Local irregularity effects on quantifying mechanical response of adherent cells to fluid flow stimuli

The local irregularity at cellular level around adherent cells was examined using the finite element method by simulating a laboratory rotational flow apparatus. An axisymmetric flow was assumed for the fluid flow and the Navier–Stokes equation in cylindrical coordinate was used to study the fluid behavior under rotational velocity. A modified hyperelastic constitutive model that incorporates the effect of the interaction between cytoplasmic motion and cytoskeleton stretching within a cell was used to describe the mechanical response of adherent cells to the external forces of fluid flow. It was found that the flow pattern around the cells deviates significantly from the laminar flow assumption for the laboratory apparatus. The local irregularities alter the behavior of fluid flow near the cell and induce nonlinearity in velocity and vorticity, which play an important role in quantifying the detachment stresses of cells. A group of curves of the detachment stress versus the radial location of cells was developed based on the finite element simulation under the rotational velocity. Subjects: Biomaterials; Biomechanics; Tribology


Introduction
Cell adhesion has been recognized as one of the fundamental mechanisms for numerous tissue organization, maintenance, and regeneration (Damsky, 1999;De Arcangelis & Georges-Labouesse, 2000;Hynes, 2002); for cell inflammatory response (Morat, 1985), metastasis (Nicolson, 1982); and for cytotoxicity (Kirkpatrick et al., 1998). Cell attachment to its substrate is usually through integrin-mediated adhesion which is highly regulated by two processes: activation of transmembrane receptors and their mechanical coupling to substrate ligands. Cell adhesion plays an important role in triggering

PUBLIC INTEREST STATEMENT
Cell is a fundamental unit in biological entities and is often called "building blocks of life." Cell involves in all the activities of a biological entities including birth, growth, mature, aging, and death that we all are familiar with. Every movement of our bodies is actually made by the contraction of muscle cells based on our decisions that are transmitted from our brains to the muscle cells through nerve cells. Most of the bacteria reside on our bodies through adhesion and seek opportunities to grow and multiply to cause diseases and make us ill. Thus, the study of cell adhesion is important for developing new and effective medicines, for curing diseases, and for cleaning surfaces.
signaling pathways that direct cellular proliferation, differentiation, migration, and apoptosis. The need for systematically understanding and further controlling of cell adhesion in vivo has stimulated many in vitro experiments devised to quantify the response of cells to externally applied mechanical stimuli. These methods include micromanipulation (Evans, Berk, & Leung, 1991;McKeever, 1974;Tozeren, Sung, & Chien, 1989), centrifugation (Channavajjala, Eidsath, & Saxinger, 1997;Lotz, Burdsal, Erickson, & McClay, 1989;McClay, Wessel, & Marchase, 1981), channel flow (Lu, Koo, Wang, Lauffenburger, Griffith, & Jensen, 2004;Ming, Whish, Hubble, & Eisenthal, 1998), rotational flow between a cone and a plate (Blackman, Barbee, & Thibault, 2000;Buschmann, Dieterich, Adams, & Schnittler, 2005;Malek, Ahlquist, Gibbons, Dzau, & Izumo, 1995), between parallel disks (Horbett, Waldburger, Ratner, & Hoffman, 1988;LaPlaca & Thibault, 1997;Ono, Ando, Kamiya, Kuboki, & Yasuda, 1991), and over a plate (Garcia, Ducheyne, & Boettiger, 1997;Reutelingsperger, Van Gool, Heijnen, Frederik, & Lindhout, 1994;Rocha, Hahn, & Liang, 2010). Along with these measurement apparatus, analytical equations to quantify the detachment forces that remove cells from their substrates were developed. Most of the analytical equations were derived from Navier-Stokes equation with assumption that the fluid flow is either laminar or turbulence in accordance with Reynolds number without considering the local irregularity around and the deformations of cells. Dewey and DePaola (1989) and Davies, Mundel, and Barbee (1995) had found that there were large variations in force magnitude and distribution of shear stresses and gradients at cellular level around cells through fluid dynamics simulation and atomic force microscopy measurements, respectively. Cell attached to a substrate induces local irregularity in flow pattern of the fluid around the cell and morphological variation between cell and its substrate under the externally applied forces. These local irregularity and morphological variation could activate or disassociate the bonding between receptor and ligand and play a significant role in the quantification of the mechanical response of cells to fluid flow stimuli.
In this paper, the cellular level variation of flow pattern of the fluid around cells was analyzed by simulating a rotational fluid flow apparatus using the finite element method. The cell is deformable and its mechanical behavior is described by a modified hyperelastic constitutive model (Zhang, 2010). The forces of the rotational fluid flow applied on the surface of cells attached to the substrate were calculated. We examined the mechanical response of cells to the applied fluid flow stimuli. Finally, a group of curves of detachment stresses as functions of the radial location of cells from the axisymmetric axis were developed based on the finite element simulation. These curves can be used either in combination with the experimentally measured critical detachment radius and rotational velocity (Rocha et al., 2010) to determine cell adhesion or in combination with cell adhesion model (Gallant & Garcia, 2007) to determine which cells would be detached from a substrate.

Rotational fluid flow
The laboratory apparatus for cell adhesion measurement, modeled by the finite element program Comsol Multiphysics (2009) is a rotating rheometer (TA Instruments, AR-G2) as shown in Figure 1(a) (Rocha et al., 2010). The rheometer consists of upper and lower plates in parallel, which are separated by a gap of 0.48 mm. The upper plate acts as a spindle rotated by a shaft controlled by a magnetic motor. The lower plate with a diameter of 22 mm is mounted on the base. The cell seeded sample is placed on the top of the lower plate. The gap is filled with phosphate buffered saline (PBS) solution. The spindle is loaded over the PBS solution to generate rotational flow with a controllable angular shear stress. The size of the gap in combination with the rotational velocity of the spindle should maintain a laminar flow in the fluid. The rotational flow is considered as an axisymmetric swirl flow and can be modeled using the 2D axisymmetric flow model in Comsol. The finite element mesh of the 2D axisymmetric fluid is shown in Figure 1(b) which has a length of 12 mm and a height of 0.5 mm. Figure 1(b) also shows the cell locations in dense meshes which were not included in the fluid dynamics analysis. The cell is represented by a half elliptical cross section with the semi-major and the semi-minor axes being 90 and 50 m, respectively. These cells are, respectively, located at 2, 4, 6, 8, and 10 mm away from the axisymmetric axis as shown in Figure 1(b) and (c). The substrate is assumed as rigid (with the elastic modulus of 2.2 ×10 9 Pa). The cell is modeled by a hyperelastic material as will be described in the later section. In this case, the computational time could be greatly minimized. The rotational velocity generated by the spindle is prescribed as moving wall on the top of the fluid. The axisymmetric condition is prescribed along z-axis. The rest of the fluid boundaries are no slip.
The axisymmetric incompressible fluid flow is described by the Navier-Stokes equations in cylindrical coordinate system as where and are density and dynamic viscosity of the fluid, respectively; u, v, and w are the velocity components in r, , and z directions, respectively; p is the fluid pressure; and F r , F , and F z are the components of the body forces in r, , and z directions, respectively. The viscous stress and the total stress tensors are defined, respectively, by in which ∇ is the gradient operator and I is the second-order identity tensor. The total fluid force on the boundary is calculated by where n is the unit normal vector of the boundary.
The fluid density and dynamic viscosity used in the current work are 1, 000 kg/m 3 and 10 −3 Pa s, respectively, and the body forces were not considered.

Wall stresses
We assumed that the density of cells seeded on the sample substrate is very high such that they can be considered as geometrically axisymmetric with respect to z-axis. In this case, the cells can be modeled as 3D solid rings (see Figure 1(c)). The simulation was divided into two steps. First, the 2D steady-state axisymmetric swirl flow analysis was carried out for the fluid (shown in Figure 1(b)) to obtain the forces (or wall stresses) at the boundaries where the cells are located. Second, the mechanical response of the cells to the wall stresses was analyzed and the reaction force at the cell base were obtained to calculate the detachment stress. Since the wall stresses obtained from 2D steadystate swirl flow analysis are three-dimensional including T r , T , and T z , the response of the cells to these applied forces has to be simulated in 3D space rather than in 2D axisymmetric case. The conversion of the 2D axisymmetric solutions into 3D space was done in Comsol Multiphysics (COSMOL, 2009). The wall stresses obtained from the 2D axisymmetric analysis were first mapped into 3D space and then applied on the 3D cell surfaces as the external forces. Such a treatment can only simulate one-way interaction, i.e. fluid flow forces applied on the cells.
Here, we extended the configurational entropy-based 8-chain constitutive model to include the effect of (1b) = −pI + ∇u + (∇u) T (2c) T = n = − pI + ∇u + (∇u) T n the interaction between cytoplasmic motion and cytoskeleton stretching in a cell to describe its mechanical behavior under externally applied forces.
In the 8-chain model developed by Arruda and Boyce (1993), we included cytoplasmic fluid in the cubic unit cell in addition to the chains arranged diagonally and deformed with the unit cell to characterize the mechanical behavior of cells. The rate form of strain energy function, in this case, consists of two parts: strain energy rate due to the configurational entropy of eight chains (Ẇ 8ch ) and entropy generation by cytoplasmic motion (Ẇ D ), i.e.
where W 8ch is the strain energy function of the eight chains, F is the deformation gradient tensor, is the absolute temperature, Ṡ fluid is the entropy generation by cytoplasmic motion, which is given by (Zhang, 2010) in which we denote V 0 =̇c hain L 0 , s = ( * L 2 0 )∕(R * 2 0 ), and where ̇c hain is the chain stretching rate, chain = � � 2 1 + 2 2 + 2 3 � ∕3 = √ I 1 ∕3 , L 0 is the macromolecular chain length, * is a reference fluid viscosity, R * is the difference between the inner and outer radius of the fluid pipe surrounding the macromolecular chain, r is its radial coordinate, v is the radial velocity, = 1, 0 is the ambient temperature in Kelvin, Λ = (2 3 V 2 0 )∕( * R * 2 ), in which 3 is a fluid material parameter (see Zhang, 2010, for details).
The second-order Piola-Kirchhoff stress tensor, R S, is given by Arruda and Boyce (1993) as, where N is the chain density, is Boltzmann's constant, K is the bulk modulus, J = det(F), B kk = I 1 = trace(B), B ij = F ik F jk , and with m being a material constant.
The second-order Piola-Kirchhoff stress tensor, D S, is determined by the entropy generation due to cytoplasmic motion (Zhang, 2010). Considering the rate of Ī 1 = I 1 ∕J 2∕3 , i.e.
Equation 4 can be written as Therefore, Thus the second-order Piola-Kirchhoff stress, S = R S + D S, is then given by The material constants used in the current work are given by K = 28.304 ⋅ 10 3 Pa, N = 1.4152 × 10 3 Pa, m = 5.15, s = 1.098 × 10 −5 , and V r = 0.01.

Fluid flow pattern
Overall, the fluid flow under prescribed rotational velocity at the top surface of the fluid is laminar. However, at cellular level around the seeded cells at the bottom, velocities, and their gradients vary abruptly. The local irregularity plays a significant role in accurately quantifying the mechanical response of cells to the fluid flow stimuli. As shown in Figure  while at the two sides of the cell, the flow is still developing and thus results in the different v r pattern from those at other z locations. Similar irregular pattern occurs in velocity v along r-axis direction as shown in Figure 4. Along the cell surface for a given r, v displays nonlinearity as function of z. However, this irregularity is very localized, as z increases beyond the cell surface range, v returns to linear variation as shown in Figure 5. Obviously, the nonlinearity in v r and v results in the local irregularity of the vorticities r , , and z .

Fluid flow stresses
The forces applied on the cell surface by the fluid flow are the total traction T r , T , and T z in r, , and z directions, respectively. The traction applied on the cell surface in r-axis direction, T r , has opposite signs on the two sides of the cell surface due to the fluid flow in r-axis direction. On the side of the cell surface facing fluid flow, T r is in compression and its magnitude decreases along the cell surface from the edge to the top and becomes zero at the top of the cell surface. On the opposite side of the cell surface, T r is in tension and increases from zero at the top of the surface to a maximum value at the cell edge (see Figure 6). The distribution of T r along the cell surface is nonlinear. Figure 6 shows that the magnitude of T r is also a function of the cell radial location with respect to z-axis. Cells closes to z-axis have larger T r value than those with longer radial distance from z-axis. The traction T r at r c = 2(mm) for ω = 4π T r at r c = 4 T r at r c = 6 T r at r c = 8 T r at r c = 10 T z at r c = 2(mm) for ω = 4π T z at r c = 4 T z at r c = 6 T z at r c = 8 T z at r c = 10 T z applied on the cell surface is in tension due to the negative pressure generated by the rotational flow as shown in Figure 7. The distribution of T z along the cell surface is parabolic with slight asymmetric due to the pressure variation on the surface along r-axis direction. Figure 7 shows the variation of T z with cell location in radial direction. T z has larger value for the cells near z-axis than those distant cells. The traction of T applied on the cell surface is shear stress in direction. Its distribution along the cell surface has similar trend as that of T z (see Figure 8). However, the variation of T with cell radial location is opposite to that of T r and T z .

Mechanical response of cells
Under the external forces of fluid flow described above, cells undergo complex response within their body. Figure 9 shows the stresses ( r , , z , rz , z ) map within a cross section of a cell. These results indicate that the variation of the normal stress components r , , z are quite similar and the magnitudes are close. This is because the magnitude of the total normal fluid forces applied on the cell surface, T r and T z , are within the same range. The variation of the magnitude of the these normal stresses within the cell is about 0.5 Pa. The variation of the shear stress rz within the cell is about 0.12 Pa, which is basically divided into two regions (higher at left and lower at right). The variation of the shear stress z within the cell is about 0.15 Pa and the distribution is basically uniform. The magnitude of the shear stress z is a little bit higher than rz .

Detachment stress
The average detachment stresses at the cell-substrate interface were calculated by dividing the reaction forces RF r , RF , and RF z applied on the base of the cell by the contact area on the substrate. The variation of these stresses with cell radial location from z-axis is shown in Figure 10. The normal stress z decreases with the increase in r parabolically (not show in Figure 10), which is mainly caused by the pressure variation along r-axis direction. The normal stress z is in tension and has two functions in removing cells from its substrate: lifting and reducing the contact area between the cell and its substrate. The two shear stress components rz and z applied on the cell-substrate interface increase with the cell radial distance from z-axis. The total shear stress on the interface is calculated by which is also shown in Figure 10. The main contribution to rz is from fluid pressure and fluid flow in r-axis direction. z , rz , and local irregularity had not been considered in most of the current analytical equations in evaluating cell detachment stress due to complexity nature. For accurate quantification of T φ at r c = 2(mm) for ω = 4π T φ at r c = 4 T φ at r c = 6 T φ at r c = 8 T φ at r c = 10 cell detachment stress, the effects of z , rz , and local irregularity should be considered. A group of -r curves can be constructed for different rotational velocities from the finite element simulation. From these curves, cell adhesion can be determined once the critical detachment radius R c and the corresponding rotational velocity measured from the experiment are given Rocha et al. (2010). These curves can also be used to determine which cells would be detached under given conditions using a cell adhesion model (Gallant & Garcia, 2007).

Conclusions
A rotational flow apparatus for the measurement of cell adhesion was simulated using the finite element method. From the simulation results, it is concluded that • The finite element simulation could capture the local irregularities at cellular level around the adherent cells and their effects on fluid flow velocities, vorticities, and fluid flow stresses applied on the cell surface.
• The finite element method is able to simulate the interaction between the fluid flow and the mechanical response of adhesive cells. Both the local irregularities around and the deformations of the cells can be considered in the simulation such that an accurate detachment force can be determined.
• A group of curves of the detachment stresses versus cell radial distance from z-axis can be constructed for different rotational velocities. From these curves, the detachment stress can be determined for a particular cell/substrate once the experimentally measured critical detachment radius and the corresponding rotational velocity are given.