2.1.

Tissue Preparation and Handling

The animal procedures were approved by the University of Auckland Animal Ethics Advisory Committee. All animal specimens were rats of Wistar breed and male. Euthanasia was performed by first anaesthetising with isoflurane and then performing cervical dislocation. A total of three nerve tissue samples were explanted, each extending the entire length of the sciatic nerve and, distal to bifurcation, as much of the tibial and peroneal branches as practicable. Samples were then stored in 0.01 M phosphate buffered solution (PBS) at 4°C for up to three days before being transported and imaged using OCT. The physiological pH and osmolarity of PBS ensured that the cellular and ultrastructural features of the sample were preserved. Several points along each tissue sample were randomly selected and marked with Davidson’s marking dye and 1% picric acid, and these points imaged with OCT. Ex vivo imaging allowed any location of the nerve to be imaged by manipulating the nerve instead of the incident light beam, which would have required significant modifications to the OCT system. In this study, we present results and images from one point each from the sciatic section of two nerves, and one point from the tibial section of one nerve.

During OCT imaging, tissue samples were suspended at two points 20 mm apart along the length. Sagging of the tissue sample was used as a visual indication that the tissue sample was not mechanically stretched. Four OCT images, at 90-deg offsets, were acquired of the dyed points within the suspended section. Between image acquisitions, 0.01 M phosphate buffered solution was applied to the outside of the nerve, using a syringe to avoid tissue drying. Postimaging, the tissue samples were returned to the phosphate buffered solution and transported to the Anatomy and Medical Imaging department at the University of Auckland for histological analysis using light microscopy.

Error introduced by thermal expansion, caused by variation in the sample temperature during imaging between 4°C and 20°C, is expected to be $<0.5\%$ using the thermal expansion coefficient of water. Error introduced from stretching, due to suspension of the tissue samples during OCT imaging, is $<0.05\%$ using a Poisson’s ratio and Young’s modulus of 0.37 and 41 MPa, respectively.²⁵

2.2.

Microsphere Samples

Three microsphere samples (Spherotech Inc.) were used in this study with concentration, and $\text{mean diameter}+/-1\text{standard deviation}$ of: (1) 5% w/v, $3.8+/-0.25\mu \mathrm{m}$; (2) 5% w/v, $5.33+/-0.25\mu \mathrm{m}$; and (3) 2.5% w/v $8.49+/-0.25\mu \mathrm{m}$. Standard deviations of their respective size distributions were calculated using measured data provided by the supplier with the samples. For each sample, the vial containing the microspheres in the solution was shaken vigorously to ensure homogeneity of the solution before a 1-mL sample was extracted and transported, using a pipette, to a separate container under the OCT system and immediately imaged.

2.3.

Optical Coherence Tomography Image Acquisition and Processing

The OCT system has a swept source centered at 1310 nm with an 80-nm bandwidth (3 dB). The system has $12.5\text{-}\mu \mathrm{m}$ axial resolution, $20\text{-}\mu \mathrm{m}$ lateral resolution, and the 6-dB drop off in air is over 12 mm. More details of the OCT system can be found in Refs. 26 and 27.

The B-scan direction was perpendicular to the length of the nerve tissue samples, spanned a physical distance of 5 mm, and contained 714 A-scans. B-scans were saved from the LABVIEW^® user interface to text files and then processed individually with MATLAB^® 2015b using the signal analysis techniques described below.

To determine the distance when calculating epineurium thickness, an average refractive index had to be used for the entire OCT image; here, we have used the refractive index of $n=1.40$, which is the average of values reported for myelin (1.455)²⁸ and bovine tendon, a collagenous connective tissue, (1.353).²⁹ The axial distance per pixel, of $10\mu \mathrm{m}$ in air, was, therefore, $7.1\mu \mathrm{m}$ in neural tissues in all subsequent calculations of epineurium thickness.

2.4.

Histological Analysis by Light Microscopy

Histology by light microscopy was performed at one location on each of the tissue samples, see Fig. 1. Tissue samples were preserved in 10% neutral buffered formalin for 24 hours, followed by 70% ethanol prior to paraffin embedding. Tissue slices of $10\text{-}\mu \mathrm{m}$ thick were obtained using a microtome then hemotoxylin and eosin were used to stain the collagen-rich epineurium, perineurium, and endoneurium tissues pink. The lipid-rich myelin was stained with Luxol Fast Blue. Images were obtained on a Leica DM500 light microscope at $4\times$, $10\times$, and $40\times$ magnifications. A uniform shrinkage of 10%, produced during the preparation process, was taken into account by multiplying all values extracted from histology by a factor of 1.1.

Fig. 1

An ADRP curve generated by averaging five adjacent A-scans from a flattened OCT image and the linear fit calculated from pixel depth range of 15 to 50 pixels (a), where a pixel depth of 0 is the surface. The residuals, calculated from Fig. 1(a), and the threshold line used to determine the epineurium inner boundary (b).

The thickness of epineurium tissues was determined using images obtained at $4\times$ magnification. In each image, a line was drawn through the tissue layers, perpendicular to the tissue outer surface, at 20 locations approximately equidistant around the fascicle boundaries. Then, the thickness of tissue along these lines was measured manually. Error introduced by the measurement procedure was estimated to be $+/-1\mu \mathrm{m}$.

Fiber diameter distribution was calculated in areas of the tissue sample images that were visually identified as containing a few artifacts from sample preparation. A grid of $100\text{-}\mu \mathrm{m}$ squares was laid over the area of interest in the $40\times$ magnification images and the fiber diameter distribution was calculated for each grid square. Fibers were typically ellipsoid shaped; therefore, the major and minor axes were used to calculate the diameter of a circle with an equivalent area. The diameter estimates contained $\sim +/-1\mu \mathrm{m}$ uncertainty. Furthermore, the method employed could not identify fiber diameters $<3\mu \mathrm{m}$ due to the very thin, or absent in the case of unmyelinated fibers, myelin sheath.

2.5.

Average Depth-Resolved Profile Technique

Two techniques to identify the fascicle boundary, one using structural data and the other phase retardation data, are presented in Ref. 18. Both techniques are applied to the ADRP of the middle 50% of A-scans of the tissue sample. The ADRP is generated by first cropping pixels outside the tissue sample, then flattening the remaining pixels so that each row in the columns of data (A-scan) correlates to the same axial depth in the tissue sample, and, finally, averaging values across each row. To estimate the fascicle boundary with higher lateral resolution, we implemented the ADRP across windows containing five A-scans, which were scanned laterally across the sample in one-pixel steps.

The ADRP-structural-data technique operates on the expectation that the anatomy of the intrafascicle volume will produce a uniform decay rate of the OCT signal.¹⁸ To identify the fascicle boundary, the ADRP-structural-data technique plots the signal-to-noise ratio (SNR) of the ADRP on a logarithmic scale against depth, then fits a linear slope to the uniformly decaying region of the curve, Fig. 1. The absolute difference between the linear fit and the ADRP curve forms the residual SNR curve. A threshold value is then calculated as the mean plus 2 standard deviations from the linear portion of the residual SNR curve. The fascicle boundary is identified by the intersection of the threshold value and the curve neighboring the uniformly decaying region. In our implementation of the five A-scans sliding window method, we fit the linear slope to data from a depth range of 15 to 50 pixels in all window ADRPs, Fig. 1. The start value of 15 pixels correlates to a depth of $113\mu \mathrm{m}$, which is outside the typical epineurium thickness values provided by histological analysis and, therefore, is highly unlikely to include epineurium tissue.

We calculated the SNR at each depth using $\mathrm{SNR}=20\log (I/\sigma )$, where $I$ is the signal intensity at that depth, and $\sigma$ is the standard deviation of the noise. To find $\sigma$, we determined the standard deviations of the five A-scans nearest to one end of the B-scan, and so known not to contain any reflectors across the depth range of interest, and then calculated the average of these standard deviations.

To identify adipose tissue, we calculated the threshold from the entire portion of the residual SNR curve, instead of just the linear portion, which increases the threshold value. In areas with epineurium, the residual-SNR curve in the uniformly decaying region was below the threshold and the initial peaks of the residual-SNR curve, associated with epineurium, were above the threshold. Conversely, in areas with adipose tissue, which contained multiple peaks and no uniformly decaying region, the residual SNR curve oscillated around the threshold.

In the ADRP-phase-retardation technique, a linear slope is fitted to the rising portion in a plot of phase retardation against depth, which occurs due to the birefringence of myelin. The residuals, threshold, and fascicle boundary are then calculated in the same way as the ADRP-structural-data technique. We did not employ the ADRP phase-retardation technique on any tissue samples as our system was not polarization sensitive.

2.6.

Autoregressive Spectral Estimation Technique

The diameter range of nerve fibers places them in the Mie scattering regime, which generates diameter-dependent spectral modifications. The AR-SE technique for estimating scatterer diameter, which statistically analyses the spectral content of backscattered light, was originally presented in Ref. 30 and then applied to biological samples, including the sciatic nerve of rabbit, in Refs. 19 and 20. The technique assumes a linear relationship between the scatterer diameter $\tilde{\mathbf{d}}$ and the eigenvectors produced by principal component analysis (PCA) of the power spectral density $\widetilde{{\mathbf{P}}_{\mathrm{xx}}}$ produced using the Burg’s method, such that $\widetilde{{\mathbf{P}}_{\mathrm{xx}}}\tilde{\mathbf{A}}=\tilde{\mathbf{d}}$ where $\tilde{\mathbf{A}}$ is a coefficient matrix that is populated using samples of known diameter, also called “training data.”

To obtain the training data, we used the three microsphere samples described above (Sec. 2.2). A window of $50\times 10\text{pixels}$ ($\text{axial}\times \text{transverse}$) in size was raster scanned across half of the OCT image at 25 and 1 pixel steps, respectively. Data were row-averaged to suppress noise and normalized to reduce variations from depth attenuation. The decay of autoregressive reflection coefficients toward zero determined the autoregressive order to use in calculating the power spectral density. A target variance of 99.99% determined the number of components to extract from weighted PCA of the power spectral density data. The remaining half of each OCT image of microspheres was used to obtain diameter estimates.

In applying the AR-SE technique to OCT data from tissue samples, larger window sizes reduce noise, but potentially remove more of the targeted detail such as the intrafascicle boundary and localized variation in fiber diameter distribution. Furthermore, the maximum autoregressive order that can be applied is one less than the number of data points, i.e., the axial window size. A window size of $10\times 10\text{pixels}$, scanned at 1 pixel steps, was selected as it produced a good balance between noise reduction and sufficient resolution. At each step, the autoregressive order to use was determined by analyzing the autoregressive reflection coefficients, with a drop below $+/-0.2$ in magnitude used as the threshold criterion. Alternatively, when the decay of autoregressive reflection coefficients toward zero was not clearly identifiable, an autoregressive order of 3 was selected as this value was common in the tissue samples and was found to minimize noise in the solution.

In the processed OCT images, each lateral and depth pixel contained a fiber diameter estimate. An area of interest was selected from the processed OCT images, which began sufficiently below the surface to pertain to the intrafascicle volume, and the diameter estimates at all pixels within the area formed the sample for the analysis of fiber diameter distribution.

2.7.

Correlation of the Derivative Spectral Estimation Technique

The CoD-SE technique was originally presented in Ref. 24, and then again with additional signal processing steps in Ref. 21. This technique uses Mie theory to generate spectra for spherical scatterers before taking the derivative and then the autocorrelation to produce the CoD. The CoD bandwidth is calculated as the first minimum, i.e., the minimum with the lowest lag value, in the CoD of the spectra. A curve is fitted to a plot of the theoretical relationship between the scatterer diameter and CoD bandwidth. In OCT data, the spectra are obtained using the Fourier transform with a Gaussian window. To lower the noise, spectra are low-pass filtered and the edges are removed. Differentiating with respect to the neighboring value followed by autocorrelation produces the CoD. CoD-SE is highly sensitive to the lateral position of the window relative to the scatterer, and to the axial window size. To address the former, an intensity threshold of 5 dB above the noise level is introduced and the highest intensity of three laterally adjacent windows is assigned to all three lateral positions. To address the latter, the axial window size, which is used to obtain sample spectra, is selected to minimize the standard deviation of the resulting scatterer diameter estimates.

We appended an additional step to the end of the original method, outlined in the previous paragraph to improve the scatterer diameter estimates produced from our microsphere samples: the window axial size was selected to minimize the “normalized” standard deviation of the scatterer diameter estimate, calculated as the standard deviation divided by the corresponding mean.

The CoD-SE algorithm was tested on three microsphere samples with same concentration and mean diameter as in the AR-SE experiments, see Sec. 2.5. We generated the theoretical curve using the MATLAB functions of Mie theory presented in Refs. 31 and 32 and refractive indices of water and polystyrene³³ ($n_{\text{water}}=1.3225+0.001i$ and $n_{\mathrm{poly}}=1.59+0.0025i$). We then fitted an exponential function of the form $f_{\mathrm{CoD}}=d^A{e}^B$, where $f_{\mathrm{CoD}}$ is the CoD bandwidth, $d$ is the diameter, and $A$ and $B$ are constants defining the curve, as this provided an excellent fit within the diameter range from 3 to $16\mu \mathrm{m}$. Unbiased methods were not identified to select (1) the standard deviation of the Gaussian window in the Fourier transform and (2) the extent to crop the edges of spectra to remove noise. A standard deviation of the Gaussian window equal to half of the window size, and cropping of 3.9 nm ($100/4096$ points) from each end of the spectral range were selected through trial and error using the microsphere samples. Diameter estimates were produced for each image pixel using 13 different axial window sizes from 3 to 15 pixels, all with a lateral size of 1 pixel, scanned at 1 pixel steps. At each step, and for each window size, the Butterworth filter (second order, zero phase, low pass) cut-off frequency was selected using Winter’s method, with resulting values in the range of $0.225+/-0.004\pi .\mathrm{rad}/\text{sample}$ ($0.899+/-0.016{\mathrm{nm}}^{-1}$).

For tissue samples, we could not find a value for the refractive index of endoneurium nor its close relative epineurium. Instead, we used bovine tendon, which is, like endoneurium, a collagen-rich connective tissue. Therefore, we generated the theoretical CoD bandwidth curve using refractive indices of myelin ($n_{\text{myelin}}=1.455$²⁸) and bovine tendon ($n_{\text{tendon}}=1.353$²⁹), to which we fitted an exponential function of the form $f_{\mathrm{CoD}}=d^A{e}^B$. Values for the standard deviation of the Gaussian window in the Fourier transform, the extent to crop the edges of spectra, and the Butterworth filter parameters were all carried over from microsphere experiments. Axial window sizes from 3 to 15 pixels were used to generate diameter estimates, and then a $3\times 3$ scanning window was used to determine the diameter estimate to use for each pixel location based on the minimization of the normalized standard deviation. Scatterer size estimates were directly assigned as fiber diameter distribution estimates.

In the processed OCT images, each lateral and depth pixels contained a fiber diameter estimate. An area of interest was selected from the processed OCT images, which encompassed the intrafascicle volume, and the diameter estimates at all pixels within the area formed the sample for analysis of fiber diameter distribution.

3.1.

Epineurium Thickness

3.1.1.

Histological analysis

With H&E staining of the fascicle, boundaries in the sciatic branches were clearly identifiable in light microscopy images of all three tissue samples, by the change from dark pink stained epineurium and perineurium tissues to the lighter pink intrafascicle environment mottled with unstained nerve fibers, Fig. 2. Adipose cells are unstained by H&E. Adipose tissue commonly appeared in clusters of several cells, each with a diameter of $>20\mu \mathrm{m}$, apposed to the nerve outer boundary in large quantities in tissue sample 1 and in small quantities in tissue samples 2 and 3, Fig. 2.

Fig. 2

Light microscopy images at $4\times$ magnification, stained with H&E, of tissue sample 1 (a) and tissue sample 2 (b) of sciatic nerves, and tissue sample 3 (c) of the tibial nerve branch. In all images the scale bar is $100\mu \mathrm{m}$. Images have been rotated in order that the incident beam in OCT, acquired for four tissue orientations, coincides with each side of the light microscopy image, where the sides: left, bottom, right, and top correspond to OCT imaging orientations: 1, 2, 3, and 4, respectively (a). After alignment with OCT, the top, bottom, and two sides of the nerve within each histological image were identified, shown with dashed lines in (b), and the middle 50% of the nerve boundary was assigned as the middle 50% in linear distance between the respective ends, shown with arrows in (b). Tissue features of peripheral nerve are labeled in (c).

The combined epineurium and perineurium thickness was estimated at $50+/-38\mu \mathrm{m}$, $45+/-32\mu \mathrm{m}$, and $31+/-25\mu \mathrm{m}$ ($\text{mean}+/-1\text{standard deviation}$) across the entire light microscopy image for tissue samples 1, 2, and 3 respectively.

Due to the small thickness of perineurium relative to epineurium, and because one layer of each are always present and adjacent to one another in the tissue layer separating the intra-fascicle volume and the outer nerve boundary, from here onward we refer to the combined epineurium and perineurium tissue layer as just the epineurium layer.

When the light microscopy images were aligned to the corresponding OCT images, and the middle 50% of the nerve boundary was analyzed to match the area imaged and analyzed by OCT, the localized standard deviations of the epineurium thickness estimates were smaller than those obtained for the entire nerve. This indicates epineurium thickness varies along the outer boundary. The distribution of the thickness estimates for tissue sample 1 orientation 3 and tissue sample 3 orientations 1 and 2 showed multiple peaks across the range of values, i.e., multimodal. All other tissue samples’ orientations were right skewed, i.e., favoring smaller thickness, see Table 1.

Table 1

Epineurium tissue layer thickness, in μm, estimated from histology, conventional and 5Ascan methods within the ADRP technique, and with a 10×10 window for the AR-SE technique. Mean (standard deviation). A refractive index of 1.41, producing 7.1-μm axial distance per pixel within nerve tissue, used to obtain ADRP and AR-SE values. Histology estimates are taken across the middle 50% of each orientation, whereas the ADRP and AR-SE signal analysis techniques are applied to the middle 50% of A-scans in each image. RS, right skewed; MM, multimodal. In the ADRP conventional column, bold indicates values within +/−1  standard deviation of the value obtained using light microscopy. Elsewhere, italic indicates agreement of medians (Wilcoxen rank sum test, α=0.05) and analysis of variance (Kruskal–Wallis test, α=0.05) with the corresponding data obtained using light microscopy.

	Light microscopy	ADRP: conventional	ADRP: five A-scan	AR-SE: 10×10
Tissue sample 1
Orientation 1	a	77	40 (37)	70 (75)
Orientation 2	a	91	42 (39)	72 (50)
Orientation 3	59 (34) MM	63	48 (28) RS	27 (24) RS
Orientation 4	19 (10) RS	85	37 (27) RS	17 (22) RS
Tissue sample 2
Orientation 1	28 (19) RS	83	44 (28) RS	43 (50) RS
Orientation 2	29 (11) RS	91	61 (32) MM	22 (23) RS
Orientation 3	44 (20) RS	91	56 (34) MM	44 (41) RS
Orientation 4	30 (25) RS	105	33 (34) RS	26 (31) RS
Tissue sample 3
Orientation 1	23 (14) MM	77	57 (38) RS	43 (43) RS
Orientation 2	41 (29) MM	46	28 (29) RS	28 (39) RS
Orientation 3	22 (12) RS	71	40 (37) RS	20 (25) RS
Orientation 4	22 (16) RS	49	51 (35) RS	27 (32) RS

^aAdipose tissue prevalent.

3.1.3.

Autoregressive spectral estimation technique

The layer of epineurium and perineurium tissues was visually identified as a contiguous area of large diameter scatterers along the surface of the nerve. The relationship between the thickness of this contiguous layer of scatterers and the thickness of the epineurium tissue was investigated using a digital phantom of an epineurium layer, which was constructed using a $50\times 50$ element array with each element equal to 0.01 and within this array several adjacent rows containing values of 1 to mimic epineurium. Application of the AR-SE algorithm to the phantom showed the epineurium thickness along each A-scan could be approximated by difference between the base width of the contiguous scatterer peak and the window axial size as shown in Fig. 3(a). In nerve samples, due to the presence of scatterers along the inside boundary of the epineurium tissue layer, Fig 3(b), we isolated the contiguous scatterer peak by removing scatterer estimates below $5\mu \mathrm{m}$. We then subtracted the scanning window axial size from the full-width-half-mean of the contiguous scatterer peak, in place of the base width, to estimate the epineurium thickness.

Fig. 3

Signal intensity and corresponding diameter estimate from the AR-SE algorithm for epineurium phantom (a) and an A-scan taken from tissue sample 1 orientation 4 (b). Using the phantom’s response to the AR-SE algorithm it was determined that epineurium thickness can be estimated from the FWHM of the contiguous scatterer peak observed at the nerve outer boundary.

Visual identification of the layer epineurium tissue, as a contiguous area of large diameter scatterers along the surface of the nerve, is an observation not reported in the original study. The distributions of thickness estimates produced with the AR-SE technique are generally right skewed with a long right-hand tail, which makes the means a poor measure for comparison, Fig. 4. Indeed, none of the AR-SE thickness estimate means were in agreement with their corresponding histology by light microscopy (two sample Kolmogorov–Smirnov test, $\alpha =0.05$). Estimate medians (Wilcoxon rank sum test, $\alpha =0.05$) and analysis of variance (Kruskal–Wallis test, $\alpha =0.05$) of tissue sample 1 orientation 4, all orientations of tissue sample 2, and tissue sample 3 orientation 4 were in agreement with their corresponding histology by light microscopy.

Fig. 4

Distribution of epineurium tissue layer thickness estimates for tissue samples (a) 1, (b) 2, and (c) 3. Within each tissue sample boxplot, estimates from histology by microscopy (black), the five A-scans window method within the ADRP technique (red), and AR-SE technique with a $10\times 10$ window (blue), are grouped together for each orientation.

It is evident in the box and whisker plots of AR-SE estimates in Fig. 4 that in all tissue samples and orientations there was at least one, and typically several, thickness estimates of $0\mu \mathrm{m}$, which is biologically impossible.

3.2.

Fiber Diameter Distribution

3.2.1.

Histological analysis

Tissue sample 1 orientation 4 contained easily identifiable landmarks—in the form of two “humps”—for alignment to corresponding OCT images, and contained two fascicles with a difference in fiber diameter distributions that was easily perceptible at $10\times$ and $40\times$ magnifications, Fig. 5(a). Tissue sample 1 orientation 4 was therefore selected for fiber diameter distribution analysis. A minor histological artifact is evident in this sample where the left fascicle has been pulled away from the right fascicle during slicing on the microtome, box and whisker plots of the fiber diameter distributions within the left (grids 1 to 9) and right (grids 10 to 18) fascicles showed higher medians, upper and lower quartiles, and maximum values in the majority of grids in the right fascicle compared with grids of the left fascicle, Fig. 5(b). Further analysis, using a one-way ANOVA with $\alpha =0.05$, revealed significant difference between the means of grids 1 and 3 within the left fascicle, and no significant difference between the means of grids in the right fascicle. The left and right fascicles possessed diameters of $6.1+/-2.1\mu \mathrm{m}$ and $10.1+/-3.0\mu \mathrm{m}$ (mean $+/-1$ standard deviation), respectively. A one-way ANOVA, $\alpha =0.05$, also revealed significant difference in fiber diameter between the two fascicles, indicating a heterogeneous fiber diameter distribution at the fascicle level.

Fig. 5

Light-microscopy image of histology at $10\times$ magnification of tissue sample stained with LF blue (a), scale bar is $100\mu \mathrm{m}$. Numbered grid used in fiber diameter distribution calculations are shown overlaid on the light microscopy image (a), with results presented in box and whisker plots (b) confirming significant difference in fiber diameter distribution between the left (grids 1 to 9) and right (grids 10 to 18) fascicles.

3.2.2.

Autoregressive spectral estimation technique

In each of the microsphere samples, the autoregressive reflection coefficients decayed rapidly toward zero, Fig. 6(a), which specified low autoregressive orders for calculating the power spectral density. Weighted PCA of the power spectral densities produced similar, but distinguishable, eigenvectors for all three microsphere samples, Fig. 6(b), where, in all cases, the first variable contributed the most to the first two principal components, and the second variable contributed the most to the third principal component. The AR-SE technique estimated the diameters of microsphere samples of 3.8-, 5.33-, and $8.49\text{-}\mu \mathrm{m}$ diameter as $3.6+/-0.90$, $5.0+/-1.2$, and $7.9+/-2.1\mu \mathrm{m}$ ($\text{mean}+/-1\text{standard deviation}$), respectively, Table 2. These results show a good diameter estimate accuracy ($t$-test, $\alpha =0.05$) but with high probability of large ($>25\%$) errors in individual measurements.

Fig. 6

(a) The first 40 autoregressive reflection coefficients, used to determine the autoregressive order when calculating the power spectral density, showing a rapid decay toward zero for all three the microsphere samples. (b) Contributions of the first three variables, $v_1$, $v_2$, and $v_3$, to the first three principal components, component 1, component 2, and component 3, from weighted PCA of the power spectral densities of (i) 3.8-, (ii) 5.33-, and (iii) $8.49\text{-}\mu \mathrm{m}$ diameter microsphere samples.

Table 2

Scatterer diameter estimates, in μm, of microsphere samples and the intra-fascicle volume of the left and right fascicles in tissue sample 1 orientation 4. Estimates from histology, a 10×10 window in the AR-SE technique, the “CoD-SE: conventional” technique, which selects each estimate based on minimization of the standard deviation, and the “CoD: normalized s.d. technique, which selects each estimate based on minimization of the normalized standard deviation. Data are presented as mean (standard deviation) in μm.

	Manufacturer’s data/histology (μm)	AR-SE (μm)	CoD-SE: conventional (μm)	CoD-SE: normalized s.d. (μm)
$3.8\text{-}\mu \mathrm{m}$ microspheres	3.8 (0.25)	3.6 (0.9)	4.3 (1.5)	4.3 (1.5)
$5.33\text{-}\mu \mathrm{m}$ microspheres	5.33 (0.25)	5.0 (1.2)	4.6 (1.5)	5.7 (1.4)
$8.49\text{-}\mu \mathrm{m}$ microspheres	8.49 (0.25)	7.9 (2.1)	6.3 (1.9)	6.3 (1.9)
Tissue sample 1 orientation 4
Left fascicle	6.9 (2.2)	7.6 (4.8)	9.7 (3.3)	10.4 (4.4)
Right fascicle	10 (3.1)	8.1 (4.6)	9.6 (3.2)	10.9 (4.4)

On OCT data from tissue sample 1 orientation 4, Fig. 7(a), the AR-SE technique predicted a heterogeneous distribution of fiber diameters in both fascicles, Fig. 7(b). A small difference between the scatterer diameter distributions in the left and right fascicles was predicted, with means of 7.6 and $8.1\mu \mathrm{m}$, respectively, Table 2, and medians of 8.2 and 9.0, respectively, Fig. 7(c). The diameter distribution estimates were significantly different to those produced from the corresponding histology by light microscopy images ($t$-test, $\alpha =0.05$; and Wilcoxon rank sum test, $\alpha =0.05$), Fig. 7(c). Therefore, in the example analysis of this image, the AR-SE technique correctly predicted a larger scatterer diameter distribution in the right fascicle but did not accurately quantify the difference between the two fascicles’ distributions.

Fig. 7

(a) OCT data of tissue sample 1 orientation 4 and (b) the same image after processing with AR-SE technique, scale bars in (a) and (b) are $100\mu \mathrm{m}$. In (b) the epineurium layer is visible as a contiguous layer of large diameter scatterers, with diameter on the color scale. The white outlined region in (b) indicate the areas where data were used to produce the fiber diameter distribution estimate with the CoD-SE algorithm for each fascicle, which are compared against the results from histology by light microscopy in a box and whisker plot (c). The AR-SE technique correctly predicted a larger scatterer diameter distribution in the right fascicle but did not accurately quantify the difference between two fascicles’ distributions.

3.2.3.

Correlation of the derivative spectral estimation technique

The CoD technique estimated the diameters of microsphere samples of 3.8-, 5.33-, and $8.49\text{-}\mu \mathrm{m}$ diameter as $2.6+/-0.7$, $2.8+/-0.5$, and $4.5+/-0.3\mu \mathrm{m}$ ($\text{mean}+/-1\text{standard}\text{deviation}$), respectively, which indicates poor diameter estimate accuracy for all microsphere diameters, Table 2. All estimates corresponded to a scanning window axial size of 4 pixels, selected to minimize the standard deviation, whereas the smallest window axial size used in the original study was 5 pixels,²¹ Figs. 8(a)–8(c). Excluding window axial sizes of 3 and 4 pixels from consideration provided a significant improvement in estimate accuracy: $4.3+/-1.5$, $4.6+/-1.5$, and $6.3+/-1.9\mu \mathrm{m}$, respectively. For all microsphere samples, the magnitude of diameter estimates showed a decreasing trend with decreasing axial window size, as did the standard deviation of estimates, which results in smaller diameter estimates being favored. Selection via minimization of the normalized standard deviation removed the effect of the downward trends, and, with window axial sizes of 3 and 4 pixels excluded, increased the estimate for $5.33\text{-}\mu \mathrm{m}$ microspheres to $5.7+/-1.4\mu \mathrm{m}$, Figs. 8(d)–8(f). Estimates for 3.8 and $8.49\mu \mathrm{m}$ microspheres remained unchanged, Table 2.

Fig. 8

(a–c) Mean and standard deviation, or (d–f) mean and normalized standard deviation, of diameter estimates produced by each window size on microsphere samples. (d–f) Normalizing the standard deviation, (a–c) removes the decreasing trend with window size which seen in standard deviations. The CoD bandwidth across the scatterer diameter range of 1 to $16\mu \mathrm{m}$, generated using refractive indices of water and polystyrene for use with microsphere samples, and refractive indices of bovine tendon and myelin for use with peripheral nerve samples (g).

On OCT data of nerve tissue, pixels containing air produced an estimate of $1.775\mu \mathrm{m}$, which corresponded to a lag of 1500(a.u.) and was the limit of considered range of the power fit to the theoretical CoD curve, Fig. 8(g). As mentioned earlier, in Sec. 3.1.4, epineurium produced estimates within the range of 2.2 to $4\mu \mathrm{m}$. Therefore, to remove air and epineurium, scatterer diameter estimates of $\le 4\mu \mathrm{m}$ were discarded in our analysis of fiber diameter distribution. On OCT data from tissue sample 1 orientation 4, Fig. 9(a), the CoD-SE algorithm predicted a heterogeneous distribution of fiber diameters in both the left and right fascicles, Fig. 9(b). A small difference between the scatterer diameter distributions in the left and right fascicles was predicted, with means of 10.4 and $10.9\mu \mathrm{m}$, respectively, Table 2, and medians of 10.0 and $10.6\mu \mathrm{m}$, respectively, Fig. 9(c). As for the AR-SE technique, the normalized CoD-SE technique correctly predicted a larger mean scatterer diameter in the right fascicle, but did not accurately quantify the difference.

Fig. 9

OCT data of tissue sample 1 orientation 4 (a) with white boxes indicating the areas of the left and right fascicles used for analysis with the CoD-SE algorithm, and the white scale bar bottom left indicating $100\mu \mathrm{m}$. The boxed areas in (a) are shown after processing with CoD-SE algorithm in (b) and (c), for the right and left fascicles, respectively, with diameter on a color scale. The diameter distribution of estimates within areas (b) and (c), with values below $4\mu \mathrm{m}$ excluded, are compared against results from histology by light microscopy in a box and whisker plot (d). The normalized CoD-SE technique correctly predicted a larger mean scatterer diameter in the right fascicle, but did not accurately quantify the difference.

3.3.

Adipose Tissue Identification

3.3.1.

Average depth-resolved profile technique

Using the 5 A-scan sliding window method and calculating the threshold from the entire portion of the residual SNR curve, regions with adipose tissue apposed to the outer nerve boundary cross the threshold several times throughout the depth, Figs. 10(b) and 10(c). In contrast, regions with epineurium crossed the threshold only once Figs. 10(a) and 10(c). Thus, layers of adipose tissue were qualitatively distinguishable from epineurium tissue, a finding not reported in the original study.¹⁸ Further studies on this effect are needed to establish the repeatability of this observation. It was not possible to identify the outer epineurium boundary beneath adipose tissue, and, as a result, the thickness of adipose tissue could not be estimated, nor was it possible to identify the uniformly decaying region indicating the intrafascicle volume beneath adipose tissue. The presence of adipose tissue, therefore, reduces the amount of information available from the ADRP technique.

Fig. 10

Residual SNR curve of epineurium (a) is clearly distinguishable from adipose tissue (b) as the threshold, when calculated from the entire residual curve, intercepts with a single large peak. A color map of tissue sample 1 orientation 2 produced by the ADRP-five A-scan method showing only above-threshold residual SNR data (c) with white scale bar bottom left indicating $100\mu \mathrm{m}$; a significant amount of adipose tissue is visible.