Figure 1.
Computational steps involved in the FPFDM.
This figure illustrates the steps involved in computing the susceptibility induced transverse relaxation rates for a 3D tissue structure using the FPFDM: (a) The tissue structure is V(x,y,z). (b) The 3D Fourier transform of (a). (c) The magnetic field from the cubic finite perturber. (d) The 3D Fourier transform of . (e) The point-wise multiplication of (b) and (d) in the Fourier domain. (f) The magnetic field shift due to the vascular structure computed as the 3D inverse Fourier transform of (e) or the convolution of (a) and (c). (g), (h) and (i) are the phase accumulation, the magnetization and the diffusion transition matrix, respectively. These are used to compute the magnetization in (j). (k) The computed MR signal. (l) The transverse relaxation rates associated with an arbitrarily shaped tissue structure.
Figure 2.
(a) FPFDM replicates the characteristic vessel size dependence of ΔR2*and ΔR2 as has been previously shown with MC methods. (b) A comparison of computed ΔR2* values as a function of sphere volume fraction and packing arrangement using MC (filled symbols) and FPFDM (open symbols) techniques, with excellent agreement between the two methods. (c) The computed ΔR2* percentage difference between MC and FPFDM decreases as the number of FPFDM structures used for averaging increases.
Table 1.
Parameters used in MC and FPFDM simulations along with total computing times to calculate ΔR2* values for 18 cylinder radii.
Figure 3.
Dependence of ΔR2* and ΔR2 on cellular shape and packing arrangement.
(a) Example of a cellular model using ellipsoid packing (left) and a 2D slice through the associated magnetic field perturbation for B0 = 1.5T and Δχ = 5×10−8 (right). (b,c) The computed ΔR2* and ΔR2 dependence on cell volume fraction and packing arrangement. For all packing arrangements, the relaxivity increases and then decreases with cell volume fraction. Ellipsoid packing yields greater relaxivity than spheres. ΔR2 exhibits qualitatively similar behavior to ΔR2* yet with a reduced magnitude.
Figure 4.
The influence of vascular morphology on ΔR2* and ΔR2.
(a–c) Sample microvascular networks simulated using a fractal tree model with increasing branching angle heterogeneity. (d) Three orthogonal slices through the magnetic field perturbation at the body center for the vascular network in (c). (e–f) Effect of branching angle heterogeneity on the concentration dependence of ΔR2* and ΔR2 computed with FPFDM (B0 = 4.7T, Δχ = 1×10−7, 2% target vascular volume fraction). Both ΔR2 and ΔR2* increase with branching angle heterogeneity.
Figure 5.
Example simulation with realistic tissue structure and contrast agent extravasation.
(a) Sample tissue structure composed of ellipsoids packed around fractal tree based vascular network. (b) Simulated Cp and Ce curves derived using 2-compartment model. (c) Example 2D map through the magnetic field perturbation computed at time t = 300 sec. (d). The time evolution of the standard deviation of the field perturbation (std ΔB) computed using B0 = 3T, Cp and Ce for the given sample structure.
Figure 6.
Dependence of DSC-MRI signals on cellular features in the presence of CA leakage.
The GE post-contrast to pre-contrast DSC-MRI signal ratio (S/S0), both in the presence (KTrans = 0.2 min−1) and absence (KTrans = 0 min−1) of CA leakage at pre-contrast T1 values of T10 = 500 ms, T10 = 1000 ms and T10 = 1500 ms, for tissue structures constructed using ellipsoids with mean radii of 5 µm (a–c) and 15 µm (d–f), respectively. The (S/S0) values were computed using input parameters of B0 = 3T, D = 1.3×10−5 cm2/s, Δt = 0.2 ms, TE = 50 ms TR = 1500 ms, α = 90°, T20* = 50 ms, r1 = 3.9 mM−1s−1, r2 = 5.3 mM−1s−1 and Pm = 0.
Figure 7.
Kidney vascular structure extracted from micro-CT.
Kidney vasculature extracted from micro-CT along with representative MR voxel-sized (1 mm3) microvascular models taken from different sections of the kidney vasculature with their respective vascular volume fractions. The existence of the bubble-like structures demonstrates the filling of glomeruli with Microfil but a higher resolution would be required to differentiate the individual capillaries.
Figure 8.
Computed kp values for vascular structure extracted from micro-CT.
(a) SE and (b) GE kp values as a function of vascular volume fraction computed using the FPFDM for the kidney microvascular models (with vascular volume fractions >0.1%) shown in Fig. 7. SE kp values ranged from 3.6–27.8 (mM–sec) −1, and GE kp values ranged from 53.8–174.3 (mM–sec) −1. Above 5% volume fraction, the GE kp values were relatively constant with a mean value of 103.3(mM–sec) −1.