Keywords
morphometry
size
dimension
fractal dimension
fractal analysis
lacunarity
size
dimension
fractal dimension
fractal analysis
lacunarity
How to Cite
Maryenko, N., & Stepanenko, O. (2022). FRACTAL DIMENSION IN MORPHOLOGY AND MEDICINE: THEORETICAL BASES AND PRACTICAL APPLICATION. Inter Collegas, 8(4), 290 - 297. https://doi.org/10.35339/ic.8.4.290-297
Abstract
Morphometry is an integral part of most modern morphological studies and the classic morphological morphometric methods and techniques are often borrowed for research in other fields of medicine. The majority of morphometric techniques are derived from Euclidean geometry. In the past decades, the principles, parameters and methods of fractal geometry are increasingly used in morphological studies. The basic parameter of fractal geometry is fractal dimension. Fractal dimension allows you to quantify the degree of filling of space with a certain geometric object and to characterize the complexity of its spatial configuration. There are many anatomical structures with complex irregular shapes that cannot be unambiguously and comprehensively characterized by methods and techniques of traditional geometry and traditional morphometry: irregular linear structures, irregular surfaces of various structures and pathological foci, structures with complex branched, tree-like, reticulated, cellular or porous structure, etc. Fractal dimension is a useful and informative morphometric parameter that can complement existing quantitative parameters to quantify objective characteristics of various anatomical structures and pathological foci. Fractal analysis can qualitatively complement existing morphometric methods and techniques and allow a comprehensive assessment of the spatial configuration complexity degree of irregular anatomical structures. The review describes the basic principles of Euclidean and fractal geometry and their application in morphology and medicine, importance and application of sizes and their derivatives, topological, metric and fractal dimensions, regular and irregular figures in morphology, and practical application of fractal dimension and fractal analysis in the morphological studies and clinical practice.
References
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8. Feder, J. (1988). Fractals. New York: Plenum Press.
9. Kronover, P.M. (2000). Fraktaly i haos v dinamicheskih sistemah. Osnovy teorii [Fractals and chaos in dynamical systems. Foundations of the theory]. Moscow: Postmarket. [In Russian]
10. Harte, D. (2001). Multifractals. London: Chapman & Hall.
11. Falconer, K. (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley.
12. Fomenko, A.T. (1993). Nagljadnaja geometrija i topologija [Descriptive geometry and topology]. — Moscow: MGU. [In Russian]
13. Mandelbrot, B.B. (1975). Les Objets fractals: forme, hasard et dimension. Paris: Flammarion. 214 p.
14. Mandelbrot, B.B. (1977). Fractals – form, chance and dimension. San Francisco: W. H. Freeman. 365 p.
15. Mandelbrot, B.B. (1982). The fractal geometry of nature. San Francisco: W.H. Freeman and Company. 470 p.
16. 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. https://doi.org/10.1177/1073858413513927
17. 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. https://doi.org/10.1177/1073858413513928
18. Rangayyan, R. M., & Nguyen, T. M. (2007). Fractal analysis of contours of breast masses in mammograms. Journal of digital imaging, 20(3), 223–237. https://doi.org/10.1007/s10278-006-0860-9
19. King, R. D., George, A. T., Jeon, T., Hynan, L. S., Youn, T. S., Kennedy, D. N., Dickerson, B., & 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. https://doi.org/10.1007/s11682-008-9057-9
20. Ţălu, Ş., Stach, S., Sueiras, V., & Ziebarth, N. M. (2015). Fractal analysis of AFM images of the surface of Bowman's membrane of the human cornea. Annals of biomedical engineering, 43(4), 906–916. https://doi.org/10.1007/s10439-014-1140-3
21. Kamiya, A., & Takahashi, T. (2007). Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature. Journal of applied physiology (Bethesda, Md. : 1985), 102(6), 2315–2323. https://doi.org/10.1152/japplphysiol.00856.2006
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23. Lorthois, S., & Cassot, F. (2010). Fractal analysis of vascular networks: insights from morphogenesis. Journal of theoretical biology, 262(4), 614–633. https://doi.org/10.1016/j.jtbi.2009.10.037
24. Tălu S. (2011). Fractal analysis of normal retinal vascular network. Oftalmologia (Bucharest, Romania : 1990), 55(4), 11–16.
25. Cross, S. S., Start, R. D., Silcocks, P. B., Bull, A. D., Cotton, D. W., & Underwood, J. C. (1993). Quantitation of the renal arterial tree by fractal analysis. The Journal of pathology, 170(4), 479–484. https://doi.org/10.1002/path.1711700412
26. Helmberger, M., Pienn, M., Urschler, M., Kullnig, P., Stollberger, R., Kovacs, G., Olschewski, A., Olschewski, H., & Bálint, Z. (2014). Quantification of tortuosity and fractal dimension of the lung vessels in pulmonary hypertension patients. PloS one, 9(1), e87515. https://doi.org/10.1371/journal.pone.0087515
27. van Beek J. H. (1997). Is local metabolism the basis of the fractal vascular structure in the heart?. International journal of microcirculation, clinical and experimental, 17(6), 337–345. https://doi.org/10.1159/000179250
28. Di Ieva, A., Grizzi, F., Gaetani, P., Goglia, U., Tschabitscher, M., Mortini, P., & Rodriguez y Baena, R. (2008). Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue. Neurosurgical review, 31(3), 271–281. https://doi.org/10.1007/s10143-008-0127-7
29. Helthuis, J., van Doormaal, T., Hillen, B., Bleys, R., Harteveld, A. A., Hendrikse, J., van der Toorn, A., Brozici, M., Zwanenburg, J., & van der Zwan, A. (2019). Branching Pattern of the Cerebral Arterial Tree. Anatomical record (Hoboken, N.J. : 2007), 302(8), 1434–1446. https://doi.org/10.1002/ar.23994
30. Stepanenko, A.Yu., & Maryenko, N.I. (2015). Fraktal'nyj analiz kak metod morfometricheskogo issledovanija poverhnostnoj sosudistoj seti mozzhechka cheloveka [Fractal analysis as a method of morphometric study of the superficial vascular network of human cerebellum]. Medytsyna syohodni i zavtra, 4(69), 50–55. [In Russian]
31. Di Ieva, A., Niamah, M., Menezes, R. J., Tsao, M., Krings, T., Cho, Y. B., Schwartz, M. L., & Cusimano, M. D. (2014). Computational fractal-based analysis of brain arteriovenous malformation angioarchitecture. Neurosurgery, 75(1), 72–79. https://doi.org/10.1227/NEU.0000000000000353
32. Glenny, R. W., Krueger, M., Bauer, C., & Beichel, R. R. (2020). The fractal geometry of bronchial trees differs by strain in mice. Journal of applied physiology (Bethesda, Md.: 1985), 128(2), 362–367. https://doi.org/10.1152/japplphysiol.00838.2019
33. Takeda, T., Ishikawa, A., Ohtomo, K., Kobayashi, Y., & Matsuoka, T. (1992). Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development. Neuroscience research, 13(1), 19–31. https://doi.org/10.1016/0168-0102(92)90031-7
34. Puškaš, N., Zaletel, I., Stefanović, B. D., & Ristanović, D. (2015). Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex. Neuroscience letters, 589, 88–91. https://doi.org/10.1016/j.neulet.2015.01.044
35. Milosević, N. T., & Ristanović, D. (2006). Fractality of dendritic arborization of spinal cord neurons. Neuroscience letters, 396(3), 172–176. https://doi.org/10.1016/j.neulet.2005.11.031
36. Fernández, E., & Jelinek, H. F. (2001). Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods (San Diego, Calif.), 24(4), 309–321. https://doi.org/10.1006/meth.2001.1201
37. Pirici, D., Mogoantă, L., Mărgăritescu, O., Pirici, I., Tudorică, V., & Coconu, M. (2009). Fractal analysis of astrocytes in stroke and dementia. Romanian journal of morphology and embryology = Revue roumaine de morphologie et embryologie, 50(3), 381–390.
38. Reichenbach, A., Siegel, A., Senitz, D., & Smith, T. G., Jr (1992). A comparative fractal analysis of various mammalian astroglial cell types. NeuroImage, 1(1), 69–77. https://doi.org/10.1016/1053-8119(92)90008-b
39. Karperien, A. L., & Jelinek, H. F. (2015). Fractal, multifractal, and lacunarity analysis of microglia in tissue engineering. Frontiers in bioengineering and biotechnology, 3, 51. https://doi.org/10.3389/fbioe.2015.00051
40. Stepanenko, A.Y., & Maryenko, N.I. (2017) Fraktal'nyj analiz belogo veshhestva mozzhechka cheloveka [Fractal analysis of the human cerebellum white matter]. World of Medicine and Biology, 3(61), 145–149. [In Russian]
41. Maryenko, N.I., & Stepanenko, O.Y. (2017) Fraktal'nyj analiz biloi' rechovyny pivkul' mozochka ljudyny [Fractal analysis of white matter of the human cerebellum hemispheres]. Ukr. ž. med. bìol. Sportu, 2 (4), 38-43. [In Ukrainian]
42. 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. https://doi.org/10.1016/j.compbiomed.2015.06.024
43. Liu, J. Z., Zhang, L. D., & Yue, G. H. (2003). Fractal dimension in human cerebellum measured by magnetic resonance imaging. Biophysical journal, 85(6), 4041–4046. https://doi.org/10.1016/S0006-3495(03)74817-6
44. 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. https://doi.org/10.1016/j.neuroimage.2009.07.042
45. Gulec, M., Tassoker, M., Ozcan, S., & Orhan, K. (2021). Evaluation of the mandibular trabecular bone in patients with bruxism using fractal analysis. Oral radiology, 37(1), 36–45. https://doi.org/10.1007/s11282-020-00422-5
46. Kato, C. N., Barra, S. G., Tavares, N. P., Amaral, T. M., Brasileiro, C. B., Mesquita, R. A., & Abreu, L. G. (2020). Use of fractal analysis in dental images: a systematic review. Dento maxillo facial radiology, 49(2), 20180457. https://doi.org/10.1259/dmfr.20180457
47. Smith, T. G., Jr, Lange, G. D., & Marks, W. B. (1996). Fractal methods and results in cellular morphology--dimensions, lacunarity and multifractals. Journal of neuroscience methods, 69(2), 123–136. https://doi.org/10.1016/S0165-0270(96)00080-5
48. Ling, E. J., Servio, P., & Kietzig, A. M. (2016). Fractal and Lacunarity Analyses: Quantitative Characterization of Hierarchical Surface Topographies. Microscopy and microanalysis : the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada, 22(1), 168–177. https://doi.org/10.1017/S1431927615015561
49. Dougherty, G., & Henebry, G. M. (2001). Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images. Medical engineering & physics, 23(6), 369–380. https://doi.org/10.1016/s1350-4533(01)00057-1
50. Yasar, F., & Akgünlü, F. (2005). Fractal dimension and lacunarity analysis of dental radiographs. Dento maxillo facial radiology, 34(5), 261–267. https://doi.org/10.1259/dmfr/85149245
51. Park, Y. W., Kim, S., Ahn, S. S., Han, K., Kang, S. G., Chang, J. H., Kim, S. H., Lee, S. K., & Park, S. H. (2020). Magnetic resonance imaging-based 3-dimensional fractal dimension and lacunarity analyses may predict the meningioma grade. European radiology, 30(8), 4615–4622. https://doi.org/10.1007/s00330-020-06788-8
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2. Hudoerkov, R.M. (2014). Metody komp'juternoj morfometrii v nejromorfologii: uchebnoe posobie (bazovyj kurs) [Methods of computer morphometry in neuromorphology: a textbook (basic course)]. Moscow: FGBU «NCN» RAMN. [In Russian]
3. Nikonenko, A.G. (2013). Vvedenie v kolichestvennuju gistologiju [Introduction to Quantitative Histology]. Kiev: Kniga-Pljus. [In Russian]
4. Speranskij, V.S., & Zajchenko, A.I. (1980). Forma i konstrukcija cherepa [Skull shape and construction]. Moscow: Medicina. [In Russian]
5. Blei, R. (2003). Analysis in integer and fractional dimensions. New York: Cambridge university press.
6. Jelinek, H. & Fernandez, E. (1998). Neurons and fractals: How reliable and useful are calculations of offractal dimensions?. Journal of Neuroscience Methods. 81. 9-18. 10.1016/S0165-0270(98)00021-1.
7. Aleksandrov, P.S., & Pasynkov, B.A. (1973). Vvedenie v teoriju razmernosti [Introduction to Dimension Theory]. Moscow: Nauka. [In Russian]
8. Feder, J. (1988). Fractals. New York: Plenum Press.
9. Kronover, P.M. (2000). Fraktaly i haos v dinamicheskih sistemah. Osnovy teorii [Fractals and chaos in dynamical systems. Foundations of the theory]. Moscow: Postmarket. [In Russian]
10. Harte, D. (2001). Multifractals. London: Chapman & Hall.
11. Falconer, K. (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley.
12. Fomenko, A.T. (1993). Nagljadnaja geometrija i topologija [Descriptive geometry and topology]. — Moscow: MGU. [In Russian]
13. Mandelbrot, B.B. (1975). Les Objets fractals: forme, hasard et dimension. Paris: Flammarion. 214 p.
14. Mandelbrot, B.B. (1977). Fractals – form, chance and dimension. San Francisco: W. H. Freeman. 365 p.
15. Mandelbrot, B.B. (1982). The fractal geometry of nature. San Francisco: W.H. Freeman and Company. 470 p.
16. 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. https://doi.org/10.1177/1073858413513927
17. 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. https://doi.org/10.1177/1073858413513928
18. Rangayyan, R. M., & Nguyen, T. M. (2007). Fractal analysis of contours of breast masses in mammograms. Journal of digital imaging, 20(3), 223–237. https://doi.org/10.1007/s10278-006-0860-9
19. King, R. D., George, A. T., Jeon, T., Hynan, L. S., Youn, T. S., Kennedy, D. N., Dickerson, B., & 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. https://doi.org/10.1007/s11682-008-9057-9
20. Ţălu, Ş., Stach, S., Sueiras, V., & Ziebarth, N. M. (2015). Fractal analysis of AFM images of the surface of Bowman's membrane of the human cornea. Annals of biomedical engineering, 43(4), 906–916. https://doi.org/10.1007/s10439-014-1140-3
21. Kamiya, A., & Takahashi, T. (2007). Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature. Journal of applied physiology (Bethesda, Md. : 1985), 102(6), 2315–2323. https://doi.org/10.1152/japplphysiol.00856.2006
22. Zamir M. (1999). On fractal properties of arterial trees. Journal of theoretical biology, 197(4), 517–526. https://doi.org/10.1006/jtbi.1998.0892
23. Lorthois, S., & Cassot, F. (2010). Fractal analysis of vascular networks: insights from morphogenesis. Journal of theoretical biology, 262(4), 614–633. https://doi.org/10.1016/j.jtbi.2009.10.037
24. Tălu S. (2011). Fractal analysis of normal retinal vascular network. Oftalmologia (Bucharest, Romania : 1990), 55(4), 11–16.
25. Cross, S. S., Start, R. D., Silcocks, P. B., Bull, A. D., Cotton, D. W., & Underwood, J. C. (1993). Quantitation of the renal arterial tree by fractal analysis. The Journal of pathology, 170(4), 479–484. https://doi.org/10.1002/path.1711700412
26. Helmberger, M., Pienn, M., Urschler, M., Kullnig, P., Stollberger, R., Kovacs, G., Olschewski, A., Olschewski, H., & Bálint, Z. (2014). Quantification of tortuosity and fractal dimension of the lung vessels in pulmonary hypertension patients. PloS one, 9(1), e87515. https://doi.org/10.1371/journal.pone.0087515
27. van Beek J. H. (1997). Is local metabolism the basis of the fractal vascular structure in the heart?. International journal of microcirculation, clinical and experimental, 17(6), 337–345. https://doi.org/10.1159/000179250
28. Di Ieva, A., Grizzi, F., Gaetani, P., Goglia, U., Tschabitscher, M., Mortini, P., & Rodriguez y Baena, R. (2008). Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue. Neurosurgical review, 31(3), 271–281. https://doi.org/10.1007/s10143-008-0127-7
29. Helthuis, J., van Doormaal, T., Hillen, B., Bleys, R., Harteveld, A. A., Hendrikse, J., van der Toorn, A., Brozici, M., Zwanenburg, J., & van der Zwan, A. (2019). Branching Pattern of the Cerebral Arterial Tree. Anatomical record (Hoboken, N.J. : 2007), 302(8), 1434–1446. https://doi.org/10.1002/ar.23994
30. Stepanenko, A.Yu., & Maryenko, N.I. (2015). Fraktal'nyj analiz kak metod morfometricheskogo issledovanija poverhnostnoj sosudistoj seti mozzhechka cheloveka [Fractal analysis as a method of morphometric study of the superficial vascular network of human cerebellum]. Medytsyna syohodni i zavtra, 4(69), 50–55. [In Russian]
31. Di Ieva, A., Niamah, M., Menezes, R. J., Tsao, M., Krings, T., Cho, Y. B., Schwartz, M. L., & Cusimano, M. D. (2014). Computational fractal-based analysis of brain arteriovenous malformation angioarchitecture. Neurosurgery, 75(1), 72–79. https://doi.org/10.1227/NEU.0000000000000353
32. Glenny, R. W., Krueger, M., Bauer, C., & Beichel, R. R. (2020). The fractal geometry of bronchial trees differs by strain in mice. Journal of applied physiology (Bethesda, Md.: 1985), 128(2), 362–367. https://doi.org/10.1152/japplphysiol.00838.2019
33. Takeda, T., Ishikawa, A., Ohtomo, K., Kobayashi, Y., & Matsuoka, T. (1992). Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development. Neuroscience research, 13(1), 19–31. https://doi.org/10.1016/0168-0102(92)90031-7
34. Puškaš, N., Zaletel, I., Stefanović, B. D., & Ristanović, D. (2015). Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex. Neuroscience letters, 589, 88–91. https://doi.org/10.1016/j.neulet.2015.01.044
35. Milosević, N. T., & Ristanović, D. (2006). Fractality of dendritic arborization of spinal cord neurons. Neuroscience letters, 396(3), 172–176. https://doi.org/10.1016/j.neulet.2005.11.031
36. Fernández, E., & Jelinek, H. F. (2001). Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods (San Diego, Calif.), 24(4), 309–321. https://doi.org/10.1006/meth.2001.1201
37. Pirici, D., Mogoantă, L., Mărgăritescu, O., Pirici, I., Tudorică, V., & Coconu, M. (2009). Fractal analysis of astrocytes in stroke and dementia. Romanian journal of morphology and embryology = Revue roumaine de morphologie et embryologie, 50(3), 381–390.
38. Reichenbach, A., Siegel, A., Senitz, D., & Smith, T. G., Jr (1992). A comparative fractal analysis of various mammalian astroglial cell types. NeuroImage, 1(1), 69–77. https://doi.org/10.1016/1053-8119(92)90008-b
39. Karperien, A. L., & Jelinek, H. F. (2015). Fractal, multifractal, and lacunarity analysis of microglia in tissue engineering. Frontiers in bioengineering and biotechnology, 3, 51. https://doi.org/10.3389/fbioe.2015.00051
40. Stepanenko, A.Y., & Maryenko, N.I. (2017) Fraktal'nyj analiz belogo veshhestva mozzhechka cheloveka [Fractal analysis of the human cerebellum white matter]. World of Medicine and Biology, 3(61), 145–149. [In Russian]
41. Maryenko, N.I., & Stepanenko, O.Y. (2017) Fraktal'nyj analiz biloi' rechovyny pivkul' mozochka ljudyny [Fractal analysis of white matter of the human cerebellum hemispheres]. Ukr. ž. med. bìol. Sportu, 2 (4), 38-43. [In Ukrainian]
42. 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. https://doi.org/10.1016/j.compbiomed.2015.06.024
43. Liu, J. Z., Zhang, L. D., & Yue, G. H. (2003). Fractal dimension in human cerebellum measured by magnetic resonance imaging. Biophysical journal, 85(6), 4041–4046. https://doi.org/10.1016/S0006-3495(03)74817-6
44. 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. https://doi.org/10.1016/j.neuroimage.2009.07.042
45. Gulec, M., Tassoker, M., Ozcan, S., & Orhan, K. (2021). Evaluation of the mandibular trabecular bone in patients with bruxism using fractal analysis. Oral radiology, 37(1), 36–45. https://doi.org/10.1007/s11282-020-00422-5
46. Kato, C. N., Barra, S. G., Tavares, N. P., Amaral, T. M., Brasileiro, C. B., Mesquita, R. A., & Abreu, L. G. (2020). Use of fractal analysis in dental images: a systematic review. Dento maxillo facial radiology, 49(2), 20180457. https://doi.org/10.1259/dmfr.20180457
47. Smith, T. G., Jr, Lange, G. D., & Marks, W. B. (1996). Fractal methods and results in cellular morphology--dimensions, lacunarity and multifractals. Journal of neuroscience methods, 69(2), 123–136. https://doi.org/10.1016/S0165-0270(96)00080-5
48. Ling, E. J., Servio, P., & Kietzig, A. M. (2016). Fractal and Lacunarity Analyses: Quantitative Characterization of Hierarchical Surface Topographies. Microscopy and microanalysis : the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada, 22(1), 168–177. https://doi.org/10.1017/S1431927615015561
49. Dougherty, G., & Henebry, G. M. (2001). Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images. Medical engineering & physics, 23(6), 369–380. https://doi.org/10.1016/s1350-4533(01)00057-1
50. Yasar, F., & Akgünlü, F. (2005). Fractal dimension and lacunarity analysis of dental radiographs. Dento maxillo facial radiology, 34(5), 261–267. https://doi.org/10.1259/dmfr/85149245
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