Characterization and Modeling of Articular Cartilage: Mathematical Formulations, Experiments and Numerical Examples

Dr. David M. Pierce (Graz University of Technology, Austria)

Thursday 11th July, 2013 14:00-15:00 Mathematics 203 (tea & coffee at 3pm)


Advances in biomechanical and biomedical engineering demand similarly advanced constitutive models for simulating soft tissue deformation. Additionally, emerging new medical imaging modalities make previously unthinkable biological and structural data available for the study of such complex tissues as articular cartilage. In light of the need for advanced computational modeling tools and the abundance of new forms of data, we discuss steps toward the development of an advanced analysis framework for cartilage, combining medical imaging, image analysis, experimental and computational mechanics.

We propose two new constitutive models capable of accepting sample/patient-specific structural data [1, 2]. The models, and 3-D large strain finite element implementations, focus on the load-bearing morphology: an incompressible, poroelastic solid matrix, reinforced by an inhomogeneous, dispersed fiber fabric, saturated with an incompressible fluid at constant electrolytic conditions residing in strain-dependent pores of the collagen-proteoglycan solid matrix. Furthermore, the inhomogeneous, dispersed fiber fabric of the solid influences the fluid permeability, as well as an intrafibrillar portion which cannot be ‘squeezed out’ from the tissue. Using representative numerical examples, we reproduce several mechanical responses which have been demonstrated experimentally in the cartilage mechanics literature [1]. Our proposed approaches, in which the material parameters have a direct physical interpretation, facilitate 3-D patient-specific simulations implementing high-resolution morphological data in a computational setting.

A promising method, multiphoton microscopy (MPM) extracts information about the microscopic structure of cartilage with submicron resolution. We imaged chicken cartilage, sectioned fresh from the medial femoral condyle of the knee and cut in three orthogonal planes (transverse, sagittal, and coronal) [3]. First, using a 10x objective, we stitched together sets of images to form large images (5 mm×5 mm), taken at 10 μm intervals to a depth of 50 μm. Second, using a 40x objective, we collected higher resolution sets of images 230 μm × 230 μm at 2 μm intervals to a depth of 100 μm. From this imaging data, we extracted local principal fiber direction and dispersion for the superficial layer using an imaging technique based on Fourier analysis. Using 2-D data extracted from three orthogonal planes, we developed a new numerical algorithm to reconstruct volumetric maps of the local collagen direction and dispersion in 3-D.

Using a second formulation for local collagen fiber dispersion, we employed ultra-high field Diffusion Tensor Magnetic Resonance Imaging data in sample-specific simulations of an indentation experiment on a human sample to test two hypotheses found in the cartilage mechanics literature: (i) the through- thickness structural arrangement of the collagen fiber fabric adjusts fluid permeation to maintain fluid pressure and optimize tissue function (Federico and Herzog 2008 Biomech. Model. Mechanobiol. 7, 367- 378), (ii) inhomogeneity of mechanical properties through the cartilage thickness acts to maintain fluid pressure at the articular surface (Krishnan et al. 2003 J. Biomed. Eng. 125, 569-577) [2]. Our numerical results support both hypotheses, although tissue inhomogeneity appears to have a larger effect on fluid pressure retention in this tissue sample, and on the advantageous pressure distribution which enhances load support, shields the solid matrix, promotes low frictional coefficients and thus reduces wear.

Our analysis framework for cartilage, combining medical imaging, image analysis, experimental and computational mechanics, enables studies of, e.g., fundamental structure-function relationships, surgical interventions, cartilage and joint integrity, disease progression, engineered cartilage replacements, imaging data analysis, and provides insight to microphysical (mechanobiological) cellular stimuli.

[1] Pierce, D.M., T. Ricken and G.A. Holzapfel, A Hyperelastic Biphasic Fiber-Reinforced Model of Articular Cartilage Considering Distributed Collagen Fiber Orientations: Continuum Basis, Computational Aspects and Applications, Comput. Methods Biomech. Biomed. Engin., (in press).

[2] Pierce, D.M., T. Ricken and G.A. Holzapfel, Modeling Sample/Patient-Specific Structural and Diffusional Responses of Cartilage Using DT-MRI, Int. J. Numer. Meth. Biomed. Engng., (in press).

[3] Lilledahl, M.B., D.M. Pierce, T. Ricken, G.A. Holzapfel and C. de Lange Davies, Structural Analysis of Articular Cartilage Using Multiphoton Microscopy: Input for Biomechanical Modeling, IEEE Trans. Med. Imaging, 30:1635–1648, 2011. 

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