university of zurich - uzh · 1) harry b. m.uylings ([email protected]) department of...
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Year: 2009
A mean three-dimensional atlas of the human thalamus:Generation from multiple histological data
Krauth, A; Blanc, R; Poveda, A; Jeanmonod, D; Morel, A; Székely, G
Krauth, A; Blanc, R; Poveda, A; Jeanmonod, D; Morel, A; Székely, G (2009). A mean three-dimensional atlas ofthe human thalamus: Generation from multiple histological data. NeuroImage:Epub ahead of print.Postprint available at:http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.ch
Originally published at:NeuroImage 2009, :Epub ahead of print.
Krauth, A; Blanc, R; Poveda, A; Jeanmonod, D; Morel, A; Székely, G (2009). A mean three-dimensional atlas ofthe human thalamus: Generation from multiple histological data. NeuroImage:Epub ahead of print.Postprint available at:http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.ch
Originally published at:NeuroImage 2009, :Epub ahead of print.
A mean three-dimensional atlas of the human thalamus:Generation from multiple histological data
Abstract
Functional neurosurgery relies on robust localization of the subcortical target structures, which cannotbe visualized directly with current clinically available in-vivo imaging techniques. Therefore, one hasstill to rely on an indirect approach, by transferring detailed histological maps onto the patient'sindividual brain images. In contrast to macroscopic MRI atlases, which often represent the average of apopulation, each stack of sections, which a stereotactic atlas provides, is based on a single specimen. Inaddition to this bias, the anatomy is displayed with a highly anisotropic resolution, leading totopological ambiguities and limiting the accuracy of geometric reconstruction. In this work we constructan unbiased, high-resolution three-dimensional atlas of the thalamic structures, representing the averageof several stereotactically oriented histological maps. We resolve the topological ambiguity bycombining the information provided by histological data from different stereotactic directions. Since thestacks differ not only in geometrical detail provided, but also due to inter-individual variability, weadopt an iterative approach for reconstructing the mean model. Starting with a reconstruction from asingle stack of sections, we iteratively register the current reference model onto the available data andreconstruct a refined mean three-dimensional model. The results show that integration of multiplestereotactic anatomical data to produce an unbiased, mean model of the thalamic nuclei and theirsubdivisions is feasible and that the integration reduces problems of atlas reconstruction inherent tohistological stacks to a large extent.
Elsevier Editorial System(tm) for NeuroImage
Manuscript Draft
Manuscript Number: NIMG-09-56
Title: A mean three-dimensional atlas of the human thalamus: Generation from multiple histological
data
Article Type: Regular Article
Section/Category: Anatomy and Physiology
Keywords:
Corresponding Author: Mr. Axel Krauth, Dipl.-Inf.
Corresponding Author's Institution: ETH Zurich
First Author: Axel G Krauth, Dipl.-Inf.
Order of Authors: Axel G Krauth, Dipl.-Inf.; Rémi Blanc, Dr.; Alejandra Poveda, M.D. Ph.D; Daniel
Jeanmonod, Prof. Dr.; Anne Morel, Ph.D.; Gábor Székely
, Prof. Dr.
Manuscript Region of Origin:
Abstract: Functional neurosurgery relies on robust localization of the subcortical target structures,
which cannot be visualized directly with current clinically available in-vivo imaging techniques.
Therefore, one has still to rely on an indirect approach, by transferring detailed histological maps
onto the patient's individual brain images. In contrast to macroscopic MRI atlases, which often
represent the average of a population, each stack of sections, which a stereotactic atlas provides,
is based on a single specimen. Moreover, the anatomy is displayed with a highly anisotropic
resolution, leading to topological ambiguities and limiting the accuracy of geometric reconstruction.
In this work we construct a high-resolution three-dimensional atlas of the thalamic structures,
representing the average of several stereotactically oriented histological maps. We resolve the
topological ambiguity by combining the information provided by histological data from different
stereotactic directions. Since the stacks differ not only in geometrical detail provided, but also in
inter-individual variability, we adopt an iterative approach for reconstructing the mean model.
Starting with a reconstruction from a single stack of sections, we iteratively register the current
reference model onto the available data and reconstruct a refined mean three-dimensional model.
The results show that the integration of multiple stereotactic anatomical data to produce a mean
model of the thalamic nuclei and their subdivisions is feasible and that the integration reduces
problems of atlas reconstruction inherent to histological stacks to a large extent.
Dear Members of the Editorial Board of “Neuroimage”, Please find enclosed our manuscript entitled “A mean three-dimensional atlas of the human thalamus: Generation from multiple histological data”, that we would like to submit for publication to “NeuroImage”. The present study presents a high-resolution three-dimensional atlas of the human thalamic structures representing the average of several brain specimens and stereotactically oriented histological maps. The results show that the integration of multiple stereotactic anatomical data to produce a mean model of the thalamic nuclei and their subdivisions reduces to a large extent problems of atlas reconstruction inherent to histological stacks. As our work is an application of image analysis methods onto anatomical data, we suggest two reviewers with a more anatomical / clinical background and two reviewers with a more technical background. This work, as a whole or part of it, has not been published nor is under consideration for publication elsewhere, and has been approved by all of the authors. Corresponding author is Axel Krauth, Sternwartstr 7, 8092 Zurich, Switzerland (tel 0041-44-632-6818, fax 0041-44-632-1199, [email protected]). Yours sincerely, Axel Krauth
* 1. Cover Letter
1) Harry B. M.Uylings ([email protected])
Department of Psychiatry and Neuropsychology
Division Brain and Cognition
Maastricht University Medical Center
Maastricht
The Netherlands
2) Jones EG ([email protected])
Center for Neuroscience
University of California
Davis, CA 95616
USA
3) Wieslaw Nowinski ([email protected])
4) Sebastien Ourselin ([email protected])
* 2. Reviewer Suggestions
A mean three-dimensional atlas of the human
thalamus: Generation from multiple histological data
Axel Krautha,∗, Remi Blanca, Alejandra Povedab, Daniel Jeanmonodb,Anne Morelb, Gabor Szekelya
aComputer Vision Laboratory, ETH Zurich, SwitzerlandbFunctional Neurosurgery, University Hospital Zurich, Switzerland
Abstract
Functional neurosurgery relies on robust localization of the subcortical target
structures, which cannot be visualized directly with current clinically available
in-vivo imaging techniques. Therefore, one has still to rely on an indirect ap-
proach, by transferring detailed histological maps onto the patient’s individual
brain images. In contrast to macroscopic MRI atlases, which often represent
the average of a population, each stack of sections, which a stereotactic atlas
provides, is based on a single specimen. Moreover, the anatomy is displayed
with a highly anisotropic resolution, leading to topological ambiguities and lim-
iting the accuracy of geometric reconstruction. In this work we construct a
high-resolution three-dimensional atlas of the thalamic structures, representing
the average of several stereotactically oriented histological maps. We resolve
the topological ambiguity by combining the information provided by histologi-
cal data from different stereotactic directions. Since the stacks differ not only
in geometrical detail provided, but also in inter-individual variability, we adopt
an iterative approach for reconstructing the mean model. Starting with a re-
construction from a single stack of sections, we iteratively register the current
reference model onto the available data and reconstruct a refined mean three-
dimensional model. The results show that the integration of multiple stereo-
tactic anatomical data to produce a mean model of the thalamic nuclei and
∗Corresponding author
Preprint submitted to Elsevier January 9, 2009
* 3. ManuscriptClick here to view linked References
their subdivisions is feasible and that the integration reduces problems of atlas
reconstruction inherent to histological stacks to a large extent.
1. Introduction
Among different applications, anatomical atlases are used to identify and lo-
calize structures to be targeted in neurosurgical treatment of therapy-resistant
neurological disorders (e.g. Parkinson’s disease, neurogenic pain or neuropsy-
chiatric disorders). The targets are localized deep in the brain, mainly in the
thalamus and the basal ganglia, and their coordinates are determined on a
stereotactic anatomical atlas and then transferred onto the patient’s anatomy
using a common stereotactic reference system. With recent developments in
neuroimaging techniques, some structures can be visualized in vivo to a certain
extent. This is the case for the subthalamic nucleus which is commonly targeted
in Parkinson’s disease (Slavin et al., 2006; Breit et al., 2006; Yelnik et al., 2003),
although additional criteria are usually necessary to verify for the target. For the
intrathalamic structures, such as the ventral lateral posterior (or VIM) nucleus,
routine clinical imaging methods provide insufficient contrast for visual detec-
tion (Patil et al., 1999; Nowinski et al., 2006; Mercado et al., 2006). To surpass
these limitations, segmentation of thalamic nuclei with a clustering approach
based on a specific MRI-sequence (Deoni et al., 2007) or on the connectivity pat-
tern of the individual constituents (Behrens et al., 2003; Johansen-Berg et al.,
2005) has been used. Jonasson et al. (2005) and Wiegell et al. (2003) employed
the characteristic fiber orientations provided by DTI-MRI to differentiate be-
tween individual areas. However, authors report problems with detecting small
nuclei and properly differentiating between individual kernels. Indeed, pre-
cise visualization of such fine structures is still beyond the capabilities of cur-
rent clinical imaging techniques, and even postmortem MRI does not allow for
it(Fatterpekar et al., 2002).
In functional neurosurgery, the target structures have to be localized in the
patient with high accuracy. This accuracy is even more essential in case of
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minimally or non-invasive surgical interventions when there is no electrophys-
iological control for the target position, such as in Gamma Knife radiosurgery
(Friehs et al., 2007). In addition to surgical treatment, neuroscience research
also relies on robust identification of individual thalamic structures for func-
tional neuroimaging studies (e.g Devlin et al. (2006)) or for localizations of trau-
matic lesions such as thalamic infarcts (Van der Werf et al., 2003; Montes et al.,
2005).
The original, paper-based atlases (Schaltenbrand and Wahren, 1977; Morel et al.,
1997; Mai et al., 2004; Morel, 2007) are limited by their presentation as stacks
of two-dimensional images. Therefore, several electronic, three-dimensional digi-
talizations have been built (St-Jean et al., 1998; Niemann et al., 2000). Yelnik et al.
(2007) and Chakravarty et al. (2006) focused on the removal of deformation
artefacts introduced during histological processing. Three-dimensional models
of subcortical structures have been built from stereotactic atlases. Kimura and Otsuki
(1993) extracted all surfaces via surface tiling of the contours. Niemann et al.
(2000) and Ganser et al. (2004) interpolated additional sections prior to surface
extraction.
Comparison of the individual stacks of histological sections, of which each at-
las consists of, has shown inconsistencies between them (Niemann and van Nieuwenhofen,
1999; Nowinski et al., 2006). Based on their evaluation, Niemann et al. (2000)
normalize the horizontal and coronal series of the Morel atlas (Morel et al.,
1997) to the sagittal series in a linear fashion. However, stacks of sections are
neither averaged nor merged into a single representation. Likewise, each pro-
vided surface reconstruction is based on a single specimen. As has been noted by
Pitiot and Guimond (2008), such a reconstruction might display the anatomy
only to a partial extent. Moreover, using a single stack in an application intro-
duces a bias towards the underlying specimen.
To overcome these limitations, anatomical atlases should be based on a pop-
ulation instead. Evans et al. (1993) averaged 305 MRIs to construct such a
mean atlas. Their work was improved through the use of non-linear registra-
tion algorithms (Seghers et al., 2004; Lorenzen et al., 2005; Christensen et al.,
3
2006). Building an atlas from a population also allows to capture the anatomical
variability of the structures of interest (Pohl et al., 2004; Styner et al., 2003).
Nonetheless, these works are all based on data acquired by MRI, whose resolu-
tion and visualization capabilities limit their use for targeting finer structures.
The aim of this work is to construct a high-resolution three-dimensional
model of the thalamic structures by combining information contained in his-
tological data from several postmortem brains. This way the model is based
on several specimens instead of a single one. To this end, we used series of
maps of the human thalamus (Morel et al., 1997; Morel, 2007) for the follow-
ing advantages over other commonly used atlases (Schaltenbrand and Wahren,
1977; Mai et al., 2004). Firstly, the structures of interest (thalamus, basal gan-
glia, subthalamic fiber tracts) are displayed in three planes orthogonal to each
other and oriented parallel or perpendicular to the reference stereotactic plane
passing through the centers of the anterior and posterior commissures (ac-pc).
Secondly, the delineation of the structures, based on multiarchitectonic criteria,
provides a high histological resolution, and, thirdly, the maps are provided at
small and regular intervals. The results demonstrate the necessity, caused by the
anisotropic resolution of the individual data sets, to combine stacks of sections
from different stereotactic directions to reconstruct an accurate, topologically
consistent 3-D model.
2. Material and Methods
2.1. Stereotactic anatomical atlas
A stereotactic anatomical atlas consists of series of maps derived from stacks
of histologically processed brain sections. Each series displays the subcortical
structures, most importantly the basal ganglia and the thalamus, in one of
the stereotactic axes. In the used atlas, the reference horizontal stereotactic
axis is defined as the line through the centers of the anterior and the posterior
commissures in the midsagittal plane. This is different to the definition by
Talairach (Talairach et al., 1957), who defines the axial axis to go through the
dorsal border of the anterior and the ventral border of the posterior commissures.
4
The spacing between maps does not allow for the same resolution in the direction
normal to the map compared with the resolution within each map.
Six stereotactic stacks of histologically processed sections were available for
the current study. In addition to the three constituting the first Morel atlas
(Morel et al., 1997), a coronal and a sagittal stack of sections from the left and
the right hemispheres of the same brain with an intercommissural distance of
26mm, which have been introduced in the second Morel atlas (Morel, 2007),
and an additional horizontal stack of sections of a brain with an intercommis-
sural distance of 25mm were used. Postmortem MRIs are available for three of
the stacks (Morel, 2007). Thalamic and basal ganglia subdivisions have been
delimited on the basis of multiple histological criteria provided by staining of
sections for Nissl, myelin, calcium-binding proteins, the non-phosphorylated
neurofilament protein (with SMI-32) and acetylcholinesterase. The final maps,
showing drawings of thalamic and basal ganglia subdivisions, are presented at
regular and close (0.9mm or 1.0mm) intervals. One series (horizontal) has been
complemented to provide higher spatial resolution (0.45mm intervals). Point
coordinates are specified in millimetres relative to the posterior commissure (for
antero-posterior axis), the intercommissural plane (for dorso-ventral axis) and
interhemispheric or ventricular border (for latero-medial axis) (see also Figure 1
for exemplary drawings). The atlas follows the revised Anglo-Saxon nomen-
clature (Table 1) (Jones, 1990). From each stack of sections, the outlines of
each relevant structure have been extracted. Stacks which stem from a right
hemisphere have been mirrored along the midsagittal plane to represent a left
hemisphere. Anatomical areas present have been arranged according to the
neuroanatomical hierarchy (see Table 1).
2.2. Topological landmarks
For a sufficiently good correspondence establishment (see Section 3.2), it
is necessary to extract as much information as possible from the individual
stacks of sections. Each stack provides not only the outlines of the individual
structures of interest, in our case the thalamic nuclei, but also the topological
5
Table 1: Anatomical hierarchy of reconstructed structuresName Abbreviation Name Abbreviation
Thalamic structures Lateral group
Medial group Ventral posterior lateral nucleus VPL
Mediodorsal nucleus MD anterior part VPLa
magnocellular part MDmc posterior part VPLp
parvocellular part MDpc Ventral posterior medial nucleus VPM
Medioventral nucleus MV Ventral posterior inferior nucleus VPI
Central lateral nucleus CL Ventral lateral nucleus VL
Central medial nucleus CeM Ventral lateral anterior nucleus VLa
Centromedian nucleus CM Ventral lateral posterior nucleus VLp
Paraventricular nucleus Pv dorsal part VLpd
Habenular nucleus Hb ventral part VLpv
Parafascicular nucleus Pf Ventral anterior nucleus VA
Subparafascicular nucleus sPf magnocellular part VAmc
Posterior group parvocellular part VApc
Medial pulvinar PuM Ventral medial nucleus VM
Inferior pulvinar PuI Anterior group
Lateral pulvinar PuL Anterior dorsal nucleus AD
Anterior pulvinar PuA Anterior medial nucleus AM
Lateral posterior nucleus LP Anterior ventral nucleus AV
Medial geniculate nucleus MGN Lateral dorsal nucleus LD
Suprageniculate nucleus SG Other structures
Limitans nucleus Li Red nucleus RN
Posterior nucleus Po mammillothalamic tract mtt
Lateral geniculate nucleus LGN Subthalamic nucleus STh
relationships between them. According to our experience we can safely assume
that these relationships are constant. Therefore, generally speaking, the surface
of a nucleus n can be robustly divided into surface patches r1, . . . , rl, each of
which being the region where n is adjacent to another nucleus. The border lines
between neighbouring patches, which result from three nuclei being adjacent,
form a grid on the surface of the nucleus. We define the endpoints of the border
lines as primary, the lines themselves as secondary and the surface patches as
tertiary landmarks (TLM) of the nucleus n. As it is well-known from stereology,
3-D objects of dimension 1 ≤ k ≤ 3 almost surely appear as k − 1 dimensional
when intersected by a section plane, and objects of dimension 0 (i.e. points) are
almost never (i.e. with probability 0) intersected by a section. Consequently,
in the proposed model, tertiary landmarks become lines, secondary landmarks
become points and primary landmarks will not be visible in a section. The visible
landmarks are extracted during preprocessing of the histological sections.
2.3. Anisotropy of histological data
The relatively sparse sampling of the anatomy along the cutting direction
of a stack of sections gives rise to reconstruction problems such as holes or
topological inconsistencies. Indeed, with parallel section planes, the probability
for a secondary or tertiary landmark to be intersected becomes low when it is
6
(a) (b) (c)
Figure 1: Example of different geometrical detail available in each stack of section. SectionsD9.9 (a) and D10.8 (b) belong to a horizontal stack and section L12.7 (c) to a sagittal stack.Anterior and posterior commissure levels are indicated by ac and pc (in red). The straightlines in the sagittal section mark the approximate location of the horizontal sections andvice versa. The transition from the lateral dorsal nucleus (LD) to the medial pulvinar (PuM)demonstrates the anisotropy of stereotactic data. For the horizontal stack, it falls between thetwo adjacent sections shown. In contrast to this, the transition is clearly visible in the sagittalsection. Abbreviations: St = stria terminalis, Cd = caudate nucleus, PuT = putamen, ic= internal capsula, R = reticular nucleus, GPi = globus pallidus (internal segment), GPe =globus pallidus (external segment), iml = internal medullary lamina, ZI = zona incerta, SNc= substantia nigra (pars compacta), SNr = substantia nigra (pars reticulata), ot=optic tract.For other abbreviations see Table 1.
nearly parallel to the section planes, as in the classical Buffon’s needle problem
(Mantel, 1953). This is exemplified by the lateral dorsal nucleus (see Figure 1),
which is separated from the underlying pulvinar and CL nuclei by a thin space.
This space is nearly parallel to the horizontal axis, as can be seen on the sagittal
section (Figure 1). However, it is hard to discern in the horizontal stereotactic
direction. Therefore, it is necessary to make use of the topological and geo-
metrical information provided by stacks from different stereotactic directions to
reconstruct a 3-D model.
In order to test whether the used data show all structures in sufficient reso-
lution and to which extent the aforementioned problems arise, we evaluated the
distribution of tertiary landmarks in the three stacks of the first Morel atlas. A
tertiary landmark was assumed to be shown with reliable resolution if it showed
up in at least two adjacent sections. In that case we considered the two struc-
tures, for which the TLM is the common surface patch, to be adjacent in that
stack. Each stack of sections is analyzed separately for the structures present
and their adjacency as indicated by the tertiary landmarks in that stack. For
7
each stack, the results of this analysis were encoded into a graph. The vertices
represent the individual structures and two vertices are connected by an edge
if the tertiary landmark shows up with reliable resolution. The intersection of
these graphs yields the set of structures and tertiary landmarks available in all
atlases; the union of these graphs are those landmarks, which are discernible
in at least one stereotactic direction. As can be seen directly from the statis-
tics (Table 2), each stereotactic stack shows only partial information (Figure 2).
A surface reconstruction based on a single stack will thus not represent the
correct topology or it contains TLM for which no information is available in
the stack. Therefore, it seems appropriate to reconstruct the atlas not only
from one, but from several data, which should include at least one element
per stereotactic direction. Furthermore, direct comparison of three-dimensional
information from different stereotactic directions has to deal with partial or
missing information. In addition to the problems described above, both the
anterior dorsal and the paraventricular nucleus are missing in the sagittal stack
of sections as sections close to the midsagittal plane cannot be aligned robustly
with the rest of the stack.
Table 2: Statistics of the adjacency graphs of the stack of sections contained in the atlas1
graph coronal stack horizontal stack sagittal stack intersection union♯ nodes 37 37 35 35 37♯ edges 103 110 99 70 138
3. Bootstrap approach for the construction of a mean atlas
A classical approach for creating a mean model from an exemplary set is to
choose one elements as an initial reference, register it onto each of the other ex-
amples and use the average of this registered set as a mean model (Evans et al.,
1993; Guimond et al., 1998). This process can be iterated using the new av-
1Figure 2 illustrates the adjacency graph of the horizontal stack as well as the union of theadjacency graphs.
8
Figure 2: A single stack of sections exhibits only partial information about the topologicalrelationship of thalamic structures. The nodes represent the individual structures. Twonodes are connected with a solid line when they are adjacent in the first horizontal stack.Two structures are connected by a dashed edge if they are adjacent in another stack but notin the horizontal one. See section 2.3 for the definition of adjacency of two structures in astack.
erage as reference until the resulting model does not differ significantly from
the previous one. In our case, this would mean to reconstruct a reference from
a single stack of sections and deform it iteratively. As stacks of sections from
different stereotactic directions display the anatomy to a different extent, their
registration is quite imprecise. Furthermore, the resulting average model would
still exhibit artifacts related to the stack of sections chosen as the initial refer-
ence.
To overcome these limitations, we propose to embed this process of mean model
generation into a second loop, which reconstructs a new reference model. As the
registration maps all examples into a common reference space, the anatomical
knowledge discernible in all stacks of sections can be used during reconstruction.
This new loop allows to effectively merge the topological information contained
in the different stereotactic directions. This process is repeated until conver-
gence. As a consequence, the resulting reference model shall reflect all available
information and be independent on the choice of initial stack of sections.
9
The algorithm is initialized with a reference which is reconstructed from a single
stack of sections. Convergence criteria for both loops is the symmetric Hausdorff
distance (Chou et al., 2007) between the old and new reference. The complete
procedure is presented as a pseudo-code in algorithm 1. The reconstruction and
registration procedures are detailed in the subsequent sections.
Algorithm 1 Bootstrap approach
Ai := stacks of sectionsTi := identity transforms Ti transforms Ai onto the current mean modelmean := reconstruct A0 (see Section 3.1)repeat
old mean := meanrepeat
old average := meanfor all Ai do
Ti := register (mean , Ai ) (see Section 3.2)end for
mean := average of T−1i (mean)
until d(old average , mean ) ≤ ǫ
mean := reconstruct (Ai, Ti)i=1...n (see Section 3.1)until d(old mean , mean ) ≤ ǫ
3.1. Reference reconstruction from registered stacks of sections
First, consider the problem of reconstructing a three-dimensional mesh of a
single structure of interest from a single stack of sections. Each structure is given
as a set of outlines. Due to anisotropy, only few outlines may be available and
outlines retrieved from adjacent sections may differ greatly. Both problems are
exemplified by a tract running almost parallel to the slicing plane. Reconstruct-
ing multiple structures adds further requirements for an appropriate algorithm.
Foremost, the union of all thalamic structures should fill up the whole volume
outlined by the thalamic surface. Moreover, reconstructing a tertiary landmark
on two neighbouring nuclei should result in the same patch on the surfaces of
both. Finally, reconstructing from several, registered stacks requires that the
approach is capable of dealing with non-perfectly overlaid structures.
Surface-tiling (Fuchs et al., 1977) reconstructs meshes by triangulating outlines
of adjacent sections. While the algorithm produces topologically correct meshes
10
for thin or elongated structures, it relies on the fact that the outlines are paral-
lel. It is thus not capable of incorporating contours from different stereotactic
directions. Another approach would be to fit surface patches onto the outlines.
However, this requires that the structure is present in a few sections, which is
not necessarily the case. Moreover, ensuring proper topology for patch fitting is
non-trivial, especially when dealing with multiple stacks. Another approach is
to first construct a three-dimensional implicit surface representation and then
reconstruct a mesh from this (Lorensen and Cline, 1987). For each structure its
signed three-dimensional distance map can be interpolated (Raya and Udupa,
1990) from its two-dimensional contours. Distance maps of registered structures
can be merged to yield an average. Reconstructing multiple structures can be
achieved by integrating the distance maps of all structures into a single one
and applying specialized surface extraction algorithms (Bischoff and Kobbelt,
2006). However, for highly elongated structures, like tracts, outlines in adja-
cent sections do not necessarily overlap and therefore shape-based interpolation
does not yield a meaningful implicit representation in which the outlines are
connected. Likewise, thin structures are likely to contain topological errors as
well (see Figure 3 for an example).
We have thus opted for an implicit surface approach for surface reconstruction
and correction of potential topological errors in an additional step. A standard
approach for topology correction (Bertrand and Malandain, 1994; Kriegeskorte and Goebel,
2001) would solve the topological problems of this method with minimal changes
but does not take into account the anatomical a-priori knowledge. Therefore, we
propose to use meshes produced via surface tiling to correct for larger topological
errors and then apply a standard topology correction algorithm to handle any
remaining errors. In order to ensure that the reconstruction of the individual
structures yields no intermittent spaces, we embed this process in a top-down
strategy, which first finds the segmentation of the whole thalamus and then
recursively distributes the internal voxels to the substructures.
11
3.1.1. High-resolution voxelization
For each individual structure in each stack of sections, its three-dimensional
distance transform is approximated via shape-based interpolation (Raya and Udupa,
1990) from the two-dimensional distance transforms of the structure’s outline
in each section. For thin and elongated structures, which have been manually
selected, a mesh is created by surface-tiling of the original contours. The im-
plicit representation for each such structure is the pointwise minimum of the
interpolated distance transform and the distance transform of the mesh created
by surface-tiling. For all other structures, the interpolated distance transform is
used as their implicit representation. In the case of reconstructing the initial ref-
erence, this distance transform is used further on; in the case of reconstructing
a new reference surface from multiple stacks, the registered distance transforms
are averaged to produce a mean distance map. The average of several distance
maps is not necessarily a distance map itself; however, this did not cause diffi-
culties during the processing, probably due to the fact that the input distance
maps were aligned. The segmentation of a top-level structure is given by all
voxels with a non-positive value for its distance transform. To subdivide a
structure, each voxel is assigned to that substructure on the next hierarchical
level with the lowest distance at this point. Finally, a standard topology cor-
rection is applied to each structure in turn to eliminate any remaining handles
or holes (Kriegeskorte and Goebel, 2001).
3.1.2. Surface reconstruction
To reconstruct the surface of the reference volume, an approach which ex-
pands singular vertices and edges, was used (Bischoff and Kobbelt, 2006). This
algorithm also allows to incorporate a priori information by considering that
two structures are delimited by another one. The resulting mesh was smoothed
using surface nets (Gibson, 1998) and decimated with a topology preserving al-
gorithm (Schroeder et al., 1992). Landmarks according to the proposed model
(see Section 2.2) are built during reconstruction.
12
(a) (b)
Figure 3: Problem associated with surface reconstruction for thin or elongated structures.Both images show the same part of a surface reconstruction of CL based solely on sectionsD2.7 and D3.6 of the first horizontal series of the Morel atlas. Extracting the surface directlyfrom the distance map generated by shape-based interpolation yields topological errors (a).Correcting the distance map first based on the surface of the CL generated by surface tilingyields a topologically sound reconstruction (b). Mesh vertices are colored according to whichtype of landmark they belong to (red indicates a secondary, green a tertiary landmark).
3.2. Alignment of stacks of sections with the reference model
The alignment has been performed in two subsequent steps, a global affine
registration followed by a B-spline controlled elastic deformation (Rueckert et al.,
1998), which guarantees a smooth mapping between two adjacent histological
sections. As a tertiary landmark is a surface in the reference model, while it is
a set of lines in a stack of sections, it is more stable to register the stack onto
the reference.
For estimating the optimal affine registration, the centers of gravity of each
thalamic substructure were chosen as corresponding points. As the center of
gravity can be robustly estimated only for sufficiently wide-streched volumes,
only those structures which show up in at least three sections were considered.
The deformation and its optimization are described in the following sections.
3.2.1. Cost function
The non-rigid warping T : R3 → R3 seeks to optimize the overlap of cor-
responding tertiary landmarks further, while avoiding large local deformations.
To evaluate the overlap of tertiary landmarks, the lines of the tertiary land-
marks in the stack are first sampled equidistantly. The fitting term f(T ) of a
13
landmark is defined as the summed squared distance of these samples to the cor-
responding TLM surface in the reference model. The smoothness (enforced by
the regularization term R(T )) of the deformation field is assessed by |J(x)|, the
Jacobian’s determinant (Sdika, 2008). These two terms are linearly combined
with a weighting parameter λ into one cost function:
C(T ) = f(T ) + λR(T ) (1)
=
∑
TLM l
∑
p∈l
dl(T (p))2
+ λ
∫
| log(|JT (x)|)|dx.
As computing the exact distance dl(p) for each point p is computationally pro-
hibitive, a distance map is precomputed for each topological landmark. To
ensure a smooth cost function, the distance is approximated by trilinear in-
terpolation. The deformation is optimized with a multi-resolution strategy,
starting with a coarse control point grid and successively subdividing it. Due
to the large number of control points at the finest resolution, a gradient-descent
optimization scheme is employed.
3.2.2. B-spline deformation
The B-spline deformation T : R3 → R3 is defined by a regular, finite grid of
control points Φ(i,j,k) ∈ R3, with a spacing of δ between adjacent control points.
Φ(i,j,k) is located at (iδ, jδ, kδ); for i, j, k < 0 or i ≥ nl, j ≥ nm, k ≥ nn it is
fixed. T transforms a = (x, y, z) via
T (a) = a +
3∑
l,m,n=0
Bl(u)Bm(v)Bn(w)Φi+l,j+m,k+n (2)
with i = ⌊xδ⌋−1,j = ⌊ y
δ⌋−1,k = ⌊ z
δ⌋−1,u = x
δ−⌊x
δ⌋, v = y
δ−⌊ y
δ⌋,w = z
δ−⌊ z
δ⌋.
The B-spline basis functions are defined as
B0(t) =(1 − t)3
6B1(t) =
3t3 − 6t2 + 4
6(3)
B2(t) =−3t3 + 3t2 + 3t + 1
6B3(t) =
t3
6,
in the following notation B′p =
dBp(t)dt
will be used.
14
3.2.3. Gradient estimation for the regularization term
The transposed Jacobian of the B-spline deformation T at point x is given
by
JTT (x) = (ai,j)1≤i,j≤3 = 1 + (4)
+
3∑
l,m,n=0
δB′l(u)Bm(v)Bn(w)
δBl(u)B′m(v)Bn(w)
δBl(u)Bm(v)B′n(w)
Φi+l,j+m,k+n
Suppose we want to compute the Jacobian for a control point Φ(i′,j′,k′), whose
location in the deformation’s domain is given by i = i′− 1, j = j′− 1, k = k′− 1
and u = v = w = 0. As B′1(0) = 0 and B3(0) = B′
3(0) = 0, the first vector in
equation 4 is non-zero solely for direct neighbors of Φ(i′,j′,k′). Most notably, it
is independent of the control point’s value. Moving an adjacent control point
Φ(i′+l,j′+m,k′+n) by p changes the Jacobian and likewise its determinant in a
linear fashion:
|JT (x) + JT (x)| = |JT (x)| + ω · p, (5)
where ω = (ν, µ, θ) is given by
ν = αa1,1 a1,2
a2,1 a2,2
+ βa2,0 a1,0
a2,2 a1,2
+ γa1,0 a2,0
a1,1 a2,1
µ = αa2,1 a0,1
a2,2 a0,2
+ βa0,0 a2,0
a0,2 a2,2
+ γa2,0 a0,0
a2,1 a0,1
θ = αa0,1 a1,1
a0,2 a1,2
+ βa1,0 a0,0
a1,2 a0,2
+ γa0,0 a1,0
a0,1 a1,1
.
with
α = B′l(0)Bm(0)Bn(0), β = Bl(0)B′
m(0)Bn(0),
γ = Bl(0)Bm(0)B′n(0).
An optimal choice for p for a direct neighbor minimizes | log(|JT (x)+JT (x)|)|
to zero. Any p with ω · p = −|JT (x)| fulfills this requirement. Thus, for
15
each control point Φ(i′,j′,k′), one linear constraint for each of its 26 adjacent
neighbours is derived from the Jacobian matrix in Φ(i′,j′,k′). As a result, 26
linear equations are available in each control point. This yields a least squares
problem, whose solution is used as the gradient of the regularization term in
Φ(i′,j′,k′).
4. Results
The initial reference mesh (see Figure 4 (a),(c)) was reconstructed from the
horizontal stack of sections, for which sections at a small interval (0.45mm) are
available. It contains the structures described in Table 1. After three iterations
of the outer loop, the process of mean model generation converged. The final
mesh consists of 134789 points, 325450 triangles, and 471 primary, 864 secondary
and 439 tertiary landmarks (see Figure 4 (b),(d)). The adjacency graph of the
reference mesh contains the union graph, presented in Section 2.3 and Figure 2,
as a subgraph. It contains some comparatively small additional landmarks;
on average they have an area of 2.1mm2, whereas a landmark in the union
graph has an average area of 31.6mm2. For each stack of sections the B-spline
deformation was initialized with an isotropic control point grid spacing of 7.2mm
and subdivided three times. The tertiary landmarks were optimized using the
described approach. Points were excluded from matching if they originated from
a landmark for which no corresponding landmark exists in the reference model.
For the deformation of the initial mesh, the weighting factor λ = 0.2 proved to
offer a good tradeoff between fitting the landmarks and obtaining an invertible
deformation. During subsequent refinements, this could be lowered to λ = 0.08.
Exemplary results of deforming the reference onto stacks of sections are shown
in Figure 5. Histological maps overlaid with the corresponding cut through the
deformed reference mesh are shown in Figure 6.
5. Discussion
When creating a model of the thalamus from histological sections, the most
difficult issues are related to the anisotropy of the individual stacks, which
16
(a) (b)
(c) (d)
Figure 4: Views of the initial (left column) and the final atlas (right column). The upper rowshows the atlas from the medial direction with the caudo-rostral axis going from left to right.The lower row shows the atlas from the lateral direction with the caudo-rostral axis goingfrom right to left.
(a) (b) (c)
Figure 5: View of reference deformed on stack of sections from the lateral direction. Caudo-rostral axis goes from right to left. Shown are the result for a sagittal series (a), a coronal series(b) and a horizontal series (c). See Figure 4 for color codes and Figure 6 for corresponding2D cuts.
17
(a) (b)
(c)
Figure 6: Fitting the reference model onto a stack of sections. Section L10.9 from the sagittalseries (a), section A4.5 from the coronal series (b) of the first Morel atlas and section D7.8 fromthe additional horizontal series (c) overlaid with the corresponding cut through the deformedreference mesh (original section in red; deformed initial model in green; deformed final modelin blue). Structures with no overlaid cut are not included in the reference.
18
Figure 7: Histograms for each data set showing the distribution of squared distances (in mm)between corresponding points at the final fit. Upper row from left to right: sagittal, coronaland horizontal series from the first Morel atlas. Lower row from left to right: sagittal andcoronal from the second Morel atlas and the additional horizontal series.
could lead to serious problems in the construction process. Each stack shows
the complete three-dimensional anatomy only to a partial extent. As the in-
tervals between sections in Schaltenbrand and Wahren (1977) and Mai et al.
(2004) are larger and/or more irregular than in the used Morel atlas, one can
expect similar problems in those as well. For large structures such dramatic
topological errors as shown in Figure 1 are unlikely to happen. However, geo-
metrical details necessary for an accurate three-dimensional reconstruction are
still lost. Warping the current atlas onto all available data captures the inter-
individual anatomical variability. This has two advantages. First of all, it allows
computing an average atlas. Secondly, the inverse warping, casting all stacks
into a common space, effectively removes the inter-individual variability, which
enables us to incorporate the geometrical and topological details provided by
each stacks during reconstruction, thereby reducing the effects of anisotropy in
the produced atlas.
The reconstruction proceeds in two steps. Firstly, an isotropic implicit rep-
resentation is generated of each structure in each stack. For thin or elongated
structures, the distance maps may contain topological errors. One could disre-
gard this and assume that averaging the distance maps alleviates the problem.
However, this would ignore the available anatomical information. As a sec-
ond step, the registered implicit representations are merged and integrated to
produce a segmentation. Averaging the distance maps merges the geometrical
19
and topological information provided by each stack and it is feasible even if the
maps are not perfectly aligned. In the current implementation, distance maps
are sampled and stored on a regular grid. The required memory limits the
volume which can be processed, as fine details require a dense sampling. This
constraint could be lifted by replacing the regular by an adaptive grid. Employ-
ing solely an implicit or a tiling approach for surface reconstruction would not
deal completely with problems inherent to reconstruction from multiple histo-
logical data (see Section 3.1). It should be noted that the smoothing algorithm
used shrinks the surface. This effect can, however, be neglected, as a subsequent
registration and averaging step removes the shrinkage.
Alignment has to be done in a manner which is robust against the problems
associated with anisotropy. Within the proposed landmark model, the primary
landmarks would provide a good starting point for establishing correspondence
between a stack and the atlas. However, these landmarks are not visible in
the stacks. In contrast, tertiary landmarks are observed as lines in histological
maps. Nonetheless it may happen (see Figure 1) that a tertiary landmark falls
between two adjacent sections and is thus not visible at all. Such a situation
may also misguide the registration as the optimizer tries to fit landmarks not
clearly visible in the reference model. However, we experienced this behaviour
only with the initial model, which was built from a single stack. In subse-
quent registrations, with reference models built from several data, this effect
diminishes. Furthermore, these subsequent registrations do not need to be reg-
ularized as strong as the registrations of the initial atlas. This indicates that
the reconstruction process successfully integrates geometrical information from
each stack. Likewise, visual comparison of the fitting results (see Figure 6)
supports this. While the proposed landmark model introduces more anatomi-
cal knowledge into the registration process, it also renders the deformation less
robust, as non-existing parts of a tertiary landmark are registered onto wrong
structures. As the histograms (Figure 7) show, outliers, although very few ones,
are present in each data set. The warping would be more robust if these outliers
were detected and properly handled.
20
The presented atlas improves the previous work on thalamic model recon-
struction in several aspects. Firstly, while those models are based on the ge-
ometry seen in a single stack, our model incorporates topological and geometric
details from different stacks and different stereotactic directions. Secondly, it
represents the average anatomy of several specimen instead of a single one. Both
aspects should improve atlas-based identification of thalamic structures.
6. Conclusions and Future Work
A stereotactic atlas of the mean anatomy of the thalamic structures from
different histological stacks of sections has been generated, in spite of the stack
dependent anisotropy. The proposed approach consists of two nested loops. The
inner loop finds optimal deformations between the individual stacks of sections
and the current reference mesh. Like for MRI-based atlases, this allows the con-
struction of an average template. The outer loop uses the optimal deformations
found in the inner loop to construct an improved reference. The reconstruc-
tion of a topologically consistent model of the thalamic structures was shown
for a single stack and for multiple, registered stacks. Establishment of a neu-
roanatomically meaningful, dense three-dimensional deformation field from the
information provided by the sections was obtained not only for the structures
but also for the topological relationship between them.
The established correspondence between the presented mean model and the
individual stacks allows to interpolate a MRI for the mean model from post-
mortem MRIs available for the histological data. Yelnik et al. (2007) has re-
cently demonstrated that such an interpolated MRI could be used to transfer
the mean model onto a patient’s MRI.
Acknowledgements
This work has been supported by the Swiss National Science Foundation,
grants No. 31-68248.02 (AM) and the National Center of Competence in Re-
search (NCCR) on “Computer Aided and Image Guided Medical Interventions”
(CO-ME).
21
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