There is potential for characterising venous health and metabolic demand with oxygen extraction fraction (OEF) measurements¹, using quantitative susceptibility mapping² (QSM). To calculate OEF it is necessary to create a venogram by segmenting the venous voxels. Binary venograms are common and suggest no voxel contains both vein and non-vein components.

This can lead to underestimation of OEF in small veins due to partial volume effects. Venogram erosion or taking the largest QSM value over a segment of vein may help overcome this, but both approaches throw away information by eliminating voxels making them more susceptible to noise. Further, they rely on an entirely venous voxel occurring, and may exclude entire veins.

In this work we present a technique for estimating partial volume by post-processing binary venograms. Trigonometric properties are exploited to fit a cylindrical model using all voxels in vein cross sections. The vessel geometry, partial volume maps, and venous QSM estimates are found to be highly accurate and reliable on simulated data.

The binary mask is processed one cross-sectional plane at a time along the vein axis. Vessel centre point and radius is estimated, and smoothed along the vein. A partial volume map is derived to calculate venous QSM.

The vessel centre point (and radius) is estimated along an axis, using two grid lines. It can be shown that the fraction of vessel area (signal) on the lesser side of a grid line obeys Equation [1].

$$\frac{S_1}{S}=\frac{1}{2\pi}\left(\theta-sin\theta\right) \>\>\>\>\> [1]$$

where $$$\frac{S_1}{S}$$$ is the ratio of QSM signal on the lesser side of a grid line, and $$$\theta$$$ is the angle between the vein centre and the two points at which the vein edge intersects the grid line (Figure 1).

By using Equation [1] to calculate $$$\theta$$$ for two separate grid lines ($$$\theta_1,\theta_2$$$ respectively), and the spacing between grid lines ($$$\Delta x$$$), the centre point along the axis and the radius ($$$R$$$) is derived.

$$R=\frac{\Delta x}{cos\left(\frac{\theta_1}{2}\right)+cos\left(\frac{\theta_2}{2}\right)} \>\>\>\>[2]$$

This process is used for both in-plane axes to provides the coordinates of the centre point and two estimates of vessel radius which are averaged.

This approach ignores the tissue signal which surrounds the vein, and results in an initial overestimation of vessel radius. This is corrected iteratively by estimating the tissue signal, $$$S_{tissue}$$$, using the voxels with zero partial volume, then removing the tissue component using the compliment of the partial volume map. Once the tissue component has been removed, the vessel radius is recomputed. The process is repeated until the change in vessel radius is below a tolerance (0.01 voxels).

Finally the venous QSM signal, $$$S_{vein}$$$, is calculated as the gradient of a least-squares linear fit of Equation [3] for voxels with non-zero partial volume.

$$ S_i - S_{tissue} (1 - p_i) = S_{vein} p_i \>\>\>\>[3]$$

where $$$S_i$$$ and $$$p_i$$$ are the QSM signal and partial volume for voxel $$$i$$$.

Uniformly randomized centre lines, radii and venous QSM for 100 vein segments were generated on 10x10x3 grids (radius=[0.75,2] voxels, venous QSM=[0.02,0.12] ppm). QSM maps were derived using 0.01 ppm QSM in tissue, and Gaussian noise was added (μ=0, σ=0.01).

The radius and centre point were estimated for all slices. To simulate the sliding window smoothing, the mean estimates from all three slices were used to calculate a partial volume map for the middle slice (z=2). The partial volume map was then used in Equation [3] to calculate the vein only QSM value, $$$S_{vein}$$$.

The error between venous QSM estimates and known truth was small, and no systematic bias was found (Figure 2). The error in vein radius estimation was minimal in most cases (Figure 3). The mean-squared error of estimated partial volume maps was less than 0.05 in 92% of cases (Figure 4). QSM values from partial volume corrected vein voxels were found to be highly consistent along the vein. The technique performed accurately even on thin veins with low vein QSM signal and low contrast-to-noise (Figure 5).

The ability to post-process binary venograms and produce accurate partial volume maps has applications in many fields. The proposed method has been shown to be robust and accurate for simulated veins which are perpendicular to the acquisition plane. It is important to note this work does not address non-axially aligned vessels, and simulated data does not explicitly incorporate the point-spread function or non-uniformity of brain parenchyma. Future work will be focused on generalizing the technique to the elliptical cross-sections of non-axially aligned veins, addressing curvature, and validating the technique on in-vivo data.