Methods: The development of linear matrix representations of processing operations makes it achievable to determine the exact spatiotemporal covariance (cov) structure of the reconstructed images after processing2. In such a framework, the vector of the processed time series images, yT, can be calculated by yT=OTsT, where sT=[s1,s2,…,sn]T is a stack of n k-space signal vectors, with each of the n vectors representing a 2p×1 image frequency vector, and OT is the product of the applied operators. With an estimate of the inherent cov in acquired data, Γ, the image-space cov matrix can be computed by cov(yT)=Σ=OTΓOT T. The image time series correlation (corr) matrix, Σρ, (representing the spatial corr) and the voxel time series corr matrix, Σv (representing the temporal corr), which are both considered in fcMRI studies can then be estimated from Σ. The conventional fMRI models detect activations on a voxel-by-voxel basis with the assumption of an identity cov structure. In order to incorporate the spatiotemporal cov matrix, Σ, into the final analysis, the CV-fMRI model3 can be expanded as given in Eq. 1, where Ci and Si are matrices with the cosine and sine of the ith voxel’s modeled phase along the diagonal, Xi and βi are the design matrix and the magnitude regression coefficient vector of the ith voxel. This model can be represented in matrix form as yT= JXβ+ηT, where ηT~N(0,Σ). The model’s parameters can then be derived through a CV weighted least squares estimation. The matrix representations of processing operations can be very computationally intensive, requiring large amounts of memory. An efficient implementation with the use of matrix partitioning, sparse matrix multiplication techniques, and utilization of the block diagonal forms in matrix multiplication was used to present the results in this study. To theoretically compute the induced corrs by smoothing (Sm), Gaussian Sm with fwhm of 3 pixels; and temporal filtering (TF), band pass filtering from 0.01 and 0.1 Hz, a single slice 64×64 image was considered in a time series of 84 repetitions. To illustrate the effects of such processing in fMRI activation statistics, a 64×64 slice is selected with two ROIs designated to have bilateral activation similar to a finger tapping experiment (20s rest, 4 epochs of 8s on, 8s off). CV fMRI data is generated by a multiple regression model with i.i.d. noise N(0,0.01) for 1000 simulations.

Results: The induced spatial corrs (in real data), Σρ, of the center voxel (cv) by a) Sm, b) TF, c) Sm and TF in the presence of a nonidentity spatial k-space cov are given Fig. 1. The spatiotemporal corr plots of the cv of a 12×12 slice in 16 repetitions are illustrated in Figs. 2a-c. Figs. 3a-b show the activation z-statistics of the processed data (Sm and TF) from voxel-based CV-fMRI model3, and the generalized CV-fMRI model in Eq. 1.

Discussion: Figs. 1a-1c show that Sm and inherent spatial corr result in induced corrs in the neighborhood of the cv. As expected, Sm induces non-negligible spatial corrs whereas TF induces only temporal corrs as given in Figs. 2a and 2b. When combined, Sm and TF induce notable spatiotemporal corrs in the originally spatially correlated data. Not accounting for the induced corrs in the fMRI model produces false negatives as presented in Fig. 3a, and such errors can be avoided with the above generalized fMRI model that accounts for these effects.

Conclusion: The proposed model with its application to fcMRI and fMRI analyses enables researchers to analytically quantify artificial correlations induced by data processing, and draw more accurate and reliable functional connectivity and cognitive brain activity results. References: 1. Nencka et al. J. Neurosci. Meth 2009;181:268-282. 2. Karaman et al. ISMRM. 2013;21:2232. 3. Rowe, Logan. NeuroImage 2004;23:1078-1092.

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