Inverse problems in reduced order models of cardiovascular haemodynamics: aspects of data assimilation and heart rate variability

Inverse problems in cardiovascular modelling have become increasingly important to assess each patient individually. These problems entail estimation of patient-specific model parameters from uncertain measurements acquired in the clinic. In recent years, the method of data assimilation, especially the unscented Kalman filter, has gained popularity to address computational efficiency and uncertainty consideration in such problems. This work highlights and presents solutions to several challenges of this method pertinent to models of cardiovascular haemodynamics. These include methods to (i) avoid ill-conditioning of the covariance matrix, (ii) handle a variety of measurement types, (iii) include a variety of prior knowledge in the method, and (iv) incorporate measurements acquired at different heart rates, a common situation in the clinic where the patient state differs according to the clinical situation. Results are presented for two patient-specific cases of congenital heart disease. To illustrate and validate data assimilation with measurements at different heart rates, the results are presented on a synthetic dataset and on a patient-specific case with heart valve regurgitation. It is shown that the new method significantly improves the agreement between model predictions and measurements. The developed methods can be readily applied to other pathophysiologies and extended to dynamical systems which exhibit different responses under different sets of known parameters or different sets of inputs (such as forcing/excitation frequencies)

Hemodynamics in a giant intracranial aneurysm characterized by in vitro 4D flow MRI

Experimental and computational data suggest that hemodynamics play a critical role in the development, growth, and rupture of cerebral aneurysms. The flow structure, especially in aneurysms with a large sac, is highly complex and three-dimensional. Therefore, volumetric and time-resolved measurements of the flow properties are crucial to fully characterize the hemodynamics. In this study, phase-contrast Magnetic Resonance Imaging is used to assess the fluid dynamics inside a 3D-printed replica of a giant intracranial aneurysm, whose hemodynamics was previously simulated by multiple research groups. The physiological inflow waveform is imposed in a flow circuit with realistic cardiovascular impedance. Measurements are acquired with sub-millimeter spatial resolution for 16 time steps over a cardiac cycle, allowing for the detailed reconstruction of the flow evolution. Moreover, the three-dimensional and time-resolved pressure distribution is calculated from the velocity field by integrating the fluid dynamics equations, and is validated against differential pressure measurements using precision transducers. The flow structure is characterized by vortical motions that persist within the aneurysm sac for most of the cardiac cycle. All the main flow statistics including velocity, vorticity, pressure, and wall shear stress suggest that the flow pattern is dictated by the aneurysm morphology and is largely independent of the pulsatility of the inflow, at least for the flow regimes investigated here. Comparisons are carried out with previous computational simulations that used the same geometry and inflow conditions, both in terms of cycle-averaged and systolic quantities.ISSN:1932-620

PC-MRI parameters.

<p>PC-MRI parameters.</p

A schematic of the experimental setup.

<p>The black solid lines indicate the flow circuit, whereas the red dashed lines indicate the electronics connections. Note that the components are not to scale.</p

Flow structure at: (a) systole (0.25 ≲ <i>t</i>/<i>T</i> ≲ 0.31); (b) cycle-averaged state (0 ≤ <i>t</i>/<i>T</i> ≤ 1); (c) steady state with the Reynolds number at the systolic peak, <i>Re</i> = 651; (d) steady state at <i>Re</i> = 1000.

<p>The velocity iso-surface thresholds are , , , and , respectively.</p

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Normalized velocity magnitude () at three selected cut planes in the aneurysm sac.

<p>Each row shows one cardiac phase. Note that for better visibility, every third vector in each direction is shown.</p

Input flow rate waveform along with the binned input waveform, and the inflow waveform reconstructed from PC-MRI at the inlet of the aneurysm.

<p>The flow rate and pressure waveforms measured upstream of the aneurysm using the ultrasonic flow meter and pressure transducer respectively are shown for comparison.</p

Columns from left to right: Iso-surface of the velocity magnitude at , streamlines, vortex lines, and the measurement phase.

<p>Five measurement phases from top to bottom rows are early systole, systolic peak, late systole, dicrotic notch, and late diastole, respectively. Streamlines are colored by the normalized velocity magnitude (), and vortex lines are colored by the normalized vorticity magnitude ().</p

Normalized relative pressure, , and normalized velocity magnitude, , at the phantom centerline.

<p>Each blue line represents velocity distribution at the centerline in one phase from diastole (light blue) to systole (dark blue) with the dashed blue line showing the cycle-averaged velocity. The red color coding shows the pressure distribution at the centerline in the same fashion. The centerlines shown at the top right and bottom right are respectively colored by the local systolic velocity and systolic pressure from blue (low) to red (high).</p