where the supremum is taken over all data pairs (x1; y1),

FIGURE 2 Pig pulmonary embolus model. (A) Axial CT image shows pulmonary embolus (arrow) in right pulmonary artery. (B) Anteroposterior view of three-dimensional surface rendering of main pulmonary arteries. (C) Closeup view of right pulmonary artery showing indentation in surface reconstruction due to presence of embolus. (D) Virtual angioscopic view showing narrowing of vessel lumen by embolus. Red coloring indicates area detected by curvature-based algorithm. Dataset kindly provided by Dr. James Brink, Yale University School of Medicine. See also Plate 137.

FIGURE 2 Pig pulmonary embolus model. (A) Axial CT image shows pulmonary embolus (arrow) in right pulmonary artery. (B) Anteroposterior view of three-dimensional surface rendering of main pulmonary arteries. (C) Closeup view of right pulmonary artery showing indentation in surface reconstruction due to presence of embolus. (D) Virtual angioscopic view showing narrowing of vessel lumen by embolus. Red coloring indicates area detected by curvature-based algorithm. Dataset kindly provided by Dr. James Brink, Yale University School of Medicine. See also Plate 137.

0*2j Yi) that lie within the grid square (x + e, y + e). The e-variation of f is defined as the average of Vf over all (x, y):

With discrete data on an N xN grid, the e-variation of f is

where un and bn are the maximum and minimum values of the function f within the grid squares bounded by the grid indices (i — kn, j — kn) and (i + kn, j + kn), where 1 < kn<R and 0 < i, j<R, and the data are grouped into R1 bins, where R<N. en = kn/R is the scale over which the variation is

FIGURE 3 Images derived from CT scans of insufflated human ileocecal autopsy specimen. (A) Three-dimensional surface rendering of terminal ileum and cecum. (B) Virtual colonoscopic view of cecum. Nodular areas shown in red were detected by curvature-based lesion detection algorithm and correspond with foci of lymphoid hyperplasia in patient with aplastic anemia and fungal typhlitis. See also Plate 138.

FIGURE 3 Images derived from CT scans of insufflated human ileocecal autopsy specimen. (A) Three-dimensional surface rendering of terminal ileum and cecum. (B) Virtual colonoscopic view of cecum. Nodular areas shown in red were detected by curvature-based lesion detection algorithm and correspond with foci of lymphoid hyperplasia in patient with aplastic anemia and fungal typhlitis. See also Plate 138.

computed. The fractal dimension D is the slope of the plot of log Vf /e3 vs log 1/e.

We found that the variation method tends to overestimate the fractal dimensions of test surfaces with low fractal

FIGURE 4 Fractal curve. Fractal dimension (D) of this curve varies continuously from its ideal topological dimension (D = 1, left side of curve) to the Euclidean dimension of a surface (D = 2, right side of curve). Amplitude of the curve is arbitrary. As its fractal dimension increases, the curve appears to fill more and more of the adjacent space and appears rougher. Fractal dimension is widely used to quantitate roughness of curves and surfaces. Adapted from [50].

dimensions and underestimate the fractal dimensions of surfaces with high fractal dimensions [25]. However, the computed estimates of the surfaces' fractal dimensions did show a monotonic relationship to the true fractal dimension; this indicates that the relative roughness of two surfaces can be compared using this method. In other words, the method provides an ordinal measure of roughness.

4.1 Clinical Application of Fractal Measures of Roughness

Application of this method to virtual bronchoscopy in a patient with a subcarinal tumor is shown in Fig. 6. The portion of

FIGURE 5 Demonstration of synthetic fractal surfaces generated using midpoint displacement method. Fractal dimension of surfaces are (A) D = 2.1 and (B) D = 2.3. Analogous to the situation for fractal curves (Fig. 4), rougher fractal surfaces have greater fractal dimensions. These surfaces, although generated using a mathematical algorithm, have texture that mimics texture of some abnormal anatomic surfaces at virtual endoscopy.

FIGURE 4 Fractal curve. Fractal dimension (D) of this curve varies continuously from its ideal topological dimension (D = 1, left side of curve) to the Euclidean dimension of a surface (D = 2, right side of curve). Amplitude of the curve is arbitrary. As its fractal dimension increases, the curve appears to fill more and more of the adjacent space and appears rougher. Fractal dimension is widely used to quantitate roughness of curves and surfaces. Adapted from [50].

FIGURE 5 Demonstration of synthetic fractal surfaces generated using midpoint displacement method. Fractal dimension of surfaces are (A) D = 2.1 and (B) D = 2.3. Analogous to the situation for fractal curves (Fig. 4), rougher fractal surfaces have greater fractal dimensions. These surfaces, although generated using a mathematical algorithm, have texture that mimics texture of some abnormal anatomic surfaces at virtual endoscopy.

FIGURE 6 Fractal analysis of virtual bronchoscopy reconstruction. Mediastinal melanoma metastasis in a 34-year-old man. (A) Axial CT image shows subcarinal mass ("M"). (B) Three-dimensional reconstruction shows mass effect on carina (large arrow) and inferior wall of right mainstem bronchus. Segment of left mainstem bronchus is occluded (small arrows). Virtual bronchoscopy views of (C) carina and (D) right mainstem bronchus show irregular wall due to mass ("M"). Fractal dimension of carina (2.38 + 0.05) was greater than that of lateral, smooth wall (*) of right mainstem bronchus (2.26 + 0.04).

the airway wall involved by tumor (D = 2.38 + 0.05) was found to be rougher than an adjacent uninvolved wall (D = 2.26+0.04). The error estimates are computed from the linear regression of the log-log plot and likely underestimate the true error. In a pilot study comparing a magnetic resonance virtual angioscopy of an atherosclerotic aorta to an aorta from a normal volunteer we found that the (visually) rougher atherosclerotic aorta (D = 2.14 + 0.03) had a slightly higher fractal dimension than the normal aorta (D = 2.09 + 0.05) in a region in the ascending aorta [25]. We attributed the small difference between these fractal dimensions to the limited data resolution. Additional studies will be necessary to test the clinical viability of these techniques and to better characterize the error estimates.

Fractal dimension measurements are exquisitely sensitive to the noise level and resolution of the data. As CT and MRI

scanners improve and generate datasets with greater resolution and signal-to-noise ratios, VE data will be more amenable to precise fractal analysis.

We have shown how morphometric methods can be applied to VE reconstructions to quantitate surface roughness and improve detection of abnormalities. Curvature and roughness analyses can be used to automatically detect potential lesions on VE, thereby improving efficiency and accuracy and enable detection of other endoluminal abnormalities such as atherosclerosis or invasion by tumor. The algorithms described in this review are logical approaches to shape and roughness analyses of VE reconstructions. There is little question about the useful role of VE for evaluation of gross lesions. However, rugosity is a much newer application and its value is an open question.

As the technology and clinical applications of VE progress, methods such as those described here to improve efficiency and accuracy and facilitate interpretation will likely become more important. Newer and faster scanners are becoming available that will improve the quality (reduced motion artifact) and resolution of the imaging data [49].

Andrew Dwyer is thanked for helpful comments and review of the manuscript. Nikos Courcoutsakis and David Kleiner are thanked for assistance with autopsy specimens. Lynne Pusanik provided computer programming support. James Malley is thanked for helpful discussions. James Brink kindly provided the pig pulmonary embolus dataset.

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