Abstract.
Based on fractal properties and spatial auto-correlation, the measures of complexity lacunarity, Moran's I and Geary's C index are defined for 3D image analysis. Their abilities to investigate translational invariance, characteristic length scales, spatial correlation and shapes of 3D micro-structures are demonstrated on proto-typical examples. Finally, using these measures of complexity, 3D images of trabecular bone are analysed. The main findings are that the complexity of the trabecular structure decreases and the plate-like shapes of the trabeculae change to mainly rod-like shapes during bone loss. These results and the proposed measures could have a great impact for medicine and for space exploration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T.-J. Chen, K.-S. Chuang, J. Wu, S.C. Chen, I.-M. Hwang, M.-L. Jan, Phys. Med. Biol. 48, N131 (2003)
T.-J. Chen, K.-S. Chuang, J. Wu, S.C. Chen, I.-M. Hwang, M.-L. Jan, J. Digital Imag. 16, 210 (2003)
K.-S. Chuang, H.K. Huang, Phys. Med. Biol. 37, 357 (1992)
G. Delling, M. Amling, Nephrol. Dial. Transplant. 10, 601 (1995)
G. Dougherty, G.M. Henebry, Med. Eng. Phys. 23, 369 (2001)
Y. Gefen, A. Aharony, B.B. Mandelbrot, J. Phys. A 17, 1277 (1984)
S.A. Goldstein, R. Goulet, D. McCubbrey, Calcified Tissue 53 (Suppl. 1); S127 (1993)
X.E. Guo, C.H. Kim, Bone 30, 404 (2002)
T. Hildebrand, A. Laib, R. Müller, J. Dequeker, P. Rüegsegger, J. Bone Min. Res. 14, 1167 (1999)
J.H. Kinney, J.S. Stölken, T.S. Smith, J.T. Ryaby, N.E. Lane, Bone 36, 193 (2005)
B.B. Mandelbrot, The fractal geometry of nature (Freeman, San Francisco, 1982)
L. Mosekilde, A. Viidik, L. Mosekilde, Bone 6, 291 (1985)
A.M. Parfitt, C.H.E. Mathews, A.R. Villanueva, M. Kleerekoper, B. Frame, D.S. Rao, J. Clin. Invest. 72, 1396 (1983)
R.E. Plotnick, R.H. Gardner, W.W. Hargrove, K. Prestegaard, M. Perlmutter, Phys. Rev. E 53, 5461 (1996)
L. Pothuaud, B. Van Rietbergen, L. Mosekilde, O. Beuf, P. Levitz, C.L. Benhamou, S. Majumdar, J. Biomech. 35, 1091 (2002)
S. Prouteau, G. Ducher, P. Nanyan, G. Lemineur, L. Benhamou, D. Courteix, Eur. J. Clin. Invest. 34, 137 (2004)
C.S. Rajapakse, J.S. Thomsen, J.S. Espinoza Ortiz, S.J. Wimalawansa, E.N. Ebbesen, L. Mosekilde, G.H. Gunaratne, J. Biomech. 37, 1241 (2004)
J.-Y. Rho, L. Kuhn-Spearing, P. Zioupos, Med. Eng. Phys. 20, 92 (1998)
P.I. Saparin, W. Gowin, J. Kurths, D. Felsenberg, Phys. Rev. E 58, 6449 (1998)
P.I. Saparin, J.S. Thomsen, S. Prohaska, A. Zaikin, J. Kurths, H.-C. Hege, W. Gowin, Acta Astronaut. 56, 820 (2005)
J.S. Thomsen, E.N. Ebbesen, L. Mosekilde, Bone 30, 502 (2002)
B. van Rietbergen, S. Majumdar, W. Pistoia, D.C. Newitt, M. Kothari, A. Laib, P. Rüegsegger, Technol. Health Care 6, 413 (1998)
L. Vico, P. Collet, A. Guignandon, M.-H. Lafage-Proust, T. Thomas, M. Rehailia, C. Alexandre, The Lancet 355, 1607 (2000)
M. Vogel, M. Hahn, G. Delling, Bone 14, 199 (1993)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marwan, N., Saparin, P. & Kurths, J. Measures of complexity for 3D image analysis of trabecular bone. Eur. Phys. J. Spec. Top. 143, 109–116 (2007). https://doi.org/10.1140/epjst/e2007-00078-x
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00078-x