Spin echo formation in muscle tissue

Determination of the spin echo signal evolution and of transverse relaxation rates is of high importance for microstructural modeling of muscle tissue in magnetic resonance imaging. So far, numerically exact solutions for the NMR signal dynamics in muscle tissue models have been reported only for the gradient echo free induction decay, with spin echo problems usually solved by approximate methods. In this work, we modeled the spin echo signal numerically exact by discretizing the radial dimension of the Bloch-Torrey equation and expanding the angular dependency in terms of even Chebyshev polynomials. This allows us to express the time dependence of the local magnetization as a closed-form matrix expression. Using this method, we were able to accurately capture the spin echo local and total magnetization dynamics. The obtained transverse relaxation rates showed a high concordance with random walker and finite-element simulations. We could demonstrate that in cases of smaller diffusion coefficients, the commonly used strong collision approximation significantly underestimates the true value considerably. Instead, the limiting behavior in this regime is correctly described either by the full solution or by the slow diffusion approximation. Experimentally measured transverse relaxation rates of a mouse limb muscle showed an angular dependence in accordance with the theoretical prediction.

Figure 1

Krogh capillary model. The blood vessel inside the dephasing domain has radius $R_{\text{C}}$ and the circular outer border has radius $R_{\text{D}}$. The cylindrical model approximates the hexagonal muscle tissue geometry.

Figure 2

Illustration of the time evolution of the total magnetization for an exemplary spin echo sequence with $\eta =0.1$ and $\tau \delta \omega =10$.

Figure 3

Total magnetization for a spin echo sequence depending on $t_{\text{E}}/\tau$ with $\eta =0.1$ and $\tau \delta \omega$ from ${10}^{-1}$ to ${10}^3$ according to Eq. (61).

Figure 4

Transverse relaxation rate $R_2/\delta \omega$ for a spin echo sequence determined at a single echo time depending on $\tau \delta \omega$ with $\eta =0.1$ for different values of $t_{\text{E}}/\tau$ according to Eq. (66)

Figure 5

Transverse relaxation rate $R_2/\delta \omega$ according to the mean relaxation time approximation for a spin echo sequence depending on $\tau \delta \omega$ with $\eta =0.1$. Full solution (black solid line) according to Eq. (64), strong collision (red dash-dotted line) according to Eq. (15), motional narrowing (green dashed line) according to Eq. (20), and slow diffusion (blue dotted line) according to Eq. (27).

Figure 6

Transverse relaxation rate $R_2/\delta \omega$ according to the mean relaxation approximation in the strong collision approximation for a spin echo sequence depending on $\tau \delta \omega$ with $\eta =0.1$. The exact strong collision approximation (red solid line) according to Eq. (15) coincides with the red dash-dotted line in Fig. 5. The monoexponential approximation (purple dashed line) is according to Eq. (19). The motional narrowing limit (green dotted line) according to Eq. (20) also coincides with the green dashed line in Fig. 5. In the static dephasing limit, the strong collision approximation [blue dash-dotted line, according to Eq. (32)] was used, which fits better than both the monoexponential approximation [blue short-dashed line, according to Eq. (33)] and the single term approximation [brown dash-dot-dotted line, according to Eq. (34)].

Figure 7

Comparison of the full solution with simulation results. Black solid line: full solution from Eq. (66), red dashed line: slow diffusion approximation from Eq. (28), blue dotted line: motional narrowing approximation as derived from Eq. (53) in [37] or from Eq. (20). Black squares: random-walker simulation, blue triangles: finite-element simulation. All calculations for $\eta =0.05$ (a) or $\eta =0.10$ (b), $t_{\text{E}}=50$ ms, and $\delta \omega =269{\text{s}}^{-1}$.

Figure 8

Illustration of the measurement setup. Axial slice of the tube containing the upper mouse limb at $t_{\text{E}}=10$ ms (a) and $t_{\text{E}}=60$ ms (b), with the exemplary region of interest in red. (c) Sample holder with grooves corresponding to different tilt angles $\theta =0^{\text{o}}$, ${30}^{\text{o}}$, ${60}^{\text{o}}$, and ${90}^{\text{o}}$ for reproducibly, placing the sample tube inside the magnet. Measured azimuthal angle $\delta$ (d) and polar angle $\theta$ (e).

Figure 9

Calculated transverse relaxation rate $T_2$ plotted depending on the angle between muscle fibers and external magnetic field. The black solid line corresponds to a fit of the full solution for the transverse relaxation time according to the total magnetization from Eq. (61), and the red dashed line corresponds to the strong collision approximation according to the total magnetization from Eq. (9).