INTRODUCTION

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THE DIAPHRAGM HAS A CURVED shape, which is vital for converting muscle tension into transdiaphragmatic pressure (P_di) and muscle shortening into volume displacement. Knowledge of the quantitative relationships among tension, muscle length, shape, and P_di is important for understanding 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 (11) 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 analyze accurately physiological behavior of the intact diaphragm, and the data available on the mechanical properties of the diaphragm muscle under passive biaxial loading are limited (1, 17). Understanding the complex geometry and mechanics of the intact diaphragm is crucial to understanding its physiological function. In particular, a realistic shape of the diaphragm subjected to physiological pressures was crucial to compute tension distribution in the diaphragm. Knowledge of tension distribution and regional muscle length changes can be used in determining regional length-tension relationships.

In this study, we tested the hypothesis that diaphragm muscle is anisotropic with smaller stiffness in the fiber direction than transverse to the fibers and therefore tension distribution is nonuniform, whereas shape change is restricted across physiological ranges of P_di. We developed in vitro preparations of both the intact and the excised rat diaphragm to investigate relationships among tension distribution, P_di, shape, and muscle length. We report the length-tension relationship of the intact rat diaphragm muscle and the length-tension relationship of the excised muscle under uniaxial loading conditions. We also report the three-dimensional shape of the passive diaphragm in vitro under different P_di. Finally, we computed the stress distribution in the pressurized membrane of the passive diaphragm using finite element analysis.

We found that the diaphragm muscle sheet is less compliant and considerably less extensible in the direction transverse to the muscle fibers than in the direction along the fibers. We also found that the intact diaphragm is less extensible during pressure loading than those muscle sheets loaded uniaxially along the fibers. Tension in the diaphragm was nonuniform, and the central tendon experienced tension that was generally larger than those measured in the muscular portion. Furthermore, tension in the muscle was greater along the fibers than transverse to the fibers.

	METHODS

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Muscle preparation. Eight male Sprague-Dawley rats (250-300 g) were maintained according to the National Institute of Health Guide for the Care and Use of Laboratory Animals, and the protocol was approved in advance by the Institutional Review Board of Baylor College of Medicine. The rats were anesthetized with intraperitoneal injection of pentobarbital sodium (40 mg/kg) followed by intramuscular injection of ketamine (35 mg/kg). Each animal was tracheostomized and mechanically ventilated with 100% O₂ during surgical removal of the diaphragm. The diaphragm was removed intact with associated ribs and a portion of the lumbar spine and placed in Krebs-Ringer solution (containing, in mM, 137 NaCl, 5 KCl, 2 CaCl₂, 1 MgSO₄, 1 NaH₂PO₄, and 24 NaHCO₃) equilibrated with 95% O₂-5% CO₂ (pH 7.18 at 23°C). Nondiaphragm tissue was trimmed in Krebs-Ringer solution after excision.

View larger version (31K): [in this window] [in a new window]   Fig. 1.   A schematic of the in vitro muscle apparatus used to measure the passive properties of the whole rat diaphragm. The muscle mold and fluid chamber are detachable components located within the fluid overflow chamber. A variable speed pump controls a continuous fluid flow, and a fraction of the input flow is diverted to wet continuously the surface of the diaphragm. A TransPac transducer monitors the pressure within the fluid chamber. The displacement of surface markers is captured by video cameras and recorded by a VCR. Note that each camera is set at a 45° angle relative to the diaphragm and at 90° to each other to satisfy the conditions necessary to calculate the 3-dimensional representation of the diaphragm surface.

View larger version (66K): [in this window] [in a new window]   Fig. 2.   An inflated rat diaphragm attached to the mold at its insertions on the rib cage and spine at a pressure of 15 cmH2O. An array of 0.05-cm markers is surgically implanted on the thoracic surface of the diaphragm at about 4- to 8-mm intervals along muscle fibers at the midline and in the ventral, middle, and dorsal regions of the left and right costal diaphragm muscles.

Measurements of length-tension relationships. Costal hemidiaphragms from three rats of the Sprague-Dawley strain (weight 170-195 g, and 1 mo old) were used in these experiments. After the rats were anesthetized and the left hemidiaphragm was excised, muscle was quickly submerged in Krebs-Ringer solution bubbled with 95% O₂-5% CO₂ at 25°C. Muscles were placed in a biaxial in vitro apparatus. Along each of two orthogonal axes forces were measured by a force transducer. Two pairs of surgical silk thread markers were sutured along two neighboring fibers, one pair along each fiber, on the abdominal side of the midcostal region. The four markers formed a square whose sides were aligned either parallel or perpendicular to the two orthogonal axes of the biaxial system. The muscle was clamped along both axes, and each axis was driven by micrometer to lengthen passively and shorten the muscle. Force data were collected at a sample rate of 10 Hz using a data acquisition board (model Lab-PC-1200/AI, National Instruments) and LabVIEW software (v 5.0) and analyzed. Displacement of the position markers was recorded (Sony SLV-620HF) using a CCTV type camera (HV-7200 by Hitachi). The recorded video was digitally captured (Captivator PC by VideoLogic) at a sample rate of 1 Hz and analyzed using Image Tool (v 2.0). Two-dimensional coordinates were obtained for each marker, and displacement was computed using MATLAB (v 5.2) software. To compute the length-tension relationship, muscles were lengthened and shortened along as well as transverse to the fibers. Muscles were passively lengthened from unstressed length (0.68 L_o) to about 1.1-1.2 L_o, where L_o is optimal muscle length or the length at which twitch force is maximal. Muscles were then passively shortened until passive force was negligible.

L_o. Absolute length of the midcostal diaphragm at L_o was determined using fiber bundles isolated from 10 Sprague-Dawley rats weighing 267 g. Each rat was anesthetized via intraperitoneal injection of pentobarbital sodium (40 mg/kg) followed by intramuscular injection of ketamine (35 mg/kg). The rat was then tracheostomized and mechanically ventilated using 100% O₂ during surgical removal of the left hemidiaphragm. Within 5-10 s after severing vascular connections, the muscle was rinsed in oxygenated Krebs-Ringer solution and placed in a dissection dish at room temperature. A muscle fiber bundle inserting on the 8th or 9th ribs was dissected from the lateral costal region and placed in a muscle bath containing 0.025-mM D-tubocurarine chloride in Krebs-Ringer solution at 25°C through which 95% O₂-5% CO₂ bubbled continuously. This fiber bundle was secured between platinum-plate stimulating electrodes (4 × 37 mm) by use of 2-0 silk suture. The central tendon was tied to a rigid support in the bath, and the rib was tied to an isometric force transducer (Grass FT-03D) mounted on a micrometer by which muscle length could be adjusted. Transducer output was amplified and displayed on a storage oscilloscope from which force was recorded. Muscle was stimulated directly by the use of supramaximal current density (550 mA at 140 V) and a pulse duration of 0.2 ms. Muscle length was adjusted to L_o, whereupon length and width were measured. The bundle was then trimmed of bone and connective tissue, blotted dry, and weighed. Bundle cross-sectional area was computed using the methods described by Close (6). It is important to note that optimal twitch length is about 5% longer than optimal tetanic length (15).

Unstressed diaphragm thickness. To compute regional stress in the diaphragm we measured regional thickness of unstressed muscles from three Sprague-Dawley rats (weighing 257 ± 4 g). Each rat was anesthetized, tracheostomized, and mechanically ventilated with 100% O₂ during surgical removal of the intact diaphragm. Bundles were excised from the ventral, middle, and dorsal regions of the left and right costal hemidiaphragm as well as the crural diaphragm (see Fig. 6A). The excised bundles were placed in a bath with continuously circulating Krebs-Ringer solution bubbled with 95% O₂-5% CO₂. Muscle thickness was estimated by dividing bundle weight by the product of the bundle length times the bundle width (1.8 mm) times muscle density (1.06 g/cm³).

Global shape, local curvature, and local stress. To compute diaphragm shape all markers on a pressurized diaphragm, including the ones near the insertion on the chest wall, were fitted to a best-fit plane (the plane) by minimizing the sum of the squares of the perpendicular distances between the marker locations and the plane. A new coordinate system was determined, in which the  and  axes were in the best-fit plane, and the  axis was normal to the plane. Positions of the markers in the coordinate system were computed. The global shapes were determined using a two-dimensional spline technique such that  = g(, ) (18). To compute local curvature and stresses in the midcostal region of the diaphragm three neighboring curved lines in the midcostal region along the fitted surface of the left hemidiaphragm were used to compute the two principal curvatures of the surface for that local region. The shape of this local region matched closely that of a cylinder with maximum principal direction of curvature nearly in the direction of the muscle fibers and a small curvature in the transverse fiber direction. Curvature was computed in the plane of maximum curvature, and maximum principal tension (₁) was computed from the Laplace equation using maximum curvature, applied P_di, and membrane thickness. We validated the accuracy of our system by measuring two principal curvatures of a cylinder of known dimensions. We digitized markers on the cylinder using a personal computer-based marker tracking system. Three-dimensional coordinates of the markers were determined and fitted to a theoretical surface using the spline technique, and markers along the principal curvature were fitted to a circle in the plane of maximum curvature. The fitted radius of the cylinder was 1.045 cm, compared with the actual radius of 1.00 cm.

Finite element modeling. The muscular portion and the central tendon of the diaphragm were assumed to act as a membranous structure. Therefore, we modeled the passive diaphragm using membrane theory, and we used membrane elements STIF41 in the ANSYS software to generate the finite element model of the diaphragm. The finite element used in this analysis was a three-dimensional element having membrane (in-plane) stiffness but no bending (out-of-plane) stiffness. The 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 , , and  directions, and element geometry was described in terms of the global coordinates for _ni for n = 1-4 and i = 1-3 of the four points. Tension distribution within the diaphragm was computed from the knowledge of the shape and applied pressure (10). A very stiff isotropic membrane was used to model both muscle and central tendon (Young's modulus = 1 × 10⁴ cmH₂O), and maximum displacement of any point on the model was less than 1 × 10³ mm. Membrane thickness was assumed to be uniformly 1 mm, which is about the same thickness as that of the unloaded left midcostal fiber bundle.

	RESULTS

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The orientations of the best-fit planes to the markers at different pressures were essentially the same, indicating that the predominant displacement of the diaphragm is in the direction perpendicular to the best-fit plane of the markers. In other words, the predominant displacement is in the  direction, perpendicular to the plane, which is parallel to the plane of insertion on the chest wall. Contours that describe passive diaphragm shape when loaded with a P_di of 0.2 P₀, 0.4 P₀, and P₀, where P₀ is about 15 cmH₂O, are shown for a representative muscle in Fig. 3. The diaphragm is viewed from the thoracic side, and the axes indicate the distance of markers in centimeters from the center of the diaphragm. The most cephalic region of the lung-apposed surface conforms generally to the V-like shape of the central tendon. A reference plane at the same height across different P_di values is shown by a dotted line, and the diaphragm is elevated relative to the reference plane by increasing P_di. A depression in the crural region is more pronounced at higher pressures, and the curvature transverse to fiber direction is smaller than along the fibers. These data show that diaphragm shapes are similar in the range of applied pressures.

View larger version (33K): [in this window] [in a new window]   Fig. 3.   Contours that describe the global shapes of the passive diaphragm, determined by a 2-dimensional spline technique, are shown for transdiaphragmatic pressure (Pdi) = 2.9, 5.7, and 15 cmH2O in A, B, and C, respectively. Note the intersection of the diaphragm with a reference plane of fixed position (dotted line). The diaphragm is elevated relative to the reference plane by increasing Pdi. A depression in the crural region is more pronounced at higher pressures, but the global shape of the fitted surface of the diaphragm does not change greatly by the increase in Pdi. The isopleths are 0.52 mm apart. Slope is space between isopleths, and curvature is change in the slope. In the midcostal region of the diaphragm the curvature transverse to fiber direction appears to be smaller than along the fibers. Numbered circles represent position markers, and units along the axes are distances in centimeters from the midline of the diaphragm.

The mean ± SD of L_o was found to be 18.5 ± 1.51 mm. Uniaxial length-tension curves along and transverse to the muscle fibers as well as a passive length-tension curve during pressure loading are shown in Fig. 4. These data demonstrate that there is a shift of the length-tension curve of the intact diaphragm compared with that of the uniaxial length-tension curve along the muscle fibers. Therefore, the diaphragm is less extensible during pressure loading than during uniaxial passive stretching in the direction of the muscle fibers. The data in Fig. 4 also demonstrate a significant leftward shift of the length-tension curve in the transverse direction to the muscle fibers compared with the length-tension curve along the fibers, indicating a greater extensibility of the muscle in the fiber direction than transverse to the fiber direction. Furthermore, at lower levels of stresses the slope of the length-tension curve transverse to the muscle fibers is much steeper than the slope of the length-tension curve along the fibers, indicating a greater muscle passive stiffness in the transverse fiber direction than along the muscle fibers.

View larger version (31K): [in this window] [in a new window]   Fig. 4.   In vitro biaxial passive stress-length relationship of the left midcostal diaphragm bundle of the intact rat diaphragm under pressure load () compared with uniaxial passive length-tension curves () along and transverse () to the fibers of the midcostal region of the left hemidiaphragm of the rat. All muscle lengths were normalized to the unstressed excised muscle length of the midcostal diaphragm bundle. Stresses were computed for the intact pressurized diaphragm from the Laplace equation, using a radius of curvature along the fiber of 1.9 cm, a muscle thickness of 0.096 cm, and assuming a negligible curvature in the direction transverse to the muscle fibers. These data show that muscle extensibility is lowest when muscle is loaded uniaxially transverse to the muscle fibers, and muscle extensibility is greatest when muscle is loaded uniaxially along the muscle fibers. When the diaphragm is loaded with pressure, the muscle is less extensible than when loaded uniaxially along muscle fibers but more extensible than when muscle loaded uniaxially transverse to fibers. It is important, however, to note that the leftward shift of the length-tension relationship during pressure loading may not be entirely related to the effect of transverse loading but may also be because of increase in collagen content in the older biaxially loaded mice diaphragms relative to the younger uniaxially loaded diaphragms.

Means ± SD of the curvature of the midcostal region of the left hemidiaphragm muscle in the plane and muscle maximum principal stress as a function of the applied pressure are shown in Fig. 5. In general, both radius of curvature and stress increase as a function of pressure. There was no significant difference between radii of curvature at P_di of 3 and at 6 cmH₂O. However, radius of curvature was greater at P_di of 15 cmH₂O than at lower pressures. Stress differed across the three pressure values and was greatest at the largest applied pressure.

View larger version (23K): [in this window] [in a new window]   Fig. 5.   Means ± SD of radius of curvature and stress of the midcostal left hemidiaphragm at 3 different Pdi values. Stresses were computed from the Laplace equation using a muscle thickness of 0.1 cm. *Radius of curvature was significantly greater at Pdi of 15 than at Pdi of 3 cmH2O (P < 0.03). +Stress was significantly different among the 3 values of pressure (P < 0.01).

Averages ± SD of the regional thickness of the excised unstressed diaphragm from three rats are shown in Fig. 6. Muscle thickness is not uniform and appears to be greater in the crural than the costal diaphragm. In contrast to the right hemidiaphragm, the dorsal region of the left costal diaphragm appears to be thinner than midcostal region.

View larger version (17K): [in this window] [in a new window]   Fig. 6.   A: schematic of diaphragm of a rat viewed from thoracic surface showing locations of measurements of muscle thickness. LV, RV: left and right ventral regions; LL, RL: left and right lateral regions; LD, RD: left and right dorsal regions; LC, RC: left and right crural regions. B: means ± SD of muscle thickness of different regions of costal and crural diaphragms from 3 rats. Letters identify regions from sketch in Fig. 6A.

Magnitudes and directions of maximal principal tension (₁) and minimal principal tension (₂), computed for P_di = 15 cmH₂O are shown in Fig. 7. ₂ is oriented at right angles to ₁ and is nearly always in the plane of the membrane. ₁ in the muscle varies between 0.06 and 0.12 N/cm for the right hemidiaphragm and from about 0.1 and 0.2 N/cm for the left hemidiaphragm. In the central tendon, ₁ is about 0.1 to 0.15 N/cm, compared with ₂ of about 0.04 to 0.1 N/cm. Tension in the central tendon is generally larger than in the muscle, and regions of muscle where tension is highest are located at the edges of the diaphragm, where the muscle was attached to the mold. In the costal regions ₁ is approximately parallel to the muscle fibers, and tensions in the direction transverse to the fibers (₂) are relatively small.

View larger version (27K): [in this window] [in a new window]   Fig. 7.   Magnitudes and directions of maximum and minimal principal tensions (1 and 2), computed using the ANSYS finite element program, are shown in A-C for the shape that corresponds to the maximum Pdi. Dotted lines are plotted along muscle fibers and the boundary of the central tendon. Uniform Pdi of 15 cmH2O and uniform thickness of 1 mm were used in the analysis. Tensions in the central tendon are generally larger than those in muscle and are nearly isotropic (c). The maximum tension lies approximately parallel to the muscle fibers, and stresses transverse to the fibers are smaller compared with stresses along the fibers.

	DISCUSSION

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Our data confirm the hypothesis that diaphragm muscle is anisotropic, with greater stiffness in the transverse fiber direction than along muscle fibers, and tension distribution in the intact diaphragm is nonuniform, whereas shapes of the diaphragm are similar across physiological range of P_di. Furthermore, our data suggested that the diaphragm is stiffer during pressure loading than during uniaxial loading along the muscle fiber direction. The passive shape of the diaphragm is determined by the material properties of the muscle and the load applied to it. We assumed that the passive diaphragm including the central tendon acts as a membrane. That is, the diaphragm is capable of carrying significant stresses in two dimensions but is too thin to carry significant bending moments or shear perpendicular to its plane. A membrane is a statically determinant structure; the tensions that it carries for a given loading can be computed without knowing its material properties. Tension parallel to the muscle fibers is the stress along the fiber direction divided by local muscle thickness. Tension in the direction transverse to the muscle fibers must balance the local P_di, local fiber stress, and the local principal radii of curvature.

In the simplest model of the diaphragm, it is assumed that tension is uniform and that the diaphragm has a hemispherical shape (19). Therefore, the relationship among pressure, tension, and curvature is governed by the simplest algebraic form of the Laplace law, P_di = 2 T, where  and T are curvature and tension of the membrane, respectively. Kim et al. (9) used direct measurements of P_di and tension in dogs to compute radius of curvature from the Laplace equation. They found that curvature changes little or increases at large lung volumes rather than decreasing, as theory would predict, and concluded that since curvature changed very little the action of the diaphragm resembles that of a piston. Gates et al. (8) developed a model that assumed an axisymmetric-shaped membrane of the diaphragm composed of anisotropic elastic material. Whitelaw et al. (19) presented a model that relaxed the hypothesis of axisymmetry of Gates et al. but assumed the diaphragm to be isotropic with uniform tension as in a soap film. Their membrane model was supported on a boundary shaped into the outline of a transverse section of the human thorax and was loaded by hydrostatic pressure similar to that in the abdomen. Whitelaw and colleagues pointed out that a soap film inflated from an elliptical ring formed an indentation across its minor axis similar to the one caused by the spine in the transverse cross-section of the human thorax. This soap film has the shape of a double hump, resembling that of the human diaphragm.

We developed a finite element hemispherical membrane model of the passive diaphragm (3). Our model relaxed the hypothesis of homogeneous anisotropic elastic membrane used by Gates et al. (8). 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. In particular, we tested the hypothesis that muscle anisotropy might serve to limit the repertoire of shapes available to the diaphragm irrespective of the P_di distribution within the physiological range. Using finite element technique we developed a hemispherical pressurized membrane model with an inextensible cap, simulating the central tendon and either isotropic or anisotropic skirt simulating the muscle. The anisotropic muscle has more compliance in the direction of the fibers than transverse to the fibers. Our results demonstrated that when the membrane is inflated, anisotropic muscle of the diaphragm changed curvature less than isotropic muscle. We concluded that changes of diaphragm shape are restricted because of the unique properties of the central tendon and muscle, i.e., the inextensibility of the tendon and anisotropy of the muscle. In previous studies of intact canine diaphragm we reported that curvature of the midcostal diaphragm was uniform along muscle fibers (2, 4) and changed little with changes in lung volume (5). Results of the current study show similarity of the shapes computed in a range of P_di between 3 and 5 cmH₂O. Our preparation, however, did not include rib cage, and therefore the zone of apposition did not affect diaphragm shape.

The diaphragm may be unique among skeletal muscles in that it supports stress in the direction transverse to the muscle fibers. Trunk skeletal muscle, such as intercostal and abdominal muscles, are in multiple muscle layers so that they can support a pressure without transverse loading, although stress has never been directly measured in these muscles. Even if stress in the diaphragm is greater parallel to the muscle fibers, there is stress in the in-plane transverse direction to the fibers as well. It is likely that this transverse stress is supported by a connective tissue that might have a structure different from those of other skeletal muscles. Because the diaphragm is subjected to pressure rather than uniaxial loads in vivo, data from uniaxial in vitro experiments may not accurately represent physiological behavior of the diaphragm muscle. There are very limited physiological data available on the canine diaphragm under biaxial loading in vitro (1, 17). In particular, Boriek et al. (1) reported stress strain data obtained from biaxial loading tests of muscular sheets excised from the midcostal region of the dog diaphragm. Stresses in the direction along and transverse to the muscle fibers were measured for different combinations of strains in the two directions. The results of Boriek et al (1) demonstrated that the muscle sheet is more compliant and considerably more extensible in the direction of the muscle fibers, and that is consistent with our data of both the excised and intact diaphragm of the rat. However, the measured transverse stiffness was not high enough to explain the observation that strains in the transverse direction in vivo were near zero in the Boriek et al. (1) study (5), and therefore it was concluded that passive transverse stress in vivo is small. It is possible, however, that during muscle activation transverse stiffness is high enough to cause in vivo negligible transverse fiber strain.

The data in Fig. 4 demonstrate that diaphragm muscle is less extensible during pressure loading than during uniaxial passive stretching of muscle sheets. These data show that the diaphragm muscle is more compliant and much more extensible in the fiber direction than transverse to the fibers, and therefore these data are consistent with published data on diaphragm properties of the dog (1, 10, 17). As noted by Smith and Loring (16), the nonlinearity of the length-tension curves are reflected by the disproportionate increases in compliance at low lung volumes. They reasoned that such nonlinear elastic behavior is related to a recruitment phenomenon: as extension increases, unstressed fibrous elements are progressively recruited and contribute to stiffness in parallel.

Because markers were physically attached to the muscle we assumed that the average interbead distances are proportional to the average changes in muscle sarcomere length. Our data on the length-tension relationships during pressure loading reflect a total length change of muscle length of about 21% from optimal length. Previous published work demonstrated that sarcomere length in the rat is about 2.8-3.0 µm at optimal length (13, 14). Therefore, by applying P_di of 15 cmH₂O we stretched the muscle in the range of sarcomere lengths of about 2.9 to about 3.5 µm.

Margulies et al. (10) used the finite element program, ANSYS, to compute diaphragm stress at functional residual capacity (FRC) in supine dogs under special conditions of a pneumothorax and fluid-filled abdomen so that P_di could be assessed accurately. Their study demonstrated that for a given fiber length stresses were two- to threefold larger along fibers under biaxial vs. uniaxial conditions. These results were not surprising because most materials require a larger tensile force to hold a sheet at a given stretched length when there is an additional tensile force acting orthogonal to that of the applied load than under uniaxial load where the orthogonal direction is unconstrained (7). Therefore, the leftward shift of the biaxial length-tension curve relative to the uniaxial length-tension curve in Fig. 4 is consistent with the data of Margulies et al. (10) of the dog diaphragm.

When the diaphragm is in static equilibrium, its unloaded state determines its shape, the material properties of the membrane, and the boundary conditions, i.e., all applied forces. 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 and 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 the study of Margulies et al. (10) have used this strategy to compute tension distribution in the diaphragm. Margulies et al. demonstrated that tension in the diaphragm of the dog is nonuniform and that the greatest in-plane tension was essentially aligned with the direction of muscle bundles and is two to four times larger than the tension in the direction transverse to the muscle fibers. 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. In the study of Margulies et al., 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 our study we report tension values computed from the finite element analysis in grams per centimeter. We used a P_di of 15 g/cm² and a uniform thickness of 0.1 cm, which is about the same thickness as that of the unloaded left midcostal fiber bundle (Fig. 6B). Tension in the costal muscle of the 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 (10) and in vitro (1) studies of the dog passive diaphragm.

Placement of markers on the central tendon 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 about one-tenth 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 0.1 cm, the thickness of unstressed 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. Our results demonstrated that muscle thickness was nonuniform between the costal and crural regions. 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.

Investigators have predicted tension in the diaphragm by using measurements of curvature from X-ray projections, providing assumptions about the P_di gradient and making use of the Laplace law. The estimate of tension by Smith and Loring (16) was 0.05 and 0.4 N/cm in upright and supine postures, respectively. Lower tensions of 0.1-0.2 N/cm were computed by Whitelaw et al. (19). Paiva et al. (12) predicted tensions of 0.32-0.54 N/cm in supine humans. Our finite element analysis of the rat diaphragm predicted tensions of 0.003-0.1 N/cm in the muscle and even higher values of 0.1-0.15 N/cm in the central tendon. We confirmed the finite element results by using the Laplace equation to compute maximum principal stress using pressure and maximum principal curvature of the midcostal muscle of the diaphragm. The finite element results in Fig. 7 are in agreement with the analytic results in Fig. 5, obtained using the Laplace equation. For example, at pressure of 15 cmH₂O, the stress in the midcostal muscle computed from Laplace was about 12.5 g/cm², whereas the stress computed from the finite element analysis ranged between 10 and 15 g/cm².

In summary, we developed in vitro preparations of both the intact and excised diaphragms to test the hypothesis that diaphragm muscle is anisotropic, with smaller compliance in transverse fiber direction than along fibers, and therefore, shape change may be small across physiological range of P_di. In particular, we determined the relationships among tension distribution, P_di, shapes for various conditions of loads, and muscle fiber length. Our results show that the overall shapes of the diaphragm and orientation of principal curvature were similar among the examined P_di, although local curvatures were different between P_di of 3 and 15 cmH₂O. Principal curvature was greater along muscle fibers than transverse to it. In the central tendon, tensions were nearly isotropic, and because tendon thickness was much smaller than muscle, stresses in the central tendon were much higher than the muscle. In the costal region, tensions were anisotropic and the largest principal tension was nearly aligned with fiber direction. Our results of the length-tension relationships demonstrated that the costal diaphragm muscle is anisotropic with greater extensibility and greater muscle compliance in the direction of fibers than in the direction transverse to the fibers. Furthermore, the diaphragm muscle is stiffer during pressure loading than during passive uniaxial stretching in the direction of muscle fibers.

Perspectives