Cell contraction forces in scaffolds with varying pore size and cell density

Abstract

The contractile behavior of cells is relevant in understanding wound healing and scar formation. In tissue engineering, inhibition of the cell contractile response is critical for the regeneration of physiologically normal tissue rather than scar tissue. Previous studies have measured the contractile response of cells in a variety of conditions (e.g. on two-dimensional solid substrates, on free-floating tissue engineering scaffolds and on scaffolds under some constraint in a cell force monitor). Tissue engineering scaffolds behave mechanically like open-cell elastomeric foams: between strains of about 10 and 90%, cells progressively buckle struts in the scaffold. The contractile force required for an individual cell to buckle a strut within a scaffold has been estimated based on the strut dimensions (radius, r, and length, l) and the strut modulus, E(s). Since the buckling force varies, according to Euler's law, with r(4)/l(2), and the relative density of the scaffold varies as (r/l)(2), the cell contractile force associated with strut buckling is expected to vary with the square of the pore size for scaffolds of constant relative density. As the cell density increases, the force per cell to achieve a given strain in the scaffold is expected to decrease. Here we model the contractile response of fibroblasts by analyzing the response of a single tetrakaidecahedron to forces applied to individual struts (simulating cell contractile forces) using finite element analysis. We model tetrakaidecahedra of different strut lengths, corresponding to different scaffold pore sizes, and of varying numbers of loaded struts, corresponding to varying cell densities. We compare our numerical model with the results of free-floating contraction experiments of normal human dermal fibroblasts (NHDF) in collagen-GAG scaffolds of varying pore size and with varying cell densities.

Fig. 1

(a) Scanning electron micrograph of a collagen-GAG scaffold (reprinted from Pek et al., 2004, with permission from Elsevier). (b) Tetrakaidecahedral model of the structure. (c) Compressive stress-strain curve for the collagen-GAG scaffold (courtesy of Brendan Harley). (d) Compressive stress-strain curve for an open-cell polyurethane foam (reprinted from Gibson and Ashby, 1997, with permission).

Fig. 2

(a) Optical micrographs of a dermal fibroblast buckling a strut in a collagen-GAG scaffold. Scale bar 50 □ m. (b-e) Schematic of cell buckling the scaffold strut (reprinted from Freyman et al., 2001b, with permission from Elsevier.)

Fig. 3

Finite element results for buckling force per cell plotted against pore size, for different numbers of cells attached. The buckling force varies with the square of the pore size (eqn 8).

Fig. 4

(a) Finite element results for the buckling force per cell, on a single, isolated tetrakaidecahedron, plotted as a function of the number of cells attached per pore, for 4 different model pore sizes. The equations for the best-fit curves are given in the text (eqn 9). (b) The total buckling force plotted as a function of the number of cells attached per pore. (c) The buckling force per cell plotted as a function of the number of cells per matrix. The equations for the best-fit curves are given in the text (eqn 13). (d) The total buckling force plotted as a function of the number of cells per matrix.

Fig. 5

NHDF adhesion and proliferation with the 96 micron pore size collagen-GAG scaffolds. Data points indicate the average cell counts for all matrices seeded at a given cell density on a given day. Bars indicate one standard deviation. Lines show the best-fit curves for the equation $C=C_o\left(1-\exp \left(-\frac{t}{\tau }\right)\right)$. C₀ and □ are given in the text.

Fig. 6

Cell contraction results. Net strain in the scaffolds plotted against time for (a) 96 □ m (b) 110 □ m (c) 121 □ m and (d) 151 □ m pore size scaffolds.

Fig. 7

Experimental results for buckling force per cell plotted against pore size, for different number of cells attached. The best-fit equations are given in the text (eqn 17). Individual data points for each pore size are plotted. The average number of cells per pore is indicated with arrows next to the corresponding data points. The lines are best-fit curves extrapolated from the data set.

Fig. 8

(a) The force per cell plotted as a function of the number of cells attached per pore, for 4 different scaffold pore sizes at the point the scaffolds buckle. The best-fit equations are given in the text (eqn 18). (b) The total force plotted as a function of the number of cells attached per pore. (c) The force per cell plotted as a function of the number of cells per matrix. The best-fit curve for the cumulative data is (eqn 21) $F_{cell}=8.86\times {10}^5{N}_{matrix}^{-1}$, R² = 1. (d) The total force plotted as a function of the number of cells per matrix: F_total = 0.886mN.

Fig. 9

(a) The force per cell plotted as a function of the number of cells attached per pore, for 4 different scaffold pore sizes at the end of the experiment. The best-fit equations are given in the text (eqn 23). (b) The total force plotted as a function of the number of cells attached per pore. (c) The force per cell plotted as a function of the number of cells per matrix. The best-fit curve for the cumulative data is (eqn 25) $F_{cell}=4.04\times {10}^6{N}_{matrix}^{-1.02}$, R² = 0.989. (d) The total force plotted as a function of the number of cells per matrix.

Fig. 10

The force per cell plotted against the number of cells per matrix. The graph shows the finite element results (eqn 15, denoted FEA) as well as the experimental results, both at the onset of buckling (at 10% strain) (Exp10%□) and at the end of the experiment (ExpEnd).