# Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Mathematical Methods

#### 2.2.1. Fractals and Fractal Dimensions

_{i}(r

_{i}) × (r

_{i})

^{Df}= N

_{j}× (r

_{j})

^{Df}= Constant, where N

_{i}is the number of boxes at length scale r

_{i}, and N

_{j}is the number of boxes at length scale r

_{j.}The average fractal dimension D

_{f}is then calculated by refining the grid through a box-counting algorithm. The equation used in this algorithm can be derived from the above definition as an average slope of ln(N(r)) vs. r curve, with varying length scales r.

_{f}is the ensemble averaged fractal dimension of the structure since the box-counting method accounts for several different realizations.

#### 2.2.2. Calculating Fractal Dimension using Microscopic Images

#### 2.3. Physical Set-Up

#### 2.3.1. TMA Samples

#### 2.3.2. Microscope Setup and Imaging

_{t}of a biological tissue sitting on a glass slide of an incident intensity I

_{0}can be expressed as the transmission through a thick sample, and it follows the following equations:

_{tissue+glass}is the effective refractive index of the tissue over the glass system, n

_{air}is the refractive index of the air, K is the effective absorption coefficient of the tissue over the glass, and d is the thickness of the slide with the tissue sample. For a TMA tissue sample, I

_{0}, K, and d are constants (for dried tissue, the absorption is quite negligible), and there is a linear relationship between transmission intensity and change in the refractive index (n

_{tissue+glass −}n

_{air}).

_{t}α (n

_{tissue+glass}− n

_{air}) α change in tissue mass density

#### 2.4. Analysis of Images

#### 2.5. Statistical Analysis

## 3. Results

#### 3.1. Pancreatic Cancer

#### 3.2. Breast Cancer

#### 3.3. Colon Cancer

#### 3.4. Prostate Cancer

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Theiler, J. Estimating fractal dimension. JOSA A
**1990**, 7, 1055–1073. [Google Scholar] [CrossRef] [Green Version] - Fernández-Martínez, M.; Sánchez-Granero, M.A. Fractal dimension for fractal structures. Topol. Its Appl.
**2014**, 163, 93–111. [Google Scholar] [CrossRef] - Losa, G.A.; Merlini, D.; Nonnenmacher, T.F.; Weibel, E.R. Fractals in Biology and Medicine: III; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1994; Volume 3. [Google Scholar]
- Fielding, A. Applications of fractal geometry to biology. Bioinformatics
**1992**, 8, 359–366. [Google Scholar] [CrossRef] [PubMed] - Losa, G.A.; Nonnenmacher, T.F. Self-similarity and fractal irregularity in pathologic tissues. Mod. Pathol.
**1996**, 9, 174–182. [Google Scholar] [PubMed] - Davies, N.A.; Harrison, N.K.; Morris, R.H.K.; Noble, S.; Lawrence, M.J.; D’Silva, L.A.; Evans, P.A. Fractal dimension (df) as a new structural biomarker of clot microstructure in different stages of lung cancer. Thromb. Haemost.
**2015**, 114, 1251–1259. [Google Scholar] [CrossRef] [Green Version] - Etehad Tavakol, M.; Lucas, C.; Sadri, S.; Ng, E.Y.K. Analysis of breast thermography using fractal dimension to establish possible difference between malignant and benign patterns. J. Healthc. Eng.
**2010**, 1, 27–43. [Google Scholar] [CrossRef] - Huang, P.W.; Lee, C.H. Automatic classification for pathological prostate images based on fractal analysis. IEEE Trans. Med. Imaging
**2009**, 28, 1037–1050. [Google Scholar] [CrossRef] [PubMed] - Chan, A.; Tuszynski, J.A. Automatic prediction of tumor malignancy in breast cancer with fractal dimension. R. Soc. Open Sci.
**2016**, 3, 160558. [Google Scholar] [CrossRef] [Green Version] - Uthayakumar, R.; Jayalalitha, G. Border detection of skin cancer cells with fractal dimension. Fractals
**2009**, 17, 171–180. [Google Scholar] [CrossRef] - Bhandari, S.; Choudannavar, S.; Avery, E.R.; Sahay, P.; Pradhan, P. Detection of colon cancer stages via fractal dimension analysis of optical transmission imaging of tissue microarrays (TMA). Biomed. Phys. Eng. Exp.
**2018**, 4, 065020. [Google Scholar] [CrossRef] - Adhikari, P.; Binu, A.P.; Bhandari, S.; Khan, S.; Jaggi, M.; Chauhan, S.C. Optical Detection of Fractal Dimensions of MUC13 Stained Pancreatic Tissues for Cancer Diagnostics. arXiv
**2018**, arXiv:181210883. [Google Scholar] - Wulfkuhle, J.D.; Liotta, L.A.; Petricoin, E.F. Proteomic applications for the early detection of cancer. Nat. Rev. Cancer
**2003**, 3, 267–275. [Google Scholar] [CrossRef] [PubMed] - Loeb, L.A.; Springgate, C.F.; Battula, N. Errors in DNA Replication as a Basis of Malignant Changes. Cancer Res.
**1974**, 34, 2311–2321. [Google Scholar] - Lingen, M.W.; Kalmar, J.R.; Karrison, T.; Speight, P.M. Critical evaluation of diagnostic aids for the detection of oral cancer. Oral Oncol.
**2007**, 44, 10–22. [Google Scholar] [CrossRef] [Green Version] - Shergill, M.I.S.; Shergill, N.K.; Arya, M.; Patel, H.R.H. Tissue microarrays: A current medical research tool. Curr. Med. Res. Opin.
**2004**, 20, 707–712. [Google Scholar] [CrossRef] - Jawhar Nazar, M.T. Tissue Microarray: A rapidly evolving diagnostic and research tool. Ann. Saudi Med.
**2009**, 29, 123–127. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cancer Statistics. 2019–Siegel–2019–CA: A Cancer Journal for Clinicians–Wiley Online Library [Internet]. Available online: https://onlinelibrary.wiley.com/doi/full/10.3322/caac.21551 (accessed on 30 October 2019).
- MacMahon, B. Epidemiology and the causes of breast cancer. Int. J. Cancer
**2006**, 118, 2373–2378. [Google Scholar] [CrossRef] [PubMed] - Giovannucci, E. Epidemiologic Characteristics of Prostate Cancer. Cancer
**1995**, 75, 1766–1777. [Google Scholar] [CrossRef] - Giovannucci, E. Willett WC. Dietary Factors and Risk of Colon Cancer. Ann. Med.
**1994**, 26, 443–452. [Google Scholar] [CrossRef] - Bunde, A.; Havlin, S. Fractals in Science; Springer: Berlin/Heidelberg, Germany, 2013; 317p. [Google Scholar]
- Li, J.; Du, Q.; Sun, C. An improved box-counting method for image fractal dimension estimation. Pattern Recognit.
**2009**, 42, 2460–2469. [Google Scholar] [CrossRef] - Panigrahy, C.; Seal, A.; Mahato, N.K. Quantitative texture measurement of gray-scale images: Fractal dimension using an improved differential box counting method. Measurement
**2019**, 147, 106859. [Google Scholar] [CrossRef] - de Arruda, P.F.F.; Gatti, M.; Junior, F.N.F.; de Arruda, J.G.F.; Moreira, R.D.; Murta, L.O. Quantification of fractal dimension and Shannon’s entropy in histological diagnosis of prostate cancer. BMC Clin. Pathol.
**2013**, 13, 6. [Google Scholar] [CrossRef] - Davies, H.G.; Wilkins, M.H.F. Interference Microscopy and Mass Determination. Nature
**1952**, 169, 541. [Google Scholar] [CrossRef] - Image-j. Available online: https://imagej.nih.gov/ij/ (accessed on 1 November 2021).
- Preston-Martin, S.; Pike, M.C.; Ross, R.K.; Henderson, B.E. Epidemiologic evidence for the increased cell proliferation model of carcinogenesis. Environ. Health Perspect
**1993**, 101 (Suppl. 5), 137–138. [Google Scholar] [PubMed] [Green Version] - Sahai, E. Mechanisms of cancer cell invasion. Curr. Opin. Genet. Dev.
**2005**, 15, 87–96. [Google Scholar] [CrossRef] [PubMed] - Scampicchio, A.; Tura, A.; Sbrignadello, S.; Grizzi, F.; Fiorino, S.; Blandamura, S. Assessment of the Fractal Dimension of Images Derived by Biopsy of Pancreatic Tissue: Implications for Tumor Diagnosis. In Proceedings of the XIV Mediterranean Conference on Medical and Biological Engineering and Computing 2016, Paphos, Cyprus, 31 March–2 April 2016; Kyriacou, E., Christofides, S., Pattichis, C.S., Eds.; Springer International Publishing: Berlin/Heidelberg, Germany, 2016; pp. 393–398. [Google Scholar]
- Losa, G.A.; Castelli, C. Nuclear patterns of human breast cancer cells during apoptosis: Characterisation by fractal dimension and co-occurrence matrix statistics. Cell Tissue Res.
**2005**, 322, 257–267. [Google Scholar] [CrossRef] - Krasowska, M.; Grzywna, Z.J.; Mycielska, M.E.; Djamgoz, M.B.A. Patterning of endocytic vesicles and its control by voltage-gated Na+ channel activity in rat prostate cancer cells: Fractal analyses. Eur. Biophys. J.
**2004**, 33, 535–542. [Google Scholar] [CrossRef] [PubMed] - Park, Y.; Depeursinge, C.; Popescu, G. Quantitative phase imaging in biomedicine. Nat. Photonics
**2018**, 12, 578–589. [Google Scholar] [CrossRef] - Kastl, L.; Isbach, M.; Dirksen, D.; Schnekenburger, J.; Kemper, B. Quantitative phase imaging for cell culture quality control. Cytom. Part A
**2017**, 91, 470–481. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Majeed, H.; Sridharan, S.; Mir, M.; Ma, L.; Min, E.; Jung, W.; Popescu, G. Quantitative phase imaging for medical diagnosis. J. Biophotonics
**2017**, 10, 177–205. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Popescu, G.; Tangella, K.V.; Balla, A. Tissue refractive index as marker of disease. J. Biomed. Opt.
**2011**, 16, 116017. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The step-by-step process for calculating the fractal dimension of a microscopic image is outlined.

**Figure 2.**Pancreatic cancer: (

**a**,

**b**) are the brightfield images of the normal and stage III pancreatic TMA. (

**a’**,

**b’**) are the corresponding binary images. (

**c**) is the bar graph of the average fractal dimension of the pancreatic tissue samples (n = 24, 3–5 subjects per stage). The results show the fractal dimension of cancer stage I increases by 4%, stage II by 6%, and stage III by 9% with respect to the normal (p-values < 0.05).

**Figure 3.**Breast cancer: (

**a**,

**b**) are the brightfield images of the normal and stage III Breast TMA. (

**a’**,

**b’**) are the corresponding binary images. (

**c**) is the bar graph of the average fractal dimension of the Breast cancer tissue microarrays (TMA) samples (n = 24, 3–5 subjects per stage). The results show the fractal dimension of cancer stage I increases by 4%, stage II by 7%, and stage III by 12% with respect to the normal (p-values < 0.05).

**Figure 4.**Colon cancer: (

**a**,

**b**) are the brightfield images of the normal and stage III Colon TMA. (

**a’**,

**b’**) are the corresponding binary images. (

**c**) is the bar graph of the average fractal dimension of the Colon cancer tissue microarrays (TMA) samples (n = 24, 3–5 subjects per stage). The results show the fractal dimension of cancer stage I increases by 5%, stage II by 7%, and stage III by 9% with respect to the normal (p-values < 0.05).

**Figure 5.**Prostate cancer: (

**a**,

**b**) are the brightfield images of the normal and stage III Prostate TMA. (

**a’**,

**b’**) are the corresponding binary images. (

**c**) is the bar graph of the average fractal dimension of the prostate cancer tissue microarrays (TMA) samples (n = 24, 3–5 subjects per stage). The results show the fractal dimension of cancer stage I increases by 2%, stage II increases by 4%, and stage III by 7% with respect to the normal (p-values < 0.05).

**Figure 6.**The plot of the systematic increases in the fractal dimension with the progress of the cancer stages. The x-axis steps are false steps that are measured by an equal distance unit of 1 for increasing stages, for better presentation. It can be seen that the fractal dimension on average increases with the stage; however, they take slightly different paths. Note: only mean values are shown, standard errors are the same as shown in bar graph plots.

Normal | Stage I | Stage II | Stage III | |
---|---|---|---|---|

Pancreatic | 1.5984 | 1.6673 | 1.6866 | 1.7047 |

Breast | 1.5448 | 1.6126 | 1.6631 | 1.7283 |

Colon | 1.5551 | 1.6393 | 1.6652 | 1.7004 |

Prostate | 1.5737 | 1.5981 | 1.6302 | 1.6798 |

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**MDPI and ACS Style**

Elkington, L.; Adhikari, P.; Pradhan, P.
Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy. *Biophysica* **2022**, *2*, 59-69.
https://doi.org/10.3390/biophysica2010005

**AMA Style**

Elkington L, Adhikari P, Pradhan P.
Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy. *Biophysica*. 2022; 2(1):59-69.
https://doi.org/10.3390/biophysica2010005

**Chicago/Turabian Style**

Elkington, Liam, Prakash Adhikari, and Prabhakar Pradhan.
2022. "Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy" *Biophysica* 2, no. 1: 59-69.
https://doi.org/10.3390/biophysica2010005