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TECHNICAL PAPERS: Soft Tissue

Effect of Indenter Size on Elastic Modulus of Cartilage Measured by Indentation

[+] Author and Article Information
Narendra K. Simha, Mellanie L. Hall, Sidharth Chiravarambath

Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, Minnesota 55455

Hui Jin

Medtronic, Inc., Minneapolis, Minnesota 55432-5604

Jack L. Lewis1

Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, Minnesota 55455lewis001@umn.edu

1

Corresponding author.

J Biomech Eng 129(5), 767-775 (Feb 16, 2007) (9 pages) doi:10.1115/1.2768110 History: Received March 31, 2006; Revised February 16, 2007

Abstract

Our preliminary indentation experiments showed that the equilibrium elastic modulus of murine tibial cartilage increased with decreasing indenter size: flat-ended $60deg$ conical tips with end diameters of $15μm$ and $90μm$ gave $1.50±0.82MPa$ ($mean±standard$ deviation) and $0.55±0.11MPa$, respectively $(p<0.01)$. The goal of this paper is to determine if the dependence on tip size is an inherent feature of the equilibrium elastic modulus of cartilage as measured by indentation. Since modulus values from nonindentation tests are not available for comparison for murine cartilage, bovine cartilage was used. Flat-ended conical or cylindrical tips with end diameters ranging from $5μmto4mm$ were used to measure the equilibrium elastic modulus of bovine patellar cartilage. The same tips were used to test urethane rubber for comparison. The equilibrium modulus of the bovine patellar cartilage increased monotonically with decreasing tip size. The modulus obtained from the $2mm$ and $4mm$ tips $(0.63±0.21MPa)$ agreed with values reported in the literature; however, the modulus measured by the $90μm$ tip was over two and a half times larger than the value obtained from the $1000μm$ tip. In contrast, the elastic modulus of urethane rubber obtained using the same $5μm–4mm$ tips was independent of tip size. The equilibrium elastic modulus of bovine patellar cartilage measured by indentation depends on tip size. This appears to be an inherent feature of indentation of cartilage, perhaps due to its inhomogeneous structure.

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Figures

Figure 7

The factor κfec for the 190–650μm tips (flat-ended conical with cone angle=60deg) indenting urethane (thickness t=2mm, ν=0.5). For h∕b=0, κfec=κcyl.

Figure 1

SEM image of 15μm diameter flat-end diamond conical tip with 60deg included angle. The tip is diamond attached to stainless steel mount (not shown).

Figure 2

SEM image of 310μm diameter flat-end conical tip with 60deg included angle. The tip is stainless steel. Stainless steel tips generally had more geometrical imperfections than diamond tips.

Figure 3

Schematic of indentation of an elastic layer by a flat-ended cone of included angle α

Figure 4

Urethane indentation results for flat-ended conical tips with end diameters of 90–310μm. Measurements (solid line, using Nano XP) show an initial toe region, followed by a linear region (displacement<tipradius) where the flat-ended conical tip behaves like a cylindrical tip; the effects of the conical sides can be seen for the 90μm tip for larger displacements. The fit values (symbols, dashed curve) were obtained from Eq. 1 or 2 using the offset h0 that minimizes the least squares error (Eq. 3) and the corresponding modulus E (Eq. 4).

Figure 5

Equilibrium indent force versus tip displacement for bovine patellar cartilage obtained using 60deg conical tip with flat-end diameter of 190μm (Sample B). Fit values (open symbol, dashed curve) with offset h0=95μm show good comparison with measurements (filled rectangle, solid curve).

Figure 6

Equilibrium elastic modulus measured with tips ranging from 5μmto4mm for urethane and bovine patella cartilage (Experiment 1, samples A–C; Experiment 2, samples D–G). Young’s modulus of urethane (triangles, filled-ELF 3100, open-Nano XP) is scattered within 8% of the mean value (4.41MPa) but not monotonically dependent on tip size. Equilibrium elastic modulus of bovine patella cartilage increases as tip size decreases (Experiment 1, rectangles, filled-ELF 3100, open-Nano XP; Experiment 2, open diamonds and circles, NanoXP). Error bars are ±s.d.

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