Diffusion broadening of the point spread function in steady-state MRI

Purpose: To demonstrate how diffusion blurs the longitudinal magnetization in steady-state gradient-echo imaging, quantify resolution dependence on flip angle and repetition time, and the effects on relaxometry at microscopic resolution. Methods: Diffusion was incorporated into the Bloch-Torrey equation in $k$ space to determine how the point spread function (PSF) of $T_1$-weighted spoiled gradient-echo images varies with flip angle. Theoretical predictions were validated numerically and extended to b-SSFP. Results: Diffusion acts as a low-pass filter whose width depends on the diffusion rate ($D$), repetition time (TR), $T_1$ and flip angle ($\alpha$). The blurring length spans from $\sqrt{D\cdot \mathrm{T}\mathrm{R}\mathrm{/}2}$ at large $\alpha$ to $\sqrt{D{T}_1}$ as $\alpha \to 0$, equaling $\sqrt{D{T}_1\mathrm{/}2}$ at the Ernst angle. Consequently, multi-angle acquisitions have mismatched resolution (PSF widths of 39 $\mu$m at $\alpha ={10}^{\circ }$ versus 11 $\mu$m at $\alpha ={80}^{\circ }$ for TR = 50 ms). For the flip-angle pair optimizing variable-flip-angle $T_1$ mapping, the effective resolutions differ by a factor of 2.4. In simulated DESPOT1 maps the $R_1$ of a 30 $\mu$m feature is reduced to 1.80 s${}^{-1}$ (true value 2.00 s${}^{-1}$). In b-SSFP, blurring depends on $T_1$, $T_2$, and $\alpha$ but is independent of TR, with a minimum of $\sqrt{D{T}_2}$. Conclusion: The PSF of a steady-state sequence depends on the flip angle and $T_1$. Combining images acquired at different flip angles implicitly assumes identical spatial resolution, but that is not valid at microscopic scales, affecting the accuracy of quantitative relaxometry. The theory identifies a new limitation on the ultimate resolution of MRI set by diffusion.