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Computation of Local Fractal Dimension Values of the Human Cerebral Cortex

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DOI: 10.4236/am.2014.512166    3,005 Downloads   3,764 Views   Citations
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Objectives: The purpose of this paper is to describe a technique for computing the local fractal dimension of the human cerebral cortex as extracted from high-resolution magnetic resonance imaging scans. Methods: 3D models of the human cerebral cortex were extracted from high resolution magnetic resonance images of 10 healthy adult volunteers using FreeSurfer. The local fractal dimension of the cortex was computed using a custom-written cube-counting algorithm. The effect of constraining the maximum region size on the measured value of local fractal dimension was examined. A proof of principle was demonstrated by comparing an individual with Alzheimer’s disease to a healthy individual. Results: Local values of cortical fractal dimension can be obtained by constraining the size of the region over which the cube counting is performed. Cubic regions of intermediate size (30 × 30 × 30 mm) yielded a profile that demonstrated greater regional variability compared to smaller (15 × 15 × 15 mm) or larger (60 × 60 × 60 mm) region sizes. Conclusions: Local fractal dimension of the cerebral cortex is a novel measure that may yield additional, quantitative insight into the clinical meaning of cortical shape changes.

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The authors declare no conflicts of interest.

Cite this paper

King, R. (2014) Computation of Local Fractal Dimension Values of the Human Cerebral Cortex. Applied Mathematics, 5, 1733-1740. doi: 10.4236/am.2014.512166.


[1] Mandelbrot, B.B. (1982) The Fractal Geometry of Nature. W.H. Freeman, San Francisco.
[2] Mandelbrot, B.B. (1977) Fractals: Form, Chance, and Dimension. W. H. Freeman, San Francisco.
[3] Bullmore, E., Brammer, M., Harvey, I., et al. (1994) Fractal Analysis of the Boundary between White Matter and Cerebral Cortex in Magnetic Resonance Images: A Controlled Study of Schizophrenic and Manic-Depressive Patients. Psychological Medicine, 24, 771-781.
[4] Free, S.L., Sisodiya, S.M., Cook, M.J., Fish, D.R. and Shorvon, S.D. (1996) Three-Dimensional Fractal Analysis of the White Matter Surface from Magnetic Resonance Images of the Human Brain. Cereb Cortex, 6, 830-836.
[5] Hofman, M.A. (1991) The Fractal Geometry of Convoluted Brains. Journal für Hirnforschung, 32, 103-111.
[6] Im, K., Lee, J.M., Yoon, U., et al. (2006) Fractal Dimension in Human Cortical Surface: Multiple Regression Analysis with Cortical Thickness, Sulcal Depth, and Folding Area. Human Brain Mapping, 27, 994-1003.
[7] Kiselev, V.G., Hahn, K.R. and Auer, D.P. (2003) Is the Brain Cortex a Fractal? Neuroimage, 20, 1765-1774.
[8] Majumdar, S. and Prasad, R.R. (1988) The Fractal Dimension of Cerebral Surfaces Using Magnetic Resonance Imaging. Computers in Physics, 2, 69-73.
[9] Ha, T.H., Yoon, U., Lee, K.J., et al. (2005) Fractal Dimension of Cerebral Cortical Surface in Schizophrenia and Obsessive-Compulsive Disorder. Neuroscience Letters, 384, 172-176.
[10] Sandu, A.L., Rasmussen Jr., I.A., Lundervold, A., et al. (2008) Fractal Dimension Analysis of MR Images Reveals Grey Matter Structure Irregularities in Schizophrenia. Computerized Medical Imaging and Graphics, 32, 150-158.
[11] Casanova, M.F., Daniel, D.G., Goldberg, T.E., Suddath, R.L. and Weinberger, D.R. (1989) Shape Analysis of the Middle Cranial Fossa of Schizophrenic Patients. A Computerized Tomographic Study. Schizophrenia Research, 2, 333-338.
[12] Casanova, M.F., Goldberg, T.E., Suddath, R.L., et al. (1990) Quantitative Shape Analysis of the Temporal and Prefrontal Lobes of Schizophrenic Patients: A Magnetic Resonance Image Study. The Journal of Neuropsychiatry and Clinical Neurosciences, 2, 363-372.
[13] Wu, Y.T., Shyu, K.K., Jao, C.W., et al. (2010) Fractal Dimension Analysis for Quantifying Cerebellar Morphological Change of Multiple System Atrophy of the Cerebellar Type (MSA-C). Neuroimage, 49, 539-551.
[14] Cook, M.J., Free, S.L., Manford, M.R., et al. (1995) Fractal Description of Cerebral Cortical Patterns in Frontal Lobe Epilepsy. European Neurology, 35, 327-335.
[15] Pirici, D., Mogoanta, L., Margaritescu, O., et al. (2009) Fractal Analysis of Astrocytes in Stroke and Dementia. Romanian Journal of Morphology and Embryology, 50, 381-390.
[16] Thompson, P.M., Lee, A.D., Dutton, R.A., Geaga, J.A., Hayashi, K.M., Eckert, M.A., et al. (2005) Abnormal Cortical Complexity and Thickness Profiles Mapped in Williams Syndrome. The Journal of Neuroscience, 25, 4146-4158.
[17] Wu, Y.T., Shyu, K.K., Chen, T.R. and Guo, W.Y. (2009) Using Three-Dimensional Fractal Dimension to Analyze the Complexity of Fetal Cortical Surface from Magnetic Resonance Images. Nonlinear Dynamics, 58, 745-752.
[18] King, R.D., George, A.T., Jeon, T., Hynan, L.S., Youn, T.S., Kennedy, D.N., et al. (2009) Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis. Brain Imaging and Behavior, 3, 154-166.
[19] King, R.D., Brown, B., Hwang, M., Jeon, T. and George, A.T. (2010) Fractal Dimension Analysis of the Cortical Ribbon in Mild Alzheimer’s Disease. Neuroimage, 53, 471-479.
[20] Walhovd, K.B., Westlye, L.T., Amlien, I., Espeseth, T., Reinvang, I., Raz, N., et al. (2011) Consistent Neuroanatomical Age-Related Volume Differences across Multiple Samples. Neurobiology of Aging, 32, 916-932.
[21] Raz, N., Lindenberger, U., Rodrigue, K.M., Kennedy, K.M., Head, D., Williamson, A., et al. (2005) Regional Brain Changes in Aging Healthy Adults: General Trends, Individual Differences and Modifiers. Cereb Cortex, 15, 1676-1689.
[22] Raz, N., Ghisletta, P., Rodrigue, K.M., Kennedy, K.M. and Lindenberger, U. (2010) Trajectories of Brain Aging in Middle-Aged and Older Adults: Regional and Individual Differences. Neuroimage, 51, 501-511.
[23] Dale, A.M., Fischl, B. and Sereno, M.I. (1999) Cortical Surface-Based Analysis. I. Segmentation and Surface Reconstruction. Neuroimage, 9, 179-194.
[24] Fischl, B., Sereno, M.I. and Dale, A.M. (1999) Cortical Surface-Based Analysis. II: Inflation, Flattening, and a Surface-Based Coordinate System. Neuroimage, 9, 195-207.
[25] Fischl, B. and Dale, A.M. (2000) Measuring the Thickness of the Human Cerebral Cortex from Magnetic Resonance Images. Proceedings of the National Academy of Sciences of the United States of America, 97, 11050-11055.
[26] Fischl, B., van der Kouwe, A., Destrieux, C., Halgren, E., Ségonne, F., Salat, D.H., et al. (2004) Automatically Parcellating the Human Cerebral Cortex. Cereb Cortex, 14, 11-22.
[27] Han, X., Jovicich, J., Salat, D., van der Kouwe, A., Quinn, B., Czanner, S., et al. (2006) Reliability of MRI-Derived Measurements of Human Cerebral Cortical Thickness: The Effects of Field Strength, Scanner Upgrade and Manufacturer. Neuroimage, 32, 180-194.
[28] Esteban, F.J., Sepulcre, J., de Mendizabal, N.V., Goñi, J., Navas, J., de Miras, J.R., et al. (2007) Fractal Dimension and White Matter Changes in Multiple Sclerosis. Neuroimage, 36, 543-549.
[29] Lee, J.M., Yoon, U., Kim, J.J., Kim, I.Y., Lee, D.S., Kwon, J.S., et al. (2004) Analysis of the Hemispheric Asymmetry Using Fractal Dimension of a Skeletonized Cerebral Surface. IEEE Transactions on Biomedical Engineering, 51, 1494-1498.
[30] Zhang, L., Dean, D., Liu, J.Z., Sahgal, V., Wang, X. and Yue, G.H. (2007) Quantifying Degeneration of White Matter in Normal Aging Using Fractal Dimension. Neurobiology of Aging, 28, 1543-1555.
[31] Zhang, L., Liu, J.Z., Dean, D., Sahgal, V. and Yue, G.H. (2006) A Three-Dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain. Journal of Neuroscience Methods, 150, 242-253.

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