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A Different Approach to the Mechanics of the Human Pelvis: TensegrityINTRODUCTION The paradigm According to conventional wisdom, the human spine behaves as an architectural column or pillar and transfers the superincumbent weight through the sacrum, to the ilium, through the hips and down the lower extremities. The pillar holds the base in place with the pressing weight of gravity. In this model, the sacrum, as the base, locks into the pelvis, either as a wedge or by some other gravity-dependent closure. The anomalies Architectural pillars orient vertically and function only in a gravity field and are rigid, immobile, base-heavy, and unidirectional. Pillars and columns resist compression forces well but need reinforcement when stressed by bending moments and shear. Stressed by internal shear, they are high energy consuming structures. Rigid Newtonian mechanical laws, such as Hooke's law, Euler's formula, Galileo's square-cube law, and Poisson's ratio govern conventional columns (Box 1). Yet, if biologic systems conformed to these laws, the human bony spine would bend with less than the weight of the head on top of it (Morris & Lucas 1964) and the vertebral bodies would crush under the leverage of a fly rod held in a hand. Animals larger than a lion would continually break their bones, and dinosaurs and mastodons larger than a present-day elephant would have crushed under their own weight. Urinary bladders and pregnant uteri would burst when full and, with each hearbeat, arteries would lengthen enough to crowd the brain out of the skull (Gordon 1978).
While it is a teleological conceit that the human spine acts as a column, phylogenetic and ontogenetic development of the human spine was not in the form of a column, but as some form of a beam. It would not be an ordinary beam, a rigid bar, but an extraordinary beam composed of rigid body segments connected by flexible connective tissue elements that floated the segments in space (Fielding et al 1976). During human gestational development and during the first year or so of life, the human spine does not function as a column but as such a beam. It must be recognized that in many postures the adult human spine does not function as a column or even a simple beam. When the spine is horizontal, the sacrum is not a base of a column but the connecting element that ties the beam to the pelvic ring. Even when upright, the vertebral blocks are not fixed by the weight of the load above, as they must be in an architectural pillar. S-shaped curves can create intolerable loads and instability in a column, particularly if it is an articulated column that has flexible, frictionless joints, as the spine does. With each breath, the interconnected bodies translate, some forward, some backward. While architectural columns bear loads from above, the human spine can accept loads from any direction with arms and legs cantilevered out in any way. The hallmark of a spine is flexibility and movement. Movement of an articulated column, even along a horizontal, is more challenging than moving an upright Titan missile to its launch pad. The spine can bend forward so a person can touch his toes and bend backward almost equally well. It can twist and bend simultaneously. It can perform intricately controlled movements in space, as in gymnastics, dance, aquatic diving, or basketball. The spine is flexible, mobile, functionally independent of gravity, and has property behavior inconsistent with an architectural column or beam. In all studies, the spine, unlike columns and beams, is a low energy consumer. The individual components of the spine, and the structure as a whole, behave non-linearly and do not conform to the standard linear Newtonian mechanical laws that govern columns and beams (Fox 1988, While & Panjabi 1978). Yet, in an attempt to make complex problems simple, bioengineers have converted non-linear complexities to linear mathematics models. The new science of non-linear dynamic systems (such as the weather) has shown the fallacy of that process (Gleick 1988). The alternative If, instead of a column, the spine is considered to be a series of rigid bodies tied together by the discs and soft tissues, with the sacrum as the connecting link to the pelvis, what locks the sacrum in place so that the spine is supported in all its functions? An omnidirectional mechanical system exists that can function in any posture and be capable of transferring considerable loads, coming from any direction, through the pelvis and to the lower extremities. Such a system must be consistent with evolutionary theory. It must also be structurally hierarchical so that in any instant in its ontological development it is mechanically functional and stable. (Embryos and fetuses do not fall apart either in or out of the womb.) KINEMATICS The kinematics of the pelvis must take into account mechanical laws that affect a free body in space. Until the work of White and Panjabi (1978), movements of anatomic structures were usually described in anatomic, rather than mechanical, terms such as forward-bending or side-bending. Planes of movement were also described in anatomic terms, such as "coronal" or "sagittal". This terminology worked as a barrier to the precise description of movements in biologic systems. For example, forward-bending is more precisely described in terms of rotations and translations of vertebral bodies in space and in relation to each other, which would be difficult to describe in anatomic terminology (Fryette 1954). In mechanical terminology, a rigid body in space is described as having six degrees of freedom of movement in a three-dimensional Cartesian coordinate system. White and Panjabi (1978) adapted the Cartesian coordinate system to the biomechanical description of the spine (Fig. 1). Cartesian coordinates are now widely used in the biomechanical literature to describe joint movements. Although in classical mechanics there are six degrees of freedom, others have considered that describing 12 degrees of freedom -- six positive and six negative -- may be more useful. It is easy to plot these coordinates and generate computer graphics. This system seems suitable for describing the complex movements of the sacrum. However, before we can discuss the dynamics of the sacrum or any other structure, we should understand the statics of that structure. How is the sacrum stabilized in its position in the body?
STATICS To fix in space a body that has 12 degrees of freedom, it seems logical that there need to be 12 restraints. Fuller (1975) proves this. One restraint will allow the body to move in a place around an axis, and three restraints fix the body but allow movement in a line along an axis. Four restraints, configured as corresponding to the vertices of a tetrahedron, are the minimum required to fix a point in space (Fig. 2). However, this would still allow turbining positively and negatively on three axes. According to Fuller's proof, to be rigidly fixed a total of 12 restraints would be necessary.
This principle is demonstrated in a wire-spoked bicycle wheel. A minimum of 12 tension spokes rigidly fixes the hub in space (anything more than 12 is a fail-safe mechanism) (Fig. 3). In a bicycle wheel, tension-loaded spokes transmit compressive loads from the frame and the ground. The hub remains suspended in its tension network, and the compression loads distribute around the rim. The compression elements are discontinuous and behave in a counterintuitive way. Rather than becoming the primary support elements of the system, as they would be in a pillar or wagon wheel model, the compression elements become secondary to the tension support network. Fuller (1975) calls these structures "tensegrity" structures, a contraction of "tension integrity". Other familiar tensegrity structures are tennis rackets, which transmit the compression force of the racket frame to the ball through the strings, snow shoes, and Buckminster Fuller geodesic domes (which are high-frequency icosahedrons.) Tensegrity structures transmit loads through tension and compression only. Because they are fully triangulated, there are no bending moments in these structures, nor is there shear. By linking the hubs of front and rear wire bicycle wheels by its frame, we create a hierarchical system in which the load on the bicycle suspends in a tension network. This network works even when the bike is doing "wheelies" (rearing-up on one wheel) and transfers all the load to one wheel.
It is generally accepted that the sacrum hangs from the ilia by its ligaments (Grant 1052, Kapandji 1974). A ligamentous tension system for support and stability is consisten with the known anatomy. If we use a bicycle wheel tensegrity structure as our model for the pelvis, the pelvic ring would be the rim and the sacrum would be the hub of the pelvis. The many tension elements of ligaments and muscles attached to the sacrum stabilize it (Fig. 4). The sacrum suspends as a compression element within the musculoligamentious envelope and tranfers its loads through that tension network. Even when a person stands on one leg, the sacrum sits within its tension network, just as does the bicycle hub when doing wheelies. This tension network provides omnidirectional structural stability, independent of gravity and hierarchy. The rim could distribute its load, rather than locally loading the forces at a point.
In a compressive loading pelvis system, as exists in the column model, the heads of the femurs would, with each step, smash into the soft cancellous bone of the acetabulum. In a tensegrity system, the forces generated at the hip would not concentrate in the acetabulum but be efficiently distributed throughout the pelvic bones and soft tissue. The sacrum would remain suspended in its soft tissue envelope (Willard 1995) and transmit the loads above and the forces below through the pelvic ligaments and muscles. Suspended in its tension network, it does not require gravity to hold it in place, as does a keystone model. The tensegrity-modeled sacrum functions right side up, upside down, or sidewars. A tension-fixed sacrum works equally well for the upright or space-walking human, the horizontal horse, the flying bat, or the swimming otter. It is the most widely adaptable, and therefore the most likely, pelvic model. DYNAMICS As a hub suspended by its spokes, the tension system must have a dynamic balance of the tension structures. A load on the wheel hub does not change its relative position within the rim. If the tension of the spokes remains constant and the spokes do not distort, the hub does not move at all. Ligaments of the body, likewise, have a high tensile strength and do not distort much when loaded. Assuming a minimum of 12 properly vectored restraints, as with the bicycle model, the sacrum cannot translate or rotate in any direction. It is fixed in position as is the hub of a wheel. Some of the restraints would have to be altered to allow pistoning or rotation to occur. However, if the sacrum moves in tandem with the other bones of the pelvis, so that the ligaments remain at the same length, tension-coupled movement patterns occur. The body does have this coupled movement option available. It is present in the double tie bar hinge mechanism that is the model for the dynamics of knee movement (Dye 1987, Mueller 1983). This type of movement occurs in the "Jacob's ladder" (Fig. 5), a 2000-year-old children's toy, which is itself a tensegrity structure. It is a series of tiles connected by crossed ribbons under tension. Flipping one of the tiles creates a controlled tumble. If the end tiles are held apart so that the entire structure is held in tension, the coupled tumbling can occur from top to bottom, bottom to top, or sideways. This crossed ligament pattern, clearly evident in the knee, also exists in the spine, at the disc, ligament, and muscle level (Gracovetskky 1988, Kapandji 1974). It explains the coupled motion observed in the spine (White & Panjabi 1978). It is also evident in other joints, such as the capsular ligaments of the hip and the crossed patterns of ligaments and muscles of the back. This crossed tie bar pattern is present at the sacroiliac joints (SIJs) with the crossing patterns of the sacrospinous and sacrotuberous ligaments, the iliolumbar ligaments, the ventral, interosseous, and long and short dorsal sacroiliac ligaments, the piriformis, iliopsoas, coccygeus, and other muscles and soft tissues of the pelvis-spine-hip complex (Kapandji 1974). The crossed tie bar mechanism at the SIJ would account for the "click-clack" phenomenon of the sacrum recognized by Snijders et al. By rotating the ilia, as we do when we walk, the sacrum is forced to tumble and the movement transmits, Jacob's ladder-like, up the spine and to the limbs. Both the static and the dynamic mechanics of the pelvic structures are explained with tensegrity modeling.
THE EVOLUTION OF THE STRUCTURE To fully understand pelvic mechanics and its integration in body mechanisms, it must be placed in its proper context. The tensegrity pelvic system is not creationist in design but is created by the physics of evolution (Fox 1988, Levin 1982, 1986, Prigogine & Stengers 1984). For biologic structures to exist as entities, they must be inherently stable as self-contained, not only when fully developed, but also at each instant of their existence. Only triangulated structures are inherently stable (Pearce 1978). Structures that are not fully triangulated have joints that must be rigidly fixed to keep from collapsing. These joints generate torque and bending moments and have high energy requirements. Triangles are stable with flexible joints and have no torque or bending moments at the joints (Fig. 6). There are only tension and compression members in a triangle, so triangulated structures are low energy consumers. Truss systems made from triangles are used by engineers for constructing buildings and bridges because of their load distribution and high strength-to-weight ratios. Engineers will frequently build structures that mix triangulated and non-triangulated components, but to take maximum advantage of the construction properties of triangles, the trusses must be constructed only of triangles. The only fully triangulated, three-dimensional trusses are the polyhedra the tetrahedron, octahedron, and icosahedron (Fig. 7). All three-dimensional, fully triangulated trusses are some combination or permutation of these polyhedra. Thompson (1965), and later Gordon (1978), used truss systems to model biologic structures. Since only trusses are stable when their joints are flexible, it follows that if a structure has flexible joints and is stable, it must be triangulated. Thus biologic structures, stuck together by surface tension at the cellular level and freely jointed at the organism level, must be hierarchical, fully triangulated constructs composed of tetrahedrons, octahedrons, or icosahedrons.
Intimately related to the laws of triangulation are the laws of close-packing (Pearce 1978). In a planar arrangement of structures, the space- and energy-efficient configuration is hexagonal cllose-packing, as in a beehive (Fig. 8). Graphite is a hexagonal close-packed array of carbon atoms in sheet and is the first structural form of carbon. The laws of close-packing are independed of scale and apply to the molecular level as well as to plate tectonics.
In three-dimensional packing, the close-packed structure would also have to be a fully triangulated polyhedron. Diamonds, the hardest known materials, are a close-packed array of carbon atoms in a tetrahedral form and the second known structural form of carbon. Methane and water molecules are configured as tetrahedrons. A pile of oranges stacks as a tetrahedron. So do grains of sand and boulders to form mountains. Any oranges, sand grains, or boulders sticking out of the tetrahedron are unstable appendages and will fall if not held by friction or other force. Spherical forms of close-packed structures would have to be icosahedrons, which are mathematically the most spherical and symmetric polyhedra as well as being fully triangulated. Carbon atoms' third molecular structural arrangement is as icosahedral-shaped fullerenes that are the roundest of all round molecules. Proteins lumped together to form viruses are icosahedral in shape. So are the silica shells of radiolaria (which are minute marine organisms), pollen grains, and blowfish. Of the three triangulated polyhedra, the icosahedron has several attributes that are advantageous for biologic structures. It is the most spherical and has the largest volume for its surface area. In a planar arrangement, icosahedra pack as neatly as billiard balls. In spherical close packing, 12 icosahedra will close-pack around a central icosahedron-shaped space and form a stable icosahedron in a hierarchical construction (Fig. 9). Icosahedra also can be parts of fractals (Mandelbrot 1983), polymerizing and combining with other icosahedra. (A fractal dimension is a dimension greater than one and less than two, or freater than two and less than three, etc. It is the structural equivalent of conjoined twins, where one cannot exist on its own but must be part of another, each having components contributing to the whole.) Since it packs in stable arrays, it is self-generating (Kroto 1988). Tetrahedral- and octahedral-based trusses are not omnidirectional in form and function. They have a smaller volume for their surface area than do icosahedra, are not fractal generators and do not close-pack. They would be less suitable for biologic structures, most of which require these properties (Barnesley et al 1987).
The mechanical laws of close-packing and triangulation apply to the multicelled embryo. Four cells will array as a tetrahedron, with four triangular faces. Eight cells will for an octrahedron. Close-packing of cells will tend toward a spherical organization, the morula. The sphere is considered to be the optimum convex shape as it encloses the largest volume for its surface area. Composed of close-packed blastomeres, they could not be a perfect sphere but a convex polyhedron that is the most spherical, an icosahedron, with 20 triangular faces. Mathematically, it is impossible to create a convex polyhedron with more than 20 equilateral triangles (Pearce 1978). All higher-level stable spherical structures are just higher-frequency icosahedra. The icosahedron is a regular solid with 20 triangular faces and 30 edges. Twelve vertices are created where 3 edges meet. Pressure on any point transmits along the 30 edges, some under tension, others under compression. It is possible to transfer all compression away from the outer edges by connecting opposite vertices of the icosahedron by compression rods. These rods do not pass through the center of the icosahedron but are eccentric and oddly angled; they hold the opposite corners away from each other. The outer shell of 30 edges is now entirely under tension, and the compression rods float within this tension shell like an endoskeleton (Fig. 10). A load applied to this structure causes a uniform increase in tension around all the edges and this distributes compression loads evenly to the six compression members. The mechanical properties of a tensegrity icosahedron are that they are omnidirectional structures, with the compression members and tension elements always maintaining their respective properties regardless of the direction of applied load, just as the wire spokes of a bicycle wheel are always under tension and the hug is always being compressed. Like the bike wheel, they can exist independent of gravity and are local load-distributing. They have a unique structural property of behaving non-linearly, as does the spine and its components, and most biologic tissue (Gordon 1988).
Fuller (1975) has shown that tensegrity icosahedra can link in an infinite array with any external form, as shown in Fig. 11. When linked, these structures can function as a single icosahedron in a hierarchical system. This model has been used to model endodkeletal structures, such as an upper extremity and cervical spine (Levin 1990), with the bones functioning as the compression rods and the soft tissues as the tension elements.
If we apply these evolutionary structural concepts to the sacrum, we can see how the tensegrity sacropelvic model develops. The sacrom, fixed in space by the tension of its ligaments and fascial envelope, functions as the connecting link between the spine and upper (or forequarter) extremities, and the pelvis and lower (hindquarter) extremities. It evolved ontogenetically, directed not only by phylogenetic forces, but also by the physical forces of embryologic development. Wolff (1892) and Thompson (1965) state that the structure of the body is essentially a blueprint of the forces applied to these structures. Carter (19991) theorizes that the mechanical forces in utero are the determinants of emryologic structure that, in turn, evolves to fetal and then newborn structure. From the physicalist and biomechanics viewpoint, as well as from Darwinian theory, the evolution of structure is an optimization problem (Fox 1988, Hildebrandt and Tromba 1984). At each step of development, the evolving structure optimizes so that it exists with the least amount of energy expenditure. At the cellular level, the internal structure of the cells, the microtubules, together with the cell wall, must resist the crushing forces of the surrounding milieu and the exploding forces of its internal metabolism. Following Wolff's law, the internal skeleton of the cell aligns itself in the most efficient way to resist those forces. Ingber and colleagues (Ingber & Jamieson 1985, Wang et al 1993) have shown that the internal microtubular skeletal structure of a cell is a tensegrity icosahedron. Other subcellular structures, such as viruses, cletherins, and endocysts, are icosahedra (de Duve 1984, Wildy & Home 1963). A hierarchical construction of an organism would use the same mechanical laws that build the most basic biologic structure and use it to generate the more complex organism. Not only is the beehive an icosahedron, but so also is the bee's eye. Many other organelles and organisms look like and/or function as icosahedra (Levin 1982, 1986, 1990). Following the concepts of Carter (1991), Wolff (1892), and Thompson (1965), a tensegrity-structured pelvis will build itself. Since the fetus develops upside down in a gravity-independent environment, as do fish eggs in water, the pelvis develops as a tensegrity ring, which is the most efficient structure to do that job. It does not develop as a structure to resist superincumbent weight-bearing. If it did, it would not function during its initial role in life of resisting in utero forces. It would also crush during delivery. Ontogeny recapitulates phylogeny. The one-celled organism evolves as a series of stepwise mechanical accidents that are the most energy efficient and most adaptable, into a complex, energy efficient, symbiotic, multicelled organism. The different phyla get off the evolutionary ladder at different steps. To believe otherwise is to be a "creationist" rather than a believer in Darwinian evolution. The development of a pelvis is not a "design" but an evolutionary accident that worked in creating an energy-efficient, ambulating creature that could survive better in a gravity environment on land and could take advantage of the already evolved lungs that allowed breathing beyond the confines of the sea. It is the marvel of tensegrity structures that they are remarkably adaptable and can resist loads in a gravity-oriented environment equally well as they do when not affected by gravity (perhaps adding a few more trabeculae and ossifying some cartilage in accordance to Wolff's "law"). The pelvis is cancellous bone because the distributed loads require nothing more, nothing less. Evolved to resist crushing forces from any direction, or exploding forces from within, the pelvis can adapt to unidirectional forces that are applied at two, three, or more points and distribute the load through the tension network of soft tissues and compression network of bones. Icosahedral tensegrity structures are self-organizing space frames that are hierarchical and evolutionary (Kroto 1988). They will build themselves, conforming to the laws of triangulation, close-packing, and, in biologic constructs, Wolff's Law. The pelvic wheel is a self-organizing structure that is part of a larger, fractal, space-frame, tensegrity construct with each part integrated into the whole. Simplicity and complexity intertwine in what Pearce (1978) calls "minimum inventory, maximum diversity." CONCLUSIONS
REFERENCES
© 2003 Dr. Stephen M. Levin. All Rights Reserved. |
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