Partial volume correction of quantitative susceptibility maps for oxygen extraction fraction measurements.

Phillip G. D. Ward^{1,2}, Audrey P. Fan^{3}, Parnesh Raniga^{1}, David G. Barnes^{2,4}, David L. Dowe^{2}, and Gary F. Egan^{1,5}

There is potential for characterising venous health and metabolic demand with oxygen extraction fraction (OEF) measurements^{1}, using quantitative susceptibility mapping^{2} (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).

1. Christen, T., Bolar, D.S., Zaharchuk, G., 2013. Imaging Brain Oxygenation with MRI Using Blood Oxygenation Approaches: Methods, Validation, and Clinical Applications. American Journal of Neuroradiology 34, 1113–1123. doi:10.3174/ajnr.A3070

2. Liu, C., Li, W., Tong, K.A., Yeom, K.W., Kuzminski, S., 2015. Susceptibility-weighted imaging and quantitative susceptibility mapping in the brain. J. Magn. Reson. Imaging 42, 23–41. doi:10.1002/jmri.24768

Figure 1: Diagram of vein cross section, depicted as a circle, with overlaid grid lines. Two triangles are drawn. The first is from the vein center to the two points at which one grid line intersects the vein wall, with angle $$$\Theta_1$$$. The second is identical, using the other grid line, with angle $$$\Theta_2$$$. The radius is labelled $$$R$$$, and the spacing between grid lines $$$\Delta x$$$.

Figure 2: Scatter plot of actual vein QSM vs. estimated vein QSM for each vein segment. The vein QSM is calculated from a linear fit between partial volume and non-tissue QSM voxel values. A linear fit and r-squared value are included.

Figure 3: Scatter plot of actual vein radius vs. estimated vein radius from trigonometric model fit for each vein segment. A linear fit and r-squared value are included.

Figure 4: Scatter plot of contrast-to-noise ratio of vein segments vs. mean-squared error of partial volume map for each vein segment. 92% of vein segments fall below the red line at 0.05 mean-squared error. Contrast-to-noise ratio is calculated as the difference between an entirely venous voxel and an entirely tissue voxel divided by the standard deviation of the added noise.

Figure 5: Cross-sectional slice of vein overlaid with actual and estimated vein center point and vein wall. Vein radius = 1 voxel, Venous QSM = 0.04 ppm.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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