Free induction decay caused by a dipole field

We analyze the free induction decay of nuclear spins under the influence of restricted diffusion in a magnetic dipole field around cylindrical objects. In contrast to previous publications no restrictions or simplifications concerning the diffusion process are made. By directly solving the Bloch-Torrey equation, analytical expressions for the magnetization are given in terms of an eigenfunction expansion. The field strength-dependent complex nature of the eigenvalue spectrum significantly influences the shape of the free induction decay. As the dipole field is the lowest order of the multipole expansion, the obtained results are important for understanding fundamental mechanisms of spin dephasing in many other applied fields of nuclear magnetic resonance such as biophysics or material science. The analytical methods are applied to interpret the spin dephasing in the free induction decay in cardiac muscle and skeletal muscle. A simple expression for the relevant transverse relaxation time is found in terms of the underlying microscopic parameters of the muscle tissue. The analytical results are in agreement with experimental data. These findings are important for the correct interpretation of magnetic resonance images for clinical diagnosis at all magnetic field strengths and therapy of cardiovascular diseases.

Figure 1

Regular arrangement of parallel capillaries in Krogh's capillary model and stepwise simplification to a single bloodfilled capillary [blue (gray) circle] with radius $R_C$, which is surrounded by the dephasing cylinder (black) with radius $R_D$.

Figure 2

Real part (a) and imaginary part (b) of the expansion coefficients $d_{nm}$ obtained from Eq. (15). Beyond the branching point the corresponding two adjacent expansion coefficients form complex conjugated pairs, e.g., $d_{10}=d_{11}^{*}$ for $\delta \omega R_C^2/D\ge 2.94$.

Figure 3

Eigenvalues ${\kappa }_n$ obtained from Eq. (25) and expansion coefficients $G_n$ obtained from Eq. (26) in dependence on the volume fraction $\eta$.

Figure 4

Dependence of the transverse relaxation rate $R_2^{*}$ on diffusion coefficient $D$ in comparison with data from Kennan et al. [24] for simulation parameters: volume fraction $\eta =0.05$, capillary radius $R_C=2.5\mu \text{m}$, frequency shift $\delta \omega =269{\text{s}}^{-1}$, and intrinsic relaxation time $T_2=\infty$. The black line is the exact analytical solution obtained from Eq. (32) and the green (dash-dotted) line represents the strong collision approximation obtained from Eq. (35). Limiting cases are the motional narrowing regime [red (dotted) line from Eq. (29)] and the static dephasing regime [blue (dashed) line from Eq. (30)].

Figure 5

Anatomical image of the hind limbs of a rat in axial view (a) with the selected PRESS voxel. The black solid line in panel (b) is the measured time evolution of the free induction decay, obtained from the selected voxel in the hind limb. The blue (gray) solid line is the exact signal decay obtained from Eq. (14) with the typical parameters given in Table 2 and the measured intrinsic relaxation time $T_2=22.3\mathrm{ms}$.

Figure 6

Anatomical image of the heart in short axis view (a) with the left ventricle $(\mathrm{LV},*)$, the right ventricle $(\mathrm{RV},†)$, and the septum [encircled with a magenta (medium gray) line], surrounded by lung tissue $(+)$. (b) The measured values (black diamonds) are compared with the exact signal decay [blue (gray) solid line] obtained from Eq. (14) with the typical parameters given in Table 2. Also, the walls of the left ventricle are highlighted in colors as anterior wall [blue (dark gray)], lateral wall [red (medium gray)] and posterior wall [yellow (light gray)]. The dashed-green image segment is used in Fig. 7 for the visualization of voxelwise relaxation time maps.

Figure 7

Quantitative $T_2$ map (a) and $T_2^{*}$ map (b) of the myocardium corresponding to the green-dashed image segment of the anatomical image Fig. 6. A significant reduction of the $T_2^{*}$ relaxation time due to background gradients at the boundary between heart and surrounding lung tissue is clearly visible (see white arrows). This effect is not observed in the septal region.

Figure 8

Real part and imaginary part of the Fourier coefficients $A_0^{\left(2m\right)}$.