Summary of the sensitivity of the
1. Introduction
The human motor system benefits from remarkable muscular redundancies: A motor task is normally performed with the simultaneous involvement of more muscles than strictly necessary. Furthermore, this same task may be performed in multiple ways, with different muscle combinations. From the mechanical viewpoint the musculoskeletal system is indeterminate, whereby the number of unknown muscle forces exceeds the number of available equations. We address in this Chapter the biomechanics of the lower limbs in long-distance running under conditions of developing fatigue. In long-distance running the running speed may result in more than 300 foot-strikes per leg per kilometer. Each such foot-strike evokes an impact loading that results in a vertical shock impulse transmitted upwards through the body and carries with it the potential for injuries in the bone and joint tissues.
Fatigue, or stress, fractures occur in bones in response to repetitive stresses over multiple cycles, when the body’s ability to adapt is exceeded [1,2]. An important factor which affects the incidence of bone stress injury, is exposure to abrupt changes in the bone loading [1], and consequent alteration in the strain distribution [3] with insufficient recovery periods [4]. Other factors include footwear, terrain and intensity of activity or training [1].
Two of the major factors responsible for impulse attenuation at foot- or heel-strike are the shock absorption capacity of the active muscle in the lower limbs, and the cushioning effect of the foot heel-pad tissue. In previous reports we have shown that in long distance running the impact shock load on the lower limbs increases with progressing fatigue [5-8]. One additional question is whether, as a result of fatigue, an imbalance between the activities of the plantar and dorsi flexor muscles of the ankle develops. Such an imbalance would compromise the protective action provided by the muscles to the shank [9-11].
The goal of this research was to characterize the heel-strike shock propagation and attenuation in running by means of a biomechanical model, and to examine changes taking place as a result of running fatigue.
2. Biomechanical modeling of the lower limb
This section deals with the modeling of the heel-strike event. With the development of biomechanical models of human body motion, it has become possible to simulate vertical landing, such as occurring during running, in order to gain insight into intermuscular coordination and to elucidate control strategies of the musculoskeletal system. A common method to deal with this type of problems is to lump together elements of the human body e.g., muscles, tendons, ligaments, bones and joints so that the overall musculoskeletal system is represented as a damped elastic mechanism. Several models describing vertical landing can be found in the literature [12-18]. These models are usually characterized by the presence of elastic springs and viscous dampers, with constant properties and provide a reasonable prediction of the maximal vertical foot/ground reaction force.
Indeterminacy of the locomotor system can be addressed by adopting the lumping method, whereby the material elements of the human body e.g., muscles, tendons, ligaments, bones and joints are lumped together so that the overall musculoskeletal system is represented as a multi-degree-of freedom damped elastic mechanism, interconnecting the masses of the body segments.
The foot- or heel-strike period during landing from fall, during hopping or during the stance phase of running has been generally modeled using one-dimensional models along the vertical direction. [13-15,19-20].
In this study we represent the body segments during heel-strike by a four degree-of-freedom elastically-damped uni-axial biomechanical model. The model thus includes 4 masses connected by elastic stiffnesses with parallel damping elements, as shown in Figure 1. In more details, the masses
3. Model equations
For the above model, the force diagram, as presented in Figure 2, yields the following model equations.
with initial conditions:
and gravitational acceleration
These values rely on reported landing velocities between -0.8 m/s to -1.2 m/s for running speeds of 3.5 m/s (comparable to the speeds of this study), while wearing various types of running shoes [21-23].
The above masses are expressible in terms of the total body mass from anthropometric data [24].
From the simultaneous recording of the foot ground reaction forces and accelerations on the masses
4. Experimental setup
Information about the impulsive loading along the skeletal elements in long-distance running can be non-invasively obtained from the foot-ground reactive forces [25] and, more directly, by measuring the transient accelerations on the shank caused by impact.
4.1. Impact accelerations
Non-invasive in vivo measurements of acceleration and impact transmission along the human body were made by externally attaching light-weight, high-sensitivity accelerometers at strategic points including bony prominences, such as the tibial tuberosity below the knee area, the greater trochanter near hip level and the sacrum area at the lower back [13,15,22,26-29].
In this study, each subject was instrumented with two light-weight (4.2 grams) uniaxial (Kistler PiezoBeam, type 8634B50, Kistler, Winterthur, Switzerland), skin-mounted accelerometers connected to a coupler (Kistler Piezotron, type 5122). One was attached on the tibial tuberosity, and the second - on the sacrum. To achieve good reliability of the measurements by means of bone-mounted accelerometers, the accelerometers were pressed onto the skin in closest position to the bony prominences of the tibial tuberosity and the sacrum, by means of two elastic belts passed in a horizontal plane around the shank and the waist, respectively. The tensions of the belts were well above the level in which the acceleration trace for a given impact force became insensitive to the accelerometer attachment force, thus ensuring stability of the accelerometer as well as consistency of the readings and reproducibility of the data [13,30].
The shank accelerometer was aligned with the axis of the tibia to provide the axial component of the tibial acceleration and the accelerometer on the sacrum was oriented along the spine. These accelerometers allowed us to acquire the shock accelerations propagated in the longitudinal directions of the tibia and the spine. As earlier reported, such attachment is suitable for faithfully measuring the amplitude of shock acceleration [5-8].
Force platforms, type Kistler Z-4035, were used for the simultaneous recordings of the foot-ground reaction forces and acceleration.
4.2. Running fatigue tests
An overview of the experimental setup is shown in Figure 3. For examining the effect of global fatigue due to running, the subjects were asked to run on a Quinton Q55 treadmill.
Global, or metabolic fatigue is associated with the development of metabolic acidosis following an endurance exercise and is accompanied by a decrease in the end tidal carbon dioxide pressure (PETCO2) [31]. In long distance running metabolic fatigue is reached when the running speed exceeds the anaerobic threshold [31].
Running was thus for a duration of 30 min and at a speed exceeding the anaerobic threshold speed of each subject by 5%. Before the test a 15 min warming up running on the treadmill was performed. In this study, the average running speed for all 14 subjects was 3.53 m/s (SD, 0.19). It should be noted, however, that in addition to global fatigue, local fatigue in a muscle may also take place as a result of an intensive activity of this muscle. This type of fatigue is reflected by certain changes in its electromyogram (EMG) signal in the time and/or frequency domains [32]. Local fatigue was not considered in this study.
Respiratory data were collected from a Sensor-Medics 4400 device and included
The respiratory data were evaluated at each of the 1st, 5th, 10th, 15th, 20th, 25th and 30th min of running and the accelerometer and force platform data were online sampled at 1667 Hz sampling rate. The model parameters were estimated, however, at the 1st, 15th and 30th min of running.
The dynamics of acceleration build-up at heel-strike is shown in Figure 4 where the simultaneous recordings of the tibial tuberosity acceleration and the ground reaction force (GRF) are shown in two time scales: complete running cycle (panel a) and zooming-in on the heel-strike event (panel b). In this case the build-up time to the tibial tuberosity peak acceleration was ~ 30 ms. It is also noted that the ground reaction force exhibits two peaks: a smaller one shortly after heel-strike and a larger one (~ 2.5 body weights), towards the middle of the running cycle.
Typical accelerometer traces for the tibial tuberosity (right leg) and sacrum are shown in Figure 5, for a complete running cycle, i.e., from heel-strike of the right foot till the next heel-strike of the same foot. Two major differences should be noted between the traces: (a) intensity of ~ 8 g in the tibial tuberosity versus less than 3 g in the sacrum; (b) while the tibial tuberosity exhibits one major peak within the first 50 ms of the running cycle from heel-strike and originating from the heel-strike of the right foot, the sacrum acceleration, due to its central location, exhibits two comparable positive peaks within the running cycle, reflecting each of the right and left heel-strikes.
5. Parameter estimation of the model
The mechanical properties of biological material are, in general, multiple variable-dependent. Specifically stiffness, in addition to its being non-linear e.g. strain dependent, often depends on the deformation rate. This is also the case with bones [33], tendons and ligaments [34], cartilage [35] and muscle [36]. Damping too may be position-dependent.
Due to nonlinearity of the stiffness/damping properties of the joints of the leg [e.g. 20,37], we were not generally able to estimate the model parameters while assuming that they remain constant over the heel-strike period. Thus, the heel-strike period was divided into two equal periods (22 ms each) and the parameters were estimated separately for each of these periods, using the Gauss-Marquardt [38-39] method of non-linear estimation. For the first period the initial conditions were as prescribed in equation (2), and for the second period the initial conditions used were the values reached at the end of the first period.
Figure 6 shows the model prediction of the shank mass (
6. Model results and model sensitivity analysis
Table 1 shows the sample results of the stiffness and damping parameters for the two time zones. A sensitivity analysis can provide an indication to the quality of the estimation of parameters. Thus, sensitivity of the
In cases of low- or no-sensitivity the parameters were varied, up and down, by one order of magnitude with the results shown in parentheses. The one order of magnitude variation in the parameter values did not evoke sensitivity beyond that of the twofold variation, except for
Due to the fact that the parameter estimation of the model coefficients was performed in short time intervals, fractions of the heel strike event, the damping coefficients disclosed a high variability. Better estimations would have probably resulted if the period of estimation was higher. Accordingly, the stiffness parameters
Difficulties in estimating the damping coefficients are not unusual due to their expected low values. It has been reported that in repetitive physical activity, such as in running, the subject bounces on the ground in a spring-like manner [17, 40-45]. Depending on the range of joint flexion and on the frequency of motion, a considerable amount of elastic energy can be stored and re-used. It has been shown that the dissipated energy in muscles increase when the amplitudes of joint movement are increased [46]. It has also been also commented that the utilization of stored elastic energy depended on the shortness in latency between the stretch and shortening phases of the muscles [47]. Accordingly, during the ground-contact period of running, the leg was modeled as a one-dimensional four-degree-of-freedom piece-wise linear spring, with no damping. During heel-strike, the joints did not have a damping effect, to contribute to energy dissipation.
Summary of the values of these parameters for the two solution time zones is given in the rightmost column of Table 2. The values, averages for 10 subjects (SD), are calculated at each of the 1st, 15th and 30th min of running to evaluate the effect of fatigue. The asterisks indicate a significant change
It should be noted that the stiffness
The
The obtained average knee stiffness
Further exploration of the effect of fatigue in the course of running was performed by correlating, using linear regression, the tibial tuberosity peak acceleration to the parameter values obtained from the model results. Figure 7 shows the correlation for which the Pearson’s coefficient was statistically different from zero (with 95% significance). The parameters are
It has been shown that in running with shoes, the foot is restricted from bulging sideways, thus limiting the vertical deformation to an average of 5.5 mm, as opposed to 9 mm when running barefoot [48,56]. This explains the higher stiffness during the first time zone compared to bare foot running. It also explains the lower stiffness during the second time zone compared to bare foot running. It has also been shown by that better energy absorption and impact shock attenuation is associated with lower stiffness [51].
The correlation found between low stiffness in the first time zone to the high stiffness in the second time zone is obvious from the anatomy of the heel pad, which consists of nearly closed collagen cells filled with fatty cells [27,48]. The vertical orientations of these cells, together with the high viscosity of the fat tissue are the major factors responsible for the absorption of the impact energy at heel strike. Initially, the fat flows sideways and small loads result in high deformation (low stiffness). In the second time zone, after the heel pad has already considerably deformed, further increase in deformation provokes a high load, thus high stiffness. The effect of fatigue could be explained by means of the heating effect during the course of running. With nearly 80 heel-strikes per min, the whole running duration of 30 min results in some 2500 heel-strikes, each of which causing a rapid deformation of the heel pad and during which the fatty tissue frictions while squeezed out of the collagen cells. The accumulated heat due to friction reduces the viscosity and the vertical displacement is accelerated, causing a reduction in stiffness during the first zone of heel strike. In the second zone, however, the thinner remaining tissue together with the underlying bone evokes an increase in stiffness.
7. Conclusion
Stress fractures in long bones of the lower limbs are believed to originate from repetitive and/or excessive loading, such as may take place in long-distance running at a speed exceeding the anaerobic threshold. In the present study the average running distance per test was 6.30 km (30 min of running at the average speed of 12.6 km/h) in agreement with the definition of ‘long distance’ [57]. We have measured and analyzed the following: respiratory data to monitor global fatigue; and accelerometry, to provide quantitative information on loading of the major segments of the lower limb. While providing accelerometer boundary values for the model system, accelerometry is an advantageous method due to its being non-invasive.
We have addressed a major fatigue-related factor taking part in exposing the shank to stress fractures risk: the decline in end tidal carbon dioxide pressure, the latter expressing metabolic fatigue [31,58]. The mechanical consequence of fatigue in long-distance running is two-fold: enhanced impact acceleration due to global fatigue and muscle activity imbalance due to local fatigue before and during foot contact, resulting in the development of excessive shank-bone bending stresses and higher risk of stress injury [11].
While departing from the stiffness constancy concept, the model revealed that a correct and sufficient variability of the joint stiffness is a two-region piece-wise constant stiffness indicating that a higher order of non-linearity is not necessary. This result should be considered meaningful in those problems where the constant stiffness representation is not sufficient and in cases where the system’s representation has to be improved. Joint stiffness is dominated by muscular activation [59-60] and as the joints stiffen, they undergo smaller angular displacements during the ground-contact phase, also resulting in smaller excursion of the hip and higher leg stiffness. Thus, since stiffness is related to muscle activation, the piece-wise constant stiffness obtained solution also provides, through the obtained stiffness profiles, an insight into the patterns of the muscular activation in the legs’ joints.
The fact that the simple model of a piece-wise constant stiffness can predict major features of the running exercise makes it an effective tool for future designing of artificial legs and robots and also for the development of more accurate control strategies.
Acknowledgement
This Chapter is partly based on an MSc Thesis of second author DD, carried out in JM’s Biomechatronics Laboratory, Department of Biomedical Engineering, Technion – Israel Institute of Technology, under the joint supervision of JM and Prof. Yacov Ben-Haim.
References
- 1.
An Aetiological Review for The Purposes of Guiding Management. Sports MedicineBeck Tibial B. R. Stress Injuries. 26 1998 265 279 . - 2.
Burr DB. Bone, Exercise and Stress Fracture. In: Holloszy JO. (ed.) Exercise and Sport Sciences Review. Baltimore (MD): Williams and Wilkins;1997 171 194 . - 3.
The Effects of Muscle Fatigue on Bone Strain.Yoshikawa Mori T. Santiesteban S. Sun A. J. Hafstad T. C. Chen E. Burr J. D. B. 188 1994 217 233 . - 4.
Sports MedReeder Dick M. T. Atkins B. H. Probis J. K. Martinez A. B. Stress J. M. Fractures Current. Concepts of. Diagnosis Treatment 22 1996 198 212 . - 5.
The Influence of Fatigue on EMG and Impact Acceleration in Running. Basic Appl. Myol.Mizrahi Voloshin J. Russek A. Verbitsky D. Isakov O. E. 7 1997 111 118 . - 6.
Shock Accelerations and Attenuation in Downhill and Level Running. Clinical BiomechMizrahi Verbitsky J. Isakov O. E. 15 2000 15 20 . - 7.
Dynamic Loading on the Human Musculoskeletal System- Effect of Fatigue.Voloshin Mizrahi A. Verbitsky J. Isakov O. E. 13 1998 515 520 . - 8.
Shock Absorption and Fatigue in Human Running. ,Verbitsky Mizrahi O. Voloshin J. Treiger A. Isakov J. E. 14 1998 300 311 . - 9.
The Journal of Bone and Joint SurgeryBaker Frankel J. Burstein V. H. Fatigue A. Fractures Biomechanical. Considerations 1972 1345 1346 . - 10.
Basic Biomechanics of the Musculoskeletal System. Philadelphia (PA): Lea and Febiger,Nordin Frankel M. Biomechanics V. of Bone. In Nordin. M. Frankel V. (eds 1989 3 29 . - 11.
Fatigue-Related Loading Imbalance on the Shank in Running: A Possible Factor in Stress Fractures. .,Mizrahi Verbitsky J. Isakov O. Fatigue E. 28 2000 463 469 . - 12.
Greene PR, McMahon TA. Reflex Stiffness of Man’s Anti-Gravity Muscles During Kneebends while Carrying Extra Weights. J. Biomechanics,12 1979 881 891 . - 13.
In-Vivo Elastic and Damping Response of the Human Leg to Impact Forces. J. Biomech. Engng.Mizrahi Susak J. In Z. 104 1982 63 66 . - 14.
An Experimental and Analytical Study of Impact Forces during Human Jumping. J BiomechanicsOzguven Berme H. N. N. 21 1988 1061 1066 . - 15.
Modeling of Heel Strike Transients during Running. Human Movement Science,Kim Voloshin W. AS Johnson S. H. 13 1994 221 244 . - 16.
Leg Stiffness and Stride Frequency in Human Running. J. Biomechanics,Farley Gonzalez C. T. O. 29 1996 181 186 . - 17.
Farley C T, Morgenroth D C. Leg Stiffness Primarily Depends on Ankle Stiffness during Human Hopping. Journal of Biomechanics32 1999 267 273 . - 18.
ASME Journal of Biomechanical EngineeringSpagele Kistner T. Gollhofer A. Modeling A. Simulation Optimization of. a. Human Vertical. Jump A. S. M. 32 1999 521 530 . - 19.
McMahon TA, Green PR. The Influence of Track Compliance on Running. J Biomechanics,12 1979 893 904 . - 20.
Constant and Variable Impedance of the Leg Joints in Human Hopping. ASME Journal of Biomechanical EngineeringRapoport Mizrahi S. Kimmel J. Verbitsky E. Isakov O. E. 125 2003 507 514 . - 21.
Cavanagh PR, Valiant GA, Misevich KW. Biological Aspects of Modeling Shoe/Foot Interaction during Running. In: Frederick EC. (ed.) Sport Shoes and Playing Surfaces. Champaign: Human Kinetics;1984 24 46 . - 22.
Mc Mahon Valiant T. A. Frederick G. Groucho E. C. Running J. Appl Physiol. 62 1987 2326 2337 . - 23.
Nigg BM. Experimental Techniques Used in Running Shoe Research. In: Nigg BM. (ed.). Biomechanics of Running Shoes. Champaign: Human Kinetics Publishers;1986 27 62 . - 24.
Winter DA. Biomechanics and Motor Control of Human Movement,2 Ed., John Wiley & Sons; 1990.51 74 . - 25.
Dickinson JA, Cook SD, Leinhardt TM. The Measurement of Shock Waves Following Heel Strike while Running. J. Biomech.18 1985 415 422 . - 26.
The Effect of Varied Stride Rate and Length upon Shank Deceleration during Ground Contact in Running.Clarke Cooper T. Clark L. Hamill D. C. 1983 170. - 27.
Valiant GA. Transmission and Attenuation of Heelstrike Accelerations. In: P.R Cavanagh (Ed.), Biomechanics of Distance Running. Champaign, IL: Human Kinetics;1990 225 247 . - 28.
Shock Attenuation and Stride Frequency during Running. Human Movement Science,Hamill Derrik J. Holt T. R. K. G. 14 1995 45 60 . - 29.
Dominant Role of Interface over Knee Angle for Cushioning Impact Loading and Regulation Initial Leg Stifness.Lafortune MA Henning E. MJ Lake 29 1996 1523 1529 . - 30.
The Response of the Lower Extremity to Impact Forces. I. Design of an Economical Low Frequency Recording System for Physiologic Waveforms. Bulletin of the Hospital for Joint DiseasesStreitman Pugh A. J. 1978 XXXIX(1): 63-73. - 31.
Determinants and Detection of Anaerobic Threshold and Consequences of Exercise above It. CirculationWasserman K. 1987 76(suppl VI), VI-29. - 32.
Edwards R H. Human Muscle Function and Fatigue. Human Muscle Fatigue: Physiological Mechanisms. London: Pitman Medical, (Ciba Foundation symposium 82)1981 1 18 . - 33.
Wright TM, Hayes WC. Tensile Testing of Bone Over a Wide Range of Strain Rates: Effects of Strain Rate, Micro-Structure and Density” Medical and Biological Engineering and Computing1980 671 680 . - 34.
The Effects of Strain Rate on the Biomechanical Properties of the Medial Collateral Ligament: A Study of Immature and Mature Rabbits. Transactions of the Orthopedic Research SocietyPeterson Gomez R. H. MA Woo-Y S. L. 198712 127 - 35.
Li JT, Armstrong CG, Mow VC. The Effects of Strain Rate on Mechanical Properties of Articular Cartilage in Tension, Proc Biomechanical Symposium ASME AMD198356 117 120 . - 36.
Validation of Optimization Models that Estimate the Forces Exerted by Synergistic Muscles. Journal of BiomechanicsHerzog Leonard W. T. R. 1991 31-39. - 37.
The Dynamics of the Subtalar Joint in Sudden Inversion of the Foot, J Biomech. Engng,Mizrahi Ramot J. Susak Y. Z. 112 1990 9 14 . - 38.
Marquardt DW. An Algorithm for Least Squares Estimation of Nonlinear Parameters. SIAM11 1963 431 441 . - 39.
Bard Nonlinear Y. Parameter Estimation. Academic Press. Inc 1974 83 217 . - 40.
Alexander RMcN. The Spring in Your Step: the Role of Elastic Mechanism in Human Running. Amsterdam: Free University press,1988 17 25 . - 41.
Locomotion Energetics of the Ghost Crab. II. Mechanics of the Centre of Mass during Walking and Running”, Journal of Experimental BiologyBlickhan Full R. R. J. 130 1987 155 -174. - 42.
The Spring-Mass Model for Running and Hopping. Journal of BiomechanicsBlickhan R. 22 1989 1217 1227 . - 43.
Mechanical Work in Running. Journal of Applied PhysiologyCavagna Sailbene G. A. Margaria F. P. R. 19 1964 249 256 . - 44.
Cavagna GA, Heglund NC, Taylor CR. Mechanical Work in Terrestrial Locomotion: Two Basic Mechanisms for Minimizing Energy Expenditure. American Journal of Physiology1977 233: R243 -R261. - 45.
McMahon TA, Cheng GC. The Mechanics of Running: How Does Stiffness Couple with Speed? J Biomechanics,23 1990 65 78 . - 46.
Evaluations Indirecte de l’Energie Elastique Utilisee dans l’Impulsion des Sauts. Schweizerischen Zeitschrift fur Sportsmedizin,Thys H. 4 1978 169 177 . - 47.
European Journal of Applied PhysiologyBosco Komi C. Mechanical P. V. Characteristics Fiber Composition. of Human. Leg Extensor. Muscles 41 1979 275 284 . - 48.
The Mechanical Characteristics of the Human Heel Pad during Foot Strike in Running: An In-Vivo Cineradiographic Study. J. Biomechanics,De Clercq Aerts D. Kunnen P. M. 27 1994 1213 1222 . - 49.
Bennett MB, Ker RF. The Mechanical Properties of the Human Subcalcaneal Fat Pad in Compression", J Anat.,171 1990 131 138 . - 50.
Deformation Characteristics of the Heel Region of the Shod Foot During a Simulated Heel Strike: The Effect of Varying Midsole Hardness", J Sports Set,Aerts De Clercq P. D. 11 1993 449 461 . - 51.
In vivo examination of the dynamic properties of the human heel pad. Med.,Kinoshita Ogawa H. Kuzuhara T. Ikuta K. K. 14 1993 312 319 . - 52.
The Mechanical Properties of the Human Heel Pad: A Paradox Resolved. J Biomechanics,Aerts Ker P. De Clercq R. F. Ilsley D. Alexander D. W. Mc R. N. 28 1995 1299 1308 . - 53.
Alexander RMcN, Bennet MB, Ker RF. Mechanical Properties and Function of the Paw Pads of Some Mammals.1 Zoo1., 1986;A209 405 419 . - 54.
Differential Shock Transmission Response of the Human Body to Impact Severity and Lower Limb Posture. J Biomechanics,Lafortune MA Lake M. J. Henning E. 29 1996 1531 1537 . - 55.
Mechanics of Running under Simulated Low Gravity. J. Appl. Physiol.,He Kram J. Mc Mahon R. T. A. 71 1991 863 870 . - 56.
Significance of Heel Pad Confinement for the Shock Absorption at Heel Strike. Int. J Sports Med.,Jorgensen Ekstrand U. J. 9 1988 468 473 . - 57.
Ward-Smith AJ. The Bioenergetics of Optimal Performances in Middle-Distance and Long-Distance Track Running. J Biomech,32 1999 461 465 . - 58.
Anaerobic Threshold and Respiratory Gas Exchange during Exercise. J. Appl. Physiol.,Wasserman Whipp K. Koyal B. J. Beaver S. N. W. L. 35 1973 239 243 . - 59.
Quantification of Cerebellar Ataxia in Movements of the Hand. In, Seminar 8, Göteburg, Sweden,Nielsen Bisgård J. Arendt-Nielsen C. Jensen L. T. S. 1994 157 166 . - 60.
Weiss PL, Hunter IW, Kearney RE. Human Ankle Joint Stiffness Over the Full Range of Muscle Activation Levels. J. Biomechanics,21 1988 539 544 .