Journal of Computational Design and Engineering 3 (2016) 385–397
Geometric and mechanical properties evaluation of scaffolds for bone tissue
applications designing by a reaction-diffusion models and manufactured
with a material jetting system
Marco A. Velasco
a,b, Yadira Lancheros
d, Diego A. Garzón-Alvarado
c,na
Servicio Nacional de Aprendizaje SENA, Centro Metalmecánico, GICEMET Research Group, Bogotá, Colombia b
Studies and Applications in Mechanical Engineering Research Group (GEAMEC), Universidad Santo Tomás, Bogotá, Colombia cBiomimetics Laboratory: Group of Mechanobiology of Organs and Tissues, and Numerical Methods and Modeling Research Group (GNUM),
Instituto de Biotecnología (IBUN), Universidad Nacional de Colombia, Bogotá, Colombia
dServicio Nacional de Aprendizaje SENA, Centro de Manufactura Textil y del Cuero, CMTC Research Group, Bogotá, Colombia
Received 25 July 2015; received in revised form 17 June 2016; accepted 26 June 2016 Available online 30 June 2016
Abstract
Scaffolds are essential in bone tissue engineering, as they provide support to cells and growth factors necessary to regenerate tissue. In addition, they meet the mechanical function of the bone while it regenerates. Currently, the multiple methods for designing and manufacturing scaffolds are based on regular structures from a unit cell that repeats in a given domain. However, these methods do not resemble the actual structure of the trabecular bone which may work against osseous tissue regeneration. To explore the design of porous structures with similar mechanical properties to native bone, a geometric generation scheme from a reaction-diffusion model and its manufacturing via a material jetting system is proposed. This article presents the methodology used, the geometric characteristics and the modulus of elasticity of the scaffolds designed and manufactured. The method proposed shows its potential to generate structures that allow to control the basic scaffold properties for bone tissue engineering such as the width of the channels and porosity. The mechanical properties of our scaffolds are similar to trabecular tissue present in vertebrae and tibia bones. Tests on the manufactured scaffolds show that it is necessary to consider the orientation of the object relative to the printing system because the channel geometry, mechanical properties and roughness are heavily influenced by the position of the surface analyzed with respect to the printing axis. A possible line for future work may be the establishment of a set of guidelines to consider the effects of manufacturing processes in designing stages.
& 2016 Society of CAD/CAM Engineers. Publishing Servies by Elsevier. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Bone scaffold; Reaction-diffusion models; Additive manufacturing; Porous structures
1. Introduction
Bones are formed by connective tissue with specialized cells as osteocytes, osteoblasts, and osteoclasts immersed in an extracellular matrix composed mainly by minerals (hydroxyapatite), proteins
(collagen, cytokines, osteonectin, osteopontin, osteocalcin,
osteoinductive proteins, sialoproteins, proteoglycans, phosphopro-teins, and phospholipids) and water. Osseous tissue has not only mechanical but synthetic and metabolic roles. Bones must provide
protection for internal organs, support and define the shape of
mammals and interact with muscles and tendons to generate
movement [1]. The synthetic function includes the production of
blood cells [2] and the metabolic function is to storage calcium,
phosphorus, growth factors and fat[3]. Osseous tissue is prone to
suffer multiple conditions triggered by extreme mechanical loads or
hormonal insufficiencies, among other causes. In those cases,
osseous tissue has a high capacity of healing without scar material
[4]. Medical procedures such as immobilization and surgery
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nCorresponding author.
E-mail addresses:mavelascope@misena.edu.co(M.A. Velasco),
eylancheros@misena.edu.co(Y. Lancheros),
dagarzona@unal.edu.co(D.A. Garzón-Alvarado).
suffice, most of the time, to achieve regeneration. However, about 10% of the cases require the use of additional support such as autografts, allografts orfixation devices to help the healing process
[5]. Whereas these procedures require donors, are expensive or
imply certain risks, bone tissue engineering is necessary.
Scaffolds, cells seeding and growth factors are the main
strategies used in tissue engineering [6]. Considering the
mechanical function of bones, scaffolds are a very important component to successfully maintain, replace and improve bone tissue functions when necessary. Particular mechanical proper-ties that characterize bones are modulus of elasticity, tensile strength, fracture toughness, Poisson's ratio, elongation per-centage, etc. For all these reasons, scaffolds should be as close as possible to the replaced tissue from a mechanical point of view. This is necessary to avoid problems like osteopenia due to the use of bone grafts that are stiffer than the original bones
[7]or new fractures due to low strength.
One of the major objectives of bone tissue engineering is to achieve similar geometries to those of trabecular bone. To date, there are several methodologies for scaffold design based on
regular and irregular structures [8]. The most constructive
approaches are based on regular arrangements: an internal
geometry filled with a periodic distribution of unit cells. Unit
cells are constructed with computer-aided design (CAD) tools using design primitives like cylinders, spheres, cones, blocks
organized in rectangular or radial layouts[9]. The advantages of
such regular porous structures allow for easier modeling, physical simulation and manufacturing. Besides the unit cell approach, there are parameterized models using mathematical functions to generate implicit surfaces with porosity gradients
[10]. It should be noted that those methodologies can be
complemented by optimization techniques to improve the
mechanical strength or permeability[11–16]. Although, periodic
or regular porous structures can be relatively sophisticated, they are still limited to represent the structures present in nature
[8,11]. On the other hand, irregular structures are obtained from
fractal curves or clinical images. For example, space-filling
curves, continuous fractal curves that cover domains like planes or tridimensional spaces can create irregular architectures that are not generally achieved in models made with unit cell
approaches [17]. Furthermore, variable porous structures can
be acquired directly from computed tomography bone scans
[18–20]. Finally, designed structures can be fabricated using
additive manufacturing methods such as SLA, SLS, FDM, 3D
Printing and many others[21].
Although most of the work on bone scaffold modeling is supported on regular porous structures, it is important to consider that some studies show that imposing the same shear stresses might not be adequate for bone scaffold regeneration. Regeneration and remodeling of osseous tissue is not only caused by a high value of stress or strain but also by differences or gradients of those signals in nearby sections of bone tissue [22,23]. This raises the possibility that irregular structures may be better to stimulate tissue regeneration due to less uniform stress distributions than those observed in regular structures. Another possible explanation for the irregular structure visible in trabecular bone is that regular structures have a tendency to
exhibit catastrophic failure opposite to irregular structures[24]. As a result, biomimetic design has been introduced and widely used as an alternative method for irregular porous structure modeling. However, a faithful reproduction of the structures present in nature, in most cases, is not strictly necessary: a simpler approach to the achievement of a biomimetic design to mimic tissues or organs functionality is the global variation of porosity of the different regions according to a natural reference model[8,25].
The reaction-diffusion models proposed by Turing [26]are
some of the best known and widely used mathematical theories to describe processes of patterns formation in biology. They describe geometric patterns generation from at least two substances called morphogens; these substances have the capacity to diffuse in the environment and react with each other. Compared to other models that only consider diffusion processes, the inclusion of the reaction term between morpho-gens allows patterns formation regardless of the initial
distribution of the substances that generate them [27].
Reaction-diffusion models are used to model behavior pattern
formation in living organisms [27,28] and bone formation
processes [29–31]. On the other hand, it is possible that
reaction-diffusion systems can be implemented in numerical methods to provide faster pattern generation and geometry
control than those supported on particle-based methods[32].
This paper explores the modeling of geometric structures by reaction-diffusion models and the geometric and mechanical properties of these structures produced by additive manufac-turing processes. For this purpose, porous structures are obtained from a Schnakenberg reaction-diffusion model. Later, those structures are fabricated using an additive manufacturing system and the capacity of these systems to reproduce the modeled geometry is estimated. Finally, structure stiffness is established by measuring its modulus of elasticity. Results show that different geometrical characteristics of porous structures are possible by varying the parameters of the reaction-diffusion system. Additive manufacturing systems can reproduce complex geometries depending on the geome-trical feature size; and, the modulus of elasticity is determined
by the structure geometry and specific characteristics of the
selected manufacturing process.
2. Materials and methods
The design and fabrication process of a porous structure that can be used as scaffold in bone tissue engineering is illustrated
in Fig. 1. Based on a reaction-diffusion model, a geometric
pattern is generated in a specific domain. In this work, the
domains used were cube-shaped, cylindrical or wedge-shaped. More complex shapes that resemble complete bones or portions of them may be used if required. From the pattern obtained, solid (resembling trabeculae) or void (representing
pores) portions inside the domain are defined. Once this
process is completed, a model of the scaffold geometry is obtained and it can be taken to manufacture using an additive manufacturing process.
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2.1. Geometry generation
A basic reaction-diffusion model is composed by two substances, a and b, which are diffused in space at a different speed and react between themselves. The general form of a reaction-diffusion system can be as shown in equation form: ∂a
∂t ¼ Fða; bÞþDa∇2a
∂b
∂t ¼ Gða; bÞþDb∇2b ð1Þ
Eq.(1)expresses that the concentration of the substance a in a
time given depends on a function F which, in turn, depends on the concentrations of a and b plus the diffusion of a from the
vicinity. The constant Da measures the diffusion speed of a,
while the Laplacian describes the concentration of a in a point with respect to the concentration of the substance in the vicinity. Thus, if the concentration of a in the vicinity of a point is greater than at the point, the Laplacian will be positive and diffusion occurs to the point. On the contrary, if the concentration is lower in the vicinity, the Laplacian is negative and diffusion will occur towards the vicinity. Pattern formation in a reaction-diffusion model occurs when a variation of the initial concentrations of the substances that make the system, which was initially unstable, reach a steady state in which the concentration of reactive substances or morphogens vary in space.
The reaction-diffusion systems represented in Eq.(1) have
multiple variants: one of them is the Schnakenberg model, which is based on a hypothetically simple but chemically feasible reaction for two chemical concentrations, which allows a wide range of possible patterns. Besides, the model
was employed for modeling bone growth[33]:
∂u
∂t ∇
2u ¼γðauþu2vÞ
∂v
∂td∇2v ¼γðbu2vÞ ð2Þ
where u and v are the chemical substances concentrations,∇2u
and d∇2v are the diffusive terms, f ðu; vÞ ¼ ðauþu2vÞ and
gðu; vÞ ¼ ðbu2vÞ reactive terms. On the other hand, γ is a
dimensionless constant and a and b are constant parameters of
the model. The variation of the values of γ, a and b allows
obtaining different porosity forms[28].
For the study of the possible geometric patterns in the space that this model can generate, homogeneous Neumann conditions have been imposed in order to simplify the solution of the model
assuming noflow of morphogen substances outside the domain.
The initial conditions assume small disturbances of 710%
around the steady state of each of the reactive terms f ðus; vsÞ ¼ 0
and gðus; vsÞ ¼ 0, were applied as described in Garzón-Alvarado
et al.[31]. In addition, a variable Tais considered, representing
the time during which the reaction-diffusion process has occurred until a labyrinth pattern emerges.
The analysis of generated geometric patterns will be made in three domains: a cube that has 3 mm in its side represented in a mesh 46,656 nodes and 42,875 cubic elements; a cylinder with a radius of 1.5 mm, 3.0 mm height represented in a mesh of 36,312 nodes and 33,462 elements; and a wedge of minor height of 2.5 mm, major height of 3.5 mm, 4 mm length, and 3 mm width represented in a mesh of 29,791 nodes and 27,000 elements. Mesh elements are hexaedrical 8-node elements. The frontal views of the
meshes are shown inFigs. 2a,3a and4a respectively.
In order to solve the system of equations in the proposed
domains using the finite element method along with the
Newton-Raphson method programmed in FORTRAN. To obtain the porous structure from the previous process, elements
that have a concentration value v above threshold value vtare
selected. They are defined by trial and error depending on the
porosity desired.
Highest concentrations of u are indicated with blue color and higher v concentrations are indicated for the cube, wedge
and cylinder inFigs. 2b,3b and4b respectively. To determine
the porous structures a threshold value of concentration vrefis
taken. vrefvalues were obtained using a trial and error method
as described in[34]in order to obtain porosities (void volume/
domain volume) close to 50%, which is the minimum porosity value recommended for bone scaffold applications. Porosity is Fig. 1. Graphic description of the stages of the proposed process from scaffold design to manufacture.
calculated as Vo/Vf, where V0is the void space volume inside
the Vf domain volume. The mesh elements that have a
concentration value of v greater than vref are considered as
solids. Finally, the resulting mesh of finite elements of the
items selection process is displayed and exported to a STLfile
using the visualization software VisIt (Lawrence Livermore National Laboratory, Livermore, California, USA).
2.2. Scaffold printing
The structure is manufactured using an Object Eden 260 (Stratasys, USA). This system uses a material jetting (Polyjet) process, which uses a mobile jetting head on a horizontal plane to spray drops of photocuring liquid polymers in order to print the workpiece material and the support material. This deposi-tion is done on a tray that moves vertically (z axis) to complete the three-dimensional construction.
The materials used are Fullcure RGD720 (Modulus of
elasticity E ¼ 2000–3000 MPa, Tensile Strength 55–60 MPa)
for the structure material and Fullcure 705 for support material. Although the RGD720 material is not a biomaterial, it has mechanical properties similar to Fullcure MED610 that can be used in the material jetting system and complies with ISO standard ISO 10993-1: 2009. It also has similar mechanical properties of modulus of elasticity compared with PDLLA (E ¼ 1900 MPa), PLGA (E ¼ 2000 MPa) or calcium phosphate
and chitosan-gelatin composites (E ¼ 3940 MPa to
10,880 MPa) [35,36]. Fullcure materials are specific to this
additive manufacturing technology. The fabrication is done at
a resolution of 42mm, 42 mm and 16 mm in x, y and z axis
respectively.
Scaffold fabrication is based on STL files and is done at
different scales (1x, 5x and 10x) regarding the measures of the domains initially proposed for mechanical and geometric evaluation. The jetting head moves by placing and curing Fig. 2. Modeling and printing of a porous structure from a cubic domain. (a) Front view of thefinite element mesh, (b) concentration distributions u and v, (c) Determination of the solid elements (orange) from the reference value, and (d) three-dimensional model.
M.A. Velasco et al. / Journal of Computational Design and Engineering 3 (2016) 385–397 388
through ultraviolet light the photopolymer and the support material that forms superimposed layers as the tray descends vertically. Finally, the material is removed using water and air in a high-pressure water cleaner (waterjet unit). Later on, curing of the workpiece material is not required.
2.3. Verification of mechanical properties and roughness
The measurement of the scaffolds geometric features is carried out using a SEM microscope (Scanning Electron Microscopy JSM-6010LA, JEOL, USA) and Outside Micro-meter 103-138 (MicroMicro-meter 103-138, Mitutoyo, Japan) and the mechanical properties in a material testing machine (H5KT single column materials testing machine, Tinius Olsen, USA). Surface roughness was determined using a surface roughness tester (Surftest SJ-301, Mitutoyo, Japan).
In the case of mechanical properties, the scaffolds were subjected to compression tests according to the standard
ISO604 Plastics – Determination of Compressive Properties,
applying a load of 890 N (approx. 200 lbf). Stress strains were graphed in order to characterize the scaffolds behavior to compression loads and to obtain the variable elasticity module.
3. Results
3.1. Reaction-diffusion model
Morphogens concentrations after a certain reaction time Ta
form labyrinth distributions for each one of the proposed domains. In all cases the model parameters are d ¼ 8,6123,
a ¼ 0,1, γ¼346,3578, b¼0.9. A value of Ta¼2 is used
considering it is the minimum time for pattern occurrence. In
addition, a computational parameter of Δt¼0.01 with a total
number of increments of 2000 is used for solving the proposed equations system in (2). Elements with high concentrations of u will be considered void spaces and elements with high concentrations of v will be considered solid components in order to obtain a porous structure that resembles a porous scaffold for bone tissue engineering applications. The max-imum and minmax-imum values for u and v concentrations are
displayed inTable 1.
In the cubic domain, the morphogen v forms a pattern with
dots and stripes irregularly arranged over the domain (Fig. 2b)
similar to a high density trabecular bone [37]. To define the
elements considered as solid scaffold components a threshold Fig. 3. Modeling and printing of a porous structure from a domain in a wedge shape: (a) Front view of thefinite element mesh, (b) distributions u and v in the wedge domain, (c) determination of the solid elements (orange) from the reference value, and (d) three-dimensional model.
value vref¼0.9008 is defined. Thus, the elements having a
higher concentration than value vrefconforms a geometry made
of plateau and rods components (visible as orange elements in Fig. 2c). Therefore, a structure with channels and interconnected trabeculae is achieved. The mesh is subsequently exported to an
STL file (Fig. 2d). The solid obtained has a volume of
13.53 mm3, a porosity of 57.54% considering the volume of
the cubic domain of 27 mm3, a surface of 259,63 mm2, 70,652
facets and 33,618 nodes.
The morphogen v forms a pattern with dots and stripes
irregularly arranged over the wedge domain (Fig. 3b) similar to
that of the previous cube. To define the elements considered solid
scaffold components, a threshold value vref¼0.8997 is defined.
Geometry with predominant plateau components oriented along
the x axis is obtained (visible as orange elements inFig. 3c). The
solid obtained has a volume of volume 17,56 mm3, a porosity of
48.79% considering the wedge volume of 36 mm3, a surface of
315.06 mm2, 52264 facets and 24145 nodes (Fig. 3d).
In the cylinder domain, the morphogen v forms a pattern
with dots and stripes irregularly arranged over the domain (Fig.
4b) similar to that of previous domains. To define the elements
considered as solid, a threshold value vref¼0.9001 is defined.
Geometry with plateau components is obtained (visible as
orange elements inFig. 4c). The solid obtained has a volume
of volume 10,80 mm3, a porosity of 50.93% considering the
volume of the cylinder of 21.21 mm3, a surface of
204.59 mm2, 55048 facets and 26,143 nodes were obtained
(seeFig. 4d).
Fig. 4. Modeling and printing of a porous structure from a domain in the form of cylinder: (a) Front view of thefinite element mesh, (b) distributions u and v in the cylinder domain, (c) Determination of the solid elements (orange) from the reference value, and (d) three-dimensional model.
Table 1
Concentrations u and v for cubic, cylindrical and wedge domains.
Concentration Cube Cylinder Wedge
u 0.9603–1.044 0.9615–1.044 0.9603–1.044
v 0.08837–0.9179 0.8834–0.9169 0.8745–0.9248
M.A. Velasco et al. / Journal of Computational Design and Engineering 3 (2016) 385–397 390
Additionally, to verify the possibility of controlling the porous structure, a different case over the same cylinder domain is proposed with parameters d ¼ 8.5737, a ¼ 0.1,
γ¼700.4675, b¼0.9 solved in value of Ta¼15 in order to
obtain a Turing pattern following the methodology described
in [38]. A porosity of 50.93% is achieved when a threshold
value vref¼0.9012 is defined. It is observed that channels and
trabeculae are oriented in parallel to the cylinder base as
showed in Fig. 5.
3.2. Geometric properties
Using the models in STL format, different scaffolds were fabricated using the material jetting additive manufacturing system to make an analysis of geometric properties. The cube-shaped scaffold was printed at 1x, 5x and 10x scales to review the ability of the printing system to reproduce details of the designed
geometry (see Fig. 6). Assessment of designed models and parts
obtained is made by visual inspection. A summary of obtained results is showed inTable 2.
Fig. 5. Modeling and printing of a porous oriented structure from a domain in the form of cylinder.
Fig. 6. Comparison between designed geometry and material jetting technology made cubic scaffold geometry at 10x scale. a) and b) Nominal measurements details of trabeculae and channel in the STL model, c) trabecular detail at the corner, d) L shape pore, e) trabecular measurements in a vertex half (compare with STL detail in b).
Table 2
Geometrical properties of cubic manufactured scaffolds at 5x and 10x scales. Scaffold scale External variation (mm) Trabecular oversize (mm) Concave radius (lm) Convex radius (lm) 5x 0.05 on z-axis þ0.10 250 360 10x þ0.10 on x and y axis 230 330
External dimensions vary slightly depending on the surfaces orientation regarding the printing direction. Perpendicular dimensions to the printing direction were on average 0,12 mm greater than the design dimensions, while dimensions in the x axis were 0.05 mm smaller than the nominal dimension.
The measurement of solid elements in the scaffold at 10x scale shows that trabeculae can be up to 0.1 mm wider in the x and y directions inside scaffolds faces, and up to 0.05 mm at the trabeculae close to the vertexes (Fig. 6). This makes the width of the channels smaller than in the designed dimension. For example, when the geometry is printed to a 1x scale, the channels having the width of an element of the domain (0.086 mm) are smaller than the space of 0,1 mm estimated to be occupied by sobrematerial and therefore they are not reproducible (Fig. 8).
It is not possible to obtain right angles in concave and convex details of the geometry. There are rounding radii
among 250mm and 360 mm according to the face orientation in
regard to the z axis and edges proximity. Straight corners have been rounded by effect of systems used for the removal of the support material, which can cause wear on the details with straight angles. Wear causes a decrease in dimensions as can be seen in trabeculae close to the scaffold edge. Rounded
corners can be seen in cubic scaffold at 10x scale (Fig. 6) as
well as in cubic scaffold at 5x scale (Fig. 7).
The designs of STLfiles and printed elements have different
degrees of similarity depending on the printing scale. The general form of geometric features as trabeculae and channels are visible at 5x and 10x printing scales; but, at 1x scale, it is not possible to appreciate the internal structure of the scaffold. It seems that material occupies the whole space of the channels. Therefore, the printed scaffold does not replicate
the details of trabeculae and pores properly (Fig. 8).
Fig. 7. Details offinished surface of material jetting technology made cubic scaffold at 5x scale: (a) Trabecula at the corner of the scaffold in the STL model, (b) Trabecula details at the middle of a vertex in the STL model, (c) Trabecula at the corner of the scaffold in the manufactured scaffold, and (d) Trabecula at the middle of a vertex in the manufactured scaffold.
Fig. 8. Micrograph of material jetting technology made cubic scaffold at 1x scale.
Fig. 9. Detail of the scaffold surface on a surface parallel to the z axis direction.
M.A. Velasco et al. / Journal of Computational Design and Engineering 3 (2016) 385–397 392
3.3. Surface roughness
The surface roughness has a defined orientation and its value
depends on the surface position with respect to the z axis,
which is the printing axis (Fig. 9). Parallel faces to the printing
axis have an average roughness Ra ¼ 4.55μm. The profile
surface is not uniform and has peaks and valleys making the Fig. 10. Surface textures and roughness profiles of 10x scaffold: (a) surface parallel to the z axis direction, (b) surface perpendicular to the z axis direction.
Table 3
Measured roughness at surfaces of 10x scaffold.
Surface Ra (lm) Rz (lm)
Parallel to the z axis 4.55 30.28
Perpendicular to the z axis 0.40 4.02
Fig. 11. Stress - Strain curves by applying axial compression load on the on x, y and z axes of the cubic scaffold at 10x scale.
Table 4
Modulus of elasticity on x, y and z axes of cubic scaffold with random pores at 10x scale.
Scaffold Ex(MPa) Ey(MPa) Ez(MPa)
Random pores at 5x scale 74 69 28
Fig. 12. Stress - Strain curves by applying axial compression load on the z axis of the cylindrical scaffolds with random and oriented pores at 5x and 10x scales.
maximum height of the roughness profile Rz raise to 30.28 mm
(Fig. 10a). These faces have grooves clearly oriented in the
direction where material layers were deposited. On the other hand, perpendicular faces to the printing axis show a smoother
surface with an average roughness Ra of 0.40μm although
there are crests that cause the total height of valley-to-peak
roughness to increase to Rz ¼ 4.02mm (Fig. 10b). A summary
of measured roughness at 10x scaffold is inTable 3.
3.4. Mechanical properties
The mechanical behavior evaluation of the designed scaf-folds is made by the following assessments: test compression of the cubic scaffold printed at 5x scale in each of its three axes to evaluate the anisotropy of the element, test compression of cylindrical scaffolds at 5x scale and 10x scale to check the size effect on the properties and compression tests of cylindrical scaffold with oriented channels at 5x scale to evaluate the effect of pore geometry.
The cubic solid at 5x scale exhibits an anisotropic elasto-plastic behavior with a strain hardening stage at the end of the stress-strain curve. Modulus of elasticity in the linear portion
of the stress-strain curves corresponding to Ex¼74 MPa,
Ey¼69 MPa and Ez¼28 MPa (Fig. 11 and Table 4) in x, y
and z axis respectively. It can be seen that modulus of elasticity in the z-axis is close to 50% of that measured at the other two axes.
A size effect in the mechanical properties of cylindrical scaffolds is evident. There is a non-linear behavior in scaffold printed at 5x scale, widely differing from the almost linear behavior exhibited for the same geometry printed at 10x scale.
Besides, an increase in the modulus of elasticity from Ez¼84
MPa to Ez¼119 MPa on each scaffold respectively (Fig. 12
and Table 5) is visible.
Pore size and orientation change the mechanical behavior of cylindrical scaffolds. Oriented pores scaffold exhibit a
mod-ulus of elasticity of Ez¼12 MPa at the beginning of the
stress-strain curve that is lower than the modulus of elasticity of scaffold at 5x scale, even though they have the same external size and similar porosity. A strain hardening behavior is also evident (Fig. 12).
4. Discussion
Results show that the reaction-diffusion systems are able to form irregular structures where features such as porosity, channel and trabeculae widths can be manipulated. Shape control is achieved by varying reaction-diffusion system variables such as
d, a, γ, b and Ta. Additionally, an analysis of the stability
conditions of the reaction-diffusion system allows periodic
structures as seen in the scaffold ofFig. 5. On the other hand,
the porosity can be controlled selecting a threshold value Vref. The
patterns obtained are consistent with previous work on reaction-diffusion systems[39,40].
The studied reaction-diffusion system generates an intercon-nected network of channels and trabeculae. Although the components of the structure are dispersed in the domain trabeculae and channels, width is relatively uniform. This may
be due to local activation– lateral inhibition principle proposed
by Gierer and Meinhardt[41]. From this principle it is possible
to generate a periodic distribution of morphogens forming an
arrangement that resembles dots or lines, thosefigures combined
can form different configurations. These patterns may be
irregular as described in [42] or periodic minimal surfaces
similar to those studied by De Wit et al. [43]. In bone tissue
engineering, controlling conditions of the reaction between substances has been used to obtain different pore sizes as seen
in [44] and minimum periodic surfaces for the modeling of
scaffolds[13,45]. This suggests that our study can support these lines of work by allowing to understand the processes and parameters that help to obtain certain geometries.
In addition to reaction-diffusion systems parameters, the shape of porous structures is determined by the size of mesh elements. A smaller element size can result in a geometric pattern that resembles more closely the visible distribution pattern of morphogens before the application of threshold
value vref to define solid elements. Loss of geometric features
like curved details appear in all domains due to a voxelization
effect when the threshold value is applied to define the solid
portion of the initial volume. The smaller size of the elements decreases the voxelization effect but has the disadvantage of greater computational cost. To improve this aspect, smoothing surface from voxels can be obtained using techniques such as
marching cubes [46] and additional work must be made in
order to determine the appropriate size of the elements of the mesh to achieve a balance between geometry resolution and computational cost to solve the reaction-diffusion system. It is also important to study how the element size affects the size of the trabeculae or channels generated. For now, it is suggested that the element size should not exceed the recommended
maximum pore size, close to 800mm[34,47].
Pores may have an orientation according to the geometry of the domain boundary. This allows to generate wider pores on scaffold surface as evidenced in the cylindrical scaffold of
Fig. 5. A porosity gradient can prevent obstruction of the
scaffold surface by tissue formation only in the periphery as
described in[10]. Moreover, the proper definition of pore size
and shape serves to develop scaffolds to fit specific patient
needs[48].
Considering the manufacturing process, it is appreciated that the material jetting system adequately reproduces geometry details. This can be attributed to the printing system used in
this work, which allows a high resolution of up to 16mm
compared to 100mm of material extrusion (FDM) process or
binder jetting (3DP) and powder bed fusion (SLS) 80mm
Table 5
Modulus of elasticity on z axis of scaffold with random pores at 5x scale.
Scaffold Ez(MPa)
Random pores H ¼ 15 mm D ¼ 15 mm (5x scale) 74
Oriented pores H ¼ 15 mm D ¼ 15 mm (5x scale) 69 Random pores H ¼ 30 mm D ¼ 30 mm (10x scale) 28
M.A. Velasco et al. / Journal of Computational Design and Engineering 3 (2016) 385–397 394
manufacturing process resolution [49]. Similar to these sys-tems, variations in the geometry and mechanical properties are
influenced by the orientation of the printing axis [14,50].
Dimensional errors, which can be up to 0.1 mm, are consistent with other additive manufacturing systems such as
SLS [14,19] but the intricate geometry obtained by the
reaction-diffusion system complicates the deletion of support material. This issue limits the minimum pore size which can be used for geometry design. It was also evident that the methods for support material removal cause wear, especially at the
edges of the scaffold as shown inFig. 7. This raises the need
for studies on how to design the scaffold considering not only the effects of orientation and size of the channels, but the effects of cleaning the support material.
The surface roughness of a scaffold helps increase the surface where cells can be attached, thus promoting prolifera-tion. It is estimated that the roughness must be similar to the
size of osteoblasts (2–8 mm) in order to influence attachment
and growth rates [51]. Roughness of Ra ¼ 4.55mm present in
surfaces parallel to the printing axis is similar to the roughness
of scaffolds surfaces with other polymers[52,53]. This
rough-ness could facilitate cell adhesion and differentiation of
mesenchymal cells into osteoblasts [54]. On the other hand,
in the case studied, the roughness is the feature most affected by the orientation of the surfaces relative to the axis printing. This phenomenon could be used to control cell growth, as the
preferential orientation of roughness profile can influence cell
proliferation direction and osteoblasts depend on the roughness
of the surface for proper adhesion[55,56].
The results of Fig. 11 show a scaffold with transversely
isotropic or orthotropic behavior. Similar modules elasticity in the x and y axis were obtained, which differ from those obtained in the z axis. This may be due to the material orientation by layer printing process or that mechanical properties of the scaffolds are affected by pores orientation. The mechanical properties obtained are similar to those of scaffolds based on regular structures
[57,58], scaffolds designed from periodic minimal surfaces [13]
or bone tissue with properties conditioned by the orientation of the collagenfibers that compose it[59,60]. Furthermore, the strain
hardening observed in the stress-strain curves ofFigs. 11and12
may prevent apoptosis by high stress or deformation [61] or
prevent deformations so high that may promote the formation of cartilage instead of bone[27,28].
Orthotropic materials could be useful because mechanical properties affect tissue regeneration processes in two ways: The tissue will grow faster in areas with higher mechanical
stimulus that will correspond to greater stress or strain [62].
Moreover, stem cells differentiate from osteoblasts in more rigid areas and chondrocytes in less rigid areas[63,64]. That is, bone regeneration tends to follow the direction of apparent maximum stiffness; therefore, if pores are oriented it is possible to modulate the rate and type of created tissue. Computer simulations and experimental tests must be made to quantify the contribution of design and printing system factors in the anisotropy of the observed mechanical proper-ties. Subsequent work may include the effect on mechanical properties due to geometric features and the distribution of
different materials blends taking advantage of material jetting manufacturing system.
The pore's shape and orientation variation modifies the
scaffold mechanical properties. These changes alter the cross-sectional area and length of load-bearing trabeculae. In addition, there is a change in orientation of the trabeculae regarding the applied load. For example, circular scaffolds of 15 mm height show different properties. The oriented-pores
scaffold (Fig. 5) has a greater trabeculae length than the
non-oriented pores scaffold (Fig. 4d). The configuration of
elon-gated and oriented pores causes an increase in the deformation
of the scaffold as seen by comparing different stress–strain
curves in Fig. 12.
The print scale significantly affects geometry and
mechan-ical properties of the printed parts. The fidelity with which
manufactured scaffolds reproduce details of designed geome-tries depends on the ratio between the width of the channels and dimensional error due to the printing method. Besides, an increase in print scale tends to cause a linear behavior in the
stress-strain curve as shown inFig. 12.
Finally, results of modulus of elasticity for the manufactured scaffolds are between 10 and 1000 MPa reported for trabecular
bone[65]. More precisely, the modulus of elasticity is similar
to those available for trabecular tissue in tibia and vertebrae
according to the data available on Lakatos[60], so they could
be used for scaffolds to replace trabecular bone tissue in those bones.
5. Conclusion
This paper presents a method for generating irregular porous geometries that could be used in scaffold design for bone tissue engineering applications and an evaluation of geometric and mechanical properties after they were manufactured using a material jetting manufacturing system. The difference between the STL model geometry and the manufactured might be due to characteristics inherent to the manufacturing processes used such as resolution, printing scale and cleaning methods to remove support material. It is also observed that the orientation of the surfaces with respect to the direction of deposition is a very important factor affecting the mechanical and geometric proper-ties of the produced parts. All the above factors cause an anisotropic behavior that can be taken into account in a more realistic modeling of the scaffold geometric features and physical properties.
Conflict of interest statement
The authors have no conflict of interest to report.
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