Fractal analysis as a method of morphometric study of linear anatomical objects: modified Caliper method

Keywords: fractal analysis, fractal dimension, morphometry, Caliper method, Richardson’s method.


The purpose of the study was to develop an original modification of the Caliper method of image fractal analysis to determine the fractal dimension of linear anatomical objects. To develop the method, the linear contour of the outer surface of the cerebral cortex was chosen as the object of study. Magnetic resonance brain images in coronal projection were used. The original modification of the Caliper method includes image analysis using Adobe Photoshop CS5 software or its analogues. The linear contour of the studied object is selected, followed by stepwise smoothing of the contour with different smoothing radius. At the 1st stage of fractal analysis smoothing is not applied, at the 2nd stage the smoothing radius is 2 pixels, the 3rd – 4 pixels, the 4th – 8 pixels, the 5th – 16 pixels. At each stage, the contour length in pixels is measured (P). The size of the fractal measurement unit (G) at the 1st stage of fractal analysis is 1 pixel, the 2nd stage – 2 pixels, the 3rd stage – 4 pixels, the 4th stage – 8 pixels, the 5th stage – 16 pixels. The contour smoothing radius, the size of the fractal measurement units and the number of stages of fractal analysis can be changed depending on the characteristics of the studied structure, size, scale and image resolution. Based on the values of the perimeter and the size of the fractal measurement units, the number of fractal measurement units covering the studied object (N) is calculated: N=P/G. The fractal dimension value is calculated based on the N and G values. The modification of the Caliper method described in this paper is automatized and does not require much time required for manual calculation. In addition, compared to the classic Caliper method, this modification is more accurate because the measurement is performed automatically. The main limitation of the developed modification is the ability to determine the fractal dimension of only closed contours of studied structures or closed linear structures, because this method involves determining the length of the closed perimeter of the selected image area. The modified Caliper method of image fractal analysis described in this paper can be used in morphology and other fields of medicine for fractal analysis of linear objects: external and internal linear contours of different anatomical structures (cerebellum, cerebral hemispheres) and pathological foci (tumors, foci of necrosis, fibrosis, etc.).


[1] Akar, E., Kara, S., Akdemir, H., & Kırış, A. (2015). Fractal dimension analysis of cerebellum in Chiari Malformation type I. Computers in Biology and Medicine, 64, 179-186. doi: 10.1016/j.compbiomed.2015.06.024
[2] Carey, S. & Maria, A. & Sigurdsson, H. (2000). Use of fractal analysis for discriminaion of particles from primary and reworked jökulhlaup deposits in SE Iceland. Journal of Volcanology and Geothermal Research, 104, 65-80. doi: 10.1016/S0377-0273(00)00200-6
[3] Di Ieva, A., Esteban, F. J., Grizzi, F., Klonowski, W., & Martín-Landrove, M. (2015). Fractals in the neurosciences, Part II: clinical applications and future perspectives. The Neuroscientist: a review journal bringing neurobiology, neurology and psychiatry, 21(1), 30-43. doi: 10.1177/1073858413513928
[4] Di Ieva, A., Grizzi, F., Jelinek, H., Pellionisz, A. J., & Losa, G. A. (2014). Fractals in the Neurosciences, Part I: General Principles and Basic Neurosciences. The Neuroscientist: a review journal bringing neurobiology, neurology and psychiatry, 20(4), 403-417. doi: 10.1177/1073858413513927
[5] Feder J. (1988) Fractals. New York: Plenum Press, 284.
[6] Fernández, E, & Jelinek, H. F. (2001) Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods, 24(4), 309-321. doi: 10.1006/meth.2001.1201
[7] Jelinek, H. F., & Fernandez, E. (1998). Neurons and fractals: how reliable and useful are calculations of fractal dimensions? Journal of Neuroscience Methods, 81(1-2), 9-18. doi: 10.1016/s0165-0270(98)00021-1
[8] King, R. D., Brown, B., Hwang, M., Jeon, T., George, A. T., & Alzheimer’s Disease Neuroimaging Initiative (2010). Fractal dimension analysis of the cortical ribbon in mild Alzheimer’s disease. NeuroImage, 53(2), 471-479. doi: 10.1016/j.neuroimage.2010.06.050
[9] King, R. D., George, A. T., Jeon, T., Hynan, L. S., Youn, T. S., Kennedy, D. N. … the Alzheimer’s Disease Neuroimaging Initiative (2009). Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis. Brain Imaging and Behavior, 3(2), 154-166. doi: 10.1007/s11682-008-9057-9
[10] Kiselev, V. G., Hahn, K. R., & Auer, D. P. (2003). Is the brain cortex a fractal? NeuroImage, 20(3), 1765-1774. doi: 10.1016/s1053-8119(03)00380-x
[11] Lee, K. I., Choi, S. C., Park, T. W., & You, D. S. (1999). Fractal dimension calculated from two types of region of interest. Dento Maxillo Facial Radiology, 28(5), 284-289. doi: 10.1038/sj/dmfr/4600458
[12] Manabe, Y., Honda, E., Shiro, Y., Sakai, K., Kohira, I., Kashihara, K., …. & Abe, K. (2001). Fractal dimension analysis of static stabilometry in Parkinson’s disease and spinocerebellar ataxia. Neurological research, 23(4), 397-404. doi: 10.1179/016164101101198613
[13] Mandelbrot, B. B. (1983) The fractal geometry of nature. N.Y.: W. H. Freeman&Co, 468.
[14] Mandelbrot, B. B. (1967) How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, New Series, 3775(156), 636-638.
[15] McLachlan, C. S., Jelinek, H. F., Kummerfeld, S. K., Rummery, N., McLachlan, P. D., Jusuf, P. …. Yin, J. (2000). A method to determine the fractal dimension of the cross-sectional jaggedness of the infarct scar edge. Redox Report: communications in free radical research, 5(2-3), 119-121. doi: 10.1179/135100000101535401
[16] Rajković, N., Krstonošić, B., & Milošević, N. (2017). Box-Counting Method of 2D Neuronal Image: Method Modification and Quantitative Analysis Demonstrated on Images from the Monkey and Human Brain. Computational and Mathematical Methods in Medicine, 2017, 8967902. doi: 10.1155/2017/8967902
[17] Ristanović, D., Stefanović, B. D., & Puskas, N. (2013). Fractal analysis of dendrites morphology using modified Richardson’s and box counting method. Theoretical Biology Forum, 106(1-2), 157-168.
[18] Ristanović, D., Stefanović, B. D., & Puškaš, N. (2014). Fractal analysis of dendrite morphology of rotated neuronal pictures: the modified box counting method. Theoretical Biology Forum, 107(1-2), 109-121.
[19] Shrout, M. K., Potter, B. J., & Hildebolt, C. F. (1997). The effect of image variations on fractal dimension calculations. Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontics, 84(1), 96-100. doi: 10.1016/s1079-2104(97)90303-6
[20] Stoa, R. (2020). The Coastline Paradox. Rutgers University Law Review, 72(2), 351-400. doi: 10.2139/ssrn.3445756
[21] Wu, Y. T., Shyu, K. K., Jao, C. W., Wang, Z. Y., Soong, B. W., Wu, H. M., & Wang, P. S. (2010). Fractal dimension analysis for quantifying cerebellar morphological change of multiple system atrophy of the cerebellar type (MSA-C). NeuroImage, 49(1), 539-551. doi: 10.1016/j.neuroimage.2009.07.042
[22] Zaletel, I., Ristanović, D., Stefanović, B. D., & Puškaš, N. (2015). Modified Richardson’s method versus the box-counting method in neuroscience. Journal of Neuroscience Methods, 242, 93-96. doi: 10.1016/j.jneumeth.2015.01.013
How to Cite
Maryenko, N., & Stepanenko , O. (2021). Fractal analysis as a method of morphometric study of linear anatomical objects: modified Caliper method. Reports of Morphology, 27(4), 28-34.