Bone Remodeling Process as an Optimal Structural Design ...elasticity theory and PID control) that...

Sapienza University of Rome

Ph.D. Thesis Department of Mechanics and Aeronautics Engineering

Bone Remodeling Process as an

Optimal Structural Design

Michele Colloca Supervisor Prof. Ugo Andreaus

Tutor Prof. Giovanni Santucci

Rome, Italy, November 2009

To Simona

A human being is part of a whole, called by us Universe. We experience ourselves as something separated from the rest…an optical delusion of our consciousness. This delusion is a kind of prison for us.

Albert Einstein

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ABSTRACT The functional adaptation of bone to mechanical usage implies the existence of a physiological control process: once the structure has sufficiently adapted, the feedback signal is diminished and further changes to shape and properties are stopped. In bone tissue it has been widely accepted that mineral component is resorbed in regions exposed to low mechanical stimulus, whereas new bone is deposited where the stimulus is high. This process of functional adaptation is thought to enable bone to perform its mechanical functions with a minimum of mass and with the strength necessary to support mechanical loads associated with daily activity and to protect internal organs. This premise can be expressed as a global, multi-objective optimization problem in which stiffness and mass are conflicting goals. Maximizing stiffness is equivalent to minimize the compliance or minimize the strain energy in the bone. From this point of view, the bone remodeling process is analogous to the topology optimization in structural design including the cellular automaton (CA) technique and the solid isotropic material with penalization (SIMP) approach. The process of bone remodeling can be analytically described, integrated with the finite element method and numerically simulated. With a proper control strategy, an iterative process drives the overall modelled structure to an optimal configuration. The optimized controllers in this investigation regard proportional, integral and derivative strategies. A local remodeling rule iteratively updates the value of the modulus (or relative mass) of the cellular automata, in which the structure is discretized, individually based on the difference between a current stimulus value and a target value, relative to the external load. The purpose is to obtain a constant,

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optimal value for the strain energy per unit bone mass, by adapting the mass density. In medical applications, a major problem threatening the long-term integrity of total hip replacement is the loss of proximal bone often found around non-cemented, press-fitted and bonded implants. The aim of this thesis is to perform a two-step procedure of control and optimization in order to solve an optimization problem of lightweight stiffened structures; various objective functions and constraints are considered so that different design requirements are compared. The selection of the optimized parameters in the evolution rules, not yet faced in an in-depth study, is successfully studied and the convergence is improved. The set of optimal parameters includes the control gains, the target of the error signal and the weight of the cost index J1, defined as the sum of the total energy and mass of the domain of interest. Two-dimensional bone samples, subjected to an in-plane constant and linear loading, are analyzed. The bone samples are discretized in 25, 625 and 1250 cellular automata, but the proposed model is suitable to be used not only in bone mechanics but in many other fields of artificial materials discretized in whatever number of elements. The contents of the Ph.D. thesis appear as follows: in Chapter 1 the fundamental cellular mechanisms responsible for bone remodeling are briefly described; Chapter 2 contains an overview on several important bone remodeling models (optimization, phenomenological and mechanistic models) of the last forty years; Chapter 3 is made up of theoretical and numerical tools (FEM analysis, topology optimization of structures, cellular automaton model, adaptive elasticity theory and PID control) that are transferred into a unified numerical code and used for modeling bone adaptation effects. Chapter 4 includes the complete control and optimization procedure that predicts an optimal distribution of mass and energy of a bone structure under specific constraints and loading conditions. To conclude, final remarks of the proposed study are reported.

I would like to thank my advisor, Prof. Ugo Andreaus, for his guidance and for providing the funding necessary to perform this investigation. I would also like to extend my gratitude to Daniela Iacoviello, Assistant Professor, for her support and ideas.

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CONTENTS ABSTRACT (i) NOMENCLATURE (v) CHAPTER 1 Cellular Mechanisms in Bone

Remodeling 1.1 Introduction (1) 1.2 Bone Cells (2) 1.3 Proposal for Bone Adaptation Process (6)

CHAPTER 2 Overview on Several Important Bone

Remodeling Models 2.1 Introduction (10) 2.2 Optimization, Phenomenological and Mechanistic

Models (11) 2.3 Influence of Mechanical Stimuli on Adaptation

Response (12) 2.4 State of Art (14)

CHAPTER 3 Theoretical and Numerical Tools for Bone Tissue

Adaptation 3.1 Introduction (40) 3.2 FEM Analysis (41) 3.3 Finite Element-Based Optimization Approach :

Topology Optimization of Structures (44) 3.3.1 Non Linear Constrained Optimization Problems (49)

3.4 Cellular Automaton Model (50) 3.4.1 Cellular Automata as Osteocytes: Phenomenological

and Optimization Approaches (54) 3.5 Adaptive Elasticity Theory (55)

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3.6 PID Controller (56)

CHAPTER 4 Control and Optimization Procedure for Bone Lightweight Stiffened Structures

4.1 Introduction (64) 4.2 Materials and Methods (65)

4.2.1 Generalities (65) 4.2.2 Control Strategy (69) 4.2.3 Optimization Strategy (74) 4.2.4 2-D Models (79) 4.3 Numerical Results (82)

4.3.1 Bone Sample (25 cellular automata) (83) 4.3.2 Bone Sample (625 cellular automata) (86) 4.3.3 Michell-type Structure (1250 cellular automata) (89)

4.4 Discussion (94) CONCLUSIONS (98) APPENDIX Implementation of the Control and Optimization

Procedure (100)

v

Nomenclature

n Number of cellular automata

N Number of the neighbouring cellular automata

E Young’s modulus

iE Young’s modulus of a cellular automaton at the discrete location i

0E Initial value of the Young’s modulus

minE Minimum value of the Young’s modulus

maxE Maximum value of the Young’s modulus

iy Relative elastic modulus of a cellular automaton at the discrete location i

ν Poisson’s ratio

ρ Apparent density

iρ Apparent density of a cellular automaton at the discrete location i

minρ Minimum value of the apparent density

maxρ Maximum value of the apparent density

0ρ Initial value of the apparent density

γ Power relating Young’s modulus and bone apparent density

x Relative mass

0x Initial value of the relative mass

ix Relative mass of a cellular automaton at the discrete location i

M Total mass

0M Initial value of the total mass

maxM Maximum value of the total mass

maxm Maximum value of a cellular automaton mass

im Mass of a cellular automaton at the discrete location i

S Mechanical stimulus *S Equilibrium value (target) of the mechanical stimulus

F Objective function

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SED Strain energy density *SED Target of the strain energy density

iSED Strain energy density of a cellular automaton at the discrete location i

optSED* Optimal value of the strain energy density target

U Total strain energy

0U Initial value of the total strain energy

iU Strain energy of a cellular automaton at the discrete location i

1J Cost index sum

2J Cost index product

optJ1 Optimal value for the cost index 1J

optJ2 Optimal value for the cost index 2J

ω Weight parameter of the cost index 1J

optω Optimal value for the weight parameter of the cost index 1J

Pc Proportional gain

Ic Integral gain

Dc Derivative gain

ρPc Proportional gain for the apparent density evolution rule

ρIc Integral gain for the apparent density evolution rule

ρDc Derivative gain for the apparent density evolution rule

PEc Proportional gain for the Young’s modulus evolution rule

IEc Integral gain for the Young’s modulus evolution rule

DEc Derivative gain for the Young’s modulus evolution rule

Pxc Proportional gain for the relative mass evolution rule

Ixc Integral gain for the relative mass evolution rule

Dxc Derivative gain for the relative mass evolution rule

Pyc Proportional gain for the relative elastic modulus evolution rule

Iyc Integral gain for the relative elastic modulus evolution rule

Dyc Derivative gain for the relative elastic modulus evolution rule

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optPc Optimal value of the proportional gain

optIc Optimal value of the integral gain

optDc Optimal value of the derivative gain

ie Error signal of a cellular automaton at the discrete location i

je Error signal of a cellular automaton at the discrete location j

ie Effective error signal of a cellular automaton at the discrete location i

iη Strength of error signal of a cellular automaton at the discrete location i

jη Transmission efficiency of error signal from a cellular automaton at the discrete location j to a cellular automaton at the discrete location i

yσ Yield stress

Misesσ Mises stress

uσ Ultimate stress

ijσ Stress tensor

ijε Strain tensor

ε Threshold for test convergence

p Penalization power

l Half width of the equilibrium zone

P Penalty term in the cost indices 1J and 2J

Chapter 1 Cellular Mechanisms in

Bone Remodeling

1.1 Introduction 1.2 Bone Cells 1.3 Proposals for Bone Adaptation

Process

1.1 Introduction Bone cells, like many other cell types, respond to alterations in their external environment. The physiological mechanism by which the mechanical loading applied to bone is sensed by the tissue and the mechanism by which the sensed signal is transmitted to the cells that accomplish deposition, removal, and maintenance have not been identified. The mechanosensing processes seem to sense and to respond to physical loadings. These processes include stretch and voltage activated ion channels, processes cyto-matrix sensation transduction, cyto-sensation by fluid shear stresses, cyto-sensation by streaming potentials, and exogenous electric field strength. The term mechanosensory is employed to mean both mechanoreception and mechanotransduction phenomena. Mechanoreception is the term used to describe the process that transmits the informational content of an extracellular mechanical stimulus to a receptor cell. Mechanotransduction is the term used to describe the process that

2 Bone Remodeling Process as an Optimal Structural Design

transforms the mechanical stimuli content into an intra-cellular signal.

1.2 Bone Cells Bone is a multifunctional, highly dynamic mineralized connective tissue that undergoes significant turnover. Osteoprogenitor differentiation is one of the key processes responsible for bone formation and remodeling. During this process, a subpopulation of mesenchymal progenitors undergoes osteoblast lineage commitment and matures through a series of differentiation steps. In response to appropriate signals the progenitor cells proliferate, then secrete extracellular matrix that will mineralize, embedding the cells within the matrix. The osteocytes, engulfed in this mineralized matrix, represent the terminal differentiation stage of osteoblast. They are the most abundant cellular component of mature mammalian bones and constitute as much as 95% of all bone cells. Osteocytes are thought to be mechanosensors and may coordinate the remodelling process carried out by osteoblasts and osteoclasts. Frost (1990) summarizes these differences by describing four different bone envelopes (trabecular, periosteal, endocortical, and osteonal) that represent different biological environments. The adaptation response of bone within these four bone envelopes could be expected to be different, both in terms of the remodeling equilibrium values and the speed of the adaptive responses (Hart 2001). At the microscopic level there are five types of bone cells associated with the growing and adult skeleton: 1) the osteocytes; 2) the osteoprogenitor cells; 3) the osteoblasts; 4) the bone-lining cells; 5) the osteoclasts. Osteoblasts and the osteocytes (Fig. 1.1) are extensively interconnected by the cell processes forming a Connected Cellular Network (CCN).

Cellular Mechanisms in Bone Remodeling

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Fig. 1.1 Scanning electron image. Connected Cellular Network: osteocytes, housed in the lacunae of the mineralized matrix, communicate each other with several cytoplasmatic processes. (From Lu et al. 2007) The CCN is the site of intracellular stimulus reception, signal transduction, and intercellular signal transmission. Osteocytes derive from osteoblasts which are entrapped in their own matrix and survive there, mutually connected by a network of canaliculi. This network also connects with the lining cells, which also derive from osteoblasts that cover the trabecular surface. Hence, osteoblasts, osteocytes and lining cells form a syncytium, which is well-equipped for signal transduction. Osteoclasts are derived by osteoprogenitor cells. They are multinucleated giant cells responsible for the resorption of bone (Fig. 1.2).


Fig. 1.2 Osteoclasts are multinucleated cells. They have several cytoplasmatic protruding filaments to mediate extracellular resorbing of bone matrix. (From Arnett and Dempster 1986)

The osteoblasts directly regulate bone deposition and maintenance, and indirectly regulate osteoclastic resorption. They are cells of mesenchymal origin, responsible for bone accretion though a tightly regulated spatiotemporal sequence that includes a regulated proliferation of pre-osteoblasts (Fig. 1.3).


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Fig. 1.3 SEM image of a dense 3D tissue about 22 µm tick. Osteoblasts synthesize bone matrix and become osteocytes. They also induce osteoclast differentiation. (From Krishnana et al. 2009) The true biological stimulus, although much discussed, is not precisely known. The osteocyte has been suggested as the stimulus sensor, the receptor of the stimulus signal histologic and physiologic data are consistent with this suggestion. In particular, fluid flow in the bone canaliculi may be the source of strain-generated potentials (Cowin et al. 1995) that may affect the


membranes as osteocytes, osteoblasts, and bone-lining cells. It is not yet clear which one or combination of these mechanics is primarily responsible for converting mechanical usage to cellular activities. Despite these uncertainties, simulation models require assumptions about the mechanical stimulus and are often simply cast in terms of portions of the strain or stress history. Thus, they do not generally address the crucial steps involved in the transduction of the mechanical signals to cellular stimuli and responses although they may help guide future experimental investigations (Hart 2001).

1.3 Proposals for Bone Adaptation Process The bone tissue domain or region over which the stimulus is felt is called the sensor domain. When an appropriate stimulus parameter exceeds or falls below threshold values, loaded tissues respond by the triad of bone adaptation processes: deposition, resorption, and maintenance (Cowin and Doty 2007). In Huiskes et al. (2000) the proposed regulatory process involves: a) strain-energy density (SED) rate in the mineralized tissue, as produced by a recent loading history that triggers feed-back from the external forces to bone metabolism is a typical (Mosley and Lanyon 1998; Carter et al. 1987; Mullender and Huiskes 1995; Turner et al. 1995; Rubin and Lanyon 1987) b) osteocytes that react to the loading in their local environments by producing a biochemical messenger in proportion to the typical SED rate (Lanyon 1996; Skerry et al. 1989; Klein-Nulend et al. 1995; Burger and Klein-Nulend 1999; Cowin et al. 1991) c) biochemical messenger produced by the osteocytes causes signals to be dissipated through the osteocytic network towards the bone surface, where they create an osteoblast recruitment stimulus (Burger and Klein-Nulend 1999). The strength of this signal, which is produced by all osteocytes in the environment, stimulates osteoblast


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recruitment and bone formation as long as it exceeds a threshold value. The nature of such a signal is not well known, although both electric current and ion transport have been suggested (Burger and Klein-Nulend 1999; Cowin et al. 1991) d) osteoclast activation which is considered to be regulated either by the presence of microcracks within the bone matrix or by disuse. Osteoclasts are activated by cytokines, which are produced by osteoblasts, and hence probably also by lining cells at the bone surface (Chambers and Fuller 1985). This activation process could originate from a signal of mechanical disuse in the bone matrix, sent by osteocytes (Burger and Klein-Nulend 1999; Noble et al. 1997). It is likely that disuse and microcracks factors act concurrently in attracting and activating osteoclasts (Huiskes 2000). The following chapter contains an overview on several important bone remodeling models (optimization, phenomenological and mechanistic models) of the last forty years.

References Arnett TR and Dempster DW, 1986. Effect of pH on Bone Resorption by Rat Osteoclasts in vitro, Endocrinology 119: 119-124. Burger EH and Klein-Nulend J, 1999. Mechanotransduction in bone—role of the lacuno-canalicular network, FASEB J. 13, S101–S112. Carter DR, Fyhrie DP and Whalen RT, 1987. Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy, J. Biomech 20, 785–794. Chambers TJ and Fuller K, 1985. Bone cells predispose bone surfaces to resorption by exposure of mineral to osteoclastic contact, J. Cell Sci. 6, 155–165. Cowin SC and Doty SB, 2007. Tissue Mechanics, New York : Springer.


Cowin SC, Moss-Salentijn L and Moss ML, 1991. Candidates for the mechanosensory system in bone, J. Biomech. Eng. 113, 191–197. Cowin SC, Weinbaum S and Zeng Y, 1995. A case for bone canaliculi as the anatomical site of strain generated potential, J. Biomech, 28(11), 1281 – 1297. Frost HM, 1990. Skeletal structures adaptations to mechanical usage (SATMU): 2.Redefining Wolff’s law: the remodeling problem, Anat. Rec., 226(4), 414-422. Hart RT, 2001. Bone modeling and remodeling: Theories and computation. In S. C. Cowin (ed.) Bone Mechanics Handbook, CRC Press. Huiskes R, 2000. If bone is the answer, then what is the question?, J. Anat. 197, 145–156. Huiskes R, Ruimerman R, Van Lenthe GH and Janssen JD, 2000. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone, Nature 405, 704-706 Klein-Nulend J, Van der Plas A, Semeins CM, Ajubi NE, Frangos JA, Nijweide PJ and Burger EH, 1995. Sensivity of osteocytes to biomechanical stress in vitro. FASEB J. 9, 441–445. Krishnan V, Dhurjati R, Vogler EA and Mastro AM, 2009. Osteogenesis in vitro: from pre-osteoblasts to osteocytes, In Vitro Cell Dev Biol-Animal DOI 10.1007/S11626-009-9238 Lanyon LE, 1996. Using functional loading to influence bone mass and architecture: objectives, mechanisms and relationship with estrogen of the mechanically adaptive process in bone, Bone 18, 37S–43S. Lu Y, Xie Y, Zhang S, Dusevich V, Bonewald LF and Feng JQ, 2007. DMP1-targeted Cre Expression in Odontoblasts and Osteocytes, J Dent Res 86(4):320-325. Mosley JR and Lanyon LE, 1998. Strain rate as a controlling influence on adaptive modeling in response to dynamic loading of the ulna in growing male rats, Bone 23, 313–318. Mullender MG and Huiskes R, 1995. A proposal for the regulatory mechanism of Wolff ’s law, J. Orthop. Res. 13, 503–512. Noble BS, Stevens H, Reeve J and Loveridge N, 1997. Identification of apoptotic changes in osteocytes in normal and pathological human bone, Bone 20, 273–282.


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Rubin CT and Lanyon LE, 1987. Osteoregulatory nature of mechanical stimuli: function as a determinant for adaptive bone remodeling, J. Orthop. Res. 5, 300–310. Skerry TM, Bitensky L, Chayen J and Lanyon LE, 1989. Early strain-related changes in enzyme activity in steocytes following bone loading in vivo, J. Bone Miner. Res. 4, 783–788. Turner CH, Owan I and Takano Y, 1995. Mechanotransduction in bone: role of strain rate, Am. J. Physiol. 269, E438–E442.

Chapter 2 Overview on Several

Important Bone Remodeling Models

2.1 Introduction 2.2 Optimization, Phenomenological

and Mechanistic Models 2.3 Influence of Mechanical Stimuli on

Adaptation Response 2.4 State of Art

2.1 Introduction Living bone is continually undergoing processes of growth, reinforcement, and resorption that are collectively termed strain adaptation, or bone remodeling or functional adaptation. The functional adaptation processes in living bone are the mechanisms by which the bone adapts its overall structure to changes in its load environment. Understanding and predicting the adaptation properties of living bone are particularly important for proper design of prosthetic devices that contact bone tissue. These devices include fracture fixation plates, surgical screws, and artificial joints. A prosthesis places a stress on adjacent bone tissue, and if this stress is different from the stress that the bone tissue is accustomed to, the bone will


remodel. It is possible that the remodeled bone tissue will be weaker and lead to failure of the surgical procedure. It has been suggested that femoral resorption could be the predominant factor in implant stem fracture of a total hip replacement. The objective of a theory of tissue adaptation is to provide numerical models or the basis for boundary problems that will predict the adaptive behavior of the tissue under altered environmental load. The adaptation of bone to environmental loads is a fundamentally nonlinear process because the properties of the bone are changing as the loading of the bone changes. The changing load and changing bone properties interact, and the interaction is the source of the nonlinearity. This nonlinearity characterizes all tissue adaptation processes and therefore all models of tissue adaptation processes (Cowin and Doty 2007).

2.2 Optimization, Phenomenological and Mechanistic Models Hart (2001) writes in his paper that the models that have been developed for simulating the functional adaptation of bone can be classified into three distinct groups: optimization, phenomenological and mechanistic modelling. The studies with the application of optimization theory give insight into bone as a mechanical structure – and may especially relevant for simulating evolutionary adaptation but when they are applied to an individual’s skeletal adaptation, few deficiencies become apparent. For instance, optimization assumes that the adaptation not only seeks to achieve a stated goal (e.g. minimize mass, maximize strength) but that there are physiological processes in place that are directed toward achieving the stated goals. This assumption implies a high level of coordination and environmental awareness without a strong foundation of evidence. Then, implementation of optimization strategies focuses upon achievement of adaptive goals, but not upon the physiological process of the adaptation. Therefore, they are ill-suited to providing strategies that can develop and address questions to advance understanding or manipulation of adaptive processes. Finally, by

Overview on Several Important Bone Remodeling Models 12

subscribing to an objective strategy to study bone adaptation, fundamental questions – including the degree of site and time dependence – cannot critically be investigated (Hart 2001) The phenomenological theories are based on the postulate that exists a causal relationship (the constitutive equation) between the rate of deposition or resorption of the bone matrix at any point and a stimulus, generally a measure of the mechanical loading history at that point in the bone matrix (Cowin and Doty 2007). These theories do not attempt to explain the cellular mechanisms by which the bone senses the mechanical loading and begins to deposit or resorb tissue (Cowin and Doty 2007) but they are useful in developing simulations and predictions of the adaptive bone consequences based on changed mechanical loading to improve implant design or to treat some patients (Hart and Fritton 1997). Finally, the mechanistic modelling goes beyond the cause – and – effect – based phenomenological models and instead seeks a detailed understanding of the steps and chemical and biological mechanisms involved in changing bone properties and architecture (Hart 2001). In this sense, in the study of bone remodeling, it is important to elucidate the interplay of genetics, hormones, drug therapy and the mechanical environment. A cellular level physiological or mechanistic understanding of the bone tissue mechanosensory system will provide insight into the maintenance of the long-term stability of bone implants, the physiological mechanism underlying osteoporosis, and the long-duration spaceflights and long-term bed rest (Cowin and Doty 2007).

2.3 Influence of Mechanical Stimuli on Adaptation Response Cowin (1984) argues that the stimulus is based on strain – a primary and directly measurable physical quantity representing local deformation – and not on stress, which is a well-defined but not directly measurable quantity. In addition, the concept of a fading memory for the mechanical strain history may be needed to model the


time-dependent effects because it seems that the more recent events in the loading environment of the bone are more important than events into the past. More generally, the mechanical stimulus almost certainly depends upon some aspect of the “strain history” – a concept that incorporates all aspects of time-dependent, local deformation measures. Included under this general description would be many of the specific stimuli that have been proposed as important regulators of functional adaptation: peak strain, strain rate, average strain, spatial strain gradients, and strain energy density. Finally, as an alternative to the strain - or stress-based measures, a possible mechanical manifestation of function that leads to bone adaptation is the local damage of bone. In that case, damage in the form of micro-cracks is repaired so that functional adaptation might be considered as a special case of the general fracture repair process of bone (Burr et al. 1990; Cowin and Doty 2007). The measure of the mechanical loading history at a point that is the true biological stimulus, although much discussed and written about, is not precisely known. Two dozen possible stimuli were compared in a combined experimental and analytical approach (Brown et al. 1990). The data supported strain energy density, longitudinal shear stress, and tensile principal stress or strain as stimuli; no stimulus that could be described as rate dependent, nor any related to cracks and microdamage, were among the two dozen possible stimuli considered in this study. The case for strain rate as a bone adaptation stimulus has been building over the last quarter century (Cowin and Moss 2000). Animal studies and experiments have quantified the importance of strain rate over strain as a strain adaptation stimulus. The case for fatigue microdamage as a bone adaptation stimulus is an old suggestion that has been given renewed impetus by the recent discovery of microcracking in bone using histological staining techniques. On the other hand, another study of fifty different mechanical parameters (six local strain components, six local stress components, three principal strains, three principal stresses, three strain invariants, three stress invariants, maximum shear strain, maximum


shear stress, strain energy, twenty-one spatial strain gradients of the six strain gradients and dilatation in the three local directions, the mechanical intensity scalar, Ê, and a signed strain energy density) shows that many of these parameters that have been used in adaptation simulations are highly correlated to each other (Oden and Hart 1995). This correlation of mechanical parameters helps masking the true regulators of the adaptation process so that distinct starting assumptions about the nature of the mechanical parameters can result in similar simulation solutions. To be more useful, the modelling assumptions should be more tightly coupled to biological parameters that can be histologically measured and manipulated (Martin 1985). So, modeling mechanical stimuli – which occur on the order of seconds or tenths of a second, including information about the loading repetitions and frequency, and accounting for changes that take place over a period of weeks or months – is a challenge (Cowin and Doty 2007). 2.4 State of Art Many of the theoretical models use the concept of an error signal to drive the adaptation process. Implicit in the use of an error-driven theory is the assumption that there is a remodeling equilibrium condition in which the mechanical environment of bone tissue is suitably adapted and not undergoing a process of net deposition or resorption. Only deviations in the mechanical environment from this remodeling equilibrium condition would initiate adaptive activity. Some models have used a single value to represent the mechanical remodeling equilibrium state; others have used the concept of a minimum signal or a lazy zone to represent a range of values for which the bone would also be quiescent (Frost 1983; Carter 1984; Huiskes et al. 1987).


Fyhrie and Carter (1986) developed a theory to predict the apparent density (or volume fraction) of bone and the trabecular orientation for a continuum description of trabecular bone material. They assumed that bone is an anisotropic material that tends to optimize its structural integrity while minimizing the amount of bone present. For trabecular adaptation simulations, they introduced the notion of an objective function dependent upon the apparent density ρ, the orientation θ, and the stress T. Then, two special cases were examined: the case where the net bone remodeling objective is based upon the strain energy and the case where the objective is based upon the failure stress. They developed the apparent level mathematical conditions that must be satisfied by the cancellous bone structure for it to be mechanically optimal for a given loading environment. However, the concept was not developed to the point of relating loading to the adaptation process as had been done in the case of bone density (Hart 2001). Frost (1987) introduced the mechanostat theory that predicts when modeling and remodeling will be activated. The mechanostat theory begins with the definition of a minimum effective strain (MES) to be exceeded in order to activate an adaptive response (Frost 1983). Frost suggests that there is an equilibrium range of strain values (dead zone or lazy zone) which will evoke no response. Strains above this range will increase bone mass and strains below this range will cause bone loss. The equilibrium range was defined as between 200 and 2500 µm/m for compression and between 200 and 1500 µm/m for tension. In this model, strains over 4000 µm/m (tension and compression) cause damage and, consequently, woven bone formation. Carter et al. (1987) argue that the role of mechanical influence upon the biological processes is more important than has previously been recognized. In their work the goals are to introduce approaches to predict bone apparent density using a general method of defining loading history which encompasses the single loading strength


optimization, a technique based on fatigue damage accumulation, and a bone strain energy density interpretation of loading history. They hypothesize that bone functional adaptation is the consequence of a cellular response to the density of strain energy density transferred to the biological tissues. The derived relationships between apparent density and stress history or strain energy density history are:

[ ] mmiin 2

1

∑∝ σρ stress approach

[ ] kkiiUn

1

∑∝ρ strain energy approach

[ ] mmenergyin 2

1

∑∝ σρ strain energy approach

[ ] bbiin 2

1

∑ ∆∝ σρ fatigue damage approach

where ni is the number of loading cycles (with i designating a specific loading case), σi is the effective stress, m, k, b are constants, Ui is the strain energy density and σenergy is the energy (effective) stress. These equations are more general than the relationship based on fatigue damage since the power exponents are, as of yet, undetermined values. The applications of the multiple loading criteria represented by equations above described are currently hampered by the lack of knowledge concerning the value of m (or k). In this paper the considerations of loading history and energy transfer density in biological tissues were made in the context of the functional adaptation of trabecular bone but the same concepts can be applied to any situation in which repeated mechanical loading may influence the growth, adaptation, injury, or aging of living tissue. A phenomenological approach for shape changes has been developed by Huiskes et al (1987). The subject of this study was the development and application of computer-simulation methods to predict stress-related adaptive bone remodeling, in accordance with


Wolff’s law. Their models were applied to investigate the relation between stress shielding and bone resorption in the femoral cortex around intramedullary prostheses, such as in Total Hip Arthroplasty (THA) (Huiskes et al. 1987). The authors proposed the following description of modulus changes:

−−

+−=

])1([

0

])1([

ne

ne

UsUC

UsUC

dt

dE

n

nn

n

UsU

UsUUs

UsU

)1(

)1()1(

)1(

−<+≤≤−

+>

where E is the elastic modulus for the point considered, U is the current strain energy, Un is the homeostatic value of strain energy density, Ce (unknown) is a arbitrary remodeling rate constant, and 2s is the width of the lazy zone representing a range of strain energy density values near the homeostatic value in which there is no net remodeling response (Hart 2001). The strain energy density (SED) is a suitable mechanical stimulus for both surface and internal remodeling. The choice for SED as the feed-back control variable is a well-defendable on physical grounds (Carter et al. 1987). It is a physical quantity for which it is possible to conceive mechanisms for its measurement by bone. It has also a relation to both rigidity and strength. From a methodological standpoint it has the advantage of being a scalar variable and an invariant of the strain tensor, which implies that only one remodeling coefficient (Ce) must be established, as opposed to the many coefficients needed by Cowin and associates in the original theory of adaptive elasticity (Cowin and Hegedus 1976; Cowin and Firoozbakhsh 1981). The theory is a static one, only considering the absolute SED value, not their rate of change. Although it is unlikely that static load-changes result in adaptive bone reactions (Rubin and Lanyon 1984), the philosophy is to consider the SED amplitudes as general representations of the actual recent loading history (Huiskes et al. 1987). The most important evidences from this study are the falsifiable, quantitative hypotheses relative to the predicted effects of stem rigidity and degree of fixation. The results are compatible with the


animal experiments data on similar intramedullary configurations reported by Miller and Kelebay (1981) (Chen et al. 1983; Dallant et al. 1986; Turner et al. 1986). In particular the finding of a completely resorbed cortex around canal-filling, well-fixed prostheses by Miller and Kelebay (1981) compares well with their predictions for a such configuration. The analyses show that the stress shielding is a transient phenomenon, and that a homeostatic strain configuration can in principle be obtained by the bone, provided that the stem is not too rigid, ad provided that the bone resorption process takes place mainly at the periosteal surface rather than inside the cortex. The results also predict that the adaptive remodeling process is quite different in the case of loose prostheses as compared to well-fixed (ingrown) prostheses. This finding is in agreement with the animal-experimental results of Turner et al. (1987). To hat extent, precisely, the process will be different in a realistic situation because the straight stem model is not very good to describe the proximal part of the loose-stem THA configuration (Weinans et al. 1987). In any case, from the analyses it appears that an increase of cortical thickness near the tip of the stem, as often reported in the literature, is associated with relatively rigid stems, which are not well-bonded to the bone (Huiskes et al. 1987). Carter et al. (1989) used a two-dimensional finite-element model of the proximal human femur as the basis for the application of skeletal morphogenesis theories. Their general approach to solving for the distribution of bone morphology was to consider the bone as an initially isotropic and homogeneous structure in which the apparent density and modulus could subsequently vary as a function of position as their computer programs remodeled the bone. As the bone apparent density changed during remodeling, the elastic modulus, E (MPa), was calculated as a power function of the apparent density ρ (g/cm3) as in Carter and Hayes (1977)

33790ρ=E


There is evidence that for a wide range of cancellous bone, the bone modulus is small closely proportional to the square, rather than the cube, of the apparent density (Harrigan et al. 1981; Gibson 1985). The basic idea is to begin with a model in which all elements have a uniform density distribution, to then apply loads and iterate the density of each finite element based on a remodeling rule, and compare the calculated density pattern to in vivo patterns. Motivated by a desire to reconcile single-loading theories to other ongoing studies relating physical activities to bone mass (Whalen et al. 1988), and to account for additional biological stimuli created by loading in different directions, Carter et al. expanded the single load approach for predicting bone density to encompass the multiple-loading history of the bone over a specified time period. In this approach, the bone loading histories for an average day are characterized in terms of stress magnitudes or cyclic strain energy density and the number of loading cycles. They used finite-element models with multiple loads, noting that a single loading condition cannot reasonably be expected to be the stimulus for the full trabecular architecture. They used a relationship between element density and an effective stress that was hypothesized in their recent work (Carter et al. 1987) as:

)2/1(

1

Mc

i

MiinK

= ∑=

σρ

where c is the number of discrete loading conditions, K and M are constants, n is the number of loading and σ continuum model cyclic peak effective stress (scalar quantity). In this model the effective stress is the energy stress, defined as

UEavgenergy 2=σ

where E is the continuum model elastic modulus and U is the continuum model strain energy density. The calculated density


distributions are similar to those seen in the proximal femur. The best correspondence was observed after the third iteration of the computational implementation and in subsequent iterations, the density distribution progressed to non physiological distributions. Interestingly, as a consequence of the use of multiple loadings to determine the trabecular architecture, the trabecular orientations are no longer perpendicular. This finding is at odds with Wolff’s firmly held presumption, and his consequently flawed observation, that the trabeculae can only intersect at right angles (Cowin 1997). The predictive scheme used in his study differs from that applied by others (Hart et al. 1984; Huiskes et al. 1987). The bone remodels toward a state in which the mechanical stimulus to the tissue is independent of its anatomical position within the bone (Fyrie and Carter 1986; Carter et al. 1987) and is determined by the cumulative influence of many loading cycles and loads from different directions. The basic procedures demonstrated in this study can be generally applied to study loading history and morphology relationships in any bone of the appendicular skeleton. The techniques can be used to predict bone functional in response to stress history changes caused by alterations and physical activities or by the use of orthopaedic appliances or implants (Carter et al. 1989). Beaupré et al. (1990a,b) developed a time-dependent approach for emulating bone modeling and remodeling in response to the daily loading history. They presented a unified, time dependent approach for periosteal bone modeling that takes into account the amount of bone surface area on which osteoclasts and osteoblasts can operate. They used the finite element method to apply this technique to predict the time-dependent distribution of density in the proximal femur (Fig. 2.3). The essence of the bone adaptation theory presented is that bone needs a certain level of mechanical stimulation to maintain itself. If bone tissue experiences excess stimulation, additional bone will be deposited. If bone tissue experiences insufficient stimulation, it will resorb. They postulate that bone does not know the appropriate level of stimulus necessary for bone


maintenance from any intrinsic knowledge of its three-dimensional spatial location, but rather the appropriate level is set by systemic influences and by the local biochemical influences of adjacent tissues. The difference between this appropriate level and the actual imposed level of daily mechanical stimulation determines the impetus and speed of modeling or remodeling. The initial assumption on which this theory is based is that for a constant daily loading history, the stress stimulus at the level of the bone tissue tends toward a uniform value. The model is based upon using a daily tissue level stress stimulus with a continuum level measure, ψ , defined as :

)/1( m

day

miin

= ∑ σψ

where ni is the number of cycles of load type i, iσ is the continuum

level stress defined as EU2=σ , and m is an empirical constant. A finite element model is used to determine the values of iσ at each location in the bone, where each location in the bone corresponds to a finite element in the computational model. Then, an error function is defined to modify the apparent density of each element and drive the mechanical stimulus toward the attractor state bASψ . This error represents the driving force for bone remodeling. The relationship between the rate of surface bone remodeling and stress stimulus for mature bone is idealized by a piecewise linear function:

−+−−−

−+−

=

2434

3

2

1211

)()(

)(

)(

)()(

wccc

c

c

wccc

r

bASb

bASb

bASb

bASb

ψψψψψψ

ψψ

&

)(

)0(

)0(

)(

2

2

1

1

w

w

w

w

bASb

bASb

bASb

bASb

+>−+<−≤<−≤−

−<−

ψψψψ

ψψψψ


where bASψ is the attractor state stress stimulus (for remodeling equilibrium), bψ is the actual stress stimulus, c1, c2, c3 and c4 are empirical rate constants, and w1 and w2 define the width of the normal activity region (Figs. 2.1, 2.2).

Fig. 2.1 Rate of surface remodeling as a function of the tissue level stress stimulus for three bone regions. (From Beaupré et al. 1990b)

Fig. 2.2 Idealized, piecewise linear rate relation. (From Beaupré et al. 1990b)


In addition Beaupré et al. write a consistent equation for the rate of change of density (net internal remodeling):

tvSr ρρ && = where Sv is the bone surface area density, and ρt is the true density of the bone tissue, assumed to be the density of fully mineralized tissue and ρ& is the time rate of change of the apparent density.

Fig. 2.3 Predicted distribution of bone apparent density after 40 and 50 time increments (330 and 412.5 days, respectively) using the nonlinear (dead zone) rate relation. (From Beaupré et al. 1990a) The implementation of this two dimensional finite-element model is in contrast to the results obtained by Carter et al. (1989). The solution is convergent with calculated density distribution remarkable similar to physiological distributions. The results are “consistent with the hypothesis that similar stress-related phenomena are responsible for both normal morphogenesis and functional adaptation in response to changes in the bone loading”. In its preliminary form, certain refinements and biological complexities that are essential to simulate actual bone remodeling have not been incorporated into this work. The underlying theory does not take into account the initial osteoclastic resorption in


cortical bone that takes place prior to remodeling changes. This sequence is precisely what is included in the BMU approach developed by Frost (1986). Additional features that have not been considered include material anisotropy and a mechanism to account for bone maturation (Hart 1988). Moreover, a parametric examination of the rate constants that examines differential rates for bone apposition and resorption has not been conducted. In 1992, Weinans et al. proposed a study in order to obtain a better understanding of the bone–remodeling simulation. They used a time-dependent rule for regulation of density as follows:

−= kU

Bdt

d a

ρρ

, cbρρ ≤<0 (*)

where the apparent density is represented by ρ = ρ(x,y,z), ρcb is the maximum density of cortical bone, B and K are constants, Ua

represents the sum of the apparent strain energy density for the number of loading configurations being considered:

∑=

=n

iia U

nU

1

1

The choice of strain energy as the remodeling stimulus was based on the fact that it is an easily interpretable physical scalar which is related to both stress and strain. To calculate the isotropic stiffness, the apparent density was used with the power law relation between density and Young’s modulus (Carter and Hayes 1977):

γρcE =


with γ = 2, c = 100 (MPa)(gxcm-3)-2. They chose to use dense meshes of finite elements and apply (*) to each element. Each element during simulations could then reach one of three outcomes: become completely resorbed (ρ ~ 0); become cortical (ρ ~ ρcb), or remain cancellous with intermediate densities (0 < ρ ≤ ρcb). The simulations were performed for two dimensional models of a proximal femur and of a simple two-dimensional plate model with a linear loading distribution. The convergence results for the plate model produced a trabecular-like structure where most of the elements had either maximum or minimum density values. In the two-dimensional proximal femur model, they found a density distribution similar to the one observed in the real femur after few iterations; however, no equilibrium convergence could be obtained. For these simulations, the solution, in terms of the discrete structures produced, was shown to be dependent upon the methods of post processing (averaging results gave the appearance of continuum solutions) and upon mesh refinement (Hart 2001). Generally speaking, the results are inherent to the concept of bone as a self-optimizing material, rather than to the way in which this concept is described in a model. This concept implies that bone mass is regulated by multiple local units, which work independently. If the control mechanism in the simulation model is characterized by a self-enhancing system (a positive feedback), the occurrence of discontinuities is inevitable. For an adequately refined mesh, in which the dependence of element dimensions is restricted to small parts only, the geometric characteristics of the solution would depend only on the characteristics of the loads, the value of the objective k and the maximal attainable trabecular elastic modulus (hence, the maximal trabecular density or rather the degree of mineralization). In this study it is also shown that the discontinuous end configuration is dictated by the nature of the differential equations describing the remodeling process. This process can be considered as a nonlinear dynamical system with many degrees of freedom, which behaves divergent relative to the objective, leading to many possible solutions.


To summarize, Weinans et al. 1) hypothesized that the bone is a self-optimizing material which produces a self-similar trabecular morphology, a fractal, in a chaotic process of self-organization, whereby uniform SED per unit mass, or a similar mechanical signal, is an attractor, 2) that the morphology has qualities of minimal weight, and 3) that its morphological and dimensional characteristics depend on the local loading characteristics, the maximal degree of mineralization, the sensor density and on the attractor value. This implies that the all characteristic morphological differences between location and species could be explained by a variation in the above mentioned parameters (Weinans et al. 1992). In the middle of 1990s the emphasis of research switched to study of mechanosensation in bone, that is to say, efforts to determine the cellular mechanisms involved in sensing mechanical loading on a bone and signaling other cells to deposit or resorb tissue. In 1994, Mullender et al. justify a biological averaging technique based on a spatial influence function. The technique eliminates the checker boarding by distinguished the sensor grid (the spatial arrangement of the osteocytes) from the actor cells (the osteoblasts and osteoclasts) and the finite-element mesh. In 1995 Mullender and co-workers (Mullender and Huiskes 1995) reexamine the hypothesis of bone as a self-optimizing structure, as proposed by Roux (1881). They hypothesize that osteocytes act as sensors of a mechanical signal or mechanoreceptors (Cowin et al 1991; Lanyon 1993; Marotti et al. 1990) and regulators of bone mass by mediating the osteoblasts and osteoclasts. The hypothesis that bone contains mechanoreceptors implies that the regulation of bone mass by actor cells is governed locally by sensor cells. It is assumed that bone regulation occurs at a local level, which is typical for a self-organizational control process (Yates 1987). The mathematical model proposed to simulate this control process uses the strain energy density as the mechanical signal that the


osteocytes appraise (Huiskes et al. 1987). The osteocytes, distributed through the bone in a particular pattern, emit a stimulus in their environment equivalent to the difference between the local strain energy density and a constant reference value. The actor cells regulate bone density in their area between zero and maximal density, dependent on the total stimulus they receive from the osteocytes, whereby the influence of an individual osteocyte stimulus diminishes exponentially according to its distance from the actor cell concerned (Mullender et al. 1994; Mullender and Huiskes 1995). This approach introduces the concept of sensor (osteocyte) density, independent of the FE mesh, and of sensor influence range (Cowin et al. 1991) as a model for the numerous interconnections of osteocytes and actor cells. The model has been used to investigate the trabecularization of bone in response to mechanical loading (Fig. 2.4). By using a tissue-level approach they show that initial idealized geometries with both uniform density distribution and a lattice structures can give very similar trabecularized geometries during the computational simulations if they are loaded in the same fashion. Each osteocyte i measures its local mechanical signal Si(t) assumed to be the strain energy density, and influences all other actor cells (osteoclasts and osteoblasts) with decreasing influence based on the distance from the osteocyte. The spatial influence function is written as:

]/)([)( Dxdi

ief −=x where di(x) is the distance between osteocyte and location x, D is the distance from an osteocyte at which its effect is e-1, so that the influence of an individual osteocyte, i, decreases exponentially with distance from the cell (Fig. 2.4). The spatial influence function solves the computational checker boarding artifact. The influence function to modify the stimulus amplitude, Si, summed from each osteocyte results in the stimulus value, Φ(x,t) :


( )∑=

−=ΦN

ii kSf1i

)(xt),(x

where k is the reference value of the strain energy density. Then the regulation of the density, ρ (x,t) is

),(),(

tdt

tdx

x Φ= τρ for 0<ρ(x,t)≤ ρcb

where τ is a rate constant, and the elastic modulus at location x, given by

γρ ),(),( tCtE xx =

where C and γ are constants.

Fig. 2.4 Influence parameter D on the end configuration. With use of the lattice structure as the starting configuration, the osteocyte influence parameter D was varied. Top row: initial morphology on the left and equilibrium morphology with D = 25 µm on the right. Bottom row: equilibrium morphology with D = 100 µm on the left and equilibrium morphology with D = 50 µm on the right. Increasing values of D result in a lower pore density and less and thicker struts. (From Mullender and Huiskes 1995)


Mullender at al. show that changing the direction of loading for the trabecularized geometries results in realignment of the trabeculae in accordance with the magnitude and the directions of the external principal stresses, and that severing a strut in the structure leads to its resorption and complete removal (Hart 2001). Hence, the osteocytes would not need information about the local strain orientation in order to form an anisotropic structure. After the orientation of the principal stress was changed, the architecture transformed to resist the new pattern of stress. In the newly formed structure, the orientations of the trabeculae approximate the new principal stress orientation. These predictions are consistent with Wolff's trajectorial hypothesis. Moreover, the results show that predicted trabecular morphology, i.e. sizes and branching of strut, is dependent on the actual relationship between local load, sensor density and range influence. Differences in trabecular morphology in various species can be explained by variations in these parameters. Mullender and coworkers conclude that the method described above is suitable to study the effect of the various parameters presumably controlling their process and may be used to estimate physiological parameters and the results prove that Roux's hypothesis was realistic: morphogenesis, maintenance, and adaptation of bone can be explained by a local, cell-based control process. Other studies (Cowin et al. 1992) examine in more detail the stability and uniqueness of the solutions that give discrete structures, even though the methodology is initially dependent upon a continuum assumption. In these cases, the checkerboarding was attributed to “poor numerical modelling of the stiffness of checkboards by lower-order finite elements” and mesh-dependency problem was eliminated using ideas borrowed from image processing. Later studies (Jog and Haber 1996; Sigmund and Peterson 1998) have shown that checkerboard patterns are due to numerical instabilities in the model. The use of higher order finite elements and image filtering techniques are standard procedures used in topology


optimization to avoid checkerboard patterns. As an alternate means to eliminate the checkboarding pattern, Jacobs et al. (Jacobs et al. 1995) suggested the use of a nodal approach (density calculated at each node) as opposed to an element approach (density calculated at the element centroid). Now it is widely recognized that the checkerboarding is simply a numerical artifact. Jacobs et al. (1995) have proposed a different method for simulation of net trabecular remodeling that can account for both changes in the volume fraction and in the alignment of the material. They have chosen to operate directly upon the anisotropic stiffness tensor that relates the stress and the strain. In this case, no a priori assumptions are needed for describing the elastic symmetry of the continuum approximation for trabecular bone behaviour. Although trabecular bone may be well characterized at the continuum level using orthotropic material properties (9 of the 21 constants are independent) during active realignment, a more general degree of anisotropy may be needed. Unfortunately, finding the 21 needed constants may not be practical, and the problem is now to assemble the structural stiffness matrix in its entirety. Thus, the advantage of separating out the measurable material properties (the bone matrix material properties) and the measurable structural properties (such as the fabric tensor) from the structural stiffness is lost. In their study the results are very similar to those obtained using an isotropic rule for density evolution (Beaupré et al. 1990a,b; Hart 2001) Huiskes et al. (2000) presented a modified version of the Mullender’s model (Mullender et al. 1994) in which the local change of relative bone density is expressed as

troctr ktxPforrktxPdt

dm >−−= ),(),(τ


troc ktxPforrdt

dm ≤−= ),(

where τ is a proportionality constant, P(x,t) (in mol mm-2day-1) is the osteoblast recruitment stimulus at the surface location x as a function of time t, ktr is its threshold level, roc is the relative amount of mineral resorbed by osteoclasts per day in the volume considered. The osteoblast recruitment stimulus is derived from:

∑=

=n

itiii tRxftxP

1

)()(),( µ

where Rti(t) (in Jx(m-3)xs-1) is the SED rate in the location of osteocyte i; µi is the mechanosensitivity of osteocyte i; fi is an exponential decay function describing the attenuation of the bone formation stimulus between osteocyte i and surface location x and n is the number of osteocytes in the neighbourhood of the surface location considered. The computer simulation was conducted as an iterative process, during which the relative density m was regulated per element between 0.01 (no bone) and 1.0 (fully mineralized bone). Per iteration, a bone-surface patch had a p per cent chance of becoming subject to resorption according to :

%10),( =txp spatially random

aPiftxPactxp <−= )],,([),( strain dependent

aPiftxp ≥= 0),( where c and a are constants (Figs. 2.5, 2.6, 2.7). The parameters in the model can be linked to metabolic factors. The authors conclude in this study that the mechanical feed-back can be a potent and stable regulator of the complex biochemical metabolic machinery towards lasting optimality of form.


Fig. 2.5 The simulation process starts from a regular grid (spatially random resorption: bone is black, marrow is white). After 2,000 iterations a homeostatic configuration is obtained, in which bone remodelling continues without changing the architecture at large. Trabecular orientation lines up with the external load. (From Huiskes et al. 2000)

Fig. 2.6 The simulation process starts from another initial configuration, the homeostatic architecture is similar after 3,000 iterations. (From Huiskes et al. 2000)

Fig. 2.7 After rotating the external loads, the architecture reorients until adapted to the new loading directions. Homeostatic architecture after strain-controlled resorption (a = 1.6, c = 12.5) is shown on the right side. (From Huiskes et al. 2000)

Jacobs (2000) in his review describes current advances in understanding of the mechanobiology of cancellous bone structure made by attempting to identify the quantitative mathematical relationship between the mechanical environment and the biologically mediated processes that are ultimately responsible for trabecular architecture. The apparent level approach to


understanding mechanical behaviour is analogous to the biological approach of studying the net effect of cellular activity on a tissue without needing to account for each cell individually. It is possible to relate the apparent level mechanism to the apparent level biological response and vice versa to arrive at an apparent level description of mechanobiological adaptation. He utilizes a mathematical approach that is capable of combining some existing rules describing mechanobiological adaptation of density with an assumption that at each point the cancellous bone is mechanically optimized to support the mechanical load at the particular point. This approach is mathematically similar to the free material approach from the field of the mechanical optimization. As bone densities become very high, the free material approach leads to predictions of extreme orientation, much higher than can be realized with actual bone tissue. Jacobs indeed proposes in the formulation of his model the microstructural approach, adopted by Fernandes et al. 1999. In this model a finite element model is created for each element in the main or primary model. The secondary models are then used to determine the optimal mechanical trabecular pattern given the apparent density and internal mechanical stress directions. This is repeated over for each element in the primary model. In the predicted results, most of the cancellous architecture includes perpendicular intersecting trabeculae, but in some regions the intersection occurs at an oblique angle. Even though the unrealistic microstructures reflect some mechanical performance patterns of the internal bone structure, it is unclear if they reflect dimensional aspects of the trabecular architecture. The results, although optimal microstructures seem to reflect some important aspect of in vivo trabecular structure in terms of mechanical performance, are at best idealizations. This suggests that some additional factors play an important role in determining trabecular morphology such as the process of morphogenesis, nutrient diffusion, or additional cellular or mechanical requirements. Lekszycki (2005), on the basis of his previous studies (Lekszycki 1999; Lekszycki 2002), wrote a paper in which the functional


adaptation of bone can be considered as a modified optimal control problem. Lekszycki makes the hypothesis that the optimal response of bone enables derivation (instead of postulation) the remodeling rules from a very general and global assumption. The link between the postulated local adaptation rules and those derived from the global assumption is also discussed. The bone attempts to make the changes in its configuration within actual constraints to approach the best solution which is never achievable since it varies due to conditions variable in time. The assumptions of this study are that the considered effects are slow and the inertia terms are negligible; the theory of small displacements and velocities is valid; the control function µ(x,t) characterizing the bone structure is selected. The derived remodeling law relates the velocities ( )), txµ& to variable in time states of the bone. Then, the functional G(µ(x,t)) :

∫Ω

Ω= dSG ),( µu

is defined; Ω denotes a domain, u is the displacement field. The functional depends on a set of time-variable control functions determining the bone configuration. For the hypothesis of optimal response of the bone the rates ( )t,xµ& assures the extremum of the objective functional which is represented by the rate of the functional G(µ(x,t)) :

∫Ω

Ω= dSdt

dG),( µu&

The global and local constraints can take into account different mechanical and non-mechanical effects. From the stationarity condition of the objective functional, with constraints attached to it by means of Lagrange multipliers, the optimality conditions follow. Some of them can be interpreted as the remodeling law.


The optimal control approach enables the control of the total amount of material via global and local constraints, what can be useful in many situations as, for example, in cases of osteoporosis or bone growth. The formulation in this paper should describe important biological effects that are involved in the control process, e.g., the interactions between different types of cells. In the next chapter some theoretical and numerical tools, involved in the study of the bone remodeling process, will be explained. They will be also unified in an implemented procedure with the aim to propose an enhanced model and to predict an optimal distribution of mass and energy of a bone structure under specific constraints and loading conditions. References Beaupré GS, Orr TE, Carter DR, 1990a. An approach for timedependent bone modeling and remodeling-application: a preliminary remodeling simulation, J Orthop Res 8:662-670. Beaupré GS, Orr TE and Carter DR, 1990b. An approach for time-dependent bone remodeling-theoretical development, J Orthop. Res., 8(5), 651-661. Brown TD, Pedersen DR, Gray ML, Brand RA, et al. 1990. Toward an identification of mechanical parameters initiating periosteal remodeling: a combined experimental and analytic approach, J Biomech 23:893–905. Burr DB, Martin RB, Schaffler MB and Radin EL, 1990. Bone remodeling in response to in vivo fatigue microdamage, J Biomech, 23(1), 11-14. Carter DR and Hayes WC, 1977. The compressive behaviour of bone as a two-phase porous structure, J. Bone Joint Surg. [Am], 59 (7), 954–962. Carter DR, 1984. Mechanical loading histories and cortical bone remodeling, Calcif. Tissue Int., 36 (Suppl.I), S19-24.


Carter DR, Fyhrie DP and Whalen RT, 1987. Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy, J Biomech. 20,785–794. Carter DR, Orr TE and Fyhrie DP, 1989. Relationship between loading history and femoral cancellous bone architecture, J Biomech., 22(3), 231-244. Chen PQ, Turner TM, Ronningen H, Galante J, Urban R and Rostocker W, 1983. A canine cementless total hip prosthesis model. Clin. Orthop. Rel. Res. 176, 24-33. Cowin SC and Doty SB, 2007. Tissue Mechanics. New York: Springer. Cowin SC and Firoozbakhsk DH, 1981. Bone remodeling of diaphyseal surfaces under constant load: theoretical prediction, J Biomech 14, 471-484. Cowin SC and Hegedus DH, 1976. Bone remodeling I: theory of adaptive elasticity, J Elasticity 6, 313-326. Cowin SC and Moss ML, 2000. Mechanosensory mechanisms in bone. In Textbook of tissue engineering, 2nd ed., ed. R Lanza, R Langer, W Chick, pp. 723–738. San Diego: Academic Press. Cowin SC, 1984. Mechanical modelling of the stress adaptation process in bone, Calcif. Tissue Int., 36 (Suppl. I), S98-103. Cowin SC, 1997. The false premise of Wolff’s law, Forma, 12, 247-262. Cowin SC, Arramon YP, Luo GM and Sadegh AM, 1993. Chaos in the discrete-time algorithm for bone-density remodeling rate equations, J Biomech 26: 1077-1089. Cowin SC, Moss-Salentijn L and Moss ML, 1991. Candidates for the mechanosensory system in bone, J Biomech Eng 113:191-197. Dallant P, Meunier A, Guillemin G, Christel P, Crolet JM and Sedel L, 1986. Cortical bone response to orthopaedic implant rigidity: an experimental study. Biological and Biomechanical Performance of Biomaterials (Edited by Christel, P., Meunier, A. and Lee, A. J. C.). pp. 441-446. Elsevier Science, Amsterdam. Fernandes P, Rodrigues H and Jacobs C, 1999. A model of bone adaptation using a global optimization criterion based on the trajectorial theory of Wolff. Comp Meth Biomech Biomed Eng 2 :125-38.


Frost HM, 1983. A determinant of bone architecture. The minimum effective strain, Clin. Orthop., 175, 286-292. Frost HM, 1986. Intermediary Organization of the Skeleton, Vol. 1, Boca Raton, FL, CRC Press. Frost HM, 1987. Bone “mass” and the ”mechanostat”: a proposal. Ant. Rec. 219, 1–9. Fyhrie DP and Carter DR, 1986. A unifying principle relating stress to trabecular bone morphology, J Orthop. Res., 4(3), 304-317. Gibson LG, 1985. The mechanical behaviour of cancellous bone, J Biomech, Vol 18, Issue 5, 317-328. Harrigan TP, Jasty M, Mann RW and Harris WH, 1981. The influence of apparent density and trabecular orientation on the elastic modulus of cancellous bone, Trans. 27th Orthop. Res. Soc. 6, 277. Hart R, Davy D and Heiple K, 1984. A computational method for stress analysis of adaptive elastic materials with a view toward applications in strain-included bone remodeling. J Biomech. Eng 106, pp. 342–350. Hart RT and Fritton SP, 1997. Introduction to finite element based simulation of functional adaptation of cancellous bone, Forma, 12, 277-299. Hart RT, 1988. Remodeling finite element calculations of strain-induced in vivo bone remodeling, Trans Orthop Res Soc 13:99. Hart RT, 2001. Bone Modeling and Remodeling: Theories and Computation. In S. C. Cowin (ed.) Bone Mechanics Handbook, CRC Press. Huiskes R, Ruimerman R, van Lenthe GH and Janssen JD, 2000. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone, Nature (405), 704-706. Huiskes R, Weinans H, Grootenboer HJ, Dalstra M, Fudala B and Slooff TJ, 1987. Adaptive bone-remodeling theory applied to prosthetic-design analysis, J Biomech 20:1135-1150. Jacobs CR, 2000. The mechanobiology of cancellous bone structural adaptation, J Rehab Res and Dev 37(2), 209–216.


Jacobs CR, Levenstone ME, Beaupré GS, Simo JC and Carter DR, 1995. Numerical instabilities in bone remodeling simulations: the advantages of a node-based finite element approach, J Biomech 28(4), 449-459. Jacobs CR, Simo JC, Beaupré GB and Carter DR, 1995. Comparing on optimal efficiency assumption to a principal stress-based formulation fro the simulation of anisotropic bone adaptation to mechanical loading, in Bone Structure and Remodeling, A. Odgaard and H. Weinans, Eds., World Scientific, Singapore, 225-237. Jog CS and Haber RB, 1996. Stability of finite element models for distributed parameter optimization and topology design, Comp. Meth. Appl. Mech. Eng. 130, 203–226. Lanyon LE, 1993. Osteocytes, strain detection, bone modeling and remodeling, Calcif Tissue Int 53(S1): S102-S106. Lekszycki T, 1999. Optimality conditions in modeling of bone adaptation phenomenon, J. Theoret. Appl. Mech., 37, 3, 607-623. Lekszycki T, 2002. Modeling of bone adaptation based on an optimal response hypothesis, Meccanica, 37, 343-354. Lekszycki T, 2005. Functional Adaptation of bone as an optimal control problem, J Theoret. Appl. Mech, 43, 3, 555-574. Marotti G, Cane V, Palazzini S and Palumbo C, 1990. Structure-function relationships in the osteocyte, Ital J Miner Electrolyte Metab 4:93-106. Martin RB, 1985. The usefulness of mathematical models for bone remodeling, Yearbook Phys. Anthropol., 28, 227-236. Miller JE and Kelebay LC, 1981. Bone ingrowth-disuse osteoporosis, Orthop. Trans. 5, 380. Mullender MG and Huiskes R, 1995. Proposal for the regulatory mechanism of Wolff’slaw, J Ortop Res., 13(4), 503-512. Mullender MG, Huiskes R and Weinans H, 1994. A physiological approach to the simulation of bone remodeling as a self-organizational control process, J Biomech., 27(11), 1389-1394. Oden ZM and Hart RT, 1995. Correlation between mechanical parameters that represent bone’s mechanical milieu and control functional adaptation, in Bone Structures and


remodeling: Recent Advances in Human Biology, vol.2,A.Odgaard and H. Weinans, Eds. World Scientific, Singapore. Roux W, 1881. Der Kampf der Theile im 0rganismu.s, Leipzig. Engelmann. Rubin CT and Lanyon LE, 1984. Regulation of bone formation by applied dynamic loads, J Bone Jt Surg 66A, 397-402. Sigmund O and Peterson J, 1998. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. Optim. 16, 68–75. Turner TM, Sumner DR, Urban RM and Galante JO, 1987. Femoral remodeling associated with uncoated and porous coated cementless total hip arthroplasty. Proceedings 33rd Annuul Meerina Ortkoordic Resrurck Socielr. p: 408. Turner TM, Sumner DR, Urban RM, Rivero DP and Galante JO, 1986. A comparative study of porous coatings in a weight-bearing total hip arthroplasty model. J. Bone Jr Surg. 68-A, 1396-1409. Weinans H, Huiskes R and Grootenboar HJ, 1992. The behaviour of adaptive bone-remodeling simulations models, J Biomech., 25(12), 1425-1441. Weinans H, Huiskes R and Grootenboer H, 1987. The modeling and mechanical. consequences of fibrous-tissue formation around femoral hip prostheses. Biomechanics Symposium (Edited by Butler DL and Torzilli PA) AM-D-Vol. 84. pp. 89-92. American Society of Mechanical Engineers, New York. Whalen RT, Carter DR and Steele DR, 1988. Influence of physical activity on the regulation of bone density, J Biomech 21: 825-837. Yates, 1987. Control of self-organization, in Self-organizing Systems: the Emergence of Order, Plenum Press, New York.

Chapter 3 Theoretical and Numerical

Tools for Bone Tissue Adaptation

3.1 Introduction 3.2 FEM Analysis 3.3 Finite Element – Based Optimization

Approach : Topology Optimization of Structures

• Non Linear Constrained Optimization Problems

3.4 Cellular Automaton Model • Cellular Automata as Osteocytes :

Phenomenological and Optimization Approaches

3.5 Adaptive Elasticity Theory 3.6 PID Controller

3.1 Introduction The development and utilization of technological products raise the question of how their quality and reliability can be improved at a minimum cost, that is, how to optimize the structures. Due to the wide range of applications, structural optimization has become a common multidisciplinary research field.


The design of complex mechanical systems and of extremely light weight constructions has enhanced the request for very precise calculation methods. Hence, computer aided procedures have become indispensable to answer to the increasing demand of even more reliable and efficient structures. In phenomenological models, the underlying premise of the remodeling process is that bones distribute their mass in an efficient way. The remodeling process enables bones to perform their mechanical functions by providing both minimum mass and maximum rigidity. In other words, the bone remodeling process is analogous to topology optimization in optimal structural design. In fact, optimization approaches assume the existence of an optimum apparent density distribution in bone that satisfies the conflicting requirements of mass and stiffness. This chapter contains the theoretical and numerical tools (FEM analysis, topology optimization of structures, cellular automaton model, adaptive elasticity theory and PID control) adopted in this study for imitating the bone tissue adaptation. They are incorporated into a unified algorithm written in MATLAB 7.0.4 which is displayed in the Appendix. 3.2 FEM Analysis Numerical approximation methods are usually employed for quantification of the mechanical usage because of the complexity of bone geometry, material property description and distribution, and time-varying boundary conditions and mechanical loading. The most common computational implementations for bone remodeling theories are based on finite element methods, which use computational models described by the subdivision of complex shapes into small finite elements which constitute the mesh. Approximate solutions for displacements, strains and stresses can be found at any point of the structure, with the accuracy of the solution controlled by the mesh refinement (Cook et al. 1989; Hart 1989)

Theoretical and Numerical Tools for Bone Tissue Adaptation 42

Accurate three-dimensional finite element models of intact and implanted human femurs, based on computed tomography images, can be used to investigate, for instance, the structural behaviour of hip joint prostheses under physiological loadings and boundary condition (Fig. 3.1). Walking and stair climbing activities are encountered with the highest frequencies during daily living and, in particular, stair climbing is a critical task for primary stability of the prosthesized femur with respect to the activity of walking. In fact, the conditions for micromotion initiation at the bone–stem interface and the role of stair climbing versus gait in promoting incipient slipping deserve attention. Finite element analysis allows the evaluation of the strain and stress distributions both in the femur and in the stem, the shear strains at the contact between femoral bone and stem (Fig. 3.2) and the stress shielding (Andreaus and Colloca 2009; Andreaus et al. 2008a; Andreaus et al. 2008b; Andreaus and Colloca 2006).

Fig. 3.1 First principal stress (tensile) [Pa] during walking (a) and stair climbing (b). (From Andreaus et al. 2008a)


Fig. 3.2 Shear strain in xy plane [rad] during walking (a) and stair climbing (b). (From Andreaus and Colloca 2009) To simulate the adaptive response, the finite element models can be incrementally updated. Because the updated geometry or material properties change the structural behaviour, the updated structure must be reanalyzed to find the new displacements, strains, and stresses. The iterative process continues until the adaptation is complete, or the user decides to terminate the simulation. The choice for controlling the incremental changes can be time-dependent if the time course of the adaptation is of interest, or may be independent of time as in the case of optimization-based simulations (Hart 2001).


3.3 Finite Element – Based Optimization Approach : Topology Optimization of Structures There are three types of finite element-based optimization approaches: sizing optimization, shape optimization and topology optimization. The first one is a technique developed for truss structures and consists in determining the optimum number, position and geometric parameters (cross-section and thickness) of the structural members. For example the optimal thickness distribution minimizes (or maximizes) a physical quantity such as the mean compliance (external work), peak strain, deflection, etc., while equilibrium and other constraints on the state and design variable are satisfied. Shape optimization and topology optimization have been developed for continuum domains. The former aims at determining the optimal profile or boundary of the design domain while the latter is concerned with the optimal distribution of the material within the domain (number, location and shape of holes, and connectivity of the domain). The topology optimization process selectively removes from the domain those unnecessary elements in which the structure is discretized. When a material distribution problem is faced within the topology optimization field, the most commonly used approaches to determine the optimal placement of a given isotropic material in space are the SIMP (solid isotropic material with penalization) approach (Bendsøe 1989) and the homogenization approach (Bendsøe and Sigmund 2003). In order to explain these approaches, a design parameterization is a fundamental starting point: Ω is the reference domain, the aim is to determine the optimal Ωmat of material points. The optimization problem defined for a minimum compliance design (maximum global stiffness) takes the form:

ufu

T

Ei

min,

adii EEts E,)(:.. ∈= fuK (3.1)


where u and f are the displacement and load vectors, respectively. The stiffness matrix K depends on the stiffness Ei in element i = 1,2….,n and it is written in the form :

∑=

=n

iii E

1

)(KK

with Ki the global element stiffness matrix. The design parameterization implies that the set Ead of admissible stiffness tensors consists of those tensors for which:

ΩΩ∈Ω∈

== ΩΩ mat

mat

ijklijklxif

xiflElE matmat

\0

1,0

(3.2)

∫Ω

Ω≤Ω=Ω VVoldl mat

mat )( (3.3)

where V is a limit on the amount of available material so that the minimum compliance design is for a fixed volume. The tensor 0

ijklE is the stiffness tensor for a given isotropic material (Bendsøe and Sigmund 2003). Generally to solve this problem the integer variables are used in place of the continuous variables and a penalty that steers the solution to discrete 0-1 values is introduced. The design problem for the fixed domain is then formulated as a sizing problem by modifying the stiffness matrix so that it depends continuously on a function which interpreted as a density of material. This function is then the design variable. In the SIMP approach, as above anticipated, the material distribution problem is parameterized by the elastic modulus of the discrete isotropic finite elements. It is defined as follows:

1,)( 0 >= pExE ijklp

ijkl ρ

(3.4)


∫Ω

Ω∈≤≤≤Ω xxVdx ,1)(0;)( ρρ (3.5)

where ρ(x) is the design function. The density interpolates between the material properties 0 and 0

ijklE . This approach requires the optimum structure to be a design consisting of regions with material,

0)1( ijklijkl EE ==ρ and regions without material, 0)0( ==ρijklE .The optimal topologies based on this approach are the so-called black and white structures. Intermediate values for the elastic modulus (gray colours) have to be penalized (Rozvany 2001). The SIMP-model can be considered as a material model if the power p satisfies that:

)3(21

1

2

3,

57

115max

)2(1

4,

1

2max

0

0

0

0

00

Dinp

Dinp

−

−−

−−≥

−

+−≥

νν

νν

νν (3.6)

where ν0 is the Poisson ratio of the given base material with stiffness tensor 0

ijklE . When a structure is discretized in n elements, the heuristic relationships are often used:

)1()( 0 >= pxExE piii

(3.7)

)10()( 0 ≤≤= iiii xxx ρρ (3.8) where, for i = 1,2…n, ρ0 is the density of the solid material, ρi is a variable density and p is the penalization power. The design variable in this approach is the relative density xi. The power p is used to penalize intermediate relative density values and drive the design to a black and white structure (Fig. 3.3).


Fig. 3.3 A black-and-white minimum compliance design for a loaded knee structure obtained with the SIMP model. (From Bendsøe and Sigmund 2003) In the homogenization approach, the finite elements of the structure are made of a composite material consisting of an infinite number of infinitely small holes periodically distributed throughout the solid base. The topology optimization problem is consequently transformed to the form of a sizing problem where the design variables are the material densities of the finite elements. Also in this case, ρ = 0 is corresponding to a void, ρ = 1 to material and 0 < ρ < 1 to the porous composite with voids at a micro level. There are thus a set of admissible Ead stiffness tensors given in the form:

)](),...,(),([)(

)](),...,(),([~

)(

xxxx

xxxExE ijklijkl

ϑγµρρϑγµ

=

=

(3.9)

∫Ω

Ω∈≤≤≤Ω xxVdx ,1)(0;)( ρρ

where ijklE

~ are the effective material parameters for the composite

and )(),(...., Ω∈Ω∈ ∞∞ LL ϑγµ angle are the geometric variables (Bendsøe and Sigmund 2003). Topology optimization has been extended to the analysis of plates and shells in bending (Diaz et al. 1994; Krog and Olhoff 1999; Olhoff


2001) and to 3D continuum structures (Cherkaev and Palais 1996; Diaz and Lipton 1997; Beckers 1999; Olhoff et al. 1998). In addition, eigen-frequencies of structural vibrations and buckling eigenvalue topology optimization have been reported. Topology optimization regarding wind and snow loading, hydrostatic pressure and fluid flow where loadings change with the structural design should be also mentioned. An exhaustive, although not up to dated account on the subject, can be found in the extensive review of Eschenauer and Olhoff (2001) and in Rozvany (2001). In topology optimization, thousands of elements and, therefore, of design variables may be involved, depending on the size of the design domain and on the desired resolution of the final results. This situation, in conjunction with the fact that the cost of the function call increases with the number of elements, makes sometimes the cost of the computer analysis prohibitive. Specialised numerical methods have been therefore implemented such as approximation techniques (Schmit and Farsi 1974; Vanderplaats and Salajegheh 1989), optimality criteria (Rozvany et al. 1990; Saxena and Ananthasuresh 2000), method of moving asymptotes (MMA) (Svanberg 1987), the evolutionary structural optimization (ESO) approach (Xie and Steven 1993) and the cellular automaton (CA) technique (Inou et al. 1994; Kita and Toyoda 2000; Hajela and Kim 2001; Tatting and Gürdal 2000; Abadalla and Gürdal 2002; Tovar 2004; Tovar et al. 2007; Patel et al. 2008) which simulates the behaviour of dynamic systems. More recently, in a comparative study of topology optimization techniques, Patel et al. (2008) have shown that, for three sample problems investigated, the cellular automaton method generally converges more rapidly to a solution in comparison with the optimality criteria method and the method of moving asymptotes. Still within the frame of optimization analysis, Jang and Kim (2008) have performed a computational study of Wolff’s law in the proximal femur for three loading conditions corresponding to different daily activities. They have found that the intersections of the trabecular patterns in the femoral neck are not orthogonal in contrast with Wolff conjecture. This was already argued by Carter et al. (1989) and also measured by Skedros and Baucom (2007) thus


confirming the strong influence of the shear stress on the trabecular patterns. 3.3.1 Non Linear Constrained Optimization Problems As already pointed out, in this thesis an optimization problem of lightweight stiffened structures is faced. If Ω is the domain of interest, Ωopt is the optimal domain in which the stiffness is maximized, that is, the strain energy is minimized. This goal can be accomplished by defining and minimizing two cost indices:

001 )1(

M

M

U

UJ ωω −+=

(3.10)

and

002 M

M

U

UJ ⋅=

(3.11)

where U and M are the total strain energy and total mass of the domain of interest, that are normalized with their initial values, U0 and M0 respectively. Total mass and global stiffness are conflicting goals. The weight ω for the cost index J1 is included between 0 and 1. J1 and J2 are functional of the strain energy and mass, which are functions of several variables: the control gains and the target of the error signal. Such parameters will be shown in more details in the following chapter. The coefficients of the proportional-integral-derivative control, along with the error signal (difference between the strain energy density of each element of the discretized domain and the target) update the Young’s modulus of each element by the evolution rule of the relative density. In other words, the effect of the structural change is translated into changes of mechanical properties and, therefore, changes in the mechanical stimulus (the strain energy density). The adaptation process continues until equilibrium is


reached (the strain energy density for each cell is equal to the target). The evaluation of an optimal choice for the control parameters, for the target and also for the weight parameter ω will be performed in order to minimize the cost indices (3.10) and (3.11) in the form:

( ) ( ) ( ) ( )

+−=

0

*DIP

0

*DIP

,,,,1

,,,,,,1

*DIP U

SEDcccU

M

SEDcccMMinJMin

SEDcccωω

ω (3.12)

and

( )( ) ( )

⋅=

0

*DIP

0

*DIP

,,,2

,,,,,,*

DIP U

SEDcccU

M

SEDcccMMinJMin

SEDccc (3.13)

The set of optimization variables is constrained:

*0, 0, 0, 0, 0 1p I Dc c c SED ω≥ ≥ ≥ > ≤ ≤ (3.14) Problems (3.12) and (3.13) with conditions (3.14) are non linear constrained optimization problems; their analytical solution is not possible since the closed-form expressions of the total mass and of the total energy are not available. Therefore a numerical solution will be considered. The constraints (3.14), along with the characteristics of the objective function, allow the application of the Weierstraβ Theorem, guaranteeing the existence of an optimal solution. 3.4 Cellular Automaton Model First introduced by Weiner and Rosenblunth (1946) and later formalized by von Neumann, a cellular automaton model is a collection of cells with distinct states on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighbouring cells. The rules are then applied iteratively for as many time steps as desired.


The distinct states of each cell may assume must also be specified. This number is typically an integer. For a binary automaton, colour 0 is commonly called white and colour 1 is commonly called black. However, cellular automata having a continuous range of possible values may also be considered. In addition to the grid on which a cellular automaton lives and the colours its cells may assume, the neighbourhood over which cells affect one another must be also specified. The simplest choice is nearest neighbours in which only cells directly adjacent to a given cell may be affected at each time step. Two common neighbourhoods in the case of a two-dimensional cellular automaton on a square grid are the so-called Moore neighbourhood (a square neighbourhood, Fig. 3.4) and von Neumann neighbourhood (a diamond-shaped neighbourhood, Fig. 3.4)1.

Fig. 3.4 Cellular automaton model with number of the selected neighbourhoods for each cell. The premise of a CA model is that overall global behavior can be computed by local rules that operate on cells that only know local conditions. Rules depend on states of the cell and its neighbours

1

Weisstein, Eric W. Cellular Automaton. From MathWorld, A Wolfram Web Resource. http://mathworld.wolfram.com/CellularAutomaton.html


within certain proximity. Since the computations are limited to neighbourhoods and the local rules are identical for the whole lattice, CAs have been proven to have an inherent massive parallel computation capability (Wolfram 2002). In other words, the macroscopic behaviour of a system is composed of many interacting elements which very little depend on the microscopic details of their interaction. At the microscopic level, interactions may be governed by complicated potentials and, sometimes, the use of quantum mechanics. At the macroscopic level, the complexity of the macroscopic world disappears, and a new behavior dominated by collective properties emerges. Cellular automata models replace equations by the simple step-by-step procedures of computer programs. In these algorithms time and space are not considered as continuous but as digital steps. The continuum is replaced by a grid or lattice. Wolfram (2002) develops the concept of the cellular automata which provide an alternative method to describe, understand and simulate the behavior of dynamic systems. Cellular automata are an idealization of a physical system in which space and time are discrete. For instance, the time evolution of physical quantities is often governed by non-linear partial differential equations. In many cases, the solutions of these dynamic systems can be very complex and strongly sensitive to initial conditions. When the cellular automaton approach is applied to the solution of partial differential equations, it seems to be superior to traditional numerical approaches because it is sometimes not clear if a feature of a solution is a consequence of the partial differential equation or merely a reflection of the discretization procedure. Since cellular automata are fully discrete models, they are never subjected to a discretization procedure and are therefore not open to this source of error (Cowin and Doty 2007). Among the authors who have developed a CA-like approach are mentioned Inou et al. (1994) which use the elastic moduli of the cells as the design variables. A local rule iteratively updates the value of the modulus of each cell based on the difference between a current stress value and a target value. Evolutionary rules based on the resorption/deposition procedure are used to fine-tune the structure.


Cells with low elastic modulus are removed, while cells with high elastic modulus create a new cell in an empty surrounding space. This approach led to structures that are similar to the ones observed in bird bones (Inou et al. 1998). Later, Kita and Toyoda (2000) have used the concept of CA for topology optimization using the thickness as the design variable. The local design rule is derived from the optimality condition of a multi-objective function, in which both the weight of the structure and the deviation between the yield stress and the equivalent stress in the cellular automata have to be minimized. The finite element method is used to evaluate and update the strain energy densities in each cellular automaton during each iteration of their algorithm. This method is a finite element-based approach and reduces the residual between external work and internal energy to zero at every iteration. The algorithm they have proposed presents problems of convergence since hundreds of iterations are required even in the solution of simple problems. Tatting and Gürdal (2000) have presented a CA model implemented with simultaneous analysis and design (SAND) approach where both design and state variable are simultaneously updated. Convergence is unfortunately very slow requiring hundreds of thousands of iteration to be attained. Improvements are reported by using multigrid and full multigrid strategies (Kim et al. 2004). In Tovar et al. (2004, 2007) a different approach to topology optimization termed hybrid cellular automaton (HCA) has been developed for the study of the bone tissue. The two different time scale involving change in mass via remodeling and the change in stress/strain fields after structural modification have been incorporated into a single algorithm thus combining the cellular automaton paradigm with the finite element method. The automata apply the remodeling rule at the tissue level with local control rules modifying the design variables, while the finite element analysis performs structural investigation at the continuum level. This approach reduces numerical instabilities and allows for determining directly the anisotropic structure of the trabecular bone.


3.4.1 Cellular Automata as Osteocytes: Phenomenological and Optimization Approaches The functional adaptation of bone to mechanical usage implies the existence of a physiological control process. Essential components for the control process include sensors for detecting mechanical usage and transducers to convert the usage measures to cellular responses. The cellular responses lead to gradual changes in bone shape or material properties and, once the structure has sufficiently adapted, the feedback signal is diminished and further changes to shape and properties are stopped (Hart 2001). The simulation of bone functional adaptation could be performed by applying the cellular automaton model. For instance, a cellular automaton lattice is proposed to model the connected cellular network in bones. The osteocytes are represented as cellular automata ideally distributed along the design domain. Each cellular automaton is an osteocyte. Mullender et al. (1996) reported data of 15000 osteocytes per cubic millimeter of bone. Marotti et al. (1990) found 500-3000 osteocytes per square millimeter in 20-30 µm histological slices from several species. The osteocytes act as sensors of mechanical and they activate the local formation and resorption of bone tissue according to an error between the mechanical stimulus and the equilibrium state. Consequently, the change in mass modifies the stress/strain field in the bone and, therefore, the stimulus operating on the cellular automata. This process continues until the error signal is zero or no possible mass change can be made. After this process it is possible to predict physiologically plausible trabecular architectures. The automata apply the remodeling rules, while the finite elements perform the structural analysis and evaluate the mechanical stimulus. Prendergast and Taylor (1994) have developed a more mechanistic method based on the hypothesis that bone adapts to attain an optimal distribution of the mass (minimum mass and maximum rigidity) by regulating the damage generated in its microstructural elements. They hypothesized that there is some damage at


remodeling equilibrium, and that the rate of repair is associated with the damage rate.

3.5 Adaptive Elasticity Theory The theory of adaptive elasticity (Cowin and Hegedus 1976; Cowin and Firoozbakhsh 1981) was developed to describe the remodeling behaviour of cortical bone from one loading configuration to another, rather than predicting the optimal structure of normal bone, as in the theory of self-optimization. For this purpose it is assumed that cortical bone tissue has a site-specific natural or homeostatic equilibrium strain state. A change of load or, in fact, an abnormal actual strain state stimulates the bone tissue to adapt its mass in a such way, that the equilibrium strain state is again obtained. In the theory, the rate of adaptation is coupled to the difference between the equilibrium and the actual strain states. Following a suggestion by Frost (1964), Cowin and co-workers separate internal and surface (or external) remodeling. In the first case, the bone has only the option of adapting its density, thereby, assuming continuum theory to be valid, adapting its elastic modulus according to:

( )0ijijijA

dt

dE εε −= (3.15)

where E is the local modulus of elasticity, εij the actual strain tensor, ε0ij the equilibrium strain tensor, and Aij a matrix of remodeling coefficients. In the case of external remodeling the bone can only add or remove material on the periosteal and endosteal surfaces, stimulated by the strain state at those surfaces, according to:

( )0ijijijB

dt

dX εε −= (3.16)


where X is a characteristic surface coordinate perpendicular to the surface, and Bij again a matrix of remodeling coefficients. Cowin et al. (1981, 1985) used the strain tensor as the mechanical signal to be the driving force of bone remodeling and also considered quadratic relations between strain and rate of adaptation, and performed a number of studies to determine possible values of the remodeling coefficients. Recently, using the theory of external remodeling to simulate animal experiments, they found agreement between experimental results and theoretical predictions. This theoretical basis suggested the use of two equations in order to evaluate the change of the material properties in a bone structure discretized in n elements:

( )*i

i SEDSEDcdt

d−= ρ

ρ

(3.17)

( )*i

i SEDSEDcdt

dEE −= (3.18)

for i = 1,2….n, ρi is the apparent density of the element at the discrete location i, SEDi is the strain energy density, Ei is the Young’s modulus and SED* is the target. In this case the cρ and cE constants and SEDi scalars are used in place of the matrices of remodeling coefficients and strain tensor. The adaptive elasticity theory doesn’t describe how local mechanical signals are detected and how they are translated to bone formation and resorption. 3.6 PID Controller A proportional–integral–derivative (PID) controller attempts, in a generic loop, to correct the error between a measured design variable and a desired target by evaluating a corrective action that can adjust the process to keep the error minimal rapidly. It involves three


separate parameters: the proportional gain determines the reaction to the current error, the integral gain determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. By tuning the correct values of the proportional, integral and derivate gains, the algorithm improves its efficiency in terms of the number of iterations. The PID action allows users to control the convergence of the algorithm but does not guarantee optimal control of the system or system stability. A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable. On the other hand, a small gain results in a small output response to a large input error, and a less responsive controller. The integral term accelerates the movement of the process towards target and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cross over the target and then create a deviation in the other direction. The derivative term is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. The derivative control slows the rate of change of the controller output and this effect is most noticeable close to the controller target. In contrast, this term is highly sensitive to noise in the error term, and a process can become unstable if the noise and the derivative gain are sufficiently large. There are several methods for tuning the control gain: manual tuning, Ziegler-Nichols tuning (Åström and Hägglund 1995) and software tools which incorporate control and optimization procedures. A problem faced with PID controllers is that they are linear. For this reason, the performance of PID controllers in non-linear systems is variable. In this study, in order to determine the geometrical or material changes for the adopted 2-D models, the (3.12) and (3.13) remodeling


rates, described in the previous paragraph, are integrated with a constant time step ∆t = 1, using a local control rule, whereby, the change in apparent density ρi or, alternatively, in elastic modulus Ei in each step follows from

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectttt

iiiiiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

ρρρ τρρρ

(3.19)

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectEttEEt

EEEEiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑

=

1,10

IDPτ

τ

(3.20)

or equivalently

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectxttxxt

ixiixixiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

τ

(3.21)

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectyttyyt

iyiiyiyiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

τ

(3.22)

where yi = Ei/Emax is the relative elastic modulus with 0 ≤ yi ≤ 1 and ēi

is the effective error. The concept of the effective error will be developed in more details in the next chapter. The coefficients cP, cI, cD are the proportional, integral and derivative gains and they determine the remodeling rate as explained above. Chapter 4 includes most of the original elements of this work. It represents the control and optimization procedure that solves a structural problem including two objective functions. A set of optimized parameters guarantees the optimization and safe-control of a considered structure with respect to conflicting goals – total mass and global stiffness.


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Chapter 4 Control and Optimization

Procedure for Bone Lightweight Stiffened

Structures

4.1 Introduction 4.2 Materials and Methods

• Generalities

• Control Strategy • Optimization Strategy

• 2-D Models

4.3 Numerical Results • Bone Sample (25 cellular automata)

• Bone Sample (625 cellular automata)

• Michell-type Structure (1250 cellular automata)

4.4 Discussion

4.1 Introduction The computer model developed in this chapter is based on the use of the Finite Element Method (FEM) in combination with the theory of


Self-Optimization (SO) and an alternative formulation of the theory of Adaptive Elasticity (AE). The aim is to perform a two-step procedure of control and optimization in order to solve an optimization problem of lightweight stiffened structures; various objective functions and constraints were included so that different design requirements were compared. The selection of the optimized parameters in the evolution rules, not yet faced in an in-depth study, has successfully been studied in this thesis and the convergence was improved. The set of optimal parameters includes the control gains, the target of the error signal and the weight of the cost index J1, defined in Chapter 3. Two-dimensional bone samples, subjected to an in-plane loading condition, have been analyzed. The reference value of mass density and elastic modulus are typical of bone tissue but the enhanced model proposed herein becomes consistent in many other fields of engineering materials.

4.2 Materials and Methods 4.2.1 Generalities Osteocytes in the bone can detect a mechanical stimulus that, with its magnitude, causes local material adaptations. This process can be described with a generic mathematical expression, using the apparent density as the characterization of the internal morphology.

The rate of change of the apparent density of the structure dρ/dt,

with ρ = ρ(x,y,z) at a particular location P(x,y,z), can be described as an objective function F, which depends on a particular stimulus at location P(x,y,z). It is assumed that this stimulus is directly related to the local mechanical load in the structure and can be determined

from the local stress tensor σij(x,y,z), the local strain tensor εij(x,y,z),

and the apparent density ρ = ρ(x,y,z):

( ) max0,, ρρρεσρ <<= ijijFdt

d (4.1)

Control and Optimization Procedure for Bone 66 Lightweight Stiffened Structures

where ρmax is the maximal density of the considered material. When the objective function F reaches zero, the system is in equilibrium. Such a generic relationship can be specified to

( ) max* 0 ρρρ

ρ <<−= SScdt

d (4.2)

where cρ is a constant, S = S(x,y,z) is the mechanical stimulus and S*

(x,y,z) is the reference value. If the structure is discretized into n elements, S is usually expressed per element (cellular automaton CA). In that case it is, in fact, assumed that there is precisely one sensor point per CA. Eq. (4.2) means that the stimulus strives to become equal to the reference value S*, which can either be site-specific (the reference state of the stimulus depends on location) or non-site-specific (the reference state of the stimulus is constant). This implicates that the normal stimulus distribution S = S*(x,y,z) must be known or determined from a normal equilibrium density distribution, in order to predict the material adaptation process to an abnormal situation, e.g. after a change in loading or geometry of the bone structure. The strain energy density (SED):

ijij21 εσ=SED (4.3)

is used as a feed-back control variable for adaptive remodeling. This choice is well-supported on physical grounds (Carter et al. 1987). It has a relation to both rigidity and strength. From a methodological standpoint it has the advantage of being a scalar variable and an invariant of the strain tensor, which implies that only one remodeling coefficient (cρ) must be established (see eq. (4.2)). Thus, the difference between the local actual SED(x,y,z,t) and a non-site-specific homeostatic equilibrium SED*, is assumed as the driving force for adaptive activity. When SED > SED* or SED < SED*, adaptive activity initiates and the rate of adaptation is proportional to cρ.


Alternatively, the changes in the apparent density ρ in each CA are converted to changes in elastic modulus E by using:

( ) ( )[ ]γρ tzyxCtzyxE ;,,;,, = (4.4)

where C and γ are constants. As the material density changes during remodeling, the elastic modulus, E (MPa), at the location P(x,y,z) is

calculated as a power function of the apparent density ρ (kg/m3). The relation between density and Young’s modulus is taken from

Carter and Hayes (1977) and, in this case, ρ is the apparent density of the bone tissue. The use of a cubic power relationship is in accordance with experimental data from Currey (1988). The basic idea is to begin with a model in which all elements have a uniform density distribution, to then apply loads and iterate the density of each CA based on a remodeling rule. In the presentation which follows, the material of the structure is initially considered as isotropic and homogeneous. The apparent density and elastic modulus subsequently vary as a function of position as the computer programs remodels the structure. To summarize, the material has the option of adapting its density or its elastic modulus according to:

( )*SEDSEDcdt

di

i −= ρρ

(4.5)

( )*SEDSEDcdt

dEiE

i −= (4.6)

for i = 1,2….n, where n is the number of cellular automata in which the structure is discretized and SEDi is the strain energy density of the CA of the discrete location i. The same idea is valid for ρi and Ei. In this way new estimates of elastic constants, eq. (4.4), could be incorporated into the new heterogeneous finite element model. With the updated properties calculated after each iteration, the finite element analyses are repeated and new estimates of the density


distribution are calculated. This iterative process continues and the progressive remodelling of the material density distribution is achieved. The total strain energy of the discretized structure can be approximated as:

∑=

=n

iiUU

1

(4.7)

where Ui is the strain energy stored in each CA. The total mass of the structure M is given by :

∑∑==

==n

ii

n

ii xmmM

1max

1

(4.8)

where mi is the variable mass, mmax the maximum mass and xi is the relative mass. The same concept for the apparent density:

ii xmaxρρ = (4.9)

The relative mass xi can range between 0 (no mass) and 1 (full mass). Following Mullender et al. (1994) and Mullender and Huiskes (1995) it is possible to set:

( ) ( ) γγγγγγ ρρρmax

maxmaxmax E

ExxExCxCCE i

iiiiii =⇒==== (4.10)

γ

1

axi

=

m

i

E

Ex (4.11)

where Ei is the variable Young’s modulus of each CA that constitutes the domain, Emax is the maximum value of the Young’s modulus, and

2 ≤ γ ≤ 3 is an empirical penalization factor.


Furthermore, a relationship between the ultimate stress σu of bone and the apparent density is enforced; the most common relation used is from the early work of Carter and Hayes (1977):

( ) 2max

2u 6868 xρρσ == (4.12)

This equation was developed to describe the strength of pooled cortical and trabecular bone specimens loaded at low strain rates

(≤1s-1). Comparing σu with the Mises stress σMises is used to check the safety of the considered structure. 4.2.2 Control Strategy The control process begins with the evaluation of the actual mechanical stimulus SEDi in each CA by the FEM analysis once the external loads, the geometry, the boundary conditions of the structure have been defined and a reference mechanical stimulus SED* has been selected. The choice of the target value SED* depends on the problem at hand. It is possible to choose, for example, SED* as the SED average of the initial design, which is the current choice. The same value is assigned to each CA. Alternatively, the target value SED* can be determined by the failure criterion with SED* =1/2x [(σy2)/mE], where σy is the yield stress of the material and m an adequate safety factor. The error signal in each CA (Huiskes et al. 1987, Beaupré et al. 1990a,b) is calculated by means of

−<−−

+≤≤−

+>+−

=**

**

**

)1()()1()(

)1()()1(0

)1()()1()(

)(

SEDltSEDforSEDltSED

SEDltSEDSEDlfor

SEDltSEDforSEDltSED

te

ii

i

ii

i (4.13)

where 2l is the width of the equilibrium zone. When ei(t)=0 no remodeling occurs and the structure is in equilibrium under the


given external loads. For many purposes it is possible to take l=0. Then eq. (4.13) simplifies into

*)( SEDSEDte ii −= (4.14)

Further investigation to account for the influence of different values of l doesn’t seem to be worthwhile to be carried out due to its insignificant influence on the final results as shown in Tovar (2004). Moreover, local evolutionary rules are inspired by error-based phenomenological models utilized in bone structure functional adaptation analysis (Carter 1987; Huiskes et al. 1987; Mullender et al. 1994, Mullender and Huiskes 1995; Huiskes et al. 2000). The local rules make use of the effective error signal defined by

]1

)()()(

)[()( 1

+

+=

∑=

N

tette

tte

N

jjji

ii

ηη

(4.15)

where ei(t) is the error signal of the CAi, ej(t) is the error signal of the neighbouring CAj (j = 1, 2, …N), N is the number of the neighbouring cells (N can be equal to 0, 4, 8, 12, 24 in two dimensional problems), ηi(t) represents the strength of the signal and ηj(t) denotes the efficiency of the transmission of the signal from cell j to cell i. In this analysis there is no deterioration in the remodeling signal and therefore ηi(t)=1 is taken. In addition also ηj(t)=1 is valid since values

of 0 ≤ ηj(t) ≤ 1 seem to be more suitable to describe the effect of microfracture on bone structure behaviour. Thus eq. (4.15) simplifies into

1

)()(

)( 1

+

+=

∑=

N

tete

te

N

jji

i (4.16)


The idea of using a neighbourhood of the CAi can be thought of a filtering technique which prevents the appearance of checkerboard patterns and mesh dependency (Patel et al. 2008). To determine the geometrical or material changes, the remodeling rates, Eqs. (4.5) or (4.6), as anticipated Chapter 3, are integrated with a constant step ∆t = 1, using a local PID control rule, whereby, the change in apparent density or, alternatively, in elastic modulus in each iteration follows from

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectttt

iiiiiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

ρρρ τρρρ (4.17)

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectEttEEt

iEiiEiEiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

τ (4.18)

or equivalently

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectxttxxt

ixiixixiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

τ (4.19)

( ) ( ) ( ) ( ) ( )[ ] ( ) nitectetectectyttyyt

iyiiyiyiii ≤≤−⋅+−−⋅+⋅=−∆+=∆ ∑=

1,10

IDPτ

τ (4.20)

where yi = Ei/Emax is the relative elastic modulus and 0 ≤ yi ≤ 1. The coefficients cP, cI, cD are referred to the proportional, integral and derivative gains and they determine the remodeling rate. The proportional gain determines the direct reaction to the current error. With the derivative control, the change is proportional to the derivative of the error with respect to a unitary change in time e(t)-e(t-1). With the integral control, the change is proportional to the sum of all previous errors. The stress analyses are done using a standard linear finite element code, COMSOL MULTIPHYSICS 3.5. At each iterative step, results from these analyses are the input to specially written algorithms by MATLAB 7.0.4 code which are used to implement Eqs. (4.18) and


(4.19). New density and modulus values are evaluated for each CA. The updated values for the Young’s modulus are then input to the comsol model and the next iterative calculations are performed. The iterative process is repeated until the stopping rule

( ) ( 1)max

( 1)

M t M t

M tε

∆ − ∆ − ≤ ∆ − (4.21)

if the relative mass is evolving according to the relationship (4.19), or

( ) ( 1)max

( 1)

E t E t

E tε

∆ − ∆ − ≤ ∆ − (4.22)

if the Young’s modulus is evolving according to eq. (4.18) are satisfied for a chosen threshold 0ε > . It means that no more significant changes in the mass density/elastic modulus distribution

occur. In addition, a saturation density of ρmax = 1740 kg/m3 is assumed so that the model can not predict a density greater than the cortical bone density (Carter et al. 1989). Computer programs, evaluating the changes in modulus and geometry and based on a change in SED, are developed and combined with the FEM code, as illustrated in the block diagram of Fig. 4.1.


Fig. 4.1 Schematic representation of the adaptive remodeling program, integrated with the finite element method (FEM) code. The feed-back control variable is the difference between the actual SED and the homeostatic SED*.

Each CA has, in principle, three possibilities to converge and, hence to reach the remodeling equilibrium: (1) the material is completely

resorbed (ρ = ρmin and E = Emin); (2) the material reaches the

maximum value of the density (ρ = ρmax and E = Emax); or (3) the structure has an apparent density satisfying Eqs. (4.17) or (4.18) and, hence, SED = SED*. The optimized control gains aim at realizing a structure having maximum stiffness and minimum weight; maximizing stiffness means minimizing the strain energy and this goal, as anticipated in Chapter 3, can be accomplished by defining the cost indices:

001 )1(

M

M

U

UJ ωω −+= (4.23)

and


002 M

M

U

UJ ⋅= (4.24)

where U0 and M0 are respectively the total strain energy and the mass when xi = x0 (for i = 1,2,…n) of the domain to be optimized and

0 ≤ ω ≤ 1. In the cost index J1 the influence of the mass and the energy terms depends on the weight parameter ω. In the control procedure, the behaviour of the enhanced model is

studied for variations of initial density distributions ρ0, the PID coefficients and the multipliers of normalized total mass and energy in the cost index. These parameters are optimized in the subsequent optimization procedure. To solve for the approximate density distribution within a structure with a specific loading condition, homogeneous material properties and an associated apparent density are first assigned to all cellular automata. Then the strain energy density for each CA is calculated for the assumed loading condition. 4.2.3 Optimization Strategy The last consideration in the previous section, as already suggested, permits considering the following optimization problem: the evaluation of an optimal choice for the control parameters, for the target and also for the weight parameter ω in order to minimize the cost indices (4.23) and (4.24):

( ) ( ) ( ) ( )

+−=

0

*DIP

0

*DIP

,,,,1

,,,,,,1

*DIP U

SEDcccU

M

SEDcccMMinJMin

SEDcccωω

ω (4.25)

and

( )( ) ( )

⋅=

0

*DIP

0

*DIP

,,,2

,,,,,,*

DIP U

SEDcccU

M

SEDcccMMinJMin

SEDccc

(4.26)


The relative mass distribution x is obtained for the optimal choice of the parameters cP, cI, cD, SED* and ω. The set of optimization variables is constrained:

*0, 0, 0, 0, 0 1p I Dc c c SED ω≥ ≥ ≥ > ≤ ≤ (4.27)

Problems (25) and (26) with conditions (27) are non linear constrained optimization problems; their analytical solution is not possible since the closed-form expressions of the total mass and of the total energy are not available. Therefore a numerical solution is essential. To speed up the convergence and to avoid local minima, it is convenient to constraint more strictly the control parameters and the target. The choice of the limits of the constraints may be based on the control strategy in which the above parameters were fixed. Therefore the constraints considered for the numerical solution of problems (25) and (26) are:

cPmin ≤ cP ≤ cPmax

cImin ≤ cI ≤ cImax

cDmin ≤ cD ≤ cDmax

SED*min ≤ SED* ≤ SED*max

0 ≤ ω ≤ 1

(4.28)

Such constraints, along with the characteristics of the objective function, allow the application of the Weierstraβ Theorem, guaranteeing the existence of an optimal solution. Furthermore, the safety condition establishes that the following relation has to hold:

01 >−u

Mises

σσ

(4.29)

In order to avoid the violation of this condition a penalty term P is added to the cost indices (23) and (24):


* *( , , , , ) ( , , , , )p I D p I DJ c c c SED P c c c SEDω ω+ (4.30)

Different choices may be considered for the penalty term; at each step, the maximum of the absolute value attained by ineq. (4.29) is assumed for P in the cellular automata where the safety condition is violated:

( )[ ] ( )[ ] 01when1 iuMisesiuMisesn,1i

≤−

−⋅=

=σσσσMaxAP (4.31)

with A = 10. In this study, the optimization process allows the

finding of a set of parameter values (cP, cI, cD, SED *, ω) that guarantee a safe structure. In Fig. 4.2 the block diagram relative to the optimization strategy is shown.


Fig. 4.2 Block diagram of the optimization strategy. The function fmincon of ©Matlab allows the finding of a constrained minimum of a function of several variables.

When also the parameter optimization is considered another step should be added to the previous control scheme (Fig. 4.1). An initial condition for the parameters to be optimized is assigned; the FEM algorithm along with the remodeling one returns a value for the chosen cost index. Successively the optimization procedure suitably modifies the set of parameters. For the new set of parameters the FEM and remodeling algorithms return a new value for the cost index (Fig. 4.3). And so on.


Fig. 4.3 C

omplete control and optimization procedure. Sch

eme of the optimal iden

tification of

the control parameters, of the target and of the weight

ω for the cost index

J1.

In the optimization algorithm block of Fig. 4.2 the function fmincon of ©MATLAB has been used; it permits finding of a constrained minimum of a function of several variables. This function uses sequential quadratic programming (SQP) method (Fletcher 1987). In this method, the function solves a quadratic programming


subproblem (Gill et al. 1981) at each iteration in which a positive definite quasi-Newton approximation of the Hessian of the Lagrangian function is updated using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method (Battiti 1992). This method is derived from the Newton’s method in optimization field. Initial conditions and constraint limits for the set of parameters are chosen; any admissible value for initial conditions does work. The optimization procedure stops the iterations when the first order optimality conditions are satisfied to the specified tolerance of 1×10-

6. 4.2.4 2-D Models The method proposed in the previous paragraphs is tested in a two-

dimensional FEM model of a square domain of 5×5 mm with a

thickness of 0.04 mm (40 µm), as applied earlier by Weinans et al. (1992). The bone sample is loaded by a compressive load distribution, decreasing linearly over the top edge. It is divided in 25 and 625 cellular automata (Figs. 4.4, 4.5), each one discretized by quadratic triangular Lagrange elements with six nodes at the corners and side midpoints. In another case, the investigated Michell-type structure is considered as bone sample (Fig. 4.6). To solve for the mean SED distribution within the structure with a specific loading condition, homogeneous mass density and associated material properties are first assigned to all elements. The strain energy density in each CA is calculated for the assumed loading condition. Therefore, the target SED* could be approximated as the surface average SED of the initial design

Area

U

Area

USED

∑===

n

1ii

* (4.32)


by dividing the total strain energy U, eq. (4.7), for the area of the domain. In more details, the first examined case is the bone sample discretized in 25 cellular automata (Fig. 4.4).

Fig. 4.4 Two-dimensional bone structure of 5×5 mm (40 µm thick) discretized in 25 cellular automata (5x5) and subjected to a compressive load.

Two uniform initial density distributions of ρ0 = 0.9 × ρmax = 1566

kg/m3 and ρ0 = 0.5 × ρmax = 870 kg/m3 are used with the maximal

density ρmax equal to 1740 kg/m3. The maximum modulus Emax is equal to 17 GPa (femoral cortex longitudinally loaded in compression (Weinans et al. 1992). The relation between Young’s modulus E and

the apparent density ρ is taken as eq. (4.10). The local elastic properties are calculated from the local relative density with use of a cubic power relationship in accordance with experimental data from

Currey (1988). Therefore, the Young’s moduli are equal to E0(ρ0 =

1566) = 12.393 GPa and E0(ρ0 = 870) = 2.125 GPa, respectively. The

Poisson’s ratio is constant (ν = 0.3) and does not change when the

0.001 m

0.005 m

107 N/m2

0.005 m 0.001 m

4x10-5 m


local bone apparent density changes. The material is assumed to be initially homogeneous and isotropic. The minimal density is

0.01xρmax, and represents complete resorption of a CA. The reference

values of the strain energy density for the two initial densities (ρ0 =

1566 kg/m3 and ρ0 = 870 kg/m3) are SED* = 7.14 J/m2 and SED* = 41.72 J/m2. The second examined case is the bone sample discretized in 625 cellular automata (Fig. 4.5). A uniform initial density distribution of

ρ0 = 870 kg/m3 is assumed. The initial Young’s modulus is equal to E0 = 2.125 GPa. The selected target for this case is SED* = 25 J/m2.

Fig. 4.5 Two-dimensional bone structure of 5×5 mm (40 µm thick) discretized in 625 cellular automata (25x25) and subjected to a compressive load.

The third investigated case is the Michell-type structure, discretized in 1250 cellular automata (Fig. 4.6). The initial density is uniformly equal to 1740 kg/m3. The initial Young’s modulus is equal to E0 = 17 GPa. The selected target is SED* = 2.5 J/m2.

0.005 m

0.005 m

4x10-5 m 107 N/m2


Fig. 4.6 Michell-type structure discretized in 1250 cellular automata (50x25). The lower edge is subjected to a tensile load.

Two comparative analyses are performed in the following section by taking the rate of change of the relative mass xi and of Young’s modulus Ei as free variables and by expressing them as a function of the effective error signal ēi(t) in the form of a PID control.

4.3 Numerical Results

In this section the numerical results originated by the two-step procedure (control and optimization procedure), applied to the 2D models and above illustrated, are presented. In the first stage, the

control parameters, the target and the weight ω for the cost index J1 are fixed, after trial experiments, whereas in the second stage the mentioned quantities are refined by a numerical optimization procedure. To evaluate the performance of the remodeling algorithm, the cost index J1 and J2 are assumed and evaluated for each iteration. In the optimization stage, the control gains cP, cI, cD, the target SED* and the

0.05 m

0.001 m

0.025 m

100 N


weight ω are obtained by the minimization problems (25) and (26) with constraints (27). For each type of evolution, both the cost indices J1 and J2 are minimized with different initial conditions. In addition, for each adopted model the evolutions of the total optimized mass and energy are displayed. The initial conditions and constraint limits of the parameters to be optimized are summarized in several tables. The results provided by the optimization procedure are illustrated, in both cases of mass and stiffness implementation. To conclude, the initial and final configuration of the relative mass distribution for each bone sample are shown. 4.3.1 Bone Sample (25 cellular automata) The evolutions of the total optimized mass and energy are displayed in Fig. 4.7, assuming the initial mass uniformly equal to 0.9 and the evolution governed by eq. (4.19). In the same initial conditions the corresponding evolutions of the cost indices J1 and J2 are shown in Fig. 4.8 and evaluated with the optimized values of the involved parameters.

0 5 10 15 20 250.75

0.8

0.85

0.9

0.95

1

1.05

1.1

ITERATIONS

Optmized

Mass

Optimized

Energy

Fig. 4.7 Evolution of total optimized mass and energy under the implemented relative mass rule. The initial relative mass x0 is uniformly assumed equal to 0.9.

U/U0 and

M/M0


0 5 10 15 20 250.75

0.8

0.85

0.9

0.95

1

ITERATIONS

Cost Index Sum

Cost Index Product

Fig. 4.8 Evolution of the cost indices J1 and J2, minimized by the optimal parameters. The relative mass rule is implemented. The initial relative mass x0 is uniformly assumed equal to 0.9. Table IV.1 and IV.2 collect the initial conditions, the constraint limits of the parameters at the beginning of the optimization process and the optimal parameters at the end of the optimization process. The implemented rule regards the relative mass distribution with x0 = 0.9.

Table IV.1 Initial conditions and constraint limits of the parameters to be optimized.

Parameter x cP cI cD SED * [J/m2] ω

t0 0.9 5x10-5 5x10-5 5x10-5 7.14 0.5

min 0 3x10-5 3x10-5 3x10-5 2 0

max 1 4x10-4 4x10-4 4x10-4 8 1

J1 and J2


Table IV.2 Optimal parameters when the relative mass evolution rule is implemented.

optPc opt

Ic optDc SED*opt [J/m2] optω

optJ1 4x10-4 4x10-4 3x10-5 3.57 0.9

optJ 2 4x10-4 4x10-4 3x10-5 3.57

Finally, in Figs. 4.9 and 4.10 the configurations of the initial and final relative mass distributions are sketched. They are obtained at the end of the optimization process by applying the evolution rule of the relative mass starting from x0 = 0.9 and x0 = 0.5, respectively.

Fig. 4.9 Initial mass distribution with x0 = 0.9 and final optimal mass distribution. The bone structure is initially considered homogeneous and isotropic. At the end of the optimization process the bone structure has become heterogeneous and anisotropic.

INITIAL RELATIVE MASS DISTRIBUTION

107 N/m2

FINAL RELATIVE MASS DISTRIBUTION


Fig. 4.10 Initial mass distribution with x0 = 0.5 and final optimal mass distribution. The bone structure is initially considered homogeneous and isotropic. At the end of the optimization process the bone structure has become heterogeneous and anisotropic. 4.3.2 Bone Sample (625 cellular automata) The evolutions of the total optimized mass and energy are illustrated in Fig. 4.11, assuming the initial mass uniformly equal to 0.5 and the evolution governed by eq. (4.19). In the same initial conditions the corresponding evolutions of the cost indices J1 and J2 are shown in Fig. 4.12 and evaluated with the optimized values of the involved parameters.

INITIAL RELATIVE MASS DISTRIBUTION

FINAL RELATIVE MASS DISTRIBUTION

107 N/m2


0 20 40 60 80 100 120 1400.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ITERATIONS

Optmized

Mass

Optmized

Energy

Fig. 4.11 Evolution of total optimized mass and energy under the implemented relative mass rule. The initial relative mass x0 is uniformly assumed equal to 0.5.

0 20 40 60 80 100 120 1400.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ITERATIONS

Cost Index Sum

Cost Index Product

Fig. 4.12 Evolution of the cost indices J1 and J2, minimized by the optimal parameters. The relative mass rule is implemented. The initial relative mass x0 is uniformly assumed equal to 0.5.

U/U0 and

M/M0

J1 and J2


It should be noted in Figs. 4.7 and 4.11 that the unit value in the ordinate axis simply means M (0)/M0 = 1, where M0 = x0 Mmax. Table IV.3 and IV.4 summarize the initial conditions, the constraint limits of the parameters at the beginning of the optimization process and the optimal parameters at the end of the optimization process. The implemented rule regards the relative mass distribution with x0 = 0.5.



t0 0.5 5x10-5 5x10-5 5x10-5 25 0.5

Min 0 2x10-6 2x10-6 2x10-6 10 0

Max 1 8x10-4 8x10-4 8x10-4 50 1


optPc opt

Ic optDc SED*opt

[J/m2] optω

optJ1 1.2x10-5 1.2x10-5 1.2x10-5 31.3 0.76

optJ 2 1.28x10-5 1.17x10-5 1.17x10-5 31.3

Finally, in Fig. 4.13 the configurations of the initial and final relative mass distributions are illustrated. They are obtained at the end of the optimization process by applying the evolution rule of the relative mass starting from x0 = 0.5.


Fig. 4.13 Initial mass distribution with x0 = 0.5 and final optimal mass distribution. The bone structure is initially considered homogeneous and isotropic. At the end of the optimization process the bone structure has become heterogeneous and anisotropic. 4.3.3 Michell-type Structure (1250 cellular automata) The evolutions of the total optimized mass and energy are displayed in Figs. 4.14 and 4.16, assuming the initial mass uniformly equal to 1 and the evolutions governed by Eqs. (4.18) and (4.19). In the same initial conditions the corresponding evolutions of the cost indices J1

and J2 are shown in Figs. 4.15 and 4.17 and evaluated with the optimized values of the involved parameters.

INITIAL RELATIVE MASS

DISTRIBUTION

FINAL RELATIVE MASS

DISTRIBUTION

107 N/m2


0 10 20 30 40 50 60 70 800.5

1

1.5

2

ITERATIONS

Optimized

Mass

Optimized

Energy

Fig. 4.14 Evolution of total optimized mass and energy under the implemented relative mass rule. The initial relative mass x0 is uniformly assumed equal to 1.

0 10 20 30 40 50 60 70 80 900.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

ITERATIONS

Cost Index Product

Cost Index Sum

Fig. 4.15 Evolution of the cost indices J1 and J2, minimized by the optimal parameters. The relative mass rule is implemented. The initial relative mass x0 is uniformly assumed equal to 1.

U/U0 and

M/M0

J1 and J2


In Table IV.5 and IV.6 the initial conditions, the constraint limits of the parameters at the beginning of the optimization process and the optimal parameters at the end of the optimization process are summarized. The implemented rule regards the relative mass distribution with x0 = 1.



t0 1 0.0525 0.00275 0.00275 2.5 0.5

min 0 0.005 5x10-4 5x10-4 1 0

max 1 0.1 0.05 0.05 5 1


optPc opt

Ic optDc SED*opt [J/m2] optω

optJ1 0.02 0.001 0.001 2.18 0.23

optJ 2 0.01 0.001 0.001 2


0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

ITERATIONS

Optimized

Mass

Optimized

Energy

Fig. 4.16 Evolution of total optimized mass and energy under the implemented Young’s modulus rule. The initial relative mass x0 is uniformly assumed equal to 1.

0 10 20 30 40 50 60 700.7

0.8

0.9

1

1.1

1.2

1.3

1.4

ITERATIONS

Cost Index Product

Cost Index Sum

Fig. 4.17 Evolution of the cost indices J1 and J2, minimized by the optimal parameters. The Young’s modulus rule is implemented. The initial relative mass x0 is uniformly assumed equal to 1.

U/U0 and

M/M0

J1 and J2


In Table IV.7 and IV.8 the same parameters are indicated, as in previous Tables, but the implemented rule regards the Young’s modulus distribution with x0 = 1.


Parameter x cP

[Pa(m/J2)]

cI

[Pa(m/J2)]

cD

[Pa(m/J2)]

SED *

[J/m2] ω

t0 1 2.52e8 2.52e8 2.52e8 2.5 0.5

min 0 5.5e6 5.5e6 5.5e6 1 0

max 1 5e8 5e8 5e8 5 1

Table IV.8 Optimal parameters when the Young’s modulus evolution

rule is implemented.

optPc

[Pa(m/J2)]

optIc

[Pa(m/J2)]

optDc

[Pa(m/J2)] SED*opt [J/m2]

optω

optJ1 10e7 10e7 10e7 2.18 0.24

optJ 2 15e7 10e7 8e7 2.18

In conclusion, in Fig. 4.18 the configurations of the initial and final relative mass distributions are designed. They are obtained at the end of the optimization process by applying the evolution rule of the relative mass starting from x0 = 1.


Fig. 4.18 Initial mass distribution with x0 = 1 and final optimal mass distribution. The bone structure is initially considered homogeneous and isotropic. At the end of the optimization process the bone structure has become heterogeneous and anisotropic.

4.4 Discussion From the results presented in the previous section derive many considerations. For the Bone Sample with 25 cellular automata, the optimization process with the initial condition x0 = 0.9 required twenty iterations to converge to an optimal state (Figs. 4.7, 4.8), and exhibited an evolution rate which was fairly faster than that one for x0 = 0.5, which required a number of iterations as larger as 50%. The optimized mass slightly augmented with respect to the initial value but a gain of 24 % in stiffness was reached. The optimized decrements achieved under mass rule were generally larger than those ones under stiffness rule. Table IV.2 shows that the

optimal value for ω is equal to 0.9 and this implied that the term weighted more significantly is the energy one. With mass implementation and elastic modulus implementation the optimal

INITIAL RELATIVE MASS

DISTRIBUTION

FINAL RELATIVE MASS

DISTRIBUTION


weight was equal to 0.9 when the initial mass was equal to 0.9, whereas when the initial mass was equal to 0.5 the optimal weight was equal to 0.1; in this case the mass term was weighted more significantly with respect to the energy one. The cost index J2 decreased faster than the cost index J1, whatever the weight value was. The best target value was about the half of the value obtained as SED* average of the initial design, when x0 = 0.9, both for mass and stiffness implementation. It was found that the solution obtained was a discontinuous patchwork (Figs. 4.9, 4.10). For a two-dimensional domain, this patchwork showed a good resemblance with the density distribution of the model tested by (Weinans et al. 1992). The discontinuous final configuration was dictated by subdividing the domain in a small number of cellular automata and by the nature of the local evolution equations describing the remodelling process, even mitigated by the effective error signal averaged over the neighbouring elements. For the Bone Sample with 625 cellular automata, the optimization process with the initial condition x0 = 0.5 required 117 iterations to converge to an optimal state (Figs. 4.11, 4.12). The optimized mass slightly augmented with respect to the initial value but a gain of 70 % in stiffness was reached. Table IV.4 shows that the optimal value for

ω is equal to 0.76 and this implied that the term weighted more significantly was the energy one. The cost index J1 and J2 followed the same path but J2 assumed values slightly lower than the J1 ones. The optimal target was a little higher than the initial one. Fig. 4.13 displays the initial mass distribution with x0 = 0.5 and the final optimal mass distribution. The bone structure was initially considered homogeneous and isotropic. At the end of the optimization process the bone sample has become heterogeneous and anisotropic and was consistent in applied boundary and loading conditions. For the Michell-type Structure with 1250 cellular automata, when the relative mass rule was implemented, the optimization process with the initial condition x0 = 1 required 80 iterations to converge to an

optimal state (Figs. 4.14, 4.15). The optimal value of the weight ω was


0.23, so that the mass term was weighted more significantly. In fact, Fig. 4.14 showed a gain of 50 % for the mass. When the Young’s modulus rule was implemented, the optimization process with the initial condition x0 = 1 required 60 iterations to converge to an

optimal state (Figs. 4.16, 4.17). The optimal value of the weight ω was 0.24, so also in this case the mass term was weighted more significantly. In fact, Fig. 4.16 shows a gain of 50 % for the mass. For both cases, the cost index J2 revealed the best behaviour in the plots. The optimal target was equal to 2.18 J/m2 for both implemented relative mass and Young’s modulus rules, and it was lower than the initial value. Fig. 4.18 exhibits the initial mass distribution with x0 = 0.5 and the final optimal mass distribution. The bone structure was initially considered homogeneous and isotropic. At the end of the optimization process the bone sample has become heterogeneous and anisotropic and was consistent in applied boundary and loading conditions. General speaking, for an adequately refined domain discretization, the geometric characteristics of the solution would depend only on the characteristics of the loads, the value of the target and the maximal attainable elastic modulus (hence, the maximal density).

References Battiti R, 1992. First and second order methods for learning: Between steepest descent and Newton's method, Neural Computation, vol. 4, no. 2, pp. 141-166. Beaupré GS, Orr TE and Carter DR, 1990b. An approach for time-dependent bone remodeling-theoretical development, J Orthop. Res., 8(5), 651-661. Beaupré GS, Orr TE, Carter DR, 1990a. An approach for timedependent bone modeling and remodeling-application: a preliminary remodeling simulation, J Orthop Res 8:662-670. Carter DR and Hayes WC, 1977. The compressive behaviour of bone as a two-phase porous structure J. Bone Joint Surg. [Am], 59(7), 954–962.


Carter DR, Fyhrie DP and Whalen RT, 1987. Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy, J Biomech. 20,785–794. Carter DR, 1987. Mechanical Loading History in Skeletal Biology, J Biomech, 20, 1095-1105. Carter DR, Orr TE and Fyhrie DP, 1989. Relationships between Loading History and Femoral Cancellous Bone Architecture, J Biomech, 22: 231-244. Currey JD, 1988. The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone, J Biomech 21:131-139. Fletcher R, 1987. Practical Methods of Optimization,John Wiley and Sons. Gill PE, Murray W and Wright MH, 1981. Practical Optimization, London, Academic Press. Huiskes R, Ruimerman R, van Lenthe GH and Janssen JD, 2000. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone, Nature (405), 704-706. Huiskes R, Weinans H, Grootenboer J, Dalstra M, Fudala M and Slooff TJ, 1987. Adaptive Bone Remodelling Theory Applied to Prosthetic-Design Analysis, J Biomech, 20, 1135-1150. Mullender MG and Huiskes R, 1995. Proposal for the regulatory mechancism of Wolff’s law, J. Ortop. Res., 13(4), 503-512. Mullender MG, Huiskes R and Weinans H, 1994. A Physiological Approach to the Simulation of Bone Remodelling as a Self-Organizational Control Process, J Biomech, 27, 1389-1394. Patel NM, Tillotson D, Renaud JE, Tovar A and Izui K, 2008. Comparative Study of Topology Optimization Techniques, AIAA J., 46 (8), 1963-1975. Tovar A, 2004. Bone Remodelling as a Hybrid Cellular Automaton Optimization Process, PhD Thesis, University of Notre Dame, Aerospace and Mechanical Engineering, Notre Dame, Indiana. Weinans H, Huiskes R and Grootenboer HJ, 1992. The behaviour of adaptive bone-remodelling simulations models, J Biomech, 25(12), 1425-1441.

CONCLUSIONS The process of functional adaptation enables bone to perform its mechanical functions with a minimum of mass and with the strength necessary to support mechanical loads associated with daily activities and to protect internal organs. This thesis achieved the goal of designing two-dimensional bone samples having maximum stiffness and minimum weight in an efficient computational way. Maximizing stiffness is equivalent to minimize the strain energy but total mass and global stiffness are conflicting objectives. The aim was fulfilled by implementing a computer program which allowed the interaction between evolution rules of bone density and structural analysis of the model under given loads. Theoretical and numerical tools (FEM analysis, topology optimization of structures, cellular automaton model, adaptive elasticity theory and PID control) were adopted in this study for trying to imitate the bone tissue adaptation. The novelty of the proposed approach is three-fold, concerning the chosen alternative evolution rules, the implemented consecutive control and optimization procedures, and the considered cost indices, among which the product of mass and energy originally introduced. Two different implemented evolution rules (for relative mass and Young’s modulus) assumed a PID control. The process comprised two stages: the first stage was constituted by a control (trial and error) procedure, which provided refined initial values of the parameters (the control gains, the target and the weight of the cost index J1) to start the subsequent optimization procedure (the second stage), which - in turn - led to the optimal solution. The automatic code was characterized by rapid convergence at low computational cost in terms of computation time and memory occupation; these advantages allowed comparing different cost indices, namely the linear combination of relatively weighted mass


and strain energy, and the product of mass and strain energy. A good compromise seemed to be enclosed in the latter one. The bone samples have been discretized in 25, 625 and 1250 cellular automata, but the proposed model is suitable to be used not only in bone mechanics but in many other fields of artificial materials discretized in whatever number of cellular automata. The mechanical behaviour of the structure and the shape of bone has become a main expanding interest in the scientific world. This thesis is an example of multidisciplinary work, combining different research fields. Future developments could concern the optimization of three-dimensional bone samples, the incorporating of the mechanical load into the set of parameters to be optimized and the introducing of experimental data, i.e., the comparison between data deriving from the complete optimization procedure designed in this investigation and data from measuring of bone mineral density by Dual Energy X-Ray Absorptiometry (DEXA).

Appendix Implementation of the

Control and Optimization Procedure

In this section the part of the complete control and optimization procedure written in MATLAB 7.0.4. code is synthesised. The following algorithm encloses the stress analysis by a standard linear finite element code, COMSOL MULTIPHYSICS 3.5 and the implementation of the evolution rule, of the optimization problem and, finally, of the function fmincon. % COMSOL Multiphysics Model M-file % COMSOL version clear vrsn vrsn.name = 'COMSOL 3.5'; vrsn.ext = ''; vrsn.major = 0; vrsn.build = 405; fem.version = vrsn; garr = geomimport('C:\Documents and Settings\sample.mphbin'); [g1]=deal(garr:); % Analyzed geometry clear s s.objs=g1; s.name='CO1';


s.tags='g1'; fem.draw=struct('s',s); fem.geom=geomcsg(fem); % Initialize mesh fem.mesh=meshinit(fem, ... 'hauto',5); % Application mode 1 clear appl appl.mode.class = 'SmePlaneStress'; appl.module = 'SME'; appl.gporder = 4; appl.cporder = 2; appl.assignsuffix = '_smps'; clear pnt pnt.Hy = 0,1,1; pnt.Hx = 0,1,0; pnt.ind = [2,1,1,1,1,1,3,1,1,1,1,1,3,1,1,1,1,1,3,1,1,1,1,1,3,1,1,1,1,1, ... 3,1,1,1,1,1]; appl.pnt = pnt; clear bnd bnd.loadtype = 'length','area'; bnd.Fy = 0,'(200*x-1)*10e7'; bnd.ind = [1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1, ... 1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1]; appl.bnd = bnd; clear equ equ.rho = 1740; equ.thickness = 4e-5; equ.E = 2.125e9; equ.ind = [number of CAs]; appl.equ = equ; fem.appl1 = appl; fem.frame = 'ref'; fem.border = 1;

Implementation of the Control and Optimization Procedure 102

clear units; units.basesystem = 'SI'; fem.units = units; % Multiphysics fem=multiphysics(fem); % Extend mesh fem.xmesh=meshextend(fem); % Solve problem fem.sol=femstatic(fem, ... 'solcomp','u','v', ... 'outcomp','u','v'); % Save current fem structure for restart purposes fem0=fem; for i=1:n I(i)=postint(fem,'Ws_smps', ... 'unit','J', ... 'dl',[i], ... 'edim',2, ... 'intorder',4, ... 'geomnum',1, ... 'solnum','end', ... 'phase',0); end SED=I' % -----FMINCON FUNCTION-------- function f=myfunction(c) c omega=c(5) %---------COST INDICES---------- Corr=4.35*10^(-5); omega3=1;


Mass0; Energy0; f11 f22 f3=omega3*(Pena) f=(1-omega)*f11+omega*f22+f3 f=f11*f22+f3 A=[ ]; b=[ ]; TT; %-------------GEOMETRY---------------- pp LX Area %------INITIAL CONDITIONS : CONTROL GAINS, TARGET AND OMEGA ----- cp0 ci0 cd0 T0 omega_0 %------INFERIOR LIMITS------------ cpm cim cdm Tm= omegam %-----SUPERIOR LIMITS---------------- cpM ciM cdM TM omegaM %------------------------------------ c0=[cp0 ci0 cd0 T0 omega_0];


lb=[ cpm cim cdm Tm omegam ]; ub=[cpM ciM cdM TM omegaM]; %----OPTIMIZATION PROBLEM-------- [c fval]=fmincon(@miafunzione, c0, A,b, [ ], [ ],lb, ub) c fval % -----FROM FMINCON------------------ cp=c(1) ci=c(2) cd=c(3) Target=c(4) omega=c(5) %------------------------------------- SED=[ ]; MISES=[ ]; %-------GEOMETRY AND INITIAL CONDITIONS---------------------- Area pp LX pen E0 x0 Ei=[ ] Etarget %----------STRAIN ENERGY DENSITY--------------------- [dr, ds]=size(SEDELEM); %-------------SEDNEIGHBORHOOD----------------------------------- SEDNEIGHBOR=[ ]; %----------------EFFECTIVE ERROR-------------- Error=SEDNEIGHBOR-Target; [s1e s2e]=size(Error); Efferr=[ ]; %--------------CONTROLLER---------------------- ErrorInt=[ ]; Efferr0=zeros(res,ces);


ErrorInt=Efferr+Efferr0 ; Up=cp*Efferr; Ud=cd*(Efferr-Efferr0); Ui=ci*(ErrorInt); U=Up+Ud+Ui ; %---------LOCAL RULE -------------------------- x=x0+U; %--------SATURATION--------------------------- for l=1:LX if (x(l)>1) x(l)=1; end if(x(l)<0) x(l)=0.001; end end SC=abs((Efferr)/Target); IT=0 %------------------------------------------------- if max(SC(:))<epsilon %--------------------------------------- 'THE END' else [rx, cx]=size(x); for ii=0: rx-1 x1(ii+1,:)=x(rx-ii,:); end x=x1; %------------------------------- equ.E = Young's modulus of each CA; equ.ind = [CAs number]; for ii=1:LX equ.Eii=Etarget*x(ii)^pen; end [rx, cx]=size(x); for ii=0: rx-1


x1(ii+1,:)=x(rx-ii,:); end x=x1; end %-----------PENALIZATION----------------- V=68*1740^2; MISES Penalization=1-MMISES./(V*x); Pe=Penalization; for ii=1:rx for jj=1:cx if Penalization(ii, jj)>0 Pe(ii,jj)=0; end end end PE=max(abs(Pe(:))) ETOT=(sum(SED(:))/(LX*Area))/Energy0; EnergyTot=[EnergyTot;ETOT]; MassTot=[MassTot;sum(x0(:))/Mass0]; mass=MassTot; f11=sum(x0(:))/Mass0 f22=ETOT

Bone Remodeling Process as an Optimal Structural Design ...elasticity theory and PID control) that...

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