Computer simulation of an adaptive damage-bone remodeling law applied to three unit-bone bars structure

Computers in Biology and Medicine 34 (2004) 259 – 273 www.elsevierhealth.com/locate/compbiomed

Computer simulation of an adaptive damage-bone remodeling law applied to three unit-bone bars structure S. Ramtania;∗ , J.M. Garciab , M. Doblareb a

Universit e Paris Val de Marne, Facult e des Sciences et Technologie, Laboratoire de Biom ecanique et Biomat eriaux Osseux et Articulaires/CNRS UMR 7052, 61, avenue du g en eral De Gaulle. 94010 Cr eteil c edex, France b University of Zaragoza, Centro Polit ecnico, Maria de Luna, Structural Mechanics Division, Mechanical Engineering Department, 50015 Zaragoza, Spain Received 28 June 2002; accepted 7 May 2003

Abstract It is well admitted that the mechanical loading plays an important role in the growth and maintenance of our skeleton, and that microdamage (i.e.: microcracks) occurs naturally when the bone is overloaded during day-to-day activities. It is also argued, from experimental and theoretical viewpoint, that the cells which built and rebuilt the skeleton are sensitive for both strain and microdamage. The recent damage-bone remodeling theory is employed here to study the mechanical response of the three unit-bone bars that simulate bone trabeculae in the form of truss. It is shown that under constant load, such a structure exhibit inhomogeneous strain and it’s response to external applied load depends strongly upon the manner in which the microdamage is distributed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Bone remodeling; Bone microdamage; Damage repair; Computer simulation

1. Introduction The bone remodeling process is generally viewed as bone material response to functional demands and muscle attachments by continual process of growth, reinforcement and resorption which occur in living situation [1–5]. It is also suggested that bone remodeling achieves three goals [6]: First, it provides a way for the body to alter the balance of essential minerals by increasing the concentration of these in serum. Second, it provides a mechanism for the skeleton to adapt to its mechanical ∗

Corresponding author. Tel.: +33-1-4517-1441; fax: +33-1-4517-1433. E-mail addresses: [email protected] (S. Ramtani), [email protected] (J.M. Garcia), [email protected] (M. Doblare). 0010-4825/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0010-4825(03)00057-X

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environment, reducing the risk for fracture and increasing the organism’s chances for passing its genes to the next generation. And third, it provides a mechanism to repair the damage as it is well known that everyday activities damage bone. It is also well argued that rapid generation of damage can exceed the repair rate and then leads to fractures in response to minor trauma [7] or acts as a stimulus for bone remodeling [8,9] and adaptation [5,10]. Frost [11] originally proposed that remodeling would occur to repair microdamage in bone. He suggested that disruption of canalicular connections that occurred when cracks crossed them could provide the stimulus to initiate remodeling. The work of Burr et al. [8,12] and Bentolila et al. [13] has also shown that microdamage in cortical bone is associated with increasing activation of remodeling, and it is generally that the same is true in cancellous bone. As it is reported in [6], the most compelling evidence that microdamage induces osteocyte apoptosis, which signals for a repair response, comes from Verborgt et al. [14] and ShaDer and Verborgt [15] which used the in vivo rat ulna loading model to induce damage. Following loading, osteocyte integrity was assessed using tunnel staining, and the number of apoptotic osteocytes was evaluated in regions with and without damage. A time course study evaluated the subsequent increase in bone resorption and its location with respect to damage and osteocyte apoptosis. They showed strong associations between microdamage and osteocyte apoptosis, and between apoptosis and subsequent activation of bone remodeling in regions of microdamage, suggesting that osteocyte apoptosis may contribute to the mechanism signaling for damage repair. Carter and Hayes [16] have hypothesized that fatigue damage acts to produce a bone maintenance stimulus. This hypothesis was developed by Prendergast and Taylor [10] to predict the time-course of bone adaptation, by accounting for the accumulative nature of the damage with the concept of a continuous damage rate. Recently, Prendergast and Huiskes [17] have proposed a microstructural Gnite element analysis to explore the relationship between damage formation and local strain of osteocyte-containing lacunae for various types of damage. Their results have given theoretical support to the experimental studies that have shown a correlation between damage and the initiation of resorption as a Grst step in bone remodeling. In particular, they showed that one eHect of high local strain around lacunae is that bands of stress concentration are formed between lacunae providing sites for crack nucleation and paths for crack growth eHectively unloading the lacunae adjacent to a damaged region. By the use of linear fracture mechanics (LFM) concepts, Taylor [18] has shown how information on the fatigue behavior of microcracks can be obtained by the analysis of stiHness changes measured during cyclic loading. In particular, implications for the role of microcracks in remodeling and adaptation phenomena are discussed. Garcia et al. [19] have proposed and implemented in a Gnite element code a model which combines an anisotropic internal remodeling rule based on continuum “damage-repair” theory and an external adaptation approach that follows the Computer Aided Optimization method. In particular, a realistic example, taken from surgeon experiment, is numerically investigated in order to see if an excessive bone resorption or formation may appear as a consequence of a femoral total hip replacement by an Exeter type hip prosthesis. More recently, Ramtani and Zidi [20,21] have considered a scalar damage state variable and have shown from theoretical point of view that bone remodeling stimulus is not exclusively damage [10] or strain [2,3] but rather that both operate together in a process of maintaining mass and avoiding failure. To illustrate the damage eHect on the bone adaptation process, Ramtani and Zidi [22] have taken back a speciGc example which is relevant to a deGnite physiological situation previously proposed in [3]. The region of interest is the central portion of a long bone such as the femur (region away from the joints), which has essentially the shape of a hollow circular cylinder, the axis of the cylinder coinciding with the long central

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axis of the bone. A situation in which this region of a long bone is subject to constant compressive stress magnitude P in the direction of the long axis of the bone and no other boundary stresses for t ¿ 0 was considered. An analytical solution is stated and it is clearly shown that damage plays a great role during the remodeling process in that way that it reduces the remodeling time constant (material characteristic time), which have in turns an no negligible impact upon the strain Geld. From the above studies, it is suggested that damage in bone is detected by osteocytes, which then signal for possible repair, even if it is still unclear how this signaling system works, or how much of bone remodeling is directed toward damage repair [6,17,23].

2. Damaged bone remodeling process Schematically, bone contains primarily three types of cells: (1) osteoblasts, which create (deposit) bone; (2) osteoclasts, which destroy (resorb) bone; and (3) osteocytes which are converted osteoblasts that become trapped in the bone matrix and then play a part in regulating the turnover of bone matrix. The bone resorption involves hydrolysis of collagen and the dissolution of bone mineral. In the bone deposition, the most obvious function is to synthesize osteoid, collagen and to control its subsequent mineralization. Although the networks of osteocytes do not themselves secrete or erode substantial quantities of matrix, they probably play a part in controlling the activities of the other cells and act as a body-wide ion exchange column helping to stabilize the concentration of calcium in the blood plasma. The presence of microcracks within the bone may be regarded as a break in the mineralized matrix which exposes a new surfaces lined by calcium ions [5]. In compact Haversian bone for example, it has been proposed that repair by osteonal remodeling is not a hit-or-miss process, but rather that the presence of microcracks stimulate the bone remodeling sequence [24] and evidence to support this hypothesis has been provides by experiments [8,25]. The study of Lee et al. [26], shows that the process of adaptation to altered applied load is preceded by a signiGcant rise in the incidence of microcracks adjacent to sites of bone modeling and remodeling. These microcracks are found principally in interstitial lamellae near the periosteal surface, where resorption lacunae and reGlling osteons are subsequently observed. What is known is that at the ultrastructural level, damage is hypothesized to be related with debonding of the collagen-hydroxyapatite composite [27]. At the microstructural level, damage is associated with slipping of lamellae along cement lines [28], cracking along cement lines or lacunae [29], shear cracking in cross-hatched patterns [30] and progressive failure of the weakest trabeculae [31]. At the macroscopic level, damage is hardly visible before a large crack and subsequent global failure occurs, but may involve a substantial alteration in mechanical function at early stages. Many possible pathways for transducing mechanical loads into adaptive response to both damage and strain have been proposed [9,17,23]. The one proposed by Prendergast et al. [17] is more pertinent for the current theory and is given below (Fig. 1): The alteration in load is responsible for an unphysiological strain distribution and unphysiological amounts of microdamage will be generated depending on the responsiveness of the repair rate. Therefore, after a change in load both strain equilibrium and damage equilibrium will be lost. Microdamage may activate remodeling itself by the release of growth factors embedded in bone or disruption of the canalicular network. It may also change the local strains to initiate a strain adaptive response.

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ALTERED LOAD

ACCUMULATED MICRODAMAGE

MICROSTRUCTURAL STRAIN CHANGE

BIOCHEMICAL SIGNAL

ADAPTATION Fig. 1. Possible pathways for transducing mechanical loads into adaptive response to both strain and damage [17].

3. Summary of basic equations 3.1. Continuum model The general theory of adaptive damaged-elastic materials proposed in [20] as a model for the physiological process of damaged-bone remodeling and specialized to the case of small strains in isothermal processes [21,22] will now be summarized. The three dimensional cartesian coordinates of a spatial point x are denoted by xi and time by t. The corresponding components of the displacement vector are denoted by ui (x; t), the components of the strain tensor by ij (x; t), the components of the stress tensor by Tij (x; t), the reference bone volume fraction of the elastic material by 0 (x; t), the change in bone volume fraction by e(x; t) = − 0 , the damage variable of the elastic bone material by d(x; t), the constant density of the bone elastic material by , and the components of the body force per unit mass by bi . The governing equations are: • the strain-displacement relation for small strain 1 ij = (ui; j + uj; i ); 2 • the equilibrium condition in terms of stress Tij; j + (1 − d)( 0 + e)bi = 0;

(1)

(2)

where the comma followed by a lower case Latin letter index indicates partial spatial diHerentiation. • the constitutive equation for stress which corresponds to a modiGed form of Hooke’s law in that the proportionality between stress and strain is dependent upon the damage d and the volume fraction of the material present e Tij = (1 − d)2 ( 0 + e)Cijkl (e)kl ; where Cijkl (e) is the adaptive elastic tensor and,

(3)

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• the remodeling rate equation which represents the mass balance resulting from a complex system of chemical reactions which turn body Ouids into solid bone matrix and vice versa [2,3,20,21] and, speciGes the rate of change of the bone volume fraction e˙ =

1 ˙ [a(e; d) + Aij (e; d)Kijkl (e; d)Tkl + ( 0 + e)d]; (1 − d)

(4)

where the superimposed dot indicates the material time derivative. Note that there is no experimental data on the material properties functions, a(e; d) and Aij (e; d), particularly in the situation where damage occurs, and are supposed to be independent of the damage variable [20–22]. We have also assumed that, there exists a fourth rank tensor Kijkl (e; d) which veriGes the condition (1 − d)2 ( 0 + e)Kijkl Cklrs (e) = ir js

(5)

such that the stress-strain relations (3) can be inverted for the strains ij = Kijkl (e; d)Tkl :

(6)

To terms of order e the inverse tensor Kijkl , the relation (6) gives Kijkl (e; d) =

1 0 1 0 [K 0 − eKijrs Crspq Kpqkl ]; (1 − d)2 ijkl

(7)

0 1 0 , Cijkl , are elastic constants representing the material properties, and Kijkl is deGned as where Cijkl [3,21] 0 0

0 Kijkl Cklpq = ip jq :

(8)

3.2. Three unit bone bars model An important prediction of the theory of small damage-strain adaptive elasticity can be inferred from the general expression (4) for the remodeling rate in presence of damage when the treated problem is stress-based formulation. As an application (Fig. 2), we consider the adaptive truss which consists of three vertical unit-members of equal lengths A
A, B
B, C
C, articulated to a Gxed horizontal rigid frame by three joints A
, B
, C
, and Gxed to horizontal rigid frame by three joints A, B, C. Distances A
B
and B
C
(AB and BC) are equal. The loading is constituted by a vertical force of intensity Q applied in D, middle of BC. The characteristics of the three bars are identical by their cross sectional area S, and original length ‘. Let the normal force and the elongation in each bar i be denoted by Ni and i , respectively. Then, the global form of the equilibrium equation (2) is as follows N1 + N2 + N3 = Q; 3N1 + N2 − N3 = 0:

(9)

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l

A′

B′

C′

1

2

3

A

B

D

C

Q Fig. 2. Three unit-bone bars structure.

For the damaged adaptive elastic truss member i, the behavior law (6) is derived as i =

Ni ‘ ; Ei (1 − di )2 S

(10)

where di is the damage that occurs in the ith bone bar, and Ei is the adaptive elastic modulus derived from Eq. (8) and, for which the following expression is proposed by [3] 1 1 = (1 − 11ei ): Ei E0

(11)

The geometrical compatibility condition (1) is now 1 − 22 + 3 = 0; and when damage rate neglected, the bone remodeling rule (4) becomes
 Ni 1 2 a0 + a1 ei + a2 ei + (1 + ei )Ai ; e˙ i = (1 − di ) Ei (1 − di )2 S

(12)

(13)

with a0 , a1 , a2 and Ai representing the material properties constants. Damage can be deGned in a various number of ways depending on the analysis involved, and there are many physical measures of damage including microscopy (number/density of cracks), ultrasonic waves, density changes, micro-hardness measurements, changes in electrical resistance, changes in relative amplitude of stress and strain or reduction in elastic modulus during fatigue. The damage law can be: (a) calculated using the remaining life approach, (b) from monotonic loading-unloading experimental test, (c) derived directly by means of a damage criterion, damage consistency condition and loading/unloading conditions. This last option has been analyzed here, considering two possible damage evolution laws. Taking advantage of previous works done for the study of materials which exhibit strain-softening behavior [33], a hypothetical exponential form for the damage evolution is

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proposed as (1 − di )2 =




"0 "i

#

Exp[ − $("i − "0 )];

265

(14)

where Yi represents the adaptive damage force (adaptive damage energy release rate) deGned by [21] as the variable associated to the damage variable di and, "i = (Yi =E0 )1=2 . Another damage evolution law has been studied, following usual associated plasticity as in nonliving materials. In both cases, the damage evolution requires the deGnition of a damage criterion for each bar i with the following functional form [21,32,33] gi (Yi ; i ) ≡ Yi (t) − i (t) 6 0;

t ∈ R+ ;

(15)

where i (t) signiGes damage threshold (energy barrier) at current time t (i.e. the radius of the damage surface). If i (0) denotes the initial damage threshold before any loading is applied in each bar, condition (15) then states that damage in the material is initiated when the damage energy release rate Yi (t) exceeds the initial damage threshold i (0). In the Grst case, in which damage repair can occur following equation (14), the hardening term is considered constant for each bar i, i (0) = Yi (0) = 12 E(e0 )02 . However, in classical damage theory, the hardening term increases following the next expression [21,33]  

i (t) = max i (0); max Yi (s) : (16) s∈[−∞; t]

Moreover in this case, as in plasticity theory, the damage loading function gi (Yi ) and the rate of damage growth have to satisfy the discrete Kuhn–Tucker conditions for each bar i [19,21,32,33] gi (Yi ) 6 0;

d˙i ¿ 0;

gi d˙i = 0:

(17)

The problem’s equations, including the nonlinear diHerential bone remodeling rule (13) are solved by using Newton–Raphson method. 4. Model simulation Before we proceed to the analysis of the results obtained with the simulations of the previously discussed model, it is clarifying to indicate that in all the cases here analysed, the third bar supports the maximum percentage of the external load Q, whereas the other two share the rest of the load, fulGlling the equilibrium (1) and compatibility (2) equations. In the Grst case, we assume that microdamage is known beforehand and induced by the resorption cellular activity rather than by externally applied mechanical loading. The adaptive truss structure is subjected to compressive loading, the initial values of microdamage introduced in bars nos. 3, 2 and 1 are given in Figs. 3–5. Under these conditions, the change in volume fraction is positive, highest in the bar with highest strain (Figs. 3a and b): clearly bar 3, later bar 2 and Gnally bar 1. When the third bar is damaged and the damage level is increased, this bar’s stiHness is signiGcantly reduced, which induces the porosity 1 to decrease (Fig. 3a). This eHect of bone mass increase is more important 1

It is clear that porosity is directly related to remodeling variable e(t), because  = 1 − = 1 − ( 0 + e).

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S. Ramtani et al. / Computers in Biology and Medicine 34 (2004) 259 – 273 Bar no.1 Bar no.2 Bar no.3

Bar no.1 Bar no.2 Bar no.3; d3=0.25

Bar no.1 Bar no.2 Bar no.3; d3=0.5

9.00E-02

Remodeling Variable e(t)

8.50E-02

8.00E-02

7.50E-02

7.00E-02

6.50E-02 0

200

400

600 800 Time (Days)

0

200

400

600

(a)

1000

1200

1400

0.00E+00 800

1000

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1400

-2.00E-05

Strain

-4.00E-05

-6.00E-05

-8.00E-05

-1.00E-04

Bar no.1 Bar no.2 Bar no.3

Bar no.1 Bar no.2 Bar no.3; d3=0.25

Bar no.1 Bar no.2 Bar no.3; d3=0.5

-1.20E-04

(b)

Time (Days)

Fig. 3. (a) Remodeling variable evolution for initially damaged bone-bar no. 3. (b) Strain evolution for initially damaged bone-bar no. 3.

in the third bar, whereas in the other bars it is much smaller. Moreover, in Fig. 3b we can observe that at the end of the simulation, all bars achieve a uniform strain distribution range (−5 e − 6; −2 e − 5) (Figs. 3b, 4b and 5b). So the Gnal strains in all bars are very sim-

S. Ramtani et al. / Computers in Biology and Medicine 34 (2004) 259 – 273 Bar no.1 Bar no.2 Bar no.3

Bar no.1 Bar no.2; d2=0.25 Bar no.3

267

Bar no.1 Bar no.2; d2=0.5 Bar no.3

9.00E-02

Remodeling Variable e(t)

8.50E-02

8.00E-02

7.50E-02

7.00E-02

6.50E-02 0

200

400

(a)

600

800

1000

1200

1000

1200

1400

Time (Days) 0

200

400

600

800

1400

-5.00E-06

Strain

-1.50E-05

-2.50E-05

-3.50E-05

-4.50E-05 Bar no.1 Bar no.2 Bar no.3

Bar no.1 Bar no.2; d2=0.25 Bar no.3

Bar no.1 Bar no.2; d2=0.5 Bar no.3

-5.50E-05

(b)

Time (Days)

Fig. 4. (a) Remodeling variable evolution for initially damaged bone-bar no. 2. (b) Strain evolution for initially damaged bone-bar no. 2.

ilar, whether they are damaged or not. This equilibrium zone, normally known as the “dead zone”, has been proposed by several authors in other bone remodeling theories [34,35].

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S. Ramtani et al. / Computers in Biology and Medicine 34 (2004) 259 – 273 Bar no.1 Bar no.2 Bar no.3

Bar no.1; d1=0.25 Bar no.2 Bar no.3

Bar no. 1; d1=0.5 Bar no.2 Bar no.3

8.50E-02 8.30E-02 8.10E-02 Remodeling Variable e(t)

7.90E-02 7.70E-02 7.50E-02 7.30E-02 7.10E-02 6.90E-02 6.70E-02 6.50E-02 0

200

400

(a)

600

800

1000

1200

1400

Time (Days)

0.00E+00 0

200

400

600

800

1000

1200

1400

-5.00E-06 -1.00E-05

Strain

-1.50E-05 -2.00E-05 -2.50E-05 -3.00E-05 -3.50E-05 -4.00E-05

Bar no.1 Bar no.2 Bar no.3

Bar no.1; d1=0.25 Bar no.2 Bar no.3

Bar no.1; d1=0.5 Bar no.2 Bar no.3

-4.50E-05

(b)

Time (Days)

Fig. 5. (a) Remodeling variable evolution for initially damaged bone-bar no. 1. (b) Strain evolution for initially damaged bone-bar no. 1.

When the second bar is damaged (Fig. 4a and b), the porosity in all bars is reduced, remaining higher in the third bar and more similar in the other two. However, in this case the evolution of the volume fraction is diHerent when the damage level is not equal to zero. So, initially, the volume

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fraction in bar 2 increases more quickly than in bar 3, but when bar 3 achieves a suTcient stiHness, then volume fraction begins to decrease slightly. In this case, when the level of damage is increased, the changes in porosity are higher in bars 1 and 2 (Fig. 4a). Similar conclusions can be obtained when the damaged bar is the Grst one (Fig. 5a and b). Actually, when the right part of the structure is damaged (bar 3) the model predicts a higher bone response in this bar in comparison with the undamaged case. However, if the left part is damaged (bar 1 or 2) then the higher modiGcation in the bone response in comparison with the undamaged situation is produced in bars 1 and 2, even one of them is not damaged. We can conclude that when the microstructure is damaged a higher bone response is produced, increasing the bone mass in order to increase the stiHness lost by damage. Then, although in this case microdamage is not associated to external mechanical loading, it is clear that microdamage may induce a stronger remodeling response. Moreover, microdamage not only induces bone remodeling in the site where it occurs as several authors propose, but also in other regions of the bone tissue. Indeed, when microdamage occurs in some trabeculae, an alteration of the global structural equilibrium is produced, inducing bone remodeling in intact and damaged trabeculae. Thus, the simple example shown in this paper is representative of this situation. Finally, we assume that microdamage is induced by external applied mechanical loading and driven by the adaptive damage force. The adaptive truss structure is subjected to compression loading, and microdamage is calculated in each bar following the two possible damage evolution laws proposed previously: repair law or classical damage law. The Grst case requires metabolic support, whereas the second one is a classical case of positive mechanical dissipation. We could say that the Grst law allows us to study the bone response in a healthy bone, which has the capacity of repairing microdamage, whereas with the second law we can analyze the diHerent bone response in non-healthy bone, whose behavior is closer to an inert material. In Fig. 6a, we can observe how bone mass response caused by damage is higher when bone has no possibility of repairing. Meanwhile, it is possible to observe how the strain equilibrium is also achieved under both conditions (Fig. 6b). The damage evolution is very high on the Grst day, increasing quickly in bars 2 and 3, whereas no damage occurs in the bar 1 (Fig. 6c). This is due to the fact that the damage force Y in the bars 2 and 3 exceeds the initial damage force Y0 . From 30 days, in the case of repair, damage in bar 1 increases until the end of the process. However in bars nos. 2 and 3, damage is continuously reduced, because the damage forces in these bars are greater than the initial damage force Y0 , being a decreasing time function (Fig. 6d). When no repairing capacity is considered, the damage in bars 2 and 3 remains constant, causing a stronger damage concentration in bar no. 1 (Fig. 6c). 5. Discussion In this work we have investigated the dynamic behavior of a new bone remodeling model, in which bone remodeling is activated in response to compressive strain mechanical level and microdamage density. To study the model’s intrinsic responses under non-uniform strain distribution a very easy example has been designed, which consists of three bars that simulate bone trabeculae in the form of truss. Most computational models that try to analyse the coupling between damage and mechanical load use examples with homogeneous strain distribution, which in turn is not a very real situation. In fact, bones usually have non-uniform strain distributions.

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S. Ramtani et al. / Computers in Biology and Medicine 34 (2004) 259 – 273 Bar no.1 Bar no.2 Bar no.3

Bar no.1, Damage Bar no.2, Damage & Repair Bar no.3, Damage & Repair

Bar no.1 Bar no.2 Bar no.3

Bar no.1, Damage Bar no.2, Damage & no Repair Bar no.3, Damage & no Repair

Bar no.1, Damage Bar no.2, Damage & Repair Bar no.3, Damage & Repair

Bar no.1, Damage Bar no.2, Damage & no Repair Bar no.3, Damage & no Repair

0.00E+00

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0

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Remodeling Variable e(t)

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-3.00E-05

7.50E-02 -4.00E-05

7.00E-02

-5.00E-05

6.50E-02

(a)

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0

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600 800 Time (Days)

Bar no.1, Damage Bar no.2, Damage & Repair Bar no.3, Damage & Repair

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1200

1400

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(b) Bar no.1 Bar no.2 Bar no.3

Bar no.1, Damage Bar no.2, Damage & no Repair Bar no.3, Damage & no Repair

Bar no.1, Damage Bar no.2, Damage & Repair Bar no.3, Damage & Repair

Bar no.1, Damage Bar no.2, Damage & no Repair Bar no.3, Damage & no Repair

1.60E+02

2.50E-01

1.40E+02 1.20E+02 Damage Force Y(e)

Damage d

2.00E-01

1.50E-01

1.00E-01

1.00E+02 8.00E+01 6.00E+01 4.00E+01

5.00E-02

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400

600 800 Time (Days)

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1200

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1400

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400

600

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Fig. 6. (a) Remodeling variable evolution for damaged bone-bars structure following hypothetical damage evolution law. (b) Strain evolution for damaged bone-bars structure following hypothetical damage evolution law. (c) Damage evolution in the bone-bars structure. (d) Damage force evolution in the bone-bars structure.

In this study, we have demonstrated that bone remodeling response strongly depends on the microdamage distribution. Actually, we have obtained very diHerent responses in the intact case and the damaged one, with the damage level as the determining factor. Moreover we have also analysed the inOuence of two extreme repair patterns of bone response, concluding that bone mass and damage distribution are completely dependent on these patterns, although the Gnal strain distribution achieved is nearly constant in all the cases. Although the results shown in this paper are representative of what happens when microdamage occurs, our remodeling theory could be improved, by including several facts that are known experimentally, such as bone remodeling activated by disuse [23] (Table 1).

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Table 1 Parameters of the model Model a0 a1 a1 A Q 0 # $ ‘ E0 (GPa) S Parameters (s−1 ) (s−1 ) (s−1 ) (s−1 ) (MPa) (mm) (mm2 ) −8 −7 −7 −5 −6 2 10 −2:5 × 10 15 × 10 −2:75 × 10 −5:0 −6:65 × 10 0.25 10 1 18.4 1

6. Summary It has been well argued that damage in bone is detected by osteocytes, which then signal for possible repair, even if it is still unclear how this signaling system works, or how much of bone remodeling is directed toward damage repair. The process of adaptation to altered applied load is preceded by a signiGcant rise in the incidence of microcracks adjacent to sites of bone modeling and remodeling. These microcracks are found principally in interstitial lamellae near the periosteal surface, where resorption lacunae and reGlling osteons are subsequently observed. At the microstructural level, damage is associated with slipping of lamellae along cement lines, cracking along cement lines or lacunae, shear cracking in cross-hatched patterns and progressive failure of the weakest trabeculae. At the macroscopic level, damage is hardly visible before a large crack and subsequent global failure occurs, but may involve a substantial alteration in mechanical function at early stages. In this study, we have demonstrated that the adaptive response of the three unit-bone bars structure strongly depends on the microdamage distribution. Actually, we have obtained very diHerent responses in the intact case and the damaged one, with the damage level as the determining factor. Moreover we have also analysed the inOuence of two extreme repair patterns of bone response, concluding that bone mass and damage distribution are completely dependent on these patterns, although the Gnal strain distribution achieved is nearly constant in all the cases. The obtained results in this study are representative of what happens when microdamage occurs. In particular, the contrast between bone behavior when there is a repair capability for microdamage compared with that of non-healthy bone lacking such a capability is an important result from the proposed model. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

J.L. WolH, The Law of Bone Remodeling, Springer, Berlin, 1986. S.C. Cowin, D.M. Hegedus, Bone remodeling I: theory of adaptive elasticity, J. Elast. 6 (1976) 313–325. D.H. Hegedus, S.C. Cowin, Bone remodeling II: small strain adaptive elasticity, J. Elast. 6 (1976) 337–352. R. Huiskes, Simulation of Self-Organization and Functional Adaptation in Bone, Springer, Berlin, 1997. T.C. Lee, L. Noelke, G.T. McMahon, J.P. Mulville, D. Taylor, Functional adaptation in bone, in: P. Pedersen, M.P. Bendsoe (Eds.), Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers, Dordrecht, 1998. D.B. Burr, Damage detection and behavior in bone, 12th Conference of the European Society of Biomechanics, Dublin, 2000, pp. 38–39. D.B. Burr, C. Milgrom, R.D. Boyd, W.L. Higgins, G. Radinel, Experimental stress fractures of the tibia-biological and mechanical aetiology in rabbit, J. Bone Joint Surg. 72B (1990) 370–375. S. Mori, D.B. Burr, Increased intracortical remodeling following fatigue damage, Bone 14 (1993) 103–109. R.B. Martin, Toward a unifying theory of bone remodeling, Bone 26 (2000) 1–6. P.J. Prendergast, D. Taylor, Prediction of bone adaptation using damage accumulation, J. Biomech. 27 (1994) 1067–1076.

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[11] H.M. Frost, Presence of microscopic cracks in vivo in bone, Bull. Henry Ford Hosp. 8 (1960) 25–35. [12] D.B. Burr, M.K. Forwood, D.P. Fyhrie, R.B. Martin, M.S. SchaDer, C.H. Turner, Bone microdamage and skeletal fragility in osteoporotic and stress fractures, J. Bone Min. Res. 12 (1997) 6–15. [13] V. Bentolila, T.M. Boyce, D.P. Fyhrie, R. Drumb, T.M. Skerry, M.S. SchaDer, Intracortical remodeling in adult rat long bones after fatigue loading, Bone 23 (1998) 275–281. [14] O. Verborgt, G.J. Gibson, M.B. SchaDer, Loss of osteocyte integrity in association with microdamage and bone remodeling after fatigue in vivo, J. Bone Mineral Res. 15 (2000) 60–67. [15] M.B. SchaDer, O. Verborgt, Mechanisms of microdamage repair in bone, 12th Conference of the European Society of Biomechanics, Dublin, 2000, p. 43. [16] D.R. Carter, W.C. Hayes, Compact bone fatigue damage-I. Residual strength and stiHness, J. Biomech. 10 (1977) 325–337. [17] P.J. Prendergast, R. Huiskes, Mathematical modeling of microdamage in bone remodeling and adaptation, in: A. Odgaard, H. Weinans (Eds.), Bone Structure and Remodeling, Recent Advances in Human Biology, Vol. 2, World ScientiGc, Singapore, 1995, pp. 213–223. [18] D. Taylor, Microcrack growth parameter for compact bone deduced from stiHness variations, J. Biomech. 31 (1998) 587–592. [19] J.M. Garcia, M.A. Martinez, M. DoblarUe, An anisotropic internal-external bone adaptation model based on a combination of CAO and continuum damage mechanics technologies, Comput. Methods Biomech. Biomed. Eng. 34 (2001) 471–479. [20] S. Ramtani, M. Zidi, Damaged-bone remodeling theory: thermodynamical approach, Mech. Res. Commun. 26 (1999) 701–708. [21] S. Ramtani, M. Zidi, A theoretical model of the eHect of continuum damage on a bone adaptation model, J. Biomech. 34 (2001) 471–479. [22] S. Ramtani, M. Zidi, 2001. Damaged-bone adaptation under steady homogeneous stress, ASME J. Biomech. Eng. 124 (2002) 1– 6. [23] S.J. Hazelwood, R.B. Martin, M.M. Rashid, J.J. Rodrigo, A mechanistic model for internal bone remodeling exhibits diHerent dynamic responses in disuse and overload, J. Biomech. 34 (2001) 299–308. [24] R.B. Martin, D.B. Burr, The Structure, Function and Adaptation of Cortical Bone, Raven Press, New York, 1989. [25] D.B. Burr, R.B. Martin, M.B. SchaDer, E.L. Radin, Bone remodeling in response to in vivo fatigue microdamage, J. Biomech. 18 (1985) 189–200. [26] T.C. Lee, Detection and accumulation of microdamage in bone, M.D. Thesis, University of Dublin, 1997. [27] J.F. Mammone, S.M. Hudson, Micromechanics of bone strength and fracture, J. Biomech. 26 (1993) 439–446. [28] R. Lakes, S. Saha, Cement line motion in bone, Science 204 (1979) 501–503. [29] K. Choi, S.A. Goldstein, A comparison of the fatigue behaviour of human trabecular and cortical bone tissue, J. Biomech. 25 (1992) 1371–1381. [30] K. Choi, D.P. Fyhrie, M.B. SchaDer, Failure mechanisms of compact bone in bending: a microstructural analysis, Transactions of 40th Orthopaedic Research Society, 1994, p. 425. [31] X.E. Guo, Finite element modeling of damage accumulation in trabecular bone under cyclic loading, J. Biomech. 27 (1994) 145–155. [32] J.L. Chaboche, Continuum damage mechanics. Part I, General concepts, Part II, Damage growth, crack initiation and crack growth, J. Appl. Mech. 55 (1988) 233–247. [33] S. Murakami, K. Kamya, Constitutive and damage evolution equations of elastic-brittle materials based on irreversible thermodynamics, Int. J. Mech. Sci. 39 (1997) 473–486. [34] D.R. Carter, D.P. Fyhrie, R.T. Whalem, Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy, J. Biomech. 20 (1987) 785–794. [35] R. Huiskes, H. Weinans, H.J. Grootenboer, M. Dalstra, B. Fudala, T.J. Sloof, Adaptive bone-remodeling theory applied to prosthetic-design analysis, J. Biomech. 20 (1987) 1135–1150. M. Doblar"e is Full Professor of Structural Mechanics of the Department of Mechanical Engineering, Zaragoza University, Spain since 1984. He is presently the Director of the AragUon Institute for Engineering Research. Doctorate at Mechanical Engineering at the Polytechnic University of Madrid in 1981 and Doctorate “Honoris Causa” at the Technical University

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of Cluj-Napoca (Romania). Visiting Scholar at the Universities of Southampton, U.K., New York University, U.S.A. (Fulbright grant) and Stanford, U.S.A. His current research interests in the Geld of Bioengineering are: Computational biomechanics and FEA, hard and soft tissues modelling and interactions with biomaterials, including constitutive laws, mechanobiological processes, interface behavior, damage mechanics. More than 50 international journal papers, conference proceedings, reports and diHerent oral presentations in Europe and USA. J.M. Garc"%a-Aznar became research assistant of the Department of Mechanical Engineering of the University of Zaragoza (Spain) in 1996. He achieved his Ph.D. in Computational Mechanics at the University of Zaragoza in 1999 and spent a post-doctoral stay at the Center for Science and Technology in Medicine (University of Keele, England) in 2001. He is member of group of Structures and Material Modelling of the AragUon Institute for Engineering Research (I3A). His current research is related to Computational Biomechanics, mainly in the Gelds of Mechanics of Hard Tissues, Mechanobiology of Skeletal Tissue Regeneration, Tissue Growth and Development, Orthopaedic Biomechanics and Fractures of Long Bones. S. Ramtani received a Ph.D. in Mechanics of Solids and Structures from the University of Paris VI Pierre & Marie Curie in 1990. The topic of his diploma was about Continuum Damage Mechanics and Thermodynamics applied to concrete behavior. In 1994, he obtained a position at the University Paris XII Val-de-Marne as Professor of Mechanical Engineering. His main research topics remain in the Geld of Biomechanics and modeling in Mechanobiology. His research centers on damage-bone remodeling and Mechanobiology of wound healing both from theoretical, numerical and experimental viewpoint.