Geometry Design Optimization of Functionally Graded Scaffolds for Bone Tissue Engineering: A Mechanobiological Approach

Abstract

Functionally Graded Scaffolds (FGSs) are porous biomaterials where porosity changes in space with a specific gradient. In spite of their wide use in bone tissue engineering, possible models that relate the scaffold gradient to the mechanical and biological requirements for the regeneration of the bony tissue are currently missing. In this study we attempt to bridge the gap by developing a mechanobiology-based optimization algorithm aimed to determine the optimal graded porosity distribution in FGSs. The algorithm combines the parametric finite element model of a FGS, a computational mechano-regulation model and a numerical optimization routine. For assigned boundary and loading conditions, the algorithm builds iteratively different scaffold geometry configurations with different porosity distributions until the best microstructure geometry is reached, i.e. the geometry that allows the amount of bone formation to be maximized. We tested different porosity distribution laws, loading conditions and scaffold Young's modulus values. For each combination of these variables, the explicit equation of the porosity distribution law-i.e the law that describes the pore dimensions in function of the spatial coordinates-was determined that allows the highest amounts of bone to be generated. The results show that the loading conditions affect significantly the optimal porosity distribution. For a pure compression loading, it was found that the pore dimensions are almost constant throughout the entire scaffold and using a FGS allows the formation of amounts of bone slightly larger than those obtainable with a homogeneous porosity scaffold. For a pure shear loading, instead, FGSs allow to significantly increase the bone formation compared to a homogeneous porosity scaffolds. Although experimental data is still necessary to properly relate the mechanical/biological environment to the scaffold microstructure, this model represents an important step towards optimizing geometry of functionally graded scaffolds based on mechanobiological criteria.

Full text

RESEARCH ARTICLE
Geometry Design Optimization of
Functionally Graded Scaffolds for Bone
Tissue Engineering: A Mechanobiological
Approach
Antonio Boccaccio
1
*, Antonio Emmanuele Uva
1
, Michele Fiorentino
1
, Giorgio Mori
2
,
Giuseppe Monno
1
1 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, 70126, Bari, Italy,
2 Dipartimento di Medicina Clinica e Sperimentale, Università di Foggia, 71122, Foggia, Italy
* a.boccaccio@poliba.it
Abstract
Functionally Graded Scaffolds (FGSs) are porous biomaterials where porosity changes in
space with a specific gradient. In spite of their wide use in bone tissue engineering, possible
models that relate the scaffold gradient to the mechanical and biological requirements for
the regeneration of the bony tissue are currently missing. In this study we attempt to bridge
the gap by developing a mechanobiology-based optimization algorithm aimed to dete rmine
the optimal graded porosity distribution in FGSs. The algorithm combines the parametric
finite element model of a FGS, a computational mechano-regulation model and a numerical
optimization routine. For assigned boundary and loading conditions, the algorithm builds
iteratively different scaffold geometry configurations with different porosity distributions until
the best microstructure geometry is reached, i.e. the geometry that allows the amount of
bone formation to be maximized. We tested different porosity distribution laws, loading con-
ditions and scaffold Young’s modulus values. For each combination of these variables, the
explicit equation of the porosity distribution law–i.e the law that describes the pore dimen-
sions in function of the spatial coordinates–was determined that allows the highest amounts
of bone to be generated. The results show that the loading conditions affect significantly the
optimal porosity distribution. For a pure compression loading, it was found that the pore
dimensions are almost constant throughout the entire scaffold and using a FGS allows the
formation of amounts of bone slightly larger than those obtainable with a homogeneous
porosity scaffold. For a pure shear loading, instead, FGSs allow to significantly increase the
bone formation compared to a homogeneous porosity scaffolds. Although experimental
data is still necessary to properly relate the mechanical/biological environment to the scaf-
fold microstructure, this model represents an important step towards optimizing geometry of
functionally graded scaffolds based on mechanobiologica l criteria.
PLOS ONE | DOI:10.1371/journal.pone.0146935 January 15, 2016 1/20
OPEN ACCESS
Citation: Boccaccio A, Uva AE, Fiorentino M, Mori G,
Monno G (2016) Geometry Design Optimization of
Functionally Graded Scaffolds for Bone Tissue
Engineering: A Mechanobiological Approach. PLoS
ONE 11(1): e0146935. doi:10.1371/journal.
pone.0146935
Editor: Jie Zheng, University of Akron, UNITED
STATES
Received: November 2, 2015
Accepted: December 25, 2015
Published: January 15, 2016
Copyright: © 2016 Boccaccio et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
Data Availability Statement: All relevant data are
within the paper.
Funding: The authors have no support or funding to
report.
Competing Interests: The authors have declared
that no competing interests exist.
Abbreviations: t, length of the side of the basis of
the scaffold prismatic model; h, height of the scaffold
prismatic model; V
TOT
, total volume of the scaffold
model;
F
V
, compression force acting on the scaffold

Introduction
Functionally Graded Scaffolds (FGSs) for bone tissue engineering are porous biomaterials
where the porosity changes with a specific gradient in space. The gradation of porosity enables
FGSs to combine together the best mechanical properties of the denser material with those of
the more porous one and the resulting material exhibits performances higher than those of the
single constitutive material s. Low porosity regions offer high mechanical strength, high poros-
ity regions promote, instead, cell adhesion and support cell growth, proliferation and differe nti-
ation [1–2].
Such scaffolds have been successfully utilized in the most variegated domains including the
repair of long bone [1,3] and osteochondral [4–5] defects, the maxillofacial [6–7] and the spinal
[8] surgery, the cranial reconstruction [9] and the drug delivery systems [1 ,10]. A large number
of studies [11–13] are reported in the literature on the manufacturing processes that can be
adopted to fabricate these biomaterials. Among the others, the strategy based on the integration
of add itive manufacturing or rapid prototyping techniques with computer-aided design models
seems to be one of the most efficient [2, 14]. The possibility of building any scaffold architecture
with any type of porosity gradation and the experimental evidence that the geometry of porous
scaffolds significantly influences the cellular response and the rate of bone tissue regeneration
[15–17] led research community to find the possible models that relate the scaffold gradient to
the mechanical and biological requirements for the regeneration of the bony tissue [2]. How-
ever, to date such models have not been developed yet.
In this article, we attempt to bridge the gap and propose a mechanobiology-driven optimi-
zation algorithm that, based on the boundary and loading conditions acting on the scaffold,
identifies the best porosity distribution that allows the bone formation to be maximized. Other
studies reported in the literature utilized optimization techniques to determine the best scaffold
geometry [18–23] but none of them adopted mechanobiological criteria and determined the
optimal porosity gradient in FGSs. In a previous study [24], the algorithm was utilized to deter-
mine the optimal pore dimension in regular structured open-porous scaffolds with homoge-
neous porosity. In the present study, the model was further developed to include a functionally
graded porosity. In particular, three different variables have been investigated: the porosity dis-
tribution law, the loading conditions and the scaffold Young’s modulus; for each combination
of the three variables, the algorithm determines the explicit equation of the porosity distribu-
tion law (i.e. the law that describes the pore dimensions in function of the spatial coordinate),
that allows the largest volume of the scaffold to be occupied by bone.
Materials and Methods
Parametric model of an open-porous functionally graded scaffold
The parametric finite element model of an open-porous functionally graded scaffold was cre-
ated in ABAQUS CAE
1
Version 6.12 (Dassault Systèmes, France). The model has a prismatic
geometry with a square t × t = 2548 μm × 2548 μm base and a h = 3822 μm height. The scaffold
(represented in yellow, Fig 1A) includes circular pores with a parametric radius A that was
assumed to change only along the y direction and remain constant along x and z direction (Fig
1A and 1B). According to Byrne et al. [25], the scaffold pores were hypothesized to be occupied
by granulation tissue (represented in red, Fig 1B). The finite element mesh includes tetrahedral
biphasic poro-elastic elements. 4-node linear coupled pore pressure elements (C3D4P) avail-
able in ABAQUS were utilized to model both, the scaffold (Fig 1C) and the granulation tissue
(Fig 1D). The approximate element size was fixed equal to 40 μm.
Optimization of Functionally Graded Scaffolds
PLOS ONE | DOI:10.1371/journal.pone.0146935 January 15, 2016 2/20
and producing a vertical distributed load;
F
H
, shear
force acting on the scaffold and producing a
horizontal distributed load;
F
M
, mixed compression-
shear force acting on the scaffold; p
pore
, pore
pressure acting on the outer surfaces of the
granulation tissue; E, scaffold Young’s modulus; A,
pore radius; A
i
(i = 1, 2, 3, 4), pore radius at specific y
locations; y
max
, y
min
, y
int
, y
int1
, y
int2
, specific y
locations where the pore radius was determined; m,
m
i
(i = 1, 2, 3), gradient of porosity distribution laws;
S, biophysical stimulus regulating the differentiation
process; ɣ, octahedral shear strain; ʋ, interstitial fluid
flow; ε
I
, ε
II
, ε
III
, principal strains; a, empirical constant
a = 3.75%; b, empirical constant b =3μms
-1
; n
resorb
,
n
mature
, c, boundaries of the mechano-regulation
diagram; A
lower
, lower bound of the pore radius
variability range; A
upper
, upper bound of the pore
radius variability range; V
i_bone
, volume of the generic
element where the formation of mature bone is
predicted to take place; n
b
, number of elements
where the formation of mature bone is predicted to
take place; V
BONE
, total volume of the elements
where the formation of mature bone is predicted to
take place; BO
%
, percentage of scaffold volume that
is occupied by bone; Ω, objective function to optimize;
PVPD, Percent Variation of the Pore Dimension; A
H
,
A
L
, highest and lowest value of A along the y-axis,
respectively; BO
%_tri-linear
, percentage of volume
occupied by bone predicted for the tri-linear porosity
distribution law; BO
%_constant
, percentage of volume
occupied by bone predicted for the constant porosity
distribution law; iBO
%
, increment of BO
%
.

Material properties implemented in the finite element model of the granulation tissue are
the same as those utilized in previous studies [24, 26–27]. In detail, the Young’s modulus was
set equal to 0.2 MPa; the permeability to 1×10
−14
m
4
/N/s; the Poisson’s ratio to 0.167; the
porosity to 0.8; the bulk modulus grain to 2300 MPa; the bulk modulus fluid to 2300 MPa. In
order to evaluate the effect of the scaffold mechanical properties on the optimal porosity distri-
bution, three different values of the Young’s modulus E were hypothesized: 500, 1000 and 1500
MPa which are the same as those utilized in a previous study [24 ].
The nodes of the bottom surface of the model were clamped (Fig 1E, 1F and 1G) while those
of the upper surface were tied to a rigid plate (represented in blue, Fig 1E, 1F and 1G). For the
outer nodes of the granulation tissue the pore pressure was fixed equal to 0 MPa which indi-
cates that the liquid can freely exudate while applying the load. Three different loading condi-
tions were hypothesized in the study: (a) a compression force
F
V
producing a vertical
distributed load of F
V
/(t × t) = 1 MPa (Fig 1E); (b) a shear force F
H
producing an horizontal
distributed load of F
H
/(t × t) = 0.5 MPa (Fig 1F); (c) a mixed compression-shear force F
M
Fig 1. Parametric finite element model of the functionally graded scaffold utilized in the study. CAD model (A-B) and finite element mesh (C-D) of the
scaffold (A, C) and granulation tissue (B, D). Circular pores with variable radius A have been modelled. The nodes of the bottom surface of the model were
clamped (E-G) while those of the upper surface were tied to a rigid plate (represented in blue). Three different loading conditions were hypothesized:a
compression force
F
V
(E); a shear force F
H
(F); a mixed compression-shear force F
M
(G). The pore pressure p
pore
on the outer surfaces of the granulation
tissue was set equal to zero to simulate the free exudation of fluid.
doi:10.1371/journal.pone.0146935.g001
Optimization of Functionally Graded Scaffolds
PLOS ONE | DOI:10.1371/journal.pone.0146935 January 15, 2016 3/20

given by the sum F
M
¼ F
V
þ F
H
(Fig 1G). The choice of setting F
H
= 0.5 × F
V
was done
because scaffolds are primarily designed to undergo to compression loading [25]. In all the
hypothesized loading conditions, force was ramped over a time period of 1 s that is the possible
time in which, a human body motion (such as to assume the erect position or to perform any
motion of anatomical regions where a FGS can be implanted), can be completed. The same
time interval was utilized in previous studies [24, 28–29].
Porosity distribution laws
The dimension of the circular pores was controlled by the parametric radius A (Fig 1) that was
hypothesized to change along the y-direction according to different porosity distribution laws.
The coefficients of these distribution laws and hence their gradients were determined via the
optimization algorithm described below. The porosity distribution laws considered in the
study are the following: constant, linear, bi-linear and tri-linear.
• Constant law. All the pores have the same dimensions (Fig 2A). In this case, the optimization
algorithm determines just one coefficient, i.e. A
1
, that is the pore radius of all the scaffold
pores.
• Linear law. The dimensions of pore change linearly with y. Two coefficients have to be deter-
mined by the optimization algorithm: A
1
and A
2
that are the pore radii at y = y
min
and y =
y
max
, respectively (Fig 2B).
• Bi-linear law. The pore radius A changes in the ranges [y
min
y
int
] and [y
int
y
max
] with two dif-
ferent linear laws that assume the same value for y = y
int
. The coefficients to optimize are
three: A
1
, A
2
and A
3
(Fig 2C).
• Tri-linear law. The dimensions of the pore change in the intervals [y
min
y
int1
], [y
int1
y
int2
],
[y
int2
y
max
] with three different linear laws. The laws defined in the first and second and those
defined in the second and third interval assume the same value for y = y
int1
and for y = y
int2
,
respectively. In this case, the optimization algorithm determines four coefficients: A
1
, A
2
, A
3
and A
4
(Fig 2D).
The specific values of y
max
, y
min
, y
int
, y
int1
and y
int2
are reported in Fig 2. It is worthy to note
that, once the coefficients A
i
(i = 1, 2, 3, 4) have been determined, the explicit equation of the
best porosity distribution, i.e. the equation that describes how the pore radius A changes with
y, can be obtained by simply implementing the obtained coefficients in the relationships
reported in Table 1.
Computational mechano-regulation model
Once the mesenchymal stem cells invade the scaffold and spread through its pores, the bone
regeneration process starts. After dispersal, cells will differentiate. The biophysical stimulus S
that regulates the differentiation process was hypothesized to be a function of the octahedral
shear strain ɣ and interstitial fluid flow ʋ in the extracellular environment of the cells. In detail,
let ε
I
, ε
II
, and ε
III
be the principal strains, the octahedral shear strain ɣ can be defined as:
g ¼
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðε
I
 ε
II
Þ
2
þðε
II
 ε
III
Þ
2
þðε
III
 ε
I
Þ
2
q
ð1Þ
Calling a and b two empirical constants defined as in Huiskes et al. [30], and given by
a = 3.75% and b =3μms
-1
, the biophysical stimulus S can be expressed, according to
Optimization of Functionally Graded Scaffolds
PLOS ONE | DOI:10.1371/journal.pone.0146935 January 15, 2016 4/20

Fig 2. Porosity distribution laws analyzed in the study. (A) constant; (B) linear; (C) bi-linear; (D) tri-linear. The specific coefficients A
i
(i =1,2,3,4)of
these laws were determined via the optimization algorithm.
doi:10.1371/journal.pone.0146935.g002
Optimization of Functionally Graded Scaffolds
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Prendergast et al. [31], as:
S ¼
g
a
þ
v
b
ð2Þ
Mesenchymal stem cells differentiate into different cell phenotypes according to the follow-
ing inequalities:
if S > c ) fibrogenesis ) fibroblasts ) fibrous tissueformation
if 1 < S < c ) condrogenesis ) chondrocytes ) cartilagineous tissueformation
if n
mature
< S < 1 ) osteogenesis ) osteoblasts ) immature bonetissueformation
if n
resorb
< S < n
mature
) osteogenesis ) osteoblasts ) maturebonetissueformation
if 0 < S < n
resorb
) osteoclasts ) bone resorbtion
ð3Þ
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
where n
resorb
= 0.01, n
mature
= 0.53 and c = 3 represent boundaries of the mechano-regulation
diagram the values of which are the same as those utilized in other studies [28, 32–33].
Optimization algorithm
The FGS parametric finite element model, the computational mechano-regulation model
above described and a numerical optimization routine were combined together in an algorithm
written in Matlab
1
(v. R2011b) (Fig 3) that aims to determine, for each of the hypothesized
scaffold Young’s moduli, loading conditions and porosity distribution laws, the equations of
the best porosity distribution that allows the bone formation to be maximized. Considering
that 3 scaffold Young’s modulus values (i.e. 500, 1000 and 1500 MPa), 3 loading conditions
(i.e.
F
V
,F
H
, and F
M
) and 4 porosity distribution laws (i.e. constant, linear, bi-linear and tri-lin-
ear) have been hypothesized, it follows that a total of 3 × 3 × 4 = 36 optimization analyses have
been performed in the study.
As a first step, the algorithm requires to select (Block [1]) one of the porosity distribution laws
(Block [2]). The initialization of coefficients A
i
follows (Block [3]), the user can assign to A
i
initial
values that fall within the interval [A
lower
A
upper
], where A
lower
=5μmandA
upper
=300μmhave
been taken the same as those utilized in a previous study [24]. The algorithm implements the
specified initial values of A
i
into a PYTHON script (Block [4]) that is given in input to ABAQUS.
The PYTHON script, based on the values A
i
, defines in function of the coordinate location y the
Table 1. Porosity distribution laws implemented in the study.
Porosity distribution Coefficients to optimize Equation Gradient
Constant law A
1
A = A
1
0
Linear law A
1,
A
2
A = A
2
+m(y−y
max
)
m ¼
A
2
A
1
y
max
y
min
Bi-linear law A
1,
A
2,
A
3
for y2[y
min
y
int
])A = A
2
+m
1
(y−y
int
)
for y2[y
min
y
int
])A = A
3
+m
2
(y−y
max
)
m
1
¼
A
2
 A
1
y
int
 y
min
m
2
¼
A
3
 A
2
y
max
 y
int
Tri-linear law A
1,
A
2,
A
3,
A
4
for y2[y
min
y
int1
])A = A
2
+m
1
×(y−y
int1
)
for y2[y
int1
y
int2
])A = A
3
+m
2
×(y−y
int2
)
for y2[y
int2
y
max
])A = A
4
+m
3
×(y−y
max
)
m
1
¼
A
2
 A
1
y
int1
 y
min
m
2
¼
A
3
 A
2
y
int2
 y
int1
m
3
¼
A
4
 A
3
y
max
 y
int2
doi:10.1371/journal.pone.0146935.t001
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Fig 3. Schematic of the algorithm implemented in Matlab environment to optimize the porosity distribution law in functionally graded scaffolds.
doi:10.1371/journal.pone.0146935.g003
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dimension of each pore. The module ABAQUS CAE builds the CAD model of the functionally
graded scaffold (Block [5]) with the computed pore dimensions, and after applying the boundary
and (one of) the (three) loading conditions above described, generates the finite element mesh
(Block [6]). The finite element analysis follows that accounts for geometrical and material non-
linearities (Block [6]). For each element occupying the scaffold pores, i.e. the elements repre-
sented in red in Fig 1D,ABAQUSprints(Block[7]) the values of the principal strains ε
I
, ε
II
and
ε
III
and of the interstitial fluid flow ʋ that the algorithm utilizes to compute, through the eqs (1)
and (2), the magnitude of the biophysical stimulus S (Block [8]). Then, the relationships eq (3)
are implemented and for those elements for which the inequality
n
resorb
< S < n
mature
ð4Þ
is satisfied, i.e. for those elements where the formation of mature bone is predicted to take place,
the volume V
i_bone
is stored (Block [9]). If n
b
is the number of elements where inequality eq (4) is
satisfied, the algorithm calculates the total volume of these elements as:
V
BONE
¼
X
nb
i¼1
V
i bone
ð5Þ
If V
TOT
is the total volume of the scaffold model V
TOT
= t × t ×
h =2548μm × 2548 μm × 3822 μm = 24.814 mm
3
, the algorithm determines the percentage of
scaffold volume BO
%
that is occupied by bone as (Block [10]):
BO
%
¼
V
BONE
V
TOT
 100 ð6Þ
and calculates the value of the objective function O as (Block [11]):
O ¼ð1ÞBO
%
ð7Þ
At this point, the algorithm formulates an optimization problem that includes the coeffi-
cients A
i
as design variables and that aims to minimize the value of the objective function O or,
equivalently, to maximize the percentage BO
%
of volume occupied by bone. It can be claimed, in
fact, that the greater the efficiency of the scaffold, the larger the amount of bone produced by
the scaffold itself. In an ideal scaffold, 100% of its volume is occupied by bone. The inverse prob-
lem described with the eqs (6) and (7) was solved with the Sequential Quadratic Programming
(SQP) method available in Matlab, an iterative method for nonlinear optimization. The number
of iterations performed by the method can be controlled by means of specific stopping criteria
that can be selected by the user and that include a number of tolerances. As one of these stop-
ping criteria is meet, the optimization process ends and after implementing the optimal coeffi-
cients A
i
in the relationships of Table 1, the optimal porosity distributions are traced in function
of y (Blocks [14] and [15]). If no stopping criteria are satisfied, the optimization algorithm
assigns new values to A
i
thus generating new candidate solutions (Block [13]). The optimization
process terminates when one of the selected stopping criteria is satisfied (Block [12]).
The biophysical stimulus S on which the objective function O depends, was computed based
on the hypothesis that the dispersal of mesenchymal stem cells has already taken place and that
the only granulation tissue, with the mechanical properties above described, occupies the scaf-
fold pores.
All the computations were performed on a HP Z620- Intel
1
Xeon
1
Processor E5-2620—
16Gb RAM. The most expensive optimization analyses were those implementing the tri-linear
law that took around 300 hours of computations.
Optimization of Functionally Graded Scaffolds
PLOS ONE | DOI:10.1371/journal.pone.0146935 January 15, 2016 8/20

Results
In the case of the compression loading F
V
the predicted pore dimension experiences small
changes (Fig 4A, 4C and 4E) along the y-axis and is almost constant. Independently from the
scaffold Young’s modulus E, A does not change by more than 15 μm. The general trend (with
the exception for the porosity distribution obtained implementing the constant law) that can
be observed is that the pore radius in the vicinity of the clamps (i.e. for high values of y) and of
the load (i.e. for small values of y) slightly decreases. For increasing values of E, the pore radius,
on average, increases. For instance, in the case of E = 500 MPa, the average pore radius is about
190 μm, for E = 1500 MPa, instead, becomes about 220 μm. The percentage of volume occupied
by bone BO
%
increases as we move from the constant to the tri-linear porosity distribution (Fig
4B, 4D and 4F). Furthermore, increasing values of BO
%
were predicted for increasing values of
the scaffold Young’s modulus (Fig 4B, 4D and 4F).
More interesting appears the porosity distribution predicted by the algorithm in the case of
the shear load F
H
(Fig 5) where important changes of the pore dimensions are predicted along
the y-axis (Fig 5A, 5C and 5E). The highest values of A are predicted in the vicinity of the load
(i.e. for small values of y) while the pore dimensions tend to decrease as we move towards the
clamped region. Also in this case BO
%
increases as we move from the constant to the tri-linear
porosity distribution, however, the change of BO
%
is more significant than in the case of com-
pression load. For increasing levels of E, the average value of BO
%
increases too (Fig 5B, 5D
and 5F).
In the case of mix ed load F
M
, the pore radius A experiences changes that are less important
than those predicted in the case of shear load F
H
but that are certainly larger than those com-
puted in the case of compression load F
V
(Fig 6A, 6C and 6E). As in the previous case, the pore
dimension decreases for increasing values of y. BO
%
increases as we move from the constant to
the tri-linear law and its average value increases for increasing values of the scaffold Young’s
modulus E (Fig 6B, 6D and 6F). For a fixed value of E and porosity distribution law, the values
of BO
%
predicted in the case of mixed load F
M
are smaller than those predicted for the other
hypothesized loading conditions (Figs 4B, 4D, 4F, 5B, 5D, 5F, 6B, 6D and 6F).
In order to quantify (i) the change of the pore dimensions with y and (ii) the “usefulness” of
utilizing a functionally graded scaffold instead of a scaffold with a homogenous porosity distri-
bution we introduced two parameters. The first one, denoted as PVPD, represents the Percent
Variation of the Pore Dimension and is defined as:
PVPD ¼
ðA
H
 A
L
Þ
A
L
 100 ð8Þ
where A
H
and A
L
are the highest and the lowest value of A along the y-axis, respectively (Fig
7A). In general, the higher the PVPD, the larger are the changes of the pore dimension A. The
highest values of PVPD have been found in the case of the shear loading F
H
(Fig 7) where
changes of A also by more than 25–30% were predicted (Fig 7C). Slightly lower are the values
of PVPD found in the case of the mixed load F
M
(Fig 7D) and yet less significant those com-
puted in the case of the compression load F
V
(Fig 7B). Averagely, it appears that PVPD does
not depend neither on the scaffold Young’s modulus E, nor on the porosity distribution law
but does depend on the loading conditions. For the constant law, regardless of the type of load
considered, the value of PVPD is zero and is not shown in Fig 7.
In general, it appears that as we move from the constant to the linear, bi-linear and, finally,
tri-linear porosity distribut ion law the percentage of volume occupied by bone BO
%
increases
(Figs 4B, 4D and 4F, 5B, 5D and 5F, 6B, 6D and 6F). In particular, the highest valu es of BO
%
have been found for the tri-linear law while the lowest ones for the constant law. Therefore, it
Optimization of Functionally Graded Scaffolds
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Fig 4. Computed values of A and BO
%
in the case of compression loading. Pore radius A (A, C, E) (vs. location y) and percentages of the scaffold
volume occupied by bone BO
%
(B, D, F) predicted by the optimization algorithm in the case of compression loading F
V
for different scaffold Young’s moduli
and after implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer.
All the values of BO
%
reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
doi:10.1371/journal.pone.0146935.g004
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Fig 5. Computed values of A and BO
%
in the case of shear loading. Pore radius A (A, C, E) (vs. location y) and percentages of the scaffold volume
occupied by bone BO
%
(B, D, F) predicted by the optimization algorithm in the case of shear loading F
H
for different scaffold Young’s moduli and after
implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer. All the
values of BO
%
reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
doi:10.1371/journal.pone.0146935.g005
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Fig 6. Computed values of A and BO
%
in the case of mixed load. Pore radius A (a, c, e) (vs. location y) and percentages of the scaffold volume occupied
by bone BO
%
(b, d, f) predicted by the optimization algorithm in the case of mixed load F
M
for different scaffold Young’s modulus values and after
implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer. All the
values of BO
%
reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
doi:10.1371/journal.pone.0146935.g006
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Fig 7. Computed values of PVPD for different loading conditions. Percent Variation of the Pore Dimension (PVPD) for the compression F
V
(B), the shear
F
H
(C) and the mixed F
M
(D) load and for all the hypothesized scaffold Young’s modulus values. (A) reference schematic utilized to calculate the parameter
PVPD. Note: A
H
and A
L
are the highest and lowest value of A that can be located in correspondence of any value of y and not necessarily, as reported in the
figure, of the furthest values y =0μm and y = h = 3822 μm.
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makes sense to introduce the second parameter, denoted as iBO
%
and defined as the increment
of BO
%
when we move from the constant to the tri-linear law. If BO
%_tri-linear
is the percentage
of volume occupied by bone predicted for the tri-linear porosity distribution and BO
%_constant
the percentage predicted with the constant one, iBO
%
can be expressed as:
iBO
%
¼ BO
% trilinear
 BO
% constant
ð9Þ
As is clear, the higher the values of iBO
%
, the more “useful” is the utilization of a function-
ally graded scaffold instead of a homogeneous porosity scaffold. In the limit case where iBO
%
=
0%, the use of a FGS does not make sense and a homogeneous porosity scaffold has the same
potentialities of generating bone as the FG one. On average, the highest values of iBO
%
were
computed in the case of shear loading F
H
followed by the mixed load F
M
and the compression
load F
V
, respectively (Fig 8). In particular, among the hypothesized scaffold Young’s moduli,
the highest values of iBO
%
were predicted for E = 1000 MPa.
A three-dimensional view of the optimal scaffold geometry predicted for the tri-linear
porosity distribution (that is the law with which the highest values of BO
%
have been obtained)
and the shear loading F
H
is shown in Fig 9. As it can be seen, the pore dimensions change sig-
nificantly along the y-axis and, on average, increase for increasing values of E.
Discussion
This article presented an optimization algorithm based on mechanobiological criteria and
aimed to determine the best porosity distribution in functionally graded scaffolds for bone tis-
sue engineering.
Four porosity distribution laws, three loading conditions and three scaffold Young’s moduli
were hypothesized. For each combination of these three variables, the optimal microstructure
geometry was determined. It was shown that all these variables have a critical effect on the
amounts of bone predicted to form within the scaffold pores.
Regarding the porosity distribution law, it was found that designing FGSs with a tri-linear
law allows the largest amounts of bone to be generated (Figs 4–6) compared to bi-linear, linear
and constant laws. In general, the use of porosity distribution laws with increasing complexity
level (i.e. with increasing number of coefficients A
i
) leads the scaffold geometry to be better tai-
lored to the specific boundary and loading conditions acting on the construct thus allowing the
bone formation to be maximized. Increasing the complexity level of a porosity distribution
means, in other words, to include a larger number of design variables and hence, to increase
the probability that the optimizer will find a geometry that allows larger amounts of bone to be
generated.
More critical appears the effect of the loading conditions. For a pure compression loading,
the changes of the pore dimension A are marginal (Figs 4, 7 and 9) and using a FGS allows the
formation of amounts of bone slightly larger than those obtainable with a homogeneous poros-
ity scaffold (Fig 8). For a pure shear loading, instead, FGSs allow to significantly increase the
bone formation compared to a homogeneous porosity scaffolds (Figs 5B, 5D, 5F and 8) and the
pore dimensions change (vs. y) also by more than 20–25% (Figs 7 and 9). This behavior can be
justified with the following argument. In the case of pure shear loading, strains increase as we
move from the loaded towards the clamped region and hence, the stimulus S, that is a function
of the strain, changes in the same manner. In order to maximize the number of elements for
which inequality eq (4) is satisfied, the optimization solver tends to reduce the dimension of
the pores subjected to higher strain and increase that of the pores subjected to lower strain. In
the case of pure compression, instead, (from the macroscopic point of view) the scaffold model
is subjected to an uniaxial stress state (with the only exception of the regions close to the loaded
Optimization of Functionally Graded Scaffolds
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Fig 8. Computed values of iBO
%
for compression (A), shear (B) and mixed (C) load.
doi:10.1371/journal.pone.0146935.g008
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and the clamped surfaces where the stress state becomes tri-axial) and then to a more or less
uniform distribution of the stimulus S, which explains the approximately uniform dimensions
of the pores. The mixed load F
M
leads to an intermediate situation between the pure compres-
sion and the pure shear. Changes of A as well increments of BO
%
are more important than
those predicted in the case of compression force F
V
but less relevant than those computed with
the shear load F
H
(Figs 6–8).
Finally, regarding the scaffold Young’s modulus it appears that the average pore dimension
A increases for increasing values of E (Fig 9). This can be justified with the argument that as
the Young’s modulus increases, the global scaffold stiffness increase too and the optimizer
tends to increase the dimensions of the pores to include larger amounts of bone.
To determine the optimal porosity distribution in FGSs some assumptions were made. First
of all, the temporal variable was neglected. It was assumed that the scaffold pores are occupied
only by granulation tissue, the processes of diffusion of the mesenchymal stem cells and of tis-
sue differentiation were not simulated and the optimization of the porosity distribution was
carried out based on the values of the biophysical stimulus registered at the initial time instant.
Furthermore, the algorithm does not include scaffold resorption potential [25].
Including the time variable would certainly allow to carry out more accurate predictions on
the best porosity distribution but would lead to a dramatic increase of the computational time
thus making the algorithm practically not implementable in a “clinical” context. Other aspects
such as angiogenesis [34–36] and growth factors [37] involved in the process of bone regenera-
tion were not modelled. This model neglects the effect of loads during the initial development
of a tissue on a scaffold, i.e. during the phase in which cell attach to the scaffold surface. The
scaffold surface is a 2D environment while the model utilized in this study is based on volumet-
ric strains. A model to predict the effect of mechanical signals on cells seeded on the surface of
a scaffold has been reported [38]. Another limitation of the model is that a deterministic
approach was adopted to determine the biophysical stimulus S, —on the definition of which
the optimal porosity distribution law is calculated,—which neglects any possible genetic
Fig 9. 3D view of the best geometrical configurations (tri-linear porosity distribution) predicted by the optimization algorithm for the shear loading
condition.
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variability in animal populations. A more general and complete approach would be the proba-
bilistic one and would take into account this variability.
However, despite these limitations, the predictions of the model are consistent with the
results of experimental studies. For instance, the patterns of bony tissue predicted in the case of
a pure compression load, constant porosity distribution, E = 1000 MPa, are consistent with
those of new tissue gen erated in circular matrix channels observed in histological analyses [39].
In vitro, it was found that, bone forms from the channel walls and tends to growth towards the
center of the pore. This same behavior was observed in the numerical model (Fig 10). The grey
Fig 10. Patterns of bone predicted in the case of: (i) compression loading; (ii) scaffold Young’s modulus E = 1000 MPa; (iii) porosity distribution
law: constant. Elements in gray are representative of the regions within the scaffold pores where the algorithm predicts bone formation. Interestingly, the
predicted bony tissue patterns appear consistent with those of new tissue formed in three-dimensional matrix channels observed in an in vitro study [39].
Bone formation starts from the pore walls and propagates towards the pore center.
doi:10.1371/journal.pone.0146935.g010
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elements shown in Fig 10 represent the volumes of the model where the mechano-regulation
model predicts the formation of bone. Furthermore, as demonstrated in previous studies a
minimum pore size of about 100 μm is required to guarantee a successful bone regeneration
process in scaffolds [40]. The pore dimensions predicted by the model are all above this thresh-
old value and well fall within the range of the typical dimensions of the pores of scaffolds for
bone tissue engineering [41]. Other studies report that the rate of bone regeneration in scaffold
is a function of the scaffold mechanical properties [42]. This is also consistent with the predic-
tions of the present model where the amounts of bone BO
%
change for changing values of the
scaffold Young’s modulus (Figs 4–6 and 9).
Conclusions
A mechanobiology-driven optimization algorithm was presented to determine the optimal
porosity distribution in functionally graded scaffolds. The results presented in this paper show
that the loading conditions are pivotal in determining optimal porosity distribution. For a pure
compression loading, it was predicted that the changes of the pore dimension are marginal and
using a FGS allows the formation of amounts of bone slightly larger than those obtainable with
a homogeneous porosity scaffold. For a pure shear loading, instead, FGSs allow to significantly
increase the bone formation compared to a homogeneous porosity scaffold. Increasing pore
dimensions are predicted for increasing values of the scaffold Young’s modulus. Increasing the
number of coefficients that define a porosity distribution law allows to design more performing
scaffolds capable of generating larger amounts of bone.
The model predictions appear reasonably consistent with what is observed in vitro.
Although experimental data is still necessary to properly relate the mechanical/biological envi-
ronment to the scaffold microstructure geometry, this model represents an important step
towards optimizing geometry of functionally grade d scaffolds and/or stimulation regimes
based on mechanobiological criteria.
Author Contributions
Conceived and designed the experiments: AB AEU MF. Wrote the paper: AB. Conceived and
designed the algorithm: AB. Edited the algorithm: AEU MF. Wrote the paper: AB. Edited the
manuscript: AEU MF G. Mori. Supervised the study and the article writing: G. Monno.
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