3.2. Simulation Results of Brain Functional Sensing
As we mentioned in the introduction, the application of spectroscopic OCT in brain functional sensing is usually to quantify three parameters: total hemoglobin concentration
C, blood oxygen saturation
Y, and scattering coefficient
S. After obtaining the OCT signals of different wavelengths using MMC-OCT,
C,
Y, and
S can be calculated using the quantitative method introduced in
Section 2.1. In this section, we studied brain functional sensing through theoretical analysis and MMC-OCT simulation. Specifically, we explored factors that affect the quantitative accuracy of
C,
Y, and
S using three wavelengths.
In the visible light waveband (500–600 nm), the absorption coefficient of hemoglobin is much larger than that in the near-infrared waveband, as shown in
Figure 2, so functional sensing is very advantageous. Therefore, we chose a visible waveband to simulate functional sensing of the brain tissue. To obtain parameters
C,
Y, and
S, we usually need to solve the following three-variable linear equations:
in which subscripts 1, 2, and 3 represent three different wavelengths. ln(10) is multiplied to
εO and
εD. The extinction coefficients
μt1,
μt2, and
μt3 that are obtained by exponential fitting of the backscattered signals include the experimental error and data fitting error. The solutions to Equation (8) are
and
where
, and the parameters with (′) as a superscript are the results of subtracting the corresponding parameters of the second wavelength from the parameters of the third wavelength. The variation of the three parameters
C,
Y, and
S caused by the errors of
μti (
i = 1, 2, and 3) can be expressed as
and
in which
,
, and
all represent errors of
μti, and their signs and magnitudes are uncertain in practice.
It can be seen from Equations (12)–(14) that the calculation errors of C and S have nothing to do with Y and are only related to the errors of μti and the absorption parameters of the selected wavelengths. However, the calculation error of Y varies with μti, that is, it is related to C, Y, and S. In addition, we find that the change trends of |dC| and |dS| are the same, while that of |dY| is opposite to the other two. In other words, we have no way to choose the optimal three wavelengths to minimize the errors of the C, Y, and S simultaneously, but can only make compromises according to the actual situation. For example, if the requirement for the accuracy of C is higher in practical application, the denominator of |dC| can be larger and the numerator smaller by choosing appropriate wavelengths, while if the requirement for the accuracy of Y is higher, the opposite choice can be made.
Figure 6 shows the errors of
C,
Y and,
S under two different wavelength choices, and the corresponding absorption parameters are shown in the pictures. It should be noted that these parameters are not actual values shown in
Figure 2, but are obvious cases set to explain the above rules. In the simulation,
C was 100 μM,
Y was 70%, and
S was 7 mm
−1. The
C of the brain tissue is approximately 100 μM [
19]. Here, for convenience, we assumed that
μt1 had error d
μt1, and
μt2 and
μt3 had no errors. In
Figure 6a, when
dμt1 is 0.1 mm
−1,
dC/
C = 0 and
dS/
S = 0.028, while
dY/
Y is up to 0.9. In
Figure 6b, when
dμt1 is 0.1 mm
−1,
dC/
C = 3.3 and
dS/
S = 1.16, while
dY/
Y is only 0.2. Therefore, we can choose the appropriate wavelengths to solve the scattering and absorption parameters according to the actual needs.
Next, functional sensing of brain tissue was simulated using MMC-OCT. The most popular components of brain tissue for researchers are often the cerebral cortex (i.e., gray matter) and blood vessels, so we obtained backscattered signals at different wavelengths of vascular tissue. The blood vessel had a diameter of 50 μm and was embedded in the gray matter tissue with a depth of 150 μm. Firstly, the appropriate wavelengths were selected based on the spectral data shown in
Figure 2. We set a light source with a center wavelength of 570 nm and a spectral width of 50 nm. The window function of the STFT has a width in the spectral domain of 15 nm, and there is no overlap between the window functions at the two adjacent wavelengths. Between 540 nm and 600 nm, we used Equations (12)–(14) to calculate the theoretical error of the three parameters obtained using different wavelength combinations, and each combination consists of three wavelengths with an interval of 15 nm. The error for each
μt is a random number within 0–0.1 mm
−1;
Y = 0.7,
C = 2 mmol, and
S = 70 mm
−1. The curve in
Figure 7a shows the sum of the errors of the three parameters calculated using different wavelength combinations, and the abscissa is the minimum of the three wavelengths. The number of calculations is 200.
Within the spectral bandwidth of the source, the optimal and worst combinations are 560 nm, 575 nm, 590 nm and 553 nm, 568 nm, 583 nm, respectively.
Figure 7a–c shows the calculation errors of
C,
Y, and
S, respectively. Since the error of the extinction coefficient is random, we compare the maximum of the 200 calculation errors of the two combinations. The maximum values of the
dC/C,
dY/Y, and
dS/S of the optimal combination are reduced by 17.6, 5.8, and 14.5 times, respectively, compared to those of the worst combination. The optical parameters of the two combinations are shown in
Table 2, and
Figure 8 shows the simulation results using MMC-OCT. When using the method described in [
6], it can be approximated that the scattering coefficient is not affected by the wavelength, so we set the scattering coefficient for each wavelength to 70 mm
−1.
The simulation results for the 560 nm wavelength are summarized in
Figure 8.
Figure 8a shows the continuous wave (CW) fluence extracted at plane
y = 0 μm, and the black dotted box in the figure indicates the position of the blood vessel. The blue curve in
Figure 8b is the backscattered signal, and the signal in the depth range of 70 μm–100 μm is selected to calculate the extinction coefficient by exponential fitting, the results of which are shown in
Figure 8c. The fitting coefficients and goodness are shown in the picture, and the
R2 value of the fitting results is 0.9969. The fitting results of two combinations and
C,
Y, and
S calculated from the fitting results are listed in
Table 3. The parameters marked “The” in
Table 3 are the theoretical calculation values, and those labeled “Rea” are the simulation values. It can be seen that the two are basically the same, and the maximum error is 0.06 mm
−1. Of course, this does not mean that the error in the actual experiment is also at this level, but rather illustrates the feasibility of the MMC-OCT in simulating the process of obtaining the extinction coefficient.
The errors of C, Y, and S obtained by the optimized combination of wavelengths are much smaller than those obtained by the worst combination, and the calculation errors of C, Y, and S are reduced by 160, 16.75, and 238.8 times, respectively.
The results in
Table 3 show that we can obtain very accurate optical parameters of blood vessels by reasonably selecting three wavelengths. However, the error between the theorical extinction coefficient and the simulation result is very small, not exceeding 0.06 mm
−1, so the result is ideal. In actual experiments, when the method described in [
34] is used, the accuracy of the exponentially fitted extinction coefficient can reach 0.8%, which is about 0.72 mm
−1 for blood. Therefore, we set the maximum error of extinction coefficient to 1 mm
−1. At
Y = 70%,
dC/C,
dY/
Y, and
dS/
S calculated by the optimized combination are 0.0859, 0.0829, and 0.0296, respectively. Accordingly, we can conclude that when imaging vascular structure, the optimization scheme shown in
Figure 7 can be used to reasonably select three wavelengths to calculate the optical parameters of the tissue while ensuring high spatial resolution.
It should be noted that the above optimization process was performed under the condition that Y = 0.7. The hemoglobin concentration of the blood vessel is high, and the absorption coefficient in the visible-light waveband is large, so the optimization results for different Y are approximately the same.
Next, the functional sensing of the cerebral cortex was discussed, where the spatial resolution requirements are reduced compared to imaging small-sized blood vessels. In the simulation, wavelengths of 540 nm, 546 nm, and 576 nm were selected, and the window function of the STFT has a width of 6 nm in the spectral domain. Optical parameters of gray matter at these wavelengths are listed in
Table 4. There were two reasons for this choice. First, the scattering coefficients of these three wavelengths were approximately the same, as shown in
Figure 2; second, we made a compromise by considering the calculation errors of
C and
Y. It should be noted that this selection of wavelengths was obtained under the condition that
Y = 70% using the optimization method described in
Figure 7. The
n and
g of tissue were 1.37 and 0.9, respectively. The scattering coefficients
S in the
Table 4 were obtained by dividing the scattering coefficients in
Figure 2 by 10 so that they were approximately the same as the scattering coefficients in gray matter [
35].
The simulated gray matter was a 300 × 300 × 500 (μm) cube, and simulation results are shown in
Figure 9.
Figure 9a shows the CW fluence extracted at plane
y = 0 μm. The black data points in
Figure 9b represent backscattered signals, and the interval of
z is 25 μm. The green curve in
Figure 9b is the fitting result, and the fitting coefficients and goodness are shown in the picture. The
R2 value of the fitting results is 0.9908. The fitting result in
Figure 9b was obtained after removing the class II photons. If class II photons are not removed, the fitting
μt is small, as shown in
Figure 9c because the optical path of the class II photon is larger. For comparison, the data in
Figure 9c were normalized. The theoretical calculation values and simulation values of the three parameters are shown in
Figure 9d. The points marked “The” in the legend are the theoretical calculation values, and the points labeled “Rea” are the simulation values.
It can be seen from
Figure 9d that the simulation values are basically in accordance with the theoretical calculation values except that the deviation of
Y at
Y = 10% is a little large, which is because the calculation error of
Y varies with the value of
μt, as described in Equation (13). The results of
Figure 9e are the differences between the theoretical calculation values and simulation values. The pre-estimated calculation errors in the case that
Y = 10% and
Y = 70% are shown in
Figure 9e, which can be seen to be consistent with the results of the actual simulation. If we predict that the
Y of tissues is relatively high, then the choice of wavelengths is reasonable now. On the contrary, the current wavelengths selection is unreasonable if we predict that the
Y of tissues is relatively low, and other wavelengths should be selected to make the errors of
C,
Y, and
S smaller. When optimized for the case that
Y = 10%, the wavelengths of 546 nm, 552 nm, and 570 nm were selected. When the error of the extinction coefficient is the same,
dC/
C,
dY/
Y, and
dS/
S were 0.235, 0.036, and 0.082, respectively, which were reduced by 1.25, 14.7 and 1.28 times, respectively, compared with the results of
Figure 9e.
Therefore, in practical application, if we have a prior rough estimate of Y and C of the brain tissue to be measured, we can then reasonably choose the wavelengths to improve the measurement accuracy.
The simulation results show that the correct optical parameters of tissues with low concentrations of hemoglobin can be obtained by rational selection of three wavelengths. However, the error of the extinction coefficient obtained by the simulation is very small, and the maximum value does not exceed 0.05 mm−1. The error of C reaches about 30% when the error of μt is 0.1 mm−1. Therefore, when the sample has a low hemoglobin concentration and does not require high spatial resolution, there is no doubt that more accurate results can be obtained by fitting the data of multiple wavelengths.
In summary, we briefly describe the optimization strategy for functional sensing of brain tissue as follows: When imaging small-sized blood vessels, the optimization scheme shown in
Figure 7 can be used to reasonably select three wavelengths to resolve
C,
Y, and
S. When imaging the cerebral cortex and the spatial resolution requirement is not high, the method of fitting multi-wavelength data should be used; when the spatial resolution requirement is high, it is necessary to roughly estimate the range of
Y and
C in advance, and reasonably select three wavelengths to resolve
C,
Y, and
S.
At the end of this section, as a little supplement to this work, functional sensing in a near-infrared waveband was simulated. The simulated tissue was gray matter with 100-μm depth in which a blood vessel with a 100-μm diameter is embedded. Similarly, considering the errors of
C and
Y simultaneously, we chose three wavelengths at 750 nm, 800 nm, and 900 nm for calculation. The window function of the STFT has a width of 50 nm in the spectral domain. Their optical parameters are shown in
Table 5. The parameters of blood were the same as those in
Figure 2, and the
S values of gray matter in
Table 5 were still obtained by dividing the scattering coefficients in
Figure 2 by 10. In the near-infrared waveband, the scattering coefficient is approximately linear with the wavelength, as described in Equation (7). The
C values of gray matter and vessel were 200 μM and 4 mM, respectively, and the
Y values were both 70%.
The simulation results are shown in
Figure 10.
Figure 10a shows the CW fluence extracted at plane
y = 0 μm. The structure parameters of the simulated tissue are also indicated. The blue curve in
Figure 10b is the backscattered signal, and the fitting results of the gray matter and vessel are shown in the picture. The final fitting results and calculated
C,
Y, and
S from the fitting results are listed in
Table 6. We can see that the calculated
C and
Y are incorrect because the absorption of gray matter is too small in the near-infrared waveband. By contrast, the calculation results for blood vessels are much more accurate.
In the simulations of this paper, we did not simulate the OCT signals of all wavelengths within the spectral bandwidth of the light source but thought that the extinction coefficient is homogeneous within a certain spectral width, which undoubtedly leads to certain errors. In the visible-light waveband, the maximum wavelength bandwidth set in this work was 15 nm, which has proven to be reasonable [
36]. In addition, the impacts of the STFT process on functional sensing were not studied. We will optimize the MMC-OCT program to simulate the complete imaging process of the spectroscopic OCT in future researches.