Digital Blood

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Thesis

Reference

Digital Blood

KOTSALOS, Christos

Abstract

This thesis aims at building high-fidelity models for the simulation of blood at both the microscopic and the macroscopic scales. Our work focuses on creating a digital replica of human blood through various tools of multi-scale/multi-physics nature, and on contributing novel pieces towards the realisation of a digital lab. We then use these tools for investigating cases that span from fundamental research to problems of clinical relevance. Focal point has been the understanding of the underlying mechanisms of platelet transport. For this, we followed a bottom-up approach, i.e. started from fully resolved blood flow simulations (microscopic scale) and moved towards calibrated stochastic models (macroscopic scale).

Our results help clarify platelet transport physics, leading to more accurate modelling and design of PLT function tests (e.g. Impact-R device). Given the limited prognostic capacity of these tests, we believe that our models could lead in next generation tests with higher clinical relevance/readiness.

KOTSALOS, Christos. Digital Blood . Thèse de doctorat : Univ. Genève, 2020, no. Sc. 5536

DOI : 10.13097/archive-ouverte/unige:148411 URN : urn:nbn:ch:unige-1484112

Available at:

http://archive-ouverte.unige.ch/unige:148411

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Départment d’informatique Professeur B. Chopard

Professeur J. Latt

Digital Blood

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention informatique

par

Christos KOTSALOS de

Athènes (Grèce)

Thèse N^o 5536

GENÈVE

Repro-Mail - Université de Genève 2020

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Toujours agité, jamais abattu

— Vincent Perdonnet

To Ypatia

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Acknowledgements

Firstly, I would like to express my sincere gratitude to my supervisors, Professor Bastien Chopard and Professor Jonas Latt, for their continuous support and guidance throughout my PhD. Bastien and Jonas were true mentors in this journey. Our endless discussions and hands- on meetings had a huge impact on my research and on the quality of my work. Moreover, their enthusiasm kept my motivation at the highest levels, even during some difficult periods. Their anthropocentric approach made my PhD a great and extremely pleasant experience. Words cannot express my appreciation and admiration to them.

I would like to thank Professor Igor V. Pivkin and Professor Gábor Závodszky for allowing me to defend my thesis and for their stimulating feedback. I am honoured that you have accepted to be part of my PhD committee.

During my PhD I collaborated with great scientists across various disciplines. Notably, I would like to thank Professor Karim Zouaoui Boudjeltia for his passion and endless energy for exploration. I had a great time visiting his lab in Charleroi.

The PhD would not be that fun without my colleagues (past and present) of the Scientific and Parallel Computing (SPC) group. Especially, I would like to thank Francesco Marson, Gregor Chliamovitch, Orestis Malaspinas, Dimitris Kontaxakis, Sha Li, Yann Thorimbert, Aziza Merzouki, Anthony Boulmier, Franck Raynaud, Christophe Coreixas (also for translating my thesis abstract to French), Raphaël Conradin, Jonathan Lemus, Rémy Petkantchin, Pierre Kunzli, Christophe Charpilloz, and Jean-Luc Falcone for their support and fantastic moments.

I will miss our breaks and our geeky/scientific discussions.

Of course, all these would not be possible without the support from CompBioMed (H2020 program). During these years, I had the opportunity to meet and exchange ideas with very interesting people. Moreover, my research would not be realised without the use of high-end supercomputing systems. For this reason, I would like to acknowledge the enormous help from the Swiss National Supercomputing Centre (CSCS), the National Supercomputing Centre in the Netherlands (Surfsara), and the HPC Facilities of the University of Geneva.

Finally, special thanks goes to my family: Makis, Maria, Konstantinos and Alexandros, for always loving, supporting and encouraging me. Furthermore, my life would not be that complete without the support and love from my family in law. Many thanks to Vaggelis, Fotini, Sofia, Nasos, Efthimis and Fotini Jr. Last but not least, I would like to thank my wife Ypatia for everything.

Geneva, December 2020 C. K.

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Abstract

This thesis aims at building high-fidelity models for the simulation of blood at both the microscopic and the macroscopic scales. Our work focuses on creating a digital replica of human blood through various tools of multi-scale/multi-physics nature, and on contributing novel pieces towards the realisation of a digital lab. We then use these tools for investigating cases that span from fundamental research to problems of clinical relevance.

Regarding the micrometre/microscopic scale, we propose a computational framework (called Palabos-npFEM) for the simulation of blood flow with fully resolved constituents, i.e. red blood cells (RBCs) and platelets (PLTs). Palabos-npFEM deploys a modular approach that consists of a lattice Boltzmann solver for the blood plasma, a novel finite element based solver (npFEM) for the deformable bodies (both trajectories and deformations), and an immersed boundary method for the fluid-solid interaction. Palabos-npFEM provides, on top of a CPU- only version, the option to simulate the deformable bodies on GPUs, thus the code is tailored for the fastest supercomputers. The software is integrated in the open-source Palabos library, and it is available on the Git repository https://gitlab.com/unigespc/palabos. Furthermore, we prove that high-fidelity FEM solvers (such as the npFEM solver) can be a viable solution for the simulation of large systems of deformable bodies in comparison to the more simplified mass-spring-systems. Performance-wise, Palabos-npFEM competes closely with other state- of-the-art libraries. Towards the validation/verification of our solver, we have performed an extensive comparison of single-RBC in silico experiments with their in vitro counterparts, while for multiple blood cells test cases, we have performed numerous simulations along specially designed in vitro experiments.

We later use the Palabos-npFEM framework along with in vitro experiments to study the effect of spherized RBCs on PLT transport. RBCs of patients with chronic obstructive pulmonary disease (COPD) are more spherical than healthy volunteers. Both the numerical and the in vitro experiments show an increase of platelet transport towards the blood vessel walls for the COPD patients. However, the in vitro study does not allow us to know if the observed effect is linked to the decrease in the electronegativity of RBCs or to an effect simply linked to the shape of RBCs. This is where the numerical counterpart kicks-in and proves that just the shape-change can explain the increased PLT transport. Therefore, this study shows the excellent complementarity between experimentations and numerical simulations to explore

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complex dynamic systems.

Inspired by the research of Chopard et al. [29], we prove that the random part of PLT velocities is governed by a fat-tailed probability distribution, usually referred to as a Lévy flight. While the transport of platelets in blood is commonly assumed to obey an advection-diffusion equation (Brownian-like random walk), we show that this assumption is not valid, and instead of a Gaussian velocity probability distribution function (pdf ), a power law velocity pdf with exponentα≤2 describes far better the motion of PLTs. This result comes from a careful statistical analysis of 64 direct numerical simulations (DNS) of blood using Palabos-npFEM.

Following, we develop stochastic models (macroscopic view of blood) based on the power law velocity pdf (α∼1.5) and simulate a PLT function analyser (impact-R device), comparing and validating our new models with the in vitro counterpart [29].

To summarise, focal point of this thesis has been the understanding of the underlying mechanisms of platelet transport. For this, we followed a bottom-up approach, i.e. started from fully resolved cellular blood flow simulations (microscopic scale) and moved towards calibrated stochastic models (macroscopic scale) based on the DNS. Our results help clarify platelet transport physics, leading to more accurate modelling and design of PLT function tests, like the impact-R device. Given the limited prognostic capacity of these tests (contradicting results and lack of consensus), we strongly believe that our newly-introduced stochastic models could lead in next generation PLT function tests with higher clinical relevance/readiness.

Keywords:

1. red blood cells, fully resolved blood flow, nodal projective FEM, projective dynamics, lattice Boltzmann, immersed boundary

2. npFEM, Palabos, GPUs, cellular blood flow, platelet transport

3. platelets, anomalous transport, fat-tailed distributions, power law behaviour, random walks, stochastic model

4. Palabos-npFEM, digital blood

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Résumé

Cette thèse a pour but la construction de modèles haute-fidélité pour la simulation d’écou- lements sanguin à des échelles microscopiques et macroscopiques. Pour ce faire, plusieurs outils de nature multi-échelles et multi-physiques sont développés afin de créer une réplique numérique de ce type d’écoulement, constituant ainsi une avancée importante vers le déve- loppement d’un laboratoire numérique destiné à la simulation de phénomènes bio-physiques à l’échelle du corps humain. Les outils développés sont par la suite utilisés dans des études de complexité croissante, en commençant par des configurations plus fondamentales, puis en se rapprochant progressivement de cas proches des conditions cliniques.

Dans le but de reproduire aux mieux les phénomènes observables aux différentes échelles d’intérêt, une extension de la libraire Palabos (Palabos-npFEM) a été développée (https:

//gitlab.com/unigespc/palabos). Elle regroupe toutes les fonctionnalités nécessaires à la simulation d’écoulement sanguin parfaitement résolu, c’est-à-dire, incluant l’évolution spatio- temporelle des globules rouges (RBC) et des plaquettes (PLT). Palabos-npFEM est une approche modulaire s’appuyant sur (1) un solveur lattice Boltzmann pour l’évolution du plasma sanguin, (2) une représentation par éléments finis innovante des objets déformables (npFEM), et (3) une description des interactions fluide-structure grâce au formalisme de frontières im- mergées. En plus de son utilisation sur des processeurs d’ordinateurs (CPU), Palabos-npFEM est compatible avec le formalisme multi-GPU qui permet l’accélération des simulations d’objets déformantes grâce à l’utilisation de cartes graphiques. Cette libraire est donc parfaitement adaptée aux architectures des super-calculateurs dernier cri.

Même si les modèles simplifiées (masse-ressort) sont souvent préférés pour la simulation de systèmes composés d’un grand nombre d’objets déformables, ce travail de thèse montre que les solveurs basés sur des éléments finis (tel que npFEM) n’ont pas à rougir de la comparaison, que ce soit en terme de précision ou bien de stabilité. Du point de vue de la performance, Palabos-npFEM est très proche des résultats obtenus avec d’autres méthodes issues de l’état de l’art. En ce qui concerne sa validation, nous avons commencé par comparer de façon détaillée les résultats obtenus lors de la simulation in silico d’un seul globule rouge avec ceux tirés d’expériences in vitro. S’en est ensuite suivi la simulation d’un ensemble de globules rouges, et de la comparaison avec les expérience in vitro correspondantes.

Ces deux outils (expériences in vitro et simulations basées sur Palabos-npFEM) ont égale- v

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ment permis d’étudier l’impact de la sphérification des globules rouges sur le transport des plaquettes. Cette sphérification est généralement observée chez les patients atteints d’obs- truction chronique des voies pulmonaires. Pour ces derniers, les expériences numériques et in vitro montrent que les plaquettes ont tendance à être plus déviées de leur trajectoire, ce qui mènent à une agglomération leur au niveau des parois des vaisseaux sanguins. Néanmoins, les données in vitro ne permettent pas de conclure quant à la cause principal de ce phénomène de migration vers les parois, c’est-à-dire, baisse de l’électro-négativité des globules rouges ou bien leur sphérification. Au contraire, les simulations numériques donnent accès à des résultats plus détaillés, et elles permettent de trancher en faveur de la sphérification. Cette étude illustre parfaitement l’intérêt des outils de simulation développés au cours de cette thèse, et qui complètent parfaitement les expériences in vitro de systèmes dynamiques et complexes.

Dans la continuation des travaux menés par Chopard et al [29], il est démontré que les fluc- tuations de vitesses mesurées pour les plaquettes sont gouvernées par une distribution de probabilité à queue lourde, aussi connue sous le nom de processus de Lévy. Alors qu’il est généralement admis que le transport de plaquettes suit une équation d’évolution de type advection-diffusion (mouvement Brownien), les résultats obtenus montrent que cette hypo- thèse n’est pas valide. Au lieu de la traditionnelle loi Normale, une loi de puissance (exposant α≤2) décrit de façon plus fidèle le phénomène de migration des plaquettes, cette dernière étant paramétrisée via une étude détaillée basée sur plus de 64 simulations d’écoulement sanguin réalisées avec la libraire Palabos-npFEM. Cette méthodologie est finalement utilisée pour le développement de modèles stochastiques permettant une description macroscopique des écoulements sanguins. Une loi de puissance enα∼1.5 est notamment utilisée pour modéliser la réponse plaquettaire (similaire à ce qui est fait avec l’outil « impact-R »), et se compare favorablement avec les données in vitro [29].

Au final, ce travail de thèse se sera focalisé sur la compréhension des mécanismes sous-jacents au transport de plaquettes. Pour ce faire, une approche ascendante (bottom-up) a été mise en place en s’appuyant sur des simulations résolues des interactions entre globules rouges et plaquettes (échelle microscopique), pour, au final, aboutir à des modèles stochastiques (échelle macroscopique) calibrés sur lesdites simulations entièrement résolues. Les résul- tats obtenus mettent en évidence les phénomènes physiques rencontrés lors de l’étude du transport de plaquettes, et permettent d’améliorer la modélisation et le design de tests de réactivité plaquettaire (tel que le système « impact-R »). Néanmoins, l’efficacité de ces tests est particulièrement décriée, et notamment, à cause des résultats contradictoires obtenus avec ce genre d’approche. C’est pourquoi l’utilisation des modèles stochastiques semble être une alternative intéressante pour le développement d’une nouvelle génération de test de réactivité pour les plaquettes.

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Résumé

Mots clés:

1. Globules rouges, écoulement sanguin parfaitement résolu, nodal projective FEM, projective dynamics, lattice Boltzmann, frontières immergées

2. npFEM, Palabos, GPUs, cellular blood flow, transport de plaquettes

3. plaquettes, transport anormale, distribution à queue lourde, loi de puissance, marche aléatoire, modèle stochastique

4. Palabos-npFEM, écoulement sanguin numérique

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Résumé

Statement

The present thesis is a collection of five articles:

1. C. Kotsalos, J. Latt, and B. Chopard. Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow. Journal of Computational Physics, 2019. doi: 10.1016/j.jcp.2019.108905.

2. C. Kotsalos, J. Latt, J. Beny, and B. Chopard. Digital blood in massively parallel CPU/GPU systems for the study of platelet transport. Interface Focus, The Royal Society, 2020. doi:

10.1098/rsfs.2019.0116.

3. K. Z. Boudjeltia, C. Kotsalos, D. Ribeiro, A. Rousseau, C. Lelubre, O. Sartenaer, M. Piag- nerelli, J. Dohet-Eraly, F. Dubois, N. Tasiaux, B. Chopard, and A. V. Meerhaeghe. Spher- ization of red blood cells and platelets margination in COPD patients. Annals of the New York Academy of Sciences, 2020. doi: 10.1111/nyas.14489.

4. C. Kotsalos, K. Z. Boudjeltia, R. Dutta, J. Latt, and B. Chopard. Anomalous platelet transport & fat-tailed distributions. https://arxiv.org/abs/2006.11755.

5. C. Kotsalos, J. Latt, and B. Chopard. Palabos-npFEM: Software for the Simulation of Cellular Blood Flow (Digital Blood). https://arxiv.org/abs/2011.04332.

completed by an introduction and a conclusion.

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1 Introduction

Blood plays a vital role in living organisms, transporting oxygen, nutrients, waste products, and various kinds of cells, to tissues and organs. Human blood is a complex suspension of red blood cells (RBCs or erythrocytes), platelets (PLTs or thrombocytes), and white blood cells (WBCs or leukocytes), submerged in a Newtonian fluid, the plasma. Blood plasma, which constitutes 55% of blood volume, is an aqueous solution containing 92% water, 8% proteins (mainly albumin), and trace amounts of other materials (e.g. glucose, amino/fatty acids, carbon dioxide). At physiological hematocrit (RBCs volume fraction), i.e. 35-45%, in just a blood drop (about amm³) there are a few million RBCs£

O(10⁶)¤

, a few hundred thousand PLTs£

O(10⁵)¤

, and a few thousand white blood cells£

O(10³)¤

. Therefore, the most abundant cells in vertebrate blood are red blood cells. These contain haemoglobin, an iron-containing protein, which facilitates oxygen transport by reversibly binding to this respiratory gas and greatly increasing its solubility in blood. In contrast, carbon dioxide is mostly transported extracellularly as bicarbonate ion transported in plasma [148].

An adult person has on average five litres of blood, and the cardiovascular system spans a length of 100, 000 km, 80% of which consists of the capillaries (smallest blood vessels).

Additionally, our blood vessels are characterised by a variety of scales, i.e. the diameter of arteries/veins ranges from few millimetres (mm) to few centimetres (cm), the diameter of arterioles/venules ranges from few micrometres (µm) to few hundred micrometres, and the capillaries are about the size of a RBC diameter (about eight micrometres). Blood is circulated around the body through blood vessels by the pumping action of the heart. In animals with lungs, arterial blood carries oxygen from inhaled air to the tissues of the body, and venous blood carries carbon dioxide, a waste product of metabolism produced by cells, from the tissues to the lungs to be exhaled. Figure 1.1 presents our cardiovascular system and its main function [149].

It is obvious from the description above (existence of fluid/solid phases, characteristic lengths from micrometres to centimetres), the multi-physics/multi-scale nature of blood flow. One can numerically approach blood as a continuum (like a Newtonian fluid), e.g. [97, 84, 89, 90, 91], or as a complex suspension (particulate description), e.g. [73, 75, 158, 46, 119]. An in-between 1

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Figure 1.1 – The pulmonary circuit moves blood from the right side of the heart to the lungs and back to the heart. The systemic circuit moves blood from the left side of the heart to the head and body and returns it to the right side of the heart to repeat the cy- cle. The arrows indicate the direction of blood flow, and the colours show the relative levels of oxygen concentration. (Illustration from Anatomy & Physiology, OpenStax Col- lege. This figure is licensed under the Creative Commons Attribution 3.0 Unported license:

https://creativecommons.org/licenses/by/3.0/deed.en)

approach would be the stochastic description [82, 4, 74] of blood through random walks for the cells of interest, e.g. platelets. Of course, the investigated case drives the aforementioned choice. In more details, when the characteristic size of the studied system is much larger than the size of the largest blood cell, then a continuum or stochastic approach is deemed more suitable in terms of computational efficiency and scale relevance. On the other hand, when the system size is comparable to the size of the blood cells, then a fully resolved (cellular) approach is the direction to follow. Let us clarify that by fully resolved simulations, we mean the tracking of the individual trajectories and deformations of blood cells and it has nothing to do with the numerical discretisation/resolution of the various phases.

Many tools for the simulation of blood across scales have been developed the last two decades, and they can readily be accessed as open-source projects or they are based on open-source components. Indeed, these tools complement in vitro/vivo experiments and have become an essential part for in-depth investigations. Regarding the microscopic scale, widely used (and available) frameworks are Hemocell [158, 157], Palabos-LAMMPS [133], the DPD/GPU- 2

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accelerated library [15], OpenFSI [154], Mihreo (DPD: dissipative particle dynamics & GPU support) [5] and Palabos-npFEM [73, 75] (the tool introduced in this thesis - for more see chapter 6). Regarding the continuum scale, Palabos [84, 1] and HemeLB [97] are the two main open-source CFD (Computational Fluid Dynamics) libraries with rich background in the domain of computational biomedicine. In particular, these two libraries have a proven record on problems of clinical relevance, such as flow diverter/stent simulation [84, 91], vertebroplasty, study of aneurysms, drug delivery devices [97] and simulation of the vasculature. Regarding the stochastic approach, any of the continuum scale libraries can be used to provide the velocity field of blood, while for the random part of the velocities of the investigated particles (e.g. platelets) any valid underlying distribution can complete the simulation approach. As we discuss later in this thesis (chapter 5), the distribution function bears a very critical role, leading to completely different transport physics. A complete state-of-the-art discussion on the relevant scale/approach is continued at every chapter of this thesis, since each of them deals with blood at different scale and clinical relevance.

Why to simulate at the micrometre level?Certainly, the continuum or stochastic modelling of blood is much less computationally expensive and it can readily reach scales of clinical relevance in comparison to the simulations with fully resolved constituents, that lag behind in these aspects (computational intensity and clinical relevance). However, many of the intriguing properties of blood originate from its cellular nature. Therefore, simulating the collective behaviour of blood cells can provide explanation to the most fundamental transport phenomena in blood [158], such as the non-Newtonian viscosity [158], PLT margination [140], thrombus/clot formation [53], the Fåhræus effect [8], the formation of the cell free layer [140], and the red blood cell (RBC) enhanced shear-induced diffusion of platelets [98, 159]. Apart from physiological conditions (health), numerical simulations at the micrometre scale have significantly assisted the understanding of pathological conditions (disease) [47, 92, 25, 21], as they offer a controlled environment for testing a large number of parameters and classifying their effect on blood rheology. In more details, pathological alterations in RBC deformability have been associated with various diseases [134] such as hereditary disorders (like spherocytosis, elliptocytosis, and stomatocytosis), metabolic disorders (like diabetes, hypercholesterolemia, and obesity), malaria, sickle anaemia, chronic obstructive pulmonary disease.

Furthermore, if we consider the role of platelets, these are involved in multiple physiological and pathophysiological processes such as haemostasis, thrombosis, clot retraction, vessel constriction and repair, inflammation including promotion of atherosclerosis, host defence, and even tumour growth/metastasis [65]. Additionally, with the occurrence of more mature experimental (in-vitro) tools, there is an increased focus on developing/analysing lab-on-a- chip systems [121, 80] and drug delivery systems [124, 139]. Consequently, viewing/simulating blood at this level is of paramount importance, given the rich application range and the need to map the impact of blood cells on blood rheology in both health and disease.

Of course, the various software tools aiming at different scales can collaborate and thus provide a more holistic approach for the simulation and understanding of such a complex flow. For example, software simulating blood at the cellular level can be used for calibrating/tuning 3

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continuum and stochastic solvers (e.g. on-going work and publication under preparation, building on top of Dutta et al. [41] and Chopard et al. [29]). Indeed, the continuum solvers need data such as apparent viscosity [158], stochastic solvers need to know the underlying distributions of the moving particles (e.g. platelets) [74], and these can be readily provided by solvers with fully resolved constituents. Chapter 5 provides an in-depth study on PLT transport physics and reveals the synergy from the combination of microscopic and stochastic modelling of blood.

Currently, the direct numerical simulation of blood, i.e. the resolution of the trajectories and deformations of individual cells, is an extremely challenging problem in terms of computational cost. Even the most efficient numerical models deployed at the fastest supercomputers can partially (or not even) deal with applications of clinical relevance, i.e. blood volumes ranging from fewmm³to fewcm³for physical time ranging from few seconds to few minutes.

Thus, the optimal approach is to use tools at various scales synergistically, trying to capture from different perspectives the investigated phenomena. A complete overview on High Perfor- mance Computing (HPC) libraries and performance metrics of Palabos-npFEM (and other competing libraries) is given in chapter 3.

It is worth mentioning, that the state-of-the-art cellular blood flow solvers [158, 46, 121, 119]

deal only with mechanical processes and completely disregard biochemical phenomena (e.g.

platelet activation/adhesion/aggregation), due to the extreme computational cost. Obviously, the incorporation of biochemical processes into cellular blood flow solvers could potentially shed more light on poorly understood phenomena. Hopefully, some biochemical processes can be implicitly taken into account through purely mechanical operations, such as the electrophoretic mobility of RBCs/PLTs (surface electrical charge) [68, 12, 63] that can be potentially encoded into the repulsive forces of the mechanical models (for more consult chapter 5).

Thesis Objectives & Outline

This thesis aims at building high-fidelity models for the simulation of blood, not limited to the micrometre scale but also to the macroscopic level (mainly stochastic approach), trying to deal with questions that span from fundamental research to problems of clinical relevance.

Our work focuses on creating a digital replica of human blood, through various tools of multi- scale/multi-physics nature. This thesis contributes tools towards the realisation of a digital lab. The ultimate goal of a human digital twin demands a plethora of digital tools of multi- disciplinary nature (bio-chemical/mechanical processes), and with this thesis we manage to add few novel pieces into this toolset. Our tools are then used for gaining valuable insights on blood rheology along with in vitro/vivo experiments, leading to more in-depth investigations of the studied phenomena.

Focal point of our research endeavours has been the understanding of the underlying mechanisms of platelet transport. It is a complicated transport phenomenon because it is extremely 4

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sensitive to multiple parameters such as hematocrit, flow conditions (e.g. shear rate), RBC characteristics (material, shape), PLT characteristics (material, shape), geometry, etc. Here, we propose a disruptive view on explaining the transport mechanisms, questioning the widely/commonly used models (chapter 5). By standard/common models we mean the “red blood cell enhanced” Brownian motion of PLTs [99] or equivalently the application of the advection-diffusion equation [82] for the description of their motion. Through a careful statistical analysis of platelet velocities, we show the limited applicability of the advection-diffusion equation (equivalent to Gaussian random walks) for the description of PLT transport, and we propose an alternative interpretation based on fat-tailed distributions. With this thesis we hope to open a discussion with the community on the re-examination of the standard models on describing PLT transport, and additionally we strongly believe that our newly introduced models (at the stochastic level) will impact applications of clinical relevance, e.g. redesign of PLT function tests.

The remaining of this thesis is structured as follows:

• In chapter 2, we introduce our computational framework for fully resolved cellular blood flow simulations. This tool combines the lattice Boltzmann method (LBM) for the resolution of blood plasma, a novel finite element method (FEM) solver for the resolution of the trajectories and deformations of blood cells (main focus on RBCs &

PLTs), and the immersed boundary method (IBM) for the coupling of the fluid and solid phases. Our framework focuses on models of high fidelity without compromising the computational cost. The challenging part is to deploy methods that are robust, accurate and performant, at the same time. While it may seem a trivial task, it involves many decisions both in the physical and numerical sides, and thus we propose a framework composed of carefully selected solvers (fluid, solid, and fluid-solid interaction -FSI-) capable of performing fast and high-fidelity blood flow simulations.

• In chapter 3, we enhance the performance of our computational framework by utilising general purpose graphics processing units (gpGPUs) for the resolution of the deformable blood cells. This chapter focuses on the high performance computing (HPC)-centric design of our software tool. We provide an exhaustive study of performance measures, deploying our code in the flagship system of the Swiss National Supercomputing Centre (Piz Daint supercomputer), one of the fastest supercomputers worldwide.

• In chapter 4, we present a collaborative study involving both in vitro and in silico experiments. In more details, we study the effect of RBC shape on PLT transport. The RBC shape-change is linked to patients suffering from chronic obstructive pulmonary disease (COPD). This chapter constitutes a proof of the benefits coming from the combination of both physical and numerical experiments. Specifically, the in vitro experiments showed that the more spherical RBCs (coming from COPD) induce an increased platelet transport to the walls. However, the in vitro study does not allow us to know if the observed effect is linked to the decrease in the electronegativity of RBCs or to an effect simply 5

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linked to the shape of RBCs. This is where the numerical counterpart is used and proves that just the shape-change can explain the increased PLT transport (only mechanical interactions and no biochemical processes). Thus, this collaborative project shows the excellent complementarity between experimentations and numerical simulations for the exploration of complex dynamical systems.

• In chapter 5, we build upon the research by Chopard et al. [29], and prove that the random part of PLT velocities is governed by a fat-tailed probability distribution, usually referred to as a Lévy flight. In more details, while the transport of platelets in blood is commonly assumed to obey an advection-diffusion equation (Brownian-like random walk), we show that this assumption is not valid (or at least has a limited applicability), with serious implications on underestimating the flux of PLTs towards the vessel walls.

We start our analysis from direct numerical simulations and through a careful statistical processing of PLT velocities (tracking individual PLTs), we validate the hypothesis of fat-tailed distributions. However, the in vitro experiments performed by Chopard et al.

[29] deal with a physical domain at the order of a cubic millimetre of blood studied for at least 20 seconds of physical time. As discussed earlier, a direct numerical simulation of this in vitro experiment is not possible as the spatio-temporal scales are still too hard to reach, even on the fastest supercomputers. For this reason, we develop stochastic models for the description of platelet motion, and calibrate them with the help of the fully resolved simulations (using the results of the statistical analysis). Thus, we are able to bridge this gap of scales by deploying the optimal model per scale.

• In chapter 6, we give an in-depth presentation of our computational framework for fully resolved blood flow simulations, called Palabos-npFEM. Main focus of this chapter is the software side of our research. This framework is open-sourced and we have integrated it into the Palabos core library [84, 1] (computational fluid dynamics software based on the lattice Boltzmann method). We strongly believe in transparent and reproducible research, and this chapter makes our computational tool open to the community for discussion, feedback, correction and amelioration.

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2 Bridging the computational gap between mesoscopic and continuum modelling of red blood cells for fully resolved blood flow

In this chapter, we present a computational framework for the simulation of blood flow with fully resolved red blood cells (RBCs) using a modular approach that consists of a lattice Boltz- mann solver for the blood plasma, a novel finite element based solver for the deformable bodies and an immersed boundary method for the fluid-solid interaction. For the RBCs, we propose a nodal projective FEM (npFEM) solver which has theoretical advantages over the more commonly used mass-spring systems (mesoscopic modelling), such as an unconditional stability, versatile material expressivity, and one set of parameters to fully describe the behaviour of the body at any mesh resolution. At the same time, the method is substantially faster than other FEM solvers proposed in this field, and has an efficiency that is comparable to the one of mesoscopic models. At its core, the solver uses specially defined potential energies, and builds upon them a fast iterative procedure based on quasi-Newton techniques. For a known material, our solver has only one free parameter that demands tuning, related to the body viscoelasticity. In contrast, state-of-the-art solvers for deformable bodies have more free parameters, and the calibration of the models demands special assumptions regarding the mesh topology, which restrict their generality and mesh independence. We propose as well a modification to the potential energy proposed by Skalak et al. 1973 [129] for the red blood cell membrane, which enhances the strain hardening behaviour at higher deformations. Our viscoelastic model for the red blood cell, while simple enough and applicable to any kind of solver as a post-convergence step, can capture accurately the characteristic recovery time and tank-treading frequencies. The framework is validated using experimental data, and it proves to be scalable for multiple deformable bodies.

2.1 Introduction

Blood is a complex suspension of red blood cells (RBCs), white blood cells and platelets, submerged in a Newtonian fluid, the plasma. Especially, the RBCs (else, erythrocytes) cover 40-45% of the whole blood volume, and their behaviour has a direct impact on blood rheology.

This chapter is based on the article entitled “Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow” by Kotsalos et al. [73].

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The accurate modelling of the collective transport of the cells in the plasma is of paramount importance since it can help us decipherin vivophenomena, e.g. blood clotting, margination of platelets, or the cell-free layer (Mountrakis 2015 [102]). In addition, the deformability of RBCs is strongly linked to some pathological conditions, e.g. hereditary disorders (like spherocytosis, elliptocytosis, and stomatocytosis), metabolic disorders (like diabetes, hypercholesterolemia, and obesity), malaria, or sickle anaemia as described by Tomaiuolo 2014 [134]. In more details, a human RBC has a biconcave discocyte shape with a liquid interior (cytoplasm, haemoglobin solution) of dynamic viscosity 6−10cP. A healthy red blood cell has an average surface area of 134µm², a volume of 94µm³, a diameter of 7.82µmand a varying thickness of 0.81µmat the dimple to 2.57µmat the periphery. An analytical formula for the RBC shape is proposed in Evans and Skalak 1980 [44]. The RBC membrane is a complex multi-layered structure consisting of an external lipid bilayer attached to an underlying cytoskeleton (spectrin-link network). These complicated anucleate structures transfer oxygen between blood and tissues thanks to their ability to undergo substantial deformations while maintaining their area and volume.

The absence of a universal numerical model capable of describing the viscoelastic behaviour of a RBC is the motivation of this study/chapter. A universal model should fulfil a number of criteria, such as generality, robustness, accuracy and performance. By generality and accuracy, we mean the ability of the model to simulate any material and all the expected behaviour as close as possible to the observed data (experiments). By robustness, we refer to the capability of the model to adapt to extreme physical circumstances such as large deformations, confined flows, or high flow shear rates, and to numerically unfavourable settings like large time steps. Last but not least, performance is a key requirement in the field of computational biophysics, since the model should be extended to cases including millions or even billions of deformable bodies. To give an example, a volume of 1mm³of blood at a hematocrit 40% contains approximately five million RBCs. For the modelling of RBCs, two main approaches are proposed in the literature. The first is a continuum-level approach, that views the membrane as a continuous medium which obeys specific partial differential equations and constitutive laws. The discretization of these equations is commonly done through a finite element method (FEM). Many researchers have successfully used this approach for the modelling of membranes, including Krüger et al. 2011 [78], MacMeccan et al. 2009 [96], Charrier et al. 1989 [26], Shrivastava and Tang 1993 [126]. Klöppel et al. 2011 [71] employed a two-layer model for an even more accurate description of the lipid bilayer and the underlying cytoskeleton. The aforementioned models use constitutive laws for the description of the membrane material, such as the material introduced by Skalak et al. 1973 [129], Neo-Hookean materials proposed in [16] or the material described by Yeoh 1990 [156]

(for a thorough investigation of material models the reader is referred to Dimitrakopoulos et al.

2012 [36] and Siguenza et al. 2017 [128]). Among the many advantages of this approach is the guaranteed mesh-independence, which is a product of the way the equations are discretized.

This modelling approach satisfies many of the above-mentioned criteria, namely generality, robustness and accuracy. However, they do not perform well in simulations of multiple

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2.1. Introduction

deformable bodies due to their high computational cost. The second modelling approach operates at a mesoscopic level, and represents surface properties through a mass-spring system. More precisely, mesoscopic modelling of the RBCs spans from the spectrin-link to the coarse-grained spectrin-link levels. The spectrin-level approach tries to imitate the physics of the cytoskeleton of a RBC as a fine-grained mass-spring model. The spectrin level was introduced by Discher et al. 1998 [37] and extended further by Dao et al. 2003 & 2006 [34, 35]

and Li et al. 2005 [87]. Though, this approach is limited by the high computational cost, given that it requires 23, 867 surface vertices [116] for the simulation of the whole spectrin network.

For this reason, the coarse-grained models of Dupin et al. 2007 [40], Pivkin and Karniadakis 2008 [116], Fedosov et al. 2010 [46], and Reasor et al. 2011 [119] have gained substantial attention during the last years, where the surface discretization demands about 500 vertices.

Currently, state-of-the-art solvers like Hemocell [158, 157] and the solver of Fedosov [46] are based on this modelling approach. These coarse-grained methods outperform all the others in terms of computational time. Nevertheless, since they are mass-spring systems they inherit limitations on generality, robustness and accuracy.

As far as the simulation of the blood plasma is concerned, there exists a plethora of mature CFD approaches. For our study, we make use of the lattice Boltzmann method (LBM) [79]

which indirectly solves the Navier-Stokes equations. Another candidate method widely used for the simulation of blood flows, is dissipative particle dynamics (DPD) [116, 46]. Both are particle-based methods which deploy either collision and streaming operations or Newtonian laws, respectively, for the time advancement of the fluid.

The coupling of the fluid with the solid phase is a critical point for an accurate and stable simulation. Peskin 1972 [109] developed the immersed boundary method (IBM) to model blood flow in combination with moving heart valves. The strength of the IBM is that the fluid solver does not have to be modified except from the addition of a forcing term, and the fluid mesh does not need to be adjusted dynamically. Moreover, the deformable body and its discrete representation do not need to conform to the discrete fluid mesh, which leads to a very efficient fluid-solid coupling. In the context of lattice Boltzmann, an alternative solution would be the deployment of moving bounce back nodes [79]. With this approach, the computational cost however steeply increases, given the need to track the lattice nodes that transition from solid to fluid state.

In a typical numerical framework for blood flow, the computational time is dominated by the structural solver for the deformable RBCs. Even with the relatively cheap mass-spring systems, Závodsky et al. 2017 [157] report for example that the deformable bodies solver constitutes over 95% of the total computational time. Since the numerical models for Newtonian fluid flow and for the fluid-solid coupling are more mature than the ones for the physics of RBCs, our main focus is to develop a novel solution which strictly fulfils the criteria of generality, robustness, accuracy and performance without any compromise. We propose a novel approach for deformable viscoelastic bodies based on nodal projective finite elements method (npFEM).

Inspired by the research of Liu et al. 2017 [94] and Bouaziz et al. 2014 [20], our solver extends

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these theories to the field of computational biophysics. The expression “nodal” refers to the mass lumping technique, in which both the masses and the forces are lumped on the vertices of the discretized body, and therefore the finite elements act like generalised viscoelastic springs. The term “projective” stands for the use of specially designed potential energies that help us build a fast solver based on quasi-Newton optimisation techniques [9, 51]. Lastly, we discretize the continuous potential energies using linear basis functions, leading to the standard Galerkin method [117] in the field of FEM. Our solver inherits the versatility and robustness of FEM and is almost as fast as plain mass-spring systems. To our knowledge, this is the first solver bridging the gap between the two approaches, continuum and mesoscopic, in the field of RBC modelling.

The present chapter is organised as follows: In section 2.2 we thoroughly introduce the npFEM method focusing on the modelling of RBCs, the lattice Boltzmann method and the immersed boundary method. Following, in section 2.3 we put all the different pieces together and show how the complete computational framework operates. Finally, in section 2.4 we present extensive validations of our computational framework and give insights about its performance and scalability to systems of multiple blood cells.

2.2 Methods

The focus of this study/chapter is to build a computational framework for fully resolved 3D blood flow. The constituents of this framework are the solid body solver (section 2.2.1), the fluid solver (section 2.2.2), and the fluid-structure interaction (FSI) (section 2.2.3). The solid body solver based on the nodal projective finite elements method (npFEM) is our main contribution to the field. Given the volume fraction of red blood cells compared to the other cells (white, platelets), and consequently their importance on blood rheology, we develop the npFEM solver around RBC simulations. The fluid solver, which is used for the simulation of the blood plasma, is based on the lattice Boltzmann method (LBM) as implemented in the open source software Palabos [1]. The coupling between the two solvers is done via an immersed boundary method known as multidirect forcing scheme [108].

2.2.1 Nodal projective FEM (npFEM)

This section provides a complete review of the methods introduced in the research of Liu et al.

2017 [94] and Bouaziz et al. 2014 [20], and some novel extensions proposed to adapt them to the field of RBC modelling. A validation of these adaptations is provided in later sections, see results section 2.4. A red blood cell is represented by its triangulated membrane, and its total mass is lumped on the membrane vertices (including the interior fluid). This section presents how we realise the dynamics of deformable bodies, as well as all the potential energies that describe the behaviour of a red blood cell, namely local area and global volume conservation, bending and material.

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2.2. Methods

Let us assume a surface mesh consisting ofnvertices with positionsx∈Rⁿ^×³and velocities v∈Rⁿ^×3. The evolution of the system in time follows Newton’s law of motion. The external forces are defined asFext∈R^n×3and come from the fluid-solid interaction, while the internal forces areF_{i nt}∈Rⁿ^×³. The internal forces are position dependent, i.e.F_{i nt}(x)= −P

i∇E_i(x), whereEi(x) is a scalar discrete elemental potential energy. Moreover, the summation of all the elemental potential energies of any type results in the total elastic potential energy of the body. Conservative internal forces, i.e. derived from potentials, are closely related to the assumption of hyperelastic materials [127]. Theimplicit Euler time integrationleads to the following advancement rules (subscriptsnandn+1 refer to timet andt+h, respectively):

v_n+1=x_n+1−x_n

h , (2.1)

F_{i nt}(x_n+1)+F_ext(x_n)−Cv_n+1=Mv_n+1−v_n

h , (2.2)

whereM∈R^n×n is the mass matrix,his the time step andC∈R^n×n is the damping matrix acting like a proxy for viscoelasticity (Rayleigh damping, see section 2.2.1.4). The mass matrix is built by lumping the total mass of the body on the mesh vertices, resulting in a diagonal structure. The lumping can be done either by equally distributing the mass or by weighting, based on the corresponding area per vertex. Combining the above equations we derive

M(xe _n+1−yn)=h²Fi nt(x_n+1), (2.3) whereMe =M+hCandyn=xn+hMe⁻¹Mvn+h²Me⁻¹Fext. Equation (2.3) can be turned into an optimisation problem [20] as

minx_n+1

1 2h²

°

° eM¹²(x_n+1−yn)°

°

2 F+X

i

Ei(x_n+1), (2.4)

where|| · ||F is the Frobenius norm. Indeed, setting the derivative of equation (2.4) to zero (thereby minimising the objective function), we recover Newton’s second law of motion. The solution to the minimisation problem givesx_n+1. The choice of implicit Euler integration serves a dual purpose. Firstly, it provides unconditional stability for arbitrary time step.

Secondly, this integration scheme is characterised by some numerical dissipation linked to the time step (the smaller the time step the smaller the numerical dissipation). This dissipation enhances the stability of the coupling (fluid & solid) by reducing energy oscillations that could potentially lead to instabilities. This dissipation term acts like the viscous dissipation provided by Rayleigh damping (matrixC).

A remark at this point is that the variational form of implicit Euler (2.4) is just a reformulation of Newtonian dynamics, which in their turn can be viewed from the Hamiltonian dynamics perspective. Nevertheless, to accurately capture the trajectory and deformation of the bodies (dynamics) we need to guarantee the conservation of linear & angular momenta. For this, one has to be cautious with the selection of the potential energies and the viscoelastic terms.

According to Noether’s theorem, momenta are conserved when the elastic energies are rigid

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motion invariant. For the viscoelastic terms, we give an extended discussion on the conservation of momenta in section 2.2.1.4. Taking into consideration all the above, we guarantee that we resolve as accurately as possible the dynamics of the simulated bodies, reproducing the correct physics as dictated by Newton’s laws.

2.2.1.1 Extended projective dynamics method and quasi-Newton optimisation

Projective dynamics method [20] requires a special form of potential energies, based on the notion of constraint projection:

E_i(x)= min

Sip∈Mi

Ee_i(x,Sip), Ee_i(x,z)= kGix−zk²_F, (2.5) whereMiis a constraint manifold defined by a desired undeformed state,pi=Sip∈R^m^×3 is an auxiliary projection variable onMi andGi is a mapping fromx to an element-wise deformation representation which depends on the mesh topology and the initial configuration.

For an elastic body, there are many elemental energies of various types, therefore various projections. For compactness, we stack all the projections intop∈R^m×3and define binary selector matricesSi∈R^m^×^msuch thatp_i=Sip. The dimensionmis equal toP

ic_i(summation over all quadratic energies), whereciis one for the bending energy, two for the local area energy andnfor the global volume conservation, i.e. it depends on the corresponding energy (see section 2.2.1.2). More in details, the matrixGi∈R^m^×ⁿformulates the deformation descriptor and places it in the involved rows and columns, where the columns correspond to the involved vertices of thei-th energy and the rows refer to the corresponding projectionpi. The term

“constraint” is used because the energy penalises the deviation from the undeformed state, and thusconstraintsthe body from deformation, and the term “projection” corresponds to finding the minimum distance from the current state onto the manifold.

Generalising the above, all the possible undeformed configurations define the so-called constraint manifoldM (Cartesian product of the individual constraint manifoldsMi), which mathematically is formulated as the zero level-set of the total potential energy. From a geometric point of view, the potential energy is a distancedwhich quantifies how far the deformed stateGxis from the manifoldM (rest state). The projection of the current state ontoMis denoted byp. There can be many user-defined manifolds. For example, a manifold for volume preservation does not guarantee area conservation and therefore, we have to define another manifold to guarantee the latter constraint. Recapitulating the above, the total potential energy can be formalised as

E(x,p)=min

z∈Md⁰(Gx,z)=°

°Gx−p°

°

2

F. (2.6)

The minimum is achieved whenpis the projection ofGxontoM. Figure 2.1 summarises the definition of the distance and its correspondence to a potential energy. The generic potential energy of equation (2.6) refers to a discretized energy. Later, we show how from continuous

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2.2. Methods

energies using linear basis functions, we retrieve this generic formulation.

Gx

p d(Gx, p)

E = min d’(Gx, z) = d(Gx, p) z

d’(Gx, z)

Figure 2.1 –Geometric representation of a potential energy. The more the distance from the undeformed stateMthe higher the energy stored in the body. The body seeks to minimise its potential energy throughGx=p.

The key idea of projective dynamics is to exploit these specialised potential energies and reformulate thevariational form of implicit Euleras

g(x_n+1)= 1

2h²t r((x_n+1−y_n)^TM(xe _n+1−y_n))+X

i

w_i µ

Siminp∈Mi

Ee_i(x,Sip)

¶

, (2.7)

wheregis the objective function whose minimum corresponds tox_n+1andwi=Aikiis a weighting term acting like a “stiffness” that converts the distance to an energy term (Ai is the area of the element andk_i is the stiffness factor). As a reminder,||A||²_F=t r(A^TA). The projectionspare essentially functions ofxand thus developing even further equation (2.5) into the variational form (2.4), we get

g(x_n+1)= 1 2h²t r¡

(x_n+1−yn)^TM(xe _n+1−yn)¢ + 1

2t r¡

x^T_n₊₁Lxn+1¢

−t r¡

x_n^T₊₁Jp(xn+1)¢ +1

2t r¡

p^T(xn+1)Sp(xn+1)¢

| {z }

projective dynamics potential energies: quadratic formulation

, (2.8)

whereL=P

wiG^T_iGi,J=P

wiG^T_i SiandS=P

wiSiS^T_i .

Seeking for greater material expressivity, and thus more general potential energies, we can

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straightforwardly extend equation (2.7) as g(x)= 1

2h²t r((x−y)^TM(xe −y))+X

i

wiE^{P D}_i (x)+X

i

E^non_i ^{−P D}(x), (2.9) where PD stands for energies that follow the quadratic formulation of projective dynamics, see equation (2.6), and non-PD for arbitrary energies. A remark is the absence of a weighting term in the non-PD energies, which is because the stiffness is already integrated into them. The minimisation ofgis performed using a quasi-Newton technique with a well-suited Hessian approximation, which is deeply rooted in the projective dynamics part. The first step is to compute the gradient of the objective functiong(x) as defined by equation (2.9). An in-depth derivation of the gradient (regarding the PD part) and an explanation on why the derivative of

1 2t r¡

p^T(x)Sp(x)¢

vanishes can be found in [94, 93]. The derivative is given by

∇g(x)= 1 h²Me¡

x−y¢

+Lx−Jp(x)

| {z }

−F_{i nt}^{P D}

+X

i

∇xE^{non−P D}_i (x)

| {z }

−F_{i nt}^{non−P D}

. (2.10)

The projective dynamics energies demand first the computation ofpfor a givenx. This step can be massively parallelized given the small computational cost of the projection per element.

The objective function could be equivalently minimised by Newton’s method. Newton’s method would proceed by computing the Hessian matrix∇²g(x) and then using it to compute a descent direction asd= −(∇²g(x))⁻¹∇g(x). However, Liu et al. 2017 [94] suggested replacing the Hessian byHe =M/he ²+Land thus opting for a quasi-Newton approach avoiding the expensive computation of the Hessian at every time step and the possible definiteness fixes to guarantee thatdis a descent direction. Using only quadratic energies, the minimisation ofg could be realised by an alternating (local/ global) solver [20], which reveals howHe is formed. A detailed presentation can be found in Appendix A.1. The approximated Hessian plays a major role in the fast convergence of our solver and in the overall computational efficiency of the npFEM solver without the expensive computation of the true Hessian. Furthermore, matrixHe is constant, symmetric positive definite and thus can be prefactorized once at the beginning, e.g. using Cholesky factorisation, allowing for very fast computation of the descent direction d= −(H)e ⁻¹∇g(x).

The non projective dynamic energies could contribute as well to the formulation of the Hessian approximation by considering their contribution toPwiG^T_iGi, whereGiis the deformation gradient operator for thei-th triangular element andwi=Aiki(Aiis the area of the element andki a stiffness factor which can be computed as described in [94]). If on the other hand there is no contribution from the non-PD part, then the iterations for convergence are slightly increased.

The overall convergence is affected by the selected potential energies. The more the PD energies, the faster the convergence, which is the case for our study since out of the four

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2.2. Methods

different potentials (area, volume, bending, material) only the material energy does not fall into the PD category, see sections 2.2.1.2 & 2.2.1.3 for more details. The reason for this, is that the Hessian approximation is mainly built upon the PD part.

It is worth noting that the size of our matrices isn×ninstead of 3n×3n, typically found in other FEM solvers [96], leading to an at least 3×speedup compared to them.

Consequently, moving towards the descent direction, this iterative procedure guarantees to converge to the minimum ofg, i.e. xn+1. A recapitulation of the algorithm is presented in section 2.2.1.5.

2.2.1.2 Quadratic Energies to simulate RBC membrane

In this section, we develop potential energies suitable for projective dynamics and critical for the simulation of the red blood cell membrane, namely local area conservation, bending and global volume preservation. The analysis reveals that the above-mentioned energies can be written in the quadratic form described by equation (2.5). The goal is to develop discrete energies and compute the auxiliary variables, i.e. the projections, in order to quantify the energy term itself. Since the current statexis known, the computation ofp results in the quantification of the energyEfrom equation (2.5).

Let the undeformed surface be a differentiable 2-manifold surfaceSembedded inR³. Let us define piecewise linear coordinate functions of the undeformed and deformed surface,g:S→ R³and f :S→R³, respectively. By introducing a setMof desired point-wise transformations T, we can formulate a potential energy [20] as

E(f,T)=k 2 Z

S

°

°∇Sf −T∇Sg°

°

2

Fd A, (2.11)

where∇Sis the gradient operator onS. The setMdetermines all allowed rest configurations T∇Sg. The energyE¡

f,T¢

is discretized over triangular elements using piecewise linear hat functions [19], and the integral is transformed to a sum over triangle potentials:

Ei(x,T)=ki

2 Ai||DsDm−1−T||²_F, (2.12) whereAi is the triangle area,Ds=[xj−xi,x_k−xi]∈R^2×2contains the triangle edges of the deformed state embedded in 2D andDmcorresponds to the undeformed state. At this point, we have introduced a generic quadratic energy similar to what is described by equation (2.5).

Based on the FEM reminder in Appendix A.2, the generic quadratic energy defined above can be rewritten as

E_i(x,T)=k_i

2 A_i||F−T||²_F, (2.13)

whereF∈R²^×²is the deformation gradient of thei-th triangular element. In this case, the deformation gradient acts like a deformation descriptor, andTcorresponds to the projection

Digital Blood

Thesis

Reference

Digital Blood

KOTSALOS, Christos

KOTSALOS, Christos. Digital Blood . Thèse de doctorat : Univ. Genève, 2020, no. Sc. 5536

DOI : 10.13097/archive-ouverte/unige:148411 URN : urn:nbn:ch:unige-1484112

Digital Blood

Acknowledgements

Abstract

Résumé

Statement

Contents

1 Introduction

Thesis Objectives & Outline

2 Bridging the computational gap between mesoscopic and continuum modelling of red blood cells for fully resolved blood flow

2.1 Introduction

2.2 Methods