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 = Ioe - ∫ u(x,y) ds

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

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

pQCT calibrates HU into density: g/cm3

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 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

xi,yi

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) = Po(x,y,z) < t ? 0 : Po(x,y,z)

Image segmentation

binarization:

P(x,y,z) = Po(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/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

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:

PB = bone perimeter length (mm)

AB = bone area (mm2)

AT = tissue section area (mm2) (bone + marrow)

2D stereology

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

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 (xi, yi) = (cos(θi), sin(θi)), i = 1→N Tb.N(θi) = A xi

2 + 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

x

y

θL1L2 L1L2 L1L2

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

( xi, yi, zi ) = ( sin(θi)cos(φi), sin(θi) sin(φi), cos(θi) )

Tb.N( θi, φi ) = A xi2 + B yi

2 + C zi2 + D

xiyi + E xizi + F yizi

θ

φx

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: L3 / L1, L2 / L1

L2

L3

L1

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 = 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

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

a shell… and its inflated surface transition of a rod to a

plate… perforation of a plate…

h:r = 10, SMI = 2.97h:r = 5, SMI = 3.02h:r = 1, SMI = 2.61h:r = 0.5, SMI = 2.00h:r = 0.05, SMI = 0.35r:R = 0, SMI = 0.35r:R = 0.05, SMI = 0.39r:R = 0.25, SMI = 0.49r:R = 0.5, SMI = 0.69r:R = 0.75, SMI = 1.16r:R = 0.87, SMI = 1.70r: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-

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