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A second gradient continuum model accounting for some eﬀects

of micro-structure on reconstructed bone remodelling

A. Madeo

a,d

D. George

T. Lekszycki

b,d

M. Nierenberger

Yves R´mond

c,d

Department

LGCIE, Universit´ de Lyon–INSA, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France

of Machinery Design and Biomedical Engineering, Warsaw University of Technology, 85 Narbutta street,

02-524 Warsaw, Poland

IMFS, Universit´ de Strasbourg - CNRS, 2 rue Boussingault 67000, Strasbourg, France

International Research Center M&MOCS, University of L’Aquila, Cisterna di Latina, Italy

Abstract

We propose a second gradient, two-solids, continuum mixture model with variable masses to describe

the eﬀect of micro-structure on mechanically-driven remodelling of bones grafted with bio-resorbable ma-

terials. A one-dimensional numerical simulation is addressed showing the potentialities of the proposed

generalized continuum model. In particular, we show that the used second gradient model allows for the

description of some microstructure-related size eﬀects which are known to be important in hierarchically

heterogeneous materials like reconstructed bones. Moreover, the inﬂuence of the introduced second gra-

dient parameters on the ﬁnal percentages of replacement of artiﬁcial bio-material with natural bone tissue

is presented and discussed.

KeyWords :

Second gradient continuum mixture model ; hierarchically heterogeneous materials ;

microstructure-related size eﬀects ; natural bone tissue ; bio-resorbable material ; bone resorption and

synthesis ; load-induced replacement of artiﬁcial material with natural bone tissue ; numerical simulations.

1. Introduction

It is nowadays well established in scientiﬁc literature that interactions between mechanics and biology

are crucial to correctly interpret and describe the behavior of growing tissues (see e.g. [5], [6], [29], [36]).

This is strictly related to the fact that Nature has developed “optimization methods which, given the

applied external loads, allow to obtain a proper resistance against mechanical failure with a minimum use

of material. Since the external applied loads unceasingly vary during life, living tissues must continuously

be resorbed and synthesized in order to be able to resist to the actual loads with the minimum possible

quantity of matter. The functional adaptation of bone to mechanical usage implies the existence of a

physiological control process. Essential components for the control process include sensors for detecting

mechanical usage and transducers to convert the usage measures to cellular responses. The cellular re-

sponses lead to gradual changes in bone shape and/or material properties and, once the structure has

adapted suﬃciently, the feedback signal is diminished and further changes to shape and properties are

stopped. Although the just presented description of biological phenomena occurring in growing tissues is

certainly incomplete, it includes a number of processes which are deﬁnitely relevant when the observed

mechanical adaptation takes place.

The amazing problem of bone adaptation and remodeling stimulated the minds of biologists and me-

chanicians ever since 1892 when Wolﬀ (see [49], and [50]) observed that “internal architecture and external

conformation of bones changes in accordance with mathematical laws. The twentieth century has then

seen an explosion of the research in the ﬁeld and diﬀerent mathematical models have been proposed to

describe functional adaptation and bone remodeling.

It is nowadays well esthablished that the cells which are involved in functional adaptation of bone are

divided in two main big classes: the sensor cells, (or osteocytes) which are able to detect the external

mechanical stimulus and to transduce it in a suitable biological signal which can be decoded by other cells,

and the actor cells (osteoblasts and osteoclasts) which detect the signal emitted by the osteocytes and

respectively synthesize and resorb bone tissue depending on the state of mechanical excitation. Various

mechanical stimuli have been proposed as triggers for bone adaptation, including strain, (see [28]) strain

energy density (see e.g. [29], [36], tissue damage (see [41]), daily stress stimulus (see [5]) and diﬀerent

forms of eﬀective stress (see [20]). In this paper we follow the idea of adopting the strain energy density

(SED) as the principal trigger of bone remodeling. More precisely, we assume that the osteocytes are well

placed within bone porosity to function as “strain gauges”, and to emit a signal (stimulus) the intensity of

which is proportional to the measured strain energy and to bone apparent density. Moreover, we assume

that a threshold value of the stimulus exists such that osteoblasts (bone synthesis) are activated if the

value of stimulus is higher than this threshold, while osteoclasts (bone resorption) are activated if the

value of stimulus is lower than the threshold itself. The hypothesis of existence of such a threshold value

is based on the idea that a high value of the deformation energy is associated to a need of a more compact

bone, while a low value of strain energy may be associated to a “surplus of material at a given location

which can then be resorbed and reused in other locations subjected to higher mechanical solicitations.

The idea of using SED as the principal trigger of bone remodeling is very well known in the scientiﬁc

literature and has been validated by several experimental observations (see e.g. [29]). Also the problem of

considering remodelling as a compromise between two conﬂicting objectives, namely weight and stiﬀness

has been investigated in [2] and [3]) where it is approached as an optimal control problem assuming SED

as mechanical stimulus.

The understanding of the profound interactions between mechanics and biology in functional bone

adaptation and remodeling naturally led to the production and current use in medical practice of artiﬁcial

bio-resorbable scaﬀolds which initially have the function of sustaining external mechanical loads, but that

are progressively resorbed by osteoclasts and partially or completely replaced by natural bone tissue (see

e.g. [25], [37], [44], [45]).

While continuum models for the description of natural bone regeneration are widely spread in the

scientiﬁc literature (see among many others [31], [7],[8], [23], [28]), the conception of rigorous models

allowing for the description of bio-materials resorption and of their gradual replacement by natural bone

tissue is still an open challenge. One of the main objectives of this paper is, following what done in [36],

to introduce an innovative continuum mixture model which allows for describing phenomena of natural

bone remodeling on one hand and, on the other hand, phenomena of resorption of bio-resorbable artiﬁcial

materials followed by a gradual substitution with natural bone tissue. A model of this type would be of

interest for the optimal design of bio-resorbable prostheses currently used in bone reconstructive surgery.

The continuum mixture theory developed here will also take into account the possibility of describing

the eﬀect of micro-structure on the overall mechanical behavior of both bone and bio-materials by means

of the use of a second gradient approach which has been widely recognized to be useful for these purposes.

It is indeed very well established in scientiﬁc literature (see e.g. [4], [27], [32], [40], [52]) that classical

Cauchy-type continuum theories do not allow for the correct prediction of the mechanical behavior of

bone, when considering suﬃciently small scales and/or particular loading conditions. These scale-eﬀects

are related to the fact that bone is a hierarchically heterogeneous material, i.e. it can be considered

as homogeneous at the scale of the millimeter, but it starts presenting heterogeneities at the scale of

the micron (see e.g. [48], [30]). Indeed, at this scale quasi-periodic circular structures (osteons) can be

detected which confer highly heterogeneous properties to the material itself. It is for this reason that

classical continuum theory does not allow to describe the correct behavior of bone at such small scales

where the prediction made by Cauchy continuum theory starts being far away from the experimental

evidence. In [4] a clear presentation of some size eﬀects observed in bone is given and explained by means

of experimental evidences. The authors explain how these size eﬀects can be accounted for by using a

Cosserat-type continuum theory which allows to perform much more precise predictions. The fact of

modelling the mechanical behavior of bone by means of Cosserat-type theories is well established in the

scientiﬁc literature as shown by the numerous theoretical and experimental publications on this subject

(see e.g. [4], [51], [53], [54]).

The idea which is developed in this paper is to account for the discussed size eﬀects by using a second

gradient continuum theory (see also [22]) instead of a Cosserat-type theory which needs the introduction

of a more complicated kinematics accounting for micro-rotations. A second gradient theory allows for the

description of some microscopic features of the considered material, even if remaining in the framework

of macroscopic continuum theories (see among many others [39]) and is able to correctly describe a

considerable number of the cited size eﬀects (see e.g. [1], [21]). Indeed, second gradient theories permit

to obtain a generalization of classical Cauchy continuum theory in the sense that they introduce the

possibility of considering the existence of particular internal contact actions exchanged between sub-

bodies of the continuum which are more general than those considered in classical ﬁrst gradient theories

(see [24], [12], [13], [14], [15], [18], [46]). In particular, while in classical continuum theories only forces

per unit area proportional to the normal to the Cauchy cut are described, in second gradient theories

one has the possibility to introduce more complicated contact actions which depend e.g. on the curvature

of the Cauchy cut (so-called “double forces” following the nomenclature adopted by Germain in [24]).

As an example, we can say that second gradient theories allow for the description of couples per unit

area exchanged between sub-bodies of the continuum, while classical theories do not. Such a possibility

of describing more complicated contact actions allows for the description of some mechanical responses

of the continuum which are directly related to its micro-structure (see e.g. [39]).

The proposed second gradient continuum mixture model is ﬁnally applied to a simple one-dimensional

case in which natural bone and bio-resorbable material are originally separated by a material discontinuity

surface. A traction external load is applied to the considered specimen and the eﬀect of the introduced

second gradient parameters on the ﬁnal percentage of replacement of artiﬁcial bio-material with natural

bone tissue is investigated. The presented preliminary results show how the introduced second gradient

model allows for describing some scale eﬀects wich are directly related with bone and bio-material micro-

structure. Since it aims to describe extremely complicated bio-mechanical phenomena, the presented

model introduces a huge number of biological and mechanical parameters which therefore need to be

carefully analysed. A rigorous parametric study will be performed in a subsequent paper in order to

study the inﬂuence of each of these parameters on the overall process of reconstructed bone remodelling.

Particular attention will be paid to the eﬀect of the second gradient parameters on the ﬁnal percentage

of replacement of artiﬁcial bio-material with natural bone tissue.

2. Equilibrium Equations for a Second Gradient Two-Solids Mixture accounting for Mass

Creation and Dissolution as driven by Bio-Mechanical Coupling

In this section we start by introducing the mechanical equations suitable for describing the deformation

of a second gradient continuum. We will then propose a set of ordinary diﬀerential equations accounting for

phenomena of creation and dissolution of mass as driven by the coupling between the external mechanical

excitation and the resulting biological stimulus. The proposed set of mechanical and biological equations

will be proven to be suitable for describing remodelling phenomena in reconstructed bone as well as some

scale eﬀects related to the heterogeneous micro-structure of both bone and bio-material.

2.1.

Second Gradient Equilibrium Equations

Following standard procedures of mixture theory (see e.g. [42]), we describe the deformation of a two

solids continuum mixture by introducing a Lagrangian (or reference) conﬁguration

⊂

and a suitably

regular kinematical ﬁeld

χ(X,

which associates to any material point

∈

its current position

time

The kinematics of the system is completed by introducing two Lagrangian densities

∗

(X,

and

∗

(X,

which represent the density of the natural bone tissue and of the artiﬁcial material respectively.

We explicitly remark that, in our mixture model, we associate to each material particle

two diﬀerent

densities which can evolve with time. The image of the function

gives, at any instant

the current

shape of the body

(t): this time-varying domain is usually referred to as the Eulerian conﬁguration

of the mixture and, indeed, it represents the system during its deformation. Since we will use it in the

following, we also introduce the displacement ﬁeld

u(X,

χ(X,

−

the tensor

∇χ

and the

Green-Lagrange deformation tensor

:= (F

−

I)/2.

Let

∗

(ε,

∇ε,

∗

, ρ

∗

) be the strain energy of a second gradient continuum mixture which is assumed to

depend on the Green-Lagrange deformation tensor

ε,

on its gradient and also on the Lagrangian apparent

mass densities of both bone and bio-material. It can be shown by means of a variational principle (see e.g.

the methods presented in [?], [43]) that, neglecting external body forces, the bulk equilibrium equation

for such a system reads

[ (

∗

(

))]

∂U

∗

div

−

div

(1)

∂ε

∂∇ε

together with the following duality conditions valid on any discontinuity surface Σ

⊂

of such a

continuum

[|t

δu|]

= 0,

[|τ

(δu)

= 0,

[|f

δu

= 0.

(2)

The ﬁrst two of these conditions are valid on Σ while the last one is valid on the edges of Σ, if any. In

the previous formulas we set

[ (

∗

(

))]

(

)

∂U

∗

∂U

∗

−

div

−

div

(3)

∂ε

∂∇ε

(

)

(

)

∂U

∗

∂U

∗

�½,

(4)

∂∇ε

where,

is the unit normal vector to the surface Σ,

div

stands for the surface divergence operator on

Σ, if the edge is regarded as the border of a surface then

�½

is the normal vector to the considered edge

which is tangent to the surface,

δu

is the variation of the displacement ﬁeld and (δu)

∇(δu) ·

stands for the normal derivative of the variation of the displacement ﬁeld. Finally, given a quantity

deﬁned everywhere and having continuous traces

and

−

on the two sides of Σ respectively, we have

set [|a|] :=

−

(with a slight abuse of notation, we use the same symbol for the jump across edges).

We explicitly remark that the vector

represents the so called “generalized force” which, contrarily to

what happens in classical Cauchy theory, explicitly depends on the “shape” of Σ. Moreover, following the

notation introduced by Paul Germain, the vector

is the so-called “double-force”, i.e. a special type of

non-local contact action which expends power on the normal derivative of velocity. Finally,

represents

a contact action per unit line which can be exchanged by two subbodies of the considered body across

the edges (if any) of the Cauchy cut.

2.2.

Bone Remodelling Equations

We now want to formulate a well-posed evolutionary problem for the introduced kinematical ﬁelds

which is able to catch the most important features of the remodelling processes occurring in bone tissue

after initial healing and in presence of bio-resorbable grafts initially functioning as a bone tissue ﬁller

and support. Before introducing the diﬀerential equations which we believe to be suitable to accomplish

this task, we recall here some basic biological facts which led us to use them in our modelling. We can

start distinguishing two types of cells (which belong to the common class of so-called “actor cells”),

namely the osteoblasts (specialized in new bone tissue formation) and the osteoclasts (which are able

to resorb both natural bone and bio-resorbable material). We assume that these two kinds of cells are

present everywhere in both the living bone tissue and artiﬁcial material under the unique condition that

a suitable porosity is present. In other words, we do not consider here those cases in which the bio-

resorbable scaﬀold is so extended that actor cells cannot survive in the ﬁrst phases after healing due to

an insuﬃcient vascularization. The biological equations presented here must therefore be limited to study

those cases in which the bio-resorbable grafts are small enough to assure the survival of actor cells also

in the inner regions of the scaﬀold. The activity of osteoblasts and/or osteoclasts is regulated by the

instructions of a signal generated by a third kind of cells called osteocytes: this signal is proportional

to the deformation which the osteocytes can measure at a given point (for this reason they are also

called “sensor cells”). In order to measure deformation energy at a given point, the sensor cells do not

move and spend all their life in that point. Sensor cells originate from osteoblasts when these latter

have accomplished their task of synthesizing new bone around them: when an osteoblast is completely

surrounded by new natural bone tissue it changes its nature and becomes an osteocyte, i.e. a sensor cell

which starts to measure deformation and to emit a signal proportional to its measured value. This brief

and simpliﬁed description of the biology which is known to take place behind the process of reconstructed

bone remodelling is suﬃcient to justify the remodelling equations which we choose to use in this paper.

For a more detailed description of the biological phenomena occurring in natural bone and bio-material

remodelling we refer to [36]. Given the just described coupled biological and mechanical phenomena, we

are now able to postulate a proper set of diﬀerential equations which are able to catch the main features

of the described bio-mechanical phenomena. We choose the evolutionary equations for apparent densities

to be simply ﬁrst order ordinary diﬀerential equations with respect to time. In formulas, we assume that

∂ρ

∗

∂ρ

∗

(φ

, S

∗

(φ

, S

∗

(5)

∂t

where

∗

is the porosity of the considered continuum mixture and

∗

is the biological stimulus. We will

duly explain in the following how the porosity

∗

and the stimulus

∗

are assumed to constitutively

depend on the introduced basic kinematical ﬁelds.

These equations are similar to those studied, among others, in [33], [34], [35] and [19]. However, the

analysis which lead to the “bifurcation chart” in ﬁgure 4 of [33] cannot be easily repeated here since the

study of bifurcation in the presented instance should account for the possible variations of resorption and

synthesis parameters, which indeed dramatically inﬂuence the ﬁnal form and structure of reconstructed

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