1 MGH/MIT/Harvard Medical School Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, MGH, 13th Street, Charlestown, MA 02129, USA, 2 Artificial Intelligence Laboratory, MIT, 13th Street, Charlestown, MA 02129, USA, 3 Laboratoire dAnatomie, Faculté de Médecine/CHRU, Tours, France, 4 INSERM U316, Tours, France, 5 Harvard Department of Psychiatry at Massachusetts Mental Health Center, 13th Street, Charlestown, MA 02129, USA, 6 Harvard Department of Psychiatry at Massachusetts General Hospital, 13th Street, Charlestown, MA 02129, USA, 7 Harvard Institute of Psychiatric Epidemiology and Genetics, 13th Street, Charlestown, MA 02129, USA, 8 Center for Morphometric Analysis, Department of Neurology, MGH, 13th Street, Charlestown, MA 02129, USA
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Abstract |
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Introduction |
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The inclusion of prior information is a critical feature of a cortical parcellation algorithm. The reason why this is the case is that the divisions that are useful in a cortical parcellation scheme can be based on properties of the brain other than cortical geometry. Many disparate pieces of information, such as knowledge of structurefunction relationship, and cytoarchitectonic or receptor labeling properties of regions, may be used by a neuroanatomist in generating a cortical parcellation information that is not directly available to an automated parcellation procedure from magnetic resonance imaging (MRI) data. A trivial example of this is the fact that many unbroken sulci change names as lobar boundaries are crossed. In addition, functional heterogeneity can make it desirable for adjacent areas to be assigned different labels despite the absence of macroscopic cortical features to distinguish them. Conversely, there are situations in which the secondary folding structure of the cortex (i.e. folds within folds) carries information, such as the location of the hand area in primary motor cortex, which is well predicted by a posterior-pointing secondary fold in the precentral gyrus (Boling et al., 1999).
Various approaches have been taken to the problem of labeling of cortical features. For example, Sandor and Leahy (1997) use a manually labeled atlas brain, which is then warped into correspondence with an individual subjects surface model. The individual subject surface then inherits the labels from the atlas. Similar surface-fitting approaches have been developed by Thompson et al. (1996b
), Valiant et al. (1996
), Lohmann (1998
) and Lohmann and von Cramon (1998
, 2000). A graph-based approach is taken in Goualher et al. (1999
), in which sulci are represented by vertices in the graph and arcs connecting them represent their relationship to one another. A manual training set is then used in order to generate a semi-automated procedure for assigning neuroanatomical labels to the detected sulci. Another graph-based approach was taken in Mangin et al. (1995
), in which a 3-D skeletonization of cortical sulci that is robust to topological errors is generated and used to construct a graph for sulcal labeling. A somewhat different technique was developed by Lohmann (1998
). In this approach, sulci are not represented as abstract quantities such as vertices in a graph, but instead are derived using a watershed transform on an MR volume. This work was extended by Rettmann et al. (2002
) to the use of surface models. The segmented regions identified by the watershed transform are then manually labeled by a trained neuroanatomist.
Here we present a technique for using manually labeled data as the basis for an automated parcellation procedure, using an intrinsically cortical coordinate system to store prior statistics and class-conditional densities (Fischl et al., 1999a,b). The use of the manual parcellation as a training set allows the technique to incorporate neuroanatomical convention into the parcellation in regions in which geometry alone is not predictive of a parcellation label. The procedure models the parcellation labels as a first order anisotropic non-stationary Markov random field (MRF), allowing it to capture the spatial relationships between parcellation units that are present in the training set. The anisotropy separates label probabilities in the first and second principal curvature directions, in order to encode the increased probability of differing labels in the direction of maximal curvature. This type of modeling allows the procedure to automatically encode information such as precentral gyrus frequently neighbors central sulcus in the direction of high curvature (moving across the sulcus), but not in the direction of low curvature (along the banks of the sulcus). This type of parcellation has commonly been generated by a having a trained anatomist or technician manually label some or all of the structures in the cortex, a procedure that can take up to a week for high-resolution images. Here, we use the results of the manual labeling using the validated techniques of the Center for Morphometric Analysis (CMA) (Rademacher et al., 1992
; Caviness et al., 1996
) to automatically extract the information required for automating the parcellation procedure. In addition, in order to demonstrate that the technique is sufficiently flexible to encode a different set of neuroanatomical conventions, we show comparable results using an entirely different surface-based (SB) parcellation scheme (Destrieux et al., 1998
).
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Materials and Methods |
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Thus, we base the cortical parcellation procedure on a number of pieces of information incorporated into a space-varying classification procedure. That is, class statistics (e.g. means and covariance matrices) are tabulated regionally throughout an atlas space, using a registration procedure that optimally aligns cortical folding patterns (Fischl et al., 1999b). Prior probabilities are computed via a frequency histogram in the atlas space, allowing the calculation of the probability that each parcellation label occurs at every atlas location. Finally, the prior probability of a given spatial arrangement of parcellation labels is incorporated into the final parcellation procedure. These priors are also computed from a training set, examples of which are given in Figure 1, for each point in the atlas by modeling the parcellation as an anisotropic non-stationary Markov random field, resulting in a procedure that is comparable in terms of accuracy to manual labeling.
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The problem of automatically labeling cortical gyri and sulci can be naturally phrased within the framework of Bayesian parameter estimation theory. In this approach, one can relate the probability of a parcellation P given the observed surface model S to the probability p(S|P) of the geometry of the surface model occurring given a certain parcellation, together with the prior probability of the segmentation:
The primary advantage of the Bayesian approach is that it allows for the explicit incorporation of prior information via the p(P) term in equation (1). In order to render the problem more tractable in the face of the large degree of variability in cortical folding patterns, both the priors on P and the conditional probability of observing the surface given the classification p(S|P) can be expressed within an atlas space, allowing them to vary as a function of position on the cortical surface (hence making them non-stationary). The advantage of using an atlas space is that coordinates in the atlas have more anatomical meaning than the native coordinate system of the image (Bajcsy et al., 1983; Bookstein, 1989
; Miller et al., 1993
; Gee et al., 1994
; Vannier et al., 1994
; Christensen et al., 1996
; Collins et al., 1996
; Ashburner et al., 1997
; Collins and Evans, 1997
; Woods et al., 1998
; Fischl et al., 1999b
; Thompson et al., 2000
). Classifiers can then be distributed throughout the atlas, allowing each one to focus on only the small number of classes that may occur within the region for which the classifier is responsible. The number of classes that occur within a region of space is then directly related to the accuracy of the atlas coordinate system. That is, P(P(r) = c) will be 0 for all but a few values of c at each atlas location r. In practice, if the classifiers are reasonably dense in the atlas space then the number of classes at each location is typically relatively small. In this way, the intractable problem of classifying each surface node into one of 80 or so labels is decomposed into a set of tractable problems of classifying the nodes in each region of the surface model into only a small number of labels.
The definition of the atlas requires the calculation of a function f(r), which takes native image coordinate as input, and returns the coordinate of the corresponding point in the atlas. For f to be useful in this context, the coordinates it returns should be related to the anatomical location of r. This type of mapping therefore provides the ability to meaningfully relate coordinates across subjects. In the most general case, we wish to maximize the joint probability of both the parcellation P and the atlas function f:
The terms p(S|P,f) and p(P|f) in equation (2) provide a natural means for incorporating atlas information into the parcellation procedure. The first term encodes the relationship between the class label at each atlas location and the predicted surface geometry. Using the atlas space, we can allow the class statistics to vary as a function of location, allowing the within-class variations in surface geometry, such as secondary and tertiary cortical folds, to be incorporated into the classification procedure. The second term allows the expression of prior information regarding the spatial structure of the parcellation labels. Finally, the term p(f) provides a means for constraining the space of allowable atlas functions (e.g. continuity, differentiability and invertibility). In the following, we will use the atlas function f described in Fischl et al. (1999b), which establishes a spherical surface-based coordinate system for each cortical hemisphere.
Atlas Construction
In general, two different approaches have been taken to the construction of anatomical atlases from neuroimaging data. The first is to use an individual as a template, and either manually or automatically estimate a transformation that aligns each new dataset with the individual template (e.g. Talairach et al., 1967; Talairach and Tournoux, 1988
; Van Essen and Drury, 1997
; Van Essen et al., 1998
). The alternative is to compile a probabilistic atlas based on the anatomy of a large number of subjects (e.g. Collins et al., 1994
; Fox et al., 1994
; Mazziotta et al., 1995
; Thompson et al., 1997
, 2000). Each of these approaches has strengths and weaknesses. The former allows one to represent anatomical structures at as fine a scale as the neuroimaging technology allows, but the atlas is then biased by the idiosyncrasies of the individual anatomy chosen as the template. The latter technique resolves this problem by averaging across anatomies, thus only retaining that which is common in the majority of subjects. Nevertheless, the cross-subject averaging removes potentially useful information in the atlas. Here, we preserve the advantages of each technique by using a group of subjects to construct an atlas that retains information about each parcellation unit at each point in space. Given the atlas function f, and a group of N manually parcellated subjects, we first estimate the prior probability of parcellation label c occurring at each atlas location, independent of all other locations:
The geometry of the observed surface S at each point is summarized by two quantities encoded as a vector G(r) at each point in the surface (note that here r refers to spherical atlas coordinates, and f(r) refers to individual subject spherical coordinates). The first of these is the average convexity as defined in Fischl et al. (1999a), while the second is the mean curvature of the surface (do Carmo, 1976
). Additional quantities such as MRI derived variables (e.g. T1 and T2), Gaussian curvature and cortical thickness can easily be incorporated into this framework. The likelihood of observing the surface geometry G given the parcellation label P(r) is modeled as a Gaussian, the parameters of which are computed from the manually labeled training set:
where Gi are geometric parameters extracted from each of the set of M surfaces for which label c occurs at location f(r) in the corresponding manually labeled surface models Si [i.e. Si(f(r)) = c]. The covariance matrix for class c at location r in the atlas is then given by:
Information about surface geometry is thus maintained separately for each parcellation label at every location in the atlas, obviating the need to average geometric information across classes. Finally, we also estimate the pairwise probability that parcellation label c2 is the neighbor at ri when parcellation label c1 is the label at r, for ri N(r), where N(r) is a neighborhood function of r.
As before, this information is stored separately for each atlas location. It is important to note that the probabilities are stored separately for each pair of classes as well as for each neighborhood location ri. While it may seem that this would lead to combinatorial explosion and intractable memory requirements, in practice the space is sparse as relatively few configurations of parcellation labels actually occur. Two examples of the manual labeling that is the basis for the atlas are given in Figure 1. The spherical atlas space is made up of a pair of super-sampled icosahedra. The priors are stored at essentially the same resolution as the individual surface models, using an icosahedral model with 163 842 vertices at 1 mm spacing, while the class conditional densities are stored 2562 vertices at
4 mm spacing, reflecting the slower variation of surface geometry for each label.
Parcellation
As mentioned above, the Bayesian approach allows us to incorporate prior information that is necessary for the parcellation procedure. This prior information takes two forms. The first makes use of the global spatial information provided by f and the atlas in order to express the probability that a given parcellation label occurs at a particular location in the atlas, independent of the local surface geometry. The second encodes the local spatial relationship between parcellation labels, allowing information such precentral gyrus can be anterior, superior or inferior to central sulcus, but never posterior to it to be automatically detected and incorporated into the segmentation.
Formally, we compute the maximum a posteriori (MAP) estimate of the parcellation P given an input surface geometry G, and the nonlinear spherical transform f. The MAP estimate can be expressed as maximizing p(P|G,f), the probability distribution of the parcellation given the observed surface geometry and the atlas function. Using Bayes rule we can relate this to the product of the probability of observing the surface geometry with the prior probability of a given spatial configuration of labels p(P):
Assuming the noise at each vertex is independent from all other vertices in the surface model, we can rewrite p(G|P,f) as the product of the distribution at each vertex over the surface:
Note that in the more general case in which the spatial correlation structure of the noise is constant across space, the equality in equation (8) should be replaced by proportionality (Worsley et al., 1992; Thompson et al., 1996a
), which does not effect any of the subsequent derivations. The distribution of the geometric properties of each class at each location in the atlas is modeled as a Gaussian, the mean vector µc(r) and covariance matrix
c(r) of which are computed using equations 4 and 5. The probability of observing the geometry at G(f(r)) is then expressed as:
All that remains is to find an expression for the prior probability of a given parcellation P. Here we assume that the spatial distribution of labels can be well approximated by an anisotropic non-stationary Markov random field. This allows one to encode prior information about the relationship between labels as a function of location within the cortex (i.e. non-stationary), as well as with local direction (i.e. anisotropic). Formally, the Markov assumption can be expressed as:
That is, the prior probability of a label at a given vertex r is only influenced by the labels within some neighborhood of r. The locality restriction imposed by the Markov model permits the probability of the entire parcellation to be written in terms of neighborhood or clique potentials Vc(P) via the HammersleyClifford theorem (Besag, 1974). That is, the probability p(P) can be equivalently characterized by a Gibbs distribution:
where Z is a normalizing constant and will be dropped in the following, and U(P) is an energy function that can be written in the form:
The clique potentials Vc(P) encode the energy associated with a certain configuration of labels within the cth clique. Choosing Vc(P) to be log(p(P(r)|P(r1), P(r2) ... P(rK)), where r is the central vertex of the cth clique, allows us write the probability of the entire parcellation as the product of the probability of the label at each vertex, given its neighborhood:
Using Bayes rule, we can rewrite this as:
Equation (14) allows the probability of a given label to be modulated by any configuration of neighboring labels. While this would be extremely useful, it is unfortunately not computationally tractable to implement, as one would need to compute separate prior probabilities for every combination of neighboring labels that occur. Instead, we make the simplifying assumption that only the first order conditional dependence is important. That is, that the dependence of a label on its neighbors can be expressed as the product of the probability given each of the neighbors:
where again we have explicitly included the dependence on the neighbor location ri to emphasize that the probability densities are maintained separately for each neighbor position in N(r). Using this assumption, we arrive at an expression for the prior probability of the full parcellation:
Equation (16) allows two types of prior information to be incorporated into the segmentation procedure. The approximate location a surface feature may occupy within the cortex is given by p(P(r)), which is computed and stored in the atlas using equation (3). The local relationship between parcellation labels is encoded in p(P(ri)|P(r),ri) using equation (6). We currently let the neighborhood function N(r) include the two principal directions on the surface at each location in the atlas space. This allows the parcellation procedure to store information separately for the direction of highest curvature from that of the direction of lowest curvature, as the majority of borders occur in the former.
Directly computing the global MAP estimate of P in equation (7) using the Markov model of equation (16) is computationally intractable. Instead, we employ the iterated conditional modes (ICM) algorithm proposed by Besag (Besag, 1986). In this approach, the parcellation is initialized with the MAP estimate assuming p(P(ri)|P(r),ri) is uniform, as no label has yet been assigned to each vertex. The parcellation is then sequentially updated at each location by computing the label P(r), that maximizes the conditional posterior probability p(P(r)|P(ri),G,ri):
Equation (17) is then iteratively applied until no vertices are changed. Snapshots of the evolution of this procedure are given in Figure 2. As can be seen convergence is rapid, resulting in a procedure that requires 3 min on a 1.5 GHz Pentium III. A small amount of and post-processing is carried out on the results of the labeling to relabel small isolated patches of label to the most likely of the neighboring labels.
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Results and Discussion |
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The data used for the CMA parcellation are part of an ongoing study of schizophrenia (Seidman et al., 1997; Goldstein et al., 1999
; Seidman et al., 1999
; Goldstein et al., 2002
). As part of this study, 36 MRI volumes (2 MP-RAGE scans per subject, motion corrected and averaged) have been manually parcellated by trained technicians. (Note that as the study is not yet complete, we are blinded to diagnosis.) Cortical models were reconstructed for each of the subjects using previously presented techniques, including non-rigid surface-based alignment to a previously constructed spherical atlas (Dale and Sereno, 1993
; Dale et al., 1999
; Fischl et al., 1999a
,b; Fischl and Dale, 2000
; Fischl et al., 2001
). The manual parcellations were then sampled onto the cortical models at the midpoint of the cortical ribbon using the thickness estimates described in Fischl and Dale (2000
). For the SB parcellation, cortical surface models were reconstructed for 12 subjects. Surface-based drawing tools that are part of the FreeSurfer software package (http://surfer.nmr.mgh.harvard.edu) were then used to generate a complete parcellation for all 24 cortical hemispheres. In order to validate the accuracy of the automated parcellation algorithm, we employed a jackknife/leave-one-out technique in which one dataset is removed from each training set, an atlas is constructed, and used to parcellate the left-out set. The error is then quantified point-by-point as the percentage of points at which the manual and automated parcellation are in disagreement. Figure 3 presents the results of this study on an average inflated surface (dark gray are sulcal regions, and light gray are gyral ones). Note that the vast majority of the surface is below threshold (i.e. accuracy >75%) for both the CMA parcellation on the left as well as the SB parcellation on the right. With the median accuracy at
81% for the left hemisphere and 83% for the right for the CMA parcellation, and 80% (left) and 79% (right) for the SB (the slightly lower accuracy for the SB parcellation is no doubt due to the greater number of parcellation units in it, some of which are quite small). In addition, the low-accuracy regions are clustered at the boundaries between labels, the precise positions of which are somewhat arbitrary in the manual training set.
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Conclusion |
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While manual methods exist for assessing this type of change, the process of manually parcellating an entire high-resolution structural MR volume requires on the order of a week for a trained neuroanatomist or technician. This makes the routine analysis of large patient and control populations untenable. Finally, manual labeling procedures do not generalize well to the use of multidimensional inputs such as cortical thickness measures, intrinsic MRI tissue parameters and cortical geometry, which can be easily incorporated into the automated parcellation procedure.
The automated method described in this paper for assigning a neuroanatomical label to every point in the cortex has been shown to be comparable in terms of accuracy to a previously validated method of manual parcellation. The accurate labeling of a large number of cortical structures is enabled through the use of both global and local spatial information. The global information is encoded by distributing classifiers throughout an atlas in a surface-based coordinate system and maintaining class statistics on a per-class, per-location basis, allowing the classifiers to be robust to variations cortical folding patterns. Local information is incorporated into the classification procedure by modeling the parcellation as a non-stationary anisotropic Markov random field. The introduction of anisotropy and non-stationarity into the parcellation model allows the spatial relationships of parcellation units to one another to be incorporated into the procedure in a principled fashion.
The automatic parcellation procedure can also be used to automatically define regions of interest (ROIs) for use in functional imaging studies. Specifically, this will allow one to generate average time courses by cortical region, or even parts of regions, facilitating for example the comparison of the response of the motor cortex to that of somatosensory cortex, or anterior superior temporal sulcus (STS) to the posterior STS.
While the results presented here are comparable to those achievable by current manual labeling techniques, we intend to extend them in a number of ways. Primarily, we will investigate the relationship between the manual and automated labelings of each brain. In particular, the CMA parcellation technique is defined in the volume using slice-based viewers, and is currently being updated to use surface-based drawing and visualization tools. When complete, we expect the surface-based extension of the CMA parcellation to be significantly more accurate than the labeling based on the volumetric parcellation. As part of this process, we will perform structure-by-structure analysis of variability both across manual labelings, and between manual and automated labelings for each of the manual training sets. This analysis should allow us to identify regions of low reliability, which will require either updating the definition of the parcellation labels in the case that neither the automated nor the manual labelings is reliable, or the inclusion of additional information into the automated parcellation procedure. This may include the relationship of the cortical regions to subcortical structures, as well as additional cortical information (e.g. regional thickness measures, intrinsic tissue parameters).
The automated nature of the methods described here, in contrast with existing manual or semi-automated techniques, allows for their routine application in large-scale studies. Having access to this type of detailed morphometric information for large populations including various disorders as well as a spectrum of normal controls should facilitate the characterization of the anatomical signatures associated with specific disorders. Ultimately, this may provide a more accurate and sensitive tool for early diagnosis of brain disorders.
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Appendix |
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AG angular gyrus
BFsbcmp basal forebrain
CALC intracalcarine cortex
Cga cingulate cortex, anterior part
CGp cingulate cortex, posterior part
CN cuneus
CO central operculum
F1 superior frontal gyrus
F2 middle frontal gyrus
F3o inferior frontal gyrus, pars opercularis
F3t inferior frontal gyrus, pars triangularis
FMC frontomedial cortex
FO frontal operculum
FOC fronto-orbital cortex
FP frontal pole
H1 Heschls gyrus
INS insula
LG lingual gyrus
MW medial wall
OF occipital fusiform gyrus
Oli occipital lateral gyri, inferior part
Ols occipital lateral gyri, superior part
OP occipital pole
PAC paracingulate cortex
PCN precuneus
Pha parahippocampal gyrus, anterior part
PHp parahippocampal gyrus, posterior part
PO parietal operculum
POG postcentral gyrus
PP planum polare
PRG precentral gyrus
PT planum temporale
SC subcallosal cortex
SCLC supracalcarine cortex
Sga supramarginal gyrus, anterior part
SGp supramarginal gyrus, posterior part
SMC supplementary motor cortex or juxtapositional lobule cortex
SPL superior parietal lobule
T1a superior temporal gyrus, anterior part
T1p superior temporal gyrus, posterior part
T2a middle temporal gyrus, anterior part
T2p middle temporal gyrus, posterior part
T3a inferior temporal gyrus, anterior part
T3p inferior temporal gyrus, posterior part
Tfa temporal fusiform gyrus, anterior part
TFp temporal fusiform gyrus, posterior part
TO2 middle temporal gyrus, temporo-occipital part
TO3 inferior temporal gyrus, temporo-occipital part
TOF temporal fusiform gyrus, temporo-occipital part
TP temporal pole
UN unknown
Labels for surface-based parcellation
Corpus_callosum
G_and_S_Insula_ONLY_AVERAGE
G_cingulate-Main_part
G_frontal_inf-Opercular_part
G_frontal_inf-Triangular_part
G_frontal_superior
G_insular_long
G_occipital_inferior
G_occipital_superior
G_occipit-temp_med-Lingual_part
G_orbital
G_parietal_inferior-Angular_part
G_parietal_superior
G_precentral
G_rectus
G_subcentral
G_temp_sup-Lateral_aspect
G_temp_sup-Planum_tempolale
G_temporal_middle
Lat_Fissure-ant_sgt-ramus_horizontal
Lat_Fissure-post_sgt
Pole_occipital
S_calcarine
S_central_insula
S_cingulate-Marginalis_part
S_circular_insula_inferior
S_collateral_transverse_ant
S_frontal_inferior
S_frontal_superior
S_intermedius_primus-Jensen
S_intraparietaland_Parietal_transverse
S_occipital_inferior
S_occipital_superior_and_transversalis
S_occipito-temporal_medial_and_S_Lingual
S_orbital_medial-Or_olfactory
S_paracentral
S_pericallosal
S_precentral-Inferior-part
S_subcentral_ant
S_suborbital
S_supracingulate
S_temporal_superior
S_transverse_frontopolar
Unknown
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Notes |
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Address correspondence to Bruce Fischl, MGH/MIT/Harvard Medical School, Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital/Harvard Medical School, Bldg 149, 13th Street, Charlestown, MA 02129, USA. Email: fischl{at}nmr.mgh.harvard.edu
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