Characterization of nanosensitive multifractality in submicron scale tissue morphology and its alteration in tumor progression

Abstract. Significance: Assessment of disease using optical coherence tomography is an actively investigated problem, owing to many unresolved challenges in early disease detection, diagnosis, and treatment response monitoring. The early manifestation of disease or precancer is typically associated with subtle alterations in the tissue dielectric and ultrastructural morphology. In addition, biological tissue is known to have ultrastructural multifractality. Aim: Detection and characterization of nanosensitive structural morphology and multifractality in the tissue submicron structure. Quantification of nanosensitive multifractality and its alteration in progression of tumor. Approach: We have developed a label free nanosensitive multifractal detrended fluctuation analysis(nsMFDFA) technique in combination with multifractal analysis and nanosensitive optical coherence tomography (nsOCT). The proposed method deployed for extraction and quantification of nanosensitive multifractal parameters in mammary fat pad (MFP). Results: Initially, the nsOCT approach is numerically validated on synthetic submicron axial structures. The nsOCT technique was applied to pathologically characterized MFP of murine breast tissue to extract depth-resolved nanosensitive submicron structures. Subsequently, two-dimensional MFDFA were deployed on submicron structural en face images to extract nanosensitive tissue multifractality. We found that nanosensitive multifractality increases in transition from healthy to tumor. Conclusions: This method for extraction of nanosensitive tissue multifractality promises to provide a noninvasive diagnostic tool for early disease detection and monitoring treatment response. The novel ability to delineate the dominant submicron scale nanosensitive multifractal properties may also prove useful for characterizing a wide variety of complex scattering media of non-biological origin.


Introduction
Early disease progression in living tissues expected to exhibit nanosensitive structural alteration at the submicron scale. It is highly desirable to develop noninvasive, label-free techniques to detect nanoscale changes in biological tissue for early diagnosis and better treatment. Recently, many optical nanoscopic techniques were developed based on labeling [1][2][3][4][5][6] and are limited to superficial imaging. [7][8][9][10][11][12][13] It is a challenging task for researchers to develop a diagnostic system that can provide label-free depth-resolved detection. There are few early developments that demonstrated averaged nanosensitive structural detection over a volume-rather than depth-resolved detection. 14,15 These methods can identify overall nanosensitive changes rather than depth-resolved alteration, which is crucial to visualize subtle changes of local submicron structure for better diagnosis. In this regard, our research group actively engaged to develop nanosensitive optical coherence tomography (nsOCT) to detect depth-resolved submicron scale structure with few nanometer accuracy. [16][17][18] We have recently demonstrated label-free nsOCT-based imaging technique to visualize few nanometer structural changes 19,20 and its application in cornea crosslinking, 21 and wound healing study. 22 There is a recently demonstrated application of nsOCT for in vivo detection of nanosensitive changes of the human tympanic membrane in otitis media. 23 In addition, biological tissue is known to have submicron structural multifractality. [24][25][26][27][28] Although, these studies are based on superficial detection and do not provide underlying tissue multifractality. Multifractality is a special class of self-similarity where multiple scaling exponents (generalized Hurst exponents) are extracted to quantify existing multifractality in a complex system. 29,30 For both the fundamental study of biological processes and early diagnosis of pathological processes, detection of multifractality in depth-resolved nanosensitive tissue submicron structural morphology is important. Therefore, we have developed a label-free nanosensitive multifractal detrended fluctuation analysis (nsMFDFA) technique in combination with multifractal analysis and nsOCT to extract nanosensitive multifractal parameters in biological tissue. Recently, we have numerically and experimentally validated nsOCT approach on synthetic submicron axial structure with few nanometer accuracy. 20 Here we have numerically validated our proposed nanosensitive multifractal analysis approach in combination of nsOCT simulation and multifractal analysis in a tissue-like randomized synthetic phantom. This approach demonstrated its applicability to measure depth-resolved nanosensitive multifractality in submicron structure in biological tissue. After successful validation, we have applied nsOCT method to construct depth-resolved en face images of dominant submicron structure with nanometer scale sensitivity in murine MFP. Subsequently, we have deployed two-dimensional multifractal detrended fluctuation analysis (2D-MFDFA) 31-33 approach to extract depth-resolved nanosensitive multifractality in MFP. In an initial ex vivo study on murine tissue, we found interesting change in depth-resolved nanosensitive multifractality in submicron structures after tumor formation in breast tissue samples. This method for extraction of nanosensitive tissue multifractality promises to develop a noninvasive diagnosis tool for the detection of cancer development. This newly developed method offers exciting depth-resolved ultrastructural detection for better treatment and monitoring if there is a tumor response to treatment.

Nanosensitive Optical Coherence Tomography
Flowchart of nsOCT is shown in Fig. 1. Recorded interference spectra divided into number of windows before applying Fourier transform for nsOCT construction. Then identifying spatial frequency corresponding to maximum contributed spectral window at different depths of constructed A-line. Subsequently, identified maximum spatial periods map as nsOCT at different depths. In nsOCT approach, detectable spatial period (Hz ¼ λ∕2 n; n = refractive index of the medium) depends on wavelength range of the broadband source (λ ¼ 1176 to 1413 nm). 18 Therefore, in case of biological tissue, we can detect axial structure range from 420 to 504 nm with few nanometer accuracy. 18 It is worth mentioning here that biological tissue may have axial structures ranging from 0 to ∞ (or 0 to tissue physical size). However, our nsOCT approach has a limitation of detection from 420 to 504 nm, which is limited by wavelength range and finding tissue structural differences based on this specified axial scale range only. There may be other axial structures that also exist in biological tissue beyond our detection range that need further study with a much more sophisticated system having a higher wavelength band. In this direction, our research group developing new ultra-high wavelength band nsOCT system.

Numerical Simulations for Nanosensitive Multifractal Detrended Fluctuation Analysis in Randomized Submicron Structure
In our recent publications, 20 35,36 to construct nsOCT. The MATLAB 2019b (MathWorks ® ) has been used for implementation of simulation. The OCT signal was constructed as an interference spectrum of the reflected light from different layers of synthetic sample with the reflected light from a gold mirror. We have also added suitable noise in detector and source spectra to mimic experimental reality. The signal-to-noise ratio was 86 dB in this simulation. The inverse Fourier transform was performed to form A-lines nsOCT of a synthetic volume as each lateral position of 1024 × 1024 pixels. Then we have performed 2D-MFDFA on synthetic and nsOCT constructed en face images to compare.

Multifractal Analysis on Nanosensitive Submicron Structural En Face Images
We have followed our established nsOCT methodology to construct depth-resolved dominant structures of synthetic and tissue volume. Flowchart of nsOCT processing displayed in Fig. 1. Subsequently, we have applied 2D-MFDFA on nsOCT constructed en face images to extract nanosensitive multifractal parameters namely, Hurst exponent [hðq ¼ 2Þ or correlation and width of singularity spectra (Δα) or strength of multifractality. To study the hidden nanosensitive multifractal properties in the submicron scale surface morphology, a 2D-MFDFA 31-33 is implemented here. This method is simple yet innovative and has easy computer implementation. This proposed method is the modification of one-dimensional MFDFA, which is implemented to study the multiple scaling exponents of one-dimensional signals and for identification of long-range correlations in non-stationary time series. 30 The detailed 2D-MFDFA approach can be found in Ref. 31. We have briefly discussed about 2D-MFDFA steps here.
Step 3: The local fitG m;n ði; jÞ for each G m;n ði; jÞ is calculated by fitting it with a bivariate polynomial function as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 6 8 9G m;n ði; jÞ ¼ ai þ bj þ c; (1) where a, b, and c are free parameters and determined by the least square fitting in nanosensitive subsurface G m;n at different locations i; j ¼ 1; 2; : : : ; s in constructed en face nsOCT. The residual nanosensitive axial size variation or detrended subsurface is given by y m;n ði; jÞ at different location as i; j ¼ 1; 2; : : : ; s as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 6 0 9 y m;n ði; jÞ ¼ G m;n ði; jÞ −G m;n ði; jÞ: (2) Step 4: The detrended fluctuation function Fðm; n; sÞ for the segment X m;n is defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 5 6 4 Here s 2 is the number of pixels in segmented square with size s. The q'th order fluctuation function for a nsOCT mapped en face image at each depth is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 4 9 3 : Here M s N s is the total number of segmented square surface in each en face image with size s. The q is order of moments varies from −3 to þ3 with 0.5 interval. At q ¼ 2, the above fluctuation function represents variance of the en face nsOCT map. Note that, in principle, we can calculate generalized Hurst exponents for q ¼ −∞ to þ∞. But here most of variation of hðqÞ happening within q ¼ −3 to þ3. Therefore, we have not extended analysis for other q values which does not provide significance multifractality [variation of hðqÞ] and are computationally expensive.
Step 5: The generalized Hurst exponents [hðqÞ] can be extracted for multiple order of moments (q ¼ −3∶0.5∶ þ 3) by considering long-range power law behavior of this calculated fluctuation function F q ðsÞ as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 6 ; 3 2 3 F q ðsÞ ∝ s hðqÞ : Here in this present study, we have found detected nanosensitive structure follow power law behavior over length scale range s ¼ 4 to 32. Therefore, values of scale s optimize to varies from 4 pixel to 32 pixels (8 to 64 μm) in this nsMFDFA analysis to extract mutifractality. From this above equation, the scaling exponent hðqÞ is obtained by calculating slopes of linear fitting on ln F q ðsÞ versus log s plots. The hðqÞ is known as the generalized Hurst exponents and H ¼ hðq ¼ 2Þ is called the Hurst index of the en face nsOCT surface.
The classical multifractal scaling exponent τðqÞ corresponding to every q value is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 2 0 0 where D f is the fractal dimension of the geometric support of the multifractal measure and D f ¼ 2 in this study.
The two-scaling exponent hðqÞ and τðqÞ along with singularity spectrum fðαÞ can completely characterize any multifractal surface. The singularity spectrum fðαÞ, which characterizes the singularity strength or multifractality of en face nsOCT surface is related to τðqÞ via a Legendre transformation as the Holder exponents, E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 6 ; 9 5 αðqÞ ¼ τ 0 ðqÞ ¼ hðqÞ þ qh 0 ðqÞ; and the singularity spectrum, E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 ; 1 1 6 ; 7 2 3 fðαÞ ¼ qαðqÞ − τðqÞ ¼ q½α − hðqÞ þ 2: Here fðαÞ measures global singularity and αðqÞ characterizes the local singularity of the en face image. The width of the singularity spectrum fðαÞ as a measure of multifractality strength as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 1 1 6 ; 6 6 8 where α max ¼ maxfαðqÞ; q ∈ ½−3; 3g and α min ¼ minfαðqÞ; q ∈ ½−3; 3g. Δα measures the nanosensitive submicron scale multifractality at each en face images. The higher value of Δα in the submicron scale en face indicates higher strength of multifractality. The Hurst scaling exponents: hðq ¼ 2Þ ¼ 0.5, >0.5, and <0.5 correspond to uncorrelated, long-range correlated, and anti-correlated fluctuations, respectively, in nanosensitive en face images at different depths.
In this study, we have characterized (a) Hurst exponent [hðq ¼ 2Þ] and (b) width of the singularity spectrum fðαÞ, (Δα) to measure correlation and multifractality, respectively, on detected nanosensitive submicron scale en face images.

Tissue Sample Preparation
All animal procedures were performed in accordance with the Guidelines for Care and Use of Laboratory Animals of the "Animal Care Research Ethics Committee (ACREC), National University of Ireland Galway (NUIG)" and approved by the "Health Product Regularity Authority (HPRA), Ireland".
Female BALB/c mice (Charles River Laboratories Ltd.) aged between 6 and 8 weeks were employed. A mouse received a mammary fat pad (MFP, 4th inguinal) injection of 1 × 10 5 4T1 breast cancer cells suspended in 100 μl RPMI medium. The early stage tumor was detected by palpation after seven days of injection and was visually inspected. Tumor growth was monitored using calipers measurement. The tumor size was 715 mm 3 . Animals were sacrificed by CO 2 inhalation. Tumor tissue and healthy portion were harvested and placed in PBS solution for transfer to the OCT imaging facility for ex vivo analysis. Harvested samples were taken out from PBS and mounted on a glass slide to bring them under the objective of spectral domain OCT system (Telesto III, Thorlabs Inc.) to record OCT images.

Numerical Validation of Nanosensitive Multifractal Detrended Fluctuation Analysis in Synthetic Submicron Scale Volume Structures
We have recently demonstrated an experimental and numerical approach for nsOCT validation and detection of submicron structure with few nanometer accuracy. 19 Here we have simulated nsOCT in synthesized volume (1024 × 1024 × 400 voxels) phantom composed of randomized submicron structures throughout the volume. Figure 2(a) displays synthesized en face map of submicron structure at ∼150 μm depth. Figure 2(b) displays corresponding nsOCT detected en face map of submicron structure. Figure 2

Application of Validated Nanosensitive Multifractal Analysis Technique on Pathologically Characterized MFP Tissue Samples
After successful validation of nsMFDFA approach, we have applied this technique on ex vivo tissue with healthy MFP and tumor portion. Figures 3(a)   tissue. Figures 3(c) and 3(f) display volume nsOCT where dominant axial submicron structure mapped onto healthy MFP and tumor tissue, respectively. Measured overall average maximum spatial period for healthy MFP is 447.5 nm and tumor tissue is around 454.6 nm. Although, we can see a difference in healthy and tumor tissue based on this overall volume average nanosensitive measurement, but they may not have differences at different depths. Figure 3(g) displays depth-resolved nanosensitive structural averages at each en face for healthy [blue plot extracted from volume in Fig. 3(c)] and tumor [red plot extracted from volume in Fig. 3(f)]. It shows that at 0.35 mm depth, the average value of dominant structure is almost the same for healthy and tumor tissue. Here it is difficult to differentiate tumor from healthy tissue based on local nsOCT measurement only. It is also unable to provide quantitative and depth-specific nanosensitive submicron scale tissue morphological complexities. In these regards, it is known that tissues have multifractality in submicron structure. Therefore, it is always interesting to have depth-resolved quantitative submicron scale multifractal parameters for better understanding. Quantitative submicron scale multifractal parameters can also help to develop computer-assisted automated differentiation of tumor tissue from healthy tissue. In this direction, we have performed 2D-MFDFA on depth-resolved nsOCT constructed en face images to find nanosensitive multifractal parameters.
In Figs Fig. 4(d)]. This multifractality reflected in singularity spectrum fðαÞ plots with larger width of singularity spectrum in case of tumor tissue [red color plot in Fig. 4(h)]. For further verification and confirmation, we have applied this extraction method of nanosensitive multifractality on 10 healthy MFP and 10 tumor volume images in different areas of a tissue sample. We found consistence differences of nanosensitive correlation and strength of multifractality over different depths of tissue. Figures 5(a) and 5(b) represent mean Hurst exponent [hðq ¼ 2Þ using Eq. (5)] and strength of multifractality [Δα using Eq. (9)] over 10 healthy MFP (blue line plot) and tumor (red line plot) tissue on en face images at different depths. Vertical lines at each depth represent standard deviation from mean trends over 10 samples. In Fig. 5(a), reduction trends of nanosensitive Hurst exponent indicate decrease of correlation of dominant submicron structural distribution over en face images as tumor progress. In Fig. 5(b), increase of nanosensitive multifractality indicates increase of strength of multifractality or distortedness of dominant submicron structural distribution over en face images as tumor progress.

Conclusions
A novel approach to quantify submicron scale nanosensitive multifractality in combination with nsOCT and multifractal analysis has been demonstrated. We validated the nsOCT technique numerically on synthetic submicron scale axial structures. We developed a novel nanosensitive submicron scale multifractal analysis technique to characterize tissue depth-resolved ultrastructural morphology. Reduction of the Hurst exponent [hðq ¼ 2Þ] from healthy MFP to tumor indicates reduction of correlation or self-similarity in dominant submicron structures. Increase of width of singularity spectrum (Δα) from healthy MFP to tumor indicates increase of multifractality or roughness in dominant nanosensitive submicron scale tissue structures. This newly developed method promises early disease detection for better treatment guidance and monitoring response to treatment in cancer patients. Results promise to detect nanosensitive multifractality in the submicron scale structural distribution and its alteration in deep tissue as tumor progress. This ability to delineate nanosensitive self-similarity may provide a noninvasive measuring tool for characterization of biological tissue and nonbiological media. Observed differences in the submicron scale nanosensitive multifractality between healthy and tumor tissue show considerable promise as potential biomarkers for cancer detection. The ability to probe and quantify nanosensitive self-similarity and change of multifractality related to development of cancer using backscattering mode FD-OCT bodes well for in vivo deployment. Exploiting the interference spectra recorded from tissue depths with the reference mirror, in vivo applications of this approach should be realized with a fiber optic-based handheld probe assisted with scanning lens and galvo mirror. Finally, the developed nsMFDFA method represents as a novel approach with much potential for in vivo detection of cancer initiation and other non-biological application remain to be rigorously evaluated.

Disclosures
The authors have no other relevant financial interest in this article and no other potential conflicts of interest to disclose.