5 Multiphasic models of cell mechanics

1989; Dong et al., 1991; Needham and Hochmuth, 1992; Karcher et al., 2003). In other approaches, the elastic behavior of the cell has been described using structural models such as the “tensegrity” approach (Ingber, 2003). Most such models have employed constitutive models that assume cells consist of a single-phase material (that is, fluid or solid), as detailed in Chapters 4, and 6. However, a number of recent studies have developed models of single cells and cell–matrix interactions that use multiphasic constitutive laws to account for interactions among solid, fluid, and in some cases, ionic phases of cells. The goals of such studies have been to characterize the relative importance of the mechanisms accounting for empirically observed phenomena such as cell volume change under mechanical or osmotic loading, the mechanistic basis responsible for cell viscoelasticity, and the coupling of various mechanical, chemical, and electrical events within living cells. The presence of these behaviors, which arise from interactions among different phases, often cannot be described by single-phase models.

The cell cytoplasm may consist of varying concentrations of water, charged or uncharged macromolecules, ions, and other molecular components. Furthermore, due to the highly charged and hydrated nature of its various components (Maughan and Godt, 1989; Cantiello et al., 1991), the cytoplasm’s gel-like properties have been described under several different contexts see for example Chapter 7 and Pollack (2001). Much of the supporting data for the application of multiphasic models of cells has come from the study of volumetric and morphologic changes of cells in response to mechanical or osmotic loading. The majority of work in this area has been performed on cells of articular cartilage (chondrocytes), likely due to the fact that these cells are embedded within a highly charged and hydrated extracellular matrix that has been modeled extensively using multiphasic descriptions. For example (see Fig. 5-1), chondrocytes in articular cartilage exhibit significant changes in shape and volume that occur in coordination with the deformation and dilatation of the extracellular matrix (Guilak, 1995; Guilak et al., 1995; Buschmann et al., 1996). By using generalized continuum models of cells and tissue, the essential characteristics of cell and tissue mechanics and their mechanical interactions can be better understood. In this chapter, we describe several experimental and theoretical approaches for studying the multiphasic behavior of living cells.

Biphasic (solid–fluid) models of cell mechanics

Viscoelastic behavior in cells can arise from both flow-dependent (fluid–solid interactions and fluid viscosity) and flow-independent mechanisms (for example, intrinsic viscoelasticity of the cytoskeleton). Previous studies have described the cytoplasm of “solid-like” cells as a gel or as a porous-permeable, fluid-saturated meshwork (Oster, 1984; Oster, 1989; Pollack, 2001) such that the forces within the cell exhibit a balance of stresses arising from hydrostatic and osmotic pressures and the elastic properties of the cytoskeleton. This representation of cell mechanical behavior is consistent with the fundamental concepts of the biphasic theory, which has been used to represent the mechanical behavior of soft hydrated tissues as being that of a two-phase material. This continuum mixture theory approach has been adopted in several studies to model volumetric and viscoelastic cell behaviors and to investigate potential mechanisms

86 F. Guilak et al.

Fig. 5-1. Three-dimensional reconstructions of viable chondrocytes within the extracellular matrix before (left) and after (right) compression of the tissue to 15% surface-to-surface tissue strain. Significant changes in chondrocyte height and volume were observed, showing that cellular deformation was coordinated with deformation of the tissue extracellular matrix.

responsible for cell mechanical behavior (Bachrach et al., 1995; Shin and Athanasiou, 1999; Guilak and Mow, 2000; Baaijens et al., 2005; Trickey et al., 2006).

Modern mixture theories (Truesdell and Toupin, 1960; Bowen, 1980) provide a foundation for multiphasic modeling of cell mechanics as well as of soft hydrated tissues. The biphasic model (Mow et al., 1980; Mow et al., 1984), based on Bowen’s theory of incompressible mixtures (Bowen, 1980), has been widely employed in modeling the mechanics of articular cartilage and other musculoskeletal tissues, such as intervertebral disc (Iatridis et al., 1998), bone (Mak et al., 1997), or meniscus. (Spilker et al., 1992). In such models, the cell or tissue is idealized as a porous and permeable solid material that is saturated by a second phase consisting of interstitial fluid (water with dissolved ions). Viscoelastic behavior can arise from intrinsic viscoelasticity of the solid phase, or from diffusive drag between the solid and fluid phases.

In the biphasic theory, originaly developed to describe the mechanical behavior of soft, hydrated tissues (Mow et al., 1980), the momentum balance laws for the solid and fluid phases, respectively, are written as:

where σs and σ f are partial Cauchy stress tensors that measure the force per unit mixture area on each phase. The symbol Π denotes a momentum exchange vector that accounts for the interphase drag force as fluid flows past solid in the mixture. Note that, in biphasic models of cells or cartilage, the contribution of inertial terms to the momentum balance equations is negligible, as the motion is dominated by elastic deformation and diffusive drag and occurs at relatively low frequencies. The mixture is assumed to be intrinsically incompressible and saturated, so that:

us is the solid displacement, v f is the fluid velocity, and φs is the solid volume fraction. For example, under the assumption of infinitesimal strain, with isotropic solid phase and inviscid fluid phase, while the momentum exchange is described by Darcy’s Law, the resulting constitutive laws are:

σs = −φs pI + λs tr (e)I + 2µs e, σ f = −φ f pI, Π = K (v f − u˙ s ) (5.3)

where I is the identity tensor, p is a pore pressure used to enforce the incompressibility constraint, e = 1/2[ us + ( us )T ] is the infinitesimal strain tensor, λs, µs are Lam´e coefficients for the solid phase, and K is a diffusive drag coefficient. The Lam´e coefficients λs, µs are associated with “drained” elastic equilibrium states that occur under static loading when all fluid flow has ceased in the mixture. An alternate set of elastic moduli are the Young’s modulus E s and Poisson ratio νs (0 ≤ νs < 0.5) where

µs = E s s is the solid phase shear modulus and λs =

2(1+ν )

For this linear biphasic model, by substituting Eq. 5.3 into Eq. 5.1, the governing equations Eq. 5.1 and Eq. 5.2 constitute a system of seven equations in the seven unknowns us, v f, p. The fluid velocity v f is commonly eliminated to yield a “u-p formulation” consisting of the four equations:

where k = (φ f )2/K is the permeability. The fluid velocity is then given by:

The governing equations Eqs. 5.4–5.5 illustrate a common formulation of the linear isotropic biphasic model. Within the framework of Eqs. 5.1–5.2, this fundamental model can be extended to account for additional mechanisms via modification of the constitutive relations in Eq. 5.3. Such mechanisms have included transverse isotropy of the solid phase, large deformation, solid matrix viscoelasticity, nonlinear strain-dependent permeability, intrinsic fluid viscosity, and tension-compression nonlinearity (Lai et al., 1981; Holmes et al., 1985; Cohen et al., 1998).

Biphasic poroviscoelastic models of cell mechanics

In other approaches, both the flow-dependent and flow-independent viscoelastic behaviors have been taken into account to describe transient cell response to loading. This type of approach has been used previously to separate the influence of “intrinsic” viscoelastic behavior of the solid extracellular matrix of tissues such as articular cartilage from the timeand rate-dependent effects due to fluid–solid interactions in the tissue (Mak, 1986a; Mak, 1986b; Setton et al., 1993; DiSilvestro and Suh, 2002).

For example, in modeling the creep response of chondrocytes during both full and partial micropipette aspiration (Baaijens et al., 2005; Trickey et al., in press), it was found that an elastic biphasic model cannot capture the time-dependent response of chondrocytes accurately (Baaijens et al., 2005). To examine the relative contributions of intrinsic solid viscoelasticity (solid–solid interactions) as compared to biphasic viscoelastic behavior (fluid–solid interactions), a large strain, finite element simulation

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of the micropipette aspiration experiment was developed to model the cell using finite strain incompressible and compressible elastic models, a two-mode compressible viscoelastic model, a biphasic elastic, or a biphasic viscoelastic model.

Assuming isotropic and constant permeability, the governing equations Eq. 5.1 and 5.2 may be rewritten (Mow et al., 1980; Sengers et al., 2004) as:

· v − · k p = 0

where v denotes the solid velocity. If a viscoelastic model is used to investigate the time-dependent behavior, a two-mode model may be used. The stress tensor is split into an elastic part and a viscoelastic part:

If a finite strain formulation is used, a suitable constitutive model for the compressible elastic contribution can be given by

respect to the reference configuration, the volume ratio is given by J = det(F), and the right Cauchy-Green tensor by B = F ·FT . The material parameters κ and G denote the compressibility modulus and the shear modulus, respectively. The viscoelastic response is modeled using a compressible Upper Convected Maxwell model:

where the operator denotes the upper-convected time derivative (Baaijens, 1998), and Dd is the deviatoric part of the rate of deformation tensor, defined by:

Gv is the modulus and λ is the relaxation time of the viscoelastic mode.

Multiphasic and triphasic models (solid–fluid–ion)

In response to alterations in their osmotic environment, cells passively swell or shrink. The capability of the biphasic model to describe this osmotic response is limited to the determination of effective biphasic material parameters that vary with extracellular osmolality. The triphasic continuum mixture model (Lai et al., 1991) provides a framework that has the capability to more completely describe mechanochemical coupling via both mechanical and chemical material parameters in the governing equations. This model has been successfully employed in quantitative descriptions of mechanochemical coupling in articular cartilage, where the aggrecan of the extracellular matrix gives rise to a net negative fixed-charge density within the tissue. Similar approaches have been used to describe other charged hydrated soft tissues (Huyghe et al., 2003).