David Marlevi ^{1,2}, Bram Ruijsink^{3}, Maximilian Balmus^{3}, Desmond Dillon-Murphy^{3}, Daniel Fovargue^{3}, Kuberan Pushparajah^{3,4}, Pablo Lamata^{3}, C. Alberto Figueroa^{3,5}, Massimiliano Colarieti-Tosti^{1,6}, Matilda Larsson^{1}, Reza Razavi^{3,4}, and David A. Nordsletten^{3}

4D-flow-MRI enables the non-invasive assessment
of cardiovascular pressure drops; however, estimation accuracy depends on vascular
topology and acquisition noise. Here, we present a method that minimizes the
impact of these by using virtual fields to isolate and probe hemodynamic pressure
drops. We show that, *in-silico*, the method accurately assesses pressure drops over multiple
segments of a patient-specific co-arcted aorta (average error below 22%),
independent of anatomical bifurcations. Additionally, the method compares
successfully against catheter measurements, using 4D-flow-MRI *in-vivo* (average
error at peak systole below 15%). With this, the method represents a refined
tool for hemodynamic analysis of cardiovascular flow.

The
proposed method originates from a work-energy formulation assessing relative
pressure drops from a flow field, $$$v$$$^{6}.
Here we propose an alteration by introducing a virtual field, $$$w$$$,
analyzing the virtual work of $$$w$$$ within a vascular domain in order to
isolate the pressure drop, $$$\Delta p$$$. By so, influence from measurement inaccuracies, low flow magnitudes or bifurcations associated with $$$v$$$
are minimized.

Here, $$$w$$$ is chosen as the solution to a Stokes flow boundary value problem,
with flow permitted over the inlet and outlet of the desired pressure drop, and
with flow restricted at any other surface (vascular wall or secondary
bifurcations). Using our previously proposed derivation^{6} in
conjuction with $$$w$$$, $$$\Delta p$$$ can be identified by:

$$ \Delta p=-\frac{1}{Q_w} \left(\frac{\partial}{\partial t}K_w + A_w + V_w \right) $$

with
$$$K_w$$$, $$$A_w$$$, and $$$V_w$$$ being the kinetic, advective, and viscous
energy components when using $$$w$$$ as the weighting velocity field, and
$$$Q_w$$$ being the flow of $$$w$$$ over the domain inlet (for details, see^{6}).

To
assess method performance, two setups are evaluated. First, a challenging patient-specific *in-silico* model
of a co-arcted aorta^{9} is
used to generate simulated image data (spatial/temporal resolution: 2 mm^{3}, 43 ms, SNR: 10 and 30), with temporal pressure drops compared to the
simulated pressure fields. Second, 4D-flow-MRI (Philips Achieva, 1.5T,
spatial/temporal resolution: 2.1 mm^{3}, 31 ms) is acquired from 6 adolescent
Fontan patients, with $$$v$$$ reconstructed and corrected using GTFlow
(GyroTools). Flow-field-based pressure drop estimates are compared against
invasive catheter measurements acquired during the same imaging session. $$$w$$$ is solved using a finite-difference scheme on the vascular
segment of interest using a right-preconditioned iterative GMRES-solver and a
BFBT-preconditioner^{10-11}. Spatial and
temporal noise filtering is introduced by polynomial interpolation and
frequency-domain low-pass filtering, respectively.

Figure 1 shows the results of the
*in-silico* setup. The method estimation follows the true pressure drop throughout
the cardiac cycle, with minor noise-levels differences. Specifically, the
pressure drop is captured with an average error of 22%, equaling 1.9 mmHg at
peak systole. Importantly, the strength of the method is shown in the preserved
accuracy from aortic inlet to coarctation, unaffected by the bifurcations along
the aortic arch, or of the diastolic low-magnitude-flows.

Figure
2 summarizes the results of the *in-vivo *setup.
Again, the method captures temporal changes in relative pressure, with the virtual
field isolating the vascular region-of-interest. The peak systolic pressure is
captured with an average error of 15%, corresponding to an accuracy of 1.3
mmHg. The method seems robust to anatomical bifurcations or
low-magnitude-flows, even though minor fluctuations seem visible in certain
phases.

For
both setups, our method performs favorable to conventional Bernoulli-based estimates
as well as to previous work-energy based measures^{6} (average error above 200%), in part due to that these methods are not developed to handle vascular bifurcations or diastolic flow-magnitudes.

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