**by Paul Thompson**

One of our on-going projects involves the construction of a probabilistic atlas of the human
brain, which retains information on how brain structure and function
vary in large populations. This growing atlas will ultimately contain
data from 7,000 subjects,
together with algorithms to identify clinically and neurobiologically important
structural and functional patterns in whole populations. The theory behind the atlas
is explained below. Please
feel free to
contact me if you have any questions!

Design of appropriate reference systems for human brain data presents considerable challenges, since these systems must capture how brain structure and function vary in large human populations, across age and gender, in different disease states, across imaging modalities, and even across species.

Due to pronounced anatomic variability between individual human brains, any atlas or clinical diagnostic system based on a single subject's anatomy cannot succeed fully. To realize the quantitative potential of digital atlases, data from single subjects must be extendable to populations (Mazziotta et al., 1995). Atlasing considerations suggest that a statistical confidence limit, rather than an absolute representation of neuroanatomy, may be more appropriate for representing particular subpopulations.

Probabilistic atlasing is a research strategy whose goal is to generate anatomical templates that retain quantitative information on inter-subject variations in brain architecture (Mazziotta et al., 1995). A digital probabilistic atlas of the human brain, incorporating precise statistical information on positional variability of important functional and anatomic interfaces, may rectify many current atlasing problems, since it specifically stores information on the population variability.

Methods to create probabilistic brain representations currently fall into three major categories, each differing slightly in its conceptual foundations. The three methods are: the density-based, label-based, and deformation-based approaches. Benefits of each approach are outlined below.

**1. Density-Based Approaches.** Initial approaches to population-based
atlasing concentrated on generating 'average' representations of anatomy by
intensity averaging of multiple MRI scans (Evans et al., 1992; Andreasen et
al., 1994). A large number of MRI scans are each linearly transformed into
stereotaxic space, intensity-normalized and averaged on a voxel-by-voxel
basis, producing an average intensity MRI dataset. The average brains that
result have large areas, especially at the cortex, where individual
structures are blurred out due to spatial variability in the population.
While this blurring limits their usefulness as a quantitative tool, the
templates can be used as targets for the automated registration and mapping
of MR and co-registered functional data into stereotaxic space (Evans et
al., 1994).

**2. Label-Based Approaches.** In label-based approaches (Evans et al.,
1994; also known as SPAM approaches, short for 'statistical/probabilistic
anatomy maps'), large ensembles of brain data are manually labeled, or
'segmented', into sub-volumes, after mapping individual datasets into
stereotaxic space. A probability map is then constructed for each segmented
structure, by determining the proportion of subjects assigned a given
anatomic label at each voxel position in stereotaxic space (Evans et al.,
1994; Otaky et al., 1995; Paus et al., 1996). The prior information which
these probability maps provide on the location of various tissue classes in
stereotaxic space has been useful in designing automated tissue classifiers
and approaches to correct radio-frequency and intensity inhomogeneities in
MR scans (Zijdenbos and Dawant, 1994). In our laboratory, we have also used
SPAM probabilistic maps to constrain the search space for significant
activations in PET and SPECT imaging experiments (Dinov et al., 1998; Mega
et al., 1998). Statistical data on anatomic labels and tissue types normally
found at given positions in stereotaxic space provide a vital independent
source of information to guide and inform mathematical algorithms which
analyze neuroanatomic data in stereotaxic space.

**3. Deformation-Based Approaches.** As noted earlier, when applied to
two different 3D brain scans, a
non-linear registration or warping algorithm
calculates a deformation map which matches up brain structures in
one scan with their counterparts in the other. The deformation map indicates
3-dimensional patterns of anatomic differences between the two subjects. In
probabilistic atlases based on deformation maps (Thompson and Toga, 1997;
Thompson et al., 1997), statistical properties of these deformation maps are
encoded locally to determine the magnitude and directional biases of
anatomic variation. Encoding of local variation can then be used to assess
the severity of structural variants outside of the normal range, which may
be a sign of disease (Thompson et al., 1997). A major goal in designing this
type of pathology detection system is to recognize that both the magnitude
and local directional biases of structural variability in the brain may be
different at every single anatomic point (Thompson et al., 1996). In
contrast to the intensity averaging of other current approaches (Evans et
al., 1992; Andreasen et al., 1994), an anisotropic random vector field
framework is introduced to encode directional biases in anatomic variability
and map out abnormalities in new subjects (Thompson et al.,1997b).

The three major approaches for probabilistic atlas construction differ only in the attribute whose statistical distribution is modeled and analyzed. Random vector fields (i.e. vector distributions of deformation vectors at each point in space) are analyzed in approaches based on deformation maps, while random scalar fields are used to model MR intensity statistics in the density-based approach, and to model the incidence of binary labels in space in the label-based approach.

**Encoding Brain Variation.** Realistically complex mathematical
strategies are needed to encode comprehensive information on structural
variability in human populations. Particularly relevant is 3-dimensional
statistical information on group-specific patterns of variation, and how
these patterns are altered in disease. This information can be
encoded so that it can be exploited by expert diagnostic systems, whose goal
is to detect subtle or diffuse structural alterations in disease. Strategies
for detecting structural anomalies can leverage information in databased
anatomic data by invoking encoded knowledge on the variations in geometry
and location of neuroanatomic regions and critical functional interfaces,
especially at the cortex.

**Probabilistic Atlases of Cortical Patterns.** The random vector field
approach is a general strategy to construct population-based atlases of the
brain (Thompson and Toga, 1997). Briefly, given a 3D MR image of a new
subject, a high-resolution parametric surface representation of the cerebral
cortex is automatically extracted (Fig. 3). The algorithm then calculates a
set of high-dimensional volumetric maps, elastically deforming this surface
into structural correspondence with other cortical surfaces, selected one by
one from an anatomic image database. The family of volumetric warps so
constructed encodes statistical properties of local anatomical variation
across the cortical surface. Specialized strategies elastically deform the
sulcal patterns of different subjects into structural correspondence, in a
way which matches large networks of gyral and sulcal landmarks with their
counterparts in the target brain (Fig. 3). Differences in the serial
organization of cortical gyri prevent exact gyrus-by-gyrus matching of one
cortex with another, but an important intermediate goal has been to match a
comprehensive network of sulcal and gyral elements which have consistent
topology across subjects (Drury et al., 1996; Thompson et al., 1997). A
probability space of random transformations, based on the theory of
anisotropic Gaussian random fields, is then used to encode information on
complex variations in gyral and sulcal topography from one individual to
another (Fig. 5). Confidence limits in stereotaxic space are determined, for
cortical surface points in the new subject's brain, enabling the creation of
color-coded probability maps to highlight and quantify regional patterns of
deformity in the anatomy of new subjects.

**Pathology Detection.** In view of the clear clinical and research
applications, mathematical methods have been developed to create
probabilistic measures of anatomic variation which are capable of detecting
pathology. We (Thompson et al., 1996b) proposed a simple type of anomaly
detection framework. Deformation fields are calculated which match elements
of the deep surface anatomy in a range of N normal subjects. After affine
components of the deformation fields are factored out, deviations from the
mean deformation field are modeled, for small N, as a Hotelling's T-squared
distributed random field, or for very large N, as a chi-squared distributed
random field with 3 degrees of freedom, defined at nodes (u,v) in parametric
mesh models of the anatomy of new subjects. Thompson et al. (1997) and Cao
and Worsley (1998) proposed the use of a Hotelling's T-squared random field
to detect abnormal deformations between groups, modeling the 3D deformation
field, at each location, by a trivariate normal distribution with an
arbitrary covariance tensor that allows for correlation between the
coordinates. A T-squared or F statistic which indicates evidence of
significant difference in deformations between the groups is calculated at
each lattice location in a 3D image or parameterized 3D surface, to form a
statistic image. Under the null hypothesis of no abnormal deformations, the
statistic image is approximated by a T-squared random field. The global
maximum of the random field can be used to test the hypothesis of no change
(Worsley, 1994a,b; Cao and Worsley, 1998). Random field approaches, some of
which are now widely used in packages for analyzing functional brain images
(Friston et al., 1995), use the Euler characteristic (EC) of the excursion
sets of a random field as an estimator of the number of local non-zero
signal components, above a given threshold in a statistic image. They also
use the expected value of the EC as an approximate p-value for the local
maximum (Worsley, 1994a,b). Probabilistic atlases based on random
deformation fields have been used to assess gender-specific differences in
the brain (Cao and Worsley, 1998), and to detect structural abnormalities in
neurodegenerative disorders such as Alzheimer's disease (Thompson
et al., 1997).

**Shape Theory Approaches.** Deformation fields expressing neuroanatomic
differences have also been analyzed using Procrustes methods, developed for
the statistical analysis of biological shape (Bookstein, 1989; 1997). In
Procrustes methods, affine components of neuroanatomic difference are
factored out not by stereotaxic alignment, but by rotating and scaling
configurations of point landmarks in each subject into least-squares
correspondence with a Procrustes mean shape. Residual deformations which
reflect individual change or anatomic difference are then expressed in terms
of an orthogonal system of principal deformations derived from the bending
energy matrix of the operator which governs the deformation (Bookstein,
1997). Of particular relevance are methods used to define a mean shape in
such a way that departures from this mean shape can be treated as a linear
process. Linearization of the pathology detection problem, by constructing
Riemannian shape manifolds and their associated tangent spaces, allows the
use of conventional statistical procedures and linear decomposition of
departures from the mean to characterize shape change. These approaches have
been applied to detect structural anomalies in schizophrenia (DeQuardo et
al., 1996; Bookstein, 1997).

**Pattern-Theoretic Approaches.** In a related approach based on pattern
theory (Grenander and Miller, 1994), a spectral approach to representing
anatomic variation is developed. This approach, like the approaches
described above, builds on the framework of deformable atlases by
representing variation in terms of probabilistic transformations applied to
deformable neuroanatomic templates. Deformation maps expressing variations
in normal anatomies are calculated, with a non-linear registration procedure
based on continuum-mechanics. In this formulation, the deformational
behavior of each subject's anatomy, driven into correspondence with other
anatomies, is expressed as a system of partial differential equations. The
equations are governed by a differential operator controlling the way in
which one anatomy is deformed into the other, and its properties can be used
to make the deformation reflect the mechanical properties of deformable
elastic or fluid media. Common choices of the differential operator are the
Laplacian (Joshi et al., 1995), biharmonic (Bookstein, 1989) and
Cauchy-Navier operator (Bajcsy and Kovacic, 1989; Gee et al., 1993, 1995;
Miller et al., 1993; Christensen et al., 1996; Davatzikos, 1996; Thompson
and Toga, 1998). Each deformation map is then expanded in terms of the
eigenfunctions of the governing operator, and Gaussian probability measures
are defined on the resulting sequences of expansion coefficients. Currently
being tested as a framework for representing anatomic variation, this
pattern-theoretic approach builds on the framework of deformable atlases and
shows promise in the automated detection of pathology.

Paul Thompson

73-360 Brain Research Institute

UCLA Medical Center

10833 Le Conte Avenue

Westwood, Los Angeles CA 90095-1761, USA.

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