A Finite Element Prediction of Strain on Cells in a Highly Porous Collagen-Glycosaminoglycan Scaffold

Confocal microscopy of live imaged cells (highlighted by CMFDA cell tracker) in a hydrated GAG-scaffold (highlighted by Alexa Fluor 633). Cell attachment was quantified based on (A) single-strut attachment or (B) multiple strut attachment. (C) and (D) illustrate the respective FE representation.

A graphical representation of GAG-scaffold shrinkage relative to time. Growth factors used for chondrogenesis were 10ng∕mlTGF‐β1, 50μm ascorbic acid, and 100nm dexamethasone; the samples were seeded on each side with 8×105 cells.

(a) GAG-scaffold idealized in the form of a 3D tetrakaidecahedral lattice using beam elements, (b) an exploded view of a regular pore, (c) an irregular pore following a distortion of the vertices (range of distortion±pore size∕5), and (d) the effective representation of the strut elements (nine elements per strut), with variation in element thickness along strut lengths. Note that (a)–(c) in this figure depict beam elements as standardized lines, whereas in reality, the beam elements possessed a thickness in accord with the strut thicknesses observed in Fig. 1; the summation of the element volumes were used to determine the solid fraction in the porosity calculations.

A log-log plot illustrating the relationship of the density with the macroscopic-microscopic moduli ratio of a GAG-scaffold. Presented here is foam theory (and FE̱MODEḺUNIFORM), the experimental results observed by Harley (37) and the FE model(s). FE̱MODEḺDV refers to a tetrakaidecahedral lattice FE mesh with distorted vertices (±pore size∕5), and FE̱MODEḺDV̱VST refers to a tetrakaidecahedral lattice FE mesh with distorted vertices and variable strut thicknesses (range=5:1). Note the more irregular and inhomogeneous the FE model, the closer to the experimental results observed by Harley (37).

A comparison of the von Mises stress (Pa) gradients post 10% uniaxial strain for Simulations 1 and 4; note the highly disorganized and deformed architecture of Simulation 4.

Histograms illustrating the distribution of cellular strain induced by 10% tensile straining of a GAG-scaffold idealized in the form of a tetrakaidecahedral lattice with distorted vertices and long-term in situ stress (pore size=57.89μm; range=13.4μm at t=7); note that the strain reported is the maximum absolute value of the maximum or minimum principal strain depending which has the largest magnitude, (a) depicts two-noded-elements (mean=1×106, max=45×106, and min=−40×106), while (b) illustrates three-noded elements (mean=0.02, max=0.49, and min=−0.26).

Histograms to illustrate the distribution of cellular strain induced by 10% tensile straining of a GAG-scaffold, note the strain reported is the maximum absolute value of the maximum or minimum principal strain depending which has the largest magnitude; (a) idealized in the form a tetrakaidecahedral lattice (Simulation 1: mean=0.025±0.07 STD, max=0.30, and min=−0.22) and (b) idealized in the form of a tetrakaidecahedral lattice with distorted vertices and long-term in situ stress (Simulation 4: mean=0.01±0.06 STD, max=0.49, and min=−0.26).

A graphical comparison of the salient cellular strain statistics calculated by the four different simulations. The strains reported are (i) the mean value of the maximum magnitude principal strains, (ii) the highest maximum principal strain, (iii) the lowest minimum principal strain, and (iv) the two standard deviation value of the combined maximum and minimum principal strains.

An illustration of the loading imparted by the struts onto the respective cells during macroscopic deformation of the scaffold; (a) represents a cell attached to a single-strut, (b) depicts a cell bridging two or more struts, (c) shows the statistics of cellular strains for a single-strut configuration, while (d) demonstrates the effect of cell length and pore size on the cellular strain, and (e) illustrates a tetrakaidecahedron in an undeformed state and a post 10% uniaxial tensile strain state with von Mises (Pa) stress gradients. The strain presented is the absolute value of the maximum or minimum principal strain. Note that the current FE investigation used an effective cell length ∕ pore size ratio of ∼0.3 (cell length=17.2μm; pore size=59μm).