Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 236

c ð i Þ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i ÞÞ represent the spine curve that passes through the centers

of vertebral bodies. If the transverse planes of measurement are image-based, i.e.
orthogonal to axis z of the image-based coordinate system, then the axial vertebral
rotation uð

Þ ¼ u z ð Þ

can be determined by considering r z ð

Þ ¼

c z ð

Þ

i i i i as:

r x ð i Þ c x ð i Þ

u z ð i Þ ¼ arctan ; ð 13 Þ

r y ð i Þ c y ð i Þ

which for every i corresponds to the angle between the line connecting the
anatomical reference point with the center of the vertebral body, and the line
representing the reference sagittal plane (i.e. the line in the direction of axis y of
the image-based coordinate system). However, because vertebrae can be sagittally
or coronally inclined against axis z, the centers of vertebral bodies and the corre-
sponding reference anatomical points may not represent corresponding anatomical
locations along the longitudinal vertebral axes. On the other hand, if the transverse
planes of measurement are spine-based, i.e. orthogonal to axis w of the spine-based
coordinate system and therefore orthogonal to cðiÞ, ð i Þ , then the axial vertebral rotation

uð i Þ ¼ u w ð i Þ is measured at corresponding anatomical locations and can be

determined as:

;

arccos ð r ð

i Þ c ð i ÞÞ e Iy ð i Þ

Þ ¼t e Iy t u w ð i Þ ¼ ; ~ e Iy ð i ð i Þ ^ ð i Þ ð 14 Þ

k r ð i Þ c ð i Þ k

where e Iy ð is the unit vector in the direction of the projection of e Iy ¼½ i Þ 0 ; 1 ; 0 I to

ned by the unit tangent vector t^ðiÞ ð i Þ the plane orthogonal to the spine curve, de as the

normal of that plane.

2.2.3 Transformation from Image-Based to Spine-Based Coordinate
System

The continuous transformation from the image-based coordinate system (Fig. 7 )to
the spine-based coordinate system (Fig. 8 ) is possible by having a continuous
description of the spine curve C (Eq. 1 ) and axial vertebral rotation U (Eq. 12 ) that
are parameterized by the same variable i representing the location on the spine:

f C ; Ug: fð i Þ ¼ ð c ð i Þ; uð i ÞÞ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i Þ; uð i ÞÞ; i ¼½ i sp ; i ep ; ð 15 Þ

where i ¼ i sp and i ¼ i ep represent the locations of the spine curve and axial ver-
tebral rotation at the start and end point of observation on the spine, respectively.
The Frenet-Serret frame (Eq. 7 ) describes the geometrical properties of the curve
and can be therefore applied to the spine curve cðiÞ. ð i Þ . However, the spine-based

3

coordinate system ð u ; v ; w Þ2 R

S has to represent also the course of the axial vertebral

. The unit tangent vector t^ðiÞ rotation uð

Þ ð Þ

nes the unit vector e Sw ¼½

; ; S

i i de 0 0 1