A Model of Fatigue and Recovery in Paraplegic’s Quadriceps Muscle Subjected to Intermittent FES

Abstract

The objective of this paper was to propose a mathematical model for the fatigue and recovery phases of a paraplegic’s quadriceps muscle subjected to intermittent functional electrical stimulation (FES). The model is based on in vivo, noninvasive, recording of fatigue related metabolic parameters recorded during stimulation and recovery. Records of the time variations of the muscle’s phosphorus metabolites, particularly the phosphocreatine (PCr) and inorganic phosphorus (Pi), obtained from 31 P magnetic resonance spectroscopy (MRS), were used to calculate the intracellular pH level in the muscle and this latter parameter was incorporated in a musculo-tendon model. The fatigue-recovery model allows the transition from the fatiguing phase to the recovery phase as soon as the stimulation terminates and vice versa. This model was incorporated into a Huxley type muscle model expressing the dynamics of the muscle. Two ordinary differential equations describing the musculo-tendon dynamics and the dynamics of the activation were solved simultaneously and records of the force trajectory during intermittent stimulations were obtained. Study cases ranging from 5 to 30 s for each of the stimulation and recovery alternating phases were stimulated. The force and the total impulse in the modeled quadriceps muscle were computed. It was found that the greatest impulse was produced in intermittent stimulation of 40-50 s duty cycle, with a 50 percent ratio between the stimulation and recovery intervals. An additional series of six runs, including two contractions, one of 3 min and one of 1 min, separated by rest periods of 3, 6, 9, 12, 15, and 30 min was performed. From the predicted force trajectories obtained, the maximal force values served for comparison with measured values made on one patient.