Ramin Jafari^{1}, Pascal Spincemaille^{2}, Martin R. Prince^{2}, Liangdong Zhou^{1}, and Yi Wang^{1}

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is a noninvasive

7 patients with liver lesions including metastasis (n=1), hemangioma (n=1), focal nodular hyperplasia (n=2), and hepatocellular carcinoma (n=3) were included in this study after removing all patient identifiers. T1-weighted images using 3D stack of spiral sequence (1) were acquired at 1.5 Tesla (GE Healthcare) before and after the injection of contrast agent (gadoxetate). Image parameters were: scan time 54s, frame rate 2.6s, 15º flip angle, 1.76×1.76×5 mm3 voxel size, TR/TE 6/0.6ms. Slices with the most prominent tumors were selected in aorta (AO) along with portal vein (PV) ROIs as the inputs. Signal intensity was converted to relative enhancement (2) and Kety’s based dual input single CM was used:

$$(k_a,k_p,k_2 )=argmin\sum_{t}||\dot{C}(t)-k_aC_a(t)-k_pC_p(t)+k_2C(t)||_2^2, [1]$$

With $$$\dot{C}(t)$$$ the temporal derivative of concentration $$$C(t)$$$, $$$C_a(t)$$$ AO concentration, $$$C_p(t)$$$ PV concentration, inflow rate constants $$$ (k_a,k_p)$$$ and outflow rate $$$(k_2)$$$. A linear formulation combined with a conjugate gradient solver allowed a significant speeding up of the perfusion parameter mapping (3).

QTM is based on the observation that the transport of molecular particles is governed by the Fokker-Planck (FP) equation that describes the temporal and spatial evolution of the probability density function of the velocity of the particles (4). In our study, the FP equation was solved in a Bayesian setting by regularization the gradient of the velocity field (5,6):

$$\textbf{U}=argmin\sum_{t}||\dot{C}(t,\textbf{r})-\triangledown C(t,\textbf{r}).\textbf{U}(\textbf{r})||_2^2+λ(||\triangledown u(\textbf{r})||_2^2+(||\triangledown v(\textbf{r})||_2^2+(||\triangledown w(\textbf{r})||_2^2), [2]$$

Where the first term is the FP based data fidelity, $$$ \textbf{U}(\textbf{r})={u}(\textbf{r})\widehat{x}+{v}(\textbf{r})\widehat{y}+{w}(\textbf{r})\widehat{z}$$$ the 6D velocity vector field, $$$\triangledown$$$ spatial gradient, and $$$λ$$$ regularization parameter. The velocity is assumed to be constant over time and the flow is incompressible. Velocity magnitude ($$$ |\textbf{U}|=\sqrt{u^2+v^2+w^2}$$$) is calculated from velocity vector field.

A correlation was performed between the two methods. To test sensitivity of CM to choice of input function (IF) ROIs (liver, lesion) were selected in one FNH case. Since liver is mostly supplied by the PV and the lesion by AO, in liver only the PV input function was changed by selecting different slices on which the PV input function ROI (n=10) was drawn. In the lesion case, only the AIF (n=10) was changed similarly.

In Figure 1, lesions are shown with red arrows. The corresponding total flow $$$(k_a+k_p)$$$ from CM is compared with the results from QTM (velocity magnitude). In metastasis, HCC, and hemangioma cases, elevated flow and velocity is observed compared to background liver. In FNH, both higher and lower total flow and velocity is observed, with the two methods in agreement. The mean Pearson value (r=0.52) shows moderate correlation between these two methods with the highest correlation in metastasis (r=0.71) and lowest correlation in FNH (r=0.46). Figure 2 shows the sensitivity of the CM to the choice of IF ROI. In the lesion maximum relative difference of 7.4% was observed while in the liver this number was 24.7%.

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2. Schabel MC, Parker DL. Uncertainty and bias in contrast concentration measurements using spoiled gradient echo pulse sequences. Phys Med Biol 2008;53(9):2345-2373.

3. Jafari R, Chhabra S, Prince MR, Wang Y, Spincemaille P. Vastly accelerated linear least-squares fitting with numerical optimization for dual-input delay-compensated quantitative liver perfusion mapping. Magn Reson Med 2017;79(4):2415-2421.

4. Darve E, Solomon J, Kia A. Computing generalized Langevin equations and generalized Fokker-Planck equations. Proc Natl Acad Sci U S A 2009;106(27):10884-10889.

5. Spincemaille P, Zhang Q, Nguyen TD, Wang Y. Vector Field Perfusion Imaging. Proc. Intl. Soc. Mag. Recon. Med. 25, 2017.

6. Zhou L, Spincemaille P, Zhang Q, Nguyen TD, Wang Y. Vector Field Perfusion Imaging: A validation Study by using Multiphysics model. Proc. Intl. Soc. Mag. Recon. Med. 26, 2018.

Figure 1. Comparison of parameters in compartment modeling approach (total flow) with the proposed QTM method (velocity) in malignant (metastasis, HCC) and benign (FNH, hemangioma) hepatic lesions.

Figure 2. Scatter and box plots showing sensitivity of compartment modeling method to the choice of input function. In liver, while arterial input function was kept identical, Portal venous input function was selected on different slices and the total flow in the liver with new input functions was compared with the total flow from original input functions. In the lesion similar analysis was performed except portal venous input function was kept identical and different ROIs were selected for arterial input function.