Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction

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Abstract

The purpose of this study was to evaluate a Hill-based mathematical model of muscle energetics and to disclose inconsistencies in existing experimental data. For this purpose, we simulated iso-velocity contractions of mouse fast twitch EDL and slow twitch SOL fibers, and we compared the outcome to experimental results. The experimental results were extracted from two studies published in the literature, which were based on the same methodology but yielded different outcomes (B96 and B93). In the model, energy cost was modeled as the sum of heat and work. Parameters used to model heat rate were entirely independent of the experimental data-sets. Parameters describing the mechanical behavior were derived from both experimental studies. The model was found to accurately predict the muscle energetics and mechanical efficiency of data-set B96. The model could not, however, replicate the energetics and efficiency of SOL and EDL that were found in data-set B93. The model overestimated the shortening heat rate of EDL but, surprisingly, also the mechanical work rate for both muscles. This was surprising since mechanical characteristics of the model were derived directly from the experimental data. It was demonstrated that the inconsistencies in data-set B93 must have been due to some unexplained confounding artifact. It was concluded that the presented model of muscle energetics is valid for iso-velocity contractions of mammalian muscle since it accurately predicts experimental results of an independent data-set (B96). In addition, the model appeared to be helpful in revealing inconsistencies in a second data-set (B93).

Introduction

Mathematical modeling and computer simulation of the human musculo-skeletal system have become important tools in the analysis of whole body human movement. In the previous three decades models have been developed to analyze the motor control and mechanical performance of all-out motor tasks such as jumping, kicking or sprint-cycling (Hatze, 1977; Audu and Davy, 1985; Pandy et al., 1990; van Soest and Bobbert, 1993; van Soest and Casius, 2000; Bobbert and van Soest, 2001). Human movement, however, frequently entails endurance tasks. Successful execution of these tasks not only depends on the mechanical output of the musculo-skeletal system, but also on metabolic energy production and mechanical efficiency. In contrast to the mechanical behavior of the muscle, however, the energy cost of muscular contraction has received relatively little attention in musculo-skeletal modeling.

In musculo-skeletal modeling, the mechanical behavior of the muscle is frequently represented by the phenomenological Hill-type muscle model. In this model muscle force depends on active state, contractile element length and shortening velocity (Hill, 1938). Studies on the energetics of muscle contraction indicate that energy consumption can be expressed as a function of these same variables (Hill, 1938; Mommaerts, 1969; Woledge et al., 1988). Hill-based mathematical models describing the relation between muscle contraction and energy cost have been formulated in the past (Chapman and Gibbs, 1972; Hatze and Buys, 1977; Schutte et al., 1993; Ettema, 2001) and have been applied in several biomechanical studies, primarily to provide a cost function for optimization (Hatze and Buys, 1977; Davy and Audu, 1987; Schutte et al., 1993; Anderson and Pandy, 2001). The equations and constants that appear in these models have been derived from different studies using a wide variety of animal species (amphibian and mammalian) and using contractions according to diverse protocols (varying in stimulation, contraction types, duration, etc.) and under diverse circumstances (physiological solutions, temperature, etc.). The diversity in the sources on which the multiple parameters in each of these models are based might compromise their validity, especially when these models are used to simulate human movement. Surprisingly, however, these models have hardly been cross-validated, by comparing their predictions to a single and independent physiological data-set, in order to test their predictive value. Only recently Bhargava et al. (2004) performed such a cross-validation. However, they compared their model predictions with data on amphibian muscle at 0 °C ambient temperature.

Despite the wealth of data on muscle energetics, direct data on energy consumption of intact muscle fibers, which can be used for the evaluation of energetic muscle models, are scarce in the literature. This is especially true for data on mammalian muscles, which are most attractive if one is interested in modeling human movement. In recent years, Barclay and co-workers (Barclay et al., 1993; Barclay, 1996) have performed measurements of muscle work and heat using fast and slow twitch muscle fibers of the mouse. Their experiments provide adequate data to evaluate a Hill-type muscle model for energy consumption during iso-velocity contractions. However, the results from these experiments seem to be inconsistent. Despite apparently identical experimental protocols Barclay et al. (1993) found an equal maximal mechanical efficiency for the slow twitch m. soleus (SOL) and fast twitch m. extensor digitorum longus (EDL), but in the later study of Barclay (1996) a significantly higher efficiency for SOL compared to EDL was found. The maximal efficiency of fast twitch and slow twitch fibers is a controversial issue in muscle physiology (Goldspink, 1978; Woledge et al., 1988), but the discrepancy between the results of Barclay et al. (1993) and Barclay (1996) has never been addressed nor resolved. Unfortunately, such a discrepancy in experimental results could preclude attempts to develop a valid model.

The purpose of the current study was to evaluate a Hill-based model of muscle energetics by comparing its predictions of muscle energetics and mechanical efficiency with independent experimental results. In addition, the model was used to demonstrate the ability and usefulness of the systematic approach of mathematical modeling in detecting and resolving the inconsistency in experimental data mentioned above, which might be obvious in retrospect but has remained undetected for more than a decade.

Section snippets

Muscle model

The complete model of muscle behavior used in this study consisted of a combination of two phenomenological models, one describing the mechanical output of the muscle and one describing the energy consumption of the muscle.

The mechanical behavior of the muscle was modeled using a classical Hill-model described in full detail by van Soest and Bobbert (1993). In short, this model consisted of a contractile element (CE) in series with an elastic element (SEE). A parallel elastic element (PEE) was

Results

Rates of energy cost during contraction and initial mechanical efficiency are presented in Fig. 1, Fig. 2, respectively, as a function of relative contraction velocity. Note that the experimental data of B96 in Fig. 1 originate from a single muscle fiber preparation while the data of B93 represent the group average of that study. Relevant mass specific data for both experimental group averages and simulation results are provided in Table 3.

Model predictions of the energy cost and initial

Discussion

In this study, a Hill-based muscle model of the energy cost of muscle contraction was evaluated using independent physiological data on iso-velocity contractions of mammalian muscles. In addition, this model was used to examine the apparent discrepancy in results between two seemingly similar experimental studies on the difference in maximal efficiency between fast and slow twitch muscle fibers. It needs to be stressed again that the model equations and constants used to model energy

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