Abstract:

Cardiovascular diseases are the major cause of death around the world. It is estimated that 20% of the population is affected by elevated arterial wall stiffness which increases the risk of aneurysms rupture. Also, in the case of narrowed vessels due to arterial wall remodelling the blood flow is disturbed which may lead to an increased hemodynamic pressure gradient and increased cardiac load. Therefore, the parameters describing the wall stiffness and pressure gradient are used as indicators helping with diagnosis of disease severity [1]. Computational models of hemodynamics such as lumped parameter models (referred to as 0D), 1D models and 3D Fluid-Structure Interaction models (FSI) are tools for studying the cardiovascular system and may be further applied to non-invasive diagnostics and disease research [2, 3].

The parameter estimation problem may be solved by variational or sequential approach. The sequential approach is iterative in nature and requires many simulations of the forward problem which may be prohibitive in the case of FSI simulations. With a sequential approach based on the Kalman Filter, the model prediction is improved at every time step by measuring the discrepancy between model output and measurements. It was shown that the total computational time for the sequential approach is of the same order of magnitude as the CPU time needed for one forward simulation [4].

For FSI models input parameters like wall stiffness and boundary conditions strongly affect the solution. Therefore, the challenge in constructing a model is the determination of parameters which make the simulation results agree with clinical data. The Kalman filter and its various modifications are used as methods to estimate unknown parameters that may not be directly observable [2].

The goal of this work is to explore the applications of Kalman filters to estimate the unknown parameters and boundary conditions for hemodynamic models and apply it to 1-way and 2-way FSI model of the carotid artery to estimate the Young’s modulus.

Acknowledgments:

The research leading to these results is funded by the Norwegian Financial Mechanism 2014-2021 operated by the National Science Center, PL (NCN) within GRIEG programme under grant UMO 2019/34/H/ST8/00624, project non-invasivE iN-vivo assessmenT Human aRtery wALls (ENTHRAL, www.enthral.pl)

[1] D. Nolte, C. Bertoglio, Inverse problems in blood flow modeling: A review, International Journal for Numerical Methods in Biomedical Engineering 38 (8) (2022) e3613.

[2] C. J. Arthurs, N. Xiao, P. Moireau, T. Schaeffter, C. A. Figueroa, A flexible framework for sequential estimation of model parameters in computational hemodynamics, Advanced modeling and simulation in engineering sciences 7 (1) (2020) 1–37.

[3] A. Quarteroni, A. Veneziani, C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Computer Methods in Applied Mechanics and Engineering 302 (2016) 193–252.

[4] Bertoglio, Cristóbal, Philippe Moireau, and Jean‐Frederic Gerbeau. "Sequential parameter estimation for fluid–structure problems: application to hemodynamics." International Journal for Numerical Methods in Biomedical Engineering 28.4 (2012): 434-455.

Aortic root connects the left ventricle to the ascending aorta and houses the aortic valve (AV) ensuring one-direction flow of blood during systole. The AV is normally composed of three leaflets, known as tricuspid aortic valve (TAV), but 1-2% of the population is born with only two leaflets, known as bicuspid aortic valve (BAV). The patients with BAV are considered at high risk of developing aneurysms and eventually dissection. The biomechanics of aortic root tissues are hypothesized to play an important role in the disease development. In this study, we use in-vivo echocardiographic images from TAV and BAV patients to analyze the differences in the biomechanics of aortic root tissues.

3D transesophageal echocardiographic (TEE) images of the aortic root were retrospectively acquired from 16 patients with the approval of the Institutional Review Board at the University of Pennsylvania. The images were segmented, registered, and converted into a medial model as presented in a previous study. The medial models were remeshed with an quadrilateral elements. Two methods were used for the biomechanical analysis: 1) patient-specific 3D inverse finite element (FE) modeling, and 2) population-level Bayesian inference based on radius variations.

The two approaches provided distinct advantages. The first, patient-specific approach preserves geometric details, but the effect of diastolic pressure and opening angle could not be accounted form. The second, Bayesian approach allowed us to calculate the population-level differences between TAVs and BAVs, but it discarded part of the information available from the images. The biomechanical differences we found in this work indicate that the aortic root tissue in BAV patients experience different intramural stresses that might be linked to the higher risk of aneurysm development. Future work will include implementation of growth and remodeling framework to further establish this link.

The primary function of arteries is as conduits to allow the heart to efficiently deliver blood throughout the entire body. The stiffness of arteries is a key functional parameter that can alter this efficiency, and increased arterial stiffness is a reliable predictor of cardiovascular risk [1]. However, directly determining the stiffness of the arterial wall is essentially impossible in vivo, and proxy measurements such as pulse wave velocity and total arterial compliance are the most feasible clinical measures of arterial stiffness. These, however, only reflect the average stiffness of a region of the arterial network and do not provide information about the local stiffness of arteries. As arterial stiffness is determined by changes in the tissue composition at local levels and diseases like atherosclerosis, aneurysms and dissections occur in relatively localized regions of the arteries, methods to provide accurate information about the local material properties are desirable to enable further research and novel clinical approaches.

We present our implementation of an inverse solver for estimation of local arterial stiffness with a 1D fluid-structure-interaction model of the artery. The model consists of a axisymmetric domain representing a human common carotid artery. The fluid is modeled as a Newtonian fluid with an assumed parabolic flow profile throughout the domain. We investigate two arterial wall models based on the theory of linear elasticity. The first derives from the application of Laplace’s law and the simplifying assumptions of a thin wall and is one of the most widely applied models for pulse wave propagation models of the arteries [2]. The second model is a novel implementation based on thick-walled cylinder theory. We implemented a least-squares procedure to estimate the Young’s modulus parameter in both models, and then evaluated this on experimental data collected from a laboratory phantom. Two sets of boundary conditions were compared. First, direct experimental data of measuremed inlet flow rate and outlet pressure were used. Second, a more general approach of applying a parameterized common carotid inflow and Windkessel outlet was applied. The Young’s modulus estimated with the thin walled approach is in general smaller than that from the thick walled approach. The different boundary conditions produce some what different time courses of pressure and flow as well as a difference in estimate Young’s modulus. Further work will compare the estimated Young’s modulus with stiffness determined through direct tension testing. Additionally, the method will be applied to estimate local arterial stiffness of the common carotid artery based pressure measured by applanation tonometry and flow and geometry by ultrasound imaging.

Acknowledgments

The research leading to these results is funded by the Norwegian Financial Mechanism 2014-2021 operated by the National Science Center, PL (NCN) within GRIEG programme under grant UMO-2019/34/H/ST8/00624, project non-invasivE iN-vivo assessmenT Human aRtery wALls (ENTHRAL, www.enthral.pl)

References

[1] Laurent et al., Eur Heart J; 27(21), 2588-2605 (2006).

[2] Boileau et al., Intl J Num Meth Biomed Eng; 31(10), (2015)

Arterial stiffness is an established biomarker of cardiovascular health [1]. By combining non-invasive measurements and computational models, arterial stiffness can be inferred through solving an optimization problem. However, the non-invasive measurements are hampered by measurement errors, and some parameter values in the optimization problem must be assumed which introduces additional uncertainties. To apply a novel computational model in clinical diagnostics, uncertainty quantification needs to be performed [2]. Model parameters which lead to a large variation in the model prediction can be identified through a subsequent sensitivity analysis. A large number of model evaluations are needed to estimate these sensitivity indices, thus, limiting the application of such analysis to computationally expensive models and allowing relatively few uncertain inputs. Using the Multifidelity Monte Carlo Method (MFMC) [3, 4], we estimate Sobol main and total effect sensitivity indices of a common carotid artery (CCA) 3D-fluid-structure interaction (FSI) model by offsetting the computational burden to computationally affordable 1D- and 0D- models. Computational resources are thus distributed over the three levels of fidelity such that a few 3D-FSI model evaluations ensure accuracy of the sensitivity indices while the lower fidelity models are leveraged to reduce the computational costs of the sensitivity analysis.

We will fist consider the situation where the CCA is modeled as an idealized, straight tube. The same geometric and material parameters as well as boundary conditions are applied to all models. At the inlet, a parabolic physiological flow rate and wave form is prescribed and at the outlet, a three-element Windkessel model mimics the downstream vasculature. In the 1D-model, the artery consists of nodes along a straight line, and in the 0D-model, the artery is represented with a resistor and a capacitor as an electrical analog. The arterial wall is modeled as a linear elastic material and blood is assumed to be a Newtonian fluid. Uncertain model parameters are the vessel diameter, arterial wall thickness, and material parameters for the arterial wall. The uncertainty and sensitivities of the pulse pressure, average pressure, and the diameter change are assessed. We will present the method and implementation we have developed for the CCA model and preliminary results comparing the sensitivity indices estimated through the MFMC approach with the ones estimated with the same computational budget through Monte Carlo simulation of the 3D-FSI model.

Time permitting, results will be shown for hyperelastic material models and patient-specific anatomies.

[1] Laurent et al., Eur Heart J; 27(21), 2588-2605 (2006).
[2] FDA, Assessing the Credibility of Computational Modeling and Simulation in Medical Device Submissions (2021).
[3] Qian et al. , J Uncertainty Quantification; 6(2), 638-706 (2018).
[4] Gorodetsky et al., J Comput. Phys. 408, 109257 (2020).

Acknowledgments:
The research leading to these results is funded by the Norwegian Financial Mechanism 2014-2021 operated by the National Science Center, PL (NCN) within GRIEG programme under grant UMO-2019/34/H/ST8/00624, project non-invasivE iN-vivo assessmenT Human aRtery wALls (ENTHRAL, www.enthral.pl).