effect of muscl e length on the force-velocity …...effect of muscl e length on the force-velocity...

Effect of Muscle Length

on the Force-Velocity Relationshipof Tetanized Cardiac MuscleBy Robert Forman, Lincoln E. Ford, and Edmund H. Sonnenblick

ABSTRACTCat papillary muscles were tetanized with repetitive electrical stimulation in

the presence of 10 ITIM caffeine and 10 mM calcium. Velocities were measuredduring the plateau of tetanus with quick releases to isotonic loads. The courseof isotonic shortening was independent of time in the contraction cycle for atleast 2 seconds after the attainment of peak isometric force. The force-velocityrelationships were measured at different muscle lengths that had beencorrected for series elastic extension. These lengths ranged from 75% to 90% ofthe passive length from which maximum force was developed. The data werefitted by a least-squares method with hyperbolas described by the Hillequation, each for a constant corrected muscle length. The extrapolatedmaximum velocities and isometric forces diminished together in almostdirect proportion to muscle shortening. Corrections for the load borne by theparallel elastic elements did not significantly change the relationships betweenisometric force, maximum velocity, and muscle length. The results can beaccounted for by two mechanisms: (1) an internal load and (2) deactivationof the contractile elements.

KEY WORDS parallel elastic elements series elastic elementscaffeine hyperbolic fit calcium internal load cat papillary muscle

• The level of cardiac muscle activationappears to change during the isometric twitchcontraction cycle (1-3) and to be affected bythe muscle length. Previous studies of theforce-velocity relationships in cardiac prepara-tions were difficult because of an inability totetanize the muscle: technically it is hard tomeasure muscle velocity under different loadsat exactly the same instant in the contractioncycle and at the same sarcomere length.Extrapolations (4) or approximations (1) to

From the Cardiovascular Division, Peter BentBrigham Hospital, Harvard Medical School, Boston,Massachusetts 02115.

This study was supported in part by U. S. PublicHealth Service Grant HE 11306 and Training GrantHL-5890 from the National Heart and Lung Instituteand by a grant from the American Heart Association.

Dr. Forman's current address is CardiopulmonaryUnit, Groote Schuur Hospital, Cape Town, SouthAfrica. Dr. Ford's current address is Department ofPhysiology, University College London, Gower Street,London, W.C. 1, England.

A brief account of some of these data has beenpresented in Circulation 44(suppl. II):II-89, 1971.

Received January 20, 1972. Accepted for publica-tion June 2, 1972.

this ideal have been made from studiesinvolving twitch contractions, and assump-tions have been made which allow measure-ments to be taken at different contractileelement lengths and different times in thecontraction cycle (3, 5). The recent descrip-tion of a method for tetanizing rat papillarymuscle (6) suggested that force-velocitystudies might be carried out under conditionswhere time in the contraction cycle could beignored as a variable. Accordingly, the meth-od for tetanizing rat muscles was modified topermit cat muscle tetanization. The effects ofmuscle length on the force-velocity relation-ship were studied in this preparation, and theresults were interpreted to distinguish be-tween the various mechanisms which mightaffect this relationship. The effect of theparallel elastic element on the results isanalyzed in the Appendix.

MethodsPapillary muscles were dissected from die right

ventricles of adult cats anesthetized with sodiumpentobarbital (150—250 mg, ip). The muscles were

Circulation Riittreb, Vol. XXXI, August 1972 195

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196 FORMAN, FORD, SONNENBLICK

mounted in physiologic buffer of pH 7.4 with amillimolar composition of CaCl2 2.5, KC1 4.7,MgSO4 2.4, KH;,PO4 1.2, NaCl 118, dextrose 5,and NaHCO3 2.4. The solution was bubbled witha mixture of 95* O2-5!t CO2 and maintained at22-24°C.

APPARATUS

The apparatus described previously (7) wasmodified. The muscles were mounted verticallywith the upper tendonous end held in a stainlesssteel clip attached to an isotonic lever system(8). Counterweights were suspended from thelever by an elastic band used to isolate the inertiaof the weights from the lever (9). The equivalentmass of the lever and clip was 95 mg. A proximalportion of the lever interrupted a light beamfalling on a photodiode that was arranged in aWheatstone bridge circuit to measure displace-ment. The velocity of displacement was obtainedby differentiating the length trace with anoperational amplifier in a circuit having Rd = 1.0Mohm, R, = 3.0 Mohms, Crt = 1.0 yxfarad, andC, = 0.003 /iiarads (10). Following a stepchange in the input voltage, the output of thedifferentiator returned to zero with a timeconstant of 2.8 msec.

The length and velocity traces were displayedon the screen of a Tektronix 564 memoryoscilloscope, using a sweep rate of 200 msec/cm,and recorded photographically.

Initially the lever was held against a stop by anair jet. The tetanized muscle was released to anisotonic load by interrupting the flow of air with asolenoid-controlled valve. The release was slowedto occur over approximately 5 msec by adjustingthe capacitance of the tubing connecting thevalve with the jet. This procedure slowed theinitial rapid shortening that occurred •with serieselastic recoil and thereby reduced the oscillationsof the lever following the recoil without impedingthe lever's subsequent movement. Velocities werenever read earlier than 40 msec after quickrelease.

The nontendonous end of the muscle was heldin a spring clip attached to a Statham Gl-4-250force transducer by a rod passing through thebottom of the bath. Force was monitored on aHewlett-Packard recorder, but only isometricforce measurements were made from theserecordings. Isotonic force was always determinedfrom the counterweights on the lever. Thecompliance of the whole system, excluding themuscle, was 12 ftm/ g.

The muscles were stimulated through 15-mmplatinum wires placed parallel to the muscle inthe bath with 15-ma pulses of 5-msec duration.The timing of events was controlled with digitallogic modules (Digi-bits, BRS-Foninger, Belts-vi]Ie, Maryland).

PROCEDURE FOR MEASURING FORCE-VELOCITYRELATIONSHIPS

Caffeine and calcium were added to the bath tobring the final concentration of each to 10 mM.Muscles were stimulated to produce twitchcontractions at a rate of 3/min and tetanized for3 seconds at intervals of 3—5 minutes, allowing theisometric twitch force to recover fully betweentetani. During the plateau of tetanus, muscleswere released to an isotonic load. With eachsuccessive tetanus, the load was decreased by 0.5g, starting from the isometric force. In alternatestudies, the load was progressively increased by0.5 g.

ResultsTETANUS

Smooth tetani could be obtained withrepetitive electrical stimulation in the presenceof both 10 mM caffeine and 10 mM calcium(Fig. 1). Attempts to tetanize cat papillarymuscles were unsuccessful in the presence ofeither caffeine or increased calcium concentra-tion alone, although either of these conditionsalone worked for rat muscles (6). Thefrequency of stimulation was critical: if it wasgreater than 12/sec, force reached an initialpeak and then declined; if it was less than3/sec, incomplete fusion resulted.

LENGTH-TENSION RELATIONSHIP

To obtain physiologically identifiablelengths, the length-tension relationship wasderived for each muscle from isometric twitchcontractions. The length-tension curves didnot have a sharp peak of developed force, butrather 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 recordproduced by repetitive electrical stimulation in thepresence of 10 mm caffeine and 10 mn calcium. ThemilUmolar concentrations of other solution componentswere NaCl 118, KCl 4.7, MgSOi 2.4, KH,POi 1.2,dextrose 5, and NaHCO, 22.4. 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.

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LENGTH AND FORCE-VELOCITY RELATIONSHIP 197

the range of length where passive force rosesteeply. The shortest resting length fromwhich developed force reached a maximumwas designated Lp, and all other lengthswere referred to as a percent of this length.Muscles were not, in general, extendedbeyond Lp, because overstretching themproduced an irreparable decline in developedforce. The length-tension curves were quitereproducible provided the muscles were notoverextended. The twitch length-tension rela-tionship for the muscle used to illustrate theforce-velocity data below is shown in Figure 2.

Caffeine and increased calcium concentra-tion greatly potentiated the twitch contrac-tions 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 twitchlength-tension relationship. The two bottom curvesrepresent passive tension, and the two top curvesrepresent developed tension. Squares indicate 2.5 mxcalcium, 0 caffeine, and circles indicate 10 mm calci-um, 10 mM caffeine. Frequency of stimulation was 12/min in the absence of caffeine and 3/min in the pres-ence of caffeine. The lower passive force in the presenceof caffeine is probably due to slight irreversible exten-sion of the noncontractHe portions of the muscle. Lp

marks the shortest rest length from which a maximumforce was reached. Muscle length at Lp = 7.0 mm, andmuscle weight = 8.4 mg. This muscle was used toobtain the data for all subsequent figures except Fig-ures 3 and 8.

CircuUtwn Ructrcb. Vol. XXXI, Aufuit 1972

the potentiated twitch force (4.9 ±0.9% SEincrease in 20 muscles measured near Lp).After the first several tetani, developed forcegenerally declined by about 101 and thenremained constant for the rest of the experi-ment, while force-velocity studies were made.Passive tension was not increased by caffeineand increased calcium concentration (Fig. 2),indicating that contracture had not occurred.

Both passive and developed force weregreatest immediately after an increase inlength and declined to steady values overseveral minutes. This finding was attributed tostress relaxation of a viscous element in serieswith the muscle (11). The length-tensioncurves were derived from the steady, ratherthan the initial, values.

COURSE OF MUSCLE SHORTENING

To study the reproducibility of isotonicshortening at different times during thetetanus, length changes were compared whenmuscles were allowed to reach full isometricforce and released to an isotonic load atprogressively later intervals in a series ofsuccessive tetani (Fig. 3A). Length andvelocity traces derived during the differenttetani were superimposed on the screen of amemory oscilloscope (Fig. 3B) and comparedduring the successive tetani. No detectabledifference was found in any of the traces if therelease was made during a 2-second periodimmediately after the attainment of peaktension. This suggests that the course ofshortening could be independent of the timein the contraction cycle.

After the quick release there was an initialrapid shortening followed by a slower lengthchange which progressed with decreasingvelocity until the muscle had reached anequilibrium position (Fig. 3B). Frequentlythe muscle oscillated around this equilibriumlength with a period of 0.5-2.0 seconds. Theamplitude of these oscillations decreased withincreasing calcium concentration and wasbelow 1* of the muscle length when thecalcium concentration in the bath was 10 mM.There was no significant change in forceduring these oscillations since the muscle wasconnected to an isotonic lever. The apparatus



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FIGURE 3

Course of isotonic shortening. A: The muscle wasstimulated to develop 8.0 g of tetanic force and re-leased to a 2.0-g isotonic load at progressively latertimes in seven consecutive tetani. The stimulus arti-facts are shown under the force records. B: Superim-posed length and velocity tracings from all seventetani showing that the time of release did not affectthe course of muscle shortening. Muscle length =6.5 mm, and muscle weight = 8.5 mg.

was investigated for possible sources of theseoscillations, and the only slow resonance thatcould be found was that of the counterweightssuspended from the elastic band. The elasticband was therefore extended to a fixed lengthwithout the weights to eliminate this resonat-ing system. In spite of this, the oscillationspersisted. This observation together with thefinding that the amplitude of the oscillationswas affected by the calcium concentration inthe bath suggests that the oscillations weregenerated in the muscles.

Because the tetani were reproducible, theseries elastic extension during isometric forcedevelopment was considered to be the same in

all tetani at the instant of quick release. Theamount of internal shortening (discussedbelow) was subtracted from the initial musclelength to determine the contractile unitlength. The initial rapid shortening wasattributed to series elastic recoil (12) withoutcontractile unit shortening. The amount ofrecoil is plotted against isotonic load in Figure4 for the same muscle used in the experimentillustrated in Figure 2. This muscle is used as atypical example in describing the force-velocity measurements and the corrections forthe parallel elastic element in the followingdiscussion.

FORCE-VELOCITY RELATIONSHIP

Muscles were stimulated to develop maxi-mum tetanic force from LD and released tovarying isotonic loads. The series elastic recoillength for each isotonic load was subtractedfrom the overall muscle shortening on theoscilloscope records to determine the course ofcontractile unit shortening (Fig. 5). Thelengths at which the contractile units stoppedshortening (Po) under six different loads were

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SERIES ELASTIC RECOIL (tmuicle length)

FIGURE 4

Series elastic recoil. The isotonic load is plotted on alogarithmic scale against the amount of rapid shorten-ing immediately after release to that load.

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FIGURE 5

Force, length, and velocity tracings recorded during cardiac muscle tetanus. A: Tetanic forcerecords 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 dottedhorizontal lines on the oscilloscope records indicate the amount of shortening associated withseries elastic recoil following releases to each load. Zero shortening and zero velocity areindicated by the short horizontal lines at the left of the tracings recorded during isometriccontractions. These lines were used to obtain the zero points for the length and velocity ordi-nates. Note that the oscilloscope traces were indistinct when the isotonic lever was movingvery rapidly during series elastic recoil. The long arrow (AL) indicates the length at whichshortening stopped under a 6-g load (top record). The velocities at the same amount ofshortening (AL) under lighter loads, marked by short arrows, were increasingly higher withlighter loads. C: These velocities were plotted as a function of load. (They are also plotted inFigure 6 on the curve having a Po equal to 6 g.) The contractile unit length at which thesevelocity points were measured was 87.2% of Lp. This length was calculated by subtractingthe isotonic shortening (0.46 mm or 6.6% of Lp) and the internal shortening (6.2% of Lv) fromthe resting muscle length. The internal shortening was derived from Figure 4 assuming thatthe series elastic elements were extended by the 8.5-g tetanic force from their length at0.8 g of tension, the rest force.

measured. The velocities at the same correctedlengths as these Po lengths were read from theoscilloscope tracings and plotted as a functionof load (Fig. 6). The force-velocity points fora given contractile unit length were fitted by aleast-squares method to hyperbolas of the Hillequation (13): (P + a) • V = b • (Po - P).Good fits were generally obtained, except thatthe measured Po points consistently lay belowthe curves (Figs. 6, 8). All hyperbolas hadnearly constant values of a and b, but theextrapolated maximum shortening velocitiesvaried in almost direct proportion to muscleshortening 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 tworeasons: (1) The contractile units underwentabout a 5-7% internal shortening due to serieselastic extension during the isometric phase oftension development. (2) A minimum of 40msec was allowed to elapse after quick releasebefore any measurements were made. Duringthis time, the muscles would shorten 2-A%under light loads.

LOAD BORNE BY THE PARALLEL ELASTIC ELEMENTS

The data are presented in the exampleabove assuming that all of the elasticity in themuscle is in series with the contractileelements (two-element model). A parallel



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FIGURE 6

Force-velocity relationships at six different contractileunit lengths. Velocities at the same contractile unitlength and different loads are plotted as a function ofload. The data for each separate length were fittedby a least-squares method to the hyperbolas of theHill equation. The muscle length used in the velocitydimension is Lp. Contractile unit lengths decrease pro-gressively from 90% of Lp for the top curve to 75% ofLp for the bottom curve. The parameters of the Hillequation derived from fitting the curves are given inFigure 7.

elastic element in these muscles appears to beunder significant tension over the range oflength studied, and the correction for thisparallel elasticity depends on the modelassumed. In the Appendix, the force-velocitycurves from the muscle illustrated above werecorrected for two types of models (14): (1)the Voigt model, in which all the parallelelasticity is in series with the series elasticityand (2) the Maxwell model, in which all theparallel elasticity is in parallel with the serieselasticity. The muscle is probably best approx-imated, however, by an intermediate modelhaving a portion of the parallel elasticity inseries with the series elasticity and the remain-der in parallel. Thus, force-velocity curves ofthe true contractile units would lie betweenthe curves corrected for the Voigt and Max-

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FIGURE 7

Parameters of the Hill equation. Measured isometricload from Figure 6 is plotted as a heavy solid lineagainst the contractile unit length at which shorteningstopped under that load (Po). The lighter lines showthe values of a (solid line), b (dotted line), and maxi-mum velocity (Vmal) (dashed line) derived by fittingthe curves in Figure 6 to the Hill equation.

well models. At the longest lengths studied,where the parallel elastic element is undergreatest tension, the corrections producechanges in maximum velocity of 103>. Further-more, the corrections for the two models pro-duce deviations in opposite directions. Thissuggests that any intermediate model wouldproduce results similar to those of the two-element model.

A Voigt model was used in calculating theseries elastic extension during isometric forcedevelopment. The extensibility of the serieselastic element at loads lighter than 0.5 g isnot known because of the exponential rela-tionship between extension and force, It wastherefore assumed that the series elasticelement was under a tension equal to thepassive load at rest. If a portion of the serieselasticity bore no load at rest, then the serieselastic element would have been extended togreater lengths during isometric force devel-opment and all contractile unit lengths wouldhave been shorter by a constant amount. Theshapes of the curves in Figure 7 would be the

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same, but they would be displaced toward theordinate by a few percent of muscle length.

AVERAGE RESULTS FROM SEVERAL MUSCLES

It is not practical to rigorously compare theeffect of length on the force-velocity curvesfrom muscles of different sizes. Force-velocitydata from five muscles of approximately thesame size were chosen and the averagedresults are shown in Figure 8. These musclesdeveloped maximum forces ranging from 8.5to 9.5 g with rest forces ranging from 0.8 to1.2 g. Cross-sectional areas, calculated bydividing muscle weight by length, werebetween 1.19 and 1.33 mm2. The resultssupport the conclusions that the force-velocitydata can be fitted with hyperbolas and thatthe extrapolated maximum velocities areapproximately proportional to isometric force.One exception to this generalization is that themeasured Po points almost always lay belowthe curve-fitted hyperbolas. At the veryshortest length, with only four data points,this deviation sometimes makes it impossibleto fit a reasonable hyperbola, as in Figure 8.

Discussion

This study of the effect of muscle length onthe force-velocity relationship in cardiac mus-cle was made possible through the develop-ment of a tetanized preparation, which allowscomparable measurements to be made atdifferent times during the contraction cycle.The results indicate that the extrapolatedmaximum shortening velocity is proportionalto muscle shortening, at least over most of therange where force falls with shortening. Thiscorrelation is similar to that found by Abbottand Wilkie in skeletal muscle (15), as is theobservation that the values of a and b in theHill equation do not change with shortening.Isometric force in this preparation falls morewith shortening than was found by Gordon etal. (16) in skeletal muscle working over thesame range of length. As discussed below,force may fall even more sharply withshortening under physiological conditions.

When muscle length was corrected forvariations in series elastic extension, the force-velocity curves could be fitted with hyper-

Cirndnion Kuttrcb, Vol. XXXI, AuguU 1972

S 0.4

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FIGURE 8

Force-velocity data averaged from jive muscles. Force-velocity relationships were measured at six lengthsdetermined as the lengths at which the muscles stoppedshortening under loads ranging from 2 to 7 g. Thereare five values for every point except for the 1.0-gpoint on the curve where Po = 7 g, which is theaverage of four points. Error bars indicate $E. Hyper-bolas of the Hill equation could be used to fit all thedata except for the length at which Po = 2 g, as dis-cussed in the text.

bolas, as found by Edman and Nilsson (1).The Hill equation (13) was used to fit thedata, rather than some other function, becausethe equation can be analyzed to distinguishamong various factors known to influence theforce-velocity relationship in muscle. Such ananalysis is dependent on the type of modelassumed and therefore cannot include allfactors which might affect muscle function. Asshown below, however, two mechanisms weresufficient to produce the changes seen. Theseare (1) deactivation of the contractile ele-ments and (2) increasing internal load.

DEACTIVATION OF THE CONTRACTILE ELEMENTS

Taylor and Riidel (17) have shown thatskeletal muscle myofilaments become deacti-vated at short sarcomere lengths, possiblythrough an influence of muscle length on theactivating system. Gordon et al. (16) havesuggested that overlapping myofilaments



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might directly interfere with each other'sability to produce force. The effect of eitherprocess is to reduce the number of activatedforce-generating cross-bridges in parallel. Thisreduction should not, by itself, change thecontractile function of the remaining cross-bridges. In the Hill equation, both b and a/P0

would therefore remain constant so thatmaximum velocity would also be constant, asis found for muscles of different cross-sectionalareas. When the number of cross-bridge sitesin parallel is reduced in skeletal muscle bydecreasing the overlap of the myofilaments(16), maximum velocity remains constantprovided the sarcomere lengths are greaterthan 1.9 ^im. Podolsky and Teichholz (18)have recently shown that decreasing thenumber of cross-bridges by directly decreasingactivation also does not decrease maximumvelocity; however, similar experiments byJulian (19) directly contradict this finding andshow that maximum velocity decreases withdecreasing activation. If Julian's findings arecorrect, deactivation alone might be sufficientto account for the decrease in both isometricforce and maximum velocity at shorter musclelengths. If, on the other hand, maximumvelocity remains constant with deactivation, asfound by Podolsky and Teichholz (18), otherfactors must be found to explain the presentdata. As shown below, an internal load couldcause the observed changes in the force-velocity curves.

INTERNAL LOAD

A muscle that contracts to less than its restlength reextends during relaxation, probablyas the result of relengthening of a compliantelement that becomes compressed duringshortening. This internal compression exempli-fies an internal load on the contractileelements which increases as the muscleshortens. There might also be another type ofload which remains constant during shorten-ing. With either type of internal load, theforce on the contractile elements is greaterthan the external force on the muscle by anamount, i, equal to the internal load. Becausethe contractile elements cannot be fullyunloaded, the true maximum velocity cannot

be achieved. If deactivation decreases Powithout changing b or a/P0, velocity in theHill equation will be determined by therelative load on the contractile elements, P/Po.In the presence of an internal load, thisrelative load becomes (P -I- i) / (Po + i). Atzero external load, the relative load on thecontractile elements, i/(P0 + i), increases asPo diminishes. Thus, the maximum velocityobtained with zero external load diminisheswith deactivation although the true maximumvelocity may remain constant. The curveswould have constant values of b, and a/Pn

would increase with shortening, as found inthe experiments presented in this paper. Thus,a combination of deactivation with shorteningand internal load is sufficient to account forthe present data.

EFFECTS OF CAFFEINE

Riidel and Taylor (20) have reported thatcaffeine inhibits the deactivation that occurswith shortening of skeletal muscle. Theobservation that the muscles used here devel-op greater force at all lengths in the presenceof caffeine and increased calcium concentra-tion suggests that deactivation was partiallyinhibited by these agents. Isometric force andmaximum velocity might therefore decreasemore sharply with shortening in the absenceof caffeine and of increased calcium concen-tration. This suggests that under physiologicalconditions, in the absence of caffeine, withoutincreased calcium, and at lengths shorter thanLp, cardiac muscle is significantly deactivated.Increasing the level of activation would thusprovide a ready means by which inotropicagents could increase force and velocity.

OSCILLATIONS

The small length oscillations at the end ofisotonic shortening were not studied in greatdetail here, and their cause was not found.They may have been due to incomplete fusionof tetanus. The observation that the musclesbecome deactivated with shortening suggestsanother mechanism. If there was a significantdelay between the time a muscle shortened toa length and the time activation reached alevel appropriate to that length, oscillations

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would occur. A delay in deactivation withshortening might also account for the devi-ation of the Po points from the hyperboliccurves. All of the other points on thecurves would have been obtained at a level ofactivation greater than was appropriate forthe length. The Po points, on the other hand,were obtained after sufficient time hadelapsed for the muscle to stop shortening andat a time when the level of activation mayhave been closer to its equilibrium value forthat length.

SERIES ELASTICITY

Muscle lengths were corrected for serieselastic extension so that velocities underdifferent loads could be related at the samesarcomere length and therefore at the sameamount of overlap of thick and thin myofila-ments. The correction was made assumingthat all the series elasticity was outside thesarcomere, in such structures as the cell to cellconnections, the pieces of tissue held in theclips at the ends of the muscle, and theequipment. If some of the series elasticity waswithin the contractile apparatus itself andtherefore inside the sarcomere, as shown forskeletal muscle by Huxley and Simmons (21),then the contractile unit length used in thispaper would be overcorrected. The observa-tion that the series elastic elements in theseexperiments were more than five times morecompliant than those in the preparation ofHuxley and Simmons suggests that theamount of overcorrection was not great.

MAXIMUM SHORTENING VELOCITY

Brutsaert et al. (22) have recently reportedthat maximum velocity is independent ofmuscle length over a range of lengths between81% and 100% of the resting muscle length atthe peak of the length-tension curve. Theexperiments presented in this paper could nottest this conclusion but indicate that maximumvelocity decreased linearly with muscle short-ening at lengths below 90% of LD. Investiga-tions at longer lengths were not made in theseexperiments because shortening always beganafter full tetanic force had been achieved andthe contractile elements had shortened from

Circmlstion Resurcb, Vol. XXXI, Autu.it 1972

their rest length to their contracted length atthe peak of the length-tension curve. The ob-servations of Brutsaert et al. were made earlyin the contraction cycle before full isometriccontractile unit shortening would have oc-curred. Their studies were therefore carriedout in the range of contractile element lengthwhere a plateau of force would have beenproduced if there was no internal shorteningand where Gordon et al. (16) have shownthat skeletal muscle maximum velocity re-mains constant.

AppendixThe force-velocity data of Figure 6 were

corrected for the load borne by the parallel elasticelements for both the Voigt and the Maxwellmodel, and the results are presented in Figures 9and 10. The corrections for a, b, isometric force,and maximum velocity are relatively small overthe range of lengths studied here, suggesting thatthe error incurred by assuming a two-elementmodel is not great. Furthermore, since thecorrections for a and maximum velocity are ofdifferent sign for the Voigt and Maxwell modelsand since the muscle is best approximated by amodel intermediate between these two, it seemslikely that the values given for the two-elementmodel are closer to those of the true contractileunits than either of the corrected values. Themethods of making the corrections are describedbelow.

VOIGT MODEL

The parallel elastic element of the Voigt modelshares the muscle load with the contractileelements but not with the series elastic elements.The only correction that need be made issubtraction of the load borne by the parallelelastic element from the total load. The passivelengdi-tension curve reflects extension of bothseries elastic and parallel elastic elements. Thecorrect parallel elastic extension curve could heobtained by subtracting the series elastic exten-sion from the passive length-tension curve.However, the series elastic extension is onlyknown for forces greater than 0.5 g (Fig. 4) andthe parallel elastic element was extended by loadsranging from 0 to 0.8 g (Fig. 2), making thesubtraction impossible over the most criticalrange. The passive length-tension curve wastherefore taken as being equal to the parallelelastic extension curve, keeping in mind that thiswould produce an overcorrection. The amount tobe subtracted from the total load was read fromthe passive length-tension curve, allowing thatthe parallel elastic element had shortened during



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FOICt (g)

FIGURE 9

Force-velocity data corrected for load borne by the parallel elastic element. A: Data fromFigure 6 corrected for Voigt model. B: Data corrected for Maxwell model. Symbols are thesame as in Figure 6.

both isometric and isotonic contractile unitshortening but not during the time of series elastic

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FIGURE 10

Corrected values of a and maximum velocity. Solidlines represent the values for the two-element modeltaken from Figure 7. The three bottom curves areplots of a, and the three top curves are plots of maxi-mum velocity. Dashed lines are values corrected forthe Voigt model, and dotted lines are values correctedfor the Maxwell model.

recoil. This amount was constant for each musclelength and displaced the curves toward theordinate without changing their shapes. Thus-, thevalues of b in the Hill equation remainedunchanged, the apparent values of a increased,and the maximum velocities decreased (Fig. 10).The magnitude of these corrections is, at themost, 10* and may well be smaller because thecompliance of the parallel elastic element wasoverestimated.

MAXWELL MODEL

The parallel elastic element of the Maxwellmodel bears all of the passive load and transfers itto the contractile element only when the wholemuscle shortens, i.e., during both quick releaseand isotonic shortening but not during isometricforce development. The load on the parallel elasticelement was therefore based on the overall musclelength, and the amount to be subtracted from thetotal load was read from the passive length-ten-sion curve.

Because the passive load is transferred to theseries elastic element as well as to the contractileelement, the series elastic element becomesextended during shortening, and the contractileelement velocity is greater than the overall musclevelocity. The contractile element velocities werecalculated from the following equation (23,24):

(dF/dL)P EVCE = V.

where VCE is the contractile element velocity, Vm

is the overall muscle velocity, (dF/dL) g E is theCirculation Rtsurcb, Vol. XXXI, August 1912



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stiffness of the series elastic element, and(dF/dL)P B is the stiffness of the parallel elasticelement. The stiffnesses were obtained by differen-tiating equations fitted to the series elastic exten-sion curve and the passive length-tension curve.The equation (dF/dL) P E = 1.2 exp [1.4(L —7)],where L is the overall muscle length in millimetersand F the force in grams, was used to describe thestiffness of the parallel elastic element for musclelengths greater than 80$ of L,,, and (dF/dL) P E= 0 for lengths less than 80$ of Lp. Since theparallel elastic element shares the muscle load withthe series elastic element, the load on the parallelelastic element was subtracted from the serieselastic load of the curve in Figure 4. The deriva-tive of the equation fitting the resulting data wasgiven by (dF/dL)S E = 6.76F, where L is theseries elastic extension in millimeters and F isgreater than 0.2 g.

The corrected force-velocity data were fittedwith hyperbolas (Fig. 9B) having changes inboth a and b. The values of b were smallercompared with those of the two-element modelbut did not chance significantly with musclelength, remaining between 0.20 and 0.22 musclelengths/sec over the entire range of lengthstudied. The values of a for these corrected curveswere less than those of the two-element model,and the maximum shortening velocities werehigher (Fig. 10).

It is not possible to define the contractile unitlength for the Maxwell model corrections. ThereLs no force on the series elastic element at rest andthe extensibility of the series elastic elementcorrected for the Maxwell model below 0.2 g isnot known. As a result, the full amount of thecontractile unit shortening during isometric forcedevelopment could not be calculated. Because ofthis uncertainty, the values of a and maximum ve-locity are plotted as functions of isotonic shorten-ing rather than contractile unit length in Fig-ure 10.

The contractile unit length at which thevelocity measurements were made has beenconsidered to be identical for all points on anysingle force-velocity curve. This is valid whetherthe data were analyzed for either a two-elementor a Voigt model. In the Maxwell model,however, this is not strictly correct. Here theprecise contractile unit length depends on theload transfer from the parallel elastic element tothe series elastic element during isotonic shorten-ing and the compliance of the series elasticelement. However, the differences in contractileunit lengths did not exceed 156 of Lp for any ofthese curves.

References1. EDMAN, K.A.P., AND NILSSON, E.: Mechanical

CircuUtio* Rtstarcb, Vol. XXXI, August 1972

parameters of myocardial contraction studiedat a constant length of the contractile element.Acta Physiol Scand 72:205-219, 1968.

2. BRADY, A.J.: Time and displacement dependenceof cardiac contractility: Problems in definingthe active state and force-velocity relations.Fed Proc 24:1410-1420, 1965.

3. SONNENBLJCE, E.H.: Determinants of active statein heart muscle: Force, velocity, instantaneousmuscle length, time. Fed Proc 24:1396-1409,1965.

4. NOBEL, M.I.M., BOWEN, T.E., AND HEFNER, L.L.:Force-velocity relationship of cat cardiacmuscle, studied by isotonic and quick-releasetechniques. Circ Res 24:821-833, 1969.

5. SONNENBLICX, E.H.: Implications of musclemechanics in the heart. Fed Proc 21:975-990,1962.

6. HENDERSON, A.H., FORMAN, R., BRUTSAERT, D.L.,AND SONNENBLICK, E.H.: Tetanic contractionin mammalian cardiac muscle. Cardiovasc Res5(suppl. l):98-100, 1971.

7. SONNENBLICK, E.H.: Active state in heartmuscle: Its delayed onset and modification byinotropic agents. J Gen Physiol 50:661-676,1967.

8. NORHIS, C , AND CARMECI, P.: Isotonic muscletransducer. J Appl Physiol 20:354-356, 1965.

9. BLJX, M.: Die Lange und die Spannung desMuskels. Skand Arkh Physiol 3:295-318,1892.

10. Applications Manual for Operation Amplifiers.Philbrick/Nexus Research Co., Dedham, Mass.,1968.

11. SONNENBLICK, E.H., Ross, J., JH., COVELL, J.W.,AND BRAUNWAUD, E.: Alterations in restinglength-tension relations of cardiac muscleinduced by changes in contractile force. CircRes 19:980-988, 1966.

12. HILL, A.V.: Series elastic component of muscle.Proc R Soc Lond [Biol] 137:273-280, 1950.

13. HILL, A.V.: Heat of shortening and the dynamicconstants of muscle. Proc R Soc Lond [Biol]126:136-195, 1938.

14. BUCHTAL, F., AND KAEZER, E.: Rheology of cross-striated muscle fibre with particular referenceto isotonic conditions. Dan Videns Sels BiolMed 21:1-318, 1951.

15. ABBOTT, B.C., AND WILKIE, D.R.: Relation be-tween velocity of shortening and the tension-length curve of skeletal muscle. J Physiol(Lond) 120:214-223, 1953.

16. CORDON, A.M., HUXLEY, A.F., AND JULIAN, F.J.:

Variation of isometric tension with sarcomerelength in vertebrate muscle fibres. J Physiol(Lond) 184:170-192, 1966.

17. TAYLOR, S.R., AND RUDEL, R.: Striated musclefibres: Inactivation of contraction induced byshortening. Science 167:882-884, 1970.



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18. PODOLSKY, R.J., AND TEICHHOLZ, L.E.: Relationbetween calcium and contraction kinetics inskinned muscle fibres. J Physiol (Lond)211:19-35, 1970.

19. JULIAN, F.J.: Effect of calcium on the force-velocity relation of briefly glycerinated frogmuscle fibres. J Physiol (Lond) 218:117-145,1971.

20. RUDEL, R., AND TAYLOR, S.R.: Striated musclefibres: Facilitation of contraction at shortlengths by caffeine. Science 172:387-388, 1971.

21. HUXUEY, A.F., AND SIMMONS, R.M.: Proposed

mechanism of force generation in striatedmuscle. Nature (Lond) 233:533-538, 1971.

22. BflUTSAERT, D.L., CLAES, V.A., AND SONNENBLICK,E.H.: Velocity of shortening of unloaded heartmuscle and the length-tension relation. CircRes 29:63-75, 1971.

23. HEFNER, L.L., AND ROWEN, T.E.: Elastic compo-nents of cat papillary muscle. Am J Physiol212:1221-1227, 1967.

24. POLLACK, G.H.: Maximum velocity as an index ofcontractility in cardiac muscle: Critical evalu-ation. Circ Res 26:111-127, 1970.

Ctrctdation Rtst.nh, Vol. XXXI, August 1972



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Robert Formon, Lincoln E. Ford and Edmund H. SonnenblickEffect of Muscle Length on the Force-Velocity Relationship of Tetanized Cardiac Muscle

Print ISSN: 0009-7330. Online ISSN: 1524-4571 Copyright © 1972 American Heart Association, Inc. All rights reserved.is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231Circulation Research

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