Finite Element Analysis of Bone Remodeling. Implementation of a Remodeling Algorithm in MATLAB and ANSYS

Master's Thesis 2006 108 Pages

Medicine - Biomedical Engineering


Table of Contents

List of Figures

List of Tables

1 Introduction
1.1 Motivation
1.2 Aim

2 Bone as Material
2.1 Composition of Bone
2.2 Difference between Cortical and Cancellous Bone
2.3 Material Properties of Bone Tissue
2.3.1 Cortical Bone Tissue
2.3.2 Cancellous Bone Tissue

3 Bone Remodeling - From Nature to Model
3.1 Remodeling Theory
3.1.1 Difference between Modeling and Remodeling
3.1.2 The Remodeling Process B asic M ulticellular U nit Remodeling Cycle Duration
3.1.3 Types of Remodeling
Osteonal Remodeling
Trabecular Remodeling
Endosteal and Periosteal Remodeling
3.2 Theories from the Beginning up to now
3.2.1 Mechanically Excited Bone Adaption Theories (1865 -1920)
3.2.2 Bone Adaptation: General Relationships of Mechanics to
Bone Physiology (1920 - 1970)
3.2.3 Bone Adaptation: Experimental Study of Mechanically Mediated Bone (1970 - 1984)
3.2.4 Theories of Bone Adaptation: Numerical Simulations (1985 to present)

4 Simulation of Remodeling
4.1 Implementation of Optimization Algorithm
4.1.1 Reference System
Material Properties
4.1.2 Optimization Part
Adaptation Functions
Convergence Criteria
4.2 Results
4.2.1 Reference System 1
4.2.2 Reference System 2
4.2.3 Convergence behavior

5 Applications
5.1 Modeling of Human Proximal Femur
5.1.1 Load Definitions
5.1.2 Initial Configuration and Boundary Conditions
5.2 Results
5.2.1 Proximal Femur with Stepwise Adaptation Function
Initial Homogeneous Density Distribution
Initial Stochastic Density Distribution
5.2.2 Proximal Femur with Linear Adaptation Function
Initial Homogeneous Density Distribution
Initial Stochastic Density Distribution
5.2.3 Convergence

6 Discussion
6.1 Approach
6.2 Details of Algorithm
6.2.1 Building the Model
6.2.2 Development of Bone Structure
6.2.3 Convergence Behavior
6.3 Conclusion and Outlook



A Implementation in MATLAB and ANSYS A

A.1 MATLAB main file run.m

A.2 ANSYS macro

A.3 MATLAB function dr.m

B Basic Anatomic Terminology L

List of Figures

2.1 Sketch of some important features of typical long bone, from [93]

2.2 SEM micrograph of ground trabecular vertebra bone, from [76]

2.3 3D reconstruction of trabecular bone (4 × 4 × 4 mm 3 cube) from [4]

2.4 The influence of loading rate on the tensile strength and modulus of cortical bone from [92]

2.5 Typical stress strain curves for trabecular bone of different densities, from [43]

2.6 Stress against strain of trabecular bone specimen under compression from [76]

2.7 Experimental determination of Young’s modulus E against density ρ. From [85, 14, 2, 57, 64, 79]

3.1 Photomicrograph of an osteonal basic multicellular unit

3.2 Schematic sketch of an osteonal BMU. Cross-sectional view at the bottom right

3.3 The Six Phases of an Osteon’s Lifetime. a) Activation AC, b) Resorption RES, c) Reversal REV, d) Formation FO, e) Mineralization MI, f) Quiescence QU. from Ott with permission [75]

3.4 BMU activation rate vs. age for human ribs. (From data by [34])

3.5 Stress trajectories in curved Culmann crane (left) compared with a schematic representation of the trabecular pattern in the proximal femur, from Wolff,

3.6 Change of trabecular structure in post-fracture , from Wolff,

3.7 Frost’s description of the different adaptive responses for the adolescent and the adult skeleton

3.8 Density distribution in the femoral head by Fyhrie

3.9 Adaptation function according to equation (3.6)

4.1 Simulation algorithm

4.2 Reference systems

4.3 Higher order 2-D element, from Zienkiewicz [94]

4.4 Young’s modulus against discretized density ρ according to equaion (4.2)

4.5 Defined load steps i against time steps

4.6 Overview of implemented adaptation functions

4.7 Resulting material (density) distribution of reference system 1. Figure 4.7(o) is done with Ole Sigmud’s code from [86]

4.8 Resulting material distribution of reference system 2. Figure 4.8(o) is done with Ole Sigmud’s code from [86]

4.9 Convergence plot of reference system

4.10 Convergence plot of reference system

5.1 Anatomy of the human proximal femur. From [89]

5.2 3-D femur model with section plane

5.3 2-D Finite element mesh of the proximal femur with 7124 elements

5.4 Element FLUID79 from [1]

5.5 Proximal femur with muscles

5.6 Hip contact force against time for human normal walking from [7]

5.7 Overview load cases for normal walking

5.8 Initial configurations with two different density distributions

5.9 cm 3 Remodeling ratio coefficient B(n)in (g )2 MPa −

5.10 Remodeling progress in human proximal femur with a stepwise adaptation function and initial homogeneous density distribution

5.11 Comparison of v. Mises stresses at initial and converged state

5.12 Comparison of principal stresses at initial and converged state

5.13 Remodeling progress in human proximal femur with a stepwise adaptation function and initial stochastic density distribution

5.14 Comparison of v. Mises stresses at initial and converged state

5.15 Comparison of principal stresses at initial and converged state

5.16 Remodeling progress in human proximal femur with a linear adaptation function and initial homogeneous density distribution

5.17 Remodeling progress in human proximal femur with a linear adaptation function and initial stochastic density distribution

5.18 Sum of averaged strain energy density u ∗

5.19 Difference | q ˙ | in linear-linear scale

5.20 Difference | q ˙ | in linear-logarithmic scale

6.1 Comparison of converged results with X-ray plot of proximal femur

B.1 Anatomic planes with labels from [61]

List of Tables

2.1 Strength of femoral cortical bone from [43]. Mean values from [78]

2.2 Moduli of femoral cortical bone, from [43]. Mean values from [78]

2.3 Mean values of modulus and ultimate strength for various anatomic

5.1 Joint and muscle forces during normal walking according to Bergmann [7]

B.1 Body components

B.2 Terms of position

Chapter 1 Introduction

1.1 Motivation

Bone is a living material which has its main function in building the skeleton and therefore enabling locomotion and protection of the organism. It is subjected to permanent and transient loads caused by the daily active or special events like accidents. In contrast to inert materials from standard mechanics, this tissue is able to react adaptively to its environment. Aside from skeletal growth and fracture healing, which are of temporary character, the internal bone structure is maintained and adapted continuously. This process is termed remodeling.

Within this process microdamage is removed, leading to an increase of the fatigue life of bone tissue. Furthermore, the structural adaptation to changes in the mechanical environment plays an important role in conjunction with implants and prostheses. In fact, latest developments of such devices have been analyzed numerically in order to predict the long-term reaction of the tissue to this impact.

Osteoporosis, nowadays a widespread bone disease, underlies similar concepts as the remodeling process. Due to the enormous social damage caused by such diseases on one side and failure of implants and prostheses on the other side, an advance in the understanding and computer simulation of remodeling is of great importance.

Therefore, the interest in this phenomenon has been increasing in the last century. Especially in the last twenty years, many numerical algorithms have been developed in order to simulate this process. Although early models were capable of predicting good approaches to the real behavior, a quantitative analysis has been impossible and many biologic aspects have been neglected.

Recently, new methods have been published which take into account aspects of the microstructure and cell activities. These models provide good results for predictions of bone loss due to osteoporosis but suffer deficiencies from a mechanical point of view.

In the following, the composition of bone, the difference of cortical and trabecular bone and its mechanical properties are described in chapter 2. Afterwards the process of remodeling will be outlined in chapter 3 coming with a historical review of research work done in this field from the of the 19th century up to now. In chapter 4 the implementation of the remodeling algorithm is described and simulations for evaluation are shown. The implemented algorithm is applied to a proximal femur FE-model with different initial configurations in chapter 5. Finally, the results are discussed in chapter 6 and the appendix provides the source code for implementation in MATLAB and ANSYS.

1.2 Aim

The aim of this work is to implement an easy to use and extendable numerical algorithm which can build up the remodeling process of bone due to mechanical stimulus. Besides mechanical stimulus, a huge number of items, such as age, race, gender, possible diseases etc. play an import role in this context. These highly complex and mostly underdetermined boundary conditions of the optimization, however, shall not be taken into account in this approach. Therefore, this thesis will be mainly focused on bone’s internal structural changes in response to a change in the mechanical environment.

Chapter 2 Bone as Material

2.1 Composition of Bone

Bone is not an uniform material, it is composed of collagen, water, hydroxyapatite mineral and small amounts of proteglycans and noncollagenous proteins . [67]

Collagen is a structural protein, that can spontaneously organize itself into strong fibers. More than a dozen types of collagen have been identified. Type I collagen is the most abundant collagen of the human body. It is present in scar tissue, the end product when tissue heals by repair. Beside in bone, it is also found in tendons, ligaments and skin. Collagen is responsible for bone’s flexibility and tensile strength. It also provides loci for the nucleation of bone mineral crystals, which give bone rigidity and compressive strength.

Mineral in bone consists almost entirely of hydroxyapatite crystals, Ca 10(PO 4)6(OH)2. The individual crystals are rods with hexagonal symmetry, measuring about 50 × 50 × 400 angstroms (1 [ µ m] = 10,000 angstroms [Å]). Bone mineral is impure, containing many structural substitutions (e.g., carbonate, fluoride, citrate). These impurities are governed by the composition of body fluids and in turn affect the solubility of the bone mineral.

Ground substance of bone consists of proteglycans. In particular, decorin and biglycan are small species of proteglycans found in bone. Although the specific role of the proteglycans is not known exactly. Decorin is known to modulate collagen fibril assembly. Proteglycans may also act to control the location or rate of mineralization in through their calcium-binding properties.

Noncollagenous proteins contain quite a few molecules whose functions are also unclear. The most abundant noncollagenous protein is osteocalcin, which is secreted by osteoclasts and appears to be important in the mineralization of new bone. It also is a chemoattractant for bone cells and or its serum concentration are an excellent method of noninvasively determining rates of bone turnover. Other noncollagenous proteins in bone include osteopontin and osteonectin.

Water appears in the calcified bone matrix in two different conditions, one part is free and the other part is bound to other molecules. The mineralization of osteoid (the organic portion of extracellular bone) displaces part of its water. Therefore, the water content of new bone tissue changes as it mineralizes.

2.2 Difference between Cortical and Cancellous Bone

In principle there are two types of bone, as determined by porosity (volume fraction of soft tissues). The porosity of bone can vary step less from zero to 100%, but most bone tissues are of either very low or very high porosity and just a small part of intermediate porosity. These two types of bone tissue are referred to compact or cortical bone and trabecular or cancellous bone, respectively as shown in Figure

2.1. This combination of trabecular and cortical bone forms a sandwich-type structure, well known in engineering for its optimal structural properties. [27]

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Figure 2.1: Sketch of some important features of typical long bone, from [ 93 ]

Trabecular bone is the spongy, porous type of bone as shown in Figure 2.2 and can be found in the cuboidal bones, the flat and irregular bones, such as the sternum, pelvis and spine (vertebra) and at the end of all long bones [54]. Its porosity varies from 75% to 95%. The interconnected pores, which scale is on the order of 1 mm, are filled with marrow. The bone matrix is in the shape of plates or struts called trabeculae each with about 200 µ m in diameter. The arrangement of the trabeculae is not unique. Mostly they build a randomly oriented meshwork. Sometimes they appear to be organized into orthogonal arrays.

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Figure 2.2: SEM micrograph of ground trabecular vertebra bone, from [76 ]

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Figure 2.3: 3D reconstruction of trabecular bone ( 4 × 4 ×

4 mm 3 cube) from [4 ]

Compact bone is the dense bone crop up in shafts of long bones and forming a cortex or shell around vertebral bodies and other cancellous bones. Its porosity varies from 5% to 10% and its pores consist of spaces categorized as follows

Haversian canals are approximately aligned to the long axis of the bone. They are about 50 µ m in diameter and contain nerves and capillaries. Haversian canals are named after an English physician, Clopton Havers (1691).

Volkmann’s canals are short, transverse canals connecting Haversian canals to each other and to the outside surfaces of the bone. These canals also include blood vessels and presumably nerves. They are named after Richard von Volkmann (1830-1889), a surgeon and early advocate of Lister’s antiseptic surgical methods.

Resorption cavities are temporary spaces caved by osteoclasts in the initial stage of remodeling, described in 3.1.2. These cavities are about 200 µ m in diameter.

It is important to keep in mind that bone is a dynamic porous structure. Its porosity may change as a result of a pathologic condition or in a normal adaptive response to a mechanical or physiologic stimulus. This leads to higher density in trabecular bone, or to lower density in compact bone. Such changes strongly affect bone’s mechanical properties.

Besides trabecular and compact bone, two further aspects how bone can be characterized should be mentioned. Namely lamellar vs. woven bone and primary and secondary bone. More details can be found in [67, 72].

2.3 Material Properties of Bone Tissue

To determine the mechanical properties of bone tissue, small uniform specimens are loaded under well-defined conditions. In homogeneous materials, such testing conditions lead to uniform stresses throughout the specimen. The resulting deformation can be measured and the stress-strain relationship can be established. With several load types, material properties in tension, compression, bending and torsion can be determined [8, 10, 9, 17, 26, 25, 78, 77, 85].

2.3.1 Cortical Bone Tissue

The material properties of cortical bone are influenced by several factors. One is the rate at which the bone tissue is loaded. Rapid loading of cortical bone specimens leads to increased elastic moduli and ultimate strength as compared to specimens loaded more slowly. To quantify the rate of deformation, one can refer to the strain rate [ s − 1] to which the tissue is exposed. In normal activities, bone is subjected to strain rates generally below 0.01 s − 1. Materials like bone, are said to be timedependent or viscoelastic, as shown in Figure 2.4. This means that the stress-strain

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Figure 2.4: The influence of loading rate on the tensile strength and modulus of cortical bone from [92 ].

characteristics and strength properties depend on the applied strain rate. According to [32], however, this rate dependency is relatively weak.

The stress-strain behavior of cortical bone is also strongly dependent on the orientation of the bone microstructure with respect to the loading direction. Cortical bone is stronger and stiffer in longitudinal direction (direction of osteon orientation) than in transverse direction. In addition, bone specimens loaded in a direction perpendicular to the osteons tend to fail in a more brittle manner, with little nonelastic deformation after yielding. Materials such as bone, for which elastic and strength properties are dependent on the direction of applied loading, are said to be anisotropic materials . Therefore, both strain rate and direction of applied load has to be specified when describing material behavior.

Table 2.1: Strength of femoral cortical bone from [43 ]. Mean values from [ 78 ].

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In Table 2.1 the ultimate strength values of adult femoral cortical bone under various modes of loading, in both the longitudinal and transverse direction, are summarized: These indicate that the material strength of bone tissue depends on the type of loading as well as on the loading direction. The compressive strength is greater than the tensile strength in both longitudinal and transverse directions. Transverse specimens are weaker the longitudinal specimens in both tension and compression. The shear strength (determined by torsion tests about the longitudinal axis and reflection shear stresses along transverse and longitudinal planes) is about one-third of the compressive strength. The modulus values for adult femoral cortical bone are shown in Table 2.2.

Table 2.2: Moduli of femoral cortical bone, from [43 ]. Mean values from [ 78 ].

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The longitudinal elastic modulus is about 50% greater than the transverse elastic modulus. The shear modulus for torsion about the longitudinal axis is about one-fifth of the longitudinal modulus.

2.3.2 Cancellous Bone Tissue

Cancellous bone is a complex material with significant heterogeneity. Its elastic and strength properties vary across anatomic sites, with aging and disease. Trabecular bone is classified from an engineering materials perspective as a composite, anisotropic, open porous cellular solid. Like many biological materials, it displays viscoelastic behavior, as well as damage susceptibility during cyclic loading [54, 29].

A critical issue that distinguishes trabecular bone from many other biological tissues is its substantial heterogeneity, which leads to wide variations in mechanical properties. This heterogeneity results from underlying variations in volume fraction, architecture and tissue properties, in that order of importance. Across sites and species, mean values of modulus and strength can differ by more than an order of magnitude (Table 2.3). Substantial loss of mechanical properties also occurs with aging in humans. For example, ultimate stress is reduced by almost 7% and 11% per decade for the human proximal femur and spine, respectively, from ages 20 - 100 [68, 71, 70]. Strength does not decrease significantly until after about age 30 [71, 70]. Trabecular bone is anisotropic in both modulus and strength [71]. Compared with such materials as fiber-reinforced composites, the extent of anisotropy is mild, but its biomechanical significance in terms of whole bone strength or boneimplant performance remains to be quantified. With increasing porosity for example, the compressive strength of trabecular bone becomes more anisotropic.

Table 2.3: Mean values of modulus and ultimate strength for various anatomic sites.

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As depicted in Figure 2.5, the elastic modulus and failure stress of trabecular bone depends primarily on apparent density, the product of volume fraction and trabecular tissue density, see equation (2.1)

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Figure 2.5: Typical stress strain curves for trabecular bone of different densities, from [43 ].

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with ρ tissue being the trabecular tissue density and BV being the bone volume fraction (bone volume over total volume).

However, the precise form of this relationship remains controversial. It appears that most of the controversy stems from the dependence of this relationship on anatomic site [50] and loading direction [21], and from the imprecision introduced by ignoring anisotropy and end-artifact effects [52, 51, 55, 63, 74, 73, 62, 48] in the mechanical tests. End artifacts arise from damage incurred at the ends of machined specimens when they are tested in compression between platens with no other means of attachment to the load frame. If strains are computed from the relative displacement of the platens, substantial systematic underestimation and random errors can occur [55].

The stress-strain properties shown in Figure 2.6 are markedly different from those of cortical bone and are similar to the compressive behavior of many porous engineering materials that are used to absorb energy on impact [38]. The stress-strain curve for trabecular bone exhibits an initial elastic region followed by yield. Yielding occurs as the trabeculae begin to fracture. After that a plateau region follows, which is created as progressively more and more trabeculae fracture. The fractured trabeculae begin to fill the marrow spaces. Further loading of trabecular bone after pore closure is associated with a marked increase in specimen modulus [72].

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Figure 2.6: Stress against strain of trabecular bone specimen under compression from [76 ].

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Figure 2.7: Experimental determination of Young’s modulus E ag ainst density ρ .

From [85 , 14 , 2 , 57 , 64 , 79 ].

Chapter 3 Bone Remodeling - From Nature to Model

Bone is not like inert engineering materials. It undergoes substantial changes in structure, shape and composition according to the mechanical and physiological environment. This adaptive process is known as bone remodeling [30]. Remodeling removes older bone and replaces it with newly formed bone. Microscopic damage is repaired by this process and it prevents accumulation of fatigue damage that could lead to fatigue fracture. It has also been hypothesized that remodeling tunes the skeleton in a way to its mechanical efficiency.

3.1 Remodeling Theory

3.1.1 Difference between Modeling and Remodeling

As the long bones develop, their shafts must grow in diameter as well as length. Growth in diameter is accomplished by periosteal intramembranous ossification. However, as growth occurs, it is not enough just to add material to make the bone organs longer and larger. The bones must also be shaped in various ways. That means that bone must be removed in some places while it is added in others. This

sculpting, combining osteoclastic activity (bone resorption) in some places and osteoblastic activity (bone formation) in others, has become known as modeling. In addition, fatigue damage must be repaired inside the bones and their internal architecture must be adjusted to varying load conditions, both while the skeleton is developing and throughout the remainder of one’s life. This repair involves removal and replacement of bone in particular places by the coupled actions of osteoclasts and osteoblasts working at the same site. This replacement process has become known as remodeling.1

Thus, modeling and remodeling refer to the actions of osteoclasts and osteoblasts in reshaping and replacing portions of the skeleton. They are distinguished in several ways:

1. Modeling involves independent actions of osteoclasts and osteoblasts. Remodeling involves sequential, coupled actions by the two types of cells.
2. Modeling results in change of bone size, shape or both. Remodeling does not usually affect size and shape.
3. The rate of modeling depends on age and is greatly reduced after skeletal maturity. Remodeling is independent of age and occurs throughout life, although it too is substantially reduced after growth stops.
4. Modeling at a particular site is continuous and prolonged, whereas remodeling is episodic, with each episode having a definite beginning and ending.

3.1.2 The Remodeling Process

Bone remodeling only occurs on the internal surfaces of the bone matrix (trabecular surfaces of cancellous bone and the Haversian system of compact bone). Bone can

only be added or removed by bone cells on these surfaces. There are four types of bone cells , which can be classified according to their functions [30].

Osteoblasts are the differentiated mesenchymal cells that produce bone.

They are created at the periosteum layer or stromal tissue of bone marrow.

Osteoclasts remove bone, demineralizing it with acid and dissolving collagen with enzymes. These cells originate from the bone marrow.

Bone lining cells are inactive osteoblasts that are not buried in new bone. They remain on the surface when bone formation stops and can be reactivated in response to chemical and/or mechanical stimuli [69].

Osteocytes are, like bone lining cells, former osteoblasts that are buried in the bone matrix. They are located in lacunae [22] and communicate with the rest of cells via canaliculi.

B asic M ulticellular U nit

The remodeling process is not performed individually by each cell but by groups of cells acting as organized units [30]. These units are named basic multicellular units by Frost [33] or BMUs.

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Figure 3.1: Photomicrograph of an osteonal basic multicellular unit

Each BMU consists of about ten osteoclasts (OC) and several hundred osteoblasts (OB) working together to accomplish the remodeling process. There are three principal stages in a BMU´s lifetime: activation, resorption and formation (ARF). Activation occurs when a chemical or mechanical signal, up today it is not known exactly, causes osteoclasts to form and begin to remove bone somewhere on or in the skeleton. The osteoclasts resorb a volume of bone in the form of a ditch on bone surfaces or tunnel in compact bone about 200 µ m in diameter that proceeds along the surface or through the cortex, moving about 40 µ m/day. After the osteoclasts have passed a certain point, osteoblasts are differentiated from mesenchymal cells over a period of several days and begin to replace the resorbed tissue. Formation is much slower than resorption. In humans the total remodeling period is about 4 months, about 3 weeks for resorption and 3 months for refilling, respectively.

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Figure 3.2: Schematic sketch of an osteonal BMU. Cross-sectional view at the bottom right.

The package of new bone produced by an BMU is called a bone structural unit, whether the remodeling occurs in cortical or trabecular bone or on an endosteal or periosteal surface. Thus, a secondary osteon in cortical bone is a bone structural unit. There is no common name for trabecular bone structural units, although they are sometimes called trabecular osteons or bone packets.

The Six Phases of an Osteon’s Lifetime The A-R-F process may be further divided into six separate phases, which are always sequential in normal bone remodeling. These phases are depicted in

Activation Osteonal BMUs may originate on periosteal or endosteal bone surfaces, or along the walls of Haversian canals. Various activation signals are thought to be present in the skeleton, including chemical, mechanical, and electrical signals, but just how or where they affect cells, and what sorts of intermediate messengers may exist, are questions still being resolved as already mentioned. Differentiated cells must be recruited from precursor cell populations before any bone resorption or bone formation can occur. This process occurs only in the origin of the BMU. The osteoclasts produced at this time appear to survive for weeks, and perhaps for the entire duration of the tunneling by the BMU. Thus the "opening up" of a resorption space on a cross section is the passage of osteoclasts through a particular plane of observation and not actually "activation" because it does not involve the initial recruitment of osteoclasts. The original activation process occurred earlier, in either case, however, the time required is about 3 days.

Resorption Bone resorption is primarily the task of newly created osteoclasts, which are mobile, multinucleated cells closely related to macrophages. They attach themselves to a bone surface, form a peripheral seal, and break down the bone within the sealed area by means of enzymes and other chemicals. Moving longitudinal osteoclasts resorb bone at a rate of 40 µ m/day on a so-called cutting cone, an ellipsoidal surface about 200 µ m in diameter and about 300 µ m long. However, it is difficult to be certain where this region stops and the following region begins, so the length of each is debatable. The final stages of the dissolution of the bone tissue appear to be carried out by mininuclear cells following along behind the osteoclasts. Nevertheless, osteoclasts are the primary resorbing cells within the BMU. Although osteoclasts exhibit considerable mobility at trabecular bone surfaces, apparently ranging over resorption territories several times their contact area, in osteonal cutting cones they seem to dig straight ahead in a tightly bunched configuration.

Reversal The transition from osteoclastic to osteoblastic activity takes several days and results in a cylindrical space lying between the resorptive region and the refilling region. The length of this region my vary considerably, depending on the lag between the resorptive and formative phases. In a complete secondary osteon, the cement line coincides with the location of the bone surface during the reversal period, the cement line is also known as the reversal line. in humans, about 30 days are required for the resorption and reversal periods combined.

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Figure 3.3: The Six Phases of an Osteon’s Lifetime. a) Activation AC, b) Resorption RES, c) Reversal REV, d) Formation FO, e) Mineralization MI, f) Quiescence QU. from Ott with permission [75 ]

Formation Osteoblasts appear around the periphery of the tunnel and refilling begins. Osteoblasts lay down concentric lamellae at a decreasing rate as refilling progresses, but the average radial closure ratio is 1-2 µ m/day. Because osteoclasts and osteoblasts require nourishment, the tunnel is not completely refilled, but leave a central passageway for a vascular loop to support the metabolism of the BMU and bone matrix osteocytes, and to carry calcium and phosphorus to and from the bone when necessary. This passageway is the Haversian canal, and it is typically 40-50 µ m in diameter in humans. The formation phase in adult humans averages about 3 months. In some respects, the refilling process is more amenable to analysis than the resorption process, partly because newly mineralizing bone surfaces can be labeled with flourochrome markers. The basic processes of bone formation within BMUs are clear, but many questions remain about the details.

Mineralization Following deposition of the unmineralized organic matrix of bone, or osteoid, mineral is deposited within and between the collagen fibers. This process is delayed by a period of time known as the mineralization lag time, which is normally about 10 days. This delay is manifested by a layer of osteoid between the osteoblasts and mineralized bone. Once begun, approximately 60% of the mineralization of osteoid occurs during the first few days (Parfitt, 1983); this is called primary mineralization. The remainder of the mineral is added at a decreasing rate for 6 months or so during secondary mineralization. The bone in new, incompletely mineralized osteons can exhibit very different mechanical properties from that in older osteons.

Quiescence After the tunneling and refilling processes are completed, the osteoclasts disappear from the scene, and the osteoblasts become osteocytes or Haversian canal lining cells or disappear. A period of relative inactivity ensues during which the secondary osteon and its associated cells carry on their mechanical, metabolic, and homeostatic functions.

Remodeling Cycle Duration

Because the wall of the tube of new bone laid down within the excavated tunnel is about 80 µ m thick, and the average accretion rate is about 1 µ m/day, the time to refill is roughly 80 days. That is, if it were possible to observe a bone cross section in vivo, it would take about 80 days for the initial resorption cavity to be converted into a completed osteon. Before refilling, about 30 days would have passed between the time when the osteoclasts initially began to open the resorption cavity and the appearance of the osteoblasts. Historically, these resorption and refilling times have been called sigma times because Frost originally used the symbol to represent them. The total sigma is approximately 4 months in human cortical bone.

3.1.3 Types of Remodeling

The remodeling process can be distinguished depending on the location or on the type of bone, where it occurs.

Osteonal Remodeling

When a BMU, tunnels through compact bone, secondary osteon is created. Figure 3.1 shows the head of an osteonal BMU in longitudinal section, tunneling to the left. One can see a few osteoclasts on the resorbing surface (OK), osteoblasts (OB) in the center and osteoid line the refilling surface. The front end of a BMU contains a capillary "bud" (B) to supply nutrients, and probably to supply the progenitor cells for osteoclasts and osteoblasts. Because osteonal BMUs become isolated deep within the cortex, this vascular supply must be maintained. Therefore, the tunnel cannot be entirely refilled, and each BMU leaves a new Haversian canal in the bone. In each Haversian canal there are two capillaries: a "supply" and a "return" vessel. These vessels connect with the vasculature in the medullary canal or on the periosteal surface.

In human adults, about 5% of compact bone are replaced by osteonal BMUs each year. Figure 3.4 shows how the rate of remodeling changes with age in the human rib. It is very high in children, is reduced in young adults, then rises and falls again in older individuals.

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Figure 3.4: BMU activation rate vs. age for human ribs. (From data by [ 34 ])

Trabecular Remodeling

Trabeculae are remodeled similarly, except that the BMUs work on their surfaces, digging and refilling trenches. Imagine the lower half of the BMU in Figure 3.1 moving over the surfaces of the trabeculae. Because a BMU is about the same diameter as a trabecula, it cannot tunnel longitudinally within an trabecular strut without constantly breaking out along its sides. Human adult trabecular bone is replaced by BMUs at a relatively high rate, about 25% each year. In fact, the rate of bone turnover varies widely throughout the skeleton.

Endosteal and Periosteal Remodeling

In principle, remodeling can also occur on endosteal and periosteal surfaces. This kind of remodeling could be responsible for the observation that long bones expand radially with age in adults as well as children. This expansion would be accomplished by arranging for formation to exceed resorption on the periosteal surface, and vice versa on the medullary canal surface.

The Rodan Theory Rodan (1992) developed a theory of BMU activation based on the hypothesis that bone lining cells are responsible for initiating new BMUs. That is not to say that lining cells differentiate to osteoclasts, but that they may be the target of various activation signals, responding in ways that lead to the appearance of osteoclasts. This idea springs from the observation that osteoblasts-turned-lining cells possess receptors for parathyroid hormone and 1 − α , 25 (OH)2 vitamin D 3, but osteoclasts and their precursors apparently do not. Furthermore, lining cells respond to such signals in ways consistent with osteoclast activation. First, they change shape, contracting so as to expose the bone surface to the osteoclasts. Also, in cell culture mechanical and electrical stimuli increase cyclic adenosine miniphosphate (cAMP) activity in osteoblast-like cells, which stimulates them to release prostaglandins, a potent intercellular messenger. Thus, the Rodan theory provides an interesting way for osteoclasts to respond to hormonal and physical signals to which they are not sensitive. It is also a theory that appeals to a sense of symmetry and simplicity; it provides a cytologic loop in which the final cell in the remodeling sequence becomes the initiator of the next remodeling cycle.

3.2 Theories from the Beginning up to now

3.2.1 Mechanically Excited Bone Adaption Theories (1865 - 1920)

Theories of how mechanical stimulus affected bone adaptation begin in the middle of the 19th century. Von Meyer, a German anatomist, studied trabecular orientation in the proximal femur in 1867. Karl Culmann, a Swiss engineer (1821 - 1881), observed basic similarities between Von Meyer’s trabecular drawings and his principal stress lines of a crane, Figure 3.5. Together they postulated trajectorial theory of trabecular bone: Trabeculae are oriented along principal.

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Figure 3.5: Stress trajectories in curved Culmann crane (left) compared with a schematic representation of the trabecular pattern in the proximal femur, from Wolff, 1870

Up to this point there was no attempt to state that the creation of trabecular structure is influenced by mechanical stimulus, only that the orientation of the trabeculae coincides with principal stress directions [46].

In addition to Von Meyer and Culmann, Julius Wolff, a German anatomist (1836 - 1902), postulated not only that the trabeculae were aligned with principal


1 These terms were coined by Frost.


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Finite Element Analysis Bone Remodeling Implementation Algorithm MATLAB ANSYS Master Science Computational Mechanics



Title: Finite Element Analysis of Bone Remodeling. Implementation of a Remodeling Algorithm in MATLAB and ANSYS