our previously
reported data by Thompson and Holland (2011), although
somewhat lower than those computed using sKL-MI.
Results
Numerical inverse consistency
To show that our inverse-consistent registration algorithm ic-sKL-
MI indeed created maps that are inverse-consistent, we made a map
of the inverse consistency error, ICE=||x−h⁎h−1(x)|| where h is the
mapping from one time point to another, and h−1is the mapping
computed in reverse (i.e., by the algorithm applied to the same scans,
but with the order of the scans switched). A typical map is shown in
Fig. 2(a), showing that the ICE is around 0.005 mm or lower,
throughout the brain, with higher values in the scalp or other non-
brain regions where signals are not important and contrast is less
consistently controlled over time. The backward mapping is within a
fewthousandthsof a millimeter oftheinverseof theforwardmapping
across the entire 3D volume of the brain. Since many voxels undergo
very small deformations, it is also instructive to assess the relative
inverse consistency error, or ICE/||x−h(x)||. Fig. 2(c) shows that
relative error is well below 5% of the measured change and much less
in most voxels.
As it is difficult to relate error in displacement fields to error in
Jacobian determinants directly, we estimated the effect of ICE on
Jacobians empirically. Fig. 2(b) shows the map of ||Dh|−|D(inv(h−1)||;
here, the outer brackets mean absolute value, and inner brackets mean
determinant. Note that using the direct numerical inversion inv()
creates additional numerical error, so this estimate is pessimistic. Still,
the map shows that within the brain, absolute error is on the order of
0.1% change or less. When one integrates over a large ROI, this error is
likely to be reduced substantially due to averaging effects.
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Improved linearity of cumulative brain atrophy using inverse-consistent
registration, with reduced offset
As shown in Fig. 3, brain changes recovered by the ic-sKL-MI
method show substantially reduced offsets. By extrapolating back
from the 6–12 month interval, the offset in the change measures is
very small — around 0.28% for the temporal lobe gray matter ROI, and
statistical ROI (Fig. 3), compared with the 1.2–1.4% offset for brain
change measures computed with sKL-MI (Fig. 1). Clearly, this is
greatly reduced, and amounts to a displacement field error of a few
thousandths of a millimeter in a 1-mm MRI voxel. As any biological
sample is heterogeneous, a linearized plot through the mean atrophy
rate data might not run through zero exactly.
In Fig. 4, offsets are measured based on best linear fit to all the data
points at 6, 12, and 24 months in the control group only (n=152), as
their trajectory is thought to be linear. The fitted intercepts and the
95% confidence intervals based on control subjects are (a) statistical
ROI: 0.29% [0.15, 0.44], (b) temporal lobe gray matter: 0.28% [0.15,
0.42], (c) statistical ROI including the known data point at baseline:
0.12% [0.05, 0.19], and (d) temporal lobe gray matter including the
known data point at baseline: 0.11% [0.04, 0.18].
Statistical power
Power estimates based on our inverse consistent measures are
shown in Table 1. To demonstrate a 25% slowing of atrophy rates with
80% power, 62 AD and 129 MCI subjects would be required for a 2-
year trial, and 91 AD and 192 subjects for a 1-year trial. These are 1.6–
2.4 times higher than our previously estimated sample sizes, but not
5–16 times higher as alleged in the Thompson and Holland (2011)
analysis. The difference is accounted for by using inverse-consistent
registration to compute brain changes.
As shown in Table 2, across the same set of subjects in AD, MCI, and
CTL, longer intervals led to greater amountof atrophy measured in the
statistical-ROI and temporal gray matter, resulting in smaller sample
size estimates.
Assessment of transitivity errors
As shown in Fig. 5, we computed a map of the voxelwise
transitivity errors. To define these, we label the 0, 12, and 24 month
scans in a given subject as A, B, and C, respectively; we compute the
deformation mappings between these time points: hAB, hACand hBC.
The transitivity error, at each point in the brain, is defined as the
difference between the Jacobians of the direct mapping (from A to C)
and the composed mapping (from A to C via B):
TE = J hAC
ðÞ–J hBC?hAB
ðÞ
As shown in Fig. 5, the transitivity error is small in all areas of the
brain, around 20 times smaller than the estimate of the true change.
We were able to confirm that the mean transitivity error was typically
around 0.2–0.4%, regardless of whether a standard anatomical or
statistical ROI is used (see Fig. 6). This error accounts for most of the
remaining offset in the data of around 0.28% in Fig. 3. As this error is
weakly correlated with the true biological change, subtracting it may
even reduce the discriminative power of the measures.
Discussion
First, we are grateful to Thompsonand Holland (2011) for pointing
out thenonlinearoffsetof 1.2–1.4%in our previously reportedatrophy
rate measures. Although some of this offset may result from biological
sources, we showed that the intercept from all sources (including
biological departures from linearity) is only 0.28% when using
inverse-consistent registration to estimate the brain changes. In-
verse-consistency errors in our new measures of change were
effectively zero throughout the brain (Fig. 2). With these measures
of change, our power estimates for clinical trials were competitive
withothersin the literature. To demonstratea 25% slowingof atrophic
rates with 80% power, 62 AD and 129 MCI subjects would be required
for a 2-year trial, and 91 AD and 192 subjects for a 1-year trial. These
are 1.6–2.4 times higher than our prior sample size estimates, but not
5–16 times higher as alleged in the Thompson and Holland (2011)
analysis of our prior numeric summaries. Power was re-gained by
using inverse-consistent registration to compute the brain changes.
Some re-interpretation of past results is warranted in the light of
these new estimates, and we suggest that our new table (Table 2)
should be consulted first, in order to appreciate the very wide range of
the confidence intervals on these sample size estimates. Although the
notion of inverse consistency is important, it is noteworthy how little
the sample size estimates have changed relative to the very large
Fig. 2. Measures of inverse consistency. (a) Map of inverse consistency error. The
inverse consistency error is less than a few thousandths of a millimeter, except outside
the brain, in a typical, representative mapping from a control individual scanned twice
with a 12-month interval. (b) Map of ||Dh|−|D(inv(h−1)||, for the image in (a). On
average, the error within the brain is on the order of 0.1% change, likely to be reduced
during Jacobian integration over an ROI. Thus, ICE is unlikely to be the major factor
contributing to the remaining nonlinearity in the data, which is very small and may
reflect sampling, biological nonlinearities, or other factors. (c) Map of relative inverse
consistency error, ICE/||x−h(x)||. Relative error is under 5%, and much lower in the
vast majority of the image. This upper bound on the inverse consistency error is well
below 5% of the measured displacement all over the brain, regardless of whether the
displacement is small or large.
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known uncertainty of the estimator itself, which is given by its 95% CI.
Any future clinical trial using this kind of estimate would be expected
to base their power predictions on both the upper and lower
confidence limits. Specifically, in almost all of our prior TBM papers,
including those initially published in ADNI (Leow et al., 2006, 2009;
Hua et al., 2008a; Hua et al., 2008b), we used a 3D inverse-consistent
elastic registration method to measure change, known as 3DMI (Leow
et al., 2005). As shown formally and with substantial empirical data in
Leow et al. (2007), inverse consistency is numerically enforced. More
recently, there were four published papers in which we changed our
registration method to sKL-MI (Yanovsky et al., 2009), because it
appeared to offer more desirable properties such as formal mathe-
matical symmetry. An independent study by Tagare et al. (2006,
2009) noted that sKL, as we formulated it in Yanovsky et al. (2009) is
advantageous, as it is an inherently symmetric cost function. We
reported sample size estimates from sKL-derived registrations in two
papers: Hua et al. (2009, 2010), so these estimates may need to be
revised upwards. In addition, there is reference to sKL-derived power
estimates in Kohannim et al. (2010) and Ho et al. (2010). Our new
findings regarding registration asymmetry would suggest that the
sample size estimates in those papers should be roughly doubled,
while bearing in mind that there is still a 4–5 fold difference in the
upper and lower confidence limits, reported here and in the past, so
the measures should not be treated as if they are precise in any case.
Symmetry and inverse-consistency
The results of our registration methods are inverse-consistent, i.e.,
symmetrical: findings are the same regardless of the order of the
images. In general, a registration algorithm will not automatically be
symmetric; to achieve symmetry, it requires either equivalent
perturbation methods (which we used), or a full space–time (4D)
optimization for every pair of images (as is done by the SyN algorithm
by Avants et al., 2008, for example).
Transitivity
Transitivity error is another source of error in maps of brain
change. In our experiments, this contributes about 0.28% to the
observed changes, or a few thousandths of a millimeter in a typical 1-
mm MRI voxel. Further reducing transitivity errors requires elaborate
registration schemes that include even more penalty functions to
adjust the registrations based on more than two input images — such
as registering sets of images in groups of three (Geng, 2007).
Transform reconciliation (Woods et al., 1998), and group-wise
Fig. 3. With inverse-consistent registration, the offset is greatly reduced, to around 0.1–0.3% for a statistical region of interest in the temporal lobes (left panel), and 0.15–0.25% for the
temporal lobe gray matter (Temporal-GM; right panel). This offset is explainable interms of statistical variability in the sample (Fig.4), and transitivity error,which is low (Figs. 5 and
6). Cumulative atrophy in both ROIs is roughly linear. Wedo not expect numerical summaries ina pre-defined ROI to give entirely linear trajectories as the focus of atrophy defined at
any one time point does not remain identical over time — it spreads out. For that reason, a statistical ROI based on the 1-year follow-up may catch greater atrophy over intervals that
lie within that one year, and lesser ongoing atrophy thereafter; to boost statistical power, it is created with a deliberate selection approach to detect voxels with greatest atrophy
occurring over one year. In ADNI, controls and AD patients are not scanned at the 18-month time-point.
Fig. 4. Cumulative atrophy shows a linear trend in controls (n=152) — the same data
set of controls as used in Fig. 3. The black line shows the best linear fit to all the data
points using a linear mixed effects model. The top panels are based on measures of
cumulative atrophy at 6, 12, and 24 months inside the statistical ROI (a) and temporal
lobe gray matter (b). Regressions inthis first row leave out the known data point of zero
change at baseline, to see what intercept would be inferred from the other data points.
The bottom panels (c), (d) include the known data point at baseline (where the change
is zero by definition) but do not force the line through the origin. Intercept estimates
are (a) statistical ROI: 0.29% (95% CI [0.15, 0.44]), (b) temporal lobe gray matter: 0.28%
(95% CI [0.15, 0.42]), (c) statistical ROI including the known data point at baseline:
0.12% (95% CI [0.05, 0.19]), (d) temporal lobe gray matter including the known data
point at baseline: 0.11% (95% CI [0.04, 0.18]). The red dotted lines show the 95%
confidence intervals onthe regression lines. The plots are generated by Rusing the lme4
package. This plot demonstrates the heterogeneity of any biological sample. Some
outliers may influence the intercept (see the lowest and highest points at 6 months).
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registration methods (Leporé et al., 2008), compute a set of mappings
between all N brains in a study, and use the internal consistency
among mappings as a means to reduce errors, or simply to
redistribute the mean error among all the mappings. In Leporé et al.
(2008), we proposed a method called multi-atlas tensor-based
morphometry; this uses groupwise registration to reduce the error
and boost statistical power in a cross-sectional TBM study. At the
expense of very high computational times, we mapped every brain in
the study to all others, and used arithmetic relations among mappings
to reduce errors. Such highly CPU-intensive groupwise registration
methods are more appropriate for small studies, and not yet realistic
to apply to datasets the size of ADNI. One promising groupwise
registration method is hierarchical correspondence detection by
clustering (Wu et al., 2010a,b). This identifies which correspondences
in a set of subjects are robust, and uses them to guide anatomical
correspondence detection among all subjects, and across time-points.
A second way to achieve formally transitive registration is to use a
registration target different from all brains in the study, and compute
mappings between brains by concatenating the maps to this target
and their inverses (Škrinjar et al., 2008). Such a method is formally
transitive, yet borders on being exhaustive algorithmically and may
result in more overall error in the individual mappings.
Biological nonlinearity
By expecting linear trajectories for the measures of atrophy over
time, as is implied by Thompson and Holland (2011), it is assumed
that the atrophy rate remains constant, at least in aggregate, across a
group. In 39 healthy controls (aged 31–84), a paper by another
research group on a different sample, Scahill et al. (2003), found that
rates of changes accelerated, especially after 70 years of age, in the
ventricles (pb0.001) and hippocampi (p=0.01). At the time of
writing (December 2010), to the best of our knowledge, there are no
other voxel-based brain mapping studies from ADNI that use more
than two time-points. Schuff et al. (2009) and McEvoy et al. (2009)
published the only two studies we know of that examined more than
two ADNI time-points. Schuff et al. (2009) examined hippocampal
volume in 112 normal elderly, 226 MCI and 96 AD patients who all
had at least three successive MRI scans at 0, 6 and 12 months. In
both MCI and AD (p=0.0001), but not in normal controls, rates of
hippocampal loss were slightly faster in the 6–12 than the 0–6 month
interval. McEvoy et al. (2009) reported changes in various ROIs
between 0–6 and 6–12 months. In that paper, if lines were drawn
connecting the mean values at the 6 and 12 month time points, and
extrapolated back to zero, many would not pass through zero (see
Fig. 7 in that paper). Depending on the ROI chosen, the changes in the
second six months are between half and double the changes occurring
over the first six months. This suggests caution in ascribing too much
meaning to small intercepts that are extrapolated using linear
assumptions, based on data that clearly depend on the sample of
subjects assessed and the region chosen.
Longitudinal MRI studies at multiple time-points indicate that
overall brain volume loss, in general (Chan et al., 2003; Carlson et al.,
2008), and hippocampal volume loss, in particular (Ridha et al., 2006;
Jack et al., 2008), may accelerate in patients with MCI and Alzheimer's
disease, but many of these studies have follow up intervals as long as
10 years. As this acceleration effect is not detected in our data, people
with accelerating atrophy may either (1) participate in ADNI in lower
proportions or drop out in higher proportions than those with linear
or decelerating atrophy, or (2) be less likely to have a full-time series
of scans every 6 months for 2 years due to their rapidly accelerating
disease progression. Analysis of later ADNI time-points with multiple
methods should shed light on this unresolved question.
The Yushkevich and Tagare effects
Yushkevich et al. (2010) noted one potential source of bias in
longitudinal image analysis, arising from differences in interpolating
baseline and follow-up images after global normalization. In our own
registration pipelines (here and previously), this specific problem
noted by Yushkevich et al. (2010) was not an issue, as our baseline
and follow-up images were treated equivalently during re-alignment,
re-sampling and interpolation. Related to our argument in Leow et al.
(2007), Tagare et al. (2009) noted that almost all registration
algorithms compute correspondences between images by summing
up quantities in the coordinate system of one image (the source), the
other (the target) or both. He noted that our original cost function,
sKL-MI, introduced in Yanovsky et al. (2009), is formally symmetric,
while many others – that are now widely used – are not. He proposed
a numerical scheme to guarantee symmetry by computing all
quantities (including the intensity matching term) in a coordinate
system that is weighted using the Jacobian determinant (Tagare,
2010). He also advocates using a specific differential form (a concept
from exterior calculus) when computing registration cost functions,
such as intensity correspondence and smoothness of the warp (cf.
Cachier and Rey, 2000). We tried this in our nonlinear registration
work by using the square-root of the Jacobian to weight volumetric
integrals (Leow et al., 2007; cf. Noblet et al., 2008). We have not
explored its empirical consequences here, but it may boost power in
clinical applications of TBM. Extremely computationally demanding
4D methods have also been proposed, that use a subject's entire 4D
time-series to infer a continuous evolution of shapes or “hyper-
Table 1
Sample size estimates for drug trials. Here weshow the estimated sample sizes required
to detect a 25% slowing in the amount of atrophy over 12 months or 24 months, with
80% power, for AD and MCI subjects. Sample sizes are smaller for AD than MCI, and not
very different for statistical versus standard ROIs. These are highly competitive with
other methods.
24 months,
using a
statistical ROI⁎
12 months,
using a
statistical ROI⁎
24 months,
temporal lobe
gray matter
12 months,
temporal lobe
gray matter
AD (N=91)
MCI (N=188)
62
129
91
192
86
161
118
251
⁎ ImpliesusingasinglestatisticalROIbasedonthe12-monthchangemaps,thresholded
atpb0.0001,uncorrected.ThesamestatisticalROIwasappliedtoallscans,forconsistency.
Table 2
Numerical values of the cumulative atrophy, as a percentage, as a mean loss (x) – the negative sign is omitted here – and standard deviation (s). The 95% confidence interval (c) for
the n80 sample size measure was estimated from 10,000 bootstrapped samples.
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templates” from a set of observations of the same subject (Durrleman
et al., 2009; Avants et al., 2010; Khan et al., 2010). When Lorenzi et al.
(2010) analyzed ADNI data from 8 MCI subjects at 4 time-points, they
noted extremely erratic trajectories for brain change (Fig. 3 of that
paper) that they smoothed by fitting a velocity field through all the
images. Use of a full time-series for hundreds of subjects is computa-
tionally difficult and has not been attempted on datasets the size of
ADNI; it also requires re-processing of all time-points when a new
scan comes in from one subject.
Summary
In addition to offering high power to assess factors influencing
brain change, TBM provides 3D anatomical maps showing the region
and rate of brain changes, which are not necessarily provided by other
numeric summary methods. As noted by Scahill et al. (2002) in their
early work on AD with fluid registration, having maps of changes is
advisable for treatment trials, in case treatments show region-specific
effects, or beneficial effects in regions not surveyed or anticipated
when focusing on a volume measure for a pre-selected region.
Therefore, it seems reasonable to use TBM for longitudinal estimation
of atrophy, so long as possible confounds and sources of error are
recognized when interpreting the estimated changes.
Acknowledgments and author contributions
We thank Wes Thompson and Dominic Holland for noticing
surprising aspects of our prior data that we address here. Data
collection and sharing for this project was funded by the Alzheimer's
Disease Neuroimaging Initiative (ADNI) (National Institutes of Health
Grant U01 AG024904). ADNI is funded by the National Institute on
Aging, the National Institute of Biomedical Imaging and Bioengineer-
ing, and through generous contributions from the following: Abbott,
AstraZeneca AB, Bayer Schering Pharma AG, Bristol-Myers Squibb,
Eisai Global Clinical Development, Elan Corporation, Genentech, GE
Healthcare, GlaxoSmithKline, Innogenetics, Johnson and Johnson, Eli
Lilly and Co., Medpace, Inc., Merckand Co., Inc., Novartis AG, Pfizer Inc,
F. Hoffman-La Roche, Schering-Plough, Synarc, Inc., and Wyeth, as
well as non-profit partners the Alzheimer's Association and Alzhei-
mer's Drug Discovery Foundation, with participation from the U.S.
Food and Drug Administration. Private sector contributions to ADNI
are facilitated by the Foundation for the National Institutes of Health
(www.fnih.org bhttp://www.fnih.orgN). The grantee organization is
the Northern California Institute for Research and Education, and the
study is coordinated by the Alzheimer's Disease Cooperative Study at
the University of California, San Diego. ADNI data are disseminated by
the Laboratory of Neuro Imaging at the University of California, Los
Angeles. This research was also supported by NIH grants P30
AG010129, K01 AG030514, and the Dana Foundation. Algorithm
development and image analysis for this study was funded by grants
to P.T. from the NIBIB (R01 EB007813, R01 EB008281, and R01
EB008432), NICHD (R01 HD050735), and NIA (R01 AG020098).
Author contributions were as follows: XH, BG, CB, PR, AL, AK, IY, AT,
and PT performed the image analyses, algorithm developments and
evaluations; CJ, NS, GE, KC, ER, and MW contributed substantially to
the image and data acquisition, study design, quality control,
Fig. 6. Transitivity errors (TE) plotted versus the true change (true change=J(hAC)−TE)
in both temporal lobe and statistical ROIs. The transitivity error (y-axis) is plotted on a
scale 10 times smaller than the true change (x-axis). This error is typically a very small
contributor to the overallchange,with typicalvaluesofaround0.2%–0.3%,or7–10% of the
overall change. This small error accounts for the remaining offset in the data. Clearly, the
TE is weakly correlated with the true change, so subtracting it may even reduce the
discriminative power of the measures. It can be eliminated with highly CPU-intensive
group-wise registration schemes (see Discussion).
Fig. 5. Transitivity error is small and sufficient to account for the remaining intercept of
up to 0.3%. Here we show maps of the average change over 2 years. Blue colors (top left)
show cumulative atrophy of around 2–10% in AD, and red colors show ventricular
expansion of around 10%. The transitivity error map is typically around 0.2–0.3%,
accounting for under one tenth of the signal; regional summaries are shown in Fig. 6.
The transitivity error is generally positive, showing that the direct mapping (which is
used to estimate atrophy) shows slightly less change than the composition of mappings
from 0 to 12, and 12 to 24 months. This is natural, as there will be some errors
associated with each of the components of the composed mappings. There may also be
biological departure from perfect linearity in the anatomical deformations, making the
direct mapping to the 2-year time point shorter than constructing the mapping via the
1-year time-point.
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calibration and pre-processing, databasing and image analysis. We
thank Anders Dale for his contributions to the image pre-processing
and the ADNI project. We thank Jason Stein, Neda Jahanshad, and
Sarah Madsen for their comments on this manuscript.
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