2.1.

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 ($Hz=\lambda /2n$; $n$ = refractive index of the medium) depends on wavelength range of the broadband source ($\lambda =1176$ 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 $\infty$ (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.

Fig. 1

Flowchart: construction of nsOCT.

2.2.

Numerical Simulations for Nanosensitive Multifractal Detrended Fluctuation Analysis in Randomized Submicron Structure

In our recent publications,^20,34 we have experimentally and numerically demonstrated the implementation of nsOCT to image a commercially fabricated sample with 431.56 nm axial periodic structures. In Ref. 34, we have demonstrated that we can detect random submicron dominant structures at different depths with $\sim 5\mathrm{nm}$ accuracy. In this study, we have numerically synthesized randomized 3D structure with submicron scale axial structure varies from 445 to 485 nm. The dimension of synthetic volume is $1024\times 1024\times 400\text{voxels}$. We have followed standard Fourier domain OCT theory and simulation strategy^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 $1024\times 1024\mathrm{pixels}$. Then we have performed 2D-MFDFA on synthetic and nsOCT constructed en face images to compare.

2.3.

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 [$h(q=2)]$ or correlation and width of singularity spectra ($\mathrm{\Delta }\alpha$) 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.

The classical multifractal scaling exponent $\tau (q)$ corresponding to every $q$ value is given by

The two-scaling exponent $h(q)$ and $\tau (q)$ along with singularity spectrum $f(\alpha )$ can completely characterize any multifractal surface. The singularity spectrum $f(\alpha )$, which characterizes the singularity strength or multifractality of en face nsOCT surface is related to $\tau (q)$ via a Legendre transformation as the Holder exponents,

In this study, we have characterized (a) Hurst exponent [$h(q=2)$] and (b) width of the singularity spectrum $f(\alpha )$, ($\mathrm{\Delta }\alpha$) to measure correlation and multifractality, respectively, on detected nanosensitive submicron scale en face images.

2.4.

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 $1\times {10}^5$ 4T1 breast cancer cells suspended in $100\mu \mathrm{l}$ 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 $715{\mathrm{mm}}^3$. Animals were sacrificed by ${\mathrm{CO}}_2$ 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.

3.1.

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 ($1024\times 1024\times 400\mathrm{voxels}$) phantom composed of randomized submicron structures throughout the volume. Figure 2(a) displays synthesized en face map of submicron structure at $\sim 150\mu \mathrm{m}$ depth. Figure 2(b) displays corresponding nsOCT detected en face map of submicron structure. Figure 2(b) demonstrates that nsOCT can detect submicron scale dominant structure with 5-nm accuracy (comparing scale bar of synthetic submicron structure: 448 to 482 nm and nsOCT constructed scale bar: 443 to 482 nm). To check multifractality and correlation in synthesized en face and in nsOCT detected en face, we have applied state of the art 2D-MFDFA on each en face images throughout the depth. Figure 2(c) shows plots of generalized Hurst exponents [$h(q)$ versus $q$] calculated from synthetic en face in Fig. 2(a) (blue color) and from nsOCT constructed en face in Fig. 2(b) (red color). Variation of $h(q)$ versus $q$ in Fig. 2(c) indicates that synthetic submicron structural en face has a multifractality (blue color plot) and can be detected with almost no error (red color plot). In addition, Hurst exponent [$h(q=2)$] in synthetic and nsOCT constructed en face is almost equal in values confirms our capability to detect submicron scale structural correlation within a complex tissue sample. Similarly, Fig. 2(d) shows plots of singularity spectrum [$f(\alpha )$ versus $\alpha$] calculated from synthetic en face in Fig. 2(a) (blue color) and from nsOCT constructed en face in Fig. 2(b) (red color). The width of singularity spectrum ($\mathrm{\Delta }\alpha$) is a measure of multifractality is almost equal in synthetic and nsOCT extracted en face image proved our capability to detect submicron scale structural multifractality within a complex tissue sample.

Fig. 2

Detection of randomized submicron structural multifractality using nsMFDFA. (a) En face map of submicron synthetic structure at $150\mu \mathrm{m}$ depth. (b) En face map of submicron dominant structure at $\sim 150\mu \mathrm{m}$ depth extracted via nsOCT simulation. (c) Generalized Hurst exponents [$h(q)$] extracted at $q=-3$ to $+3$ from synthetic en face structure [blue color plot from Fig. 2(a)] and nsOCT detected en face structure [red color plot from Fig. 2(b)]. Extracted Hurst exponent, $h(q=2)=0.45$ from synthetic surface and $h(q=2)=0.44$ from nsOCT simulated surface. (d) Singularity spectrum [$f(\alpha )$ versus $\alpha$) extracted from synthetic en face structure [blue color plot from Fig. 2(a)] and from nsOCT detected en face structure [red color plot from Fig. 2(b)]. Extracted width of singularity spectrum, $\mathrm{\Delta }\alpha =1.43$ from synthetic surface and $\mathrm{\Delta }\alpha =1.42$ from nsOCT simulated surface. (e) Depth-resolved self-similarity and multifractality. Depth-resolved Hurst exponents [$h(q=2)$] extracted from depth-resolved synthetic submicron scale en face structures (black color plot) and from nsOCT detected depth-resolved submicron scale en face structures (blue color plot). Depth-resolved width of singularity spectrum ($\mathrm{\Delta }\alpha$) extracted from synthetic submicron scale en face structures (green color plot) and from nsOCT detected submicron scale en face structures (red color plot).

The lower part of Fig. 2(e) displays 2D-MFDFA extracted depth-resolved Hurst exponent [$h(q=2)$] from synthetic (black color plot) and from nsOCT detected (blue color plot) submicron scale en face structures. The upper part of Fig. 2(e) displays 2D-MFDFA extracted depth-resolved width of singularity spectrum ($\mathrm{\Delta }\alpha$) from synthetic (green color plot) and from nsOCT detected (red color plot) submicron scale en face structures. Detected depth-dependent Hurst exponent [$h(q=2)$] and strength of multifractality ($\mathrm{\Delta }\alpha$) from synthetic and nsOCT simulated en face structure is very close to each other. These results validated our capability to detect submicron scale structural multifractality through the proposed nsMFDFA methodology from a tissue-like complex submicron scale multifractal structure with greater accuracy.

3.2.

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) and 3(d) display hematoxylin and eosin (H&E) stained microscopic images of healthy MFP and tumor tissue, respectively. Figures 3(b) and 3(e) represent conventional OCT images from healthy MFP and tumor tissue, respectively. Note that average intensity values are 32.04 and 33.32, respectively, in healthy MFP and tumor tissue volume. Therefore, it is very difficult to differentiate OCT of healthy MFP and tumor 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.

Fig. 3

Hematoxylin and eosin (H&E) stained histological images ($500\mu \mathrm{m}\times 500\mu \mathrm{m}$) of (a) normal MFP and (d) tumor tissue. (b), (e) 3D volume ($500\mu \mathrm{m}\times 500\mu \mathrm{m}\times 360\mu \mathrm{m}$) OCT images of healthy and tumor tissue. Gray scale bar represents OCT intensity for Figs. 3(b) and 3(e). (e), (f) Dominant submicron axial structural mapped 3D volume OCT images of healthy and tumor tissue, respectively. Color bar represents size of dominant submicron axial structures for Figs. 3(c) and 3(f). (g) Depth-resolved nanosensitive structural averages at each en face for healthy MFP [blue plot extracted from volume in Fig. 3(c)] and tumor [red plot extracted from volume in Fig. 3(f)].

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. 4(a) and 4(e), we have displayed maximum spatial period en face images at $150\mu \mathrm{m}$ depth corresponding to volume images in Figs. 3(c) and 3(f). Figures 4(b) and 4(f) represent submicron structural histogram corresponding to Figs. 4(a) and 4(e). Figures 4(c) and 4(g) represent local nanoscale variation after subtracting local trends with detrending scaling window $s=4$ [see Eqs. (3) and (4)]. Figure 4(d) displays generalized Hurst exponent $h(q)$ versus $q$ for healthy (blue color) en face and tumor (red color) en face image. Figure 4(h) displays singularity spectrum $f(\alpha )$ versus Holder exponent, $\alpha$ for healthy MFP (blue color) en face, and tumor (red color) en face image. Variation of $h(q)$ in Fig. 4(d) indicates multifractality and greater variation of $h(q)$ indicates larger multifractality in tumor tissue [red color plot in Fig. 4(d)]. This multifractality reflected in singularity spectrum $f(\alpha )$ plots with larger width of singularity spectrum in case of tumor tissue [red color plot in Fig. 4(h)].

Fig. 4

En face nsOCT image at $150\mu \mathrm{m}$ depth for (a) healthy and (e) tumor tissue corresponding to volume image in Figs. 3(c) and 3(f). Color bar represents size of dominant submicron axial structures. Panels (b) and (f) represent submicron structural histogram corresponding to Figs. 4(a) and 4(e). (c), (g) Corresponding detrended en face images with window size $s=8\mu \mathrm{m}\times 8\mu \mathrm{m}$. Color bar represents nanosensitive variation of local submicron axial structures. (d) Generalized Hurst exponent $h(q)$ versus $q$ for healthy (blue color) en face and tumor (red color) en face. (h) Singularity spectrum $f(\alpha )$ versus Holder exponent, $\alpha$ for healthy (blue color) en face and tumor (red color) en face.

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 [$h(q=2)$ using Eq. (5)] and strength of multifractality [$\mathrm{\Delta }\alpha$ 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.

Fig. 5

Overall multifractality in dominant submicron structure over en face images at different tissue depths at 10 different areas of the tissue sample. (a) Hurst exponent [$h(q=2)$] represents correlation of submicron structural distribution over en face for healthy (blue) and tumor (red) tissue at different depths. (b) The width of singularity spectrum ($\mathrm{\Delta }\alpha$) represents randomness of submicron structures over en face images at different depths for healthy (blue) and tumor (red) tissue. Vertical black lines, and vertical green lines are standard deviations at each depth.