title: Mathematical models use varying parameter strategies to represent paralyzed muscle force properties: a sensitivity analysis.
abstract: BACKGROUND
  Mathematical muscle models may be useful for the determination of appropriate musculoskeletal stresses that will safely maintain the integrity of muscle and bone following spinal cord injury. Several models have been proposed to represent paralyzed muscle, but there have not been any systematic comparisons of modelling approaches to better understand the relationships between model parameters and muscle contractile properties. This sensitivity analysis of simulated muscle forces using three currently available mathematical models provides insight into the differences in modelling strategies as well as any direct parameter associations with simulated muscle force properties.
  
  
  METHODS
  Three mathematical muscle models were compared: a traditional linear model with 3 parameters and two contemporary nonlinear models each with 6 parameters. Simulated muscle forces were calculated for two stimulation patterns (constant frequency and initial doublet trains) at three frequencies (5, 10, and 20 Hz). A sensitivity analysis of each model was performed by altering a single parameter through a range of 8 values, while the remaining parameters were kept at baseline values. Specific simulated force characteristics were determined for each stimulation pattern and each parameter increment. Significant parameter influences for each simulated force property were determined using ANOVA and Tukey's follow-up tests (alpha <or= 0.05), and compared to previously reported parameter definitions.
  
  
  RESULTS
  Each of the 3 linear model's parameters most clearly influence either simulated force magnitude or speed properties, consistent with previous parameter definitions. The nonlinear models' parameters displayed greater redundancy between force magnitude and speed properties. Further, previous parameter definitions for one of the nonlinear models were consistently supported, while the other was only partially supported by this analysis.
  
  
  CONCLUSIONS
  These three mathematical models use substantially different strategies to represent simulated muscle force. The two contemporary nonlinear models' parameters have the least distinct associations with simulated muscle force properties, and the greatest parameter role redundancy compared to the traditional linear model.
authors.affiliation: Graduate Program in Physical Therapy and Rehabilitation Science, 1-252 Medical Education Bldg,, The University of Iowa, Iowa City, IA, USA. laura-freylaw@uiowa.edu
authors.forename: Laura A
authors.initials: LA
authors.lastname: Frey Law
authors.forename: Richard K
authors.initials: RK
authors.lastname: Shields