Mechanical models for malaria infected erythtocytes 4

Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes 4.1 Introduction The human erythrocyte is generally believed to behave like an elastic body. Since the haemoglobin enclosed by the erythrocyte membrane is a liquid, the cell shape recovery after the removal of external forces that induces the shape change is associated with the erythrocyte’s membrane elasticity (Evans 1983). The elastic rigidity of the cell membrane is associated to the change in free energy caused by both the stretch and the bending of the erythrocyte membrane (Evans et al. 1979). If the curvature of a shell is changed by deformation, we must consider the bending stiffness (Fung 1993). The bending stiffness of the erythrocytes was reported to be in the range of 1.8 ~ x 10 -19 J (Evans 1983; Strey et al. 1995; Sleep et al. 1999). In the last chapter, a two-component finite element model was developed to study the effect of initial membrane shear modulus on the deformation of malaria infected erythrocytes in micropipette aspiration and optical tweezers stretching experiments. In this chapter, the effect of bending stiffness on the infected erythrocytes will be discussed using the same model. Parametric studies will be done to determine whether bending stiffness has an impact on the cell deformation during micropipette aspiration and optical tweezers stretching experiments. 76 Chapter 4.2 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Material Constitutive Relations Similar to that in Chapter 3, the erythrocyte was modeled as a Newtonian fluid enclosed by a homogeneous hyperelastic membrane. The haemoglobin was modeled as an incompressible or nearly incompressible fluid with a hydraulic fluid model within a fluid-filled cavity. The constitutive relation of the material used to model the erythrocyte’s membrane is expressed in the form of strain energy potential. The strain energy potential is defined in Yeoh form (Yeoh 1990), which was given in Equation 3.8. As derived in the last chapter (Eq. 3.1 ~ 3.14), Eq. 4.1 can be transformed into U 0 2h0   22  32  3  C20  12  22  32  3  C30  12  22  32  3 (4.1) where U is the strain energy potential, µ0 is the initial shear modulus [N/m] of the material, h0 [m] is the initial thickness of the membrane, λ1, λ2 , λ3 are the principal stretch ratios corresponding to principal axes x1, x2, x3, and C10, C20, C 30 are the temperature-dependent material parameters. . The shear modulus can be also expressed as the following equation (Dao et al. 2003) µ0 = G0 h0 (4.2) where G0 is the shear modulus [Pa or N/m2]. 77 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Since   2C10 h0 (4.3) as derived in Eq. 3.1~3.13. The shear modulus G is therefore G0 = C10 (4.4) For homogeneous isotropic materials, the Young’s modulus E can be calculated using shear modulus G and Poisson’s ratio ν, E = 2G (1+ν) (4.5) Since we assume the membrane to be incompressible, the Poisson’s ratio is given by  (4.6) The initial Young’s modulus of the material is therefore E0= 3G0= C10 (4.7) When the membrane is treated as a thin plate, its bending stiffness is given by (Fung 1993; Lu et al. 2001) D Eh3 12(1  ) (4.8) where h is the membrane thickness. 78 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Combining Eq. 4.6~4.8, we can calculate the initial bending stiffness of the erythrocytes’ membrane using the following equation D0  C10 h03 (4.9) Therefore, the strain energy potential can be expressed as U 3D0   22  32  3  C20  12  22  32  3  C30  12  22  32  3  2h0 (4.10) 4.3 Simulation of Micropipette Aspiration of Erythrocytes In this section, simulation of micropipette aspiration will be done using the finite element two-component model developed in Chapter 3. The geometric description, boundary and loading conditions, finite element mesh will be introduced. The simulations were done using the finite element analysis program ABAQUS. A parametric study of bending stiffness’s influence on the erythrocytes’ deformation during micropipette aspiration will be introduced. 4.3.1 Geometry, Boundary and Loading Conditions The malaria infected erythrocyte model consists of a shell and an enclosed fluid. The average normal RBC shape measured by Evans and Fung (1972) was given in Equation 3.15. To study the effect of bending stiffness on the cell deformation in micropipette aspiration, we performed simulations of malaria infected erythrocytes at different 79 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes infection stages. The cell radius and pipette radius of each experiment were measured individually. In this section, we are going to present two sets of experiment data and their corresponding computational simulations. One is for an uninfected erythrocyte, and the other is for a ring stage malaria infected erythrocyte. The uninfected erythrocytes refer to the cells that are cultured together with infected erythrocytes but have not been invaded by the parasite. For both of these two cases, the erythrocytes maintain their biconcave shape, which is calculated using Equation 3.15. The geometric sketch is shown in Figure 3.3. The cell radius Rcell of these two samples were both measured to be 2.67 µm. The pipette radii Rp were 0.67 µm and 0.8 µm, respectively. The fillet radius of the pipette edge was 0.4 µm. Due to the axisymmetric loading condition and cell geometry, the simulation was simplified as an axisymmetric problem. The boundary and loading conditions are shown in Figure 3.5. Same as was described in Chapter 3, the finite element model was created in the 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 within the aspiration area of the fixed pipette. The blue cross shown in Figure 3.5 represents the reference node of the analytical rigid pipette. 4.3.2 Finite Element Analysis Using ABAQUS The model was analyzed in ABAQUS 6.4. The finite element mesh used in this model was the same as was described in Chapter 3. 80 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Similar to what was described in Chapter 3, the aspiration pressure started with an initial value of Pa, and increased constantly to 200 Pa in the simulation. In Chapter 3, a method was introduced in Figure 3.7 to determine whether the experiment data can be fitted with the simulation curve. This method can also help us to find out the initial aspiration pressure P0 for each experiment. For each set of experiment data, we used the hemispherical cap model to calculate the initial shear modulus of the cell membrane µ0 first. Then we chose suitable C10 and h0 values that can fit µ0 (refer to Equation 4.4) and at the same time lead to a bending stiffness value (refer to Eq. 4.9) in the range we found in literature review (Evans 1983; Strey et al. 1995; Sleep et al. 1999). (a) Effect of bending stiffness on the erythrocyte deformation in micropipette aspiration. The initial bending stiffness D0 used in different simulation curves equalled 1/8, 1/4, 1/2, 1, and times of Di, where Di = 2.6x10-19 J, µ0 = 13.1 µN/m, Rp = 0.67 µm. 81 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes (b) Effect of bending stiffness on the erythrocyte deformation in micropipette aspiration. The initial bending stiffness D0 used in different simulation curves equalled 1/8, 1/4, 1/2, 1, and times of Di, where Di = 2.6x10-19 J, µ0 = 18.9 µN/m, Rp = 0.80 µm. Figure 4.1. Effect of bending stiffness on the RBC deformation in micropipette aspiration. This combination of C10 and h0 were used in the finite element model which was simulated using ABAQUS. We then used the method described in the last chapter (Figure 3.7 and 3.8) to check whether the experiment data could be fitted with the simulation curve. The two sets of data used here for analyzing the effect of bending modulus were the data presented in Figure 3.10 (b) & (c). As we discussed in Section 3.3.5, the simulation results were in agreement with the results calculated using hemispherical cap model. The initial aspiration pressure of the experiments could therefore be determined. We then applied fixed initial shear modulus and initial aspiration pressure, with different bending stiffness 82 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes by changing the values of C10 and h0 . In order to obtain a fixed initial shear stiffness with a changing bending stiffness, both C10 and h0 have to be varied at the same time ( Equations 4.3 & 4.9). The results were shown in Figure 4.1. In Figure 4.1(a), an uninfected erythrocyte with a radius of 2.67 µm were aspirated into a pipette of Rp = 0.67 µm. Its initial shear modulus µ0 was 13.1 µN/m, calculated using hemispherical cap model. In Figure 4.1 (a), the initial shear modulus µ0 were the same in all the six curves, while the initial bending stiffness D0 used in different simulation curves equalled 1/8, 1/4, 1/2, 1, and times of D i, where Di was set to be 2.6 x 10 -19 J to make sure that the range of bending stiffness tested in this parametric study was larger than the one reported by previous studies. Therefore, the initial bending stiffness ranged from 3.3 x 10 -20 J to 1.3 x 10 -18 J. This range of bending stiffness was wider than what was reported by previous studies. In Figure 4.1 (b), a ring stage malaria infected erythrocyte with a radius of 2.67 µm were aspirated into a pipette of which Rp = 0.8 µm. Its initial shear modulus µ0 was 18.9 µN/m, calculated using hemispherical cap model. Similar to Figure 4.1 (a), the bending stiffness was set to be 2.6 x 10 -19 J for the first simulation curve, which was illustrated using the red dashed line (Di) in the graph. The initial shear modulus µ0 were the same in all the six curves, while the initial bending stiffness D0 was changed from 3.3 x 10 -20 J to 1.3 x 10 -18 J. The initial aspiration pressure P0 corresponding to the zero pressure difference (∆P = Pa) could not be detected in the experiments. In this parametric study, it was 83 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes determined by fitting the simulation curve (red dashed line Di) for each experiment. P0 was found to be Pa in Figure 4.1(a), and Pa in Figure 4.1(b). For comparison, these P0 values were used for other curves to study the effect of different bending stiffness. 4.3.3 Discussion From Figure 4.1, we can observe that with fixed initial shear modulus and initial aspiration pressure, simulation curves obtained using different bending stiffness had similar trend and slope. But the erythrocytes’ deformation became smaller when a larger bending stiffness was used. In micropipette aspiration of a thin membrane, the extensional deformation was small everywhere, but the large curvature change was concentrated in small areas which were aspirated into the pipette. The simulation curves shown in Figure 4.1 differed from each other because the curvature change of the aspirated membrane area was affected by the membrane bending stiffness. When the membrane bending stiffness became larger, the curvature change of the erythrocyte membrane induced by micropipette aspiration became smaller, resulting in a smaller projection length. However, if we assume a bigger initial aspiration pressure value for these curves, which means that we shift the curves along the horizontal axis in the negative direction, the simulation curves with large bending stiffness can still be fitted to the experiment data. 84 Chapter 4.4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Simulation of Optical Tweezers Stretching In this section, the effect of bending stiffness on the malaria infected erythrocytes’ deformation during optical tweezers stretching will be studied using the same finite element two-component model similar to that in Chapter 3. 4.4.1 Geometric Description Similar to that in Chapter 3, the two-component finite element model was reduced and represented by only one-eighth of the cell, because of the erythrocyte’s plane symmetric geometry and the axial loading conditions. As shown in Figure 4.2, the three-dimensional model was estimated to be 3.78 μm in direction x and 3.91 μm in direction z. The contact surface where the cell attached to the silica micro beads is modelled as a flat oval region of μm in width and 0.55 μm in height. The cell shape was calculated using Equation 3.14. 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 Chapter 4.4.2 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Boundary and Loading Conditions Similar as in the last chapter (Figure 3.13), the initial boundary conditions are as follows: the edge in y-z plane: U1=UR 2=UR3=0, the edge in z-x plane: U2=UR 3=UR1=0, the edge in x-y plane: U3=UR1=UR2=0. where U1, U2, U3 are the displacement in x, y, z direction, respectively, and UR , UR 2, UR3 are the rotation to x, y, z axes. The coordinates are illustrated in Figure 4.2. The displacement in direction x was applied on the flat oval surface to stretch the erythrocyte to model the stretching of the erythrocyte by the silica beads. 4.4.3 Finite Element Analysis using ABAQUS The model was analyzed in ABAQUS 6.4. The finite element mesh was similar to that modelled in Chapter 3. To study the effect of bending stiffness on the erythrocytes deformation in optical tweezers stretching, we used ABAQUS to conduct finite element analysis, and obtained the diameter changes of the erythrocyte as a function of stretching force. The illustration of the simulation can be found in Chapter (Figure 3.14). 86 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Here we use the initial shear modulus we found in Figure 3.15 for the normal, uninfected, and ring stage malaria infected erythrocytes. As discussed in Chapter 3, the transverse diameter changes of the erythrocyte observed in experiments might not be as accurate as the axial diameter changes. The initial shear moduli used in this chapter were obtained by fitting the axial deformation of the erythrocytes with simulations. 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. In Figure 4.3, a parametric study was done with a fixed initial shear modulus µ0 = 7.6 µN/m, while the initial bending stiffness D0 ranged from 3.75 x 10 -20 J to 1.5 x 10 -18 J, which covered a wider range than the bending stiffness reported by earlier 87 Chapter research works. Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes 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 to make sure that the range of bending stiffness tested in this parametric study was larger than the one reported by previous studies. We can see that with different bending stiffness in this range, the relationship between axial diameter of the cell and the stretching force applied on the cell would not be changed obviously, and it can be fitted with the average experimental data of normal erythrocytes. Figure 4.4. Effect of bending stiffness on the normal erythrocytes’ transverse 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. 88 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Similar test was done on the transverse diameter change of the erythrocyte 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. The bending stiffness of the material showed little effect on the axial diameter change of the cell, but significantly influenced the transverse diameter change of the erythrocytes, as shown in Figure 4.4. The simulation with high bending stiffness and large stretching force did not complete properly due to converging difficulties. The possible explanations would be given in Section 4.4.4. 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 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Similarly, another parametric study was done with a fixed initial shear modulus µ0 = 15.2 µN/m, while the initial bending stiffness D0 also ranged from 3.75 x 10 -20 J to 1.5 x 10 -18 J, as shown in Figure 4.5. The relationship between axial diameter of the cell and the stretching force applied on the cell also did not change with the bending stiffness. All the six curves can be fitted with the average experimental data of uninfected erythrocytes. 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. However, the transverse diameter change of the cell was also affected by the initial bending stiffness, as shown in Figure 4.6. The simulation curves also used 90 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, and times of Di, respectively, where D i was set to be x 10 -19 J. 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. In Figure 4.7 & 4.8, the initial shear modulus µ0 = 17.1 µN/m was used, with the same range of initial bending stiffness as the earlier examples. All the six curves in Figure 4.7 can be fitted with the average experimental data of ring stage malaria infected erythrocytes. Similar to the earlier examples, the bending stiffness had little 91 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes effect on the axial diameter change but more effect on the transverse diameter change, as shown in Figure 4.8. 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. 4.4.4 Discussion From Figures 4.3~4.8, we can observe that with fixed initial shear modulus, the initial bending stiffness had little effect on the relationship between axial cell 92 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes diameter and stretching force. But the relationship between transverse cell diameter and stretching force varies with the changing initial bending stiffness. (a) (b) (c) (d) (e) (f) 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 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes The simulation images with an initial bending stiffness that ranged from 3.75 x 10 -20 J to 1.5 x 10 -18 J were given in Figure 4.9. When the cell was stretched along the axial direction, the cell tended to fold in the transverse direction. Since the folding in transverse direction changed the curvature of the membrane, it would be influenced by the bending stiffness. The higher the initial bending stiffness was, the more difficult the cell seems to be able to fold, resulting in a smaller transverse diameter change of the stretched erythrocyte. When the bending stiffness is high and the stretching force is large, the cell could not fold easily along transverse direction, therefore simulations with these conditions could not complete properly. This might be able to explain why the bending stiffness had a larger effect on the transverse diameter change than on the axial diameter change of the erythrocytes. 4.5 Conclusions In this chapter, the two-component model introduced in Chapter was used to study the effect of bending stiffness on the erythrocytes deformation in micropipette aspiration and optical tweezers stretching. The simulation was performed using the finite element analysis program ABAQUS. The numerical results were found to be insensitive to the mesh parameter changes. In micropipette aspiration, parametric studies were done using a range of 3.3 x 10 -20 J to 1.3 x 10 -18 J for the initial bending stiffness D0 , with fixed initial shear modulus. This range covered the bending stiffness of erythrocytes reported by other researchers. The bending stiffness was found to have little effect on the trend and 94 Chapter Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes slope of the simulation curves, which made all the curves able to be fitted with the experimental data by changing the preset initial aspiration pressure. In optical tweezers stretching, parametric studies were done in the range of 3.75 x 10 -20 J to 1.5 x 10 -18 J for the initial bending stiffness D 0, with fixed initial shear modulus. This range also covered the bending stiffness of erythrocytes reported by other researchers. The bending stiffness was found to have little effect on the relationship between axial diameter and stretching force, but influenced the relationship between transverse diameter and stretching force. 95 [...]... diameter changes of the erythrocyte as a function of stretching force The illustration of the simulation can be found in Chapter 3 (Figure 3. 14) 86 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Here we use the initial shear modulus we found in Figure 3.15 for the normal, uninfected, and ring stage malaria infected erythrocytes As discussed in Chapter 3, the transverse... transverse diameter change of the erythrocytes, as shown in Figure 4. 4 The simulation with high bending stiffness and large stretching force did not complete properly due to converging difficulties The possible explanations would be given in Section 4. 4 .4 Figure 4. 5 Effect of bending stiffness on the uninfected erythrocytes’ axial deformation in optical tweezers stretching The initial shear modulus... curves in Figure 4. 7 can be fitted with the average experimental data of ring stage malaria infected erythrocytes Similar to the earlier examples, the bending stiffness had little 91 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes effect on the axial diameter change but more effect on the transverse diameter change, as shown in Figure 4. 8 Figure 4. 8 Effect of... 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 However, the transverse diameter change of the cell was also affected by the initial bending stiffness, as shown in Figure 4. 6 The simulation curves also used 90 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes different initial bending stiffness, equalled 1/8, 1 /4, 1/2,... stiffness on the ring stage erythrocytes’ transverse deformation in optical tweezers stretching The initial shear modulus µ = 17.1 µ N/m 0 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 4. 4 .4 Discussion From Figures 4. 3 ~4. 8, we can observe that with fixed initial shear modulus,... had little effect on the relationship between axial cell 92 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes diameter and stretching force But the relationship between transverse cell diameter and stretching force varies with the changing initial bending stiffness (a) (b) (c) (d) (e) (f) Figure 4. 9 Effect of bending stiffness on the shape change in the simulation...Chapter 4 4 .4. 2 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Boundary and Loading Conditions Similar as in the last chapter (Figure 3.13), the initial boundary conditions are as follows: the edge in y-z plane:... 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 89 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes Similarly, another parametric study was done with a fixed initial shear modulus µ = 15.2 µ N/m, while the initial bending stiffness D0 also ranged from 3.75 0 x 10 -20 J to 1.5 x 10 -18 J, as shown in Figure 4. 5 The relationship... effect on the trend and 94 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes slope of the simulation curves, which made all the curves able to be fitted with the experimental data by changing the preset initial aspiration pressure In optical tweezers stretching, parametric studies were done in the range of 3.75 x 10 -20 J to 1.5 x 10 -18 J for the initial bending... 4. 5 The relationship between axial diameter of the cell and the stretching force applied on the cell also did not change with the bending stiffness All the six curves can be fitted with the average experimental data of uninfected erythrocytes Figure 4. 6 Effect of bending stiffness on the uninfected erythrocytes’ transverse deformation in optical tweezers stretching The initial shear modulus µ = 15.2 . Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes 76 Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes 4. 1. cell deformation in micropipette aspiration, we performed simulations of malaria infected erythrocytes at different Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected. ) Eh D    (4. 8) where h is the membrane thickness. Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes 79 3 0 10 0 2 3 D C h Combining Eq. 4. 6 ~4. 8,