From MathWorld – A Wolfram Web Resource
- Weisstein EW. Riemann Sum
- Available at: http://mathworld.wolfram.com…
High-spatial-resolution functional MRI (fMRI) can enhance image contrast and improve spatial specificity for brain activity mapping. As the voxel size is reduced, an irregular magnetic fieldmap will emerge as a result of less local averaging, and will lead to abnormal fMRI signal evolution with respect to the image acquisition TE. In this article, we report this signal turbulence phenomenon observed in simulations of ultrahigh-spatial-resolution blood oxygenation level-dependent (BOLD) fMRI (voxel size of less than 50 × 50 × 50 µm³). We present a four-level coarse-to-fine multiresolution BOLD fMRI signal simulation. Based on the statistical histogram of an intravoxel fieldmap, we reformulate the intravoxel dephasing summation (a form of Riemann sum) into a new formula that is a discrete Fourier transformation of the intravoxel fieldmap histogram (a form of Lebesgue sum). We interpret the BOLD signal formation by relating its magnitude (phase) to the even (odd) symmetry of the fieldmap histogram. Based on multiresolution BOLD signal simulation, we find that the signal turbulence mainly emerges at the vessel boundary, and that there are only a few voxels (less than 10%) in an ultrahigh-resolution image that reveal turbulence in the form of sparse point noise. Our simulation also shows that, for typical human brain imaging of the cerebral cortex with millimeter resolution, TE < 30 ms and B₀ = 3 T, we are unlikely to observe BOLD signal turbulence. Overall, the main causes of voxel signal turbulence include a high spatial resolution, high field, long TE and large vessel.