Mechanical models for malaria infected erythtocytes 1

MECHANICAL MODELS FOR MALARIA INFECTED ERYTHROCYTES JIAO GUYUE (B.Sc., FUDAN UNIVERSITY) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Summary Mechanical properties play an important role in the physiology of cells. For example, human erythrocytes play a crucial role in oxygen exchange in the human body and an important property of normal human erythrocyte is that it can undergo extremely large deformation while passing through small capillaries. However, when erythrocytes are infected by malaria parasites such as Plasmodium falciparum (P.f.), it is extremely deadly due to its ability to cause cerebral malaria. It is believed that the increase in stickiness and decrease in deformability of the malaria (P.f.) infected erythrocytes will lead to capillary blockage and obstruction of blood flow. Hence in this thesis, a twocomponent model and a multi-component model were developed to quantitatively investigate the decrease in deformability of malaria (P.f.) infected erythrocytes. The elastic rigidity of the cell membrane is associated to the change in free energy caused by both the stretching and bending of the erythrocyte membrane. Finite element analysis was done using the finite element program ABAQUS to simulate erythrocyte deformation in micropipette aspiration and optical tweezers stretching. The effect of shear modulus and bending stiffness on the erythrocyte deformability was analyzed using the two-component model. The model was able to predict the cell deformation in both micropipette aspiration and optical tweezers stretching. By comparing simulation results with experimental results, the model was also able to quantify the increase of shear modulus with the progression of disease stage. The numerical results were found to be insensitive to the mesh parameter changes. The effect of bending stiffness on the deformation of the cell, ranged from 3.3 x 10 -20 J to 1.5 x 10 -18 J, which covered the range of bending stiffness reported by other researchers, was also quantified using the two-component model. The two-component model proposed in this thesis was also able to study the validity of the commonly used hemispherical cap model. The hemispherical cap model was popular in analyzing cell deformation in micropipette aspiration, due to its simplicity in calculating membrane shear modulus. The model and method proposed in this thesis allowed us to test the validity of hemispherical cap model by applying the model to analyze the simulation curves with known values of membrane shear modulus, and to obtain the valid range of pipette radius used in micropipette aspiration as a function of cell radius. The multi-component model proposed in this thesis was developed based on the two-component model. It was able to quantitatively analyze the effect of parasitophorous vacuole membrane (PVM) which enclosed the parasite within the host cell, on cell deformation in micropipette aspiration and optical tweezers stretching. It is hoped that the computational models proposed in this thesis will contribute to more accurate evaluations of the mechanical properties of malaria (P.f.) infected erythrocytes and a better understanding of the underlying pathophysiology. Acknowledgements This thesis involves the joint efforts of many people. First, I would like to thank my supervisors Prof. Lim Chwee Teck and Associate Prof. Sow Chorng Haur for their help in improving my scientific thinking and writing. They introduced me to the research field in biomechanics, and provided me with ample freedom in pursuing my own research interest. I am also grateful to all the people from Nano Biomechanics Lab: Zhou Enhua, Li Ang, Fu Hongxia, Vedula S.R.K., Eunice Tan, Hairul N.B.R., Kelly Low, Tan Lee Ping, Chong Ee Jay, Li Qingsen, Shi Hui, Ng Sin Yee, Anthony Lee, Cheng Tien Ming (National Taiwan University), Zhong Shaoping, Zhang Yousheng, Sun Wei, Yuan Jian, Tan Swee Jin, Hou Han Wei, Earnest Mendoz, Yow Soh Zeom and Lim Tong Seng. Without their kind encouragement and support, I could not have finished the work in this thesis. I would like to especially thank Rosemary Zhang and Lee Yeong Yuh for sharing their experimental data. My sincere thank also goes to all the people in SMART Infectious Diseases group, especially to Dr. Ming Dao who instructed me and offered great help in my research works. I also want to express my heartfelt thanks to Sandra Ho and Lena Lui for their warm friendship and advice in scientific writing. i I would also like to thank all the staff from Dr. Kevin S.W. Tan’s lab in Department of Microbiology, especially Yin Jing, Alvin Chong and Vivien Loon for their help in cell culture. I’m also gratitude to NUS and NUSNNI for their research scholarship, which financially supported me in my research life. I’m also grateful to Singarpoe-MIT Alliance and the SMART Center for their funding support to our project. Last but not least, to my parents, I really appreciate your constant love, support and understanding. ii Table of Contents Acknowledgements .i Table of Contents . iii Summary . viii List of Tables x List of Figures xi List of Symbols xix Chapter Introduction 1.1 Background 1.1.1 The Infectious Human Disease—Malaria .1 1.1.2 Structure and Functions of an Erythrocyte 1.1.3 Life Cycle of the Malaria Parasite .2 1.1.4 Connections between Cell Mechanics & the Pathogenesis of Malaria .3 1.1.5 Structural Changes of the Malaria Infected Erythrocytes 1.1.6 Proteins Secreted to the Infected Erythrocyte’s Membrane .6 1.2 Objectives and scope of work .7 1.3 Thesis Organization .8 Chapter iii Literature Review .10 2.1 Experimental Works on Probing Mechanical Property Changes in MalariaInfected Erythrocytess 10 2.1.1 Experimental Works on Probing Overall Cell Deformability .11 2.1.2 Experimental Works on Probing Membrane Mechanical Property Change .15 2.2 Modelling of Erythrocytes .18 2.2.1 Mechanical Models of Living Cells .19 2.2.2 Modeling of Normal Erythrocytes .26 2.2.3 Modeling of Plasmodium falciparum Infected Erythrocytes .34 2.2.4 Modeling of Optical Tweezers Stretching of Erythrocytes 36 2.2.5 Modeling of Cell Entrance into Capillaries .37 2.2.6 Multicomponent Models of Cells .38 2.3 Deformation Models for Micropipette Aspiration .40 2.3.1 Hemispherical Cap Model 40 2.3.2 Homogenous Half-space Model .42 2.3.3 Comments 42 Chapter A Two-Component Model of Malaria Infected Erythrocytes .43 3.1 Introduction 43 3.2 Material constitutive relations .44 3.3 Simulation of Micropipette Aspiration .48 3.3.1 Geometric Description of Micropipette Aspiration 49 3.3.2 Boundary and Loading Conditions 51 iv 3.3.3 Finite Element Mesh .52 3.3.4 Finite Element Analysis Using ABAQUS 52 3.3.5 Comparison between Two Different Computational Models 56 3.4 Simulation of Optical Tweezers Stretching 61 3.4.1 Geometric Description 61 3.4.2 Boundary and Loading Conditions 63 3.4.3 Finite Element Mesh .65 3.4.4 Finite Element Analysis using ABAQUS .65 3.4.5 Comparison of Deformed Cell Shapes between Simulation and Experiments .66 3.4.6 Computational Results and Comparison with Other Works 73 3.5 Conclusions 74 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes ……………………………………………………………………………… … 76 4.1 Introduction 76 4.2 Material Constitutive Relations .77 4.3 Simulation of Micropipette Aspiration of Erythrocytes 79 4.3.1 Geometry, Boundary and Loading Conditions .79 4.3.2 Finite Element Analysis Using ABAQUS 80 4.3.3 Discussion .84 4.4 Simulation of Optical Tweezers Stretching 84 4.4.1 Geometric Description 85 4.4.2 Boundary and Loading Conditions 86 v 4.4.3 Finite Element Analysis using ABAQUS .86 4.4.4 Discussion .92 4.5 Conclusions 94 Chapter Study of the Valid Range of Pipette Radius Used in Micropipette Aspiration of Erythrocytes 96 5.1 Introduction 96 5.2 Material Constitutive Relations .97 5.3 Methodology 98 5.4 Geometric Description .99 5.4.1 Boundary and Loading Conditions 101 5.4.2 Finite Element Mesh .102 5.4.3 Finite Element Analysis Using ABAQUS 103 5.5 Discussion .107 5.6 Membrane Shear Modulus of Malaria Infected Erythrocytes Calculated Using Hemispherical Cap Model .111 5.7 Conclusions……………………………………….…………………………….112 Chapter A Multi-component Model for the Malaria Infected Erythrocyte .114 6.1 Introduction 114 6.2 Material Constitutive Relations .115 6.3 Simulation of Micropipette Aspiration .116 6.3.1 Geometric Description of Micropipette Aspiration 116 vi 6.3.2 Boundary and Loading Conditions 117 6.3.3 Finite Element Mesh .118 6.3.4 Finite Element Analysis Using ABAQUS 118 6.3.5 Effect of Enclosed Fluid .121 6.3.6 Effect of Parasite Location .122 6.3.7 Effect of PVM Rigidity 127 6.4 Simulation of Optical Tweezers Stretching 129 6.4.1 Geometric Description of Optical Tweezers Stretching .130 6.4.2 Boundary and Loading Conditions 131 6.4.3 Finite Element Mesh .132 6.4.4 Finite Element Analysis using ABAQUS for Optical Tweezers Stretching 132 6.4.5 Effect of Erythrocyte Membrane Stiffness, Interaction between Erythrocyte Membrane and PVM, and PVM Sizes on the Cell Deformation Undergoing Optical Tweezers Stretching .134 6.4.6 Effect of PVM Stiffness on the Erythrocyte Deformation undergoing Optical Tweezers Stretching 137 6.5 Conclusions 138 Chapter Conclusions and Future Work .140 7.1 Conclusions 140 7.2 Future Works 142 References………………………………………………………………………….144 vii curves that were fitted to the average, maximum and minimum value of experimental data. .68 Figure 3.18. Relationship between axial and transverse diameters of deformed ring stage malaria infected erythrocytes and the applied stretching force for both experimental data and simulation results. The values of initial shear modulus µ0 of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. .68 Figure 3.19. Relationship between axial and transverse diameters of deformed trophozoite stage malaria infected erythrocytes and the applied stretching force for both experimental data and simulation results. The values of initial shear modulus µ0 of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. 69 Figure 3.20. Relationship between axial and transverse diameters of deformed schizont stage malaria infected erythrocytes and the applied stretching force for both experimental data and simulation results. The values of initial shear modulus µ0 of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. .70 Figure 3.21. Sketch of experimental observation of the cell shape change under a microscope from (a) top view, (b) cross section, which illustrates the possible tilting or twisting of the erythrocyte stretched by optical tweezers .71 Figure 3.22. Comparison among the results obtained from this presented model and the earlier works done by others for simulation of optical tweezers stretching .73 Figure 4.1. Effect of bending stiffness on the RBC deformation in micropipette aspiration………… 82 Figure 4.2. The three-dimensional model of erythrocytes in the simulation of optical tweezers stretching experiments. The flat oval surface represents the contact area between erythrocyte and silica beads. 85 Figure 4.3. Effect of bending stiffness on the normal erythrocytes’ axial deformation in optical tweezers stretching. The initial shear modulus µ0 = 7.6 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J. 87 Figure 4.4. Effect of bending stiffness on the normal erythrocytes’ transverse deformation in optical tweezers stretching. The initial shear modulus µ0 xv = 7.6 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J…………… .…… .88 Figure 4.5. Effect of bending stiffness on the uninfected erythrocytes’ axial deformation in optical tweezers stretching. The initial shear modulus µ0 = 15.2 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J. …………… .…….89 Figure 4.6. Effect of bending stiffness on the uninfected erythrocytes’ transverse deformation in optical tweezers stretching. The initial shear modulus µ0 = 15.2 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J. ………… …….….90 Figure 4.7. Effect of bending stiffness on the ring stage erythrocytes’ axial deformation in optical tweezers stretching. The initial shear modulus µ0 = 17.1 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J. ……… ……….….91 Figure 4.8. Effect of bending stiffness on the ring stage erythrocytes’ transverse deformation in optical tweezers stretching. The initial shear modulus µ0 = 17.1 µN/m. The six simulation curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where Di was set to be x 10 -19 J. …………… .…….92 Figure 4.9. Effect of bending stiffness on the shape change in the simulation of optical tweezers stretching, (a) original shape, (b) D0 = 3.75 x 10-20 J, (c) D0 = 7.5 x 10-20 J, (d) D0 = 3.0 x 10-19 J, (e) D0 = 6.0 x 10-19 J, (f) D0 = 1.5 x 10-18 J. .93 Figure 5.1. The workflow in the study of a valid range of pipette radius for the hemispherical cap model. .99 Figure 5.2. The geometric description of the model used in the study of the valid range of pipette radius in hemispherical cap model. 100 Figure 5.3. Boundary and loading conditions of simulation of erythrocyte’s deformation in micropipette aspiration. A rigid hemispherical cap was introduced to push the cell surface. ………………………………….101 Figure 5.4. Cell deformation and final position of hemispherical cap in the simulation of micropipette aspiration at different . .104 xvi Figure 5.5. Effect of pipette radius Rp on the erythrocyte deformation in micropipette aspiration, with fixed initial shear modulus µ0 = 4.8 µN/m. 105 Figure 5.6. Effect of pipette radius Rp on the erythrocyte deformation in micropipette aspiration, with fixed initial shear modulus µ0 = 9.6 µN/m. 106 Figure 5.7. Effect of pipette radius Rp on the erythrocyte deformation in micropipette aspiration, with fixed initial shear modulus µ0 = 14.4 µN/m. .107 Figure 5.8. Effect of the ratio between pipette radius and cell radius on the validity of the hemispherical cap model. 108 Figure 5.9. Membrane shear modulus of malaria infected erythrocyte calculated using hemispherical cap model, tested at room temperature and body temperature, respectively. . ……………………… .111 Figure 6.1. The geometric sketch of the simulation of cell deformation during micropipette aspiration using multi-component model. .116 Figure 6.2. Comparison between experiment image and simulation image of a trophozoite stage malaria (P.f) infected erythrocyte with a radius of 3.83 µm being aspirated into a pipette with Rp = 0.61 µm. .118 Figure 6.3. Relationship between projection length Lp and pressure difference ∆P of a trophozoite stage malaria (P.f) infected erythrocyte with a radius of 3.83 µm being aspirated into a pipette with Rp = 0.61 µm, analyzed using multi-component model. …… .………………………………119 Figure 6.4. Experiment Image of a schizont stage malaria erythrocyte obtained from experiment, Rp = 0.73 µm, Rcell = 3.43 µm, RPVM = 2.52 µm. 120 Figure 6.5. Finite element model of a schizont stage malaria erythrocyte being aspirated into a micropipette, Rp = 0.73 µm, Rcell = 3.43 µm, RPVM = 2.52 µm 120 Figure 6.6. Relationship between projection length Lp and pressure difference ∆P of a trophozoite stage cell, analyzed using multi-component model. .121 Figure 6.7. Effect of enclosed fluid on a trophozoite stage cell’s deformation in micropipette aspiration (Rp = 0.61 µm, Rcell = 3.83 µm, RPVM = 1.92 µm). .122 Figure 6.8. Illustration of probing position: section A and section B defined for micropipette aspiration. 123 xvii Figure 6.9. Movement and deformation of PVM within the host cell as the aspiration pressure increased. …………………………….… .…….123 Figure 6.10. The effect of probing positions on the calculated shear modulus using the hemispherical cap model in micropipette aspiration. .124 Figure 6.11. The effect of probing locations on studying the same trophozoite stage malaria infected erythrocyte. The cell radius was measured as 3.09 µm. The radius of the PVM was measured as 1.67 µm. The initial shear modulus of the host cell µ0 (cell) = 15.78 µN/m. ……………………………………… .…………………… .125 Figure 6.12. The experimental and simulation images of the micropipette aspiration of a trophozoite stage malaria infected erythrocyte (Rp = 0.67 µm, Rcell = 4.23 µm, RPVM = 2.52 µm). ……………………………… .… .127 Figure 6.13. Effect of μ0(PVM) on the host cell deformation, with a fixed μ0(cell) value, (a) μ0(PVM)/μ0(cell) varied from 1~100, (b) an enlarged plot of μ0(PVM)/μ0(cell) varied from 1~5. 129 Figure 6.14. The three-dimensional multi-component model of erythrocytes in the simulation of optical tweezers stretching experiments .130 Figure 6.15. Finite element analysis of the multi-component model’s deformation undergoing optical tweezers stretching. 132 Figure 6.16. Finite element simulation of the multi-component model deformation undergoing optical tweezers stretching, when the membrane contact with PVM. .133 Figure 6.17. Effect of membrane stiffness, interaction between PVM and cell membrane, and PVM sizes on the erythrocyte deformation in optical tweezers stretching. The PVM radii R1, R2, R3, R4 and R5 shown in the figure were 2.5 µm, µm, 3.3 µm, 3.4µm, and 3.48µm, respectively. 135 Figure 6.18. Effect of host cell membrane and PVM initial shear modulus on the cell deformation in the simulation of optical tweezers stretching. The host erythrocyte radius was 3.5 µm for all the simulation curves. 137 xviii List of Symbols a Radius of the contraction A0 Initial area Af Final area b Power C Constant C10 , C 20 , C 30 Temperature-dependent material parameters E Young’s Modulus D Bending stiffness D0 Initial bending stiffness Di A preset value of bending stiffness G Shear modulus G0 Initial bulk shear modulus h Membrane thickness h0 Initial membrane thickness I Deviatoric strain invariant k1 , k Elastic constants L Total length of the narrow contraction lc Cell length in modeling of cell entrance into capillaries Lp Projection length ∆P Pressure Drop Rcell Radius of the cell Rp Pipette Radius xix Rpvm Radius of the PVM P0 Initial aspiration pressure Smax Maximum shear strain T0 Static in-plane isotropic tension corresponding to zero shearing and dilatory rates T1 , T2 In-plane principal stress resultants which are perpendicular to each other Ts Membrane shear stress t Entrance time U Strain energy potential per unit of initial volume ui Displacement components in Cartesian coordinate system Va Rates of dilation vi Velocity components in Cartesian coordinate system Vs Rates of shear Wm Strain energy per unit of initial area c Characteristic shear rate  ij Engineering strain components m Mean shear rate γmax Maximum shear stress s Shear strain δ Gap between the cell and the capillary wall  i (i  1, 2,3)  Principal strains Coefficient of viscosity for shear xx c Characteristic viscosity  ij Kronecker delta  Area dilatational modulus i (i  1, 2,3) Principal stretch ratios  Membrane shear modulus 0 Initial shear modulus µ0 (cell) Initial shear modulus of the host cell µ0 (PVM) Initial shear modulus of the PVM f Third stage large deformation modulus l Second stage large deformation modulus m Viscosity of the matrix in which the cell is suspended  i (i  1, 2,3) Principal stresses  ij Whole stress components  ij Deviatoric stress components υ Poisson’s Ratio xxi Chapter Introduction Chapter Introduction 1.1 Background 1.1.1 The Infectious Human Disease—Malaria Malaria is a life-threatening tropical parasitic human disease transmitted by the female Anopheles mosquitoes. In 2006, among 3.3 billion people at risk, the number of malaria cases was estimated to be 247 million. In 2008, there were 109 endemic countries for malaria. It is responsible for nearly million deaths each year, mostly children under the age of (World-Health-Organization 2008). Malaria is induced by a one-cell parasite called the Plasmodium. Human red blood cells (RBCs), also called erythrocytes can be infected by four different species of the Plasmodium (P.): P. falciparum, P. ovale, P. malariae and P. vivax. Among them, P. vivax and P. falciparum are the most common, but the malaria induced by the latter is the most severe due to its ability to cause cerebral malaria (Miller et al. 2002). P. falciparum (P.f.) can invade the erythrocytes and change the membrane skeleton. 1.1.2 Structure and Functions of an Erythrocyte The volume of blood in the human body is about liters, with liters consisting of plasma and the rest consisting of cells, 99% being erythrocytes (Hill 2002). Healthy erythrocytes are shaped like a biconcave disk under static and isotonic condition. They are filled with hemoglobin, but not have a nucleus or cytoplasmic organelles. They can easily undergo extremely large deformation as they flow through Chapter Introduction narrow capillaries. Their major function is to deliver oxygen from the lungs to the various organs and to remove carbon dioxide from organs back to the lungs. Figure 1.1 shows a human erythrocyte membrane structure. It consists of transmembrane proteins, lipid bilayer and the underlying spectrin network (Dao et al. 2003). Figure 1.1. The structure of a human erythrocyte membrane, consisting of the transmembrane proteins, the lipid bilayer, and the spectrin network, reprinted from (Dao et al. 2003) with permission. 1.1.3 Life Cycle of the Malaria Parasite Figure 1.2 illustrates the life cycle of the malaria parasite. Human is infected after being bitten by a female Anopheles mosquito. While the mosquito is taking a blood meal, parasitic sporozoites are transported via the insect's saliva into the peripheral circulation. These sporozoites will then move to the liver, where they can invade a hepatocyte. About to days later, one sporozoite can develop into thousands of merozoites and these merozoites will then be released back to peripheral circulation to invade and infect other healthy erythrocytes. On encountering an erythrocyte, the merozoite adheres to the cell surface and invades into the erythrocyte. Once inside the host cell, the merozoite consumes hemoglobin as their source of food, and matures from the ring stage to the mid-stage trophozoites, and eventually to the late-stage schizonts within about 48 hours. After maturing, the schizont stage infected Chapter Introduction erythrocyte will rupture and release about 12-20 merozoites back into the blood and perpetuates the cycle. In the meantime, some invading merozoites differentiate into male (microgametocyte) and female (macrogametocyte) sexual forms. If another mosquito feeds on the blood containing such sexual form of parasites, it becomes infected. Figure 1.2. Life cycle of the malaria parasites, reprinted from (Miller et al. 2002) with permission. 1.1.4 Connections between Cell Mechanics & the Pathogenesis of Malaria Malaria first begins as a flu-like illness 7-9 days after infection, causing fever, headache, chills, shivering, joint pain and repeated vomiting. The parasites living in erythrocytes rupture the hosts eventually, and results in anemia (World-HealthOrganization 2000). The pathophysiological mechanism of severe malaria is still not clear. But based on clinical evidences, several hypotheses have been proposed, among Chapter Introduction which the mechanical hypothesis is well accepted and has been studied (WorldHealth-Organization 2000). Healthy erythrocytes are extremely deformable and have weak interactions with other blood cells and endothelium. But when the erythrocyte is infected by the P.f. parasites, the following changes occur:  Altered membrane transport mechanism;  Digestion of hemoglobin to pigment;  Decreased deformability and other mechanical and rheological changes;  Expression of variant surface neo-antigens;  Development of electron-dense protuberances or knobs beneath the surface membrane; and  Development of cytoadherent and rosetting properties resulting in sequestration of infected erythrocytes containing late trophozoites and schizonts in deep vascular beds. The increase in cytoadherence and the decrease in deformability of infected erythrocytes are often considered as crucial contributing factors in malaria pathology. The severity of P.f. infection is a function of the extent of capillary blockage caused by infected erythrocytes in organs such as spleen, liver, kidney and brain. Such blockage will result in obstruction of blood flow, and can lead to a decrease in the distribution of oxygen and nutrients to vital organs and the removal of toxic waste products like lactic acid. Chapter Introduction The extreme flexibility of a healthy erythrocyte is a result of its biconcave shape which gives rise to a high surface to volume ratio (Weed 1970). This surfacevolume ratio may be decreased in the case of a malaria infected erythrocyte. Malaria may affect the property of erythrocyte membrane, and the enclosed parasite itself may result in an increased internal viscosity of the cell (Miller et al. 1971). The less flexible membrane and/or increased internal viscosity might be the cause of the decrease in cell deformability. Proteins transported from the parasite to the cell membrane may also contribute to the loss of deformability. However, whether the loss of deformability is mainly due to the cell membrane or the components within the cell is still unclear. 1.1.5 Structural Changes of the Malaria Infected Erythrocytes Figure 1.3. Development of parasites in a malaria-infected erythrocyte, reprinted from (Marti et al. 2005) with permission. After parasite invasion, the host cell undergoes severe structural changes (Marti et al. 2005). Figure 1.3 shows the development of parasites in a malariainfected erythrocyte. “N” indicates the position of the parasite nucleus. At about 0-5 Chapter Introduction hours after parasite invasion, the parasitophorous vacuole (PV) encloses the parasites, forming a ring-like structure. The parasitophorous vacuole membrane (PVM) is semipermeable which allows the acquisition of nutrient and the secretion of parasite. This infection stage is called the ring stage. At about 5-10 hours after parasite invasion, which is called the early trophozoite stage, the membranous extensions of PVM induced by parasites form erythrocyte membrane-tethered Maurer’s clefts (MCs) (Haldar et al. 2002) and a tubulovesicular network (TVN). The hemozoin crystals deposit in the food vacuole (FV), making the hemoglobin digestion products visible. At about 10-20 hours after parasite invasion, the parasites continue growing and occupy nearly 40% of the host cell’s volume. Knobs (K) are induced on the cell surface. The last stage is called schizont stage. The parasite divides itself and forms 16-32 daughter merozoites surrounded by the PVM. These daughter merozoites will burst out and invade uninfected erythrocytes. 1.1.6 Proteins Secreted to the Infected Erythrocyte’s Membrane Various proteins are derived from the malaria parasite, among which at least five are related to the membrane of the host cell, as shown in Figure 1.4 (Ho et al. 1999). When the parasite progresses into the trophozoite stage, knobs appear on the membrane surface of the host cell, associated with proteins such as P.f. erythrocyte membrane protein & (PfEMP1 & PfEMP2) and knob-associated histidine-rich protein (KAHRP or PfHRP1). PfEMP2 and KAHRP or PfHRP1 exist on the inner surface of the cell membrane, associated with electron-dense material (EDM). PfEMP1 appears on the outer surface of the cell membrane, mediating cytoadherence of the infected erythrocytes to various host cell receptors (Magowan et al. 1988). Chapter Introduction Figure 1.4. Distribution of P.f. proteins on the surface of an infected erythrocyte membrane. RBCM, red blood cell membrane; EDM, electron-dense material; PfEMP, P.f. erythrocyte membrane protein; PfHRP, P.F. histidine-rich protein, also known as knobassociated histidine-rich protein (KAHRP); RESA, ring-infected erythrocyte surface antigen, reprinted from (Ho et al. 1999) with permission. 1.2 Objectives and Scope of Work In view of the above, the mechanical properties of the malaria infected erythrocytes change with the progression of disease states and have important effects on the pathophysiological outcomes. Hence, it is necessary to quantify the contribution of the subcellular components such as the parasites within the infected cells in order to more accurately determine the change in overall cellular deformability. The objectives of this project are: 1. To propose a computational model for the P.f. erythrocytes at their different stages of parasite maturation, especially at the mid- and late-stage, for which no accurate computational models have been proposed yet, and 2. To quantitatively investigate the loss of deformability in P.f. infected erythrocytes arising from the subcellular components such as the parasites as well as structural changes occurring within the cell. Chapter Introduction Ultimately, we hope to obtain a deeper understanding of the relationship between malaria progression and the change in mechanical properties of infected erythrocytes. To fulfill the objectives, the scope of this project will include the following: 1. Using ABAQUS to develop computational models of erythrocytes at their different malaria infection stages, including uninfected, ring stage, trophozoite stage and schizont stage. These models will incorporates both the membrane and structural changes in the infected erythrocytes. 2. Identifying suitable mechanical and material property parameters for the infected erythrocyte computational models by comparing simulation with experimental results obtained from optical tweezers and micropipette aspiration experiments previously done by others. 1.3 Thesis Organization The organization of the remainder of this thesis is as follows: Chapter contains the literature review on existing experimental works and modeling on mechanical property changes in malaria infected erythrocytes. Chapter introduces a two-component model of malaria infected erythrocytes and studies the effect of membrane shear modulus on the erythrocyte’s deformation in micropipette aspiration and optical tweezers stretching. Chapter reports the study on the effect of bending stiffness on the erythrocyte’s deformation when modeling micropipette aspiration and optical tweezers stretching experiments. Chapter Introduction Chapter reports the study on the effect of pipette radius on validity of hemispherical cap model used to model micropipette aspiration experiments. Chapter introduces a multi-component model of malaria infected erythrocytes and studies the effect of parasite inclusion on the infected erythrocyte’s deformation as observed in micropipette aspiration and optical tweezers stretching experiments. Chapter concludes the thesis by summarizing the main contributions and indicating potential future directions. [...]... stiffness, equalled 1/ 8, 1/ 4, 1/ 2, 1, 2 and 5 times of Di, respectively, where Di was set to be 3 x 10 -19 J…………… …… 88 Figure 4.5 Effect of bending stiffness on the uninfected erythrocytes’ axial deformation in optical tweezers stretching The initial shear modulus µ 0 = 15 .2 µ N/m The six simulation curves were obtained using different initial bending stiffness, equalled 1/ 8, 1/ 4, 1/ 2, 1, 2 and 5 times... 17 .1 µ N/m The six simulation curves were obtained using different initial bending stiffness, equalled 1/ 8, 1/ 4, 1/ 2, 1, 2 and 5 times of Di, respectively, where Di was set to be 3 x 10 -19 J …………… …….92 Figure 4.9 Effect of bending stiffness on the shape change in the simulation of optical tweezers stretching, (a) original shape, (b) D0 = 3.75 x 10 -20 J, (c) D0 = 7.5 x 10 -20 J, (d) D0 = 3.0 x 10 -19 ... temperature, respectively ……………………… 11 1 Figure 6 .1 The geometric sketch of the simulation of cell deformation during micropipette aspiration using multi-component model 11 6 Figure 6.2 Comparison between experiment image and simulation image of a trophozoite stage malaria (P.f) infected erythrocyte with a radius of 3.83 µ being aspirated into a pipette with Rp = 0. 61 µm .11 8 m Figure 6.3 Relationship... was set to be 3 x 10 -19 J …………… …….89 Figure 4.6 Effect of bending stiffness on the uninfected erythrocytes’ transverse deformation in optical tweezers stretching The initial shear modulus µ 0 = 15 .2 µ N/m The six simulation curves were obtained using different initial bending stiffness, equalled 1/ 8, 1/ 4, 1/ 2, 1, 2 and 5 times of Di, respectively, where Di was set to be 3 x 10 -19 J ………… …….….90... erythrocytes’ axial deformation in optical tweezers stretching The initial shear modulus µ 0 = 17 .1 µ N/m The six simulation curves were obtained using different initial bending stiffness, equalled 1/ 8, 1/ 4, 1/ 2, 1, 2 and 5 times of Di, respectively, where Di was set to be 3 x 10 -19 J ……… ……….…. 91 Figure 4.8 Effect of bending stiffness on the ring stage erythrocytes’ transverse deformation in optical...  i (i  1, 2,3) Principal stresses  ij Whole stress components  ij Deviatoric stress components υ Poisson’s Ratio xxi Chapter 1 Introduction Chapter 1 Introduction 1. 1 Background 1. 1 .1 The Infectious Human Disease Malaria Malaria is a life-threatening tropical parasitic human disease transmitted by the female Anopheles mosquitoes In 2006, among 3.3 billion people at risk, the number of malaria cases... 6 .13 Effect of μ0(PVM) on the host cell deformation, with a fixed μ0(cell) value, (a) μ0(PVM)/μ0(cell) varied from 1~ 100, (b) an enlarged plot of μ0(PVM)/μ0(cell) varied from 1~ 5 12 9 Figure 6 .14 The three-dimensional multi-component model of erythrocytes in the simulation of optical tweezers stretching experiments .13 0 Figure 6 .15 Finite element analysis of the multi-component model’s deformation... (a) a normal cell (Pipette Diameter = 1. 222 µ (b) an uninfected cell (Pipette Diameter = 1. 3 41 µm) , (c) a ring m), stage cell (Pipette Diameter = 1. 6 01 µ m), (d) a trophozoite stage cell (Pipette Diameter = 0.990 µm) and (e) a schizont stage cell (Pipette Diameter = 1. 476 µm) 60 Figure 3 .11 Geometry of one-eighth of an erythrocyte finite element model for the simulation of optical tweezers... following: 1 Using ABAQUS to develop computational models of erythrocytes at their different malaria infection stages, including uninfected, ring stage, trophozoite stage and schizont stage These models will incorporates both the membrane and structural changes in the infected erythrocytes 2 Identifying suitable mechanical and material property parameters for the infected erythrocyte computational models. .. defined for micropipette aspiration 12 3 xvii Figure 6.9 Movement and deformation of PVM within the host cell as the aspiration pressure increased …………………………….… …… .12 3 Figure 6 .10 The effect of probing positions on the calculated shear modulus using the hemispherical cap model in micropipette aspiration 12 4 Figure 6 .11 The effect of probing locations on studying the same trophozoite stage malaria . Chapter 1 Introduction 1 1. 1 Background 1 1. 1 .1 The Infectious Human Disease Malaria 1 1. 1.2 Structure and Functions of an Erythrocyte 1 1. 1.3 Life Cycle of the Malaria Parasite 2 1. 1.4 Connections. Multi-component Model for the Malaria Infected Erythrocyte 11 4 6 .1 Introduction 11 4 6.2 Material Constitutive Relations 11 5 6.3 Simulation of Micropipette Aspiration 11 6 6.3 .1 Geometric Description. Literature Review 10 2 .1 Experimental Works on Probing Mechanical Property Changes in Malaria- Infected Erythrocytess 10 2 .1. 1 Experimental Works on Probing Overall Cell Deformability 11 2 .1. 2 Experimental