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Research Papers

A Quantitative Interpretation of the Response of Articular Cartilage to Atomic Force Microscopy-Based Dynamic Nanoindentation Tests

[+] Author and Article Information
Matteo Taffetani

MOX, Politecnico di Milano and Fondazione
CEN–Centro Europeo di Nanomedicina,
Piazza Leonardo da Vinci, 32,
Milano 20133, Italy
e-mail: matteo.taffetani@polimi.it

Roberto Raiteri

Department of Informatics, Bioengineering,
Robotics, and System Engineering,
Università di Genova,
via Opera pia, 13,
Genova 16145, Italy
e-mail: roberto.raiteri@unige.it

Riccardo Gottardi

Ri.MED Foundation,
Palermo 90133, Italy
Department of Orthopaedic Surgery,
University of Pittsburgh,
Pittsburgh, PA 15260
e-mail: RIG10@pitt.edu

Dario Gastaldi

Department of Chemistry,
Materials and Chemical Engineering,
Politecnico di Milano,
Piazza Leonardo da Vinci, 32,
Milano 20133, Italy
e-mail: Dario.gastialdi@polimi.it

Pasquale Vena

Mem. ASME
Department of Chemistry Materials
and Chemical Engineering,
Politecnico di Milano,
Piazza Leonardo da Vinci, 32,
Milano 20133, Italy
IRCCS—Istituto OrtopedicoGaleazzi,
P.zzaR.Galeazzi4,
Milano 20161, Italy
e-mail: pasquale.vena@polimi.it

1Corresponding author.

Manuscript received December 9, 2014; final manuscript received March 15, 2015; published online June 2, 2015. Assoc. Editor: Amir Abbas Zadpoor.

J Biomech Eng 137(7), 071005 (Jul 01, 2015) (8 pages) Paper No: BIO-14-1615; doi: 10.1115/1.4030175 History: Received December 09, 2014; Revised March 15, 2015; Online June 02, 2015

In this paper, a quantitative interpretation for atomic force microscopy-based dynamic nanoindentation (AFM-DN) tests on the superficial layers of bovine articular cartilage (AC) is provided. The relevant constitutive parameters of the tissue are estimated by fitting experimental results with a finite element model in the frequency domain. Such model comprises a poroelastic stress–strain relationship for a fibril reinforced tissue constitution, assuming a continuous distribution of the collagen network orientations. The identification procedure was first validated using a simplified transversely isotropic constitutive relationship; then, the experimental data were manually fitted by using the continuous distribution fibril model. Tissue permeability is derived from the maximum value of the phase shift between the input harmonic loading and the harmonic tissue response. Tissue parameters related to the stiffness are obtained from the frequency response of the experimental storage modulus and phase shift. With this procedure, an axial to transverse stiffness ratio (anisotropy ratio) of about 0.15 is estimated.

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Figures

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Fig. 1

Schematic side-view of the AFM-DN setup representing the microcantilever, the spherical indenter, and the soft sample. The controlled displacement of the fixed end of the cantilever (h0), the cantilever deflection (hc), and the indentation depth (hs) are reported.

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Fig. 2

Overview of the numerical approach based on two models working in sequence and described in the central column: the abaqus model, where the equilibrium indentation depth is reached through a nonlinear simulation, and the matlab model, where the harmonic displacement is applied using a linearized constitutive relationship. The lateral columns report the AC constitutive equations used to feed the models: the transversely isotropic model and the continuum mixture model, respectively. It is worth noting that, for both the constitutive relationships, the two steps share the same mesh; in the case of the continuum mixture model, also the stiffness tangent matrix obtained at the end of the abaqus step is passed to the integration points in the matlab step.

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Fig. 3

Experimental data in terms of (a) the storage modulus versus frequency and (b) the tangent of the phase shift versus frequency for the three indentation depths analyzed. Mean values and standard deviation bars are plotted.

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Fig. 5

Best fitting results for all the analyzed experimental data, in terms of (a) E'*(f) and (b) tan(φf) when the fibril distribution model is used. Gray lines define experimental data (mean curves only) whereas black lines indicate the numerical results obtained by using the parameters of model AFIBERS.

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Fig. 4

Best fitting results for all the analyzed experimental data, in terms of (a) E'*(f) and (b) tan(φf) when the transversely isotropic model is used. Data for the three penetration depths are reported: black lines indicate the numerical results and gray lines represent the experimental data (mean curves only).

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