University of Groningen
Inverse problems in elastography and displacement-flow MRI
Carrillo Lincopi, Hugo
DOI:
10.33612/diss.112422123
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Publication date: 2020
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Carrillo Lincopi, H. (2020). Inverse problems in elastography and displacement-flow MRI. Rijksuniversiteit Groningen. https://doi.org/10.33612/diss.112422123
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Inverse Problems in Elastography and
displacement-flow MRI
The work in this thesis has been carried out at the Faculty of Physical and Mathematical Sciences, University of Chile and at the Bernoulli Institute, Uni-versity of Groningen. It was financially supported by the Chilean govern-ment through the Comisi´on Nacional de Investigaci´on Cient´ıfica y Tecnol´ogica
(CONICYT, grant number 21151645). Copyright c 2020 Hugo Carrillo Lincopi
ISBN 978-94-034-2402-6 (printed version) ISBN 978-94-034-2403-3 (electronic version)
Inverse Problems in
elastography and
displacement-flow MRI
PhD thesis
to obtain the degree of PhD at the University of Groningen
on the authority of the Rector Magnificus Prof. C. Wijmenga
and in accordance with the decision by the College of Deans
and
to obtain the degree of PhD at the University of Chile,
Faculty of Physical and Mathematical Sciences.
Double PhD degree
This thesis will be defended in public on
Thursday 30 January 2020 at 09:00 a.m.
by
Hugo Patricio Anner Carrillo Lincopi
born on 30 October 1989 in Angol, Chile
Supervisors Prof. A. Osses Prof. R.W.C.P. Verstappen Co-supervisors Dr. C.A. Bertoglio Dr. A.M.S. Waters Assessment committee Prof. B. Jin Prof. J. Ortega
Prof. A. van der Schaft Prof. C. Stolk
UNIVERSIDAD DE CHILE
FACULTAD DE CIENCIAS F´ISICAS Y MATEM ´ATICAS ESCUELA DE POSTGRADO
INVERSE PROBLEMS IN ELASTOGRAPHY AND DISPLACEMENT-FLOW MRI
TESIS PARA OPTAR AL GRADO DE DOCTOR EN CIENCIAS DE LA INGENIER´IA,
MENCI ´ON MODELACI ´ON MATEM ´ATICA
EN COTUTELA CON LA UNIVERSIDAD DE GRONINGEN
HUGO PATRICIO ANNER CARRILLO LINCOPI
PROFESORES GU´IAS: AXEL OSSES ALVARADO
ROEL VERSTAPPEN
MIEMBROS DE LA COMISI ´ON: BANGTI JIN
JAIME ORTEGA PALMA ARJAN VAN DER SCHAFT
CHRISTIAAN STOLK
SANTIAGO DE CHILE 2020
Contents
Contents 7
1 Introduction 11
1.1 Motivation . . . 11
1.2 Research objectives . . . 12
1.3 Magnetic Resonance Imaging . . . 13
1.3.1 Generalities of MRI . . . 13
1.3.2 Velocity Encoding . . . 14
1.3.3 Magnetic Resonance Elastography (MRE) . . . 15
1.4 Hybrid Inverse Problems in Elasticity . . . 19
1.4.1 Hybrid Inverse Problems . . . 19
1.4.2 Linear Elasticity Equations . . . 20
1.5 Thesis Overview . . . 21
2 Optimal Dual-VENC in Phase-Contrast MRI 25 2.1 Introduction . . . 25
2.2 Theory . . . 26
2.2.1 Classical PC-MRI . . . 26
2.2.2 Dual-VENC approaches . . . 27
2.2.3 Least-squares formulation of the single-VENC problem . 28 2.2.4 The dual-VENC least squares problem . . . 30
2.2.5 Choice of β . . . . 30
2.2.6 The optimal dual-VENC (ODV) algorithm . . . 32
2.3 Methods . . . 33 2.3.1 Synthetic data . . . 33 2.3.2 Phantom data . . . 33 2.3.3 Volunteer data . . . 34 2.4 Results . . . 34 2.4.1 Synthetic data . . . 34 2.4.2 Phantom data . . . 37 2.4.3 Volunteers data . . . 38 2.5 Discussion . . . 40 7
8 CONTENTS
2.6 Conclusion . . . 41
2.A Supplementary material . . . 43
3 Dual-encoding in harmonic MRE 49 3.1 Introduction . . . 49
3.2 Theory . . . 49
3.2.1 Harmonic displacement encoding (Henc) . . . 49
3.2.2 Cost Functionals . . . 50
3.2.3 Dual encoding strategy . . . 50
3.3 Methods . . . 51
3.4 Results . . . 52
3.4.1 Results for a fixed time . . . 52
3.4.2 Results for the discrete Fourier transform in time . . . . 53
3.5 Discussions and conclusions . . . 55
4 Hybrid Inverse Problems in Elasticity 57 4.1 Introduction . . . 57
4.2 Notation . . . 59
4.3 Preliminaries on Over-determined Elliptic Boundary-Value Problems . . . 61
4.4 Linear elasticity with elastic energy density measurements . . . 64
4.4.1 Ellipticity arguments in dimension 2 . . . 64
4.4.2 Lopatinskii condition . . . 70
4.4.3 Stability estimates . . . 73
4.4.4 Injectivity . . . 74
4.4.5 Fixed-point algorithm . . . 76
4.5 Model with generic forcing term f (u) . . . . 81
4.5.1 Ellipticity and Lopatinskii condition . . . 82
4.5.2 Injectivity . . . 82
4.5.3 Fixed point algorithm . . . 85
4.6 Linear Elasticity with internal measurements, incompressible case 87 4.6.1 Ellipticity . . . 87
4.6.2 Lopatinskii condition . . . 89
4.6.3 Local injectivity . . . 91
4.7 Nonlinear Elasticity (Saint-Venant model) with internal mea-surements . . . 91
4.7.1 Ellipticity . . . 92
4.7.2 Lopatinskii condition and local injectivity . . . 93
4.7.3 Algorithm . . . 94
5 Conclusion 97 5.1 Conclusion . . . 97
CONTENTS 9 Summary 101 Samenvatting 103 Resumen 105 Aknowledgements 107 Curriculum vitae 109 Bibliography 111
