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Research Papers

Uncertainty Quantification in a Patient-Specific One-Dimensional Arterial Network Model: EnKF-Based Inflow Estimator

[+] Author and Article Information
Andrea Arnold

Department of Mathematics,
North Carolina State University,
2108 SAS Hall, 2311 Stinson Drive, Box 8205,
Raleigh, NC 27695-8205
e-mail: anarnold@ncsu.edu

Christina Battista

DILIsym Services, Inc.,
Six Davis Drive,
Research Triangle Park, NC 27709
e-mail: cbattista@dilisym.com

Daniel Bia

Department of Physiology,
Universidad de la República,
Montevideo 11800, Uruguay
e-mail: dbia@fmed.edu.uy

Yanina Zócalo German

Department of Physiology,
Universidad de la República,
Montevideo 11800, Uruguay
e-mail: yana@fmed.edu.uy

Ricardo L. Armentano

Department of Biological Engineering,
CENUR Litoral Norte—Paysandú,
Universidad de la República,
Montevideo 11800, Uruguay
e-mail: armen@favaloro.edu.ar

Hien Tran

Department of Mathematics,
North Carolina State University,
2108 SAS Hall, 2311 Stinson Drive, Box 8205,
Raleigh, NC 27695-8205
e-mail: tran@ncsu.edu

Mette S. Olufsen

Department of Mathematics,
North Carolina State University,
2108 SAS Hall, 2311 Stinson Drive, Box 8205,
Raleigh, NC 27695-8205
e-mail: msolufse@ncsu.edu

1Corresponding author.

Manuscript received March 30, 2016; final manuscript received January 31, 2017; published online February 22, 2017. Assoc. Editor: Tina Morrison.

J. Verif. Valid. Uncert 2(1), 011002 (Feb 22, 2017) (14 pages) Paper No: VVUQ-16-1008; doi: 10.1115/1.4035918 History: Received March 30, 2016; Revised January 31, 2017

Successful clinical use of patient-specific models for cardiovascular dynamics depends on the reliability of the model output in the presence of input uncertainties. For 1D fluid dynamics models of arterial networks, input uncertainties associated with the model output are related to the specification of vessel and network geometry, parameters within the fluid and wall equations, and parameters used to specify inlet and outlet boundary conditions. This study investigates how uncertainty in the flow profile applied at the inlet boundary of a 1D model affects area and pressure predictions at the center of a single vessel. More specifically, this study develops an iterative scheme based on the ensemble Kalman filter (EnKF) to estimate the temporal inflow profile from a prior distribution of curves. The EnKF-based inflow estimator provides a measure of uncertainty in the size and shape of the estimated inflow, which is propagated through the model to determine the corresponding uncertainty in model predictions of area and pressure. Model predictions are compared to ex vivo area and blood pressure measurements in the ascending aorta, the carotid artery, and the femoral artery of a healthy male Merino sheep. Results discuss dynamics obtained using a linear and a nonlinear viscoelastic wall model.

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Figures

Grahic Jump Location
Fig. 1

(Left) Mock circulation including a pneumatic pump, a perfusion line connected to the chamber with the mounted vessel segment, a resistance modulator (R), and a reservoir. Blood pressure (P) is measured with a microtransducer while the diameter (D) is measured with a pair of ultrasonic crystals using sonomicrometry. (Middle) The single, straight vessel segment shown with inflow and outflow boundary conditions. (Right) Time series data for area and pressure at the center of the vessel.

Grahic Jump Location
Fig. 2

Arterial network representation showing (from left to right) the carotid artery (CA), the ascending aorta (AA), and the femoral artery (FA). Time series data for area and pressure, along with the corresponding area versus pressure loop, are shown at each vessel location.

Grahic Jump Location
Fig. 3

Illustration of how to form an initial ensemble of inflow profiles by assigning probability distributions at discretized points. Here, for example, a Gaussian distribution is assigned at each of six equispaced time points over the interval [0, 0.53] s. (Left) The assigned distributions have means given by the discretized points on the curve and fixed standard deviations. (Right) At each discretized time point, the corresponding distribution is sampled (here, the sample size is three) and the sample points are connected in an ascending manner to form physically plausible inflow profiles. These profiles are then smoothed with a cubic spline to enforce regularity (not pictured).

Grahic Jump Location
Fig. 4

Results with AA data using the linear Kelvin wall model (top row), the linear Kelvin wall model with increased model innovation (middle row), and the nonlinear sigmoid wall model (bottom row). EnKF estimated inflow profiles (solid red) and ±2 standard deviation curves (dashed red) obtained from an initial ensemble of inflow curves (gray cloud) are shown in ((a), (e), (i)). Predicted area and pressure curves using the EnKF estimated inflow profiles and ±2 standard deviation curves are shown in ((b), (f), (j)) and ((c), (g), (k)), respectively. Corresponding area versus pressure curves obtained using the EnKF estimated inflow profiles are shown in ((d), (h), (l)). Area and pressure data are plotted in black.

Grahic Jump Location
Fig. 5

Predicted area, pressure, and flow at the center of the vessel obtained using the EnKF estimated inflow profiles and ± 2 standard deviation curves for single vessels at different locations in the arterial network. Results are shown for (from left to right) the carotid artery (CA), the ascending aorta (AA), and the femoral artery (FA). Corresponding area versus pressure curves obtained using the EnKF estimated inflow profiles are also shown at each vessel location. Area and pressure data are plotted in black.

Grahic Jump Location
Fig. 6

Results using the linear Kelvin wall model (first and third rows) and the nonlinear sigmoid wall model (second and fourth rows) with synthetic AA data generated using the Kelvin model (first and second rows) and the sigmoid wall model (third and fourth rows). EnKF estimated inflow profiles (solid red) and ±2 standard deviation curves (dashed red) obtained from an initial ensemble of inflow curves (gray cloud) are shown in ((a), (e), (i), (m)). Predicted area and pressure curves using the EnKF estimated inflow profiles and ± 2 standard deviation curves are shown in ((b), (f), (j), (n)) and ((c), (g), (k), (o)), respectively. Corresponding area versus pressure curves obtained using the EnKF estimated inflow profiles are shown in ((d), (h), (l), (p)). The area between ±2 standard deviation uncertainty bounds is shaded in light blue. The true inflow profile and corresponding area and pressure data are plotted in black.

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