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  • Review Article
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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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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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