Both diaphragm shape and tension contribute to transdiaphragmatic pressure, but of the three variables, tension is most difficult to measure. We measured transdiaphragmatic pressure and the global shape of the in vivo canine diaphragm and used principles of mechanics to compute the tension distribution. Our hypotheses were that 1) tension in the active diaphragm is nonuniform with greater tension in the central tendon than in the muscular regions; 2) maximum tension is essentially oriented in the muscle fiber direction, whereas minimum tension is orthogonal to the fiber direction; and 3) during submaximal activation change in the in vivo global shape is small. Metallic markers, each 2 mm in length, were implanted surgically on the peritoneal surface of the diaphragm at 1.5- to 2.0-cm intervals along the muscle bundles at the midline, ventral, middle, and dorsal regions of the left costal diaphragm and along a muscle bundle of the crural diaphragm. Postsurgery, a biplane videofluoroscopic system was used to determine the in vivo three-dimensional coordinates of the markers at end expiration and end inspiration during quiet breathing as well as at end-inspiratory efforts against an occluded airway at lung volumes of functional residual capacity and at one-third maximum inspiratory capacity increments in volume to total lung capacity. A surface was fit to the marker locations using a two-dimensional spline algorithm. Diaphragm surface was modeled as a pressurized membrane, and tension distribution in the active diaphragm was computed using the ANSYS finite element program. We showed that the peak of the diaphragm dome was closer to the ventral surface than to the dorsal surface and that there was a depression or valley in the crural region. In the supine position, during inspiratory efforts, the caudal displacement of the dorsal region of the diaphragm was greater than that of the dome, and the valley along the crural diaphragm was accentuated. In contrast, at lower lung volumes in the prone posture, the caudal displacement of the dome was greater than that of the crural region. At end of inspiration, transdiaphragmatic pressure was ∼6.5 cmH2O, and tensions were nonuniform in the diaphragm. Maximum principal stress σ1 of central tendon was found to be greater than σ1 of the costal region, and that was greater than σ1 of the crural region, with values of 14–34, 14–29, and 4–14 g/cm, respectively. The corresponding data of the minimum principal stress σ2 were 9–18, 3–9, and 0–1.5 g/cm, respectively. Maximum principal tension was approximately parallel to the muscle fibers, whereas minimum tension was essentially orthogonal to the longitudinal direction of the muscle fibers. In the muscular region, σ1 was ∼3-fold σ2, whereas in the central tendon, σ1 was only ∼1.5-fold σ2.
- respiratory muscle mechanics
- chest wall mechanics
- finite element modeling
- membrane mechanics
the shape of the diaphragm is vital for converting muscle tension into transdiaphragmatic pressure (Pdi) and muscle shortening into volume displacement. It is important to know the quantitative relations among tension, muscle length, shape, and Pdi to attain a comprehensive understanding of diaphragm mechanics. The bulk of our knowledge on in vitro muscle mechanics of the diaphragm comes from observations on uniaxial length-tension relationships; for example, McCully and Faulkner (15) measured passive and active uniaxial length-tension relationships. However, in vivo, the diaphragm is under pressure loading and, therefore, is subjected to biaxial rather than uniaxial loads. That is, the diaphragm experiences loads both along and transverse to the direction of the fibers. Therefore, data from uniaxial loading cannot be extrapolated to accurately analyze physiological behavior of the intact diaphragm, and the data available on the mechanical properties of the diaphragm muscle under passive biaxial loading are limited (2, 19). Previously, we measured shape and tension in the intact pressurized rat diaphragm, and other investigators measured these variables in the dog (6, 14). These measurements were, however, conducted in the passive diaphragm. Our current study is the first one to provide measurements of global tension and shape in the active diaphragm during quiet breathing. We also measured global shape of the diaphragm during forceful submaximal activation of the diaphragm. The complex geometry and mechanics of the intact diaphragm are crucial to its physiological function. In particular, a realistic shape of the diaphragm subjected to physiological pressures was crucial to compute tension distribution in the diaphragm.
In this study, we tested the hypotheses that 1) tension in the active diaphragm is nonuniform with greater tension in the central tendon than in the muscular regions; 2) in the active diaphragm muscle maximum principal tension is essentially oriented in the muscle fiber direction, whereas minimum principal tension is orthogonal to the fibers; and 3) change in the in vivo global shape of the diaphragm is small during submaximal activation. We report the three-dimensional shape of the passive and active diaphragm in vivo under different physiological states in the supine and prone positions. Finally, we computed the magnitude and orientation of the tension distribution of the active diaphragm using finite element analysis.
Dogs were maintained according to the National Institutes of Health Guide for Care and Use of Laboratory Animals, and all procedures were approved in advance by the Institutional Review Board of Baylor College of Medicine.
Three bred-for-research beagle dogs (9.9–10.5 kg) were deeply anesthetized with pentobarbital sodium (60–80 mg/kg) and surgically prepared using the same methods of our earlier studies (1, 7). After midline laparotomy, 22 two-mm silicon-coated lead spheres were sewn on the peritoneal surface along the muscle bundles in the costal region of the left hemidiaphragm, along the junction between the dorsal costal and the crural diaphragm, along the crural muscle near the midplane of the dog, and on the central tendon along the midline (Fig. 1). The anteroposterior and lateral roentgenogram of a prone dog at functional residual capacity (FRC) are shown in Fig. 2, A and B, respectively. Figure 2 shows the position of four markers along muscle bundles from the insertion on the rib cage to the central tendon in the costal and crural regions. Markers 1, 3, 7, 11, 15, 18, and 19 are located at the insertion of the diaphragm on the chest wall. Markers 2, 6, 10, 14, and 17 are located at the junction of diaphragm muscle with central tendon. Markers 2, 20, 21, and 22 lie approximately near the midplane of the dog. After 3-wk postoperative recovery, the animals were lightly anesthetized with pentobarbital sodium (30 mg/kg), intubated with a cuffed endotracheal tube, and placed in the supine or prone posture in a radiolucent body plethysmograph situated in the test field of a high spatial (±0.5 mm) and temporal (30 Hz) resolution biplanar video roentgenographic recording system. We positioned balloon-tipped catheters in the stomach and esophagus using fluoroscopy. We verified the catheter placement by demonstrating 1) increasing abdominal pressure and decreasing esophageal pressure during a spontaneous breath and 2) equal decrease in esophageal and airway pressures during an occluded inspiratory effort at FRC. We inflated the lungs to its total capacity, and the airway was then occluded at TLC and at steps of 1/3 inspiratory capacity down to FRC. When the change of airway pressure reached a plateau, usually during the fifth or sixth inspiratory effort at each lung volume, biplanar images continuously recorded displacements of the radio-opaque metallic markers. After rotating the animal to the prone posture, we repeated the experimental protocol.
Shape of the global diaphragm.
The passive and active global shape of the diaphragm was computed using a two-dimensional spline technique (20). With the use of this technique, the positions of the markers were fitted to a surface at each lung volume in the anatomical Cartesian coordinates. The X-axis is downward (supine: dorsal-ventral; prone: ventral dorsal), the Z-axis is cephalad, and the Y-axis is in the lateral medial direction.
Finite element modeling.
The diaphragm was modeled as a pressurized membrane. A finite element network of 1,576 membrane elements and 876 nodal points was built using the fitted surface to model the active diaphragm. Tension distribution was computed using ANSYS finite element program. Transdiaphragmatic pressure is determined by both diaphragm shape and tension. Therefore, we assumed that the muscular portion and the central tendon of the diaphragm act as a membranous structure. Therefore, principles of mechanics can be used to compute the tension from the shape and pressure loading. In particular, the bending moments and out-of-plane shear forces can be neglected; therefore the problem of solving for the tension distribution is greatly simplified because membranes are statically determinant structures, and a solution can be obtained with membrane theory (21). Accordingly, the internal forces or tensions of a membrane are independent of the intrinsic stiffness and depend only on external load, shape, and geometric boundary constraints. We used membrane elements in the ANSYS software (STIF41) to generate the finite element model of the diaphragm. This is a three-dimensional element with membrane (in plane) stiffness but no bending (out of plane) stiffness. The geometric boundary of the membrane model simulated the insertion of the diaphragm on the rib cage. Nodal points that lay at the edge of the membrane models were restrained to zero translational displacements. Each element had three degrees of freedom at each node corresponding to translations in the x, y, and z directions, and element geometry was described in terms of the global coordinates of the four points. Tension distribution within the diaphragm was computed from the knowledge of the shape and applied pressure (13). A very stiff isotropic membrane was used to model both muscle and central tendon (Young's modulus = 1 × 104 cmH2O). Maximum displacement of any point on the model was less than 1 × 10−3 mm. Uniform thickness of the membrane element of 0.25 cm was used.
Contours that describe the shapes of the passive diaphragm, determined by a two-dimensional spline technique, are shown for a supine dog at FRC (Fig. 3A) and end of inspiration during spontaneous quiet breathing (Fig. 3B), as well as end of inspiratory effort against an occluded airway at lung volumes equal to FRC (Fig. 3C), FRC + inspiratory capacity (IC) (Fig. 3D), FRC + IC (Fig. 3E), and at total lung capacity (Fig. 3F). Figure 4, A–F, shows corresponding data in the prone dog. If the diaphragm were axisymmetric around the midplane of the dog, the contours would be circles with centers at the axis of symmetry. Each contour shown in either of Fig. 3 or Fig. 4 represents an intersection of the diaphragm surface with a plane at a constant cephalocaudal position. In Figs. 3 and 4, the height difference between two consecutive contours is ∼4 mm. A reference plane is shown as a dashed line and at the same height across all figures. This facilitates comparisons between the shapes of the diaphragm between the different maneuvers. Relative to the reference plane, the diaphragm dome has descended during inspiratory efforts. The most cephalad region of the lung-apposed surface lies within the boundaries of the V-shaped central tendon but includes less of the central tendon in the prone than corresponding maneuvers in the supine. In both supine and prone, the peak of the diaphragm dome is closer to the ventral surface than to the dorsal surface. In the prone, the depression or valley of the crural region extends less ventrally than in the supine. At the end of a spontaneous inspiration or at the end of inspiratory effort against an occluded airway, the valley of the crural region of the prone is therefore less pronounced than in the supine dog.
Diaphragm membrane principal tension contours, σ1 and σ2, computed using ANSYS finite element program, are shown in Fig. 5, A and B. This is corresponding to the shape of the diaphragm at end of inspiration during spontaneous breathing for a mean transdiaphragmatic pressure (Pdi) of ∼6.5 cmH2O. In the costal region, the mean value of maximum principal tension is ∼3-fold the minimum principal tension. Tensions transverse to the fibers in the costal region of the diaphragm (Fig. 5B, contour B: σ2) are small. Furthermore, the magnitude of maximum tension in the central tendon is significantly greater than that in the muscular portion. There are relatively high tensions in the ventral region of the crural diaphragm. Maximum principal stress σ1 of central tendon was found to be greater than σ1 of the costal region, and that was greater than σ1 of the crural region, with values of 14–34, 14–29, and 4–14 g/cm, respectively. The corresponding data of the minimum principal stress σ2 are 9–18, 3–9, and 0–1.5 g/cm, respectively. Magnitude and direction of principal tension vectors are shown in Fig. 6, A and B. In general, principal tension vectors lie approximately along and transverse to the muscle fiber direction. There is a cloud of large tension vectors in the ventral lateral costal region (contour E, σ1; contour H, σ2). Minimum principal tension σ2 is oriented at right angles to σ1 and is nearly always in the plane of the membrane.
We have computed the detailed in vivo shape of the diaphragm, and using the principles of mechanics we computed tension distribution of the active diaphragm. With increasing diaphragmatic muscle contraction, the dome of the diaphragm is more pointed, and the valley formed by the crural diaphragm deepens. The lateral region of the costal diaphragm demonstrates smaller principal curvature in the direction transverse to the muscle fibers than along the fiber direction. The diaphragm has nonuniform tension with those in the central tendon generally larger than those computed in the muscular portion. Maximum and minimum principal tensions lie along and transverse to the long axis of the muscle fibers, respectively, with the muscle having greater tension along the fibers than transverse to the fibers.
The simplest model of the diaphragm assumes a hemispherical diaphragm shape with uniform tension (22). Such shape is governed by the Laplace law, Pdi = 2κT, where κ is the curvature and T is the tension of the membrane governing the relationship between pressure, tension, and curvature. Kim et al. (11) directly measured Pdi and tension in dogs, and using the Laplace equation, they computed radius of curvature. In theory, curvature decreases at large lung volumes, but Kim et al. (11) demonstrated little change in curvature at those volumes. The model developed by Gates et al. (10) assumed an axisymmetric, anisotropic elastic membrane of the diaphragm and was modified by Whitelaw et al. (22) with an isotropic membrane of uniform tension supported on a transverse section of the human thorax and loaded by hydrostatic pressure. The Whitelaw model resembled the double hump shape of human diaphragm with an indentation similar to that caused by the spine. We determined the shape and tension in the intact rat diaphragm using in vitro muscle preparation (6) and found greater tension in central tendon compared with muscle and smaller curvature in the direction transverse to muscle fibers compared with along muscle fibers. In previous studies of intact canine diaphragm, we reported that curvature of the midcostal diaphragm was uniform along muscle fibers and changed little with changes in lung volume (3, 7). Furthermore, we developed a finite element hemispherical membrane model of the passive diaphragm (5) by relaxing the hypothesis of homogeneous anisotropic elastic membrane used by Gates (10). Instead, we examined the effect of anisotropic properties of the muscle and inextensibility of the central tendon on the displacement and curvature of a pressurized hemispherical membrane. From our model, we concluded that the change in shape of the diaphragm in response to tensile stresses produced by the pressure is restricted when the central tendon is inextensible and the muscle is anisotropic, having a smaller stiffness in the longitudinal direction of the muscle fibers than in the transverse direction to the muscle (5).
Supporting physiological data.
Although stress has never been directly measured in trunk skeletal muscle, e.g., intercostal and abdominal muscles, they can support a pressure without transverse loading with their multiple layer arrangement. In contrast, the diaphragm uniquely supports stress in the direction transverse to the muscle fibers even if greater stress were present in the direction of muscle fibers. Data from uniaxial in vitro experiments may not accurately represent physiological behavior of the diaphragm muscle because it is being subjected to a pressure load in vivo, and very limited physiological data exist on the canine diaphragm under biaxial loading in vitro (2, 19). Recently, we conducted biaxial loading tests of muscular sheets excised from the midcostal region of the dog diaphragm (2) and found that the muscle sheet is more compliant and considerably more extensible in the direction of the muscle fibers. Those data are consistent with that of both the excised and intact diaphragm of the rat (6). However, measured passive stiffness in the perpendicular direction (2) was not high enough to explain our earlier observation (8) that strains in the perpendicular direction in vivo were nearly zero. We therefore concluded that passive transverse stress in vivo is small although the possibility of high transverse stiffness may cause negligible transverse fiber strain in vivo as well. Margulies et al. (13) used the finite element program ANSYS to compute diaphragm stress at FRC in supine dogs and demonstrated two- to threefold larger stress along fibers under biaxial vs. uniaxial conditions for a given fiber length. These results were expected because with an additional tensile force acting orthogonal to that of the applied load, most materials require a larger tensile force to maintain a given stretched length than with the orthogonal direction unconstrained (9).
Effect of shape on tension distribution.
We determined global shape in the diaphragm and found it to be complex (1a, 14). However, the computation of such shape was restricted to the passive state at FRC. In the current analysis, we assumed symmetry around the midplane of the dog, and therefore, shape and consequently tension of the right hemidiaphragm is identical to that of the left. The assumption of symmetry was made only for convenience, and the data presented here should only be relevant to the left hemidiaphragm. Our current measurements of diaphragm shape include global shape of the submaximally activated diaphragm. When the diaphragm is in static equilibrium, the internal forces or stresses are independent of the material properties and depend only on the external load (Pdi), the shape of the structure, and its boundary constraints. The tension distribution in such an elastic system would be the same as in a rigid structure with the same deformed shape as that of the elastic diaphragm that is subjected to the same loads. Although the unloaded shape and material properties of the diaphragm are not known, if the deformed shape at static equilibrium and the loads are known, finite element analysis can be used to compute the stress distribution. The current study and Margulies' study (13) have used this strategy to compute tension distribution in the diaphragm. Margulies et al. (13) demonstrated that tension in the passive diaphragm of the dog is nonuniform and that the greatest in-plane tension was essentially aligned with the direction of muscle fascicles and is two to four times larger than the tension in the direction transverse to the muscle fibers. In Margulies' study, however, markers were not placed on the central tendon, and therefore the central tendon and muscle domains could not be distinguished from each other. In the central tendon of the canine diaphragm, collagen fibers have a random orientation near the midline, but near the insertion of the muscle, collagen fibers are preferentially oriented parallel to the muscle fibers. If collagen fibers were oriented in the direction of greatest stress, this would imply that near the central tendon in the muscular region of the diaphragm there is greater stress along the muscle fibers than transverse to the fibers.
Tension and stress from finite element analysis.
In our study we report tension values computed from the finite element analysis in grams per centimeter. We used a Pdi of 6.5 g/cm2 and a uniform thickness of 0.25 cm, which is about the same thickness as that of the unloaded left midcostal region of the diaphragm. Tension in the costal muscle of the dog diaphragm was nonuniform, and tension along the fibers was severalfold greater than tension in the transverse direction. This is consistent with earlier predictions of small in vivo passive transverse stress inferred from in vivo (8) and in vitro (2) studies of diaphragm mechanics in the dog. In the current study, we placed markers on the central tendon as well as the muscular portion of the diaphragm, and this allowed us to distinguish between the central tendon and the muscle domains. Tension is greater in the central tendon than in the muscle, and because thickness of the central tendon is ∼ that of the costal muscle, stresses in the central tendon should be even greater than that of the muscle. Stress in the midcostal muscle is computed by dividing tension in the midcostal region of the membrane by 0.25 cm, the average thickness of unstressed midcostal diaphragm muscle. The dorsal costal region is thinner than other regions of the diaphragm muscle, so that equal tension in the membrane will produce larger stress in the muscle of the dorsal region. Muscle thickness of the canine diaphragm is nonuniform (4, 12). This indicates that both stress in the muscle and tension in the diaphragm membrane cannot be uniform in these regions. If tension were uniform in either the crural or the costal diaphragm, then stress would vary inversely with thickness. Therefore, muscle thickness data can be used to estimate stresses from tension. In Fig. 5A, tension contours D and C lie in the dorsal costal region, whereas tension contours B and D lie in the medial crural region. Averaging values for these tension contours therefore provides approximations for σ1 in their corresponding regions, yielding tensions of 16.9 and 14.2 for costal and crural region, respectively. Taking into account thickness of 0.30 cm in the crural muscle and 0.17 cm in the dorsal region of the costal diaphragm (4, 12), our results yield predicted stress of 56.3 and 83.5 g/cm2, respectively. Skeletal muscle is incompressible and does not change volume as it shortens. Therefore, to a first approximation, the sum of the three principal strains must be zero. Our previous data demonstrated that strain in the transverse fiber direction is negligible (8). This implies that the diaphragm must thicken by nearly the same fraction as in its fractional shortening. During inspiratory efforts, however, the diaphragm exhibits nonuniform muscle shortening (18) and consequently nonuniform muscle thickening. Therefore, stress calculations using tension should take into account muscle thickening during inspiratory efforts. For example, the midcostal diaphragm shortens by ∼18% and therefore thickens by about the same percentage at end of inspiration during quiet breathing in supine dogs (8). Therefore, the true stress that the diaphragm experiences in the midcostal region is ∼18% smaller than those values computed using membrane tension and unloaded thickness of the diaphragm.
Validation of finite element results.
Previous investigators predicted tension in the diaphragm by measuring curvature from X-ray projections, providing assumptions about the Pdi gradient, and making use of the Laplace law (16, 17, 22). Whitelaw et al. (22) predicted lower tensions of 10–20 g/cm in the upright position. Smith and Loring (17) computed tension of 5 and 40 g/cm in upright and supine postures, respectively. Paiva et al. (16) estimated tensions of 32–54 g/cm in supine humans. Our finite element analysis predicted maximum tensions of 14–29 g/cm in the muscle along the direction of fibers and a much smaller tension in the direction transverse to the muscle fibers. The published data (16, 17, 22) remain consistent with values predicted by our finite element model. Diaphragmatic curvature in the direction transverse to muscle fibers is negligible, and it is therefore reasonable to ignore the contribution of tension in that direction. Our published data on diaphragm mechanics in dogs demonstrated radius of curvature in the midcostal region of the diaphragm ranging from 4.8 to 5.3 cm with little effect of lung volume or level of muscle activation (7). Assuming a Pdi of 6.5 g/cm2, and 5 cm radius of curvature, the predicted tension along the direction of costal muscle fibers would be 32.5 g/cm, consistent with current data in Fig. 5A showing maximal tension along the direction of costal muscle fibers reaches a peak of 29 g/cm.
This work was supported by National Heart, Lung, and Blood Institute Grants HL-46230, HL-072839, and HL-63134.
We are grateful to Dr. T. Wilson for discussion of the data. We thank Q. Lin and D. Zhu for technical assistance.
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- Copyright © 2005 the American Physiological Society