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Research Papers

Alignment Localization in Nonlinear Biological Media

[+] Author and Article Information
Leonard M. Sander

Professor of Physics & Complex Systems,
Randall Laboratory,
Department of Physics,
University of Michigan,
Ann Arbor, MI 48109
e-mail: lsander@umich.edu

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received December 13, 2012; final manuscript received April 3, 2013; accepted manuscript posted April 13, 2013; published online June 11, 2013. Assoc. Editor: Edward Sander.

J Biomech Eng 135(7), 071006 (Jun 11, 2013) (5 pages) Paper No: BIO-12-1614; doi: 10.1115/1.4024199 History: Received December 13, 2012; Revised April 03, 2013; Accepted April 13, 2013

Cells imbedded in biopolymer gels are important components of tissue engineering models and cancer tumor microenvironments. In both these cases, contraction of cells attached to the gel is an important phenomenon, and the nonlinear nature of most biopolymers (such as collagen) makes understanding the mechanics of the contraction a challenging problem. Here, we investigate a unique feature of such systems: a point source of contraction leads to substantial deformation of the environment, but large strains and large alignment of the fibers of the gel are confined to a small region surrounding the source. For fibroblasts in collagen-I, we estimate that the radius of this region is of order 90 μ. We investigate this idea using continuum estimates and a finite element code, and we point out experimental manifestations of the effect.

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Figures

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Fig. 1

Left panel: Schematic representation of the stress-strain relationship for collagen-I redrawn after Ref. [5]. Many biopolymers have this sort or strain stiffening. The break in the curve, marked σc in the figure, occurs at a strain of about 10% and a stress of about 1 Pa for collagen-I. Right panel: The small strain regime on a log scale from the experiments reported in Ref. [6]. Both figures are for 2 mg/mL.

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Fig. 2

Invasion of a collagen gel by green fluorescent protein-labeled glioma cells starting from two tumor spheroids about 250 μ in diameter. This figure was kindly provided by T. Demuth and M. E. Berens. The cells invade preferentially between the spheroids instead of the more usual isotropic pattern. The fibers are strongly aligned in this region because of the cell traction of the two spheroids. For direct imaging of the alignment in a geometry of this sort, see Refs. [7] and [9].

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Fig. 3

A visualization of collagen-I fibers near a U87 tumor spheroid (upper left). The scale of the picture is 400 μ. Near the spheroid, the fibers are substantially aligned by the traction of the cells on the spheroid surface. Fibers are imaged by confocal reflectance. The tumor spheroid is the solid region, upper left. This figure was kindly provided by D. Vader and D. Weitz. For details on these and similar experiments, see Ref. [7].

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Fig. 4

The scheme for estimating the critical radius. The cell pulls in on the cavity of radius a, and the medium is aligned out to Rc.

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Fig. 5

Three-dimensional plots of the collagen network attached to two walls that represent tumor spheroids. Distances in microns. Left: Network of collagen fibers as measured before stress is applied. The size of the sample is 120 μ. Right: Result of finite element elasticity calculation after the wall at x = 0 is displaced by 80 μ.

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Fig. 6

Our representation of a “cell” interacting with the simulated network. The heavy lines are the filopods, which are rigid rods attached to the network at random points near the cell. Then, the rods are decreased in length by a fixed fraction and the network allowed to come to equilibrium. Simulations of this type give the results in Fig. 7.

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Fig. 7

Log-log plot of the energy stored in the fibers of the simulated collagen network as a function of distance from the center of stress. Here, ro = 1 μ. Left: stretching energy of the fibers. Right: bending energy of the fibers, showing individual data points for all the simulations used. All the energies fall off very quickly at around 70–90 μ.

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