Mechanical models for malaria infected erythtocytes 5

The simulation results of micropipette aspiration provided the range of membrane shear moduli for each stage of infection, while the commonly used hemispherical cap model provided a sing

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Chapter 5 Study of the Valid Range of Pipette Radius Used in

Micropipette Aspiration of Erythrocytes

5.1 Introduction

In Chapter 3, a two-component model was developed to study the malaria

infected erythrocyte deformation in micropipette aspiration and optical tweezers stretching The simulation results of micropipette aspiration provided the range of membrane shear moduli for each stage of infection, while the commonly used hemispherical cap model provided a single value of membrane shear modulus for each of these stages When we applied the value calculated using hemispherical cap model to the finite element simulation, the simulation curve could be fitted with the experimental data However, due to the wide range of pipette sizes we used in the experiments, it is necessary to study the range of pipette sizes with which the shear modulus was calculated using the hemispherical cap model can be compared with the range of shear modulus obtained from finite element simulation before we proceed to

a more advanced complex model, which will be introduced in Chapter 6

Therefore, the validation range of micropipette radius will be studied and validated in this chapter Parametric studies will be done for different pipette radius and membrane shear modulus

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5.2 Material constitutive relations

The erythrocyte’s model used in this chapter is similar to the one in Chapter 3

The difference was that the material used for modelling the cell membrane was simplified, so that we could reduce the parameters in defining material properties and focus more on the effect of pipette radius

The erythrocyte was again modeled as a Newtonian fluid enclosed by a homogeneous hyperelastic membrane The cytoplasm was modeled as an 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 given in Equation 3.8, and the initial shear modulus of the material is given in Equation 3.13

In this chapter, we simplified Eq 3.8 by setting C 20 = C 30 = 0, where C 20 , C 30

are the temperature-dependent material parameters The strain energy potential was therefore simplified as

0

2

h



where C 10 is the temperature-dependent material parameters, λ 1 , λ 2 , λ 3 are the

principal stretch ratios corresponding to principal axes x 1 , x 2 , x 3, µ 0 is the initial shear

modulus of the material, h 0 is the initial thickness of the membrane

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5.3 Methodology

In our experiments, the radii of micropipettes fabricated were in the range of 10~50% of the cell radius Therefore, in this chapter, we will present the parametric

studies done for p

cell

R

R = 0.1, 0.2, 0.3, 0.4 and 0.5, where Rp is the pipette radius and

Rcell is the cell radius

In hemispherical cap model, when Lp/Rp is about or more than 1, the deformation is directly proportional to the increase in pressure and the relationship

can be approximated using Equation 2.28 (Shu et al 1978)

The workflow chart is given in Figure 5.1 The basic idea was to input an initial shear modulus value for the erythrocyte membrane and run a simulation, and then use hemispherical cap model to analyze the simulation curve and obtain the shear modulus Then we compared the input and output value to determine whether the result of hemispherical cap model was in agreement with the initial value we put into the finite element analysis Different combinations of shear modulus and pipette radius were tested according to the workflow

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Figure 5.1 The workflow in the study of a valid range of pipette radius for the

hemispherical cap model

5.4 Geometric Description

To study the validity range of pipette radius in micropipette aspiration, a series

of test was done for p

cell

R

R = 0.1, 0.2, 0.3, 0.4 and 0.5 Rcell was assumed to be 3.91 µm,

according to the average normal RBC shape measured by Evans and Fung (1972), given in Equation 3.15

Therefore, the pipette radii presented in this chapter would be 0.391 µm, 0.782

µm, 1.173 µm, 1.564 µm, and 1.955 µm

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Figure 5.2 The geometric description of the model used in the study of the valid range of pipette radius in hemispherical cap model

To improve the convergence and save computation time, a small hemispherical cap was introduced to push the cell surface, as shown in Figure 5.2

The radius of the hemispherical cap was in consistence with the pipette radius, which were 0.391 µm, 0.782 µm, 1.173 µm, 1.564 µm, and 1.955 µm, respectively

The details about the usage of the cap are going to be introduced in the next section The fillet radius of the pipette end was 0.4 µm

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5.4.1 Boundary and Loading Conditions

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 5.3 Similar to what was described in the previous chapter, the finite element model was created in x-y plane Points located on the symmetric axis were only allowed to move in direction y

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

Noted that L p /R p ≥ 1 ( where L p is the projection length and R p is the pipette radius) is the requirement to use hemispherical cap model to calculate the membrane

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shear modulus (Shu et al 1978), in the first simulation step, the nearly rigid small hemispherical cap would push the cell surface until the projection length L p into the

micropipette reached the pipette radius R p The small cap would then stop moving and detach from the inner surface of the membrane

In the following steps, a uniformly increasing aspiration pressure was uniformly applied on the part of cell surface that was within the aspiration area of the fixed rigid pipette RP-1 shown in Figure 5.3 represents the reference node of the analytical rigid pipette, which was set to be fixed

5.4.2 Finite Element Mesh

In this model, there were three components: outer membrane, inner cytoplasm, and the small hemispherical cap

The membrane was represented by 449 plane stress axisymmetric 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)

The hemispherical cap was represented by 100 plane stress axisymmetric elements (SAX1)

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5.4.3 Finite Element Analysis Using ABAQUS

Rp/Rcell = 0.1

Rp/Rcell = 0.2

Rp/Rcell = 0.3

Rp/Rcell = 0.4

Erythrocyte Surface

Hemispherical Cap

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Rp/Rcell = 0.5

Figure 5.4 Cell deformation and final position of hemispherical cap in the simulation of micropipette aspiration at different p

cell

R

R

The model was analyzed in ABAQUS 6.4 Similar as we described in Chapter

3, the aspiration pressure started with an initial value of 0 Pa, and increased constantly

to 200 Pa in the simulation The ratio between pipette radius and cell radius ranged from 0.1 to 0.5 Since the average cell radius was reported to be 3.91 µm (Evans and

Fung 1972), the pipette radius varied from 0.391 ~ 1.955 µm The simulation images

were shown in Figure 5.4

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Figure 5.5 Effect of pipette radius Rp on the erythrocyte deformation in micropipette aspiration, with fixed µ0 = 4.8 µN/m, D 0 = 2.6 x10 -19 J

In Figure 5.5, the initial shear modulus µ0 was preset to be 4.8 µN/m The

initial bending modulus D0 was preset to be 2.6 x 10-19 J The material properties remained the same for all the five curves, while the ratio between pipette radius and cell radius varied from 0.1 to 0.5 Since the hemispherical cap pushed the cell surface

to where Lp/Rp = 1, the beginning portion of the curve was parallel to the horizontal axis With the increase of pipette radius, the protrusion length decreased with the same aspiration pressure

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Figure 5.6 Effect of pipette radius Rp on the erythrocyte deformation in micropipette

aspiration, with fixed µ0 = 9.6 µN/m, D 0 = 2.6 x10 -19 J

Similar test was done with the initial shear modulus µ0 set at 9.6 µN/m and

14.4 µN/m, as shown in Figure 5.6 and 5.7 The initial bending modulus D0 was preset to be 2.6 x 10-19 J The ratio between pipette radius and cell radius also varied from 0.1 to 0.5, while the material properties remained the same for all the five curves

From the results, we can see that the protrusion length of the erythrocyte aspirated into the pipette decreased with the increase of pipette radius

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Figure 5.7 Effect of pipette radius Rp on the erythrocyte deformation in micropipette aspiration, with fixed µ0 = 14.4 µN/m, D 0 = 2.6 x10 -19 J

5.5 Discussion

From Figure 5.5~5.7, we can observe that with fixed initial shear modulus, bending stiffness and aspiration pressure, simulation curves obtained using different pipette radius had different slope between the threshold pressure and critical point of aspiration The erythrocyte deformation became smaller when the pipette had a larger radius The next step is to analyze the simulation curves using hemispherical cap model, and determine whether the result matched the input value

The comparison results were shown in Figure 5.8 The horizontal axis represents different ratios between pipette radius and cell radius The blue diamonds

represented the data we analyzed with an input value of 4.8 µN/m The red squares

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represented the data analyzed with an input value of 9.6 µN/m The green triangles represented the data analyzed with an input value of 14.4 µN/m The vertical axis

represents the membrane shear modulus obtained by analyzing simulation curves using the hemispherical cap model

Figure 5.8 Effect of the ratio between pipette radius and cell radius on the validity of the hemispherical cap model

For easier understanding, the blue, red and green dashed lines were drawn for

the constant vertical axis value of 4.8, 9.6 and 14.4 µN/m, respectively The nearer the

data points were to the dashed line in their same colour, the better the hemispherical cap model were able to predict the shear modulus of the membrane The difference between shear modulus calculated using hemispherical cap model and the initial shear

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modulus of the membrane is shown in Table 5.1, the percentage is calculated as

(OutputModulus 1) 100%

InputModulus  

Table 5.1 The difference between shear modulus calculated using hemispherical

cap model and the initial shear modulus of the material

p cell

R R

Input Modulus

Among the blue diamonds (input modulus = 4.8 µN/m), the ones for p

cell

R

R =

0.1, 0.2 and 0.3 all obtained an output modulus that were only 4.5 ~ 4.9 % smaller

than the input modulus The one for p

cell

R

R = 0.4 obtained an output modulus that was

0.40% bigger than the input modulus But the one for p

cell

R

R = 0.5 obtained an output

modulus that was 33.8 % smaller than the input modulus

Among the red squares (input modulus = 9.6 µN/m), the ones for p

cell

R

R = 0.1 ~

0.4 obtained the output modulus that were at most 2.0 % bigger or 2.8% smaller than

the input value, while the one for p

cell

R

R = 0.5 obtained an output modulus that was

33.99 % smaller than the input modulus

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Similarly, among the green triangles (input modulus = 14.4 µN/m), the ones

for p

cell

R

R = 0.1 ~ 0.4 obtained the output modulus that were at most 2.3 % bigger or

2.8% smaller than the input value, while the one for p

cell

R

R = 0.5 obtained an output

modulus that was 33.6 % smaller than the input modulus

The range of micropipette sizes we used in experiments were p

cell

R

R = 0.1 ~ 0.5

It had been shown that when p

cell

R

R = 0.5, the hemispherical cap model were not able

to predict the shear modulus of the material within a 30% error range When p

cell

R

R =

0.1 ~ 0.4, the difference between the value calculated using hemispherical cap model and the input shear modulus of the material did not exceed 5% of the input modulus

It is probably caused by the large pipette size in relation to the aspirated erythrocytes

Therefore, the validity range of micropipette radius used in hemispherical cap

model can be obtained as p

cell

R

R = 0.1 ~ 0.4

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5.6 Membrane Shear Modulus of Malaria Infected Erythrocytes

Calculated Using Hemispherical Cap Model

According to the range of micropipette radius discussed in the last section, we can select experimental data done with suitable pipette sizes and analyze them using hemispherical cap model

Figure 5.9 Membrane shear modulus of malaria infected erythrocyte calculated using

hemispherical cap model, tested at room temperature and body temperature,

respectively

The results of membrane shear modulus of malaria (P.f.) infected erythrocytes are given in Figure 5.9 The black colume and white colume represent the P.f infected

erythrocytes tested at room temperature (20oC) and body temperature (37oC),

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respectively We can see that with the progression of disease stage, the shear modulus

of the cell membrane increased In the schizont stage, the P.f infected erythrocytes

appears to be as 3.8 and 4.6 times as stiff as the uninfected erythrocytes, at room temperature and body temperature, respectively The difference between these two temperatures did not result in significant difference in the membrane shear modulus of

the P.f infected erythrocytes

5.7 Conclusions

In this chapter, the two-compartment model introduced in Chapter 3 was used

to study the effect of pipette radius on the validity of hemispherical cap model in calculating shear modulus of the cell in micropipette aspiration The simulations were done using finite element analysis program ABAQUS The numerical results were found to be insensitive to the mesh parameter changes

As mentioned earlier, the finite element model provides a range of initial shear modulus of the erythrocyte membrane for each experiment, while the hemispherical

cap model provides a single value for each experiment The range of 0.1 ≤ p

cell

R

R ≤ 0.4

was proved valid for using hemispherical cap model in micropipette aspiration For cells that can be assumed as a liquid enclosed by incompressible membrane, it is easier to apply hemispherical cap model due to its simplicity in calculation, and the

results will be valid as long as 0.1 ≤ p

cell

R

R ≤ 0.4