Predicting human chronically paralyzed muscle force: a comparison of three mathematical models

Abstract

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.

Fig. 1

Schematic representation of the stimulation patterns, showing only the 10-Hz series and the doublet ramp used for parameter determination for each model. CT, constant-frequency train; DT, single doublet train; DDT, dual doublet train.

Fig. 2

Representative examples of the raw data from a single subject (subject 4) resulting from the doublet ramp (A), the 5-Hz DT (B), the 10-Hz DDT (C), and the 20-Hz CT (D). EMG, electromyogram. *Doublet pulses.

Fig. 3

Representative example of the optimized best fits for the 3 models to the experimental doublet ramp (solid line) train for subject 1: linear (dotted line), 2nd-order nonlinear (NL; dash-dot line), and the Hill-Huxley-type NL (dashed line).

Fig. 4

Experimental force (first row) and modeled forces (second to fourth rows) for chronically paralyzed soleus muscle (subject 4) at 5, 10, and 20 Hz. The linear model is shown in row 2, 2nd-order NL model in row 3, and Hill-Huxley-type model in row 4. CT and DDT stimulation patterns are displayed at left and right, respectively.

Fig. 5

Overall %error between modeled and experimental forces. A: mean (SD) %error by subject. B: mean (SE) overall %error. C: mean (SE) %error by frequency. D: mean (SE) %error by stimulation pattern.

Fig. 6

Overall %peak force (PF) error between modeled and experimental forces. A: mean (SD) PF %error by subject. B: mean (SE) overall PF %error. C: mean (SE) PF %error by frequency. D: mean (SE) PF %error by stimulation pattern. Note that mean errors may be approaching zero but have clearly visible variances.

Fig. 7

Overall %force time integral (FTI) error between modeled and experimental forces. A: mean (SD) FTI %error by subject. B: mean (SE) overall FTI %error. C: mean (SE) FTI %error by frequency. D: mean (SE) FTI %error by stimulation pattern. Note that mean errors may be approaching zero but have clearly visible variances.

Fig. 8

Specific force time errors between modeled and experimental forces by frequency. A: mean (SE) time to peak tension (TPT) for 5 Hz only. B: mean (SE) half relaxation time (1/2RT). C: mean (SE) relative fusion index (RFI). D: mean (SE) doublet difference %error (DDiff), an indicator of the catch-like property of muscle.