# Porous Scaffold Design Based on Minimal Surfaces: Development and Assessment of Variable Architectures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

^{3}is iteratively refined to minimise the area of the polygonal mesh spanning the boundary. This refinement continues until no single vertex of the triangulation can be moved further to decrease its area. The phase-field is another boundary method allowing the generation of a triply periodic surface with a constant mean curvature satisfying a constraint on the volume fractions of the regions that the surface separates.

^{3}. The minimal surface is then defined as the locus of points at which the implicit function f(x,y,z) takes a zero value. Since any periodic surface can be described by the sum of an infinite number of Fourier terms, a surface derived from the Schwarz minimal surface (P-surface in the following) can be described, to the first order of approximation, by the following nodal equation:

^{3}. For a cubic scaffold with a number of cells along one side equal to N, the domain of the function is given by the boundary conditions: x = [−Nπ, Nπ], y = [−Nπ, Nπ], and z = [−Nπ, Nπ].

- -
- the gradient of k function along x and y must be as large as possible;
- -
- the values of k function must be such as to preserve the integrity of the P-cell and that of its lateral openings. This requirement is essential to allow the cells to gradually and progressively populate the ducts of the scaffold, regenerating the bone tissues. The satisfaction of this aspect requires that ranges of variability, suitable for the two parameters k and s, are identified. Since in this paper s has been assumed equal to 1, k must vary between −1 and 1 into the function domain identified by the boundary conditions x = [−7π, 7π] and y = [−7π, 7π].

## 3. Results and Discussion

^{3}cells; in fact, the difference between the value of convergence of the simulation and the calculated value is, in percentage, negligible. The models made of a limited number of cells provide the additional benefit of reduced computational cost in the Finite Elements FE simulations compared with larger models.

_{eff.}), given by the ratio between the homogenised stress and the applied strain, was evaluated for each variable architecture considered in comparison with the scaffold obtained by replicating a unit P-cell. The mechanical properties of a porous structure can be determined via FEA [25] based on Hooke’slaw, and the effective elastic modulus of a porous structure can be expressed by means of the relationship:

_{eff}. = (F

_{R}/A)/e

_{A}

_{R}is the reaction force calculated by the FE solver, A is the total cross-sectional area, and e

_{A}is the applied strain. Since the applied strain is known, the parameter can be estimated. This parameter of stiffness represents an average value for the structures with variable architectures; however, it is useful to globally evaluate them. Figure 11 shows a diagram which summarises the results obtained.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Levine, B. A new era in porous metals: Applications in orthopaedics. Adv. Eng. Mater.
**2008**, 10, 788–792. [Google Scholar] [CrossRef] - Noyama, Y.; Miura, T.; Ishimoto, T.; Itaya, T.; Niinomi, M.; Takayoshi, N. Bone loss and reduced bone quality of the human femur after total hip arthroplasty under stress-shielding effects by titanium-based implant. Mater. Trans.
**2012**, 53, 565–570. [Google Scholar] [CrossRef] - Coelho, P.G.; Hollister, S.J.; Flanagan, C.L.; Fernandes, P.R. Bioresorbable scaffolds for bone tissue engineering: Optimal design, fabrication, mechanical testing and scale-size effects analysis. Med. Eng. Phys.
**2015**, 37, 287–296. [Google Scholar] [CrossRef] [PubMed] - Sallica-Leva, E.; Jardini, A.L.; Fogagnolo, J.B. Microstructure and mechanical behavior of porous Ti–6Al–4V parts obtained by selective laser melting. J. Mech. Behav. Biomed. Mater.
**2013**, 26, 98–108. [Google Scholar] [CrossRef] [PubMed] - Weißmann, V.; Bader, R.; Hansmann, H.; Laufer, N. Influence of the structural orientation on the mechanical properties of selective laser melted Ti6Al4V open-porous scaffolds. Mater. Des.
**2016**, 95, 188–197. [Google Scholar] [CrossRef] - Horn, T.J.; Harrison, O.L.A.; Marcellin-Litte, D.J.; West, H.A.; Lascelles, B.D.X.; Aman, R. Flexural properties of Ti6Al4V rhombic dodecahedron open cellular structures fabricated with electron beam melting. Addit. Manuf.
**2014**, 1–4, 2–11. [Google Scholar] [CrossRef] - De Viteri, V.S.; Fuentes, E. Titanium and titanium alloys as biomaterials. In Tribology-Fundamentals and Advancements; Gegner, J., Ed.; InTech: London, UK, 2013; pp. 155–181. [Google Scholar]
- Harrysson, O.L.A.; Cansizoglu, O.; Marcellin-Little, D.J.; Cormier, D.R.; West, H.A., II. Direct metal fabrication of titanium implants with tailored materials and mechanical properties using electron beam melting technology. Mater. Sci. Eng. C
**2008**, 28, 366–373. [Google Scholar] [CrossRef] - Ahmadi, S.M.; Yavari, S.A.; Wauthle, R.; Pouran, B.; Schrooten, J.; Weinans, H.; Zadpoor, A.A. Additively manufactured open-cells porous biomaterials made from six different space-filling unit cells: The mechanical and morphological properties. Materials
**2015**, 8, 1871–1896. [Google Scholar] [CrossRef] [PubMed] - Giannitelli, S.M.; Accoto, D.; Trombetta, M.; Rainer, A. Current trends in the design of scaffolds for computer aided tissue engineering. Acta Biomater.
**2014**, 10, 580–594. [Google Scholar] [CrossRef] [PubMed] - Yanez, A.; Cuadrado, A.; Martel, O.; Afonso, H.; Monopoli, D. Gyroid porous titanium structures: A versatile solution to be used in bone defect reconstruction. Mater. Des.
**2018**, 140, 21–29. [Google Scholar] [CrossRef] - Ambu, R.; Morabito, A.E. Design and analysis of tissue engineering scaffolds based on open porous non-stochastic cells. In Advances on Mechanics, Design Engineering and Manufacturing; Eynard, B., Nigrelli, V., Olivieri, S., Peris-Fajarnes, G., Rizzuti, S., Eds.; Lecture Notes in Mechanical Engineering; Springer International Publishing: Cham, Switzerland, 2017; pp. 777–787. [Google Scholar]
- Lord, E.A.; Mackay, A.L. Periodic minimal surfaces of cubic symmetry. Curr. Sci.
**2003**, 85, 346–362. [Google Scholar] - Montazerian, H.; Davoodi, E.; Asadi-Eydivand, M.; Kadkhodapour, J.; Solati-Hashjin, M. Porous scaffolds internal architecture design based on minimal surfaces: A compromise between permeability and elastic properties. Mater. Des.
**2017**, 126, 98–114. [Google Scholar] [CrossRef] - Rajagopalan, S.; Robb, R.A. Schwarz meets Schwann: Design and fabrication of biomorphic and durataxic tissue engineering scaffolds. Med. Image Anal.
**2006**, 10, 693–712. [Google Scholar] [CrossRef] [PubMed] - Yan, C.; Hao, L.; Hussein, A.; Young, P. Ti-6Al-4V triply periodic minimal surface structures for bone implants fabricated via selective laser melting. J. Mech. Behav. Biomed. Mater.
**2015**, 51, 61–73. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Melchels, F.P.W.; Bertoldi, K.; Gabbrielli, R.; Velders, A.H.; Feijen, J. Mathematically defined tissue engineering scaffold architectures prepared by stereolithography. Biomaterials
**2010**, 31, 6909–6916. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sudarmadji, N.; Tan, J.Y.; Leong, K.F.; Chua, C.K.; Loh, Y.T. Investigation of the mechanical properties and porosity relationships in selective laser-sintered polyhedral for functionally graded scaffolds. Acta Biomater.
**2011**, 7, 530–537. [Google Scholar] [CrossRef] [PubMed] - Gabbrielli, R.; Turner, I.G.; Bowen, C.R. Development of modelling methods for materials to be used as bone substitutes. Key Eng. Mater.
**2008**, 361–363, 903–906. [Google Scholar] [CrossRef] - Afshar, M.; PourkamaliAnaraki, A.; Montazerian, H.; Kadkhodapour, J. Additive manufacturing and mechanical characterization of graded porosity scaffolds designed based on triply periodic minimal surface architectures. J. Mech. Behav. Biomed. Mater.
**2016**, 62, 481–494. [Google Scholar] [CrossRef] [PubMed] - Khoda, A.K.M.B.; Ozbolat, I.T.; Koc, B. Engineered tissue scaffolds with variational porous architecture. J. Biomech. Eng.
**2011**, 133, 011001. [Google Scholar] [CrossRef] [PubMed] - Andersson, S.; Hyde, S.T.; Larsson, K.; Lidin, S. Minimal surfaces and structures: From inorganic and metal crystals to cell membranes and biopolymers. Chem. Rev.
**1988**, 88, 221–242. [Google Scholar] [CrossRef] - Smith, M.; Guan, Z.; Cantwell, W.J. Finite element modelling of the compressive response of lattice structures manufactured using the selective laser melting technique. Int. J. Mech. Sci.
**2013**, 67, 28–41. [Google Scholar] [CrossRef] - Maskery, I.; Aremu, A.O.; Parry, L.; Wildman, R.D.; Tuck, C.J.; Ashcroft, I.A. Effective design and simulation of surface-based lattice structures featuring volume fraction and cell type grading. Mater. Des.
**2018**, 155, 220–232. [Google Scholar] [CrossRef] - Cahill, S.; Lohfeld, S.; McHugh, P.E. Finite element predictions compared to experimental results for the effective modulus of bone tissue engineering scaffolds fabricated by selective laser sintering. J. Mater. Sci. Mater. Med.
**2009**, 20, 1255–1262. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 2.**Different models of the P-cell at the varying of the k parameter and s = 1 (The Schwarz Primitive is indicated in red).

**Figure 3.**Different models of P-cells at the varying of the s parameter and k = 0 (the Schwarz Primitive is indicated in red): (

**a**) P-cells for s ≥ 1; (

**b**) Half-P-cells for s ≤ 1.

**Figure 5.**Main steps of the methodology used for the modelling of a scaffold with variable architecture: (

**a**) Scaffold surface modelling and scaling to a cubic structure of given side; (

**b**) Offset with an assigned thickness; (

**c**) Planar cuts and scaffold surface closing.

**Figure 9.**Iso-colour representation of the compressive stress: (

**a**) radial scaffold; (

**b**) quadratic scaffold.

Type of Scaffold | k(x,y,z) |
---|---|

Linear | C∙z |

Radial | C∙(x^{2} + y^{2}) |

Quadratic | C∙(x^{2} − y^{2}) |

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**MDPI and ACS Style**

Ambu, R.; Morabito, A.E.
Porous Scaffold Design Based on Minimal Surfaces: Development and Assessment of Variable Architectures. *Symmetry* **2018**, *10*, 361.
https://doi.org/10.3390/sym10090361

**AMA Style**

Ambu R, Morabito AE.
Porous Scaffold Design Based on Minimal Surfaces: Development and Assessment of Variable Architectures. *Symmetry*. 2018; 10(9):361.
https://doi.org/10.3390/sym10090361

**Chicago/Turabian Style**

Ambu, Rita, and Anna Eva Morabito.
2018. "Porous Scaffold Design Based on Minimal Surfaces: Development and Assessment of Variable Architectures" *Symmetry* 10, no. 9: 361.
https://doi.org/10.3390/sym10090361