MRI Measurement of Heart Wall Motion - Advance Image Processing in Magnetic Resonance Imaging - page 346

imaged at ED (shortly after tag creation) move off the imaging plane at subsequent
times due to through-plane components of heart motion, so their displacements
are not imaged. However, material points imaged at ES (or any other time after
ED) are located on the same tag saturation plane as their corresponding tracked
image stripe points at ED.
Assuming the ED tag planes (shortly after tag creation) are approximately
flat and orthogonal to the image plane, the component of displacement normal
to the tag plane can be calculated as the distance from the material point position
at ES to its projection on the original tag plane:

d

⊥ =⋅ −

nx ( x ) (10.8)

ES ED

where n is the normal to the original tag plane (i.e., a unit vector in the direction
of the tagging gradient g ) and x ES and x ED are the positions of the tracked stripe
point at ES and ED respectively. All three components of the displacement field
from ES to ED can, therefore, be recovered by fitting displacement components
from all images simultaneously, minimizing

∑

ε= S () u + ( n ⋅ (( ) u ξ − u )

2

(10.9)

i i i

i

ξ i )

where u i is the 3-D displacement of the tracked stripe point ( x ES − x ED ) and u (

is the corresponding model 3-D displacement.
The previous procedure can be used to reconstruct the 3-D displacements of
all image stripe points in all desired frames. In order to calculate Lagrangian
strain in each frame referred to the ED state, a single ED model must be deformed
to each subsequent frame. This can be done using a least-squares fit of the ED
model to each subsequent frame, minimizing the objective function defined in
Equation 6. Strain can then be calculated at each frame using Equation 7 and
Equation 4.
Validation experiments with a deformable MR phantom were performed to
determine the magnitude of errors in motion and strain reconstruction using this
method [19]. A silicone gel in the shape of a cylindrical annulus was deformed
by rotating the inner cylinder with respect to the outer cylinder. This resulted in
a well-controlled nonhomogeneous deformation field which could be calculated
analytically using a universal solution of the finite elasticity equations. The dis-
placement field for this problem is independent of the material stiffness of the gel.
The model reconstructed the displacement field to less than 0.5 mm. The average
root-mean-square (RMS) errors in strain were 6% in shear and 16% in the radial
axial strain.
Subsequently, this method was extended to interactively reconstruct the 3-D
motion and strain directly from the images, without the need for stripe tracking
[26]. A set of “model tags” was embedded into a finite element model of the left
ventricular geometry at ED. This geometry can be interactively determined in 3-D