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Human Performance Laboratory, Faculty of Kinesiology, University of Calgary, Calgary, Alberta, Canada
Submitted 23 September 2004 ; accepted in final form 9 December 2004
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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force-length relationship; models of skeletal muscle; series elasticity; skeletal muscle
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With the model illustrated in Fig. 1A, herein referred to as model A, it is reasonable to subtract passive force measured in the resting state at the fascicle or muscle-tendon length at which the contraction occurs (see Fig. 1C). The inherent assumption is that the contractile element shortens during a fixed-end contraction, stretching the series elastic element, without affecting the length and force contributions of the parallel elastic component. However, in the model shown in Fig. 1B (model B), when the contractile element shortens against the series elastic component, the passive force contributed by the parallel elastic structures will decrease (see Fig. 1D), and the actual active force will be greater than the difference between peak total force and initial passive force. The choice of model has potentially serious implications on the estimate of active force by a whole muscle, particularly when passive force is substantial (i.e., at long muscle lengths). However, there is a limited scientific basis available to permit selection of one of these models as the more appropriate one.
Historically, muscle physiologists have tried to restrict the conditions of their studies to situations where passive force is minimal. This is done by working at relatively short sarcomere lengths in isolated single fibers. However, in situ studies using whole muscle often necessitate working at lengths where passive force is present. Determination of the length dependence of active force of the medial gastrocnemius muscle of the rat is such a situation (see Fig. 2). Like the heart, intact skeletal muscle of this type has a steep passive force-length relationship.
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Knowing that much of the passive force in an intact muscle is provided by interstitial connective tissue and that costameres that align adjacent myofibrils at the Z disk extend into the connective tissue (25), it seems reasonable to expect passive force contribution to decrease during contraction, as sarcomeres shorten against a series elastic structure (presumably tendon and aponeurosis). Similarly, the contribution of titin to passive force would decrease as sarcomeres shorten against the series elastic structures. These structural associations with passive elastic elements are consistent with model B. However, this intuitive approach is not sufficient to permit selection between the two models.
In most cases in which passive force is evident, it is subtracted as measured at the muscle length at which the contraction occurred (see, e.g., Refs. 24, 29). Often, this manipulation is implied by reference to active or developed force without specific description of how this was calculated (19, 20). The alternative, subtracting the passive force corresponding with the fascicle (or sarcomere) length that occurs at the peak of the contraction, is technically challenging. Three approaches have been described. Laser diffraction has been used to monitor sarcomere length during contraction of cardiac muscle, permitting subtraction of a passive force corresponding with the sarcomere length reached at the peak of the contraction (27). Sonomicrometry has been used to monitor fascicle length, and passive force associated with the fascicle length reached at the time of peak total force was subtracted from peak total force (26). Others have used a photographic technique to estimate fascicle length during the contraction (2). In the identified cases in which passive force associated with conditions at the time that the peak of the contraction occurs has been estimated, there has been no attempt to explain why this approach was used.
There were two purposes to this study. The first was to determine whether the choice of model, and therefore how passive force will be dealt with, has a serious impact on the length dependence of active force for whole muscle. The second was to determine which of these models is more appropriate for whole skeletal muscle. We hypothesized that use of model B would result in a longer optimal length and a broader curve relating active force to length. The results of this study confirm the hypotheses and provide evidence to support model B (Fig. 1B). Additional experiments were done to obtain a reasonable way to correct for this problem when fascicle length measurements were not available. This was done to permit interpretation of repeated determination of the length dependence of active force in circumstances where fascicle length had not been measured.
The term "length dependence of force" is used in this report rather than "force-length relationship" because we are dealing with submaximal contractions (23). The term "force-length relationship" should be reserved for conditions of maximal activation.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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In Situ Preparation
The medial gastrocnemius muscle of the rat was used for these experiments. Rats (180–250 g) were anesthetized with a ketamine-xylazine mixture [100 mg/ml ketamine and 100 mg/ml xylazine (85:15), at a dose of 0.11 ml/100 g body mass] for the duration of the experiment and were euthanized by anesthetic overdose at the end of the experiment. All procedures were approved by a University of Calgary Animal Care Committee. Results from 14 rats are presented. The advantage of using the in situ preparation is that it provides information regarding whole muscle behavior with known stimulation parameters.
The left medial gastrocnemius muscle was surgically isolated at its insertion, as previously described (16, 17), and connected to a force transducer by a thin stainless steel wire. The sciatic nerve was dissected to clear a long segment, which was draped across a pair of stainless steel electrodes. Distal branches of the nerve, not innervating the medial gastrocnemius muscle, were severed. Steel shafts were inserted perpendicularly into the femur and longitudinally into the tibia, and these were affixed to the myograph base to prevent movement at the muscle origin. Piezoelectric crystals (1-mm diameter) were placed in the muscle at each end of a fascicle, identified by direct microstimulation. A pocket was formed with a needle (18 gauge), and the crystal was sealed in place with glue (Vetbond, 3M, St. Paul, MN). The skin of the hindlimb was pulled up around the muscle to form a bath that was filled with paraffin oil, kept at 37°C.
Fascicle length was determined by emission and detection of ultrasound pulses, controlled and measured by a sonomicrometer (Sonometrics, London, ON). This technique is based on measurement of the time from emission to detection and calculation of distance based on an assumed speed of transmission of ultrasound in muscle tissue (1,540 m/s). This technique has been extensively used for detection of cardiac and skeletal muscle fascicle lengths (e.g., see Ref. 26).
The force transducer was mounted on a translation table that could be moved by a stepper motor (Arrick Robotics Systems, Hurst, TX; MD2 dual-stepper motor) with custom software. This permitted accurate setting of muscle lengths over a broad range. Stimulation of the sciatic nerve was with supramaximal square pulses, 50 µs in duration (Grass model S88 or S48 stimulator, Grass Instruments, Quincy, MA). The muscle was initially set at a reference length (RL), defined as the length that permitted the largest apparent active force for double-pulse stimulation (delay = 5–10 ms). Apparent active force is defined here as peak total force obtained in the contraction minus the passive force at that muscle-tendon length as measured before the contraction. Double-pulse stimulation was used in this study to avoid use of forceful tetanic contractions. In preliminary experiments, we observed substantial changes in passive force after a series of 200-ms tetanic contractions. Apparent optimal length is the same for double-pulse contractions as it is for longer duration tetanic contractions (21).
Impact of Model Choice
To determine the impact of model choice on observations of muscle contractions, a series of contractions were obtained at a variety of muscle lengths with double-pulse (delay = 5–10 ms) stimulation. The response to this stimulation is referred to as a double-pulse contraction. Contractions were obtained in the following order: RL (referred to as 0), RL – 1.2, RL – 2.4, RL – 3.6, RL – 4.8, RL – 4.8, RL + 1.2, RL + 2.4, and RL + 3.6 mm of muscle-tendon length change. Computer control of the motor was used to move the force transducer to the desired position, and then a contraction was obtained and the muscle was returned to RL. This approach prevented substantial stress relaxation from occurring while contractions were obtained at long lengths. The second contraction at RL – 4.8 mm was obtained without moving the transducer back to the RL position. This was done to allow the muscle to be repositioned to the desired length with less slack and resulted in a shorter initial fascicle length and higher peak total force on the second contraction at this length. The first of these two contractions was ignored.
Determination of active force across different lengths. Passive and peak total force were measured at each test length. Corresponding fascicle length was also determined. Regression analysis (fourth-order polynomial) was used to obtain an equation relating passive force to fascicle length. This equation was used to estimate the passive force that would be expected at the fascicle length at which peak total force occurred. Active force was calculated according to the assumptions of either model A or model B. In the case of model A, it was assumed that the passive force observed at the resting fascicle length was maintained throughout the contraction, as illustrated in Fig. 1C; active force was calculated as peak total force minus the passive force. For model B, it was assumed that the force contributions of the parallel elastic component decreased to the passive force associated with the fascicle length observed at the peak of contraction, as shown in Fig. 1D; active force was calculated as peak total force minus this fascicle length-dependent passive force.
Estimation of optimal length. Active force, calculated in the two ways described above, was plotted against fascicle length observed at the peak of the contraction, and these data were fit to a third-order polynomial equation. The length corresponding to the peak of this relationship (slope = 0) was named optimal length A or optimal length B, corresponding to the method of calculating active force.
Which Model is Appropriate?
If the results from the experiment described above showed little or no effect of using the alternative model, then further experiments would not have been needed. However, since the choice of model had a substantial effect on the length dependence of active force, it is imperative to know which model is appropriate.
To permit determination of which model was appropriate for use, a simple test was devised. It is known that passive force decreases when a skeletal muscle is held at a long length. This decrease is referred to as stress relaxation (1, 9). If it is assumed that stress relaxation results from loss of elasticity of either the parallel or series elastic components (or both), then the consequence of this change in either model A or model B can be anticipated. It should be kept in mind that passive force of model A is contributed solely by the parallel elastic structures. If model A is correct, then stress relaxation cannot result in changes to length of the contractile element, and active force would be expected to remain constant as passive force decreases. For active force to remain constant, peak total force would have to decrease by the same magnitude as passive force.
In model B, the passive force is borne by both the parallel and series elastic components. If model B is correct, then stress relaxation could result from changes in elasticity of either the parallel or series elastic components (or both), and length of the contractile element may or may not change. With an increase in compliance of the series elastic element, the contractile element will shorten at the same muscle-tendon unit length. If the muscle is on the descending limb of the force-length relationship, then this should result in an increase in active force. Peak force will not decrease as much as passive force. With an increase in compliance of the parallel elastic component, the contractile element will be longer at the same muscle-tendon unit length. This should result in a change in active force; peak force would be expected to decrease more than passive force. It is important to note that the relevant passive force for model A is the passive force measured at rest at the muscle-tendon length at which the contraction will be performed. The appropriate passive force for model B is the passive force associated with the fascicle length at which peak force occurs.
A second approach was used to evaluate which model was appropriate. It had been observed in a study of the length-dependent properties in muscle fatigue that passive force decreased due to repeated stretching of the muscle (unpublished observations). Control experiments (no fatigue) from that study are presented here, because these contractions provide additional evidence relevant to the selection of model A or B. In these experiments, double-pulse contractions were obtained at a variety of muscle-tendon lengths, in the same way as described above. In addition, twitch and 50-Hz contractions were obtained at these lengths. The results of these will not be presented here, but they are mentioned because these measurements required repeated stretching of the muscle. The response to double-pulse stimulation at these lengths was reevaluated after 60 min of rest. The repeated manipulation of length resulted in decreased passive force at any long length. If model A was appropriate, peak total force would be expected to decrease by the same magnitude as the decrease in passive force. This assumes that the 60 min of rest allowed the contractile mechanisms to produce the same response in the second evaluation of the length dependence of force for double-pulse contractions.
Corrections in the Absence of Fascicle Length Measurements
To permit retrospective analysis of data available without fascicle length measurement, it was undertaken to determine an appropriate method to estimate the decrease in passive force during a series of isometric contractions. Apparent compliance was measured as the length change of the fascicles divided by the change in force during contractions with double-pulse stimulation. This value was obtained for contractions at the RL and longer in each animal. This range of lengths was used, because, below the RL, the apparent compliance increases, but this is probably due to viscosity effects preventing the full shortening of the fascicles on passive repositioning of the muscle to shorter lengths. Furthermore, it is unnecessary to correct the contractions at lengths shorter than the RL because passive force is essentially zero. These measurements were obtained before and after fatigue in three of the animals and just before fatigue in two more. The relationship between fascicle length and muscle length was found to be linear above the RL. The average slope of this relationship was used to estimate the apparent muscle length change that would correspond with a given fascicle length change. This relationship was then used to estimate the passive force contribution during the peak of the contraction, when only muscle length was available.
Assessment of compliance. The compliance of the connections between the tendon and the force transducer was measured separately. This was done by tying the wire to a fixed bar at one end and a pneumatic lever at the other end. The position of the lever and the corresponding force were measured at progressively higher levels of pressure. From these data, the slope of the force-length relationship gave compliance.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Stimulation with two pulses, 5–10 ms apart, results in a very brief tetanic contraction that has an apparent active force (peak force minus passive force at the same muscle length) that is 2.5–3 times greater than apparent active force of a twitch contraction. Sample double-pulse contractions at test lengths are shown in Fig. 3 along with the corresponding fascicle length measurements.
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The passive force-length relationship of the rat medial gastrocnemius muscle curves upward at lengths longer than the RL (see Fig. 2). Considering the steepness of this curve, a small amount of fascicle shortening during a fixed-end contraction could substantially reduce the passive force contribution, if the structures corresponding to the parallel elastic component are in series with the series elastic component. In this situation, active force would be much greater than the estimate obtained if the passive force before the contraction were subtracted from the total force.
Estimation of active force with the assumptions inherent in model B resulted in higher values for active force at long lengths than when the assumptions inherent in model A were used. There was no apparent difference between these estimates for lengths shorter than RL, but differences were significant at RL (4.2 ± 0.4 vs. 4.6 ± 0.4 N for active A and active B, respectively) and at longer lengths (see Fig. 4). There was a shift in the optimal length to a longer fascicle length and a broadening of the curve depicting the length dependence of force. Mean optimal fascicle length A was 13.1 ± 1.2 mm, and optimal length B was 14.0 ± 1.1 mm (n = 4). This difference was significant (P < 0.01).
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The outcome with respect to passive and peak total forces for double-pulse contractions, while the muscle was held at a constant length, is presented in Table 1. Passive force decreased substantially between the first and second contractions and more slowly after this, when the muscle was held at 2.4 or 3.6 mm longer than RL. Peak total force decreased as well but not enough to keep the difference constant; the difference actually increased. Fascicle length was measured in two of these experiments (see Table 2). There was no indication of a change in fascicle length at the peak of these contractions. There is no explanation for a substantial increase in active force that would be evident if model A were used. These observations are consistent with model B.
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It is apparently inappropriate to calculate active force by subtracting the passive force measured at the isometric muscle-tendon length from the peak total force obtained for that contraction. The reasonable alternative is to subtract the passive force associated with the fascicle length at which peak total force occurs. Because fascicle length is not always measured, it would be useful to be able to estimate the magnitude of fascicle length change associated with a given contraction so that an accurate estimate of active force can be obtained. It should be pointed out that the estimate of fascicle length change obtained in our experiments can only be applied to our experimental setup. However, the approach used here can be used to obtain appropriate estimates of changes in fascicle length and passive force for other whole muscle experiments.
The average change in fascicle length during double-pulse contractions at the RL and longer was 0.33 ± 0.02 mm/N. This estimate was based on 32 contractions in five animals. Fascicle length changed 0.5 mm for each millimeter of passive length increase of the muscle-tendon unit at the RL. When the passive muscle-tendon force-length relationship is known, the fascicle passive force change during a contraction can be estimated: change in fascicle length = 0.33 x DF, where DF is the difference in force from passive to peak total during a contraction. The passive force during the contraction will be equal to the passive force associated with a muscle-tendon length that is two times this change in fascicle length.
Using this average compliance and the apparent muscle length change associated with a given fascicle length change, an estimated change in passive force was calculated for contractions at each length tested. This was compared with the measured fascicle length change and corresponding estimated passive force to evaluate the precision of this approach. Figure 7 shows the resulting active forces calculated with passive force estimated in the two ways. There was very good agreement between the estimates of active force with and without fascicle length measurement. The open symbols show the active force calculated simply by subtraction of passive force measured at the initial muscle-tendon length. This active force departs from the line of identity for contractions obtained at long lengths.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Presumably, sarcomere length is the underlying structural influence on the length dependence of force. In these experiments, sarcomere length was not measured but fascicle length was. If it is assumed that sarcomere length changes are reasonably uniform along the length of each cell, then fascicle length change would reflect sarcomere length changes. It is also assumed that the measured fascicle length changes would reflect an average fascicle length within the whole muscle. This is a reasonable assumption, considering how consistent these measurements were between muscles. If fascicles across the muscle behaved in a substantially different way, then we would have expected more variability between muscles.
Most studies concerned with contractile properties of whole muscle use the approach dictated by model A; passive force measured at the initial muscle-tendon length is subtracted from the peak total force to obtain active force. In cases in which active force has been estimated at lengths where there is substantial passive force, active force would have been underestimated. Furthermore, the length that is considered optimal would also have been underestimated. Ironically, many investigators use twitch contractions to obtain optimal length. Twitch contractions continue to increase in active force beyond the length at which peak tetanic active force is obtained (20). This is known as length-dependent activation and is attributed to a length-dependent increase in Ca2+ sensitivity (15). This may be a case where two errors cancel each other. The use of twitches will result in an overestimation of optimal length, whereas subtraction of the initial passive force will result in underestimation of optimal length.
The results of these experiments indicate that the assumptions inherent in model B are more appropriate than those of model A for calculation of active force in muscles with a substantial series elasticity. There were two lines of evidence in support of this conclusion. When the muscle was held at a constant length for repeated contractions at 20-s intervals, passive force decreased due to stress relaxation, but peak total force did not decrease to the same extent (Table 1). Considering that peak total force actually increased when the muscle was held at the RL, the small increase in estimated active force that resulted with model B at longer length was not surprising. Activity-dependent potentiation could account for this small increase in active force (18). However, this could not account for the large increase that was obtained when model A was used to calculate active force, particularly considering that activity-dependent potentiation is decreased at long sarcomere lengths (22).
The second line of evidence supporting model B was associated with repeated assessment of the length dependence of contractile response when passive force had decreased. Under these circumstances, active force calculated with model A had an unexplained increase at long lengths, despite no increase in peak total force. These experiments provide strong evidence against the practice of subtracting the passive force measured at the initial muscle-tendon length from the peak total force to obtain active force. Model B would appear to be more appropriate for this purpose.
Model B is consistent with the assumption that the parallel elastic component is contributed by structures within or adjacent to the muscle fibers; these structures change in length corresponding to a length change of the fibers. This would include interstitial connective tissue, sarcolemma, and titin. Titin is a macromolecule that connects the ends of the myosin filament to the Z disk (5, 6). Although titin is structurally in series with a contractile filament, myosin, it is thought to contribute substantially to passive force at long lengths (13, 14). When sarcomere length decreases during a fixed-end contraction, strain of titin will decrease, thereby contributing less passive force. Labeit et al. (12) have shown that passive tension of skinned fibers is higher at a given sarcomere length during slow stretch in the presence of high Ca2+ (10–4 M). The possibility of cross bridges generating tension in their experiments was eliminated by dissolving actin filaments with gelsolin or treatment with 2,3-butanedione monoxime, an inhibitor of myosin ATPase. These results can be interpreted as an increased stiffness of titin during activation. If this is the case, then passive force contribution of titin would actually decrease more during an isometric contraction than we have estimated, based on the passive force-length relationship. It should be kept in mind that, in a whole muscle like the rat medial gastrocnemius muscle, several factors in addition to titin contribute to the passive tension.
Structures that are typically associated with the series elasticity of a whole muscle include the tendon and aponeurosis. Additional series elasticity is contributed in these experiments by the structures connecting the tendon to the force transducer and the force transducer itself. The magnitude of these effects could potentially be quite different between investigators and warrants further exploration. Tendon is considered relatively noncompliant, but aponeurosis contributes substantially to in-series elasticity (11). Because fibers connect to the aponeurosis across its full area, it is not intuitively obvious how this structure would behave with respect to models A and B. The results of these experiments suggest that the aponeurosis behaves as a series elastic structure, because the external connections cannot explain total fascicle shortening during a fixed-end contraction. The total compliance of these parts of our apparatus was 0.10 mm/N. Observed fascicle length change during fixed-end contractions was 0.33 mm/N. The magnitude of series elastic stretch during an isometric contraction was at least 0.23 mm/N. Additional stretch would have resulted in change of pinnation angle.
Overall, muscle-tendon length change can be accommodated by a combination of fascicle length change and change in angle of pinnation, and, in the case of model B, a change in series elastic component length can also contribute. If it is assumed that the contractile element is freely extensible, then model A would predict that muscle-tendon length change would not affect series elastic component length. Considering that fascicle length change in our experiments could only account for one-half of the length change of the whole muscle, the contribution of pinnation angle change would have to be substantial for model A. However, some of the length change could have been contributed by elastic structures in series with the fascicle, if model B is considered. This includes aponeurosis, tendon, and external connections. These structures would have been stretched as the muscle was passively elongated. Considering that the total series elastic compliance was 0.33 mm/N, a length change that increased passive force by 1 N (i.e., from RL + 2.4 mm to RL + 3.6 mm in Fig. 5) would increase series elastic component length by 0.33 mm for model B. Since fascicle length change associated with a whole muscle length change of 1 mm was 
0.5 mm, this means that pinnation angle would have changed enough to account for 0.17 mm of muscle length change. Considering that all of the difference between fascicle length change and muscle-tendon length change (0.5 mm in this example) would have to be accounted for by pinnation angle change if model A is considered, this difference lends further support to selecting model B.
These experiments were done using double-pulse stimulation. It is important to consider what implications this study has for a longer duration tetanic contraction. At the RL, the apparent active force of a 200-ms tetanic contraction is double that of the double-pulse contraction, but, at 3 mm longer than this length, the amplitude of the double-pulse contraction is only 40% less than the longer duration tetanic contraction (21) when calculated with model A. If Fig. 5 is used as a reference, then peak force for a tetanic contraction at the longest length would be 7.7 N. It can be estimated that fascicle shortening would be considerably greater in the longer tetanic contraction than in the double-pulse contraction. Therefore, the appropriate passive force to subtract from the longer duration tetanic contraction would be 0.2 N. The estimated active force for the tetanic contraction would be 5.7 N with model A and 7.5 N with model B. The choice of model could potentially have a greater impact on the length-dependent active force of tetanic contractions than we have demonstrated for the double-pulse contractions.
It is clearly important to know which model is appropriate. Model selection has serious implications for identifying whole muscle length at which active force is highest and in knowing the shape and dimensions of the relationship between active force and muscle length. These experiments have demonstrated that model B is more appropriate than model A. This means that muscle research that is conducted under conditions where passive force is subtracted should subtract the passive force associated with the sarcomere or fascicle length at which contractile force reaches a peak. This cannot be measured, so it can only be inferred by measurement of the relationship between passive force and fascicle or sarcomere length in an inactive muscle.
The results of this study support the conclusion that model B more closely represents the actual situation in whole skeletal muscle. However, it must be conceded that we know very little about the dynamics of passive force. The contributions of passive force by various structures may be different during a fixed-end contraction than during adjustment through the same range of lengths in the inactive state. This may be true for titin and could also be true for other structures contributing to the passive force. Despite this element of unknown, subtraction of the passive force associated with the fascicle length at which peak force is reached is a more appropriate method of calculating active force attributable to the contractile process.
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GRANTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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M. B. MacNaughton and B. R. MacIntosh Reports of the length dependence of fatigue are greatly exaggerated J Appl Physiol, July 1, 2006; 101(1): 23 - 29. [Abstract] [Full Text] [PDF]
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) and peak total (
) force are plotted against fascicle length measured at the start and peak of double-pulse contractions obtained at a variety of muscle lengths for a representative experiment. Compliance is represented by the slope of the dashed lines joining the passive and peak total forces of some of the contractions. The vertical solid lines represent the method of estimating passive force for a given contraction. The passive force at the fascicle length at the peak of the contraction was used to calculate active force. The bracket on the right indicates the magnitude of the correction in passive force and hence the magnitude of error that would be obtained by subtracting the passive force associated with the fascicle length at which the muscle began the contraction (for the contraction at the longest length). This error would decrease as length decreases to the reference length.
). It can be seen that calculation of active force according to model B resulted in a shift in optimal length to a longer fascicle length and a broadening of the curve.
