Predicting the Elastic Properties and Deformability of Red Blood Cell Membrane Using an Atomistic-Continuum Approach

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1 roceedings of the nternational MultiConference of Engineers and Computer Scientists 26 Vol, MECS 26, March 6-8, 26, Hong Kong redicting the Elastic roperties and Deformability of Red Blood Cell Membrane Using an Atomistic-Continuum Approach A.S. Ademiloye, Member, AENG, L.W. Zhang, and K.M. Liew Abstract This paper employs the gradient theory to study the elastic properties and deformability of red blood cell (RBC) membrane using the first-order Cauchy-Born rule as an atomistic-continuum hyperelastic constitutie model that directly incorporates the microstructure of the spectrin network. The well known Cauchy Born rule is etended to account for a three-dimensional (3D) reference configuration. Using the strain energy density function and the deformation gradient tensor, the elastic properties of the RBC membrane were predicted by minimizing the potential energy in the representatie cell. This etended formulation was then coupled with the meshfree method for numerical modeling of the finite deformation of the RBC membrane by simulating the optical tweezer eperiment using a self written MATLAB code. The results obtained proide new insight into the elastic properties and deformability of RBC membrane. n addition, the proposed method performs better when compared with those found in literature in terms of prediction accuracy and computation efficiency. nde Terms Red blood cells, Cauchy Born rule, elastic properties, finite deformation, meshfree method, Optical tweezers eperiment B. NTRODUCTON LOOD is a bodily fluid in animals that deliers necessary substances such as nutrients and oygen to the cells and transports metabolic waste products away from the same cells. t is a special fluid, which can be iewed as a suspension of red blood cells (RBCs) or erythrocytes, white blood cells (WBCs) or leukocytes, platelets and blood plasma in a Newtonian fluid. The RBCs alone account for more than 99 % of the particulate matter in blood and about 4 45 % of the blood by olume. The RBCs are biconcae, Manuscript receied December 23, 25. The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administratie Region, China (roect No , CityU 2894), and National Natural Science Foundation of China (Grant No and Grant No ). A. S. Ademiloye is with the Department of Architecture and Ciil Engineering, City Uniersity of Hong Kong, Tat Chee Aenue, Kowloon, Hong Kong (phone: ; fa: ; ademiloye.adesola@my.cityu.edu.hk). L. W. Zhang is with the College of nformation Technology, Shanghai Ocean Uniersity, Shanghai 236, R China ( zlwen@hotmail.com). K. M. Liew is a Chair rofessor in the Department of Architecture and Ciil Engineering, City Uniersity of Hong Kong, Tat Chee Aenue, Kowloon, Hong Kong (phone: ; fa: ; e- mail: kmliew@cityu.edu.hk). non-nucleated elastic particles with diameter and thickness of about 8 µm and nm, respectiely. Their elastic and mechanical properties are essential for sustaining cell functions. Changes in these properties may eentually lead to the manifestation of arious blood related hereditary and non-hereditary diseases. Recent studies hae reealed that the deformability of the red blood cell (RBC) membrane can be assessed eperimentally, using atomic force microscopy (AFM), micropipette aspiration technique, optical tweezers, and microfluidic eperiments [] [4], and numerically, through microscopic, continuum or atomistic continuum models [5] [8]. Oer the years, the optical tweezers eperiment has become a powerful tool for carrying out important biomechanical characterization. n this study, we carried out numerical simulations to inestigate the elastic properties as well as the finite deformation behaior of RBC membrane. n order to map the cell structure in the reference configuration to that in the current deformed configuration directly, the standard Cauchy Born rule was employed. By using a 3D framework, the in-io three-dimensional deformation of RBC membrane can be fully captured. Since this current study inoles large deformation, the use of meshfree method is ustified due to limitations of mesh-based method [9], []. The displacement field comprises of only three nodal displacement degree of freedom because meshfree approimations possess intrinsic non-local properties, leading to real rotation-free approimation hence displacements are the only nodal freedoms present [].. HYERELASTC CONSTTUTVE MODEL The Cauchy Born rule [2] establishes a connection between the deformation of the lattice ector of an atomistic system and that of a continuum displacement field, and plays an important role in the deelopment of continuum constitutie models of atomic lattices. n this section, we describe a hyperelastic constitutie model that is deried from the first-order Cauchy Born rule by using the coarsegrained Helmholtz free energy density to describe the atomic interactions. SBN: SSN: (rint); SSN: (Online) MECS 26

2 roceedings of the nternational MultiConference of Engineers and Computer Scientists 26 Vol, MECS 26, March 6-8, 26, Hong Kong F 2 X. (4) 3 The optimal alues of the geometrical parameters in (3) and (4) can be obtained by minimizing the strain energy density using the Newton s method. Fig. Representatie cell of the RBC membrane Considering a representatie cell that is composed of si spectrin links J (J =,, 6) as shown in Fig aboe, the deformation of each spectrin links can be approimated using the Cauchy-Born rule, as follows, r FR, () J J where F= FiJ ei e, J R and J r denotes the first-order J deformation gradient tensor, the undeformed and deformed spectrin link length between unction complees and J, respectiely. The strain energy density in this representatie cell can be epressed in terms of energy contributions from the in-plane energy, bending energy, surface and olume conseration energy terms [3], as V W WF VWLC ( rj ) C / A Vbending( kb, ) 2 J 3 J a (, ) (, ), Vsurface rj ka Volume rj k constraint constraint where k, k, and k are the area, olume constraint and bending coefficients, and b (2) denotes the area of the representatie cell calculated as 3L 2 2. For more details, interested readers are referred to the following literature and the references cited therein [7], [4], [5].. ELASTC ROERTES OF RBC MEMBRANE n order to determine the elastic properties of RBC membrane, we assume that the elastic deformation of an infinitesimal RBC membrane patch from an initially undeformed planar sheet (Fig 2a) to a deformed planar sheet (Fig 2b) embedded in a 3D space is a simple shear deformation. This shear deformation can be described using the four geometrical parameters, 2, 3, and, whose deformation map and gradient can be epressed as follows, X X X, (3) X Fig 2. Simple shear deformation of an infinitesimal patch on the RBC membrane surface (a) reference configuration, (b) current configuration Once the minimization step is completed, the first-order iola Kirchhoff stress tensor and the tangent modulus, M FF are the first- and second-order deriatie of the strain energy density with respected to F, respectiely can be calculated using, W r W =, J F F rj 2 2 FF W rj W rj M = 2 2 F F r J F The Young s modulus E, oisson s ratio, shear modulus µ h, area compression modulus K, and bending modulus B of the RBC membrane can be obtained using the epressions below: E M ( M / M ), M FF 2 FF / M, ( M M ), h K.5 ( M M ), B 2.5 ( M2222 M22 ), where is an elastic length scale, taken as.28 μm [6]. The microstructure parameter set used in the study are as follows: initial equilibrium spectrin length L = 87 nm, persistence length p = 8.5 nm, maimum spectrin etension length L = 238 nm, number of triangular face on RBC ma membrane surface N f = 492, bending coefficient k = 2.77 b -9 J, and temperature T = 3 K. Using the models gien in (6) and the algorithm described aboe, we computed the elastic properties of the RBC membrane as shown in Table below. T. (5) (6) SBN: SSN: (rint); SSN: (Online) MECS 26

3 roceedings of the nternational MultiConference of Engineers and Computer Scientists 26 Vol, MECS 26, March 6-8, 26, Hong Kong TABLE VALUES OF ELASTC ROERTES OF RBC MEMBRANE N COMARSON WTH THOSE N LTERATURE FOR DFFERENT VALUE OF k AND k. Elastic ka k roperties ξ = ξ = [ (5) 3] (units) References E (μn/m) (9.3) (ξ = ) [7], [8], [] (.45) (ξ = ) [7], ~.5 [9],.49 [2]. µh (μn/m) (6.7) (ξ = ) [7], [2]. K (μn/m) (7.65) (ξ = ) [7], [2]. 9 B ( J ) (5.25) [22], ~ 9. [23]. V. ATOMSTC-CONTNUUM MESHFREE METHOD Based on the constitutie relations in section, a computational scheme can be established to perform the numerical simulation of RBC membrane deformation. The meshfree method is a computational technique that has some ecellent adantages oer the classical finite element method. n particular, meshfree approimations hae nonlocal properties, and satisfy the higher order continuity requirement []. The intrinsic non-local property of meshfree interpolation leads to real rotation-free approimation, and displacements can thus be used as the only nodal freedoms []. mproed moing least squares (MLS) approimation [24] is used to construct the meshfree interpolation and can be combined with the Ritz minimization procedure to form the MLS-Ritz meshfree method. n the computation, we considered the RBC as a body that initially occupies a region and deforms to a new configuration t. The deformation of a material particle X at a time t is described by X (, t) through the mapping functions ϕ as ( X, t) = u( X, t) + X (7) where u is the displacement of this material particle. The first-order deformation gradients F can therefore be defined as i ui Fi i, i,,2,3. (8) X X or u F, X X (9) where is the kronecker delta property. i a where u denotes the nodal parameter, Φ ~ the meshfree interpolation (or shape) function deried using the MLS approimation at the th node and N is the total number of nodes coered by the compact support domain. By treating the RBC membrane as a 2D surface embedded in a 3D space, all nodes hae three degree of freedom. The 3D MLS shape function with linear basis and circular support domain were employed. The first-order deformation gradients F corresponding to the displacement step aboe are approimated by using (9) and () aboe, as Φ N N, F ( Φ. u ) Φ. u +, () where X,2 Φ,3 ~ Φ, K (K =,2,3) is the first-order deriatie of the meshfree shape function with respect to the reference configuration. The total energy of the system can be calculated as: W F d ub d ut ds, (2) where W( F) hw ( F ) is the strain energy density function adopted in order to incorporate the effect of membrane thickness h, t represent the first-order stress tractions on the surface of the domain. Contribution from body force term was neglected. The Ritz parameters are determined (or adusted) such that δπ =. For simulations in this section, contributions from area and olume conseration energy were ignored for the sake of simplicity. Upon discretizing (8) aboe using the well-known meshfree and finite elasticity framework, the resulting incremental system equations can be written as K u f (3), n n n where u contains the nodal parameters of all meshfree n nodes. K and n f represents the global stiffness matri n and non-equilibrium force ector, and they are gien by K n T FF [ B M B ] d, (4) T FF [( LΦ) M ( LΦ ) ] d et int T T fn fn fn Φ t ds B n d, (5) = Φ t ds ( LΦ) d T T where denotes assembled stress ector at the time t when n the stress components are known and L is an assembled differential operator matri with respect to the undeformed cell configuration, n The displacements relatie to the undeformed cell configuration are approimated as N u Φ ( ) u ( t) Φ( ) u () SBN: SSN: (rint); SSN: (Online) MECS 26

4 roceedings of the nternational MultiConference of Engineers and Computer Scientists 26 Vol, MECS 26, March 6-8, 26, Hong Kong X X 2 X3 X L X 2. X3 X X2 X 3 (6) The stiffness matri in (4) and the internal force ector in (5) are integrated using the Gauss quadrature integration techniques while the eternal force ector in (5) was computed using the direct nodal force application technique. n the present study, a simple technique is used to ensure that the solution conerges to the minimum point. This was achieed by replacing K with K n n, where is a positie number that is slightly larger than the magnitude of the most negatie eigenalue of K, and is an identity n matri. After a few cycles of replacements, K becomes n positie definite and the standard Newton method continues. force ector f is assembled from the total stretching et n forces, +F and F. Within each loading step, iteration was performed until the solution conerges, after which the net loading step starts. At the end of the final loading steps, the undeformed RBC geometry was updated to obtain the final deformed geometry. The deformed aial diameter, D A was then computed as, where ma min ma is the maimum coordinate among the N ertices, and min is the minimum among N ertices. Also, the deformed transerse diameter, D T was obtained using the epression 2 2 2ma ( y ) ( z ), where y and z are coordinate... N alue of each ertices in y and z directions, and N represents the total number of meshfree nodes discretized on the problem domain as shown in Fig 4 below. V. FNTE DEFORMATON OF THE RBC MEMBRANE Using the present meshfree method, we hae studied the finite deformation behaior of the RBC membrane. This was achieed by simulating the optical tweezers techniques, in which two silica beads trapped by laser beams are attached to left and right ends of the cell. The trapped cell was stretched by fiing one of the beads and pulling the other in an aial, as illustrated in Fig 3. Fig 4. Full geometry of RBC membrane (problem domain) showing meshfree node distributions and triangular meshes as isualization aid. Fig 3. Optical tweezer eperiment setup and computational configuration adopted in this study. The aial (D A) and transerse (D T) diameters of the stretched cell, corresponding to the applied stretching force were recorded. The total stretching force, +F within the range 8 pn was applied to N N ertices of the problem domain with the largest -coordinates in the positie -direction, then a corresponding equal but opposite force, F was eerted on N N ertices with the smallest -coordinates in the negatie -directions. The alue of was taken to be.5 which represents the number of nodes in contact with the attached silica beads. The simulations for each gien stretching force within the force range were performed lambda (λ) times, where λ correspond to the maimum number of loading steps adopted. The eternal This simulation was performed using 62 meshfree nodes, quadrilateral background cell for numerical integration, cubic spline weight function, meshfree scaling parameter dma.462, 7 loading step and RBC membrane parameter set: RBC diameter D 7.82 µm, L = 87 nm, p 8.5 nm, Lma 238 nm, temperature T = 3 K, and assumed membrane thickness t = 2 nm. Fig 5 below shows the comparison between our current meshfree simulation, spectrin leel modeling, quasicontinuum meshfree simulation as well as the optical tweezer eperiment. From the plotted figure, the performance of the newly implemented approach is obiously better as the obtained alues for stretched aial and transerse diameters for each stretching force are ery close to those obsered in the eperiment. SBN: SSN: (rint); SSN: (Online) MECS 26

5 roceedings of the nternational MultiConference of Engineers and Computer Scientists 26 Vol, MECS 26, March 6-8, 26, Hong Kong Fig 5. Aial and transerse diameter of RBC membrane as a function of stretching force in comparison with optical tweezer eperiment [25], quasicontinuum meshfree simulations [8] and spectrin leel modeling [7]. V. CONCLUSON The elastic properties and deformability of the RBC membrane were studied in this paper using a gradient theory based on the standard Cauchy Born rule. The elastic moduli of the RBC membrane were obtained using the strain energy density function and deformation gradient tensor by minimizing the energy in the representatie cell. An atomistic-continuum meshfree method was also implemented to study the three-dimensional finite deformation of the RBC membrane by simulating the optical tweezers eperiment. 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