Respiration Physiology, 87 (1992) 141-155 © 1992 Elsevier Science Publishers B.V. All rights reserved. 0034-5687/92/$05.00
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Analysis of the contraction of series and parallel muscles working against elastic loads G. Supinski, A.F. DiMarco, F. Hussein, R. Bundy and M. Altose MetroHeaith Medical Center. Department of Medicine, Pulmonary Division, Case Western Reserve University Cleveland. OH 44106, U.S.A.
(Accepted 8 October 1991) Abstract. The purpose of the present study was to analyze the manner in which series and parallel arrangements of respiratory muscles, contracting together, augment the forces and displacements applied to external elastic loads over those produced by a single muscle contracting alone. We first developed a series of mathematical expressions to describe the behavior of various arrangements of muscles contracting against elastic loads. We then compared the predictions ofthese equations with the results from experiments in which the forces and displacements produced by simple arrangements of muscles were measured. Both theoretical and experimental results indicate that, against high elastic loads, parallel arrangements of muscle strips produce greater forces and greater displacements than do single muscles; parallel arrangements do not, however, significantly increase the displacement or force applied to low elastic loads. Conversely, series arrangements result in greater forces and greater displacement of low loads, but are no better than single muscles when contracting against high loads. Against moderate loads parallel and series arrangements of muscles appear to be equivalent in generating forces and displacements during contraction. This analysis suggests that a major determinant of the effects of contraction of various networks of inspiratory muscles is the magnitude and character of the respiratory impedance against which these muscles must work. The primary difference between series and parallel arrangements of muscles is that muscles arranged in series are most effective against low elastic loads and muscles in parallel act most effectively against high loads.
Elastic load, respiratory muscles; Respiratory muscles, force, series vs parallel arrangement
The manner in which the various inspiratory muscles interact with each other and with the passive impedances provided by the lung and chest wall is extremely complex (Zamel, 1983; Macklem, 1985). These muscles produce movement both by insertional actions (e.g., the pull of the costal diaphragm on the rib cage at its point of insertion into this structure) as well as by altering the pressures developed within adjacent body compartments (e.g., the costal diaphragm increases abdominal pressure and reduces Correspondence to: G. Supinski, MetroHealth Medical Center, 3395 Scranton Road, Cleveland, OH 44109, U.S.A.
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(3. SUPINSK! et al.
pleural pressure during contraction). The various inspiratory muscles appear to differ in their geometrical arrangements, fiber composition, length-tension relationships and patterns of activation during resting and stimulated breathing. In addition, the chest cage has multiple degrees of freedom and can assume a variety of configurations in response to contraction of different sets of muscles. Several authors have mathematically modeled the inspiratory muscle and chest cage in an effort to attain a simplified description of the effects of contraction of the inspiratory muscles (Hillman and Finucane, 1987; Loring and Mead, 1982; Macklem et al., 1983). For example, the model of Macklem etaL (1983) proposes that the respiratory system can be viewed as a network of force generators, the inspiratory muscles, arranged in parallel and series with one another and contracting against elastic loads (the lung, the passive elements of the chest wall). According to this modal, the costal diaphragm is arranged so that it is mechanically in parallel with the intercostal muscles and at FRC, in parallel with the crural diaphragm. While each of these models has been successful in describing some aspects of chest wall behavior, none of these has dealt with all the complexities of this system. In particular, no model has made an allowance for the length tension relationships of the respiratory muscles or of the chest wall impedances. The purpose of this report is to provide an analysis of skeletal muscle contraction that takes into account the forcelength characteristics of the muscle themselves and of the elastic loads against which these muscles may contract. In keeping with the work of Macklem et al. (1983), we will assume networks of muscles can be viewed as contractile elements arranged in parallel and in series with each other. We will first describe series of mathematical expressions that should apply to any parallel and/or sorios arrangem~:nt of skeletal muscles contracting against elastic loads, Subsequently, we will also present the results of experiments in which the forces and displacements produced by various arrangements of skeletal muscles contracting against elastic loads wore measured and compared to the predictions of this model. Finally, the implications of this model will be discussed.
Model for analysis of the effects of muscle contraction on elastic loads
We will first develop equations to predict the force applied to an elastic load by a single muscle contracting alone. To simplify this analysis, we will first assume that muscle activation is maximal throughout contraction. When a single muscle contracts against an elastic load it shortens, moving down its length-tension curve. This is shown diagrammatically in the left-hand panel of Fig. 1. Before contraction, muscle length is close to the length at which the isometric tension generating ability ofthe muscle is maximum (Li); with contraction, the muscle shortens, moving to a new length that is closer to Lz (Lz is the muscle length at which the sarcomeres are so short that the crosslinkages of the action myosin filaments no longer overlap, thereby making active tension generation impossible). To illustrate more directly how muscle force output varies as a function ofthe degree of muscle shortening,
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
143
Force
I
Lz
Li
0
Li-Lz
Length ~ Length from L I Fig. I. length-tension relationship for skeletal muscles. The left panel shows tension as a function of absolute length; the right panel shows tension as a function of muscle length relative to a reference length, Li. Arrow indicates direction of change in length during contraction.
we have redrawn, in the right hand panel of Fig. 1, the skeletal muscle length tension relationship to express maximum muscle force output as a function of the degree to which muscle length has decreased below Li. As this muscle shortens, the elastic spring lengthens, developing an elastic force that tends to resist further stretching. The magnitude of the elastic force developed by at, ideal spring is linearly related to spring length, as shown in Fig. 2. Before stretching, the force developed by the spring is zero: with stretching, the elastic force developed by the spring increases. Muscle shortening will continue until the force exerted by the elaslic recoil of the spring equals the force generated by the contracting muscle. The force applied to the spring by the muscle at this equilibrium point will be termed Feq, while the degree of spring lengthening and muscle shortening at equilibrium will be termed Leq.
tiff" Spring
Feq,A
.__
A
I~
Feq,B
& Leq,A & Leq,B Fig. 2. Comparison of the equilibrium force and displacements resulting from contraction of a single muscle against high and low elastic loads. A represents point of equilibrium for a single muscle contracting against a high load; B for a single muscle contracting against a low load.
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G. SUPINSKIetal.
The magnitude of the force and displacement of the spring at the equilibrium point depends upon the elastic constant of the spring. A "loose' spring will undergo greater displacement when acted on by a given muscle but the force applied by the muscle will be less (as represented diagrammatically in Fig. 2 as point B). Conversely, the force applied to the stiff spring by a given muscle will be greater but the displacement will be less than for a loose spring (point A, Fig. 2). These relationships can be expressed mathematically. The elastic force developed by a stretched spring can be represented as: FL = KAL
(1)
where FL is the force developed by the elastic load, AL is the displacement of the load, and K is the spring constant. Assuming that the ascending limb of the muscle forcelength curve can be approximated as a straight line, the force developed by a contracting muscle is simply the equation of the line shown in the right-hand panel of Fig. 1: Fm = - a ( L i - L z - AL)
(2)
where Li is the precontraction muscle length, Lz the length at which the muscle can no longer develop tension, AL is the amount of muscle shortening, and a is the slope of the muscle Force-length relationship. During contraction, the muscle and spring reach equilibrium when: FL
=
Fm
Substituting from Eqs. (1) and (2) we obtain: KAL = - a ( L i - Lz)+ aAL Rearranging, we can calculate the equilibrium displacement, ALeq, which represents the amount of spring lengthening produced by muscle contraction: ALeq = a(Li - Lz) a- K
(3)
We can also calculate the equilibrium Force: Feq =
Ka(Li- Lz) a-K
(4)
A similar analysis can be applied to two muscles working in parallel against elastic loads, as shown graphically in Fig. 3. The y-intercept of the isometric force-length relationship for two muscles working in parallel is twice that for a single muscle. Since,
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
145
••o ~)~B
Muscles in Parallel
Single Muscle
/1 Length Fig. 3. Comparison ofthe equilibrium forces and displacements resulting from contraction of single muscles and two identical muscles in parallel. A and C represent points ofequilibrium for a single muscle contracting against high and low loads, respectively; B and D are equilibrium points for parallel muscles contracting against these loads.
however, both muscles are taken to be identical, their precontraction lengths (Li) and the length at which contractile protein crossbridges no longer overlap (Lz, the length at which tension development is no longer possible) are equal. This constrains the force-length relationship ofboth parallel and single muscles to intersect the x-axis at the same point. Based on this model, two muscles working in parallel against a stiff or high elastic load will develop greater force, at the point of equilibrium, than will a single muscle working alone (Fig. 3, points A, B). In other words, Feq at point B, the point of equilibrium for two muscles contracting in parallel against a stiff spring, is greater than Feq at point A, the point of equilibrium for a single muscle co~tr~cting against the same load. This force is, however, less than the sum of the forces developed by each muscle when contracting alone. In addition, the load displacement produced by two muscles working in parallel will be almost twice that produced by a single muscle working alone. In effect, there is partial summing of both force and displacement. When working against a low elastic load, however, parallel muscles produce virtually the same forces and displacements as a single muscle working alone (Fig. 3, points C, D). That is, the displacements and forces at equilibrium for muscles contracting in parallel against a low load (Fig. 3, point D) are nearly the same as for a single muscle contracting against the same load (point C). These relationships can also be expressed mathematically. For two identical muscles contracting in parallel, the isometric force that can be generated should be twice that applied by a single muscle at the same length. As a result, the Force equation (Fpar) for two muscles is" Fpar = 2 F m - - 2 a ( L i - L z - AL)
(5)
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At equilibrium: Fpar-- FL Substituting from Eqs. (1) and (5), we obtain: KAL - - 2 a ( L i - L z - AL) Rearranging, we obtain the equilibrium displacement, that is, the spring lengthening resulting from simultaneous contraction of two muscles arranged in parallel: ALeq,par =
2a(Li- Lz)
(6)
2a- K
with the force applied to the spring at equilibrium" Feq,par =
2Ka(Li- Lz) 2a- K
(7)
A similar analysis can be used for two muscles working in series. When two muscles of equal size contract in series, the absolute change in length, for the pair contracting together, is twice the absolute shortening of each individual muscle. The force generated by the pair can therefore be expressed as: Fser
=
- a(Li - Lz - AL/2)
(8) i
where AL is the total shortening of the pair of muscles. At equilibrium: Fser = FL Substituting Eqs. (!) and (8) and rearranging, we obtain the equilibrium displacement: A Leq,ser = a(Li - Lz)
a/2- K
(9)
and the equilibrium force: Feq,ser = Ka(Li- Lz) a/2- K
(10)
These relationships are shown graphically in Fig. 4. The y-intercept for the forcelength relationship of two muscles working in series is equal to that of either muscle working alone because the isometric tension generating ability of the two muscles are
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
14"/
Two Muscles
Single Muscle /1 Length Fig. 4. Comparison ofthe equilibrium forces and displacements resulting from contraction of single muscles
and from two identical muscles in series. A and C represent equilibrium points for a single muscle contracting against high and low loads, respectively, B and D for muscles in series contracting against these
loads. equal. The absolute range of lengths over which two muscles in series can generate tensions, however, is twice as great as for a single muscle working alone, and, as a result, the x-intercept of the series arrangement is different from that of a single muscle. Adding a muscle in series does little to increase either the force applied to a high elastic load or the degree of stretch of the load. This is illustrated, in Fig. 4, by the fact that the forces and displacements at equilibrium (point B) for two muscles working in series against a stiff load are approximately the same as those produced by a single muscle (point A). in contrast, adding a muscle in series results in substantial increases in both the displacement and force applied to low elastic loads, Le., force and displacement are greater at point D than at point C in Fig. 4. The displacement produced by two muscles contracting together in series is somewhat less, however, than the sum of the displacements produced by the two muscles when contracting individually. In effect, there is partial summing ofboth force and displacement for muscles working in series against low elastic loads.
Parallel Series
j~
I AL
•ries ~ ~
Parallel
K K Fig. 5. Theoretical plots of the forces and displacements produced by single, series, and parallel arrange-
ments of muscles contracting against elastic loads of varying magnitude.
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G. SUPINSK! et al.
According to this model, partial summing of both the displacement of and the force applied to external loads can be produced by both parallel and series arrangements of muscles. The principal distinction between these two muscle arrangements is the range of loads over which such summing occurs. As shown in Fig. 5, series arrangements produce both greater force and dispi'ficetnent for elastic loads with low spring constants, while parallel arrangements have greater effects than series arrangements for loads with high spring constants. Moreover, for medium-sized elastic loads, almost identical increases in the forces applied to and displacements of these loads are produced by adding muscles in either series or parallel. These relationships were derived assuming a constant level of maximal muscle activation. Providing sufficient time is provided to reach equilibrium conditions, however. the activation history of muscles should have no effect on the final equilibrium displacement, which should be determined by the terminal level of muscle activation during the contraction. These equations can also be modified to account for less than maximal levels of activation by modifying Eq. (2) so that this expression becomes: Fm - - a A ( L i - Lz - AL)
(lO)
where A represents the level of muscle activation at the end ofthe contraction. The force generated by a single muscle, contracting isometrically at a length of Li (AL - O) then becomes: F m = = aA(Li = Lz)-- A (constant)
(ll)
if we t:lke A -ffi 1 to represent maximal tctanic activation, the constant in Eq. (11) becomes simply the maximal isometric tension at Li, which will be denoted as Fimu,,. Equation (10) then becomes: Fm -- A(Fi,,,,,,)- aAAL
(12)
Using this latter expression, ALeq and Feq at equilibrium for single, parallel and series arrangements of muscles contracting against elastic loads are: Single: ALeq = AFire"" K - Aa
Feq = KAFi,,,,.~ K - Aa
(13)
Parallel: ALeq = 2AFi,,,,,, K - 2Aa
Feq = 2AKFi,,~,, K - 2Aa
(14)
Series:
Feq = KAFim"' K - Aa/2
(15)
ALeq = AFi,,,,~ K - Aa/2
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
149
Experiments In the above analysis, we have assumed that the tension of skeletal muscles can be expressed for lengths below Lo (length of peak tension) as a linear function of muscle length. To the extent that this relationship actually deviates from linearity, the actual displacements and forces achieved by muscles contracting against elastic loads may vary from those predicted by our derived equations. To provide some assessment of the degree to which the contraction of real muscles might 'fit' the behavior predicted by these equations, we performed a small number of experiments examining in situ canine diaphragm muscle strips contracting against externally applied elastic loads. Our choice of using in situ diaphragm strips for this purpose is based on our familiarity with this preparation. In situ diaphragm strips were prepared in anesthetized animals as described in detail in a recent publication (Supinski et al., 1986). In brief, strips of costal diaphragm were dissected from the hemidiaphragm with the ribs transected so that diaphragm strips were free of connection to the chest wail. The rib ends of these strips were attached to a metal T-piece, which, in turn was connected to a Grass isometric force transducer via a linkage of adjustable length (Supinski et al., 1986). Two types oflinkage were used: rigid metal bars and elastic loads (i.e., metal springs). Strips were electrically stimulated to contract tetanically (50 Hz) using intramuscular electrodes. In four studies, we compared the forces and displacements produced by two diaphragm strips of approximately equal size, dissected from adjacent portions of the same hemidiaphragm and contracting in parallel against elastic loads, with the forces and displacements produced by one of the pair contracting alone. In an additional four experiments, we compared the force and displacement produced by an intact diaphragm strip from one hemidiaphragm with the force and displacement generated by the lower half of the same strip (i.e., the top half of the strip was excised). In these latter studies, we took the results obtained during contraction of the lower half of the strip as a single muscle, and the output of the whole strip to represent two single muscles working in series. Each muscle arrangement (single, series, and parallel) was stimulated to contract tetanicaUy while connected to a rigid metal bar and to each ofthree different highly linear metal springs having spring constants of 0.06, 0.42 and 1.67 kg/cm. Muscle length was adjusted to Lmax, Le., the length at which active isometric tension was maximal, prior to each tetanic stimulus. The force generated by muscle contractions was measured with an isometric force transducer, and the amount of spring lengthening produced by each contraction was measured with a linear displacement transducer (Metripak, Brush Instruments). Tensions were calculated as previously described (Close, 1972)and were expressed as a fraction of the maximum tetanic tension of a single muscle, Tma x. The tensions generated during contractions of pairs of muscles (series or parallel arrangements) were expressed as a fraction of the Tma x of one of the pair contracting alone. Cross sectional area for these muscles averaged approximately I cm 2. The displacements produced by single muscles were reported as a fraction of the
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G. SUPINSKI etal. 150,
60-
A
E
..I
E u. 100
~40-
|
III oL _
o
20
50
U.
~5 I
I
,
I
I
I
I
High Mad Low
High Med Low
Spring Constant (K)
Fig. ~. Displacements (right panel) and tensions (left panel) produced by single muscles (closed circles) and by two muscles in parallel (open circles) contracting against elastic loads with a range of elastic spring constants, Symbols represent means + ! SE.
precontraction muscle length (Lmax). Displacements produced during the contraction of pairs of muscles were expressed as a fraction of the Lm,,, of one of the pair. Comparisons between single muscles and pairs of muscles were made using Student's t-tests (Snedecor and Cochran, 1967). Results are expressed as mean values + 1 SE.
Comparison of experimental results and model predictions Figure 6 compares the experimentally derived forces and displacements generated during contractions of pairs of parallel diaphragmatic muscles with those produced by contraction of single muscles, while Fig. 7 compares pairs of series to single muscles. These results are in keeping with the predictions of our mathematical analysis. Specifically, the parallel muscle arrangement was no better than single muscles in augmenting the force applied to or displacement of low elastic loads, but significantly increased
8O
"
so
m
eo 40
|
20
~ 2o 5 !
I
!
High Med Low
'
I
I
High Med Low
Spring Constant (K) Fig, 7, Displacements (right panel) and tensions (left panel) produced by single muscles (closed circles) and by two muscles in series (open circles) contracting against elastic loads.
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
151
//Parallel ~J
.u
U~ uJ
Load Magnitude (K) Fig. 8. Theoretical plot of elastic work performed as a function of elastic load magnitude (K) for a single muscle and for parallel and series muscle arrangements. The peak of the elastic work-load relationship occurs at a higher K for the parallel arrangement than for a single muscle. The peak for the series arrangement is at a lower K than for a single muscle. displacement of high elastic loads (Fig. 6). In contrast, series arrangements augmented the displacement of and force applied to low elastic loads, but had no greater effect than single muscles against high loads. Against moderate loads, both series and parallel arrangements of muscle had very similar effects, with both arrangements producing significant and fairly similar increases in both force and displacement. To produce a more quantitative comparison of the experimental results with the predictions of our equations, we used Eqs. ( 13)-(15) to generate predicted values of Feq TABLE I Forces (Feq) and displacement (&Leq) applied to elastic loads of different magnitudes (K). g
ALeq (% Lm) Predicted
Feq (% Fm) Observed
!.5 0.42 0.06
9
7 + I
23 42
Parallel muscles
1.5 0.42 0.06
Series muscles
1.5 0.42 0.06
Single muscles
Predicted
Observed
17 + 2 42 + 3
81 54 15
86 + 2 66 + 3 19 +__2
16 3! 46
15 + 2 27 + 3 44 + 2
136 75 16
143 + 12 100 + 15 17 + 4
10 29 73
10 + 2 25 + 3 70 + 7
90 70 26
88 + 3 66 + 2 22 + 3
Comparison is made of Feq and ALeq observed in experimentsperformed on canine costal diaphragm strips with values predicted based on calculation from Eqs. (13)-(15).
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G. SUPINSKI etal.
and ALeq for diaphragm strips. Maximum isometric tension of the strips used (Fire,,,) was 2.1 + 0.2 kg/cm 2 (mean + SE). Lm,,, was 12.1 + 0.8 cm for these strips, and precontraction length was set to Lm.~,,; as a result, Li was approximately 12.1 cm. Since diaphragm fiber force output falls to zero when fiber length is approximately 50% of the length at which tension is maximal (Farkas et aL, 1985; Supinski and Kelsen, 1982), Lz can be taken to be 6 cm. We can calculate a from the expression" a - Fima,,/ ( L i - Lz); doing so, we find that a is approximately -0.35 kg/cm for the canine diaphragm per cm 2 cross-sectional area. By inserting these constants into Eqs. (13) and (14), one obtains the predicted ALeq and Feq values shown in Table 1. As shown in this table, experimental results were in good agreement with the predicted values.
Discussion Potential implications. Several previous papers have described models of the respiratory system. Perhaps the best known model is that ofMacklem et al. (1983, 1985). These authors have modeled the ~spiratory system as a network of force generators, i.e., the inspiratory muscles, arranged in parallel and in series and working against the elastic loads provided by the lung, rib cage, and abdominal compartments. According to this model, the crural diaphragm is mechanically in series with the intercostal muscles while the costal diaphragm is mechanically in parallel with the intercostal muscles and, at FRC, primarily in parallel with the crurai portion of the diaphragm. In this model, the pressures generated during contraction of various combinations of muscles is taken as a reflection of muscle force output, while the tidal volumes generated during inspiration are a reflection of the degree of shortening of the elastic load provided by the lungs. The present analysis extends this previous model, examining in some detail the possible utility that various networks of series and parallel muscles may play in adjusting muscle performance to load conditions. An important point raised by the present analysis is that specific magnitudes of load are best accommodated by specific muscle configurations. A muscle normally confronted by a large load would not be assisted significantly by recruiting an agonist in series with the first muscle while muscles normally confronted with small elastic loads would not benefit from the recruitment of additional muscles in parallel. Moreover, having a network of a number of muscles in series and parallel with each other would make sense only if the principal elastic load confronted by the primary muscle in the network were of a 'moderate' magnitude with respect to this muscle, since only contraction agaanst moderate loads would be appreciably assisted by the recruitment of both parallel and series agonists. This raises the question as to whether the normal elastic loads faced by the respiratory muscles fall into the 'low', 'moderate' or 'high' range. To provide a rough estimate of the 'normal' load magnitude placed on the canine costal diaphragm, we performed an experiment to compare diaphragmatic shortening during exogenous loading with that observed during contraction against the normal endogenous load ofthis muscle. We first
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
153
placed sonomicrometry crystals along the length of this muscle in an anesthetized dog via a midline abdominal incision, closed the incision and stimulated the left phrenic nerve in the neck with supramaximal impulses at several stimulation frequencies (i.e., 10, 20, 50, 100 Hz) and, thereby, assessed the degree of costal diaphragm shortening during contraction against its -'normal' load. We then opened the abdomen, created a diaphragm strip around the site of sonomicrometry placement (this strip was dissected so as to be of similar size and position to those used to generate the data in Table 1) and stimulated this muscle (10, 20, 50, 100 Hz) to contract against elastic loads with spring constants of 0.06, 0.42 and 1.67 kg/cm. The magnitude of the shortening observed for strips contracting against the 0.06 kg/cm load was much greater, shortening against the 1.67 kg/cm load was much less, and shortening against the 0.42 kg/cm load was virtually identical to that observed in the intact diaphragm (50 Hz shortening was equal to that listed in Table 1). While this is a crude experiment, it suggests that the costal diaphragm of the normal dog contracts against what would be considered a 'moderate' elastic load. This finding also has important implications with respect to the work that the costal diaphragm is capable of performing when contracting against its 'normal' elastic load. This can be shown by calculating the work done by a muscle contracting against an elastic load: (16)
Work = j'~ ~ t.at'bF" dL
whore L is muscle length, F is the instantaneous force generated at any given length, La is the initial length and Lb is the final length. When contracting against an elastic load to an equilibrium position, F = KL. If we taken initial length as zero and final length as ALeq, then Eq. (16) becomes: Work =
t. ~ At.eq $ KLdL L ~o
K(ALoq) 2 = 2
or
Work = Feq x ALeq 2
(17)
This expression should apply to both single, series, and parallel arrangements of muscles. Graphically, we can construct a plot of work done versus load applied for different muscle configurations as shown in Fig. 8. Note that each muscle arrangement has a characteristic elastic load for which work output is maximal. The magnitude of this load can be determined, for any given muscle configuration, by differentiating work with respect to load magnitude (K) and setting this equal to zero, i.e. dW/dK - o
(18)
1:54
G. SUPINSK! et al.
Substituting for Feq and ALeq in Eqs. (17) and (18) using Eqs. (13)-(15), we find that maximal work output for a single muscle should occur when the elastic load, K equals - Aa. For parallel and series muscles, maximal work output should occur when K = - 2Aa and K = - Aa/2, respectively. For the maximally activated canine costal diaphragm (A = 1), this last equation would predict that maximal work output should occur when K - - a or K = 0.35 kg/cm. It is interesting to note that this value is fairly similar to the normal load (0.42 kg/cm) placed upon the diaphragm, as estimated from our experiment using sonomicrometry crystals. This analysis would suggest that the normal costal diaphragm may contract against an elastic load for which it is well suited, and that this muscle may normally function on an advantageous portion of its work-elastic load curve. This analysis also shows that recruitment of muscles in parallel with the costal diaphragm should help to compensate for bigger elastic loads both by increasing the work done for a given load and by shifting the peak of the muscle work-elastic load relationship to higher absolute loads (see Fig. 8). Recruitment of either or both muscles in series and parallel with the costal diaphragm should dramatically increase the work done on its normal elastic load. The predictions outlined in this paper are not restricted to the respiratory muscles, but should also apply to any group of skeletal muscle agoni~ts contracting against an elastic load. in addition, this notion can be extended to describe the interaction of groups of individual fibers within a single contracting muscle. The number of sarcomeres in series along tho length of a fiber could be viewed as a number of series elements and fibers in parallel as parallel elements. Tho position of the peak and tho shapo of the work-elastic load curvo of a given musclo would then be a function of the number of such s0ries and parallel elements. Our analysis would suggest that, for a given muscle mass, olastic load and desired work output, thero should b o a uniquo musclo configuration (i.e,, number of s0rios and parallol clcmonts) that would generate that level of work with the lowest level of muscle activation. In summary, this analysis indicates an intimate relationship between the structure of a given muscle or network of muscles and its capability to work against elastic loads, Our data also suggests that the canine costal diaphragm may normally function near tho peak of its work-elastic load relationship and that recruitment of additional agonists in sories and parallel with this muscle should be extremely effective in producing additional work against its normal elastic load.
Acknowledgement,Supported by NIH grant HL 38926.
References Close, R.I, (1972), The dynamic properties of mammalian skeletal muscles. Physiol. Rev. 52: 129-197. Farkas, G,A,, M, DeCramer, D.F, Rochester and A. DeTroyer (1985). Contractile characteristics of intercost~l muscles and their functional significance.J. Appl. Physiol, 59: 528-535.
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Hillman, D.R. and K.E. Finucane (1987). A model ofthe respiratory pump. J. Appl. Physiol. 63: 951-961. Loring, S.H. and .I. Mead (1982). Action of the diaphragm on the rib cage inferred from a force-balance analysis. J. Appi. Physiol. 53: 756-760. Macklem, P.T., D.M. Macklem and A. De Troyer (1983). A model of inspiratory muscle mechanics. J. Appl. Physiol. 55: 547-557. Macklem, P.T. (1985). Interaction ofthe Respiratory Muscles in the Thorax; edited by C. Roussos and P.T. Macklem. New York: Marcel Dekker. Snedecor, G. S. and W. G. Cochran (1967). Statistical Methods, 6th ed. Ames: Iowa State University Press. Supinski, G.S. and S.G. Kelsen (1982). Effect of elastase-induced emphysema on the force-generating ability of the diaphragm. J. Clin. Invest. 70: 978-988. Supinski, G. S., H. Bark, A. Guancial¢ and S.G. Kelsen (I 986). Effect of alterations in muscle fiber length on diaphragm blood flow. J. Appl. Physiol. 60: 1789-1796. Zamel, N. (1983). Normal lung mechanics. In: Textbook of Pulmonary Diseases; edited by G. L. Baum and E. Wolinsky. Boston: Little, Brown, pp. 85-116.
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Analysis of the contraction of series and parallel muscles working against elastic loads G. Supinski, A.F. DiMarco, F. Hussein, R. Bundy and M. Altose MetroHeaith Medical Center. Department of Medicine, Pulmonary Division, Case Western Reserve University Cleveland. OH 44106, U.S.A.
(Accepted 8 October 1991) Abstract. The purpose of the present study was to analyze the manner in which series and parallel arrangements of respiratory muscles, contracting together, augment the forces and displacements applied to external elastic loads over those produced by a single muscle contracting alone. We first developed a series of mathematical expressions to describe the behavior of various arrangements of muscles contracting against elastic loads. We then compared the predictions ofthese equations with the results from experiments in which the forces and displacements produced by simple arrangements of muscles were measured. Both theoretical and experimental results indicate that, against high elastic loads, parallel arrangements of muscle strips produce greater forces and greater displacements than do single muscles; parallel arrangements do not, however, significantly increase the displacement or force applied to low elastic loads. Conversely, series arrangements result in greater forces and greater displacement of low loads, but are no better than single muscles when contracting against high loads. Against moderate loads parallel and series arrangements of muscles appear to be equivalent in generating forces and displacements during contraction. This analysis suggests that a major determinant of the effects of contraction of various networks of inspiratory muscles is the magnitude and character of the respiratory impedance against which these muscles must work. The primary difference between series and parallel arrangements of muscles is that muscles arranged in series are most effective against low elastic loads and muscles in parallel act most effectively against high loads.
Elastic load, respiratory muscles; Respiratory muscles, force, series vs parallel arrangement
The manner in which the various inspiratory muscles interact with each other and with the passive impedances provided by the lung and chest wall is extremely complex (Zamel, 1983; Macklem, 1985). These muscles produce movement both by insertional actions (e.g., the pull of the costal diaphragm on the rib cage at its point of insertion into this structure) as well as by altering the pressures developed within adjacent body compartments (e.g., the costal diaphragm increases abdominal pressure and reduces Correspondence to: G. Supinski, MetroHealth Medical Center, 3395 Scranton Road, Cleveland, OH 44109, U.S.A.
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(3. SUPINSK! et al.
pleural pressure during contraction). The various inspiratory muscles appear to differ in their geometrical arrangements, fiber composition, length-tension relationships and patterns of activation during resting and stimulated breathing. In addition, the chest cage has multiple degrees of freedom and can assume a variety of configurations in response to contraction of different sets of muscles. Several authors have mathematically modeled the inspiratory muscle and chest cage in an effort to attain a simplified description of the effects of contraction of the inspiratory muscles (Hillman and Finucane, 1987; Loring and Mead, 1982; Macklem et al., 1983). For example, the model of Macklem etaL (1983) proposes that the respiratory system can be viewed as a network of force generators, the inspiratory muscles, arranged in parallel and series with one another and contracting against elastic loads (the lung, the passive elements of the chest wall). According to this modal, the costal diaphragm is arranged so that it is mechanically in parallel with the intercostal muscles and at FRC, in parallel with the crural diaphragm. While each of these models has been successful in describing some aspects of chest wall behavior, none of these has dealt with all the complexities of this system. In particular, no model has made an allowance for the length tension relationships of the respiratory muscles or of the chest wall impedances. The purpose of this report is to provide an analysis of skeletal muscle contraction that takes into account the forcelength characteristics of the muscle themselves and of the elastic loads against which these muscles may contract. In keeping with the work of Macklem et al. (1983), we will assume networks of muscles can be viewed as contractile elements arranged in parallel and in series with each other. We will first describe series of mathematical expressions that should apply to any parallel and/or sorios arrangem~:nt of skeletal muscles contracting against elastic loads, Subsequently, we will also present the results of experiments in which the forces and displacements produced by various arrangements of skeletal muscles contracting against elastic loads wore measured and compared to the predictions of this model. Finally, the implications of this model will be discussed.
Model for analysis of the effects of muscle contraction on elastic loads
We will first develop equations to predict the force applied to an elastic load by a single muscle contracting alone. To simplify this analysis, we will first assume that muscle activation is maximal throughout contraction. When a single muscle contracts against an elastic load it shortens, moving down its length-tension curve. This is shown diagrammatically in the left-hand panel of Fig. 1. Before contraction, muscle length is close to the length at which the isometric tension generating ability ofthe muscle is maximum (Li); with contraction, the muscle shortens, moving to a new length that is closer to Lz (Lz is the muscle length at which the sarcomeres are so short that the crosslinkages of the action myosin filaments no longer overlap, thereby making active tension generation impossible). To illustrate more directly how muscle force output varies as a function ofthe degree of muscle shortening,
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
143
Force
I
Lz
Li
0
Li-Lz
Length ~ Length from L I Fig. I. length-tension relationship for skeletal muscles. The left panel shows tension as a function of absolute length; the right panel shows tension as a function of muscle length relative to a reference length, Li. Arrow indicates direction of change in length during contraction.
we have redrawn, in the right hand panel of Fig. 1, the skeletal muscle length tension relationship to express maximum muscle force output as a function of the degree to which muscle length has decreased below Li. As this muscle shortens, the elastic spring lengthens, developing an elastic force that tends to resist further stretching. The magnitude of the elastic force developed by at, ideal spring is linearly related to spring length, as shown in Fig. 2. Before stretching, the force developed by the spring is zero: with stretching, the elastic force developed by the spring increases. Muscle shortening will continue until the force exerted by the elaslic recoil of the spring equals the force generated by the contracting muscle. The force applied to the spring by the muscle at this equilibrium point will be termed Feq, while the degree of spring lengthening and muscle shortening at equilibrium will be termed Leq.
tiff" Spring
Feq,A
.__
A
I~
Feq,B
& Leq,A & Leq,B Fig. 2. Comparison of the equilibrium force and displacements resulting from contraction of a single muscle against high and low elastic loads. A represents point of equilibrium for a single muscle contracting against a high load; B for a single muscle contracting against a low load.
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G. SUPINSKIetal.
The magnitude of the force and displacement of the spring at the equilibrium point depends upon the elastic constant of the spring. A "loose' spring will undergo greater displacement when acted on by a given muscle but the force applied by the muscle will be less (as represented diagrammatically in Fig. 2 as point B). Conversely, the force applied to the stiff spring by a given muscle will be greater but the displacement will be less than for a loose spring (point A, Fig. 2). These relationships can be expressed mathematically. The elastic force developed by a stretched spring can be represented as: FL = KAL
(1)
where FL is the force developed by the elastic load, AL is the displacement of the load, and K is the spring constant. Assuming that the ascending limb of the muscle forcelength curve can be approximated as a straight line, the force developed by a contracting muscle is simply the equation of the line shown in the right-hand panel of Fig. 1: Fm = - a ( L i - L z - AL)
(2)
where Li is the precontraction muscle length, Lz the length at which the muscle can no longer develop tension, AL is the amount of muscle shortening, and a is the slope of the muscle Force-length relationship. During contraction, the muscle and spring reach equilibrium when: FL
=
Fm
Substituting from Eqs. (1) and (2) we obtain: KAL = - a ( L i - Lz)+ aAL Rearranging, we can calculate the equilibrium displacement, ALeq, which represents the amount of spring lengthening produced by muscle contraction: ALeq = a(Li - Lz) a- K
(3)
We can also calculate the equilibrium Force: Feq =
Ka(Li- Lz) a-K
(4)
A similar analysis can be applied to two muscles working in parallel against elastic loads, as shown graphically in Fig. 3. The y-intercept of the isometric force-length relationship for two muscles working in parallel is twice that for a single muscle. Since,
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
145
••o ~)~B
Muscles in Parallel
Single Muscle
/1 Length Fig. 3. Comparison ofthe equilibrium forces and displacements resulting from contraction of single muscles and two identical muscles in parallel. A and C represent points ofequilibrium for a single muscle contracting against high and low loads, respectively; B and D are equilibrium points for parallel muscles contracting against these loads.
however, both muscles are taken to be identical, their precontraction lengths (Li) and the length at which contractile protein crossbridges no longer overlap (Lz, the length at which tension development is no longer possible) are equal. This constrains the force-length relationship ofboth parallel and single muscles to intersect the x-axis at the same point. Based on this model, two muscles working in parallel against a stiff or high elastic load will develop greater force, at the point of equilibrium, than will a single muscle working alone (Fig. 3, points A, B). In other words, Feq at point B, the point of equilibrium for two muscles contracting in parallel against a stiff spring, is greater than Feq at point A, the point of equilibrium for a single muscle co~tr~cting against the same load. This force is, however, less than the sum of the forces developed by each muscle when contracting alone. In addition, the load displacement produced by two muscles working in parallel will be almost twice that produced by a single muscle working alone. In effect, there is partial summing of both force and displacement. When working against a low elastic load, however, parallel muscles produce virtually the same forces and displacements as a single muscle working alone (Fig. 3, points C, D). That is, the displacements and forces at equilibrium for muscles contracting in parallel against a low load (Fig. 3, point D) are nearly the same as for a single muscle contracting against the same load (point C). These relationships can also be expressed mathematically. For two identical muscles contracting in parallel, the isometric force that can be generated should be twice that applied by a single muscle at the same length. As a result, the Force equation (Fpar) for two muscles is" Fpar = 2 F m - - 2 a ( L i - L z - AL)
(5)
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G. SUPINSKI et aL
At equilibrium: Fpar-- FL Substituting from Eqs. (1) and (5), we obtain: KAL - - 2 a ( L i - L z - AL) Rearranging, we obtain the equilibrium displacement, that is, the spring lengthening resulting from simultaneous contraction of two muscles arranged in parallel: ALeq,par =
2a(Li- Lz)
(6)
2a- K
with the force applied to the spring at equilibrium" Feq,par =
2Ka(Li- Lz) 2a- K
(7)
A similar analysis can be used for two muscles working in series. When two muscles of equal size contract in series, the absolute change in length, for the pair contracting together, is twice the absolute shortening of each individual muscle. The force generated by the pair can therefore be expressed as: Fser
=
- a(Li - Lz - AL/2)
(8) i
where AL is the total shortening of the pair of muscles. At equilibrium: Fser = FL Substituting Eqs. (!) and (8) and rearranging, we obtain the equilibrium displacement: A Leq,ser = a(Li - Lz)
a/2- K
(9)
and the equilibrium force: Feq,ser = Ka(Li- Lz) a/2- K
(10)
These relationships are shown graphically in Fig. 4. The y-intercept for the forcelength relationship of two muscles working in series is equal to that of either muscle working alone because the isometric tension generating ability of the two muscles are
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
Force
14"/
Two Muscles
Single Muscle /1 Length Fig. 4. Comparison ofthe equilibrium forces and displacements resulting from contraction of single muscles
and from two identical muscles in series. A and C represent equilibrium points for a single muscle contracting against high and low loads, respectively, B and D for muscles in series contracting against these
loads. equal. The absolute range of lengths over which two muscles in series can generate tensions, however, is twice as great as for a single muscle working alone, and, as a result, the x-intercept of the series arrangement is different from that of a single muscle. Adding a muscle in series does little to increase either the force applied to a high elastic load or the degree of stretch of the load. This is illustrated, in Fig. 4, by the fact that the forces and displacements at equilibrium (point B) for two muscles working in series against a stiff load are approximately the same as those produced by a single muscle (point A). in contrast, adding a muscle in series results in substantial increases in both the displacement and force applied to low elastic loads, Le., force and displacement are greater at point D than at point C in Fig. 4. The displacement produced by two muscles contracting together in series is somewhat less, however, than the sum of the displacements produced by the two muscles when contracting individually. In effect, there is partial summing ofboth force and displacement for muscles working in series against low elastic loads.
Parallel Series
j~
I AL
•ries ~ ~
Parallel
K K Fig. 5. Theoretical plots of the forces and displacements produced by single, series, and parallel arrange-
ments of muscles contracting against elastic loads of varying magnitude.
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G. SUPINSK! et al.
According to this model, partial summing of both the displacement of and the force applied to external loads can be produced by both parallel and series arrangements of muscles. The principal distinction between these two muscle arrangements is the range of loads over which such summing occurs. As shown in Fig. 5, series arrangements produce both greater force and dispi'ficetnent for elastic loads with low spring constants, while parallel arrangements have greater effects than series arrangements for loads with high spring constants. Moreover, for medium-sized elastic loads, almost identical increases in the forces applied to and displacements of these loads are produced by adding muscles in either series or parallel. These relationships were derived assuming a constant level of maximal muscle activation. Providing sufficient time is provided to reach equilibrium conditions, however. the activation history of muscles should have no effect on the final equilibrium displacement, which should be determined by the terminal level of muscle activation during the contraction. These equations can also be modified to account for less than maximal levels of activation by modifying Eq. (2) so that this expression becomes: Fm - - a A ( L i - Lz - AL)
(lO)
where A represents the level of muscle activation at the end ofthe contraction. The force generated by a single muscle, contracting isometrically at a length of Li (AL - O) then becomes: F m = = aA(Li = Lz)-- A (constant)
(ll)
if we t:lke A -ffi 1 to represent maximal tctanic activation, the constant in Eq. (11) becomes simply the maximal isometric tension at Li, which will be denoted as Fimu,,. Equation (10) then becomes: Fm -- A(Fi,,,,,,)- aAAL
(12)
Using this latter expression, ALeq and Feq at equilibrium for single, parallel and series arrangements of muscles contracting against elastic loads are: Single: ALeq = AFire"" K - Aa
Feq = KAFi,,,,.~ K - Aa
(13)
Parallel: ALeq = 2AFi,,,,,, K - 2Aa
Feq = 2AKFi,,~,, K - 2Aa
(14)
Series:
Feq = KAFim"' K - Aa/2
(15)
ALeq = AFi,,,,~ K - Aa/2
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
149
Experiments In the above analysis, we have assumed that the tension of skeletal muscles can be expressed for lengths below Lo (length of peak tension) as a linear function of muscle length. To the extent that this relationship actually deviates from linearity, the actual displacements and forces achieved by muscles contracting against elastic loads may vary from those predicted by our derived equations. To provide some assessment of the degree to which the contraction of real muscles might 'fit' the behavior predicted by these equations, we performed a small number of experiments examining in situ canine diaphragm muscle strips contracting against externally applied elastic loads. Our choice of using in situ diaphragm strips for this purpose is based on our familiarity with this preparation. In situ diaphragm strips were prepared in anesthetized animals as described in detail in a recent publication (Supinski et al., 1986). In brief, strips of costal diaphragm were dissected from the hemidiaphragm with the ribs transected so that diaphragm strips were free of connection to the chest wail. The rib ends of these strips were attached to a metal T-piece, which, in turn was connected to a Grass isometric force transducer via a linkage of adjustable length (Supinski et al., 1986). Two types oflinkage were used: rigid metal bars and elastic loads (i.e., metal springs). Strips were electrically stimulated to contract tetanically (50 Hz) using intramuscular electrodes. In four studies, we compared the forces and displacements produced by two diaphragm strips of approximately equal size, dissected from adjacent portions of the same hemidiaphragm and contracting in parallel against elastic loads, with the forces and displacements produced by one of the pair contracting alone. In an additional four experiments, we compared the force and displacement produced by an intact diaphragm strip from one hemidiaphragm with the force and displacement generated by the lower half of the same strip (i.e., the top half of the strip was excised). In these latter studies, we took the results obtained during contraction of the lower half of the strip as a single muscle, and the output of the whole strip to represent two single muscles working in series. Each muscle arrangement (single, series, and parallel) was stimulated to contract tetanicaUy while connected to a rigid metal bar and to each ofthree different highly linear metal springs having spring constants of 0.06, 0.42 and 1.67 kg/cm. Muscle length was adjusted to Lmax, Le., the length at which active isometric tension was maximal, prior to each tetanic stimulus. The force generated by muscle contractions was measured with an isometric force transducer, and the amount of spring lengthening produced by each contraction was measured with a linear displacement transducer (Metripak, Brush Instruments). Tensions were calculated as previously described (Close, 1972)and were expressed as a fraction of the maximum tetanic tension of a single muscle, Tma x. The tensions generated during contractions of pairs of muscles (series or parallel arrangements) were expressed as a fraction of the Tma x of one of the pair contracting alone. Cross sectional area for these muscles averaged approximately I cm 2. The displacements produced by single muscles were reported as a fraction of the
150
G. SUPINSKI etal. 150,
60-
A
E
..I
E u. 100
~40-
|
III oL _
o
20
50
U.
~5 I
I
,
I
I
I
I
High Mad Low
High Med Low
Spring Constant (K)
Fig. ~. Displacements (right panel) and tensions (left panel) produced by single muscles (closed circles) and by two muscles in parallel (open circles) contracting against elastic loads with a range of elastic spring constants, Symbols represent means + ! SE.
precontraction muscle length (Lmax). Displacements produced during the contraction of pairs of muscles were expressed as a fraction of the Lm,,, of one of the pair. Comparisons between single muscles and pairs of muscles were made using Student's t-tests (Snedecor and Cochran, 1967). Results are expressed as mean values + 1 SE.
Comparison of experimental results and model predictions Figure 6 compares the experimentally derived forces and displacements generated during contractions of pairs of parallel diaphragmatic muscles with those produced by contraction of single muscles, while Fig. 7 compares pairs of series to single muscles. These results are in keeping with the predictions of our mathematical analysis. Specifically, the parallel muscle arrangement was no better than single muscles in augmenting the force applied to or displacement of low elastic loads, but significantly increased
8O
"
so
m
eo 40
|
20
~ 2o 5 !
I
!
High Med Low
'
I
I
High Med Low
Spring Constant (K) Fig, 7, Displacements (right panel) and tensions (left panel) produced by single muscles (closed circles) and by two muscles in series (open circles) contracting against elastic loads.
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
151
//Parallel ~J
.u
U~ uJ
Load Magnitude (K) Fig. 8. Theoretical plot of elastic work performed as a function of elastic load magnitude (K) for a single muscle and for parallel and series muscle arrangements. The peak of the elastic work-load relationship occurs at a higher K for the parallel arrangement than for a single muscle. The peak for the series arrangement is at a lower K than for a single muscle. displacement of high elastic loads (Fig. 6). In contrast, series arrangements augmented the displacement of and force applied to low elastic loads, but had no greater effect than single muscles against high loads. Against moderate loads, both series and parallel arrangements of muscle had very similar effects, with both arrangements producing significant and fairly similar increases in both force and displacement. To produce a more quantitative comparison of the experimental results with the predictions of our equations, we used Eqs. ( 13)-(15) to generate predicted values of Feq TABLE I Forces (Feq) and displacement (&Leq) applied to elastic loads of different magnitudes (K). g
ALeq (% Lm) Predicted
Feq (% Fm) Observed
!.5 0.42 0.06
9
7 + I
23 42
Parallel muscles
1.5 0.42 0.06
Series muscles
1.5 0.42 0.06
Single muscles
Predicted
Observed
17 + 2 42 + 3
81 54 15
86 + 2 66 + 3 19 +__2
16 3! 46
15 + 2 27 + 3 44 + 2
136 75 16
143 + 12 100 + 15 17 + 4
10 29 73
10 + 2 25 + 3 70 + 7
90 70 26
88 + 3 66 + 2 22 + 3
Comparison is made of Feq and ALeq observed in experimentsperformed on canine costal diaphragm strips with values predicted based on calculation from Eqs. (13)-(15).
152
G. SUPINSKI etal.
and ALeq for diaphragm strips. Maximum isometric tension of the strips used (Fire,,,) was 2.1 + 0.2 kg/cm 2 (mean + SE). Lm,,, was 12.1 + 0.8 cm for these strips, and precontraction length was set to Lm.~,,; as a result, Li was approximately 12.1 cm. Since diaphragm fiber force output falls to zero when fiber length is approximately 50% of the length at which tension is maximal (Farkas et aL, 1985; Supinski and Kelsen, 1982), Lz can be taken to be 6 cm. We can calculate a from the expression" a - Fima,,/ ( L i - Lz); doing so, we find that a is approximately -0.35 kg/cm for the canine diaphragm per cm 2 cross-sectional area. By inserting these constants into Eqs. (13) and (14), one obtains the predicted ALeq and Feq values shown in Table 1. As shown in this table, experimental results were in good agreement with the predicted values.
Discussion Potential implications. Several previous papers have described models of the respiratory system. Perhaps the best known model is that ofMacklem et al. (1983, 1985). These authors have modeled the ~spiratory system as a network of force generators, i.e., the inspiratory muscles, arranged in parallel and in series and working against the elastic loads provided by the lung, rib cage, and abdominal compartments. According to this model, the crural diaphragm is mechanically in series with the intercostal muscles while the costal diaphragm is mechanically in parallel with the intercostal muscles and, at FRC, primarily in parallel with the crurai portion of the diaphragm. In this model, the pressures generated during contraction of various combinations of muscles is taken as a reflection of muscle force output, while the tidal volumes generated during inspiration are a reflection of the degree of shortening of the elastic load provided by the lungs. The present analysis extends this previous model, examining in some detail the possible utility that various networks of series and parallel muscles may play in adjusting muscle performance to load conditions. An important point raised by the present analysis is that specific magnitudes of load are best accommodated by specific muscle configurations. A muscle normally confronted by a large load would not be assisted significantly by recruiting an agonist in series with the first muscle while muscles normally confronted with small elastic loads would not benefit from the recruitment of additional muscles in parallel. Moreover, having a network of a number of muscles in series and parallel with each other would make sense only if the principal elastic load confronted by the primary muscle in the network were of a 'moderate' magnitude with respect to this muscle, since only contraction agaanst moderate loads would be appreciably assisted by the recruitment of both parallel and series agonists. This raises the question as to whether the normal elastic loads faced by the respiratory muscles fall into the 'low', 'moderate' or 'high' range. To provide a rough estimate of the 'normal' load magnitude placed on the canine costal diaphragm, we performed an experiment to compare diaphragmatic shortening during exogenous loading with that observed during contraction against the normal endogenous load ofthis muscle. We first
MUSCLE CONTRACTION AGAINST ELASTIC LOADS
153
placed sonomicrometry crystals along the length of this muscle in an anesthetized dog via a midline abdominal incision, closed the incision and stimulated the left phrenic nerve in the neck with supramaximal impulses at several stimulation frequencies (i.e., 10, 20, 50, 100 Hz) and, thereby, assessed the degree of costal diaphragm shortening during contraction against its -'normal' load. We then opened the abdomen, created a diaphragm strip around the site of sonomicrometry placement (this strip was dissected so as to be of similar size and position to those used to generate the data in Table 1) and stimulated this muscle (10, 20, 50, 100 Hz) to contract against elastic loads with spring constants of 0.06, 0.42 and 1.67 kg/cm. The magnitude of the shortening observed for strips contracting against the 0.06 kg/cm load was much greater, shortening against the 1.67 kg/cm load was much less, and shortening against the 0.42 kg/cm load was virtually identical to that observed in the intact diaphragm (50 Hz shortening was equal to that listed in Table 1). While this is a crude experiment, it suggests that the costal diaphragm of the normal dog contracts against what would be considered a 'moderate' elastic load. This finding also has important implications with respect to the work that the costal diaphragm is capable of performing when contracting against its 'normal' elastic load. This can be shown by calculating the work done by a muscle contracting against an elastic load: (16)
Work = j'~ ~ t.at'bF" dL
whore L is muscle length, F is the instantaneous force generated at any given length, La is the initial length and Lb is the final length. When contracting against an elastic load to an equilibrium position, F = KL. If we taken initial length as zero and final length as ALeq, then Eq. (16) becomes: Work =
t. ~ At.eq $ KLdL L ~o
K(ALoq) 2 = 2
or
Work = Feq x ALeq 2
(17)
This expression should apply to both single, series, and parallel arrangements of muscles. Graphically, we can construct a plot of work done versus load applied for different muscle configurations as shown in Fig. 8. Note that each muscle arrangement has a characteristic elastic load for which work output is maximal. The magnitude of this load can be determined, for any given muscle configuration, by differentiating work with respect to load magnitude (K) and setting this equal to zero, i.e. dW/dK - o
(18)
1:54
G. SUPINSK! et al.
Substituting for Feq and ALeq in Eqs. (17) and (18) using Eqs. (13)-(15), we find that maximal work output for a single muscle should occur when the elastic load, K equals - Aa. For parallel and series muscles, maximal work output should occur when K = - 2Aa and K = - Aa/2, respectively. For the maximally activated canine costal diaphragm (A = 1), this last equation would predict that maximal work output should occur when K - - a or K = 0.35 kg/cm. It is interesting to note that this value is fairly similar to the normal load (0.42 kg/cm) placed upon the diaphragm, as estimated from our experiment using sonomicrometry crystals. This analysis would suggest that the normal costal diaphragm may contract against an elastic load for which it is well suited, and that this muscle may normally function on an advantageous portion of its work-elastic load curve. This analysis also shows that recruitment of muscles in parallel with the costal diaphragm should help to compensate for bigger elastic loads both by increasing the work done for a given load and by shifting the peak of the muscle work-elastic load relationship to higher absolute loads (see Fig. 8). Recruitment of either or both muscles in series and parallel with the costal diaphragm should dramatically increase the work done on its normal elastic load. The predictions outlined in this paper are not restricted to the respiratory muscles, but should also apply to any group of skeletal muscle agoni~ts contracting against an elastic load. in addition, this notion can be extended to describe the interaction of groups of individual fibers within a single contracting muscle. The number of sarcomeres in series along tho length of a fiber could be viewed as a number of series elements and fibers in parallel as parallel elements. Tho position of the peak and tho shapo of the work-elastic load curvo of a given musclo would then be a function of the number of such s0ries and parallel elements. Our analysis would suggest that, for a given muscle mass, olastic load and desired work output, thero should b o a uniquo musclo configuration (i.e,, number of s0rios and parallol clcmonts) that would generate that level of work with the lowest level of muscle activation. In summary, this analysis indicates an intimate relationship between the structure of a given muscle or network of muscles and its capability to work against elastic loads, Our data also suggests that the canine costal diaphragm may normally function near tho peak of its work-elastic load relationship and that recruitment of additional agonists in sories and parallel with this muscle should be extremely effective in producing additional work against its normal elastic load.
Acknowledgement,Supported by NIH grant HL 38926.
References Close, R.I, (1972), The dynamic properties of mammalian skeletal muscles. Physiol. Rev. 52: 129-197. Farkas, G,A,, M, DeCramer, D.F, Rochester and A. DeTroyer (1985). Contractile characteristics of intercost~l muscles and their functional significance.J. Appl. Physiol, 59: 528-535.
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Hillman, D.R. and K.E. Finucane (1987). A model ofthe respiratory pump. J. Appl. Physiol. 63: 951-961. Loring, S.H. and .I. Mead (1982). Action of the diaphragm on the rib cage inferred from a force-balance analysis. J. Appi. Physiol. 53: 756-760. Macklem, P.T., D.M. Macklem and A. De Troyer (1983). A model of inspiratory muscle mechanics. J. Appl. Physiol. 55: 547-557. Macklem, P.T. (1985). Interaction ofthe Respiratory Muscles in the Thorax; edited by C. Roussos and P.T. Macklem. New York: Marcel Dekker. Snedecor, G. S. and W. G. Cochran (1967). Statistical Methods, 6th ed. Ames: Iowa State University Press. Supinski, G.S. and S.G. Kelsen (1982). Effect of elastase-induced emphysema on the force-generating ability of the diaphragm. J. Clin. Invest. 70: 978-988. Supinski, G. S., H. Bark, A. Guancial¢ and S.G. Kelsen (I 986). Effect of alterations in muscle fiber length on diaphragm blood flow. J. Appl. Physiol. 60: 1789-1796. Zamel, N. (1983). Normal lung mechanics. In: Textbook of Pulmonary Diseases; edited by G. L. Baum and E. Wolinsky. Boston: Little, Brown, pp. 85-116.
