Most of us view the skeleton as the frame upon which the soft tissues are draped. The post-and beam construction of a skyscraper is the favored model for the spine’ and is used for all biologic structures – the upright spine is regarded as the highest biomechanical achievement. The soft tissues are regarded as stabilizing “guy wires,” similar to the curtain walls of steel-framed buildings (Fig. 1).
FIG.1 – Adult thoracolumbar ligamentous spine, fixed at the base and free at the top, under vertical loading, and restrained at midthoracic and midlumbar levels in the anteroposterior plane. A, before loading; B, during loading; C, stability failure occurring under a load of 2.04 kg.; D, lateral view showing anteroposterior restraints. (From Morris M, Markolk K Biomechanics of the lumbar spine. In American Academy of Orthopaedic Surgeons: Atlas of Orthotics: Biomechanical Principles and Application, St. Louis, Mosby, 1975: with permission.)
Skyscrapers are immobile, rigidly hinged, high-energy-consuming, vertically oriented structures that depend on gravity to hold them together. The mechanical properties are Newtonian, Hookian, and linear.’,’ A skyscraper’s flagpole or any weight that cantilevers off the building creates a bending moment in the column that produces instability. The building must be rigid to withstand even the weight of a flag blowing in the wind. The heavier or farther out the cantilever, the stronger and more rigid the column must be (Fig. 2). A rigid column requires a heavy base to support the incumbent load. The weight of the structure produces internal shear forces that are destabilizing and require energy just to keep the structure intact (Fig. 3).
FIG.2-Bending stresses in a beam. (From Galileo: Discorsidimonstrazioni matematiche intorno a due nuove scienze. Leiden, 1638.)
FIG.3-When simple compressive load is applied, both compressive and shear stresses must exist on planes that are oriented obliquely to the line of application to the load.
Biologic structures are mobile, flexibly hinged, low-energy-consuming, omni directional structures that can function in a gravity-free environment. The mechanical properties are non-Newtonian, non-Hookian, and nonlinear.’ If a human skeletal system functions as a lever, reaching out a hand or casting a fly at the end of a rod is impossible. The calculated forces with such acts break bone, rip muscle, and deplete energy (Fig. 4). A post-and-beam cannot be used to model the neck of a flamingo, the tail of a monkey, the wing of a bat, or the spine of a snake (Fig. 5). Because invertebrates do not have bones, there is no satisfactory model to adequately explain the structural integrity of a worm. Post-and-beam modeling in biologic structures could only apply in a perfectly balanced, rigidly hinged, upright spine (Fig. 6).Mobility is out of the equation. The forces needed to keep a column whose center of gravity is constantly changing and whose base is rapidly moving horizontally are overwhelming to contemplate. If we add that the column is composed of many rigid bodies that are hinged together by flexible, almost frictionless joints, the forces are incalculable.2 The complex cantilevered beams of horizontal spines of quadrupeds and cervical spines in any vertebrate require tall, rigid masts for support2 that are not usually available.
FIG.4-A log of 200 kg located 40 cm from the fulcrum requires a muscle reaction force of 8 x 200 = kg. The erectores spinae group can generate a force of about 200400 kg, which is only a quarter to half of the force that is necessary. Therefore, muscle power alone cannot lift such a load, and another supporting member is required. (Courtesy of Serge Gracovetsky, Ph.D.)
FIG.5-Bird Skeleton. (Courtesy of California Academy of Sciences, San Francisco.)
FIG.6-Balancing compressive loads
Since post-and-beam construction has limited use in biologic modeling, other structural models must be explored to determine if a more widely applicable construct can be found. Thompson14 and, later, Gordon4 use a truss system similar to those used in bridges for modeling the quadruped spine. Trusses have clear advantages over the post-and lintel construction of skyscrapers as a structural support system for biologic tissue. Trusses have flexible, even frictionless hinges with no bending moments about the joint. The support elements are either in tension or compression only. Loads applied at any point are distributed about the truss as tension or compression (Fig. 7). In post-and-beam construction, the load is locally loaded and creates leverage. There are no levers in a truss, and the load is distributed throughout the structure. A truss is?fully triangulated, inherently stable, and cannot be bent without producing large deformations of individual members. Since only trusses are inherently stable with freely moving hinges, it follows that any stable structure with freely moving hinges must be a truss. Vertebrates with flexible joints must therefore be constructed as trusses.
FIG.7-Loading a square and a triangular (truss) frame.
When the tension elements of a truss are wires or ropes, the truss usually becomes unidirectional (see Fig. 7); the element that is under tension will be under compression when turned topsy-turvy. The tension elements of the body (the soft tissues – fascia, muscles, ligaments, and connective tissue) have largely been ignored as construction members of the body frame and have been viewed only as the motors. In loading a truss the elements that are in tension can be replaced by flexible materials such as ropes, wires, or in biologic systems, ligaments, muscles, and fascia. Therefore, the tension elements are an integral part of the construction and not just a secondary support. However, ropes and soft tissue can only function as tension elements, and most trusses constructed with tension members will only function when oriented in one direction. They could not function as mobile, omnidirectional structures necessary for biologic functions. There is a class of trusses called tensegrity3 structures that are omnidirectional so that the tension elements always function in tension regardless of the direction of applied force. A wire bicycle wheel is a familiar example of a tensegrity structure. The compression elements in tensegrity structures “float” in a tension network just as the hub of a wire wheel is suspended in a tension network of spokes.
To conceive of an evolutionary system construction of tensegrity trusses that can be used to model biologic organisms, we must find a tensegrity truss that can be linked in a hierarchical construction. It must start at the smallest sub cellular component and must have the potential, like the beehive, to build itself. The structure would be an integrated tensegrity truss that evolved from infinitely smaller trusses that could be, like the beehive cell, both structurally independent and interdependent at the same time. This repetition of forms, like in a hologram, helps in visualizing the evolutionary progression of complex forms from simple ones. This holographic concept seems to apply to the truss model as well.
Architect Buckminster Fuller3 and sculptor Kenneth Snelson13 described the truss that fits these requirements, the tensegrity icosahedron. In this structure, the outer shell is under tension, and the vertices are held apart by internal compression “struts” that seem to float in the tension network (Fig. 8).
FIG.8-A, tetrahedron; B, octahedron; C, icosahedron; and D, tension-vectored icosahedron with compression elements within the tension shell.
The tensegrity icosahedron is a naturally occurring, fully triangulated, three dimensional truss. It is an omnidirectional, gravity-independent, flexibly hinged structure whose mechanical behavior is nonlinear, non-Newtonian, and non-Hookian. Independently, Fuller and Snelson use this truss to build complex structures. Fuller’s familiar geodesic dome is an example, and Snelson12 has used it for artistic sculptures that can be seen around the world. Ingber’? 16 and colleagues use the icosahedron for modeling cell construction. Research is underway to use this structure in more complex tissue modeling.16 Naturally occurring examples that have already been recognized as icosahedra are the self generating fullerenes (carbon6o organic molecules),” viruses,17 cells,” radiolaria 6 pollen grains, dandelion balls, blowfish, and several otherbiologic structures’ (Fig. 9).
FIG.9-The icosahedral structure of a virus.
Icosahedra are stable even with frictionless hinges and, at the same time, can easily be altered in shape or stiffness merely by shortening or lengthening one or several tension elements. Icosahedra can be linked in an infinite variety of sizes or shapes in a modular or hierarchical pattern with the tension elements (the muscles, ligaments and fascia)forming a continuous interconnecting network and with the compression elements (the bones) suspended within that network (Fig. 10). The structure would always maintain the characteristics of a single icosahedron. A shaft, such as a spine, may be built that is omnidirectional and can function equally well in tension or compression with the internal stresses always distributed in tension or compression. Because there are no bending moments within a tensegrity structure, it has the lowest energy costs.
FIG.10-Indefinitely extensive array of tensegrity icosahedra. (From Fuller RB:Synergetics. New York, Macmillan, 1975; with permission)
Viewed as a model for the spine of humans or any vertebrate species, the tension icosahedron space truss (Fig. 11) with the bones acting as the compressive elements and the soft tissues as the tension elements will be stable in any position, even with multiple joints. They can be vertical or horizontal and assume any posture from ramrod straight to a sigmoid curve (Fig. 12). Shortening one soft tissue element has a rippling effect throughout the structure. Movement is created and a new, instantly stable shape is achieved. It is highly mobile, omnidirectional, and consumes low energy. Tension icosahedrons are unique structures whose constructs, when used as a biologic model, would conform to the natural laws of least energy, laws of mechanics, and the distinct characteristics of biologic tissues. The icosahedron space truss is present in biologic structures at the cellular, sub cellular, and multi cellular levels. Recent research on the molecular structures of organisms such as viruses, sub cellular organelles, and whole organisms has shown them to be icosahedra. The very building block of bone, hydroxyapatite, is an icosahedron. In the spine, each subsystem (vertebrae, disks, soft tissues) would be subsystems of the spine meta system. Each would function as an icosahedron independently and as part of the larger system, as in the beehive analogy.
FIG.12/E-C column. (Courtesy of Kenneth Snelson.)
The icosahedron space truss spine model is a universal, modular, hierarchical system that has the widest application with the least energy cost. As the simplest and least energy-consuming system, it becomes the metasystem to which all other systems and subsystems must be judged and, if they are not simpler, more adaptable, and less energy consuming, rejected. Since this system always works with the lowest energy requirement, there would be no benefit to nature for spines to function sometimes as a post, sometimes as a beam, sometimes as a truss, or to function differently for different species, conforming to the minimal inventory-maximum diversity concept of Pearce10 and evolutionary theory.
The icosahedron space truss model could be extended to incorporate other anatomic and physiologic systems. For example, as a “pump” the icosahedron functions remarkably like cardiac and respiratory models, and, so, may be an even more fundamental meta system for biologic modeling. As suggested by Kroto,b the icosahedron template is “mysterious, ubiquitous, and all-powerful.”
Visit Dr. Levin’s web site at: biotensegrity.com
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