2.1.

Theoretical Consideration

When a light impulse is incident on the surface of a semiinfinite homogeneous medium (object), the time-resolved intensity of reflected light at time $t$ can be expressed as²⁹

However, in the case where the object is an inhomogeneous medium (and the reference is a homogeneous one), $A^{\mathrm{diff}}\left(t\right)$ is no longer linear to $t$ even if $S^{\mathrm{diff}}\left(t\right)$ can be ignored. When ${\mu }_a^{\mathrm{diff}}$ varies due to the layer structure, $\mathrm{d}{A}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ reflects the depth dependence in ${\mu }_a^{\mathrm{diff}}$ . Here we introduce time-dependent apparent absorption coefficient, ${\mu }_a\left(t\right)$ , given by ${\sum }_i{\mu }_{ai}{l}_i\left(t\right)∕L\left(t\right)$ , where the subscript $i$ refers to the layer number in the object medium, $l_i\left(t\right)$ is the mean partial path length in the $i$ ’th layer for the photons detected at time $t$ , and $L\left(t\right)$ is total path length defined as $L\left(t\right)=\mathit{ct}$ . Then we define time dependent difference in apparent absorption coefficient ${\mu }_a^{\mathrm{diff}}\left(t\right)$ , which is the difference between ${\mu }_a\left(t\right)$ and ${\mu }_{a,R}$ at each time. In the case of two-layered media, the $i$ ’th layer simply corresponds to either upper or lower layer and $\mathrm{d}{A}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ is expressed as follows:

Fig. 1

Number of scattering events in the upper part of the depth from $2\text{to}36\mathrm{mm}$ for a homogeneous semiinfinite medium ( ${\mu }_a=0{\mathrm{mm}}^{-1}$ , ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$ , $\rho =30\mathrm{mm}$ , $n=1.37$ ), predicted by Monte Carlo simulation. The vertical axis represents the number of scattering events per photon detected at time $t$ .

2.2.

Time-Segmented ${\mu }_a$ and Method for Obtaining ${\mu }_a$ of the Lower Layer

Next, we introduced mean absorption differences obtained by $\Delta A^{\mathrm{diff}}\left(t\right)∕\Delta t$ instead of $\mathrm{d}{A}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ in consideration of errors by getting the regression curve and to simplify an analysis process. For this aim, we divided the temporal profile of detected light intensity into segments to extract time-dependent ${\mu }_a$ (time-segment analysis). When we assume that ${\mu }_s^{\prime }$ and refractive index are the same for reference and object,

In the following, a calculation process is explained for the case of the two-layered object and homogeneous reference. First, from the two sets of data for the object and reference, we calculate $A^{\mathrm{diff}}\left(t\right)$ profile and divide it into segments. Then $A^{\mathrm{diff}}\left(t\right)$ at each time segment is approximated linear to time and its slope (time-segmented slope) is estimated by linear regression. From the time-segmented slope, ${\mu }_a^{\mathrm{seg}}$ can be obtained from ${\mu }_{a,R}$ and $c$ based on Eq. 4. In this study, time range of $500\mathrm{ps}$ was selected for the segment size considering the reliability of regression analysis of the time-segmented slope. Before calculations of $A^{\mathrm{diff}}\left(t\right)$ , temporal profiles of light intensity measured by TRS are deconvoluted by the pulse response of the system that can be determined in every experiment. Deconvolution can be performed by the Fourier transform, and higher frequency components over $3∕4$ of the maximum are eliminated as they are judged to be noise.

3.1.

Monte Carlo Simulation

To analyze the light propagation and calculate the reflectance from a two-layered semiinfinite medium, we used the Monte Carlo code developed by Wang and Jacques.³¹ The code was modified to correspond to our measurement system. The number of scattering events in layers ranging from $z$ to $z+2\mathrm{mm}$ in depth, and reflectance for the medium were calculated with the time step of $10\mathrm{ps}$ . Source photons were perpendicularly irradiated on the surface of the semiinfinite media, and photons that were emitted from a detector position on the surface were all detected. The source-detector distance was $30\mathrm{mm}$ . The calculation was repeated until the number of the detected photons reached 1,000,000. For each layer separately, ${\mu }_s^{\prime }$ were given as 1.0 and $1.5{\mathrm{mm}}^{-1}$ and ${\mu }_a$ were varied at the range of $0.005\text{to}0.02{\mathrm{mm}}^{-1}$ . These values were chosen close to the optical properties of the tissues in human heads, ${\mu }_s^{\prime }$ and ${\mu }_a$ of the head³² were 0.7 to 1.0 and $0.012\text{to}0.016{\mathrm{mm}}^{-1}$ at $825\mathrm{nm}$ , and ${\mu }_s^{\prime }$ and ${\mu }_a$ of gray matter³³ were 0.4 to 0.7 and $0.018{\mathrm{mm}}^{-1}$ at $849\mathrm{nm}$ , respectively. To compare with the results of phantom experiments described in the following, a case for smaller ${\mu }_a$ value $\left(0.005{\mathrm{mm}}^{-1}\right)$ was also examined. We compared some reflectance profiles simulated in an anisotropic setting (scattering coefficient ${\mu }_s=10{\mathrm{mm}}^{-1}$ , anisotropy parameter $g=0.9$ ) with those in an isotropic one $({\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1})$ and confirmed that the difference between the two was small enough to be ignored. Therefore, isotropic scattering was assumed to reduce the calculation time. A refractive index of 1.37 was used for both the upper and lower layers.

3.2.

Phantom Preparation

Homogeneous and two-layered gelatin phantoms, the sizes of which were $230\times 230\times 55\text{to}70\mathrm{mm}$ , were prepared. The thicknesses of the upper layers in two-layered phantoms were 10 and $15\mathrm{mm}$ . The base material of phantoms was an $8\mathrm{wt}\%$ gelatin (Wako Pure Chemical Industries, Ltd., Osaka, Japan) solution. The gelatin solution itself was transparent and its ${\mu }_a$ is $0.003{\mathrm{mm}}^{-1}$ at $760\mathrm{nm}$ . We adjusted ${\mu }_s^{\prime }$ and ${\mu }_a$ to 1.0 to 1.5 and $0.003\text{to}0.02{\mathrm{mm}}^{-1}$ at $760\mathrm{nm}$ , respectively, by adding Intralipid (Fresenius Kabi AB, Upsala, Sweden) as scatterers and ink (greenish brown ink; Chugai Kasei Co., Musashino, Japan; ${\mu }_a$ at 760, 800, and $830\mathrm{nm}$ were 0.247, 0.180, and $0.125{\mathrm{mm}}^{-1}$ , respectively, in $1\mathrm{vol}\%$ aqueous solution) as an absorber. A homogeneous phantom without ink (S0; ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.003{\mathrm{mm}}^{-1}$ ) was also prepared as a reference for analysis. We determined ${\mu }_s^{\prime }$ and ${\mu }_a$ of each phantom by curve fitting the solution of photon diffusion equation to time-resolved data (the details are described in Sec. 3(D)). To confirm the reproducibility of the results, we prepared eight homogeneous phantoms on different days and/or at different times and compared their optical properties. The thickness of the upper layer and optical properties of phantoms are listed in Table 1 .

Table 1

Sets of optical properties at 760nm of gelatin phantoms used in experiment, where μa and μs′ are theoretical values based on the concentration of ink and Intralipid, respectively.

3.3.

Instrumentation

A single-channel TRS instrument³⁴ (TRS-10, Hamamatsu Photonics KK, Hamamatsu, Japan) was employed in this study. In the TRS-10, three laser diodes with different wavelengths (760, 800, and $830\mathrm{nm}$ ) generate optical light pulses having duration of around $100\mathrm{ps}$ (full width at half maximum; FWHM) at the repetition rate of $5\mathrm{MHz}$ . After adjusting the intensity by an optical attenuator, light pulses are delivered to the sample through an optical light guide [GI type, $200\mu \mathrm{m}$ core diameter, numerical aperture $\left(\mathrm{NA}\right)=0.25$ ]. The strongest power irradiated to the sample is around $30\mu \mathrm{W}$ at each wavelength. The light pulses passing through the sample are collected by a fiber bundle ( $3\mathrm{mm}$ diameter, $\mathrm{NA}=0.21$ ), and transmitted to a high-speed photomultiplier tube with S-25 photocathode (H6279-MOD, Hamamatsu Photonics KK, Japan) for single-photon detection in the NIR light region. A circuit for time-resolved measurement based on time-correlated single-photon counting (TCSPC) method measures the temporal profile. Minimal data acquisition time is $100\mathrm{ms}$ .

3.4.

Time-Resolved Measurement and Determination of Optical Properties of Phantoms

In phantom experiments, as shown in Fig. 2 , incident and detecting light guide with a separation of $30\mathrm{mm}$ were placed on the upper surface of a phantom. Measurements with an accumulation time of $10\mathrm{s}$ were performed at more than three different positions within an area $40\mathrm{mm}$ apart from the edge of the phantom to avoid the distortion of photon diffusion due to the edge. The count rate was adjusted to $100\text{to}150\mathrm{kcps}$ at each wavelength by optical attenuator to prevent the pile-up distortion. We avoided specular reflection and light leakage by employing a black light-guide holder. In every experiment, the instrumental responses of the TRS-10 were measured, facing the input and receiving fibers each other through a neutral density filter in a black tube. The instrumental response was around $150\mathrm{ps}$ FWHM at each wavelength.

Fig. 2

Schematic of the measurement for samples. It performed by reflectance mode, in which both light guides for radiation and detection are arranged in parallel. A mat black plate that has holes for light guides is on the samples as a light guide holder to decrease the error due to specular reflection and light leakage.

Optical properties in homogeneous phantoms and in each layer of the two-layered phantoms were determined by fitting an analytical solution of the photon diffusion equation to the measured temporal profile with nonlinear least squares regression. The data in a time range of $0\text{to}5400\mathrm{ps}$ were selected for fitting. In obtaining the analytical solution, the extrapolated boundary condition of reflectance mode^{35, 36} was employed. The theoretical profile was convoluted with the measured incident pulse shape in the fitting process. Light velocity in gelatin was assumed to be $0.225\mathrm{mm}∕\mathrm{ps}$ , corresponding to its refractive index of 1.33. The same values of these were also used for the calculation of ${\mu }_a^{\mathrm{seg}}$ .

4.1.

Monte Carlo Simulation

4.1.1.

Temporal variations of ${\mu }_a^{\mathrm{seg}}$

First we examined the temporal variations of ${\mu }_a^{\mathrm{seg}}$ under several conditions in which the ${\mu }_a$ of each layer were changed. As a matter of convenience, the segments were numbered as I, II, III, etc. in order of time starting from $t=500\mathrm{ps}$ (e.g., the segment by time range from $500\text{to}1000\mathrm{ps}$ was denoted as segment I and that from $2500\text{to}3000\mathrm{ps}$ was as segment V, and so on). Figure 3 shows the variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ ( $K=\text{segment}$ number) of media, the upper layer thickness of which was $10\mathrm{mm}$ and ${\mu }_s^{\prime }$ of both layers were $1{\mathrm{mm}}^{-1}$ . The ${\mu }_a^{\mathrm{seg}}\left(K\right)$ are expressed as the ratio to the real ${\mu }_{a,\text{lower}}$ . A homogeneous medium was used as a reference profile ( ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ ). Roughly, three types of the ${\mu }_a^{\mathrm{seg}}\left(K\right)$ curves were observed. The ${\mu }_a^{\mathrm{seg}}\left(K\right)$ in the case of the identical ${\mu }_a$ between the two layers were almost constant in all the time segments. In the case where ${\mu }_{a,\text{upper}}$ was larger than ${\mu }_{a,\text{lower}}$ , the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ was above 1 at $0.5\text{to}1\mathrm{ns}$ (segment I) and gradually decreased with time, converging on 1 in $2.5\text{to}3.5\mathrm{ns}$ (segments V and VI). On the other hand, in the case where ${\mu }_{a,\text{upper}}$ was smaller than ${\mu }_{a,\text{lower}}$ , ${\mu }_a^{\mathrm{seg}}\left(\mathrm{I}\right)$ was below 1 at $0.5\text{to}2\mathrm{ns}$ and increased with time. Even in a later time period, ${\mu }_a^{\mathrm{seg}}\left(K\right)$ did not reach the ${\mu }_{a,\text{lower}}$ .

Fig. 3

Temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of the two-layered media, where the upper layer thickness was $10\mathrm{mm}$ and ${\mu }_s^{\prime }$ of both layers were $1{\mathrm{mm}}^{-1}$ , predicted by Monte Carlo simulation. The vertical axis represents the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ . A homogeneous medium was used as a reference profile ( ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ ).

4.1.2.

Effects of the difference in scattering on ${\mu }_a^{\mathrm{seg}}$

Next, using two additional two-layered models in which upper ${\mu }_s^{\prime }$ was different from the lower ${\mu }_s^{\prime }$ , we examined the effects of the difference in ${\mu }_s^{\prime }$ between the two layers on the ${\mu }_a^{\mathrm{seg}}\left(K\right)$ . Figure 4(a) shows the temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of the media with different ${\mu }_s^{\prime }$ between the two layers as well as those of media with the same ${\mu }_s^{\prime }$ . As the reference profile, a homogeneous medium in which ${\mu }_s^{\prime }$ was the same as that of the upper layer ( ${\mu }_s^{\prime }=1$ or $1.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0{\mathrm{mm}}^{-1}$ ) was used. The vertical axis represents the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ . Even though upper and lower ${\mu }_s^{\prime }$ values were different from each other in the range from $1.0\text{to}1.5{\mathrm{mm}}^{-1}$ , the temporal variation of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ was very similar to that in the case in which both layers had the same ${\mu }_s^{\prime }$ .

Fig. 4

(a) Temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of two-layered media with different ${\mu }_s^{\prime }$ between the two layers together with those of media with the same ${\mu }_s^{\prime }$ predicted by Monte Carlo simulation. The upper layer thickness was $10\mathrm{mm}$ . As a reference profile, a homogeneous medium in which ${\mu }_s^{\prime }$ was the same as that of the upper layer ( ${\mu }_s^{\prime }=1$ or $1.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ ) was used. The vertical axis represents the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ . (b) Temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of two-layered media (upper layer: ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.015{\mathrm{mm}}^{-1}$ , lower layer: ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ ; upper layer $\text{thickness}=10\mathrm{mm}$ ) predicted by Monte Carlo simulation and analyzed using three different homogeneous references [ ${\mu }_s^{\prime }=0.8$ (▵), 1.0 (∎), $1.2{\mathrm{mm}}^{-1}$ (엯); ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ ].

Figure 4(b) shows the effects of the ${\mu }_s^{\prime }$ difference between an object and a reference on the temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ . Each ${\mu }_a^{\mathrm{seg}}\left(K\right)$ was calculated by a reference in which ${\mu }_s^{\prime }$ was the same $\left(1.0{\mathrm{mm}}^{-1}\right)$ as, larger $\left(1.2{\mathrm{mm}}^{-1}\right)$ than, and smaller $\left(0.8{\mathrm{mm}}^{-1}\right)$ than that of upper layer of an object. When we used a reference with larger ${\mu }_s^{\prime }$ , ${\mu }_a^{\mathrm{seg}}\left(K\right)$ in the earlier time segments were larger than ${\mu }_a$ estimated by a reference with the same ${\mu }_s^{\prime }$ . In contrast, when a reference with smaller ${\mu }_s^{\prime }$ was used, ${\mu }_a^{\mathrm{seg}}\left(\mathrm{I}\right)$ was smaller than that in the case of the same ${\mu }_s^{\prime }$ . The ${\mu }_a^{\mathrm{seg}}\left(\mathrm{VI}\right)$ calculated by a $1.2\text{-}{\mathrm{mm}}^{-1}$ reference and the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{V}\right)$ and (VI) by the $0.8\text{-}{\mathrm{mm}}^{-1}$ reference were almost equal to that calculated by the reference with the same ${\mu }_s^{\prime }\left(1.0{\mathrm{mm}}^{-1}\right)$ .

4.1.3.

Effect of the ${\mu }_a$ difference between the upper and lower layer on the ${\mu }_a^{\mathrm{seg}}$ in later segments

Figure 5 shows the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{IV}\right)$ (V), and (VI) of two-layered media (upper layer $\text{thickness}=10\mathrm{mm}$ ) predicted by Monte Carlo simulation as a function of the ratio of ${\mu }_{a,\text{upper}}$ to ${\mu }_{a,\text{lower}}$ $\left({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}\right)$ . The data obtained in the cases in which ${\mu }_s^{\prime }$ was equal (the models in Sec. 4.1.1) and different (the models in Sec. 4.1.2) between the two layers were plotted together. As the reference profile, a homogeneous medium in which ${\mu }_s^{\prime }$ was the same as that of the upper layer was used. When the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ was equal to or more than 1, the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{IV}\right)$ , (V), and (VI) nearly represented the real ${\mu }_a$ . The error was less than 5.0% in segment VI. On the other hand, the ${\mu }_a^{\mathrm{seg}}\left(K\right)$ even in these later time periods were smaller than the real ${\mu }_a$ in the range of the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ less than 1. The deviation grew larger as the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ decreased.

Fig. 5

The ${\mu }_a^{\mathrm{seg}}$ (IV) (◻), (V) (◆), and (VI) (▵) of two-layered media predicted by Monte Carlo simulation as a function of the ratio of ${\mu }_{a,\text{upper}}$ to ${\mu }_{a,\text{lower}}\left({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}\right)$ . Upper layer thickness was $10\mathrm{mm}$ . The data in both cases that ${\mu }_s^{\prime }$ was identical or different between the two layers were plotted together. A homogeneous medium in which ${\mu }_s^{\prime }$ was the same as that of the upper layer was used as the reference profile.

4.2.

Phantom Experiment

4.2.2.

Temporal variation of ${\mu }_a^{\mathrm{seg}}$

Whichever wavelength (760, 800, and $830\mathrm{nm}$ ) we used, the results obtained from the following analyses of phantoms were almost the same. Thus, we present the data at $760\mathrm{nm}$ as representative results. In the phantoms with values of ${\mu }_a$ larger than $0.015{\mathrm{mm}}^{-1}$ , the profiles of $A^{\mathrm{diff}}\left(t\right)$ were reliable only in the time range of $0.5\text{to}2\mathrm{ns}$ . Therefore, we mainly show the results of phantoms with ${\mu }_a$ of $0.003\text{to}0.013{\mathrm{mm}}^{-1}$ , for which our instrument shows adequate performance, although they were smaller than that of human head generally estimated. Figure 6 shows the typical time-resolved reflectance [Fig. 6(a)] and $A^{\mathrm{diff}}-t$ curve [Fig. 6(b)] obtained by phantom measurements. In this condition, the data were reliable in the time period from $0.35\text{to}3\mathrm{ns}$ . In the early time period before $0.25\mathrm{ns}$ and in the later time period after $3\mathrm{ns}$ , data were noisy because of the interaction of incident light and/or the decrease in detected photon numbers. Therefore, analysis for slope of $A^{\mathrm{diff}}\left(t\right)$ (calculation of ${\mu }_a^{\mathrm{seg}}$ ) was performed in the range of $0.5\text{to}3\mathrm{ns}$ . The segments were numbered as I, II, III, etc. in order of time in the same manner as the case in Sec. 4(A).

Fig. 6

Typical time-resolved reflectance and $A^{\mathrm{diff}}-t$ curve obtained by phantom measurements: (a) temporal profiles of incident light (IN), intensity of the detected light through a homogeneous $\left[I_R\left(t\right)\right]$ and a two-layered medium $\left[I\left(t\right)\right]$ and (b) $A^{\mathrm{diff}}-t$ curve obtained from the data in (a) in the time period of $0\text{to}4\mathrm{ns}$ . The data of detected light intensity through the phantoms are deconvoluted by the temporal profile of the incident light and $A^{\mathrm{diff}}$ at each time was calculated. The Roman numerals show numbers of segments of the curve parted every $0.5\mathrm{ns}$ to get the ${\mu }_a^{\mathrm{seg}}$ .

Figure 7 shows temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ in two-layered phantoms (measured optical properties of L1: ${\mu }_s^{\prime }=1.14\pm 0.02{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{upper}}=0.0143\pm 0.0001{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{lower}}=0.0088\pm 0.0002{\mathrm{mm}}^{-1}$ ; L2: ${\mu }_s^{\prime }=1.12\pm 0.01{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{upper}}=0.0030\pm 0.0001{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{lower}}=0.0081\pm 0.0001{\mathrm{mm}}^{-1}$ ) with the upper layer thickness of $10\mathrm{mm}$ as well as that of the homogeneous phantom (S1: ${\mu }_s^{\prime }=1.06\pm 0.02{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.0084\pm 0.0002{\mathrm{mm}}^{-1}$ ). The ${\mu }_a^{\mathrm{seg}}\left(K\right)$ are expressed as the ratio to the real ${\mu }_{a,\text{lower}}$ . The ${\mu }_s^{\prime }$ of the reference was the same as that of the upper and lower layers of these phantoms. The ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of S1 were almost constant in all the time segments, and the ratio to the real ${\mu }_{a,\text{lower}}$ was $0.96\pm 0.06$ . As for L1 $({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}=1.66)$ , the ratio of ${\mu }_a^{\mathrm{seg}}\left(\mathrm{I}\right)$ to the real ${\mu }_{a,\text{lower}}$ was $1.37\pm 0.05$ . It gradually decreased with time to $0.98\pm 0.07$ in segment IV and $0.91\pm 0.01$ in segment V. On the other hand, in L2 $({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}=0.40)$ , ${\mu }_a^{\mathrm{seg}}\left(\mathrm{I}\right)$ was $0.56\pm 0.04$ and increased with time to $0.80\pm 0.05$ in $\sim 3\mathrm{ns}$ (segment V).

Fig. 7

Temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of two-layered phantoms (L1 and L2), of which upper layer thickness was $10\mathrm{mm}$ , and homogeneous phantom of S1 (L1: ${\mu }_s^{\prime }=1.14\pm 0.02{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{upper}}=0.0143\pm 0.0001{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{lower}}=0.0088\pm 0.0002{\mathrm{mm}}^{-1}$ ; L2: ${\mu }_s^{\prime }=1.12\pm 0.01{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{upper}}=0.0030\pm 0.0001{\mathrm{mm}}^{-1}$ , ${\mu }_{a,\text{lower}}=0.0081\pm 0.0001{\mathrm{mm}}^{-1}$ ; S1: ${\mu }_s^{\prime }=1.06\pm 0.05{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.0084\pm 0.0002{\mathrm{mm}}^{-1}$ ). A homogeneous phantom S0 ( ${\mu }_s^{\prime }=1.01\pm 0.04{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.0030\pm 0.0001{\mathrm{mm}}^{-1}$ ) was used as a reference. The plots represent mean $\pm \mathrm{SD}$ (standard deviation) ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ (L1, L2: $n=3$ positions $\times 2$ samples; S1: $n=3$ positions $\times 8$ samples). Top dotted line and bottom broken line indicate the real ${\mu }_{a,\text{upper}}$ shown as the ratio to the real ${\mu }_{a,\text{lower}}$ in L1 and L2, respectively.

Figure 8 shows the temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of phantoms with an upper layer thickness of $15\mathrm{mm}$ . The optical properties of each layer in $\mathrm{L}{1}^{\prime }$ and $\mathrm{L}{2}^{\prime }$ were the same as those of L1 and L2, respectively. For comparison, ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of S1 is also shown. Like in the phantoms with an upper layer thickness of $10\mathrm{mm}$ shown in Fig. 7, the pattern of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ was dependent on ${\mu }_{a,\text{upper}}$ . When ${\mu }_{a,\text{upper}}$ was larger than ${\mu }_{a,\text{lower}}\left(\mathrm{L}{1}^{\prime }\right)$ , the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{V}\right)$ indicated ${\mu }_{a,\text{lower}}$ .

Fig. 8

Temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ of two-layered phantoms ( $\mathrm{L}{1}^{\prime }$ and $\mathrm{L}{2}^{\prime }$ ), of which the upper layer thickness was $15\mathrm{mm}$ , and homogeneous phantom of S1 at $760\mathrm{nm}$ using S0 ( ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.003{\mathrm{mm}}^{-1}$ ) as a reference. The optical properties of each layer in $\mathrm{L}{1}^{\prime }$ and $\mathrm{L}{2}^{\prime }$ are the same as L1 and L2. The notation of the vertical axis is the same as that of Fig. 7. Values are means $\pm \mathrm{SD}$ ( $\mathrm{L}{1}^{\prime }$ , $\mathrm{L}{2}^{\prime }$ : $n=3$ positions $\times 2$ samples, S1: $n=3$ positions $\times 8$ samples). Top dotted line and bottom broken line represent the real ${\mu }_{a,\text{upper}}$ shown as the ratio to the real ${\mu }_{a,\text{lower}}$ in $\mathrm{L}{1}^{\prime }$ and $\mathrm{L}{2}^{\prime }$ , respectively.

The effects of the scattering difference on the temporal variations of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ were found in a similar manner to the results by Monte Carlo simulation (data not shown).

4.2.3.

Effect of the ${\mu }_a$ difference between the upper and lower layer on the ${\mu }_a^{\mathrm{seg}}$ in later segments

Figure 9 shows the ${\mu }_a^{\mathrm{seg}}$ in segment IV [Fig. 9(a)] and segment V [Fig. 9(b)] as a function of ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ by measurements of phantoms in Table 1. As in the Monte Carlo simulation, the ${\mu }_a^{\mathrm{seg}}\left(K\right)$ in later time periods depended on the difference in ${\mu }_a$ values between the upper and the lower layers: it was smaller than the real ${\mu }_{a,\text{lower}}$ in the range of the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ less than 1, while it almost represented the real ${\mu }_{a,\text{lower}}$ when the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ was equal to or more than 1. In the case in which upper layer thickness was $15\mathrm{mm}$ , the ${\mu }_a^{\mathrm{seg}}$ was much smaller in the range of the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ below 1, while it was larger than the real values in the range of the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ above 1 in segment IV, but not in segment V. When the ${\mu }_s^{\prime }$ value of the lower layer was different from that of the reference, there were no significant changes in the relationship between the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ and the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{IV}\right)$ or (V).

Fig. 9

Plots of ${\mu }_a^{\mathrm{seg}}$ in (a) segment IV and (b) segment V in reference to the ratio of ${\mu }_{a,\text{upper}}$ to ${\mu }_{a,\text{lower}}$ $\left({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}\right)$ of three kinds of phantoms (Table 1); upper layer thickness is $10\mathrm{mm}$ and ${\mu }_s^{\prime }$ of both layers are the same as the reference (∎), upper layer thickness is $10\mathrm{mm}$ and ${\mu }_s^{\prime }$ of the lower layer is different from the reference (◻), and upper layer thickness is $15\mathrm{mm}$ and ${\mu }_s^{\prime }$ of both layers are the same as the reference (●). The vertical axis represents the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ . Each plot was average value by several measurements for one sample.

5.1.

Relationship Between ${\mu }_a^{\mathrm{seg}}$ and Real ${\mu }_a$ of Each Layer

Time segment analysis, dividing the temporal profile of detected light intensity into segments, leads to some findings on path distribution in the direction of depth of photons detected at each time for layered turbid media with absorbers.

The results obtained from the analyses for Monte Carlo simulation data were generally confirmed by those of phantom experiments data. As one of the mutual results, it was found that the ${\mu }_a^{\mathrm{seg}}$ values in later segments depended on the difference in real ${\mu }_a$ between two layers. This represents that in later time, the partial path length in each layer depends on both ${\mu }_a$ values of the layers because the probability that the photons mainly passed in the larger ${\mu }_a$ layer are detected declines. When ${\mu }_{a,\text{upper}}$ is smaller than ${\mu }_{a,\text{lower}}$ , $l_{\text{upper}}∕l_{\text{lower}}$ at a later time is relatively high and $\Delta l_{\text{upper}}$ cannot be ignored even in later time segments. Thus, ${\mu }_a^{\mathrm{seg}}$ was underestimated by the effect of ${\mu }_{a,\text{upper}}$ . In contrast, when ${\mu }_{a,\text{upper}}$ is equal to or larger than ${\mu }_{a,\text{lower}}$ , $l_{\text{lower}}$ of photons detected later in time is sufficiently longer than $l_{\text{upper}}$ and steadily increase with $t$ , that is, $(\Delta l_{\text{upper}}∕\Delta L)$ can be considered nearly zero in later time periods, as expected. The presented results, for which ${\mu }_a^{\mathrm{seg}}$ were underestimated if ${\mu }_{a,\text{upper}}$ was smaller than ${\mu }_{a,\text{lower}}$ , were in agreement with a study by Hielscher, ²² who employed a curve-fitting method with homogeneous diffusion equation to later part of the time-resolved reflectance.

Similarly, ${\mu }_a^{\mathrm{seg}}\left(\mathrm{I}\right)$ did not give the real ${\mu }_{a,\text{upper}}$ values correctly, which were also affected by the difference in ${\mu }_a$ between two layers (data not shown). As shown in Fig. 1, the photons detected at $500\text{to}1000\mathrm{ps}$ have a low probability of passing through layers lower than the depth of $10\mathrm{mm}$ . Although $\Delta l_{\text{upper}}$ is dominant to $\Delta L$ , $\Delta l_{\text{lower}}∕\Delta L$ cannot be taken as zero even in this period, especially when the thickness of the upper layer is less than $10\mathrm{mm}$ . To estimate ${\mu }_{a,\text{upper}}$ more properly, it might be effective to use an earlier time segment of $A^{\mathrm{diff}}-t$ curve (e.g., at $350\text{to}700\mathrm{ps}$ ). But due to the interference of incident light, it seems difficult to select the optimal time period in practical measurements.

5.2.

Estimation of ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ by the Ratio of ${\mu }_a^{\mathrm{seg}}$ at an Early Time to that at a Later Time

In spite of the discrepancy of ${\mu }_a^{\mathrm{seg}}$ from the real ${\mu }_a$ , our method gives us enough information to know the difference in ${\mu }_a$ between the upper and lower layers. Moreover, based on the relationship between the extent of underestimation of ${\mu }_a$ and the ratio of the real ${\mu }_a$ of two layers (Fig. 5), we would derive ${\mu }_{a,\text{lower}}$ by correction of ${\mu }_a^{\mathrm{seg}}$ under the condition where ${\mu }_s^{\prime }$ of the upper layer and that of a reference are the same and upper layer thickness is almost known. For judgment of the ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ , we attempted to use the ${\mu }_a^{\mathrm{seg}}$ at an early time (segment I) and at a later time (segment V) $\left({\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}\right)$ . First, under the condition where ${\mu }_s^{\prime }$ of the reference was the same as that of the upper layer of the medium, we examined the ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ of various media with different ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ predicted by the Monte Carlo simulation. Figure 10 shows the plots of ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ to ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ . There was a strong linear correlation between these values $(r^2=0.99)$ . Then we checked the relationship between the ratio of the ${\mu }_a^{\mathrm{seg}}$ (V) to the real ${\mu }_{a,\text{lower}}$ and the ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ for both data sets by Monte Carlo simulation [Fig. 11(a) ] and by phantom experiment [Fig. 11(b)]. In the range of this ratio below 1, the ${\mu }_a^{\mathrm{seg}}\left(\mathrm{V}\right)∕{\mu }_{a,\text{lower}}$ linearly related to the ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ , i.e., we could express them as ${\mu }_a^{\mathrm{seg}}\left(\mathrm{V}\right)∕{\mu }_{a,\text{lower}}=a\times {\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}+b$ , where $a$ and $b$ are constants. By regression analysis, the relationships and the correlation coefficients were $a=0.85$ , $b=0.16$ , and $r=0.80$ in the plot with Monte Carlo data, and $a=0.84$ , $b=0.18$ , and $r=0.96$ in that by the phantom experiment, which were almost in agreement. Segment IV and segment VI could also be used for the estimation of ${\mu }_{a,\text{lower}}$ . But under the condition of this study, segment V was the most adequate to analyze ${\mu }_{a,\text{lower}}$ , because segment IV included more information of the upper layer [cf. the case of the upper thickness of $15\mathrm{mm}$ in Fig. 9(a)] and segment VI had higher noise than segment V.

Fig. 11

Plots of ${\mu }_a^{\mathrm{seg}}$ (V) with reference to the ratio of the ${\mu }_a^{\mathrm{seg}}$ (I) to (V) $\left({\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}\right)$ in (a) Monte Carlo simulation data and (b) two-layered phantoms with an upper layer thickness of $10\mathrm{mm}$ (◻) and $15\mathrm{mm}$ (●), analyzed by a reference of which ${\mu }_s^{\prime }$ was the same as that of the upper layer. The vertical axis represents the ratio of ${\mu }_a^{\mathrm{seg}}\left(K\right)$ to the real ${\mu }_{a,\text{lower}}$ . Broken lines are regression lines within the range of ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ below 1.

Fig. 10

Plots of the ratio of ${\mu }_{a,\text{upper}}$ to ${\mu }_{a,\text{lower}}$ ( ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ ) in reference to the ratio of ${\mu }_a^{\mathrm{seg}}$ (I) to (V) ( ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ ) for the Monte Carlo simulation data.

Consequently, we could evaluate ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ from ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ and we could estimate the ${\mu }_{a,\text{lower}}$ by dividing ${\mu }_a^{\mathrm{seg}}\left(K\right)$ in later time by the value of $(a\times {\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}+b)$ . In Fig. 11, the data in both cases in which ${\mu }_s^{\prime }$ was equal and different between the two layers within the range of $1.0\text{to}1.5{\mathrm{mm}}^{-1}$ were plotted together. Therefore, the equation for the relation between ${\mu }_a^{\mathrm{seg}}\left(\mathrm{V}\right)∕{\mu }_{a,\text{lower}}$ and ${\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}$ holds its validity at least under the condition that ${\mu }_s^{\prime }$ is $1.0\text{to}1.5{\mathrm{mm}}^{-1}$ and ${\mu }_a$ is $0.003\text{to}0.02{\mathrm{mm}}^{-1}$ of each layer in the two-layered media.

5.3.

Effect of the Scattering Coefficient on Estimation of ${\mu }_a$ of the Lower Layer

As shown in Fig. 4(a), although the ${\mu }_s^{\prime }$ of the lower layer was different from that of the reference, the ${\mu }_a^{\mathrm{seg}}$ values were almost identical with those for the case where the ${\mu }_s^{\prime }$ of the lower layer was the same as that of the reference. In contrast, as shown in Fig. 4(b), the difference in ${\mu }_s^{\prime }$ between the upper layer and the reference significantly influenced the ${\mu }_a^{\mathrm{seg}}$ in earlier time segments. These results indicate that $S^{\mathrm{diff}}\left(t\right)$ in Eq. 2 is almost equal to the difference in scattering function only between the shallower part of the media and the reference. In addition, our results showed that the dependence of ${\mu }_a^{\mathrm{seg}}$ after $2.5\mathrm{ns}$ on the difference in ${\mu }_s^{\prime }$ between the upper layer and the reference was weak. These results are explained by the analytical solution derived by Patterson [Eqs. (7) and (8), Ref. 18]. In the case of homogeneous media, time differential of $S^{\mathrm{diff}}\left(t\right)$ in Eq. 2 [or in the first line of Eq. 3] in this paper can be represented as follows: $\mathrm{d}{S}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t=[{\rho }^2∕\left(4D_R{c}_R\right)-{\rho }^2∕\left(4D_{O}c\right)]∕t^2$ , where $D_R$ and $D_O$ are the diffusion coefficients for the reference and the object, respectively. As $[{\rho }^2∕\left(4D_R{c}_R\right)-{\rho }^2∕\left(4D_{O}c\right)]$ is constant, $\mathrm{d}{S}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ is in proportion to the reciprocal of the square of time. Thus, the $\mathrm{d}{S}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ is negligible at the later time. Similarly, in the case of the two-layered media, it can be considered that $\mathrm{d}{S}^{\mathrm{diff}}\left(t\right)∕\mathrm{d}t$ is very small at the later time, i.e., the effect of the ${\mu }_s^{\prime }$ difference on the ${\mu }_a^{\mathrm{seg}}$ at the later time segments can be negligible. In contrast, the scattering term cannot be ignored in the segment I.

5.4.

Determination of Proper Reference with Time Segment Analysis

We have showed so far the results mainly in the case where ${\mu }_s^{\prime }$ of the reference was the same as that of the upper layer of the medium. However, for proper analysis with the correction in Sec. 5(B), we must select an adequate reference with ${\mu }_s^{\prime }$ very close to that of the upper layer of the target medium that was actually unknown. We can show one of the methods to determine the adequate reference.

As for a homogeneous medium, if ${\mu }_s^{\prime }$ of an object and a reference are the same, which implies $S^{\mathrm{diff}}\left(t\right)$ is zero in Eq. 2, the $A^{\mathrm{diff}}-t$ relation becomes linear. That is, if we select a reference that makes $A^{\mathrm{diff}}\left(t\right)$ a straight line, the ${\mu }_s^{\prime }$ of the reference is the same as that of the object.³⁷ In layered media, as mentioned in Sec. 5(C), $S^{\mathrm{diff}}\left(t\right)$ mainly depended on the difference in ${\mu }_s^{\prime }$ between the upper layer and the reference. Thus, we applied the precedings concept that $S^{\mathrm{diff}}\left(t\right)$ is estimated based on the degree of linear relation for $A^{\mathrm{diff}}-t$ curve to two-layered media to select an adequate reference. Figure 12(a) shows $A^{\mathrm{diff}}-t$ curves of two-layered medium (upper layer, ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ , and lower layer, ${\mu }_s^{\prime }=1.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.015{\mathrm{mm}}^{-1}$ , upper layer $\text{thickness}=10\mathrm{mm}$ ) in $400\text{to}1000\mathrm{ps}$ , predicted by Monte Carlo simulation and analyzed using five references that had different ${\mu }_s^{\prime }$ values within the range of $0.6\mathrm{to}1.4{\mathrm{mm}}^{-1}$ $({\mu }_a=0{\mathrm{mm}}^{-1})$ . We examined the coefficient of determination $r^2$ of each curve by linear regression analysis. It could be found that when a reference was used whose ${\mu }_s^{\prime }$ was the same as that of the upper layer of an object, $r^2$ was extremely close to 1. It indicates that when thickness of the upper layer is larger than $10\mathrm{mm}$ , the change in ${\mu }_a^{\mathrm{diff}}\left(t\right)$ with $t$ is so small as compared with that of $S^{\mathrm{diff}}\left(t\right)$ in time segment I and in time earlier than segment I. Thus, using this method of investigating linear relation for $A^{\mathrm{diff}}-t$ at an earlier time, we can select an appropriate reference. Figure 12(b) shows estimated ${\mu }_a$ values of the lower layer and the errors for this model analyzed by various references already described. The correction of ${\mu }_a^{\mathrm{seg}}$ (V) with equation in Sec. 5(B) was used to estimate these ${\mu }_a$ . It was found that when a reference was used whose ${\mu }_s^{\prime }$ value was the same as that of the upper layer of an object, the estimated ${\mu }_a$ was almost the same as the real ${\mu }_a$ . We also found that when the difference in ${\mu }_s^{\prime }$ between a reference and the upper layer of an object was smaller than $0.5{\mathrm{mm}}^{-1}$ , the error of estimated ${\mu }_a$ was less than 10%. In other words, we should prepare a series of reference in which ${\mu }_s^{\prime }$ were changed by every $0.5{\mathrm{mm}}^{-1}$ within the range of the human extracerebral tissue to estimate reliable ${\mu }_a$ by this method.

Fig. 12

(a) The $A^{\mathrm{diff}}-t$ curves of a two-layered medium (upper layer: ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ ; and lower layer: ${\mu }_s^{\prime }=1.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.015{\mathrm{mm}}^{-1}$ ; upper layer $\text{thickness}=10\mathrm{mm}$ ) in $400\text{to}1000\mathrm{ps}$ , predicted by Monte Carlo simulation and analyzed using five references that had various ${\mu }_s^{\prime }$ values within the range of $0.6\text{to}1.4{\mathrm{mm}}^{-1}$ $({\mu }_a=0{\mathrm{mm}}^{-1})$ . (b) Estimated value of ${\mu }_{a,\text{lower}}$ and the error caused by analyzing with each reference which ${\mu }_s^{\prime }$ was within the range of $0.8\text{to}1.6{\mathrm{mm}}^{-1}$ . The correction of ${\mu }_a^{\mathrm{seg}}$ (V) with the equation in Sec. 5(B) was used to estimate these ${\mu }_a$ values.

5.5.

Possibility of Applying the Present Method to Human Head Measurements

The time segment analysis of time-resolved reflectance in two-layered media enabled us to simply acquire information on the ${\mu }_a$ of each layer and was implemented for a reflectance by measurement with a single source-detector distance. Moreover, we found that the estimated value of ${\mu }_{a,\text{lower}}$ depended on the value of ${\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}$ , i.e., ${\mu }_a^{\mathrm{seg}}$ were underestimated if ${\mu }_{a,\text{upper}}$ was smaller than ${\mu }_{a,\text{lower}}$ , which was almost the same as the results in the case of fitting analysis for the later part of time-resolved reflectance by a solution of diffusion equations for homogeneous media.²² In our method, however, the difference in ${\mu }_a\left({\mu }_a{\text{ratio}}_{\text{upper}∕\text{lower}}\right)$ can be evaluated using ${\mu }_a^{\mathrm{seg}}$ both at earlier time and at later time $\left({\mu }_a^{\mathrm{seg}}{\text{ratio}}_{\mathrm{I}∕\mathrm{V}}\right)$ .

For the adult human forehead, the total thickness of scalp and skull is^{28, 38} about $10\mathrm{mm}$ . Thus, the ${\mu }_a$ of the brain may be estimated by ${\mu }_a^{\mathrm{seg}}$ (IV) to (VI) if the human head could be approximated to two-layered model. On the other hand, the human head is a multilayered structure and the scattering phenomenon must be more complex. Therefore we require further experiments with a more sophisticated model of the human head to confirm the utility and limitations of this method. For example, relation between the discrepancies of estimated ${\mu }_a$ from the real ${\mu }_a$ and the ratio of ${\mu }_a^{\mathrm{seg}}$ at an earlier time to that at a later time in multilayered models, and dependence of the layer thickness on the relationship should be examined. As the segment size, a time range of $500\mathrm{ps}$ was chosen in this study. For a multilayered model, however, we should evaluate the segment size to obtain better results. Moreover reliable estimation requires determining ${\mu }_s^{\prime }$ values of the scalp and skull, which vary with individuals and each position on the head.