Abstract
A tissue engineering scaffold provides a proper environment to support physiological loads, and enhance cell migration and delivery for re-modeling of regenerating tissue. Hence, in the design of scaffolds, it is required to control the scaffold architecture with mechanical and mass transport properties simultaneously. In this paper, a level set-based topology optimization method will be developed to systematically generate three dimensional (3D) microstructures for tissue engineering scaffolds, with the prescribed properties for mechanical stiffness, fluid porosity and permeability. To create the internal architecture for scaffolds with desired properties, the numerical homogenization method will be used to evaluate the effective properties of the microstructure for building the periodic composite media, and a parametric level set method will be introduced to find the optimized shape and topology of the microstructure. Several numerical examples are used to demonstrate the effectiveness of the proposed method in achieving scaffolds with desired multifunctional properties, within the numerically estimated cross-property bounds between the effective bulk modulus and permeability under different porosities.
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References
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363–393
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, Berlin
Boschetti F et al (2004) Biomechanical properties of human articular cartilage under compressive loads. Biorheology 41:159–166
Challis VJ (2010) A discrete level-set topology optimization code written in Matlab. Struct Multidiscip Optim 41:453–464
Challis VJ, Roberts AP, Wilkins AH (2008) Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int J Solids Struct 45:4130–4146
Challis VJ, Guest JK, Grotowski JF, Roberts AP (2012) Computationally generated cross-property bounds for stiffness and fluid permeability using topology optimization. Int J Solids Struct 49:3397–3408
Chen YH, Zhou SW, Li Q (2009) Computational design for multifunctional microstructural composites. Int J Mod Phys B 23:1345–1351
Chen YH, Zhou SW, Li Q (2011) Microstructure design of biodegradable scaffold and its effect on tissue regeneration. Biomaterials 32:5003–5014
de Kruijf N, Zhou S, Li Q, Mai YW (2007) Topological design of structures and composite materials with multiobjectives. Int J Solids Struct 44:7092–7109
Goulet RW et al (1994) The relationship between the structural and orthogonal compressive properties of trabecular bone. J Biomech 27:375–389
Guedes J, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83:143–198
Guedes JM, Rodrigues HC, Bendsøe MP (2003) A material optimization model to approximate energy bounds for cellular materials under multiload conditions. Struct Multidiscip Optim 25:446–452
Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43:7028–7047
Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140
Hollister SJ (2005) Porous scaffold design for tissue engineering. Nat Mater 4:518–524
Hollister SJ, Maddox RD, Taboas JM (2002) Optimal design and fabrication of scaffolds to mimic tissue properties and satisfy biological constraints. Biomaterials 23:4095–4103
Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the babuška-brezzi condition. Comput Methods Appl Mech Eng 59:85–99
Kang Z, Wang Y (2011) Structural topology optimization based on non-local Shepard interpolation of density field. Comput Methods Appl Mech Eng 200:3515–3525
Kang H, Lin CY, Hollister SJ (2010) Topology optimization of three dimensional tissue engineering scaffold architectures for prescribed bulk modulus and diffusivity. Struct Multidiscip Optim 42:633–644
Kelly DJ, Prendergast PJ (2006) Prediction of the optimal mechanical properties for a scaffold used in osteochondral defect repair. Tissue Eng 12:2509–2519
Lanza R, Langer R, Vacanti JP (2011) Principles of tissue engineering. Academic Press
Lin CY, Kikuchi N, Hollister SJ (2004) A novel method for biomaterial scaffold internal architecture design to match bone elastic properties with desired porosity. J Biomech 37:623–636
Luo Z, Wang MY, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76:1–26
Luo Z, Zhang N, Gao W, Ma H (2012a) Structural shape and topology optimization using a meshless Galerkin level set method. Int J Numer Methods Eng 90(3):369–389
Luo Z, Zhang N, Ji J, Wu T (2012b) A meshfree level-set method for topological shape optimization of compliant multiphysics actuators. Comput Methods Appl Mech Eng 223–224:133–152
Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimization of structures using meshless density variable approximants. Int J Numer Methods Eng 93:443–464
Osher S, Fedkiw RP (2002) Level set methods and dynamic implicit surface. Springer, New York
Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49
Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University
Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489–528
Sigmund O (2002) On the optimality of bone microstructure. Springer, Neverland
Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24:359–373
van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437–472
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246
Wang YQ, Luo Z, Zhang N, Kang Z (2014) Topological shape optimization of microstructural metamaterials using a level set method. Comput Mater Sci 87:178–186
Wang YQ, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586
Wendland H (2006) Computational aspects of radial basis function approximation. Stud Comput Math 12:231–256
Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896
Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89:197–224
Acknowledgments
This research is supported in part by the Australian Research Council (ARC)-Discovery Project (DP160102491, DP150102751), the National Natural Science Foundation of China (51575204), and the Science and Technology Support Program of Hubei Province of China (2015BHE026), as well as by the Open Research Foundation of State Key Lab. of Digital Manufacturing Equipment & Technology (DMETKF2015010), Huazhong University of Science & Technology, Wuhan, China.
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This paper is submitted for possible publication in Structural and Multidisciplinary Optimization. It has not been previously published, is not currently submitted for review to any other journals, and will not be submitted elsewhere during the peer review. It is noted that this manuscript has been submitted in a style of “Your Paper Your Way” only for the convenience of peer-review.
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Wang, Y., Luo, Z., Zhang, N. et al. Topological shape optimization of multifunctional tissue engineering scaffolds with level set method. Struct Multidisc Optim 54, 333–347 (2016). https://doi.org/10.1007/s00158-016-1409-2
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DOI: https://doi.org/10.1007/s00158-016-1409-2