Sensitivity of frequency-domain optical measurements to brain hemodynamics: simulations and human study of cerebral blood flow during hypercapnia

Abstract

This study characterizes the sensitivity of noninvasive measurements of cerebral blood flow (CBF) by using frequency-domain near-infrared spectroscopy (FD-NIRS) and coherent hemodynamics spectroscopy (CHS). We considered six FD-NIRS methods: single-distance intensity and phase (SDI and SDϕ), single-slope intensity and phase (SSI and SSϕ), and dual-slope intensity and phase (DSI and DSϕ). Cerebrovascular reactivity (CVR) was obtained from the relative change in measured CBF during a step hypercapnic challenge. Greater measured values of CVR are assigned to a greater sensitivity to cerebral hemodynamics. In a first experiment with eight subjects, CVR_SDϕ was greater than CVR_SDI (p < 0.01), whereas CVR_DSI and CVR_DSϕ showed no significant difference (p > 0.5). In a second experiment with four subjects, a 5 mm scattering layer was added between the optical probe and the scalp tissue to increase the extracerebral layer thickness (L_ec), which caused CVR_DSϕ to become significantly greater than CVR_DSI (p < 0.05). CVR_SS measurements yielded similar results as CVR_DS measurements but with a greater variability, possibly resulting from instrumental artifacts in SS measurements. Theoretical simulations with two-layered media confirmed that, if the top (extracerebral) layer is more scattering than the bottom (brain) layer, the relative values of CVR_DSI and CVR_DSϕ depend on L_ec. Specifically, the sensitivity to the brain is greater for DSI than DSϕ for a thin extracerebral layer (L_ec < 13 mm), whereas it is greater for DSϕ than DSI for a thicker extracerebral layer.

1. Introduction

Regulation of blood flow in the brain (cerebral blood flow, CBF) is critically important to maintain proper brain function and tissue viability. Measurements of CBF during hypercapnic challenges yield the cerebrovascular reactivity (CVR), which is defined as the percent change of CBF per unit change in arterial carbon dioxide pressure (${\text{PaCO}}_2$). This measurement of CVR is indicative of the ability of cerebral vessels to dilate in response to hypercapnia, and is a well-established biomarker to elicit cerebral hemodynamic responses and assess cerebrovascular functions [1].

Commonly used techniques for assessing CBF are magnetic resonance imaging (MRI) [2–4] and transcranial Doppler ultrasound (TCD) [5]. Optical techniques such as near-infrared spectroscopy (NIRS) and diffuse correlation spectroscopy (DCS) have also become more relevant for brain imaging in both healthy human subjects and patients with brain injury [6–8]. This is due to their advantages of providing continuous, noninvasive, bedside measurements, and being sensitive to local tissue hemodynamics. Some NIRS studies used changes in the hemoglobin concentration difference $\mathrm{\Delta }[O(t)-D(t)]$ as a surrogate of CBF changes (with $O(t)$ and $D(t)$ being the concentrations of oxy- and deoxy-hemoglobin in tissue, respectively) [9]. Another method is based on a hemodynamic model of coherent hemodynamics spectroscopy (CHS) to convert frequency-domain (FD) NIRS measurements into relative changes in CBF [cbf$(t)$], under a condition of negligible changes in cerebral metabolic rate of oxygen (${\mathrm{CMRO}}_2$) [10,11]. As compared to $\mathrm{\Delta }[O(t)-D(t)]$, this so-called FD-NIRS-CHS method allows for more accurate measurements of CBF dynamics by taking into account the effects of cerebral blood volume (CBV) changes and blood transit times in the capillary and venous compartments [11]. Previous studies using FD-NIRS-CHS to measure CBF [10,11] have only exploited intensity (I) data collected with one source and one detector (single-distance intensity, SDI, at a source-detector distance of 20 to 40 mm) to measure relative changes in hemoglobin concentrations. In those studies, FD-NIRS phase ($\varphi$) data, along with FD-NIRS modulated intensity amplitude data, have only been used to calculate absolute hemoglobin concentrations [12].

Measurements with SDI have been known to be prone to extracerebral tissue contamination [13]. Thus, several studies have focused on using $\varphi$ data or the slope information of both I and $\varphi$ to improve depth sensitivity of FD-NIRS measurements to the brain. For instance, single-distance $\varphi$ (SD$\varphi$) could provide deeper sensitivity than SDI [14,15]. The use of the slope of multi-distance I or $\varphi$ measurements versus source-detector distances has also been shown to provide increased sensitivity to cerebral oxygenation [15,16]. This method is referred to as single-slope (SS) when it uses either a single source or detector. The SS method features a lower sensitivity to uniform superficial absorption changes than SD but may yield misleading results when there are localized hemodynamic changes (especially close the medium boundary) [17]. A method called dual-slope (DS) [13,17,18] has been recently proposed, which involves using the average of two single-slopes in a special source-detector arrangement to enhance sensitivity to deeper tissue regions and reduce sensitivity to uniform or localized superficial inhomogeneities. In a study on human subjects with induced hemodynamics in both scalp and cortical tissues by systemic perturbations in arterial blood pressure (ABP), DS using $\varphi$ data (DS$\varphi$) was shown to be more sensitive to cerebral hemodynamics than SDI, SD$\varphi$ and DS using I data (DSI) [18]. DS measurements are also less sensitive to instrumental drifts and changes in optical couplings between the optical probe and tissue; which comes from the idea of the self-calibrating configuration [19,20].

The spatial distribution of regional sensitivity of NIRS data have been reported using diffusion theory for a homogeneous medium containing localized absorption changes. In particular, SDI and SD$\varphi$ feature banana-shaped sensitivity regions, with maximal sensitivity of SDI closer to the source and the detector in the superficial layer, and maximal sensitivity of SD$\varphi$ more uniform and extending deeper into the tissue [13,21–23]. DSI and DS$\varphi$ feature nut-shaped sensitivity regions and show a maximal sensitivity deeper in the tissue than SD measurements, especially for DS$\varphi$ [13,17,18]. Although models based on homogeneous media may be used to get a general sense of the shape of the region of sensitivity and may be representative of a variety of biological tissues, the case of the highly heterogeneous adult head requires more suitable medium inhomogeneity. In fact, optical measurements based on homogeneous tissue models were shown to underestimate absolute hemoglobin concentrations in adult human brain by about 30$\mathrm{\%}$ [24]. To overcome the limitation of homogeneous models, two-layered models have been developed in the FD [25,26], with experimental validations on tissue-like layered phantoms and on human subjects [24,27,28]. Multi-layered medium models using either analytical solutions to the diffusion equation or numerical approaches (finite element method (FEM) or Monte Carlo simulations) have been applied to the study of the depth sensitivity of CW-NIRS SDI [29–31], FD-NIRS SDI and SD$\varphi$ [14], time-domain (TD)-NIRS SD data [32], and TD-NIRS data with a subtraction method applied to two SDI measurements [33]. The sensitivity of FD-NIRS DSI and DS$\varphi$ measurements in heterogeneous turbid media has not yet been investigated.

The present study aims to elucidate the sensitivity of FD-NIRS data from different measurement methods (SDI, SD$\varphi$, SSI, SS$\varphi$, DSI, and DS$\varphi$) to cerebral hemodynamics in response to a rapid step increase in end-tidal carbon dioxide pressure (${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$). Our study is under a hypothesis of a significant increase in CBF [3] and negligible scalp blood flow changes [34,35] during hypercapnia. We note that the changes in scalp blood flow during the hypercapnic response could be more complicated in actual experiments and could be varied across subjects, experimental conditions, tissue geometry, etc [36]. In the simulations, we considered different conditions for tissue heterogeneity between extracerebral tissues and the brain, by modeling the measured tissue as made of two layers: the top layer representing superficial extracerebral tissues (scalp, skull, cerebrospinal fluid CSF), and the bottom layer representing the cortical tissue in the brain. We addressed two cases: either the baseline optical properties of the two layers are the same (a semi-infinite homogeneous medium at baseline) or they are different (a two-layered medium at baseline). We present analytical simulations of the sensitivity of SDI, SD$\varphi$, SSI, SS$\varphi$, DSI, and DS$\varphi$ data to absorption changes occurring only within the cerebral layer, which represent CBF changes during the hypercapnic stimulus. The sensitivities computed in these simulations serve as a basis to guide the interpretation of the in vivo measurements of CVR on human subjects as a function of extracerebral layer thickness ($L_{ec}$). This work also presents dynamic CBF measurements with FD-NIRS-CHS based on DSI and DS$\varphi$ data for the first time in vivo.

We use a number of acronyms in this paper; for the benefit of the reader we provide a table of acronyms, as well as a table of symbols, in Appendix A.

2. Methods

2.1. In vivo human measurements during hypercapnia

2.1.1. Subjects, experimental setup and measurement protocol

A complete experiment included: 5 minutes of normocapnic baseline when the subject breathed medical air (21$\mathrm{\%}$ ${\mathrm{O}}_2$, 79$\mathrm{\%}$ ${\mathrm{N}}_2$), 3 minutes of hypercapnia when the subject breathed medical air mixed with 5$\mathrm{\%}$ ${\mathrm{CO}}_2$ (5$\mathrm{\%}$ ${\mathrm{CO}}_2$, 21$\mathrm{\%}$ ${\mathrm{O}}_2$, 74$\mathrm{\%}$ ${\mathrm{N}}_2$), and 5 minutes of normocapnic recovery when the subject breathed medical air. During the experiment, gas mixtures (Airgas, Billerica, MA) were delivered from air tanks to a variable volume gas reservoir (to allow for cyclic respiration) and then to the subject through a facemask (AFT25, BIOPAC Systems, Inc., Goleta, CA) (Fig. 1(A)). A gas flow controller (FMA3206, Omega Engineering, Inc., Stamford, CT) was used to keep the air flow rate at 10 L/min. An infrared-based ${\mathrm{CO}}_2$ monitor module (CO2100C, BIOPAC Systems, Inc., Goleta, CA) was connected to the facemask through a sampling tube to measure breath-by-breath ${\mathrm{CO}}_2$ content. Examples of CO${\mathrm{CO}}_2$ content and extracted ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$ (in mmHg) are shown in Fig. 1(B). FD-NIRS data were collected with a commercial instrument (Imagent, ISS, Inc., Champaign, IL) that operates at wavelengths of 690 and 830 nm and at a modulation frequency ${\text{f}}_{\text{mod}}=140.625$ MHz. From each source-detector distance pair, $\varphi$ and I data were collected. The Tufts University Institutional Review Board approved the experimental protocol, and the subjects provided written informed consent prior to the experiment.

Fig. 1.

(A) Experimental setup with the optical probe on the subject’s right side of the forehead. (B) Examples of breath-by-breath ${\mathrm{CO}}_2$ content and extracted end-tidal ${\mathrm{CO}}_2$ (P${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$, indicated by the red line). Grey shaded area indicates the 5$\mathrm{\%}$ ${\mathrm{CO}}_2$ period. (C) Schematic of the optical probe used in experiment 1. The sources are labeled in numbers (1 and 2), and the detectors are labeled in letters (A and B). The probe has: two single-distance (SD) measurements at 35 mm (1B and 2A), two single-slope (SS) measurements with 25 and 35 mm distances (1AB and 2BA), and one dual-slope (DS) measurement with two 25 mm and two 35 mm distances (1AB2). (D) Schematic of the optical probe used in experiment 2. The probe has two DS measurements: one touching the head directly on the scalp (1AB2), and one touching a 5 mm thick phantom layer placed in between the fibers and the scalp (3CD4).

In the first experiment (experiment 1), eight healthy subjects (3 females, 5 males, age range: 23-55 years) participated in the study. An optical probe was placed on the right side of the subject’s forehead. The probe consisted of a linear array of two source optical fiber pairs and two detector fiber bundles symmetrical about the midline between the two sources (Fig. 1(C)). This arrangement allowed for two SD measurements at 35 mm, two SS measurements with 25 and 35 mm, and one DS measurement with two 25 mm and two 35 mm distances. NIRS data were collected at a sampling rate of 9.93 Hz.

In the second experiment (experiment 2), three of the eight subjects (subjects 2, 3, and 5) came back for additional measurements on a different day. We also recruited a new subject (subject 9). Therefore, we had a total of four subjects in experiment 2 (1 female, 3 males, age range: 23-34 years). The optical probe was placed at about the same forehead area sensed by the probe in experiment 1. This optical probe had two DS arrangements separated by 10 mm laterally (Fig. 1(D)): one touching the head directly on the scalp (experiment 2a), and one touching a 5 mm thick layer of scattering phantom placed in between the fibers and the scalp (experiment 2b). Each DS arrangement also had two 25 mm and two 35 mm source-detector distances. The optical properties of the phantom layer were chosen within the ranges of the extracerebral layer properties [27] as: absorption coefficients ${\mu }_a=0.05$ ${\mathrm{cm}}^{-1}$ and $0.05$ ${\mathrm{cm}}^{-1}$, and reduced scattering coefficients ${\mu }_s^{\prime }=11$ ${\mathrm{cm}}^{-1}$ and $13$ ${\mathrm{cm}}^{-1}$ at 830 and 690 nm, respectively. The phantom layer was cut off from a solid block having a size of about $150\times 150\times 150$ mm made by mixing silicone (Smooth-On, Macungie, PA), titanium dioxide powder (AEE, Upper Saddle River, NJ), and acrylic India ink (Daler-Rowney, Bracknell, England). Optical properties of the phantom block were measured by multi-distance FD-NIRS with a fixed detector and linearly scanned light sources [37]. We perforated the phantom layer to allow some of the optical fibers to pass through the layer and get in contact with the scalp tissue (see Fig. 1(D)), and we secured the combined probe and phantom layer setup in place using a Velcro strap wrapped around the head. The purpose of adding the phantom layer in the second DS arrangement is to assess the effect of a greater extracerebral tissue thickness $L_{ec}$ on the FD-NIRS data from a given tissue region. This additional 5 mm thickness is strictly associated with no dynamic changes. In this second experiment, NIRS data were collected at a sampling rate of 4.97 Hz.

For eight out of nine subjects (all except subject 4), we performed ultrasound imaging to measure skull thickness using a SonoSite ${\mathrm{S-Nerve}}^{\text{TM}}$ ultrasound system (FUJIFILM SonoSite Inc, Bothell, WA) equipped with a multi-frequency, broadband, 50 mm linear array transducer probe (HFL50x, frequency range: 6-15 MHz). Ultrasound gel was layered onto the probe, and then the probe was lightly applied to the scalp in parallel with the midsagittal line for 30 s at most. Scans were performed on three to four positions on the forehead area measured by the optical probe, and still ultrasound images were recorded. An example of an individual scan and processing steps to extract skull thickness can be seen in the Supplementary materials, Fig. S1. Average and standard deviation of skull thickness values measured across different scan positions were obtained. We added to the average skull thickness values an assumed value of 3.5 $\pm$ 0.5 mm for scalp thickness (based on literature [38]) to obtain measurements of $L_{ec}$.

2.1.2. Measurements of baseline absolute optical properties and hemoglobin concentrations

The self-calibrating method was used to find baseline tissue optical properties ${\mu }_{a,0}$ and ${\mu }_{s,0}^{\prime }$ at two wavelengths by using combined DSI and DS$\varphi$ data, while suppressing optical coupling and instrumental factors and removing the need for calibration. As explained in Appendix B, given a DS configuration with two sources (1, 2) and two detectors (A, B) as shown in Fig. 1(C), the two quantities of $\frac{1}{2}\ln [({\text{I}}_{\text{1B}}{\text{I}}_{\text{2A}})/({\text{I}}_{\text{1A}}{\text{I}}_{\text{2B}})]$ and $\frac{1}{2}({\varphi }_{\text{1B}}+{\varphi }_{\text{2A}}-{\varphi }_{\text{1A}}-{\varphi }_{\text{2B}})$ are equivalent to $\ln [\text{I}({\rho }_L)/\text{I}({\rho }_S)]$ and $\varphi ({\rho }_L)-\varphi ({\rho }_S)$, respectively. $\text{I}({\rho }_L)$, $\text{I}({\rho }_S)$, $\varphi ({\rho }_L)$ and $\varphi ({\rho }_S)$ are the theoretical intensity and phase at long (${\rho }_L$) and short (${\rho }_S$) source-detector distances that are free of instrumental and other experimental confounds. To measure baseline optical properties, we fit these measured quantities to the complex reflectance ($\tilde{R}$) for semi-infinite media with extrapolated boundary conditions in the FD [17,18]. Specifically, $\frac{1}{2}\ln [({\text{I}}_{\text{1B}}{\text{I}}_{\text{2A}})/({\text{I}}_{\text{1A}}{\text{I}}_{\text{2B}})]$ was fitted to $\ln [|\tilde{R}({\rho }_L)|/|\tilde{R}({\rho }_S)|]$ and $\frac{1}{2}({\varphi }_{\text{1B}}+{\varphi }_{\text{2A}}-{\varphi }_{\text{1A}}-{\varphi }_{\text{2B}})$ was fitted to $\text{Arg}[\tilde{R}({\rho }_L)]-\text{Arg}[\tilde{R}({\rho }_S)]$. The fitting procedure was performed by numerical optimization (fminsearch, MATLAB, Mathworks Inc., USA) to minimize the sum of squared differences between the prediction and data, normalized by the measurement errors. The procedure was tested on phantom data to ensure accuracy, from which we found an estimation error of less than 10$\mathrm{\%}$ for ${\mu }_a$ and less than 5$\mathrm{\%}$ for ${\mu }_s^{\prime }$.

The measured ${\mu }_{a,0}$ at two wavelengths were translated into absolute baseline tissue concentrations of oxy-, deoxy- and total-hemoglobin concentrations ($O_0$, $D_0$, and $T_0$, respectively) using their known extinction coefficients [39], with an assumed 70$\mathrm{\%}$ water volume fraction [40].

2.1.3. Measurements of relative changes in absorption and hemoglobin concentrations

We detected and removed motion artifacts from raw SDI and SD$\varphi$ data. In particular, we visually detected the step-like motion artifacts (any rapid transient step changes in signals with characteristic times < 1 s) and corrected them by adding the offset to signals following the step changes. Then we applied a discrete wavelet-based motion artifact correction method (described by Molavi and Dumont [41]) to remove any spikes from signals. Here we used the Daubechies 2 (db2) wavelet for all measurements. We further applied our proposed technique of joint detrending on the raw data. This technique aims to remove linear temporal drifts in the phase and logarithmic amplitude by targeting drifts of instrumental origin from both sources and both detectors simultaneously; see Appendix B for a complete description of the procedure. We note that this joint detrending procedure will remove linear trends from SD and SS measurements but has no impact on DS measurements, which are intrinsically insensitive to instrumental drifts.

SDI and SD$\varphi$ were translated into changes in the absorption coefficients at two wavelengths by using the differential pathlength factors (${\text{DPF}}_{\text{I}}$ for I and ${\text{DPF}}_{\varphi }$ for $\varphi$) for a semi-infinite medium geometry under extrapolated boundary conditions. Similarly, SS and DS data were converted into temporal changes in absorption using differential slope factors (${\text{DSF}}_I$ for I and ${\text{DSF}}_{\varphi }$ for $\varphi$). The calculations of DPF and DSF are described in Ref. [18] and require the input of ${\mu }_{a,0}$ and ${\mu }_{s,0}^{\prime }$, as found by using the self-calibrating approach. A DS measurement is an average of two paired SS measurements. To a first approximation, SSI refers to the linear dependence of $\ln [({\rho }^{2}I)/(\sqrt{3{\mu }_a{\mu }_s^{\prime }}+1/\rho )]$ on the source-detector distance $\rho$, whereas SS$\varphi$ refers to the linear dependence of $\varphi$ on $\rho$. Exact expressions of the linear functions on $\rho$ associated with I and $\varphi$ can be found from Eqs. (12.35) and (12.36) in Ref. [39]. We note that the equation Eq. (12.35) in Ref. [39] is for direct current (DC) intensity, but here we used alternating current (AC) amplitude data instead to reduce contamination from room light. AC and DC data are similar at this range of ${\text{f}}_{\text{mod}}$ [42]. The diffusive medium was considered to be the head in experiment 1, and the phantom layer together with the head in experiment 2. Thus, we considered the coordinates of sources and detectors of the optical probe at the boundary between the optical probe and the diffusive medium for experiment 1 and 2b; and at 5 mm inside the diffusive medium for experiment 2a.

Relative absorption changes obtained with SDI, SD$\varphi$, SSI, SS$\varphi$, DSI and DS$\varphi$ at two wavelengths were translated into measurements of hemoglobin concentration changes with respect to normocapnic baseline ($\mathrm{\Delta }O(t)$, $\mathrm{\Delta }D(t)$, and $\mathrm{\Delta }T(t)$).

Figure 2 shows a comparison between original data, data obtained from the joint detrending and from the independent detrending approaches. Examples are shown for SDI, SSI and DSI measurements of $\mathrm{\Delta }O$ from subject 3, experiment 1. Independent detrending is a typically used approach in which every SD signal is subtracted by a linear trend of their own initial baseline. It is clear from the figure that the original SDI data are affected by instrumental drifts, while those drifts are removed in the DSI measurements. Although both independent and joint detrending techniques remove linear trends in SD measurements, only the joint detrending approach preserves the DS signal obtained from original data. We also note that the instrumental drifts do not significantly affect SS in this particular example, which indicates that the dominant coupling/instrumental effects could have originated from the sources.

Fig. 2.

Example time traces of relative oxyhemoglobin concentration change ($\mathrm{\Delta }O(t)$) from Subj. 3, experiment 1, measured with single-distance intensity (SDI) at 25 mm (1A and 2B), SDI at 35 mm (1B and 2A), single-slope intensity (SSI, 1AB and 2BA) and dual-slope intensity (DSI, 1AB2). Signals are shown for original data (blue dashed lines), independently detrended data (yellow lines) and jointly detrended data (red lines). All signals are lowpass filtered to 0.05 Hz. Grey shaded area indicates 5$\mathrm{\%}$ CO${\mathrm{CO}}_2$ period.

2.1.4. Relative cerebral blood flow changes and cerebrovascular reactivity

The obtained time traces of hemoglobin concentration were low-pass filtered by using a linear phase finite impulse response (FIR) filter based on the Parks–McClellan algorithm [43] (MATLAB function "firpmord", with a passband-edge frequency of 0.05 Hz and a stopband-edge of 0.1 Hz). Filtered $\mathrm{\Delta }O(t)$ and $\mathrm{\Delta }D(t)$ for each method were then translated into $\text{cbf}(t)$ (defined as $100\times \mathrm{\Delta }\text{CBF}(t)/{\text{CBF}}_0$, in units of $\mathrm{\%}$) by using the CHS model, as described in Ref. [11]. The calculation is under negligible changes in CMRO${\mathrm{CMRO}}_2$ during hypercapnia [36,44]. Model parameters were assumed based on reported values in healthy adults [11,45] as: capillary blood transit time $t^{(c)}=1$ s, venous blood transit time $t^{(v)}=5$ s, rate constant of oxygen diffusion $\alpha =0.8$ $s^{-1}$, arterial baseline blood volume fraction ${\text{CBV}}_0^{(a)}/{\text{CBV}}_0=0.3$, and arterial to venous baseline blood volume ratio ${\text{CBV}}_0^{(a)}/{\text{CBV}}_0^{(v)}=1$. Compartmental blood volume changes (for the arterial ($a$), capillary ($c$) and venous ($v$)) were related to total blood volume changes as $\mathrm{\Delta }{\text{CBV}}^{(a)}(t)=0.78\mathrm{\Delta }\text{CBV}(t)$, $\mathrm{\Delta }{\text{CBV}}^{(c)}(t)=0$, and $\mathrm{\Delta }{\text{CBV}}^{(v)}(t)=0.22\mathrm{\Delta }\text{CBV}(t)$ based on values in literature for hypercapnic response [3,46,47].

We obtained CVR as a scaling factor between the measured $\text{cbf}(t)$ and a convolution of the recorded step change in ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$, $\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2(t)$, with a hemodynamic response function, $\text{HRF}(t)$ [4,48]:

$$\text{cbf}(t)=\text{CVR}\cdot [\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2\ast \text{HRF}(t)],$$ (1)

where CVR is in units of $\mathrm{\%}$ relative change in CBF per mmHg increase in ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$, and * denotes the convolution operator. The HRF is given by $\text{HRF}(t)=(e^{-t/\tau })/({\int }_{t_0}^{\mathrm{\infty }}{e}^{-t/\tau }dt)$ for $t\ge t_0$ and $\text{HRF}(t)=0$ for $t<t_0$, where $\tau$ is the time constant of response, $t_0$ is the time delay between the initial rise of $\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2(t)$ and that of $\text{cbf}(t)$. We fit for CVR, along with $\tau$ and $t_0$, by numerical optimization (fminsearchbnd, MATLAB, Mathworks Inc., USA) to minimize the sum of the squared differences between the model prediction and the measured cbf$(t)$. The fitting procedure was performed over a time window from the initial rise in $\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2(t)$ to the end of the decline in $\text{cbf}(t)$ following the return of ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$ to normocapnia. In some cases when $\text{cbf}(t)$ did not return to baseline as expected, the time window is then limited to the end of the hypercapnia interval. While some studies are interested in $\tau$ and $t_0$ [4,48], this study only focused on the steady-state CVR values. A one-tailed paired Student’s t-test was used to investigate differences in CVR obtained from the analysis of $\varphi$ versus I data from different measurement methods, with the statistical significance defined as $p<0.05$.

2.2. Analytical simulations of regions of sensitivity for FD-NIRS measurements

We performed analytical simulations of the regional sensitivity of FD-NIRS measurements by considering a two-layered medium that mimics the superficial extracerebral tissue (top layer) and the deeper brain tissue (bottom layer). Specifically, when absorption changes occur independently within the entire extracerebral and cerebral layers ($\mathrm{\Delta }{\mu }_{a,ec}$ and $\mathrm{\Delta }{\mu }_{a,c}$, respectively), the resulting measured absorption change $\mathrm{\Delta }{\mu }_{a,\text{meas}}$ can be expressed as follows, assuming linearity [17]:

$$\mathrm{\Delta }{\mu }_{a,\text{meas}}=\mathrm{\Delta }{\mu }_{a,ec}\sum _{j=1}^{N_{ec}}S({\mathbf{\text{r}}}_j)+\mathrm{\Delta }{\mu }_{a,c}\sum _{j=1}^{N_c}S({\mathbf{\text{r}}}_j)=\mathrm{\Delta }{\mu }_{a,ec}{S}_{ec}+\mathrm{\Delta }{\mu }_{a,c}{S}_c,$$ (2)

where $S({\mathbf{\text{r}}}_j)$ is the sensitivity to a local absorption change $\mathrm{\Delta }{\mu }_{a,j}$ at voxel $j$ centered at the position vector ${\mathbf{\text{r}}}_j$. $S_{ec}=\sum _{j=1}^{N_{ec}}S({\mathbf{\text{r}}}_j)$ and $S_c=\sum _{j=1}^{N_c}S({\mathbf{\text{r}}}_j)$ are the sensitivity to extracerebral and cerebral layers, respectively, and $N_{ec}$ and $N_c$ are the numbers of voxels in the extracerebral and cerebral layers, respectively. In this study, we focus on the sensitivity of FD-NIRS measurements to the cerebral layer for a homogeneous change in blood flow occurring only in the brain, thus mimicking the hypercapnic response. When the absorption change is negligible in the extracerebral layer (i.e., $\mathrm{\Delta }{\mu }_{a,ec}=0$), the measured absorption change can be calculated as $\mathrm{\Delta }{\mu }_{a,\text{meas}}=\mathrm{\Delta }{\mu }_{a,c}{S}_c$.

The regional sensitivity $S({\mathbf{\text{r}}}_j)$ for SD, SS and DS methods using I and $\varphi$ data can be obtained from a complex total generalized optical pathlength $⟨\tilde{L}⟩$ and a complex partial generalized optical pathlength $⟨\tilde{l}⟩$ [15]. We note that the DS sensitivity is the average sensitivity of two paired SSs.

• •For the simulation of a homogeneous medium at baseline (i.e., the optical properties at baseline of the extracerebral and cerebral layers are the same), $⟨\tilde{L}⟩$ and $⟨\tilde{l}⟩$ were obtained from diffusion theory for a semi-infinite medium with extrapolated boundary conditions, as reported in Refs. [17,18]. Optical properties of the medium were varied within the ranges of $0.08$ to $0.15$ ${\mathrm{cm}}^{-1}$ for ${\mu }_a$ and $5$ to $11$ ${\mathrm{cm}}^{-1}$ for ${\mu }_s^{\prime }$, based on reported ranges of multi-distance FD-NIRS measurements on adults [12,15,27,40]. The indices of refraction $n$ of the outer medium and the homogeneous medium were set at 1 and 1.4, respectively.
• •For the simulation of a two-layered medium at baseline (i.e., the optical properties at baseline of the two layers are different), $⟨\tilde{L}⟩$ and $⟨\tilde{l}⟩$ were computed using the solution of the diffusion equation for two-layered media (details are provided in Appendix C). Optical properties of the medium were varied as: $0.06$ to $0.10$ ${\mathrm{cm}}^{-1}$ for ${\mu }_{a,ec}$, $0.14$ to $0.20$ ${\mathrm{cm}}^{-1}$ for ${\mu }_{a,c}$, $11$ to $13$ ${\mathrm{cm}}^{-1}$ for ${\mu }_{s,ec}^{\prime }$, and $2$ to $4$ ${\mathrm{cm}}^{-1}$ for ${\mu }_{s,c}^{\prime }$ based on values from healthy human adults [24,27,28,49]. $n=1$ for the outer medium and $n=1.4$ for both layers.

We simulated a turbid medium with a size of $90\text{mm}\times 90\text{mm}\times 50$ mm ($x\times y\times z$). The medium extends from $-45$ to $45$ mm along the $x$- and $y$-axis, and from $0$ to $50$ mm along the $z$-axis (the depth coordinate). The extracerebral layer is confined to $z$-coordinates that are less than $L_{ec}$. Thus, $S_c$ was calculated as the sum of all regional sensitivity values at voxels deeper (i.e. having $z$-coordinates greater) than $L_{ec}$. We used a voxel size of $1$ mm $\times$ $1$ mm $\times$ $0.5$ mm for the simulation of a homogeneous medium, and a voxel size of $1$ mm $\times$ $1$ mm $\times$ $1$ mm for the simulation of a two-layered medium. A larger voxel size for the two-layered medium simulations was chosen because of the longer time and greater computational cost with respect to the homogeneous medium simulations. However, this difference in voxel size is not expected to affect the results of this paper. In all the calculations, the angular modulation frequency ${\omega }_{\text{mod}}$ was calculated as ${\omega }_{\text{mod}}=2\pi {\text{f}}_{\text{mod}}$ where ${\text{f}}_{\text{mod}}=140.625$ MHz, and the speed of light in vacuum was set as $c=2.998\times {10}^{10}$ cm/s.

3. Results

3.1. Absolute optical properties

Table 1 reports absolute baseline optical properties, ${\mu }_{a0}$ and ${\mu }_{s0}^{\prime }$, with corresponding $O_0$, $D_0$, and $T_0$ measured on the head of human subjects with the self-calibrating approach. Results are reported as means and standard deviations over eight subjects in experiment 1 and four subjects in experiment 2. In experiment 2, hemoglobin concentrations were calculated only from the measurements with the optical fibers placed directly on the scalp.

Table 1. Average absolute baseline absorption coefficients $({\mu }_{a0})$ and reduced scattering coefficients (${\mu }_{s0}^{\prime }$) at two wavelengths (690 and 830 nm), with corresponding oxy- ($O_0$), deoxy- ($D_0$), and total-($T_0$) hemoglobin concentrations, obtained from experiment 1 (8 subjects) and experiment 2 (4 subjects, experiment 2a for optical probe on the scalp, and experiment 2b for a 5 mm scattering layer between optical probe and scalp). Values are reported as mean $\pm$ standard deviation.

3.2. Measurements of CVR from different FD-NIRS methods and effects of increasing the extracerebral layer thickness on CVR

From experiment 1 and 2, ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$ increased significantly from $35\pm 5$ mmHg at normocapnia to $53\pm 4$ mmHg during the last minute of the 5$\mathrm{\%}$ ${\mathrm{CO}}_2$ interval ($p<0.001$, $N=12$).

Figure 3 displays cbf, the best fit by using HRF model, and $\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2$ from two subjects from experiment 1 (Fig. 3(A), (B)) and one subject from experiment 2 (Fig. 3(C), (D)). Signals are shown in an 8 minute interval starting from 30 seconds before the 5$\mathrm{\%}$ ${\mathrm{CO}}_2$ interval. Every subplot in Fig. 3 includes cbf traces from I and $\varphi$ data from each method of SD, SS, and DS. In the two experiments, ${\text{cbf}}_{\text{SD}\varphi }$ shows a greater increase during hypercapnia than ${\text{cbf}}_{\text{SDI}}$ in most cases. By contrast, ${\text{cbf}}_{\text{SS}\varphi }$ and ${\text{cbf}}_{\text{DS}\varphi }$ show a greater increase during the hypercapnic response than ${\text{cbf}}_{\text{SSI}}$ and ${\text{cbf}}_{\text{DSI}}$, respectively, in subject 3 ($L_{ec}$ = 14 mm), while the opposite behavior was observed in subject 6 ($L_{ec}$ = 12 mm). In experiment 2, ${\text{cbf}}_{\text{SSI}}$ and ${\text{cbf}}_{\text{DSI}}$ hypercapnic responses are greater than ${\text{cbf}}_{\text{SS}\varphi }$ and ${\text{cbf}}_{\text{DS}\varphi }$, respectively, for measurements with optical probe placed directly on the scalp (experiment 2a), whereas the opposite occurs for measurements with the 5 mm scattering layer between probe and scalp (experiment 2b). These results indicate that SD$\varphi$ measurements are typically more sensitive to cerebral hemodynamics than SDI measurements, whereas the relative brain sensitivity of intensity and phase slopes depends on the subject, and more specifically on the value of $L_{ec}$.

Fig. 3.

(A) and (B)-Experiment 1: cbf$(t)$ measurements from two subjects [(A): Subj. 3; (B): Subj. 6)] for: two single-distance intensity (SDI) and phase (SD$\varphi$) (1B and 2A), two single-slope intensity (SSI) and phase (SS$\varphi$) (1AB and 2BA), and dual-slope intensity (DSI) and phase (DS$\varphi$) (1AB2). (C) and (D)-Experiment 2: cbf$(t)$ measurements from Subj. 2 [(C): experiment 2a; (D): experiment 2b] for: four SDI and SD$\varphi$ (directly on the scalp: 1B and 2A, through the scattering layer: 3D and 4C), four SSI and SS$\varphi$ (directly on the scalp: 1AB and 2BA, through the scattering layer: 3CD and 4DC), and two DSI and DS$\varphi$ (directly on the scalp: 1AB2, through the scattering layer: 3CD4). ${\text{cbf}}_{\text{I}}$ are shown in magenta and ${\text{cbf}}_{\varphi }$ are shown in blue. Data are shown by thin solid lines, and the best fits with the hemodynamic response function (HRF) are shown by thick solid lines. Grey line is the recorded $\mathrm{\Delta }{\text{P}}_{\text{ET}}{\text{CO}}_2$, while the grey shaded area indicates the 5$\mathrm{\%}$ ${\mathrm{CO}}_2$ inhalation period.

Figure 4 and Table 2 present median CVR values along with their 95$\mathrm{\%}$ confidence interval (C.I.) from the two experiments. CVR${\mathrm{CVR}}_{\text{SDI}}$ was significantly lower than ${\mathrm{CVR}}_{\text{SD}\varphi }$ in experiments 1 and 2b ($p<0.01$ and $p<0.05$, respectively) but not in experiment 2a ($p>0.1$). There was no statistical difference between ${\mathrm{CVR}}_{\text{DSI}}$ and ${\mathrm{CVR}}_{\text{DS}\varphi }$ for those measurements with optical probe placed on the scalp, as shown in experiments 1 and 2a ($p>0.5$ and $p>0.1$, respectively). For those measurements with the scattering phantom layer added to the optical probe, ${\mathrm{CVR}}_{\text{DSI}}$ became significantly lower than ${\mathrm{CVR}}_{\text{DS}\varphi }$, as shown in experiment 2b ($p<0.05$). We also observed a similar behavior in for SS measurements. Individual measurements of CVR from each subjects are shown in the Supplementary materials, Fig. S2.

Fig. 4.

Box plots of cerebrovascular reactivity (CVR) measurements from experiment 1 (Exp. 1) and experiment 2 with optical probe directly on the scalp (Exp. 2a) and with the scattering layer between optical probe and scalp (Exp. 2b). Data are shown for (A) single-distance (SD), (B) single-slope (SS), and (C) dual-slope (DS) methods. Medians are shown by horizontal lines (magenta for ${\mathrm{CVR}}_{\text{I}}$ and blue for ${\mathrm{CVR}}_{\varphi }$), 95$\mathrm{\%}$ confidence intervals are shown by shaded areas, and the whiskers extend to the most extreme data points excluding the outliers. Individual data points are depicted with black circles.

Table 2. Medians and [25$\mathrm{\%}$, 75$\mathrm{\%}$] quartiles of cerebrovascular reactivity (CVR, in $\mathrm{\%}$/mmHg) measured in the two experiments, and the corresponding $p$-values obtained by paired t-test for the difference between phase ($\varphi$) and intensity (I)-based CVR values.

Parameters	Exp. 1	Exp. 2a	Exp. 2b
${\mathrm{CVR}}_{\text{SDI}}$ ($\mathrm{\%}/$mmHg)	$0.30$ $[0.22,0.43]$	$0.11$ $[0.09,0.20]$	$0.08$ $[0.05,0.10]$
${\mathrm{CVR}}_{\text{SD}\varphi }$ ($\mathrm{\%}/$mmHg)	$0.43$ $[0.29,0.51]$	$0.16$ $[0.06,0.27]$	$0.13$ $[0.09,0.18]$
$p$-value	$<0.01$ ($N=16$)	$>0.1$ ($N=8$)	$<0.05$ ($N=8$)
${\mathrm{CVR}}_{\text{SSI}}$ ($\mathrm{\%}/$mmHg)	$0.45$ $[0.25,0.73]$	$0.25$ $[0.20,0.37]$	$0.20$ $[0.17,0.27]$
${\mathrm{CVR}}_{\text{SS}\varphi }$ ($\mathrm{\%}/$mmHg)	$0.36$ $[0.28,0.54]$	$0.37$ $[0.28,0.56]$	$0.41$ $[0.34,0.65]$
$p$-value	$>0.5$ ($N=16$)	$>0.05$ ($N=8$)	$<0.01$ ($N=8$)
${\mathrm{CVR}}_{\text{DSI}}$ ($\mathrm{\%}/$mmHg)	$0.42$ $[0.29,0.65]$	$0.3$ $[0.2,0.5]$	$0.19$ $[0.17,0.27]$
${\mathrm{CVR}}_{\text{DS}\varphi }$ ($\mathrm{\%}/$mmHg)	$0.41$ $[0.28,0.48]$	$0.34$ $[0.33,0.45]$	$0.4$ $[0.3,0.6]$
$p$-value	$>0.5$ ($N=8$)	$>0.1$ ($N=4$)	$<0.05$ ($N=4$)

3.3. Comparison between in vivo human measurements and theoretical simulations

Figure 5 presents examples of bottom layer (i.e. cerebral) sensitivity of SDI, SD$\varphi$, SSI, SS$\varphi$, DSI and DS$\varphi$ for homogeneous and two-layered media at baseline. In this example, optical properties of the homogeneous medium were set as: ${\mu }_a=0.11$ ${\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=7$ ${\mathrm{cm}}^{-1}$. Optical properties of the two-layered medium were set as ${\mu }_{a,ec}=0.1$ ${\mathrm{cm}}^{-1}$, ${\mu }_{s,ec}^{\prime }=12$ ${\mathrm{cm}}^{-1}$, ${\mu }_{a,c}=0.2$ ${\mathrm{cm}}^{-1}$, and ${\mu }_{s,c}^{\prime }=3$ ${\mathrm{cm}}^{-1}$. Here, we also report ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ to compare $S_{c,\text{SDI}}$ versus $S_{c,\text{SD}\varphi }$, $S_{c,\text{SSI}}$ versus $S_{c,\text{SS}\varphi }$, and $S_{c,\text{DSI}}$ versus $S_{c,\text{DS}\varphi }$, respectively. Negative (positive) values represent lower (greater) $S_{c,\text{I}}$ than $S_{c,\varphi }$, and values of -1 (+1) represent lower (greater) $S_{c,\text{I}}$ versus $S_{c,\varphi }$ by a factor of 1/2 (2). We note that for laterally homogeneous media, which is the case of our simulations in this study, the two SS give the same regional sensitivity values. Therefore, $S_c$ values are the same for SS and DS. In the homogeneous medium, we observe a greater $S_c$ for $\varphi$ over I regardless of the method (SD, SS, DS) and $L_{ec}$. In the two-layered medium, the situation is more complex. For SD measurements, $S_{c,\text{SD}\varphi }$ is greater than $S_{c,\text{SDI}}$ (i.e., negative ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$) regardless of $L_{ec}$, but with a less pronounced difference than in the homogeneous medium model. For DS measurements (and also for SS measurements), $S_{c,\text{DSI}}$ is greater than $S_{c,\text{DS}\varphi }$ (i.e., positive ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$) for $L_{ec}<10.5$ mm (thin extracerebral layer), and $S_{c,\text{DS}\varphi }$ becomes greater than $S_{c,\text{DSI}}$ (i.e., negative ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$) for $L_{ec}>10.5$ mm (thicker extracerebral layer). $S_{c,\text{DSI}}$ and $S_{c,\text{DS}\varphi }$ are both greater than $S_{c,\text{SDI}}$ and $S_{c,\text{SD}\varphi }$ for all $L_{ec}$ values. We extended the plot in Fig. 5 to different values of optical properties within physiological ranges, as shown in Fig. 7, Appendix D .

Fig. 5.

Sensitivity to absorption changes in the bottom layer (i.e. brain tissue) for single-distance intensity ($S_{c,\text{SDI}}$), single-distance phase ($S_{c,\text{SD}\varphi }$), single-slope intensity ($S_{c,\text{SSI}}$), single-slope phase ($S_{c,\text{SS}\varphi }$), dual-slope intensity ($S_{c,\text{DSI}}$) and dual-slope phase ($S_{c,\text{DS}\varphi }$) as a function of top layer (i.e. extracerebral tissue) thickness ($L_{ec}$) from $1.5$ to $20$ mm. Values of $S_c$ are the same for SS and DS. Results are shown from simulations with (A) a homogeneous medium (optical properties: absorption coefficient ${\mu }_a=0.11$ ${\mathrm{cm}}^{-1}$; scattering coefficient ${\mu }_s^{\prime }=7$ ${\mathrm{cm}}^{-1}$) and (B) a two-layered medium (optical properties of the top extracerebral layer: ${\mu }_{a,ec}=0.1$ ${\mathrm{cm}}^{-1}$ and ${\mu }_{s,ec}^{\prime }=12$ ${\mathrm{cm}}^{-1}$; optical properties of the bottom cerebral layer: ${\mu }_{a,c}=0.2$ ${\mathrm{cm}}^{-1}$ and ${\mu }_{s,c}^{\prime }=3$ ${\mathrm{cm}}^{-1}$). (C) The ratios of sensitivity to the brain of I versus $\varphi$ for SD, SS and DS are shown in base-2 logarithmic scale (${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$, ${\log }_2[S_{c,\text{SSI}}/S_{c,\text{SS}\varphi }]$, and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$, respectively).

Figure 6 presents the comparison between simulations of $S_c$ and in vivo human measurements of CVR. Specifically, for each of the three methods of SD, SS and DS, we compared the simulated $S_{c,\text{I}}$ versus $S_{c,\varphi }$ ratios (${\log }_2[S_{c,\text{I}}/S_{c,\varphi }]$) with ${\mathrm{CVR}}_{\text{I}}$ versus ${\mathrm{CVR}}_{\varphi }$ (${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$) as a function of $L_{ec}$ value obtained from ultrasound imaging in the human study. The simulations of ${\log }_2[S_{c,\text{I}}/S_{c,\varphi }]$ with homogeneous and two-layered media are shown by shaded areas, which cover all values obtained from different combinations of optical properties within the ranges considered (see Appendix D for more details). Measurements of ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$ are plotted from eight subjects from experiment 1 and 2 (excluding subject 4), with those from experiment 2b having $L_{ec}$ increased by 5 mm to account for the scattering phantom layer thickness. As illustrated in Fig. 6(C), ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ are consistent with the range of expected ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ for a two-layered medium, while they are outside the corresponding range for a homogeneous medium, as described in the following points:

• •In some subjects with a thinner skull ($L_{ec}$<13 mm), ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ measurements can take positive values indicating a better sensitivity of DSI than DS$\varphi$ measurements to the brain. This is consistent with Table 2 and Fig. 4 that reported no statistical difference between ${\mathrm{CVR}}_{\text{DSI}}$ and ${\mathrm{CVR}}_{\text{DS}\varphi }$ for experiment 1 and 2a.
• •As $L_{ec}$ increases, ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ becomes more negative, indicating a better sensitivity of DS$\varphi$ than DSI measurements to the brain. From experiment 2b, all the values of ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ are negative. This is an evident effect of increasing $L_{ec}$ by adding the 5-mm scattering layer. This explains the statistical test in Table 2 that reported significantly lower ${\mathrm{CVR}}_{\text{DSI}}$ than ${\mathrm{CVR}}_{\text{DS}\varphi }$ in experiment 2b.

Fig. 6.

Comparison between simulation of sensitivity to absorption changes in the brain ($S_c$) and in vivo cerebrovascular reactivity (CVR) measurements from eight subjects in the two experiments. For single-distance (SD, A), single-slope (SS, B), and dual-slope (DS, C), values are plotted as intensity (I) versus phase ($\varphi$) $S_c$ ratios (${\log }_2[S_{c,\text{I}}/S_{c,\varphi }]$) for the simulations and CVR ratios (${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$) for the human measurements as functions of extracerebral layer thickness ($L_{ec}$). Simulation results are shown in green and orange shaded areas for homogeneous and two-layered media, respectively. In vivo data are shown with $\times$ symbol for experiment 1 (experiment 1), with $\bullet$ symbol for experiment 2 with optical probe on the scalp (experiment 2a), and with $\circ$ symbol for experiment 2 with optical probe on the phantom layer (experiment 2b).

CVR measurements from SD and SS methods are less obviously consistent with the two-layered medium simulations, also as a result of their greater variance with respect to DS measurements (Fig. 6(A),(B)). The greater variance in SD and SS data with respect to DS data is attributed to the better suppression of motion and instrumental artifacts, as well as a reduced sensitivity to tissue heterogeneity, in DS data versus SD and SS data. Specifically, we found the variance of ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$ for SD, SS and DS to be 0.5, 0.9 and 0.4, respectively. For SD, 7 out of 30 values of ${\log }_2[{\text{CVR}}_{\text{SDI}}/{\text{CVR}}_{\text{SD}\varphi }]$ reported in Fig. 6(A) are positive, while both homogeneous and two-layered medium simulations predict negative values. Nevertheless, both SD and SS measurements are clearly more consistent with the two-layered simulations than they are with the homogeneous medium simulations.

Regarding the comparison between the two individual SD measurements, a paired t-test across all subjects showed that there is no significant difference between ${\log }_2[{\text{CVR}}_{\text{SDI}}/{\text{CVR}}_{\text{SD}\varphi }]$ for the two individual SD measurements ($p>0.1$). On the other hand, there is a significant difference in ${\log }_2[{\text{CVR}}_{\text{SSI}}/{\text{CVR}}_{\text{SS}\varphi }]$ for the two individual SS measurements when conducting an analogous paired t-test on SS data ($p<0.05$).

4. Discussion

In this study, we have shown that the FD-NIRS measurements of CVR are more consistent with the sensitivity to absorption changes in the brain $S_c$ predicted by theoretical simulations using a two-layered medium than those using a homogeneous medium. The two-layered model qualitatively predicts that: (1) ${\mathrm{CVR}}_{\text{SD}\varphi }$ is typically greater than ${\mathrm{CVR}}_{\text{SDI}}$; (2) ${\mathrm{CVR}}_{\text{DSI}}$ may be greater than ${\mathrm{CVR}}_{\text{DS}\varphi }$ in the presence of a thin extracerebral tissue layer; and (3) ${\mathrm{CVR}}_{\text{DS}\varphi }$ becomes greater than ${\mathrm{CVR}}_{\text{DSI}}$ in the presence of a thick extracerebral tissue layer, achieved by adding a scattering layer between the optical fibers and the scalp. In vivo measurements of ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$ aligned well with the simulated ranges of ${\log }_2[S_{c,\text{I}}/S_{c,\varphi }]$ as a function of extracerebral thickness $L_{ec}$, especially for DS measurements (see Fig. 6). We used CVR as an in vivo validation metric of $S_c$; specifically, a higher measured CVR implies higher brain sensitivity. This application assumes negligible hemodynamic changes in the scalp tissue, which has been previously reported during hypercapnia [29,34,35,50]. Under this condition, a measured CVR is representative of cerebral hemodynamics only. We note that our CVR values are smaller than values reported for gray matter using MRI techniques, which are around 4 to 6% for a similar increase in ${\mathrm{P}}_{\text{ET}}$${\mathrm{CO}}_2$ [51,52]. This difference could be due to the so-called partial volume effect in non-invasive optical measurements, which results from the fact that brain tissue makes up only a portion of the interrogated tissue volume [29]. However, we stress that in this work we focused on the relative values of ${\mathrm{CVR}}_{\text{I}}$ and ${\mathrm{CVR}}_{\varphi }$, measured with different FD-NIRS methods, rather than on absolute CVR values. These relative values of CVR highlight the different CBF sensitivities of $\varphi$ versus I measurements using different FD-NIRS methods.

The main finding of this paper is that the sensitivity of DSI to the brain could be greater than DS$\varphi$ depending on the extracerebral layer thickness $L_{ec}$. By varying the optical properties of simulated homogeneous and two-layered media within reasonable ranges in Fig. 7 (Appendix D), we found that such a situation cannot be achieved by a homogeneous medium at baseline regardless of $L_{ec}$. Figure 7 also shows that the relative sensitivity between DSI and DS$\varphi$ as a function of $L_{ec}$ is primarily attributed to the assumption of greater scattering in the top layer than in the bottom layer within the two-layered model. This assumption is based on a number of human studies that applied two-layered models to measure ${\mu }_s^{\prime }$ in the brain and extracerebral tissue [27,28,49]. A low ${\mu }_s^{\prime }$ in the second layer (from $2$ to $4$ ${\mathrm{cm}}^{-1}$) may have an anatomical origin from the contribution of low-scattering CSF to the optical properties of the bottom layer.

As illustrated in Fig. 6, the detrended SD data appeared to be less obviously consistent with the two-layered simulations. In particular, while ${\mathrm{CVR}}_{\text{SD}\varphi }$ was significantly greater than ${\mathrm{CVR}}_{\text{SDI}}$ in experiments 1 and 2b, as predicted by the simulations, the situation is reversed in experiment 2a (see Table 2). Additionally, some of the individual ${\log }_2[{\text{CVR}}_{\text{SDI}}/{\text{CVR}}_{\text{SD}\varphi }]$ measurements yielded positive values (as shown in Fig. 6(A)), which indicates a greater ${\mathrm{CVR}}_{\text{SDI}}$ than ${\mathrm{CVR}}_{\text{SD}\varphi }$. We assign these inconsistencies between theory and experiment to the imperfect removal of instrumental artifacts by the joint detrending technique. This technique assumes that the coupling effects are linear functions of time, which is a simplification that is unable to remove non-linear or variable instrumental contributions to the measurements. We stress again that these instrumental artifacts only partially affect SS measurements, and that they are fully canceled out in DS measurements. These considerations further highlight the advantage of DS (and, to a lesser extent SS) data over SD data to effectively suppress experimental and instrumental artifacts from the measurements. We also note that any extracerebral tissue hemodynamics during hypercapnia, which were not addressed by our simulations, may have a stronger impact on SD measurements than SS and DS measurements. An increase in scalp blood flow, possibly linked to changes in heart rate and systemic blood pressure caused by the hypercapnic challenge, has been recently reported [36]. Such an increase in scalp blood flow may result in a greater contribution to ${\mathrm{CVR}}_{\text{SDI}}$ than ${\mathrm{CVR}}_{\text{SD}\varphi }$, since SDI is expected to be more sensitive to superficial tissue than SD$\varphi$.

When the absorption changes during hypercapnia are laterally homogeneous within the extracerebral and cerebral layers, $S_{c,\text{SSI}}$ and $S_{c,\text{SS}\varphi }$ should be identical to $S_{c,\text{DSI}}$ and $S_{c,\text{DS}\varphi }$, respectively, by definition. Indeed, our in vivo data have confirmed that ${\mathrm{CVR}}_{\text{SS}}$ is generally similar to ${\mathrm{CVR}}_{\text{DS}}$. Specifically, our ${\mathrm{CVR}}_{\text{SS}}$ measurements showed that (1) ${\mathrm{CVR}}_{\text{SSI}}$ and ${\mathrm{CVR}}_{\text{SS}\varphi }$ are not significantly different in experiments 1 and 2a, but they became significantly different in experiment 2b due to the effect of adding the scattering layer (as shown in Table 2); and (2) ${\log }_2[{\text{CVR}}_{\text{SSI}}/{\text{CVR}}_{\text{SS}\varphi }]$ is qualitatively consistent with the theoretical predictions of the two-layered model for both SS and DS measurements (see Fig. 6(B)). However, we note that SS showed a larger variation in ${\log }_2[{\text{CVR}}_{\text{SSI}}/{\text{CVR}}_{\text{SS}\varphi }]$ than DS. This variation may be caused by inhomogeneity in the baseline optical properties or by inhomogeneity in tissue hemodynamics in the forehead [15,17]. Additionally, we note that SS measurements are still potentially affected by experimental confounds (those associated with the detectors if the SS measurements use a single source) that DS measurements avoid (as discussed in Section 2.1.3).

Parameters for the CHS model were assumed in the calculations of cbf$(t)$ and CVR (Section 2.1.4). In Appendix E, we varied CHS parameters ($t^{(c)}$, $t^{(v)}$, and $\mathrm{\Delta }{\text{CBV}}^{(a)}(t)/\mathrm{\Delta }{\text{CBV}}^{(v)}(t)$) and calculated CVR from SD and DS measurements. In particular, decreasing $t^{(c)}$ will decrease CVR by up to a factor of 2, and decreasing $\mathrm{\Delta }{\text{CBV}}^{(a)}(t)/\mathrm{\Delta }{\text{CBV}}^{(v)}(t)$ will increase CVR by up to a factor of 3. $t^{(v)}$ does not have a significant effect on CVR values. These results are consistent with our previous study [11]. All variations of CHS parameters resulted in negative values for ${\log }_2[{\text{CVR}}_{\text{SDI}}/{\text{CVR}}_{\text{SD}\varphi }]$, positive or near-zero values for ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ in experiment 2a (with optical probe on the scalp), and negative values for ${\log }_2[{\text{CVR}}_{\text{DSI}}/{\text{CVR}}_{\text{DS}\varphi }]$ in experiment 2b (with optical probe on the phantom layer). Therefore, we conclude that the ratio of ${\mathrm{CVR}}_{\text{I}}$ to ${\mathrm{CVR}}_{\varphi }$ is not significantly impacted by the choice of values of CHS parameters for either SD or DS methods. Future studies may include adding induced ABP oscillations [53] or a transient change in ABP [11] to find individual values of CHS model parameters, with an additional benefit of obtaining absolute CBF values (${\text{CBF}}_0$).

In addition to the CHS parameters, the calculations of cbf$(t)$ and CVR were also based on the absolute baseline optical properties (as reported in Table 1), obtained under the assumption of a homogeneous semi-infinite geometry. Modeling tissues as homogeneous semi-infinite media is an approximation that is typically made in order to apply analytical solutions of the diffusion equation and to obtain effective, average optical properties of tissue [12]. This approximation is particularly limiting in the case of non-invasive brain measurements, given the highly inhomogeneous layered structure (scalp, skull, CSF layer, brain cortex, etc.). The measurements reported in tab:absprop show an agreement with results reported in the literature [15,27,40] under similar conditions, and are intended to be representative of the effective, or average, optical properties of the probed tissue (which includes multiple tissue layers). Regarding the specific objectives of this work, the absolute effective optical properties of tissue are used for calculating $T_0$ and differential pathlength and slope factors (${\mathrm{DPF}}_{\text{I}}$, ${\mathrm{DPF}}_{\varphi }$, ${\mathrm{DSF}}_{\text{I}}$, ${\mathrm{DSF}}_{\varphi }$). In the calculation of ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$, $T_0$, being a factor in the equation for cbf calculation [11], cancels out. In a previous paper (Ref. [15], Table A2), we showed that a change in ${\mu }_a$ or ${\mu }_s^{\prime }$ results in changes that are in the same direction for ${\mathrm{DPF}}_{\text{I}}$ and ${\mathrm{DPF}}_{\varphi }$, as well as for ${\mathrm{DSF}}_{\text{I}}$ and ${\mathrm{DSF}}_{\varphi }$. Therefore, a systematic error in the optical properties as a result of approximating tissue as a semi-infinite homogeneous medium will not significantly impact our results, which are based on relative values of ${\mathrm{CVR}}_{\text{I}}$ and ${\mathrm{CVR}}_{\varphi }$.

We have shown that the two-layered model can predict the experimental data more accurately than the homogeneous model. However, our previous studies have shown a good correlation between the homogeneous model and experimental data collected during systemic induced oscillations in ABP [15,18]. These previous studies are based on a hypothesis that arterial blood volume oscillations are dominant in the scalp (i.e., $O$ and $D$ oscillations are in phase, and $O$ exhibits a larger amplitude), and blood flow oscillations are dominant in the brain (i.e., $O$ and $D$ oscillations are almost in opposition of phase with more comparable amplitudes). This is in contrast to induced hypercapnic conditions where the measured blood flow change is expected to be mostly due to a change in CBF. Systemic ABP oscillation stimulations can induce changes in both scalp/skull and brain hemodynamics, causing the measured quantity to include a combination of blood flow and blood volume changes in both cerebral and extracerebral tissue. Further studies will focus on characterization of sensitivity of FD-NIRS methods to extracerebral and cerebral hemodynamics during systemic ABP oscillations using two-layered or multi-layered models. This will lead to a better understanding of the blood volume and blood flow dynamics in extracerebral and cerebral tissues, allowing for the development of depth dependent CHS.

The model used in the present study is based on an assumption of negligible changes in extracerebral blood flow during hypercapnia, which may not always be valid [36]. We note that some DS measurements of cbf$(t)$ did not recover back to baseline following the hypercapnic challenge, which could be due either to brain physiology (CBF not recovering immediately after hypercapnia) or to the contributions of scalp blood flow. Nevertheless, we have found that our simulation with a two-layered model was able to explain our in vivo CVR data based on DS methods. Further studies with a larger sample of subjects could employ a pneumatic tourniquet to cause scalp ischemia for further validations of SD and DS sensitivity, as well as incorporate short (< 15 mm) source-detector SD measurements to probe the changes in extracerebral hemodynamics. In future studies we also plan to perform optical measurements at different locations on the subject’s head to investigate the effect of anatomical differences in extracerebral tissue thickness on the same subject. Finally, simulations could be further extended to more complex geometries to better mimic the heterogeneity in biological tissues. Further investigations of multi-layered geometry with varying layer thicknesses and optical properties will enhance our understanding of the sensitivity of non-invasive FD-NIRS measurements to the brain.

5. Conclusion

FD-NIRS measurements of hemoglobin concentrations can be converted into relative CBF changes under the assumption of negligible changes in ${\mathrm{CMRO}}_2$ by using the CHS model. Yet optical measurements are acquired on the scalp, and therefore contain contributions from hemodynamics in the extracerebral vasculature. Characterization of the sensitivity of measured signals to the brain is important to assess the ability of FD-NIRS to selectively measure cerebral hemodynamics. Theoretical calculations of the sensitivity of data collected with six methods (SDI, SD$\varphi$, SSI, SS$\varphi$, DSI, and DS$\varphi$) to the bottom layer of a two-layered medium demonstrate a better agreement with experimental data in human subjects with respect to calculations based on a homogeneous model. Our results indicate that in the presence of a thin extracerebral tissue layer (less than 8-13 mm) that is more scattering than the underlying cerebral tissue, slope intensity measurements (SSI, DSI) may be more sensitive to the brain than slope phase measurements (SS$\varphi$, DS$\varphi$). However, it is important to observe that this study was based on a hypercapnia protocol that is expected to induce greater hemodynamic changes in the brain than in extracerebral tissue. The presence of significant extracerebral hemodynamics may impact SSI and DSI measurements to a greater extent than SS$\varphi$ and DS$\varphi$ measurements. This is a critically important point that will be investigated in future studies to more fully characterize the potential of DS FD-NIRS in non-invasive brain studies. The combination of (1) strong sensitivity to brain tissue, (2) weak sensitivity to extracerebral tissue, (3) minimal impact from instrumental drifts, and (4) insensitivity to motion artifacts are crucial requirements in non-invasive optical studies of the human brain. This work presented significant results for the characterization of FD-NIRS data toward the development of better tools for optical monitoring of cerebral hemodynamics, with the overall goal to achieve the above four requirements.

A. Appendix A: acronyms and symbols

Lists of acronyms and symbolic notation used in this paper are provided in Table 3 and Table 4, respectively.

Table 3. Acronyms used in this paper.

Table 4. Notation used in this paper.

B. Appendix B: self-calibrating approach and joint detrending technique for SD measurements

In an actual experiment, SDI and SD$\varphi$ measurements are affected by the opto-mechanical coupling between optical probe and tissue, the alleviation of which has been one of the primary motivations for the original work to develop the self-calibrating method for absolute measurements [19]. Given two sources (1, 2) and two detectors (A, B) in the configuration shown in Fig. 1(C), we can express the set of $\varphi$ measurements using a system of equations:

$$\begin{array}{cc}{\varphi }_{1A}& =\varphi ({\rho }_S)+C_1+C_A\\ {\varphi }_{1B}& =\varphi ({\rho }_L)+C_1+C_B\\ {\varphi }_{2A}& =\varphi ({\rho }_L)+C_2+C_A\\ {\varphi }_{2B}& =\varphi ({\rho }_S)+C_2+C_B\end{array},$$ (3)

where $C_1$, $C_2$, $C_A$, and $C_B$ are optical coupling terms originating from random or systematic temporal fluctuations, drifts in source or detector characteristics, displacement of the optical probe, etc., and the quantities $\varphi ({\rho }_S)$ and $\varphi ({\rho }_L)$ denote the ground-truth $\varphi$ values that would be measured at short (${\rho }_S$) and long (${\rho }_L$) source-detector distances in the absence of noise or experimental confounds [17]. From Eq. (3), SS method can remove the coupling effects originating only from the source (if using one source and two detectors), as follows:

$$\begin{array}{c}{\varphi }_{1B}-{\varphi }_{1A}=\varphi ({\rho }_L)-\varphi ({\rho }_S)+(C_B-C_A)\\ {\varphi }_{2A}-{\varphi }_{2B}=\varphi ({\rho }_L)-\varphi ({\rho }_S)+(C_A-C_B)\end{array}.$$ (4)

By combining the two SSs, the DS measurement should in principle be immune to all the coupling effects from sources and detectors. That is,

$$\frac{{\varphi }_{1B}-{\varphi }_{1A}+{\varphi }_{2A}-{\varphi }_{2B}}{2}=\varphi ({\rho }_L)-\varphi ({\rho }_S).$$ (5)

Though we do utilize DS measurements in this work, we would still like to make effective use of our SD measurements as well. In this section we also describe the joint detrending procedure we have employed to estimate and remove the instrumental effects from our SD measurements. We assume that systematic coupling effects arise mainly from instrumental drifts in source or detector characteristics on measurements, which could be modelled as linear trends with time, such that

$$\begin{array}{cc}{C}_1(t)& ={\gamma }_{1,0}+{\gamma }_{1,1}t\\ C_2(t)& ={\gamma }_{2,0}+{\gamma }_{2,1}t\\ C_A(t)& ={\gamma }_{A,0}+{\gamma }_{A,1}t\\ C_B(t)& ={\gamma }_{B,0}+{\gamma }_{B,1}t\end{array},$$ (6)

where the various $\gamma$ parameters are fixed in time. Selecting only the initial 5-min baseline prior to the onset of hypercapnia, we jointly fit this eight-parameter model to the data using standard least-squares minimization. The best-fitting parameters are used to construct linear $\varphi$ trends that we then subtract from the full time span of SD$\varphi$ data. Note that the form of our instrumental model in Eq. (3) explicitly preserves the features of DS$\varphi$, such that the DS$\varphi$ constructed from the jointly detrended SD data is identical to that constructed from the original data.

We further note that the procedure described in this section, though expressed in terms of $\varphi$ measurements, is equally applicable to logarithmic amplitude measurements under the substitution of $\varphi$ with $\ln (\text{I})$.

C. Appendix C: complex generalized optical pathlength in the two-layered medium with hybrid boundary conditions

The solution of the diffusion equation for a two-layered cylindrical medium in frequency domain was derived by Ref. [26], where the first cylindrical layer has a thickness of $L_1$ and the second one is infinitely extended along $z$ axis (which contains the axis of the two cylinders). Both layers have radius $a$ (which in this study was set to 150 mm). The laser beam is impinging at the center of the top surface of the first layer (along the $z$ axis) and the extrapolated-boundary condition (EBC) is applied at the interface between this surface and the outer medium. The zero-boundary condition (ZBC) is applied between the lateral boundaries of the two cylinders and the outer medium. The photon fluence ${\tilde{\mathrm{\Phi }}}_k({\mathbf{\text{r}}}_s,\mathbf{\text{r}})$ at the cylindrical point $\mathbf{\text{r}}=(\rho ,\theta ,z)$ in the $k^{th}$ layer of the medium is given by:

$${\tilde{\mathrm{\Phi }}}_k({\mathbf{\text{r}}}_s,\mathbf{\text{r}})=\frac{1}{\pi a^2}\sum _{n=1}^{\mathrm{\infty }}{G}_k(s_n,z,\omega )J_0(s_n\rho )J_1^{-2}(a^{\prime }{s}_n)$$ (7a)

Here the point source at ${\mathbf{\text{r}}}_s$ (equivalent to the laser beam) has coordinates ($0$, $0$, $1/{\mu }_{s1}^{\prime }$); $J_m$ is the $m$-order first kind Bessel function; $s_n$ is the positive root of the 0-order first kind Bessel function divided by $a$. In this study, $2000$ roots were used. $G_k$ ($k=1$ or $2$ for a two-layered medium) is given by:

$$\begin{array}{ll}{G}_1(s_n,z,\omega )& =\frac{\exp [-{\alpha }_1|z-z_0|]-\exp [-{\alpha }_1(z+z_0+2z_{b1})]}{2D_1{\alpha }_1}\\ & +\frac{\sinh [{\alpha }_1(z_0+z_{b1})]\sinh [{\alpha }_1(z+z_{b1})]}{D_1{\alpha }_1\exp [{\alpha }_1(L_1+z_{b1})]}\\ & \times \frac{n_1^2{D}_1{\alpha }_1-n_2^2{D}_2{\alpha }_2}{D_1{\alpha }_1{n}_1^2\cosh [{\alpha }_1(L_1+z_{b1})]+D_2{\alpha }_2{n}_2^2\sinh [{\alpha }_1(L_1+z_{b1})]}\end{array}$$ (7b)

$$G_2(s_n,z,\omega )=\frac{n_2^2\sinh [{\alpha }_1(z_0+z_{b1})]\{\exp [{\alpha }_2(L_1-z)]\}}{D_1{\alpha }_1{n}_1^2\cosh [{\alpha }_1(L_1+z_{b1})]+D_2{\alpha }_2{n}_2^2\sinh [{\alpha }_1(L_1+z_{b1})]}$$ (7c)

$${\alpha }_k=\sqrt{\frac{{\mu }_{ak}}{D_k}+s_n^2+\frac{i{\omega }_{mod}}{D_k(c/n_k)}}$$ (7d)

where $D_k=1/(3{\mu }_{sk}^{\prime })$; $z_{b1}=2D_1(1+R_{eff})/(1-R_{eff})$; $R_{eff}$ is the fraction of photons that are internally diffusely reflected at the cylinder boundary [54]; and $z_0=1/({\mu }_{s1}^{\prime })$.

Even though the local complex generalized optical pathlength requires the calculation of two reflectances [17], here we used only the fluence. Usually the correct definition of generalized local pathlength and the one adopted in this work coincide within a few percent ($<10\mathrm{\%}$). In other words, the complex partial generalized optical pathlength $⟨\tilde{l}⟩$ can be estimated from the photon fluence $\tilde{\mathrm{\Phi }}$ in the medium as:

$$⟨\tilde{l}⟩({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_i,{\mathbf{\text{r}}}_d)\cong \frac{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_i)\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_i,{\mathbf{\text{r}}}_d)}{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d)}{V}_i$$ (8)

where $V_i$ is the volume of the defect $i$ centered at the position vector ${\mathbf{\text{r}}}_i$. In Eq. (8), $\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_i)$ is the fluence from the point-like source at ${\mathbf{\text{r}}}_s$ to the field point at ${\mathbf{\text{r}}}_i$, while $\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_i,{\mathbf{\text{r}}}_d)$ is the fluence from ${\mathbf{\text{r}}}_i$ to the detector at ${\mathbf{\text{r}}}_d$. The denominator is $\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d)$, which is the fluence from the source at ${\mathbf{\text{r}}}_s$ to the detector at ${\mathbf{\text{r}}}_d$. Because $\tilde{\mathrm{\Phi }}$ are in the units of inverse length squared and $V_i$ is in the unit of length cubed, we note that $⟨\tilde{l}⟩$ is in the unit of length (therefore the word "generalized pathlength" is used, even though it is associated with a complex number).

The complex total generalized optical pathlength $⟨\tilde{L}⟩$ can be estimated from the relative changes in $\tilde{\mathrm{\Phi }}$ when there exists a small local change $\mathrm{\Delta }{\mu }_a$ in each layer (with $\mathrm{\Delta }{\mu }_a$ set as ${10}^{-6}$ ${\mathrm{cm}}^{-1}$ in this study):

$$⟨\tilde{L}⟩({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d)\approx -\frac{(\frac{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1}+\mathrm{\Delta }{\mu }_a,{\mu }_{a,2})-\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1},{\mu }_{a,2})}{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1},{\mu }_{a,2})}+\frac{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1},{\mu }_{a,2}+\mathrm{\Delta }{\mu }_a)-\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1},{\mu }_{a,2})}{\tilde{\mathrm{\Phi }}({\mathbf{\text{r}}}_s,{\mathbf{\text{r}}}_d,{\mu }_{a,1},{\mu }_{a,2})})}{\mathrm{\Delta }{\mu }_a}$$ (9)

D. Appendix D: effects of varying optical properties on the sensitivity of SD and DS measurements to absorption changes in the brain

Figure 7 presents ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ (the same for ${\log }_2[S_{c,\text{SSI}}/S_{c,\text{SS}\varphi }]$) as functions of $L_{ec}$ by varying the optical properties of the homogeneous and two-layered media within reasonable ranges.

Fig. 7.

Dependence of intensity (I) versus phase ($\varphi$) ratio of sensitivity ($S_c$) for single-distance (${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$) and dual-slope (${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$) as functions of extracerebral layer thickness ($L_{ec}$) on absorption (${\mu }_a$) and scattering (${\mu }_s^{\prime }$) optical properties of homogeneous and two-layered media at baseline. In the homogeneous medium, (A) and (B) show the effects of varying ${\mu }_a$ and ${\mu }_s^{\prime }$ on SD and DS methods, respectively. Different line styles represent different ${\mu }_a$ values. Different colors represent different ${\mu }_s^{\prime }$ values. In the two-layered medium, (C) and (E) show the effects of varying the ratio of extracerebral to cerebral ${\mu }_s^{\prime }$ (${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$), as well as extracerebral and cerebral ${\mu }_a$ (${\mu }_{a,ec}$ and ${\mu }_{a,c}$) on SD and DS methods, respectively. ${\mu }_{s,ec}^{\prime }$ was set at 12 ${\mathrm{cm}}^{-1}$. Different colors represent different ${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ values. Different line styles represent different combinations of ${\mu }_{a,ec}$ and ${\mu }_{a,c}$ values. (D) and (F) show the effects of varying ${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ and ${\mu }_{s,ec}^{\prime }$ on SD and DS methods, respectively, while setting ${\mu }_{a,ec}=0.06$ ${\mathrm{cm}}^{-1}$ and ${\mu }_{a,c}=0.2$ ${\mathrm{cm}}^{-1}$. Different colors represent different ${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ values, with lighter colors indicating lower ${\mu }_{s,ec}^{\prime }$ values and vice versa.

In the homogeneous medium (Fig. 7(A),(B)), the optical properties were varied as: ${\mu }_a$ from $0.08$ to $0.14$ ${\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }$ from $5$ to $11$ ${\mathrm{cm}}^{-1}$. We observe that $S_{c,\text{SDI}}<S_{c,\text{SD}\varphi }$ and $S_{c,\text{DSI}}<S_{c,\text{DS}\varphi }$ regardless of $L_{ec}$ and optical properties (as shown by the negative ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$).

In the two-layered medium (Fig. 7(C)-(F)), we varied these following parameters: the ratio of extracerebral to cerebral ${\mu }_s^{\prime }$ (${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$) from $2.4$ to $6$, ${\mu }_{a,ec}$ from $0.06$ to $0.1$ ${\mathrm{cm}}^{-1}$, ${\mu }_{a,c}$ from $0.14$ to $0.2$ ${\mathrm{cm}}^{-1}$, and ${\mu }_{s,ec}^{\prime }$ from $11$ to $13$ ${\mathrm{cm}}^{-1}$. We observe that the relative sensitivity between DSI and DS$\varphi$ depends mostly on the discrepancy between scattering coefficients of the layers. In particular, ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ becomes significantly more positive as the ratio ${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ increases (i.e., ${\mu }_{s,ec}^{\prime }$ is greater than ${\mu }_{c,c}^{\prime }$), as shown in Fig. 7(E) and (F). Similar behavior is also observed for ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$, as shown in Fig. 7(C) and (D), but ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ values are always negative. The absolute values of ${\mu }_{s,ec}^{\prime }$ and ${\mu }_{s,c}^{\prime }$ have less effects on both ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ than their mismatch (${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ ratio), as shown in Fig. 7(D) and (F). Regarding the effects of ${\mu }_a$, Figs. 7(C) and (E) show that both ${\log }_2[S_{c,\text{SDI}}/S_{c,\text{SD}\varphi }]$ and ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]$ decrease as ${\mu }_{a,ec}$ increases from $0.06$ to $0.10$ ${\mathrm{cm}}^{-1}$ and ${\mu }_{a,c}$ decreases from $0.20$ to $0.14$ ${\mathrm{cm}}^{-1}$. However, the effect of ${\mu }_a$ is also smaller than the effect of ${\mu }_{s,ec}^{\prime }/{\mu }_{s,c}^{\prime }$ ratio. With all the considered combinations of extracerebral and cerebral optical properties, the cross-over point where ${\log }_2[S_{c,\text{DSI}}/S_{c,\text{DS}\varphi }]=0$ have a range of critical $L_{ec}$ values from from approximately $8$ to $13$ mm.

E. Appendix E: Effects of varying CHS parameters to CVR measurements

We consider SD and DS measurements as 1B, 1AB2 and 3CD4 from subject 2, experiment 2. 1B and 1AB2 are SD and DS measurements with optical probe on the scalp (experiment 2a), and 3CD4 is DS measurement with optical probe on the phantom layer (experiment 2b). We initially start with the set of CHS parameters as assumed in Sec. 2.1.4, then vary one of these following parameters $t^{(c)}$, $t^{(v)}$ and the relative arterial-to-venous contributions to CBV dynamics ($\mathrm{\Delta }{\text{CBV}}^{(a)}(t)/\mathrm{\Delta }{\text{CBV}}^{(v)}(t)$). CVR and ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$ from SD and DS measurements were then re-calculated with the new set of parameters, as seen in Table 5.

Table 5. Effects of varying CHS parameters ($t^{(c)}$, $t^{(v)}$, and $\mathrm{\Delta }{\text{CBV}}^{(a)}(t)/\mathrm{\Delta }{\text{CBV}}^{(v)}(t)$ on SD and DS measurements) of ${\mathrm{CVR}}_{\text{I}}$, ${\mathrm{CVR}}_{\varphi }$, and ${\log }_2[{\text{CVR}}_{\text{I}}/{\text{CVR}}_{\varphi }]$).

Funding

National Institutes of Health10.13039/100000002 (R01 NS095334).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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