Simple method to generate and fabricate stochastic porous scaffolds

Simple method to generate and fabricate stochastic porous scaffolds

Materials Science and Engineering C 56 (2015) 444–450 Contents lists available at ScienceDirect Materials Science and Engineering C journal homepage...

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Materials Science and Engineering C 56 (2015) 444–450

Contents lists available at ScienceDirect

Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec

Simple method to generate and fabricate stochastic porous scaffolds Nan Yang ⁎, Lilan Gao, Kuntao Zhou Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System, Tianjin University of Technology, Binshuixi Road No. 391, Xiqing District, Tianjin 300384, China

a r t i c l e

i n f o

Article history: Received 14 November 2014 Received in revised form 4 March 2015 Accepted 22 June 2015 Available online 27 June 2015 Keywords: Stochastic porous scaffold Function-based method Additive manufacturing

a b s t r a c t Considerable effort has been made to generate regular porous structures (RPSs) using function-based methods, although little effort has been made for constructing stochastic porous structures (SPSs) using the same methods. In this short communication, we propose a straightforward method for SPS construction that is simple in terms of methodology and the operations used. Using our method, we can obtain a SPS with functionally graded, heterogeneous and interconnected pores, target pore size and porosity distributions, which are useful for applications in tissue engineering. The resulting SPS models can be directly fabricated using additive manufacturing (AM) techniques. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Porous structures have recently become increasingly important, particularly for medical applications. In contrast to solid structures, porous structures typically have larger internal surface areas and higher strength-to-weight ratios [1], their interconnected pores can govern cell attachment [2], and they provide efficient cell seeding into a scaffold [3], the mass transfer metabolites [4], the delivery drugs or growth factors [5], and a sufficient regeneration space for newly formed tissue [6]. Traditional methods were used to fabricate stochastic porous scaffolds, such as salt leaching [7], gas-foaming [8], freeze-drying [9], and so on. However, these methods failed to accurately control pore properties including interconnectivities, sizes, and porosities [10]. To overcome these limitations, there has recently been some research on computer-aided design method for constructing stochastic porous structures (SPSs), but the methods were complex. Kou and Tan [11] used Voronoi vertices as the control points of a closed B-spline curve to create a cell. This method first obtained convex shaped cells duo to the properties of Voronoi diagram, and then required merging some adjacent cells to create a concave shaped cell. Thus, their algorithm was quite complex and might not be sufficiently flexible for constructing a SPS with a complex external shape. Nachtrab et al. [12] also used a Voronoi-based method for constructing a SPS. The solid phase of a network solid was defined as the set of all those points in space whose distance to the nearest network edge was less or equal to the cylinder radius, but this method may not flexibly construct functionally gradient porosities. Feng et al. [13] reconstructed two-phase composite materials

⁎ Corresponding author. E-mail address: [email protected] (N. Yang).

http://dx.doi.org/10.1016/j.msec.2015.06.039 0928-4931/© 2015 Elsevier B.V. All rights reserved.

using Gaussian random fields. The resulting structure met a binaryvalued marginal probability distribution function. Currently, a simple function-based method is used to describe pore geometries: a triply periodic minimal surface (TPMS) [14–19]. Compared with the above complex CAD methods, this method can simply use trigonometric implicit functions to derive a complex porous structure. However, based on current efforts, this method focuses on regular porous structures (RPSs) and RPSs with very simple functionally gradient pores. Furthermore, RPSs are quite different from the natural counterparts that are intended to be replaced and did not have some desired properties which SPSs can provide, including good mechanical, thermal, or other physical properties [20]. In this short communication, we extended our previous work [21] for constructing and fabricating SPSs. We could generate a SPS with (1) heterogeneous and interconnected pores, (2) functionally gradient pores, and (3) target porosity and pore size distributions, and keep the overall operations simple. When imported as STL files, the resulting models could be directly fabricated using AM techniques [22–24]. 2. Material and methods The core of generating a SPS is replacing the fixed parameters of a RPS model with random functions. 2.1. Functions representing a RPS A RPS is defined by   ϕ ax; by; cz; ex ; ey ; ez ; d ≤0:

ð1Þ

This inequality defines the 3D solid part for the structure. a, b, and c control the pore sizes in the x, y, and z directions, respectively. For

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will be deformed. To construct desired pore size or porosity distributions, random functions need to be generated within given value ranges. A 3D random function f(X) can be generated by f ðX Þ ¼

N X

λi θ −

kX−X i k2

i¼1

Fig. 1. Relationship between parameter d and porosity. Porosity linearly increases with increasing d for types 1, 2, 3, and 4 TPMS-based RPSs. These four types were defined by ϕ1(x,y,z) = cos(ax)sin(by) + cos(by)sin(cz) + cos(cz)sin(ax) + d ≤ 0 (also type G in [17]); ϕ2(x,y,z) = sin(ax)sin(by) + sin(by)sin(cz) + sin(cz)sin(ax) + d ≤ 0 (also type L); ϕ3(x,y,z) = cos(ax)cos(by) + cos(cz) + d ≤ 0 (also type P); ϕ4(x,y,z) = cos(ax)cos(by)cos(cz) − sin(ax)sin(by)sin(cz) + d ≤ 0 (also type D).

example, in x direction, “a” means that we arrange a cell per 2π/a unit length. d controls porosity, which is shown in Fig. 1 for the four types of TPMS-based porous structures. ex, ey, and ez are the offset distances in the x, y, and z directions which control the location of the entire structure. For example, the type P structure can be represented in the full form     ϕ ax; by; cz; d; ex ; ey ; ez ¼ cosðax þ ex Þ þ cos by þ ey þ cosðcz þ ez Þ þ d:

2.2. Generating a SPS In a RPS, a, b, c, ex, ey, ez, and d are all constants. But if we replace these parameters with random functions (such as f(X)), then the RPS

δ2

! ð2Þ

where X = (x,y,z) is the spatial coordinate, and Xi(i = 1,…,N) are the N random points generated within a given 3D scaffold domain D3. λi is a weight coefficient for point Xi, θ(.) is a basis function, and δ is an adjustable parameter. Xi and λi are generated randomly. However, we cannot assure if the random function generated using Eq. (2) is within a given range. This needs to normalize the function f in the given range f(X) ∈ [ f1, f2] for desired pore sizes or porosities. For example, if we intend to control the porosity from 20% to 60% for type 4 structure, then the value range of random function d(X) for ϕ4 should be from −0.2 to 0.2 as shown in Fig. 1. Thus, we define operation T[.] for f as T ½ f ðX Þ ¼ k f ðX Þ þ t

ð3Þ

where k = ( f2 − f1)/( fmax − fmin), t = f2 − kfmax, and fmax, fmin are, respectively, a global maximum and minimum of f that are determined by solving MaxðMinÞf ðX Þ s:t:X∈D3 :

ð4Þ

Operation T [.] can be regarded as the optimization processes for target pore sizes and porosities. In this way, we obtain random functions that are all within their own given ranges, such as, a′(X) ∈ [a 1 ,a 2 ], b′(X) ∈ [b 1 ,b 2 ], c′(X) ∈ [c 1 ,c2 ], d′(X) ∈ [d 1 ,d 2 ], e x ′(X) ∈ [e x1 ,e x2 ], e y ′(X) ∈ [e y1 ,e y2 ], and e z ′(X) ∈ [e z1 ,e z2 ] where a′(X), b′(X), c′(X), ex′(X), ey′(X), ez′(X), and d′(X) are the results of the T[.] operation of random functions a(X), b(X), c(X), ex(X), ey(X), ez(X), and d(X) generated by means of Eq. (2).

Fig. 2. (a) A 2D regular TPMS-based structure, (b) porous structure with gradient pore sizes, (c) porous structure with gradient porosities, and (d) local deformation; SPSs with different randomized degrees obtained by ex,ey ∈ [−3,3](e), ex,ey ∈ [−6,6](f), and ex,ey ∈ [−9,9](g).

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Fig. 3. (a) Cubic SPS and its cross-section (unit: mm), (b) the 3D products of a RPS (left) and a SPS (right).

Finally, a SPS is generated by substituting a′(X), b′(X), c′(X), ex′(X), ey′(X), ez′(X), and d′(X) into Eq. (1) as

  0 0 ϕ a0 ðX Þx; b ðX Þy; c0 ðX Þz; ex 0 ðX Þ; ey 0 ðX Þ; ez 0 ðX Þ; d ðX Þ ≤0:

ð5Þ

2.3. Fabrication Our proposed method can be executed using Wolfram Mathematica 9.0 software. This software provides various types of pseudo random numbers, an optimization package, and a 3D plot package. The 3D structures can also be exported as STL files using the same software.

Fig. 4. SPSs with gradient porosities along x direction (a) and radial direction (b).

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Fig. 5. Solid sphere shell with a SPS surface: (a) bottom view, (b) top view and locally magnified view.

Based on these STL models, we used stereolithography as an additive manufacturing means, and the steps were similar to Refs. [18] and [24]. Here, we simply introduce the processes and parameters. The resins consisted of methacrylated poly(ε-caprolactone) (PCL) macromers mixed with photoinitiator-TPO-L (3 wt.%), Orasol Orange G dye (0.1 wt.%), and bioactive glass (10 wt.%). Layers of 50 μm thick were cured using an exposure time of 12 s, a light intensity of 16 mW cm−2, and a lens focal distance of 75 mm. Excess macromer and photoinitiator were extracted from the scaffolds with acetone. Uncured resin was washed out. Then, the products were dried under a vacuum. 3. Results 3.1. Deforming a RPS Here, we used 2D examples for visualization purposes. A regular TPMS-based structure (type P) was defined by    cosðaðx þ ex ÞÞ þ cos a y þ ey þ d≤0: By setting a = 5π/20, d = 0, and ex = ey = 0, a RPS with 5 × 5 unit pores and porosity of 50% on the region D2:x ∈ [−20,20], y ∈ [−20,20] was constructed, as shown in Fig. 2a. Below, we would show how the RPS was deformed by changing parameters. Parameter a affects the pore size along x direction. By setting a = (5 + 0.125(x + 20))π/20, d = 0, and ex = ey = 0, we obtained a porous structure with a constant porosity of 50% and gradient pore sizes from 5 unit pores (x = −20) to 10 unit pores (x = 20), as shown in Fig. 2b. Parameter d affects the porosity. By setting a = 5π/20, d = (x + 20)/40 − 0.5, and ex = ey = 0, we obtained a porous structure with gradient porosities from 35% (x = − 20) to 60% (x = 20), as shown in Fig. 2c. Parameter ex,ey,ez can deform a part of the RPS. By setting a = 5π/20, d = 0, ex ¼ −10e−0:1ðx þy Þ, and ey = 0, only a part of the structure near the original point was deformed along x direction, as shown in Fig. 2d. Based on this, we could obtain a SPS when setting these parameters with random functions. For example, ex,ey were generated based on the method proposed in Section 2.2. The details of ex,ey in this case are shown in Appendix A. By setting a = 5π/20 and d = 0, we obtained three porous structures with different randomized degrees by ex,ey ∈ [− 3,3], ex,ey ∈ [− 6,6], as ex,ey ∈ [−9,9] as shown in Fig. 2e–g. Thus, a large value range of the terms ex,ey yielded a more disordered pore arrangement. 2

2

3.2. Three-dimensional SPS 3.2.1. Constructing and fabricating a cubic SPS Using our method, a 3D cubic SPS (263 mm3) with porosity of about 50% was generated using a type P TPMS-based function. This SPS and its

middle cross-section are shown in Fig. 3a, in which heterogeneous shapes for pores/solids are obtained. This SPS model was saved as an STL file and manufactured using AM techniques. The products of a cubic SPS with 49.98% porosity and a regular porous counterpart (type P) with 51.08% porosity are shown in Fig. 3b. 3.2.2. SPS with functionally gradient porosities Our method can also be used to generate a SPS with functionally gradient porosities when using random functions and deterministic functions at the same time. For example, also for a type P structure, cos(x + ex) + cos( y + ey) + cos(z + ez) + d ≤ 0, if ex,ey,ez were set as x , then random functions, but d was set as a linear function d ¼ 0:5 13 we obtained a SPS with gradient porosity along x direction as shown in Fig. 4a. Moreover, if we set d ¼ 0:5− 12 ðx2 þ y2 Þ, then a structure 13

with gradient porosity along the radial direction was generated as shown in Fig. 4b. Interestingly, the radial increasing porosity can mimic a femur structure that has a dense cortical shell and a cancellous interior. 3.2.3. An application of SPS As an application for femoral head replacement, in contrast to RPS, a SPS can provide better conditions to govern cell attachment and efficient cell seeding into an implant. Thus, we constructed a solid sphere shell with a SPS surface for this purpose as shown in Fig. 5. The thickness of the porous surface is about 3 mm, the porosity is around 67%, and the pore sizes are between 1 mm and 1.5 mm. This SPS surface was conjoint with the solid part (see Fig. 5a) and had interconnected pores (see the locally magnified view in Fig. 5b). 4. Discussion In this short communication, the porosity of a porous model was calculated by the volume of the pores divided by the overall volume of the region of interest (cubic region): Z Z Z p ¼ 1−

ðϕðx;y;zÞ ≤ 0Þ∩ðjxj ≤ 2L Þ∩ðjyj ≤ 2L Þ∩ðjzj ≤ 2L Þ

L3

d xdydz ð6Þ

where L was the length (width or height) of the cubic region, and p was the porosity. Rather than the random functions or other parameters, d (in Eq. (1)) is still a major parameter to control the porosity for a SPS. For example, in Section 3.2.1, by setting d = 0, we intended to construct a SPS of 50% porosity based on type P TPMS-based regular structure (see line “3” in Fig. 1). As a result, we obtained the SPS of 49.98% porosity that was measured using Eq. (6). In our experiments, we always obtained desired

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Table 1 The relationship between η and porosity for SPS and RPS of type P. Porosity

50%

60%

70%

74%

80%

84%

η

100% 100%

100% 100%

100% 100%

93.6% 100%

19.4% 25.6%

1.8% 0.79%

SPS RPS

porosity according to the relationship between d value and porosity in Fig. 1. The porosity of a SPS constructed using this method has an upper limit. Exceeding this limit, the solid phase begins to become disconnected. Here, the connectivity η was simply defined by the volume of the maximum connected solid phase divided by the volume of the overall solids. We investigated the connectivity η (using Rhinoceros software) of a SPS and a RPS of type P by increasing their porosities, and the data were shown in Table 1. Exceeding the porosity of 74%, the solid phase of the SPS was disconnected, and the value of η decreased rapidly. Even the RPS had a limit of porosity. Exceeding that limit (between 74% and 80%), all the struts that joined the spheres were broken, and only those isolated spheres were remained (see Fig. 6). Therefore, the RPS had a smaller η value (0.79%) than that of the SPS (1.8%) at 84% porosity. This also implied that if the solid volume fraction exceeded a given value, then the void (pore) phase might be disconnected. By interchanging the solid/void phase (because Rhinoceros software can handle solid conveniently), we found that the interconnectivity of the void phase had the same regularity when increasing the solid volume fraction. As limitations of this short communication, we did not investigate the porosity, pore size distribution and interconnectivity of fabricated scaffolds. Using micro-CT techniques, the structural data of fabricated scaffolds can be imported into computers. After reconstruction, those morphological analysis can be performed using Feldkamp algorithm [18]. Moreover, the influences of different material properties on the scaffold structure also need to be investigated. These will be a future work.

Fig. 6. Broken struts and isolated spheres in a P type RPS of 84% porosity.

5. Conclusion In this short communication, we have proposed a straightforward method for constructing SPSs. Replacing the constant parameters of a RPS model with random functions immediately yields a SPS. We can easily derive heterogeneous pore morphologies and keep them interconnected. This method provides for constructing a SPS with target pore size and porosity distributions. When imported as an STL file, the resulting model can be directly fabricated using AM techniques.

Acknowledgments This work was supported by the China Post-doctoral Foundation No. 2012M520572, Tianjin Municipal Education Commission Grant No. 20120401, and Tianjin Municipal Science and Technology Commission Key Grant No. 14JCZDJC39500. We also thank Dr. Duaine Jackola for proofreading this article.

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Appendix A. Functions ex, ey mentioned in Section 3.1.

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