A probabilistic rule of mixtures for elastic moduli

A novel approach is developed for mathematically modeling the variability observed in experimentally determined elastic moduli of longitudinally oriented fibrous tissues such as ligaments and tendons. The elastic modulus of these tissues is modeled with a rule of mixtures (ROM) where each parameter (fibril and matrix moduli and fibril volume fraction) is assumed to be an independent random variable. A joint density function formed from the independent densities results in a probabilistic ROM (pROM). This pROM is used to generate a distribution of moduli which agrees well with moduli determined from tests of rabbit medial collateral ligaments (Woo and Ohland, 1994, Unpublished experimental data as gift). Minimizing the error between the pROM and experimental distributions resulted in an integrated error of 9% for a constrained set of independent distribution parameters derived from the literature. This pROM thus incorporates microstructural observations (fibril and matrix moduli and fibril volume fraction) to partially explain the experimentally observed variability in a macroscopic property (tissue modulus).