- Email: [email protected]
Prediction of summation in incompletely fused tetanic contractions of rat muscle
ARTICLE IN PRESS
Journal of Biomechanics 40 (2007) 1066–1072 www.elsevier.com/locate/jbiomech www.JBiomech.com
Prediction of summation in incompletely fused tetanic contractions of rat muscle Brian R. MacIntosha,, David Jonesa, Andrea N. Devromea, Dilson E. Rassierb a
Human Performance Laboratory, Faculty of Kinesiology, University of Calgary, Calgary, Alta, Canada T2N 1N4 Department of Kinesiology and Physical Education, Faculty of Education, McGill University, Montreal, Que., Canada
b
Accepted 23 April 2006
Abstract Summation is the accumulating contractile force resulting from sequential activations applied to a muscle without sufficient interval to permit complete relaxation. The purpose of this study was to evaluate summation in the rat medial gastrocnemius muscle, and to determine if the contractile responses during summation could be predicted from the relationship between force and activation pattern. In the first part of this study, the consistency of summation in the rat gastrocnemius muscle was assessed and prediction equations were derived. The second part compared predicted summation with actual contractions obtained in a new set experiments. Summation was assessed by calculation of the contractile response, per stimulation, for up to five stimulating pulses at these frequencies: 20, 40, 60 and 80 Hz. This was done by subtraction of the force transient for j1 pulses of stimulation (where j ¼ 1–5 pulses) from the force response with j pulses of stimulation. Each of these force differences was evaluated for peak rate of force development, contraction time and half-relaxation time. Contraction and half-relaxation times changed by only a small magnitude from values obtained for the twitch. Peak rate of force development was proportional to the active force for all force transients obtained by subtraction. The force per activation increased from the first to the fifth stimulus, and was dependent on interpulse delay. In the second series of experiments, the predicted force was related to the actual force for brief tetanic contractions at 40, 50 and 60 Hz (r2 ¼ 0.875). These experiments demonstrate that the force response to sequential activations is consistent and predictable. Summation can be predicted, knowing only the amplitude of the twitch contraction and the relationship between delay and force for each activating stimulus. r 2006 Elsevier Ltd. All rights reserved. Keywords: Muscle; Contractile response; Force–frequency relationship; Twitch
1. Introduction The twitch is a transient rise and fall of active force in response to a single stimulus, and is considered to be the elementary contractile response in skeletal muscle (MacIntosh and Gardiner, 1987; MacIntosh et al., 2006). When a second stimulus is given shortly after the first, a similar twitch-like force transient is elicited Corresponding author. Tel.: +001 403 220 3421; fax: +001 403 220 0105. E-mail address: [email protected] (B.R. MacIntosh).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.04.009
(Stein and Parmiggiani, 1981; MacIntosh et al., 1988). This contractile response to the second stimulus is added to the declining force of the first, a process called summation. The twitch-like force transient obtained with any stimulus in a train is not the same as that observed for a twitch (Raikova et al., 2006). It is generally greater in magnitude. Summation is therefore a nonlinear process (Burke et al., 1976; Zajac and Young, 1980; Stein and Parmiggiani, 1981). It would be desirable if the force response for a given train of activating pulses of stimulation could be predicted. This nonlinearity makes prediction of force summation more
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
challenging, but if the pattern of change in magnitude of force transient is consistent, then prediction should be feasible. The magnitude of the contractile response can be presented as either the force produced by the muscle, or the force–time area. Burke et al. (1976) evaluated the contractile response of cat medial gastrocnemius muscle motor units, using the force–time area as the criterion measure of the contractile response. Their rationale for this approach was that the force–time area of a skeletal muscle contraction is thought to be proportional to the energy demand of the contraction. This has been shown to be the case for cardiac muscle (Alpert and Mulieri, 1982). However, the energetics of cardiac and skeletal muscle are very complex and the force–time index is not likely the best correlate of the energy cost of contraction in skeletal muscle (Hisano and Cooper, 1987; Ameredes et al., 1998). Instead, the number of activating pulses is a reasonable predictor of the energy cost (Fales et al., 1960). This removes a potential advantage for using the force–time area as a measure of the contractile output. According to Burke et al. (1976), because the force– time area is also known as the impulse, or the potential for disturbing the motion of an object on which the force is acting, this method of quantifying the contractile response is useful. However, this notion can be misleading; any impulse that relates to the potential change in motion of an object must refer to a dynamic contraction. Therefore it would not be appropriate to consider the force–time area of an isometric contraction to be related to the potential for disturbing motion. Furthermore, a given force–time area can be obtained with a high force acting for a short time, or a lower force acting for a longer time. This complexity would make prediction of this contractile output during trains of stimulation difficult, because there would be an indeterminate number of possibilities. It would be more appropriate to consider force as the primary output variable; the duration of force production can be modulated by changing the duration of stimulation, but often it is the force resulting from muscle activation that is targeted in voluntary muscle control. There have been limited studies dealing with force summation in a quantitative way, and none have attempted to predict the contractile force from the relationships between interpulse interval and number of stimulating pulses. This is surprising, as prediction of the contractile response for a given stimulation pattern is important to skeletal muscle modeling (Wexler et al., 1997; Ding et al., 2000) and functional electrical stimulation (Stein, 1999). The purpose of this study was to evaluate summation in the rat medial gastrocnemius muscle, and determine if the contractile response could be predicted from the relationship between force and activation pattern.
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2. Methods Sprague–Dawley rats weighing 190–260 g were anesthetized with ketamine/xylazine (85 and 15 mg/kg, respectively), and the gastrocnemius muscle was surgically isolated in situ for attachment of the Achilles tendon to an isometric force transducer. The plantaris and the soleus muscles were cut from the Achilles tendon and peeled back to prevent their force contribution. The distal stump of the cut sciatic nerve was placed over a pair of stainless steel electrodes. The femur and tibia were immobilized by pins that were affixed to the myograph base to prevent motion at the origin of the muscle. Additional anesthetic was given during the experiment if needed. At the end of the experiments, rats were killed by anesthetic overdose. The Achilles tendon was attached to a force transducer that was mounted on a rack and pinion device that permitted precise control of muscle length, and this was set at the length that yielded the greatest apparent active force (difference between peak total force and passive force with double-pulse stimulation, delay ¼ 5 ms). The experiments were conducted at this reference length. The loosened skin was pulled up around the muscle, forming a container that was filled with warmed paraffin oil. This oil and the rectal temperatures were kept near 37 1C with radiant heat. Stimulation of the sciatic nerve was performed with supramaximal 50 ms square pulses, at the specified intervals (Grass model S88). In the first series of experiments (n ¼ 7), trains of stimulation consisted of 1–5 pulses, and the interpulse intervals ranged from 12.5 to 50 ms, corresponding to stimulation frequencies of 20, 40, 60 and 80 Hz. The time between sequential contractions was 1 min to avoid complicating issues associated with fatigue and activity dependent potentiation (MacIntosh and Willis, 2000). These contractions were used to estimate the expected force per stimulus (see analysis below) and to see if the force transients associated with sequential stimulation resulted in a predictable change in the time-course of the force transient per activation. In a second series of experiments (n ¼ 14), similar contractions with 1–5 pulses of activation were obtained at frequencies of 40, 50 or 60 Hz. The force of these contractions was compared with the estimated active force for simulated contractions, that were based on an equation for the twitch contraction and the relative force per activation as estimated from the first series of experiments. The equation for the twitch was based on MacIntosh and Heinemeyer (1994): Active force ¼ kf ðt^ ðaf 1Þ e^ ðt=bf ÞÞ þ ks ðt^ ðas 1Þ e^ ðt=bs ÞÞ,
ð1Þ
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
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Table 1 Constants for twitch contraction simulation Fiber type
Fast
Slow
k b a
3.23106 4.50103 3.80
20.5 2.0102 2.20
where k, a and b are constants, t is time; subscripts f and s refer to fast and slow-twitch motor units, respectively. The force of the simulated twitch was essentially the sum of the force of fast and slow-twitch motor units. The force contribution of the slow-twitch motor units was set at 7% of the peak force, because it has been reported that about 7% of the cross-sectional area of the rat gastrocnemius muscle is composed of slow-twitch muscle fibers (Ariano et al., 1973). Table 1 presents the values for the constants used in the equation. 2.1. Data analysis Contractions obtained in the first series from each sequence of 1–5 stimulating pulses at a given frequency were analyzed to obtain the active force per activating pulse. This was done by subtracting the force profile of a contraction with j1 stimulating pulses from the force profile with j stimulating pulses (see Fig. 1) where j refers to the number of stimulating pulses. These twitch contractions and active force per activation obtained in the first series were fit to the following equation for each sequential activating pulse: force ¼ ða dÞ=ð1 þ ðt=cÞ^ bÞ þ d,
(2)
where a, b, c and d are constants, obtained by least squares fitting of the data and t is time interval or delay between stimulations. The data for second, third, fourth and fifth force transients were fit to this equation for the frequencies ranging from 20 to 80 Hz. The resulting equations were used to estimate the individual contraction amplitude for the second, third, fourth and fifth stimulation and these were added with appropriate delay to simulate contractions for sequential activation at 40, 50 and 60 Hz. The predicted force transients were created by scaling the standard equation for the twitch to the amplitude of the observed twitch, and scaling the sequential responses according to Eq. (2) for the appropriate delay (corresponding to 40, 50 and 60 Hz). The result was then added with the corresponding delay to create a predicted contraction for comparison with the contractions obtained in the second series of experiments. The force transients per activation that were obtained by subtraction (Series 1, see above) of the contraction with 1–4 pulses of stimulation from the contraction with one more pulse of stimulation were evaluated for
Fig. 1. Five contractions (from 1, 2, 3, 4 and 5 pulses) are superimposed in A and C. These contractions were obtained with 25 (A) or 16.67 (C) ms delay between pulses of stimulation. Subtraction of the force profile of the twitch from the force profile of the doublet, the doublet from the triplet, etc. yields the force transient for the second pulse, the third pulse, etc. as shown in B and D. Note that sequential activations at 60 Hz elicit much greater force increments than sequential activations at 40 Hz.
contraction time, half-relaxation time and peak rate of force development. This was done to see how consistent these parameters were for transients derived from sequential activations. If these parameters of the force profile changed substantially within an unfused tetanic contraction, then prediction of force in the manner proposed (vertical scaling of the twitch) would not provide a good estimate of peak active force. In fact, this method would only be effective if the amplitude of the force transient remained proportional to the peak rate of force development. Contraction time was measured from the beginning of force development to the peak of the contractile response. Half-relaxation time was measured from the peak of the contractile response until force decreased to 50% of the active force. Peak rate of force development was measured as the greatest change in force across 2 ms on the rising phase of the force transient. Statistical evaluation was performed with linear and non-linear regression analyses, and single factor ANOVA with repeated measures. When significant differences were detected (a ¼ 0.05), for contraction time and half-relaxation time, Newman Keuls test was used to identify the specific locations of these differences.
3. Results Sample contractions of the gastrocnemius muscle are shown in Fig. 1 for 1–5 pulses of activation at 40 Hz (1A) and 60 Hz (1C). Five contractions are superimposed to illustrate the reproducibility of the peaks
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
Fig. 2. Contraction time (time to peak force, open symbols) and halfrelaxation time (solid symbols) for the force transients resulting from each of 1–5 pulses of stimulation at 20–80 Hz are shown. Only minor changes were evident. Results are mean values with some standard errors to illustrate variability.
within a series of stimulations. The lower panel (1B and 1D) in each case shows the force for each activation, as obtained by subtraction of the active force for j1 stimulations from the active force for j stimulations. Contraction time for force transients obtained for 1–5 pulses of activation at 20–80 Hz is shown in Fig. 2. Significant changes were observed for 40 Hz (transient number 1 vs. 4 and 5 as well as 2 vs. 5) and 60 Hz (transient number 3 vs. 5) stimulation, while contraction time for force transients at other frequencies did not change. Although these differences were significant at 40 and 60 Hz, the magnitude of change was less than 10%. Half-relaxation time for force transients obtained for 1–5 pulses of activation at 20 to 80 Hz is shown in Fig. 2. Values for half-relaxation time vary from about 10.6 ms for the first contraction transient to 11.4 or 9.7 ms for later transients. This difference was significant only for 20 Hz, and only for the difference between the fifth transient and the first or second transient. The maximum magnitude of difference was less than 10%. Whereas there were only minor changes to the contraction time and half-relaxation times, there were substantial and predictable changes to the peak rate of force development. Changes in this parameter would appear to be the primary mechanism by which twitch force is modulated in sequential activations. Peak rate of force development changed in proportion to the developed force across all frequencies and for all five transients within a contraction (Fig. 3). The amplitude of a force transient within a contraction was dependent on the delay between sequential stimuli as well as the number of stimuli preceding. This is illustrated in Fig. 4. The constants that best fit the
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Fig. 3. The relationship between peak rate of force development and active force of the force transient for each of 1–5 sequential activations delivered with 16.67–50 ms delay. The strong linear relationship typically observed for twitch contractions (MacIntosh and Gardiner, 1987) is preserved for force transients within an incompletely fused tetanic contraction.
Fig. 4. Active force of the force transients resulting from the 2nd (square), 3rd (diamond), 4th (cross), and 5th (circle) pulse of sequential activation plotted against delay. Symbols represent mean values and lines represent the best fit according to the equation: Force ¼ ða dÞ=ð1 þ ðt=cÞ^ bÞ þ d and the constants presented in Table 2.
data to the equation are given in Table 2. These equations, with the appropriate constants, were used to predict the force for 40, 50 and 60 Hz contractions and these predicted contractions were compared with contractions obtained in the second series of experiments. The input variable was the amplitude of the observed twitch, and the predicted force was compared with measured force at common frequencies. The force per activating pulse increases with the number of stimulating pulses, and decreases with increase in the
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
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delay (interpulse interval). Fig. 5 presents a series of modeled contractions at 40, 50 and 60 Hz and the corresponding actual contractions that these simulated contractions are intended to predict. The result of the comparison between predicted and actual contractions is shown in Fig. 6. The correlation (r) for this relationship was 0.93. This represents significant (po0.05) agreement between the predicted and actual contractions. 4. Discussion This study has revealed a unique relationship between active force and delay between sequential activations for up to 5 stimulating pulses, similar to that which has been shown for the force–time integral (Zajac and Young, 1980). This information was then used to obtain a reasonable estimate of the active force for incompletely fused tetanic contractions. This study confirmed that the force transient associated with each activation in a train of pulses can be determined by subtraction of the force for a contraction elicited with one less stimulation. These force transients have a shape that is similar to that of a twitch contraction, and the contraction and half-relaxation times remain reasonably constant over the range of frequencies and for the train duration
evaluated here. The magnitude of these force transients varied with the time interval between activations and the number of pulses but remained proportional to the peak rate of force development. This meant that prediction of summation could be accomplished with vertical scaling of a standard twitch. The fact that summation is more than the linear addition of identical twitches is referred to as nonlinear summation. This term has been used for two phenomena in the study of skeletal muscle: summation of sequential activations and parallel summation of motor unit activations. The nonlinear summation of parallel motor unit activations is a consequence of complex connective tissue structure and parallel as well as series elastic properties of muscle (Troiani et al., 1999; Sandercock, 2000). However, this is unlikely the factor associated with nonlinear summation of sequential activation, as activation at 40 Hz, which begin at progressively higher levels, would not have a similar magnitude, as shown in Fig. 4.
Table 2 Constants for prediction equationsa Pulse
2nd
3rd
4th
5th
a b c d
4.65 26.4 25.4 2.95
5.84 13.6 24.0 3.01
6.60 13.1 22.6 3.04
7.27 9.4 20.5 3.07
a
The following equation was used to predict the amplitude of force transients: Force ¼ (ad)/(1+(t/c)b)+d, where force is absolute active force in newtons and a, b, c and d are from the table and t is delay (ms) between sequential stimulations. These values for the constants were derived from test contractions at 20, 40, 60 and 80 Hz. The average active force of the initial transient (twitch) was 3N.
Fig. 6. Predicted active force and corresponding measured active force for brief incompletely fused tetanic contractions at 40, 50 and 60 Hz. The regression (green) line is very close to the line of identity (black), indicating agreement between the predicted and measured contractions.
Fig. 5. Predicted (light line) and sample contractions at 40, 50 and 60 Hz, from one experiment are shown to illustrate the agreement between actual contractions (dark lines) and those based on the standard twitch (Eq. (1)), and subsequent responses scaled according to the amplitude values predicted from Eq. (2). These contractions are scaled relative to the initial force transient (what would otherwise be the amplitude of the twitch).
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
The mechanism for nonlinear summation of sequential activations is not known, but it seems reasonable to speculate that it is related to factors associated with Ca2+ handling: release, binding and uptake. The increment in Ca2+ bound to troponin C for a given stimulation will depend on the initial free [Ca2+], the amount of Ca2+ released from the sarcoplasmic reticulum, the occupation of alternative binding locations (Ca2+ buffer sites) and the binding of Ca2+ to troponin C. A given amount of Ca2+ released from the terminal cisternae causes more or less increment in binding to troponin C depending on the Ca2+ binding sites that are already occupied and the rate at which Ca2+ is removed from the cytoplasm. Twitch active force, contraction time and halfrelaxation time were highly reproducible during our experiments. These parameters did not change substantially over the course of 5 stimulating pulses, for frequencies corresponding to: 20, 40, 60 and 80 Hz. This fact is an important prerequisite for prediction of force using the approach described in this study. If the contraction or relaxation times had changed, then the standard twitch used in this study could not have been simply scaled according to Fig. 4 to sum and create the incompletely fused tetanic contractions. It is quite likely that half-relaxation time and contraction time would change with additional activating pulses (more than 5), since they change with repeated activation at 10 Hz (MacIntosh et al., 1994). Changes in the contraction time and half-relaxation time for sequential activations within incompletely fused tetanic contractions of motor units have been reported (Raikova et al., 2006). Fast fatigable motor units changed in one direction while changes were in the opposite direction in the fast fatigue resistant motor units (Raikova et al., 2006). This may explain why we saw only minor changes in contraction time and halfrelaxation time over the first five activations. In contrast with our approach, Raikova et al., 2006 did not obtain sequential force transients by subtraction of contractions with one less activation; they used an iterative modeling procedure to obtain an estimate of the force transient for each activation. A limitation of our model is that it cannot account for modulation of the force transient when sequential incompletely fused tetanic contractions occur within a short period of time. Staircase, a form of activity dependent potentiation would be expected if contractions were repeated at short intervals (MacIntosh and Willis, 2000), and fatigue would be a likely consequence of continued intermittent contractions (MacIntosh and Rassier, 2002). It is known that contraction and relaxation times change with continued repetitive stimulation (MacIntosh et al., 1994), and that would cause the approach used here to give incorrect values. Activity dependent potentiation and fatigue were
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intentionally avoided by permitting a minute rest between contractions in this study. Further work will be needed to permit these factors to be accounted for in a model of this nature. The synchronous activation of motor units described in this study is quite different from the asynchronous activations of motor units typically seen in voluntary motor tasks. Our results provide insight into the nature of summation of sequential activations for whole muscle, but additional work with single motor units is needed to better understand the differences between fiber types and how these motor units sum together to give a smooth voluntary contraction. Some progress in this effort has been made by Celichowski and colleagues (Celichowski et al., 2005; Celichowski et al., 1999; Raikova et al., 2006). The approach taken in this paper is useful to characterize the properties of summation of any muscle of interest, and the result can be used to predict the force that would be expected for a given pattern of stimulation. This information would be useful in modeling contractile response, and possibly for pre-calculation of anticipated force in functional electrical stimulation. Functional electrical stimulation is a technique that has been used to overcome paralysis since the 1970s, but there are still several barriers to its effective use, including a limited understanding of summation in skeletal muscle. The current information could be used to formulate the appropriate number of pulses and corresponding frequency of activation required to achieve a given force. Acknowledgements This research was supported by NSERC, Canada.
References Alpert, N.R., Mulieri, L.A., 1982. Heat, mechanics and myosin ATPase in normal and hypertrophied heart muscle. Federation Proceedings 41, 192–198. Ameredes, B.T., Brechue, W.F., Stainsby, W.N., 1998. Mechanical and metabolic determination of VO2 and fatigue during repetitive isometric contractions in situ. Journal of Applied Physiology 84, 1909–1916. Ariano, M.A., Armstrong, R.B., Edgerton, V.R., 1973. Hindlimb muscle fiber populations of five mammals. Journal of Histochemistry and Cytochemistry 21, 51–55. Burke, R.E., Rudomin, P., Zajac III, F.E., 1976. The effect of activation history on tension production by individual muscle units. Brain Research 109, 515–529. Celichowski, J., Grottel, K., Bichler, E., 1999. Differences in the profile of unfused tetani of fast motor units with respect to their resistance to fatigue in the rat medial gastrocnemius muscle. Journal of Muscle Research and Cell Motility 20, 681–685. Celichowski, J., Pogrzebna, M., Raikova, R.T., 2005. Analysis of the unfused tetanus course in fast motor units of the rat medial gastrocnemius muscle. Archieves Italiennes de Biology 143, 51–63.
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Ding, J., Wexler, A.S., Binder-Macleod, S.A., 2000. Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains. Journal of Applied Physiology 88, 917–925. Fales, J.T., Heisey, S.R., Zierler, K.L., 1960. Dependency of oxygen consumption of skeletal muscle on number of stimuli during work in the dog. American Journal of Physiology 198, 1333–1342. Hisano, R., Cooper, G.L., 1987. Correlation of force-length area with oxygen consumption in ferret papillary muscle. Circulation Research 61, 318–328. MacIntosh, B.R., Gardiner, P.F., 1987. Posttetanic potentiation and skeletal muscle fatigue: interactions with caffeine. Canadian Journal of Physiology and Pharmacology 65, 260–268. MacIntosh, B.R., Heinemeyer, R., 1994. Slowing of relaxation in fatigued muscle. Canadian Journal of Physiology and Pharmacology 72. MacIntosh, B.R., Rassier, D.E., 2002. What is fatigue? Canadian Journal of Application Physiology 27, 42–55. MacIntosh, B.R., Willis, J.C., 2000. Force–frequency relationship and potentiation in mammalian skeletal muscle. Journal of Applied Physiology 88, 2088–2096. MacIntosh, B.R., Roberge, M.C., Gardiner, P.F., 1988. Absence of staircase following disuse in rat gastrocnemius muscle. Canadian Journal of Physiology and Pharmacology 66, 707–713. MacIntosh, B.R., Grange, R.W., Cory, C.R., Houston, M.E., 1994. Contractile properties of rat gastrocnemius muscle during staircase, fatigue and recovery. Experimental Physiology 79, 59–70.
MacIntosh, B.R., Gardiner, P.F., McComas, A.J., 2006. Skeletal Muscle: Form and Function, second ed. Human Kinetics Publishers, Champaign, ILL. Raikova, R., Celichowski, J., Pogrzebna, M., Aladjov, H., Krutki, P., 2006. Modeling of summation of individual twitches into unfused tetanus for various types of rat motor units. Journal of Electromyography and Kinesiology. Sandercock, T.G., 2000. Nonlinear summation of force in cat soleus muscle results primarily from stretch of the common-elastic elements. Journal of Applied Physiology 89, 2206–2214. Stein, R.B., 1999. Functional electrical stimulation after spinal cord injury. Journal of Neurotrauma 16, 713–717. Stein, R.B., Parmiggiani, F., 1981. Nonlinear summation of contractions in cat muscles. I. Early depression. Journal of General Physiology 78, 277–293. Troiani, D., Filippi, G.M., Bassi, F.A., 1999. Nonlinear tension summation of different combinations of motor units in the anesthetized cat peroneus longus muscle. Journal of Neurophysiology 81, 771–780. Wexler, A.S., Ding, J., Binder-Macleod, S.A., 1997. A mathematical model that predicts skeletal muscle force. IEEE Transactions on Biomedical Engineering 44, 337–348. Zajac, F.E., Young, J.L., 1980. Properties of stimulus trains producing maximum tension-time area per pulse from single motor units in medial gastrocnemius muscle of the cat. Journal of Neurophysiology 43, 1206–1220.
Journal of Biomechanics 40 (2007) 1066–1072 www.elsevier.com/locate/jbiomech www.JBiomech.com
Prediction of summation in incompletely fused tetanic contractions of rat muscle Brian R. MacIntosha,, David Jonesa, Andrea N. Devromea, Dilson E. Rassierb a
Human Performance Laboratory, Faculty of Kinesiology, University of Calgary, Calgary, Alta, Canada T2N 1N4 Department of Kinesiology and Physical Education, Faculty of Education, McGill University, Montreal, Que., Canada
b
Accepted 23 April 2006
Abstract Summation is the accumulating contractile force resulting from sequential activations applied to a muscle without sufficient interval to permit complete relaxation. The purpose of this study was to evaluate summation in the rat medial gastrocnemius muscle, and to determine if the contractile responses during summation could be predicted from the relationship between force and activation pattern. In the first part of this study, the consistency of summation in the rat gastrocnemius muscle was assessed and prediction equations were derived. The second part compared predicted summation with actual contractions obtained in a new set experiments. Summation was assessed by calculation of the contractile response, per stimulation, for up to five stimulating pulses at these frequencies: 20, 40, 60 and 80 Hz. This was done by subtraction of the force transient for j1 pulses of stimulation (where j ¼ 1–5 pulses) from the force response with j pulses of stimulation. Each of these force differences was evaluated for peak rate of force development, contraction time and half-relaxation time. Contraction and half-relaxation times changed by only a small magnitude from values obtained for the twitch. Peak rate of force development was proportional to the active force for all force transients obtained by subtraction. The force per activation increased from the first to the fifth stimulus, and was dependent on interpulse delay. In the second series of experiments, the predicted force was related to the actual force for brief tetanic contractions at 40, 50 and 60 Hz (r2 ¼ 0.875). These experiments demonstrate that the force response to sequential activations is consistent and predictable. Summation can be predicted, knowing only the amplitude of the twitch contraction and the relationship between delay and force for each activating stimulus. r 2006 Elsevier Ltd. All rights reserved. Keywords: Muscle; Contractile response; Force–frequency relationship; Twitch
1. Introduction The twitch is a transient rise and fall of active force in response to a single stimulus, and is considered to be the elementary contractile response in skeletal muscle (MacIntosh and Gardiner, 1987; MacIntosh et al., 2006). When a second stimulus is given shortly after the first, a similar twitch-like force transient is elicited Corresponding author. Tel.: +001 403 220 3421; fax: +001 403 220 0105. E-mail address: [email protected] (B.R. MacIntosh).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.04.009
(Stein and Parmiggiani, 1981; MacIntosh et al., 1988). This contractile response to the second stimulus is added to the declining force of the first, a process called summation. The twitch-like force transient obtained with any stimulus in a train is not the same as that observed for a twitch (Raikova et al., 2006). It is generally greater in magnitude. Summation is therefore a nonlinear process (Burke et al., 1976; Zajac and Young, 1980; Stein and Parmiggiani, 1981). It would be desirable if the force response for a given train of activating pulses of stimulation could be predicted. This nonlinearity makes prediction of force summation more
ARTICLE IN PRESS B.R. MacIntosh et al. / Journal of Biomechanics 40 (2007) 1066–1072
challenging, but if the pattern of change in magnitude of force transient is consistent, then prediction should be feasible. The magnitude of the contractile response can be presented as either the force produced by the muscle, or the force–time area. Burke et al. (1976) evaluated the contractile response of cat medial gastrocnemius muscle motor units, using the force–time area as the criterion measure of the contractile response. Their rationale for this approach was that the force–time area of a skeletal muscle contraction is thought to be proportional to the energy demand of the contraction. This has been shown to be the case for cardiac muscle (Alpert and Mulieri, 1982). However, the energetics of cardiac and skeletal muscle are very complex and the force–time index is not likely the best correlate of the energy cost of contraction in skeletal muscle (Hisano and Cooper, 1987; Ameredes et al., 1998). Instead, the number of activating pulses is a reasonable predictor of the energy cost (Fales et al., 1960). This removes a potential advantage for using the force–time area as a measure of the contractile output. According to Burke et al. (1976), because the force– time area is also known as the impulse, or the potential for disturbing the motion of an object on which the force is acting, this method of quantifying the contractile response is useful. However, this notion can be misleading; any impulse that relates to the potential change in motion of an object must refer to a dynamic contraction. Therefore it would not be appropriate to consider the force–time area of an isometric contraction to be related to the potential for disturbing motion. Furthermore, a given force–time area can be obtained with a high force acting for a short time, or a lower force acting for a longer time. This complexity would make prediction of this contractile output during trains of stimulation difficult, because there would be an indeterminate number of possibilities. It would be more appropriate to consider force as the primary output variable; the duration of force production can be modulated by changing the duration of stimulation, but often it is the force resulting from muscle activation that is targeted in voluntary muscle control. There have been limited studies dealing with force summation in a quantitative way, and none have attempted to predict the contractile force from the relationships between interpulse interval and number of stimulating pulses. This is surprising, as prediction of the contractile response for a given stimulation pattern is important to skeletal muscle modeling (Wexler et al., 1997; Ding et al., 2000) and functional electrical stimulation (Stein, 1999). The purpose of this study was to evaluate summation in the rat medial gastrocnemius muscle, and determine if the contractile response could be predicted from the relationship between force and activation pattern.
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2. Methods Sprague–Dawley rats weighing 190–260 g were anesthetized with ketamine/xylazine (85 and 15 mg/kg, respectively), and the gastrocnemius muscle was surgically isolated in situ for attachment of the Achilles tendon to an isometric force transducer. The plantaris and the soleus muscles were cut from the Achilles tendon and peeled back to prevent their force contribution. The distal stump of the cut sciatic nerve was placed over a pair of stainless steel electrodes. The femur and tibia were immobilized by pins that were affixed to the myograph base to prevent motion at the origin of the muscle. Additional anesthetic was given during the experiment if needed. At the end of the experiments, rats were killed by anesthetic overdose. The Achilles tendon was attached to a force transducer that was mounted on a rack and pinion device that permitted precise control of muscle length, and this was set at the length that yielded the greatest apparent active force (difference between peak total force and passive force with double-pulse stimulation, delay ¼ 5 ms). The experiments were conducted at this reference length. The loosened skin was pulled up around the muscle, forming a container that was filled with warmed paraffin oil. This oil and the rectal temperatures were kept near 37 1C with radiant heat. Stimulation of the sciatic nerve was performed with supramaximal 50 ms square pulses, at the specified intervals (Grass model S88). In the first series of experiments (n ¼ 7), trains of stimulation consisted of 1–5 pulses, and the interpulse intervals ranged from 12.5 to 50 ms, corresponding to stimulation frequencies of 20, 40, 60 and 80 Hz. The time between sequential contractions was 1 min to avoid complicating issues associated with fatigue and activity dependent potentiation (MacIntosh and Willis, 2000). These contractions were used to estimate the expected force per stimulus (see analysis below) and to see if the force transients associated with sequential stimulation resulted in a predictable change in the time-course of the force transient per activation. In a second series of experiments (n ¼ 14), similar contractions with 1–5 pulses of activation were obtained at frequencies of 40, 50 or 60 Hz. The force of these contractions was compared with the estimated active force for simulated contractions, that were based on an equation for the twitch contraction and the relative force per activation as estimated from the first series of experiments. The equation for the twitch was based on MacIntosh and Heinemeyer (1994): Active force ¼ kf ðt^ ðaf 1Þ e^ ðt=bf ÞÞ þ ks ðt^ ðas 1Þ e^ ðt=bs ÞÞ,
ð1Þ
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Table 1 Constants for twitch contraction simulation Fiber type
Fast
Slow
k b a
3.23106 4.50103 3.80
20.5 2.0102 2.20
where k, a and b are constants, t is time; subscripts f and s refer to fast and slow-twitch motor units, respectively. The force of the simulated twitch was essentially the sum of the force of fast and slow-twitch motor units. The force contribution of the slow-twitch motor units was set at 7% of the peak force, because it has been reported that about 7% of the cross-sectional area of the rat gastrocnemius muscle is composed of slow-twitch muscle fibers (Ariano et al., 1973). Table 1 presents the values for the constants used in the equation. 2.1. Data analysis Contractions obtained in the first series from each sequence of 1–5 stimulating pulses at a given frequency were analyzed to obtain the active force per activating pulse. This was done by subtracting the force profile of a contraction with j1 stimulating pulses from the force profile with j stimulating pulses (see Fig. 1) where j refers to the number of stimulating pulses. These twitch contractions and active force per activation obtained in the first series were fit to the following equation for each sequential activating pulse: force ¼ ða dÞ=ð1 þ ðt=cÞ^ bÞ þ d,
(2)
where a, b, c and d are constants, obtained by least squares fitting of the data and t is time interval or delay between stimulations. The data for second, third, fourth and fifth force transients were fit to this equation for the frequencies ranging from 20 to 80 Hz. The resulting equations were used to estimate the individual contraction amplitude for the second, third, fourth and fifth stimulation and these were added with appropriate delay to simulate contractions for sequential activation at 40, 50 and 60 Hz. The predicted force transients were created by scaling the standard equation for the twitch to the amplitude of the observed twitch, and scaling the sequential responses according to Eq. (2) for the appropriate delay (corresponding to 40, 50 and 60 Hz). The result was then added with the corresponding delay to create a predicted contraction for comparison with the contractions obtained in the second series of experiments. The force transients per activation that were obtained by subtraction (Series 1, see above) of the contraction with 1–4 pulses of stimulation from the contraction with one more pulse of stimulation were evaluated for
Fig. 1. Five contractions (from 1, 2, 3, 4 and 5 pulses) are superimposed in A and C. These contractions were obtained with 25 (A) or 16.67 (C) ms delay between pulses of stimulation. Subtraction of the force profile of the twitch from the force profile of the doublet, the doublet from the triplet, etc. yields the force transient for the second pulse, the third pulse, etc. as shown in B and D. Note that sequential activations at 60 Hz elicit much greater force increments than sequential activations at 40 Hz.
contraction time, half-relaxation time and peak rate of force development. This was done to see how consistent these parameters were for transients derived from sequential activations. If these parameters of the force profile changed substantially within an unfused tetanic contraction, then prediction of force in the manner proposed (vertical scaling of the twitch) would not provide a good estimate of peak active force. In fact, this method would only be effective if the amplitude of the force transient remained proportional to the peak rate of force development. Contraction time was measured from the beginning of force development to the peak of the contractile response. Half-relaxation time was measured from the peak of the contractile response until force decreased to 50% of the active force. Peak rate of force development was measured as the greatest change in force across 2 ms on the rising phase of the force transient. Statistical evaluation was performed with linear and non-linear regression analyses, and single factor ANOVA with repeated measures. When significant differences were detected (a ¼ 0.05), for contraction time and half-relaxation time, Newman Keuls test was used to identify the specific locations of these differences.
3. Results Sample contractions of the gastrocnemius muscle are shown in Fig. 1 for 1–5 pulses of activation at 40 Hz (1A) and 60 Hz (1C). Five contractions are superimposed to illustrate the reproducibility of the peaks
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Fig. 2. Contraction time (time to peak force, open symbols) and halfrelaxation time (solid symbols) for the force transients resulting from each of 1–5 pulses of stimulation at 20–80 Hz are shown. Only minor changes were evident. Results are mean values with some standard errors to illustrate variability.
within a series of stimulations. The lower panel (1B and 1D) in each case shows the force for each activation, as obtained by subtraction of the active force for j1 stimulations from the active force for j stimulations. Contraction time for force transients obtained for 1–5 pulses of activation at 20–80 Hz is shown in Fig. 2. Significant changes were observed for 40 Hz (transient number 1 vs. 4 and 5 as well as 2 vs. 5) and 60 Hz (transient number 3 vs. 5) stimulation, while contraction time for force transients at other frequencies did not change. Although these differences were significant at 40 and 60 Hz, the magnitude of change was less than 10%. Half-relaxation time for force transients obtained for 1–5 pulses of activation at 20 to 80 Hz is shown in Fig. 2. Values for half-relaxation time vary from about 10.6 ms for the first contraction transient to 11.4 or 9.7 ms for later transients. This difference was significant only for 20 Hz, and only for the difference between the fifth transient and the first or second transient. The maximum magnitude of difference was less than 10%. Whereas there were only minor changes to the contraction time and half-relaxation times, there were substantial and predictable changes to the peak rate of force development. Changes in this parameter would appear to be the primary mechanism by which twitch force is modulated in sequential activations. Peak rate of force development changed in proportion to the developed force across all frequencies and for all five transients within a contraction (Fig. 3). The amplitude of a force transient within a contraction was dependent on the delay between sequential stimuli as well as the number of stimuli preceding. This is illustrated in Fig. 4. The constants that best fit the
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Fig. 3. The relationship between peak rate of force development and active force of the force transient for each of 1–5 sequential activations delivered with 16.67–50 ms delay. The strong linear relationship typically observed for twitch contractions (MacIntosh and Gardiner, 1987) is preserved for force transients within an incompletely fused tetanic contraction.
Fig. 4. Active force of the force transients resulting from the 2nd (square), 3rd (diamond), 4th (cross), and 5th (circle) pulse of sequential activation plotted against delay. Symbols represent mean values and lines represent the best fit according to the equation: Force ¼ ða dÞ=ð1 þ ðt=cÞ^ bÞ þ d and the constants presented in Table 2.
data to the equation are given in Table 2. These equations, with the appropriate constants, were used to predict the force for 40, 50 and 60 Hz contractions and these predicted contractions were compared with contractions obtained in the second series of experiments. The input variable was the amplitude of the observed twitch, and the predicted force was compared with measured force at common frequencies. The force per activating pulse increases with the number of stimulating pulses, and decreases with increase in the
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delay (interpulse interval). Fig. 5 presents a series of modeled contractions at 40, 50 and 60 Hz and the corresponding actual contractions that these simulated contractions are intended to predict. The result of the comparison between predicted and actual contractions is shown in Fig. 6. The correlation (r) for this relationship was 0.93. This represents significant (po0.05) agreement between the predicted and actual contractions. 4. Discussion This study has revealed a unique relationship between active force and delay between sequential activations for up to 5 stimulating pulses, similar to that which has been shown for the force–time integral (Zajac and Young, 1980). This information was then used to obtain a reasonable estimate of the active force for incompletely fused tetanic contractions. This study confirmed that the force transient associated with each activation in a train of pulses can be determined by subtraction of the force for a contraction elicited with one less stimulation. These force transients have a shape that is similar to that of a twitch contraction, and the contraction and half-relaxation times remain reasonably constant over the range of frequencies and for the train duration
evaluated here. The magnitude of these force transients varied with the time interval between activations and the number of pulses but remained proportional to the peak rate of force development. This meant that prediction of summation could be accomplished with vertical scaling of a standard twitch. The fact that summation is more than the linear addition of identical twitches is referred to as nonlinear summation. This term has been used for two phenomena in the study of skeletal muscle: summation of sequential activations and parallel summation of motor unit activations. The nonlinear summation of parallel motor unit activations is a consequence of complex connective tissue structure and parallel as well as series elastic properties of muscle (Troiani et al., 1999; Sandercock, 2000). However, this is unlikely the factor associated with nonlinear summation of sequential activation, as activation at 40 Hz, which begin at progressively higher levels, would not have a similar magnitude, as shown in Fig. 4.
Table 2 Constants for prediction equationsa Pulse
2nd
3rd
4th
5th
a b c d
4.65 26.4 25.4 2.95
5.84 13.6 24.0 3.01
6.60 13.1 22.6 3.04
7.27 9.4 20.5 3.07
a
The following equation was used to predict the amplitude of force transients: Force ¼ (ad)/(1+(t/c)b)+d, where force is absolute active force in newtons and a, b, c and d are from the table and t is delay (ms) between sequential stimulations. These values for the constants were derived from test contractions at 20, 40, 60 and 80 Hz. The average active force of the initial transient (twitch) was 3N.
Fig. 6. Predicted active force and corresponding measured active force for brief incompletely fused tetanic contractions at 40, 50 and 60 Hz. The regression (green) line is very close to the line of identity (black), indicating agreement between the predicted and measured contractions.
Fig. 5. Predicted (light line) and sample contractions at 40, 50 and 60 Hz, from one experiment are shown to illustrate the agreement between actual contractions (dark lines) and those based on the standard twitch (Eq. (1)), and subsequent responses scaled according to the amplitude values predicted from Eq. (2). These contractions are scaled relative to the initial force transient (what would otherwise be the amplitude of the twitch).
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The mechanism for nonlinear summation of sequential activations is not known, but it seems reasonable to speculate that it is related to factors associated with Ca2+ handling: release, binding and uptake. The increment in Ca2+ bound to troponin C for a given stimulation will depend on the initial free [Ca2+], the amount of Ca2+ released from the sarcoplasmic reticulum, the occupation of alternative binding locations (Ca2+ buffer sites) and the binding of Ca2+ to troponin C. A given amount of Ca2+ released from the terminal cisternae causes more or less increment in binding to troponin C depending on the Ca2+ binding sites that are already occupied and the rate at which Ca2+ is removed from the cytoplasm. Twitch active force, contraction time and halfrelaxation time were highly reproducible during our experiments. These parameters did not change substantially over the course of 5 stimulating pulses, for frequencies corresponding to: 20, 40, 60 and 80 Hz. This fact is an important prerequisite for prediction of force using the approach described in this study. If the contraction or relaxation times had changed, then the standard twitch used in this study could not have been simply scaled according to Fig. 4 to sum and create the incompletely fused tetanic contractions. It is quite likely that half-relaxation time and contraction time would change with additional activating pulses (more than 5), since they change with repeated activation at 10 Hz (MacIntosh et al., 1994). Changes in the contraction time and half-relaxation time for sequential activations within incompletely fused tetanic contractions of motor units have been reported (Raikova et al., 2006). Fast fatigable motor units changed in one direction while changes were in the opposite direction in the fast fatigue resistant motor units (Raikova et al., 2006). This may explain why we saw only minor changes in contraction time and halfrelaxation time over the first five activations. In contrast with our approach, Raikova et al., 2006 did not obtain sequential force transients by subtraction of contractions with one less activation; they used an iterative modeling procedure to obtain an estimate of the force transient for each activation. A limitation of our model is that it cannot account for modulation of the force transient when sequential incompletely fused tetanic contractions occur within a short period of time. Staircase, a form of activity dependent potentiation would be expected if contractions were repeated at short intervals (MacIntosh and Willis, 2000), and fatigue would be a likely consequence of continued intermittent contractions (MacIntosh and Rassier, 2002). It is known that contraction and relaxation times change with continued repetitive stimulation (MacIntosh et al., 1994), and that would cause the approach used here to give incorrect values. Activity dependent potentiation and fatigue were
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intentionally avoided by permitting a minute rest between contractions in this study. Further work will be needed to permit these factors to be accounted for in a model of this nature. The synchronous activation of motor units described in this study is quite different from the asynchronous activations of motor units typically seen in voluntary motor tasks. Our results provide insight into the nature of summation of sequential activations for whole muscle, but additional work with single motor units is needed to better understand the differences between fiber types and how these motor units sum together to give a smooth voluntary contraction. Some progress in this effort has been made by Celichowski and colleagues (Celichowski et al., 2005; Celichowski et al., 1999; Raikova et al., 2006). The approach taken in this paper is useful to characterize the properties of summation of any muscle of interest, and the result can be used to predict the force that would be expected for a given pattern of stimulation. This information would be useful in modeling contractile response, and possibly for pre-calculation of anticipated force in functional electrical stimulation. Functional electrical stimulation is a technique that has been used to overcome paralysis since the 1970s, but there are still several barriers to its effective use, including a limited understanding of summation in skeletal muscle. The current information could be used to formulate the appropriate number of pulses and corresponding frequency of activation required to achieve a given force. Acknowledgements This research was supported by NSERC, Canada.
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