“Twisting and bending” model. (a) The twisting model assumes that the 3D carotid image (shown as the gray block) is a part of a cylindrical shaft. The model parameters defines the “twist–bend center,” which does not move during the deformation, where is the twisting torque and is the twisting angle at the plane normal to the axis passing through . (b) The bending model assumes that 3D carotid image (shown as the gray block) is a part of a rectangular homogeneous shaft with the center axis going through the twist–bend center . A new coordinate system, is defined at the twist–bend center with the axis defined in the “bending direction” at an angle to the axis. Then, the bending is defined in coordinate system, where the length of the neutral plane does not change during bending, and bending moment, creates a bending angle due to the curvature .
Images showing preprocessing of 3D MR and US images for registration. The 3D MR and US images have been sliced to show longitudinal views of the branches of the carotid artery: (a) MR, (b) US, and (c) and (d) corresponding views after masking. The user-selected bifurcation is also shown.
Registration initialization: (a) User-defined points, bifurcation points (BF), center of the common carotid, , at the bottom and centers of internal, , and external, , carotid arteries at the transverse plane through the bifurcation, and (b) initial rotation angles on the three orthogonal planes.
Registration error calculation using the distance between corresponding points, on the target and registered surfaces at the direction, , on each plane, , normal to axes at every in carotid section, .
Examples of registration results showing vessel boundaries after twisting and bending model-based nonrigid registration using: (a) 3D US with MR, (b) 3D US with MR. The color-coded registration errors, , have been mapped on the registered MR surfaces using: (c) 3D US with MR and (d) 3D US with MR.
Average for different carotid sections, , and MRE for whole vessel of all samples for intrasubject rigid and twisting and bending model-based nonrigid registration of (a) 3D US with MR and (b) 3D US with MR images of carotid artery, where an asterisk indicates statistically significant (paired t-test, ) reduction in registration error.
Average of planes perpendicular to the axis for intrasubject rigid and twisting and bending model-based nonrigid registration of (a) 3D US with MR and (b) 3D US with MR images of carotid artery, where is the distance along the axis from the user-defined carotid bifurcation in the carotid section, .
Magnetic resonance imaging parameters for T1-weighted double inversion recovery scans at 1.5 and .
Distance-based mean registration error for twisting and bending model-based nonrigid registration of US with MR and US with MR for the : common, 2: internal, and 3: external carotid artery segments, as well as MRE for the whole vessel. Note that the 3D images of the whole vessel here used in the registration procedure. The last row shows the overall mean (and standard deviation in parentheses) for all of the image sets.
Comparison of the difference in (mm) and MRE for rigid and twisting and bending model-based nonrigid registration results with MR images of both field strengths 1.5 and .
Maximum registration error of different carotid sections for both rigid (R) and twisting and bending model-based nonrigid (NR) registration and MR field strengths of 1.5 and .
Comparison of the difference in (mm) and MRE for US–MR twisting and bending mode-based nonrigid registration results with MR images of field strengths 1.5 and . There were no statistically significant (paired t-test, , , ) changes in registration error.
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