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Simulation of Near-Infrared Light Absorption Considering Individual Head and Prefrontal Cortex Anatomy: Implications for Optical Neuroimaging

Functional near-infrared spectroscopy (fNIRS) is an established optical neuroimaging method for measuring functional hemodynamic responses to infer neural activation. However, the impact of individual anatomy on the sensitivity of fNIRS measuring hemodynamics within cortical gray matter is still unknown. By means of Monte Carlo simulations and structural MRI of 23 healthy subjects (mean age: years), we characterized the individual distribution of tissue-specific NIR-light absorption underneath 24 prefrontal fNIRS channels. We, thereby, investigated the impact of scalp-cortex distance (SCD), frontal sinus volume as well as sulcal morphology on gray matter volumes () traversed by NIR-light, i.e. anatomy-dependent fNIRS sensitivity. The NIR-light absorption between optodes was distributed describing a rotational ellipsoid with a mean penetration depth of considering the deepest of light. Of the detected photon packages scalp and bone absorbed and absorbed of the energy. The mean volume was negatively correlated () with the SCD and frontal sinus volume () and was reduced by in subjects with relatively large compared to small frontal sinus. Head circumference was significantly positively correlated with the mean SCD () and the traversed frontal sinus volume (). Sulcal morphology had no significant impact on . Our findings suggest to consider individual SCD and frontal sinus volume as anatomical factors impacting fNIRS sensitivity. Head circumference may represent a practical measure to partly control for these sources of error variance.


Introduction
Functional near-infrared spectroscopy (fNIRS) is an optical neuroimaging technique increasingly used to investigate neural processing in healthy subjects and in psychiatric patients [1]. fNIRS relies on the differential absorption of near-infrared (NIR) light by oxygenated and deoxygenated hemoglobin (O2-Hb; H-Hb) and the change in the concentration of O2-Hb and H-Hb mediated by the neurovascular coupling following neural activation. However, other molecules in biological tissue such as melanin also absorb NIR-light [2], but since their concentrations are constant the relative change in NIR-light absorbance during the hemodynamic responses is not impacted by these chromophores [3]. However, due to the highly scattering properties of the heterogeneous biological tissues, a plethora of paths of the emitted NIR-light are possible. As a consequence of NIR-light scattering, only about 10 {3 to 10 {4 of the light reaches the detector, which is commonly positioned in a distance of 3 cm on the head surface of adults. The average path of the emitted NIR-light has been reported to describe a ''banana-shaped'' [4,5] or ellipsoid path [6] with an approximate head penetration depth of about 2 cm to 3 cm [4,6]. Importantly, fNIRS is a reliable technique [7][8][9] to indirectly detect and quantify neural activity by measuring changes in NIR-light absorption due to O2-Hb and H-Hb concentration changes [3,10]. Neural signals are mainly integrated, processed and reliably coupled to functional hemodynamic responses within gray matter. Thus, functional neuroimaging commonly focusses on this tissue type.
NIR-light penetration depth has been shown to depend on the scalp-cortex distance (SCD) using spatial distributions of correlations between hemodynamic responses concurrently measured with fNIRS and functional magnetic resonance imaging (fMRI) [6]. However, an impact of individual anatomy and tissue composition on the traversed gray matter volume and its variability between subjects as well as fNIRS-channels, has not been investigated yet. Moreover, while fNIRS measurements of functional hemodynamics within anterior orbitofrontal cortex (OFC) have been conducted [11], the feasibility of fNIRS to measure functional hemodynamics within gray matter underneath frontal sinus sites is still unclear.
The frontal sinus is a fronto-polar air-filled cavity, whose morphology is highly inter-individually variable and may differ among races (mean volume standard deviation (SD): (7:4+5:5) cm 3 , assessed in adult Caucasians [12]; (3:5+0:8) cm 3 assessed in (young) adult Asians [13]). The frontal sinus is mostly located rostrally to parts of the OFC. Since air-filled frontal sinus has previously been suggested to affect NIR-light propagation [14,15], we systematically assessed the relationship between frontal sinus volume and NIR-light penetration depth as well as gray matter volume absorbing NIR-light.
In order to provide a numerical solution for the forward problem of photon migration in highly scattering biological tissues, we implemented a Monte Carlo simulation (MCS) of a large number of photon packages. MCS have repeatedly been used to describe characteristics of photon scattering and migration using head or tissue layer models [16][17][18][19]. Also, MCS has been applied to segmented MRI data to compute fNIRS sensitivity spread functions within the visual cortex [20][21][22]. Hitherto, a MCS of spatial tissue-specific distributions of NIR-light absorbance in the prefrontal cortex has not been conducted.
The MCS implemented in our study considered the following basic principles. The light attenuation between an emitter-detector pair is decribed by A~{ln(I de =I em ) where I em is the intensity of light eradiated by the emitter and I de is the intensity measured by the detector. The rate of absorption and scattering of a certain tissue t depends on the absorption coefficient m at and the scattering coefficient m st . Derived from the Radiation Transport Equation the Beer-Lambert law can be used to calculate the attenuation A between arbitrarily arranged optodes in an optically homogeneous medium (m a (r r)~m a ) [23], where SLT is the mean pathlength of all photons and A 0 is the fraction of attenuation solely caused by scattering and not by absorption [24]. The total attenuation of light traveling through different homogeneous tissue types, can be calculated by the sum of the partial attenuations. The differential of the total attenuation over N (t~1 . . . N) tissue types is proprotional to fNIRS signal changes during a measurement and can be described by: Neural activity is followed by a local increase in blood flow, predominantly in gray matter, which relatively increases O 2 -Hb and decreases H-Hb concentrations in the blood, thereby changing the absorption coefficient of a given wavelength and thereby the NIR-light attenuation dA. However, dA is proportional to the mean pathlength of the light SL gray T through gray matter. Thus, the sensitivity of fNIRS to detect changes in O 2 -Hb and H-Hb concentrations directly depends on the traversed gray matter volume and the mean partial pathlength through this tissue type. In the present study, we applied a MCS to 24 prefrontal channel positions of a typical fNIRS probe-set using threedimensional segmented structural MRI data of 23 subjects. We simulated channel-wise NIR-light spatial absorbance maps to systematically quantify the impact of forehead anatomy on gray matter volumes traversed by the NIR-light before detection, i.e. the fNIRS sensitivity. Specifically, we studied the impact of individual and inter-channel anatomical variability of frontal sinus, SCD as well as sulcal geometry on these gray matter volumes.
Slices through three individual channel cubes (subject #20) exemplify the spatial distribution of different tissues and NIR-light absorbing voxels in (figure 1a, b). Slices of channel-cubes show an absorption pattern similar to an ellipsoid shape as slices move through the emitter-detector plane. The simulated individual absorption distributions reached gray matter in all channel-cube simulations, except in one case (subject #14, channel #18), and had similar spatial characteristics compared to the layer model simulations.
Based on the assumption of a rotational ellipsoid traverse of NIR-light through the tissue [6], we derived a function describing the relationship between SCD and V gray . Here, V gray is represented by the ellipsoid calotte and V gray as a function of SCD is solely dependent on the ellipsoid parameter a (see scheme of figure 2). V gray (SCD) can thus be calculated by: where b is the ellipsoid radius at the optode level and E~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 {b 2 p =a is the ellipsoid function.
The parameter a is a function of the cumulative probability distribution (P depth ) of the spatial distance of absorbing voxels within the channel-cubes (figure 3). The cumulative absorption distribution with increasing penetration depth of all channel-cubes is shown in (figure 4). We used these distributions to gain a value of a producing the best fit of the function V gray (SCD) to the data, which was obtained for a~20:3mm (i.e., the mean distance of the 82:6% most proximal absorbing voxels calculated over all subjects and channels). This value of a generating a rotational ellipsoid best describing the traversed depth of all channels and subjects we refer to as ''relevant'' penetration depth (D relevant ). The 5% (arbitrary threshold) most distal reaching pathways have a mean penetration depth of (23:6+0:7)mm.
However, since mere distance from scalp to cortex may not fully explain the variance of V gray , we investigated, if (1) scalp, skull and CSF, respectively, and (2) SCD-dependent proportional changes in tissue type compositions would underlie the profound reduction in V gray as SCD increases. First, we used a stepwise multiple linear regression of V gray and the tissue type layer thickness as independent variables. Layer thickness of skull (b~{:61,pv10 {4 ), scalp (b~{:43,pv5 : 10 {3 ) as well as CSF (b~{:37,pv:05) layer thickness significantly predicted V gray . Second, we plotted the mean depth distribution of scalp, scull, CSF and gray matter tissue against SCD. The linear increase in CSF layer thickness with increasing SCD had a slope twice as steep compared to scalp and skull (figure 5). Previously, the presence of low scattering CSF has been shown to increase the partial pathlength through brain tissue compared to a head model with the same SCD but lacking CSF [18]. Therefore, the ellipsoid callote model as a function describing the relationship between SCD and V gray (figure 6) may be affected by the overproportional increase in CSF layer thickness d csf with increasing SCD. To test this, we correlated the difference between simulated (MCS) and analytically determined (ellipsoid function) V gray with the proportional ratio of CSF/(scalp+skull) layer thickness. These variables were significantly correlated (r~:46,pv:05; figure 6). Thus, the overproportional increase in CSF may attenuate the reduction of V gray with increasing SCD.

Frontal sinus and NIR-light absorption
In order to investigate the feasibility of fNIRS measuring functional hemodynamics within neural structures behind the frontal sinus, we assessed the impact of its volume on the NIR-light penetration depth as well as V gray generated by the MCS.
Derived from the relationship between cumulative absorption distribution and penetration depth, we assessed the distance of the 82:6% of absorbing voxels for each channel and subject to relate this distance (D relevant ) to the volume of air-filled frontal sinus traversed by the simulated photon packages. In channel #1 these variables were correlated (r~:48,pv:05); i.e. the larger the frontal sinus the deeper the penetration.

Gyrification and NIR-light absorption
To investigate the potential influence of gyrification and sulcal geometry we calculated the mean standard deviation of gray matter surface vectors in x, y, z direction, respectively. The lenght of the resulting standard deviation vector indicated the magnitude of gyrification. We selected all gray matter surface voxels within ellipsoid volumes, which were individually selected by a~SCDz5mm. Three examples of gray matter surface vector constellations, ellipsoid slices and the corresponding standard deviations are given in (figure 7B). The gyrification estimates for each fNIRS channel (averaged over subjects) are shown in   . The relationship between scalp to cortex distance (SCD) and light absorbing gray matter volume V gray averaged over channels (crosses) or averaged over subjects (asterisks). In order to define a curve (continuous line) describing this relationship, we fit the depth parameter a(P depth ) from the analytical expression for V gray to the data, whereby P depth is the percentage of light absorption beyond a certain depth according to figure 4. For a(P depth~1 7:4% the best fit (least square) of the ellipsoid is achieved, which corresponds to 20:3mm referred to as the relevant penetration depth D relevant . doi:10.1371/journal.pone.0026377.g003 (figure 7A). Channel #4 and #11 located over the interhemispheric sulcus as well as channel #2 (probably over the orbital part of the middle frontal gyrus) showed increased gyrification. The mean gyrification estimate per channel was not correlated with the corresponding mean V gray (r~{:16,pw:1).

Discussion
In the present study, we implemented a Monte Carlo simulation of NIR-light traversing individual forehead tissue compositions. We investigated the impact of individual and inter-channel variability in anatomy on the tissue-specific distribution of NIRlight absorption from emitter to detector. The present study replicates two previous findings: (1) While NIR-light traverses biological tissues of the head (and is partly absorbed),the average pathway of the light and the average absorption distribution are best described by an elliptical curve [6] [16]. (2) Penetration depth of NIR-light (of about 2 cm to 3 cm at 3 cm interoptode distance) is negatively correlated with SCD, and SCD increases from right lateral to medial PFC [6].
Moreover, this study shows that (3) not only penetration depth, but also the simulated V gray is highly negatively correlated with SCD. (4) While NIR-light absorption within skull and scalp mainly underlie the reduction of V gray with increasing SCD, the low scattering CSF layer becomes overproportionally thick as SCD increases, thereby attenuating the V gray reduction. (5) Penetration depth is larger for fNIRS channels located over air-filled frontal sinus, but V gray is significantly reduced. (6) The magnitude of gyrification has no significant impact on V gray .
These findings are discussed with regard to previous studies and their implications for fNIRS as a functional neuroimaging technique in human neuroscience and clinical research. MCS have repeatedly been used to provide numerical solutions for the forward problem of photon migration in highly scattering biological tissues [16,18,20,21] to infer the NIR-light migration within head and tissue models. Another approch is to map NIRlight pathways using the spatial distribution of correlations between functional hemodynamic responses using simultaneous fNIRS and fMRI [6]. Our findings are in line with these previous studies, which were (relatively) consistent regarding shape (banana, ellipsoid) of NIR-light pathways and penetration depth estimates (about 2 cm to 3 cm). Complementing these studies we investigated a relatively large number (n~23) of individual segmented MRI data-sets to assess the impact of individual anatomy on NIRlight pathways and tissue-specific absorption profiles.
By comparing the spatial sensitivity profile of simulated NIRlight between a virtual head phantom (132 averaged T1 weighted MR images, segmented) and a simplified layered slab model [25] with uniform superficial tissue layer thickness, Kawaguchi et al. (2007) [26] showed that the uneven distribution of superficial tissue layer thickness in the head phantom impacts the mapping of simulated hemodynamics in the temporal area. While our study investigated frontal and not temporal areas, our results confirm and extend these findings by simulating the NIR-light penetration and absorption individually considering anatomical parameters for a large number of subjects. Moreover, we suggest, that a rotational ellipsoid of NIR-light absorption, that reaches to a variable degree (e.g. depending on SCD) into gray matter (see figure 4 and 5, equation 3) might be an illustrative explanation for this result.
Although biological tissues absorb relatively little of the NIRlight, the exponential attenuation of light intensity along a given pathway explains the differences in tissue-wise absorption (scalp and bone: 96%, gray matter: 3%). The total energy absorbed from photon packages reaching the detector amounts to (83+18)% leaving some variability in absorption, i.e. fNIRS signal changes due to a functional increase of oxygenated and decrease of deoxygenated hemoglobin, respectively. However, fNIRS sensitivity is dependent on the traversed volume of gray matter in which this absorption change occurs. Since several sources of error variance may be part of the fNIRS-signal (e.g. extracranial blood flow, motor-artefacts, blood pressure), V gray might be critically small in some subjects and/or channels associated with large SCD. We found, that especially skull layer thickness (b~{:61) represents a good predictor for V gray reduction. Previously, a layer model with thicker skull layer instead of low scattering CSF has been shown to decrease partial optical pathlength in brain tissue compared to a model with the same SCD including CSF [18]. We show, that skull, scalp as well as CSF contribute to the reduction of V gray in healthy adult subjects. However, an overproportional CSF layer thickness may attenuate the reduction, thus, not the mere SCD but the partial pathlengths in specific tissue types have to be considered for the estimation of V gray or anatomy-dependent fNIRS-sensitivity.
Since structural MRI might not be available for most participants of fNIRS studies, we suggest to consider head circumference as a practical predictor of SCD and V gray , thus, the fNIRS sensitivity. Head circumference could easily be accounted for when introduced as a covariate of fNIRS data or could be matched when comparing groups. Also, large differences in head circumference may affect the reliability of the assignment and functional measurement of putative neural structures spatially associated with a certain fNIRS channel. In this regard, mean head circumference has been shown to significantly differ between men with (58:5+2:0)cm and women witch (55:7+1:6)cm [27], which may play an important role in the light of our findings.
A possible explanation for the frontal sinus to decrease V gray could be the presence of an air-filled cavity, which represents an extreme case of differential tissue layer composition. Regarding the transition of the light between skull and CSF (transition 1, normal case) on the one hand and between skull and air (transition 2, only possible at frontal sinus) on the other hand the most striking difference is the ratio of refractive indices (table 2), i.e. difference of optical impedance is higher for transition 2 than for transition 1. Using Snell's Law and equation (9) and considering total internal reflection the probability for the light to be reflected at transition 1 (42%) and transition 2 (62%) was calculated. Thus, the light reflection increased by ca. 50% at the frontal sinus (transition 2) compared to the transition (transition 1) without frontal sinus. By means of Monte Carlo simulation Seunghwan et al. (2005) [28] investigated the effect of an refractive index mismatch between tissue and CSF and found that such a mismatch can lead to changes in measured light intensity.
The air-filled frontal sinus mostly covers parts of the OFC. As the OFC has been shown to be centrally involved in decisionmaking and the processing of the subjective evaluation of reward, emotional, social, and sensory information [29], future fNIRS measurements will probably aim to further study its function. Our findings suggest, that a reduction in V gray , restricted or in extreme cases lack of fNIRS sensitivity due to large frontal sinus volume may have to be considered as sources of error variance in studies aiming to measure OFC activity. While we found traversed frontal sinus volume to be correlated with the head circumference Figure 6. The ratio between the CSF layer thickness d csf and the thickness of the superficial tissue layer d superficial~dscalp zd skull plotted against the difference of the simulated light absorbing gray matter volume V gray,meas and the analytically determined gray matter volume V gray,ana (see figure 3). The significant positive correlation indicates, that the simple ellipsoid model underestimates V gray when the CSF layer is relatively thick compared to the superficial tissue layer. doi:10.1371/journal.pone.0026377.g006 (r~:43), the morphology of the frontal sinus has been shown to be individually highly variable [12]. Future fNIRS studies aiming to measure the OFC may, thus, benefit from additional anatomical information of individual CT or MRI scans.
Although we found no significant impact of the gyrification estimate on the simulated V gray in prefrontal cortex, we cannot exclude the possibility that extreme cases of sulcal geometry as in the area of the fronto-temporal junction may affect fNIRS sensitivity.
Currently, the feasibility of fNIRS for single subject analyses of functional hemodynamics to derive an individual psychiatric diagnosis of e.g. depression, biploar disorder or schizophrenia is controversially debated [30]. Here, our findings suggest cautious interpretation of fNIRS signals on a single subject basis, because individual anatomy regarding SCD and frontal sinus volume may profoundly impact fNIRS sensitivity. Aforementioned psychiatric illnesses themselves are associated with anatomical changes such as a reduction in gray matter layer thickness [31][32][33]. Also, the impact of cortical atrophy and increased CSF volumes after longterm alcohol abuse or in patients with Alzheimer's disease on fNIRS sensitivity has not been investigated yet. Because the participating subjects were young and healthy, this question could not be addressed in the present study. Thus, further functional measurements and simulation studies of anatomical factors modulating fNIRS signals are necessary to quantify the influence anatomical differences exert on fNIRS sensitivity.

Participants
A total of 23 adult healthy subjects (12 male) with a mean age of (25:0+2:8) years including 4 left-handed subjects (2 male) were included in the study. One subject (#10) was excluded due to noisy structural MRI. To exclude any history of Axis I or II pathology, neurological disorders or psychoactive medication all subjects were screened by questionnaires based on DSM-IV criteria for Axis I mental disorders. All participants gave written informed consent. The study was in accordance with the latest version of the Declaration of Helsinki and approved by the Ethics Committee of the University of Wuerzburg.

fNIRS optode positions
We precisely defined optode locations using a typical 4|4 fNIRS probe arrangement (ETG-4000, Hitachi Medical Systems, Japan), which was positioned over the right prefrontal cortex. A set of 4|4 optodes, which correspond to 24 fNIRS recording channels each consisting of an NIR-light emitter-detector pair, was used in the present study. The probe set was placed in respect to the Fpz surface marker of the International EEG 10-20 system ( figure 8). Specifically, the light emitter of the NIRS-channel #1 and #4, respectively, was placed on Fpz while the bottom row of NIRS optodes of the probe set was located on a line from Fpz to F8. A previous pilot fNIRS-fMRI measurement confirmed feasibility of identification of precise optode positions as probeholder of each optode and/or slight skin indentation by optode tips were visible in the structural image. This positional information defined native spatial coordinates of the 24 NIRS channels in each subject using the MRIcro software [34]. The channel coordinates between two optodes were normalized to MNI-space using SPM8 normalization routines. The normalized coordinates were then spatially averaged over all subjects and  Optical coefficients of different tissues used for the MCS. Refraction coefficients were taken from [39] except for scalp: [40]. Scatter, absorption and anisotropy coefficients of skin and bone were from [17]. Since coefficient values for gray and white matter differed between sources [41][42][43], the respective values were averaged. The light wavelengths used for assessing these coefficients were between 650 nm and 800 nm. doi:10.1371/journal.pone.0026377.t002 indicated by blobs of 5 voxels diameter using MRIcro. The channels #1, #2 and #4 are mostly associated with parts of the frontal sinus with most volume underneath channel #1. The normalized and averaged probe set is displayed in figure 8.

Head circumferences
The head circumference of each subject was measured using the individual Fpz marker point (International 10-20 system) and inion head surface position. From these two points the head circumference was assessed by a mearsuring tape.
Automatic anatomical tissue segmentation was performed using the SPM8 tool New Segment, that is an extension of the default unfied segmentation [35]. The new segmentation tool was used because it is possible to obtain probability images for scalp and skull and not only for background ( = air), CSF, gray and white matter. The algorithm classfies different tissue types by using their respective intensity distribution and a segmented template. Because voxel intensities are used for tissue classification the structural images must be corrected for modulations of intensity caused by noise. In low intensity regions (such as frontal sinus) the noise after segmentation was suppressed by smoothing the individual probability images with a 1:5mm Gaussian kernel. These probability images are used to create an individual tissue model for each subject. To summarize the results of the segmentation table 3 and figure 7 give an overview of the tissue layer thickness of scalp, skull and CSF.
However, the segmentation may have been improved by using multi-spectral data, e.g. additonal T 2 weigthed images [35]. Unfortunately, such additional structural information was not available in the current study.

Monte Carlo simulation
The purpose of the Monte Carlo simulation (MCS) implemented in this study is to describe the interaction of light within different tissue types in individually segmented human heads. We thereby characterize the spatial distribution of absorbed energy in voxels along NIR-light pathways from emitter to detector. As a guideline for implementation served the MCS by [36]. The source code of the MCS was written in C++ and compiled with MinGW 4.4.0. Data preparation and analyses of simulation data were performed using custom software written in MATLAB 7.9 (The MathWorks, Natick, MA, USA). Statistical analyses were performed using SPSS 17.0 (SPSS Inc., Chicago, IL, USA).

Photon packages
The MCS consecutively and perpendicularly launches photon packages from the emitter into the individual foreheads. For the packages of photons an energy amount referred to as weight w was defined and initialized with w~1. These packages moved through the native space of the segmented brains, which was described by a three dimensional vectorr r in carthesian coordinates [mm]. The direction of the photon package migration was described by a unit   vectorŝ s, that holds the three direction cosines in its components. A positional change fromr r tor r 0 with a given step size l was given by: In order to reduce computational load of the channel-wise MCS, a cube of 50|50|50 voxels was selected comprising the emitterdetector positions on the scalp and segmented tissue underneath.
In the following, we refer to these cubic volumes of interest as channel-cubes.

Probabilistic light interactions
Because the absorption and scattering coefficient m a and m s were defined as rates, the probability of a photon interacting after a step size l follows an exponential decay and the normalized probability distribution p ia (l) was calculated by: After the photon traveled this random step size l, the current weight was reduced due to absorption: The scattering caused a direction change fromŝ s toŝ s 0 . To describe this process a coordinate system was created, in which the unit vector of the z-axis was equal toŝ s. Thereafter, the new directionŝ s 0 was specified by the two angles Q and h: Due to the cylindrical symmetry of the scattering process, the angle Q is uniformly distributed between 0 and 2p. The probability distribution of the angle h is given by the Henyey-Greenstein phase function [37]: where the parameter g~ScoshT quantifies the anisotropy of the scattering. For values of g close to 1 as in biological tissues (g&0:9) forward scattering strongly prevails.

NIR-light reflection and refraction
Depending on the difference in refractive indices (n 1 =n 2 ) between two tissues, a photon hitting the phase transition with an incident angle a i is either reflected with the angle a t~ai or transmitted with a refraction angle, which was calculated using Snell's law (n 1 sin a i~n2 sin a t ). To obtain the incident angle of the photon relative to a boundary between two tissues, a normal vector was attached to all voxels of this boundary. These normal vectors were defined as unit vectors perpendicular to the surface while considering 3 to 6 neighboring surface voxels in two perpendicular directions, respectively.
The acute incident angle of a photon hitting a surface was calculated by the dot product between the photon directionŝ s and the normal vectorn n at the corresponding voxel: cos a i~j Sŝ s,n nTj. By including the refraction angle a t from Snell's law, the reflection probability R was obtained using the Fresnel equations under the assumption of equally distributed polarization directions: The probability T that a photon is refracted was: T~1{R.

Random processes
The step size l with its probability distribution from equation (5) was given by where X [½0,1 is an equally distributed pseudorandom number. Wheather a photon hitting a phase transition was reflected or transmitted was determined by the test X vR (R from equation (9)). If the test is true, the photon was reflected, otherwise it was transmitted. The cosine of the scattering angle h was given by: using the distribution from equation (8).

Measured quantities
We simulated the spatial distribution of voxel-wise absorbed energy of NIR-light photon packages reaching the detector within the channel-cubes. Each weight loss Dw at a locationr r a was registered to the corresponding location in the cubic voxel-space (ñ n~floor(r r)). By using the tissue type information of each voxel, the magnitude as well as the tissue volume contributing to Dw were calculated. Location data of all simulated light-matter interactions were collected and used to generate NIR-light pathways reaching the detector. Then, the partial pathlength through gray matter was calculated for each photon package. To quantify the penetration depth the distances fDg~ER R{fñ n a gE between each voxel contributing to Dw and a reference pointR R centered between respective channel optodes on the scalp were calculated.

MCS configuration
Optical properties of each tissue type were described by four coefficients taken from the literature: Absorption coefficient m a , scattering coefficient m s , anisotropic parameter g and refraction index n. The coefficient values are listed in table 2.
Depending on g the coefficient m s is reduced, and deduced from the diffusion approximation of the radiative transport equation m For g close to 1 the coefficient m s is strongly reduced. Thereby, the mean step size SlT increases by the factor a reducing the number of interactions. In order to reduce MCS calculation time by the factor 6, we used g-dependent m 0 s , but instead simulated isotropic (g~0) instead of anisotropic (g~0:9) scattering. Another advantage of this approach is that the calculation of the scattering angle h following equation (8) is faster and not prone to errors as described by [38]. A possible impact of this procedure on the distribution of absorption was ruled out by applying the simulation to a simple layer model ( figure 9a, b). Due to variability of absorption coefficients (in the literature) we used the simple tissue layer model to investigate the impact of coefficent value variations on the simulated absorption pattern. We found that the absorption distributions of a run with original coefficients and a run with scattering coefficients decreased by factor 10 still overlapped by 87%. Thus, the simulation was stable under coefficient variations.
To estimate how many photons are required for the MCS to obtain stable absorption distributions, the MCS was conducted for channel #1 and #9 of two subjects with increasing number of emitted photon packages N E . We used seven equal photon package number increments (from 10 4 to 10 7 ) and calculated the overlap of the normalized absorptions between two runs. The absorption overlap comparing 5 : 10 6 and 10 7 emitted photons was greater 99:8%, while approximately 1000 and 8000, respectively, were detected. Therefore, we simulated each channel until 1000 photons hit the detector. In total, we conducted simulations of 24 channels|23 subjects|approximately 5 million photon packages &2,760,000,000 photon package simulations.

Scalp-cortex distance
The SCD was individually assessed for each subject and NIRS channel. We calculated the radial distance between the reference point on the head surface and the closest volume per mille of gray matter within a channel cube. The layer thickness of scalp, skull and CSF layers (d scalp , d skull , d csf ) were obtained analogously.