Mapping three-dimensional stress and strain fields within a soft hydrogel using a fluorescence microscope

Abstract

Three-dimensional cell culture is becoming mainstream as it is recognized that many animal cell types require the biophysical and biochemical cues within the extracellular matrices to perform truly physiologically realistic functions. However, tools for characterizing cellular mechanical environment are largely limited to cell culture plated on a two-dimensional substrate. We present a three-dimensional traction microscopy that is capable of mapping three-dimensional stress and strain within a soft and transparent extracellular matrix using a fluorescence microscope and a simple forward data analysis algorithm. We validated this technique by mapping the strain and stress field within the bulk of a thin polyacrylamide gel layer indented by a millimeter-size glass ball, together with a finite-element analysis. The experimentally measured stress and strain fields are in excellent agreements with results of the finite-element simulation. The unique contributions of the presented three-dimensional traction microscopy technique are: 1), the use of a fluorescence microscope in contrast with the confocal microscope that is required for the current three-dimensional traction microscopes in the literature; 2), the determination of the pressure field of an incompressible gel from strains; and 3), the simple forward-data-analysis algorithm. Future application of this technique for mapping animal cell traction in three-dimensional nonlinear biological gels is discussed.

Figure 1

Microsphere indentation method. Schematics of a microsphere indenting on a thin polyacrylamide gel substrate. The contact point of the sphere with the un-deformed gel is defined as the origin or (0,0,0) coordinates of the system with z axis being in the vertical direction. Fluorescent beads embedded in the gel are displaced from their original positions (red dots) to their final positions (green dots) upon the indentation of the microsphere.

Figure 2

Three-dimensional defocused particle-tracking method. (A) A ray-tracing diagram of light traveling through a spherical lens. A point source of light in the focal plane emits rays that are bent by the lens and converge to a point at the image plane (left side). If the point source is displaced a distance z_fo from the focal plane, its rays do not converge and they produce a defocused ring on the image plane with diameter d (right side) due to spherical aberration. Representative images of a point source under each condition are inserted (bottom). (B) Experimentally derived calibration curve of fluorescent bead z_fo versus defocused-ring diameter d. Images of defocused rings at different z_fo are inserted (top).

Figure 3

Displacement field in the indented gel. (A) Combined images of fluorescent beads embedded in the un-deformed and deformed polyacrylamide gel. The red defocused rings indicate the original bead position in un-deformed gel; (green rings) positions of fluorescent beads within the deformed gel. (White arrows) The xy displacement of each bead in x-y plane upon indentation. Increase in ring size indicates displacement in the negative z direction. (B) Experimentally measured three-dimensional bead displacements. The lengths of the arrows are scaled by a factor of 0.7 for better illustration. (Circles) Initial position of beads; (ends of the arrows) final positions of the beads after the gel is indented. (C) Continuous displacement field of u_z/h and u_x/h at cross section y = 0 (side view). The displacement fields are derived from the discrete displacements shown in panel B using MLSIM.

Figure 4

Continuous displacement fields from experiments and FEM calculations. Contour plots of the continuous displacement field u_x/h (A and B) and u_z/h (C and D) at the cross section y = 0 and x > 0. The same color map is used for experiments and FEM calculations.

Figure 5

Strain fields from experiments and FEM calculations. Contour plots of strain field ε_xx (A and B), ε_yy (C and D), and ε_zz (E and F) at the cross section y = 0 and x > 0. The same color map is used for experiments and FEM calculations.

Figure 6

Stress field from experiments and FEM calculations. Contour plots of stress-field σ_xx/E (A and B) σ_zz/E (C and D), and σ_xz/E (E and F) at the cross section y = 0 and x > 0. The same color map is used for experiments and FEM calculations. (G) The normalized stresses versus x at the contact interface (y = 0 and z = 0). (Symbols) Experimental results; (lines) FEM results. (H) The normalized normal stress σ_zz/E, and 2με_zz/E at the contact interface (y = 0 and z = 0). (Symbols) Experimental results; (lines) FEM results.