Assimilation of velocity data into fluid dynamic simulations, an application to computational hemodynamics Open Access

Abstract

Cardiovascular applications recently fostered the development of numerical methods for fluid dynamics. Furthermore, thanks to new precise measurement devices and efficient image processing techniques, medicine is experiencing a tremendous increment of available data, inevitably affected by noise. Beyond validation, these data can be combined with numerical simulations in order to develop mathematical tools, known as data assimilation (DA) methods, of clinical impact. In the context of hemodynamics, accuracy and reliability of assimilated solutions are particularly crucial in view of possible applications in the clinical practice. Hence, it is fundamental to quantify the uncertainty of numerical results.

In this thesis, we propose a robust DA technique for the inclusion of noisy velocity measures, collected from magnetic resonance imaging, into the simulation of hemodynamics equations, namely the incompressible Navier-Stokes equations (NSE). The technique is formulated as a control problem where a weighted misfit between velocity and data is minimized under the constraint of the NSE; the optimization problem is solved with a discretize then optimize approach relying on the finite element method. The control variable is the normal stress on the inflow section of the vessel, which is usually unknown in real applications. We design deterministic and statistical estimators (the latter based on a Bayesian approach to inverse problems) for the estimation of the blood velocity and its statistical properties and of related variables of medical relevance, such as the wall shear stress. We also derive conditions on data location that guarantee the existence of an optimal solution.

Numerical simulations on 2-dimensional and axisymmetric 3-dimensional geometries show the consistency and accuracy of the method with synthetic noise-free and noisy data. Simulations on 2-dimensional geometries approximating blood vessels demonstrate the applicability of the approach for hemodynamics applications.

Table of Contents

Glossary 1
1 Introduction 3
1.1 Methods for DA in fluid geophysics . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Objective analysis and optimal interpolation . . . . . . . . . . . . 6
1.1.2 Kalman filter and its extensions . . . . . . . . . . . . . . . . . . . 7
1.1.3 Nudging method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Variational approaches . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Recent advances in DA for the cardiovascular system . . . . . . . . . . . 15
1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Preliminary analysis 21
2.1 Candidate methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Splitting techniques . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Control based methods . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 Dynamic relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Further analysis of the DO method . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 The software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 A deterministic approach to DA 43
3.1 Mathematical formulation of the DA problem . . . . . . . . . . . . . . . 44
3.2 Discretize then optimize method for the linear problem . . . . . . . . . . 46
3.2.1 Non-singularity of the reduced Hessian . . . . . . . . . . . . . . . 49
3.2.2 Forcing optimality via interpolation . . . . . . . . . . . . . . . . . 54
3.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Iterative procedure for the nonlinear problem . . . . . . . . . . . . . . . 65
3.3.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Towards real geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 A statistical approach to DA 79
4.1 The multivariate normal distribution . . . . . . . . . . . . . . . . . . . . 80
4.1.1 The multivariate normal PDF . . . . . . . . . . . . . . . . . . . . 80
4.1.2 Properties of the multivariate normal distribution . . . . . . . . . 82
4.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1 The formulation for the nonlinear NSE . . . . . . . . . . . . . . . 86
4.2.2 Statistical point estimators . . . . . . . . . . . . . . . . . . . . . . 87
4.2.3 Statistical spread estimators . . . . . . . . . . . . . . . . . . . . . 89
4.3 Interpolation of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4.1 Point estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.2 Test case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.3 Test case III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.4 Test case IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.5 Spread estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.6 Towards real geometries . . . . . . . . . . . . . . . . . . . . . . . 102

5 DA for the unsteady NSE 109
5.1 The linearized problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 Non-periodic formulation . . . . . . . . . . . . . . . . . . . . . . 112
5.1.2 Periodic formulation . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Formulation for the nonlinear problem . . . . . . . . . . . . . . . . . . . 122
5.2.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Conclusion 131

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Computer Science Engineering, Biomedical Mathematics
Keyword	Inverse Problems Data Assimilation Hemodynamics Computational Fluid Dynamics Optimization
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Benzi, Michele, Emory University Nagy, James, Emory University

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