Modeling of porous scaffolds based on the triply periodic structures P and G
DOI:
https://doi.org/10.46502/issn.2710-995X/2021.5.04Keywords:
Hybrid porous scaffold, triply periodic minimal surfaces, Mathematica, multilinear regression.Abstract
In this work we study the modeling of hybrid porous scaffold structures for bone tissue regeneration based on the triply periodic minimal surfaces Gyroid (G) and Schwarz´s primitive (P). The design of hybrid prismatic probes, with dimensions according to the norm ASTM D695_15, is made from the equations defining each of the structures using the sigmoid function with k=0.5 within the CAS software Wolfram Mathematica v 11.2. The issues related to the use of Mathematica as a tool for the design of probes are discussed in detail. The constants within each structure are taken as factors in a 22 factorial design in order to study its effect on porosity and pore size. The model equations relating dependent variables with factors are obtained from multilinear regression analysis and discussed. It is concluded that a bilinear model is adequate to describe the response variables what justifies the factorial design chosen.
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References
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D´Agostino, R. B., & Stephens, M. A. (Eds.). (1986). Goodness-of-fit techniques. United States: Marcel Dekker, Inc.
Farah, S., Anderson, D. G., & Langer, R. (2016). Physical and mechanical properties of PLA, and their functions in widespread applications — A comprehensive review. Advanced Drug Delivery Reviews, 107, 367-392. doi:https://doi.org/10.1016/j.addr.2016.06.012
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Hollister, S. J. (2005). Porous scaffold design for tissue engineering. Nature Materials, 4(7), 518-524. doi:10.1038/nmat1421
Jung, Y., & Torquato, S. (2005). Fluid permeabilities of triply periodic minimal surfaces. Physical Review E, 72(5), 056319.
Kanczler, J. M., Wells, J. A., Gibbs, D. M. R., Marshall, K. M., Tang, D. K. O., & Oreffo, R. O. C. (2020). Bone tissue engineering and bone regeneration. In R. Lanza, R. Langer, J. P. Vacanti, & A. Atala (Eds.), Principles of Tissue Engineering (5th ed.). United States of America: Elsevier Inc.
Knychala, J., Bouropoulos, N., Catt, C., Katsamenis, O., Please, C., & Sengers, B. (2013). Pore geometry regulates early stage human bone marrow cell tissue formation and organisation. Annals of Biomedical Engineering, 41(5), 917-930.
Lee, J. S., Cha, H. D., Shim, J. H., Jung, J. W., Kim, J. Y., & Cho, D. W. (2012). Effect of pore architecture and stacking direction on mechanical properties of solid freeform fabrication?based scaffold for bone tissue engineering. Journal of Biomedical Materials Research Part A, 100(7), 1846-1853.
Malinauskas, M., Skliutas, E., Jonušauskas, L., Mizeras, D., Šešok, A., & Piskarskas, A. (2015). Tailoring bulk mechanical properties of 3D printed objects of polylactic acid varying internal micro-architecture (Vol. 9505), SPIE.
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Meeks III, W. H. (1990). The theory of triply periodic minimal surfaces. Indiana University Mathematics Journal, 877-936.
Melchels, F. P. W., Bertoldi, K., Gabbrielli, R., H., V. A., Feijen, J., & Grijpma, D. W. (2010). Mathematically defined tissue engineering scaffold architectures prepared by stereolithography. Biomaterials, 31(27), 6909-6916. doi:https://doi.org/10.1016/j.actbio.2010.06.012
Montazerian, H., Davoodi, E., Asadi-Eydivand, M., Kadkhodapour, J., & Solati-Hashjin, M. (2017). Porous scaffold internal architecture design based on minimal surfaces: A compromise between permeability and elastic properties. Materials & Design, 126, doi: 10.1016/j.matdes.2017.04.009
Moroni, L., de Wijn, J. R., & van Blitterswijk, C. A. (2006). 3D fiber-deposited scaffolds for tissue engineering: Influence of pores geometry and architecture on dynamic mechanical properties. Biomaterials, 27(7), 974-985. doi:https://doi.org/10.1016/j.biomaterials.2005.07.023
Naing, M. W., Chua, C. K., Leong, K. F., & Wang, Y. (2005). Fabrication of customised scaffolds using computer?aided design and rapid prototyping techniques. Rapid Prototyping Journal, 11(4), 249-259. doi:10.1108/13552540510612938
Nikolova, M. P., & Chavali, M. S. (2019). Recent advances in biomaterials for 3D scaffolds: A review. Bioactive Materials, 4, 271-292. doi:https://doi.org/10.1016/j.bioactmat.2019.10.005
Norato, J. A., & Johnson, A. J. W. (2011). A Computational and Cellular Solids Approach to the Stiffness-Based Design of Bone Scaffolds. Journal of Biomechanical Engineering, 133, 091003-091001:8091003-8091001. doi:10.1115/1.4004994
Restrepo, S., Ocampo, S., Ramírez, J. A., Paucar, C., & García, C. (2017). Mechanical properties of ceramic structures based on Triply Periodic Minimal Surface (TPMS) processed by 3D printing. Journal of Physics: Conference Series, 935(012036). doi:10.1088/1742-6596/935/1/012036
Saito, E., Kang, H., Taboas, J. M., Diggs, A., Flanagan, C. L., & Hollister, S. J. (2010). Experimental and computational characterization of designed and fabricated 50:50 PLGA porous scaffolds for human trabecular bone applications. Journal of Materials Science: Materials in Medicine, 21(8), 2371-2383. doi:10.1007/s10856-010-4091-8
Shapiro, S. S., Wilk, M. B., & Chen, H. J. (1968). A Comparative Study of Various Tests for Normality. Journal of the American Statistical Association, 63(324), 1343-1372. doi:10.2307/2285889
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730-737. doi:10.2307/2286009
Wu, D., Spanou, A., Diez-Escudero, A., & Persson, C. (2019). 3D-printed PLA/HA composite structures as synthetic trabecular bone: A feasibility study using Fused Deposition Modelling. Journal of the Mechanical Behavior of Biomedical Materials, 103, doi: https://doi.org/10.1016/j.jmbbm.2019.103608
Yang, N., Quan, Z., Zhang, D., & Tian, Y. (2014). Multi-morphology transition hybridization CAD design of minimal surface porous structures for use in tissue engineering. Computer-Aided Design, 56, 11-21. doi:https://doi.org/10.1016/j.cad.2014.06.006
Yang, N., & Zhou, K. (2014). Effective method for multi-scale gradient porous scaffold design and fabrication. Materials Science and Engineering: C, 43, 502-505. doi:https://doi.org/10.1016/j.msec.2014.07.052
Zadpoor, A. A. (2014). Bone tissue regeneration: the role of scaffold geometry. Biomaterials Science, 1-34.
Amirkhani, S., Bagheri, R., & Zehtab Yazdi, A. (2012). Effect of pore geometry and loading direction on deformation mechanism of rapid prototyped scaffolds. Acta Materialia, 60(6), 2778-2789. doi:https://doi.org/10.1016/j.actamat.2012.01.044
Anderson, D., Davis, H., Nitsche, J., & Scriven, L. (1990). Periodic surfaces of prescribed mean curvature. In Physics of amphiphilic layers (pp. 130-130): Springer.
Anderson, T. W., & Darling, D. A. (1952). Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes. The Annals of Mathematical Statistics (2), 193-212, 120.
Bidan, C. M., Wang, F. M., & Dunlop, J. W. C. (2013). A three-dimensional model for tissue deposition on complex surfaces. Computer methods in biomechanics and biomedical engineering, 16(10), 1056-1070. doi:10.1080/10255842.2013.774384
Blanquer, S. B., Werner, M., Hannula, M., Sharifi, S., Lajoinie, G. P., Eglin, D., Grijpma, D. W. (2017). Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds. BioFa, 9(2), 025001.
Bobbert, F., Lietaert, K., Eftekhari, A. A., Pouran, B., Ahmadi, S., Weinans, H., & Zadpoor, A. (2017). Additively manufactured metallic porous biomaterials based on minimal surfaces: A unique combination of topological, mechanical, and mass transport properties. Acta Biomaterialia, 53, 572-584.
Corder, G. W., & Foreman, D. I. (2009). Chapter 2. Testing data for normality. In Nonparametric statistics for non-statisticians: a step-by-step approach. United States: John Wiley & Sons, Inc.
Cuan-Urquizo, E., Barocio, E., Tejada-Ortigoza, V., Pipes, R. B., Rodriguez, C. A., & Roman-Flores, A. (2019). Characterization of the mechanical properties of FFF structures and materials: A review on the experimental, computational and theoretical approaches. Materials, 12(6), 895.
Cuan-Urquizo, E., Yang, S., & Bhaskar, A. (2015). Mechanical characterisation of additively manufactured material having lattice microstructure. Paper presented at the IOP Conf. Series: Materials Science and Engineering.
D´Agostino, R. B., & Stephens, M. A. (Eds.). (1986). Goodness-of-fit techniques. United States: Marcel Dekker, Inc.
Farah, S., Anderson, D. G., & Langer, R. (2016). Physical and mechanical properties of PLA, and their functions in widespread applications — A comprehensive review. Advanced Drug Delivery Reviews, 107, 367-392. doi:https://doi.org/10.1016/j.addr.2016.06.012
Gandy, P. J. F., Bardhan, S., Mackay, A. L., & Klinowski, J. (2001). Nodal surface approximations to the P,G,D and I-WP triply periodic minimal surfaces. Chemical Physics Letters, 336(3), 187-195. doi:https://doi.org/10.1016/S0009-2614(00)01418-4
Giannitelli, S. M., Accoto, D., Trombetta, M., & Rainer, A. (2014). Current trends in the design of scaffolds for computer-aided tissue engineering. Acta Biomaterialia, 10(2), 580-594. doi:https://doi.org/10.1016/j.actbio.2013.10.024
González, C. G., Liste, A. V., & Felpeto, A. B. (2013a). Apéndice I. Tablas Estadísticas: Tabla de Durbin-Watson. In Tratamiento de datos con R, STATISTICA y SPSS (pp. 875). España: Editorial Díaz de Santos.
González, C. G., Liste, A. V., & Felpeto, A. B. (2013b). Capítulo XI. Regresiones. In Tratamiento de datos con R, STATISTICA y SPSS (pp. 460). España: Editorial Díaz de Santos.
Han, L., & Che, S. (2018). An Overview of Materials with Triply Periodic Minimal Surfaces and Related Geometry: From Biological Structures to Self-Assembled Systems. Advanced Materials, 30(17), 1705708. doi:10.1002/adma.201705708
Hollister, S. J. (2005). Porous scaffold design for tissue engineering. Nature Materials, 4(7), 518-524. doi:10.1038/nmat1421
Jung, Y., & Torquato, S. (2005). Fluid permeabilities of triply periodic minimal surfaces. Physical Review E, 72(5), 056319.
Kanczler, J. M., Wells, J. A., Gibbs, D. M. R., Marshall, K. M., Tang, D. K. O., & Oreffo, R. O. C. (2020). Bone tissue engineering and bone regeneration. In R. Lanza, R. Langer, J. P. Vacanti, & A. Atala (Eds.), Principles of Tissue Engineering (5th ed.). United States of America: Elsevier Inc.
Knychala, J., Bouropoulos, N., Catt, C., Katsamenis, O., Please, C., & Sengers, B. (2013). Pore geometry regulates early stage human bone marrow cell tissue formation and organisation. Annals of Biomedical Engineering, 41(5), 917-930.
Lee, J. S., Cha, H. D., Shim, J. H., Jung, J. W., Kim, J. Y., & Cho, D. W. (2012). Effect of pore architecture and stacking direction on mechanical properties of solid freeform fabrication?based scaffold for bone tissue engineering. Journal of Biomedical Materials Research Part A, 100(7), 1846-1853.
Malinauskas, M., Skliutas, E., Jonušauskas, L., Mizeras, D., Šešok, A., & Piskarskas, A. (2015). Tailoring bulk mechanical properties of 3D printed objects of polylactic acid varying internal micro-architecture (Vol. 9505), SPIE.
Maskery, I., Sturm, L., Aremu, A. O., Panesar, A., Williams, C. B., Tuck, C. J., . . . Hague, R. J. M. (2017). Insights into the mechanical properties of several triply periodic minimal surface lattice structures made by polymer additive manufacturing. Polymer. doi:10.1016/j.polymer.2017.11.049.
Meeks III, W. H. (1990). The theory of triply periodic minimal surfaces. Indiana University Mathematics Journal, 877-936.
Melchels, F. P. W., Bertoldi, K., Gabbrielli, R., H., V. A., Feijen, J., & Grijpma, D. W. (2010). Mathematically defined tissue engineering scaffold architectures prepared by stereolithography. Biomaterials, 31(27), 6909-6916. doi:https://doi.org/10.1016/j.actbio.2010.06.012
Montazerian, H., Davoodi, E., Asadi-Eydivand, M., Kadkhodapour, J., & Solati-Hashjin, M. (2017). Porous scaffold internal architecture design based on minimal surfaces: A compromise between permeability and elastic properties. Materials & Design, 126, doi: 10.1016/j.matdes.2017.04.009
Moroni, L., de Wijn, J. R., & van Blitterswijk, C. A. (2006). 3D fiber-deposited scaffolds for tissue engineering: Influence of pores geometry and architecture on dynamic mechanical properties. Biomaterials, 27(7), 974-985. doi:https://doi.org/10.1016/j.biomaterials.2005.07.023
Naing, M. W., Chua, C. K., Leong, K. F., & Wang, Y. (2005). Fabrication of customised scaffolds using computer?aided design and rapid prototyping techniques. Rapid Prototyping Journal, 11(4), 249-259. doi:10.1108/13552540510612938
Nikolova, M. P., & Chavali, M. S. (2019). Recent advances in biomaterials for 3D scaffolds: A review. Bioactive Materials, 4, 271-292. doi:https://doi.org/10.1016/j.bioactmat.2019.10.005
Norato, J. A., & Johnson, A. J. W. (2011). A Computational and Cellular Solids Approach to the Stiffness-Based Design of Bone Scaffolds. Journal of Biomechanical Engineering, 133, 091003-091001:8091003-8091001. doi:10.1115/1.4004994
Restrepo, S., Ocampo, S., Ramírez, J. A., Paucar, C., & García, C. (2017). Mechanical properties of ceramic structures based on Triply Periodic Minimal Surface (TPMS) processed by 3D printing. Journal of Physics: Conference Series, 935(012036). doi:10.1088/1742-6596/935/1/012036
Saito, E., Kang, H., Taboas, J. M., Diggs, A., Flanagan, C. L., & Hollister, S. J. (2010). Experimental and computational characterization of designed and fabricated 50:50 PLGA porous scaffolds for human trabecular bone applications. Journal of Materials Science: Materials in Medicine, 21(8), 2371-2383. doi:10.1007/s10856-010-4091-8
Shapiro, S. S., Wilk, M. B., & Chen, H. J. (1968). A Comparative Study of Various Tests for Normality. Journal of the American Statistical Association, 63(324), 1343-1372. doi:10.2307/2285889
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730-737. doi:10.2307/2286009
Wu, D., Spanou, A., Diez-Escudero, A., & Persson, C. (2019). 3D-printed PLA/HA composite structures as synthetic trabecular bone: A feasibility study using Fused Deposition Modelling. Journal of the Mechanical Behavior of Biomedical Materials, 103, doi: https://doi.org/10.1016/j.jmbbm.2019.103608
Yang, N., Quan, Z., Zhang, D., & Tian, Y. (2014). Multi-morphology transition hybridization CAD design of minimal surface porous structures for use in tissue engineering. Computer-Aided Design, 56, 11-21. doi:https://doi.org/10.1016/j.cad.2014.06.006
Yang, N., & Zhou, K. (2014). Effective method for multi-scale gradient porous scaffold design and fabrication. Materials Science and Engineering: C, 43, 502-505. doi:https://doi.org/10.1016/j.msec.2014.07.052
Zadpoor, A. A. (2014). Bone tissue regeneration: the role of scaffold geometry. Biomaterials Science, 1-34.
Published
2021-11-23
How to Cite
González González, A., Rivas Santana, M., Quiza Sardiñas, R., Paz Estévez, E. A., & Pla Pérez, A. (2021). Modeling of porous scaffolds based on the triply periodic structures P and G. Orange Journal, 3(5), 30–41. https://doi.org/10.46502/issn.2710-995X/2021.5.04
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