2.1.1.

Homogeneous medium

First, with the homogeneous medium, we studied the impact on the sensitivity of the technique to periodic perturbations of the optical properties of the medium of the main operational parameters: the SDD, the average total number of photons used to generate the DTOF (${\overline{N}}_{\mathrm{tot}}$), the measurement length ($T_{\mathrm{meas}}$), and the sampling rate ($f_s$). To limit the complexity of our model, we only considered oxygenated hemoglobin (${\mathrm{O}}_2\mathrm{Hb}$) and deoxygenated hemoglobin (HHb) as the main absorbing chromophores in human tissues in the wavelength range of interest, i.e., 600 to 900 nm. As for their baseline concentrations, we assumed the values of 30 and $20\mu \mathrm{M}$, respectively.²³ Sinusoidal perturbations with a fixed amplitude of 1% of the baseline value and fixed frequency of 1 Hz were imposed to create temporal variations in both hemodynamic parameters. The 1 Hz frequency was selected to mimic the effect of the cardiac muscle contraction on the probed tissue, a phenomenon previously observed in several CW fNIRS studies,^24,25 and a relative amplitude of 1% was used to represent the minimum amplitude expected for physiological noise components from the previous literature.^24,25 To account for the lack of consensus in the literature regarding the relative phase of oscillations of the same origin between chromophores or across different medium domains, a random phase shift was applied to each perturbation.^26,27 Then, by exploiting the known absorption spectra of ${\mathrm{O}}_2\mathrm{Hb}$ and HHb, the hemodynamic parameters were translated, by Beer’s law, in the absorption coefficient (${\mu }_a$) at two wavelengths, 690 and 830 nm. The reduced scattering coefficient was set according to the empirical approximation of the Mie theory (${{\mu }_s}^{\prime }=a({\lambda }_0/\lambda {)}^{-b}$), with $a=10.0{\mathrm{cm}}^{-1}$, $b=1.0$, and ${\lambda }_0=690\mathrm{nm}$. Finally, the solution of the diffusion equation for a 5 cm-thick, laterally infinite homogeneous slab with a 1.4 refractive index was used to calculate the dataset of DTOFs²⁸ in the range 0 to 5 ns, with a time resolution of 8 ps. First, three values of ${\overline{N}}_{\mathrm{tot}}$ were considered: ${10}^4$, ${10}^5$, and ${10}^6$ photons (ph), while maintaining the other parameters fixed: $T_{\mathrm{meas}}=15\min$, $f_s=20\mathrm{Hz}$ (case H_$N_{\mathrm{tot}}$). Then, three values of $T_{\mathrm{meas}}$ were considered: 5, 10, and 15 min, while maintaining ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$, and $f_s=20\mathrm{Hz}$ (case H_$T_{\mathrm{meas}}$). Finally, three values of $f_s$ were considered: 5, 10, and 20 Hz, while maintaining ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$, and $T_{\mathrm{meas}}=15\min$ (case H_$f_s$). In these simulations, two SDDs, 1 and 4 cm, were used. An additional simulation was performed using SSD from 1 to 6 cm, with steps of 1 cm, with the other parameters fixed (${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$, $T_{\mathrm{meas}}=15\min$, $f_s=20\mathrm{Hz}$) to better explore the effect of SSD (case H_SDD). In Table 1, all of the parameters used in each simulation case for the homogeneous medium (H_$N_{\mathrm{tot}}$-H_SDD) are summarized.

Table 1

Parameters used in each simulation case for the homogeneous medium.

2.1.2.

Bilayer medium

The bilayer medium allowed for the evaluation and comparison of approaches for depth-selectivity enhancement, both in CW and TD fNIRS. This comparison aimed to assess the ability of the techniques to localize perturbations originating at different depths and effectively reject superficial extra-cerebral contributions. The bilayer medium is composed of two stacked cylinders with a radius of $R=10\mathrm{cm}$ with a total thickness of 5 cm and a refractive index of 1.4. The superficial layer (UP) had a thickness of 1 cm, and the deeper layer (DW) had a thickness of 4 cm. Both layers had identical baseline hemoglobin concentrations and optical properties so the two layers are only identified by different and independent perturbations of the hemoglobin concentrations in time. The time duration of measurements $T_{\mathrm{meas}}=15\min$ and the sampling frequency $f_s=20\mathrm{Hz}$ were fixed based on the results obtained for the homogeneous medium (Sec. 2.1.1). The average total number of photon counts ${\overline{N}}_{\mathrm{tot}}$ was varied as before by considering three values of ${10}^4$, ${10}^5$, and ${10}^6\mathrm{ph}$ and two SDDs of 1 and 4 cm. Preliminary simulations investigated the effect of a single oscillatory component imposed in either the UP layer (case B_UP) or the DW layer (case B_DW) while keeping the concentrations in the other layer constant at baseline values. The same perturbation as in the homogeneous medium was used: an oscillation of both ${\mathrm{O}}_2\mathrm{Hb}$ and HHb concentrations with a frequency of 1 Hz and an amplitude of 1% of the baseline value. In addition, we focused on evaluating the effects generated through the superimposition of multiple Fourier contributions simultaneously originating in the UP and DW layers. Four cases were considered:

These simulations were repeated with the applied perturbations in-phase or out-of-phase. The solution of the diffusion equation for the bilayer geometry was used to calculate the dataset of DTOFs²⁸ in the range 0 to 5 ns, with a time resolution of 20 ps to limit the computational time. Table 2 summarizes the parameters used in each simulation case when the bilayer medium was used (B_UP-B_AmpFreq).

Table 2

Parameters used in each simulation case for the bilayer medium.

2.2.1.

Estimate of optical and hemodynamic parameters

For the simulations carried out using the homogeneous medium, the DTOFs were numerically time integrated to retrieve the measured light intensity, represented as the number of total photons detected within the measurement window, which can be considered an equivalent of the CW fNIRS signal.

When the bilayer geometry was considered, a time-windowing approach was used to exploit the depth-selective information encoded in the DTOFs. For our analysis, we used 10 time windows (gates), each with a width of 500 ps, covering the time window between 0 and 5 ns (full width of the largest curve) following the IRF temporal position. Successively, the diffusion equation solution for a bilayer medium²⁸ was used as a model for photon migration in a home-made software²² for non-linear fitting (based on the Levenberg–Marquardt minimization algorithm²⁹). After calculating the baseline optical coefficients ${\mu }_a$ and ${{\mu }_s}^{\prime }$ using a homogeneous model fit, the time-dependent mean partial pathlength (TMPP) method³⁰ was used to estimate the variations $\mathrm{\Delta }{\mu }_{a,j}$ from the baseline value from each DTOF, to extract the absorption coefficients (${\mu }_a+\mathrm{\Delta }{\mu }_{a,j}$) in the two layers, $j=1$ or 2, of the medium at every time point. The TMPP method, presented by Zucchelli et al.,³⁰ is based on a time gating of the DTOF, which is used to estimate the time-dependent mean pathlength $L_j(t)$ traveled by photons in each domain $j$ of the medium with high accuracy. Once $L_j(t)$ is known, the variations $\mathrm{\Delta }{\mu }_{a,j}$ can be retrieved by inverting the following formulation of the time-resolved reflectance curve: $R(t)=R_0(t)e^{-\sum _j\mathrm{\Delta }{\mu }_{a,j}{L}_j(t)}$. Finally, Beer’s law was used to derive the hemodynamic parameters from the absorption coefficient.²⁸

To enable a direct comparison between TD and CW fNIRS data, the DTOFs were again integrated over the measurement window. The variations of hemoglobin concentrations for the CW fNIRS configuration were then calculated by means of the modified Lambert–Beer law.^28,31 Increases or decreases in hemoglobin concentration were expressed relative to a baseline, calculated as the average intensity over the entire measurement. The depth-selectivity in CW fNIRS is generally implemented using multiple acquisition points at different SDDs and exploiting the relationship between the SDD and the photon penetration depth.³² Signals acquired with an SDD of 1 and 4 cm were thus compared in this phase.

2.2.2.

Frequency-domain analysis

The PSD of the intensity and hemodynamic signals were calculated using a uniform algorithm across all data, implemented in Matlab (Release 2022a, The MathWorks Inc., Natick, Massachusetts, United States). First, the signals were detrended using a third-order polynomial fit to remove the slow component.^10,26 Then, the periodogram estimation of the PSD was performed by employing the Cooley–Tukey fast Fourier transform algorithm.^33,34 Because our aim was to evaluate the PSD with the maximum frequency resolution without any variance reduction and all the data series met the stationarity requirements, no windowing was applied.

In Fourier-domain analysis, evaluating the significance of spectral peaks is crucial: to discern peaks attributable to underlying oscillatory phenomena from Poisson noise background, an ad-hoc contrast function was introduced. The contrast function $C(f)$ was defined to compare the PSD amplitude at frequency $f$ to the average amplitude of the PSD in a noise-only interval, $\epsilon$

In Sec. 3, to quantify the contrast value efficiently, it is expressed in decibels, as needed

Due to the characteristics of our synthetic signals, the PSD average noise $\epsilon$ was typically calculated by averaging the PSD values for frequencies higher than 5 Hz. However, for the case $f_s=5\mathrm{Hz}$, a modification in the definition of $\epsilon$ was necessary, as the sampling rate did not permit reconstruction of the PSD for frequencies higher than 2.5 Hz. In this case, $\epsilon$ was calculated as the average value of the PSD in the range $2<f\le 2.5\mathrm{Hz}$ to avoid interference from the second harmonic perturbation at 2 Hz, if present.

3.1.1.

Effect of the average total number of photon counts ${\overline{N}}_{\mathrm{tot}}$

Figure 1 shows the effect of varying ${\overline{N}}_{\mathrm{tot}}$ on the spectra obtained in case H_$N_{\mathrm{tot}}$. Panels (a) and (b) show the obtained PSD for the SSD of 1 and 4 cm, respectively. Data are shown for a single wavelength (690 nm) as the results of the other wavelengths (830 nm) are similar. The peak at 1 Hz is clearly visible for every value of ${\overline{N}}_{\mathrm{tot}}$ (different colors). Increasing ${\overline{N}}_{\mathrm{tot}}$, higher amplitudes are obtained for the whole spectrum. Figures 1(c)–1(h) show the amplitude of the spectral peak at 1 Hz, the amplitude of the average noise, and the 1 Hz peak contrast as a function of ${\overline{N}}_{\mathrm{tot}}$ in logarithmic–logarithmic plots, for SSD = 1 cm (4 cm) and both wavelengths. A power law fit is applied to each data series, always returning an $R^2$ value higher than 0.999. It is interesting to note that a second harmonic of the 1 Hz peak (at 2 Hz) is visible for the longer SDD and the higher values of ${\overline{N}}_{\mathrm{tot}}$ [Fig. 1(b), in green and purple] but remains absent in the short SSD signals [Fig. 1(a)]. Comparing the difference in amplitude between the first and second harmonic (where visible) and assuming this difference to be consistent for all measurements, we confirmed that the second harmonic is covered by noise for all measurements in which it is not visible in the PSD. The results obtained by increasing ${\overline{N}}_{\mathrm{tot}}$ can be summarized as follows: (i) the 1 Hz peak amplitude increases with ${({\overline{N}}_{\mathrm{tot}})}^2$; (ii) the average noise amplitude increases with ${\overline{N}}_{\mathrm{tot}}$; and (iii) the 1 Hz peak contrast value increases with ${\overline{N}}_{\mathrm{tot}}$.

Fig. 1

Effect of ${\overline{N}}_{\mathrm{tot}}$ (case H_$N_{\mathrm{tot}}$). The two columns represent signals acquired with SSD of 1 cm (left) and 4 cm (right). (a), (b) Examples of PSDs obtained at 690 nm. (c), (d) Amplitude of the spectral peak at 1 Hz; (e), (f) amplitude of the average noise; (g), (h) peak contrast. The results are reported for 690 nm, in blue, and 830 nm, in orange. A power law fit is applied to each dataset, all giving $R^2\ge 0.999$.

3.1.2.

Effect of the measurement time $T_{\mathrm{meas}}$

Figure 2 shows the effect of varying the $T_{\mathrm{meas}}$ on the spectra obtained in case H_$T_{\mathrm{meas}}$. Panels (a) and (b) show the PSD obtained for the SSD of 1 and 4 cm, respectively. Data are shown for a single wavelength (690 nm) as the results for the other wavelength (830 nm) are similar. The smaller box on the right of each graph represents a zoom on a higher-frequency region, showing the noise level in the three signals. Figures 2(c)–2(h) show the amplitude of the spectral peak at 1 Hz, the amplitude of the average noise, and the 1 Hz peak contrast as a function of $T_{\mathrm{meas}}$, for SSD = 1 cm (4 cm) and both wavelengths. A power law fit was applied to data shown in Figs. 2(c), 2(d), 2(g) and 2(h), resulting in $R^2$ values higher than 0.999. In Figs. 2(e) and 2(f), horizontal lines indicating the 3% variation around the data mean value were added to highlight that the average noise amplitude is invariant with respect to $T_{\mathrm{meas}}$.

Fig. 2

Effect of $T_{\mathrm{meas}}$ (case H_$T_{\mathrm{meas}}$). The two columns represent signals acquired with SSD of 1 cm (left) and 4 cm (right). (a), (b) Examples of PSDs obtained at 690 nm. The smaller boxes show a zoom of the PSD noise region. (c), (d) Amplitude of the spectral peak at 1 Hz; (e), (f) amplitude of the average noise; (g), (h) peak contrast. The results are reported for 690 nm, in blue, and 830 nm, in orange. A power law fit is applied to each dataset in panels (c), (d), (g), and (h), all giving $R^2\ge 0.999$. In panels (e) and (f), the horizontal lines represent a 3% variation around the mean value.

The results obtained by increasing $T_{\mathrm{meas}}$ can be summarized as follows: (i) the 1 Hz peak amplitude increases with $T_{\mathrm{meas}}$; (ii) the average noise amplitude is invariant with respect to $T_{\mathrm{meas}}$; and (iii) the 1 Hz peak contrast value increases with $T_{\mathrm{meas}}$.

3.1.3.

Effect of the sampling rate $f_s$

Figure 3 shows the effect of varying the $f_s$ on the spectra obtained in case H_$f_s$. Panels (a) and (b) show the PSD obtained for the SSD of 1 and 4 cm, respectively. Data are shown for a single wavelength (690 nm) as the results for the other wavelength (830 nm) are similar. The smaller box on the right of each graph represents a zoom on a higher-frequency region, showing the noise level in the three signals. Figures 3(c)–3(h) show the amplitude of the spectral peak at 1 Hz, the amplitude of the average noise, and the 1 Hz peak contrast as a function of $f_s$, for SSD = 1 cm (4 cm) and both wavelengths. In Figs. 3(c) and 3(d), horizontal lines indicating the 3% variation around the data mean value were added to highlight that the 1 Hz peak amplitude is invariant with respect to $f_s$. A power law fit was applied to the data shown in Figs. 3(e)–3(h), resulting in $R^2$ values higher than 0.99.

Fig 3

Effect of $f_s$ (case H_$f_s$). The two columns represent signals acquired with SSD of 1 cm (left) and 4 cm (right). (a), (b) Examples of PSDs obtained at 690 nm. The smaller boxes show a zoom of the PSD noise region. (c), (d) Amplitude of the spectral peak at 1 Hz; (e), (f) amplitude of the average noise; (g), (h) peak contrast. The results are reported for 690 nm, in blue, and 830 nm, in orange. A power law fit is applied to each dataset in panels (e), (f), (g), and (h), all giving $R^2\ge 0.99$. In panels (c) and (d), the horizontal lines represent a 3% variation around the mean value.

The results obtained by increasing $f_s$ can be summarized as follows: (i) the 1 Hz peak amplitude is invariant with respect to $f_s$; (ii) the average noise amplitude decreases with ${(f_s)}^{-1}$; and (iii) the 1 Hz peak contrast value increases with $f_s$.

3.1.4.

Comparison among different SSDs

Comparing the results obtained for the two SDD values in Secs. 3.1.1–3.1.3, an increase in the peak amplitude and contrast can be observed when increasing the SDD (Figs. 1–3). This effect is mainly due to the increase in the length of the average optical path traveled by the detected photons within the medium.²⁸ To better understand the effect of the SDD on the retrieved PSDs, we performed additional simulations varying SDD (H_SDD). The relative results are shown in Fig. 4. Panels (a)–(c) respectively show the amplitude of the spectral peak at 1 Hz, the amplitude of the average noise, and the 1 Hz peak contrast as a function of SDD, for the two wavelengths. In Fig. 4(b), horizontal lines indicating the 3% variation around the data mean value were added to highlight that the average noise amplitude is invariant with respect to SDD. A power law fit was applied to the data shown in Figs. 4(a) and 4(c), resulting in $R^2$ values higher than 0.999.

Fig. 4

Effect of SDD (case H_SDD). (a) Amplitude of the spectral peak at 1 Hz, (b) amplitude of the average noise, and (c) peak contrast. The results are reported for 690 nm, in blue, and 830 nm, in orange. A power law fit is applied to each dataset in panels (a) and (c), all giving $R^2\ge 0.999$. In panel (b), the horizontal lines represent a 3% variation around the mean value.

The results obtained by increasing SDD can be summarized as follows: (i) the 1 Hz peak amplitude increases approximatively with the square of SDD; (ii) the average noise amplitude is invariant with respect to SDD; and (iii) the 1 Hz peak contrast value increases approximatively with the square of SDD.

3.2.1.

Time-windowing of the DTOFs

A time-windowing of the DTOFs was applied using 10 gates of 500 ps width, covering a 5 ns time window starting from the IRF temporal position (Fig. 5).

Fig. 5

Example of DTOFs as recorded at SDD = 1 cm (dark blue, crosses) and SDD = 4 cm (light blue, circles) for ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$ and $\lambda =690\mathrm{nm}$. The applied time-windowing is represented by means of black vertical lines, positioned at the edges of each gate.

Figure 6 provides a comprehensive summary of the results obtained for cases B_UP, B_DW, and B_UPDW, showcasing the contrast of the 1 Hz peak in the PSD of the number of photon counts within each gate, as a function of the gate number, for ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$, both SDDs in Figs. 6(a) and 6(b), respectively, and a single wavelength (690 nm). The results are denoted with green crosses for case B_UP, with yellow triangles for case B_DW, and with purple circles for case B_UPDW. The results obtained for the CW fNIRS signals are also shown on the right. The contrast values are expressed in decibels, and the red horizontal line represents a peak significance threshold of 15 dB, tailored specifically for this analysis, which is the maximum contrast in the noise frequency range (Sec. 2.2.2).

Fig. 6

Contrast value of the 1 Hz peak as a function of the gate number for case B_UP (green, crosses), case B_DW (yellow, triangles), and case B_UPDW (purple, circles), with ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$. (a) Results obtained for SSD = 1 cm; (b) results obtained for SDD = 4 cm. In both graphs, the red horizontal line represents the peak significance threshold of 15 dB.

Remarkably, a significant peak at 1 Hz was observed in the PSD of each gate that contained a not null portion of the reflectance curve (DTOFs retrieved for the short SDD decayed to zero around 3 ns, leaving the last four gates with only Poisson noise, see Fig. 5) for case B_UP (green crosses) and case B_UPDW (purple circles). This is a notable result because a significant peak is found even in gates covering the very end of the DTOF tail [from around gate #5 in Fig. 6(a) and gate #10 in Fig. 6(b)], in which the average number of photon counts is lower than 20 (Fig. 5). The only exception is represented by the first gate in case B_DW (yellow triangles), for both SDDs. In addition, the curves exhibit differences in peak and barycenter position and slope in the last gates. Another remarkable result is that the PSD calculated for the CW fNIRS signal always shows a substantial peak at the perturbation frequency. Of particular importance is the sensitivity shown by the CW fNIRS signal recorded at 1 cm to the perturbation imposed in depth [Fig. 6(a), case B_DW, yellow triangles], where, even if less pronounced, the peak at 1 Hz remains indeed significant.

3.2.2.

Estimated hemodynamic parameters

For TD fNIRS, the TMPP method was applied to retrieve the hemodynamic parameters in the two layers of the tissue. The obtained results are shown in Fig. 7 for case B_UP, in Fig. 8 for case B_DW, and in Fig. 9 for case B_UPDW. The results are presented for ${\overline{N}}_{\mathrm{tot}}={10}^5\mathrm{ph}$ and both SDDs.

Fig. 7

PSDs of the hemodynamic parameters obtained from TD fNIRS for case B_UP, with ${\overline{N}}_{\mathrm{tot}}={10}^5\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm, superficial layer (UP); (b) signals acquired at SDD = 4 cm, superficial layer; (c) signals acquired at SDD = 1 cm, deeper layer (DW); (d) signals acquired at SDD = 4 cm, deeper layer.

Fig. 8

PSDs of the hemodynamic parameters obtained from TD fNIRS for case B_DW, with ${\overline{N}}_{\mathrm{tot}}={10}^5\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm, superficial layer (UP); (b) signals acquired at SDD = 4 cm, superficial layer; (c) signals acquired at SDD = 1 cm, deeper layer (DW); (d) signals acquired at SDD = 4 cm, deeper layer.

Fig. 9

PSDs of the hemodynamic parameters obtained from TD fNIRS for case B_UPDW, with ${\overline{N}}_{\mathrm{tot}}={10}^5\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm, superficial layer (UP); (b) signals acquired at SDD = 4 cm, superficial layer; (c) signals acquired at SDD = 1 cm, deeper layer (DW); (d) signals acquired at SDD = 4 cm, deeper layer.

In case B_UP (Fig. 7), oscillations in the hemodynamic parameters of both layers were accurately reconstructed for each SDD as the PSD of both hemoglobin species exhibited a visible peak at the imposed perturbation frequency (1 Hz) in the superficial layer [UP, Figs. 7(a) and 7(b)], but not in the deeper layer [DW, Figs. 7(c) and 7(d)], as expected. In case B_DW (Fig. 8), similar results were observed for the long SDD [4 cm, Figs. 8(b) and 8(d)], with a visible peak at the perturbation frequency in the PSD of the hemoglobin concentrations calculated in the DW layer [Fig. 8(d)], but not in the UP layer [Fig. 8(b)], as expected. However, when the short SDD [1 cm, Figs. 8(a) and 8(c)] was used, different results were obtained. In this case, although the UP layer static behavior was correctly reconstructed [Fig. 8(a)], the perturbation peak in the PSD was no longer visible in the DW layer [Fig. 8(c)], where it should be present. The same observations regarding the lower sensitivity of short SDD measurements to variations in the deeper DW layer can also be done for case B_UPDW [Fig. 9(c)]. When considering SDD = 4 cm, Figs. 9(b) and 9(d) demonstrate a flawless reconstruction of the PSD for both species and both layers. The superimposition of the two contributions had no discernible effects, evident from the perfect overlap between PSDs in Figs. 7(b) and 9(b) [Figs. 8(d) and 9(d)]. Comparing the above PSDs, the amplitude of the 1 Hz peak is indeed equal for both species, except for the variability due to the background noise. Analogous results were obtained by imposing oscillations with different frequencies and/or amplitudes in the two layers of the medium (cases B_Amp, B_Freq, and B_AmpFreq) while considering them either in-phase or out-of-phase (data not shown).

For CW fNIRS, the results are shown in Fig. 10 for case B_UP, Fig. 11 for case B_DW, and Fig. 12 for case B_UPDW. The results are shown for ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$ to account for the higher intensities usually characterizing CW fNIRS signals compared with TD fNIRS ones.

Fig. 10

PSDs of the hemodynamic parameters obtained from CW fNIRS for case B_UP, with ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm; (b) signals acquired at SDD = 4 cm.

Fig. 11

PSDs of the hemodynamic parameters obtained from CW fNIRS for case B_DW, with ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm; (b) signals acquired at SDD = 4 cm.

Fig. 12

PSDs of the hemodynamic parameters obtained from CW fNIRS for case B_UPDW, with ${\overline{N}}_{\mathrm{tot}}={10}^6\mathrm{ph}$. (a) Signals acquired at SDD = 1 cm; (b) signals acquired at SDD = 4 cm.

In all of the above cases, a visible peak is present at the perturbation frequency for both species and both SDDs. This is particularly interesting in case B_DW (Fig. 11) when the perturbation is imposed solely in the DW layer. In Fig. 11, the CW fNIRS signals recorded at both SDDs show sensitivity to the perturbation imposed in the DW layer of the medium for both species. In Sec. 3.2.1, the sensitivity shown by the signal recorded at 1 cm [Fig. 11(a)] is a notable and unexpected result.