Effect of Muscle Length on the Force-Velocity Relationship of Tetanized Cardiac Muscle

Transcription

1 Effect of Muscle Length on the Force-Velocity Relationship of Tetanized Cardiac Muscle By Robert Forman, Lincoln E. Ford, and Edmund H. Sonnenblick ABSTRACT Cat papillary muscles were tetanized with repetitive electrical stimulation in the presence of 10 ITIM caffeine and 10 mm calcium. Velocities were measured during the plateau of tetanus with quick releases to isotonic loads. The course of isotonic shortening was independent of time in the contraction cycle for at least 2 seconds after the attainment of peak isometric force. The force-velocity relationships were measured at different muscle lengths that had been corrected for series elastic extension. These lengths ranged from 75% to 90% of the passive length from which maximum force was developed. The data were fitted by a least-squares method with hyperbolas described by the Hill equation, each for a constant corrected muscle length. The extrapolated maximum velocities and isometric forces diminished together in almost direct proportion to muscle shortening. Corrections for the load borne by the parallel elastic elements did not significantly change the relationships between isometric force, maximum velocity, and muscle length. The results can be accounted for by two mechanisms: (1) an internal load and (2) deactivation of the contractile elements. KEY WORDS parallel elastic elements series elastic elements caffeine hyperbolic fit calcium internal load cat papillary muscle The level of cardiac muscle activation appears to change during the isometric twitch contraction cycle (1-3) and to be affected by the muscle length. Previous studies of the force-velocity relationships in cardiac preparations were difficult because of an inability to tetanize the muscle: technically it is hard to measure muscle velocity under different loads at exactly the same instant in the contraction cycle and at the same sarcomere length. Extrapolations (4) or approximations (1) to From the Cardiovascular Division, Peter Bent Brigham Hospital, Harvard Medical School, Boston, Massachusetts This study was supported in part by U. S. Public Health Service Grant HE and Training Grant HL-5890 from the National Heart and Lung Institute and by a grant from the American Heart Association. Dr. Forman's current address is Cardiopulmonary Unit, Groote Schuur Hospital, Cape Town, South Africa. Dr. Ford's current address is Department of Physiology, University College London, Gower Street, London, W.C. 1, England. A brief account of some of these data has been presented in Circulation 44(suppl. II):II-89, Received January 20, Accepted for publication June 2, this ideal have been made from studies involving twitch contractions, and assumptions have been made which allow measurements to be taken at different contractile element lengths and different times in the contraction cycle (3, 5). The recent description of a method for tetanizing rat papillary muscle (6) suggested that force-velocity studies might be carried out under conditions where time in the contraction cycle could be ignored as a variable. Accordingly, the method for tetanizing rat muscles was modified to permit cat muscle tetanization. The effects of muscle length on the force-velocity relationship were studied in this preparation, and the results were interpreted to distinguish between the various mechanisms which might affect this relationship. The effect of the parallel elastic element on the results is analyzed in the Appendix. Methods Papillary muscles were dissected from die right ventricles of adult cats anesthetized with sodium pentobarbital ( mg, ip). The muscles were Circulation Riittreb, Vol. XXXI, August

2 196 FORMAN, FORD, SONNENBLICK mounted in physiologic buffer of ph 7.4 with a millimolar composition of CaCl 2 2.5, KC1 4.7, MgSO 4 2.4, KH;,PO 4 1.2, NaCl 118, dextrose 5, and NaHCO The solution was bubbled with a mixture of 95* O 2-5!t CO 2 and maintained at C. APPARATUS The apparatus described previously (7) was modified. The muscles were mounted vertically with the upper tendonous end held in a stainless steel clip attached to an isotonic lever system (8). Counterweights were suspended from the lever by an elastic band used to isolate the inertia of the weights from the lever (9). The equivalent mass of the lever and clip was 95 mg. A proximal portion of the lever interrupted a light beam falling on a photodiode that was arranged in a Wheatstone bridge circuit to measure displacement. The velocity of displacement was obtained by differentiating the length trace with an operational amplifier in a circuit having R d = 1.0 Mohm, R, = 3.0 Mohms, C rt = 1.0 yxfarad, and C, = /iiarads (10). Following a step change in the input voltage, the output of the differentiator returned to zero with a time constant of 2.8 msec. The length and velocity traces were displayed on the screen of a Tektronix 564 memory oscilloscope, using a sweep rate of 200 msec/cm, and recorded photographically. Initially the lever was held against a stop by an air jet. The tetanized muscle was released to an isotonic load by interrupting the flow of air with a solenoid-controlled valve. The release was slowed to occur over approximately 5 msec by adjusting the capacitance of the tubing connecting the valve with the jet. This procedure slowed the initial rapid shortening that occurred with series elastic recoil and thereby reduced the oscillations of the lever following the recoil without impeding the lever's subsequent movement. Velocities were never read earlier than 40 msec after quick release. The nontendonous end of the muscle was held in a spring clip attached to a Statham Gl force transducer by a rod passing through the bottom of the bath. Force was monitored on a Hewlett-Packard recorder, but only isometric force measurements were made from these recordings. Isotonic force was always determined from the counterweights on the lever. The compliance of the whole system, excluding the muscle, was 12 ftm/ g. The muscles were stimulated through 15-mm platinum wires placed parallel to the muscle in the bath with 15-ma pulses of 5-msec duration. The timing of events was controlled with digital logic modules (Digi-bits, BRS-Foninger, Beltsvi]Ie, Maryland). PROCEDURE FOR MEASURING FORCE-VELOCITY RELATIONSHIPS Caffeine and calcium were added to the bath to bring the final concentration of each to 10 mm. Muscles were stimulated to produce twitch contractions at a rate of 3/min and tetanized for 3 seconds at intervals of 3 5 minutes, allowing the isometric twitch force to recover fully between tetani. During the plateau of tetanus, muscles were released to an isotonic load. With each successive tetanus, the load was decreased by 0.5 g, starting from the isometric force. In alternate studies, the load was progressively increased by 0.5 g. Results TETANUS Smooth tetani could be obtained with repetitive electrical stimulation in the presence of both 10 mm caffeine and 10 mm calcium (Fig. 1). Attempts to tetanize cat papillary muscles were unsuccessful in the presence of either caffeine or increased calcium concentration alone, although either of these conditions alone worked for rat muscles (6). The frequency of stimulation was critical: if it was greater than 12/sec, force reached an initial peak and then declined; if it was less than 3/sec, incomplete fusion resulted. LENGTH-TENSION RELATIONSHIP To obtain physiologically identifiable lengths, the length-tension relationship was derived for each muscle from isometric twitch contractions. The length-tension curves did not have a sharp peak of developed force, but rather they were characterized by a plateau in. I I I II I I I I I I II I I i l l ; ' FIGURE 1 Tetanus of cat papillary muscle. Isometric force record produced by repetitive electrical stimulation in the presence of 10 mm caffeine and 10 mn calcium. The milumolar concentrations of other solution components were NaCl 118, KCl 4.7, MgSO i 2.4, KH,PO i 1.2, dextrose 5, and NaHCO, Solution ph was 7.4, and temperature was 23 C. S.A. = stimulus artifact. Stimuli consisted of 15-ma square-wave pulses of 5- msec duration. Circulttioa Rete*rcb, Vol. XXXI, August 1972

3 LENGTH AND FORCE-VELOCITY RELATIONSHIP 197 the range of length where passive force rose steeply. The shortest resting length from which developed force reached a maximum was designated Lp, and all other lengths were referred to as a percent of this length. Muscles were not, in general, extended beyond L p, because overstretching them produced an irreparable decline in developed force. The length-tension curves were quite reproducible provided the muscles were not overextended. The twitch length-tension relationship for the muscle used to illustrate the force-velocity data below is shown in Figure 2. Caffeine and increased calcium concentration greatly potentiated the twitch contractions over all ranges of length (Fig. 2). Tetanic force was only slightly greater than MUSCLE LENGTH FIGURE 2 Effect of caffeine and increased calcium on the twitch length-tension relationship. The two bottom curves represent passive tension, and the two top curves represent developed tension. Squares indicate 2.5 mx calcium, 0 caffeine, and circles indicate 10 mm calcium, 10 mm caffeine. Frequency of stimulation was 12/ min in the absence of caffeine and 3/min in the presence of caffeine. The lower passive force in the presence of caffeine is probably due to slight irreversible extension of the noncontracthe portions of the muscle. L p marks the shortest rest length from which a maximum force was reached. Muscle length at L p = 7.0 mm, and muscle weight = 8.4 mg. This muscle was used to obtain the data for all subsequent figures except Figures 3 and 8. CircuUtwn Ructrcb. Vol. XXXI, Aufuit 1972! the potentiated twitch force (4.9 ±0.9% SE increase in 20 muscles measured near Lp). After the first several tetani, developed force generally declined by about 101 and then remained constant for the rest of the experiment, while force-velocity studies were made. Passive tension was not increased by caffeine and increased calcium concentration (Fig. 2), indicating that contracture had not occurred. Both passive and developed force were greatest immediately after an increase in length and declined to steady values over several minutes. This finding was attributed to stress relaxation of a viscous element in series with the muscle (11). The length-tension curves were derived from the steady, rather than the initial, values. COURSE OF MUSCLE SHORTENING To study the reproducibility of isotonic shortening at different times during the tetanus, length changes were compared when muscles were allowed to reach full isometric force and released to an isotonic load at progressively later intervals in a series of successive tetani (Fig. 3A). Length and velocity traces derived during the different tetani were superimposed on the screen of a memory oscilloscope (Fig. 3B) and compared during the successive tetani. No detectable difference was found in any of the traces if the release was made during a 2-second period immediately after the attainment of peak tension. This suggests that the course of shortening could be independent of the time in the contraction cycle. After the quick release there was an initial rapid shortening followed by a slower length change which progressed with decreasing velocity until the muscle had reached an equilibrium position (Fig. 3B). Frequently the muscle oscillated around this equilibrium length with a period of seconds. The amplitude of these oscillations decreased with increasing calcium concentration and was below 1* of the muscle length when the calcium concentration in the bath was 10 mm. There was no significant change in force during these oscillations since the muscle was connected to an isotonic lever. The apparatus

4 198 FORMAN, FORD, SONNENBLICK / : r1. I t / I - i, 1 : 1 i i 1 tec all tetani at the instant of quick release. The amount of internal shortening (discussed below) was subtracted from the initial muscle length to determine the contractile unit length. The initial rapid shortening was attributed to series elastic recoil (12) without contractile unit shortening. The amount of recoil is plotted against isotonic load in Figure 4 for the same muscle used in the experiment illustrated in Figure 2. This muscle is used as a typical example in describing the forcevelocity measurements and the corrections for the parallel elastic element in the following discussion. 1 sec FIGURE 3 0 > Course of isotonic shortening. A: The muscle was stimulated to develop 8.0 g of tetanic force and released to a 2.0-g isotonic load at progressively later times in seven consecutive tetani. The stimulus artifacts are shown under the force records. B: Superimposed length and velocity tracings from all seven tetani showing that the time of release did not affect the course of muscle shortening. Muscle length = 6.5 mm, and muscle weight = 8.5 mg. was investigated for possible sources of these oscillations, and the only slow resonance that could be found was that of the counterweights suspended from the elastic band. The elastic band was therefore extended to a fixed length without the weights to eliminate this resonating system. In spite of this, the oscillations persisted. This observation together with the finding that the amplitude of the oscillations was affected by the calcium concentration in the bath suggests that the oscillations were generated in the muscles. Because the tetani were reproducible, the series elastic extension during isometric force development was considered to be the same in FORCE-VELOCITY RELATIONSHIP Muscles were stimulated to develop maximum tetanic force from L D and released to varying isotonic loads. The series elastic recoil length for each isotonic load was subtracted from the overall muscle shortening on the oscilloscope records to determine the course of contractile unit shortening (Fig. 5). The lengths at which the contractile units stopped shortening (P o ) under six different loads were o cr O SERIES ELASTIC RECOIL (tmuicle length) FIGURE 4 Series elastic recoil. The isotonic load is plotted on a logarithmic scale against the amount of rapid shortening immediately after release to that load. CircuUtio* Rtsurcb, Vol. XXXI, August 1972

5 LENGTH AND FORCE-VELOCITY RELATIONSHIP 199 FIGURE 5 Force, length, and velocity tracings recorded during cardiac muscle tetanus. A: Tetanic force records showing isometric force development and release to 6-, 4-, and 2-g isotonic loads. B: The corresponding length and velocity records made during isotonic shortening. The dotted horizontal lines on the oscilloscope records indicate the amount of shortening associated with series elastic recoil following releases to each load. Zero shortening and zero velocity are indicated by the short horizontal lines at the left of the tracings recorded during isometric contractions. These lines were used to obtain the zero points for the length and velocity ordinates. Note that the oscilloscope traces were indistinct when the isotonic lever was moving very rapidly during series elastic recoil. The long arrow (AL) indicates the length at which shortening stopped under a 6-g load (top record). The velocities at the same amount of shortening (AL) under lighter loads, marked by short arrows, were increasingly higher with lighter loads. C: These velocities were plotted as a function of load. (They are also plotted in Figure 6 on the curve having a Po equal to 6 g.) The contractile unit length at which these velocity points were measured was 87.2% of Lp. This length was calculated by subtracting the isotonic shortening (0.46 mm or 6.6% of Lp) and the internal shortening (6.2% of Lv) from the resting muscle length. The internal shortening was derived from Figure 4 assuming that the series elastic elements were extended by the 8.5-g tetanic force from their length at 0.8 g of tension, the rest force. measured. The velocities at the same corrected lengths as these Po lengths were read from the oscilloscope tracings and plotted as a function of load (Fig. 6). The force-velocity points for a given contractile unit length were fitted by a least-squares method to hyperbolas of the Hill equation (13): (P + a) V = b (P o - P). Good fits were generally obtained, except that the measured P o points consistently lay below the curves (Figs. 6, 8). All hyperbolas had nearly constant values of a and b, but the extrapolated maximum shortening velocities varied in almost direct proportion to muscle shortening over the range of lengths studied (Fig. 7). Measurements could not be made at Circulation Research, Vol. XXXI, August 1972 lengths much greater than 90% of L,, for two reasons: (1) The contractile units underwent about a 5-7% internal shortening due to series elastic extension during the isometric phase of tension development. (2) A minimum of 40 msec was allowed to elapse after quick release before any measurements were made. During this time, the muscles would shorten 2-A% under light loads. LOAD BORNE BY THE PARALLEL ELASTIC ELEMENTS The data are presented in the example above assuming that all of the elasticity in the muscle is in series with the contractile elements (two-element model). A parallel

6 200 FORMAN, FORD, SONNENBLICK E 0.4- o UJ > 0.2 \ * L m a * \ o 90.1 \\ 87,3 \ \ =84 7 i \ \ B \ \ \ " 2 \ \ \ * 74 6 UJ o cr O o a: O " O o 2 2 x <2 1 ^ T*"* -^ *- FORCE (g) FIGURE 6 Force-velocity relationships at six different contractile unit lengths. Velocities at the same contractile unit length and different loads are plotted as a function of load. The data for each separate length were fitted by a least-squares method to the hyperbolas of the Hill equation. The muscle length used in the velocity dimension is L p. Contractile unit lengths decrease progressively from 90% of L p for the top curve to 75% of L p for the bottom curve. The parameters of the Hill equation derived from fitting the curves are given in Figure 7. elastic element in these muscles appears to be under significant tension over the range of length studied, and the correction for this parallel elasticity depends on the model assumed. In the Appendix, the force-velocity curves from the muscle illustrated above were corrected for two types of models (14): (1) the Voigt model, in which all the parallel elasticity is in series with the series elasticity and (2) the Maxwell model, in which all the parallel elasticity is in parallel with the series elasticity. The muscle is probably best approximated, however, by an intermediate model having a portion of the parallel elasticity in series with the series elasticity and the remainder in parallel. Thus, force-velocity curves of the true contractile units would lie between the curves corrected for the Voigt and Max CONTRACTILE UNIT LENGTH (tmmcle length) FIGURE 7 Parameters of the Hill equation. Measured isometric load from Figure 6 is plotted as a heavy solid line against the contractile unit length at which shortening stopped under that load (P o ). The lighter lines show the values of a (solid line), b (dotted line), and maximum velocity (V mal ) (dashed line) derived by fitting the curves in Figure 6 to the Hill equation. well models. At the longest lengths studied, where the parallel elastic element is under greatest tension, the corrections produce changes in maximum velocity of 103>. Furthermore, the corrections for the two models produce deviations in opposite directions. This suggests that any intermediate model would produce results similar to those of the twoelement model. A Voigt model was used in calculating the series elastic extension during isometric force development. The extensibility of the series elastic element at loads lighter than 0.5 g is not known because of the exponential relationship between extension and force, It was therefore assumed that the series elastic element was under a tension equal to the passive load at rest. If a portion of the series elasticity bore no load at rest, then the series elastic element would have been extended to greater lengths during isometric force development and all contractile unit lengths would have been shorter by a constant amount. The shapes of the curves in Figure 7 would be the Circulation Ruttrcb, Vol. XXXI, August 1972

7 LENGTH AND FORCE-VELOCITY RELATIONSHIP 201 same, but they would be displaced toward the ordinate by a few percent of muscle length. AVERAGE RESULTS FROM SEVERAL MUSCLES It is not practical to rigorously compare the effect of length on the force-velocity curves from muscles of different sizes. Force-velocity data from five muscles of approximately the same size were chosen and the averaged results are shown in Figure 8. These muscles developed maximum forces ranging from 8.5 to 9.5 g with rest forces ranging from 0.8 to 1.2 g. Cross-sectional areas, calculated by dividing muscle weight by length, were between 1.19 and 1.33 mm 2. The results support the conclusions that the force-velocity data can be fitted with hyperbolas and that the extrapolated maximum velocities are approximately proportional to isometric force. One exception to this generalization is that the measured P o points almost always lay below the curve-fitted hyperbolas. At the very shortest length, with only four data points, this deviation sometimes makes it impossible to fit a reasonable hyperbola, as in Figure 8. Discussion This study of the effect of muscle length on the force-velocity relationship in cardiac muscle was made possible through the development of a tetanized preparation, which allows comparable measurements to be made at different times during the contraction cycle. The results indicate that the extrapolated maximum shortening velocity is proportional to muscle shortening, at least over most of the range where force falls with shortening. This correlation is similar to that found by Abbott and Wilkie in skeletal muscle (15), as is the observation that the values of a and b in the Hill equation do not change with shortening. Isometric force in this preparation falls more with shortening than was found by Gordon et al. (16) in skeletal muscle working over the same range of length. As discussed below, force may fall even more sharply with shortening under physiological conditions. When muscle length was corrected for variations in series elastic extension, the forcevelocity curves could be fitted with hyper- Cirndnion Kuttrcb, Vol. XXXI, AuguU 1972 S 0.4 FORCE g) FIGURE 8 Force-velocity data averaged from jive muscles. Forcevelocity relationships were measured at six lengths determined as the lengths at which the muscles stopped shortening under loads ranging from 2 to 7 g. There are five values for every point except for the 1.0-g point on the curve where P o = 7 g, which is the average of four points. Error bars indicate $E. Hyperbolas of the Hill equation could be used to fit all the data except for the length at which P o = 2 g, as discussed in the text. bolas, as found by Edman and Nilsson (1). The Hill equation (13) was used to fit the data, rather than some other function, because the equation can be analyzed to distinguish among various factors known to influence the force-velocity relationship in muscle. Such an analysis is dependent on the type of model assumed and therefore cannot include all factors which might affect muscle function. As shown below, however, two mechanisms were sufficient to produce the changes seen. These are (1) deactivation of the contractile elements and (2) increasing internal load. DEACTIVATION OF THE CONTRACTILE ELEMENTS Taylor and Riidel (17) have shown that skeletal muscle myofilaments become deactivated at short sarcomere lengths, possibly through an influence of muscle length on the activating system. Gordon et al. (16) have suggested that overlapping myofilaments

8 202 FORMAN, FORD, SONNENBLICK might directly interfere with each other's ability to produce force. The effect of either process is to reduce the number of activated force-generating cross-bridges in parallel. This reduction should not, by itself, change the contractile function of the remaining crossbridges. In the Hill equation, both b and a/p 0 would therefore remain constant so that maximum velocity would also be constant, as is found for muscles of different cross-sectional areas. When the number of cross-bridge sites in parallel is reduced in skeletal muscle by decreasing the overlap of the myofilaments (16), maximum velocity remains constant provided the sarcomere lengths are greater than 1.9 ^im. Podolsky and Teichholz (18) have recently shown that decreasing the number of cross-bridges by directly decreasing activation also does not decrease maximum velocity; however, similar experiments by Julian (19) directly contradict this finding and show that maximum velocity decreases with decreasing activation. If Julian's findings are correct, deactivation alone might be sufficient to account for the decrease in both isometric force and maximum velocity at shorter muscle lengths. If, on the other hand, maximum velocity remains constant with deactivation, as found by Podolsky and Teichholz (18), other factors must be found to explain the present data. As shown below, an internal load could cause the observed changes in the forcevelocity curves. INTERNAL LOAD A muscle that contracts to less than its rest length reextends during relaxation, probably as the result of relengthening of a compliant element that becomes compressed during shortening. This internal compression exemplifies an internal load on the contractile elements which increases as the muscle shortens. There might also be another type of load which remains constant during shortening. With either type of internal load, the force on the contractile elements is greater than the external force on the muscle by an amount, i, equal to the internal load. Because the contractile elements cannot be fully unloaded, the true maximum velocity cannot be achieved. If deactivation decreases Po without changing b or a/p 0, velocity in the Hill equation will be determined by the relative load on the contractile elements, P/P o. In the presence of an internal load, this relative load becomes (P -I- i) / (P o + i). At zero external load, the relative load on the contractile elements, i/(p 0 + i), increases as Po diminishes. Thus, the maximum velocity obtained with zero external load diminishes with deactivation although the true maximum velocity may remain constant. The curves would have constant values of b, and a/p n would increase with shortening, as found in the experiments presented in this paper. Thus, a combination of deactivation with shortening and internal load is sufficient to account for the present data. EFFECTS OF CAFFEINE Riidel and Taylor (20) have reported that caffeine inhibits the deactivation that occurs with shortening of skeletal muscle. The observation that the muscles used here develop greater force at all lengths in the presence of caffeine and increased calcium concentration suggests that deactivation was partially inhibited by these agents. Isometric force and maximum velocity might therefore decrease more sharply with shortening in the absence of caffeine and of increased calcium concentration. This suggests that under physiological conditions, in the absence of caffeine, without increased calcium, and at lengths shorter than L p, cardiac muscle is significantly deactivated. Increasing the level of activation would thus provide a ready means by which inotropic agents could increase force and velocity. OSCILLATIONS The small length oscillations at the end of isotonic shortening were not studied in great detail here, and their cause was not found. They may have been due to incomplete fusion of tetanus. The observation that the muscles become deactivated with shortening suggests another mechanism. If there was a significant delay between the time a muscle shortened to a length and the time activation reached a level appropriate to that length, oscillations Circulation Rtsurch, Vol. XXXI, August 1972

9 LENGTH AND FORCE-VELOCITY RELATIONSHIP 203 would occur. A delay in deactivation with shortening might also account for the deviation of the Po points from the hyperbolic curves. All of the other points on the curves would have been obtained at a level of activation greater than was appropriate for the length. The Po points, on the other hand, were obtained after sufficient time had elapsed for the muscle to stop shortening and at a time when the level of activation may have been closer to its equilibrium value for that length. SERIES ELASTICITY Muscle lengths were corrected for series elastic extension so that velocities under different loads could be related at the same sarcomere length and therefore at the same amount of overlap of thick and thin myofilaments. The correction was made assuming that all the series elasticity was outside the sarcomere, in such structures as the cell to cell connections, the pieces of tissue held in the clips at the ends of the muscle, and the equipment. If some of the series elasticity was within the contractile apparatus itself and therefore inside the sarcomere, as shown for skeletal muscle by Huxley and Simmons (21), then the contractile unit length used in this paper would be overcorrected. The observation that the series elastic elements in these experiments were more than five times more compliant than those in the preparation of Huxley and Simmons suggests that the amount of overcorrection was not great. MAXIMUM SHORTENING VELOCITY Brutsaert et al. (22) have recently reported that maximum velocity is independent of muscle length over a range of lengths between 81% and 100% of the resting muscle length at the peak of the length-tension curve. The experiments presented in this paper could not test this conclusion but indicate that maximum velocity decreased linearly with muscle shortening at lengths below 90% of L D. Investigations at longer lengths were not made in these experiments because shortening always began after full tetanic force had been achieved and the contractile elements had shortened from Circmlstion Resurcb, Vol. XXXI, Autu.it 1972 their rest length to their contracted length at the peak of the length-tension curve. The observations of Brutsaert et al. were made early in the contraction cycle before full isometric contractile unit shortening would have occurred. Their studies were therefore carried out in the range of contractile element length where a plateau of force would have been produced if there was no internal shortening and where Gordon et al. (16) have shown that skeletal muscle maximum velocity remains constant. Appendix The force-velocity data of Figure 6 were corrected for the load borne by the parallel elastic elements for both the Voigt and the Maxwell model, and the results are presented in Figures 9 and 10. The corrections for a, b, isometric force, and maximum velocity are relatively small over the range of lengths studied here, suggesting that the error incurred by assuming a two-element model is not great. Furthermore, since the corrections for a and maximum velocity are of different sign for the Voigt and Maxwell models and since the muscle is best approximated by a model intermediate between these two, it seems likely that the values given for the two-element model are closer to those of the true contractile units than either of the corrected values. The methods of making the corrections are described below. VOIGT MODEL The parallel elastic element of the Voigt model shares the muscle load with the contractile elements but not with the series elastic elements. The only correction that need be made is subtraction of the load borne by the parallel elastic element from the total load. The passive lengdi-tension curve reflects extension of both series elastic and parallel elastic elements. The correct parallel elastic extension curve could he obtained by subtracting the series elastic extension from the passive length-tension curve. However, the series elastic extension is only known for forces greater than 0.5 g (Fig. 4) and the parallel elastic element was extended by loads ranging from 0 to 0.8 g (Fig. 2), making the subtraction impossible over the most critical range. The passive length-tension curve was therefore taken as being equal to the parallel elastic extension curve, keeping in mind that this would produce an overcorrection. The amount to be subtracted from the total load was read from the passive length-tension curve, allowing that the parallel elastic element had shortened during

10 204 FORMAN, FORD, SONNENBLICK FOICt (g) FIGURE 9 Force-velocity data corrected for load borne by the parallel elastic element. A: Data from Figure 6 corrected for Voigt model. B: Data corrected for Maxwell model. Symbols are the same as in Figure 6. both isometric and isotonic contractile unit shortening but not during the time of series elastic 0.8 -» 0.6 E X t I 0.2 IS 10 ISOTONIC SHORTEN ING{Xmu»ele length) FIGURE 10 Corrected values of a and maximum velocity. Solid lines represent the values for the two-element model taken from Figure 7. The three bottom curves are plots of a, and the three top curves are plots of maximum velocity. Dashed lines are values corrected for the Voigt model, and dotted lines are values corrected for the Maxwell model. recoil. This amount was constant for each muscle length and displaced the curves toward the ordinate without changing their shapes. Thus-, the values of b in the Hill equation remained unchanged, the apparent values of a increased, and the maximum velocities decreased (Fig. 10). The magnitude of these corrections is, at the most, 10* and may well be smaller because the compliance of the parallel elastic element was overestimated. MAXWELL MODEL The parallel elastic element of the Maxwell model bears all of the passive load and transfers it to the contractile element only when the whole muscle shortens, i.e., during both quick release and isotonic shortening but not during isometric force development. The load on the parallel elastic element was therefore based on the overall muscle length, and the amount to be subtracted from the total load was read from the passive length-tension curve. Because the passive load is transferred to the series elastic element as well as to the contractile element, the series elastic element becomes extended during shortening, and the contractile element velocity is greater than the overall muscle velocity. The contractile element velocities were calculated from the following equation (23, 24): V CE = V. (df/dl) PE where V CE is the contractile element velocity, V m is the overall muscle velocity, (df/dl) ge is the Circulation Rtsurcb, Vol. XXXI, August 1912

11 LENGTH AND FORCE-VELOCITY RELATIONSHIP 205 stiffness of the series elastic element, and (df/dl) PB is the stiffness of the parallel elastic element. The stiffnesses were obtained by differentiating equations fitted to the series elastic extension curve and the passive length-tension curve. The equation (df/dl) PE = 1.2 exp [1.4(L 7)], where L is the overall muscle length in millimeters and F the force in grams, was used to describe the stiffness of the parallel elastic element for muscle lengths greater than 80$ of L,,, and (df/dl) PE = 0 for lengths less than 80$ of L p. Since the parallel elastic element shares the muscle load with the series elastic element, the load on the parallel elastic element was subtracted from the series elastic load of the curve in Figure 4. The derivative of the equation fitting the resulting data was given by (df/dl) SE = 6.76F, where L is the series elastic extension in millimeters and F is greater than 0.2 g. The corrected force-velocity data were fitted with hyperbolas (Fig. 9B) having changes in both a and b. The values of b were smaller compared with those of the two-element model but did not chance significantly with muscle length, remaining between 0.20 and 0.22 muscle lengths/sec over the entire range of length studied. The values of a for these corrected curves were less than those of the two-element model, and the maximum shortening velocities were higher (Fig. 10). It is not possible to define the contractile unit length for the Maxwell model corrections. There Ls no force on the series elastic element at rest and the extensibility of the series elastic element corrected for the Maxwell model below 0.2 g is not known. As a result, the full amount of the contractile unit shortening during isometric force development could not be calculated. Because of this uncertainty, the values of a and maximum velocity are plotted as functions of isotonic shortening rather than contractile unit length in Figure 10. The contractile unit length at which the velocity measurements were made has been considered to be identical for all points on any single force-velocity curve. This is valid whether the data were analyzed for either a two-element or a Voigt model. In the Maxwell model, however, this is not strictly correct. Here the precise contractile unit length depends on the load transfer from the parallel elastic element to the series elastic element during isotonic shortening and the compliance of the series elastic element. However, the differences in contractile unit lengths did not exceed 156 of L p for any of these curves. References 1. EDMAN, K.A.P., AND NILSSON, E.: Mechanical CircuUtio* Rtstarcb, Vol. XXXI, August 1972 parameters of myocardial contraction studied at a constant length of the contractile element. Acta Physiol Scand 72: , BRADY, A.J.: Time and displacement dependence of cardiac contractility: Problems in defining the active state and force-velocity relations. Fed Proc 24: , SONNENBLJCE, E.H.: Determinants of active state in heart muscle: Force, velocity, instantaneous muscle length, time. Fed Proc 24: , NOBEL, M.I.M., BOWEN, T.E., AND HEFNER, L.L.: Force-velocity relationship of cat cardiac muscle, studied by isotonic and quick-release techniques. Circ Res 24: , SONNENBLICX, E.H.: Implications of muscle mechanics in the heart. Fed Proc 21: , HENDERSON, A.H., FORMAN, R., BRUTSAERT, D.L., AND SONNENBLICK, E.H.: Tetanic contraction in mammalian cardiac muscle. Cardiovasc Res 5(suppl. l):98-100, SONNENBLICK, E.H.: Active state in heart muscle: Its delayed onset and modification by inotropic agents. J Gen Physiol 50: , NORHIS, C, AND CARMECI, P.: Isotonic muscle transducer. J Appl Physiol 20: , BLJX, M.: Die Lange und die Spannung des Muskels. Skand Arkh Physiol 3: , Applications Manual for Operation Amplifiers. Philbrick/Nexus Research Co., Dedham, Mass., SONNENBLICK, E.H., Ross, J., JH., COVELL, J.W., AND BRAUNWAUD, E.: Alterations in resting length-tension relations of cardiac muscle induced by changes in contractile force. Circ Res 19: , HILL, A.V.: Series elastic component of muscle. Proc R Soc Lond [Biol] 137: , HILL, A.V.: Heat of shortening and the dynamic constants of muscle. Proc R Soc Lond [Biol] 126: , BUCHTAL, F., AND KAEZER, E.: Rheology of crossstriated muscle fibre with particular reference to isotonic conditions. Dan Videns Sels Biol Med 21:1-318, ABBOTT, B.C., AND WILKIE, D.R.: Relation between velocity of shortening and the tensionlength curve of skeletal muscle. J Physiol (Lond) 120: , CORDON, A.M., HUXLEY, A.F., AND JULIAN, F.J.: Variation of isometric tension with sarcomere length in vertebrate muscle fibres. J Physiol (Lond) 184: , TAYLOR, S.R., AND RUDEL, R.: Striated muscle fibres: Inactivation of contraction induced by shortening. Science 167: , 1970.

12 206 FORMAN, FORD, SONNENBLICK 18. PODOLSKY, R.J., AND TEICHHOLZ, L.E.: Relation between calcium and contraction kinetics in skinned muscle fibres. J Physiol (Lond) 211:19-35, JULIAN, F.J.: Effect of calcium on the forcevelocity relation of briefly glycerinated frog muscle fibres. J Physiol (Lond) 218: , RUDEL, R., AND TAYLOR, S.R.: Striated muscle fibres: Facilitation of contraction at short lengths by caffeine. Science 172: , HUXUEY, A.F., AND SIMMONS, R.M.: Proposed mechanism of force generation in striated muscle. Nature (Lond) 233: , BflUTSAERT, D.L., CLAES, V.A., AND SONNENBLICK, E.H.: Velocity of shortening of unloaded heart muscle and the length-tension relation. Circ Res 29:63-75, HEFNER, L.L., AND ROWEN, T.E.: Elastic components of cat papillary muscle. Am J Physiol 212: , POLLACK, G.H.: Maximum velocity as an index of contractility in cardiac muscle: Critical evaluation. Circ Res 26: , Ctrctdation Rtst.nh, Vol. XXXI, August 1972