Mechanical models for malaria infected erythtocytes 6

In Chapter 3, 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 micropipett

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As introduced in Chapter 1, the human erythrocyte is generally believed to

behave like an elastic body, consisting of haemoglobin and erythrocyte membrane, but structural changes do occur within the host erythrocyte after the invasion of

malaria (P.f) parasite In the ring stage, the parasitophorous vacuole (PV) encloses the

parasites and forms a ring-like structure The parasitophorous vacuole membrane (PVM) is semi-permeable allowing the parasite to secrete and gain nutrient In the trophozoite stage, the parasite continue growing and occupy nearly 40% of the host erythrocyte’s volume The membranous extensions of PVM form erythrocyte

membrane-tethered Maurer’s clefts (MCs) (Haldar et al 2002) and a tubulovesicular

network (TMN) The hemozoin crystals deposit in the food vacuole (FV), making the hemoglobin digestion products visible Knobs (K) are induced on the cell surface In the schizont stage The parasite divides itself and forms 16-32 daughter merozoites surrounded by the PVM These daughter merozoites will burst out and invade uninfected erythrocytes

In Chapter 3, 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, and quantified the initial shear modulus of erythrocytes at the different infection

Chapter 6

A Multi-component Model for the Malaria Infected

Erythrocyte

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stages In this chapter, a multi-component model will be developed to account for the structural changes occurring within the host cell due to the parasite invasion Parametric studies will be done to investigate the effect of parasite inclusion on the cell deformation during micropipette aspiration and optical tweezers stretching experiments

Different from earlier chapters, a multi-component model was used The haemoglobin was also modeled as an incompressible or nearly incompressible fluid with a hydraulic fluid model within a fluid-filled cavity

The constitutive relation of the materials used to model the erythrocyte’s membrane and PVM was expressed in the form of strain energy potential The strain energy potential was defined in Yeoh’s form (Yeoh 1990), which was given in Equation 3.8

As derived in the Chapter 3 (Eq 3.1 ~ 3.13), if we assume that the material is incompressible and has a constant surface area, the initial shear modulus of the material was given in Equation 3.13

As in the previous chapter, the material used for modelling the cell membrane and PVM was simplified, so that we could reduce the parameters in defining material properties and focus more on the effect of parasite inclusion We reduced the

parameters 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

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0

1 2 3 0

32

U

h

   

where λ 1 , λ 2 , λ 3 are the principal stretch ratios corresponding to principal axes x1, x 2 , x 3,

µ 0 is the initial shear modulus of the material, h0 is the initial thickness of the

membrane

In this section, simulation of micropipette aspiration will be done using the finite element multi-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 effect of parasite location, PVM size and rigidity and the inclusion of cytoplasm on the erythrocyte deformation during micropipette aspiration will be introduced

6.3.1 Geometric Description of Micropipette Aspiration

Figure 6.1 The geometric sketch of the simulation of cell deformation during micropipette aspiration using multi-component model

Figure 6.1 is an illustration of the model’s basic geometry The radius of the host cell, PVM and pipette will be measured and resized for each specific experiment The location of PVM also varies from case to case The fillet radius of the pipette is 0.4 µm

6.3.2 Boundary and Loading Conditions

Similar to the two-component model, due to the axisymmetric loading condition and cell geometry, the simulation was simplified as an axisymmetric problem The finite element model was created in the x-y plane The nodes located on the symmetric axis 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 rigid and fixed pipette The interaction between membrane and PVM will be discussed later in the chapter

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6.3.3 Finite Element Mesh

Similar to the two-component model, the membrane was represented by 449 plane stress axisymmetric elements (SAX1), and the fluid cavity was represented by

449 2-node linear hydrostatic fluid elements (FAX2)

The PVM was also represented by 449 plane stress axisymmetric elements (SAX1)

6.3.4 Finite Element Analysis Using ABAQUS

The model was analyzed in ABAQUS 6.4 Similar to what was described in

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

uniformly to 200 Pa in the simulation When the membrane of the host cell deformed, the relative position of the PVM to the membrane was fixed A method was

introduced in Chapter 3 (Figure 3.7) to determine whether the experiment data can be

fitted with the simulation curve

Figure 6.2 shows the comparison between experiment image and simulation

using ABAQUS A trophozoite stage malaria (P.f) infected erythrocyte with a radius

of 3.83 µm were aspirated into a pipette with R p = 0.61 µm Its shear modulus µ0 was

33.75 µN/m, calculated using hemispherical cap model The relationship between projection length Lp and pressure difference ∆P is plotted in Figure 6.3 for both experimental data and simulation curve The simulation was done by assuming that µ0

of the host cell membrane was equal to the value given by hemispherical cap model

The µ0 of PVM varied from 1 µN/m to nearly rigid, and the results showed that the

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PVM with this size and location did not affect the deformation of the host erythrocyte

in micropipette aspiration

Figure 6.2 Comparison between experiment image and simulation image of a trophozoite

stage malaria (P.f) infected erythrocyte with a radius of 3.83 µm being aspirated into a pipette with R p = 0.61 µm

Figure 6.3 Relationship between projection length L p and pressure difference ∆P of a trophozoite stage malaria (P.f) infected erythrocyte with a radius of 3.83 µm being aspirated into a pipette with R p = 0.61 µm, analyzed using multi-component model

is plotted in Figure 6.6 for both experimental data and simulation curve

Figure 6.4 Experiment Image of a schizont stage malaria erythrocyte obtained from

experiment, Rp = 0.73 µm, Rcell = 3.43 µm, RPVM = 2.52 µm

Figure 6.5 Finite element model of a schizont stage malaria erythrocyte being aspirated

into a micropipette, Rp = 0.73 µm, Rcell = 3.43 µm, RPVM = 2.52 µm

Figure 6.6 Relationship between projection length L p and pressure difference ∆P of a

trophozoite stage cell, analyzed using multi-component model

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6.3.5 Effect of Enclosed Fluid

The enclosed fluid in the multi-component model also affects the model’s deformation in the simulation of the micropipette aspiration An example is shown in Figure 6.7 Assuming that the experiment started at the same initial aspiration pressure, the deformation of the cell without enclosed fluid was bigger than the one with enclosed fluid In the given example, the shear modulus of PVM did not affect the deformation of the host cell in micropipette aspiration

Figure 6.7 Effect of enclosed fluid on a trophozoite stage cell’s deformation in micropipette aspiration (R p = 0.61 µm, R cell = 3.83 µm, R PVM = 1.92 µm)

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6.3.6 Effect of Parasite Location

Figure 6.8 Illustration of probing position: section A and section B defined for micropipette

Increasing Aspiration Pressure

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In this section, the multi-component model will be used to study the deformation of the cells that had a parasite located near to the pipette To discuss this problem clearly, we define the section of the membrane that is very close to the PVM

as section B, and the section away from the PVM as section A, as illustrated in Figure 6.8

From the experiments, it can be observed that if we probe section B of the host cell surface, with increase in aspiration pressure, the PVM also deformed within the host cell An example is shown in Figure 6.9

Noticing the PVM’s deformation in micropipette aspiration, we did a series of tests to probe the trophozoite and schizont stage cells at both section A and section B The results calculated using hemispherical cap model are shown in Figure 6.10

Figure 6.10 The effect of probing positions on the calculated shear modulus using the hemispherical cap model in micropipette aspiration

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Every cell was probed at both section A and section B The results showed that if we probe the same cell at section B, the shear modulus of the host cell membrane given by hemispherical cap model was always higher than the ones probed

at section A One possible contributing factor was that the PVM was also aspirated into the pipette and affected the deformation of the host erythrocytes, which can be proven by Figure 6.9

Figure 6.11 The effect of probing locations on studying the same trophozoite stage malaria infected erythrocyte The cell radius was measured as 3.09 µm The radius of the PVM

was measured as 1.67 µm The initial shear modulus of the host cell µ 0 (cell) = 15.78

µN/m

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To study the effect of probing location of the cell surface on studying the same cell, the multi-component model was used to simulate this process, as shown in Figure 6.11 The example was given for a trophozoite stage malaria infected erythrocyte The cell radius was measured as 3.09 µm The radius of the PVM was

measured as 1.67 µm The inner and outer radius of the micropipette was measured as

0.71 µm and 1.41 µm, respectively When we probed this cell at section A, the

hemispherical cap model gave a shear modulus value of 15.78 µN/m When we probe

the same cell at section B, the hemispherical cap model gave a shear modulus value of 26.11 µN/m In this graph, the preset initial shear modulus of the host cell µ 0 (cell) = 15.78 µN/m, given by calculation using hemispherical cap model at section A The initial shear modulus of the PVM µ 0 (PVM) changed from 1 ~ 10 times of µ 0 (cell)

In this simulation, all the four curves obtained by simulation of probing section A overlapped each other, and the simulation agreed well with that of the experimental probing the same section This implied that the initial shear modulus of the host cell µ 0 (cell) obtained by probing section A was suitable for describing the

host cell membrane stiffness Therefore, we applied this value obtained by probing section A to the finite element simulation of probing the host cell at section B Similar

as in Chapter 5, a hemispherical cap model was used to push the host cell ( probing

section A) or PVM ( probing section B) into the pipette until Lp = Rp The PVM and

host cell membrane elements did not separate after contact

The simulation results showed that if the host cell was probed at section B, the PVM affect the deformation of the host cell The black solid curve and red dashed curve shared the same mechanical properties of the host cell membrane and PVM, but

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they differed from each other due to their different probing location during the

simulation When the µ 0 (PVM) value ranged from 1 ~ 2 times of µ 0 (cell), the simulation curves can be fitted with the experimental data When the µ 0 (PVM) is 5 or

10 times of the µ 0 (cell), there was no obvious cell deformation in the simulation

From these results, we can see that if the cell was probed at section B, the PVM affected the cell deformation, and thus affected the validity of using the hemispherical cap model, which assumed the cell to be a fluid enclosed by membrane only But if the cell was probed at section A, hemispherical cap model was still able to predict the shear modulus of the host cell membrane

6.3.7 Effect of PVM Rigidity

Figure 6.12 The experimental and simulation images of the micropipette aspiration of a

trophozoite stage malaria infected erythrocyte (Rp = 0.67 µm, Rcell = 4.23 µm, RPVM = 2.52

µm)

In section 6.3.6, it was shown that with different probing locations on the host cell surface, the PVM affected the cell deformation, and thus affected the validity of using hemispherical cap model When we probed the host cell membrane at where the

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membrane and PVM were very close to each other (defined as section B), the PVM had an influence on the host cell’s deformation, and the deformation would be

changed with different µ 0 (PVM) value

Therefore, in this section, more computational tests would be done to study the effect of PVM stiffness on the cell’s deformation in micropipette aspiration The cell

radius was measured as 4.23 µm The radius of the PVM was measured as 2.52 µm The inner and outer radius of the micropipette was measured as 0.67 µm and 1.63 µm,

respectively, as shown in Figure 6.12 When we probed this cell at section A, the

hemispherical cap model gave a shear modulus value of 23.33 µN/m, and this value will be used for µ 0 (cell) in this parametric study

The simulation data or probing section B of this cell was plotted in the form of



 The data shown the projection length obtained when

the aspiration pressure reached 100 Pa Since the µ 0 (cell) was fixed with the value we obtained probing section A, only µ 0 (PVM) was changed in this parametric study



 varied from 1~100 The projection length

(aspiration pressure = 110 Pa) decreased with the 0

0

PVM cell



 equals to 5 and 100, the cell deformation

reached an Lp of 0.76 µm and 0.71 µm, respectively The decreasing rate of projection



 dropped as shown in Figure 6.13

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Figure 6.13 Effect of μ 0(PVM) on the host cell deformation, with a fixed μ 0 (cell) value, (a)

μ 0 (PVM)/μ 0 (cell) varied from 1~100, (b) an enlarged plot of μ 0 (PVM)/μ 0(cell) varied from 1~5

In this section, simulation of optical tweezers stretching will be done using the finite element multi-component model The geometric description, boundary and

(a)

(b)