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Design and Compressive Behavior of Controllable Irregular Porous Scaffolds: Based on Voronoi-Tessellation and for Additive Manufacturing Guanjun Wang, Lida Shen, Jianfeng Zhao, Huixin Liang, Deqiao Xie, Zongjun Tian, and Changjiang Wang ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.7b00916 • Publication Date (Web): 16 Jan 2018 Downloaded from http://pubs.acs.org on January 18, 2018

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ACS Biomaterials Science & Engineering

Design and Compressive Behavior of Controllable Irregular Porous Scaffolds: Based on VoronoiTessellation and for Additive Manufacturing Guanjun Wang1,2, , Lida Shen1, , Jianfeng Zhao1,*, Huixin Liang1, Deqiao Xie1, Zongjun Tian1, ‡

‡

and Changjiang Wang3 College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and

1

Astronautics, 29 Yudao Street, Nanjing 210016, PR China. Suzhou Kangli Orthopedics Instrument Co.Ltd, Luyuan Tangqiao Town, Zhangjiagang

2

Suzhou, 215600, PR China. 3

Department of Engineering and Design, University of Sussex, Sussex House, United Kingdom. E-mail: [email protected]

*

KEYWORDS: Porous scaffold, Voronoi-Tessellation, Selective Laser Melting, Stress shielding, Elastic modulus, Gradient pore structure. ABSTRACT: Adjustment of the mechanical properties (apparent elastic modulus and compressive strength) in porous scaffolds is important for artificial implants and bone tissue engineering. In this study, a top-down design method based on Voronoi-Tessellation was

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proposed. This method was successful in obtaining the porous structures with specified and functionally graded porosity. The porous specimens were prepared by selective laser melting technology. Quasi-static compressive tests were conducted as well. The experiment results revealed that the mechanical properties were affected by both porosity and irregularity. The irregularity coefficient proposed in this study can achieve good accommodation and balance of ‘irregularity’ and ‘controllability’. The method proposed in this study provides an efficient approach for the bionic design and topological optimization of scaffolds.

1. Introduction Long-term aseptic loosening of artificial implants and refracture are fatal complications within the clinical setting1-2. The main issue within artificial implants is ‘stress shielding’. Stress is unable to adequately transfer from the metal implant device to the bone. This is caused by the large difference between their elastic moduli. Osteocytes that do not have adequate stress stimulation will die and be absorbed, which results in the loosening of implants and sometimes fracturing occurs. Porous structures were introduced to the implant devices to stop these occurrences. These structures could reduce the apparent elastic modulus to the level of human bones (1 Gpa-10 Gpa)2. The porous structures also provide necessary space for osteocyte and transport pathways of tissue fluid. This is conductive to biological fixation between the bone and implant3-5. Metal implants with porous structures could be fabricated rapidly and precisely as additive manufacturing (AM) has become more developed. The selective laser melting (SLM) and the electron beam melting (EBM) are the most popular methods within AM. Designing porous scaffolds that could meet both structural and mechanical properties requirements has become necessary6.


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The morphology of these porous structures are broadly divided into regular or irregular structures. The primary construction methods of the regular porous structures are the cell unit method7 and the triply periodic minimal surface method8. The regular porous scaffolds have regular pore morphology, good connectivity, and controllable mechanical properties. This results in their wide application9. Irregular porous scaffolds are typically implemented by computer programs and mathematical models10-12. The irregular porous scaffolds enable geometric and mechanical parameters to be designed precisely or distributed in a gradient manner. Furthermore, irregular porous scaffolds have the ability to simulate the complex and anisotropic microstructures of bone tissues. This provides additional freedom within bionic design and topological optimization13-15. The design approach for irregular porous structure is another key technology that requires additional research. Unfortunately, most of previous studies on irregular porous structures ignored the effect of irregularity and its evolution. Uncontrollable irregularity leads to poor repeatability of morphology and the mechanical properties. In this study, we proposed a top-down design method (probability sphere method) based on the Voronoi-Tessellation method to construct controllable porous scaffolds. The probability sphere method used a regular three-dimensional point array that was agitated by probability spheres. This generated the irregular but controllable porous scaffold. In order to evaluate the porous scaffolds, we prepared two sets of specimens by SLM. We then measured their geometric parameters with industrial CT (Computer Tomography). Their mechanical parameters were tested with quasi-static compressive experiments.

2. Materials and Methods 2.1 Modeling and characterization of porous scaffolds


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The Voronoi-Tessellation is a method of space partition, based on the seed point16-18. The Voronoi diagram is defined below: For a set of points on the m-dimensional Euclidean space m

P  { p 1 , , p n }  R , 2  n  , p i  p j ,i , j  I n  {1, , n}

(1)

The line between pi and pj divides the space into two parts, Hi(pipj) is the part including pi, then, V ( p i )  {x x  p i  x  p j }  

jI n \{i}

H(p i, p j)

(2)

is the m-dimensional Voronoi generated by pi on Rm space , γ(p) = {V (pi),..., V(pn)} is called the Voronoi diagram and pi is the seed point. Voronoi diagram is determined by the amount and the distribution of seed points. The control of the seed points is paramount to successful irregular porous scaffolds modeling. Previous researches have preferred to generate three dimensional points in random manner19-21. However, there is little control of these random points and the morphology and the mechanical properties have poor repeatability. In this study, the controllable irregular porous structures were designed using CAD software Grasshopper version 0.9.0076 (http://www.grasshopper3d.com). First, a cube region was generated and the volume of the cube region was the outer volume of the porous structure. Second, an ordered cube lattice containing n layers was generated within the cube region. The distance between the two adjacent points within Layer i was bi, as seen in Figure 1. Denote the point on Line m and Column n in Layer i as Pmn_i. Denote the space between Layer (i-1) and Layeri as ai. The spherical regions that were centered at each point and generated with the radius of Ri. Third, a new point P’mn_i was then generated randomly in each spherical region to replace their predecessors. This was the new lattice. P’mn_i was generated as follows:


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 x  x  R  rand  sin(  rand )  cos(2  rand )   y  y  R  rand  sin(  rand )  cos(2  rand )  z   z  R  rand  sin(  rand )  cos(2  rand ) 

(3)

Where, rand is a random function with uniform distribution in [0, 1].

Figure 1. Regular lattice. (a) Perspective view. (b) Front view. (c) Top view of Layer i. Irregularity coefficient was defined as ε:



1  N mni ,,

 ， Pmn_iP mn_i ai

(4)

Where, N is the number of points in the irregular lattice generated previously, 0 ≤ ε < 1. The radius of spherical region Ri was constrained to increase the controllability of the irregular lattice by 0  Ri 

1 a . The Voronoi cells were generated in Grasshopper following the 2 i

irregular lattice generation. The struts were generated based on the edges of the Voronoi cells. Then, a Boolean operation was used to form a porous scaffold with a specific shape, as shown in Figure 2. The size of the pores and struts were controlled by introducing the scale coefficient K.


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This was the ratio of a pore area to the corresponding surface area of the cell K  S ci , as S pi

shown in Figure 3. K controls both diameters of the pores and the struts. When the seed points are given, the diameters of the pores and struts could be adjusted by K.

Figure 2. The design principle of porous scaffold. (a)Designing regular lattice. (b)Generating probability spheres. (c)Obtaining irregular lattice. (d)Generating voronoi cells based on irregular lattice. (e)Constructing porous structures.

Figure 3. The definition of scale coefficient K. (a) One of the Voronoi cell. (b) Porous structure based on (a).


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The gradient distribution of the points in the Z axis were obtained. This allowed one to obtain the gradient distribution of porosity in the direction of Z axis, as shown in Figure 4. K was then replaced with linear variables that related to cell position:

K

Zt  Zb Z  Zb  Kb  t  Zb Kt  K b Kt  K b

(5)

Where, Zb and Zt denoted the minimum and the maximum of the Z axis coordinate of body center of all the voronoi cells. Kb and Kt represented the minimum and maximum of the value of K. In order to fit the gradient distribution of points, the radius of probability sphere was set as Ri 

1 a . Based on the probability sphere method, Figure 5 shows the process of constructing 2 i

the artificial bone with porous gradient structures.

Figure 4. The modeling method of porous scaffold with gradient porosity. (a) The regular lattice. (b) The irregular lattices. (c) Porous scaffolds with gradient distribution of porosity (Kt = 0.9, Kb = 0.7).


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Figure 5. The method of constructing the artificial bone with porous gradient structures: Firstly, offset the surface by incremental distances; Secondly, generate spherical regions based on the intersection points of UV isocurves; Thirdly, generate irregular lattice then construct porous gradient structures. The parameters of the porous scaffold included pore size, porosity, and strut diameter. The 

V 





porosity in the geometric model was calculated by the formula   1  P   100% , where Φ is V porosity, VP is the volume of the porous scaffold, and V is outer volume of the porous scaffold. The sections at the midpoint of the strut sections were extracted. The equivalent diameter method was used to calculate the pore diameters when the area was equivalent to the sections. The equivalent diameter method was used to calculate the aperture, as shown in formula(6) .  1 n 4Si D= n    D i 1  n 4A i d = 1   n i 1  d 

(6)

Where, nD is the number of pores, nd is the number of struts, D is the equivalent aperture, Si is the hole area, d is the equivalent strut diameter, and Ai is the cross-sectional area of strut.


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2.2 Fabrication and compressive testing The porous specimens were fabricated with an SLM machine (M290, EOS GmbH, Germany) using the optimized processing parameters, as shown in Table 1. The materials were commercial Ti6Al4V ELI powders, supplied by EOS GmbH. Table 1. SLM process parameters

Parameter

Value

Laser power

Scanning speed

Hatch Spacing

Layer thickness

/W

/ mm/s

/mm

/mm

180

1350

0.1

0.03

Laser focus /mm 0.08

Atmosphere

Ar

The factors that influenced the mechanical properties of the porous scaffolds included porosity, pore diameter, strut diameter, and shape of the strut section. Porosity is thought to be the most influential factor22. Irregularity is another important parameter within irregular porous scaffolds. In this paper, two sets of experiments were conducted to investigate the effects of porosity and irregularity on the mechanical properties of porous scaffolds. The design parameters of the porous scaffold are shown in Table 2. Three identical samples were prepared for each model and the test results were average values of three samples. Table 2. Design parameters of porous scaffolds Group Porosity series

N

125

K

R

0.65

1

0.7

1

0.75

1


9


Irregularit y series

125

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0.8

1

0.85

1

0.9

1

0.8

0.4

0.8

0.8

0.8

1.2

0.8

1.4

0.8

1.6

0.8

1.8

X-ray computed tomography (XCT) is a effective nondestructive testing method in the field of AM. Resolutions from 9.68 µm to 15.34 µm were performed by an industrial CT (XTH225, Nikon, Japan) to scan the porous scaffolds. A 3D model of the porous scaffolds was reconstructed to measure the porosity in software, as shown in Figure 6.

Figure 6. Reconstruction of 3D model by XCT.


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The compression tests of specimens were conducted with a mechanical testing machine (CMT5105, MTS System Corporation, America). The crosshead displacement speed was fixed at 0.25 mm/min.The apparent elastic modulus (E) of porous specimens was characterized by quasielastic gradient (ISO13314:2011), and the compressive strength (S) was characterized by ultimate compressive stress, as shown in Figure7.

Figure 7. Characterization of stiffness and compressive strength. In order to fit scattered data of mechanical properties, we used the Gibson-Ashby model22. The Gibson-Ashby model is a typical model that simplified the irregular porous structure into the regular structure built by the stacking of a special basic unit. This model discovered the following functions between porosity, elastic modulus, and compressive strength: Φ m   S  k1 (1  100 ) S 0   E  k (1  Φ ) n E 2 0  100

(7)

Where, S0 and E0 is the compressive strength and elastic modulus of substrate materials, k1 and k2 are constants related to mechanical properties of substrate materials.


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3. Results and discussion 3.1 Controllability of geometric parameters Figure 8 shows lattices and corresponding porous scaffolds with different regularities. Figure 9 depicts the relationship between the irregularity (ε) and (R=a) of probability spheres. The porous scaffolds increase in irregularity as irregularity grows. There was a good linear relationship between R=a and irregularity. This suggests that the probability sphere method could control and characterize the degree of irregularity of porous scaffold. Figure 10 shows the influence of R/a on porosity. When R/a is more than 0.2, porosity stay stable with the increase of R/a. In theory, the irregular coefficient ε defined in this paper does not fully characterize the true  degree of irregularity of the points. For example, if all vectors pi p’i

were unidirectional,

although the value of ε could be huge, the degree of irregularity of the lattice would not change. However, there is little prospect of such special case occurring in practical experiments. As a result, the irregular coefficient ε meets the application requirements.


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Figure 8. Porous structures with different irregularities. (a) ε = 0.06. (b) ε = 0.25. (c) ε =0.43. (K = 0.9, N = 512).

Figure 9. Relationship between R=a and ε.

Figure 10. Relationship between R/a and porosity. Figures 11 and 12 show the effect of the amount of the seed points (N) and scale coefficient (K) on geometric parameters. seed points had little effect on porosity. The overall fluctuation range was less than 10%. The larger seed point number was, the less the effect of it had on porosity. A power relation existed between seed points and diameters of pores and struts. There were


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strongly linear relation between scale coefficient and porosity, as well as diameters of pores and struts. Scale coefficient had greater effect on geometric parameters than seed points. The primary factor that affected the geometric parameters of porous scaffolds was scale coefficient.The standard for the proper size of bone cell ingrowth is 200 µm to 1200 µm23. The method presented in this paper adjusted seed points, scale coefficient and R=a to be within this scope.

Figure 11. The relation between points and (a) porosities, and (b) diameters of pores and struts. (K = 0.8, R = 0.5).

Figure 12. The relation between scale coefficients and (a) porosities, and (b) diameters of pores and struts. (N = 216, R = 0.5).


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3.2 Controllability of gradient porosity Table 3 shows the four groups of Kb and Kt (see Figure 4). These groups were set to explore the influence of different combinations on the gradient change of porosity. The slope of the best fit line was defined as the porosity gradient G. Figure 13 shows the gradient variation of the porosity as it related to height. As seen in Figure 10, the porosity changed linearly, in the height direction which corresponded with K. When Kt remained constant, the maximum porosity remained constant as well (the porosity at the top of the structure). This further confirmed that porosity was primarily determined by K in 3.1. Table 3. Scale coefficients for gradient porosity Series

Kt

Kb

Ⅰ

0.9

0.6

Ⅱ

0.9

0.7

Ⅲ

0.9

0.8

Ⅳ

0.9

0.9

Figure 13. Porosity gradient corresponding to the different sets of scale coefficients.


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3.3 Controllability of manufacture precision by SLM Figure 14 illustrates the porous structures of SLMed specimens and reconstructions by XCT. All struts, both inside and outside were fabricated precisely. Table 4 lists the contrast between the design porosity and the as-built porosity. The porosity deviations between the models and the specimens were less than 4%. This met the requirements of high-precision manufacturing of porous scaffolds.

Figure 14. Porous structures with different porosities and irregularities. (a)Porosity series,ε=0.41. (b)Irregularity series, Φ=85.5%±2%. Table 4. Porosity of CAD models and specimens of porous structure


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Porosity/%

Series

Deviation/%

Model

Specimen

1

64.77

63.18

2.45

2

71.83

69.68

2.99

Porosity

3

78.46

75.57

3.69

Series

4

85.66

82.43

3.77

5

90.24

87.50

3.03

6

93.91

91.86

2.18

1

86.26

84.89

1.59

2

85.75

83.92

2.14

Irregularity

3

85.50

83.66

2.16

Series

4

85.44

84.05

1.63

5

85.50

83.83

1.96

6

85.69

83.67

2.36

3.4 Controllability of mechanical properties Figure 15 shows the compressive curves of these two series. Figure 16 shows the the relationship between porosity Φ, apparent elastic modulus E, and compressive strength S. Figure 17 shows the relation between irregularity, elastic modulus, and compressive strength.


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Figure 15. The compressive curves. (a) Porosity series, ε=0.50. (b) Irregularity series, Φ=85.5% ±2%.

Figure 16. Relation between porosity and (a) elastic modulus, and (b) compressive strength.

Figure 17. Relation between irregularity and (a) elastic modulus, and (b) compressive strength.


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Figure 18. The gradient pore structure and compressive test curve. There was a power function relationship between the compressive strength and porosity, as well as a strong linear relationship between apparent elastic modulus and porosity. With application of the Ashby-Gibson model, the elastic modulus and compressive strength of the porous structure were fitted as:  1.54   E  11.83(1  100 )   S  1400.62(1   ) 2.38  100

(8)

Due to the variation of irregularities, the elastic modulus showed a sharp downward trend. Meanwhile, the compressive strength exhibited a wide range of fluctuation with the increase of irregularity, which was similar with previous work. 10 As shown in Figure 13 (a), the stiffness of porous structure is highly dependant on its porosity and irregularity. On one hand, the decrease of porosity, caused by the decrease of strut diameters and controlled by K, resulted in the decay


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of struts stiffness, which made the porous structure deformed more easily; On another hand, the increase of irregularity make more struts change from vertical or horizontal to tilted position, which also reduce the structure stiffness. The special mechanical behaviors of irregular porous structures, reflected in Figure 17(b), were attributed to the increase of irregularity and the SLM process. Previous study 10believed that small levels of irregularity resulted in a decrease in strength that is caused by the initial destabilization of the unit cell on which the structure was based. When the irregularity exceeded a certain value, a new stress balance was built and compressive strength tended to be stable. In addition, over the SLM processing, the pores and cracks occurred in specimens inevitably24-25. These cracks , related to the orientation of the struts , had a significant influence on the compressive strength26-27. This leaded to the fluctuation in compressive strength test data as well. When the irregularity of porous scaffolds is given, the mechanical properties are in accordance with the Gibson-Ashby model. However, when the irregularity is accounted for, the GibsonAshby model is no longer applicable. The Gibson-Ashby model is based on the regular assumption, ignoring the influence of irregularity in principle. It cannot reflect the change trend of the irregular effect on the mechanical properties. The results from the current study demonstrate that when the influence of irregularity is considered, the mechanical properties of porous structures should be at least three variable function parameters of the solid material (porosity, irregularity, and mechanical). As a result, further studies should focus on the mechanism of this phenomenon and explore more precise theoretical models. It is worth mentioning that, the structures that we prepared and tested are less stiff than bone. According to previous work2, the range of apparent elastic moduli of bone is 1-10 GPa. The values depend on the sample and method strongly, and different researchers have different


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results28. For example, patients with osteoporosis tend to have loose bones, low trabecular elastic modulus. In this case, low apparent modulus of elasticity is significant and necessary. As shown in Figure 16, adjusting K and R can change the stiffness of the structure. The stiffness of porous structures increases with decreasing K. According to Figure 17, the stiffness of the structure is higher when the irregularity is lower (ε=0.11) or higher (ε=0.50). In addition, by optimizing the test methods and process parameters, the apparent elastic moduli of the porous structures can be further enhanced. Figure 18 shows the gradient pore structure and the compression test curve. These structures were designed according to the parameters in Table 3 and Figure 4. The mechanical behavior of the gradient pore structures are significantly different from uniform porous structures. In the low strain stage, the compression curve shows a linear trend similar to a uniform random porous structure. In platform stage, the stress platform is no longer horizontal distribution, but showing upward trend. And as the gradient increases, the angle of the platform curve increases. This is different from a uniform porous structure. This is because the porosity of the porous structure continuously decreases in the height direction, and the struts of the pore gradually deformed and failed in the direction of increasing porosity. The larger the porosity gradient, the greater the load required for the destruction, which is in agreement with the trend disclosed in Figure 16 in the manuscript. Gradient porous structure not only has potential application value in mechanics, but also has broad application prospect in tissue engineering scaffold. A realistic application is that it provides an integrated scaffold for studying the osteosynthesis capabilities of different pore sizes.

4. Conclusion


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This study proposed a probability sphere method in order to construct irregular structures based on the Voronoi-Tessellation method. The porous specimens were fabricated by SLM. Their compressive behaviors were investigated. Our findings are summarized as follows: (1) This method allowed us to obtain not only an accurate design of the specific porosity, as well as the gradient distribution of porosity. Porous scaffolds with special porosities ranging from 60% to 95 % and pore size ranging from 200 µm to 1200 µm were designed precisely and conveniently. A porosity gradient ranging from 0.03 to 0.54 was obtained. (2) The proposed irregular coefficient could control the irregularity of the porous structures and achieve good accommodation and balance of ‘irregularity’ and ‘controllability’. (3) The compressive properties were affected by both porosity and irregularity. The compressive strength decayed at low levels of irregularity. The compressive strength was enhanced at high levels, which enabled us to obtain porous structure with high strength but low stiffness. (4) The compressive properties of porous structures could be adjusted to the level of cortical bone. The apparent elastic moduli of porous specimens had high irregularity (0.50) that varied from 0.14 GPa to 2.37 GPa. The compressive strengths varied from 1.94 MPa to 116.61 MPa. (5) The mechanical behavior of the gradient pore structures are significantly different from uniform porous structures in the platform stage. The compressive stress platform of gradient pore structure tends to rise. The larger the porosity gradient, the larger the angle of ascent. *Corresponding Author Phone:+86 025 84892520 Author Contributions


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The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally. Funding Sources The authors would like to thank the Key Research and Development Plan of Jiangsu Province (No., BE2016010-3), Science and Technology Support Program (The Industrial Part), Jiangsu Provincial Department of Science and Technology of China (No., BE2014009-1), the Science and Technology Supporting Plan (The Industrial Part) of Zhangjiagang (No., ZKG1615), the Science and Technology Supporting Plan (Social Development Part) of Zhangjiagang (No., ZKS1606), the Nation Nature Science Fund of China (No., 51475238). REFERENCES 1.Geetha, M.; Singh, A. K.; Asokamani R.; Gogia, A. K. Ti based biomaterials, the ultimate choice for orthopaedic implants- A review. Prog. Mater. Sci. 2009, 54, 397-425. DOI:10.1016/j.pmatsci.2008.06.004. 2. Choi, K.; Kuhn, J. L.; Ciarelli, M. J.; Goldstein, S. A. The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J Biomech. 1990, 23, 1103-1113, DOI:10.1016/0021-9290(90)90003-L. 3. Habibovic, P.; Yuan, H.; van der Valk, C. M.; Meijer, G.; van Blitterswijk, C. A.; de Groot, K. 3D microenvironment as essential element for osteoinduction by biomaterials. Biomaterials 2005, 26, 3565-3575. DOI:10.1016/j.biomaterials.2004.09.056.


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