We used biophysical modelling to investigate the contribution of cortical changes in HbO and HbR to the NIRS signal taken from

*in vivo* NIRS-BOLD measurements. The Obata model (

Obata et al., 2004), a refined version of the original Balloon model (

Buxton et al., 1998), describes the fluctuations in the BOLD signal as a function of the changes in deoxyhemoglobin (HbR) concentration and cerebral blood volume (CBV) in a given voxel:

All the parameters involved in the Obata model are summarized in .

| **Table 1**Parameters of the Obata model |

The general idea behind our method is the following: NIRS can measure independently variations in cerebral blood volume and in the concentration of deoxyhemoglobin. Based on the Obata model, these are the two physiological phenomena giving rise to the BOLD signal. Therefore, one could potentially predict the BOLD signal from the NIRS measurements. This idea has been investigated previously by

Huppert et al. (2006b,

2009). However, the NIRS signal is contaminated by pial vein washout (

Culver et al., 2005;

Dehaes et al., 2011). While the pial compartment cannot be extracted from the NIRS measurements, this component can be removed from the fMRI data because of the high spatial resolution of BOLD-fMRI. Because NIRS suffers from pial contamination, the NIRS-predicted BOLD signal will agree with the measured BOLD signal only if the NIRS data are corrected for pial vein washout.

To apply the above methodology, the Obata model must be modified to account for two discrepancies between BOLD-fMRI and NIRS. (1) Continuous-wave NIRS cannot measure relative changes in hemoglobin but rather a quantity proportional to absolute variations. (2) NIRS measures variations in total hemoglobin (HbT) rather than CBV. The Obata model as written in

Eq. (1) contains only dimensionless variables i.e. all the variables are normalized by their value at rest. Making use of the relation

where Hct represents the hematocrit and MW

_{Hb} the molecular weight of hemoglobin, the Obata model can be re-written in terms of absolute hemoglobin variations which can be measured by NIRS. The new model will be referred to the NIRS-adapted Obata model:

with

and

The definition and value of the parameters involved in this new version are summarized in .

| **Table 2**Parameters of the NIRS-adapted Obata model |

Generally, the pathlength of the light in the head can be separated into scalp and skull layers, non-activated brain tissue and activated brain tissue (

Strangman et al., 2003). In this paper, we further separate the activated brain tissue into two regions: the cortical tissue and the pial vasculature. More specifically, the quantity PVC represents a partial volume correction extracting the signal coming from the activated brain tissue from the signal coming from the rest of the head (i.e. skin/skull and non-activated brain tissue crossed by the light). Finally, the variable γ represents the fraction of the activated brain tissue signal that is coming from cortical tissue while 1 − γ represents the fraction originating from pial vein oxygenation changes.

The coeffcients

*a*_{1} and

*a*_{2} in

Eq. (3) were estimated from the multimodal data set using a standard least-square method. ΔBOLD/BOLD was extracted from the fMRI data while ΔHbT and ΔHbR were computed from the NIRS data. Once

*a*_{1} and

*a*_{2} were recovered, the cortical contribution of HbR relative to HbT defined as γ

^{HbR}_{r} = γ

^{HbR}/γ

^{HbT} was computed using the following relation:

where

*a*_{1} and

*a*_{2} were obtained from the least-square fit and the value for the rest of the parameters taken from the literature.

In our model, we separately account for the volume fraction of the activated brain tissue (with PVC) and the cortical vs pial composition of the activated tissue (with γ). With the above definition, PVC depends on the wavelengths of the NIRS sources and γ does not.

Strangman et al. (2003) showed that measurements performed at 690 nm and 830 nm minimize the cross-talk between HbT and HbR introduced by incorrect values of PVC. Since these wavelengths were used in our measurements, crosstalk between HbR and HbT was negligible and it was reasonable to assume PVC

^{HbT}=PVC

^{HbR}. As such, these two factors cancel out in the estimation of

with

Eq. (6), as do Hct and MW

_{Hb}. Our estimation of

was therefore independent of the values assumed for these parameters. Under these assumptions,

Eq. (6) reduces to

Culver et al. (2005) showed that the spatial extend of the activation region measured by HbT is smaller than the one measured by HbO or HbR. The larger activation region measured by HbO and HbR was attributed to potential washout of the deoxyhemoglobin in the pial vasculature during the activation. The same phenomenon of extended pial vein washout is handled by the parameter γ in our model. Pial veins exhibit only very small volume changes following brain activation (

Culver et al., 2005;

Hillman et al., 2007;

Drew et al., 2011). Therefore, the value of γ

^{HbT} is very close to 1 indicating that most of the HbT signal is coming from the cortical region. Under this assumption, a pure washout of HbR is observed and thus the amplitude of the pial increase in HbO matches the amplitude of the pial decrease in HbR:

where (1 − γ) indicates the pial fraction of the signal. One can derive the expression for the cortical weighting factor for HbO (γ

^{HbO}) from

Eq. (8):

The parameter γ

^{HbR} alone cannot be extracted from the

*in vivo* data but under the assumption that γ

^{HbT} ≈ 1, the approximation

holds. We will refer to

as the estimation of γ

^{HbO} under this assumption by substituting γ

^{HbR} by

in

Eq. (9):

where the subscript “r” emphasizes the assumption of no pial volume changes.

Conversely, in the cortical tissues the amplitude of the increase in HbO does not match the decrease in HbR because vascular dilation gives rise to a change in blood volume (

Buxton et al., 1998). Therefore, the cortical fraction of the signal, γ, given by the ratio of the cortical signal over the total activated tissue signal

, will be di erent for HbR and HbO. Since the amplitude of the cortical ΔHbR is lower than the amplitude of the cortical ΔHbO, the relative cortical weighting factor will be lower for HbR compared to HbO (

).