Assessment of Bone
Quality from pQCT Images
Dean Inglis, Ph.D.
Assistant professor (adjunct)
Department of Civil Engineering
McMaster University
Overview
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CT image source, formation and characteristics
Image segmentation
Bone morphometry
2D stereology: basic principles, assumptions
3D stereology: mean intercept lengths, Eigen
analysis, interpretation
Model independent measures
Topology: Euler number, Structure Model Index
Summary
What is Peripheral
Computed Tomography?
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pQCT (2D), hr-pQCT (3D)
CT imaging techniques that target
peripheral sites
use computer controlled X-ray source +
detector system
multiple X-ray 1D/2D projections
reconstructed into 2D slice/3D volume
images
CT basic principles
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electron beam strikes tungsten target
and generates polychromatic X-ray
beam
spectrum
source
CT basic principles
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X-rays pass through a sample and are
attenuated:
I = Ioe - ∫ u(x,y) ds
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I = intensity at the detector
Io= intensity of the source
u(x,y) = attenuation characteristics of the
sample: depend on atomic number (density)
attenuation is integrated along a ray
CT basic principles
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emergent X-rays detected by a phosphor
detector coupled to a CCD camera
CT image formation
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detection of many rays results in a
projection (silhouette) of the sample
many projections are generated by
rotating the source and detector around
the sample
image is reconstructed using convolution
back-projection
CT image formation
CT image formation
CT image characteristics
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raw CT data represent linear attenuation
coefficients
coefficients are converted to CT
numbers, Hounsfield Units (HU), in the
reconstruction process
pQCT calibrates HU into density: g/cm3
Image characteristics
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an image in its most basic sense is a matrix of numbers
a 2D matrix has topology consisting of pixels (picture
elements) 8-connected to their neighbours
images have a spatial origin, eg. (0,0,0) mm, and finite
spacing between their pixel centers, eg. 0.5×0.5×0.5 mm3
spacing partly governs ability to resolve small features
accurately
pQCT resolution: 0.2×0.2×0.5 mm3 (non-isotropic)
hr-pQCT resolution: 0.08×0.08×0.08 mm3 (isotropic)
Topology example: 6x5 image
7
4
8
3 xi,yi 1
6
2
5
Image characteristics
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a 3D image can be considered as a stack of 2D
images having thickness
pixels are now called voxels (volume elements)
and are 27-connected topologically
Image segmentation
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segmentation is the task of classifying
pixels/voxels based on their value and
topological affinity
segmentation is used to isolate features of
interest (bone) in an image
Image segmentation
Image segmentation
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thresholding:
P(x,y,z) = Po(x,y,z) < t
? 0 : Po(x,y,z)
Image segmentation
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binarization:
P(x,y,z) = Po(x,y,z) < t
?0:1
Image segmentation
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some problems to consider…
how do we pick “t” without bias?
how do we pick one bone from another?
how do we pick bone constituents
(cortex vs trabeculae)?
Image segmentation
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bone images consist of 2
pixel groups: bone and soft
tissue (or background): a
histogram of a bone image
appears bimodal
segment bone from nonbone using an automated
thresholding scheme to
determine “t”
Otsu’s method minimizes
the error of misclassifying a
non-bone pixel as bone and
vice versa by minimizing the
within-class variance of the
two groups
Otsu : t
Image segmentation
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at low resolution Otsu fails for bone within
bone:
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cortical bone vs. trabecular bone
trabecular bone vs. marrow
Image segmentation
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many other schemes exist:
livewire tracing, active contours, level
sets
desirable characteristics of any method:
simple, fast, reproducible, automated,
gets the job done!
Bone morphometry
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given a segmented image of bone,
what can be measured?
HU’s represent attenuation: analog for
density
calibration allows volumetric BMD
(g/cm3):
BMD = ∑ [Pi != 0 ? m×Pi + b : 0 ]
segmentation provides volume (cm3):
V = [ ∑ Pi != 0 ? 1 : 0 ]×dx×dy×dz
BMC = BMD × V (g)
Bone morphometry
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what is structure and is it important?
3 plank beam: σ = My/I
I-beam / block ~ 4 for L / t = 5
in addition to density (stiffness), the
spatial arrangement of material
(structure) contributes to strength
BMD/BMC is limited:
no information on spatial arrangement
Bone morphometry
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how can structure be measured?
before CT, samples were embedded in resin,
sliced and polished, and photomicrographed
2D images: area, perimeter length, number
more information (e.g., thickness, spacing)
can be inferred using stereology:
mathematical science based on geometric
probability
2D stereology
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Parfitt et. al. developed the “parallel plate
model” for analyzing 2D images
(J. Clin. Invest. 1983, v72, 1396-1409)
key assumptions:
-trabecular bone comprised mainly of
interconnected plates
-tissue is isotropic
-sample is uniformly randomly obtained
2D stereology
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basic 2D quantities:
PB = bone perimeter length (mm)
AB = bone area (mm2)
AT = tissue section area (mm2)
(bone + marrow)
2D stereology
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bone volume fraction (%):
TBV = BV/TV = AB / AT
Bone surface density (mm2/mm3):
Sv = BS/TV = PB / AT
bone surface to volume ratio (mm2/mm3):
S/V = BS/BV = PB / AB
mean trabecular plate thickness (mm):
MTPT = Tb.Th = 2 AB / PB
mean trabecular plate density (/mm):
MTPD = Tb.N = BV/TV / Tb.Th = PB / (2 AT)
mean trabecular plate separation (mm):
MTPS = Tb.Sp = 1 / Tb.N – Tb.Th = 2 (AT – AB) / PB
3D stereology
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trabecular bone is a highly
organized 3D oriented structure
3D provides additional metrics:
surface area, volume, orientation
a stereologic technique using a 3D
array of line probes provides BV/TV,
Tb.Th, Tb.N, and Tb.Sp
3D stereology
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considering the 2D case, focus
on the boundary between bone
and marrow within a circular
ROI
overlay an array of test lines
spaced δ apart
the sum of test line lengths, L,
is orientation independent
this is only true with uniform
sampling: circular ROI
3D stereology
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consider the intercepts between test
lines and boundaries
the number of intercepts, Tb.N(θ),
depends on orientation
the sum of intercept lengths, ∑I, is
orientation independent as δ→0
BV/TV = ∑I / L
mean intercept length, a.k.a. Tb.Th:
MIL(θ) = ∑I / Tb.N(θ)
the number of intercepts in marrow,
M.N(θ), is not equal to Tb.N(θ)
Tb.Sp(θ) = ( L - ∑I ) / M.N(θ)
3D stereology
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in 2D, an ellipse can be fit to data from N
orientations
Let (xi, yi) = (cos(θi), sin(θi)), i = 1→N
Tb.N(θi) = A xi2 + B xiyi + Cyi2
least squares fitting gives A,B and C
arranging A, B, C into a 2×2 matrix:
A ½B
½B C
Eigen analysis gives the orientation and
lengths of the principle axes of the
ellipse
anisotropy is defined as the ratio of the
axes’ lengths: L2 / L1
y
L2
Lθ1
x
3D stereology
θ
z
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in 3D, a 3D array of parallel test lines
probes the image uniformly within a
spherical ROI
“uniformly” means equal area partitions
of the surface of a unit sphere or many
x
random orientations
orientation of the lines is defined in
terms of two angles: θ, φ
( xi, yi, zi ) = ( sin(θi)cos(φi), sin(θi)
sin(φi), cos(θi) )
Tb.N( θi, φi ) = A xi2 + B yi2 + C zi2 + D
xiyi + E xizi + F yizi
φ
y
3D stereology
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least squares fitting gives A,B,C,D,E,F
A,B,C,D,E,F are arranged to form a 3×3 matrix
Eigen analysis gives the orientation and
lengths of the 3 principle axes of the ellipsoid
anisotropy is defined by the ratios of the axes’
min to max lengths: L3 / L1, L2 / L1
z
L2
x
L3
L1
y
Model independent
measures
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Tb.Th and Tb.Sp can be
measured without model
assumptions
find the medial axes (2D) or
surface (3D) of the bone
(marrow)
fit maximal non-overlapping
spheres within bone (marrow)
analyze the histogram of
spherical diameters
works for any ROI shape
Topology
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the Euler Number is an index of
connectivity of trabecular bone
measures redundant connectivity: the
degree to which parts of the object are
multiply connected:
Χ = β0 – β1 – β2
β0 is the number of isolated objects = 1
for bone
β1 is the connectivity
β2 is the number of enclosed cavities =
0 for bone
β1 is calculated by analyzing the local
neighbourhood connectivity of each
voxel representing bone
works for any ROI shape
Topology
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the Structure Model Index, SMI, is a
measure of the degree of convexity of a
structure
in bone, it indicates the relative
prevalence of rods and plates
SMI is calculated by differential analysis
of the triangulated surface of the bone:
SMI = 6 BV ( dBS/dr ) / BS2
dBS/dr is estimated by translating the
surface by a small distance, dr, in its
normal direction:
dBS/dr = (S´ - S) / dr
an ideal plate, cylinder (rod) and sphere
have SMI values of 0, 3, and 4
Topology
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a shell…
and its inflated surface
transition of a rod to a
plate…
perforation of a plate…
h:r == 0.05,
r:R
5,
1,
0.5,
10,
0.25,
0.75,
0.87,
0.95,
0, SMI
0.05,
0.5,
SMI
SMI
SMI
SMI
SMI
SMI
====
3.02
2.61
=
0.35
=
2.97
=2.00
0.69
0.35
0.49
1.16
1.70
2.09
0.39
Summary
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pQCT is an X-ray tomographic imaging
modality
pQCT provides high resolution 2D / 3D
images
images of trabecular (and cortical) bone
can be digitally partitioned into
bone/non-bone
bone (quality) can be numerically
characterized in terms of BMD and
structure
structure can be quantified using
stereological and topological methods
stereological methods may have
embedded assumptions / limitations
model independent measures
Finis!
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further reading:
http://www.scanco.ch/support/generalfaq.html#c781
 http://www.stratecmed.com/en/prod_xct2000.php
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