Carson Anthony Hoffman^{1}, Oliver Wieben^{1,2}, and Gabe Shaughnessy^{1}

4D flow magnetic resonance imaging (MRI) can provide a way to analyze both the anatomical and hemodynamic properties related to complex vessel networks. Using basic principles related to flow conservation the entire vessel networks data can be used to help improve local flow calculations. A Bayesian approach is utilized with a Markov Chain Monte Carlo where flow conservation is enforced to obtain, for a complete vascular network, estimates of mean flow and flow uncertainty. The estimated data results in a lower flow uncertainty overall and can allow for localization of potential erroneous branches in the initial data.

Figure 1: The
Initial segmentation is completed with a global threshold then manual
segmentation in order to get a vessel branching that is expected to conserve
flow. Average flow calculations, flow uncertainties, and the directed adjacency
matrix are fed into a Bayesian fit which applies a Markov Chain Monte Carlo
algorithm. Visualization and quantification of output data is then completed by
three different methods: bar graph plots, color and data encoded node networks,
and color encoded anatomical vessels.

Figure
2 Color and data encoded branch networks provided a way to present all of the
computed variables in a single figure. The circles represent branches and the
red lines indicate the connections between branches. The color intensity
represents the level of deviation from the initial input flow value. The red or
blue coloration indicates if the variation was in a positive or negative
direction. All of the input and estimated flow values with associated statistics
are presented as text in each circle.

Figure 3 The
bar charts show the comparison between input flow values and best fit flow
values with an ordering which is based on the initial flow estimate. The pull is computed as the difference of
input and fit flow values divided by the initial standard deviation. The
uncertainty improvement is the ratio of the initial standard deviation to the
fit standard deviation, and is a measure of the improvement afforded from the
Bayesian fit. The fractional deviation is the relative change of the fit flow
estimate relative to the initial flow.

Figure
4: A color encoded mask for (a) input flow and (b) best fit flow estimations were
completed for all cases to localize where major flow changes were occurring. The
initial standard deviation of flow for each branch (c) and the uncertainty
estimate from the fit (d) were computed to highlight initial branches with
large variation in flow through the vessel. A global reduction in uncertainty
in flow measurements can clearly be seen between (c) and (d).

Table 1: The
average improvement in uncertainty and pull root mean square (RMS) was computed
for all cases. It is expected that the pull RMS should be not much larger than
unity, indicating that the potential flow measurements are balanced by the
statistical uncertainty. Over the cases, an overall uncertainty improvement was
found to be 63.2%.