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TECHNICAL PAPERS: Bone/Orthopedic

# A Mechanical Composite Spheres Analysis of Engineered Cartilage Dynamics

[+] Author and Article Information
Sean S. Kohles1

Kohles Bioengineering, 1731 SE 37th Avenue, Portland, OR 97214-5135; Reparative Bioengineering Laboratory, Dept. of Mechanical & Materials Engineering, Portland State University, Portland, OR 97207-0751; Department of Surgery, Division of Plastic & Reconstructive Surgery, Oregon Health & Science University, Portland, OR 97239-3098;ssk@kohlesbioengineering.com

Christopher G. Wilson

Bioengineering Program, Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0535

Lawrence J. Bonassar

Sibley School of Mechanical and Aerospace Engineering, Department of Biomedical Engineering, Cornell University, Ithaca, NY 14853-5201

1

Corresponding author.

J Biomech Eng 129(4), 473-480 (Nov 22, 2006) (8 pages) doi:10.1115/1.2746366 History: Received September 27, 2005; Revised November 22, 2006

## Abstract

In the preparation of bioengineered reparative strategies for damaged or diseased tissues, the processes of biomaterial degradation and neotissue synthesis combine to affect the developing mechanical state of multiphase, composite engineered tissues. Here, cell-polymer constructs for engineered cartilage have been fabricated by seeding chondrocytes within three-dimensional scaffolds of biodegradable polymers. During culture, synthetic scaffolds degraded passively as the cells assembled an extracellular matrix (ECM) composed primarily of glycosaminoglycan and collagen. Biochemical and biomechanical assessment of the composite (cells, ECM, and polymer scaffold) were modeled at a unit-cell level to mathematically solve stress-strain relationships and thus construct elastic properties ($n=4$ samples per seven time points). This approach employed a composite spheres, micromechanical analysis to determine bulk moduli of: (1) the cellular-ECM inclusion within the supporting scaffold structure; and (2) the cellular inclusion within its ECM. Results indicate a dependence of constituent volume fractions with culture time $(p<0.05)$. Overall mean bulk moduli were variably influenced by culture, as noted for the cell-ECM inclusion ($Kc‐m=29.7kPa$, $p=0.1439$), the cellular inclusion ($Kc=5.5kPa$, $p=0.0067$), and its surrounding ECM ($Km=373.9kPa$, $p=0.0748$), as well as the overall engineered construct ($K=165.0kPa$, $p=0.6899$). This analytical technique provides a framework to describe the time-dependent contribution of cells, accumulating ECM, and a degrading scaffold affecting bioengineered construct mechanical properties.

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## Figures

Figure 1

Hierarchical development of the mechanical composite sphere models. Here, scanning electron micrographs (SEMs) from a similar engineered cartilage design are shown as fill images (26) (courtesy of Mary Ann Liebert, Inc.). The applied and reaction loads at the construct scale (scaffold fibers at ×30 magnification) during confined compression are extended to a spherical element (cell, ECM, and scaffold constituents at ×1000 magnification). A single 10‐μm-diameter chondrocyte becomes the centerpiece for the unitcell construction characterized by either a cell-ECM inclusion surrounded by a homogeneous scaffold (Model 1) or a three phase extension where the single cell inclusion is surrounded by synthesized ECM and supported within a homogeneous scaffold mesh (Model 2). All boundary and continuity conditions are reduced to the unit cell when defining the stress-strain relationship during the radial stress state (σ0). In Model 2, a far-field perspective with external boundaries sufficiently distant from the inclusion is assumed to dominate the effective construct properties (Ks=K).

Figure 2

Division of engineered cartilage constructs at harvest. The initial 12.7‐mm-diameter construct was divided into three parts: a 6‐mm-diameter punch taken along a diametric axis, and two halves of the remaining piece. SEM fill image shown (×30 magnification).

Figure 3

Development of engineered cartilage in culture as described by time-dependent volume fractions (p<0.0001) from the composite spheres models. Mean ± standard deviation (SD) values are shown for n=4 samples per culture time point. Curves represent the application of dynamic composition models (3) which predict ECM synthesis concomitant with scaffold degradation.

Figure 4

(A) Mean biosynthesis rates of GAG content (± SD) indicated a variability in production throughout culture time (p<0.0001). (B) The dependency of synthesis rate on ECM content is demonstrated in both composite sphere models. Here, ECM content as defined by two volume fraction approaches indicates direct or inverse proportional effects depending on how ECM is included in the model. Trendlines and correlation coefficients are shown as applied to mean values (error bars are indicated in (A)).

Figure 5

(A) Time-dependent bulk modulus values (p=0.1439) of the cell-ECM inclusion as determined from the composite spheres model (Eq. 5) where the assumed Poisson ratios are νconstruct=0.4 and νscaffold=0.4. Mean ± standard error (SE) values are shown for n=4 samples per culture time point. (B) Construct modulus (K) including the variability due to the influence of νconstruct=0.1 or 0.4 (mean ± SE). There is a 95% confidence interval of the cell-ECM bulk modulus (Kc‐m) due to iterative variations in νconstruct=0.1 or 0.4 and νscaffold=0.1 or 0.4 as indicated by the gray bar, smoothed through time.

Figure 6

Variation of cell (p=0.0067) and ECM (p=0.0748) constituent bulk moduli over time. Comparison of Poisson ratio variability (νconstruct=0.1 or 0.4) on cell and ECM bulk moduli as calculated from Eqs. 8,9 (mean values, Km+SE and Kc−SE). The ECM modulus is statistically greater than the cell modulus at all analyzed values of νconstruct(p<0.05). Note, Km=0 at t=0 indicates that ECM was not present at cell seeding.

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