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Simulation of the force enhancement phenomenon in muscle
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SIMULATION OF THE FORCE ENHANCEMENT PHENOMENON IN MUSCLE ERIC J. SPRKXNGS Biomechanics
Laboratory.
(Received
College of Physical Education, University Saskatoon. Sask. S7N OWO. Canada 13 November
1985; in rrrised
form
of Saskatchewan.
10 _Apr~l 1986)
Abstract-This paper demonstrates that the mathematical modeling of the mechanical force enhancement phenomenon in muscle can be achieved using the simple rheological model proposed by Houk [l]. The simulation offorce enhancement for this model requires that the pre-stretch of the muscle be entered as an input function to the rheological model’s equations. The results of the mathematical simulation suggest that the main characteristics of the force enhancement phenomenon could quite conceivably be explained in terms of a greater induced compliance of the series elastic component as a result of a stretch placed on the muscle by an external load Simulation Muscle stretch
Mechanical
force enhancement
Muscle modeling
Rheological
INTRODUCTION The simple rheological model (Fig. 1) proposed by Houk [l], has been chosen by many researchers 12-61 as representing a bare bones mechanical model of muscle. In this type of tn’o component model, the muscle is envisaged to consist of only a contractile component (CC) and a series elastic component (SEC). In Fig. 1 the CC is represented rheologically as a parallel combination of a constant force generator and a dashpot. The SEC is represented rheologically as a Hookean spring. It is this model’s mathematical and programing simplicity that makes it most attractive to researchers who wish to incorporate the main mechanical elements of muscle contraction in their simulation of human movement. Although it has been shown by Chapman and Harrower [4] that this linear rheological model is somewhat limited in its description of a dynamic muscle contraction, it does have the capacity to provide a reasonable first approximation to a muscle’s force output profile during both concentric and eccentric muscle contraction. What has not been evident about this particular model is that it also has the capacity to display the so-called force enhancement phenomenon that characterizes a muscle that has just previously been stretched. The purpose of this present paper is to show that the mechanical force enhancement phenomenon of muscle can be simulated using the simple rheological model proposed by Houk [l].
Fig. 1. Houk’s rheological muscle model consisting of a constant force generator (F,), a viscous damper with damping coefficient of viscosity B. and a Hookean spring of stiffness K. _Y, and y2 represent the absolute displacement of the two node points. 423
424
FORCE
ENHANCEMENT
PHENOMENON
Force enhancement was a term used by Edman rt ul. [7] to represent the phenomenon they observed in active single muscle fibers that had just previously been subjected to stretch. They observed that the isometric tension of the muscle fiber after stretch remained elevated above the normal isometric tetani tension produced by the muscle of the same sarcomere length. Their experimental results indicated that the output tension of the muscle fiber was enhanced in both magnitude and duration by increases in the amplitude of stretch and the length to which the muscle fiber was stretched [7]. In order to explain this observed force enhancement phenomenon. Edman et ul. proposed two theories: (1) that after stretch, an additional elastic component is recruited in parallel with the shortening elements of the contractile system, or (2) that stretch during contraction alters the properties of the contractile component (CC) in such a way as to allow it to produce a greater force than could be attained during ordinary tetanus. However, it was evident from their subsequent discussion that they could not support either theory with any firm experimental evidence. Prior to the work of Edman et ill.. similar stretch experiments on isolated muscle fibers had been performed by both Cavagna et ~1. [8, 91 and Sugi [I I]. While it is certainly true that these researchers observed the same stretch phenomenon as did Edman et crl., they did not refer to it specifically as “force enhancement”. Cavagna and his colleagues observed in their work [8] that a muscle which had just previously been stretched was able to perform more positive work than a similar muscle which was released from a state of isometric contraction, even when differences in initial force values were taken into account. Their subsequent research [9] on this phenomenon lead them to conclude that stretching of the active muscle temporarily modified the elastic characteristics of the muscle in the sense that a greater amount of mechanical energy was released for a given fall of force during subsequent muscle shortening. They hypothesized [lo] that this greater compliance ofthe elastic characteristics of the muscle was due to a small population of cross-bridges between the actin and myosin myofilaments which were more strained and unstable than some of the neighboring cross-bridges. They further hypothesized that these heavily strained cross-bridges would recoil, releasing mechanical energy if the muscle were allowed to shorten immediately after stretching, but would themselves break down if the muscle were kept active at the stretched length. The published results of Sugi’s work [ 1I] were similar to those later reported by Edman et al. Of particular relevance to this present paper was his brief mention of a proposal suggested to him by A. F. Huxley on the possible cause of the so-called “slip” phenomenon in muscle which accounts for observed changes in muscle tension during stretch. Huxley stated that there was experimental evidence indicating that the A-band was extended when a muscle was extremely stretched during tetanus. The theoretical consequences of this observation was that the “slip” phenomenon could be considered to be related to the extension of themyofilament:, or the cross-bridges themselves, and not to the actual breaking of the cross-links. This observation lends support to the hypothesis proposed by Cavagna et ~1. [lo] which stated that the extra positive work performed by a muscle after stretch was due in part to a greater elastic compiiance that developed temporarily within the muscle system. The temporary change in compliance of the muscle system after stretch, as suggested by Cavagna et al., would seem to be a property of the series elastic component (SEC) since it has been shown in the work by Huxley and Simmons [12], that the SEC mainly resides within the cross-bridges themselves. METHOD Mathemuticnl
modeling of‘ the jtirce enh~mcement phenomenon
usitty (I Two component
tnotlel
Figure 1 is Houk’s rheological representation of the two component muscle model. This model, comprised of a parallel combination of a force generator and dashpot coupled in series with a Hookean spring, has been chosen by many researchers as representing a bare bones mechanical model of skeletal muscle. The model’s mathematical and programing
Simulation
of the force enhancement
phenomenon
tn muvzle
425
t
I
I-i,
7. Di\placement
hiator!
slope=c of node
1 asa result
of an external
strctchinp
force.
simplicity makes it very attractive to researchers who wish to incorporate the main characteristics of a muscle’s output tension in their simulation of human motion. In the model shown in Fig. 1, F‘,, is considered to be able to supply a constant level of internal force instantaneously at the start of the test period. B and h’ are constants representing the damping coethcient of the dashpot. and the stiffness of the spring respectively. From Fig. 1, it is evident that the following two equations can be written: F,,, - .\.2B - K(.Xl - Xl) = 0 F(t) = k’(r,
(1) - Xi).
(2)
In order to test the mode1 to see if it is capable of exhibiting the force enhancement phenomenon, it was necessary to be able to subject the spring (i.e. SEC) to a stretch by an external force during the time that the force generator F,,, is considered to be active. The graphical representation of such a stretch is shown in Fig. 2. In Fig. 2, the time period “a” represents the period of isometric tension build up when node 1 is not permitted to move. The time period (b - tl) represents the period of eccentric muscle contraction when the muscle is being stretched out by an external load. The time period that follows time h is again a period of isometric muscle contraction but at a new muscle length. The problem then was one ofsolving for the output force history of the muscle, F(t), during the stages of stretch shown in Fik‘. 3. The key to the solution of this problem was to find a method of entering the displacement history of I, as an input function to equations (1) and (2). This was most easily achieved by rewriting the displacement history of x1 in terms of a series of Heaviside unit functions [I?]. .X!(f) = + C’r-:(1 - u) (f - LI) ~~(’ L’(t ~ h) (r - hi where:
L’= negative
(3)
slope
The Heaviside representation of .ui(t), as shown in equation (3), was then substituted into equation (I). The resulting first order linear differential equation in terms of x2(t) was then solved by means of a Laplace transformation. The resulting expression for x2(t), and the He:iviside expression for xi(t) shown in equation (3). were then substituted into equation (2) in order to obtain the desired expression for f(t). The resulting expression is shown below: F(r 1= f,,,l I - exp( - Kt/B)) + U(r - a) cB expl - K(r - LI)!B) - r,B + cK(t - u) ~ U(t ~ A) cB exp( - K(r - h)‘B) - c,B + c,K(r - b) - Kc L’(t - a) (t - LI) + Kc, U(f - h) (t - h).
In this form, equation
(4) is somewhat
(4) difficult to visualize.
but if we examine
the three
426
EKIC J.
Fig. 3. Relationship
between
SPRIGINGS
the displacement
of node
I and the measured external force.
phases indicated in Fig. 3 separately, equation (4) begins to take on meaning. Phase A. This phase represents the build-up of the isometric tension at the start of the test period when node 1 of the SEC is not being stretched by an external force. During this phase: (1 - a) < 0 :. U(t - IL)= 0 and (r - h) < 0 Therefore
equation
:, U(f - h) = 0.
(4) reduces to F(t) = F, (1 - exp( - M/B)).
Phase B. This phase represents is actively stretching the SEC. During this phase:
(5)
the time during the test period time when the external
(f - a) > 0
:. U(t - u) = 1
(t - b) < 0
;. U(r - h) = 0.
force
and
Therefore
equation
(4) reduces to
F(r) = F,, (I - exp( - KtIB)) - cB (1 - exp( - K(t - a)/@) (Note:
(6)
c < 0).
Phase C. This phase represents the time at the end of the test period when the external no longer stretches the SEC any further out. During this phase: (r - a) > 0
:. U(t - a) = 1
(r - b) > 0
:. U(r - b) = 1.
force
and
Therefore
equation
(4) reduces to
F(r) = F, (1 - exp( - Kt/B)) - CB (1 - exp( - K(t - a)/@) + CB (1 - exp( -K(t (Note:
c < 0).
- b)/B))
(7)
327
Simulation or the forceenhancement phenomenon m muscle Simulation qf the,force
enhancement
phenomenon
It is during phase C that we would expect to see the force enhancement phenomenon resulting from a prior stretch of the muscle. The actual simulation of this phenomenon, from a mathematical point of view, is quite easily done. All that is required mathematically, is that the time constant (7) of the decaying exponential curve take on a greater value during phase C. However from a modeling point of view we must have a valid reason, based on solid empirical evidence. for suggesting that this time constant actually does change during phase It is evident from close examination of equation (7) that the observed during phase C are actually due to the very last term only. That is: A, F(t) = CB (1 - exp( - k’(r - h)/B)). The reason for this is that accounted for in equations value of phase C. Since the portion of the curve during B and k’. More specifically
changes
in F(t)
(8)
the first two major terms of equation (7) have already been (5) and (6) and are totally responsible for determining the initial term shown in equation (8) represents the exponential decaying phase C, it is evident that the time constant is a function of both
f = B,K
(9)
Thus either an increase in B or a decrease in K during phase C would lead to an increase in 7 N hich in turn would decrease the rate of decay of the external output force F(t), that in turn would produce the phenomenon known as force enhancement. From the review of literature, there was strong support from the work of Cavagna et al. [9, 10, 141, that the compliance of the SEC actually does momentarily increase as a result of a prior stretch placed on the muscle. Thus to simulate the force enhancement phenomenon, one merely has to increase the compliance of the SEC during phase C. The reader is reminded that compliance is actually the inverse of the spring stiffness K. Thus to increase the compliance of the SEC, the value of K must be reduced. RESULTS
AND
DISCUSSION
The equations for the model where programed in FORTRAN and run on a DEC 20/60. It should be noted however that the programing requirements are such that the model could eastly be implemented on any microcomputer. The program requires input values for F,,, K, B, c, a, h, as well as the total time of the test pertod, and the desired calculation interval. Representative values for K(31,OOO kg/s’) and B(4000 kg;s) were found in the work by Bach et cd. [SJ and are shown in brackets. For the other constants, values of40 N, - 0.0125 m/s, 1 s and 2 s were used respectively for F,, c, a and h. Four seconds was used as the total time of the test period and a complete set of calculations was performed after every 0.05 s of the stretch history. The results of increasing the compliance of the SEC during phase C are shown in Fig. 4. The curve with K set equal to 31.000 N/m represents the force pattern that would exist if there was no change in the compliance of the SEC. As can be seen, increases in the compliance of the SEC decreased the rate of decay of the output muscle force F(t) during phase C. This decrease in the rate of decay of F(t) allows more work to be performed by the muscle during this interval than would be the case if there had been no change in the compliance of the SEC. This increase in the work capabilities of a muscle during phase C has been well documented experimentally by Cavagna et al. [S, 10. 141. The results of subjecting the model to different rates of stretch, while at the same time keeping the amplitude of stretch constant. are shown in Fig. 5. It will be noted that the velocities of stretch are in the negative direction with respect to the absolute coordinate system established at node I in Fig. 1. As can be seen. the peak magnitude of the output muscle force F(t) during the stretch phase is dependent on the rate of stretch. This is certainly in agreement with the experimental measurements reported by Komi [15] made on eccentric
ERIC J.
428
SPRIGINGS
I h
2100
5 ;
20
*, I
a 0
i OL.
,
I 1.
0.
2’. 1.5
-5
3.
4.
2.5
CONTRACTION
3.5
TIME
(S)
Fig. 4. Force enhancement phenomenon after stretch [K = 31,OOON/m (U); K = lO,OOON,im(~?);K = SOOON,‘m(A)].
type contractions. It is also evident from Fig. 5, that the rate of decay is unaltered by the rate of muscle stretch. This result is in agreement with the experimental evidence provided by Edman et ul. [7]. However, Edman et al. did find that the force enhancement phenomenon was dependent on the amplitude of stretch. This would seem to suggest that the momentary changes in compliance are due to the length the muscle is stretched to, and not with the velocity of stretch. A tentative hypothesis is now offered by this writer as a possible explanation for the greater compliance observed in a muscle after stretch. The first step is to visualize the cross-bridges as being tiny parallel springs that are lumped together to make up the major portion of the SEC element. Now if these cross-bridges are stretched to a sufficient amplitude by an external force, some of the less stable cross-bridges may break which means that there are fewer tiny springs acting in parallel than there were before. This will result in the remaining cross-bridges, or springs, being stretched out further as they attempt to develop a force which can resist the externally applied load. It is this increased stretch for a given external load that
~+ 0.
1. -5
2. 1.5
CONTRACTION
Fig. 5. Force output constant.
.-- ..,
3.
4.
2.5 TIME
.-i
3.5 (S)
profile when the rate of stretch was varied but the amplitude of stretch was held [V = -0.01 m/s (0); F = -0,0125m/s (PI: I~ = -O.O2m/s (A)].
Simulation
of the force enhancement
phenomenon
in muscle
429
would be interpreted as an overall increase in the compliance of the lumped SEC element. During the time of this muscle stretch, the less stable cross-bridges, which have been broken during the earlier stages of the stretch, will attach to new actin sites at some non-stretched cross-bridge length and develop a tension equivalent to the isometric maximum. Ifthe muscle is held in this stretched out position for any length of time, the maximally strained cross-bridges will eventually break and they too will reattach at some non-stretched cross-bridge length such that the overall output tension of the muscle will decay to that of the isometric maximum. It is this reduced rate of decay of the resultant output tension of the muscle back down to its isometric maximum that is responsible for enhancing the work performance capability of a previously stretched muscle. Ifthe muscle is stretched out further. the whole process would repeat itself. In conclusion, the results of this mathematical simulation suggest that the force enhancement phenomenon could quite conceivably be explained in terms of a greater induced compliance of the SEC as a result of a stretch placed on the muscle by an external lobAd. SUMMARY The purpose of this paper was to demonstrate that the simple linear muscle model proposed by Houk [I] was capable of exhibiting the phenomenon referred to by Edman (~2 (11 [7] as force enhancement. While Edman suggests that the phenomenon results from either: (a) a recruitment of an additional elastic element in parallel with the shortening elements of the CC, or(b) an increased force producing capacity by the CC itself, it is possible that momentary changes in the SEC stiffness, as a result of muscle stretch. may account for most of the force enhancement phenomenon. The rheological model of Houk consists of a parallel combination of a constant force generator and dashpot in series with a Hookean spring. The model’s mathematical and programing simplicity makes it very attractive to researchers who wish to incorporate the main elements of muscle contraction in their simulation of human movement. The programing procedure used in this study was to enter the externally induced stretch of the SEC as an input function. This was implemented by means of expressing the externally induced displacement history of the SEC as a series of Heaviside unit functions. In addition. the model was programed to account for the greater compliance of the SEC at the end of the stretch phase of the active muscle as suggested by the experimental evidence of Cavagna et (Ii. [9, 10. 141. The mathematical model was subjected to similar test conditions as those implemented by Edman or ul. [7]. The resulting external force history curves generated by the model were in agreement with the force enhancement results published by Edman YI nl. [7].
REFERENCES A mathematical model of the stretch reflex III human muscle systems. M.S. Thesla. M.I.T.. Cambridge. MA (1963). G. L. Gottlieb and G. C. Agarhal. Dynamic relationship between Isometric muscle tension and the electromyogram in man. J. appl.Physid. 30, 346 (1971). T. W. Calvert and A. E. Chapman, The relationship between the surface EMG and force-transients in muscle: simulation and experimental studies. Pnw. IEEE 65. 683 (1977). A. E. Chapman and P. T. Harrower. Linear approximations ofmuscle mechanics In isometric contractions. Rio/. Cyhtww~. 27, 1-Z (1977). T. M. Bach. A. E. Chapman and T. W. Calvert. Mechanical resonance of the human body during %rtlunrary oscillation about the ankle joint. J. Birwwch. 16, 85 (1983). S Niku and J. M. Henderson. Determination ofthr parameters for an athetotic arm model. .I. Bi~mec~h. 18, 210 (1985). K. A. I’. Edman. G. Elzinga and H. I. M. Noble. Enhancement of mechanical performance by stretch during tctanic contractions of vertebrate skeletal muscle fibres. J. Phy.wd. Loml. 381, 139 ( 1978). G. A. Cavagna. B. Dusman and R. Margarla. PositiLr work done hy a previously stretched muscle. ./. nppl. PI~wro/. 24, 2 I ( 1968). Cr. A. C‘avagna and G. Citterio, Effect of stretchingon the elastic characteristics and the contractile component of frog striated muscle. J. Physiol. Lontl. 239, I 2 (I 974)
I. J. c‘. Houk.
2. 3. 4.
5. 6. 7. 8. 9.
b o-
EKIC J. SPKIGINGS
430
10. G. A. Cavagna, Aspects of efficiency and inefficiency of terrestrial locomotion, in Bmntechunic.\ C7.4. p. 18. University Park Press, Baltimore (1978). 11. H. Sugi. Tension changes during and after stretch in frog muscle fihres. J. PI~y.tiol.hnd. 225, 737 (1972). 12. A. F. Huxley and R. M. Simmons. Mechanical properties of the cross-bridges of frog striated muscle. .I. Phr.stol. Loud. 218, 59960 (1971). 13. M. R. Spiegel. Lap/ace Transforms Schaum‘s Outline Serves. No 8. McGraw-Hill, Toronto (1965). 14. G. A. Cavagna, Storageand utilization ofelasticenergy in skeletal muscle, E.~n~i.~Spc~r~.r Sci. Rrr. 5. I21 (1977). 15. P. V. Komi, Measurement of the force-velocity relatronship in human muscle under concentric and eccentric contractions. In Btomeclrunic~s 111. p. 227. University Park Press. Baltimore 227 11973).
JOHN SPRIGINGS was horn in Montreal, Quebec on 12 February 1946. He received the BSc. degree in chemistry and mathematics from Mount Allison University in 1968, and the MSc. and Ph.D. degrees in hiomechanics from the University of Alberta in 1972 and 1974 respectively. He is presently a member of both the Internatronal and Canadian Societies of Biomechanics. Dr. Sprigings has worked at the University of Saskatchewan smce 1974 and IS presently an associate professor in the College of Physical Education. His present research actrvities center around the mathematrcal modeling of fundamental movement patterns found m the sports of divmg and gymnastics.
About the Author-ERtc
SIMULATION OF THE FORCE ENHANCEMENT PHENOMENON IN MUSCLE ERIC J. SPRKXNGS Biomechanics
Laboratory.
(Received
College of Physical Education, University Saskatoon. Sask. S7N OWO. Canada 13 November
1985; in rrrised
form
of Saskatchewan.
10 _Apr~l 1986)
Abstract-This paper demonstrates that the mathematical modeling of the mechanical force enhancement phenomenon in muscle can be achieved using the simple rheological model proposed by Houk [l]. The simulation offorce enhancement for this model requires that the pre-stretch of the muscle be entered as an input function to the rheological model’s equations. The results of the mathematical simulation suggest that the main characteristics of the force enhancement phenomenon could quite conceivably be explained in terms of a greater induced compliance of the series elastic component as a result of a stretch placed on the muscle by an external load Simulation Muscle stretch
Mechanical
force enhancement
Muscle modeling
Rheological
INTRODUCTION The simple rheological model (Fig. 1) proposed by Houk [l], has been chosen by many researchers 12-61 as representing a bare bones mechanical model of muscle. In this type of tn’o component model, the muscle is envisaged to consist of only a contractile component (CC) and a series elastic component (SEC). In Fig. 1 the CC is represented rheologically as a parallel combination of a constant force generator and a dashpot. The SEC is represented rheologically as a Hookean spring. It is this model’s mathematical and programing simplicity that makes it most attractive to researchers who wish to incorporate the main mechanical elements of muscle contraction in their simulation of human movement. Although it has been shown by Chapman and Harrower [4] that this linear rheological model is somewhat limited in its description of a dynamic muscle contraction, it does have the capacity to provide a reasonable first approximation to a muscle’s force output profile during both concentric and eccentric muscle contraction. What has not been evident about this particular model is that it also has the capacity to display the so-called force enhancement phenomenon that characterizes a muscle that has just previously been stretched. The purpose of this present paper is to show that the mechanical force enhancement phenomenon of muscle can be simulated using the simple rheological model proposed by Houk [l].
Fig. 1. Houk’s rheological muscle model consisting of a constant force generator (F,), a viscous damper with damping coefficient of viscosity B. and a Hookean spring of stiffness K. _Y, and y2 represent the absolute displacement of the two node points. 423
424
FORCE
ENHANCEMENT
PHENOMENON
Force enhancement was a term used by Edman rt ul. [7] to represent the phenomenon they observed in active single muscle fibers that had just previously been subjected to stretch. They observed that the isometric tension of the muscle fiber after stretch remained elevated above the normal isometric tetani tension produced by the muscle of the same sarcomere length. Their experimental results indicated that the output tension of the muscle fiber was enhanced in both magnitude and duration by increases in the amplitude of stretch and the length to which the muscle fiber was stretched [7]. In order to explain this observed force enhancement phenomenon. Edman et ul. proposed two theories: (1) that after stretch, an additional elastic component is recruited in parallel with the shortening elements of the contractile system, or (2) that stretch during contraction alters the properties of the contractile component (CC) in such a way as to allow it to produce a greater force than could be attained during ordinary tetanus. However, it was evident from their subsequent discussion that they could not support either theory with any firm experimental evidence. Prior to the work of Edman et ill.. similar stretch experiments on isolated muscle fibers had been performed by both Cavagna et ~1. [8, 91 and Sugi [I I]. While it is certainly true that these researchers observed the same stretch phenomenon as did Edman et crl., they did not refer to it specifically as “force enhancement”. Cavagna and his colleagues observed in their work [8] that a muscle which had just previously been stretched was able to perform more positive work than a similar muscle which was released from a state of isometric contraction, even when differences in initial force values were taken into account. Their subsequent research [9] on this phenomenon lead them to conclude that stretching of the active muscle temporarily modified the elastic characteristics of the muscle in the sense that a greater amount of mechanical energy was released for a given fall of force during subsequent muscle shortening. They hypothesized [lo] that this greater compliance ofthe elastic characteristics of the muscle was due to a small population of cross-bridges between the actin and myosin myofilaments which were more strained and unstable than some of the neighboring cross-bridges. They further hypothesized that these heavily strained cross-bridges would recoil, releasing mechanical energy if the muscle were allowed to shorten immediately after stretching, but would themselves break down if the muscle were kept active at the stretched length. The published results of Sugi’s work [ 1I] were similar to those later reported by Edman et al. Of particular relevance to this present paper was his brief mention of a proposal suggested to him by A. F. Huxley on the possible cause of the so-called “slip” phenomenon in muscle which accounts for observed changes in muscle tension during stretch. Huxley stated that there was experimental evidence indicating that the A-band was extended when a muscle was extremely stretched during tetanus. The theoretical consequences of this observation was that the “slip” phenomenon could be considered to be related to the extension of themyofilament:, or the cross-bridges themselves, and not to the actual breaking of the cross-links. This observation lends support to the hypothesis proposed by Cavagna et ~1. [lo] which stated that the extra positive work performed by a muscle after stretch was due in part to a greater elastic compiiance that developed temporarily within the muscle system. The temporary change in compliance of the muscle system after stretch, as suggested by Cavagna et al., would seem to be a property of the series elastic component (SEC) since it has been shown in the work by Huxley and Simmons [12], that the SEC mainly resides within the cross-bridges themselves. METHOD Mathemuticnl
modeling of‘ the jtirce enh~mcement phenomenon
usitty (I Two component
tnotlel
Figure 1 is Houk’s rheological representation of the two component muscle model. This model, comprised of a parallel combination of a force generator and dashpot coupled in series with a Hookean spring, has been chosen by many researchers as representing a bare bones mechanical model of skeletal muscle. The model’s mathematical and programing
Simulation
of the force enhancement
phenomenon
tn muvzle
425
t
I
I-i,
7. Di\placement
hiator!
slope=c of node
1 asa result
of an external
strctchinp
force.
simplicity makes it very attractive to researchers who wish to incorporate the main characteristics of a muscle’s output tension in their simulation of human motion. In the model shown in Fig. 1, F‘,, is considered to be able to supply a constant level of internal force instantaneously at the start of the test period. B and h’ are constants representing the damping coethcient of the dashpot. and the stiffness of the spring respectively. From Fig. 1, it is evident that the following two equations can be written: F,,, - .\.2B - K(.Xl - Xl) = 0 F(t) = k’(r,
(1) - Xi).
(2)
In order to test the mode1 to see if it is capable of exhibiting the force enhancement phenomenon, it was necessary to be able to subject the spring (i.e. SEC) to a stretch by an external force during the time that the force generator F,,, is considered to be active. The graphical representation of such a stretch is shown in Fig. 2. In Fig. 2, the time period “a” represents the period of isometric tension build up when node 1 is not permitted to move. The time period (b - tl) represents the period of eccentric muscle contraction when the muscle is being stretched out by an external load. The time period that follows time h is again a period of isometric muscle contraction but at a new muscle length. The problem then was one ofsolving for the output force history of the muscle, F(t), during the stages of stretch shown in Fik‘. 3. The key to the solution of this problem was to find a method of entering the displacement history of I, as an input function to equations (1) and (2). This was most easily achieved by rewriting the displacement history of x1 in terms of a series of Heaviside unit functions [I?]. .X!(f) = + C’r-:(1 - u) (f - LI) ~~(’ L’(t ~ h) (r - hi where:
L’= negative
(3)
slope
The Heaviside representation of .ui(t), as shown in equation (3), was then substituted into equation (I). The resulting first order linear differential equation in terms of x2(t) was then solved by means of a Laplace transformation. The resulting expression for x2(t), and the He:iviside expression for xi(t) shown in equation (3). were then substituted into equation (2) in order to obtain the desired expression for f(t). The resulting expression is shown below: F(r 1= f,,,l I - exp( - Kt/B)) + U(r - a) cB expl - K(r - LI)!B) - r,B + cK(t - u) ~ U(t ~ A) cB exp( - K(r - h)‘B) - c,B + c,K(r - b) - Kc L’(t - a) (t - LI) + Kc, U(f - h) (t - h).
In this form, equation
(4) is somewhat
(4) difficult to visualize.
but if we examine
the three
426
EKIC J.
Fig. 3. Relationship
between
SPRIGINGS
the displacement
of node
I and the measured external force.
phases indicated in Fig. 3 separately, equation (4) begins to take on meaning. Phase A. This phase represents the build-up of the isometric tension at the start of the test period when node 1 of the SEC is not being stretched by an external force. During this phase: (1 - a) < 0 :. U(t - IL)= 0 and (r - h) < 0 Therefore
equation
:, U(f - h) = 0.
(4) reduces to F(t) = F, (1 - exp( - M/B)).
Phase B. This phase represents is actively stretching the SEC. During this phase:
(5)
the time during the test period time when the external
(f - a) > 0
:. U(t - u) = 1
(t - b) < 0
;. U(r - h) = 0.
force
and
Therefore
equation
(4) reduces to
F(r) = F,, (I - exp( - KtIB)) - cB (1 - exp( - K(t - a)/@) (Note:
(6)
c < 0).
Phase C. This phase represents the time at the end of the test period when the external no longer stretches the SEC any further out. During this phase: (r - a) > 0
:. U(t - a) = 1
(r - b) > 0
:. U(r - b) = 1.
force
and
Therefore
equation
(4) reduces to
F(r) = F, (1 - exp( - Kt/B)) - CB (1 - exp( - K(t - a)/@) + CB (1 - exp( -K(t (Note:
c < 0).
- b)/B))
(7)
327
Simulation or the forceenhancement phenomenon m muscle Simulation qf the,force
enhancement
phenomenon
It is during phase C that we would expect to see the force enhancement phenomenon resulting from a prior stretch of the muscle. The actual simulation of this phenomenon, from a mathematical point of view, is quite easily done. All that is required mathematically, is that the time constant (7) of the decaying exponential curve take on a greater value during phase C. However from a modeling point of view we must have a valid reason, based on solid empirical evidence. for suggesting that this time constant actually does change during phase It is evident from close examination of equation (7) that the observed during phase C are actually due to the very last term only. That is: A, F(t) = CB (1 - exp( - k’(r - h)/B)). The reason for this is that accounted for in equations value of phase C. Since the portion of the curve during B and k’. More specifically
changes
in F(t)
(8)
the first two major terms of equation (7) have already been (5) and (6) and are totally responsible for determining the initial term shown in equation (8) represents the exponential decaying phase C, it is evident that the time constant is a function of both
f = B,K
(9)
Thus either an increase in B or a decrease in K during phase C would lead to an increase in 7 N hich in turn would decrease the rate of decay of the external output force F(t), that in turn would produce the phenomenon known as force enhancement. From the review of literature, there was strong support from the work of Cavagna et al. [9, 10, 141, that the compliance of the SEC actually does momentarily increase as a result of a prior stretch placed on the muscle. Thus to simulate the force enhancement phenomenon, one merely has to increase the compliance of the SEC during phase C. The reader is reminded that compliance is actually the inverse of the spring stiffness K. Thus to increase the compliance of the SEC, the value of K must be reduced. RESULTS
AND
DISCUSSION
The equations for the model where programed in FORTRAN and run on a DEC 20/60. It should be noted however that the programing requirements are such that the model could eastly be implemented on any microcomputer. The program requires input values for F,,, K, B, c, a, h, as well as the total time of the test pertod, and the desired calculation interval. Representative values for K(31,OOO kg/s’) and B(4000 kg;s) were found in the work by Bach et cd. [SJ and are shown in brackets. For the other constants, values of40 N, - 0.0125 m/s, 1 s and 2 s were used respectively for F,, c, a and h. Four seconds was used as the total time of the test period and a complete set of calculations was performed after every 0.05 s of the stretch history. The results of increasing the compliance of the SEC during phase C are shown in Fig. 4. The curve with K set equal to 31.000 N/m represents the force pattern that would exist if there was no change in the compliance of the SEC. As can be seen, increases in the compliance of the SEC decreased the rate of decay of the output muscle force F(t) during phase C. This decrease in the rate of decay of F(t) allows more work to be performed by the muscle during this interval than would be the case if there had been no change in the compliance of the SEC. This increase in the work capabilities of a muscle during phase C has been well documented experimentally by Cavagna et al. [S, 10. 141. The results of subjecting the model to different rates of stretch, while at the same time keeping the amplitude of stretch constant. are shown in Fig. 5. It will be noted that the velocities of stretch are in the negative direction with respect to the absolute coordinate system established at node I in Fig. 1. As can be seen. the peak magnitude of the output muscle force F(t) during the stretch phase is dependent on the rate of stretch. This is certainly in agreement with the experimental measurements reported by Komi [15] made on eccentric
ERIC J.
428
SPRIGINGS
I h
2100
5 ;
20
*, I
a 0
i OL.
,
I 1.
0.
2’. 1.5
-5
3.
4.
2.5
CONTRACTION
3.5
TIME
(S)
Fig. 4. Force enhancement phenomenon after stretch [K = 31,OOON/m (U); K = lO,OOON,im(~?);K = SOOON,‘m(A)].
type contractions. It is also evident from Fig. 5, that the rate of decay is unaltered by the rate of muscle stretch. This result is in agreement with the experimental evidence provided by Edman et ul. [7]. However, Edman et al. did find that the force enhancement phenomenon was dependent on the amplitude of stretch. This would seem to suggest that the momentary changes in compliance are due to the length the muscle is stretched to, and not with the velocity of stretch. A tentative hypothesis is now offered by this writer as a possible explanation for the greater compliance observed in a muscle after stretch. The first step is to visualize the cross-bridges as being tiny parallel springs that are lumped together to make up the major portion of the SEC element. Now if these cross-bridges are stretched to a sufficient amplitude by an external force, some of the less stable cross-bridges may break which means that there are fewer tiny springs acting in parallel than there were before. This will result in the remaining cross-bridges, or springs, being stretched out further as they attempt to develop a force which can resist the externally applied load. It is this increased stretch for a given external load that
~+ 0.
1. -5
2. 1.5
CONTRACTION
Fig. 5. Force output constant.
.-- ..,
3.
4.
2.5 TIME
.-i
3.5 (S)
profile when the rate of stretch was varied but the amplitude of stretch was held [V = -0.01 m/s (0); F = -0,0125m/s (PI: I~ = -O.O2m/s (A)].
Simulation
of the force enhancement
phenomenon
in muscle
429
would be interpreted as an overall increase in the compliance of the lumped SEC element. During the time of this muscle stretch, the less stable cross-bridges, which have been broken during the earlier stages of the stretch, will attach to new actin sites at some non-stretched cross-bridge length and develop a tension equivalent to the isometric maximum. Ifthe muscle is held in this stretched out position for any length of time, the maximally strained cross-bridges will eventually break and they too will reattach at some non-stretched cross-bridge length such that the overall output tension of the muscle will decay to that of the isometric maximum. It is this reduced rate of decay of the resultant output tension of the muscle back down to its isometric maximum that is responsible for enhancing the work performance capability of a previously stretched muscle. Ifthe muscle is stretched out further. the whole process would repeat itself. In conclusion, the results of this mathematical simulation suggest that the force enhancement phenomenon could quite conceivably be explained in terms of a greater induced compliance of the SEC as a result of a stretch placed on the muscle by an external lobAd. SUMMARY The purpose of this paper was to demonstrate that the simple linear muscle model proposed by Houk [I] was capable of exhibiting the phenomenon referred to by Edman (~2 (11 [7] as force enhancement. While Edman suggests that the phenomenon results from either: (a) a recruitment of an additional elastic element in parallel with the shortening elements of the CC, or(b) an increased force producing capacity by the CC itself, it is possible that momentary changes in the SEC stiffness, as a result of muscle stretch. may account for most of the force enhancement phenomenon. The rheological model of Houk consists of a parallel combination of a constant force generator and dashpot in series with a Hookean spring. The model’s mathematical and programing simplicity makes it very attractive to researchers who wish to incorporate the main elements of muscle contraction in their simulation of human movement. The programing procedure used in this study was to enter the externally induced stretch of the SEC as an input function. This was implemented by means of expressing the externally induced displacement history of the SEC as a series of Heaviside unit functions. In addition. the model was programed to account for the greater compliance of the SEC at the end of the stretch phase of the active muscle as suggested by the experimental evidence of Cavagna et (Ii. [9, 10. 141. The mathematical model was subjected to similar test conditions as those implemented by Edman or ul. [7]. The resulting external force history curves generated by the model were in agreement with the force enhancement results published by Edman YI nl. [7].
REFERENCES A mathematical model of the stretch reflex III human muscle systems. M.S. Thesla. M.I.T.. Cambridge. MA (1963). G. L. Gottlieb and G. C. Agarhal. Dynamic relationship between Isometric muscle tension and the electromyogram in man. J. appl.Physid. 30, 346 (1971). T. W. Calvert and A. E. Chapman, The relationship between the surface EMG and force-transients in muscle: simulation and experimental studies. Pnw. IEEE 65. 683 (1977). A. E. Chapman and P. T. Harrower. Linear approximations ofmuscle mechanics In isometric contractions. Rio/. Cyhtww~. 27, 1-Z (1977). T. M. Bach. A. E. Chapman and T. W. Calvert. Mechanical resonance of the human body during %rtlunrary oscillation about the ankle joint. J. Birwwch. 16, 85 (1983). S Niku and J. M. Henderson. Determination ofthr parameters for an athetotic arm model. .I. Bi~mec~h. 18, 210 (1985). K. A. I’. Edman. G. Elzinga and H. I. M. Noble. Enhancement of mechanical performance by stretch during tctanic contractions of vertebrate skeletal muscle fibres. J. Phy.wd. Loml. 381, 139 ( 1978). G. A. Cavagna. B. Dusman and R. Margarla. PositiLr work done hy a previously stretched muscle. ./. nppl. PI~wro/. 24, 2 I ( 1968). Cr. A. C‘avagna and G. Citterio, Effect of stretchingon the elastic characteristics and the contractile component of frog striated muscle. J. Physiol. Lontl. 239, I 2 (I 974)
I. J. c‘. Houk.
2. 3. 4.
5. 6. 7. 8. 9.
b o-
EKIC J. SPKIGINGS
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10. G. A. Cavagna, Aspects of efficiency and inefficiency of terrestrial locomotion, in Bmntechunic.\ C7.4. p. 18. University Park Press, Baltimore (1978). 11. H. Sugi. Tension changes during and after stretch in frog muscle fihres. J. PI~y.tiol.hnd. 225, 737 (1972). 12. A. F. Huxley and R. M. Simmons. Mechanical properties of the cross-bridges of frog striated muscle. .I. Phr.stol. Loud. 218, 59960 (1971). 13. M. R. Spiegel. Lap/ace Transforms Schaum‘s Outline Serves. No 8. McGraw-Hill, Toronto (1965). 14. G. A. Cavagna, Storageand utilization ofelasticenergy in skeletal muscle, E.~n~i.~Spc~r~.r Sci. Rrr. 5. I21 (1977). 15. P. V. Komi, Measurement of the force-velocity relatronship in human muscle under concentric and eccentric contractions. In Btomeclrunic~s 111. p. 227. University Park Press. Baltimore 227 11973).
JOHN SPRIGINGS was horn in Montreal, Quebec on 12 February 1946. He received the BSc. degree in chemistry and mathematics from Mount Allison University in 1968, and the MSc. and Ph.D. degrees in hiomechanics from the University of Alberta in 1972 and 1974 respectively. He is presently a member of both the Internatronal and Canadian Societies of Biomechanics. Dr. Sprigings has worked at the University of Saskatchewan smce 1974 and IS presently an associate professor in the College of Physical Education. His present research actrvities center around the mathematrcal modeling of fundamental movement patterns found m the sports of divmg and gymnastics.
About the Author-ERtc