a physical model of muscle working to evaluate the work ability

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A physical model of muscle working

to evaluate the work ability

G. VermiglioT, M.G. Tripepi, V. Vermiglio,

C. Sansotta, B. Testagrossa Department of Environmental, Sanitary, Social and Industrial Protection-University of Messina, Italy

Abstract. It was well known that work ability included a wide kind of related aspects, involving

both mental and physical well-being. When mechanical works were taken into account, it was

necessary to refer to muscular structure and its consistency, for assessing the work ability and

employability; i.e. efficiency of performances of the muscles has to be expressed in terms of

physiological parameters representing the best expression of physical health. To enhance the

existence of idoneous relationships between the muscle operation and the work ability, it should be

necessary to point out a specific physical model of muscle motor enabling to correlate physiological

parameters involved in the muscle operation to the physical quantities usually employed for

estimating the work made by a machine. The authors have evaluated a mechanical model of the

muscle motor, which allowed them to include the isometric and isotonic contractions, the primitive

role of osmotic exchanges and the presence of energy dissipation, also in the presence of elastic

forces, avoiding the so constructed model to have conservative nature. The so developed model,

virtually animated using specific interactive software to reproduce the effective muscular behaviour

during limbs’ movements, has appeared extremely suitable both for educational and scientific aim.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Muscle model; Mechanical work; Muscular contraction; Work ability

1. Introduction

Despite numerous efforts, formulating an accurate definition of work ability has proved

a difficult task. However, as several studies have consistently indicated, there are three

0531-5131/ D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.ics.2005.02.076

T Corresponding author. Dipartimento di Protezionistica Ambientale, Sanitaria, Sociale ed Industriale,

Policlinico Universitario bG. Martino Q , V.le Gazzi, 98127 GAZZI Messina, Italy. Tel.: +39 902213031; fax:

+39 902212664.

E-mail address: [email protected] (G. Vermiglio).

International Congress Series 1280 (2005) 238–243

www.ics-elsevier.com



fundamental aspects that contribute to its definition: mental, social and physical aspect [1].

To estimate the physical work ability and the performance efficiency it was necessary torealise a physical model of the muscle, able to reproduce as true as possible the behaviour

of an equivalent physiological system. In fact, realizing a mechanical model of the

musculo-skeletal system is a subject of importance to fundamental research as well as to

many practical engineering and medical applications [2]. In this paper, we have elaborated

a new physical–chemical model that introduced an osmotic chamber as a new element

describing the different forms of energy involved during the muscular contraction. The

realistic evaluation and analysis of all the mechanical quantities allowed describing the

behaviour of the muscle during its activity, and, successively, quantitatively determining

the efficiency and power of the muscle contraction, that depended on the workers’ age and

could be related to the physical work ability.

The musculo-skeletal apparatus consists of basic motion units which represent machines

that convert the biochemical energy into mechanical energy and heat. The energy transfers

in biological bmachines Q are complex although the efficiency of whole body movements

and the muscle action involved can be explained in rather simple terms [3].

Anatomically, the muscle appears as ba fibres bundle, not more than 30 cm in length

and 0.05–0.15 mm in diameter, joined by connective tissue Q , where all the fibres bare made

up of a membrane, a lot of nuclei spread out along the fibre under the membrane, and of

thousand of internal filaments that have the same length as the fibre (myofibrils)

constituting the cytoplasm: the basic motion unit is composed of all the motion nervouscells (motion neurons) and of the muscular fibres that these cells innervate, from one to

over thousand in number Q (Fig. 1). The muscle contraction consists of two phases: the

isometric phase and the isotonic one that is characterised by a shortening or lengthening of

the muscle due to the interaction of the actin/myosin filaments [4].

The muscle spends energy during both the phases and the power is related to the

different modalities of the energetic expense.

2. Materials and methods

The physical analysis of the contraction mechanism clearly highlighted the difficulties

in developing a model able to justify the contractile activity.

The most simple fibre structure has been represented as a combination of elastic and

contractile components, the well known Voigt element (Fig. 2) [5]. So, it could be possible

Fig. 1. The muscular structure: it is possible to distinguish the different part that constitute it [4].

G. Vermiglio et al. / International Congress Series 1280 (2005) 238–243 239



to distinguish three different elements that mechanically interacted during the muscle

contraction:

1. Parallel elastic components, that simulated the behaviour of contiguous tissues,

included the membranes of muscular fibres and bundles;

2. Consecutive elastic components, describing the muscular chordas;

3. Contractile components characterised by the overlapping of the actin/myosin filaments.

Successively, Saito and Tamura [6,7], measured the response of human muscles against

the applied torque and compared the results with the simulated model of their system, that

consisted of an inverted pendulum with a Voigt element moved by a contractile

component. In addiction, the most recent technological improvements relative to the

artificial muscles, such as actuators, nematic polymers (Fig. 3), polymeric gels or

conducting polymers (Fig. 4), that change their length after receiving a stimulus, employ

different muscle contraction models [8,9].

To satisfy the fundamental requirements of the muscular movement, which had to

reproduce the isometric and the isotonic phases, in the right sequencing, to imply an

energetic dissipation during both the phases, to take into account the amount of energy

spent during the motion activity to do mechanical work and the converted one into heat,

involving also a physical–chemical energy, we introduced an osmotic chamber as the

element developing the force in order to better describe the movement of the muscular

fibres. This model could be described as:

an elastic element coupled to an inextensible one (elastic-inextensible composite unit);

Fig. 2. The fibre structure and its components.

Fig. 3. The muscle contraction model constituted by nematic molecules (A) that stretched out after receiving a

luminous stimulus (B) [8].

G. Vermiglio et al. / International Congress Series 1280 (2005) 238–243240



an osmotic unit consisting of a chamber divided into two sections that guaranteed the

fibre movement. Every section contained solutions variable in concentration; so that a

porous airtight septum easily slid inside because of concentration change;

a viscous frictional chamber to take into account energetic dissipation.

The above described model, which has also been virtually animated using specific

interactive software to reproduce the muscular behaviour during limbs’ movements, for

educational applications, is represented in Fig. 5, where the inextensible element was

considered of fixed length, whereas the elastic one was defined by its elastic constant or

Youngs modulus. The membrane inside the septum was characterised by its permeability

coefficient and the solution could acquire different concentration values during

contraction: so, the presence of the osmotic unit properly reproduced the osmotic fluxes

and the biochemical energy of the motion unit. At last, also the viscous frictional force,

which always intervened in the evaluation of the efficiency and of the power developed

during the muscle activity, was included.

In this manner, the force exerted by the basic motion unit, was related to the

mechanical, anatomical and physiological conditions which depended upon the age.

3. Results

In Fig. 5 it has been simulated the behaviour of our motion unit when joined at the ex-

tremities of two bony stiff segments, during the muscle shortening to lift up an applied load.

At first, a lowering occurred unless the motion unit intervened, by activating the osmotic

chamber (Fig. 6) that, following the variation of concentration, caused a shifting of the

inextensible element together with the septum. The shifting gave rise to the elongation of the spring that developed the elastic force necessary to balance the load and, therefore,

maintain the equilibrium position. The viscous forces were always present in a joint, but

only a part of the chemical energy developed by the osmotic unit was used for balancing

Fig. 4. The muscle contraction model constituted by polymeric gels (A) or conducting polymers (B) [8].

Fig. 5. The muscle contraction described through our model.




them; so, even if the load was eliminated and the motion unit came back, an energetic

expense with heat production occurred also during restoring of the spring elastic energy.

If the motion activity went on, the load was lifted up because of a further shifting of the

porous septum and the elastic-inextensible composite unit, that was equal to the effective

system displacement.

This kind of action could be analysed by energetic quantities.

The work done by the muscle unit was expressed by the osmotic work; so, it wasrelated to the variations of the concentration inside the chamber:

d L ¼ nRT dc=cð Þ ð1Þ

where n is the mole of the solutes inside the osmotic chamber, c represents the

concentration value, R the gas constant and T the constant value of the temperature.

This work included the energy involved during the isometric phase and that one relative

to the isotonic one, respectively named d LA and d LB.

To calculate the energy during isometric phase, three quantities had to be taken into

account. The work done by the elastic component that was related to the elastic energy

stored by the spring, the kinetic energy of the elastic-inextensible composite unit, and theenergy dissipated by the viscous force, the last one related to the isometric square velocity.

Therefore, the isometric energy could be written as:

d LA ¼ k Dl Ad Dl Að Þ þ M v Adv A C xS q v Að Þ2=2d Dl Að Þ ð2Þ

where k is the elastic constant of the spring, Dl A the displacement from the spring

equilibrium position, M the mass of the piston included both elastic unit and inextensible

one composite system, v A the isometric velocity, C x

the aerodynamic coefficient of the

septum, S its surface and q the solution density within osmotic chamber.

The isotonic energy was also related to three quantities, but, in this case, it had toconsider the energetic amount necessary to lift up the load, expressed as the kinetic energy

of the load, while the elastic energy did not take part in. So, the isotonic work was equal to:

d LB ¼ mv VBdv VB þ M v Bdv B C xS q v Bð Þ2=2d Dl Bð Þ ð3Þ

where the subscript B indicates the analogous isotonic quantities, m is the mass of the load

and v B V its velocity. So, the total work was expressed as

d L ¼ k Dl Ad Dl Að Þ þ M v Adv A C xS q v Að Þ2=2d Dl Að Þ þ mv B V dv B V þ M v Bdv B

C xS q v Bð Þ

2

=2d D

1Bð Þ ð4Þ

At this point, the above expression (Eq. (4)) could be correlated to unit motion

characteristics by evaluating muscle efficiency and muscle power.

Fig. 6. The osmotic chamber.

G. Vermiglio et al. / International Congress Series 1280 (2005) 238–243242



The efficiency of the muscle contraction g was the ratio between the useful work and

the total work:

g ¼ mv B V dv B V Þ=d L;ð ð5Þ

where d L represents the motor (osmotic) work.

The muscle power P was the ratio between the mechanical or useful work and the time

necessary to perform it:

P ¼ mv B V dv B Vð Þ=dt ¼ d L=dt ð Þ k Dl A d Dl Að Þ=dt ½ M v A dv A=dt ð Þ þ C xS q v Að Þ2

=2 d Dl Að Þ=dt ½ M v B dv B=dt ð Þ þ C xS q v Bð Þ2=2 d D1Bð Þ=dt ½

ð6Þ

It was evident that Eqs. (5) and (6) constituted a system of two differential equations

whose solution could provide quantitative information about the muscle activity, byinserting idoneous antropometric parameters.

4. Discussion

The efficiency and the power of the muscle activity represented two non-correlated

quantities, which contributed together in the determination of the physical work ability. In

fact, it was our opinion that solution of such a system could be numerically carried out and

directly related to the physical work ability and to the workers’ age.

It was evident that presence in the Eqs. (5) and (6) of both physical and biological

elements permitted the evaluation of the efficiency degree of the musculo-skeletal systemof workers, only using proper values of the biophysical quantities that characterized all the

elements involved in the description of our model, namely the elastic constant, the mass of

the elastic-inextensible composite unit, the muscle contraction velocity, etc., which

represented the anthropometrical and physiological parameters that appeared on the

equation system, depending also upon the workers’ age.

In this way, it seemed possible to obtain, by carrying out numerical simulation on the

basis of the above stated equations and the energetic quantities involved in the muscular

contraction, useful information to properly evaluate a statement for physical work ability

and its dependence upon age.

References

[1] O. Korhonen, Fitness and work ability. http://www.ttl.fi/Internet/English/Information/Electronic+journals/

Tyoterveiset+journal/1999-01+Special+Issue/09.htm 1999.

[2] J. Rasmussen, M. Damsgaard, M. Voigt, J. Biomech. 34 (2001) 409 – 415.

[3] M. Voigt, et al., J. Biomech. 28 (3) (1995) 281 – 291.

[4] J.L. Andersen, P. Schjerling, B. Saltin, Sci. Am. 283 (2000) 48 – 55.

[5] M. Kawai, P.W. Brandt, J. Muscle Res. Cell Motil. 1 (1980) 279– 303.

[6] Y. Tamura, M. Saito, Adaptive Compliant Control of Pseudo-Muscular Visco-Elastic Actuator: Simulation

and Experiment. 11th Conference Of the ESB, July 8–11 98, Toulouse, France: 73.

[7] Y. Tamura, M. Saito, J. Biomech. 35 (2002) 1273 – 1277.[8] P. Auroy, Muscoli artificiali, Le Scienze 407 (2002) 72 – 78.

[9] D. Beebe, et al., Nature 404 (2000) 588– 590.


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a physical model of muscle working to evaluate the work ability

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