Supplementary Material for the Article Minimal Surface Scaffold Designs for Tissue Engineering

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1 Supplementary Material for the Article Minimal Surface Scaffold Designs for Tissue Engineering Sebastian C. Kapfer, Stephen T. Hyde, Klaus Mecke, Christoph H. Arns, and Gerd E. Schröder-Turk July 4, 20

2 A Voxelization of Minimal Surface Scaffolds The majority of minimal surfaces analyzed in this study are generated using area minimization with the Surface Evolver software []. The P, G, I-WP and D triangulated surfaces are derived from their respective Weierstraß parametrization [2, 3]. Voxelized representations of the bicontinuous minimal surface scaffolds, i. e. approximations by cubic voxels, are obtained by discretizing the EDM field of the surfaces. For each surface, the EDM field is evaluated on a simple cubic lattice corresponding to the desired voxel size, from a copy of the oriented translational unit cell of the surface. All surfaces are orientable, and consequently we can define a signed EDM field such that SEDM > 0 in one labyrinth, and SEDM < 0 in the complement. Network solids are then produced by setting all voxels with SEDM > 0 to one (solid). Sheetlike materials are created by setting all voxels with SEDM < d to one, with d chosen to yield a volume fraction of 50%. All other voxels are void voxels. Triangulated surface representations of branched minimal surfaces (3srs and 4srs in this work) may also be generated using Surface Evolver, but are nonorientable. The EDM field is computed in the same way as for bicontinuous surfaces, and sheetlike scaffolds are generated by setting all voxels with EDM < d to one, with d chosen to yield a volume fraction of 50%. The source code of the voxelization software and the voxelized structures are available from B Computation of Mechanical Properties Both the morphological characteristics and the mechanical properties are computed on the voxelized representations of the materials. The former is described in section C; for the mechanical properties, a parallelized finite-element method is implemented [4] which operates directly on the cubic voxels. Each voxel is considered to be either a void voxel or filled with an isotropic linear-elastic material with stiffness tensor C 0 ijkl = ( κ µ 0) δij δ kl + µ 0 (δ ik δ jl + δ il δ jk ) () with bulk modulus κ 0 and shear modulus µ 0 of the material. The elastic energy U = 2 eij ( r)c ijklχ( r)e 0 kl ( r), (2) ijkl is then minimized using a conjugate-gradient scheme, where e is the strain tensor, and χ is the indicator function of the solid domain, which is unity in a solid voxel, and vanishes in the void domain; the angular brackets denote spatial averaging, d 3 rf( r) f( r) =, (3) d3 r 2

3 the integrals over the whole scaffold volume. The strain tensor is subject to the constraint e ij = 2 ( iu j + j u i ), where u( r) is a displacement field. The principal degrees of freedom are the displacements of the voxel corners, and the displacement field is linearly interpolated across each voxel. Boundary conditions are periodic and fix the net strain u( r) in the structure. This process is repeated for three distinct net strains, yielding the raw effective elastic moduli: Net strain u( r) Modulus 3 δ ij bulk modulus κ 2 (δ i3δ j3 δ i δ j ) shear modulus µ 2 (δ iδ j3 + δ i3 δ j ) shear modulus µ From discretizations ranging from 64 3 to 52 3, we extrapolate to the continuum limit of vanishing voxel size. For reliable results, the minimum domain diameter should correspond to at least 5 7 finite element voxels. Young s modulus E( n) can be computed from κ, µ and µ, and the direction n of compression [5]. We give, in table of the main article the orientationally averaged Young s modulus E defined from E := d 3 n 4πE( n) = 9κ + 2 5µ + 5µ, (4) where the integration is over the unit sphere. The source code of the finite-element code to compute effective elastic moduli are available from C Morphological Analysis The Euclidean distance map, also known as Euclidean distance transform [6, 7, 8], specifies, for each point p in a domain D, the distance to the boundary of the domain, formally EDM(D, p) := min p D p p, (5) where D is the solid or void domain. A discrete version of the EDM, which is accurate to at least one voxel diagonal, is easily evaluated on the voxelized scaffolds [9], and permits the unambiguous definition of both a minimum and maximum domain radius, R perc (D) and R max (D) respectively. For the latter, the maximum value of the EDM field R max (D) = max EDM(D, p), (6) p D measures the largest occurring strut radius or the largest occurring pore radius. R max has been used previously to characterize domain widths in space partitions (see ref. [0] and refs. therein). 3

4 While the maximum value of the EDM is a useful quantity, the minimum value is always zero. Moreover, in network or sheetlike geometries as discussed here, the distribution of EDM values is dominated by small values contributed by points on or close to the solid-void interface. A useful measure for the minimal thickness is, however, given by the percolation critical radius R perc (D) []. It may informally be defined as the maximum radius of a sphere that may move from z = to z = + while being confined to the domain D. Using the language of mathematical morphology [2], it is equivalent to define R perc (D) as the largest radius b such that the b-eroded domain D S(b) := ( D c S(b) ) c, (S(b) a radius b sphere centered at the origin, the Minkowski sum, and D c denoting the complement domain) remains percolating in the z direction. In principle, the percolation radii may be different for the x and y direction. For the cubic scaffolds considered in this work, the three radii are identical. The percolation critical radius can be determined from the EDM on a single copy of the translational unit cell by successive erosions and Hoshen-Kopelman percolation tests. A slight complication occurs because for percolation of the infinite periodic scaffold, a path in the translational unit cell connecting two diametrically opposed faces of the unit cell is not sufficient. Proper connectivity is required in order to form a percolating path, and not a closed loop in the infinite periodic scaffold. References [] Brakke K. The surface evolver. Exp Math. 992;(2):4 65. [2] Nitsche J. Vorlesungen über Minimalflächen. Springer-Verlag, Berlin; 975. [3] Hyde ST, Andersson S, Larsson K, Blum Z, Landh T, Lidin S, et al. The Language of Shape. st ed. Amsterdam: Elsevier Science; 997. [4] Garboczi EJ. Finite element and finite difference programs for computing the linear electric and elastic properties of digital images of random materials. NIST; 998. NISTIR [5] Walpole LJ. The elastic shear moduli of a cubic crystal. J Phys D Appl Phys. 986;9: [6] Danielsson P. Euclidean distance mapping. Computer Graphics and Image Processing. 980;4: [7] Saito T, Toriwaki J. New algorithms for Euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognit. 994;27(): [8] Cuisenaire O, Macq B. Fast Euclidean distance transformation by propagation using multiple neighborhoods. Comput Vis Image Underst. 999;76(2):

5 [9] Felzenszwalb PF, Huttenlocher DP. Distance transforms of sampled functions. Cornell Computing and Information Science; TR Available from: pdf. [0] Schröder-Turk GE, Fogden A, Hyde ST. Local v/a variations as a measure of structural packing frustration in bicontinuous copolymer mesophases, and prediction of an alternating Im3m (I-WP) phase in block-copolymers with polydispersity. Eur Phys J B. 2007;59():5 26. [] Katz AJ, Thompson AH. Quantitative prediction of permeability in porous rock. Phys Rev B. 986;34(). [2] Soille P. Morphological Image Analysis Principles and Applications. Springer-Verlag;

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