Assessment of Bone
Quality from pQCT Images
Dean Inglis, Ph.D.
Assistant professor (adjunct)
Department of Civil Engineering
McMaster University
Overview
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?
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
spectrum
CT basic principles
electron beam strikes tungsten target
and generates polychromatic X

ray
beam
source
CT basic principles
X

rays pass through a sample and are
attenuated:
I = I
o
e

∫
u
(x,y) ds
I = intensity at the detector
I
o
= 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
emergent X

rays detected by a phosphor
detector coupled to a CCD camera
CT image formation
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
raw CT data represent linear attenuation
coefficients
coefficients are converted to CT
numbers, Hounsfield Units (HU), in the
reconstruction process
p
QCT calibrates HU into density: g/cm
3
Image characteristics
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 mm
3
spacing partly governs ability to resolve small features
accurately
pQCT resolution: 0.2
×
0.2
×
0.5 mm
3
(non

isotropic)
hr

pQCT resolution: 0.08
×
0.08
×
0.08 mm
3
(isotropic)
Topology example: 6x5 image
x
i
,y
i
5
1
2
3
4
6
7
8
Image characteristics
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
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
thresholding:
P(x,y,z) = P
o
(x,y,z) < t
? 0 : P
o
(x,y,z)
Image segmentation
binarization:
P(x,y,z) = P
o
(x,y,z) < t
? 0 : 1
Image segmentation
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
bone images consist of 2
pixel groups: bone and soft
tissue (or background): a
histogram of a bone image
appears bimodal
segment bone from non

bone 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
at low resolution Otsu fails for bone within
bone:
cortical bone vs. trabecular bone
trabecular bone vs. marrow
Image segmentation
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
given a segmented
image of bone,
what can be measured?
HU’s represent attenuation: analog for
density
calibration allows volumetric BMD
(g/cm
3
):
BMD = ∑ [P
i
!= 0 ? m
×
P
i
+ b : 0 ]
segmentation provides volume (cm
3
):
V = [ ∑ P
i
!= 0 ? 1 : 0 ]
×
dx
×
dy
×
dz
BMC = BMD
×
V (g)
Bone morphometry
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
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
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
basic 2D quantities:
P
B
= bone perimeter length (mm)
A
B
= bone area (mm
2
)
A
T
= tissue section area (mm
2
)
(bone + marrow)
2D stereology
bone volume fraction (%):
TBV = BV/TV = A
B
/ A
T
Bone surface density (mm
2
/mm
3
):
S
v
= BS/TV = P
B
/ A
T
bone surface to volume ratio (mm
2
/mm
3
):
S/V = BS/BV = P
B
/ A
B
mean trabecular plate thickness (mm):
MTPT = Tb.Th = 2 A
B
/ P
B
mean trabecular plate density (/mm):
MTPD = Tb.N = BV/TV / Tb.Th = P
B
/ (2 A
T
)
mean trabecular plate separation (mm):
MTPS = Tb.Sp = 1 / Tb.N
–
Tb.Th = 2 (A
T
–
A
B
) / P
B
3D stereology
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
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
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
in 2D, an ellipse can be fit to data from N
orientations
Let (x
i
, y
i
)
= (cos(
θ
i
), sin(
θ
i
)), i = 1
→
N
Tb.N(
θ
i
) = A x
i
2
+ B x
i
y
i
+ Cy
i
2
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: L
2
/ L
1
x
y
θ
L
1
L
2
L
1
L
2
L
1
L
2
3D stereology
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
random orientations
orientation of the lines is defined in
terms of two angles:
θ
,
φ
( x
i
, y
i
, z
i
) = ( sin(
θ
i
)cos(
φ
i
), sin(
θ
i
)
sin(
φ
i
), cos(
θ
i
) )
Tb.N(
θ
i
,
φ
i
) = A x
i
2
+ B y
i
2
+ C z
i
2
+ D
x
i
y
i
+ E x
i
z
i
+ F y
i
z
i
θ
φ
x
y
z
3D stereology
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: L
3
/ L
1,
L
2
/ L
1
L
2
L
3
L
1
y
z
x
Model independent
measures
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
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
=
β
0
–
β
1
–
β
2
β
0
is the number of isolated objects = 1
for bone
β
1
is the connectivity
β
2
is the number of enclosed cavities =
0 for bone
β
1
is calculated by analyzing the local
neighbourhood connectivity of each
voxel representing bone
works for any ROI shape
Topology
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 ) / BS
2
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
a shell…
and its inflated surface
transition of a rod to a
plate…
perforation of a plate…
h:r = 10, SMI = 2.97
h:r = 5, SMI = 3.02
h:r = 1, SMI = 2.61
h:r = 0.5, SMI = 2.00
h:r = 0.05, SMI = 0.35
r:R = 0, SMI = 0.35
r:R = 0.05, SMI = 0.39
r:R = 0.25, SMI = 0.49
r:R = 0.5, SMI = 0.69
r:R = 0.75, SMI = 1.16
r:R = 0.87, SMI = 1.70
r:R = 0.95, SMI = 2.09
Summary
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!
further reading:
http://www.scanco.ch/support/general

faq.html#c781
http://www.stratec

med.com/en/prod_xct2000.php
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