*3.1.1. EV/IV signal separation*

By calculating the EV and IV signal portions separately and their convex linear combination in Eq. (11), we present the EV/IV signal behaviors with respect to a span of echo time (*TE*=[0,

**Figure 7.** Intravascular (IV) and extravascular (EV) voxel signal simulations. The IV signal evolves drastically for a long *TE* and its contribution to the full voxel signal is relatively small.

60] ms) and for a range of field strength (*B*0=[1.5, 3, 4, 7, 9] Tesla). It is seen in **Figure 7** that the IV signal changes quickly with a long echo time. However, the drastic IV signal changes are greatly suppressed in the voxel signals by the dominant EV signals. In particular, the IV signal may be developed into phase wrapping phenomenon for a long echo time (see **Figure 7(b2)**). With the dominance of EV signals in a large voxel, a voxel signal remains as a linear phase accrual with echo time (see **Figure 7(b3)**).

## *3.1.2. Multiresolution voxel signal behavior*

As a voxel size decreases, the voxel space contains less (or none) vessels, and there is less voxel average effect. In **Figure 8**, the four-level voxel subdivision and multiresolution voxel signal behaviors are demonstrated. At level =1, the parent voxel contains a clutter of vessels where the complex voxel signal appears as a short line-segment trajectory (with respect to *TE*). As the voxel is decomposed into an 8 × 8 × 8 array at level = 4, the subvoxel only contains a single vessel, and the voxel signal becomes turbulent due to the high field values for rapid Larmor precession [14, 23].

**Figure 8.** Multiresolution complex-valued voxel signals due to voxel subdivision. As the voxel size is dyadically re‐ duced, the smaller voxels contain less vessels, and the voxel signal may become turbulent at vessel boundary (Adapted from [23]).

#### *3.1.3. Diffusion effects on magnitude and phase signals*

The numerical simulations on the diffusion effect on MR magnitude and phase are presented in **Figure 9** for a span of *TE* = [0, 60] ms with different field strengths (in terms of Δχ*B*0 = [0.1, 3] ppmT). The results show that the diffusion has more effect on low field magnitude than on high field magnitude [20]. Nevertheless, the diffusion has little effect on MR phase signals.

**Figure 9.** Effects of diffusion and field strength on voxel signal magnitude and phase. It is seen that the diffusion has more effects on magnitude signal than on phase signal and that the diffusion effect decreases as the field strengths increases (Adapted from [20]).
