Assessment of Predicting Blood Flow and Atherosclerosis in the Aorta and Renal Arteries

Assessment of predicting blood flow and atherosclerosis in the aorta and renal arteries

Alexander Fuchs

August 2020 Technical Reports KTH Royal Institute of Technology Department of Engineering Mechanics SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 28:e augusti 2020 klockan 14:00 i Sal F3, Lindstedtsvägen 26, Stockholm

ISBN: 978-91-7873-585-3 TRITA-SCI-FOU 2020:23

Cover: Time-averaged Wall Shear-Stress (TAWSS) distribution over a human aorta and in the main arteries branching from the thoracic and abdominal aorta. Heart-rate 60 BPM (beats per minute) and cardiac output of 5 LPM (liters per minute).

Tryck: Universitetsservice US AB, 2020

To Louise, Gunnar and Bernard

Assessment of predicting blood flow and atherosclerosis in the aorta and renal arteries Alexander Fuchs KTH, Dept. Engineering Mechanics

Abstract Cardiovascular diseases (CVD) are the most common cause of death in large parts of the world. Atherosclerosis (AS) has a major part in most CVDs. AS is a slowly developing disease which is dependent on multiple factors such as genetics and life style (food, smoking, and physical activities). AS is primarily a disease of the arterial wall and develops preferentially at certain locations (such as arterial branches and in certain vessels like the coronary arteries). The close relation between AS sites and blood flow has been well established over the years. However, due to multi-factorial causes, there exist no early prognostic tools for identifying individuals that should be treated prophylactically or followed up. The underlying hypothesis of this thesis was to determine if it is possible to use blood flow simulations of patient-specific cases in order to identify individuals with risk for developing AS. CT scans from patients with renal artery stenosis (RAS) were used to get the affected vessels geometry. Blood flow in original and “reconstructed” arteries were simulated. Commonly used wall shear stress (WSS) related indicators of AS were studied to assess their use as risk indicators for developing AS. Divergent results indicated urgent need to assess the impact of simulation related factors on results. Altogether, blood flow in the following vessels was studied: The whole aorta with branches from the aortic arch and the abdominal aorta, abdominal aorta as well as the renal arteries, and separately the thoracic aorta with the three main branching arteries from the aortic arch. The impact of geometrical reconstruction, employed boundary conditions (BCs), effects of flow-rate, heart-rate and models of blood viscosity as function of local hematocrit (red blood cell, RBC, concentration) and shear-rate were studied in some detail. In addition to common WSS-related indicators, we suggested the use of endothelial activation models as a further risk indicator. The simulations data was used to extract not only the WSS-related data but also the impact of flow-rate on the extent of retrograde flow in the aorta and close to its walls. The formation of helical motion and flow instabilities (which at high flow- and heart-rate lead to turbulence) was also considered. Results A large number of simulations (more than 100) were carried out. These simulations assessed the use of flow-rate specified BCs, pressure based BCs or so called windkessel (WK) outlet BCs that simulate effects of peripheral arterial compliance. The results showed high sensitivity of the flow to BCs. For example, the deceleration phase of the flow-rate is more prone to flow instabilities (as also expressed in terms of multiple inflection points in the streamwise velocity profile) as well as leading to retrograde flow. In contrast, the acceleration phase leads to uni-directional and more stable flow. As WSS unsteadiness was found to be pro-AS, it was important to assess the effect flow-rate deceleration, under physiological and pathological conditions. Peaks of retrograde flow occur at local temporal minima in flow-rate. WK BCs require ad-hoc adjusted parameters and are therefore useful only when fully patient

4 specific (i.e. all information is valid for a particular patient at a particular point of time) data is available. Helical flows which are considered as atheroprotective, are formed naturally, depending primarily on the geometry (due to the bends in the thoracic aorta). Helical flow was also observed in the major aortic branches. The helical motion is weaker during flow deceleration and diastole when it may locally also change direction. Most common existing blood viscosity models are based on hematocrit and shear-rate. These models show strong variation of blood (mixture) viscosity. With strong shear-rate blood viscosity is lowest and is almost constant. The impact of blood viscosity in terms of dissipation is counter balanced by the shear-rate; At low shear-rate the blood has larger viscosity and at high shear-rate it is the opposite. This effect and due to the temporal variations in the local flow conditions the effect of blood rheology on the WSS indicators is weak. Tracking of blood components and clot-models shows that the retrograde motion and the flow near branches may have so strong curvature that centrifugal force can become important. This effect may lead to the transport of a thrombus from the descending aorta back to the branches of the aortic arch and could cause embolic stroke. The latter results confirm clinical observation of the risk of stroke due to transport of emboli from the proximal part of the descending aorta upstream to the vessels branching from the aortic arch and which lead blood to the brain. Conclusions The main reasons for not being able to propose an early predictive tool for future development of AS are four-folded: i. At present, the mechanisms behind AS are not adequately understood to enable to define a set of parameters that are sensitive and specific enough to be predictive of its development. ii. The lack of accurate patient-specific data (BCs) over the whole physiological “envelop” allows only limited number of flow simulations which may not be adequate for patient- specific predictive purposes. iii. The shortcomings of current models with respect to material properties of blood and arterial walls (for patient-specific space- and time-variations) are lacking. iv. There is a need for better simulation data processing, i.e. tools that enable deducing general predictive atherosclerotic parameters from a limited number of simulations, through e.g. extending reduced modeling and/or deep learning. The results do show, however, that blood flow simulations may produce very useful data that enhances understanding of clinically observed processes such as explaining helical- and retrograde flows and the transport of blood components and emboli in larger arteries.

Key words: Blood flow simulations, Atherosclerosis, Wall shear stress (WSS), blood rheology models

Sammanfattning Hjärt- och kärlsjukdomar är den vanligaste dödsorsaken i stora delar av världen. Åderförkalkning (atheroscleros) spelar betydande roll för denna grupp av sjukdomar. Åderförkalkning utvecklas under lång tid och beror på många olika faktorer såsom genetiska och livsstilsrelaterade (exempelvis kost, rökning och fysisk aktivitet). Åderförkalkning drabbar blodkärlens, artärernas, väggar och formas oftare på vissa lokaler än andra (t ex vid kärlförgreningar och i särsklida kärl som exempelvis hjärtats kranskärl). Det nära sambandet mellan blodflöde och åderförkalkning är väl etablerat sedan många år. P.g.a. den multifaktoriella genesen existerar inget tillförlitligt test för att tidigt upptäcka individer i behov av förebyggande behandling eller uppföljning vid atherosclerotisk hjärt- och kärlsjukdom. Den bakomliggande hypotesen för denna avhandling var att bestämma om blodflödessimuleringar i patient-specifika fall kan identifiera risk för atherosclerosutveckling. Datortomografiundersökningar från patienter med njurartärstenos användes för att framställa blodkärlens utseende (geometri). Blodflödet simulerades i de drabbade kärlen samt efter att de ”rekonstruerats till originalskick”. Vanligt använda indikatorer baserade på väggskjuvspänningen (VSS) studerades för att bedöma risk för atherosclerosutveckling. Bitvis spretiga resultat visade på stort behov att bedöma känsligheten för simuleringsberoende faktorer. Blodflödet har huvudsakligen studerats i hela stora kroppsålderna (aorta) inklusive dess stora grenar i bröstkorgen och buken, bröstkorgens aorta med tre stora tillhörande grenar samt bukaorta inklusive njurartärerna. Inverkan av kärlgeometrin, dess rekontruktion, randvillkor, hjärt-minut-volym, puls och blodets viskositet (den sistnämnda beroende på röda blodkropparnas volymfraktion (hematokrit) och den lokala skjuvhastigheten) studerades. Utöver vanligen använda VSS-baserade parameter har även modeller för aktivering av endotelceller prövats som riskindikator för åderförkalkning. Därtill användes resultaten från simuleringarna till att kvantifiera backflöde (både inne i aorta och vid dess vägg). Förekomst av helikalt (spiralformat) flöde och instabiliteter (vilka leder till turbulens vid hög puls och hjärt-minut-volym) betraktades också. Resultat: Ett stort antal simuleringar (över 100) har utförts. Utvärdering har gjorts av flödesspecifika randvillkor, blodtryckbaserade randvillkor och s.k. ”windkessel (WK)”-randvillkor på kärlens utlopp för att simulera effekterna av kärlträdets perifera eftergivlighet. Resultaten visar betyande känslighet för randvillkoren. Decelerationsfasen i hjärtcykeln ger förutsättningar för flödesinstabiliteter och bakåtflöde. Motsvarande accelerationsfasen ses istället mer stabilt flöde huvudsakligen utmed kärlriktningen. Ostadigheter i VSS verkar driva atherosclerosutveckling och är därför viktiga för att bedöma decelerationsfasens effekter under både normala och sjukliga flödesförhållanden. Backflödet är som störst under den tidpunkt då inflödet från hjärtat är som minst. WK-randvillkor kräver ”ad hoc”-justeringar (anpassade till tillgänglig information vid varje tidpunkt) för att bli patientspecifika. Heliska flöden är ansedda att skydda mot åderförkalkning och formas som en naturlig följd av kärlens geometri (p.g.a. krökarna i aortabågen). Heliska flöden förekommer i alla större artärgrenar och försvagas under deceleration samt diastole då helixen kan byta riktning. De vanligaste modellerna för blodets viskositet baseras på hematokrit och skjuvhastighet. Modellerna visar stora inbördes variationer. Vid höga skjuvhastigheter är viskositeten låg och nästan konstant. Viskositetens effekter på energiförluster agerar motvikt till variationer i

6 skjuvhastighet. Vid låg skjuvhastighet är viskositeten hög och tvärtom. Detta fenomen ger dock liten effekt på de VSS-relaterade indikatorerna. ”Spårning” av partiklar (t ex celler, proteiner och blodproppar) i det strömmande blodet visar att backflödet i aorta och nära dess förgreningar kan ge upphov till stark krökning och därmed betydande centrifugalkrafter på partiklarna. Dessa kan då slungas från nedåtstigande aorta in i dess större grenar från aortabågen vilket i fallet med blodpropp kan ge upphov till stroke. Det sistnänmnda fenomenet har observerats kliniskt. Slutsatser: Det som ommöjliggör tidig upptäckt och prgonos av åderförkalkning vid blodflödessimuleringar i stora kärl kan tillskrivas 4 huvudorsaker: i. I nuläget är mekanismerna bakom åderförkalkning inte tillräckligt bra kvantitativt beskrivna för att definiera parametrar vilka både är tillräckligt känsliga och specifika att beskriva processens utveckling. ii. Bristen på tillräckligt noggranna patientspecifika mätningar av hjärtats, blodets och kärlens fysiologi räcker inte som underlag till de randvillkor som erfodras för simuleringar som skall ge förutsägelser i individuella fall. iii. Nuvarande modeller för kärlväggens och blodets materialegenskaper (samt deras interaktion och variationer över rum och tid) har flera tillkortakommanden. iv. Med hänsyn till i-iii erfodras bättre verktyg för databearbetning i form av verktyg som från ett begränsat antal simuleringar härleder mer generella parmetrar beskrivande atheroscleros genom s.k. modellreducering eller djup maskininlärning (s.k. ”deep learning”). Blodflödessimuleringar kan dock med fördel användas för att dra mer generella slutsatser om flödets effekter på kliniskt observerbara fenomen som exempelvis bakåtflöden vid t ex hjärt- och klaffsjukdom samt transport av blodprodukter och -proppar i stora kärl. Nyckelord: Blodflödessimulering, Åderförkalkning, Väggskjuvspänning, Blodreologiska modeller.

Summary of the papers and contributions The thesis deals with blood flow simulation in the human aorta and renal arteries. The overall aim of the work was to assess the possibility to use wall shear stress (WSS) based indicators with the ability to predict risk for future development of atherosclerosis. Sclerotic renal arteries were “reconstructed to original shape” and used as test bed for studying three commonly used WSS indicators. The results were summarized in Paper 1. Strong dependence, of the results, on geometrical and modeling related parameters was observed and hence the following studies focused on clarifying the impact of boundary conditions (BCs, Paper 2); The impact of heart-rate and flow-rate and its temporal variation (Paper 4) and the impact of blood rheological modeling (Paper 5). Paper 3 deals with the mechanisms that lead to formation of large scale structures (retrograde and helical flow) and their dependence of the temporal variation of flow-rate as well as the formation of small scale instabilities that my lead to transitional flow. Finally, the transport of blood components (cells, certain proteins and emboli) from the aorta is discussed in the thesis and Paper 6.

Division of work between authors: The main advisor for the research is Lisa Prahl Wittberg (LPW). The research was done in close collaboration with a former fellow PhD student (Niclas Berg, NB). The contribution of Alexander Fuchs (AF) and co-authors to the publications is given below. 1. Paper 1: Fuchs, A. & Berg, N. & Prahl Wittberg, L. Stenosis Indicators Applied to Patient-Specific Renal Arteries without and with Stenosis. Fluids. 4.26. 2019. DOI:10.3390/fluids4010026. AF wrote the ethical approval applications (for using patient CTA data). The CTA data was segmented by AF who also generated the different grids. The simulations were done using a version of OpenFoam which was modified and compiled by NB. The initial set-up was done by AF with the help of NB. Post-processing was done by AF using paraview and python scripts written by AF and/or NB. The initial version of the paper was written by AF and reviewed by NB and LPW. AF was the corresponding author and coordinated the response to reviewers and revision of the paper.

2. Paper 2: Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). Pulsatile aortic blood flow – A critical assessment of boundary conditions. Accepted for publication in ASME Journal of Engineering and Science in Medical Diagnostics and Therapy. AF set up the problem (segmentation of the thoracic aorta, generating five different grids) with nine inlet-flow profiles and different outlet BCs. The windkessel BC was implemented by NB. The simulations and post-processing was done by AF. Post-processing was done by python scripts written by either NB or by AF. The initial version of the paper was written by AF. The paper was reviewed before submission by LPW and NB. AF was the corresponding author, coordinating paper revision and response to reviewers.

3. Paper 3: Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020) - Fluid mechanical aspects of blood flow in the thoracic aorta. Submitted for journal publication.

AF set-up the problem of the flow in the thoracic aorta. The simulations and post- processing was done by AF. Post-processing was done by python scripts written by either AF or NB. The initial version of the paper was written by AF. The paper was reviewed before submission by LPW and NB. AF was the corresponding author coordinating paper revision.

4. Paper 4: Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020) - The impact of heart-rate and cardiac output on the flow in the human thoracic aorta. Submitted for journal publication. AF used published data to define the different cases of physiological and pathological heart-rate and flow rates. AF carried out the simulation using the OpenFoam modules by NB. AF wrote the draft of the paper, which was revised by LPW. AF submitted the paper for journal publication as corresponding author.

5. Paper 5: Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020) - Blood rheology modeling effects in aortic flow simulations. Submitted for journal publication. AF set up the different cases with the aim of understanding the impact of the rheological model on the WSS indicators. AF carried out the simulation and the post processing. The different mixture and transport models were implemented in OpenFoam by NB. AF wrote the draft of the paper. The paper was reviewed by LPW and was submitted for journal publication with AF as corresponding author.

6. Paper 6: Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020) - On the modelling of cell and lipoprotein transport in the thoracic aorta. To be submitted. AF set up the problem with the aim of understanding the transport of cells and lipoproteins in the thoracic aorta. A possible clinical application was to explore potential risk for stroke due to embolism from thrombi in the descending aorta. AF carried out the simulations using OpenFoam modules written by NB. The draft of the paper was written by AF. The internal review process is on-going. Some of the results are included in Chapter 5 of the thesis.

List of Abbreviations

0D/1D/2D/3D/4D Zero-, one-, two-, three-, four-dimensional ARAS Atherosclerotic Renal Artery Stenosis BC/BCs Boundary condition/Boundary conditions BCA Brachiocephalic artery CO Cardiac output (LPM) CT/CTA Computed tomography/CT-angiography CVD Cardiovascular disease DNS Direct Numerical Simulation EAI-N Endothelial Activation Index-Nobili EAI-S Endothelial Activation Index-Soares ECMO Extra Corporal Membrane Oxygenation FKE Fluctuating Kinetic Energy HDL High Density Lipoprotein HF Heart failure HR Heart rate (beats/minutes, BPM, 1/60 s-1) HU Hounsfield unit LCCA Left Common Carotid artery LDL Low Density Lipoprotein LPM Litres per minute (10-3/60 m3/s) LPT Lagrangian Particle Tracking LRA Left renal artery LSCA Left subclavian artery MKE Mechanical kinetic energy MRI Magnetic Resonance Imaging NCD Non-Communicable Diseases OSI Oscillatory Shear Index (eq. 4.8) PAS Platelet Activation State (eq. 4.10--4.12) PDEs Partial Differential Equations RAS Renal Artery Stenosis RBC Red Blood Cell(s) RNWSS Relative negative WSS (eq. 4.6b) ROS Reactive oxygen species RRA Right renal artery RRT Relative Residence Time (eq. 4.9) RVRF Relative volumetric retrograde flow (eq. 4.6a) STL Stereo lithography TAWSS Timer-Averaged WSS (eq. 4.7) TKE Turbulent kinetic energy US Ultrasound VLDL Very Low Density Lipoprotein VWF/vWF von Willebrand Factor WBC White Blood Cell(s)

WK Windkessel WSS Wall shear-stress WSS Wall shear-stress tensor

List of Symbols:

D Diameter De Dean number

Db Diffusion coefficient (eq. 3.10) j Mass flux (Zydney & Colton, eq. 3.11) J Mass flux ji Mass flux (Leighton & Acrivos, eq. 3.12) L Characteristic length p pressure Q Flow rate (3.14-3.20)

Re, Rep Reynolds number, particle Reynolds number

Rp, Rc, C Parameters of WK-3 (3.14-3.20) ui velocity component in the i-th direction

Greek  Hematocrit, Womesley number  shear-rate (norm)  dynamic viscosity  kinematic viscosity (=)  density

ij shear-stress tensor (ij component), also magnitude ||  vorticity

Dimensionless numbers: 0.5 Dean number De = Re(D/Dc) ; D,Dc pipe and bend diameters, respectively Kundsen number Kn = /L;  mean free path/mean distance between particles

Péclet number (mass) Pe = L U /m ; m mass diffusivity Reynolds number Re = U L /), U, L are characteristic velocity and length scale.

Schmidt number Sc=/m

Stokes number St=p/F ; particle and fluid times, respectively. p=1/(3  d U ).

Strouhal number S  D f /U f – frequency ( f) Weissenberg number, Wi Shear-rate * Relaxation time D  Womersley number   2 

Acknowledgments

First I like to thank my main supervisor associate professor Lisa Prahl Wittberg for bringing me back in to KTH and for all your support and never failing optimism. I also thank my co-supervisors: Professor Örjan Smedby for all valuable advice in research as well as bringing me into the world of medical imaging; Associate Professor Chunliang Wang for all your help with and teaching about blood vessel segmentation and providing the software Mialab to perform the segmentations; Professor Anders Persson for all your support, particularly with the renal artery project. I would like to give a special thanks to Dr. Niclas Berg whose earlier work and our joint work on blood flow simulations was invaluable for making this thesis come true. Thank you also for everything you thought me about, programing, numerical analysis, art and music and more. The computations have largely been carried out with resources from the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre at Linköping University and at High Performance Computing Center North (HPC2N) at Umeå University. I would like to acknowledge all support from the Department of Radiology in Linköping University Hospital: Head of department Mathias Axelsson for giving me the opportunity to combine research and clinical work and Johan Blomma for all practical support and adapting a proper schedule. My colleagues in the MSK-section (Maria Lindblom, Layth George, Marcus Casselgren, Lena Törnqvist, Bengt-Åke Hedén, Jafar Yakob, Aleksandar Komnenov and Per Widholm). My clinical advisor during residency Anders Knutsson, Linköping University Hospital. All other colleagues (residents and senior), technicians, nurses and administrative staff in the radiology departments in Linköping and Norrköping that I got the wonderful opportunity to work with for a total of 6 years. I also thank all my clinical/radiological colleagues in Karolinska University Hospital, in particular Dr Amar Karalli for recruiting me and giving support to perform research as well as Natalia Luotsinen for invaluable last minute support with time for finalizing the thesis. I also like to thank all the many people I worked with (former colleagues, nurses, heads of department and other staff) in the hospitals I served at in Värmland, Örebro and Kalmar county for giving me an irreplaceable experience. Many big thanks to all other people with whom I’ve had the wonderful opportunity to share office in the Mechanics Department: Dr. Lukas Shickhofer, Asuka Gabriele Pietroniro, Dr. Valeriu Dragan, Dr. Elias Sundström, Dr. Shyang Maw Lim, Emilie Trigell, Francesco Fiusco, Roberto Mosca and Frida Nilsson. I also thank the other people in our group in the Mechanics Department: Associate Professors Anders Dahlkild and Mihai Mihaescu, as well as, Dr. Song Chen, Dr. Julien Lemétayer, Federico Rorro, Gustavo Mori Romero, Ghulam Majal, and Gaia Cairelli. I thank everybody who I ever had the opportunity to play music with, in particular: All musicians (past, present and provisional members) in the Royal Institute of Technology’s, KTH, Academic Chapel Orchestra (KTHAK) and the Nordic Youth Orchestra (NUO).

Many unprecedented thanks to my in-laws Eva and Hans Olof Rixon for taking care of our children when dealing with the thesis work. Thanks to my brothers-in-law Johan and Gustav for being good friends. Thanks to my brother Gabriel for many stimulating discussions. My most profound thanks goes to my parents Ilona and Laszlo for giving me more in life than I could ever repay. Finally, I thank my wife Louise for all love and support in the ordeals of the last years and for giving birth to our wonderful boys (Gunnar and Bernard).

Contents i. Abstract i ii. Summary of the papers and contribution v iii. List of abbreviations and symbols vii iv. Acknowledgements ix

1. Introduction-background 1 1. Epidemiology – medical challenges 2. Scientific/modeling/understanding challenges 3. Diagnostic/Radiological challenges 4. Computational challenges 5. Summary/disposition & achievements of the thesis 2. Physiology and pathology of arterial flows 10 1. The cardiovascular system 2. Cardiovascular diseases 3. Fluid mechanics of arterial flows

3. Blood flow in arteries and its modeling 28 1. Blood rheology-challenges 2. Mathematical modeling of blood flow 3. Blood as a transport medium 4. Boundary conditions

4. Simulation methods and data analysis 48 1. Segmentation and computational domain 2. Governing equations and BCs 3. Transport of blood components 4. Analysis of the results

5. Results 63 Introduction to the simulated results 1. Renal artery stenosis-background 2. Whole aorta simulations 3. Thoracic aorta simulation

6. Summary and conclusions 91 7. Future perspective 94

8. References 95

List of Papers:

14 a. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2019). Stenosis Indicators Applied to Patient- Specific Renal Arteries without and with Stenosis. Fluids. 4.26. DOI:10.3390/fluids4010026. b. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). Pulsatile aortic blood flow – A critical assessment of boundary conditions. Accepted for publication. c. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). - Fluid mechanical aspects of blood flow in the thoracic aorta. Submitted for journal publication. d. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). - The impact of heart-rate and cardiac output on the flow in the human thoracic aorta. Submitted for journal publication. e. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). - Blood rheology modeling effects in aortic flow simulations. Submitted for journal publication. f. Fuchs, A. & Berg, N. & Prahl Wittberg, L. (2020). - On the modelling of cell and lipoprotein transport in the thoracic aorta. Paper draft, to be submitted.

Conference contributions and presentations: a. Fuchs, A., Berg, N., & Prahl Wittberg, L. - Stenosis in renal arteries – a numerical study. 8th World Congress of Biomechanics, July, 2018, Dublin, Ireland. b. Fuchs, A., Berg, N., Smedby Ö. & Prahl Wittberg, L. - Blodflödets inverkan på Arteriosclerosutveckling. Röntgenveckan (Radiology week by the Swedish Society of Radiology), September 2018, Örebro, Sweden. c. Fuchs, A., Berg, N., & Prahl Wittberg, L. - Temporal and spatial wall shear stress characterization at the renal artery branching site. XXVII Congress of the International Society of Biomechanics (ISB2019), July 2019, Calgary, Canada.

Chapter 1: Background and introduction

Epidemiology – medical challenges Cardiovascular diseases (CVDs) is an umbrella term for conditions that affects the circulatory system (heart and blood vessels). CVD encompasses several separate and at least partially independent types of pathology. Many of these different pathologies have in common that they reside or originate in a blood vessel. Most often, CVD refers to atherosclerotic disease of the blood vessel wall and related conditions such as ischemia. In general, CVDs also includes other pathologies including, arterial and heart valve stenosis, bleeding, thrombi, emboli, heart failure (HF), arrhythmias and congenital disorders. Indirect impact of such pathologies leads often to tissue and organ damages. In severe cases, the damages are irreversible and may lead to death. The World Health Organization (WHO) estimates that in 2016 about 17.6 million people worldwide died from CVDs (31% of total death). Of these 85% due to heart attack and stroke. WHO also notes that this figure amounted to an increase of 14.5% from 2006. The age-adjusted death rate per 100 000 was 278, which represents a decrease of 14.5% from 2006. In Sweden, the estimate is that CVDs and other Non-Communicable Diseases (NCDs) are the primary cause of death with 35% and 22%, respectively (WHO, https://www.who.int/nmh/countries/swe_en.pdf). WHO defines NCDs very vaguely: “No communicable diseases (NCDs), including heart disease, stroke, cancer, diabetes and chronic lung disease, are collectively responsible for almost 70% of all deaths worldwide. Almost three quarters of all NCD deaths, and 82% of the 16 million people who died prematurely, or before reaching 70 years of age, occur in low- and middle-income countries. 15

The rise of NCDs has been driven by primarily four major risk factors: tobacco use, physical inactivity, the harmful use of alcohol and unhealthy diets. (https://www.who.int/ncds/en/). However, the major risk factors for NCDs are the same as for CVDs and therefore it is not always possible to distinguish between a pure CVD from an NCD. More commonly it is customary to attribute 50% or more of mortality in western countries to CVDs. As noted, stroke and atherosclerosis are the most common CVDs and pathological complications of atherosclerosis are important cause of mortality in the western world [cf Chatzizisis et al. (2007), Lozano et al. (2012)]. CVDs imply not only suffering for the individual patient and its kin but as lifesaving treatment is constantly improving and as curing treatment legs behind, the burden on society increases. Typical example of such tendencies are the treatment of heart attack, stroke and heart failure (HF). When patients with heart attack or stroke can get adequate (invasive) treatment within a short period of time (a few hours), life can be saved and secondary complications substantially reduced. Heart attack treatment may include coronary artery dilation and stenting or even Extracorporeal Membrane Oxygenation (ECMO) treatment during a shorter period of time. Advanced stroke treatment includes thrombectomy [Mokin et al. (2015)]. Treatment of HF was traditionally rather frustrating as it was mainly symptomatic (diuretics and inotropic drugs and/or Angiotensin-Converting-Enzyme Inhibitors or Angiotensin II Receptor Blockers) or ultimately palliative care (for detail see: American Heart Association Therapy Guidelines; http://www.ksw- gtg.com/hfguidelines/pdfs/HFGuidelinesAlgorithm.pdf). However, recent advances in mechanical assist device technology opens new options in addition to the commonly preferred heart transplantation [Seco et al. (2017)]. Using mechanical devices solves also a serious limiting factor for heart transplantation, namely the limited access to organs. The development of mechanical assist devices depends strongly on reducing the risks for thromboembolic events. In order to make advances in these areas better understanding of the processes involved and developing modeling tools are essential items. In addition to the above mentioned organ pathologies, CVDs are strongly associated with pathological changes in the arteries and their walls. These changes initially manifest as buildup of lipid material and later inflammatory process, progressing into atheromatous and fibrous plaques and finally stenosis. The atherosclerotic artery tends to calcify at later stages and as its mechanical strength reduces, it may develop aneurysms which in turn may rupture. The different pathologies have predilection sites due to anatomical and hemodynamic reasons: Stenosis often forms at bifurcations and downstream of arterial branching sites in arteries of certain size (3-6 mm) and in those carrying relatively large amount of blood (around 1 LPM or more). Examples of such arteries are the larger arteries bifurcating from the aorta: Brachiocephalic artery (BCA), the left common carotid artery (LCCA) and the left subclavian (LSCA) and the (right and left) renal arteries (RRA and LRA) as depicted in Fig 2.5. Hemodynamic factors are believed to be the most important cause for arterial stenosis due to the focal nature of the disease, [Vanderlaan et al. (2004) and Berk (2008)]. Numerical simulation of the flow, as has been done in this thesis, are aimed at improving the understanding blood flow and relate the flow to some of the clinical observations of atherosclerosis.

Scientific challenges Biological systems are complex by nature as they combine multiple physical and chemical phenomena and processes with strong influence from genetical factors and personal history effects. Many of these entities are not commonly addressed by simplified models used in physics or engineering. The genetical set-up of a cell, or genotype, is not always translated into observable expressions (phenotypes). The genes may be activated or suppressed as a reaction to certain states or stimulus whereby the genetics are only one of the parameters controlling the biological processes. As a result of such multi-facetted set of coupled physical and biochemical conditions, it is impossible to define a pathological process like atherosclerosis in a simple unique and deterministic manner. It has 16 so far not been possible to find a single test for determining the risk for developing atherosclerosis or monitoring treatment of the disease by following a marker. Traditional (measurable) markers that have been correlated with atherosclerotic CVD includes diabetes (elevated blood glucose levels), hypertension and body mass index (BMI). Nowadays, several more markers have been identified, such as family history and hs-CRP (an inflammatory reaction protein) [Yeboah et al. (2016)]. Most markers, not only in the case of atherosclerosis, may or may not directly have to do with the disease. In general, markers should not be associated with the formation of the pathology, unless evidence exists for such a relation. Similar objection may be raised against using observation of pathologies (form, location and progress) and relate these to hemodynamical observations. Uncertainties are associated not only with observation and measuring concentration of markers but also in measuring for example size of anatomical structures (arteries) in vivo and not least determining the flow through the arteries. Mathematical models are commonly used to extend understanding of phenomena which are only partially known and understood. Models are optimally knowledge based and reflect the understanding of the underlying processes. An example of such a model is the set of Partial Differential Equations (PDEs) that express conservation of mass and momentum in a continuous space-time domain. However, when such understanding is too limited one has to rely on models with expressions that are more or less “curve fitting” to experimental data. In the field of blood flow simulations, there are several crucial models of the latter type rather than the former. Measured rheological properties of blood were used to calibrate models of different types (e.g. power law, gradient based, particle based). Uncertainties in the empirical data used for calibration contribute to the shortcoming of models. Thus, a major scientific challenge is to extend the understanding which would lead to improved modeling that enables further improvement of the understanding and so on (that is what research is all about). In over more than a decade there has been a large number of publications (google count is in the order of a million) and attempts to use flow simulations as a potentially clinical tool. Some typical attempts can be found in Capelli et al. (2018) on congenital heart disease; Zhong et al. (2018) on coronary and intra-cardiac flow simulation. The latter paper list a number of such patient-specific publications during the past 20 years. The main challenges associated with blood flow simulation in arteries stem from the shortcomings of modeling the physical properties of the blood, the arterial wall properties and applying boundary conditions that can emulate the shifting character of the flow in arteries of a living individual. A less critical challenge, yet not negligible, is related of replicating a living artery into computational model and accounting for the uncertainties generated during the process.

Diagnostic challenges The diagnosis of CVDs is based on clinical findings, laboratory test and medical imaging. In most cases, a combination of these is necessary to make a diagnosis. Imaging can be used both for initial detection of a certain diagnosis in the CVD group and as follow up for managing disease progression. A common example is heart failure that is conveniently assessed with cardiac ultrasound. Laboratory markers include proteins found in the cardiac muscles (but released in the blood in case of muscle injury), e.g. troponin and creatine kinase (CK). These markers are relatively specific since they are found in muscular tissue. Other markers include concentration of lipids/lipoproteins and inflammatory markers as mentioned above. These markers are unfortunately very non-specific, the former are found also in “normal” blood and hence their presence is indicative at elevated concentrations. Inflammatory parameters could be elevated for different reasons (e.g. infection). Imaging tools for CVD include ultrasound, conventional angiography, computed tomography (CT), magnetic resonance imaging (MRI) and nuclear medicine studies. The different imaging techniques

(referred to as modalities in the radiologic community) can be used to visualize both the heart and blood vessels, each one with certain qualities and limitations. Ultrasound in general has limited resolution in space but good time resolution compared to the other imaging modalities. Therefore, it is used to assess motion of the heart (muscles and valves). The theoretical maximal spatial resolution is about 0.1 mm (based on a frequency of 7 MHz and speed of sound of 1540 m/s) [Ashley et al. (2004)]. In practice, the maximal resolution is even more limited (in clinical practice an order of magnitude higher) due to noise (because of tissues attenuation of the sound waves), multiple sound pulses that has to be transmitted and wide beam of the sound wave (focused only in one depth) [Ng et. al. 2011]. With Doppler ultrasound it is possible to measure the speed of the flowing blood along the axis of examination, hence only 1D or actually 2D (1 spatial dimension over time). The flow could be shown qualitatively (color Doppler) or quantitatively (pulse wave or spectral Doppler) showing the variation in axial velocity over time (Figure 1.1). Other limitations of ultrasound include the reflection and dispersion of sound waves by certain tissues making the deepest part of cardiovascular organs less suitable for examination. Ultrasound is however easily available and non-invasive making it suitable for many clinical purposes.

Figure 1.1: Ultrasound image with Doppler. A longitudinal view over the right radial artery. The gray scale image shows the blood vessel in the center of the upper image (arrow?) with surrounding soft tissues. Color Doppler shows the direction of the flowing blood (red corresponding to the arterial flow towards the transducer (upper part of the image) and blue the venous flow away from the transducer). Note the color bar in the upper right corner. The axial speed (the 60° angled component) of the arterial flow is shown in the bottom of the image with variations in time corresponding to the heart beats/cycle. (A. Fuchs).

Conventional angiography (or angiogram) can be used to visualize a vessel stenosis (Fig 1.2). Angiograms are performed invasively by inserting a catheter into the blood vessel of interest. Via this catheter, visualization is performed by injecting a contrast agent in the blood stream, thereby increasing the x-ray density. With the development of the other modalities mainly CT and MRI, angiograms are nowadays primarily used as guide for intravascular procedures (e.g. dilation, stenting, thrombectomy etc.) after a diagnosis has been made by other means. Before treatment, it is possible to perform invasive measurement e.g. blood pressure (by inserting a pressure-guidewire via the catheter), a feature not available by other imaging tools. The invasive character is both the main advantage and drawback of angiography.

Figure 1.2: Angiogram of the renal artery showing a narrow stenosis at the branching site (left image, A). The same artery is shown after treatment with dilation and placement of a metal stent. Reproduced from [Arteriosclerotic renal artery stenosis: conservative versus interventional management, Haller, C.; 88(2), 193-197, Copyright 2002 by Heart] with permission from BMJ Publishing Group Ltd.

During the last decades, there has been a significant technological development in medical imaging particularly in CT and MRI making both significant improvements in both spatial and temporal resolution and discovering new means of visualization. This development has enabled much more detailed diagnostics of CVD with both qualitative and quantitative data. Moreover, it has made possible to replace angiography as the main diagnostic method since CT and MRI are only minimally invasive. CT utilizes X-rays to create 2D and 3D images of the human body. The heart and blood vessels are well visualized by injecting a contrast agent in the blood stream. Most of these agents contain Iodine which is x-ray dense, has a suitable x-ray energy spectrum for imaging, and could be incorporated in molecules which are relatively well tolerated by the human body [Lee et al. (2015)]. The iodine contrast agent will “highlight” the inside of the heart chambers and the blood vessels lumen (Fig 1.3). A CT scan using contrast agent to visualize blood vessels (most commonly arteries) is called CT Angiography (or CTA). The equivalent term when using MRI in a similar way is MRA.

Figure 1.3: Two axial CT images of the upper abdomen before injection of a contrast agent (upper image) and after (lower image). Note how the abdominal aorta (AA) with surrounding arteries as well as right and left kidneys (LK and RK) gets brighter due to enhancement, that is obtaining a higher x-ray density because of the contrast agent in the blood. (A. Fuchs).

The maximum resolution of most clinical CT scanners used today is in the order of magnitude of 0.1 mm, with isotropic voxels from 0.1-0.3 mm. To reduce noise, the images are usually reconstructed about 0.75-5 mm thick. However, with the emerging technique of photon counting CT, the resolution could today be down to (0.07 x 0.07 mm2) in each cross sectional image [Willemink et al. (2018)]. Theoretically, this could be pushed at least one order of magnitude more if a suitable photon detector is available. Doing this would set the resolution in the order of magnitude of cells. CT is with current technology not able to perform hemodynamical measurements but the high resolution anatomical data would be well suited for deriving patient specific geometrical shapes needed for blood flow simulations [Doost et al. (2016)]. Recent development in dual-energy and multi-energy as well as phase contrast CT opens up the quantitative tool of spectroscopy. Because different tissues/substances show different energy spectrums (at different X-ray energies), it is possible to quantify tissue contents, e.g. fat, water, blood, iodine and by this differ atherosclerotic plaques with various compositions, e.g. lipid-rich plaques from fibrotic plaques [Wang et al. (2008)]. MRI utilizes the phenomenon nuclear magnetic resonance (NMR) to obtain images based on the spin of protons (or equivalently hydrogen atoms). Hydrogen atoms are present everywhere in the body in different concentration, hence very suitable for diagnostics. Just like CT, MRI is useful to provide 3D images of the cardiovascular system but can also measure blood flow in 4D (based on the movement of protons in the blood and phase shifts in the radio waves used to detect these). In cardiovascular MRI, the limits of the spatial and temporal resolution are dependent of each other and a trade-off has

20 to be made between them. In cardiac MRI, the resolution is down to about 1.5 x 1.8 mm2 with maximum temporal resolution of 50 ms [Saaed et al. (2015)]. For the aorta and carotid arteries, the corresponding figures are about 0.5-1 x 0.5-1m2 and 30-60 ms, respectively [Potters et al. (2015)8]. The difference is due to variation in size and the relative movement of the heart and arteries. Heart movement is relatively large and thereby giving more “noise”. Although the theoretical spatial resolution could be several orders of magnitude lower, it is limited by impractically long scanning times (several hours), a general problem in MRI imaging. So called 4D-MRI captures MRI images in sequence at a rate of up to about 40 images/second, whereby flow visualization up to 40Hz is possible. Fig 1.4 depicts helical flow in the ascending aorta. There has been recent attempts to extract turbulence data from 4D-MRI sequences [cf Ha et al. (2018)].

Figure 1.4: Examples of helix grade levels. 4D flow MRI-generated streamlines showing (A) grade 2 helical flow, (B) grade 1 helical flow, and (C) grade 0 helical flow in the ascending aorta (AAo), left ventricle (LV). Allen, Bradley D.; Choudhury, Lubna, Three-dimensional haemodynamics in patients with obstructive and non-obstructive hypertrophic cardiomyopathy assessed by cardiac magnetic resonance, Eu heart J cardiovasc Imag, 2014, 16(1), 29-36, by permission of Oxford University Press/European Society of Cardiology.

In summary, medical imaging, provides both qualitative and quantitative means for diagnosing CVD. The various modalities have advantages and disadvantages relating mostly to limits in spatial and temporal resolution, cost and availability.

Summary of the scientific challenges Understanding the fluid mechanical mechanisms behind the atherosclerotic process would not only explain the reasons for development of atherosclerotic pathology, but could also enable assessing the risk for developing the pathology and possibly to take measure to prevent or at least delay the manifestation of the pathology. Blood flow and formation of atherosclerotic pathologies in the aorta and the renal arteries are the topics of this thesis. The scientific challenges are substantial as the results ultimately should boil down to deduce generally valid theories and models that enable the understanding of observation and measured data, with the goal that better understanding may lead improved clinical treatment.

Summary of the thesis The thesis is structured with the aim to give more detailed background material to the publications related to the thesis and some further results not included in the publications. Chapter 2 gives a description of the physiology and pathology of the cardiovascular system (CVS). The emphasis is, however, more focused around the items that are treated in the papers enclosed to the thesis. This includes CVS pathologies with emphasis to atherosclerosis (AS), but also some accounts for related diseases, such as the impact on arterial aneurysms and myocardial infarction (MI). The impact of HF on the pumping properties of the heart and its relation to the CO is also discussed shortly. Chapter 3 deals with modeling blood flow in large arteries. The chapter describes blood composition and challenges in modeling its rheology. The mathematical framework for modeling the flow of blood in the arteries is described, along with the advantages and limitation of different options. Different inlet and outlet conditions are stated shortly as these are studied in Paper 2. The rheology and its modeling are discussed along with potential limitations of different approaches. Different possible approaches are discussed in terms of advantages and disadvantages. Chapter 4 shortly provides the numerical methods and post-processing tools that were used in the framework of this thesis. Chapter 5 summarizes the results of the thesis complimentary to the results presented in Papers 1-5. Further details and results are given in cases that did not fit into journal papers which commonly set limits the number of figures/tables that are allowed to be included. The additional data also reflects potential utilization of the computed data for deeper post-processing to allow further and better understanding the questions under consideration. Chapters 6 and 7 give a short summary/conclusions of the results and propose a possible future path to further research. The background to and the development of the thesis work can be summarized as follows: The thesis work started with the hypothesis that atherosclerosis develops as an outcome of unfavorable blood flow conditions leading to “activation” of endothelium, with associated response of the immune and coagulation systems. Renal artery stenosis is a “silent” disease, such that it is discovered often rather late and commonly as an outcome of hypertension inquiry. The basic approach was to evaluate the blood flow in the abdominal aorta and the proximal segments of the renal artery in patients with renal artery stenosis (RAS). Computed Tomography Angiography (CTA) data form these patients provided a starting geometry of the involved arteries. Blood flow was simulated using models and methods described in the thesis. In further simulations the stenosis was “removed” from the pathological renal arteries. The impact of stenosis on the Wall Shear Stress (WSS) and on parameters commonly used to identify regions with risk for developing stenosis were evaluated (Paper 1). The outcome of the work showed that the results can be sensitive to the models used for blood flow simulations. Paper 2 considered the impact of choosing inflow- and outflow-boundary conditions applied to the thoracic aorta, using two different inlet geometries: one with aortic sinus and a second without aortic sinus but instead having a straight cylinder simulating the left ventricle without the effect of the aortic root. At the aortic and vessel outlets, specified flow rates, pressure, pressure gradient and a 3-element Windkessel model were evaluated. Any of these approaches require further knowledge and assumptions. The sensitivity of the results to boundary effects implies that such simulations can be very useful as foundation for understanding hemodynamically relevant phenomena but not directly for predicting blood flow under normal conditions which show large variability in terms of heart rate and cardiac output. The variations also make it difficult to use simulations predictive for development of pathologies in a deterministic sense. On the other hand, the simulations can be useful in statistical sense with determination of the uncertainty due to boundary effects.

Paper 3 explores simulated data for exposing the underlying mechanism for the formation of larger scale flow structures in the thoracic aorta; namely, helical and retrograde flow structures in the aortic lumen along with formation of negative WSS at the wall. The relation between the driving flow-rate and retrograde flow was clearly shown (extended to more than two dozen cases in Paper 4). Helical motion is primarily the result of the geometry of the vessel (bend and torsion) and spatial distribution of the inlet velocity profile. Helical flow is common also in bifurcating branches of the aorta due to strong streamline curvature leading to secondary flow formation. Generation of vortical structures and the risk for formation of turbulence was further discussed in Paper 3. Paper 4 gives further details on the WSS and related potentially predictive parameters for different heart pumping variations; Heart-rate (HR), flow-rate (Cardiac Output, CO) and cardiac contraction/relaxation rate variations. The different cases emulate normal, healthy cases at rest and at exercise as well as cardiac pathological cases such as aortic stenosis and heart failure. Paper 5 discusses the impact of blood rheology on the simulations. The transport of cells (RBC, White Blood Cells (WBC), platelets), some lipoproteins (Chylomicrons, Very Low Density Lipoproteins (VLDL), High Density lipoproteins (HDL)) and von Willebrand Factor (VWF) as well as emboli in the thoracic aorta are discussed in the draft of Paper 6. The results show that the retrograde flow that is associated with the deceleration phase of the cardiac cycle may lead to upstream transport of small blood clots which, during systole will enter the arteries leading to the head which in turn may result in stroke.

Chapter 2: Physiology and pathology of arterial flows

2.1 Cardiovascular System

The cardiovascular system has several important functions for maintain necessary conditions for life. Blood has multiple essential functions: it carries oxygen, “fuel” and substances needed for converting our food into chemical compounds needed for building the body. Blood also transports hormones to cells throughout the body and removes metabolic wastes (carbon dioxide, nitrogenous wastes). Blood plays an important role in maintaining the right environment for life (pH and temperature) and generate compounds (e.g. ATP) that are used by different muscles in the body. Blood is also the carrier of major parts of the immune and inflammatory systems which are needed to defend the body against foreign microbes and toxins. Clotting mechanisms are also present, protecting the body from blood loss when injured. Blood flow is regulated, as described below, enabling the body to optimize the flowing blood for the needs; for example, during exercise more blood is needed for active muscles, the heart and the lungs. To maintain body temperature, capillary flow can be regulated to increase or decrease blood flow in the skin. After a meal, more blood is directed towards the intestine and liver. In emergency, the most important measures to take are to maintain blood circulation, free airways and sustained breathing.

Figure 2.1: The main arteries of the systematics circulation loop. In this thesis we consider the thoracic aorta together with the three main branches of the neck/head/arm. Blood flow in the region of the bifurcation of the renal arteries from the abdominal aorta is also studied.

Original image title: Aorta branches. Link to original image: https://commons.wikimedia.org/wiki/File:Aorta_branches.j pg – 2020-07-13. The author to whom credit is given for publishing the image under a creative commons license: Mikael Häggström(https://commons.wikimedia.org/wiki/User:Mik ael_H%C3%A4ggstr%C3%B6m – 2020-07-13). License: The image was reused in its original form under the Creative Commons Attribution-Share Alike 3.0 Unported license (https://creativecommons.org/licenses/by-sa/3.0/deed.en).

The cardiovascular system consists primarily of two coupled loops through which the blood flow is driven by the left and right parts of the heart, respectively. The systematic circulation includes the left heart ventricle, the aorta, with its branches to the head, neck, upper and lower limbs, as well as the thorax, abdomen and pelvis. The arterial blood, rich in oxygen, flows through the capillary system in the different tissues, whereby nutrition and oxygen is delivered. The de-oxygenated blood flows back through veins converging in the right atrium, the right ventricle, through the lungs and, after re- oxygenation, back to the left atrium and the left ventricle.

Figure 2.2: The pressure in the systematic and the pulmonary circulations. Note the increase in the pulse pressure in the large arteries due to the property of the arterial wall. Original image title: Circulation pressures V1. Link to original image: https://commons.wikimedia.org/wiki/File:Circulation_pressures_v1.tif – 2020-07-13. The author to whom credit is given for publishing the image under a creative commons license: Adh30 (https://commons.wikimedia.org/w/index.php?title=User:Adh30&action=edit&redlink=1 - 2020-07-13). License: The image was reused in its original form under the under the Creative Commons Attribution- Share Alike 4.0 International license (https://creativecommons.org/licenses/by-sa/4.0/deed.en)

The arterial part of the circulation system is characterized by higher pressure (about 120 mmHg, 16kPa) as compared to the vein side (about 20 mmHg, 2.7kPa) (Fig 2.2). The blood pressure in the systematic circulation loop reaches the same peak as the peak pressure in the heart in systole (about 100-120 mmHg at rest). During diastole, the pressure is lower (the change in pressure is the pulse pressure which is defined as the difference in pressure between systole and diastole) by about 40 mmHg, whereby the aorta maintains a mean pressure of about 100mmHg throughout the cardiac cycle. To understand the functionality of the major arteries, it is instructive to consider their histology. A schematic summary of the sizes and the composition is depicted in Table 2.1. More detailed structure of the arterial wall and a histological picture are given Figs 2.3 and 2.4. The arteries differ histologically from the veins as they are adjusted to handle the higher pressure. The arteries as all vessels, have a single layer of cells (endothelium) placed upon a base membrane under which smooth muscle cells along with connective tissue cells (fibroblasts) are found. The latter generate fibers and interstitial material. The arterial wall can be divided into three layers.  Tunica Intima consists of a continuous single cell layer endothelium supported by an elastic and one collagenous layer (membrane).  Tunica Media consists of a layer composed of variable portions of muscular and elastic components  Tunica Adventitia has varying thickness composed of collagen with interspersed bundles of elastin. Blood supply to larger arteries are through vessels inside the vessel wall (vasa vasorum).

A schematic picture of the arterial wall is depicted in Fig 2.3 and a corresponding histological picture in Fig 2.4. The histology of the artery reflects its function. The three layer allows the major arteries to respond to blood pressure and lead to minimal losses as blood flows thorough them.

Table 2.1: Typical characteristics of arteries, veins and capillaries (cf. Wnek & Bowlin (2008)). Arteries Veins Capillaries Function Oxygenated blood to Return blood to heart Solvent exchange body Pressure 120 mmHg - aorta 20 mmHg 50-20 > 60 mmHg - arteriole Lumen diameter 25-30 mm (aorta) 30 mm (v. cava) 1-8 m 5-10 mm (medium) 5 mm 20-50m (small arteriole, 20-30 m (venule) sphincter) Wall thickness Thick Thin Extremely thin Wall layers Tunica intima Tunica intima Tunica intima Tunica media - thick Tunica media - thin Tunica adventitia - thick Tunica adventitia - thin Muscles and elastic Large amounts Small amount None fibers Valves No Yes No

Figure 2.3: The structure of the arterial wall. Original image title: Artery. Link to original image: https://commons.wikimedia.org/wiki/File:Artery. svg – 2020-07-13. The author to whom credit is given for publishing the image under a creative commons license: Kelvinsong (https://commons.wikimedia.org/wiki/User:Kelvi n13 - 2020-07-13). License: The image was reused in its original form under the Creative Commons Attribution- Share Alike 3.0 Unported license (https://creativecommons.org/licenses/by-

sa/3.0/deed.en).

Figure 2.4: Histology of common carotid artery. The arrow pointing on the (tunica intima) which consist of the endothelium placed on an internal elastic lamina. The middle (tunica media) and the outer (tunica adventitia) layers are marked by the black and green brackets, respectively. Original image title: Fig 1: Histology of common carotid

artery. Link to original image: https://www.nepjol.info/index.php/NHJ/article/view/197 05/16251 – 2020-07-13. The amount of elastin is largest in the The author to whom credit is given for publishing the aorta whereas smooth muscles image under a creative commons license: Murari Prasad dominate medium sized arteries Barakoti (such as the main branches from the (https://www.nepjol.info/index.php/NHJ/article/view/197 aorta; e.g. Brachiocephalic, 05 - 2020-07-13). Common carotid, subclavian, and License: The image was reused in its original form under renal arteries). The elastin in the the under the Creative Commons Attribution-Share Alike 4.0 International license arterial walls gives the compliance (https://creativecommons.org/licenses/by-sa/4.0/deed.en) needed to modulate the pulsatile heart pumping into a strong mean flow with pulsation of only about 10%. The smooth muscles have a regulatory function through which the body can increase or decrease the pressure in parts of the arterial system leading to improved perfusion in required organs. Such a regulation is carried out normally rather frequently during the day; more blood is supplied to the stomach after a meal, more to the skeletal muscle at work/exercise, etc. Systematic regulation of the systematic circulation depends not only on the smooth muscles of the medium sized arteries, but also on major sensory organs. Systematic pressure is regulated by local, humoral, or neural reflex mechanisms. Baroreceptors are found in the carotid body (glomus cells are innervated both by sensory and autonomic fibers mostly from the carotid sinus nerve), the aortic arch, the right atrium and kidneys. The cardiovascular center also receives data from chemoreceptors, i.e. sensory neurons that monitor levels of CO2 and O2. These neurons alert the cardiovascular center when levels of O2 drop or levels of CO2 rise (which result in a drop in pH). Chemoreceptors are found in carotid bodies located near the carotid sinus, aortic arch and in the kidney (juxtamedullary glomeruli cells). The cardiac center stimulates cardiac output by increasing heart rate and contractility. These nerve impulses are transmitted over sympathetic cardiac nerves. Additionally, the cardiac center inhibits cardiac output by decreasing heart rate mediated by the parasympathetic Vagus (X-th cranial) nerve branches. Local vasomotor regulation of artery diameter is exercised by sympathetic motor neurons (vasomotor) nerves that innervate smooth muscles in arterioles. The hormonal regulation of blood pressure includes the kidney and the liver/lung through management of blood volume. Hypoxia-mediated renal and carotid body afferent signaling triggers activation of the renin‐angiotensin‐aldosterone system (RAAS). In response to rising blood pressure, the juxtaglomerular cells in the kidneys secrete renin into the blood. Renin converts the plasma protein angiotensinogen to angiotensin I, in turn converted to angiotensin II by enzymes from the lungs. Angiotensin II acts throughout the body by constricting blood vessels and thereby raising blood pressure. Constricted blood vessels reduce the amount of blood delivered to the kidneys, which decreases the excretion of water through the kidneys and thereby raising blood pressure by increasing blood volume. Additionally, Angiotensin II stimulates the adrenal cortex to secrete aldosterone. Aldosterone is a hormone that reduces urine volume by increasing retention of water and Na+ by the kidneys.

Blood pressure may also be regulated by natural (body owned) substances and artificial analogs. Epinephrine and norepinephrine, are body own hormones secreted by the adrenal medulla, can raise blood pressure by increasing heart rate and the contractility (inotrope effect) of the heart muscles and by causing vasoconstriction of arteries and veins. These two hormones, called sometimes as “stress” hormones, are released as stress response. Antidiuretic hormone (ADH), is also a hormone (produced by the hypothalamus and released by the posterior pituitary gland) as the name says, acts on kidney to retain water. Water retention leads to increase in blood volume and thereby increasing also blood pressure. NO (nitric oxide) is a very small molecule acting as local hormone in several life maintaining circuits. NO is released among others by endothelial cells causing local vasodilatation. Alcohol lowers blood pressure by inhibiting the vasomotor center leading directly to vasodilation and by inhibiting the release of ADH and increase diuresis. Nicotine raises blood pressure by stimulating sympathetic neurons to increase vasoconstriction and by stimulating the adrenal medulla to increase secretion of epinephrine and norepinephrine. The inner walls of the (young) arteries is smooth, with small viscous losses. The dispensability (compliance) of the arterial wall allows the cross-sectional area and its length of the artery to adjust to the pressure in the artery. Through this functionality, the larger arteries expand in their volume and can maintain the relatively high pressure also during diastole. Thus, the larger arteries and in particular the aorta and its main branches act a flexible vessel that store parts of the energy supplied during systole by the heart and releases that energy during diastole. Unfortunately, the composition of the arterial wall changes with age. As the larger arteries becoming stiffer, the heart has to produce higher pressure to maintain the same functionality (i.e. flow) as the arteries had at younger age. Over time, partially as consequence of the elevated pressure, a sclerotic process takes place in the arteries which results in a more permanent hypertension.

2.2 Cardiovascular diseases Atherosclerosis Normal aging of the body implies a general loss of elastic substances in various tissues, including the skin and blood vessels. Lower elasticity and reduced compliance of the larger arteries also leads to elevated systemic blood pressure. Yet, atherosclerosis is initiated in certain locations. The disease is an outcome of multiple factors. In spite of its multifactorial genesis, its location is usually distinct: The process predominantly located to arterial branches and bifurcation of arteries of certain size and range regions (Fig 2.5 from (VanderLaan et al. 2004) and only those arteries which carry a certain volume of blood. Flow in a bifurcation region leads to streamline curvature having a radius of curvature of the same order or possibly even small than the diameter of the bifurcating vessel. Therefore, it is not surprising that in addition to vessel bifurcation also vessel with stronger curvature may be sites for arterial pathologies (Gimbrone et al. 2000)). An example of such an “artery” is the Circle of Willis, which is prone to develop flow induced aneurysms.

Figure 2.5: Atherosclerosis occur often in curved or bifurcating arterial regions. 1. Aortic sinus/aortic valve 2. Ascending aorta 3. Aortic arch - lesser curvature 4. Aortic Arch - greater curvature 5. Brachiocephalic artery (BCA) 6. Right common carotid artery (RCCA) 7. Left common carotid artery (LCCA) 8. Left subclavian artery (LSCA) 9. Descending aorta with intercostal artery branches, 10. Right and left renal arteries (RRA and LRA) 11. Celiac trunk & superior/inferior mesenteric arteries 12. Right and left common iliac arteries.

From VanderLaan, P. A., Reardon, C. A., & Getz, G. S., Site specificity of atherosclerosis: site-selective responses to atherosclerotic modulators. Arteriosclerosis, thrombosis, and vascular biology, 24, 1,

12-22. https://www.ahajournals.org/doi/full/10.1161/01.ATV.0000105054.43 The shape of the arterial wall in- 931.f0 vivo can be observed using Reused with permission from Wolters Kluwer Health Inc./American common radiological methods, Heart Association. such as ultrasound (US), computational tomography (CT) of magnetic resonance or common radiological methods. Fig 2.6 depicts typical radiological findings of arterial renal artery stenosis (RAS), using CT scan with contrast. A constriction of the aorta may occur as rest of the embryonal period when the function of ductus arteriosus becomes redundant and it is “dismantled” leaving a rest at the insertion into the aorta.

Figure 2.6: Volume rendering of a CTA scan of the abdominal aorta and the renal arteries. Note how the left renal artery is affected by a narrow stenosis (encircled in red) near the branching from the aorta while the right renal artery has a normal width. The CT scan used a contrast agent which also enables detailed visualization of the kidneys. (A Fuchs).

The most common of atherosclerotic sites are close to arterial branching or highly curved vessels. In these region the blood elements follow a path-line that is highly curved, leading to unsteady separated regions near the walls. A separated flow region forms a hemodynamical constriction (partial stenosis) leading to elevated speed in the streamwise direction and elevated WSS opposite to the flow separation region. Sharp change of the flow direction implies generation of vorticity that may lead to (intermittent) formation of turbulence. Atherosclerosis evolves over long period of time (decades). Arterio-venous fistula that are used for hemodialysis often develop, during periods of weeks/months and sometimes days, lesions alike atherosclerotic plaques. As the regular development of the process is slow, it has been difficult to pinpoint to a simple/single cause for the development of atherosclerosis. The most commonly stated risk factors are obesity; smoking; nutrition, genetics and infection. High blood pressure (hypertension) is associated strongly with CVDs due to arterial response to elevated BP in form of thicken and stiffen arterial walls. Elevated levels of LDL cholesterol (low density lipoprotein cholesterol) increases the risk of CVD is by increasing the fatty deposits in blood vessels, leading to atherosclerosis. Ganjia et al. (2003) suggested that LDL cholesterol activates endothelial cells to express adhesion molecules that speed-up the process of atherosclerosis. Low physical activity, alcohol intake, nutrition (blood sugar levels and risk for diabetes and elevated salt level combined with kidney dysfunction may lead to elevated BP). Other less obvious factors include age, gender and diseases such as diabetes (Bakhru & Erlinger 2005; Farmer & Gotto 1997; Ferdowsian & Barnard 2009; Sinha et al. 2009; Streppel et al. 2009). The details of the mechanisms of atherogenesis are known only roughly. The process is very slow and normally takes decades before the patient notices clinical effects. The process is multifactorial since it involves several well-regulated sub-systems, such as the immune, inflammatory and to some extent also the coagulation systems. Nevertheless, there is a very rich literature describing components of atherogenesis. Large scale population studies provided data about risk factors for developing atherosclerosis. Each of the risk factors alone may not cause the disease, but it may be the result of the disease or co-finding to the disease. Atherogenesis was described in many publications (e.g. Libby et al. (2013); Weissberg (1999)], to name some). The atherosclerotic process is closely associated with inflammation and it is now claimed that atherosclerosis is a lipid-driven inflammatory disease of the arterial intima [Bäck et al. (2019)]. Fat carrying lipoproteins infiltrate the intimal part of the arterial wall. The lipoproteins are taken care of by macrophages leading to formation of lipid-filled foam cells. This initial step in the process proceeds when apoptotic cells and foam cells are not removed due to inadequate inflammation resolution. Activation of endothelial cells follows and entails extending molecules (selectins and adhesion molecules) from the surface that attracts and captures inflammatory cells (macrophages, T- cells and mast cells) in the circulation [Weissberg (1999)]. The process continues as modified lipoproteins and cholesterol crystals accumulate in the arterial intima and induce foam cell formation and inflammation. This in turn leads to accumulation of necrotic macrophages and foam cells. Defective efferocytosis is a sign of failure in the resolution of inflammation, a process that is mediated by specialized pro-resolving lipid mediators and proteins. Smooth muscle cells are recruited into the intima and subsequently proliferate. A fibrous cap is formed maintaining the stability and protection of the plaque against rupture. The inflammatory process leads to inhibition of further proliferation of smooth muscle cell leading to weakening or erosion of the fibrous cap. The erosion or rupture of the cap leads to platelet accumulation and activation that enhances the activity of the coagulation system. This leads to formation of fibrin that may led to formation of a thrombus. A thrombus may block partially or completely the lumen of the vessel that may lead to infarction of the affected organ. The thrombus, as the cap may separate from the lesion and results in an embolus. Arterial wall pathologies (atherosclerosis with stenosis or rupture) are found in different organs. Myocardial infarction (MI) is commonly an end-stage of atherosclerosis resulting in blocking blood flow to parts of the cardiac muscles. A corresponding blocking of an artery in the brain results in stroke. In both cases blocking may also be a results of an embolus (i.e. a blood clot that follows the

30 blood flow and then trapped in a vessel that is too small to allow its passage). Both the heart and the brain have arteries with multiple collaterals that can compensate to certain extent for the acute blockage of an artery. Bleeding may be caused by rupture of the weakened atherosclerotic affected arterial wall. Both bleeding and blocking of an artery leads to activation of the inflammatory and the coagulation systems. In the case of an embolus, active disintegration of it takes place through fibrinolysis. A stenoted/blocked artery may be treated by stenting (a net supporting the arterial wall) or by removing surgically the blocking part of the artery. Thrombectomy implies removing a thrombus mechanically using catheter [Mokin et al. (2015)]. Medical alternative to thrombectomy is thrombolysis using natural (in the past streptokinase) or synthetic thrombolytic substances. Certain organ failures are caused or associated with atherosclerosis such as heart failure (HF) and chronic kidney disorder (CKD). Atherosclerosis implies also loss of elasticity of the arterial walls which was commonly observed in for example coronary arteries and in the ostium of the heart-valves. The latter case may imply valve stenosis or regurgitation which in turn leads to increasing load on the heart that ultimately may lead to HF. About half of deaths in patients with CKD on dialysis are attributable to cardiovascular disease [Foley et al. (1998)]. It is not only that CKD can be caused by atherosclerosis, the inverse seems to be valid as well. Valdivielso et al. (2019) refers to data suggesting a contribution of CKD itself to subclinical atherosclerosis. Furthermore, progression of atherosclerosis was closely related to CKD progression. Fig 2.7 (Valdivielso et al. (2019)) depicts a normal progression of atherosclerosis due to common risk-factors and an accelerated progress due to CKD. Advanced CKD must be treated, as it may be fatal. Kidney transplantation can restore the patient to practically normal life quality. However, due to limited access to organs, hemodialysis is used as interim treatment. Hemodialysis implies that the blood is purified from certain substances in an extracorporeal (diffusion based) apparatus. The treatment must be done 2-4 times a week taking some 2-3 hours each time. As large amount of blood has to be circulated, an arterio-venous fistula (AV-fistula) has to be created. Brachiocephalic AV-fistula is a typical hemodialysis fistula, based on connecting the brachial artery with the cephalic vein in the upper arm. The venous side of the fistula enlarges due to the higher arterial pressure relative to the low venous pressure. The enlarged size of the fistula enables the larger volume of blood flow needed for hemodialysis. The venous part of the fistula changes not only in size but also the morphology of the vessel wall changes. The changes in the vessel wall resemble naturally acquired atherosclerosis with the infiltration of immune cells, wall thickening and formation of connecting tissue matrix. This iatrogenic process is fast (days and weeks) as compared to the natural process which takes place over years and decades. This observation was the basis behind a side project initiated by the candidate, although not reported in this thesis. As part of that concept, the effect of removing renal artery stenosis (which may lead to CKD) on blood flow was studied initially, within the thesis work (Chapter 5).

Endothelial cell (EC) activation process As discussed above, it is commonly accepted that hemodynamical factors play a role in atherogenesis. Hemodynamics may facilitate or contra-act the processes since it is a carrier of most of the components leading to the disease. Some central components in the process (such as endothelial and platelet factors) are produced locally. Endothelial cells or platelets producing atherogenic factors are termed as being “activated” as normally such factors are not produced. The literature in this field is extensive and it is not covered in the (very short) following discussion of some of the hemodynamic related cellular processes related to endothelial (and platelet) activation. Morphological changes in the endothelial cells (ECs) were observed when the cells were exposed to shear-stress. Fig 2.8 depict typical results of such morphological changes. ECs align themselves in the flow direction. When subject to unsteady flow, the EC response was dependent on the strength and frequency of the flow oscillation. Dai et al. (2004) showed that specific endothelial phenotypes could be invoked by arterial

31 waveforms derived from certain regions of human arteries. The regions selected were typical to pro- atherosclerotic or non-atherosclerotic regions. The paper describes two arterial waveforms, “athero- prone” and “athero-protective”, typically found in the carotid artery; namely the carotid sinus and the distal internal carotid artery, respectively. The two waveforms were applied to EC culture. The results showed distinct phenotypic modulation in response to the wall shear stresses.

Figure 2.7: A schematic model of normal atherosclerotic process (top) caused by common risk factors (such as smoking, age and overweight). Chronic kidney disease (CKD) itself may be a factor leading to accelerated formation of atherosclerosis. From: Valdivielso JM, Rodriguez-Puyol D, Pascual J, Barrios C, Bermudez-Lopez M, Sanchez-Nino MD, Perez-Fernandez M, and Ortiz A., Atherosclerosis in Chronic Kidney Disease More, Less, or Just Different?, Arteriosclerosis, Thrombosis, and Vascular Biology, 39,10, 1938-1966. https://www.ahajournals.org/doi/10.1161/ATVBAHA.119.312705 Reused with permission from Wolters Kluwer Health Inc./American Heart Association.

Figure 2.8: Endothelial cell culture under static conditions (A, top) and under laminar shear stress (13.5 dynes/cm2, after 30 h). The morphological change was found to be reversible. From DePaola N, Davis PF, Pritchard WF Jr, Florez L, Harbeck N, and Polacek DC. - Spatial and temporal regulation of gap junction connexin43 in vascular endothelial cells exposed to controlled disturbed flows in vitro. PNAS, 96, pp. 3154–3159, 1999. Copyright (1999) National Academy of Sciences.

Guo et al. (2007) showed that steady laminar flow in the straight parts of the arterial tree is athero- protective, whereas disturbed flow with oscillation in branch points and the aortic root were athero- prone, in part, because of the distinct roles of the flow patterns in regulating the cell cycle of vascular ECs.

There seems to be some consensus about the athero-protective effect of normal physiological low shear stress (approximately 15 dynes/cm2 (= 1.5 Pa) on the endothelium. White et al. (2011) compared transcriptomics and related functional analyses of human endothelial cells under laminar shear stress of 1.5 Pa or 7.4 Pa. The higher level of shear stress led to modified gene expression, reduced ROS levels, and reduced cAMP levels. The impact on the phenotype of the endothelium was not established. Oscillatory WSS was recognized to promote plaque transformation to a more vulnerable phenotype. Timmins et al. (2017) combining clinical and computational approach, studied the association between oscillatory WSS, in combination with WSS magnitude, and coronary atherosclerosis progression. The study showed that in patients with non-obstructive CAD, in regions subjected to low and oscillatory WSS regression of total plaque, fibrous and fibrofatty tissue was observed. On the other necrotic core and dense calcium regions did how progression. To increase the understanding of the atherosclerotic process, numerical simulation of different processes believed to be important for the pathology have been carried out. Such simulations range from molecular processes at cellular level, larger scale processes that include the transport and diffusion of species known to participate in atherogenesis and finally mechanical forces that act on cells and change their response and phenotype. Kwak et al. (2014) discuss in an ESC Position Paper on biomechanical factors in atherosclerosis, the state of the art on the interaction between mechanical forces and atherosclerotic plaque and the possibilities in integrating clinical imaging techniques with finite element modelling techniques allows for detailed examination of local morphological and biomechanical characteristics of atherosclerotic lesions.

2.3 Fluid mechanics of arterial flows Blood flow in arteries is very similar to water flow in bend pipes. The main features of such flows include the formation of flow structures that are observed clearly when small light-reflecting particles are present in the carrier fluid. The main flow structures that could be observed are vortices in cross- sectional planes, potentially spiraling flows or with time-varying driving pressure also retrograde flow. When the flow-rate in the pipe is large enough, the flow may become turbulent and chaotic. The flow in human aorta is non-turbulent in resting, healthy individuals. A short discussion of the fluid mechanical structures found in pipes and arteries is given the following.

Pipe flows and Cross-plane vortices Flow of a Newtonian fluid in a pipe was probably one of the first applications of theoretical studies of fluid motion. Steady, axi-symmetric flow through a straight pipe was solved independently by both Hagen and Poiseuille already around 1840. Some 40 years later, Osborne Reynolds (1883) published his observation about the development of turbulent flow in a straight pipe. Reynolds name is now associated with a dimensionless number (Reynolds number, Re) characterizing the relative importance of inertia over viscous effects. This simple flow problem attracted attention for many years, not only because its applicability but due to the fact that it was difficult to have consensus about the value of “the critical” Reynolds number (Re=U D /), with U, D and  being the characteristic velocity, pipe diameter and the kinematic viscosity, respectively). Different pipe flow experiments showed different results, observed transition at Re about 1900 and laminar flow up to about 30000. Linear theory could not either settle the question as the Poiseuille flow is linearly stable for all Re. Fully resolved Direct Numerical Simulations (DNS), enable computing the details of the flow, but also reveal several interesting features. Most pipe flow DNS are based on the assumption of pipe periodicity, whereby the results (often) depend on the length of the periodic pipe. Another observation

33 was the presence of very long (large) scale structures with a length of about 12 times the pipe radius [Zanoun et al. (2017)]. The origin and impact and behavior of such structures under unsteady conditions is not yet understood. Steady flow in a curved pipe was first studied by Dean (1927, 1928). The analytical solution was computed, for small curvatures, in a toroidal coordinate system. As part of the solution Dean defined a dimensionless number that relates inertia and viscous to viscous force. The Dean number, De, is 0.5 related to the Reynolds number through: De=Re(D/Dc) , where D and Dc are the pipe diameter and its curvature diameter. The fluid in a pipe flow follows a curved path and is, is subject to a centrifugal force driving the fluid from the inner wall towards the outer walls of the curve. The solution of Dean for two cases of small curvatures are given in Fig 2.9.

Figure 2.9: Typical Dean vortices in a cross- sectional plane of a bend pipe. The outer curvature is the top part of the figure. The figure is the first POD mode of turbulent pipe flow (courtesy of Fredrik Hellström).

Womersley (1955) studied unsteady flow in a straight pipe recognizing that at low Re number unsteady cases, the flow is determined by a parameter (i.e. the Womersley number, () measuring the relative importance of the driving pressure oscillation rate as compared to the viscous response rate. Thus 2=Transient-contribution/viscous contribution. The driving pressure with angular frequency  has a contribution of U -1 and the viscous contribution isU/D2, where U is characteristic velocity and D is the diameter of the pipe. Hence  is defined as D    (2.1) 2  The range of Womersley number in the human body vary in the order of 10-25 in the aorta down to about 4 in the renal arteries. The corresponding range of Reynolds number is below 8-10 103 and 500- 1000, for the aorta and renal arteries, respectively. In Table 2.2 Re and Womersley number for human arteries are provided. The steady and unsteady flow in curved pipes was investigated for larger values of pipe curvatures. Horlock (1956) investigated the secondary flow through a pipe with sinusoidal shaped centerline. He found that in such a geometry the secondary flow may be enhanced or weakened along the pipe. Barua (1963) examined curved tubes at a large Dean number and found fair agreement between theory and experiments.

Table 2.2: Order of magnitudes of the Reynolds and Womersley number in human arteries Reynolds umber Womersley number Ascending aorta ~ 4000 ~ 10 - 30 Thoracic aorta ~ 2500 ~ 7 - 20 Brachiocephalic artery ~ 1000 ~ 3 - 7 Left common carotid artery ~ 800 ~ 2 - 6 Small arteries ~ 100 < 1 Capillary ~ 0 ~ 0

Unsteady flow in a straight pipe was studied by Lyne. It was found that for large Womersley number a secondary flow in the interior of the pipe was in the opposite direction to that was observed for a steady pressure gradient. Sudo et al. (1992) studied experimentally and computationally oscillating flows through a curved pipe at various Dean- and Womersley-numbers, and found (Fig 2.10) five categories for the secondary flow: i. Dean circulation type; ii. Deformed Dean circulation, iii. Intermediate circulation between Dean and Lyne, iv. Deformed Lyne circulation, and finally, v. Lyne circulation. The lines of B and D in Fig 2.10 show the Womersley number or the relation of the Womersley and Dean numbers at which the secondary flow changes its pattern from type iii to iv. These lines were given by = 14 (D < 100) and 2.8/D=11.3 (D > 200). Lines H and G in Fig 2.10 denote when the time-averaged velocity toward the inner bend at the pipe axis takes the maximum value. These lines were for = 17 (D < 100) and 3/D = 30 (D > 200). Lines H and G mark the rough border between secondary flow of types iv and v. As noted the shape of the secondary flow in the cross-plane is sensitive to flow conditions. Increasing the Womersley number implies that the unsteady inertia dominates over viscosity. A second effect of the Womersley number is that it determines the extent of retrograde flow. Also, the streamwise velocity profile as measured and computed by Sudo et al. (1992) for the curved pipe show the presence multiple inflection points. This fact was not commented or analyzed in the paper. In the experiments of Sudo et al. (1992), the Reynolds number roughly equals 2.8 De. Thus, the results of that paper are in the right range of Womersley number (O(10)), but too low in terms of De (Re) as compared to the thoracic aorta. For comparison, our simulation at heart-rates above 120BPM lie at the upper edge or beyond the Womersley number range (20 <  < 35) and in or beyond the upper edge of the Dean numbers (103 3 < De < 3.510 ) in Fig 2.10.

Figure 2.10: Dean number (De) - Womersley number () map with the five types of secondary flow in the curved pipe. From Sudo, K., Sumida, M. and Yamane, R., Secondary motion of fully developed oscillatory flow in a curved pipe, Journal of Fluid mechanics, 237 (1992): 189-208. Reproduced with permission of Cambridge University Press.

Tada et al. (1996) carried out numerical simulations of pulsatile flow in curved pipes using a toroidal system with streamfunction vorticity formulation, assuming plane symmetry. The computations 35 included a range of Womersley- and Dean-numbers, along with variations in the amplitude parameter (. The latter parameter was defined as the ratio between flow rate and the imposed oscillatory component. The ranges of parameters that were 2<  < 50; 15 < De < 265 and 0.5 <  < 2, including physiological flows in the aorta. The secondary flow patterns were classified into four groups directly related to the corresponding dimensionless numbers: i. Viscosity-dominated (small De or Re or ratio of pipe radius to curvature and low a) where unsteady and the centrifugal effects are small; ii. Inertia- dominated (large , Re) with an oscillatory boundary layer with a central core; iii. Convection- dominated (large De) with the secondary flow vortex shifts from the inner to the outer side of the pipe cross section due to the influence of the centrifugal force. iv. The forth pattern had mixed character as two of the three typical flows are mixed. These regions lie in between the different regimes in the  vs De map (Fig 2.10). It was also found that when =1, four to six secondary flow vortices were present at high Dean numbers, and that for stronger oscillations (>0.5) the Lyne-type flow patterns disappear. Komai and Tanishita (1997) simulated fully developed flow through a curved tube, assuming plane symmetry, using a physiological pulse at De = 393, and = 4-27, curvature ration, = 1/2, 1/3 and 1/7 and intermittency parameters (defined as the ration of systolic to diastolic periods) 0-0.5. The underlying assumptions were similar to that of Tada et al. (1996) but employing primate variables instead of stream function-vorticity formulation. The results showed that the secondary flow persists throughout diastole and affected the flow in the coming cycle. The simulations were in physiological range, but using simple pipe geometry and the assumption of plane-symmetry are considerable limitations in terms of the relevance of the result to physiological flows. Siggers and Waters (2008) studied unsteady flow in a curved pipe. For small pipe curvature and Womersley numbers (), analytical methods were used, whereas for larger curvature and Womersley numbers numerical simulation were used. Sinusoidal driving pressure gradient (with Dean number, De = 0), three distinct classes of stable symmetric and periodic or stable asymmetric solutions were identified, depending on Re. Driving pressure gradient with zero steady component and D > Dcrit the periodic asymmetric solutions became spatially symmetric. The transition between solutions is dependent on the curvature. When the Dean number increases an unstable solution can become stable again. The effects of finite curvature can lead to substantial quantitative differences in the wall shear stress distribution. Further fundamental studies of pulsatile flow were carried out by for example the experimental Bulusu & Plesniak (2013) in an 1800 pipe bend. The work was extended and included non-Newtonian fluid and to include also numerical simulation using Quemada viscosity mode [van Wyk et al. (2015)]. In these experiments and simulations, the different types of secondary flows were observed. Furthermore, effects of flow instability of Görtler type (effect of combination of curved wall with centrifugal force) with formation of smaller scale unsteady structures was observed.

Retrograde motion As noted above, periodic driving pressure of flow in a pipe implies a periodic flow which may be locally non-unidirectional. The axial pressure gradient driving the flow has to be large enough to maintain the flow. The cross-sectional variations in the pressure are commonly small (i.e. the pressure variations in the cross-sectional plane can be neglected). This observation is the basis of in-vivo measuring pressure in a vessel, and therefore the location of the pressure gauge within the vessel is not critical for the accuracy of the measured value. When the negative pressure gradient becomes less negative, the same force per unit area is acting on the fluid within a cross-sectional plane, leading a reduction in fluid momentum. As the near wall region has lower momentum, the reduction of momentum may imply retrograde (back) flow. Considering a volume element of fluid placed near the wall of the vessel and when the same element is placed near the center of the pipe. As the pressure decreases during the pulsation, the same force acts on the fluid elements throughout the given cross- 36 section. The flow element near the walls changes direction before the similar effect takes place closer to the central parts of the pipe due to the lower momentum of the near wall element. This behavior is the results of the larger inertia of the fluid at the central parts compared to the low inertia fluid near the walls. As the frequency of pulsation (a) increases the amplitude of inflow oscillations decreases. This phenomenon was observed already in Dean’s solution and later both experimentally and in numerical simulation (e.g. Sudo et al. (1992)) and our results for the thoracic aorta (Papers 3&4). Thus, retrograde flows found in pipes of arbitrary cross-sectional shapes, is a result of the change in pressure gradient, not necessarily requiring a positive pressure gradient. In addition to pure fluid mechanical aspects, retrograde flows are of clinical interest. It was found [Kronzon and Tunick (2008); Wehrum et al. (2015)] that the descending arotic plaques (> 4 mm thick, ulcerated or containing thrombi) in the thoracic aorta were a major source of when they are located downstream of the outlets of the major arteries branching from the aortic arch (i.e. the common carotid arteries, Fig 2.1). The plaques are sites with slow and irregular flow that promote formation of thrombi. Such a thrombus may be washed upstream by the retrograde flow near the aortic wall and become an embolus which in the following systole may enter into the carotid arteries. This phenomenon was studies in-vivo by US (or rather Transesophageal Echocardiography, TEE) and 4D flow MRI (e.g. Wehrum et al. (2015) and (2018). The clinical study of Wehrum et al. (2015) included stroke 48 patients underwent Doppler examinations of the transition zone between the aortic arch and the descending aorta. To quantify the retrograde and anterograde velocity, Velocity-time-integrals (VTI) was used. VTI was defined [cf Caruthers et al. (2003); Waters et al. (2005)] as the time-integral during systole, of the peak velocity at a given plane. Wehrum et al. (2015) calculated VTI for anterograde and retrograde phases. The retrograde pathlines were visualized using 4D flow MRI. Diastolic retrograde blood flow within the descending aorta was observed in all 48 patients. Typical values of VTI anterograde and retrograde were approximately 21 cm and 11 cm, respectively, with the measuring plane located 3 cm downstream of the distal edge of the left subclavian artery branching from the aorta. As shown in Chapter 5, our simulation show that particles injected at similar location during flow deceleration may end up in the arteries leading to the brain, supporting the hypothesis presented by Wehrum et al. (2015).

Helical flows An unconfined axisymmetric jet at low Reynolds numbers maintains its symmetry. However, this state is unstable at higher Reynolds numbers and the jet may change state to a family of helical (spiraling, swirling) motions with right and left rotation direction and integer increments of pitch distances [cf Strange and Crighton (1983)]. A confined jet maintains the property of having helical states, but as the flow enters the confinement it may lose its free-jet like shape. If the pipe has a second bend which is out of plane of the first bend, the second Dean like vortex forms into a swirling motion (as depicted in Fig 2.11 for an aorta). Practically all arteries are vessels with multiple bends. Hence it was natural to expect that helical flows form in arteries. The mechanism of formation depends on geometrical aspects and the inlet conditions (or the geometry upstream the segment under consideration). Helical flows are often found in bifurcating arterial branches due to the flow curvature and the vorticity that is generated through the vortex stretching mechanism. von Spiczak et al. (2015) introduced the vorticity in the thoracic aorta as a quantitative measure for vortical flow. 4D phase contrast MRI data of nine healthy and three patients, were extracted and used to compute the vorticity in six regions of interests. The data was also used to identify the vortex core and its development during the cardiac cycle. The authors defined strength, elongation and radial expansion of 3D vortex cores at three instances in time during the cardiac cycle. Vortex flow that developed late in mid/end systole close to the bulb and no physiological helix was found in the aortic arch.

Helical flows can be nicely observed in patients using MRI, as shown in Fig 2.11. The figures show generation of helical motion due to aortic valve disease. Swirling flow through the aortic orifice may initiate the helical flow. Such large swirling motion can be seen already in the left ventricle during the ventricle filling phase. This flow structure changes its shape over time and it may lose its swirling component during systole. The mechanism of formation of helical flow can also be independent on the inlet conditions. Helical motion may be formed in a non-straight vessel also under steady state conditions. Fig 2.12 depicts the secondary flow in a pipe with double bend (simulations by Fredrik Hellström). It is interesting to note that helical flow is commonly observed in the descending part of the thoracic aorta, due to the aorta shape itself (as is depicted in Fig 2.12). In fact, it is the non-straight shape of the aorta that may lead to the formation of a helical component. The ascending aorta has a cranial and slightly posterior direction while the aortic arch has a bend in the sagittal plane together with a weaker bend in the posterior-anterior direction. Fig 2.13 depicts the two common bends of the thoracic aorta. The figure depicts a typical CT scan of the thoracic aorta with the red lines indicating the aortic curvatures which resemble that of the double bend pipe of Fig 2.12, though with less sharp bands.

Figure 2.11: Helical motion in the aorta. Time-resolved 3-dimensional magnetic resonance phase contrast imaging (4D Flow MRI) reveals altered blood flow patterns in the ascending aorta of patients with valve-sparing aortic root replacement. Left image: Reproduced from [Multimodality imaging in heart valve disease, John B Chambers, Saul G Myerson, Ronak Rajani, Gareth J, Morgan-Hughes and Marc R Dweck, Open Heart 2016;3:e000330, 1-7, Copyright 2015 by Open Heart] with permission from BMJ Publishing Group Ltd. Right image: Reprinted from The Journal of thoracic and cardiovascular surgery, 159(3), Oechtering TH, Sieren MM, Hunold P, Hennemuth A, Huellebrand M, Scharfschwerdt M, Richardt D, Sievers H-H, Barkhausen J and Frydrychowicz A, Time-resolved 3-dimensional magnetic resonance phase contrast imaging (4D Flow MRI) reveals altered blood flow patterns in the ascending aorta of patients with valve-sparing aortic root replacement, 798-810, Copyright (2019), with permission from Elsevier/The American Association for Thoracic Surgery.

a b Figure 2.12: Flow in a pipe with double bend leading to a swirling motion (left frame). The inlet to the pipe is normal to the inlet plane with no in-plane component. Numerical simulation by Fredrik Hellström. The right frame shows the instantaneous flow in the proximal region of the descending aorta. The helical motion is superimposed on (multiple) streamwise vortices. The coloring is related to the axial velocity component. The blue color indicates retrograde flow.

Kilner et al. (1993) studied helical and retrograde secondary flows in the aortic arch in ten healthy volunteers, using 3D MRI. Right-handed helical flow was observed in nine of the ten subjects. Further downstream, in the upper descending aorta, the helical motion varied between subjects. End-systolic retrograde flow was observed usually along inner wall curvatures. In addition to the formation of helical flow the paper reports only possibly weak turbulence. Turbulence was considered to be detectable with MRI when signal loss was observed [cf Kilner et al. (1994), Iwamoto et al. (2014)].

Figure 2.13: The shape of the thoracic aorta in oblique sagittal (left) and transaxial (right) projections. The red lines depict the cranio-caudal bend of the aortic arch (left) and the anterior- posterior torsion of the thoracic aorta (right). (A. Fuchs)

Vortical flow in systematic circulation Turbulence is associated with enhanced mixing which is achieved on the cost of losses. With this in mind, it would be surprising if natural evolution of homo sapiens, under normal conditions, would lead to turbulent flow in arteries where no mixing is needed. On the other hand, turbulent flows are common with for example valvular heart diseases and other vitia cordis and obstructive flow in the upper airways. Such turbulent flows can be detected easily by listening with a simple stethoscope. In these cases, turbulence is generated through the production of vorticity and the non-linear vortex stretching mechanism. The growth rate of instabilities in different shear-layers vary considerably.

Perturbation grow very quickly in jets, such that a jet with laminar profile at the orifice becomes turbulent at some 4-5 jet diameters downstream. On the other hand, boundary layers over a plane or in a pipe amplify perturbations with considerably slower rate. Acoustic signals can be heard only if they are strong enough and lie within a certain range of frequencies. Therefore, most commonly, pathologies that are heard by stethoscopes are generated by (more or less circular) jets. Such jets occur in stenotic heart valves during systole or heart valve insufficiency (leakage) during diastole. Arterial stenosis generates wakes which may generate vibrations and/or sound throughout the cardiac cycle or intermittently, depending on the severity of the stenosis. With complete occlusion there is no flow and hence no sound is detectable. This type of behavior is the foundation of measuring blood pressure by occluding arteries in the arm and listening to the (Kortokoff) sounds when the occlusion is relaxed. Given this background it is not expected to observe turbulence in larger arteries under normal conditions but possible at higher cardiac flows. Historically, there has been some controversy with regard to the presence of turbulence under physiological conditions. In the literature one may find references asserting that aortic flows are turbulent whereas others claim the flow is mostly laminar. As a measure for the presence of turbulence, pipe flows were used as reference. The seminal Reynolds experiments indicated turbulent flow at (continuous) pipe flow with Reynolds number (Re) exceeding some critical value (commonly stated at about 2100). However, the critical Reynolds number depends on the presence of disturbances that are being amplified by the flow itself and thereby maintaining turbulence. The aorta is not a straight pipe and the flow in it is pulsatile. Therefore, it is highly doubtful that a critical Reynolds number determined in steady turbulent pipe flow experiments is applicable to a particular aortic flow. Additionally, some confusion may also be attributed to the lack of a rigorous definition of “turbulence”, since turbulence has not been defined strictly rather has in some studies been only characterized. Moreover, in the literature the term turbulence was sometimes used for vortical flows containing non-random structures. Aortic flows may have some intermittently transitional flow properties (during parts of systole and early diastole) and therefore fully developed turbulent flows, similar to those observed at high Reynolds number pipe flow, are not encountered in the aorta of healthy individuals and under normal conditions. Stein and Sabbah (1976) and Sabbah and Stein (1976) studied the aortic flow and the presence of turbulence in patient normal and diseased aortic valves. Their catheter based measurements had a data rate that enabling capturing of Fourier components of velocity of significant magnitude up to 320Hz. In patients with aortic stenosis, one observed a turbulent kinetic energy level that was more than 10 times larger than the corresponding level in normal subjects. In the former cases, turbulence extended throughout the ascending aorta and into the brachiocephalic artery (BCA), whereas in other cases turbulence was dissipated before reaching the BCA artery. Turbulence was also observed in the ascending aorta of normal subjects at elevated heart-rate. Using 3D-PTV and time-of-flight magnetic resonance angiograms (TOF-MRA) Gülan et al. (2018) studied the flow and losses in vitro compliant, silicone aortic models under physiological flow conditions. The unsteady flow was generated by a mechanical pump, using a fluid mimicking the viscosity of blood (glycerol/water mixture). The measured velocity was decomposed into the (phased averaged) mean and fluctuating component. The viscous and turbulent kinetic energy and dissipation rates were computed, using an eddy viscosity type model. The two measuring techniques indicated that the contribution of turbulence to the total kinetic energy and the dissipation rates was minor. Stalder et al. (2011) using MRI, studied in-vivo the flow in the aorta probing of flow instabilities for physiological pulsatile blood flow. Flow instabilities were prominent in the ascending and descending aorta but considerably less in the aortic arch where the mean Reynolds number was least. In the group of healthy volunteers, it was found that the presence of flow instabilities was higher for men than for women. The results of the study used empirical relation for the critical peak Reynolds c 0.83 0.27 number: Repeak  169 St ( and St being the Womersley and Strouhal numbers, respectively), derived by Peacock et al. (1998) using data measured in a pulsatile flow in a straight tube.

Gallo et al. (2014) used an MRI-derived silicon-made rigid phantom for in vitro experiments and as input for a CFD study. The 3D-PTV data was compared to CFD results and was also used to assess the level of turbulence. The ratio of the turbulent kinetic energy to the mean flow kinetic energy was small during flow rate acceleration but grew with flow rate deceleration. The level of TKE diminished towards the end of the flow cycle. Estimating the losses is often divided into the two components; one related to direct losses due to molecular dissipation and a part due to turbulence. It should be emphasized that the turbulent losses depend also on the molecular viscosity of the fluid. One may divide the kinetic energy in the flow into the laminar part (so called mechanical kinetic energy, MKE) and turbulent kinetic energy (TKE). Direct measurement of the two components of the kinetic energy of the flow is possible using modern PIV technique. PIV requires optical access and adequate temporal and spatial resolution. From a temporally resolved instantaneous velocity fields it is possible to compute the dissipation rate of MKE and TKE, respectively. Gulan et al. (2017) carried out in- vitro measurements using 3D-PTV and MRI to estimate kinetic energy losses in the flow in an aortic geometry. The MKE and TKE were estimated directly from the measured velocities whereas the turbulent dissipation rate was estimated from an eddy viscosity type model. In contrast to PIV based in-vitro experiments, existing 4D-MRI techniques do not have the corresponding resolution and hence one has to estimate the turbulent kinetic energy and the corresponding losses. Such efforts were reported by several research groups [cf. Kvitting et al. (2004), Markl et al. (2011), von Spiczak et al. (2015), Casas et al. (2016), Ziegler et al. (2017), Ha et al. (2018)]. Markl et al. (2011) report a review of 4D MRI study of the heart and aorta. The methodology was later refined and attempts to include turbulence were made. Some of these 4D-MRI studies do state the presence of turbulence in the aorta in healthy human individuals. Ziegler et al. (2017) reported using MR-estimated turbulence quantities in order to assess turbulence effects on the vessel wall. The approach employed computational results which were used to simulate the MR-estimates. The results indicated that MRI processing itself did not contributed to the overall error in estimating turbulent wall shear stress (tWSS). The results also showed that TKE, estimated near the wall, had a linear relationship to the tWSS. As the authors pointed out direct estimation of tWSS from 4D MRI is challenging due to limitations in spatial resolution. Ha et al. (2018) utilized 4D flow MRI data gathered from a group of younger and elderly males. The data was used to quantify the turbulent kinetic energy (TKE) in the aorta. All healthy subjects developed turbulent flow in the aorta, with total TKE of 3–19 mJ which is of the same order as the MKE. The paper also reported that the increase in TKE in the ascending aorta in an elderly patient group was “associated with age-related dilation of the ascending aorta which increases the volume available for turbulence development”. Further results along the similar lines were published by Ha et al. (2019). The paper assessed the effects of normal functioning and stenotic artificial heart valves on the flow. In these cases, the estimated levels of turbulence were somewhat smaller than those found in the younger patient group reported in the earlier paper. Fredriksson et al. (2018) suggested high TKE levels associated with the complications seen in Tetralogy of Fallot (ToF). Gulan et al. (2017) proposed a novel approach, which relies on TKE and dissipation of MKE for the assessment of aortic valve stenosis severity. Zajac et al. (2015) used TKE for determining diastolic dysfunction in the left ventricle of patients with dilated cardiomyopathy. Similarly, Dyverfeldt et al. (2011) utilized total kinetic energy levels to estimate the severity of mitral regurgitation. Binter et al. (2013) showed that TKE may provide complementary information to echocardiography, assisting to distinguish within the heterogeneous population of patients with moderate to severe aortic stenosis. As discussed above and in particular in Chapter 5 and Paper 3, MRI based turbulence data are probably an overestimate of levels of fluctuations. As was also pointed out, fluctuations in larger arteries are normally not turbulent in the sense that the flow is not fully chaotic in character. So when turbulence occurs it has a very limited range of scales due to the low Reynolds number of flow.

Chapter 3: Blood flow in arteries and its modeling

A major role of blood is transport of substances and cells needed by the different parts of the body. This transport must be quick enough and therefore the main mechanism is convection with the blood stream. With this understanding it is obvious that there has been much effort, and in particular in recent years to utilize the blood as carrier for targeted drugs. Therefore, transport processes by blood was considered in different ways in the literature. In the following, an overview of the physiological background is given followed by a discussion of appropriate methodologies for accounting for the transport process.

3.1 Blood rheology Blood rheology describes the physical properties of streaming blood on a macroscopic scale, commonly expressed in terms of apparent viscosity. Blood viscosity in turn is a function not only of the blood composition and the mechanical properties of blood constituents, but also of the flow itself and the properties of the vessel carrying the blood. The following sections is aimed at discussing blood viscosity in terms relevant to flow in larger arteries. Mechanical properties of the blood play an important role in many ways. As blood is the main transporter of O2, nutrients and substances needed to maintain life, blood circulation must be maintained. An average healthy human at rest circulates about 5 liters per minute (LPM) blood with a pressure drop of about 100 mmHg (about 13 kPa), which implies that the left ventricle of the heart has to generate about at least 1W of mechanical power. Thus, during about one minute under rest conditions, the blood volume in the body (5-6 liters) is circulating once through body during each minute. During exercise the heart has to generate up to 5-7 times as much net mechanical work. The heart must provide larger mechanical work as it has to maintain the lung circulation and work against gravitational (irreversible) effect. For comparison, the total rate of energy consumption by a human has been estimated to be between 50 to 100 W. The flow in the circulatory system imply irreversible losses due to blood viscosity. As blood viscosity determines also the local mixing, it has an impact on the biochemical processes in the blood and the arterial walls. Water may be considered to be a simple fluid in the sense that applying a certain shear yields a stress proportional to the shear (i.e. the proportionality coefficient is the viscosity). The viscosity of Newtonian fluids, such as water, is independent of the flow conditions, implying that the rheology is property of the fluid alone. Blood viscosity, on the other hand depends not only on the properties (i.e. composition) of the blood but also on the local instantaneous flow conditions. Often one defines an equivalent (apparent) viscosity for blood (may be defined as the viscosity computed from Poiseuille's law when the flow rate and pressure drop are measured). Blood viscosity as function of shear-rate is depicted in Fig 3.1a (following Cowan et al. (2012). The figure also marks the shear ranges that the blood is subject to during the cardiac cycle. The figure should be seen as conceptual, as more detailed blood viscosity variations in space and time are given in the results chapter (Chapter 5) in this thesis. Whole blood rheology affects the transport of lipoproteins due to the interaction with RBC and plasma. It was suggested by Cowan et al. (2012) to use advanced measuring techniques to determine whole blood viscosity (WBV) over a wide range of shear-rates and to use the data as a predictor of cardiovascular diseases. Fig 3.1b relates the shear rate to the cardiac cycle and to the apparent viscosity of blood. During diastole (low shear-rate) blood viscosity is up to 20 times larger than that of water. During systole and highest shear-rate the viscosity is only about 4 times larger than that of water. The whole blood viscosity and the contribution of some blood components is depicted in Fig 3.1c. The figures also show that blood is a shear-thinning fluid but tends asymptotically to a Newtonian fluid (i.e. constant viscosity) at high shear-rates.

a b

c Figure 3.1: The shear-thinning viscosity of whole blood (left-upper frame, after Cherry & Eaton (2013)) and the corresponding range of hear during the heart cycle (right-upper frame, after Cowan et el (2012)). The viscosity of whole blood (red and black lines are for two healthy main with 45% hematocrit) and viscosity of different blood substances vs shear-rate (lower frame). a Reprinted from Cherry, Erica M., and John K. Eaton. "Shear thinning effects on blood flow in straight and curved tubes." Physics of Fluids 25:7 (2013):073104, with the permission of AIP Publishing. b and c: Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Cardiovascular drugs and therapy 26.4 (2012): 339-348. Importance of blood rheology in the pathophysiology of atherothrombosis, Cowan, Aimee Q., Daniel J. Cho, and Robert S. Rosenson., COPYRIGHT (2012).

The behavior of whole blood viscosity as function of composition (mainly RBC concentration), shear- rate and flow in micro-tubes, have been measured rather extensively. The different measurements showed the difficulties with the meaning of apparent viscosity, since the measured values depend not only on blood composition (which is a variable) but also on the particular apparatus used in the measurements. However, some general tendencies of blood viscosity has been determined. Blood viscosity reduces, given a fixed hematocrit (RBC concentration) with increasing shear, i.e. blood is shear-thinning as depicted in Fig 3.1. The figure (3.1a) compiles several measurements over a period of about 20 years. It may be noted that the spreading of the data is larger at lower shear-rate and the viscosity tends to a constant value with increasing shear-rate. Brooks et al. (1970), Fig 3.2a examined the dependence of viscosity on shear rate and hematocrit, and found that whole blood behaves in a Newtonian manner up to a hematocrit level of about 12%. For larger hematocrit levels the viscosity

43 increased, but as the shear rate increased the viscosity decreased again reaching an asymptotic value above shear rates of approximately 100 s−1.

a b Figure 3.2: The dependence of whole blood viscosity as function of hematocrit (a, left image, after Brooks et al. (1970)). Apparent viscosity of the blood flowing in capillary tubes (b, right image, after Pries et al. (1992)). The apparent blood viscosity depends also in the size of the vessel, in particular when the diameter to RBC size ratio is not large enough. The data of Nichols & O’rourke (2005) and Fig 3.2b, show that the apparent viscosity is large when the RBC size is of the same size as the tube itself, it decreases when the RBC is aligned in the tube and is allowed to make change alignment angle. As the tube diameter not much larger than RBC diameter the RBC motion is limited. With increasing tube diameter, RBC motion stronger and thereby also WBV increases and ultimately reaches an asymptotic value.

Figure 3.3; Aarts et al. (1988) measured platelet concentration in a pipe without (left) and in the presence of ghost (RBC analog) particles at different concentration (20%, 40% and 60%, right frames, top to bottom). Note the low platelet concentration in the central parts of the pipe and high concentration near the walls. From Aarts, P. A., Van Den Broek, S. A., Prins, G. W., Kuiken, G. D., Sixma, J. J. and Heethaar, R. M., Blood platelets are concentrated near the wall and red blood cells, in the center in flowing blood., Arteriosclerosis: An Official Journal of the American Heart Association, Inc., 8(6), 819- 824. https://www.ahajournals.org/doi/abs/10.1161/01.atv.8.6.819

Reused with permission from Wolters Kluwer Health Inc./American Heart Association.

Closely related to the RBC-pipe diameter to the apparent viscosity is the concept of “Cell Free Layer” (CFL), sometimes referred to as a cell depletion layer. The appearance of the CFL is expressed as the Fåhraeus effect and the Fåhraeus-Lindqvist effects. The former implies reduction of the hematocrit with decreasing vessel diameter (Fig 3.2b) whereas the latter implies the reduction of apparent viscosity with decreasing diameter [Popel & Johnson (2005)]. Obviously, the first effect (reduction of local hematocrit) leads to the latter effect (reduction of apparent viscosity, [Aarts et al. (1988)], (cf Fig 3.3). Although this effect is important in simple tube flow studies acting as a lubrication layer, it is unknown how strong the influence is in large arteries due to the enhanced effect of convection. Insight into the motion of RBC and platelets was gained in recent year through detailed numerical simulation of blood flow in micro-tubes and channels. For example, Vahidkhah et al. (2014) carried 3D simulation of RBC and platelet interaction and migration in a near-wall shear layer. The paper shows that the RBC distribution of whole blood in a micro-channel becomes anisotropic forming clusters and cavities. As blood cells are elastic and both the cell membrane as the cytoskeleton may absorb and store energy, their response to forcing is time-dependent, leading to the thixotropic property of the blood. Thus, the observable or measured viscosity may depend on history or may have a stress with shearing or straining (so called viscoplastic behavior). One of the reasons for the non-Newtonian behavior of blood is due to the tendency of RBC to aggregate at low shear which leads to a higher viscosity as energy is absorbed in the RBC rouleaux (chain of RBC aligned into a longer chain). At higher shear the RBC chain breaks up and the apparent viscosity of the blood reduces. Due to its memory property blood viscosity is time-dependent shear-thinning (i.e. thixotropic). A major challenge was and still is understanding the details of the motion of the cells and other “particles” of blood and how that motion affects the apparent viscosity. An associated challenge is expressing such an understanding in form of a model which should be simple enough so that it can be utilized for simulating blood flow in human arteries of clinical interest.

Blood composition Blood is fluid composed mainly of water. However, it contains blood cells, lipoproteins, different types of macromolecules along with smaller molecules and ions (salts) making blood mechanically very different form regular water. Blood cells have a size range from a few m to about 20 m characterized by different mechanical properties that may change depending on the physiological conditions of the barrier. Lipoproteins are similar to micelles with a hydrophilic outer shell and a hydrophobic inner part. Therefore, lipoprotein are carriers of hydrophobic lipids and are believed to play an important role in formation of atherosclerosis. Certain macromolecules are part of different life-maintaining systems such as immune-, inflammatory- and/or hemostasis-systems (as discussed below). Large molecules, such as albumin are carriers of for example Ca2+ and other smaller molecules. The cells and the different blood components have a finite life time and the turn-over may be of the order of minutes (e.g. consumption of substances needed for blood clotting) to months Red Blood Cells, RBC, with life length of about 120 days). The blood as mixture can be seen as a living tissue that is changing its composition and character continuously. Whole blood (about 8% of body weight) is composed of blood plasma occupying about 55% of the volume and RBC (almost 45%). The plasma is composed of water (91.5%) proteins (7%) and other solute (about 1.5%). Albumin constitute about 54% of all plasma protein by weight, whereas globulins account for about 38%, Fibrinogen 7% and all others to about 1%. Albumin is a large protein of 70 kDa and is responsible to maintain the osmotic pressure in the kidney, needed to

45 maintain body water after initial filtration in the kidney. Albumin is also a major carrier and reservoir of Ca+2, which cannot be transported as an ion. Globulins have a central role for the defense of the body against foreign particles/proteins. These proteins are important not only in the immune system, but may have a role in inflammatory reactions as well and unfortunately also in auto induced allergic and autoimmune reactions. Fibrinogen is an essential component in the coagulation system. When fibrinogen is converted enzymatically (thrombin) to fibrin it can together with platelets and RBC form a fibrin network based blood clot. The “rest” proteins in the plasma include regulatory substances, such as those responsible for maintaining blood pressure and regulating blood redistribution in the arterial tree. Proteins that regulate bleeding are part of the coagulation system and most of them circulate throughout the body to enable quick detection and reaction to local bleeding. Electrolytes (mainly sodium and chloride) and in plasma solved bicarbonate help in maintaining a relative narrow window of base pH (pH about 7.45 ± 0.05). RBCs, as already stated occupies about 45% of the blood volume. RBCs have a bi-concave shape (usually referred to as discoid) that has a larger surface area for a given volume compared to a spherical form. This shape also has the advantage of a higher packing volume and allows the transport of RBCs in capillaries, where the RBC may deform substantially. The diameter of RBC 6-8 m and 0.8-1 m over the thinnest part. The cell volume is about 90 fL and there are some 5 million RBCs per microliter. The RBCs carries hemoglobin which binds oxygen molecules. These are released in the peripheral tissue. The RBC’s shape is maintained by a cytoskeleton that enable cell deformation without destruction. Under stronger shear stress RBC may deform and also undergo hemolysis (due to rupture of the cell membrane). Such hemolysis occurs more often in artificial circulation, e.g. blood pumps, and more commonly compact centrifugal pumps running at high rotation rate. RBCs are produced in the bone marrow and have a life length of the order of 120 days. Hemoglobin released from “old” or destroyed RBC is taken care of in the liver and then transported to the bone marrow. The recirculation of iron is impressively highly efficient resource management whereby the amount of additional iron needed by the body is minimal. Platelets are often recognized as cells, though they, just like RBCs, have no nucleus and no DNA. A more consistent designation would be cell residues. Platelets are also formed in the bone marrow derived from other much larger cells (megakaryocytes) where only parts of the cytoplasm follow into the platelets. These cell fragments do have mRNA and are hence able to synthetize some proteins necessary for the cells normal function [cf Rowley et al. (2012)] for a short review). Platelets are nearly oblate spheroids with semi-axis ration of 8:2 and 2-3 m in greatest diameter. The number of platelets varies largely, normally in the range of 150-400 thousand per microliter. Too low number of platelets (thrombocytopenia) may lead to bleeding and too high value may lead to enhanced tendency for thrombus formation. Life-length of platelets is about a week. Platelets as the name thrombocytes reveals are essential for the formation of thrombi. Platelets may have different shapes and sizes under normal and pathological conditions. Such variations affect also the functionality of the platelets. Platelets carry substances in different types of granule, that are active in the coagulation system, or lead to aggregation of more platelets, white blood cells and von Willberand Factor (vWF). Platelets activation leads to a chain-reaction with quick mobilization of cells, their activation and generation of fibrin and/or platelet based blood clots. The shape of a clot and its mechanical properties depend on the mechanism that lead to its formation. Shear stress generated by the flowing blood plays a central role not only in platelet activation and clot formation but also the transport process of molecules and cells needed to the clot formation. White blood cells (WBC) include a heterogeneous group of cells with a common primary function, namely defense of the body. About 60-70% of the WBC group is neutrophils, 20-25% are so called le leukocytes, 3-8% are monocytes and smaller portion of eosinophils and basophils (named after

46 staining properties). Neutrophils have phagocytic properties and release agents which destroy/digest bacteria. Eosinophil cells are for parasitic defense and are also involved in the allergic response, including the release histaminase to slow down inflammation caused by basophils. Basophil cells release heparin, histamine and serotonin and enhance the inflammatory response and account for allergic reaction. Monocytes can enter into various tissues and differentiate (change shape and character) into phagocytic macrophages. T-lymphocytes (Thymus origin) are responsible for cell mediates immunity whereas B-lymphocytes (Bone marrow origin) produce the immunoglobulin mentioned above, and which can bind to antigen as part of the neutralization of that antigen. Size variation in blood cells and in particular RBC and platelets may occur due to different pathologies. Silva et al. (2019) studied the impact of infection on RBC and platelets in infected mice. The main finding was that the inflammatory process triggered by bacterial infection induced pathological changes in RBCs and platelets activation. However, the infected mice blood rheological behavior was alike the non-infected blood. On the other hand, Piagnerelli et al. (2003) and Voerman et al. (1989a, 1989b)) did find that RBC rheology can be influenced by many factors, including sepsis and white blood cells which produce reactive oxygen species (ROS). Shortage of iron may lead to smaller RBC and with sickle cell anemia the life span of the cells is shortened. Infections may hamper the functionality of the bone marrow which may lead to shortage of all blood cells (for example CMV infection leading to pancytopenia [Koukoulaki et al. (2010)].

Hematopoiesis Hematopoiesis is the process leading to the formation of (cellular) blood components which takes place in the marrow of long bones (for example, the humerus and femur), flat bones (e.g. ribs and skull), vertebrae, and pelvis. Within the red bone marrow, hematopoietic stem cells divide to produce various “blast” cells. Each of these cells matures and becomes a particular formed element. Erythropoiesis begins with the formation of proerythroblasts from hematopoietic stem cells. Over three to five days, several stages of development follow as ribosomes proliferate and hemoglobin is synthesized. The nucleus is ejected, producing the typical shape of the RBC. Young erythrocytes (i.e reticulocytes), contain some ribosomes and endoplasmic reticulum, mature into erythrocytes after a day or two days. RBC production is stimulated by erythropoietin (EPO) which is produced in the kidneys. Low oxygen levels stimulate production of EPO (e.g. stay at high altitude). As the number of RBC is large (45% of blood volume) and life-span of about 120 days requires production rate of RBC at an impressive level of 2 million RBCs per second. Erythropoiesis requires iron, vitamin B12, and folic acid. A lack of either vitamin B 12 or folic acid can result in pernicious anemia which manifest in large, fragile, immature erythrocytes (megaloblasts). Leukopoiesis is stimulated by hormones produced by mature white blood cells. The development of each kind of white blood cell begins with the division of the hematopoietic stem cells into one of the following “blast” cells: Myeloblasts divide to form eosinophilic, neutrophilic, or basophilic myelocytes, which lead to the development of the three kinds of granulocytes. Monoblasts lead to the development of monocytes and lymphoblasts lead to the development of lymphocytes.

Table 3.1: Physiochemical properties of lipoproteins [Chaudhary et al. (2019)]. Chylomicrons VLDL LDL HDL Density (g/mL) < 0.95 0.95-1.006 1.019-1.063 1.063-1.210 Diameter (nm) > 75 30-80 18-25 7-14 Protein 1-2 8-10 20-25 52-60 Tri-glyceride (%) 80-95 45-65 4-8 2-7 Cholesterol (%) 1-3 4-8 6-8 3-5 47

Phospholipid (%) 3-6 15-20 18-24 26-32 Cholesteryl ester 2-4 6-10 45-50 15-20

Thrombopoiesis begins with the formation of megakaryoblasts from hematopoietic stem cells. The megakaryoblasts divide and become megakaryocytes with multilobed nucleus. The megakaryocytes then fragment into un-nucleated thrombocytes. Thrombocytes (platelets) contain some mRNA and able to synthesize certain proteins which are stored in granules. Lipoproteins have an important role in the body beyond delivering hydrophobic fats (triglycerides, phospholipids and cholesterol). These lipids are used as precursors for hormones and as building elements in cell membrane or as energy source. The composition and the origin of different lipoproteins is primarily in the liver. Lipoproteins constitute a heterogeneous group of different size, of spherical shape. The density, size and composition of different lipoproteins is given in Table 3.1. As seen in the table, the content, size and density of different lipoproteins may differ considerably. The physical properties of lipoproteins are important for their transport (as discussed in Paper 6 of this thesis). Low density lipoproteins (LDL, VLDL), in the popular press, are referred to as "bad cholesterol” since they are associated with atherosclerosis. High density lipoproteins (HDL) are considered as “good”. In recent years there has been an increasing effort to utilize lipoprotein for targeted delivery of drugs, and in particular hydrophobic ones. Zhu and Xia (2017), Upadhyay (2018) and Chaudhary et al. (2019) give a review of this topic.

Hemostasis A major role of blood is to maintain a “friendly” environment of the cells in the different organs of the body. Therefore, it is essential that the circulation of blood is maintained. Hemostasis is the process that maintains the circulation by stopping bleeding. Hemostasis may require the initiation of several sub-systems: i. Local vascular spasm of the damaged blood vessel at the site of injury. Vasoconstriction is initiated by the smooth muscle of the blood vessel. ii. Bleeding is stopped by a platelet plug which is composed of cross-linked platelets that fill the hole in the damaged blood vessel. Platelet plug formation by platelet adhesion after exposure to collagen fibers in the damaged blood vessel wall. Platelets granule release different substances (ADP, serotonin and thromboxane A2). These substances activate further platelets and the other parts of the coagulation system (proteins) in chain reaction form, leading to quick (1-2 min) formation of blood clot. Following its formation, a clot is further strengthened by a process called clot retraction, in which fibrin strands in the clot form a more tightly sealed patch. Fibrinolysis is the inverse process in which the clot is disintegrated, once the damaged blood vessel is repaired. The healthy endothelial tissue that replaces the damaged areas of the blood vessel secretes tissue plasminogen activator (t‐PA), converting plasminogen into its active form, plasmin, which breaks down fibrin and leads to the dissolution of the clot.

Rheological modelling As discussed above a major challenge was and is creating a model that can account for the apparent viscosity of whole blood flow in large and small arteries and under different flow conditions. Such attempts were made over decades, and is probably the very proof of the size of the challenge. The simplest approach to handle the blood is to assume that it is a homogenous mixture and it is a fluid with certain physical properties. The most relevant property from arterial flow point of view would be the viscosity of blood. However, the complexity of blood viscosity implies that measured values depend strongly not only on the measuring device but also on the blood sample itself and how

48 it was acquired and handled since the sample was taken. In addition to the uncertainties in the measured viscosity of blood, there is no thorough understanding of the processes that occur during the measurements. The combination of lack of good understanding of the biophysics and the disparity in experimental data lead to a relative large number of models that attempt to account for blood viscosity under different conditions. A summary of such models was given by Yilmaz and Gundogdu (2008). An up to date review may be found in Hund et al. (2019). In the following, the models most relevant to this thesis, are discussed. These constitute a small sub-set of the different models listed by Hund et al. (2019). Models with an eponymous name may have different versions in the literature making direct comparison of results from different papers more intricate or rather uncertain. The simplest model that is used rather often is of Newtonian type but with lager viscosity coefficient than of water. The viscosity is commonly in the range of 3-4 10-3 Pa s. Such as approximation is reasonable for higher shear-rate (>103 s-1). As noted above, blood viscosity is not only property of the blood but also of the flow, implying that a Newtonian model is not applicable over a range of flows. A large group of models are based on power law type of curve fitting, with hematocrit (i.e. RBC volume fraction, ) as a parameter appearing in form of non-linear contribution. Typical model in this category are those of (Casson 1959, Carreau, 1972 and Walburn & Schneck 1976),

Rheological models of blood as a mixture A critical assessment of common shear-thinning models was carried out recently by Gallagher et al. (2019). The models of Bird, Carreau, Cross and Yasuda were studied in order to address the problem of inferring model parameters by fitting to experiments. The study found families of parameter sets that fit the data equally well. However, by varying parameter sets, very different flow profiles were predicted. The authors conclude that such parameters cannot be used to draw conclusions about physical properties of the fluids, such as zero-shear viscosity or relaxation time of the fluid, or flow behavior. By introducing small perturbations to model parameters, it was concluded that the issues that were identified, have been inherent to the models that were studied. This class of models were used extensively in the literature, some of them were used also by us and they are detailed in the following.

Power law based models The simplest power law related the apparent, effective viscosity can be expressed as n1 eff  k (3.1) Where k and n are model parameters and  is the magnitude of the local shear-rate. k may depend on the hematocrit, but when it is a constant the model reduces to Ostwald de Waele model. The power of the shear rate must be negative as blood is shear-thinning, i.e. n < 1. For example, Cole et al. (2002) used the following parameter values: k=0.042 kg m-1 sn-2 and n= 0.61. An extension of the model above is to allow k = k(). Further extension of the model was by having k = k(), such that  and  are the hematocrit and shear rate, respectively. The Walburn-Schneck model as implemented in our code has the following form:

2 C24 C TMPA/ C3 eff  max(, min())min1 C e e (3.2)

The empirical constants of the model, are given by C1 = 0.000797 Pa s; C2 = 0.0608, C3 = 0.00499, C4 = 14.585 liter/g. TPMA is the Total Proteins Minus Albumin concentration for normal human blood (25 g/liter). The shear rate validity for this model has been reported as being in the broad range greater than approximately 0.01 s-1 [Zydney et al. (1991)].

Casson’s model Casson (1959) suggested a model and based on a calibrated power law concept. The following form and parameters were used herein. 2  kCy()()  min, (3.3a) effmax    Where 2  p B A/2 kCy; (  )  ((1   )  1) (3.3b) (1 )A A The numerical values of the parameters were as follows: p=0.00132 Pa s; max=0.04 Pa s; A= 1.387133; B=0.1965353. The mode parameters were derived for =0.45 (Cokelet et al. 1963); Perktold et al. 1999). The Casson model has been considered to express well blood viscosity over a wide range of shear rates, larger than 1 s-1 [Charm and Kurland (1972); Zydney et al. (1991)]. Variants of the Casson model were applied to different arterial flows such as intracranial aneurysms [Suzuki et al. (2017)].

Quemada model The Quemada model [Quemada (1977, 1978)], include two variables, namely the hematocrit () and the shear-rate (). The model is non-linear in terms of the variables and includes several parameters, depending on the details of the formulation: 2 k(,) p 1 (3.4a) 2 -3 the plasma viscosity p =1.32 ∙ 10 Pa s,  is the magnitude of the shear-rate tensor 1 휕푢푖 휕푢푗 훾푖푗 = ( + ) and 2 휕푥푗 휕푥푖 kk()()/() k(,)  0  c (3.4b) 1/() c

The parameters c, k0 and k1 are the critical shear rate and non-dimensional intrinsic viscosities related to low and high shear rates, respectively. With vanishing shear, the viscosity is the same as that for blood plasma which is an underestimate as at low shear, RBC form rouleaux leading to larger viscosity. Furthermore, the original version of Quemada had singularities at some values. This effect was remedied by modifying the model in different ways. For example, Cokelet (1987) introduced the following parameters: 3.874 1041  13.8 23  6.738  ke0  (3.4c) 1.3435 2.803  2.711 23  0.6479  ke  (3.4d) 6.1508 27.923 25.63.67923 c ()e (3.4e) Das et al. (1988) proposed a modified version yields lower viscosity at low shear-rates (below -1 < 1 s ) by replacing k0 with another expression: 2 ka00 with a0= 0.275363 and a1 = 0.100158 a1 

3.2 Mathematical modeling of blood flow Rheological models of different types were developed to be used within simulation tools for blood flow. Flowing fluids exhibit dual (particles versus continuum) character as the small scale character is that of particles whereas the larger scale character is that of continuum. There is a fundamental difference between a particle perspectives of matter as compared to a continuum approach. Particles have mass (m) and momentum (p) and thereby it is possible to determine their velocity. If the volume of the particle (V) is known and if it has a uniformly distributed mass, it possible to compute its density (p). The dimeter of spherical particles (rp) can be determine from its volume. When the particles are statistically uniformly distributed, it is possible to define the particle number density (n) from which the mean distance between the particles (R) may be defined. Moving particles have kinetic energy (1/2 m U2), but lack several other physical quantities that we associate with fluids. For example, the temperature of a fluid is a macroscopic property of an ensemble of fluid particles (molecules). For the ensemble of particles/molecules in a small but arbitrary volume it is possible to define a mean (space averaged) velocity and from that define a fluctuating (random) velocity associated with Brownian motion. The kinetic energy due to random motion can be expressed in terms of the macroscopic temperature for the ensemble. In a similar way it is possible to define the density of the ensemble (, particle mass times the number density divided by the volume under consideration). The random motion leads to collisions between particles and to momentum exchange between the particles. The collisions imply that two particles that are close to each other at a certain time (t=0), are at some distance X after some time (t=). At macroscopic level the transfer of momentum within the volume is seen as diffusion of momentum. Quantitatively, the flux of momentum () is defined =  grad(u) where u is the mean velocity and  is the diffusion coefficient of momentum (=X2/), aka kinematic viscosity of the fluid. The viscosity of the fluid (i.e. diffusion of momentum) is given by =

Table 3.2: Diffusion coefficients of substances found in cells and the blood. After http://book.bionumbers.org/what-are-the-time-scales-for-diffusion-in-cells/ - 2020-07-20 Molecule Measured context Diffusion coefficient (m2/s)

H2O water 2000 H+ water 7000

O2 water 2000 tRNA (20kDa) water 100 Protein (30kDa) Eukaryoric cell cytoplasm 30 Protein (30kDa) E. coli cytoplasm 7-8 Fluorescent dye (Fluorescein) Cell wall 30 mRNA HeLa cell nucleus 0.03-0.1

Fluids having viscosity that is independent of space and fluid flow is called Newtonian. It should be noted that as particles/molecules are moving around they also transport their kinetic energy (=temperature). Heat flux (q) may be defined along the same lines as momentum flux; q=k grad(T) (=Fourier’s law, with k being the diffusion coefficient of heat). Correspondingly, in an ensemble of particles with different properties (size, mass), it is possible to define mass flux (J) and a corresponding diffusion coefficient; J=D grad(), Fick’s law, with D and are the diffusion coefficient of mass and density, respectively). A list of diffusion coefficient for blood related

51 substances is given in Table 3.2. As noted, the range of diffusion coefficient in water, may vary from 2000 m2/s for a small molecule down to 0.04 m2/s, for large molecules. For comparison, the mass diffusivity of small molecules is still much smaller (by about three orders of magnitudes) than the diffusion of momentum in water. The A crucial question is how quantities such as temperature, density and the different diffusion coefficients behave as the volume under consideration is reduced. Obviously, the (control) volume cannot be too small as the statistics will be inaccurate if the number of particles in the volume is too small. In the following, we continue the discussion in terms of blood properties. As described above, blood contains cells, lipoproteins and molecules of different size, varying from some nm to around 20m. The blood is flowing in vessels of varying diameter from the order of a few cm down to capillaries of the order of 1 m and the corresponding flow rate also varies by order of magnitudes. Naturally, one would like to resolve all the scales involved in arterial blood flow. Unfortunately, this is not possible now and most probably it will not be possible in the near future. With the large variation in scales, it is not obvious which scales that have to be resolved and which are small enough so that locally averaged values are adequate to describe the processes of interest. The most common procedure is to average the fluid molecules/particles in small volumes and characterize the value of interest at any point in space as the average of the fluid properties around that point. Such an average is considered to be meaningful and representative when the molecules/particles are in equilibrium (i.e. no quick variation in that control volume) and that the distance between the molecules/particles is small relative to the scales of the flow (i.e. Knudsen number, Kn << 1). This limitation is to ensure the above mentioned equilibrium which also requires that the temporal variations in the flow are slower than the mean time of collision/interaction among molecules/particles. Furthermore, when the molecule/particle size (r) and inter-particle distance (R) are much smaller than the length scales found in the flow (L), one may assume that the fluid and the flow can be considered as continuum in the mathematical sense. Thus, for continuum assumption we require commonly that R/L << 1. A more precise requirement may be defined by computing space averages of different properties (such as velocity, density, pressure, etc) in a control volume. When the averaged values converge as the control volume diminishes in size, it is possible to define the relevant variable at a mathematical point (i.e. zero volume). The continuum assumption implies that the defined variables are continuous and their derivatives exist and adequate number of derivatives are continuous to allow to formulate a well posed set of differential equations. These equations are derived from basic physical principles related to conservation of certain quantities. The requirements for continuum are satisfied easily for water under normal conditions, since the mean distance between water molecules is less than 1 nm (=/c, where is the kinematic viscosity (10-6 m2/s) and c is the speed of sound (1500 m/s)). This distance is far smaller than all the length scales that are expected to occur in blood flow in human arteries. When it comes to RBC, the mean distance between the cells is the same order as the RBC size (about 4 10-6 m). This distance is the same order as the size of small arteriole or capillaries and hence is will be difficult to have a limiting value for the space average of RBC in small vessels. Hence, the continuum approach is not valid for whole blood flow in arteriole or capillaries. On the other hand, the continuum assumption may be a good enough approximation for blood vessels that are about 1000 times larger than the RBC. This effect is also observed in Fig 3.2b, where the effective viscosity is shown to converge with increasing tube diameter. In the following, we discuss formulating the basic physical laws in form of partial differential equations (PDEs) for multiphase formulations of blood flows. The starting point is defining the plasma and the RBC phases as continuum interacting phases (so called Euler-Euler). A simplification is introduced by making assumptions about the RBC phase, whereby the blood as considered as a mixture with an apparent (effective) viscosity. Expressions for the viscosity alike equations (3.1) - (3.4) are all based on calibrating model parameters by fitting to available experimental data. Further

52 simplifying assumptions have to be introduced in order to make the solutions possible or faster. Of course, results obtained with simplifications must be checked for consistency against the underlying assumptions. The same procedure must also be carried out for the boundary conditions needed for solving the PDEs. As shown in Paper 2, such a consistency check is not always done which may result in non-realistic data.

Euler-Euler blood flow model The plasma and the RBC phases have their own physical properties (velocity, density, viscosity and local volume-fraction occupied by these two species). Both phases occupy the same domain, though with the possibility of having vanishing volume fraction. Each phase is assumed to be incompressible, though the mixture may have variable viscosity as is shown by the definition of the mixture density (equations (3.7)). In the following, the equations for multiple species (N) are formulated, valid for example for both platelets and LDL. Each of the phases is conserved independently of the other. Hence, the conservation of mass of each of the phases can be written in an arbitrary control volume. Using the divergence theorem one obtains the PDE for the conservation of mass of species (phase) q.  N ()()()qqqqq ipqqpq ummS,  (3.5) txi p1

Where subscript p and q represent species p and q, respectively. q, q and uq,i are the volume fraction, density and velocity in the i-direction of species q, respectively. mpq and mqp are the mass exchange rate between species p and q and q and p, respectively. Sq is the rate of local production of species q and N is the number of species. The corresponding momentum equation may be expressed by

N  p  ij ()()()q  quu q ,, iq uRMMFq ,,,, q i q jqpqpq iqp iq i  (3.6a) txxx jij p1 Where the two terms on the left hand side describe the rate of change of momentum (per unit volume) of species q in the i-direction. The terms on the right hand side include the pressure gradient in the i- direction multiplied by the volume fraction of the species q. The next term is the contribution of the stresses to the force on the fluid (species q). The third term describes the momentum exchange between species q and p in the i-direction. Rpq is an interaction between the phases and Mpq,i and Mqp,i are the momentum exchange due to mass exchange. The last term on the right hand side (3.6a) includes the sum of all the forces acting on species q (also per unit volume). Equation (3.6a) has to be complemented by a constitutive equation for a stress-strain relation:

uuuq,,, iq jq k 2 qq  qq qqki ()() (3.6b) xxxjij 3 Further closure relationships are needed for the interfacial interactions. Rpq depends on pressure, friction and possibly other factor (such as electrical charge) with the constrain that Rpq=- Rqp, expressing Newton’s 3rd law. Commonly, the interaction term includes only the drag effect between the phases. The expression for the drag follows closely the drag on an object, and not in line with the assumption of having two continuum fluids. In fact, expressing the interaction between the phases is a weak link of the Euler-Euler model.

Modelling blood as a mixture

For blood flow, an alternative and simpler approach is to assume that the blood is a mixture of plasma and RBC (N=2 in equations (3.5 and 3.6), having a mixture density and viscosity. The density of the RBC and plasma phases are denoted by RBC and p, respectively. Having only two phases, the volume fraction of the RBC phase is denoted by , the volume fraction of the plasma phase is then 1 -. The mixture density can be expressed so that the continuity equation for the mixture follows directly from equation (3.5):

 RBCp (1) (3.7) Similarly, one may define the mixture velocity and pressure (dropping the subscript from the variables) leading to the common single phase governing equations:  u i 0 (3.8a) txi

uu()uu p ii  ij  (3.8b) txxxx jijj where  are the density and viscosity of the mixture, respectively. p is the pressure and ui is the Cartesian velocity component in the i-direction. The system of PDEs (3.8) requires an expression for the mixture viscosity. Typical examples for such expressions are assuming that the mixture is Newtonian (with for example =0.0036 Pa s), or satisfying a relation like those given in equations (3.1)-(3.4). When the constitutive relation for the viscosity depends on the local hematocrit, the RBC transport has been modelled. Transport of any substance by a fluid is through convection and some mixing process, most commonly termed as diffusion. The transport equation for the RBC volume fraction () can be written as:  ujii() (3.8c) txx ii

Equation (3.8c) describes the transport of RBC hematocrit by convection (with the velocity ui) and other processes resulting in a flux-rate of  expressed by the “diffusive” flux vector j. Different types for the origin of flux-rate can be motivated. The classical gradient driven diffusion (Fickian type law) implies that the flux-rate is proportional to the gradient of the concentration () with a diffusion coefficient (often taken to be a constant). As the blood contains high volume fraction of RBC, RBC- diffusion is low. On the other hand, the RBC do have a relatively much stronger rotational motion, leading enhanced diffusivity of substances immersed or dissolved in the plasma, as compared to shear-induced, collision caused diffusivity. Casa and Ku cite the expression of Zydney and Colton (1988) and Jordan et al. (2004) for the effective diffusivity (Deff): 20.8 DDaeffsf  0.15(1) (3.9) where Dsf is the diffusivity of the species of interest in a quiescent fluid, a is the mean radius of the -13 2 RBC. Based on the Stokes-Einstein relation Dsf is of the order of 10 m /s The second term on the right hand side of equation (3.9) is of the order of 10-8 m2/s, for =104 s-1.

The simplest possible model is to express the diffuse flux as being Fickian;  jDib (3.10) xi with Db the Brownian diffusivity. The value of Db can be computed from equation (3.9) and is taken -8 2 to be Db=10 m /s Further cases were computed using also larger values of diffusivity constants: -7 2 -6 2 Db=10 m /s and Db=10 m /s.

Zydney and Colton (1988) formulated an expression (similar to equation (3.9), expressed in terms of the flux instead of diffusion coefficient) for the diffusivity of deformable particle suspension: 2 n jDKab  (1) (3.11)

Where   2SSijij being the norm of the shear-rate and a being the mean of the RBC radius, also in this formulation Db is the Brownian diffusivity, Kα and n are model parameters. The following -8 -6 parameter values were used in the simulations in this thesis: Db=10 ; a=410 ; Kα=0.15; n=0.8. A generalization of the above model may be derived if assuming that diffusion may be driven by forces due to inhomogeneity in different parameters of the mixture. An example is expressing the flux in terms of three components related to the interaction through direct collision between the particles (denoted as jc), shear induced interaction (j) and interaction through momentum exchange (j), which is related to viscosity of the carrier fluid. j = jc + j + j Based on scaling arguments Leighton and Acrivos (1987), Phillips et al. (1992) propose a model of the following form:

22222 1 jDKaKaKaib() (3.12) xxxiii  The model parameters are Kα, Kγ and Kμ are non-dimensional constants, determining the contribution of the gradients in hematocrit, shear-rate and viscosity, respectively. The following parameter values: -8 -6 Db=10 ; a=410 ; Kα=0.43; Kγ=0.43; Kμ=0.65. The results in the thesis include simulation with each of the viscosity models with fixed hematocrit or in combination with one of the transport models (3.9)-(3.12).

3.3 Blood as a transport medium Transport processes and atherosclerosis Braunersreuther and Mach (2006) studied leukocyte recruitment in atherosclerosis, due to atherosclerosis being an inflammatory process that involves recruited (inflammatory) cells and substances in addition to the participation of local endothelial and sub-endothelial cells. Recruitment of leukocytes from the blood stream into the vessel intima was found to be a crucial step for the development of the disease. The study reviewed known steps in the process, such as leukocyte arrest and transendothelial migration. Tranendothelial migration, as discussed below, was often modeled as gradient driven process. In contrast, Braunersreuther and Mach (2006) discussed another, non- mechanical mechanism of migration and transendothelial transport. Chemokines are small chemoattractant proteins known to induce leukocyte migration, growth, activation and to regulate leukocyte during inflammation. Chemokines play a role in platelet activation and contribute in “capturing” macrophages at a given site. In a recent review Jones et al. (2017) discussed experimental evidence of activation of leukocytes via inflammatory mediators such as chemokines, cytokines, and adhesion molecules. The study discussed also leukocyte activation and the mechanisms of chemokine-mediated recruitment in atherosclerosis. Sub-endothelial accumulation of LDL in coronary artery disease was studied by Olgac et al. (2011). The blood was considered Newtonian and assumed to be steady. The endothelium was considered as porous surface that yields the local blood plasma and LDL fluxes as a function of the luminal WSS and pressure fields. The main finding was that locally elevated subendothelial LDL concentration correlated well with subsequent plaque formation at the same location. Cejkova et al. (2016) discussed adhesion and migration of monocytes occur at the beginning of atherosclerotic plaque formation. These cells later differentiate into tissue macrophages leading to the development and transformation into foam cells. Monocyte recruitment depends on endothelial activation and possibly dysfunction, accumulation of LDL and HDL and the local activity of chemokines and cytokines. Further evidence of relation between atheroscleroris and

55 transport processes was provided by Sharifi and Niazmand (2015) analyzing the flow and the LDL concentration on Non-Newtonian effects in the internal carotid artery. The time-dependent incompressible along with a convection-diffusion equation for the LDL, using a given flux rate into the arterial wall, were solved numerically. The main results showed that there may be elevated risk depending on the bends of the artery. The risky sites strongly depend on the specific shape of internal carotid artery.

Modeling transport processes Over the years, the transport of lipoproteins was studied as these, and in particular HDL, LDL and VLDL were considered to counteract atherosclerosis (HDL) or promote it (LDL and VLDL). In relation to the disease, the transport of WBC and platelets was studied. Tripolino et al. (2017) studied lipid accumulation product (LAP), body adiposity index (BAI) and body shape index (ABSI) relation to obesity. Tripolino et al. (2017) show that LAP index is strongly associated to blood viscosity. The chemical and the physical mechanisms for this effect are not clear as yet. Sun et al. (2009) considered computationally steady and time-dependent flow, LDL and albumin transport in a human right coronary artery. The LDL concentration was assumed to be steady-state and its transport through the endothelium was modeled by Kedem–Katchalsky equation (i.e. the mass flux being proportional to the driving pressure). The study concluded that it is essential to account for pulsatility when modeling shear-dependent macromolecular transport. Moreover, LDL and albumin were found to accumulated at low WSS regions. The review paper of Tarbell (2003) focused on the mass transport processes leading to local accumulation of lipid in specific location in arteries. The review discussed the role of the blood flow itself, the role of the endothelial junctions and the transport across intercellular junctions and into the sub-endothelial intima and media. However, the details of the transport process are not well understood and hence the modeling of these process is still only in early stage. In analogy to natural transport, targeted drug delivery is a highly active research field. The idea is that by introducing at well-chosen positions, “particles” with appropriate physical properties, target drug delivery can be improved. Packing substances that cannot be dissolved in blood plasma (hydrophobic) into lipoproteins allows the drug to be transported by the blood to the target [e.g. Zhu and Xia (2017)], Lehninger Principles of Biochemistry and Chaudhary et al. (2019)). Lipoproteins are also associated with the formation of cancer [Chuadhary et al. (2019)] and references therein). Thus, the transport and distribution of lipoproteins in the circulatory system is of interest from different points of view. Sarhadi et al. (2019) reviewed the advantages of using HDL as drug delivery of antitumor drugs. HDL is small in size, has proper surface properties, along with long circulation time, without stimulating the immune system. The anti-tumor drug, commonly toxic, is not exposed to the body as long as it resides within the HDL. An earlier work of Liu et al. (2009) numerically simulated LDL transport in the aorta under steady-state flow conditions, in order to test the hypothesis that LDL concentration may be suppressed by helical aortic flow. The authors stated that helical flow reduced the LDL surface concentration in the aortic arch, speculating that this effect protects from atherogenesis. Liu et al. (2011) conducted a numerical study to assess the effects of blood rheology and flow pulsation on the distributions of luminal surface LDL concentration and oxygen flux along the wall of the human aorta. The results [Liu et al. (2009, 2011)] showed that under pulsatile flow conditions in certain disturbed regions, in contrast to steady-state cases, the level of near wall LDL was significantly reduced along with enhanced oxygen flux. The authors concluded that in the atherogenic-prone areas, luminal surface LDL concentration was high and oxygen flux is low. In a later publication, Liu et al. (2013) investigated modulating intracellular calcium levels and concentration of nucleotides on the luminal surface of the human thoracic aorta. The effects of rate of ATP flow pulsation were simulated numerically. The results show absence of ATP in the main

56 blood stream whereas relatively low nucleotides concentration was observed in disturbed flow regions on the aortic surface. Tracking substances and drugs in-vivo is possible, but requires resources and procedures that ethically may be very doubtful. Experimental data is a necessary component for developing models. Yeh and Eckstein (1994) used latex beads in a study measuring lateral transport of platelet-sized objects in flows of blood suspensions with WSS ranging from 250 to 1220 s-1. Stronger lateral transport occurred when the suspensions contained RBC and suspension with 40% hematocrit exhibited strongest lateral transport. No simple quantitative relation between shear-rate and lateral transport could be established. From a modelling perspective two types of options are available, namely, a continuum or a particle approach, as shortly discussed above. Blood components (in addition to RBC) may be considered as an additional phase and assumed as continuum. The discussion above shows that the underlying assumption for a fluid to be continuum is not met by any of blood components (except RBC in larger vessel as shown also in Fig 3.2). The continuum approach (using equations (3.5-3.6)) was used by e.g. Tandon et al. (2015) for accounting for the motion of RBC towards the center of the vessel. Melka et al. (2018) assumed also WBC to be a continuum, solving equations (3.5-3.6) with three components; the plasma, RBC and WBC, respectively. The more common approach is to use the mixture model described above and apply an additional convection-diffusion model (assuming continuum with specified or modeled diffusion) for the blood component under consideration. Hund and Antaki (2009) suggested an extended convection-diffusion model which include, in addition to common gradient diffusion (of platelets or WBC) and a second term containing the gradient the RBC, counteracting classical concentration driven diffusion. The model was evaluated for 2D channel and pipe flows. However, the success of the model require prescribing RBC concentration distribution. The extended diffusion model was based on several formulations, such as that of Keller (1977), Zydney and Colton (1988) and Sorensen et al. (1999a, 1999b). The results of the extended model were found to match experimental data. Lantz and Karlsson (2012) considered blood flow and LDL surface concentration changed during a cardiac cycle. The numerical simulations used LES and the relationship between WSS and LDL surface concentration was investigated. The boundary conditions on the LDL transport assumed a fully developed turbulent flow model. Also, the flux of LDL into the wall was assumed to be gradient driven with diffusivity equal to that of the blood. The main finding was regions of low WSS corresponded to regions of increased LDL concentration and vice versa. The instantaneous LDL values changed significantly during a cardiac cycle; during systole the surface concentration was low due to increased convective fluid transport, while in diastole there was an increased accumulation of LDL on the surface. Also, LDL accumulation was found to be sensitive to near-wall flow conditions. Iasiello et al. (2016) considered steady-state flow in a bifurcating artery. Four rheological models were considered and their effects on LDL transport was analyzed with a multi-layer model. The study considered the steady state near wall concentration and the influx of LDL into the arterial wall, using Staverman–Kedem–Katchalsky model combined with porous media equations. The non-Newtonian effect on LDL deposition was found to be negligible and the Newtonian assumption considered valid for medium and large arteries. At high Reynolds numbers a non-Newtonian fluid model was found to possibly have more impact due to the presence of recirculation. Mpairaktaris et al. (2017) also considered steady state flow and LDL transport in the thoracic aorta, using the Kedem-Katchalsky equation for handling the LDL transport into the thoracic wall. It was found that for hypertensive conditions, the LDL concentration at the endothelium/media was about 4.5 times lower than the corresponding luminal concentration. Low WSS was associated with high LDL near wall concentration as well as high near-wall LDL concentration. In a recent study, De Nisco et al. (2018) analyzed the impact of initial LDL concentration and inflow boundary conditions on the computed LDL blood-to-wall transfer in aorta using a convection- diffusion model. A constant diffusion coefficient (D=5.98310-12 m2 s-1), leading to a Peclet number 57 of 6.5108, was used. Boundary condition at the arterial wall was set to be proportional to the sum of water filtration velocity into the wall and (molecular) diffusion flux as within the domain a variant of the Kedem- Katchalsky model. The main conclusion of De Nisco et al. (2018) was the need of imposing realistic 3D velocity at the inlet and that different LDL concentration initial conditions lead to markedly different patterns of LDL transfer. The conclusion related to the inlet velocity was the similar to the conclusions of this thesis. A third option (in addition to the two continuum approaches; Euler-Euler and the convection- diffusion formulations) is using a “particle” approach for the dilute phase. The components of the dispersed phase are in the following termed as “particles”. These particles are transported in the blood (assumed to be a continuum) by the forces that fluid exerts on the particles. The convection (termed also as advection) term in the above mentioned models is accounted for applying a drag force that the particle is subject to. The drag vanishes when the particle follows completely the carrier fluid. Diffusion effect is the result of other forces that the fluid exerts on the particle. The forces on the particles act also on the fluid (with opposite sign). As in the convection-diffusion model, the later forces are neglected (so called one-way interaction). When the forces on a particle are known at any instance of time, it is possible to compute the particle motion by using Newton’s 2nd law. Thus, the particle is tracked in a Lagrangian framework (following the particle) and hence it is termed as Lagrangian Particle Tracking (LPT). As this approach was used in this thesis, further details were included in Section 4.3 (and Paper 6). Our simulations were aimed at proving insight into the mechanical transport process of blood cells and macromolecules to near wall regions, where the endothelium can be activated and allow maintaining the inflammatory process.

3.4 Boundary conditions for PDE The system of equations (3.8) requires boundary conditions (BCs). The character of the PDEs determine where and the number of conditions that must be specified to possible have a solution. Equation (3.8c) is a typical convection-diffusion equation of second degree. If the Db is constant, the symbol of equation (3.8c) with (3.10) is the Laplacian, which requires specifying one condition on all boundary points. The parabolic character (time-derivative and the Laplacian) imply also the initial conditions has to be specified. The fact the diffusivity is non-constant does not alters the required BCs. The system of PDEs (3.8a) and (3.8b) are of more complex character. For constant density, the steady state case (vanishing time-derivatives) is elliptic of degree twice the dimension of the problem. Thus, the system requires, for a three dimensional flow, three conditions on all boundary points. The time- dependent problem is termed as partially parabolic (since the continuity equation with constant density lacks the time-derivative). Keeping the density as variable would require an additional equation (equation of state) and a small time-step to be able to follow the acoustic waves (which have infinite speed in the incompressible limit). In addition to the BCs required by the PDEs, further conditions may be needed. For example, when the velocity vector is specified, the total mass flux must be such that the total mass in the system is conserved. In blood flow simulations, only parts of the circulation system are treated. Inlet conditions are often easier to determine as it is possible to measure (4D-MRI) the time-dependent inlet velocity profile. Often, the measured data is not a detailed spatial- and temporal-velocity distribution, but rather the flow-rate versus time. The details of the spatial distribution are important, since that determines possible presence of shear-layers that may act as amplifiers for perturbations. Additionally, non- uniform spatial distribution may initiate large scale structures, such as retrograde flow and helical motion and its rotation-direction. Inlet conditions and other BCs are discussed in Chapter 4. In the

58 following we discussed lumped (simplified) models for the governing equation, which under certain circumstances may be used as outlet BCs.

Lumped models Defining outlet BCs may be more difficult than on other boundaries since the flow in the segment under consideration also depends on the flow in downstream vessels/organs. The downstream vessel/organs act as reservoirs for blood (mass and energy) as it can store some of the flow energy and release it at some later time during the cardiac cycle. This is the Windkessel effect, which cannot be included into the basic PDEs (3.8) when using rigid walls. In certain situations, the flow close to the outlet develops such that the variations in the flow in the streamwise directions are slow as compared to the cross-sectional directions. Under such conditions it is possible to introduce a set of simplifications that may be used as approximations to the governing equation or alternatively act as outlet BCs. In the following, outlet BCs based on systematic simplifications that allow taking into account the physics of the flow are considered. If it is assumed that the section of the aorta in proximity to inlet boundary has no branches, the vessel curvature is mild and the flow in the streamwise direction has a much larger length scale relative to the length scales in the spanwise directions. As noted above, the momentum equations require BCs on all boundary points. It can be shown that if the flow direction is laminar and uni-directional (streamwise) the effects of the downstream data is limited to a distance proportional to the square root of the Reynolds number. This effect occurs when the length scale in the streamwise direction is considerably larger than in the spanwise directions. By using such a scale relation, the size of the different terms in the PDEs can be estimated. Two main types of approximations may be derived. One is based on neglecting the smallest terms and the second is based on maintaining the largest terms in the governing equations. Further assumptions on two set of approximations leads ultimately to a simpler 1D and 0D approximation. Some further details about the assumptions and derivation of simplified equations can be found in the appendix of Paper 2. Windkessel model (0D)

2 Simplest 0D models imply that the total pressure PtPt ( )( tQtA )( )/  is independent of x. It is assumed that, downstream the vessel, there exists a system to which a force is exerted on the boundary. Thus, the right-hand side of equations (A6 in Paper 2) would be non-zero; QQAA(/) dPQ2  KxxF  () (3.13) txdxA  BC where (x-xBC) is the Dirac delta function, and F is the force acting on the boundary point xBC. It can be argued that F should contain a pressure term and its temporal derivative, and no spatial derivative (to be 0D). This assumption may be clarified by considering the 1D equation. If Q(t) is expressed as Q(t) = q eit then the time-derivative of Q and Q itself (in 3.16) are out of phase relative to each other and can be balanced only at steady state. To balance the equation, F should be proportional to (p- p0)+ dp/dt where p0 and  are model parameters. This assumption allows for satisfying the simplified 1D equation for time-dependent flows with Q=Q(t). A more common derivation of some 0D models are so called Windkessel model, originally developed in analogy to passive electrical circuits. This approach gained many applications over the past 1-2 decades (cf. [Grinberg and Karniadakis (2008)], [Tsanas et al. (2009)], [Du et al. (2015)], [Pirola et al. (2017)], [Madhavan and Kemmerling (2018)], [Boccadifuocoa et al. (2018)]). The model based on passive component (resistor) of an electrical circuit it is possible to regulate the flow rate (for a

59 given driving pressure). By adding further components (capacitors and inductors) it is possible to regulate the phase difference between the flow (current) and pressure (voltage). This property may simulate the windkessel property of the vessels/organs downstream of the outlet boundary. Figure 3.4 depict four electrical circuits that were reported in the literature.

Figure 3.4: Four typical electrical analogues for outlet BCs for the PDEs (3.8)

Fig 3.4a shows the one component circuit. The flow (Q(t), current through a resistor, R) to a pressure loss (p(t), voltage difference)

ptR()() Qt R (3.14) The analogy of equation (3.14) to fluid is clear when considering Hagen-Poisseuille flow in a pipe of length L. The pressure drop=p/L is balanced by the viscous loss = 8Q/r4 where is the fluid viscosity of the fluid, Q is the volume flow-rate and r is the pipe radius. The “resistance” of the pipe “circuit” is then: R= Q/p = 8L/r4. A capacitor which accumulates charges may represent the blood accumulation (compliance) by the downstream organs/vessels. The increase in volume V=Cp. The flow rate (current) equals to the rate of change of the volume: dVdp QtC() (3.15) c dtdt Fig 3.4b depicts the addition of a capacitor in parallel to the resistance of the peripheral organs. The flow rate (current) is the sum of flow into the capacitor and through the resistor. p dp Q()()() t QRC t  Q t   C (3.16) Rp dt The two-element model allows a phase shift between the pressure and the flow. However, by specifying the phase shift the circuit is locked and the peripheral resistance must be specified independently. This may be done by adding another resistance (Rc) according to Fig 3.4c. The three component circuit (Fig 3.4c) leads to the pressure-flow relation (cf [Grinberg and Karniadakis (2008)], [Pirola et al. (2017)], [Madhavan and Kemmerling (2018)], [Youssefi et al. (2018)]): dp dQ p()() t R C  R  R Q  R R C (3.17) pdt c p p c dt

. By taking Fourier transform of (3.20): ~~~~ PRCiPRRQRRkpkkcpkpckk  CiQ()´ leading to:

~~ RRiRRCcpkpc  PQkk (3.18) ´11iRCiRCkpkp The parameters of the 3-component model can be expressed by the following three parameters, and. 111 R  ;;1 c (3.19)  CRcpccp RRCRR  An important difference between the two component (RC) and the three component (RCR) circuits is that the high-frequency waves (e.g., any numerical noise) in flow rate are not sufficiently filtered by the RCR circuit for any values of Rc and Rp. Clearly, when Rc << Rp the RCR and RC circuits alike, except for the response for very large values of k. For not small values of Rc noise added to Q(t) is transferred to p(t) which may lead also to numerical problems. Stergiopulos et al. (1996) found that the three-element windkessel (WK-3), did not lead to correct estimates of the arterial parameters such as the aortic characteristic impedance and total arterial compliance. The authors suggested therefore instead of using WK-3, a version of a four-component winkessel (WK-4) was used,. The circuit equation of the four-component model (Fig 3.4d) can be expressed by: dpdQd Q 2 pR CRR QRR()() CLLC (3.20) pcppcdtdtdt 2 As noted, the presence of the inductance allows further flexibility, regarding of shifting the phase lag between pressure and flow. Segers et al. (2008) assessed a WK-3 and two variants of WK-4 models. These three models were fitted to data measured non-invasively in 2404 healthy subjects. The results showed that the WK-4 performed best for a group of individuals with low blood pressure and wave reflection. For this group the WK-3 and the other WK-4 models led to increased overestimation of total arterial compliance and underestimation of characteristic impedance. In the majority of the cases the different models were very similar to each other. The main conclusion of the paper was that “the debate about which lumped-parameter model is the better approximation of the arterial tree is therefore still not fully resolved”. In summary, the role of the Windkessel models is to determine the phase angle between the pressure and the flow-rate. This function, the phase angle, must be determined by measurement, a major drawback since it is an ad-hoc parameter that may vary from patient to patient and from time to time. However, once it is known it can be incorporated directly in the 1D, 0.5D or the 0D models.

Chapter 4: Simulation methods and data analysis

The methodologies presented in Chapter 3 were used in this thesis. Specific problem set-ups, including boundary conditions and methodologies for data analysis are presented in the following. Chapter 3 describes some options for simulating blood flows. In this chapter the details of the methods that were used in our simulations are described. Additionally, further details about the methods used to extract the results that were presented in the enclosed papers and in Chapter 5. To facilitate the reading of the thesis, relevant equations are given not only in terms of equation numbers (from Chapter 3) but are copied over while retaining their numbering. The blood is assumed to be a mixture satisfying the conservation of mass and momentum (equations 3.8a and 3.8b). The mixture viscosity was modeled by one of the four models; namely; i. Newtonian fluid (=3.5 10-3 Pa s); ii. Walburn-Schneck model (equation (3.2)); iii. Casson’s model (equations (3.3)); iv. Quemada model (equations (3.4)). The three non-Newtonian viscosity model (ii-iv) use the local strain-rate () and the RBC-volume fraction (). The transport equation for  is equation (3.8c) with one of the models for the diffusion flux for  (equations (3.9) - (3.12)). In Paper 5, however, the hematocrit was set to a constant, with = 0.45. The mixture density is computed through equation (3.7): the mixture density is RBCp , where RBC and p are the density of the RBC and the plasma, respectively. Chapter 3 contains a general discussion about BCs for PDEs and those that were commonly used in the literature for blood flows. In the following we specify the boundary conditions that were employed in the simulations considered in this thesis. Equations (3.8) require three values to be given at all boundary points. Three types of boundary points exist in the cases under consideration: i. Arterial walls; ii. Inlet; iii. Outlets.

4.1 Segmentation and computational domain The thesis considers blood flow simulations in the thoracic and abdominal aorta, with emphasis on the flow into the and within the proximal parts of the renal arteries. CTA images were used to define the aortic geometries. Siemens Somatom Flash Scanners, having a limit of 0.3 mm isotropic voxels, was used to gather the CTA data with slice thicknesses of 1 mm and 0.75 mm. The reason for the difference is advances in imaging software improving quality and noise reduction for the image rendering carried out in 2015 and 2017, respectively. For image reconstruction Siemens SAFIRE I30 (iterative reconstruction algorithm) was applied. Iodine contrast used for the CT angiography showed a concentration in abdominal aorta of about 300-500 HU. The process of generating the three-dimensional shape of the arteries is termed segmentation. The shape of a sclerotic artery has randomly distributed irregularities making it difficult to distinguish these from noise in the CTA data with currently available resolution. Here, the software Mialab by Wang and Smedby (2010) was applied to perform the segmentations, utilizing centerline and threshold based techniques. The effect of segmentation approach on arteriosclerotic indicators was investigated by Berg et al. (2019). It was shown that smoothing of the images was needed in order to remove over estimation of surface irregularities. Such surface irregularities may lead to local separation regions with unsteady WSS. As the WSS is the basis of atherosclerotic indicators, overestimating surface irregularities may lead to overestimating the risk for atherosclerotic lesions. On the other hand, extensive smoothing of the arterial wall may lead to underestimation of these risks. In the simulations we used three aortic segments. Figure 4.1 shows the geometry of the complete aorta, including the major arteries branching form the aortic arch and the abdominal aorta. Implementing outflow boundary on the many branches is an issue as discussed below and therefore

62 we carried out simulations separately for the thoracic aorta with its branches and the abdominal aorta and its branches. The reconstruction of the geometry was made using two different methods. Either an edge detection technique or for one of the renal arteries and the aortic sinus, or when the reconstruction was not meaningful due to the character of the CTA data (and the stenosis location) the reconstructions were created manually using Blender (freeware). No attempt was made to reshape the artery into a circular cross-section.

Figure 4.1: The geometry of the whole aorta, including the braches included in the different simulations. Note, the straight extensions added (about 5 diameters in length) to the inlet and the outlets. These extensions were added to reduce the ambiguity in imposing BCs.

The aortic geometry use in this study was derived from a CTA data of a healthy patient. Segmentation of the CTA Dicom data was carried out using a combination of centerline based and threshold methods with manual adjustments afterwards to segment as much of the vessel lumen as possible. The walls of geometry were smoothed. Three aortic geometries were derived: one of the entire thoracic and abdominal aorta down to the iliac bifurcation and two composing of solely the thoracic aorta. Only the main arterial branches of the aorta were maintained; brachiocephalic, left common carotid, left subclavian, celiac trunk, superior mesenteric as well as the right and left renal arteries. The two geometries including the thoracic aorta, both had the same shape and branches as the complete aorta but differed in the inlet segment. One variant was identical to the entire aorta, including an extension segment at the inlet, with the shape of the sinotubular junction extended as a straight tube of five cross sectional diameters. The second thoracic aorta included the aortic sinus without extension. The different geometries are shown in the result section (Figs. 5.1, 5.11 and 5.18, respectively). All arterial branches were extended further (at least five mean diameters) by straight tubes with the shape corresponding to the CTA derived shape of each branch. The thoracic aorta outlet plane positioned at level of the diaphragm was left without extension in order to study potential difficulties associated with different boundary conditions at that outlet The domain of interest must be discretized into small elements. The shape of the elements can be in principle of any multidimensional polygonal. The size of these elements has to be small enough such that the smallest length-scale of interest is resolved by several intervals/cells (the more the better). The cell nodes are organized in an unstructured mode. The advantage of this approach is that it applicable for arbitrary geometries. The disadvantage is that need for rather extensive “book- keeping”, implying larger and slower computer codes. In the simulation presented in the thesis a mix of tetrahedral and hexagonal elements were used. The grid was generated using STAR CCM+ software after importing the (smoothed) geometry as an stl file. For each of the geometries, detailed in Chapter 5, a sequence of grids was generated. For each of the cases a grid with adequate resolution was chosen. The largest number of simulations were carried out for the thoracic aorta (two different 63 geometries) for which a grid sequence composed of about 1, 2.5, 5, 14 and 20 million computational nodes were included. In most simulations the 5 million grid was used, though for low HR also the results on the 2.5 million grid were accurate enough for the WSS-related parameters and flow structures (helicity and retrograde flow).

4.2 Applied boundary conditions

Arterial walls The BC for equations (3.8a) and (3.8b) was no slip (i.e. the velocity vector was set to 0). The discretization, using collocated grids, and the solution algorithm (solving a Poisson equation) require also BC on the pressure, for which the pressure gradient was set to 0. Further conditions were needed for the transport equation of RBC volume fraction (3.8c). Different cases used different conditions. In some cases,  was set to a constant, =0.45 or =0.25. The latter value was used to simulated the so called “cell free layer” (CFL). In some other cases the -flux into the wall (i.e. the normal derivative of ) was set to 0 which is commonly used as no RBC passes through the arterial walls. For the Lagrangian transport of cells and different “particles” (described below), slip-rebound or zero flux were employed. The specific type of BC is given explicitly for the different cases along with the results in Chapter 5.

Inlet The velocity vector has to be specified all over the inlet surface. 4D-MRI data can provide the velocity vector into the artery segment of interest, though with limited resolution. The temporal resolution is limited to below 40 Hz, implying a rather poor resolution and higher HR where turbulence scales may occur. The spatial resolution is also limited with about 20-50 pixels across the artery (with a resolution of about 0.5 mm). The corresponding resolution in numerical simulations is often much finer as refinements are used in regions with large gradients. Computations use temporal resolution below 1 ms (often 10-50 times smaller than MRI) and spatial resolution at least a factor of 2-3 when coarser to medium girds are used. Ultrasound, in-vivo measurements may provide volume flow data with better temporal resolution, but with limited spatial resolution. Thus, inlet boundary conditions often use some interpolation or additional assumption applied to the measured data. Due to the lack of adequate spatial resolution, several papers examined using different inlet spatial distributions (e.g. [Madhavan and Youssefi et al. (2018)], [Morbiducci et al. (2013)]) with a given time-dependent influx. Commonly used spatial distributions include a flat profile (“plug flow”), parabolic flow profile (valid for slow stationary flow in pipes) or Womersley type profiles (valid for laminar time-dependent flow in a circular pipe). Both Youssefi and Morbiducci found significant differences in hemodynamic results when different inlet conditions were used. However, it was also found that single-component (plane average) inflow profiles were rather close to the results obtained by the use of three-component inflow profiles. The importance of temporal variation of the inlet profile was not examined in detail in the thesis. Fig 4.2 depict inlet flow rate profiles from the literature. In our simulations the measured flow-rate was either assumed to have a top-hat (so called “plug-flow”) profile or in a couple of cases a 3D distribution was used to simulate stenosis, assuming a smoothed plug-flow. The smoothed profile was defined by a distance function from the wall such that |was the distance from the wall (located at . The axial flow profile was multiples by a function f = U0(1-tanh((|)|/))/2, where regulates the thickness of the shear-layer and U0 is the centerline velocity. Of course, the integral of f over the inlet must equal the specified CO. The published data was digitized and smoothed (e.g. Fig 4.2a).

In this thesis further modifications of the measured inlet profile were done in order to explore the effect of the deceleration phase on the different WSS-related and large scale structure parameters. Different heart-rate (HR) and flow-rates (cardiac output, CO) were derived using the measurements presented by Munch et al. (2011), Rodeheffer et al. (1984) and Gemignani et al. (2008). The two former references measured the HR and CO under different loading conditions for three age groups. For a given CO, HR was larger the younger the group. The end-diastolic volume decreased with CO for the younger group, maintained the volume in the intermediate group and mildly increased the end diastolic volume in the elderly group, showing mild HF. The stroke volume increased with increasing CO until about 13 LPM for the younger group where after it was unchanged as CO increased to over 18 LPM. For the elderly group the stroke volume increased slowly even for the large range of CO. Gemignani et al. (2008) measured the timing and timing ratio between systole and diastole in 103 patients. For low HR (60 BPM) the length of diastole was about twice as long as systole. This ratio decreased to almost one already at about 120 BPM. At higher HR both timing decrease at the same rate, and at HR = 150 BPM, both systole and diastole have a duration of about 200ms. We used the timing of systole/diastole for some of the cases presented in Paper 4 and the corresponding results in Chapter 5, while the simulations in the other papers used the scaling of Munch et al. (2011). If a constant (often =0.45) is set at the inlet and the boundaries, the solution of convection- diffusion equation (3.8c) is a constant value for  (=0.45) as no sources are present in that equation. In order to have a nontrivial solution,  should be of non-constant value at least by the inlet. The inlet condition that was applied in such cases was as for the inflow with stenosis; namely = (1- tanh((|)/))/2, where here regulates the thickness of the shear-layer and 0 is the centerline RBC volume fraction with the integral of  over the inlet being equal to the hematocrit. | is the distance from the wall. The inlet was also used to release the WBC, platelets, HDL, VLDL, Chyolomicros, VWF and RBC. These “particles” were released at a given rate in each time step and then followed in the fluid. The number of each of these particles leaving the different vessels at a given instant of time was monitored.

a b

c d

e f

g h

i j

k l m Figure 4.2: Inlet flow rate profiles derived from published data that were used in the simulation. a) Benim et al. (2011); b) Munoz et al. (2013); c) Bertelsen et al. (2016); d) van Wyk et al. (2015) e) Frydrychowicz et al. (2009); f) Hope-Liu (2007); g) Analytic (Taylor et al. (2002); h) Bertelsen et al. (2016). i) Rengier et al. (2012), j) Youssefi et al. (2017). k) and l) Frydrychowicz et al. (2009) with shorter diastole; m) Frydrychowicz et al. (2009) with quick deceleration phase;

Outlet Outflow BCs is a major issue in simulating blood flow in general and so called patient-specific simulations. The reasons for the difficulty lies in the fact that the flow at the outlet boundary depends on condition (“resistance” and “compliance”) downstream of the outlet boundary. The model equations on the other hand, when assuming incompressibility, do not allow upstream propagation of information unless there is backflow. The only exception is upstream propagation of information by viscosity. This upstream propagation is limited in distance, which is inversely proportional to the square root of the Reynolds number. Numerical BCs (for the Poisson equation for the pressure) may allow specifying the pressure gradient. This data is, however, unknown in general and not for patient- specific simulations in particular. As mentioned above, this is circumvented by adding a WK model, which add 2-4 “free” parameters allowing the “tuning” of the flow-pressure relation. Hence, there are three classes of options (used in this thesis) for the outlet BCs: I. Outflow rates based in the inlet flow-rate, see Table 4.1. II. Pressure (fixed at one outlet) and pressure gradient at the others (along the lines of reduced models in Section 3.3. The simplest version implies setting p  0 , where is the streamwise  direction). III. Windkessel type models based on analogy of the distal circulatory system with electrical circuit as describe in Section 3.3.

Measurements of the time-dependent flow-rate at different segments of the aorta and its branches providing data that can be applied as BC by assuming the spatial profile (e.g. plug flow or a parabolic profile) have been carried out. However, it should be noted that such data is valid for the given patient and for the specific time when the measurements were performed. Benim et al. (2011) and Madhavan and Kemmerling (2018) both presented reference values for flow-rate distribution in the aortic branches, reproduced in Table 4.1. For a given flow-rate, the outflow profile is commonly assumed to vary weakly over the cross section (i.e. plug-like profile).

Table 4.1: Flow-rate at aortic branches as percentage of the inlet flow-rate. Benim et al. (2011) assign also 5% to the flow in the coronary arteries. In our simulation we assign 70% to the flow out of the thoracic aorta. The distribution of the flow due to exercise showing the increase in flow into the arteries supplying blood to the arms, and corresponding reduction into the abdominal aorta. The results with the distribution used by Liu et al. (2011) and Pirola et al. (2017) is not presented in this thesis and is shown in the table only as reference. Brachiocephalic Left Carotid Left subclavian Outlet Thoracic aorta Benim et al. (2011) 15% 7.5% 7.5% 70% Madhavan and Kemmling (2018) 19.3% 5.2% 6.4% 69.1%

Fernandez 17% 8% 10% 65% Fernandez-exercise 25% 5% 11% 59%

Liu et al. (2011) 5% 5% 5% 85% Pirola et al. (2017) 11.6% 4.7% 3.6% 80.1%

Often, the inlet and outlet sections are extended in order to allow for the use of general (generic) velocity profiles and eliminating the risk for retrograde flow at an outlet boundary. Under such conditions it may be justified to impose simpler type of BC, such as assumed spatial distribution of the velocity vector with given flow-rate. Simplified BC are justifiable when the length scale (L) in the streamwise direction is much larger than in the cross-plane (l). When l/L<<1, allowing different simplifications to the system of equations (3.8) as discussed in Section 3.3. A common approach for outlet BC that have gained support in recent years is based in patient-specific adaption of 2 or 3 element electric circuit analogues (Vignon-Clementel et al. (2010)). The 3-element electrical-circuit analog uses an outlet resistor (representing the peripheral arterial resistance) in serial with parallel resistor/capacitor element (corresponding to the vessel compliance and distal resistance). The pressure and flow-rate are related (Madhavan and Kemmerling (2018)) through the electrical- circuit analog (equation 3.17). The three model coefficients (Rc, Rp and C) are determined by fitting the model parameters to available data. Paper 2 describes the details of the results. These results show that not all the Windkessel parameters that were suggested in the literature satisfy the 0-dimesnional model assumption. These details are discussed further in Paper 2.

Flow solver The discretization of the governing equations was done using available schemes in OpenFoam (of formally second order schemes). The governing equations represent conservation of mass and momentum in any control volume. Therefore, the same conservation laws must also be satisfied on each computational cells. The flux (defined the quantity under consideration multiplied with the velocity vector projected on the normal to the surface element) is computed by linear interpolation. The formal order of accuracy of this scheme is two. The temporal integration employed also an implicit (second order) scheme along with a PISO-type scheme available within OpenFoam. The models of Walburn-Schneck, Casson and Quemada as well as the diffusivity models of Zydney and Colton, Leighton-Acrivos were added to the OpenFoam package.

4.3 Transport of blood components As discussed in Section 3.3 the most common approach for modelling transport of blood components is in analogy to that of the transport of RBCs. The used, one-way interaction implies, that the blood flow affects the transport of the species but not vice versa. As the volume fraction of the different species that are considered in the following (WBC, platelets, HDL, VLDL, Chylomicrons and VWF) is low, the mean distance among the particles is much larger than the particle diameter (with local exceptions) the underlying assumption for continuum are not valid. Hence, the particles are tracked individually in a Lagrangian framework; so called Lagrangian Particle Tracking, LPT. When local particle concentration increases, LPT may be still useful, though the inter-article interaction must be accounted for. Similar measure has to be added, however, all multi-phase models. The basic LPT is simple in principle which is a major advantage. A freely moving particle (assumed to be spherical in the following) with a mass mp, and velocity up and that is placed in a flow field with local velocity u is subject to forces. It is rather obvious that water and a solution based on water satisfy the requirement for being considered as a continuum. On the other hand, non-solvent components of blood, with the exception of RBC, have very low concentration and if each component is considered as an own phase, the condition for continuum will not be met. Therefore, it is natural to consider the different non-solvent components of blood as individual particles. The interaction between particles and the carrier fluid takes place in terms of forces that the fluid exerts on the particles implying that the particle exerts the same force on the fluid. Additionally, the particles may interact with each other directly through collisions, or indirectly by perturbing the flow condition around the particle. In that case neighboring particles are affected of the modified flow field. Depending on the strength of the contribution of different interactions simplifying assumption can be made. Situation where the interaction between the fluid and particles and between the particles is taken into account is called as “four-way interaction”. When the inter-particle interaction is neglected, the interaction if called a “two-way interaction”. When the particles are small in size and in number they hardly affect the flow, leaving only the flow to affect the particle motion. This situation is termed as “one-way interaction”. When the particles are small the particles follow the flow (almost) as a fluid “particle”. The condition for this situation is that the particle can adjust (almost) instantly to the local flow conditions (particle velocity equals local fluid velocity). This conditions can be estimated by assuming that the small particle is subject to a uniform flow and only to drag. The particle Reynolds number can be defined by Rep=d U/, with d, U and  are the particle diameter, the difference between fluid and particle velocity and kinematic viscosity of the fluid, respectively. When Rep <<1 the (Stokes) drag on the particle is FD=3  d U , where  is the viscosity of the fluid (). The fluid time may be estimated to be F=d/U, and the particle acceleration time (to adjust to the local flow conditions) is p can be

69 found from Newton’s second law; FD=(U-0)/p, implying that p=1/(3  d U ). The particle will follow the fluid when the Stokes number, St=p/F, is small; i.e. St<<1.

When St is not small, the drag force becomes important and the particle does not follow the fluid anymore. Moreover, the particle may be subject to additional forces. If the particle size and the gradient in the flow in the particle neighborhood are of the same order, the particle experiences a lift force. The forces are often decomposed into a drag force parallel to the streamwise direction and a lift force normal to it (may lead to some ambiguity). For small particles in a steady, slow flow field the drag is the so called Stokes drag. For larger particle Reynolds numbers (Rep=(u-up)Dp/, with and Dp being the kinematic viscosity of the fluid and particle diameter, respectively). Stokes drag, FS (=3(up- u)Dp) is valid for small particle Reynolds number. Often the drag is expressed in terms of drag 2 coefficient, Cd (Fd/(0.5 V A), where Fd and A are the drag force and object cross-sectional area, respectively. V=|u-up| are the fluid density, and the velocity of the particle relative to that of the fluid, respectively. Experimental data showed (Fig 4.3) that the drag (or as expressed in Fig 4.3 as the drag coefficient, Cd) is reduced with increasing Rep. Extension of the Stokes drag for small but finite Rep the drag is given by a Stokes component along with a Reynolds number correction: 241 Cd  (4.1a) Re p 3 1Re p 16 Experimental data (as depicted also in Fig 4.3, the drag coefficient reduced until regime 3 where it is flat (i.e. independent of flow speed). At even larger speeds the drag drops suddenly (regime 5) where after it increases again. The sudden drop in Cd is due to the on-set of turbulence which reduces the long wake of the laminar flow. At even larger speeds the drag increases again as the losses increase with increasing turbulence. For a sphere with rough surface (e.g. golf ball) the drag is lower also at 6 lower speeds. Morrison (2013) fitted an expression to the experimental data (up to Rep<10 ): 7.94 Rep   Re p   Re p  2.6  0.411 56  0.25   24 5   2.6310   10  Cd  1.52  8  (4.1b) Re Re Re Re p p pp    1 11  5  6 5   2.6310  10 The first term also in this relation is the Stokes drag coefficient. Commonly, in blood flow the particle Reynolds number is small and often equation (4.1a) is adequate. The error in assuming the Stokes drag only may be estimated by determining the contribution to the Stokes drag, namely; the correction term, 3/16 Rep. The Stokes drag is viscosity dependent. Viscosity may also cause a lift force acting normal to the drag force (and the particle velocity). Lift force may act on a spherical particle when the particle is subject to shear. The shear lift originates from the inertia effects in the viscous flow around the particle and is fundamentally different from aerodynamic (pressure difference) lift force. The expression for shear lift was first obtained by Saffman (1965, 1968). Saffman’s expression was derived with and assumption of a small Reynolds number Re << 1. Additionally, a Reynolds number based on the 2 shear-rate was defined: Res = |∇u| D /μ. The assumption is also Res << 1. However, the ratio of the 1/2 two Reynolds numbers has to be non-small; i.e. =(Res) /Re >> 1. Assuming that |∇u| is scaled by 1/2. -1/2 the shear-layer thickness, , than, =(/(u)) or alternatively = (Res) which is satisfied.

1/2 du du F1.615 D2 ( u u ) sgn( ) (4.2a) Spdy dy

The expression in generalized form is

2 ()uux p  FDwithxuS 1.615   (4.2b) 

Extension of the basic expression may be found in Mei (1992). Extension of Stokes drag and Saffman lift force expressions to non-spherical particles may be found in Zastawny et al. (2012).

Figure 4.3: Drag coefficient of a sphere as a function of Reynolds number. The blue line corresponds to equation (4.1a). Original image title: Drag coefficient of a sphere as a function of Reynolds number. Link to original image: https://commons.wikimedia.org/wiki/File:Drag_coefficient_of_a_sphere_as_a_function_of_Reynolds_numb er.png – 2020-07-13. The author to whom credit is given for publishing the image under a creative commons license: Bernard de Go Mars (https://commons.wikimedia.org/wiki/User:Bernard_de_Go_Mars - 2020-07-13). License: The image was reused in its original form under the Creative Commons Attribution-Share Alike 3.0 Unported license (https://creativecommons.org/licenses/by-sa/3.0/deed.en).

Non-viscous dependent forces may also act on a particle moving in a fluid. These may be induced by particle rotation (Magnus effect) or by gravity (in the case of density difference between that of the particle and the fluid). Particle in a shear flow may induces particle rotation whereby a lift (Magnus) force acts on the particle. The Magnus force on a spinning sphere in contrast to the Saffman lift and the Stokes drag is not dependent on the viscosity of the fluid. The Magnus lift force is proportional to the vector product of the (angular) rotation speed () and the flow speed (U); i.e. FMagnus=A xU, with A being a constant depending on the flow regime (cf Bluemink et al. (2009)). Another non- viscous lift force may be related centrifugal effect. The force is proportional to the density difference between that of the particle and that of the fluid. Such a force may occur not only in a rotating system but also when the particle is subject to centrifugal force due to highly curved particle path. Instead of shear-induced lift, the simulations showed that close to branches, the curvature of particle paths could be smaller than the size of the branches. This may be obvious since the blood has to turn also from outside of near wall layer, in order to maintain the required flow rate. High path-curvature may lead to a lift force if there is a non-negligible density difference between the blood and the particle. From the particle trajectory its curvature () can be computed. The lift-force (Fc) due to centrifugal effect is computed by:

u2 FVt () (4.3) cpbp 

Where, Vp is the particle volume,  is the path curvature, ut is the particle velocity, p and b are the density of the particle and blood, respectively. The direction of the force depends on the density difference between the particle and the blood. Lighter particles will be attracted towards the inner part of the curved trajectory whereas the heavier particles experience a force in the opposite direction. In this thesis the particles are assumed to be spherical and that these particles are small as compared to the length scale of the flow. These assumptions lead to neglecting the rotational motion and shear- induced force (depending on the square of particle size). Legendre and Magnaudet (1997) obtained a theoretical expression for the lift force that, at small Reynolds numbers, improved on the one obtained by Saffman (1965). Later,ation Legendre and Magnaudet (1998) applied numerical computations to explain lift force behavior, also deriving an expression to calculate the lift force for a range of relevant Reynolds numbers. The ratio of drag to lift, for Rep of about up to about 20, is about 10:1 for a range of dimensionless shear-rate (S, defined dividing the shear-rate by u/D) for S<0.5. For larger Re and up to Re=103 the ratio of drag to lift (scaled by 1/S) increases in proportion to Re (in logarithmic scales). In terms of the shear-layer thickness, S=D/. Thus, for the lower Re range the Lift/Drag force ratio is 0.1 D/. As it is assumed that the particles are small and are much smaller than the thickness of the shear layer, it would be inconsistent with the size assumption to use a lift-force in the LPT simulations herein. Once the forces are known, the particle path can be integrated through: du dx mFFFp  and p  u (4.4) pdS dt dt p where mp is the mass of the particle. When the particle is not spherical, in addition to the forces, the particle may be subject to a torque T. The torque may be computed by integrating the tangential force to the surface of the particle over the surface. The torque may be used to compute the angular acceleration and angle (in analogy to equations (4.4). The discussion above was relevant to Newtonian fluid. When it comes to non- Newtonian fluids, additional force and in particular a torque may act on the sphere. One of the few papers dealing with the forces acting on a sphere in shear flow of a non-Newtonian fluid, was reported recently by Gavrilov et al. (2018). As noted, the LPT of non-interacting particle is in principle rather simple. It is important keeping in mind the underlying assumptions about the particles (small size as compared to flow scales, low particle number density such that the total particle volume locally is small as compared to computational cells), particles do not interact with each other and that particles are non-deforming. A numerical drawback of LPT is the need to keep track of each individual particle, its velocity and the local blood velocity. Thus, the book-keeping and overhead computations of the LPT approach becomes large with increasing number of particles. In our simulations we used up to 2 million particles, mostly only a few hundreds of thousands. When it comes RBC, the loading if so high (normally 45% of the whole blood volume) that the inter- RBC distance is at most the order of the RBC size (a few m). Thus, the inter-RBC interaction is strong and a four-way interaction is needed if the Eulerian-Lagrangian approach is to be employed. However, due to the high number density, no simple expressions exist to describe the forces acting on an object of the shape of RBC. On the other hand, the short inter-RBC distance motivates the use of Euler-Euler approach. However, the lack of information on the interphase interaction set a limit also to the Euler-Euler modeling.

4.4 Analysis of the results The computed flow field was used to assess flow-structures such as helical and retrograde flow structure and WSS-related parameters used as atherosclerosis risk indicators. Moreover, the vorticity transport equation can be used to assess the dynamics of the flow. The computed flow field was also used to compute the transport of cells and blood components in the thoracic aorta and into its branches. The flowing text includes definition of several parameters that are used for evaluating and comparing results from different simulations. The parameters were defined also in some of the papers and these are given here for the same of completeness.

Helical motion and helicity The data was analyzed with respect to the presence of helical flow and its relation to retrograde flow and to WSS. The helicity (h) is defined as the scalar product of the local velocity vector (u) with the vorticity vector (), i.e. h=u∙ The helicity data was also reduced [cf De Nisco et al. (2018)] by averaging in space in order to assess the temporal development of helicity relative to the cardiac cycle. The space- and time-average of the helicity and its absolute value (hs and ha, respectively) were defined (Paper 4) as:

1 T 1 T huddt() huddt() (4.5a) sii  aii  T 0 T 0 Where T and  are the integration time and aortic volume, respectively. An indicative index for directionality of rotation was suggested [cf De Nisco et al. (2018)] by defining normalized parameters. Here, we introduce a parameter to assess temporal oscillation in helicity, i.e. the Helicity Oscillatory Index, denoted in the following by hosi. This parameter (equation (4.5b) is analogous to the Oscillatory Shear Index (OSI) commonly used to characterize the oscillatory character of the wall shear stress.

hhss1  hr and h osi 1  (4.5b) hhaa2 

For flows with unidirectional helical motion hs = ha implying that hr=1 and hosi=0. For highly oscillating helicity hs=0, and hence hr=0 and hosi=0.5.

Retrograde flow Non-stationary retrograde flow leads to non-stationary negative WSS (in the sense defined below). The WSS in plane of the arterial wall can be defined as the projection of the WSS tensor into the plane tangent to the wall at a given point. If n is the normal vector to the wall at a given point, the projected WSS tensor (i.e. n∙WSS) is a vector in plane tangent to the wall. For convenience this vector is denoted in the following, also as WSS. Two parameters related to retrograde flow are defined (Paper 4): - Relative Volumetric Retrograde Flow (RVRF), defined as the volume of the aortic lumen where the local velocity component (u) projected on the tangent to the centerline (T) is negative, relative to the total aortic volume (V). The tangential velocity component is defined as utan = u∙T. The parameter RVRF is defined as the volumetric integral of negative utan normalized by the total aortic volume: 73

HudV()  e 00u  RVRFtHu( );()V tan (4.6a) e 10u   dV tan V

- Relative Negative WSS (RNWSS): First, wall points where WSS is negative (WSSneg) are defined: WSSneg= T∙n∙WSS < 0. RNWSS is defined on the aortic wall surface (S) by: HWSSdS()  e 00WSS  RNWSS( tHWSS );()S neg (4.6b) e 10WSS   dS neg S

WSS related indictors As the WSS varies in space and time, it become customary to use WSS related parameters to characterize the WSS in different cases. The most common parameters are as follows:

Time Averaged Wall Shear Stress (TAWSS) (cf. [Suo et al. (2008)] and [Chen et al. (2016)]): 1 T TAWSSWSSdt (4.7)  i T 0 TAWSS is a local value time averaged value of WSS. Spatial variation of WSS give an idea about the level of WSS and its spatial non-uniformity of WSS (i.e. WSS-gradient), but no information about WSS temporal variation. Oscillatory Shear Index (OSI) (cf [He et al. (1996)], [Chen et al. (2016)]), on the other hand, measures high-frequency variations of WSS. OSI is defined as T ||WSSdt 1  i OSI 1 0 (4.8) 2 T WSSdt  i 0 OSI varies between 0 and 0.5. When WSSi is positive (oscillatory or non-oscillatory) OSI = 0. When WSSi changes signs such that the integral of the positive and negative sequences are equal, OSI gets the value 0.5. Values of 0 < OSI < 0.5 indicate an oscillatory WSS field, but the indicator is not able to indicate the amplitude nor frequency of the oscillations. The time during which extreme WSS (high or low) exists appears to be essential for the development of atherosclerosis. Low TAWSS (< 0.4 Pa) and significant OSI (> 0.15) are risk factors for incidence and progression of atherosclerosis ([Zarins et al. (1983)], [LaDisa et al. (2011)]).

Residence time of relevant chemical compounds and cells at a given location is essential for the development of pathologies or treating these. By combining OSI and WSS and diffusion of momentum (i.e. viscosity) one may obtain a time scale termed as Relative Residence Time (RRT) which is given by: 1 RRT  (4.9) 1 T WSS (1 2) OSIdt  i T 0  For OSI=0 the time scale is proportional to the viscosity and inversely proportional to TAWSS (for constant viscosity). A different definition of RRT is also common (cf [Rikhtegar et al. (2012)], [Gallo et al. (2012)]) by eliminating the viscosity in equation (6), thereby RRT is dimensionless. In the following we do use RRT as defined by equation (6).

In an earlier paper [Fuchs et al. (2019)] we proposed to use an approach to model activation of endothelial cells, in analogy to activation of platelets as was proposed by Nobili et al. (2008). In the platelet activation model, the target variable was defined as the Platelet Activation State (PAS). PAS was defined as the fraction of platelets that were activated relative to all platelets in the tested blood. Thus, PAS is a number varying between 0 and 1. The model has three empirically determined parameters; a, b and c (c.f. [Nobili et al. (2008)]). Here, the target variable is denoted Endothelial Activation Index (EAI), using the same numerical values for the model parameters as suggested by the original model for platelets. Nobili’s model is a power law based expression, assuming that PAS is a function of the scalar shear-stress () and time (t). The stress is a symmetric tensor with components ij and the scalar stress () is defined as the Frobenius norm of the tensor. d PASd D caDb/1/ aab a  ; (4.10) dtdt The model parameters used here were: a=1.3198, b=0.6256, and c=10-5. The numerical values were determined by measurements using the Hemodynamic Shearing Device (HSD), cf. [Xenos et al. (2010)]. Soares et al. (2013) proposed a model that also includes explicitly the effects of the temporal variations of the stress. In its simplest form, with constant coefficients, the basic form of activation akin chemical reactions is exponential, leading to asymptotically to a final PAS value: d PAS KPAS(1) (4.11) dt 0 However, K0 is a function of PAS itself and the stress is calculated along the platelet path. In other words, K0 is a function of time and the local conditions at the endothelium. For a constant K0, PAS vanishes at time zero but increases exponentially and approaches asymptotically a final value of PAS=1 with time. Here, K0 is assumed to be composed of three components: S, F and G, all functions of time and the stress along the platelet path, such that: d PAS ()SFGPAS (1) (4.12a) dt The three terms were defined by:

SPASHSPAS(,)( )( ) trt  tHt (4.12b)

1 1 F(,)()() PAS C  PAS t   t  (4.12c)

1 1   G(,)( PASCPAS )( )  ttr (4.12d)

−7 −7 −4 Where Sr =1.5701 10 ; C=1.4854 × 10 ; =1.4854; =1.4401; Cr=1.3889 10 ; =0.5720, = t  0.5125. Ht is the (time) accumulated stress. Htsds()()  ;  was defined by Soares et al. (2013) t 0 as the average of the absolute value of time-derivative of the stress. We propose instead, to use a low pass filtered value (i.e. below 30Hz) of the scalar stress.

Flow dynamics-vorticity generation and growth In order to analyze the results in terms of formation of vortical structures, the vorticity transport equation was considered, obtained by taking the curl of equation (1a). The transport equation for the vorticity vector (i) is given by:

2 iiii u u j (1/)p u jjiijk  2 (4.13) txxxxxx jjjjkj  With the assumptions that the fluid mixture is incompressible with constant density, the 2nd and 3rd terms on the right hand side vanish identically. The main driving term for the generation of vorticity is the 1st term on the right-hand side of the equation (i.e. the so-called vortex stretching (VS) term), whereas the last term is viscosity related that commonly has a smaller effect compared to the vortex stretching term. Equation (4.13) is used for analysis in Paper 3.

Chapter 5: Results

In this chapter, the main results and findings during the course of the thesis work are presented. In the sections below we discuss the simulated results for Atherosclerotic Renal Artery Stenosis (ARAS), the whole aorta and in particular pulsatile blood flow under different conditions in the thoracic aorta.

5.1 Renal artery stenosis – a short background Renal artery stenosis (RAS) is a “quiet” disease, having no clinical manifestation over longer periods of time. It can be detected by coincidence when blood pressure is measured and severely elevated. Abdominal CT is often used as part of RAS investigation. The risk for complication from RAS is not only risk for bleeding due to hypertension but also elevated risk for renal failure. For this study, three patients who had underwent Computed Tomography Angiography (CTA) were selected (shown in Fig 5.1): 1. Two patients with a left sided ARAS (at common locations for the stenosis). 2. A previously healthy patient undergoing an CTA study of the aorta, where no pathological changes were found (except minor atherosclerotic changes that could be considered as a “normal” finding considering the patient’s age). The patients having ARAS underwent conventional angiography and treatment with dilatation and/or stenting of the stenosis. The CTA of the two patients with RAS (Figs 5.1b and 5.1c) was used to “smooth-out” the stenosis. The “reconstruction” of the artery included in addition to the stenosis itself also the post-stenotic dilatation. The original CTAs and the reconstructed arteries were segmented. The case in Fig 5.1c required manual segmentation due to the proximity of the stenosis to the artery bifurcation which caused also deformation of parts of the junction with the abdominal aorta. The quality of the segmentation and its smoothness was assessed by computing the statistics of surface curvature. The computational geometries were used to generate a sequence of grids. Grid sensitivity study was carried so as to reduce the impact of computational resolution on the important parameters under consideration.

a b c Figure 5.1: Three abdominal aortas with the branches that were considered in this study (derived from CTA). Case 5.1a shows non-stenotic arteries, 5.1b shows a stenosis of the left renal artery. Fig 5.1 c shown multiple stenosis both renal arteries and also at the branches of the celiac trunk and the superior mesentery artery.

Figure 5.2: The shape of the left renal arteries in two RAS patients (corresponding to Fig 5.1b and 5.1c, upper and lower rows, respectively). The left frames show the original shapes, the mid-frames show a reconstructed artery and the right-most a smoother surface shape. The upper mid-frame shows the presence of not-reconstructed post-stenotic dilatation. Such a shape leads to flow acceleration and enhanced WSS is not characteristic for a healthy artery.

The simulations were based on the methodology and BC describe in Chapter 4. The inlet profiles were either the ones shown in Fig 4.2d or 4.2j. The flow rate into the abdominal aorta was taken to 70% of the CO and the different branches carried the following flow-rates; 23% to the renal arteries (each), 22%, 22% 33% used for the celiac trunk, superior mesenteric artery and the aortic outlet. An overview of the flow in the artic segment is depicted in Fig 5.3 for four time instances in the cardiac cycle. The stream paths are colored by the absolute value of the local velocity. The figures show the lack of clear helical motion in the aortic section except towards late diastole.

a b

c d Figure 5.3: Stream-paths colored by the magnitude of the velocity in the abdominal aortic segment containing the renal arteries. The left most frame (a) is close to peak systole. Each frame corresponds to an increment of 20% in the cardiac cycle. Thus, frames

Paper 1 of the thesis shows the effect of grid refinement and the two “reconstruction” approaches. Here, we expand on the relation between the surface curvature and WSS and related atherosclerotic indicators. Fig 5.4 depict the flow in some cross-sections before and after (left) renal artery reconstruction. The stenosis generates a strong jet with a large unsteady, separated flow region (post-stenotic dilatation).

a b

c d

e f Figure 5.4: The flow in the original (top row) and reconstructed (middle and low rows) left renal artery. Note the strong jet due to the stenosis (a) and the formation of Dean vortex downstream. The reconstructed artery shows a regular flow with non-uniformity due to the mild curvature of the artery. The stronger curvature seen in frame e leads a helical cross-sectional flow as shown in frame f.

The reconstructed artery shows a smoother flow field leading to different WSS as reflected in the WSS-related parameters (shown in Figs 9 and 10 in Paper 1). In the following, the surface curvature related to the WSS parameters and in particular the endothelial activation model are considered. Fig 5.5 depict a case of stenoted renal artery (Case 2.1 in Paper 1). The surface curvature is as smaller than 0.5 mm. The enodthelial activation model shows strong activation detects the strong WSS that is generated by the stenosis. The corresponding reconstructed-smoothed case is depicted in Fig 5.5c. As noted the activation model shows an equally strong activation, although the wall is much smoother. The reconstructed (before smoothing) case maintains the post-stenotic dilatation region. Fig 5.6b depict the strong reaction of the endothelial model to the non-regularity (smoothness) of the arterial wall. Fir comparison, Fig 5.6c depicts that endothelial activation model for the normal renal artery. The results show that the WSS-based models are sensitive (i.e. they do indicate WSS irregularities) but not specific enough to be used as risk indicator for future atherosclerosis.

a b c Figure 5.5: The geometry of a stenosed artery (Case 2 in Paper 1). The curvature of the arterial wall (a) shows sub mm radius of curvature (at the stenosis it is about 510-4 m). The endothelia activation model applied to the stenotic artery (b) and the reconstructed-smoothed (c).

a b c Figure 5.6: A reconstructed stenoted artery (Case 1.2). The surface curvatures (a, left) and the corresponding reconstructed artery marked with the endothelial activation model (b). The response of the endothelial activation model for the normal left renal artery.

A more detailed information may be obtained by considering the inner part of the abdominal aorta near the junction with the left renal artery. Fig 5.7 (normal aorta and renal arteries) depict in four time instances, the WSS components parallel to the aortic wall colored by the absolute value of the WSS. The flow near the ostium and into the left renal artery changes significantly during the cardiac cycle; from a clockwise rotation to a counter-clockwise direction with stronger WSS during the deceleration phase. Two other flow structures are also be observed. Fig 5.7b shows a stagnation at the wall downstream of the ostium. This stagnation point persists for part of the cycle but disappears in the presence of retrograde flow. Another flow structure may be observed in Fig 5.7c where a clockwise rotating vortex is observed downstream of the ostium. Both structures underpin the strong unsteadiness of the flow and the WSS. Corresponding flow visualization in a stenoted renal artery case is depicted in Fig 5.8. The inflow into the renal arteries is helical in both cases, though weaker in the pathological case. In the latter case there is a surface irregularity at the location of the stagnation region in the normal aorta case. This irregularity has a major impact on the flow and in particular on the local WSS (Figs 5.9 and 5.10).

a b c d Figure 5.7: Near wall flow (WSS parallel to the arterial wall (normal aorta) for 4 time-instances in the cardiac cycle.

a b c d Figure 5.8: Section of the abdominal aorta around the ostium of the left renal artery. The projection of WSS tensor parallel to the arterial wall shows the flow direction close to the wall in 4 time instances of the cardiac cycle. L

Figs 5.9 and 5.10 compare six parameters of the so called normal aorta with that of Case 1 (stenoted left renal artery). The surface curvature of the latter case is by roughly an order of magnitude larger (i.e. more rough) and in particular neat the ostium to the renal artery and the presence of the structure noted also in Fig 5.8. The WSS gradient is more regular and lower at the ostium on the case of normal aorta. As a consequence the endothelial activation model shows a false negative behavior around the renal artery ostium. Similar behavior was observed for the other three WSS-related parameters (TAWSS, RRT and OSI).

a b c

d e f Figure 5.9: Normal aorta: Surface curvature (a); WSS (magnitude, b); Enodthelial activation model (d); RRT (d); TAWSS (e); OSI (f)

a b c

d e f

Figure 5.10: Stenoted left renal artery: Surface curvature (a); WSS (magnitude, b); Endothelial activation model (d); RRT (d); TAWSS (e); OSI (f)

5.2 Whole aorta simulations One of the outcome of the renal artery simulation was the need for realistic and robust but relevant BCs. In the literature one may find a considerable number of flow-rate profiles measured at the aortic root. Hence, it was natural to simulate the flow in the whole aorta, including the same arteries as included in the abdominal aorta simulations. The final grid chosen for the simulations included 7.2 million cells. The basic geometry (Fig 5.11a) was derived from CTA data (Chapter 4). The inlet and outlet sections were extended by a straight pipe of the same diameter is the section itself and a length of at least five diameters. Fig 5.11b depicts the corresponding computational domain with the extensions at the inlet and outlet.

Figure 5.11: The segmented aorta (a, left frame) and the computational domain with the extensions added to the inlet and outlets (b, right frame).

The flow in the whole aorta was simulated using different inlet conditions and specified flow conditions on the outlets, with the exception for the renal arteries where some options were tested (as discussed below). The reason for these simulation was that it enabled direct comparison between the limited simulation in the abdominal aorta and the renal arteries with those in the whole aorta. In the following only some of the results are presented. In particular the impact of the shape of the whole aorta on the WSS parameters, helicity, retrograde flow and the flow in the segment which corresponds 84 to the inlet section to the abdominal aorta (presented in Section 5.1 above) and outlet section of the thoracic aorta (presented in Section 5.3, below). The inlet flow-rate profiles into the whole aorta are depicted in Fig. 5.12. In all cases the flow rate was 5LPM at 60BPM.

Figure 5.12: Inlet flow rate profile for the whole aorta. The left frame depicts three profile that were used for the whole aorta simulations. These profiles can be compared to those used for the thoracic aorta simulations in Fig 4.2. The flow-rate in all three cases was 5LPM at 60BPM.

To get an overview picture of the flow in the whole aorta, a sequence of images of the helicity and vorticity are depicted in Figs 5.13 and 5.14, respectively. The vorticity is large throughout the cycle in the abdominal vessels. The vorticity increases during systole with start close to the walls due to the formation of a stronger and stronger boundary layer during the anterograde flow phase. The flow exhibits a clockwise helical motion. Towards the end of systole (d) and beginning of diastole (f) strong vorticity is formed in the ascending aorta and the aortic arch, due to the retrograde flow. The vorticity weakens and the descending aorta lateral wall upstream to the abdominal branching vessels. Weak vorticity was found downstream of the abdominal branching vessels throughout diastole. The strong vorticity around the branching (and within the proximal parts of the vessels themselves) is maintained by the strong curvature of the fluid. The situation here is similar to the observations made for the renal artery flow (Figs 5.7-5.9).

a b c d

e f g h Figure 5.13: Stream-paths colored by vorticity. Frames a-h correspond to 5-85% of the cardiac cycle. (Profile3)

The helicity (defined as the scalar product of the velocity and the vorticity vectors contains information about the alignment of these two vectors and the three-dimensional character of the flow. Helicity may change sign when the flow changes direction or when the vorticity changes direction. Such effects may be observed locally during the cardiac cycle. The helicity is weak at early systole, becoming strong positive in the ascending aorta and the aortic arch throughout systole. The sign of helicity is negative near the inner curvature of the ascending aorta and close to the junction of the inlet cylindrical extension. It is also negative in the branches from the aortic arch, the suprarenal part and the infra-renal segment of the abdominal aorta. During early diastole the clockwise helical motion is observed clearly (5.14e). The helical motion gets weaker during diastole (5.14e-5.14h). The helical motion changes direction (5.14f) in the proximal part of the descending aorta and restores back to the original but much weaker helical motion (5.14g and 5.14h). The helicity has mixed sign in the abdominal aorta around the branching of the abdominal vessels as the flow is dominated by the branches rather than having a unidirectional motion.

The flow in the renal artery cases (Fig 5.3) is significantly different from that shown in Figs (5.13) and (5.14). These results demonstrate the short coming of using simplified BCs. The lack of temporal- spatial data at the boundaries on segments of an artery, making such “patient-specific” simulations of limited value for decision making for that particular patients. Yet, the results shown above are useful for understanding the effect of surface details (stenotic patient) not explored in the whole aorta simulations (healthy individual).

Next, we consider the impact of the flow on the WSS-related indicators; namely OSI, RRT, TAWSS, EAI-N and in some cases also EAI-S.

a b c d

e f g h Figure 5.14: Stream-paths colored by helicity. Frames a-h correspond to 5-85% of the cardiac cycle. (Profile3, Fig 5.12)

For the discussion below about the WSS-rated indicators we focus on the cases of Profiles 1 and 2 in Fig 5.12. Fig 5.15 depict the spatial distribution of OSI (Figs 5.15a-5.15c); RRT ((Figs 5.15d-5.15f); TAWSS (Fig 5.15g-5.15i) and EAI-Nobili (Figs 5.15j-5.15k). The OSI indicator, as its name indicates is sensitive to temporal fluctuations (value close to 0.5). Larger temporal fluctuations can be noted at the inner wall of the ascending aorta, near the branching of vessels in the aortic arch and the abdominal vessels. In particular, OSI close to 0.5 is located just downstream of the renal arteries. This behavior can be explained by the formation of unsteady flow structured as explained in relation to Fig 5.7.

a b c

d e f

g h i

j h k

l m n

Figure 5.15: Spatial distribution of the WSS-indicators; OSI (a-c), RRT (d-f); TAWSS (g-i); EAI- Nobili (j-k) and EAI-Soares (l-n) using inlet profile is Profile1 (Fig 5.12)

a b c d

e f g h

i j k l

m n o p Figure 5.16: WSS-indicators using another inlet profile (denoted as “Profile2” in Fig 5.12). This particular inlet flow-rate profile has stronger amplitude variations during the cycle as compared to the other profiles. OSI (frames a-d); RRT (frames e-h); TAWSS (frames (i-l); EAI-Nobili (m-p).

Instead of graphical assessment of the surface distribution of the WSS-indicators, the use of statistical data of the spatial distribution of these indicators, is explored. Fig 5.17 depict three simulations with different inlet profiles comparing the statistics of the five WSS-indicators (OSI RRT, TAWSS, EAI- Nobili and EAI-Soares). Obviously, there are graphical differences between the different cases. The OSI distribution of the first two inlets has higher frequency counts are found for smaller OSI values. The counts for larger OSI is largest for inlet case (“Profile3”) indicating stronger (temporal) fluctuations. In general, the OSI values are small, indicating presence of “hot spots” rather than general “smooth” distribution of OSI. The behavior of RRT as depicted in Fig 5.15 and 5.16 is similar 90 to OSI for the two cases, respectively. On the other hand, the RRT distribution seem to have two modes; one following OSI with larger frequency at low RRT values and a second with a wider RRT distribution with peak at 0.008, 0.008 and 0.012, respectively. The latter case has also a longer “tail” implying that there is non-negligible probability for the presence of larger RRT values as compared to the two former cases. TAWSS distribution has a peak at about 0.5 Pa but the “tail” being shorter in the case “Profile3”.

The EAI models were suggested by us and as such less tested and evaluated in the literature. The activation models exhibit somewhat different behavior as compared to RRT or TAWSS. The peak of the EAI-Nobili occurs for lower activation values along with a shorter “tail” as compared to the EAI- Soares. The difference is due to the contribution of the (low-pass filtered) WSS temporal derivative. It should be emphasized that the frequency (i.e. probability) of the largest EAI values are of interest and not the distribution at lower values. On the other hand, such information may be indicative to the effect of the stress-transients on endothelial activation.

OSI

RRT

TAWSS

EAI-N

EAI-S

Profile1 Profile2 Profile3

Figure 5.17: Frequency distribution of OSI, RRT, TAWSS, EAI-Nobili and EAI-Soares, for three inlet flow rates (Profile1-Profile3, depicted in Fig 5.12).

5.3 Thoracic aorta

The thoracic aorta has been used in many simulations as it combines a well studied geometry along with large range of flow rates across the aortic valve (i.e. large number of patient data, though mostly given in terms of flow-rates). Patient data shows that the shape of the aorta is like many other organs, are literarily patient specific. Thus, if general understanding of the aortic hemodynamics is sought a particular specific case is not of prime interest. The basic geometry (Fig 5.11a) was used with two modifications at the inlet. The first version (V1) had the same extension as used in the whole aorta simulations (Fig 5.11b). In the second version (V2) the inlet extension was replaced by the aortic sinus section. Both versions were discretized using different number of cells, which was used to determine the required resolution for the large number of simulations. The sequence of grids contained from 1 million cells and up to 20 million cells. For the thoracic aorta mostly 5 million cells were used, with a resolution close to the 7.2 million cells used for the whole aorta. The finer grids are mostly useful for cases with high flow- and heart-rates when the flow may be turbulent. Fig 5.18 depict the two geometries under consideration. The branches from the aorta with their extensions are the same as for the whole aorta cases discussed in Section 5.2.

Figure 5.18: The computational geometries of the thoracic aorta. The arrows show a cross plane used to show the secondary flow (Fig 5.21). The pressure range was generated automatically and shows the pressure span for the two cases.

A large number of simulations were carried out for combinations of different: a. Geometries V1 and V2 (Paper 2) b. Inlet flow-rate profiles (nine in Paper 2) c. Outlet BCs: fixed flow-rate, or windkessel (WK) models (Paper 2) d. A range of heart-rates (60BPM – 150BPM) and low-rates (5LPM and up 18LPM) (Paper 4) e. Effects of rheological models (four models for mixture viscosity; Newtonian, Quemada, Wallburn-Scneck and Casson model) without (Paper 5) or with combination with RBC flux transport models (such as power-law, Zydney-Colton and Leighton-Acrivos). (Paper 6) The results were summarized in terms of WSS-indicators: TAWSS, OSI and RRT, equations (4.7) - (4.9), respectively. For most cases, we added the Endothelial Activation Indicator (equations (4.1) - (4.12)), based on Nobili et al. (2008) or Soares et al. (2013) platelet activation models. The behavior of the in-plane WSS and its spatial and temporal variation were analyzed. As volume parameters characterizing the time-dependent flow, the helicity and the extent of retrograde flow (equations (4.5) and (4.6). In the following we summarize shortly the results which are presented in Papers 2-6 and point out some of the main findings. The impact of the added inlet segment is shown in Figs 5.19-5.21. WSS-indicators show rough qualitative similarity (Fig 5.19). However closer inspection shows that these commonly used parameters do depend on the details of the inlet. The spatial probability distribution of these parameters are depicted in Fig (5.20). As noted, the distributions of probabilities clearly differ. The general trend of OSI being larger in the case with the aortic sinus, reflects the fact that the aortic sinus leads to the formation of separated, unsteady flow. The effect of inlet shape is not limited to the proximal region. The distributions of the other WSS-indicators exhibit a tendency to larger values in the presence of the aortic sinus. The extension pipe allows for slow development of the flow which is also expressed in longer RRT in the extended section (Fig 5.19).

OSI

RRT

TAWSS

Figure 5.19: WSS-indicators (OSI, RRT and TAWSS) with the aortic extension at the inlet (left frames) and with aortic sinus (right frames).

Figure 5.20: Probability (spatial) distribution of the WSS-indicators. The wider and the larger values of the indicators for the (right frames) case with the aortic sinus, indicate the stronger, unsteady recirculating flow.

The impact of the extension is not limited to the proximal region. Also in a downstream plane (depicted in Fig 5.18) the in-plane flow differs, as depicted in Fig 5.21. The order of the frames is related to the cardiac cycle. In both cases, helical motion can be noted. The two cases differ in terms of location and strength of streamwise vortices. However, it is not possible to relate in a simple way the differences in the results to the inlet shape.

15%

35%

55%

75%

95%

V1 V2 Figure 5.21: Instantaneous secondary flow in a cross-plane (marked by arrow in Fig 5.18). The two columns correspond to geometry V1 (with extension at the inlet, left frames) and V2 (with the carotid sinus shape, right frames). The time-sequence (from top to bottom) are for 15%, 35%, 55%, 75% and 95% of the cardiac cycles. The arrows are colored by the strength of the vorticity.

Effects of heart-rate and flow-rate In the following, the WSS indicators for two heart- and flow rates are compared. Fig 5.22 depict the three common WSS indicators for 150BPM/15LPM and 120BPM/10LPM cases, respectively. A general observation is that the RRT is larger while TAWSS is smaller for the lower HR/CO case. This effect is seen more clearly in Fig 5.23 where the peak probability and the distribution are depicted. The wider distribution of OSI (larger probability for larger OSI values) indicates that the lower heart- and flow rate case (120BPM/10LPM) has more temporal oscillation. The larger OSI values leads also to larger RRT (by definition (equation (4.9)). The definition of TAWSS sums-up the magnitude of the WSS and therefore TAWSS is larger for the high flow rate case.

OSI

RRT

TAWSS

Figure 5.22: The distribution of WSS indicators for two cases: 150BPM/15LPM (left frames) and 120BPM/10LPM (right frames). Note the different scales for the three variables.

Fig 5.23 also depict the probability distribution of the endothelial activation indicators using the Nobili and the Soares models (EAI-N and EAI-S, respectively). EAI-N has lower peak but wider distribution for the high flow-rate case. On the other hand, EAI-S shows similar behavior for the

98 lower flow-rate. This clear difference demonstrates the role of the time-derivative of the stress, included in the EAI-S but not in EAI-N. The (low-pass) filtering introduced in the EAI-S model filters out high frequencies to which cells cannot respond, but maintain low-frequencies that were eliminated from the EAI-N model. A lower heart- and flow-rate (90BPM/9LPM) is depicted on Fig 5.24. The right frames correspond to the same rheological model used in the cases depicted in Fig 5.23. The trend that was observed in Figs 5.22 and 5.23 are maintained with decreasing flow-rate, implying larger RRT and smaller TAWSS for lower flow-rates. The OSI distribution has a tendency to be larger probability for small values and both EAI-N and EAI-S decrease with lower flow-rate. For the lower flow rate the difference between both EAI is smaller as the contribution from temporal WSS variations decrease at lower flow-rate.

Figure 5.23: The WSS indicator probability distribution for the same cases as in Fig 5.22; 150BPM/15LPM (left column) and 120BPM/10LPM (right column).

Effects of rheological models

The effects of viscosity models are described in more details in Paper 5. Here we depict only the shape of the distributions of the WSS-indicators for the Newtonian (constant viscosity of 3.3 10-6 Pa s) and for the Quemada mixture model (equations 3.4) complimented with Leighton-Acrivos RBC transport model (equations 3.12). The effect of blood rheology is observed in the distribution of the probability of the spatial distribution of the five WSS indicators. The OSI parameter has more even distribution with the Quemada mixture model indicating the damping effect of the rheological models. The distribution of the local viscosity shows that at regions of low shear-rate the blood viscosity can be larger by a substantial factor (in some cases up to an order of magnitude). However, the dissipation effect of viscosity depends not only on the viscosity but also on the shear-rate. As the viscosity increases at low-shear-rate region viscous dissipation remains low. Thus, there is a canceling effect in terms of viscous dissipation as viscosity and shear-rate counteract each other under normal rest conditions. At high shear-rate the viscosity decreases (and approached a constant value similar to the one used in the Newtonian case simulations, for shear-rates of larger than 103 s-1, Fig 3.1) hence viscous dissipation increases. In terms of WSS indicators the impact of the different rheological models is small. The Newtonian model is adequate unless the local instantaneous effects are important, as for example in the case of tracking small particles in which the forces acting on the particle depend both on local viscosity and local density.

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Figure 5.24: Comparison of the WSS indicator probability distribution

Particle and cell transport The distribution of cells and lipoproteins/macromolecules (RBC, WBC, platelets, three lipoproteins and vWF. The lipoproteins (HDL, VLDL, Chylomicrons) were studied, using methods described in Section 4.3. In the following, particle tracking from two source area discussed shortly: In the first case particles are released at the inlet and in the second the particles were released at a plane in the proximal part of the descending aorta. The particles are assumed to be solid and of spherical shape with physical properties (size and density) as given in Table 5.1

Table 5.1: Particle sizes (m) and densities (g/l) RBC WBC Platelet HDL VLDL Chylomicrons vWF Diameter (m) 8 20 2 0.02 0.1 2 1 Density (g/L) 1125 1080 1075 1200 950 900 1102

The data related lipoproteins may be found in Chaudhary et al. (2019) and Upadhyay (2018). Cell densities follow data by Norouzi et al. (2017). In addition to the effects of rheological models on the wall parameters and larger flow structures, also the impact of blood rheology on the transport of the species listed in Table 5.1 was considered. Fig 5.25 depict three instantaneous picture of the distribution of two “particles” (Chylomicrons and WBC) in the aorta. The three frames are for different time instances in the cardiac cycle. The last frame is displayed since it shows an instant when the injection of particles has stopped. The example shows qualitatively that there are differences in the distribution of the particles. The two particles differ in size and density. As seen in the right most frame (Fig 5.25c), the Chylomicrons are captured in the recirculation region of the aortic sinus over longer time as compared to the WBC. The other two frames show a qualitative picture of situations in the cardiac cycle where one may observe accumulation of and dilution of the two particles at different location at the two time-instances that are depicted in Figs 5.25a and 5.25b.

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a b c Figure 5.25: Instantaneous distribution of Chylomicrons (a) and platelets (b), both with the same size of 2 m but differ in density, during three time instances during the cardiac cycles. The right frame (c) corresponds to a case where no further particles are injected after a certain time, leading to successive particle dilution.

a b c Figure 5.26: Instantaneous distribution of the seven “particles” at different time instances during the cardiac cycle. Note the variations in concentrations in the different frames.

When all the seven particles are tracked, the instantaneous picture becomes less clear, as depicted in Fig 5.26. Nevertheless, one may observe the non-homogenous particle distribution in each of the frames, with shift in concentration along the aorta and at different times. In order to quantify the differences among the transport and location of the different particles some averaging steps has to be taken. The results in Paper 6 show the accumulated portion of particles leaving each of the exits from the aorta, namely the BCA, LCCA, LSCA and the inlet to the abdominal aorta (i.e. outlet from the thoracic aorta).

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A hypothesis put forward (cf [Kronzon and Tonic (2006)] and [Wehrum et al. (2015)]) was that transport of thrombi from the descending aorta into arteries leading to the brain, might occur. In order to assess the hypothesis through our simulations, particles were released in a plane about 3 cm downstream of the branching of the left subclavian artery (LSCA). Several different cases were considered; instantaneous injections during systole and at the end of systole. Spherical particles (5500 in number) of different size (8 m to 0.5 mm) and density (between 950 to 1250 g/l) were released at different parts of the cardiac cycle. In one of the simulation, the released particles had the same density and size as RBC (8 m in diameter) and had an initially zero velocity. These particles were transported with the fluid over 3 cardiac cycles. Fig 5.27 depict the particle paths colored by particle age (residence time after release) injected at different phases of the cardiac cycle. As noted, when injected at early systole, the particles remain in the descending aorta. When injected at peak retrograde flow, the particles were transport upstream in the aorta. During the simulation period some of the particles (about 6%) ended up in the subclavian artery (Fig. 5.27, mid-frame). The results correlate well with the findings in the literature (e.g. [Kronzon and Tonic (2006)] and [Wehrum et al. (2015)]) reporting the correlation between stroke and the presence of atherosclerotic plaque (> 4mm) in the descending aorta. Patient with strong retrograde flow had higher risk for developing cerebral embolism. When a large particle (0.5 mm) is injected at the same location at end systole (150BPM/15LPM) particle are transported not only into the left subclavian artery but also into the left common carotid artery.

Figure 5.27: Transport of particles (colored by particle age) from a plane (marked as a white line in the mid-frame). The left frame shows the injection of spherical particles of 8 m in diameter and density of 1125 g/l at the beginning of systole whereas the mid-frame depicts the same case when the injection takes place at peak retrograde flow phase. The right frame depicts the injection of larger particles (0.5 mm in diameter) at end systole at high heart- and flow-rate (150BPM/15LPM).

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Chapter 6: Summary and conclusion

The starting point of the thesis work was the hypothesis that by simulating blood flow in healthy arteries, it would be possible to predict if and where an atherosclerotic lesion would develop. An even more ambitious aim was to predict its progress of development. A surrogate question associated with the hypothesis was to evaluate if the available models and tools are adequate for handling the issues related to the aims. In fact, the thesis ended-up being focused on determining the appropriateness of existing models and numerical tools. By being appropriate we infer that the outcome of the results can be interpreted in terms of the hypothesis (i.e. predicting future development of atherosclerotic lesions). In retrospective, it was found that the original task was too ambitious for mainly two reasons: i. Complex pathophysiology ii. Expressing patient-specific data in terms of a computational model The first item includes the complexity of the physiological and pathological processes which are known only partially and hardly understood. The atherosclerotic process involves several interacting systems each with own, “independent” feedback loop. The term independent reflects a previously common understanding that the inflammatory, immune and coagulation system react each independently. In recent years it become evident that the division lines between the systems is artificial and one system may activate and interact with others. A pathological specimen of an atherosclerotic lesion may include immune cells (e.g. macrophages), inflammation of the arterial wall layers near the lesion and sometimes presence of blood clots and/or bleeding. Progress of the pathological process may lead to a stenosis which at some point may give clinical symptoms and lead to the detection of the lesion. In addition to hypertension, RAS may lead ultimately to kidney failure and the need for hemodialysis and/or transplant. Therefore, early detection of RAS can avoid the progress degradation of the kidneys and eliminate/reduce the risks associated with hypertension. The pathophysiology of RAS leading to hypertension is due to the blood pressure regulation of the kidney (through re-update of water and salts in the distal tubuli and through the renal-angiotensin system which regulate arterial wall tension and thereby blood pressure). The biochemical details on formation of the arterial pathology is largely unknown and therefore it is not obvious either which parameters should be used as early warning parameters for such a process. As discussed in the introduction, it has been generally accepted that WSS plays a role in the process. Different WSS-related expressions were proposed (cf equation (4.8)—(4.12)). As is shown also in this thesis, there is a correlation between the location of stenosis and the indicators. However, this correlation seems to be non-exclusive. The main short coming of the common WSS-related parameters is due to the fact that these are often averaged in different ways. A more important factor is that the atherosclerotic process is very slow (under normal conditions). Possibly, simulations over periods of seconds may be completely inadequate as they do not capture the very slow process and small changes that develop. In spite of these limitations, it is believed that the knowledge gained by the simulations are essential as it yields a puzzle bit that can and should be added to other puzzle bits; namely radiological data (MRI, CT and US) and not forgetting the clinical story (including chemical-lab data) and its temporal development. The second item (of expressing patient-specific data in terms of a computational model) is also a complex story. Each of the steps taken from the patient CT data to the post-processing of the numerical simulation data, is associated with assumptions that are directed by some (often intuitive) background knowledge whereby some assumptions and approximation are adopted and at the same

105 time certain uncertainties are introduced. This combination makes any of the simulations as a point in the multi-dimensional parameter space. The patient parameters (operating conditions) vary continuously in the parameter space. Hence, it should be clear that a single or few simulations (points in the parameter hyperspace) do not provide a complete picture of a specific patient. Yet, the few simulations may provide understanding on the relative importance of the different processes whereby indicating in which direction future research should go. The results of the work are presented in the papers with more detailed background and additional information given in Chapters 1-5. The outcome of the research work may be summarized in terms of contributions. The main achievements of the research in this thesis are as follows:  The vision of developing a screening tool for predicting early the risks for developing atherosclerosis in general and renal artery stenosis in particular, was shown to be difficult to achieve with the current state of the art knowledge and modelling ability. Traditional (averaged) WSS related parameters, such as TAWSS, OSI and RRT are not sensitive nor specific enough to for the screening purpose (Papers 1 and 4). The reason is due lack of adequate knowledge of the underlying mechanism for atherosclerosis and how these mechanism manifest at early stages. In addition to the issue of clinical relevant metrics, the thesis demonstrated the shortcomings in terms of modeling (e.g. segmentation, BCs, rheological models) contribute to the difficulty in attaining the original goal. The seemingly negative results is in fact instructive as it indicates that averaged quantities as those commonly used to assess the effects of WSS have to extended to include temporal variations, as was done in the Endothelial Activation Index (EAI-S).  The shortcomings in defining patient-specific “profiles” is associated with large uncertainties making the prediction unreliable in terms of an individual patient. On the other hand, the prediction can be adequate in terms of risk factors (i.e. statistically probable causes for potential pathology). A complete patient-specific “profile” implies given patient-specific geometry and BCs. Unfortunately, no complete patient-specific data exist and hence uncertainties in data become essential. The effects of uncertainty due to segmentation and boundary conditions were addressed in Papers 1 and 2.  The underlying mechanisms for generation helical motion, retrograde flow and the possible formation of turbulence was exposed. Turbulence was identified through the presence of non- harmonic fluctuations generated by vortex stretching (Paper 3).  The flow in thoracic aorta is laminar or transitional in healthy individuals at rest and moderate exercise. In the presence of aortic valve stenosis, turbulence (identified by having a frequency range in the spectrum that contain non-harmonic, non-tonic fluctuations) may develop during systole. Turbulence is dissipated during diastole, unless the heart-rate is high and the fluctuation dissipation rate not strong enough to eliminate the higher frequency components associated with turbulence. (Paper 3).  Normal variability in heart-rate, flow-rate and inflow profiles were studied through different simulations whereby the effect of each parameter on helical and retrograde flow structures was identified. Similarly, the impact of negative and oscillatory WSS on wall parameters was determined (Paper 4). The results indicate also that it is inadequate to draw conclusions about aortic hemodynamics from a small number of simulations.  Endothelial cell activation models were introduced, utilizing models developed for platelet activation. The models account for temporal variations in the WSS and were modified so as to smooth out highly oscillatory components to which cells cannot respond. Therefore, all

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temporal variations with frequencies above 30Hz (i.e. < 33 ms) were filtered out by a low- pass filter (Paper 4).  Three non-Newtonian models (Casson, Walburn-Schneck and Quemada) and a Newtonian model for blood viscosity were assessed and compared in terms of several parameters. The Casson and Quemada models behave similarly and close to the Newtonian case with sufficiently large shear-rate (order of 100 s-1 or more). At low shear-rates, the non-Newtonian models differ more strongly form each other and from Newtonian behavior. Paper 5 concludes that the choice of rheological model should be determined by the flow under consideration and in particular the parameters and variables that are the target of the study. For the thoracic aorta, the total effect of the different rheological models was less than changing the flow rate by about 10%.  In the framework of continuum of transport models for cells and other blood components, three models were evaluated. The contribution of substance concentration gradient, density gradient and viscosity gradient was assessed, showing that the commonly using concentration gradient has least contribution when applied to RBC transport (Paper 6). It is also shown that the contribution of RBC concentration and shear-rate to local viscosity may be significant, but the effect on the flow is not important for inertia dominated flows (Papers 5 and 6).  LPT simulations, with one-way interaction, was used compute the RBC distribution to the different branches of the thoracic aorta, showing good agreement with the continuum model in terms of mean concentration. The LPT simulation yield additional information in terms of RBC residence time and path lengths, showing large variation of these two parameters (Paper 6). The results indicate that the dynamics of particle motion can be essential and relying bon mean concentration may be misleading (for example when certain locations are targeted by highly potent drugs).  LPT was also used to analyze the distribution of leukocytes, platelets, HDL, VLDL, chylomicrons and von Willebrand factor (vWF) into the different branches of the thoracic aorta and near the wall regions. Tracking small blood clots in the proximal part of the descending aorta show that clots may be transported upstream with the retrograde flow into the cervical branches during systole, ending up in the intracranial vessels (Chapter 5.3 and Paper 6). As far as we know, no such simulation results were reported in the literature, although such mechanism has been clinically suspected  The effect of strong path-line curvature for particles having density differing from that of the carrier fluid, leading to centrifugal force acting on cells and particles suspended in the blood was assessed. The centrifugal forces are significant in the vicinity of arterial branches due to strong curvature (Papers 1 & 6). This phenomenon may lead to physiological hemodilution and may have impact on the concentration of cells/substances in different parts in the circulatory system.

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7. Future perspective The starting point of this research was an attempt to be able predicting the risks for developing atherosclerosis before it occurs. This ultimate goal was not achieved due to the lack of knowledge required to define appropriate metrics and the difficulties in simulating blood flow for a specific patient. To get a better understanding of the atherosclerotic process, a prospective study with ultrasound/elastography of vessel wall remodeling directly after arteriovenous fistula (AVF) operation was initiated. The underlying reason for this study were observations that morphological wall changes in the fistula resemble atherosclerotic changes in systematic arteries. The fact that the wall changes in the AVF develop quickly, within days to weeks, may give an opportunity and have greater potential to more easily identify dominating components and details of the wall remodeling process. With these aspects as background, it would be natural to carry out a prospective and retrospective study of the flow and vessel properties using combined US and simulation tools. An ethical approval has been granted for the study. Data from a group of patients was gathered. A natural continuation of the project is expected to shed light on both the wall remodeling and the several tools used the thesis: Converting clinical geometrical (and volumetric flow) data into a simulation tool that is useful for clinical work. In addition to identifying clinical relevant metrics, future work should focus on being able to compute patient specific “operational maps” rather than isolated cases in the parameter space. Such a “map” should include the behavior of the flow under the whole range of relevant parameters; covering the full range of heart-rate and flow-rate combinations along with variation of mixture viscosity due to normal and/or pathological conditions (cell and protein concentration in the blood). Such an “operational map” cannot be achieved by computing each operating point in the parameter space individually. Instead, one may use reduced modelling that would enable determining “preliminary” data and enable one to carry out more detailed (conventional) simulations at points of interest or at critical points where the results change character. A similar and complimentary approach would be to use a smaller number of simulations (coarse parameter space) and use deep learning techniques to determine points of interest for detailed simulation. The process should be repeated iteratively to improve accuracy at lower computational cost. A patient specific “operational map” is meaningful only if it can be translated to determining risk levels for CVD. Thus, it is necessary that more advanced simulation tools are developed and utilized in close connection to clinical activities.

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