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The size-dependent spectral variations, predicted by Mie theory, have already been considered as a contrast enhancement mechanism in optical coherence tomography. In this work, a new spectroscopic metric, the bandwidth of the correlation of the derivative, was developed for estimating scatterer size which is more robust and accurate compared to existing methods. Its feasibility was demonstrated using phantoms containing polystyrene microspheres as well as images of normal and cancerous human colon. The results are very promising, suggesting that the proposed metric could be utilized for measuring nuclear size distribution, a diagnostically valuable marker, in human tissues.
© 2017 Optical Society of America
Corrections
1 June 2017: A typographical correction was made to the title.
1. Introduction
A large proportion of all human cancers (~90%) arises in epithelial tissues where they remain confined, sometimes for years. These cancers become exponentially more lethal as soon as they invade through their underlying basement membrane and begin to spread to other tissues. At the pre-invasive stage, cancers are readily treatable but these early lesions are often almost impossible to detect [1]. A seminal histopathologic feature, that is crucial in the diagnosis of early cancer, is the variability in the cell nuclei which become pleomorphic, crowded, and enlarged. While the diameter of non-dysplastic cell nuclei is typically 5-10 μm, dysplastic nuclei can be as large as 20 μm across. A tool that could non-invasively detect such changes, in vivo and in situ, could be extremely valuable in the early diagnosis of cancer.
In the last two decades, research efforts have focused on measuring the size of epithelial cell nuclei using Light Scattering Spectroscopy (LSS) [1–4]. Recently, LSS has been combined with low-coherence interferometry (LCI), offering the possibility of depth localization of the LSS signal [5–8]. OCT has also been introduced as an excellent candidate for spectroscopic, depth-resolved, imaging. Spectroscopic OCT (SOCT) extracts the localized spectra that are inherently available in the OCT signal [9,10]. In order to fully exploit the potential of SOCT, there has been considerable effort to find spectroscopic metrics that could accurately measure the size of cell nuclei. Most of the methods proposed have been based on the assumption that epithelial cell nuclei can be considered as spheroidal scatterers whose interactions with light are described by Mie theory [11,12]. One approach, to get their nuclear size, has been to curve-fit the backscattered spectra, obtained from the OCT or LCI signal, to the theoretical prediction curves [13–15]. The drawback of these methods is that they require an exhaustive search through many possible scattering sizes and precise knowledge of the refractive index of the scatterer and the surrounding medium [16]. Moreover, they do not adequately account for the spectral shifts that may occur in the experimental measurements if the incident beam waist spot size is small, or comparable to the wavelength, since Mie theory was not developed for Gaussian beams but rather for plane waves [17]. On the other hand, many studies have been based on the observation, derived again from Mie theory, that as the particle size increases, so does the oscillation “frequency” imparted on the backscattered spectrum. Wax et al [6] proposed some pre-processing of the backscattered spectrum followed by a Fourier transform. The location of the maximum of this function characterizes the dominant scattering features in the analyzed region. This method, along with many studies on phantoms and biological tissues that followed, yielded quite accurate estimates of the scatterer sizes [6,14,16–20]. However, when the light source spectrum is limited, it does not allow for a sufficient number of oscillations to be captured in the backscattered spectrum and becomes increasingly difficult to distinguish the maximum peak from other, low-frequency, components [21]. Another approach for estimating the scatterer size, proposed by Adler et al., relied on measuring the autocorrelation width of the backscattered spectrum [22]. The key observation was that backscattered spectra with high spectral modulation produce autocorrelation functions that fall of rapidly away from the central point, while the autocorrelation function of spectra with low spectral modulation is broader. Using this information, the contrast of OCT images was enhanced. The advantage of this spectroscopic analysis technique is that it does not depend on the distribution of optical power over absolute wavelength and thus it is insensitive to major sources of spectroscopic noise. However, it does not always result in an accurate prediction of the scatterer size. Variations of the above techniques were adopted in other studies. For example, Oldenburg et al. used the autocorrelation width of backscattered spectra, at 80% of the peak value, in order to enhance the contrast of OCT images of macrophages and fibroblasts [21], whereas Kartakoullis et al. and Jaedicke et al. combined the spectral information with principal component analysis (PCA) and clustering algorithms with the intention of differentiating phantom samples consisting of microspheres with different diameters [23,24]. In addition, Tay et al. suggested that the use of multiple bandwidths can improve the sensitivity of scatterer size estimates, demonstrating the technique in spectroscopic OCT images where solutions of 0.5 and 45 μm microspheres could be clearly distinguished [25].
In this study, Mie theory was used to create a new metric for SOCT, the bandwidth of the correlation of the derivative (COD). The feasibility, accuracy and robustness of this new method, in estimating scatterer size, has been validated using images from microsphere phantoms and gastrointestinal normal and cancerous tissue. These preliminary results have proven that scatterer size can be accurately estimated and further investigation of the technique, as a diagnostic tool for cancer, is warranted.
2. Theory and methods
2.1 The bandwidth of the correlation of the derivative from Mie theory
Mie theory was used to develop a new metric for the prediction of scatterer size. Figure 1(A)-1(C) illustrate the backscattered spectra for 6.0, 10.0 and 16.0 μm scatterers in the wavelength range of a common OCT system which operates between 1230 and 1390 nm. As expected, there are characteristic oscillations in the spectra. In order to extract the scatterer size, the first derivative of the spectrum was taken followed by its autocorrelation. The use of the derivative before calculating the autocorrelation is critical since its self-normalizing properties eliminate differences in the peak intensities of the spectra and, therefore, allow for more emphasis on the oscillations. The lag location of the first minimum of the autocorrelation is used as a metric for scatterer size estimation and is referred to as the bandwidth of the Correlation of the Derivative (COD). The COD bandwidth was found, from Mie theory, to be related to the scatterer size. Figure 1(D)-1(F) illustrate the COD function for the case of 6, 10 and 16 μm diameter scatterers. The red arrows indicate the COD bandwidth. Figure 2 is the graph of the COD bandwidth as a function of the scatterer size for diameters in the range of 1 to 20 μm. The Mie-derived curve (blue line) exhibits a strong and nearly monotonic relationship between the two parameters for diameters larger than 4 μm. Therefore, this relationship can be used to estimate the scatterer size from the COD bandwidth of an OCT spectroscopic image. However, reliable scatterer estimates using the proposed method can only be performed for scatterers above 4 μm (e.g. for cell nuclei but not for mitochondria). For practical purposes, especially during the experimental verification process, the 4th order approximation of this curve (Fig. 2, red line) was used to estimate the scatterer diameter corresponding to each spectrum.
Fig. 1 Backscattering Mie Spectra for (A) 6 μm, (B) 10 μm and (C) 16 μm scatterers with medium and sphere refractive indices set at 1.47 and 1.59 respectively. The parameters for the calculations were chosen according to the specifications of the light source and the microsphere samples used in the experiments. Graphs D-F show the Correlation of the Derivative (COD) with the red dot indicating the first minimum and the red arrow indicating the bandwidth of the COD
Fig. 2 Correlation of the Derivative (COD) bandwidth plotted as a function of scatterer size. The blue line corresponds to the theoretical curve and the green line to the 4th order approximation curve. Reliable scatterer estimation can be performed only in the region above 4 μm which demonstrates a fairly monotonic relationship (solid red line).
It should be noted that the relation between the COD bandwidth and the scatterer size depends on the wavelength and the spectral range of the light source and the refractive indices of the medium and scatterer. These dependencies are illustrated in Fig. 3. For ease of visualization only the approximation curves are shown. In addition to the degree of dependency, one can also observe, in these figures, changes in the monotonicity of the curves which imply that the technique may not be applicable in certain combinations of sources and scatterers since the slope of the prediction curve is either too small or even inverted.
Fig. 3 Curves illustrating the COD bandwidth dependence on matrix index of refraction (A), source bandwidth (B), and source wavelength (B & C). Reliable scatterer estimation can be performed only in the region above 4 μm (solid lines).
2.2 OCT images of microsphere phantoms
To evaluate the feasibility of the COD bandwidth as a spectroscopic metric for estimating the scatterer size, phantom samples were created containing polystyrene microspheres (Polybead®, Polysciences, Inc., Warrington, Pennsylvania) with 6 ± 0.58, 10 ± 3.04 and 16 ± 2.56 μm nominal diameters, embedded in an acrylamide gel. The microsphere phantoms were constructed using a process previously described in the literature [23]. The refractive index of the matrix was found to be 1.47 whereas the refractive index of the microspheres, according to manufacturer, was 1.59. Hence, the relative refractive index (1.08) approximates biological conditions, where the relative refractive index of subcellular organelles in cytoplasm is in the range between 1.03 and 1.10 at visible wavelengths [12]. The OCT images of the microsphere samples were acquired using a Santec IVS-300 Swept-Source OCT system (Santec Corp., Komaki, Japan). The system has a center wavelength of 1310 nm with 160 nm full range (90 nm FWHM), axial and lateral resolutions of 12 and 22 μm respectively and an imaging depth range of 4 mm (in air). The incident power on the samples was ~1 mW while the system sensitivity was about 100 dB.
2.3 Spectral processing
To extract the depth-resolved spectrum from the OCT images, a short time Fourier transform (STFT) of a moving Gaussian window was used. The Gaussian window isolates the portion of the signal that corresponds to the neighborhood under observation and reduces the side-lobes of the resulting FFT peaks (compared to e.g. a rectangular window). Subsequently, the depth-resolved spectrum was divided by the source spectrum (obtained from an OCT image of a mirror) so as to obtain the backscattering profile of the sample at each corresponding spatial region. The spectra are, then, smoothed by median and low pass (LPF) filtering to avoid noise amplification during the differentiation process. A zero-phase, 2nd-order, low-pass, Butterworth filter, with a cutoff at 0.07 nm−1, was used. Moreover, the edges of the spectra were removed since the borders of the source spectrum are lower in intensity compared to the center spectral region and therefore the edges of the depth-resolved spectra are more susceptible to noise than the central part. Noise can be very detrimental to scatterer estimation since it can affect the estimation of the first minimum of the correlation of the derivative by as much as half a period which in terms of scatterer size is 20-60% error depending on the slope of the curve of Fig. 2. Differentiation is performed by obtaining the difference between adjacent values in the spectrum. Figure 4 shows an OCT image of a 16 μm microsphere sample with a Gaussian window used in the spatial/spectral analysis and the depth-resolved spectrum derived from that particular region before and after dividing by the source spectrum. The OCT image of Fig. 4, suffers from ghost artifacts which are probably caused by the existence of point-spread function (PSF) side-lobes and/or internal reflections in the microspheres [26]. However, the artifacts do not significantly affect the spectral calculations.
Fig. 4 (A) Portion of an OCT image of a 16μm microsphere phantom. The green rectangle shows the extent of the Gaussian window used while the shorter red rectangle denotes the Gaussian standard deviation of this window (B) Depth-resolved spectrum corresponding to the region marked in (A). (C) Spectrum of the light source. (D) The ratio of the depth-resolved spectrum and the source spectrum corresponding to the backscattering spectrum of the sample in the examined spatial region. (E) Derivative of the spectrum in D. (F) Autocorelation of the derivative in E. (The red dot indicates the first minimum.)
It should be noted that the location of the Gaussian window with respect to the scatterer may affect the spectrum. When the window is shifted with respect to depth, it affects the intensity but does not change the shape of spectrum. Figure 5 shows example spectra with the Gaussian window at different depths. In this case, regardless of the window position in the axial direction, as long as the Gaussian width includes the interface between the scatterer and the matrix, the spectrum will have the same shape which, in the specific example, consistently leads to a correct estimation of the diameter (~10 μm). When the window includes the entire axial length of the scatterer and the interface at both the upper and lower sides, the signal results in a spectrum of the highest intensity. However, there is a more significant dependence of the depth-resolved spectrum on displacements in the lateral direction. Figure 6 illustrates the spectra obtained from three different but adjacent lateral locations. The maximum intensity spectrum occurs when the spatial window is in the center of the scatterer and as the window moves to adjacent positions, the intensity falls off and also, more importantly, the waveform shape changes slightly. This is expected since, as the beam is displaced, only a portion of the sphere is within the focal volume, thus the beam crosses a smaller effective diameter. The remedy to this problem was to apply an intensity threshold (5 dB above the noise floor), below which the spectra were ignored, and also compare 3 adjacent spectra and assign to each location the spectrum with the highest intensity.
Fig. 5 Dependence of the backscattering spectrum on the axial location of the Gaussian window. On the top are portions of OCT images of a phantom with 10 μm diameter microspheres and at the bottom are the resulting backscattering spectra. Centering on (A), above (B), and below (C) the microsphere only affects the intensity but not the shape of the spectrum.
Fig. 6 Dependence of the backscattering spectum on the lateral location of the Gaussian window. On the top are portions of OCT images of a phantom with 10 μm diameter microspheres and at the bottom are the resulting backscattering spectra. Centering on (A), left (B), and right (C) of the microsphere affects both the intensity and the shape of the spectrum.
2.4 Application of the COD bandwidth metric to gastrointestinal images
To demonstrate the applicability of the novel COD bandwidth metric to human tissues, the technique was applied to ex vivo OCT images of normal and cancerous colon obtained immediately post excision from patients who were scheduled for surgical removal of their tumors. These very preliminary results are included purely as a demonstration of the applicability of the proposed technique to highly scattering tissue samples. Eleven normal and 14 abnormal images, as confirmed by histology, were included in this preliminary study. The technique described above was used to calculate the scatterer size from each OCT image using different Gaussian windows (50-300 μm). The mean and median values from each image were used as a feature set input to a Linear Discriminant Analysis (LDA) algorithm. All window values were used so as to avoid introducing any bias based on the selection of the window while at the same time include the window size that provides the optimized estimation. The samples were, thus, classified as normal or abnormal using LDA and leave-one-out-cross-validation (LOOCV). The result was a normal vs. abnormal classification which was compared to the tissue stage, as determined by histology, to provide a correct classification rate. For comparison the samples were also classified using the autocorrelation width, described in the literature, as a classification feature [22].
3. Results and discussion
The feasibility of this technique was initially demonstrated on phantom samples containing polystyrene microspheres, embedded in an acrylamide gel. For the spectral estimation, the short time Fourier transform (STFT) with a Gaussian window function was used. The COD bandwidth calculated for each portion of the image (50-300 μm window) was compared to the curve of Fig. 2 to estimate the scatterer size. The window size was variable and was optimized on a per sample case to minimize the standard deviation of the resulting scatterer size distribution. This was deemed necessary since the mean and standard deviation of the scatterer distribution varied slightly with window size but resulted in the best estimate when the window was selected as described. The scatterer size was encoded in the color map of the OCT images as a hue pseudocolor scale [24]. Figure 7 shows typical spectroscopic OCT images of microspheres created with the COD bandwidth technique as well as histograms of the scatterer size distribution. The median and the standard deviation of these distributions are in agreement with the expected values (6.0 ± 0.7 μm, 9.0 ± 1.2 μm and 16.6 ± 0.9 μm for the 6, 10 and 16 μm microsphere samples respectively).
Fig. 7 Spectroscopic images of the microsphere samples. SOCT images of the (A) 6 μm, (B) 10 μm, and (C) 16 μm microsphere samples with corresponding histograms of scatterer size distribution (D-F).
Preliminary results of the application of the COD bandwidth technique on human colon images, from normal and cancerous regions, show distinct differences in the scatterer size with a mean change from ~10.1 ± 0.7 μm to ~13.9 ± 1.4 μm (Figs. 8, 9, and 10). The distribution of the scatterer diameters for the two populations, normal and adenocarcinoma, are statistically significant with a t-test p-value of 2.5x10−8. Both the increased diameter and the increased standard deviation are consistent with the expected nuclear neoplastic changes in cancer (i.e. increased nuclear size and pleomorphism.) In addition, the classification of the samples as normal and abnormal using LDA and LOOCV resulted in 96% correct classification. If classification is performed using only one, fixed, window size, the results are significantly degraded since the scatterer estimation is not optimized. For example, using only the 50 μm window the correct classification rate is reduced to 80%. When using the same LDA and LOOCV procedure with the width of the aurocorrelation metric instead, the correct classification rate was only 70%.
Fig. 8 (A) Portion of an OCT image of normal colon tissue. (B) The filtered scatterer spectrum corresponding o the Gaussian window indicated in A (Green line: extend, Red: standard deviation). (C) The derivative of the spectrum in B. (D) The autocorrelation of the derivative in C (Red dot: the first minimum). The Gaussian window used was 75 μm. (E-H) The same as above for colon adenocarcinoma with a Gaussian window of 225 μm.
Fig. 9 OCT image (A) and corresponding histology (B), the spectroscopic image (C) and histogram of scatterer size distribution (D) of normal human colon. Adenocarcinoma of the colon images are also shown (E-H).
Fig. 10 Histograms of the distribution of the mean scatterer size of the normal and fourteen tissue images.
4. Conclusions
In conclusion, a new spectroscopic metric for OCT analysis, the COD bandwidth, was developed. This metric derives information regarding the modulation of depth-resolved spectra and estimates the size of the scatterers in a sample based on a relationship predicted by Mie theory. The effectiveness of this metric was demonstrated on OCT images of microsphere samples, with diameters of 6, 10 and 16 μm, and normal and abnormal human colon images. The performance of the COD bandwidth is more robust to noise and artifacts than other techniques thus providing an accurate method for calculating the scatterer size and, also, classifying human tissue. The success of this preliminary study warrants further investigation to both optimize the methodology for in vivo use and to verify its diagnostic potential using a larger number of images from tissues of varying degrees of pathology spanning from normal to invasive disease.
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