Investigation of the sensitivity of functional near-infrared spectroscopy brain imaging to anatomical variations in 5- to 11-year-old children

Abstract. Functional near-infrared spectroscopy (fNIRS) is a noninvasive brain imaging technique that uses scalp-placed light sensors to measure evoked changes in cerebral blood oxygenation. The portability, low overhead cost, and ability to use this technology under a wide range of experimental environments make fNIRS well-suited for studies involving infants and children. However, since fNIRS does not directly provide anatomical or structural information, these measurements may be sensitive to individual or group level differences associated with variations in head size, depth of the brain from the scalp, or other anatomical factors affecting the penetration of light into the head. This information is generally not available in pediatric populations, which are often the target of study for fNIRS. Anatomical magnetic resonance imaging information from 90 school-age children (5 to 11 years old) was used to quantify the expected effect on fNIRS measures of variations in cerebral and extracerebral structure. Monte Carlo simulations of light transport in tissue were used to estimate differential and partial optical pathlengths at 690, 780, 808, 830, and 850 nm and their variations with age, sex, and head size. This work provides look-up tables of these values and general guidance for future investigations using fNIRS sans anatomical information in this child population.


Introduction
Functional near-infrared spectroscopy (fNIRS) is a noninvasive neuroimaging technique that uses low levels of red to near-infrared light (650 to 950 nm) to measure changes in the optical absorption of tissue due to hemoglobin. In this region of wavelengths, often referred to as the "optical window," light can propagate up to several centimeters through tissue, which is deep enough to reach parts of the cerebral cortex from optical emitters and detectors typically placed on the surface of the scalp. Using a grid of these sensors placed within a head cap and worn by the participant, the underlying changes in evoked cerebral hemodynamic responses can be spatially and temporally recorded. The precise penetration of light into the brain, however, depends on a number of individual factors such as optical scattering and the anatomical structure of the brain and extracerebral layers. Variations between individuals in the cortical folding of the brain, head-size, skull thickness, or layers of cerebral spinal fluid (CSF) can influence the recorded fNIRS signals and the sensitivity to underlying brain activity. This is a limiting situation since one of the advantages of fNIRS technology is often cited as its portability, low cost, and ability to record brain activity from pediatric or other special subject populations for whom magnetic resonance imaging (MRI) may be difficult or contraindicated. The objective of this current work is to investigate how these factors affect fNIRS measurements in school-age children and to systematically examine the quantitative effect of age, head-size, and sex on fNIRS measurements. Specifically, we examine the effects on the optical differential pathlength factor (DPF) and partial pathlength factor (PPF) using Monte Carlo modeling of the optical transport model. A dataset of structural MRI volumes from 90 children (58 to 131 months) is examined in this work.

Functional Near-Infrared Spectroscopy Imaging
Over the last 40 years, since it was first demonstrated by Jobsis, 1 fNIRS has been applied to a growing number of applications in psychology, psychiatry, and brain development. [2][3][4][5][6] In particular, the application of fNIRS in child and infant populations has been successfully demonstrated by numerous researchers. [7][8][9][10][11][12] Compared to functional magnetic resonance imaging (fMRI), fNIRS recordings can be made in a nonrestrictive environment and do not require a specialized scanning room nor the participant to lie in a motionless supine position. During fNIRS imaging, the participant can sit or even stand while wearing the fNIRS head cap, allowing reasonable movement of the participant and interactions with other people or the environment.
While a number of researchers have used fNIRS to examine group-level changes in brain activity with respect to subject age or sex in the context of child development, a persistent underlying confound of such work is the potential for systematic differences in the underlying structure of the head, brain, and other factors that affect the magnitude of the fNIRS signal. While light in the near-infrared window can penetrate up to several centimeters of tissue due to low intrinsic absorption in this range, light passing through the tissue is highly scattered resulting in the diffusion of the light through the tissue. Thus, the path of this scattering depends on the structure of the head, particularly the boundaries of the brain, skull, and CSF. In these layers, systematic differences with age or sex would result in a bias in the reported magnitude of the fNIRS recordings. For instance, in older adults, atrophy of the frontal and temporal cortices 13 could result in a decrease in the sensitivity and reported magnitude of measured brain activation as the distance increases between the brain and the surface of the scalp where the fNIRS sensors are positioned. 14 Similarly, in children, the growth of the head and/ or brain could cause similar biases. Previous work by Beauchamp et al. 15 examined changes in the structure of the brain over a range of pediatric structural MRI volumes in 71 children ages 0 to 12 year old. While this work documented the changes in the scalp-brain distance with age, the direct quantitative impact on fNIRS was not examined. In particular, these changes would have an effect on the optical path of light in tissue and the fraction of the signal coming from the actual brain compared to the superficial layers.

Modified Beer-Lambert Law
Cope et al. 16 introduced the concept of the modified Beer-Lambert law (MBLL) as a way to approximate the effect of scattering on the propagation of light in tissue and the resulting increase in the effective distance (optical pathlength) that light travels as it moves through the tissue. The modified Beer-Lambert relationship is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 6 3 ; 3 7 0 ΔOD λ ≅ where ΔOD is the change in optical density (absorption) measured between an fNIRS source-to-detector pair, ε is the extinction coefficient at a particular wavelength (λ) and for a particular i'th chromophore, and c i is the concentration of that chromophore. In the case of fNIRS, the two chromophores of interest are typically oxy-and deoxy-hemoglobin (HbO 2 and Hb, respectively). In the original Beer-Lambert law, optical density is proportional to the optical pathlength through the sample. However, in the MBLL, this is replaced by an effective pathlength to account for scattering of the light and the diffuse path that photons will travel in the tissue. The effective pathlength through the tissue is approximated by the product of the distance along the surface between a source-detector pair (L) and a wavelength correction term called the DPF, which is a unitless scalar that adjusts for scattering. As an example, for a scalp distance (L) of 3 cm between an fNIRS source-detector pair, the photon will typically travel an effective distance of ∼18 cm as it scatters back and forth though the tissue. In this case, the DPF would be 6 (18 cm ¼ 3 cm × 6). However, of this 18 cm, most of this is through the extracerebral skin, skull, and CSF layers that are of little interest to fNIRS. Thus, the partial volume factor (PVF) in Eq. (1) is applied to adjust for the fraction of this path that is actually in the brain. The resulting correction to the MBLL (L · DPF λ · PVF λ ) represents the effective pathlength through the brain volume of interest specifically. The PPF is defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 3 2 6 ; 5 1 0 giving the brain-specific MBLL as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 3 2 6 ; 4 6 8 Thus, PPF is a wavelength specific, unitless scalar that adjusts for the effective pathlength in the brain. In contrast, DPF adjusts for the pathlength through all tissues. Since the optical measurements between a source-to-detector pair (ΔOD) are proportional to PPF and the magnitude of the signal change in the brain, the PPF term is directly relevant in examining systemic differences with age, sex, or head size.

Subject Population
All MRI data were collected at 3 Tesla on a Siemens TIM TRIO scanner using T1-wieghted MPRAGE imaging. All subjects had Note: 3-T MRI data from a total of 90 subjects was used in this study. Age (months), height, and weight were taken at the time of scan. Head circumference and the arc length in the anterior-posterior (AP; nasion to inion) and right-left (between preauricular points) directions were computed post hoc from the structural MRI data as described in the text.
Neurophotonics 011009-2 Jan-Mar 2018 • Vol. 5 (1) participated in one of the several imaging studies between 2012 and 2015 as part of the healthy/control cohort at the University of Pittsburgh and provided written IRB consent via parent/ guardian proxy. 17 Structural MR images from a total of 95 subjects were used in this work. Five of the subjects were removed from analysis due to low MRI quality and/or errors in the anatomical registration or segmentation algorithms. Of the 90 remaining subjects, 46 were female. The age range was 58 to 131 months (mean 95 months; SD 17 months). The demographics of these subjects are provided in Table 1.

Magnetic Resonance Imaging Processing and Segmentation
The structural T1-weighted MR images were processed through the FreeSurfer-based 18 HCP structural pipelines. 19 This automated processing pipeline is designed to produce minimally distorted structural volumes for each subject both in "native" space and standardized Montreal Neurological Institute (MNI) space. T1-weighted images were internally cropped to a smaller field of view to remove the neck using FSL's "robustfov" tool, then aligned to the MNI template space using a 12 degree-offreedom affine FMRIB's linear image registration tool. A brain mask was applied and then a 6-degree-of-freedom transform was used to align the anterior commissure (AC), posterior commissure (PC), and AC-PC line. The AC-PC aligned brain extracted images were then registered linearly and nonlinearly to the MNI template. These warps were inverted and the template brain masks were brought back into the AC-PC aligned space.
The aligned T1-weighted images were then intensity normalized and FreeSurfer's "recon-all" function was run to generate white matter and pial surfaces. Pial surfaces were generated using Gaussian parameters (3 standard deviations above and below gray matter mean intensity). Morphometric measurements of volumes, surface areas, and thicknesses were then computed from these surfaces.
A secondary segmentation was performed on the skin, skull, and CSF layers using the FreeSurfer watershed algorithm "mri_watershed." 20 The watershed segmentation algorithm was used to determine the intensity values for white matter, gray matter, and CSF. An elliptical surface was fitted to the brain and the shape of the surface fit was evaluated against a previously derived template. The brain surface file was then grown outward to generate an inner skull surface. A fifth-order icosahedral surface was fit around the outer edge of the volume and smoothed to make the skin surface. Finally, this skin surface was shrunk to make the outer skull surface. Following automated segmentation using the FreeSurfer and "mri_watershed" algorithms, the brain head-layer surfaces were visually examined. A total of five subjects were discarded due to either poor segmentation, registration, or general data quality leaving 90 final datasets that were used in analysis.

Functional Near-Infrared Spectroscopy Modeling
Each of the 90 subjects' segmented anatomical volumes was used to model the sensitivity and characteristics of theoretical fNIRS measurements using Monte Carlo simulations. 21,22 A four-layer (skin, skull, CSF, and brain) anatomical meshbased model was created from the segmented boundaries of the skin, outer skull, inner skull, and pial surface of the brain using the iso2mesh program from Fang and Boas. 23 Monte Carlo simulations were run at five wavelengths (690, 780, 808, 830, and 850 nm). The optical properties for these tissues and wavelengths are given in Table 2. In this work, gray and white matter brain tissues were assigned the same optical properties similar to the earlier work in Strangman et. al. 24 The optical properties were computed from the general tissue models given by Jacques 25 and optical scattering (μ 0 s ) and absorption (μ A ) was computed by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 3 2 6 ; 2 5 0 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 3 2 6 ; 2 0 6 μ A ¼ B · S · μ A;HbO2 þ B · ð1 − SÞ · μ A;Hb þ W · μ A;water þ F · μ A;fat ; (5) where the parameters for these two equations were compiled from various empirical sources in the literature as given in Jacques. 25 For scattering, the parameters for the skin (a ¼ 4.6 mm −1 , b ¼ 1.421), bone (a ¼ 2.29 mm −1 , b ¼ 0.716), and brain (a ¼ 2.42 mm −1 , b ¼ 1.611) were used. For the brain, we assumed oxygen saturation ðSÞ ¼ 70%, blood volume fraction ðBÞ ¼ 2.2% ½¼ 50 μM × ð66;458 gm∕molÞ∕ð150 gm Hb∕L bloodÞ, water fraction ðWÞ ¼ 70%, and fat fraction ðFÞ ¼ 0%. For the other tissues, we used values given by Jacques. 25 Optical properties were adopted from Jacques. 19 μ A is the optical absorption coefficient, μ 0 s is the reduced scattering coefficient, N is the index of refraction, and g is the anisotropy coefficient.
For each subject, we modeled the optical sensitivity (forward model) using a mesh-based Monte Carlo method. 22 Functional NIRS sources were modeled from 346 positions on the head for each subject at the international 10-5 coordinate positions (see Fig. 1). For each source and wavelength, 5 × 10 7 photons were simulated. Thus, over the whole cohort, 157,000 simulations were run for a total of 72,000 CPU-hours of computing on a 240 CPU high-performance computing cluster. For each simulation, the exiting photons were monitored at all other positions on the surface of the head yielding a 346 × 2564 matrix of virtual fNIRS source-detector combinations from around each source position.
For each simulated source-detector pair, the optical pathlength through the entire tissue (DPF) and partial pathlength through the brain (PPF) was recorded. The DPF and PPF were computed for each 10-5 position by fitting the ∼120 measurements between 10 and 40 mm around each source to a linear regression model using an iterative robust regression algorithm (MATLAB function "robustfit").

Topographic Projections
Each subject was registered to a 10-5 head coordinate space by aligning the left/right preauricular ear points and nasion point in the MRI data to an elliptical template of 10-5 points included in the SPM software ("ext1020.sfp") 26 followed by an iterative closest point registration and refinement using the head surface. Based on this head coordinate registration, the head circumference (at 10% up along the arc length between preauricular ear points) and the arc lengths from nasion to inion and between preauricular ear points was computed retrospectively.
To compare estimated parameters (cortical depth, optical pathlength, etc.) across subjects, values of interest were first projected along the surface normal to the nearest position on the scalp. The scalp positions were then projected using a Clarke's twilight azimuthal projection using a normalized head radius into a two-dimensional topographic map and interpolated onto an equidistant polar grid. This projection allowed data from subjects to be compared in a standardized space independent of the subject head circumference and size. Except where noted, the median value of the parameters was computed across subjects in this polar (10-5 coordinate) space.

Characterization of Anatomical Variations
Using the anatomical MRI volumes from the 90 children, we computed the median skull thickness and CSF layer thickness according the 10-5 space projected values. The median value for the male and female groups is shown in Fig. 2. The skull thickness ranged between 1.1 and 9.6 mm {2.51 AE 0.92 ½median AE 0.67499 × ðmedian absolute deviationÞ} in the females and 1.1 to 8.3 mm (2.66 AE 0.92) in the males. In both sexes, the skull thickness at the crown and posterior of the head (posterior of 10-5 position FpZ and lateral out to positions Cp3/Cp4) was about 2 to 3 times thicker (3.2 to 5 mm) than the other frontal or temporal regions as shown in Fig. 2.
The CSF thickness (defined as the distance between the inner surface of the skull and the pial surface of the brain) was  between 3 and 9 mm in both sexes (male 5.12 AE 0.95; female 5.05 AE 0.89) and greatest (7 to 9 mm) at the slightly anterior to the crown of the head around 10-5 position Fz and extending posterior along the sagittal sulcus. The area of this thicker CSF region was larger in the males than the females. In our segmentations, we did not consider the venous dural sinuses, and some of this CSF thickness at the sagittal sulcus is likely due to the sagittal sinus, which is labeled as CSF in the segmentations. The depth of the surface of the cortex relative to the scalp is shown in Figs. 3(a) and 3(b) for both sexes. The cortical depth ranged from 6.2 to 14.7 mm in females (median 10.2 AE 1.8) and 6.6 to 14.5 mm in males (median 10.13 AE 1.73). This was lowest (7 to 9 mm) along the frontal cortex and bilateral lateral regions. This depth was about 50% deeper along the top of the head down the midline where it ranged from about 11 to 15 mm. This pattern reflected the same regions that had been observed to have thicker skull or CSF layers shown in Fig. 2 for the two sexes. Figure 3(c) shows the difference map of the cortical depth of the females verses the males. Regions in red color show parts of the cortex that were deeper in the females compared to the males. On average, the cortex was 0.1 mm (range [−5.6 to 4.89]; p < 1 × 10 −15 ; F 30420;1 . two-way ANOVA with position and group) deeper in the female participants. Notably, the medial and right frontal/medial cortex was between 2 and 3.5 mm deeper in the females compared to the males. The deeper depth of the cortex in the females means that fNIRS measures will be less sensitive to changes in brain activity in the females compared to the males. This could result in underestimation of brain activity in the female group from these regions and a sex-related bias in the activity in group-level analysis models.
We also examined variability in the location of the cortical folding pattern (mapped according to MNI space) relative to the 10-5 head coordinate system. Using FreeSurfer, the cortical surfaces are extracted and registered as surfaces into the "fsaverage" space where parcellation labels are assigned. In this space, the equivalent anatomical regions can be morphed from one subject's anatomy onto another. Figure 4 shows the median displacement in the equivalent fsaverage cortical positions relative to the surface of the head in both the males and females. The most conserved anatomical regions between subjects [shown in blue-green colors in Figs. 4(a) and 4(b)] were in the lateral frontal, temporal, and posterior/occipital regions and varied from about 5 to 30 mm median displacements. The most anatomically variable regions were along the midline and superior frontal regions of the head which varied up to 20 to 30 mm. Specifically, these regions of high variability in the underlying structure of the cortical folding are the regions where the alignment of fNIRS measurements according to the 10-5 positioning on the scalp would be more sensitive to unknown variability in the underlying brain region.
Finally, in Fig. 5, we looked at the effect of head size and age on the cortical depth. The registered polar (10-5 coordinate) space data for all 90 subjects across both sexes was pooled and regressed using the head circumference or subject age in a robust regression model. In Fig. 5, we show the t-statistic map for these two models. As shown in Fig. 5(a), we found that head circumference was more correlated to the depth of the brain than age. This is because over our group of subjects in this school-age range, we found that there was a considerable spread in head size, height, and weight, which was not well predicted by age alone and reflected different growth rates for the children. Specifically, head circumference was only marginally related to age (R 2 ¼ 0.176). We found that the depth of the brain at the top of the head was positively related to head circumference [ Fig. 5(a)], which indicates that this depth was greater for children with larger head sizes. This also indicates that the head is probably growing faster than the brain over this age range resulting in an increasing depth.

Variation in Optical Properties
As previously described in the methods, a Monte Carlo-based model was used to simulate fNIRS measurements for each of Fig. 3 Gender differences in the scalp-to-brain depth. (a) and (b) The median distance normal to the surface from the scalp to the nearest position on the pial cortical surface in the subject normalized polar (10-5 coordinate) space. (c) The difference between the depth of the females compared to the males. In most regions, the brain of the female subjects up to 3-mm deeper. Underlying variability in brain (MNI) space measured from scalp-based fNIRS sensors. This figure shows the root-mediansquared displacement in underlying registered MNI coordinates of the surface of the brain as measured from each position on the scalp. Red color indicates higher variability in the underlying structure and folding of the brain beneath each position and indicates areas where fNIRS measurements would have lower precision due to individual differences in the folding of the cortex. The results for the (a) female and (b) male subjects are shown with the same color scale.
Neurophotonics 011009-5 Jan-Mar 2018 • Vol. 5 (1) the 90 subjects at all 346 of the 10-5 positions on the head for wavelengths 690, 780, 808, 830, and 850 nm. These wavelengths were selected as the five most widely used wavelengths in currently commercially available fNIRS systems. At each 10-5 head position, virtual "photons" from an fNIRS emitter were simulated and mapped to all surface points symmetrically around the emitter. For each emitter position, the DPF was computed by a weighted least-squares fit of the detectors from 10 to 40 mm around a source. The differential pathlength (e.g., how far did the light travel within the tissue) is equal to the product of the DPF and the emitter-detector distance along the arc of the scalp [Eq. (1)]. Similarly, the PPF as defined in Eq. (2) was computed as the pathlength through only the cortex layer of the model. Both DPF and PPF are unitless scaling factors, which adjust the fNIRS emitter-detector distance in the MBLL. Figure 6 shows the topographic maps of the estimated DPF for both sexes at all five simulated wavelengths. The DPF was highest for the 690 nm wavelength in both sexes; DPF ranged from 5.53 to 6.82 in the females (median ¼ 6.00) and from 5.38 to 6.84 in the males (median ¼ 6.04). Although lower in comparison, in all other wavelengths (780, 808, 830, and 850 nm), the median DPF between sexes was similar; 5.76, 5.71, 5.67, and 5.64 in females and 5.84, 5.79, 5.76, and 5.74 in the males, respectively). The DPF was slightly higher in the males by 1.3% to 1.6% compared to the females. Spatially, the DPF was symmetric across hemispheres and was highest along the top of the head along precentral sulcus and superior frontal cortex. The average DPF for several head regions is given in Table 3. Tables 5-15 for the DPF for both sexes and at each of the 346 10-5 coordinate positions are given in the Appendix. Figure 7 shows the spatial distribution of the PPF for the two sexes and five simulated wavelengths. The PPF has less variation between wavelengths compared to DPF, but has a larger difference between the two sexes. The PPF was higher in the males by 15 Fig. 3(c). The observed 2 to 3 mm increased depth in the females and corresponding decrease in the PPF directly translates to an expected 26% underestimation in fNIRS measurements of brain activity in these regions relative to males. The PPF across the head is summarized in Table 3. Due to space constraints, both males and females have been combined in Table 3. However, for both sexes, Tables 5-15 describing the 346 10-5 coordinate positions are given in the Appendix.
Similar to the analysis of the scalp-to-brain depth with age and head circumference, we also examined DPF and PPF with these two regression models. Figure 8 shows the statistical result (t-statistic) from the regression model pooling data from all 90 subjects. Both head circumference and age had the biggest effect on the DPF in the frontal and occipital regions of the head with positive correction (e.g., increasing DPF with increased age or head circumference). For PPF, age was positively correlated with PPF in the frontal regions. Head circumference seemed to matter only along the equator of the head, but decreased   Note: This table shows the DPF and PPF for several fNIRS-accessible regions on the head. Data has been combined for both male and female subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.
Neurophotonics 011009-7 Jan-Mar 2018 • Vol. 5 (1) with head circumference in most other regions. In Figure 8, variability in DPF and PPF, seen as a noticeable ring near the equator of the Clarke projection, is due to the inferior surface of the brain cutting in where it rests on the cranium floor. This is also evident in the positional variability maps shown in Fig. 4.

Regions-of-Interest
The segmented MRI volumes for each of the 90 subjects were parcellated into 70 anatomical regions of the cortical surface using FreeSurfer. 27 These parcellation labels were created for each subject and then projected into the registered polar (10-5 coordinate) space. Figure 9(a) shows the most frequent (mode) label across the head from all the subjects. Male and female subjects had only minor differences (see the Appendix). Similarly, we used the automatic anatomical labeling dataset 28 to find Brodmann area labels on the cortical surface in MNI space [ Fig. 9(b)]. As a note, the Brodmann areas from the automatic anatomical labeling dataset are defined in threedimensional space and cover parts of the sulci as well as the gyri folds. Therefore, these regions extend to slightly deeper areas compared to the FreeSurfer gyri labels. In Table 4, we provide a list of the cortical depth and closest 10-5 head coordinate points for a subset of the most accessible of these regions. Tables 5-15 are provided in the Appendix.

Discussion
In this paper, we used 90 segmented MRI volumes from children ages 5 to 11 years to model the intersubject variations in head and brain anatomy and their effects on the sensitivity of fNIRS measurements. The overall objective of this paper was to quantitatively examine how variations in anatomy with age, sex, and head-size affected the optical properties of the DPF and PPF. These two factors are used in the MBLL [Eqs.
(1)-(3)] to correct for the effective pathlength of light in the total tissue and brain, respectively, which in turn determines the quantitative accuracy of fNIRS. In most cases using continuous wave (CW) fNIRS recordings, the absolute quantification of the signals is not a concern. However, the potential that DPF and/or PPF could vary across spatial regions or subject demographics creates a confound that could bias group-level statistical comparisons. Namely, when DPF or PPF is ignored as a scaling factor in CW-fNIRS recordings, this assumes that these terms are constant over space/channels and subjects. This term appears in both the numerator and denominator of a t-test used in accessing brain activity from a linear model. 29 This term would cancel only in the case of a first level model (a statistical test within a single subject and per optical channel), but not for comparing the magnitudes across channels in region-of-interest or grouplevel models. As an example from our results, although the exact value of PPF may be less of a concern, the finding that the PPF in the frontal cortex varied by about 20% in the frontal cortex  between males and females means that there would be an expected bias toward underestimation of the magnitude of the brain signals specifically in the female group. Thus, if the magnitude of the hemodynamic response was exactly the same in the brain/cortex space of the two groups, the resulting optical measurements in the female group would be 20% smaller in these regions. While we caution against interpreting the results of statistical tests performed in fNIRS channel space as related to true differences in brain activity compared to systematic variations in anatomy, we believe that this work can provide valuable guidance toward how these biases may be addressed at the group level. Specifically, we suggest careful examination of sex difference in fNIRS datasets in future research projects.

Anatomical Variations in Subjects
In the first part of this paper, we examined differences in the structural anatomy of the brain with sex, head-size, and subject age based on segmentation and registration of the anatomical MRI data. In both the male and female populations, we found quite a bit of spatial variation in the depth of the brain relative to the scalp surface. While skull and CSF thicknesses were comparable across the two groups, we found that on average the cortical depth was slightly deeper in the female subjects. This difference was greatest in the frontal regions where we believe a larger sinus cavity in the female subjects is responsible for the 2 to 3 mm increase depth of the brain in these regions. As the distance between the brain and the scalp increases, the fraction of light reaching these brain regions decreases, therefore, we observed a lower PPF value in these regions for the female subjects. This is particularly relevant for fNIRS measurements that target these frontal regions due to accessibility of the forehead, absence of hair, and scientific interest in the underlying cognitive functional domains (e.g., Brodmann areas 10, 46, and 44). We also found an asymmetry in the cortical depth, with the right side being slightly deeper than the left side of the frontal lobe. This is consistent with the structural asymmetries noted previously in a similar analysis by Beuachamp et al. 15 When we looked at head size and age, we found that head size was the better predictor of underlying anatomy rather than age in this range of 5-to 11-year-old children. Thus, a recommendation to future fNIRS studies might be to report head sizes as well as age when listing subject demographics. In Fig. 4, we also examined variations in the cortical folding of the brain with respect to the scalp-based 10-5 mapping. We found that certain regions, including the crown of the head in both sexes and the left frontal in specifically males, tended to show more variability in the underlying cortical regions that would be measured from fNIRS sensors. For example, based on the results shown in Fig. 4, fNIRS sensors positioned on 10-5 location Fp1 (left superior frontal) in two subjects would, on average, be measuring from two different cortical regions displaced by 3 cm in registered cortical space. Conversely, regions along the lateral frontal area (e.g., around AF8/7-F8/7) are more conserved across subjects; fNIRS measurements from these regions would be expected to come from more similar cortical regions. This implies that studies focused on group-level analyses from these more variable regions should expect a lower effect size in brain activation and consequently should plan on larger sample sizes to counteract the variability in the cortical regions relative to the fNIRS sensors. Fortunately, we found that this effect was not sex  specific and both males and females had similar areas of variability ( Fig. 4). This means that sex is not expected to affect precision of fNIRS measurements, although we found evidence of a clear bias in the magnitude (accuracy) of the measurements due to cortical depth.

Effects of Variations on Optical Properties
Based on the segmented structural MRI data, we also ran Monte Carlo simulations to look at how PPF and DPF varied spatially and across age and sex. The regions that showed the greatest variations in cortical depth between sexes were also the most variable in the partial pathlength (PPF). Since optical PPF is defined as the multiplication factor to estimate the pathlength through specifically the brain [e.g., L brain ¼ PPF × L source-detector ; Eq. (2)] within the MBLL, these spatial variations have a direct impact on the quantitative report of the magnitude of estimated brain signals. In particular, we found about a 13% to 26% spatial difference going from the frontal or lateral regions of the head to superior regions (Fig. 7). This means that for the same change in hemoglobin within these brain regions, the measured optical signals in the lateral regions would predict a lower magnitude change compared to those from near the top of the head. We also noted significant sex differences in the optical PPF, particularly in the frontal cortex (Fig. 7). Specifically, the females had ∼13% to 18% smaller PPF values compared to the males. This seems to be the effect of the cortical depth of about 2 to 3 mm in the females in this region and we believe reflects a slightly larger sinus cavity at the front of the head. Unfortunately, this PPF difference implies that for the same underlying magnitude of hemoglobin changes in the brain, the optical measurements in the females are expected to be smaller. This introduces a systematic bias in group-level statistical analysis, which looks at sex as a covariate. Since the dominant source of noise in fNIRS measurements is probably physiological signals in the skin layers, which would be expected to be similar in both sexes, the lower PPF likely directly relates to a lower contrast-to-noise ratio and lower statistical effect sizes in this group. Even if a sex-adjusted DPF is applied in analysis, this difference in the expected statistical effect would have an effect on group-size and power estimates for analysis. Optical DPF did not have nearly as much spatial variation and showed only very little variation across the same regions. DPF is the multiplication factor between the source-detector distance and the total pathlength through the tissue. Although DPF can be measured directly by time-domain or frequency-domain fNIRS technologies, PPF cannot and must be modeled. Although DPF and PPF varied over space, within a given region, they do not vary much across the five optical wavelengths examined in this work. This means that the potential for crosstalk in the separation of HbO 2 and Hb does not seem to vary across space, although we did not explicitly model this factor. The work by Strangman et al. 24 examined the effect of cross talk in the separation of HbO 2 and Hb in the choice of wavelengths and how this cross-talk is affected by differences in the DPF/PPF values at different wavelengths. Our current work seems consistent with their findings in that we did not see much evidence that the level of cross-talk would be expected to vary much across space or by subject age/sex. Tables 3 and 4 as well as Tables 5-15 in the appendix can be used to design future fNIRS studies. Table 3 provides estimates of the relative sensitivities of various brain regions to fNIRS measurements taken from the normalized count of photons able to reach these brain regions. This depended on both the spatial extent and depth of these regions. As a general guidance, based on our own experience, we believe that regions down to about −40 dB are fNIRS-accessible based on current instrument sensitivities. For example, the TechEN Inc. (Milford MA) system has an instrument noise floor of about 68 dB. 30 Table 3 provides these values for the FreeSurfer anatomical parcellations of the cortex for a source-detector spacing of 30 mm. Tables 5-15 for Brodmann areas and by sex are also presented in the Appendix. Likewise, Table 4 provides a list of the nearest 10-5 head coordinate point to specific accessible Brodmann areas, which could be used to guide head cap designs for future studies.

Limitations of this Study
In this study, we used Monte Carlo modeling of the fNIRS forward solution to attempt to quantify the effects of head anatomy in this population of 5-to 11-year-old children. One limitation of this work is that our results depend on the parameters and optical properties that we used in these simulations. In particular, we used bulk optical properties for the skin, skull, CSF, and brain [gray/white] layers, but did not consider any subject or spatial variability in these values. Although both gray and white matters were segmented, the optical properties used for these two layers ( Table 2) were the same. In addition, there is considerable variability in these values in the literature. 25 Several previous papers have attempted to experimentally quantify these values using time-resolved or frequency-domain fNIRS methods. Duncan et al. 31 used frequency-domain fNIRS on 283 subjects (ages 0 to 50 year old) in the frontal-temporal regions and examined the relationship of DPF with subject age.
In that work, an equation was derived to predict DPF as a function of subject age. Additional work by Cooper 32 examined DPF in neonates and several papers have also examined this in adults. 13,14,[33][34][35][36] The work by Scholkmann and Wolf 37 provides a review and summary of these methods and offers a generalized model for the DPF in the frontal cortex. In comparison to this previous literature, our results are generally consistent with DPF being between about 4 to 6.5. Of note however, the work by Bonnery et al. 14 did suggest some discrepancies between timeresolved experimental fNIRS measurements of DPF and Monte Carlo simulations. Therefore, we caution anyone trying to use the values provided in this work to provide quantitative accuracy. This work is intended as guidance for the design and interpretation of fNIRS studies in school-aged children. For example, we used the same optical absorption and scattering for all simulations, but it is possible that these values might change over age or the values used may offer some bias in the results. This study is not intended to substitute for direct measurements of DPF and optical properties that could be obtained by timedomain or frequency-domain fNIRS techniques. Further experimental work is still needed to verify these results. A second limitation of this work is that our analysis of DPF and PPF in this study was based on the collection of data from simulated measurements of emitter-detectors spacing from 10 to 40 mm. The MBLL [Eq. (2)] assumes that the pathlength correction is linear with this spacing, however, this is probably not entirely true in regions with thick CSF layers such as the frontal pole. In particular, for short spacings or areas where the brain is deeper, the PPF term would be particularly sensitive to nonlinearities.

Appendix:
Provided below are the tables describing the results of the Monte-Carlo simulations. Tables 5-14 provide information by either Brodmann area or anatomical region.  Note: This table shows the differential (DPF) and PPF for several fNIRS-accessible Brodmann areas in the head. Data have been combined for both male and female subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 27 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.
Neurophotonics 011009-13 Jan-Mar 2018 • Vol. 5(1)    Note: This table shows the DPF and PPF for several fNIRS-accessible Brodmann areas in the head in female subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.   Note: This table shows the DPF and PPF for several fNIRS-accessible Brodmann areas in the head in male subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.   Note: This table shows the DPF and PPF for several fNIRS-accessible anatomical regions in the head in female subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.   Note: This table shows the DPF and PPF for several fNIRS-accessible anatomical regions in the head of male subjects. The sensitivity of the region to fNIRS is provided in decibels (dB; 10 × log 10) and indicates the fraction of photons normalized to the incidence power reaching this region from the scalp at a source-detector distance of 30 mm.
Neurophotonics 011009-23 Jan-Mar 2018 • Vol. 5(1)   Note: This table provides the depth (median and range) of several Brodmann areas in female subjects. For each region, the nearest three 10-5 coordinate positions and the depth of the region to this position is provided as guidance for the placement of fNIRS sensors.   Note: This table provides the depth (median and range) of several Brodmann areas in male subjects. For each region, the nearest three 10-5 coordinate positions and the depth of the region to this position is provided as guidance for the placement of fNIRS sensors.   Note: This table provides the depth (median and range) of several anatomical regions in female subjects. For each region, the nearest three 10-5 coordinate positions and the depth of the region to this position is provided as guidance for the placement of fNIRS sensors.
Neurophotonics 011009-30 Jan-Mar 2018 • Vol. 5(1)  Note: This table provides the depth (median and range) of several anatomical regions in male subjects. For each region, the nearest three 10-5 coordinate positions and the depth of the region to this position is provided as guidance for the placement of fNIRS sensors.
Neurophotonics 011009-32 Jan-Mar 2018 • Vol. 5 (1) Where Table 3 in the text describes the sensitivity, DPF, and PPF of each wavelength by select anatomical regions, Table 5 describes these values for select Brodmann areas. Similarly, where Table 4 in the text describes the nearest 10-5 locations for select Brodmann areas, Table 6 describes these locations  for select anatomical regions.  Tables 3-6 contain our findings for select areas/regions for  both sexes and Tables 7-10 are separated by sex. Tables 7-10 are the complete results of our sensitivity, DPF, and PPF findings specifically divided by sex and explained by either Brodmann areas or anatomical regions. Tables 11-14 are the complete results of the nearest 10-5 locations divided by sex and explained by either Brodmann areas or anatomical regions.
Finally, Table 15 contains the results of our generalized equation for DPF for each 10-5 location.

Appendix A: Generalized Model of the Differential Pathlength Factor
The earlier works by Scholkmann and Wolf 37 and Duncan et al. 31 have suggested equations to estimate the DPF as a function of                     the subject age. Using our simulations from the N ¼ 90 subjects in this work, we attempted to fit a similar equation to the work of Scholkmann and Wolf to additionally incorporate head circumference and gender. The DPF values from each of the 320 10-5 positions on the head by 90 subjects were fit to a linear regression model using up to a third power of head circumference, age, and wavelength. A stepwise regression of the entire dataset was used to find the best set of covariates given the inclusion criteria of p < 0.05 for the change in F-statistic of the increased model. Based on this, we model with a total of seven coefficients given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; x 2 ; 6 3 ; 3 1 1 DPF ¼ β 0 þ β 1 ðcircÞ þ β 2 ðcircÞ 2 þ β 3 ðcircÞ 3 þ β 4 1 if female 0 þ β 5 · ðageÞ þ β 6 · ðλÞ; where circ is the head circumference in millimeters, age is in months, and the wavelength is in nanometeres units.

Disclosures
None of the authors has any financial conflicts of interest to disclose related to this work.