The mechanical advantage (μ) of a respiratory muscle is defined as the respiratory pressure generated per unit muscle mass and per unit active stress. The value of μ can be obtained by measuring the change in the length of the muscle during inflation of the passive lung and chest wall. We report values of μ for the muscles of the canine diaphragm that were obtained by measuring the lengths of the muscles during a passive quasistatic vital capacity maneuver. Radiopaque markers were attached along six muscle bundles of the costal and two muscle bundles of the crural left hemidiaphragms of four bred-for-research beagle dogs. The three-dimensional locations of the markers were obtained from biplane video-fluoroscopic images taken at four volumes during a passive relaxation maneuver from total lung capacity to functional residual capacity in the prone and supine postures. Muscle lengths were determined as a function of lung volume, and from these data, values of μ were obtained. Values of μ are fairly uniform around the ventral midcostal and crural diaphragm but significantly lower at the dorsal end of the costal diaphragm. The average values of μ are −0.35 ± 0.18 and −0.27 ± 0.16 cmH2O ⋅ g−1 ⋅ kg−1 ⋅ cm−2in the prone and supine dog, respectively. These values are 1.5–2 times larger than the largest values of μ of the intercostal muscles in the supine dog. From these data we estimate that during spontaneous breathing the diaphragm contributes ∼40% of inspiratory pressure in the prone posture and ∼30% in the supine posture. Passive shortening, and hence μ, in the upper one-third of inspiratory capacity is less than one-half of that at lower lung volume. The lower μ is attributed primarily to a lower abdominal compliance at high lung volume.
- respiratory muscles
- chest wall
in physics, the term “mechanical advantage” is defined as the ratio of the force delivered to a load (FL) to the force applied at a remote point on a machine (F1). For a simple lever, this ratio FL/F1equals the ratio of the lever arms. If the position of the fulcrum were unknown, the ratio of the lever arms could be determined by measuring the displacement of the lever arm (x 1) per unit displacement of the load (x L). That is, FL = μF1, where μ =x 1/x L. As stated by Maxwell’s reciprocity theorem, the ratio of forces and the ratio of displacements for any multiple-degree-of-freedom linear elastic system are similarly related.
Wilson and De Troyer (28) defined the respiratory mechanical advantage (μ) as respiratory pressure per unit muscle mass (m) and per unit active stress (ς). That is Equation 1where ΔPao is the change in airway pressure that occurs when a muscle is activated with the airway occluded. They modeled the chest wall as a linear elastic system and applied Maxwell’s reciprocity theorem to obtain the following relation between μ and change in muscle length during inflation of the relaxed chest wall Equation 2where (dL/L odVl)relis the fractional change in muscle length (dL/L o) per unit change in lung volume (dVl) during the relaxation maneuver. Equation 2 states that μ can be determined by measuring muscle shortening during a relaxation maneuver when the lung and chest wall are expanded by an externally applied pressure. In the derivation of Eqs.1 and 2 , muscle volume was replaced by m, and the numerical relations are valid for the variables described in the following familiar, but mixed, units: ΔPao (cmH2O), μ (g), ς (kg/cm2), and Vl (liters). That is, the left side of Eq. 1 with units of centimeters water per gram equals the right side, with μ expressed per liter and ς in kilograms per square centimeter.
In a series of studies, De Troyer and colleagues have given substance to this theory by demonstrating that the respiratory effects of a number of respiratory muscles, the parasternal internal intercostals (10, 17), the sternomastoids and scalenes (16), and the triangularis sterni (9), are well described by Eqs.1 and 2 . That is, they measured the change in length of each of these muscles during inflation of the passive respiratory system to obtain (dL/L odVl)reland, hence, μ. They then measured ΔPao when the muscle was maximally activated, excised the activated muscle, and measured its mass. They found that ΔPao/m is proportional to μ, with a constant of proportionality of ∼3 kg/cm2, a value at the upper end of the range of values of maximum ς measured in vitro (26). In addition, they tested the validity of the linear model of the respiratory system by testing one of the fundamental properties of linear systems, i.e., superpositon (18). For different combinations of intercostal muscles in different interspaces, they found that, to within 10%, ΔPao produced by simultaneous activation of two muscle groups equals the sum of the ΔPao values produced by each group activated alone.
As a result of the work of De Troyer et al., Eq.2 can now be used with some confidence to obtain the values of μ of muscles for which ΔPao/m cannot be measured directly. That is, for muscles that cannot be activated to produce a known ς in a known m, μ can be obtained by measuring (dL/L odVl)rel. The diaphragm falls in this category. To be sure, the diaphragm can be activated by stimulating the phrenic nerves. However, the chest wall is severely distorted when the diaphragm is maximally activated (14), and it is likely that diaphragm function and stress are altered by the change in geometry and muscle length that occurs. Submaximal activation is possible, but ς in the submaximally activated diaphragm is unknown.
We report values of diaphragm muscle shortening during passive inflation of the lung and chest wall. From these data andEq. 2 , we obtain values of μ for the diaphragm.
Four bred-for-research dogs were studied. The body mass of the dogs was 10.1 ± 0.3 kg, and the inspiratory capacity (IC) was 0.7 ± 0.1 liter. The video-fluoroscopic method for measuring diaphragm muscle length has been described previously (1, 5, 24, 25). Briefly, in a preparatory surgical procedure, silicone-coated lead spheres and cylinders were stitched to the peritoneal surfaces of the left hemidiaphragms of the four dogs. The pattern of placement in one dog is shown in Fig. 1. Three or four markers were placed at intervals of ∼1 cm along each of six muscle bundles at different points from the ventral to the dorsal end of the costal diaphragm, and three or four markers were placed along each of two muscle bundles of the crural diaphragm. The animals were allowed to recover for ≥3 wk. The animals were anesthetized with pentobarbital sodium (30 mg/kg), intubated with a cuffed endotracheal tube, placed in the prone or supine position in a radiolucent body plethysmograph situated in the test field of an orthogonal biplane fluoroscopic system, and mechanically ventilated. The dog was switched from the ventilator to a supersyringe, and IC was determined by manually inflating the lungs to total lung capacity (TLC), defined as volume at an airway pressure of 30 cmH2O. Biplane fluoroscopic images were taken at TLC and at three equally spaced volumes down to functional residual capacity (FRC).
The coordinates of the markers in the two orthogonal images were determined, and the three-dimensional coordinates of the markers were calculated from their coordinates in the two orthogonal images. The lengths of the six muscle bundles in the costal diaphragm and two muscle bundles in the crural diaphragm were computed by adding the distances between adjacent markers along each bundle. This sum of the chord lengths is >95% of the length of a smooth curve through the points (25).
Muscle lengths at FRC of the eight muscle bundles in the two postures are shown in Fig. 2.
Lengths of costal bundle 3 in the prone and supine postures are plotted vs. Vl in Fig.3. This plot is representative of the data for most muscle bundles in both postures. That is, length decreased with increasing volume by about the same amount in the two volume steps at lower Vl, but the change in length in the volume increment to TLC was smaller. In the prone posture the change in length in the step near TLC was about one-half of that at the lower volumes. In the supine posture the same was true for the more ventral bundles, but the change at the last volume step was larger for the more dorsal bundles.
Fractional changes in muscle length per unit volume change for the volume interval from FRC to FRC + IC are shown in Fig.4.
The radiopaque marker technique was adapted to the measurement of diaphragm muscle length by Rodarte and colleagues (1, 3-5, 24,25). In particular, Sprung et al. (25) report passive shortening of three bundles in the costal diaphragm and one bundle in the crural diaphragm each of five dogs. Their values of passive shortening from FRC to TLC averaged ∼25%, considerably smaller than the values obtained earlier by sonomicrometry (22). Here we report values of fractional shortening per liter rather than fractional shortening per IC. The average IC of our dogs was 0.7 liter, and thus the values of fractional shortening per liter shown in Fig. 4, ∼0.35 liter−1, are consistent with the data of Sprung et al. The number of markers used in our study is larger than in the study of Sprung et al., and more comprehensive data on passive muscle shortening were obtained.
As shown in Fig. 2, muscle lengths were slightly larger in the supine than in the prone posture. The distribution of muscle lengths around the circumference of the diaphragm in vivo (Fig. 2) is similar to the distribution obtained by Boriek and Rodarte (2) from measurements in diaphragms that had been excised and laid flat. Muscle length is greater in the midcostal region and smaller and more variable near the dorsal end of the costal diaphragm.
The values of the quantity (dL/L odVl)relshown in Fig. 4 constitute the primary results of this study, i.e., the values of μ for the diaphragm in the lower two-thirds of the IC. The implications of these data are discussed below.
Circumferential distribution of μ.
The distribution of μ is fairly uniform around the diaphragm. However, at the dorsal end of the costal diaphragm, μ is lower than the mean. In the prone dog the region of low μ is confined to the most dorsal muscle bundle sampled. In the supine dog the region of lower μ includes the two or three most dorsal bundles. De Troyer et al. (8, 10, 15, 17) found that the distributions of mass and activation of the parasternals mirror the distribution of μ; muscles with smaller μ are thinner, and their activation during spontaneous breathing, as a fraction of maximum activation, is lower. Data in the literature show a similar correlation between the distributions of diaphragm m and the distribution of μ. The thickness of the muscle layer in the dorsal region of the costal diaphragm is about two-thirds of the thickness in the mid- and ventral regions (20). Also, blood flow to the dorsal region is smaller (6). This implies that the activation is lower in the dorsal region. Thus mass and activation are correlated with μ in the diaphragm as in the parasternals.
Relative μ values of the diaphragm and intercostal muscles.
In the lower two-thirds of IC, μ of the canine diaphragm is −0.35 ± 0.18 and −0.27 ± 0.16 liter−1 in the prone and supine postures, respectively. This is considerably larger than the values for the inspiratory muscles of the rib cage. De Troyer and colleagues (10, 17) report that the values of μ for the parasternal intercostals vary axially with interspace number and laterally within each interspace. The maximum occurs near the sternum in the second or third interspace, and the maximum value in supine animals is ∼0.10 liter−1. To compare the values for the diaphragm and parasternals, the size of the animals should be taken into account. One would expect that fractional length change per IC would be independent of animal size, IC would be proportional to body mass, and fractional shortening per liter would therefore be inversely proportional to IC and body mass. If this were true and if m were proportional to body mass, ΔPao would be independent of animal size. The body mass of the animals studied by De Troyer et al. ranged from 14 to 25 kg, whereas the average body mass of our dogs was 10 kg. Scaling the data on the parasternals to animals of our size yields 0.2 liter−1 for the parasternals. Therefore, the μ of the diaphragm is estimated to be 1.5–2 times the maximum μ of the parasternals.
According to Eq. 1 , the contribution of the diaphragm to ΔPao can be calculated by multiplying μ bym and ς. Margulies (20) reported the masses of the diaphragms of dogs as a function of body mass. For 10-kg dogs, the size of our animals, diaphragm mass is ∼56 g, and most of this is m. Therefore, for a maximum ς of 2.2 kg/cm2 (12), the maximum ΔPao for the diaphragm is approximately −43 and −33 cmH2O in the prone and supine postures, respectively. ΔPao values of −30 to −40 cmH2O are indeed generated during coordinated inspiratory efforts, but part of ΔPao is contributed by muscles of the rib cage. The diaphragm is never maximally activated during spontaneous inspiratory efforts (4, 13), and its contribution to ΔPao is less than the value computed for maximum activation.
Although the computed value of maximum ΔPao for the diaphragm has no functional significance, the value of μ can be used to infer the value of a quantity with physiological interest, i.e., the fraction of the total inspiratory pressure that is contributed by the diaphragm during coordinated inspiratory effort. First, ΔPao, as described byEq. 1 , must be distinguished from transdiaphragmatic pressure (Pdi), which is often used to describe diaphragm function. ΔPao is the change in airway pressure produced by ς in the muscle. The ς is converted to ΔPao by a complicated mechanism that involves the entire chest wall. Although this mechanism cannot be traced in detail, its effect is summarized by the value of μ. Pdi also depends on stress in the diaphragm, but the mechanism that transforms muscle stress to Pdi is a local mechanism that can be described in detail (4). In the midcostal region, where the diaphragm has the shape of a right circular cylinder, Pdi is proportional to τ/r, where τ is membrane tension (kg/cm) and r is the radius of curvature of the sheet. In turn, τ = ςt, wheret is the thickness of the muscle sheet. Thus Equation 3The factor 103 in Eq.3 is required to convert ς (kg/cm2) to pressure (cmH2O).
During a coordinated inspiratory effort, the diaphragm and the muscles of the rib cage contribute to ΔPao. These two contributions are denoted ΔPaodi and ΔPaorc. The total is denoted ΔPaotot, and ΔPaotot = ΔPaodi + ΔPaorc. Pdi is the difference between gastric pressure and pleural pressure. During an inspiratory effort against an occluded airway, the change in gastric pressure is small compared with the change in pleural pressure, and the change in pleural pressure equals ΔPao. Therefore, Pdi ∼ −ΔPaotot, and the fraction f of ΔPaotot that is contributed by the diaphragm is Equation 4Substituting for ΔPaodi fromEq. 1 and Pdi fromEq. 3 yields the following equation for f Equation 5Substituting values from the literature, m = 56 g (20), r = 5 cm (4),t = 0.25 cm (2, 20), and our values for μ yields the values for f of ∼0.4 and ∼0.3 for the prone and supine postures, respectively. We conclude that during spontaneous breathing in the prone dog the diaphragm contributes ∼40% of inspiratory pressure and the muscles of the rib cage contribute 60%. For a coordinated effort that produces a ΔPao of −30 cmH2O, the contribution of the diaphragm is −12 cmH2O. This is ∼30% of ΔPaodi at maximum ς, and therefore ς is ∼30% of maximum. It should be emphasized that some assumptions were made in deriving Eq.5 . In particular, it was assumed that the inspiratory effort was a coordinated effort and that the change in gastric pressure was small compared with ΔPao. Also, the values of the parametersm, r, andt that were substituted intoEq. 5 came from a variety of sources. As a result, the value of f should be taken as an estimate.
Volume dependence of μ.
In the upper one-third of IC, passive diaphragm muscle shortening per unit increase in Vl is less than one-half of that in the lower two-thirds of IC. Thus the μ of the diaphragm drops sharply at high Vl. We looked for the change in coupling between the diaphragm and the lung that would account for this decrease in diaphragm function at high volume. We focused on the muscle bundles of the midcostal diaphragm that we studied previously (1, 4,5), i.e., costal bundles 2–4. The coordinates of the markers on these bundles were transformed to a local ξ-η-ζ coordinate system that we used to describe this region. A plane was fit to the 12 markers. A quadratic was fit to the distances of the markers from this plane, and the directions of the principal curvatures of the quadratic were determined. The ξ-axis was chosen to lie along the direction of minimum principal curvature, and the ζ-axis was chosen to lie parallel to the midplane of the dog. As we found previously, the minimum principal curvature of the diaphragm surface is small in this region, and the muscle bundles of the midcostal diaphragm lie in planes that are nearly parallel to the η-ζ plane shown in Fig. 5. The average values of the η- and ζ-coordinates of corresponding markers in the three bundles in the four dogs were computed (Fig.6); i.e., each point in Fig. 6 is the average of 12 values. For example, the point on the chest wall is the average coordinate of 12 markers: 1 for each marker on the chest wall of the 3 bundles in 4 dogs.
The data shown in Fig. 6 have properties that are similar to those found in earlier studies (2, 4). In this region the muscle bundles follow a curved path from the chest wall to the central tendon, and the zone of apposition is not apparent in the shape of the bundles. With increasing Vl, material points in the muscle move nearly parallel to the sagittal midplane of the dog as the dome of the diaphragm descends and the muscle bundles shorten.
Two features of the data shown in Fig. 6 are pertinent to the volume dependence of μ. First, the point on the chest wall moves laterally as Vl increases. As the cross-sectional area enclosed within the zone of apposition increases, the dome must descend and the muscles shorten to maintain the volume under the diaphragm constant. Thus a lateral displacement of the zone of apposition causes muscle shortening at constant abdominal volume. As Vl increases and the zone of apposition becomes narrower (3), less shortening is required to maintain constant volume. However, the lateral displacements are small, ∼0.5 cm in the supine posture and less in the prone posture, and this is not the primary cause of muscle shortening. The primary cause of muscle shortening is the volume displacement of the abdominal compartment of the chest wall. We know of no data on the volume dependence of rib cage and abdominal compliance in the dog; in humans, however, abdominal compliance falls at high Vl, whereas rib cage compliance does not (23). It can be seen from Fig. 6 that the displacement of the dome of the diaphragm in the volume step to TLC is about one-half of the displacements in the other two volume steps. Presumably, a smaller fraction of the increase in Vl is taken up by the abdomen in the last volume step than in the first two. Thus we conclude that the decrease in diaphragm shortening at high Vl is primarily due to a decrease in abdominal displacement. For both mechanisms for decreased passive shortening, a corresponding mechanism for decreased respiratory effect can be identified. In the present model for the diaphragm (11,19), abdominal pressure acting across the zone of apposition has an inspiratory effect on the rib cage, and this inspiratory effect decreases with decreasing height of the zone of apposition. A decrease in abdominal compliance would also cause a decrease in the inspiratory effect of the diaphragm; for a given ς and a given increase in Pdi, gastric pressure would rise more and pleural pressure would fall less.
The data shown in Fig. 6 have an unexpected feature: the line of insertion of the diaphragm on the chest wall moves caudally as Vl increases. Although we know of no data on the displacement of the caudal ribs during passive inflation, the more cranial ribs move cranially during passive inflation (21), and the more caudal ribs also move cranially during active inspiration (7). We therefore expected that the caudal ribs and the line of insertion would move cranially as Vl increased. Perhaps the location of the line of insertion is affected by tensions in the diaphragm and abdominal muscles. If that were the case, the caudal displacement with increasing Vlmay be the result of decreasing tension in the diaphragm and increasing tension in the abdominal muscles.
The geometric effect of the caudal displacement of the line of insertion on muscle length is clear. Roughly speaking, shortening is proportional to the relative displacement between the dome and the line of insertion. For a given caudal displacement of the dome, the muscle shortens less if the line of insertion moves caudally than it would if the line of insertion moved cranially. The mechanical consequences of this geometric relationship are described by Eq.2 . Greater passive shortening implies a greater inspiratory effect per unit ς. However, there are competing effects of shortening on muscle function. Greater shortening would cause a greater displacement along the length-tension curve and a greater change in diaphragm shape. If the line of insertion moved cranially, maximum tension and diaphragm curvature would decrease more rapidly with increasing Vl, and the range of Vl values over which the diaphragm could exert an inspiratory force would be reduced. For example, if the axial displacement of the line of insertion shown in Fig. 6 were reversed, muscle length at TLC would be ∼50% of length at FRC and the diaphragm would be flat. Maximum tension approaches zero at that length (12), and tension is not converted to pressure if the diaphragm is flat (4). Of course, this tradeoff between μ and volume range is simply a result of the fact that the length-tension curve sets a constraint on the work that the muscle can deliver.
The primary results of this study are as follows.
1) The value of μ for the canine diaphragm is −0.35 liter−1 in the prone posture and somewhat smaller, i.e., −0.27 liter−1, in the supine posture. This is 1.5–2 times the μ of the intercostal muscles with greatest μ. In addition, the value of μ is uniform around most of the costal and crural diaphragms but lower near the dorsal end of the costal diaphragm. The distributions ofm and activation are correlated with the distribution of μ. Finally, we estimate that the diaphragm contributes ∼40% of inspiratory pressure in the prone posture. We also estimate that, during a coordinated inspiratory effort that produces an inspiratory pressure of 30 cmH2O, ς in the diaphragm is ∼30% of maximum.
2) The value of μ decreases at higher Vl, and we attribute this to the decrease in the height of the zone of apposition and, more importantly, to a decrease in the volume expansion of the abdominal compartment at high Vl. In addition, the line of insertion of the midcostal diaphragm moves caudally as Vl increases. As a result, the value of μ is smaller, but the volume range of diaphragm function is larger.
This work was supported by National Heart, Lung, and Blood Institute Grants HL-45545 and HL-46230.
Address for reprint requests: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455.
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- Copyright © 1998 the American Physiological Society