Biol. Cybernetics 23, 61 72 (1976) Biological Cybernetics �9 by Springer-Verlag 1976

Estimation of Muscle Active State*

G. F. ]nbar and D. Adam Biological Control Laboratory, Faculty of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel

Abstract. A method is presented for the estimation of the complete time course of muscle active state. The method is based on the selection of a proper model for the muscle, consisting of linear and non-linear components, and on the estimation of its parameters from a simple experiment. The model's parameters are estimated, using the least square method, from measure- ments of a tetanized muscle's response to a change of its length. The time course of the active state is cal- culated from an isometric twitch tension response of the same muscle. The twitch tension response is taken as the system's output, and the active state as its input. The latter can be estimated since the system parameters have already been estimated from the tetanized muscle experiment. Experiments were performed on the gastrocnemius muscle of frogs and cats. Results are given for the whole active state time course of these muscles. The results show that the peak active state force does not reach tetanic value, and a negative force is generated during the relaxation period. Additional experiments were carried out with the purpose of verifying the existence of this force; however, no con- clusive results were obtained.

I. Introduction

The active state (AS) of muscles has been the subject of many investigations, which so far have been unable to give a complete description of one of the muscles' most important roles, i.e. its ability to bear a load (Wilkie, 1967). The problem with evaluating a muscle's AS was always the inability to measure it directly, since its effect can be detected only at the muscle tendon, where it is already masked by the viscoelastic properties of the tissues transmitting it from the contractile machinery.

* This research was supported by the Julius Silver Institute of Bio-Medical Engineering Sciences, Grant 050-304

In most studies, as well as in the one presented here, a two component model of the muscle is assumed, containing an elastic component in series with a contractile component. The characteristics of the elastic component can be easily evaluated; however, those of the contractile element can not be measured directly. Thus the different descriptions of the AS time course were calculated from indirect measurements, at times when the masking effect of the elastic com- ponent could be overcome.

Most of the existing methods for calculating a muscle's AS suffer from one deficiency or another. The main deficiency is their inability to predict the full time course (rise time, peak, and complete decay char- acteristics), in addition to the disturbances which most methods impose upon the contractile element, i.e. quick stretch or quick muscle release.

It was proposed (Gasser and Hill, 1924) that, by supplying a quick stretch to an isometrically contract- ing muscle, the elastic component can be increased to the extent that its tension equals the force developed by the contractile component at that instant. By apply- ing quick stretches at different instants during the development of the twitches, a time course of the force generated by the contractile component was obtained. The peak AS force during a twitch, as calculated using this method, is equal to the muscle's tetanus level. Using this method, only the declining portion of the AS was obtained, since Hill assumed that the quick stretch had no effect on the contractile component. As this component is stretched during the quick stretch, and as it has been found to be sensitive to the velocity of change of its length, Hill's conclusions were subject to criticism. Some of his measurements even contradicted his conclusions, since the measured force during a quick stretch was nearly 25 % higher than the maximal tetanic force.

The same idea was applied in a different method (Ritchie, 1954), wherein the muscle was released for

62

a fixed length at different times after a single stimulus. The tension which at first falls, rises again because the contractile component is still active. The peaks of the isometric tension, where dT/dt = 0, lie on the AS curve. This method is limited, since only part of the decay of the AS can be investigated.

Ritchie's method was improved, however, (Edman, 1970, 1971), and for the first time the rise time of the AS was obtained. After the redevelopment of tension, as in Ritchie's experiment, another stimulus was given. A new set of extremum points this time minimal tension points--were obtained. The line connecting these points provide the rising phase of the AS curve. The rise time, 3-4 ms long, and the decay time were both found to be independent of length, and only the AS peak was length dependent. However, in Edman's method the peak and the end of the decay are not measured and can only be extrapolated. The time of the peak, its shape and its amplitude are unknown.

The comparison of the time course of tension development between an isometric twitch and tetanus was proposed (McPherson and Wilkie, 1954) as a new method for the AS evaluation. The instant that the two curves diverge was suggested to be the beginning of the decay of the AS. The resolution of the measure- ments is greatly improved by differentiating the signal s , enabling more accurate determination of the point of separation. This method does not measure any of the other previously defined characteristics of the AS.

The AS phenomenon was also studied (Jewell and Wilkie, 1960) by utilizing the muscle's ability to shorten. The measurements were carried out under isotonic conditions, and the release was done at different times during an isometric twitch. As in Ritchie's work, the peaks, where dx/dt = 0, provide points along the falling phase of the AS curve. The measurements were re- peated for different loads, and the AS curve was obtained at Vo (velocity of shortening with no load) and Po (the force at zero velocity). The latter should have coincided with Ritchie's result from the isometric measurements. Since the curves were not similar, Ritchie's experiment was repeated with a smaller release. The similarity of the results emphasizes the importance of both methods; however, in both cases only the decay of the AS is produced. Other methods have been developed (Ritchie, 1954; Gable, 1968), based on the same principles.

An analytic synthesis of the muscle's dynamic behaviour was developed (Bahler et al., 1967, 1968) as a new approach to the AS dynamics, since it re- presents the contractile component as a force generator shunted by a velocity dependent internal load. The AS time course can be calculated from the isometric tension-time curve of a twitch, the strain-stress curve of the series elastic component, and the isometric

force-velocity curve of tetanus. This technique yields most of the AS time course, but does not include the rise time and the end of the decay. The result differs from that of Hill in that: a) The maximal intensity reached is less than the tetanic tension (0.92 Po); b) There is no plateau, but one single peak; c) The curve reaches its peak much later than described by Hill.

The present work suggests a new method for estimating the complete AS time course. First a muscle model is chosen and its parameters are identified from experimental results (Inbar et al., 1970). Using the results of a single twitch experiment and the identified muscle parameters, the AS time course is calculated.

Obviously the coupling tissue, the mechanical filter for the force generated internally by the muscle, has a great influence on the measured force. The model selected for this filter in the new method will be shown greatly to influence the shape of the AS obtained.

II. Methods The experiments were performed in vitro on the gastrocnemius muscles of 22 frogs--"Rana Esculenta". Medium sized frogs were chosen, with an average muscle length of 1 o-~ 25 mm. The muscle was bathed at a controlled room temperature of 24 ~ C in a pH-~ 6.9 Ringer solution (In mM/1.: NaCI-116.5; Na2HPO~-2.55; KC1-2.5; CaC1.2H20-1.8; NaH2PO4.HzO-0.45; HIO (dist.)- 117.1). Carbogen gas (95 % 02; 5 % CO2) was continuously bubbled through the solution.

The muscle's tendon was connected by a silk thread to the force transducer, while the proximal part of the Tibio-Fibula, still attached to the muscle; was connected by a stainless steel hook and rod to the electromechanical puller.

In order to fulfil the special requirements of the experiment, a new mechanical system was designed. This system, which can perform perfectly and easily all the common muscle experiments, is based on linear movement, using a loudspeaker-like force generator (M.B. Electronics, type-MB 2250 system). The puller has a large displacement amplitude (0.5"), can produce great amounts of force and has a moving part of very low mass. All these qualities are reinforced by connecting the puller in a closed feedback loop. The feedback element is a linear displacement transducer (H.P. Model 7DCDT-250) which enables a system linearity of _+ 0.5 %, a frequency response from D.C. to 120 Hz, and a mechanical waveform pro- portional to the input voltage, which was in the parameter identifica- tion (PI) experiment trapezoidal. The system's stiffness was measured to be better than 5 gr/gm, its velocity better than 1000 mm/s and its amplitude _+ 6 mm (at specified linearity).

The stimulation was by a Grass $88 stimulator, which also supplied the synchronization pulses to the waveform generator and the oscilloscope. Pulse duration varied between 0.05 and 0.2 ms, while the rate was between 40 to 90 p.p.s, for tetanus. The amplitude was increased until supermaximal stimulation was achieved. Since high levels of stimulation were needed in order to keep the muscle at tetanus for long periods, massive Ag-AgC1 electrodes were chosen. The polarity was reversed after each stimulation.

Data from the force transducer (Grass F.T.-03) and the displace- ment transducer were recorded simultaneously on an F.M. tape recorder (Philips ANA-LOG 7) and a U.V. strip chart recorder (S.E. Lab. 3006). Both had frequency responses of D.C. to 3 kHz.

The procedure for the PI experiment and for the evaluation of the AS time course was simple: after fixing the muscle in the bath between the force transducer and the puller at length lo, an isometric twitch was induced. The muscle was then stimulated continuously for a tetanus of 2-3 s. During its contraction the muscle was stretched, held at that length, and then released. The change of length was 1.5-*2% of l o (10--*15% of the physiological range). The rate of change was chosen in order to contain components in the full physiological frequency range- -up to 100 Hz. At the end of this tetanus-stretch experiment a single twitch was performed, for com- parison with the first one. (Some typical results are shown in Fig. 2.) The whole experiment was terminated when the tetanic tension decreased more than 20 % from one contraction to the next.

A comparative study was also carried out on the gastrocnemius muscles of eight decerebrated cats, each weighing between 2 and 4 kgm. Contrary to the frog experiments, these experiments were carried out in vivo for the purpose of lengthening the stimulation time without causing great changes in the muscle's parameters. The cat was fastened to the table, and the muscle's tendon was cut and connected to the mechanical system. The stimulation was done by Ag-AgC1 electrodes, with supermaximal pulses of 0.2 ms duration, at a rate between 40 to 70 p.p.s. The procedure of the experiment was exactly the same as that for the frog, but longer contraction times were attained (5 10 s).

Il l . Muscle Models

Fundamental to the new method for estimating the AS is the assignment of a muscle model, and herein lies its greatest difficulty. Most of the muscle models constructed to date are based on Hill's equation. Since the properties of the series elastic component are easily measured, cach model differs in its description of the contractile component. This component exhibits non-linearities, and, therefore, the model for it should include components which represent the physiological elements responsible for these characteristics.

The approach used herein was to choose simple tractable models, lumped and not distributed, constant parameters when possible. Simplifying assumptions were incorporated to gain a simple tool, for estimating the overall shape of the AS time course. The starting point was Brown's (1959) general equation, which describes the external dynamic behavior of muscles. However, his equation has variable parameters which must be mapped independently for each muscle, and therefore is impractical for our purposes. The method used here assumes constant parameters, or parameter dependency on muscle states, i.e. on muscle length, its velocity or its tension. It should be noted that the parameter dependency on muscle states yields non- linear equations.

The general form for the non-linear differential equation which relates muscle tension to muscle length is as follows (Brown, 1959):

k 1 T = - X + f X - - T (1) "c "c

63

where, total muscle length, is in our calculations the muscle displacement above the initial length 10. The resting tension at 10, being very low relative to the tetanic tension, is being neglected in the calculations. T is therefore the measured muscle tension above its resting level, k, f and r are determined by T and X, thus making the equation a non-linear one. It is easy to interpret this equation in its linear form, when k, f, z are all assumed to be constant. In this case we get a model of the same form as that of Figure 1.

Using this model a method has been suggested (Inbar et al., 1970) wherein all of the components of a specified model are computed simultaneously from one simple experiment, by means of a Parameter Identifica- tion (P.I.) technique (Kalman, 1958; Hsia and Bailey, 1968). Bearing in mind that assigning a model to the muscle is the first step in determining the AS time course, this method was chosen because of its simplicity, accuracy, and ability to separate the active force generator from the passive components of the muscle's model. Several linear and non-linear models were tested and,, for each, an attempt was made to provide a plausible physiological explanation. The models pro- posed for the passive muscle (Inbar et al., 1970) were utilized again in this work for the active muscle, in accordance with Brown's suggestion (Brown, 1959). A few of the models tested are given below.

a. Simple Linear Model

The equations obtained from the linearized model can be written as the equilibrium of forces at node (a) in Figure 1 :

K X 1 = C 1 X 2 4- Bf(. 2 + P (2)

X = X 1 -t- X 2 . (3)

ic,

(a)

(b)f K

I--IB X~

Fig. 1. A mechanical model of a mnscIe. The simple linear model is obtained by deleting the non-linear component, C 2. X I and X 2 are taken as the displacement about resting length

64

Where P is the active force generating component, assumed to remain constant during tetanus indepen- dent of muscle length.

Substituting (3) into (2) and assigning T = K X I , we get the model's differential equation:

J" = b lX q- bzX q- b 3 T + b4P , (4)

where b~ are linear transformations of the parameters of the model's equation. This last equation relates the tension developed by the muscle to its total length, the two variables which are recorded continuously at the muscle's tendon. This general equation holds for the passive muscle as well as the active one, and for the muscle in a closed loop reflexive mode. Only the parameters take different values, as has already been shown (Brown, 1959; Inbar et al., 1970). In other words, the general topology, the shape of the model, or the order of the equation remains the same.

In the linear model, and the models which follow, the force generator which generates the AS is assumed to be in parallel with the passive components C1 (parallel elastic element) and B (viscosity element), while K represents the series elastic element which lies in the bridges of the contractile machinary.

b. Complex Linear Model

It is known that an increase in the order of a model will usually improve the quality of its fit to the experi- mental data. Physiological justification for proceeding beyond a second order equation which incorporates the effects of acceleration and muscle mass could not be found. Such a model can be described by the follow- ing generalized equation:

T=blJ~ + bz~2 + b3X + b4J'+ bsT + b6P. (5)

c. Tension Feedback Model

Since the muscle responds differently to stretch and release, a new force component which resists lengthening and whose magnitude is proportional to muscle tension was added. Its action is similar to a negative tension feedback within the contractile com- ponent itself, and is justified by physiological findings (Jewell and Riiegg, 1966; Hill, D.K., 1968). The new model's force equation is:

T=KXI=C1X2q-Bf f2q-C3T.pgn( f (2 )q -P (6)

where

/1:22~ 0 pgn(X2)= [0:22 <0 .

The generalized differential equation of the tension- feedback model is therefore:

J~=b12+b2X+b3T+b4T . pgn(22)+bsP. (7)

d. Length and Velocity Feedback Model

Instead of the tension feedback component, another non-linear component was tested (Inbar et al., 1970).

This model's force equation is:

T = K X 1 = CxX 2 + BX 2

+ C3X. ggn(X2)+ C4X. pgn(22)+ P (8)

where

I I : X 2 > 0 ggn(f(2)= [0:22 =< 0

and its generalized differential equation is:

7" = blX + b2 X + b 3 T + b4X . ggn(X2) + bsX" pgn(X2) + b6P. (9)

Here it is assumed that a force is generated in the muscle, opposing its extension, which is a function of muscle length and the velocity of its extension. This force is assumed to exist during lengthening of the muscle only, and vanishes upon its shortening.

The search for the "best" model is inevitable, and it reflects our inability to assign a unique mathematical expression to the dynamic behaviour of the muscle, without getting into very complex non-linear partial differential equations. To keep the equations tractable and as physiologically meaningful as possible, the simpler lumped model approach was preferred. Al- though the results are not unique, because of the modelling problem, they are clearer and; as will later be demonstrated, yield the general shape of the whole active state, regardless of the particular selected model, within a reasonable range.

IV. Results

A. Experimental Results

Two short experiments are sufficient for the estimation of the AS time course. In the first experiment the muscle is isometrically driven into a "Steady-State" condition of tetanus by means of high repetition supermaximal stimulation. The muscle system is then disturbed by a change of length and the resultant change of tension is measured as the output. Typical results of such an experiment are given in Figure 2, for the frog's muscle. The slow exponential decay of the muscle tension justifies the assumption that the model's transfer function does not contain distant poles, i.e. does not contain high frequency components, and therefore can be identified with the aid of a low- frequency harmonic input signal. The input signal was programmed accordingly as a trapezoidal change of length, which excluded frequencies outside of the muscle's physiological operating range. Since the

65

Fig. 2. The recorded tension output of a frog gastrocnemius muscle, as a result of an isometric twitch, followed by induced tetanus, which is maintained during an externally induced trapezoidal change in length and then removed. Upper trace: muscle tension. Lower trace: muscle length

tension decay following the change of length contains the muscle's response to low frequencies, the tendency was to stimulate the muscle for the longest possible period. The experiments with the cat's muscles were performed in vivo at optimal conditions, and t h e muscles could therefore tolerate longer tetanus stimula- tions than those of the frog.

It is evident from the muscle's response that the muscle contains non-linear components. The muscle does not respond similarly to positive (stretch) and negative (release) stimuli. This phenomenon reappeared in all muscle studies (22 frogs, 8 cats), which led to the idea of adding nonlinear components to the model.

The accuracy of the measurements was a function of the linearity of the force and displacement trans- ducers, since noise was at avery low level. The displace- ment transducer was accurate to 1% of full scale, while the accuracy of the force transducer was dependent on the type of experiment. In the frog experiments the tension measurements were accurate to 0.5 %, and for the cats they were better than 1%. The digitization was done at a rate higher than 2- f~ (fc--the upper frequency limit desired) and added very little inaccuracy (better than 0.5 % of the digitized signal's maximal amplitude).

The results, as seen in Figure 2, allow a direct estimation of some components of the muscle models. The series elastic component K can be calculated for all models, assuming that the applied stretch is quick enough compared with the time constant of the viscous component. Its value is derived from the ratio of the change of tension from the tetanus tension level, to

the change of length which caused it, i.e. K - rpeak X

both measured at the instant of stretch. Other compo-

nents can be calculated for the linear models only. For the simple model this can easily be done by assum- ing that, as a result of the stretch, the tension rises quickly and then decays to a new steady state level. Therefore, the value of the parallel elastic component, C1, can be calculated by utilizing the ratio of T~ s, the difference between the new tension level and the tetanus tension level, to X, total change of length, and substituting the value for K computed above:

Ca �9 K Ts s - ( 1 0 )

C I + K X

or

KTsjX KT~ C I -

K - TJx K X - T~

In the case in which this assumption does hold, i.e. for quick stretches, the time constant of the decay, r, can be measured and together with the values of the elastic components, C 1 and K be used to compute the value of the viscous component, B:

B r = , B = ( K + C 1 ) . r . (11)

( K + C1)

Since the measured decay is not exactly a single exponential, the calculated value is only an estimation. A comparison of some values for the linear muscle model components, computed directly, as above, and by the parameter identification technique (see part B) is given in Table 1 for one set of experiments on the frog and on the cat. The measured time constants for these experiments are: ~ = 0.23 s for frog and z = 1.02 s for the cat.

66

Table 1. Comparison of values of linear muscle model components, computed directly and by parameter identification

Value of component computed by parameter identification

Value of component computed directly

Type of component

Experiment Symbol

1460.1 1471.8 [cg~] frog 5303. 5035. K cat

1514.9 1119.6 [cg_~rm] frog 2039.6 2137. C1 cat L -J

225.1 595.7 B I gr sl frog 2009.6 7315.3 L cm] cat

Series elastic component

Parallel elastic component

Viscosity

The values of the viscous and parallel elastic com- ponents differ from those calculated by parameter identification, since these components alone are not sufficient to describe the muscle's dynamic charac- teristics and the least square method averages the differences between the model and experimental re- sults for all the parameters. When non-linear com- ponents are added to the model, the values of these two components are usually reduced drastically.

The direct estimation method is convenient, how- ever, as a first approximation of the muscle model components. It can be used for checking the results of the parameter identification or for comparing the parameters of different muscles.

B. Identification Results

In order to identify the parameters of the differential equations of the models mentioned above, a P.I. scheme had to be chosen. The use of digital computation requires sampling of the experimental results and descretization of the model's differential equations. The P.I. method chosen is the Least Mean Squared Error Method (Inbar, 1970), because of its simplicity and reliability in getting a result without stopping at a local minimum. It is applicable just as well to linear

and non-linear systems, and also to systems with transport lag. Noise with zero mean is cancelled and does not affect the results. One drawback of this method is that it does not supply a numerical result for the quality of the fit, which can be judged only by visual inspection. A numerical criterion of mean squared error was invoked,

S= _ _ 1 N

N ~ - 1 ~ (Tdata-- Tsimulated)2 ' i=0

which, although compatible with the decisions made by the visual inspection, could not supply specific information about the fitting of amplitudes and time constants or about other discrepancies between the model and experimental results.

The identified parameters for the different models described earlier, for the frog and the cat muscles, appear in Table 2. The results of the simple linear model agree well with those reached by direct calcula- tions, except for the viscous component. The parallel elastic component, in both linear models, is found to be as stiff as the series one. In the non-linear models, the addition of the non-linear components reduced the values of both the parallel and viscous components by more than an order of magnitude. The parameters

Table. 2. Summary of the model equation parameters for various models, fitted to the frog's and cat's gastrocnemius muscle (same experimental data as in Table 1)

t::l [gr] ode, S-Meansquared K C1 cm B [cm] paration

e r r o r

76.23 1460.1 1514.9 225.11 frog linear 52357. 5303.0 2039.6 2009.6 cat

73.67 1275.8 1766. 33.12 2.09 frog linear 50871. 7344.0 2860. 1956.7 146.65 cat (2 poles)

52.16 1472.8 202.16 39.63 0.864 frog tension 117180. 8903.9 232.65 365.09 0.8643 cat F.B.

64.04 1382.0 - 3208.5 - 147.23 - 25.47 2258. frog length 49280. 3142.2 - 2252.2 247.9 222.41 2684. cat vel. F.B.

67

180

120 "T

~-~ 6O

0

-60 0

Time course of measured and generated tension of tetanized frog's gastrocnemius muscle:tinear model

b ~ . ~ (3 parameters )

(a)

0.'20 0140 0.'60 0180 1.'00 1.20 1AO 1.60 Time (sec)

180 Time course of measured and generated tension of tetanized frog' s gastrocnemius muscle :linear model

E 120

g 60

b -

0

-60 ~ ~ ~ ~ 0.20 o. o o.;o o'8o .2o 16o

Time (sec)

180

~120

g 60 C

o

-60 0

Time course of measured and generated tension of 180 tetanized frog's gastrocnemius muscle : non-linear

k (4 parameters) modet 120

i ; n

q | ~ 0

0.~20 0.40 0.60 0.80 1.00 1.20 1.40 1.'60 Time (sec)

~. Time course of measured and generated tension of \ . t �9 �9 ~. tetanlzed frog s gastrocnemlus muscle : non-hnear ~, ( 5 parameters ) model

(d) I I I I

0.20 0.40 0.60 0.80 1.00 1.~20 1.40 1.60 Time (sec)

Fig. 3. Response of various models of a tetanized frog's gastrocnemius muscle to a trapezoidal stretch. (a) Linear model response; (b) Com- plex linear model response; (e) Tension feedback model response; (d) Length and velocity feedback model response. Tens ion--above tetanus level--in grams; time in seconds. The model's response (x) is compared to the measured muscle tension output (0). Displacement is 0.1 cm

which were identified for the length feedback model of the frog had negative values, which cannot be ex- plained physiologically. For the cat length feedback model, only the parallel elastic component had a negative value, which can be interpreted physically as being an amplifier, but cannot be explained physiologi- cally. In spite of that, the fitting of this model to the cat's muscle was better than the others.

The different final models found for the frog's and cat's muscles were simulated on a digital computer, using the C.S.M.P. programme. Each model was fed with the same input, and the outputs were compared with the experimental results for the corresponding muscle (Fig. 3 and Fig. 4). It can be seen from these figures that even the linear models produced quite a good fit. The addition of the non-linear components generally improved the quality of fit. The addition of the tension-feedback component much improved the fit of the gastrocnemius muscle of the frog in the active mode. The model which best suited the cat's gastroc- nemius muscle was the length feedback model. The quality of the fit was judged by comparing the peak, the decay time and the steady state levels of tension developed by the model to those obtained experi- mentally in response to the same stretch.

V. Results: The AS Curve

Since the AS curve is a property of the internal force generator of the muscle, it cannot be measured directly,

but has to be calculated. Some assumptions must be made during these calculations, and, therefore, the AS time course is only an estimation. After a proper model has been found, the following assumptions must be made:

a) The parameters remain constant during tetanus, while they are being identified, and during twitch, where they are used for the AS estimation.

b) The parameters do not vary with muscle length, which requires their evaluation for small length perturbations only, preferably about 1 o.

The AS time course is estimated from the results of an isometric twitch experiment. The twitch is the muscle's measured output force, while the AS is the calculated internal input force. The muscle's length is constant during this experiment; therefore,

X = 0 ; X t = - X 2 (12)

where X is the change of length from rest. The time course of P as a function of T can now be derived from the model's equation.

For the simple linear model, the differential equa- tion is

J" = b i n + b2X + b 3 T + b4P (13)

and the AS time course can be derived from the relation

P= l ( J ~ - b3r) (14) be

68

3O

x

g o

-15

-30

Time course of measured and generated tension of tetanized cat's gastrocnemius muscle :linear model

' (3 parameters )

(a}

Time (sec)

30 c~ 0

~ . 1 5 I ~n g 0,

# - 1 5

-30

Time course of measured and generated tension of tetunized cat's gastrocnemius muscle :lineGr model

, ~ ( 5 parameters )

(b) I ~ I I I I I [

1 3 4 5 6 7 8 Time (sec)

. 3 0

"7

'i! -3

Time course of measured and generated tension of tetanized cat's gastrocnemius muscle :non- [bear

(4 parameters ) model

i

Time (sec)

Time course of measured and generated tension of tetanized cat's gastrocnemius muscle : non-linear

( 5 parameters ) model

o 30

~- 15 ~r

-15

(d)

Time (see)

Fig. 4. Responses of various models of a tetanized cat's gastrocnemius muscle to a trapezoidal stretch. (a) Linear model response; (b) Complex linear model response; (c) Tension feedback model response; (d) Length and velocity feedback model response. Tension above tetanus level--in grams; time in seconds. The model's response (x) is compared to the measured muscle tension output (0). Displacement is 0.8 cm

where T is the time course of the muscle tension during an isometric twitch. For the other models, the equa- tions are derived similarly. The complex linear model:

1 P= ~66 (J~-bcT-bsT). (15)

The tension-feedback model:

P= ~(J~-b3T-bcT. pgn(X2)). (16)

The length and velocity feedback model:

1 P= ~(T-b3T ). (17)

These equations were simulated on a computer using C.S.M.P., where the input to the system was the recorded time dependent muscle twitch tension, T, and the output was the desired AS time course. Some results of these simulations appear in Figure 5 and Figure 6.

From these results it is evident that the models which did not fit the muscle well also generated peak AS which exceeded by a large amount the tetanus level and the peak AS found by others (Bahler, 1967; Taylor, 1969). For the frog's muscle the tension feed-

back model should generate the most accurate time course, since it best fitted the muscle. In this case, the peak amplitude of the generated AS time course was 1.65 times that of the twitch tension, and only 72% of that of tetanus. The rise-time is quick at the beginning, and then the tension rises slowly towards a "plateau", which has been defined in previous work (Gasser and Hill, 1924). The AS force then falls through the peak of the twitch to a negative level and finaly rebounds to the resting level (see discussion).

From the results of the cat's muscle model, it is evident that none were as good as the ones generated by the frog's tension feedback model. However, the length feedback model gave reasonable results, and the estimation of AS meets most of the known physiologi- cal criteria. Its maximum amplitude is 3 times higher than the twitch and only 80 % of the tetanus amplitude for the same muscle.

VI. Discussion

In the previous sections a new method for the estima- tion of the whole AS time course was outlined and applied to the gastrocnemius muscle of both the frog and the cat. It is evident that, since no perfect mathe- matical model could be found for characterizing

69

500

400

~-~300

~ 200

100

0

-100

lime course of the calculated active -state and measured ten- sion of a twitch

k x x / - -x "- 'xxz

( a ) , \ ~ - ~ - ' , , , 0.05 0.10 0.15 0.20 0.25

Time (see)

100

80

o 60 i

4o

g 2o

-2%

Time course of the calculated active-state and measured ten-

"~ sion of a twitch

x

0.05 0.10 0.15 0.20 0.25 Time (sec)

250

200

150 ~T ~-~ 100

Time course of the calculated active-state and measured tension of a twitch

II / < , . ,

-5~ o.io o.15 o.~o o.~5 Time (see)

Fig. 5. The estimated Active State time course (x) obtained from the frog twitch experiments (zl), and calculated from: (a) The linear model; (b) The complex linear model; (c) The tension feedback model

100

8o

.~ 40

t i Time course of the calculated active-state and measured ten- sion of a twitch

201/,1-,..~,, "~" ...... ~. ....

0"- ~ __~ ~ - -

-20 (a) ~ , / / " , , , 0 0.~05 .* 0.15 0.20 0.25 • (sec)

500[ L Time course of the calculated 75[ active-state and measured ten-

4001- sion of a twitch 50 r~ 0 7

-~300 45

200 c 3C

= 100 ~ 15

> - " ~ - ' ~ - - ~ " 0 •

- 100 0.~10 , r -15 0 0.;5 0.15 0/20 0.25 0 Time (sac)

Time course of the calculated active-state and measured tension of a twitch

~A'~�9 ~. A ~,

0.05 0.10 0.15 0.20 0.25 Time (sac)

Fig. 6. The estimated Active State time course (x) obtained from the cat twitch experiments (A), and calculated from: (a) The linear model; (h) The tension feedback model; (e) The length velocity feedback model

muscle dynamics, the AS curve obtained by this method is only a qualitative representation of the actual AS. However, within the limitations of this method, it is assumed that the obtained AS does represent the real shape of the whole AS time course.

It is possible to compare the results presented here only with those works which follow the same defini- tion of the AS given here. Two characteristics of the AS obtained here stand out when making these com- parisons: (i) the low amplitude of the AS peak, obtained from the best fitting muscle model, and (ii) the negative level of generated AS force at the end of the decay. The former phenomenon is supported by other works (Bahler, 1967; Taylor, 1969), while the latter has never been observed before.

Contrary to Hill's result (Hill, 1970), the AS maxi- mal level was found here to be 72% (for frog) or 81% (for cat) of the tetanus level. These results imply that, during a twitch, the AS level does not reach the tetanic level, possibly because the relaxation processes start to function immediately. In other methods (Bahler, 1967; Taylor, 1969), it has been found that the AS peak is lower than the tetanic level. It must be emphasized that the results obtained in the present work can be only qualitatively compared with previous works, and the differences are not of great importance. The main result, that the AS peak is lower than the peak tetanus force, indicates the existence of a non-linear saturation phenomenon in the generation of force in the muscle. It is exhibited by the dependence of the AS on the

70

immediate past history of muscle stimulation. Stimula- tion at tetanic rate produces tetanic force, which means that each AS is reaching a saturation value, its maxi- mum under these conditions. Stimulation at lower rates yields lower peaks for each AS, and a minimum value is reached for an AS peak during a single twitch, as evaluated here. Bahler's AS curve is similar to the present one in some more aspects, since in both cases there is no plateau but a single peak, and the decay time is nearly the same.

The decay of the AS obtained here is very similar to the most accepted shape (Ritchie, 1954; Edman, 1970, 1971). The ratio of the rise time to decay time is similar in these works and in the present work, and, in all, the AS decay passes through the peak isometric twitch tension. Only with the length-feedback model is there a slight difference in the decay shape, since the non-linear component has a large value and does not vanish at the time when dT/&=O, causing the AS to decay somewhat before the twitch reaches its peak.

Of all the methods mentioned, however, none has succeeded in estimating two phases of the AS which are described here--the whole peak and the end of the decay which are produced here without any disturbance of the muscle during the AS measure- ment.

The fall of the AS to a negative level below its resting force was not detected by other methods, because these methods were not seeking to calculate the end of the decay. This phenomenon is difficult to explain and only an hypothesis can be given. The contraction process is usually explained today by the sliding filaments theory. During the peak of activation, distor- tions occur in the molecular structure, since the forces acting on the filaments are very high. As the protein filaments move back, they might be generating force in their attempt to reach the more stable state of equilibrium, which they occupy when not stimulated. This return to the normal structure might cause the discussed phenomenon.

Another explanation might be as follows. When the muscle is resting, active force is being generated without electrical stimulation, by a steady state level of calcium ions. This phenomenon agrees well with Hill's "filamentary resting tension" which represents active interaction between the filaments even in a resting muscle, under zero tension (Hill, 1968). When electrically stimulated, the calcium ion concentration rises and the active force increases with it. If a single pulse is applied, the back pumping of the calcium ions into the sarcoplasmic reticulum, which starts simul- taneously with the ion release, might temporarily drive the ion level below its steady state level towards the end of the AS time course. This undershoot in ion

concentration will take its course and the steady state level will be re-established with the AS following it. This explanation depends on the establishment of active back pumping of calcium ions and its time course in response to a single pulse, or vice versa. The above applies specifically, to the tension feedback model, which incorporats in it a force component proportional to the tension existing in the muscle.

The method presented here requires the charac- terization of the muscle by a mathematical model. The development of the models and the motivation for selecting each particular model were only briefly discussed, since they have been presented elsewhere (Inbar et al., 1970). A more recent work (Huxley and Simmons, 1971) supports the general contention that the model of the muscle contains an elastic element in series with a visco elastic force generation contractile component. The series elasticity is attributed to the crossbridges themselves, and is believed to be an essential part of the force generating mechanism. It is only interesting to note that the length-feedback model, which best fitted the active cat muscle, also fitted well the same passive one (Inbar et al., 1970). This result seems to verify the assumption (Brown, 1959) that a model fits a muscle independently of muscle tension or activity (although its parameters change), and that the AS shape can be generated from the model, in spite of the fact that its parameters were identified at different tension conditions.

Two basic works, dealing with muscle modelling, are connected to the present work: Julian (1969) developed mathematical relationships between the activation mechanism and muscle dynamics. However, these relationships cannot yield the AS time course directly, and the classical "quick release" experiments (Ritchie, 1954) must be performed in order to calculate it. In the work presented here, the AS is an integral part of the model, which can be estimated only from the measured muscle output. In this sense the two methods differ markedly, and this provided the motiva- tion for suggesting the new method. It should be mentioned that the activation mechanism could be used as the input signal to another model whose output will be the estimated AS time course.

Huxley et al. (Huxley and Simmons, 1971) have used the measured external muscle dynamics in order to describe how the force between the thick and thin muscle filaments may be generated. In contrast, the models presented here do not indicate the precise molecular interactions responsible for the AS genera- tion. As with Julian's work, it is possible that an intermediate model can be used to relate the molecular structure model to the model presented here.

The present method suffers from some difficulties, as has already been mentioned. Some can be solved

71

at the expense of clarity and physiological tractability, while others are inherent in the method. The major problems are: a) Due to the non-linearities of the system and the distribution in parameter values among the various motor units, the model presents only an approximation of dynamic muscle behaviour. Since the models are not, and can hardly ever be, perfect, the estimation of the AS is not exact, b) The assump- tions made have to be confirmed, i.e. whether the model is always independent of the tension level, and how closely the parameters are independent of length around the resting length l o. c) The muscle response at low frequencies can only be checked by a long tetanus. It is nearly impossible to obtain a steady tension level during a long tetanus because the muscle tension starts to deteriorate, d) The generation of the AS time course was based on the twitch tension time course. This information had to be digitized and its derivative used in the AS calculations. In this way an element of error was added to the results, as can be seen in Figure 5 and 6. This error may be reduced by digitizing at a higher rate, or by averaging the results over several twitches.

The proposed method for the estimation of the AS time course has some advantages over the other known methods: a) The two necessary experiments are simple and short. The experiment for the identification of the parameters requires a means of introducing and recording the desired disturbance, but then the entire procedure involves only tetanizing the muscle and stretching it, followed by a second experiment .con- sisting of a simple isometric twitch, for AS est~matmn. b) The mathematical analysis is simple, although quite a large computer memory core is needed, c) No other method is known to produce the whole AS time course, from its beginning to the end of its decay. The other

methods are not even able to measure the whole peak of the AS, or to measure the end of the AS decay.

Appendix: Experimental Verification

It is obvious that such a theoretical finding as a negative, below steady state (BSS), active state level requires an independent ex- perimental verification. To date we have failed to devise such an experiment; however, the results obtained from the best attempt so far are shown in Figure 7. Ira BSS level exists, then it can be assumed that a second twitch appearing during the BSS period have a lower peak than the first twitch, when measured externally. This would be the case if: A. The two AS responses sum linearly, or at least do not have non-linearities. B. The summat ion of forces generated by the individual motor units in the muscle does not mask the BSS as a result of the spread in the time responses among the various motor units.

Bearing these limitations in mind, a double pulse experiment was carried out on the soleus and gastrocnemius muscles of the cat. The experiment was run in a similar set-up as described in Methods, but with anesthesized cats. The nerve supplying the muscle under study was dissected free and supermaximally stimulated with a pair of hook electrodes. Double pulses were applied while the muscle was held at l o and its tension recorded. The pulse interval (At) was shortened between successive pairs, in order to bring the second pulse within the BSS period of the first pulse in the pair. At least 5 s elapsed between each pair of pulses, so that the first twitch response always had the same amplitude (within 1%). In this way it was hoped that if the BSS exists the second twitch response will be smaller than the first one when falling into the BSS period of its predecessor.

The following results can be observed in Figures 7 (1) and (2): a) From At = 400 ms down to A t = 200 ms, the second twitch response is higher than the first one by A T = 3 %. b) From about At = 200 ms down to A t = l l 0 m s , the rise in A T is much smaller than with 110> At. Although this At is more than could have been predicted from the'results shown in Figure 5 and Figure 6 for the BSS period, the slump in the A T rise for l l 0 < A t < 2 0 0 may be explained by the BSS. c) With At< 110 ms the A T kept rising smoothly down to A t = 2 5 m s , where a saturation phenomena is observed. It was observed during the experiment that, for A t = 2 0 m s , the second twitch response is more than twice as high as the first one (i.e. A T > 100%) by about 10%. This indicates the existence of at least

i

51

48

4 5 ?

42

3 9

g 36 x 33

.~

~ 27

24

18

15

12

9

6

3

0

~ A t

3OO At(msec)

Fig. 7. Percent rise in second twitch response over that of first twitch response, A T / T ~ , as a function of interspike interval, At. (1) Soleus muscle stimulated with double pulses. (2) Gastrocnemius muscle stimulated as in 1. (3) Soleus stimulated with pulses of continuously varying frequency. Insert: Tension response of cat soleus muscle to double pulse stimulation

72

some slight non-linearities, as speculated upon in condition A, since a linear summation should have yielded A T = 100 %.

In the soleus case, in addition to the double pulse method, the interspike intervals of the stimulating pulses were changed continuously at a slow rate from one pulse per second up to 50 pulses per second. In addition to the variability in muscle twitch response to stimulation under similar experimental conditions, as observed previously, the possibility of fatigue caused by continuous stimulation cannot be ignored, and, consequently, did not allow this experiment to be carried out above 30 pps. In addition, con- tinuous stimulation causes continuous potentiation of the succeeding pulses, which is the reason for the right hand shift in the Ar curve in this case, compared with the double pulse method. The results are shown in Figure 7 (3). In this mode of experimentation, despite the mentioned limitations, the same slump in the A T rise, in the vicinity of At= 150 ms, is seen, similar to the double pulse experiment.

As was mentioned, this experiment did not prove, and could not disprove, the existence of BSS in the AS response, except for some positive indication as was demonstrated by the AT slump in the l lO<At<200 range. It is possible that applying this experi- ment to single motor units, which is difficult, or to single fibers might demonstrate this phenomenon. No other tests have been suggested to date to verify the BSS.

Acknowledgement. This work was supported by the Julius Silver Institute of Bio-Medical Engineering Sciences, Grant 050-304.

References

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Bahler, A.S., False, J.T., Zierler, K.L.: The dynamic properties of mammalian skeletal muscle. J. gen. Physiol. 51, 369-384 (1968)

Bahler, A. S.: Modelling of mammalian skeletal muscle. IEEE Trans. Biomed. Eng. BMD-15, 249

Brown, A.C.: Analysis of the myotatic reflex. Ph. D. Diss. Univ. of Washington (1959)

Carlson, F. F.: Kinematic studies on mechanical properties of muscle. In: Tissue elasticity, Ed. by Washington, R.J.W.: Amer Physiol. Soc. pp. 55-72 (1957)

Edman, K. A. P.: The rising phase of the active state in single skeletal muscle fibers of the frog. Acta physiol, scand. 79, 167=173 (1970)

Edman, K.A.P., Kiessling, A.: The time course of the active state in relation to sarcomere length and movement studied in single skeletal fibers of the frog. Acta physiol, scand. 81, 182-196 (1971)

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Hill, D. K. : Tension due to interaction between the sliding filaments in resting striated muscle. J. Physiol. (Lond.) 199, 637-683 (1968)

Hill, A:V.: First and last experiments in muscle mechanics. Cam- bridge: University Press 1970

Hsia, T.C., Baily, A.L.: Learning model approach for non-linear system identification. IEEE Systems Science and Cybernetics Conf. Rec. 228 232 (i968)

Huxley, A. F., Simmons, R. M.: Proposed mechanism of force genera- tion in striated muscle. Nature (Lond.) 233, 533-538 (1971)

Inbar, G.F., Hsia, T.C., Baskin, R.J.: Parameter identification anal- ysis of muscle dynamics. Math. Biosci. 7, 61-79 (1970)

Jewell, B.R., Wilkie, D.R.: The mechanical properties of relaxing muscle. J. Physiol. (Lond.) 152, 3~47 (1960)

Jewell, B.R., Ruegg, J.C.: Oscillatory contraction of inset fibrillar muscle after glycerol extraction. Proc. Roy. Soc. B-164, 428-459 (1966)

Julian, F.J.: Activation in a skeletal muscle contraction model with a modification for insect fibrillar muscle. Biophys. J. 9, 547 570 (1969)

Kalman, R.E.: Design of a self-optimizing control system. Trans. Amer. Soc. ASME 80, 468 478 (1958)

MacPherson, L., Wilkie, D.R.: The duration of the active state in a muscle twitch. J. Physiol. (Lond.) 124, 292-299 (1954)

Ritchie, J.M.: The effect of Nitrate on the active state of muscle. J. Physiol. (Lond.) 126, 155-168 (1954)

Ritchie,J.M.: The duration of the plateau of full activity in frog muscle. J. Physiol. (Lond.) 124, 605-612 (1954)

Taylor, C.P.S.: Isometric muscle contraction and the active state, an analog computer study. J. Biophys. 9, 759 780 (1969)

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Received: November 28, 1975

Dr. G. F. Inbar Faculty of Electrical Engineering Technion, ITT Haifa, Israel