Fractional-order elastic models of cartilage: A multi-scale approach Richard L. Magin a, * , Thomas J. Royston b a University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607, USA b University of Illinois at Chicago, 842 West Taylor St., Chicago, IL 60607, USA article info Article history: Received 22 April 2009 Received in revised form 4 May 2009 Accepted 4 May 2009 Available online 12 May 2009 PACS: 43.80.Cs 46.35.+z 87.61.c 45.10.Hj Keywords: Mathematical models Mechanical stress Medical systems Stress Tissues Hysteresis abstract The objective of this research is to develop new quantitative methods to describe the elas- tic properties (e.g., shear modulus, viscosity) of biological tissues such as cartilage. Carti- lage is a connective tissue that provides the lining for most of the joints in the body. Tissue histology of cartilage reveals a multi-scale architecture that spans a wide range from individual collagen and proteoglycan molecules to families of twisted macromolecular fibers and fibrils, and finally to a network of cells and extracellular matrix that form layers in the connective tissue. The principal cells in cartilage are chondrocytes that function at the microscopic scale by creating nano-scale networks of proteins whose biomechanical properties are ultimately expressed at the macroscopic scale in the tissue’s viscoelasticity. The challenge for the bioengineer is to develop multi-scale modeling tools that predict the three-dimensional macro-scale mechanical performance of cartilage from micro-scale models. Magnetic resonance imaging (MRI) and MR elastography (MRE) provide a basis for developing such models based on the nondestructive biomechanical assessment of car- tilage in vitro and in vivo. This approach, for example, uses MRI to visualize developing proto-cartilage structure, MRE to characterize the shear modulus of such structures, and fractional calculus to describe the dynamic behavior. Such models can be extended using hysteresis modeling to account for the non-linear nature of the tissue. These techniques extend the existing computational methods to predict stiffness and strength, to assess short versus long term load response, and to measure static versus dynamic response to mechanical loads over a wide range of frequencies (50–1500 Hz). In the future, such meth- ods can perhaps be used to help identify early changes in regenerative connective tissue at the microscopic scale and to enable more effective diagnostic monitoring of the onset of disease. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction For the repair and regeneration of load-bearing tissues, such as bone, spinal disks, or articular cartilage, knowledge of the mechanical strength and stiffness is critical. Such biomechanical properties depend, in the case of cartilage, on the distribu- tion and micro-scale architecture of the component elastomer and protein adhesion molecules (collagen, elastin, proteogly- can and hyaluronan) that form the extracellular matrix (ECM). Chondrocytes, the cells of cartilage that mold the ECM, function at the microscopic scale by creating nano-scale networks of elastomers, while the tissue biomechanics of cartilage is expressed at the macroscopic scale through the free movement and shock absorbing properties of the articulating joints of the body [1]. Consequently, in designing knee or hip joint replacements from either synthetic or natural cell-based materials, 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.05.008 * Corresponding author. Tel.: +1 312 996 2331. E-mail addresses: rmagin@uic.edu (R.L. Magin), troyston@uic.edu (T.J. Royston). Commun Nonlinear Sci Numer Simulat 15 (2010) 657–664 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns