Chapter A Two-Component Model for Malaria Infected Erythrocytes Chapter A Two-Component Model of Malaria Infected Erythrocytes 3.1 Introduction As reviewed in Chapter 2, various experimental techniques have been used to test the stiffening effect of P.f. infected erythrocytes, and have shown that the membrane shear modulus of the cell changes with the progression of parasite maturation. In this chapter, in order to quantify the membrane shear modulus of the infected cell at their different infection stages, finite element models were used to simulate the cell deformation in two experiments: micropipette aspiration and optical tweezers stretching. The cell was considered as a liquid enclosed by a solid membrane, for all the infection stages. The initial membrane shear moduli obtained from the finite element analysis were compared with the ones calculated from the commonly used hemispherical cap model to verify the validity of the model. The inclusion of parasite will be further discussed in Chapter using a more advanced and complex model. 43 Chapter A Two-Component Model for Malaria Infected Erythrocytes 3.2 Material Constitutive Relations Based on available literature which was reviewed, the cortical-shell liquidcore model was adopted. The cytoplasm was represented by a homogeneous viscous Newtonian fluid. The membrane was defined as a hyperelastic material, which is described in terms of strain energy potential U, which defines the strain energy stored in the material per unit volume in the initial configuration as a function of the strain at that point in the material. The most general stress-strain relationship for a two-dimensional generalized plane-stress field in an isotropic material is given by 1  1 (   ),   (  1 ) E E (3.1) where ε1 , ε are the principal strains, σ1, σ2 are the principal stresses corresponding to principal axes x1, x2, the elastic constants E and υ are functions of the strain invariants 1   , 1 . The strains ε1, ε2 measured in the plane of the membrane can be defined using Green’s strains (Fung 1993): 1  (12  1),   (22  1) (3.2) where λ1, λ2 are the principal stretch ratios corresponding to ε1 , ε2 , respectively. If the membrane material is incompressible, 44 Chapter A Two-Component Model for Malaria Infected Erythrocytes 1 2 3  (3.3) where λ3 is the stretch ratio in the direction perpendicular to the membrane. If the area of the RBC membrane is a constant, 12  (3.4) Since the area of the RBC membrane is assumed not to change during deformation, the thickness of the cell membrane remains almost constant (Fung 1993). According to Eq. (3.2), 1 -  (12  22 ) (3.5) Using Mohr’s circle, we can get the maximum shear strain:  max  1  22 (3.6) The maximum shear stress resultant is given by S max   max (3.7) where µ is the membrane shear modulus. Several existing material constitutive laws commonly used to model cell membranes were introduced in Chapter 2. In my current model, the strain energy potential is defined in Yeoh’s form (Yeoh 1990), which is given by 45 Chapter A Two-Component Model for Malaria Infected Erythrocytes       U  C10 12  22  32   C 20 12  22  32   C30 12  22  32  (3.8) where U is the strain energy per unit of initial volume, C10, C20, C30 are temperaturedependent material parameters. If the stretch is uniaxial, the uniaxial membrane stresses of the hyperelastic thin membrane are given by (Dao et al. 2003) T1  h U , T2  1 (3.9) where h is the current membrane thickness. The membrane shear stress is given by Ts  (T1  T2 ) (3.10) And the membrane shear modulus is given by  (1 )  Ts  s (3.11) where  s is the shear strain. According to Eqs. (3.7) and (3.10)-(3.13), the membrane shear stress and shear modulus can be calculated, which are given by 46 Chapter A Two-Component Model for Malaria Infected Erythrocytes Ts (1 )   (1 )  h0 (C10  A  2C20  AB  3C30  AB )  h0 C10  C  2C20 ( A2  BC )  3C30 (2 AB  B 2C ) D  A  2(1  13 ), B  12  1  2, C   61 , D  2(1  13 ), (3.12) where h0 is the initial membrane thickness. From Eq. (3.14), we can obtain the initial shear modulus of the membrane, which is given by   2C10 h0 (3.13) which is consistent with the equation given in the ABAQUS manual (ABAQUS v6.4). Therefore, Eq. (3.8) becomes U 0 2h0   22  32  3  C20  12  22  32  3  C30  12  22  32  3 (3.14) If we assume the area of the RBC membrane is a constant, the initial shear modulus of the membrane µ would be 75% of the value without enforcing this assumption (Dao et al. 2003). 47 A Two-Component Model for Malaria Infected Erythrocytes Membrane Shear Modulus Membrane Shear Stress Chapter Shear Strain (a) Shear Strain (b) Figure 3.1. (a) The stress-strain relationship of the hyperelastic thin membrane under uniaxial stretch, & (b) the relationship between membrane shear modulus and shear strain, the shear strain illustrated in the figure is expressed in term of 2γs. The stress-strain relationship of this material can be calculated from Eq. (3.14), as plotted in Figure 3.1 (a), while the relationship between shear modulus and shear strain is illustrated in Figure 3.1 (b). The cytosol was modeled as an incompressible or nearly incompressible fluid with a hydraulic fluid model within a fluid-filled cavity. 3.3 Simulation of Micropipette Aspiration In this section, simulation of micropipette aspiration will be done using a finite element two-component model. The geometric description, boundary and loading conditions, and finite element mesh will be introduced. The simulations were done using the finite element analysis program ABAQUS. The simulation results will be compared with the ones calculated using the commonly used hemispherical cap model. 48 Chapter 3.3.1 A Two-Component Model for Malaria Infected Erythrocytes Geometric Description of Micropipette Aspiration The healthy, uninfected and ring stage malaria infected erythrocyte model consists of a shell and an enclosed fluid. The measured average normal RBC shape (Evans and Fung 1972) is given by, y = 0.5[1-x2 ]1/2(0.207161+2.002558x2-1.122762x4) z = 3.91y, r = 3.91x (-1≤ x ≤1) (3.15) where x,y,z is the geometry parameters of the erythrocyte, and r is the distance from the axis of symmetry. According to these equations, a cell membrane was modelled with a biconcave shape, as shown in Figure 3.2. This geometry was used in modelling biconcave shaped cells for both the micropipette aspiration and laser tweezers stretching experiments. Figure 3.2. Model of a normal erythrocyte for finite element analysis, the shape was measured by Evans and Fung (1972). 49 Chapter 3.3.1.1 A Two-Component Model for Malaria Infected Erythrocytes Normal, Uninfected, Ring Stage and Trophozoite Stage Infected Erythrocytes In the early stages, the infected erythrocytes maintain their biconcave shape, as shown in Figure 3.3. Due to the axisymmetric loading condition and cell geometry, the simulation was simplified as an axisymmetric problem. The cell shape was calculated according to the average normal RBC shape using Equation 3.15. The inner/outer pipette radius and cell radius were measured from different experiments, and were changed according to the specific experiment being modeled. The uninfected erythrocytes refer to the cells that are cultured together with infected erythrocytes but have not been invaded by the parasite. Figure 3.3. Schematic geometry for micropipette aspiration of healthy, uninfected, ring stage and trophozoite stage infected erythrocytes. Rcell is the radius of erythrocyte and Rp is the radus of pipette. 50 Chapter 3.3.1.2 A Two-Component Model for Malaria Infected Erythrocytes Schizont Stage Infected Erythrocytes Figure 3.4. Schematic geometry for the micropipette aspiration of a schizont stage infected erythrocyte. In the late stage, the infected erythrocytes can be assumed as spherical (Suresh et al. 2005). Similar as the earlier stage cells, it was simplified as an axisymmetric model as shown in Figure 3.4. 3.3.2 Boundary and Loading Conditions The boundary and loading conditions are shown in Figure 3.5. The axisymmetric model was created in x-y plane. Point A & B were only allowed to move in y-direction. A constantly increasing aspiration pressure was uniformly applied on the part of cell surface that was inside the rigid and fixed pipette. The blue cross shown in this figure represents the reference node of the analytical rigid pipette. 51 Chapter A Two-Component Model for Malaria Infected Erythrocytes Figure 3.5. Boundary and loading conditions for the simulation of micropipette aspiration of erythrocyte. 3.3.3 Finite Element Mesh In this model, the erythrocyte consisted of only two components: outer membrane and inner cytoplasm. Since the geometry and loading condition of the model is axisymmetric, the membrane was represented by 449 axisymmetric shell elements (SAX1), with a higher mesh density in the half of the cell that was nearer to the pipette, and a lower mesh density in the other half. The cytoplasm was represented by 449 2-node linear hydrostatic fluid elements (FAX2), which are provided to model fluid-filled cavities. This type of fluid element consists of nodes and shares a common cavity reference node. 3.3.4 Finite Element Analysis Using ABAQUS From micropipette aspiration, we can obtain aspiration pressure (∆P), pipette radius (Rp ) and the projection length (Lp ), as shown in Figure 3.6. 52 Chapter A Two-Component Model for Malaria Infected Erythrocytes (e) The plot of projection length vs pressure difference for a schizont stage malaria infected erythrocyte (Pipette Diameter = 1.476 µm). Figure 3.10. Comparison between the two-component finite element model and the hemispherical cap model of (a) a normal cell (Pipette Diameter = 1.222 µm), (b) an uninfected cell (Pipette Diameter = 1.341 µm), (c) a ring stage cell (Pipette Diameter = 1.601 µm), (d) a trophozoite stage cell (Pipette Diameter = 0.990 µm) and (e) a schizont stage cell (Pipette Diameter = 1.476 µm). It can be seen that if we ignore the internal components of structural changes occurring within the infected erythrocytes, we can duplicate the results of commonly used hemispherical cap model by adopting this two-component finite element model. Therefore, this finite element model and its material constitutive relations are suitable for modelling the infected cell membrane deformation. However, the hemispherical cap model has its own assumptions and limitations, as reviewed in Chapter 2. If we take into account the parasite growth within the host erythrocyte, especially during the trophozoite and schizont stages, the hemispherical cap model might not be accurate This will be discussed in Chapter 6. Due to the wide range of pipette sizes used in the experiments, more work will be done in the Chapter to study the range of pipette sizes that can be used in the 60 Chapter A Two-Component Model for Malaria Infected Erythrocytes hemispherical cap model to replicate the simulation results before we proceed to an advanced complex model. 3.4 Simulation of Optical Tweezers Stretching In this section, the malaria infected erythrocytes deformation during optical tweezers stretching will be studied using a finite element two-component model. The contents include the geometric descriptions, boundary and loading conditions, the finite element mesh, the simulation using ABAQUS, the comparison between experiments and simulations, and the comparison with other different works. 3.4.1 3.4.1.1 Geometric Description Normal, Uninfected, Ring Stage and Trophozoite Stage Erythrocytes (a) 61 Chapter A Two-Component Model for Malaria Infected Erythrocytes (b) Figure 3.11. Geometry of one-eighth of an erythrocyte finite element model for the simulation of optical tweezers stretching experiments, (a) 3D drawing, (b) side view. Owing to the plane symmetry of both the geometry of the erythrocyte and the intended axial loading conditions, the computational model was reduced and represented by only one-eighth of the cell, as shown in Figure 3.11. The model was estimated to be 3.91 μm in direction z, and 3.78 μm in direction x. The contact surface is flat oval of μm in width and 0.55 μm in height. This actually modelled the erythrocyte with a small flat oval region of 1.7 μm2 at its diametric opposite ends to signify the attachment of the silica micro beads which were assumed to be rigid spheres. 62 Chapter 3.4.1.2 A Two-Component Model for Malaria Infected Erythrocytes Schizont stage infected erythrocytes Figure 3.12. Geometry of one-eighth of a schizont stage malaria infected erythrocyte model used in the simulation of optical tweezers stretching experiments. Similar to the model used for normal, P.f. uninfected, ring stage and trophozoite stage erythrocytes, the model for schizont stage cell was simplified to only one-eighth of the cell. The model was estimated to be 3.5 μm (Suresh et al. 2005) in direction y and z, and 3.38 μm in direction x due to the cell’s attachment to the silica beads. The contact surface is flat oval of 0.89 μm in width and height, estimated according the experimental images. This actually modelled the erythrocyte with a small flat region of 2.4 μm2 at its diametric opposite ends to signify the attachment of the silica micro beads which were assumed to be rigid spheres. 3.4.2 Boundary and loading conditions According to the symmetric cell geometry and loading conditions in the experiment, the initial boundary conditions of the three-dimensional model are shown 63 Chapter A Two-Component Model for Malaria Infected Erythrocytes in Figure 3.13 (a). The displacement in direction x was applied on the flat surface to stretch the cell, as shown in Figure 3.13 (b). (a) Initial boundary conditions of the simulation of optical tweezers stretching. (b) Boundary conditions of the simulation of optical tweezers stretching in step 1. Figure 3.13. Boundary and loading conditions of the simulation of optical tweezers stretching, in (a) step 0, and (b) step 1. 64 Chapter 3.4.3 A Two-Component Model for Malaria Infected Erythrocytes Finite Element Mesh The model was analyzed using ABAQUS. Two types of elements were used. As shown in Figure 3.11, the outer surface of the cell, which represented the membrane, used 3000 S4R shell elements (4-node doubly curved shell finite membrane strains elements). The inner surface and contact surface of the cell was composed of 3000 F3D4 elements (4-node linear 3-dimensional quadrilateral hydrostatic fluid element), which represented the cytoplasm. 3.4.4 Finite Element Analysis using ABAQUS (a) (b) Contact Area Figure 3.14. Simulation of optical tweezers stretching of a healthy RBC, (a) ¼ of the RBC without the flat oval surface, (b) 1/8 of the RBC with the flat oval surface. 65 Chapter A Two-Component Model for Malaria Infected Erythrocytes Figure 3.14 shows the finite element simulation of optical tweezers stretching. Both axial and transverse diameter changes of the cell and the reaction forces were obtained from the simulation using ABAQUS. The results were compared with that of experimental data (Suresh et al. 2005). 3.4.5 Comparison of Deformed Cell Shapes between Simulation and Experiments If we apply the finite element model illustrated in Figure 3.11 to the normal, uninfected, ring stage infected erythrocytes to fit the experiment data (Suresh et al. 2005), we can compare the axial and transverse diameter changes as a function of stretching force, as illustrated in Figure 3.15. Figure 3.15. Comparison between simulation curves and average experimental data of the normal, uninfected and ring stage infected erythrocytes using the same original diameter. 66 Chapter A Two-Component Model for Malaria Infected Erythrocytes The comparison shows that the initial shear modulus of the membrane increases with the progression of the malaria infection (Figure 3.15). However, the average radius of uninfected and ring stage erythrocytes tested in the experiments were smaller than that of the normal cells. More simulations were done considering both the size deviation and radius change in cells at different stages, except for those in schizont stage, as shown in Figure 3.16 ~ 3.19. Different simulation curves were fitted to the average, maximum and minimum value of experimental data for both axial and transverse diameters, and the value of initial shear modulus µ of each simulation was marked beside the corresponding curve. Thus the range of initial shear modulus was obtained for each type of cell. Figure 3.16. Relationship between axial and transverse diameters of deformed normal erythrocytes and the applied stretching force for both experimental data and simulation results. The values of initial shear modulus µ of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. 67 Chapter A Two-Component Model for Malaria Infected Erythrocytes Figure 3.17. Relationship between axial and transverse diameters of deformed uninfected erythrocytes and the applied stretching force for both experimental data and simulation results. The values of initial shear modulus µ of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. 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 µ of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. 68 Chapter A Two-Component Model for Malaria Infected Erythrocytes 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 µ of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. The in-plane initial shear modulus of the normal erythrocyte membrane ranges from 4.8 µN/m to 12.4 µN/m, with an average value of about µN/m ( Figure 3.16). While the initial shear moduli of the uninfected erythrocytes and ring stage infected erythrocyte membrane range from 10.0 to 18.1 µN/m and from 11.9 to 24.7 µN/m, respectively (Figure 3.17 & 3.18). The average initial shear moduli of uninfected erythrocytes and ring stage infected erythrocyte membrane are larger than the ones given in Figure 3.16 because of the different initial cell dimension used in the finite element model. The initial shear mosulus of trophozoite stage infected erythrocytes ranges from 22.4 µN/m to 57.1 µN/m (Figure 3.19). These results show obvious increasing in the initial shear modulus of erythrocyte membrane during the early progression of parasite infection. 69 Chapter A Two-Component Model for Malaria Infected Erythrocytes 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 µ of each simulation were marked beside the corresponding curves that were fitted to the average, maximum and minimum value of experimental data. When the parasite grows into schizont stage, the host erythrocyte becomes spherical, with accompanying size changes. So the schizont stage infected erythrocyte was modelled as 1/8 of a sphere, as illustrated in Figure 3.12. The simulation results are illustrated in Figure 3.20. Similarly, different simulation curves were fitted to the average, maximum and minimum value of experimental data for both axial and transverse diameters, and the value of initial shear modulus µ of each simulation was marked beside the corresponding curve. Thus the range of initial shear modulus was obtained for the schizont stage cell, which was 36.8 ~ 99.7 µN/m. 70 Chapter A Two-Component Model for Malaria Infected Erythrocytes 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. . When we compare the initial shear modulus obtained from axial diameter changes of the erythrocyte with those obtained from transverse diameter changes of the erythrocyte, we can see that the latter were a little smaller than the former. 71 Chapter A Two-Component Model for Malaria Infected Erythrocytes This disagreement in the fitting of the axial and transverse diameter is probably caused by the twisting or tilting of the cell during experiments, as illustrated in Figure 3.21. When the cell was stretched by silica beads, we were able to observe the cell shape change under the microscope from the top. However, the cross section of the cell was not observed. It is possible that the cell was not placed horizontally at the beginning, or it twisted or tilted during the stretching process. Therefore, the observed transverse diameter from the microscope might be smaller than the cell’s real diameter, and hence resulted in a bigger transverse diameter change observed in experiments than its real change. As a result, we need to use a softer material to simulate the transverse diameter change of the erythrocytes observed in experiments, and obtained a smaller value than what we obtained from fitting of axial diameter change of the erythrocytes. From both axial and transverse diameter changes, the membrane shear modulus we obtained increased with the progression of infectious stages. The initial shear modulus obtained from average axial diameter changes of the erythrocytes were 7.6 µN/m, 14.3 µN/m, 16.6 µN/m, 32.8 µN/m and 65.1 µN/m for normal, P.f. uninfected, ring, trophozoite and schizont stage, respectively, showing the increasing of membrane stiffness. 72 Chapter 3.4.6 A Two-Component Model for Malaria Infected Erythrocytes Computational Results and Comparison with Other Works As reviewed in Chapter 2, computational works have been done by other researchers to study the mechanical properties of malaria infected erythrocytes membrane stretched using optical tweezers (Suresh et al. 2005; Dao et al. 2006). The comparison between my current model and the earlier works done by others is given in the following figure. 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. All these three works assumed the cell to be a Newtonian fluid enclosed by hyperelastic membrane. However, in the computational simulations discussed in this 73 Chapter A Two-Component Model for Malaria Infected Erythrocytes chapter, the deviation of original cell radius and the errors of experimental data were considered, which made the evaluation of membrane shear modulus more accurate and comprehensive. These three models all showed the increase in initial membrane shear modulus of the P.f. infected erythrocytes with the progression of disease stage. The values of membrane shear modulus were close among the work given by (Suresh et al. 2005), (Dao et al. 2006) and the current model introduced in this chapter, and there were no statistically significant differences between them. 3.5 Conclusions In this chapter, a two-compartment model was introduced. Finite element analysis was done using ABAQUS to simulate the deformation of malaria infected erythrocytes. The numerical results were found to be insensitive to the mesh parameter changes. This model was capable of predicting the cell deformation induced by optical tweezers stretching and micropipette aspiration. The finite element analysis obtained similar values of initial shear modulus of normal erythrocytes, when compared with other works. Both of these two models assumed that the cell was composed of incompressible membrane and Newtonian fluid, and that it was frictionless between the pipette surface and the cell membrane. For normal and early stage malaria infected erythrocytes, these assumptions are sufficient for the computational modelling. 74 Chapter A Two-Component Model for Malaria Infected Erythrocytes However, whether this two-component model is suitable for the mid and late stage malaria infected erythrocytes due to their obviously changed internal structures will be discussed in Chapter 6. 75 [...]... N/m, 13. 1 µ N/m, 18.9 µ N/m, 35 .3 µ N/m and 39 .2 µ N/m, respectively 57 Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes (a) The plot of projection length vs pressure difference for a normal erythrocyte (Pipette Diameter = 1.222 µ m) (b) The plot of projection length vs pressure difference for an uninfected erythrocyte (Pipette Diameter = 1 .34 1 µm) 58 Chapter 3 A Two-Component Model for. .. Two-Component Model for Malaria Infected Erythrocytes (c) The plot of projection length vs pressure difference for a ring stage malaria infected erythrocyte (Pipette Diameter = 1.601 µ m) (d) The plot of projection length vs pressure difference for a trophozoite stage malaria infected erythrocyte (Pipette Diameter = 0.990 µ m) 59 Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes (e)... the comparison with other different works 3. 4.1 3. 4.1.1 Geometric Description Normal, Uninfected, Ring Stage and Trophozoite Stage Erythrocytes (a) 61 Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes (b) Figure 3. 11 Geometry of one-eighth of an erythrocyte finite element model for the simulation of optical tweezers stretching experiments, (a) 3D drawing, (b) side view Owing to the plane... 3 3.4.1.2 A Two-Component Model for Malaria Infected Erythrocytes Schizont stage infected erythrocytes Figure 3. 12 Geometry of one-eighth of a schizont stage malaria infected erythrocyte model used in the simulation of optical tweezers stretching experiments Similar to the model used for normal, P.f uninfected, ring stage and trophozoite stage erythrocytes, the model for schizont stage cell was simplified... spheres 3. 4.2 Boundary and loading conditions According to the symmetric cell geometry and loading conditions in the experiment, the initial boundary conditions of the three-dimensional model are shown 63 Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes in Figure 3. 13 (a) The displacement in direction x was applied on the flat surface to stretch the cell, as shown in Figure 3. 13 (b)... step 1 Figure 3. 13 Boundary and loading conditions of the simulation of optical tweezers stretching, in (a) step 0, and (b) step 1 64 Chapter 3 3.4 .3 A Two-Component Model for Malaria Infected Erythrocytes Finite Element Mesh The model was analyzed using ABAQUS Two types of elements were used As shown in Figure 3. 11, the outer surface of the cell, which represented the membrane, used 30 00 S4R shell... trophozoite stage infected erythrocytes ranges from 22.4 µ N/m to 57.1 µ N/m (Figure 3. 19) These results show obvious increasing in the initial shear modulus of erythrocyte membrane during the early progression of parasite infection 69 Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes Figure 3. 20 Relationship between axial and transverse diameters of deformed schizont stage malaria infected. .. 14 .3 µ N/m, 16.6 µN/m, 32 .8 µ N/m and 65.1 µ N/m for normal, P.f uninfected, ring, trophozoite and schizont stage, respectively, showing the increasing of membrane stiffness 72 Chapter 3 3.4.6 A Two-Component Model for Malaria Infected Erythrocytes Computational Results and Comparison with Other Works As reviewed in Chapter 2, computational works have been done by other researchers to study the mechanical. .. of 30 00 F3D4 elements (4-node linear 3- dimensional quadrilateral hydrostatic fluid element), which represented the cytoplasm 3. 4.4 Finite Element Analysis using ABAQUS (a) (b) Contact Area Figure 3. 14 Simulation of optical tweezers stretching of a healthy RBC, (a) ¼ of the RBC without the flat oval surface, (b) 1/8 of the RBC with the flat oval surface 65 Chapter 3 A Two-Component Model for Malaria Infected. .. Two-Component Model for Malaria Infected Erythrocytes The simulation using ABAQUS gives the range of initial shear modulus that can be valid for a single experiment, while the commonly used hemispherical cap model gives a specific value for a single experiment The comparison between these two methods is discussed in the next section 3. 3.5 Comparison between Two Different Computational Models Figure 3. 9 Flow chart . in Yeoh’s form (Yeoh 1990), which is given by Chapter 3 A Two-Component Model for Malaria Infected Erythrocytes 46       3 2 3 2 2 2 130 2 2 3 2 2 2 120 2 3 2 2 2 110 33 3   CCCU . (ABAQUS v6.4). Therefore, Eq. (3. 8) becomes       23 2 2 2 2 2 2 2 2 2 0 1 2 3 20 1 2 3 30 1 2 3 0 3 3 3 2 U C C h                       (3. 14) If we assume the. Model for Malaria Infected Erythrocytes 47 )32 ( 2 1 )( 2 30 201001 ABCABCAChT s     )2 (3) (2)( 2 30 2 2010 0 1 CBABCBCACCC D h   ),(2 ,62 ,2 ),(2 3 11 4 1 2 1 2 1 3 11             D C B A