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Computational Mechanical Characterization of Geometrically Transformed Schwarz P Lattice Tissue Scaffolds Fabricated via Two Photon Polymerization (2PP)

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Schwarz P unit cell-based tissue scaffolds comprised of poly(D,L-lactide-co- ε -caprolactone)(PLCL) fabricated via the additive manufacturing technique, two-photon polymerisation (2 P P) were found to undergo geometrical transformations from the original input design. A Schwarz P unit cell surface geometry CAD model was reconstructed to take into account the geometrical transformations through CAD modeling techniques using measurements obtained from an image-based averaging technique before its implementation for micromechanical analysis. Effective modulus results obtained from computational mechanical characterization via micromechanical analysis of the reconstructed unit cell assigned with the same material model making up the fabricated scaffolds demonstrated excellent agreement with a small margin of error at 6.94% from the experimental mean modulus (0.69 ± 0.29 MPa). The possible sources for the occurrence of geometrical transformations are discussed in this paper. The inter-relationships between different dimensional parameters making up the Schwarz P architecture and resulting effective modulus are also assessed and discussed. With the ability to accommodate the geometrical transformations, maintain efficiency in terms of time and computational resources, micromechanical analysis has the potential to be implemented in tissue scaffolds with a periodic microstructure as well as other structures outside the field of tissue engineering in general.
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Additive Manufacturing
journal homepage: www.elsevier.com/locate/addma
Computational mechanical characterization of geometrically transformed
Schwarz P lattice tissue scaffolds fabricated via two photon polymerization
(2PP)
Adi Z. Zabidi
a,
, Shuguang Li
b
, Reda M. Felfel
a,d
, Kathryn G. Thomas
a
, David M. Grant
a
,
Donal McNally
c
, Colin Scotchford
a
a
Advanced Materials Research Group, Faculty of Engineering, University of Nottingham, United Kingdom
b
Composites Research Group, Faculty of Engineering, University of Nottingham, United Kingdom
c
Bioengineering Research Group, Faculty of Engineering, University of Nottingham, United Kingdom
d
Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
ARTICLE INFO
Keywords:
Tissue scaffold
Schwarz P TPMS structure
Two-photon polymerization (2PP)
Micromechanical analysis
Beam bending
Column buckling
ABSTRACT
Schwarz P unit cell-based tissue scaffolds comprised of poly(D,L-lactide-co-
-caprolactone)(PLCL) fabricated via
the additive manufacturing technique, two-photon polymerisation (2PP) were found to undergo geometrical
transformations from the original input design. A Schwarz P unit cell surface geometry CAD model was re-
constructed to take into account the geometrical transformations through CAD modeling techniques using
measurements obtained from an image-based averaging technique before its implementation for micro-
mechanical analysis. Effective modulus results obtained from computational mechanical characterization via
micromechanical analysis of the reconstructed unit cell assigned with the same material model making up the
fabricated scaffolds demonstrated excellent agreement with a small margin of error at 6.94% from the experi-
mental mean modulus (0.69
±
0.29 MPa). The possible sources for the occurrence of geometrical transformations
are discussed in this paper. The inter-relationships between different dimensional parameters making up the
Schwarz P architecture and resulting effective modulus are also assessed and discussed. With the ability to
accommodate the geometrical transformations, maintain efficiency in terms of time and computational re-
sources, micromechanical analysis has the potential to be implemented in tissue scaffolds with a periodic mi-
crostructure as well as other structures outside the field of tissue engineering in general.
1. Introduction
Technological advancements over recent years have enabled the
implementation of computer aided design (CAD) beyond conventional
applications demonstrated in automotive, civil and aeronautical en-
gineering. With developments in imaging technologies as well as re-
verse engineering techniques, CAD can now be implemented in the
realms of biomedical engineering with applications in clinical medicine
through to the manufacturing of patient specific customized implants in
a new field known as computer aided tissue engineering (CATE) [1].
Implementation of CAD in CATE reduces experimental testing stages
and shortens the duration of the tissue scaffold design process through
the use of computer techniques based on mechanical design [2]. By
pairing the above with additive manufacturing techniques, tissue scaf-
folds can be easily fabricated and reproduced while also allowing
precise control over scaffold micro-structural parameters such as pore
size, shape and interconnectivity [3].
The implementation of two-photon polymerization (2PP) has been
widespread with applications found in photonic crystals, microfluidic
devices, biomedical science, micro-optics, dielectrics and metamaterials
[4]. Unlike traditional 3D prototyping techniques such as stereo-
lithography, inkjet printing and laser sintering, 2PP is able to fabricate
3-dimensional structures with feature sizes smaller than the diffraction
limit of an applied laser length at submicron resolutions [5–7]. Fem-
tosecond lasers used has been reported to be powerful for advanced
materials processing at a micro- and nano- scale level compared to
traditional laser processing techniques with its attribution of ultrashort
pulse widths and extremely high peak intensities [8]. Through 2PP,
CAD model files can be directly transferred to the structure in its
physical form out of photopolymer in tens of minutes. The whole
https://doi.org/10.1016/j.addma.2018.11.021
Received 14 September 2018; Received in revised form 6 November 2018; Accepted 17 November 2018
Corresponding author.
E-mail address: eaxaz@nottingham.ac.uk (A.Z. Zabidi).
Additive Manufacturing 25 (2019) 399–411
Available online 22 November 2018
2214-8604/ © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/BY/4.0/).
T
fabrication process of a structure consists of a few steps and can be
performed under one hour. Both processes mentioned can be repeated
immediately as required and/or if parameters of the structure require
modifications before undergoing the fabrication process again thus
demonstrating the repeatability and reproducibility of 2PP [9]
The need for experimental evaluation of mechanical and/or biolo-
gical responses can be reduced by implementing mathematical models.
The mathematical models can be implemented in the form of finite
element simulations with tissue scaffold CAD models which thus en-
abled studies to evaluate their mechanical properties and biomimetic
properties [2,10]. One such model, micromechanical analysis involves
reducing a tissue scaffold consisting of periodic microstructures to a
representative unit cell through derivations of boundary conditions
made based on geometrical symmetries.
Triply periodic minimal surfaces (TPMS) topologies have found a
place in various applications such as heat transfer, fluid permeability
and acoustic attenuation [11–13]. The architecture of TPMS surfaces
exhibit properties which make them ideal in systems that can be im-
proved upon through the implementation of fluid dynamics [14]. In the
field of tissue engineering, TPMS has particularly been implemented in
applications for the design of biomorphic scaffolds due to their
smoothly curved surfaces which provide a feasible environment for
recuperation and regeneration of damaged tissue cells [15,16]. Adding
to this, each TPMS structure has no self-intersecting or enfolded sur-
faces which enable them to be patterned into a 3-dimensional space
[17]. Previous literature has investigated the use of nine different TPMS
architecture for tissue scaffold design [18]. The Schwarz P (Primitive)
TPMS structure makes one of them [19,20]. The Schwarz P structure is
characterized to exhibit a high load bearing capacity due to its inherent
strain distributions [21]. Investigations characterizing the mechanical
properties of a Schwarz P structure through compressive loading has
determined optimal stress distributions compared to other TPMS
structures [22]. In addition, the Schwarz P structure has also been
characterized to have the largest fluid permeability compared to other
TPMS structures [23]. In due regards to this, tissue scaffolds im-
plementing the Schwarz P architecture has the potential to fulfill bio-
logical design criteria in terms of transport properties. With inherent
circular cross-sectional surfaces, the Schwarz P architecture makes an
ideal pore morphology for scaffold fabrication [16].
This study investigates the computational mechanical character-
ization of Schwarz P unit cell-based tissue scaffolds fabricated via two-
photon polymerization (2PP) in the context of geometrical transfor-
mations. These tissue scaffolds have been fabricated from methacry-
lated poly(D,L-lactide-co-
-caprolactone) (PLCL) copolymer with a
lactide (L) to caprolactone (C) molar ratio of 16:4 i.e. LCM3, via two-
photon polymerisation (2PP) as part of the European Union Research
and Innovation 7
th
Framework Program under grant agreement no.
2633 63 (INNOVABONE) with TETRA-Society for Sensoric, Robotics
and Automation GmbH (Ilmenau, Germany) using an input stereo-
lithography file of a Schwarz P unit cell model. Previous investigations
performed under the INNOVABONE project has found geometrical
transformations between the intended original Schwarz P unit cell de-
sign and the unit cells comprising the physical fabricated tissue scaf-
folds. This study seeks to justify the implementation of micro-
mechanical analysis as a reliable approach to computationally
characterize the mechanical properties of Schwarz P unit cell-based
tissue scaffolds despite the geometrical transformations that have oc-
curred. Validating the data against experimental results and subse-
quently justifying the approach would emphasize the flexibility of mi-
cromechanical analysis in accommodating potential occurrences of
geometrical transformation as a result of fabrication via additive
manufacturing. In cases where the geometrical transformations can be
predicted, micromechanical analysis can be simply employed to char-
acterize resulting mechanical properties of a tissue scaffold with a
periodic microstructure. This would effectively shorten mechanical
testing stages to make time for biological characterization if the me-
chanical properties were found to be desirable.
Only a few studies have been performed in regards to computational
mechanical characterization of tissue scaffolds based on the Schwarz P
TPMS architecture. Two such studies, performed by Shin et al (2012)
considered a phase-field model; and a more recent study by Huang et al
which considered a multiscale modeling approach [22,24]. This study
investigates Schwarz P unit cell-based tissue scaffolds in the context of
their fabrication via 2PP and computational mechanical characteriza-
tion via micromechanical analysis.
2. Materials and methods
2.1. Scaffold design and fabrication
A Schwarz P unit cell packaged into a stereolithography (.STL) file
was used as an input to fabricate its formation into a lattice to form
tissue scaffolds via two-photon polymerization (2PP). Limitations of the
file format for use in computational models required the. STL file to be
‘reverse engineered’ into a useable surface geometry CAD model.
Through Altair Hyperworks 14.0 (Altair Engineering Inc., Michigan,
USA), surfaces were manually created by utilizing vertices between
different triangles comprising the triangulations of the. STL model. A
comparison of the. STL file and the surface geometry CAD model is
shown in Fig. 1.
2PP involves the initiation of femtosecond laser pulses on photo-
sensitive or photoresistive materials resulting in two-photon absorption
and subsequent material polymerization. Schwarz P unit cell-based
lattice tissue scaffolds have been fabricated using materials based on
methacrylated poly(D,L-lactide-co-
-caprolactone) (PLCL) copolymer
with a lactide (L) to caprolactone (C) molar ratio of 16:4 i.e. LCM3.
Fig. 1. Surface geometry CAD model (right) of a Schwarz P unit cell after 'reverse engineering' its stereolithography model (left).
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
400
Syntheses of the liquid precursor making up the photoresist for scaffold
material considered have been detailed in previous literature [21].
A femtosecond laser source with 800 nm wavelength pulsating at
140 fs at 80 MHz repetition rate (VISION II, Coherent, Scotland) was
used as part of the 2PP apparatus (M3DL, LZH Hannover, Germany).
The focus of the laser beam was equipped with an x63 objective lens
and a numerical aperture (NA) of 0.75 (LD-Plan-NEOFLUAR, Zeiss,
Germany). During laser pulses into the photoresist, a tri-axial posi-
tioning stage dynamically moves in tolerance of nanometers according
to the .STL model of the Schwarz P unit cell with the support of a
galvano-scanner (Aerotech, USA). The result is an array of 4
×
9
×
4
unit cells in the x, y and z directions respectively [21]. A graphical
schematic of the 2PP process is shown in Fig. 2. An SEM micrograph of
the fabricated tissue scaffold is shown in Fig. 3.
2.2. Scaffold experimental mechanical characterization
Mechanical characterization of fabricated Schwarz P unit cell-based
lattice tissue scaffolds involved compression experiments performed
using a Hounsfield testing machine. Scaffold samples were compressed
to 20% strain from its original height at a crosshead speed of 0.5 mm
min
−1
with a 5 N load cell. The compression was restricted to 20%
strain to prevent permanent deformation of the scaffolds and allow for
full recovery. The stiffness i.e. compressive modulus, representative of
the mechanical property was determined based on the equation of a
linear region formed at low strain values of the stress-strain curve ob-
tained from the compression experiment.
2.3. Scanning Electron microscope (SEM) microscopy
2.3.1. Scanning procedure
Scaffold samples that had not undergone compression were coated
with platinum using a Polaron SC7640 coater (Quorum Technologies,
UK) at 2.2 kV for 90 s A JEOL JSM-6490LV SEM microscope was then
used to image the top and side surfaces of the samples. Each view
perspective i.e. the top and two sides of the scaffold were assigned a
specific directional plane axis. This allows the nomination of a di-
mensional parameter which will be referenced for a particular mea-
surement for each directional axis on a particular plane axes i.e. hor-
izontal and vertical. Due to the nanoscale size and soft nature of the
samples, it was not possible to make cross-sectional cuts without any
accidental perturbations to the original structure. Therefore, measure-
ments were restricted to observable unit cells on the external sides of
the sample.
2.3.2. Measurement analysis
SEM micrographs obtained were implemented for use in an aver-
aging technique to obtain a micrograph containing an average of all
observable unit cells with respect to different view perspectives i.e.
plane axes. Considering a SEM micrograph taken at a respective plane
axes, each observable unit cell were ‘cut’ through ImageJ (National
Institute of Health, USA) and saved as a separate graphic file without
any changes in resolution. A collection of unit cell micrographs was
thus obtained respective of the plane view considered. A custom Python
algorithm was used to transpose the center point of each unit cell mi-
crograph (defined by the center point of the internal pore diameter) on
a blank canvas to standardize the different image dimensions. A sepa-
rate custom Python algorithm was then used to average the unit cell
micrographs with black backgrounds and an averaged unit cell SEM
micrograph was subsequently obtained. Dimensional parameters which
include the height, width, length and wall thickness were measured
using available tools in ImageJ based on the nominations made during
the scanning procedure. A complete guideline in performing the SEM-
based averaging has been appended in Supplementary Material S.1.
2.4. Schwarz P unit cell CAD model reconstruction
CAD modeling techniques were applied to reconstruct the Schwarz
P unit cell model based on measurements of dimensional parameters
obtained from the averaged unit cell SEM micrograph. This gives a unit
cell which approximately represents an average of observable unit cells
comprising the fabricated tissue scaffold while also taking into account
geometrical transformations. Implementing a computational approach
for mechanical characterization on the reconstructed unit cell model
should thus give a more accurate representation of unit cells comprising
the fabricated tissue scaffolds and subsequently allow a more faithful
comparison to experimental mechanical characterization.
Reconstruction was generally performed on the Schwarz P unit cell
surface geometry CAD model through available tools in Altair
Hyperworks (Altair Engineering Inc., Michigan, USA).
Fig. 2. Schematic presentation of the two-photon polymerization (2PP) fabri-
cation process of Schwarz P unit cell lattice tissue scaffolds. A positioning stage
moves in the x and y direction while the focus beam translates in the z direc-
tion. Polymerization of the liquid precursor occurs in accordance to the com-
puter model i.e. STL model of the Schwarz P unit cell with the support of a
galvano-scanner. The result of the 2PP fabrication process in a 4 × 9 × 4 unit
cell array in the x, y and z direction respectively [21].
Fig. 3. SEM micrograph of a fabricated physical Schwarz P unit cell-based
tissue scaffold.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
401
2.5. Micromechanical analysis
Micromechanical analysis has its foundation in idealizing a struc-
ture consisting of a periodic microstructure as a crystal packing system.
Geometrical symmetries can be subsequently exploited to determine a
representative unit cell i.e. Voronoi cell. This would thus reduce me-
chanical characterization of a tissue scaffold with a periodic unit cell
microstructure to the analysis of a single representative unit cell
through derivations of boundary conditions. Macroscopic loads in terms
of stresses or strains can be applied along with the boundary conditions
before evaluation of the effective properties of the structure in question.
2.5.1. Selection for crystal packing system & representative unit cell
Several typical packing systems have been examined in previous
literature [25]. The fabrication of scaffolds considered in this study
involved a simple 3-dimensional translation of the Schwarz P unit cell
model. Based on the aforementioned literature in regards to the above
and taking into account the uniform height, width and length inherent
in the Schwarz P architecture, the tissue scaffold can be idealized as a
‘simple cubic packing’ system with the Schwarz P unit cell themselves
the representative ‘cubic’ unit cell. However, in the context of
geometrical transformations, unit cells comprising the fabricated tissue
scaffold consists of non-uniform height, width and length. Therefore, a
cuboidal variation of the ‘simple cubic packing’ system i.e. ‘simple cu-
boidal packing’ system with the reconstructed unit cell model as the
representative ‘cuboidal’ unit cell is more appropriate for idealization.
2.5.2. Derivations for boundary conditions
The reconstructed unit cell represented as a ‘cuboid’ in a ‘simple
cuboidal packing’ system is shown in Fig. 4.
Through translational symmetries, the application of macroscopic
stresses or strains on a single unit cell is applied identically to other unit
cells as representative images of the unit cell being considered. With
this, the macroscopic strains and the relative displacements at a parti-
cular point in a unit cell is mathematically related to the image of that
point in another unit cell in accordance to Eq. (1).
= + +
= +
=
u u x x y y z z
v v y y z z
w w z z
( ) ( ) ( )
( ) ( )
( )
xxy xz
yyz
z
0 0 0
0 0
0
(1)
for which,
Fig. 4. (a) Example of a unit cell with non-uniform height,
width and length i.e. cuboidal dimensions fitting into a cuboid
as a representative unit cell; (b) Simple cuboid representing
the reconstructed unit cell model taking height, width and
length measurements obtained from measurement analyes
made from the averaged unit cell SEM micrograph (respective
of the plane axes). Translating the cuboid i.e. reconstructed
unit cell into a 3-dimensional array gives a more realistic re-
presentation of fabricated tissue scaffold as shown in Fig. 3;
(c) Simple cuboidal packing system consisting of an array of
cuboids/reconstructed unit cells translated in a 3-dimensional
direction.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
402
Constraining the displacements where
at any arbi-
trary point in a unit cell eliminates rigid body translations in a 3-di-
mensional direction and thus allow Eq. (1) to be obtained. Rotations of
the x-axis about the y-axis and the z-axis and the rotations of the y-axis
about the x-axis are constrained such that
= = = 0
w
x
v
x
w
y
at
= = =x y z 0
. With the above, Eq. (1) can then be implemented further
to derive the necessary displacement boundary conditions in conjunc-
tion to a selected packing system.
Consider point Pin a representative ‘cuboidal’ unit cell and its
image,
P
in another representative unit cell translated in a 3-dimen-
sional direction as presented in Fig. 4(c). Point
P
can be determined by
translational symmetry transformations as shown in Eq. (2).
= + + +x y z x ia y jb z kc( , , ) ( 2 , 2 , 2 )
(2)
for which
i
,
j
and
k
are the number of representative cuboidal unit cells
in which
P
is translated from
P
in the
x
,
y
and
z
directions respectively.
The faces of a representative cuboidal unit cell can thus be paired by
transformations tabulated in Table 1. Substituting the equations es-
tablished in Table 1 into Equation (1), the displacement boundary
conditions for each of faces is tabulated in Table 2.
The displacement boundary conditions tabulated in Table 2 are ap-
plied to nodes of the faces of the discretized reconstructed unit cell model
wherein the notations
|y z,
,
|x z,
and
|x y,
indicate common directional axes of
tessellated faces between each pair. Since no two faces share a common
edge in a Schwarz P unit cell and the reconstructed unit cell mode, no
further boundary conditions need to be established in regards to this.
Therefore, only the displacement boundary conditions for the faces are
required for implementation of micromechanical analysis. Implementation
of the displacement boundary conditions along with applied macroscopic
stress would give results in terms of average strains and average shear
strains. Based on the results, the effective properties can subsequently be
determined in accordance to previous literature [25].
2.5.3. Validation method for correct implementation of displacement
boundary conditions
Correct implementation can be assessed through the results for average
strain and average shear strain results. Tabulating the results for average
strain values should give symmetric results along the diagonal with differing
values in the diagonal. The same also applies for tabulated results of average
shear strain with the exception that the values lying outside the diagonal can
be approximated to zero due to their small values. Tabulated representation
of the above is appended in Appendix A (Tables A1 and Table A2).
2.5.4. Reconstructed unit cell model discretization and material model
assignment
The reconstructed unit cell model was discretized into second-order
tetrahedral, C3D10 elements. With displacement boundary conditions
being established between pairs of faces, discretization of a pair of faces
lying along a common directional axes must be identical. This was ob-
tained through tools available in Altair Hyperworks. The reconstructed
unit cell model was assumed to take homogenous, isotropic and linearly
elastic material properties of LCM3. The lower limit (4.39 MPa), averaged
(5.74 MPa) and the upper limit (6.65 MPa) modulus values of LCM3 was
obtained from compression experiments of bulk LCM3 cylinders while the
Poisson’s ratio was taken to be 0.3. The bulk material data has been ap-
pended as Supplementary material in Supplementary Material S.2.
3. Results
3.1. Experimental characterisation of tissue scaffold
Graphical presentation of stress-strain data for all five specimens
obtained from compression experiments are presented in Fig. 5.
Based on consideration of the linear region (lying immediately after
the ‘toe region’), the stiffness i.e. modulus of fabricated Schwarz P unit
cell-based tissue scaffolds were averaged at 0.69
±
0.29 MPa with an
obtainable maximum of 1.14 MPa and a minimum of 0.40 MPa.
Table 1
Equations defining the pairing between opposite faces of a representative cu-
boidal unit cell. For cubic unit cells in a simple cubic packing system, the values
of a, b and c are equivalent i.e. a = b = c.
Faces A and B
=x a
for face A
=x a
for face B
= = =i j k1; 0
Faces C and D
=y b
for face C
=y b
for face D
= = =j i k1; 0
Faces E and F
=z c
for face E
=z c
for face F
= = =k i j1; 0
Table 2
Displacement boundary conditions for simple cuboidal packing system. Similar
to a simple cubic packing system with a cubic representative unit cell, 3 dif-
ferent pairs of faces are obtained from 6 different faces. In the case of a simple
cubic packing system, the variables a, b and c are equivalent.
Pair faces Displacement boundary conditions
A and B
=u u a| | 2
A B y z x
,0
=v v| | 0
A B y z,
=w w| | 0
A B y z,
C and D
=u u b| | 2
C D x z xy
,0
=v v b| | 2
C D x z y
,0
=w w| | 0
C D x z,
E and F
=u u c| | 2
C D x z xz
,0
=v v c| | 2
C D x z yz
,0
=w w c| | 2
C D x z z
,0
Fig. 5. Stress-strain curves for all 5 specimens of fabricated Schwarz P unit cell-
based tissue scaffolds obtained from compression experiments. Compression
was only performed at a maximum 20% strain from its original height to pre-
vent permanent deformation of scaffolds.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
403
3.2. SEM measurement analysis & reconstructed unit cell model
With the SEM measurement analysis performed, it can be confirmed
that geometrical transformations have occurred between the input
Schwarz P unit cell design and the unit cells comprising the fabricated
tissue scaffold. More specifically, the geometrical transformations that
have occurred are in the form of shrinkage. Measurements also showed
non-uniformity between height, width and length along with increased
wall thickness compared to the original design. A comparison of mea-
surements for the parameters mentioned between the original Schwarz
P unit cell design and the unit cell averaged from the fabricated tissue
scaffold is shown in Tables 3 and 4.
To take into account the geometrical transformations, the mean
values for height, length and wall thickness measurements were used to
reconstruct the Schwarz P unit cell surface geometry CAD model
through the software Altair Hyperworks. This gives a unit cell that
approximately represents the average of unit cells comprising the fab-
ricated tissue scaffold. It shall be noted here that the reconstruction was
performed with the assumption that the average unit cell is of quarter
symmetry to allow the implementation of micromechanical analysis.
The surface geometry CAD model of the reconstructed unit cell model is
shown in Fig. 6.
3.3. Micromechanical analysis
Application of micromechanical analysis in the context of a linear
material model assigned to the reconstructed unit cell (model A) gave
results in terms of average strain and average shear strains. The results
for average strains and average shear strains comply with the validation
methods for correct implementation of displacement boundary condi-
tions. Tabulated results for this in regards to the compliance for vali-
dation can be referred to in Appendix B (Tables B1 and Table B2). The
effective modulus was determined based on the average strain results
obtained and is graphically presented in Fig. 7 along with the experi-
mental modulus range obtained. The effective modulus of the original
Schwarz P unit cell (model O) is also presented. Since physical tissue
scaffolds were compressed height-wise (along the y-axis) before de-
termining the experimental stress-strain curve and subsequent modulus,
Table 3
Comparison of height, width and length measurements of the averaged unit cell
SEM micrograph (averaged from unit cell micrographs comprising the fabri-
cated tissue scaffold) and the intended Schwarz P unit cell design. Averaging of
the SEM micrographs produces a single micrograph with respect to the plane
view in consideration. However, one dimensional parameter can be represented
in two different plane view i.e. two micrographs. Calculating the mean average
for that particular dimensional parameter would also give the standard devia-
tion.
Parameter Measurement (Mean
±
Standard deviation) (mm)
Average of unit cells comprising
fabricated scaffolds
Intended Schwarz P unit
cell design
Height 0.389
±
0.008 0.520
Width 0.403
±
0.018 0.520
Length 0.413
±
0.008 0.520
Table 4
Comparison of wall thickness measurements between the averaged unit cells
comprising the fabricated tissue scaffold (obtained from measurements analysis
of the averaged SEM micrograph) and the intended Schwarz P unit cell design.
Plane Wall Thickness Measurement (Mean
±
Standard deviation) (mm)
Average of unit cells comprising
fabricated scaffolds
Intended Schwarz P unit cell
design
YX 0.0280
±
0.018 0.0139
YZ 0.0288
±
0.016 0.0139
XZ 0.0450
±
0.000 0.0139
Fig. 6. Graphical presentation of the reconstructed unit cell
model. The Schwarz P unit cell model was reconstructed based
on mean measurements previously established with the dis-
cussed CAD modeling techniques. (a) General view of the re-
constructed unit cell; (b) YZ plane view of the reconstructed
unit cell; (c) YZ plane view of the reconstructed unit cell.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
404
it shall be noted here that only the effective modulus with respect to the
y-axis is considered.
The effective modulus of the reconstructed unit cell model (model
A) falls within the experimental range of modulus values for which
model A over-estimates the mean experimental modulus value by
6.94%. On the other hand, the original Schwarz P unit cell (model O) on
the other hand differs by 86.05% from the experimental mean. To
further understand the relationships between different dimensional
parameters and the resulting effective modulus, several parameters
were derived to allow comparison between model A (which represents
unit cells comprising the fabricated tissue scaffolds after subsequent
geometrical transformations) and the original Schwarz P unit cell de-
sign, model O. The dimensional volume (simple product of height,
width and length) is normalized to that of model O. The same applies
for the wall thickness and radius of curvature with respect to different
cross-sectional planes (graphically presented in Fig. C1 of Appendix C)
from which measurements are made. The methodology for measure-
ments of the wall thickness and radius of curvature are also detailed in
Appendix C. Comparison of different derived parameters between
model A and O are presented in Fig. 8 and Fig. 9.
Considering Fig. 8(a), the porosity of the original Schwarz P unit
cell model is compared to the reconstructed unit cell model. The same
also applies in terms of the dimensional volume as implied from the
normalized dimensional volume in Fig. 8(b). Comparisons show that
the porosity and normalized dimensional volume of the reconstructed
unit cell is smaller compared to the original Schwarz P unit cell model.
Model A having a smaller normalized dimensional volume compared to
model O implies that the reconstructed unit cell has a smaller overall
size compared to the original Schwarz P unit cell while the low porosity
of model A suggests lesser amount of pore space within the confined of
the unit cell construct. The normalized wall thickness measured at
different cross-sectional planes of the reconstructed unit cell model is
consistently thicker compared to the original unit cell model as shown
in Fig. 9(a). However, the opposite is observed in consideration of the
normalized radius of curvature. The normalized wall thickness mea-
sured at different cross-sectional planes was determined to be con-
sistently lower compared to the original unit cell model as shown in
Fig. 9(b). The dimensional parameters considered would theoretically
influence the resulting stiffness of the unit cell and subsequently its
array into a tissue scaffold construct. In addition, these parameters can
also compensate with one another to influence the resulting effective
modulus. This will be discussed further in the discussions.
4. Discussion
4.1. Geometrical transformations
Initial investigations performed under the INNOVABONE project
has found geometrical transformations between the intended Schwarz P
Fig. 7. Average effective modulus determined from average strain results (in
the 2nd degree of freedom) between the reconstructed unit cell (model A) and
the experimental modulus values obtained in Sub-section 4.1. The average ef-
fective modulus of the original Schwarz P unit cell (model O) is also presented
in the chart. The error bars represent the respective lower and upper limit ef-
fective modulus values obtained. The experimental modulus results are re-
presented as horizontal lines according to the lower limit, average and upper
limit values obtained from compression experiments of fabricated tissue scaf-
folds.
Fig. 8. Graphical comparison between (a) porosity and; (b) normalized di-
mensional volume between the reconstructed unit cell model (model A) and the
original Schwarz P unit cell (model O).
Fig. 9. Graphical comparison of the (a) normalized wall thickness and; (b)
normalized radius of curvature between the reconstructed unit cell (model A)
and the original Schwarz P unit cell (model O). The wall thickness and radius of
curvature measurements with respect to different cross-sectional planes for the
reconstructed unit cell model are normalized against the original Schwarz P
unit cell model. Hence, the normalized wall thickness and normalized radius of
curvature for model O is equivalent to 1.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
405
unit cell design and the unit cells comprising the 2PP-fabricated tissue
scaffold structure [21]. Measurement analyses made on averaged unit
cell SEM micrograph of the scaffolds in this study complements the
aforementioned investigation. Emphasizing further, it was determined
based on the measurements that the geometrical transformations show
the occurrence of shrinkage. The 2PP fabrication process parameters
can generally be considered as the main source of discrepancy i.e.
geometrical transformations and shrinkage.
Polymerization is known to be accompanied by a reduction in the
total volume of material in a phenomenon known as volumetric or
polymerization shrinkage [26]. Despite shrinkage being an important
role in the fabrication of tiny structure with high resolution features, it
does become an obstacle in the fabrication of complex structures [9]. In
the context of implementing scaffolds with seeded cells, tissue scaffolds
comprising of porous structures undergoing shrinkage would conse-
quently lead to smaller pores for the movement of cells throughout the
confines of scaffolds. This would lead to insufficient cell growth and
cellular activity when cells are seeded within the tissue scaffold con-
struct. 2PP works in a lay-by-lay method which involves the stacking of
the fundamental element – voxels [4]. A voxel (also known as a volu-
metric pixel) is defined as the unit volume of a material cured by 2PP
[7]. Hence, the accuracy of a structure fabricated via 2PP is thus de-
pendent on the voxels and their respective feature size [9]. During
fabrication, the activation energy i.e. polymerized threshold fluence
exceeds the threshold dose of a photoresist after a threshold exposure
time to form polymerized voxels around the light intensity maximum of
the focal spot [6,27]. These polymerized voxels expand in an aniso-
tropic direction where the expansion is greater along the axial direction
compared to that along the focal plane [6]. Shrinkage also occurs due to
the solid phase having a higher mass density during the liquid phase
during polymerization [4]. With the above, the effects between the
anisotropic voxel growth and the differences in mass density can be
implied to occur in tandem of one another and/or in isolation. Despite
the overall shrinkage found from measurement analyses of the averaged
unit cell, the measurements also showed that the height is smaller
compared to the width and length. This would potentially imply the
compounding effect mentioned previously.
Shrinkage has been reported to be a side effect of working in the
sub-diffraction regime when the polymerized volume is minimized from
the reduction in monomer-to-polymer conversation rate suggesting the
influence of the degree of polymerization on the shrinkage rate [9]. The
degree of polymerization rate on the other hand is subsequently influ-
enced by both laser power and writing speed. Studies investigating the
dependency of laser power on the degree of polymerization have shown
that the laser power is indicative of the amount of energy supplied for
polymerization i.e. polymerized threshold fluence which affects the
resulting overall shrinkage of a fabricated structure. One previous study
investigating the shrinkage of Zr-based hybrid photosensitive materials
produced by 2PP found that at laser powers below a certain threshold
resulted in shrinkage of the overall dimensions of fabricated structures
[28]. This complemented another study which concluded that the ul-
timate shrinkage and shrinkage rate are both dependent on the amount
of energy required for polymerization [29].
The influence of writing speed is implied through the time between
pulses of femtosecond laser i.e. exposure time in which increased
writing speeds would corresponds to lower exposure times [30]. The
growth of polymerized voxels in terms of diameter and height has been
mathematically expressed to be a function of power and time; and ex-
perimentally characterized through line writing experiments [30–33].
Line writing experiments evaluating the expansion of voxels in terms of
width with respect to writing speed has generally shown that higher
writing speeds resulted in thinner polymerized line widths. This com-
plements investigations performed under the INNOVABONE project
[34]. Therefore, the writing speed potentially has some contribution to
the shrinkage. It is possible that writing speeds higher than ideal sub-
sequently resulted in thin polymerized line widths and eventually
smaller than intended gross build-up of voxels making up the fabricated
structure.
4.2. Influence of dimensional parameters
The wall thicknesses and radius of curvature defining the pre-
dominant surface curvature inherent in the Schwarz P architecture
provides bending resistance for the structure under compression and
subsequently affects the resulting effective modulus. Bending resistance
for surface curvatures connecting two adjacent pores can be described
in term of Castigliano’s theorem for curved thin beams [35]. Here, the
resulting deflection is dependent on the ratio between the cube of the
radius of curvature,
R3
and the moment of inertia,
I
(particularly re-
lated to the wall thickness). Hence compensation for effective modulus
can be implied to occur between these two parameters. A high
R
at
constant
I
gives higher amount of deflections and subsequently lower
bending resistance i.e. lower effective modulus. Thicker walls give
higher values of
I
which compensate to give lower deflections and
therefore increasing bending resistance i.e. higher effective modulus.
Bending resistance for surface curvatures connecting two opposite
pores can be described in terms of Euler buckling theorem for initially
curved beams [35,36]. Here, the radius of curvature,
R
is a scaling
factor contributing to the resulting maximum lateral deflection for
which higher values of
R
gives higher deflections. Higher values of
deflections give way to lower bending resistance and subsequently
lower effective modulus values.
A ranking system was established in regards to the above to show
the expectations of how one model succeeds another in terms of the
effective modulus in conjunction to the parameters considered. The
reconstructed unit cell and the original Schwarz P unit cell model was
ranked between 1 and 2, with the ranking numbers indicating the ex-
pectancy for lower values of deflections i.e. high modulus values.
Hence, a ranking of 2 implies low deflections and a resulting higher
modulus values. The rankings are tabulated in Table 5.
Based on the rankings in Table 5, it can be observed that the nor-
malized wall thickness and/or normalized radius of curvature of the
original Schwarz P unit cell model consistently succeeds the re-
constructed unit cell model for high deflections at different cross-sec-
tional planes. It can be deduced here the original Schwarz P unit cell is
expected to have a lower effective modulus than the reconstructed unit
cell.
Further adding to the above, the porosity and dimensional volume
also affects the resulting effective modulus. At constant porosity be-
tween different unit cell models, a smaller dimensional would imply
smaller amount of void space within the confines of the unit cell and
subsequently overall tissue scaffold construct available for compression.
With constant application of load across the different models consisting
of the same material, the unit cell with the smaller dimensional volume
would give higher modulus values. The original Schwarz P unit cell
design has a larger dimensional volume and a porosity value that is
25.96% higher compared to that of the reconstructed unit cell. Hence, it
can be deduced that the original Schwarz P unit cell is expected to have
Table 5
Evaluation of ratios between cube of normalized radius of curvature and the
normalized wall thickness along plane YZ and YX; and the normalized radius of
curvature along the split plane for the reconstructed unit cell and the original
Schwarz P unit cell model. Rank numberings (showed in brackets) indicate the
expectancy for higher values of deflections i.e. low modulus values.
Model Ratio between cube of normalised radius of
curvature and the normalised wall thickness
Normalised radius of
curvature
Plane YZ Plane YX Split plane
O 1 (1) 1 (1) 1 (1)
A 0.3940 (2) 0.3493 (2) 0.638 (2)
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
406
a lower effective modulus than the reconstructed unit cell. This adds
into effect with the theoretical expectation that the original design
should have a lower effective modulus due to its attribution in terms of
the wall thickness and radius curvature in regards to the inherent sur-
face curvatures. Results obtained from micromechanical analysis of the
reconstructed unit cell and original Schwarz P unit cell confirm the
theoretical expectations discussed above. Since the unit cell was re-
constructed to approximately represent the average of all unit cells
comprising the fabricated tissue scaffold, the experimental model also
indirectly confirms the theoretical expectations.
5. Conclusion
Geometrical transformations as a result of non-uniform shrinkage
have been taken into account by reconstructing the Schwarz P unit cell
surface geometry model to approximately represent the average of unit
cells comprising the fabricated tissue scaffolds. Micromechanical ana-
lysis of the reconstructed unit cell assigned with the same material
model making up the fabricated Schwarz P unit cell-based tissue de-
monstrated excellent agreement to experimental mechanical char-
acterization in terms of modulus with only a small margin error of
6.94%; a stark comparison to the original intended design which gave
an error of 86.05% from the experimental mean modulus. Emphasizing
this, micromechanical analysis was able to be implemented despite the
geometrical transformations by simply taking them into account in
derivations for boundary conditions and in terms of CAD modeling.
The above would thus validate the use of micromechanical analysis
in the context of the tissue scaffold considered in this study. With
various different crystal packing systems which can be used to idealize
structures with periodic microstructures, micromechanical analysis can
be applied to other regular tissue scaffold structures. Furthermore, the
approach can also be applied to computationally characterize the me-
chanical properties of structures outside the field of bone tissue en-
gineering depending on their idealization to crystal packing systems.
Considering that the micromechanical analysis has a foundation in re-
ducing a structure to a periodic microstructure to the analysis of a
single representative unit cell based on derivations of boundary con-
ditions, efficiency in terms of time and computational resources is
maintained.
Acknowledgements
The authors would like to thank our colleagues at the TETRA
Society for Sensoric, Robotics and Automation GmbH (Germany) and
the Institute for Bioprocessing and Analytical Measurement Techniques
(Germany) for providing all the samples involved in this study. This
research project is funded by the European Union’s 7
th
Framework
Program under grant agreement no. 2633 63 (INNOVABONE). This
work was supported by the Engineering and Physical Sciences Research
Council [grant number EP/K029592/1]; and the EPSRC Centre for
Innovative Manufacturing in Medical Devices (MeDe Innovation).
Appendix A
The average strains and average shear strain results are used to validate correct implementation of the displacement boundary conditions for
micromechanical analysis. For boundary conditions derived for a simple cuboidal packing system, the tabulated average strains and average shear
strains results should be as shown in Table A1 and Table A2 respectively. The average strains are symmetric along the diagonal with differing values
within the diagonal. The same applies for average shear strains however, the values lying outside the diagonal can be approximated to zero due to
their small values.
Appendix B
Micromechanical analysis of the reconstructed unit cell model gave average strains and average shear strains that complied with the validation
methods for correct implementation of displacement boundary conditions. The average strains and average shear strains are tabulated in Table B1
and Table B2 respectively.
Table A1
Tabulated representation of average strain values from micromechanical analyses to validate displacement boundary conditions established for a
simple cuboidal packing system.
Degree of freedom of load/displacement application Average strain
x
y
z
x
0
1
2
3
y
0
( )
4 2
5
6
z
0
( )
7 3
( )
8 6
9
Table A2
Tabulated representation of average shear strain values from micromechanical analyses to validate displacement boundary conditions established for
a simple cuboidal packing system.
Degree of freedom of load/displacement application Average shear strain
x
y
z
yz
0
1
0 0
xz
0
0
2
0
xy
0
0 0
3
A.Z. Zabidi et al.
Additive Manufacturing 25 (2019) 399–411
407
Appendix C
Cross-sectional planes were used as references to measure the wall thickness and radius of curvature. The cross-sectional planes are shown in Fig.
C1.
For each plane view, line segments can be constructed based on cross-sections of the surface geometry that also lie on the cross-sectional planes
presented in Figure C.1 to make measurements for the radius of curvature. These line segments are highlighted on a Schwarz P unit cell as a
representative example in Fig. C2. In creating line segments for the reconstructed unit cell model on the split plane, intersection points formed
between lines originating from 3 adjacent pores are selected. This is graphically presented as yellow dots in the same aforementioned figure.
A midline within the enclosed line segments can be constructed before subsequently determining its length using Altair Hyperworks as shown in
Fig. C3 By approximating the midline to take a circular arc, the arc angle,
can be determined via ImageJ. The radius of curvature can subsequently
be determined using the equation,
=s r
, for
s
is the length of the midline/curvature. In regards to the wall thickness, the average length of the two
ends of the enclosed line segments for each cross-sectional plane is tabulated. This should not be confused with the average wall thickness measured
from unit cell micrographs at each plane view.
Table B1
Average strain values obtained from micromechanical analysis of discretised reconstructed unit cell model assigned with 3 different LCM3 material models i.e. lower
limit (specimen 5), the average and upper limit (specimen 3). The unit cell was reconstructed to approximately represent an average of unit cells comprising the
fabricated tissue scaffold. The average strains for each material model are symmetric along the diagonal with differing values within the diagonal. This thus complies
with the validation methods for correct implementation of displacement boundary conditions derived for the simple cuboidal packing system in revised micro-
mechanical analysis.
Material model Degree of freedom of load
application
Average strain
x
y
z
Specimen 5 (Lower
limit)
x
0
0.102 0.0591 0.0123
y
0
0.0591 0.135 0.0527
z
0
0.0123 0.0527 0.0897
Average
x
0
0.0787 0.0454 0.00945
y
0
0.0454 0.104 0.0405
z
0
0.00945 0.0405 0.0689
Specimen 3 (Upper
limit)
x
0
0.0677 0.0391 0.00813
y
0
0.0391 0.0894 0.0348
z
0
0.00813 0.0348 0.0593
Table B2
Average shear strain values obtained from micromechanical analysis of discretised reconstructed unit cell model assigned with 3 different LCM3 models i.e. specimen
5 (for the lower limit), the calculated average and specimen 3 (for an upper limit). The average shear strains for each material model are symmetric along the
diagonal with differing values within the diagonal. Values outside the diagonal can be approximated to zero. This thus complies with the validation methods for
correct implementation of displacement boundary conditions derived for the simple cuboidal packing system in revised micromechanical analysis.
Material model Degree of freedom
of load application
Average shear strain
x
y
z
Specimen 5
(Lower limit)
yz
0
0.122 4.754E-8
( 0)
2.088E-7
( 0)
xz
0
4.754E-8
( 0)
0.176 3.782E-7
( 0)
xy
0
2.088E-7
( 0)
3.782E-7
( 0)
0.125
Average
yz
0
0.0939 3.653E-8
( 0)
1.605E-7
( 0)
xz
0
3.653E-8
( 0)
0.135 2.906E-7
( 0)
xy
0
1.605E-7
( 0)
2.906E-7
( 0)
0.0957
Specimen 3
(Upper limit)
yz
0
0.0807 3.141E-8
( 0)
1.380E-7
( 0)
xz
0
3.141E-8
( 0)
0.116 2.499E-7
( 0)
xy
0
1.380E-7
( 0)
2.499E-7
( 0)
0.0823
A.Z. Zabidi et al.
Additive Manufacturing 25 (2019) 399–411
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Fig. C2. Line segments on a Schwarz P unit cell. (Red) Line segment on the YZ plane; (Green) Line segment on the YX plane; (Violet) Line segment on the split plane;
The line segment on the split plane is created between intersections points formed by line segments originating from 3 different adjacent pores. These intersections
points are represented as yellow dots in the figure.
Fig. C1. Cross-sectional planes used as reference to measure the radius of curvature of reconstructed unit cells; (a) Plane YZ; (b) Plane YX; (c) Plane YX/YZ i.e. Split
plane.
A.Z. Zabidi et al. Additive Manufacturing 25 (2019) 399–411
409
Appendix D. Supplementary data
Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.addma.2018.11.021.
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... Over the past few decades, this technology has experienced extensive improvements in terms of the printing technique and materials compatibility ( Matias and Rao 2015 ). Today, various 3D printing techniques such as Stereolithography (SLA) ( Eng et al., 2017 ;Melchels et al., 2010 ), Digital Light Processing (DLP) ( Bagheri Saed et al. 2020 ;Hong et al., 2020 ;Kwak et al., 2020 ), Fused Deposition Modelling (FDM) ( DePalma et al., 2020 ;Garzon-Hernandez et al., 2020 ;Mohamed et al., 2015 ;Solomon et al., 2020 ), Polyjet Pugalendhi et al., 2020 ;Quan et al., 2020 ), Selective Laser Sintering (SLS) ( DePalma et al., 2020 ;Yuan et al., 2019 a) and others ( Balli et al., 2017 ;Tumbleston et al., 2015 ;Wu et al., 2020 ;Zabidi et al., 2019 ;Ziaee and Crane 2019 ) are available for polymer printing. It has also expanded beyond polymers to electronics ( Saengchairat et al., 2017 ;Tan et al., 2020 ), metals ( Kuo et al., 2020 ;Tan et al., 2015 ), composites ( Grady et al., 2015 ;Valino et al., 2019 ;Yu et al., 2019 ), and even living cells bioprinting ( Ackland et al., 2017 ;Derakhshanfar et al., 2018 ;Murphy and Atala 2014 ;Ng et al., 2019 ). ...
... It is undeniable that the AM technology has experienced huge development in all aspects over the past decade. Today, AM is capable of printing a variety of materials such as polymers ( Bagheri Saed et al. 2020 ;Balli et al., 2017 ;DePalma et al., 2020 ;Eng et al., 2017 ;Garzon-Hernandez et al., 2020 ;Hong et al., 2020 ;Kwak et al., 2020 ;Lee et al., 2017 a;Melchels et al., 2010 ;Mohamed et al., 2015 ;Pugalendhi et al., 2020 ;Quan et al., 2020 ;Solomon et al., 2020 ;Tumbleston et al., 2015 ;Wu et al., 2020 ;Yuan et al., 2019 a;Zabidi et al., 2019 ;Ziaee and Crane 2019 ), metals ( Kuo et al., 2020 ;Tan et al., 2015 ), concretes ( Labeaga-Martínez et al., 2017 ;Lim et al., 2012 ;Tay et al., 2016 ;Wu et al., 2016 ) and biomaterials ( Ackland et al., 2017 ;Derakhshanfar et al., 2018 ;Murphy and Atala 2014 ;Ng et al., 2019 ). This section will focus on the different ways AM can be used to fabricate prototypes, and also discuss the suitability of various AM techniques that can be used for membrane-based water treatment. ...
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Over the past decade, 3D printing or additive manufacturing (AM) technology has seen great advancement in many aspects such as printing resolution, speed and cost. Membranes for water treatment experienced significant breakthroughs owing to the unique benefits of additive manufacturing. In particular, 3D printing's high degree of freedom in various aspects such as material and prototype design has helped to fabricate innovative spacers and membranes. However, there were conflicting reports on the feasibility of 3D printing, especially for membranes. Some research groups stated that technology limitations today made it impossible to 3D print membranes, but others showed that it was possible by successfully fabricating prototypes. This paper will provide a critical and comprehensive discussion on 3D printing specifically for spacers and membranes. Various 3D printing techniques will be introduced, and their suitability for membrane and spacer fabrication will be discussed. It will be followed by a review of past studies associated with 3D-printed spacers and membranes. A new category of additive manufacturing in the membrane water industry will be introduced here, known as hybrid additive manufacturing, to address the controversies of 3D printing for membrane. As AM technology continues to advance, its possibilities in the water treatment is limitless. Some insightful future trends will be provided at the end of the paper.
... Recently, triply periodic minimal surfaces (TPMSs) have emerged as a powerful tool for designing porous TE scaffolds [28]. Owing to their smooth surfaces, highly interconnected structures, and conveniently controllable porous features, the designed TPMS scaffolds exhibit outstanding performances as compared to conventional scaffolds [29]. ...
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