2.1.

Multidistance Method for Obtaining Optical Properties of the Semi-infinite Medium

The propagation of light in tissue is best described by the radiative transport theory. As a result of the high scattering and low absorption of biological tissues in the near-infrared region, one can simplify the transport equation by using the diffusion approximation to the transport equation. The diffusion equation has been solved analytically for simple geometries such as infinite, semi-infinite, and two-layer media; slabs; spheres; and cylinders, providing solutions for a pointlike light source.¹³ Fantini et al.;¹¹ have obtained the analytical solution for the frequency-domain parameters, such as the phase shift, mean light intensity, and modulation amplitude, as functions of the distance in a semi-infinite geometry boundary condition using the diffusion approximation. The measured phase and amplitude can be fit to the analytical solution in order to estimate μ_a and μ_s ^′ for both an infinite and the semi-infinite medium. In particular, the solution yields expressions for the derivatives:

The solution of the diffusion equation for a two-layer medium in the frequency domain was derived by Kienle et al.;¹² The first layer of the two-layer medium has a thickness l and the second layer is semi-infinite. The intensity, modulated amplitude, and the phase (I, M, and ϕ) can be obtained by using the equation for the spatially resolved reflectance from two-layer media R(ρ,ω), which is derived in Ref. 12.

2.2.

Estimation of Physiological Parameters

In the near-infrared region, the primary chromophores in biological tissues are oxyhemoglobin and deoxyhemoglobin molecules. Depending on their oxygenation state, the absorption spectra of hemoglobins are different. Assuming that no other chromophores significantly contribute to the absorption other than oxyhemoglobin and deoxyhemoglobin, one can calculate their concentrations using the extinction coefficients at wavelength λ by applying the Beer-Lambert law:

From [O ₂ Hb] and [HHb], one can obtain other physiologically important parameters. For instance, total hemoglobin concentration ([tHb]) and oxygenation (Ox) are defined as

3.1.

Instrumentation

We used a multichannel frequency-domain near-infrared spectrometer (Imagent, ISS, Champaign, Illinois), which is described in Ref. 11. Briefly, the instrument employs laser diodes (LD) modulated at the frequency of 110 MHz and photomultiplier tubes (PMT) whose gain is modulated at a slightly offset frequency of 110.005 MHz to heterodyne the high frequency down to the frequency of 5 kHz. In this study, 32 LDs (16 at 758 nm and 16 at 830 nm) and four PMTs were used. The LDs were multiplexed so that two LDs with the same wavelength and at the same location were on simultaneously. The light from the LDs was guided by optical fibers with a diameter of 400 μm to the tissue surface and the photons reemitted from the tissue were collected simultaneously by the bundles with a diameter of 5.6 mm, placed several centimeters apart from the source fibers. The collected light was carried to the PMTs and then the signals from the PMTs were digitally processed to yield the average intensity (I), modulation amplitude (M), and phase difference (ϕ).

3.2.

Optical Sensor

In order to recover the layered structure of the forehead, a large-range optode separation from 1 to 8 cm in 0.5 cm increments was employed. The arrangement of sources and detectors is shown in Fig. 1. Two laser diodes were turned on synchronously to double the intensity of light from remote sources. Neutral-density filters were installed at the end of these source fibers closer to the detector fibers to reduce excessive light and balance the overall detected light intensity. In order to provide a stable contact between fibers and the skin, right-angle prisms were attached to the ends of the fibers and a flexible black silicon pad was molded to hold fibers and prisms.

Figure 1

Geometric arrangement of the long-ranged, multidistance probe containing the sources and detectors for the frequency-domain spectrophotometer. The detectors are labeled as A, B, C, and D, respectively, according to their distances from the sources.

3.3.

Calibration

Equations (3) and (4) are valid only if all the sources have equal intensity, modulation, and phase and if all the detectors have equal sensitivity. Different source intensities and phases and nonuniform optical coupling with the PMTs require an instrument calibration using a phantom with known optical properties. The calibration phantom was made of optically transparent silicone resin and its optical properties were set using black carbon and TiO ₂ powders to mimic absorption and scattering of tissue, respectively. The optical properties of the phantom were made similar to those of the human forehead in order to provide an accurate calibration for the conditions in the tissue. To simulate the curvature of the human head, the phantom had a cylindrical shape with a radius of curvature of 10 cm. The sides of the phantom were covered with aluminum foil to simulate a semi-infinite medium by reflecting photons. We found that the curvature of the phantom is not a significant factor for errors when we calibrate on a phantom with a curvature similar to that of the head.

3.4.

Human Subjects

Fifteen female and 15 male healthy volunteers (from 20 to 50 years) were included in the study. A written informed consent was obtained from each subject. The subject, sitting in a dark room, was asked to relax during the 2-min baseline of measurement. The probe was positioned on three areas of the forehead (left, middle, and right), close to the hairline to avoid the channeling effect of the frontal sinuses.

3.5.

Data Analysis

The raw optical data for M and ϕ were converted into values of μ_a and μ_s ^′ using Eqs. (3) and (4) after calculating the slopes of ln(ρ²M) and ϕ with respect to the source–detector distance ρ. To obtain values of μ_a and μ_s ^′ corresponding to a specific ρ by using the multidistance method, we used data from five neighboring sources, which spanned a distance of 2 cm. Since the noise in ϕ increases exponentially as ρ increases, owing to the exponential decrease in the number of photons reaching the detector, we used the robustfit function in MATLAB 6.0. This function utilizes a weighted least-squares algorithm, with a lower weight for points that exhibit higher noise. In addition, ϕ was smoothed by a moving average.

In order to characterize light penetration into the head, we evaluated the penetration depth z and the differential path length factor (DPF) for each ρ using the following expressions¹⁵:

In order to investigate the regional dependence of oxygenation and the lateralization of total hemoglobin concentration, respectively, we defined normalized quantities L _Ox and L _tHb as

4.1.

Optical Properties of the Forehead as a Function of Source–Detector Distance

Studying the distance dependence of the optical properties, μ_a and μ_s ^′ of the adult forehead, we found that the traces of μ_a and μ_s ^′ as a function of ρ typically have a sigmoidal shape. Figure 2 shows global averages of μ_a and μ_s ^′ over 30 subjects plotted with respect to the penetration depth z at wavelengths of 758 and 830 nm. Because of declining intensity readings at long distances in some subjects, we could not obtain enough signal-to-noise ratio at all the distances in some subjects. Each sigmoidal curve in Fig. 2 asymptotically approaches its lower and upper limits corresponding to μ_a and μ_s ^′ values of scalp/skull and brain. The ensemble averages were performed only on the points with a sufficient signal to noise ratio. At intermediate depths (approximately z=0.8 to 1 cm), the information from both layers is mixed. In order to specify the penetration depth at which the curve approximately approaches the limit corresponding to the brain value, we defined the second turning point of z as a critical penetration depth z_c. For both wavelengths, z_c for μ_a and μ_s ^′ is approximately 0.9 cm, which is about the thickness of the top layer.

Figure 2

Cross-subject averages (circles) and standard deviations (dotted lines) of the optical properties μ_a (a) and (c) and μ_s ^′ (b) and (d), measured on the right side of the adult forehead, plotted with respect to z at two different wavelengths, λ₁=758 nm and λ₂=830 nm, respectively. The solid lines indicate the fitted curves of the average of μ_a and μ_s ^′ obtained by a Gauss-Newton nonlinear least-squares fit.

4.2.

Comparison with the Two-Layer Model

It is important to determine whether the saturation values of μ_a and μ_s ^′ at long ρ correspond to the optical properties of the brain. In order to examine this issue, we compared the solutions of the two-layer diffusion model derived by Kienle et al.; [Eqs. (17) and (18) in Ref. 12] with the experimental results from the adult head. Measurements of I, M, and ϕ were taken from the forehead of a male aged 38 whose scalp/skull thickness was measured using magnetic resonance imaging (MRI), and the values of μ_a and μ_s ^′ at both the scalp/skull and brain were determined using Eqs. (3) and (4). Since the optical properties of the scalp/skull are similar to each other in the near-infrared region, we consider the scalp/skull as one layer. We compared I, M, and ϕ evaluated from the two-layer model with those values obtained from the adult head. The input parameters for R (ρ,ω) consisted of the measured μ_a ¹ and μ_s ^′1 of the first layer, and μ_a ² and μ_s ^′2 of the second layer, and the thickness of the first layer. From the asymptotic values at short distances from 1 to 3 cm, μ_a ¹ and μ_s ^′1 were determined to be 0.095±0.005 cm⁻¹ and 12±0.2 cm⁻¹, respectively. And at long distances from 5 to 7 cm, μ_a ² and μ_s ^′2 were 0.14±0.005 cm⁻¹ and 4±0.2 cm⁻¹, respectively. l was determined to be 1.1 cm from the MRI scan of the subject. Figure 3 compares the spatial distribution of M and ϕ obtained from the measurements and those obtained by two-layer model. There is good agreement between the experimental data and the prediction of the two-layer model. This suggests that μ_a ² and μ_s ^′2 extracted from the sigmoid curves represent the optical values of the bottom layer in the two-layer model.

Figure 3

A comparison of the frequency-domain parameters between experimental results and theoretical predictions based on the two-layered model. (a) The modulation amplitude, M and (b) the phase shift, ϕ were obtained from the right side of the forehead of a 38-year-old male with respect to ρ. The circles represent the measured values of M and ϕ, and the lines are the values predicted using the diffusion equation for the two-layer model. The optical coefficients of the top and bottom layers for the two-layered model were determined using the diffusion equation for the semi-infinite medium at short and long distances, respectively: μ_a=0.095 cm ⁻¹, μ_s ^′=12 cm ⁻¹ for the top layer and μ_a=0.14 cm ⁻¹, μ_s ^′=4 cm ⁻¹ for the bottom layer. The thickness of the top layer measured by MRI was 11 mm. The good agreement between the measured and the calculated data shows the internal consistency of our results.

Our goal was to determine the optical properties of the brain from the reflectance data using a nonlinear regression of M and ϕ to the solution of the two-layer model. Kienel et al.;¹² retrieved the optical properties of both layers within a tolerable error using nonlinear regression to a known l. However, when the l was unknown, the optical coefficients were not accurately retrieved, implying that the minimizing functions in the nonlinear regression have nonsingular local minima with respect to the five unknown parameters (μ_a ¹, μ_s ^′1, μ_a ², μ_s ^′2, and l).¹² Measurements of the modulation and phase can be used to derive μ_a ² and μ_s ^′2 for given μ_a ¹, μ_s ^′1, and l values.

Figure 4 shows graphically the relationship between μ_a ² and μ_s ^′2 and the experimental parameters of ∂_ρ ln[ρ²M(ρ)] and ∂_ρϕ for different top layer thicknesses of l when the optical properties for the first layer are given. To illustrate the influence of l, two examples are superimposed: l=10 and 15 mm. We see clearly in Fig. 4 that μ_a ² and μ_s ^′2 cannot be determined uniquely from the slopes of ln[ρ²M(ρ)] and ϕ if l is unknown. Therefore, the following procedure was used for brain measurements: (1) Determine μ_a ¹ and μ_s ^′1 from ln[ρ²M(ρ)] and ϕ(ρ) for small values of ρ (i.e., ρ=1 to 3 cm). (2) Obtain the anatomical information of l from an MRI scan. (3) Apply a nonlinear regression algorithm to iterate until one finds the parameters that best fit ln[ρ²M(ρ)] and ϕ(ρ) to the solution of the two-layer model. We used the asymptotic values of μ_a and μ_s ^′ at long distances as initial values of μ_a ² and μ_s ^′2 for the iteration algorithm to obtain μ_a ² and μ_s ^′2 more effectively and precisely. Table 1 shows μ_a ² and μ_s ^′2 retrieved from the nonlinear regression and from the multidistance method for the same subject in Fig. 3. The good agreement confirms the absolute measurement of the brain.

Figure 4

The slopes of M and ϕ noted as ∂_ρ log(ρ²M) and ∂_ρϕ, which are fdNIRS parameters, are shown for different optical properties of the brain, μ_a ² and μ_s ^′2 at two different top layer thicknesses of l. The optical properties of the skin/skull are given as μ_a ¹=0.1 cm ⁻¹ and μ_s ^′1=10.0 cm ⁻¹. For given fdNIRS parameters, the optical properties of the brain cannot be uniquely determined when the other parameters are unknown. The slightly vertically tilted lines are isoscattering coefficient lines, and the slightly horizontally tilted lines are isoabsorption coefficient lines.

Table 1

4.3.

Statistics of Physiological Parameters Across the Population of Subjects

A quantitative analysis of physiological parameters such as hemoglobin concentrations ([O ₂ Hb], [HHb], and [tHb]) in the scalp/skull and brain, extracranial tissue oxygen saturation (S _t O ₂), and intracranial (cerebral) tissue oxygen saturation (cS _t O ₂) was performed in 30 healthy volunteers. The parameters were obtained using Eqs. (6), (7), and (8). Table 2 summarizes the results and Fig. 5 displays the respective histograms. For the extracranial and intracranial layers, we used a range of source–detector distances of 1 to 3.5 cm and 4.5 to 7 cm, respectively. In the distributions of the S _t O ₂, [O ₂ Hb], [HHb], [tHb], and DPF in both extracranial and intracranial layers shown in Fig. 5, there is a larger variation in all parameters for the deeper layer. In addition, for the same depth, the histograms of the population of hemoglobin parameters ([O ₂ Hb], [HHb], and [tHb]) show a broader distribution than that of S _t O ₂. This large intersubject variation in the hemoglobin concentrations may be due to anatomical variations among the subjects. However, it is interesting that for S _t O ₂, the between-subject variability is 3 for the extracranial layer and 6 for the brain.

Figure 5

Histograms of calculated S _t O ₂, [O ₂ Hb], [HHb], [tHb], and DPF corresponding to a population of 30 adult foreheads. The average ρ for the extracranial and intracranial layers are 2.7 and 5.7 cm, respectively. The vertical axis in each histogram is the number of counts in each bin. Each subject was measured three times on the right, middle, and left sides.

Table 2

In Fig. 6, the standard errors in the oxygen saturation and the total hemoglobin concentration are plotted with respect to the standard error of the number of photons reemitted from the tissue during data acquisition of the baseline that we obtained from the left side of a randomly chosen subject. The standard error of the photon count is inversely proportional to the square root of the intensity I according to the Poisson distribution of noise and increases exponentially as ρ increases because of the exponential decrease of the number of photons reaching the detector at ρ. In Fig. 6, the variations in S _t O ₂ and [tHb] follow the trends of photon noise. Therefore the larger variation in the deeper layers is consistent with errors that are due to the noise on the reflectance intensity at a larger ρ.

Figure 6

Noise level of the photon counts versus (a) standard error σ_M of S _t O ₂ in percent and (b) σ_M of [tHb] in micromoles as a function of ρ.

Figure 7 shows the histogram of L _Ox and L _tHb These values contain the information on the normalized regional differences in the oxygen saturation and the tissue blood volume, respectively. Owing to individual differences in anatomy, L _tHb is more broadly distributed than L _Ox . The L _Ox values concentrated to zero indicate that the oxygen saturation is symmetric in both hemispheres.

Figure 7

Histograms of calculated L _Ox in (a) the scalp/skull and (b) the brain, and L _tHb in (c) the scalp/skull and (d) the brain, corresponding to population of adult subjects. The average ρ’s for the extracranial and intracranial layers were 2.7 and 5.7 cm, respectively. The vertical axis is the number of subjects in each bin.