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HARD TISSUE . APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING

Lecture 2. Phenomenological models of bone remodelling

Manuel Doblaré

Aragón Institute of Engineering Research (I3A) University of Zaragoza (Spain)

mdoblare@unizar.es

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

Lecture 2. Phenomenological models of bone remodelling 2.1. Introduction

2.2. Experimental facts

2.3. Phenomenological models

2.4. An anisotropic phenomenological model of bone remodelling 2.5. Phenomenological models of external bone modelling

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Lecture 2. Phenomenological models of bone remodelling 3 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

2.1. Introduction

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

From Evolution. Savage, J. M. 1969 (2nd Edition).

Holt, Rhinehart and Winston, Inc. p. 20.

Charles Darwin

TISSUE ADAPTATION

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Lecture 2. Phenomenological models of bone remodelling 5 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

FIGURE 1. Galileo's illustration of the same bone (femur) from animals of different sizes. Whereas the lengths of the bones differ by about 2.5 times, the width of the bones differ over tenfold.

Discorsi e Dimostrazioni Matematiche in torno a due nuove scienze, 1638

ƒ Construction and reconstruction of tissues is the combined result of two factors: phylogenetic and ontogenetic processes.

ƒ Phylogenetic processes (species evolution) involve random genetic variation and natural selection of the best fitted.

ƒ Ontogenetic (individual) adaptation is accomplished by appropriate mechano-regulatory rules.

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ These processes are controlled by a combination of genetic and epigenetic factors (Edelman, Topobiology. An Introduction to Molecular Embriology. Basic Books, 1988).

ƒ Genes direct the formation of the basic building blocks, including proteins, extracellular matrix and adhesion molecules.

ƒ Epigenetic factors, including chemical agents and mechanical strain, influence which genes are expressed and how these blocks are assembled into tissues.

ƒ The mechano-regulatory processes can be evolved themselves, so epigenetic (environmental) factors can have also important influence in long-term evolution of species.

(Carter, D.R., Mikic, B. and Padian, K. “Epigenetic mechanical factors in the evolution of long bone epiphyses”. Zool. J. Linnean Soc. 123 (1998) 163-178)

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Lecture 2. Phenomenological models of bone remodelling 7 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ These processes are involved in many biological aspects like growth, healing, oseointegration, or pathological problems like osteoporosis or tissue damage.

ƒ Their good understanding is essential in the progress of techniques as tissue regeneration, tissue engineering or design of synthetic organs.

Bone fracture healing

Wound healing Induced osteoporosis

by osteolysis

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009 ƒ Therefore, tissues, organs and the whole living body adapt

themselves to the long-term history of their specific mechanical, metabolical and physiological environments.

ƒ They continually undergo processes of growth, reinforcement and resorption that evolve all along their lives and are termed collectively as functional adaptation or tissue remodelling.

ƒ Bone remodeling for instance provides:

a way for the body to alter the balance of essential minerals by increasing their concentration in serum.

a mechanism for the skeleton to reduce the risk for fracture.

a mechanism to repair damage.

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Lecture 2. Phenomenological models of bone remodelling 9 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

Wolff, J. “Das Gesetz der inneren Transformation der Knochen bei pathologischen Veränderungen der äusseren Knochenform”. Sitzungsber Preuss. Akad. Wisse, 22;475-496, 1884

Wolff, J. The Law of Bone Remodelling.

Springer Verlag, 1986

BONE REMODELLING. FIRST THEORIES

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ “Weak form” of the Wolff’s law: dependence between bone structure and the load that it supports. Cowin also argued that the macroscopic anisotropic structure and properties could be aligned with principal stress directions and not the actual trabeculae. Wolff did not indicated any idea about the possible processes responsible of this effect. This has to be credited to Roux, who proposed that bone adapted itself in order to support stresses in an optimal way with minimum mass, We should therefore talk about “Roux’s theory” instead of “Wolff’s law”.

ƒ However, and although intensively discussed and reformulated, Wolff’s Law must be considered as the start of the tremendous development of remodeling models during the second half of the XXth century.

(Cowin, S.C. “The false premise in Wolff’s law”, in Bone Mechanics. 2nd Ed. (Cowin, S.C. ed.) Editorial, 30;1-15, año)

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Lecture 2. Phenomenological models of bone remodelling 11 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

Architectural adaptation

Surface growth (internal) Surface growth (external) Interstitial growth

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

(Skalak, R. Proceedings of the IUTAM Symposium on Finite Elasticity. (Carlson, D.E., Shields, R.T. eds.) Martinus Nijhoff Publ, 347-355, 1981)

30ºFLEXIÓN

Without prestress

With prestress

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Lecture 2. Phenomenological models of bone remodelling 13 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

2.2. Experimental facts

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Different bones have different functions and are subjected to different loads. Thus they likely respond differently to the same stress (Meade, J.B. “The adaptation of bone to mechanical stress: experiments and current concepts”. in Bone Mechanics 1stEd. (Cowin, S.C. ed.) CRC Press 211-251, 1989)

ƒ Load-bearing bones likely adapt to minimize bending. (Currey, J.D.

“The adaptation of bone to stress”. J. Theor. Biol. 20; 91-106, 1968).

ƒ For a given change in load magnitude, the rate of atrophy is grater than the rate of hypertrophy (Woo, SL.Y. “The relationships of changes in stress levels on long bone remodelling”. Biomechanics of Bone (Cowin, S.C. ed.) 107-129, 1981).

ƒ Cancellous bone microstructure align at macroscopic level along the directions of principal stress (Hayes, W.C. and Snider, B. Towards a quantitative formulation of Wolff´s Law in trabechular bone”. in Mechanical Properties of Bone (Cowin, S.C. ed.) ASME 43-68,1981)

ƒ Electrical and electromagnetical stimulation have shown to have also some influence on bone remodelling.

EXPERIMENTAL FACTS

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Lecture 2. Phenomenological models of bone remodelling 15

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009 ƒ Bone is in remodelling equilibrium over a range of compresive

strain levels. Outside this range, decreased compressive stress induced loss of bone mass and increased compressive stress causes increased mass. For instance:

ƒ In long-term space flight there is a considerable loss of bone mass of up to 7% of skeletal mass. (Collet, P. et al. Effects of 1 and 6 month space flight on bone mass and biochemistry in two humans. Bone, 20;547-551, 1997)

ƒ This reduction was proved to be prevented by the use of a high frequency, short-term mechanical stimulus. (Goodship, A.E. et al. Bone loss during long term space flight is prevented by the application of a short term impulsive mechanical stimulus, Acta Austronaut., 43;65-75, 1998).

ƒ As few as four cycles of osteogenic strain applied once per day are enough to prevent bone loss. (Rubin, C.T. And Lanyon, L.E. Regulation of bone formation by applied dynamic loads. J. Bone Joint Surg. 66a, 397-402, 1984)

ƒ Loss of bone mass associated to mechanical isolation is inhibited by the application of subphysiological levels of deformation applied at a specific frequency of about 30 Hz. (Qin, Y.X. et al. Nonlinear dependence of loading intensity and cycle number in the maintenance of bone mass and morphology.

J. Orthop. Res., 16;482-489, 1998)

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009 „ Two dozen possible mechanical stimuli were compared in a

combined experimental and analytic approach. The data supported strain energy density, shear stress and tensile principal stress or strain as stimuli; no stimulus that could be described as rate dependent was considered in this study.

(Brown et al., Toward an identification of mechanic parameters initiating periosteal remodelling: a combined experimental and analytic approach. J. Biomechanics, 23;893- 905, 1990)

„ Cowin argued that strain is probably the mechanical growth factor, since cells can measure strain but not stress with specialized receptors. (Cowin, S.C. “Mechanical modelling of the stress adaptation process in bone”. Calcif. Tissue Int. 36; S98-S103,1984)

„ Rodriguez et al. however, pointed out that strain depends on the choice of a reference configuration whereas the Cauchy stress does not. It is likely that tissues never experience the zero-stress state in vivo and so there is no basis for sensing an absolute measure of strain. (Rodriguez, E.K. et al. “Stress-dependent finite growth in

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Lecture 2. Phenomenological models of bone remodelling 17 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

„ Stress and strain rates also are possibilities that do not have this drawback if defined in an Eulerian sense, while strain could be other appropriate choice for bone that undergoes small deformation (Cowin, S.C. and Firoozbakhsh, K. “Bone remodelling of diaphysial surfaces under constant load: theoretical prediction”. J. Biomech. 14;471-484, 1981)

ƒ There is a clear correlation between strain rate and bone hypertrophy.

(O’Connor, J.A. et al. The influence of strain rate on adaptive bone remodelling, J.

Biomech. 15;767-781, 1982)

ƒ Intermitent loads are more effective remodelling stimulus than are static loads (Lanyon,L.E. and Rubin, C.T. “Static versus dynamic load as an influence on bone remodelling”. J. Biomech. 17; 897-905,1984)

ƒ Trabechular bone density increases in regions of high shear (Hayes, W.C. and Snider, B. Towards a quantitative formulation of Wolff´s Law in trabechular bone”. in Mechanical Properties of Bone (Cowin, S.C. ed.) ASME 43- 68,1981) and as the loading rate incresases (Goldstein,S.A. et al.

“Trabechular bone remodelling: an experimental model”. J. Biomech. 24;Suppl. 1 135-150,1991)

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Some authors investigated whether or not the damage in the bone matrix would affect the strain felt by the osteocyte. They found that osteocyte deformation was reduces by the growth of microcraks.

(Prendergast, P.J. and Huiskes, R.”Microdamage and osteocyte lacuna strain in bone: a microstructural finite element analysis”. J. Biomech. Eng. 118;240-246, 1996)

ƒ Internal microcracks are associated with regions of bone resorption, the first step in the remodelling process (Verbogt, O. et al.

Loss of osteocyte integrity in association with microdamage and bone remodelling after fatigue in vivo. J. Bone Miner. Res. 15, 60-67, 2000)

ƒ Bone remodelling can occur if the strain distribution is abnormal, even if the strain magnitudes are normal.(Lanyon, L.E. et al. “Mechanically adaptive bone remodelling”. J. Biomech. 17; 897-905 1982)

ƒ There is also a strong influence in of age, hormonal modulation, pathological conditions or dietary effects.

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Lecture 2. Phenomenological models of bone remodelling 19 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

2.3. Phenomenological models

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009 ƒ Mathematical models and their solution via computational methods

have opened a new via to analyze biological processes like functional adaptation.

ƒ We can talk about Biomechanical models whose goal is to predict the mechanical behaviour (movement, strains and stresses) of a tissue or organ, taking into account the acting loads, its microstructure and the constraints imposed by other organs.

„ Or about Mechanobiological models that, on the contrary, try to predict the evolution of the microstructure and biological constitution of a tissue or an organ as consequence of the mechanical environment.

ƒ BONE REMODELLING MODELS HAVE TO BE CONSIDERED WITH THIS DEFINITON AS MECHANOBIOLOGICAL MODELS.

ƒ They require in general the modelling of coupled fields (Thermomechanics, Chemistry, Biology) with different spatial and

ƒ Mathematical models and their solution via computational methods have opened a new via to analyze biological processes like functional adaptation.

ƒ We can talk about Biomechanical models whose goal is to predict the mechanical behaviour (movement, strains and stresses) of a tissue or organ, taking into account the acting loads, its microstructure and the constraints imposed by other organs.

Deformed shape (x10) – Fracture AO-C31 of the pelvis García-Aznar, J.M. et al.,2001)

ƒ Mathematical models and their solution via computational methods have opened a new via to analyze biological processes like functional adaptation.

ƒ We can talk about Biomechanical models whose goal is to predict the mechanical behaviour (movement, strains and stresses) of a tissue or organ, taking into account the acting loads, its microstructure and the constraints imposed by other organs.

„ Or about Mechanobiological models that, on the contrary, try to predict the evolution of the microstructure and biological constitution of a tissue or an organ as consequence of the mechanical environment.

ƒ Mathematical models and their solution via computational methods have opened a new via to analyze biological processes like functional adaptation.

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Lecture 2. Phenomenological models of bone remodelling 21 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Main objective: To develop mathematical models able to predict the long-term behaviour of bone under physiological and pathological loads.

ƒ Most models admit the existence of a certain mechanical stimulus that produces bone apposition or resorption such us, by this process, that stimulus tends to a certain physiological level (homeostasis).

ƒ Mechanical stimuli:

Strains (maximum principal, equivalent or effective strains)

Stresses (maximum principal, von Mises, deviatoric invariants,..)

Strain deformation energy (complete, deviatoric,..)

Damage

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Internal variables, directly associated to the microstructure, easily measurable and able to reflect the evolution of the bone mass (including the directional properties in the case of architectural remodelling) and controlling the evolution of the stiffness of the tissue.

ƒ Criterion of mass production, resorption and homeostatic equilibrium.

ƒ Rate of net bone mass production.

ƒ Correlation of the internal variables with tissue stiffness.

ƒ Although usually coupled in reality, it is usually distinguished between internal and external remodelling.

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Lecture 2. Phenomenological models of bone remodelling 23 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

PHENOMENOLOGICAL BONE REMODELLING MODELS

ƒ Models based on global optimality criteria: Rodrigues, 1998;

Fernandes, Rodrigues and Jacobs, 1998; Terrier and Rakotomanana, 1997,..).

ƒ Models based on achieving a homeostatic state of strain or stress: Pauwels and Kummer 1965, 1972; Cowin and Hegedus, 1977; Carter et al., 1987; Huiskes et al. (1987), Beaupré, 1990a, 1990b; Cowin et al. (1992), Jacobs, 1994, 1997; Fyhrie y Schaffler, 1995; Terrier et al, 1997a; Mikic y Carter, 1995;

Petermann et al, 1997; Stülpner et al, 1997; Turner et al., 1997;

Luo et al., 1995, Doblaré and García, 2001, and many others.

ƒ Models based on damage repairing: Prendergast and Taylor, 1994; Ramtani, 2000,2001.

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

MODEL OF PAUWELS & KUMMER

ƒ Powels (1965) proposed that the maximal principal stress controls bone mass production. Values over a certain value produce new mass production, while values under other level give rise to resorption. This was formulated in Kummer (1972).

ƒ Kummer, who agreed with the trajectorial hypothesis of Wolff, drew the alignment of

) - ( ) - ( ) -

( σ σu σ σn σ σo dt c

dm =

dt dm

σu

σn

σo σ

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Lecture 2. Phenomenological models of bone remodelling 25 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Cowin & Hegedus (1976) proposed a thermodynamically consistent formulation of bone remodelling based on Continuum Mechanics.

ƒ They considered bone tissue as a poroelastic material with a solid matrix surrounded by interstitial fluid. They considered bone remodelling as a consequence of chemical reactions between the solid matrix and the fluid.

ƒ This theory (Cowin & Hegedus, 1976; Hegedus & Cowin, 1976; Cowin &

Firoozbakhsh, 1981) was developed for cortical bone. It assumed the existence of a natural equilibrium state without remodelling.

ƒ Following initial ideas of Frost (1964), they proposed a coupled internal-external bone remodelling law:

) - t (

d E

d =A ε ε0 ( - ) t

d X

d =B ε ε0

THEORY OF ADAPTIVE ELASTICITY OF COWIN et al.

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Cowin et al. (1992) extended this theory to include anisotropy and reorientation of the trabechular architecture as a function of mechanical actions (strains).

ƒ They introduced the fabric tensor, H as a quantitative stereological measure of the trabechular architecture and the directional arrangement of the microstructure of cancellous bone (Cowin, 1985; Cowin, 1986; Turner y Cowin, 1987; Turner et al., 1990;

Cowin et al., 1992).

ƒ H is a second-order positive definite tensor, whose principal directions are coincident with the principal directions of the microstructure and its principal values are proportional to the amount of bone mass along each principal direction.

ƒ The internal parameters considered were the internal porosity v and the fabric tensor, that was determined by Cowin by the MIL method.

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Lecture 2. Phenomenological models of bone remodelling 27 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ The fabric tensor is normalized imposing trH=1, so it is uniquely determined by its deviatoric part, K, which now measures the degree of anisotropy of cancellous bone.

ƒ They correlated the fabric tensor with the elastic constants of bone tissue, obtaining the first mathematical description of Wolff’s law

σ =C(K,e) ε or ε = D(K,e) σ

where e is the porosity from a reference value v0, (e = v - v0).

They proposed

σ = β11 + β2ε + β3K + β4K2 + β5(Kε + εK) + β6(K2ε + εK2)

ƒ The equilibrium state is achieved when the strain is inside a predefined range and the principal directions of bone (fabric tensor) align with those of the strain tensor, becoming also aligned with the ones of the stress tensor.

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

• They proposed an evolution law both for porosity and the fabric tensor:

= α11 + α2K + α3K2+ α4ε + α5(Kε + εK)

The values of αiandβiare arbitrary functions of e, trK2, y trK3, trε, trεK y trεK2, to be determined experimentally.

ƒ This model was too complex and was drastically simplified for its computational implementation.

ƒ In addition, the model did not include any biological effect what makes it difficult to validate experimentally.

K&

e) , tr , tr , tr , tr , tr ( e

e& =& ε K2 K3 εK εK2

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Lecture 2. Phenomenological models of bone remodelling 29 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ In a series of papers (Carter et al., 1987; Whalen et al., 1988; Carter et al., 1989). Carter and co-workers proposed a new mathematical formulation for the functional adaptation of cancellous bone, based on the concept of auto-optimization.

ƒ In agreement with Frost & Pauwels they proposed that a certain mechanical stimulus S should be present in bone tissue, such as when this value is achieved a quasi-stationary of null remodelling is achieved.

ƒ Carter et al. proposed a strimulus proportional to a certain effective stress as

=

σ

N

1 i

m i

ni

S

CARTER’S CONCEPT OF AUTO-OPTIMIZATION

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Probably the most important contribution of this model was the consideration of the influence of different load cases (i = 1 … N) and the number of load cycles for each of these cases ni

ƒ The effective stress was defined in terms of the local mechanical state as a function of the strain energy density U and the local apparent density ρ.

ƒ They assumed that functional adaptation maximizes structural integrity (stiffness) with the minimum bone mass provision.

(Fyhrie & Carter, 1986). From this assumption, they proposed a relation between the apparent density and the effective stress as

U ) E(

2 ρ

= σ

) m 2 / 1 N (

1 i

m i

ni

⎜ ⎞

⎛ σ

ρ

=

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Lecture 2. Phenomenological models of bone remodelling 31 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ They also extended the model to include the effect of anisotropy. Carter et al. (1989) proposed that trabechulae tend to orientate according to a directional quantity and related to stresses, that they named normal equivalent stress, σn*, based as well on the contributions of different load cases.

ƒ This model was implemented into a finite element program and several examples were solved to predict the evolution of the apparent density distribution on the proximal femur. As initial state, a homogeneous intermediate distribution was considered and three load cases on the femoral head and the associated reactions on the abductor muscle were responsible of the mechanical state (Carter et al, 1987).

( )

m / N 1

1 i

m i t

* i

n n

) n

( ⎥

⎢ ⎤

⎡ ⎟⎟⎠ • •

⎜⎜ ⎞

= ⎛

σ

=

n n

n σ

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

ƒ Following the ideas of Cowin in the adaptive theory of elasticity, Huiskes et al. (1987) established a model of bone remodelling, using the strain energy density U(e) as the mechanical stimulus.

ƒ The assumed a linear relation between the adaptive response of bone tissue and the stimulus, including the idea of Carter (1984) that assumes a certain “laziness” of bone tissue, defining a “dead zone” or equilibrium region in the stimulus space.

MODEL OF HUISKES et al.

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Lecture 2. Phenomenological models of bone remodelling 33 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ With this, the evolution law for the coupled process of internal- external remodelling was established as:

ƒ They detected some problems like negative values of the elastic modulus or excessive geometry distortion, but they solved several examples like the external remodelling of a cantilever beam made of bone tissue, the Carter’s problem and a simple model of a prosthesis in a bone predicting the well- known effect of “stress shielding” (Huiskes et al., 1987)

⎥⎥

⎢⎢

<

+

+

>

+

=

0 0

e

0 0

0 0

e

U s) - (1 U si

), s)U (1 - U ( C

U s) (1 U s)U - (1 si

, 0

U s) (1 U si ),

s)U (1 - U ( C t d

E d

⎥⎥

⎢⎢

<

+

+

>

+

=

0 0

x

0 0

0 0

x

U s) - (1 U si

), s)U (1 - U ( C

U s) (1 U s)U - (1 si

, 0

U s) (1 U si ),

s)U (1 - U ( C t

d X d

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ In later works, Weinans and Huiskes et al. (1989,1990,1991, 1992); Mullender et al. 1994), characterized the mechanical behaviour of bone tissue using the apparent density as internal variable like other authors (Fyhrie and Carter, 1986;

Carter et al. 1989; Beaupré et al., 1990b).

ƒ The change of apparent density is written in terms of the strain energy density taking into account the influence of different load cases:

with ρ = ρ(x,y,z) the apparent density, B and k model constants, and ncthe number of load cases.

ρ

≤ ρ

⎟⎟ <

⎜⎜ ⎞

= ρ

ρ U -k ,con 0 ˆ

t B d

d a

=

= nc

1 i

i c

a U

n U 1

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Lecture 2. Phenomenological models of bone remodelling 35 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Based on the previous work of Carter and co-workers.

ƒ It considers that bone tissue need a certain level of mechanical stimulus and auto-regulated to maintain that level.

ƒ The objective is to homogenise the local mechanical stimulus in values near that homeostatic level (Homeostatic condition).

with the stimulus the one of Carter

ƒ The remodelling or stimulus error is defined as:

* t t

= ψ ψ

Diary stress stimulus at tissue level Homeostatic value

STANFORD ISOTROPIC MODEL

N 1/m

1

ni

=

= i

m t

t σi

ψ

* t 2

* t

t ˆ -

-

e ψ ψ

ρ ψ ρ

ψ ⎟⎟

⎜⎜

=

=

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

ƒ Characterization of load history:

Mechanical loads generates a stress state in bone that varies in a complex manner in short time periods.

Bone response is much slower than the load that induces such response.

Due to the different time scales involved it is convenient to characterize the load history with stress and strain tensors that do not change in the short-term scale of load application.

For instance, it is possible to formulate the theory in terms of the range of variation of the cyclic loads over a certain period of time.

{

( )| t- t t

}

-min

{

( )| t- t t

}

max

(t)= σ τ Δ <τ < σ τ Δ <τ <

σr

{

( )| t- t t

}

-min

{

( )| t- t t

}

max

(t)= ε τ Δ <τ < ε τ Δ <τ <

εr

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Lecture 2. Phenomenological models of bone remodelling 37 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

Instantaneous stress and strain are substituted by quasi-static quantities that vary at much longer time scales, but include the load history properties relevant for the remodelling process.

The time step used to average stress and strain has a negligible effect on the results of remodelling, if it is not very long (weeks) or short (min).

Load history is summarized in a small number of similar activities that conform load groups. Only the most important for remodelling are considered; the rest are discarded.

Fundamental hypothesis: the order of application of the different loads has no influence on the adaptive response of bone tissue, due to the very different time scales of both processes (loads and remodelling). Jacobs (1994) demonstrated that the results were very similar when loads were applied sequentially instead of simultaneously.

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

Example: Scheme of load history treatment

N 1/m

1

ni

=

= i

m t

t σi

ψ ψt =n1/mc σt

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Lecture 2. Phenomenological models of bone remodelling 39

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009 ƒ Hypothesis: the bone matrix is completely mineralised, and its

density corresponds to the one of the tissue with null porosity

ƒ Relation between the variables at continuum level (macro) and variables at the tissue level (micro). Let A be any scalar function at continuum level, we can write with R(n) such as R(0)=1 (bone completely mineralised).

ƒ From experimental estimates:

ƒ Therefore, the mechanical stimulus is written as

with

ρˆ

( )

ρˆ

-ρ 1 M

) ρˆ / -(M V 1

-V V 1

V V V n V

T T T

M T

M T T

huecos= = = =

=

At

A =R(n)

2

) ˆ ( ⎟⎟

⎜⎜

= ρ ρ ρ R

t

t σ

ψ =n1/mc σ

ρ ψ n ρˆ

2 1/m

c ⎟⎟

⎜⎜

=

t

U E = 2 σ

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

ƒ They define the relation between the mechanical stimulus (remodelling error) and the surface remodelling rate: production (resorption) of bone volume in bone surfaces per unit of active surface and time.

ƒ Different curves have been established for different bone regions.

ƒ Simplified functions were used in this model:

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Lecture 2. Phenomenological models of bone remodelling 41 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Evolution of the apparent density:

surface remodelling rate that quantifies the volume of bone produced or resorbed per available surface and unit time (mm3/(mm2.day) );

Sv available bone suface for remodelling per unit volume (mm2/mm3);

density of completely mineralised bone with null porosity (new bone is assumed to be in this model completely mineralised);

k percentage of available surface that is active for remodelling. This model considers the simplification that all the available surface is active (k=1).

ρ ρ & = k & r S

v

ˆ

ρˆ

r&

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Martin (1984) determined experimentally the available surface for each bone in term of the porosity.

5 4

3

2 134n -101n 28.8n n

93.9 - n

32.3 + +

v = S

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Lecture 2. Phenomenological models of bone remodelling 43 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Elastic constants are experimentally correlated to the apparent density (isotropic behaviour):

ƒ Numerical implementation:

=

g/cc.

1.2 si , 1763

g/cc.

1.2 si , 2014

2 . 3

5 . 2

ρ ρ

ρ E ρ

=

g/cc.

1.2 si , 0.32

g/cc.

1.2 si , 0.2

ρ ν ρ

⎟⎟

⎜⎜

+

= + 1 1 I

C (1 ) (1-2 ) E

ν ν ν

(1) Dado σn ,εn, tn ,ρn ,Δεny Δtn

(2) ρn+1=ρn+Δt ρ&(σn ,εn ,ρn) (3) E = B(ρn+1) ρβn+1

(4) ν=νˆ( ρn+1)

(5) ⎟⎟

⎜⎜

+

ν ν ν

= +

+ 1 1 I

C (1 ) (1-2 ) 1 E

n

(6) σn+1=Cn+1 : (εn+Δεn)

(7) Se devuelve ρn+1 ,σn+1y Cn+1

ECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO EERING-BIOMAT-Granada June-2009

ƒ The bone adaptation process in cancellous bone produces changes not only in the aparent density, but also in the orientation of the trabeculae. Isotropic models are insuficient to predict bone evolution.

ƒ Two models were proposed by Jacob et al. Both use as independent variables:

The apparent density

The elasticity tensor C

ƒ Model based on energy: It follows the principles of Continnum Damage Mechanics of Simo&Ju (1987)

ƒ Model based on stresses: Bone orients its microstructre according to Wolff’ law (similar to adaptive elasticity of Cowin)

ANISOTROPIC MODELS OF JACOBS et al.

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Lecture 2. Phenomenological models of bone remodelling 45 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS T IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Inert materials can be considered as closed thermodynamic

systems. On the contrary, living materials requiere metabolic energy to control different cellular processes.

Total energy dissipation (D) = Mechanical dissipation (DM) + Metabolical dissipation (DE)

Resorption =>

Formation =>

ƒ This model assumes that the internal energy is a function of strain, the elasticity tensor and the apparent density, these two latter are therefore considered as internal variables.

ƒ The evolution of the internal energy is computed therefore as:

0 DM

0 DM

) ( : 2 : ) 1 , ,

( ρ ρ

ψ ε C = ε C ε+Θ

ρ ρ

ψ& = &+ :&: +Θ′&

2 : 1 ) , ,

(ε C σ ε ε C ε

MODEL BASED ON ENERGY

COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009

ƒ Mechanical dissipation is defined as the difference bewteen the power introduced in the system by the external environment (loads) and the variation of internal and kinetic energy:

where:

and operating:

⎥⎦

⎢⎣ + ( ))

= ext

(U) t

M t

d(

K) dt

-d P

D φ φ

)) ( ( , ))

( ( ,

) ( )

(

Ω

>

<

+ Ω

>

<

=

∫ ∫

t

U t

U ext

t t

d d

P ρ φ δφ

δφ φ

v t v

v b 2 v, 1ρ

= K

)) ( ( : ))

( ( : ) ( ))

( ( ) ( : ))

( ( : ) (

)) ( ( ) , ( ))

( ( , ))

( ( ,

) ( )

( )

( )

(

) ( )

( )

(

Ω +

Ω

= Ω +

Ω

= Ω

>

<

= Ω

>

<

= Ω

>

<

t U

t U

t U

t U

t U

t U

t U

t t

t t

t t

t

d d

div d

div d

div

d div

d d

φ φ

φ φ

φ δφ

δφ

φ φ

φ φ

φ δφ

δφ

ε v

v v

v n

v v

t

&

σ σ

σ σ

σ σ

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