MYOCARDIAL. MECHANICS AND CONTRACT {LITY Thesis for the Degree 0‘ M. S. MICHIGAN STATE UNIVERSITY Chang-Yi Wang 1976 _ JillHHIIUHIIHHHHIIHUIIHHIHHIHUHIIHHIHIUW 3 1293 10381 3428 '-‘ *‘ Kw’b “'W‘.‘ 3.1%.!" mué I .L‘n‘ ‘ I! I r , n «L; 4 " I. , a: ,_ - .g —..' ‘.‘-.;"‘I"’ .-.., ‘ a ".t .' ‘ g. 5 T.“ v ’ ”w b ‘Wu‘n I- "a' I .t . “must:— *1} ‘9 vi? 5-. ' ”Wrap MYOCARDIAL MECHANICS AND CONTRACTILITY BY Chang-Yi Wang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physiology 1976 ABSTRACT MYOCARDIAL MECHANICS AND CONTRACTILITY BY Chang—Yi Wang This work is a study of the mechanics of cardiac con- traction and its quantification. A comprehensive review of both theorectical and experi- mental reports shows that differences of Opinions exists regarding the basic characteristics of the contractile prop- erties of the myocardium. Controversy still rages as to which index best describes contractility. A theoretical analysis, based on Hill's three element models, 1. yields the following result: The maximum velocity of shortening of muscle at no load (Vmax), obtained from isotonic contraction experiments, does not characterize the contractility of cardiac muscle. The only index of contractility that is theoretically sound is the maximum velocity of shortening of the contractile element at no load (VmCE). The index V calculated from the Maxwell model, mCE’ is less dependent on preload than that calculated from the Voigt model. A CKNOWLEDGMENT S The author wishes to express sincere appreciation to Professor Jerry B. Scott, who has given encouragement and guidance throughout the course of this research, and to ProfessorsLester F. Wolferink and Donald K. Anderson who served as members of the guidance committee. The author also expresses his gratitude for the support of Professor Joseph E. Adney, Chairman of the Department of Mathematics. Thanks are due to Dr. John C. Yeager and Mr. Ronald Kienitz for their help in obtaining the experimental data, to Dora Wang for the reading of the manuscript and to Jill Shoemaker for the typing. ii ACKNOWLEDGMENTS . . TABLE OF CONTENTS . LIST OF TABLES . . LIST OF FIGURES . . I. II. III. IV. INTRODUCTION REVIEW OF LITERATURE . . . . . TABLE OF CONTENTS Functional Anatomy of Muscle The Law of the Heart and the Relation . Resting Tension MUSCLE MECHANICS . Mechanics and Modelling Hill's Basic Models . . . . The Force—Velocity Relation Contraction Length-Tension The Elastic Properties of Cardiac Muscle The Active State ANALYSIS OF THE THREE ELEMENT MODELS . . . 'The Three Element Models . . Proof that Vmax (of Myocardium) is Length Dependent Calculation of V Elastic Properties from the Resting Tension CE Curve and Quick Release Experiments . . Some Important Deductions from the Analysis. for Three Element Models. ii iii vi 12 12 l3 19 29 33 4O 4O 4O 44 46 48 V. MECHANICS OF THE INTACT HEART Geometry . . . . . . . . . . . . . . . . Properties of the Intact Ventricle . . . Problems Associated with the Intact Heart VI. CONTRACTILITY . . . . . . . . . . . . . . . The Index of Contractility . . . . . Theoretical Basis of V for the Intact Heart . . . . . . . . .ng. . . . . . . . Some Criticisms of the Simplified Theory Indices Using the Pressure Tracing“ . . . Discussion and Other Methods . . . . VII. AN EXPERIMENT ILLUSTRATING THE CALCULATIONS Purpose of Experiment . . . . . . . . . Materials and Methods . . . . . . . . . . .Result of Experiment . . . . . . . . } . Discussion . . . . . . . . . . . . . . . VIII. CONCLUSIONS AND SUGGESTIONS . . . . . . . . APPENDIX A. EXPERIMENTS ON ISOLATED MUSCLE . . . 'Isometric Contraction . . . . . . . . Isotonic Contraction . . . ... . . . Quick Release Methods . . . . . . . . Length Clamp on CE . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . iv 53 53 54 56 59 59 63 66 69 73 76 76 76 81 82 84 86 86 91 91 93 95 LI ST OF TABLES Table 1 Comparison of three indices rf a... P1 fir. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1" \IO‘U'IIwa 10 ll 12 l3 14 15 16 17 Al A2 A3 A4 A5 LIST OF FIGURES Sarcomere length and active tension Comparison of resting and total tensions Hill's two element model Hill's three element models Length response during isotonic twitch The force-velocity curve Comparison of Vmax and Vm (Parmley et a1 1972) CE Load clamp (Brutsaert 1971) Techniques in studying the active state Effect of quick-release on active state (Jewell and Wilkie 1960) Stress—strain relations Modelling the ventricle Extrapolation for VmCE Effect of epinephrine on p and dp/dt Effect of volume infusion on p Effect of (dp/dt) /p (dp/dt) /p epinephrine on Effect of volume infusion on Tension development for isometric tetany Isotonic contraction setup Isotonic response Isometric quick-release Isotonic quick-release vi and dp/d t p. ‘ int str thr the til RV Q» CHAPTER I INTRODUCTION The sole function of the heart is to maintain adequate circulation through its pumping action. The output of the heart is directly preportional to the frequency of contrac- tion, which is regulated through the nervous system, and the stroke volume. The stroke volume is in turn dependent on three factors: the degree of filling at end diastole (preload), the arterial pressure at systole (afterload), and the contrac- tility of the myocardium. ’ The present investigation is concerned with the contrac- tility of the heart, an inherent property of the heart muscle. It should be independent of extrinsic factors such as loading or heart rate, Which fluctuate with Changes in postural move- ments and in physical and/or psychological environment. How can we tell whether a certain heart is intrinsically strong or weak? In other words, what is contractility and hOW'dO we measure or calculate it? The quantification of contractility orzn1index of contrac- tility is important in the clinical evaluation of heart patients. Because of the various compensatory mechanisms in the human body, severe cardiac dysfunction may not be detected until late in the course of the heart disease. Also, some criterion is needed to assess the recovery of post—surgical heart patients. C81 and + I ~lli. Unfortunately, such an index is still elusive in spite of the voluminous literature published on the subject. About twenty indices have been proposed, most of which appeared in papers published in the past decade. Some are better accepted than others, but none has gained universal accep— tance. The profusion of papers published on contractility (one hundred in the past five years) erupted about fifteen years ago when it was discovered that the analysis of skeletal muscle mechanics developed by Hill in the late thirties did not apply to cardiac muscle. The complex properties of both the myo- cardium and the intact heart, alone with difficulties in physical measurements, have resulted in confused and contra— dictory opinions. The experiments are seldom reproducible; the theories are based on assumptions which are inadequately justified; and the conclusions seldom agree with each other, not only between different groups but also within the same group, published in different years. Indeed, if a single re- port is read, the proposed index, at a first glance, seems to be perfect both in theoretical and in experimental justifica- tion. But after a year or two the same index is invariably refuted and another "perfect" index is proposed. This does not, however, imply carelessness or the incompetence of the researchers, but rather reflect the complexity of the problem and lack of precision in the definition of the term "contrac- tility". fol It is not the object of the present report to add to the controversy by proposing another "better" contractility index. Instead, the purpose of this work is to present the following: 1) 2) 3) 4) 5) A modern review of Hill's theory of muscular con— traction incorporating more recent work based on Huxley's sliding filament hypothesis, The differences and similarities between skeletal and cardiac muscle, Derivation and comparison of the principle medhan- ical models. A critical objective review of the different con— tractility indices pr0posed in the recent literature, and, The proposal of theoretically sound indices and methods of measurement. Therefore, this thesis reports the necessary analysis which must precede attempts to theoretically and experimentally determine the elusive, and hopefully ideal, index. Perhaps the present study will help clarify the current state of the art regarding controversial subject. CHAPTER II REVIEW OF LITERATURE Functional Anatomy of Muscle Contraction Under the electron microscope, cardiac muscle differs from skeletal muscle only slightly. Due to the greater energy requirements, cardiac muscle contains a proportionally larger amount of mitochondria and capillaries. Cardiac muscle cells also appear to form a syncytium, in that their outlines are indistinct and they seem to fuse with one another. A closer look reveals this syncytium is only a functional one. The cells are actually distinct, separated from one another by sarcolemma (cell membrane) and by intercalated disks. Unlike skeletal muscle, cardiac muscle has the ability to generate spontaneous, rhythmical contractions. After stimula- tion , a longer delay is required (40 msec) before contraction starts in cardiac muscle, as compared with skeletal muscle (3 msec). Also the absolute refractory period is much longer, lasting throughout systole. Because of the long absolute re- fractory period, tetanic contractions cannot be attained by artificially increasing the frequency of stimulation. As we shall see later, this fact restricts the determination of certain myocardial properties. A recent paper (Ford and Forman, 1974) showed that tetany can be attained with large amounts of caffeine and calcium, however, the side effects of such drugs on other pertinent properties of the myocardium have not been delineated. 811) was ext and the act the the: COT: Cardiac and skeletal muscle are both striated. The basic unit is the sarcomere, composed of the thicker myosin (A band) and the thinner actin (I band). On the basis of a series of x—ray diffraction and electron microsc0pic studies, Huxley and Hanson (1954), Hanson and Huxley (1955), Huxley (1957) proposed the "sliding filament" hypothesis. The theory 'was originally applied to akeletal muscle but was subsequently extended to cardiac muscle (Huxley 1961) and subsequently verified (Stenger and Spiro 1961). The sliding filament theory is based on the observation that the actual lengths of both actin and myosin filaments are constant, both at rest and during contraction. With the activation of the sarcomere, the proposed "cross bridges" on the myosin begin to propel actin filaments further into the A band, causing contraction. Assuming the strength of the sarcomere is porportional to the amount of overlap between actin and myosin fibers, one can then relate muscle length to developed tension in what is commonly called the length-tension relationship. The Law of the Heart and the Length-Tension Relation The fact that muscles contract more forcibly when stretched ‘was known well before Starling's time. Notable experiments on the frog heart was done by Roy (1879) and Frank (1895) who laid down the fundamental concepts. Howell and Donaldson (1884), working on the heart of the dog, constructed the ascending limb of "Starlings curve". Starling's own work did not appear until 1912 (Knowlton and Starling 1912, Patterson and Starling 1914, Tension developed C U _-—=‘—-—- L 50 l O 150 2.1g Sarcomere length % optimum actin {- mYOS in Fig. l Sarcomere length and active tension Howe in c Law 1 law". can k curve Starling 1918). Although his experiments were more refined, Starling's works appear almost 30 years later than that of Howell and Donaldson. Perhaps we should define the increase in contraction due to the stretching of heart muscle as the Law of the Heart instead of the conventional term "Starling’s law", of the "Frank-Starling relation". It is now generally accepted that the Law of the Heart can be explained by the ascending portion Of the length-tension curve constracted by Huxley's sliding filament theory. This was done by Sonnenblick et a1 (1963), Spiro and Sonnenblick (1964), Hanson and Lowy (1965), Gordon et al (1966). Fig. 1 shows the length-tension relation of a single sarcomere. Maximum active tension is developed at sarcomere lengths be- tween 2.0 and 2.2 microns for both skeletal and cardiac muscles. At these lengths the maximum interaction of cross bridges occurs. The active tension decreases for shorter sarcomere lengths due to interference of actin fibers. It again decreases when the sarcomere is stretched beyond 2.2 microns where the amount of overlap (and thus the number of cross bridges en- gaged) becomes less. For a population of sarcomeres, as in the whole muscle, the length—tension curve is similar but rounded at the corners. The theory agrees well with experi— ments on tetanized, isometrically contracting muscles (See Appendix A for dexcription). With the development of increasingly accurate instruments, the sliding filament theory was obscured by several new experi- ments published in recent years. Rfidel and Taylor (1971) found that the tension produced at short sarcomere lengths were greater than the theoretical predictions set forth by Gordon et a1 (1966). Furthermore, Close (1972) found that the de— scending part of the curve changes lepeau22.8 microns. These facts lead to the speculation that there exists another, hitherto unknown, mechanism of contraction besides that intro— duced by the sliding filament theory. Most experiments, how— ever, were done on skeletal muscle and no evidence of comparable quality was available from studies of cardiac muscle, although it would be natural, albeit presumptuous, to assume that the sliding filament theory would apply to all striated muscles. The reason for the poor cardiac muscle data lies in the fact that tetany is impossible for cardiac muscle, the contraction measurements are highly time-dependent, and the degree of acti— vation is questionable (Close 1972). Resting tension One of the greatest differences between the mechanical properties of skeletal muscle and those of cardiac muscle is resting tension. As distinguished from active tension which is caused by the stimulated contraction of myofibrils, resting tension represents the elasticity of the fiber itself and can be measured by pulling the unstimulated sarcomere to different lengths. The total tension, or the maximum force produced by the fiber after stimulation, is the sum of active tension and resting tension. tension tension Fig. 2 length, length, 9 skeletal muscle cardiac muscle active tension -— — - resting tension ----- natal tension Comparison of resting and total tensions 10 In skeletal muscle, the resting tension (resistance to passive pulling) is negligible until the sarcomere is pulled to 2.42 microns (110% Of optimum length Of contraction). Thus, the total tension is equal to the active tension in most phys— iological stiuations. Unlike skeletal muscle, there is con- clusive evidence that the myocardium develops considerable tension When stretched (Sonnenblick 1962, Brady 1965, Parmley and Sonnenblick 1967, Edman and Nilsson 1968, Grimm et al 1970, Brutsaert et al 1971). Fig. 2 shows a comparison between the resting tensions Of skeletal and cardiac muscle adapted from the results Of Spiro and Sonnenblick (1964). Although active tension is similar (but not identical) for both muscles, resting tension is much larger for cardiac muscle. Grimm et a1 (1970) showed that when cardiac muscle is pulled beyond 2.2 microns (85% Of Optimum length of sarcomere), resting tension cannot be ignored. After 2.2 microns, resting tension contributes to an appreciable amount to total tension, reaching nearly one half of total tension at 110% of optimum length. Since, in normal intact hearts the sarcomere lengths are usually increased to 92% of Optimum and sometimes beyond Optimum (Braunwald etefl. 1968), resting tension becomes an important factor in the modelling of the myocardium. Due to resting tension, the shapecxfthe total tension curve is also quite different for skeletal and cardiac muscles. While.the total tension Of the myocardium rises monotonically with initial length, that Of the skeletal muscle shows a Charac— teristic decrease between Optimum length and 120% of Optimum. 11 Measurements on skeletal muscle are thus easier since the slope of total tension is zero near optimum length, making contractile forces insensitive to initial stretching in the neighborhood Of Optimum. Cardiac muscle does not have this property. Alt} 3101'} my 120‘; l. P; ‘lti. ica ml... 0. pH» is 7",}: t. .PL fit) ‘IK «NW A .w CHAPTER III MUSCLE MECHANICS Mechanics and Modelling The steady state or time independent behavior Of muscle has been discussed in Chapter II where the maximum force developed was related to the initial length Of the fiber. Steady states occur when the muscle is at rest or at tetany. Although skeletal muscle shows some steady state behavior, normal cardiac muscle functions without any discernable steady state. This time dependence greatly complicates both the ex- perimental and the theoretical investigations. During unsteady motion, the relationship among time, force and velocity is called mechanics. The first step in formulating a mechanical theory is to utilize experimental Observations to construct a simple mechan— ical model which may describe the behavior. From the model an equation linking force, displacement or velocity is written. The solution of this equation is then cheCked with the original experiments. Modifications Of the model is usually necessary to eliminate gross discrepencies between theory and experiment. There are times when the phenomenon is so complicated that no simple model is able to describe the behavior. Difficulty with the experimental isolation Of the effects of certain fac— tors is usually a deterrent to the construction Of a successful model. l3 Hill's Basic Models Based on a series Of experiments using the frog sar— torius muscle, Hill (1939, 1950, 1953) laid the foundation Of muscle mechanics by constructing mechanical models to des— cribe skeletal muscle behavior. He idealized the muscle into passive, elastic elements and active, contractile elements. These elements exist as separate entities only functionally. There is no structural or anatomical separation in the muscle itself. For instance, the elastic behavior Of the elastic element may represent the sum Of the elasticity due to the sarcolemma, the connective tissues between the muscle fibers, the crossbridges and the proteins in the myofibrils. Similarly, the contractile element may be composed of the action part Of the crossbridges, say, in the sliding filament theory and/Or some other hitherto unknown contractile mechanism discussed previously. Since the muscle does show active contraction when stim- ulated, there is no doubt about the existence Of a (functional) contractile element (CE). The existence of an elastic element (SE) in series with the contractile element is suggested by the following experiments. Firstly, when the muscle is tetanized isometrically the force developed does not rise instantaneously to the maximum value determined by the length—tension curve. Some delay (milliseconds) is needed for the tension to rise gradually from zero (Fig. Al, Appendix A). This indicates the contractile element may be pulling some spring—like material in 14 Fig. 3 Hill's two element model SE SE .1 PE PE I CE H CE "Voigt" model "MaxwelI'model Fig. 4 Hill's three element models #91. 1r. Y. nu. .NJ tk. .\ 16 the muscle. Furthermore, if a muscle at isometric tetanus is suddenly released to a smaller constant load (isotonic quick release), there is an initial instantaneous shortening followed by a slower shortening Of the fiber (Fig. A5, Appendix A). The initial shortening is attributed to the elasticity Of an elastic element in series with the contractile element which reacts more slowly. Some Oscillation in the length is also Observed, suggesting a spring-like behavior. Appendix A des- cribes some of these experiments. 2 Hill then suggested the two element model for skeletal muscle (Fig. 3). Since the sliding filaments in the contrac— tile element cannot sustain any tension in the unstimulated steady state, because the "crossbridges" are unattached, the two-element model can not produce any resting tension. In order to take into account the resting tension development by both skeletal and cardiac muscles, Hill further suggested the three element models (Voigt and Maxwell) in Figure 4, Where a parallel elastic element (PE) has been added for the resistance to passive stretch. These three element models are discussed in more detail in the next chapter. The manner in which the stimulated model would respond to external loads depends primarily on three factors: 1) The shortening prOperties of the CE defined by a "force-velocity" relation at a given instant, 2) The elastic properties Of SE, PE defined by "stress- strain" relations Of these "springs“. shortening total load 2 gm Fig. 5 time length response during isotonic twitch Vmax \ extrapolation 0 experimental data preload total load Fig. 6 The force—velocity curve 19 3) The time allowed for the force to develop, defined by an "active state" of contraction. These factors shall be discussed subsequently. The Force-Velocity Relation The basic experiment which relates the load and the velocity of shortening Of CE is the isotonic contraction experiment described in Appendix A. By increasing the weight Of the afterload (and thus the total load),mthe amount of shortening of the muscle fiber after a twitch can be repre- sented as a series Of time dependent curves shown in Fig. 5. In these experiments, the initial length is the same, determined by the same small preload. It is seen that the larger the afterload, the longer the delay in shortening Of the fiber. The amount Of shortening is also less. There exists a maximum total load Pb beyond which the muscle is unable to show any shortening after stimulation. Such contractions are entirely isometric. The maximum slope for each curve in Fig. 5 (the maximum velocity) is usually at the initial instant Of shortening. This is plotted against total load P in Fig. 6. Since the minimum total load is the preload, which maintains the initial length and cannot be zero, the curve has tO be extrapolated to zero load by mathematical or visual extension Of the curve. This is the famous force—velocity curve of Hill (1938) showing the inverse relationship between load and maximum velocity of shortening. as fo devel cross short tach Since time, decre Hill ening ening short the m liHEa We ha 20 Conceptually, this inverse relationship can be explained as follows. From the sliding filament theory, the tension developed at any instant is prOportional to the number of crossbridges attached. When an active muscle is allowed to shorten, some crossbridges must be allowed to detach and re- tach as the myosin and actin filaments slide past each other. Since the process of detachment and retachment takes finite time, the number Of attached (force producing) crossbridges decreases in proportion to the shortening velocity. By measuring the heat produced in isotonic contractions, Hill (1938) found that the extra heat produced during short— ening is independent Of both the load and the time Of short— ening. It is, however, directly proportional to the distance shortened. Furthermore, the rate of extra energy released by the muscle during isotonic contraction is found to be decreasing linearly with increasing tension. Since the first law of ther- modynamics states that _d_ the rate of extra energy released = dt (work done + (1) extra heat released) we have 16(120— p) = 5%(Px-I-ax) (P+a)V. (2) Here a,b are proportional constants, x is the distance shortened, and V is the initial velocity of shortening When most of the heat is released. rn 21 From the values Of a and b Obtained from experiments on the frog sartorious, Hill found that the theoretical force— velocity curve (Eq. 2) agreed well with that from the iso— tonic contraction experiments. Rearranging Eq. 2 we find the curve is actually a hyperbola: (P4—a)(V-+b) = b(Pb-ta) = constant (3) Since the resting tension is negligible for skeletal muscle at physiological lengths, Hill used the two element model to explain the contractile behavior in his experiments. During the isometric phase after a stimulated twitch, tension rises because the contractile element pulls against the series elastic element. When tension exceed the total load, the muscle shortens with a jerk. The isotonic phase then starts ‘with no further changes in the length of the series element, for the toad it sustains is now constant. With the passing of the active state Of the muscle, the shortening becomes slower and finally the muscle begins to relax and the CE can no longer lift the load. The important result for the two—element model is that Vmax, defined as the maximum velocity at zero load, is independent Of both preload and afterload. It is thus an inherent property of the muscle. Fig. 3 shows that without load, the length Of SE would remain the same and Vmax of the whole muscle would be identical to the max velocity of CE itself. It is thus a measure Of contractility., The similarities in structure between skeletal and cardiac muscles led naturally to the application of Hill's concepts to 22 the myocardium. In 1959, Abbott and Mommaerts concluded that the behavior Of cardiac muscle was similar to that of skeletal muscle, on the basis that cardiac muscle also seemed to show a hyperbolic force-velocity relation. Two years later, Sonnenblick (1961, 1962) conducted experiments on the papillary muscle Of the cat and observed that the force- velocity relation for different initial fiber lengths could be extrapolated to a single Vmax at zero load. Following Sonnenblick, numerous papers were written using Vmax thus Obtained as an index Of contractility. The differences be- tween cardiac and skeletal muscle were not seriously appreci— ated unitl the appearance Of a paper by Pollack (1970) dis- puting the use of Vmax (from the force velocity curve) as an index of contractility. Although Pollack's work was quite convincing, contemporary texts in cardiovascular physiology (Folkow and Neil 1971, Berne and Levy 1972, Montacastle 1974) still carry Sonnenblick's earlier analysis. The reason is perhaps partly due to a lag in the spread Of knowledge and partly due to the fact that there exists no universally accepted substitute index to gain insight into the contractility of car- diac muscle. Pollack (1970) recognized that the resting tension in cardiac muscle could not be ignored and that one must go back to Hill's three element models which have extra parallel elastic elements (PE). There is, thus, a distinction between the force-velocity curve Of the cardiac muscle and the force— velocity curve Of the contractile element Of cardiac muscle. 23 Pollack then used Sonnenblick's (1962) original data to calculate a new force-velocity curve for the CE alone. It was found that for both three-element models (Fig. 4), the maximum velocity Of CE at no load (V mCE dependent of preload or initial fiber length. Conceptually ) is not in— from Fig. 4, it is apparent that the velocity Of CE is definitely altered by the elastic properties Of PE. The mathematical proof is given in the next chapter. Pollack's work, being theoretical, isfldevoid of the un- certainties encountered in experimental reports. Although Noble et a1 (1969) experimentally refuted the independence of Vmax earlier, his quick—release methods are now under criti- cism. It is apparent that the contractile element of one model is not the same as the contractile element Of another model. Pollack's recalculation of Sonnenblick's data shows a smaller variation Of Vm with respect to preload for the Voigt model CE than for the Maxwell model. However, even using the Voigt model the variation is still about 50%6 There are two points one should keep in mind in regards to Pollack's analysis. Firstly, the cardiac muscle data Of Sonnenblick (1962) is atypical Of the many experiments published by this group (Sonnenblick 1970). This does not, however, seriously affect Pollack's conclusions. Secondly, in the analysis, the maximum velocity Of shortening is assumed to be immediately after the initiation of isotonic contraction. Studies by Parmley and Sonnenblick (1967), r: .P—sl~ a Try c:— 13 -H AU % Of maximum 100 80 7O 0 Vmcs A V11“ ax L 1 1 n 88 92 96 100 Muscle length % of Optimum Fig. 7 Comparison of Vmax and VmCE (Parmley et a1 1972) Parsley shorten Ta account ments a maximum 25 Parmley et a1 (1972) indicated that a significant amount of shortening had already taken place at the time of peak velocity. Taking the above two criticisms Of Pollack's work into account, Parmley et a1 (1972) conducted the isotonic experi— ments again, using on—line differentiation to Obtain the maximum velocity Of shortening. Both Vmax and VmCE were calculated, using Pollack's method. The result for the Maxwell model is shown in Fig. 7. For some reason, similar calculations for the Voigt model was not done. It seems V is a better mCE index than Vmax due to the following: 1) VmCE" the maximum velocity of CE at no load, is independent Of the elastic elements by definition. It is, thus, theoretically length independent. Vmax is not. 2) As shown in Fig. 7, V is experimentally less mCE dependent on initial length or preload than Vmax, especially at higher loads. .3) The calculation Of Vfi does not require extrapola— CE tion to zero total load. This extrapolation, neces— sary in the calculation Of Vmax, induces a good deal or error. In some cases Vmax can not be calculated for muscle lengths longer than Optimum length, because the force—velocity curve is sufficiently removed from the usual hyperbolic form, making extrapolation im- possible. 26 Also evident from Fig. 7 is the fact that even the better index, VmCE’ ideal index is thus still elusive. shows 20% variation with preload. The Let us return to the shape Of the force—velocity curve. Hill showed, from simple thermodynamics and semi-empirical results, that the curve should be hyperbolic (Eq. 3). Perhaps this hyperbolicity has been overly emphasized by later workers. As far as we know, there exists no theoretical proof showing that the force-velocity curve should be hyperbolic. Further- more, whether the curve is hyperbolic or not has no relation to Hill's mechanical models. Consider the following: 1) There exist other forms of the force—velocity re- lation. Penn and Marsh (1935) and Aubert (1956) used exponential relations of the type P = Poe—bv - cV (4) or P: (P0+c)e—bV-c (5) Polissar (1952) found l-P/PO P/PO-l V = C(a - b ) (6) By adjusting the constants a,b,c,P , all the above forms fit the experimental ata perfectly. Hill, however, has thermodynamic data to support his con— stants. 2) 3) 4) 27 In Hill's original (1938) experiments, the heat Of shortening was found tO be independent Of load. This point was investigated again by Hill (1964) using more SOphisticated instruments. It was found that the heat of shortening was in fact proportional to both Pb and P. This conclusion effectively invalidates the thermodynamic support Of his earlier hyperbolic relation. Fung (1970) showed theoretically that if Hill's equation were used, the time to reach the maximum force P0 in an isometric contraction would be in- finite. This would contradict the experimental re— sults. Fung suggested using a force-velocity relation of the type b(PO-P)a= (P+a)V (7) where a is an exponent between zero and one. Experiments on cardiac muscle showed that the force— velocity relation deviated appreciably from Hill's equation. 'Unlike Hill and Sonnenblick (1961), Brady's (1965) investigations showed no such hyperbolic re- lation. Similarly, Yeatman, Parmley and Sonnenblick (1965), using isotonic and isometric experiments; and Noble, Bowen and Hefner (1969), Brutsaert and I Sonnenblick (1969), using quick-release experiments showed hyperbolic relation only at lower loads. Hefner and Bowen (1967) even found bell-shaped force—velocity curves . 28 Finally, we note that Hill's force-velocity relation was obtained from skeletal muscle which can be tetanized. The contractile force was maximally activated before the load was lifted. For cardiac muscle, the degree Of activa— tion Of contraction is very much dependent on the transient qualities of the active state of the twitch. Sonnenblick (1965, 1967), Civan and Podolsky (1966), Parmley and Sonnenblick (1967), Parmley et a1 (1972) showed that there was a delay and a complex velocity transient before a steady velocity Of short- ening was achieved. Due to this delay, measurements of peak velocities of lightly loaded cardiac muscles would register lower values than those more heavily loaded, since the former are less activated at the time they are measured. Another Objection to the force—velocity curve was raised by Jewell and Wilkie (1958). According to Hill's theory, the transient rise Of tension during isometric tetranus of skeletal muscle must be precisely governed by the force-velocity curve and the force-length curve of the series element. Now caterers: EE‘det‘dLV (8) or t=‘f'a'§-;L——GF (9) ._._..V dL Here F is the force, L is the length and the slope dF/dL can be Obtained from the prOperties of SE as a function of the load F (assuming a two—element model) and the velocity 29 V is also a function Of F in the force—velocity curve. Jewell and Wilkie compared the rise of tension from experi— ment and from Eq. (9). It was found that the actual rise was not as fast as the theory predicted (50% error). It was suggested that the velocity Of shortening at maximum activa- tion depended not only on the load but also on the load history Of the muscle. The Elastic Properties of Cardiac Muscle Since the active contractile element cannot be physically isolated, the calculation Of the properties of CE requires the knowledge Of the passive elastic elements SE and PE. The basic assumptions about the elements Of the mechancial models,are: l. The contractile element, like the shock absorber on a screen door, does not resist extension but resists the extension rate. In other words, it is compliant to slow stretching and does not contribute to resting tension. For quick stretches and quick releases CE behaves as if it were rigid. This assumption is not imcompatible with the sliding filament theory. 2. The stiffness Of the elastic elements is a function Of extension only. It is not effected by either the velocity Of extension or the contractile state or inotropic drugs. This is an assumption of convenience. As we shall see later, the elastic elements in the muscle are not analogue to a mechanical spring. {’7 ll} " f 30 Assuming both assumptions to be valid, one can see that the resting tension curve discussed in Chapter II represents the stiffness Of PE in the Maxwell model. The result from quick release experiments yields the stiffness Of SE in the Voigt model. PrOperties of SE Of the Maxwell model and that of PE Of the Voigt model may be Obtained from mathe- matical calculation from the above experiments. Instead of force-length relations, it is more practical to use stress— strain relations for elastic material. Chapter IV gives the details Of the calculations. The series elastic element (Voigt model) has been in— vestigated through the isotonic quick—release method by several authors (Abbott and Mommaerts 1959, Sonnenblick 1964, Parmley and Sonnenblick, 1967, Yeatman et a1 1969, 1971). In all cases, the stress—strain relations were found to be nonlinear—- the stiffness, or the slope or stress as a function Of strain, increases nonlinearly with increasing strain. An empirical exponential curve can be fitted to the data: M - 1) (10) _.E o — k(e or = ko-tc (ll) Qslfln m Q Here 0 is the total stress, 6 is the natural strain and k,c are constants. From Eq. (11), we see that k must be nondimensional, while c has the dimension of stress (force/ area). For cat papillary muscle, Sonnenblick (1964), Yeatman 31 (1971) found k to be approximately 32; Hefner and Bowen (1967) Obtained the value 34; While Parmley et a1 (1967) gave k = 40. In general, each investigator agreed that Eq. (11) is a linear function, independent Of preload. It is probably true that k is indeed a constant, implying an exponential stress—strain curve. The variation Of reported values may have been due to the different experimental appa- ratus used. According to Parmley and Sonnenblick (1967), equipment distensibility error may amount to 30%. The value Of c was Obtained from the slope of the stress-strain curve at zero load. Since all experimental results can only be extrapolated tO zero load, this value is more uncertain. Yeatman (1971) gave the value Of c (at 37°C) as 20 gm/mm2 or 147 mm Hg. Assuming the magnitude of c to be of this order, and the pressure during systole to be equal to the stress in the muscle (i.e., wall thickness approximately equal to the mean inner radius of left ventricle), then the value of ko ranges from 640 to 3800 mm Hg and c can be ignored in comparison with k6, except at very low pressures. There seems to be comparatively less discussion of the empirical equation for the passive stress-strain curve Of the resting tension in the literature. Earlier works plotted the data in terms Of force and length which is quantitatively use— less. For instance, on a thicker muscle, a larger force would be needed to achieve the same extention. Since the cross- sectional area and the resting length were not given, it isrmH: possible to translate the force—length relation into the 32 stress-strain relation. Qualitatively the curve is still exponential, about half as stiffas the series element (Parmley et a1 1972). In experiments on the myocardium, the papillary muscle was invariably used. This was probably because strips of ventricular wall muscleéflmeimpossible to separate without destroying contractile behavior, and papillary muscle is more uniformly oriented. The elastic prOperties of the papillary muscle, however, do not reflect that of the ven— tricular wall muscles. Furthermore, it is not known whether there is a difference between the muscle Of the domesticated cat or dog and the muscle of the human heart. 113 the series element truly passive as assumed? ‘Wildenthal et a1 (1969) found that stiffness was affected by ihypertonic solutions. Yeatman et a1 (1969) determined the influence of temperature on SE. These factors, however, can be controlled in the experiments. The in situ conditions are probably constant. However, the prOperties of SE and also PE are found tO be dependent on velocity. Although velocity dependence (viscosity) of muscle had been discounted by Hill (1938), more recent studies (Sonnenblick et a1 1966, Little and Wead, 1971) clearly demonstrated stress relaxation (a fall in tension following sustained extension) in both SE and PE elements. Pollack et a1 (1972), Noble and Else (1972) showed that the "passive" properties Of SE 1 .LJ .a o o H m > time o m 0 '1 time Fig. 8 load clamp (Brutsaert 1971) tension shortening(muscle) shortening CE Hill (1949) Ritchie (1954) Brady (1968) /“\ ”\‘\ I \ I \ \ I x ‘ ' \ \ I \ \ I \\ \ I ¥ \\ *— L l—\ W ' Fig. 9 Techniques in studying the active state 37 3) Length clamp technique (Brady 1968). This method requires the determination Of the elastic properties Of the series element using the isotonic quick stretch method. A mechanical model is then chosen and the amount of stretch is calcu- lated through a computer such that the length of CE is kept constant. Using the length clamp tech— nique, Brady found that the tension Of CE rises slowly, in a similar fashion but somewhat faster than the isometric curve. The method Of course assumes all the inadequacies Of the mechanical modes. 4) Load clamp technique (Brutsaert el al 1971, a,b). By using abrupt load clamp during isotonic shortening, Brutsaert was able to show that the velocity Of shortening was dependent on the instan- taneous load and length and independent or prior load history. By switching the isotonic load it different values at different times, it was found that the velocity-length curve for any giVen load can be re- produced (Fig. 8). This independence of prior changes, occuring from onset Of contraction until peak Of shortening, led Brutsaert to conclude that maximum active state occurs immediately and is steady at maximum for a considerable portion Of the time. Fig. 9 depicts some typical results Of various techniques. It was found that all agreed that the activity Of CE occurs /.isotonic, light preload shortening time(sec) Fig. 10 Effect of quick release on active state (Jewell and Wilkie 1960) 39 earlier and the tension develOped is stronger than that developed by isometric stimulation. Both length clamp and load clamp are sophisticated techniques developed relatively recently. However, they yield conflicting results as to the rise Of the active state. In both cases, the papillary mus- cles are subjected to non—physiological maneuvers. The question is whether these maneuvers affect the active state itself. For instance, the quick—release method, so important in determining the elastic prOperties Of SE and active state properties, has been shown to alter the active state. Jewell and Wilkie (1960) studied length changes following isotonic release against the same load at different times during a twitch. Fig. 10 shows the results. It is seen that for the same twitch at the same time (at 0.5 sec, say) the muscle is found to be relaxing if release is early, and con— tracting if release is late. In other words, if a muscle is allowed to shorten, the duration Of active state becomes less. Brutsaert et a1 (1972), using velocity clamping, found that as clamped velocity decreases, the duration Of active state increases, reaching a maximum at isometric contraction. The question Of "active state" is still unsettled. We may safely say confusion still exists in its definition, determination, and interpretation. CHAPTER IV ANALYSIS OF THE THREE ELEMENT MODELS The Three Element Models The three element models were proposed by Hill (1938) to account for the resting tension in muscle. Later, authors used the names "Voigt" model and "Maxwell" model (Fig. 4). These are actually misnomers, for although Voigt and Maxwell both worked on similar two element models in viscoelasticity (contractile element replaced by a dashpot), it was Kelvin who devised the three-element viscoelastic model. The most apprOpriate model for cardiac muscle is still unclear. Using quick release experiments to test the prOp- erties of SE and PE, investigators favored either the Voigt model (Yeatman et a1 1969) or the Maxwell model (Hefner and Bowen 1967) or both (Parmley and Sonnenblick 1967). Brady (1967) found that it was necessary to classify each muscle in— dividually into either "Voigt" Of "Maxwell" types. However, Fung (1971) showed mathematically that by changing the para- meters and allowing force-length relations to be velocity de- pendent, in general, one model would be identical to another in performance. Thus, any one model would be sufficient to describe the muscle. Proof that erax (Of Myocardium) is Length Dependent Presented here is Pollack's (1970) refutation of Sonnenblick's (1962) claim that Vmax (of myocardium) is length (and load) 41 independent. original version, but the idea remains the same. define the following symbols F = force L = length V = velocity 51E S=dL The proof is slightly different from the Let us Vmax = maximum velocity Of muscle at no load on muscle .--—.. VmCE = max1mum veloc1ty of CE and subscripts CE = contractile element PE = parallel elastic element M muscle P = preload A afterload First, consider the Voigt model (Fig. 4). that Sonnenblick plotted FM against VM’ alone and FCE During isotonic contractions, FM arises from CE instead. V = 0. SE Therefore, should be plotted against V at no load on CE It is evident (1.2) (13) However, "contractility" CE is constant and 42 v = v (15) If the parallel element is absent (as in the two element model), then muscle force and velocity is identical to CE force and velocity. However, if resting tension is considerable, then Eq. (14) shows that a correction of -FPE should be made to FM. Now in Sonnenblick's work, the velocity of muscle is plotted as a function Of force on muscle: vM = f(FM) (16) Thus, using Eqs. (14—15), V = f(FCE-+F CE ) (17) PE In these experiments, Pollack estimated that FPE is maximum when the muscle is unstimulated. The elastic elements should also have no resistence to compression. Therefore, O‘g FPE_g FP (18) At the initial instant Of shortening, however, FPE = FP and FCE = 0. Therefore, VmCE = f(FP) (19) i.e., the maximum velocity Of CE is a function of preload or initial length. 43 For the Maxwell model, the analysis is more complicated. From Fig. 4 we see FM = FSE + FPE = FCE + FPE (20) VM = VPE = VSE + VCE (21) During isotonic contractions, FM is constant. Differentiating Eq. (20) with respect to time, 0 = dFSE + dFPE = dFSE dLSE + dFPE dLPE (22) (it dt dLSE dt dLPE dt °r O = SSEVSE + SPEVPE = SSEVSE + SPEVM (23) Using Eq. (21), S V PE M V + V (24) M SSE CE S = PE and VCE (1 + §——)VM . (25) SE Then Eq. (16) gives SPE ’ VCE = (1 + §—)f(FCE + FPE) (26) SE At the initial instant, again FPE = FP’ FCE = 0.. Also, the ratio of the slopes at the initial instant S is a func— PE SE tion Of the initial length or a function Of preload. SPE/SSE = 9(FP) (27) 44 Thus, vaE = [1 + gmpnfmp) (28) This is again a function heavily dependent on preload or length. Calculation Of VCE for Three Element Models Presented here is the general form for V The analysis CE' is similar to those suggested by Hefner and Bowen (1967) and Pollack (1970). First, consider the Voigt model. From Eq. (13), we have dL _ _ = _ SE VCE " VM VSE VM dt = VM ' diFsiiit (29' SE SE = V .. dFM/dt M - SSE For isotonic contraction, dFM = O and Eq. (29) reduces to Eq. (15) found previously. For isometric contraction, VM = 0 and dF VCE = " Elm—<17:fl (30) SE SSE is difficult to Obtain for this model, since the force- length relation for resting tension measures the sum of the lengths Of SE and PE. However, due to the assumption that CE is not responsive to velocity changes, quick stretch or quick release experiments should show the response of SE 45 alone. It seems the Voigt model is more suited for isotonic measurements, since VCE is in simple form, VCE ___ VM (31) As discussed previously (Parmley 1972), maximum V may CE not occur at the initial instant. If so, Eq. (19) does not hold and Vm may Occur during the isometric phase. CE For the Maxwell model, in general, Eq. (21) gives dL ___ _ = __ SE VCE VM VSE VM dt = v _ aFSE/at M dFSE/dLSE _ dFM/dt - dFPE/dt " VM ' s SE _ dFM/dt - SPEdLPE/dt _- VM _ s (32) SE = V _ __L 351 + 332V M SSE dt SSE M _ SPE 1 dFM — (1 + T)VM " E— 75+.— SE SE In the case of isotonic contractions, the last term is zero and Eq. (32) reduces to the correct form, Eq. (25). For an isometric contraction, Eq. (32) yields the same form as Eq. (30). VCE = " "é.“— "a? (33) Although the force—length relation for PE Of the Maxwell model can be Obtained by passively stretching the muscle by 46 varying the preload, the property Of SE must be Obtained by first using quick stretch or release to get the sum of the force response Of both SE and PE, and then subtracting the response Of PE from the resting tension. Whether quick stretch would alter the properties of PE and SE is still in question. Elastic Properties from the Resting Tension Curve and Quick Release Experiments The mechanical behavior Of elastic material is best described by the stress-strain relation, which is the normal— ized form of the force-length relation. Stress (o) is de- fined as the tension divided by the cross sectional area of the muscle, and Strain (6) is defined as the increase in length divided by the original length. Two experiments are then performed. The muscle is first slowly passively stretched and the following stress—strain relation is obtained for the resting tension curve 01 = G(El) (34) or inversely e = G‘1—c 40 p (nhtEg) of volume Cl: *0 infusion 81 The effect of the epinephrine decayed in one minute. Another control tracing was taken and then 6%)Dextran-7O (in 0.9%»NaC1 solution) was pumped continuously into the right fermoral vein. The pumping rate was 284 cc/min, until one liter of infusion had been accumulated, after which the rate was decreased to 170 cc/min, until a total of two liters had been infused. The end diastolic pressure had been rising steadily throughout the infusion. Result of Experiment Figure 14 shows pressure tracings with and without epinephrine. The lepe (or dp/dt) is seen to rise several fold due to the inotropic effect. Fig. 15 shows a typical pressure tracing with a 500 c.c. Dextran infusion. Minimum pressure and especially end diastolic pressure increased with increasing volume infusion. The flattened tops of the pressure curves are probably due to the amplitude limitation of the re- cording pen. This did not affect the result significantly, since most of the isovolumic phase had been recorded. Using a magnifying glass the pressure tracings Of (dp/dt) and p were read Off for various times during sys— tole. The ratio (dp/dt)/p was then plotted against pressure. The error involved from the reading was about 10%» This can be minimized by using an on-line computer to calculate the ratio. The effect of epinephrine is shown if Fig. 16. It is; seen that the positive inotropic effect of epinephrine greatly increases the ratio. Figure 17 Shows that volume 82 infusions lower the peak (dp/dt)/p, and the curves shift to the right. The end diastolic pressures (the intercepts on the abscissa) also increase with volume infusion. Discussion we were unable to calculate any index which involved time explicitly. This was because an electrocardiogram had not been done and the "R" peak of ECG was needed to set the starting time. It is also evident from Figs. 16—17 that extrapolation to zero load was impossible due to the large distances involved. We were, however, able to calculate that (dp/dt)max, (dp/dt)max/p, [(dp/dt)/p] = V ° k. These are max pm listed in Table 1. Table 1. Comparison of three indices (HE) max %(%E) max k me ) (mmHg/sec) (sec-1) (sec-1) Control 4500 52 82 epinephrine 11000 110 118 De§§5a2,c, 4700 90 68 500 c.c. 4700 94 56 1000 c.c. 5100 96 50 2000 c.c. 5900 100 —— All three indices were sensitive to norepinephrine (inotropic effects). All three were also affected, although not as much, by Dextran infusion (hypervolema, including 83 effects of preload increase). Since volume infusion should not influence contractility, these indices are not perfect- measures of contractility. Some significance, however, can be deduced from Fig. 16. The falling parts of the curves, those due to isovolumic contractions, seem to coalesce. This is the same region upon which VmCE depended, before the extrapolation to zero load. Thus, VmCE can indeed be independent of preload, as pre- dicted by the theory. One must stop here before making further inferences. The experiment was done on ggg_dog under less than ideal con— ditions. Each curve was laboriously plotted by hand over a single contraction. Statistically the curves do not lead to conclusions that can be generalized. The use of 7 to 14 dogs, as practiced by investigators, is not statistically significant either. CHAPTER VIII CONCLUSIONS AND SUGGESTIONS The forgoing review of the literature, and particularly the analysis of the three element models, results in the following conclusions of physiological importance. 1) The index Vmax, Obtained by traditional isotonic experiments, can not represent cardiac contractility. Contemporary texts on medical physiology still carry the earlier erroneous conclusion that cardiac muscle is similar to skeletal muscle, thus making Vmax a measure of contractility for both muscles. 2) Most current indices advocated in the literature are empirical, at best. The only index which has theoretical support is VmCE' All other indices investigated in this thesis cannot be justified theoretically. 3) The present work shows that VmCE calculated from the Maxwell model is better than that calculated from the Voigt model. I.e. the index is less de- pendent on preload when the Maxwell model is used. 4) A period of pure isotonic or pure isometric phase contraction is not necessary for the calculation of VmCE' 85 Suggestions for further work include the following: 1) 2) The theory for VmCE can be improved by incorpor— ating some of the criticisms discussed earlier. More experiments must be done on the intact heart in order to obtain viable elastic constants for the theory. The extrapolation process must be circumvented before VmCE can be extrapolated with confidence. A theory may be formulated utilizing the existing data on the descending isovolumic curve. APPENDIX A EXPERIMENTS ON ISOLATED MUSCLE Isometric Contraction Isometric means constant length. The basic idea is to hold the muscle at the ends between two fixed supports, one of which is attached to a tension transducer. The muscle is then simultaneously stimulated electrically at various points to assure synchronous contraction. Tetany (for skeletal mus— cle) may be achieved when the frequency of stimulation is above a certain "fusion frequency". A typical time response is shown in Fig. Al. The resulting maximum tension, plotted against the length of the sarcomere, is called the length— tension curve. The experiment is not as simple as it may seem. Huxley and Peachey (1961) found that sarcomere lengths in single muscle fibers are not constant along the length of the fiber, being shorter at the ends. Krueger and Pollack (1975) found that during cOntraction, the degree of contraction differs for individual sarcomeres, and some sarcomeres are actually being stretched by the shortening of others. In order to assure a truly isometric contraction, Gordon et al (1966) used an elaborate electro—Optical feedback system on a small seg- ment of the muscle, where the sarcomeres are almost identical in length and behavior. The feedback mechanism adjusts the supports to ensure constant sarcomere length in the segment during contraction. tetany tension CO \) Fig. A1 Tension develOpment for isometric tetany movable transducer Fig. A? Isotonic contraction setup afterload preload A. ~ [A shortening A tension #Jl/li isometric isotonic Fig. A3 Isotonic response :0 tension time Fig. A4 Isometric quick release 91 Isotonic Contraction Isotonic means constant load or tension. There are two kinds of loading. In free—loaded experiments, the muscle at rest is loaded and then stimulated. Since the initial length cannot be controlled, this kind of loading is seldom used. In after—loaded experiments, the initial length of the muscle is first determined by a suitable pre- load on a balance shown in Fig. A2. The st0p and tension transducer are then adjusted. The muscle is not loaded at rest, but must lift a load in order to shorten. The movement of the lever, which reflects changes in muscle length, is then recorded. A typical response of a muscle twitch is shown in Fig. A3. The maximum velocity of shortening (the slope at A) is then plotted against different total loads. This is called the force-velocity curve. Since the force—velocity curve is the result of a time dependent dynamic phenomena, care must be taken to minimize the error due to acceleration. This is usually done by short- ening the lever arm on the side of the loads. Quick Release Methods When a muscle at isometric tetanus is suddenly shortened to a new constant length, the tension drops abruptly then rises to a new maximum determined by the length-tension curve (Fig. A4). This is called isometric quick release. The rise of tension with respect to time from quick release experiments, however, is not identical to the rise of tension starting from rest at the same shortened length. AF l tension T . 40L shortening 1 Fig. AS Isotonic quick release 93 Another method is isotonic quick release. The apparatus is similar to that shown in Fig. A2, except there is a re— lease relay stOp below the lever on the muscle side. At the beginning of stimulation, the muscle contracts isometrically and the tension rises to its maximum value. When the release stop is withdrawn, the tension abruptly falls to a lower level which is equal to the loads. A typical response is shown in Fig. A5. The shortening of the length is abrupt at first, followed by some oscillations, and then proceed at a slower, constant rate. Also used are isometric and isotonic quick stretdhes where the muscle is suddenly lengthened. All the above methods not only help to determine the transient properties of the muscle, but also the elastic properties of SE, since CE is assumed to be non—responsive to quick changes in length. Length Clamp on CE This experiment is used by Brady (1968) to determine the transient force response of the contractile element at a fixed length of CE. Since during isometric contraction, CE shortens at the expense of SE, the method calls for active pulling of the muscle to compensate for CE shortening. The amount of pulling is dependent on time and the elastic prop- erties of SE and PE. To illustrate this idea, Hill's two element model can be used. Suppose the elastic properties of SE if found by quick stretch or other methods, giving the following relation be- tween the length of SE and the force on SE 94 LSE = fn(FSE) = fn(FM) (Al) then \ = = LM(t’ LSE+LCE fn(FM(t)) + LCE (A2) Since LCE is held constant, one can obtain the instantaneous muscle length (LM) from the instantaneous force on muscle (FM) through Eq. (A2). Brady used an elaborate on-line com- putorized feedback system to give the proper amount of pull. A similar relation between LM and FM can be derived for constant CE length in the case of the three element models. The method is very much dependent on the model selected. A typical response is shown in Fig. 9. It is seen that in com— parison to isometric twitch, the active state occurs earlier When CE length is fixed. 10. 12. 13. REFERENCES Abbott, B.C. and D.G. Gordon, A commentary on muscle mechanics, Cir. Res. 36: l, 1975. Abbott, B.C. and W.H.F.M. 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