Haider Mansoor A, Guilak Farshid
Department of Mathematics, North Carolina State University, Raleigh 27695-8205, USA.
J Biomech Eng. 2002 Oct;124(5):586-95. doi: 10.1115/1.1504444.
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, 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 epsilon). 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, epsilon/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 epsilon/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 epsilon/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 epsilon/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.
近年来，微吸管抽吸技术已被广泛用于测量活细胞的力学性能。在本研究中，采用具有二次单元的边界积分公式来预测三维不可压缩球形连续体细胞模型（杨氏模量E）在微吸管抽吸接触问题中的弹性平衡响应。与半空间模型不同，球形细胞模型考虑了细胞响应中的非线性，这些非线性是由于考虑了包括有限细胞尺寸（半径R）、细胞边界曲率、细胞 - 微吸管接触区域的演变以及微吸管边缘曲率（内半径a，边缘曲率半径ε）等几何因素而产生的。边界元法的效率有助于在参数a/R = 0.4 - 0.7、ε/a = 0.02 - 0.08的范围内，根据可以通过视频显微镜量化的两种测量方法，将细胞响应量化为缩放压力p/E的函数。这两种测量方法分别是抽吸长度，它测量细胞向微吸管内的投影，以及特征应变，它测量沿对称轴的拉伸。对于细胞响应的这两种测量方法，发现抽吸阻力随着纵横比a/R和曲率参数ε/a值的增加而减小，并且细胞响应中的非线性在抽吸测试的早期部分最为明显。发现抽吸长度对纵横比a/R的敏感性低于对曲率参数ε/a的敏感性，而特征应变提供了更实际的整体细胞刚度测量方法，对纵横比a/R表现出敏感性。球形细胞模型中的抽吸阻力最初小于半空间模型，但最终超过半空间预测，并且随着参数ε/a减小，两个模型之间的偏差增大。给出了半空间模型预测的杨氏模量E的调整因子，并且当在响应曲线上局部测量时，发现与球形细胞模型的偏差高达35%。在实际中，偏差将小于最大值，但其精确值将取决于实验中可用的数据点数量和具体的曲线拟合程序。球形细胞模型允许对微吸管抽吸接触问题进行高效且更实际的模拟，并量化细胞响应的两种可观察测量方法，使用视频显微镜可以有助于确定各种细胞群体的杨氏模量，同时提供一种评估连续体细胞模型有效性的方法。此外，该数值模型可以很容易地扩展以考虑更复杂的几何形状、细胞特性的不均匀性或细胞更复杂的本构描述。
