Objective:

Modern clinical MRI collects millimeter scale anatomic information, but scalp electroencephalography (EEG) source localization (ESL) is ill-posed, and cannot resolve individual sources at that resolution. Dimensionality reduction in the space of cortical sources is needed to improve computational and storage complexity, yet volumetric methods still employ simplistic grid coarsening that eliminates fine scale anatomic structure. We present an approach to extend near-arbitrary spatial scaling to volumetric localization.

Methods:

Starting from a voxelwise brain parcellation, sub-parcels are identified from local cortical connectivity with an iterated graph cut approach. Spatial basis functions in each parcel are constructed using either a decomposition of the local leadfield matrix, or spectral basis functions of local cortical connectivity graphs.

Results:

We present quantitative evaluation with extensive simulations, and use multiple sets of real data to highlight how parameter changes impact computed reconstructions. Our results show that volumetric basis functions can improve accuracy by as much as 30%, while reducing computational complexity by over two orders of magnitude. In real data from epilepsy surgical candidates, accurate localization of seizure onset regions is demonstrated.

Conclusion:

Spatial dimensionality reduction with volumetric basis functions improves reconstruction accuracy while reducing computational complexity.

Significance:

Near-arbitrary spatial dimensionality reduction will enable volumetric reconstruction with modern computationally intensive algorithms and anatomically driven multi-resolution methods.

A. Bioelectric Forward Modeling

The forward model for source localization is the so-called leadfield matrix, which describes the linear relationship between cortical current sources and measured electrode voltages. This relationship can be written as [8]:

where$\mathbf{\text{y}}\in {ℝ}^{N_e\times 1}$ is the measured electrode data atN_e electrodes,$\overline{\mathbf{A}}\in {ℝ}^{N_e\times 3\ast N_v}$ is the lead field matrix, with three columns for each of theN_v source locations voxels corresponding to the dipole activity along each of the principal axes. The vector$\overline{\mathbf{x}}\in {ℝ}^{3\ast N_v\times 1}$ contains the unknown dipole magnitudes to be reconstructed, and$\mathbf{\text{n}}\propto N\left(0,{\sum }_n\right)$ represents Gaussian distributed model and measurement noise.

The clusters of pyramidal neurons primarily responsible for the generation of the EEG signal are oriented orthogonal to the cortical surface [21]. Local cortical geometry can thus be used to constrain dipole orientations, which we implement using a model based approach, [33]. The orientation constrained leadfield matrix can be recast as:$\mathbf{\text{A}}=\overline{\mathbf{A}}\mathbf{\text{D}}$. Here,A, the constrained lead field matrix, is of sizeN_e ×N_v, andD ∈ 3 *N_v ×N_v is a matrix with unit column norms encoding the dipole orientations at each voxel. This produces the modified measurement model:

where$\overline{\mathbf{x}}=\mathbf{\text{Dx}}$ uses the matrixD to project the dipole magnitudes inx to full vector dipoles in the original image space. While orientation constraints reduce the dimensionality of the solution space by two thirds, volumetric models constructed from high resolution MRI scans can contain millions of cortical voxels, and the dimensionality of the inverse problem remains significantly higher than desirable, motivating the need for further dimensionality reduction.

B. Dimensionality Reduction

Modern clinical MRI is often collected at a 1mm isotropic resolution [34]. At this resolution, the leadfield has approximatelyN_v ≃ 10⁶ − 10⁷ voxels in the segmented cortical region. Scalp source localization has limited ability to resolve individual source at 1mm resolution, making representation with reduced dimensionality desirable both to reduce required computation and improve storage efficiency. The latter is particularly important when computing spatiotemporal localizations, as storing solutions for many timepoints rapidly becomes memory limited when the number of unknowns in the image at each timepoint is large.

A principled approach to reduce the dimensionality of the cortical activationx is to project the image into a subspace of dimensionN_c, described by a set of basis functions forming the columns of a matrix$\mathbf{\text{C}}\in {\mathbf{\text{R}}}^{N_v\times N_c}$. For the work here, we assume that this basis set is orthonormal. Orthogonality is desirable because it prevents imposing additional correlations in the cortical source space. Normality is more complicated; If regularization is applied in the space of reduced coefficients, variable amplitudes can be used to produce directly comparable coefficients that compensate for differences in patch size and leadfield sensitivity. However, here we regularize in the original voxelized space, and orthonormality will not impact the solutions.

This is a generalized approach that can be used to implement a range of downsampling and basis selection strategies. When applied to the constrained leadfield matrixA, this produces a new leadfield in the reduced dimensional subspace$\tilde{\mathbf{A}}=\mathbf{\text{AC}}$, and an associated measurement model that employs the reduced dimensional encoding:

Each of theN_c columns ofC describes a projection from the original 1mm isotropic source space to a reduced dimensional subspace of dimensionN_c.

The remainder of this work presents and evaluates an approach for constructingC, using a combination of cortical subdivision into non-overlapping parcels, and one or more spatially varying basis functions computed within the spatial support of each parcel. Using an iterated graph cut approach, we cluster voxels in the segmented cortical region intoN_p non-overlapping parcels. Then, within each parcel, we computeN_b spatially varying basis functions using two different approaches. The matrixC, withN_c =N_p *N_b, can then be constructed from these basis functions.

C. Construction of a Cortical Connectivity Graph

Our strategies for both parcellation and basis function identification are based on a graph of local cortical connectivity. We construct this undirected graph using a local neighborhood around voxel labelled as cortex. This graph has an adjacency matrix$\mathbf{\text{W}}\in {\mathbf{\text{R}}}^{N_v\times N_v}$, with individual elements:

Here,v_i is the voxel at the ith location, andN₁₈(v_j) represents an 18-neighborhood (taxicab distance of 2) around voxelv_j. This graph describes the local structural connectivity of the cortex, and satisfies theundirected assumption with no self-loops because W_ij >= 0 ∀i ≠ j andW_ii = 0 ∀i.

In addition toW, our method requires the computation of the diagonal graph degree matrixD, with elements:

that represent the total weight of connections to each node. Finally, the graph Laplacian can then be computed fromW andD asL =D − W. These three matrices are the required inputs to the binary normalized graph cut algorithm of Shi and Malik [35], and for computation of graph spectral basis functions.

We note that voxelwise statistical segmentation of the cortex does not guarantee this graph will be fully connected. In practice, however, we have found that only a small number of isolated individual voxels are separated from the main connected component. In this analysis, we disregard these additional voxels and remove them from the space of potential cortical sources.

D. Cortical Parcellation Strategy

Spatial dimensionality reduction in the cortical space is motivated by the inability of scalp EEG to localize individual sources at the full resolution of available bioelectric models. Dimensionality reduction is commonly obtained by selecting a subset of source locations lying on a coarse grid of positions. While simple to implement, this approach fails to incorporate high resolution anatomic structure available from MRI. In place of this grid downsampling, we propose clustering of individual voxels in the segmented MRI into larger scale features. This permits a reduction of the spatial dimensionality, while retaining the underlying fine scale anatomic information. Starting with an MR brain parcellation, we subdivide the parcels using normalized graph cuts applied to the local cortical connectivity graph constructed above. Below, we detail the specific steps of our subdivision approach. For further details on graph cuts and graph spectral theory, we refer the reader to Shi and Malik’s seminal paper [35].

1. Determine Target Parcel Size: We define the “resolution” of the downsampling asN_target, the target number of parcels in the final subdivision. Assuming an even size distribution, the parcels should each contain, on average,S_p =N_v/N_target voxels. This value will be used to construct a stopping criterion when performing the iterated graph cut approach.

2. Precluster with an Anatomic Parcellation: To reduce computational overhead, we initially parcellate the cortical region into 129 parcels using an automated atlas based parcellation of the MRI [36]. Treating that parcellation as an initial set of graph cuts reduces the complexity of the initial eigenvalue problems in Step 4 fromN_v ~ 10⁷ toN_v ~ 10⁵. This greatly accelerates the initial parcellation process: the complexity of the eigenvalue problem scales with the square of the number of nodes in the graph, and solving 129 reduced dimensional problems (One for each region in the MRI parcellation) requires less computation than applying the initial cuts to the full cortical connectivity graph.

3. Select a Parcel: Given the current parcellation of the cortex, select a parcel which has not yet met the stopping criterion.

4. Perform Binary Cut: Using the method of Shi and Malik [35], we perform a binary normalized graph cut using the eigendecomposition of the generalized system: (D −W)x = λDx. The cut is computed as a threshold applied to the eigenvector associated with the second smallest eigenvalue (The smallest eigenvalue will be 0), with the threshold level selected to minimize the total cost of the associated cut. This cost is the sum of the connection weights that will be removed by the cut.

5. Determine if Resulting Parcels Require Further Subdivision: Given the target parcel sizeS_target, we determine whether each parcel has been sufficiently subdivided by applying a fixed threshold to the size of the parcel currently under consideration. If the parcel satisfies the conditionS_i> αS_target, we apply a binary cut to subdivide that parcel. For this work, we usedα = 2, which was empirically determined to provide final parcellation numbers approximating the target number.

6. Iterate Steps 3–5: These steps are repeated until all parcels have met the stopping criterion.

7. Merge small parcels to improve size distribution: The above graph cut procedure can produce parcels containing only a small number of voxels. To produce more evenly distributed parcel sizes, we iterate across all parcels, and identify those of sizeS_i< βS_target, withβ = 0.25. Each parcel is then merged into its smallest neighboring parcel, to produce a single parcel closer to the target size.

E. Basis Function Identification

Once a final parcellation has been computed, one or more basis functions are needed to describe cortical activity within each of those parcels. This can be as simple as assuming piecewise constant source activations, or it can incorporate additional basis functions to allow spatially varying activation within each parcel. While piecewise constant functions are computationally appealing, at coarse spatial resolutions they can produce reconstructions that are qualitatively less appealing. Using additional basis functions within each parcel improves visual reconstruction quality, often accompanied by improved quantitative accuracy as well.

We investigated two approaches to computing these basis functions. Our first method uses the singular value decomposition, as previous work with surface based cortical models has proposed this approach for identifying basis functions [37]. Additionally, in keeping with the graph theoretic framework of our parcellation strategy, we also employ a graph spectral approach to basis computation.

While the SVD provides a numerically attractive approach for identifying spatial basis functions, it is based purely on the structure of the forward model, and does not incorporate prior information about the expected patterns of cortical activation. To generate measurable scalp EEG signals, large patches of cortex must be synchronously active, and it is reasonable to assume that these activations should be spatially smooth. Our graph spectral approach addresses this by identifying spatially smooth basis functions on each parcel.

F. Image Reconstruction

In our evaluations, we compute reconstructions using a minimum norm approach, regularized at the full 1mm resolution:

where the regularization matrixL is chosen as a local Laplacian operator to encourage smooth solutions, and is similar to the regulariation employed by the LORETA reconstruction approach [38], [19]. Selection of the regularization parameterλ was performed using the L-Curve [39].

A. Comparison Metrics

We employ two metrics to compare the reconstruction performance of the identified sets of cortical basis functions: the relative difference metric (RDM) and the magnitude difference (MAG) [45]. RDM provides a measure of the topographic difference between two images, while MAG measures the overall difference in observed magnitude. For two signalsu_a andu_b, RDM is computed as:

Here,u_a andu_b are the two quantities being compared, both being vectors of lengthm. This provides a measure of the difference in topography between the two signals, and evaluates how closely the shape of each of the two regions matches. The RDM has a bounded range (RDM(u_a,u_b) ∈ [0, 2]), is symmetric (RDM(u_a,u_b) = RDM(u_b,u_a), and is minimized when RDM(u_a,u_b) = 0.

While the RDM provides a measure of topographic distance, it provides no information about the magnitude difference between the two vectors. To examine these changes, we use the magnitude metric (MAG):

which computes the ratio of lengths, or amplitudes, between the two vectors. The range of MAG is unbounded (MAG(u_a,u_b) ∈ [0, ∞]), has inversely-symmetric relationship (MAG(u_a,u_b) = 1/ MAG(u_b,u_a)), and is minimized when MAG(u_a,u_b) = 1. To compute statistics that accurately represent the mismatch in magnitudes, we apply a logarithmic transform to the computed values:

|MAG| takes values ∈ [1, ∞], and is minimized at 1 when |u_a| = |u_b|.

B. Identified Clusters and Spatial Basis Functions

Spatial voxel clusters were identified with both the iterated graph cut approach detailed above, and a standard regular grid downsampling of the brain segmentation. When performing the grid downsampling, voxels at the coarsened scales were labeled using a majority voting approach based on the constituent 1mm voxels. The spatial clusters identified by these procedures are illustrated inFigure 1. Matching sagittal brain slices are shown from parcellations with a target number of parcels between 200 and 20000. While the grid based downsampling rapidly eliminates fine scale cortical structure, particularly in sulci and regions such as the fonto-temporal junction, the graph based parcellation retains this information.

Figure 2 displays the first five basis functions computed using both the SVD and graph spectral approaches. Using our graph spectral approach, the first basis function models uniform activation across the parcel. The remaining bases provide smoothly varying spatial activations. By contrast, the first five right singular vectors of the associated submatrix of the leadfield show an increase in spatial complexity as compared to the graph spectral approach. These differences are likely due to variation in cortical orientation, as well as variation in model sensitivity due local anatomic differences such as the presence of CSF. Note that in both sets of images, the apparent “missing” voxels have magnitudes close to zero, and are made transparent during the visualization overlay.

C. Simulation Results

We performed a series of simulations to evaluate how reconstruction accuracy was impacted by the number of parcels in the final subdivision, and the type and number of basis functions computed on each parcel. These simulations were computed at a spatial resolution of 1mm, using a subject specific head model generated for a 15yo epilepsy patient (Image size: 192 × 256 × 256 voxels, each 0.90 × 0.78 × 0.78mm, or 0.55mm³). Electrode locations for the Electrical Geodesics (EGI) 128-lead geodesic head net were identified using photogrammetry, and coregistered to the subject’s scalp [44]. For each simulation,n ∈ [1 … 3] active regions were identified. Each active region was constructed as a three dimensional patch centered on a randomly selected cortical location. Each patch was assigned a diameterd ∈ {2mm, 6mm, 10mm, 20mm, 30mm}, and activity within each patch as assigned as:

wherer is the distance from the center point of the patch through the cortical ribbon (And thus respecting cortical topography).

Within each simulation, the same spatial size was used for all active regions. Confounding background cortical activity was added as white Gaussian noise with a variance selected to provide an overall signal to noise ratio (SNR) of 0dB. Scalp electrode measurements were then simulated using the 1mm scale model, and additional white Gaussian measurement noise added at a level of 5 dB.

A total of 1500 simulations were computed. This consisted of 500 simulations with each of one, two and three active regions. For each set of 500 simulations, 100 were computed with Gaussian defined active regions with radii of 2mm, 6mm, 10mm, 20mm, and 30mm. The source locations used for each simulation were randomly selected from all available cortical voxels, with the constraint that locations must located more than three standard deviations from other sources.

For each simulation, 110 reconstructions were computed. Eleven different parcellations were considered: three using grid based downsampling at factors of 2×, 4×, and 8× the original 1mm resolution (Voxel volumes of 8mm³, 64mm³, and 512mm², respectively). With each parcellation, ten reconstructions were computed. Five of these used SVD basis functions, withN_b ∈ {1, 2, 3, 4, 5}, while the other five used graph spectral basis functions, again withN_b ∈ {1, 2, 3, 4, 5}.

Figure 3 summarizes the RDM and MAG results across all simulations, parcellations, and basis function selection methods. Values of each comparison metric are plotted on the Y-axis, while the X-axis displays the total computational complexity of the inverse problem (i.e. The total number of basis functions,N_c =N_p *N_b). Each plotted point is the mean value across all simulations for a particular choice of parcellation and basis function. Points connected with a line use the same parcellation and basis selection method, whileN_b, the number of basis functions per parcel, is varied from 1 to 5. Lines plotted in blue use grid downsampling, while those in black use our graph cut approach. The red line corresponds to solutions with the original atlas based parcellation of the MRI, used to initialize the graph cut process. A circle marker is used to denote solutions computed using SVD bases, while a diamond denotes graph spectral basis functions.

For all parcellation levels, reconstruction with graph spectral basis functions improves the accuracy with which the activation topography is resolved, as measured with RDM. Reconstruction with the graph cut based parcellation also improves accuracy as compared to grid downsampling. Using grid downsampling, increasing the resolution of the grid is associated with an improvement in topographic accuracy; Increasing the number of basis functions does not significantly alter reconstruction performance and can actually produce a decrease in accuracy when using a fine scale grid. For parcels containing only a few voxels, some of the first singular vectors may be associated with small singular values, allowing the solution to overfit to noise in the signal and degrade reconstruction performance.

Reconstruction performance with the graph cut parcellation and graph spectral basis functions is largely determined by the total model complexity (Number of parcels times number of basis functions per parcel). While reconstruction performance continues to improve as the total number of basis functions is increased to ~ 10⁵, the majority of the benefit is obtained by the timeN_c =N_b *N_p ~ 10³. We also note that for total model complexity ≤~ 10³, there is a benefit seen from increasing parcellation density that is not observed for high model complexities. Additionally, for low complexities the topographic error in the reconstructions increases exponentially, as the parcels become too large to accurately reconstruct the activation regions.

Magnitude differences are largely similar across all reconstruction methods forN_c> 10³. For large parcels with a small number of basis functions, mean MAG differences> 10 are produced; These are not plotted inFigure 3. While improvements in topography (RDM) are mostly achieved for complexities of 10³ and higher, optimal magnitude matching requires complexities of approximately 10⁴ and above. In all cases, total magnitude is overestimated as a result of the ill-posed nature of the inverse problem.

Figure 4 shows a similar set of plots, with each plot showing the results for a specific patch radius used to generate the simulated activations. These demonstrate a clear relationship between the size of the target region and the performance benefit obtained through the use of graph spectral basis functions. Atr = 2mm, there is little to no topographic benefit, although magnitude errors are reduced as compared to SVD bases. As the radius of the active patch increases, the topographic benefit of graph bases becomes increasingly apparent. A similar set of plots, illustrating minimal change in error metrics as the number of targets is varied from one to three, is presented in asupplementary figure.

D. Interictal Spike Localization

To illustrate how reconstruction accuracy varies with changes to the downsampling and basis selection process, we examined interictal spike localization in two epilepsy patients who received surgical resections at Boston Children’s Hospital. In each subject, interictal spike data was collected using a high density 128 lead electrode net (Electrical Geodesics, Eugene OR) under a hospital approved IRB. Electrode locations were identified using photogrammetry, and coregistered to the scalp surface identified from MRI [44]. Individualized head models were constructed for each subject using a multi-atlas segmentation of the brain. Reconstructions were computed at a point 75% up the rising slope of the averaged interictal spike using the same 110 combinations of parcellation and basis function selection as used for the simulations.

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