Triply Periodic Minimal Surfaces Sheet Scaffolds for Tissue

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Triply Periodic Minimal Surfaces Sheet Scaffolds for Tissue Engineering Applications: An Optimization Approach towards Biomimetic Scaffold Design Sanjairaj Vijayavenkataraman, Lei Zhang, Shuo Zhang, Jerry Ying Hsi Fuh, and Wen Feng Lu ACS Appl. Bio Mater., Just Accepted Manuscript • Publication Date (Web): 27 Jul 2018 Downloaded from http://pubs.acs.org on July 27, 2018

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ACS Applied Bio Materials

Triply Periodic Minimal Surfaces Sheet Scaffolds for Tissue Engineering Applications: An Optimization Approach towards Biomimetic Scaffold Design Sanjairaj Vijayavenkataraman1#*, Lei Zhang1#, Shuo Zhang1, Jerry Ying Hsi Fuh1, Wen Feng Lu1 1

Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576

#

These authors contributed equally to this work. (*Corresponding author: [email protected])

Abstract Biomimetic scaffold design is gaining attention in the field of tissue engineering of late. Recently, Triply Periodic Minimal Surfaces (TPMS) have attracted the attention of tissue engineering scientists for fabrication of biomimetic porous scaffolds. TPMS scaffolds offer several advantages that include a high surface area to volume ratio, less stress concentration, and increased permeability compared to the traditional lattice structures, thereby aiding in better cell adhesion, migration, and proliferation. In literature, several design methods for TPMS scaffolds have been developed, which considered some of the important tissue-specific requirements, such as porosity, Young’s modulus, and pore size. However, only one of the requirements of a tissue engineering scaffold was investigated in these studies and not all the requirements were satisfied simultaneously. In this paper, we develop a design method for TPMS sheet scaffolds which are able to satisfy multiple requirements including the porosity, Young’s modulus, and pore size, based on a parametric optimization approach. Three TPMS namely the Primitive (P), Gyroid (G), and Diamond (D) surfaces with cubic symmetry are considered. The versatility of the proposed design method is demonstrated by three different applications, namely tissue-specific scaffolds, scaffolds for stem cell differentiation and functionally graded scaffolds with biomimetic functional gradients.

Keywords: triply periodic minimal surfaces; functionally graded scaffolds; gradient property; design optimization; tissue engineering scaffolds; 3D printing

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1. Introduction Tissue engineering scaffolds act as templates for the cells to attach, grow, proliferate and mature into a tissue construct. The properties of the scaffolds, such as the porosity, pore morphology, stiffness, biocompatibility, and biodegradability influence the cell adhesion, proliferation, and differentiation 1. Traditional two-dimensional (2D) culture fail to provide the accurate representation of the in situ cell / tissue microenvironment as the native tissue microenvironment is three-dimensional (3D) in nature 2. Hence, researchers in the life sciences domain prefer working with 3D porous scaffolds than the 2D culture systems. Many fabrication technologies such as solvent casting, particulate leaching, gas foaming, phase separation, freeze-drying, and electrospinning were used for the fabrication of 3D porous scaffolds 3. In addition to the other limitations of these processes such as the use of toxic solvents, poor pore interconnectivity, limited porosity range, and presence of residual porogen particles or toxic solvents, all these processes suffer from a common limitation which is the geometric freedom or flexibility. The scaffold morphology, pore size, porosity, and pore architecture cannot be precisely controlled in any of these methods, resulting in inconsistent 3D structures with poor repeatability 4. However, one of the main requirements of tissue engineering scaffolds is biomimicry 5. Scaffolds should mimic the native tissue microenvironment, in terms of structure, mechanical properties such as stiffness and strength, and other chemical and biological cues. Additive manufacturing (AM), also called as 3D printing or rapid prototyping, is a relatively new technology that has been used in the field of healthcare and tissue engineering 6. This technology possesses many advantages compared to the other fabrication methods mentioned above. AM offers great geometric freedom, with which complex biomimetic structures which are otherwise not possible to fabricate by traditional manufacturing methods, could be fabricated. Hence, with the advent of AM, tissue engineering scientists focus on biomimetic scaffold design with complex shapes and structures. Of all the AM techniques, the three widely used methods for fabrication of tissue engineering scaffolds are fused deposition modeling (FDM), stereolithography (SLA), and selective laser sintering (SLS) 4. Complex biomimetic scaffold designs such as human mandibular cancellous bone scaffold and a femur bone segment with functional gradients 7 and hybrid scaffold designs for vascularization are some examples 4, 8. Minimal surfaces are surfaces that have zero mean curvature and are of immense interest to mathematicians, engineers, and biologists 9. Recently, triply periodic minimal surfaces (TPMS) have attracted the attention of tissue engineering scientists for the design and fabrication of biomimetic porous scaffolds. The surface area to volume ratio of TPMS based scaffolds is very high compared to the conventional lattice scaffolds. Particularly, the TPMS sheet structures which are created by thickening the surfaces exhibit significantly large surface areas. This higher surface area of TPMS scaffolds contribute to enhanced cell adhesion, migration, and proliferation


10

. Many cellular and

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biological functions such as ion exchange, oxygen diffusion, and nutrient transport occur at the surface and hence, TPMS scaffolds with a higher surface area to volume ratio could provide better biological cues to the cells that are cultured on them. Furthermore, the infinitely continuous surface with smooth joints ensures less stress concentration and enhanced mechanical properties compared to the regular lattice structure scaffolds 11. The increased permeability of TPMS scaffolds not only helps in better cell penetration but also the better mass transport of nutrients and growth factors 9, 12. 13

Though minimal surfaces were known to us for more than 200 years

, the application of such

structures to tissue engineering scaffolds was relatively very recent. There are a few published works on TPMS based scaffolds. In one of the earlier works 14, Primitive (P) surface based scaffolds were fabricated by AM methods from poly(propylene fumarate) which is a biodegradable polymer. On seeding the cells on the fabricated scaffold, high cell viability (>95%) was achieved. Melchels et al. 15 designed TPMS scaffolds with Gyroid (G) architectures fabricated by Stereolithography. Mesenchymal stem cells (MSCs) were seeded on the G-surface based scaffolds. Results revealed that the TPMS scaffolds had larger cell population and cell distribution than the scaffolds fabricated by the traditional salt leaching method, with a 10-fold higher permeability. Although there are not many biological studies on TPMS scaffolds, the few studies reported give a positive outlook on the use of TPMS scaffolds for tissue engineering. However, further detailed cell studies are required to prove the effectiveness of TPMS scaffolds on cell attachment, viability, proliferation, growth, and differentiation. The architectures of TPMS scaffolds play an important role in their overall properties. In order to find the optimal geometries of TPMS scaffolds, many design and optimization studies have been conducted regarding the geometrical, mechanical, and fluid properties. Blanquer et al.

16

studied the

curvature distributions of different types of TPMS scaffolds, which can help to tailor a scaffold to the desired surface curvature. In

17-20

, the compressive modulus and strength of TPMS scaffolds were

investigated with respect to relative density. The plastic deformation and failure mechanism of TPMS scaffolds made of an ABS plastics were studied by Kadkhodapour et al.

21

. The effect of architectures

and pore size of TPMS scaffolds on their permeability was characterized in

17-18, 22

. In above studies,

single or multiple properties were investigated separately, namely the scaffold was designed to fulfil one of the requirements. However, a desired scaffold should satisfy all the requirements simultaneously. To this end, developing a design method incorporating multiple properties, such as porosity, pore size, and Young’s modulus, concurrently is of great importance. This paper focusses on the design of TPMS sheet scaffolds based on a parametric optimization approach incorporating multiple tissue-specific requirements, leading to better biomimicry. In this paper, three TPMS sheet scaffolds composed of three types of surfaces namely the Primitive (P), Gyroid (G), and Diamond (D) surfaces with cubic symmetries are studied. The Young’s modulus,



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porosity, and pore size of the TPMS sheet scaffolds are of interest, which can be controlled by the cell size and shell thickness. An optimization approach is developed to find the optimized cell size and shell thickness so that the TPMS scaffolds satisfy all the above requirements. The effectiveness of the proposed design method is validated by three biomedical examples: designing tissue-specific scaffolds, scaffolds for stem cell differentiation and functionally graded scaffolds.

2. Materials and Methods 2.1 Design of TPMS sheet scaffolds TPMS are a group of 3D periodic surfaces which divide the space into two or more sub-spaces. In this work, we make use of TPMS to create thin-walled structures, namely TPMS sheet scaffolds, in which TPMS are thickened and assigned with a solid material. Another type of TPMS-based structures are generated by filling one sub-space separated by TPMS with the solid material, i.e. skeletal TPMS structures

12, 20

. TPMS sheet scaffolds are studied in this work because of the superior mechanical

properties and the significantly large surface area. Implicit method which represents surfaces by using nodal equations and their zero-valued surfaces, are commonly used to approximate TPMS structures

23

. The P, G, and D surfaces with cubic

symmetries are described by the following nodal approximations,

Φ P ( x, y, z ) = cos(ω x) + cos(ω y ) + cos(ω z ) = 0 Φ D ( x, y, z ) = sin(ω x ) sin(ω y ) sin(ω z ) + cos(ω x) sin(ω y ) sin(ω z ) + sin(ω x) cos(ω y ) sin(ω z ) + sin(ω x ) sin(ω y ) cos(ω z ) = 0 Φ G ( x, y, z ) = cos(ω x) sin(ω y ) + cos(ω y ) sin(ω z ) + cos(ω z ) sin(ω x) = 0

, with ω =

2π (1) a

where x, y, z are spatial coordinates and a denotes the periodicity, representing the size of a unit cell. The surfaces defined in Eqs. (1) are thickened to formulate TPMS sheet structures. Figure 1 shows the unit cell and 3 × 3 × 3 tessellating arrays of TPMS sheet scaffolds. Finally, the stereolithography (STL) format files are generated for AM. Considering the TPMS scaffolds composed with thin shells, the relative densities are proportional to the shell thickness, t, by

ρ (a, t ) = S (a)t / a 3

(2)

S(a) denotes the surface area of TPMS, which depends on the cell size. The pore size is defined as the diameter of a maximum sphere that is able to pass through the smallest channel of the scaffold. Figure 1 (a-c) illustrates the pore size definition and the maximum sphere of P, G, and D scaffolds. The pore size, p, depends on both the cell size and shell thickness by


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p(a, t ) =

p(a0 )a −t a0

(3)

where p(a0) is the pore size of a scaffold with the cell size of a0 and the shell thickness of 0.

Figure 1. TPMS scaffolds with the pore size definition. Unit cells of scaffolds composed with (a) Psurface, (b) G-surface, (c) D-surface, in which the cell size and pore size are labeled by a and p, respectively. The corresponding 3 × 3 × 3 repeated unit cells (d-f). 2.2 Finite element (FE) modeling FE analysis was conducted to investigate the structure-property relationships for TPMS scaffolds. The commercial FE software Abaqus/Standard was employed to compute Young’s modulus of the scaffolds. TPMS sheet scaffolds were modeled using 3D linear triangular shell elements (S3R) as the scaffolds are comprised of thin-walled structures. The simulation considered a linear elastic material and geometric linearity. When considering periodic structures, one can focus on a unit cell of the periodic structure with appropriate periodic boundary conditions in simulations. This simulation method was adopted in a number of studies, leading to several advantages such as low computational costs and the elimination of boundary effects

24-26

. Considering the macroscopic uniaxial loading, the periodic boundary

conditions can be stated by following equations 24,



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u '− u = ( x '− x)ε x0 v '− v = (y'− y)ε y0

(4)

w '− w = ( z '− z )ε z0 where (x, y, z) and (x’, y’, z’) are coordinates of points periodically located at the boundary of unit cells; (u, v, w) and (u’, v’, w’) are the corresponding displacements. ε x0 , ε y0 , ε z0 denote the macroscopic strains. Eqs. (4) were implemented in Abaqus using Equation constraints. The details of the formulation of periodic boundary conditions in Abaqus was described in 24. The unit cells of TPMS scaffolds were subjected to a small compressive strain, ε z , in the z-direction. The effective Young’s 0

modulus, E, was computed by,

E = FR / ( Aε z0 )

(5)

where FR is the reaction force in the z-direction on the top face, and A = a2 is the face area of the unit cell. The analytical models for cellular structures predict that their Young’s moduli are determined by the base material’s Young’s modulus, Es, and the relative density, following the power law by 26-28 n

E = CEs ρ

(6)

where the coefficient C and exponent n depend on the unit cell geometries. Although this analytical model is developed based on lattice structures, it is also suited for TPMS-based cellular structures 17, 20, 29

. It is noted that the Young’s modulus of scaffolds is independent of the cell size at the mm scale.

Therefore, we modelled the unit cells with a cell size of 1 mm and the average element size of 0.02 mm and then compute the relative Young’s modulus, E/Es, to reveal the structure-property relationship for TPMS scaffolds without loss of generality. 2.3 Optimization methodology The aim of using optimization approach to design tissue engineering scaffolds is to incorporate multiple requirements simultaneously, leading to better biomimicry. In this work, the porosity, Young’s modulus, and pore size of TPMS scaffolds are taken into account. For a particular application, generally a scaffold with a given base material should have Young’s modulus and pore size that are within target ranges, and the porosity,

φ , should be as high as possible. To achieve this,

the cell size and shell thickness are considered as design variables. The optimization problem can be stated by


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min : ρ (a, t ) = 1 − φ (a, t ) a ,t

s.t. : E > Emin pmin < p < pmax

(7)

al ≤ a ≤ au tl ≤ t ≤ tu where ρ (a, t ) = 1 − φ (a, t ) denotes the relative density of the scaffolds. Emin is the minimum allowable Young’s modulus. pmin and pmax describe the target range for the pore size. The lower and upper bounds of design variables are denoted by al , tl and au , tu , respectively. It should be noted that the maximum allowable Young’s modulus, Emax, is not included in Eq. (7), because the relative density is correlated with Young’s modulus. Minimizing the relative density would lead Young’s modulus approaching Emin; therefore Emax is omitted in Eq. (7). Computing the Young’s modulus for each individual design during the optimization processes is numerical expensive. We employ a reduced-order model to reduce the computational costs 30, in which Young’s modulus of each design is estimated by linear interpolation on the structure-Young’s modulus relationship established by the FE simulations. In order to balance the computational efficiency and interpolation accuracy, the Young’s modulus of each type of TPMS scaffolds with 15 key relative densities is calculated by FE analyses for the interpolation. The optimization procedures are implemented in an interconnected Matlab-Abaqus framework. The function ‘fmincon’ is used to solve the nonlinear and multi-variable optimization problem (7), which adopts the interior-point algorithm and finite difference method. The details of this optimization framework are described in the authors’ previous work 31. 2.4 Selection of base materials Despite the TPMS micro-architectures, the properties of the base materials also play important roles on the properties of the resultant scaffolds. According to Eq. (6), the Young’s modulus of a scaffold is proportional to the Young’s modulus of the base material Es. Therefore, the selection of the base material is important to satisfy the requirements of the porosity, Young’s modulus and the pore size of scaffolds. Too soft base materials result in relative low porosities of scaffolds; while too stiff materials lead to extreme high porosities and small shell thicknesses which are difficult to fabricate. In order to achieve feasible designs, the upper and lower bounds of the base material’s Young’s modulus should be determined based on the prescribed TPMS type as well as the requirements for scaffolds. Making use of Eq. (6), the lower bound of the base material, Esl, can be calculated according to the minimal allowable porosity, i.e. ϕmin=70% in this paper, and the minimal allowable Young’s modulus, Emin, of the scaffold, by



Esl = Emin / (C (1 − φmin ) n )

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(8)

For the upper bound of base materials, the direct cause is the minimal allowable shell thickness depending on the printing resolution of the 3D printer. Note that the constraint of the minimal shell thickness may not be imposed directly in Eq. (7) as it can lead to infeasible designs. This constraint can be satisfied by the proper selection of base materials, i.e. finding to upper bound of the base material’s Young’s modulus. However, calculating the upper bound based on the minimal shell thickness is complicated because the Young’s modulus, cell size and pore size of scaffolds are involved, and the relationship is highly non-linear. Alternatively, the constraint can be relaxed by imposing the maximal porosity of scaffold (such as 99%) and thereafter the upper bound of base materials, Esu, can be estimated by

Esu = Emax / (C (1 − φmax ) n )

(9)

where ϕ and Emax are the maximal allowable porosity and the maximal allowable Young’s modulus of the scaffold, respectively. In Eqs. (8) and (9), the coefficient C and exponent n are determined by the prescribed TPMS type. With this feasible range of the Young’s modulus of the base materials obtained, appropriate base materials whose Young’s modulus are within this range could be selected. 2.5 Design of functionally graded TPMS scaffolds One of the key steps of creating functionally graded TPMS scaffolds is to combine different scaffolds with smooth shape transitions. Using the mathematical definition and implicit modeling method of TPMS, multiple sub-scaffolds can easily be merged without any sharp features that are commonly observed from the results of the Boolean union operation. A large number of design and modeling methods were developed for functionally graded TPMS scaffolds. The simplest method was to spatially vary the constant on the right-hand side of Eqs. (1) (the value is fixed to zero in this paper). This method is able to achieve pore size and porosity gradients 12; however, the size and topology of unit cells are constrained. In 19, 32, interpolation techniques were employed combine multiple scaffolds, in which the weighting function could be linear, sigmoid, or Gaussian radial basis functions. The studies in

33

proposed a method for generating functionally graded scaffolds from the give sub-

scaffolds and boundaries based on sigmond function. In this work, we adopt the method described in 33

to generate functionally graded TPMS scaffolds because the boundaries and the transitions of two

zones can be controlled with ease. To be specific, two sub-scaffolds can be combined by

Φ ( x, y , z ) =

1

1+ e

k ⋅b ( x , y , z )

Φ1 ( x, y, z ) +

1

1+ e

− k ⋅b ( x , y , z )


Φ 2 ( x, y , z )

(10)

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where Φ1 ( x, y, z) , Φ2 ( x, y, z) , and

Φ(x, y, z) represent

the two sub-scaffolds and the resultant

scaffold. The boundary is described by the function b(x, y, z) . The factor k controls the transition zone near the boundary; a big k-value results in a small transition zone and rapid shape changes, and vice versa. In addition, Eq. (10) can be recursively applied for combining more than two substructures. Using this method, multiple scaffolds with different cell size and topologies can be combined, resulting in graded scaffolds including controllable boundaries and smooth shape transitions. Eq. (10) is used to generate the surface geometry of functionally graded scaffolds. Then the sub-scaffolds are thickened based on their shell thicknesses.

3. Results and Discussion 3.1 Mechanical and geometrical properties The mechanical properties of cellular structures are determined by the unit cell geometry, relative density, and the properties of the base material 27-28. In order to study the contributions of the unit cell geometry and relative density on the scaffolds’ property, the relative Young’s moduli of P, D, and G scaffolds are calculated and plotted with respect to the porosity ( φ = 1 − ρ ) as shown in Figure 2. The high porosity scaffolds, i.e. above 70%, are studied in this paper. The porosities of TPMS sheet scaffolds are controlled by varying the shell thicknesses. The relative Young’s moduli decrease monotonously with increasing porosity. The exponents in the power law Eq. (6) are computed by the least square method (see Figure 2). The FE predictions reveal that Young’s moduli of the TPMS sheet scaffolds are nearly linear with the relative density. This indicates that the TPMS sheet scaffolds exhibit the stiffness comparable to stretch-dominated lattices, showing high mechanical stiffness. The D scaffolds have higher Young’s moduli than P and G scaffolds with same base material and porosity.



Figure 2. The normalized Young’s modulus versus porosity of TPMS scaffolds. The pore size of TPMS sheet scaffolds is mainly determined by the cell size, while the shell thickness plays a minor role. This is because the wall thickness is significantly small compared to the cell size. P, D, and G scaffolds have distinct pore morphologies and locations (see Figure 1). Since D scaffolds have more complex geometries and larger surface area, their pore size is smaller than those of P and G scaffolds of the same cell size. The P and G scaffold exhibit large pore size, facilitating the permeability of scaffolds. Based on the assumption of high porosity and thin shells, the surface area of TPMS scaffolds is considered twice of that of TPMS and is independent of the shell thickness. The surface area per volume of TPMS scaffolds is determined by the geometry and size of the unit cell as shown in Figure 3. The surface area per volume (S/V) is scale dependent, which is inverse proportional to the cell size, i.e. S/V ∝1/a. The D scaffolds display greatest surface area per volume. Compared with the TPMS skeleton structures whose surface areas are same as those of TPMS, the TPMS sheet scaffolds exhibit larger surface area due to the thin-walled features.


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Figure 3. The surface area per volume versus cell size of TPMS sheet scaffolds.

3.2 Tissue-specific scaffold optimization The scaffold morphology and mechanical properties influence the cell behavior to a great extent 34

35

15

.

1

Pore size , porosity , and effective Young’s modulus are three most important scaffold properties that determine the cell infiltration, growth, proliferation, and differentiation. There is always a tradeoff between the scaffold porosity and its mechanical properties

36

. A high porosity is desirable for

tissue engineering scaffolds for better permeability of cell media, nutrients, and growth factors. However, highly porous scaffolds tend to be weaker and structurally unstable. Hence, an optimal balance between these two parameters has to be achieved. From the literature, a porosity of greater than 70% is generally acceptable for a tissue engineering scaffold 37. Therefore, it is suggested that the scaffold structure should have a porosity of greater than 70%. Apart from porosity, different tissues or cell types prefer different pore size

37

. The scaffold is also expected to have the same or similar

mechanical properties in order that the cells experience their native in vivo microenvironment. The desired pore size and Young’s modulus of various tissues / cell types have been collated from several previous studies as listed in Table 1. The optimization method is applied to find the suitable scaffold architecture, with a maximum porosity, having the pore size and Young’s moduli in the desirable range (according to the tissue type). Despite the unit cell geometry and porosity, the selection of the base material also plays an important role in the optimization results to satisfy the requirement of Young’s modulus. Using a soft base material may result in a small porosity that is smaller than the requirement of 70%. On the contrary,



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using a stiff base material leads to a scaffold with very high porosity, which is also not desired because the shell thickness may be extremely small. Therefore, three different base materials were used depending on the desirable mechanical properties. For soft tissues with a Young’s modulus in the range of several kPa, a soft material such as hydrogel (Es = 110 kPa)

38

is used; for a medium

range of stiffness in several MPa, an intermediate material such as polycaprolactone (PCL, Es = 400 MPa)

39-40

is used; and for hard tissues such as bone with an Young’s modulus of several GPa, stiff

materials such as titanium alloy (Es=114 GPa) 41 is used. Table 2 shows the optimized scaffolds that maximize the porosity and satisfy the requirements of Young’s modulus and pore size. As expected, all the optimized Young’s moduli are very close to the minimal allowable Young’s modulus. This implies that the scaffold’s Young’s modulus monotonously decreases with increasing its porosity and the optimization problem formulation is reasonable. To achieve the same Young’s modulus with the same base material, the D scaffolds have greater porosity than P and G scaffolds, indicating the D structures are mechanically stiff. Using Dsurface structures to create scaffolds would reduce the weight and increase the porosity. In addition, it can be seen in Table 2 that the optimized cell size is linked to the target range of pore size. According to Eq. (3), a large pore size leads to a greater cell size for a specific unit cell architecture. Since the D scaffolds have a relatively smaller pore size, the optimized cell sizes are greater compared to P and G scaffolds. Most of the optimized porosities in Table 2 are greater than 70%; however, this condition is not strictly guaranteed as there exist some optimized porosities smaller than 70%, e.g. the P scaffold for heart tissue. According to Figure 2, there exist an upper bound of E/Es for each TPMS scaffold with the porosity higher than 70%, which are 0.06, 0.10, and 0.12 for P, G, and D scaffolds, respectively. Since soft base materials would yield the E/Es beyond these upper bounds, the output porosities are smaller than 70%. In order to achieve a valid design, a stiffer base material is suggest to position the relative Young’s modulus in an appropriate range. On the contrary, there are some optimized scaffolds including very thin shell thickness due to the ultra-low relative Young’s modulus, which is also undesired because of the manufacturing constraints. To address this issue, one could set the lower bound of the shell thickness in Eq. (7) based on the manufacturing resolutions, or make use of a softer base material. 3.2 Scaffold optimization for stem cell differentiation Stem cells have become an integral part of tissue engineering 42. Though the use of stem cells faces several challenges and ethical concerns

43-44

, stem cells are indispensable for tissue engineering of

continuously self-renewing tissues, such as skin. Stem cells, given the appropriate cellular microenvironment, can differentiate into different lineages

45

. The mechanical properties of the

substrate are crucial factors in determining the stem cell differentiation

45-46

. The use of TPMS

scaffolds for stem cell differentiation has not been reported previously. In this study, the proposed


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method has been used to design a TPMS scaffold of different stiffness in order to affect stem cell differentiation into different lineages. It is widely known from the literature 1, 45-46 that certain matrix stiffness promotes differentiation of stem cells into certain lineages. The preferred stiffness ranges for adipogenic, neurogenic, myogenic, and osteogenic differentiation are 10-50 Pa, 0.1-1 kPa, 8-17 kPa, and 25-40 kPa respectively

1, 45-46

. The constraints applied in the optimization are porosity of greater

than 70%, and pore size between 100-300 µm, with the required Young’s modulus as stated previously. Two different materials were used depending on the desirable mechanical properties. For adipogenic and neurogenic lineage with E desirable in the range of several Pa, a soft material (Es=40 kPa) is used, and for myogenic and osteogenic lineage with E required in the range of several kPa, relatively stiff materials (Es=110 and 500 kPa) are used. The P, D, and G scaffolds containing four different regions with different stiffness are shown in Figure 4. Since the desired pore sizes for all regions are same, the surface type and cell size are considered constant in each scaffold for simplicity. In this example, the cell size is fixed to 0.5 mm, namely, the four optimized sub-scaffolds are characterized by the shell thickness and base material. The optimization problem is formulated as Eq. (7) by setting the upper and lower bounds of cell size to 0.501 and 0.499, respectively. In order to avoid extremely thin shell thicknesses, the minimal allowable Young’s moduli for adipogenic and neurogenic scaffolds are set to be the upper limits of the stiffness range. The parameters and properties of the optimized TPMS scaffolds are given in Table 3. The optimized scaffolds maintain the pore size in the range of 100-300 µm. The Young’s modulus and pore size are tailored by varying the shell thickness of the scaffolds. Since all sub-scaffolds have same cell sizes but different shell thicknesses, the thickness mismatch leads to sharp changes around the boundaries. This discontinuity could be alleviated by smoothing operations. Compare to P and G scaffolds, D scaffolds display the maximal porosity of the same Young’s modulus, contributing to materials saving and better permeability. However, the high porosity leads to very small shell thickness, which may cause manufacturing difficulties.



Figure 4. TPMS scaffolds containing four different regions with different stiffness so as to study the effect of scaffold stiffness on stem cell fate determination. (a) P surface in isometric view, (b) P surface in zoomed plane view, (c) D surface in isometric view, (d) D surface in zoomed plane view, (e) G surface in isometric view, (f) G surface in zoomed plane view. Green, purple and red colors represent the base material with Es=40, 110 and 500 kPa, respectively. In addition to the inherent advantage of the TPMS scaffolds in offering a high surface area to volume ratio enabling higher cell adhesion and proliferation, a TPMS scaffold with multi-stiffness regions as shown in Figure 4 would be useful to study the effect of substrate stiffness on stem cell differentiation within the same scaffold 1. It could also be used as in vitro models for metastatic cancer studies. Cancer metastasis is a phenomenon in which the tumor cells from primary cancer invade other parts of the body causing new tumor formations. Based on the primary cancer site, the possible site of metastatic cancer could be known. For instance, bone is the most frequent site of metastasis in breast


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cancer patients 47. A single scaffold with two distinct biomimetic regions depicting the primary cancer tissue site and the possible site of metastasis could aid in the study of the metastatic mechanism, which in turn would help in related drug discoveries for preventing the same. 3.3 Functionally Graded Scaffolds (FGS) The native tissues generally contain structural and functional gradients across a spatial volume

48

.

Hence, in order to engineer a biomimetic tissue, it is important that the scaffolds also exhibit biomimetic gradient structures 49. There are several tissues possessing structural gradients, of which the zonal cartilage architecture is the most studied. The zonal architecture of articular cartilage has three distinct zones namely superficial, middle and deep zones 50, with the deep zone connected to the subchondral bone. There is Young’s modulus gradient increasing from inner to the outer region, with individual preferred pore size for each region

50-52

, and a porosity of greater than 70% (as shown in

Table 4). Since the range of Young’s modulus throughout the graded scaffold is large, two different materials were used depending on the desirable mechanical properties, one for the three zones of cartilage (Es = 5 MPa) and one for the subchondral bone (Es = 5 GPa). The optimized P scaffolds are obtained by individually solving optimization problem Eq. (7) (see Table 4). Using the method described in section 2.4, the FGS for zonal cartilage architecture are generated as shown in Figure 5. The boundary functions in Eq. (8) are b( x, y, z ) = y − yi (i = 1, 2,3) where y1 = 0.5, y2 = 1.5, y3 = 3 (unit: mm), defining the planar boundaries between the subchondral and deep zones, the deep and middle zones, the middle and superficial zones, respectively. The factor that controls the shape transitions in Eq. (10) is set to k = 20 in order to achieve small transition zones meanwhile avoid sharp features. In order to demonstrate the effectiveness of combining sub-scaffolds of different size and shapes of the proposed method, two TPMS combinations are considered to design FGS. The FGS composed of single P scaffolds are shown in Figure 5 (b), in which the cell size of P scaffolds varies through the four regions. Another design incorporating different types of TPMS scaffolds (P-G-D-P from the subchondral bone to superficial zone) is shown in Figure 5 (c). The FGS include four sub-scaffolds with large cell size, scaffold shape and shell thickness variations (see Table 4). As can be seen in Figure 5, the size and shape of the FGS smoothly change along the boundaries of two zones. Since the sub-scaffolds have different wall thicknesses, the thickness mismatch is observed at the boundaries of different zones. This issue could be addressed by post-processing operations such as smoothing. The fabrication of multi-material scaffolds is challenging. In this example, the FGS includes two base materials in order to achieve the large Young’s modulus variation. While multi-material 3D printing is possible with multi-nozzle printers, only different types of polymers can be printed. With the two extreme range of materials considered, fabrication of such multi-material components is difficult with the current methods. Possible solution could be development of polymers with high mechanical



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stiffness (Es in the order of several GPa) by nanoparticle / fiber reinforcement and suitable rheological properties that could be printed using extrusion-based AM methods.

Figure 5. TPMS scaffolds with four distinct regions representing the zonal architecture of the articular cartilage. (a) Zonal architecture of articular cartilage, (b) scaffold consisting of P surface only, (c) scaffold consisting of P, D, and G surfaces. Green and blue colors represent the base material with Es=5 MPa and 5 GPa, respectively. Another application of FGS is in the area of bone implants. An implant with a homogenous structure and mechanical property might result in a problem called ‘stress-shielding’

53

. A radial gradient

structure with a decreasing stiffness from inside out is proposed to reduce this stress-shielding effect 54

. In this example, a radial FGS is designed by the proposed method for bone implants with an intent

of mitigating the stress-shielding effect, including two different zones, namely, inner and outer zones. Titanium alloy Ti-6Al-4V with Es = 114 GPa is considered. Using the same design method in section 2.3 and 2.5 with k = 5, the optimized sub-scaffolds with different types of TPMS and two designs of FGS are obtained. The properties of the optimized scaffolds are shown in Table 5 and the radial FGS for the bone implant are shown in Figure 6. Both single P surface (Figure 6 (a, b)) and multiple TPMS (D and G from inner to outer zone, Figure 6 (c, d)) can be smoothly connected to generate FGS. Two sub-scaffolds with cylindrical boundaries described by b( x, y, z ) = x 2 + y 2 − 1.52 (unit: mm) are included in the FGS, and the porosity increases while Young’s modulus decreases from the inner to the outer zone. The pore size of both subscaffolds are similar, while their cell sizes decrease from inside out due to the decreasing thicknesses.


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As can be seen in Figure 6, the size and shape of the radial FGS transit smoothly throughout the scaffold, and the thickness mismatch is smoothed.

Figure 6. Radial FGS based on TPMS with two distinct regions for bone implants. (a) P surface in 3D view, (b) P surface in top view, (c) G and D surfaces in 3D view, (d) G and D surfaces in top view.

3.4 Discussion A desired FGS requires perfectly smooth shape transitions at the boundaries of sub-scaffolds, allowing smooth property variations and no stress concentrations. The defects of FGS at the boundaries of two adjacent sub-scaffolds are mainly caused by two reasons: the first one is the shell thickness mismatch, the other one is the sharp transitions around the boundaries, which can lead to abrupt change of scaffold’s properties. For the first issue, the smoothing operations may not eliminate large thickness mismatch properly, which can be seen in Figures 4-6. This issue can be addressed by introducing shell thickness variations. Using this method and Eq. (10), the shell thickness of an FGS can be locally adjusted around the boundaries to ensure smooth thickness transitions. Figure 7 shows smooth thickness transitions throughout the FGS including periodicity changes, in which no thickness mismatch can be found, and no post smoothing process is required. Instead of using Eqs. (1), the TPMS sheet structures with thickness variations are described by the following approximations,



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Φ P ( x, y , z ) = (cos(ω x) + cos(ω y ) + cos(ω z )) 2 − c 2 = 0 Φ D ( x, y, z ) = (sin(ω x) sin(ω y ) sin(ω z ) + cos(ω x) sin(ω y ) sin(ω z ) + sin(ω x) cos(ω y ) sin(ω z ) + sin(ω x) sin(ω y ) cos(ω z )) 2 − c 2 = 0

(11)

Φ G ( x, y, z ) = (cos(ω x) sin(ω y ) + cos(ω y ) sin(ω z ) + cos(ω z ) sin(ω x)) 2 − c 2 = 0 where parameter c is able to control the porosity, pore size, and the average thickness of a TPMS unit cell. It should be noted that Eqs. (11) describe 3D solid models with non-uniform shell thickness, which is different from Eqs. (1) that define surface models only. Analyzing the mechanical and geometrical properties of variable thickness TPMS sheet structures increases the complexity and is out of the scope of this paper. Future studies on variable thickness TPMS sheet scaffolds are recommended.

Figure 7. P-type FGS with thickness variations and smoothly changed cell sizes The method described in Eq. (10) defined the FGS based on sigmond function, which is able to generate smooth boundary transitions with appropriate parameters. To further improve the smoothness of shape transitions when combining TPMS scaffolds with different cell sizes, a smooth function for cell size, a(x,y,z), can be defined within a scaffold domain. For simplicity, we use a 1D function to describe the cell size variations,


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a1   3 a −a  3(a − a ) x x a ( x, y , z ) =  2 1 ( − 3 ) + 2 1 4 2 δ 3 δ  a2 

x < x0 − δ x0 − δ ≤ x < x0 + δ x ≥ x0 + δ

(12)

where a1 and a2 are cell sizes for the two adjacent sub-scaffolds with the boundary defined by x=x0, and δ denotes transition width. In this example, we take the transition width to be the average cell size δ = (a1+a2)/2. As shown in Figure 7, the FGS with smoothly changed periodicity generated using Eqs. (10) and (12) exhibit no abrupt shape change and smoothly changed cell size and pore size.

4. Conclusion TPMS scaffolds provide several advantages over the conventional lattice-based scaffolds. In this work, an optimization approach for the design of biomimetic TPMS sheet scaffolds is investigated. Unlike the previous publications on scaffold optimization taking only one requirement at a time, all the three important scaffold parameters namely porosity, Young’s modulus, and pore size are simultaneously considered in the process of optimization. Our results reveal that the Young’s moduli of the TPMS scaffolds are nearly linear with the relative density, indicating their high mechanical stiffness. The relative Young’s moduli decrease monotonously with increasing porosity. The P scaffolds display the largest pore size, showing the high permeability. The D scaffolds have higher Young’s moduli than P and G scaffolds with same base material and porosity while the D scaffolds display the greatest surface area per volume. Using D surface to create scaffolds would reduce the weight and increase the porosity. The optimization process is applied to three different applications. In the first application of tissue-specific scaffold optimization, the desired pore size, and Young’s moduli are obtained from the literature that served as the constraints and the porosity is maximized. In the second application, P scaffolds with different stiffness were designed to effect in stem cell fate determination. These could also be used as in vitro models for metastatic cancer studies. Thirdly, functionally graded TPMS scaffolds (P) are designed using this approach to design a zonal cartilage architecture scaffold that mimics the native tissue with a functional gradient, and a bone implant with a stiffness gradient along its radial direction to mitigate the effect of stress shielding. The proposed approach could be used for designing biomimetic TPMS scaffolds, including those having a functionally gradient architecture, for tissue engineering, regenerative medicine, in vitro models for studying stem cell differentiation, cancer mechanisms, and drug discovery and development.



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Table 1. Desirable properties of tissue engineering scaffolds for various tissue types Cell / Tissue Type

Desirable Pore Size

Young’s modulus

References

Cancellous bone

500-1000 um

0.02-0.5 GPa

55-56

Cortical bone

Triply Periodic Minimal Surfaces Sheet Scaffolds for Tissue

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