Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging
Autocorrelation function of a test-line in a TB image. (a) An image slice from a 3D data with ; (b) original density function and its shifted counterpart (black) where arrow indicates ; (c) shows maximum overlap with at resulting in a local maximum in the ACF; (d) ACF: FWHM of the ACF estimates the TB Th while corresponds to TB Sp.
Determination of threshold to differentiate bone from marrow in a image of voxel size. (a) The threshold is selected as the intensity at which the minimum between bone and bone marrow modes of the image intensity histogram occurs. The resulting binary image (c) closely resembles the complement image of (a) with marrow having high intensity.
Mean intercept-length calculation along the test line within the set of uniformly spaced lines parallel to the direction , . Bone entry points and exit points are marked along one test line.
(a) Axial image of the distal tibia at resolution used in image shading analysis. Bottom to top corresponds to lateral-medial and left to right to antero-posterior. Zoomed view of image indicates slice of . (b) Manually masked region of TB within the cortical shell used to calculate the coil sensitivity map; (c) computed coil sensitivity where each pixel is equal to 20th percentile of the marrow and bone intensities within a ten-voxel neighborhood. Dark dashed box indicates location of used in inhomogeneity analysis and fainter boxes mark the locations of four regions of the inhomogeneity applied to .
Effect of Rician noise addition in a dataset on , and , . (a) Plot of , the angle between the largest ellipsoid axis and the axis perpendicular to the transverse image plane, vs SNR; (b) plot of , and , vs SNR. The MIL-derived ellipsoids become unstable at SNR=3.9 while the corresponding values for the ACF are robust down to SNR .
Effect of noise on the determination of bone intercepts and FWHM for a 1D profile in a image of a TB specimen from the human radius: (a) image (SNR=4.8) with test line indicated; (b) profile along test line where THR=100 provides effective Sp of bone marrow (high) and bone (low) intensities, resulting in six bone intercepts and a mean Th of ; (c) The ACF is smooth with a FWHM of ; (d) image with added Rician noise, SNR=3.0; (e) with the same threshold (THR=100), 11 bone intercepts are identified, leading to the reduction in MIL-derived Th to . (f) The parent peak of becomes sharper due to the presence of the spike in the ACF, resulting in an artifactual reduction in FWHM to .
Effect of image resolution on (a) and , (b) and , and (c) , , and . It is noted that and are not affected by resolution. By contrast, and deviate due to the error in identifying bone and marrow in partially volumed voxels after thresholding. Resolution is estimated for each dataset as where the denominator is the -space cutoff frequency.
Effects of partial volume blurring on ACF and MIL-derived TB Th measurements: (a) High resolution image of TB at voxel size along with test line whose intensity function is shown in (c); (b) of profile in (a), and ; (c) MIL identifies six pairs of bone intercepts along the profile at the threshold chosen resulting in and ; (d) image with an effective resolution of obtained by low-pass filtering of image in (a); (e) at this resolution becomes smoother resulting in and ; (f) using the same threshold, four pairs of bone intercepts are detected resulting in artifactual increases to and of Th and Sp.
Effect of intensity variations on MR image of TB obtained in vivo (a); (b) 1D profile along line in (a) in the absence (solid) and presence (dotted) of an intensity gradient caused by inhomogeneous signal reception of a surface coil. The intensity gradient affects the intercepts detected (arrow, (b) but has negligible effect on ACF-derived FWHM and (c).
Means and corresponding standard deviations for structural parameters derived by MIL and ACF in the distal tibia of healthy women: (a) anisotropy, (b) angular offset of largest principal axis.
Fabric tensor computed from in vivo MR images obtained with both techniques; (a) central slice from reconstructed raw data acquired at voxel size showing registered position of VOI for all ten subjects; (b) 3D rendering of the VOI rotated clockwise by 45° around the longitudinal axis; (c) ellipsoids for FWHM (solid surface w/ dark axes) and MIL Th (mesh w/ gray axes) indicating longitudinal fabric direction. Note greater anisotropy obtained with ACF than with MIL. (d) ACF Sp (mesh and dark axes) and MIL Sp (solid and gray axes) ellipsoids have matching orientations but differ substantially in anisotropy: and . Plot units are pixels .
Effect of intensity threshold on derived structural anisotropy measures: (a) Intensity histogram of VOI shown in Fig. 11(b) (w/ inverted bone/bone marrow contrast) after bone-volume fraction mapping and subvoxel processing with two thresholds indicated at 25 and 40; (b) MIL Th fabric tensors for the two values of the threshold, 25 (outer mesh ellipsoid with gray axes) and 40 (inner solid ellipsoid with dark axes). Increasing the threshold causes a uniform reduction of MIL Th with and largely unaffected. (c) MIL Sp tensors for the two thresholds, 25 (inner solid ellipsoid with dark axes) and 40 (outer mesh ellipsoid with gray axes). A reduction in (from 0.732 to 0.657) occurs for the larger threshold while decreases from 7.0° to 4.9°.
Average rms difference for angular offset for the ellipsoid’s maximum principal axis relative to the axis and anisotropy.
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