A validated model of passive muscle in compression

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Abstract

A better characterisation of soft tissues is required to improve the accuracy of human body models used, amongst other applications, for virtual crash modelling. This paper presents a theoretical model and the results of an experimental procedure to characterise the quasi-static, compressive behaviour of skeletal muscle in three dimensions. Uniaxial, unconstrained compression experiments have been conducted on aged and fresh animal muscle samples oriented at various angles from the fibre direction. A transversely isotropic hyperelastic model and a model using the theory of transverse isotropy and strain dependent Young's moduli (SYM) have been fitted to the experimental data. Results show that the hyperelastic model does not adequately fit the data in all directions of testing. In contrast, the SYM gives a good fit to the experimental data in both the fibre and cross-fibre direction, up to 30% strain for aged samples. The model also yields good prediction of muscle behaviour at 45° from the fibre direction. Fresh samples show a different behaviour than aged tissues at 45° from the fibre direction. However, the SYM is able to capture this difference and gives a good fit to the experimental data in the fibre, the cross-fibre and at 45° from the fibre direction. The model also yields good prediction of muscle behaviour when compressed at 30° and 60° from the fibre direction. The effect of the time of test after death has also been investigated. Significant stiffening of muscle behaviour is noted a few hours after death of the subject.

Introduction

The use of virtual human modelling has increased in the last few years, in particular in the field of impact biomechanics (Forbes et al., 2005; van Rooij et al., 2003; Verver et al., 2004; Ward et al., 2005). Finite element models of the human body can be used to predict deformations during transient loading. These models require both a description of hard and soft tissues geometries, as well as the determination of their material properties under large deformations, as appearing during impacts. However, the properties of soft tissues continue to be poorly classified. In particular, muscle tissue presents difficulties. As with most biological soft tissue, muscle presents a viscoelastic behaviour and has anisotropic properties due to its fibre-oriented structure. It is subject to large deformations in vivo and the stress state in a given muscle is the result of passive and active contributions. Furthermore, parameters such as age, gender and species are also influential. During the past century, numerous investigations have been conducted to determine muscle structure and function. Experimental studies have been performed to determine muscle properties. However, most of these studies have considered deformation in one dimension only (the longitudinal direction) and significant variability between data sets is present (see Fig. 1). In this, passive force–length curves obtained by tensile tests conducted variously on rat, cat or rabbit muscles are presented (Davis et al., 2003; Gareis et al., 1992; Hawkins and Bey, 1994, Hawkins and Bey, 1997; Muhl, 1982; Woittiez et al., 1984). Grieve and Armstrong (1988) and Bosboom et al., 2001a, Bosboom et al., 2001b published experimental data on the compressive behaviour of passive skeletal muscle in the transverse direction. To the authors knowledge, these are the only experimental study characterising muscle compressive behaviour. One-dimensional models have been developed to reproduce muscle global mechanical behaviour observed in the experiments. Most of these models (see for example Best et al., 1994; Cole et al., 1996; de Jager, 1996) are based on the famous model proposed by Hill (1938).

Three-dimensional constitutive models of nonlinear anisotropic tissue exist (see Humphrey et al., 1990; Li et al., 2001; Limbert and Middleton, 2004; Weiss et al., 1996). They have been developed to reproduce the tensile behaviour of soft tissues with fibre-oriented structure such as tendons and ligaments. However, only the model proposed by Humphrey et al. (1990) has been used to reproduce muscle behaviour (Martins et al., 1998) and the paucity of experimental data does not permit one to determine the validity of the model.

The objectives of this study are to provide an experimental as well as a theoretical characterisation of passive skeletal muscle properties in three dimensions. Particular attention is placed on muscle compressive behaviour, as this is the relevant deformation mode in human/machine impacts and seated postures. Emphasis is also placed on model simplicity and ease of parameter determination.

This paper presents the results of an experimental procedure developed to determine the quasi-static properties of muscle tissue, as well as a simple model which reproduces the behaviour observed during the experiments.

Section snippets

Theoretical models

Skeletal muscle has a fibre-oriented structure: it is composed of fascicles containing bundles of fibres, themselves composed of parallel bundles of myofibrils (Gaudin, 1997). From a modelling point of view, muscle tissue is thus often considered as a nonlinear, unidirectional composite consisting of parallel fibres embedded in a matrix (see Fig. 2 for illustration). Unidirectional composites are transversely isotropic; however, applying engineering material technology to biological material is

Test samples and experimental set-up

Uniaxial unconfined compression experiments have been conducted on porcine, bovine and ovine muscle samples. The bovine and a first set of porcine samples were obtained from a local butcher and had, therefore, been bled and hung (aged samples). No control over age, gender, time of death or methods of storage was available for these samples. In contrast, the ovine and the second set of porcine samples were obtained immediately after death from unbled animals (fresh samples). The ovine samples

Aged tissue

Fig. 4 presents experimental stress–strain curves obtained by compression of aged porcine samples oriented in (a) the cross-fibre, (b) fibre and (c) 45° from the fibre direction. Four samples were tested in each direction and mean curves with standard deviation are presented. Fig. 4(d), in which mean curves in the three directions of testing are plotted, shows that the cross-fibre direction is stiffer than the 45° direction, which in turn is stiffer than the fibre direction.

The theoretical

Discussion

Three-dimensional models describing the behaviour of nonlinear, transversely isotropic soft tissues exist, but little experimental data was previously available to judge their applicability to model passive skeletal muscle, as existing data is mostly limited to tensile longitudinal behaviour. In this study, we provide an experimental procedure and a nonlinear model to characterise the three-dimensional compressive behaviour of passive skeletal muscle. Data on porcine muscle tissue tested in

Acknowledgements

This project is funded by the Programme for Research in Third Level Institutions (PRTLI), administered by the Irish Higher Education Authority (HEA).

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