Mechanical models for malaria infected erythtocytes 4

76 Chapter 4 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 bod

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Chapter 4 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 ~ 7 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

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4.2 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

0

0

2

h

(4.1)

where U is the strain energy potential, µ 0 is the initial shear modulus [N/m] of the

material, h 0 [m] is the initial thickness of the membrane, λ 1 , λ 2 , λ 3 are the principal

stretch ratios corresponding to principal axes x 1, x 2 , x 3 , and C 10 , C 20 , C 30 are the

temperature-dependent material parameters

The shear modulus can be also expressed as the following equation (Dao et al 2003)

where G 0 is the shear modulus [Pa or N/m2]

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0 10

0 2C h



Since

(4.3)

as derived in Eq 3.1~3.13 The shear modulus G is therefore

G0 = 2 C10 (4.4)

For homogeneous isotropic materials, the Young’s modulus E can be calculated using

shear modulus G and Poisson’s ratio ν,

Since we assume the membrane to be incompressible, the Poisson’s ratio is given by

1

2

The initial Young’s modulus of the material is therefore

When the membrane is treated as a thin plate, its bending stiffness is given by

(Fung 1993; Lu et al 2001)

3 2

Eh D





where h is the membrane thickness

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3

2 3

D  C h

Combining Eq 4.6~4.8, we can calculate the initial bending stiffness of the erythrocytes’ membrane using the following equation

(4.9)

Therefore, the strain energy potential can be expressed as

0

3

0

3

2

D

(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

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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 R p 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

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Similar to what was described in Chapter 3, the aspiration pressure started

with an initial value of 0 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 C 10 and h 0 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

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(b) Effect of bending stiffness on the erythrocyte deformation in micropipette aspiration

Figure 4.1 Effect of bending stiffness on the RBC deformation in micropipette aspiration

This combination of C 10 and h 0 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

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by changing the values of C 10 and h 0 In order to obtain a fixed initial shear stiffness

with a changing bending stiffness, both C 10 and h 0 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 R p = 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 D 0 used in different simulation curves

equalled 1/8, 1/4, 1/2, 1, 2 and 5 times of D i , where D i 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 R p = 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 D 0

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 = 0 Pa) could not be detected in the experiments In this parametric study, it was

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determined by fitting the simulation curve (red dashed line D i) for each experiment

P0 was found to be 5 Pa in Figure 4.1(a), and 7 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

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4.4 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 1 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

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4.4.2 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=UR2=UR3=0,

the edge in z-x plane: U2=UR3=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 UR1, UR2,

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

3 (Figure 3.14)

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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

curves were obtained using different initial bending stiffness, equalled 1/8, 1/4, 1/2, 1, 2

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 D 0 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

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research works The six simulation curves were obtained using different initial

bending stiffness, equalled 1/8, 1/4, 1/2, 1, 2 and 5 times of D i , respectively, where D i

was set to be 3 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

simulation curves were obtained using different initial bending stiffness, equalled 1/8,