The micropipette aspiration technique has been used extensively in recent years to measure the mechanical properties of living cells. In the present study, a boundary integral formulation with quadratic elements is used to predict the elastic equilibrium response in the micropipette aspiration contact problem for a three-dimensional incompressible spherical continuum cell model (Young’s modulus E). In contrast to the halfspace model [19], the spherical cell model accounts for nonlinearities in the cell response which result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region and curvature of the edges of the micropipette (inner radius a, edge curvature radius ε). The efficiency of the boundary element method facilitates the quantification of cell response as a function of the scaled pressure p/E, for the range of parameters a/R=0.4-0.7,ε/a=0.02-0.08, in terms of two measures that can be quantified using video microscopy. These are the aspiration length, which measures projection of the cell into the micropipette, and a characteristic strain, which measures stretching along the symmetry axis. For both measures of cell response, the resistance to aspiration is found to decrease with increasing values of the aspect ratio a/R and curvature parameter ε/a, and the nonlinearities in the cell response are most pronounced in the earlier portion of the aspiration test. The aspiration length is found to exhibit less sensitivity to the aspect ratio a/R than to the curvature parameter ε/a, whereas the characteristic strain, which provides a more realistic measure of overall cell stiffness, exhibits sensitivity to the aspect ratio a/R. The resistance to aspiration in the spherical cell model is initially less than that of the half space model but eventually exceeds the halfspace prediction and the deviation between the two models increases as the parameter ε/a decreases. Adjustment factors for the Young’s modulus E, as predicted by the halfspace model, are presented and the deviation from the spherical cell model is found to be as large as 35%, when measured locally on the response curve. In practice, the deviation will be less than the maximum figure but its precise value will depend on the number of data points available in the experiment and the specific curve-fitting procedure. The spherical cell model allows for efficient and more realistic simulations of the micropipette aspiration contact problem and quantifies two observable measures of cell response that, using video microscopy, can facilitate the determination of Young’s modulus for various cell populations while, simultaneously, providing a means of evaluating the validity of continuum cell models. Furthermore, this numerical model may be readily extended to account for more complex geometries, inhomogeneities in cellular properties, or more complex constitutive descriptions of the cell.
Dependence of Young’s modulus of vertically aligned carbon nanotubes on their aspect ratio
