THÈSE DE DOCTORAT DE « Clémentine SHAO »

THÈSE DE DOCTORAT DE

L’UNIVERSITE DE RENNES 1 COMUE UNIVERSITE BRETAGNE LOIRE

Ecole Doctorale N°601 Mathématique et Sciences et Technologies de l’Information et de la Communication Spécialité : Signal, Image, Vision Par « Clémentine SHAO » « Images and models for decision support in aortic dissection surgery »

«Images et modèles pour l’aide à la décision clinique de la chirurgie de la dissection aortique»

Thèse présentée et soutenue à RENNES , le 16/12/19 Unité de recherche : LTSI, Inserm U1099 Thèse N° :

Rapporteurs avant soutenance : Frans Van De Vosse, PR, Eindhoven University of Technology (TU/e), Netherlands Alain Lalande, MCU-PH, Université de Bourgogne Franche-Comté, France Composition du jury : Président : Examinateurs : Alain Lalande, MCU-PH, Université de Bourgogne Franche-Comté, France Nadjia Kachenoura, CR INSERM, Sorbonne Universités, France Frans Van De Vosse, PR, Eindhoven University of Technology (TU/e), Netherlands Jean-Philippe Verhoye, PU-PH, CHU de Rennes, France Dir. de thèse : Pascal Haigron, PR, Université de Rennes 1, France Co-dir. de thèse : Gabriele Dubini, PR, Politecnico di Milano, Italie

Invité(s)

Michel Rochette, Directeur Technique, Ansys France, France

ACKNOWLEDGEMENT

Je tiens à remercier I would like to thank. my parents.. J’adresse également toute ma reconnaissance à ......

LIST OF ABBREVIATIONS

AA Ascending Aorta AD Aortic Dissection BC Boundary Condition BT Brachiocephalic Trunk CFD Computational Fluid Dynamics CT Computed Tomography CVS Cardiovascular System DA Descending Aorta DOE Design Of Experiment DNS Direct Numerical Simulation DSC Dice Similarity Coefﬁcient FL False Lumen FSI Fluid-Structure Interaction HD Hausdorff Distance HED Holisticallynested Edge Detect LCC Left Common Carotid LES Large Eddy Simulation LS Left Subclavian MRI Magnetic Resonance Imaging PC MRI Phase-Contrast Magnetic Resonance Imaging RANS Reunold Averaged Navier-Stokes RBC Red Blood Cell RMS Root Mean Square RNG Renormalization Group ROM Reduced Order Model SAS Scale Adaptative Simulation TL True Lumen WSS Wall Shear Stress

iii

TABLEOF CONTENTS

List of abbreviations iii

List of Figures ix

List of Tables xvii

Résumé en Français xix

Introduction 1

1 Clinical context 5 1.1 Cardiovascular system ...... 5 1.1.1 The heart ...... 6 1.1.2 Pulmonary and systemic circulations ...... 8 1.1.3 Aorta anatomy ...... 11 1.2 Aortic dissection ...... 12 1.2.1 Pathology, symptoms and causes ...... 13 1.2.2 Classiﬁcation of ADs ...... 14 1.2.3 Treatment ...... 15 1.3 Conclusion ...... 16

2 Image segmentation of aortic dissection 19 2.1 Context and difﬁculties of the segmentation of type B AD ...... 19 2.2 State of the art ...... 22 2.3 Semi-automatic method for type B aortic segmentation ...... 24 2.3.1 Pre-processing ...... 25 2.3.2 Initialization of the two lumina ...... 27 2.3.3 Segmentation of the two lumina ...... 30 2.3.4 Post processing ...... 32 2.3.5 From segmentation to surface mesh ...... 33

v TABLE OF CONTENTS

2.3.6 Implementation of the method ...... 34 2.4 Results ...... 36 2.4.1 CT Image acquisition ...... 37 2.4.2 Parameters setup ...... 37 2.4.3 Segmentation error criteria ...... 38 2.4.4 Quantitative analysis ...... 38 2.4.5 Qualitative analysis ...... 43 2.4.6 Comparison with deep-learning based methods ...... 45 2.4.7 Comparison of CFD results using different smoothing methods . 48 2.5 Discussion ...... 51 2.6 Conclusion ...... 53

3 Patient specific CFD simulation of the aorta 55 3.1 State of the arts of CFD for AD ...... 55 3.1.1 Fluid modeling ...... 56 3.1.2 Model of the aorta wall ...... 58 3.1.3 Boundary conditions ...... 59 3.1.4 Summary ...... 60 3.2 0D models of the CVS as BCs for 3D CFD ...... 62 3.2.1 0D model of the systemic and pulmonary circulation ...... 63 3.2.2 0D model of the heart ...... 65 3.2.3 Multiscale coupling ...... 65 3.2.4 Personalization of the 0D model ...... 66 3.3 Method ...... 67 3.3.1 Geometry and meshes ...... 68 3.3.2 Creation of the hybrid 0D model of the aorta ...... 69 3.3.3 Initialization of the 0D model of the CVS ...... 75 3.3.4 Personalization of the 0D model of the CVS ...... 80 3.3.5 Setup of the 3D patient specific CFD ...... 83 3.4 Results ...... 84 3.4.1 Response surface from the static simulations ...... 84 3.4.2 Optimization processes ...... 85 3.4.3 3D transient fluid simulations of the AD ...... 88 3.4.4 Focus on the tear regions of the AD ...... 97

vi TABLE OF CONTENTS

3.5 Discussion ...... 97 3.6 Conclusion ...... 101

4 Dynamic ROM based on CFD 103 4.1 State of the arts ...... 103 4.2 Method ...... 105 4.2.1 SVD to obtain 3D dynamic ROM ...... 105 4.2.2 Dynamic ROM method ...... 107 4.2.3 Generation of the learning and validation data for the ROMs . . . 108 4.2.4 Integration of the ROM in the 0D model of the full CVS ...... 109 4.2.5 Computation of different clinical scenarios ...... 109 4.3 Results ...... 111 4.3.1 0D dynamic ROMs ...... 112 4.3.2 Final coupled 0D models ...... 115 4.3.3 ROMs computing the pressure at the wall ...... 119 4.3.4 ROMs computing the WSS ...... 126 4.3.5 ROMs computing the velocity vectors ...... 130 4.3.6 Simulation of clinical scenarios for AD ...... 135 4.4 Discussion ...... 139 4.5 Conclusion ...... 141

Conclusion 143

Bibliography 167

vii

LISTOF FIGURES

1 Stucture de l’aorte. a) Différentes parties de l’aorte. b) Couches de la paroi de l’aorte...... xx 2 Dissection aortique. a) Déchirure de l’intima. b) Classiﬁcation de Stan- ford...... xx 3 Schéma des différentes étapes de la méthodes de segmentation. . . . . xxii 4 Schéma des différentes étapes de la méthode de simulation dynamique spéciﬁque patient...... xxiv 5 Résultat de la simulation dynamique de l’aorte au pic de la systole. a) Pression à la paroi. b) Contraintes de cisaillement. c) Champs de vitesse...... xxv 6 Schéma de l’approche basée sur les modèles d’ordre réduit pour le cas de la dissection aortique...... xxvii

1.1 Scheme of the cardiovascular system...... 6 1.2 Heart anatomy...... 7 1.3 Cardiac cycle phases...... 8 1.4 Vessels composing the pulmonary and systemic circulations...... 9 1.5 Structure of the wall of the veins and arteries...... 10 1.6 Structure of blood vessels. The values for medium arteries, arterioles, venules, and veins are merely illustrative because the dimensions can range widely. The drawings of the vessels are not to scale...... 11 1.7 Anatomy of the aorta. a) Structure of the aorta. b) Branches of the aorta. 12 1.8 AD with a zoom on the entry tear...... 13 1.9 Stanford and De Bakey AD classiﬁcations...... 15 1.10 Type B dissection and endovascular devices (Stent-graft and delivery system) [105] ...... 16

ix LIST OF FIGURES

2.1 CT slices of ADs with various shapes. a) Typical case. b) Difference of contrast between the two lumina. c) Noise giving the flap blurry. d) True lumen completely surrounded by the false lumen. e) Moving flap. f) Partially thrombosed false lumen. g) Aneurysm in the aortic arch. h) Aneurysm and separated lumina. i) Stent in one lumen...... 21 2.2 Image processing pipeline of the segmentation method...... 25 2.3 Preprocessing steps of the image. a) Ct-scan image. b) Smoothed image. c) Gradient image. d) Sigmoid image...... 27 2.4 Pathlines of the two lumina : initialization points and different parts of TL. 28 2.5 Segmentation steps of the two lumina. a) Result from the fast-marching method for the true lumen. b) Result from the active contour geodesic method for the true lumen. c) Result from the fast-marching method for the false lumen. d) Result from the active contour geodesic method for the false lumen...... 32 2.6 Steps of the Laplacian smoothing. a) Original points. b) Center of gravity calculation G. c) Moving the initial point in the direction of G by a fraction λ of the distance. d) New point position...... 33 2.7 Screenshot of the interface of 3D Slicer...... 35 2.8 Screenshot of the graphic user interface of the developed segmentation module...... 35 2.9 Screenshot of the final segmentation results...... 36 2.10 Example of the uncertainty of the manual segmentation on a CT slice with noise. a) CT scan slice. b) First example of segmentation. c) Second example of segmentation...... 39 2.11 Distance error for patient 2. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Seg- mentation results on the slice with the highest distances. c) Correspon- ding CT slice...... 41 2.12 Distance error of patient 4.1. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Seg- mentation results on slice 1. c) CT slice of slice 1. d) Segmentation results on slice 2. e) CT slice of slice 2...... 42

x LIST OF FIGURES

2.13 Distance error of patient 7.1. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Seg- mentation results on slice 1. c) CT slice of slice 1. d) Segmentation results on slice 2. e) CT slice of slice 2...... 43 2.14 CT slice of the case where the true lumen was too thin and the segmentation failed...... 44 2.15 Segmentation on various geometries of the lumina. a) Typical AD slice b) Case where three lumina appear on the same slice. c) Case with a partially thrombosed FL. d) Case with an aneurysm in the aortic arch. e) Case with an aneurysm and a stent. f) Case of a dissection in the iliac arteries. g) Case with a stent in one lumen. h) Case with noise...... 45 2.16 Comparison results from the three segmentation methods. a) Blurry flap. b) Iliac arteries. c) Connections between the two lumina. d) Aneurysm in the aortic arch. e) Contrast problem...... 47 2.17 Geometries generated using three different smoothings. a) Laplace with 80 iterations. b) Laplace with 40 iterations. c) Taubin...... 49 2.18 Mass flow profile applied at the inlet...... 50 2.19 Comparison of the velocity fields in the three planes of the aortic dissection computed on geometries obtained with different smoothing techniques : Laplacian with 80 iterations, Laplacian with 40 iterations and Taubin (from top to bottom)...... 51

3.1 Basic components of 0D models (resistance, inductance and capacitance). 63 3.2 Main steps of the method to compute patient speciﬁc transient CFD of the aorta...... 67 3.3 Surfaces rendering of patient 1 and 2. a) Aorta of patient 1. b) Aorta of patient 2...... 68 3.4 0D model of the full cardiovascular circulation used in this thesis. . . . . 70 3.5 Division of the 3D geometry of the aorta in segments. a) Patient 1 (without dissection). b) Patient 2 (with dissection)...... 70 3.6 Inputs and outputs of the response surfaces. a) Patient 1. b) Patient 2. . 72 3.7 0D hybrid models of the aortic cross. a) Patient 1. b) Patient 2...... 74 3.8 Coupling of the hybrid 0D model and the 0D model of the CVS for patient 1...... 75

xi LIST OF FIGURES

3.9 Coupling of the hybrid 0D model and the 0D model of the CVS for patient 2...... 76 3.10 Functions applied on the pressure computed from the coupled hybrid 0D model and 0D model of the CVS. a) Function applied to the pressure at diastole. b) Function applied to the pressure at systole...... 81 3.11 Comparison of the flow at each outlet between 2D PC MRI data and the coupled hybrid 0D model/0D model of the CVS of patient 1. a) Flow in the BT. b) Flow in the LCC. c) Flow in the LS. d) Flow in the DA. . . . . 87 3.12 Pressure at the different outlets and inlets from the final optimized 0D model of patient 1...... 88 3.13 Comparison of the flow at each outlet between 2D PC MRI data and the coupled hybrid 0D model/0D model of the CVS of patient 2. a) Flow in the BT. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL...... 89 3.14 Pressure at the different outlets and inlets from the final optimized 0D model of patient 2...... 90 3.15 Mass flow rate at each outlet from the 3D simulation over a single cardiac cycle of the AD...... 90 3.16 Comparison of the flow at each outlet between 3D simulation results, 0D simulation results and 2D PC MRI data. a) Flow in the BC. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL ...... 92 3.17 Velocity fields of the flow of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole...... 93 3.18 Streamlines from the inlet of the flow of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole...... 94 3.19 Pressure at the wall of patient 2. a) Mid-systole. b) Peak systole. c) Di- crotic Notch d) Mid-diastole...... 95 3.20 WSS of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole...... 96 3.21 Velocity fields in planes of the four tears (entry tear 1, entry tear 2, re- entry tear 1 and re-entry 2 from top to bottom). a) Mid-systole. b) Peak systole. c) Dicrotic Notch...... 98 3.22 WSS near the areas of the four tears. a) Mid-systole. b) Peak systole. c) Dicrotic Notch...... 99

xii LIST OF FIGURES

4.1 Main steps of the computation of the dynamic 0D ROM and the 3D ROMs.106 4.2 Inputs and outputs of ROMs. a) 0D dynamic ROM of patient 1. b) 0D dynamic ROM of patient 2. c) 3D dynamic ROMs of patient 1. d) 3D dynamic ROMs of patient 2...... 108 4.3 Coupling of the 0D model of the CVS with the 0D dynamic ROM of patient 2...... 110 4.4 Comparison between the pressure computed from the simulation and the 0D dynamic ROM for validation case 10 a) BT. b) LCC. c) LS. d) DA. 113 4.5 Difference between the pressure computed from the simulation and the 0D dynamic ROM a) Validation case 9. b) Validation case 10. c) Valida- tion case 11. d) Validation case 12...... 114 4.6 Pressure computed from the simulation and the 0D dynamic ROM for validation case 11 a) BT. b) LCC. c) LS. d) TL. e) FL...... 116 4.7 Pressure computed from the simulation and the 0D dynamic ROM for validation case 15 a) BT. b) LCC. c) LS. d) TL. e) FL...... 117 4.8 Comparison of the flow at each outlet between 2D PC MRI data, the coupling between 0D model of the CVS and the hybrid 0D model and the coupling between the 0D model of the CVS and the 0D dynamic ROM for patient 1. a) Flow in the BC. b) Flow in the LSS. c) Flow in the LC. d) Flow in the DA...... 118 4.9 Comparison of the flow at each outlet between 2D PC MRI data, the coupling between 0D model of the CVS and the hybrid 0D model and the coupling between the 0D model of the CVS and the 0D dynamic ROM for patient 2. a) Flow in the BC. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL ...... 120 4.10 RMS error depending on the number of modes from the SVD of the pressure at the wall of patient 1...... 121 4.11 Relative errors (%) on each mode coefficients for the validation cases for the ROM computing the pressure at the wall of patient 1. The errors higher than 10 % are cropped for clarity purpose...... 121 4.12 Comparison of the pressure at the wall computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid diastole...... 122

xiii LIST OF FIGURES

4.13 RMS error depending on the number of modes of the SVD of the pressure at the wall of patient 2...... 123 4.14 Relative errors (%) on each mode coefficients for the validation cases of the ROM computing the pressure at the wall of patient 2. The errors higher than 10 % are cropped for clarity purpose...... 124 4.15 Comparison of the pressure at the wall computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid-diastole...... 125 4.16 RMS error depending on the number of modes used for the SVD on the WSS of patient 1...... 126 4.17 Relative errors (%) during one cardiac cycle between the projected snapshot of the WSS of patient 1 using 20 modes and the snapshot from the simulation ...... 127 4.18 Relative errors (%) at peak systole on each mode coefficients for the validation cases of the ROM computing the WSS of patient 1. The errors higher than 10 % are cropped for clarity purpose...... 127 4.19 Comparison of the WSS of patient 1 computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid diastole...... 128 4.20 RMS error depending on the number of modes for the SVD on the WSS of patient 2...... 129 4.21 Relative errors (%) at peak systole on each mode coefficients for the validation cases for the ROM computing the WSS of patient 2. The errors higher than 10 % are cropped for clarity purpose...... 130 4.22 Comparison of the WSS of patient 2 computed from the ROM and the simulation (on left and right respectively) a) Validation case 1. b) Valida- tion case 2...... 131 4.23 RMS error depending on the number of modes for the SVD computing the velocity fields of patient 1...... 132 4.24 Relative errors (%) on each mode coefficients for the validation cases at peak systole for the ROM computing the velocity fields of patient 1. The errors higher than 10 % are cropped for clarity purpose...... 133

xiv LIST OF FIGURES

4.25 Comparison of the velocity fields of patient 1 computed from the ROM and the simulation (on left and right respectively) a) Whole aorta. b) Plane in the aortic arch. c) Plane in the DA...... 134 4.26 RMS error depending on the number of modes for the SVD on the velocity fields of patient 2...... 134 4.27 Relative errors (%) on each mode coefficients for the validation cases at peak systole for the ROM computing the velocity fields of patient 2. . . . 135 4.28 Comparison of the velocity fields computed from the ROM and the simulation (at top and bottom respectively for subfigures a. and b. and on left and right respectively for subfigures c. and d.) a) Plane including the first entry tear. b) Plane including the second entry tear. c) Plane including the first re-entry tear. d) Plane including the second re-entry tear. . . . . 136 4.29 Comparison of the pressure at the wall results between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia...... 137 4.30 Comparison of the WSS results between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia...... 138 4.31 Comparison of the velocity fields in the first entry tear between the patient’s initial state and the three clinical scenarios a) Initial state. b) Control- led pressure. c) Hypertension caused by vasoconstriction. d) Hyperten- sion caused by hypervolemia...... 139 4.32 Comparison of the velocity fields in the second entry tear between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hy- pertension caused by hypervolemia...... 140 4.33 Comparison of the velocity fields in the second re-entry tear between the patient initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia...... 140

LISTOF TABLES

2.1 Parameters of the proposed method...... 37 2.2 The obtained distances (in mm) between the segmentations from the proposed method and the manual segmentation : Dice coefﬁcient, Haus- dorff distance, mean absolute distance and standard deviation from the semiautomatic segmentation to the manual one and vice versa. The average values across all datasets are displayed in the last row of the table. 40 2.3 Cardiac outputs at each outlet for the outﬂow boundary conditions. . . . 50

3.1 Summary of the different modeling assumptions of ADs in the literature. 61 3.2 Variables equivalent of the hydraulic/electric analogy where l corresponds to the length of the vessel, r the radius and µ the ﬂuid dynamic viscosity...... 62 3.3 Boundaries of the different input parameters of the response surfaces. . 72 3.4 Value of the parameters L (P a.s2.m−3) of each branch of each patient. . 73 3.5 Values of R (P a.s.m−3), L (P a.s2.m−3) and C (m2.P a−1) for the beginning of the AA of patient 1 and of the end of the FL for patient 2...... 77 3.6 Time parameters of the elastance function for patient 1 and 2...... 78

−3 2 −1 −3 3.7 Values of R1 (P a.s.m ), C (m .P a ) and R2 (P a.s.m ) in the upstream and downstream vasculatures for patient 1 and 2...... 80 3.8 Evaluation results for the response surface computed from static ﬂuid simulations of the aortic arch of patient 1...... 84 3.9 Evaluation results for the response surface computed from static ﬂuid simulations of the aorta of patient 2...... 85 3.10 Optimized set of parameters (R in P a.s.m−3, L in P a.s2.m−3 and C in m2.P a−1) of the 0D models of the CVS for patient 1 and 2...... 86

xvii LIST OF TABLES

4.1 Range of variation of the BCs of patient 1 and 2 for the generation of learning and validation data for the computation of the ROM where P is the pressure applied at the DA and TL for patient 1 and 2 respectively −1 in mm Hg, Q the mass ﬂow rate at the inlet (in kg.m ) and Tc is the duration of the cardiac cycle...... 109 4.2 Relative Error (RE) (%) and Maximum Absolute Error (MAE) (mm Hg) on all outputs of the 0D dynamic ROM of patient 1...... 112 4.3 Relative error (%) and maximum absolute error (mm Hg) on all outputs of the 0D dynamic ROM of patient 2...... 115

xviii RÉSUMÉEN FRANÇAIS

La dissection aortique est une pathologie initiée par la présence de déchirures dans la paroi interne de l’aorte entraînant un écoulement du sang entre les couches de la paroi aortique. Ainsi, un deuxième chenal, appelé faux chenal, se forme le long de l’aorte où le sang peut circuler. La décision clinique concernant le traitement de cette maladie est encore actuellement controversée dans certaines configurations spécifiques et dépend actuellement de la géométrie de la dissection. La simulation numérique est en- visagée pour obtenir des informations sur l’hémodynamique de la dissection aortique de manière non-invasive. Ces données pourraient se réveler pertinentes pour prédire l’évolution de cette pathologie. L’objectif de cette thèse est de développer un modèle de simulation patient spécifique de la dissection aortique pour l’aide à la décision clinique.

Contexte clinique

L’aorte est l’artère principale du corps humain qui délivre le sang dans tous les organes. Elle naît au niveau du coeur et monte sur environ 5cm en une partie appelée l’aorte ascendante. Elle décrit ensuite une courbe lui permettant de passer au dessus des vaisseaux pulmonaires appelée l’arche aortique, puis elle descend jusqu’aux ar- tères illiaques. Cette dernière partie est l’aorte descendante. La paroi de l’aorte est composée de trois couches avec des propriétés physiologiques et mécaniques diffé- rentes. Les parois interne, medium et externe sont respectivement l’intima, la media et l’adventitia. La figure 1 illustre les différentes structures de l’aorte (figure 1 a.) et de sa paroi (figure 1 b.). La dissection aortique est une maladie de l’aorte causée par la présence de dé- chirures dans l’intima. Le sang peut s’infiltrer dans ces déchirures et s’écouler entre les parois de l’aorte créant ainsi un deuxième chenal de sang appelé le faux chenal. Par opposition, le chenal initial est appelé le vrai chenal. La paroi entre le vrai et le faux chenal est le flap intimal et les déchirures établissant les connexions des deux chenaux sont appelées les entrées. On peut distinguer différents types de dissections aortiques en fonction de la localisation anatomique du faux chenal. Si la dissection

xix Résumé en Français

(a) (b)

FIGURE 1 – Stucture de l’aorte. a) Différentes parties de l’aorte. b) Couches de la paroi de l’aorte. est présente dans l’aorte ascendante, c’est une dissection de type A (selon la classification de Stanford) [82]. Dans ce cas, une opération chirurgicale est immédiatement requise en raison du risque élevé de rupture de l’aorte. Sinon, on parle de dissection de type B et le jugement clinique quant au traitement de la dissection est plus compli- qué. La figure 2 illustre la formation d’une dissection aortique (figure 2 a.) et les types de dissections selon la classification de Stanford (figure 2 b.).

(a) (b)

FIGURE 2 – Dissection aortique. a) Déchirure de l’intima. b) Classiﬁcation de Stanford.

La période de 14 jours suivant l’admission du patient à l’hôpital est appelée la phase

xx Résumé en Français aigue. Durant cette période, si le patient souffre de malperfusion des organes ou d’un risque de rupture de l’aorte, la dissection est classifiée comme compliquée et un traitement endovasculaire thoracique est la référence [18]. Sinon, le patient est suivi régu- lièment et reçoit un thérapie médicale pour baisser sa tension cardiaque. Cependant, 20% à 30% de ces patients vont développer des complications nécessitant une opéra- tion chirurgicale. Dans ces cas, la réalisation de la chirurgie quand la dissection n’est pas encore compliquée présenterait moins de risques. La prédiction de l’évolution des dissections aortiques de type B non-compliquées est actuellement incertaine. Les facteurs cliniques existant, basés sur des mesures géométriques [12], sont insuffisants. La simulation numérique permet d’avoir accès à des données telles que la pression au niveau de la paroi de l’aorte qui pourrait aider la prédiction de l’évolution des dissections aortiques de type B. L’objectif de cette thèse est de développer un modèle précis et spécifique patient de l’aorte.

Segmentation de la dissection aortique

Afin d’obtenir un modèle numérique spécifique patient de la dissection aortique, la géométrie de l’aorte du patient est nécessaire. Cette géométrie peut être extraite des images scanner du patient. De nombreuses difficultés liées à la segmentation de la dissection aortique existent. Le flap intimal est très fin et souvent flou car il peut bouger sous la pression du sang durant le temps d’acquisition de l’image. De plus, de nombreux patients avec des dissections aortiques ont d’autres problèmes de type cardiovasculaire entraînant la présence de dispositif métallique ou d’anévrisme dans l’aorte. Enfin, la géométrie du vrai et du faux chenal est très variable entre les patients. Une seule étude a été réalisée sur la segmentation du vrai et du faux chenal, dans l’optique de la simulation fluide, mais les méthodes proposées ne permettent pas d’obtenir une segmentation distincte du vrai et du faux chenal. Nous avons développé une méthode de segmentation semi-automatique du vrai et faux chenal de la dissection aortique basée sur une méthode de propagation de front compétitive (de type fast marching) et de contour actif géodésique. Dans un premier temps, on applique un prétraitement sur les images du patient pour réduire le bruit et réhausser le contour du vrai et du faux chenal. À partir de points placés par l’utilisateur, une ligne passant dans chaque chenal est créee. Ces lignes permettent d’avoir une première approximation de la géométrie du vrai et du faux chenal et d’obtenir des

xxi Résumé en Français

FIGURE 3 – Schéma des différentes étapes de la méthodes de segmentation. points d’initialisations pour l’application de la méthode de segmentation. À partir de ces lignes et de l’image issue du prétraitement, une méthode de propagation de front compétitive est appliquée dans chaque chenal. Le seuil, indiquant quand arrêter le front de propagation dans chaque chenal, est optimisé à chaque coupe de l’image. On recherche la valeur de seuil maximale pour que les deux fronts de propagation ne se chevauchent pas. La segmentation est ensuite complétée par une méthode de contour actif géodésique. Un post-traitement, visant à enlever les zones partagées par les deux chenaux qui ne sont pas des entrées, est ensuite appliqué. On transforme la segmentation ﬁnale en une surface 3D avec la méthode des Marching cubes et des opérations de lissage. La ﬁgure 3 résume les différentes étapes de la méthode de segmentation. La méthode de segmentation a été évaluée sur 21 scanners de patients avec des dissections aortiques de types B. En moyenne, les valeurs de la distance de Hausdorff, la distance moyenne et la déviation standard sont de 6.21, 0.34 and 0.67 mm. On ob-

xxii Résumé en Français tient une bonne segmentation du vrai et faux chenal sur 20 cas. Sur le cas restant, le vrai chenal n’a pas été segmenté sur une petite partie de l’aorte car le chenal était trop ﬁn. La méthode a été comparée avec deux méthodes de segmentation basées sur des approches de réseaux de neurones convolutifs [131]. La première est une approche 2D tandis que la deuxième est une approche 3D combinée avec une méthode d’extraction de contours. Ces approches sont plus rapides d’éxecution et automatiques. Cependant, la méthode proposée est plus robuste contre la présence d’anévrisme, de bruit ou la présence de dispositifs métalliques et permet la segmentation de dissections aortiques avec des géométries variées.

Simulation numérique de la dissection aortique

Non-invasive, la simulation numérique par élément fini a été envisagée pour obtenir des données sur l’hémodynamique de l’aorte. Dans la littérature, il existe une grande variété dans les choix de modélisation de l’aorte et plus particulièrement de la dissection aortique [157]. Il est encore difficile de modéliser précisément les conditions limites du domaine fluide car les données cliniques requises sont mesurées de manière invasive. Dans cette thèse, une nouvelle méthode pour la définition des conditions limites spécifiques patient est proposée. En se basant sur la géométrie obtenue avec la méthode de segmentation, des simulations dynamiques spécifiques patient de l’aorte ont été réalisées. On utilise un modèle équivalent électrique de la circulation cardiovasculaire complète. Ce modèle décrit le coeur, la circulation pulmonaire et systémique et permet d’obtenir le flux et la pression aux niveaux des entrées et sorties fluides de l’aorte. Les différentes parties de la circulation (artères, artérioles, capillaires et veines) sont représentées avec des résistances, des bobines et des condensateurs. Le couplage direct entre un mo- dèle 0D et 3D est un processus compliqué et instable. Pour pallier cette difficulté, un modèle 0D hybride de l’aorte basé sur des simulations statiques spécifiques patient et des composants électriques a été développé. Ce modèle est ensuite couplé au modèle 0D de la circulation cardiovasculaire. Le modèle 0D hybride de l’aorte est spécifique patient mais ce n’est pas le cas du modèle de la circulation cardiovasculaire. On doit personaliser les paramètres des différents composants électriques. Ces paramètres sont dans un premier temps initialisés en utilisant les données du patient (données CT, d’imagerie par résonance magnétique et pression systolique et diastolique) et les va-

xxiii Résumé en Français

FIGURE 4 – Schéma des différentes étapes de la méthode de simulation dynamique spécifique patient. leurs issues de la littérature. Ensuite, un processus d’ optimisation de type Levenberg Marquardt est mis en oeuvre en utilisant les données de flux d’imagerie par résonance magnétique et les pressions systolique et diastolique artérielles du patient. Une fois le modèle optimisé, on peut extraire les courbes de pression et flux du couplage du modèle 0D de la circulation cardiovasculaire et du modèle 0D hybride de l’aorte pour réaliser la simulation dynamique spécifique patient dans le cadre de la dissection aortique. Les différentes étapes de la méthode sont résumées dans la figure 4. Différentes hypothèses ont été utilisées pour notre modèle. Les parois de l’aorte sont considérées rigides. Le sang est modélisé avec un modèle de fluide Newtonien incompressible. Trois secondes de cycle cardiaque sont simulées avec un pas de temps de 1 ms. La méthode a été appliquée sur une aorte de volontaire sain et sur un cas de dissection aortique. La pression à la paroi, les contraintes de cisaillement et les champs de vitesse du sang ont été analysés à différentes étapes du cycle cardiaque. Les résultats au pic de la systole sont présentés dans figure la 5 pour le cas de dissection aortique. On observe une différence de pression d’environ 15 mm Hg entre le vrai et faux chenal à la systole. Des zones avec une contrainte de cisaillement élevée sont pré- sentes à côté des entrées de la dissection. C’est aussi le cas dans les parois du faux chenal en face des entrées. En effet, le sang s’infiltre dans le faux chenal à travers les entrées où il accélère en direction de la paroi du faux chenal. On peut observer que les zones où le sang a la vitesse la plus élevée se situent aux niveaux des entrées proches de l’artère sous clavière gauche. Les temps de calcul pour les simulations sur un ordinateur 64-bits 16 cores sont d’environ 17h et 24h pour le cas d’aorte saine et

xxiv Résumé en Français

(a) (b) (c)

FIGURE 5 – Résultat de la simulation dynamique de l’aorte au pic de la systole. a) Pression à la paroi. b) Contraintes de cisaillement. c) Champs de vitesse. de dissection aortique respectivement.

Modèle réduit de la dissection aortique

Une des principales limitations pour l’application des méthodes de simulations nu- mériques dans la routine clinique est le temps de calcul. Les approches de type réduc- tion de modèles permettent d’obtenir une approximation du modèle original avec un temps de calcul moindre. Peu de modèles d’ordre réduit des artères ont été dévelop- pés dans la littérature. Nous avons dévelopé des modèles réduits à partir des simulations ﬂuides transi- toires. Le procédé de création des modèles réduits s’appuie sur une méthode d’apprentissage permettant de résoudre les équations non-linéaires reliant des entrées dynamiques aux sorties du modèle numérique. Les données d’apprentissage et de validation requises sont obtenues à partir de simulations numériques utilisant le procédé décrit dans la section précédente. Les variables 3D sont pré-traitées avec la méthode de décomposition en valeurs singulières pour réduire leur complexité. Le modèle réduit résulte de l’entrainement d’un réseau de neurones sur les coefﬁcients des modes de la décomposition. Deux types de modèles réduits ont été créés : un modèle réduit 0D dynamique et des modèles réduits 3D. Le modèle réduit 0D dynamique retourne la pression aux dif-

xxv Résumé en Français férentes sorties de l’aorte. Ce modèle a été utilisé pour remplacer le modèle 0D hybride de l’aorte. En effet, il offre une meilleure précision puisqu’il est basé sur des simulations dynamiques et non statiques. À partir du modèle réduit 0D dynamique couplé avec le modèle 0D de la circulation complète, on applique un nouveau processus d’optimisation affin de rendre ce dernier patient spécifique en utilisant les données IRM du patient. Trois modèles réduits 3D ont été créés retournant respectivement la pression à la paroi, les contraintes de cisaillements et les champs de vitesse. La figure 6 montre l’intégration du modèle 0D dynamique dans le modèle de la circulation complète et son utilisation pour les modèles réduits 3D pour le cas de la dissection aortique. On obtient une bonne correspondance entre les résultats de la simulation par élé- ments fini et les résultats par modèles d’ordre réduit. L’erreur relative pour les mo- dèles réduits 0D dynamiques, spécifique à chaque patient, est inférieure à 2% pour les données d’apprentissage et de validation. De même, on obtient une bonne correspondance pour le modèle réduit 3D retournant la pression aux parois de l’aorte. On obtient une erreur relative inférieure à 1 % pour le processus de décomposition en valeurs propres et une erreur relative inférieure à 2 % sur l’apprentisage des premiers modes pour les données d’apprentissage et de validation. Concernant les modèles réduits 3D de l’aorte pour les contraintes de cisaillement et les champs de vitesses, ces variables 3D sont difficilement décomposables durant la diastole. Durant cette pé- riode du cycle cardiaque, le flux est turbulent et les variables en question ont une faible variation. En conséquence, on ne s’intéresse aux résultats de ces modèles réduits qu’au pic de la systole. Les modèle réduits retournant les contraintes de cisaillement décrivent correctement les zones avec des valeurs élevées. De même, au pic de la systole, les modèles réduits 3D décrivant les champs de vitesse retrouvent les mêmes configurations de l’écoulement du sang les simulations par éléments finis. Une fois le système paramétré, l’approche modèle réduit proposée prend au plus 2 minutes pour simuler 1h réelle de cycles cardiaques. Ce temps comprend l’écriture de fichiers et peut être réduit. L’approche des modèles d’ordre réduit permet d’agir sur les paramètres du mo- dèle 0D de la circulation cardiovasculaire pour simuler des scnénarios cliniques. Nous avons modélisé trois scénarios cliniques différents : la baisse de la pression arté- rielle à travers la prise de béta-bloquants, l’hypertension artérielle causée par la vasoconstriction et l’hypervolémie. On constate une baisse des contraintes de pressions et de cisaillements dans le cas de la pression contrôlée et une augmentation de ces

xxvi Résumé en Français

FIGURE 6 – Schéma de l’approche basée sur les modèles d’ordre réduit pour le cas de la dissection aortique.

xxvii Résumé en Français contraintes dans les cas d’hypertensions conﬁrmant la plausibilité des modèles déve- loppés.

Contributions

Nos travaux ont porté sur la simulations fluide numérique pour l’aide à la décision clinique dans le cadre de l’évolution des dissections aortiques de type B. Dans cette optique, une méthode de segmentation semi-automatique de la dissection aortique pour la simulation a été développée. Cette méthode a démontré la capacité de seg- menter correctement une large variété de géométries de dissections aortiques. Une nouvelle méthode pour la définition des conditions limites a été proposée afin de réa- liser des simulations spécifiques patient. On obtient une bonne correspondance des flux aux sorties de l’aorte entre les calculs et les données patients. Les courbes de pressions correspondantes présentent des formes cohérentes avec des courbes de pression issues de la littérature et mesurées habituellement de manière invasive. Les simulations numériques permettent d’observer les zones de la paroi de l’aorte pré- sentant des pressions ou des contraintes de cisaillement élevées. Une étude sur une cohorte de patients plus large avec des données restrospectives permettrait d’évaluer des facteurs d’impacts sur l’évolution des dissections aortiques de type B. Une approche de simulation spécifique patient fondée sur la modélisation d’ordre réduit a été proposée. Elle permet de diminuer considérablement le temps de calcul afin d’autori- ser la simulation de l’évolution de la dissection aortique à long terme. On peut obtenir rapidement des données sur la pression à la paroi, les contraintes de cisaillement et les champs de vitesse du sang pour n’importe quel jeu de paramètres du modèle de la circulation cardiovasculaire. Dans le futur, une étude simulant plusieurs mois de cycles cardiaques dans différentes configurations (pression contrôlée avec la prise de béta- bloquants par exemple) est envisagée. Un modèle de fatigue à la paroi de l’aorte devra être ajouté pour souligner les zones présentant un risque de rupture.

Publications

Shao C, Lv T, Tomasi J, Zhao X, Verhoye JP, Yang G, Chen Y, Haigron P. A comparison of conventional and deep learning methods of image segmentation on aortic dissection. Proceedings of the 33rd Congress CARS 2019 - Computer Assisted Radio-

xxviii Résumé en Français logy and Surgery, Rennes, France, June 18–21, 2019. IJCARS 14(1) : S28-29.

Shao C, Rochette M, Morgenthaler V. SYSTEM AND METHOD OF MODELING VASCULATURE IN NEAR REAL-TIME. US Patent pending application NO. 16/578,140 ﬁled September 20, 2019

Shao C, Morgenthaler V, Lederlin M, Verhoye JP and Haigron P. Reduced order model for patient specific fluid transient simulation of blood flow in aortic cross. 44ème Congrès de la Société de Biomécanique - Poitiers 28-30 octobre 2019

xxix

INTRODUCTION

Aortic Dissection (AD) is a pathology of the aorta initiated by the formation of tears in the innermost layer of the aortic wall. The blood flows through the tears inducing the cleavage of the blood bed in two lumina (true and false lumina). It occurs at an estimated rate of 3 per 100 000 people per year. The occurrence is relatively rare but the consequences can be very serious. 20% of patients die before the admission to the hospital and a further 30% die during admission [89]. The choice of treatment is critical for the remaining patients and depends on the anatomy of the dissection. If the false lumen is located in the ascending aorta, the dissection is classified as Stanford type A [82]. An immediate surgery is required because of the high risk of intra-pericardial rupture. If the dissection is located in the descending aorta, it is a Stanford type B dissection and the clinical decision is still controversial. During the 14 days following the formation of the dissection, if the patient presents risk of malperfusion of the organs or of aortic wall rupture, the dissection is classified as complicated type B. Thoracic endovascular procedure is the reference treatment. Otherwise, the dissection is classified as uncomplicated type B dissection and the patient receives medical therapy and is monitored. Nevertheless, 20% to 30% of initially uncomplicated type B dissections become complicated because of an extension of the dissection or due to the formation of aneurysms. Consequently, some organs might be malperfused and the risk of aortic wall rupture might increase. In these cases, the patient will also require a thoracic endovascular treatment. With regard to this type of evolution, the treatment is less risky if practiced when the dissection is uncomplicated. Nevertheless, nowadays no reliable factors exist to predict the evolution of uncomplicated type B dissection. Computational approaches is an emerging field in medical research for prediction or clinical decisions. Despite the progress in medical imaging and non-invasive measurements, it is still challenging to acquire in vivo patient data with sufficient accuracy to fully understand the patient condition. Moreover, the acquisition of such data is only available for a patient at the resting state. Modeling approaches such as Computatio- nal Fluid Dynamics (CFD), using patient specific features such as the geometry or flow measurements, can yield the access to data such as the pressure or the Wall Shear

1 Introduction

Stress (WSS) at the patient aorta. Such data could be relevant with regard to the outcomes of initially uncomplicated AD of type B. Furthermore, a patient specific model provides the possibility to investigate various treatment options such as the intake of medication to lower the aortic pressure. The main objective of this thesis is to develop an accurate patient specific model of AD from non-invasive measurements to support clinical treatment decision. This work was a collaboration between ANSYS France, LTSI (Laboratoire Traitement du Signal et de l’Image – Université de Rennes 1, INSERM) and the University Hospital of Rennes. The accuracy of patient specific simulation of ADs relies on many factors. First, an accurate geometry of the patient is required. Contrast enhanced Computed Tomogra- phy (CT) is currently a standard procedure for patient with AD. It allows to obtain a visualization of the aorta for medical diagnosis. However, extracting the accurate geometry of the aorta is challenging since the layer, between the true and false lumina, is very thin and numerous complex features related to AD exist. Moreover for simulation purpose, a strict separation of the lumina must be ensured where there is no tear. Few studies have been reported concerning the segmentation of aortic dissection. It is still challenging to obtain a robust segmentation method against complex structures related to aortic dissection. The segmentation of aortic dissection for CFD is the first issue addressed in this work. Modeling assumptions, such as the model of the flows, are critical to obtain an accurate model. Numerous studies have investigated these issues [157]. One of the main difficulties is to properly assess patient specific Boundary Conditions (BCs). Re- lated works generally report the use of invasive measurements or retrospective data. The definition of patient specific BCs using only non-invasive measurements is a open issue that has to be tackled for clinical exploitation of CFD. One of the main limitations for the integration of CFD in daily clinical routine is the high computational cost. Reduced Order Models (ROMs) for large arteries exist but they do not allow to obtain dynamic 3D variables such as the velocity fields. Devising ROM approach to compute the pressure at the wall, the WSS and velocity fields of the patient aorta in real time from transient fluid simulations results is of main interest to assess different clinical scenarios. The document is organized as follows : — Chapter I introduces the human anatomy related to the cardiovascular system and more specifically the aorta. The aortic dissection pathology is also featured

2 Introduction

with a description of the symptoms, diagnosis and treatments. — Chapter II presents a novel approach for the segmentation of type B dissections oriented toward CFD. The difficulties of the segmentation and an analysis of the state of the art are presented. The proposed method is described followed by the results of the segmentation process. — Chapter III deals with patient specific simulation of ADs. The different modeling assumptions are discussed in this Chapter. A novel method to obtain patient specific BCs is described. Simulations were performed on one case of healthy aorta and one case of AD. — Chapter IV presents a method to obtain ROMs from patient specific simulations. The method and the accuracy of the computed ROMs are presented. The developed ROMs were combined with a patient specific 0D model of the full cardiovascular circulation that can be parametrized to assess different clinical scenarios.

CHAPITRE 1

CLINICALCONTEXT

This ﬁrst chapter introduces the human anatomy related to this thesis and the clinical context about type B ADs. An overview of the functions and structures of the cardiovascular system is presented. The heart, the systemic and pulmonary circulations are detailed, followed by a description of the aorta anatomy. The diagnosis, symptoms and treatments of AD are described.

1.1 Cardiovascular system

The main functions of the cardiovascular system are the transport of nutrients to the body cells, the protection of the body and the maintain of the homeostasis by balancing the body temperature and pH.

The cardiovascular system is a complex structure composed of blood, vessels and the heart. The blood provides vital nutrients and oxygen to the body tissues. At the same time, the body removes its wastes, like carbon dioxide, by passing them to the blood. These exchanges are realized through vessels composing a network connecting all the organs of the body. The blood circulation can be decomposed in two parts : the pulmonary circulation and the systemic circulation. The pulmonary circulation refers to the blood circulation inside the lungs while the systemic circulation refers to the blood circulation in the rest of the body. The heart links those two parts by pumping the blood from one to the other. The global structure of the cardiovascular is illustrated in ﬁgure 1.1.

5 Partie , Chapitre 1 – Clinical context

FIGURE 1.1 – Scheme of the cardiovascular system. Source: 2011 Pearson Education, Inc.

1.1.1 The heart

The function of the heart is to pump the blood between the pulmonary and systemic circulations. It is a muscle located in the chest between the two lungs. The heart measures approximately the size of a fist. A fibrous membrane called the pericardium wraps the heart. The heart is composed of four chambers (figure 1.2) : the upper chambers (called the left and right atria) and the lower chambers (called the left and right ventricles). On the left part, called the left heart, the atrium receives the oxygenated blood from the pulmonary veins. Then, the blood is passed to the left ventricle which is the biggest cavity of the heart. The left ventricle pumps the blood to the rest of the body through the aorta. On the right part, called the right heart, the atrium receives deoxygenated blood from the systemic veins (inferior and superior vena cava) and discharges it to the right ventricle. The blood is then transported to the lungs through the pulmonary arteries. The left and right hearts are separated by a wall of muscles called the septum. Each cavity of the heart is connected to a blood vessel and to another cavity. The junctions between each part are called valves. They are fibrous flaps of tissues and are located either in between the cavities or in the blood vessels. They ensure that the blood flows in only one direction. The valves between the atria and ventricles are the right and left atrioventricular valves (also known respectively as the tricuspid and mitral

6 1.1. Cardiovascular system

FIGURE 1.2 – Heart anatomy. Source: Queensland Cardiovascular Group valves). The aortic and pulmonary valves are situated at the base of the aorta and of the pulmonary trunk respectively. There is no valve between the veins and the atria, the atria are continuously filled with blood. The sequence of events resulting in one heart beat is called the cardiac cycle. Two phases compose the cardiac cycle : the systole and the diastole. The systole is the pumping phase. First, the atria contract to fill the ventricles. The atrioventricular valves close to prevent the blood from returning to the atria. The ventricles contract to eject the blood from the heart increasing the pressure of the blood. The relaxing phase, when the atria are filled with blood, is the diastole. During this phase the blood pressure decreases. As the ventricular pressure drops, the flow tends to flow backward while the valves in the aorta and the pulmonary trunk close. This phenomenon is called the dicrotic notch. Figure 1.3 presents the pressure and volume curves of the ventricles, atria and aorta during the cardiac cycle. The electrocardiogram, showing the electrical activity of the heart, and the phonocardiogram, showing the recording of the sound of the heart, are also presented in this figure.

7 Partie , Chapitre 1 – Clinical context

FIGURE 1.3 – Cardiac cycle phases. Source: [62]

1.1.2 Pulmonary and systemic circulations

As previously mentioned, the blood circulation is divided in the pulmonary and systemic circulations. The pulmonary circulation aims to transport the blood between the heart and the lungs while the systemic circulation is a loop between the heart and the rest of the body. These two structures are composed of different vessels with different sizes and functions. In this section, we ﬁrst present the anatomical structure of a blood vessel. The different types of blood vessels and their characteristics are then described. Figure 1.4 is a scheme of the different structures composing the pulmonary and systemic circulations.

8 1.1. Cardiovascular system

FIGURE 1.4 – Vessels composing the pulmonary and systemic circulations. Source: Urgo medical

The inner part of a vessel, where the blood flows, is called the lumen. The walls of the arteries and the veins are composed of three distinct layers with different mechanical and biological properties. The intima, the innermost layer of a blood vessel, has mainly a role in regulating agents from the blood flow to the middle layer, the media. It ensures the impermeability of the wall and prevents blood from coagulation. The intima is composed of monolayers of endothelial cells attached to an internal elastic membrane. The media is composed of layers of smooth muscle tissues in the framework of the extra-cellular matrix (constituted of elastin and collagen). The smooth muscle cells can contract to decrease the diameter of the vessel and relax to increase it. The main function of the media is to ensure the elasticity of the vessel. Finally the outer layer, the adventitia, contains fibroblasts embedded in a network of collagen. It ensures the

9 Partie , Chapitre 1 – Clinical context rigidity of the vessel. Figure 1.5 shows the structure of the walls of the arteries and veins.

FIGURE 1.5 – Structure of the wall of the veins and arteries. Source: [134]

The blood vessels of the systemic and pulmonary circulations are the arteries, arterioles, capillaries, venules and veins. The arteries start from the heart and become progressively smaller vessels called the arterioles. The arterioles then split into the capillaries. Thus, the blood can flow to the venules and after to the veins. Arteries can be classified into different scales at macroscopic and microscopic levels. The biggest arteries (the pulmonary arteries and the aorta) start from the heart and have a diameter bigger than 10 mm. They carry blood to the peripheral arteries which then split into the arterioles (with a diameter measuring between 40 to 110 µm). The blood pressure in the cardiovascular network reaches its maximum in the arteries. The blood needs to be ejected from the heart with enough pressure to travel back to the atria, despite the total peripheral resistances caused by the reduction of the vessels diameter, the rigidity of the vessels wall and the viscosity forces. Consequently, the biggest arteries have elastic walls allowing them to expand under the blood pressure. They are called the compliant arteries. They dilate during the systole and contract during the diastole to ensure the blood flow continuity. It allows to maintain the blood pressure through the cardiac cycle. On the contrary, arterioles have rigid muscular walls increasing the blood pressure. They have the greatest influence on the overall blood pressure and participate mainly in the total peripheral resistance. After the arterioles, the vessels divide again into smaller vessels at a microscopic

10 1.1. Cardiovascular system level called the capillaries. The capillaries are the place where the nutrients and wastes are exchanged between the blood cells and the body cells. Those transfers can be done thanks to the permeable walls of the capillaries. Once the exchanges are done, the process of vessels becoming progressively smaller is reversed. After the blood leaves the capillaries, it enters the small venules, which then become progressively larger vessels called veins. The vein walls are thinner than the artery walls, since the blood pressure is much lower in this part of the circulation. They have elastic walls that allow them to dilate to accommodate the increased blood volume. The physical structures of the different vessels of the cardiovascular system are represented in ﬁgure 1.6.

FIGURE 1.6 – Structure of blood vessels. The values for medium arteries, arterioles, venules, and veins are merely illustrative because the dimensions can range widely. The drawings of the vessels are not to scale. Source: Medical Physiology, 3rd Edition Elastic Properties of Blood Vessels

1.1.3 Aorta anatomy

The aorta is the main artery of the human body in the systemic circulation. Be- ginning from the left ventricle, the aorta then ascends from it. This ﬁrst part is called the Ascending Aorta (AA) and measures approximately 5 cm with a diameter of about 27 mm. At the root of the AA, a structure called the aortic sinuses contains the origin of the coronary arteries which supply the heart with nutrients. Then, the aorta makes

11 Partie , Chapitre 1 – Clinical context a hairpin turn called the aortic arch. The Brachiocephalic Trunk (BT), the Left Com- mon Carotid (LCC) and Left Subclavian (LS) arteries, which supply blood to the head and upper body, branch from it. The aorta then extends down to the abdomen, as the Descending Aorta (DA), where it splits into the common iliac arteries. The DA is composed of two parts : the thoracic aorta, located above the diaphragm, and the abdominal aorta below. The thoracic aorta has small branches supplying blood to the ribs and some chest structures. The abdominal aorta peripheral branches are the lumbar and musculophrenic arteries, the renal and middle suprarenal arteries, and the visce- ral arteries (the celiac trunk, the superior mesenteric artery and the inferior mesenteric artery). Figure 1.7 illustrates the different structures of the aorta (ﬁgure 1.7 a.) and the secondary arteries branching from it (ﬁgure 1.7 b.).

(a) (b)

FIGURE 1.7 – Anatomy of the aorta. a) Structure of the aorta. b) Branches of the aorta. Source: www.feghalicardiology.com

1.2 Aortic dissection

Aortic dissection is a severe pathology of the aorta and occurs at an estimated rate of 3 per 100 000 people per year [104, 32]. Thus, this occurrence may be underestimated as many patients, particularly the elderly persons, are not autopsied and their death are classiﬁed as cardiovascular failure. In this section, we present the pathology,

12 1.2. Aortic dissection symptoms and causes of AD followed by a description of the existing classiﬁcations of AD. This section is completed by the treatment options for this disease.

1.2.1 Pathology, symptoms and causes

ADs are caused by the formation of a tear in the intimal layer of the aortic wall which can be initiated either by cleavage formation in the intimal layer or by an intra- mural haematoma. The blood flow can enter the media layer through the tear and the high pressure of the blood induces a cleavage of the media layer. This separation can propagate along the aorta, in both longitudinal and circumferential directions, resulting in the creation of a second lumen called the False Lumen (FL) as opposed to the initial lumen called the True Lumen (TL). The layer separating the true and false lumina is called the intimal flap and is composed of the intimal layer and a part of the media layer. The blood flow in the FL can initiate the formation of secondary tears, called re-entry tears, where the blood can re-enter the TL. AD can therefore be communicating or non- communicating. AD is non-communicating, if there is no re-entry tear, and the blood in the FL is almost stationary. If the AD is communicating, the blood flows through the FL parallel to the TL. Figure 1.8 illustrates a case of AD with a zoom on the entry tear.

FIGURE 1.8 – AD with a zoom on the entry tear. Source: [100]

The main symptom of AD initiation is sudden severe chest or upper back pain often described as a tearing sensation. Other symptoms related to typical consequences of AD, such as the malperfusion of the organs or a rupture of the wall, can appear. For

13 Partie , Chapitre 1 – Clinical context example, a weakness present on one side of the body, shortness of breath or difﬁculties to speak are also found in AD patients.

AD mostly concerns the elderly persons, it involves mainly men between 40 and 70 years. Hypertension is the most common cause of AD initiation and has a preva- lence of around 62-78 % on patients with AD [70]. A higher blood pressure increases the stress on the aorta wall and weakens it, inducing an increased risk of cleavage formation. A number of aortic diseases have also been observed to increase ADs occurrence. Up to 30% of patients with AD are found to be suffering from prior aortic conditions. The concerned aortic diseases are an excessive dilatation of the aorta and aortic root, aortic aneurysm, anuloartic ectasia, aortic arch hypoplasia, coarctation of the aorta, aortic arteritis and bicuspid aortic valve [70, 15]. Connective tissue disorders can also cause AD formation and propagation. Marfan syndrom is a hereditary condition causing abnormal aortic distensibility and stiffness. It is responsible for about 5% of AD patients and mostly affects the ascending aorta [151, 54]. Trauma to the aorta resulting from previous surgical procedures, such as aortic valve repair, can also be responsible for AD formation. Around 5% of ADs are caused by this kind of injuries [70, 32]. Finally, substance abuse such as cocaine, inducing an elevated blood pressure, and physiological changes associated with pregnancy could lead to AD.

1.2.2 Classiﬁcation of ADs

ADs can be classified depending on the anatomical location of the dissection. The Stanford classification [82] distinguishes dissections involving the ascending aorta (type A) and dissections involving only the descending aorta (type B). A more specific classification called the DeBakey classification exists. It subdivides dissections in three types : the type I when the dissection involves the ascending aorta and the descending aorta, the type II when only the ascending aorta is involved and the type III when the dissection is present only in the descending aorta. We can notice that the Stanford type A classification regroups the DeBakey type I and II while the Stanford type B is equivalent to the DeBakey type III. In this thesis, we will use the Stanford classification. Figure 1.9 shows the Stanford and DeBakey classifications.

14 1.2. Aortic dissection

FIGURE 1.9 – Stanford and De Bakey AD classiﬁcations. Source: [100]

1.2.3 Treatment

In case of type A dissection, the risk of intra-pericardial rupture is too high and a surgical replacement of the AA is the reference treatment [18]. The clinical decision is simple. In case of type B AD, the decision concerning the treatment is still controversial and depends on subjective clinical judgment. From the moment the dissection happens, the following 14 days are called the acute phase [32]. During this period if the AD presents risks of malperfusion of the organs or of aortic wall rupture, the aortic dissection is classiﬁed as complicated type B dissection. In these cases, the reference treatment is thoracic endovascular procedure if the patient conditions are favorable. It concerned 24% of type B ADs [160]. Figure 1.10 illustrates the implementation of a stent in a case of type B dissection. This surgery presents risks and should not be applied if not needed. If the dissection is not complicated the patient receives medical therapy to reduce the arterial blood pressure as well as an active monitoring. The systolic arterial blood pressure should be under 120 mm Hg and is reduced through the use of beta blockers. This

15 Partie , Chapitre 1 – Clinical context

FIGURE 1.10 – Type B dissection and endovascular devices (Stent-graft and delivery system) [105] . treatment extends after the acute phase called the chronic phase. Images of the AD are regularly acquired to monitor the evolution of the dissection. Nevertheless despite the medical therapy in the chronic phase, 20 to 30% of ADs will develop aneurysms or an extension of the dissection inducing secondary malperfusions or risks of rupture. These dissections become complicated and require a surgical procedure. For these patients, an early surgical procedure in the acute phase would have been less risky since the dissection was not complicated then. Some anatomic factors exist to predict which initially uncomplicated dissections will develop chronic complications : an initial aortic diameter superior to 40 mm, a false lumen diameter superior to 22 mm, an eliptic true lumen associated with a round false lumen or a tear situated less than 5 cm from the origin of the LCC artery [12]. However defining a sub-population that will benefit from an early endovascular procedure in uncomplicated situations remains a debate. The prediction of the evolution of initially uncomplicated type B dissection is a difficult issue.

1.3 Conclusion

Aortic dissection is a life threatening condition. Nowadays, it is difﬁcult to predict the evolution of initially uncomplicated type B ADs. Deﬁning accurate factors allowing the

16 1.3. Conclusion prediction of the outcomes of the AD is of high importance for early medical decision. CFD could allow the clinician to assess data such as the wall shear stress in the patient aorta in a non-invasive way. The patient speciﬁc simulation of ADs could help to obtain a better understanding of the pathology. Relevant factors indicating the evolution of the disease could be computed. To perform patient speciﬁc 3D simulations, the accurate geometry of the true and false lumina of the dissection is required to compute a mesh suitable for CFD. In the next chapter, we present a method for the segmentation of type B aortic dissection.

CHAPITRE 2 IMAGE SEGMENTATION OF AORTIC DISSECTION

An accurate geometry of the true and false lumina as well as their connections is required to create a 3D mesh suitable for patient-specific fluid simulations. Image segmentation is the process of dividing a digital image into multiple set of pixels. It is widely used in the medical domain for clinical decision support and can be used to obtain the 3D surface of aortic dissections from CT images. In this chapter we present a semi-automatic segmentation method for ADs of type B targeted for computational fluid simulations. An overview of the difficulties related to the segmentation of type B dissections is introduced. The state of the art is presented. The main steps of the proposed method are described followed by the qualitative and quantitative results obtained from CT data.

2.1 Context and difﬁculties of the segmentation of type B AD

Aortic dissection studies investigated both idealized geometry [1, 57, 144, 123, 71] and patient specific geometry [4, 24, 23, 28, 27, 36, 108, 130, 11, 138, 159, 170, 153, 67, 68] reconstructed from CT and MRI. With regard to ideal geometries, simplifications are commonly made such as constant lumen diameter from the ascending arch to the aortic arch [57, 123, 1], straight vessel [1, 144, 123] or the occlusion of the upper arteries (BT, LCC and LS) [144, 123]. Nevertheless, in reality ADs are tortuous, and tapering occurs from the AA to the DA, these geometrical particularities impact the velocity fields, pressure and WSS along the aorta. As well, the occlusion of secondary arteries induces an overestimation of the flow in the TL and FL. Moreover, the characteristics of the tears such as size, posi-

19 Partie , Chapitre 2 – Image segmentation of aortic dissection tion or orientation have an important impact on the flow behavior of the AD. Therefore, patient specific geometries are the most commonly used in the literature. CT imaging modality has the highest spatial resolution and is the most used. Studies using patient-specific geometries oriented toward the investigation of the hemodynamics of AD often used simple segmentation methods. Most of these studies considered only one patient and the extraction of the AD geometry was based on manual segmentation and region growing techniques. Enhanced contrast CT-scan is currently a standard procedure for patients with AD of type B. By observing the images of the patient aorta, the clinicians have insight on the type of the dissection and the procedure to follow. Nevertheless, the global geometry of the AD cannot be visualized directly from the images of the CT-scan. The extraction of this geometry for the creation of a 3D surface could help the visualization of the clinicians as well as the construction of a 3D mesh for CFDs. Manual delineation is highly time consuming, so semi-automatic and automatic approaches have been developed over the last decades. Several difficulties are encountered when segmenting ADs. The main challenges are the following and are detailed in this section : the geometries of the true and false lumina are very complex and variable, patients with AD can be subject to other cardiovascular issues such as aneurysms and problems related to the quality of image acquisition are common. Figure 2.1 presents slices of ADs with variable shapes. A typical AD CT slice presents a round aorta separated in the middle by the intimal flap. The intimal flap is very thin since its structure measures several voxels. Moreo- ver, it can move under the pressure of the blood giving it a blurry edge or duplicated appearance (see figure 2.1 e.). There is a high variability of the geometry of the lumina among the patients. The dissection can start in the aortic arch and terminate in the common iliac arteries (resulting in a high variation of the diameter of the lumina) as well as measure few centimeters. The size and form of the true and false lumina change a lot. For example, some patients can have their true lumen almost surrounded by the false lumen (see figure 2.1 d.) and other can have completely disjointed lumina (see figure 2.1 f. and h.). Some difficulties are caused by the natural evolution of type B ADs. As said in chapter I, some patients present small or large aneurysms (see figure 2.1 g. and h.). Moreover, patients who have been diagnosed with type B AD for a long time can have a partially thrombosed false lumen (see figure 2.1 f.) due to the healing process of the

20 2.1. Context and difﬁculties of the segmentation of type B AD

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

FIGURE 2.1 – CT slices of ADs with various shapes. a) Typical case. b) Difference of contrast between the two lumina. c) Noise giving the ﬂap blurry. d) True lumen completely surrounded by the false lumen. e) Moving ﬂap. f) Partially thrombosed false lumen. g) Aneurysm in the aortic arch. h) Aneurysm and separated lumina. i) Stent in one lumen.

21 Partie , Chapitre 2 – Image segmentation of aortic dissection wall. Aortic dissections mainly concern elderly patients, which can be suject to other cardiovascular issues, leading to the presence of metallic devices in some cases such as stents (see figure 2.1 i.). Finally, problems inherent to image segmentation like noise (see figure 2.1 c.) or contrast artifacts are also frequently encountered. Moreover, the contrast between the two lumina can be very different since the velocity of the blood, and therefore the contrast product, is dissimilar from one lumen to the other (see figure 2.1). The segmentation of ADs is therefore a challenging problem and has been investigated in several studies.

2.2 State of the art

Several methods have been proposed for the segmentation of ADs. We can distin- guish methods aiming for the segmentation of the lumina from methods targeting the extraction of the intimal flap. A global visualization of the two lumina can help the sur- geon to classify the dissection and perform quantitative analysis. For example, geometrical features, such as the size of the radius of the false lumen, are relevant information for clinical treatment decision. The geometry of the intimal flap can help the prediction of the landing of prosthesis. For both purposes, approaches based on machine learning have been investigated since they are automatic and fast as well as conventional segmentation methods. First, a review of the methods aiming for the segmentation of the lumina is presented. Two methods based on machine learning approaches have been developed. Lee et al. developed a generative-discriminative model matching [79]. The aorta is first segmented and preprocessed with a multi-scale wavelet analysis to extract edges information. A discriminative learning model is then used to reduce the false positive rate while a generative model is computed to model the appearance and information of the object. The final segmentation is obtained by matching the two models. The model training was performed on a data set of 10 patients. The second study was developed by Li et al. An approach based on multi-task deep convolutional neural network segmenting the full aorta and the true and false lumina [84] was established. A hybrid loss function using the Dice Similarity Coefficient (DSC) and Hausdorff Distance (HD) (described afterward in the results section) was used. A total data set of 254 cases was

22 2.2. State of the art generated for the training. Methods based on machine learning are highly dependent on the training data. It must include all the scenarios described previously to obtain a correct and robust segmentation. It can be difficult to obtain such a large data set since it requires the manual segmentation of all dissections, which is laborious and time consuming. Due to the great variability of patient geometry and complex features such as aneurysms or the presence of metallic devices, this type of approach has not yet demonstrated its ability to overcome the difficulties posed by the segmentation of aortic dissections [131]. Fetnaci et al. developed a conventional method to segment the true and false lumina separately [41]. This approach was based on a fast-marching method using a modified speed function to avoid propagation of the contour of one lumen to the other through the tears. This study reported results only on one standard AD case. Several studies focused on the segmentation of the intimal flap. The most common method begins by segmenting the entire aorta and then, extracts the membrane [76, 77, 168]. Kovacs et al. [76] proposed a method based on the Hough transformation that automatically segmented the aorta and then extracted the intimal flap using the Desco- teaux’s sheetness measure [33]. Krissian et al. segmented the aorta by first computing the central line of the aorta in a semi-automatic way and then by applying an active contour process [77]. The intimal flap was extracted by an algorithm based on the zero crossing of two vector fields. Finally, Duan et al. used an active shape model to extract the aorta followed by a method based on Hessian matrix combined with the Bayesian theory of spatial continuity to obtain the intimal flap [168]. Those three methods depend on the first segmentation of the aorta which is sensitive to the contrast intensity changes, aneurysms or the presence of stent in one lumen. Moreover, it is challenging to correctly detect the membrane near the wall or if the two lumina are separated. Two other methods targeting the extraction of the intimal flap have been developed. Based on the segmentation method of the true and false lumina described above [41], Lohou et al. applied mathematical morphology operators to extract the intimal flap by retrie- ving the volume between the two lumina [86]. Trujillo-Pino [150] developed a method based on a subpixel edge detector followed by a denoising process to extract the intimal flap. However, for CFD purpose, we are interested in the segmentation of the fluid domain rather than the segmentation of the intimal flap. To perform patient specific CFD on AD, a suitable geometry of the patient aorta is required. Too much details such as small gaps can slow down the computation wi-

23 Partie , Chapitre 2 – Image segmentation of aortic dissection thout adding any accuracy. A compromise between accuracy and smoothness must be achieved. To our knowledge, only Anderson et al. investigated the problem of AD segmentation for CFD purpose [5]. They developed a semi-automatic segmentation method based on an extended maxima transform. This method segmented the two lumina as one part. They were not separated by the intimal flap and a manual segmentation was required on 20-25% of the slices. For our purpose, the true and false lumina must be segmented separately and an accurate segmentation of the tears connecting them is also required. Chen et al. showed that blood flow with highest velocity occurs near tears [28] and Wan Ab Naim et al. showed that the presence of re-entry tears changes the magnitude and direction of the blood flow [158]. The segmentation has to respect the position of the tears in the CT-scan to obtain accurate fluid simulation results. Since it is difficult, even for clinicians, to be certain of the position of the tears in the CT-scan, finding the tears in an automatic way is complicated. For simulation purpose, a control at each slice, to ensure that the true and false lumina are not connected (where there is no tear) and that no gaps are present in the lumina, is mandatory. We developed a semi-automatic method based on user’s inputs to overcome the issue of the position of the tears and the fact that there is a high variability of the geometry of the dissection between the patients. The method relies on a competitive fast marching process completed by an active contour process ensuring that the two lumina do not overlap where there is no tear.

2.3 Semi-automatic method for type B aortic segmentation

In this section, the steps of the proposed method are detailed. Pre-processing steps were applied on the CT-scan to denoise it and enhance the visibility of the aorta wall and the intimal ﬂap. To segment the different geometries of type B aortic dissection, the proposed method was partially controlled by the user to ensure that the algorithm recognized the different lumina. The user must select three sets of input points to deli- mitate and localize the true lumen, the false lumen and the tears. It allowed to segment very different forms of aortic dissections. The two lumina were segmented separately using the user input points and the preprocessed image. A competitive fast marching process was applied to extract the region roughly included in each of the two lumina.

24 2.3. Semi-automatic method for type B aortic segmentation

FIGURE 2.2 – Image processing pipeline of the segmentation method.

The evolution of the true and false lumina was controlled at each slice so the two lumina do not coincide. This segmentation was then completed by an active contour geodesic process. Some post-treatments were applied to make the segmentation suitable for simulation : separation of the two lumina and creation and smoothing of the ﬁnal 3D surface mesh. The image processing pipeline for segmenting the aortic dissection to generate a CFD simulation is shown in ﬁgure2.2.

2.3.1 Pre-processing

CT images were ﬁrst cropped roughly based on the minimum and maximum heights of the user input points to include only the slices containing the dissection and to minimize the calculation time. The following preprocessing steps were performed to smooth the true and false lumina of the dissection while enhancing the visibility of the aortic wall and the intimal ﬂap.

25 Partie , Chapitre 2 – Image segmentation of aortic dissection

First, the image was denoised while preserving specific image features such as the aortic wall and the intimal flap. Anisotropic diffusion methods were used. Those methods are based on the diffusion equation, which is the general case of the heat equation. Perona & Malik [113] introduced the flux function to constrain the diffusion process in homogeneous regions while not crossing edges. Let’s consider an image U(x) and the conductance term C, the diffusion equation is given as :

dU(x) = C(x)∆U(x) + ∇C(x)∇U(x). (2.1) dt This equation can undergo a "negative" diffusion which enhances the contrast of edges. To counter this problem, a variation of the function called the curvature anisotropic diffusion method [163] was used. It resolves the modified curvature diffusion equation which does not exhibit the edges. The diffusion is always positive, so finer structures of the image are preserved. The conductance term, which controls whether the edges are preserved, was set so that the aorta wall and the intimal flap were still clearly visible. All the parameters setting used during the different steps of the method are presented in table 2.1. To delineate the contour of the intimal flap and of the aortic wall, the gradient magnitude recursive Gaussian method was used. It computes the magnitude of the gradient of the image after smoothing it with a Gaussian kernel. The standard deviation σ was set to regulate the sensitivity of the noise and ensure the intimal flap was still visible in all cases. To enhance the contours while smoothing the lumen regions, the following sigmoid function was applied :

1 y = (M − m) −x−β + m, (2.2) 1 + e α where x is the value resulting from the image preprocessed with the gradient operator. The minimum and maximum values of the sigmoid function, m and M, were set to 0 and 1 respectively, so that the final preprocessed image represented the normalized contour propagation speed. The parameter β can be seen as an offset determined according to the voxel values of the structures of interest (the intimal flap and aorta wall in our case). β was set empirically to 8 since the minimal values of the gradient of the wall and intimal flap were approximately 10 for all the considered data. α controls the width of the input intensity range. Consequently, the values of the resulting image (sigmoid

26 2.3. Semi-automatic method for type B aortic segmentation image) were close to one inside the lumen and close to zero at the wall and the ﬂap. It was used as a speed function for the next steps of the segmentation methods. Figure 2.3 illustrates the different steps of the preprocessing on a CT slice of AD.

(a) (b) (c) (d)

FIGURE 2.3 – Preprocessing steps of the image. a) Ct-scan image. b) Smoothed image. c) Gradient image. d) Sigmoid image.

2.3.2 Initialization of the two lumina

Some input points must be selected by the user in each of the following structures : the true lumen, the false lumen and the tears. From these points and the preprocessed image, a line passing roughly within each lumen, to obtain a ﬁrst estimation of their geometry, was computed. The line in the true lumen, TL, was initialized from three input points : one in the ascending aorta (TLinit[0]), one at the beginning of the descending aorta (TLinit[1]) and one at the end of the descending aorta (TLinit[2]). The line in the false lumen, FL, was initialized from one input point at the beginning of the dissection

(FLinit[0]) and one input point at the end of the dissection (FLinit[1]). TL and FL did not have to be centered in their respective lumen provided that they were in the correct lumen. Since TL and FL initialized the fast marching process, it was preferable that they were placed far enough apart, especially in the areas surrounding the tears. TL was composed of three parts : the ascending aorta (TLaa), the aortic arch (TLarch) and the descending aorta (TLda). FL was composed of only one part. Figure 2.4 shows the localization of the initialization points and the decomposition of the two lines.

27 Partie , Chapitre 2 – Image segmentation of aortic dissection

FIGURE 2.4 – Pathlines of the two lumina : initialization points and different parts of TL.

. The computation of FL is ﬁrst described. Each new point was calculated from the −→ direction vector (V ) given by the two previous points. The two initial points of FL were

FLinit[0] and a point placed at the same position two CT slices below. We imposed −→ the vector V always pointing downwards. If the value at the line point in the sigmoid image was greater than to 0.5, the line point was kept. Otherwise, the algorithm looked for points (N) around the calculated position in the same slice ﬁrst, and then in the next slice below. This operation was iterated until the position of the last calculated point was below the end of the dissection ( FLinit[1]). Algorithm 1 presents the different steps for the computation of FL.

TLda was calculated using the same method. The initial point was TLinit[1] and the computation of the line stopped when the last calculated point was below TLinit[2].

Concerning the computation of the points of TLaa and TLarch, the method was similarly −→ but the initial points, the computation of V and the stop criterion were different.

For TLaa, the two initial points were TLinit[0] and a point at the same position two −→ slices above. The direction of V was imposed to be toward the up and the algorithm stopped when the last calculated point was above TLinit[2].

Finally, for TLarch, the ﬁrst point was the last point of TLaa. Then each new point

28 2.3. Semi-automatic method for type B aortic segmentation

Algorithm 1 Calculate the points in the false lumen

Input : Sigmoid image, FLinit[0], FLinit[1] Output : Points of the line of the false lumen (FL) Initialization : 1: FL[0] = FLinit[0] 2: FL[1] =FLinit[0]-(0,0,2) 3: i=1 4: for points in FLinit do 5: while FLZ [i] < F Linit,Z [1] do −→ −−−−−−−−−−→ 6: V = FL[i]FL[i − 1] 7: if (VZ > 0) then 8: Vz = −1 9: end if −→ 10: normalize(V ) −→ 11: FL[i + 1] = FL[i] + ( V ) 12: while value(FL[i + 1]) ≤ 0.5 do 13: for neighbourNofF L[i + 1] do 14: if value(N) ≤ 0.5 then 15: FL[i + 1] = N 16: end if 17: end for 18: FLZ [i + 1]+ = 1 19: end while 20: i+ = 1 21: end while 22: i+ = 1 23: FL[i + 1] = FLinit[1] 24: end for

29 Partie , Chapitre 2 – Image segmentation of aortic dissection was calculated using the normalized vector obtained from the difference between the last calculated point and TLinit[1]. The algorithm stopped when the distance between the last calculated point and TLinit[1] was inferior to 2 mm. If TL (resp. FL) deviated from its lumen, the user could add points in the initial set of points of the corresponding lumen. The algorithm integrated the points in the computation of TL (resp. FL) and corrected the trajectory by adding them as intermediate stopping points. For TL, the intermediate stopping points were sorted out in function of the part of TL they belong (TLaa or TLda). In TLaa (resp. TLda), if the last calculated point was above (resp. below) than one of the intermediate stopping points, the algorithm stopped, and TL continued to be calculated from the corresponding intermediate stopping point. For FL, the algorithm worked the same as for TLda. If the dissection terminated in the iliac arteries, instead of the input points at the end of each lumen, the user must select three points : one at the beginning of the bifurcation and two at the ends. The points after the bifurcation were calculated from the bottom of the iliac arteries to the beginning of the bifurcation similarly to the points of TLaa. Finally, the user must select a point in each of the different tears of the aortic dissection.

2.3.3 Segmentation of the two lumina

The segmentation of the true and false lumina must be carried out using the computed lines and the sigmoid image. Since the intimal ﬂap is very thin, a sensitive method was required for the segmentation.The geodesic active contour method [21] was applied to segment the true and false lumina as well as the tears. It is a level set segmentation approach combining active contour method and the computation of geodesic distance curves. The classic active contour models or "snakes" are based on the deformation of an initial contour towards the boundary of the object to detect. The ﬁnal deformation is obtained by minimizing the energy function. The energy function is driven by two main components : one controlling the curvature of the contour (curvature parameter) and one depending on a speed function attracting the contour near the boundaries of the object (advection parameter). Geodesic active contour methods are also based on the computation of geodesic distance curves. This method allows stable boundaries detection for geometry including gaps. It is therefore suitable for the

30 2.3. Semi-automatic method for type B aortic segmentation extraction of the true and false lumina since the intimal flap is not always visible and that tears connect them. A first contour estimation was needed to apply this process. Fast-marching method can provide this initialization from the preprocessed image and the lines passing through each lumen. Fast marching method solves the nonlinear Eikonal equation. It describes the evolution of a contour moving at a given speed function F (the sigmoid image in our case) and returns an arrival time surface. It is a time-crossing map indicating for each voxel, how much time it would take for the contour to reach the voxel location at the velocity given by the speed image. The segmentations of the true and false lumina were carried out separately. One front in each lumen was computed from the fast marching method using the points of TL and FL. Near tears, the front of one lumen tended to propagate to the other lumen through the tears on several slices above and below the slices including the tears. The fast-marching method was applied locally (on the 2D image) on the structure near the tears and globally (on the 3D image) elsewhere. The value of the maximum arrival time needed to be fixed and was equivalent to a threshold fixing when to stop the propagation of the front. The tuning of the thresholds indicating when to stop the fronts was particularly critical since the diameters of the true and false lumina were often different and changed along the aorta. The false lumen was often thin at the beginning of the dissection and increased thereafter and the dissection could start in the aortic arch and finish in the iliac arteries. An overestimation of the threshold led to an overlapping of the lumens while an underestimation of the threshold caused bad segmentation of the large structures such as the aortic arch. To counter this issue, the threshold varied along the aorta. At each slice, a high threshold was first applied for the two lumens. If the two fronts from each lumen coinci- ded, then a smaller threshold was applied. This process was iterated until the applied threshold was small enough such that two final fronts did not coincide on the slice of interest. This method allowed us to correctly segment small structure, such as the iliac arteries, as well as larger structures, such as the aortic arch. Moreover, we ensured that one lumen did not overlap the other. Nevertheless, it was difficult to obtain automatically a final correct segmentation of the true and false lumina using only the fast-marching method since the tuning of the threshold was highly sensitive. The geodesic active contour method was therefore applied to complete the segmentation. This method allowed to control the curvature of

31 Partie , Chapitre 2 – Image segmentation of aortic dissection

(a) (b) (c) (d)

FIGURE 2.5 – Segmentation steps of the two lumina. a) Result from the fast-marching method for the true lumen. b) Result from the active contour geodesic method for the true lumen. c) Result from the fast-marching method for the false lumen. d) Result from the active contour geodesic method for the false lumen. the final contour to avoid the overlap of the lumina and a set of parameters suitable for all dissections could be found to obtain a fully automatic segmentation. We used a set of 3 CT-scans to manually optimize the value of the parameters of the method. To avoid infiltration from one lumen to the other, we set a higher value for the curvature parameter than for the advection parameter. The geodesic active contour method was applied using as input the speed image issued from the preprocessing steps and the outputs of the fast-marching processes. These two segmentation steps of the true and false lumina are illustrated in figure 2.5. At this stage, the described process allowed the segmentation of the true and false lumens but not the tears as regions. The segmentation of the tears had yet to be carried out. From the set of user input points corresponding to the tears, the tears were extracted using the fast marching method with a fixed threshold and the geodesic active contour technique as for the segmentation of the lumens. The parameters of the active contour method were different from those for the segmentation of the lumens since the geometry of the tears presents a higher curvature (see Table 2.1).

2.3.4 Post processing

At this stage, a segmentation of both lumina and the tears was carried out. Next step was to ensure that both lumina did not share points except at tears position. During the application of the geodesic active contour method, there was no condition concerning the separation of the true and false lumina. Even if the initialization fronts from the fast

32 2.3. Semi-automatic method for type B aortic segmentation

(a) (b) (c) (d)

FIGURE 2.6 – Steps of the Laplacian smoothing. a) Original points. b) Center of gravity calculation G. c) Moving the initial point in the direction of G by a fraction λ of the distance. d) New point position. marching method were separated, some shared points could remain. A binary dilatation filter was applied, with a ball kernel size of 1, on both lumen segmentations and the remaining shared points were removed. This step was important since the two lumina were often connected at several locations by a few voxels which created spike shape geometries in the final surface model. This imperfections generated irregular surfaces which must be approximated by a large number of small mesh elements increasing the computation time. Also, unwanted connections affects the hemodynamics results of the 3D fluid simulations.

2.3.5 From segmentation to surface mesh

At this stage, the geometry of the fluid domain of the aorta was segmented. For fluid simulation purpose, a 3D surface was needed to generate a mesh. First, the well-known marching cubes algorithm [164] was applied to create triangle models of constant density surface from the 3D segmentation. At this stage, the generated surface presented many bumps. A smoothing step was necessary. The two most common techniques for this kind of medical application are the Laplacian and Taubin smoothing filters. The Laplacian filter moves the position of each vertex of the mesh depending on the local center of gravity by a scale factors λ as illustrated in figure 2.6. The main limitation of the Laplacian filter is the loss of topological information due to the shrinkage induced by the algorithm. Since each point is moved in the direction of the center of gravity of its neighbors, all the geometry tends to move toward the center of gravity of the initial segmentation. The Taubin smoothing filter preserved better the volume of the initial surface. An iteration including a negative scale-factor µ is added at

33 Partie , Chapitre 2 – Image segmentation of aortic dissection each iteration of the Laplacian filter. This algorithm implements a low-pass filter with a bandwidth of [0 ;k pb] with k pb = 1/λ + 1/µ > 0. Usually, λ and µ are set so k pb = 0.1 [145]. A trade-off between accuracy and smoothness is required when choosing the number of iterations for each smoothing filter. This parameter was chosen for each segmentation depending on the quality of the segmentation. We tested both smoothing filters with different number of iterations on one case of AD and evaluated its impact on the results from fluid simulation in section 2.4.6. Once the model was smoothed, the resulting surface was decimated. The purpose was to reduce the number of triangles constituting the 3D mesh. Therefore, the number of triangles was only reduced by 15% to limit detrimental effects on the mesh topology. The final step consisted in cutting the different outlets to obtain a planar surfaces.

2.3.6 Implementation of the method

The segmentation method was implemented in the 3D Slicer framework. 3D Slicer is an open source software available on multi-platform (Linux, windows and MacOSX) (http ://www.slicer.org)[40]. It has been built over the past two decades through the support of the National Institutes of Health and the contribution of numerous institutes around the world. This software allows to display and to manipulate imaging data from various types (MRI, CT scan, etc). It is designed for medical images analysis and visualization. Numerous applications like registration, interactive segmentation tools or volume rendering are included. 3D Slicer is contributing and extensible with a powerful plug-in capability for adding applications. An easy access to the Insight Toolkit [169] and Vizualisation Toolkit (http ://www.vtk.org) libraries and the development of fully interactive custom interface in C++ or python are available. In this context, we developed a python module integrated in 3D Slicer to segment semi-automatically aortic dissection. This module only performs the segmentation part, the user must use the modules Model Maker and Models of 3D slicer to create the 3D surface meshes from the final segmentation and obtain the 3D visualization. The graphic user interface of the implemented method is presented in the figure 2.7. The graphic user interface is separated in two parts. The former is dedicated to the user inputs and the preprocessing and the latter to the segmentation (figure 2.8). The user must first load the input volume corresponding to the CT data of the considered

34 2.3. Semi-automatic method for type B aortic segmentation

FIGURE 2.7 – Screenshot of the interface of 3D Slicer.

FIGURE 2.8 – Screenshot of the graphic user interface of the developed segmentation module.

35 Partie , Chapitre 2 – Image segmentation of aortic dissection dissection. Then, the input points must be placed using the module Markups integrated in 3D Slicer. Once the section Input points true lumen, Input points false lumen and Input points for the tears are ﬁlled with the correct ﬁducial points list, the user can launch the preprocessing step by clicking on the Apply button in the User Inputs and Preprocessing section after selecting the option Dissection in the iliac arteries if required. The different images resulting from the preprocessing steps are generated so the user can visualize them and check for any issue. The lines passing through each lumen can now be computed. They are displayed in the 3D view and the user can check if the trajectories of the lines are correct and add points otherwise. Once the lines are correctly computed, the user can click on the Apply button on the Segmentation section. The rest of the segmentation is automatically carried out and all the images from the intermediary steps are also available. From this segmentation, some manual post processings can be carried out from the module Editor if necessary. Finally, the 3D surface mesh can be created from the ModelMaker module and can be visualized in the 3D view.

FIGURE 2.9 – Screenshot of the ﬁnal segmentation results.

2.4 Results

This section presents the results of the proposed segmentation method. First, the data base, on which the method was tested, is presented. The parameters of the me-

36 2.4. Results thod are shown. Then, a quantitative and qualitative analysis are presented followed by a comparison of the method with deep-learning based segmentation methods. The section is concluded by an evaluation of the smoothing methods on simulation results.

2.4.1 CT Image acquisition

The proposed method for aortic dissection segmentation was tested on 21 CT data from 10 patients with postoperative, presurgical and follow up data. The CT data sets typically consisted of 400-650 slices of 512*512 pixels, with pixel size of 0.6–0.8 mm and an effective slice thickness of 0.625 mm. Of these 21 CT-scans, 10 presents metallic devices such as stents. Aortic calciﬁcations and aneurysms can also be seen on some of the volumes.

2.4.2 Parameters setup

The parameters setup used to segment all the geometries is presented in table 2.1.

Step Filter Parameters Value Pre-processing Curvature Anisotropic Conductance 10 diffusion Time step 5 Gradient Sigma 0.9 Sigmoid M 1 m 0 β 8 α -1 Segmentation of the lumen Geodesic active Propagation 10 contour Curvature 100 Advection 1 Segmentation of the tears Geodesic active Propagation 10 contour Curvature 10 Advection 10

TABLE 2.1 – Parameters of the proposed method.

37 Partie , Chapitre 2 – Image segmentation of aortic dissection

2.4.3 Segmentation error criteria

For each CT-scan, the comparison between the segmentations obtained from the proposed method and from the manual segmentation was carried out. Different error measures were used.

Dice Similarity Coefﬁcient

The DSC is a relative index of similarity between two segmentations and is deﬁned as : 2|L ∩ SA| DSC = , (2.3) |L| + |SA| where L is the set of voxels from the manual segmentation, SA is the set of voxels from the semi-automatic segmentation and |·| represents the volume of the given region. The maximal value of the DSC is one and represents the case where the two segmentations are the same. Since this value represents an overall overlap between the two regions, it is mostly representative of the segmentation of the global aorta and does not represent accurately the segmentation of the intimal ﬂap. Further measures are needed to obtain more information on the quality of the segmentation of the method.

Distance criterion

The distance between the manual and computed surfaces can be deﬁned by the distance between each vertice by projecting them to the other surface. This measure is asymmetric and depends on the image used as reference and the one used to be projected. The two conﬁgurations were computed. The mean distance, the standard deviation and the Hausdorff Distance (HD) were calculated for each image.

2.4.4 Quantitative analysis

The average computation time, on an Intel(R) Xeon(R) processor, is 205 seconds for the preprocessing, 3 seconds for the computation of the curves and 270 seconds for the segmentation steps, which makes a total of approximately 8 minutes. Since the manual selection of the initialization points takes a few more minutes depending on the complexity of the dissection and the experience of the user, the full processing of the data sets takes about 12 minutes.

38 2.4. Results

Results

Table 2.2 presents the comparison of the different segmentation errors for the 21 considered cases. The segmentation scores of our method are globally satisfying. We obtained an average DSC of 0.96 meaning that the global aorta is well segmented. Globally the average distance and HD when projecting the vertices of each surface from the proposed method (resp. manual segmentation) to the manual segmentation (resp. semi- automatic segmentation) are of 0.21 and 5.47 (resp. 0.34 and 6.21). The cases with the worst results are detailed below. The manual segmentation is operator dependent. Data sets with noise can induce a blurred intimal ﬂap leading to an incertitude on the geometry of the lumina. One example of this situation is illustrated in ﬁgure 2.10 with two possible corresponding segmentations of the true and false lumina.

(a) (b) (c)

FIGURE 2.10 – Example of the uncertainty of the manual segmentation on a CT slice with noise. a) CT scan slice. b) First example of segmentation. c) Second example of segmentation.

Kovács performed an evaluation of the manual segmentation of ADs [75] to assess the quality and reproducibility of the method. In average, the DSC was of 95.13% between manual segmentations realized by three medical experts and the ground truth (deﬁned as the average of the compared segmentations) showing the difﬁculties to correctly assess the geometry of aortic dissection.

39 Partie , Chapitre 2 – Image segmentation of aortic dissection

Patient Manual → Semi-Auto Semi-auto → Manual CT DSC HD Mean SD HD Mean SD Patient 1 1 0.95 6.66 0.34 0.60 4.86 0.21 0.37 2 0.95 5.88 0.37 0.49 3.25 0.25 0.33 3 0.94 6.58 0.46 0.79 5.16 0.24 0.42 Patient 2 0.97 8.88 0.28 0.58 5.77 0.17 0.38 Patient 3 0.97 4.86 0.25 0.53 4.97 0.18 0.42 Patient 4 1 0.93 8.64 0.63 1.00 5.53 0.29 0.47 2 0.95 5.99 0.20 0.47 3.41 0.12 0.20 Patient 5 1 0.97 4.49 0.25 0.43 4.32 0.17 0.30 2 0.98 4.47 0.18 0.35 9.20 0.12 0.24 Patient 6 1 0.97 5.87 0.25 0.66 4.84 0.13 0.44 2 0.97 5.62 0.25 0.59 4.32 0.12 0.27 Patient 7 1 0.92 7.18 0.52 1.01 12.25 0.51 1.38 2 0.91 7.08 0.57 0.80 6.49 0.47 0.71 3 0.96 5.46 0.31 0.63 5.51 0.16 0.36 Patient 8 1 0.99 4.29 0.18 0.40 3.27 0.11 0.26 Patient 9 1 0.99 4.58 0.13 0.39 9.19 0.09 0.33 2 0.94 6.43 0.50 0.77 3.40 0.24 0.42 3 0.95 6.38 0.37 0.65 2.80 0.18 0.36 4 0.99 7.12 0.32 0.61 8.29 0.25 0.52 Patient 10 1 0.97 7.08 0.43 0.78 4.30 0.27 0.43 2 0.97 7.12 0.32 0.61 3.75 0.23 0.40 Average 0.96 6.21 0.34 0.67 5.47 0.21 0.48

TABLE 2.2 – The obtained distances (in mm) between the segmentations from the proposed method and the manual segmentation : Dice coefﬁcient, Hausdorff distance, mean absolute distance and standard deviation from the semiautomatic segmentation to the manual one and vice versa. The average values across all datasets are displayed in the last row of the table.

40 2.4. Results

Comment on patient 2 and 4

Figure 2.11 (resp. 2.12) depicts the absolute distance between the surface meshes obtained from the proposed segmentation method and the manual segmentation for patient 2 (resp. patient 4.1). Patient 2 obtained the highest distance when projecting the semi-automatic segmentation on the manual segmentation and patient 4.1 obtained the maximum average distance.

(b)

(a) (c)

FIGURE 2.11 – Distance error for patient 2. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Segmentation results on the slice with the highest distances. c) Corresponding CT slice.

On both patients, the dominant color being blue, the segmentation score is globally accurate. We can observe that the segmentation from the proposed method can be inaccurate for lumina with crescent moon like shape or thin corners near fuzzy intimal flap. Blurred intimal flap occurs near tears. On these slices, the threshold computed for the fast marching method is low since the true and false lumina propagation fronts easily cross the intimal flap. Moreover, the curvature parameter of the geodesic active contour was set high to avoid the propagation of the segmentation from one lumen to the other and therefore impacts the segmentation of thin and sharp structures.

41 Partie , Chapitre 2 – Image segmentation of aortic dissection

(b) (d)

(a) (c) (e)

FIGURE 2.12 – Distance error of patient 4.1. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Segmentation results on slice 1. c) CT slice of slice 1. d) Segmentation results on slice 2. e) CT slice of slice 2.

42 2.4. Results

Comment on patient 7.1

Figure 2.13 depicts the absolute distance between the surface meshes obtained from the semi-automatic and manual segmentation for the data set presenting the highest HD value (patient 7.1).

(b) (d)

(a) (c) (e)

FIGURE 2.13 – Distance error of patient 7.1. a) Absolute distance between the manually segmented dissection and the semi-automatically extracted one. b) Segmentation results on slice 1. c) CT slice of slice 1. d) Segmentation results on slice 2. e) CT slice of slice 2.

For this data set, two main problems lead to poorly segmented regions. First, there was a huge contrast artifact in the aortic arch preventing the algorithm from segmenting correctly the aortic arch. Secondly, the contrast did not allow to differentiate enough the aorta from the background on some slices.

2.4.5 Qualitative analysis

In this section, a qualitative analysis of the results of the segmentations from the proposed method is presented.

43 Partie , Chapitre 2 – Image segmentation of aortic dissection

FIGURE 2.14 – CT slice of the case where the true lumen was too thin and the segmentation failed.

We succeeded to segment separately the true and false lumina as well as the tears on all data set except one. For this case, the true lumen was too thin on a small part of the dissection and the algorithm could not detect it (see figure 2.14). Most of the cases (16 out of 21) did not need the addition of intermediate stopping points for the computation of the lines in the lumina. Additional points (up to 4) were required for cases including large aneurysm or tortuous lumina. Two cases required more additional points (10 and 12). Numerous tears with blurry intimal flap on several slices above and below appeared on these cases and the CT data were very noisy. Dissections with variable geometries were successfully segmented. Typical cases were correctly delimited (figure 2.15 a.). Aortae with aneurysm (figure 2.15 d. and e.), inhomogeneous contrast (figure 2.15 a. and c.) or the presence of stents (figure 2.15 e. and g.) were well handled. These are difficult cases for methods segmenting first the entire aorta. As well, dissections where three lumina appear on the same slice (figure 2.15 b.) or cases where the two lumina are separated (figure 2.15 c.) were correctly segmented which can be challenging for algorithms segmenting the intimal flap. Globally, a good segmentation can be achieved from the aortic arch to the iliac arteries (figure 2.15 f.). The user interaction allows the method to segment all kinds of geometries since, in case of bad segmentation, points can easily be added to correct mishandled regions. The method was also robust enough to handle correctly noise (figure 2.15 b.).

44 2.4. Results

(a) (b) (c) (d)

(e) (f) (g) (h)

FIGURE 2.15 – Segmentation on various geometries of the lumina. a) Typical AD slice b) Case where three lumina appear on the same slice. c) Case with a partially thrombosed FL. d) Case with an aneurysm in the aortic arch. e) Case with an aneurysm and a stent. f) Case of a dissection in the iliac arteries. g) Case with a stent in one lumen. h) Case with noise.

2.4.6 Comparison with deep-learning based methods

Over the last decade, deep learning architectures have achieved remarkable advances in computer vision task. Convolutional Neural Network (CNN) methods have

45 Partie , Chapitre 2 – Image segmentation of aortic dissection demonstrated the ability to perform tasks in the medical imaging field like classification, segmentation or registration [85]. Our conventional method was compared with 2 methods based on Convolutional Neural Network (CNN) developed in the South-East University of Nanjing (China) in the Laboratory of Image Science and Technology by Tianling Lv. The first method was a 3D CNN using the Unet architecture [122]. The image was first resized into a smaller one to save GPU memory and clipped into a 256*256 size image. A linear interpolation along the z-direction was then applied to unify the volume size. This volume was inputted to the 3-D Unet model resulting in a volume containing aorta confidence between 0 and 1. Then, the confidence volume was up-sampled to retrieve its original shape and a threshold of 0.5 was applied to obtain the final segmentation. The second method was a 2D CNN based on PSPnet [171]. An edge extraction based on the Holisticallynested Edge Detect (HED) network was added to extract the aorta boundaries. The two branches were then fused using a network reduced to three convolutional layers. The deep-learning-based methods, the models were trained on 35 CT-scans from Chinese hospitals (excluding the ones used for testing) with more than 18000 slices in total. The three segmentation methods results were compared on a set of 10 CT data : 5 from the French hospital and 5 from the Chinese hospitals. The results are presented in figure 2.16. Deep learning methods are fully automatic and much faster than conventional methods (25 seconds for the 3D CNN and 83 seconds for the 2D CNN). Nevertheless, the segmentation results based on deep learning methods are highly dependent on the training data set. AD cases including aneurysm for example were more difficult to handle for the CNN based methods (figure 2.16 d.). The presence of metallic devices like stents can also lead to a bad segmentation using the deep learning based methods. It was also complicated to impose that the two lumina were separated for the methods based on deep learning. We can observe cases where the two lumina were connected for the 2D CNN based method (figure 2.16 b. and c.). Cases, where the contrast between the two lumina was different, were globally well handled by the three methods but we can observe some cases where the segmentation from the 3D CNN based methods failed (figure 2.16 e.). Difficult cases, like when the intimal flap were blurry, were globally handled equivalently by the three methods (figure 2.16 a.).

46 2.4. Results

CT-scan 2D CNN 3D CNN Conventional

FIGURE 2.16 – Comparison results from the three segmentation methods. a) Blurry ﬂap. b) Iliac arteries. c) Connections between the two lumina. d) Aneurysm in the aortic arch. e) Contrast problem.

47 Partie , Chapitre 2 – Image segmentation of aortic dissection

2.4.7 Comparison of CFD results using different smoothing methods

The proposed method is targeted toward fluid simulations. To assess the quality of the segmentation method, we ran fluid simulations on one case. The post-processing steps are essential to obtain a good quality mesh suitable for fluid simulation. Tthe strict separation of the lumina avoids holes and spikes resulting in skewed elements in the final mesh. The smoothing step improves the overall quality of the final surface. As said previously, we chose to compare the two most common meshing methods : the Laplacian and Taubin smoothing filters. We created two geometries using the Laplacian method with two different number of iterations (40 and 80) and one geometry using the Taubin smoothing. The selected dissection started in the aortic arch and finished just before the iliac arteries. The final geometries had one entry tear in the aortic arch and one re-entry tear in the descending aorta. The beginning of the brachiocephalic artery, left subclavian artery, left common carotid, celiac trunk, superior mesenteric artery and renal arteries were also segmented. Figure 2.17 illustrates the final geometry. The three 3D surfaces were meshed with polyhedral elements and 8 prism layers on the wall. We used a local refinement on areas with high curvature. The mesh of the geometry from the Laplacian smoothing with 80 iterations contained 955 516 cells, the one using 40 iterations had 2 003 525 cells and the one from the Taubin smoothing had 3 081 182 cells. The high difference in the number of cells between the different meshes is explained by the curvature refinement applied during the meshing. Meshes sensitivity tests were carried out for each geometry. The average relative error on the pressure at the wall at systole was of 1% between the described meshes and finer meshes. 3D transient fluid simulations using the same setup were carried out using these meshes. The blood was modeled as an incompressible Newtonian fluid with constant density (ρ =1056 kg.m3) and viscosity ( µ =0.0035 Pa*s). Turbulence was modeled with a scale adaptative simulation model. The walls were considered rigid. A simple outflow boundary condition was used for the outlets and a mass flow profile was applied at the inlet. The flow distribution between the cardiac outputs in each terminal vessel are presented in table 2.3 and were taken from [139]. Figure 2.18 is the curve of the mass flow rate applied at the inlet.

48 2.4. Results

(a) (b) (c)

FIGURE 2.17 – Geometries generated using three different smoothings. a) Laplace with 80 iterations. b) Laplace with 40 iterations. c) Taubin.

49 Partie , Chapitre 2 – Image segmentation of aortic dissection

Outlet % cardiac output Brachiocephalic 10.41 Left common carotid 2.14 Left subclavian 8.27 Celiac 13.24 Superior mesenteric 15.97 Righ renal 13.15 Left renak 13.15 Descending aorta 23.68

TABLE 2.3 – Cardiac outputs at each outlet for the outﬂow boundary conditions.

FIGURE 2.18 – Mass ﬂow proﬁle applied at the inlet.

We compared the difference of flow velocity in three different planes : one in each tear and one in between the two tears. The different geometries are illustrated in figure 2.17. Figure 2.19 illustrates the velocity fields at systole in the different planes for the three geometries.

50 2.5. Discussion

FIGURE 2.19 – Comparison of the velocity ﬁelds in the three planes of the aortic dissection computed on geometries obtained with different smoothing techniques : Laplacian with 80 iterations, Laplacian with 40 iterations and Taubin (from top to bottom).

The results were globally similar, we can observe the same ﬂow patterns. For example, whirlpools were formed inside the false lumen on planes including tears. Since the Laplacian method reduced the volume of the initial segmented geometry, the tears were smaller in the geometry smoothed by the Laplacian process with 80 iterations. It induced a higher velocity ﬂow inside the tears.

2.5 Discussion

The proposed method for the segmentation of type B aortic dissections is a semiautomatic method incorporating user input points and a competitive fast marching method supplemented by a geodesic active contour process. The segmentation of aortic dissections is challenging because : of the thinness of the intimal ﬂap, the high variability of the geometry among the patients and the frequency of aneurysms, metallic devices

51 Partie , Chapitre 2 – Image segmentation of aortic dissection and contrast problems in aortic dissection CT data. We tested our algorithm on 21 clinical cases with various forms of type B dissections. Our method proved to be robust against complex configurations including aortic aneurysms, aortic calcifications and the presence of metallic devices. The algorithm successfully segmented the true and false lumens in 20 cases. Several features related to the dissection might limit the performance of the segmentation. In some cases, it is difficult to correctly segment the regions near tears. The flap can move during image acquisition giving it a blurry or duplicated appearance. Since the geometry of the flap is not fixed in those cases, an approximation can be acceptable for fluid simulation purposes. Second, crescent moon shape lumens cannot be segmented completely. The advection parameter of the geodesic active contour method is set low to ensure that the lumens do not overlap each other but this can prevent the segmentation from extracting sharp corners. The proposed method focuses on the segmentation of the lumens. The geodesic active contour method setup was chosen to prevent the lumens from overlapping, setting a high curvature parameter therefore also prevent the process from segmenting the secondary arteries. For simulation purposes, ideally the beginnings of the brachiocephalic, the left subclavian, the left carotid, the celiac trunk, the superior mesenteric and bilateral renal arteries should be segmented. Several segmentation methods have already been reported in the literature. They can easily be combined with the proposed method since the image processing pipeline is compatible with a straightforward integration of additional segmentation processes and results [44, 80]. The final segmentation depends on the position of the user input points. An inaccurate initialization might induce an incorrect segmentation. Although, it is easy to verify if the points are correctly placed within their lumen. Moreover, the acceptable areas for the placement of the initialization points are large. An automatization of the method could be investigated. The proposed method was compared to CNN based methods. The proposed method demonstrated a better ability to segment complex features such as aneurysm. However, deep learning based methods are fully automatic and fast. Fluid simulations were carried out on one of the aortic dissection cases to assess the influence of the smoothing methods on the simulation results. Differences in the velocity fields near the tears were observed because Laplacian methods induce a shrinkage of the geometry decreasing the size of the tears. The Taubin smoothing method

52 2.6. Conclusion seems more appropriate for simulation purposes.

2.6 Conclusion

In this chapter, we introduced a new method for the reconstruction of a 3D geometrical model of type B aortic dissection for patient specific fluid simulation based on a competitive segmentation process. The method proved the ability to correctly segment the true and false lumina of ADs and to be robust against complex structures related to ADs such as aneurysm or the presence of metallic devices. Transient fluid simulations were computed to demonstrate that the proposed method is suitable for CFD application. In the next chapter, we present a novel method to compute patient specific simulations in the context of ADs.

CHAPITRE 3 PATIENT SPECIFIC CFD SIMULATION OF THE AORTA

Over the past decades, due to the numerous advances in computing speed, models of aortae have been developed to help understanding the physics induced by the aorta pathologies. Patient-specific simulations can provide clinicians relevant data like the velocity of the blood and the pressure on the aorta wall in a non-invasive way [172] for the evaluation and diagnostic of the diseases or surgical and treatment planning [3], [17]. In this chapter, our purpose is to develop an accurate 3D dynamic fluid model of aortic dissection using only data from CT-scan and 2D PC MRI. A new method to obtain accurate patient specific BCs at the inlet and outlets was developed based on a 0D model of the full cardiovascular system. The state of the art related to the simulation of large arteries, and more specifically aortic dissections, is discussed in this chapter. An analysis of the challenges related to the usage of 0D models as BCs for CFD is presented. The proposed method is then described considering the case of healthy aortic arch and the case of AD. At last, the results of the simulations are presented.

3.1 State of the arts of CFD for AD

Numerous methods have been developed to simulate the aorta and more specifically aortic dissections. Models with different levels of complexity, from 0D to 3D, exist to describe the aorta [133]. 0D models, or lumped parameters models, provide global cardiovascular dynamics data like the general pressure or the flow-rate changes in the local circulation and can be used as boundary conditions for 3D model [7, 106]. The 1D models can describe the wave transport effect since the pressure and flow changes are represented into the full length of the vessel studied. 2D models are able to model

55 Partie , Chapitre 3 – Patient specific CFD simulation of the aorta the radial changes of the vessels. Finally, 3D models are based on the computation of the Navier-Stokes equations applied on a finite element discretization of the full 3D geometry. 3D models are able to describe locally the pressure and velocity fields as well as the WSS and are necessary to describe complex features like bifurcation or aneurysm. 3D models are the most accurate and interesting for the clinicians but have a high computational cost due to their complexity. Also, the validation of the 3D models can be difficult due to the lack of available measures and 3D models are not always realistic. In this thesis, we want to obtain an accurate 3D patient specific model of aortic dissection. The results of the simulation depend on various modeling assumptions like the model of the flow or the definition of the BCs. In this section, an analysis of the state of the arts is presented. We focus on 3D simulations for large arteries in general with a more particular focus on aortic dissection simulations.

3.1.1 Fluid modeling

Material properties of the blood

The blood is a complex fluid mainly composed of plasma (55%) and blood cells (red and white blood cells and platelets). The plasma is a yellow liquid containing 90% of water and can be considered as an incompressible Newtonian fluid. However, the presence of the blood cells, complicates its material properties. Red Blood Cells (RBCs) are one of the major constituents of the blood (40%) which is about four times more viscous than water. Therefore, the viscosity of the blood depends on the local concen- tration of the RBCs which varies depending on the type of blood vessels [116]. The blood in large arteries, when measured in constant shear rheometers, exhibits non- Newtonian properties. It has a behavior of the shear-thinning type meaning that the viscosity decreases when the shear rate increases. Some studies investigated the blood flow properties in large arteries [146] but there remains little consensus as to the most realistic model. Three studies about the simulation of AD used the Carreau-Yasuda blood flow model [138, 3, 11] which model the viscosity depending on the shear rate. Cheng et al [27] compared the effect of using a Newtonian model against the Quemada model, which is haematocrit and shear dependent, and demonstrated that the non-Newtonian model caused a reduction in the maximum WSS value (' 8%) and pressure magnitude (' 12%). Hou et al. [57] used

56 3.1. State of the arts of CFD for AD a three-phases model (modeling the plasma, RBCs and leukocytes) to investigate the effect of the particles interaction and of biomechanical factors. A separation of RBCs in the FL was observed with an increase in haematocrit in the distal FL and a decrease in the proximal FL. However, several books [47, 112] reported that modeling the blood as a Newtonian incompressible ﬂuid is sufﬁcient. This assumption is based on the following arguments : the effect of particles can be negligible in large arteries, the shear rate is high in large arteries and it increases computation time. Therefore, many studies assume the blood to have Newtonian properties in large arteries.

Steady and transient simulation

The flow simulation can be steady (time-independent) or transient. In the smaller arteries of the human body far from the pumping heart, the assumption of steady flow can be correct. The flow in the aorta is pulsatile and a transient approach is needed to accurately capture time-dependent features. Steady model can be used to obtain an approximation of the hemodynamics in the aorta.

Turbulence modelization

Another key issue is to decide whether the flow is laminar, transitional or turbulent. In a laminar flow, the fluid layers slide in parallel with no disruptions like eddies or swirls between them. A turbulent flow is characterized by erratic changes in the velocity of the fluid particles. The passage from laminar to turbulent flow is called transition state. The dimensionless Reynolds number is the ratio of inertial forces to viscous forces within the fluid. It aims to describe the global behavior of the fluid. When the viscous forces are dominant, the Reynolds number is low, and the flow tends to be laminar. On the contrary, when the fluid is driven by the inertial forces, it tends to produce flow instabilities and turbulent flow occurs. But the Reynolds number does not describe the flow locally and laminar and turbulent regimes can cohabit in the same flow. The limit between laminar and turbulent flow is not well defined but is normally assumed to be above the critical Reynolds number. In a steady pipe, the transition to turbulence happens at a critical Reynolds number around 2000. However, in case of pulsatile flow, the critical Reynolds number is much higher since during the systole the pulse tends to keep the flow structured while disturbances and chaotic features appear during

57 Partie , Chapitre 3 – Patient specific CFD simulation of the aorta the decelerating phase. Many studies assumed the blood flow to be laminar in aorta simulations since the aorta is a large vessel and the mean flow velocity is low resulting in a relatively low Reynolds number. Based on this assumption, most of the studies about the simulation of aortic dissection considered the flow to be laminar [23, 153, 26, 64, 67, 65, 68, 159, 38, 123, 10, 11]. However, even if the blood flow is generally laminar in the aorta, turbulent flow has been observed in healthy aorta and is even more prevalent in case of aortic diseases [143, 71] like aortic dissections due to the complex geometry. Turbulent flow exists at different scales and the description of all the scales is very time consuming. Three main approaches have been developed to model turbulent flows : Reynolds Averaged Navier-Stokes (RANS), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). RANS and LES methods aim to capture the effects of turbulent flow features without resolving all the details while DNS methods compute all turbulent scales. RANS turbulence models compute the effect of the turbulence on the mean flow quantities but the details about the turbulent fluctuations are not solved. LES models resolve the large scales turbulent motion of the flow and the turbulence scales smaller than a grid size are solve using a subgrid-scale model but are more time consuming than RANS methods. Several studies compared different turbulence models for the CFD simulations of the arteries. Miyazaki et al. compared laminar scheme, renormalization group (RNG) k − and LES models to evaluate the turbulence computations in the aortic arch and found better correspondence to 4D MRI data using the RNG k − and LES turbulence models [95]. A study by Tan et al. [143] demonstrated that the transitional model SST Tran [93] gives a better correspondence of the flow patterns compared to MRI data than laminar flow model. Based on this findings, several studies of ADs [27, 66, 3, 28] used the correlation based transitional version of Menter’s hybrid k − /k − ω SST Tran model. Finally, other studies of AD assumed the flow to be turbulent [1, 71, 57] and used the Spalart-Allmaras, the Wilcox’s k − ω and SST models.

3.1.2 Model of the aorta wall

The material properties of the aortic wall of AD are complex. The structure of the intimal ﬂap is different from the structure of the outer walls surrounding the TL and FL. Moreover, due to the tissue healing process, some parts of the aortic wall become

58 3.1. State of the arts of CFD for AD more rigid than the initial wall. Consequently, it is complicated to properly assess the material properties of the wall since it contains both healthy and diseased tissues. Most of the studies of AD simulations considered the wall to be rigid. They justified this choice by a report from Ganten et al. showing a reduced vessels distensibility in ADs [48] and the little wall movements observed in time resolved imaging data [24, 27, 68]. However, the study from Gaten et al. reported a reduction of only 12% of the distensibility of the vessels compared to healthy aortae. Moreover, in the study from Karmonik et al. [66] the intimal flap was observed to contract (3.5 ± 0.9 mm) and extend (3.0 ± 2 mm) depending on the difference of pressure between the true and false lumina. Several studies used Fluid-Structure Interaction (FSI) methods to model the motion of the wall and intimal flap [1, 3, 71]. Afkari et al. [1] and Alimohammadi et al. [3] modeled the wall as one layer with hyperelastic properties while Khanafer et al. [71] differentiated the different layers of the aortic wall and used different material properties for each one. Alimohammadi et al. [3] compared the results of simulations using rigid and hyperelastic walls and concluded that modeling the wall motion is a necessary component of the model. Afkari et al. [1] also showed that the motion of the wall affects the WSS. The three-layered model of the aortic wall study [71] reported a larger stress in the media layer than in the intima and adventitia. Nevertheless, FSI methods are highly time consuming and the coupling between the fluid and structure parts can be unstable.

3.1.3 Boundary conditions

Despite the numerous advances in medical imaging, it is still impossible to obtain the description of the full vasculature. 3D fluid simulations can only be performed on a small portion of the cardiovascular system and boundary conditions providing the physical behavior at the fluid limits of the considered domain (called inlet and outlets) are required. Most of the information needed is usually unavailable due to the difficulty to make in vivo flow measurements on patients or because some measures like the pressure in the arteries are invasive. Studies have shown that different formulations of the boundary conditions at the inlet and outlets can change quantitatively the velocity of the flow or the WSS [87, 98, 99, 114]. The most common boundary conditions applied at the outlets are fixed flow division, prescribed pressures and Windkessel models

59 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

[114]. With regard to AD, most studies used prescribed pressure with either a Zero- pressure outlets (often combined with flow split outlets) [28, 27, 108, 130] or pressure profiles from the literature [138, 144, 71, 123]. Fixed flow division and Zero-pressure outlets conditions are not able to reproduce realistic pressure curve [114]. Pressure profiles cannot be acquired for the patient in a non-invasive way. Moreover, there is a phase shift difficult to calculate in between the outlets since the flow does not arrive at the same time everywhere depending on the distance of the outlet from the heart and the velocity of the flow. Pirola et al., Morbiducci et al. and Alimohammadi et al. [114, 99, 2] compared imposed flow rate and Windkessel models and showed that Wind- kessel models were able to reproduce realistic flow conditions. Windkessel models are 0D models based on the analogy between electric system and hydraulic system. This analogy is detailed in the next paragraph. Westerhof et al [162] presented in detail the different existing Windkessel models and their physical and physiological meanings. Several studies about AD simulations used the Windkessel models ([4, 3, 36, 11]). In case of models with multiple outlets (like models of the full aorta), the outlets are not correlated which can lead to numerical instabilities if the parameters of the Windkes- sel models are not correctly calibrated. Moreover, Windkessel models assume that the pressure in the veins is nil and do not provide information for the flow at the inlet. On opposite to the Windkessel model, the cardiovascular system is a closed network. Clo- sed loop 0D models of the full cardiovascular system should be considered to impose a coherence between the different outlets of the model. In this study, we investigated the use of a closed loop model of the full cardiovascular circulation to obtain the boundary conditions for the 3D fluid simulations. When it comes to the BC at the inlet, most studies prescribed velocity or flow at the inlet with values either from the literature, experimental measures or patient MRI data. The existing 0D models and the challenges related to their coupling with 3D models are detailed in section 3.2.

3.1.4 Summary

Table 3.1 reports a summary of the different assumptions that were made in the simulations of ADs by the different researchers on the geometry, blood and wall model and BCs.

60 3.1. State of the arts of CFD for AD Autoregulation and Impedance Pressure Fixed-flow ratepressure and literature FSI Windkessel Rigid Zero-gradient condition Rigid Value from the literature FSI Windkessel Rigid Patient dataRigid and fixed flow rate Patient data wtonian Rigid Literature Blood model Wall BC’s at the outlets Newtonian FSI Carreau- Yasuda Multi-phase flow Carreau- Yasuda Ne Carreau- Yasuda Newtonian FSI Literature Newtonian Rigid Windkessel Newtonian Rigid Zero-Pressure Newtonian Rigid/ Literature value t x’s ω − − Turbulence model Spalar k Wilco –Allmaras k SST Laminar Newtonian Rigid Literature y atient Specific Laminaratient Specific SST Newtonian tran atient Specific SST Tranatient specific Rigid Newtonian Laminar Windkessel atient Specificatient SST Specific Newtonian Tran Rigidatient SST Specific Tran Newtonian Zero-pressure + fixed-flow / rate Newtonian Rigid Rigidatient Specific Zero-pressure Rigidatient Laminar Specific Zero-pressure / atient Newtonian Specific Laminar atient Specificatient / Laminar Specificatient / specific Newtonianatient Zero-pressure Laminar Specificatient Laminar specific Newtonian Rigid / / Fixed-flow rate Rigid Fixed-flow rate atient Specific Laminar Geometr Ideal P P P P P P P Ideal P P P P P P P P Ideal Ideal Ideal P . 1] 4] 3] 24] 23] 28] 27] 36] 57] 108] 130] 138] 159] 170] 170] 67] 68] 144] 71] 123] 11] Ref [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

TABLE 3.1 – Summary of the different modeling assumptions of ADs in the literature.

61 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

Electrical variable Hydraulic equivalent Charge q [C] Volume V [m3] Current I [A] Volumetric ﬂow rate Q [m3/s] Current density j [A/m2] Velocity v [m/s] Potential V [V ] Pressure P [P a]

∆V π∗r4 Ohm’s law i = R Poiseuille’s law Q = 8lµ ∗ ∆P

TABLE 3.2 – Variables equivalent of the hydraulic/electric analogy where l corresponds to the length of the vessel, r the radius and µ the ﬂuid dynamic viscosity.

3.2 0D models of the CVS as BCs for 3D CFD

In this section, a description of the existing 0D models of the cardiovascular circulation system is presented. An analogy can be made between hydraulic and electrical systems. The electric current is equivalent to the volume flow rate, the charge is equivalent to the fluid volume and the potential difference is equivalent to the pressure drop. This analogy is based on the assumption that the physiological variables of interest (e.g. pressure, flow and volume) are uniformly distributed in space. By following this idea, the pipe (or vessel) is comparable to a conducting wire. At node or junction, the Kirchhoff’s law is applied in the electrical circuit and is equivalent to the mass conservation of an incompressible flow. Also, the Ohm’s law for electrical current and the Poiseuille’s law, for the flow of a fluid in a horizontal pipe in a laminar stationary state, are of the same form. Table 3.2 summarizes the different analogies between hydraulic and electric systems. As well, other basic electrical components are comparable to physic situations in a hydraulic system and more precisely in the cardiovascular system. Those models have become an established method for understanding the underlying mechanisms in the cardiovascular system and are known as lumped parameters models. The systemic and pulmonary vessels can be described by a combination of resistances, inductances and capacitances. The resistance main function is to oppose a resistance to the electrical current. It induces a potential drop. In hydraulic systems, a resistance can be represented by a

62 3.2. 0D models of the CVS as BCs for 3D CFD constriction in the ﬂuid pipe. In the cardiovascular system, resistance to the ﬂow is induced by the reduction of the vessel diameter, especially when the wall of the vessel presents a high rigidity like in the arterioles, and by the frictional losses due to viscous effects. It is mathematically formulated as follows :

∆P = RQ. (3.1)

In the hydraulic analogy, inductances model the blood inertia in the vessel circulation. It is mathematically formulated as follows :

dQ ∆P = L ∗ . (3.2) dt

In the blood circulation, capacitances describe the fact that vessels can expand their diameter under the pressure and store a larger amount of blood. It is formulated as follows : dP Q = C ∗ . (3.3) dt The different basic components of the electrical scheme are illustrated in ﬁgure 3.1.

FIGURE 3.1 – Basic components of 0D models (resistance, inductance and capacitance).

3.2.1 0D model of the systemic and pulmonary circulation

A wide range of 0D models have been developed to describe the systemic vasculature depending on the level of complexity of study of the interested. They are classiﬁed in two main categories : the mono-compartment models and multi-compartment models.

63 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

The mono-compartment description simulates the whole vasculature using only a combination of R, L and C components. The first mono-compartment model was proposed by Stephen Hales in 1733 and then formulated by Otto Frank in 1899 [83]. It was the two-elements Windkessel model and consisted of a capacitor representing the storage of the large arteries in parallel with a resistance modeling the resistance of the small vessels like the arterioles and capillaries. This model is based on the assumption that the veins can be neglected and be represented as a zero-pressure sink. This model can describe the decay in aortic pressure during the diastole with the time constant τ = RC. Despite the fact that the two-elements Windkessel model only has one time constant and is therefore unable to describe the high frequency caused by the pressure reflections in the vessels, it can be used as a boundary condition for 3D transient fluid simulations. Numerous models have been developed during the following decades to obtain a more accurate representation of the systemic vasculature. For example, the Westkessel model or RCR model adds a resistance, describing the characteristic impedance of the arterial network allowing a better description of the high frequencies [161]. Further models with more complexities also exist and are described in the review of Shi et al. [133]. Mono-compartment models are unable to describe the local flow and pressure in a precise portion of the systemic vasculature. Multi-compartment models divide the systemic vasculature into vessel segments being described by a combination of capacitances, resistances and inductances depending on their properties. Each segment are connected to form the global systemic circulation. Depending on the part of the network which is studied, multi-compartment models can describe more or less precisely parts of the network by adding segment locally and using more simple lumped model to describe the rest of the network. Most studies divide the systemic circulation in the following parts : aorta, arteries, capillaries, arterioles and veins. As said in Chapter I, the aorta and arteries are compliant vessels and the blood flow is pulsatile, therefore RLC components are required to describe correctly the physics in these segments. Arterioles and capillaries vessel walls are much more rigid, and a single resistance is enough to represent them since the flow is steady and the frictional loss is the main impact factor. The veins have elastic walls and the flow is steady in the venous part of the network thus the inertial effects can be neglected. A capacitance and a resistance are considered enough to describe the flow and pressure characteristics.

64 3.2. 0D models of the CVS as BCs for 3D CFD

More complex models exist, for example Noordergraaf et al. [147] or Avolio [7] have developed models where each segment of the arterial tree is described. However, a more complex model induces a higher number of parameters which are difﬁcult to measure.

3.2.2 0D model of the heart

Numerous studies tried to quantify the heart as a pump. The heart can be divided in several parts, each of them being represented by a component : the ventricles, atria and valves. Numerous models have been developed to simulate the ventricles, valves and the interaction between the different elements, differing in their complexity. One model that has been widely adopted by the scientiﬁc community is the time varying elastance model from Suga et al. [141] for the ventricle. This model describes the ventricle pressure as a linear function of the ventricle volume and elastance. The time dependent elastance function is based on in vivo measurements of the ventricular activity over the cardiac cycle. Concerning the valves, an easy way to model them is to use a simple electric diode that is either fully closed or open depending on the pressure on each side of the valve. More complex models exist taking into account non-linear effects, external interactions or neuroregulation. Shi et al. [133] presented a review of the different existing models for more details.

3.2.3 Multiscale coupling

0D model of the full vasculature can be coupled with a 3D model to obtain correct boundary conditions at the inlet and outlets. Multiscale models, coupling 3D arteries with closed-loop model, have been carried out mainly to investigate the hemodynamics of coronary arteries [125, 51, 154]. Coronary arteries simulations need specific boundary conditions due to the influence of ventricular contraction resulting in the ne- cessity to model the 3D aortic arch and the coupling to closed loop model. Other studies have been made to study the treatment of congenital heart diseases [94, 31] or left pulmonary artery stenosis [126]. The direct coupling with 0D models can be difficult and unstable process. First, it takes time (5/6 cardiac cycles) for the 0D simulation to converge (in the sense that the flow and pressure are the same from one cardiac cycle to the other) due to the loading of the capacitances and the stabilization with CFD.

65 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

Secondly, if the boundary conditions linking the outlets and the inlet are not coherent, unphysical solution will be computed leading to divergence in the solution. Most of the authors are concentrating their effort on stabilizing the boundary conditions via the usage of under-relaxation, additional damping terms in the Navier-Stokes equations at boundary conditions, specific numerical schemes or a redefinition of the coupling functions to ensure conservation using more advanced pseudo monolithic methods [119, 43, 37, 59, 52]. Most of the time the instabilities are triggered by inproper initial solution in an ill-posed problem. The existing methods are not representative of the equilibriums in between the 0D model and the 3D model leading to unbalanced boundary conditions. Even in such a case the methods listed above help to sustain such an artificially created flow, their convergences will be long if even possible.

3.2.4 Personalization of the 0D model

In order to obtain a patient specific model, the parameters of the 0D model (inductances, resistances, capacitances and elastance parameters) must be calibrated. Several methods have been developed to personalize 0D models, so that they are patient specific. Some values can be determined directly using clinical data like the global resistance or geometrical parameters [55, 121]. The most common method is to use data from the patient or the literature to initialize the parameters and to apply an optimization method to refine the initialization process [149, 20, 56]. These studies used invasive clinical measurements like the pressure in the aorta [55, 115, 142], values from the literature or retrospective patients [149, 56], or invasive measures from animal or a combination of these data that hinder eventual incorporation of those approaches into clinical routines. Thus, some studies investigated methods using medical data from non-invasive methods [111, 69]. CFD allows to access data like the pressure in the aorta in a non-invasive way. Some studies computed 3D fluid simulations to obtain the correct pressure at each iteration of the optimization process, but it is time consuming [110]. The optimization algorithm is constrained to converge in a few iterations and to reduce the number of parameters to optimize. Nevertheless, most of the previous studies did not coupled 3D and 0D models.

66 3.3. Method

FIGURE 3.2 – Main steps of the method to compute patient speciﬁc transient CFD of the aorta.

3.3 Method

We propose a method to compute patient specific simulations of the aortic arch using only CT and 2D PC MRI data. The method was applied on a healthy aortic arch and on a case of AD. The geometry of the patients can be extracted from the CT data and meshed to create a 3D domain. As explained in the previous section, one of the main difficulties in the CFD simulation of the aorta is to obtain realistic boundaries conditions. We used a 0D model of the full cardiovascular system, i.e. closed loop 0D model, to describe the flow and pressure at the inlets and outlets. To counter unstability problems induced by multiscale coupling, we created a hybrid 0D model of the aortic arches based on steady simulations and electrical components. These models allow to obtain an approximation of the behavior of the flow and pressure in the branches of the two aortae. They were integrated and coupled with the 0D models of the full cardiovascular system. Patients data were used to personalize the resulting coupled 0D models so the different parameters (resistances, inductances, capacitances and heart parameters) were patient specific. From these final 0D models, coherent and realistic boundaries conditions were extracted to compute transient fluid simulations. The main steps of the method are represented in the figure 3.2.

67 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b)

FIGURE 3.3 – Surfaces rendering of patient 1 and 2. a) Aorta of patient 1. b) Aorta of patient 2.

3.3.1 Geometry and meshes

The geometries of the two aortae were extracted from CT data. For the healthy aorta (patient 1), the CT was performed without contrast product, so a manual segmentation was carried out to obtain the 3D surface. For the aortic dissection case (patient 2), the method described in Chapter II was applied. All geometries included a part of the AA, the BT, the LCC, the LS arteries and the beginning of the DA. The different inlets and outlets were cropped to obtain planar surfaces suitable for CFD. The dissection of patient 2 was an uncomplicated type B dissection beginning just after the LS artery. This patient had undergone a Bentall procedure several years ago because he was suffering from an aneurysm in the aortic root and an aorta insufﬁ- ciency. This procedure consists in replacing the aortic valve, the aortic root and the beginning of the ascending aorta by a composite graft. Consequently, patient 2 had a small coarctation in the ascending aorta in addition to the type B dissection. The dissection had two entry tears near the LS artery and two re-entry tears in the DA. Figure 3.3 illustrates the ﬁnal 3D domains for patient 1 and 2. For the present study, meshing was carried out using ANSYS FLUENT MESHING. The mesh sizing was controlled with a minimum element size of 0.5 mm. Automatic

68 3.3. Method mesh refinement close to regions of significant curvature was enabled. Boxes surrounding the small outlets and the regions of the tears were created and an automatic mesh refinement was also enabled in these boxes. These regions of high interest must be crossed by at least 5 elements to obtain a correct description. A surface wrapping was then applied on the surfaces from the segmentation process to remove portions of the geometries irrelevant for the fluid simulation as well as close small gaps or holes. The wrapping process was based on the mesh sizing described above. The fluid volumes were then extracted from the wrapped surfaces. The obtained volumes were meshed with polyhedral elements and 8 prism layers on the wall with a ratio of 1.2 (meaning that each layer is 1.2 time thicker than the preceding layer). The meshes were refined until the maximum skewness was below 0.8. The final mesh of patient 1 had 193 037 cells and 534 160 nodes. The final mesh of patient 2 had 374 293 cells and 1 021 558 nodes.

3.3.2 Creation of the hybrid 0D model of the aorta

The description of the BCs at the inlets and outlets was obtained by a closed loop 0D model of the full cardiovascular circulation. We chose to use the model developed by Shi et al. [74]. The aorta is included in the systemic arteries compartment. This model is simple, and all the important parts of the systemic circulation are represented. The model of the heart chambers is the one from Suga et al. and is described later. The valves are modeled with a simple diode model. Figure 3.4 illustrates the 0D model of the full Cardiovascular System (CVS). From the 3D geometry of the aortic arch of each patient, an equivalent 0D model based on static simulations and electrical components was first created to approximate the 3D fluid simulations. This hybrid 0D model can be coupled with the 0D model of the CVS and dynamic boundaries conditions for the transient CFD can be extracted. The solution presented here mainly focuses on providing the system with an initial solution that will be closer to the final solution in order to infer the right response of the coupled system and avoid as much as possible the instabilities. The 3D geometries were divided in branches representing the inlets and outlets : the AA, the BT, the LCC, the LS arteries and the beginning of the DA (the true and false lumina in the case of patient 2). A proximal plan was created at the end of the

AAs named Pprox. The division is illustrated in ﬁgure 3.5.

69 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

FIGURE 3.4 – 0D model of the full cardiovascular circulation used in this thesis.

(a) (b)

FIGURE 3.5 – Division of the 3D geometry of the aorta in segments. a) Patient 1 (without dissection). b) Patient 2 (with dissection).

70 3.3. Method

For each branch, the relation between the pressure and the flow must be determined. In the literature, a branch of the aorta is usually represented in 0D with a resistance, a capacitance and an inductance in serial. In our model, we made the assumption that the walls were rigid so we did not use a capacitance. The relation between the flow and pressure in the resistances does not depend on the time while inductances are based on a time dependent equation. Static simulations of a 3D aorta provide a better description of the flow and pressure in the branches of the aorta than a simple resistance model. To improve the accuracy of the usual 0D representation, the resistances were replaced by a response surface computed from the results of static simulations. The use of a response surface rather than direct static simulation allowed to reduce computation time. The inductances were kept in the representation of each branch to model the transient effects.

Static model

This section describes the method used to obtain a response surface from static simulations of the aorta. Numerous patient speciﬁc steady simulations must be computed for each patient to generate enough learning data for the computation of an accurate response surface. The ANSYS Workbench software was used for the computation of the static simulations (Ansys Fluent) and the response surface (Ansys DesignXplo- rer). The following steady simulation setup was used for the two patients. The meshing method is the same as described later in section 3.2.1. The blood was modeled as an incompressible Newtonian ﬂuid with constant density ( ρ = 1056 kg/m3) and viscosity ( µ = 0.0035 Pa*s). The SST k − ω model was set for the turbulence. For the BCs, pressure at the inlets and velocity at the outlets were prescribed. The inputs of the response surface were the BCs applied at the inlet and outlets.

The outputs were the pressure at the outlets and in Pprox. Figure 3.6 shows the inputs and outputs of the response surface for patient 1 and 2. One simulation took on average 10 minutes to compute with a 6 cores processor which is too long for a direct coupling with the 0D model of the CVS. Numerous simulations were launched to compute the response surface. A Design Of Experiment (DOE) containing 500 sampling points was generated from the range of variations of the input parameters. Let’s call Ω the design space. The points (xi) were added in an incremental way by placing them as far as possible from the existing points to optimally

71 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b)

FIGURE 3.6 – Inputs and outputs of the response surfaces. a) Patient 1. b) Patient 2.

Patient number Parameters min max 1 2 PAA(mmHg) 37.5 158 −1 QBT (m.s ) -0.8 0.3 −1 QLCC (m.s ) -0.8 0.3 −1 QLS(m.s ) -0.8 0.3 −1 1 QDA(m.s ) -2 0.6 −1 2 QT L(m.s ) -1.5 0.3 −1 QF L(m.s ) -0.5 0.3

TABLE 3.3 – Boundaries of the different input parameters of the response surfaces.

ﬁll Ω as following :

xn+1 ∈ arg max min ||x − xi||. (3.4) x∈Ω i∈1..n

The design space was based on the values from the 2D PC MRI of the two patients (the min and max were extended so the model can handle larger variations). The range of variations of the input parameters are given in table 3.3.

Once the DOE was generated the corresponding static ﬂuid simulations were computed and the pressures at each outlets and at Pprox were saved.

From the results of the simulations of all the points of the DOE, a response surface was computed to obtain results for any set of parameters. The genetic aggregation method was used. It selects, conﬁgures and generates the best type of response surface for each output parameters from the following available methods : full 2nd order polyno- mial, non-parametric regression, kriging and moving least square. A surface response using neural network was also generated but it could not achieve a better accuracy.

72 3.3. Method

Inductance parameters

As said in the introduction of this section, each branch of the aorta was represented using a response surface based on static simulations and an inductance. The value of the parameter L of the inductance can be calculated using the geometrical assumption that each segment is a perfect cylinder and the law of Poiseuille with the following formula :

9ρl L = , (3.5) 4πr2 where l is the length of the segment, r the radius and ρ the blood density. The values calculated for each branch of each patient are given in table 3.4.

Branch Patient 1 Patient 2 AA 82900 625 BT 421000 1000000 LCC 3260000 2120000 LS 632000 973000 DA 754000 TL 171000000 FL 2940000

TABLE 3.4 – Value of the parameters L (P a.s2.m−3) of each branch of each patient.

Final hybrid 0D model and coupling with the 0D model of the CVS

The ﬁnal 0D hybrid models for patient 1 and 2 are illustrated in ﬁgure 3.7. The response surface was integrated as a pressure source and put in series with an inductance for the representation of each branch.

73 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b)

FIGURE 3.7 – 0D hybrid models of the aortic cross. a) Patient 1. b) Patient 2.

To allow the coupling between the 0D model of the CVS and the 3D aortic arch, the 0D models of the CVS were adapted to include the different branches of the aortic cross. In this model, the systemic circulation is represented as one branch and must be split to ﬁt our problem. For the model of the healthy aorta, the branches representing the arteries of the systemic circulation were divided in four branches. Each branch was connected to the outlets of the hybrid 0D model to represent the distal arteries that were not part of our model of the aortic cross. The three branches corresponding to the arteries of the up-stream vasculature were connected to one branch composed of one resistance (modeling the arterioles and capillaries) in serial with one capacitance and one resistance (modeling the veins). The branch representing the arteries after the DA was also connected to a branch composed of the same components representing the downstream vasculature. The 3D geometry of patient 1 did not include all the ascending aorta, so an RLC compartment was added between the aortic sinus block and the inlet branch of the 0D hybrid model to represent the beginning of the ascending aorta. For the geometry of patient 2, the beginning of the ascending aorta was included in the 3D geometry so the RLC block was not added and the inlet of the hybrid 0D model was directly connected to the aortic sinus block. The initial artery branch of the systemic circulation was divided in ﬁve branches since in this model the DA is

74 3.3. Method

FIGURE 3.8 – Coupling of the hybrid 0D model and the 0D model of the CVS for patient 1. composed of the FL and TL. The upstream vasculature was modeled like for patient 1. For the downstream vasculature, the two FL and TL branches were connected to a block modeling the downstream arterioles, capillaries and veins. Figure 3.8 and 3.9 show the ﬁnal 0D models of the CVS coupled with the hybrid 0D models of the aorta for patient 1 and 2 respectively. Colors meaning is explained in the next section.

3.3.3 Initialization of the 0D model of the CVS

At this stage of the method, we have computed an initial patient specific hybrid 0D model of the aorta and integrated it in the 0D model of the CVS. For each patient, the parameters of the 0D model of the CVS (resistances, capacitances, inductances and elastances) must be determined and tuned to be patient specific. For each patient, 2D PC MRI data were available. They gave us the flow in the section of the different inlets and outlets of the aortae during one cardiac cycle. First, the parameters of the 0D models of the CVS were initialized using patient data and values from the literature. Then, through an optimization process, the parameters were set so the flow at each outlet fit the values from the 2D PC MRI and that the systolic and diastolic pressures were the same as the patients. The parameters to be optimized are in color in figures 3.8 and 3.9.

75 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

FIGURE 3.9 – Coupling of the hybrid 0D model and the 0D model of the CVS for patient 2.

Beginning of the AA for patient 1 and of the arteries compartment connected to FL for patient 2

The components we consider in this section are in green in ﬁgures 3.8 and 3.9. The beginning of the ascending aorta of patient 1 was modeled with an RLC compartment. To initialize the parameters of the electrical components, we made the assumption that the represented branch was a perfect cylinder (with a length l and a radius r) like for the inductor parameters of the hybrid 0D models. The length and radius of the structure of interest were approximated using the CT data. The values of the inductors and resistances were approximated by the Poiseuille steady-state formula. The equation giving L was equation 3.4 and the equation giving R was the following :

8µl R = , (3.6) πr4 where µ was the ﬂuid viscosity. We considered the wall to be linear elastic with a thickness h of 1.20e−3 m. With this assumption, it can be shown that the compliance C, with a Young’s bulk modulus of elasticity E (taken as 40 000 Pa), was given by :

3πr3l C = . (3.7) 2Eh

76 3.3. Method

The parameters of the block representing the arteries of patient 2 connected to the FL were calculated with the same method since the CT data gave us the geometry of the entire FL. The calculated parameters are given by table 3.5.

Patient RLC Patient 1 (AA) 8810 168000 3.31e-08 Patient 2 (FL) 2024 14300 2.52e-08

TABLE 3.5 – Values of R (P a.s.m−3), L (P a.s2.m−3) and C (m2.P a−1) for the beginning of the AA of patient 1 and of the end of the FL for patient 2.

Parameters of the arteries block

The components we consider in this section are in yellow in ﬁgures 3.8 and 3.9. The parameters RLC of the arteries blocks connected to the outlets of the 0D hybrid model were taken from the literature [7].

Parameters of the elastance of the heart

The components of interest are in blue in ﬁgures 3.8 and 3.9. The heart chambers of these models were described with the pressure-volume relation of the widely used Suga et al. variable elastance model. The chamber volume depended on the ﬂow-rate difference between the inlet and the outlet of the chamber :

dV c = Q − Q . (3.8) dt in out The time-varying ventricle elastance was a function of the characteristic elastance

Ec,s and Ec,d and an activation function ec(t) :

E − E e (t) = E + c,s c,d e (t), (3.9) c c,d 2 c where c indicated the heart chamber of interest (v corresponded to the ventricles, a to the atria and l and r precise the left and right part respectively), s the systole and d the diastole. The activation function was deﬁned as follows for the ventricles :

77 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

 t  1 − cos( π), if 0 ≤ t < Ts1  Ts1 t−Ts1 ev(t) = 1 + cos( ), if Ts1 ≤ t < Ts2 Ts2−Ts1   0, if Ts2 ≤ t < T

where T was the duration of one cardiac cycle, Ts1 was the peak systole and Ts2 the end of the systolic phase. The activation function of the atria was deﬁned as follows :

  0 if 0 ≤ t < T  pwb  t−Tpwb ea(t) = 1 + cos( T ) if Tpwb ≤ t < Tpwb + Tpww  pww   0 if Tpwb + Tpww ≤ t < T where Tpww was the beginning of the pressure wave in the electrocardiogram and Tpwb was the duration of the pressure wave. Finally the pressure was given by :

Pc = Pc,0 + ec(Vc − Vc,0), (3.10) where the subscript 0 indicated the initial value. The different parameters to be set for the elastance function of the heart were mainly time parameters like the duration of a cardiac cycle and the time of the systole. Those data were extracted from the MRI data. The time parameters of the elastance are described in table 3.6. The other parameters were taken from the article [74].

Patient 1 Patient 2 T (s) 0.914 0.7875

Ts1(s) 0.238 0.25

Ts2(s) 0.381 0.40

Tpww(s) 0.814 0.71

Tpwb(s) 0.067 0.09

TABLE 3.6 – Time parameters of the elastance function for patient 1 and 2.

Parameters of the compartment modeling the arterioles, capillaries and veins.

The components of interest are in purple in ﬁgures 3.8 and 3.9. The compartments modeling the arterioles, capillaries and veins were composed of one resistance (R1),

78 3.3. Method

one capacitance (C) and another resistance (R2) in serial similarly as the three elements Windkessel model. Since the Windkessel model represents the vasculatures below the outlets on which they are connected, we made the assumption that the parameters of these components were close. We initialized the parameters of the compartment of interest using the same method from [167] for the initialization of the Wind- kessel parameters. Given a target diastolic pressure (Pd) and systolic pressure (Ps) and a ﬂow rate (Qaa(t)), the total peripheral resistance (resistance of the arterioles and capillaries) was calculated as :

1 Pd + 3 (Ps − Pd) RT = ¯ , (3.11) QAA ¯ where QAA was the mean value of the ﬂow in the ascending aorta (given by the 2D

PC MRI data). The individual proximal resistances (R1) were assumed to be equal to : ρc R = , (3.12) 1 A where c and A were respectively the wave speed. c was calculated as follows :

s β 1 4√ c = A 4 , β = πEh. (3.13) 2ρA 3

The resistances of the downstream and upstream vasculatures were calculated so the ﬂow distribution was correct and satisﬁed :

1 2 1 = X , (3.14) j J RT j=1 R1 + R2 where j corresponded to the branches of interest (downstream and upstream vasculature). The total compliance was calculated using the time constant τ = 1.79s from the exponential fall-off of pressure during diastole :

τ CT = . (3.15) RT

i j Let’s call C0D the compliance of the N vessels of the patient aorta and C the i compliance of each compartment representing the compliance of the veins. C0D was

79 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta calculated as : Aili Ci = , (3.16) 0D ρi(c)2 and Cj was given as follows :

N j X i RT C = (CT − C0D) j . (3.17) i=1 R2

Table 3.7 gives the calculated parameters for each patient.

Patient Branch R1 CR2 Patient 1 Downstream 2.0e07 1.3e−08 1.3e08 Upstream 1.7e09 5.8e−09 2.9e08 Patient 2 Downstream 2.4e07 1.3e−08 1.3e08 Upstream 3.2e10 2.8e−08 1.3e08

−3 2 −1 −3 TABLE 3.7 – Values of R1 (P a.s.m ), C (m .P a ) and R2 (P a.s.m ) in the upstream and downstream vasculatures for patient 1 and 2.

Remaining parameters.

For the remaining parameters of the 0D model of the CVS (components in black in ﬁgure 3.8 and 3.9), the values were taken from [74].

3.3.4 Personalization of the 0D model of the CVS

The initialization methods described in the previous section used numerous ap- proximations. A more accurate setting of the parameters was required to obtain patient speciﬁc 0D models of the CVS. An optimization process using MRI data and the systolic and diastolic arterial pressures of the patient was used to complete this initialization. The systolic and diastolic pressures for both patients were of 80 and 120 mm Hg respectively. The accuracy of the blood pressure test used to measure the pressures is clinician dependent. Moreover, depending on the state of the patient, the pressure can increase due to white coat hypertension, the measurement at a single time point cannot be reliable. Therefore, the values of the simulated pressure taken into the error function

80 3.3. Method underwent a simple preprocessing. The non linear functions presented in ﬁgure 3.10 were applied at the values of the systolic and diastolic pressures from the 0D models. They allowed to have a less strict condition on the pressure : the pressure interval at diastole is between 75 and 85 mm Hg and the pressure interval at systole is between 115 and 125 mm Hg.

(a) (b)

FIGURE 3.10 – Functions applied on the pressure computed from the coupled hybrid 0D model and 0D model of the CVS. a) Function applied to the pressure at diastole. b) Function applied to the pressure at systole.

The set of parameters to be optimized were the parameters of the resistors, capa- citors and inductors for the following compartments : systemic aortic sinus, systemic arteries and systemic arterioles, capillaries and veins. We supposed that the influence of the parameters of the pulmonary circulation was negligible. The total number of parameters to optimize were 21 for patient 1 and 24 for patient 2. Let’s call β the set of parameters to be optimized for each patient. The range of variation of each parameter was as follows : the maxima were set as twice the values of the initial parameters and the minima were the initial values divided by two. If the minimum or maximum of a parameter was reached at the end of the optimization process, the boundaries were extended (provided that the values were always positive) and the optimization process was relaunched. All the parameters were normalized at the beginning of the process. The function to minimize was set so the final 0D model was patient specific. For each patient, the curve of the flow from the MRI data at each outlet and inlet (QMRI (t)) m m as well as the systolic and diastolic pressures (Ps and Pd ) were used. The error

81 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta function to minimize was deﬁned as :

N tinit+Tc X X MRI 0D 2 m 0D 2 m 0D 2 = (Qi (t) − Qi (t, β)) + α((Ps − Ps (β)) + (Pd − Pd (β)) ), (3.18) i=1 t=tinit where N was the number of inlet and outlets of the patient, Tc the duration of the cardiac cycle and α a weight to equalize the importance of the pressure and ﬂow in the error function. tinit was set so the variables from the 0D simulations were stable in the sense that they did not change from one cardiac cycle to another. For simpliﬁcation purpose, let’s call yj the value from the patient, f(xj, β) the value from the simulation of the 0D model using β as parameters and (xi, yi) the number of empirical datum pairs. This is a classical optimization problem which can be expressed with the form of :

m X 2 argminβ [yi − f(xi, β)] . (3.19) i=1

The Levenberg-Marquardt algorithm was chosen. This method solves non-linear least squares problems and has shown good performance for the minimization of least square curve ﬁtting. This method is an iterative process and started using the βinit obtained during the initialization process. At each time step, a new set of parameters β was calculated minimizing the error function. The distance δ of the vector β between two time steps was calculated by the linearization of f(xi, β) as follows :

f(xi, β + δ) ' f(xi, β) + Jiδ, (3.20) where ∂f(x , β) J = i , (3.21) i ∂β was the gradient of f in function of β. The error function reached its minimum at a 0 gradient with respect to β. Using the linearization of f(xi, β), we obtained :

m X 2 ' [yi − f(xi, β) − Jiδ] . (3.22) i=1

So if we take the derivative of in function of δ and set it to 0, the result is :

(J T J)δ = J T [y − f(β)], (3.23)

82 3.3. Method

where J was the Jacobian matrix where the i−th row was equal Ji and and where f(β) and y were vectors with i-th component f(xi, β) and yi respectively. This equation was a set of linear equations that can be solved for δ. A coefﬁcient was applied to compute δ so the resulting β + δ stayed in the imposed range of variation. If the coefﬁcient was inferior to 0.01, the parameter outside its range of variation was removed and the new δ was computed. If the solution of the equation did not minimize the error by half of what the results would have been if f(β) was linear, the method solved the linear equations by replacing the equation by a "damped version" as :

(J T J + λI)δ = J T [y − f(β)], (3.24)

where I was the identity matrix. λ was a positive damping factor adjusted at each iteration. The value of λ was taken successively in [0.01, 0.05, 0.1] until the error was minimized by at least half the value it would have been if the function was linear.

3.3.5 Setup of the 3D patient speciﬁc CFD

At this stage, the hybrid 0D model of the aorta as well as the 0D model of the CVS were tuned to correspond to the state of the patient when the MRI data were acquired. From the coupling of both models, patient specific BCs were extracted to perform transient fluid simulations. The blood was modeled as an incompressible Newtonian fluid with constant density ( ρ = 1056 kg/m3) and viscosity ( µ = 0.0035 Pa*s). The Scale Adaptative Simulation (SAS) method was set to model the turbulence. This formulation is based on the introduction of the von Karman length-scale into the turbulence scale equation. It allows the resolution of different turbulence scales in unstable flow condition. The von Kar- man length-scale provides information on the stability of the local flow allowing the SAS model to dynamically adjust depending on the local structure behavior. In unsteady regions of the flow field, the SAS model has a LES-like behavior while in stable flow regions standard RANS capabilities are used. We supposed that the walls were rigid for simplification purpose. Pressure profiles were applied at the outlets and mass flow rate curves were applied at the inlets. The different curves were provided by the 0D models of the CVS coupled with the hybrid 0D models analysis. The profiles of the BCs were registered as soon as the 0D simulation was stable from one cardiac cycle to the other. 3 seconds of the cardiac cycle were

83 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

Outputs Pprox PBC PLCC PLS PDA Mean quadratic error mmHg (best value = 0) Validation points 0.01 0.01 0.02 0.02 0.01 Learning points 0.01 0.01 0.01 0.01 0.01 Relative absolute maximal error (best value = 0%) Validation points 0.04 0.13 0.13 0.14 0.12 Cross validation on learning points 0.20 0.40 0.78 0.57 0.34 Relative absolute mean error (best value = 0%) Validation points 0.01 0.04 0.04 0.04 0.02 Cross validation on learning points 0.02 0.05 0.07 0.07 0.04

TABLE 3.8 – Evaluation results for the response surface computed from static ﬂuid simulations of the aortic arch of patient 1.

simulated with a time step of 1 ms.

3.4 Results

In this section the results of the response surfaces computed for patient 1 and 2 are ﬁrst presented. Then, the results of the optimization process on the coupled hybrid 0D models/ 0D models of the CVS are described. At last, the simulations results of patient 2 using the patient speciﬁc BCs from the coupled 0D model are shown. The dynamic simulation results for patient 1 are not detailed since this model was created to test the developed method.

3.4.1 Response surface from the static simulations

Response surfaces from static simulations were computed to obtain hybrid 0D models of the aortae. Initial DOEs of 500 points were computed for each aorta and the respective static ﬂuid simulations were computed. 10 validation points were generated to assess the quality of the response surface. The mean quadratic errors, the relative absolute maximal errors and the relative mean absolute errors are reported for each pressure output for the validation points and the cross validation on the learning points in table 3.8 (resp 3.9) for patient 1 (resp. patient 2).

84 3.4. Results

Outputs Pprox PBC PLCC PLS PTL PFL Mean quadratic error mmHg (best value = 0) Validation points 0.03 0.20 0.11 0.10 0.14 0.07 Learning points 0.03 0.13 0.16 0.11 0.18 0.12 Relative absolute maximal error (best value = 0%) Validation points 0.14 1.09 0.54 0.60 0.80 0.27 Cross validation on learning points 0.70 1.71 2.30 1.20 3.68 1.56 Relative absolute mean error (best value = 0%) Validation points 0.05 0.38 0.20 0.16 0.21 0.16 Cross validation on learning points 0.05 0.21 0.25 0.20 0.27 0.21

TABLE 3.9 – Evaluation results for the response surface computed from static ﬂuid simulations of the aorta of patient 2.

The maximum predicted error on all outputs is of 0.60 mm Hg. The response surface is able to accurately replace the static ﬂuid simulations.

3.4.2 Optimization processes

To obtain patient specific BCs from our final 0D models, the parameters of the 0D models of the full CVS were optimized to fit the clinical data of the patients. The final set of parameters as well as the optimized curves of the flows and pressures are presented in this section. Table 3.10 shows the final set of parameters from the optimization processes of patient 1 and 2.

Patient 1

Figure 3.11 shows the comparison of the ﬂow at the outlets between the coupled 0D model and the MRI data of patient 1. Globally, there is a good correspondence between the simulation and the patient data, especially for the biggest outlets. The ﬂow at the LCC artery from the simulation presents many bumps. The value L computed for the arteries following this branch is very high compared to the other inductances.

85 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

Patient 1 Patient 2 R 1.7e07 1.0e05 Aortic sinus L 7.1e05 1.7e03 C 3.5e−09 2.3e−09 R 5.0e07 4.3e07 Arteries after BT L 1.0e03 2.3e03 C 1.1e−09 4.5e−10 R 2.6e07 1.0e08 Arteries after LCC L 2.0e07 1.0e06 C 1.2e−10 1.3e−10 R 1.7e07 4.3e08 Arteries after LS L 1.0e04 1.0e05 C 1.0e−10 1.5e−10 R 1.8e07 Arteries after DA L 1.4e06 C 1.6e−08 R 1.3e06 Arteries after TL L 8.0e05 C 1.0e−11 R 7.1e07 Arteries after FL L 5.0e03 C 9.6e−10 09 07 R1 1.8e 2.4e Downstream vasculatures C 3.7e−09 4.0e−08 08 07 R2 1.0e 1.9e 08 07 R1 2.4e 5.5e Uptream vasculatures C 1.9e−10 2.6e−09 07 07 R2 7.1e 5.9e

TABLE 3.10 – Optimized set of parameters (R in P a.s.m−3, L in P a.s2.m−3 and C in m2.P a−1) of the 0D models of the CVS for patient 1 and 2.

86 3.4. Results

(a) (b)

FIGURE 3.11 – Comparison of the ﬂow at each outlet between 2D PC MRI data and the coupled hybrid 0D model/0D model of the CVS of patient 1. a) Flow in the BT. b) Flow in the LCC. c) Flow in the LS. d) Flow in the DA.

Figure 3.12 shows the pressure at the outlets from the optimized coupled 0D model. As for the ﬂow at the outlets, the pressure at the LCC artery presents many bumps. The pressure wave forms at the other outlets are in good agreement compared to reported curves of arterial pressure in the literature. The pressure at peak systole and at diastole in the AA are of 132 mm Hg and of 75 mm Hg respectively.

Patient 2

Figure 3.13 represents the curves of the flow at the outlets from the 2D PC MRI compared to the flow at the outlets obtained from the final 0D model for patient 2. Figure 3.14 illustrates the pressure curves at the inlet and outlets. The systolic and diastolic obtained pressures are of 127 and 73 mm Hg respectively. The physiological pressure wave forms are in good agreement compared to arterial pressure curves measured from invasive measures. We can observe that the pressure in the FL lags compared to the pressure at the other outlets. It also presents a higher

87 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

FIGURE 3.12 – Pressure at the different outlets and inlets from the ﬁnal optimized 0D model of patient 1.

maximum than the pressure in the TL.

3.4.3 3D transient ﬂuid simulations of the AD

On average the computation time of one simulation (for 3s of cardiac cycle) on 64- bit 16 core (Intel® Xeon® 3.10GHz) machine with 64.0 GB shared memory was of 17h and 24h for patient 1 and 2 respectively. In this section, an analysis of the results from the simulation using the computed patient specific BCs of patient 2 is presented. The flow characteristics, the velocity fields, the pressure at the wall and the WSS are detailed at different stages of the cardiac cycle. Note that the scale of the variables in the sub-figures are different for clarity purpose. In this section a global analysis is provided and section 3.4.4 details the areas near the tears.

88 3.4. Results

(a) (b)

(e)

FIGURE 3.13 – Comparison of the ﬂow at each outlet between 2D PC MRI data and the coupled hybrid 0D model/0D model of the CVS of patient 2. a) Flow in the BT. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL.

89 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

FIGURE 3.14 – Pressure at the different outlets and inlets from the ﬁnal optimized 0D model of patient 2.

Flow Characteristics

FIGURE 3.15 – Mass ﬂow rate at each outlet from the 3D simulation over a single cardiac cycle of the AD.

Figure 3.15 shows the calculated mass flow rate through each outlet for one cardiac cycle. The peak systolic flow phase is indicated with a vertical dashed line. Since the parameters of the 0D models were optimized using the flow data of the patient, the pro- portion of flow received by all outlets corresponds to commonly reported value [153]. In the descending aorta, 13% of the flow goes through the FL and 87% goes through the TL. All of the supra-aortic branches (BC, LCC and LS) and FL have an early peak phase compared to the flow in the AA while the peak flow of the TL lags the AA peak. We can observe a backflow on the LCC, the LS, the TL and the FL. The reversed flows of the LCC, the LS and the TL are negligible compared to the FL. Figure 3.16 give the

90 3.4. Results comparison between the mass ﬂow rate at each outlet between the MRI data, the 0D model and the 3D CFD.

The flow at the outlets from the transient fluid simulations and the 0D model of the CVS coupled with the hybrid 0D model are very close, particularly in the systolic phase. We can observe that the reversed flow phenomenon is underestimated with the hybrid 0D model for all outlets except the true lumen.

Figure 3.17 shows the velocity vectors in the aorta of patient 2 at different stages of the cardiac cycle. At mid-systole (figure 3.17 a.) the velocity reaches the highest values in the BC, LS and LCC arteries. As seen in figure 3.15, the supra-aortic branches have an early peak systole phase compared to the FL and TL. The highest velocity occurs in LS and LCC due to the narrowing of the cross section areas. We can also observe that higher velocities occur near the entry tears. At the peak systole (figure 3.17 b.), the highest velocity vectors are near the two entry tears. Also, the velocity increases along the aorta in the FL near the re-entry tear 1 and decreases after the re-entry 2. The lowest velocity vectors are in the FL between the entry and re-entry tears. The coarctation in the AA caused by the Bentall surgery increases the velocity of the blood. At the dicrotic notch (figure 3.17), the velocity vectors increase in the TL after the first re-entry. The velocity of the flow is globally higher in the TL than in the FL at this point of the cardiac cycle. At mid-diastole, the velocity is relatively uniform in the whole domain (' 0.2m.s−1) even if high velocity vectors are present near the LS and LCC arteries.

Figure 3.18 shows the streamlines in the AD at different times of the cardiac cycle. At mid-systole, relatively uniform streamlines can be observed along the AA, the TL, the FL and the supra-aortic branches (figure 3.18 a.). At peak systole (figure 3.18 b.), the flow is still relatively uniform. However, the flow enters the FL though the entry tears and impinges the wall (opposite to the second entry tear) creating helical flow. Chaotic flow patterns are present near all the re-entry tears. At the dicrotic notch (figure 3.18 c.), we can observe that almost all the flow passes through the TL. The flow patterns are also chaotic along the aorta starting after the small coarctation in the AA and after the first re-entry tear. At mid-diastole (figure 3.18 d.), the velocity is globally low compared to the other moments of the cardiac cycle. The flow is highly chaotic along the aorta in the FL and TL, especially in the FL.

91 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b)

(e)

FIGURE 3.16 – Comparison of the ﬂow at each outlet between 3D simulation results, 0D simulation results and 2D PC MRI data. a) Flow in the BC. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL .

92 3.4. Results

(a) (b)

FIGURE 3.17 – Velocity ﬁelds of the ﬂow of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole.

93 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b)

FIGURE 3.18 – Streamlines from the inlet of the ﬂow of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole.

94 3.4. Results

Pressure at the wall

Figure 3.19 shows the pressure at the wall of the aorta at different stages of the cardiac cycle. At mid-systole (figure 3.19 a.), the pressure drop between the AA and the FL (resp. TL) is of 10 mm Hg (resp. 15 mm Hg). The pressure also decreases toward the different outlets. The pressure is globally higher in the TL along the aorta. There is approximately a difference of 5 mm Hg along the central line of the aorta between the TL and FL. At systole (figure 3.19 b.), the difference of pressure between the TL and FL is higher with the value of 20 mm Hg. The pressure drop between the AA and the TL is of 12 mm Hg and of 15 mm Hg between the AA and FL. We can observe that the pressure decreases in the coarctation in the AA. At the dicrotic notch (figure 3.19 c.), the pressure is higher in the FL than in the TL which generates the reversed flow observed in figure 3.15. The difference of pressure is approximately of 5 mm Hg. The pressure is globally uniform in the TL and FL. We can observe that, in the wall opposite to the first entry tear in the FL, the pressure is elevated. At mid-diastole (figure 3.19 d.), the pressure is globally uniform in the whole aorta.

(a) (b)

FIGURE 3.19 – Pressure at the wall of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch d) Mid-diastole.

95 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

WSS

Figure 3.20 shows the WSS of the aortic wall. The results are generated with a transparency of 0.5 for a better visualization of the areas near the intimal flap. Globally the WSS is low except near the tears. The WWS near the tears will be detailed in the next section. The WSS at mid-systole is high in the supra-aortic branches. We can notice that the WSS increases at mid-systole and peak systole (figure 3.20 a. and b. respectively) near the coarctation in the AA. At peak systole, the WSS increases in the TL between the entry and re-entry tears. At dicrotic notch, the WSS is globally higher in the TL. We can observe areas (like in the red bow of figure 3.20 c.) near the intimal flap where the WSS is higher which can be potential weak areas with increased rupture risk. At mid-diastole the WSS is very low (< 3 Pa) in the aorta.

(a) (b)

FIGURE 3.20 – WSS of patient 2. a) Mid-systole. b) Peak systole. c) Dicrotic Notch. d) Mid-diastole.

96 3.5. Discussion

3.4.4 Focus on the tear regions of the AD

In this section, a focus on the areas near the tears of the AD is presented. Figure 3.21 shows the velocity vectors in the plane of the four tears at different stages of the cardiac cycle. At mid-systole (figure 3.21 a.), the flow enters the FL from the TL through the four tears. The blood has a high velocity in the tears compare to the velocity of the blood in the lumina. We can also observe that the velocity is higher in the entry tears than in the re-entry tears. At peak systole (figure 3.21 b.), the flow enters the TL from the FL through all tears. At the first entry tear, the velocity is very high and oriented toward the opposite wall of the tear in the FL. At mid-diastole (figure 3.21 c.), the flow passes from the FL through the TL this time. At the first entry tear, we can observe the formation of a whirl near the tear. Figure 3.22 presents the WSS near the entry and re-entry tears. At mid-systole (figure 3.22 a.), the WSS is high at the contour of the tears. There also are areas of high WSS at the base of the LSS. At peak-systole (figure 3.22 b.), the WSS is also high at the contour of the tears. There are also circle areas of high WSS at the opposite wall of the first entry and re-entry tears in the FL. For the first entry tear, it corresponds to an area near the formation of a whirl like that shown in figure 3.18. At the second entry tear, the area with an elevated WSS is extended compared to mid-systole. Considering the second re-entry tear, the WSS with high values is located only around the tear. At dicrotic notch (figure 3.22 c.), there is also an area with high WSS in the wall at the opposite of the first entry tear but in the TL, since at this time of the cardiac cycle the blood flow from the FL to the TL. At the first re-entry tear, the area with high WSS extends in the TL.

3.5 Discussion

A novel method for the computation of patient specific BCs for fluid simulation was developed. A hybrid 0D model of the aorta was first computed from static simulations and electrical components. This model was coupled to a 0D model of the entire cardiovascular system. The resulting model’s parameters were optimized using patient MRI data and the patient arterial pressure at systole and diastole. With this method, we obtained patient specific BCs corresponding to the resting state of the patient and we avoided the instabilities problems generated from the coupling between 0D and 3D

97 Partie , Chapitre 3 – Patient speciﬁc CFD simulation of the aorta

(a) (b) (c)

FIGURE 3.21 – Velocity ﬁelds in planes of the four tears (entry tear 1, entry tear 2, re-entry tear 1 and re-entry 2 from top to bottom). a) Mid-systole. b) Peak systole. c) Dicrotic Notch.

98 3.5. Discussion

(a) (b) (c)

FIGURE 3.22 – WSS near the areas of the four tears. a) Mid-systole. b) Peak systole. c) Dicrotic Notch.

99 Partie , Chapitre 3 – Patient specific CFD simulation of the aorta models. The hybrid 0D models offer a good approximation of the behavior of the pressure and flow in the aorta compared to the results from 3D CFD. The flows at the outlets generated from the coupling of the 0D model of the CVS and the hybrid 0D model were very close to the flow computed from the fluid simulations particularly in the systolic phase. The proposed method allows to obtain patient specific flow at the inlet and outlets of the aorta. Moreover, the corresponding pressure curves are realistic compared to reported curves from invasive measures and have minimum and maximum coherent with patient measures. The proposed method was applied to one case of AD and the results were detailed. The flow in the aorta is almost laminar during the systole phase except near the tears where areas of turbulence are formed on the wall of the FL opposite to the tear. At diastole, the flow is highly turbulent in the TL and FL. The pressure at the wall is different in the FL and TL except during diastole. At systole the difference of pressure is approximately 15 mm Hg with a higher pressure in the TL. At dicrotic notch, the pressure at the wall is higher in the FL than in the TL. At this moment the pressure in the FL is the highest and this period could be of interest for the prediction of the evolution of ADs. The simulations revealed region of high WSS around the tears. In the tears, the flow increases and can pass from the TL to the FL (at systole) or from the FL to the TL (at the dicrotic notch). The flow tends to impinge the wall opposed to the tear and it creates area of high WSS which could potentially be areas with increased rupture risk. The base of the LS artery is also an area with high WSS during the mid-systole. A retrospective study on a case of AD with a clinical follow up of the geometry could be interesting to asses the impact of elevated region of WSS and pressure wall on the geometry. Also, the application of the method on a larger data set of ADs could highlight factors of the evolution of initially uncomplicated type B AD. Nevertheless, several simplification hypothesis were made in this study. The walls were considered to be rigid. The wall of AD is generally more rigid than healthy aortic wall but the intimal flap, which is thin, can move under the pressure of the blood. It is difficult to assess correctly the material properties of the aortic wall and the intimal flap in a patient specific way. A FSI model taking into account the different material properties of the different regions of the aortic wall would offer a better description of the hemodynamics of AD. The blood was considered to be an incompressible Newtonian

100 3.6. Conclusion

ﬂuid. Different models of the blood exist that could offer a more realistic description and were detailed in the state of the art section. At last, the geometry of the AD was cropped before the end of the dissection. The false lumen of the considered patient ﬁnished just above the iliac arteries. Ideally, the coeliac trunk, the mesenteric and renals arteries should be modeled.

3.6 Conclusion

A method to compute patient specific simulation of AD, and more generally of the aorta, has been reported in this chapter. The method is based on a novel method for the computation of patient specific BCs. The pressure at the outlets of the patients were computed using only non-invasive patient data. Patient specific fluid simulations corresponding to the resting state of the patients were carried out for one healthy case of aorta and one case of AD. The simulation allows to observe the flow patterns during the cardiac cycle and to highlight areas with elevated WSS or pressure. Nevertheless, the computation time of one simulation is too heavy for clinical application. In the next chapter, we address the issue of reduced order modeling.

101

CHAPITRE 4 DYNAMIC ROM BASEDON CFD

One of the main limitations of dynamic fluid simulations for clinical applications is the heavy computational cost. ROM methods refer to the methods aiming to reduce the computational complexity of numerical simulations. They can be used for problems requiring real-time results or numerous simulations such as system control and optimization process. ROMs have been developed for many fields like electronics or structural mechanics. In this Chapter, we focus on transient fluid simulation applications. These simulations are known to be complex to reduce because they exhibit strong nonlinearities [78]. It is particularly the case when investigating unsteady flow or complex structures such as the study of the hemodynamics of AD. In this chapter, ROMs of the aortae of patient 1 and 2 were developed from the results of 3D simulations computed using the method described in chapter III. In this chapter, a state of the arts on the existing ROMs of large arteries is presented. The proposed method is described followed by the results of the different developed ROMs of the aortae. At last, the effects of the different clinical scenarios on the AD of patient 2 are described.

4.1 State of the arts

Different approaches have been developed to reduce models of arteries. These models are often used to simplify 3D FSI simulations [30] or unstable multi-scale cou- plings which are complex and time consuming. As seen in the previous chapter, the spatial dimension of the 3D geometry can be reduced to obtain 0D, 1D or 2D models of the arteries. The existing 0D models were already described in the previous chapter. 1D models, in the same way as 0D models, are often used to prescribe boundaries conditions for 3D models. 1D models consist in modeling the vessel by axi-symmetric 1D vessel segments. They have been used to address many issues like the modeling

103 Partie , Chapitre 4 – Dynamic ROM based on CFD of coronary arteries [60] or aorta coarctation [61]. Shi et al. [133] described the different existing 1D models of the arteries. Finally, 2D models also exist to simulate the large arteries. They can take into account radial variations of the vessel. Ghigo et al. [49] proposed a 2D nonlinear multiring model to compute the hemodynamics in large elastic arteries. This model was proposed to simplify FSI simulations. However, our purpose is to compute a ROM of the aorta allowing to access 3D variables such as the WSS. Approaches other than the reduction of geometrical dimension must be investigated to solve this issue. However, few applications of such approaches have been tested on models of large arteries. A common used method is the reduced basis method [88, 117, 118] . This approach consists in creating a reduced basis of a lower dimensional space of the problem. The partial differential equations are then projected to this space. Chang et al. [22] used the proper orthogonal decomposition together with the Galerkin projection [120] method to assess the WWS of an abdominal aorta aneurysm depending on the inflow angle. Buoso et al. [13] investigated the coronary arteries using also the proper orthogonal decomposition to assess the pressure and velocity fields. Ballarin et al. [8] used similar methods to study coronary artery bypass. These methods are considered and called intrusive because the resolution of the equilibrium equations is solved in the physics solver on the reduced basis directly. These methods can be unstable, and a high accuracy is required. Other methods called non-intrusive methods have also been developed. Over the last decades, they have been a subject of interest because they avoid instabilities and overcome nonlinearities issues encountered in intrusive methods [166, 109, 127]. They are also easier to implement since the access to the source code of the solver is not required. Although, training and validation data must be computed or measured to apply these types of methods. Köppl et al. [73] built a ROM to study the stenosis in peripheral arteries. They used a combined 1D model and machine learning on kernel method. The solution presented in this study is a linear combination of kernel functions like the Gaussian or the Wendland kernel and the coefficients are determined using learning and validation data computed from 3D CFD using a machine learning process. The final model allows the prediction of pressure and flow-rate curves in function of the degree of stenosis. However, the outputs of this model are not 3D variables. To our knowledge, no study developed non-intrusive ROMs of large arteries giving 3D results. In this thesis, we implemented a non-intrusive method combining reduced bases and neural networks to compute the dynamic ROM.

104 4.2. Method

4.2 Method

Our purpose in this chapter is to obtain 3D simulations results for any set of parameters of the 0D model of the full CVS in a few seconds. For this purpose, we created different set of parameters of the 0D model and generated the respective BCs profiles for transient fluid simulations. The different results of the corresponding simulations were saved and used to create learning and validation data for the computations of ROMs. Several ROMs were computed from the simulation results : 0D dynamic ROMs and 3D dynamic ROMs. The dynamic 0D model of the aorta computed the pressure at the outlets depending on the pressure at the inlets and the mass flow rate at the outlets. This model was more accurate than the hybrid 0D model from the previous chapter. It was coupled with the 0D model of the CVS instead of the hybrid 0D model. From this updated 0D model, the parameters of the electrical components and of the elastance functions of the 0D model of the CVS were changed to simulate clinical scenarios such as hypertension or the intake of beta-blockers. 3D results of the aorta were computed using the ROMs for the corresponding scenarios. The 3D dynamic ROMs computed the following 3D variables : pressure at the wall, WSS and velocity fields. These 3D variables were first preprocessed with the Singular Value Decomposition (SVD) method. The coefficients of the modes were then learned using a method based on neural network. The input of the 3D ROMs were the dynamic BCs at the inlet and outlets of the aorta computed from the coupled 0D model of the CVS and the hybrid 0D model. Figure 4.1 summarizes the main steps of the method for the computation of the ROMs. In this section, we present the method to compute the different ROMs and the method to simulate different clinical scenarios.

4.2.1 SVD to obtain 3D dynamic ROM

To obtain output data suitable for the computation of the ROMs, the SVD process was applied at each of the 3D physical variables of interest (pressure at the wall, WSS and velocity ﬁelds). It allows to obtain a "compressed" approximation of the 3D variables. In this section, we summarize the SVD method. The SVD of a m ∗ n matrix L is deﬁned as :

L = UDV|, (4.1)

105 Partie , Chapitre 4 – Dynamic ROM based on CFD

FIGURE 4.1 – Main steps of the computation of the dynamic 0D ROM and the 3D ROMs. where : — U is a m ∗ m orthonormal matrix (U |U = I) — V is a n ∗ n orthonormal matrix (V|V = I)

— D is a m ∗ n diagonal matrix D = diag(σ1, σ2, ...) where σ1 ≥ σ2... ≥ σi ≥ ... ≥ 0.

σi are the real singular values of L. We can notice that while D is unique, it is not the case for U and V. Let’s r be the rank (the number of non-zero singular values) of the matrix L. L can be decomposed as :

Lij = σ1ui1vj1 + σ2ui2vj2 + ... + σruirvjr (4.2)

The matrix L can be truncated to k (< r) terms as follow :

(Lk)ij = σ1ui1vj1 + σ2ui2vj2 + ... + σkuikvjk, (4.3)

where the singular values of higher order than k are not considered. Lk is an approximation of the initial matrix. k is the number of modes in which the matrix is decomposed. A compromise must be made between the accuracy and the complexity of the model when choosing the number of modes to describe the 3D variables. Since the singular values are listed in increasing order, the ﬁrst modes contain more information on the initial 3D variables than the modes with higher ranks. The coefﬁcient of the

106 4.2. Method modes (singular values) can be used as the learning data from the computation of the 3D dynamic ROMs.

4.2.2 Dynamic ROM method

In this section, we describe the method used to compute the different ROMs of the aortae. The different computed ROMs in this study are also presented.

Method to compute the ROM

The dynamic ROM method used is implemented in the software Twin Builder 2019 R2 (Ansys) as DynamicROM. The dynamic ROM method presented here computes one or more outputs xˆ(t) from one or more dynamic inputs g(t). It is a non-intrusive method based on deep neural networks. To compute the ROMs, the same initial conditions must be applied on all learning and validation data. The purpose of the computation of the dynamic ROMs is to build a model x(ˆt) from one or more dynamic outputs provided by the solver x(t) corresponding to the dynamic inputs g(t) within a requested accuracy kx(t) − xˆ(t)k < , where is the deﬁned error. The model consists in the resolution of a nonlinear function using deep neural network methods.

ROMs computed in this study

Different ROMs were computed for patient 1 and 2 with different outputs. As explained in the introduction of section 4.2, we can differentiate what we called 3D dynamic ROMs and 0D dynamic ROMs. Three 3D dynamic ROMs returning respectively the pressure at the wall, the WSS and the velocity fields of the aortae were computed for each patient. The inputs to the 3D dynamic ROMs were the BCs (mass flow rate at the inlet and pressure at the outlets) computed from the 0D model. The 0D dynamic ROMs return the pressure at the outlets and in Pprox depending on the pressure at the inlet and mass flow rate at the outlets for each patient. The inputs to the 0D dynamic ROMs are different from the inputs of 3D dynamic ROMs. This choice is explained in section 4.2.4. Figure 4.2 summarizes the different computed ROMs.

107 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.2 – Inputs and outputs of ROMs. a) 0D dynamic ROM of patient 1. b) 0D dynamic ROM of patient 2. c) 3D dynamic ROMs of patient 1. d) 3D dynamic ROMs of patient 2.

4.2.3 Generation of the learning and validation data for the ROMs

The learning and validation data for the computation of the ROMs were generated using 3D fluid simulations. All simulations started at an initial state with zero pressures and flows at the inlets and outlets. The 3D simulations were computed using the same setup as described in Chapter III except that the BCs profiles were different for each learning and validation case. A case corresponds to the simulations of 3 s of cardiac cycle with a constant time step of 1 ms. The pressure at the outlets was saved at each time step while the WSS, the pressure wall and the velocity fields were saved every 10 time steps in the format of snapshot. 14 different 3D simulations were computed for each patient.

The different BCs profiles were generated as in Chapter III, using the coupling between the 0D models of the full CVS and the hybrid 0D models. The parameters of the 0D models were changed empirically to obtain different pressure and flow profiles. The range of variation of the different generated BCs for patient 1 and 2 are shown in table 4.1..

108 4.2. Method

Pmax Pmin Qmax Qmin ∆P ∆QTc(s) max 143 105 0.99 0.07 83 0.49 1.2 Patient 1 min 77 35 0.17 -0.27 6 0.12 0.5 max 239 152 0.51 0.03 99 0.49 1.2 Patient 2 min 18 8 0.05 -0.04 5 0.04 0.5

TABLE 4.1 – Range of variation of the BCs of patient 1 and 2 for the generation of learning and validation data for the computation of the ROM where P is the pressure applied at the DA and TL for patient 1 and 2 respectively in mm Hg, Q the mass ﬂow −1 rate at the inlet (in kg.m ) and Tc is the duration of the cardiac cycle.

4.2.4 Integration of the ROM in the 0D model of the full CVS

The 0D dynamic ROMs (of patient 1 and 2), returning the pressure at the outlets, offered a better description of the pressure in the aorta than the hybrid 0D model described in the previous chapter, since they were computed from transient simulation results rather than static simulation results. The hybrid 0D models were replaced by the new 0D dynamic ROM for each patient. We observed that prescribing an imposed pressure is more stable than imposing a flow in the coupled 0D models. Consequently, the 0D dynamic ROMs inputs were the pressure at the inlet and the mass flow rate at the outlets and the outputs were the pressure at the outlets. Once the coupled 0D models were updated, the same optimization processes as in the previous chapter were launched to ensure the parameters of the 0D models were patient specific since the results of the hybrid 0D models and of the 0D dynamic ROMs were slightly different. Figure 4.3 is an illustration of the 0D model of the CVS coupled with the 0D dynamic ROM for patient 2 and of the different 3D dynamic ROMs.

4.2.5 Computation of different clinical scenarios

The parameters of the 0D models of the CVS were optimized so the model outputs ﬁt the clinical data for the patient at rest. Our purpose was to change the parameters of 0D model to model different clinical scenarios : controlled pressure, hypertension due to vasoconstriction and hypertension due by hypervolemia. In case of uncomplicated type B aortic dissection, as said in chapter I, the patient is monitored and receives a medical therapy to lower his/her arterial pressure. We chose

109 Partie , Chapitre 4 – Dynamic ROM based on CFD

FIGURE 4.3 – Coupling of the 0D model of the CVS with the 0D dynamic ROM of patient 2.

110 4.3. Results to simulate the intake of betablockers which are the main antihypertensive medicine used for type B AD. This medicine mainly acts on the heart by reducing the heart rate and increasing left ventricular stroke volume index and left ventricular ejection fraction [35]. The purpose of administrating beta-blockers is to reduce the systolic arterial pressure under 120 mm Hg. To model the effects of beta-blockers, we ﬁrst decreased the heart rate of the patient in the elastance functions of the heart chambers. We also decreased the parameter Elv,s of the patient elastance function of the left ventricle. Consequently, the volume stroke of the left ventricle increased and the pressure in it decreased (Plv = Plv,0 + elv(Vlv − Vlv,0)). Hypertension can have detrimental effect on the evolution of uncomplicated type B AD and is the main cause of the initiation of AD. A patient is considered with hypertension if the systolic arterial pressure is above 150 mm Hg. We chose to simulate this scenario by two different approaches. The former was to simulate the chronic high blood pressure with the vasoconstriction of the arterioles. This phenomenon was modeled by simply increasing the resistances of the arterioles in the 0D models of the CVS. The resistances of the arterioles of the upstream and downstream vasculatures were increased proportionally until a systolic arterial pressure superior to 150 mm Hg was reached. The latter approach to simulate a hypertension scenario was to model the hypervolemia in the veins [58]. Hypervolemia is caused by an excess of sodium and water in the extracellular ﬂuid which leads to an expansion of the blood volume. To model it, we increased the value of the capacitances modeling the veins. The upper and downstream veins capacitances were increased proportionally until the systolic arterial pressure was superior to 150 mm Hg.

4.3 Results

In this section, we present the results of the different computed ROMs compared to simulation results and of the different simulated clinical scenarios. ROMs allow to obtain similar results in a much reduced time. Once the BC’s are generated from the parametrized 0D models, 1 real hour of cardiac cycle took up to 100s.

111 Partie , Chapitre 4 – Dynamic ROM based on CFD

4.3.1 0D dynamic ROMs

As explained in the section 4.2.4, a 0D dynamic ROM computing the pressure drop in each branches depending on the pressure at the inlet and the ﬂow at the outlet was computed for patient 1 and 2.

Patient 1

For patient 1, 10 learning cases were used to compute the 0D dynamic ROM and for 4 cases were used for validation. Several tests were made to obtain the setup with the smallest the error between the ROM and the simulation results. Table 4.3 presents the relative and maximum absolute error on all the pressure outputs of the 0D dynamic ROM of patient 1.

Error Learning cases Validation cases 1 2 3 4 5 6 7 8 9 10 11 12 13 14 RE 5.53 4.16 0.8 0.69 1.42 1.01 1.06 1.28 1.45 1.18 1.4 2.8 0.93 1.06 MAE 25.30 15.02 2.26 1.19 5.56 1.97 4.60 5.46 5.56 4.29 3.23 7.40 2.58 3.81

TABLE 4.2 – Relative Error (RE) (%) and Maximum Absolute Error (MAE) (mm Hg) on all outputs of the 0D dynamic ROM of patient 1.

For the learning and validation cases the relative errors were on average 1.86 % and 1.55 % respectively and the maximum absolute errors were on average 7.14 mm Hg and 4.26 mm Hg. Figure 4.4 shows the comparison between the pressure computed from the ﬂuid simulation and the 0D dynamic ROM for the validation case with the worst error. It corresponds to the validation case 10 which is also the validation case with the highest systolic pressure and magnitude of pressure. We can observe that the global wave forms of the pressures are well retrieved. There is a small shift phase between the pressure computed from the simulation and the 0D dynamic ROM. Figure 4.5 shows the error between the pressure at each outlet for all validation cases. The gray dashed line indicates on each sub-ﬁgure the peak systole. We can observe that the diastole part is well described by the 0D dynamic ROM with a difference of pressure below 2 mm Hg. The highest error is often just before the peak systole when the pressure increases. The pressures are overestimated just

112 4.3. Results

(a) (b)

FIGURE 4.4 – Comparison between the pressure computed from the simulation and the 0D dynamic ROM for validation case 10 a) BT. b) LCC. c) LS. d) DA.

113 Partie , Chapitre 4 – Dynamic ROM based on CFD before the peak systole and underestimated after. As seen in ﬁgure 4.4, there can be a small shift phase between the pressures computed by the ROM and the simulations. We can also observe that the outlets with the highest relative errors are the LCC and LS arteries which are the smallest outlets.

(a) (b)

FIGURE 4.5 – Difference between the pressure computed from the simulation and the 0D dynamic ROM a) Validation case 9. b) Validation case 10. c) Validation case 11. d) Validation case 12.

Patient 2

For patient 2, 8 learning cases were used for the training and 6 validation cases for the evaluation. As for patient 1, several tests were made to obtain the setup minimizing

114 4.3. Results the error between the ROM and the simulations results. Table 4.3 presents the relative and maximum absolute error on all the pressure outputs of the 0D dynamic ROM of patient 2.

Error Learning cases Validation cases 1 2 3 4 5 6 7 8 9 10 11 12 13 14 RE 0.14 0.11 0.16 0.06 0.19 0.26 0.10 0.19 0.19 0.37 1.1 0.44 0.23 1.76 MAE 1.90 0.47 1.22 0.44 3.16 2.24 0.71 1.67 1.16 3.16 0.49 3.35 2.81 13

TABLE 4.3 – Relative error (%) and maximum absolute error (mm Hg) on all outputs of the 0D dynamic ROM of patient 2.

On average, the relative errors were of 0.15 % and 0.68 % and the maximum absolute errors were of 1.48 mm Hg and 4.00 mm Hg for the learning and validation cases respectively. The results were better than for patient 1. Figure 4.6 shows the comparison between the pressures computed by the simulation and the 0D dynamic ROM for the validation case 11. The pressures computed by the ROM are very close to the one computed by the simulation, there is no shift phase. At dicrotic notch, the pressure at the outlets has a little bump which is well retrieved by the ROM. Figure 4.7 shows the comparison between the pressure computed by the simulation and the ROM for the validation case 15. This case presents a high magnitude ( ' 93 mm Hg in the AA). The increase of the pressure at the beginning of the systole is earlier in the ROM results than in the simulation results. It explains the high absolute error of validation case 15. Otherwise, the form of the curve and the minimum and maximum are well retrieved.

4.3.2 Final coupled 0D models

Once the 0D dynamic ROM was computed, it was used to replace the hybrid 0D model of the aorta in the 0D model of the CVS. The same optimization process as in chapter III was used to personalize the parameters of the 0D model of the CVS for each patient. In this section we present the results of the optimization processes.

115 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

(e)

FIGURE 4.6 – Pressure computed from the simulation and the 0D dynamic ROM for validation case 11 a) BT. b) LCC. c) LS. d) TL. e) FL.

116 4.3. Results

(a) (b)

(e)

FIGURE 4.7 – Pressure computed from the simulation and the 0D dynamic ROM for validation case 15 a) BT. b) LCC. c) LS. d) TL. e) FL.

117 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.8 – Comparison of the ﬂow at each outlet between 2D PC MRI data, the coupling between 0D model of the CVS and the hybrid 0D model and the coupling between the 0D model of the CVS and the 0D dynamic ROM for patient 1. a) Flow in the BC. b) Flow in the LSS. c) Flow in the LC. d) Flow in the DA.

Patient 1

Figure 4.8 shows the ﬂow at the outlets from the MRI data, the 0D model of the CVS coupled with the hybrid 0D model and the 0D model of the CVS coupled with the 0D dynamic ROM.

The ﬂows at the outlets from MRI data present a better agreement with the results from the 0D model of the CVS coupled with the 0D dynamic ROM than with the 0D model of the CVS coupled with the hybrid 0D model. The worst results are on the LCC artery which presents the lowest ﬂow. The pressures at systole and diastole in the AA are of 115 mm Hg and 75 mm Hg respectively.

118 4.3. Results

Patient 2

Figure 4.9 shows the flow at the outlets from the MRI data, the 0D model of the CVS coupled with the hybrid 0D model and the 0D model of the CVS coupled with the 0D dynamic ROM. A good agreement is found between the curves from the model coupled with the 0D dynamic ROM and the MRI data. The period when the flow decreases after the peak systole is better described by the 0D dynamic ROM than by the hybrid 0D model. However, in the supra-aortic branches, the flow is reversed at the end of the systole in the MRI data and the 0D dynamic ROM coupled with the 0D model of the CVS fails to reproduce it. The pressures at systole and diastole are of 135 mm Hg and 75 mm Hg respectively.

4.3.3 ROMs computing the pressure at the wall

This section presents the results of the SVD and corresponding ROM of patient 1 and 2 computing the pressure at the wall. The SVD was realized on the snapshots from the learning cases.

Patient 1

Figure 4.10 shows the Root Mean Square error (RMS) of error between the pressure wall from the simulations and the projections from the SVD process depending on the number of modes used for the SVD. 20 modes were chosen to describe the pressure at the wall of patient 1 resulting in an error inferior to 0.1%. The average maximum absolute error between the initial snapshot and the projected snapshots using 20 modes was of 3.07 mm Hg and the maximum average relative projection error in L2 norm was of 0.31 %. 10 learning cases and 4 validation cases were used. The relative error of the validation cases on each mode coefficients is given by figure 4.11. The error superior to 10% were cropped to obtain a better visualization of the curves. We can observe that the first modes have lower errors compared to the highest modes. Nevertheless, most of the information on the pressure wall is contained in the first modes. Figure 4.12 shows the comparison of the pressure at the wall given by the ROM and the simulation for validation case 1.

119 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

(e)

FIGURE 4.9 – Comparison of the ﬂow at each outlet between 2D PC MRI data, the coupling between 0D model of the CVS and the hybrid 0D model and the coupling between the 0D model of the CVS and the 0D dynamic ROM for patient 2. a) Flow in the BC. b) Flow in the LCC. c) Flow in the LS. d) Flow in the TL. e) Flow in the FL

120 4.3. Results

FIGURE 4.10 – RMS error depending on the number of modes from the SVD of the pressure at the wall of patient 1.

FIGURE 4.11 – Relative errors (%) on each mode coefﬁcients for the validation cases for the ROM computing the pressure at the wall of patient 1. The errors higher than 10 % are cropped for clarity purpose.

121 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.12 – Comparison of the pressure at the wall computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid diastole.

122 4.3. Results

Globally, a good correspondence is achieved between the ROM and the simulation. At mid-systole (figure 4.12 a.) the pressure computed from the ROM is overestimated in the ascending aorta by approximately 7 mm Hg. At systole (figure 4.12 b.), the areas with higher pressure (the areas in the red squares) are well retrieved by the ROM even if they are a little underestimated. Elsewhere, there is a good correspondence between the results of the ROM and the simulation. At dicrotic notch (figure 4.12 c.), the pressure increases along the DA, the BT and the LS arteries. The ROM describes correctly these changes in the pressure. At mid-diastole (figure 4.12 d.), the pressure is homogeneous in the aorta (' 73 mm Hg) and the pressure from the ROM is in the correct range. Like for the 0D dynamic ROM of patient 1, the ROM have more difficulties to learn the part of the systole where the pressure increases.

Patient 2

Figure 4.13 shows the RMS error between the pressure at the wall from the simulations and the projections from the SVD process depending on the number of modes used for the SVD.

FIGURE 4.13 – RMS error depending on the number of modes of the SVD of the pressure at the wall of patient 2.

We chose to use 16 modes to obtain an accuracy below 1% as for patient 1. The maximum absolute error between all the snapshots from the simulations and the projections of the SVD was of 29 mm Hg and the average relative projection error in L2

123 Partie , Chapitre 4 – Dynamic ROM based on CFD norm was of 0.01 %. The ROM learned the coefficient of the 16 modes. 10 learning cases were used to build the ROM of the pressure at the wall for patient 2 and 6 validation cases wer used for the evaluation. The relative error of the validation cases on each mode coefficients is given by figure 4.14 (the error superior to 10% were cropped for visualization purpose). As for patient 1, the first modes have small errors (except for validation case 3 where the third mode have an error of 9%). The relative error on the two first modes is below 0.7 %. From the 13th mode, the error is frequently higher than 10%. Figure 4.14 shows the result computed from the simulation and the ROM for the validation case 4 at different stage of the cardiac cycle.

FIGURE 4.14 – Relative errors (%) on each mode coefﬁcients for the validation cases of the ROM computing the pressure at the wall of patient 2. The errors higher than 10 % are cropped for clarity purpose.

At mid-systole and systole (figure 4.15 a. and b. respectively), the pressure in the TL and FL is well described by the ROM. In both cases, there is an underestimation of the pressure in the AA compared to the results from simulation. At dicrotic notch (figure 4.15 c.), the pressure near the outlets of the BT, FL and TL is underestimated by the ROM and the pressure in the beginning of the AA is overestimated. Far from the inlet and outlet, the pressure given by the ROM and the simulation is similar. There is an increase of the pressure on the wall of the FL opposite to the first re-entry tear present in both simulation and ROM results. At mid-diastole (figure 4.15) the range of variation of the pressure is of approximately 5 mm Hg. The results from the ROM have the same range of variation.

124 4.3. Results

(a) (b)

FIGURE 4.15 – Comparison of the pressure at the wall computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid-diastole.

125 Partie , Chapitre 4 – Dynamic ROM based on CFD

4.3.4 ROMs computing the WSS

In this section the results of the ROM computing the WSS for patient 1 and 2 are presented. The SVD processes were realized using the snapshots of the learning cases.

Patient 1

Figure 4.16 shows the RMS error between the WSS from the simulation results and the projections from the SVD process depending on the number of modes used for the SVD.

FIGURE 4.16 – RMS error depending on the number of modes used for the SVD on the WSS of patient 1.

Differently from the SVDs for the pressures at the walls, the WSS snapshots are complicated to compress with an error inferior to 1%. Therefore, the values of the WSS during the diastole part are very low and it is complicated for the SVD to correctly describe this part. If we look at the projection error at each time step of the cardiac cycle using only 20 modes, the error is high during the diastole part and low during the systole. Figure 4.17 shows the relative error between the snapshots projected by the SVD using 20 modes and from the simulation for one learning case. The gray dashed line indicates the peak systole. We can see that near the peak systole the relative error is below 5%. Using 20 modes to describe the WSS vectors, on average the error between the snapshots from the simulations and the projections using the SVD is of 10% on the learning cases and 5% on the validation cases at peak systole.

126 4.3. Results

FIGURE 4.17 – Relative errors (%) during one cardiac cycle between the projected snapshot of the WSS of patient 1 using 20 modes and the snapshot from the simulation

10 learning cases were used for the learning of the ROM returning the WSS for patient 1 and 4 validation cases were used for the evaluation. Figure 4.18 shows the relative error between the coefﬁcients of the 20 modes computed by the SVD and the dynamic ROM for the validation cases. The ﬁrst three modes have a relative error below 2%.

FIGURE 4.18 – Relative errors (%) at peak systole on each mode coefﬁcients for the validation cases of the ROM computing the WSS of patient 1. The errors higher than 10 % are cropped for clarity purpose.

Figure 4.19 shows the WSS computed from the ROM and the simulation for patient 1 at different stages of the cardiac cycle on validation case 3. We can observe that at mid-systole, the areas with high WSS are well described by the ROM. In the BT artery,

127 Partie , Chapitre 4 – Dynamic ROM based on CFD the variations of the WSS were correctly captured. At the beginning of the artery, the WSS decreases on the side of the AA and increases on the area opposite to the LCC artery. At peak systole, the variations of the WSS are also correctly reproduced by the ROM. The area in between the LS and LCC arteries presents a high WSS. The WSS in the BT artery has a lot of variations and is correctly reproduced by the ROM. At dicrotic notch, we can observe more differences between the WSS computed by the ROM and the simulation. For example, the results from the simulation present an increase of the WSS below the BT artery in the AA which is not present in the results computed from the ROM. At mid-diastole, the variation of the WSS is relatively low (between 0 and 2 mm Hg). Globally, the value computed from the ROM are closed to the one computed from the simulation since the variation of the WSS is low. But, there is an area at the end of DA (from simulation) where the WSS increases that is not present in the results from the ROM. Nevertheless, in the context of the analysis of the prediction of the evolution of non-complicated type B AD, we are interested in areas with high WSS. The WSS is the highest during systole. We will only consider the results during systole for the WSS.

(a) (b)

FIGURE 4.19 – Comparison of the WSS of patient 1 computed from the ROM and the simulation (on left and right respectively) a) Mid-systole. b) Peak systole. c) Dicrotic notch. d) Mid diastole.

128 4.3. Results

Patient 2

Figure 4.20 shows the RMS error between the simulations and the projections of the SVD process.

FIGURE 4.20 – RMS error depending on the number of modes for the SVD on the WSS of patient 2.

As for patient 1, the WSS snapshots are difficult to compress correctly using a small number of modes. But as for patient 1, the high error is induced by the small values of the WSS in diastole. We chose to use 30 modes to describe the WSS. On average the relative errors are of 15% and 14 % for the learning and validation cases respectively at peak systole. Here we will consider only the results of the SVD and ROM processes at peak systole. The ROM returning the coefficients of the modes was computed using 11 learning cases and 3 validation cases were used for the evaluation. Figure 4.21 shows the relative errors on the coefficients of the 30 modes computed by the SVD and the ROM at peak systole for each validation case. The error is below 5% until the 22th mode. Figure 4.22 shows the WSS on the AD of patient 2 from the simulations and ROM at peak systole for validation case 1 (figure 4.22 a.) and validation case 2 (figure 4.22 b.). The areas presenting a high WSS are on the wall of the FL opposite to the tears and near the entry tears. These areas are well retrieved by the ROM and SVD processes. On the results from the ROM, the WSS near the second entry tear is overestimated for validation cases 1 and 3. Along the TL, there are some areas where the WSS

129 Partie , Chapitre 4 – Dynamic ROM based on CFD

FIGURE 4.21 – Relative errors (%) at peak systole on each mode coefﬁcients for the validation cases for the ROM computing the WSS of patient 2. The errors higher than 10 % are cropped for clarity purpose. increases before the re-entry tears. These areas are well described by the ROM for both validation cases. We can observe that the WSS increases in the coarctation in the AA and is well described by the ROM.

4.3.5 ROMs computing the velocity vectors

In this section, we present the result of the ROMs computing the velocity ﬁelds of patient 1 and 2. The SVD was realized using the snapshots from the learning cases.

Patient 1

Figure 4.23 shows the RMS error between the snapshots computed from the simulation and the snapshots computed from the projection of the SVD process. As for the WSS, it is difficult to compress correctly the velocity fields using a small number of modes. 874 modes are necessary to obtain an error below 1%. Therefore, as for the WSS, the error is induced by the velocity fields at diastole. During this stage of the cardiac cycle, the values of the vectors are very low and are highly turbulent. It results in the creation of complex flow patterns with a small velocity that are difficult to correctly compress. Consequently, we will only consider the results of the ROM at peak systole. 50 modes were used to create the SVD.

130 4.3. Results

(a)

(b)

FIGURE 4.22 – Comparison of the WSS of patient 2 computed from the ROM and the simulation (on left and right respectively) a) Validation case 1. b) Validation case 2.

131 Partie , Chapitre 4 – Dynamic ROM based on CFD

FIGURE 4.23 – RMS error depending on the number of modes for the SVD computing the velocity ﬁelds of patient 1.

The computation of the ROM was realized using 10 learning cases and the evaluation was realized using 4 validation cases. The relative projection error is on average of 6.2% for the learning cases and 3.13% for the validation cases at peak systole. Figure 4.24 shows the relative error at peak systole for the coefficients of the modes computed by the ROM and the SVD for the validation cases. On the eight first modes and between the 13th and 28th modes, the relative error is below 5%. Figure 4.25 shows the comparison between velocity fields computed by the ROM and simulation in : the whole aorta, a plane in the aortic arch and a plane in the DA for validation case 1. The velocity fields magnitude and patterns are well retrieved by the ROM. In the whole aorta, (figure 4.25 a.) the velocity increases along the AA toward the BT and in between the LS and LCC arteries. In the aortic arch (figure 4.25 b.) the areas where the flow increases are similar in both results. A layer of flow along the wall with a lower velocity is present in the results from the ROM and the simulation. In the plane in the DA (figure 4.25 c.), a swirl is formed near the wall. The ROM was able to reproduce this pattern.

Patient 2

Figure 4.26 shows the RMS error between the results from the simulations and the projections from the SVD process depending on the number of modes. As for the WSS and the velocity ﬁelds of patient 1, it is difﬁcult to compress the

132 4.3. Results

FIGURE 4.24 – Relative errors (%) on each mode coefficients for the validation cases at peak systole for the ROM computing the velocity fields of patient 1. The errors higher than 10 % are cropped for clarity purpose. velocity fields of patient 2. For the same reasons, we only consider the peak systole. 50 modes were used for the SVD process. On average the relative error between the snapshots computed from the simulation and projected by the SVD is of 12% for the learning cases and 8% on the validation cases at peak systole. The ROM computing the coefficients of the 50 modes used 10 learning cases for the learning and 4 validation cases for the evaluation. Figure 4.27 shows the relative errors for the coefficients of the modes computed by the SVD and ROM at peak systole for all validation cases. The first 3 modes present an error below 2%. Figure 4.28 shows the velocity fields in the tears of patient 2 computed by the simulation and the SVD and ROM processes. The scale of the velocity vectors is the same for all planes. In the first entry tear (figure 4.28 a.) the flow enters from the TL toward the FL and increases in the FL. A swirl is created near the wall in the FL. Those structures are well described by the ROM. In the second entry tear (figure 4.28 b.), the flows in the TL and FL are oriented toward the tear and meet in the FL creating an area with high variations of the flow magnitude. This phenomenon is well retrieved by the ROM. The same phenomenon happens in the first and second re-entry tears (figure 4.28 c. and figure 4.28 d. respectively) and is also well described by the ROM. However, the flow in the FL in the planes of the re-entry tears is slightly underestimated near the wall opposed to the tear in the FL.

133 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a)

(b) (c)

FIGURE 4.25 – Comparison of the velocity ﬁelds of patient 1 computed from the ROM and the simulation (on left and right respectively) a) Whole aorta. b) Plane in the aortic arch. c) Plane in the DA.

FIGURE 4.26 – RMS error depending on the number of modes for the SVD on the velocity ﬁelds of patient 2.

134 4.3. Results

FIGURE 4.27 – Relative errors (%) on each mode coefﬁcients for the validation cases at peak systole for the ROM computing the velocity ﬁelds of patient 2.

4.3.6 Simulation of clinical scenarios for AD

Three clinical scenarios were simulated for patient 2 (with AD) by changing the parameters of the 0D model of the CVS coupled with the 0D dynamic ROM : controlled pressure, hypertension caused by vasoconstriction and hypertension caused by hypervolemia. To model the clinical scenario corresponding to a controlled pressure, we simulated the intake of beta-blockers by the patient to obtain a systolic pressure below 120 mm Hg. According to the MRI data, the patient initially had a stroke volume of 69 ml and a cardiac cycle duration of 0.78 s. The cardiac cycle duration was increased to 1 s. The parameter Elv,s was divided by 8.33 to keep the same stroke volume of 69 ml. Using these new parameters, the systolic arterial pressure reached the value of 112 mm Hg and the diastolic arterial pressure was of 55 mm Hg. To simulate a hypertension caused by vasoconstriction, the values of the resistances of the arterioles were increased. The arterioles resistances were multiply by 6.5 to obtain an arterial pressure at systole of 165 mm Hg and a pressure at diastole of 92 mm Hg. The ﬂow at peak systole was of 0.39 kg/s. To simulate a hypertension caused by a hypervolumia, we increased the capacitances in the veins of the upstream and downstream vasculatures. The veins capacitances were multiply by 2.5 to obtain an arterial pressure at systole of 165 mm Hg and

135 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.28 – Comparison of the velocity fields computed from the ROM and the simulation (at top and bottom respectively for subfigures a. and b. and on left and right respectively for subfigures c. and d.) a) Plane including the first entry tear. b) Plane including the second entry tear. c) Plane including the first re-entry tear. d) Plane including the second re-entry tear.

136 4.3. Results a pressure at diastole of 88 mm Hg. It increased the mass ﬂow rate at peak systole to 0.50 kg/s. Figure 4.29 shows the pressure at the wall for the three clinical scenarios and the patient initial state (corresponding to the state of the patient during the MRI acquisition) at peak systole. The pressure in the scenario controlled pressure is the lowest. The pressure in the TL in the FL is approximately of 110 mm Hg and of 70 mm Hg respectively. In the initial case, the pressure in the TL is approximately of 135 mm Hg and of 100 mm Hg in the FL. For the hypertension scenarios the pressure in the TL is approximately of 165 mm Hg. In the FL, the pressure for the hypertension caused by the vasoconstriction is of approximately 130 mm Hg and for the hypertension caused by hypervolemia of 150 mm Hg. In the case of the controlled pressure, we can observe that the pressure at the wall near the re-entry tears decreased. In the case of the hypertension caused by hypervolemia, the pressure at the outlets is higher than for the scenario of hypertension caused by vasoconstriction since the ﬂow was increased.

(a) (b)

FIGURE 4.29 – Comparison of the pressure at the wall results between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia.

Figure 4.30 shows the WSS for the three clinical scenarios and the patient initial

137 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.30 – Comparison of the WSS results between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia.

state at peak systole. We can observe that the areas with high WSS near the entry tears are smaller for the controlled pressure scenario. As well the WSS in the FL wall opposed to the re-entry tears is lower for the controlled pressure case and higher for the hypertension scenarios. The case of hypertension caused by vasoconstriction has higher WSS in the wall of the FL opposed to the re-entry tears than the scenario of hypertension caused by hypervolemia while the WSS near the entry-tears and at the basis of the LS artery is lower. It is probably induced by the fact that the flow is higher in the case of hypertension caused by hypervolemia and more flow passes through the entry tears. The WSS in the TL between the entry and re-entry tears is also the highest for this case. Figure 4.31 shows the velocity fields in the first entry tear for the three clinical scenarios and the patient initial state at peak systole. The velocity patterns are similar for all the scenarios. The velocity is the lowest for the scenario of controlled pressure and the highest for the scenarios of hypertension (especially for the scenario of hypertension caused by hypervolemia).

138 4.4. Discussion

(a) (b)

FIGURE 4.31 – Comparison of the velocity ﬁelds in the ﬁrst entry tear between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia.

Figure 4.32 shows the velocity fields in the second entry tear for the three clinical scenarios and the patient initial state at peak systole. The velocity of the flow in the FL is the lowest for the initial scenario and the hypertension caused by vasoconstriction. For the two other scenarios, the velocity of the flow near the wall of the FL opposed to the tear increases. Figure 4.33 shows the velocity fields in the re-entry tears for the three clinical scenarios and the patient initial state at peak systole. The velocity of the flow is the lowest in the case of controlled pressure and the highest for the case of hypertension caused by hypervolemia. Otherwise the flow patterns are similar. The flow in the TL presents a highest velocity in case of controlled pressure than hypertension caused by vasoconstriction.

4.4 Discussion

The method to compute dynamic ROMs allows the accurate retrieval of simulation results in a few seconds. The 0D dynamic ROMs average relative errors are below 2 % and allow one to obtain accurate 0D models of the aorta. With regards to the 3D

139 Partie , Chapitre 4 – Dynamic ROM based on CFD

(a) (b)

FIGURE 4.32 – Comparison of the velocity ﬁelds in the second entry tear between the patient’s initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia.

(a) (b) (e) (f)

FIGURE 4.33 – Comparison of the velocity ﬁelds in the second re-entry tear between the patient initial state and the three clinical scenarios a) Initial state. b) Controlled pressure. c) Hypertension caused by vasoconstriction. d) Hypertension caused by hypervolemia.

140 4.5. Conclusion

ROMs, the pressure at the wall and its variations are correctly retrieved for patient 1 and 2 at all stages of the cardiac cycle. Nevertheless, the SVD process for the WSS and the velocity fields fails to correctly describe the variables during the diastole period. During diastole, the flow is turbulent and has small values compared to the systole. It is complicated for the SVD process to correctly compress the snapshots during this period. Other reduction basis methods such as the principal component analysis could be investigated. Here, only the results at peak systole are considered. Since, the areas presenting the highest WSS are present at peak systole, for the study of the evolution of AD, considering only the peak systole is correct. At peak systole, the variations of the WSS along the aortae of patient 1 and 2 are correctly described by the 3D ROMs. As well, at peak systole, the patterns and the magnitude of the flow in the aorta are accurately described by the 3D ROMs. Even the regions of the tears where the flow from the TL and FL met are correctly described by the 3D ROMs. The areas presenting the highest error with the ROMs are near the inlet and outlets. These areas are the interface between the 3D simulations and the prescribed flows and pressures and are sensitive regions for the simulation. This side effect make the learning of the ROM more difficult in these areas. The ROMs allow to obtain 3D results within seconds. Nevertheless, in order to compute the ROM, several 3D simulations must be computed to obtain validation and learning data. This process takes a lot of computation time and must be repeated for each new geometry. Even though this learning step is transparent to the user, a solution would be to parametrize the geometry of the different aortae and to compute ROMs taking as input the geometrical parameters. Different clinical scenarios were computed using the approach based on ROMs. : controlled pressure and hypertension. Globally in the controlled pressure scenario, the WSS, the pressure and the velocity fields are lower while in case of hypertension the variables increase. The results suggest a good reliability of the proposed approach.

4.5 Conclusion

In this chapter, different ROMs were computed from the results of transient ﬂuid simulations of patient 1 and 2. 0D dynamic ROM returning the pressure at each outlet depending on the pressure at the inlet and the ﬂow at the outlets was created and used to replace the hybrid 0D model in the 0D model of the CVS. These models allow to have

141 Partie , Chapitre 4 – Dynamic ROM based on CFD a more accurate description of the aorta while maintaining a patient specific 0D model of the rest of the CVS. 3D ROMs returning the pressure at the wall, the WSS and the velocity fields, for different sets of parameters of the 0D model of the CVS, were also computed using the SVD method. The compression of the variables of interest can be complicated in the diastole part since turbulence occurs during at this cardiac phase and the range of variations of the values of interest is small. However, at peak systole we obtained a good correspondence between the results from the simulations and the ROMs. We can obtain the pressure at the wall, WSS and velocity fields for any set of parameters of the 0D model of the CVS in a few seconds. Different clinical scenario were simulated using the different computed ROMs : intake of beta-blockers to reduce the arterial pressure, hypertension caused by vasoconstriction and hypertension caused by hypervolemia. Reducing the arterial pressure with beta-blockers induces a global decrease of the pressure at the wall, WSS and velocity of the flow. On the opposite, the two scenarios simulating hypertension globally increase the pressure at the wall, the WSS and the velocity of the flow. Impact factors concerning the evolution of AD for a patient taking medicine could be assessed from the simulation methods.

142 CONCLUSION

This thesis dealt with the issue of the simulation of type B aortic dissections for computer-aided clinical treatment decision. The complications related to ADs of type B were presented. We highlighted the difficulties to predict the evolution of initially uncomplicated ADs of type B and the consequences on treatment decisions. Numerical simulation approaches were investigated to assess the hemodynamics behavior of ADs. To this aim, an accurate geometry of the fluid domain of AD was required. Quite a few studies have been published to answer this problem, but none proposed an accurate CFD oriented approach. A semi-automatic approach for the segmentation of AD from CT data, targeted toward simulation, based on a competitive fast-marching process and geodesic active contour method was proposed. A strict separation of the two lumina, where there is no tear, was imposed so the final geometry was suitable for the construction of a mesh. The method proved to be robust against complex features related to AD as well as to be suitable for simulation. We analyzed the CFD implementation in the context of ADs. The literature presents a huge variability in CFD modeling of large arteries. A trade-off between complexity and accuracy must be achieved. The BCs at the fluid domain limits are complicated to assess properly. Most of the data needed are usually unavailable due to the difficulty to make in vivo flow measurement or because some data can only be assessed in an invasive way. A novel approach for the definition of the BCs at the fluid limits was proposed based on a closed-loop 0D model of the full cardiovascular system and static simulations. Patient-specific BCs were obtained using only non-invasive data from 2D PC MRI. The flow distribution computed by the simulation was close to the patient data. Moreover, the pressure curves at the inlet and outlets had realistic wave forms compared to values reported from the literature. One of the main limitations in the implementation of CFD in clinical routine is the heavy computation cost. Few studies have been reported concerning ROM for the aorta in general and none was based on non-intrusive methods returning 3D dynamic variables. An innovative approach based on ROMs was proposed to replace 3D dynamic simulations. ROMs were computed from 3D dynamic results and returned the

143 pressure at the wall, the WSS and the velocity fields for healthy or dissected aorta. A good correspondence between the results of the ROMs and the simulations was achieved during the systolic phase. In our final model, 3D simulation results can be assessed in a few seconds for a setting of the 0D model of the full cardiovascular simulation. The proposed approach allows to assess the influence of clinical scenarios such as the intake of beta-blockers to reduce the arterial pressure or hypertension. In this work, we proposed a method to assess patient specific dynamic results in real time. However, several assumptions were made and limit the accuracy of the proposed models. More detailed investigations are necessary to further analyze factors for the prediction of initially uncomplicated AD of type B. The proposed segmentation method was compared to manual AD segmentation. The segmentation was based on user input points. An automatic localization of the points could be considered. The accuracy of the simulations of the blood flow in the aorta depends on the modeling assumptions made. These include morphological aspects, such as the inclusion of the end of the descending aorta, fluid aspects, such as non-Newtonian viscosity and turbulence, and the definition of BCs. In this work, we addressed a number of these issues for clinical applications for which both accuracy and stability were considered. Nevertheless, only the upper part of the aorta was considered despite the fact that the dissection of the considered patient finished just above the iliac arteries. Ideally, the coeliac trunk, the mesenteric and renal arteries should be modeled. The SAS model was used to model the turbulence. To evaluate the accuracy of this model, the comparison with other turbulence models confronted to 2D PC MRI data could be carried out. One main assumption was made in this study : the walls were considered rigid despite the fact that the wall and intimal flap can move. An accurate model of the wall, taking into account the different material properties of the arterial wall and intimal flap, coupled with the proposed fluid model could be investigated to assess the influence of the motion of the wall. The proposed method allowed to assess areas with increased pressure and WSS. Nevertheless, only one case of AD was investigated in this study. A retrospective study including several patients with follow up data could be interesting to assess more accurately impact factors for the evolution of type B ADs. ROMs of the aorta were created from the 3D simulations data to obtain 3D results rapidly. Once the 0D model was parametrized, the ROM allowed to simulate 1h of cardiac cycle within 2 minutes. Although the solution may currently provide valuable

144 information, the methodology has some limitations. Numerous 3D computations were required to compute the ROMs. Moreover, the whole process must be repeated for each new geometry of the aorta. The geometry could be parametrized and the geometrical parameters could be included for the learning of the ROMs so it would be suitable for every patients. Even though the ROMs computing the pressure at the wall and at the outlets provided accurate results compared to the simulations at all stage of the cardiac cycle, the accurate approximation of the velocity ﬁelds and WSS during the diastole could not be achieved using a low number of modes. Other basis reduction approaches could be investigated to improve the proposed method. The ROM method allows to simulate the aorta for a long period of time like several months. Neverthe- less, in the proposed model, the simulation results were not changing from one cardiac cycle to the other. The description of the motion of the wall through FSI methods could allow to assess the fatigue at artery the wall during a long period of time. Future works include performing patient speciﬁc simulations of the aorta with a fatigue model for the prediction of the outcomes of type B AD using the proposed ROM based approach.

145

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Titre : Images et modèles pour l’aide à la décision clinique pour la chirurgie de la dissection aortique

Mot clés : Dissection aortique, Segmentation d’image, Simulation ﬂuide, Modèle d’ordre réduit

Resumé : La décision clinique, concer- simulation fluide ; ii) implémenté des si- nant le traitement de la dissection aor- mulations spécifiques patient basées sur tique de type B, est encore actuellement une nouvelle méthode pour la définition controversée dans certaines configura- des conditions aux limites ; iii) créé des tions. La simulation numérique est envisa- modèles d’ordre réduit à partir des si- gée pour obtenir des informations sur l’hé- mulations fluides dynamiques. Ces mo- modynamique de manière non-invasive. dèles permettent de calculer rapidement Nous avons ; i) proposé une méthode de les contraintes de pression et de cisaille- segmentation semi-automatique pour la ment pour différents scénarios cliniques.

Title : Images and models for decision support in aortic dissection surgery

Keywords : Aortic dissection, Image segmentation, CFD, Reduced order Model

Abstract : The clinical decision concer- plemented a CFD model using a novel ning the treatment of type B aortic dis- method for the definition of the boundary section is still controversial in some confi- conditions ; iii) created reduced order mo- gurations. CFD approachs were investiga- dels from 3D dynamic fluid simulations. ted to assess the hemodynamics in a non- These models allow to calculate in real invasive way. In this context, we : i) pro- time the wall shear stress and pressure for posed a semi-automatic method for the different clinical scenarios. segmentation of aortic dissections ; ii) im-