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Human red blood cells (RBCs) are responsible to transport oxygen and carbon dioxide for human bodies. The physiological functions of RBCs are greatly influenced by their mechanical properties. When RBC is infected by Malaria parasite called Plasmodium falciparum, it shows progressive changes in mechanical properties and loses its deformability. The infected red blood cells (IRBCs) develop properties of cytoadherence (stickiness) and rosetting (the binding of non-infected RBCs to parasitized RBCs). In this paper to analyze the mechanical properties and deformability of the IRBC, we applied stress-stretch ratio relation of its biomembrane .To express this constitutive relation, we proposed a mathematical model (Neo-Hookean model) based on membrane theory. On this model, we present continuous stress-stretch ratio curves for the relation derived from the model for different intracellular developmental stages of the parasite, to determine the mechanical properties of IRBC. The analytical results obtained from the mathematical model are more closed with the experimental data [1] which demonstrates the validity of the model. By restricting our attention to spherically symmetric deformation in the final schizont stage of parasite development, the pressure-extension ratio relation curve also adapted from the proposed strain energy function. The change in osmotic pressure versus volumetric ratio has been also considered for IRBC before hemolysis.

The human RBC with a biconcave shape which allows more and more oxygen molecules to come in contact with the cells surface.RBC has an average diameter of about 8µm and a typical life span of 120 days. The physiological functions of RCBs are greatly influenced by their mechanical properties. Studies of the mechanical properties of the human RBC and membrane [

Particularly, mechanical deformability and biorheology of the human RBC are known to play a vital role in influencing organ function as well as states of overall health and disease system [

The functions of RBC are determined by the mechanical properties of its biomembrane. Cell membrane mechanics and the deformation characteristics of human RBCs have received considerable attention [

In this paper we proposed a mathematical model to analyze the mechanical characterization of IRBC membrane arising from malaria infection disease. The model is based on membrane theory and utilizes a hyperelastic material to describe the deformation behavior of RBC biomembrane. Membrane is fundamental structure that strongly influences cell functions.

Blood is a suspension of formed elements (red blood cells, white blood cells and platelets) in plasma. An RBC consists of cytoplasm enclosed by a thin and elastic membrane. Hence, we applied the principle of mechanics regarding to elastic continuum for the present study. To do so, we need to relate the stress in the body to some measure of deformation, which will be accomplished through the introduction of constitutive equations. Because constitutive equations describe stress-strain relations of soft biological tissues require parameters such as the strain energy functions or their derivatives [

The spectrin network which underlies the phospholipid bilayer of human RBC membrane is genenerally considered to impart shear induced resistance to the cell membrane although the bilayer itself has little resistance to shear deformation [_{s} (expressed in units of force per unit length) and the principal stretch ratios and is given by [

and (1b)

where T_{1} and T_{2} are the in-plane principal membrane tensions, and _{ }are the in-plane principal Green’s strains components of the membrane, γ_{s} is the shear strain and µ the membrane shear modulus (assumed to be constant and expressed in units of force per unit length). Equation (1c) reflects the assumption that the total membrane area is constant during deformation in normal condition. By combining (1a) and (1b) we obtained as,

For large deformation characteristics of a membrane material, when the biomembrane is assumed to be a three dimensional elastic continuum, the principal tensions T_{1}and T_{2 }can be expressed [

Where h is the final thickness of the deformed membrane, and µ are material properties. The Green (material) strain tensor components can be defined as,

and (4)

Living cells in the human body are constantly subjected to mechanical simulations throughout life. These stresses and strains can arise from both the external environmental and internal physiological conditions. Any deviation in the structural and mechanical properties can result in the breakdown of these physiological functions and may possibly lead to disease [

At the schizont stage the infected red cell is found to exhibit a viscoelastic solid like behavior which is in contrast to the liquid drop behavior demonstrated by healthy RBC and early stage of IRBC [

where G_{0} is the initial value of bulk shear modulus, (i = 1,2,3) are the principal stretches. The incompressibility condition implies that.The potential function U, defines the non-linear elastic stress-strain behavior. When the initial membrane thickness is, the constitutive description of equation (5) results in the initial in-plane membrane shear modulus [

where (i = 1,2,3). The principal tensions of the cell membrane can be written as

The membrane tension resultants from the proposed strain energy potential function are expressed as follows,

If the strain energy density function is supposed to be independent of the second invariant, single test such as a uniaxial tension test is needed for material response [

Assume D, be diameter of the cell. In the undeformed state this quantity shall have the value D_{0} taken to be the same as the biconcave model which is approximately 8μm. The principal stretch ratio is defined as λ_{1} = D/D_{0}, where D is deformed in the axial or transverse direction. By substituting this value in (10), we obtained T_{1} as a function of diameter as follows.

Since T_{1} is expressed under uniaxial tension, variations of axial and transverse diameter of the cell are considered. In uniaxial tension (stretch) [1,2,17] the axial diameter of the cell increase and the transverse diameter of the cell becomes reduced. Therefore, (11) presents continuous increasing and decreasing force-deformation curves for the deformation response of the RBC for different developmental stages of the parasite.

The erythrocytic developmental stages of the parasite are broadly classified as the ring stage (pf-R-IRBC), trophozoit stage (pf-T-IRBC) and the schizont stage (pf-S-IRBC). Single cell mechanical property measurements performed using the micropipette aspiration method [12,23] and the laminar shear flow method [

As the parasite develops in RBC, the IRBC becomes more spherical with increase in its volume as the parasite multiply within it [1,8]. The possible increased pressure through plasmodium biochemical process, might lead to a higher erythrocyte volume and therefore to the increased erythrocyte fragility [

One of the factors reduced deformability besides to molecular changes in the cell membrane or cytoplasm among red cells is a purely geometric effect, viz., the decrease in the surface-to-volume ratio of the red cell due to the increased volume resulting from parasite growth [_{H} and subsequently, P_{H} the osmotic pressure when the cell hemolyzed. This calculation is based on the Van’t Hoff, relation given by

where V_{HA} andV_{0A} are the apparently osmotically active volumes of V_{H} and V_{0} respectively, P_{0} isotonic osmotic pressure.V_{0} the isotonic volume of the cell. When the parasite growth in cell membrane, there is a change in osmotic pressure, denoted by, after the cell become spherical, is given by Canham and Parkinson [

where volume of the cell when it first become a sphere, 0.42V_{0} osmotically inactive fraction of the cell volume in isotonic condition. When the parasite develops and multiplies with in RBC, both the pressure and volume increases in the parasitized cell. According to Magowan et al. [_{p} + V_{0} = 0.3V_{0} + V_{0}, where V_{p} is volume of the parasite inside the cell membrane. Based on these assumptions (13) can be written as,

The change in osmotic pressure versus volumetric inflation ratio due to the development of parasite inside the cell membrane is the gradient of osmotic pressure which the cell must have either due to stiffness or allowing its membrane to be stretched or by loosing osmotically active solute.

In the schizont stage, the plasmodium falciparum parasite IRBC exhibits a change in shape, which is spherical compared to the biconcave shape of HRBC [_{0}). As Skalak, R. et al., [_{1} acts in the meridonal directions and T_{2} is the circumferential or hoop stress. The stretches (extension ratios) λ_{1} and λ_{2} are obviously given by the ratio (r/r_{0}) of the circumferences of the sphere of the IRBC at any point in the inflated and uninflated state, so that by isotropic tension orientation (T_{1} = T_{2}), we understood that, therefore, the tension distribution on the sphere is given by

The pressure difference in the direction of the outward normal across the spherically deformed cell membrane surface can be calculate from the equation of equilibrium and is written as [

Where r is the radius of the spherical shape of IRBC and it can be expressed as.As pressure increases, the value of the stretch ratio increases [

whereis the initial radius of the cell and μ_{0} is the parameter in the process of sphering of RBCs for both cases HRBC and IRBC at the final schizont stage of the parasite development.

In

with uniaxial stretch progressively reduced in the advanced stages of intracellular parasite compared to the healthy RBC. This is shown in the

The analytical result presents, that the HRBC is easily deformable with low stretch force than the parasitized RBC. Furtherly, the coefficient (shear modulus) of function of IRBC is drastically increased for the period from ring stage to schizont stage. Our analytical solution simulation as shown in

As can be seen curves from the graph of the proposed mathematical model, through the constitutive relation, the interpretation from the curves leads to different mechanical properties according to different developmental stages of the parasite in RBC. Thus, this study indicates that the proposed mathematical model can be used to predict the deformation characteristics of RBCs during parasite development in RBC and may have potential biomedical applications such as characterizing rheological properties and distinguishing infected cells from normal ones. The membrane tension distribution versus deformation in uniaxial tension is an increasing function of deformation in axial diameter and decreasing function in transverse diameter as shown in

(IRBC) at the schizont stage and HRBC which is spherical shape with the Neo-Hookean model in an isotropic tension .we obtain a non-monotonic curves with non linear behavior, which have a zero value of course at (λ = 1). The inflated pressure rises steeply with circumferential stretch. At the schizont stage the infected cell exhibits almost solid-like behavior and increased rigidity [

In this study, we proposed a mathematical model to predict the mechanical response or mechanical behavior of the RBC membrane which infected by plasmodium falciparum parasite. Our work is the new method to employ mathematical model regarding to continuum mechanics approach to analyze mathematically and quantify the mechanical behavior of RBC membrane in malaria infection. The model is based on the continuum approach membrane theory to describe the deformation behavior of RBC membrane. Since RBC membrane is a hyperelastic material, it is possible to introduce strain energy function defined with respect to the undeformed state. The unique analytical result (solution) in uniaxial stretch test based on the model makes this study differs from the other previous works on mechanical properties of RBC during the developmental stages of the parasite. With this model, as the curves indicate in

Moreover, as observed in Figure1 (a) and (b) the analytical result obtained from the mathematical model compared with the experimental results of Suresh et al. [

In

Since membrane is a constitutive material, we will attempt to study by using other constitutive models, in order to analyze and determine mechanical properties of malaria infected RBCs in our future work.