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  • Review Article
  • Published:

Lung cancer—a fractal viewpoint

Nature Reviews Clinical Oncology volume 12, pages 664–675 (2015)Cite this article

Subjects

Key Points

  • Cancer-related structural alterations in lung tissue and individual cells can often be readily observed, but can be difficult to quantify using conventional metrics, such as length or volume

  • Fractals are mathematical constructs that appear infinitely self-similar over a range of scales

  • Many biological entities, including the lung, can be considered as fractals within a limited scaling range known as a 'scaling window'

  • A fractal dimension (FD) is a non-integer value that relates how the detail and complexity of an object changes with scale

  • FD can be used to quantify complex shapes and patterns in a range of clinical and biological images, including those illustrating DNA, cellular architectural, histopathological, and radiological features

  • Fractal dimension can detect subtle changes in images and could potentially provide clinically useful information relating to tumour type, stage, and response to therapy

Abstract

Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed.

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Figure 1: Examples of biological and mathematical fractal patterns.
Figure 2: The box-counting method of calculating FD.
Figure 3: Lacunarity.
Figure 4: Fractal analysis of DNA sequences.
Figure 5: Fractal analysis of lung cancer histology.
Figure 6: Lung cancer progression and fractal dimension.
Figure 7: Treatment response and FD.

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References

  1. American Cancer Society. Cancer Facts and Figures 2015 [online], (2015).

  2. Siegel, R., Ma, J., Zou, Z. & Jemal, A. Cancer statistics, 2014. CA Cancer J. Clin. 64, 9–29 (2014).

    Article  PubMed  Google Scholar 

  3. Mozley, P. D. et al. Change in lung tumor volume as a biomarker of treatment response: a critical review of the evidence. Ann. Oncol. 21, 1751–1755 (2010).

    Article  CAS  PubMed  Google Scholar 

  4. Mandelbrot, B. B. The Fractal Geometry of Nature (W. H. Freeman & Co. Ltd, 1982).

    Google Scholar 

  5. Peitgen, H.-O., Ju¨rgens, H. & Saupe, D. Chaos and fractals: New Frontiers of Science 2nd edn (Springer-Verlag, 2004).

    Book  Google Scholar 

  6. Legner, P. Fractals . Mathigon—World of Mathematics [online], (2015).

    Google Scholar 

  7. Ristanovic´, D. & Milosevic´, N. T. Fractal analysis: methodologies for biomedical researchers. Theor. Biol. Forum 105, 99–118 (2012).

    PubMed  Google Scholar 

  8. Mandelbrot, B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156, 636–638 (1967).

    Article  CAS  PubMed  Google Scholar 

  9. Eghball, B., Hergert, G. W., Lesoing, G. W. & Ferguson, R. B. Fractal analysis of spatial and temporal variability. Geoderma 88, 349–362 (1999).

    Article  Google Scholar 

  10. Lopes, R. & Betrouni, N. Fractal and multifractal analysis: a review. Med. Image Anal. 13, 634–649 (2009).

    Article  CAS  PubMed  Google Scholar 

  11. Dubuc, B., Quiniou, J. F., Roques-Carmes, C., Tricot, C. & Zucker, S. W. Evaluating the fractal dimension of profiles. Phys. Rev. A 39, 1500–1512 (1989).

    Article  CAS  Google Scholar 

  12. Jelinek, H. F. & Fernandez, E. Neurons and fractals: how reliable and useful are calculations of fractal dimensions? J. Neurosci. Methods 81, 9–18 (1998).

    Article  CAS  PubMed  Google Scholar 

  13. Karperien, A., Ahammer, H. & Jelinek, H. F. Quantitating the subtleties of microglial morphology with fractal analysis. Front. Cell. Neurosci. 7, 3 (2013).

    Article  PubMed  PubMed Central  Google Scholar 

  14. Nonnenmacher, T. F., Baumann, G., Barth, A. & Losa, G. A. Digital image analysis of self-similar cell profiles. Int. J. Biomed. Comput. 37, 131–138 (1994).

    Article  CAS  PubMed  Google Scholar 

  15. Smith, T. G. Jr, Lange, G. D. & Marks, W. B. Fractal methods and results in cellular morphology—dimensions, lacunarity and multifractals. J. Neurosci. Methods 69, 123–136 (1996).

    Article  PubMed  Google Scholar 

  16. Iannaccone, P. M. & Khokha, M. (eds) Fractal Geometry in Biological Systems: An Analytical Approach (CRC Press, 1996).

    Google Scholar 

  17. Peleg, S., Naor, J., Hartley, R. & Avnir, D. Multiple resolution texture analysis and classification. IEEE Trans. Pattern Anal. Mach. Intell. 6, 518–523 (1984).

    Article  CAS  PubMed  Google Scholar 

  18. Tolle, C. R., McJunkin, T. R. & Gorsich, D. J. An efficient implementation of the gliding box lacunarity algorithm. Physica D 237, 10 (2008).

    Article  Google Scholar 

  19. Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K. & Perlmutter, M. Lacunarity analysis: a general technique for the analysis of spatial patterns. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53, 5461–5468 (1996).

    CAS  PubMed  Google Scholar 

  20. Borys, P., Krasowska, M., Grzywna, Z. J., Djamgoz, M. B. & Mycielska, M. E. Lacunarity as a novel measure of cancer cells behavior. Biosystems 94, 276–281 (2008).

    Article  CAS  PubMed  Google Scholar 

  21. Weibel, E. R. What makes a good lung? Swiss Med. Wkly 139, 375–386 (2009).

    PubMed  Google Scholar 

  22. Iber, D. & Menshykau, D. The control of branching morphogenesis. Open Biol. 3, 130088 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  23. Kitaoka, H., Takaki, R. & Suki, B. A three-dimensional model of the human airway tree. J. Appl. Physiol. (1985) 87, 2207–2217 (1999).

    Article  CAS  Google Scholar 

  24. Glenny, R. W. Emergence of matched airway and vascular trees from fractal rules. J. Appl. Physiol. (1985) 110, 1119–1129 (2011).

    Article  Google Scholar 

  25. Fleury, V., Gouyet, J.-F. & Léonetti, M. (eds) Branching in Nature: Dynamics and Morphogenesis of Branching Structures, From Cell to River Networks (Springer-Verlag, 2001).

    Book  Google Scholar 

  26. West, B. J. Physiology in fractal dimensions: error tolerance. Ann. Biomed. Eng. 18, 135–149 (1990).

    Article  CAS  PubMed  Google Scholar 

  27. Nelson, T. R., West, B. J. & Goldberger, A. L. The fractal lung: universal and species-related scaling patterns. Experientia 46, 251–254 (1990).

    Article  CAS  PubMed  Google Scholar 

  28. Alencar, A. M. et al. Physiology: dynamic instabilities in the inflating lung. Nature 417, 809–811 (2002).

    Article  CAS  PubMed  Google Scholar 

  29. Suki, B. et al. Mechanical failure, stress redistribution, elastase activity and binding site availability on elastin during the progression of emphysema. Pulm. Pharmacol. Ther. 25, 268–275 (2012).

    Article  CAS  PubMed  Google Scholar 

  30. Bates, J. H. & Suki, B. Assessment of peripheral lung mechanics. Respir. Physiol. Neurobiol. 163, 54–63 (2008).

    Article  PubMed  PubMed Central  Google Scholar 

  31. Boser, S. R., Park, H., Perry, S. F., Ménache, M. G. & Green, F. H. Fractal geometry of airway remodeling in human asthma. Am. J. Respir. Crit. Care Med. 172, 817–823 (2005).

    Article  PubMed  Google Scholar 

  32. Gehr, P., Bachofen, M. & Weibel, E. R. The normal human lung: ultrastructure and morphometric estimation of diffusion capacity. Respir. Physiol. 32, 121–140 (1978).

    Article  CAS  PubMed  Google Scholar 

  33. Losa, G. A. The fractal geometry of life. Riv. Biol. 102, 29–59 (2009).

    PubMed  Google Scholar 

  34. Landini, G. & Rippin, J. W. Quantification of nuclear pleomorphism using an asymptotic fractal model. Anal. Quant. Cytol. Histol. 18, 167–176 (1996).

    CAS  PubMed  Google Scholar 

  35. Bancaud, A. et al. Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin. EMBO J. 28, 3785–3798 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  36. Lieberman-Aiden, E. et al. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science 326, 289–293 (2009).

    CAS  PubMed  PubMed Central  Google Scholar 

  37. Grosberg, A. Y. Nechaev, S. K. & Shakhnovich, E. I. The role of topological constraints in the kinetics of collapse of macromolecules. J. Phys. (France) 49, 2095–2100 (1988).

    Article  CAS  Google Scholar 

  38. Metze, K. Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis. Expert Rev. Mol. Diagn. 13, 719–735 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Peng, C. K. et al. Fractal landscape analysis of DNA walks. Physica A 191, 25–29 (1992).

    Article  CAS  PubMed  Google Scholar 

  40. Peng, C. K. et al. Long-range correlations in nucleotide sequences. Nature 356, 168–170 (1992).

    Article  CAS  PubMed  Google Scholar 

  41. Arakawa, K. et al. Genome Projector: zoomable genome map with multiple views. BMC Bioinformatics 10, 31 (2009).

    Article  PubMed  PubMed Central  Google Scholar 

  42. Jeffrey, H. J. Chaos game representation of gene structure. Nucleic Acids Res. 18, 2163–2170 (1990).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  43. Almeida, J. S. Sequence analysis by iterated maps, a review. Brief. Bioinform. 15, 369–375 (2014).

    Article  PubMed  Google Scholar 

  44. Tsai, I. J., Otto, T. D. & Berriman, M. Improving draft assemblies by iterative mapping and assembly of short reads to eliminate gaps. Genome Biol. 11, R41 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  45. Peng, C. K. et al. Quantifying fractal dynamics of human respiration: age and gender effects. Ann. Biomed. Eng. 30, 683–692 (2002).

    Article  CAS  PubMed  Google Scholar 

  46. West, B. J. Fractal physiology and the fractional calculus: a perspective. Front. Physiol. 1, 12 (2010).

    PubMed  PubMed Central  Google Scholar 

  47. Mutch, W. A., Graham, M. R., Girling, L. G. & Brewster, J. F. Fractal ventilation enhances respiratory sinus arrhythmia. Respir. Res. 6, 41 (2005).

    Article  PubMed  PubMed Central  Google Scholar 

  48. Gutierrez, G. et al. Decreased respiratory rate variability during mechanical ventilation is associated with increased mortality. Intensive Care Med. 39, 1359–1367 (2013).

    Article  PubMed  Google Scholar 

  49. Seely, A. J. et al. Do heart and respiratory rate variability improve prediction of extubation outcomes in critically ill patients? Crit. Care 18, R65 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  50. Lee, L. H. et al. Digital differentiation of non-small cell carcinomas of the lung by the fractal dimension of their epithelial architecture. Micron. 67, 125–131 (2014).

    Article  PubMed  Google Scholar 

  51. Vasiljevic, J. et al. Application of multifractal analysis on microscopic images in the classification of metastatic bone disease. Biomed. Microdevices 14, 541–548 (2012).

    Article  PubMed  Google Scholar 

  52. US National Institutes of Health. ImageJ [online], (2015).

  53. Karperien, A. FracLac for ImageJ. US National Institutes of Health [online], (2013).

    Google Scholar 

  54. Fudenberg, G., Getz, G., Meyerson, M. & Mirny, L. A. High order chromatin architecture shapes the landscape of chromosomal alterations in cancer. Nat. Biotechnol. 29, 1109–1113 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  55. Misteli, T. Higher-order genome organization in human disease. Cold Spring Harb. Perspect. Biol. 2, a000794 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  56. Irinopoulou, T., Rigaut, J. P. & Benson, M. C. Toward objective prognostic grading of prostatic carcinoma using image analysis. Anal. Quant. Cytol. Histol. 15, 341–344 (1993).

    CAS  PubMed  Google Scholar 

  57. Streba, C. T. et al. Fractal analysis differentiation of nuclear and vascular patterns in hepatocellular carcinomas and hepatic metastasis. Rom. J. Morphol. Embryol. 52, 845–854 (2011).

    CAS  PubMed  Google Scholar 

  58. Shtivelman, E. et al. Molecular pathways and therapeutic targets in lung cancer. Oncotarget 5, 1392–1433 (2014).

    PubMed  PubMed Central  Google Scholar 

  59. Hayano, K., Yoshida, H., Zhu, A. X. & Sahani, D. V. Fractal analysis of contrast-enhanced CT images to predict survival of patients with hepatocellular carcinoma treated with sunitinib. Dig. Dis. Sci. 59, 1996–2003 (2014).

    Article  CAS  PubMed  Google Scholar 

  60. Kido, S., Kuriyama, K., Higashiyama, M., Kasugai, T. & Kuroda, C. Fractal analysis of internal and peripheral textures of small peripheral bronchogenic carcinomas in thin-section computed tomography: comparison of bronchioloalveolar cell carcinomas with nonbronchioloalveolar cell carcinomas. J. Comput. Assist. Tomogr. 27, 56–61 (2003).

    Article  PubMed  Google Scholar 

  61. Michallek, F. & Dewey, M. Fractal analysis in radiological and nuclear medicine perfusion imaging: a systematic review. Eur. Radiol. 24, 60–69 (2014).

    Article  PubMed  Google Scholar 

  62. Miwa, K. et al. FDG uptake heterogeneity evaluated by fractal analysis improves the differential diagnosis of pulmonary nodules. Eur. J. Radiol. 83, 715–719 (2014).

    Article  PubMed  Google Scholar 

  63. Al-Kadi, O. S. Assessment of texture measures susceptibility to noise in conventional and contrast enhanced computed tomography lung tumour images. Comput. Med. Imaging Graph. 34, 494–503 (2010).

    Article  PubMed  Google Scholar 

  64. Dimitrakopoulou-Strauss, A. et al. Prediction of short-term survival in patients with advanced nonsmall cell lung cancer following chemotherapy based on 2-deoxy-2-[F-18]fluoro-D-glucose-positron emission tomography: a feasibility study. Mol. Imaging Biol. 9, 308–317 (2007).

    Article  PubMed  Google Scholar 

  65. Dimitrakopoulou-Strauss, A., Pan, L. & Strauss, L. G. Quantitative approaches of dynamic FDG-PET and PET/CT studies (dPET/CT) for the evaluation of oncological patients. Cancer Imaging 12, 283–289 (2012).

    Article  PubMed  PubMed Central  Google Scholar 

  66. Al-Kadi, O. S. & Watson, D. Texture analysis of aggressive and nonaggressive lung tumor CE CT images. IEEE Trans. Biomed. Eng. 55, 1822–1830 (2008).

    Article  PubMed  Google Scholar 

  67. Hayano, K., Lee, S. H., Yoshida, H., Zhu, A. X. & Sahani, D. V. Fractal analysis of CT perfusion images for evaluation of antiangiogenic treatment and survival in hepatocellular carcinoma. Acad. Radiol. 21, 654–660 (2014).

    Article  PubMed  Google Scholar 

  68. Doubal, F. N. et al. Fractal analysis of retinal vessels suggests that a distinct vasculopathy causes lacunar stroke. Neurology 74, 1102–1107 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  69. Lee, J., Zee, B. C. & Li, Q. Detection of neovascularization based on fractal and texture analysis with interaction effects in diabetic retinopathy. PLoS ONE 8, e75699 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  70. Talu, S. Fractal analysis of normal retinal vascular network. Oftalmologia 55, 11–16 (2011).

    PubMed  Google Scholar 

  71. Di Ieva, A. et al. Computer-assisted and fractal-based morphometric assessment of microvascularity in histological specimens of gliomas. Sci. Rep. 2, 429 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Di Ieva, A. et al. Fractal dimension as a quantitator of the microvasculature of normal and adenomatous pituitary tissue. J. Anat. 211, 673–680 (2007).

    Article  PubMed  PubMed Central  Google Scholar 

  73. Di Ieva, A. et al. Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue. Neurosurg. Rev. 31, 271–281 (2008).

    Article  PubMed  Google Scholar 

  74. Di Ieva, A., Grizzi, F., Sherif, C., Matula, C. & Tschabitscher, M. Angioarchitectural heterogeneity in human glioblastoma multiforme: a fractal-based histopathological assessment. Microvasc. Res. 81, 222–230 (2011).

    Article  PubMed  Google Scholar 

  75. Goutzanis, L. P. et al. Vascular fractal dimension and total vascular area in the study of oral cancer. Head Neck 31, 298–307 (2009).

    Article  PubMed  Google Scholar 

  76. Al-Kadi, O. S. A multiresolution clinical decision support system based on fractal model design for classification of histological brain tumours. Comput. Med. Imaging Graph. 41, 67–79 (2014).

    Article  PubMed  Google Scholar 

  77. Ferro, D. P. et al. Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma. PLoS ONE 6, e20706 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  78. Pasqualato, A. et al. Shape in migration: quantitative image analysis of migrating chemoresistant HCT-8 colon cancer cells. Cell Adh. Migr. 7, 450–459 (2013).

    Article  PubMed  PubMed Central  Google Scholar 

  79. Pantic, I., Harhaji-Trajkovic, L., Pantovic, A., Milosevic, N. T. & Trajkovic, V. Changes in fractal dimension and lacunarity as early markers of UV-induced apoptosis. J. Theor. Biol. 303, 87–92 (2012).

    Article  CAS  PubMed  Google Scholar 

  80. Fuseler, J. W., Millette, C. F., Davis, J. M. & Carver, W. Fractal and image analysis of morphological changes in the actin cytoskeleton of neonatal cardiac fibroblasts in response to mechanical stretch. Microsc. Microanal. 13, 133–143 (2007).

    Article  CAS  PubMed  Google Scholar 

  81. Park, S. H. et al. Texture analyses show synergetic effects of biomechanical and biochemical stimulation on mesenchymal stem cell differentiation into early phase osteoblasts. Microsc. Microanal. 20, 219–227 (2014).

    Article  CAS  PubMed  Google Scholar 

  82. Qian, A. R. et al. Fractal dimension as a measure of altered actin cytoskeleton in MC3T3-E1 cells under simulated microgravity using 3-D/2-D clinostats. IEEE Trans. Biomed. Eng. 59, 1374–1380 (2012).

    Article  CAS  PubMed  Google Scholar 

  83. Qi, Y. X., Wang, X. D., Zhang, P. & Jiang, Z. L. Fractal and Image Analysis of Cytoskeletal F-Actin Orgnization in Endothelial Cells under Shear Stress and Rho-GDIα Knock Down in 6th World Congress of Biomechanics (WCB 2010): In Conjunction with 14th International Conference on Biomedical Engineering (ICBME) and 5th Asia Pacific Conference on Biomechanics (APBiomech). IFMBE Proceedings Vol. 31 (eds Lim, C. T. & Goh, J. C.) 1051–1054 (Springer, 2010).

    Google Scholar 

  84. Di Ieva, A. Fractal analysis of microvascular networks in malignant brain tumors. Clin. Neuropathol. 31, 342–351 (2012).

    Article  PubMed  Google Scholar 

  85. Brodatz, P. Textures: A Photographic Album for Artists and Designers (Peter Smith Publisher, Incorporated, 1981).

    Google Scholar 

  86. Florindo, J. B., Landini, G. & Bruno, O. M. Texture descriptors by a fractal analysis of three-dimensional local coarseness. Digit. Signal Process. 42, 70–79 (2015).

    Article  Google Scholar 

  87. Jimenez, J. et al. A Web platform for the interactive visualization and analysis of the 3D fractal dimension of MRI data. J. Biomed. Inform. 51, 176–190 (2014).

    Article  CAS  PubMed  Google Scholar 

  88. Vakoc, B. J. et al. Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging. Nat. Med. 15, 1219–1223 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  89. Véhel, J. L. & Legrand, P. Signal and image processing with FracLab in Thinking in Patterns: Fractals and Related Phenomena in Nature (ed. Novak. M. M.). 321–322 (World Scientific, 2004).

    Book  Google Scholar 

  90. ThéMA. Fractalyse—Fractal Analysis Software [online], (2015).

  91. Silijkerman, F. Ultra fractal 5 [online], (2014).

  92. Reuter, M. Image Analysis: Fractal Dimension—FDim [online], (2015).

    Google Scholar 

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Acknowledgements

The work of the authors is supported in part by the NIH National Cancer Institute (grant P30 CA014599 to the University of Chicago Cancer Research Foundation). The work of R.S. is supported by the Mesothelioma Applied Research Foundation, the Guy Geleerd Memorial Golf Invitational–V-Foundation for Cancer Research.

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Authors and Affiliations

  1. Section of Hematology/Oncology, University of Chicago, 5841 South Maryland Avenue, MC 2115, Chicago, 60637, IL, USA

    Frances E. Lennon, Everett E. Vokes & Ravi Salgia

  2. Department of Pathology, University of Chicago, 5841 South Maryland Avenue, MC 2115, Chicago, 60637, IL, USA

    Nicole A. Cipriani

  3. Department of Radiology, University of Chicago, 5841 South Maryland Avenue, MC 2115, Chicago, 60637, IL, USA

    Hannah J. Zhang, Chin-Tu Chen & Michael W. Vannier

  4. Department of Medicine, University of Chicago, 5841 South Maryland Avenue, MC 2115, Chicago, 60637, IL, USA

    Septimiu D. Murgu

  5. Department of Cell and Molecular Biology, Feinberg School of Medicine, Northwestern University, 303 East Chicago Avenue, Chicago, 60611, IL, USA

    Gianguido C. Cianci

  6. NorthShore University Health System, 2650 Ridge Avenue, Evanston, 60201, IL, USA

    Thomas A. Hensing

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F.E.L., G.C.C., N.A.C., T.A.H., M.W.V. and R.S. wrote the article and contributed to all stages of the preparation of the manuscript for submission. In addition, H.J.Z. and C.-T.C. contributed to researching data for the article, and S.D.M. and E.E.V. made substantial contributions to discussion of content. All authors reviewed/edited the manuscript before submission.

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Supplementary information

Sierpinski gasket chaos game.

This video demonstrates the Sierpinski triangle chaos game. The game unfolds as follows: the corners of a triangle are labelled with the numbers 1–6, assigning two numbers to each corner. Starting at a random point plotted within the triangle, a die is cast and a new point is plotted halfway from the starting point towards the corner labelled with the number rolled; this process is repeated for multiple iterations, plotting a new point for each roll of the die. After sufficient iterations, the ensemble of points visited during the game forms a fractal image, in this case the Sierpinski gasket, which is shown in the final frame. (MOV 692 kb)

Self-similarity across multiple scales of the fractal image generated by chaos game representation of the chromosome 2 DNA sequence.

The fractal pattern of a chaos game was generated using the DNA sequence of chromosome 2, as shown in Figure 4b. In this chaos game, each corner of a square was assigned a DNA base (either A, C, G, or T), and starting at point in the centre of the square, a new point was plotted at half the distance from this point towards the corner corresponding to the first base in the DNA sequence of chromosome 2; the next point was plotted halfway between this position and the corner corresponding to the second base. This process was repeated for each base in the DNA sequence. In this video we zoom in on the resulting fractal image, to illustrate a defining feature of a fractal: self-similarity over a range of scales. (MOV 12433 kb)

Supplementary Figure 1

Method of calculating the fractal dimension of histological samples. (DOC 2588 kb)

Supplementary Figure 2

Method of calculating the fractal dimension of lung tumour–stroma interface on clinical contrast-enhanced CT images. (DOC 287 kb)

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Lennon, F., Cianci, G., Cipriani, N. et al. Lung cancer—a fractal viewpoint. Nat Rev Clin Oncol 12, 664–675 (2015). https://doi.org/10.1038/nrclinonc.2015.108

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