Bone 57 (2013) 142–154
Contents lists available at ScienceDirect
Bone
journal homepage: www.elsevier.com/locate/bone
Original Full Length Article
A quantitative framework for the 3D characterization of the osteocyte
lacunar system
Kevin Scott Mader a,b, Philipp Schneider c, Ralph Müller c, Marco Stampanoni a,b,⁎
a
b
c
Swiss Light Source, Paul Scherrer Institut, Villigen 5232, Switzerland
Institute for Biomedical Engineering, University of Zurich and ETH Zurich, Zurich 8006, Switzerland
Institute for Biomechanics, ETH Zurich, Zurich 8006, Switzerland
a r t i c l e
i n f o
Article history:
Received 13 December 2012
Revised 21 May 2013
Accepted 21 June 2013
Available online 18 July 2013
Edited by: Harry Genant
Keywords:
Osteocyte lacunae
3D morphology
Cortical bone
Cell shape
Cell distribution
Cell alignment
a b s t r a c t
Assessing the role of osteocyte lacunae and the ways in which they communicate with one another is important
for determining the function and viability of bone tissue. Osteocytes are able to play a significant role in bone
development and remodeling because they can receive nourishment from, interact with, and communicate
with other cells. In this sense the immediate environment of an osteocyte is crucial for understanding its function.
Modern imaging techniques, ranging from synchrotron radiation-based computed tomography (SR CT) to confocal laser scanning microscopy, produce large volumes of high-quality imaging data of bone tissue on the micrometer scale in rapidly shortening times. These images often contain tens of thousands of osteocytes and their
lacunae, void spaces which enclose the osteocytes. While theoretically possible, quantitative analysis of the osteocyte lacunar system is too time consuming to be practical without highly automated tools. Moreover, quantitative morphometry of the osteocyte lacunar system necessitates clearly defined, robust, and three-dimensional
(3D) measures. Here, we introduce a framework for the quantitative characterization of millions of osteocyte
lacunae and their spatial relationships in 3D. The metrics complement and expand previous works looking at
shape and number density while providing novel measures for quantifying spatial distribution and alignment.
We developed model, in silico systems to visualize and validate the metrics and provide a concrete example
of the attribute being classified with each metric. We then illustrate the applicability to biological samples in a
first study comparing two strains of mice and the effect of growth hormone. We found significant differences
in shape and distribution between strains for alignment. The proposed quantitative framework can be used in
future studies examining differences and treatment effects in bone microstructure at the cell scale. Furthermore,
the proposed strategy for quantitative bone cell morphometry will allow investigating structure–function relationships in bone tissue, for example by linking cellular morphometry to bone remodeling.
© 2013 Elsevier Inc. All rights reserved.
Introduction
Over the last years, sound evidence has demonstrated that osteocytes, the most abundant cells in bone, play a role in mechanosensation
[29,27], mineral homeostasis [62,25], and bone mass regulation [30,64].
Groups of osteocytes form a cellular network through their slender cell
processes called canaliculi, which link the individual osteocytes together and with other bone cells. This cellular network, embedded inside the
bone tissue, forms the lacuno-canalicular network (LCN) [52]. The LCN
consists of microstructural porosity elements, which enclose the osteocytes and their cellular processes. The traditional approach for studying
the osteocyte network and the LCN in three dimensions (3D) uses confocal laser scanning microscopy (CLSM), which, if combined with
appropriate fluorescent labels, allows for functional bone imaging. Yet,
⁎ Corresponding author at: WBBA/216, CH-5232 Villigen, Switzerland.
E-mail addresses: kevinmader@gmail.com (K.S. Mader), pschneider@ethz.ch
(P. Schneider), ram@ethz.ch (R. Müller), marco.stampanoni@psi.ch (M. Stampanoni).
8756-3282/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.bone.2013.06.026
this advantage comes at a price since the samples require extensive
preparation and that only a limited depth can be probed. These factors ultimately limit the number and size of samples to be studied and make
studies with more than 10 samples very time consuming. Standard 3D
imaging techniques with higher penetration power, such as desktop
micro-computed tomography (μCT) or magnetic resonance imaging
(MRI) still lack the ability to visualize the LCN with typical dimensions
of osteocyte lacunae 10 μm and volumes ranging from 200–600 μm3 in
a stable, time-efficient manner [52]. More recent developments have
further increased the resolution of desktop μCT scanners, but issues
with stability, and extremely long measurement times (N10 h) make
such techniques impractical for larger studies. Higher spatial resolution
techniques (b 1 μm), such as electron microscopy, capable of visualizing
the canaliculi are impractical because their range of depth is typically limited to a few micrometers for bone tissue [22]. In addition, coherent X-ray
experiments can be used to visualize canaliculi, but they are usually
time and sample size prohibitive [52,25]. Furthermore post-processing
techniques [41] in combination with high-resolution phase sensitive
K.S. Mader et al. / Bone 57 (2013) 142–154
tomographic imaging [33] have recently been able to extract the canaliculi from small regions, but while they offer a many fold improvement
on speed, they still have limited fields of view and require cutting the
sample. Given the rapid pace of development, this technique may soon
be practical for more detailed studies of the osteocyte lacunar network
on a larger scale.
High-resolution 3D imaging technologies, such as synchrotron
radiation-based CT (SR CT), have been applied on bone tissue to assess
thousands of lacunae in cortical bone microstructure with millimetersized field of views and sub-micron voxel-sizes [2,7,10,52,33,26]. Although measurements and analyses have already been conducted
within many biological studies, there has been little work to develop a
consistent, well-defined, technique and scale independent set of quantitative metrics for describing the osteocyte lacunar network. In order
for results to be comparable and more abstract notions like spatial
distribution, organization, and alignment to be quantified, a number of
new metrics need to be introduced. Since many of these ideas are new
and initially abstract, a set of in-silico lacunae where shape and position
are specified, is required to show how the metrics characterize changes
in the lacunae and provide a baseline for comparing the results with
random values.
While the images from the discussed techniques are impressive,
appropriate and automated quantitative tools for evaluation of the
LCN are needed to take full advantage of a number of new avenues
in imaging at the cell scale. For osteocytes and osteocyte lacunae,
several works have addressed the task of providing descriptive metrics about overall shape, orientation, global number densities, and
occupancy ratio [7,10,53,60,65,67]. The work of Ascenzi et al. [2]
has gone further, looking locally and quantifying analytically the
properties of individual osteocyte lacunae in human secondary osteons.
Specifically, the osteocyte lacunar orientation with respect to lamellae
was calculated by fitting an ellipse to lacuna images obtained by confocal microscopy. The limitations of this study are that it is a twodimensional (2D) approach and the need for substantial user interaction, which is problematic for reporting on reproducible results. More
recent works such as Ref. [25] have examined a large number of metrics
related to connectivity and spatial distribution, but only on small samples. With such large data sets, a clear, flexible, and power set of tools
are required to analyze the data well. Following the guidelines put
forth in recent “Reproducible Research” initiatives, the quantitative
analysis should be performed using open or as open as possible software
packages, and statistical analysis should be made publicly available, a
task which is currently not possible with many packages [44,57,40].
The availability of the raw or cached data and analyses performed allows
detailed inspection of the results obtained and the possibility to easily do
further analyses.
BoneJ, a toolbox that was developed by Doube et al. [15] for standard
bone measures in the form of an ImageJ plug-in (http://rsb.info.nih.gov/
ij/; U.S. National Institutes of Health, Bethesda, Maryland, USA), makes a
major step towards these goals by offering an open-source technique
for consistently calculating several whole-bone, trabecular, and osteocyte lacunar measures. While the tools are principally designed for
lower resolution investigation looking at trabecular bone and entire
specimens, a basic 3D shape analysis tool is provided for extracting osteocyte lacunar measures, such as surface area, volume, and orientation
for segmented osteocyte lacunae. However, this toolset lacks the flexibility to perform a number of different analyses, necessary for large-scale
studies and detailed analysis of distribution, alignment, and other osteocyte lacunar measures. The analysis of osteocyte lacunae in a given bone
specimen presents several challenges, particularly in scale, because even
a small tissue volume of 1 mm3 contains between 10,000 and 100,000
individual osteocyte lacunae [52]. Consequently, novel image postprocessing methods for quantitative morphometry, measuring shape,
orientation, and characterizing distribution of osteocyte lacunae must
be fully automated. Yet, until now, there have only been limited efforts
to systematically define, verify, and validate the 3D shape, orientation,
143
and distribution measures for the quantification of osteocyte lacunae
on the same level as well-established quantitative bone morphometry
metrics [6,43]. Therefore, the main goal of this study was to introduce,
validate, and apply a novel framework to quantitatively assess 3D
shape, orientation, and distribution of osteocyte lacunae in a fully automated, high-throughput manner. Furthermore, we wish to provide a set
of easily visualized metrics to illustrate the quantitative differences between groups using a few simple figures. Inspired by existing measures
in the field of bone research, material science, and soft matter physics,
we adopt some of the pertinent metrics to provide a comprehensive
and robust scheme for characterizing osteocyte lacunae in bone. A
wider aim of the study is to introduce a consistent instrumentationindependent pipeline for taking microstructural 3D imaging data to
meaningful quantitative information on the sample. Due to the abstract
nature of the metrics and the oversimplification of synthetic data, we
present a study, where we assess the lacunar descriptors for femoral
bones from two different mouse strains with and without growth
hormone measured using SR CT. The study shows the utility of the
given metrics to quantify real systems and insight into which metrics
are conserved between strains and how growth hormone might affect
microstructure.
Materials and methods
To establish a framework for the quantitative assessment of osteocyte
lacunae, we first define mathematically the osteocyte lacunar descriptors
(definition). We then verify the descriptors using in silico models (verification). We then use the osteocyte lacunar descriptors to compare femoral bones from two different inbred mouse strains as measured using
SR CT (application). In the Supplementary materials, we provide extensive details about the generation of the in silico models and the comparisons made between model generation parameters and numerically
calculated osteocyte lacunar descriptors. We provide as well more details
on the examinations of resolution dependencies and comparisons between statistical to physical osteocyte lacunar measures.
Definition of osteocyte lacunar descriptors
The following descriptors for the osteocyte lacunae are defined with
a view to quantify their individual morphology, their spatial relationship
to each other, and their position within the context of the whole-bone
geometry. To this end, the osteocyte lacunar descriptors are grouped
into three categories (Fig. 1): shape and orientation, local environment,
and global environment. This hierarchical organization allows analysis
of the osteocyte lacunar system beginning from the cellular level
(shape and orientation) going up to the whole-bone scale (global environment). The shape descriptors characterize the dimensions, orientation, and anisotropy of each lacuna individually. The local environment
measures provide ways to collectively characterize the surroundings of
a particular lacuna including its neighbors and the amount of surrounding bone. The metrics provide the ability to determine the relationship
between lacunae and their immediate neighboring lacunae and the corresponding distances and orientations. Finally, the distribution and
alignment metrics provide the ability to assess the entire bone or regions
of interests with a single metric. For the summary statistics (e.g. average
stretch), the numbers represent first finding the average lacuna and
then calculating the metric for that average lacuna, which reduces the
effect of measurement error and noise.
We start by establishing a tensor framework for addressing this problem, which provides 3D descriptors that are independent of the imaging
frame. The first tensor is the shape tensor (S) describing the individual lacuna, which will be discussed in Shape and orientation section. We then
introduce two tensors in Section Local environment for characterizing
the collective metrics for a group of lacunae. The distribution tensor (D)
describes the spatial distribution and spacing between lacunae and the
alignment tensor (T) describes the collective orientation.
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K.S. Mader et al. / Bone 57 (2013) 142–154
Shape
Orientation
Local
Environment
Global
Environment
Prolate
Oblate
Fig. 1. Here we show a schematic of the three categories and scales of metrics introduced in this manuscript and how they can be utilized to better visualize bone structure. The descriptors
range from a single lacuna to tissue (local) and organ (global) level characterization. (Left panel) Shape and orientation: Here we show a voxel representation of a single object (white
boxes) and the shape tensor (pink ellipse). The vectors (V1, V2, V3) represent the semi-axes of the shape tensor. Additionally we show the dimensions as measured through caliper measurements [] and the shape-tensor (lambda). Finally, the inset indicates the various extremes of the Lc.Ob metric as a prolate and oblate ellipsoid respectively. (Center) Local environment:
The multicolored ellipsoidal objects are lacuna, and the green lines surrounding show the territory for each lacuna boundary for each lacuna. The original gray-level image from the CT
measurement is used as the background. (Right panel) Global-environment: An axial view of all lacunae in a cortical bone sample. Each lacuna is represented by an arrow pertaining
! to its secondary orientation Lc: O r2 . The colors of the arrows from blue to red show the angle between the given lacuna's orientation and the mean orientation as determined by the
alignment tensor.
Shape and orientation
Shape tensor. The shape tensor describes the shape and the orientation of
a single osteocyte lacuna. Osteocyte lacunae generally resemble elongated pancakes. This is a geometrical shape, which requires 2 vectors and 3
scalar quantities to be fully determined. Therefore, we used a tensorbased analysis for the characterization of osteocyte lacunar shape and orientation. For any 3rd order tensor, a number of 3D representations are
possible. For osteocytes, an ellipsoidal representation represents a suitable choice. Furthermore, ellipsoidal models have been used in earlier
works [4,7,10,49] to describe the 3D shape of osteocyte lacunae. Provided
that individual osteocyte lacunae have been assessed and segmented in
3D, a shape tensor can be created for each lacuna according to Ref. [39],
by creating a 3 × 3 matrix for the covariance of the x, y, and z positions
of the constitutive voxels. For a lacuna consisting of m voxels, we denote
!
each voxel by v ðiÞ ¼ vx ðiÞ; vy ðiÞ; vz ðiÞ for i between 0 and m. For convenience, we define for each voxel pertaining to a given lacuna the relative
m !
!
!
position p ðiÞ ¼ v ðiÞ−m1 ∑ j¼0 v ð jÞ as the displacement of that voxel from
the lacuna's center of volume. The resulting shape tensor is then given by
2
S¼
p ðiÞpx ðiÞ
1 Xm 4 x
py ðiÞpx ðiÞ
i¼0
m
p ðiÞp ðiÞ
z
x
3
px ðiÞpy ðiÞ px ðiÞpz ðiÞ
py ðiÞpy ðiÞ py ðiÞpz ðiÞ 5:
pz ðiÞpy ðiÞ pz ðiÞpz ðiÞ
ð1Þ
This (symmetric) shape tensor can be diagonalized to provide the
! ! ! (orthogonal) eigenvectors
V S;1 ; V S;2 ; V S;3 defining the ellipsoid
axes and the corresponding eigenvalues (λ2S,1,λ2S,2,λ2S,3), which correspond to the covariance about each principal axis. We denote the eigenvalues as squares to preserve the intuitively applicable unit of length for
the eigenvalues, which makes later analyses simpler. The extent of the
ellipsoid semi-axes (Lc.Le.1, Lc.Le.2, Lc.Le.3), which we call lacuna
lengths, can be directly calculated by scaling the respective eigenvalues
by 2√5 [39]. The lengths of the semi-axes are thus based on the spatial
distribution of voxels, in contrast to caliper measures derived from actual extent of the lacunae along a given axis (Feret diameter) [39].
Lacuna stretch (Lc.St). The lacunar stretch (Lc.St) describes the difference
between the largest and smallest eigenvalues of the shape tensor, compared to the largest one
Lc:St ¼
λS;3 −λS;1
λS;3
ð2Þ
where (λ2S,1,λ2S,2,λ2S,3) represent the eigenvalues of the shape tensor in
ascending order of their magnitude. Lc.St is defined in contrast to common non-normalized values for stretch, such as the aspect ratio (ratio
between the longest and the shortest semi-axis' length). The reason
for using the given definition of Lc.St for describing osteocyte lacunar
stretch is based on our observation (data not shown) that normalized
differences as given in Eq. (15) are less sensitive to spatial resolution,
particularly when a particular lacuna was composed of a few voxels
only (low resolution). Lacunar stretch takes values in the range [0,1],
where 0 corresponds to a perfectly spherical, and 1 to an infinitely
stretched object.
Lacuna oblateness (Lc.Ob). Lacuna oblateness (Lc.Ob) quantifies the lacunar shape in more detail and is defined as
Lc:Ob ¼ 2
λS;2 −λS;1
−1
λS;3 −λS;1
ð3Þ
where (λ2S,1,λ2S,2,λ2S,3) are the eigenvalues of the shape tensor in ascending order of their magnitude. The measure provides information
on the nature of the geometrical anisotropy of osteocyte lacunae,
in addition to stretch. Oblateness mathematically defined as in
Eq. (3), but called ‘plateness’, has been used earlier to classify trabeculae into rod-like and plate-like structures [66]. Oblateness takes
values in the range [− 1,1], where − 1 corresponds to a perfect rod
(strongly prolate), and 1 to a perfect plate (strongly oblate). Here,
we explicitly exclude singularities (λS,3 − λS,1 = 0) in Eq. (3),
which represent degenerate osteocyte lacunae (spheres) and since
oblateness describes how an object is anisotropic, it is meaningless
for isotropic objects. Furthermore even with the limited resolution,
K.S. Mader et al. / Bone 57 (2013) 142–154
the number of lacuna with less than 0.1 (10% stretch) was less than
0.1% of all lacunae, which means they do not have a significant impact on the oblateness calculation and can be ignored.
Lacuna orientation. To quantify lacuna orientation, we define two different orientations for each lacuna
!
!
Lc: O r1 ¼ V S;3
!
!
Lc: O r2 ¼ V S;2 :
ð5Þ
Local environment
To characterize the vicinity of the osteocyte lacunae we introduce
local environment measures. These measures describe the spatial relationships each lacuna has with its proximal neighbors. Specifically
these metrics provide quantitative definitions for qualitative statements
such as ‘clustered in groups’ or ‘randomly scattered’ and ‘crystalline’ or
‘randomly positioned’. These measures describe lacuna number density,
information on proximal lacunae, lacunar distribution, and lacunar orientation. Narrow distributions (for number density and information on
proximal lacunae) and high values (for distribution and orientation)
might be suggestive of the rules or arrangements, which govern osteocyte lacunar location.
Lacuna territory volume (Lc.Te.V). The term ‘territory’ is used here in line
with the idea introduced by Canè et al. [9], where the concept of osteocyte territory in 2D was used to evaluate histological sections from canine bone samples by light microscopy and again in Beno et al. [4],
where the idea was used for assessing bone porosity. We extend this
definition to 3D by defining the territory of each lacuna by performing
a Voronoi tessellation [17] on the osteocyte lacunar network enclosed
within the bone tissue. A Voronoi tessellation basically grows all objects
within a volume of interest (VOI) until they consume the entire volume
in such a way. This is done in such a manner that each voxel within the
VOI is assigned to the object, which is closest to it in terms of the Euclidean distance metric. A visual (2D) example for a Voronoi tessellation of
the osteocyte lacunar network can be found in the middle panel of Fig. 1.
Given a section of bone with bone volume BV containing N.Lc lacunae,
the sum of all territories is equal to the total bone volume (BV). Furthermore, the inverse of the lacuna number density (N.Lc/BV) is identical to
the mean territory volume (bLc.Te.VN)
1 XN:Lc
1
BV:
Lc:Te:VðiÞ ¼
i¼0
N:Lc
N:Lc
proximal group of lacunae. This strategy has already been successfully
implemented in various disciplines, such as neuroscience [18,42], physics [34,71,70], and applied mathematics [58], and it is used to calculate
number density without standard pitfalls, such as arbitrary selection
of region of interests (ROIs) or scales required by most approaches.
Accordingly, we define lacuna territory density for lacuna i as the inverse
of the corresponding territory volume
ð4Þ
The first corresponds to the direction of the longest semi-axis (primary lacuna orientation) and the second to the secondary semi-axis
(secondary lacuna orientation). The importance to describe osteocyte
lacunar shape by the second orientation is directly related to the degree
of oblateness. In a strongly prolate lacuna (λS,1, λS,2 ≪ λS,3) the first
lacuna orientation is the only meaningful direction to be reported,
while for a largely oblate lacuna (λS,1, λS,2 ≫ λS,3), both the first and
second lacunar orientations are important as they define the plane
which describes the orientation of the pancake-shaped lacunae.
Thus in a situation where all lacunae are strongly oblate, only the
third eigenvector need be considered since it's direction alone provides
the orientation of the plate.
hLc:Te:Vi ¼
145
ð6Þ
Lacuna territory density. Lacuna territory density is a measure to quantify the number density of a single lacuna as given by its immediate
neighbors. It provides a local density metric for every lacuna in the specimen, which can then be considered as a lacuna attribute. This perspective allows density to be directly compared with other parameters such
as shape, size, and relative position, whereas density, as it is traditionally defined (count per unit volume), is only defined for a spatially
1
:
ð7Þ
Lc:Te:VðiÞ
It should be noted that the mean of the individual lacunar territory
densities is generally not the same as the lacuna number density
(N.Lc/BV)
h1=Lc:Te:VðiÞi ¼
N:Lc
1 X
1
N:Lc i¼0 Lc:Te:VðiÞ
N:Lc
N:Lc
:
¼
1=hLc:Te:VðiÞi ¼ XN:Lc
BV
Lc:Te:VðiÞ
i¼0
ð8Þ
ð9Þ
except for the degenerate case, where all lacunae have identical territory volumes.
Lacunar Delaunay neighbors (Lc.DN and Lc.ND). For each lacuna, we refer
to the concept of a neighborhood. If the canaliculi could be directly
observed, this metric would be derived from the physical connectedness of two lacunae. Since with the given field of view and resolution
this is not possible and proxy measures need to be used instead. The
neighborhood is made up of the immediate spatially proximal neighbors. To characterize the relation of a particular osteocyte lacuna with
its neighbors, we define here a set of neighborhood measures. They
quantify the relation of individual lacunae to their nearest neighbor(s)
and reveal ensemble properties on larger scales up to the whole osteocyte lacunar system. Those descriptors can be grouped into two different classes, which reflect either individual or ensemble properties of a
particular lacuna, respectively. These measures describe for example
nearest neighbor distance (individual), number of nearest neighbors
(individual), grouping and self-avoidance (ensemble), or degree of
alignment (ensemble). Nearest neighbor distance, Lc.ND, is defined by
the Euclidean distance between the centroids of neighboring lacunae,
where the respective centroids are derived from the best-fit ellipsoids
based on the spatial distribution of their constituting voxels (see definition of shape tensor in Section Shape and orientation). We compute this
metric, also referred to as natural neighbors [12,13,50], using Delaunay
triangulation of the lacunae's centroids. This method is well established
in structural dynamics and phase identification in material science
[8,56,63], and for our study, it serves to provide a scale-free metric on
the spatial organization of the osteocyte lacunae. A more biologically
oriented explanation is the following: Lc.DN provides a proxy measure
for the number of other lacunae with which a given lacuna could be in
contact with (no other lacuna would block the direct connection).
This neighborhood count serves as a measure for examining network
properties and it provides a method to characterize arrangements in
an ensemble of similar objects [11,19]. Given the mathematical duality
of the Voronoi tessellation and the Delaunay triangulation [51], we
can calculate the Lacunar Delaunay neighbors (Lc.DN) using the territory volume (Lc.Te.V) defined earlier. More precisely, the Delaunay triangulation for a set of lacunae can be calculated by interconnecting
the center of volumes of osteocyte lacunae whose territories are directly
touching. The metric Lc.DN is thus equivalent to the number of
other territories a given lacuna territory has contact with. Accordingly,
Lc.DN ranges from 0 for a completely isolated lacuna to Lc.N-1 for the
degenerate case, where the territory of a lacuna is touching the territory
of all other lacunae in the network (imagine a bicycle wheel hub
connected to each part of the rim). When used to examine many
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K.S. Mader et al. / Bone 57 (2013) 142–154
lacunae, the mean and standard deviation of Lc.DN are indications for
the style and degree of arrangement, to which the collection of lacunae
best resembles, as it is done with quasi-crystalline structures in solid
state physics [5,34]. A combination of multiple modes in a given osteocyte lacunar system would result in a multimodal distribution of Lc.DN.
Lacunar grouping (Lc.Gp). In order to characterize the grouping of
osteocyte lacunae, we introduce a descriptor called lacunar grouping
(Lc.Gp). This descriptor provides a value characterizing how clustered
or closely packed a group of lacunae is, given in terms of the (local) lacuna number density (N.Lc/BV). Such a measure, also known as selfavoidance [37], has been shown to be useful in the domain of neuroscience to investigate the architecture and hierarchy of neural networks
and thus, potential functional pathways within those networks. By comparing osteocyte lacunar distributions obtained from experiments with
those retrieved from randomly placed lacunae, one can calculate and
display the characteristics of the osteocyte lacunar distribution in the
form of hubs (regions where clustering is high or self-avoidance is
low, respectively) and spokes (regions where self-avoidance values
are high or clustering is low, respectively). Following this idea, we formally define lacunar grouping by comparing lacuna nearest neighbor
distance (Lc.ND) with the mean nearest neighbor distance (r0,overall) of
a system with feature centroids, which follow a Poisson distribution
with the same spatial density (i.e. N.Lc/BV)
r 0;overall ¼
1=3
BV
2πN:Lc
ð10Þ
according to Ref. [45]. We combine the concept of lacuna territory to
define r0 for each lacuna individually by substituting lacuna number
density (N.Lc/BV) by lacuna territory density (1/Lc.Te.V), so that
Lc:Te:VðiÞ 1=3
r 0;individual ðiÞ ¼
:
2π
ð11Þ
A direct comparison of the actual nearest neighbor distance (Lc.ND)
to the expected value in a random system (r0,overall) provides then the
lacunar grouping
Lc:Gr ¼
Lc:ND
Lc:ND
¼
1=3 :
BV
r 0;overall
ð12Þ
2πN:Lc
The mean lacunar grouping (b Lc.GpN) typically falls in the range [0,2],
where 0 is the minimum and represents an infinitely densely packed or
clustered osteocyte lacunar system (Lc. ND ≪ r0,overall), 1 refers to a randomly distributed system (Lc. ND = r0,overall) and values above 1 indicate
a self-avoiding osteocyte lacunar system (Lc. ND N r0,overall) [32,55].
Global environment
To characterize quantitatively the spatial distribution and alignment
of a given set of osteocyte lacunae, we introduce several global environment descriptors. These measures aim to portray the anisotropy of the
spatial lacunar distribution, their spatial grouping characteristics, as
well as their alignment as regards to a laboratory frame of reference
(e.g. the longitudinal axis of a long bone).
Distribution tensor. The distribution tensor provides a fully 3D set of metrics for quantification of the spatial distribution and structure of a given
osteocyte lacunar system. For example, the distribution tensor can serve
to calculate the spacing between lacunae, with the aim to be better understand other osteocyte lacunar metrics in the context of a 3D bone system, such as the lacuna number density (N.Lc/BV). The calculation of the
distribution tensor parallels the shape tensor and done using the so
called network, determined in the Lacunar Delaunay Neighbors section.
The distribution tensor is calculated using the Delaunay triangulation
mentioned in the previous section (Local environment). The distribution
tensor can be defined for a single lacunae (with respect to its Delaunay
neighbors) or group of lacunae, but it is applied more typically to an appropriately sized ROI inside the specimen. We shall present the calculation of D in the most generic form here, where the steps are visualized in
Fig. 2 (columns a–c). The first step is to perform the Delaunay triangulation of the lacunae's centroids as discussed in the previous section. We
first select k edges from this triangulation and we shall call these
edges ‘lacunar links’ and number
them i = 0…k. Each edge can be rep
!
resented by a vector l ðiÞ ¼ lx ðiÞ; ly ðiÞ; lz ðiÞ defined by the centroids of
two neighboring lacunae in the laboratory frame of reference (x, y, z).
The distribution tensor is then calculated by taking the covariance of
the set of edge vectors. More detail and the full formulas are available
in the Supplementary materials. Similar to the analysis done previously
for the shape tensor (see Section Shape and orientation), the distribution tensor is first transformed by a principal axis transformation. The
!
!
! result is a new set of normalized eigenvectors V D;1 ; V D;2 ; V D;3 ,
representing the primary directions (the best-fit ellipsoid axes) and
the corresponding eigenvalues (λD,1,λD,2,λD,3). Since the distribution
tensor encodes both topological and spatial information through the lacunar links, there is no direct relation between the distribution tensor
given here and the territory unit described earlier (see Section Local
environment). Given a fixed topology, changes in the distribution tensor
correspond to changes in territory shape. Correspondingly, the shape of
the territory shape will be quantified in the following based on measures
derived from the distribution tensor. We define lacunar distribution
stretch (Lc.Dt.St) — similar to lacuna stretch (Lc.St), as the ratio of the
difference between the largest and smallest eigenvalues of the distribution tensor, compared to the largest one
Lc:Dt:St ¼
λD;3 −λD;1
λD;3
ð13Þ
where (λD,1,λD,2,λD,3) represent the eigenvalues of the distribution tensor
in ascending order of their magnitude. Lc.Dt.St takes values in the range
[0,1]. A value of 0 means the distribution is completely isotropic
(Fig. 2d). A value of 1 indicates an infinitely anisotropic distribution
(Figs. 2e, f), where lacunae are closely spaced in one direction and far
from each other in perpendicular directions. We define lacunar distribution oblateness (Lc.Dt.Ob) — similar to lacuna oblateness (Lc.Ob) as the oblateness of the ellipsoid described by the principal axis transformation of
Lc:Dt:Ob ¼ 2
λD;2 −λD;1
−1
λD;3 −λD;1
ð14Þ
where (λD,1,λD,2,λD,3) are the eigenvalues of the distribution tensor.
Lc.Dt.Ob takes values within [−1,1] and further classifies the anisotropy
of the spatial lacunar distribution. Lc.Dt.Ob close to 1 represents a highly
oblate distribution (Fig. 2b), where lacunae are organized in strand-like
structures, i.e., closely spaced along one direction and further apart in
the other perpendicular directions. Lc.Dt.Ob close to −1 indicates a very
prolate distribution (Fig. 2c), where lacunae are organized in plane-like
structures, i.e., very closely spaced in two dimensions and further in the direction perpendicular to this plane.
Alignment tensor. A high degree of osteocyte lacuna alignment is observed in most bone tissues [2,31,39,65,67]. While the eigenvectors
! ! ! V S;1 ; V S;2 ; V S;3 of the shape tensor provide a good measure for the
orientation of a single lacuna, simply averaging these vectors provides
a very limited view of the collective orientation of many lacunae. The
method of texture analysis, in contrast, allows describing the distribution of the collective orientation of many osteocyte lacunae. Furthermore texture analysis-based descriptors are computationally efficient
and independent from the coordinate system selected [36,48]. As for
the calculation of the distribution tensor, the derivation of the alignment tensor is displayed in Figs. 2d–f. We first perform the texture
K.S. Mader et al. / Bone 57 (2013) 142–154
Alignment
Distribution
Spacing
147
a
b
c
1:1:1
1:2:2
1:1:2
Lacunae
with
Links
d
Theta
e
f
10%
80%
95%
91%
63%
Lacunae
Main Axis
Directions
Link
Directions
Alignment
Tensor
Distribution
Tensor
Oblate
Distribution
Anisotropy
1%
78%
Prolate
90%
Lacunar
Alignment
29%
Fig. 2. In this figure, we show the progression from lacuna to alignment and distribution tensors for a series of synthetically generated data sets. For the alignment tensor, we show how the
collective orientation of many lacunae can be expressed as a tensor. We start with 3 synthetic data sets (columns a–c) with a progressively increasing degree of randomness of lacuna
orientation. The top row shows the range of the uniformly distributed angles used to generate the synthetic data. The second row shows a subset of the lacunae themselves as simple prolate ellipsoids. The third row shows a plot of the main axis of each lacuna in set. The fourth row shows the alignment tensor as an ellipsoid calculated from the main axes of the lacunae. The
last row shows the anisotropy of this tensor. For the distribution tensor, we show how the lacuna positions affect the anisotropy and oblateness. We show here 3 configurations of lacuna
created by adjusting the mean spacing along the cardinal directions in a lattice of lacunae jittered slightly to show the positions of all lacunae better and to demonstrate that the method is
not grid-based (columns d–f). The first row shows the ratio of the cardinal spacings (x:y:z). The second row shows the lacuna depicted as red colored spheres and the links indicated by
green cylinders. The next row shows the distribution of just the link elements. The covariance of these forms the basis for the distribution tensor. The bottom row shows the distribution
tensor as an ellipsoid and its degree of anisotropy as a percentage (under) and whether or not the distribution is oblate/strand-like (e) or prolate/sheet-like (f).
analysis by defining the texture tensor as a covariance matrix (similar to
the steps taken for the distribution tensor and shape) from the individ ! ual primary lacuna orientation Lc: O r1 of many (i = 1 … m) different
lacunae in the measurement frame of reference (x, y, z). More detail and
the full formulas are available in the Supplementary materials. The principal axis transformation of T provides the three eigenvectors
! ! ! V T;1 ; V T;2 ; V T;3 , representing the primary directions (the best-fit ellipsoid axes) and the corresponding eigenvalues (λT,1,λT,2,λT,3). The
! principal eigenvector V T;3 represents the average lacuna orientation,
where the average has been taken over all lacunae (i = 1 … m) exam! ined. On the other hand, V T;3 indicates the direction of the biggest de! viation from the primary orientation V T;3 . The eigenvalues and the
ratio between them reflect the distribution of orientations within
the group of the studied lacunae. If the biggest eigenvalue (λT,3) of
the average lacunar orientation is dominant compared to the
others (λT,3 ≫ λT,2, λT,1) most of the lacunae are aligned close to
(λT,1). If, on the other hand, the three eigenvalues are approximately
similar (λT,1 ≈ λT,2 ≈ λT,3) the lacunae are more or less randomly
oriented. We define lacunar alignment (Lc.Al) as the difference between
the largest and the smallest eigenvalues of the alignment tensor, normalized by the largest eigenvalue
Lc:Al ¼
λT;3 −λT;1
:
λT;3
ð15Þ
Lacunar alignment falls in the range [0,1]. For a set of lacuna with
random orientations, Lc.Al is 0 and for a set, where all lacunae are oriented in the same direction, Lc.Al is 1 as shown in Figs. 2d–f.
Implementation of osteocyte lacunar descriptors
The quantitative framework for the 3D characterization of the osteocyte lacunar system was implemented using a custom in house framework written in Java. The various components are assembled together
into a workflow (Fig. 3) using the Jython scripting language. The matrix
math calculations, specifically the eigentransforms were done using the
JAMA mathematical library (http://math.nist.gov/javanumerics/jama/;
U.S. National Institutes of Standards and Technology, Gaithersburg,
Maryland, USA). The analysis was performed on a HPC Merlin4 cluster
at the Paul Scherrer Institute (PSI), consisting of 30 machines each
with 12 cores and 2–4 GB of memory per core. Nevertheless, the analysis could be run on any major operating system, provided sufficient
RAM is available (1024 × 1024 × 1024 voxels required approximately
20 GB of RAM). Given the large number of osteocyte lacunae analyzed
in the current study (N500,000) and the intention to perform genomic
scale studies (N1000 bone samples) in the future, we have developed
a database-driven tool for managing the results. The database contains
three tables: one includes all samples, the second contains all osteocyte
lacunae of those samples, where the third table is a summary of ensemble statistics (distribution, alignment, etc.) based on the second table's
entries. The database is run using a MySQL database engine, where
queries are executed using Python and R and standard SQL language.
The setup was designed to be as independent of specific software as
148
K.S. Mader et al. / Bone 57 (2013) 142–154
Measurement
Reconstruction
Threshold and
Morphology
Inversion
Contouring
Subtraction
(Porosity)
Distance Map
(Bone Surface
Distance)
Component
Labeling
(Lacunae)
Component
Labeling
(Canals)
Voronoi
Tesselation
(Lc.Territory)
Voronoi
Tesselation
(Ca.Distance,
Ca.Territory)
1.5mm
Shape Analysis
Fig. 3. In this figure, we show the data processing workflow from acquisition to final shape analysis and how the different steps connect to form the final result. The top shows the measurement and reconstruction step, which result in a density map of sorts of the imaged bone. The threshold and morphology steps result in the calcified tissue. The inversion steps provides
inverse of the calcified tissue and thus all air in and outside of the bone. The contouring step provides the mask indicating the bone boundaries. The subtraction (inversion from contouring)
provides the porosity inside of the bone, which can then be classified through component labeling into lacunae and canals. These classified and labeled structures are subsequently given to
the tessellation operations for territory and relative position analysis, which are finally given to the shape analysis tool.
possible to ensure scalability to a higher-performance commercial or
cloud-based database server in the future.
Biological study
As a first application of the proposed quantitative framework for the
3D characterization of the osteocyte lacunar system, we present a biological study, where we derived the proposed lacunar descriptors for
femoral bones extracted from bones of four different mouse strains
used in earlier studies [54,53]. The four mouse strains were namely
C57BL/6J (B6) and C3H/HeJ (C3H) with a mutation in the growth hormone releasing hormone receptor (Ghrhr, allele symbol = lit) [14]:
C57BL/6J-Ghrhrlit/J (B6-lit/+), C57BL/6J-Ghrhrlit/Ghrhrlit (B6-lit/lit),
C3H/HeJ-Ghrhrlit/J (C3H-lit/+), and C3H/HeJ-Ghrhrlit/Ghrhrlit (C3Hlit/lit). Due to sample availability the groups are not equally sized and
the first two groups have 3 mice each while the C3H-lit/lit has 5 and
C3H-lit/+ has 6. The mice were raised and sacrificed at the Jackson Laboratory, and all animal procedures were approved by the local IACUC.
3D measurements of the osteocyte lacunar system
The osteocyte lacunar measures we propose here are not limited to a
specific imaging modality. For the current study, we employed SR CT at
the TOMCAT beamline of the Swiss Light Source (SLS) to image the middiaphysis of murine femora similar to our previous work [35,53]. The CT
setup at the TOMCAT beamline provides a high-degree of existing automation, which is a prerequisite for high-throughput studies. The 17
femoral bone samples were automatically mounted and aligned, and
their mid-diaphysis was automatically found and scanned according
to Ref. [35]. The data were acquired and reconstructed at a 1.4 μm isotropic voxel size within a field of view of 1.4 × 1.4 × 1.4 mm3 and at
an X-ray energy of 17.5 keV. For each scan 1500 projections were
taken over 180° with an exposure time of 100 ms. The details of
the image processing pipeline are shown in Fig. 3. A histogrambased global threshold was applied to all samples to identify the calcified bone tissue. The segmentation of the cortical bone region
(mask) from the data sets was done similar to Ref. [53] using a custom ImageJ-based tool chain utilizing morphological operations (dilations, erosions, contouring, and component labeling). Once the
cortical mask has been created, which encompasses the cortical
bone tissue, it was reoriented along its principal axis. The
intracortical porosity was then extracted by subtracting the calcified
tissue from the cortical mask and elements smaller than 1000 μm3
were classified as osteocyte lacunae. This decision criterion was correct for more than 95.5% of all classified intracortical porosity elements when compared with classification by hand as shown by a
separate study on another 10 femur samples examining the largest
100 contiguous porosity objects in each sample (results not
shown). In the next step, the lacuna descriptors were calculated
for each sample. For each of the 17 data sets, consisting of
1024 × 1024 × 1024 voxels, not more than 24 GB of RAM and 6 h of
computation time on 4 cores was required. The code is parallelized,
but not every operation can take advantage of multiple cores. Furthermore bandwidth limitations make it difficult to estimate the total CPU
K.S. Mader et al. / Bone 57 (2013) 142–154
usage and performance on other systems. For quantitative analysis,
the results were stored in the central database, described in
Section Implementation of osteocyte lacunar descriptors.
Statistical analysis
The study design represents a 2 × 2 factorial study comparing two
strains with and without growth hormone. The results were than analyzed using 2-way ANOVA analysis correcting for the multiple comparisons performed. The analysis was done in R [61] using the RMySQL [24]
and compareGroups [59] packages. The visualizations were made using
ggplot2 and plyr [69,68]. In compliance with the open-initiative the raw
shape data (http://dx.doi.org/10.6084/m9.figshare.695235) and statistical analysis are available here (http://dx.doi.org/10.6084/m9.figshare.
695233) [1].
Results
The results like the study are divided into two parts the in-silico validation and the biological study. The validation results show the comparison between model data generated in-silico and the computed
values using the analytical tools developed. Much of the more detailed
work on the validation and in-silico data sets are placed in the Supplementary materials. The biological study shows these analytical tools
applied to a bone samples and the various comparisons that can be
performed with these data.
Validation of osteocyte lacunar descriptors
The validation was done on the shape and neighborhood metrics. The
dimensions and orientations extracted from the shape tensor were compared with those used to create the model and all metrics matched the
input data within a tolerance of 1 voxel/1voxel side length. Furthermore,
the derived Delaunay triangulation matched the input arrangement of lacunae perfectly (every link in the input structure had a 1 to 1 match in the
output structure). Since distribution is derived from the center of volume
and the neighbors determined by Delaunay triangulation, its validation is
completed by association. Alignment as it is based on the collection of orientations is only dependent on the orientations being determined accurately, which was validated in the shape tensor validation. Finally, all
the derived tensor metrics (lacuna stretch, oblateness, etc) are calculated
directly from the tensor data and thus the accuracy of the tensor calculation dictates the accuracy of these metrics. Specifics on the validation and
structures used to validate are available in Supplementary materials.
149
Biological study
Summarized in Table 1, and graphically in Fig. 4, we present the results for the morphometric analysis conducted on two inbred strains of
mice, using the novel osteocyte lacunar descriptors introduced in
Section Definition of osteocyte lacunar descriptors. We shall highlight
here the most pronounced differences observed in the results.
Lit/lit mutation/growth-hormone effect
The effect of growth hormone serves to increase the size of the lacunae without significantly affecting shape or density. In the case of C3H,
the growth hormone had a strong effect on the distribution of the osteocyte lacuna and consequently the Lc.Dt.St and Lc.Dt.Ob while the B6
remained largely unchanged.
Mouse strain differences
The differences between the groups were generally smaller than the
effect produced by the lit mutation but still present. The lacuna lengths
and consequently volume is higher in C3H but shape and oblateness
were consistent. Neighbor count and nearest neighbor distance were
both significantly different while the difference in the density was
below the significance threshold. Finally the alignment, a metric calculated on all the lacunae in a sample, was significantly different between
strains, being higher in C3H independent of growth hormone action.
Grouping (Lc.Gp)
The grouping metric was consistent in all 4 groups independent of the
other metrics and in every sample around 1.3. The value indicates a more
self-avoiding than random organization. Since the metric was calculated
by examining all lacunae in the sample the significance of the difference
was particularly notable. Furthermore, the standard deviation (available
in the Supplementary materials) of the metric within groups was low
(0.2) but comparably higher than the difference between groups (0–
0.05). The variation being higher within samples than between samples
indicates that some how the global value or mean behavior is conserved
despite spatial heterogeneity within the sample.
Distribution difference in C3H due to growth hormone
As noted in the main body of the manuscript, the differences between the lit/lit and lit/+ of the C3H mice are exceptionally large. We
aim to shed additional light on these differences and provide detailed
explanations for a sample taken from each group. We provide Fig. 5 to
graphically depict the differences. The distribution tensor is calculated
by examining all of the so-called links between neighboring lacunae. A
difference in the distribution tensor therefore implies a difference in either the distribution of links or the links themselves. We show the links
Table 1
Selected results from the biological study showing the differences between the two strains examined (B6, C3H) and with or without the little mutation (lit/lit vs lit/+). The units of length,
width, height, and Lc.ND are μm. The units of Lc.V and Lc.Te.V are μm3 and the units of density are kLc/mm3.
Lc.N (#)
Length (μm)
Width (μm)
Height (μm)
Lc.V (μm3)
Lc.St (%)
Lc.Ob
Lc.Te.V (μm3)
Density (kLc/mm3)
Lc.ND (μm)
Lc.DN (#)
Lc.Gp
Lc.Dt.St
Lc.Dt.Ob
Lc.Al (%)
B6lit/lit
B6lit/+
C3Hlit/lit
C3Hlit/+
N=3
N=3
N=6
N=5
26,063 (4560)
15.2 (0.09)
8.29 (0.12)
5.16 (0.19)
317 (16.2)
0.64 (0.01)
−0.31 (0.03)
22,456 (2064)
44.8 (4.15)
19.4 (0.10)
7.34 (0.06)
1.27 (0.03)
0.29 (0.01)
0.61 (0.09)
0.87 (0.01)
32,514 (2885)
17.2 (0.59)
9.96 (0.91)
5.89 (0.14)
469 (45.6)
0.64 (0.01)
−0.21 (0.13)
26,247 (3253)
38.5 (4.53)
20.4 (0.34)
7.26 (0.08)
1.27 (0.03)
0.29 (0.04)
0.20 (0.30)
0.84 (0.03)
33,799 (13,822)
16.9 (0.92)
8.92 (0.36)
5.24 (0.16)
378 (41.8)
0.67 (0.01)
−0.33 (0.05)
28,333 (3518)
35.7 (4.22)
20.7 (0.17)
7.42 (0.04)
1.26 (0.04)
0.26 (0.03)
0.55 (0.11)
0.91 (0.01)
53,287 (3742)
19.4 (0.25)
10.1 (0.18)
6.21 (0.12)
577 (15.4)
0.66 (0.01)
−0.36 (0.02)
25,687 (1150)
39.0 (1.77)
21.2 (0.25)
8.00 (0.08)
1.33 (0.01)
0.74 (0.04)
0.84 (0.02)
0.88 (0.01)
p.lit/lit B6 vs C3H
p.lit/+ B6 vs C3H
p.B6 lit vs +
p.C3H lit vs +
0.59
0.01
0.24
0.87
0.08
0.01
0.98
0.04
0.02
b0.01
0.42
0.93
0.81
0.91
b0.01
0.02
b0.01
0.97
0.04
b0.01
0.10
0.02
0.99
1.00
b0.01
b0.01
0.08
b0.01
b0.01
b0.01
0.78
0.01
b0.01
b0.01
b0.01
0.96
0.21
0.31
0.18
b0.01
0.37
1.00
1.00
0.01
0.13
0.01
b0.01
b0.01
b0.01
b0.01
0.10
0.77
0.36
0.45
0.02
b0.01
0.01
b0.01
0.02
0.01
150
K.S. Mader et al. / Bone 57 (2013) 142–154
a
b
c
Shape
Lacunae
d
Alignment
Distribution
Stretch/
Anisotropy
C3H lit/lit
1
B6 lit/lit
0.8
0.5
C3H lit/+
0.2
B6 lit/+
0
Fig. 4. Biological study. Here we show the segmented lacunae and analysis from each of the samples examined. The rows represent the four groups, which were investigated in this paper.
The columns in each panel represent the 3 different samples (for the groups with more than 3, the first 3 ordered by measurement time were taken) from each group. a) shows the segmented lacunae from a birds-eye perspective. b–d) are shown with z in the vertical direction, y on the horizontal direction and x coming out of the page. The ellipsoidal representations are
normalized in each panel so that they do not overlap. b) shows the average shape tensor for the lacunae independent of orientation colored by anisotropy calculated from the eigenvalues.
c) shows the distribution tensor calculated on all of the lacunae in each sample colored by anisotropy calculated from the eigenvalues. d) shows the alignment tensor calculated on all of the
lacunae in each sample and colored by anisotropy calculated from the eigenvalues.
for both samples in Fig. 5a and while both samples have a distribution of
links in all directions the Growth Hormone sample has longer links at
approximately 45° to the X axis indicating that the lacuna are spaced
further apart in this direction. We then show the local distribution by dividing the samples up into 20 × 20 × 20 grids with isotropic block size.
We then calculate the distribution tensor for every block in this grid. We
see in Fig. 5c that the lower half of the samples are nearly identical but
the upper-half of the distribution is strongly stretched and oriented like
the final result. We have determined by looking at the detailed porosity
structures in the bone samples that the difference is caused by increased
canal structure and a stronger horizontal orientation of the canals in this
area of the bone and we show a rendering of the canal structures in
Fig. 5d. We had intentionally left the analysis of canal structures from
the manuscript, but it appears in many aspects that the lacunae and canals are strongly linked. In particular, for lacuna distribution, canals play
an important role because they physically prevent particular spatial distributions of lacunae.
Discussion/conclusion
The methods developed and metrics introduced provide a powerful,
quantitative tool for morphological analysis at the cellular level. Considering previous works, which have presented analysis of osteocyte shape
[21,29,28,53,62,65], the techniques had limited field of view, sample
counts, scalability, and automation. Furthermore in studies looking at
densities [9,21,46,47,53], a variety of different counting techniques
have been suggested, ranging from computing the average over the entire bone to counting lacunae which are within a given radius of a canal,
which are fundamentally incomparable. Furthermore, the results however are very sensitive to the counting method and normalization
used. The territory metric introduced in this paper is scale and model independent and additionally provides a graphical explanation and significantly more flexibility. It can be coupled with nearest neighborhood
parameters to better quantify not just the density, but also the variations in density for a given region or sample. Finally our approach enables the application of analytical tools inherited from soft-matter
science such as the strain tensor [3,20]. The strain tensor, typically
used to quantify deformation in cellular structures like foam and ice,
could drastically simplify the analysis of the structural changes in
bone occurring during loading and fracture.
Shape
Oblateness has several advantages over other similar measures. The
metric is very easy to calculate requiring only the dimensions from the
diagonalized shape tensor of the lacunae. There is a degree of familiarity
in the bone scientists as it has already been defined, validated, and
applied on trabecular bone. Finally, the parameter provides a scale free
measurement of shape as applicable to murine bone as elephants. An
observation we had during validation was that, if the lacunae were thinner than 3 voxels, it was not possible to determine with this technique
the degree of anisotropy and the variations seen are more likely to be directly caused by variations in orientation. Additionally surface area and
surface area to volume parameters, which would certainly be practical
for examining fluid flow, stress concentration, remodeling area and numerous other biologically significant metrics, were not included in this
study. While the metrics were calculated for all samples, they were
found to be extremely resolution-dependent changing by more than
50% with a 30% change in voxel size (data not shown). It might be
possible to compare the metrics for samples acquired at a given resolution, we decided however that such extreme variability was undesirable and including such an analysis at this size scale would be
detrimental. On the other hand, a model-based surface-area from
the fitted ellipsoid might serve as a robust way of estimating such
properties in future studies.
Density
The territory volume shows high correlation with both nearest neighbor distance and number of neighbors within 50 μm on both the synthetic and measured data (N 500,000 lacunae). However, it has the advantage
that it is threshold-free and thus independent of the scale of the system.
Furthermore, nearest neighbor distance provides alone no information
about the overall lacuna count and in very anisotropic or clustering systems will be skewed from the actual distribution. The metrics used in
this paper have also been used in papers in other fields from sociology
to neuroscience to material science [18,38,42,58,71,70] because of their
flexibility and low-scatter. We define scatter in this context as the sensitivity of the metric to small changes in field of view or sample position for
a visually homogenous or nearly homogenous group.
K.S. Mader et al. / Bone 57 (2013) 142–154
a
b
c
d
151
C3H lit/lit
C3H lit/+
475,000
~7 per lacuna
930,000
~8 per lacuna
Link
Distribution
Global
Distribution
Local
Distribution
Canal
Structures
Fig. 5. Here we show a detailed analysis of the significant differences in distribution between the lit and +C3H mouse strains. We select one of the mice from each group and examine
several different aspects of the structure, which affect the distribution. In (a) we show the link distribution for the links determined by the Delaunay triangulation, which are used to calculate the distribution tensor. Here we already see a large number of links, which are strongly biased in the diagonal left-right direction (like the distributions themselves (b)). In (c) we
show the distribution tensor calculated on the local scale. We did this by dividing the bone up into 20 × 20 × 20 isotropic blocks and calculated the distribution based on the lacunae inside
of these blocks. Here we see the two samples match quite well in the bottom half (z) of the axes, but the + mouse has noticeable changes in the upper half. In (d) we show the segmented
canal structures for both samples. The + samples has many more canals and in particular the top-half has a higher density of nearly horizontal canals, which likely cause the drastic change
in the distribution tensor.
Neighbor count
Although the neighbor count and distribution tensor showed differences in the biological study conducted, the preliminary findings are too
weak to understand the biological significance. A future study looking
carefully into these osteocytes and the canalicular network connecting
them might provide insight into the communication within and hierarchy of cortical bone and a deeper understanding for these changes.
Grouping
The grouping factor (Lc.Gp) was found to be very consistent among
all four groups. The consistency is actually even higher than it appears in
the result since the metric for each sample represents the average over
30,000 lacunae. The factor, being around 1.3, indicates a self-avoidance
in the spatial distribution of lacunae that is higher than would be
expected by random coincidence. Furthermore, we found that the
grouping remained nearly constant despite the relatively large differences between groups when looking at other local and global environment metrics. In our in-silico data (not shown), we found changes in
the parameters like spacing and consequently density to coincide with
changes in grouping factor. This might be an indication that the biology
actively works to maintain this level of grouping despite variation in
other metrics. Furthermore, the grouping being self-avoiding reinforces
the idea that each osteocyte has a territory it maintains and other osteocytes are somehow prevented or discouraged from settling within this
territory.
Distribution tensor
The distribution tensor showed very strong changes in one of
the groups where the tensor and its stretch and oblateness were significantly consistently different from the other 3 groups (Fig. 4, Table 1).
Although it is difficult to make a full interpretation of this number, it
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K.S. Mader et al. / Bone 57 (2013) 142–154
appears that for these samples other microstructural elements (canals)
are in such a high density that they disrupt the lacunar structure causing
so many of the neighbors, particularly in the off diagonal direction to be
much further away than would normally be expected and thus strongly
changing this tensor.
Alignment
The quantification of osteocyte lacuna alignment is new in the bone
field and thus there are few references for comparison. In one study the
tool of Mean Intercept Length (abbreviated as MIL and commonly used
for characterizing trabecular structure) was applied to several lacunae in
an effort to examine their alignment [65]. In all other bone studies, such
analysis has either been left out entirely [21,46,53] or described qualitatively using images and wording such as “the concentric alignment of
the osteocytic network around the Haversian canal”, “osteocyte lacunae
in the lamellar areas are aligned in individual layers”, “a correspondence
between the arrangement of the osteocytic network and the collagen
matrix orientation is obvious”, “in woven bone, there is almost no mutual alignment between osteocytes”, “interestingly, the osteocytic network
throughout much of the fibrolamellar cortex also shows such a layer-like
organization”, “a highly oriented network of osteocytes aligned in nearly
concentric cylindrical layers” [26], and “reduced alignment of osteocyte
lacunae” [23]. All these qualitative descriptions can now be well quantified using the texture-analysis techniques we present. One study uses
the term “alignment” to refer to what we call orientation in this paper
rather than the ensemble analysis of the orientation of many lacunae
[65]. MIL further provides a useful metric for the characterization of
the spatial distribution of lacunae in bone and pores of other porous
media; however, it is difficult to determine exactly, which changes
have occurred as it encompasses size, anisotropy, spacing, and alignment into a single analysis. Discerning which underlying morphological
change is responsible for a given change in MIL can be quite difficult especially in highly irregular media like bone. Additionally, in our in silico
systems the MIL anisotropy parameter was much less sensitive to the
changes between the most aligned and most disordered system (17%
change compared to 50% change). Furthermore, an increase in the anisotropy of the lacuna had no effect on the alignment metric, but caused
the MIL metric to increase for equally unaligned lacunae. MIL is also incapable of examining the orientation of lacuna in respect to other objects (such as canals or bone surfaces) as it only works in the Cartesian
coordinates defined for the sample. Finally MIL is a very time consuming
metric to compute and must be recomputed every time the region of interest examined is changed. Our alignment metric is very computationally efficient and can be calculated quickly (several million lacunae can
be done per second) using only the analyzed lacuna shape tensor
whereas MIL requires the entire image data. Other methods for measuring alignment include the averaging of either vectors or angles. Texture
analysis has the advantage over averaging vectors since it is orientation
independent (it does not differentiate between V3 and −V3, which
given the lack of head/tail features on an osteocyte lacuna is a positive
feature). It has the advantage over angles in that it does not require a
model or frame of reference to measure the angles from (slightly offaxis sample mounting will not affect texture analysis results). In other
biological fields alignment has been characterized in very specific
cases by measuring the number of cells oriented in a particular direction
[16], but this provides only limited information for more complicated
structures like bone and requires a-priori knowledge of orientation.
These techniques however are not useful when trying to determine
alignment in larger samples with 10s of thousands of lacunae or for
comparing the relative degree of alignment between samples. Alignment as defined in this paper works very well if the objects under investigation are aligned in the same direction, but would yield deceptively
poor results in other configurations, specifically if lacunae were to be
radially-oriented away from a canal or cylindrically-oriented around a
canal, the alignment definition we use would not appropriately account
for this. A careful examination of the samples is required before
asserting that a given sample is more poorly or randomly aligned as
the alignment might in fact be in respect to another structure. Alignment
was, as previously stated, examined in both the primary lacuna direction
(V3) and the secondary direction (V2). While these values correlated
very strongly, the secondary direction seems to be more spatially dependent and possibly anatomically sensitive than the primary direction. A
more detailed investigation of both metrics around Haversian canals
and woven bone might be of particular interest.
Limitations
The primary shortcomings of these tools as applied were from the
measurements themselves. Given the lack of large, high-resolution detectors, measurements must always make a trade-off between field of
view and resolution. Unfortunately, in order to see the entire crosssection of the femur, a resolution was chosen which was too coarse to
accurately assess surface area or estimate canalicular structure. Furthermore, while synchrotron-based computed tomography provides very
fast acquisition times without destroying the sample or requiring special preparation, it provides purely morphological information and the
occupation or biological contents of lacunae and canals is unknown.
Histological data, which provides such information through staining,
would serve as a good accompanying measurement. Unfortunately
most histological techniques are still limited to 2D data sets which require slicing and the direction in which the slice is taken can drastically
change the observed metrics [9]. It would however still provide valuable
connections between structure and function and enhance the biological
meaning of the morphological parameters. In general, if the objects investigated do not resemble ellipsoids or have very rough surfaces (for
example micro-cracks) an ellipsoidal model poorly characterizes the
shape of the object. Additionally, with higher resolution data the processes or canaliculi are visible and should be removed for an accurate
characterization of the body of the cell. If, however, the criteria are
met, the ellipsoidal model should prove to be very useful for a wide variety of different shape characterizations from nano- to macro-scale.
While the techniques introduced greatly increase the scale and scope of
possible experiments, they are ultimately not a complete replacement
for hand-examination and visualization of data. The tools presented will
always give quantifiable output, but the meaning and correct interpretation of these numbers is crucial to further understanding of the underlying biology.
Biological outlook
On our particular biological samples, there are a number of metrics,
which are very consistent between the strains and growth hormone
groups of mice, illustrating the large degree of homogeneity of the osteocyte lacunae. Of particular interest was the grouping (Lc.Gp), which
might indicate similar processes of higher level organization. Since the
field of view allows the viewing of the entire cross-section of the bone
specimen, it is possible to examine carefully the distribution of lacuna
and canals and its anatomical dependence. Using the territory parameter to color the bone volume, the variations in local number density are
made clear. In all the samples measured there seems to be a preferred
accumulation of lacunae on the posterior side of the femur. This accumulation is not evenly distributed but rather much stronger in a stripe
near the middle of the sample, which might indicate higher compression loads or forces. A previous study [7] has shown changes in density
and volume in active and immobilized bones in rats and the application
of our analysis tools could provide additional spatial descriptors to
quantify the differences and better understand the changes and compare the two groups. We observe that the spatial variation within a sample is quite large and there is potentially some evidence of higher-level
cellular organization. Unfortunately, the volume dependence for regions of interest for any spatial variation analysis makes selecting a
K.S. Mader et al. / Bone 57 (2013) 142–154
specific volume for comparison difficult. Further work could conduct a
more thorough analysis of these data and potentially quantitatively
identify evidence of recent growth or turnover. The development of
in-vivo scanners with submicron resolution would allow for testing of
this hypothesis through a study scanning the same animal at a number
of different ages and observing the changes in density. Alternatively, a
number of animals could be studied at each age and the trends inferred
from this information. If a large enough population were examined it
might even be possible to determine what selection rules are used
for the addition or filling of lacunae. The techniques presented lay
the groundwork for future larger-scale studies where the quantitative
assessment of ultrastructural parameters in bone can be directly compared. Additionally these definitions are instrumentation and softwareimplementation independent and thus applicable, assuming proper calibration, to the results from other tools, which are sensitive to different
biological or chemical properties of bone such as SEM or CLSEM. With a
standard system of metrics, it is possible to not only validate studies
but also coalesce studies to optimize the understanding of bone growth,
sensing, repair, and disease.
Funding sources
National Competence Center for Biomedical Imaging in Lausanne.
Disclosures
All authors state that they have no conflicts of interest.
Acknowledgments
We would like to thank the Paul Scherrer Institute for beamtime
at the TOMCAT Beamline of the Swiss Light Source and the National
Competence Center for Biomedical Imaging in Lausanne for funding.
We also thank Leah Rae Donahue and the Jackson Laboratory for providing the mouse specimens.
Appendix A. Supplementary data
Supplementary data to this article can be found online at http://dx.
doi.org/10.1016/j.bone.2013.06.026.
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