Journal of Applied Physiology
Vol. 82, No. 5, pp. 1626-1636, May 1997
GAS EXCHANGE, MECHANICS, AND AIRWAYS

Effects of transverse fiber stiffness and central tendon on displacement and shape of a simple diaphragm model

Aladin M. Boriek and Joseph R. Rodarte

Pulmonary Section, Department of Medicine, Baylor College of Medicine, Houston, Texas 77030

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

APPENDIX

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

Boriek, Aladin M., and Joseph R. Rodarte. Effects of transverse fiber stiffness and central tendon on displacement and shape of a simple diaphragm model. J. Appl. Physiol. 82(5): 1626-1636, 1997.Our previous experimental results (A. M. Boriek, S. Lui, and J. R. Rodarte. J. Appl. Physiol. 75: 527-533, 1993 [Medline] and A. M. Boriek, T. A. Wilson, and J. R. Rodarte. J. Appl. Physiol. 76: 223-229, 1994 [Medline] ) showed that 1) costal diaphragm shape is similar at functional residual capacity and end inspiration regardless of whether the diaphragm muscle shortens actively (increased tension) or passively (decreased tension); 2) diaphragmatic muscle length changes minimally in the direction transverse to the muscle fibers, suggesting the diaphragm may be inextensible in that direction; and 3) the central tendon is not stretched by physiological stresses. A two-dimensional orthotropic material has two different stiffnesses in orthogonal directions. In the plane tangent to the muscle surface, these directions are along the fibers and transverse to the fibers. We wondered whether orthotropic material properties in the muscular region of the diaphragm and inextensibility of the central tendon might contribute to the constancy of diaphragm shape. Therefore, in the present study, we examined the effects of stiffness transverse to muscle fibers and inextensibility of the central tendon on diaphragmatic displacement and shape. Finite element hemispherical models of the diaphragm were developed by using pressurized isotropic and orthotropic membranes with a wide range of stiffness ratios. We also tested heterogeneous models, in which the muscle sheet was an orthotropic material, having transverse fiber stiffness greater than that along the fibers, with the central tendon being an inextensible isotropic cap. These models revealed that increased transverse stiffness limits the shape change of the diaphragm. Furthermore, an inextensible cap simulating the central tendon dramatically limits the change in shape as well as the membrane displacement in response to pressure. These findings provide a plausible mechanism by which the diaphragm maintains similar shapes despite different physiological loads. This study suggests that changes of diaphragm shape are restricted because the central tendon is essentially inextensible and stiffness in the direction transverse to the muscle fibers is greater than stiffness along the fibers.

respiratory muscles; chest wall; diaphragm mechanics

INTRODUCTION

THE DIAPHRAGM is a thin sheet composed of a central tendon and a fringe of costal and crural muscle. Diaphragmatic muscle fibers originate from the central tendon and insert on the rib cage and spine. Curvature of the muscle fibers is essential for converting tension developed by the muscle into transdiaphragmatic pressure (Pdi) and muscle shortening into diaphragmatic displacement. The diaphragm can be categorized in mechanics as a membrane: a structure capable of carrying significant stresses in two dimensions but too thin to carry either bending moments or shear forces out of the plane of the membrane.

Current models of the diaphragm include those developed by Gates et al. (7) and Whitelaw et al. (14). The former assumed the diaphragm to be an axisymmetric membrane composed of homogeneous anisotropic material. The anisotropy was determined by in-plane stiffnesses in both the fiber direction (meridional) and in the direction transverse to the fibers (circumferential). Gates et al. (7) measured the in-plane strains along the fiber direction and transverse to the fibers, the radii of curvature, and diaphragm thickness at three different prescribed values of Pdi. Their results demonstrated that passive diaphragm stiffness is greater in the direction transverse to the fibers than along the fibers.

Whitelaw et al. (14) relaxed the hypothesis of axisymmetry by Gates et al. (7) but assumed the diaphragm to be isotropic, having uniform tension similar to a soap film. Whitelaw and colleagues (14) computed the shape of a membrane with this tension distribution supported on a boundary with the shape of a transverse section of the human thorax; the membrane was loaded by hydrostatic pressure applied to one surface. When inflated, the dome of the model developed a double hump similar to that seen in the human diaphragm.

Our previous experimental results (1) showed that shape of the midcostal region of the dog diaphragm is similar at functional residual capacity (FRC) and end inspiration. This is true whether the muscle shortens actively or passively. Furthermore, we demonstrated that the diaphragm dimension changes are insignificant in the direction transverse to the muscle fibers, suggesting that the muscle is essentially inextensible in that direction (3). These findings are consistent with those of Gates et al. (7) and Strumf and associates (12), who measured the biaxial stress-strain relationship of the passive canine diaphragm.

We developed a heterogeneous membrane model to explore the effects of anisotropic muscle sheet and an inextensible central tendon on diaphragm curvature. This model is based on two fundamental parameters: 1) the ratio of stiffness transverse to the muscle fibers and stiffness along the fibers, and 2) size of the central tendon. Effects of these parameters were tested by using both uniform pressure and hydrostatic pressure. By investigating the effects of central tendon size, we tested the hypothesis that tendon inextensibility severely restricts the change of diaphragm curvature in response to Pdi. By investigating the effects of transverse fiber stiffness, we tested the hypothesis that a greater stiffness in the transverse direction to the muscle fibers compared with the stiffness along the fibers may limit the shape change of the diaphragm in response to Pdi.

METHODS

The diaphragm was modeled by using a hemispherical membrane of uniform thickness, which was generated by rotating a planar curve called the meridian around a line in the plane of the meridian. This plane contains the principal radius of curvature of a muscle fiber. Therefore, the shape or curvature of the lines simulating muscle fibers is the same everywhere in the model, and the orientation of the fiber is restricted to be along a meridional line. Either a uniform pressure or a hydrostatic pressure was applied to the inner surface of the membrane. Hydrostatic pressure simulated the forces in a recumbent animal, in which hydrostatic gradient of abdominal pressure is larger than the pleural pressure gradient.

Curvature is defined at a point on a curve as the inverse of the radius of the best fit circle (6). Using the least squares methods, we fitted the nodal points along a muscle fiber in three-dimensional Euclidean space by using Cartesian coordinates X, Y, and Z. We used the following quadratic polynomial

(1)

where a, b, c, d, and e are constants.

Because our model is axisymmetric, each curved muscle fiber lies in an arbitrary meridional plane, e.g., the XZ plane. Therefore, curvature () of a muscle fiber is given by

(2)

Curvature  was computed for each of the finite element models that are subjected to either uniform or hydrostatic pressure gradient.

Three classes of models were used to test our hypotheses. These were 1) homogeneous (uniform properties throughout) and isotropic (material properties are not functions of orientation), 2) homogeneous and orthotropic [the membrane has elastic moduli that are different in two orthogonal directions (along the muscle fibers and transverse to the fibers)], and 3) heterogeneous (properties are functions of position in the membrane) and orthotropic. In this study, the term "heterogeneous" is used in a particular sense. The model has two homogeneous compartments: the central tendon, which is a homogeneous isotropic material, and the muscle, which is a homogeneous orthotropic material.

In using the term "orthotropic," we are referring to a material as anisotropic in a particular sense, as there are orthogonal principal axes of the stress-strain coefficient tensor. Strains in response to uniform stresses along those principal axes are different. In this special class of anisotropy, a two-dimensional material is described by four elastic constants, only three of which are independent. The structure of the diaphragm, in particular the orientation of muscle fibers, suggests that the diaphragm is an anisotropic structure. Therefore, we oriented the principal axes of our anisotropic membrane model in the meridional and circumferential directions. The meridional direction is the direction along the muscle fibers; the circumferential direction is transverse to the muscle fibers.

To facilitate comparison, identical loading conditions (uniform or hydrostatic pressure) and geometric boundary conditions (no displacements at the chest wall insertions) were applied to all models. A schematic of the hemispherical membrane model of the diaphragm simulating a supine posture is shown in Fig. 1. The anisotropic skirt of the model has a modulus of elasticity [in the direction transverse to the muscle fibers (E _fiber) that was greater than along the muscle fibers (E_fiber)]. Poisson's ratio (_ij) is the ratio of transverse strain in the j direction to the longitudinal strain when stressed in the i direction; that is, _ij = _j/_i, where _j and _i are the lateral contraction and the longitudinal extension, respectively, during simple uniaxial tension. According to the reciprocal law of Betti (13), ₁₂/E_fiber = ₂₁/E _fiber is valid. Therefore, there are only three independent material constants. The cap simulating the central tendon is isotropic and has stiffness at least two orders of magnitude greater than transverse muscle stiffness.

 fiber

A three-dimensional finite element membrane model of the diaphragm was developed with the use of ANSYS software (4). The model was discretized as a number of warped triangular elements having straight boundaries. A total of 402 isoparametric nonlinear membrane elements and 439 nodes were used to generate the model shown in Fig. 2; the methods are detailed in APPENDIX.

Displacements of the nodal points along a muscle fiber relative to their unstressed positions were computed. The normal (radial) component and the meridional component of the displacement were computed in the direction perpendicular to the tangent plane of the muscle fiber and also in the direction along the fiber. The meridional angle in a supine posture was defined as zero at the apex of the hemispherical membrane, 90° at the dorsal extreme (chest wall insertion), and 90° at the ventral extreme (opposite insertion).

To investigate the accuracy of our finite element model, we compared the finite element solution with the Laplace solution of a uniformly pressurized hemispherical membrane. The principal tensile stresses (₁ and ₂) at points along a muscle fiber produced by uniform pressure in a stiff isotropic membrane model (E_membrane = 10⁶ P, where P is pressure) as a function of the meridional angle along with the Laplace solution of an isotropic pressurized hemispherical model are plotted in Fig. 3. The use of a stiff model was essential to maintain the hemispherical shape. Stresses were normalized by dividing their numerical value by 50 P. Figure 3 shows that stress of an isotropic membrane was essentially uniform and that ₁ and ₂ were similar. The maximum departure of the finite-element solution from the exact Laplace solution of 4% occurs at the boundary. The finite-element solution is, therefore, in good agreement with the analytic solution of the Laplace law (for P = 1.0 cmH₂O, t = 0.1 cm, ₁ = ₂  = 0.1 1/cm, ₁ = ₂ =  = P/2t = 50 cmH₂O, where P is pressure imbalance across the membrane, t is thickness of the membrane, ₁ = ₂ =  are the principal curvatures, and  is the tensile stress along principal curvatures in the plane of the membrane).

RESULTS

We first compared our homogeneous anisotropic models (E _fiber / E_fiber = 5, 10, 15, 20) with isotropic models having the same dimensions, geometry, and loading conditions. We assumed stiffness of the isotropic model to be the same as stiffness in the direction along the fibers of the anisotropic model. Models were subjected to either uniform pressures or hydrostatic pressure gradients; gradient pressures varied from 0.5 Pmean at meridional angle +90° to 1.5 Pmean at meridional angle 90°, where Pmean is the transmembrane pressure in the uniformly pressurized models. Deformed shapes of isotropic and anisotropic (E _fiber / E_fiber = 5) models subjected to uniform pressure are shown in Fig. 4. At and near the chest wall insertions, the curvature of the isotropic model is greater than the undeformed shape. Near the pole, the curvature is nearly the same as that of the undeformed shape. A muscle fiber in the deformed anisotropic model appears to have a similar shape to that of the undeformed shape near the points of insertion of the diaphragm on the chest wall. Therefore, the radius of curvature of a muscle fiber in that region should be similar to that of the undeformed state. There is a change of shape, however, near the apex where an anisotropic muscle fiber departs from the undeformed hemispherical shape.

 fiber

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Normal and meridional displacements are plotted against meridional angle for uniformly pressurized isotropic and anisotropic models. For anisotropic models, the ratio of stiffness in the direction transverse to the muscle fibers to stiffness along the fibers varies between 5 and 20 (Fig. 5). A point on an isotropic muscle fiber displaces minimally along the fiber, and such a displacement is independent of the angle between the line of insertion on the chest wall and the point on the fiber. Normal displacement, however, increased from zero at the line of insertion on the chest wall to a nearly constant value by ±60°.

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Deformed shapes of uniformly pressurized heterogeneous anisotropic models are shown in Fig. 6. The central tendon in the stressed state has virtually the same radius of curvature as that of the undeformed state. This observation indicates that the shape change of the central tendon in response to pressure was dramatically restricted. Minimal shape change resulting from greater sizes of central tendon is illustrated by comparing Fig. 6, A-C. Surface area of the central tendon in the dog is ~20% of total surface area of the diaphragm (1). This varies among mammals, however, so we modeled central tendon sizes that varied from 21 to 76% of total surface area in undeformed models.

Normal and meridional displacements for heterogeneous anisotropic models are shown in Fig. 7, A and B, respectively. Effect of central tendon size on displacement of a point on the membrane in response to a uniform pressure is shown as a function of cap angle. The greater the size of central tendon, the smaller the normal and meridional displacements. Normal displacement of the isotropic fiber is greater than the anisotropic fiber (Fig. 7A). Models with isotropic muscle fibers have smaller meridional displacement when compared with the anisotropic muscle fibers. The apex has essentially no meridional component of displacement (Fig. 7B).

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The effect of the hydrostatic pressure gradient on the deformation of homogeneous isotropic and orthotropic membranes is illustrated in Fig. 8. The deformed state of the isotropic model in the ventral diaphragm (low-pressure region) has a shape that is similar to that of the orthotropic model. In contrast, at the dorsal region, the deformed state of the isotropic model departs significantly from the orthotropic model and sags in the gravitationally dependent region of the diaphragm (dorsal in the supine position). The dependent part of the orthotropic membrane is very similar to the nondependent region and the undeformed shape. When the uniform and hydrostatic pressure gradient load cases are compared, the orthotropic model has very similar shapes with different pressure distributions, in contrast to the isotropic model.

 fiber

 fiber

Normal and meridional displacements of hydrostatic pressure gradient models as a function of the meridional angle are shown in Fig. 9. It can be seen that, at a low-pressure region (ventral), a point on a muscle fiber in the isotropic model had essentially identical displacement along the fibers as that of the anisotropic models. At the high-pressure region (dorsal), the isotropic model had much greater normal and meridional displacements than did the anisotropic models of the diaphragm.

 fiber

 fiber

Effects of hydrostatic pressure gradient on shape and displacement of heterogeneous anisotropic models are shown in Fig. 10, A-C. Inextensible caps, of the same sizes as in Fig. 6, simulating central tendon, markedly reduce the displacement of the apex of the membrane. Deformed state of the central tendon has virtually the same radius of curvature as that of the undeformed state. Effects of inextensible central tendon in reducing the displacement of the apex of the membrane are more pronounced in the pressure gradient model than in the uniform pressure model.

Normal and meridional displacements of heterogeneous models subjected to hydrostatic pressure as a function of the meridional angle are shown in Fig. 11, A and B, respectively. In general, the greater the size of the central tendon, the smaller the displacement of the membrane. As illustrated in Fig. 11A, at the low-pressure region of the membrane, an isotropic muscle fiber has slightly greater normal displacement than that of an anisotropic fiber. However, at the high-pressure region (dorsal), the isotropic model has a much greater normal displacement than does the anisotropic models of the diaphragm. It can be seen that, at a low-pressure region (ventral), a point on a muscle fiber in the isotropic model has essentially identical displacement along the fibers (meridional displacement) as that of the anisotropic models (Fig. 11B).

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The curvature along a muscle fiber is plotted against the meridional angle for isotropic and anisotropic homogeneous models subjected to uniform pressure (Fig. 12A) and hydrostatic pressure gradient (Fig. 12B), compared with the curvature of the undeformed shape of a curved fiber that lies on a perfect hemispherical model. Because our model assumes a radius of curvature of 10 cm for the undeformed shape, the curvature is 0.1 1/cm. In Fig. 12A, for meridional angles less than 45°, the deformed anisotropic models with stiffness ratios E _fiber / E_fiber of 5 and 10 have less change of curvature from the undeformed model, as compared with the isotropic model. Models with greater stiffness ratios, E _fiber / E_fiber of 15 and 20, show a change of curvature from the undeformed model of similar magnitude but different sign than the isotropic model. At greater meridional angles (45°), the curvature of the deformed isotropic model converges to similar values to the undeformed shape. In contrast, curvature of the anisotropic models departs significantly from that of the undeformed model. Data in Figure 12B show that under a hydrostatic pressure gradient all anisotropic material properties of the deformed fibers have less change in curvature from the undeformed model than from the isotropic model.

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DISCUSSION

It is important to differentiate the mechanics of the diaphragm as a membranous structure from the mechanical properties of its three-dimensional muscle fiber-connective tissue composite structure. In mechanics, a membrane is a structure that is capable of carrying significant stresses in two dimensions; however, it is too thin to carry significant bending moments or shear stresses in the direction transverse to the plane of the membrane. Pressure load applied to the membrane can produce internal stresses. Basic membrane resistance forces are tension, compression, and shear in the plane of the membrane. When a region of the muscle is activated, equilibrium muscle length is determined by 1) the active length tension properties of the muscle, 2) the load against which the muscle is working, and 3) the degree to which the muscle is activated. The equilibrium length of the muscle at different levels of activation determines the final shape of the diaphragm. However, the displacement of the diaphragm having changes in muscle activation is determined by the equilibrium shape of the entire membrane under the varied load and by elastic material properties.

Experimental results of Gates et al. (7) showed that the passive diaphragmatic muscle of the dog in situ behaves as an anisotropic linearly elastic material for loads in the physiological range. Using the assumption of axisymmetry and incompressibility combined with the law of Laplace, Gates et al. computed the elastic moduli along the fiber and transverse to the fiber directions. They arrived at a surprisingly low orthotropic ratio of 1.6 (E _fiber / E_fiber = 1.6). Gathered by Strumf et al. (12), in vitro biaxial data of the passive diaphragmatic muscle of the dog demonstrated an anisotropic ratio of ~5 (E _fiber / E_fiber =  5). Our anisotropic model, having a stiffness ratio (E _fiber / E_fiber) of 5, exhibited, in response to pressure, the greatest limitation of shape change when compared with models of higher stiffness ratios. Our data suggest that a moderate value of the stiffness ratio is sufficient to limit shape change in response to a pressure.

Loring and Mead (8) conceptualized the diaphragm as a cylindrical zone of apposition (Zap) capped by a lung-apposed dome. On the basis of this geometry, they believed that Pdi in the Zap should be small because curvature along a fiber would be expected to be greater in the dome than in the Zap. Their analysis did not consider curvature in the direction transverse to the muscle fibers. However, our previous study (1) showed that curvature in the direction transverse to the fibers was negligible. Furthermore, no abrupt change of curvature was apparent along the muscle fibers at the point where the diaphragm peels away from the rib cage at the top of Zap. The curvature was approximately uniform along the muscle fibers and about the same at all lung volumes. By shortening and rotating in their plane of maximum curvature around the chest wall insertion, muscle fibers displace volume without a resultant change in curvature.

Results of these numerical experiments show the effects of anisotropic material properties on the curvature of the diaphragm in the direction of muscle fibers when subjected to either uniform or hydrostatic pressure gradient. The diaphragm is treated as a hemispherical membraneous structure, and the muscle fibers have the same curvature and lie along the meridional lines. In the present study, we hypothesized that the increase in transverse stiffness of muscle fibers and the inextensibility of the central tendon provide a mechanism by which diaphragm curvature could be similar irrespective of pressure distribution. Our finite-element models of the diaphragm were built to test this hypothesis.

We now consider an alternative mechanism by which it is conceivable that curvature could remain constant during active and passive ventilation. This mechanism assumes that Pdi is proportional to tension in the diaphragm (Tdi) at the upper margin of the Zap. If the upper margin of the Zap is approximately planar, the resultant force of Pdi on the lung-apposed region of the diaphragm is the product of Pdi and the area of the enclosed plane. This force is balanced by the component of membrane tension perpendicular to the plane times the circumference of the enclosed plane. During tidal breathing, neither the circumference nor the projected area change considerably compared with the changes of Pdi and Tdi. Therefore, the dome of the diaphragm could have a constant curvature during active and passive shortening because the ratio of Pdi to tension could remain constant.

If one considers the diaphragm as a homogeneous isotropic elastic membrane stretched by uniform tension, then a change of tension should cause uniform strain away from the chest wall insertions. When Pdi decreases during passive inflation, away from the chest wall insertions, the membrane should contract uniformly, and principal strains in the plane of the diaphragm should be the same. However, our recent work (3) showed that during both passive and active inflation the in-plane principal strain in the transverse direction to the muscle fibers was substantially smaller than the in-plane principal strain along the fiber direction. Excluding the unlikely possibility that transverse stress decreases during active muscle contraction and remains constant during passive shortening, these data suggest that the diaphragm muscle is an anisotropic structure. Stiffness in the transverse direction to the muscle fibers may be greater than the stiffness in the direction along the fibers. More recently (11), we found that, during efforts against occluded airways, the strains in the direction transverse to the fibers are small. Therefore, the diaphragm may be essentially inextensible in that direction. We argued that the relative inextensibility of the diaphragm in the direction transverse to the muscle fibers is advantageous teleologically (3). As the muscle contracts and Pdi and stress in the diaphragm increase, the resultant extension in the perpendicular direction would reduce the volume displacement of the diaphragm, and part of the pressure volume work of the muscle contraction would be expended on the elastic extension in the perpendicular direction.

Previous anatomic measurements of the excised diaphragmatic muscle of the dog demonstrated that the ratio of the surface area of the central tendon to the total area of the diaphragm was 0.2 ± 0.025. However, these measurements were not corrected for the areas where the vena cava traverses the central tendon or where the esophagus and aorta traverse the crural diaphragm. In our models, the sizes of the caps simulating the central tendon cover a range that includes the anatomic measurements (20-66%).

Hydrostatic pressure gradient produced a maximum pressure difference that is equal to Pmean; in a supine dog, with a diaphragm height of 16 cm, there is a difference between abdominal pressure and pleural pressure. Thus, if a gradient of 0.5 cmH₂O/cm height is assumed, a gradient of Pdi from 0.5 to 1.5 Pmean should correspond to 8 cmH₂O at mid height. This is in the range measured by Margulies and associates (9) in supine dogs having fluid in the abdomen, which could have increased the Pdi slightly.

Results in Figs. 4, 6, 8, and 10 demonstrate that the increased stiffness in the transverse direction to the muscle fibers limits the shape change of the membrane near the insertions on the chest wall. Additionally, these figures demonstrate that the presence of an essentially inextensible cap simulating the central tendon causes the membrane to have essentially the same shape under uniform pressure (Fig. 6) or hydrostatic pressure gradient (Fig. 10). It is clear from Fig. 8 that volume swept by hydrostatic pressure is greater than that swept by uniform pressure. However, the isotropic model would allow the diaphragm to splay back in the region above the chest wall insertion, whereas in vivo there would be restriction by the chest wall. Splaying back is only pronounced in the gravitationally dependent region of the hydrostatic pressure-gradient model.

As compared with isotropic models, the uniformly pressurized anisotropic models with stiffness ratio of 5 and 10 demonstrated a limitation of shape change from the undeformed shape, as demonstrated in Fig. 12A. Higher stiffness ratio, however, showed reduction of curvature as compared with the isotropic model. In contrast, all hydrostatically pressurized anisotropic models demonstrated a smaller shape change from the undeformed shape compared with the isotropic model as shown in Fig. 12B. These curvature data clearly show that changes of diaphragm model shape are restricted because of the membraneous anisotropic material properties.

We previously demonstrated that in the dog markers along a muscle fiber at FRC can be fitted to a circular arc and to an arc having the same radius, but having its center shifted and its arc length reduced fitted the physiological data at total lung capacity (TLC) well (3). Displacement trajectories that would carry points from the longer arc (at FRC) to the shorter arc (at TLC) were nearly parallel lines. These trajectories are similar to those generated by the displacement of the points on an anisotropic muscle fiber in that these trajectories are parallel lines. The anisotropic model having transverse inextensibility is, therefore, consistent with the major features of the physiological data on curvature, strain, and displacement of the diaphragm.

Our finding that moderate increase in transverse stiffness dramatically limits the shape changes of the diaphragm model may contribute to the remarkable similarities of diaphragm shapes at FRC, end inspiration, and TLC. The change in shape of the diaphragm in response to tensile stresses produced by the pressure is restricted even more when the central tendon is inextensible and the muscle is anisotropic, having a greater stiffness in the transverse direction to the muscle fibers than the stiffness in the direction along the fibers. Our model, therefore, provides a potential mechanism for the constant curvature along the diaphragmatic muscle fibers in vivo.

ACKNOWLEDGEMENTS

The authors thank Drs. Theodore Wilson, John Akin, and Michael Reid for discussions about the data and the manuscript. The authors are grateful to Q. Lin, Y. Rui, and A. M. Doneski for their technical assistance.

FOOTNOTES

Address for reprint requests: A. M. Boriek, Pulmonary Section, Suite 520B, Dept. of Medicine, Baylor College of Medicine, 1 Baylor Plaza, Houston, TX 77030.

Received 11 August 1995; accepted in final form 2 January 1997.

APPENDIX

Generally, a largely deformed state of elastic body cannot be analytically solved because of the nonlinear relation between strain and displacement. Thus the continuous diaphragm model was practically treated as a finite element model in this study. We used nonlinear isoparametric (5) membrane elements in the ANSYS software (SHELL41) to generate the finite-element model of the diaphragm. Each element has three nodes. The planar edge of the model simulates the insertion of the diaphragm on the chest wall. Therefore, the nodal points that lie at the planar edge of the models were restrained to zero translational displacements.

In models where the internal pressure (abdominal pressure) was assumed to be uniform, the pressure was applied as an axisymmetrical load. In models where hydrostatic pressure gradient was assumed, pressure was applied as a nonaxisymmetrical load. In the uniform-pressure models, lines of symmetry coincided with two meridional lines (muscle fiber directions) at 90° to each other. In the nonuniform-pressure models, the line of symmetry coincided with a muscle fiber that spans 180° and lies in ventral dorsal plane. Appropriate displacement constraints along the line of symmetry were applied. Nodal points on the symmetry line were restricted from displacement in the direction perpendicular to the meridional line in the plane of the membrane. No rotation constraints were applied because the membrane element SHELL41 has only translational degrees of freedom.

The finite element used in this analysis was a three-dimensional element having membrane (in-plane) stiffness but no bending (out-of-plane) stiffness. Each element has three degrees of freedom at each node, corresponding to translations in the X , Y, and Z directions. The displacements at the three corner nodes comprise a total of nine global degrees of freedom with respect to the global rectangular coordinate system x_i. The element geometry is completely described in terms of the global coordinates x_ni for n = 1-3 and i = 1-3 of the three points.

In our analysis, anisotropic material directions corresponded to the element coordinate directions. The element coordinate system was rotated to be along the fiber direction and in the direction transverse to the fibers. We assumed large deflection, which is treated in ANSYS as a nonlinear problem. Faster convergence was obtained by breaking the load step into multiple smaller steps, each having several iterations. A triangle element was formed by defining duplicate K and L node numbers. Triangular shape is required for large deflection analysis because four-node elements may warp during deflection (10). Out-of-plane deflection within the element was allowed. This deflection may cause an instability in the displacement solution. To counteract this instability, a slight normal stiffness was added to the element (stress stiffening). The stress stiffening (also called geometric stiffening, incremental stiffening, initial stress stiffening, or differential stiffening) is the stiffening of a structure caused by its stressed state. Physically, it represents coupling between the in-plane and transverse deflections within the membrane. This coupling is the mechanism used by a thin membrane to carry lateral loads. As in-plane tensile stresses increase, the capacity of the system to carry lateral loads also increases (5).

REFERENCES