2.1.

Qualitative Comparison of the Depth Sensitivity of CW and Time-Gated Measurements with Monte Carlo Simulations

Figure 1 shows the results of Monte Carlo (MC) simulations modeling the transport of light in the head. A description of the MC method can be found in Ref. 29. The medium of propagation is a segmented head obtained from a magnetic resonance imaging (MRI) scan. Five tissue types are distinguished: skin, skull, cerebrospinal fluid (CSF), gray matter, and white matter. They are all considered homogeneous tissues with optical properties typical of those found in the literature, as described in Ref. 30 (at 830 nm, scalp, μ_a=0.191 cm ⁻¹ and μ_s ^′=6.6 cm ⁻¹; skull, μ_a=0.136 cm ⁻¹ and μ_s ^′=8.6 cm ⁻¹; CSF, μ_a=0.026 cm ⁻¹ and μ_s ^′=0.1 cm ⁻¹; gray matter, μ_a=0.186 cm ⁻¹ and μ_s ^′=11.1 cm ⁻¹; and white matter, μ_a=0.186 cm ⁻¹ and μ_s ^′=11.1 cm ⁻¹) . The five tissue types appear in five different shades of gray. One source and one detector are set 4.4 cm apart on the scalp. The contours of the photon visit probability maps between the source and the detector, as given by the MC simulations, are superimposed on the structure of the head (black contours). This shows the region where the detected photons have traveled.

Figure 1

Coronal view of the photon visit probability maps in the head for a source-detector separation of 44 mm, obtained by MC simulations, with continuous light (a), and with a time-gated technique at 1 (b) and 2 ns (c). The black contours are drawn every order of magnitude of the probability profiles. In all figures, the value of the first contour was chosen so that the probability profile reaches the skull.

Figure 1(a) shows the propagation of light in the CW case: the signal is dominated by the light coming from the superficial layers (scalp, skull, and CSF). Figures 1(b) and 1(c) show the TD results, for a short delay (1 ns) and for a long delay (2 ns), respectively. Selecting early photons yields a probability profile close to that of the CW system. On the contrary, for the 2-ns delay, the visit probability map goes deeper inside the brain [Fig. 1(c)]. By choosing the late gate, we can thus increase sensitivity to photons coming from the brain without being dominated by light coming from superficial layers. Judging from these simulations, it is rather intuitive that TD systems can improve the sensitivity to deep tissues of the head.

2.2.

Analytical Model for the Depth Sensitivity of cw and Time-Gated Measurements

The MC simulations already presented suggest that TD systems improve the sensitivity to deep tissues relative to CW signals. Here, we estimate the improvement in contrast offered by TD, and study how it varies with the depth of the probed tissues, as well as with the source-detector separation.

2.2.1.

Model geometry

For our simulations, we chose to approximate the head by a homogeneous semi-infinite medium (see Fig. 2), of absorption coefficient μ_a,0=0.2 cm ⁻¹ and reduced scattering coefficient μ_s,0 ^′=15 cm ⁻¹. The medium contains a small absorbing perturbation, which simulates an activated region of the brain, at depth z below the surface. The inclusion has an absorption coefficient μ_a,0+Δμ_a. This absorbing perturbation is assumed to be small enough so that we can estimate its effect by using the Born approximation. It has been observed experimentally that physiological signals in the skin (heart beat, respiration) could impede the detection of deeper signals.^{31 32} To include this observation in our model, we define a superficial layer of thickness 5 mm, which is typical of the scalp thickness (see Refs. 27 and 33, for instance). This layer has an increased absorption coefficient μ_a,0+Δμ_a,back (see dashed region in Fig. 2), which creates a background signal simulating the physiological signal clutter.

Figure 2

Geometry and notations used for the analytical model.

The medium is probed with one source, emitting an incident power of 10 mW at 800 nm, and one detector, of diameter 1 mm and numerical aperture 0.3 (characteristic of optical fibers typically used to collect light). They are placed at a distance ρ from each other with the inclusion in the middle, as shown in Fig. 2. We study the sensitivity of both CW and TD systems to the inclusion as a function of its depth z, and the source-detector separation ρ. To this effect, we define the following parameters: signal, contrast, background, and noise.

2.2.3.

Noise model

The noise is evaluated with realistic signal to noise models adapted from our experimental systems. Figure 3 shows the evolution of the SNR for both systems as a function of the incident power (the signal). For CW systems, a commonly used detector is an avalanche photodiode (APD). There are mainly two sources of noise: the photoelectron shot noise and the electronic noise. The bandwidth of the detector was set to 1 Hz. The gain, sensitivity and noise equivalent power (NEP) of the APD are given by the manufacturer and were checked by experimental measurements on phantoms for systems we routinely use in our laboratory.³⁶ At a low light level (typically below 5 pW), the noise is dominated by the constant electronic noise: the SNR is proportional to the signal. At a high light level, the noise becomes shot-noise limited, leading to an SNR almost linear with the square root of the signal [see Fig. 3(a)]. The noise model for the TD system was based on our own experimental device, for which the detector is an intensified CCD (ICCD) camera. The noise model³⁷ takes into account the photon shot noise, the amplification stages, the dark noise of both the intensifier and the CCD camera, and the effect of on-chip and postprocessing pixel binning. We considered a camera exposure time of 100 ms, with a postaveraging of 10 camera frames, for a fair comparison with the 1-Hz bandwidth of the CW system. The SNR evolution being dependent on the intensifier gain, we chose an intensifier voltage of 400 V, which is typical of our experimental measurements. An on-chip binning of 8×8 pixels was assumed, followed by a postprocessing binning of 4×4 pixels. The combination of all sources of noise finally leads to a dependence of SNR with signal that is very close to a square root [see Fig. 3(b)].

Figure 3

Experimental SNR as a function of the incident power for the cw and TD systems routinely used in our laboratory. The darkened regions of the plots correspond to the saturation of the detectors for the chosen experimental parameters.

In both plots, we darkened the region where the detector saturates for the chosen experimental parameters. In practice, we can measure higher incident power by reducing either the detector gain or the integration time. However, the noise model we present here would vary accordingly.

2.3.

Optimum Delay in the TD Results

In the CW case, only two parameters are effectively changed: the depth of the perturbation and the source-detector separation. In the TD case, the signal, noise, contrast, and background also depend on the measurement gate delay. Figure 4(a) shows the signal, noise, and contrast-to-noise ratio (CNR) as a function of time delay for a source-detector separation of 2 cm and an inclusion at a depth of 1.5 cm below the surface. The curve is typical of the signal recorded on the head with a TD system.

Figure 4

(a) Evolution of the signal, noise, and CNR with delay for an inclusion at depth z=1.5 cm below the surface and a source-detector separation of 2 cm. The signal and noise units are watts, the CNR unit is mm mm⁻³ (CNR per unit absorption coefficient and unit volume of the activation). (b) Variation of the optimum delay τ _CNR (solid lines) that maximizes the CNR, and of the delay τ _sig (dotted lines) yielding the maximum signal, as a function of the depth of the inclusion and the source-detector separation. The delay values indicated on the contours are given in nanoseconds.

The maximum signal and the maximum CNR happen at different delays. We call the delay τ _CNR at which the CNR is maximal the optimum delay: it corresponds to the delay where we are experimentally most sensitive to the inclusion. Figure 4(b) shows the optimum delay τ _CNR as a function of z and ρ. For comparison, the delay τ _sig yielding the maximum signal (the peak of the TPSF) is also plotted in dotted lines on the same curve. Note that the latter is independent of the depth of the inclusion since we neglected the effect of the small inclusion on the signal, although this would produce only a small change for typical perturbations. Note that the dependence of the optimum delay on the source-detector separation is rather small. Contrary to CW measurements, where deep layers of the brain can be reached only by using a large source-detector separation, TD measurements have little dependence on the source-detector separation. The depth sensitivity is essentially obtained through the time delay. The optimum delay is strongly dependent on the depth of the inclusion, as expected. The deeper inclusion is probed more by later arriving photons. As a consequence, the delay between τ _CNR and τ _sig increases dramatically with the depth of the inclusion. For an inclusion 3 cm below the surface, the optimum delay would be 2 ns after τ _sig , i.e., in the late tail of the TPSF.

We compare the depth sensitivity of CW and TD systems by examining the dependence of the CNR and the contrast-to-background ratio (CBR) on the depth of the inclusion and source-detector separation. Ideally, we would consider only these values and try to maximize them. However, to apply our theoretical study to experimental conditions, we must take into account the dynamic range of the detector, defined as the ratio between the largest and the smallest signals we can detect. For instance, if the CNR is maximum at a delay where the contrast is smaller than the minimal detectable signal, then we cannot experimentally detect it. In the following figures, we choose the ideal situation and present the TD results at the optimum delay τ _CNR . However, we always discuss the relevance of the results in consideration of the dynamic range of the detector.

2.4.

SNR and CNR versus the Depth of the Absorption Change

We first compare the depth sensitivity of CW and TD systems by analyzing their respective CNRs. However, as mentioned previously, the SNR cannot be ignored since it gives information relative to the dynamic range of the system. Figure 5 presents the dependence of both the CNR and the SNR on the depth of the inclusion and source-detector separation for the CW domain [Fig. 5(a)] and the TD [Fig. 5(b)]. Note that by taking the ratio of the CNR and SNR, the contrast-to-signal ratio is immediately obtained. As mentioned, the values are given per unit absorption coefficient and unit volume (mm mm⁻³) and can also be considered without units for an absorbing inclusion of volume 1 cm³ and Δμ_a=0.01 cm ⁻¹. For instance, in the CW case [Fig. 5(a)], an inclusion of this size located at a depth 12 mm below the surface, and for a source-detector separation of 3 cm, the contrast is calculated to be approximately 1% of the signal (small × on the CW curves). This value is in good agreement with typical experimental values.³⁶ The CW figure also confirms an empirical observation: for each depth of the inclusion, there is an optimum source-detector separation that maximizes the CNR. This optimum separation was plotted as a dotted line in the same figure. As expected, the deeper the activation in the brain, the farther apart the source and detector must be to probe the activated region. Although the 2-D graph shows depth values starting at 0, only depths higher than typically 10 mm, i.e., underneath scalp and skull, are characteristic of cortical activation (for typical values of skin and bone thicknesses, see, for instance, Refs. 27 and 33).

Figure 5

Evolution of the CNR (solid lines) and the SNR (dotted lines) with depth of the inclusion and source-detector separation in (a) the cw case and (b) the TD case. The CNR values are shown in decimal logarithm per unit absorption and unit volume of the inclusion (mm mm⁻³). The SNR values are in decimal logarithm. The gray regions of the graphs correspond to the saturation of the detectors for the chosen experimental parameters, i.e., the region where the noise model is no longer valid.

The TD results are more surprising [Fig. 5(b)]. Let us first insist on the fact that TD values are plotted for the delay τ _CNR yielding the maximum CNR. This explains the fact that the SNR curves, which are independent of the inclusion depth for the CW case, are bent for the TD case: when the inclusion depth increases, the optimum delay increases as well, and the signal decreases because we move toward the tail of the TPSF. The CNR curves show an unexpected result. Whatever the depth of the inclusion, the maximum CNR is always obtained for the smallest source-detector separation. However, this result obtained from a simplified model cannot directly be applied to experimental setups. First, the model itself is wrong when the source-detector separation becomes small: the diffusion approximation is no longer valid. Furthermore, as mentioned earlier, the dynamic range of the detector was not considered. The case where the source and detector are at the same position cannot be experimentally implemented because the detector would be saturated by the peak of the TPSF.

This last remark is relevant for both plots. As mentioned at the end of Sec. 2.2, our noise model is true only for the chosen parameters in the region where the detectors do not saturate. We present this additional information in the figure by darkening the region where the theoretical model is not valid because of detector saturation. Measurements in this region of the graphs are possible, but the noise model is no longer valid because the experimental parameters must be changed.

The comparison between the CW and TD systems shows a very similar CNR, provided that the CW system is used at the optimum source-detector separation. From these simulations, we cannot expect TD systems to improve the CNR at the source-detector separations typically used (around 3 cm). This can be explained by the fact that TD systems yield a better contrast to deep activation, but are also limited by a higher noise level. At small source-detector separations (e.g., 1 cm), TD systems yield a better CNR at the optimum delay than CW systems. However, this might no longer be true in experimental conditions if we choose to resolve the whole TPSF, because the detector dynamic range will not enable us to measure the signal at the optimum delay without being saturated by the signal at the peak of the TPSF.

2.5.

CBR versus Depth of the Absorption Change

In this section, we study the CBR, which characterizes the ability of the system to distinguish the relevant information (activation in deep layers) from the signal clutter (background systemic physiological signals from the superficial layers). Figure 6 shows the evolution of the CBR with the depth of the inclusion and the source-detector separation. The good sensitivity of TD systems to deep activation despite the presence of a superficial background signal is put into evidence, and the improvement over CW techniques appears clearly (typically a factor of 10 for a 20-mm-deep activation). The deterioration of the CBR with the depth of the inclusion is dramatic in the CW case, whereas it is relatively slow for TD systems. This result is a quantitative description of the phenomenon evoked previously with the MC simulations in Fig. 1: CW systems mostly detect light from superficial layers, whereas TD systems can reject this light and select late photons, which have probed deeper in the tissue.

Figure 6

Evolution of the CBR as a function of the depth of the activation and of the source-detector separation for a cw system and a TD system. The CBR values are given in decimal logarithm.

2.6.

Evolution of Contrast, Noise, and Background with Delay for a Superficial or a Deep Activation

So far we presented the performances of both CW and TD systems as a function of the depth of the inclusion and of the source-detector separation. In experimental conditions, we measure an activation at a fixed unknown depth, and we have the option to vary the source-detector separation and the time-gate delay. In this section, we show the evolution of signal, contrast, CNR, and CBR with delay for a fixed depth of the activation. The source-detector separation is set to 2 cm. We showed previously that TD results are only weakly dependent on the source-detector separation. The results are presented in Fig. 7. We treat two different cases: a signal coming from a uniform superficial layer of thickness 5 mm (Fig. 7, top), and a cortical focal activation at a depth of 15 mm (Fig. 7, bottom). The behaviors in both cases are very different.

Figure 7

Evolution of signal, contrast-to-signal ratio, CNR, and CBR versus delay for a superficial activation (top) and a deep activation (bottom), as predicted by our analytical model.

The signal is the same in both cases, since we neglected the small influence of the inclusion. The contrast-to-signal ratio (or percentage change in the signal) increases with delay over more than 2 ns for a deep inclusion, whereas it saturates for a superficial activation. As we showed previously, the optimum delay for the CNR is strongly dependent on the depth of the inclusion. For a superficial activation, the CNR reaches its maximum a couple of picoseconds after the TPSF peak. For a deep inclusion, the maximum CNR is more than 1 ns after the maximum signal of the TPSF. Finally, we show the CBR variation with delay for only a deep activation. The CBR is no longer a relevant parameter for a superficial signal, since background and contrast represent the same value. For a deep activation, the CBR increases with delay, since late gates select photons coming from deep tissue and reject early light that is mainly responsible for the background.

This strong difference in the behavior with delay can be used experimentally to distinguish between a deep activation and a superficial one, as discussed in Ref. 20. These results are used in Sec. 3 to interpret our experimental results.

3.1.

TD Setup

We developed a TD system based on a pulsed laser and an ICCD camera (see Fig. 8). The source is a Ti:sapphire pulsed laser (MaiTai, Spectra Physics) emitting in the near-IR region (750 to 850 nm) with a repetition rate of 80 MHz. The laser beam is coupled into an optic fiber of diameter 400 μm, which delivers the light to the subject’s head. Various source positions can be used by scanning the laser beam over an array of fibers with a galvanometer. The diffuse light remitted by the tissue is detected through optical fibers which deliver light to the ICCD camera (PicoStar, LaVision) from the subject. The fiber tips are imaged on the intensifier and the amplified image is in turn recorded on a CCD camera. The intensifier acts as a fast shutter, whose opening is triggered relative to the laser pulses. The temporal gate width of the shutter was adjusted to 500 ps, and its delay relative to the laser pulse can be adjusted with a precision of 25 ps. The CCD camera was set to record the intensified image for 100 ms. Each camera frame corresponds to one relative delay between the laser pulse and the gate opening, and is the image of all the fiber tips simultaneously, as can be seen in the example shown in Fig. 8. The signal for each fiber is extracted from the CCD frame by spatially integrating the intensity on all pixels corresponding to this fiber. All the camera frames are recorded on a PC for later processing. This system enables a parallel recording of all the detectors, which are imaged at the same time on the CCD array. To record the TPSF on each detector, the delay must be scanned electronically, typically from 0 to 4 ns in 50-ps steps. The drawback of this method is the slowness of the electronic switching of the delay (∼0.5 s/step). The time required to record a complete TPSF (up to a couple of minutes) is incompatible with functional imaging where we want to study behaviors on the time scale of seconds.

Figure 8

Experimental time-gated setup and geometry of the probe used for the functional imaging. The PC screen shows a typical CCD frame where the 63 fiber tips are visible: Int, image intensifier; CCD, charge-coupled device camera; G, galvanometer.

To overcome this difficulty, we chose to add an optical delay to the electronic one. Each detector position consists of seven fibers of different lengths. In that way, the seven fibers record the same TPSF but with a relative delay depending on their length. This is illustrated in Fig. 9, which shows the seven normalized TPSFs corresponding to one detector at a distance 2 cm from the source, measured on a static phantom. It appears clearly that every fiber records the same information, with a delay of approximately 500 ps between two fibers, corresponding to the 10-cm increasing length of each fiber. If we fix the electronic delay, the seven fibers will see seven different parts of the TPSF simultaneously.

Figure 9

Normalized TPSFs recorded by seven fibers of different lengths positioned at the same distance of 2 cm away from the source on a static phantom.

3.2.

Brain Function Protocol

We performed experiments to demonstrate the ability of our system to measure the functional activity of the brain. We acquired measurements over the motor cortex and studied the hemodynamic response to a finger-tapping task. The study was approved by the Massachusetts General Hospital Institutional Review Board. All subjects gave written informed consent. The probe consisted of a single source adjacent to nine detectors positioned 1, 2, and 3 cm from the source. The configuration of the probe is shown in Fig. 8. To compensate for the large dynamic range of the light level over the different distances, we used fibers of diameter 62.5, 200, and 400 μm, respectively. The length increment between two fibers was also adapted to the distance between source and detector. Farther away from the source the measured TPSF is broader. Therefore, the time delay sampling points must be farther apart. We used length increments of 5, 7, and 10 cm, which correspond to delays of 250, 350, and 500 ps, respectively. The probe was located on the subject’s primary motor cortex, on the side contralateral to the tapping hand. For each subject, the experiment was divided in two parts: the preliminary adjustment part and the functional measurement during motor brain activation. In the preliminary stage, the probe is first adjusted on the subject’s head. We then record a complete TPSF by scanning the electronic delay, and adjust the intensifier gain so that the CCD camera is close to or slightly above saturation at the peak of the signal. This first recording also enables us to determine the optimum electronic delay that will give a good sampling of all the seven TPSFs recorded at each detector position. An experimental example is discussed shortly and shown in Fig. 10.

Figure 10

Experimental results for a source-detector separation of 2 cm: top, representation of the seven gates corresponding to the seven fibers; bottom, intensity change at each gate during the brain activation, after averaging over 16 successive blocks. The thick horizontal bar indicates the tapping task. The last curve of the line shows the estimated cw signal obtained by integrating all TD delay gates.

During the second part of the experiment, which is the activation measurement itself, we fix the electronic delay, and we “continuously” grab CCD frames at 8 Hz, i.e., as fast as the camera exposure time enables us. The measurement consists of a succession of runs, each run lasting between 5 and 10 min. During this time, the subject is alternatively performing a contralateral finger-tapping exercise for periods of 10 s, and then resting for periods of 20 s.

The raw data—the intensities for all fibers as a function of time—are low-pass filtered at 0.5 Hz (moving average filter over a 2-s window) to remove high-frequency noise and the heart beat oscillation. We then subtract the slow drifts of the signals, obtained using a moving average filter over a 50-s window. The signals are then averaged over the 20 successive blocks of motor stimuli. Due to our high-pass filter, early and late times usually present some artifacts and must be removed from the block averaging. The actual blocks taken into consideration are typically only the 16 successive middle ones for each 10-min run.

3.3.

Experimental Results

For better clarity, we show in Fig. 10 the typical averaged data obtained for one subject, recorded by the seven fibers at one detector position, 2 cm away from the source. The upper part of Fig. 10 shows the seven TPSFs recorded by the seven fibers during the preliminary part of the experiment, while the subject was resting. As shown previously on a phantom (Fig. 9), the seven fibers receive the same information, which is simply delayed by ∼350 ps as designed for the fibers placed at a 2-cm separation. The time gates used for data collection during brain activation are indicated on each of the seven TPSFs.

The lower portion of the figure shows the block-averaged result of the activation measurements from averaging 16 successive stimuli. For each time gate, we plot the percentage change in the signal versus time. The error bars represent the noise, calculated from the standard deviation of the raw intensity for each fiber. The 10-s thick horizontal bar indicates the duration of the stimulus. The response to the brain activation appears clearly with a good CNR: the measured intensity decreases from 1 to 6% during the motor task, increasing with longer time-gate delays, and goes back to its initial level approximately 10 s after the end of the stimulus.

To compare our results with those given by a CW system, we estimated the CW signal by integrating the TD signal over all delays. We expected this CW estimation to give performances inferior to what a designated CW system would achieve. However, this estimation based on the time integration of TD data gives an SNR of 500 for the CW system. We therefore believe that our approximation of the cw signal is a fair estimation.

3.3.1.

Variation of signal and contrast with delay

It appears clearly in Fig. 10 that the response to the motor task increases with late delays. This result is summarized on Fig. 11(b), where we plotted the evolution of the contrast-to-signal ratio with delay. For reference, the evolution of signal with delay is shown in Fig. 11(a). The signal is the mean value of the intensity over the time of the experiment. The contrast was calculated as follows. We defined the baseline signal, which is the average of the intensity during the 5 s before the tapping task, and the activation signal, which is the average of the intensity during the last 5 s of the stimulus. We only take into account the last 5 s of the two states to remove the transition states. The contrast-to-signal ratio is given by the difference between the activation signal and the baseline signal, divided by the baseline signal.

Figure 11

Experimental results for a source-detector separation of 2 cm: (a) variation with the time delay of the signal; (b) the contrast-to-signal ratio, i.e., the percentage change during the activation task; (c) the CNR, and (d) the CBR. The values of these parameters for the estimated cw experiment are plotted as dashed lines in all graphs.

The contrast-to-signal ratio increases with delay, which is characteristic of a deep activation, as was shown in Sec. 2.6 and Fig. 7. A superficial activation would lead to a saturation of the contrast-to-signal ratio with delay. The contrast-to-signal ratio at the last delay presents a high standard error, because the signal level on the CCD camera is only a few counts. For comparison, the equivalent contrast-to-signal ratio for a CW system is plotted as a dotted line in the same figure. The maximum contrast for the TD is more than twice that of the CW domain. As a matter of fact, the CW contrast is mainly due to the early high-signal gates (delays 2 and 3), where the contrast is relatively small.

3.3.4.

Depth discrimination of brain activation and superficial clutter

We observed previously that the contrast-to-signal ratio is increasing with later delays. On the contrary, other features in the response to the stimulus are constant with delay. In particular, we observe a secondary peak 10 s after the cessation of the stimulus, indicated with a small arrow on Fig. 10, whose amplitude is the same at all time gates. This type of 10-s oscillatory signal was discussed previously in the literature^{38 39} and was suggested to arise from an intrinsic blood pressure oscillation that phase locks to a stimulus. Given that this response is believed to be a systemic blood pressure oscillation, the signal origin is likely to arise more superficially in the scalp than intracranially. This secondary peak has an almost constant amplitude with delay, which is characteristic of a superficial signal, as was shown on Fig. 7(b) in Sec. 2. It thus appears clearly that our measured response to the stimulus is the sum of contributions from signals originating from different depths. Their variation with delay gives information about their depth, and therefore led us to implement a depth reconstruction of the signal.

The capability of TD systems to achieve depth resolved measurements was shown previously by Steinbrink et al.²⁴ They presented a 15-layer model of the brain to deduce depth-resolved absorption changes during motor stimulation. They used MC simulations to compute the changes in intensity at different time gates for an absorption change in each layer. We used a simpler model consisting of three layers of respective thickness 5 mm (scalp), 7 mm (skull), and 50 mm (brain, large enough so that we consider the medium to be semi-infinite). At a particular delay τ, the percentage change in the measured optical density (OD) is given by the sum of the contributions in the three layers:

Eq. (8)

$$\Delta \text{OD}(\tau )=l_1(\tau )\Delta {\mu }_{a,1}+l_2(\tau )\Delta {\mu }_{a,2}+l_3(\tau )\Delta {\mu }_{a,3},$$

where Δμ_a,i is the change in absorption in the i’th layer, and l_i(τ) is the partial path length in the i’th layer at delay τ. Writing Eq. (8) for all the seven delays leads to the matrix form:

Eq. (9)

$$\Delta OD=\mathbf{L}\cdot \Delta {{\mu }}_a,$$

where Δμ _a is the 3×1 vector of the absorption changes in the three layers, and ΔOD the 7×1 vector of the intensity changes at all delays. The matrix L was obtained through finite difference simulations as described in Sec. 2.2.4, adapted to a three-layer model. Given the measurements ΔOD, the least-squares solution of Eq. (9) is given by

Eq. (10)

$$\text{Δ}{\mu }_a={({\mathbf{\text{L}}}^T\cdot \mathbf{\text{L}})}^{-1}\cdot {\mathbf{\text{L}}}^T\cdot \text{Δ}\mathbf{\text{OD}}.$$

Figure 12 shows the changes in the absorption coefficient for the three layers versus time deduced from the experimental data in Fig. 10. The first layer shows a large oscillatory change in the absorption coefficient, probably a systemic response to the stimulus. The absorption coefficient in the second layer shows very little variation with time. The third layer, corresponding to the cortex, shows a change in the absorption coefficient that rises during the 10-s stimulus and then slowly returns to baseline following the stimulus. Importantly the peak that was observed 10-s poststimulus has been resolved in the upper (scalp) layer of the model, indicating the discrimination of superficial systemic signals from the cerebral signals. This discrimination should lead to a more accurate estimate of the cerebral absorption change with a better CBR. A more detailed experimental investigation of this possibility is left for future work.

Figure 12

Change in the absorption coefficients of the first 5-mm-thick layer (dotted line), the second 7-mm-thick layer (thin solid line), and the semi-infinite third layer (thick solid line) during the finger-tapping exercise. The absorption changes were calculated for all time points, then block-averaged over 16 blocks. The thick horizontal bar indicates the tapping task.