2.1.1.

Head structure and baseline properties

We used a 3-D head structure obtained from an MRI scan of a healthy adult.³⁴ The internal structure was segmented into four tissue types:³⁵ scalp and skull, cerebrospinal fluid (CSF), gray matter, and white matter. These were subsequently combined into two compartments: brain (CSF, gray, and white matters), and extracerebral tissue (scalp and skull), sometimes simply referred to as “scalp” below. For the simulations, unless specified otherwise, the baseline total hemoglobin concentrations were fixed at ${[\mathrm{HbT}]}_{\text{Scalp}}=30\mu \text{M}$ and ${[\mathrm{HbT}]}_{\text{Brain}}=75\mu \text{M}$, in the extracerebral tissue and the brain, respectively, with oxygen saturation set at 65% in both compartments. Different studies in the literature report a wide range of ${[\mathrm{HbT}]}_{\text{Brain}}$ as estimated by time-domain or frequency-domain NIRS, with values as low as 40 μM and as high as 120 μM.^36–39 We chose a value approximately in the middle of this range. Water content was set to 75% in both tissue types.⁴⁰ Using published extinction coefficients for hemoglobin⁴¹ and water,⁴² we obtained the corresponding absorption coefficients at the wavelengths of interest in our experimental data, namely 690 and 830 nm for NIRS and 785 nm for DCS. The resulting optical properties are reported in Table 1. The reduced scattering coefficient was set to $12{\mathrm{cm}}^{-1}$ in the whole head for all wavelengths.

Table 1

Baseline physiological and optical parameters for the Monte Carlo simulations.

For DCS, it was necessary to assign baseline values for blood flow in the extracerebral and cerebral regions. The flow of moving scatterers can be characterized by the effective Brownian motion coefficient $D_{\mathrm{B}}$, previously shown to best describe the DCS signal.^6,13,31,43 DCS yields a blood flow index $\mathrm{BFI}=\alpha D_{\mathrm{B}}$, where $\alpha$ is the ratio of moving scatterers to total scatterers. BFI is expressed in ${\mathrm{mm}}^2{\mathrm{s}}^{-1}$ and has been demonstrated to correlate well with CBF (traditionally expressed in $\mathrm{mL}/100\mathrm{g}/\min$) as assessed by other modalities.^11–15 Based on positron emission tomography (PET) data in healthy adults,⁴⁴ we used a relative brain-to-scalp flow ratio of 6, with BFI set to ${1\times 10}^{-6}{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ in the extracerebral layer and to ${6\times 10}^{-6}{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ in the brain.

We also investigated the dependence of our results on these baseline parameters. Specifically, we varied the brain baseline total hemoglobin ${[\mathrm{HbT}]}_{\text{Brain}}$ from 50 to 100 μM, with a constant brain-to-scalp blood volume ratio of 2.5. We varied CBF index ${\mathrm{BFI}}_{\text{Brain}}$ from ${2\times 10}^{-6}$ to ${10\times 10}^{-6}{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, with a constant brain-to-scalp blood flow ratio of 6. Finally, at constant baseline cerebral total hemoglobin ${[\mathrm{HbT}]}_{\text{Brain}}=75\mu \text{M}$ and flow ${\mathrm{BFI}}_{\text{Brain}}{=6\times 10}^{-6}{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, we varied the brain-to-scalp baseline blood volume ratio ${[\mathrm{HbT}]}_{\text{Brain}}/{[\mathrm{HbT}]}_{\text{Scalp}}$ from 1 and 4, and the baseline flow ratio ${\mathrm{BFI}}_{\text{Brain},0}/{\mathrm{BFI}}_{\text{Scalp},0}$ from 1 and 10.

2.1.2.

Monte Carlo simulations of NIRS and DCS signals

We considered eight probe locations, four on the forehead and four on the left temporal region, each consisting of one source and three aligned detectors approximately 1, 2, and 3 cm away. The precise locations of the optodes were obtained from a larger probe wrapped onto the surface of the head as described in a previous study.⁴⁵ The median values of the true SD direct-line separations, i.e., without following the curvature of the head, were 0.98, 1.96, and 2.94 cm. The true individual direct-line SD separations were used for each subject in the subsequent data analysis.

We used a Monte Carlo code modified in-house from the version described by Boas et al.⁴⁶ The partial pathlength in each tissue type, i.e., “brain” and “scalp,” was recorded for every detected photon, enabling the postprocessing computation of the fluence⁴⁶ at each detector for any absorption coefficient value ${\mu }_{\mathrm{a}j}$ in tissue type $j$. For the simulation of DCS data, we also recorded the total momentum transfer, summed over all scattering events, for each photon.^5,47 This allows the postsimulation computation of the electric field autocorrelation function $g_1(\tau )$, where $\tau$ is the autocorrelation delay for any value of the Brownian diffusion coefficient ${\mathrm{BFI}}_j$ characterizing moving scatterers in tissue type $j$. We launched ${10}^9$ photons at the source, resulting in approximately ${2\times 10}^6$, ${2\times 10}^5$, and ${7\times 10}^4$ photons detected at the three SD separations, respectively.

2.1.3.

Sensitivity to scalp and brain

The first metric we computed is the sensitivity of both NIRS and DCS to brain versus extracerebral layers. Specifically, we simulated a true variation in the parameter of interest (blood volume for NIRS, blood flow for DCS) in the brain only, and in the extracerebral tissue only. Next, we fit this simulated data as if it were a semi-infinite medium instead of a layered medium, as is done in our experimental analysis, in order to compute a bulk measured change in the parameter of interest. Finally, we defined the sensitivity to each region (brain and scalp) as the ratio of the measured change over the true change. We also studied the relative brain-to-scalp sensitivity $S_{\text{Brain}/\text{Scalp}}$, defined as the ratio of sensitivities to both compartments.

For NIRS, we independently simulated in each tissue type (brain or scalp) a 5% increase in blood volume with no change in oxygen saturation, so that ${\mathrm{\Delta }[\mathrm{HbT}]}_{\mathrm{Tiss}}/{[\mathrm{HbT}]}_{0,\mathrm{Tiss}}{=\mathrm{\Delta }[\mathrm{HbO}]}_{\mathrm{Tiss}}/{[\mathrm{HbO}]}_{0,\mathrm{Tiss}}{=\mathrm{\Delta }[\mathrm{HbR}]}_{\mathrm{Tiss}}/{[\mathrm{HbR}]}_{0,\mathrm{Tiss}}$, where ${[\mathrm{HbT}]}_0$, ${[\mathrm{HbO}]}_0$, and ${[\mathrm{HbR}]}_0$ are the baseline concentrations in total, oxy-, and deoxy-hemoglobin, respectively, and $\mathrm{\Delta }[\mathrm{HbT}]$, $\mathrm{\Delta }[\mathrm{HbO}]$, and $\mathrm{\Delta }[\mathrm{HbR}]$ are their corresponding absolute changes from baseline. We computed the corresponding change in measured optical density at the two wavelengths of interest for the NIRS signal, i.e., 690 and 830 nm. Finally, using the modified Beer–Lambert law,^48,49 we retrieved the measured changes ${\mathrm{\Delta }}_{\mathrm{m}}{[\mathrm{HbT}]}_{\text{Brain}}$ and ${\mathrm{\Delta }}_{\mathrm{m}}{[\mathrm{HbT}]}_{\text{Scalp}}$ in [HbT], for a brain-only change and for a scalp-only change, respectively. We used differential pathlength factors (DPFs) of 6.5 at 1 cm, 7 at 2 cm, and 7.5 at 3 cm, at both wavelengths. These values were obtained from the Monte Carlo simulations as the average of the true DPF over all probe locations, for the default baseline optical properties. The NIRS sensitivity to brain $S_{\mathrm{CW},\text{Brain}}$ and to scalp $S_{\mathrm{CW},\text{Scalp}}$ was finally computed as $S_{\mathrm{CW},\text{Brain}}={\mathrm{\Delta }}_{\mathrm{m}}{[\mathrm{HbT}]}_{\text{Brain}}/\mathrm{\Delta }{[\mathrm{HbT}]}_{\text{Brain}}$ and $S_{\mathrm{CW},\text{Scalp}}{=\mathrm{\Delta }}_{\mathrm{m}}{[\mathrm{HbT}]}_{\text{Scalp}}/\mathrm{\Delta }{[\mathrm{HbT}]}_{\text{Scalp}}$.

For DCS, we simulated a true 20% relative increase in blood flow ${r\mathrm{BFI}}_{,\mathrm{Tiss}}=\mathrm{\Delta }{\mathrm{BFI}}_{\mathrm{Tiss}}/{\mathrm{BFI}}_{0,\mathrm{Tiss}}$ in the brain or in the scalp only. We fit the resulting $g_1(\tau )$ curve with the analytical solution of the correlation diffusion equation for a semi-infinite homogeneous medium,⁵ at each SD separation independently, at baseline and after the flow increase. This resulted in the measured blood flow index relative change $r_{\mathrm{m}}\mathrm{BFI}$. The sensitivities to brain and to scalp for DCS were defined as $S_{\mathrm{DCS},\text{Brain}}=r_{\mathrm{m}}{\mathrm{BFI}}_{\text{Brain}}/r{\mathrm{BFI}}_{\text{Brain}}$, and $S_{\mathrm{DCS},\text{Scalp}}=r_{\mathrm{m}}{\mathrm{BFI}}_{\text{Scalp}}/r{\mathrm{BFI}}_{\text{Scalp}}$. Additionally, we compared the sensitivity of DCS when fitting the whole autocorrelation curve (maximal delay ${\tau }_{\mathrm{Max}}=10\mathrm{ms}$), or only the early delays, as illustrated in the insets of Fig. 1. For the latter, we restricted the fit to the early portion of the curve for which the normalized function $g_1(\tau )$ was greater than 0.7. Restricting the fit to the early delays $\tau$ is known to give more weight to longer photon pathlengths and to higher blood flow, therefore increasing the cerebral contribution to the signal.¹⁴

Fig. 1

Sensitivity of near-infrared spectroscopy (NIRS) (a) and diffuse correlation spectroscopy (DCS) (b) as simulated with Monte Carlo simulations on an MRI-based head structure, for source–detector separations of 1, 2, and 3 cm. We present the sensitivity to scalp (measured-over-true scalp change), sensitivity to brain (measured-over-true brain change), and the relative brain-to-scalp sensitivity. The NIRS sensitivity is based on a blood volume change only, and that of DCS is based on a blood flow change only. For DCS, we present the cases where the whole autocorrelation curve $g_1(\tau )$ is fit (magenta, see first inset), and when only the early delays corresponding to $g_1(\tau )>0.7$ are fit (cyan, see second inset). In all cases (NIRS and DCS), the reconstruction is done using a homogeneous semi-infinite model. The bar heights present the median value across all eight probe locations, and the error bars extend from the 25th to 75th percentile of the values.

For all sensitivities, we present the median, and 25th and 75th percentiles of the data over the eight probe locations.

2.1.4.

Contrast-to-noise ratio

Our second metric is the CNR for a physiologically realistic change in CBF and volume. We chose two representative physiological activations, namely functional activation and hypercapnia.

2.2.1.

Protocol

CW-NIRS and DCS data were simultaneously acquired on all subjects. For the NIRS measurements, we used a laser-diode based TechEn device (TechEn Inc., Medford, Massachusetts). The NIRS probe was located on the left side of the forehead, and consisted of one source (690 and 830 nm) and two detectors along a line at 0.8 and 3 cm. The NIRS data were acquired at 50 Hz. The DCS device is custom-built with one 785-nm long coherence length laser source (CrystaLaser, Reno, Nevada) and eight single photon counting avalanche photodiode detection channels (PerkinElmer/Excelitas, Waltham, Massachusetts, SPCM-AQRH series).^6,60 The DCS probe was located on the right side of the forehead and consisted of one source and two detection locations along a line at 0.8 and 3 cm. Seven single-mode fibers were bundled together and placed at the 3 cm separation in order to improve the signal-to-noise ratio at this location. The DCS autocorrelation curves were acquired over an integration time of 2 s.

The recordings analyzed in this study were part of a longer protocol that we do not describe here. Relevant to this study is the hypercapnic runs. The subjects breathed through a mouthpiece connected to a specialized breathing circuit that minimized spontaneous breath-to-breath fluctuations in end-tidal ${\mathrm{CO}}_2$.⁶¹ End-tidal ${\mathrm{CO}}_2$ (${\mathrm{etCO}}_2$) was monitored continuously, and recorded synchronously with the optical data. Periods of hypercapnia were achieved by adding ${\mathrm{CO}}_2$ to the inspired air. Each subject underwent two 11-min trials that comprised six hypercapnic episodes during each trial. Each hypercapnic episode was sustained for 30 s, during which inspired ${\mathrm{CO}}_2$ was briskly increased to elevate ${\mathrm{etCO}}_2$ to either 4 or 8 mm Hg above each subject’s habitual baseline ${\mathrm{etCO}}_2$, in a predefined order (4-4-8-4-8-8 mm Hg). Each hypercapnic episode of 4 and 8 mm Hg was followed by a recovery/wash-out period of 60 and 90 s, respectively.

2.2.2.

Data analysis

From the NIRS intensities at 690 and 830 nm for both SD pairs, we computed the relative changes in optical density at each wavelength, applied a wavelet motion correction algorithm,⁶² low-pass filtered the motion-corrected optical density at 0.5 Hz, and converted them to relative changes in oxy- and deoxy-hemoglobin using the modified Beer–Lambert law. We used a DPF of 6 at both separations and both wavelengths. We did not apply any partial volume correction factor. Finally, we applied a 6-s sliding window averaging of the hemoglobin time series for consistency with the temporal processing of the DCS data described below.

For DCS, a weighted average of the seven autocorrelation curves for the 3 cm SD separation was computed, using weights equal to the integrated intensities of the DCS signal for each fiber. The DCS autocorrelation curves were further averaged using a moving window of 6 s (three curves) prior to fitting. This resulted in two time series of autocorrelation curves, for 0.8 and 3 cm. The intensity autocorrelation curves were fit at each time point with the analytical solution of the correlation diffusion equation for a semi-infinite homogeneous medium. We fixed the unknown absorption and reduced scattering coefficient values to ${\mu }_{\mathrm{a}}=0.15{\mathrm{cm}}^{-1}$ and ${{\mu }_{\mathrm{s}}}^{\prime }=10{\mathrm{cm}}^{-1}$, respectively. We compared three different delay ranges: $g_1{(\tau )}^2>0.01$, $g_1{(\tau )}^2>0.1$, and $g_1{(\tau )}^2>0.5$ [i.e., $g_1(\tau )>0.1$, $g_1(\tau )>0.3$, and $g_1(\tau )>0.7$, respectively). The whole intensity autocorrelation curve $g_2(\tau )=1+\beta$. $g_1{(\tau )}^2$ was first fitted, then the threshold delay was computed on the fitted $g_1{(\tau )}^2$ curve to reduce noise. Time points for which the short separation curve fit resulted in a $R^2$ below 0.95 were discarded from the time series. The resulting BFI time traces were interpolated to the same time points as the NIRS signal before further analysis.

We computed the hemodynamic response function (HRF) to the hypercapnic episodes for each modality and each hypercapnic strength (4 and 8 mm Hg). The time of onset of each hypercapnic event was manually identified on the ${\mathrm{etCO}}_2$ time traces. Block averages over a period of $-20$ to $+85\mathrm{s}$ from stimuli onsets of both runs were performed on [HbO], [HbR], and [HbT] for NIRS, and on relative changes in BFI for DCS. At each time point, we also computed the standard deviation of the signal over the six successive blocks of a specific hypercapnic strength. We computed the CNR of the 4- and 8-mm Hg HRFs for each subject. The contrast was defined as the change in the block-averaged signal between a baseline period (time $-20$ to 0 s from hypercapnia onset), and an “active” period (time $+45$ to $+65\mathrm{s}$ from onset). The noise was defined as the average of the HRF standard deviation over all time points.

3.1.1.

Sensitivity to brain and scalp

The sensitivity to brain and scalp obtained from the Monte Carlo computations are shown in Fig. 1, where the bar height is the median, and the error bars extend from 25th to 75th percentile of the values over all eight simulated probe locations. The NIRS sensitivities at 1, 2, and 3 cm are 95%, 88%, and 88%, respectively, for scalp, and 0.6%, 3%, and 8% for brain. This results in brain-to-scalp sensitivities of 0.7%, 4%, and 10% at these same separations. For DCS, the sensitivities to scalp when fitting the whole autocorrelation curve are 100%, 94%, and 80%, and the sensitivities to brain are 0.4%, 6%, and 19%, resulting in relative brain-to-scalp sensitivities of 0.4%, 6%, and 24%. When fitting only the early portion of the curve [$g_1(\tau )>0.7$], the DCS sensitivity to scalp decreases to 99%, 89%, and 69%, and the sensitivity to brain increases to 1%, 12%, and 30%, resulting in higher brain-to-scalp sensitivities of 1%, 13%, and 43% at the three separations.

The NIRS Monte Carlo simulations rely on the Beer–Lambert law, which assumes a linear relationship between the optical density change and the absorption change. This assumption holds true for small variations in blood volume. For our computation, we used a 5% increase in blood volume, but the resulting sensitivity to brain and scalp is independent of this value. Note that, at each wavelength, the relative sensitivity to brain and scalp is simply the ratio of the partial pathlengths in both tissue types. Our computations are performed for the sensitivity in HbT, incorporating two wavelengths, as opposed to optical density sensitivity at a single wavelength. For comparison, the brain-to-scalp ratios of partial pathlengths at these same separations were 0.9%, 4%, and 10% at 690 nm, and 0.7%, 3%, and 9% at 830 nm.

Contrary to the NIRS signal simulations, the DCS signal has a nonlinear dependence on blood flow changes, and the sensitivity to each compartment depends weakly on the amplitude of the flow change. In particular, the relative contribution of scalp is slightly stronger for higher relative flow changes in the brain. The results of Fig. 1 correspond to a 20% increase in CBF, but the orders of magnitude of the sensitivities remain the same for a large range of relative flow increase. Specifically, we tested CBF increases of 1% and 100%, and the brain sensitivity varied from 0.4% to 0.3% at 1 cm, 6% to 5% at 2 cm, and 20% to 17% at 3 cm, while the scalp sensitivity varied from 100% to 100% at 1 cm, 94% to 93% at 2 cm, and 80% to 78% at 3 cm (data not shown).

Figure 2 shows the dependence of the sensitivities of both modalities on the baseline parameters. At a constant brain-to-scalp blood volume ratio, the sensitivity to brain of both NIRS and DCS decreases with increasing brain blood volume (Fig. 2, column 1). This can be intuitively understood by the increased absorption of the whole head, which reduces the depth of photon penetration and, therefore, the sensitivity to the brain. Note that at low CBV, NIRS presents a sensitivity to scalp above 100% for the 1 cm SD separation. This can be simply explained by the fact that we kept the same DPF as was computed for ${[\mathrm{HbT}]}_{\text{Brain}}=75\mu \text{M}$. Changes in the scalp are, therefore, slightly overestimated because the DPF is slightly underestimated. However, the choice of DPF has no effect on the relative brain-to-scalp sensitivity. For DCS, we do not correct for the change of absorption due to blood volume change in the autocorrelation fits, i.e., we assume constant absorption. For cerebral total hemoglobin increasing from 40 to 120 μM, the relative brain-to-scalp sensitivity is approximately divided by two for both modalities.

Fig. 2

Dependence on the baseline physiological parameters of NIRS and DCS sensitivities: (a) varying CBV, at constant brain-to-scalp volume $\text{ratio}=2.5$; (b) varying brain-to-scalp volume ratio, at constant $\mathrm{CBV}=75\mu \text{M}$; (c) varying CBF, at constant brain-to-scalp $\text{flow ratio}=6$; (d) varying brain-to-scalp flow ratio, at constant ${\mathrm{CBF}=6\times 10}^6{\mathrm{mm}}^2{\mathrm{s}}^{-1}$. The dotted lines show the sensitivity to scalp, and the solid lines the sensitivity to brain. For DCS brain-to-scalp sensitivity, we present the results when the whole autocorrelation curve is fit (thick magenta lines), and when only the early delays corresponding to $g_1(\tau )>0.7$ are included (thin cyan lines).

At constant CBV (${[\mathrm{HbT}]}_{\text{Brain}}=75\mu \text{M}$), varying the brain-to-scalp blood volume ratio from 1 to 4 (i.e., decreasing the scalp blood volume from 75 to 18.8 μM) slightly increases the sensitivity of NIRS to both scalp and brain (Fig. 2, column 2). For DCS, the sensitivity to scalp slightly increases and that to brain slightly increases. The resulting impact of decreasing scalp blood volume depends on the SD separation for NIRS: at 1 and 2 cm, the relative brain-to-scalp sensitivity increases from 0.3% to 0.9% and from 2.6% to 3.9%, respectively, while at 3 cm it decreases slightly from 10% to 9%. For DCS, the effect of decreasing the scalp blood volume is a slight increase in the relative scalp-to-brain sensitivity.

For a constant brain-to-scalp blood flow ratio of 6, varying the baseline CBF has no effect on the sensitivity of NIRS to cerebral or extracerebral layers (Fig. 2, column 3). This is expected since the sensitivity of this modality depends solely on scattering and absorption properties, and not flow parameters. The variation of baseline CBF also has a negligible effect on the sensitivity of DCS to brain and scalp, provided that the brain-to-scalp flow ratio is kept constant.

Similarly, at constant CBF, varying the scalp flow has no effect on NIRS sensitivity (Fig. 2, column 4). On the contrary, the sensitivity of DCS to scalp and brain depends strongly on their relative flow values. As an example, at a typical SD separation of 3 cm, the relative brain-to-scalp sensitivity is only 8% for identical baseline flows in both compartments. In contrast, when considering a brain flow 10 times that of scalp, the relative brain-to-scalp sensitivity increases almost four times, up to 30%. This relative brain-to-scalp sensitivity further increases to 48% when fitting only the early portion of the autocorrelation curves.

3.1.2.

Contrast-to-noise ratio

Figure 3 shows the contrast and CNR of both modalities in response to a simulated functional activation and to hypercapnia. We simulated a similar flow change in both cases, 50% and 60% CBF increase, respectively, which leads to a similar DCS contrast: for the 3 cm channel, a flow increase of about 10% is detected (when taking into account the absorption change in the autocorrelation fitting). When this change in absorption is not taken into account, the retrieved change in flow decreases from 10% to 6.5% for both physiological events (data not shown). Similarly, the modeled increase in CBV being almost identical for both physiological events, the NIRS contrast on [HbT] is identical in both cases, with a detected increase of approximately 0.25 μM at 3 cm. On the contrary, the contrast on [HbO] and [HbR] differs between the simulated functional activation and hypercapnia because oxygen consumption increases during functional activation but not during hypercapnia. For the functional activation, the measured increase in [HbO] ($+0.38\mu \text{M}$ at 3 cm) is three to four times the amplitude of the [HbR] decrease ($-0.11\mu \text{M}$ at 3 cm). In contrast, in the case of hypercapnia, the measured [HbO] increase ($+0.62\mu \text{M}$ at 3 cm) is only twice the amplitude of the [HbR] decrease ($-0.33\mu \text{M}$ at 3 cm).

Fig. 3

Monte Carlo simulations of the contrast (top) and CNR (bottom) of NIRS and DCS, in response to functional activation (left) and hypercapnia (right). The bar heights show the median value across the eight probe locations. For each parameter, the three bars of increased darkness in the same color show the results at 1, 2, and 3 cm, respectively. For DCS, we present the cases where the whole autocorrelation curve $g_1(\tau )$ is fit (magenta), and when only the early delays corresponding to $g_1(\tau )>0.7$ are fit (cyan).

When considering the NIRS and DCS noise levels, we find a very similar CNR for both modalities during cerebral activation. Specifically, the CNR on [HbO] ($\mathrm{CNR}=1.9$ at 3 cm), [HbR] ($\mathrm{CNR}=2.3$ at 3 cm), and BFI ($\mathrm{CNR}=2.1$ at 3 cm) are all within 20% of each other, with a slight advantage to [HbR]. Because of lower contrast on [HbT], the resulting CNR is also lower ($\mathrm{CNR}=1.3$ at 3 cm).

For the simulated hypercapnia, our model results in high CNR on [HbR] ($\mathrm{CNR}=6$ at 3 cm), two times larger than that for [HbO] ($\mathrm{CNR}=3$) and four times larger than that for [HbT] ($\mathrm{CNR}=1.5$). The CNR on the DCS retrieved flow ($\mathrm{CNR}=2.5$ at 3 cm) is similar to that on [HbO] and lower than that of [HbR].

3.2.1.

End-tidal ${\mathsf{C}\mathsf{O}}_{\mathsf{2}}$ data

Table 2 shows, for each subject, the mean increase in ${\mathrm{etCO}}_2$ from their individual baseline, across all 4 and 8 mm Hg hypercapnic episodes. The ${\mathrm{etCO}}_2$ increase as manually identified on the ${\mathrm{etCO}}_2$ time traces is slightly higher than the targeted values.

Table 2

Mean increase in end-tidal CO2 for each subject from their individual baseline, as manually identified on the etCO2 time traces.

3.2.2.

Hemodynamic response functions

Figure 4 shows the hemodynamic response we observed with NIRS (upper row) and with DCS (lower row) on subject 1, after independently block-averaging all 4 mm Hg events (left) and all 8 mm Hg events (right). For DCS, the three curves were obtained with the three different ranges of delays corresponding to $g_1{(\tau )}^2>0.01$, $g_1{(\tau )}^2>0.1$, and $g_1{(\tau )}^2>0.5$. For this subject, we observed a response with both modalities at 3 cm to both grades of hypercapnia, specifically an increase in blood flow as measured with DCS, as well as a decrease in [HbR] and increase in [HbO] as measured by CW-NIRS. No significant change in [HbT] was detected. The changes are qualitatively identical for both hypercapnic strengths but present higher amplitude at 8 mm Hg.

Fig. 4

Hemodynamic response function (HRF) to hypercapnia measured in subject 1, with NIRS (top) and DCS (bottom). The targeted hypercapnia duration is indicated with the shaded gray area. For DCS, the three colors indicate different range of delays included in the fit, corresponding to different thresholds on $g_1{(\tau )}^2$ (0.01, 0.1, and 0.5).

Figure 5 displays the HRF at 3 cm for the 8 mm Hg hypercapnia for all four subjects. Although an increase in flow was detected by DCS in all subjects (bottom row), albeit with varying amplitude, changes in the NIRS signals were not consistently observed (top row). In fact, while two subjects show the expected response described above, one subject displays the inverse response (decrease in [HbO] and small increase in [HbR]) and the CNR in one subject was too low to observe any hemoglobin change.

Fig. 5

NIRS and DCS HRF to hypercapnia (8 mm Hg above baseline) measured at 3 cm for all four subjects.

3.2.3.

Contrast and contrast-to-noise ratio

Figure 6(a) reports the median contrast for both modalities over the four subjects. For the 4 mm Hg hypercapnia, we observed an increase in blood flow at 0.8 and 3 cm of 9% and 15%, respectively, an increase in [HbO] of $+0.23$ and $+0.44\mu \text{M}$, a decrease in [HbR] of $-0.11$ and $-0.29\mu \text{M}$, and a small increase in HbT of $+0.14$ and $+0.17\mu \text{M}$. For the 8 mm Hg hypercapnia, we observed increase an in blood flow at 0.8 and 3 cm of 11% and 27%, respectively, an increase in [HbO] of $+0.08$ and $+0.25\mu \text{M}$, a decrease in [HbR] of $-0.07$ and $-0.25\mu \text{M}$, and a small increase in [HbT] of $+0.01$ and $+0.03\mu \text{M}$. For both hypercapnia responses, when fitting only the early portion of the autocorrelation curves, the flow response decreases slightly at 0.8 cm and increases slightly at 3 cm.

Fig. 6

Median across four subjects of contrast (top) and CNR (bottom) of the HRF to 4 mm Hg hypercapnia (left) and 8 mm Hg hypercapnia (right). For each parameter, we present the results at 0.8 cm (light color) and at 3 cm (dark color). The DCS results are presented for two different ranges of delays incorporated in the fit, corresponding to two thresholds of the autocorrelation curve: $g_1{(\tau )}^2>0.01$ (“All $\tau$,” magenta), and $g_1{(\tau )}^2>0.5$ (“Short $\tau$,” cyan).

The corresponding median CNR is shown in Fig. 6(b). For both hypercapnic strengths, the CNR was low ($<1$) for HbO and HbT at both separations, as well as for the 0.8 cm HbR signal and DCS signal. In contrast, at the long 3 cm separation, the CNR on [HbR] was almost 2 at both hypercapnic strengths, and the CNR on flow was about 2 for 4 mm Hg and above 3 for 8 mm Hg. Even though the contrast itself was slightly increased when fitting only the early portion of the curve [$g_1{(\tau )}^2>0.5$], the CNR on the 3 cm DCS blood flow nonetheless decreased to about 1.5 and 1.8 for 4 and 8 mm Hg, respectively, because of increased noise.

4.6.1.

NIRS measure of hypercapnia

Interestingly, our NIRS experimental results are consistent with the previous hypercapnia studies that reported low or inconsistent responses to hypercapnia^79,80 for a 3 cm SD separation. Leung et al. combined NIRS and TCD ultrasound measurements in healthy adults during hypercapnia (about a 6% increase in inspired ${\mathrm{CO}}_2$).⁷⁹ Over 14 subjects, they report an average flow increase of 15%, but only a 1% increase in the tissue hemoglobin index representative of CBV. In fact, 13 out of 14 subjects showed an increase in CBF as measured by TCD, but four presented a decrease in CBV during hypercapnia. Virtanen et al.⁸⁰ presented NIRS measurements of hypercapnia (2% to 3% ${\mathrm{CO}}_2$ increase from rest level). For a 3 cm SD separation, the response to hypercapnia is barely visible in the HbO signal, but appears in the HbR time trace. However, at a 5-cm interoptode distance, the response to hypercapnia is clearly visible in both species. A rough estimate from their figures shows a CNR of about 1 on HbR and 0.3 on HbO at 3 cm, and about 1.5 and 0.7, respectively, at a 5 cm separation. Similarly to our own NIRS experimental findings, these results show higher CNR on HbR than on HbO, and lower CNR absolute values than we modeled. Alderliesten et al. observed a consistent response to hypercapnia in a study combining NIRS and fMRI imaging.⁸¹ They used a 4-cm SD separation which could explain their higher sensitivity to brain. However, they also found that the NIRS measure of CBV changes was systematically about four times lower than its fMRI-derived measure. Smielewski et al.⁸² consistently observed a response to hypercapnia as measured by CW-NIRS with a 6-cm interoptode separation. Successful measures of responses to hypercapnia with NIRS in adults have, therefore, been obtained with long SD separations ($>4\mathrm{cm}$), while a shorter (3 cm) interoptode distance produced inconsistent results similar to the present study.

The question arises of why our simplified model, while qualitatively consistent with experimental results, fails to quantitatively describe the NIRS signal reported by multiple hypercapnia studies. Specifically, it predicts a response on HbO, and even more so on HbR, much higher than was experimentally observed. The flow and volume responses we model are based on the Grubb relationship, and are consistent with multiple PET and MRI studies during hypercapnia.^54,56 Reported values for the Grubb coefficient vary widely, typically from 0.12 to 0.38.^52–56 We used a relatively low value of 0.25. Chen et al. even report a lower value, specific to venous-weighted fMRI, of 0.12 in the frontal cortex in response to hypercapnia (value not significantly different from the rest of the head value of 0.17). If we use a lower value of 0.15 for the Grubb coefficient, our simulations result in a lower HbT contrast, in better agreement with our experimental results. However, because we assume a constant ${\mathrm{CMRO}}_2$ in the relationship, such as that of Maydew et al. [Eq. (1)], this in turn results in a higher contrast on HbR, further away from our experimental results. Conversely, using a higher $\alpha$ value in our simulations decreases the contrast on HbR, in better agreement with our experimental results. Our simplified model lumps all microvasculature into a single compartment to which NIRS is homogeneously sensitive. It is possible that we are more sensitive than modeled to the arterial compartment, where we expect a smaller decrease in HbR. For instance, Sakadzic et al. observed in a rat model a smaller increase in ${\mathrm{SO}}_2$ in the arterioles than in the venules in response to hypercapnia.⁸³

4.6.2.

DCS measure of hypercapnia

We report in this study a high increase in blood flow in response to a short ($\sim 30\mathrm{s}$) period of approximately 5% inspired ${\mathrm{CO}}_2$ (${\mathrm{etCO}}_2$ about 8 mm Hg above baseline), with a median value of 27% before applying any partial volume correction, or 3.4% per mm Hg of ${\mathrm{etCO}}_2$. From our simulations, a correction factor of approximately 5 is required on the adult head. This would correspond to an actual CBF increase of 135%, or 17% per mm Hg of ${\mathrm{etCO}}_2$. This value is physiologically too high compared to numerous other studies of response to hypercapnia as assessed by other modalities for a comparable amount of inspired ${\mathrm{CO}}_2$.^45,47,84 We expect an increase in CBF on the order of 60% for 5% inspired ${\mathrm{CO}}_2$. Note, however, that our results are consistent with one publication that reported DCS measurements of hypercapnia in adults.⁸⁵ Durduran et al. observed an average 2.4% increase in CBF per mm Hg of ${\mathrm{etCO}}_2$ in five adult subjects. Buckley et al. performed DCS measurements in children undergoing hypercapnia and validated their results through comparison with phase-encoded velocity mapping MRI.¹⁴ They observed an average increase in blood flow of 49% in response to a 30-min long hypercapnia episode (approximately a 3% fraction of the inspired ${\mathrm{CO}}_2$). Note that these results cannot be directly compared to our simulations and experimental data as the subjects were children with a thinner extracerebral layer than adults.

One explanation for the high CBF increase we observe is the possible contribution of some extracerebral systemic blood flow increase. Indeed, the short separation DCS measurement (0.8 cm) shows a small blood flow increase (11% on average) which probably arises from the scalp, since the sensitivity to brain at 1 cm is very low ($<1\%$). This hypothesis is also supported by the fact that subjects with the highest flow increase at 3 cm also have a higher contribution at 0.8 cm (data not shown). This observation is, however, in contrast to the results of Durduran et al.,⁸⁵ who reported no change in scalp flow during hypercapnia as measured with laser Doppler flowmetry. It is also surprising that we did not observe a stronger response with NIRS if there is indeed a systemic superficial contribution to the signal. Further studies incorporating a more rigorous multilayer model will be needed to distinguish the cerebral and superficial contributions to the DCS signal.