Computer Methods in Biomechanics and Biomedical Engineering 2

J. Middleton, Gyan Pande, M. L. Jones

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000159455
• Topic: Education
• Subtopic: Early Childhood Education

Contains papers presented at the Third International Symposium on Computer Methods in Biomechanics and Biomedical Engineering (1997), which provide evidence that computer-based models, and in particular numerical methods, are becoming essential tools for the solution of many problems encountered in the field of biomedical engineering. The range of subject areas presented include the modeling of hip and knee joint replacements, assessment of fatigue damage in cemented hip prostheses, nonlinear analysis of hard and soft tissue, methods for the simulation of bone adaptation, bone reconstruction using implants, and computational techniques to model human impact. Computer Methods in Biomechanics and Biomedical Engineering also details the application of numerical techniques applied to orthodontic treatment together with introducing new methods for modeling and assessing the behavior of dental implants, adhesives, and restorations. For more information, visit the "http://www.uwcm.ac.uk/biorome/international symposium on Computer Methods in Biomechanics and Biomedical Engineering/home page, or "http://www.gbhap.com/Computer_Methods_Biomechanic s_Biome dical_Engineering/" the home page for the journal.

1. MULTIBODY SYSTEMS AND JOINT MODELS

ANATOMICAL MODELS OF DIARTHRODIAL JOINTS: RIGID MULTIBODY SYSTEMS AND DEFORMABLE STRUCTURES

J.H. Heegaard¹

1. ABSTRACT

Computer models of diarthrodial joint are commonly represented by a set of constraints limiting the possible motion between limb segments. The nature of these constraints determines the joint motion and the forces and stresses acting across the joint. This paper presents a brief review of commonly used mathematical methods to model diarthrodial joints. Merits and limitations of each method are also presented and possible trends for future research in computational joint biomechanics are briefly discussed.

2. INTRODUCTION

Diarthrodial joints provide mobility to the skeletal system. At the same time they must also provide stability to the skeletal assemblage. For example, mobility at the knee is required to ensure clearance during the swing phase of the leg, while stability at the stance leg knee ensures proper support of the body. Mobility and stability of the skeletal system are antagonistic features which are realized by subjecting diarthrodial joints to high loading. Most joints in the leg bear several times body weight during daily activities (1).

Computer models of diarthrodial joints have been recognized as effective tools to better understand the relationship existing between joint anatomy, joint kinematics and loading at the joint(2, 3). Each limb segment can be modeled as a rigid body resulting in a finite dimensional description of its kinematics. If stresses need to be evaluated, the limb segments must be modeled as deformable continua, requiring numerical methods, such as the finite element method, to discretize the segment’s kinematics into a finite dimensional space.

From a mathematical point of view, joints can be viewed as constraint equations limiting the range of motion of the connected limb segments. The most common types of joint used in computer models include hinges, linkages, and contact between anatomical articular surfaces. The objective of this paper is to briefly review these different joint models, and to discuss their capabilities and limitations.

In Section 3 we model limb segments as rigid bodies and express their dynamics using Lagrange’s equations of motion. We further derive the constraint equations used to model the most commonly used types of joints. In Section 4 we model limb segments as deformable continua and discuss the constraint equations used to represent deformable anatomical joints. The features of these joint models are illustrated with a 3D model of the knee used to calculate the motion of the patella and the stresses in the adjacent tissues during knee flexion. We conclude in Section 5 with a few remarks concerning computer models for diarthrodial joints and possible improvements to current models.

Keywords: Diarthrodial joint, Contact, Dynamics, Continuum mechanics

¹ Professor, Mechanical Engineering Department, Stanford University, Stanford, CA 94305-4040

3. RIGID BODY MODELS

3.1. Equations of motion

In this Section we treat each limb segment as a rigid body. The kinematics of the system can therefore be expressed with a finite number N of generalized coordinates q₁,…,q_N. The Lagrangian L of the system is defined as

$$L=T-V$$

(1)

where T and V represent respectively the kinetic energy and the potential energy. The N Lagrange’s equations of motion are given by

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_r}-\frac{\partial L}{\partial q_r}=Q_r$$

(2)

where Q_r is the rth. generalized force

$$Q_r={\displaystyle \sum _{\beta }{F}_{\beta }\frac{\partial r_{\beta }}{\partial q_r}}$$

(3)

corresponding to those forces F_β not already included in V.

3.2. Motion constraints

Additional constraints may limit the possible motion of each segment. For instance a joint between two segments B₁ and B₂ may prevent some configurations to occur.

An important motivation for using Lagrange’s equations is the possibility to eliminate constraint equations by choosing a proper set of generalized coordinates. In some circumstances however, it may be helpful not to eliminate the constraint equations. Such may be the case when handling difficult to express constraints, or when constraint forces need to be evaluated. In those cases, additional constraint equations f_s(q, t) = 0; s = 1, …, C must be introduced to ensure consistency between the system configuration and the motion constraints.

The equations of motion take then the usual form

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_r}-\frac{\partial L}{\partial q_r}+{\displaystyle \sum _{s=1}^C{\lambda }_s}\frac{\partial f_s}{\partial q_r}=Q_r$$

(4)

where the λ_s are Lagrange multiplier associated to the generalized c...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Contributors
• 1. Multibody Systems and Joint Models
• 2. Hip Replacements: Prosthesis/Cement/Bone Analysis
• 3. Bone Adaptation, Structural Models and Architecture
• 4. Spine and Vertebra Mechanics
• 5. Reconstructive Surgery, Virtual Reality and Implant Analysis
• 6. Soft Tissue Structures, Contact and Biofluid Mechanics
• 7. Dental Materials, Behaviour and Biomechanics
• 8. Craniofacial Mechanics and Diagnostic Methods