Effect of respiratory THEODORE
A. WILSON
muscle tension on lung volume AND
ANDRE
DE TROYER
Department of Aerospace Engineering- and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455; and Laboratory of Cardiorespiratory Physiology, Brussels School of Medicine, 1070 Brussels, Belgium
WILSON,THEODORE A., AND ANDREDE T~~~~~.Effectof respiratory muscle tension on lung volume. J. Appl. Physiol. 73(6): 2283-2288,1992.-The chest wall is modeledas a linear systemfor which the displacementsof points on the chest wall are proportional to the forces that act on the chest wall, namely, airway openingpressureand active tension in the respiratory muscles.A standard theorem of mechanics,the Maxwell reciprocity theorem, is invoked to showthat the effect of active muscletension on lung volume, or airway pressureif the airway is closed,is proportional to the change of musclelength in the relaxation maneuver. This relation wastested experimentally. The shortening of the cranial-caudal distance between a rib pair and the sternum wasmeasuredduring a relaxation maneuver. These data were usedto predict the respiratory effect of forcesapplied to the ribs and sternum. To test this prediction, a cranial force wasapplied to the rib pair and a caudal force was applied to the sternum, simulating the forces applied by active tension in the parasternal intercostal muscles.The change in airway pressure, with lung volume held constant, was measured. The measuredchange in airway pressure agreed well with the prediction. In some dogs,nonlinear deviations from the linear prediction occurred at higher loads. The model and the theorem offer the promisethat existing data on the configuration of the chest wall during the relaxation maneuver can be usedto compute the mechanical advantage of the respiratory muscles.
modeling; chest wall; mechanicaladvantage
THERESPIRAT~RYMUSCLEPUMP iscomplex,and
the act of breathing involves many muscles. The diaphragm is clearly the primary muscle of inspiration in mammals. However, a number of electromyographic recordings in anesthetized cats (5,6) and dogs (2,4) have clearly established that inspiration is also associated with activation of three groups of intercostal muscles, namely, the internal intercostals of the parasternal region (the so-called parasternal intercostals), the external intercostals of the rostra1 interspaces, and the levator costae. Selective activation of any one of these three muscles by electrical stimulation causes elevation of the ribs into which it inserts (2, 3). These three groups of intercostal muscles thus play a major role in the act of breathing: they contract together to overcome the expiratory effect of pleural pressure on the rostra1 part of the rib cage and to elevate the ribs during inspiration. Recent experiments in dogs have also demonstrated that the relative contributions of the parasternal intercostals, external intercostals, and levator costae to the inspiratory elevation of the ribs are not equal (1). When the parasternal intercostals are selectively denervated, 0161-7567/92
$2.00 Copyright
0
1992
the inspiratory cranial displacement of the ribs is markedly reduced even though the inspiratory activation of the external intercostals and levator costae increases substantially. On the other hand, when the parasternal
intercostals are left intact, removing the external intercostals induces only a moderate reduction in the inspiratory cranial displacement of ribs. Thus, in the dog breathing at rest, the parasternal intercostals play a greater role than the external intercostals and the levator costae in causing the inspiratory cranial displacement of the ribs; a simple model analysis based on these observations has estimated the parasternal intercostal contribution to be =80% (I). So far, however, it has not been possible to quantify the contribution of these different muscles to lung volume or pressure during breathing. Here, a standard theorem of mechanics, the Maxwell reciprocity theorem, is used to show that the effect of active muscle tension on lung volume or pleural pressure is proportional to the change of muscle length during the relaxation maneuver. This relationship was tested experimentally by measuring the displacement of the ribs and sternum and applying external forces on the ribs and the sternum to simulate the action of the inspiratory intercostal muscles. The experimental observations agreed well with the theoretical predictions. These results thus offer the promise that existing data on the configuration of the chest wall during relaxation can be used to compute the effect of the different respiratory muscles on lung volume or pressure during breathing. MODELING AND ANALYSIS
The analytic
method will be illustrated
by applying
it
to simple examples, and then it will be generalized and applied to the chest wall. The first example, shown in Fig. lA, is a simple lever. If a force F, is applied at the end of the lever, the displacement of the end X, is F,lk, and the displacement X, is smaller by the ratio of the lever arms L,/L,
x, = [(L,IL,)Ik]F,, F, = 0 (0 If a force F, is applied at point I, the displacement of the end x2 is [ (L,/L& lk]F, because of the difference between the lever arms to points I and 2 x, = [(L,IL,)Ik]F,, F, = 0 (2) Equations 1 and 2 illustrate the reciprocity between points 1 and 2: the displacement at point 1 due to a force applied at point 2 equals the displacement at point 2 due to a force applied at point 1. 2283 the AmericanPhysiological Society
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2284
EFFECT
OF RESPIRATORY
MUSCLE
TENSION
FIG. 1. Simple mechanisms that illustrate Maxwell’s reciprocity theorem. A: displacement x2 caused by force F, equals displacement x1 caused by force F,. B: force at end of lever arm is produced by pressure (P) in cylinder of cross-sectional area A. Proportionality between change in cylinder volume (VL) and F, equals proportionality between X, and cylinder pressure P. C: diagonal element between lever arms represents a muscle with a passive elastic element and a capability of generating active tension (T). Ratio of shortening of relaxed muscle (s) to airway pressure P equals proportionality between VL and T. In this case, displacement of levers that accompanies a volume change produced by P is different from displacement that accompanies a volume change produced by T. By Maxwell’s reciprocity theorem, effect of active muscle tension on lung volume can be determined from measurements of muscle shortening during inflation of relaxed chest wall. k, constant; L,, L2, lever arms; t, length.
The second example, shown in Fig. lB, is like the first, except the force on the lever arm at point 2 is applied by a pressure P acting on a piston of area A. The displacement of the piston is VLIA, where VL is the change of the volume of the cylinder x, = [(L,IL,)AI(k,
VL = [(L,IL,)Al(k,
+ k,)]P, + kL)]F1,
F, = 0
(3)
P = 0
(4)
This example illustrates the objective of the analysis. If the change in lung volume that is produced by a force F, is of interest but the values of the geometrical and mechanical parameters of the system are unknown, the proportionality between VL and F, can be determined by measuring the displacement produced by pressure applied to the cylinder. The third example, shown in Fig. lC, is more complex, primarily because it is a system with two degrees of freedom rather than one. The diagonal element represents a muscle. It contains a passive spring and an element that generates active tension, denoted T. The length of the diagonal element is 1. The position shown, with the levers horizontal, is the reference position. In this position, T = 0, P = 0, L = I,, and the forces in all the springs are zero. The variable s = I, - Zis introduced to describe the shortening of the diagonal element from its reference length. The equations of static equilibrium of the lever arms can be solved to obtain the following relations between VL and s and the applied forces P and T
s = [(LJLJA + { [(k,lk,)
cos WAIT
(5)
cos O/A]P + (L;/L;)]cos’
xi = ZjC’ijFj
(7)
If the displacements xi and forces F; are chosen as complementary pairs, that is, each pair -is chosen so that the incremental work done by the force Fi for an incremental displacement d36iis Fi&i, the matrix of coefficients is symmetrical c ij = Cji
This symmetry
(8)
condition
was stated by Maxwell in 1864, reciprocity theorem,” because it implies that the displacement at point 1 caused by a force applied at point 2 equals the displacement at point 2 caused by a force applied at point 1. It can be proved in several ways. For example, it can be shown that if C, # C”i, there is a cyclic loading sequence for and it is known as the “Maxwell
VL = ([l + (k,lk,)cos2 0]A2/A}P + [(L,IL,)A
If T = 0, VL is proportional to P and the coefficient of P in Eq. 5 is the compliance of the passive system. If P = 0, VL is proportional to T. The array of coefficients in these equations is symmetrical, and the proportionality between VL and T for P = 0 is the same as the proportionality between s and P for T = 0. Roughly speaking, the leverage exerted on VL by muscle tension equals the leverage exerted on muscle length by P. Therefore a measurement of the shortening of the muscle during expansion of the passive system by airway pressure yields the coefficient that describes the volume change per unit active force exerted by that muscle. The symmetry of the coefficients in the relations between displacements and forces for the systems shown in Fig. 1 are, of course, not peculiar to these examples. For any system for which displacements are proportional to the forces that act on the system, the relation between displacements and forces has the form
@/AIT
A = k, + km[(kLlkl) + (L;/L;)]cos2
(6)
0
In this case, the equations that describe the effect of both forces on both displacements have been presented.
which work would be obtained from the system, and the system could drive a perpetual motion machine (7).
This general method can be applied to the chest wall. The first displacement-force pair is chosen to be lung
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
EFFECT
OF RESPIRATORY
volume and airway pressure. The remaining pairs are the displacements xi of points on the chest wall and forces F; that act at those points VL = C,,P + C,,F, + CL2T2 + = C,,P + C,,F, + C,,F, +
Xl
l
l
. 4
l
a (9
)
l CW
l l .
Equation 9a describes the effect of P and force on lung volume. The coefficient of the first term in Eq. 9a is the static compliance of the respiratory system (Crs). The remaining terms describe the effect of forces applied to the chest wall. The values of the coefficients CLi are needed. Those coefficients are equal to C,, the coefficients of the first terms in the equations for xi. In the equations for xi, the coefficients CiL describe the displacement during the relaxation maneuver with F; = 0 for all i. The ratio of displacement to airway pressure in the relaxation maneuver is denoted (dx#P),,, c Li
=
c iL
= (dX,ldP),,,
Because dVL = CrsdP during the relaxation the value of CLi is also given by C Li = C,fj(dxilavLJ~~~
N-0 maneuver, (10
If the airway is open and airway pressure is zero, Eq. 9u, with the substitution for CLi given by Eq. I I, ‘yields the change in lung volume produced by an applied force
VL(P = 0) = C,,(d3cildVL),,1Fi
(12)
If the airway is closed and lung volume is fixed, Eq. 9a can be solved to obtain the change in airway pressure produced by an applied force P(VL = 0) = - (dxJdVL) RelFi
(13)
If more than one force is applied, the values of VL(P = 0) and P(VL = 0) are the sums of the contributions of each force. If the forces that act on the chest wall are active tensions in the respiratory muscles, the force-displacement pairs are chosen to be muscle shortening (sJ and active muscle tension (Ti). The equations that describe the respiratory effect of the respiratory muscles are identical to Etp. 12 and 13, with xi replaced by si and Fi replaced by Ti. Experimental tests of the relation between force and airway pressure given by Eq 13 are described below. MATERIAL
AND
METHODS
Five dogs with body masses that ranged from 14 to 19 kg were studied. Each animal was anesthetized by an intravenous injection of pentobarbital sodium (initial dose 25 mg/kg), placed supine in a V-shaped trough, and intubated. The parasternal region of the rib cage was then exposed on both sides of the sternum from the second through the seventh interspace, as shown in Fig. 2. A screw was inserted in the sternum, and screws were inserted near the tips of the sixth bony ribs. In dogs 2-5, additional screws were also inserted in the fourth ribs. A
MUSCLE
TENSION
2285
string was attached to the sternal screw and led caudally over a pulley at the foot of the table and through a displacement transducer. A yoke was attached to the screws in the ribs and connected to a string that was led over the head of the dog, over a pulley at the head of the table, and down through a displacement transducer. Second strings were attached to both the sternal screw and the rib yoke and led caudally and cranially, respectively, along lines nearly colinear with the first strings, over bars, and down to baskets in which weights could be placed to load the sternum and ribs. After completion of the surgical procedures, the animal was paralyzed by an injection of pancuronium and ventilated mechanically. The ventilation was stopped, and the chest wall was allowed to relax to equilibrium. The tracheal cannula was connected to a large syringe, and lung volume was increased in five steps of 100 ml. Values of x,, the cranial displacement of the rib screw yoke, x,, the caudal displacement of the sternal screw, and P, airway opening pressure, were recorded. This procedure was repeated three times, after which the tracheal cannula was closed. Several 200-g weights were then placed sequentially in the basket attached to the rib yoke up to a total load of 1 kg. Airway opening pressure and rib displacement were recorded. This procedure was also repeated three times. In dogs 2-5, the rib yoke was transferred to the rib screws in the other ribs and loaded, following the same procedure. A second loading was applied with the procedure similar to the first, except weights were added simultaneously to both the rib and sternum baskets. The data for the three repetitions of each procedure were quite consistent, and the data from the three repetitions were averaged. RESULTS
The data obtained in the five animals studied are shown in Figs. 3 and 4. Figure 3, top, shows the displacement of the rib yoke x, as a function of lung volume during inflation of the passive respiratory system. The lines show the best linear fit to the data. The slopes of these lines are the values of In Fig. 3, bottom, the lines show the relation between airway pressure and force applied to the rib yoke predicted from Eq. 13 by use of the experimentally determined value of (dx,/dVL),,,, and the circles show the measured values of airway pressure during loading of the rib yoke. Figure 4 describes the results when equal and opposite loads were applied to the ribs and sternum, a loading geometry that is similar to the loads applied by active tension in the parasternal intercostal muscles. The sum of the rib and sternal displacements x, + x2 during passive inflation of the respiratory system is shown in Fig. 4, top. During passive inflation, the sternum moves cranially (3) so that x2 is negative, and x1 + x2 is less than x1 shown in Fig. 3. The lines in Fig. 4, top, show the best linear fit to the data. The slopes of these lines are the values of 8(x, + x,)&W or ds/dV~, where s is the distance between the ends of the mock parasternals. The lines in Fig. 4, bottmn, represent the predicted fall in airway pres@Jaw.,,
l
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2286
EFFECT
OF RESPIRATORY
MUSCLE
TENSION
FIG. 2. Experimental set-up for testing the method. Cranial displacement x1 of a yoke attached to screws in 4th or 6th ribs and caudal displacement X, of sternum were measured during inflation of passive respiratory system. These data were used to predict change in airway opening pressure produced by loads F1 and F2 applied to rib yoke and sternal screw.
sure with load. The measured values are shown by circles. Numerical values of the slopes of the data shown in Figs. 3 and 4 are given in Table 1. Slopes of measured airway pressure vs. load are plotted against predicted values in Fig* 5. For dogs, 2, 3, and 5, some of the data for airway pressure vs. load at higher loads show a nonlinear deviation from the trend of the data for smaller loads, and the values indicated by the asterisk in Table 1 were . *
obtained from the data for the three smaller loads. Rib displacement also plateaued at higher loads in these oases, DISCUSSION
We have assumed that the displacements of points on the chest wall are proportional to the forces that are applied to the chest wall. In dynamic maneuvers, the re4
Dog1
l
-
RIB 6
0
--*-
RIl34
0.2
0.4
Lff ng Volume (1)
Load (kg) 0.8 0.4
,8
t
FIG. 3. Comparison of predicted and measured respiratory effect of loads applied to ribs. Top: circles show measured rib yoke displacement x1 vs. lung volume VL for inflation of relaxed respiratory system. Lines are best linear fit to data. Slopes of lines are values of (dx,/dV~),, used to predict respiratory effect of a force applied to rib yoke. Bottom: circles show measured change in airway opening pressure P with airway closed vs. force F applied to rib yoke. Lines are prediction given by equation P = -(8x,/dV~).,,F.
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
EFFECT
----
OF RESPIRATORY
MUSCLE
2287
TENSION
RIB 6 RIB4
Lung “;‘,I,,, ;;) Load (kg) 0.4
- --
0.8
a FIG. 4. Comparison of predicted and measured respiratory effect of equal forces applied to ribs and sternum in simulation of forces applied by parasternal intercostal muscles. RIJX circles show sum of measured cranial displacement of rib yoke x1 and caudal displacement of sternum x2 vs. lung volume VL for inflation of relaxed respiratory system. Sternum moves cranially during passive inflation, x, is negative, and values of x, + x2 are smaller than values of X, shown in Fig. 3. Lines are best linear fit to data, and slopes of lines are values of [d(x, + n,)ldV~]~,. Bottom: circles show measured airway pressure P vs. force F applied to both ribs and sternum. Lines are predictions given by equation P = -[a(~,
+ x,)ldV~]~,F.
1. Predicted and measured slopes of airway opening pressure us. load TABLE
Rib Loading Dog
No.
k% /aw,, cm/l
?
Rib and
-aPraF,
w,
cmH,O/kg
Sternal
+ 4 /aw,,, cm/l
Loading
-dP/dF, cmH,O/kg
4th rib
2 3 4 5 Avg
1.15 1.25 0.85 1.33 1.15+0.21
0.79% 1.11* 0.78 0.93 0.903-O.16
0.73 1.36 0.95 0.72 1.25 1.00t0.30
0.64 1.02” O.BO* 0.55 0.85 0.77&O. 18
0.76 0.96 0.45 0.62 0.70t0.22
0.54 0.76” 0.51 0.64” 0.61t0.11
0.55 0.86 0.68 0.29 0.60 0.60+0.21
0.56 0.74 0.55* 0.20 0.46* 0.50&0.20
6th rib I
2 3 4 5 A% First column: (dx, /~VL),,~ slopes of Iines fit to data for rib displacement x1 vs. lung volume VL shown in Fig. 3, top;value represents predicted rate of decrease of airway pressure (P) with force (F) applied to rib. Second column: dP/dF, measured rate of decrease of P with F obtained from data shown in Fig. 3, bottom. Third column: [a(~, + x2)/ ~WFbb slopes of lines fit to data for sum of cranial rib displacement and caudal sternal displacement x2 shown in Fig. 4, top; value represents predicted rate of decrease of airway pressure with force for equal and opposite forces applied to ribs and sternum. Fourth column: dP/dF, measured rate of decrease of P with F obtained from data shown in Fig. 4, bottom. *Values obtained from data for 3 smaller loads, with data for 2 higher loads omitted.
lation between force and displacement includes a dependence on rate of displacement. In quasi-static maneuvers, the force-displacement relation shows some hysteresis and the relation between force and displacement
0.5
1 .o
Predicted
0.5
1.0
Predicted
FIG. 5. Measured values of -dP/dF from Table 1 plotted against predicted values. On average, measured values are 21% below predicted values for rib loading and 13% below predicted values for rib and sternal loading.
includes a dependence on displacement history. Also, at large displacements, the linear approximation to the force-displacement relation becomes inaccurate. Nonetheless, the linear force-displacement model is widely used, because for many purposes the benefits of its simplicity justify the approximation it entails. The Maxwell reciprocity theorem applies to any system for which displacements are proportional to forces. Therefore the relation that is used to predict the effect of active muscle tension, Eq. 13, follows directly from the assumption that the chest wall can be described as a linear elastic system; no additional assumptions are needed. The experimental results agree reasonably well with the prediction. The method works, both on the average and in tracking differences among dogs. The data also display the limitations of the model. Nearly one-half of the plots of airway pressure vs. load show nonlinearity at
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2288
EFFECT
OF RESPIRATORY
high loads. A force applied to a single rib, as in our exper-
iments, is not a natural loading, and it produces a local distortion from a natural displacement of the rib cage. It may be that the force- displace ment curve would be linear to la rger displacemen .ts if the load were distributed over several ribs. For loading of the ribs alone, the average difference between the measured and predicted airway pressure is 21%. In this case, an unbalanced cranial force was applied to the ribs. An equal and opposite force must have been exerted ‘on the dog at the points of contact between the dog and the V-shaped trough that supported the dog. Part of the force may have been applied on the dorsal parts of the rib cage. A caudal force on the ribs would have an expiratory effect that would reduce the change in airway pressure. In the second loading procedure, equal and opposite forces were applied to the rib yoke and sternal screw and no additional force was required to hold the dog stationary. For this loading, the agreement between the measured and predicted changes in airway pressure is better; the average difference between measured and predicted airway pressure is 13%. This is the size of error that might be expected, because hysteretic effects are neglected in the model. The main implication of these results is that the coefficient in the relation between active muscle force and lung volume can be determined from measurements of the change of length of the muscle during a relaxation maneuver. There exist extensive data on the configuration of the chest wall during the relaxation maneuver. The configuration of the ribs has been determined at different lung volumes, and the changes in length of both the muscles of the rib cage and the diaphragm have been measured. These data provide the information that is needed to evaluate the respiratory effect of active tension in these muscles. The analysis has another implication. The factor (ds,l aW,,, 9 the rate at which the muscle shortens as lung volume increases during expansion of the passive chest wall, describes the respiratory effect of active tension in the muscle. The value of this factor depends on muscle shortening for a special sequence of chest wall configurations, the configurations that Occur in the relaxation maneuver, but its use in describing the respiratory effect of muscle tension is not restricted to these configurations. The relaxation configurations have a special mechanical property. They are the configurations for which the elastic energy of chestwall distortion for a given lung volume is minimum. If the respiratory muscles act with the coordination required to drive the chest wall along the relaxation trajectory, they produce lung volume change with minimum work. If they do not, the work done by the muscles is greater, but the respiratory effect of the active tensions is the same. For example, if a single muscle were active, it may shorten more than it would as part of coordinated activity; the additional work done by this muscle Provides the additional energv of chest wall deformation.
MUSCLE
TENSION
but if the active force is the same, the effect on lung volume is the same as its contribution as part of coordinated activity. The linear model of the force-displacement relation provides a simple description of coordinated muscle activity. Within the range of forces and displacements for which the linear model is accurate, the resultant effect of coordinated active forces is simply the sum of the contributions of all active forces. It must be emphasized, however, that superposition applies to forces and not to activation levels. For a given activation, the force exerted by a muscle depends on its length, its length depends on the chest wall configuration, and the chest wall configuration depends on the forces exerted by other muscles. Therefore, if two muscles are activated by given stimuli, first individually and then together, the respiratory effect of both may differ from the sum of the individual effects, because the active forces when both are stimulated may differ from the forces that occur when each is stimulated individually. Nonetheless, the linear description of the effect of muscle forces offers a clear quantitative description of coordinated activity. In this model, there is no distinction between the functions of different muscles except the distinction between inspiratory and expiratory function that is determined by the sign of (as,/ ~VL),, and the distinction between the effectiveness of different muscles as described by the magnitude of (as,/ From this point of view, there are two possible benefits of coordinated activity. One is to reduce the distortion of the chest wall from its relaxation configuration at a given lung volume and thereby reduce the elastic energy of chest wall deformation. The other is to maintain muscle lengths within a favorable operating range. aVL)&?l
l
We thank Stephen Loring for guidance on the issues that are discussed in the last paragraph. This work was supported by National Heart, Lung, and Blood Institute
Grant
HL-45545.
Address for reprint requests: T. A. Wilson, 107 Akerman Union St. SE, Minneapolis, MN 55455.
Hall,
110
Received 5 December 1991; accepted in final form 5 June 1992. REFERENCES 1.
DE TROYER, A. The inspiratory elevation of ribs in the dog: primary role of the parasternals.
J. AppZ. Physiol.
70: 1447-1455,
1991.
2. DE TROYER, A., AND G. A. FARKAS. Inspiratory function of the levatar costae and external intercostal muscles in the dog. J. AppZ. Physid. 67: 2614-2621, 1989. 3. DE TROYER, A., AND S. KELLY. Chest wall mechanics in dogs with acute diaphragm paralysis. J. Appl. Physiol. 53: 373-379, 1982. 4. DE TROYER, A., AND V. NINANE. Respiratory function of intercostal muscles in supine dog: an electromyographic study. J. Appl. Physiol. 60: 1692-1699,1986. HxLAIRE,G.G., J. G.NICHOLLS,AND T.A. SEARS. Centralandproprioceptive influences on the activity of levator costae motoneurones in the cat. J. Physiol, Lond. 342: 527-548, 1983. 6, SEARS, T. A. Efferent discharges in alpha and fusimotor fibres of intercostal
nerves
of the cat. J. Physiol. Lond. 174: 295-315, in Structu?uZ Mechanics.
7. ITAUCHERT, T. R. Energy Principtes York: McGraw-Hill, 1974.
1964. New
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
A. WILSON
muscle tension on lung volume AND
ANDRE
DE TROYER
Department of Aerospace Engineering- and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455; and Laboratory of Cardiorespiratory Physiology, Brussels School of Medicine, 1070 Brussels, Belgium
WILSON,THEODORE A., AND ANDREDE T~~~~~.Effectof respiratory muscle tension on lung volume. J. Appl. Physiol. 73(6): 2283-2288,1992.-The chest wall is modeledas a linear systemfor which the displacementsof points on the chest wall are proportional to the forces that act on the chest wall, namely, airway openingpressureand active tension in the respiratory muscles.A standard theorem of mechanics,the Maxwell reciprocity theorem, is invoked to showthat the effect of active muscletension on lung volume, or airway pressureif the airway is closed,is proportional to the change of musclelength in the relaxation maneuver. This relation wastested experimentally. The shortening of the cranial-caudal distance between a rib pair and the sternum wasmeasuredduring a relaxation maneuver. These data were usedto predict the respiratory effect of forcesapplied to the ribs and sternum. To test this prediction, a cranial force wasapplied to the rib pair and a caudal force was applied to the sternum, simulating the forces applied by active tension in the parasternal intercostal muscles.The change in airway pressure, with lung volume held constant, was measured. The measuredchange in airway pressure agreed well with the prediction. In some dogs,nonlinear deviations from the linear prediction occurred at higher loads. The model and the theorem offer the promisethat existing data on the configuration of the chest wall during the relaxation maneuver can be usedto compute the mechanical advantage of the respiratory muscles.
modeling; chest wall; mechanicaladvantage
THERESPIRAT~RYMUSCLEPUMP iscomplex,and
the act of breathing involves many muscles. The diaphragm is clearly the primary muscle of inspiration in mammals. However, a number of electromyographic recordings in anesthetized cats (5,6) and dogs (2,4) have clearly established that inspiration is also associated with activation of three groups of intercostal muscles, namely, the internal intercostals of the parasternal region (the so-called parasternal intercostals), the external intercostals of the rostra1 interspaces, and the levator costae. Selective activation of any one of these three muscles by electrical stimulation causes elevation of the ribs into which it inserts (2, 3). These three groups of intercostal muscles thus play a major role in the act of breathing: they contract together to overcome the expiratory effect of pleural pressure on the rostra1 part of the rib cage and to elevate the ribs during inspiration. Recent experiments in dogs have also demonstrated that the relative contributions of the parasternal intercostals, external intercostals, and levator costae to the inspiratory elevation of the ribs are not equal (1). When the parasternal intercostals are selectively denervated, 0161-7567/92
$2.00 Copyright
0
1992
the inspiratory cranial displacement of the ribs is markedly reduced even though the inspiratory activation of the external intercostals and levator costae increases substantially. On the other hand, when the parasternal
intercostals are left intact, removing the external intercostals induces only a moderate reduction in the inspiratory cranial displacement of ribs. Thus, in the dog breathing at rest, the parasternal intercostals play a greater role than the external intercostals and the levator costae in causing the inspiratory cranial displacement of the ribs; a simple model analysis based on these observations has estimated the parasternal intercostal contribution to be =80% (I). So far, however, it has not been possible to quantify the contribution of these different muscles to lung volume or pressure during breathing. Here, a standard theorem of mechanics, the Maxwell reciprocity theorem, is used to show that the effect of active muscle tension on lung volume or pleural pressure is proportional to the change of muscle length during the relaxation maneuver. This relationship was tested experimentally by measuring the displacement of the ribs and sternum and applying external forces on the ribs and the sternum to simulate the action of the inspiratory intercostal muscles. The experimental observations agreed well with the theoretical predictions. These results thus offer the promise that existing data on the configuration of the chest wall during relaxation can be used to compute the effect of the different respiratory muscles on lung volume or pressure during breathing. MODELING AND ANALYSIS
The analytic
method will be illustrated
by applying
it
to simple examples, and then it will be generalized and applied to the chest wall. The first example, shown in Fig. lA, is a simple lever. If a force F, is applied at the end of the lever, the displacement of the end X, is F,lk, and the displacement X, is smaller by the ratio of the lever arms L,/L,
x, = [(L,IL,)Ik]F,, F, = 0 (0 If a force F, is applied at point I, the displacement of the end x2 is [ (L,/L& lk]F, because of the difference between the lever arms to points I and 2 x, = [(L,IL,)Ik]F,, F, = 0 (2) Equations 1 and 2 illustrate the reciprocity between points 1 and 2: the displacement at point 1 due to a force applied at point 2 equals the displacement at point 2 due to a force applied at point 1. 2283 the AmericanPhysiological Society
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2284
EFFECT
OF RESPIRATORY
MUSCLE
TENSION
FIG. 1. Simple mechanisms that illustrate Maxwell’s reciprocity theorem. A: displacement x2 caused by force F, equals displacement x1 caused by force F,. B: force at end of lever arm is produced by pressure (P) in cylinder of cross-sectional area A. Proportionality between change in cylinder volume (VL) and F, equals proportionality between X, and cylinder pressure P. C: diagonal element between lever arms represents a muscle with a passive elastic element and a capability of generating active tension (T). Ratio of shortening of relaxed muscle (s) to airway pressure P equals proportionality between VL and T. In this case, displacement of levers that accompanies a volume change produced by P is different from displacement that accompanies a volume change produced by T. By Maxwell’s reciprocity theorem, effect of active muscle tension on lung volume can be determined from measurements of muscle shortening during inflation of relaxed chest wall. k, constant; L,, L2, lever arms; t, length.
The second example, shown in Fig. lB, is like the first, except the force on the lever arm at point 2 is applied by a pressure P acting on a piston of area A. The displacement of the piston is VLIA, where VL is the change of the volume of the cylinder x, = [(L,IL,)AI(k,
VL = [(L,IL,)Al(k,
+ k,)]P, + kL)]F1,
F, = 0
(3)
P = 0
(4)
This example illustrates the objective of the analysis. If the change in lung volume that is produced by a force F, is of interest but the values of the geometrical and mechanical parameters of the system are unknown, the proportionality between VL and F, can be determined by measuring the displacement produced by pressure applied to the cylinder. The third example, shown in Fig. lC, is more complex, primarily because it is a system with two degrees of freedom rather than one. The diagonal element represents a muscle. It contains a passive spring and an element that generates active tension, denoted T. The length of the diagonal element is 1. The position shown, with the levers horizontal, is the reference position. In this position, T = 0, P = 0, L = I,, and the forces in all the springs are zero. The variable s = I, - Zis introduced to describe the shortening of the diagonal element from its reference length. The equations of static equilibrium of the lever arms can be solved to obtain the following relations between VL and s and the applied forces P and T
s = [(LJLJA + { [(k,lk,)
cos WAIT
(5)
cos O/A]P + (L;/L;)]cos’
xi = ZjC’ijFj
(7)
If the displacements xi and forces F; are chosen as complementary pairs, that is, each pair -is chosen so that the incremental work done by the force Fi for an incremental displacement d36iis Fi&i, the matrix of coefficients is symmetrical c ij = Cji
This symmetry
(8)
condition
was stated by Maxwell in 1864, reciprocity theorem,” because it implies that the displacement at point 1 caused by a force applied at point 2 equals the displacement at point 2 caused by a force applied at point 1. It can be proved in several ways. For example, it can be shown that if C, # C”i, there is a cyclic loading sequence for and it is known as the “Maxwell
VL = ([l + (k,lk,)cos2 0]A2/A}P + [(L,IL,)A
If T = 0, VL is proportional to P and the coefficient of P in Eq. 5 is the compliance of the passive system. If P = 0, VL is proportional to T. The array of coefficients in these equations is symmetrical, and the proportionality between VL and T for P = 0 is the same as the proportionality between s and P for T = 0. Roughly speaking, the leverage exerted on VL by muscle tension equals the leverage exerted on muscle length by P. Therefore a measurement of the shortening of the muscle during expansion of the passive system by airway pressure yields the coefficient that describes the volume change per unit active force exerted by that muscle. The symmetry of the coefficients in the relations between displacements and forces for the systems shown in Fig. 1 are, of course, not peculiar to these examples. For any system for which displacements are proportional to the forces that act on the system, the relation between displacements and forces has the form
@/AIT
A = k, + km[(kLlkl) + (L;/L;)]cos2
(6)
0
In this case, the equations that describe the effect of both forces on both displacements have been presented.
which work would be obtained from the system, and the system could drive a perpetual motion machine (7).
This general method can be applied to the chest wall. The first displacement-force pair is chosen to be lung
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
EFFECT
OF RESPIRATORY
volume and airway pressure. The remaining pairs are the displacements xi of points on the chest wall and forces F; that act at those points VL = C,,P + C,,F, + CL2T2 + = C,,P + C,,F, + C,,F, +
Xl
l
l
. 4
l
a (9
)
l CW
l l .
Equation 9a describes the effect of P and force on lung volume. The coefficient of the first term in Eq. 9a is the static compliance of the respiratory system (Crs). The remaining terms describe the effect of forces applied to the chest wall. The values of the coefficients CLi are needed. Those coefficients are equal to C,, the coefficients of the first terms in the equations for xi. In the equations for xi, the coefficients CiL describe the displacement during the relaxation maneuver with F; = 0 for all i. The ratio of displacement to airway pressure in the relaxation maneuver is denoted (dx#P),,, c Li
=
c iL
= (dX,ldP),,,
Because dVL = CrsdP during the relaxation the value of CLi is also given by C Li = C,fj(dxilavLJ~~~
N-0 maneuver, (10
If the airway is open and airway pressure is zero, Eq. 9u, with the substitution for CLi given by Eq. I I, ‘yields the change in lung volume produced by an applied force
VL(P = 0) = C,,(d3cildVL),,1Fi
(12)
If the airway is closed and lung volume is fixed, Eq. 9a can be solved to obtain the change in airway pressure produced by an applied force P(VL = 0) = - (dxJdVL) RelFi
(13)
If more than one force is applied, the values of VL(P = 0) and P(VL = 0) are the sums of the contributions of each force. If the forces that act on the chest wall are active tensions in the respiratory muscles, the force-displacement pairs are chosen to be muscle shortening (sJ and active muscle tension (Ti). The equations that describe the respiratory effect of the respiratory muscles are identical to Etp. 12 and 13, with xi replaced by si and Fi replaced by Ti. Experimental tests of the relation between force and airway pressure given by Eq 13 are described below. MATERIAL
AND
METHODS
Five dogs with body masses that ranged from 14 to 19 kg were studied. Each animal was anesthetized by an intravenous injection of pentobarbital sodium (initial dose 25 mg/kg), placed supine in a V-shaped trough, and intubated. The parasternal region of the rib cage was then exposed on both sides of the sternum from the second through the seventh interspace, as shown in Fig. 2. A screw was inserted in the sternum, and screws were inserted near the tips of the sixth bony ribs. In dogs 2-5, additional screws were also inserted in the fourth ribs. A
MUSCLE
TENSION
2285
string was attached to the sternal screw and led caudally over a pulley at the foot of the table and through a displacement transducer. A yoke was attached to the screws in the ribs and connected to a string that was led over the head of the dog, over a pulley at the head of the table, and down through a displacement transducer. Second strings were attached to both the sternal screw and the rib yoke and led caudally and cranially, respectively, along lines nearly colinear with the first strings, over bars, and down to baskets in which weights could be placed to load the sternum and ribs. After completion of the surgical procedures, the animal was paralyzed by an injection of pancuronium and ventilated mechanically. The ventilation was stopped, and the chest wall was allowed to relax to equilibrium. The tracheal cannula was connected to a large syringe, and lung volume was increased in five steps of 100 ml. Values of x,, the cranial displacement of the rib screw yoke, x,, the caudal displacement of the sternal screw, and P, airway opening pressure, were recorded. This procedure was repeated three times, after which the tracheal cannula was closed. Several 200-g weights were then placed sequentially in the basket attached to the rib yoke up to a total load of 1 kg. Airway opening pressure and rib displacement were recorded. This procedure was also repeated three times. In dogs 2-5, the rib yoke was transferred to the rib screws in the other ribs and loaded, following the same procedure. A second loading was applied with the procedure similar to the first, except weights were added simultaneously to both the rib and sternum baskets. The data for the three repetitions of each procedure were quite consistent, and the data from the three repetitions were averaged. RESULTS
The data obtained in the five animals studied are shown in Figs. 3 and 4. Figure 3, top, shows the displacement of the rib yoke x, as a function of lung volume during inflation of the passive respiratory system. The lines show the best linear fit to the data. The slopes of these lines are the values of In Fig. 3, bottom, the lines show the relation between airway pressure and force applied to the rib yoke predicted from Eq. 13 by use of the experimentally determined value of (dx,/dVL),,,, and the circles show the measured values of airway pressure during loading of the rib yoke. Figure 4 describes the results when equal and opposite loads were applied to the ribs and sternum, a loading geometry that is similar to the loads applied by active tension in the parasternal intercostal muscles. The sum of the rib and sternal displacements x, + x2 during passive inflation of the respiratory system is shown in Fig. 4, top. During passive inflation, the sternum moves cranially (3) so that x2 is negative, and x1 + x2 is less than x1 shown in Fig. 3. The lines in Fig. 4, top, show the best linear fit to the data. The slopes of these lines are the values of 8(x, + x,)&W or ds/dV~, where s is the distance between the ends of the mock parasternals. The lines in Fig. 4, bottmn, represent the predicted fall in airway pres@Jaw.,,
l
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2286
EFFECT
OF RESPIRATORY
MUSCLE
TENSION
FIG. 2. Experimental set-up for testing the method. Cranial displacement x1 of a yoke attached to screws in 4th or 6th ribs and caudal displacement X, of sternum were measured during inflation of passive respiratory system. These data were used to predict change in airway opening pressure produced by loads F1 and F2 applied to rib yoke and sternal screw.
sure with load. The measured values are shown by circles. Numerical values of the slopes of the data shown in Figs. 3 and 4 are given in Table 1. Slopes of measured airway pressure vs. load are plotted against predicted values in Fig* 5. For dogs, 2, 3, and 5, some of the data for airway pressure vs. load at higher loads show a nonlinear deviation from the trend of the data for smaller loads, and the values indicated by the asterisk in Table 1 were . *
obtained from the data for the three smaller loads. Rib displacement also plateaued at higher loads in these oases, DISCUSSION
We have assumed that the displacements of points on the chest wall are proportional to the forces that are applied to the chest wall. In dynamic maneuvers, the re4
Dog1
l
-
RIB 6
0
--*-
RIl34
0.2
0.4
Lff ng Volume (1)
Load (kg) 0.8 0.4
,8
t
FIG. 3. Comparison of predicted and measured respiratory effect of loads applied to ribs. Top: circles show measured rib yoke displacement x1 vs. lung volume VL for inflation of relaxed respiratory system. Lines are best linear fit to data. Slopes of lines are values of (dx,/dV~),, used to predict respiratory effect of a force applied to rib yoke. Bottom: circles show measured change in airway opening pressure P with airway closed vs. force F applied to rib yoke. Lines are prediction given by equation P = -(8x,/dV~).,,F.
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
EFFECT
----
OF RESPIRATORY
MUSCLE
2287
TENSION
RIB 6 RIB4
Lung “;‘,I,,, ;;) Load (kg) 0.4
- --
0.8
a FIG. 4. Comparison of predicted and measured respiratory effect of equal forces applied to ribs and sternum in simulation of forces applied by parasternal intercostal muscles. RIJX circles show sum of measured cranial displacement of rib yoke x1 and caudal displacement of sternum x2 vs. lung volume VL for inflation of relaxed respiratory system. Sternum moves cranially during passive inflation, x, is negative, and values of x, + x2 are smaller than values of X, shown in Fig. 3. Lines are best linear fit to data, and slopes of lines are values of [d(x, + n,)ldV~]~,. Bottom: circles show measured airway pressure P vs. force F applied to both ribs and sternum. Lines are predictions given by equation P = -[a(~,
+ x,)ldV~]~,F.
1. Predicted and measured slopes of airway opening pressure us. load TABLE
Rib Loading Dog
No.
k% /aw,, cm/l
?
Rib and
-aPraF,
w,
cmH,O/kg
Sternal
+ 4 /aw,,, cm/l
Loading
-dP/dF, cmH,O/kg
4th rib
2 3 4 5 Avg
1.15 1.25 0.85 1.33 1.15+0.21
0.79% 1.11* 0.78 0.93 0.903-O.16
0.73 1.36 0.95 0.72 1.25 1.00t0.30
0.64 1.02” O.BO* 0.55 0.85 0.77&O. 18
0.76 0.96 0.45 0.62 0.70t0.22
0.54 0.76” 0.51 0.64” 0.61t0.11
0.55 0.86 0.68 0.29 0.60 0.60+0.21
0.56 0.74 0.55* 0.20 0.46* 0.50&0.20
6th rib I
2 3 4 5 A% First column: (dx, /~VL),,~ slopes of Iines fit to data for rib displacement x1 vs. lung volume VL shown in Fig. 3, top;value represents predicted rate of decrease of airway pressure (P) with force (F) applied to rib. Second column: dP/dF, measured rate of decrease of P with F obtained from data shown in Fig. 3, bottom. Third column: [a(~, + x2)/ ~WFbb slopes of lines fit to data for sum of cranial rib displacement and caudal sternal displacement x2 shown in Fig. 4, top; value represents predicted rate of decrease of airway pressure with force for equal and opposite forces applied to ribs and sternum. Fourth column: dP/dF, measured rate of decrease of P with F obtained from data shown in Fig. 4, bottom. *Values obtained from data for 3 smaller loads, with data for 2 higher loads omitted.
lation between force and displacement includes a dependence on rate of displacement. In quasi-static maneuvers, the force-displacement relation shows some hysteresis and the relation between force and displacement
0.5
1 .o
Predicted
0.5
1.0
Predicted
FIG. 5. Measured values of -dP/dF from Table 1 plotted against predicted values. On average, measured values are 21% below predicted values for rib loading and 13% below predicted values for rib and sternal loading.
includes a dependence on displacement history. Also, at large displacements, the linear approximation to the force-displacement relation becomes inaccurate. Nonetheless, the linear force-displacement model is widely used, because for many purposes the benefits of its simplicity justify the approximation it entails. The Maxwell reciprocity theorem applies to any system for which displacements are proportional to forces. Therefore the relation that is used to predict the effect of active muscle tension, Eq. 13, follows directly from the assumption that the chest wall can be described as a linear elastic system; no additional assumptions are needed. The experimental results agree reasonably well with the prediction. The method works, both on the average and in tracking differences among dogs. The data also display the limitations of the model. Nearly one-half of the plots of airway pressure vs. load show nonlinearity at
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
2288
EFFECT
OF RESPIRATORY
high loads. A force applied to a single rib, as in our exper-
iments, is not a natural loading, and it produces a local distortion from a natural displacement of the rib cage. It may be that the force- displace ment curve would be linear to la rger displacemen .ts if the load were distributed over several ribs. For loading of the ribs alone, the average difference between the measured and predicted airway pressure is 21%. In this case, an unbalanced cranial force was applied to the ribs. An equal and opposite force must have been exerted ‘on the dog at the points of contact between the dog and the V-shaped trough that supported the dog. Part of the force may have been applied on the dorsal parts of the rib cage. A caudal force on the ribs would have an expiratory effect that would reduce the change in airway pressure. In the second loading procedure, equal and opposite forces were applied to the rib yoke and sternal screw and no additional force was required to hold the dog stationary. For this loading, the agreement between the measured and predicted changes in airway pressure is better; the average difference between measured and predicted airway pressure is 13%. This is the size of error that might be expected, because hysteretic effects are neglected in the model. The main implication of these results is that the coefficient in the relation between active muscle force and lung volume can be determined from measurements of the change of length of the muscle during a relaxation maneuver. There exist extensive data on the configuration of the chest wall during the relaxation maneuver. The configuration of the ribs has been determined at different lung volumes, and the changes in length of both the muscles of the rib cage and the diaphragm have been measured. These data provide the information that is needed to evaluate the respiratory effect of active tension in these muscles. The analysis has another implication. The factor (ds,l aW,,, 9 the rate at which the muscle shortens as lung volume increases during expansion of the passive chest wall, describes the respiratory effect of active tension in the muscle. The value of this factor depends on muscle shortening for a special sequence of chest wall configurations, the configurations that Occur in the relaxation maneuver, but its use in describing the respiratory effect of muscle tension is not restricted to these configurations. The relaxation configurations have a special mechanical property. They are the configurations for which the elastic energy of chestwall distortion for a given lung volume is minimum. If the respiratory muscles act with the coordination required to drive the chest wall along the relaxation trajectory, they produce lung volume change with minimum work. If they do not, the work done by the muscles is greater, but the respiratory effect of the active tensions is the same. For example, if a single muscle were active, it may shorten more than it would as part of coordinated activity; the additional work done by this muscle Provides the additional energv of chest wall deformation.
MUSCLE
TENSION
but if the active force is the same, the effect on lung volume is the same as its contribution as part of coordinated activity. The linear model of the force-displacement relation provides a simple description of coordinated muscle activity. Within the range of forces and displacements for which the linear model is accurate, the resultant effect of coordinated active forces is simply the sum of the contributions of all active forces. It must be emphasized, however, that superposition applies to forces and not to activation levels. For a given activation, the force exerted by a muscle depends on its length, its length depends on the chest wall configuration, and the chest wall configuration depends on the forces exerted by other muscles. Therefore, if two muscles are activated by given stimuli, first individually and then together, the respiratory effect of both may differ from the sum of the individual effects, because the active forces when both are stimulated may differ from the forces that occur when each is stimulated individually. Nonetheless, the linear description of the effect of muscle forces offers a clear quantitative description of coordinated activity. In this model, there is no distinction between the functions of different muscles except the distinction between inspiratory and expiratory function that is determined by the sign of (as,/ ~VL),, and the distinction between the effectiveness of different muscles as described by the magnitude of (as,/ From this point of view, there are two possible benefits of coordinated activity. One is to reduce the distortion of the chest wall from its relaxation configuration at a given lung volume and thereby reduce the elastic energy of chest wall deformation. The other is to maintain muscle lengths within a favorable operating range. aVL)&?l
l
We thank Stephen Loring for guidance on the issues that are discussed in the last paragraph. This work was supported by National Heart, Lung, and Blood Institute
Grant
HL-45545.
Address for reprint requests: T. A. Wilson, 107 Akerman Union St. SE, Minneapolis, MN 55455.
Hall,
110
Received 5 December 1991; accepted in final form 5 June 1992. REFERENCES 1.
DE TROYER, A. The inspiratory elevation of ribs in the dog: primary role of the parasternals.
J. AppZ. Physiol.
70: 1447-1455,
1991.
2. DE TROYER, A., AND G. A. FARKAS. Inspiratory function of the levatar costae and external intercostal muscles in the dog. J. AppZ. Physid. 67: 2614-2621, 1989. 3. DE TROYER, A., AND S. KELLY. Chest wall mechanics in dogs with acute diaphragm paralysis. J. Appl. Physiol. 53: 373-379, 1982. 4. DE TROYER, A., AND V. NINANE. Respiratory function of intercostal muscles in supine dog: an electromyographic study. J. Appl. Physiol. 60: 1692-1699,1986. HxLAIRE,G.G., J. G.NICHOLLS,AND T.A. SEARS. Centralandproprioceptive influences on the activity of levator costae motoneurones in the cat. J. Physiol, Lond. 342: 527-548, 1983. 6, SEARS, T. A. Efferent discharges in alpha and fusimotor fibres of intercostal
nerves
of the cat. J. Physiol. Lond. 174: 295-315, in Structu?uZ Mechanics.
7. ITAUCHERT, T. R. Energy Principtes York: McGraw-Hill, 1974.
1964. New
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 12, 2019.
