Lukas Buschle^{1}, Christian Ziener^{1}, Thomas Kampf^{2}, Sabine Heiland^{3}, Martin Bendszus^{3}, Heinz-Peter Schlemmer^{1}, and Felix Kurz^{3}

Diffusion MRI is highly influenced by diffusion boundaries created by microscopic cells and blood-filled vessels. The Gaussian phase approximation is a common method to analyze the signal evolution in the presence of diffusion. However, the Gaussian phase approximation exhibits several drawbacks as it is only valid for strong diffusion effects, it does not provide phase information and it yields only the total magnetization. In this work, we generalize the traditional Gaussian phase approximation and solve the mentioned issues. The new Gaussian local phase approximation is then applied to diffusion restricted between two slabs, within a cylinder and within a sphere.

In MR imaging, the total magnetization $$$M(t)$$$ is measured as a superposition of the local magnetization $$$m(\mathbf{r},t)$$$ over the imaging voxel $$$V$$$.

In the Gaussian phase approximation, one assumes a Gaussian distribution of the phases of the total magnetization. Thus, the magnetization $$$M(t)$$$ can be connected with the diffusion propagator $$$p(\mathbf{r},\mathbf{r}_0,t)$$$ and the local Larmor frequency $$$\omega(\mathbf{r})$$$ in the form:

$$M(t) = M_0 \mathrm{exp}\left({-\int\limits_0^t \mathrm{d}\xi [t-\xi] K(\xi)}\right),$$ with the correlation function

$$K(t) = \frac{1}{V} \int_V \mathrm{d}^3\mathbf{r} \int_V \mathrm{d}^3\mathbf{r}_0 \omega(\mathbf{r}) p(\mathbf{r},\mathbf{r}_0,\xi) \omega(\mathbf{r}_0).$$

In contrast, the Gaussian local phase approximation assumes a Gaussian distribution of the local phases and, thus, yields information on the local magnetization $$$m(\mathbf{r},t)$$$:

$$m(\mathbf{r},t) = m_0 \mathrm{e}^{\mathrm{i}\alpha(\mathbf{r},t)-\beta(\mathbf{r},t)+\alpha^2(\mathbf{r},t)/2},$$

where the phase exponent $$$\alpha(\mathbf{r},t)$$$ considers single diffusion jumps

$$ \alpha(\mathbf{r},t) = -\int_0^t \mathrm{d} \xi \int_V \mathrm{d}^3 \mathbf{r}_0 \omega(\mathbf{r}_0) p(\mathbf{r}_0,\mathbf{r},\xi)$$ and the attenuation exponent $$$\beta(\mathbf{r},t)$$$ accounts for double jumps:

$$\beta(\mathbf{r},t) = \int_0^t \mathrm{d} \eta \int_0^{t-\eta} \mathrm{d} \xi \int_V \mathrm{d}^3 \mathbf{r}_0 \int_V\mathrm{d}^3\mathbf{r}_1\omega(\mathbf{r}_0) p(\mathbf{r}_0,\mathbf{r}_1,\xi) \omega(\mathbf{r}_1) p(\mathbf{r}_1,\mathbf{r},\xi).$$

In contrast to the traditional Gaussian phase approximation, the newly developed Gaussian local phase approximation provides information on the local magnetization and its phase. A quantitative evaluation of the Gaussian local phase approximation is performed for diffusion restricted between two slabs, within a cylinder and within a sphere.

Restricted diffusion between two slabs $$$-l/2 \leq x \leq +l/2$$$ in a constant field gradient $$$\omega(x) = \delta\omega x/l$$$ is analyzed. The Gaussian phase and Gaussian local phase approximation are compared with the exact numerical solution presented in [3]. A monoexponential approximation of the total signal of the form $$$M(t) = M_0 \mathrm{e}^{-R_2^\prime t}$$$ is presented in Fig. 1: the Gaussian local phase approximation covers both limiting regimes of motional narrowing (small $$$\delta\omega l^2/D$$$) and static dephasing (large $$$\delta\omega l^2/D$$$) and qualitatively agrees with the exact solution.

Similarly, diffusion restricted to a cylindrical object is considered under the influence of a constant field gradient. The total signal in the Gaussian local phase approximation is compared in Fig. 2 with the Gaussian phase approximation. Again, the Gaussian local phase approximation agrees in both limiting regimes in contrast to the traditional Gaussian phase approximation shown in red.

The same argumentation applies for diffusion restricted to a spherical object. The local magnetization as predicted by the Gaussian local phase approximation is shown in Fig. 3. Since the $$$y$$$-component of the magnetization in this specific geometry is antisymmetric, the total magnetization is purely real as shown in Fig. 4, where the total magnetization is compared with the Gaussian phase approximation.

In this work, we introduce the Gaussian local phase approximation to solve several issues of the well-known Gaussian phase approximation. The Gaussian local phase approximation is valid in both, motional-narrowing and static-dephasing-limit and provides information on the local magnetization as well as on the signal phase. The Gaussian local phase approximation depends only on the diffusion propagator and the local magnetic field and, thus, is applicable in the same way as the traditional Gaussian phase approximation. For restricted diffusion between two slabs, within a cylinder or within a sphere, this work shows that the Gaussian local phase approximation is a better approximation than the previously used Gaussian phase approximation. The newly developed Gaussian local phase approximation is not restricted to the analysis of diffusion in a constant field gradient but can also be used to quantify diffusion in magnetic dipole fields. Finally, this work presents a novel theoretical concept to analyze restricted diffusion processes.

[1] A. L. Sukstanskii and D. A. Yablonskiy. Gaussian approximation in the theory of MR signal formation in the presence of structure-specific magnetic field inhomogeneities. J Magn Reson, 163: 236-247, 2003.

[2] A. L. Sukstanskii and D. A. Yablonskiy. Gaussian approximation in the theory of MR signal formation in the presence of structure-specific magnetic field inhomogeneities. Effects of impermeable susceptibility inclusions. J Magn Reson, 167:56-67, 2004.

[3] D. S. Grebenkov. Laplacian eigenfunctions in NMR. I. A numerical tool. Concepts Magn Reson A, 32:277-301, 2008.

Fig. 1: Relaxation rate $$$R_2^\prime$$$ for restricted diffusion between two slabs in a constant field gradient. The traditional Gaussian phase approximation (red line) agrees only for strong diffusion effects (small $$$\tau\delta\omega$$$), whereas the newly developed Gaussian local phase approximation agrees in both limits and correctly describes the qualitative behavior of the relaxation rate.

Fig. 2: Total magnetization $$$M(t)$$$ for diffusion restricted to a cylindrical object with radius $$$R$$$ in a constant field gradient. (a) Gaussian local phase and Gaussian phase approximation agree with the numerical solution for strong diffusion effects. (b) For intermediate diffusion effects, the newly developed Gaussian local phase approximation exhibits a better agreement with the exact solution. (c) Moreover, the Gaussian local phase approximation is also valid for weak diffusion strengths.

Fig. 3: Transverse components of the local magnetization $$$m_x(\mathbf{r},t)$$$ and $$$m_y(\mathbf{r},t)$$$ for diffusion restricted to a spherical object. The $$$x$$$-component for this specific geometry is symmetric, whereas the $$$y$$$-component exhibits an antisymmetric behavior. This corresponds to a purely real signal evolution as shown in Fig. 4.

Fig. 4: Signal evolution $$$M(t)$$$ for diffusion restricted to a spherical object with radius $$$R$$$ in a constant magnetic field gradient. In analogy to the one- and two-dimensional case, the Gaussian local phase approximation exhibits a better agreement with the exact solution than the traditional Gaussian phase approximation for arbitrary diffusion strengths.