Mechanical models for malaria infected erythtocytes 3

3.3 Simulation of Micropipette Aspiration In this section, simulation of micropipette aspiration will be done using a finite element two-component model.. 49 3.3.1 Geometric Description

parasite will be further discussed in Chapter 7 using a more advanced and complex

model

44

3.2 Material Constitutive Relations

Based on available literature which was reviewed, the cortical-shell core model was adopted The cytoplasm was represented by a homogeneous viscous Newtonian fluid The membrane was defined as a hyperelastic material, which is

liquid-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

where ε 1 , ε 2 are the principal strains, σ 1 , σ 2 are the principal stresses corresponding to

principal axes x 1 , x 2 , the elastic constants E and υ are functions of the strain

45

where λ 3 is the stretch ratio in the direction perpendicular to the membrane

If the area of the RBC membrane is a constant,

12

S

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

46

3 2 2 2 1 30 2 2 3 2 2 2 1 20 2

3 2 2 2 1

where h is the current membrane thickness

The membrane shear stress is given by

)(2

1

2

1 T T

T s  

(3.10)

And the membrane shear modulus is given by

s s T

( 1

(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

47

)3

2(

2

1)

2

,62

,2

),(

2

3 1 1

4 1

2 1 2

1

3 1 1

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

0 10

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

49

3.3.1 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,

Figure 3.2 Model of a normal erythrocyte for finite element analysis, the shape was

measured by Evans and Fung (1972)

50

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

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3.3.1.2 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

52

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 2 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 (R p ) and the projection length (L p), as shown in Figure 3.6

53

Figure 3.6 Schematic diagram of micropipette aspiration

The model was analyzed using ABAQUS 6.4 In the simulation, the aspiration pressure started with an initial value of 0 Pa, and increased to 200 Pa In the experiments, only pressure difference could be recorded, leaving the initial aspiration pressure unknown When the aspiration pressure was larger than a certain threshold pressure, the erythrocyte flowed continuously into the pipette The cell was not able to maintain a stable equilibrium and often involved rapid protrusion, retraction and lysis

of the cell when the aspiration pressure was larger than a certain value, which was called the critical point of aspiration (Derganc et al 2000) Therefore, to compare the simulation result with experimental images obtained, we selected the linear part before the aspiration pressure reached the critical point of aspiration (pressure difference = ∆Plinear ) of each series of experimental data and found the projection length Lpmid and pressure difference ∆Pmid of its middle point, as shown in Figure 3.7(a)

54

(a) Selecting the linear part of the plot of projection length vs pressure difference before the aspiration pressure reached the critical point of aspiration (pressure difference =

∆P linear )

(b) Fitting the period of (P mid - ½∆P linear ) to (P mid + ½∆P linear ) in the simulation curve on

to the experimental data on the plot of projection length vs aspiration pressure Figure 3.7 Schematic diagram of data selection from micropipette aspiration experimental data

55

The value of Lpmid corresponds to Pmid in the simulation curve in Figure 3.7(b) Therefore, the period of (Pmid -½∆Plinear) to (Pmid +½∆Plinear ) in the simulation curve should be selected The membrane mechanical properties were changed until the simulation curve lies in the range of experimental data

Two examples are given in Figure 3.8, for a biconcave shaped normal erythrocyte and a spherical shaped schizont stage malaria infected erythrocyte, respectively

(a)

(b)

Figure 3.8 Comparison of aspirated projection length between experimental and

simulation image for (a) a biconcave shaped normal erythrocyte, and (b) a spherical

shaped schizont stage infected erythrocyte

56

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 of comparing the finite element model with hemispherical cap model and obtaining the membrane shear modulus from simulation

As reviewed in Chapter 2, the hemispherical cap model is commonly used to analyze the membrane shear modulus of erythrocytes As we can see from Eq 2.27 &

57

2.28 (Shu et al 1978), the shear modulus can be calculated from the slope of the

experimental data where the aspiration pressure was between threshold pressure and critical point of aspiration

Before using a more complex model to analyze the complexity and influence

of the internal components of infected erythrocytes, it is important to testify whether the current two-component model is able to obtain results similar to what the commonly used hemispherical cap model produces

The work flow for the computational modeling to extract the change in shear modulus of malaria infected erythrocytes is shown in Figure 3.9 Following this work flow, we were able to obtain simulation results of projection length as a function of

the increase in pressure for normal, uninfected, ring stage, trophozoite stage and

schizont stage erythrocytes One example of each is given in Figure 3.10 where ∆P is

the increase in suction pressure, µ is the shear modulus, Rp is the pipette radius and Lp

is the projection length and C is a constant The initial membrane shear moduli of the

given examples are found to be 7.1 µN/m, 13.1 µN/m, 18.9 µN/m, 35.3 µN/m and

39.2 µN/m, respectively

59

(c) The plot of projection length vs pressure difference for a ring stage malaria infected

erythrocyte (Pipette Diameter = 1.601 µm)

(d) The plot of projection length vs pressure difference for a trophozoite stage malaria

infected erythrocyte (Pipette Diameter = 0.990 µm)

60

(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 5 to study the range of pipette sizes that can be used in the

61

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 Geometric Description

(a)

62

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 1 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

(b)

63

3.4.1.2 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

64

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

65

3.4.3 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

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

(a)

(b)

Contact Area