Maria Engel^{1}, Lars Kasper^{1,2}, and Klaas Paul Pruessmann^{1}

^{1}Institute for Biomedical Engineering, ETH Zurich and University of Zurich, Zurich, Switzerland, ^{2}Translational Neuromodeling Unit, Institute for Biomedical Engineering, University of Zurich and ETH Zurich, Zurich, Switzerland

The timing of BOLD-fMRI is typically chosen such that TE matches T_{2}*.
This assumes contrast being exclusively elicited by signal differences at TE.
While sensible for short or TE-symmetric readouts, such as EPIs, it
oversimplifies contrast generation for longer or TE-asymmetric trajectories,
such as spirals.

We propose the concept of a BOLD PSF for a more comprehensive perspective on the imaging characteristics of a functional experiment. Our findings indicate that TE can be reduced for spiral-out without sacrificing BOLD-sensitivity when compared to EPI. Furthermore, we characterize the intrinsic trade-off between specificity and resolution of the BOLD response under varying TE.

Spirals exploit the gradient system more efficiently than EPI. However, the exclusive focus on TE for the choice of timing holds back spiral capabilities in BOLD fMRI. It induces an overly long dead time for spiral-out and a resolution-limited timing inflexibility for spiral-in trajectories.

The assumption of BOLD contrast being encoded in the center of k-space is a good approximation when the acquisition time is shorter than the timescale on which BOLD contrast changes significantly. Furthermore, it is a reasonable perspective if the k-space center is sampled in the center of the acquisition window, as it is the case for EPI. Yet, it does not fully capture the situation for extended readouts that sample the k-space center at any one time within the acquisition window. Like the image itself, BOLD contrast and its spatial representation are determined by contributions from all of k-space.

In the present work, the impact of this trajectory-dependent mixture on different characteristics of the BOLD contrast is investigated by introducing the concept of a ‘BOLD point spread function (PSF)’, which is the Fourier transform (FT) of BOLD weighting across k-space. Characterizing only the imaging process, it is different from other notions of the PSF in BOLD imaging that describe the translation of neuronal activation to oxygenation change of hemoglobin

$$H_{weight}(\boldsymbol{k})=H_{T_2^*}(\boldsymbol{k})=e^{-\frac{t(\boldsymbol{k})}{T_2^*}}$$

In a BOLD experiment, the interesting output is not the images themselves but their differences, originating in temporal changes. Therefore, BOLD imaging is characterized by a differential PSF (Fig.1).\begin{equation}\label{eq2}\begin{split}PSF_{BOLD}(\boldsymbol{x},T_2^*,ΔT_2^*)&=PSF_{GE}(\boldsymbol{x},T_2^*+ΔT_2^*)-PSF_{GE}(\boldsymbol{x},T_2^*)\\&=\frac{∂PSF_{GE}(\boldsymbol{x},T_2^* )}{∂T_2^*}ΔT_2^*\end{split}\end{equation}Since FT is a linear operation, we may interchange it with the derivative and as the sampling filter does not depend on $$$T_2^*$$$, only the weighting function must be adapted:\begin{equation}H_{weight}^{BOLD}(\boldsymbol{k})=\frac{∂H_{T_2^*}(\boldsymbol{k})}{∂T_2^*}=\frac{t(\boldsymbol{k})}{{T_2^*}^2}e^{-\frac{t(\boldsymbol{k})}{T_2^*}}\end{equation}The full BOLD PSF then reads\begin{equation}PSF_{BOLD}(\boldsymbol{x},T_2^*,ΔT_2^*)=\mathcal{F}^{-1}(H_{sampling}(\boldsymbol{k})\frac{t(\boldsymbol{k})}{{T_2^*}^2}e^{-\frac{t(\boldsymbol{k})}{T_2^*}})ΔT_2^*\end{equation}Like the GE imaging PSF, the BOLD PSF is not shift-invariant due to its spatially variable dependence on $$$T_2^*$$$. However, it may be assumed that in a BOLD experiment at a given field strength, the interesting signal changes stem from regions with similar $$$T_2^*$$$.

We computed the BOLD PSF (Fig.2) for EPI and spirals (1mm and 2mm resolution). In each case, the covered k-space area and the readout duration (36ms and 11ms respectively) were matched such that the EPI had a slightly higher undersampling factor. $$$t(\boldsymbol{k})$$$ was gridded onto a rectangular grid using a sufficiently wide kernel to make up for the undersampling. The real part of the resulting PSF

*Resolution*: FWHM of the main lobe, as the diameter of a disk with area equal to the half height area of the main lobe.*Sensitivity*: integral over the main lobe inside an isoline of 20 % of its peak value.*Specificity*: relative side lobe suppression as the relation between the integral over the main lobe and the L^{2}-Norm of the side lobes.

Optimization example 1mm resolution @$$$T_2^*$$$=50ms (typical for GM at 3T

Fig. 4 illustrates effects of the BOLD PSF on exemplary artificial activation patterns.

Our model assumes a mono-exponential signal decay, which does not describe the relaxation process in brain tissue entirely accurately

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