## Differing self-similarity in light scattering spectra: a potential tool for pre-cancer detection |

Optics Express, Vol. 19, Issue 20, pp. 19717-19730 (2011)

http://dx.doi.org/10.1364/OE.19.019717

Acrobat PDF (1318 KB)

### Abstract

The fluctuations in the elastic light scattering spectra of normal and dysplastic human cervical tissues analyzed through wavelet transform based techniques reveal clear signatures of self-similar behavior in the spectral fluctuations. The values of the scaling exponent observed for these tissues indicate the differences in the self-similarity for dysplastic tissues and their normal counterparts. The strong dependence of the elastic light scattering on the size distribution of the scatterers manifests in the angular variation of the scaling exponent. Interestingly, the spectral fluctuations in both these tissues showed multi-fractality (non-stationarity in fluctuations), the degree of multi-fractality being marginally higher in the case of dysplastic tissues. These findings using the multi-resolution analysis capability of the discrete wavelet transform can contribute to the recent surge in the exploration for non-invasive optical tools for pre-cancer detection.

© 2011 OSA

## 1. Introduction

1. N. Ramanujam, “Fluorescence spectroscopy of neoplastic and non-neoplastic tissues,” Neoplasia **2**, 89–117 (2000). [CrossRef] [PubMed]

6. N. Ghosh, S. K. Majumder, and P. K. Gupta, “Polarized fluorescence spectroscopy of human tissues,” Opt. Lett. **27**, 2007–2009 (2002). [CrossRef]

7. A. S. Haka, K. E. Shafer-Peltier, M. Fitzmaurice, J. Crowe, R. R. Dasari, and M. S. Feld, “Diagnosing breast cancer by using Raman spectroscopy,” Proc. Natl. Acad. Sci. (USA) **102**, 12371–12376 (2005). [CrossRef]

8. N. N. Boustany, S. A. Boppart, and V. Backman, “Microscopic imaging and spectroscopy with scattered light,” Annu. Rev. Biomed. Eng. **12**, 285–314 (2010). [CrossRef] [PubMed]

9. J. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. **21**, 1361–1367 (2003). [CrossRef] [PubMed]

13. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. **7**, 329–340 (2002). [CrossRef] [PubMed]

8. N. N. Boustany, S. A. Boppart, and V. Backman, “Microscopic imaging and spectroscopy with scattered light,” Annu. Rev. Biomed. Eng. **12**, 285–314 (2010). [CrossRef] [PubMed]

14. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods **4**, 717–719 (2007). [CrossRef] [PubMed]

27. C.-C. Yu, C. Lau, G. O’Donoghue, J. Mirkovic, S. McGee, L. Galindo, A. Elackattu, E. Stier, G. Grillone, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Quantitative spectroscopic imaging for non-invasive early cancer detection,” Opt. Express **16**, 16227–16239 (2008). [CrossRef] [PubMed]

16. M. Kalashnikov, W. Choi, C.-C. Yu, Y. Sung, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Assessing light scattering of intracellular organelles in single intact living cells,” Opt. Express **17**, 19674–19681 (2009). [CrossRef] [PubMed]

20. L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. **80**, 627–630 (1998). [CrossRef]

23. N. Ghosh, P. Buddhiwant, A. Uppal, S. K. Majumder, H. S. Patel, and P. K. Gupta, “Simultaneous determination of size and refractive index of red blood cells by light scattering measurements,” Appl. Phys. Lett. **88**, 084101 (2006). [CrossRef]

24. N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. **40**, 176–184 (2001). [CrossRef]

8. N. N. Boustany, S. A. Boppart, and V. Backman, “Microscopic imaging and spectroscopy with scattered light,” Annu. Rev. Biomed. Eng. **12**, 285–314 (2010). [CrossRef] [PubMed]

28. İ. R. Çapoğlu, J. D. Rogers, A. Taflove, and V. Backman, “Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media,” Opt. Lett. **34**, 2679–2681 (2009). [CrossRef] [PubMed]

32. T. T. Wu, J. Y. Qu, and M. Xu, “Unified Mie and fractal scattering by biological cells and subcellular structures,” Opt. Lett. **32**, 2324–2326 (2007). [CrossRef] [PubMed]

18. R. Graf and A. Wax, “Nuclear morphology measurements using Fourier domain low coherence interferometry,” Opt. Express **13**, 4693–4698 (2005). [CrossRef]

29. M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. **97**, 138102 (2006). [CrossRef] [PubMed]

33. W. Gao, “Square law between spatial frequency of spatial correlation function of scattering potential of tissue and spectrum of scattered light,” J. Biomed. Opt. **15**, 030502 (2010). [CrossRef] [PubMed]

35. J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. **21**, 1310–1312 (1996). [CrossRef] [PubMed]

36. L. Perelman, “Optical diagnostic technology based on light scattering spectroscopy for early cancer detection,” Expert Rev. Med. Devices **3**, 787–803 (2006). [CrossRef]

39. J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, “Multifractal detrended fluctuation analysis of nonstationary time series,” Physica A **316**, 87–114 (2002). [CrossRef]

40. P. Manimaran, P. K. Panigrahi, and J. C. Parikh, “Wavelet analysis and scaling properties of time series,” Phys. Rev. E **72**, 046120 (2005). [CrossRef]

41. P. Manimaran, P. Panigrahi, and J. Parikh, “Multiresolution analysis of fluctuations in non-stationary time series through discrete wavelets,” Physica A **388**, 2306–2314 (2009). [CrossRef]

42. S. Gupta, M. Nair, A. Pradhan, N. Biswal, N. Agarwal, A. Agarwal, and P. Panigrahi, “Wavelet-based characterization of spectral fluctuations in normal, benign, and cancerous human breast tissues,” J. Biomed. Opt. **10**, 054012 (2005). [CrossRef] [PubMed]

46. A. H. Gharekhan, S. Arora, A. N. Oza, M. B. Sureshkumar, A. Pradhan, and P. K. Panigrahi, “Distinguishing autofluorescence of normal, benign, and cancerous breast tissues through wavelet domain correlation studies,” J. Biomed. Opt. **16**, 087003 (2011). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Fourier analysis and power law spectrum

### 2.2. A brief review of wavelet transforms

47. I. Daubechies, *Ten Lectures on Wavelets*, 1st ed., CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM: Society for Industrial and Applied Mathematics, 1992). [CrossRef]

50. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” *IEEE Trans. Pattern Anal. Mach. Intell.*11, 674–693 (1989). [CrossRef]

*ϕ*(

*n*) and

*ψ*(

*n*), called the father and the mother wavelets respectively, form the kernels. They satisfy the admissibility conditions: ∫

*ϕ*(

*n*)

*dn*< ∞, ∫

*ψ*(

*n*)

*dn*= 0, ∫

*ϕ*

^{*}(

*n*)

*ψ*(

*n*)

*dn*= 0, ∫ |

*ϕ*(

*n*)|

^{2}

*dn*= ∫ |

*ψ*(

*n*)|

^{2}

*dn*= 1. The scaled and translated versions of the mother wavelet

*ψ*(

*n*) are called the daughter wavelets which form a complete set and differ from the former in terms of their height and width. At the

*s*scale, the height and width of the daughter wavelet are 2

^{th}*and 2*

^{s}^{s/2}of that of the mother wavelet respectively.

*s*and

*m*are the scaling and translation parameters.

*f*(

*t*) is given by, the coefficient

*a*(called the approximation coefficient) extracts the trend and

_{m}*d*

_{s,m}(called the detail coefficient) extracts the details or fluctuations from the signal. Here,

*l*= ⌊log(

*N*)/log(2)⌋ is the upper bound for taking the maximum scale for analysis above which the edge effects corrupt the wavelet coefficients giving rise to spurious results. This mathematical artifact is explained by the cone of influence [49

49. C. Torrence and G. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. **79**, 61–78 (1998). [CrossRef]

### 2.3. Wavelet Fluctuation Power Law Analysis

47. I. Daubechies, *Ten Lectures on Wavelets*, 1st ed., CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM: Society for Industrial and Applied Mathematics, 1992). [CrossRef]

*Db*– 4 isolates a monomial trend while

*Db*– 6 isolates a quadratic trend. We first obtain the profile from the fluctuations by taking their cumulative sum:

*Y*and

*X*are profile and fluctuation signal respectively and

*N*is the data length. In this method, we first obtain the fluctuations at every scale by a wavelet reconstruction taking the approximation coefficients (using Db-4 wavelet) and subsequent subtraction from the signal. A flowchart for this fluctuation extraction is shown in Fig. 1. We then fit the Fourier power spectrum of these fluctuations to obtain the power law exponent as a function of scale, i.e.

*α*≡

*α*(

*s*) for the identification of the short and long term correlations. This technique sheds light on the scaling behavior of the fluctuations in different spatial frequency regimes.

### 2.4. Wavelet Based Multi Fractal De-trended Fluctuation Analysis

### 2.5. Correlation based analysis

## 3. Experimental materials and methods

*μm*, lateral dimension ∼ 4

*mm*×6

*mm*) were prepared on glass slides. The standard method employed for the preparation of the sections is tissue dehydration, embedding in wax and subsequent sectioning under a rotary microtome [55]. The prepared slides were then examined under the microscope by the pathologists for histopathological characterization. The normal counterparts were obtained from the adjacent normal area of the resected tissues. The tissue samples were obtained from G.S.V.M. Medical College and Hospital, Kanpur, India. A total number of fifteen tissue samples were collected from patients in the age group 35 – 60 years. Out of these, eight samples were included in the spectroscopic study. Among these, four were histopathologically characterized as dysplastic and the other four samples were pathologically normal. All the cervical tissues included in this study were squamous type.

*θ*: 10° – 150° at 10° intervals. White light output from a Xe-lamp (Newport USA, 50 – 5000 W) was collimated using a combination of lenses and were made incident on the sample kept at the center of the goniometer. The spot-size incident on the sample was controlled by a variable aperture (∼ 1

*mm*). The scattered light from the sample was collimated by a pair of lenses and was then focused into a collecting fiber probe, the distal end of which was coupled to a spectrometer (Ocean Optics HR2000).

*Db*– 4 wavelets and the minimum window size (corresponding to level 1) was 8, which in term of wavelength was ∼ 3.5

*nm*. The spectroscopic data for the wavelength range 400 −800 nm was used in fluctuation analysis.

*θ*= 10°) in our angle-resolved elastic scattering spectroscopic studies, was based on the range of scattering vector (

*λ*= 400 nm,

_{min}*λ*= 800 nm, Δ

_{max}*λ*= 400 nm;

*θ*= 10°,

_{min}*θ*= 150°; resulting in Δ

_{max}*q*∼ 29

*μm*

^{−1}). The idea was to cover the entire range of variation of Δ

*q*by varying scattering angle

*θ*. For the smallest scattering angle used in our study (

*θ*= 10°), the required angular interval Δ

_{min}*θ*was found to be ∼ 10°. Note that at larger scattering angles, one may not need to acquire spectra at angular interval of 10°, to capture the range of variation Δ

*q*. However, in order to be consistent, the angular interval of Δ

*θ*= 10° was used for the range of scattering angles studied. The scattering spectra recorded from the tissue samples at each scattering angles were normalized by the lamp spectra recorded from the glass slides alone. Since for fluctuation analysis spectral shape rather than the absolute intensity information is important, the angle-resolved scattering spectra were not normalized for absolute scattered intensities. Thus normalized angle-resolved scattering spectra were then subjected to the fluctuation analysis.

## 4. Results and discussions

*θ*= 10° and

*θ*= 150° in Figs. 3(a) and 3(b) respectively. The broader structures observed in the scattering spectra possibly originates due to contributions of the regular and large scale scattering inhomogeneities (such as epithelial cell nuclei) [15

15. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. **7**, 1245–1248 (2001). [CrossRef] [PubMed]

26. R. Drezek, M. Guillaud, T. Collier, I. Boiko, A. Malpica, C. Macaulay, M. Follen, and R. Richards-Kortum, “Light scattering from cervical cells throughout neoplastic progression: influence of nuclear morphology, DNA content, and chromatin texture,” J. Biomed. Opt. **8**, 7–16 (2003). [CrossRef] [PubMed]

*α*(

*θ*) as a function of scattering angle

*θ*for normal and dysplastic tissues are shown in this figure. The values are the average values over the sample size and the error bars represent standard deviations. Interesting differences can be noted in the values for

*α*between the normal and the dysplastic tissues. While the value for

*α*for the dysplastic tissues are higher at forward scattering angles, the reverse is the trend in backscattering angles. For example, in the forward scattering angles (10° – 70°), the dysplastic tissues show ∼ 1.186 ≤

*α*≤ 1.322, in the same region however, the normal tissues show ∼ 1.106 ≤

_{c}*α*≤ 1.242. Similarly, in the backscattering angles (130° – 150°), the values for

_{n}*α*lie in the range ∼ 1.338 ≤

*α*≤ 1.397 for normal, and ∼ 1.119 ≤

_{n}*α*≤ 1.251 for dysplastic tissues. The higher values of the power-law coefficient

_{c}*α*corresponds to higher values of the Hurst scaling exponent

*H*, indicative of “coarseness” of the fluctuations tending towards sub-fractal behavior; whereas at higher angles (130° – 150°),

*α*values are observed to be smaller in dysplastic samples than that for normal samples in the same region indicating more “roughness” in the fluctuations, signifying a trend towards extreme fractality.

*α*as a function of the scale for normal (Fig. 5(a)) and dysplasia (Fig. 5(b)) for forward scattering angles 40° and 60° and for backward scattering angles 120° and 140°. Note that, for this analysis we have not used

*α*-values for lower scales (

*s*= 1,2,3) in order to suitably extract the power law behavior of the fluctuations. The results are thus shown for scales

*s*= 4 to 9. Standard deviations for scale 6 (as representative of all scales) are shown by error-bars. This analysis was performed following the method discussed in Sec. 2.3. Apparent differences in the absolute values of

*α*for Fourier (Fig. 4) and wavelet (Fig. 5) analysis arises from the fact that, in Fourier analysis, we analyze the signal itself while in wavelet analysis, we obtain the

*α*values from the fluctuations which can be thought of as higher derivatives of the signal. We observe that the

*α*values show a strong dependence on the scattering angle

*θ*and the presence of a broad spectrum of processes such that 0.9 ≤

*α*≤ 1.3. This arises possibly from the contribution of differently sized scatterers as has been mentioned earlier. Never-the-less, the observed trends in

*α*from the wavelet analysis is qualitatively similar to that observed from the Fourier analysis. Here also, the values for

*α*of the dysplastic tissues are higher at forward scattering angles, the reverse is the case for backscattering angles. For example, in the forward scattering region, at 40°, we observe that the normal tissues show an average

*α*= 1.123 ± 0.052, while the dysplastic tissues show an average

_{n}*α*= 1.143 ± 0.104 over aforementioned scales. Similarly, in the backward scattering region, we obtain

_{c}*α*= 1.191 ± 0.034 and

_{n}*α*= 1.068 ± 0.028 for normal and dysplastic tissues respectively at 140°. In the other regions also, we found differences between normal and dysplastic samples in the

_{c}*α*values averaged over aforementioned scales.

*H*=

*h*(

*q*= 2)) as a function of the scattering angle

*θ*in Fig. 6; the error bars representing standard deviations. Once again the average values for H are found to be higher for dysplastic tissues in the forward scattering angles (0.214 ±0.003 ≤

*H*≤ 0.332 ±0.009 and 0.187 ±0.038 ≤

_{c}*H*≤ 0.308 ± 0.062). In contrast, the values for

_{n}*H*of the normal tissues are higher in the backscattering angles (0.284 ±0.027 ≤

*H*≤ 0.312 ±0.003 and 0.251 ±0.003 ≤

_{n}*H*≤ 0.264 ±0.032). This is consistent with the results presented in Figs. 4 and 5.

_{c}*f*(

*β*) is a quantitative indicator of the exact nature of the self-similarity and its width represents the strength of the multi-fractality. In Fig. 8, we have shown the

*f*(

*β*) as obtained from Eq. (7) derived following Sec. 2.4. The results are shown for a typical normal and dysplastic tissue sample. We observe that the dysplastic sample has a higher multi-fractality than the normal sample, indicated by the width of singularity spectrum. It must be noted that for a mono-fractal, the singularity spectrum is similar to a Gaussian with a very small variance. This is consistent with our observations of Fig. 7, where, the plots of

*h*(

*q*) vs

*q*for the same (as in Fig. 8) normal and dysplastic tissues at a few representative forward and backward scattering angles are shown. For mono-fractals, the

*h*(

*q*) is independent of

*q*. In Fig. 7, we observe that for the normal samples, at angles 50° – 70°, the dependence of

*h*(

*q*) on

*q*is relatively weaker (Δ

*h*(

*q*) ∼ 0.376 ± 0.114), while in the same angular region (50° – 70°), dysplastic tissues show stronger dependence on

*q*(Δ

*h*(

*q*) ∼ 0.546±0.074). The normal samples thus show a wide range of trends from mono-fractality to multi-fractality. However, dysplastic tissue show strong multi-fractality for all scattering angles. In order to quantify the strength of multi-fractality, we have also determined the width of the singularity spectrum (Δ

*β*), defined as

*β*–

_{max}*β*for all the samples. The mean and the standard deviations of Δ

_{min}*β*corresponding to scattering angles 50° – 70° (where maximum differences were observed) for the normal and dysplastic tissues were found to be 0.442 ±0.077 and 0.556 ±0.044 respectively.

## 5. Conclusion

## Acknowledgments

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52. | A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. |

53. | H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature |

54. | P. Šeba, “Random matrix analysis of human EEG data,” Phys. Rev. Lett. |

55. | J. D. Bancroft and M. Gamble, |

**OCIS Codes**

(100.7410) Image processing : Wavelets

(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine

(290.0290) Scattering : Scattering

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: July 14, 2011

Revised Manuscript: August 23, 2011

Manuscript Accepted: August 25, 2011

Published: September 23, 2011

**Virtual Issues**

Vol. 6, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Sayantan Ghosh, Jalpa Soni, Harsh Purwar, Jaidip Jagtap, Asima Pradhan, Nirmalya Ghosh, and Prasanta K. Panigrahi, "Differing self-similarity in light scattering spectra: a potential tool for pre-cancer detection," Opt. Express **19**, 19717-19730 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19717

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