Sarah Barns Thesis

NUMERICAL MODELLING OF RED BLOOD CELL

MORPHOLOGY AND DEFORMABILITY

SARAH BARNS

BEng(Mech)(Hons), GradCert(TerEd), GradDip(Math)

Submitted in fulfilment for the requirements of the degree DOCTOR OF PHILOSOPHY

School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology

in collaboration with

The Australian Red Cross Blood Service, Brisbane, Queensland, Australia

2018 Statement of Originality

The work contained in this thesis has not been previously submitted to meet requirements for an award at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

Signature: QUT Verified Signature

Date: July 28, 2018

Numerical Modelling of Red Blood Cell Morphology and Deformability Page i Keywords

. Red blood cell . Numerical modelling . Course-grained particle method . Morphology . Shape . Deformability . Discocyte . Echinocyte . AFM indentation . Optical tweezer stretching . Membrane . Mechanics . Hertz contact . Mechanical deformation

Page ii Numerical Modelling of Red Blood Cell Morphology and Deformability

Abstract

The primary function of the red blood cell (RBC) is to distribute oxygen throughout the body. This requires RBCs to squeeze through narrow blood vessels which can be just half their own diameter, making cellular deformability critical for performance. RBC deformability is predominately controlled by the properties of the cell membrane, which consists of an outer lipid bilayer with embedded proteins and cholesterol, and a spectrin-based cytoskeleton tethered beneath. The strength of the bilayer’s resistance to bending and surface area changes, as well as the strength of the cytoskeleton’s resistance to stretch and shear dictate the cell’s ability to deform under external loading.

Changes within the RBC membrane’s physical structure can reduce deformability, making the cells less efficient at moving through the body. This occurs naturally as RBCs age, but is accelerated during the storage of RBCs prior to transfusion, and also during the progression of diseases such as malaria. Although changes within the RBC membrane are well-accepted as the basis of deformability loss, the specifics of the underlying mechanisms remain unclear. Furthermore, given deformability differences can be detected experimentally, measuring RBC deformability in various experimental setups has been proposed as a potential tool for quantifying the quality of RBC units prior to transfusion and for the diagnosis of disease.

Underlying structural changes of the RBC membrane can be explored with numerical models, which allow investigation of the mechanical aspects that define RBC behaviour at a much smaller scale than is possible with experimentation, which becomes challenging and costly. Deeper insight into how the mechanical state of the membrane contributes to physical changes in RBCs will aid in developing new strategies for improving RBC storage, potentially leading to higher quality RBCs for transfusion and longer maximum storage durations. It may also contribute to understanding the mechanical basis of RBC conditions. Therefore, the impact of mechanical changes within the RBC membrane on deformability was investigated using the coarse-grained particle method (CGPM). Secondly, deformability measurements in indentation and stretching setups were explored to understand their potential for detecting differences in mechanical properties of the membrane.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page iii

As a first step of this study, the typical RBC discocyte resting shape was predicted in both 2D and 3D and validated against experimental observations from the literature. The models were then adapted to investigate the behaviour of RBCs when placed under a local compressive force to simulate indentation with atomic force microscopy (AFM), as well as when a global tensile force was applied to the cell, modelling stretching within an optical tweezer setup.

Indentation investigations revealed that changes to the membrane’s bending stiffness had the most dominant impact on deformability in this loading scenario. As bending stiffness is mainly provided by the bilayer, AFM indentation would be well- suited to detecting physical changes within this part of the RBC membrane. Force- deformation curves from RBC indentation have been historically analysed using Hertzian contact theory to extract Young’s modulus for the membrane. However numerical investigations showed that this method of analysis was extremely limited in its application to RBC indentation problems due to difficulties justifying underlying assumptions of solid contact and negligible substrate influence. Furthermore, Young’s modulus predictions were highly sensitive to the region of the cell indented and the degree of adhesion, meaning that the deformability of both the cell and substrate were being measured rather than the membrane in isolation. Therefore an alternative method of analysis is required to reconcile the comparison of measurements between different AFM setups to further mature this technique.

The stretching investigations showed that changes to the membrane’s linear stiffness had the most dominant impact on deformability in this loading scenario. Given that linear stiffness of the membrane is mainly associated with the spectrin- based cytoskeleton, the optical tweezer setup would be well-suited to detecting changes to cytoskeletal mechanical properties. To inform experimental design, it was found that optical bead size had little impact on deformability measurements, while the degree of adhesion did have a significant impact.

As a final step in this study, the capacity to predict the discocyte-echinocyte resting shape sequence was explored. This found that impacting the spontaneous curvature for local regions gave rise to the development of echinocyte morphologies. This established a baseline direction for testing echinocyte morphologies in deformability scenarios in the future when experimental data becomes available.

The CGPM models developed in this project have been shown effective in predicting the physical behaviour of RBCs under local compressive and global

Page iv Numerical Modelling of Red Blood Cell Morphology and Deformability tensile loading. They may then form the basis of future work informing the design of devices and data analysis techniques for quantifying the deformability of RBC units prior to transfusion or for the diagnosis of disease. However, as the current methodology relies on the stiffness coefficients being tailored to the specific application, improved optimisation for the stiffness coefficients and greater sophistication in the energy equations should be considered. Resolving the lack of universality of the model should enable improved performance and predictive power going forward.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page v

Acknowledgements

I would like to thank my partner Laurie for supporting me through this PhD, including the sacrifices associated with commuting between Brisbane and Melbourne for the first two years. I would also like to thank my family and friends for their ongoing support. Furthermore, special thanks to the friends who have been on the research pathway alongside me, particularly Ted Pickering, Marie Anne Balanant, Ari Bo and Chris From.

I would like to thank my supervisory team of YuanTong Gu, Emilie Sauret and Robert Flower. I appreciate YuanTong allowing me the time and space to pursue my teaching ambitions, Emilie for being an ever-reliable source of advice and direction, and Robert for welcoming me into the Blood Service community. I would also like to extend thanks to the wider Blood Service Research & Development Team, and in particular Helen Faddy who was instrumental in providing feedback through the last few months in preparing this document.

I would like to thank the undergraduate students who have worked on this project across both final year projects and the vacation research experience scheme. Finally, I would also like to acknowledge the top-up scholarship I received from the Alexander Steele Young Lions Memorial Foundation which eased the financial pressures of pursuing a PhD over the last three years.

Page vi Numerical Modelling of Red Blood Cell Morphology and Deformability

List of Publications

Peer-Reviewed Journal Articles:

. S. Barns, M. A. Balanant, E. Sauret, R. Flower, S. Saha, and Y. Gu, "Investigation of red blood cell mechanical properties using AFM indentation and coarse-grained particle method," BioMedical Engineering OnLine, vol.

16, pp. 1-21, 2017. Available from: http://doi.org/10.1186/s12938-017-0429-5 . S. Barns, E. Sauret, S. Saha, R. Flower, and Y. Gu, "Two-Layer Red Blood Cell Membrane Model using the Discrete Element Method," Applied Mechanics and Materials, vol. 846, pp. 270-275, 2016. Available from: http://doi.org/10.4028/www.scientific.net/AMM.846.270

Extended Conference Abstract:

. M. A. Balanant, S. Barns, E. Sauret, and Y. T. Gu, "Investigation of Red Blood Cell Membrane Elasticity using AFM Indentation and the Coarse- Grained Particle Method," in 10th Australasian Biomechanics Conference, Melbourne, Australia, 2016.

Poster Presentation:

. S. Barns, E. Sauret, R. Flower, and Y.T. Gu, “Numerical Investigation of Red Blood Cell Membrane Mechanical Properties on Deformability during Indentation,” in 27th Regional Congress of the International Society of Blood Transfusion, Copenhagen, Denmark, 2017.

Article in Preparation:

. S. Barns, E. Sauret, Y. Gu, “Critical Assessment of Hertz-Based Equations for RBC Indentation,” In preparation.

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Table of Contents

Statement of Originality ...... i

Keywords…… ...... ii

Abstract……...... iii

Acknowledgements ...... vi

List of Publications ...... vii

List of Figures ...... xii

List of Tables ...... xix

List of Acronyms ...... xx

Chapter 1. Introduction ...... 1

1.1 Background ...... 1

1.2 Research Objectives ...... 3

1.3 Contribution, Innovation & Significance ...... 3

1.4 Thesis Overview ...... 5

Chapter 2. Literature Review ...... 7

2.1 Cellular Structure ...... 7

2.2 Deformability ...... 10

2.3 Morphology ...... 11

2.4 Storage & Other Conditions Impacting RBC Properties ...... 14

2.4.1 Storage & Transfusion ...... 14

2.4.2 Other Conditions Impacting RBC Properties ...... 17

2.5 Physical Experiments ...... 17

2.5.1 AFM Indentation ...... 18

2.5.2 Optical Tweezer Stretching ...... 25

2.6 Modelling & Simulations ...... 26

2.6.1 Coarse-Grained Particle Method ...... 26

2.6.2 Coarse-Grained Molecular Dynamics ...... 27

2.6.3 Dissipative Particle Dynamics ...... 28

2.6.4 Bilayer-Couple Hypothesis-Based ...... 29

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2.6.5 Finite Element Methods ...... 30

2.6.6 Analytical Models for Indentation ...... 30

2.7 Summary & Implications ...... 31

Chapter 3. Development of CGPM Models for Discocyte Resting Shape .... 33

3.1 Introduction ...... 34

3.2 Methods ...... 35

3.2.1 Overview of CGPM Approach ...... 35

3.2.2 2D Implementation ...... 37

3.2.3 3D Implementation ...... 40

3.2.4 Adaptive Discretisation ...... 42

3.2.5 Selection of Stiffness Coefficients ...... 44

3.2.6 Experimental Observations of Discocyte Shape for Validation ...... 47

3.3 Results & Discussion...... 48

3.3.1 2D Model ...... 48

3.3.2 3D Model ...... 49

3.4 Summary & Conclusions ...... 51

3.5 Contribution to Research Objectives ...... 52

Chapter 4. Investigation of RBC Mechanical Properties during AFM Indentation ...... 53

4.1 Introduction ...... 54

4.2 Methods ...... 55

4.2.1 Experimental Observations for Model Validation ...... 55

4.2.2 Modelling Methodology ...... 57

4.2.3 Sensitivity Studies for Parameter Selection ...... 58

4.2.4 Validation of RBC Shape & Force-Deformation Behaviour ... 61

4.3 Results & Discussion...... 63

4.3.1 Energy & Shape through Indentation ...... 63

4.3.2 Effect of Changing Stiffness Coefficients ...... 67

4.3.3 Effect of Different Adhesion Levels on Adhered Shapes ...... 69

4.3.4 Comparison to Stiffness Coefficients of Previous Studies .... 71

4.4 Summary & Conclusions ...... 73

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4.5 Contribution to Research Objectives ...... 74

Chapter 5. Critical Assessment of Hertz-Based Equations for RBC Indentation ...... 76

5.1 Introduction ...... 77

5.2 Methods ...... 78

5.3 Results & Discussion ...... 81

5.3.1 Effect of Probe Size & Geometry ...... 81

5.3.2 Effect of Indenting Over Surface ...... 86

5.3.3 Discussion of Findings ...... 94

5.3.4 Proposed Alternative Estimation Method for Young’s Modulus ...... 98

5.4 Summary & Conclusions ...... 99

5.5 Contribution to Research Objectives ...... 100

Chapter 6. Investigation of RBC Mechanical Properties during Optical Tweezer Stretching ...... 101

6.1 Introduction ...... 102

6.2 Methods ...... 103

6.2.1 Experimental Observations for Validation ...... 103

6.2.2 Modelling Methodology ...... 105

6.2.3 Sensitivity Study for Parameter Selection ...... 107

6.2.4 Validation of RBC Shape & Force-Deformation Behaviour 108

6.3 Results & Discussion ...... 111

6.3.1 Energy through Stretching ...... 112

6.3.2 Effect of Changing Stiffness Coefficients ...... 113

6.3.3 Effect of Bead Size ...... 114

6.3.4 Effect of Contact Diameter ...... 116

6.3.5 Comparison of Stiffness Coefficients to Indentation Study . 117

6.4 Summary & Conclusions ...... 119

6.5 Contribution to Research Objectives ...... 120

Chapter 7. Prediction of Discocyte-Echinocyte Resting Shape Sequence 121

7.1 Introduction ...... 122

7.2 Methods ...... 124

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7.2.1 Modelling Approach Selection ...... 124

7.2.2 Modelling Methodology for Localised Spontaneous Curvature Change ...... 125

7.2.3 Experimental Observations of RBC Shape for Geometric Inputs & Model Validation ...... 128

7.3 Results & Discussion...... 130

7.3.1 Discocyte-Echinocyte Resting Shapes ...... 130

7.3.2 Discussion of Approach for Investigating Deformability ...... 132

7.4 Summary & Conclusions ...... 133

7.5 Contribution to Research Objectives ...... 134

Chapter 8. Conclusions & Recommendations ...... 135

8.1 Conclusions ...... 135

8.2 Limitations ...... 137

8.3 Future Direction ...... 139

List of References ...... 142

Appendix…… ...... 152

Appendix A – 2D Force Calculations for General Implementation ...... 153

Linear Interactions ...... 153

Bending Interactions ...... 154

Cross-Sectional Area Penalty ...... 157

Appendix B – 3D Force Calculations for General Implementation ...... 159

Linear Interactions ...... 159

Surface Area Incompressibility Penalty ...... 160

Bending Interactions ...... 162

Volumetric Incompressibility Penalty ...... 166

Appendix C – Proof-of-Concept Investigation for Relaxed Surface Area Difference Energy in 2D ...... 169

Relaxed Surface Area Energy Term ...... 169

Identify Inside and Outside Particle Projections ...... 170

Forces on Particles from Mechanism ...... 172

Proof-of-Concept Simulation Results ...... 175

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List of Figures

Figure 1.1 – Relationship between results chapters of thesis ...... 6

Figure 2.1 – Cross-section of the RBC ...... 7

Figure 2.2 – Cross-sectional schematic of the RBC membrane. Image adapted from Cooke et al. [26]...... 8

Figure 2.3 – (a) Electron microscopy image of large area of the RBC cytoskeleton showing triangulated network, (b) magnification of network showing junctional complexes, (c) schematic of structural elements including cross-links. Note variations in the spectrin cross-links with tetramers (Sp4), hexamers (Sp6) and double spectrin tetramers (2Sp4). Image from Liu et al. [5]...... 9

Figure 2.4 – Cross-sectional shape of RBC discocyte proposed by Evans and Fung [42] using Equation 2.1 ...... 12

Figure 2.5 – Classification of RBC morphologies by stage where dotted lines denote where transformation becomes irreversible. Image from Lim [34]...... 13

Figure 2.6 – Example of bilayer-couple hypothesis for shape transformations; a chemical agent (blue) preferentially enters the outer leaflet to produce an echinocyte morphology or the inner leaflet to produce a stomatocyte morphology ...... 14

Figure 2.7 – Deformed shapes of RBCs in different experiments ...... 17

Figure 2.8 – Force-indentation plots reconstructed from mean results reported by previous studies; Bremmell et al. [86] is not included as not enough information was provided for reconstruction; a typical tip angle of 20 degrees has been assumed for Ciasca et al. [36] as this was not stated23

Figure 2.9 – Schematic showing the application of force (퐹) through optical beads

and the axial diameter (퐷푎푥푖푎푙) and transverse diameter (퐷푡푟푎푛푠) measurements for the stretched RBC ...... 25

Figure 3.1 – Discretisation process; (a) schematic of RBC membrane structure, (b) schematic of numerical model including particles and springs ...... 35

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Figure 3.2 – Flow chart for layout of code ...... 37

Figure 3.3 – 2D schematic of initialised red blood cell numerical model with 푁 = 8 membrane particles ...... 38

Figure 3.4 – Spherical surface mesh where a particle is located at each of the 푁 = 122 vertices, a linear harmonic potential is located on each edge, and a bending spring is located between each pair of adjoining triangles ...... 40

Figure 3.5 – Adaptive discretisation process where the original shape is a pre- converged configuration which is then refined to continue the simulation from, and the referenced shape is where the relaxed properties are extracted from, (a) 2D implementation on a unit circle, and (b) 3D implementation on a unit sphere ...... 43

Figure 3.6 – Cross-sectional shape of RBC discocyte optimised from fitting average Ponder [40] measurements to Equation 2.1 ...... 47

Figure 3.7 – 2D discocyte resting shape prediction; (a) energy minimisation over time with inset showing first 10 µs, (b) shape evolution over time where steady-state shape is achieved after approximately 1 ms, (c) comparison of resting shape predicted by simulation to the experimentally determined cross-sections of Evans and Fung [42] and Ponder [40] ...... 48

Figure 3.8 – 3D discocyte resting shape prediction; (a) energy minimisation over time with inset showing more clearly how energy changed over the first 2 ms, (b) shape evolution over time where steady-state shape is achieved after approximately 60 ms, (c) 3D resting shape prediction viewed from three angles ...... 50

Figure 3.9 – (a) Comparison of 3D resting shape to the experimentally determined cross-sections of Ponder [40] and Evans and Fung [42], (b) comparison of 2D and 3D numerically predicted cross-sections ...... 51

Figure 4.1 – Comparison between experimental data and Hertz equation modified by Dimitriadis et al. [89] for a typical RBC where 퐸 = 9.83 kPa ...... 56

Figure 4.2 – Sensitivity study of particle number for resting and adhered shapes in (a) 2D and (b) 3D ...... 59

Numerical Modelling of Red Blood Cell Morphology and Deformability Page xiii

Figure 4.3 – Sensitivity study for contact stiffness coefficient for indentation in 2D and 3D ...... 60

Figure 4.4 – Contact between cell and probe at 200 nm indentation depth for typical case in (a) 2D and (b) 3D – negligible penetration of the membrane and probe is observed ...... 61

Figure 4.5 – 2D and 3D energy minimised shapes with optimised stiffness coefficients at rest, adhered to substrate, and compared to a typical confocal image obtained from the experiments ...... 62

Figure 4.6 – Force versus deformation curves comparing the numerical model’s prediction using optimised stiffness coefficients to the experimental reference curve (modified Hertz equation with mean Young’s modulus from experiment) ...... 63

Figure 4.7 – 2D indentation results; (a) energy stored in each mechanism through the indentation stroke, (b) change in energy between adhered state and when indented 200 nm, (c) deformed shapes at selected indentation depths ...... 64

Figure 4.8 – 3D indentation results; (a) energy stored in each mechanism through the indentation stroke, (b) change in energy between adhered state and when indented 200 nm, (c) change in energy from resting state, (c) deformed cross-sectional shapes at selected indentation depths ...... 65

Figure 4.9 – Results of parametric study in 2D measuring force to indent to nominal depth of 100 nm and 200 nm when varying the stiffness coefficients ... 67

Figure 4.10 – Results of parametric study in 3D measuring force to indent to nominal depth of 100 nm and 200 nm when varying the stiffness coefficients ...... 68

Figure 4.11 – RBC shapes predicted using 2D model when adhered to the substrate to varying degrees ...... 70

Figure 4.12 – RBC cross-sectional shapes predicted using 3D model when adhered to the substrate to varying degrees ...... 71

Page xiv Numerical Modelling of Red Blood Cell Morphology and Deformability

Figure 4.13 – Results for indentation using stiffness coefficients from previous 2D studies; force-indentation curves plotted against the best-fit effective Young’s modulus in the Hertz equation modified by Dimitriadis et al. [89] (Equation 4.1) with adhered cell shape inset for (a) Tsubota et al. [104], (b) Wang et al. [105] minimum values, and (c) Wang and Xing [107] maximum values ...... 72

Figure 5.1 – Example showing how curvature of RBC surface was found for indentation at a point of interest (4 µm from the centre here) by fitting a circle to the three closest particles, zoomed in at right ...... 79

Figure 5.2 – Geometry of conical tips represented in the model ...... 80

Figure 5.3 – Indentation with spherical probes of varying diameter; (a) selected deformed cell shapes at 200 nm indentation depth, (b) force-deformation predictions, (c) best-fit Young’s modulus predictions using various Hertz- based equations ...... 82

Figure 5.4 – Indentation with conical probes with 20° cone angle and varying tip radius; (a) deformed cell shapes at 200 nm indentation depth, (b) best- fit Young’s modulus predictions using various Hertz-based equations .. 84

Figure 5.5 – Indentation with conical probes with 10 nm tip radius and varying cone angle; (a) deformed cell shapes at 200 nm indentation depth, (b) best-fit Young’s modulus predictions using various Hertz-based equations ...... 85

Figure 5.6 – Zoomed in view of contact between probe and membrane for

푟푐표푛푒 = 10 nm and 휃푐표푛푒 = 20° showing that only spherical part of probe is contacting the membrane (red line) ...... 86

Figure 5.7 – Deformed shapes of dome-shaped cell at increasing distances from centre at 200 nm indentation depth ...... 87

Figure 5.8 – Indentation over surface of dome-shaped cell at increasing distances from centre; (a) force-deformation predictions, (b) best-fit Young’s modulus predictions using various Hertz-based equations, (c) radius of cell curvature and effective radius of contact ...... 88

Figure 5.9 – Central indentation of biconcave-shaped cell; (a) force-deformation measurements, and (b) deformed RBC shapes at selected depths ...... 90

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Figure 5.10 – Deformed shapes of cells with biconcave upper surface profile at increasing distances from centre at 200 nm indentation depth ...... 92

Figure 5.11 – Indentation over surface of cell with biconcanve upper surface profile at increasing distances from centre; (a) applied force over surface at 100 nm and 200 nm indentation depth noting negative horizontal force means it was applied in the opposite direction, (b) best-fit Young’s modulus predictions using various Hertz-based equations, (c) radius of cell curvature and effective radius of contact ...... 93

Figure 5.12 – Schematic of stress fields created for contact against (a) an isotropic solid material, and (b) a fluid-filled membrane ...... 95

Figure 5.13 – Schematic of stress fields created for solid contact when contact region is (a) negligible in comparison to size of bodies and (b) significant in comparison to size of bodies ...... 96

Figure 6.1 – Images of RBCs stretched with increasing force. Image taken from Dao et al. [98]...... 104

Figure 6.2 – Experimental measurements for axial and transverse diameter of RBCs stretched using optical tweezers. Experimental data recreated from Dao et al. [98]...... 104

Figure 6.3 – Typical stored energy versus axial diameter plot used to identify the minimum energy configuration with attached beads ...... 106

Figure 6.4 – Sensitivity study for contact stiffness coefficient for optical tweezer stretching in 3D ...... 107

Figure 6.5 – Contact between membrane and bead when RBC stretched 7 µm from resting shape with beads attached, showing negligible offset distance between the bodies ...... 108

Figure 6.6 – 3D shape predictions with optimised stiffness coefficients; (a) at rest without beads, (b) at rest with beads, (c-f) at increasing stretch distances, dimensions in micrometres...... 110

Figure 6.7 – Force versus deformation curves comparing the numerical model’s prediction using optimised stiffness coefficients to the experimental reference curves from Dao et al. [98] ...... 111

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Figure 6.8 – Energy stored in each mechanism through stretch ...... 112

Figure 6.9 – Results of parametric study measuring force to stretch to 3.5 µm and 7 µm when varying stiffness coefficients...... 113

Figure 6.10 – Force required to stretch RBC using optical beads of increasing size with inset images of stretched shapes ...... 115

Figure 6.11 – Relationship between bead diameter and curvature, noting that a curvature of zero represents a flat surface ...... 115

Figure 6.12 – Force to stretch RBC when average contact diameter between the membrane and bead is varied with inset images of stretched shapes 117

Figure 6.13 – Results for stretching RBC using stiffness coefficient values optimised for indentation listed in Table 4.1 (inset zoomed). Comparison to experimental observations of Dao et al. [98] show that the simulation predicts a significantly stiffer RBC...... 118

Figure 7.1 – Selection of edges (yellow) for an impacted region, where centre particle is nominated and adjoining interactions are selected as the region is “webbed” outward ...... 126

Figure 7.2 – Typical SEM image of RBCs; unpublished image by Marie Anne Balanant ...... 129

Figure 7.3 – Examples of (a) measuring spicule length and (b) radius of curvature of a spicule; unpublished image by Marie Anne Balanant ...... 129

Figure 7.4 – Discocyte-echinocyte sequence of resting RBC shapes predicted using the local spontaneous curvature change methodology in 3D compared to SEM images ...... 131

Figure 7.5 – Influence of membrane skeleton for large spontaneous curvatures; (a) when the cytoskeleton is not considered, buds of the characteristic radius form with narrow necks; (b) when the cytoskeleton is considered, this behaviour is replaced with spicule formation due to the high shear experienced in the neck region. Image taken from Lim et al. [27]...... 132

Figure A.1 – Isolated linear interaction in 2D showing notation for particles and their position ...... 153

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Figure A.2 – Isolated bending interaction in 2D showing notation for particles and their position ...... 154

Figure A.3 – In 2D (a) Illustration of 훼푖 and 훽푖 for convex and concave angles, and (b) schematic showing how the convexity or concavity of a region was determined ...... 155

Figure A.4 – Area under the curve in 2D found from connecting a pair of adjacent particles ...... 158

Figure B.1 – Isolated linear interaction in 3D showing notation for particles and their position ...... 159

Figure B.2 – Isolated triangle in 3D showing notation for particles and their position ...... 160

Figure B.3 – Isolated pair of adjacent triangles in 3D showing notation for particles and their position ...... 162

Figure B.4 – In 3D (a) Illustration of 훼푖 and 훽푖 for convex and concave angles, and (b) schematic showing how the convexity or concavity of a region was determined ...... 163

Figure B.5 – Volume under the surface found from projecting the vertices of a triangle onto the 푥-푦 plane as shown ...... 167

Figure C.1 – 2D schematic of with 푁 = 8 membrane particles showing the inside and outside leaflet particle projections ...... 170

Figure C.2 – Isolated group of three adjoining particles in 2D showing notation for particles and their position ...... 171

Figure C.3 – Isolated group of four adjacent particles in 2D showing notation for particles and their position ...... 173

Figure C.4 – Example result from the proof-of-concept simulations using the additional relaxed surface area energy term; inputs used for this

simulation were 푁 = 80, 푟 = 3 휇푚, 훼0 = 180°, 푅퐴 = 0.6, 푁푠푒푐 = 8, 푑0 =

50 푛푚, 푘푑 = 100 퐽, Δ푃푟푒푓,푖 = 0.43 휇푚, 퐿푖 = 1.2 휇푚 and stiffness coefficients as per 2D indentation validated case in Chapter 4 ...... 176

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List of Tables

Table 2.1 – Geometric measurements of RBC discocytes where blanks are not reported in the study ...... 11

Table 2.2 – Definitions of the echinocyte and stomatocyte morphological stages ... 13

Table 2.3 – Hertz-based equations used to analyse RBC indentation problems ..... 20

Table 2.4 – Summary of studies investigating Young’s modulus (YM) of healthy RBCs using indentation ...... 21

Table 2.5 – Summary of applied forces and bead properties for experimental stretching of healthy RBCs ...... 25

Table 3.1 – Parameters for RBC simulations with the CGPM applied in flows ...... 45

Table 3.2 – Parameter values for predicting 2D discocyte resting shape ...... 46

Table 3.3 – Parameter values for predicting 3D discocyte resting shape ...... 46

Table 4.1 – Optimised stiffness coefficients for indentation in 2D and 3D ...... 62

Table 6.1 – Optimised stiffness coefficients for optical tweezer stretching in 3D ... 108

Table 7.1 – Modelling parameters used to demonstrate discocyte-echinocyte sequence in 2D and 3D ...... 127

Table 7.2 – RBC geometric properties a function of morphological stage ...... 130

Numerical Modelling of Red Blood Cell Morphology and Deformability Page xix

List of Acronyms

2D Two-Dimensions

3D Three-Dimensions

ADE Area-Difference-Elasticity

AFM Atomic Force Microscopy

CGMD Coarse-Grained Molecular Dynamics

CGPM Coarse-Grained Particle Method

DPD Dissipative Particle Dynamics

FEM Finite Element Method

HE Hereditary Elliptocytosis

HPP Hereditary Pyropoikilocytosis

HS Hereditary Spherocytosis

IBM Immersed Boundary Method

PBS Phosphate Buffered Saline

PC Phosphatidylcholine

PE Phosphatidylethanolamine

PS Phosphatidylserine

RBC Red Blood Cell

SAGM Saline-Adenine-Glucose-Mannitol

SEM Scanning Electron Microscopy

SM Sphingomyelin

SPH Smoothed Particle Hydrodynamics

YM Young’s Modulus

Page xx Numerical Modelling of Red Blood Cell Morphology and Deformability

Chapter 1. Introduction

This chapter provides an introduction to the research topic (Section 1.1) and the objectives (Section 1.2). It then details the significance of the project (Section 1.3). Finally, an overview of the document is presented to summarise the contents of this thesis (Section 1.4).

1.1 Background

The red blood cell (RBC) is composed of a membrane which surrounds haemoglobin-rich fluid, to which dioxygen binds when the cell travels through the lungs. RBCs are responsible for distributing this oxygen throughout the body, as well as removing wastes like carbon dioxide [1]. In this process, RBCs must repeatedly pass through narrow blood vessels which can be less than half their own diameter [2]. This requires a high degree of deformability to allow the cells to squeeze through capillary beds and perform their vital functions.

RBC deformability is predominately governed by the mechanical properties of the cell membrane [3, 4] which consists of two main components – an outer lipid bilayer composed mostly of phospholipids with embedded proteins and cholesterol, and a spectrin-based cytoskeleton tethered beneath [1]. From a mechanical perspective, the bilayer provides resistance to bending and changes in its surface area. The cytoskeleton facilitates stretch deformation through the folding and unfolding of spectrin proteins which form a triangulated spectrin-network surface beneath the bilayer [5].

The surface area to volume ratio of RBCs is not readily changed even under significant loading due to areal incompressibility of the bilayer and volumetric incompressibility of the internal cytoplasmic fluid. RBCs with large surface area to volume ratios are able to adopt a broad range of shapes, enabling them to pass through restrictions with relative ease. This can be contrasted against the least deformable shape, a sphere, which is unable to deform at all if maintaining a constant surface area to volume ratio [1]. RBCs recently released from the bone marrow assume a biconcave disc shape when at rest, also known as the discocyte morphology. This morphology has a high surface area to volume ratio, and is thus the preferred shape of RBCs for optimal oxygen delivery. Most RBCs are able to

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 1 maintain the biconcave shape throughout their lifetime [3], which is typically 120 days in vivo [6]. However changes within the membrane structure can cause a morphological transformation to “cup-shaped” stomatocyte morphologies or “spikey” echinocyte morphologies [7, 8]. In both cases, the cells gradually become more spherical, reducing the surface area to volume ratio and thus restricting deformability [9].

RBC aging is associated with the transformation toward echinocyte morphologies, and this is accelerated during the storage of RBCs prior to transfusion [10]. This is thought to impact transfusion outcomes as RBCs must be sufficiently deformable at the time of transfusion to elicit the desired response of increasing the oxygen- carrying capacity of the blood [1]. Consequently, there is currently a 42 day limit for the storage of RBCs before transfusion in Australia [11]. Membrane changes are also known to impact deformability for certain RBC conditions including infection with the malaria parasite, sickle cell anaemia and some hereditary diseases [12].

Although changes within the RBC membrane are well-accepted as the basis of deformability loss, the specifics of the underlying mechanisms remain unclear [13]. These mechanisms can be explored with numerical models, which allow investigation of the mechanical aspects that define RBC behaviour at a much smaller scale than is possible with experimentation, which becomes challenging and costly [14]. Studying elasticity of the RBC membrane at this level can provide insight on the state of the membrane and how structural changes and defects impact on physical characteristics of the cells [15]. Deeper understanding of the mechanics which contribute to physical changes in RBCs will aid in developing new strategies for improving RBC storage [13], potentially leading to higher quality RBCs for transfusion and longer maximum storage durations. It may also contribute to understanding the mechanical basis of RBC conditions.

Finally, the deformability difference between healthy and deteriorated RBCs can be detected experimentally using techniques such as atomic force microscopy (AFM) indentation, optical tweezer stretching and various flow experiments [16-18]. Measuring RBC deformability in different experimental setups has subsequently been proposed as a potential tool for the diagnosis of RBC conditions and for quantifying the quality of RBC units prior to transfusion. However more work is required to understand the potential and limitations of these techniques for various applications. Consequently, the present study specifically focuses on understanding how mechanical properties of the RBC membrane impact morphology and

Page 2 Numerical Modelling of Red Blood Cell Morphology and Deformability deformability, and how experimental setups can influence deformability measurements.

1.2 Research Objectives

The objectives of this thesis are to:

1. Develop a numerical model based on the coarse-grained particle method (CGPM) representing the RBC membrane as a series of particles related by a complex spring network to accurately represent RBC morphology and deformability. 2. Validate the developed CGPM model against experimental observations of RBC shape and deformability in indentation and stretching scenarios. 3. Conduct numerical experiments using the validated model to investigate the impact of mechanical property changes of the RBC membrane, in order to identify those that most significantly affect morphology and deformability. 4. Conduct numerical experiments using the validated model to investigate which aspects of indentation and optical tweezer experimental setups impact deformability measurements.

1.3 Contribution, Innovation & Significance

This research contributes to better understanding of the factors which impact RBC morphology and deformability by applying the CGPM to RBC deformability investigations. The major innovations of this research include:

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deeply assessed using experimental or numerical means despite concerns about the applicability in the literature. . Development of the CGPM to simulate RBC behaviour during optical tweezer stretching – The established methodology explicitly considers the optical beads, thus enabling control over bead size and the degree of bead- membrane contact. This improves upon the previously most advanced particle-based models for optical tweezer stretching by Shi et al. [20] and Fedosov et al. [21] which applied stretching forces directly onto selected RBC membrane particles. . Application of the established model to investigate factors influencing deformability measurements in optical tweezer stretching experiments – The effect of bead size and degree of bead-membrane contact were investigated to assess their impact on deformability measurements. These aspects of the experimental setup had not previously been evaluated using either numerical or experimental means. . Extension of the CGPM models to the prediction of RBC shapes in the discocyte-echinocyte sequence – The CGPM methodology was developed further for the prediction of echinocyte morphologies, laying the groundwork for future investigations into their deformability when experimental data becomes available. The CGPM had not been previously shown capable of predicting these morphologies.

In the future, the models developed in this thesis may be applied to understanding the factors which influence RBC physical behaviour under specific circumstances, such as during RBC storage or when impacted by conditions such as malaria or sickle cell anaemia. For storage, this may lead to new and novel approaches to slowing deterioration, leading to higher quality cells for transfusion and improved patient outcomes. Enhanced RBC storage conditions could also reduce wastage of RBC products by extending the current 42 day maximum shelf-life, thus easing inventory and supply management pressures. In 2011-2012, more than 27,000 RBC products were discarded by health care providers, representing $9.5 million [22]. Although it is inevitable for some products to be discarded to maintain a reliable blood supply, reducing the amount is a strategic priority of the National Blood Authority, the statutory body which oversees the supply of blood and blood products and services in Australia [23]. It should be noted that due to data collection limitations, it is not possible to identify the historical reasons for discards and thus how many could have reasonably been avoided [23]. Finally, the models may also

Page 4 Numerical Modelling of Red Blood Cell Morphology and Deformability form the basis of future work informing the design of devices and data analysis techniques for quantifying the deformability of RBC units prior to transfusion or for the diagnosis of disease.

1.4 Thesis Overview

An overview of the chapters of this thesis is provided below. The relationship between results chapters is also illustrated visually in Figure 1.1 below.

. Chapter 1: Introduction – This chapter has introduced the research, its significance to RBC storage and the potential implications. It has also stated the associated objectives. . Chapter 2: Literature Review – This chapter presents a review of the relevant literature including RBC structure, deformability, morphology, and previous studies investigating the physical behaviour of RBCs of both experimental and numerical focus. . Chapter 3: Development of CGPM for Discocyte Resting Shape – This chapter aims to establish the model for predicting the resting shape of the RBC discocyte morphology in both 2D and 3D, validated against experimental observations. . Chapter 4: Investigation of RBC Mechanical Properties during AFM Indentation – This chapter adapts the model to simulate RBC indentation in both 2D and 3D. The model is then applied to numerical investigations with the aim of understanding how mechanical properties of the RBC membrane impact cellular deformability. . Chapter 5: Critical Assessment of Hertz-Based Equations for RBC Indentation – This chapter extends the indentation investigations in the previous chapter with the aim of assessing the applicability of Hertz-based equations to RBC force-indentation data for the purpose of extracting Young’s modulus of the membrane. . Chapter 6: Investigation of RBC Mechanical Properties during Optical Tweezer Stretching – This chapter adapts the 3D model to simulate the stretching of RBCs in an optical tweezer setup. The model is then applied to numerical investigations with the aim of understanding how mechanical properties of the RBC membrane impact cellular deformability. Investigations are also performed to understand the impact of aspects of the experimental setup including bead size and bead-membrane adhesion level.

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. Chapter 7: Prediction of Discocyte-Echinocyte Resting Shape Sequence – This chapter adapts the model to predict the discocyte-echinocyte sequence of RBC resting shapes and validates these against experimental observations. . Chapter 8: Conclusions – This chapter concludes the work by providing a summary of the findings and identifying the limitations. Recommendations for future research directions are also provided.

Figure 1.1 – Relationship between results chapters of thesis

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Chapter 2. Literature Review

The literature review introduces the RBC structure (Section 2.1), deformability (Section 2.2), morphology (Section 2.3), storage and other conditions impacting RBC properties (Section 2.4), experimental techniques for assessing deformability (Section 2.5), and finally modelling methods employed for RBC studies (Section 2.6). A summary of the implications are discussed in the final section (Section 2.7).

2.1 Cellular Structure

RBCs have a relatively simple structure compared to other cell types as they do not contain a nucleus or other organelles in their mature form [24]. Instead RBCs consist only of a composite membrane which surrounds a cytosol of haemoglobin- rich fluid as seen in Figure 2.1. The membrane has two main components as shown in Figure 2.2 – a lipid bilayer composed mostly of phospholipids with embedded proteins and cholesterol, and a spectrin-based cytoskeleton tethered beneath [25, 26]. The bilayer has a thickness of approximately 4 nm while the cytoskeleton has a thickness of about 50 nm [27].

Membrane

Haemoglobin-rich fluid

Figure 2.1 – Cross-section of the RBC

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Outside Cell (extracellular fluid)

Outer

Lipid Bilayer Leaflet Bilayer Inner

Leaflet

Cytoskeleton

Inside Cell (haemoglobin-rich fluid) …

Figure 2.2 – Cross-sectional schematic of the RBC membrane. Image adapted from Cooke et al. [26].

The bilayer is mostly composed of phospholipids, integral proteins and cholesterol, which together form a semi-permeable barrier between the cell and its environment. The phospholipids naturally align such that their hydrophilic (water-loving) heads are on the outer surface of the bilayer and the hydrophobic (water-repelling) tails are on the inside (see Figure 2.2) [28]. These inner and outer layers are known as leaflets and contain asymmetric phospholipid concentrations. Phosphatidylcholine (PC) and sphingomyelin (SM) are mainly concentrated in the outer leaflet, while phosphatidylethanolamine (PE) and phosphatidylserine (PS) are mostly found in the inner leaflet [12, 29]. The phospholipids on the inner and outer leaflets are not directly connected meaning they are able to slide against each other [30-32].

Integral proteins, such as band 3 and glycophorin, are embedded in the bilayer (see Figure 2.2) and have several functional roles. These include control over which molecules are allowed to pass through the membrane, sensing environmental signals and surrounding cells, and forming connections with the cytoskeleton to maintain membrane stability [33]. Cholesterol contributes to membrane fluidity, which facilitates individual protein and lipid molecules having the freedom to rotate and move laterally within the membrane. Cholesterol concentration and composition, including the number of double bonds and molecular length, influence the degree of membrane fluidity [33].

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The cytoskeleton principally consists of four peripheral proteins – spectrin, actin, protein 4.1 and ankyrin [34]. Spectrin is the main skeletal protein and is composed of α and β monomers which bond to form tetramer, hexamer and octamer molecular structures. These spectrin molecules create a two-dimensional triangulated lattice structure by interacting with actin at their ends to form junctions as shown in Figure 2.3 below. On average, six spectrin ends terminate at each junction to create hexagonal-shaped complexes, however pentagon and septagon-shaped complexes occur in 3% and 8% of cases respectively [5]. Protein 4.1 stabilises each junction as the attractive forces between spectrin and actin alone are relatively weak [34].

There are two types of couplings between the bilayer and cytoskeleton. The dominant coupling uses ankyrin to form bonds between spectrin (binding site near the centre of the spectrin molecule) and band 3 integral proteins in the bilayer [33]. The secondary coupling is much weaker and occurs between actin in the junctions and glycophorin C integral protein in the bilayer [35]. Both couplings are illustrated in Figure 2.2 above.

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2.2 Deformability

Deformability is the term used to describe the ease with which RBCs are able to change shape in response to external loading [4]. It is critical to RBCs being able to move throughout the circulation to perform their vital functions. As the RBC has no internal structural components such a nucleus or cytoskeleton that crosses the inside of the cell, its physical behaviour is governed almost entirely by the outside membrane [12].

From a mechanical perspective, the bilayer provides resistance to bending deformation. It also maintains a near constant surface area at all times, despite the repeated deformation cycles [21, 34, 36]. Curvature can be induced within the bilayer by a variation in the properties of the inner and outer leaflets [30], discussed further in Section 2.3. The cytoskeleton facilitates stretch deformation through the folding and unfolding of spectrin molecules. The independent domains which make up the molecule are stretched and then unfolded under tensile force, resulting in a sawtooth pattern for force versus deformation [37, 38]. Maximum extension, known as the contour length of the molecule, is regularly approached during cell deformation cycles [38]. The average end-to-end distance of in vivo spectrin tetramers (the most common form of spectrin in the RBC membrane) is 75 nm, however the contour length is 200 nm [2]. This axial deformation takes place within the triangulated lattice structure which is able to resist shear. Thus flexibility of the membrane structure in axial, shear and bending directions controls to the ease with which the cell can change shape as required.

Cell geometry is a factor in deformability, defined by the surface area to volume ratio [3, 4]. This is because RBC surface area is not readily changed even under significant external loading due to surface area incompressibility of the bilayer and volumetric incompressibility of the internal fluid. RBCs with large surface area to volume ratios are able to adopt a broad range of shapes while maintaining a constant ratio enabling them to pass through restrictions with relative ease. This can be contrasted against the least deformable shape, a sphere, which is unable to deform at all if maintaining a constant surface area to volume ratio [1].

Finally, cytoplasmic viscosity can impact morphology but only in dynamic scenarios as it is a time-based property. Furthermore, its effect only becomes dominant at very high haemoglobin concentrations that are not observed in healthy individuals [3].

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2.3 Morphology

RBCs develop from hematopoietic stem cells in the bone marrow [39]. In their immature form they contain a nucleus and are spherical in shape. However with time, the nucleus decreases in size and is ejected. The cell then enters the circulation and transforms from a sphere to the biconcave disc shape synonymous with RBCs [3, 27]. The biconcave disc is the optimal shape for RBCs due to the high surface area to volume ratio that provides a great degree of deformability [9]. Most RBCs are able to maintain the biconcave shape throughout their lifetime in vivo [3].

The geometry of RBCs of the discocyte morphology has been studied extensively with several reports in the literature for these cells’ dimensions. Those studies measuring multiple geometric properties are summarised in Table 2.1 below. Donor- to-donor variation is a major contributor to differences reported within and between studies.

Table 2.1 – Geometric measurements of RBC discocytes where blanks are not reported in the study Min Max Surface Diameter Volume, V Reference Thickness Thickness Area, SA (µm) (µm3) (µm) (µm) (µm2) Ponder [40] 8.55 1.02 2.40

Canham and 8.07 138.1 107.5 Burton [41] Evans and 7.82 0.81 2.58 135 94 Fung [42] Jay [43] 8.03 136.9 104.2

Linderkamp and 7.88 134.1 89.8 Meiselman [44] Kugeiko and 5.2-10.1 0.1-2.5 1.5-4.1 Smunev [45]

In addition to the above dimensions, Evans and Fung [42] characterised the average cross-sectional shape for RBCs suspended in solutions of varying osmolarity. This study estimated the RBC height, 푧, as a function of radius, 푟, to generate an equation to describe the cross-sectional shape of RBCs (Equation 2.1).

This equation relies on four parameters – the cell radius, 푅0, and three constants named 퐶0, 퐶2 and 퐶4. For cells at physiological osmolarity (300 mOsm) the average radius was found to be 3.91 µm, while the constants 퐶0, 퐶2 and 퐶4 were 0.81 µm, 7.83 µm and -4.39 µm respectively. The cross-section this predicts is plotted in Figure 2.4. Equation 2.1 can be converted into a three-dimensional description of

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RBC shape using the relationship between radius, 푟, and the 푥 and 푦 positions of 푥2 + 푦2 = 푟2.

1 2 2 2 4 1 푟 푟 푟 ( 2.1 ) 푧(푟) = ± [1 − ( ) ] [퐶0 + 퐶2 ( ) + 퐶4 ( ) ] ; −푅0 ≤ 푟 ≤ 푅0 2 푅0 푅0 푅0

Figure 2.4 – Cross-sectional shape of RBC discocyte proposed by Evans and Fung [42] using Equation 2.1

RBCs are able to undergo significant shape transformations away from the discocyte morphology. A classification of RBC resting shapes was first proposed by Bessis in 1973 [46] and is illustrated in Figure 2.5. Descriptive definitions for each stage are included in Table 2.2 based on Bessis [46] and Lim [34]. It can be seen that RBCs can transform into “cup-shaped” stomatocytes, or develop spicules (small membrane protrusions) to become “spikey” echinocytes. In both cases cells gradually loose deformability through the transition. The morphological changes are reversible through Stages I, II and III of each branch, however they are irreversible once Stage IV is reached. This is due to the shedding of small membrane pieces, known as micro-vesiculation [34]. This process is thought to be used by the cell to remove defective parts of its membrane [6, 47]. The loss of micro-vesicles reduces RBC surface area more than volume and severely restricts deformability [48]. Beyond Stage IV, damage becomes so severe that cells haemolyse [49]. Stage IV of each branch may also be referred to as sphero-echinocytes and sphero- stomatocytes respectively [34]. No studies have been identified which quantify the geometry of echinocytes or stomatocytes other than Piety et al. [49] who measured RBC diameter at various morphological stages.

Page 12 Numerical Modelling of Red Blood Cell Morphology and Deformability

Figure 2.5 – Classification of RBC morphologies by stage where dotted lines denote where transformation becomes irreversible. Image from Lim [34].

Table 2.2 – Definitions of the echinocyte and stomatocyte morphological stages Morphology Stage Definition IV Sphere with irregularly contoured region on one side III Sphere with a minimal central depression Stomatocyte II Cup-shape with a deep invagination I Cup-shape with a shallow invagination Discocyte - Normal biconcave disc I Irregularly contoured discocyte II Flat cell with spicules distributed evenly over its surface Echinocyte III Sphere with 30 to 50 regularly spaced sharper spicules IV Sphere with short, needle-like spicules

The bilayer-couple hypothesis was first proposed by Sheetz and Singer [50] and is one of the most well-accepted physical explanations for RBC shape transformations. It states that the leaflets of the bilayer respond differently to shape- changing agents while remaining coupled to each other. Shape-changing agents include changes in the concentration of amphipathic drugs, cholesterol and salt, as well as changes in pH [34]. Shape-changing agents preferentially act on the inner or outer leaflet due to the phospholipid asymmetry and change the relaxed surface area difference by increasing or decreasing the number of molecules in a leaflet, or changing the projected surface area per molecule in a leaflet [27]. This leads to a difference in the relaxed surface area of the outer leaflet compared to the inner leaflet of the bilayer. Those agents causing an expansion of the outer leaflet compared to the inner favour echinocyte morphologies, and those causing a contraction of the outer leaflet compared the inner favour stomatocyte morphologies as shown in Figure 2.6.

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Outside Cell

Inside Cell

Another interpretation of this concept which has been used to explain spicule development is localised increases in membrane spontaneous curvature [47]. Spontaneous curvature is the natural curvature that the bilayer wants to form due to the difference in molecular shape between molecules of the inner and outer leaflets [34]. Changing spontaneous curvature is said to be equivalent to changing the relaxed surface area difference between leaflets with regard to the impact on cell shape, as spontaneous curvature originates from the area difference [47]. It should be noted that spontaneous curvature not only stems from a difference in physical surface area, but can also be caused by protein-lipid hydrophobic mismatches as well as assisted by proteins which impose their own curvature on the bilayer or insert themselves into one leaflet more so than the other [51].

2.4 Storage & Other Conditions Impacting RBC Properties

2.4.1 Storage & Transfusion

RBCs are regularly transfused to patients for the treatment of cancer, blood diseases and anaemia, as well as during surgery and trauma [11]. As approximately 98% of oxygen is carried through the body bound to the haemoglobin in RBCs (with the other 2% dissolved in plasma), the transfusion of RBCs increases blood’s oxygen-carrying capacity to restore tissue oxygenation, vital to keeping organs alive and fully functional [52].

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RBCs are separated from the whole blood collected from volunteer donors. They are separated from the white blood cells, plasma and platelets contained in whole blood using centrifugation processing and are stored in an additive solution to prolong their life in refrigerated storage. In Australia, RBCs are stored in a saline- adenine-glucose-mannitol (SAGM) solution at 4°C [11]. To meet international quality standards, 75% of RBCs must survive 24 hours in the body following transfusion and the level of haemolysis (cells which have ruptured) must be less than 0.8% at the end of the storage period [53]. Current Australian practices allow for a maximum of 42 days storage under these standards [11].

RBCs deteriorate more rapidly in the storage environment compared to that experienced in the body [52, 54]. The reduced lifespan of stored RBCs is a direct result of the storage conditions that lack the clearance and upkeep mechanisms present in the body [48]. The complex set of biochemical and biomechanical changes that RBCs undergo in storage are collectively known as the “storage lesion” [55]. The most significant biomechanical change resulting from the storage lesion is the accelerated morphological transformation, where RBCs transform into echinocytes by developing spicules on their surface and becoming more spherical [52, 56].

Changes to the composition and structure of the RBC membrane are well-accepted as the cause for deformability loss and the shape evolution, however the specifics of the underlying mechanisms remain unclear [13]. This is due to the complexity of the changes and their interactions. Band 3 protein changes are thought to play a key role as these are abundant in the membrane and involved in binding the cytoskeleton and bilayer together [54]. Identified changes include oxidative degradation [6, 54], crosslinking [54] and clustering [6, 57]. The latter can be promoted by haemoglobin denaturation [58]. Likewise, changes to ankyrin proteins can impact the bilayer-cytoskeleton connection [57]. With regard to lipids, their distribution between bilayer leaflets has been reported to evolve over time. In particular, an increase in the concentration of PS in the outer leaflet has been associated with cell aging and is thought to be a senescence (cell death) trigger [54]. Lipids are also lost from the membrane through vesiculation where small parts of the membrane bud off [47]. With regard to the cytoskeleton, increased crosslinking of spectrin and actin restricts movement, which may be caused by a shift in the forms of spectrin present within the cytoskeleton (tetramers, hexamers and so on) [59]. An increase in the average length of spectrin molecules within the triangulated network has also been reported to reduce deformability [15].

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The changes that occur to RBCs through the storage period are thought to impact transfusion outcomes given the reduction in functionality [54]. Several studies have reported RBC storage duration to be an independent risk factor in high-risk patient groups like those suffering from trauma and the critically ill [60, 61] as well as sickle cell disease patients who require regular transfusions to manage their symptoms [62]. Links have been made with the rate of multi-organ failure [63], infection [62, 64], deep vein thrombosis [65] and cardiovascular complications [66] (a comprehensive summary is included in Tinmouth et al. [61]). However the extent to which this impacts within the 42 day storage timeframe is not fully understood. One barrier to reaching a consensus is research design limitations which make generalisation and extrapolation of results challenging [67, 68] – studies which report a relationship between RBC storage duration and patient outcomes often suffer from small sample sizes and are based on the retrospective analysis of clinical data rather than the ideal case of large randomised clinical trials [62, 68]. Another limitation is that most RBC units are transfused well before the 42 day deadline [66], with the mean age of transfused RBCs typically between 16 and 21 days [69]. This restricts data collection for the sub-population of RBC units transfused at the very end of shelf-life, where the storage lesion may have a greater impact. This is a limitation for even a very recent large-scale international investigation designed as a randomised double-blind trial, which reported no significant difference between transfusing young and old RBC units [70]. Here the average storage duration for older units was 22 days compared to 12 days for the younger units. Therefore the extent to which the RBC storage lesion impacts transfusion outcomes remains an unresolved question.

Currently there are several directions being pursued for modifying storage conditions to reduce the impact of the RBC storage lesion. These include varying chemical constituents, concentrations, pH and osmolarity of the storage solution [55, 71], incubating the cells in a second solution following SAGM storage [48], sustaining metabolic processes by supplementing the storage solution with glucose [72], and depleting blood bags of oxygen content to reduce oxidative damage [73]. Present storage protocol is based on technology developed over two decades ago and has not been largely improved upon since [55]. Therefore there is significant potential to enhance the storage of RBCs to improve cell quality over time. This also then has the potential to increase maximum storage durations, easing supply chain management problems and reducing waste.

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2.4.2 Other Conditions Impacting RBC Properties

Some disorders of the RBC are known to cause irregular morphologies which arise from membrane abnormalities. Hereditary elliptocytosis (HE) and hereditary pyropoikilocytosis (HPP) are caused by insufficient linkages in the spectrin network, resulting in ellipse-shaped and distorted RBC morphologies respectively [12]. Hereditary spherocytosis (HS) sufferers show highly spherical RBC morphologies arising from a lack of spectrin, ankyrin and band 3 proteins which limit the formation of primary couplings. This causes separation of the bilayer from the cytoskeleton [12]. Sickle cell disease is a genetic disorder which affects haemoglobin. It changes haemoglobin’s binding characteristics, causing the formation of long and stiff rod- like fibres. These act to increase membrane rigidity and cause a “sickle” morphology [12, 74]. Malaria infection also impacts RBC properties, significantly altering mechanical behaviour by increasing membrane rigidity and reducing deformability through the insertion of parasite-encoded proteins in the membrane [26].

2.5 Physical Experiments

RBC deformability can be quantified using several experimental techniques including micropipette aspiration, optical tweezer stretching, flow scenarios and atomic force microscopy (AFM) [75]. The physical experiments induce different deformed shapes to the RBC with typical examples shown in Figure 2.7.

Micropipette Optical Tweezer Atomic Force Flow Aspiration Stretching Microscopy

Figure 2.7 – Deformed shapes of RBCs in different experiments

Micropipette aspiration is one of the oldest methods for assessing RBC membrane properties [75]. It involves sucking a small region of the cell into a very small pipe and measuring the distance it enters under this known pressure. As the technique applies force on only a specific region of the cell, local deformability of the membrane is measured. The region targeted is on the order of 1 µm diameter [76]. In contrast, AFM is able to target membrane regions on the nanoscale by applying a force through a probe [75]. This has led to AFM being used to investigate how

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 17 mechanical properties vary in different regions of the membrane [36, 77]. It is discussed in more detail below.

Flow visualisation involves placing RBCs in a flow field and observing their deformed shapes. Often of particular interest is flow through pipes on the scale of a capillary as RBCs assume parachute shapes to enable their passage [78-80]. Optical tweezer stretching also involves applying force to the overall RBC through beads attached to the membrane [75]. Consequently, it affords more fine-tuned control over the force’s application. This technique is also discussed in more detail below.

2.5.1 AFM Indentation

AFM indentation is a relatively new technique which involves a cantilevered probe applying force onto a sample while displacement is measured. The position of the probe is detected by a sensor which measures the deflection of a laser beam reflected off the cantilever. Applied force is correlated against the deformation measurement using the stiffness properties of the cantilever [75].

In order to perform indentation, cells must be adhered to the substrate to prevent them from slipping out from under the probe. Poly-lysine is a chemical typically used for this purpose [36, 74, 77, 81-83]. Depending on the protocol, RBCs can exhibit either dome-shapes or biconcave shapes on their surface once adhered. This experimental step can cause tension to develop in the membrane if the adhesion protocol is too strong [84, 85] and can also cause changes within the membrane’s structure. These effects should be minimised if the natural state is to be observed.

Another major consideration for AFM studies is probe selection, with conical [36, 82], pyramidal [74, 77, 81] and spherical tips [83, 86, 87] previously used on RBCs. There is also significant variation in the size of probes, with tip radii ranging from 10 nm [36, 77, 88] to 50 nm [81, 82] for conical and pyramidal probes, and diameters between 6 µm [86] and 15 µm [83] for spherical probes. Conical and pyramidal probes are able to target a highly specific region of the cell membrane, however their sharpness can push the membrane beyond physiological limits leading to penetration and rupture [89]. Spherical probes can overcome these risks given their smooth surface [89], however the probes tend to be larger in size. This means they cannot target as localised regions of the membrane.

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AFM force-deformation curves have been traditionally analysed using equations with roots in Hertzian contact theory [90] in order to obtain Young’s modulus of the membrane. This theory was derived for frictionless contact between elastic, isotropic solids where the size of the contact area is small compared to the size of the contacting bodies [91]. The original Hertz equation describing the force- deformation relationship for contact between two spherical bodies is given in Equation 2.2 [91], where 퐸∗ is the combined modulus of the bodies in contact and 푅 is the effective radius of the contact.

4 퐹 = 퐸∗푅0.5훿1.5 ( 2.2 ) 3

The combined modulus is given by Equation 2.3 where 퐸1 and 휈1, and 퐸2 and 휈2 refer to the Young’s modulus and Poisson’s ratio of the respective bodies in contact. When a soft body (such as a cell) is in contact with a much more rigid body (such as an AFM probe), Young’s modulus of the rigid material can be assumed infinitely large [91]. This simplifies the expression for effective modulus to Equation 2.4, where 퐸 is Young’s modulus and 휈 is Poisson’s ratio of the softer body. For soft tissues, Poisson’s ratio is typically in the range of 0.490 to 0.499 [92]. Therefore it is normally assumed 휈 = 0.5 [36, 74, 77, 82, 83]. The effective contact radius is given by Equation 2.5 where 푅1 and 푅2 are the radii of the respective bodies.

1 (1 − 휈 )2 (1 − 휈 )2 1 2 ( 2.3 ) ∗ = + 퐸 퐸1 퐸2 퐸 퐸∗ = ( 2.4 ) (1 − 휈)2 1 1 1 = + ( 2.5 ) 푅 푅1 푅2 Modifications have been made to the original Hertz equation to account for the different contact areas and pressure distributions created when probe geometries such as conical and pyramidal are used [93-96]. Those alternatives previously applied in RBC indentation studies are summarised in Table 2.3. To obtain Young’s modulus from the force-deformation experimental data, the relevant Hertz-based relationship between applied force (퐹) and indentation depth (훿) is fitted to minimise the error between the curves [36, 74, 77, 81-83, 86-88].

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Table 2.3 – Hertz-based equations used to analyse RBC indentation problems Description Equation Hertz Equation refined 2퐸tan(휃)훿2 퐹 = 휋(1 − 휈2) by Briscoe for Conical ( 2.6 ) Probe [94] 휃= angle of cone side

2 Hertz Equation refined 2퐶0퐸훿 퐹 = by Bilodeau for Four- 휋(1 − 휈2) tan(휃) ( 2.7 ) Sided Pyramidal 휃=angle of pyramidal tip side, 퐶0=probe Probe [95] geometry coefficient

Hertz Equation refined 3퐸훿2 tan(휃) 퐹 = 4(1 − 휈2) by Rico for Four-Sided ( 2.8 ) Pyramidal Probe [96] 휃=angle of pyramidal tip side

Table 2.4 below summarises the results from previous studies for Young’s modulus of the RBC membrane. It can be seen that a one thousand-fold variation in Young’s modulus has been reported – between 0.1 kPa [83] and 98 kPa [97]. It should be noted that the Lamzin and Khayrullin [88] results need to be interpreted with caution as this study appears to have unknowingly reported the combined modulus (퐸∗) rather than Young’s modulus (퐸). Using Equation 2.4 and 휈 = 0.5 to correct this, Young’s modulus would be 75% of their reported values. Furthermore, the results of Bremmell et al. [86] are difficult to compare to the other studies as Hertz-based analysis was not used, and the alternative analysis procedure lacks detail and justification.

To further compare studies, representative force-deformation results were reconstructed using the respective Hertz equations and mean reported Young’s modulus (Figure 2.8). The larger Young’s modulus values do correspond to the studies applying the largest forces despite the use of different forms of the Hertz equations. Thus the large variation in forces applied at increasing indentation depths is consistent with the great range of Young’s modulus values reported.

Page 20 Numerical Modelling of Red Blood Cell Morphology and Deformability

Table 2.4 – Summary of studies investigating Young’s modulus (YM) of healthy RBCs using indentation Probe YM, Location for Sample Analysis Adhered Cell Indentation Study Sample Size Shape & M ± σ (kPa) YM Preparation Method Shape Depth Limit Size Lekka et 4.9 ± 0.5 13 donors, Average over Poly-L-lysine Pyramidal, Hertz Equation, Not reported 500 nm al. [81] ~20 cells per cell surface for adhesion, 50 nm tip assumed same donor dried cells radius as Dulińska et al. [82] (Equation 2.2) Bremmell Early Region Not reported Not reported 3- Spherical, 6 Linear fits Biconcave Not reported et al. [86] Deformation: aminopropylt µm diameter through three 4 rimethoxysila regions of the ne for force- Later Region adhesion, deformation Deformation: resuspended curve (not Hertz- 35 – 150 in phosphate based analysis), buffered unclear how saline (PBS) regions are defined Dulińska 26 ± 3 Not reported Central area Poly-L-lysine Conical, 40- Equation 2.2 Biconcave Indented to et al. [82] for adhesion, 50 nm tip where 푅=tip 400 nm, Glutaraldehy radius radius rather Young’s de for fixing, than effective modulus fitted resuspended radius to 200 nm in PBS depth Maciaszek 1.1 ± 0.4 Not reported Average over Poly-L-lysine Pyramidal, Hertz Equation Biconcave Indented to and cell surface for adhesion, 20 nm tip refined by Rico 700 nm, Lykotrafitis resuspended radius [96], Equation Young’s [74] in PBS 2.8 modulus fitted to 250 nm depth

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 21

Probe YM, Location for Sample Analysis Adhered Cell Indentation Study Sample Size Shape & M ± σ (kPa) YM Preparation Method Shape Depth Limit Size Girasole Average: Not reported Over cell Poly-L-lysine Pyramidal, Hertz Equation Biconcave 50 – 60 nm et al. [77] 98 ± 17 surface for adhesion, 10 nm refined by Central area: Cresyl blue nominal tip Bilodeau [95], 90 ± 14 stain, May– radius Equation 2.7 Ridge area: Grundwald– 107 ± 16 Giemsa (MGG) fixing Li et al. 0.1 – 0.2 5 cells Average over Poly-L-lysine Spherical, 15 Equation 2.2 Not reported 500 nm [83] cell surface for adhesion, µm diameter resuspended in HBBS Lamzin Day 2: Day 2: Average over Dried cells Conical or Equation 2.2, Biconcave Not reported and 1.81 ± 0.4 360 cells cell surface Pyramidal however error in Khayrullin (not stated), analysis as [88] Day 35: Day 35: 10 nm tip combined 3.23 ± 0.70 945 cells radius modulus reported rather *combined than actual modulus Young’s modulus reported Ciasca et Average: 15 donors, Over cell Poly-L-lysine Conical, 10 Hertz Equation Biconcave 200 nm al. [36] 1.82 ± 1.6 cells per surface for adhesion, nm tip radius refined by donor not resuspended Briscoe [94], Central area: reported in PBS Equation 2.6 Up to 9

Edge area: Down to 0.06

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Differences in Young’s modulus between studies can be attributed in part to varying sample preparation and indentation protocols. Those studies that reported a higher Young’s modulus typically used chemicals to fix the membrane [82, 97] or dried the samples [88], which are known to have a stiffening effect. Given the lack of studies using spherical tips, it is difficult to gauge if probe shape has influenced results. Similarly, of those studies reporting the adhered shape of the RBCs, all observed biconcave surface profiles making it impossible to assess if adhered shape was a factor. Several different Hertz-based equations have been applied to extract Young’s modulus and a large range of indentation depths have been used – between 50 nm [77] and 500 nm [81, 83]. Again no correlation with Young’s modulus is apparent for these attributes, however this may be due to other factors confounding the comparison.

A factor which does appear influential in the reported Young’s modulus but has not been deeply analysed in the literature is the region of the cell surface for which Young’s modulus was measured, as it has been shown to vary based on surface location [36, 82, 97]. Most studies have reported an average Young’s modulus over the entire cell surface, with the exception of Dulińska et al. [82] who reported the

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 23 central region, Girasole et al. [97] who reported the ridge and trough of a fixed biconcave disc, and Ciasca et al. [36] who specifically investigated the variation in Young’s modulus over cell surface. Ciasca et al. [36] reported Young’s modulus significantly higher in the central region of RBCs – for a “typical cell”, Young’s modulus was as high as 9 kPa at the centre, as low as 0.06 kPa near the edge, and 1.87 kPa when averaged over the entire surface. This was attributed to the biconcave shape of the RBC and local changes in the cell’s elastic properties [36]. Thus the region of the cell for which Young’s modulus is reported also needs to be considered when comparing the results of different studies.

Despite the extensive use of Hertz-based analysis for extracting Young’s modulus of the RBC membrane, there are serious concerns in the literature about the validity of Hertzian contact theory for quantifying Young’s modulus of the RBC membrane [19]. This is due to the theory being developed specifically for solid-to-solid contact of elastic, isotropic materials where the size of the contact is negligible compared to the bodies themselves [91]. Instead RBCs are soft matter and their small size may mean that the substrate does influence their physical behaviour, especially during deep indentations. Despite studies claiming that Hertz-based analysis is accurate for analysing RBC force-indentation curves [74], no deep analysis of the accuracy of this approach has been proposed in the literature so far.

Quantifying Young’s modulus of the RBC membrane has value for developing diagnostic techniques based on indentation. This is because underlying changes in the membrane structure that impact deformability can be detected experimentally [16]. AFM force-indentation results have been shown sensitive enough to detect a difference in deformability between healthy RBCs and those from patients suffering from diabetes [36, 81] and sickle cell disease, as well as between individuals who did and did not smoke cigarettes [81]. Furthermore, differences have been detected between RBCs stored for different lengths of time [88], which provides promise for the use of this technique in evaluating stored RBC unit quality. However, the current limitations with force-indentation data analysis need to be addressed to improve confidence in comparing the Young’s modulus values between different experimental setups.

Page 24 Numerical Modelling of Red Blood Cell Morphology and Deformability

2.5.2 Optical Tweezer Stretching

Optical tweezer stretching involves attaching microscopic beads to the cell membrane which are then pulled on with lasers. The pulling force is generated by the change in momentum of the photons within the laser beam as they pass through the beads which have a high refractive index. The imparted force pushes the beads toward the focal point of the laser beam [98]. This technique can be used to stretch the cell along its axial direction, causing it to contract in the lateral direction. Force applied to the cell is found from the relationship with laser power, while critical cell dimensions are measured using optical methods [17]. A schematic of the experimental setup is shown in Figure 2.9 where 퐷푎푥푖푎푙 and 퐷푡푟푎푛푠 are the axial and transverse diameters respectively, and 퐹 is the applied force.

Figure 2.9 – Schematic showing the application of force (퐹) through optical beads and the axial

diameter (퐷푎푥푖푎푙) and transverse diameter (퐷푡푟푎푛푠) measurements for the stretched RBC

Optical tweezer stretching of RBCs has been explored in several prior studies [17, 18, 20, 98-102] with key attributes of the experimental protocols summarised in Table 2.5. It can be seen that the optical beads have ranged in diameter between 2.1 µm [101] and 5 µm [18], and applied forces have been as large as 400 pN [98]. Bead-membrane contact diameters, where reported, have been fairly consistent.

Table 2.5 – Summary of applied forces and bead properties for experimental stretching of healthy RBCs Estimated Bead Maximum Force Bead Diameter Study Contact Applied (pN) (µm) Diameter (µm) Hénon et al. [101] 60 2.1 Not Stated Dao et al. [98] 400 4.12 1-2 Mills et al. [103] 200 4.12 2 Suresh et al. [17] 200 4.12 2 Li et al. [18] 315 5 Not Stated

The force-deformation response from stretching tests has shown excellent potential for the technique to be developed a diagnostic tool. Suresh et al. [17] showed that

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 25 the force-deformation response changes markedly between healthy RBCs and those infected with the malaria parasite. Furthermore, the deformability gradually reduced through the stages of disease progression. Li et al. [18] demonstrated that the deformability of RBCs decays with storage duration. This supports the potential of this technique for quantitatively assessing the quality of stored blood prior to transfusion. Brandao et al. [102] showed that RBC deformability differed between patients with sickle cell disease, sickle cell trait and healthy individuals. Finally, Hénon et al. [101] showed that the deformability of a RBC of discocyte morphology behaved differently to one which had been swelled to be spherical. This finding would need to be considered in the development of the diagnostic tools, given that the swelling ratio may impact deformability measurements even when membrane mechanical properties are constant. Therefore further work on reconciling analysis techniques may be required to make results comparable between setups. No studies have reported a Young’s modulus for the RBC membrane using this experimental method.

2.6 Modelling & Simulations

To investigate and understand the more fundamental mechanics of RBC deformability, it is desirable to develop numerical models. This allows investigation of the mechanical aspects that define RBC behaviour at a much smaller scale than is possible with experimentation, which can become challenging and costly [14]. Studying elasticity of the RBC membrane at this level can provide insight on the state of the membrane and how structural changes and defects impact on physical characteristics of the cells [15].

2.6.1 Coarse-Grained Particle Method

The CGPM was first applied to RBCs by Tsubota et al. [104], and has since been implemented in a number of subsequent studies [14, 20, 78, 105-118]. It involves coarse-graining the membrane into particles which are interconnected by a network capable of storing energy. The preferred resting shape of the RBC is obtained by minimising total stored energy, given that this shape is easiest for the cell to maintain. The principle of virtual work (force obtained from the partial derivative of energy with respect to particle positions) and Newton’s Second Law are used to drive the energy minimisation process.

Page 26 Numerical Modelling of Red Blood Cell Morphology and Deformability

The energy-storing network is developed to model the mechanical roles of cellular components. This includes stretch resistance provided by the spectrin-based cytoskeleton, surface area incompressibility of the bilayer, bending resistance of the bilayer and volumetric incompressibility of the internal fluid. A stiffness coefficient is associated with each of these energy storing mechanisms to control its strength. This is a significant point of difference between models (discussed with respect to Table 3.1 later in Section 3.2.5.1). The CGPM has been demonstrated to efficiently predict overall resting shape for discocytes, achieving good agreement with experimental observations in both 2D and 3D [14, 20, 78, 105-118]. However, it should be noted that this method has not been used to predict stomatocyte or echinocyte morphologies.

The CGPM has almost exclusively been applied to simulating the behaviour of RBCs in flows. This is done by introducing a set a fluid particles and then coupling the CGPM with another method such as smoothed particle hydrodynamics (SPH) (e.g. [78, 111, 117]) or the immersed boundary method (IBM) (e.g. [14, 109]). Shi et al. [20] also used the CGPM to simulate stretching. Thus the CGPM has been shown to efficiently capture the large deformations that RBCs are capable of at a whole cell scale in both flow and stretching scenarios.

2.6.2 Coarse-Grained Molecular Dynamics

In coarse-grained molecular dynamics (CGMD), coarse-graining of the membrane is also completed. However, the elements are on a much smaller scale. This enables molecular complexes to be modelled with different types of particles with varying properties. For example, Li and Lykotrafitis [119] used particles of six varieties: actin junction complexes, spectrin particles, glycophorin C particles, band 3 complexes which do and do not connect with the spectrin network and lipid particles. Relationships were drawn between particles as per the RBC structure using harmonic springs and Lenard-Jones (attraction-repulsion) potentials. As in the CGPM, the energy network was relaxed using the principle of virtual work and Newton’s Second Law, until the equilibrium configuration was achieved.

CGMD models have been used to investigate defects within the RBC membrane such as deficient couplings between the bilayer and cytoskeleton, and a lack of connections at actin junctions [119, 120]. This is of relevance to RBC membrane disorders like HS, HE and sickle cell disease which experience breakdowns in certain areas of the membrane structure [12]. Particle-based methods are well-

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 27 suited to this type of investigation as local defects can be introduced easily [121]. The CGMD model has since been extended to investigate vesiculation of the RBC membrane by designating regions of the membrane to be impacted by a change in spontaneous curvature [47]. A compressive force was also applied to the membrane surface, and when energy was minimised, buds gradually formed in the designated regions [47].

A major drawback of CGMD models is that they are suited to simulating small regions of the RBC membrane, and when simplified in this manner, cannot take overall RBC shape into account [47, 119, 120]. This is due to the far more detailed representation of the membrane which means an enormous number of particles are required to discretise the system. Consequently, the simulation becomes extremely costly if the entire RBC membrane is represented at this scale. Tang et al. [121] attempted to overcome this as part of the OpenRBC project which developed several new computational techniques. This included an adaptive spatial-searching algorithm to reduce time identifying locations for short-range pairwise interactions. This made simulations achievable, however they still required significant high performance computational infrastructure and incurred a large time cost. Nonetheless, this model has been used to predict the shape of RBCs of the discocyte morphology and has been able to capture membrane wrinkling. The model has also been applied to optical tweezer stretching and vesiculation. However, the suitability of this approach needs to be evaluated against the substantial cost of performing the simulations, given that millions of particles and time steps are necessary.

2.6.3 Dissipative Particle Dynamics

Another particle-based method is dissipative particle dynamics (DPD). It was used by Fedosov et al. [21] to simulate RBCs in flow and stretching, and by Peng et al. [100] to investigate bilayer-cytoskeletal interactions of RBCs both in flow and during micropipette aspiration. DPD is similar to the CGPM, using energy terms to describe the relationships between membrane particles. The system is also evolved with the principle of virtual work. The key difference between DPD and CGPM is that a fluid phase is innately incorporated in DPD while this is not necessary for the CGPM. Only discocyte morphologies have been modelled with DPD.

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2.6.4 Bilayer-Couple Hypothesis-Based

The bilayer-couple hypothesis (introduced in Section 2.3) has been incorporated into numerical models to predict stomatocyte and echinocyte morphologies at rest.

The shape transformation is driven by Δ퐴0, the relaxed surface area difference between leaflets of the bilayer. Prediction of the range of RBC morphologies was first achieved in the work of Lim et al. [8, 27, 34].

The parameter, Δ퐴0, is integrated into the energy term that accounts for bilayer behaviour including bending (discussed in more detail in Section 7.1 later). Other energy terms correspond to those in the CGPM and DPD methods of surface area incompressibility, volumetric incompressibility and stretch resistance. Total energy in the cell is minimised in order to predict preferred resting shape. However, a Monte Carlo randomised approach is used rather than the principle of virtual work to minimise energy. In one respect, the Monte Carlo method is regarded as advantageous because it is analytically simpler. However, computation time can be longer as not all trial moves are successful in lowering the energy of the system [34].

Although a handful of studies have shown echinocyte and stomatocyte morphologies can be predicted numerically, these existing models have limitations. Significantly, studies have only predicted the resting shapes of RBCs [8, 27, 34, 122, 123] – no studies have applied external loads to evaluate deformability. Another limitation relates to the use of the Monte Carlo energy minimisation, which cannot be extended easily to take into account a fluid phase or dynamics. In the other particle-based methods, the principle of virtual work was applied to move the particles according to forces derived from the energy mechanisms which evolved with time. Given that the Monte Carlo method relies on random movement of the particles, this development is not possible.

Another avenue which has not been pursued is a parametric study on the impact of various stiffness and geometric parameters on morphology, as previous modelling work has only focused on varying relaxed surface area difference [8]. Significant examples of this include the change in surface area and volume due to vesiculation [47] as well as membrane structural defects which can accrue [10].

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 29

2.6.5 Finite Element Methods

Another method to consider for simulating RBC physical behaviour is the finite element method (FEM). It has been used to investigate RBCs in flow [124, 125], micropipette aspiration [100] and stretching [98, 100]. However, each of these studies required the shape of the RBC to be pre-defined, typically using the equation developed by Evans and Fung [42] describing the biconcave disc cross- section (Equation 2.1). This is an issue when investigating other morphologies, especially echinocytes, as these have not been quantified with equations describing their surface geometry. Therefore unsurprisingly, FEM has only been used to investigate discocyte RBCs in the various scenarios.

Simulating the behaviour of RBCs in flow using FEM-based methods is more challenging than for particle-based methods. This is because higher degrees of mesh refinement are required as the method is less suited to representing the large deformations RBCs are capable of. Additionally, more complex coupling is necessary between the solid and fluid phases for FEM, thus adding significant computational cost [125, 126]. Unlike particle-based methods, introducing heterogeneity and defects within the RBC membrane has not been attempted with FEM. This is because FEM it is not particularly well-suited to incorporating material property variations, especially at such a small scale. This means there is less potential to link the state of the membrane under different conditions to observed behaviour, and to thus develop deeper understanding for the role and importance of individual membrane components using this method.

2.6.6 Analytical Models for Indentation

From the Literature Review above, it is evident that no numerical models have previously been applied to AFM indentation. Looking more deeply into the literature, there is only one model that has simulated the overall shape of the probe and RBC during indentation in the report of Sen et al. [19]. This is an analytical membrane model focused on membrane tension, which approximates the cell as a partial sphere with constant volume. Critically, it does not consider the bending resistance of the membrane which is cited as a major limitation of the study, as bending is known to contribute significantly to RBC physical behaviour. The model is also limited in its application, as it is only relevant for indentation and only for RBCs which have formed perfectly symmetrical dome shapes when adhered to the

Page 30 Numerical Modelling of Red Blood Cell Morphology and Deformability substrate. The probe is also limited to being “sharp” and applied at the centre of the cell. The particle-based modelling methodologies appear well-suited to addressing these limitations.

2.7 Summary & Implications

From the Literature Review, the following points are summarised:

. The RBC membrane is composed of a bilayer and cytoskeleton tethered together and it is their combined mechanical properties which govern cellular deformability. Bending and surface area changes are resisted by the bilayer, while stretch and shear are resisted by the cytoskeleton. . RBCs can undergo a staged transition from discocytes to echinocytes as they age due to structural changes within the membrane. However, there is no consensus from the scientific community on the specific underlying mechanisms which trigger the changes. . AFM indentation is able to assess the local deformability of the RBC membrane while optical tweezer stretching applies force to the overall cell and thus assesses global deformability. . There is significant variation in Young’s modulus reported for the RBC membrane from AFM indentation experiments. This has been attributed to differences in experimental protocols and methods of analysis. . There are concerns about the applicability of Hertz-based equations to the RBC indentation problem due to the solid contact and negligible substrate influence assumptions, which require deeper evaluation. . Numerical modelling can be used as a tool to explore morphological and deformability changes to RBCs from a mechanical perspective at a scale which can be challenging, costly and sometimes impossible to target with experimentation. . The CGPM is well-suited to simulating the RBC at the full cell scale as it can efficiently capture the large deformations RBCs are capable of, incorporate heterogeneity and structural defects into the membrane, and shows significant versatility for simulating RBC behaviour in different scenarios. . The main drawbacks of other modelling methods are: CGMD is more computationally intensive to simulate on the same physical scale due to the finer coarse-graining; DPD automatically models a fluid phase which is more computationally intensive to simulate when this aspect of the system could

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 31

otherwise be neglected; finite element methods are not able to incorporate heterogeneity and structural defects into the membrane; analytical models are restricted in their application as they are developed only for a very specific scenario. The concept of the bilayer-couple hypothesis or spontaneous curvature changes may be considered for incorporation into the CGPM. . Although echinocytes and stomatocytes have been predicted numerically, these studies have only simulated the resting shapes by including an energy term for the relaxed surface area difference between bilayer leaflets.

This leads to the following research gaps which lay the foundation to the research objectives of this thesis:

. The CGPM has not been applied to AFM indentation and has been applied in a very limited capacity to optical tweezer stretching loading scenarios. . The applicability of Hertz-based analysis to the RBC indentation problem requires clarification due to concerns around the implied solid contact and negligible substrate influence assumptions. . There has been limited investigation into how the mechanical properties of the RBC membrane impacts morphology and deformability, especially in specific circumstances like RBC storage and during disease progression. . There is very limited understanding of how aspects of both indentation and stretching experimental setups impact deformability measurements, and how results may be normalised for fair comparison. . There is a significant gap in assessing the deformability of RBCs of the stomatocyte and echinocyte morphological stages.

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Chapter 3. Development of CGPM Models for Discocyte Resting Shape

Parts of this chapter have been published in a peer-reviewed journal article:

S. Barns, M. A. Balanant, E. Sauret, R. Flower, S. Saha, and Y. Gu, "Investigation of red blood cell mechanical properties using AFM indentation and coarse-grained particle method," BioMedical Engineering OnLine, vol. 16, pp. 1-21, 2017.

Parts of this chapter have been presented in a peer-reviewed journal article:

S. Barns, E. Sauret, S. Saha, R. Flower, and Y.T. Gu, “Two-Layer Red Blood Cell Membrane Model using the Discrete Element Method,” Applied Mechanics and Materials, vol. 846, pp. 270-275, 2016.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 33

3.1 Introduction

The overarching aim of this thesis is to develop a numerical model capable of predicting the morphology and deformability of RBCs, with the potential to investigate the impact of changing mechanical properties and aspects of the experimental setup. Consequently, selection of a numerical technique is a key consideration for enabling these investigations. From the Literature Review (Chapter 2), the advantages of the CGPM make it the most suitable for the present developments. Advantages over other methods include:

. The CGPM can efficiently predict overall cell shape by coarse-graining the membrane into relatively large interconnected regions. Although CGMD involves coarse-graining as well, the elements are on a much smaller scale, indicating an enormous number are required to discretise the system making it unsuitable for simulating the overall morphology of RBCs at this time. . The CGPM is less computationally intensive than CGMD and DPD to perform simulations at the full cell scale as it incorporates fewer particles. . The CGPM is able to incorporate heterogeneity and structural defects into the membrane unlike FEM, meaning there is greater potential for understanding how changes within the membrane structure impact on morphology and deformability in the future. . The CGPM is more versatile than analytical models which are developed for very specific scenarios which cannot be extended easily. . Incorporation of the bilayer-couple hypothesis or spontaneous curvature changes into the CGPM model may be considered for simulating echinocyte and stomatocyte resting shapes.

The first step of the modelling is to predict the typical resting shape of the RBC – the discocyte. This will be done in both 2D and 3D. The 2D model is a simplified version which has advantages of reduced computational cost and complexity, but is limited in modelling 3D phenomena. Nonetheless, it is able to show proof-of-concept where appropriate and can potentially provide accurate results where 3D effects are insignificant. To validate both 2D and 3D resting shape predictions, comparison can be made to literature reports for RBC cross-sectional shape. An equation proposed by Evans and Fung [42] (Equation 2.1) was identified for this purpose in the Literature Review (see Section 2.3). For comparison, the data published by Ponder [40] can also be considered for curve-fitting to the equation.

Page 34 Numerical Modelling of Red Blood Cell Morphology and Deformability

This chapter aims to develop and validate a CGPM model for predicting the RBC discocyte morphology. This will form the basis of future investigations into the deformability of the RBC in extended scenarios such as during indentation and stretching. Development of the foundation model will be based on previous works identified in the Literature Review including [14, 20, 78, 104-118].

3.2 Methods

3.2.1 Overview of CGPM Approach

The CGPM relies on discretisation of the membrane into particles which are interconnected by a network capable of storing energy (see Figure 3.1). Each particle represents a small region of the membrane, made up of many molecules. Mathematical relationships are drawn between these regions to model their interaction and the mechanical behaviour of cellular components such as bending resistance, stretch resistance, surface area incompressibility and volumetric incompressibility. A stiffness coefficient is associated with each of the energy storing mechanisms within the cell. Quantification of the stiffness coefficients is an absolutely critical aspect of the modelling, as they dictate accuracy of predictions [127].

(a)

(b)

Figure 3.1 – Discretisation process; (a) schematic of RBC membrane structure, (b) schematic of numerical model including particles and springs

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 35

As the preferred shape of the cell is one which requires the least amount of energy to maintain [3], minimisation of the total energy stored in the membrane can be exploited to predict preferred shape. In practice, the minimum energy shape is converged upon by moving particles according to the principle of virtual work and 푡ℎ Newton’s Second Law shown in Equation 3.1. Here the force on the 푖 particle, 퐹푖, is calculated by taking the partial derivative of total energy (expressed as a function 푡ℎ of particle positions) with respect to the 푖 particle’s position, 푠푖. A particle mass, 푚, is used to convert the force into an acceleration, 푎푖.

휕퐸 퐹푖 = − = 푚푎푖 ( 3.1 ) 휕푠푖 The total energy, 퐸, is calculated from the energy stored in each of the different forms being considered. This can include stretch resistance from the spectrin-based cytoskeleton, surface area incompressibility of the bilayer, bending resistance of the bilayer and volumetric incompressibility of the internal fluid (discussed in detail in subsequent sections).

Newton’s Second Law can be discretised for the numerical implementation by applying a time step, Δ푡, to calculate how far each particle moves over progressive iterations. This is shown in Equations 3.2 and 3.3 where 푣 is the velocity and 푐 is a damping constant. The time step, mass and damping constant do not impact steady-state shape, however they do contribute to the dynamics of how quickly the steady-state is achieved as well as maintaining numerical stability of the system. For dynamic simulations these parameters must be quantified to have physical meaning, but when predicting steady-state shapes they become purely numerical inputs.

푣푖(푡 + Δ푡) = 푐푣푖(푡) + 푎푖(푡)Δ푡 ( 3.2 )

푠푖(푡 + Δ푡) = 푠푖(푡) + 푣푖(푡 + Δ푡)Δ푡 ( 3.3 ) In this thesis the time step was treated as a variable which could be manipulated in order to accelerate progression toward the steady-state solution. At the beginning of the simulation when large forces were acting, a small time step would be chosen. This could then be increased as equilibrium was approached in order to move the particles more rapidly and thus reduce computational time. This was combined with the adaptive discretisation technique described in Section 3.2.4 to obtain a converged solution for systems with large particle numbers, as quickly as possible.

Page 36 Numerical Modelling of Red Blood Cell Morphology and Deformability

In terms of the practical implementation of the methodology, the simulations were programmed in Matlab R2017b (Mathworks, Natick, USA). The general layout of the code for all cases is illustrated in Figure 3.2 below. The only significant difference between model versions is which forces are considered (calculated at the second box in the diagram and discussed at length in the two subsequent sections).

Figure 3.2 – Flow chart for layout of code

The simulations were run on a combination of a personal computer and QUT’s high performance computing facility. The supercomputing facilities were used to enable the running of multiple simulations in parallel, with each simulation only using a single CPU. Given the focus on development, validation and exploration of the model’s potential, optimising simulation time was not a key concern in this project.

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3.2.2 2D Implementation

To initiate the 2D model, 푁 membrane particles were distributed evenly around a circle of radius, 푟, as shown in Figure 3.3. The size of the circle was chosen such that perimeter was equal to the RBC’s [104]. The mechanical roles of the cell’s components form the basis of the energy storing mechanisms – stretch resistance of the cytoskeleton and surface area incompressibility of the bilayer (퐸푙), bending resistance of the bilayer (퐸푏), and volumetric incompressibility of the internal fluid

(퐸푎) [104]. Total energy, 퐸, is the sum of these as shown in Equation 3.4.

퐸 = 퐸푙 + 퐸푏 + 퐸푎 ( 3.4 )

Figure 3.3 – 2D schematic of initialised red blood cell numerical model with 푁 = 8 membrane particles

The force-deformation behaviour of spectrin molecules follows a saw-tooth pattern at the molecular level due to the sudden folding and unfolding of the molecular domains [37]. However, when cytoskeletal behaviour is observed on a larger scale, the fluctuations can be simplified to a linearly increasing trend for force versus deformation. This is equivalent to a harmonic energy potential and is deemed suitable due to the coarse-graining of the membrane. In 2D, areal incompressibility of the bilayer is effectively combined with spectrin’s stretch resistance to oppose the relative movement of adjacent particles. Thus both mechanisms are modelled with a combined linear interaction between adjacent particles as shown in Figure 3.3. The total energy stored via these means, 퐸푙, was calculated with Equation 3.5, where 푖 is the interaction number, 푘푙 is the combined linear stiffness coefficient, 푙푖 is the actual distance between the adjacent particles, and 푙0,푖 is the relaxed distance between

Page 38 Numerical Modelling of Red Blood Cell Morphology and Deformability adjacent particles. The distance between particles in the initial circular configuration was used for the relaxed distances.

푁 푘 퐸 = ∑ 푙 (푙 − 푙 )2 ( 3.5 ) 푙 2 푖 0,푖 푖=1 The bilayer is responsible for resisting bending of the membrane. Therefore bending potentials were introduced between adjacent particles. Stored bending energy (퐸푏) was calculated with Equation 3.6, where 푘푏 is the bending stiffness coefficient, 훼푖 is the internal angle the bending interaction spans (refer to Figure 3.3) and 훼0,푖 is the spontaneous angle (that is, the natural angle for the bending spring). Previous models have performed bending calculations with the angle away from the horizontal (e.g. 180° − 훼푖) [126], which has the same effect. However it is more difficult to then consider the direction of the spontaneous angle – either convex or concave relative to the surrounding surface. This has not been an issue for previous

CGPM models as the spontaneous angle has been set to 훼0,푖 = 180° (eg. [126, 128]), implying that the membrane tries to be as flat as possible. However, control of the spontaneous angle is planned as a new development for the proposed CGPM model (explored in Chapter 7), and thus the internal angle was considered instead. Appendix A includes a more detailed discussion and calculations of the convex and concave cases for the internal angle.

푁 푘 훼 − 훼 퐸 = ∑ 푏 tan2 ( 푖 0,푖) ( 3.6 ) 푏 2 2 푖=1 A volume requirement was imposed on the RBC. In 2D, this manifests as a requirement on the cross-sectional area. A swelling ratio (푅퐴) was used to calculate the desired cross-sectional area (퐴푟푒푓) as per Equation 3.7. It is the ratio of RBC’s cross-sectional area in its mature form compared to its immature spherical form

(퐴푐푖푟푐푙푒). A penalty function enforces this cross-sectional area by storing energy (퐸푎) when actual area (퐴푡표푡) deviates from the desired area. This is shown in Equation

3.8, where 푘푎 is the area penalty coefficient. This represents a “soft” restraint given that volume can still vary, however the extent is limited if the strength of the penalty coefficient is large, thus modelling the incompressibility of the cytoplasm.

퐴푟푒푓 = 푅퐴퐴푐푖푟푐푙푒 ( 3.7 )

2 푘푎 퐴푡표푡 − 퐴푟푒푓 퐸푎 = ( ) ( 3.8 ) 2 퐴푟푒푓

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 39

If the number of particles used to represent the membrane is changed but stiffness coefficients remain steady, total energy stored in the linear and bending interactions will change. This is incorrect as it will modify how the cell responds to force.

Therefore, base coefficients of the form 푘푏푎푠푒 = 푘/푁 can be specified to normalise the stiffness coefficients associated with linear and bending energy components. It should be noted that the stiffness coefficient for maintaining cell volume does not require this treatment as it is applied globally rather than locally.

The force expressions from applying the principle of virtual work (Equation 3.1) to each of these energy components (Equations 3.5, 3.6 and 3.8) are detailed in Appendix A due to their substantial length.

3.2.3 3D Implementation

The 3D implementation is an extension of the 2D methodology described above. The main differences are the use of a triangulated mesh as well as separation of the membrane incompressibility into surface area and stretch aspects.

To initiate the computational model, 푁 membrane particles were evenly distributed on the surface of a sphere of radius, 푟. This was done with the aid of a spherical surface mesh created with Comsol Multiphysics 4.4, composed of 푁 vertices, 푁푒 edges and 푁푡 triangles. An example mesh is shown in Figure 3.4. On average, six triangles formed around each vertex, aligning with the cytoskeleton’s junctional complex structure [5]. Connectivity of the particles (mesh triangulation) was extracted from Comsol for use in the energy calculations.

훽푖

bending potential between adjoining triangular surfaces

linear potential at each edge

Page 40 Numerical Modelling of Red Blood Cell Morphology and Deformability

In the 3D model, the same mechanical roles of the cell’s components form the basis of the energy storing mechanisms. Thus total energy, 퐸, given by Equation 3.9 is the sum of stretch resistance of the cytoskeleton (퐸퐿), surface area incompressibility of the bilayer (퐸퐴), bending resistance of the bilayer (퐸퐵), and volumetric incompressibility of the internal fluid (퐸푉).

퐸 = 퐸퐿 + 퐸퐴 + 퐸퐵 + 퐸푉 ( 3.9 ) In the 3D model, the stretch resistance of spectrin and incompressibility of the bilayer can be separated. The energy stored due to the stretching of spectrin is modelled with linear harmonic potentials (Equation 3.10), as was the case in the 2D model. Each edge of the mesh gives the location of a harmonic potential as ideally on average six spectrin molecules terminate at each membrane particle with some variation to represent the non-uniformity of the real cytoskeleton structure. Here 푘퐿 푡ℎ is the stiffness coefficient for stretch, 푙푖 is the length of the 푖 edge and 푙표,푖 is this length relaxed, taken to be equal to that in the original spherical mesh.

푁푒 푘 퐸 = ∑ 퐿 (푙 − 푙 )2 ( 3.10 ) 퐿 2 푖 0,푖 푖=1 A surface area requirement was imposed on each individual triangle forming the RBC surface to model the incompressibility of the bilayer. Some previous CGPM models have also applied an additional energy term to restrict global surface area of the cell [20, 113, 114, 117, 118], however this over-constrains the problem given the split between the local and global components was arbitrarily set [116] and stiffness coefficients were selected to simply ensure negligible variation in surface area [113]. Wu and Feng [115] also omitted the global surface area constraint. Thus energy stored due to areal incompressibility is given by Equation 3.11. Here 푘퐴 is the 푡ℎ stiffness coefficient for areal incompressibility, 퐴푖 is the area of the 푖 triangle and

퐴0,푖 is the relaxed area of this triangle. The relaxed area for each triangle was set equal to that in the initial sphere.

푁푡 푘 2 퐸 = ∑ 퐴 (퐴 − 퐴 ) ( 3.11 ) 퐴 2 푖 0,푖 푖=1 The resistance of the bilayer to bending was modelled with bending energy potentials between adjoining triangular surfaces. Thus a bending interaction was located at each edge of the mesh (see Figure 3.4). The total energy stored via this means is given by Equation 3.12 where 푘퐵 is the bending stiffness coefficient, 훼푖 is

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 41 the internal angle the bending interaction spans and 훼0,푖 is the spontaneous internal angle. The internal angle between triangular surfaces (훼푖) can be found from the angle 훽푖 measured between outward-pointing normal vectors to these surfaces as shown in Figure 3.4. The conversion depends on the local convexity or cavity. See Appendix B for a more detailed discussion and calculations of convex and concave cases for the internal angle.

푁푒 푘 훼 − 훼 퐸 = ∑ 퐵 푡푎푛2 ( 푖 0,푖) ( 3.12 ) 퐵 2 2 푖=1

A volume requirement was imposed on the RBC. A swelling ratio (푅푉) was used to calculate the desired volume (푉푟푒푓) as shown in Equation 3.13. It is the ratio of RBC volume compared to that in initial spherical form (푉푠푝ℎ푒푟푒). A penalty function enforces this volume by storing energy (퐸푉) when actual volume (푉푡표푡) deviates from the desired. This is shown in Equation 3.14, where 푘푉 is the volumetric incompressibility coefficient. This again represents a “soft” restraint given that volume can still vary, however the extent is limited if the strength of the coefficient is large, thus modelling the incompressibility of the cytoplasm.

푉푟푒푓 = 푅푉푉푠푝ℎ푒푟푒 ( 3.13 )

2 푘푉 푉푡표푡 − 푉푟푒푓 퐸푉 = ( ) ( 3.14 ) 2 푉푟푒푓

Finally, stiffness coefficients for those energy mechanisms applied locally can be normalised against the number of particles, as was done in 2D. Therefore, base coefficients of the form 푘푏푎푠푒 = 푘/푁 were specified to normalise the stiffness coefficients associated with spectrin’s stretch resistance, surface area incompressibility, and bending resistance. It should be noted that the stiffness coefficient for maintaining the volume of the cell, 푘푉, does not require this treatment as it is applied globally.

The derivations of the force expressions from applying the principle of virtual work to each of these energy components (Equations 3.10, 3.11, 3.12 and 3.14) are detailed in Appendix B due to their length.

3.2.4 Adaptive Discretisation

An adaptive discretisation technique was introduced to reduce convergence time. This involved first running a simulation with few particles to convergence. The pre-

Page 42 Numerical Modelling of Red Blood Cell Morphology and Deformability converged model was then refined to have more particles, and was then run from the pre-converged state until convergence was again achieved. This process was repeated as many times as necessary to achieve the desired level of accuracy as it was found to be significantly quicker than running a simulation with a large number of particles from the circular/spherical starting configurations to convergence.

In 2D, the pre-converged shape was refined by including an additional particle at the centre of every linear interaction as demonstrated in Figure 3.5a. However, the relaxed lengths between particles (푙0,푖) were equated to what it would have been had the simulation been started from the circular configuration with the refined particle number. This ensured that no compounding error was introduced using this adaptive meshing approach. Each time the mesh was refined, the total number of particles was doubled.

(a)

(b)

Figure 3.5 – Adaptive discretisation process where the original shape is a pre-converged configuration which is then refined to continue the simulation from, and the referenced shape is where the relaxed properties are extracted from, (a) 2D implementation on a unit circle, and (b) 3D implementation on a unit sphere

In 3D, a pre-converged shape was refined by dividing each triangle into four through the midpoints of the sides (see Figure 3.5b). The relaxed lengths between particles

(푙0,푖) and relaxed areas of the triangles (퐴0,푖) were referenced back to the spherical shape had the simulation been started with this many particles originally. Again, this means no compounding error was introduced by incorporating the adaptive meshing

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 43 technique. Each time the mesh was refined, the number of triangles was quadrupled which almost quadruples the number of particles (the relationship between the number of triangles and number of vertices in a mesh is 푁푇 = 2푁 − 4).

3.2.5 Selection of Stiffness Coefficients

3.2.5.1 Review of Previously Used Values

In order to select stiffness coefficients, a review was conducted on previous studies employing the CGPM to model RBCs. All models identified and presented in Table 3.1 below only considered flows, with the exception of Shi et al. [20] who also considered stretching. The stiffness coefficients associated with individual interactions have been normalised against particle number for fair comparison (noting that variations in the forms of the energy equations in previous studies have been taken into account). The stiffness coefficients for areal incompressibility in 3D are not presented as all models except Wu and Feng [115] used both local and global constraints for which an equivalent combined areal stiffness coefficient cannot be computed for comparison. These models also selected the value to be extremely large to ensure negligible variation in the surface area, and thus the parameter does not have physical meaning. Furthermore, Wu and Feng [115] did not state 푘퐴 for their model. Two 3D studies used a variable 푘퐿 in an effort to better capture spectrin’s stretch resistance properties [114, 116] and thus cannot be easily compared to the remainder which used a constant coefficient.

The significant variation evident in the stiffness coefficients used in previous models shows there is no standard set of values for simulating RBC deformation, and thus there is still a need to resolve which values are most suitable. It suggests that either the RBC’s physical behaviour is insensitive to the stiffness coefficients selected, or that the values need to be optimised for specific applications. This will be explored in the subsequent chapters.

Therefore, for validating the resting shape here, an alternative approach was taken. For this purpose where no external forces are applied, absolute values for the stiffness coefficients do not have an impact [27]. Rather it is the ratio of the stiffness coefficients to each other that defines the minimum energy shape. For example, in 2D, if the linear and cross-sectional area stiffness coefficients are set proportionately large, it produces a case where perimeter and cross-sectional area conform to the reference targets [27], while the membrane tries to achieve the

Page 44 Numerical Modelling of Red Blood Cell Morphology and Deformability spontaneous curvature target as closely as possible. Thus this approach will be used to select stiffness coefficients to validate the 2D and 3D results, and the stiffness coefficients will be revisited in future studies to ensure that they are also validated against the force-deformation behaviour.

Table 3.1 – Parameters for RBC simulations with the CGPM applied in flows

2D Study Case N r 풌풍,풃풂풔풆 풌풃,풃풂풔풆 풌풂 (particles) (µm) (N/m/ (J/rad/ (J) particle) particle) Tsubota et al. [104] - 76 3.0 1.1E+04 6.6E-12 1.0E-05 Wang et al. [105] Min 2.5E-02 1.3E-15 1.0E-09 76 2.8 Max 2.5E-01 1.3E-14 1.0E-08 Pan and Wang Min 2.5E-03 1.3E-16 1.0E-10 76 2.8 [106] Max 7.4E-01 3.9E-14 3.0E-08 Shi et al. [14] - 76 2.8 1.2E+04 6.6E-12 1.0E-05

Wang and Xing Min 2.5E-04 1.3E-17 1.0E-11 76 2.8 [107] Max 1.2E-02 6.6E-16 5.0E-10 Tsubota and Wada Min 2.7E+01 4.2E-13 2.0E-07 48 3.0 [108] Max 2.7E+03 4.2E-11 2.0E-05 Shi et al. [109] - 76 2.8 1.2E+04 6.6E-12 1.0E-05

Polwaththe- Gallage et al. [78], - 88 2.8 1.4E+04 5.7E-12 1.0E-05 [110] Polwaththe- - 88 2.8 8.5E-01 3.4E-14 3.0E-08 Gallage et al. [111] Wang et al. [112] Min 7.4E-02 3.9E-15 3.0E-09 76 2.8 Max 7.4E-01 3.9E-14 3.0E-08 3D Study Case N r 풌푳,풃풂풔풆 풌푩,풃풂풔풆 풌푽 (particles) (µm) (N/m/ (J/rad/ (J) particle) particle) Tsubota and Wada - 2304 3.27 2.6E+04 5.6E-22 1.8E-16 [113] Nakamura et al. - Not Variable Not Cal- 3.25 4.3E-15 [114] Stated Stiffness culable Shi et al. [20] Min 7.1E-09 2.9E-21 4.7E-15 770 3.28 Max 9.7E-09 4.1E-21 4.7E-15 Wu and Feng [115] 1 Not 440 2.7E-08 3.2E-21 Stated 2.8 2 Not 1058 1.1E-08 1.3E-21 Stated Nakamura et al. Variable 2648 3.27 3.8E-06 4.4E-15 [116] Stiffness Polwaththe- - Gallage et al. [117], 954 3.1 1.6E-08 2.0E-20 3.7E-15 [118]

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 45

3.2.5.2 Values Selected for Predicting Discocyte Shape

A set of stiffness coefficients based on ratios was chosen to validate resting shape predictions. The 2D parameters (Table 3.2) were mainly sourced from Tsubota et al. [104], the first study to apply the CGPM to RBCs. The 3D parameters were chosen from preliminary investigations and are listed in Table 3.3. The relaxed angle for the bending interactions was set to 180° as per previous CGPM studies (e.g. [104, 113]).

Table 3.2 – Parameter values for predicting 2D discocyte resting shape Parameter Value Source/Justification 푁 50 particles Chosen from preliminary investigations, parametric studies performed in subsequent chapters to select 푁 for specific applications 푟 3 µm Matches cell dimensions in energy minimised shape, Tsubota et al. [104] 푅퐴 0.48 Physiological, Shi et al. [109] 5 푘푙 5.5 x 10 N/m Tsubota et al. [104] -5 푘푏 3.3 x 10 J/rad Tsubota et al. [104] -5 푘푎 1.0 x 10 J Tsubota et al. [104] 훼0 180° Tsubota et al. [104]

Table 3.3 – Parameter values for predicting 3D discocyte resting shape Parameter Value Source/Justification 푁 482 particles Chosen from preliminary investigations, parametric studies performed in subsequent chapters to select 푁 for specific applications 푅푉 0.6 Physiological, Tsubota and Wada [113] 푟 3.3 µm Matches cell dimensions in energy minimised shape, Tsubota and Wada [113] -3 푘퐿 1.8 x 10 N/m Chosen from preliminary investigations -14 푘퐵 1.8 x 10 J/rad Chosen from preliminary investigations 12 2 푘퐴 1.8 x 10 N/m Chosen from preliminary investigations -10 푘푉 1.0 x 10 J Chosen from preliminary investigations 훼0 180° Tsubota and Wada [113]

Page 46 Numerical Modelling of Red Blood Cell Morphology and Deformability

3.2.6 Experimental Observations of Discocyte Shape for Validation

Several studies have measured the geometry of discocytes (see Table 2.1) including Evans and Fung [42] who developed an equation to describe cross- sectional shape. In addition to Evans and Fung [42], Ponder [40] provided enough information for RBC cross-sectional shape to be quantified with an equation. This was completed to provide a second cross-sectional shape for experimental validation.

The mean data of Ponder [40] was fitted to Equation 2.1 by minimising the error between the curve and points on the cell boundary. This found 퐶0, 퐶2 and 퐶4 to be

0.97 µm, 7.19 µm and -4.67 µm respectively. The cell radius, 푅0, was 4.28 µm. The predicted shape using these coefficients is shown in Figure 3.6, where the crosses represent the data measurements from Ponder [40]. It can be seen that the match is extremely close. Thus both the Evans and Fung [42] and Ponder [40] observations (referred to as “Evans Data” and “Ponder Data” respectively in the below figures) can be compared to the model’s resting shape predictions.

Figure 3.6 – Cross-sectional shape of RBC discocyte optimised from fitting average Ponder [40] measurements to Equation 2.1

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 47

3.3 Results & Discussion

3.3.1 2D Model

The 2D model’s prediction of the discocyte resting shape using the parameters given in Table 3.2, is shown in Figure 3.7 below.

(a)

(b)

(c)

Page 48 Numerical Modelling of Red Blood Cell Morphology and Deformability

Figure 3.7a shows the energy versus time graph for the simulation. Energy changes rapidly at the beginning of the simulation, coinciding with the period where the cross-sectional area is adjusted to match the target. This is evident at the 0.66 µs time point in Figure 3.7b where the cross-sectional area has been reduced with no major change from the initial shape. Following this, the linear restraints become dominant, evident by the “sun” shape that occurs at the 2.7 µs mark where the cross-sectional area and perimeter targets are met. From here the bending requirement dictates, resulting in the cell reconfiguring to have a smooth surface. The steady-state shape is virtually achieved by 120 µs, with no major change in shape evident between the final two time points depicted in Figure 3.7b. The steady-state shape was compared to the experimentally observed discocyte geometries in Figure 3.7c. It can be seen they are very comparable. The predicted shape is also consistent with Tsubota et al. [104] (not shown). This validates the resting shape prediction of the 2D model.

3.3.2 3D Model

The process used to validate the 2D prediction of the discocyte resting shape was repeated for 3D. The results are shown in Figure 3.8 below.

Figure 3.8a shows the energy versus time graph when approaching the discocyte resting shape. As in the 2D case, energy changes rapidly at the beginning of the simulation, coinciding with the period where the volume is adjusted to match the target. This is evident at the 0.02 ms time point in Figure 3.8b where the volume has been reduced with no major change from the initial spherical shape. Following this, the linear and surface area restraints become dominant, evident by the “sun” shape that occurs at the 0.2 ms mark where these targets are largely met. From here the bending requirement dictates the shape evolution, resulting in the cell reconfiguring to have a smooth surface. A constant shape is achieved by 60 ms, with no major change in shape evident between the final two time points depicted in Figure 3.8b. Thus the steady-state shape is achieved.

The particle number used to predict the final shape was increased through one adaptive discretisation refinement to improve accuracy so that 푁 = 1922. The resulting predicted shape is shown in Figure 3.8c. This can then be compared to the experimental observations for RBC cross-sectional shape in Figure 3.9a. It can be seen that the numerical shape compares well with both Ponder [40] and Evans and Fung [42] data. A comparison of the 2D and 3D cross-sectional shapes is shown in

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 49

Figure 3.9b. It can be seen that these are almost identical. This validates the 3D model’s resting shape prediction.

(a)

(b)

(c)

Page 50 Numerical Modelling of Red Blood Cell Morphology and Deformability

(a)

(b)

3.4 Summary & Conclusions

This chapter detailed the development of 2D and 3D models based on the CGPM for predicting the discocyte morphology of RBCs. The following conclusions were drawn:

. The RBC membrane can be represented as a series of particles related by a complex spring network which accounts for the mechanisms of stretch resistance, bending resistance, surface area incompressibility and volumetric incompressibility. When total energy in the network is minimised, preferred resting shape is predicted. . Each mechanism is associated with a stiffness coefficient to control its strength. Previous studies have used a wide range of stiffness coefficients to simulate RBC physical behaviour, meaning further work is required to select these parameters for future deformability simulations. . It is unclear from the review whether the wide variation in stiffness coefficients is due to the CGPM being insensitive to the stiffness coefficient

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 51

selection or whether the values need to be optimised for specific applications. This will be explored in the following chapters. . Using a nominal set of stiffness coefficients, the 2D and 3D resting shape predictions showed good agreement with experimental observations for the discocyte morphology. . The methodology set out in this chapter forms the basis of the modelling which can be modified going forward to model RBC behaviour in various deformability scenarios.

3.5 Contribution to Research Objectives

This chapter has contributed to the first two objectives of the research project. First, it established the basis of the numerical model which represents the RBC membrane as a series of particles related by a complex spring network. Second, the discocyte morphology at rest has been validated against experimental observations.

Page 52 Numerical Modelling of Red Blood Cell Morphology and Deformability

Chapter 4. Investigation of RBC Mechanical Properties during AFM Indentation

Parts of this chapter have been published in a peer-reviewed journal article:

Parts of this chapter have been presented in an extended conference abstract:

M. A. Balanant, S. Barns, E. Sauret, and Y. T. Gu, "Investigation of Red Blood Cell Membrane Elasticity using AFM Indentation and the Coarse-Grained Particle Method," in 10th Australasian Biomechanics Conference, Melbourne, Australia, 2016.

Parts of this chapter have been presented in a poster presentation:

S. Barns, E. Sauret, R. Flower, and Y.T. Gu, “Numerical Investigation of Red Blood Cell Membrane Mechanical Properties on Deformability during Indentation,” in 27th Regional Congress of the International Society of Blood Transfusion, Copenhagen, Denmark, 2017.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 53

4.1 Introduction

The mechanical properties of the RBC membrane are known to evolve over time, particularly during refrigerated storage [129] and with certain conditions such as the malaria infection [17]. These changes tend to cause RBCs to become less deformable, meaning a greater amount of force is needed to create the same level of deformation [13]. This causes the cells to become less efficient at traversing narrow capillaries to deliver oxygen, resulting in their eventual removal from the circulation by the spleen [1]. Investigating the relative dominance of each mechanism resisting deformation may aid in understanding their role and importance in RBC physical behaviour. This may help pinpoint changes occurring within the specific structures of the RBC membrane, which may in turn be fed back into developing strategies for the prevention of deformability loss.

In order to explore deformability, it is necessary to measure the response of RBCs to externally applied force. As discussed in the Literature Review, micropipette aspiration and AFM indentation both involve applying a force onto a small region of the membrane and can thus explore deformability on a local level. However, from a modelling perspective, indentation is more simplistic to reproduce as a probe is used to apply force directly onto the membrane while the cell is immobilised on a substrate. In contrast, micropipette aspiration requires simulation of the contact between the cell and the cylindrical micropipette boundary surface in addition to calculating the equivalent force applied due to the pressure. AFM also shows significant promise as a diagnostic tool given that deformability differences can be measured experimentally between healthy and deteriorating RBCs [16, 36, 74, 81, 88]. Therefore modelling AFM indentation was chosen for exploring deformability on a local level.

As identified in the Literature Review, the most advanced RBC indentation model is an analytical model in the report of Sen et al. [19] which focuses on membrane tension when dome-shaped RBCs are indented at their centre with a sharp probe. Critically, it does not consider membrane bending resistance and is only applicable to this very specific indentation scenario. Although particle-based models are more versatile than this analytical model, none have previously simulated indentation. Therefore the development of a CGPM model for indentation is an innovation which will enable more detailed exploration of the mechanisms which govern the membrane’s physical behaviour during indentation than was previously possible. It

Page 54 Numerical Modelling of Red Blood Cell Morphology and Deformability should be noted that the scope will be restricted to RBCs with the typical discocyte resting morphology which most RBCs exhibit throughout their lifetime (noting echinocyte morphologies are specifically explored later in Chapter 7). Thus the aim of this chapter is to investigate how mechanical properties of the RBC impact the deformability of RBCs when a local compressive force is applied.

4.2 Methods

4.2.1 Experimental Observations for Model Validation

For validation of the numerical predictions, comparison was necessary against experimental results. Experimental data was obtained as part of a study performed alongside the numerical model’s development by another party. It will be discussed briefly to give context to the model development detailed in the subsequent section (refer to Barns et al. [130] for detailed experimental procedure and discussion).

The experimental protocol involved adhering RBCs to the substrate using poly-D- lysine which resulted in them forming dome-shapes on the surface. RBCs were lightly treated with glutaraldehyde to prevent them spreading further on the substrate during testing. In this state the cells were measured to have an average height of 2.1 µm. From confocal microscopy measurements, they were found to have an average substrate contact area of 54.7 µm2 (see Figure 4.5 below for a typical confocal image of the RBC cross-section). A NanoSurf FlexAFM with NanoSurf C3000 software (NanoSurf, Liestal, Switzerland) was used to indent the samples (n = 26 cells). Individual RBCs were indented with 5 µm spherical beads at a speed of 1 µm/s in their central region. The deformation depth was kept to less than 10% of cell height to minimise the influence of the substrate [89]. The applied force and resulting deformation were measured.

To benchmark the experimental results against similar experimental studies, the Hertz equation modified by Dimitriadis et al. [89] for spherical tip shape and finite sample thickness was fitted to the experimental data (Equation 4.1). Here 퐹 is the applied force, 퐸 is Young’s modulus, 훿 is the indentation depth, 푅 is the indenter radius and ℎ is equated to cell height. This equation was selected over the other Hertz equations identified in the Literature Review (Section 2.5.1) as it was modified to remove substrate effects for thin samples [89]. However, it should be noted that

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 55 this equation was originally developed for thin gel applications, and thus it also suffers from limitations of the Hertz analysis being applied to RBCs highlighted above (Section 2.5.1) and which will be explored in the subsequent chapter. Nonetheless, the experimental data was observed to closely fit the force- deformation trend predicted by this equation for the range tested. An example of this is shown in Figure 4.1.

16 퐹 = 퐸푅0.5훿1.5[1 + 1.133휒 + 1.283휒2 + 0.769휒3 + 0.0975휒4 9 (푅훿)0.5 ( 4.1 ) where 휒 = ℎ

Figure 4.1 – Comparison between experimental data and Hertz equation modified by Dimitriadis et al. [89] for a typical RBC where 퐸 = 9.83 kPa

The average Young’s modulus across 26 tested cells was found to be 7.57 kPa with a standard deviation of 3.25 kPa. This was within the range reported for previous studies investigating Young’s modulus of the RBC membrane (see Table 2.4) especially when considering indentation was conducted in the central region where higher values are expected [36].

Page 56 Numerical Modelling of Red Blood Cell Morphology and Deformability

4.2.2 Modelling Methodology

The 2D and 3D CGPM models established in Chapter 3 were modified to consider adherence of the RBC onto the substrate and contact between the probe and membrane. In order to validate the model against experimental results, the specific indentation scenario using spherical probes of 5 µm diameter detailed above (Section 4.2.1) was replicated.

Firstly, both AFM and confocal imaging showed that the RBCs formed dome-shapes when adhered. To incorporate this into the model, a constraint was introduced for specific membrane particles to be in contact with the substrate. In 2D, a section of the membrane corresponding to 8.5 µm in length was set to the substrate’s height, while in 3D, 50% of the particles were set to the substrate’s height. This treatment of adhesion as a constraint rather than an additional energy term saves computational cost. However, if the model was to be used to specifically study adhesion or substrate detachment, the attraction potential between the membrane and substrate should be quantified with an additional energy term. The principle of virtual work (Equation 3.1) was re-applied with the added constraint to minimise energy and thus predict adhered RBC shape.

Secondly, the 5 µm spherical probe was represented as a rigid body (incapable of deforming), given that RBCs are significantly softer. Contact between the probe and cell was modelled with a penalty function (Equation 4.2) which stored energy, 퐸푐표푛, when cell membrane particles penetrated the probe surface. Here 푁푝 is the number of membrane particles which have penetrated, 푘푐표푛 is the penalty stiffness 푡ℎ coefficient for contact and 푑푖 is the depth to which the 푖 membrane particle has penetrated the surface.

푁푝 푘 퐸 = ∑ 푐표푛 푑2 ( 4.2 ) 푐표푛 2 푖 푖=1 The penalty stiffness coefficient for contact is a numerical parameter implemented to ensure negligible cross-over of the probe and cell membrane. When it is sufficiently large to restrict the contact, it becomes independent of steady-state RBC shape and measured force. However, when too large, it can cause numerical oscillation and instability. Thus a sensitivity study must be performed to select its value (see Section 4.2.3 below).

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To simulate indentation, the probe was centred above the adhered cell and moved down a specified distance beneath the cell’s height. The adhered particle positions were fixed to immobilise the cell and thus prevent it from moving around on the substrate when probed as per the experimental conditions. The principle of virtual work was re-applied with the additional energy term, 퐸푐표푛, to minimise total energy. This treats the indentation as a “quasi-static” problem, justified by the indentation speed being slow enough that the system remains in internal equilibrium. Total contact force between the cell and probe was measured when the simulation had reached steady-state. By Newton’s Third Law, the measured contact force is equivalent in magnitude to the force applied by the probe to cause the deformation. Indentation was simulated at a series of depths between 0 and 200 nm as per the experiments. Contact force and indentation depth were plotted against each other and compared against the experimental reference curve.

4.2.3 Sensitivity Studies for Parameter Selection

Sensitivity studies were required for selecting a sufficient number of particles for discretising the membrane and an appropriate contact stiffness coefficient. These studies were subsequently carried out for both the 2D and 3D cases.

In order to select the number of particles, simulations were conducted on both the resting and adhered RBC when particle number was varied. The adaptive discretisation technique (see Section 3.2.4) was utilised to accelerate the convergence process.

Convergence of RBC dimensions and steady-state energy for a typical 2D case are shown in Figure 4.2a. From these plots, 400 particles were chosen for the 2D model as the point where dimensions and energy sufficiently stabilised – doubling the number of particles from here resulted in less than a 1% change to the critical dimensions. An equivalent sensitivity study was conducted for the 3D model with these results shown in Figure 4.2b. This found 1922 to be the most suitable number of particles for the 3D indentation simulations.

The contact stiffness sensitivity study involved simulating indentation to a nominal depth of 200 nm for a range of contact stiffness coefficients. The contact force for each case was measured with the results shown in Figure 4.3 below. It should be noted that the contact stiffness coefficient was set proportional to the bending

Page 58 Numerical Modelling of Red Blood Cell Morphology and Deformability stiffness coefficient to ensure that it remained proportionately dominant when absolute values were varied.

(a)

(b)

Figure 4.2 – Sensitivity study of particle number for resting and adhered shapes in (a) 2D and (b) 3D

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Figure 4.3 – Sensitivity study for contact stiffness coefficient for indentation in 2D and 3D

It can be seen in Figure 4.3 that as the contact stiffness is increased, the measured contact force stabilises. This is because as contact stiffness increases, displacement of the contact “springs” approach zero. However, when the coefficient becomes too large, instability issues can emerge due to oscillation of membrane particles against the contact. To overcome this, adjustments to the time step and damping constant are necessary to move the particles slower, meaning total computation time increases as a result. On the other end of the spectrum, when the contact stiffness is too small, it is not dominant over the other energy terms and significant penetration of the membrane particles into the probe occurs. Thus to 20 −1 balance these competing factors, 푘푐표푛 = 10 × 푘푏 푚 was chosen for where the contact stiffness became independent of the force in 2D. Similarly in 3D, −1 푘푐표푛 = 10 × 푘푏 푚 was selected.

The effectiveness of enforcing the contact between the probe and membrane can be confirmed by observing the cross-over distance between the probe and membrane as demonstrated in Figure 4.4. For the typical 2D and 3D cases shown at an indentation depth of 200 nm, the average distance membrane particles penetrate is just 0.0015 nm and 0.0022 nm respectively, negligible in comparison to the size of the cell (diameter ~8 µm and height ~2 µm).

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(a) (b)

Figure 4.4 – Contact between cell and probe at 200 nm indentation depth for typical case in (a) 2D and (b) 3D – negligible penetration of the membrane and probe is observed

4.2.4 Validation of RBC Shape & Force-Deformation Behaviour

The mean value for effective Young’s modulus from the experiments (7.57 kPa) was used within the Hertz equation modified by Dimitriadis et al. [89] (Equation 4.1) for validation of the force-deformation behaviour. The mean was used due to the variability in Young’s modulus measured for different cells but each closely followed the trend of Equation 4.1. Thus, in this context the equation only represents an empirical relationship for the force-deformation trend for which the performance of the numerical model can be referenced against – its grounding in Hertz theory has no impact on validating the model’s performance.

Given the immense variation in previous values used for the stiffness coefficients (refer to Section 3.2.5), an inverse method was applied to extract the stiffness coefficients that best predicted RBC shape (resting and adhered) and the force- deformation behaviour for indentation. Initial values were assumed for each stiffness coefficient which were then iteratively converged until agreement was reached between the model and experimental observations for resting shape, adhered shape and force-indentation behaviour. The optimised stiffness coefficients for the 2D and 3D models are shown in Table 4.1 along with the other inputs for clarity. The predicted resting and adhered cell shapes using these optimised values are shown in Figure 4.5. The resting shapes are almost identical to those established during the previous chapter. The adhered RBCs have a diameter and height within 5% of the mean dimensions obtained in the experiments. Qualitatively, the cross-sectional shape matches well with the confocal results (Figure 4.5).

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Table 4.1 – Optimised stiffness coefficients for indentation in 2D and 3D Parameter 2D Model 3D Model Starting Radius of 푟 = 3 휇푚 푟 = 3.3 휇푚 Circle/Sphere Number of Particles 푁 = 400 푁 = 1922 Referenced Properties 훼0,푖 = 180° 훼0,푖 = 180° 푙0,푖 = original lengths 푙0,푖 = original lengths 퐴0,푖 = original areas Reduction Ratio 푅퐴 = 0.48 푅푉 = 0.6 2 −4 Stiffness Coefficients 푘푙 = 1.2 × 10 푁/푚 푘퐿 = 2.1 × 10 푁/푚 −14 −15 푘푏 = 1.6 × 10 퐽/푟푎푑 푘퐵 = 5.3 × 10 퐽/푟푎푑 −10 11 2 푘푎 = 2.3 × 10 퐽 푘퐴 = 1.4 × 10 푁/푚 −10 푘푉 = 7.0 × 10 퐽

2D 3D

Resting Shape

Adhered Shape

Confocal

Figure 4.5 – 2D and 3D energy minimised shapes with optimised stiffness coefficients at rest, adhered to substrate, and compared to a typical confocal image obtained from the experiments

The force-deformation curves predicted by the 2D and 3D numerical models using the optimised stiffness coefficients are shown in Figure 4.6. These can be compared against the experimental reference curve. For the 2D case there is some discrepancy in the early deformation region where the model predicts a more linear trend and a larger force than in the experimental case. The 3D model is able to better capture the behaviour through this early region. However, overall good agreement is reached over the investigated range to validate the model’s predictions.

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4.3 Results & Discussion

In order to understand how the mechanical properties of the RBC impact behaviour when a 200 nm indentation was performed, several aspects were investigated. These included how energy was stored during indentation, the effect of varying stiffness properties on deformability, and the effect of the adhesion protocol on the exhibited shapes for cells with otherwise constant properties. Each of these investigations was completed with both the 2D and 3D models for comparison, in order to understand the capabilities and limitations of the 2D model going forward. Finally, the stiffness coefficients used in previous models were tested in the indentation methodology to understand whether the stiffness coefficients need to be optimised for specific applications.

4.3.1 Energy & Shape through Indentation

Figure 4.7 and Figure 4.8 show the results for energy through the indentation stroke as well as the deformed RBC shapes at selected depths from the 2D and 3D models respectively.

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(a)

(b)

(c)

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(a)

(b)

(c)

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Figure 4.7a and Figure 4.8a plot the energy stored through each mechanism as a function of indentation depth. The 3D model follows the same order as 2D for the amount of energy stored in each mechanism – bending stores the most, followed by surface area incompressibility and linear stretch (which are combined in 2D), and then volumetric incompressibility. However, the absolute values for energy stored and the relative change vary considerably between the models (Figure 4.7a-b and Figure 4.8a-b). This difference is attributed to the 2D model’s simplification which combines the surface area incompressibility and linear stretch aspects into a single mechanism. Therefore further analysis of these stored energy results focuses on the 3D predictions only. The shape predictions in Figure 4.7c and Figure 4.8c closely resemble each other. Therefore, although energy estimates are complicated by the 2D simplification, shape predictions are accurate.

Figure 4.8b shows that the cytoplasmic fluid absorbs a negligible amount of energy over the 200 nm indentation. This is consistent with its nature as an incompressible fluid, meaning it has considerably higher “stiffness” than the membrane. Therefore any potential increase in fluid pressure is quickly dissipated to prevent the compression, and this is redirected onto the membrane. This causes the stress to be taken through the bending, linear stretch and surface area mechanisms, consistent with the observations in Figure 4.8b.

The surface area and linear mechanisms are expected to become increasingly significant at larger indentation depths. This is because the internal fluid would be expected to apply increasing pressure onto the membrane as the probe penetrates further into the cell, in order to prevent its compression, thus putting the membrane into tension resulting in an increase in energy stored in the linear and surface area mechanisms.

As bending is most influential in the 2D model (Figure 4.7), the overestimation for the indentation force in the 2D model at small deformations (see Figure 4.6) is most likely caused by the underlying assumption for the quantification of bending energy. Energy developed in bending interactions is directly proportional to the tangent squared relationship of the angle, which has been used in all previous CGPM studies [14, 20, 78, 105-118]. This appears to capture the behaviour well in the 3D model, but introduces a small error in 2D. In order to better capture the trend and magnitude in the future, a variable bending stiffness coefficient may be considered to compensate. Variable stiffness coefficients have been implemented in previous

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3D models for the linear stretch mechanism [114, 116], but have not been attempted for bending before.

4.3.2 Effect of Changing Stiffness Coefficients

To explore the relative influence of individual mechanisms on RBC deformability, a parametric study was conducted for the stiffness coefficients. This involved varying each stiffness coefficient in isolation to between a tenth and ten times the baseline value established during the validation. Indentation was then simulated to nominal depths of 100 nm and 200 nm and applied force was measured.

The results for the 2D parametric study are shown in Figure 4.9. It can be seen that varying the membrane’s bending stiffness results in a substantial change in the deformability of the RBC at both the 100 nm and 200 nm depths. In fact, a roughly linear relationship exists between the force required to indent the cell and the bending stiffness coefficient. In contrast, the effect of the linear stiffness coefficient plateaus substantially as it is increased from the base multiplier of 1. This means further increases in the linear stiffness of the membrane have very little effect on the RBC’s deformability. However, it can be seen that if the linear stiffness coefficient decreases from its baseline value, it does result in a small reduction in the indentation force. The stiffness coefficient for incompressibility of the internal fluid has a negligible impact on deformability.

Figure 4.9 – Results of parametric study in 2D measuring force to indent to nominal depth of 100 nm and 200 nm when varying the stiffness coefficients

The results of the 3D parametric study are shown in Figure 4.10, where it can be seen that increasing the membrane’s bending stiffness still has the most substantial impact on reducing deformability at both the 100 nm and 200 nm depths. However its effect plateaus unlike the 2D model. The linear and areal stiffness coefficients

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 67 have a small influence when indenting to 100 nm, with the influence of the areal stiffness coefficient becoming more important at the 200 nm indentation depth. This aligns with the hypothesis discussed above (Section 4.3.1) that tension in the membrane will become increasingly significant with larger indentation depths when the internal fluid needs to redistribute causing outward pressure on the membrane. The negligible impact of the volumetric incompressibility coefficient in 3D agrees with the 2D result.

Figure 4.10 – Results of parametric study in 3D measuring force to indent to nominal depth of 100 nm and 200 nm when varying the stiffness coefficients

This parametric study demonstrates that deformability loss of the RBC in this loading scenario is most sensitive to changes in bending stiffness. As bending resistance is provided by the membrane’s outer lipid bilayer and its embedded proteins, it suggests structural changes here play a critical role in the loss of deformability observed in deteriorating RBCs. The second most dominant mechanism, surface area incompressibility, is also provided by the bilayer. This reinforces the bilayer as an important component of the cell for maintaining deformability in this loading scenario. In contrast, the cytoskeleton’s properties were shown to have a less significant impact on the cell’s deformability for the 200 nm indentation.

Structural changes within the bilayer identified in the Literature Review included oxidative degradation [6, 54], crosslinking [54] and clustering [6, 57] of band 3 proteins, as well as the change in lipid distribution between leaflets over time [54] (see Section 2.4). However the specific processes or combination thereof which may drive the increase in bending stiffness or increase in surface area incompressibility, particularly during RBC storage, remain unclear [13]. As the indentation setup has been shown to be sensitive to changes in the bending

Page 68 Numerical Modelling of Red Blood Cell Morphology and Deformability stiffness and surface area incompressibility, it suggests this experimental technique would be well-suited to detecting changes within the bilayer that impact the mechanical stiffness. This finding may be leveraged in the future to achieve the longer-term goal of developing protocols for diagnosing diseases or quantifying deterioration of stored RBCs based on the change in deformability. This may in turn lead to new approaches for preventing or reversing deformability loss.

4.3.3 Effect of Different Adhesion Levels on Adhered Shapes

RBCs can exhibit the biconcave disc shape on their top surface even when adhered to a substrate [74, 82, 83, 97]. This is hypothesised to be influenced by the adhesion protocol rather than exclusively a function of membrane properties. To test this, the adhesion level was manipulated by modifying the number of particles attached to the substrate while keeping cellular mechanical properties constant. The diameter of the contact between the membrane and substrate was used to define the adhesion level given that this is comparable between the 2D and 3D models.

The adhered shapes predicted by the 2D model for varying adhesion levels are shown in Figure 4.11. The cross-sections predicted for cells with increasing substrate adherence using the 3D model are shown in Figure 4.12. It can be seen that the 2D and 3D shapes closely resemble each other – at the smaller contact diameters, the RBCs show the biconcave upper-surface profile, and at larger contact diameters, the RBCs exhibit dome-shapes. The tipping point occurred at a contact diameter of approximately 7 µm for the 3D case, which is consistent with the 2D model where it occurred around the 7.5 µm mark.

These results demonstrate that the proportion of membrane adhered to the substrate plays a significant role in exhibited shape, independent of membrane mechanical properties. This has possible implications for the design of indentation protocols for the diagnosis of diseases and ailments affecting the RBC membrane, which has been lauded as a significant direction for future work. This is because interpreting indentation results may be complicated if the adhesion state and thus exhibited shape alters deformability measurements when the underlying membrane properties remain the same. In this case, to identify deformability changes within the membrane itself, the influence of the adhesion protocol would need to be isolated as part of the analysis process. Hertzian contact theory [90] has been used extensively for this purpose [36, 74, 81-83, 88, 97], however the validity and accuracy for this

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 69 application has been openly questioned [19]. Thus these points are explored in detail in the following chapter.

Figure 4.11 – RBC shapes predicted using 2D model when adhered to the substrate to varying degrees

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Figure 4.12 – RBC cross-sectional shapes predicted using 3D model when adhered to the substrate to varying degrees

4.3.4 Comparison to Stiffness Coefficients of Previous Studies

A review was conducted on the stiffness coefficients used in previous studies employing the CGPM in Table 3.1 of Section 3.2.5.1. It was seen that the range of previously used values is large, and that they have been used predominately in flow scenarios. To investigate whether combinations of stiffness coefficients from previous models would also be appropriate for indentation, a large, small and mid- range set of stiffness coefficients from Table 3.1 were tested for indentation in 2D. 3D values were not tested due to the difference in how the areal incompressibility was applied and the lack of stated stiffness coefficients for the Wu and Feng [115] case which did use the same method (refer to Section 3.2.5.1). The results for the selected 2D cases are shown below in Figure 4.13.

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(a)

(b)

(c)

Figure 4.13 – Results for indentation using stiffness coefficients from previous 2D studies; force- indentation curves plotted against the best-fit effective Young’s modulus in the Hertz equation modified by Dimitriadis et al. [89] (Equation 4.1) with adhered cell shape inset for (a) Tsubota et al. [104], (b) Wang et al. [105] minimum values, and (c) Wang and Xing [107] maximum values

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Using the values of Tsubota et al. [104], one of the largest input sets, while the resting and adhered shapes were reasonable, significantly more force was required to deform the cell than was found in the experiments (Figure 4.13a). This resulted in a best-fit effective Young’s modulus of 951 MPa, well in excess of experimental measurements reported in the literature (refer to Section 2.5.1) [36]. The minimum stiffness coefficients used by Wang and Xing [107] were the smallest in Table 3.1. The adhered shape predicted by these values was flatter than the confocal results and the force-deformation predictions did not follow the trend of the modified Hertz equation (Figure 4.13b). The mid-range case of Wang et al. [105] also provided little agreement between the force-deformation trend and modified Hertz equation with the best fit Young’s modulus approximately 485 kPa (Figure 4.13c).

These findings demonstrate the stiffness coefficients extracted in this study are more suitable for modelling indentation of RBCs, suggesting the stiffness coefficients in the CGPM model need to be optimised for specific applications. It shows indentation simulations are highly sensitive to stiffness coefficient selection, perhaps even more so than flow studies. This is because flow studies use qualitative comparison of RBC shapes in general flow conditions to assess deformability, whereas the indentation model uses both qualitative RBC shapes and quantitative force-deformation response. The quantitative aspect means the system’s response is more sensitive to changes in the stiffness coefficients – if only qualitative comparison of shape is considered, modification of the stiffness coefficients which have little impact on cell shape cannot be detected. These results suggest that the CGPM model currently lacks the sophistication of being a universal model, as the stiffness coefficients need to be tailored to the application. This will be explored further in Chapter 6 when another experimental scenario is investigated.

4.4 Summary & Conclusions

This chapter investigated how the mechanical properties of the RBC impacted deformability when a compressive force was applied via a spherical probe to indent the cell 200 nm at its centre. The following conclusions were drawn:

. The 2D model accurately simplifies the 3D case for the purpose of predicting adhered and indented RBC shapes as well as force-deformation measurements. However its accuracy is limited for quantifying stored energy due to the simplification of surface area incompressibility and linear stretch aspects into a single mechanism.

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. The 3D model showed that bending, stretch and surface area incompressibility mechanisms absorb energy when RBCs are indented to 200 nm. Volumetric incompressibility of the internal fluid absorbed negligible energy. . RBC deformability during indentation is most sensitive to changes in the membrane’s bending stiffness, and then secondly to the strength of surface area incompressibility. As both bending resistance and surface area incompressibility are provided by the bilayer, it suggests that AFM indentation is well-suited to detecting structural changes within this part of the membrane. Therefore the model may be leveraged in the future to aid in the development of protocols for diagnosing diseases which impact bilayer properties, as well as understanding the extent of changes that manifest in the bilayer during storage. . When a large proportion of the membrane is adhered, RBCs form dome- shapes on the substrate, but when a smaller proportion of the membrane is adhered, cells show a well-defined biconcave profile on their upper surface. The tipping point, when the surface is very flat, occurs when the cell has a contact diameter of approximately 7.5 µm. . Further investigation is required to understand if measured deformability of the cell is altered by the adhesion state even when the underlying membrane properties remain the same. This has implications for using AFM indentation for the diagnosis of diseases and ailments which impact RBC mechanical properties, as the adhesion state may need to be taken into account during the data analysis. . The stiffness coefficients used in previous CGPM studies are inappropriate for simulating indentation as the predicted force-deformation curves do not match the experimental trend. The optimised stiffness coefficients proposed in this study should be used in the future for modelling RBC indentation.

4.5 Contribution to Research Objectives

This chapter has contributed to the second and third research objectives. The 2D and 3D models were validated against experimental observations for RBCs adhered to a substrate and indented with 5 µm spherical probes at their centre. These models were then applied to numerical experiments to understand how mechanical properties of the membrane influence deformability. Most significantly, increasing membrane bending stiffness and surface area incompressibility were the most

Page 74 Numerical Modelling of Red Blood Cell Morphology and Deformability dominant mechanisms in reducing RBC deformability in this loading scenario, implying that AFM indentation is well-suited to detecting structural changes within the bilayer which provides these forms of mechanical resistance. Furthermore, the proportion of the membrane adhered to the substrate controlled whether RBCs showed dome shapes or biconcave surface profiles when adhered. This requires further exploration in the subsequent chapter to understand whether this impacts Young’s modulus predictions when using traditional Hertz-based analysis.

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Chapter 5. Critical Assessment of Hertz- Based Equations for RBC Indentation

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5.1 Introduction

As highlighted in the Literature Review (Section 2.5.1), previous AFM studies of the RBC membrane have produced a one thousand-fold variation in the value for Young’s modulus extracted from force-deformation measurements [36]. Differences can be attributed in part to varying sample preparation and indentation protocols. However, the extent of the variation suggests that there is still a need to reconcile measurement and analysis techniques. In particular, many questions have been raised about the validity of Hertz-based analysis for quantifying Young’s modulus of the RBC membrane [19]. This is due to Hertzian contact theory being developed specifically for solid-to-solid contact of isotropic materials where the size of the contact is negligible compared to the bodies themselves [131]. Challenging this, RBCs are soft matter, and their force-deformation response may be influenced by the presence of the substrate due to their small size.

Quantifying Young’s modulus of the RBC membrane has value for developing diagnostic techniques based on indentation. This is because underlying changes in the membrane structure that impact deformability can be detected experimentally [16, 36, 74, 81]. However, the current limitations with data analysis complicate the comparison of absolute Young’s modulus values between different experimental setups, and potentially even when the same cell is indented in different surface regions. In the former case, it is hypothesised that the RBC acts as a fluid-filled membrane rather than a solid leading to inaccurate estimates for the contact area and pressure distribution. In the latter case, it is hypothesised that the measured difference in stiffness in local areas of the cell is impeded by the substrate’s influence, especially given the change in height and in orientation of the contact across the cell surface. Thus the key assumptions which need to be investigated to assess the validity of the Hertz-based approach for RBC indentation problems are: (1) the RBC acts as an isotropic solid, and (2) the substrate has a negligible influence.

If Hertz-based equations are accurately extracting Young’s modulus for the RBC membrane, they would be expected to measure a consistent value as properties of the probe such as size and geometry are changed. A consistent value should also be predicted when the cell is indented in different regions when constant membrane properties are applied. However, a critical assessment of this has not been performed previously with either experimental or numerical means. Furthermore, it

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 77 has not been possible to date with the existing indentation models to compare different probe geometries and different indentation regions, given that the previously most advanced RBC indentation model by Sen et al. [19] is only applicable for sharp probes indenting dome-shaped cells at their centre.

A numerical investigation is well-suited to assessing the accuracy of Hertz-based equations given that simulations can be performed quickly and cheaply for a host of factors. Modelling also means that the properties of the RBC membrane can be kept exactly the same for many different variations, making fair comparisons possible without the need for extensive replicates. Thus, this chapter aims to evaluate the accuracy of Hertz-based equations for extracting Young’s modulus of the RBC membrane by investigating the impact of varying probe geometry and size, as well as indentation region and degree of substrate adhesion.

5.2 Methods

The indentation model established and validated in Chapter 4 forms the basis of the modelling. As the force-deformation behaviour and deformed shapes between 2D and 3D were very similar (refer to Chapter 4), the simplified 2D model was used for the investigations. This enabled simulations to be run more rapidly, with only a slight compromise in accuracy. The equivalent 3D cases may be run in the future, with the expectation of mirroring the trends found in 2D, with improved accuracy for the absolute measurements. This is supported by the findings of previous CGPM studies directly comparing 2D and 3D representations of RBCs in flow such as Polwaththe-Gallage et al. [110] and [117], as well as Shi et al. [109] and [20].

The numerical investigations were separated into those studying the impact on Young’s modulus when using different probes, and secondly when indenting cells with the same mechanical properties but with varying degrees of adhesion in different regions of the cell surface. The first investigations isolate the effect of the probe to understand if Hertz-based equations are accurately quantifying contact areas and corresponding pressure distributions to extract a consistent Young’s modulus. If the substrate is influencing results, it should be doing so to the same extent for each of the cases. The second set of investigations keeps probe properties constant to understand whether Hertz-based equations can extract a consistent Young’s modulus for the RBC membrane when indenting a cell with constant mechanical properties but varying substrate adhesion and in different regions. This can then assess the influence of the substrate and adhesion protocol.

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For each case, the chosen probe was used to simulate indentation to 200 nm which is less than 10% of cell height (~2 µm). This meets the typical rule-of-thumb that indentations must be less than 10% of sample thickness for the substrate’s influence to be considered negligible [89, 132]. Zero indentation depth was defined as the point where the probe first made contact with the cell. When indenting over the surface of the cell, the horizontal position of the probe was measured from the adhered cell’s centre. Two adhered cell geometries were considered for this investigation – dome-shaped (8.5 µm contact diameter) and biconcave-shaped (5.5 µm contact diameter).

The original Hertz equation (Equation 2.2, referred to as “Original”) and that refined by Dimitriadis et al. [89] (Equation 4.1, referred to as “Dimitriadis”) were tested for extracting Young’s modulus for spherical probe indentation. As some studies have set 푅 in these equations as the tip radius (e.g. [82]) and others the effective radius (e.g. [83]), both versions were considered in the extraction process. In order to calculate effective radius of the contact (Equation 2.5), curvature of the cell at the point of interest was measured by fitting a circle through the three nearest membrane particles as shown in Figure 5.1. For indentation over the surface, an additional variation to the Dimitriadis et al. [89] equation was considered where ℎ was set to the contact height. This is because the equation was originally developed for the indentation of thin gels which had a uniform height profile so no distinction was needed between contact height and sample height. It was assumed 휈 = 0.5 [36, 74, 77, 82, 83]. To extract Young’s modulus, the chosen Hertz equation was fitted to the simulated force-deformation curve by varying Young’s modulus to minimise the root mean square error. The best-fit Young’s modulus was reported.

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To compare RBC indentation using a spherical indenter to a sharp indenter, a conical probe was modelled. The geometry of the cone was defined by tip radius,

푟푐표푛푒, and sides at angle, 휃푐표푛푒, as shown in Figure 5.2. The cone sides were assumed to be tangent to the circular tip. The number of particles used to represent the RBC membrane was increased for the conical indentation simulations to prevent the probe from penetrating through the membrane. This was necessary due to the extremely narrow probe geometry which otherwise allowed it to pass between particles. Thus using the adaptive discretisation technique 푁 was increased to 3200 (see Section 3.2.4). In addition to the spherical Hertz-based equations trialled above, the Briscoe et al. [94] equation (Equation 2.6, referred to as “Briscoe”) was tested for conical probe cases.

Radius of tip, 푟푐표푛푒 Angle of side, 휃 푐표푛푒

Figure 5.2 – Geometry of conical tips represented in the model

For indentations performed at the centre of the cell, a horizontal interface is formed between the probe and the cell. This means that only a net vertical force is required to indent the cell. However, for indentations performed at other locations where the contact interface’s orientation is at an angle, a force with a component in both the vertical and horizontal directions must be applied to deform the cell. To calculate the total force applied in these cases, force in both the horizontal direction (퐹ℎ, parallel to the substrate) and vertical direction (퐹푣, perpendicular to the substrate) must be measured. The resultant force, 퐹푅, can then be calculated with Equation 5.1.

2 2 퐹푅 = √퐹ℎ + 퐹푣 ( 5.1 )

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5.3 Results & Discussion

5.3.1 Effect of Probe Size & Geometry

The impact of using spherical and conical shaped probes of varying size and geometry was investigated. Results are presented and discussed below.

5.3.1.1 Spherical Probes

Indentation was simulated to 200 nm for the validated dome-shaped cell using spherical probes with diameters varying between 1 and 9 µm. The deformed shapes, force-deformation measurements and extracted Young’s modulus values are shown in Figure 5.3 below. It can be seen in Figure 5.3b that at small deformation depths (up to approximately 100 nm), there is a negligible difference in the applied force when the probe size is changed. However, beyond the 100 nm depth, a small difference is evident, with the larger probes requiring more force to deform the cell. This trend is due to the contact area between the cell and probe only becoming significantly different at larger depths when the concavity is introduced and the membrane starts to take the shape of the probe (see Figure 5.3a). The larger diameter probes impose a larger contact area through this phase, meaning a larger force needs to be applied to deform the membrane.

Given that the mechanical properties of the RBC are constant, Young’s modulus would be expected to show a consistent value as probe size is varied. However, as can be seen in Figure 5.3c, there is non-linear decreasing trend of Young’s modulus with increasing probe size for each of the Hertz-based equations tested. This implies that the equations are not accurately accounting for the contact area, and thus force-deformation measurements are being incorrectly normalised into stress- strain relationships for Young’s modulus extraction. The other issue is that different values are predicted with each Hertz-based equation. The difference between predictions is lower for the smaller probes compared to the larger probes – varying by 4.9 kPa for 1 µm probes increasing to 6.9 kPa for 9 µm probes. Young’s modulus using the Dimitriadis et al. [89] equation is lower than for the original equation. Noting that the substrate is much stiffer than the cell, the lower Young’s modulus implies that the substrate’s effect has been at least partially accounted for through the modifications. Using effective radius rather than the tip radius produces a larger

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(a)

Increasing Probe Diameter

(b)

(c)

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Young’s modulus. Effective radius should be considered more accurate given this theoretically quantifies the contact area more precisely – when only tip radius is considered, it is equivalent to assuming that the sample surface is perfectly flat.

5.3.1.2 Conical Probe Geometry

Indentation was simulated to 200 nm on the validated dome-shaped cell using conical probes with varying geometry. The first set of simulations varied the tip radii between 10 nm and 100 nm, while 휃푐표푛푒 = 20° given that this angle has been used in previous experimental studies (e.g. [74]). The second set of simulations varied cone angles between 10° and 60° while maintaining a tip radius of 10 nm given that this was a common tip size identified in the Literature Review [36, 88, 97]. The deformed shapes, force-deformation measurements and extracted Young’s modulus values are shown in Figure 5.4 and Figure 5.5 respectively.

It can be seen in Figure 5.4a that the deformed RBC shapes are almost identical for each tip radius tested. This is due to the contact area differing by only a very small amount between probes, given that the tip radii are all incredibly small. This is consistent with the finding related to Figure 5.3b above where a different force was only measured when the contact area between the probe and membrane became substantially different. It can be seen that only the spherical part of the probe is in contact with the membrane (see Figure 5.6 for zoomed view), implying that the conical sides are having no influence at this indentation depth. Thus this situation would be better described as a spherical contact case with a nano-sized tip. As a result, for the cases where cone angle was varied (Figure 5.5a), the force- deformation behaviour predicted by the model was the same for each case.

Noting that cone sides do not contact the cell, it makes little sense to apply the Briscoe et al. [94] equation when conical angle is varied. However, if it is applied, it is biased toward lower Young’s modulus values at higher cone angles (Figure 5.5b), as the tangent of cone angle is inversely proportional to Young’s modulus. The other Hertz-based equations for spherical probes show a consistent Young’s modulus value – as the tip radius does not change and cone angle has no impact, contact is quantified the same resulting in the same prediction. However, the accuracy is still questionable given the previous findings that the Young’s modulus was biased toward higher values with smaller tips.

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(a)

(b)

Figure 5.4 – Indentation with conical probes with 20° cone angle and varying tip radius; (a) deformed cell shapes at 200 nm indentation depth, (b) best-fit Young’s modulus predictions using various Hertz- based equations

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(a)

(b)

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Figure 5.6 – Zoomed in view of contact between probe and membrane for 푟푐표푛푒 = 10 nm and 휃푐표푛푒 = 20° showing that only spherical part of probe is contacting the membrane (red line)

5.3.2 Effect of Indenting Over Surface

Indentation across the surface of a dome-shaped cell and biconcave-shaped cell with the same membrane stiffness properties was investigated. The results are presented and discussed below.

5.3.2.1 Dome-Shaped Cell

Indentation was simulated to 200 nm using a 5 µm spherical probe over the surface of the dome-shaped cell. The deformed shapes are shown in Figure 5.7. The force- deformation measurements, extracted Young’s modulus values and cell curvature are shown in Figure 5.8.

It can be seen in Figure 5.8a that when the RBC is indented away from the centre, a horizontal component of force needs to be applied. The further from the centre, the larger this horizontal force becomes, while the vertical force reduces. This is due to the orientation of the contact area which becomes increasingly rotated further from the centre for dome-shaped cells (see Figure 5.7).

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Figure 5.7 – Deformed shapes of dome-shaped cell at increasing distances from centre at 200 nm indentation depth

Overall, the resultant force required to indent the cell decreases at further distances from the centre. This has been argued in the past to be due to a difference in RBC membrane mechanical properties over the cell surface [36]. However, in the model identical membrane properties were applied across the surface, which means in this instance it is not the explanation. Rather, the cell showing more deformability at the edges is attributed to it having more freedom to deform in the direction of the applied force – when the cell is indented at the centre it is forced downward due to the flat contact formed. Consequently, a reaction force will be felt against the substrate directly beneath. This makes the cell appear stiffer than it actually is. In contrast, when the cell is indented near the edges there is a significant horizontal force component. However, there is no body (such as a substrate) to react against this. Thus the cell is free to deform in that direction. This makes the cell seem more deformable further from the centre, but it is actually an artefact of the substrate’s influence (or lack thereof).

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(a)

(b)

(c)

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There is generally a decreasing trend in Young’s modulus further from the centre for each of the Hertz-based equations (Figure 5.8b). This is consistent with the report of Ciasca et al. [36] who found RBCs were significantly stiffer at their centre compared to at the edge (see Section 2.5.1). In the present study, there is an exception in the decreasing trend for two cases at the 4 µm distance where Young’s modulus drifts upward. This is caused by the sudden change in cell curvature which impacts the effective radius (Figure 5.8c). The use of contact height partially compensates for this in the Dimitriadis et al. [89] equation with effective radius and contact height, so it does not drift upwards. Finally, each interpretation of the Hertz-based equations predicts a different value for Young’s modulus across the surface, varying by up to 6 kPa.

The presence of the substrate is clearly impacting the Young’s modulus results, which is in turn discrediting the validity of the half-space assumption which implies negligible substrate effects. This infers that the Young’s modulus being reported in this Hertz-based analysis is a combined stiffness of the substrate and cell, rather than just the membrane in isolation. As membrane properties were maintained constant across the cell and different Young’s modulus values were predicted, this further supports that Hertz-based equations are not suitably capable of normalising the force-deformation measurements into stress-strain relationships for the extraction of Young’s modulus.

5.3.2.2 Biconcave-Shaped Cell

Indentation was simulated to 200 nm using a 5 µm spherical probe on a biconcave- shaped cell to explore further whether extracted Young’s modulus values represented a measure of combined stiffness of the cell and substrate. The results for the central indentation case are shown in Figure 5.9.

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(a)

(b)

Figure 5.9 – Central indentation of biconcave-shaped cell; (a) force-deformation measurements, and (b) deformed RBC shapes at selected depths

It can be seen that the biconcave cell’s central indentation shows a very different force-deformation profile to the other cases explored to date, with applied force peaking at 130 nm (Figure 5.9a). Inspection of the deformed shapes shows that the cell slipped out from under the probe, and this happened just after the peak applied force at 130 nm indentation depth (Figure 5.9b). This implies that the applied force became so significant that the cell slipped out to relieve the stress. This is further confirmed by the horizontal force measurements (Figure 5.9a) which shows that a horizontal force component only exists after the force peak from 140 nm. When the deformation is symmetric, the horizontal force is equal on each side of the probe, meaning net horizontal force is negligible.

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This slippage behaviour of the biconcave-shaped cell is hypothesised to be caused by the rapid increase in applied force from the contact area between the membrane and probe dramatically increasing as the membrane wrapped around the probe (see Figure 5.9b). In fact, the length of the contact formed between the cell and probe was 2.83 µm for the biconcave cell at 200 nm indentation depth compared 0.38 µm for the dome-shaped case at the same depth. The probe also got much closer to the substrate for the biconcave case, which likely also contributed to the rapid increase in applied force, as movement toward the substrate was increasingly inhibited due to the substrate’s presence. This slippage behaviour is inconsistent with how a solid would behave in the same circumstances (and especially within the elastic deformation region). Thus this curve cannot be justified for analysis using Hertzian contact theory which assumes the contacting bodies act as elastic solids.

The deformed shapes for indentation over the surface of the biconcave cell are shown in Figure 5.10. The corresponding force-deformation measurements, extracted Young’s modulus values and cell curvature are shown in Figure 5.11 below. The central indentation case has been removed as an outlier for the purposes of testing Hertz-based analysis on these results, given the peaking behaviour. This was not observed in the other cases. Unlike the dome-shaped case there is an inconsistent change in the RBC surface profile, moving from concave to convex curvature at increasing distances from the centre (see Figure 5.10). Furthermore, the orientation angle of the contact is constantly changing – this is highlighted in the horizontal force which is shown to be virtually zero for the 3 µm case, applied in one direction for the 3.5 µm case, and then in the opposite direction for the remaining cases (Figure 5.11a). This aligns with the orientation of the contact evident in Figure 5.10 where the 3 µm case is on a flat ridge. Before the ridge the contact with the cell is on one side of the probe, and after it is on the other side.

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Figure 5.10 – Deformed shapes of cells with biconcave upper surface profile at increasing distances from centre at 200 nm indentation depth

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(a)

(b)

(c)

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In Figure 5.11b, there is significant variation in the reported Young’s modulus based on the equation used. At 0.5 µm from the centre, the difference in extracted Young’s modulus values is largest at 0.46 kPa. The smallest difference of 0.08 kPa occurs between 1.5 and 2 µm distances from the centre. The Young’s modulus trends are less predictable than for the dome-shaped case. This is due to the varying contact height and curvature of the RBC’s surface profile (see Figure 5.11c) which are considered in different ways across the various interpretations of the Hertz-based equations. None of the equations predicted a consistent Young’s modulus value. Overall, the Dimitriadis et al. [89] interpretations generally produced lower estimates for Young’s modulus, again because modifications were included to account for substrate effects.

The extracted Young’s modulus values for the biconcave-shaped cell (Figure 5.11b) are significantly lower than for the dome-shaped cell (Figure 5.8b), regardless of the extraction equation used. This is because the biconcave-shaped cells showed significantly more deformability when adhered to the surface compared to the equivalent dome-shaped cells. This is because they were less constrained in their movement, evident in Figure 5.10 where there is very little change in shape at large distances from the centre as the membrane only needs to deform slightly to escape the probe. This is analogous to the situation of RBCs with large surface area to volume ratios being able to adopt a broad range of shapes while maintaining a constant ratio, enabling them to deform easily. This is contrasted against the least deformable shape, a sphere, which is unable to deform at all if maintaining a constant ratio [1]. Here, in a similar way, when many particles are adhered, the cell’s capacity to change shape in response to the loading is greatly compromised. Thus, in addition to the substrate’s influence, the extent of the adhesion directly impacts on the measured deformability of the RBC. This again infers that the Young’s modulus being reported is a measure of the combined stiffness of the substrate and cell, rather than the membrane in isolation.

5.3.3 Discussion of Findings

This numerical investigation has shown that Hertz-based analysis in its current form is not capable of producing a reliable estimate for Young’s modulus of the RBC membrane. At the most basic level, the variation in contact area caused by changing probe diameter is not being accurately quantified, with larger probes resulting in smaller Young’s modulus predictions despite cellular mechanical

Page 94 Numerical Modelling of Red Blood Cell Morphology and Deformability properties being constant. This is attributed to the RBC not acting as an isotropic solid for which the Hertz equations were developed. Instead RBCs are soft matter. Moreover, as the RBC has no internal structural components such a nucleus or cytoskeleton crossing the inside of the cell, physical behaviour is governed entirely by the fluid-filled 2D membrane through its bending and stretch resistance as well as surface area incompressibility. This is a very different mechanism for resisting deformation than for a solid.

As a result of this difference in how a solid and a RBC deform, the stress fields developed at the contact differ substantially. As shown in the Figure 5.12 schematic, for the isotropic solid, stress propagates in all directions away from the contact region. As it is an isotropic material, it behaves the same in each direction. In contrast, for the case of the RBC, the membrane takes the stress by changing its shape – as the internal fluid is incompressible (and significantly “stiffer” in comparison to the membrane), stress is redirected from the fluid onto the membrane through outwardly applied pressure. This phenomenon was previously highlighted in Section 4.3.1 where it was shown that the internal fluid strongly resisted compression and thus stored a negligible amount of energy. Instead the bending, linear and surface area mechanisms resisted the external loading. Thus the cell is also not resisting stress the same way in all directions.

(a) (b) Rigid Probe High Stress Membrane

Isotropic Solid Incompressible Low Stress Material Fluid

Figure 5.12 – Schematic of stress fields created for contact against (a) an isotropic solid material, and (b) a fluid-filled membrane

These issues with the RBC not behaving like solid are compounded by the mobility of the internal fluid and the membrane being capable of large strains. The fluid can move freely inside the membrane and thus when a force is applied, the fluid has the ability to relocate reasonably easily to dissipate any stress. This is aided by the highly elastic nature of the membrane and was clearly observed where the cell slipped out from under the probe in order to relieve the applied pressure (Section 5.3.2.2). This represents a significant deviation from how solids behave.

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As a consequence of the RBC acting as a fluid-filled membrane rather than an isotropic solid material, the force-deformation behaviour is not being accurately mapped into a stress-strain relationship for this application. This is causing substantial variations in the Young’s modulus estimates. Therefore a new method should be developed for estimating Young’s modulus of the RBC membrane which treats the RBC as a fluid-filled membrane. This is discussed in Section 5.3.4.

The inconsistency in the Young’s modulus predictions from indenting across the surface of RBCs with constant mechanical properties has demonstrated that the reported Young’s modulus values are impacted by the presence of the substrate. Furthermore, the greater deformability shown by biconcave-shaped RBCs compared to dome-shaped RBCs with the same mechanical properties supports that the reported Young’s modulus is measuring a combined stiffness of the adhered cell and substrate rather than the membrane in isolation.

Classical Hertzian theory uses a half-space assumption, which means that the stress field created by the contact can propagate unimpeded through the material, and there is no other stress field caused by another body such as a substrate. In practical terms, this implies that the area of contact is much smaller than the body’s characteristic radius [96]. Figure 5.13a shows the stress field developed when a small indentation is performed. The contact area is small compared to the size of the body and the stress quickly dissipates through the material such that stress near the substrate is virtually unchanged. Figure 5.13b shows the stress field when a large indentation is performed where the contact area is comparable to the size of the body. In this case, the stress created by the contact has not been dissipated before the substrate interface, meaning that the substrate’s stiffness will influence behaviour in the contact region. This is not considered in the conventional Hertz equations.

(a) (b) High Stress

Low Stress

Figure 5.13 – Schematic of stress fields created for solid contact when contact region is (a) negligible in comparison to size of bodies and (b) significant in comparison to size of bodies

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The typical rule-of-thumb is that indentations must be less than 10% of sample thickness for the substrate’s influence to be considered negligible [89, 132]. The present study used 200 nm indentations which is less than 10% of cell height. However, due to the variation in height across the RBC surface, in many cases the indentation depth was significantly more than 10% of height at the contact point. Therefore, this would have introduced some error into the analysis. However the substrate’s influence goes deeper – its effect varied based on the orientation of the contact region. When a horizontally orientated contact region was formed (e.g. in the centre of the dome-shaped cell), the substrate’s effect meant more force was needed to deform the cell as it was unable to move in the direction of the applied force. In contrast, when the contact region became increasingly rotated (e.g. at the edge of the dome-shaped cell), less force was required to deform the cell as less force was directed toward the substrate. This further substrate effect is not being taken into account in the classical Hertz-based analysis.

Although the modifications included in the Dimitriadis et al. [89] equation accurately accounted for finite sample thickness in its original application for thin and soft materials (it was validated experimentally for poly vinyl alcohol gels), it has been shown here to be unsuitable for RBC studies for the purpose of extracting Young’s modulus. This is attributed to the equation being tailored specifically to materials which were homogeneous, and which covered large areas on the substrate with a uniform height profile. The uniform height means that the contact formed between the sample and probe was consistently orientated parallel to the substrate so the influence of the substrate was always equal. Furthermore, the larger material area also means that “edge effects” were avoided.

In order to minimise substrate effects, smaller indentations could be performed. However in experimental practice, this can introduce greater uncertainty around identifying the contact point and can increase the significance of tip-membrane adhesive interactions [81, 131]. In fact, in the literature, indentations have been conducted up to 2.5 times deeper than the present study to overcome these problems [81, 83]. However this is likely causing a further amplification of the substrate effects. Therefore a new method should be developed for estimating Young’s modulus of the RBC membrane which addresses the root cause and explicitly considers the substrate. This is discussed in the following section.

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5.3.4 Proposed Alternative Estimation Method for Young’s Modulus

The applicability of Hertz-based equations to extracting Young’s modulus of the RBC membrane has been shown to be extremely limited. As a result, alternative methods should be proposed for the purpose of obtaining an accurate estimate for Young’s modulus of the RBC membrane. This new method needs to be able to address the major issues identified with Hertz analysis – considering the cell as a fluid-filled membrane to accurately capture how it deforms under stress (rather than assuming it acts as a solid), as well as explicitly considering the substrate (rather than neglecting it). The combination of these aspects should mean that the force- deformation behaviour is accurately mapped into stress-strain relationships for the purpose of extracting Young’s modulus of the membrane.

An inverse FEM-based approach is recommended for addressing the highlighted issues. This technique has been used in the past to characterise the elastic properties of biological samples including soft tissues [133], human bone [134] and individual chondrocyte cells [135]. For the RBC application, it is envisioned that a single RBC would be represented as a membrane with constant volume. The initial adhered shape of the membrane could be obtained from experimental imaging techniques, or predicted using the existing CGPM method. A boundary condition would be applied to restrict movement of the membrane at the substrate interface. Young’s modulus would be inputted as a material property for the membrane. Its value would be iteratively converged upon by matching the experimental force- deformation behaviour against the numerical prediction when contact between the membrane and a probe was simulated.

This FEM approach could be taken further to propose a new equation describing the force-deformation response as a function of the probe size, cell geometric properties, Young’s modulus and so on. This could be achieved by simulating RBC behaviour under various property combinations to generate the relationships. An expression could be proposed which linked significant factors together. It would mean that the analysis procedure had the simplicity of the Hertz-based curve-fitting approach, but was grounded in appropriate theory. Despite the potential for this FEM-based avenue, it will not be explored further in the present project given that it represents a substantial divergence in the modelling methodology.

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5.4 Summary & Conclusions

This chapter investigated the accuracy of Hertz-based equations in extracting a Young’s modulus estimate from the force-deformation data obtained from AFM indentation. The following conclusions were drawn:

. Larger sized spherical probes are biased toward providing smaller estimates for Young’s modulus using Hertz-based analysis. . For small indentations (200 nm) using conical probes, tip radius is the dominant aspect of the geometry and not conical angle. . Significant variation in the extracted Young’s modulus value is observed based on the version of the Hertz-based equation applied. . The variation in contact area caused by changing probe diameter is not being accurately quantified. This is attributed to the RBC not acting as an isotropic solid for which the Hertz equations were developed, but rather a 2D fluid-filled membrane. . The substrate impacts the force-deformation behaviour which is not accounted for in the Hertz analysis, causing RBCs to generally appear more deformable at their edges compared to in the centre. This infers that the Young’s modulus being reported is the combined stiffness of the substrate and cell, rather than the membrane in isolation. . The degree of adhesion of the RBC on the substrate also impacts force- deformation measurements – smaller Young’s modulus values are extracted for a biconcave cell compared to a dome-shaped cell. This is due to the reduced adhesion constraint which enables the cell to move more freely away from the probe, meaning less force is required to cause the same level of deformation. . As the applicability of Hertz-based equations for extracting Young’s modulus of the RBC membrane is extremely limited, a new analysis method is required. An inverse FEM-based approach is recommended for this future development. . The Young’s modulus values reported by previous studies should be interpreted with caution given the above findings that the Hertz-based analysis is extremely limited in its applicability to RBC indentation problems.

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5.5 Contribution to Research Objectives

This chapter has addressed the fourth objective of this project – to apply the model in numerical experiments to investigate the factors that impact measurements of deformability. Here it has been shown that Hertz-based equations are extremely limited in their validity for the RBC indentation application as they are not accurately accounting for probe size, cell geometry, substrate influences and adhesion state. This has serious implications for existing studies which have attempted to quantify Young’s modulus of RBCs as the numerical value reported holds very little physical meaning. A new method is required which is able to address the limitations of the Hertz theory being applied to RBCs, namely treating the cell as an isotropic solid on a substrate that has a negligible impact.

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Chapter 6. Investigation of RBC Mechanical Properties during Optical Tweezer Stretching

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6.1 Introduction

In order to expand the deformability investigations to develop further understanding for how membrane properties govern the physical behaviour of RBCs, an examination of overall cellular deformability was proposed. This is contrasted against the indentation studies which investigated deformability by probing localised regions of the membrane.

As discussed in the Literature Review, both flow and optical tweezer stretching enable deformability of the cell to be observed globally, but optical tweezer stretching permits greater control over the application of the force and this force is directly quantified. For flow, force is not directly measured and instead properties of the bulk fluid such as inlet velocity and pressure are known. Thus in order to model flow numerically, an explicit representation of the fluid phase is required to accurately distribute force onto the cell. Flow has also already been investigated extensively with the CGPM [14, 20, 78, 104-118]. Therefore stretching was chosen for modelling.

Optical tweezer stretching uses beads attached to the RBC membrane and pulled in opposite directions to put the cell into tension (see Section 2.5.2). This results in the cell elongating along its axial direction and contracting along its lateral direction. The less force required to deform the cell, the more deformable it is.

Only limited investigation of RBC stretching using numerical models has occurred. In terms of particle-based modelling, Shi et al. [20] used the CGPM to model stretching in order to validate stiffness coefficients for a flow simulation, while Fedosov et al. [21] simulated stretching with a DPD model but again only showed that the numerical predictions matched experimental observations. Neither of these models explicitly represented the optical beads, instead applying stretching force directly onto selected membrane particles. In terms of FEM-based modelling, both Peng et al. [100] and Dao et al. [136] generated models which were validated against stretching observations, but the deeper mechanisms were not explored. Consequently, there is very limited understanding of the relative influence of the mechanical properties which govern the membrane’s physical behaviour in this scenario. Yet improved knowledge may aid in identifying the structural changes occurring within the RBC membrane, leading to improved strategies for preventing deformability loss. Furthermore, understanding of the factors which impact deformability measurements in this experimental setup may aid in maturing the

Page 102 Numerical Modelling of Red Blood Cell Morphology and Deformability technique for diagnostic purposes, given that differences have been shown between healthy RBCs and those suffering progressive stages of malaria infection [17], as well as between RBCs stored for increasing durations [18]. Thus the aim of this chapter was to investigate how RBC mechanical properties impact deformability, as well as how the experimental setup may influence measurements. It should be noted that the scope will again be restricted to RBCs with the typical discocyte resting morphology, with echinocytes explored in Chapter 7.

6.2 Methods

6.2.1 Experimental Observations for Validation

For validation of the numerical predictions, comparison was necessary against experimental results. A handful of data sets have been published for the stretching of healthy RBCs using optical tweezer techniques [17, 18, 98, 101, 103]. Of these data sets, Dao et al. [98] was chosen as the reference for the numerical model’s performance. This is because a large deformation range was studied, sufficient information was provided about the contact between the bead and membrane for modelling, the force-deformation data was complete including error estimates, and the results have been used for benchmarking other models (e.g. [100]), thus establishing it as a high-quality standard. This experimental study will be discussed briefly to give context to the model development detailed in the subsequent section.

Dao et al. [98] attached micro-silica beads of 4.12 µm diameter to diametrically opposite ends of isolated RBCs of the discocyte morphology. The contact between the bead and membrane was estimated to have a diameter of between 1 and 2 µm. Force of up to 400 pN was applied through the beads using laser beams, as the axial and transverse diameters of the cell were measured using optical means. The stretched shapes at various points in the experiment are shown in Figure 6.1 below. The force-deformation measurements are shown in Figure 6.2 with a line of best fit added for the mean diameter trends. These equations describing how the axial diameter, 퐷푎푥푖푎푙, and the transverse diameter, 퐷푡푟푎푛푠, vary with the applied force, 퐹, are given in Equations 6.1 and 6.2 respectively. It can be seen that the 푦-intercept for the axial diameter occurs at approximately 7.6 µm while the transverse diameter is approximately 6.8 µm. This is equivalent to the RBC resting dimensions with beads attached.

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0 pN

145 pN

280 pN

340 pN

Figure 6.1 – Images of RBCs stretched with increasing force. Image taken from Dao et al. [98].

Figure 6.2 – Experimental measurements for axial and transverse diameter of RBCs stretched using optical tweezers. Experimental data recreated from Dao et al. [98].

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퐷푎푥푖푎푙 [μm] = 0.016퐹 [pN] + 7.6165 ( 6.1 )

퐷푡푟푎푛푠 [μm] = −0.0059퐹 [pN] + 6.7801 ( 6.2 )

6.2.2 Modelling Methodology

The model established in Chapter 3 was modified for the optical tweezer setup specifically described above (Section 6.2.1). However, this was only done for the 3D case where the cell is able to extend in the axial direction while contracting in the lateral direction. In the 2D model this is not possible as the lateral direction is not represented. Instead when the 2D cell is revolved about a vertical axis through its centre to visualise the 3D case, it corresponds to a tensile force being applied outwardly around the circumference rather than in just one axial direction.

In order to use the experimental data to validate the model, the experimental conditions relating to bead adherence were replicated. Beads of 4.12 µm diameter were attached to diametrically opposite ends of the cell using contact energy terms as per the indentation methodology (Equation 4.2 in Section 4.2.2). Here 푑푖 was considered the distance between the 푖푡ℎ particle to be attached to a bead and the closest point on that bead. The contact stiffness coefficient (푘푐표푛) was selected from a sensitivity study (see Section 6.2.3 below). When the cell was orientated to lie flat on the 푥-푦 plane, the two particles furthest from the centre of the cell in the positive and negative 푥-directions were chosen as the centre of the bead-membrane contact. The bead-membrane contact in the experiment was reported to have a diameter of between 1 and 2 µm [98]. This was achieved in the model by selecting adjoining particles moving away from the centre of contact until the region encompassed was within the desired range.

To find the minimum energy configuration with beads attached, the principle of virtual work (Equation 3.1) was used to re-minimise energy with the additional contact energy term when axial diameter was varied. This produced a curve for stored energy versus axial diameter (a typical example is shown in Figure 6.3) and the minimum energy point was identified. This was considered the resting shape when beads were attached.

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Figure 6.3 – Typical stored energy versus axial diameter plot used to identify the minimum energy configuration with attached beads

To simulate stretching, the distance between the beads was increased by up to

7 µm from the resting state with beads. The meaning of distance, 푑푖, in Equation 4.2 was changed to be the distance between the 푖푡ℎ particle attached to a bead and its position relative to the bead in the minimum energy state identified in the previous step. This ensured that the geometric characteristics of the contact region were maintained even when the cell was placed under considerable stress. It was found that if the previous definition of 푑푖 was used, the contact region contracted considerably in size as the cell was stretched, inconsistent with the experimental setup.

The principle of virtual work (Equation 3.1) was re-applied with the 퐸푐표푛 energy term to re-minimise total energy. This treats the stretching as a “quasi-static” problem, justified by the stretching speed being slow enough that the system remains in internal equilibrium. Total contact force between the cell and each bead was measured when the simulation had reached steady-state. By Newton’s Third Law, the measured contact force is equivalent in magnitude to the force applied to cause the deformation. Applied force and cell dimensions (axial and transverse diameter) were plotted against each other and compared against the experimental reference data.

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6.2.3 Sensitivity Study for Parameter Selection

The particle number for representing the RBC membrane was selected from the sensitivity study completed in Section 4.2.3 based on the resting shape of the RBC. The number of particles was chosen as 1922 due to stabilisation of the resting shape dimensions and energy, and also to be consistent with the indentation work.

The contact stiffness coefficient was set proportional to the bending stiffness coefficient (푘푏) to ensure it remained proportionately dominant when absolute values were varied. Its value was chosen from a sensitivity study based on stretching a cell 7 µm from the resting position with beads. Contact force results are shown in Figure 6.4. It can be seen in that as the contact stiffness coefficient is increased, contact force stabilises. This is due to the contact “springs” approaching a length of zero, meaning the attachment is being enforced. As in the indentation simulations, when the contact stiffness coefficient becomes too large, instability issues can emerge due to oscillation of membrane particles against the contact. However, when too small, the contact is not effectively enforced causing a variation 18 −1 in the force reading. To balance these competing factors, 푘푐표푛 = 10 × 푘푏 푚 was chosen for where contact stiffness became independent of the force (Figure 6.4).

Figure 6.4 – Sensitivity study for contact stiffness coefficient for optical tweezer stretching in 3D

The effectiveness of enforcing the bead-membrane contact can be confirmed by observing the offset distance between the membrane particles and bead as

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 107 demonstrated in Figure 6.5. The average distance between attached membrane particles and their closest point on the bead surface is 0.0078 nm, which is negligible in comparison to the scale of the cell (axial diameter is 14.6 µm in Figure 6.5).

Figure 6.5 – Contact between membrane and bead when RBC stretched 7 µm from resting shape with beads attached, showing negligible offset distance between the bodies

6.2.4 Validation of RBC Shape & Force-Deformation Behaviour

Due to the significant variation in the stiffness coefficients used in previous studies (refer to Section 3.2.5.1), an inverse method was used to extract the stiffness coefficients which provided agreement between the model and force-deformation experimental results during stretching. This involved assuming values, simulating stretching, comparing the force-deformation behaviour between the experiment and simulation, and then modifying the assumed values to repeat the process until reasonable agreement was reached. The optimised stiffness coefficients are shown in Table 6.1 along with other inputs for clarity. Figure 6.6 shows the predicted shapes of the RBCs at rest and when stretched with increasing forces.

Table 6.1 – Optimised stiffness coefficients for optical tweezer stretching in 3D Parameter 3D Model Starting Radius of Sphere 푟 = 3.3 휇푚 Number of Particles 푁 = 1922 Referenced Properties 훼0,푖 = 180° 푙0,푖 = original lengths 퐴0,푖 = original areas Reduction Ratio 푅푉 = 0.6 −5 Stiffness Coefficients 푘퐿 = 2.5 × 10 푁/푚 −18 푘퐵 = 9.9 × 10 퐽/푟푎푑 9 2 푘퐴 = 2.5 × 10 푁/푚 −13 푘푉 = 2.4 × 10 퐽

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(a)

(b)

(c)

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 109

(d)

(e)

(f)

Figure 6.6 – 3D shape predictions with optimised stiffness coefficients; (a) at rest without beads, (b) at rest with beads, (c-f) at increasing stretch distances, dimensions in micrometres

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The dimensions of the RBC without beads (Figure 6.6a) align with resting shape validated in Section 3.3.2. Furthermore, the resting shape with beads has an axial diameter of 7.6 µm (Figure 6.6b) which matches the experimental resting diameter obtained from Equation 6.1. The stretched shapes (Figure 6.6c to Figure 6.6f) are qualitatively consistent with the experimental images of stretched RBCs previously shown in Figure 6.1.

The simulated force-deformation curves are shown in Figure 6.7. It can be seen that the simulated results match closely with the experimental results, with each observation within the error bars. This validates the performance of the model against the experimental behaviour.

Figure 6.7 – Force versus deformation curves comparing the numerical model’s prediction using optimised stiffness coefficients to the experimental reference curves from Dao et al. [98]

6.3 Results & Discussion

In order to understand how the mechanical properties of the RBC impact behaviour when the cell was stretched by 7 µm, several aspects were investigated. These included how energy was stored during stretching and the effect of varying stiffness properties on deformability. In order to inform experimental design, the impact of bead size and contact diameter were also investigated. Finally, the stiffness

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 111 coefficients used in the indentation study were tested in the stretching model to further evaluate the universality of the coefficients among different loading scenarios.

6.3.1 Energy through Stretching

Figure 6.8 plots the energy stored in each mechanism as a function of the distance stretched. It can be seen that the linear component stores the most energy, followed by bending, surface area incompressibility and then volumetric incompressibility. The linear mechanism has the greatest change in energy, followed by the bending mechanism. There is a negligible change in the energy stored via surface area and volume changes. The lack of change in the volumetric energy again supports its nature as an incompressible fluid. This means that any stress on the fluid is dissipated onto the membrane which is considerably softer.

Figure 6.8 – Energy stored in each mechanism through stretch

The substantial energy absorbed by the linear mechanism is consistent with the type of loading. The global stretching necessitates that the linear interactions extend in order to facilitate the significant elongation in overall shape, thus storing increasing amounts of energy. As a consequence of the shape change, some additional energy is stored in bending.

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6.3.2 Effect of Changing Stiffness Coefficients

To explore the relative influence of the individual mechanisms involved in RBC deformability, a parametric study was conducted where each stiffness coefficient was varied in isolation to between a tenth and ten times the baseline value. The cells were then stretched nominal distances of 3.5 µm and 7 µm with the applied force measured (Figure 6.9).

Figure 6.9 – Results of parametric study measuring force to stretch to 3.5 µm and 7 µm when varying stiffness coefficients

Changing the stiffness coefficients in this loading scenario had a fairly scaled impact on the force required to stretch the cell for the linear, bending and surface area mechanisms (Figure 6.9). The linear stiffness caused the largest change in required force, followed by the surface area incompressibility and then bending stiffness. Varying volumetric incompressibility had a negligible impact. As the cell was stretched globally, it is consistent that linear stiffness would have the biggest impact on the required force, as each linear interaction needs to elongate substantially to enable the overall deformation. Increasing the stiffness coefficient then increases the force required to enact this. Changing the areal incompressibility also caused a marked change in the force that needed to be applied, however its effect started to plateau at the 7 µm stretch distance. The effect of changing the bending stiffness was less apparent, especially at the 7 µm stretch distance. This is likely because the change in curvature of the cell, which causes additional energy to be stored in bending, is secondary to the overall elongation.

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This study demonstrates that deformability loss of the RBC in this loading scenario is most sensitive to changes in linear stiffness, which is directly related to the mechanical properties of the spectrin-based cytoskeleton. This suggests structural changes here play a critical role in the deformability loss observed in deteriorating RBCs. Examples of biological processes linked to a reduction in cytoskeleton deformability as identified in the Literature Review include increased cross-linking of spectrin and actin [59] and an increase in the average length of spectrin molecules within the triangulated network [15]. As optical tweezer stretching has been shown to be highly sensitive to changes in the cytoskeleton’s linear stiffness, it would be well-positioned in detecting these types of changes within the RBC membrane.

6.3.3 Effect of Bead Size

The effect of bead size in the optical tweezer setup was investigated to understand whether it influenced the deformability measurements, and thus whether it impacted the comparison of different study results and should be an important consideration in an optical tweezer setup. This has implications for developing optical tweezer stretching as a diagnostic tool, given the capacity to detect deformability differences between healthy and deteriorating RBCs [17, 18, 102]. Previous RBC studies have used beads between 2.1 µm [101] and 5 µm [18] in diameter.

The force required to stretch RBCs using beads of varying diameter is plotted in Figure 6.10. It can be seen that the force required to stretch the cell does not change substantially despite the change in bead size – the difference between the maximum and minimum force required to stretch the RBC 7 µm is just 0.06 nN.

The lack of impact from bead size on applied force is caused by the fact that the adhered position of the membrane particles is only slightly altered by the change in bead curvature – there is no significant influence on the resting diameter of the RBC or overall shape (see insets of Figure 6.10). Furthermore, as the beads become larger, the attachment surface assympototically approaches a flat plate with a curvature of zero (see Figure 6.11). Therefore using larger and larger beads will have even less effect.

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Figure 6.10 – Force required to stretch RBC using optical beads of increasing size with inset images of stretched shapes

Figure 6.11 – Relationship between bead diameter and curvature, noting that a curvature of zero represents a flat surface

As bead size has been shown to be virtually independent of deformability measurements, bead size should not be a complicating factor in comparing the results of different experimental setups reported in the literature. Furthermore, its influence could be neglected in the technique optimisation – instead bead size can

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 115 be selected based on factors like ease of use in practice, accuracy of measurement, cost and availability.

6.3.4 Effect of Contact Diameter

The effect of changing the size of the contact region due to variation in the bead- membrane adhesion protocol was investigated to understand whether it influenced deformability measurements, and thus whether it should be an important consideration in experimental design. This also has implications for developing optical tweezer stretching as a diagnostic tool.

The force required to stretch RBCs when increasing areas of the membrane were adhered to the optical beads is plotted in Figure 6.12. It can be seen that the force required to stretch the cell increases substantially with increasing adherence – there is a difference of approximately 0.8 nN to stretch the cell 7 µm when the average contact diameter is 0.5 µm compared to 2 µm. This is because increasing the membrane area adhered to the beads influences the resting position of the cell – resting axial diameter is 7.7 µm when the contact diameter is 0.5 µm, compared to 6.9 µm when the contact diameter is 2 µm. This is due to the beads being drawn together as more of the membrane becomes wrapped around the beads. Furthermore, the deformed morphologies are substantialy different between the cases due to the increasing constraints placed on membrane particles being attached to the beads (see insets of Figure 6.12). The more the cell membrane is adhered to the beads, the less deformability the RBC shows. This is because a larger portion of the membrane is restricted in its movement, giving the cell less freedom to change its shape in response to the externally applied force.

As the contact area has a significant influence on the measured deformability of the RBC, this may be a complicating factor when comparing the results reported by different experimental setups. However, the contact diameter is not always stated in past optical tweezer stretching experimental results (e.g. Hénon et al. [101] and Li et al. [18]), meaning it is hard to gauge how much this varies between all experimental designs. Nonetheless, future experimental studies should carefully consider the adhesion protocol and its influence on the deformability measurements. The contact diameter should be minimised as much as reasonably possible in order to observe the deformability of the RBCs as close to their natural state as possible.

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Figure 6.12 – Force to stretch RBC when average contact diameter between the membrane and bead is varied with inset images of stretched shapes

It may also be necessary to develop more sophisticated analysis procedures that take the degree of bead-membrane adhesion into account, similar to what is required for the indentation force-deformation analysis for extracting Young’s modulus (concluded in Chapter 5). An inverse FEM-based approach may be considered for this purpose as well, given that it could explicitly incorporate the beads. This would also allow a direct comparison of RBC membrane Young’s modulus values obtained from both experimental techniques.

6.3.5 Comparison of Stiffness Coefficients to Indentation Study

In order to understand the applicability of the stiffness coefficients optimised for indentation (listed in Table 4.1) to the stretching scenario, these were tested in the stretching methodology. Figure 6.13 shows the resulting force-deformation relationship.

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It can be seen that the simulated RBC with the stiffness coefficients optimised for indentation is significantly stiffer than the experimental observations when stretched – while the experiment showed that 400 pN were required to achieve an axial diameter of 14 µm, the simulation showed that about 13 000 pN of force was needed. This may be attributed in part to differences in the physical samples tested. In particular, the indentation experiments used RBCs which were lightly fixed, so they would be expected to be stiffer. However, the extent of the variation is too significant for this to be the only factor in the difference observed. Given that a wide range of values have been used in previous CGPM models (Section 3.2.5.1) and that values optimised for a particular application have been shown unsuitable for other applications (Section 4.3.4), a lack of universality for the current modelling methodology is the determined root cause. This is attributed to the need for improved optimisation of the stiffness coefficients, combined with the simplicity of the equations used to quantify the amount of energy stored in the various forms (bending, linear, surface area and volume). To achieve a more sophisticated model, greater complexity should be built into these equations to account for the non-

Page 118 Numerical Modelling of Red Blood Cell Morphology and Deformability linearity of the behaviour across the different levels of stress and strain. This is discussed in the context of the overall work in Section 8.2 below.

6.4 Summary & Conclusions

This chapter investigated how the mechanical properties of the RBC impacted the deformability of the RBC when stretched up to 7 µm using optical tweezers. The following conclusions were drawn:

. The linear stretch mechanism absorbs the most energy when RBCs are stretched as each linear interaction must deform in order to allow the cell to elongate to such a significant extent from the resting shape. . RBC deformability during stretching is most sensitive to changes in the membrane’s linear stretch stiffness. As this stretch resistance is provided by the spectrin-based cytoskeleton, the optical tweezer stretching technique appears well-suited to detecting structural changes impacting deformability of the cytoskeleton. . Bead size is virtually independent of the force-deformation measurements in the optical tweezer setup. This is because there is only a small change in the curvature of the attachment surface, which asymptotically approaches a flat plate as bead size is increased. Therefore bead size should not be a complicating factor in comparing the results of different studies, and does not require optimisation in experimental design. . Contact area between the bead and membrane has a significant impact on force-deformation measurements, with larger contact areas causing RBCs to require more applied force to achieve the same level of deformation. Therefore this may be a complicating factor in comparing the results of previous studies, and its effect should be carefully considered in experimental design. . It may be necessary to develop analysis techniques which are able to take into account the bead-membrane contact in order to make results between different experimental setups comparable, similar to what is required to extract a Young’s modulus value for the RBC membrane during indentation. . Stiffness coefficients need to be tailored to the specific deformability application. This is a limitation of the current modelling methodology and may be addressed by improving the optimisation process for obtaining the

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stiffness coefficients or improving the equations for quantifying stored energy.

6.5 Contribution to Research Objectives

This chapter has contributed to the second, third and fourth objectives of this project. Firstly, the model was validated against experimental observations of RBC behaviour in an optical tweezer stretching scenario. Secondly, the validated model was applied to numerical experiments investigating how mechanical properties of the RBC impacted upon deformability, finding that linear stretch resistance was most dominant in this scenario. Finally, the model was applied to numerical experiments investigating how aspects of the experimental setup impacted measurements. This found that while the bead size made little difference, the size of the bead-membrane attachment region did have a significant impact. This has implications for developing optical tweezer stretching for the purpose of quantifying the quality of RBCs or diagnosing an ailment based on deformability properties, as the adhesion level may need to be considered in the data analysis.

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Chapter 7. Prediction of Discocyte-Echinocyte Resting Shape Sequence

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7.1 Introduction

RBCs can undergo a morphological transformation from the typical discocyte shape to echinocyte morphologies during aging [10] and under certain environmental conditions [34]. This occurs progressively, with the transformation classified into four stages based on the surface profile and sphericity [34, 46] (see Table 2.2). The move away from the discocyte morphology is associated with a reduction in cellular deformability, which is why it is considered a key indicator for cell function [10]. Consequently, the effect of morphology may need to be incorporated when applying techniques like indentation, stretching or even flow to detect structural changes within the membrane. This is particularly relevant to the problem of quantifying stored RBC quality given the accelerated shape transformation experienced in these conditions [10, 129].

As introduced in Section 2.3, a well-accepted physical explanation for the RBC morphological transformation is the bilayer-couple hypothesis – that the leaflets of the bilayer respond differently to shape-changing agents while remaining coupled to each other, creating a relaxed surface area difference between the inner and outer leaflets [50, 137]. This is said to be equivalent to a change in the membrane’s spontaneous curvature in terms of the physical shape implications [47]. Consequently, there are two methods by which shape transformations have been driven from a numerical modelling perspective – targeting a relaxed surface area difference between inner and outer layers of the bilayer [8, 27, 34], and implementing localised changes in membrane spontaneous curvature [47].

The relaxed surface area difference approach has successfully shown the full range of RBC morphologies [8, 27, 34]. It used the area-difference-elasticity (ADE) expression for energy stored in the bilayer [8, 27, 34]. ADE energy (퐸퐴퐷퐸) is calculated with Equation 7.1 where it can be seen it is made up of two components

– 퐸푠푐 for energy stored due to the spontaneous curvature difference and 퐸푎푑 for energy stored due to the relaxed surface area difference. These are expanded in

Equations 7.2 and 7.3 respectively, where 퐾푏 and 퐾̅ are elastic bending moduli, 퐻 is the local mean membrane curvature, 퐶0 is the spontaneous membrane curvature, 퐴 is the membrane’s surface area, 퐷 is the membrane’s thickness, Δ퐴 is the actual surface area difference between inner and outer leaflets of the bilayer, and Δ퐴0 is the relaxed surface area difference. In Equation 7.2, the integral is taken over the surface, 푆. The spontaneous curvature component (Equation 7.2) is analogous to

Page 122 Numerical Modelling of Red Blood Cell Morphology and Deformability the existing bending energy expression (Equations 3.6 and 3.12 for 2D and 3D implementations respectively) in the present study’s CGPM approach. However the area difference component (Equation 7.3) has not been considered to date.

퐸퐴퐷퐸 = 퐸푠푐 + 퐸푎푑 ( 7.1 )

퐾푏 2 퐸푠푐 = ∮ (2퐻 − 퐶0) 푑퐴 ( 7.2 ) 2 푆 퐾̅ 휋 퐸 = (Δ퐴 − Δ퐴 )2 ( 7.3 ) 푎푑 2 퐴퐷2 0

There are two major limitations associated with this approach. Firstly, as 퐶0 and Δ퐴0 are not easily measured or otherwise established in the literature [34], these parameters were combined into a single control variable, Δ푎̅̅0̅, in the previous studies (Equation 7.4) [8]. Consequently, the two aspects of the ADE energy expression were not treated independently, reducing the value in using this compartmentalised approach. Secondly, a Monte Carlo energy minimisation method was used rather than the principle of virtual work. The Monte Caro approach cannot be easily extended to incorporate a fluid phase or model dynamic scenarios as it is based on randomly moving particles and testing for a decrease in total system energy. Conversely, the principle of virtual work can be systematically extended to include a fluid phase through coupling with SPH [78, 79, 111, 117] or IBM [14, 109], and systematically extended to consider dynamics by quantifying time-based properties such as viscosity. Therefore there is significant benefit in using the principle of virtual work to minimise energy. However, the extra energy term may become costly due to complexity of the corresponding force expressions. This would be a significant consideration in adapting this approach into the CGPM models.

Δ퐴 퐾 퐷퐶 Δ푎̅̅̅ = 0 + 푏 0 ( 7.4 ) 0 퐴 휋퐾̅ An alternative method that may be considered for integration into the CGPM models was used by Li and Lykotrafitis [47] to simulate the micro-vesiculation process of RBCs. This involved changing the spontaneous curvature of designated local membrane regions, which was said to stem from specific proteins and lipids with a high affinity aggregating within the membrane in preparation for their detachment [47]. This process was said to involve a combination of surface area difference between bilayer leaflets, protein-lipid hydrophobic mismatches and proteins imposing their own curvature on the bilayer at their interface [51], generating the localised change in spontaneous curvature. Therefore the effect of area difference

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 123 and these protein influences are rolled into a single control parameter (the local spontaneous membrane curvature) without the need for an additional and computationally costly energy term. Li and Lykotrafitis [47] were able to show the micro-vesicle formation process by combining the local spontaneous curvature change with the application of compressive forces. However, as this was only conducted within a small square section of membrane measuring 0.8 µm x 0.8 µm, its applicability to overall morphology has not been demonstrated.

If the discocyte-echinocyte sequence of shapes can be incorporated into the CGPM models, it would enable investigations into the deformability of these RBCs. This is a significant gap in modelling to date, given that those previous numerical models which predicted the range of RBC shapes have only observed the effect of the control parameter Δ푎̅̅0̅ on resting shape [8, 27, 34]. However, at present, there is extremely limited experimental data available for the deformability of RBCs within the discocyte-echinocyte sequence, as well as for their resting geometry. Therefore the aim of this chapter is to explore how the discocyte-echinocyte sequence of shapes may be predicted within the existing CGPM modelling framework to establish the basis for future numerical investigations which combine the deformability simulations with echinocyte morphologies. This can then be developed further when suitable experimental data becomes available.

7.2 Methods

7.2.1 Modelling Approach Selection

From the Literature Review, the advantages of the localised spontaneous curvature change approach make it the most suitable for the present developments. Advantages over the relaxed surface area difference approach include:

. Unlike the local spontaneous curvature change method, the relaxed surface area difference approach introduces an additional energy term comparing the surface area of the inside and outside leaflets of the bilayer. Therefore sets of particles representing the inner and outer leaflets need to be at least implicitly considered within the model in order to enable this calculation. This adds significant computational cost to the system in identifying these positions and tracking the movement. This was strongly evidenced in a preliminary proof-of-concept investigation which has been briefly documented in Appendix C for reference.

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. If the principle of virtual work is applied to the new energy equation to find the force on each particle, significant computational cost is incurred. This is because taking the partial derivative of surface area difference with respect to each particle’s position becomes a complex and lengthy process, as there are several parameters in the energy equation which are functions of particle position (refer to Appendix C where this was explored in a preliminary manner). Previous studies employing the relaxed surface difference approach have used the Monte Carlo energy minimisation method to circumvent this issue, but this negates the future extension to dynamic simulations and inclusion of a fluid phase. . Although the relaxed surface area difference approach initially separated the

effect of spontaneous curvature (퐶0) and surface area difference (Δ퐴0), these were subsequently combined into a single control parameter. Therefore there is no major advantage in using this approach over the localised spontaneous curvature change at this time. In the future, if the influence of spontaneous curvature and relaxed surface area difference are to be studied separately, then it would be worth revisiting this point.

7.2.2 Modelling Methodology for Localised Spontaneous Curvature Change

The discocyte resting shape model established and validated in Chapter 3 formed the basis of the modelling. The present developments were completed in both 2D and 3D. This enables the 2D model to serve as a low-cost proof-of-concept tool which may be leveraged in the future when developing the methodology further.

7.2.2.1 2D Implementation

To simulate the formation of spicules, the spontaneous curvature of designated membrane regions was impacted. The number of regions was set equal to the number of spicules (푁푠푝푖푐), and these regions were roughly evenly spaced over the cell surface. The spicule perimeter (퐿푠푝푖푐) controlled the length of the impacted regions. The spontaneous angle measured internal to the cell, 훼0,푖, was adjusted for bending interactions within these impacted regions as per Equation 7.5 where 푙0 is the relaxed separation distance between particles and 푟푠푝푖푐 is the radius of curvature for the spicule. The geometric parameters of 푁푠푝푖푐, 퐿푠푝푖푐 and 푟푠푝푖푐 vary for each morphological classification (see Section 7.2.3 below).

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−1 푙0 훼0,푖 = 2 cos ( ) ( 7.5 ) 2푟푠푝푖푐

The force expressions for this method remain the same as those detailed in Appendix A given that only the spontaneous angle is changed for designated bending interactions. Total energy in the membrane, given by Equation 3.4, was minimised to predict the preferred resting shape of the RBC.

7.2.2.2 3D Implementation

To simulate the formation of spicules, designated regions of the membrane were impacted with a spontaneous curvature change. The number of regions was again equal to the number of spicules (푁푠푝푖푐), and these regions were roughly evenly spaced over the cell surface. Spicule perimeter (퐿푠푝푖푐) controlled the size of the impacted regions. This was done by selecting the particle at the centre of the region and then “webbing” out until the desired region size was achieved. This process is illustrated in Figure 7.1. The number of web levels required, 푁푤푒푏, was approximated with Equation 7.6. This calculation is based on the average relaxed length of the linear interactions within the cell (푙0,푎푣푔) and the simplifying assumption that the mesh is formed of equilateral triangles of equal size.

(a) 푵풘풆풃 = ퟏ (b) 푵풘풆풃 = ퟐ (c) 푵풘풆풃 = ퟑ

Figure 7.1 – Selection of edges (yellow) for an impacted region, where centre particle is nominated and adjoining interactions are selected as the region is “webbed” outward

퐿푠푝푖푐 푁푤푒푏 ≈ √3 ( 7.6 ) (1 + ) 푙 2 0,푎푣푔

The spontaneous angle measured internal to the cell, 훼0,푖, was adjusted for bending interactions within the impacted regions. For 3D, the adjustment was as per Equation 7.7. This calculation is again based on the simplifying assumption that the mesh is formed of equilateral triangles of equal size with side lengths of 푙0,푎푣푔. The

Page 126 Numerical Modelling of Red Blood Cell Morphology and Deformability geometric parameters of 푁푠푝푖푐, 퐿푠푝푖푐 and 푟푠푝푖푐 vary for each morphological classification (see Section 7.2.3 below).

3푙 −1 √ 0,푎푣푔 훼0,푖 = 2 cos ( ) ( 7.7 ) 4푟푠푝푖푐

The force expressions for this method remain the same as those detailed in Appendix B given that only the spontaneous angle is changed for designated bending interactions. Total energy in the membrane, given by Equation 3.9, was minimised to predict the preferred resting shape of the RBC.

7.2.2.3 Stiffness Coefficient Selection

As only resting shapes are to be simulated here, absolute values for the stiffness coefficients do not impact [27] – instead the ratio of the stiffness coefficients to each other defines the minimum energy shape (refer to Section 3.2.5.1 where this was first discussed). Moreover, stiffness coefficients were earlier found to require tailoring to specific deformability applications so would need further work before being applied in these investigations. Therefore, for the purpose of simulating the resting shapes in this chapter, stiffness coefficients were initially and nominally drawn from the indentation study since it was completed in both 2D and 3D. However, it was found that for 3D, the linear and areal stiffness coefficients needed to be increased to allow appropriately-proportioned spicules to form. Thus the values chosen are summarised in Table 7.1.

Table 7.1 – Modelling parameters used to demonstrate discocyte-echinocyte sequence in 2D and 3D Parameter 2D Model 3D Model Starting Radius of 푟 = 3 휇푚 푟 = 3.3 휇푚 Circle/Sphere Number of Particles 푁 = 100 푁 = 1922 Referenced Properties 180°, NIR 180°, NIR 훼 = { 훼 = { 0,푖 Equation 7.5, IR 0,푖 Equation 7.7, IR NIR = non-impacted region, NIR = non-impacted region, IR = impacted region IR = impacted region

푙0,푖 = original lengths 푙0,푖 = original lengths 퐴0,푖 = original areas Reduction Ratio 푅퐴 = 0.48 푅푉 = 0.6 1 −3 Stiffness Coefficients 푘푙 = 3.0 × 10 푁/푚 푘퐿 = 2.1 × 10 푁/푚 −15 −15 푘푏 = 4.0 × 10 퐽/푟푎푑 푘퐵 = 5.3 × 10 퐽/푟푎푑 −10 13 2 푘푎 = 2.3 × 10 퐽 푘퐴 = 1.4 × 10 푁/푚 −10 푘푉 = 7.0 × 10 퐽

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7.2.3 Experimental Observations of RBC Shape for Geometric Inputs & Model Validation

The modelling methodology outlined above required some geometric inputs for each of the morphologies. These were the number of spicules (푁푠푝푖푐), spicule perimeter

(퐿푠푝푖푐), and spicule radius of curvature (푟푠푝푖푐). Given a lack of data of this nature was identified in the Literature Review, a sample set of experimental data was obtained from a study performed by another party [138] which involved imaging RBCs with a scanning electron microscope (SEM). This will be described briefly to give context to the model’s development.

RBC samples at day 2 and 42 of storage were fixed in glutaraldehyde (1%) for 10 minutes, then osmium tetroxide in cacodylate buffer (1%) for one hour. The cells were dried in ethanol and hexamethyldisilizane before being gold coated and imaged with a Zeiss Sigma field emission-SEM. Three different buffers were used for the imaging phase – PBS, plasma and SAGM – which were observed to favour different morphological stages of the discocyte-echinocyte transformation [138]. Figure 7.2 shows a typical SEM image obtained from this process which includes RBCs with the discocyte morphology and some early stage echinocytes.

In order to quantify key aspects of the RBC geometry as a function of the morphological stages, representative images of echinocytes were analysed. The number of the spicules was estimated separately for the 2D and 3D interpretations. In 2D, the number of spicules visible for a cross-sectional slice was used. In 3D, the number of spicules on the visible upper side was assumed to be also present on the underside. The spicule perimeter and radius were found with the aid of Matlab R2017b as illustrated in Figure 7.3. Spicule perimeter was found by tracing the surface outline and finding the total length of the line formed (Figure 7.3a). The radius of curvature was found by fitting a circle through three points on the spicule surface (Figure 7.3b). Using scale on these images, the distances measured on the images were converted into physical measurements.

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Figure 7.2 – Typical SEM image of RBCs; unpublished image by Marie Anne Balanant

(b) (a)

Figure 7.3 – Examples of (a) measuring spicule length and (b) radius of curvature of a spicule; unpublished image by Marie Anne Balanant

Three images were analysed for each of Echinocyte I, Echinocyte II and Echinocyte III classifications. The Echinocyte IV stage was not characterised as the surface area to volume ratio varies from the other stages due to the shedding of micro- vesicles [34] (refer to Section 2.3). Given the top-down view of the present SEM images, the required surface area and volume cannot be obtained and these are not otherwise established in the literature. In the future, confocal imaging may be considered for this purpose of obtaining RBC surface area and volume, given that a

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3D stacked image of the cell surface can be generated. Table 7.2 lists the key geometric properties of RBCs through the discocyte-echinocyte sequence of shapes used as geometric inputs to the models.

Table 7.2 – RBC geometric properties a function of morphological stage Morphology Average Average Average Average Spicule Spicule Spicule Spicule Number, Number, Radius, 풓풔풑풊풄 Perimeter, 푵풔풑풊풄 (2D) 푵풔풑풊풄 (3D) (µm) 푳풔풑풊풄 (µm) Discocyte 0 0 N/A N/A Echinocyte I 8 12 0.55 1.42 Echinocyte II 12 17 0.25 0.87 Echinocyte III 8 30 0.16 0.75

7.3 Results & Discussion

7.3.1 Discocyte-Echinocyte Resting Shapes

The sequence of RBC resting shapes predicted using the local spontaneous curvature change methodology in both 2D and 3D are shown in Figure 7.4 on the following page. It can be seen that a progression of increasingly severe echinocyte morphologies is realised using this modification to the modelling methodology.

The spicules produced in the 2D model are slightly more rounded at the Echinocyte III stage than the experimental observations. This is consistent with the findings of Lim et al. [27] illustrated in Figure 7.5 below. This study reported that in the absence of the cytoskeleton, large spontaneous curvature values led to rounded buds of the preferred curvature with a narrow neck (Figure 7.5a). This is as opposed to when the cytoskeleton was present, and pointed spicules were formed due to high shearing loads experienced in the neck region (Figure 7.5b). In 2D, the shear contribution of the cytoskeleton is not considered, whereas in 3D, shear is resisted through the triangulated lattice. It can be seen that this has resulted in larger-necked spicules for 3D as expected. Consequently, the predicted shapes in 3D align well with the representative SEM images at each stage.

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Echinocyte I Echinocyte II Echinocyte III

Repre- sentative SEM Image

Numerical Prediction in 2D 1 µm 1 µm 1 µm

Numerical Prediction in 3D

Figure 7.4 – Discocyte-echinocyte sequence of resting RBC shapes predicted using the local spontaneous curvature change methodology in 3D compared to SEM images

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7.3.2 Discussion of Approach for Investigating Deformability

The methodology established here bodes well to addressing the key gap identified in the Literature Review of assessing the deformability of RBCs of the echinocyte morphology. This type of study would be best suited to global deformability techniques of flow and optical tweezer stretching, rather than local deformability techniques of micropipette aspiration and indentation. This is because the global application of force evaluates holistic deformability behaviour, making it less sensitive to the individual characteristics and geometrical irregularities of single cells. This would likely be a significant factor in local deformability measurements, which would be influenced greatly by the presence of a spicule (or spicules) in the contacted region. The local influence of spicules may also become significant when attaching optical beads to the later stage echinocytes. Therefore it is recommended that deformability in flow scenarios is prioritised in terms of obtaining experimental data and then performing the corresponding numerical investigations going forward.

Once the model has been validated against deformability observations, the impact of membrane stiffness properties and geometric aspects can be assessed. This was identified as another major gap in the literature. Improving understanding of which aspects dominate the exhibited deformability of RBCs may lead new strategies for preserving these most influential characteristics in specific applications such as during RBC storage.

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The use of the principle of virtual work to evolve the system opens the door to including a fluid phase and performing dynamic simulations of RBC deformability. This is not too distant a step from what has been completed in the literature to date (e.g. [110]) given that the only variation required is to the spontaneous curvature of designated bending interactions. This is advantageous as there would be no increase in computational cost from inclusion of an additional energy term for relaxed surface area difference, meaning these simulations are achievable in the present environment. It should also be emphasised that the CGPM has already been used to assess how multiple RBCs interact in flows, and thus this aspect could also be investigated using the alternative morphologies.

7.4 Summary & Conclusions

This chapter explored how the discocyte-echinocyte sequence of RBC shapes could be incorporated into the CGPM models. The following conclusions were drawn:

. Impacting localised spontaneous curvature values is preferable to including an extra energy term for relaxed surface area difference due to its simplicity and reduced computational cost. . Changing the spontaneous curvature of designated membrane regions, combined with changing the number and size of the regions, was able to predict increasingly severe echinocyte morphologies. . The spicules produced in the 2D model at the Echinocyte III stage are slightly more rounded than experimental observations due to the lack of shear resistance from the cytoskeleton. This is considered in the 3D model resulting in larger-necked spicules as expected. . The methodology established here has the potential to address key gaps in the literature around assessing the deformability of RBCs of the echinocyte morphology, as well as the impact of membrane stiffness properties and geometric attributes. However, experimental data is first required to enable this development. . Global deformability investigations, and especially flow scenarios, would be better suited to assessing the deformability of echinocyte morphologies as they would be less sensitive to the individual characteristics and geometrical irregularities of individual cells. . The use of the principle of virtual work to minimise energy (rather than the Monte Carlo method) means only slight modifications would be needed to

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integrate the echinocyte morphologies into existing models published in the literature which have included a fluid phase (e.g. [110, 118]), as well as the extension to dynamic scenarios.

7.5 Contribution to Research Objectives

This chapter contributed to the first two objectives of the research project. First, it developed the modelling methodology to include a provision for spicule formation. Secondly, the predicted shapes within the discocyte-echinocyte sequence were validated against experimental observations. This has primed the methodology for use in future deformability investigations when experimental data becomes available.

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Chapter 8. Conclusions & Recommendations

This research employed the CGPM to simulate the physical behaviour of RBCs in response to externally applied forces from AFM indentation and optical tweezer stretching, as well as RBC resting shapes within the discocyte-echinocyte shape sequence. This final chapter provides a summary of the key conclusions (Section 8.1), the identified limitations (Section 8.2), and recommended future directions of the research (Section 8.3).

8.1 Conclusions

The key conclusions that can be drawn from this work are summarised as follows:

. CGPM method can simulate RBC indentation and stretching: The CGPM method with energy terms for stretch resistance, bending resistance, surface area incompressibility and volumetric incompressibility was capable of capturing RBC physical behaviour during both AFM indentation and optical tweezer stretching. However, the stiffness coefficients needed to be tailored to the specific application. . AFM indentation suited to detecting bilayer structural changes: Changes to the membrane’s bending stiffness had the most dominant impact on deformability for AFM indentation. As bending stiffness is mainly provided by the bilayer, AFM indentation would be well-suited to detecting physical changes within this part of the membrane as RBC mechanical properties evolve during storage or with disease. . Hertz contact theory unsuitable for RBC indentation analysis: The use of Hertzian contact theory to analyse force-deformation data needs to be rethought for RBC indentation problems due to difficulties justifying the underlying assumptions of solid contact and negligible substrate influence for this application. It was shown that Young’s modulus predictions were highly sensitive to size of the indenter, the region of the cell indented and the degree of adhesion, making the Young’s modulus estimates unreliable. . Optical tweezer stretching suited to detecting cytoskeleton structural changes: Changes to the membrane’s linear stiffness had the most

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dominant impact on deformability during optical tweezer stretching. Given that linear stiffness of the membrane is mainly associated with the cytoskeleton, the optical tweezer setup would be well-suited to detecting mechanical changes to cytoskeleton properties as RBCs evolve during storage or with disease. . Optical tweezer stretching design considerations: To inform the design of optical tweezer experimental setups, it was found optical bead size had little impact on deformability measurements, while the degree of adhesion did have a significant impact. . Prediction of echinocyte morphologies: Incorporating local spontaneous curvature changes into the CGPM model enables the prediction of the discocyte-echinocyte morphological sequence and lays the groundwork for future investigations into the deformability of these RBC morphologies.

The key original findings and contributions of this work are as follows:

. Development of the CGPM to simulate RBC behaviour during AFM indentation – The established methodology allows control over the degree of adhesion, probe geometry, and direction and position of applied force. This greatly improves upon the previously most advanced analytical model by Sen et al. [19] which was only applicable to dome-shaped cells indented with a sharp probe at their centre. . Assessment of the validity of Hertz-based analysis for RBC indentation experiments – The effect of probe size and shape, indentation region and degree of substrate adhesion were investigated to evaluate the validity of Hertz-based analysis for extracting Young’s modulus in RBC contact problems, which was an open question highlighted in the literature. From the investigations it was found that Hertz-based analysis is not applicable to RBC indentation due to the assumptions of the cell acting as a solid with negligible influence from the substrate not being met. Consequently, an alternative method of analysis involving inverse-FEA was recommended. Furthermore, this can explain the large variation in reported Young’s modulus values in the literature. . Development of the CGPM to simulate RBC behaviour during optical tweezer stretching – The established methodology explicitly considers the optical beads, thus enabling control over bead size and the degree of bead- membrane contact. This improves upon the previously most advanced particle-based models for optical tweezer stretching by Shi et al. [20] and

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Fedosov et al. [21] which applied stretching forces directly onto selected RBC membrane particles. . Application of the established model to investigate factors influencing deformability measurements in optical tweezer stretching experiments – The effect of bead size and degree of bead-membrane contact were investigated to assess their impact on deformability measurements. These aspects of the experimental setup had not previously been evaluated using either numerical or experimental means. It was found that bead size is not a significant factor in deformability measurements, but the bead-membrane contact area is. This is an important finding for progressing this experimental procedure, as it highlights the need for degree of adhesion to be taken into account during data analysis. . Extension of the CGPM models to the prediction of RBC shapes in the discocyte-echinocyte sequence – The CGPM methodology was developed further for the prediction of echinocyte morphologies, laying the groundwork for future investigations into their deformability when experimental data becomes available. These morphologies had not previously been predicted using the CGPM.

8.2 Limitations

The key limitation of this study is that although the CGPM was successfully demonstrated to simulate RBC physical behaviour across different deformability scenarios, the stiffness coefficients needed to be tailored to the specific application. This issue is not exclusive to the work presented in this thesis – it is also evident in the immense variation for stiffness coefficients used in previous CGPM models for simulating RBCs during flow (Table 3.1).

The first step in addressing this limitation would be to definitively verify whether a single set of stiffness coefficients could reasonably satisfy the force-deformation behaviour across the different experimental setups. This would involve employing an advanced optimisation technique for obtaining the stiffness coefficients, as the present project used a manual approach which was inefficient. This manual approach was consequently incapable testing a sufficiently large combination of values to rule out there being a series of coefficients which could theoretically work across the scenarios.

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If it is found that a single set of stiffness coefficients cannot satisfy the force- deformation behaviour across the different experiments, a deep examination of the energy equations is required. This is because the simplicity of the equations used to quantify the energy stored in the various forms (bending, linear, surface area and volume) may be the constraint, as these are heavily restricted in their capacity to capture the membrane’s response to externally applied forces across different levels of stress and strain. For example, the stretch energy mechanism (symbolic of how spectrin molecules behave within the triangulated lattice) was quantified such that force was linearly scaled to its displacement, regardless of the level of strain. However, in reality there is a maximal strain limit for that spectrin molecule (i.e. when it reaches the contour length), and consequently it will respond with a non- linearly increasing force as it approaches this limit. Likewise, bending energy is set proportional to the tangent squared of the curvature angle but it is unlikely that the bilayer responds according to this relationship across the full spectrum of possible curvatures.

In order to improve the model’s accuracy, more sophisticated approaches to quantifying energy may be considered. However, it must be emphasised that this comes at the cost of computational efficiency. Energy equations could be modified such that the approach moves toward a more multi-scale or molecular dynamics representation. By drawing on the more nuanced response of the membrane to various types of loading at the corresponding magnitude of stress and strain, it should allow a universal set of stiffness coefficients to be identified. These universal stiffness coefficients coupled with the improved energy equations should make the model applicable to any deformability scenario, providing greater versatility and predictive power.

Secondary limitations for the current work include:

. Quasi-static: This study made the assumption that dynamic aspects such as cytoplasmic viscosity and membrane fluidity were negligible. In cases such as rapid loading scenarios, dynamics may become significant. Therefore dynamics should be considered in the future through quantification of time- based properties such as the membrane’s damping constant. . Lack of fluid phase: This study did not explicitly model the cytoplasmic or extracellular fluid phases. This extension should be considered if the models are applied to other scenarios such as flow simulations, or for loading situations where the viscosity of the fluids becomes significant.

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. Lack of data for echinocyte morphologies: Deformability data is not presently available for echinocytes at the various morphological stages. Therefore deformability of these morphologies cannot be validated or meaningfully assessed at the present time. . Simulation time: All simulations completed in this study were capable of being run on a personal computer, however the time taken for 3D cases with large particle numbers could extend up to a day. The adaptive discretisation technique and the use of previously converged solutions as the starting configuration for new simulations were able to mitigate long simulation times, and given the focus on development, validation and exploration of the model’s potential, reducing simulation time was not a key concern in this project. However, if the modelling methodology established here is to be developed for broader applications, reducing the computational time by incorporating parallel processing and selecting a programming language which runs quicker than Matlab (such as C++) would be important considerations.

8.3 Future Direction

The work presented in this thesis has shown that the CGPM has the versatility to predict the physical behaviour of RBCs when subjected to different loading situations. Numerical experiments were able to highlight key areas of the experimental setup and data analysis. Furthermore, it was shown that with the inclusion of local spontaneous curvature changes, the discocyte-echinocyte sequence of RBC shapes could be predicted. Future work that builds on these findings is suggested as follows:

. Achieve model universality: Improvements to the stiffness coefficient optimisation and energy equations are recommended as per the discussion in Section 8.2. This should focus on incorporating more advanced and automated optimisation methods for obtaining the stiffness coefficients, to determine whether a single set of stiffness coefficients could reasonably satisfy the force-deformation behaviour across different scenarios. The second focus on improving the sophistication of the energy equations would enable the model to better capturing of how the membrane responds at different levels of stress and strain. Reconciling the energy equations to

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enable universal stiffness coefficients represents a significant development to the methodology, providing greater versatility and predictive power. . Improved AFM indentation data analysis: Numerical investigations showed that Young’s modulus predictions obtained from Hertz-based analysis of force-indentation data were highly sensitive to size of the indenter, the region of the cell indented and the degree of adhesion. Therefore a new method of analysis should be developed which considers the cell as a fluid- filled membrane and incorporates substrate effects. An inverse FEM approach is recommended for this purpose. . New equation for Young’s modulus extraction: For AFM indentation, an interesting future direction would be to develop a robust equation for extracting Young’s modulus of the RBC membrane based on factors like cell geometry, probe geometry, degree of adhesion and so on. This would be based on the new method of analysis recommended in the previous point. . New method for optical tweezer stretching data analysis: A similar approach is recommended for analysing the force-deformation data obtained from optical tweezer stretching. If an inverse FEM approach is also adopted for this application, comparison could be made between the Young’s modulus obtained during stretching and indentation. . Deformability of echinocyte morphologies: Investigation of the deformability of RBCs through the morphological transformation remains a significant gap in the literature. This may be addressed using the CGPM models established in this work which have used the principle of virtual work to evolve the system. However, more experimental data is required for validation. This includes improved quantification of the resting geometry for these cells, as well as measurements of their force-deformation behaviour under different loading scenarios. . Effect of membrane defects: The impact of membrane defects and heterogeneity can be assessed with the models by varying the stiffness of individual interactions. This can be extended to looking at specific conditions which stem from membrane abnormalities. . Aid maturation of experimental techniques: The models could be used to aid the development of techniques or devices for quantifying the quality of stored RBC units or for the diagnosis of disease, based on deformability. This potential development stems from the differences in deformability reported between healthy and deteriorating RBCs using both AFM indentation [36, 74, 81, 88] and optical tweezer stretching [17, 18, 102]

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experiments. The models may be utilised to inform experimental design as well as data analysis methodologies. For example, as indentation was shown to be more sensitive to bending stiffness, it may be better suited to detecting changes in the bilayer, whereas optical tweezer stretching was more sensitive to the linear stiffness, so it may be better suited to detecting changes in the cytoskeleton.

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Appendix

Appendices A to C detail mathematical derivations for the force on each particle as a result of the imposed energy requirements and calculated from the principle of virtual work (Equation 3.1). In order to generalise this process, the partial derivatives at intermediate steps are taken with respect to a generic particle position, 퐷푃. Here 퐷 stands for the coordinate direction, either 푥 or 푦 or 푧. Furthermore, 푃, stands for the particle identifier such as 푝1, 푝2, and so on. When derivatives are taken with respect to the specific particles and directions, it will be expressed in a matrix of the format:

휕푄 휕푄 … 휕푥 휕푥 휕푄 푝1 푝2 = 휕푄 휕푄 휕퐷푃 … 휕푦푝1 휕푦푝2 ( ⋮ ⋮ ⋱) Here 푄 is a general quantity. The first row of the matrix is partial derivatives in the 푥 direction, the second row is partial derivatives in the 푦 direction, and so on. The first column is partial derivatives with respect to particle 푝1, the second row is with respect to particle 푝2, and so on as needed.

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Appendix A – 2D Force Calculations for General Implementation

This section details the forces acting on particles in the 2D model due to the energy equations introduced in Section 0.

Linear Interactions

The energy stored in linear interactions, 퐸푙, was given by Equation 3.5. For an isolated linear interaction, 푖, between particles 푝1 and 푝2 as depicted in Figure A.1, energy can be calculated with Equation A.1. Recall 푘푙 is the combined linear stiffness coefficient, 푙푖 is the actual distance between the adjacent particles, and 푙0,푖 is the relaxed distance between adjacent particles.

푝1 (푥푝1, 푦푝1 ) 푝2 (푥푝2, 푦푝2)

Figure A.1 – Isolated linear interaction in 2D showing notation for particles and their position

푘 퐸 = 푙 (푙 − 푙 )2 ( A.1 ) 푙,푖 2 푖 0,푖 The force on each particle as a result of linear interaction 푖 can be found by applying the principle of virtual work to Equation A.1. It should be noted that here the only variable which is a function of particle positions is 푙푖. Thus the force on a generic particle from the 푖푡ℎ linear interaction can be calculated as:

휕퐸푙,푖 휕푙푖 퐹퐷푃 | 퐸푙,푖 = − = −푘푙(푙푖 − 푙0,푖) ( A.2 ) 휕퐷푃 휕퐷푃

The actual distance between the particles, 푙푖, is defined as:

√ 2 2 ( A.3 ) 푙푖 = (푥푝1 − 푥푝2) + (푦푝1 − 푦푝2)

Thus the derivative with respect to each specific particle and direction can be expressed as:

푥푝 − 푥푝 푥푝 − 푥푝 1 2 − 1 2 휕푙 푙 푙 푖 = ( 푖 푖 ) ( A.4 ) 푦푝 − 푦푝 푦푝 − 푦푝 휕퐷푃 1 2 − 1 2 푙푖 푙푖

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Substituting Equations A.3 and A.4 into Equation A.2 provides the force on each particle due to the 푖푡ℎ linear interaction. This can be repeated for every linear interaction in order to compute the total force on each particle resulting from this mechanism.

Bending Interactions

The energy stored in bending interactions, 퐸푏, was given by Equation 3.6. For an isolated bending interaction, 푖, between particles 푝1, 푝2 and 푝3 as depicted in Figure

A.2, energy can be calculated with Equation A.5. Recall 푘푏 is the bending stiffness coefficient, 훼푖 is the internal angle (relative to the cell’s interior) formed between the adjacent particles, and 훼0,푖 is this angle relaxed.

푝2 (푥푝2, 푦푝2) 푂푢푡푠푖푑푒 퐶푒푙푙

푙 12 푙23 푝 (푥 , 푦 ) 훼푖 1 푝1 푝1 푝3 (푥푝3, 푦푝3) 퐼푛푠푖푑푒 퐶푒푙푙 푙31

Figure A.2 – Isolated bending interaction in 2D showing notation for particles and their position

푘 훼 − 훼 퐸 = 푏 tan2 ( 푖 0,푖) ( A.5 ) 푏,푖 2 2 The force on each particle as a result of bending interaction 푖 can be found by applying the principle of virtual work to Equation A.5. It should be noted that the only parameter which is a function of particle positions is 훼푖. Thus when considering a 푡ℎ general particle position, 퐷푃, the force from the 푖 bending interaction can be calculated as:

훼푖 − 훼0,푖 휕퐸 푘푏 tan ( ) 휕훼 퐹 = − 푏,푖 = − 2 푖 ( A.6 ) 퐷푃 | 퐸푏,푖 훼 − 훼 휕퐷푃 2 cos2 ( 푖 0,푖) 휕퐷푃 2

In order to find 훼푖, first the angle 훽푖 is calculated. 훽푖 is defined as the angle on the inside of the triangle formed when the three particles of interest are connected together. This is shown in Figure A.3a for both the convex and concave cases.

훽푖 can be converted into the angle, 훼푖, which is internal to the cell based on the case as shown in Figure A.3a and in Equation A.7.

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Convex Concave a)

훼 = 훽 푖 푖 훽푖

훼 = 360° − 훽 푖 푖

Convex Concave

푛̂푎푣푔

푛12 푛 b) 23 푛12 푛23 푝2 푝1 푝 푛̂ 3 푎푣푔 푢̂푚푖푑

푝1 푝 푢̂푚푖푑 3 푝2

Figure A.3 – In 2D (a) Illustration of 훼푖 and 훽푖 for convex and concave angles, and (b) schematic showing how the convexity or concavity of a region was determined

훽푖, convex 훼푖 = { ( A.7 ) 360° − 훽푖, concave In order to identify whether a given bending interaction was in the convex or concave state, outward pointing normal vectors, 푛12 and 푛23, were found to the sides 푙12 and 푙23 respectively (see Figure A.2 and Figure A.3b). These normal vectors were then averaged to find the unit vector, 푛̂푎푣푔. Another unit vector, 푢̂푚푖푑, was then drawn from particle 푝2 in the direction of the midpoint of side 푙31. This is also illustrated in Figure A.3b for clarity. The difference between two sets of vectors was then computed – set 1 of 푢̂푚푖푑 and 푛̂푎푣푔 (Equation A.8) and set 2 of 푢̂푚푖푑 and

−푛̂푎푣푔 (Equation A.9). If the difference between unit vectors was smaller for set 1 then it was the concave case, and if the difference was smaller for set 2 then it was the convex case (Equation A.10).

푆1 = |푢̂푚푖푑 − 푛̂푎푣푔| ( A.8 )

푆2 = |푢̂푚푖푑 − (−푛̂푎푣푔)| ( A.9 ) 푆 > 푆 , convex Case = { 1 2 ( A.10 ) 푆1 < 푆2, concave

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Regardless of convex or concave state, 훽푖 can be found from the cos rule. This first requires the distance between the pairs of particles as follows (refer to Figure A.2):

√ 2 2 ( A.11 ) 푙12 = (푥푝1 − 푥푝2 ) + (푦푝1 − 푦푝2)

√ 2 2 ( A.12 ) 푙23 = (푥푝2 − 푥푝3 ) + (푦푝2 − 푦푝3)

√ 2 2 ( A.13 ) 푙31 = (푥푝3 − 푥푝1 ) + (푦푝3 − 푦푝1)

2 2 2 The angle is given below, where parameters 푞1 = 푙12 + 푙23 − 푙31 and 푞2 = 2푙12푙23 have been introduced to simplify the expressions which follow:

2 2 2 −1 푙12 + 푙23 − 푙31 −1 푞1 훽푖 = cos ( ) = cos ( ) ( A.14 ) 2푙12푙23 푞2

The derivative of Equation A.14 with respect to the generic particle position is thus:

휕훽푖 1 푞1 휕푞2 1 휕푞1 = ( 2 − ) 휕퐷푃 푞 2 푞2 휕퐷푃 푞2 휕퐷푃 ( A.15 ) √1 − ( 1) 푞2

The partial derivatives of 푞1 and 푞2 are thus:

휕푞1 휕푙12 휕푙23 휕푙31 = 2푙12 + 2푙23 − 2푙31 ( A.16 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푞2 휕푙23 휕푙12 = 2푙12 + 2푙23 ( A.17 ) 휕퐷푃 휕퐷푃 휕퐷푃 Finally, partial derivatives of the distance between the pairs of particles are as follows:

푥푝 − 푥푝 푥푝 − 푥푝 1 2 − 1 2 0 휕푙 푙 푙 12 = ( 12 12 ) ( A.18 ) 푦푝 − 푦푝 푦푝 − 푦푝 휕퐷푃 1 2 − 1 2 0 푙12 푙12

푥푝 − 푥푝 푥푝 − 푥푝 0 2 3 − 2 3 휕푙 푙 푙 23 = ( 23 23 ) ( A.19 ) 푦푝 − 푦푝 푦푝 − 푦푝 휕퐷푃 0 2 3 − 2 3 푙23 푙23

푥푝 − 푥푝 푥푝 − 푥푝 − 3 1 0 3 1 휕푙 푙 푙 31 = ( 31 31 ) ( A.20 ) 푦푝 − 푦푝 푦푝 − 푦푝 휕퐷푃 − 3 1 0 3 1 푙31 푙31

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The partial derivative of 훼푖 with respect to the generic particle is case dependent as shown:

휕훽 푖 , convex 휕훼 휕퐷 푖 = 푃 ( A.21 ) 휕퐷 휕훽 푃 − 푖 , concave { 휕퐷푃 Substituting Equations A.6 to A.21 into Equation A.5 provides the force on each particle due to the 푖푡ℎ bending interaction. This set of calculations can be repeated for each bending interaction in order to compute the total force on each particle resulting from this mechanism.

Cross-Sectional Area Penalty

The energy stored due to variation in the cross-sectional area from the reference target, 퐸푎, was given by Equation 3.8. This is repeated below for ease of viewing.

Recall that 푘푎 is the stiffness coefficient for maintaining cross-sectional area, 퐴푡표푡 is the cross-sectional area of the cell and 퐴푟푒푓 is the target cross-sectional area.

2 푘푎 퐴푡표푡 − 퐴푟푒푓 퐸푎 = ( ) ( 3.8 ) 2 퐴푟푒푓

The force on each particle as a result of the cross-sectional area requirement can be found by applying the principle of virtual work to Equation 3.8. It should be noted that the only parameter which is a function of particle positions is 퐴푡표푡. Thus when considering a general particle position, 퐷푃, the force can be calculated as:

휕퐸푎 푘푎 휕퐴푡표푡 퐹퐷푃 | 퐸푎 = − = − 2 (퐴푡표푡 − 퐴푟푒푓) ( A.22 ) 휕퐷푃 퐴푟푒푓 휕퐷푃

The cross-sectional area enclosed by the particles can be calculated by summing the area under the curve formed by connecting adjacent particles. This can be expressed as:

푁

퐴푡표푡 = ∑ 퐴푖 ( A.23 ) 푖=1

The derivative of 퐴푡표푡 with respect to the generic particle position is thus:

푁 휕퐴 휕퐴 푡표푡 = ∑ 푖 ( A.24 ) 휕퐷푃 휕퐷푃 푖=1

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 157

For two adjacent particles 푝1 and 푝2 as depicted in Figure A.4, the area under the curve (between the dotted lines shown) can be found using Equation A.25. In order to ensure the correct sign (positive or negative) is computed, the areas are calculated in order moving anticlockwise around the shape.

1 퐴 = 푦 (푥 − 푥 ) + (푦 − 푦 )(푥 − 푥 ) ( A.25 ) 푖 푝1 푝1 푝2 2 푝2 푝1 푝1 푝2

푦

푝2 (푥푝2, 푦푝2)

푝1 (푥푝1, 푦푝1)

푥

Figure A.4 – Area under the curve in 2D found from connecting a pair of adjacent particles

The derivative of Equation A.25 with respect to particle positions is then:

1 1 푦 + (푦 − 푦 ) −푦 − (푦 − 푦 ) 휕퐴 푝1 푝2 푝1 푝1 푝2 푝1 푖 = ( 2 2 ) ( A.26 ) 휕퐷 1 1 푃 푥 − 푥 − (푥 − 푥 ) (푥 − 푥 ) 푝1 푝2 2 푝1 푝2 2 푝1 푝2 By substituting Equations A.23 to A.26 into Equation A.22, the total force on each particle resulting from this mechanism can be calculated.

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Appendix B – 3D Force Calculations for General Implementation

This section details the forces acting on particles in the 3D model due to the energy equations introduced in Section 3.2.3.

Linear Interactions

The energy stored in linear interactions, 퐸퐿, was given by Equation 3.10. For an isolated linear interaction, 푖, between particles 푝1 and 푝2 as depicted in Figure B.1, energy can be calculated with Equation B.1. Recall 푘퐿 is the linear stiffness coefficient, 푙푖 is the actual distance between the adjacent particles, and 푙0,푖 is the relaxed distance between adjacent particles.

푝1 (푥푝1, 푦푝1, 푧푝1) 푝2 (푥푝2, 푦푝2, 푧푝2)

Figure B.1 – Isolated linear interaction in 3D showing notation for particles and their position

푘 퐸 = 퐿 (푙 − 푙 )2 ( B.1 ) 퐿,푖 2 푖 0,푖 The force on each particle as a result of linear interaction 푖 can be found by applying the principle of virtual work to Equation B.1. It should be noted that the only parameter which is a function of particle positions is 푙푖. Thus the force on the generic particle in the desired direction from the 푖푡ℎ linear interaction can be calculated as:

휕퐸퐿,푖 휕푙푖 퐹퐷푃 | 퐸퐿,푖 = − = −푘퐿(푙푖 − 푙0,푖) ( B.2 ) 휕퐷푃 휕퐷푃 The distance between the particles is defined as:

√ 2 2 2 ( B.3 ) 푙푖 = (푥푝1 − 푥푝2) + (푦푝1 − 푦푝2) + +(푧푝1 − 푧푝2)

Thus the derivative with respect to each specific particle and direction can be expressed as:

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푥푝 − 푥푝 푥푝 − 푥푝 1 2 − 1 2 푙 푙 푖 푖 휕푙 푦푝 − 푦푝 푦푝 − 푦푝 푖 = 1 2 − 1 2 ( B.4 ) 휕퐷푃 푙푖 푙푖 푧푝 − 푧푝 푧푝 − 푧푝 1 2 − 1 2 ( 푙푖 푙푖 ) Substituting Equations B.3 and B.4 into Equation B.2 provides the force on each particle due to the 푖푡ℎ linear interaction. This can be repeated for every linear interaction in order to compute the total force on each particle resulting from this mechanism.

Surface Area Incompressibility Penalty

The energy stored due to variation in surface area of the triangles forming the RBC,

퐸퐴, was given by Equation 3.11. For an isolated triangle, 푖, between particles 푝1, 푝2 and 푝3 as depicted in Figure B.2, energy can be calculated with Equation B.5.

Recall 푘퐴 is the areal stiffness coefficient, 퐴푖 is the actual surface area of the triangle, and 퐴0,푖 is the relaxed area of this triangle.

푛

푝2 (푥푝2, 푦푝2, 푧푝2)

푎

푝 (푥 , 푦 , 푧 ) 1 푝1 푝1 푝1 푏 푝3 (푥푝3, 푦푝3, 푧푝3)

Figure B.2 – Isolated triangle in 3D showing notation for particles and their position

푘 2 퐸 = 퐴 (퐴 − 퐴 ) ( B.5 ) 퐴,푖 2 푖 0,푖 The force on each particle as a result of the surface area requirement placed on triangle 푖 can be found by applying the principle of virtual work to Equation B.5. It should be noted that the only parameter which is a function of particle positions is

퐴푖. Thus the force on the generic particle can be calculated as:

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휕퐸퐴,푖 휕퐴푖 퐹퐷푃 | 퐸퐴 = − = −푘퐴(퐴푖 − 퐴0,푖) ( B.6 ) 휕퐷푃 휕퐷푃 The area of the triangle can be found from halving the magnitude of the normal vector to the surface, 푛 (Equation B.7). It should be noted that in order to account for the signs in this calculation, the normal vector was ensured to be an outward pointing vector.

|푛| 퐴 = ( B.7 ) 푖 2

The derivative of 퐴푖 with respect to a generic particle position is thus:

휕퐴 1 휕|푛| 푖 = ( B.8 ) 휕퐷푃 2 휕퐷푃 The normal vector can be computed from the cross-product of the vectors from two sides of the triangle (see Figure B.2). Ensuring that the triangle connectivity is such that the vertices are stated in the same order (for example clockwise when looking on the outward-facing surface), these side vectors, 푎 and 푏, are defined as follows:

̂ ̂ 푎 = (푥푝1 − 푥푝2)푖̂ + (푦푝1 − 푦푝2)푗̂ + (푧푝1 − 푧푝2)푘 = 푎푥푖̂ + 푎푦푗̂ + 푎푧푘 ( B.9 ) ̂ ̂ 푏 = (푥푝3 − 푥푝1)푖̂ + (푦푝3 − 푦푝1)푗̂ + (푧푝3 − 푧푝1)푘 = 푏푥푖̂ + 푏푦푗̂ + 푏푧푘 ( B.10 )

Additional variables for the components in each of the directions are introduced (푎푥,

푎푦, 푎푧, 푏푥, 푏푦 and 푏푧) to simplify the proceeding expressions. Thus the normal vector to the surface can be written as:

̂ 푛 = |푏 × 푎| = (푏푦푎푧 − 푏푧푎푦)푖̂ + (푏푧푎푥 − 푏푥푎푧)푗̂ + (푏푥푎푦 − 푏푦푎푥)푘 ̂ ( B.11 ) = 푛푥푖̂ + 푛푦푗̂ + 푛푧푘

Again additional variables are introduced for the components (푛푥, 푛푦 and 푛푧) to simplify subsequent expressions. The magnitude of the normal vector is then:

2 2 2 |푛| = √푛푥 + 푛푦 + 푛푧 ( B.12 )

The derivative of the magnitude of the normal vector with respect to the generic particle position is:

휕|푛| 1 휕푛푥 휕푛푦 휕푛푧 = (푛푥 + 푛푦 + 푛푧 ) ( B.13 ) 휕퐷푃 |푛| 휕퐷푃 휕퐷푃 휕퐷푃

The derivatives of the components of the normal vector with respect to a generic particle position can be written as:

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휕푛푥 휕푎푧 휕푏푦 휕푎푦 휕푏푧 = 푏푦 + 푎푧 − 푏푧 − 푎푦 ( B.14 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푛푦 휕푎푥 휕푏푧 휕푎푧 휕푏푥 = 푏푧 + 푎푥 − 푏푥 − 푎푧 ( B.15 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푛푧 휕푎푦 휕푏푥 휕푎푥 휕푏푦 = 푏푥 + 푎푦 − 푏푦 − 푎푥 ( B.16 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 The partial derivatives of the side vector components with respect to specific particles and directions are then:

1 −1 0 0 0 0 0 0 0 휕푎푥 휕푎푦 휕푎푧 = (0 0 0) , = (1 −1 0) , = (0 0 0) ( B.17 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 푃 0 0 0 푃 1 −1 0 −1 0 1 0 0 0 0 0 0 휕푏푥 휕푏푦 휕푏푧 = ( 0 0 0) , = (−1 0 1) , = ( 0 0 0) ( B.18 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 푃 0 0 0 푃 −1 0 1 Substituting Equations B.7 to B.18 into Equation B.6 provides the force on each particle due to the surface area requirement on the 푖푡ℎ triangle. This set of calculations can be repeated for each triangle in order to compute the total force on each particle resulting from this mechanism.

Bending Interactions

The energy stored in bending interactions, 퐸퐵, was given by Equation 3.12. For an isolated bending interaction, 푖, between particles 푝1, 푝2, 푝3 and 푝4 as depicted in

Figure B.3, energy can be calculated with Equation B.19. Recall 푘퐵 is the bending stiffness coefficient, 훼푖 is the internal angle formed between the adjacent triangular surfaces, and 훼0,푖 is this angle relaxed.

푛 푚

푝3 (푥푝 , 푦푝 , 푧푝 ) 푝2 (푥푝2, 푦푝2, 푧푝2) 3 3 3

푎 푏 푑 푐

푝4 (푥푝 , 푦푝 , 푧푝 ) 푝 (푥 , 푦 , 푧 ) 4 4 4 1 푝1 푝1 푝1

Figure B.3 – Isolated pair of adjacent triangles in 3D showing notation for particles and their position

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푘 훼 − 훼 퐸 = 퐵 푡푎푛2 ( 푖 0,푖) ( B.19 ) 퐵,푖 2 2 The force on each particle as a result of bending interaction 푖 can be found by applying the principle of virtual work to Equation B.19. It should be noted that the

only parameter which is a function of particle positions is 훼푖. Thus when considering 푡ℎ a general particle position, 퐷푃, the force from the 푖 bending interaction can be calculated as:

훼푖 − 훼0,푖 휕퐸 푘퐵 tan ( ) 휕훼 퐹 = − 퐵,푖 = − 2 푖 ( B.20 ) 퐷푃 | 퐸퐵,푖 훼 − 훼 휕퐷푃 2 cos2 ( 푖 0,푖) 휕퐷푃 2

In order to find 훼푖, first the angle 훽푖 is calculated. 훽푖 is defined as the angle between the outward pointing normal vectors for the adjoining triangular surfaces. This is

shown in Figure B.4a for both the convex and concave cases. 훽푖 can be converted

into the 훼푖, which is internal to the cell based on the case as shown in Figure B.4a. The internal angle and its derivative are expressed mathematically below (Equations B.21 and B.22).

Convex Concave a) 훽푖

훼푖 = 180° − 훽푖 훼푖 = 180° + 훽푖 훽푖

Convex Concave

푛̂푎푣푔 푛 b) 푚 푛 푝3 푚 푝1 푛̂ 푝4 푎푣푔

푢̂푚푖푑 푝1 푝4 푝3 푝2

푢̂푚푖푑 푝 2

Figure B.4 – In 3D (a) Illustration of 훼푖 and 훽푖 for convex and concave angles, and (b) schematic showing how the convexity or concavity of a region was determined

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180° − 훽푖, convex 훼푖 = { ( B.21 ) 180° + 훽푖, concave 휕훽 − 푖 , convex 휕훼 퐷 푖 = 푃 ( B.22 ) 퐷 훽 푃 푖 , concave {퐷푃 In order to identify whether a given bending interaction was in the convex or concave state, outward pointing normal vectors to the surfaces, 푚 and 푛, were used (see Figure B.3 and Figure B.4b). These normal vectors were then averaged to find the unit vector, 푛̂푎푣푔. Another unit vector, 푢̂푚푖푑, was then drawn from the midpoint of particles 푝1 and 푝2 in the direction of the midpoint between particles 푝3 and 푝4. This is illustrated in Figure B.4b for clarity. The difference vectors was then computed – set 1 of 푢̂푚푖푑 and 푛̂푎푣푔 and set 2 of 푢̂푚푖푑 and −푛̂푎푣푔. If the difference between unit vectors was smaller for set 1 then it was the concave case, and if the difference was smaller for set 2 then it was the convex case (same as 2D implementation – refer to Equations A.8 to A.10).

In Figure B.3, the triangle at right is the same as the one in Figure B.2 that was used for the surface area calculations. Therefore side vectors 푎 and 푏, and the outward pointing normal vector to the surface, 푛, can be computed using the equations detailed in the previous section (Equations B.9 to B.16). However, new calculations are required for the triangle at left in Figure B.3 in order to compute its normal vector, designated as 푚. First, side vectors 푐 and 푑 are:

̂ ̂ 푐 = (푥푝1 − 푥푝4)푖̂ + (푦푝1 − 푦푝4)푗̂ + (푧푝1 − 푧푝4)푘 = 푐푥푖̂ + 푐푦푗̂ + 푐푧푘 ( B.23 ) ̂ ̂ 푑 = (푥푝2 − 푥푝1)푖̂ + (푦푝2 − 푦푝1)푗̂ + (푧푝2 − 푧푝1)푘 = 푑푥푖̂ + 푑푦푗̂ + 푑푧푘 ( B.24 )

Additional variables for the components in each of the directions are introduced (푐푥,

푐푦, 푐푧, 푑푥, 푑푦 and 푑푧) to simplify the proceeding expressions. The outward pointing normal vector, 푚, can be written as:

̂ 푚 = |푑 × 푐| = (푑푦푐푧 − 푑푧푐푦)푖̂ + (푑푧푐푥 − 푑푥푐푧)푗̂ + (푑푥푐푦 − 푑푦푐푥)푘 ̂ ( B.25 ) = 푚푥푖̂ + 푚푦푗̂ + 푚푧푘

Again additional variables are introduced for the directional components (푚푥, 푚푦 and 푚푧) to simplify subsequent expressions. The magnitude of the normal vector is then:

2 2 2 |푚| = √푚푥 + 푚푦 + 푚푧 ( B.26 )

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The derivative of the magnitude of the normal vector is:

휕|푚| 1 휕푚푥 휕푚푦 휕푚푧 = (푚푥 + 푚푦 + 푚푧 ) ( B.27 ) 휕퐷푃 |푚| 휕퐷푃 휕퐷푃 휕퐷푃

The derivatives of the components of the normal vector with respect to a generic particle position can be written as:

휕푚푥 휕푐푧 휕푑푦 휕푐푦 휕푑푧 = 푑푦 + 푐푧 − 푑푧 − 푐푦 ( B.28 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푚푦 휕푐푥 휕푑푧 휕푐푧 휕푑푥 = 푑푧 + 푐푥 − 푑푥 − 푐푧 ( B.29 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푚푧 휕푐푦 휕푑푥 휕푐푥 휕푑푦 = 푑푥 + 푐푦 − 푑푦 − 푐푥 ( B.30 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 The partial derivatives of the side vector components with respect to the four specific particles and directions are given below (noting that Equations B.17 and B.18 from the surface area calculations have been updated to include the extra particle of interest, 푝4).

1 −1 0 0 0 0 0 0 0 0 0 0 휕푎푥 휕푎푦 휕푎푧 = (0 0 0 0) , = (1 −1 0 0) , = (0 0 0 0) ( B.31 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 0 푃 0 0 0 0 푃 1 −1 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 휕푏푥 휕푏푦 휕푏푧 = ( 0 0 0 0) , = (−1 0 1 0) , = ( 0 0 0 0) ( B.32 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 0 푃 0 0 0 0 푃 −1 0 1 0 1 0 0 −1 0 0 0 0 0 0 0 0 휕푐푥 휕푐푦 휕푐푧 = (0 0 0 0 ) , = (1 0 0 −1) , = (0 0 0 0 ) ( B.33 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 0 푃 0 0 0 0 푃 1 0 0 −1 −1 1 0 0 0 0 0 0 0 0 0 0 휕푑푥 휕푑푦 휕푑푧 = ( 0 0 0 0) , = (−1 1 0 0) , = ( 0 0 0 0) ( B.34 ) 휕퐷 휕퐷 휕퐷 푃 0 0 0 0 푃 0 0 0 0 푃 −1 1 0 0

The angle 훽푖 is defined as the angle formed between the normal vectors to the surfaces regardless of convexity or concavity (Equation B.35). Two simplifying variables of 푞3 = 푛 ∙ 푚 and 푞4 = |푛||푚| have been introduced. The partial derivative is given below (Equation B.36).

−1 푛 ∙ 푚 −1 푞3 훽푖 = cos ( ) = cos ( ) ( B.35 ) |푛||푚| 푞4

휕훽푖 1 푞3 휕푞4 1 휕푞3 = ( 2 − ) 휕퐷푃 푞 2 푞 휕퐷푃 푞4 휕퐷푃 ( B.36 ) √1 − ( 3) 4 푞4

The dot product, 푞3, and its partial derivative with respect to the generic particle are:

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푞3 = 푛 ∙ 푚 = 푛푥푚푥 + 푛푦푚푦 + 푛푧푚푧 ( B.37 )

휕푞3 휕푚푥 휕푛푥 휕푚푦 휕푛푦 휕푚푧 휕푛푧 = 푛푥 + 푚푥 + 푛푦 + 푚푦 + 푛푧 + 푚푧 ( B.38 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

The partial derivative of 푞4 is:

휕푞 휕|푚| 휕|푛| 4 = |푛| + |푚| ( B.39 ) 휕퐷푃 휕퐷푃 휕퐷푃 Substituting the above expressions into Equation B.20 provides the force on each particle due to the 푖푡ℎ bending interaction. This set of calculations can be repeated for each bending interaction in order to compute the total force on each particle resulting from this mechanism.

Volumetric Incompressibility Penalty

The energy stored due to variation in the volume of the RBC from the reference target, 퐸푉, was given by Equation 3.14. This is repeated below for ease of viewing.

Recall that 푘푉 is the stiffness coefficient for maintaining volume, 푉푡표푡 is the actual volume of the cell and 푉푟푒푓 is the target volume.

2 푘푉 푉푡표푡 − 푉푟푒푓 퐸푉 = ( ) ( B.40 ) 2 푉푟푒푓

The force on each particle as a result of the cross-sectional area requirement can be found by applying the principle of virtual work to Equation 3.14. It should be noted that the only parameter which is a function of particle positions is 푉푡표푡. Thus when considering a general particle position, 퐷푃, the force can be calculated as:

휕퐸푉 푘푉 휕푉푡표푡 퐹퐷푃 | 퐸푉 = − = − 2 (푉푡표푡 − 푉푟푒푓) ( B.41 ) 휕퐷푃 푉푟푒푓 휕퐷푃

The volume enclosed by the particles, 푉푡표푡, can be calculated by summing the volume under the triangular surface formed by connecting adjacent particles:

푁푇

푉푡표푡 = ∑ 푉푖 ( B.42 ) 푖=1 The partial derivative of total volume with respect to a generic particle is thus:

푁푇 휕푉 휕푉 푡표푡 = ∑ 푖 ( B.43 ) 휕퐷푃 휕퐷푃 푖=1

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푡ℎ For the 푖 set of particles 푝1, 푝2 and 푝3 which form a triangular surface as depicted in Figure B.5, the volume under the surface can be found using Equation B.44, ′ where 퐴푖 is the area of the triangle when projected onto the 푥-푦 axis as shown in

Figure B.5 and ℎ푎푣푔,푖 is the average height of the vertices in the 푧-direction. In order to ensure the correct sign (positive or negative) is computed, normal vectors to the surfaces are ensured to be outward-pointing as was done in the surface area incompressibility calculations above.

′ 푉푖 = 퐴푖ℎ푎푣푔,푖 ( B.44 ) 푧

푝2 (푥푝2, 푦푝2 , 푧푝2)

푝1 (푥푝1, 푦푝1, 푧푝1)

푝3 (푥푝3, 푦푝3, 푧푝3) 푛′

푦 푎′ ′ 푝2 (푥푝2, 푦푝2, 0)

′ 푝1 (푥푝1, 푦푝1, 0)

푏′

′ 푝3 (푥푝3, 푦푝3, 0)

푥

Figure B.5 – Volume under the surface found from projecting the vertices of a triangle onto the 푥-푦 plane as shown

The area of each projected triangle can be found from halving the magnitude of the normal vector to the surface, 푛′:

|푛′| 퐴′ = ( B.45 ) 푖 2 ′ The derivative of 퐴푖 with respect to a generic particle position is thus:

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 167

휕퐴′ 1 휕|푛′| 푖 = ( B.46 ) 휕퐷푃 2 휕퐷푃 The normal vector can be computed from the cross-product of the vectors from two sides of the projected triangle (see Figure B.5). Knowing that the triangle connectivity was such that the vertices were stated in the same order (for example clockwise when looking at the outward-facing surface), side vectors 푎′ and 푏′ are defined as follows:

′ ′ 푎′ = (푥푝1 − 푥푝2)푖̂ + (푦푝1 − 푦푝2)푗̂ = 푎푥푖̂ + 푎푦푗̂ ( B.47 )

′ ′ 푏′ = (푥푝3 − 푥푝1)푖̂ + (푦푝3 − 푦푝1)푗̂ = 푏푥푖̂ + 푏푦푗̂ ( B.48 )

′ Additional variables for the components in each of the directions are introduced (푎푥, ′ ′ ′ 푎푦, 푏푥 and 푏푦) to simplify the proceeding expressions. Thus the normal vector to the projected surface can be written as:

′ ′ ′ ′ ̂ 푛′ = |푏′ × 푎′| = (푏푥푎푦 − 푏푦푎푥)푘 ( B.49 )

The magnitude of the normal vector and its derivative with respect to a generic particle position are then:

′ ′ ′ ′ |푛′| = 푏푥푎푦 − 푏푦푎푥 ( B.50 )

| | 휕푎′ ′ ′ 휕푏′ 휕 푛′ ′ 푦 ′ 휕푏푥 ′ 휕푎푥 ′ 푦 = 푏푥 + 푎푦 − 푏푦 − 푎푥 ( B.51 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 The derivatives of the edge vectors with respect to specific particles are:

′ 1 −1 0 ′ 0 0 0 휕푎푥 휕푎푦 = (0 0 0) , = (1 −1 0) ( B.52 ) 휕퐷 휕퐷 푃 0 0 0 푃 0 0 0

′ −1 0 1 ′ 0 0 0 휕푏푥 휕푏푦 = ( 0 0 0) , = (−1 0 1) ( B.53 ) 휕퐷 휕퐷 푃 0 0 0 푃 0 0 0 The average height of the triangle and its derivative with respect to the specific particles can be written as:

푧푝1 + 푧푝2 + 푧푝3 ℎ = ( B.54 ) 푎푣푔,푖 3 0 0 0 휕ℎ푎푣푔,푖 0 0 0 = (1 1 1) ( B.55 ) 휕퐷푃 3 3 3 This process can be repeated for each triangle in order to compute the total force on each particle resulting from this mechanism.

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Appendix C – Proof-of-Concept Investigation for Relaxed Surface Area Difference Energy in 2D

This appendix documents exploration of incorporating an energy term for the relaxed surface area difference between inner and outer layers of the bilayer. This is considered in 2D and is based on the resting shape model established and validated in Chapter 3.

Relaxed Surface Area Energy Term

To incorporate the relaxed surface area difference into the CGPM model, an additional energy term was introduced which was of a similar form to the 퐸푎푑 term (Equation 7.3). Noting in 2D that surface area is equivalent to the perimeter of a shape, the surface area difference requirement was applied over defined sections of the membrane’s perimeter. For a section 푖, of length 퐿푖, the perimeter difference

(Δ푃푖) between the inner (푃푖푛,푖) and outer layers (푃표푢푡,푖) was computed with Equation C.1. This perimeter difference was then compared to a reference perimeter difference, Δ푃푟푒푓,푖. Energy, 퐸푑, was stored when the actual perimeter difference varied from the reference perimeter difference (Equation C.2). 푁푠푒푐 is the number of sections. A stiffness coefficient, 푘푑, was associated with the relationship in order to control the strength of the requirement.

Δ푃푖 = 푃표푢푡,푖 − 푃푖푛,푖 ( C.1 )

푁푠푒푐 2 푘푑 Δ푃푖 − Δ푃푟푒푓,푖 퐸푑 = ∑ ( ) ( C.2 ) 2 Δ푃푟푒푓,푖 푖=1 As this new energy equation relies on a difference in perimeter between outer and inner leaflets of the bilayer, an inner and outer layer was represented in the model. This was done indirectly – only the one set of membrane particles explicitly existed, with an inner and outer layer of particles implicitly represented by projecting the explicitly represented particles a specified offset distance, 푑0, as shown in Figure C.1. These projection calculations are detailed below.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 169

Membrane Particle (explicitly represented)

Outer Leaflet Particle (implicitly represented)

푑0 Inner Leaflet Particle 푑0 (implicitly represented)

Figure C.1 – 2D schematic of with 푁 = 8 membrane particles showing the inside and outside leaflet particle projections

Identify Inside and Outside Particle Projections

In order to calculate the perimeter difference, the polygon formed by connected membrane particles was projected inwards and outwards a specified distance 푑0.

This process is demonstrated based on an group of three particles – 푝1, 푝2 and 푝3 – shown in Figure C.2.

Focusing on the projections of particle 푝2, the first step was to draw lines parallel to the sides (푎 and 푏) at the specified offset distance (푑0). These are illustrated in Figure C.2 as the orange and green dotted lines. The equations of lines 푎, 1푎 and 2푎 (refer to Figure C.2) can be determined as:

푎 → 푦 = 푚푎푥 + 푐푎

1푎 → 푦 = 푚푎푥 + 푐1푎

2푎 → 푦 = 푚푎푥 + 푐2푎 ( C.3 ) 푦 − 푦 푝2 푝1 −1 where 푚푎 = , 푐푎 = 푦푝2 − 푚푎푥푝2, 휃푎 = tan (푚푎) , 푥푝2 − 푥푝1 푑0 푑푣푎 = , 푐1푎 = 푐푎 + 푑푣푎, 푐2푎 = 푐푎 − 푑푣푎 cos(휃푎)

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1푏 푏 2푏

푛23 푝2,표푢푡 (푥푝2,표푢푡 , 푦푝2,표푢푡) 푂푢푡푠푖푑푒 퐶푒푙푙

푝3 (푥푝 , 푦푝 ) 푝2 (푥푝2, 푦푝2) 3 3 푛12 퐼푛푠푖푑푒 퐶푒푙푙 푝2,푖푛 (푥 , 푦 ) 1푎 푝2,푖푛 푝2,푖푛 푎 2푎

푝1 (푥푝1, 푦푝1)

휃푏푒푡

Figure C.2 – Isolated group of three adjoining particles in 2D showing notation for particles and their position

Similarly, the equations of lines 푏, 1푏 and 2푏 are:

푏 → 푦 = 푚푏푥 + 푐푏

1푏 → 푦 = 푚푏푥 + 푐1푏

2푏 → 푦 = 푚푏푥 + 푐2푏 ( C.4 ) 푦 − 푦 푝3 푝2 −1 where 푚푏 = , 푐푏 = 푦푝3 − 푚푏푥푝3, 휃푏 = tan (푚푏) , 푥푝3 − 푥푝2 푑0 푑푣푏 = , 푐1푏 = 푐푏 + 푑푣푏, 푐2푏 = 푐푏 − 푑푣푏 cos(휃푏) This results in four intersection points between the orange and green dotted lines (see Figure C.2). The coordinates of these intersections are:

푐1푏 − 푐1푎 퐵푒푡푤푒푒푛 1푎 & 1푏 → (푥#1, 푦#1) = ( , 푚푎푥#1 + 푐1푎) 푚푎 − 푚푏 ( C.5 )

푐2푏 − 푐1푎 퐵푒푡푤푒푒푛 1푎 & 2푏 → (푥#2, 푦#2) = ( , 푚푎푥#2 + 푐1푎) 푚푎 − 푚푏 ( C.6 )

푐2푏 − 푐2푎 퐵푒푡푤푒푒푛 2푎 & 2푏 → (푥#3, 푦#3) = ( , 푚푎푥#3 + 푐2푎) 푚푎 − 푚푏 ( C.7 )

푐1푏 − 푐2푎 퐵푒푡푤푒푒푛 2푎 & 1푏 → (푥#4, 푦#4) = ( , 푚푎푥#4 + 푐2푎) ( C.8 ) 푚푎 − 푚푏

The angle, 휃푏푒푡, between outwardly pointing normal vectors to the adjacent sides,

푛12 and 푛23, was then measured (refer to Figure C.2) with Equation C.9.

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 171

−1 푛12 ∙ 푛23 휃푏푒푡 = cos ( ) ( C.9 ) |푛12||푛23|

If 휃푏푒푡 was less than 90 degrees, the closest intersection points to 푝2 corresponded to the correct set of inside and outside points. Conversely, if 휃푏푒푡 was more than 90 degrees, the intersection points furthest from 푝2 corresponded to the correct set of inside and outside points. Thus set of inside and outside points was determined.

The normal vectors of 푛12 and 푛23 were then averaged to find the unit vector, 푛̂푎푣푔.

The difference between two sets of vectors was then computed – set 1 of 푛̂푎푣푔 and the unit vector drawn between 푝2 and one inside/outside particle, and set 2 of

푛̂푎푣푔 and the unit vector drawn between 푝2 and the other inside/outside particle. The set with the smaller difference used corresponded to the outside particle. Conversely, the set with the larger difference used in the inside particle. Thus the inside (푝2,푖푛) and outside (푝2,표푢푡) particles could be distinguished from the cases given by Equations C.5 to C.8. This process was repeated for each set of adjoining particles.

Forces on Particles from Mechanism

The energy stored due to perimeter differences, 퐸푑, was given by Equation C.2. For an isolated section, 푖, energy can be calculated with Equation C.10. Recall 푘푑 is the stiffness coefficient for enforcing the perimeter difference, Δ푃푖 is the actual difference in perimeter between outer and inner layers, and Δ푃푟푒푓,푖 is the relaxed difference.

2 푘푑 Δ푃푖 − Δ푃푟푒푓,푖 퐸푑,푖 = ( ) ( C.10 ) 2 Δ푃푟푒푓,푖

The force on each particle as a result of the perimeter difference of section 푖 can be found by applying the principle of virtual work to Equation C.10. It should be noted that the only variable which is a function of particle position is Δ푃푖. Thus the force on a generic particle from the 푖푡ℎ linear interaction can be calculated as:

휕퐸푑,푖 푘푑 휕Δ푃푖 퐹퐷푃 | 퐸푑,푖 = − = − 2 (Δ푃푖 − Δ푃푟푒푓,푖) ( C.11 ) 휕퐷푃 Δ푃푟푒푓,푖 휕퐷푃

For a section 푖 made up of 푁푗 segments, the perimeter difference for the section

(Δ푃푖) is given by Equation C.12. Here 훥퐿푗 is the difference in length between the outer (퐿표푢푡,푗) and inner segments (퐿푖푛,푗) for the segment given by Equation C.13.

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푁푗

Δ푃푖 = ∑ 훥퐿푗 ( C.12 ) 푗=1

훥퐿푗 = 퐿표푢푡,푗 − 퐿푖푛,푗 ( C.13 )

The derivatives of these last two expressions with respect to the generic particle position are then:

푁푗 휕Δ푃 휕훥퐿푗 푖 = ∑ ( C.14 ) 휕퐷푃 휕퐷푃 푗=1

휕훥퐿푗 휕퐿표푢푡,푗 휕퐿푖푛,푗 = − ( C.15 ) 휕퐷푃 휕퐷푃 휕퐷푃 Focusing on a set of four adjoining particles forming set 푗, as shown in Figure C.3, the inner and outer lengths for the middle segment (between 푝2 and 푝3 and the projections) can be found with Equations C.16 and C.17.

1푏 푏 2푏 푝 (푥 , 푦 ) 2,표푢푡 푝2,표푢푡 푝2,표푢푡 푝3,표푢푡 (푥푝3,표푢푡 , 푦푝3,표푢푡) 푂푢푡푠푖푑푒 퐶푒푙푙 푝 (푥 , 푦 ) 2 푝2 푝2 푝3 (푥푝3, 푦푝3) 1푐 푐 1푎 푝 푝 푑 2,푖푛 3,푖푛 0 2푐 푎 (푥푝 , 푦푝 ) (푥푝 , 푦푝 ) 2푎 2,푖푛 2,푖푛 3,푖푛 3,푖푛

퐼푛푠푖푑푒 퐶푒푙푙 푑 푝 (푥 , 푦 ) 0 푝 (푥 , 푦 ) 1 푝1 푝1 4 푝4 푝4

Figure C.3 – Isolated group of four adjacent particles in 2D showing notation for particles and their position

√ 2 2 ( C.16 ) 퐿표푢푡,푗 = (푥푝2,표푢푡 − 푥푝3,표푢푡) + (푦푝2,표푢푡 − 푦푝3,표푢푡)

√ 2 2 ( C.17 ) 퐿푖푛,푗 = (푥푝2,푖푛 − 푥푝3,푖푛) + (푦푝2,푖푛 − 푦푝3,푖푛)

The derivatives of these lengths with respect to the generic particle position are:

휕퐿표푢푡,푗 1 휕푥푝 ,표푢푡 휕푥푝 ,표푢푡 = [(푥 − 푥 ) ( 2 − 3 ) 휕퐷 퐿 푝2,표푢푡 푝3,표푢푡 휕퐷 휕퐷 푃 표푢푡,푗 푃 푃 ( C.18 ) 휕푦푝2,표푢푡 휕푦푝3,표푢푡 + (푦푝2,표푢푡 − 푦푝3,표푢푡) ( − )] 휕퐷푃 휕퐷푃

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 173

휕퐿푖푛,푗 1 휕푥푝 ,푖푛 휕푥푝 ,푖푛 = [(푥 − 푥 ) ( 2 − 3 ) 휕퐷 퐿 푝2,푖푛 푝3,푖푛 휕퐷 휕퐷 푃 푖푛,푗 푃 푃 ( C.19 ) 휕푦푝2,푖푛 휕푦푝3,푖푛 + (푦푝2,푖푛 − 푦푝3,푖푛) ( − )] 휕퐷푃 휕퐷푃

The positions of the inner and outer particles were found above. There are four possible cases for where the particles are situated based on the intersections given by Equations C.5 to C.8. The case where Equation C.5 gives the position of the particle 푝2,표푢푡 will be detailed below. The other cases follow the same pattern. Thus it is assumed here that (푥푝2,표푢푡, 푦푝2,표푢푡) = (푥#1, 푦#1) . The derivatives of the particle positions under this assumption are given by Equations C.20 and C.21, noting that the gradients and intercepts were defined above in Equations C.3 and C.4.

휕푥푝2,표푢푡 (푐1푏 − 푐1푎) 휕푚푎 휕푚푏 1 휕푐1푏 휕푐1푎 = − 2 ( − ) + ( − ) ( C.20 ) 휕퐷푃 (푚푎 − 푚푏) 휕퐷푃 휕퐷푃 푚푎 − 푚푏 휕퐷푃 휕퐷푃 휕푦 휕푥 휕푚 휕푐 푝2,표푢푡 푝2,표푢푡 푎 1푎 ( C.21 ) = 푚푎 + 푥푝2,표푢푡 + 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃 The derivatives of the gradients with respect to each particle are:

푦푝2 − 푦푝1 푦푝2 − 푦푝1 2 − 2 0 0 휕푚 (푥 − 푥 ) (푥 − 푥 ) 푎 = 푝2 푝1 푝2 푝1 ( C.22 ) 휕퐷 1 1 푃 − 0 0 ( 푥푝2 − 푥푝1 푥푝2 − 푥푝1 )

푦푝3 − 푦푝2 푦푝3 − 푦푝2 0 2 − 2 0 휕푚 (푥 − 푥 ) (푥 − 푥 ) 푏 = 푝3 푝2 푝3 푝2 ( C.23 ) 휕퐷 1 1 푃 0 − 0 ( 푥푝3 − 푥푝2 푥푝3 − 푥푝2 ) The derivative of the 푦-intercept of the line 1푐 is given by Equation C.24, followed by other equations which are required to be substituted in.

휕푐 휕푐 휕푑 1푎 = 푎 + 푣푎 ( C.24 ) 휕퐷푃 휕퐷푃 휕퐷푃

휕푐푎 휕푦푝2 휕푥푝2 휕푚푎 = − (푚푎 + 푥푝2 ) ( C.25 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푥푝 0 1 0 0 2 = ( ) ( C.26 ) 휕퐷푃 0 0 0 0

휕푦푝 0 0 0 0 2 = ( ) ( C.27 ) 휕퐷푃 0 1 0 0

휕푑푣푎 푑0sin (휃푎) 휕휃푎 ( C.28 ) = 2 휕퐷푃 cos (휃푎) 휕퐷푃

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휕휃푎 1 휕푚푎 = 2 ( C.29 ) 휕퐷푃 푚푎 + 1 휕퐷푃

The derivative of the 푦-intercept of the line 1푏 is given by Equation C.30, followed by other equations which are required to be substituted in.

휕푐 휕푐 휕푑 1푏 = 푏 + 푣푏 ( C.30 ) 휕퐷푃 휕퐷푃 휕퐷푃

휕푐푏 휕푦푝3 휕푥푝3 휕푚푏 = − (푚푏 + 푥푝3 ) ( C.31 ) 휕퐷푃 휕퐷푃 휕퐷푃 휕퐷푃

휕푥푝3 0 0 1 0 = ( ) ( C.32 ) 휕퐷푃 0 0 0 0

휕푦푝3 0 0 0 0 = ( ) ( C.33 ) 휕퐷푃 0 0 1 0

휕푑푣푏 푑0sin (휃푏) 휕휃푏 = 2 ( C.34 ) 휕퐷푃 cos (휃푏) 휕퐷푃

휕휃푏 1 휕푚푏 = 2 ( C.35 ) 휕퐷푃 푚푏 + 1 휕퐷푃 The process followed from Equation C.20 to here can be repeated for the other cases given in Equations C.6 to C.8. Furthermore, it can be followed for the inner projection of particle 푝2 and for the inner and outer projections of particle 푝3. Substituting these into Equation C.11 provides the force on each particle due to this mechanism.

Proof-of-Concept Simulation Results

The additional energy term was trialled in proof-of-concept simulations by minimising the total energy in the system. An example result is shown in Figure C.4 where the effectiveness of the additional energy term is evident through the bumps which formed on the surface. However, to converge this shape required hundreds of thousands more time steps compared to the localised spontaneous curvature method and substantially more time. Despite this, the shape of the produced cell resembles that predicted by the simpler alternative approach (see Figure 7.4).

Numerical Modelling of Red Blood Cell Morphology and Deformability Page 175

Figure C.4 – Example result from the proof-of-concept simulations using the additional relaxed surface area energy term; inputs used for this simulation were 푁 = 80, 푟 = 3 휇푚, 훼0 = 180°, 푅퐴 = 0.6, 푁푠푒푐 = 8, 푑0 = 50 푛푚, 푘푑 = 100 퐽, Δ푃푟푒푓,푖 = 0.43 휇푚, 퐿푖 = 1.2 휇푚 and stiffness coefficients as per 2D indentation validated case in Chapter 4

From this proof-of-concept it is clear that incorporating the relaxed surface area approach into the CGPM is inferior to the localised spontaneous curvature method due to the significant computational cost incurred in identifying inner and outer layer particle positions and as well as the forces created by the relaxed surface area difference. This would be become increasingly apparent with the extension to 3D given that the inside and outside particle projections would need to be identified from the intersection of three planes, which results in eight intersection points each time. This would also be followed up with increasingly complex derivative expressions. Therefore this approach was not pursued further in this work.

Page 176 Numerical Modelling of Red Blood Cell Morphology and Deformability