Quality from pQCT Images

peachpuceΤεχνίτη Νοημοσύνη και Ρομποτική

6 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

52 εμφανίσεις

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