## Abstract

In a recent paper by Bosschaart et al. [Biomed. Opt. Express 4, 2570 (2013)]
various algorithms of time-frequency signal analysis have been tested for their performance in blood
analysis with spectroscopic optical coherence tomography (sOCT). The measurement of hemoglobin
concentration and oxygen saturation based on blood absorption spectra have been considered. Short
time Fourier transform (STFT) was found as the best method for the measurement of blood absorption
spectra. STFT was superior to other methods, such as dual window Fourier transform. However, the
algorithm proposed by Bosschaart et al. significantly underestimates values of blood oxygen
saturation. In this comment we show that this problem can be solved by thorough design of STFT
algorithm. It requires the usage of a non-gaussian shape of STFT window that may lead to an
excellent reconstruction of blood absorption spectra from OCT interferograms. Our study shows that
sOCT can be potentially used for estimating oxygen saturation of blood with the accuracy below
1% and the spatial resolution of OCT image better than 20 *μm*.

© 2014 Optical Society of America

## 1. Introduction

Spectroscopic optical coherence tomography (sOCT) is an extension of OCT allowing to obtain spatially resolved spectroscopic information on a biological tissue. It was introduced by Morgner et al. in terms of time-domain OCT [1] and Leitgeb et al. in terms of spectral-domain OCT [2]. Among different applications [3–6], several attempts of applying this technique for a blood analysis have appeared [7–9]. Other authors have also presented theoretical analyses of various methods for the time-frequency signal analysis of sOCT [10–12]. Main of the examined algorithms were short time Fourier transform (STFT), Wigner-Ville distribution (WV) and dual-window method (DW) introduced by Robles et al. [12]. Other algorithms such as wavelength transform, windowed ARMA modeling or Cohen class distributions have also been examined [11].

DW provides high resolution in both z- and k-domains [6,13]. However, in their recent paper, Bosschaart el al. showed that it may lead to an erroneous shape of the recovered spectra [14]. It is particularly important in the case of the measurement of blood oxygen saturation.

In their method, Boschaart el al. modelled the absorption spectrum of blood by equation [14]

*μ*(

_{a}*k*),

*μ*

_{a,HbO2}(

*k*) and

*μ*(

_{a,Hb}*k*) are absorption spectra of blood, oxygenated hemoglobin and non-oxygenated hemoglobin respectively, whereas

*tHb*is hemoglobin concentration in blood and

*SO*

_{2}is oxygen saturation.

The proposed measurement method consists in the determination of
*μ _{a}*(

*k*) by means of sOCT and finding parameters

*tHb*and

*SO*

_{2}by a non-linear optimization. The goal of the optimization is to minimize mean square difference between a measured spectrum

*μ*(

_{a}*k*) and theoretical one, calculated with Eq. (1). In such a measurement precise reconstruction of

*μ*(

_{a}*k*) is crucial.

In their paper, Bosschaart et al. considered the measurement of blood layer with optical
thickness 25*μm. SO*_{2} = 70.0% and
tHb=146.9 g/l were the best presented estimation of blood parameters for exact values
*SO*_{2} = 85.0% and tHb=150.0 g/l. These results
were obtained by applying STFT method with gaussian window with width of
22*μm*. In case of DW method, the best reported results were
*SO*_{2} = 45.8% and tHb=145.8 g/l [14].

The described results indicate that the measurement of blood oxygen saturation with sOCT is unreliable and inherently lead to large errors. In this comment we show that these errors are caused by gaussian-shaped window, not by the application of STFT itself. We argue that the proper choice of STFT window allows the theoretical determination of *SO*_{2} with accuracy better than 1% and tHb with accuracy better than 1 g/l even in the presence of noise.

Time-frequency signal analysis in sOCT can be performed for interferograms being a function of optical path length [1] or a function of wavenumber [2]. In further part of this manuscript we will denote them as z- and k-domain analyses respectively. Following Bosschaart et al. [14] we will focus on z-domain analysis.

## 2. Impact of STFT window on the reconstruction of light spectrum

Both STFT and DW method consist of calculating Fourier transform of interferometric signal multiplied by window function. Window function limits interferogram to a single peak in the case of TD-OCT or narrow wavelength range in the case of SD-OCT. To the best of our knowledge, application of only gaussian-shaped windows have been reported since the introduction of sOCT. The advantage of gaussian function is the fact that product of its duration and bandwidth is smaller than for any other functions. Additionally, Fourier transform of gaussian function does not possess any sidelobes.

However, gaussian window has a significant disadvantage when applied for the reconstruction of light spectra in sOCT. The value of gaussian function decreases immediately after its maximum. This causes different amplification of various parts of the peak in OCT scan and consequently changes the shape of the recovered spectrum. An example of such situation is presented in Fig. 1.

To avoid errors in are covered spectrum, the width of a gaussian window should be much greater than the actual width of a pulse in OCT scan [10]. However, this is undesirable because wider window causes the drop of scan resolution. A solution to this contradiction is the use of window with non-gaussian shape. The desirable function should be flat in the range where the energy of OCT peak is high and then rapidly drop to very low values.

Designing such windows is a well known problem of digital signal processing which can be solved by application of digital filter design methodology [15, 16]. Such methodology is especially convenient for k-domain sOCT analysis often used in SD-OCT [2]. If the shape of the window in z-domain is considered as a transmittance of digital filter, the shape of the window in k-domain would be the impulse response of such a filter.

In the following section, we present the impact of window shape on blood parameters measurement with the method described by Bosschaart et al. [14].

## 3. Impact of window choice on blood parameters measurement

We have conducted simulations similar to those presented by Boschaart et al. [14] for three different types of windows. OCT light source
with gaussian spectrum, central wavelength 546 nm and FWHM in k-domain 2.35
*μm*^{−1} was assumed. OCT scan of a single blood layer with
the optical width 45 *μm* was simulated. The parameters of blood were chosen
as tHb=150 g/l and *SO*_{2} = 85%. The absorption
spectrum of blood was calculated using Eq. (1) with
the data from [17]. OCT scan and STFT spectra
were calculated as described in [14]. Blood
parameter tHb and *SO*_{2} were then estimated by a non-linear optimization
using Levenberg-Marquardt algorithm routine from Matlab environment. Because of the low energy of
simulated light source at the edges of the spectrum, only part of the recovered absorption spectrum
from 500 nm to 600 nm was used for the non-linear optimization.

The spectra of light reflected from the first and the second boundary of the blood layers were calculated using the following windows:

- Rectangular window with width Δ
*z*= 12.8*μm*, given by:$$w(z)=\{\begin{array}{ll}1\hfill & \text{if}\hspace{0.17em}\left|z\right|<\mathrm{\Delta}z,\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$and where*z*was the maximal depth of the scan determined by a sampling frequency and_{max}*F*= 0.007 is normalized cut-off frequency. Parameter_{c}*F*is widely used in the methodology of digital filters design [15, 16], so its introduction is convenient during applying this methodology to sOCT._{c} - A smoothed rectangular window with normalized cut-off frequency
*F*= 0.007. This window was designed in k-domain by calculating the impulse response of a lowpass digital filter. The window shape in k-domain is equal to discrete Fourier transform of its shape in z-domain: where_{c}*W*(_{s}*k*) is an envelope window in k-domain and*k*= 2*π/λ*is the wavenumber. The window in z-domain*w*(*z*) is given by Fourier transform of*W*(*k*).By choosing

*W*(_{s}*k*) one can make the trade-off between the steepness of the window slope and the level of its sidelobes in z-domain. In this study parametric Chebyshev window was used as*W*(_{s}*k*). This window provides the maximal steepness of the slope for specified maximal level of sidelobes [18, 19].

The shapes of all windows in both z- and k-domain are presented in Fig. 3. Simulated OCT scan with windows in z-domain are presented in Fig. 2.

#### 3.1. Impact of window shape

The results of the blood parameter estimation for different windows are presented in Table 1. Figure 4 shows the comparison between exact spectra, spectra recovered by STFT and theoretical spectra calculated using tHb and *SO*_{2} estimated by the optimization algorithm. The simulations show that the usage of a rectangular and smoothed rectangular window provides much better reconstruction of *μ _{a}*(

*k*) than the usage of gaussian window.

The application of gaussian window caused small underestimation of tHb (by 3 g/l) and significant underestimation of *SO*_{2} (by 13%). These results are in agreement with Boschaart et al. [14].

The application of both rectangular and smoothed rectangular windows provided almost exact estimation of blood parameters. The improvement of estimation accuracy by two orders of magnitude comparing to the application of gaussian window was obtained.

#### 3.2. Resolution of the window

The definitions of Δ*z* for different types of windows (eq. (2) – (5)) are arbitrary, so it is needed to test how they correspond to real axial resolution of sOCT measurement. We performed an analysis similar to the one presented by Bosschaart et al. [14]. The third peak in OCT scans was added for depth larger than the peak corresponding to the second boundary of a blood layer. The spectra of the second and the third peak were identical and their separation varied from 5 *μm* to 40 *μm*. With these conditions gaussian window with Δ*z* = 22*μm* and both types of rectangular windows with Δ*z* = 20*μm* were examined.

Figure 5 presents an error of *SO*_{2} measurement for different values of peak separation. For peaks separated by less than 18 *μm* error of gaussian window method starts to oscillate. The error of estimation by STFT with gaussian window was higher for higher values of *SO*_{2}. This result is in agreement with Bosschaart et al. [14] where 18 *μm* was determined as the resolution of gaussian window.

The error of *SO*_{2} measurement with rectangular windows starts to increase for peak separation lower than 20 *μm*.

This indicates that the given definition of Δ*z* works well as intuitive definition of a window axial resolution and it allows to compare the accuracy of rectangular and gaussian windows.

#### 3.3. Impact of window size

In order to determine the impact of a window size on blood parameters estimation, we have repeated simulation from section 3.1. using windows with varying size. The heighest value of Δ*z* that was considered was equal to 70 *μm*. Our goal was to study only the impact of peak shape modification by the window. Therefore, the effect of covering two pulses by STFT window had to be removed and the peaks in OCT scan were separated by 100 *μm*.

Rectangular windows cover not only more energy of the useful signal, but also more energy of the noise. Therefore the simulations were repeated by adding white noise to inteferometric signal. The value of SNR=20 dB was chosen, since SNR of practical OCT systems is unlikely to drop below that level.

The results of the simulation are presented in Fig. 6. Only the estimation of *SO*_{2} was plotted, since it was much less accurate than the estimation of tHb. This is in agreement with the results presented by Boschaart et al. [14].

In the case of signal without the noise, the obtained estimation error of *SO*_{2} was smaller than 1% for rectangular and smoothed rectangular windows with Δ*z* > 13 *μm*. Te application of gaussian window with Δ*z* = 70 *μm* allowed to obtain an estimation error of *SO*_{2} equal to 1.5%.

In the case of signal with noise, bias error smaller than 1% and a standard deviation of the measurement equal to 3% were obtained for rectangular and smoothed rectangular windows with Δ*z* > 18 *μm*. The bias estimation error for STFT with gaussian window in the case with noise was at the same level as error in noiseless case. The level of bias error below 1% and the standard deviation of the measurement below 2.7% were obtained for window with Δ*z* > 60 *μm*. The bias errors and the standard deviations of the measurement have been calculated using the results of 30 simulations with different realisations of noise.

#### 3.4. Impact of oxygen saturation

We repeated the simulations from section 3.3. in the case of noiseless signal and window providing good estimation of *SO*_{2} (Δ*z* = 13 *μm* for both rectangular windows and Δ*z* = 70*μm* for gaussian window). Th exact value of *SO*_{2} was varying from 0 to 100 %.

Figure 7 shows the estimation errors of *SO*_{2} and tHb for different values of *SO*_{2}. The application of rectangular and smoothed rectangular window provided an accurate estimation of both parameters for each case.

## 4. Dual-window method

Dual-window (DW) method has been developed as a tool for the measurement of light spectra with high spectral and spatial resolution [12]. However, the results presented by Bosschaart et al. indicate that this method does not provide an accurate measurement of blood parameters by considered procedure, when gaussian window is applied [14]. In order to verify if it is also the case with the usage of rectangular windows, the following simulations have been performed.

OCT scan of a single blood layer as in section 3.1. have been simulated and blood parameters have been estimated using DW method with rectangular and smoothed rectangular windows. The width of the short window has been set to 5 *μm* and 22 *μm*. The width of the long window has been set to 50 *μm* and 75 *μm*. Values 5 *μm* and 75 *μm* have been chosen because they are either too low or too high to obtain an accurate measurement of blood parameters (the same values have been used for gaussian window in paper by Bosschaart et al.). Values 22 *μm* and 50 *μm* have been chosen because they provide an accurate measurement (Fig. 6).

Additionaly, blood parameters estimation have been performed using STFT method with each size of the window. The results of blood parameters estimation are presented in Table 2 and 3.

## 5. Discussion

The presented results indicate that the absorption spectra of blood can be accurately recovered with sOCT technique. However, a window with flat amplitude in the entire range of peak in OCT scan should be used.

It is important that the errors in absorption spectra recovered using gaussian window do not come from low resolution in neither z- or k-domain. Low resolution in z-domain causes overlapping of two peaks by the STFT window. In such a case additional, high frequency oscillation appears in recovered spectrum. Low resolution in k-domain causes the vanishing of rapid variations of the measured spectrum. In the absorption spectra of blood such rapid variations are not present, so obtaining high wavenumber resolution is not crucial. Therefore, accurate reconstruction of measured spectrum is a crucial factor in the measurement method proposed by Bosschaart et al..

It is important to notice that other types of window function used in SOCT (eg. the Hamming window) do not possess the region of flat amplitude as well as the gaussian function. Numerical experiments performed with standard windows – Hamming, Hanning and Blackman (results not shown) lead to a similar level of accuracy for *tHb* and *SO*_{2} determination as experiments with gaussian window.

To perform an accurate measurement, cut-off frequency of STFT window needs to be high enough to cover entire peak in OCT scan. On the other hand, too high cut-off frequency causes a drop of axial resolution of the system. Since the width of the OCT peak depends on the applied light source, optimal window needs to be designed for a particular OCT setup.

In the presented research, the performance of both smoothed and non-smoothed rectangular windows is almost the same. Sincethe design of smoothed window is more difficult than the design of non-smoothed one, its usage should be justified by better performance. Since the sidelobes of the window in k-domain are lower in smoothed version, a better resolution of spectrum measurement can be expected. However this is not crucial for the reconstruction of blood absorption spectrum, as discussed previously.

Tests of window axial resolution (Fig. 5) show that smoothed window works better when two peaks of OCT scans are overlapped by one window. This suggests that steepness of the window is a parameter that could be optimized for particular application of sOCT.

The tests of window resolution also show that the performance of a rectangular window drops faster than the performance of gaussian window in terms of required separation of two OCT peaks. However, *SO*_{2} error for a smoothed rectangular window examined in this research was similar or better than the error for gaussian window even for the separation of peaks as low as 10 *μm*. This again confirms that the method of blood parameter estimation presented by Bosschaart et al. requires good reconstruction of the absorption spectrum rather than a high axial resolution.

The results of blood parameters estimation using DW method and rectangular window functions (Table 2) show that this method can provide accurate measurement. However, the width of short window must be high enough to provide an accurate measurement using STFT method. For this reason, the application of DW method for the considered application does not lead to the improvement of the measurement spatial resolution. Still, results obtained with DW method for windows with width equal to 22 *μm* (short) and 75 *μm* (long) are actually more accurate than the results for STFT with 75 *μm* windows. This shows that DW method may be useful if the optimal size of the window is unknown. In such a case, if long window is too wide for a correct measurement, the drop of accuracy may be reduced by additional, shorter window.

The usage of *a priori* information about the light spectrum might be a different approach allowing for accurate estimation of blood parameters. Low-pass filtration of the measured spectrum caused by gaussian shape of the window can be predicted numerically. Using this information one may try to estimate the blood model parameters directly from this low-pass filtered spectrum taking into consideration the filtration effect. However, the absorption phenomenon is governed by Lambert-Beer law, so the light intensity is not a linear function of the absorption coefficient. For this reason, the spectrum of blood absorption coefficient obtained using the gaussian window is not simply weighted sum of low-pass filtered absorption coefficient spectra of oxygenated and deoxygenated hemoglobin. It still might be possible to overcome this problem by using more sophisticated models for blood parameters estimation using the curve fitting approach. The development of such a signal processing algorithm and testing whether it provides results better in terms of accuracy or spatial resolution might by studied in the future.

This presented study concerns only gaussian and rectangular windows. However, research on sOCT technique should not be limited to these simple choices. As an example, the optimization of signal-to-noise ratio may be concerned. A rectangular window provides the same amplification of each part of the OCT peak. However, some parts of the peak contain lower signal level with the constant level of noise. Perhaps attenuation of these parts of peak can provide better results for noisy interferograms. Therefore, the shape of the window should be fitted to the shape of the peak. This is a problem known from the adaptive filters theory [15, 20]. An application of such filters in sOCT may be a subject of future studies.

It is also important, that the analysis presented by Bosschaart et al. and repeated in this comment assumes that only two or three peaks are present in the OCT scan. However, in real case, it is highly possible that additional ambient signal from many scattering centers (eg. red blood cells) will be present near the analysed peak. If the peak is much higher than this ambient signal (eg. it comes from reflection on boundary of the blood layer) an accurate measurement may be still possible. However, if the height of the peak would be at the same level as the ambient signal (eg. it also comes from the small scattering center), the drop of accuracy should be expected. Additional signal processing methods, such as averaging many light spectra at different points of the tissue may help to overcome this problem. This may also be a subject of further studies.

## 6. Conclusion

In this comment, we have shown that changing the shape of the window in STFT can greatly improve the accuracy of measurement method proposed by Bosschaart et al. [14]. We have tested the performance of rectangular window in sOCT analysis and found that they can provide a better estimation of blood parameters than gaussian window with much bigger width. The error of *SO*_{2} estimation below 1% was obtained using rectangular window comparing to 13% obtained using gaussian window. This result may actually make the measurement method proposed by Bosschaart et al. accurate enough for clinical application.

Our result also shows that analysis of various time-frequency analysis methods in sOCT should not be limited to the choice of the algorithm and size of the applied window. Other aspects, such as the shape of the window also need to be considered. This proves that there is a need for new, more detailed studies of sOCT technique and its application in different conditions.

## Acknowledgments

This research work has been supported by The National Centre for Research and Development (NCBiR), Poland under grant no. LIDER/32/205/L-3/11 and DS program of Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology.

## References and links

**1. **U. Morgner, W. Drexler, F. X. Kartner, D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. **25**, 111–113 (2000). [CrossRef]

**2. **R. Leitgeb, M. Wojtkowski, A. Kowalczyk, C. K. Hitzenberger, M. Sticker, and A. F. Fercher, “Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography,” Opt. Lett. **25**, 820–822 (2000). [CrossRef]

**3. **C. Xu, D. Marks, M. Do, and S. Boppart, “Separation of absorption and scattering profiles in spectroscopic optical coherence tomography using a least-squares algorithm,” Opt. Express **12**, 4790–4803 (2004). [CrossRef] [PubMed]

**4. **C. Kasseck, V. Jaedicke, N. C. Nils, H. Welp, and M. R. Hofmann, “Substance identification by depth resolved spectroscopic pattern reconstruction in frequency domain optical coherence tomography,” Opt. Commun. **283**, 4816–4822 (2010). [CrossRef]

**5. **A. Karakoullis, E. Bousi, and C. Pitris, “Scatterer size-based analysis of optical coherence tomography images using spectral estimation techniques,” Opt. Express **18**, 9181–9191 (2010). [CrossRef]

**6. **Y. L. Li, K. Seekell, H. Yuan, F. E. Robles, and A. Wax, “Multispectral nanoparticle contrast agents for true-color spectroscopic optical coherence tomography,” Biomed. Opt. Express **3**, 1914–1923 (2012). [CrossRef] [PubMed]

**7. **D. Faber, E. G. Mik, M. C. G. Aalders, and T. G. van Leeuwen, “Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography,” Opt. Lett. **28**, 1436–1438 (2003). [CrossRef] [PubMed]

**8. **D. Faber, E. G. Mik, M. C. G. Aalders, and T. G. van Leeuwen, “Toward assessment of blood oxygen saturation by spectroscopic optical coherence tomography,” Opt. Lett. **30**, 1015–1017 (2005). [CrossRef] [PubMed]

**9. **D. Faber and T. G. van Leeuwen, “Are quantitative attenuation measurements of blood by optical coherence tomography feasible?” Opt. Lett. **34**, 1435–1437 (2009). [CrossRef] [PubMed]

**10. **B. Hermann, K. Bizheva, A. Unterhuber, B. Povazay, H. Sattmann, L. Schmetterer, A. F. Fercher, and W. Drexler, “Precision of extracting absorption profiles from weakly scattering media with spectroscopic time-domain optical coherence tomography,” Opt. Express **12**, 1677–1688 (2004). [CrossRef] [PubMed]

**11. **C. Xu, F. Kamalabadi, and S. A. Boppart, “Comparative performance analysis of time-frequency distributions for spectroscopic optical coherence tomography,” Appl. Opt. **44**, 1813–1822 (2005). [CrossRef] [PubMed]

**12. **F. Robles, R. N. Graf, and A. Wax, “Dual window method for processing spectroscopic optical coherence tomography signals with simultaneously high spectral and temporal resolution,” Opt. Express **17**, 6799–6812 (2009). [CrossRef] [PubMed]

**13. **F. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nature Photon. **5**, 744–747 (2011). [CrossRef]

**14. **N. Bosschaart, T. G. van Leeuwen, M. C. G. Aalders, and D. J. Faber, “Quantitative comparison of analysis methods forspectroscopic optical coherence tomography,” Biomed. Opt. Express **4**, 2570–2584 (2013). [CrossRef]

**15. **S. W. Smith, “Digital filters,” in *The Scientist and Engineer’s Guide to Digital Signal Processing*2nd ed., (California Technical Publishing, 1997), pp. 261–350.

**16. **F. J. Taylor, “Window design method,” in *Digital Filters: Principles and Applications with MATLAB*, (California Technical Publishing, 2012), pp. 71–82.

**17. **S. Prahl, “Optical Absorption of Hemoglobin,” http://omlc.ogi.edu/spectra/hemoglobin/index.html.

**18. **C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level,” Proceedings of the IRE **34**, 335–348 (1946). [CrossRef]

**19. **F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proceedings of the IEEE **66**, 51–83 (1978). [CrossRef]

**20. **F. J. Taylor, “Adaptive Filtering and Signal Analysis,” in *Adaptive Digital Filters*2nd ed., (Marcel Dekker Inc., 2001), pp. 10–23.