A flexible framework for sequential estimation of model parameters in computational hemodynamics

Fig. 1. Vascular geometry, reconstructed from MRI data, showing the PC-MRI flow acquisition planes, and the corresponding flow waveforms

Fig. 2. Top: problem setup for an idealized carotid artery. The plane where the volumetric flow and cross section-averaged pressure are observed is denoted in blue. The inflow velocities are prescribed using a typical carotid flow waveform and a parabolic velocity profile. The outflow face is coupled to a three-element Windkessel. Bottom: flow and pressure synthetic data (with added Gaussian white noise) created from the mid-vessel observation site during the forward simulation

Fig. 3. Diagram of the estimation procedure during a single time step, showing the interaction between the FSI model and the filtering algorithm. The green box identifies the constrained least squares step, described in “Estimation algorithm—ROUKF-CLS for arbitrary LPNs” section, which is only part of the ROUKF-CLS algorithm (not of the basic ROUKF procedure)

Fig. 4. Signed wall distance metric for wall motion observations. a The distance metric is the Euclidean distance between a point and the surface, and is positive in the outward-pointing unit normal direction, and negative otherwise. A distance map between points on the deformed wall boundary and the j-th data surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{j}$$\end{document}Sj is shown. b The signed metric is used to compute an interpolated distance between consecutive times where wall position data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{j}$$\end{document}Sj are available. Note that the blue curve represents the current wall boundary; the “interpolated” surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

Fig. 5. Evolution of the parameters in the idealized carotid example with synthetic data. In cases A–C (top) only a single parameter was estimated. In cases D, E (middle), two parameters were simultaneously estimated. In case F (bottom) all three parameters were simultaneously estimated. The true values, initial guesses, and final estimated values for the parameters are shown in the table. The shaded regions depict plus/minus the estimation error standard deviation

Fig. 6. Simultaneous estimation of wall stiffnesses and Windkessel compliance. Top: the simulation set-up. The vessel surface is divided into three regions, each with a different value of stiffness, E. The inflow rate is given by a prescribed waveform, and a three-element Windkessel model is coupled at the outlet. The three values of E and the Windkessel compliance, C, will be estimated. Middle: evolution of the parameters during estimation. Solid lines denote the estimates; dashed lines indicate their true values. Shaded regions depict the estimation standard deviation. Bottom: the results are summarized

Fig. 7. Spatial distribution of aortic wall properties. Left: stiffness, E. Right: wall thickness, h

Fig. 8. Boundary conditions, outlet numbering, and observation planes (depicted in blue, roman numerals) in the subject-specific aorta with synthetic data. The table shows the different sets of observations considered. In all cases, flow was observed at every branch. In case A, no pressure observations were available. Case B used a single pressure observation in the left carotid (outlet III). In Case C, pressure was observed in every branch

Fig. 9. Subject-specific aorta with synthetic data: evolution of Windkessel parameters during estimation, assuming error-free initial state variables. (Left) Case A: estimation with no pressure observations. (Middle) Case B: estimation run with a single pressure observation in the left carotid artery (cross-section III in Fig. 8). (Right) Case C: estimation run with a pressure observation in each of the branches (see Fig. 8). From top to bottom: Windkessel compliance, C, proximal resistance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_1$$\end{document}R1, and distal resistance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2$$\end{document}R2. Dashed lines denote the true parameter value used in the forward simulation and the solid lines the estimate. The shaded regions cover the standard deviation of the parameter…

Fig. 10. Subject-specific aorta with synthetic data: evolution of Windkessel parameters during estimation with initial errors introduced into the pressure field. (Left) Case A-PE: estimation run with no pressure observations. (Right) Case B-PE: estimation run with a single pressure observation in the left carotid artery (cross-section III in Fig. 8). From top to bottom: pressure waveforms at outlets 1 (innominate artery, blue lines) and 9 (left common iliac, black lines); Windkessel compliance, C; proximal resistance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_1$$\end{document}R1; and distal resistance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2$$\end{document}R2. The dashed lines denote the truth and the solid lines the estimate. The shaded regions cover the standard deviation of the parameter estimates. In both…

Fig. 11. Left: the flow observation sites (roman numerals) in the simulated aorta, which coincide with the flow acquisition planes in the PCMRI scans. Waveforms at oulets IV and VI were not measured directly with MRI, but were assumed to be identical to those measured at sites V and VII, respectively. Parameters at outlets 4 (celiac trunk) and 5 (superior mesenteric artery) were not estimated due to the lack of flow data. For simulation, three-element Windkessel models are attached at each outlet, indicated by the asterisks. Right: flow waveforms derived from PCMRI data were used in the estimation problem

Fig. 12. Subject-specific aorta with PC-MRI and applanation tonometry data: evolution of Windkessel parameters during estimation for each outlet. Top: compliance C; middle: distal resistance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2$$\end{document}R2; bottom: proximal resistance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_1$$\end{document}R1. The solid lines denote the estimated values for each parameter; the shaded regions show the range of estimated values plus/minus the the estimation error standard deviation, providing a visual representation of the estimation error