3.1.

Monte Carlo Simulation

To determine the cross talk using absorption changes in a layered model of the adult human head, MC simulations were performed.

The simulations were calculated using a model with homogeneous layers labeled by j. The medium was composed of ten layers with the upper nine layers having a thickness of 2 mm and the deepest layer being semi-infinite. Either all layers had the same reduced scattering coefficient μ_s,j ^′ (1 or 2 mm⁻¹) or they had a wavelength dependent μ_s,j ^′ as calculated for the adult head by Matcher, Cope, and Delpy¹⁴ with values given by μ_s ^′(λ)=a⋅λ+b (a=−6.5×10⁻⁴ mm ⁻¹ nm ⁻¹, b=1.45 mm ⁻¹ ). In the homogeneous head model all layers had the same (wavelength dependent) absorption coefficient μ_a,j. For μ_a,j 19 different values from 0.010–0.070 mm⁻¹ were used to cover the range expected over the NIR spectral band used in tissues.

To investigate focal changes we used the homogeneous head model and introduced absorption changes in spheres of radius 2 mm (sphere model), which were positioned at various depth and distances with respect to source and detector. In addition to the homogeneous head model, the more complex head model of Okada et al.;² was investigated.

Scattering anisotropies were disregarded. We assumed the same refractive index n=1.4 for all layers. Fresnel reflection at the tissue–air boundary was disregarded, i.e., the boundary was totally absorbing.

The MC simulation based on the variance reduction technique is described elsewhere in detail.^{8 10} In brief, the pathlength in each layer and the survival weight were calculated for each photon. The photons were detected in concentric rings differing by 3 mm in diameter. The middle radius r of these rings was varied from 6 to 60 mm. For focal changes photons were detected in a ring segment with a fractional outer circumference of 2 mm length. The time-independent partial pathlength l_j for every layer was obtained by integrating the product of the time-dependent photon pathlengths and the normalized distribution of time of flight (for the details of the averaging method see Ref. 8).

For all layers the standard deviation of l_j was determined for different μ_a values by performing 30 runs of the MC simulation and was found to be smaller than 1.5 for a source-detector distance of 30 mm. The standard deviation of MTSF _j was determined to be smaller than 7 for the layers in the depth from 4 to ∞ mm.

For the homogeneous head model 4.2×10⁷ photons were injected. For the more complex head model based on Okada et al.;,² 2×10⁸ and for the sphere model 2.5×10⁹ photons were used.

3.2.

μ _a −λ relationship

The composition of the model tissue has to be assumed because for a spectroscopic analysis l_j(λ) rather than l_j(μ_a) is needed.

A water fraction of 90 (vol.) was assumed, which is the average water concentration in the neonatal brain.⁷ Because of the small μ_a of water in the so-called “optical window” (700–900 nm) a variation of the water concentration has little effect on the resulting μ_a of the tissue. For simplicity other chromophores than hemoglobin like fat and lipids were neglected.⁷

The total hemoglobin concentration of the brain is approximately 100 μM.^{7 14} This value was adopted in the model. For the oxygen saturation Y=Oxy-Hb/(Oxy-Hb+Deoxy-Hb) values between Y=50 and 90 were regarded. Here Oxy-Hb and Deoxy-Hb symbolize the concentration of the oxygenated and deoxygenated hemoglobin. The resulting absorption coefficient μ_a is consequently

The μ_a,water was calculated from water extinction coefficients given by Ref. 15. The ϵ _Oxy-Hb and ϵ _Deoxy-Hb were taken from Ref. 16. In Figure 1(a) the resulting wavelength dependent absorption coefficients are shown for the different saturation values.

Figure 1

(a) Absorption coefficient μ_a and (b) calculated DPF for different oxygen saturations Y as a function of wavelength λ. For the tissue model a total hemoglobin concentration of 100 μM and μ_s ^′=1 mm ⁻¹ was assumed. To allow a comparison of the DPF=L/r with experimental values taken from Essenpreis et al.;¹² (open squares) a source-detector distance of r=40 mm was used.

The relative maximum of the extinction of deoxygenated hemoglobin around 760 nm can be seen in the resulting μ_a spectra. The values of μ_a at 800 nm ( μ_a∼0.021 mm ⁻¹ ) are in agreement with the literature, where values ranging from 0.016 to 0.021 mm⁻¹ are reported for the adult head.^{9 14} Bevilacqua et al.;¹⁷ measured absorption coefficients of various cortical tissues including skull and found μ_a values in the range 0.018–0.022 mm⁻¹.

A direct way to check the assumed tissue parameters is the comparison of experimental time-of-flight measurements with total pathlengths calculated from the model. The pathlength divided by the source-detector distance (called differential pathlength factor DPF=L/r ) is frequently used in the literature.

Experimental DPF values taken from Ref. 12 for r=40 mm and the calculated DPF from the model tissue using the same r and μ_s ^′=1 mm ⁻¹ are shown in Figure 1(b). The measured and the calculated wavelength dependence of the DPF are similar, especially for saturation values under physiological conditions (Y=70–90). The magnitude of the measured DPF is about 20 greater than the calculated one, however, this deviation is smaller than the intraindividual variations.¹²

4.1.

Relative Partial Pathlength ( l _j ^*) and Relative Partial Mean Time Sensitivity Factor (MTSF _j ^*)

Figure 2 shows the relative partial pathlength l_j ^* and the relative partial mean time sensitivity factor MTSF _j ^* as a function of the absorption coefficient μ_a. The quantity MTSF _j ^* is the ratio of the partial to total mean time sensitivity factor. Here a commonly used source detector distance of r=30 mm was chosen; μ_s ^′=1 mm ⁻¹ was used. The different layers are indicated by different symbols. The absorption coefficient was varied between μ_a=0.010 and 0.0360 mm⁻¹ which is approximately the range given in Figure 1(a). Based on these values and a fifth order polynomial, l_j ^* and MTSF _j ^* were approximated as a function of μ_a.

Figure 2

(a) Relative partial pathlength l_j ^* and (b) relative partial mean time sensitivity factor MTSF _j ^* as a function of absorption coefficient μ_a for different layers of 2 mm thickness. For the calculations μ_s ^′=1 mm ⁻¹ and r=30 mm are assumed.

The objective of this publication is focused on cortical absorption changes in the adult head, thus l_j ^* and MTSF _j ^* in the layers at depths of 8–12 mm are of special relevance. In this region of interest l_j ^* varies up to a factor of 2 and MTSF _j ^* by less than 25 within the range of regarded μ_a. MTSF _j ^* is very small for the layer from 0 to 2 mm and the noise level is consequently high. Therefore no concentration changes of the chromophores are determined from Δ〈t〉 for this layer.

4.2.

Homogeneous Layered Model

Figure 3 shows the determined concentration changes for induced changes of 1 μM in Oxy-Hb [Figure 3(a)] and Deoxy-Hb [Figure 3(b)]. The corresponding cross talk is plotted in Figures 4 and 5. To compare the depth sensitivity of intensity and mean time spectroscopy, chromophore changes were derived from both ΔA and Δ〈t〉. The induced changes were restricted to single layers, which are indicated by their corresponding mean depth. Based on the standard deviation of l_j (or MTSF _j ) an uncertainty in the calculated concentration changes smaller than 3 (or 14) is estimated.

Figure 3

Determined ΔOxy-Hb and ΔDeoxy-Hb from changes in attenuation ΔA and mean time-of-flight Δ〈t〉 for induced (a) ΔOxy-Hb=1 μM and (b) ΔDeoxy-Hb=1 μM. The entire wavelength range from 700 to 900 nm (multi λ) or only two wavelengths at 760 and 830 nm (2 λ) are used. The assumed saturation is Y=70. Note that in (a) ΔDeoxy-Hb and in (b) ΔOxy-Hb are scaled by a factor of 5. The gray box indicates the most likely depth for cortical activity.

Figure 4

Cross talk determined from changes in attenuation ΔA in as a function of depth of the activated layer with induced (a) ΔOxy-Hb or (b) ΔDeoxy-Hb=1 μM for different oxygen saturations Y using wavelengths at 760 and 830 nm; μ_s ^′=1 mm ⁻¹ was assumed. For the case Y=70 additionally a wavelength dependent μ_s ^′(λ) was chosen. The gray box indicates the most likely depth for cortical activity.

Figure 5

Cross talk determined from changes in mean time-of-flight Δ〈t〉 in as a function of depth of the activated layer with induced (a) ΔOxy-Hb or (b) ΔDeoxy-Hb=1 μM for different oxygen saturations Y using wavelengths at 760 and 830 nm; μ_s ^′=1 mm ⁻¹ was assumed. For the case Y=70 additionally a wavelength dependent μ_s ^′(λ) was chosen. The gray box indicates the most likely depth for cortical activity.

For the analysis we used either the entire wavelength range (700–900 nm) or two wavelengths commonly used for hemoglobin spectroscopy (760 and 830 nm). While in the latter case Eqs. (6) and (8) give an analytical solution, for the multi-wavelengths case a least squares fitting routine was employed to calculate concentration changes from spectra of ΔA(λ) or Δ〈t〉(λ). The scattering coefficient was set to μ_s ^′=1 mm ⁻¹ and a saturation of Y=70 was assumed.

For the case of the entire wavelength range, induced changes in the region of interest (8–12 mm) in Oxy-Hb of 1 μM [Figure 3(a)] leads to determined ΔDeoxy-Hb up to ∼0.006 μM and to determined ΔOxy-Hb up to ∼0.12 μM. The cross talk for these layers is below ∼11. Using only the two mentioned wavelengths for the analysis, this cross talk is about 2. For induced changes in Deoxy-Hb of 1 μM and focusing on the same layers (8–12 mm), using the entire wavelength range the magnitude of the cross talk is below ∼17. Again, based on the two wavelength analysis, the magnitude of the cross talk (∼3) is much smaller.

When determined from Δ〈t〉 rather than ΔA, the maximum of ΔOxy-Hb as well as the minimum of the magnitude of the cross talk are in deeper layers. In the layers from 2 to 8 mm the magnitude of the cross talk calculated with Δ〈t〉 is bigger than calculated with ΔA, for deeper layers it is smaller (see Figures 4 and 5).

Using ΔA, the cross talk determined for two wavelengths 760 and 830 for induced changes in Oxy-Hb [Figure 4(a)] and Deoxy-Hb [Figure 4(b)] are different in magnitude and sign. For 70 saturation the cross talk is almost zero for all layers because the difference in the absorption coefficient is small [compare with Figure 1(a)]. For saturations of 80 and 90 the cross talk is significant in deep layers for an induced change in Oxy-Hb [Figure 4(a)]. Given a change in Deoxy-Hb, the cross talk is high for saturation values of 50, 80 and 90 [Figure 4(b)]. At a depth of approximately 7 mm, the cross talk is close to zero almost independent of the saturation value.

The cross talk determined with Δ〈t〉 based on 760 and 830 nm is shown in Figure 5 for induced changes in Oxy-Hb and Deoxy-Hb. For each saturation value the magnitude of the cross talk in the region of interest (8–12 mm) is smaller than determined with ΔA. As the cross talk depends on the differences in absorption coefficient [Figure 1(a)], partial pathlength and partial mean time sensitivity factor (Figure 2), the cross talk is crucially dependent on the wavelength combination and the saturation value.

When including a reduced scattering coefficient μ_s ^′(λ) which linearly decreases with wavelength (see Method section) and assuming a saturation of Y=70, for induced Oxy-Hb [Figures 4(a) and 5(a)] and Deoxy-Hb changes [Figures 4(b) and 5(b)] the magnitude of the cross talk for all layers is smaller than 4. For the region of interest (8–12 mm) the magnitude of the cross talk determined from ΔA (or Δ〈t〉 ) with wavelength dependent μ_s ^′ is smaller than determined with a constant μ_s ^′.

Changing the magnitude of the scattering coefficient from μ_s ^′=1–2 mm ⁻¹ has only an insignificant effect on the cross talk (data not shown). For a greater source-detector separation r, the sensitivity of attenuation and mean time-of-flight measurements is in deeper layers. However, the magnitude of the cross talk remains in the same order (data not shown).

So far, only changes in Oxy-Hb and Deoxy-Hb were considered. The question arises if determined concentration changes in cytochrome-c-oxidase (ΔCyt-Ox) can be due to the cross talk artifact. During visual stimulation experiments using an analysis based on MLBL, an increase in Oxy-Hb, decrease in Deoxy-Hb and a simultaneous increase in Cyt-Ox was obtained.^{18 19} To cover the observed concentration changes in our model, we induced different values of ΔOxy-Hb (0–4 μM) while the decrease in ΔDeoxy-Hb was fixed at −1 μM. Again, the assumed saturation is 70 and μ_s ^′=1 mm ⁻¹ . From the attenuation changes (700–900 nm) induced by the hemoglobin changes, ΔOxy-Hb, ΔDeoxy-Hb as well as ΔCyt-Ox were calculated with Eqs. (2) and (3). In Figure 6 the determined ratio of ΔCyt-Ox/ΔDeoxy-Hb from changes in (a) attenuation ΔA and (b) mean time of flight Δ〈t〉 is shown as a function of the depth of the activated layer.

Figure 6

Ratio of determined Δ Cyt-Ox to ΔDeoxy-Hb from changes in (a) attenuation ΔA and (b) mean time-of-flight Δ〈t〉 as a function of depth of the activated layer for different values of induced ΔOxy-Hb and a fixed ΔDeoxy-Hb of −1 μM. Y=70, μ_s ^′=1 mm ⁻¹ and a source-detector distance of r=30 mm is assumed. A wavelength range 700–900 nm is considered. The gray box indicates the most likely depth for cortical activity.

For induced ΔOxy-Hb=1 μM the determined ΔCyt-Ox/ΔDeoxy-Hb is small for all layers—except for the semi-infinite layer determined from Δ〈t〉. At the depth of ∼6 mm (∼9 mm) ΔCyt-Ox/ΔDeoxy-Hb determined from ΔA (Δ〈t〉) is zero for all values of induced ΔOxy-Hb. For a pure Deoxy-Hb change in the tissue (ΔOxy-Hb=0 μM), a spurious Cyt-Ox change determined from ΔA as well as from Δ〈t〉 up to 20 of the Deoxy-Hb change is estimated for cortical layers (8–12 mm). For larger ΔOxy-Hb, the derived magnitude of the ratio ΔCyt-Ox/ΔDeoxy-Hb is even bigger.

Using the same model and an analysis with different combinations of three or four wavelengths, the magnitude of the determined ratio ΔCyt-Ox/ΔDeoxy-Hb is bigger than 30 for at least one of the induced Oxy-Hb values (data not shown).

4.3.

Focal Changes

In the models considered so far, the hemoglobin changes are restricted to whole layers of the head. This is certainly a coarse assumption for functional activations for which focal changes are more realistic. For this reason we also investigated induced changes in spheres. For the data of Figure 7 the center of the spheres of 2 mm radius are located at a depth of 10 mm. The different spheres are displaced 0–10 mm perpendicular to the source-detector connection line. For zero displacement the sphere is positioned under the midpoint of this connection line (source detector distance 27 mm). The same wavelengths as above (760 and 830 nm) and a saturation value of Y=70 are used. In Figure 7 ΔOxy-Hb and ΔDeoxy-Hb determined from changes in attenuation ΔA and mean time-of-flight Δ〈t〉 are presented induced by ΔOxy-Hb=1 μM in the spheres. The results for induced changes in Deoxy-Hb are qualitatively similar and therefore not shown. With increased displacement the observed concentration change gets smaller. ΔOxy-Hb determined from Δ〈t〉 is ∼1.5 times greater than determined from ΔA for zero displacement and ∼3 times greater for 10 mm displacement. The corresponding cross talk determined from ΔA (Δ〈t〉) is smaller than 2.5 (6) for all displacements chosen.

Figure 7

Determined changes of Oxy-Hb and Deoxy-Hb from changes in attenuation ΔA and mean time-of-flight Δ〈t〉 induced by ΔOxy-Hb=1 μM in a sphere as a function of the displacement perpendicular from the line between source and detector. The center of the sphere of radius 2 mm is located at a depth of 10 mm. For zero displacement the sphere is positioned under the midpoint of the line connecting source and detector (distance 27 mm). Y=70, μ_s ^′=1 mm ⁻¹ is assumed. The wavelengths used are 760 and 830 nm. Note that ΔDeoxy-Hb is scaled by a factor of 5.

4.4.

Layered Brain Model

Above, the brain is modeled in a crude fashion using the same optical properties in all layers. Okada et al.;² have developed a more realistic model including different absorption and scattering properties for the different tissues of the head. We adopted this model with the optical properties given in Table 1. For μ_a of the cerebrospinal fluid (CSF) we used the absorption coefficient of water at 830 nm instead of 0.001 mm⁻¹.²

Table 1

For this head model, the cross talk was calculated at two wavelengths (760 and 830 nm). The wavelength dependence of μ_a was included by assuming that the absorption coefficients of Table 1 are valid at λ=830 nm and due to water and the two hemoglobin fractions. Based on the same water content (90) and the same saturation Y=70, absolute hemoglobin concentrations were derived for each layer and subsequently absorption coefficients at 760 nm: μ_a=0.0381 mm ⁻¹ for skin/skull, 0.0236 mm⁻¹ for gray matter and 0.0046 mm⁻¹ for white matter. Again for CSF the absorption coefficient of water (0.0029 mm⁻¹) was used. A wavelength independent μ_s ^′ was assumed.

In Figure 8 determined ΔOxy-Hb and ΔDeoxy-Hb values from ΔA and Δ〈t〉 are shown for induced changes ΔOxy-Hb=1 μM. (For induced ΔDeoxy-Hb=1 μM similar results were obtained.) Although a change of 1 μM in the CSF layer is not a physiological plausible value, here it was chosen for consistency. As before for the layered homogeneous model, the magnitude of the calculated changes reflects the pathlength contribution of the different layers. The Oxy-Hb signal determined from ΔA peaks for the CSF. This is explained by the “light pipe” properties of the CSF.² For the skin/skull and CSF layers the determined ΔOxy-Hb from ΔA is greater than determined from Δ〈t〉, for the other layers it is smaller.

Figure 8

Determined changes in Oxy-Hb and Deoxy-Hb from changes in attenuation ΔA and mean time-of-flight Δ〈t〉 in an inhomogeneous layered brain model and induced ΔOxy-Hb=1 μM. Wavelengths at 760 and 830 nm and for source-detector distance r=27 mm were used. The optical properties of the layers are given in Table 1 for 830 nm. Note that ΔDeoxy-Hb is scaled by a factor of 5.

The magnitude of the cross talk determined from ΔA is smaller than 3 for skin and skull and the upper part of the gray matter (depth 12–14 mm). For deeper layers it goes up to 14. However, as the magnitude of the determined chromophore changes is very small for these layers, this cross talk is probably of no significance in real measurement conditions. The magnitude of the cross talk determined from Δ〈t〉 is smaller than 4 for all layers.

To test for possible Cyt-Ox artifacts, four wavelengths (750, 830, 850 and 925 nm) were included in the analysis. These are the same used by Heekeren et al.;¹⁸ and close to those of a commercial NIRS monitor (NIRO-500, Hamamatsu Photonics K.K., Japan). For 750, 850, and 925 nm the absorption coefficient was derived from Table 1 in the same fashion as for 760 nm. Inducing different ratios of ΔOxy-Hb and ΔDeoxy-Hb (as in Figure 6) in the upper gray matter layer returns a ratio ΔCyt-Ox/ΔDeoxy-Hb determined from ΔA (Δ〈t〉) which is smaller by a factor of 2 (4) compared to using the homogeneous layered model same depth 12–14 mm and same wavelengths (data not shown).