Research Papers

Determination of Cellular Tractions on Elastic Substrate Based on an Integral Boussinesq Solution

[+] Author and Article Information
Jianyong Huang, Xiaoling Peng, Lei Qin, Tao Zhu, Jing Fang

Department of Biomedical Engineering and Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, PR China

Chunyang Xiong1

Department of Biomedical Engineering and Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, PR Chinacyxiong@pku.edu.cn

Youyi Zhang

Key Laboratory of Molecular Cardiovascular Sciences of Education Ministry, Institute of Vascular Medicine, Peking University Third Hospital, Beijing 100083, PR China


Corresponding author.

J Biomech Eng 131(6), 061009 (Apr 29, 2009) (9 pages) doi:10.1115/1.3118767 History: Received June 28, 2008; Revised December 19, 2008; Published April 29, 2009

Cell-substrate interaction is implicated in many physiological processes. Dynamical monitoring of cellular tractions on substrate is critical in investigating a variety of cell functions such as contraction, migration, and invasion. On account of the inherent ill-posed property as an inverse problem, cellular traction recovery is essentially sensitive to substrate displacement noise and thus likely produces unstable results. Therefore, some additional constraints must be applied to obtain a reliable traction estimate. By integrating the classical Boussinesq solution over a small rectangular area element, we obtain a new analytical solution to express the relation between tangential tractions and induced substrate displacements, and then form an alternative discrete Green’s function matrix to set up a new framework of cellular force reconstruction. Deformation images of flexible substrate actuated by a single cardiac myocyte are processed by digital image correlation technique and the displacement data are sampled with a regular mesh to obtain cellular tractions by the proposed solution. Numerical simulations indicate that the 2-norm condition number of the improved coefficient matrix typically does not exceed the order of 100 for actual computation of traction recovery, and that the traction reconstruction is less sensitive to the shift or subdivision of the data sampling grid. The noise amplification arising from ill-posed inverse problem can be restrained and the stability of inverse solution is improved so that regularization operations become less relevant to the present force reconstruction with economical sampling density. The traction recovery for a single cardiac myocyte, which is in good agreement with that obtained by the Fourier transform traction cytometry, demonstrates the feasibility of the proposed method. We have developed a simple and efficient method to recover cellular traction field from substrate deformation. Unlike previous force reconstructions that numerically employ some regularization schemes, the present approach stabilizes the traction recovery by analytically improving the Green’s function such that the intricate regularizations can be avoided under proper conditions. The method has potential application to a real-time traction force microscopy in combination with a high-efficiency displacement acquisition technique.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

Schematic illustration of a grid of elements covering the substrate surface actuated by cell. On each small area element, consisting of 2l×2b pixels, a tangential force intensity p acts uniformly (along×direction here).

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Figure 2

Surface displacement field of the elastic substrate induced by the force intensity integrated on an element of Fig. 1. In the integration of the Boussinesq solution, we use the parameters of E=1000 Pa, υ=0.5, p=1 Pa(along×direction), l=2 μm, and b=1 μm, respectively, with the dimensions and displacements presented in unit of microns. (a) The displacement component in the x-direction, showing the field is smoothly distributed with a peak value instead of an infinite discontinuity at the origin. (b) The displacement component in the y-direction, presenting an asymmetrical distribution about the coordinates. To compare the results with the classical Boussinesq solution, (c) and (d) show the curves of displacement component u distributed along the x-axis and the y-axis, respectively, presenting that a singularity exits at the origin of the displacement field resulting from the equivalent pointlike force, but the displacements from the improved solution are smoothly distributed with finite values. The vertical lines indicate the edge positions of the element, showing the displacements from both solutions are coincided with each other when the positions leave the force element.

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Figure 3

(a) A simulation of displacement field produced by two pairs of balanced forces contracting in the opposite directions (mesh element size: 8×8 pixels). Different colors denote the absolute magnitude of the surface displacements with the unit of microns. (b) The result of traction recovery based on the proposed approach, showing that the traction intensity is fully reconstructed without any nonzero stress components exterior to the loading regions.

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Figure 4

A statistic relation between the condition number of the matrix g (or g−1) and the dimension of the grid element, resulting from computational statistics for an image with 120×100 pixels. The result shows that the 2-norm condition number has a base-2 logarithmic growth approximately with the decrease in the sampling element in size and predicts that the number is less than the order of several hundreds even when the size of sampling mesh is reduced to unit pixels.

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Figure 5

Traction recovery of a single cardiac myocyte using the proposed method. (a) A phase-contrast picture of neonatal rat ventricular myocyte, in which the underlying flexible substrate is composed of polyacrylamide gel. (b) The displacement field of the cell-substrate matter computed by the DIC processing of the fluorescence images, reflecting the substrate distortion actuated by the cell from total relaxation to complete contraction states. The colors show the absolute magnitude of the displacements in μm. The arrows denote the relative magnitude and direction of the displacement field. ((c) and (d)) The traction field calculated by the present approach and the FTTC method, respectively, showing that the stresses are mainly located on the cell edge zones, with the maximum traction stress of 321.60 Pa in (c) and that of 341.80 Pa in (d). The colors display the absolute magnitude of the tractions in Pa. The arrows show the relative magnitude and direction of the cellular tractions.




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