Design and statistical analysis of irregular porous scaffolds for orthopedic reconstruction based on voronoi tessellation and fabricated via selective laser melting (SLM)

Journal Pre-proof Design and statistical analysis of irregular porous scaffolds for orthopedic reconstruction based on Voronoi tessellation and fabricated via selective laser melting (SLM)

Yue Du, Huixin Liang, Deqiao Xie, Ning Mao, Jianfeng Zhao, Zongjun Tian, Changjiang Wang, Lida Shen PII:

S0254-0584(19)30767-9

DOI:

https://doi.org/10.1016/j.matchemphys.2019.121968

Article Number:

121968

Reference:

MAC 121968

To appear in:

Materials Chemistry and Physics

Received Date:

11 June 2019

Accepted Date:

05 August 2019

Please cite this article as: Yue Du, Huixin Liang, Deqiao Xie, Ning Mao, Jianfeng Zhao, Zongjun Tian, Changjiang Wang, Lida Shen, Design and statistical analysis of irregular porous scaffolds for orthopedic reconstruction based on Voronoi tessellation and fabricated via selective laser melting (SLM), Materials Chemistry and Physics (2019), https://doi.org/10.1016/j.matchemphys. 2019.121968

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.

Journal Pre-proof Design and statistical analysis of irregular porous scaffolds for orthopedic reconstruction based on Voronoi tessellation and fabricated via selective laser melting (SLM) Yue Du1,

3, §,

Huixin Liang1, §, Deqiao Xie1, Ning Mao1, 4, Jianfeng Zhao1, Zongjun Tian1,

Changjiang Wang2, Lida Shen1, * 1College

of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and

Astronatics, 29 Yudao Street, Nanjing 210016, PR China. 2Department

of Engineering and Design, University of Sussex, Sussex House, Brighton BN19RH,

United Kingdom. 3Suzhou

Kangli Orthopedics Instrument Co, Ltd, Suzhou, 215600, PR China.

4Suzhou

Yunzhi Medical Technology Co, Ltd, Suzhou, 215600, PR China.

§These authors contributed equally. *Corresponding Author. E-mail: [email protected] (L.D. Shen) The use of irregular porous structures in bone tissue engineering (BTE) has attracted increasing attention. An irregular porous structure is similar to that of human bone tissue and is more suitable for bone tissue growth than a regular porous structure. In this study, we propose a top-down design method for constructing irregular porous structures based on Voronoi tessellation. The model constructed with this method was fabricated by selective laser melting (SLM) and industrial computed tomography (CT) was used to measure the morphological characteristics; the mechanical properties were obtained by quasi-static compressive test. The experimental results showed that the samples had a porosity ranging from 50% to 85%, an average pore diameter ranging from

Journal Pre-proof 512 to 998 um, an apparent elastic modulus ranging from 2.13 to 3.97 Gpa, and a compressive strength ranging from 78.99 to 130.5 Mpa and met the artificial implant requirements. Three regression equations were established between the three design parameters and the porosity, apparent elastic modulus, and compressive strength using a response surface methodology (RSM). The equations allow for better control of the irregular porous structure and the prediction of the properties. Keywords: Porous scaffolds; Voronoi tessellation; SLM; Stress shielding; Young’s modulus

1. Introduction The recent success in using additive manufacturing (AM) in the field of orthopedic regenerative medicine has greatly increased interest in titanium alloy implants [1, 2]. The apparent elastic modulus of human bone ranges from 0.02 to 30 GPa depending on the type of bone tissue and the direction of measurement [3-5]. The large difference in the apparent elastic modulus between the implant and human bone results in difficulty in transferring the stress between the human bone and the implant; this is referred to as “stress shielding” [6]. The apparent elastic modulus of a porous structure implant better matches that of the human bone; therefore, the use of a porous structure implant reduces the stress shielding problem caused by artificial implants and the loosening and fracture of artificial implants in clinical applications due to stress shielding [7-9]. Porous structure implants also have a large specific surface area and good connectivity, which promotes cell migration and adhesion, nutrient and oxygen flow, and promotes biologic fixation between human bone and the porous structure implant [10-14]. Compared to traditional manufacturing, AM uses layer-by-layer processing and manufacturing based on 3D model data, which greatly increases the

Journal Pre-proof design flexibility [15-18]. The porous Ti6Al4V structure manufactured by AM using laser selective laser melting (SLM) and electron beam melting (EBM) has good biocompatibility and mechanical properties [19]. Porous structures can be categorized as regular and irregular porous structures. At present, the modeling methods of regular porous structures include the computer-aided design (CAD)-based unit cell method, the topology optimization method, and the triply periodic minimal surface method. Their common feature is that the methods are based on a unit cell and the entire model consists of unit cell geometric array. Inevitably, the porous structure obtained by these methods has periodicity and regularity [21]. Since a small change to the unit cell will result in a global change in the entire internal structure, it is difficult to apply local control to the pore shape and the pore size distribution [21]. The CT/magnetic resonance imaging (MRI)-based reverse modeling method can also accurately mimic human bone tissue and has demonstrated the superiority of an irregular porous structure [22,23]. However, the porous model obtained by the reverse modeling method is difficult to modify later. In addition, the artificial implant with a large difference from the human bone material still faces the problem of mismatch with the human skeleton. The development of a porous structure based on Voronoi tessellation recently has received increasing attention. On the one hand, because the porous structure is similar to the complex microstructure of human bone, the pore size distribution has a large range, unlike the small range of the pore size distribution of a regular structure. On the other hand, different structural and mechanical performance requirements can be met by adjusting the design parameters. At present, the irregular porous structure based on Voronoi tessellation [24-27] is limited specific modeling methods and it is not easy to control the strut diameter and the porous structure. The relationships

Journal Pre-proof between the design parameters of porous structures and the porosity, pore size distribution, and related mechanical properties are also unclear. In this study, we propose a top-down design method for constructing irregular porous structures based on Voronoi tessellation. The model constructed with this method is fabricated by SLM; industrial CT is used to determine the morphology and the mechanical properties were obtained by a quasi-static compressive test. The relationships between the structural design parameters and the porosity and the mechanical properties are determined by establishing three regression equations using the response surface methodology (RSM).

2. Materials and methods 2.1 Modeling and characterization of irregular porous scaffolds

Voronoi tessellation is a method of space partitioning. A number of points are distributed in a particular Euclidean space. These points are called seed points. Circles with an initial radius of 0 are drawn around these seed points and expand at the same speed. The two circles touch and form a line or boundary. This is continued until the circles cannot be expanded further. Finally, in the 2D space, a Voronoi diagram bounded by line segments is formed and in 3D space, a 3D Voronoi diagram bounded by a polygon plane is formed [28]. The mathematical relationship of the Voronoi tessellation is defined as follows [29]： 𝑉(𝑝𝑖) = {𝑝 𝑑(𝑝,𝑝𝑖) ≤ 𝑑(𝑝,𝑝𝑗),𝑗 ≠ 𝑖,𝑗 = 1,…,𝑛}

(1)

Where 

𝑝 = {𝑝𝑖,…,𝑝𝑛} is a set of distinct seed points located in the d-dimensional Euclidean space Rd;



𝑑(𝑝,𝑝𝑖) represents the Euclidean distance between the location 𝑝 and seed 𝑝𝑖;

Journal Pre-proof 

𝑉(𝑝𝑖) represents the ordinary Voronoi polygon associated with seed 𝑝𝑖; In this study, the controllable irregular porous structure was designed using the software

Rhinoceros 6 (McNeal, Seattle, WA, USA) with the parametric design plugin Grasshopper (v.1.0.0007). In the model design, first, a uniformly distributed set of points was generated in space as seen in Figure 1. The points are called the original points, where the distance between two points is defined as the unit distance d. Based on the original points, a series of cubes with side length 2r were generated and the original points were the area center of the bottom surface of the cube, where 0 ≤ 𝑟 ≤ 𝑑 2, 𝑖 = 𝑟 𝑑, i was defined as the irregularity.

Fig. 1 Original point lattice. (a) Top view. (b) Front view. (c) Perspective view. Second, we generated a random point in each cube area; these points were the seed points, as mentioned above. Subsequently, cells of the 3D Voronoi diagram were generated in Grasshopper by generating seed points, where each seed point corresponded to a Voronoi cell. Third, we extracted the wireframe of the 3D Voronoi diagram and created a porous structure with a strut diameter of D. A Boolean operation was used to create smooth porous scaffolds. The model generation process is shown in Figure 2.

Journal Pre-proof

Fig. 2 Model generating process. (a) Designing the original point lattice. (b) Generating cubes. (c) Obtaining seed points. (d) Generating Voronoi cells. (e) Obtaining wireframe of Voronoi cells. (f) Constructing porous structures. At this time, the three structural design parameters (strut diameter D, unit distance d, irregularity i) were used to construct the irregular porous scaffolds.

2.2 Fabrication and test

The samples designed in this study were manufactured using an SLM machine (NCL-M2120, China). The material was Ti6Al4V powder with an average particle size of 40 um. The optimum processing parameters were: laser power of 130 W, scanning speed of 1200 mm/s, hatch spacing of 0.05 mm, layer thickness of 30 um; argon was used as a protective gas. Preparing three samples of each irregular porous structure. All samples were removed from the base via wire electrical discharge machining (WEDM). In order to characterize the structural features of the irregular porous structure, we evaluated the

Journal Pre-proof porosity and average pore size because they represent the macroscopic and microscopic structural features of the irregular porous structure, respectively. With the powerful parametric design capabilities of Grasshopper, we could calculate the porosity and average pore size of the model in the software. Figure 3 shows Si and Si* on a face of the single voronoi cell. Here, the porosity and the average pore diameter correspond to Eqs. (2), (3), and (4), respectively: 𝑃 = (1 ― 𝑉𝑝 𝑉) × 100%

(2)

Where P is the porosity, Vp is the volume of the porous scaffolds, V is the outer volume of the porous scaffold. ∗ 𝑊𝑖 = 4(𝑆𝑖 ) 𝜋 1

(3) 1

𝑛 𝑛 ∗ 𝑊𝑎 = 𝑛∑𝑖 = 1𝑊𝑖 = 𝑛∑𝑖 = 1 4(𝑆𝑖 ) 𝜋

(4)

where Si is the area of the face in the Voronoi cell, Si* is the area occupied by the scaffolds on the face of the Voronoi cell.

Fig. 3 Definition of Si and Si* on a face of the single voronoi cell. X-ray industrial CT is a fast and effective non-destructive test method. Industrial computed tomography (XT H 225, Nikon, Japan) with 32 um resolution was used to scan samples at tube voltage of 192 kv. The industrial CT data were analyzed and visualized using VGSTUDIO MAX 3.1 (Volume Graphics GmbH, Germany), as shown in Figure 4.

Journal Pre-proof

Fig. 4 Sectional view of the reconstruction of the 3D model by industrial CT (D=0.6 mm, d=1.9 mm, i=0.25). Finally, a 3D model of an irregular porous structure was created to measure the porosity in the software. A quasi-static compression test was performed according to the ISO standard 13314:2011 using a mechanical testing machine (CMT 5105, MTS System Corporation, USA) with a deformation speed of 1 mm/min. The apparent elastic modulus E of each sample is the slope of the linear phase of the stress-strain curve. The maximum compressive strength S is obtained from the first stress peak of the stress-strain curve.

2.3 Statistical analysis

In order to analyze the relationship between the mechanical properties and the porosity and design parameters of the irregular porous structures, we used the Box-Behnken experimental design method for the RSM. The three structure design factors (strut diameter D, unit distance d, and irregularity i) were the independent variables. The response surface analysis was conducted for 17 series with three factors and three levels. The data analysis was conducted using Design-expert 8.0.6. In the regression models, the porosity, apparent elastic modulus, and compressive strength were the response variables.

Journal Pre-proof 3. Results and discussion 3.1. Controllability of irregular porous scaffolds

Figure 5 shows the irregular porous scaffolds samples of all series. Figure 6 shows the scatter plots of the porosity values of the irregular porous scaffolds samples of all series. Table 1 lists the average pore diameters of all series (design value). The pore diameter ranges from 512 to 998 um, Table 2 lists the porosity design value of the irregular porous structure model and the actual porosity value of the sample. The difference is less than 4%, demonstrating that the SLM technology has high manufacturing precision; the result also indicates that the difference between the actual value and the design value of the pore diameter is quite small. In the other word, the average pore diameter (actual value) of irregular porous scaffolds samples are within the optimum size of 200 to 1200 um for bone cell ingrowth [30]. The pore diameter distribution of a typical irregular porous structure (D = 0.6 mm, d = 1.9 mm, i = 0.25) is shown in Fig. 7, which shows a trend of fluctuation but overall increase, 58.62% of the pores are within the optimum size of 200 to 1200 um. For ease of labeling in the figure, we abbreviate the three design parameters of the model or sample as follows: for example, if the structural design parameters of the model or sample are strut diameter D=0.6 mm, unit distance d=1.9 mm, and irregularity i= 0.25, we abbreviated this as 0619025. Table 1 Average pore diameter of all series (design value). Series

1-5

6

7

8

9

10

11

12

13

14

15

16

17

Wa (um)

753

637

825

836

998

584

601

946

953

512

702

721

869

P*: average design value of pores diameter.

Table 2 Design values and actual values of the porosity of the samples of the irregular porous

Journal Pre-proof scaffolds. Model porosity

Sample porosity

Deviation

(%)

(%)

(%)

1 (D=0.6mm, d=1.9mm, i=0.25)

69.07

66.86

2.21

2 (D=0.6mm, d=1.9mm, i=0.25)

69.07

67.13

1.94

3 (D=0.6mm, d=1.9mm, i=0.25)

69.07

66.97

2.1

4 (D=0.6mm, d=1.9mm, i=0.25)

69.07

66.55

2.52

5 (D=0.6mm, d=1.9mm, i=0.25)

69.07

65.62

3.45

6 (D=0.5mm, d=1.6mm, i=0.25)

71.48

68.19

3.29

7 (D=0.5mm, d=1.9mm, i=0.1)

80.12

77.85

2.27

8 (D=0.5mm, d=1.9mm, i=0.4)

78.87

76.32

2.55

9 (D=0.5mm, d=2.2mm, i=0.25)

83.47

80.51

2.96

10 (D=0.6mm, d=1.6mm, i=0.1)

64.9

62.06

2.84

11 (D=0.6mm, d=1.6mm, i=0.4)

61.72

58.44

3.28

12 (D=0.6mm, d=2.2mm, i=0.1)

78.53

75.47

3.06

13 (D=0.6mm, d=2.2mm, i=0.4)

77.11

74.86

2.25

14 (D=0.7mm, d=1.6mm, i=0.25)

51.96

48.24

3.72

15 (D=0.7mm, d=1.9mm, i=0.1)

65.46

62.56

2.9

16 (D=0.7mm, d=1.9mm, i=0.4)

62.54

58.73

3.81

17 (D=0.7mm, d=2.2mm, i=0.25)

70.0

67.48

2.52

Series

Journal Pre-proof

Fig. 5 Irregular porous scaffolds samples of all 17 series.

Fig. 6 Scatter plot of porosity values the irregular porous scaffolds samples of all series.

Fig. 7 The curve for the pore diameter distribution (design value) of a typical irregular porous

Journal Pre-proof structure (D = 0.6 mm, d = 1.9 mm, i = 0.25)

3.2. Statistical analysis of the relationship between the structural design parameters and the porosity and mechanical properties

Table 3 lists the experimental results of the compression test and Figure 8 shows the compressive test curves of the three sets of irregular porous structure samples. The stress-strain curves of the compression test of the three sets of samples have three stages, i.e., the elastic stage, platform stage, and compact stage. In the elastic phase, the stress increases linearly with the strain. In the platform stage, the stress begins to fluctuate with the buckling and failure of the scaffolds. In the compact phase, the stress begins to rise rapidly until the sample is completely destroyed. Table 3 Experimental design and results. Design parameters

Experiment results

Series D (mm)

d (mm)

i

P (%)

E (Gpa)

S (Mpa)

1

0.6

1.9

0.25

66.86

2.13

109.26

2

0.6

1.9

0.25

67.13

2.33

118.05

3

0.6

1.9

0.25

66.97

2.18

111.23

4

0.6

1.9

0.25

66.55

2.25

115.43

5

0.6

1.9

0.25

65.62

2.17

112.93

6

0.5

1.6

0.25

68.19

3.97

130.5

7

0.5

1.9

0.1

77.85

3.73

96.63

8

0.5

1.9

0.40

76.32

2.96

103.86

9

0.5

2.2

0.25

80.51

3.72

84.78

10

0.6

1.6

0.10

62.06

3.12

111

11

0.6

1.6

0.40

58.44

2.88

112.89

12

0.6

2.2

0.10

75.47

2.59

79.19

Journal Pre-proof 13

0.6

2.2

0.40

74.86

3.01

78.99

14

0.7

1.6

0.25

48.24

2.59

108.98

15

0.7

1.9

0.10

62.56

2.63

90.57

16

0.7

1.9

0.40

58.73

2.58

92.49

17

0.7

2.2

0.25

67.48

2.44

80.1

Fig. 8 Compressive test curves of the three series of irregular porous structure samples. Table 4 lists the factors and levels in irregular porous scaffolds in the Box-Behnken design. An analysis of variance (ANOVA) of the porosity P was performed, as listed in Table 5. A P-value less than 0.05 indicated that the model terms were significant. The results show that all three structure parameters (strut diameter D, unit distance d, and irregularity i) affect the porosity. The high value of the coefficient of determination (R2=0.9965) indicates that 99.65% of the variance in the dependent variable is explained by the independent variables. The model has a good degree of fit and the test error is small. The quadratic equation of the porosity is shown in Eq. (5). In fact, if the SLM manufacturing capability allows for it, the three design parameters of the irregular porous structure can be freely combined over a larger range, thereby achieving a larger adjustment range of the porosity. 𝑃 = 120.48072 ― 220.49833𝐷 +22.23361𝑑 ―59.44722𝑖 +57.66667𝐷 × 𝑑 ―

38.333333

Journal Pre-proof 𝐷 × 𝑖 +16.72222𝑑 × 𝑖 +31.825𝐷2 ―9.325𝑑2 +85.36667𝑖2

(5)

Table 4 Factors and levels in irregular porous scaffolds in the Box-Behnken design Level Factor -1

0

1

Strut diameter D (mm)

0.5

0.6

0.7

Unit distance d (mm)

1.6

1.9

2.2

Irregularity i

0.1

0.25

0.4

Table 5 ANOVA for the porosity P Source

Sum of Squares

df

Mean Square

F Value

p-value

Model

1058.78

9

117.64

415.93

< 0.0001

A-D

542.19

1

542.19

1916.94

< 0.0001

B-d

471.09

1

471.09

1665.56

< 0.0001

C-i

11.50

1

11.50

40.64

0.0004

AB

11.97

1

11.97

42.33

0.0003

AC

1.32

1

1.32

4.68

0.0674

BC

2.27

1

2.27

8.01

0.0254

A2

0.43

1

0.43

1.51

0.2592

B2

2.97

1

2.97

10.49

0.0143

C2

15.53

1

15.53

54.92

0.0001

R2=0.9981 An ANOVA of the apparent elastic modulus E was performed, as listed in Table 6. The results show that the strut diameter D has a significant effect on the apparent elastic modulus E. The R2 value is 0.9398. The model has a good degree of fit and the test error is small. The quadratic equation

Journal Pre-proof of the apparent elastic modulus E is shown in Eq. (6). If the SLM manufacturing capability allows for it, the three design parameters of the irregular porous structure can be freely combined over a larger range, thereby achieving a larger adjustment range of the apparent elastic modulus, and satisfy the optimum range of the apparent elastic modulus for human bone. 𝐸 = 48.02853 ― 72.95083𝐷 ― 20.44389 𝑑 ― 20.475𝑖 + 0.83333 𝐷 × 𝑑 + 12.5𝐷 × 𝑖 2 𝐷 + 3.66667𝑑 × 𝑖 + 52.525 + 4.91944𝑑2 +10.9𝑖2

(6)

Table 6 ANOVA for the apparent elastic modulus E Source

Sum of Squares

df

Mean Square

F Value

p-value

Model

4.98

9

0.55

11.95

0.0018

A-D

2.14

1

2.14

46.30

0.0003

B-d

0.080

1

0.080

1.73

0.2300

C-i

0.051

1

0.051

1.11

0.3278

AB

2.500E-003

1

2.500E-003

0.054

0.8229

AC

0.13

1

0.13

2.80

0.1381

BC

0.11

1

0.11

2.35

0.1689

A2

1.15

1

1.15

24.75

0.0016

B2

0.84

1

0.84

18.14

0.0038

C2

0.25

1

0.25

5.31

0.0547

R2=0.9398 The ANOVA results for the compressive strength S are listed in Table 7. The R2 value is 0.9806, indicating a good degree of fit of the model with a small test error. The quadratic equation of the compressive strength S is shown in Eq. (7). By adjusting the structural design parameters, a stronger

Journal Pre-proof irregular porous structure can be obtained. 𝑆 = ―104.38542 + 416.15417 𝐷 +127.50694 𝑑 +340.47222 𝑖 + 140.33333 𝐷 × 𝑑 ― 88.5𝐷 × 𝑖 ―11.61111 𝑑 × 𝑖 ―596 𝐷2 ―70.33333 𝑑2 ―512.55556 𝑖2

　

(7)

Table 7 ANOVA for the compressive strength S Source

Sum of Squares

df

Square

F Value

p-value

Model

3760.42

9

417.82

39.39

< 0.0001

A-D

237.95

1

237.95

22.43

0.0021

B-d

2460.86

1

2460.86

231.97

< 0.0001

C-i

14.69

1

14.69

1.38

0.2778

AB

70.90

1

70.90

6.68

0.0362

AC

7.05

1

7.05

0.66

0.4418

BC

1.09

1

1.09

0.10

0.7577

A2

149.56

1

149.56

14.10

0.0071

B2

168.71

1

168.71

15.90

0.0053

C2

559.99

1

559.99

52.79

0.0002

R2=0.9806 In order to verify the relationship between the three design parameters of the irregular porous structure (Eqs. (5) - (7)) and the porosity and mechanical properties, we designed three different structures (a, b, c) using the same materials, manufacturing processes, and testing requirements. The predicted values and the measured values are listed in Table 8. A positive or negative deviation of less than 10% between the predicted value and the experimental value indicates that the three regression equations are suitable when the three independent variables are in or near the range (D ∈[0.5,0.7],d∈[1.6,2.2],i∈[0,0.5]). Table 8 The predicted values and the measured values of three series samples

Journal Pre-proof

porous scaffolds

a (0415025)

b (0617040)

c (0823045)

predictive value

experimental value

Deviation（%）

P (%)

77.25

76.34

1.19

E(Gpa)

6.34

6.12

3.59

S(Mpa)

123.81

115.64

7.07

P (%)

61.36

62.81

-2.31

E(Gpa)

2.53

2.66

-4.89

S(Mpa)

112.43

118.59

-5.19

P (%)

66.41

68.23

-2.67

E(Gpa)

5.11

4.98

2.61

S(Mpa)

32.06

35.16

-8.82

The effects of the structural design parameters (D, d, i) on the porosity P, apparent elastic modulus E, and compressive strength S under single factor conditions are shown in Figure 9, 10, and 11 respectively; the values were calculated using Eqs. (5) - (7). When the unit distance d and the irregularity i are determined, the frame of the irregular porous structure (Voronoi cell frame) has been determined and it does not change as the strut diameter D changes. As shown in Figure 9, the porosity P decreases linearly with the increase in the strut diameter D. Un-expectedly, the apparent elastic modulus E first decreases and then increases and the opposite is true for the compressive strength S. At D = 0.6 mm, E reaches the minimum value and S reaches the maximum value. There are many struts in the Voronoi cell that are close to each other. As the strut diameter increases, these struts are combined, resulting in changes in the irregular porous structure. This may cause fluctuations in the apparent elastic modulus E and compressive strength S. For the unit distance d, a change in d causes a significant change in the irregular porous structure (Voronoi cell frame). As

Journal Pre-proof shown in Figure 10, as d increases, the porosity P increases, the compressive strength S decreases, and the apparent elastic modulus E first decreases and then increases. When d = 1.9 mm, the minimum value is obtained. As shown in Figure 11, as the degree of irregularity increases, the porosity P and the apparent elastic modulus E first decrease and then increase; the minimum is reached at i=0.3. The results are opposite for the compressive strength S. This result is different from the results of a previous study [26], which may be attributed to the difference in modeling methods.

Fig. 9 The effect of the strut diameter D on the porosity P, the apparent elastic modulus E, and the compressive strength S (d=1.9 mm, i=0.25).

Journal Pre-proof

Fig. 10 The effect of the irregularity d on the porosity P, the apparent elastic modulus E, and the compressive strength S (D=0.6 mm, i=0.25).

Fig. 11 The effect of the irregularity i on the porosity P, the apparent elastic modulus E, and the compressive strength S (D=0.6 mm, d=1.9 mm).

Journal Pre-proof 3.3 Relation between porosity and mechanical properties

As a simplified porous unit, the Gibson-Ashby model is widely used for the evaluation of porous structures. The model describes the power function relationship between the porosity and the apparent elastic modulus and the compressive strength [31]. The details as follows: 𝑚

(8)

𝑛

(9)

𝐸 = k1(1 ― 𝑃 100) 𝐸0 𝑆 = k2(1 ― 𝑃 100) 𝑆0

It is evident from the Eqs. (8) and (9) that E and S decrease as the porosity P increases. Wang et al. [26] and Liang et al. [27] demonstrated that the relationship between the porosity and the apparent elastic modulus and the compressive strength are consistent with the Gibson-Ashby model when the irregularity is given. However, when the irregularity is accounted for, the Gibson-Ashby model is not applicable. In this study, we believe that the Gibson-Ashby model is not suitable for the irregular porous structure we designed, regardless of the above mentioned two aspects. When the irregularity is given (i=0.25), as the porosity increases, the apparent elastic modulus and compressive strength do not exhibit a decreasing trend in the power function (Figures 9, 10). Similarly, when the irregularity is considered and the porosity increases, the apparent elastic modulus and compressive strength also do not exhibit a decreasing trend in the power function (Figure.11). Figure 12 depicts the scatter plots between the porosity P and the apparent elastic modulus E and the compressive strength S; there are no significant correlations. Near a certain porosity value, there are two apparent elastic modulus values that differ greatly. For example, although the actual porosity of the two samples of the sixth and the 17th group are similar at 68.17% and 67.48%, the apparent elastic modulus values are 3.97 Gpa and 2.44 Gpa respectively. The compressive strength

Journal Pre-proof is 130.5 Mpa and 80.1 Mpa respectively. In fact, Eq. (5) shows that the porosity is affected by the three structural design parameters, namely, the strut diameter D, unit distance d, and irregularity i. The same porosity may be determined by different combinations of the three structural design parameters. Equations (6) and (7) also show that the apparent elastic modulus and compressive strength of the regular porous structure also change accordingly. Therefore, for the irregular porous structure designed by the proposed method, there are many different structures that have different mechanical properties for the same porosity.

Fig. 12 Scatter plots between porosity P and (a) apparent elastic modulus E and (b) compressive strength S. For the irregular porous structure we designed, the relationships between the porosity and apparent elastic modulus and the compressive strength cannot be simply summarized as a causeeffect relationship, i.e., an increase in one parameter results in a decrease in another parameter (similar to the Gibson-Ashby model). On the one hand, we can predict the porosity and mechanical properties from Eqs. (5) to (7) and compare the trends. On the other hand, in view of the complexity of the irregular porous structure based on Voronoi tessellation, more complex relationships may exist and further research is needed.

Journal Pre-proof 4. Conclusion In this study, we proposed a method to construct irregular porous scaffolds based on Voronoi tessellation by adjusting the three structural design parameters (strut diameter D, unit distance d, irregularity i). We constructed samples using SLM. Our research results based on a series of experimental evaluations are summarized as follows: (1) A method based on Voronoi tessellation was used to construct irregular porous scaffolds and the three independent structural design parameters were adjusted using the parametric design software Grasshopper. (2) The obtained irregular porous structure exhibited good properties. Porous scaffolds with an average pore diameter ranging from 512 to 998 um and a porosity ranging from 50% to 85% were fabricated precisely using SLM. Satisfactory results were obtained for the mechanical properties adjusted by the four structural design parameters. The apparent elastic modulus varied from 2.13 to 3.97 Gpa and the compressive strength varied from 78.99 to 130.5 Mpa, which basically met the bone tissue engineering requirements. (3) We determined the relationship between the structural design parameters and the porosity, apparent elastic modulus, and compressive strength. Three quadratic regression equations were developed to predict the parameters. The experimental verification indicated a good fit of the models. It was determined that the Gibson-Ashby model was not suitable for the irregular porous structure.

Acknowledgements The work was financially supported by the Advanced Research Project of Army Equipment

Journal Pre-proof Development (301020803), the Key Research and Development Program of Jiangsu (BE 2015161) , the Jiangsu Provincial Research Foundation for Basic Research, China (BK 20161476), the Science and Technology Planning Project of Jiangsu Province of China (BE 2015029) and the Science and Technology Support Program of Jiangsu (BE 2016010-3), The Nanjing University of Aeronautics and Astronautics major project cultivation plan(NP2017414),The Nanjing University of Aeronautics and Astronautics Youth Technology Innovation Fund (NT2018016).

Conflict of interest The authors declare no competing financial interest. Declarations of interest: none

Reference 1. André, L.J, Maria, A. L, Rubens, M. F, Cecília, A. de. C. Z, Luis, F. B, Carlos S. L, Dav i, R. C, Paulo, K (2014) Cranial reconstruction: 3D biomodel and custom-built implant c reated using additive manufacturing. J. Cranio-MaxilloFac. Surg 42:1877-1884. https://doi.o rg/10.1016/j.jcms.2014.07.006 2. Mertens, C, Lowenheim, H, Hoffmann. J (2013) Image data based reconstruction of the mi dface using a patient-specific implant in combination with a vascularized osteomyocutane ous scapular flap. J. Cranio-MaxilloFac. Surg 41:219-225.https://doi.org/10.1016/j.jcms.20 12.09.003 3. Wang, X.J, Xu, S.Q, Zhou, S.W, Xu, W, Leary, M, Choong, P, Qian, M, Brandt, M, Xie, Y.M, (2016) Topological design and additive manufacturing of porous metals for bone s caffolds and orthopaedic implants: A review. Biomaterials 82:127-141.https://doi.org/10.1

Journal Pre-proof 016/j.biomaterials.2016.01.012 4. Katz. JL (1980) Anisotropy of Young's modulus of bone. Nature 283:106-107. 5. Currey, JD (2004) Tensile yield in compact bone is determined by strain, post-yield behavi our by mineral content. J Biomech 37:549–556. https://doi.org/10.1016/j.jbiomech.2003.0 8.008 6. Sumner, D.R, Turner, T.M, Igloria, R, Urban, R.M, Galante, J.O (1998) Functional adaptat ion and ingrowth of bone vary as a function of hip implant stiffness. J. Biomech 31:909 -917. 7. Wilkinson, J.M, Hamer, A.J, Rogers, A, Stockley, I, Eastell, R (2003) Bone mineral densit y and biochemical markers of bone turnover in aseptic loosening after total hip arthropla sty. J. Orthop. Res 21:691-696. https://doi.org/10.1016/S0736-0266(02)00237-1 8. Geetha, M., Singh, A.K., Asokamani, R., Gogia, A.K., 2009.Ti based biomaterials, the ulti mate choice for orthopaedic implants – A review, Prog. Mater. Sci. 54 397-425. https://d oi.org/10.1016/j.pmatsci.2008.06.004 9. Choi, K, Kuhn, J. L, Ciarelli, M. J, Goldstein, S. A (1990) The elastic moduli of human s ubchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J Biomech 23:1103-1113. https://doi.org/10.1016/0021-9290(90)90003-L 10. Levine, B (2008) A New Era in Porous Metals: Applications in Orthopaedics. Adv. Eng. Mater 10:788-792. https://doi.org/10.1002/adem.200800215 11. Alvarez, K, Nakajima, H (2009) Metallic scaffolds for bone regeneration. Materials 2:790 -832. https://doi.org/10.3390/ma2030790 12. Heinl, P, Müller, L, Körner, C, Singer, R.F, Müller, F.A (2008) Cellular Ti–6Al–4V struc

Journal Pre-proof tures with interconnected macro porosity for bone implants fabricated by selective electr on beam melting. Acta Biomater 4:1536–1544. https://doi.org/10.1016/j.actbio.2008.03.013 13. Marin, E., Fusi, S, Pressacco, M, Paussa, L, Fedrizzi, L (2010) Characterization of cellula r solids in Ti6Al4V for orthopedic implant applications: Trabecular titanium. J. Mech. B ehav. Biomed. Mater 3:373–381. https://doi.org/10.1016/j.jmbbm.2010.02.001 14. Rouwkema, J, Rivron, N.C, van Blitterswijk, C.A (2008) Vascularization in tissue engine ering. Trends Biotechnol 26:434–441. https://doi.org/10.1016/j.tibtech.2008.04.009 15. Campbell, I, Bourell, D, Gibson, I (2012) Additive manufacturing: rapid prototyping com es of age. Rapid Prototyping J 18:255–258. https://doi.org/10.1108/13552541211231563 16. Braunig, J, Toppel, T, Muller, B, Burkhardt, M, Hipke, T, Drossel, WG (2014) Advanced material studies for additive manufacturing in terms of future gear application. Adv Mech Eng 8:1–10. https://doi.org/10.1155/2014/741083 17. Yang, S, Zhao, Y.F (2015) Additive manufacturing-enabled design theory and methodolog y: a critical review. Int J Adv Manuf Tech 80:327–342. https://doi.org/10.1007/s00170-01 5-6994-5 18. Jing, SK, Zhang, H, Zhou, JT, Song, G.H (2015) Optimum weight design of functionally graded material gears. Chin J Mech Eng 28:1186–1193. https://doi.org/10.3901/CJME.20 15.0930.118 19. Zhou, L.X, Yang, Q.Z, Zhang, G.R, Zhao, F.X, Shen, G, Yu, B (2014) Additive manufac turing technologies of porous metal implants. China Foundry 11:322-331. 20. Yang, Y.W, Wang, G.Y, Liang, H.X, Gao, C.D, Peng, S.P, Shen, L.D, Shuai, C.J (2019) Additive manufacturing of bone scaffolds. Int J Bioprint 5:148. http://dx/doi.org/10.18063

Journal Pre-proof /IJB.v5i1.148 21. Kou, XY, Tan, ST (2010) A Simple and Effective Geometric Representation for Irregular Porous Structure Modeling. Comput.-Aided Des 42:930-941. https://doi.org/10.1016/j.cad. 2010.06.006 22. Meyer, U, Runte, C, Dirksen, D, Stamm, T, Fillies, T, Joos, U, Wiesmann, H.P (2003) I mage–Based Biomimetric Approach to Design and Fabrication of Tissue Engineered Bon e. International Congress Series 1256:726–732. https://doi.org/10.1016/S0531-5131(03)005 22-3 23. Pattanayak, D.K, Fukuda, A, Matsushita, T, Takemoto, M, Fujibayashi, S, Sasaki, K, Nis hida, N, Nakamura, T, Kokubo, T (2011) Bioactive ti metal analogous to human cancell ous bone: Fabrication by selective laser melting and chemical treatments. Acta Biomater 7:1398–1406. https://doi.org/10.1016/j.actbio.2010.09.034 24. Fantini, M, Curto, M, De Crescenzio, F (2016) A method to design biomimetic scaffolds for bone tissue engineering based on Voronoi lattices. Virtual and Physical Prototyping 1 1:77-90. https://doi.org/10.1080/17452759.2016.1172301 25. Fantini, M, Curto, M (2018) Interactive design and manufacturing of a Voronoi-based bio mimetic bone scaffold for morphological characterization. Int J Interact Des Manuf 12：5 85–596. https://doi.org/10.1007/s12008-017-0416-x 26. Wang, G.J, Shen, L.D, Zhao J.F, Liang, H.X, Xie, D.Q, Tian Z.J, Wang, C.J (2018) Desi gn and Compressive Behavior of Controllable Irregular Porous Scaffolds: Based on Voro noi-Tessellation and for Additive Manufacturing. ACS Biomater. Sci. Eng 4:719–727. htt ps://doi.org/10.1021/acsbiomaterials.7b00916

Journal Pre-proof 27. Liang, H.X, Yang, Y.W, Xie, D.Q, Li, L, Mao, N, Wang, C.J, Tian, Z.J, Qing Jiang, she n, L.D (2019) Trabecular-like Ti-6Al-4V scaffolds for orthopedic: fabrication by selective laser melting and in vitro biocompatibility. J. Mater. Sci. Technol 35:1284-1297. https://d oi.org/10.1016/j.jmst.2019.01.012 28. Van der Putte, T (2009) Using the discrete 3D Voronoi diagram for the modelling of 3D continuous information in geosciences. Master Thesis. Utrecht University. 29. Okabe, A, Boots, B, Sugihara, K (1995) Spatial tessellations: concepts and applications of Voronoi diagrams. The College Mathematics Journal 26:79-81. https://doi.org/10.2307/26 87299 30. Stamboulis, A. G, Boccaccini, A. R, Hench, L. L (2002) Novel Biodegradable Polymer/B ioactive Glass Composites for Tissue Engineering Applications. Adv. Eng. Mater 4:105-1 09. https://doi.org/10.1002/1527-2648(200203) 31. Gibson, L. J. A.A, Ashby, M. F (2014) Cellular solids: structure and properties. Cambrid ge University Press: Cambridge,U.K

Journal Pre-proof Highlights 

Irregular porous scaffolds were constructed and samples were obtained by SLM.



Irregular porous structure met needs of bone tissue engineering.



Relationship between structural design parameters and porosity, mechanical properties were concluded.