Abstract
Angelillo, Maurizio, Aladin M. Boriek, Joe R. Rodarte, and Theodore A. Wilson. Theory of diaphragm structure and shape.J. Appl. Physiol. 83(5): 1486–1491, 1997.—The muscle bundles of the diaphragm form a curved sheet that extends from the chest wall to the central tendon. Each muscle bundle exerts a force in the direction of its curvature; the magnitude of this force is proportional to the curvature of the bundle. The contribution of this force to transdiaphragmatic pressure is maximal if the direction of bundle curvature is orthogonal to the surface and the curvature is maximal. That is, the contribution of muscle tension to transdiaphragmatic pressure is maximal if the muscle bundles lie along lines that are both geodesics and lines of maximal principal curvature of the surface. A theory of diaphragm shape is developed from the assumption that all muscle bundles have these optimal properties. The class of surfaces that are formed of line elements that are both geodesics and lines of principal curvature is described. This class is restricted. The lines that form the surface must lie in planes, and all lines must have the same shape. In addition, the orientation of the lines is restricted. An example of this class that is similar to the shape of the canine diaphragm is described, and the stress distribution in this example is analyzed.
 muscle
 mechanics
 mathematical model
recently, boriek et al. (1) described the shape of the diaphragm and the geometry of the muscle bundles in the midregion of the costal diaphragm of the dog. They found that in this midcostal region, the diaphragm has nearly the shape of a right circular cylinder and that the muscle bundles lie in the direction of the maximal principal curvature of the surface. They also found that the muscle bundles lie in planes and that the planes of the muscle bundles lie perpendicular to the tangent plane of the surface. It follows that the curvature of the muscle bundles is orthogonal to the tangent plane of the surface at every point along the bundle. The latter property is the defining property of a geodesic. Thus, in the midcoastal region, the muscle bundles lie along lines that are both lines of principal curvature and geodesics. The shape of the diaphragm is different in other regions. However, the properties of the muscle bundles that Boriek et al. (1) observed in the midcostal region have functional significance. The direct contribution of muscle tension to transdiaphragmatic pressure is maximal if the muscle bundle lies in the direction of maximal surface curvature and the direction of bundle curvature is normal to the diaphragm surface. It therefore seems plausible that, although the shape is different, the optimal structure may be preserved and the muscle bundles may lie on lines that are both geodesics and lines of principal curvature in all regions.
Not all surfaces allow this optimal structure; surfaces that are formed of lines which are both geodesics and lines of principal curvature must be a restricted class. We therefore set out to describe this class of surfaces. Surfaces were constructed by placing line elements, representing muscle bundles, along a reference line and requiring that the line elements be both lines of principal curvature and geodesics at every point on the surface. We found that, to form surfaces with the required properties, the line elements must be restricted. The lines must lie in planes, and all line elements must have the same shape. The shape of the surface is restricted because of these restrictions on the line element. In addition, the shape of the surface is restricted by a condition on the orientation of the line element with respect to the reference line. Although the class of allowed surfaces is severely limited, it includes shapes that seem compatible with the limited information that is available about the diaphragm. A simple example qualitatively similar to the shape of the canine diaphragm is described, and the stress distribution in a diaphragm with this shape is analyzed.
MODEL
Nomenclature and assumptions.
The coordinates that are used to describe the geometry of the surface are shown in Fig. 1. The surface is denotedS and the vector positions of points on S are denotedx. A family of lines that span the surface are denoted μ. A line Γ is drawn orthogonal to the lines μ and forms a reference curve in the surface. Points on the curve Γ are denotedx°(s), where s is the arc length along Γ. Points on the lines μ are described byx =x°(s) +z(s,t), where z is a vector from the intersection of Γ and μ to a point along μ andt is the arc length along μ from the reference curve Γ.
The vectors T andC shown in Fig. 1, are the tangent and curvature vectors of the line μ, respectively. They are related to the derivatives ofx(s,t) by the following equations
Two assumptions are made about the relation of the line μ to the surface S. First, it is assumed that the line is a geodesic. The defining property of a geodesic is the property that its curvature is orthogonal to the surface (6). Thus this assumption is expressed by the equation
Second, it is assumed that the line μ is a line of principal curvature of the surface. The surface is locally symmetrical around the line of principal curvature. That is, the surface falls away from the tangent plane at the same rate ahead and behind the point of tangency. As a result, the component normal to the tangent plane of the derivative of the normal vectorB along a line of principal curvature is zero
Two additional properties of the geometry follow from these assumptions. First, the lines μ must lie in planes. This property can be demonstrated by means of the following equations
ANALYSIS
In the previous section, the assumption that the lines which form the surface are both geodesics and lines of principal curvature has been expressed as equations governing the variables which describe the surface, and the assumption has been shown to imply that the lines lie in planes which are orthogonal to the tangent plane of the surface. In this section, the class of surfaces that are formed by a family of lines with these properties is described.
We begin by restating a condition on the derivative ofB that follows from the fact that the direction of B is the same at all points along the line μ
Finally, a condition on the inclination of the universal curve to the reference line can be obtained. The derivation of this condition begins with the definition of local reference vectors on Γ. The natural set of reference vectors on Γ is the Frenet triad consisting of the tangent, the normal, and the binormal. Unit vectors in these three orthogonal directions are denotede
_{i} and are defined as follows
The derivatives ofe
_{i} are given by the following equations, where κ and τ are the curvature and torsion of Γ, respectively
Equation 15
provides a formal condition on the surface, but it does not provide much help in visualizing the surface. Some help in visualizing the surface is obtained by considering a second reference line for the surface, the line of centers of the arc of the universal curve along Γ. The radius of curvature of the universal curve is constant along Γ with magnitude 1/‖C°‖. Thus the line of centers of the arc for alls, denotedx
_{c}(s), is given by
The preceding analysis provides the basis for a more pictorial method for constructing the surface. This method can be summarized as follows. The surface has two elements, a universal curve μ that lies in a plane and an arbitrary curve Γ. The surface is constructed from these elements by the following procedure. First, a given point on the universal curve is chosen, and the radius of curvature of μ at that point is computed. Then a line is constructed parallel to Γ at a distance from Γ equal to the radius of curvature of μ at the fixed point. The surface is constructed by moving the universal curve along Γ, holding the plane of μ perpendicular to Γ and holding the fixed point on Γ and the center of curvature of the fixed point on the parallel line. Alternatively, the line of centers could be chosen as the arbitrary line, and the line Γ could be constructed parallel to it. If the line μ is a circle, the second method is simpler, because, in that case, the line Γ plays no role and the surface can be constructed from the arbitrary line of centers alone.
Some familiar simple surfaces are members of the allowed class. For example, any surface of revolution is a member. For a surface of revolution, the reference line is a circle on the surface, and the line of centers reduces to a point on the axis of revolution. A torus is also a member. For a torus, the reference line is a circle on the surface, and the line of centers is a circle that runs along the axis of the sleeve of the torus. The line of centers is not restricted to a circle. A Slinky, laid on a table, with the axis of the Slinky following any curve in a plane, forms an allowable surface. The cross section of the Slinky need not be circular, but the crosssectional shape must be the same at all points along the Slinky and the orientation of the shape, relative to the vertical, must be the same at all points. That is, if the axis lies in a plane, the Slinky cannot twist around its axis. A more general example is obtained by allowing the axis of the Slinky to follow any path in space rather than restricting the axis to lie in a plane. If the Slinky has a circular crosssectional shape, the orientation around the line of centers is meaningless. However, in the most general example, a Slinky of arbitrary crosssectional shape with its axis following an arbitrary path in space, the orientation of the crosssectional shape, as a function of position along the axis, is constrained by the relation given by Eq. 15 .
A simple example.
A surface with a shape that is qualitatively similar to the shape of the canine diaphragm (5) can be constructed from simple components. This shape is shown in Fig. 3. It consists of a segment of a torus capped by segments of spheres. That is, the section ABDE is a segment of a torus with circular cross section, and the sectionBCD is a segment of a sphere. The reference line is the line ABCDE. The line of centers for the toroidal section is a circle of radiusR. For the spherical cap, the line of centers reduces to a point at the center of the sphere. The universal curves are circular arcs of radius ρ.
The surface is divided into two parts by a line that runs along the crest. Section ABC represents the costal diaphragm and section CDErepresents the crural diaphragm. In the toroidal sections,AB andDE, the planes of the universal lines are orthogonal to the reference line and the coordinatet runs along the circular arcs and ranges from t = 0 at the reference line to t = (π/2)ρ at the peak of the torus. In the spherical sections, any great circle is both a geodesic and a line of principal curvature, and the orientation of the lines on the sphere is therefore somewhat arbitrary. Different families of great circles have been chosen to represent the two parts of the spherical cap. The family for sectionBC passes through a pole that is inclined toward the midplane, and the family for sectionCD passes through a pole that is inclined away from the midplane. In sectionsBC andCD, the value oft ranges from zero at the reference line to a variable upper limit at the line of intersection of the two segments of the spherical cap.
An analytical solution can be obtained for the tension in a membrane with this shape loaded by a uniform pressure difference P. Membrane tension in the direction of the coordinatet is denoted ς_{1}, and tension in the orthogonal direction, the direction of coordinates, is denoted ς_{2}. Shear tensions are assumed to be zero. The equilibrium equations for the toroidal sections of the model are the following
The solution to these equations is the following; where the same convention on signs is used, the upper sign is for sectionAB and the lower sign is for section DE
In the spherical caps, ς_{1} = ς_{2} = ρP/2, and this familiar value of tension in a spherical shell is a useful reference value. In the toroidal section, ς_{2} equals tension in the spherical cap, and the normal tensions are equal at the boundary between the two sections. The other component of tension in the toroidal section, ς_{1}, represents tension in the direction of the muscle bundles. In sectionAB, ς_{1} is larger than ρP/2 att = 0 and increases to ρP at the peak of the torus. In section DE, ς_{1} is considerably larger than ρP/2 at t = 0 and de creases toward the peak to match the tension in sectionAB at the peak. Thus ς_{1} is smallest in the spherical caps, larger in the ventral region of the costal diaphragm, and larger still in the crural diaphragm with a maximal at the dorsal end of the crural diaphragm.
DISCUSSION
In this paper, we have presented a theory of diaphragm shape. This theory is based on assumptions about the structure and geometry of the diaphragm, namely, the assumptions that all muscle bundles of the diaphragm lie along lines which are both lines of maximal principal curvature and geodesics of the surface. We showed that these assumptions imply that the muscle bundles lie in planes that are orthogonal to the tangent planes of the surface. In fact, these two sets of properties are equivalent; one implies the other. Boriek et al. (1) studied the midcostal region of the canine diaphragm and found that the geometry of the muscle bundles in this region matches our assumptions; the muscle bundles lie in planes that are orthogonal to the surface, and they noted that the muscles lie along the lines of maximal principal curvature of the surface. In addition to this limited observational support for our assumptions, it can be argued that the assumed geometry has functional advantages. A curved muscle exerts a net force per unit length in the direction of its curvature, and the magnitude of the force is proportional to curvature. Thus the contribution of this force to transdiaphragmatic pressure is maximal if the muscle bundle lies along a line of maximal curvature and the direction of its curvature is normal to the surface. If the line of the muscle bundle were not a geodesic, a component of force would be exerted in the plane of the surface, and this force would tend to distort the surface and distort the line of the muscle bundle toward a geodesic. These arguments about the relation between shape and function are pertinent to lungapposed regions of the diaphragm; in the zone of apposition, diaphragm shape must conform to chest wall shape and transdiaphragmatic pressure is small.
The class of surface shapes that are compatible with the assumption about muscle geometry was described. Compatible shapes are subject to two restrictions. The lines that generate the surface must all have the same shape, and the orientation of the lines is restricted. These are testable predictions, and these predictions constitute the major results presented in this paper.
A simple example of an allowable surface was described. This example is qualitatively similar to the canine diaphragm, but quantitative differences between this example and the observed shape are apparent. The ratio of the ventrodorsal to lateral dimensions of the example is smaller than the observed ratio. We think that this implies that the line of centers of the costal and crural muscles does not coincide as in the example. It appears that the line of centers for the crural diaphragm lies dorsal to the line of centers of the costal diaphragm. As a result, the arcs of the crural and costal muscles do not meet, and the central tendon fills the gap between these two arcs. The gap is greatest on the midplane and narrows when moving laterally. Also, in the example, the shape of the universal curve is the same in both muscles, and the line of centers is continuous. Perhaps the costal and crural diaphragms are more distinct, with different shapes of the lines of muscle bundles and a discontinuity between the line of centers at the joint.
Although there may be quantitative differences between the example and the true diaphragm shape, we think that the example is useful for two reasons. First, it illustrates a method for describing diaphragm shape. If all muscle fibers have the same shape, as predicted, the shape of the surface is described by the shape of that universal curve and the shape of the line of centers. This provides a simple and graphical description of the surface. It also offers the promise of providing a simple method of describing diaphragm kinematics. The descent of the diaphragm could be described by a succession of universal curves and a descent of a line of centers.
Second, we think that the analytical description of the tension distribution is useful. Finiteelement methods have been used to determine the tension distribution in the diaphragm (5), but finiteelement methods are not well suited to the solution of membrane problems, and the solutions that have been reported are not readily interpretable. The analytical solution presented here provides a qualitative guide that can be used to test and interpret finiteelement solutions.
The distribution of muscle tension in the model is a result of the shape, and because this shape is qualitatively similar to the shape of the canine diaphragm, we expect the distribution of tension in the model to be qualitatively similar to the distribution of tension in the canine diaphragm. The qualitative features of the model distribution of muscle tension are the following. In the ventral and midcostal diaphragms, tension in the direction of the muscle fibers is larger than tension in the transverse direction. Curvature is also larger in the direction of the muscle fibers than in the transverse direction, and, in this region, transdiaphragmatic pressure is balanced primarily by muscle tension. At the dorsal end of the costal diaphragm and the lateral end of the crural diaphragm, curvature in the transverse direction is large and the contribution of transverse tension to transdiaphragmatic pressure is large. As a result, tension in the muscle fibers is lower. In the medial region of the crural diaphragm, transverse curvature is reversed, and muscle tension must be large in this region. In the model, muscle tension at the boundary between the toroidal and spherical sections changes abruptly because the change in geometry is abrupt. In the diaphragm, we would expect these changes to be smoother.
The distribution of muscle tensions in the model is consistent with some physiological observations. Tension ς_{1} is a membrane tension, a force per unit distance in a direction transverse to the muscle direction. Therefore, in regions in which ς_{1} is higher, either the thickness of the diaphragm must be greater or the force, or activation, of each muscle bundle must be greater, or both. In the model, muscle tension is largest in the medial region of the crural diaphragm. This is consistent with the observation that the crural diaphragm is thicker than the costal diaphragm (2, 4). Muscle tension is least at the dorsal end of the costal diaphragm, and this is consistent with the observations that the diaphragm is thinner (2, 4) and that blood flow is smaller (3) near the dorsal end of the costal diaphragm.
Acknowledgments
This work was supported by National Heart, Lung, and Blood Institute Grant HL46230.
Footnotes

Address for reprint requests: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455.
 Copyright © 1997 the American Physiological Society