A constrained von Mises distribution to describe fiber organization in thin soft tissues

Biomech Model Mechanobiol. 2012 Mar;11(3-4):475-82. doi: 10.1007/s10237-011-0326-y. Epub 2011 Jul 8.


The semi-circular von Mises distribution is widely used to describe the unimodal planar organization of fibers in thin soft tissues. However, it cannot accurately describe the isotropic subpopulation of fibers present in such tissues, and therefore an improved mathematical description is needed. We present a modified distribution, formed as a weighted mixture of the semi-circular uniform distribution and the semi-circular von Mises distribution. It is described by three parameters: β, which weights the contribution from each mixture component; k, the fiber concentration factor; and θ ( p ), the preferred fiber orientation. This distribution was used to fit data obtained by small-angle light scattering experiments from various thin soft tissues. Initial use showed that satisfactory fits of fiber distributions could be obtained (error generally < 1%), but at the cost of non-physically meaningful values of k and β. To address this issue, an empirical constraint between the parameters k and β was introduced, resulting in a constrained 2-parameter fiber distribution. Compared to the 3-parameter distribution, the constrained 2-parameter distribution fits experimental data well (error generally < 2%) and had the advantage of producing physically meaningful parameter values. In addition, the constrained 2-parameter approach was more robust to experimental noise. The constrained 2-parameter fiber distribution can replace the semi-circular von Mises distribution to describe unimodal planar organization of fibers in thin soft tissues. Inclusion of such a function in constitutive models for finite element simulations should provide better quantitative estimates of soft tissue biomechanics under normal and pathological conditions.

MeSH terms

  • Animals
  • Biomechanical Phenomena
  • Biophysics / methods
  • Collagen / chemistry*
  • Computer Simulation
  • Connective Tissue / metabolism*
  • Elasticity
  • Finite Element Analysis
  • Light
  • Models, Statistical
  • Models, Theoretical
  • Rats
  • Reproducibility of Results
  • Scattering, Radiation


  • Collagen