Hybrid cellular automaton hierarchical algorithm to predict ... - CiteSeerX

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Submitted to Journal of Biomechanics

Hybrid cellular automaton hierarchical algorithm to predict bone density distribution Andrés Tovar1

Glen L. Niebur2

Ryan K. Roeder3

John E. Renaud4

Submitted: November 10, 2005

Abstract Simulations of bone functional adaptation can be applied to improving our understanding of its structural changes due to the use of tissue-engineered devices, aging or disease. Several finite element-based optimization algorithms have been used to predict bone apparent density distribution. After a few iterations, most of the proposed algorithms are able to obtain a density distribution that is similar to the one observed in long bones. However, these algorithms converge to a black-and-white structure that does not correspond to a realistic layout. This work presents a novel hierarchical algorithm to predict bone apparent density distribution as a convergent solution. The upper (continuum) level of this hierarchical algorithm considers bone as an organ, while the lower (tissue) level simulates the functional adaptation of the mineralized bone material making use of the cellular automaton paradigm. The proposed algorithm has been applied a classical plate test problem to show its implementation. Keywords: Bone remodeling; Bone apparent density distribution; Topology optimization; Hybrid cellular automata.

1 Introduction In the last four decades, several mathematical models have been proposed to explain and predict bone functional adaptation to changes in its mechanical environment (Frost, 1964; Cowin & Hegedus, 1976; Hart et al., 1984a; Frost, 1987; Carter, 1987; Huiskes et al., 1987). Some recent models have been developed to predict changes in bone structure due to: aging or disease (van der Linden et al., 2001, 2004), the effect of antiresoptive drugs (van der Linden et al., 2003), and immobilization or microgravity exposure (Foldes et al., 1990; Cowin, 1998). In joint arthroplasties, computational models are intended to be used in preclinical tests to improve medical procedures and prosthesis design


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(Prendergast & Maher, 2001; Stolk et al., 2003). The ultimate objective of these computational implementations is to decrease the current dependence on economically and socially expensive clinical trails and animal experimentation. Functional adaptation in bones has been observed in clinical cases and experimental studies. In conjunction with these investigations, several mathematical models have been proposed to explain the relationship between mechanical loading and trabecular bone structure. According to (Hart, 2001), these models can be classified into three groups: phenomenological, mechanistic and optimization modeling. The majority of bone adaptation models have been phenomenological. These models seek to quantitatively describe the mechanical stimulus (cause) and the consequent adaptive response (effect). Two decades ago, Carter (1987) and Huiskes et al. (1987) introduced phenomenological models based on the finite element analysis. The first practical computational algorithms were developed under the assumption of isotropy of the trabecular structure at the continuum level and predicted a static equilibrium state after remodeling (Huiskes et al., 1987; Beaupré et al., 1990a; Weinans et al., 1992). Density changes were translated into changes in the material constants via a scalar remodeling rule by implicitly assuming that bone remains isotropic. After a few iterations, these algorithms were able to predict a bone apparent density distribution that was similar to the one of in vivo bone samples. However, when these algorithms converge, they produced black-and-white structure that does not correspond to a realistic layout at a continuum level. Based on the results presented by Weinans et al. (1992), Mullender et al. (1994) and Mullender & Huiskes (1995) proposed the use of a similar phenomenological algorithm to simulate bone functional adaptation at a tissue level, in which a blackand-white structure makes perfect sense. Phenomenological models have been used in engineering studies related to the design of orthopedic implants and medical applications as a guide for treatment of a variety of metabolic bone diseases (Hart, 2001). However, phenomenological models do not directly contribute to understanding of the biological basis of functional adaptation. This level of modeling requires mechanistic approaches that go beyond the cause-and-effect-based phenomenological models (Hart & Fritton, 1997). More recently, continuum-level models have been developed to consider the anisotropic nature of trabecular bone. Some of these models make use of optimization principles to predict the apparent density distribution of the bone (Jacobs et al., 1997; Fernandes et al., 1999; Bagge, 2000; Fernan-


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des et al., 2002). The application of optimization analysis for different problem formulations has resulted in a variety of interesting results for the constrained objective functions, e.g., maximizing stiffness subject to a mass constraint (Bagge, 2000; Fernandes et al., 2002). Even though these idealized microstructures reflect some approximated mechanical aspects, they are not able to represent the physiological nature of trabecular bone. These studies give insight into bone as a mechanical structure but, as Hart (2001) points out, they involve three major deficiencies. First, they rest on the assumption of a high level coordination in the body without a strong foundation of evidence. Second, they fail to consider physiological processes and, therefore, are unable to provide an understanding of the adaptive process. Third, within the framework of these studies, time dependence and other fundamental questions cannot be investigated. This work presents a novel technique that overcomes the numerical difficulties of the previous approaches. This technique is referred to as the hybrid cellular automaton (HCA) hierarchic algorithm. This algorithm predicts apparent density distribution as a convergent solution and also models the trabecular architecture with no physical nor mathematical restrictions. The HCA hierarchical algorithm incorporates tissue-level phenomenological mechanisms of bone functional adaptation into a continuum-level model. In this way, the anisotropic nature of trabecular bone will be determined by direct simulation and not by mathematical approximation. This hierarchical model combines remodeling principles at the tissue level with structural analysis at the continuum level. This technique makes use of the hybrid cellular automaton (HCA) paradigm presented by (Tovar et al., 2004a,b, 2005d) and the concepts of structural optimization presented by (Tovar et al., 2005c; Tovar, 2005). This paper describes the principles of the HCA algorithm and the methodology, implementation and application of the hierarchical model.

2 Hybrid cellular automata 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. This situation leads to what is called chaotic behavior. The same complications occur in discrete systems. Cellular automata (CAs) provide an alternative method to describe, understand and simulate the behavior of complex systems (Chopard & Droz, 1998; Deutsch & Dormann,

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2005). A CA model is a dynamical system that is discrete in space and time. The model operates on a uniform, regular lattice of cells. The premise under a CA model is that a global complex behavior can be simulated by simple local rules that operate on the cells. The solution of simple local equilibrium problems can provide a more accurate and robust procedure to solve large and complex problems. Also, the inherent parallelism of CA models makes this approach very appealing. The hybrid cellular automaton (HCA) method is intended to solve complex structural optimization problems in engineering. The premise of the HCA approach is that complex static and dynamic problems can be decomposed into a set of simple local rules that operate on a large number of CAs that only know local conditions.

2.1 Components The HCA model has three components: (1) a lattice of cells, (2) a set of states for each cell, αi (t), and (3) a set of rules associated with the set of states, Ri . The lattice of cells can be defined in an n-dimensional space but usually the models are incorporated in one, two or three dimensions. For each automaton in the model there is a set of J discrete states which can be described as ⎧ ⎪ ⎪ α1i (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ α2 (t) i αi (t) = ⎪ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ αJ (t) i

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

(1)

which are defined for the discrete location i at the discrete time t. For each state αji there is a corresponding rule Rij that defines its evolution in time. This evolution can be expressed as αji (t + 1) = Rij (αi (t), αi+∆1 (t), . . . , αi+∆Nˆ (t)),

(2)

where the αi+∆1 (t), . . . , αi+∆Nˆ (t) designate cells belonging to the neighborhood of the i-th cell. In the simplest case, a cell has a single state, α1i (t), that is defined by a single bit of information, T

{0, 1}. In the above definition, the set of rules, Ri = {Ri1 , . . . , RiJ } , is identical for all sites and is applied simultaneously to all of them which leads to a synchronous dynamic. In other words, the rule is homogeneous, that is, it does not depend on the position of the cell. However, spatial

5


(or temporal) inhomogeneities can be introduced to model a particular characteristic (i.e., boundary conditions, random phenomena, etc.) An asynchronous updating scheme is useful to model events that do not necessarily occur in parallel. Another variation is the alternative use of two (or more) rules that operate at different time steps. For example, one rule operates at even time steps and the other rule operates at odd time steps. Sometimes it is desirable to use a specific rule with a certain probability. Cellular automata whose updating rules are driven by external probabilities are called probabilistic CAs, as opposed to deterministic CAs. In the definition of a local evolutionary rule in Eq. (2) a new state at time t + 1 depends on states at time t. It is sometimes necessary to have a lingering memory and introduce a dependence on the states at time t − 1, t − 2, . . . , t − T . Depending on its definition, a rule can be or not be reversible in time. The set of local rules operate according to local information collected in the neighborhood of each cells. The neighborhood does not have any restriction on size or location, except that it is the same for all the cells. In practice, the size of the neighborhood is often limited to the adjacent cells but can also be extended. Figure 1 depicts some common neighborhood layouts. The most commonly used are the ˆ = 4) and the Moore layout that includes von Neumann layout that includes four neighboring cells (N ˆ = 8). Another possible layout is the so-called MvonN composed of twelve eight neighboring cells (N ˆ = 12). The neighborhood can also be reduced down to an empty layout (N ˆ = 0) or extended cells (N as much as the model requires. In addition to the layouts described above, this work makes use of an ˆ = 24). extended neighborhood that includes 24 cells (N

(a)

(b)

(c)

(d)

(e)

ˆ = 0), (b) Von Neumann (N ˆ = 4), Figure 1: Neighborhoods of the cellular automata. (a) Empty (N ˆ ˆ ˆ (c) Moore (N = 8), (d) MvonN (N = 12) and (e) Extended Moore (N = 24).

To define the evolutionary rule for a cell located on the boundary of the design domain, the design domain can be extended in different ways. Figure 2 depicts some types of boundary conditions obtained by extending the design domain. A fixed boundary is defined so the neighborhood is com-

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pleted with cells having a pre-assigned fixed state. An adiabatic boundary condition is obtained by duplicating the value of the cell in an extra virtual neighbor. In a reflecting boundary, the state of the opposite neighbor is replicated by the virtual cell. Periodic boundary conditions are used when the design domain is assumed to be wrapped in a torus-like shape. This work makes use of fixed boundary conditions where the extra cells are considered empty spaces without physical or mechanical properties.

X

X

0

(a)

X X

(b)

X

(c)

X

(d)

Figure 2: Boundaries of cellular automata. (a) Periodic, (b) Fixed, (c) Adiabatic and (d) Reflecting.

2.2 Optimality conditions In the HCA method, the state of the i-th cell, αi (t), is defined by a design variable xi (t) and a state variable yi (t). These variables have been defined after the solution of Roux’s maximum-minimum principle of structural tissues (Roux, 1895). This principle refers to the competing optimization problem of maximizing strength while minimizing material. Mathematically, this multi-objective optimization problem can be stated as min x

M c(x) = ω UU0 + (1 − ω) M 0

s.t.

0 ≤ x ≤ 1,

(3)

where U and M respectively represent the strain energy and the mass of the bone structure. U0 and M0 are the strain energy and mass of the solid bone material. The coefficient ω balances the relative weight of the ratios U/U0 and M/M0 in the objective function, such that 0 ≤ ω ≤ 1. The design variable xi is defined as the relative density, which is the ratio between the maximum bone density ρ0i and the actual bone density ρi at the tissue level. This can be expressed as xi =

ρi . ρ0i

(4)

The design variable xi varies between the limits 0 and 1; however, in practice, the lower boundary of

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xi is not zero but a small positive value, for example, 1 × 10−3 . This condition guards against the singularity of the stiffness matrix during the application of the finite element analysis (FEA). For the FEA at the tissue level, the solid isotropic material with penalization (SIMP) model was used. As presented by Bendsøe (1989), Zhou & Rozvany (1991) and Rozvany (2001), the SIMP model is based on the heuristic relationship Ei = xpi E0i

(5)

where p is the penalization power (p > 1), Ei is the variable elastic modulus of the cell and E0i is the elastic modulus of the mineralized tissue. As demonstrated by Tovar et al. (2005c), the optimality KKT condition for an interior point (0 < xi < 1) can be written as

(1 − ω) ρ0i v0i U0 ui = , xi ω p M0

(6)

where v0i is the constant volume of a cellular automaton. If one defines the state variable yi as ui xi

(7)

(1 − ω) ρ0i v0i U0 , ω p M0

(8)

yi = and its optimum value yi∗ as yi∗ =

then the optimality condition for an interior point can be simplified as yi = yi∗ . Tovar et al. (2005c) also demonstrated that if xi = 0 then yi ≤ yi∗ and if xi = 1 then yi ≥ yi∗ . In order to satisfy these optimality conditions, let us define the error function as ei (t) = yi (t) − yi∗ .

(9)

The state of a cellular automaton, αi (t), at the discrete location i will be defined by the design variable xi (t), described by Eq. (4), and the state variable yi (t), described by Eq. (7). This is

αi (t) =

⎫ ⎧ ⎪ ⎬ ⎨ xi (t) ⎪ ⎪ ⎭ ⎩ yi (t) ⎪

.

(10)

For illustration, using fixed boundary conditions, the state of the outer cells in the neighborhoods

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at the boundaries of the design domain is given by

αi (t) =

⎫ ⎧ ⎪ ⎬ ⎨ 0 ⎪ ⎪ ⎭ ⎩ yi∗ ⎪

.

(11)

In this way, there is no error from the outer cells and there is no change in their relative density. However, this work makes use of periodic boundary condition.

2.3 Effective variables In the HCA methodology, the local design rule makes use of an effective value of the design variable, x ¯i (t), and an effective value of the state variable, y¯i (t). The effective values are determined as the average value in the neighborhood. This can be expressed as

x ¯i (t) =

xi (t) +

k=1 xk (t)

ˆ +1 N

and y¯i (t) =

Nˆ

yi (t) +

Nˆ

k=1 yk (t)

ˆ +1 N

.

(12)

(13)

The effective values of the state variables have been used to model cellular communication in biological structures (Tovar, 2004). Similar ideas, using a spatial influence region, have been implemented in bone remodeling simulations (Mullender et al., 1994; Mullender & Huiskes, 1995). In the same way, an effective error e¯i (t) is defined as e¯i (t) =

ei (t) +

Nˆ

k=1 ek (t)

ˆ +1 N

.

(14)

where ei (t) is defined in Eq. (9). The local evolutionary rules Ri is determined according to the approaches presented in Sec. 2.5.

2.4 Algorithm The HCA algorithm, illustrated in Fig. 3, is described as follows (Tovar et al., 2004a): Step 1. Define the design domain, material properties, load conditions and initial design x(0).

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Start

?

x(0)

Initial design

x(t)

Structural analysis FEM

y(t) FE mesh

Change in mass CA rule

x(t+1) = R(x(t), y(t)) CA lattice

Convergence?

no

yes

Mass update

End

Final design

Figure 3: Hybrid cellular automaton algorithm. The algorithm starts with the definition of the design domain, material properties, load conditions and initial design x(0). Finite element analysis (FEA) is performed to determine the mechanical stimuli y(t). The mass is updated according to the set of rules, x(t + 1) = R(x(t), y(t)). The convergence is satisfied when the change in mass is small. If there is no convergence the process continues with a new FEA. Step 2. Evaluate the state variable yi (t) defined by Eq. (7), using the finite element method. Step 3. Calculate the effective state variable y¯i (t) according to Eq. (13) and the effective design variable x ¯i (t) according to Eq. (12). Step 4. Apply the local evolutionary rule Ri and update the mass fraction xi (t + 1) at each cell. Step 5. Check for convergence. If the structure does not change with respect to the previous design, the convergence criterion is satisfied; otherwise, the iterative process continues from Step 2.

2.5 Local evolutionary rules In phenomenological models, it has been hypothesized that bone acts as a control system in which bone functional adaptation is driven by the error between a mechanical set point or equilibrium and a mechanical signal or stimulus. As Roesler (1987) points out, theoretical models that use the concept of a deviation or error signal were presented as early as 1858 by Roux. These models are currently used in several phenomenological models (Frost, 2003; Xinghua et al., 2002; Turner, 1999; Huiskes,

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2000). Even though these type of models ignore non-mechanical factors such as genetics, nutrition or hormonal changes, they do describe the steps observed in mechanical adaptation (Hart, 2001). In the HCA algorithm, the change in relative density can be written as xi (t + 1) = xi (t) + ∆xi (t),

(15)

where ∆xi (t) is a function of the effective error signal e¯i (t). The concept of proportional control in bone remodeling has been used in several phenomenological approaches (Hart et al., 1984b; Beaupré et al., 1990a,b; Weinans et al., 1992; Mullender et al., 1994; Mullender & Huiskes, 1995; Turner, 1999; Huiskes, 2000). In this investigation, the concept of control in bone functional adaptation has been extended to a generalized control strategy that includes two-position, proportional, integral and derivative (PID) actions. The two-position control is the simplest strategy to modify the relative density. With this rule, the change in relative density is determined by a discrete value cT and the sign of the effective error signal. This is given by ei (t)], ∆xi (t) = cT × sgn[¯

(16)

where cT is a positive constant. With this control control strategy, if the effective error signal e¯i (t) in a cell is not zero, then the relative density changes the amount of cT . This process continues until the cellular automaton saturates, a condition that occurs when the cell reaches the maximum or minimum relative density value, xi = 1 or xi = 0. A more complex strategy incorporates the proportional-integral-derivative (PID) control actions. With the PID controller, the change in relative density can be expressed as

∆xi (t) = cP × e¯i (t) + cI ×

t

e¯i (t − τ ) + cD × [¯ ei (t) − e¯i (t − 1)].

(17)

τ =0

where cP , cI and cD are constants respectively referred to as proportional, integral and derivative gains. The inclusion of the integral and derivative strategies in the phenomenological algorithm is required to take into account the effect of loading history in bone. With integral control, the net remodeling is proportional to the accumulated (integral) error signal in time. With derivative control, the

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net remodeling is proportional to the variation (derivative) in error signal. These two types of control work under the assumption that osteocytes have memory and can remember previous loading conditions. The implementation and characteristics of these control strategies are presented in more detail by Tovar et al. (2005b).

2.6 Convergence criteria The iterative optimization process converges when no further change in mass is possible. This state can be expressed as ∆M (t) = M (t) − M (t − 1) ≈ 0.

(18)

Numerical experience with the HCA algorithm has shown that, in some applications, ∆M (t) displays a cyclic behavior in which a small change in mass is followed by a bigger change. To avoid premature convergence, the convergence criterion is defined by using the average change in two consecutive iterations. This yields |∆M (t)| + |∆M (t − 1)| ≤ ε, 2

(19)

where ε is a small fraction of the total mass of the solid structure.

3 Hierarchical algorithm The hierarchical algorithm begins with the definition of the geometry, loading conditions and apparent density distribution at the continuum level. The initial density information identifies the regions of cortical and trabecular architecture in the bone model. If these regions are not provided, the algorithm defines a homogeneous distribution of mineralized tissue throughout the entire model, i.e., constant apparent density. In this case, the macro-mechanical properties of the bone will correspond to the ones of a homogeneous, isotropic material. With the definition of the initial design, the algorithm performs a finite element analysis. The material properties of the model correspond to the ones defined for cortical and trabecular bone at the continuum level. This model is subsequently divided into a lattice of sub-models. By default, the lattice will be composed of identical sub-models, but this is not a requirement of the algorithm. The sub-models are pieces of bone in which the apparent density is considered to be homogeneous.


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Each sub-model is comprised of a regular lattice of cellular automata that represents the osteocyte network in the mineralized tissue. Each cellular automaton is able to modify its mass according to the design rules developed for the HCA algorithm (Tovar et al., 2005a; Tovar, 2005). The initial design at each sub-model is defined according to the initial apparent density distribution. The loading condition is derived from the stress field obtained with an initial finite element analysis. The effective stress at each sub-model is the average of nine sample points. The effective stresses are represented as normal and shear forces distributed on the surfaces of the sub-model. Once the loading conditions are defined, the local HCA algorithm is performed for every submodel. The resulting black-and-white structures represent the bone microstructure that defines the anisotropic mechanical properties at the tissue level. Since no symmetry is imposed on the planes of the resulting microstructure nor on the tensor of elastic coefficients in the generalized Hooke’s law, the resulting structure is fully anisotropic. The new properties of every sub-model are fed back to the global model where a new global finite element analysis is performed. The steps involved in the hierarchical algorithm, depicted in Fig. 4, are described as follows: Step 1. Define the global model: geometry, load conditions and initial apparent density distribution X(0). The initial density distribution determines the regions of cortical and trabecular bone. This distribution can be homogeneous, e.g., no distinction between cortical and trabecular regions. Step 2. Define the initial stress field. The algorithm performs a (global) finite element analysis for the initial material distribution. If the apparent density distribution is homogeneous, the finite element analysis corresponds to an isotropic material. Step 3. Create a lattice of sub-models and calculate the loading conditions at each one. Each submodel has a homogeneous density. The size of the sub-models determines the detail of the trabecular structure at the tissue level and the resolution of the apparent density distribution at the continuum level. The loading condition at each sub-model is calculated from the previous finite element analysis by averaging the values at several interior points. Step 4. Perform the local HCA algorithm at each sub-model. Since the calculations for a sub-model do not depend on other sub-models, this process can be done in parallel. Once this process is complete, every sub-model feeds into the global model the resulting (black-and-white)

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Start

? Initial design

X(0)

Local HCA algorithm (applied to a sub-model)

Initial structural analysis Local load conditions

Global FE mesh

Global sub-model lattice

Load conditions in sub-models

Local structural analysis

Local HCA algorithm

Local FE mesh

Anisotropic structural analysis

Local CA lattice

Apparent density distribution

x(t)

Change in mass CA rule

Convergence?

no

yes

Convergence? yes Final apparent density distribution

x(0)

X(T)

no Final structure

Final trabecular structure

End

Figure 4: Hierarchical algorithm flow chart. The hierarchical algorithm developed in this investigation includes a continuum-level model that is divided into tissue-level sub-models. The sub-models are defined by the apparent densities X(T ), where T is a discrete time (iteration) at the continuum level. The apparent density varies from 0 (void) to 1 (cortical bone). Intermediate values represent trabecular architecture. The stress field evaluated by the global finite element analysis defines the loading conditions operating on the sub-models. Cellular automata comprising each sub-model modify the microstructure by processes of formation/resorption using the (local) HCA algorithm. This process results in formation and adaptation of trabeculae. The (tissue-level) material of the CAs is characterized by a continuous volume fraction x(t) that varies from 0 (bone marrow) to 1 (fully mineralized tissue). The variable t is the discrete time (iteration) at the tissue level. The elastic properties at this level are quantified using a power-law relationship between Young’s modulus and volume fraction.


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microstructures and their corresponding anisotropic structural properties. The homogenized microstructure represent the apparent density of the sub-model. This density is defined as the ratio between the mass of the resulting (solid) mineralized tissue in relation to the maximum mass of completely mineralized sub-model. Step 5. Perform a global finite element analysis. A new global analysis is performed over the resulting structure to update the stress field. Step 6. Perform convergence test. If there is no change in the relative densities and there is no change in the stress field, the algorithm is converged; otherwise, the iterative, hierarchical process continues starting with Step 3.

4 Implementation To illustrate the implementation of the hierarchic algorithm, let us consider the two-dimensional plate model presented by Weinans et al. (1992). In their work, a remodeling algorithm was applied to this model to show its effect on the apparent density distribution; however, the convergent solution showed a black-and-white structure instead of a more physiologically reasonable (gray) distribution. This plate model was later used to apply different remodeling algorithms at the tissue level (Mullender et al., 1994; Mullender & Huiskes, 1995; Xinghua et al., 2002), abandoning its original application at the continuum level. In this work, the hierarchical algorithm was applied to the originally proposed plate model. With the hierarchical technique, the convergent solution shows a feasible apparent density distribution at the continuum level along with the corresponding trabecular structure at the tissue level. The model presented by Weinans et al. (1992) consists on a two-dimensional plate that is supported at the bottom while a linearly decreasing distributed load compress its upper edge. Figure 5 depicts this layout. In the hierarchical algorithm, each constitutive element in the model is a sub-model composed of a lattice of cellular automata (Fig. 5 right). The loading condition of each sub-model is determined from a finite element analysis performed at the continuum level. The local HCA algorithm is then executed to find the optimum trabecular structure at the tissue level. The resulting architecture defines the

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10 N/mm2

1

6

11

16

21

2

7

12

17

22

3

8

13

18

23

4

9

14

19

24

5

10

15

20

25

Figure 5: Plate model. The compressive load linearly decreases from a value of 10 N/mm2 . The model is composed of 5 × 5 elements as presented by Weinans et al. (1992). In the HCA hierarchical algorithm, each of the 25 elements in the global model is a sub-model composed of a lattice of 25 × 25 cellular automata. The sub-models are identified with a number from 1 to 25. The loading condition on each sub-model is determined by the finite element analysis performed at the continuum level. The material comprising the structure has a Young’s modulus of 303 MPa and a Poisson’s ratio of 0.3. new mechanical properties of the continuum-level model and also a new stress distribution. The new stresses change the loading conditions on the sub-models and the HCA algorithm is used to modify or adapt the trabecular structure. This iterative procedure continues until there is no change in the trabecular structure and no change in the stress distribution.

4.1 Initial structural analysis The objective of the structural analysis is to determine the loading conditions at every sub-model in the plate. The initial design consists of 25 sub-models of solid material. The structural analysis is then performed on this isotropic-material model using a single elastic constant. In two dimensions, the state of stress at any location within the plate is represented by two normal stresses, σx and σy , and a shear stress, σxy = σyx . In order to estimate the equivalent state of stress for a particular sub-model, the algorithm samples the stresses at nine of its interior points and determines the average (Fig. 6). The nine points correspond to the center of the sub-model, the four corners and the four middle points of the edges. Let us consider, for example, the element number 13 at the center of the plate (Fig. 5). Table 1 shows the state of stress of its nine sample points. The effective state of stress for this sub-model is the average of these points, which is σx = −1.31×10−12 , σy = −125.00 and σxy = −1.33×10−13 MPa. This effective state of stress defines the loading conditions to be used in the local HCA iterative

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6

1

2 7

5 0

3

4 8

Figure 6: Stress state sample points. The state of stress in a sub-model is determined by sampling and averaging the stresses at the nine points. algorithm. Table 2 shows the resulting state of stress for each sub-model in the plate. Table 1: State of stress for the sample points within sub-model 13 σx

Point 0 1 2 3 4 5 6 7 8

−12

−2.60 × 10 2.42 × 10−12 −2.22 × 10−12 −3.31 × 10−12 −6.82 × 10−13 −1.12 × 10−12 6.75 × 10−13 −3.13 × 10−12 −1.83 × 10−12

σy

σxy

−125.00 −125.00 −100.05 −125.00 −149.95 −100.05 −149.95 −149.95 −100.05

−4.6544 × 10−12 −4.7825 × 10−13 2.0155 × 10−12 −9.6504 × 10−13 3.3734 × 10−12 1.9642 × 10−13 −4.0993 × 10−13 −1.0315 × 10−14 −2.7329 × 10−13

4.2 First trabecular structure The effective stresses in each sub-model are represented by the distributed forces on the sides of the sub-model (Fig. 7). The resulting forces, Fx , Fy and Txy , act in the same directions as the stresses. These forces are given by Fx = σx × A, Fy = σy × A and Txy = σxy × A, where A is the area in which the force is distributed. This area corresponds to the product of the side length and its thickness, which is A = 1 × 1/25 mm2 or A = 0.04 mm2 . For example, for the sub-model 13, the resulting forces are Fx = 5.25 × 10−14 , Fy = −5.00 and Txy = −5.32 × 10−13 N. In the local HCA algorithm, each sub-model represents a bone section of 1 × 1 × 1/25 mm3 that is composed of a lattice of 25 × 25 cellular automata. In the sub-models, the relationship between the Young’s modulus Ei at the cell i and the tissue-level mass fraction xi is given by the power law defined by Eq. (5), where the penalization power p is used to drive the local mass fractions to 1 or 0 and, therefore, generate a black-and-white trabecular structure. In this work, the penalization power will be p = 3.0. The Young’s modulus of the mineralized tissue is E0i = E0I = 303 MPa, where the subindex i refers to a cells and the subindex I refers a sub-model.

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Table 2: State of stress for the initial solid sub-models Sub-model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

X

σx −11

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

−1.70 × 10 −8.87 × 10−13 4.29 × 10−13 −7.64 × 10−13 −2.85 × 10−13 −6.67 × 10−12 −2.17 × 10−12 1.49 × 10−12 6.87 × 10−12 9.79 × 10−12 −6.56 × 10−12 −4.35 × 10−12 −1.31 × 10−12 3.50 × 10−12 7.37 × 10−12 −4.74 × 10−12 −3.28 × 10−12 −1.96 × 10−12 1.05 × 10−12 5.44 × 10−12 −9.63 × 10−13 −1.76 × 10−12 −1.26 × 10−12 −8.58 × 10−13 7.45 × 10−13

σy

σxy

−225 −225 −225 −225 −225 −175 −175 −175 −175 −175 −125 −125 −125 −125 −125 −75 −75 −75 −75 −75 −25 −25 −25 −25 −25

−1.17 × 10−11 −4.31 × 10−13 −3.45 × 10−13 1.37 × 10−12 −3.14 × 10−12 −3.90 × 10−13 1.06 × 10−12 1.30 × 10−12 2.62 × 10−12 −3.36 × 10−13 −1.36 × 10−13 −7.97 × 10−14 −1.33 × 10−13 2.70 × 10−13 1.16 × 10−13 −9.29 × 10−14 −1.02 × 10−12 −9.37 × 10−13 4.55 × 10−14 1.03 × 10−12 −2.22 × 10−14 −2.80 × 10−14 −8.68 × 10−13 −4.57 × 10−13 −1.52 × 10−13

Fy Txy – Fx

Fx

–Txy –Fy

Figure 7: Load distribution in a sub-model. The forces are homogeneously distributed along the the four edges of the two-dimensional model. The sub-model is supported by a pin at the bottom left corner that prevents horizontal and vertical displacements. A roller support in the bottom right corner prevents vertical displacement.

18


The HCA algorithm with the local PID control rules, Eq. (17), is used to obtain the tissue-level structure for each sub-model, where the control gains are selected as cP = 0.10/yi∗ , cI = 0.05/yi∗ and cD = 0.05/yi∗ . These values are chosen to provide numerical efficiency to the local HCA algorithm and obtain a quick convergence. The effective error signal e¯i (t) is defined as the average in the cell neighborhood. The osteocyte communication network is modeled using the Moore layout, hence the ˆ = 8. neighborhood size is N The target mechanical stimulus is set as yi∗ = (ui /xi )∗ = 0.25 Nmm/mm2 for all the sub-models in the plate. In order to improve the continuity of adjacent sub-models, the HCA algorithm makes use of periodic boundary conditions. The convergence criterion is described by Eq. (19). In this application, the fraction value is defined as ε = 0.0001 × M0 , which is ε = 0.0625 g. If there is no convergence after 100 iterations, the iterative process is interrupted and finished. When the local HCA algorithm is executed for every sub-model, the resulting trabecular structure is composed of vertical columns that end in supporting pedestals upon which the main loads act. In some cases, there are small, horizontally-oriented struts that prevent the deformation of the bigger columns. The resulting local structures are assembled to form the global model. Figure 8 shows the final structures obtained for the sub-models in the plate.

Sub-models 1 to 5

Sub-models 6 to 10

Sub-models 11 to 15

Sub-models 16 to 20

Sub-models 21 to 25

Figure 8: Resulting trabecular structures for the plate sub-models (T = 1). The loading condition is a load homogeneously distributed along the four sides of each sub-model. The local HCA algorithm used PID control rules, the Moore neighborhood and periodic boundary conditions. The number of iterations varied between t = 40 (for sub-models 1 to 5) and t > 100 (for sub-models 21 to 25). The hierarchical algorithm puts together the resulting local trabecular structure to create a global structure that contains all the anisotropic properties and the apparent density of the sub-models. The apparent density corresponds to the relative mass of the local solid structure of a sub-model with respect to its maximum mass. Figure 9 shows the resulting tissue-level structure and apparent density distribution.


19

T =1

Figure 9: Plate trabecular structure (left) and apparent density distribution (right) obtained after the first iteration of the hierarchical algorithm (T = 1).

4.3 Second finite element analysis The initial stress distribution within the continuum-level model was determined after performing finite element analysis on a solid isotropic material. The subsequent finite element analysis seeks to update the stress information taking into account the internal changes in the structure. Three approaches have been proposed for this task. The first approach makes use of the apparent density distribution and the power law to determine an approximate Young’s modulus at each sub-model (Carter & Hayes, 1977; Currey, 1988; Rice et al., 1988). The second approach involves on determining the coefficients of the elastic tensor for each submodel to perform a fully anisotropic finite element analysis. These coefficients can be calculated by numerical experimentation on the trabecular structures. This approach requires the use of a specialized finite element software that allows such anisotropic information to be included. The third approach consists of direct structural analysis of the global microstructural architecture. Instead of approximating the anisotropic properties or determining their elastic coefficients, this approach proposes forming a global trabecular structure and perform the structural analysis with the same technique and resolution used at the tissue level. The main difficulty of this approach is the requirement of large computational capabilities. The model of one cubic millimeter of bone contains 15, 625 cellular automata and an accurate analysis requires the finite element mesh to contain a similar amount of elements. With this proportion, the model of a piece of bone of 10 × 10 × 10 cubic millimeters would contain 15, 625, 000 elements which makes this approach computationally expensive for large models. The most common approach is the first one, i.e., the use of the apparent density to approximate

20


the Young’s modulus using a power law. This method has been proposed by Carter & Hayes (1977) and experimentally evidenced by Currey (1988), Rice et al. (1988) and more recently by Turner et al. (1999). This approach was also used by Weinans et al. (1992) in the plate model. In this approach, the relationship between the Young’s modulus EI of the sub-model I and the apparent density ρI is given by EI = cργI ,

(20)

where the power γ = 2.0 and the constant c = 100 (MPa)(g/cm3 )−2.0 . The apparent density can be written in terms of the relative mass XI as ρI = ρ0I XI ,

(21)

where the maximum apparent density ρ0I corresponds to the one of cortical bone, which is ρ0I = 1.74 g/cm3 . This is also the value used by Weinans et al. (1992). Combining Eq. (21) and Eq. (20) yields EI = cργ0I XIγ ,

(22)

or simply EI = 303XIγ MPa. The apparent densities obtained from the first iteration are used to calculate the new Young’s modulus in each sub-model. The Poisson’s ratio remains constant and it is equal to νI = 0.3. Table 3 presents the new state of stress for each sub-model.

4.4 Convergence criteria The convergence criteria compare the change in relative mass (apparent density) distribution and the change in the stress field. In the hierarchical algorithm, the convergence criterion for relative mass is defined as max{|∆XI (T )|} < εX ,

(23)

where |∆XI (T )| = |XI (T ) − XI (T − 1)| is the absolute value of the change in relative mass for the sub-model I between the iterations T and T − 1. For a desired resolution of ±0.10 g/cm3 in the result of the apparent density distribution, the tolerance εX is selected as the ratio between the resolution and the maximum (cortical) density. This is εX = 0.10/1.74 = 0.06. The convergence over the stress

21


Table 3: Results for the plate sub-models with a distributed load after the second finite element analysis Sub-model

X

σx

σy

σxy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.6467 0.6467 0.6467 0.6467 0.6467 0.5123 0.5123 0.5123 0.5123 0.5123 0.4306 0.4306 0.4306 0.4306 0.4306 0.3417 0.3417 0.3417 0.3417 0.3417 0.2498 0.2498 0.2498 0.2498 0.2498

1.3861 −0.6014 0.3748 0.3542 0.1304 −6.5292 0.7801 2.2096 1.6207 0.6528 −8.6614 1.8952 3.3866 2.1065 1.3860 −7.2033 1.9198 2.4047 1.3584 0.9740 −3.0017 0.6550 0.5542 0.3042 0.1565

−239.31 −245.91 −250.31 −254.75 −253.83 −166.98 −165.50 −163.73 −162.64 −160.49 −121.62 −118.37 −116.14 −113.40 −111.40 −72.08 −70.26 −68.80 −70.04 −68.20 −23.62 −29.24 −33.40 −35.72 −35.88

−4.1184 −4.1113 −3.1423 −2.0118 −0.2098 −7.0836 −4.3061 −2.9389 −1.9819 −0.1207 −1.6251 0.8940 0.8476 0.5126 0.3635 1.1858 4.4922 2.9661 1.5708 0.5384 1.2726 2.4671 1.3249 0.6243 0.2514

22


field is defined as max {|∆σI (T )|} < εσ ,

(24)

and |∆σI (T )| = |σI (T ) − σI (T − 1)| is the absolute value of the change in stress for the sub-model between iterations T and T − 1. A change in the maximum distributed force of ∆FI = 0.5 N represents a change in the stress given by ∆σI = 0.5/0.04 = 12.5 MPa, which is the value used in this application. At the end of the first iteration, the maximum change in relative mass corresponds to |∆X| = 0.7502 (> 0.06) for sub-models 21 to 25, while the maximum relative change in stress corresponds to |∆σy | = 480 (> 12.5) for sub-model 4. Therefore, there is no convergence and the hierarchical iterative process continues.

4.5 Hierarchical iterative process From the updated stress field, the hierarchical algorithm determines the new loading conditions for each sub-model. Using the current tissue-level structure, the local HCA algorithm applies the design rules based on control and determines the new architecture. Since there is a trabecular structure already formed, the HCA algorithm makes use of the strict surface conditions. In this way, the trabecular architecture adapts to the new mechanical environment following the mechanism used by the actual bone structure. In each sub-model, the convergence of the local HCA algorithm is achieved after 20 to 60 iterations, which is about half the number of iterations obtained the first time the local HCA algorithm was run. The hierarchical algorithm (local HCA algorithm, finite element analysis and convergence check) is performed until the change in mass and stress field fits under the specified tolerances. Six iterations of the hierarchical algorithm (T = 6) are required to achieve convergence. Figure 10 shows the iterative change in the trabecular structure and apparent density distribution. Figure 11 shows the maximum change in stress (max ∆σ) and the maximum change in relative mass (max ∆X) for the first ten iterations (T = 10) of the hierarchical algorithm.


23

T =2

T =4

T =6

Figure 10: Iterative change in the plate trabecular structure (left) and apparent density distribution (right). Global convergence is achieved after six iterations, after which there is no perceptible change in apparent density distribution and stresses on the plate.

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Submitted to Journal of Biomechanics 30

0.30 max{|∆X |}

max{|∆σ y |}

0.25

20

0.20

15

0.15 max{|∆σ x |}

10

max{∆X }

max{∆σ }

25

0.10

5

0.05 max{|∆σ xy |}

0

0.00 0

2

4

6

8

10

T

Figure 11: Convergence plot of the hierarchical algorithm. The graph presents the results for the first 10 iterations. During the iterative process the maximum change in stress max ∆σ and relative mass ∆X tend to decrease. Convergence is achieved at T = 6 in which max ∆σ < 12.5 MPa and max ∆X < 0.05.

5 Conclusions and final remarks The hierarchical algorithm developed in this investigation includes a continuum-level (global) model that is divided into tissue-level (local) sub-models. The stress/strain field evaluated in the global model defines the load condition in the sub-models. Cellular automata comprising each sub-model modify the topology with the use of different control strategies that simulate processes of formation and resorption. These rules modify the cell relative density in order to achieve the zero-error condition between the state variable effective value and its target value. This local algorithm is referred to as the hybrid cellular automaton (HCA) method. It makes use of the finite element method for structural analysis and the cellular automaton paradigm to simulate the functional adaptation process. The HCA algorithm follows optimization principles derived from the KKT conditions and drives the local structure to an optimal configuration (Tovar et al., 2005c). During this iterative process, a trabecular architecture is created defining both the apparent density of the sub-model and its anisotropic properties. The apparent density at each location is determined as the total mass of the sub-model divided by its volume. This work presents three approaches to perform structural analysis over the anisotropic tissue-level structure: power law, elastic tensor and direct structural analysis.


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The numerical implementation of this algorithm is demonstrated using a simple problem consisting of a plate supporting a distributed compressive load. The structural properties are determined using the power law. Unlike other remodeling algorithms (Weinans et al., 1992; Beaupré et al., 1990b), this method predicts apparent density distribution with a convergent solution. In comparison to free material approaches (Jacobs et al., 1997), this algorithm is able to predict physiologically plausible trabecular architectures. In comparison to algorithms based on optimization (Bagge, 2000; Fernandes et al., 2002; Jacobs, 2000), the hierarchical algorithm does not assume symmetry in the tensor of elastic coefficients to obtain the apparent density information and does not require a defined density to determine the microstructure. The hierarchical algorithm takes into account physiological processes of formation and resorption and defines a realistic geometric scale. However, the time scales in the local and global models are disconnected and they do not correspond to realistic adaptation processes. Current effort is made to incorporate a more accurate dynamic model to predict bone structural changes in time.

6 Acknowledgments Support for this research has been provided by the Dirección de Investigación Sede Bogotá de la Universidad Nacional de Colombia (DIB), the Honda Initiation Grant HIG 2004 and the Defense Advanced Research Projects Agency DARPA.

References Bagge, M. 2000. A model of bone adaptation as an optimization process. J Biomech 33, 1349–1357. Beaupré, G. S., T. E. Orr & D. R. Carter. 1990a. An approach for time-dependent bone modeling and remodeling—application: a preliminary remodeling simulation. J. Orthopaedic Research 8, 662–670. Beaupré, G. S., T. E. Orr & D. R. Carter. 1990b. An approach for time-dependent bone modeling and remodeling—theoretical development. J. Orthopaedic Research 8, 651–661. Bendsøe, M. P. 1989. Optimal shape design as a material distribution problem. Struct. Optim. 1, 193– 200.


26

Carter, D. R. 1987. Michanical loading history and skeletal biology. J. Biomech. 20, 1095–1105. Carter, D. R. & W. C. Hayes. 1977. The bahavior of bone as a two-phase porous structure. J. Bone Joint Surgery 59-A, 964–962. Chopard, B. & M. Droz. 1998. Cellular Automata Modeling of Physical Systems. Cambridge University Press. Cowin, C. S. & D. H. Hegedus. 1976. Bone remodeling I: A theory of adaptive elasticity. J. Elasticity 6, 313–326. Cowin, S. C. 1998. On mechanosensation in bone under microgravity. Bone 22(Suppl. 5), 119S–125S. Currey, S. C. 1988. The effect of porosity and mineral content on the young’s modulus of elasticity of compact bone. J. Biomech. 21, 131–139. Deutsch, A. & S. Dormann. 2005. Cellular Automaton Modeling of Biological Pattern Formation. Birkhäuser Boston. Fernandes, P., H. Rodrigues & C. Jacobs. 1999. A model of bone adaptation using a global optimisation criterion based on the trajectorial theory of Wolff. Comput. Methods Biomech. Biomed. Engi. 2(2), 125–138. Fernandes, P. R., J. Folgado, C. Jacobs & V. Pellegrini. 2002. A contact model with ingrowth control for bone remodelling around cementless stems. J Biomech 35, 167–176. Foldes, I., M. Rapcsak, T. Szilagyi & V. S. Oganov. 1990. Effects of space flight on bone formation and resorption. Acta. Physiol. Hung. 75(4), 271–285. Frost, H. M. 1964. Mathematical Elements of Lamellar Bone Remodelling. Charles C. Thomas Publisher. Frost, H. M. 1987. Bone ”mass” and the ”mechanostat”: a proposal. Ant. Rec. 219, 1–9. Frost, H. M. 2003. Bone’s mechanostat: A 2003 update. Anat. Rec. Part A 2, 1081–1101. Hart, R. T. 2001. Bone modeling and remodeling: Theories and computation. In S. C. Cowin (ed.) Bone Mechanics Handbook, CRC Press.


27

Hart, R. T., D. T. Davy & K. G. Heiple. 1984a. A computational method for stress analysis of adaptive elastic materials with a view toward applications in strain-induced bone remodeling. J. Biomech. Eng. 106(4), 342–350. Hart, R. T., D. T. Davy & K. G. Heiple. 1984b. Mathematical modeling and numerical solutions for functionally dependent bone remodeling. Calcif. Tissue Int. 36(Suppl. 1), S104–S109. Hart, R. T. & S. P. Fritton. 1997. Introduction to finite element based simulation of functional adaptation of cancellous bone. Forma 12, 277–299. Huiskes, R. 2000. If bone is the answer, then what is the question? J. Anat. 197, 145–156. Huiskes, R., H. Weinans, J. Grootenboer, M. Dalstra, M. Fudala & T. J. Slooff. 1987. Adaptive bone remodelling theory applied to prosthetic-design analysis. J. Biomech. 20, 1135–1150. Jacobs, C. R. 2000. The mechanobiology of cancellous bone structural adaptation. Journal of Rehabilitation Research and Development 37(2), 209–216. Jacobs, C. R., J. C. Simo, G. S. Beaupr/’e & D. R. Carter. 1997. Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations. J. Biomech. 30(6), 603–613. Mullender, M. G. & R. Huiskes. 1995. Proposal for the regulatory mechanism of Wolff’s law. J. Orthop. Res. 13(4), 503–512. Mullender, M. G., R. Huiskes & H. Weinans. 1994. A physiological approach to the simulation of bone remodeling as a self-organizational control process. J. Biomech. 27(11), 1389–1394. Patel, N. M., J. E. Renaud & A. Tovar. 2005. Compliant mechanism design using the hybrid cellular automaton method. In 1st AIAA Multidisciplinary Design Optimization Specialist Conference, Austin, Texas. Prendergast, P. J. & S. A. Maher. 2001. Issues in pre-clinical testing of implants. J. Mater. Process. Tech. 118(1-3), 337–342. Rice, J. C., S. C. Cowin & J. A. Bowman. 1988. On the dependence of the elasticity and strenght of cancellous bone on apparent density. J. Biomech. 21, 155–168.


28

Roesler, H. 1987. The history of some fundamental concepts in bione biomechanics. J. Biomech. 20, 1025–1034. Roux, W. 1895. Gesammelte Abhandlungen u¨ ber Entwicklungsmechanick der Organismen, vol. I and II. Leipzig: Wilhem Engelmann. Rozvany, G. I. N. 2001. Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Structural and Multidisciplinary Optimization 21(2), 90–108. Stolk, J., S. A. Maher, N. Verdonschot, P. J. Prendergast & R. Huiskes. 2003. Can finite element models detect clinically inferior cemented hip implants? Clin. Orthop. Relat. R. 409, 138–150. Tovar, A. 2004. Bone Remodeling as a Cellular Automaton Optimization Process. Ph.D. thesis, University of Notre Dame. Tovar, A. 2005. Optimización topológica con la técnica de los autómatas celulares h´ıbridos. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingenier´ıa 21(4). Tovar, A., G. L. Niebur, M. Sen & J. E. Renaud. 2004a. Bone structure adaptation as a cellular automaton optimization process. In 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, California. Tovar, A., G. L. Niebur, M. Sen & J. E. Renaud. 2005a. Topology optimization using a hybrid cellular automaton approach with distributed control. AIAA Journal Submitted for review. Tovar, A., N. Patel, G. A. Letona, G. L. Niebur, M. Sen & J. E. Renaud. 2004b. Evolutionary model for bone adaptation using cellular automata. Poster at the 14th Conference of the European Society of Biomechanics, Hertogenbosch, The Netherlands. Tovar, A., N. M. Patel, G. L. Niebur, M. Sen & J. E. Renaud. 2005b. Topology optimization using a hybrid cellular automaton method with local control rules. ASME Journal of Mechanical Design Submitted for review. Tovar, A., N. M. Patel, J. E. Renaud & H. Agarwal. 2005c. Optimality of the hybrid cellular automata. In 1st AIAA Multidisciplinary Design Optimization Specialist Conference, Austin, Texas.


29

Tovar, A., W. I. Quevedo, N. Patel & J. E. Renaud. 2005d. Hybrid cellular automata with local control rules: a new approach to topology optimization inspired by bone functional adaptation. In 6th ISSMO World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil. Turner, C. H. 1999. Toward a mathematical description of bone biology: the principle of cellular accommodation. Calsif. Tissue. Int. 65(6), 466–471. Turner, C. H., J. Rho, Y. Takano, T. Y. Tsui & G. M. Pharr. 1999. The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. J. Biomech. 32(4), 437–441. van der Linden, J. C., J. S. Day, J. A. Verhaar & H. Weinans. 2004. Altered tissue properties induce changes in cancellous bone architecture in aging and diseases. J. Biomech. 37(3), 367–374. van der Linden, J. C., J. A. Verhaar, H. A. Pols & H. Weinans. 2003. A simulation model at trabecular level to predict effects of antiresorptive treatment after menopause. Calcif. Tissue Int. 73(6), 537– 544. van der Linden, J. C., J. A.a Verhaar & H. Weinans. 2001. A three-dimensional simulation of agerelated remodeling in trabecular bone. J. Bone Miner. Res. 16(4), 688–696. Weinans, H., R. Huiskes & H. J. Grootenboer. 1992. The behavior of adaptive bone-remodeling simulation models. J. Biomech. 25(12), 1425–1441. Xinghua, Z., H. Gong, Z. Dong & G. Bingzhao. 2002. A study of the effect of non-linearities n the equation of bone remodeling. J. Biomech. 35, 951–960. Zhou, M. & G. I. N. Rozvany. 1991. The coc algorithm, part II: Topological, geometry and generalized shape optimization. Comput. Methods Appl. Mech. Engrg. 89, 197–224.


Affiliation of authors 1

Associate Professor, corresponding author Department of Mechanical and Mechatronic Engineering National University of Colombia, Cr. 30 45-03, Of. 453-401, Bogota, Colombia Phone: (57)(1)316-5320, Fax: (57)(1)316-5333, E-mail: [email protected]

2

Assistant Professor Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, Indiana, 46556, USA Phone: (574)631-9052, Fax: (574)631-8341, E-mail: [email protected]

3

Assistant Professor Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, Indiana, 46556, USA Phone: (574)631-9052, Fax: (574)631-8341, E-mail: [email protected]

4

Professor Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, Indiana, 46556, USA Phone: (574)631-9052, Fax: (574)631-8341, E-mail: [email protected]

30