Vf x ye sup fx1 yf x2 y27

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

Lymphoid Hyperplasia Colonoscopy

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.

5 Conclusions

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].

Acknowledgments

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.

References

1. Vining DJ, Liu K, Choplin RH, Haponik EE Virtual bronchoscopy. Relationships of virtual reality endobronchial simulations to actual bronchoscopic findings. Chest 1996; 109:549-553.

2. Higgins WE, Ramaswamy K, Swift RD, McLennan G, Hoffman EA. Virtual bronchoscopy for three-dimensional pulmonary image assessment: state of the art and future needs. Radiographics 1998; 18:761-778.

3. Hara AK, Johnson CD, Reed JE, Ahlquist DA, Nelson H, Ehman RL, McCollough CH, Ilstrup DM. Detection of colorectal polyps by computed tomographic colography: feasibility of a novel technique. Gastroenterology 1996; 110:284-290.

4. Davis CP, Ladd ME, Romanowski BJ, Wildermuth S, Knoplioch JF, Debatin JF. Human aorta: Preliminary results with virtual endoscopy based on three-dimensional MR imaging data sets. Radiology 1996; 199:37-40.

5. Robb RA, Aharon S, Cameron BM. Patient-specific anatomic models from three dimensional medical image data for clinical applications in surgery and endoscopy. Journal of Digital Imaging 1997; 10:31-35.

6. Kimura F, Shen Y, Date S, Azemoto S, Mochizuki T. Thoracic aortic aneurysm and aortic dissection: New endoscopic mode for three-dimensional CT display of aorta. Radiology 1996; 198:573-578.

7. Fleiter T, Merkle EM, Aschoff AJ, Lang G, Stein M, Gorich J, Liewald F, Rilinger N, Sokiranski R. Comparison of real time virtual and fiberoptic bronchoscopy in patients with bronchial carcinoma: opportunities and limitations. Am J Roentgenol 1997; 169:1591-1595.

8. Ferretti GR, Knoplioch J, Bricault I, Brambilla C, Coulomb M. Central airway stenoses: preliminary results of spiral-CT-generated virtual bronchoscopy simulations in 29 patients. Eur Radiol 1997; 7:854-859.

9. Summers RM, Feng DH, Holland SM, Sneller MC, Shelhamer JH. Virtual bronchoscopy: segmentation method for real-time display. Radiology 1996; 200:857862.

10. Rubin GD, Beaulieu CF, Argiro V, Ringl H, Norbash AM, Feller JF, Dake MD, Jeffrey RB, Napel S. Perspective volume rendering of CT and MR images: applications for endoscopic imaging. Radiology 1996; 199:321-330.

11. Lorensen WE, Cline HE. Marching Cubes: A high resolution 3D surface reconstruction algorithm. ACM Computer Graphics 1987; 21:163-169.

12. McFarland EG, Brink JA, Loh J, Wang G, Argiro V, Balfe DM, Heiken JP, Vannier MW. Visualization of colorectal polyps with spiral CT colography: evaluation of processing parameters with perspective volume rendering. Radiology 1997; 205:701-707.

13. Summers RM, Shaw DJ, Shelhamer JH. CT virtual bronchoscopy of simulated endobronchial lesions: effect of scanning, reconstruction, and display settings and potential pitfalls. Am J Roentgenol 1998; 170:947-950.

14. Summers RM. Navigational aids for real-time virtual bronchoscopy. Am J Roentgenol 1997; 168:1165-1170.

15. Paik DS, Beaulieu CF, Jeffrey RB, Napel S. Virtual colonoscopy visualization modes using cylindrical and planar map projections: Technique and evaluation. Radiology 1998; 209P:429-429.

16. Paik DS, Beaulieu CF, Jeffrey RB, Rubin GD, Napel S. Automated flight path planning for virtual endoscopy. Medical Physics 1998; 25:629-637.

17. Beaulieu CF, Jeffrey RB, Karadi G, Paik DS, Napel S. Visualization modes for CT colonography: Blinded comparison of axial CT, virtual endoscopy, and panoramic-view volume rendering. Radiology 1998; 209P:296-297.

18. Wang G, McFarland EG, Brown BP, Vannier MW. GI tract unraveling with curved cross sections. IEEE Trans Med Imaging 1998; 17:318-322.

19. Hara AK, Johnson CD, Reed JE, Ahlquist DA, Nelson H, MacCarty RL, Harmsen WS, Ilstrup DM. Detection of colorectal polyps with CT colography: initial assessment of sensitivity and specificity. Radiology 1997; 205:59-65.

20. McAdams HP, Goodman PC, Kussin P. Virtual broncho-scopy for directing transbronchial needle aspiration of hilar and mediastinal lymph nodes: a pilot study. Am J Roentgenol 1998; 170:1361-1364.

21. McAdams HP, Palmer SM, Erasmus JJ, Patz EF, Connolly JE, Goodman PC, Delong DM, Tapson VF. Bronchial anastomotic complications in lung transplant recipients: virtual bronchoscopy for noninvasive assessment. Radiology 1998; 209:689-695.

22. Smith PA, Heath DG, Fishman EK. Virtual angioscopy using spiral CT and real-time interactive volume-rendering techniques. J Comput Assist Tomogr 1998; 22:212-214.

23. Summers RM, Selbie WS, Malley JD, Pusanik LM, Dwyer AJ, Courcoutsakis N, Shaw DJ, Kleiner DE, Sneller MC, Langford CA, Holland SM, Shelhamer JH. Polypoid lesions of airways: early experience with computer-assisted detection by using virtual bronchoscopy and surface curvature. Radiology 1998; 208:331-337.

24. Summers RM, Pusanik LM, Malley JD. Automatic detection of endobronchial lesions with virtual broncho-scopy: comparison of two methods. In: Medical Imaging 1998: Image Processing. San Diego, California: SPIE, 1998; 3338:327-335, http://www.cc.nih.gov/drd/two-methpc.pdf

25. Summers RM, Pusanik LM, Malley JD, Hoeg JM. Fractal analysis of virtual endoscopy reconstructions. In: Medical Imaging 1999: Physiology and Function from Multidimensional Images. San Diego, California: SPIE, 1999; 3660:258-269, http://www.cc.nih.gov/drd/fractal.pdf

26. Shepard JA. The bronchi: an imaging perspective. J Thorac Imaging 1995; 10:236-254.

27. Thirion J-P, Gourdon A. Computing the differential characteristics of isointensity surfaces. Comput Vision Image Understand 1995; 61:190-202.

28. Monga O, Benayoun S. Using partial derivatives of 3D images to extract typical surface features. Comput Vision Image Understand 1995; 61:171-189.

29. Summers RM, Beaulieu CF, Pusanik LM, Malley JD, Jeffrey RB, Glazer DI, Napel S. An automated polyp detector for CT colonography—feasibility study. Radiology 2000; in press.

30. Summers RM, Pusanik LM, Malley JD, Reed JE, Johnson CD. Method of labeling colonic polyps at CT colono-graphy using computer-assisted detection. In: Computer Assisted Radiology and Surgery (CARS). San Francisco, CA: Elsevier Science, 2000:in press.

31. Summers RM. Image gallery: a tool for rapid endobron-chial lesion detection and display using virtual bronchoscopy. J Digital Imaging 1998; 11:53-55.

32. Vining D, Ge Y, Ahn D, Stelts D, Pineau B. Enhanced virtual colonoscopy system employing automatic detection of colon polyps. Gastroenterology 1998; 114:A698-A698.

33. Vining DJ, Ahn DK, Ge Y, Stelts DR. Improved computerassisted colon polyp detection. Radiology 1998; 209P:649-649.

34. Mandelbrot BB. The Fractal Geometry of Nature. San Francisco: W.H. Freeman, 1982.

35. Zahouani H, Vargiolu R, Loubet JL. Fractal models of surface topography and contact mechanics. Mathematical and Computer Modelling 1998; 28:517-534.

36. Lestrel PE. Fourier Descriptors and Their Applications in Biology. New York: Cambridge University Press, 1997.

37. Oshida Y, Hashem A, Nishihara T, Yapchulay MV. Fractal dimension analysis of mandibular bones: toward a morphological compatibility of implants. Biomed Mater Eng 1994; 4:397-407.

38. Stachowiak GW, Stachowiak GB, Campbell P. Application of numerical descriptors to the characterization of wear particles obtained from joint replacements. Proc Inst Mech Eng [H] 1997; 211:1-10.

39. Fazzalari NL, Parkinson IH. Fractal properties of sub-chondral cancellous bone in severe osteoarthritis of the hip. J Bone Miner Res 1997; 12:632-640.

40. Talibuddin S, Runt JP. Reliability test of popular fractal techniques applied to small 2-dimensional self-affine data sets. J Appl Phys 1994; 76:5070-5078.

41. Saupe D. Algorithms for random fractals. In: M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H.-O. Peitgen, D. Saupe and R. F. Voss, ed. The Science of Fractal Images. New York: Springer-Verlag, 1988.

42. Biswas MK, Ghose T, Guha S, Biswas PK. Fractal dimension estimation for texture images: A parallel approach. Pattern Recognition Letters 1998; 19:309-313.

43. Milman VY, Stelmashenko NA, Blumenfeld R. Fracture surfaces — a critical-review of fractal studies and a novel morphological analysis of scanning-tunneling-microscopy measurements. Prog Mat Sci 1994; 38:425-474.

44. Dubuc B. On estimating fractal dimension. M. Eng. Thesis, Montreal: McGill University, 1988.

45. Dubuc B, Quiniou JF, Roques-Carmes C, Tricot C, Zucker SW. Evaluating the fractal dimension of profiles. Phys Rev A 1989; 39:1500-1512.

46. Huang Q, Lorch JR, Dubes RC. Can the fractal dimension of images be measured. Pattern Recognition 1994; 27:339349.

47. Kulatilake P, Um J, Pan G. Requirements for accurate estimation of fractal parameters for self-affine roughness profiles using the line scaling method. Rock Mechanics and Rock Engineering 1997; 30:181-206.

48. Dubuc B, Zucker SW, Tricot C, Quiniou JF, Wehbi D. Evaluating the fractal dimension of surfaces. Proc R Soc LondA 1989; 425:113-127.

49. Summers RM, Sneller MC, Langford CA, Shelhamer JH, Wood BJ. Improved virtual bronchoscopy using a multi-slice helical CT scanner. In: Medical Imaging 2000: Physiology and Function from Multidimensional Images. San Diego, California: SPIE, 2000; 3978 : 117-121, http:// www.cc.nih.gov/drd/betterVB.pdf

50. Turner MJ, Blackledge JM, Andrews PR. Fractal Geometry in Digital Imaging. San Diego: Academic Press, 1998.

0 0

Post a comment