chapter  12
24 Pages

Human Lumbar Vertebral Body: Analysis of Its Functionally Optimal Structure

WithDhanjoo N. Ghista, Sridhar Idapalapati, Ramakrishna Kotlanka

CONTENTS 12.1 Introduction and Scope.......................................................................... 357 12.2 Introduction: Concept of the Spinal Vertebral Body Being

an Intrinsically Optimal Structure ....................................................... 358 12.2.1 Optimal Dimensions of the Femur Cortical Bone............... 359 12.2.2 Spinal Vertebral Body as an Optimal Structure .................. 360

12.3 Vertebral Body Shape and Membrane Stresses ................................. 360 12.3.1 Hyperboloid Geometry of the Vertebral Body .................... 360 12.3.2 Membrane Stresses in the Vertebral Body Cortex .............. 362

12.4 Analysis for Forces in the Vertebral Body Generators under Different Loadings.................................................................................. 364 12.4.1 Stress Analysis of the Vertebral Body under Axial

Compression.............................................................................. 364 12.4.2 Vertebral Body Stress Analysis under Bending Moment .. 366 12.4.3 Vertebral Body Stress Analysis under Torsional Loading.. 368

12.5 Optimal Design ....................................................................................... 370 12.5.1 Structural Analogy of the Vertebral Body

to the Cane Stool....................................................................... 370 12.5.2 Optimization of the Hyperboloid Shape

of the Vertebral Body............................................................... 371 12.6 Overview of Vertebral Body Fixators and Impact

of the Intrinsic Design on Better Anterior Fixation .......................... 373 References ........................................................................................................... 378

Spine gives the body structure, support, and allows the body to bend with flexibility. It is also designed to protect the spinal cord. The spine is made up

of 24 small bones (vertebrae) that are stacked on top of each other to create the spinal column. Between each vertebra, there is an intervertebral disk that helps to cushion and transmit the load between the vertebrae and keeps the vertebrae from rubbing against each other [1]. The flexibility of the spine is primarily due to the intervertebral disks [2]. Each vertebra is held to the others by groups of ligaments. There are also tendons that fasten muscles to the vertebrae. The normal spine has an ‘‘S’’-like curve when looking at it from the lateral side. The ‘‘S’’ curve must have evolved to help a healthy spine to perform its role in providing stability, strength, and flexibility [3,4]. Natural structures usually evolve with larger cross section, where stresses are maximum, and leaner cross section, where stresses are minimum, thereby attaining minimum weight. Spinal biomechanical efficacy is to a large extent based on the optimal intrinsic designs of the spinal vertebral body (VB). In the VB, the load-carrying and transmitting function is primarily done

by the cortical VB, whose shape resembles a hyperboloid (HP) shell. We have hence modeled the cortical VB as an HP shell, whose geometry and composition is made up of its generators. This chapter analyzes the forces in the VB generators due to compression, bending, and torsional loadings. The unique feature of the HP geometry is that all the loadings are transmitted as axial forces in the generators of VB HP shell. This makes the VB a highstrength structure. Further, because the cortical VB material is intrinsically made up of its generators (through which all the loadings are transmitted axially), it also makes the VB a natural lightweight structure. We then analyze for the optimal HP shape and geometry by minimizing

the sum of the forces in the HP VB generators (due to its loadings) with respect to the HP shape parameter (angle b between pairs of generators). The value of b is determined to be 26.58, which closely matches with the in vivo geometry of the VB based on the magnetic resonance imaging (MRI). In other words, for the HP shape parameter b¼ 26.58, the VB generators’ forces (under the combined loadings acting on the VB) are minimal, so as to then enable it to bear maximal amounts of loadings. In this way, we have demonstrated that the VB is an intrinsically functionally optimal structure. This chapter (along with Figures 12.2-12.10) is based on our paper on the Human lumbar VB as an intrinsic functionally optimal structure [5].*

In nature, anatomical structures are customized to be functionally optimal [6,7]. If it is a load-bearing structure, then it is adroitly designed to be a lightweight and high-strength structure. For example, a long bone ismodeled

such that it can sustain maximum loading with least amount of material. Consider the case of the femur. Its shape and material density correspond to its stress trajectories under its functional loading (see Figure 12.1) as per Wolff’s law [8]. In other words, there needs to be less density of bone where the stress trajectories are apart (such as in trabecular bone) and more density of bone where the stress trajectories are closer (as in cortical bone).