1. Introduction

The monitoring of brain activity through novel, less costly, more tuneable and more versatile techniques is a major trend in neuroimaging. We concentrate here on cortical mapping, also called optical topography [1]. Unlike the more widely investigated diffuse optical tomography (DOT [2, 3]) which allows to recover three dimensional distributions of the optical parameters, cortical mapping concentrates on the surface region of the brain closest to the skull. Cortical mapping is a relatively new technique and is mainly applied to relate superficial brain activity to high cognitive processes such as in the study of mental illnesses [4, 5], developmental psychology [6–10] and linguistics [11–13].

In all optical topography/cortical mapping/near-infrared spectroscopy (NIRS) problems, the quantity which is monitored is the Hemoglobin/Total Hemoglobin ratio (HbO ₂/Hb_total) or the Deoxy-Hemoglobin/Total Hemoglobin ratio closely related to the absorption coefficient. It gives an insight on the oxygenation and can indirectly measure the activation of a region. In order to find this coefficient, the medium is illuminated with a known source and detectors are placed on the surface to record the response of the domain within the continuous wave (attenuation), frequency domain (amplitude and phase) or time domain (Temporal Point Spread Function, TPSF [14]) frameworks.

Most of the theory which allows one to infer the absorption coefficient from photon density measurements is based on the modified Beer-Lambert law [14]. Such considerations allow the use of DC-intensity, continuous wave instruments to find the oxygenation ratio in tissue. They are currently the most readily available commercially and versions with large number of source-detector couples exist [15–17].

Use of a time-resolved instrument, on the other hand, offsets a smaller number of source-detector pairs against the increased information content of the time-resolved data [18–23].

The human brain, when probed with a NIR device, behaves like a multi-layered geometry, which is often considered as a slab [24]. This biological tissue is very thick for near-infrared light to pass through. Experimentally, this value varies between 5 and 10 mm [25] or slightly more [26] up to 15–20 mm. This is very close to the size of the extracerebral layers, which varies between 8 [26] and 14 mm [27]. It means that only the superficial layer of the adult brain (cortex) is imaged.

Currently, most of the studies involving near-infrared investigation of the brain do not include an anatomical model in the reconstruction computation. Instead a reconstruction in a simplified geometry is followed by registration to an MRI based geometry as a post-processing step, and can lead to inaccurate localisation and/or quantisation.

We aim here to use the anatomical information as a constraint of the inverse problem. This objective manifests itself in the restriction of the reconstructed region to the surface of the brain and our multi-layered model.

Current successful optical topography systems are similar to devices like the Hitachi ETG-4000 which is able to image up to 52 channels at the same time [28]. They have been widely used and shown to produce results of interest in clinical and psychological research. However, despite the success of this technology, it is possible to improve many aspects of the output (localisation, spatial resolution, quantitation, repetability) by including anatomical data in the reconstruction process.

As far as accuracy is concerned, the best studies so far have relied on MRI-derived data and analysis methods. F. Abdelnour et al. [29, 30], for example, have used a Finite Element Method (FEM) model for the propagation of light through tissue in a biologically accurate 5 tissue types model of the head. The fidelity to the anatomical features was ensured by the use of a well-known MRI tool, Freesurfer [31] which contains an atlas of the adult head. However, since the FEM gives a volume solution, the surface one has to be extracted in order to perform actual cortical mapping. Our model aims to simplify a similar process by integrating the photon density on the volume within the model itself. The solution that it outputs is a surface density. Another example of a multi-modality combination is found in a study by Y. Tong and B. Frederick [32] where the fMRI (functional MRI) analysis tools have been used with the Blood-Oxygen-Level Dependent (BOLD) signal being replaced by the fNIRS (functional NIRS) data. Studies by Custo et al. [33] and Caffini et al. [34] also show the successful use of MRI-derived data for the geometry of the adult head in a similar context.

In this paper, we reconstruct an in vivo experiment uses an atlas mesh built from MRI data. Using a boundary element method (BEM), we develop a model for the mapping of photon densities between surfaces of the multilayer representation, thus reconstructing what we introduce as an Equivalent Cortical Absorption (μ ^ECA _a). Our method suggests that it is possible to overcome the fundamental limitations of functional optical topography and obtain reliable maps of the cortex with a simple and fast algorithm.

2. Diffuse optical cortical mapping

In principle, functional optical tomography can be achieved using a volume based approach and full tomographic data obtained over a large part of the head [19, 20]. In the limit of 'complete' data measured for all possible source and detector combinations over a closed surface, the inverse problem technically has a unique solution. However, the strong ill-posedness of the problem coupled with the limited amount of available data means that the reconstructed solutions are prone to artefacts and require substantial regularisation. In Diffuse Optical Cortical Mapping (DOCM) we propose instead the replacement of a volume image of activation with a surface map on the cortex. This idea is analogous with the inverse problem of Electroencephalography in which volume charges and currents in the brain are represented by an equivalent charge or dipole layer [35]. In that problem, only direct imaging of the potential arising from current sources induced by functional activation are measured and the problem is both ill-posed and non-unique; sources reconstructed onto an equivalent source layer are then forward mapped onto the cortical source for effective imaging of the equivalent cortical surface source distribution. An alternative to this strategy is to directly derive a cortical potential image by considering the mapping

where 𝒯 is known as the cortex-scalp transfer map [36,37]. It has the form of a (nonstationary) blurring operation. In the case of discrete measurments the mapping becomes

where ℳ represents the sampling of the potential U ^scalp at a set of detectors. The inversion of the linear operator ℳ𝒯 is underdetermined and ill-posed and requires regularisation. To derive the equivalent theory for DOCM, we first recall some principles of the Boundary Element Method for this application

2.1. BEM forward model theory

For a system of n + 1 nested layers, Ω₀ . . .Ω_n, with interfaces Γ₀ . . .Γ_n the BEM system is written [38, 39],

with Φ^(ℓ), J ^(ℓ) representing the vectors of photon density and photon current on interface Γ_ℓ, and Q representing the source term. The matrix blocks K_ℓℓ′ represent the discretisation of the single and double layer potential operators 𝒜 ^SL,𝒜 ^DL that couple a function and its normal derivative on the boundary of a domain with the Green's function in the interior of that domain. In particular, for the diffusion equation model used in this paper, for optical parameters defined in terms of absorption coefficient μ _a and reduced scattering coefficient μ ^′ _s we define the Green's function in the ℓ^th layer, as

with ${\varpi }_{\ell }={\left(\frac{{\mu }_{\mathsf{a},\ell }+\frac{\mathsf{i}\omega }{c}}{D_{\ell }}\right)}^{\frac{1}{2}}$ being the wave number and $D_{\ell }=\frac{1}{3({\mu }_{\mathsf{a},\ell }+{\mu }_{s,\ell }^{\prime })}$ the diffusion coefficient. For a surface Γ_ℓ with a mesh consisting of nodes { $N_{\ell ,k},k=1...n_{{\mathsf{\Gamma }}_{\ell }}$ } and basis functions { ${\varphi }_{\ell ,k},k=1...n_{{\mathsf{\Gamma }}_{\ell }}$ } the component matrices A^SL _ll′,A^DL _ll′ are given by surface integrals

from which K_ℓℓ′ are constructed by linear combination [38, 39].

For the calculation of the probe fields we consider S inward directed photon currents at source locations {_j, j = 1. . .S} so that

Solution of the BEM system Eq. (3) for each input source generates the cortical photon probe densities {Φ^cortex _j, j = 1. . .S}.

Now consider the secondary sources given by the product of the probe fields and the activation in the brain, which, using Green's identity, become

where ${\mathrm{\Phi }}_j^{\left(n\right)}\left(\mathbf{r}\prime \right)$ is the solution density in the n^th layer for the source j. When dealing with nested BEM regions representing a head, Ω_n is the brain or the innermost region. Solution of Eq. (3) with the secondary sources Q ^actvn _j results in the secondary density Φ^actvn _j. The equivalent cortical absorption is therefore modeled by

Applying the transfer matrix and the measurement operator gives the measured activation signal

The transfer operator 𝒯 does not depend on the sources or measurements. It can be calculated from the BEM matrix by repeated application of the Schur complement to eliminate all the unknowns in Eq. (3) except for the cortical and scalp photon densities. We give an explicit form of the matrix representation T for an n+1 layer layer system in the Appendix.

Combining data from all of the S sources gives rise to a stacked operator

The computational cost of constructing the linear Jacobian matrix W in Eq. (11) is the cost of the S solutions of Eq. (3) for the probe fields. We may alternatively construct W by an adjoint method by solving Eq. (3) for M adjoint sources located at the detector positions, to give adjoint fields Φ^cortex,* _i, i = 1. . .M as described in [38] and constructing the row of W corresponding to source j and detector i as

The adjoint method was employed in this paper.

Further, we consider the scenario in [20], wherein the measurements are the logarithmic amplitude lnA and phase θ of the complex valued density Φ^scalp. The experimental derivation of the data is obtained using a time-domain measurement system, followed by Fourier transformation and sampling at a single modulation frequency [20]. This gives rise to a splitting of the data (y) and Jacobian;

2.2. Inverse problem

Since the matrix W is ill conditioned we consider a regularised solution as the minimiser of the weighted norm

where C⁻¹ _y is the inverse covariance matrix of the assumed measurement noise model and R(μ ^ECA _a) is a regularising functional. In this paper we take the simple zero-order Tikhonov regulariser which leads to the linear inversion:

As in [38] we here take ${\mathsf{C}}_y^{-1}=\mathrm{diag}[\frac{1}{\overline{\ln \mathsf{A}}},\frac{1}{\overline{\theta }}]$ , where $\overline{\ln \mathsf{A}}$ and θ̅ are the average initial errors in log ampliture and phase respectively [40].

Equation (15) is solved using the MINRES solver in MATLAB with a tolerance of 10⁻⁶.

3. Results

3.1. Simulation

We begin the investigation of the accuracy of our method by testing on a simulation case. The same meshes, optical properties and optode placement were used for the simulation as in the invivo case in section 3.2. We use a 3-layer BEM mesh composed of 2806 surface nodes and 1402 surface elements for the scalp layer, 3294 surface nodes and 1646 surface elements for the skull layer, 2098 surface nodes and 1048 surface elements for the brain region. Accuracy of the BEM forward model has been tested in [38, 39]. The optical parameters were close to a neonatal human head μ ^′brain _s = 0.61mm ⁻¹, μ ^brain _a = 0.014mm ⁻¹, μ ^′skull _s = 0.625mm ⁻¹, μ ^skull _a = 0.024mm ⁻¹ and μ ^′scalp _s = 1.4mm ⁻¹, μ ^scalp _a = 0.016mm ⁻¹. The optodes set consisted of 12 sources and 30 detectors, as presented earlier in [38]. 258 measurements were kept for the analysis out of the 360 possible ones. A spherical inclusion with μ ^inclusion _a = 0.034mm ⁻¹ and of diameter 10mm was put inside the brain layer and moved along a surface normal deeper inside the mesh. The BEM-FEM model of [38] was used to generate the data and 1% of white noise was added to it. The regularisation parameter was set to α = 1.72 × Tr[W^TW].

The Figs. 1 and 2 show a trend of contrast increasing and diminishing, as the blob varies in overlap with the Photon Measurement Density Function [41] of the optodes. As it moves deeper inside the brain, the contrast decreases and the noise is reconstructed instead.

The pictures in Figs. 1 and 3 have been generated by a projection of the solution on a higher resolution brain mesh of 13498 surface nodes mesh for improved visualisation. Both the coarse and fine resolution meshes were obtained from a model using data from MRI scans of a neonate, courtesy of A. Tizzard et al. [42].

The projection between meshes was implemented using the following steps:

 

Fig. 1. Left to right and top to bottom, reconstructions of a Gaussian absorption target simulated with the FEM (1% noise). The target is moved by 2mm from the surface of the brain (top left) to 20mm below the surface.

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3.2. In vivo reconstruction

After the successful testing of the method on simulation we went on to apply it on data from an in vivo motor response study on pre-term babies as reported in [20, 43].

As the 33 weeks old neonate is sleeping, his arm is moved: this stimulus activates an area of the brain corresponding to the action, here the motor and somatosensory cortices. Such a technique allows to use the linear reconstruction theory, since we can have the data before (reference rest state) and during the movement. In the present case, it is the right arm which is moved.

 

Fig. 2. Evolution of the contrast, i.e. the difference absorption of the target divided by the background absorption. The nominal value is 1.43.

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In such an experiment, due to the setup and the nature of the subject, one only has the position of the sources and detectors relative to an absolute origin. The actual attitude of the baby has to be inferred from this data. The mesh is then warped onto the measurement cap using a method in three steps:

The details of this method can be found in an article by A. Gibson et al. [44].

In a previous publication [20], the motor cortex's activation was reconstructed with a resolution of 10mm. As it can be seen in Fig. 3, the activation has been localized on the brain surface. The regularisation parameter was set to α = 0.97× Tr[W^TW]. The lateralization of the movement manifests itself by an increase in absorption in the hemisphere involved. A spatial full-width at half maximum (FWHM) of 6mm is observed for the equivalent cortical absorption due to activation. It shows that our method is able to provide qualitatively reliable cortical mapping results, localized and characterized.

The relation between the quantitative reconstruction and the volume absorption activation is determined by the volume convolution in Eq. (8), and varies with depth.

4. Discussion and conclusions

We developed a cortical mapping method able to localize the perturbations to the NIR signal originated from activation in the brain. By testing it on simulated and in vivo data, we demonstrated its ability to accurately find the position of the part of these perturbations which is on the surface of the cortex.

With our simple surfaces-based model, we did not explicitly model the influence of the CSF, although this is also possible within a BEM framework. Instead we restricted the reconstruction to the cortex, as this is the key feature of a measurement passing through this region of the brain [24]. With the use of time-gated measurements [46, 47] or the use of the variance of the TPSF [48], the signal could be reliably restricted to the brain surface. This will be the subject of an upcoming publication.

 

Fig. 3. Side and top views of a cortical map of the activated brain - the primary motor and somatosensory cortices activation is clearly visible. This position corresponds to the arm region as it can be seen in [45].

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Setting up the BEM cortical mapping computation is simple, once anatomical data can be obtained. This can be done with the deformation of an existing atlas respective to measured fiducials during the experiment, like in our case, or with the help of individual anatomically-resolved scans (MRI, X-Ray). As soon as a surface mesh is obtained, which is usually the first step taken after segmentation, the BEM cortical mapping can be used. After the initial building of a Jacobian for the problem, subsequent measurements on the same subject could be analysed in real time, since the actual reconstruction time is less than one second.