##### Department of Mathematics,

University of California San Diego

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### Informal Seminar on Mathematics and Biochemistry-Biophysics

## Eunjung Kim

#### University of Notre Dame \\ Department of Mathematics

## Multiscale Biomechanical Models for Biological Soft Tissue

##### Abstract:

\footnotesize Articular cartilage is a resilient soft tissue that supports load joints at the knee, shoulder and hip. Cartilage is primarily comprised of interstitial water (roughly 80\% by volume) and extracellular matrix (ECM). Cells called chondrocytes are dispersed through ECM and maintain and regenerate the tissue. Chondrocytes are surrounded by a narrow layer called pericellular matrix (PCM), which is believe to be important for modulating the biomechanical environment of chondrocyte. In this study, computational models will be presented to analyze the multiscale micromechanical environment of chondrocytes.\\ Firstly, we will discuss transient finite element method (FEM) to model linear biphasic mechanics of a single cell within cartilage layer under cyclic loading. The FEM model was employed to analyze the effects of frequency on mechanical variables in cellular environment under macroscopic loading at 1\% strain and in the frequency range 0.01 0.1 Hz. In this frequency range, intracellular axial strains exhibited up to a ten-fold increase in magnitude relative to 1\% applied strain. The dynamics of strain amplification exhibited a two-scale response that was highly dependent on ratios of typical time scales in the model, such as the loading period, gel diffusion times for the cell, the PCM and the ECM. In conjunction with strain amplification, solid stress in the surrounding ECM was reduced by up to 35\%. We propose here that the computational model developed in this study has potential application in correlating mechanical variables in the cellular microenvironment to biosynthetic responses induced by cyclic loading of native cartilage or engineered cell-gel constructs.\\ Secondly, we will discuss the formulation, implementation and application of multiscale axisymmetric boundary element method (BEM) for simulating in situ deformation of chondrocyte and the PCM in states of mechanical equilibrium. The BEM was employed to conduct a multiscale continuum model to determine linear elastic properties of the PCM in situ. Taken together with previous experimental and theoretical studies of cell-matrix interactions in cartilage, these findings suggest an important role for the PCM in modulating the mechanical environment of the chondrocyte. \\ This is joint work with Mansoor Haider (NCSU), and our experimental colleague, Farshid Guilak (Duke).

Hosts: Li-Tien Cheng and Bo Li

### October 22, 2009

### 2:00 PM

### AP&M 5829

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