Veins affect voxel magnitude and phase signals in GRE acquisitions by virtue of the high magnetic susceptibility of deoxygenated blood. Estimation of vessel caliber and oxygenation from susceptibility would - along with blood velocity measurements - enable quantification of local CMRO₂ [1], but partial-volume effects confound the estimation of both. Previously, we presented a method for accurately measuring blood vessel oxygenation in the presence of partial-volume effects called Joint Utilization of Magnitude and Phase (JUMP). In this work, we present a novel approach that accurately measures both blood vessel caliber and oxygenation state and is robust to partial-volume effects, which we call Multi-Voxel Joint Utilization of Magnitude and Phase (MV-JUMP).

We assume that blood vessel segments are straight and oriented nearly parallel to the main field $$$B_0$$$. The blood vessel voxel magnitude and phase signals in a GRE acquisition both depend on the vessel oxygenation saturation ($$$Y_v$$$) and the fraction of the voxel comprised of blood ($$$\alpha$$$). This relationship can be described by a two-compartment model [2]. If we impose that all voxels in a vessel have the same $$$Y_v$$$, then we can estimate $$$Y_v$$$ and each voxel's partial-volume fraction by solving the following minimization problem: $$\min_{\{\hat{\alpha}_k\},\hat{Y}_v}\left[\sum_{k}\sum_{i}\max(\varphi_{v,ik},0.1)\cdot{\left\|\hat{M}_{v,i}(\hat{\alpha}_{k},\hat{Y}_{v})\cdot e^{j\hat{\varphi}_{v,i}(\hat{\alpha}_{k},\hat{Y}_{v})}-M_{v,ik}\cdot e^{j\varphi_{v,ik}}\right\|}^{2}\right]$$Here, $$$\hat{Y}_{v}$$$ is the estimated vessel oxygen saturation, $$$\{\hat{\alpha}_{k}\}$$$ are the estimated voxel partial-volume fractions,and $$$M_{v,ik}$$$ and $$$\varphi_{v,ik}$$$ are the measured voxel signal magnitudes and phases. $$$\hat{M}_{v,i}(\hat{\alpha}_{k},\hat{Y}_{v})$$$ and $$$\hat{\varphi}_{v,i}(\hat{\alpha}_{k},\hat{Y}_{v})$$$ describe the forward signal model in [2], and further account for the $$$Y_v$$$-dependence of blood $$${R}_{2}^{*}$$$. The sub-indices $$$k$$$ and $$$i$$$ indicate voxel and echo time ($$$TE$$$), respectively. The thresholding term limits the contribution of low-SNR voxels to the $$$\hat{Y}_{v}$$$ solution. As a consequence of the sinc-shaped voxel function implicit in k-space windowing, $$$\hat{\alpha}_k$$$ is constrained to be between -0.1628 and +1.39. Venous cross-sectional area ($$$\hat{A}_{cs}$$$) is calculated by summing $$$\hat{\alpha}_{k}$$$ from all voxels comprising an axial section of the vessel. The cross-sectional area is then: $$\hat{A}_{cs}=\hat{\alpha}_\Sigma \cdot \delta_x \cdot \delta_y \cdot \cos(\theta)$$ $$$\hat{\alpha}_{\Sigma}$$$ is the axial section $$$\hat{\alpha}_{k}$$$ sum, $$$\delta_x$$$ and $$$\delta_y$$$ are the nominal voxel dimensions along the x- and y-axes, and $$$\theta$$$ is the angle made by the vessel with $$$B_0$$$. Averaging $$$\hat{A}_{cs}$$$ from all axial sections then gives an average vessel cross-sectional area.

Numerical vessels (Figure 1) and in vivo vessels (Figure 2) were analyzed. Dual-echo ($$$TE=8.1ms, 20.3ms$$$) GRE data from [2] were used to analyze four in vivo vessels from 2 subjects. All phase images were unwrapped with the robustunwrap tool [3] and processed with SHARP to remove background phase [4]. Vessel ROIs were manually identified, and included all voxels that visibly coincided with the vessel (Figures 1-2). Vessels were then analyzed with MV-JUMP. The MV-JUMP minimization problem was solved using the MATLAB fmincon tool (initial guesses for all runs: $$$\hat{Y}_v=0.6$$$; $$$\hat{\alpha}_k=0.6$$$ if $$$\varphi_{v,k}>0$$$, $$$\hat{\alpha}_k=-0.1$$$ if $$$\varphi_{v,k}<0$$$ at $$$TE=20.3ms$$$). Figure 3 shows the measured $$$\hat{Y}_v$$$ and average $$$\hat{A}_{cs}$$$ for numerical vessels. $$$\hat{Y}_v$$$ was always within 10% of the ground truth vessel $$${Y}_v$$$ for voxel sizes under 2mm isotropic. $$$\hat{A}_{cs}$$$ deviated from ground truth by under 4% at $$$Y_{v}=0.8$$$; by under 10% at $$$Y_{v}=0.7$$$; and by under 19% at $$$Y_{v}=0.6$$$ for voxel sizes under 2mm isotropic. Figure 4 shows the measured vessel $$$\hat{Y}_v$$$ and $$$\hat{A}_{cs}$$$ for two vessels in each subject. Vessels 3 and 4 could not be properly identified in Subject 2 at 0.6mm isotropic due to reconstruction artifacts. Measured mean $$$\hat{Y}_v$$$ across acquisition resolutions varied between 0.575 and 0.701 for the 4 vessels, and all acquisitions for a vessel gave $$$\hat{Y}_v$$$ within 9% of the vessel mean. Measured mean $$$\hat{A}_{cs}$$$ across resolution varied between 0.545mm² and 2.2mm². All acquisitions for a given vessel gave $$$\hat{A}_{cs}$$$ within 16% of the vessel mean.

MV-JUMP accurately measured $$$\hat{Y}_v$$$ and $$$\hat{A}_{cs}$$$ in vessels with $$$Y_v$$$ between 0.6 and 0.8 and with a tilt angle of 20°. In in vivo experiments, MV-JUMP provided average measurements of $$$\hat{Y}_v$$$ that fell within or near this range. Also, variations in $$$\hat{Y}_v$$$ and $$$\hat{A}_{cs}$$$ for in vivo vessels at different acquisition resolutions were similar to the variations seen in simulation. This suggests that MV-JUMP accurately measured both oxygenation and caliber from the vessels analyzed in this study. This approach is currently limited to vessels oriented close to $$$B_0$$$. Increasing extravascular fields and the “magic angle” problem are ignored in the physical model used here, but become important at large tilt angles. The MV-JUMP inverse problem is capable of using a model that accounts for these concerns through modification of the form of $$$\hat{M}_{v,ik}$$$ and $$$\hat{\varphi}_{v,ik}$$$, and future work will look to develop such a model.