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TECHNICAL PAPERS: Soft Tissue

# A Structural Model for the Flexural Mechanics of Nonwoven Tissue Engineering Scaffolds

[+] Author and Article Information
George C. Engelmayr

Engineered Tissue Mechanics Laboratory, Department of Bioengineering and McGowan Institute for Regenerative Medicine, University of Pittsburgh, 100 Technology Drive, Suite 200, Pittsburgh, PA 15219

Michael S. Sacks1

Engineered Tissue Mechanics Laboratory, Department of Bioengineering and McGowan Institute for Regenerative Medicine, University of Pittsburgh, 100 Technology Drive, Suite 200, Pittsburgh, PA 15219msacks@pitt.edu

1

Corresponding author.

J Biomech Eng 128(4), 610-622 (Jan 23, 2006) (13 pages) doi:10.1115/1.2205371 History: Received November 12, 2004; Revised January 23, 2006

## Abstract

The development of methods to predict the strength and stiffness of biomaterials used in tissue engineering is critical for load-bearing applications in which the essential functional requirements are primarily mechanical. We previously quantified changes in the effective stiffness $(E)$ of needled nonwoven polyglycolic acid (PGA) and poly-L-lactic acid (PLLA) scaffolds due to tissue formation and scaffold degradation under three-point bending. Toward predicting these changes, we present a structural model for $E$ of a needled nonwoven scaffold in flexure. The model accounted for the number and orientation of fibers within a representative volume element of the scaffold demarcated by the needling process. The spring-like effective stiffness of the curved fibers was calculated using the sinusoidal fiber shapes. Structural and mechanical properties of PGA and PLLA fibers and PGA, PLLA, and 50:50 PGA/PLLA scaffolds were measured and compared with model predictions. To verify the general predictive capability, the predicted dependence of $E$ on fiber diameter was compared with experimental measurements. Needled nonwoven scaffolds were found to exhibit distinct preferred (PD) and cross-preferred (XD) fiber directions, with an $E$ ratio (PD/XD) of $∼3:1$. The good agreement between the predicted and experimental dependence of $E$ on fiber diameter $(R2=0.987)$ suggests that the structural model can be used to design scaffolds with $E$ values more similar to native soft tissues. A comparison with previous results for cell-seeded scaffolds (Engelmayr, G. C., Jr., , 2005, Biomaterials, 26(2), pp. 175–187) suggests, for the first time, that the primary mechanical effect of collagen deposition is an increase in the number of fiber-fiber bond points yielding effectively stiffer scaffold fibers. This finding indicated that the effects of tissue deposition on needled nonwoven scaffold mechanics do not follow a rule-of-mixtures behavior. These important results underscore the need for structural approaches in modeling the effects of engineered tissue formation on nonwoven scaffolds, and their potential utility in scaffold design.

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Topics: Fibers , Stiffness

## Figures

Figure 8

Predicted dependency of the (Ef)′∕Ef ratio (logarithmic scale) on the values of aeff and Λeff(a). Note that in the limit aeff goes to zero (i.e., straight fibers), Eq. 6 correctly predicts an (Ef)′∕Ef ratio of 1. Predicted dependency of E on the σ parameter of the fiber orientation distribution for 50:50 PGA/PLLA scaffolds (b).

Figure 6

Representative experimental moment-curvature plots for PD (a) and XD (b) specimens of 50:50 PGA/PLLA scaffold (엯). The experimental curves were bracketed by the lower and upper bounds calculated using Eqs. 2,5. The structural model (Eqs. 5,6) closely matched the experimental data.

Figure 7

E versus df(mean±SEM) measured for PD specimens of PGA scaffold subjected to surface hydrolysis in 1NNaOH solution. The model predictions were strikingly similar to the experimental results, thus providing additional verification of the predictive capacity of the model.

Figure 9

E results for cell-seeded nonwoven 50:50 PGA/PLLA scaffolds plotted versus collagen concentration (mean±SEM, n=6) (a). Adapted from Engelmayr (see Ref. 10) with permission by Elsevier Inc. Structural model predicted dependence of E on the fiber interbond arc length (b). Note that as the number of rigidly bonded fiber-fiber cross-over points increases (such as with P4HB dip coating or collagen accretion), the fiber interbond arc length decreases and the fiber effective stiffness (Ef)′ increases. Values of E for unseeded (short dashed line) and 9-week mesenchymal stem cell-seeded (long dashed line) nonwoven 50:50 PGA/PLLA scaffolds are indicated for comparison.

Figure 1

Representative photomicrograph of a crimped PGA fiber teased from a nonwoven PGA scaffold (40× original magnification, scale bar=100μm) (a). The image was calibrated to the fiber diameter and the intrinsic crimp wavelength Λ and amplitude a0 were measured using the Trace Measurement Mode of SigmaScan Pro (SPSS, Inc.). SEM image depicting the planar microstructure of a 50:50 PGA/PLLA scaffold (100× original magnification, scale bar=100μm) (b). The angular orientation of the PD was arbitrarily designated as Θ=0deg, with the orthogonal XD at Θ=±90deg. Schematic illustrating how the planar microstructure depicted in panel (b) relates to the three-dimensional structure of the needled nonwoven scaffold (c). The distance between the vertically oriented loops of fibers introduced by needling provides an intuitive basis for the RVE width b. The angular and numerical distribution of fibers throughout the RVE thickness t was assumed to be equal.

Figure 2

Schematic depicting the idealized structure used in quantifying the effective crimp (a). For each intervening fiber-fiber cross-over point (e.g., 1, 2, 3), a potential value for the effective crimp wavelength Λeff and amplitude aeff were determined from the lengths of the dashed line segments. Schematic showing how the idealized structure (a) was implemented to quantify Λeff and aeff in the SEM images (b).

Figure 3

Representative uniaxial stress-strain curves for single PGA (◻) and PLLA (▵) fibers (a). The longitudinal fiber stiffness was taken as the slope of the initial linear region of the stress-strain curve prior to yielding (b). The PGA fibers exhibited a significantly higher stiffness (p<0.05) and tensile strength (p<0.05) than the PLLA fibers. The PLLA fibers exhibited a significantly higher strain-to-failure (p<0.05) than the PGA fibers.

Figure 4

Mean load-deflection curves measured by three-point bending for both dry and wet PLLA (a), PGA (b), and 50:50 PGA/PLLA (c) scaffolds (n=6). Mean moment-curvature curves measured by three-point bending for both dry and wet PLLA (d), PGA (e), and 50:50 PGA/PLLA (f) scaffolds (n=6). E was calculated for each scaffold specimen via the Bernoulli-Euler moment-curvature relationship (Eq. 1) and then averaged to yield the mean value of E (Table 5). Differences between dry and wet scaffolds were not statistically significant.

Figure 5

The fiber orientation distribution (mean±SEM) for PGA (◻), PLLA (▵), and 50:50 PGA/PLLA (엯) as measured by SALS. The distributions were modeled using a normalized Gaussian distribution (Eq. 9) with the mean μ constrained to zero. Note that the distributions depicted for PGA and PLLA were scaled for presentation purposes (baseline indicated by dotted line), and that the integral from Θ=−90 to 90deg was exactly equal to 1 for each distribution.

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