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90F. Guilak et al.

configurations to the theoretical response as predicted by analytical or numerical models of each individual experimental configuration. Given the complexity of the governing equations for multiphasic or poroelastic models, analytical solutions have only been possible for simplified geometries that approximate various testing configurations. In most cases, numerical methods such as finite element techniques are used in combination with optimization methods to best fit the predicted behavior to the actual cellular response in order to determine the intrinsic material properties of the cell.

Micropipette aspiration

Cells

The micropipette aspiration technique has been used extensively to study the mechanical properties of both fluid-like and solid-like cells (Hochmuth, 2000), including circulating cells such as red blood cells (Evans, 1989) and neutrophils (Sung et al., 1982; Dong et al., 1991; Ting-Beall et al., 1993), or adhesion-dependent cells such as fibroblasts (Thoumine and Ott, 1997), endothelial cells (Theret et al., 1988; Sato et al., 1990), or chondrocytes (Jones et al., 1999; Trickey et al., 2000; Guilak et al., 2002). This technique involves the use of a small glass pipette to apply controlled suction pressures to the cell surface while measuring the ensuing transient deformation via video microscopy. The analysis of such experiments has required the development of a variety of theoretical models that assume cells behave as viscous liquid drops (Yeung and Evans, 1989), potentially possessing cortical tension (Evans and Yeung, 1989), or as elastic or viscoelastic solids (Theret et al., 1988; Sato et al., 1990; Haider and Guilak, 2000; Haider and Guilak, 2002) and specifically, to model the solid-like response of cells to micropipette aspiration, Theret et al. (1988) developed an elegant analytical solution of an associated contact problem to calculate the Young’s modulus (E ) of an incompressible cell. This elastic model was subsequently extended to a standard linear solid (Kelvin) model, thus incorporating viscoelastic cell properties (Sato et al., 1990). These models idealized the cell as an elastic or viscoelastic incom-

pressible and homogeneous half-space. Experimentally, the length of cell aspiration was measured at several pressure increments, and the Young’s modulus (E ) was determined as a function of the applied pressure ( p), the length of aspiration of the cell into the micropipette (L ), and the radius of the micropipette (a) as E = 3a p/(2π L ), where is a function of the ratio of the micropipette thickness to its inner radius (Fig. 5-2).

In recent studies, the micropipette aspiration test has been modeled assuming that cells exhibit biphasic behavior. The cell was modeled using finite strain incompressible and compressible elastic models, a two-mode compressible viscoelastic model, or a biphasic elastic or viscoelastic model. Comparison of the model to the experimentally measured response of chondrocytes to a step increase in aspiration pressure showed that a two-mode compressible viscoelastic formulation could predict the creep response of chondrocytes during micropipette aspiration (Fig. 5-3). Similarly, a biphasic two-mode viscoelastic analysis could predict all aspects of the cell’s creep response to a step aspiration. In contrast, a purely biphasic elastic formulation was not capable

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Fig. 5-2. Viscoelastic creep response of a chondrocyte to a step increase in pressure in the micropipette aspiration test. Images show the chondrocyte in the resting state under a tare pressure at time zero, followed by increasing cell displacement over time after step application of the test pressure. From Haider and Guilak, 2000.

(a)

 

(b)

 

 

 

Normalized Cell Displacement

 

 

 

 

Normalized Time

Fig. 5-3. (a) Biphasic, viscoelastic finite element mesh used to model the micropipette aspiration test. The mesh contains 906 elements and has 3243 nodes, with biquadratic interpolation for the displacement and a bilinear, continuous interpolation for the pressure. Along the left boundary, symmetry conditions are applied, while along the portion of the cell boundary within the micropipette, suction pressure is applied. Sliding contact conditions along the interface with the micropipette are enforced through the use of a Lagrange multiplier formulation. (b) Normalized aspiration length of a chondrocyte as a function of the normalized time. The solid dots indicate the experimental behavior of an average chondrocyte. The solid line corresponds to the two-mode viscoelastic model. The dashed line corresponds to the biphasic, two-mode viscoelastic model. For the latter model, the triangle marks the end of the pressure application and the start of the creep response. The creep response of the cell was well described by both the two-mode viscoelastic model and the biphasic viscoelastic model. From Baaijens et al., 2005.

92 F. Guilak et al.

Fig. 5-4. Micropipette aspiration test to examine the volumetric response of cells to mechanical deformation. Video images of a chondrocyte and micropipette before (a) and after (b) complete aspiration of the cell. Cells show a significant decrease in volume, which when matched to a theoretical model can be used to determine Poisson’s ratio as one measure of compressibility. From Trickey et al., 2006.

of predicting the complete creep response, suggesting that the viscoelastic response of the chondrocytes under micropipette aspiration is predominantly due to intrinsic viscoelastic phenomena and is not due to the biphasic behavior.

Other studies have also used the micropipette technique to determine the volume change of chondrocytes after complete aspiration into a micropipette (Jones et al., 1999). While many cells are assumed to be incompressible with a Poisson ratio of 0.5, these studies demonstrated that certain cells, such as chondrocytes, in fact exhibit a certain level of compressibility (Fig. 5-4), presumably due to the expulsion of intracellular fluid. Isolated cells were fully aspirated into a micropipette and allowed to reach mechanical equilibrium. Cells were then extruded from the micropipette and cell volume and morphology were measured over time. By simulating this experimental procedure with a finite element analysis modeling the cell as either a biphasic or viscoelastic material, the Poisson ratio and viscoelastic recovery properties of the cell were determined. The Poisson ratio of chondrocytes was found to be 0.38, suggesting that cells may in fact show volumetric changes in response to mechanical compression. The finding of cell compressibility in response to mechanical loading is consistent with previous studies showing significant loss of cell volume in chondrocytes embedded within the extracellular matrix (Guilak et al., 1995). Taken together with micropipette studies, these studies suggest that cell volume changes are due to biphasic mechanical effects resulting in fluid exudation from the cell, while cellular viscoelasticity is more likely due to intrinsic behavior of the cytoplasm and not to flow-dependent effects.

Biphasic properties of the pericellular matrix

In vitro experimental analysis of the mechanics of isolated cells provides a simplified and controlled environment in which theoretical models, and associated numerical solutions, can be employed to measure and compare cell properties via material parameters. Ultimately, however, most biophysical analyses of cell mechanics are motivated by a need to extrapolate the in vitro findings to a characterization of the physiological

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(in vivo) environment of the cell. For cells such as articular chondrocytes, it is possible to isolate a functional cell–matrix unit and analyze its mechanical properties in vitro, thus providing a link to the physiological setting (Poole, 1992). This modeling approach is briefly described here in the context of the articular chondrocyte.

Chondrocytes in articular cartilage are completely surrounded by a narrow region of tissue, termed the pericellular matrix (PCM). The PCM is characterized by the presence of type VI collagen (Poole, 1992), which is not found elsewhere in cartilage under normal circumstances, and a higher concentration of aggrecan relative to the extracellular matrix (ECM), as well as smaller amounts of other collagen types and proteins. The chondrocyte together with the pericellular matrix and the surrounding capsule has been termed the chondron (Poole, 1992; Poole, 1997). While the function of the PCM is not known, there has been considerable speculation that the chondron plays a biomechanical role in articular cartilage (Szirmai, 1974). For example, it has been hypothesized that the chondron provides a protective effect for the chondrocyte during loading (Poole et al., 1987), and others have suggested that the chondron serves as a mechanical transducer (Greco et al., 1992; Guilak and Mow, 2000).

To determine mechanical properties of the PCM, the solution of a layered elastic contact problem that models micropipette aspiration of an isolated chondron was developed. This theoretical solution was applied to measure an elastic Young’s modulus for the PCM in human chondrons isolated from normal and osteoarthritic sample groups (Alexopoulos et al., 2003). The mean PCM Young’s modulus of chondrons isolated from the normal group (66.5 ± 23.3 kPa) was found to be a few orders of magnitude larger than the chondrocyte modulus ( 1 kPa) and was found to drop significantly in the osteoarthritic group (41.3 ± 21.1 kPa, p < 0.001). These findings support the hypothesis that the PCM serves a protective mechanical role that may be significantly altered in the presence of disease. In a multiscale finite element analysis (Guilak and Mow, 2000), the macroscopic solution for transient deformation of a cartilage layer under a step load was computed and used to solve a separate microscale problem to detemine the mechanical environment of a single chondrocyte. In this study, the inclusion of a PCM layer in the microscale model significantly altered the mechanical environment of a single cell. A mathematical model for purely radial deformation in a chondron was developed and analyzed under dynamic loading in the range 0–3 Hz (Haider, 2004). This study found that the presence of a thin, highly stiff PCM that is less permeable than the chondrocyte enhances the transmission of compressive strain mechanical signals to the cell while, simultaneously, protecting it from excessive solid stress.

Using the micropipette aspiration test coupled with a linear biphasic finite element model, recent studies have reported the biphasic material properties of the PCM of articular chondrocytes (Alexopoulos et al., 2005) (Fig. 5-5). Chondrons were mechanically extracted from nondegenerate and osteoarthritic (OA) human cartilage. Micropipette aspiration was used to examine the creep behavior of the pericellular matrix, which was matched using optimization to a biphasic finite element model (Fig. 5-6). The transient mechanical behavior of the PCM was well-described by a biphasic model, suggesting that the viscoelastic response of the PCM is attributable to flow-dependent effects, similar to that of the ECM. With osteoarthritis, the mean