Journal of Biomechanics 35 (2002) 1565–1573
In vivo force–velocity relation of human muscle: a modelling from sinusoidal oscillation behaviour A. Desplantez, F. Goubel* D!epartement de Ge!nie Biologique, UMR CNRS 6600, Universit!e de Technologie, BP 20529, 60205 Compi"egne cedex, France Accepted 26 June 2002
Abstract Isokinetic tests performed on human muscle in vivo during plantar flexion contractions lead to torque–angular velocity relationships usually fitted by Hill’s equation expressed in angular terms. However, such tests can lead to discrepant results since they require maximal voluntary contractions performed in dynamic conditions. In the present study, another way to approach mechanical behaviour of a musculo-articular structure was used, i.e. sinusoidal oscillations during sub-maximal contractions. This led to the expression of (i) Bode diagrams allowing the determination of a damping coefficient (Bbode ); and (ii) a viscous parameter (Bsin ) using an adaptation of Hill’s equation to sinusoidal oscillations. Then torque–angular velocity relationships were predicted from a model based on the interrelation between Bbode and Bsin and on the determination of optimal conditions of contraction. This offers the possibility of characterizing muscle dynamic properties by avoiding the use of isokinetic maximal contractions. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Human muscle; Model; Sinusoidal perturbations; Isokinetic; Muscle power
1. Introduction It has been known for many years that force (P) developed by a fully activated muscle decreases when its shortening velocity (V ) increases (Fenn and Marsh, 1935; Hill, 1938). First evaluated in isolated amphibian muscle, a hyperbolic force–velocity relationship was also adapted for human muscle in vivo (Wickiewicz et al., 1984; Dudley et al., 1990; MacIntosh et al., 1993). In such a case, tests consist in developing maximal voluntary contraction during joint rotation effected under constant angular velocity, i.e. isokinetic contraction. Results presented in the literature show that torque decreases systematically as angular velocity increases. Despite the fact that experimental conditions for isolated muscle and human muscle in vivo are not strictly identical, the similarity between relationships obtained from both experiments led some authors to propose an interpretation of the torque–velocity rela-
*Corresponding author. Tel.: +33-3-44-23-43-90; fax: +33-3-44-2048-13. E-mail address:
[email protected] (F. Goubel).
tionship in terms of Hill’s hyperbola (Thorstensson et al., 1976; Ivy et al., 1981; Dudley et al., 1990). However, establishment of such a hyperbolic relationship on human muscle in vivo remains delicate. In fact, for low angular velocities most of the studies report a force less than that expected from Hill’s hyperbola. Some authors suggested that this may be due to an inhibition in force originating from the central nervous system (Narici et al., 1991; Harridge and White, 1993). This hypothesis is favoured by the fact that torque– velocity relationship established on human muscle in vivo under supra-maximal stimulation do not present the same force inhibition phenomenon (Thomas et al., 1987). Furthermore, some authors argued that during this kind of experiment, measured torque may also include a participation of antagonists (Cabri, 1991; Kellis and Batzopoulos, 1997). These problems cannot be easily solved when effecting maximal voluntary contractions. Nevertheless, the torque–velocity dependency is of high interest to characterize human muscle behaviour in dynamic conditions (see the review of Gulch, . 1994) notably in terms of viscous-like properties (Niku and Henderson, 1989; Martin et al., 1994; Desplantez et al., 1999).
0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 1 9 0 - 2
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Another way to approach mechanical properties of human muscle in vivo is the use of sinusoidal oscillations. This consists in applying sinusoidal oscillations either in torque (Agarwal and Gottlieb, 1977; Gottlieb and Agarwal, 1978) or in position (Hunter and Kearney, 1982; Evans et al., 1983; Cornu et al., 1997). These oscillations are imposed during isometric contractions over a given frequency range. Then, muscle behaviour is characterized in terms of a second-order model leading to consistent results expressed as inertial, elastic and viscous parameters. This indicates that sinusoidal oscillations performed during sub-maximal contractions are more reliable than isokinetic movements. The purpose of the present study was to predict torque–velocity relationship of human muscle in vivo from sinusoidal oscillation tests. With this view in mind, a mathematical model was developed. Finally, torque data obtained by using this approach were compared to that obtained during isokinetic movements within a limited range of velocities. This procedure was applied to the human ankle joint during plantar flexion efforts in order to determine the torque–velocity characteristics of the plantar flexor muscle group.
where a and b are constants in torque and angular velocity respectively and T0 is maximal voluntary torque developed under isometric conditions. This T y’ relationship can be interpreted in terms of viscous-like behaviour (Biso ) assuming that recorded torque is the sum of an internal contractile torque (T0 ) ’ and an internal viscous resisting torque (Biso y) ’ T ¼ T0 Biso y: ð2Þ On the other hand, sinusoidal oscillation tests can be performed when the subject develops a mean submaximal constant torque (Tm ). For each Tm ; averaged position-to-torque amplitude ratios (gain curve) and position-to-torque differences in phase (phase curve) are plotted on a Bode diagram (Fig. 2) that reflects their dependency on frequency. A second-order model can be fitted to both curves (Kearney and Hunter, 1990) and expressed as . DTm ¼ Ky þ Bbode y’ þ I y; ð3Þ where I is inertia, K elasticity and Bbode viscosity of the musculo-articular system; DTm is the variation in torque 5
0
2.1. Basic assumptions
-5
In human muscle, both isokinetic movements and sinusoidal oscillation tests can give information about muscle viscosity. On one hand, isokinetic tests can lead to a torque (T)’ relationship that can be described by angular velocity (y) an expression of Hill’s hyperbola (Fig. 1) in angular terms: bðT0 TÞ y’ ¼ ; ðT þ aÞ
ð1Þ
Gain (dB)
2. Model
-10
-15
-20
-25 1
10 Frequency (Hz)
1
10 Frequency (Hz)
(a)
100
0
-50
Phase (˚)
Torque (Nm)
100 80 60
-100
40 -150
20 0 0
0.5
1
1.5
2
2.5 -200
Angular velocity (rad s-1) Fig. 1. Typical torque–angular velocity relationship obtained from isokinetic experiments. Data obtained for subject 2 are fitted by a hyperbola (r2 ¼ 0:98).
(b)
100
Fig. 2. Typical Bode diagram. (a) Gain curve, (b) Phase curve. Data obtained for subject 5 (r2 ¼ 0:96).
A. Desplantez, F. Goubel / Journal of Biomechanics 35 (2002) 1565–1573
due to sinusoidal oscillations, y the angular position, y’ and y. its first and second time derivative, respectively. K and Bbode exhibit a linear relationship against Tm whereas I remains constant (Hunter and Kearney, 1982). Thus, K and Bbode can be expressed as K ¼ k1 Tm þ k2 ;
ð4aÞ
Bbode ¼ b1 Tm þ b2 ;
ð4bÞ
where k1 ; k2 ; b1 and b2 are constants. Finally, a method that is described in detail elsewhere (Desplantez et al., 1999) can be used to determine a viscous coefficient (Bsin ) characterizing muscle behaviour during small amplitude movements. Briefly, this method consists in adapting Eq. (1) to sinusoidal oscillations. A sinusoidal oscillation in position brings about alternating change in torque (DTm ) around mean constant torque Tm : Applying Eq. (1) with T equal to Tm and deriving this equation by Tm leads to the expression Dy’ ¼
ðT0 þ aÞb DTm : ðTm þ aÞ2
ð5Þ
Since a viscous parameter is a ratio between change in torque and change in velocity, Bsin can finally be expressed by the formula Bsin ¼
ðTm þ aÞ2 : ðT0 þ aÞb
ð6Þ
2.2. Interrelation between Bbode and Bsin: determination of a parameter From Eq. (1) it appears that knowledge of T0 ; a and b parameters is necessary to predict a T y’ relationship. T0 can be determined thanks to an isometric experiment. Then, the aim of the present model is to determine a and b parameters from sinusoidal oscillations owing to the fact that Bsin depends on a and b (Eq. (6)). Since Bsin is a function of Tm2 (Eq. (6)) whereas Bbode is a function of Tm (Eq. (4b)), Bsin and Bbode are linked by a function f ðTm Þ linear against Tm Bbode ðX1 Tm þ X2 Þ ¼ Bsin ;
ð7Þ
where X1 and X2 are constants defining f ðTm Þ: Solving Eq. (7) leads to determine a parameter (see Appendix A): a¼
b2 : b1
ð8Þ
However b remains unknown. To determine this parameter, optimal conditions of contraction will be considered in the following by calculating muscle power.
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2.3. Optimal conditions of contraction: determination of b parameter Mechanical power calculated from a force–velocity relationship is well known to present a parabolic behaviour against force or velocity (Hill, 1950; Harridge and White, 1993). Thus, in the present work, a power calculation is performed to determine optimal conditions of contraction corresponding to maximal power development. Then, knowledge of optimal torque (Topt ) and its corresponding velocity (y’ opt ) will allow us to determine b thanks to Eq. (1). 2.3.1. Determination of Topt Power derived from a torque–angular velocity relationship (Piso ) is expressed as the product of torque T ’ With the aid of into its corresponding angular velocity y: Eq. (1), Piso can finally be defined as ðT0 TÞb Piso ¼ T y’ ¼ T : ð9Þ ðT þ aÞ Reducing to zero the derivation of Piso by T allows to express optimal torque Topt : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Topt ¼ a þ aðT0 þ aÞ: ð10Þ 2.3.2. Determination of y’ opt Determination of y’ opt is not possible with Piso calculation because it necessitates knowledge of b U Swoap et al. (1997) indicated that, in rat muscles, optimal velocity determined from Piso –velocity relationship did not significantly differ from optimal velocity calculated thanks to a power–velocity relationship obtained from sinusoidal oscillations. Power developed during sinusoidal oscillations (Psin ) can be considered as the product between imposed pulsation (o) and torque. Such a torque can be calculated for optimal conditions of contraction by virtue of its expression in terms of contractile torque and resisting torque (see Appendix B). Thus, Psin is found to depend on o; exhibiting a maximum for a pulsation labelled oopt (Fig. 3). Finally, y’ opt is derived from the combination of cycle duration ð2p=oopt Þ and movement amplitude (jDyj). Considering that a cycle corresponds to a displacement of 2jDyj leads to the formula y’ opt ¼
2jDyj jDyj oopt : ¼ ð2p=oopt Þ p
ð11Þ
Supposing that the above-mentioned proposition of Swoap et al. (1997) is also valid for human muscle in situ leads to calculate b parameter by associating optimal conditions of contraction (y’ opt and Topt ) in Eq. (1)
b¼
y’ opt ðTopt þ aÞ : ðT0 Topt Þ
ð12Þ
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1568 4000 3500
Psin (Nm rad s-1)
3000 2500 2000 1500 1000 500 0 0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
ω (rad s-1) Fig. 3. Typical Psin o relationship obtained by application of Eq. (B.1) in the 0–295 s1 pulsation range. Data obtained for subject 3. ( ):oopt :
2.4. Modelled torque–angular velocity relationship a and b parameters being identified, it is now possible to determine torque corresponding to every angular velocity, i.e. to construct a hyperbolic torque–angular velocity relationship without any data collected during isokinetic experiments. Moreover, maximal angular velocity (y’ max ) corresponding to torque equal to zero can be calculated from Eq. (1) as bT0 : y’ max ¼ a
ð13Þ
3. Methods 3.1. Materials Tests were performed with the aid of an ankle ergometer that is described in detail elsewhere (Tognella et al., 1997). Briefly, this ergometer consists of two elements: (i) a power unit composed of an actuator, its power supply unit, position, velocity and torque transducers and associated electronics; and (ii) a driving unit composed of a 486PC-type computer equipped with a specific 12-bit A/D and timer board. This apparatus allows notably to measure torque developed by the muscle groups crossing the ankle joint for different angular velocities—including zero velocity, i.e. isometric contraction—and to apply sinusoidal oscillations. The subject is seated on a chair without back support, with the left knee and ankle joints bent to approximately 1201 and 901, respectively. The left foot is enclosed in a rigid shoe that is fixed without any gear reduction to the actuator by means of a footplate. The rotational axis of the actuator is aligned by means of vertical and horizontal settings using the mean bi-malleolar axis as a standard approximation of the ankle axis. Further-
more, surface electromyograms (EMGs) are recorded differentially, amplified and band-pass filtered (1 Hz– 1 kHz). Bipolar electrodes are placed over the belly of the gastrocnemius medialis and of the gastrocnemius lateralis and on the central part of the soleus under the gastrocnemii. Mechanical variables and EMGs are loaded directly onto a 486 PC-type computer and instructions are presented to the subject on an oscilloscope. 3.2. Protocol Seven subjects volunteered for this study. All the subjects were informed for the nature of the study and had signed informed consent forms. They realized an initial session of habituation to the technique 1 week before their effective experiment. The protocol was approved by the Hygiene, Safety and Ethics Committee of the University. Three tests were performed during an experimental session. To determine model parameters, the subject was asked to perform his maximal voluntary contraction (MVC) in plantar flexion under isometric conditions. Three trials were performed and the best result was considered to be the effective MVC. Then, sinusoidal oscillations were achieved. 31 peak-to-peak harmonic angular displacement around reference position was imposed for frequencies ranging from 4 to 16 Hz. From one trial to another, frequency was increased by steps of 1 Hz, with each trial lasting 4 s. During these perturbations, the subject had to develop four randomized plantar flexion torques (Tm ) equal to either 15%, 30%, 45% and 60% of his MVC despite instantaneous changes reflecting the driving frequency. Finally, isokinetic tests were performed to obtain classical torque– velocity relationships for comparison purposes. The subject performed maximal voluntary plantar flexion contractions under isokinetic rotations realized at five
A. Desplantez, F. Goubel / Journal of Biomechanics 35 (2002) 1565–1573
Parameters collected during sinusoidal oscillations were torque and angular displacements. The following analysis only considered torque modulated at the driving frequency in order to neglect non-linearities. Thus, additional forces due to sollicitation of muscle proprioceptors are rejected (Kearney and Hunter, 1990). For each maintained torque, a gain curve and a phase curve were plotted. As in other studies (Agarwal and Gottlieb, 1977; Hunter and Kearney, 1982), the gain curve presented a peak for the resonant frequency followed by a linear decreasing with a slope of 40 dB/ decade (Fig. 2a) and the phase curve showed a 901 phase lag for the resonant frequency (Fig. 2b). Both curves reflect a mixed mechanical contribution from inertia, elasticity and viscosity of the musculo-articular system according to Eq. (3). A second-order model including such parameters was fitted to Bode plots to quantify each contribution with the aid of a processing module using a linear identification methodology (Levy, 1959). Then, after compensation of actuator dynamics, calculations presented in Section 2 were effected to model torque-angular velocity relationships. Thus, a was determined with the aid of Eq. (8). b was calculated from Eq. (12) after determination of Topt (Eq. (10)) and y’ opt (Eq. (11)). T0 was measured during isometric experiments (MVC). Finally, torques were calculated for a series of velocities (Eq. (1)) and y’ max (Eq. (13)) was determined. Parameters recorded during isokinetic trials were angular velocity, torque and EMGs. A processing module allowed the experimenter to determine the maximal torque value developed during each isokinetic trial. Then, peak torque obtained for a given angular velocity was considered to characterize this angular velocity. This technique presents notably the advantages to give to the subject enough time to develop full activation before the torque measurement and to be more reliable than that measuring torque at a constant angle (Coyle et al., 1981). Plotting torque against angular velocity led to a torque–velocity relationship fitted by Hill’s hyperbola (Fig. 1) using a statistical software (Statgraphicss). In order to verify the constant character of the voluntary effort of the muscle group, for each trial root-mean-square (RMS) EMG values obtained for each muscle were summed. Finally,
3.4. Statistics When plotting Bode diagrams, a r2 was calculated in order to determine the scattering with regard to the theoretical behaviour of the second-order model. For hyperbolic and linear regressions, a r2 was calculated to ensure significance in the relations. As a series of comparisons between experimental torque and modelled torque were made at the six different velocities, a twoway ANOVA was carried out. This test is appropriate, since it analyses the effect of two qualitative factors (i.e. velocity and experiment/model) on one response variable (torque). For that, experimental and modelled torque data of the subjects were arranged in a 6 2 replicated factorial design. Furthermore, a one-way ANOVA was run on the differences between experimental torque and modelled torque. Given the assumption that the present model is adequate, the differences should be zero, and thus, independently of velocity. In all cases, the level of significance was set at po0:05:
4. Results 4.1. Bbode–Tm and K–Tm relationships K and Bbode were related to Tm : Fig. 4 shows examples of such relationships illustrating that Bbode
300 250
K (Nm rad-1)
3.3. Data processing
for each velocity, experimental torque was compared to that calculated from the model.
200 150 100 50 0 0
10
20
30
40
50
60
70
Tm (Nm)
(a) 0.8
Bbode (Nm s rad-1)
randomized constant angular velocities (0.52, 0.87, 1.22, 1.57 and 1.92 rad s1). The range of motion was fixed to 201 from the reference position. The subject performed three trials for each velocity. Resting periods between each trial were observed to prevent muscle fatigue. At the end of the experimental session, a measure of MVC was performed to ensure the absence of muscle fatigue induced by the experiment.
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0.6 0.4 0.2 0 0
(b)
10
20
30
40
50
60
70
Tm (Nm)
Fig. 4. Typical linear relationships between (a) K and Tm (r2 ¼ 0:98) and (b) Bbode and Tm (r2 ¼ 0:96). Data obtained for subject 1.
A. Desplantez, F. Goubel / Journal of Biomechanics 35 (2002) 1565–1573
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and K were linearly related to Tm : Thus, experimental data were fitted to linear relationships obeying Eqs. (4a) and (4b). Coefficients of correlation (r2 ) were in the range 0.90–0.99. 4.2. Determination of modelled a and b parameters For each subject, values of Topt ; a; b; y’ opt and y’ max were calculated from the model. Results are presented in Table 1. Then, for each subject and for each experimental angular velocity, torque deduced from the model (Tmodel ) was calculated with the aid of Eq. (1).
Table 2 Numerical values of a and b parameters obtained by fitting Eq. (1) to experimental torque–velocity relationships Subjects
a (N m)
b (rad s1)
r2
1 2 3 4 5 6 7
19.10 2.54 22.30 25.40 52.03 23.80 26.50
0.96 1.67 0.50 0.55 5.11 0.52 0.64
0.99 0.98 0.99 0.99 0.98 0.99 0.99
r2 is coefficient of correlation.
4.3. Experimental torque–angular velocity relationships 140
4.4. Comparison between torque–angular velocity relationships Fig. 5 illustrates typical experimental Tiso y’ and modelled Tmodel y’ relationships, which were determined in the 0–1.92 rad s1 velocity range. For the population, the effect of velocity on the experimental torque and modelled torque was tested first. The twoway ANOVA indicated that both experimental torque and modelled torque were highly affected by velocity [F (5, 77)=57.3; po0:001]. On the other hand, the ANOVA attested no statistical significant differences between experimental torque and modelled torque [F (1, 77)=1.74; p > 0:05)], indicating that the use of the experimental torque and the modelled torque should be equivalent. Secondly, the differences between experimental torque and modelled torque was tested. The one-way Table 1 Modelled values of constants of Hill’s equation (a and b), of optimal conditions of contraction (Topt and y’ opt ) and of maximal angular velocity (y’ max ) Subjects a (N m) b (rad s1) Topt (% T0 ) y’ opt ðrad s1 Þ y’ max ðrad s1 Þ 1 2 3 4 5 6 7
31.26 17.64 14.35 90.31 81.54 9.85 21.01
2.30 2.15 1.61 9.02 8.35 0.84 2.46
0.31 0.29 0.26 0.41 0.41 0.22 0.31
2.76 2.99 2.91 2.81 3.50 2.19 3.14
8.83 10.14 11.07 6.50 8.47 10.10 10.29
120 Torque (Nm)
For each subject, RMS EMGs calculations indicated that from one trial to another variation in muscle group activation was less than 10%. For each subject, peak torque (Tiso ) was calculated for each of the five angular velocities. Tiso was found to depend on y’ in a hyperbolic manner (Fig. 1). Values of a and b parameters found by fitting Eq. (1) to experimental data and coefficients of correlation (r2 ) are presented in Table 2.
100 80 60 40 20 0 -20 0
0 0.5
1.0
.5 1.5
2.0
22.5
Angular velocity (rad s-1) Fig. 5. Comparison between isokinetic (’) and modelled (&) torque– angular velocity data. Torque corresponding to 0 rad s1 is the same for the model and experiment due to the fact that this isometric torque is used to calculate modelled torque–velocity data (T0 in Eq. (1)). Differences () between isokinetic and modelled torque are also indicated. Data obtained from subject 1.
ANOVA attested that velocity had no significant effects on the differences [F (5, 36)=0.95; p > 0:05)], indicating that the proposed model was adequate (see Fig. 5).
5. Discussion To our knowledge, two former studies considered the problem of constructing force–velocity relationships from sinusoidal oscillation tests. First, Rack (1966) applied sinusoidal oscillations on tetanized mammalian muscle. Tension recorded during sinusoidal oscillations in length was separated in two parts: an elastic contribution in phase with length oscillations and a viscous contribution in opposite phase with the imposed perturbation. Elliptic tension–velocity relationships obtained for each imposed frequency were assumed to characterize contractile element behaviour. These ellipses presented a short phase of constant tension. Plotting this tension against corresponding velocity allowed to construct a force–velocity relationship. Rack (1966) indicated that this force–velocity relationship well fitted Hill’s hyperbola. Second, Venegas (1991) modelled
A. Desplantez, F. Goubel / Journal of Biomechanics 35 (2002) 1565–1573
B ¼ mb F ;
ð14Þ
where B is viscosity, F tension and mb slope of the relationship. Since B is a ratio between tension and velocity variations, integration of such a relationship leads to a force–velocity relationship fitted by an exponential function with coefficient equal to mb : Venegas (1991) indicated that this force–velocity relationship well described experimental data reported in other studies as well as the classical Hill’s equation. It can be argued that sinusoidal oscillations induce small amplitude stretch-shortening cycles in muscle whereas isokinetic contractions correspond to a shortening of large amplitude. Such differences in experimental protocol may induce differences in contraction mechanics at the cross bridge level notably in terms of attachment and detachment kinetics. In spite of that, Rack (1966) and Venegas (1991) demonstrated that it was possible to characterize force–velocity relationship by virtue of sinusoidal movements. This might be due to the fact that this hyperbolic relationship has not evident structural basis and results from a curve fitting procedure (Ettema and Meijer, 2000). However, that may be, these two methods cannot be widened to human muscle in situ. In fact, it was demonstrated by Desplantez et al. (1999) that sinusoidal oscillations applied on human muscle in vivo do not only represent muscle behaviour. This is due to the fact that structures involved in such experiments are more complex that those involved in isolated muscles. Thus, it would be erroneus to adapt calculations presented above to human muscle in vivo. However, the fact that in the present experiment K and Bbode were found to depend on muscle torque (Eqs. (4a) and (4b)) led us to hypothesize that these parameters also reflected muscle properties. The difficulty of the study was to extract muscle component from the total system response. This was solved by defining the interrelation between Bbode and Bsin as Bsin characterizes muscle behaviour during small amplitude movements. To our knowledge, there is no method available in the literature to construct torque–angular velocity relationships from sinusoidal oscillation tests performed on human muscle in vivo. This is why the model proposed in this study was used to construct a theoretical hyperbolic torque–angular velocity relationship. In order to validate this approach, the first step was to compare, for each experimental velocity, developed torque (Tiso ) and modeled torque (Tmodel ). Calculations of RMS values indicated that a similar level of muscle activation was achieved during isokinetic tests for each
of the five tested velocities and for each subject. Then, observed changes in Tiso were not ascribable to changes in muscle activation. Statistical analysis showed that there was no significant difference between Tmodel and Tiso in the velocity range used in the present experiment. This validates the range of numerical values found for Tmodel : On the other hand, a and b values calculated from fitting Eq. (1) to Tiso y’ data were questionable. For instance, a had a negative value for most of the subjects (Table 2). Using such parameters led to a torque–velocity relationship fitting correctly experimental data but exhibiting an asymptotic behaviour (Fig. 6). Consequently, a torque was still produced when velocity was largely above physiological values. Furthermore, as expected from Eq. (13), a negative value was found for y’ max : Such kind of results was also found when fitting Eq. (1) to some torque–velocity data presented in the literature (Thomas et al., 1987; Harridge and White, 1993). This means that the classical approach of the force–velocity relationship by using isokinetic tests can be meaningful in a very limited range of velocities. This problem was solved with the present model since more realistic values for y’ max were obtained (Table 1). Secondly, data obtained from the model were compared to the literature related to isokinetic plantar flexion movements performed under maximal voluntary contraction. This comparison was limited to torque– velocity data well fitted by a hyperbola leading to positive values of a. Wickiewicz et al. (1984) fitted their torque–velocity relationships to Hill’s equation and found a y’ max of 6.5 rad s1. By calculating a and b parameters from a linearization of some published torque–angular velocity relationships, it was possible to estimate y’ max with the aid of Eq. (13). Thus, y’ max was found to be equal to 8.43 rad s1 for Fugl-Meyer et al. (1979), 10.52 rad s1 for Bobbert and Van Ingen Schenau (1990) and 6.84 rad s1 for Harridge and White 100 80 60 Torque (Nm)
detrusor muscle behaviour submitted to sinusoidal oscillations by a second-order model (Eq. (3)) including a viscous coefficient which was found to be linear against tension
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40 20 0
×
-20 -40 -60
-40
-20 0 20 40 Angular velocity (rad s-1)
60
80
Fig. 6. Typical torque–velocity relationship constructed from a hyperbolic fitting to isokinetic data (’) exhibiting a negative value for y’ max ( ). Results obtained for subject 2.
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(1993). All these results are in accordance with those found in the present model since y’ max was found to vary between 6.50 and 11.07 rad s1 (Table 1). All the more, maximal power during a single contraction is of more physiological significance than maximal shortening velocity for the evaluation of the functional capacity of whole muscle because (i) all fibres in a muscle contribute to the measurement of maximal power (Brooks et al., 1990), (ii) at low forces the velocities at which the muscle shortens can be faster than the maximum shortening velocity of its slower fibres (Claflin and Faulkner, 1985) and (iii) in vivo muscles are never completely unloaded (MacIntosh et al., 1993). Thus, it is of high interest to verify that y’ opt and Topt parameters are consistent with the literature. On one hand, Fugl-Meyer et al. (1982) and Harridge and White (1993) found y’ opt equal to 3.14 and 3 rad s1, respectively. In another study, Edgerton et al. (1984) deduced from results published by Wickiewicz et al. (1984) that y’ opt would be equal to 0.35y’ max ; i.e. 2.3 rad s1. From the results of Fugl-Meyer et al. (1979) and Bobbert and Van Ingen Schenau (1990) it was possible to determine y’ opt which was found to be equal to 3.12 and 2.82 rad s1, respectively. These results are still in accordance with the present model since y’ opt was found to vary from 2.19–3.5 rad s1. On the other hand, Topt estimated from the same published studies was found to vary between 0.25 and 0.41T0 : This is in good accordance with present results since Topt was found to vary between 0.22 and 0.41T0 (Table 1). In conclusion, this paper indicates that sinusoidal oscillation experiments can be used to construct a torque–angular velocity relationship characterizing the dynamic behaviour of human muscle in vivo. When considering a limited range of velocities, this approach leads to similar results than those obtained by using isokinetic experiments. This validates the torque values obtained with the model and the choice of optimal conditions. Moreover, the present approach offers many advantages since (i) more realistic values of a and y’ max are obtained since, contrary to isokinetic data, negative values are avoided; (ii) sinusoidal experiments are performed under sub-maximal isometric contractions that are easier to perform than isokinetic maximal contractions. This offers the possibility to characterize muscle dynamic properties in patients suffering from neuromuscular disease; (iii) the risk in scattering of results induced by neurophysiological responses (e.g. inhibitions for large torques) is minimized.
Acknowledgements This work was supported in part by the Centre National d’Etudes Spatiales and the Association Fran-
c-aise contre les Myopathies. The authors thank Dr. Daniel Lambertz for his help in statistical analysis.
Appendix A. Determination of a parameter Determination of a parameter needs insertion of Eqs. (4b) and (6) in Eq. (7). This leads to the expression: ðb1 Tm þ b2 ÞðX1 Tm þ X2 Þ ¼
ðTm þ aÞ2 : ðT0 þ aÞb
ðA:1Þ
After development of Eq. (A.1), identification between terms in Tm2 ; Tm and constants gives the following formulas: b1 X 1 ¼
1 ; ðT0 þ aÞb
b1 X2 þ b2 X1 ¼
b2 X 2 ¼
2a ; ðT0 þ aÞb
a2 : ðT0 þ aÞb
ðA:2aÞ
ðA:2bÞ
ðA:2cÞ
In this system of three equations, only b1 and b2 are known thanks to the sinusoidal oscillation tests (Eq. (4b)). Solving Eq. (A.2) leads to determine a parameter (Eq. (8)).
Appendix B. Expression of Psin for optimal conditions of contraction Assuming that torque is defined as the sum of an internal contractile torque (Tm ) and an internal resisting torque (Tr ) Psin can be expressed as Psin ¼ ðTm Tr Þo;
ðB:1Þ
where o is the imposed pulsation. Using complex values for representing frequency dependent changes leads to a transformation of Eq. (3) into the expression: DTm ¼ ðK Io2 Þ þ joBbode Dy
ðB:2Þ
with j the complex value (j2 ¼ 1). Then, Tr can be determined by expressing modulus of Eq. (B.2): 1=2 ðB:3Þ Tr ¼ jDTm j ¼ jDyj ðK Io2 Þ2 þ ðBbode oÞ2 where jDyj is peak-to-peak angular displacement. K and Bbode corresponding to Topt can be calculated from Eqs. (4a) and (4b), respectively. Inserting these values in Eq. (B.3) leads to determine the optimal torque Tr : Then, substituting Tm by Topt in Eq. (B.1) allows to express Psin for optimal conditions of contraction.
A. Desplantez, F. Goubel / Journal of Biomechanics 35 (2002) 1565–1573
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