Mechanical models for malaria infected erythtocytes 2

Chapter Literature Review Chapter Literature Review Both experimental techniques and computational models have been used to study the mechanical properties of biological cells. In this chapter, several experimental works on probing the mechanical properties of malaria infected erythrocytes will be introduced. Some of the existing developed computational models for both healthy and malaria infected erythrocytes will also be presented. 2.1 Experimental Works on Probing Mechanical Property Changes in Malaria Infected Erythrocytes Since the emergence of cell mechanics in the 1960s (Fung et al. 1968; Fung 1990; Fung 1993; Fung 1997), the experimental techniques for investigating the mechanical properties of biological cells have developed very rapidly (Merkel 2001; Missirlis et al. 2002; Van Vliet et al. 2003; Huang et al. 2004). To investigate the deformability of cells, tools and techniques such as cell filtration, rheoscope, microfluidics, micropipette aspiration and optical tweezers have been applied. In this chapter, some experimental works carried out to investigate the mechanical properties of erythocytes, especially malaria-infected erythrocytes will now be introduced. 10 Chapter 2.1.1 Literature Review Experimental Works on Probing Overall Cell Deformability 2.1.1.1 Cell Filtration Flow characteristics of cells through polycarbonate sieves can be studied using the constant-pressure method of cell filtration (Gregersen et al. 1967). A polycarbonate sieve with a mean pore diameter of 5µm is placed in a filter holder connected to a pressure transducer and a syringe, which is driven by an infusion pump (Miller et al. 1971). The pressure-time curve can be recorded to study the filtration pressure under a constant flow rate. Cell filtration has been applied in studying the effect of Plasmodium knowlesi on the flow of infected erythrocytes through small sieves. The researchers plotted the relationship between pressure and time for Ringer’s solution and suspension of infected and uninfected erythrocytes, and showed that the maturation of parasite influenced the flow of erythrocytes through the polycarbonate sieves. 2.1.1.2 Rheoscope Rheoscope is able to discriminate between the erythrocyte deformability distributions (RBC-DD) of density separated cells (Dobbe et al. 2002). The basic setup is shown in Figure 2.1 (a). Between two counter-rotating parallel plates, the red blood cells are deformed by simple shear flow within the plate-plate chamber. Rheoscope has been applied in studying diseases which contain anomalous fraction of less deformable or rigid cells, such as sickle cell disease, malaria tropica and hereditary elliptocytosis. Figure 2.1 (b & c) plots the deformability distribution of control sample, infected blood sample, and the parasitized cells, illustrating that the parasitized fraction contains 53% rigid and 47% deformable cells. 11 Chapter Literature Review Figure 2.1. The basic setup of an automated rheoscope and the deformability distribution of erythrocytes in a blood sample cultured with malaria tropica. (a) A dilute erythrocyte suspension is subjected to the shear flow between two counter-rotating plates. The upper glass plate is below a water bath which controls the temperature of the suspension. The erythrocytes are observed to be elongated with the inverted microscope. (b) Control sample and infected blood sample. (c) Fraction of parasitized cells. δ: deformability index; f: probability density; n: number of red blood cells, reprinted from (Dobbe et al. 2002) with permission. The mean deformability index may decrease due to either the presence of a small portion of less deformable cells or the slight overall deformability decrease. Comparing to other measuring techniques which only provide the mean deformability index, the rheoscope is able to measure the deformability distribution of the cells, 12 Chapter Literature Review which is helpful in determining the abnormal mechanical property of the cell population. 2.1.1.3 Microfluidics Figure 2.2. Schematic illustrating the geometry of the microchannel. The white arrow represents the direction of fluid flow, reprinted from (Shelby et al. 2003) with permission. Shelby et al. used microfluidics to mimic the capillary blood flow and test the cell stiffness effect on capillary blockage by malaria-infected erythrocytes (Shelby et al. 2003). Due to their ability to replicate and control micro-environments efficiently, microfluidics have been largely applied in biology, medicine and biochemistry. To meet the need for mimicking capillary micro-environment, capillary-like channel systems have been designed in different substrates such as silicon (Sutton et al. 1997), glass (Cokelet et al. 1993), and PDMS (Duffy et al. 1998). Figure 2.2 illustrates the geometry of a microchannel made of PDMS, mimicking capillaries between to microns in diameter (Shelby et al. 2003). The elastic modulus of PDMS can be adjusted to provide a good approximation of the material properties of capillaries (McDonald et al. 2002). 13 Chapter Literature Review Figure 2.3. Passage of malaria-infected erythrocytes at different stages through microfluidic channels. (A-D) Ring stage infected erythrocytes could pass through channels of all the sizes. (E-L) early and late trophozoite stage infected erythrocytes could only pass through the and 8µm channels but not the and 4µm channels. (M-P) Schizont stage infected erythrocytes could only pass through the 8µm channel. The flow direction is indicated by the white arrows, reprinted from (Shelby et al. 2003) with permission. Being able to control pressure, temperature and flow rate, microfluidics can closely mimic the physiological micro-environment. This can be realized by integrating the system into comprehensive testing platforms. One of the works which was closely related to our research objective was the microfluidic device which was fabricated out of PDMS to study malaria infected erythrocytes, as shown in Figure 2.3. It was shown that late stage malaria infected erythrocytes had difficulty in passing through extremely narrow channels. 14 Chapter 2.1.2 Literature Review Experimental Works on Probing Membrane Mechanical Property Change (a) Micropipette Aspiration Micropipette aspiration has been applied to measure the membrane elasticity of different cells, such as leukocytes and erythrocytes (Rand et al. 1964; Dong et al. 1988; Hochmuth 2000; Shao et al. 2004). It has also been used to measure the deformability of single erythrocytes at different stages of malaria progression (Zhou et al. 2004; Lim et al. 2006) . When the cell is aspirated into a micropipette, measuring of the leading edge of the aspirated portion of the cell can be used in evaluating the elastic modulus of the cell. Micropipette aspiration has its advantage of exerting a wide range of aspiration pressure to induce deformation on a specific section of the cell membrane. The deformability of erythrocytes was found to decrease progressively as the parasite matures within the cell, as shown in Figure 2.4. Figure 2.4. Mechanical probing of the various disease states of a malaria-infected erythrocyte using the micropipette aspiration technique, reprinted from (Lim et al. 2006) with permission. 15 Chapter Literature Review In Chapter 1, malarial proteins on the membrane surface of the erythrocytes are introduced, such as KAHRP, PfEMP1 and PfEMP2. Another protein is PfEMP3, which is associated with the spectrin network of erythrocyte (Pasloske et al. 1993; Kyes et al. 1999). In order to analyze the effect of KAHRP or PfEMP3 on the decreased deformability of malaria infected erythrocytes, Glenister et al. used micropipette aspiration to measure the membrane shear elastic modulus of normal, uninfected and parasitized cells (Glenister et al. 2002), as shown in Figure 2.5. It is suggested that the decrease in deformability of uninfected erythrocytes may be due to the exo-antigens released by mature parasites. KAHRP and PfEMP3 knockouts exhibit less stiffening effects on the cell membrane when compared to normal infected erythrocytes. The uninfected erythrocytes referred to the cells that were cultured together with malaria infected cells but not invaded by parasites. Figure 2.5. Effect of proteins KAHRP and PfEMP3 on the membrane shear elastic modulus of erythrocytes infected by mature stages of P. falciparum, reprinted from (Glenister et al. 2002) with permission. 16 Chapter Literature Review (b) Optical Tweezers Stretching Optical tweezers were first developed by Arthur Ashkin and his co-workers in the early 1970s, and applied in a wide range of experiments, from cooling and trapping of neutral atoms to manipulating live bacteria and viruses (Ashkin et al. 1985; Ashkin et al. 1987). It has also been used to manipulate a single cell and probe the elasticity of cellular components such as cytoskeleton (Lenormand et al. 2003) and membrane (Sleep et al. 1999). The mechanical properties of DNA (Bustamante et al. 2003) as well as that of whole cells (Henon et al. 1999; Barjas-Castro et al. 2002; Brandao et al. 2003; Lim et al. 2004), and protein-protein interaction forces (Litvinov et al. 2002) has also been studied. Optical tweezers has been applied in testing force-displacement responses of P.f. infected erythrocytes (Lim et al. 2006), as shown in Figure 2.6. Two silica beads were attached to the opposite ends of the cell surface. Laser beams were used to control the silica beads to stretch the cell. The results showed variations in axial and transverse diameters of erythrocyte over the erythrocytic developmental stages of the parasite development: healthy, uninfected, ring stage, trophozoite and schizont (H-RBC, Pf-U-RBC, Pf-R-pRBC, PfT-pRBC and Pf-S-pRBC) in PBS solution at room temperature. The ability of the parasitized cell to deform in both the axial and transverse directions is progressively reduced with erythrocytic development of the parasite. The experimental results indicated significant stiffening of the erythrocyte with the maturation of the parasite from the ring to the schizont stage. 17 Chapter Literature Review (c) Figure 2.6. Sketch and optical image of optical tweezers stretching experiment, (a) original shape and (b) deformed shape of the erythrocytes, the axial and transverse diameters of the cell were compared between simulation and experiments as a function of stretching force. (c) Optical image of malaria-infected erythrocytes at different stages stretched using optical tweezers at room temperature, reprinted from (Lim et al., 2006) with permission. 2.2 Modelling of Erythrocytes Vast literatures are available on computational modeling in single cell mechanics. They served well to quantitatively evaluate the mechanical properties as well as to investigate the cellular responses to external stimulations. Different approaches have been used while all have a single common objective, which is how 18 Chapter Literature Review best to fit the experimentally observed phenomena with suitable choices of computational models and/or material properties. 2.2 .1 Mechanical Models of Living Cells (a) Overview The mechanical models for living cells developed by various researchers were reviewed by Lim et al. (Lim et al. 2006), as shown in Figure 2.7. Either the continuum approach or the micro/nanostructural approach can be used to develop the models. The continuum approach regards a living cell as comprising a continuum material of which the constitutive material model and related parameters are derived from experiments. While the micro/nanostructural approach regards the cytoskeleton as the main structural component. In comparison, the micro/nanostructural approach can probe a more detailed molecular mechanical change in the cell membrane and spectrin network, but the continuum approach provides an overall distribution of stress and strain on the cell, which can improve the micro/nano structural model by computing the distribution and transmission of the induced forces to cytoskeletal and subcellular level. 19 Chapter Literature Review of erythrocyte is suggested to have the cross section as shown in Figure 2.13 (a) (Fung et al. 1968), while the average erythrocyte shape was measured to have the cross section as shown in Figure 2.13 (b) (Evans et al. 1972). Figure 2.13. (a) Unstressed shape of a normal erythrocyte. (Fung et al. 1968) (b) Average normal erythrocyte shape measured (Evans et al. 1972), reprinted from (Evans et al. 1972) with permission. Early studies indicated that erythrocytes not move in an axisymmetric orientation during their passage through cylindrical capillaries (Guest et al. 1963). An edge-on orientation was reported (Skalak et al. 1969) and the shape of the erythrocyte becomes more nearly axisymmetric at high velocity (Hochmuth et al. 1970). 27 Chapter Literature Review Figure 2.14. Cell shapes show streamlines of membrane motion associated with tanktreading. (a) 5μm tube; (b) 6μm tube; (c) 7μm tube, reprinted from (Secomb et al. 1989) with permission. Based on the axisymmetric shape, erythrocytes can also be analyzed which consists of an approximately axisymmetric convex rounded front, a cylindrical region and a markedly asymmetric concave rear, as shown in Figure 2.14 (Secomb et al. 1989). 28 Chapter Literature Review (b) Constitutive response The cell can be assumed to be composed of an elastic shell filled with incompressible fluid (Zarda et al. 1977). The constitutive equations for erythrocyte membrane can be given (Skalak et al. 1973) as follows: Under small stresses, the membrane area can be assumed to be constant and T1  Wm T  1 (2.10) T2  Wm T 1  (2.11) where T is an isotropic membrane tension. 1 and 2 are the principal stretch ratios in the meridian and circumferential directions, respectively. Wm is the strain energy per unit of initial area, which can be assumed to be a function of two dimensional invariants I  12  22  (2.12) I  12 22  (2.13) With large stresses and the membrane area changes, the full strain energy function for the erythrocyte material is used: Wm  B C ( I  I  I )  I 22 (2.14) 29 Chapter Literature Review where B  0.5 102 dyn/cm and C=100 dyn/cm. If the membrane is assumed to be incompressible, Wm is given by the first term of (2.14), provided that dAf dA0  12  (2.15) where Af and A0 are final and initial areas, respectively. Other variations of continuum constitutive models have been applied to analyze the large deformation of a erythrocyte, such as the hyperelastic effective material model proposed by Lim et al. (Lim et al. 2004). An incompressible neoHookean form will be the simplest first-order formulation for such case. The strain energy function (Simo et al. 1984) is given by U G0 (1  22  32  3) (2.16) which defines a nonlinear elastic stress-strain behavior, where G is the initial bulk shear modulus and i (i  1, 2, 3) are the principal stretch ratios. If the membrane area remains constant, 1 3  . If the hyperelastic cell membrane is thin and subjected to a uniaxial stretch, the stresses can be derived from equation (2.16), T1  h U  G0 h0 (11.5  11.5 ) 1 T2  (2.17) (2.18) where h and h0 are the current and initial membrane thickness, respectively. The membrane shear stress Ts under such uniaxial stretch is 30 Chapter Literature Review Ts  1 (T1  T2 )  G0 h0 (11.5  11.5 ) 2 (2.19) The membrane shear modulus is given by  (1 )  Ts 3G0 h0 (10.5  12.5 )   s 4(1  13 ) (2.20) where  s is the shear strain. The computational simulations presented by Mills et al. (Mills et al. 2004) used a neo-Hookean hyperelastic material with the constitutive response. Before reaching the third stage modulus  f , the membrane elasticity modulus decreases from the initial shear modulus  to the second stage large deformation modulus  l . The stress-strain slope initially decreases and eventually increases rapidly with the shear strain, as shown in Figure 2.15. 31 Chapter Literature Review Figure 2.15. Hyperelastic constitutive response used in some of the computational models. (a) Uniaxial stress-strain relationships. (b) Membrane shear modulus as a function of shear strain in the first order hyperelastic model. (c) Membrane shear modulus as a function of shear strain in a higher order hyperelastic model. Reprinted from (Mills et al. 2004) with permission. Some researchers used a finite-temperature particle-dynamics simulation to establish a mechanical property model of the erythrocyte plasma membrane (Parker et al. 1999; Marcelli et al. 2005), and compared their results for shear modulus and bending modulus with experimental results obtained from different measuring technique. This comparison illustrates the possibility of relating the elastic property of erythrocytes to thermal fluctuation, as shown in Figure 2.16. 32 Chapter Literature Review Figure 2.16. Comparison between experimental results and values of shear modulus µ and bending modulus B calculated in finite-temperature particle-dynamics simulation, where k is the spring constant, reprinted from ( Marcelli et al., 2005) with permission. (c ) Comments Many early efforts have been made on modeling normal erythrocytes. It is shown that the healthy erythrocyte can be modeled as a fluid enclosed by a shell, the large deformation of which can be simulated by adopting a hyperelastic material model for the membrane. 33 Chapter Literature Review 2.2.3 Modeling of Plasmodium falciparum Infected Erythrocytes P.f. infected erythrocytes have been modeled as they are aspirated into a micropipette (Zhou et al. 2004) and as they are stretched by the optical tweezers (Suresh et al. 2005). 2.2.3.1 Shape of Infected Erythrocytes For healthy erythrocytes, ring stage P.f. infected erythrocytes and trophozoite stage P.f. infected erythrocytes, the shape was assumed to be biconcave, and the initial diameter of the cell is taken to be 7.0-7.8 m according to experimental observations. As shown in Chapter 1.1, the late-stage infected erythrocytes becomes spherical, so only the schizont stage erythrocytes is assumed to have a spherical shape (Suresh et al. 2005). 2.2.3.2 Constitutive Equations There are two kinds of models for late stage P.f. infected erythrocytes. One is to consider the cell as a membrane filled with incompressible fluid (Suresh et al. 2005). Owing to the highly nonlinear elastic response, a two-parameter third-order hyperelastic model was used in modeling the large deformation of the erythrocyte, where the strain energy potential is U G0 (1  22  32  3)  C3 (12  22  32  3) (2.21) 34 Chapter Literature Review where G is the initial bulk shear modulus and i (i  1, 2, 3) are the principal stretches. If the membrane area remains constant, λ1 λ2 λ3 = 1.Parameter C matches the experimental data best when it is equal to G /20. Another model that considered the whole cell as a homogeneous incompressible solid was adopted in the simulation of micropipette aspiration of malaria-infected erythrocytes (Zhou et al. 2004), as shown in Figure 2.18. A hyperelastic material model was used to describe the nearly incompressible material. The strain energy function of this incompressible neo-Hookean material is U G0 ( I  3) (2.22) where I1  12  22  32 is the deviatoric strain invariant, and i (i  1, 2, 3) are the principal stretches. Figure 2.17. The axisymmetric model of micropipette aspiration of a malaria-infected erythrocyte, reprinted from (Zhou et al. 2004) with permission. The dimensionless shear modulus and pressure scaled by shear modulus are described by G0  * G0 G0 1 P  t   * P G0 (2.23) 35 Chapter Literature Review The dimensionless cell, pipette and fillet radius scaled by the cell radius Rcell are given by Rcell  * Rp  * Rp Rcell e  * e Rcell (2.24) The normalized projection length scaled by Rp is given by  Lp  Lp Rp (2.25) Using this method, the elastic shear modulus of a schizont stage Pf-infected erythrocyte was found to be 119±62 Pa. 2.2.3.3 Remarks The existing models for late stage malaria-infected erythrocytes did not consider the effect of the less deformable membrane and maturation of the enclosed parasite as separate contributing factors. The first model cited only consider the change in mechanical property of cell membrane as the parasite matures, while the second one consider the cell as an entire rubber-like solid. Improvement can be made to establish a more accurate model. In addition, there is currently no accurate model for trophozoites, which are the mid-stage infected erythrocytes. To complete the model system of malaria infected erythrocytes with the progression of parasite maturation, a model of trophozoite stage infected cell is needed. 2.2.4 Modeling of Optical Tweezers Stretching of Erythrocytes Li et al. studied human erythrocyte’s equilibrium shape and deformation using spectrin level energetic (Li et al. 2005), and compared the simulation with 36 Chapter Literature Review experimental results obtained from optical tweezers experiments, as shown in Figure 2.18. Figure 2.18. Erythrocyte shape at different optical tweezers stretch forces simulated using spectrin level energetic, reprinted from (Li et al. 2005) with permission. The corresponding shear modulus are µ=8.3 µN/m, Young’s modulus E=22.1 µN/m, average bending modulus D  2.4 1019 J , Poisson’s ratio υ=1/3, and area dilatational modulus  = 16.6 µN/m. Dao et al. simulated the similar problem (Dao et al. 2003), and concluded that the shear modulus affects the shape change of stretched erythrocytes, and the existence of cytosol increases the transverse rigidity. 2.2.5 Modeling of Cell Entrance into Capillaries Newtonian and viscoelastic models were used to simulate the entrance of a neutrophil into a capillary (Zhou et al. 2007), as shown in Figure 2.19. The ordinary differential equation for the cell length lc(t) is given by: 37 Chapter Literature Review  9.9( L  l c ) cl c  dlc P   m    a2   dt  (2.26) where P is the pressure drop,  m is the viscosity of the matrix in which the cell is suspended, L is the total length of the narrow contraction, a is the radius of the contraction, δ is the gap between the cell and the capillary wall, t is the entrance time, and c is a coefficient. The numerical result was consistent with experimental data. The entrance time decreased with pressure drop, increased with cell viscosity. Figure 2.19. Meridian plane of the simulation setup. There is an 90 degree arc connecting two cylindrical tubes, reprinted from (Zhou et al. 2007) with permission. 2.2.6 Multicomponent Models of Cells The complex subcellular structures of living cells undergo deformation and stresses in response to external force applications. Constitutive mechanical continuum models use average mechanical property values to predict these deformation and stresses, but fail to construct the subcellular topography, which may play an important role in responses to force application. A multicomponent solid elastic continuum model was proposed to study intracellular stresses of an endothelial cell in a monolayer subjected to physiological fluid flow (Ferko et al. 2007). 38 Chapter Literature Review Figure 2.20. Multicomponent model: the cell monolayer is in light green and the nucleus is in dark green. The arrow indicates the direction of the flow above the monolayer, reprinted from (Ferko et al. 2007) with permission. As shown in Figure 2.20, this model includes the cell monolayer, the nucleus and flow. The cell cytoplasm and nucleus were taken as isotropic elastic materials. By changing the Young’s moduli and Poisson’s ratio of the elements representing nucleus in the finite element model, the authors were able to predict the effects of shear flow on the monolayer with or without the presence of the relatively rigid nucleus. When the moduli of the nucleus and the cytoplasm were equal, the effect of the nucleus was neglected without changing mesh settings. Furthermore, by changing the boundary conditions, the authors were able to predict the effects of focal adhesions and uniform adhesions. The results indicated that the stress was higher in the nucleus and lower in the soft cytoplasm, while the strain was higher in the cytoplasm and lower in the nucleus. The focal adhesions also affected the distributions of stress, strain and displacement. 39 Chapter Literature Review 2.3 Deformation Models for Micropipette Aspiration Various techniques have been used to measure the mechanical properties of malaria infected erythrocytes. Among them, micropipette aspiration has its advantage of exerting a wide range of suction pressure to induce specific localized deformation on a cell membrane instead of the whole cell. Hemispherical cap model and homogenous half-space model have also been developed to evaluate the elastic modulus of a whole cell undergoing micropipette aspiration by assuming the cell as a homogenous entity despite the fact that a cell is heterogeneous and can comprise various components inside the cell. 2.3.1 Hemispherical Cap Model In the hemispherical cap model, it is assumed that (1) The erythrocyte membrane is two-dimensional and incompressible; (2) The shear deformation of the membrane is not affected by the Newtonian fluid in the erythrocytes. (3) The radius of the cell is very much larger than the radius of the micropipette. (4) There is no friction between the cell and the pipette surface. Therefore, the aspiration of the membrane is an infinite plane membrane deformation into a cylindrical tube with constant area (Shu et al. 1978). 40 Chapter Literature Review Figure 2.21. Schematic diagram of the hemispherical cap model. The increase in the projection length, Lp , is not proportional to the increase in suction pressure when Lp/Rp is less than 1, where Rp is the pipette radius. 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 by (Shu et al. 1978), P  2.45 Lp  C , R p2  Lp    1 R   p  (2.27) Therefore, the shear modulus can be calculated as  2.45 d (P) , R p2 d ( L p )  Lp   1    Rp  (2.28) However, as the parasite grows inside the host cell in the case of a P.f. infection, the cell may not behave like fluid-filled, but more like a visco-elastic solid due to the presence of the rigid parasite components. Thus, the homogenous halfspace model may be a better option to represent the mechanical behavior. 41 Chapter Literature Review 2.3.2 Homogenous Half-space Model The homogenous half-space model has also been used to study the Young’s modulus of cancer cells (Ting 2007) and endothelial cells (Theret et al. 1988), which are usually larger and stiffer than normal erythrocytes. It was developed based on a homogenous visco-elastic plate with a flattened profile. The size of the inner radius of the pipette and the wall thickness are assumed to be small compared to the thickness of the plate and the curvature of the radius of the cell (Theret et al. 1988). 2.3.3 Comments Both the hemispherical cap model and the homogenous half-space model were used in previous studies to analyze the living cells’ deformation in micropipette aspiration. The hemispherical cap model was commonly used to study the elastic shear modulus of normal erythrocytes, while the homogeneous half-space model was more suitable for studying the stiffness of larger and stiffer cells, such as cancer cells and endothelial cells. Both of these two models have their assumptions and limitations. The hemispherical cap model assumes the cell to be a liquid enclosed by membrane, while the homogenous half-space model assumes the cell to be a homogenous solid. They might not be suitable for studying malaria infected erythrocytes due to the cell’s structural complexity. 42 [...]... where the strain energy potential is U G0 2 2 2 2 (1  2  3  3)  C3 (1  2  3  3) 3 2 2 2 (2. 21) 34 Chapter 2 Literature Review where G 0 is the initial bulk shear modulus and i (i  1, 2, 3) are the principal stretches If the membrane area remains constant, λ1 2 λ3 = 1.Parameter C 3 matches the experimental data best when it is equal to G 0 /20 Another model that considered the whole... function for the erythrocyte material is used: Wm  B 1 2 C 2 ( I1  I1  I 2 )  I 2 4 2 8 (2. 14) 29 Chapter 2 Literature Review where B  0.5 10 2 dyn/cm and C=100 dyn/cm If the membrane is assumed to be incompressible, Wm is given by the first term of (2. 14), provided that dAf dA0  1 2  1 (2. 15) where Af and A0 are final and initial areas, respectively Other variations of continuum constitutive models. .. aspiration of malaria- infected erythrocytes (Zhou et al 20 04), as shown in Figure 2. 18 A hyperelastic material model was used to describe the nearly incompressible material The strain energy function of this incompressible neo-Hookean material is U G0 ( I 1  3) 2 (2. 22) 2 2 where I1  1  2  3 is the deviatoric strain invariant, and i (i  1, 2, 3) are the 2 principal stretches Figure 2. 17 The axisymmetric... Wm T  2 1 (2. 10) T2  1 Wm T 1  2 (2. 11) where T is an isotropic membrane tension 1 and 2 are the principal stretch ratios in the meridian and circumferential directions, respectively Wm is the strain energy per unit of initial area, which can be assumed to be a function of two dimensional invariants 2 I 1  1  2  1 2 (2. 12) 2 I 2  1 2  1 2 (2. 13) With large stresses and the membrane... (Guilak et al 20 00; Leterrier 20 01; Shieh et al 20 02; Shieh et al 20 03) 2. 2 .2 Modeling of Normal Erythrocytes (a) Erythrocyte Shape The healthy erythrocyte has a biconcave shape which makes it able to undergo large deformation as it goes through narrow capillaries The normal unstressed shape 26 Chapter 2 Literature Review of erythrocyte is suggested to have the cross section as shown in Figure 2. 13 (a)... the large deformation of which can be simulated by adopting a hyperelastic material model for the membrane 33 Chapter 2 Literature Review 2. 2.3 Modeling of Plasmodium falciparum Infected Erythrocytes P.f infected erythrocytes have been modeled as they are aspirated into a micropipette (Zhou et al 20 04) and as they are stretched by the optical tweezers (Suresh et al 20 05) 2. 2.3.1 Shape of Infected Erythrocytes... Lp Rp (2. 25) Using this method, the elastic shear modulus of a schizont stage Pf -infected erythrocyte was found to be 119± Pa 62 2 .2. 3.3 Remarks The existing models for late stage malaria- infected erythrocytes did not consider the effect of the less deformable membrane and maturation of the enclosed parasite as separate contributing factors The first model cited only consider the change in mechanical. .. constitutive models have been applied to analyze the large deformation of a erythrocyte, such as the hyperelastic effective material model proposed by Lim et al (Lim et al 20 04) An incompressible neoHookean form will be the simplest first-order formulation for such case The strain energy function (Simo et al 1984) is given by U G0 2 2 (1  2  3  3) 2 2 (2. 16) which defines a nonlinear elastic stress-strain... is 30 Chapter 2 Literature Review Ts  1 1  (T1  T2 )  G0 h0 (1.5  1 1.5 ) 1 2 2 (2. 19) The membrane shear modulus is given by  (1 )  0  1 Ts 3G0 h0 (1 5  1 2. 5 )   2  s 4(1  1 3 ) (2. 20) where  s is the shear strain The computational simulations presented by Mills et al (Mills et al 20 04) used a neo-Hookean hyperelastic material with the constitutive response Before reaching... 1 ij ,  k  k2 2   (2. 9) where µ is the viscous constant, k1 and k2 are two elastic constants, represented by the dashpot and springs as shown in Figure 2. 12 25 Chapter 2 Literature Review Figure 2. 12 The homogeneous SLS model (a) The homogeneous linear viscoelastic solid model (b) The creep response (γ) of a standard linear viscoelastic solid subjected to a stress τ ( k1 = k 2 , µ =10 k1 , τ= . 1 2 2 2 11   I (2. 12) 1 2 2 2 12   I (2. 13) With large stresses and the membrane area changes, the full strain energy function for the erythrocyte material is used: 2 221 2 1 8 ) 2 1 ( 4 I C III B W m  . widely used in studying musculoskeletal cells (Guilak et al. 20 00; Leterrier 20 01; Shieh et al. 20 02; Shieh et al. 20 03). 2. 2 .2 Modeling of Normal Erythrocytes (a) Erythrocyte Shape The. induced forces to cytoskeletal and subcellular level. Chapter 2 Literature Review 20 Figure 2. 7. Overview of the mechanical models developed for living cells, reprinted from (Lim et al. 20 06)