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Research Papers

# A Phenomenological Approach Toward Patient-Specific Computational Modeling of Articular Cartilage Including Collagen Fiber Tracking

[+] Author and Article Information
David M. Pierce

Institute of Biomechanics, Center of Biomedical Engineering, Graz University of Technology, Kronesgasse 5-I, 8010 Graz, Austriapierce@tugraz.at

Werner Trobin

Institute for Computer Graphics and Vision, Graz University of Technology, Inffeldgasse 16-II, 8010 Graz, Austriatrobin@icg.tugraz.at

Siegfried Trattnig

Department of Radiology, Center of Excellence for High Field MR, Medical University of Vienna, Lazarettgasse 14, 1090 Vienna, Austriasiegfried.trattnig@akhwien.at

Horst Bischof

Institute for Computer Graphics and Vision, Graz University of Technology, Inffeldgasse 16-II, 8010 Graz, Austriabischof@icg.tugraz.at

Gerhard A. Holzapfel1

Institute of Biomechanics, Center of Biomedical Engineering, Graz University of Technology, Kronesgasse 5-I, 8010 Graz, Austria; Department of Solid Mechanics, School of Engineering Sciences, Royal Institute of Technology (KTH), Osquars Backe 1, 100 44 Stockholm, Swedenholzapfel@tugraz.at

Ethical approval for this study was granted by the Ethics Committee of the Medical University of Vienna.

Refined around the indentation location.

1

Corresponding author.

J Biomech Eng 131(9), 091006 (Aug 05, 2009) (12 pages) doi:10.1115/1.3148471 History: Received October 13, 2008; Revised April 27, 2009; Published August 05, 2009

## Abstract

To model the cartilage morphology and the material response, a phenomenological and patient-specific simulation approach incorporating the collagen fiber fabric is proposed. Cartilage tissue response is nearly isochoric and time-dependent under physiological pressure levels. Hence, a viscoelastic constitutive model capable of reproducing finite strains is employed, while the time-dependent deformation change is purely isochoric. The model incorporates seven material parameters, which all have a physical interpretation. To calibrate the model and facilitate further analysis, five human cartilage specimens underwent a number of tests. A series of magnetic resonance imaging (MRI) sequences is taken, next the cartilage surface is imaged, then mechanical indentation tests are completed at 2–7 different locations per sample, resulting in force/displacement data over time, and finally, the underlying bone surface is imaged. Imaging and mechanical testing are performed with a custom-built robotics-based testing device. Stereo reconstruction of the cartilage and subchondral bone surface is employed, which, together with the proposed constitutive model, led to specimen-specific finite element simulations of the mechanical indentation tests. The force-time response of 23 such indentation experiment simulations is optimized to estimate the mean material parameters and corresponding standard deviations. The model is capable of reproducing the deformation behavior of human articular cartilage in the physiological loading domain, as demonstrated by the good agreement between the experiment and numerical results ($R2=0.95±0.03$, $mean±standard$ deviation of force-time response for 23 indentation tests). To address validation, a sevenfold cross-validation experiment is performed on the 21 experiments representing healthy cartilage. To quantify the predictive error, the mean of the absolute force differences and Pearson’s correlation coefficient are both calculated. Deviations in the mean absolute difference, normalized by the peak force, range from 4% to 90%, with $40±25%$$(M±SD)$. The correlation coefficients across all predictions have a minimum of 0.939, and a maximum of 0.993 with $0.975±0.013$$(M±SD)$, which demonstrates an excellent match of the decay characteristics. A novel feature of the proposed method is 3D sample-specific numerical tracking of the fiber fabric deformation under general loading. This feature is demonstrated by comparing the estimated fiber fabric deformation with recently published experimental data determined by diffusion tensor MRI. The proposed approach is efficient enough to enable large-scale 3D contact simulations of knee joint loading in simulations with accurate joint geometries.

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## Figures

Figure 1

Schematic representation of the layered structure of the Type II collagen fibers within articular cartilage (1)

Figure 2

Overall view (a) and detail (b) of the test setup used to perform in vitro relaxation experiments on articular cartilage specimens

Figure 3

Features of the computational model used for the analysis: (a) finite element mesh; (b) collagen fiber directions as element input, from the same perspective as (a); and (c) cross-section close-up of the collagen fiber fabric taken from the cutting plane indicated in (a)

Figure 4

Schematic representation for the physical interpretation of the material parameters and subsequent extraction from experimental data

Figure 5

Reaction force versus compression response of 23 indentation tests performed on five different human articular cartilage specimens.

Figure 6

Box plots of the variation in the resulting parameter sets for human articular cartilage with k1=0.425 MPa and k2=39.8: (a) μ [MPa], (b) βm [–], (c) τm [sec], (d) βf [–], and (e) τf [sec]

Figure 7

Representative comparison: (a) experimentally measured reaction force versus time response and corresponding numerical simulation of specimen 3; (b) test performed at a corner location

Figure 8

Qualitative comparison of (a) experimentally determined fiber fabric deformation at finite strains (shown at 0%, 18%, and 29% compressions) (57); (b) related simulation results

Figure 9

Histograms providing the mean change in the collagen fiber orientation measured with respect to the normal to the cartilage surface at 18% and 29% compressions, under direct influence of the indentor: (a) for a volume considering all zones; (b) related numerical results for the middle zone only

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