OOZI-9290192SS.oO+.OO ‘(:” 1992 Pergamon Press plc J. Biomechonics Vol. 25, No. 4, pp. 421428. 1992. Printed in Great Britain MODEL OF MUSCLE-TENDON INTERACTION DURING FROG SEMITENDINOSIS FIXED-END CONTRACTIONS RICHARD L. LIEBER*, CYNTHIA G. BROWN and CHRISTINE L. TRESTIK Department of Orthopaedics and Rehabilitation, Biomedical Sciences Graduate Group, Veterans Administration Medical Center and University of California, San Diego, CA 92161, U.S.A. Abstract-A structural model was developed to explain sarcomere shortening at the expense of tendon lengthening in the frog semitendinosis (ST) muscle=-tendon system. The model was based on the data of Lieber et al. [Am. J. Physiol. 261, C&C92 (1991)], who determined the relationship between the sarcomere length, tendon load (as a fraction of maximum isometric tension) and tendon, bone-tendon junction (BTJ), and aponeurosis strain. The model was generated assuming a finite time-course of cross-bridge attachment [Huxley, Prog. Biophys. 7,255-318 (1957)], an ideal sarcomere length-tension relationship [Gordon et al., J. Physiol. 184, 170-192 (1966)] and an ideal force-velocity relationship [Katz, J. Physiol. %, 454 (1939); Edman, J. Physiol. 291,143-159 (1979)]. Functionally, sarcomeres operated on three distinct regions of the length-tension curve: (1) regions where the muscle force decreased as sarcomeres shortened (the shallow and steep ascending limbs); (2) regions where the muscle force increased as sarcomeres shortened and there was little passive tension (descending limb, where sarcomere length < 3.0 pm); and (3) regions where the muscle force increased as sarcomeres shortened and there was a significant passive tension (descending limb where sarcomere length > 3.0 Pm). Using such a physiological model, it was found that the effect of tendon compliance was to ‘skew’ the sarcomere length-tension curve to the right and to increase the operating range of the muscle-tendon unit. Thus, maximum tension in the muscle occurred at an active sarcomere length of 2.0-2.2 pm, whereas in the muscle-tendon system, the maximum tension occurred at a longer resting sarcomere length of about 2.5 Pm. The degree to which the tendon affected the muscle system depended on its material properties and dimensions. These data suggest that tendons are not merely rigid links connecting muscles to bones, but impart distinct properties to the muscular system. INTRODUCTION models exist which describe the relationship between skeletal muscle and force production. Such models range from formulations of cross-bridge attachment and detachment rates (Huxley, 1957; Squire, 1990) to phenomenological models of muscle force output as a function of length, activity, and velocity input (Hatze, 1973; Zajac, 1989). In all models muscle force varies as a function of length (Gordon et al., 1966) and velocity (Katz, 1939), which is expected for the muscle contractile component. However, a whole muscle is not simply an amplified sarcomere. Muscle has significant series elasticity within and outside it (Morgan, 1976; Rack and Westbury, 1984). In fact, recent studies demonstrated that skeletal muscle-tendon units may have unique properties compared to the properties of muscle alone (Walmsley and Proske, 1981; Zajac, 1989; Hoffer et al., 1989). Therefore, models which are useful in describing normal movement must account for both muscle and tendon properties as well as their interaction. The recent mammalian muscle-tendon model by Zajac (1989) is generic in that it was designed to apply to any muscle-tendon actuator given appropriate scaling factors. While it admirably accomplishes its purpose, it is not possible to simply apply that model to any muscle-tendon unit in any species since the issues of Numerous Received infinalform 25 July 1991. *Author to whom correspondence should be addressed. scaling and species specificity come into play (Schmidt-Nielsen, 1984). Since we were interested in the behavior of the frog semitendinosis (ST) during normal locomotion (Mai and Lieber, 1990), our purpose was to develop a model for this particular muscle-tendon actuator that was based on experimental data in order to determine the tendon’s influence in this system. A brief report of this work has appeared elsewhere (Lieber and Leonard, 1989). METHODS Biomechanical experiments The model chosen for this study was the dorsal head of the frog semitendinosis (ST) muscle-tendon unit (Rana pipiens). This model was chosen based on the muscle’s well-established sarcomere length-tension properties (Gordon et a/., 1966) and previous studies establishing the relationship between muscle and joint properties (Lieber and Boakes, 1988; Mai and Lieber, 1990). Frogs were sacrificed by double pithing (n= 14 independent experiments) and the ST-tendon unit was carefully isolated along with its attachments ko the pelvis and tibia. The bones of the bone-muscletendon (BMT) unit were clamped to specially designed fixtures which permitted viewing of the bone-tendon interface while maintaining secure contact with the BMT unit (Lieber et al., 1991). The BMT unit was submerged in chilled Ringer’s solution adjusted to pH = 7.0. One clamp was fixed to the moving arm of a 421 422 R. L. LIEBERet al. A B Contractile Component (CC) Parallel Elastic Component (PEC) Fig. 1. (A) Schematic diagram of the frog semitendinosis muscle-tendon unit drawn to scale. Values shown at the top of the figure are lengths in mm (mean + SD. for 14 specimens) of the muscle fiber, aponeurosis, tendon, and bone-tendon junction (BTJ). Note that the ratio of muscle fiber to connective tissue is 1.5 (calculated based on relative lengths as { [2.8 x 2]+ [2.1 x 23 + 5.5}/10.5), rendering this system relatively ‘stiff’ as defined by Zajac (1989). (B) Mechanical analog representing theoretical model. Muscle contractile component with ideal length-tension and force-velocity properties is represented by a schematic sarcomere. servo motor which permitted simultaneous control of force and measurement of displacement (Cambridge Technology Model 310, Watertown, MA). Dye lines (elastin stain) were applied at intervals along the BMT unit partitioning it into three regions: a region containing the bone-tendon interface (referred to as the bone-tendon junction), a region containing only the bare tendon (tendon), and a region containing the muscle-tendon junction (aponeurosis). Boundaries between these regions were defined somewhat arbitrarily based on morphological appearance. Muscle length was set to L,, the length at which twitch tension was maximal. This occurred at a nominal sarcomere length of 2.45 pm, approximately in the midpoint of sarcomere lengths achievable in the frog semitendinosis with various hip and knee joint configurations (see Fig. 4A of Mai and Lieber, 1990). Passive tension at this length was near the noise level of the transducer (about 100 pg). Following the measurement of maximum tetanic tension (P,), muscles were passively loaded to P, and the strain (a) was measured in three different regions of the connective tissue (Fig. 1A): the muscle-tendon junction (aponeurosis), the tendon, and the bone-tendon junction (BTJ). The average load-strain function for each connective tissue region was calculated (Fig. 2) and it was determined that there was no significant difference between the tendon and bone-tendon junction regions. dual-mode 0 I 2 3 1 6 6 6 Fig. 2. Average load-strain relationship for the three different connective-tissue regions studied. In this experiment, the aponeurosis was significantly more compliant than either the tendon or bone-tendon junction (P<O.O5), which were not significantly different from one another. At maximum tetanic tension, the average strain in the aponeurosis, bone-tendon junction, and tendon were 8, 3.4, and 2%, respectively. series arrangement of muscle fibers with two lengths of tendon and one length of aponeurosis (Fig. IB). The equations describing the passive properties of each of these components were: Tendon: p,(%p,)= 10(~+0.633)/1.35, Aponeurosis: ~~(%~~)=10(~+3.63)/5.66, Model assumptions In developing the model for fixed-end muscle contraction, each muscle-tendon unit was modelled as a 7 Strain (%) (I) (2) Muscle: ~,,,(~~p~)= IO~(sL-~~~O)/l.O7~-2.14, (3) Model of frog muscle-tendon where E represents the strain, %P, represents the relative maximum tetanic tension, and SL represents the sarcomere length (in pm). Since the passive properties of the connective tissue were known [Table 1, equations (l)-(3)], the remainder of the model was generated based on the following assumptions: (1) Muscle fibers have an ideal length-tension relationship as described by Gorden et al. (1966). (2) Muscle fibers have an ideal force-velocity relationship for shortening as described by Katz (1939), Edman (1979) and Morgan et al. (1982), where the force-velocity constants are: a = 0.25 and b = 0.25. (3) Connective tissue in series with muscle fibers has load-strain properties according to Lieber et al. (1991) [equations (1) and (2)]. (4) Muscles have a sarcomere-length-passivetension relationship according to Lieber et al. (1991) [equation (3)]. (5) The time course of muscle activation follows the time course of cross-bridge attachment as indicated by Huxley’s parameterf, for frog skeletal muscle at 12°C (Huxley, 1957). For the strains observed during fixedend contractions, the parameter g would be negligible. The cross-bridge attachment rate was scaled so that the muscle would be maximally activated after 100 ms according to the equation: Activation fraction = No muscle series elastic component (SEC) was incorporated into the model. This was because there is some uncertainty as to the physical location of frog muscle SEC (e.g. compare Julian and Morgan, 1981 and Ford et al., 1980). In addition, since the displacement required to drop the muscle fiber force to zero is only 3-4 nm/half sarcomere (Ford et al., 1977), we felt that muscle SEC would be dominated by tendon and aponeurosis compliance. Of course, this model structure will thus tend to underestimate sarcomere shortening slightly. We thus bias the results toward showing no effect of tendon compliance rather than overemphasizing it. However, tendon and aponeurosis strain will not be affected by this assumption. Using the above assumptions, a FORTRAN program was developed (Fig. 3) which simulated a fixedend tetanic contraction. The logical program flow proceeded as follows: following initialization of the experimental parameters such as connective tissue material properties, sarcomere length and muscle-tendon dimensional quantities (Fig. 3, box l), resting tendon, aponeurosis and muscle fiber lengths were calculated using experimentally obtained load-strain values (Fig. 3, box 2). The muscle was then activated according to the data of Huxley (1957) (Fig. 3, box 3). As the muscle developed tension appropriate to the specified sarcomere length, a certain amount of passive tension resisted sarcomere shortening according to assumption (4) (Fig. 3, boxes 46). This tension strained both the tendon and aponeurosis according interactions 423 Table 1. Muscle-tendon properties used in our model* Parameter Value Muscle properties Muscle length (mm) Fiber length (mm) Sarcomere number Maximum tetanic tension (N) Sarcomere length at L, (pm) TendonJAponeurosislBTJ 22.5_+1.7 10.5+ 1.4 4781+647 0.366+0.17 2.45 + 0.06 properties Tendon length (mm) BTJ length (mm) Aponeurosis length (mm) Tendon Young’s modulus at P, (MPa) 2.11 +0.63 2.80 +0.52 5.51 f 1.10 188+?1 *Values shown represent the mean k standard deviation for the same 14 specimens for which stress-strain data were obtained. Physiological CSA is the muscle physiological cross-sectional area as calculated using the equations of Sacks and Roy (1982). BTJ refers to the bone-tendon junction region as described in Methods section. P, is the maximum tetanic tension, Lo is the muscle length at which the tension is maximum. Data from Lieber et al. (1991). to their relative stiffnesses and was distributed accordingly (Fig. 3, boxes 4,5). This tension was compared to the isometric tension for that sarcomere length and level of activation to determine the relative isometric tension (Fig. 3, box 7). Using the force-velocity relationship (Katz, 1939), sarcomere shortening velocity was then calculated (Fig. 3, box 8) and the new sarcomere length calculated (based on the sarcomere shortening velocity and time interval of 0.01 ms). This process was repeated in 0.01 ms increments (no significant difference in the results was obtained for time intervals ranging from 0.00-1.0 ms, data not shown) and sarcomeres continued to shorten until muscle contractile force was equivalent to the resistive force of the connective tissue. At this point, the velocity was equal to zero (Fig. 3, box 10). Throughout the activation scheme, sarcomere length and velocity, muscle active tension, muscle passive tension, and tendon tension were recorded to permit complete description of sarcomere shortening at the expense of tendon lengthening during fixed-end contractions. Sarcomere shortening at the expense of tendon lengthening was calculated for sarcomere lengths ranging from 1.3-3.7 pm for the ST and, for comparative purposes, was also calculated for muscle-tendon systems with different lengths of tendon and aponeurosis. RESULTS As expected, sarcomere shortening and magnitude were dependent on initial sarcomere length. At very long sarcomere lengths, relatively little shortening occurred while at shorter sarcomere lengths, shortening often exceeded 0.2 pm (Fig. 4A). At very long sarcomere lengths where passive tension was appreciable (e.g. sarcomere lengths greater than 3.1 pm), as 424 R. L. LIEBERet al. I Initialize Parameters Q I .I 1 I Calculate Length ok Tendon Aponeurosis I Musde l l 21 l Begin Activation I l l l Tendon Aponeurosis Muscie f-l1 5 Calculate Total Tension 6 Calculate Relative Isometric Tension 7 4 Calculate Isotonic ShorteningVelocity 8 1 Calculate New SarcomereLength I Yes Fig. 3. Flow chart of muscle-tendon contraction algorithm. See text for calculation details. Model of frog muscle-tendon interactions 425 1207 i? .**. 100. . 8. 80. l . . . . s 'B 5 60. /\ tp . . . 40. a 6 20. 0 0 100 200 10 300 15 20 ++* \:. , .+.*** 2.5 30 35 40 Sarcomere Length (pm) Time (ms) Fig. 6. Sarcomere length-tension relationship for an ideal muscle fiber (solid line) and muscle fiber in series with a tendon of the properties measured (filled circles). Note that the sarcomere length-tension relationship is skewed to the right because sarcomeres which begin at lengths above the optimum (i.e. slightly longer than 2.0-2.2 pm) are allowed to shorten to higher regions of the descending limb or even onto the plateau of the sarcomere length-tension curve. Passive tension (P,,,) is shown in crosses. Sarcomere Length (pm) cause shortening Fig. 4. (A) Sarcomere length vs time for sarcomere lengths ranging from 1.3-3.7 pm in 0.2 pm increments. Ending sarcomere length is shown to the right of the trace. (B) Identical data as (A) but plotting sarcomere length vs tension. Note that sarcomeres shorten until the velocity equals zero, at which point they have ‘touched’ a point on the length-tension curve. Initial Sarcomere Length = 3.0 pm 0 loo 203 XPI Time (ms) Fig. 5. Sample time course of muscle fiber, aponeurosis and tendon length change for an initial sarcomere length of 3.0pm. Note that the aponeurosis deforms more than the tendon since it is longer and more compliant. Deformation magnitude is (AL) shown for each region to the right of the trace. sarcomeres shortened on the descending limb of the length-tension relationship, they became stronger due to the increasing filament overlap and decreasing velocity (Fig. 4B). It is interesting to note that, as sarcomeres shortened in these regions of high passive tension, the muscle contractile component (CC, Fig. 1B) was required to develop relatively large tensions to shortening since some force was lost by the of the muscle parallel elastic component (PEC, Fig. 1B). Thus, as the CC developed tension, it had to exceed the force lost by the PEC in order for shortening to occur! At shorter sarcomere lengths where, as sarcomeres shortened, they became potentially weaker (e.g. sarcomere lengths less than 2.0 pm), tension increased with shortening, primarily due to increasing activation (Fig. 4B). Using this algorithm, it was possible to plot the time course of length change in muscle, tendon, and aponeurosis for any sarcomere length (Fig. 5). Note that for a typical fixed-end contraction (e.g. at a sarcomere length of 3.0 pm) tendon strain was less than aponeurosis strain due to its higher stiffness. The relative lengthening of the two connective tissue components resulted directly from their values of relative stiffnesses which changed with load. By varying the initial sarcomere length, we constructed a sarcomere length-tension curve where sarcomere length represented that at the beginning of the contraction for a system with series compliance (Fig. 6). Note that the ‘ideal’ sarcomere length-tension curve (i.e. the one described by Gordon et al., 1966) (Fig. 6, solid line) was distorted to larger sarcomere lengths as a result of series compliance (Fig. 6, circles). Thus, in the muscle-tendon system, maximum tetanic tension (P,) would occur between sarcomere lengths of 2.3 and 2.4 pm (Fig. 6). In fact, this was not significantly different from the resting sarcomere length at P, of 2.45kO.06 pm measured by Lieber et al. (1991). At both long and short sarcomere lengths, the amount of shortening was relatively small-although for different reasons: at short sarcomere lengths, with almost no passive resistance, sarcomeres could shorten only a small amount before they entered the ‘delta state’ (Ramsey and Street, 1940). At large 426 R. L. LIEBERet al. sarcomere lengths, with high passive tension, sarcomere shortening was relatively small because sarcomeres could generate only a small amount of active tension (Pee), and because any shortening decreased muscle fiber tension due to decreased P,nc. In other words, shortening tended to decrease P,nc faster than sarcomere it increased P,,. Finally, at intermediate lengths, as sarcomeres shortened, they first increased the potential tension with shortening (on the descen- ding limb of the length-tension curve) and then decreased the potential tension with shortening (on the steep and shallow ascending limb of the length-tension curve), but in all cases shortening occurred until total muscle tension (Pc, + P,,,) was equal to the resisting force of the connective tissue (PA or PT). Thus, the relationship between the initial sarcomere length and sarcomere length change grossly resembled an inverted parabola (Fig. 7). In addition to the sarcomere length-tension curve, it was also possible to calculate a muscle-tendon length-tension curve (Fig. 8). By adding series com- I? j % 5 5 f. P 3 9 El 8 z cn o.3 . O2 . .***. l. . . . . Q’- . . . . . g . . I 1.5 * 2.0 2.5 2.0]A . . 007 1.0 pliance, the operating range of the muscle-tendon unit actually increased by about 1 mm! In other words, by having a compliant connection with the bone, the muscle-tendon unit increased its operating range to a length greater than that which would be expected based simply on the number of sarcomeres in the muscle fibers! As mentioned, the frog ST muscle-tendon actuator is considered stiff due to the short connective tissue length/fiber length ratio (1.5; Fig. 1). In order to simulate other muscle-tendon units, the ratio was varied from 1 to 15 and the resulting change in the sarcomere length (Fig. 9A) as well as the sarcomere length-tension relationship (Fig. 9B) were constructed. Note that, as connective tissue length increased, the magnitude of sarcomere shortening permitted also increased (Fig. 9A). Note also that the sarcomere length which shortened the greatest amount shifted to longer lengths as the connective tissue length increased. This is because longer sarcomeres were allowed to shorten to optimal sarcomere length (2.2 pm). Similarly, as the connective tissue length increased, the length-tension relationship skewed further to the right and narrowed as more and longer sarcomeres were allowed to shorten to or past optimal sarcomere length (Fig. 9B). 8 3.0 . . - I. ’ 3.5 4.0 Initial Sarcomere Length (pm) Fig. 7. Sarcomere length change during fixed-end contraction as a function of the initial sarcomere length. Sarcomeres which begin at very large or small sarcomere lengths do not shorten as much as sarcomeres of intermediate length. Explanation is given in text. 0 07, 20 I, 22 I. 24 I. ze I, 28 4. 30 . I.. 32 1.0 I 34 Muscle-Tendon Length (mm) Fig. 8. Muscle-tendon length-tension curve for a fiber in series with a perfectly stiff tendon (solid line) or a compliant tendon (dotted curve) as measured in the current study. The effect of tendon compliance is to increase the muscle-operating range. This is because a greater range of sarcomere lengths is allowed to shorten ‘onto’ the descending limb of the length tension curve. 1.5 2.0 2.5 3.0 3.5 40 Initial Sarcomere Length (pm) Fig. 9. (A) Sarcomere length change during fixed-end contractions as a function of the initial sarcomere length. Different curves are shown for different connective-tissue length/fiber length ratios, varying from 1 to 15 (shown above curves). As the ratio increases, a greater magnitude of sarcomere shortening is permitted. (B) Sarcomere length-tension curves corresponding to conditions shown in (A). Note that as the ratio increased, the length-tension relationship is narrowed and skewed to the right. Model of frog muscle-tendon interactions 421 Fig. 10. Sample experimental record of the error associated with the measurement of maximum tetanic tension in a muscle-tendon unit. As contractile tension develops and muscles shortening at a lower velocity is that they maintain a greater relative tension (Cans and deVree, 1987). A second consequence of tendon compliance is that the operating range of the muscle-tendon unit is increased above that which would be predicted based on the additive excursion of a number of sarcomeres which are in series. Tendon compliance allows sarcomeres to shorten ‘into’ the length range at which they can develop tension. This effect is especially pronounced for muscles with very long tendons and short muscle fibers such as the medial gastrocnemius (Sacks and Roy, 1982) or flexor carpi ulnaris (Lieber et al., 1990). This effect would probably be physiologically significant since the in viva frog ST sarcomere length at the end of the swimming stroke was reported to be 2.1 pm (Tidball and Daniel, 1986) and at the end of the hop to be 2.6 pm (Mai and Lieber, 1990). A final consequence of tendon compliance actually represents a warning to physiologists in evaluating skeletal muscle contractile components. Because compliant tendons allow muscles to shorten, they allow muscles to decrease their internal passive tension during shortening. As a result, the traditional method for the estimation of maximum tetanic tension may yield misleading information, the magnitude of which will depend on the tendon length and material properties. The traditional experiment is modelled in Fig. 10. Note that at sarcomere length 3.3 pm, some passive tension exists (P,,,) before the muscle is activated. As the muscle is activated it develops a certain ‘total tension P,,,. This total tension, which represents the sum of the active and passive tensions is used to determine the active tension (P,) according to the relationship P, = P,,, - PPss. However, there is a flaw in this logic. The flaw is that P,,, is not constant during the contraction, but actually decreases according to the passive sarcomere length-tension relation (cf. Fig. 5). As a result, P, is underestimated since Ppas overestimated (Fig. 10). The resulting was length-tension curve may be different from the actual length tension curve of the muscle’s contractile component. The magnitude of the error depends on the amount and material properties of the connective tissue. As Zajac (1989) described, a useful parameter for characterization of the muscle-tendon unit is the muscle fiber length : tendon length ratio. In mammalian systems, this ratio can vary from about 1 to about 15 (Zajac, 1989). ‘Compliant’ muscle-tendon units can be considered those with high ratios while ‘stiff’ units are those with low ratios. The frog STtendon unit has a value of 1.5 for this ratio and is, therefore, considered stiff. As the relative amount of the connective tissue increases, the relative distortion of the sarcomere properties occurs. sarcomeres shorten, internal muscle passive tension decreases (solid line at the bottom). However, if the passive tension is taken as the initial passive tension, active tension will be underestimated (open arrows) compared to the actual active tension (filled arrows). Simulation for sarcomere length = 3.3 pm and initial passive tension level = 13%P,. Acknowledgements-The authors thank Dr Felix Zajac and Scott Delp (Stanford University) for helpful discussions on different aspects of this project. This work was supported by the Veterans Administration and NIH grants AR35192 and DISCUSSION The main result of this modelling exercise was that the addition of a series compliance to a muscle, imparted properties to the muscle-tendon unit which were unique from those predicted based only on the properties of the sarcomere. These results are qualitatively similar to those predicted by Zajac (1989) and demonstrate that the relatively small but significant compliance present in the ST muscle-tendon unit has a functional significance. The assumptions of the model appear to be minimal. The assumption-that the muscle velocity at continuously changing force can be described by the force-velocity relationship has recently been confirmed in single fibers by Iwamoto et al. (1990). Numerous putative properties for tendon have been proposed from the simple transmission of muscle force to bones to the more complex storage of elastic strain energy. Several recent experimental studies have suggested that muscle-tendon units must be considered as an entity if an appropriate description of locomotion events is to be obtained. For example, Hoffer et al. (1989) reported that during level walking in cats, tendons lengthen considerabfy while muscle fibers may actually shorten! Because of the asymmetry in muscle mechanical properties (Katz, 1939) this phenomenon has important functional consequences. First, it allows the tendon to absorb much of the length change which accompanies ankle flexion. This allows the muscle fibers to remain nearly isometric, or even shorten during this yield phase. Thus, the very high tensions, which are associated with muscle fiber lengthening, are avoided. In a similar way, depending on the tendon length and material properties, as the muscle-tendon unit shortens, the tendon can actually recoil, which permits the muscle fibers to shorten at a lower velocity. It is well known that the advantage of 50 150 250 Time (ms) 428 R. L. LnZBER AR40050. The authors thank Dr Zajac for providing an advance copy of his review article. We also thank Margot Leonard for constructive comments on the manuscript. REFERENCES Edman, K. A. P. 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