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Phil. Trans. R. Soc. A (2009) 367, 2055–2078
doi:10.1098/rsta.2009.0055
A mathematical approach to bone
tissue engineering
B Y J. A. S ANZ -H ERRERA 1,2,3 , J. M. G ARCÍA -A ZNAR 1,2, *
1,2
AND M. D OBLARÉ
1
Group of Structural Mechanics and Materials Modelling (GEMM ), Aragón
Institute of Engineering Research (I3A), and 2Centro de Investigación
Biomédica en Red en Bioingenierı́a, Biomateriales y Nanomedicina
(CIBER-BBN ), University of Zaragoza, 50018 Zaragoza, Spain
3
School of Engineering, University of Seville, 41092 Seville, Spain
Tissue engineering is becoming consolidated in the biomedical field as one of the
most promising strategies in tissue repair and regenerative medicine. Within this
discipline, bone tissue engineering involves the use of cell-loaded porous biomaterials, i.e.
bioscaffolds, to promote bone tissue regeneration in bone defects or diseases such as
osteoporosis, although it has not yet been incorporated into daily clinical practice. The
overall success of a particular bone tissue engineering application depends strongly on
scaffold design parameters, which do away with long and expensive clinical protocols.
Computer simulation is a useful tool that may reduce animal experiments and help to
identify optimal patient-specific designs after concise model validation. In this paper, we
present a novel mathematical approach to bone regeneration within scaffolds, based on a
multiscale framework. Results are presented over an actual scaffold microstructure,
showing the potential of computer simulation, and how it can aid in the task of making
bone tissue engineering a reality in clinical practice.
Keywords: bone regeneration; multiscale approach; homogenization theory;
permeability computation
1. Introduction
The aim of bone tissue engineering, as a subdiscipline of tissue engineering, is to
regenerate bone mass loss due to tumour or any other physiological disorder,
especially in large bone defects, to accelerate bone fracture healing or to promote
bone growth under certain circumstances or diseases such as osteoporosis. This
subdiscipline has received an increasing interest since the early works presented
in the 1970s (Levin et al. 1975; Rejda et al. 1977). In the last decade, a wide
number of experimental works using scaffolds for bone regeneration have been
presented. The strategy commonly involves the use of cells (usually bone marrow
stromal cells (BMSC), mesenchymal stem cells (MSCs) or preosteoblasts),
combined with bone morphogenetic proteins or growth factors such as
* Author for correspondence ( [email protected]).
One contribution of 15 to a Theme Issue ‘The virtual physiological human: tools and applications I’.
2055
This journal is q 2009 The Royal Society
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J. A. Sanz-Herrera et al.
transforming growth factor-b (TGF-b), seeded within porous scaffolds (usually
made of polymeric, bioceramic or bioglass materials) to promote bone tissue after
implantation. The scaffold’s behaviour in vivo has been evaluated in many
papers, although numerical simulation may help in the understanding of such
behaviour with the use of the computer.
Very few works have been conducted within the framework of numerical
simulation in tissue engineering. It should be stated that the main objective of
mathematical modelling and numerical simulation in this field is to predict the
behaviour of scaffolds in order to get an optimal design according to its
functionality. In tissue engineering methodology (see Evans et al. 2007), scaffolds
may be preseeded with cells in vitro prior to implantation in the human body. In
this step, an appropriate pore size, pore interconnection, porosity and
permeability must be assured in order to permit cells to move through the
scaffold core and adhere onto its surface. In bone tissue engineering applications,
it is important that the scaffold is strong enough to support the mechanical loads
transmitted by muscles and joints, which will be brought about by its effective
(macroscopic) mechanical properties. Both permeability and effective mechanical
properties may be tailored individually by gaining control over some scaffold
parameters microscopically, such as mean pore size, porosity and bulk
biomaterial mechanical properties. In this context, Hollister and co-workers
(Hollister et al. 2002; Taboas et al. 2003; Lin et al. 2004) focused on the design of
bone scaffolds as an optimization problem to obtain a microstructure as similar
as possible (mechanical properties, porosity, pore size, etc.) to that of the
implanted region. For this purpose, homogenization theory was extensively
applied in their studies. Additionally, mechanical properties and permeability
were homogenized over a representative element of a bioceramic scaffold
microstructure (Sanz-Herrera et al. 2008c). The obtained results were
corroborated by an experimental set-up showing the potential of numerical
tools for the characterization of scaffold properties.
Ideally, the scaffold should degrade, giving way over time to new bone
regeneration and therefore complete scaffold disappearance. Scaffold degradation
usually takes place by chemical pathways (hydrolysis) in the case of polymeric
scaffolds. This issue has been the subject of numerical analysis by Adachi et al. (2006)
and recently by Wang et al. (2008). A proper understanding of hydrolysis kinetics can
modulate the rate of scaffold mass loss, allowing us to choose a specific polymer
biomaterial for a specific application. Other issues related to scaffold microarchitecture, such as nutrient supply and waste removal, have been modelled
computationally and may be reviewed in the work conducted by Sengers et al. (2007).
Cell–biomaterial interaction is considered to play a major role in tissue
engineering. In recent years, it has been observed that adherent cells sense and
respond to the mechanical environment they are anchored to (Discher et al. 2005;
Engler et al. 2007). The mechanical environment is mainly related to substrate
stiffness, although topological factors such as surface roughness or curvature are
of extreme importance as well (Pelham & Wang 1997; Wong et al. 2003; Engler
et al. 2004). Moreover, mechanical forces in the intra- and extracellular space
can induce a wide range of cell processes such as growth, differentiation,
apoptosis, migration, gene expression and signal transduction (Lauffenburger &
Horwitz 1996; Kitano 2002; Bischofs & Schwarz 2003; Engler et al. 2007;
Ghosh et al. 2007; Suresh 2007). In fact, cells transduce mechanical stimuli into
Phil. Trans. R. Soc. A (2009)
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Approach to bone tissue engineering
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biochemical signals and vice versa, i.e. mechanotransduction. Cell migration,
proliferation and differentiation have a prominent significance in bone tissue
engineering processes (Jerome & Maquet 1997; Ishaug-Riley et al. 1998) both
in vivo and in vitro and are mainly affected by scaffold stiffness, microarchitecture and biomaterial composition. All these arguments have drawn the
attention of mechanicians focusing their interests on another emerging research
line, namely, cell mechanics. There is today a vast amount of literature specially
relating to mathematical modelling and computer simulation (DiMilla et al.
1991; Kitano 2002; Bischofs & Schwarz 2003; Nicolas et al. 2004; Zaman et al.
2005; Maini et al. 2006; Schwarz et al. 2006; Suresh 2007; Moreo et al. 2008)
showing significant results and direct application to tissue engineering.
From all the above, it becomes clear that bone tissue regeneration in vivo
using scaffolds inherently attends to two well-differentiated spatial and temporal
scales. One scale is at the tissue level or macroscopic scale, whereas the other is
at the pore level or microscopic scale. Traditionally, mathematical modelling of
tissue growth within scaffolds has been restricted to one of these scales. For
instance, at the macroscopic scale, osteochondral defect repair using scaffolds has
been simulated by Kelly & Prendergast (2006) using a previous mathematical
model of cell/tissue differentiation (Prendergast et al. 1997). A macroscopic
approach for the simulation of bone tissue regeneration within scaffolds
considering the effect of its microstructure was introduced in our previous
work by Sanz-Herrera et al. (2008b). At the microscopic scale, Byrne et al. (2007)
presented a mechanobiological model applied to a periodic unit cell of the scaffold
microstructure. In that model, the influence of several factors such as porosity,
mechanical properties and others were analysed with regard to tissue
regeneration. At the same microscale, bone tissue regeneration within a scaffold
unit cell was numerically reproduced by Adachi et al. (2006) using concepts and
hypotheses previously established for a bone remodelling theory (Adachi et al.
1997, 2001). Recently, a parametric analysis of several factors of major
importance in tissue engineering has been presented by Sanz-Herrera et al.
(2009). The model was based on a multiscale technique applied to a bone tissue
engineering example. The underlying microstructure of the scaffold used in the
model was simplified by considering a periodic face-centred cubic repetition of
the pores in the unit cell (Brı́gido-Diego et al. 2007).
This paper focuses on the mathematical modelling and computer simulation of
bone tissue engineering events in the presence of scaffolds (see table 1 for
nomenclature). This work differs from the previous ones in that it employs a
mathematical methodology that considers two scales of analysis. This is critical
in bone tissue engineering since scaffold implantation occurs macroscopically,
although the most important biophysical phenomena and control design
parameters stand microscopically. In our model, the macroscopic level includes
the bone organ and scaffold region, whereas the microscopic scale is associated to
the pore scaffold level. Even when the presented model considers multiphysics
phenomena, the main underlying hypothesis is that bone regeneration is
regulated by mechanical factors.
The proposed multiscale model is described mathematically in §2. An example
of application is analysed in §3, showing results at both the macroscopic and
microscopic levels. The example includes a realistic microstructure of a polymeric
porous biomaterial. Finally, the paper closes with some concluding remarks
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J. A. Sanz-Herrera et al.
Table 1. Nomenclature.
s0
e
Macroscopic Cauchy’s stress tensor
0
Macroscopic strain tensor
0
Macroscopic displacement field
u
C0
Macroscopic elasticity tensor
U0S
0
UKS
Macroscopic scaffold domain
Macroscopic bone domain
D
Macroscopic diffusion coefficient matrix
c
Normalized cell population (macroscopic level)
M
J
4
Bone mass formation at the microscopic level
Microscopic mechanical stimulus at the scaffold region
U4S
Microscopic solid domain
ks
Empirical rate function of mass regeneration at the microscopic level
E
Young’s modulus
n
Poission’s ratio
4
U
Strain energy density at the microscopic level
C4i
Microscopic elasticity tensor, for iZn non-mature bone and for iZm
mature bone
d
Normalized water concentration at the microscopic solid domain
W
Scaffold molecular weight
s
4
e4
u
4
Microscopic Cauchy’s stress tensor
Microscopic strain tensor
Microscopic displacement field
u
ckh
Microscopic perturbation of the displacement field
v4
p4
Microvelocity field
v
U4F
P
0
k
0
r
U
r_
Microscopic pressure
Perturbation of the microvelocity field
Microscopic fluid domain
Macroscopic pressure gradient
Characteristic fluid velocities at the microscopic level (homogenization)
K
J
Elementary strain states at the microscopic level (homogenization)
Macroscopic Darcy’s permeability
Macroscopic mechanical stimulus at the bone organ region
Macroscopic apparent bone density
0
Strain energy density at the macroscopic level
Surface remodelling rate of the bone organ at the macroscopic level
regarding the perspectives and future directions of the bone tissue engineering
field from a clinical perspective, and how this work and computer simulation can
provide qualitative information in parallel to animal experimentation.
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Approach to bone tissue engineering
2059
2. Mathematical methodology
The mathematical model based on a multiscale methodology is presented in this
section. We first introduce the governing equations and their weak form,
distinguishing between bone organ and scaffold domain. The scaffold macrodomain is coupled with its underlying microstructure. Moreover, the internal
microstructure of the scaffold evolves as cells form new bone tissue. The
modelling of this phenomenon as well as the coupling between the micro- and
macro-scales of the problem at the scaffold region is explained in §2b. Finally, the
macroscopic model of the bone organ is introduced in §2c.
(a ) Multiscale analysis
Let us consider the macroscopic region to be composed of two subdomains,
0
namely, the scaffold region U0S and the non-scaffold region UKS
, i.e. the bone
organ. The domain labelling may be seen in figure 1, which in turn will serve as
application (§3) to the developed model. We assume that bone remodelling takes
place at the bone organ region, while microscopic bone apposition occurs at the
microsurface of the scaffold. As hypothesis, we consider that these phenomena
are governed mainly by mechanics through a mechanical stimulus, similar to
other bone remodelling theories (Pauwels 1965; Cowin & Hegedus 1976; Carter
et al. 1987; Huiskes et al. 1987; Beaupré et al. 1990; Jacobs 1994).
The macroscopic elasticity equations in the absence of body forces are posed
both at the scaffold and bone organ as
9
V$s0 Z 0;
>
>
>
>
>
1
0
0
0 T
0
0
e Z ðVu C ðVu Þ Þ; in UKS ðxÞg U S ðx; yÞ; =
ð2:1Þ
2
>
0
0 0
>
>
s ZC e ;
>
>
;
C boundary conditions;
with e(x) and s(x) being second-order tensor fields denoting the (linearized)
strain and stress, respectively, and C0 the macroscopic fourth-order elasticity
tensor defined as
( 0
)
0
ðxÞ
in
U
ðxÞ;
C
K
S
C0 Z
ð2:2Þ
C0 ðx; yÞ in U0S ðx; yÞ;
where x denotes the macroscopic scale and y the microscopic one. Note that the
elasticity tensor is a function of each macroscopic point, being therefore a
heterogeneous material. In fact, it will be a function of a field variable (apparent
density) defined in the bone organ as explained below. However, the effective
mechanical properties of the scaffold are directly derived from the evolution of
the underlying microstructure. This evolution is determined by the new bone
apposition, scaffold degradation and invasion of cells to the scaffold core. The
model of microscopic bone growth within the scaffold and biomaterial
degradation is explained below.
As pointed out, we macroscopically model the process of cell migration and to
some extent vasculogenesis in the scaffold region. This is based on the
assumption that bone regeneration follows the same sequence of events as
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J. A. Sanz-Herrera et al.
(a)
(b)
scaffold,
0
S
S
(c)
F
bone organ,
0
–S
fully clamped
Figure 1. Model of the example of application taken from Savarino et al. (2007). (a) CAD
macroscopic model of the femur and scaffold implantation, showing the boundary loads and
prescribed displacements. (b,c) The underlying scaffold microstructure in terms of RVE,
distinguishing between (b) solid and (c) fluid domains. The typical cell size was set to 1 mm.
conventional bone regeneration under normal biocompatibility conditions. First,
a haematoma is formed and inflammation occurs (Einhorn et al. 1995). At this
level, the walls of the scaffold are filled with MSCs and preosteoblastic cells. This
migration process from the surrounding bone to the scaffold core is here modelled
by a standard Fick’s law,
)
c_ Z V$ðD$V
cÞ in U0S ðx; yÞ;
ð2:3Þ
C boundary and initial conditions;
where the diffusion matrix denoted by D(x,y) is considered fully anisotropic and
cðxÞ dcðxÞ=cs 2½0; 1, c(x) being the number of cells per unit volume in the
scaffold domain and cs the number of cells per unit volume surrounding the
scaffold boundary, which is assumed to be the maximum density of MSCs
available. Note that, as with mechanical properties, the diffusion coefficient is
obtained from an analysis of the scaffold microstructure, which couples together
the macro- and micro-scales.
With (2.1) and (2.3) in hand, we can apply the variational form of these
equations as
ð
U0S
ð
0
0
UK
S
0
c_h0 dU0S ðxÞ C
0
s : e ðw Þ
Z
ð
0
vUKS
ð
0
U0S
0
dUK
S ðxÞ C
0
t^$w 0 dUK
S ðxÞ
Phil. Trans. R. Soc. A (2009)
Vx h
DVxc dU0S ðxÞ
"
ð
0
U0S
0
e ðw Þ :
1
j U4 j
c w 0 ðxÞ; h0 2Vx ;
ð
Z
ð
vU0S
q^$nh0 dU0S ðxÞ;
4 4
U4S
ð2:4Þ
#
4
C e ðu Þ
dU4S ðyÞ
dU0S ðxÞ
ð2:5Þ
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Approach to bone tissue engineering
the space of admissible variations being defined as
Vx d w 0 ðxÞ; h0 ðxÞ 2R3 ; R j wi0 ; h0 2H 1 ðU0S Þ and w 0 Z 0; h0 Z 0 on vU0S ; ð2:6Þ
with H 1(U0) being the first-order Sobolev space. The derivation of D is
explained below.
For simplification, we have considered that the bone organ and the scaffold are
fully connected. Moreover, the free boundary of the scaffold (which is not
connected with the rest of the bone organ) does not support boundary loads.
Using a homogenization procedure in (2.5), we obtain the two-scale variational
form of the mechanical problem, namely
ð
ð
ð
0
0
0
0
0
0
0 0
0
0
0
s : e ðw Þ dUKS ðxÞ C e ðw Þ : C e ðu Þ dUS ðxÞ Z
ðxÞ;
t^$w 0 dvUKS
0
UK
S
U0S
0
vUK
S
ð2:7Þ
with C0 obtained through the homogenization process as explained below.
Note that (2.7) is a direct consequence of the linearity of the problem. In
nonlinear situations, the separation between the macro- and micro-scales cannot
be accomplished unless an appropriate linearization is conducted (Terada &
Kikuchi 2001). This statement is valid in classical multiscale linear mechanics,
where no evolution of the microstructure occurs. Nevertheless, since the
considered microstructure is expected to grow and to degrade, an additional
hypothesis is introduced. We consider that both bone growth and scaffold
degradation are averaged during each time iteration. This implicitly considers
that the macroscopic apparent properties are unchanged during the time step.
A consequence of this simplification is that the time scales associated with the
application of loads and diffusive analyses are uncoupled from those of bone
ingrowth and degradation.
Equations (2.4) and (2.7) are numerically implemented in a finite-element
framework using the ABAQUS software (HKS 2001). Parallel performance is used
to accelerate the micro–macro approach. The details of the numerical
implementation are described by Sanz-Herrera et al. (2008a).
(b ) Model for the scaffold domain
The scaffold domain corresponds to the underlying microstructure of a
macroscopic material point x 2U0S . It is treated in the form of a representative
volume element (RVE) that statistically represents the microstructure of such a
macroscopic point. We distinguish between fluid and solid domains in the RVE,
such that U4S ðyÞg U4F ðyÞ h RVE, with U4S and U4F being the microscopic solid
and fluid domains, respectively (see figure 1). The microscopic scale is denoted by y.
Bone ingrowth occurs at the scaffold surface, i.e. vU4S , after MSC attachment
and subsequent proliferation, differentiation and bone matrix production. We
assume that bone formation is driven by a mechanical stimulus evaluated at the
microstructural level and given through macroscopic quantities. Furthermore,
we consider that the activity of bone cells and subsequent bone matrix
production and/or resorption happen at the scaffold pore surface. Then, we
establish that the rate of bone mass formation at the local scaffold microsurface is
ð2:8Þ
M_ Z M
_ _ ð
c; J4 Þ in vU4S ðyÞ;
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where the bone mass rate M_ is a function of a microscopic mechanical stimulus
J4 and the current macroscopic population of MSCs, c. The derivation of the
microscopic stimulus from macroscopic information is addressed below.
Additionally, we consider that the activity of the cells attached to the scaffold
surface starts when a mechanical threshold is achieved. Above this threshold,
MSCs start to differentiate into osteoblasts and deposit bone matrix on the
surface, while under that value they preferentially stay inactive (Turner et al.
1994; Tang et al. 2006). Therefore, we can express bone growth on the scaffold
surface within the scaffold domain in the following way:
(
)
ks ð
cÞ$ðJ4 KJ KwÞ if J4 KJ O w;
ð2:9Þ
M_ Z
0
otherwise;
cÞZ k$
c is an empirical rate function (k being a model parameter)
where ks ð
proportional to the macroscopic concentration of MSCs, c, currently attached to
the surface of the scaffold. The assumed microscopic mechanical stimulus J4 is
analogous to that in the previously established theories of bone growth and
remodelling (Beaupré et al. 1990), but conceptually different since we consider it
here to be a microscopic stimulus evaluated at the scaffold pore surface. This
is stated as
!1=m
N
X
4
m
J Z
ni si
;
ð2:10Þ
i Z1
where N is the number of different load cases; ni is the average number of cycles
per time unit for the load case i; m is a model parameter (Beaupré et al. 1990);
and s is an effective scalar stress expressed as
pffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2:11Þ
s Z 2EU 4 ;
with local Young’s modulus E distinguishing among scaffold (s), non-mature
bone (n) or mature bone (m). Finally, U 4Z1/2s4 : e4 is the strain energy
density at the scaffold microsurface. This is obtained through the localization
problem and subsequent distribution of strains along the scaffold microstructure
described below.
As a first simplified approach, the time evolution of the mechanical properties
of non-mature bone is considered to be linear (Adachi et al. 2006),
t
C4n Z C4m
; 0% t % Tm ;
ð2:12Þ
Tm
where C4i ; iZ n; m, are the fourth-order elasticity tensors of non-mature (n) and
mature bone (m), respectively. Note that, at the microscopic level, the mechanical
properties depend only on the mineralization ratio since the distinction between
cancellous and cortical bone does not make sense at this level. The newly formed
bone matrix is considered at this scale as isotropic, as a first approach.
The degradation mechanism is modelled as a hydrolysis process within the solid
microstructure of the scaffold. This model was presented by Adachi et al. (2006).
Briefly, the water content in the polymer chemically reacts and produces an
internal change of the polymer, causing it to detach, consequently leading to
erosion of the biomaterial bulk (Gopferich 1997). We consider that the
dimensionless spatial rate of water content d, defined as the ratio between
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Approach to bone tissue engineering
2063
the concentration of water (mol) in the bulk of the polymer and the concentration
of water (mol) at the boundary of the scaffold, is governed by the diffusion equation
(Adachi et al. 2006)
d_ Z aDd
in U4S ðyÞ;
C boundary and initial conditions;
ð2:13Þ
with aO0 being the diffusion coefficient. On the other hand, the rate of change of
the molecular weight of the polymer due to hydrolysis is assumed to depend on the
local water content (Adachi et al. 2006) (0%d%1) as
_ ðdÞ ZKbd;
W
ð2:14Þ
with bO0 being a material constant. The mechanical properties of the polymer are
assumed to be linearly related to its molecular weight (Adachi et al. 2006) as
C4 ðW ðtÞÞ Z C4 ðW0 Þ
W ðtÞ
W0
in U4S ðyÞ;
ð2:15Þ
with W0 being the reference molecular weight of the polymer and C 4 ðW0 Þ being the
associated elasticity tensor.
The bone growth process is numerically modelled by using the voxel finiteelement method (FEM; Adachi et al. 2006). Thus, the amount of bone to be
formed within the scaffold is simulated by adding voxels, representing the newly
created bone (Sanz-Herrera et al. 2008a). As a simplification, we considered that
bone formation occurs only on the surface of the scaffold. The mechanical
stimulus in equation (2.10) is computed at the centroid of each voxel to be the
candidate for bone formation, i.e. the boundary voxels. Then, equation (2.9) is
integrated by the forward Euler method and the number of voxels corresponding
to the new bone volume production Mc/rb are added, where rb is the mineral
bone density and Mc a model parameter, during each period or time step. For the
implementation, the voxel scaffold microstructure is treated as a binary thirdorder matrix with indices i, j and k (with i, j, k from 1 to the considered length of
voxels). The matrix contains ‘1’ at the spatial position ijk when a bone or scaffold
voxel is identified at that position, and ‘0’ when it corresponds to a void voxel.
Then, we identify new voxels to be formed when a shift between 0 and 1 is
computed along the scaffold microstructure. Then, the position that contains a
0 automatically changes to 1, meaning that a new bone voxel has been created.
Analogously, scaffold degradation is simulated within a voxel FEM framework.
Here, we remove the voxels where W!Wc, Wc being a model parameter, during
each time step. A scheme of this procedure can be seen in figure 2.
The remainder of this subsection is devoted to the analysis of the RVE in order
to obtain the homogenized properties C0 and D available for the implementation
of equations (2.4) and (2.7). Moreover, this analysis allows us to obtain the
microscopic distributions of the field variables involved. The linear, elastic
microscopic problem in the absence of body loads in the solid domain in U4S
reads as
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2064
J. A. Sanz-Herrera et al.
1
2
3
Figure 2. Voxel formation during bone growth. Possible candidates for new voxel formation are
found at the boundary voxel faces. The variables involved during the voxel (solid line) formation
are computed at the centroid of the voxel candidate (dashed line), whereas those involved in voxel
deletion (degradation) are computed at the centroid of boundary voxels. Voxel centroids are
represented by black dots. Courtesy of Sanz-Herrera et al. (2008a).
9
V$s4 Z 0;
>
>
>
>
1
4
4
4 T >
e Z ðVu C ðVu Þ Þ; =
2
>
4
>
s Z C4 e4 ;
>
>
>
4
0
4
0 ;
hs i Z s or he i Z e ;
ð2:16Þ
with h$i being the volume average over the RVE. The absence of boundary
conditions in (2.16) is noticeable. These boundary conditions must reproduce, as
closely as possible, the in situ state of the RVE inside the material. Therefore,
they strongly depend on the choice of the RVE itself, and especially on its size.
Two classical boundary conditions have been established traditionally,
s4 $n Z s0 $n
u4 Z e0 $y
on vU4S ;
on
vU4S ;
)
ð2:17Þ
with n being the outward normal to the boundary. Equations (2.17) refer to
uniform stresses and strains, respectively, on the boundaries of the solid domain
of the RVE. It is immediately seen that a divergence-free field s4 that satisfies
(2.17)1 also satisfies hs4iZs0, and a displacement field that satisfies (2.17)2 also
satisfies he4iZe0 (see Suquet (1987) for further reading).
If periodic media are under consideration, the boundary conditions (2.17) must
be rejected (Suquet 1987). Therefore, the periodicity conditions imply the
following statements:
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Approach to bone tissue engineering
2065
(i) the stress vectors s4$n are opposite on opposite sides of the contour
vU4S , and
(ii) the local strain e4(u4) is split into its average and fluctuating terms such that
)
e4 ðu4 Þ Z e0 C e4 ðu Þ;
ð2:18Þ
he4 ðu Þi Z 0:
Thus, the periodicity boundary conditions yield
9
s4 $n anti-periodic on vU4S ; >
>
=
4
4
0
u Z e $y C u on vUS ;
>
>
;
u periodic on vU4S :
ð2:19Þ
Let us substitute equations (2.18) into the equilibrium (divergence-free)
equation of the stress tensor (see equation (2.16)1), namely,
)
V$ðC4 e4 ðu ÞÞ ZKV$ðC4 e0 Þ in U4S ðyÞ;
ð2:20Þ
C boundary conditions;
where the boundary conditions are related to (2.17) or (2.19). By virtue of the
linearity of the problem, the solution of e4 ðu Þ in equations (2.16) for a general
macro-strain e0 may be expressed as the superposition of elementary unit strain
solutions e4(ckh) (Suquet 1987), such as
e4 ðu Þ Z e0kh e4 ðckh Þ;
ð2:21Þ
where ckh are the displacements associated with those elementary strain states
denoted by indices kh resulting from the solution of equations (2.16).
By substitution of equation (2.21) into equation (2.18)1, the microstrains are
expressed as
e4 ðu4 Þ Z e0kh ðI kh C e4 ðckh ÞÞ;
ð2:22Þ
where Ikh is the fourth-order identity tensor with components IZ I khZ
ðIij Þkh Z 1=2ðdik djh C dih djk Þ. Equation (2.22) allows us to obtain the distribution
of the strain field along the microstructural domain from the macroscopic strain
field. For that purpose, proper boundary conditions have to be previously
established according to (2.19). Then, we can obtain the strain energy density from
the microscopic stress and strain fields to evaluate the micromechanical stimulus
needed in equation (2.11) (Sanz-Herrera et al. 2008a).
Once the elementary solutions e4(ckh) through (2.22) have been obtained, the
macroscopic stress–strain relationship is obtained straightforwardly as follows:
s0 Z hs4 ðu4 Þi Z hC4 e4 ðu4 Þi Z hC4 ðI kh C e4 ðckh ÞÞi : e0 :
ð2:23Þ
Consequently, at the macroscopic scale, the elasticity tensor is identified in
equation (2.23) as
C0 Z hC4 ðI kh C e4 ðckh ÞÞi:
ð2:24Þ
Using the standard formulation, the variational form of equation (2.20) yields
(Suquet 1987)
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J. A. Sanz-Herrera et al.
ð
ð
4
4 4
e
ðwÞ
:
ðC
e
ðu
ÞÞ
dy
ZK
e4 ðwÞ : ðC4 e0 Þ dy
4
4
US
US
c wðyÞ 2VY ;
ð2:25Þ
where the space VY is defined as
VY dfw j w 2H 1 ðU4S;F Þg;
ð2:26Þ
with H1(U4) being the first-order Sobolev space. Using (2.21), equation (2.25)
can be further developed, namely
ð
ð
kl
4 vcp vui
4
4 vui
Cijpq
dUS Z 4 Cijkl
dU4S ;
ð2:27Þ
4
vyq vyj
vyj
US
US
where cklp represents the characteristic microstructure displacement in the
p-direction due to an applied kl unit strain, kZl being the normal unit strain
states and ksl being the shear unit strain states; ui represents a virtual
displacement. There are six strain states in total (three normal and three shear)
corresponding to the six linear equations above (equation (2.27)). Once these
functions have been obtained, the macroscopic stiffness tensor can be computed
through equation (2.24).
Homogenization theory is used in fluids as well to obtain the overall response
of a microscopic flow in a macroscopically porous medium, i.e. Darcy’s law, and
the microscopic distribution of the velocity field. Therefore, here we establish an
analysis similar to that of solids to analyse domain U4F . The equilibrium and
continuity microscopic equations for an incompressible viscous fluid within the
steady-state Stokes’ flow problem read as
9
KVp4 $1 C m4 V$Vv 4 C r4 f Z 0; >
>
>
>
4
=
V$v Z 0;
ð2:28Þ
v 4 Z 0 ðno-slip conditionÞ; >
>
>
>
;
Cboundary conditions;
with p4 being the fluid pressure; v4 the fluid velocity; r4 and m4 the fluid density
and dynamic fluid viscosity, respectively. As in the solid domain, we consider the
overall velocity to be the result of the macroscopic velocity field plus the
perturbation induced by the microstructure, i.e. v 4 Z v0 C 4v , with 4Zl/L
being the ratio between the micro and macro levels. The body forces vector per
unit volume is denoted by f. For brevity, we define P 4 dKVp4 $1C r4 f , with 1
being the second-order identity tensor. Since we are dealing with periodic
boundary conditions, we establish an analogous procedure to (2.19) for the
fluid problem,
)
p4 n anti-periodic on vU4F ;
ð2:29Þ
v periodic on vU4F :
As in the solid problem, the microvelocities in (2.28) admit a solution of
the form
v 4 ZKP 0j kij ;
Phil. Trans. R. Soc. A (2009)
ð2:30Þ
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Approach to bone tissue engineering
2067
with kij being the characteristic fluid velocities associated with unit pressure
gradients (Terada et al. 1998). Equation (2.30) permits us to obtain the
microvelocities along the microscopic domain through the macroscopic pressure.
After some algebraic manipulations, and using (2.28) and (2.30), functions kij
(Terada et al. 1998) must obey the following expression in the microscopic fluid
domain of the RVE:
m4 V2 k ZKP 0 ;
ð2:31Þ
in combination with the boundary conditions (2.29).
To obtain the homogenized macroscopic Darcy’s law, the microvelocities of
(2.30) are averaged over the microstructural domain (RVE) as follows:
K Z hkij i;
ð2:32Þ
with K being the macroscopic permeability matrix. To derive (2.32), the null
fluctuation of the velocity field hv iZ 0 in U4F is taken into consideration. The
Fick’s diffusion macroscopic matrix is related to the permeability as DZK$D0,
D0 being a model parameter.
The variational form of the fluid phase is developed analogously to that of
solids, as shown before. Therefore, by using periodicity assumptions in (2.29), the
variational form ofð equations (2.28) yields
ð
m4
U4F
Vg$Vv dy ZK
U4F
P 0 $g dy
c gðyÞ 2VY ;
ð2:33Þ
with the space VY being defined as earlier in this section. Using the superposition
of (2.30) and the result (2.31), we finally get the following integral expression:
ð
ð
vkij vgi
4
4
m 4
dUF Z 4 gj dU4F ;
ð2:34Þ
UF vyp vyp
UF
where kij represents the characteristic microstructure fluid velocity due to an
applied generalized pressure gradient Pi0 in the direction i and where gj
represents a virtual velocity. Once these functions have been obtained, the
macroscopic permeability Kij may be obtained from equation (2.32).
(c ) Model for the bone organ domain
The bone organ is only treated in macroscopic terms by simulating the
phenomenon of bone remodelling. Carter’s group (Carter et al. 1989; Beaupré
et al. 1990; Jacobs 1994) proposed an isotropic remodelling model with the
mechanical stimulus identified with the so-called daily tissue stress level, J0, a
scalar quantity similar to the one defined by Carter (Carter et al. 1987) but
generalized to include several load cases. It is defined as
!1=m
N
X
0
m
J Z
ni sti
;
ð2:35Þ
i Z1
where N is the number of different load cases; ni the average number of load
cycles per time step for each load case i; m an experimental parameter varying
from 3 to 8 (Whalen & Carter 1988); and sti the so-called effective stress at the
tissue level.
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2068
J. A. Sanz-Herrera et al.
The first assumption of this model is that the order of load application has no
influence on the adaptive response of the bone. This seems reasonable owing to
the difference between the time scales of causes (load frequencies) and effects
(remodelling process) as computationally proved by Jacobs (1994). Beaupré et al.
(1990) also established a relation between the stress at the continuum level s and
the one at the tissue level sti , following a standard homogenization procedure and
supported by experimental data. With these two assumptions, J0 may be
rewritten as a function of stress at the continuum level as
2 N X
^
1=m r
si ;
J Z
ni
r
i Z1
0
ð2:36Þ
with si being defined by
si Z
pffiffiffiffiffiffiffiffiffiffiffiffi
2EU 0 ;
ð2:37Þ
where E is the elasticity modulus and U 0 the internal strain energy at the
macroscopic level; r is the apparent density; and r^ is the density of the ideal bone
tissue with null porosity.
The next step corresponds to the establishment of the evolution law for
density as a function of the external stimulus. Bone resorption and apposition
occur at internal surfaces, so the mechanical stimulus is directly related to the
surface remodelling rate, r,
_ which quantifies the bone volume formed or
eliminated over the available surface per time unit. Following this idea, and as
an approximation to the experimental curves, Jacobs (1994) proposed the
following expression:
(
r_ Z
k1 $ðJ0 KJ0 KwÞ if
J0 KJ0 O w;
k2 $ðJ0 KJ0 KwÞ if
J0 KJ0 % w;
)
ð2:38Þ
ki being the slopes of the two straight lines of apposition and resorption,
respectively; J0 a reference value of the tissue stress level; and 2w the width of
the so-called ‘dead’ or ‘lazy’ zone. Consequently, the evolution law for the
apparent density is expressed as
r_ Z Krb rS
_ v r^;
ð2:39Þ
where the added or removed bone is assumed to be completely mineralized, that
is, with a maximum density r^; Sv is the specific surface; and Krb is the ratio
between the available surface for remodelling and the total internal surface
(Jacobs’ model considers that the whole existing surface is active, that is, Krb Z 1,
which leads to an excessively fast remodelling process). The elasticity modulus
and Poisson’s ratio are finally obtained by the following experimental correlation
(Jacobs 1994):
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Approach to bone tissue engineering
EðMPaÞ Z
nZ
8
< 2015r2:5
if r% 1:2 g cmK3 ;
: 1763r3:2
8
< 0:20
if rO 1:2 g cmK3 ;
: 0:32
if rO 1:2 g cmK3 :
if r% 1:2 g cmK3 ;
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
ð2:40Þ
The parameters we used regarding this model were obtained from Jacobs (1994).
3. Results
The mathematical model described above was mathematically implemented for
an application example found in the literature (Savarino et al. 2007). A healthy
left femur of a New Zealand rabbit was modelled by computer tomography (CT)
using the MIMICS software (Materialise 2006; figure 1a). On this solid model, a
virtual segmental 3!7 mm defect in the proximal third of the left femur,
laterally to the greater trochanter (Savarino et al. 2007), was performed in order
to simulate the scaffold implantation. The FE mesh of that model was obtained
using HARPOON software (Sharc 2006) with four-node bilinear tetrahedra in
both the bone and scaffold macroscopic domains. Femoral loads were applied at
the proximal part considering the rabbit hopping according to Gushue et al.
(2005). Moreover, the initial femur density was obtained by applying the
remodelling model presented above (Beaupré et al. 1990) in the healthy femur
until convergence.
Since the mathematical model accounts for two different scales of analysis, the
scaffold microstructure needed to be simulated as well. Hence, the actual
geometry of a poly-e-caprolactone scaffold was obtained through micro-CT.
An RVE of such microstructure was selected, distinguishing between the solid
(figure 1b) and fluid (figure 1c) parts. For simplification, we considered that this
RVE represents the microstructure of the entire biomaterial, such that its
microstructure may be reproduced by a periodic repetition of such RVE.
The scaffold microstructures were meshed using voxel FEs for the purpose of
simplicity (Adachi et al. 2006; Sanz-Herrera et al. 2008a). The model parameters
used for this simulation are shown in table 2, whereas the scaffold properties are
highlighted in table 3.
The results of the simulation of the example described above, obtained by
implementing the multi-scale model, are represented in figure 3. We can observe
the apparent bone density distribution, the macroscopic results in the bone
region in terms of apparent density and the microscopic new bone deposition
distribution along the scaffold architecture. The microscopic results corresponding to any point could be illustrated. For example, the results at the macroscopic
midpoint of the scaffold biomaterial are shown in figure 3. This figure also shows
bone mass evolution within the scaffold microstructure over time and the
resorption of the scaffold in the last steps of the analysis. Note that at the
microscopic level new bone formation and apposition occur at surfaces where a
higher mechanical stimulus is found, according to our model. Since this variable
depends on scaffold microstructure, it can be concluded that the scaffold
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J. A. Sanz-Herrera et al.
Table 2. Microscopic bone growth model parameters.
parameter
description
value
m
r^
J
w
N
Em
nm
Tm
D0
k
Mc
exponent that quantifies the importance of the number of cycles (K)
bone density (g cmK3)
reference value for the mechanical stimulus at the tissue level (MPa dK1)
half-width of the dead zone (MPa dK1)
number of cycles considered in the load history (cycles dK1)
Young’s modulus of the matured bone (MPa)
Poisson’s ratio of the matured bone (K)
maturation time (days)
scaling diffusion coefficient (mm2 dK1)
empirical rate constant for bone growth (mg MPaK1)
threshold mass for bone voxel formation (mg)
4.0
1.92
12.5
3.125
24 000
20 000
0.3
2
400.0
4!10K5
0.1
Table 3. Reference scaffold parameters.
parameter
description
value
F
E
n
a
b
W0
Wc
scaffold porosity (K)
bulk biomaterial Young’s modulus (MPa)
bulk biomaterial Poisson’s ratio (K)
hydrolysis model parameter (mm2 dK1)
hydrolysis model parameter (dK1)
bulk biomaterial molecular weight (Da)
molecular weight value below which the biomaterial degrades (Da)
0.85
1570
0.39
4!10K4
4000
114 000
10 000
microarchitecture is key to new bone tissue formation. Within the context of the
influence of scaffold morphology on bone regeneration, further information can be
found in the experimental studies carried out by Tsuruga et al. (1997), Kuboki
et al. (2001) and Karageorgiou & Kaplan (2005), or as recently presented by
Andreykiv et al. (2008) within the framework of the effect of peri-implant surface
geometry, from a computational perspective. One of the conclusions derived
from the analysis is that the effect of the scaffold implantation on the bone organ
(in terms of bone remodelling) is negligible, except for a very localized area
surrounding the scaffold proximity region.
The overall bone regeneration and scaffold resorption are depicted in figure 4.
The former is computed as the percentage of new bone regeneration in the
microstructure (averaged per volume unit) at each macroscopic point. The
scaffold resorption is computed analogously but considering the biomaterial
scaffold mass loss in this case. Figure 4 shows that bone regeneration starts
around day 22, which means that the critical mass Mc has been reached after that
implantation time, after which an increasing trend over time is obtained. This is
connected with many experimental works where bone regeneration increases
with implantation time. Owing to the dispersion and variety of applications, we
should only aim at validating this model qualitatively under similar conditions.
For example, Pothuaud et al. (2005) implanted 75 per cent of porous
hydroxyapatite–tricalcium phosphate scaffolds in the proportion of 4 to 6 loaded
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Approach to bone tissue engineering
(a)
(b)
(c)
(d)
(e)
density
1.92
1.72
1.52
1.32
1.12
0.91
0.71
0.51
0.31
0.11
scaffold
mature bone
non-mature bone
Figure 3. Apparent density evolution (g cmK3) of a proximal femur implanted with a scaffold in the
greater trochanter region. In each part, the bone organ (right), detail of the scaffold implantation
(bottom left) and the bone regeneration distribution onto the scaffold microsurface of the macroscopic
scaffold midpoint (top left) are shown: (a) 2 days, (b) 14 days, (c) 28 days, (d ) 42 days and (e) 56 days
after implantation.
with BMSC, in the femoral condyle of New Zealand rabbits. One group of
the scaffolds used in this experiment were functionalized with a cyclo-RGD
(arginine–glycine–aspartic acid) peptide to enhance cell adhesion onto the scaffold
surface. Results showed 12.3 and 21.3 per cent overall bone regeneration for
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J. A. Sanz-Herrera et al.
100
90
80
percentage
70
60
50
40
30
20
10
0
10
20
30
40
50
60
implantation time (days)
Figure 4. The overall ratio of bone regeneration (circles) and scaffold mass loss (squares) for the
simulated example of application.
non-functionalized and functionalized scaffolds at two weeks, respectively; and 21.1
and 18.1 per cent overall bone regeneration for non-functionalized and functionalized
scaffolds at four weeks. On the other hand, injectable calcium phosphate into the
knee joint of New Zealand rabbits was presented by Gauthier et al. (2005). New bone
apposition was evaluated through three-dimensional micro-tomographic imaging,
resulting in approximately 30 per cent of the overall bone volume formation at six
weeks after implantation. The results yielded by these experimental works are in
the range of those reported in this work, as shown in figure 4. However, the
microarchitecture, biomaterial composition and exact location of the implantation of
the scaffold simulated in this work are different from those analysed in the previously
mentioned experimental reports, which may explain the differences between the
results. The scaffold resorption shown in figure 4 is quite abrupt, starting
approximately at day 54, once the critical minimum molecular weight was
encountered at the scaffold microstructure. This trend is typical in the degradation
of polymeric materials and lies within the range of the experimental results
presented by Sung et al. (2004). Nevertheless, the degradation mechanism of
polymeric biomaterials, i.e. hydrolysis, is highly dependent on parameters regarding
the internal scaffold microarchitecture such as pore size or porosity. Based on clinical
evidence, the degradation rate of polymeric materials should be as slow as possible
(Hutmacher 2000).
4. Concluding remarks
The use of numerical techniques in tissue engineering is a promising tool to reach
clinical application and to help in the task of making tissue engineering a
clinically viable reality. It allows useful information to be obtained in a large
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Approach to bone tissue engineering
2073
number of clinical cases with the use of the computer. This is key to tissue
engineering, where the role of many factors and parameters are misunderstood
or need to be elucidated. A deep understanding of each biological phenomenon
has to be gained by modellers, and a strong collaboration of the fields involved
is required.
In this scenario, we presented in this paper a novel mathematical approach
available for modelling bone tissue regeneration using scaffolds. We also included
the simulation of some realistic experiments of scaffolds implanted in rabbits.
From these examples, we see that bone formation occurs gradually over time,
whereas scaffold resorption starts abruptly, which is consistent with some
previous experimental works (Sung et al. 2004; Gauthier et al. 2005; Pothuaud
et al. 2005). The major implicit hypothesis of this model is that bone regeneration
within the biomaterial is mainly driven by mechanical factors. This hypothesis is
in agreement with the experimental evidence of bone regeneration in the presence
of biomaterials found elsewhere (Bostrom et al. 1998; Simmons et al. 2001;
van der Donk et al. 2002; Gardner et al. 2006). Furthermore, mechanical factors
have also been considered to play an important role in bone regeneration within
scaffolds, as shown in other theoretical models (Kelly & Prendergast 2006; Byrne
et al. 2007). However, non-mechanical factors such as nutrients/oxygen supply or
the scaffold’s chemical surface, among others, may also be of extreme importance
in these applications. In order to account for some of the most important
phenomena, the model includes multiphysics as well as multiscale arguments for
the proper simulation of the main biological processes. For example, it is known
that, when a scaffold is implanted into a bone defect, it produces a haematoma in
the area surrounding the scaffold region that contains osteoprogenitor and
mesenchymal cells. Here, effects such as cell migration and vasculogenesis
take place, which were modelled as a Fickean diffusion, as a first approach.
A more concise analysis could be performed to better simulate this kind of
biological event.
The need for including two scales of analysis is due to the fact that tissue
regeneration and biomaterial degradation take place at the pore level, whereas
the overall behaviour of the scaffold implant is better observed at the organ level,
where bone–scaffold interaction occurs, and each scale determines the response of
the other. Moreover, this approach allows us to properly model both the actual
bone organ geometry and the scaffold microstructure, as demonstrated in the
present paper in contrast to other works. Indeed, we only define two model
parameters to be determined from experimental fitting. However, when actual
scaffold patterns are considered in the simulation, as presented in this work, it is
difficult to assume periodicity of the RVE, and periodic boundary conditions may
be debatable. Nevertheless, even when the RVE is non-symmetric, periodic
boundary conditions provide an overall good estimation for a proper RVE size
(Suquet 1987). Moreover, the mathematical multiscale implementation results
showed an unusual class of nonlinear macroscopic response, in the sense that
variable fields evolve over time for constant boundary conditions. This fact
comes from the evolution of the microstructure because of regeneration and
degradation phenomena, this nonlinearity being unusual in classical multiscale
methods applied to mechanics. Nevertheless, we only modelled the net amount of
regenerated bone, which is a rough approximation of reality. Experimentally,
cells form a biolayer onto the scaffold microsurface as they attach. After this,
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2074
J. A. Sanz-Herrera et al.
they differentiate into osteoblasts that fabricate primary bone, finally resulting in
mineralized mature bone. A more complete modelling should include all these
factors. Another point that may be improved is a finer geometrical approximation
of bone growth and hydrolytic resorption as presented here by the voxel FEM.
The multiscale model requires further experimental validation, even though
a great effort has been made in this work to find experimental applications
in the literature. There are many input parameters of the model presented in
this paper that are difficult to obtain from the literature, for instance the
internal microstructure of the scaffold. This makes a simulation versus
experiment comparison extremely difficult. An exhaustive validation of the
proposed models should include multidisciplinary collaboration among
physicians, veterinary surgeons, material and mechanical engineers as well as
modellers. In this framework, we could get access to the main parameters of the
experimental set-up, hence validation may be properly conducted. Once this
task is performed, this work may be used for scaffold design in a specific
patient. In this sense, the anthropometric and metabolic features of individual
patients may be taken into account, as long as the results provided are only
treated qualitatively.
Finally, it may be stated that tissue engineering is still a growing young field.
It has not yet accomplished relevant or satisfactory results to be translated to
clinical application. Many biomaterials need to be proven, protocols must be
further tested and refined, and commercial interests have to prove this field to be
economically viable. Nevertheless, given the scientific promise, potential social
impact and the young age of the field, many believe that it should be only a
matter of time until tissue engineering reaches the main stream of surgical
practice (Fauza 2003).
This work has been supported by the Ministerio de Educación y Ciencia of Spain (Project
DPI2007-65601-C03-01). Their financial support is gratefully acknowledged. The reader may view
our webpage at: http://i3a.unizar.es/gemm. There, one can find plenty of information about our
group’s activities and the models we are developing. Furthermore, the reader is kindly invited to
contact J.M.G.-A. whenever necessary, to obtain additional information about the presented models.
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