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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, krauth@vision.ee.ethz.ch). Yours sincerely, Axel Krauth

* 1. Cover Letter

1) Harry B. M.Uylings (h.uylings@np.unimaas.nl)

Department of Psychiatry and Neuropsychology

Division Brain and Cognition

Maastricht University Medical Center

Maastricht

The Netherlands

2) Jones EG (ejones@ucdavis.edu)

Center for Neuroscience

University of California

Davis, CA 95616

USA

3) Wieslaw Nowinski (wieslaw@sbic.a-star.edu.sg)

4) Sebastien Ourselin (s.ourselin@cs.ucl.ac.uk)

* 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

2

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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