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

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

Rita Ambu 1,* ID and Anna Eva Morabito 2 ID

1 Department of Mechanical, Chemical and Materials Engineering, University of Cagliari, 09123 Cagliari, Italy 2 Department of Engineering for Innovation, University of Salento, 73100 Lecce, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-070-675-5709

Received: 13 July 2018; Accepted: 17 August 2018; Published: 25 August 2018

Abstract: In tissue engineering, biocompatible porous scaffolds that try to mimic the features and function of the bone are of great relevance. In this paper, an effective method for the design of 3D porous scaffolds is applied to the modelling of structures with variable architectures. These structures are of interest since they are more similar to the stochastic configuration of real bone with respect to architectures made of a unit cell replicated in three orthogonal directions, which are usually considered for this kind of applications. This property configures them as, potentially, more suitable to satisfy simultaneously the biological requirements and those relative to the mechanical strength. The procedure implemented is based on the implicit surface modelling method and the use of a triply periodic minimal surface (TPMS), specifically, the Schwarz’s Primitive (P) minimal surface, whose geometry was considered for the development of scaffolds with different configurations. The representative structures modelled were numerically analysed by means of finite element analysis (FEA), considering them made of a biocompatible titanium alloy. The architectures considered were thus assessed in terms of the relationship between the geometrical configuration and the mechanical response to compression loading.

Keywords: tissue engineering; scaffold; design; TPMS; FEA

1. Introduction A ﬁeld of interest in tissue engineering (TE), the discipline whose main aim is to replace damaged tissue and organs, relates to bone repairing [1]. Porous biocompatible structures, generally termed as engineering scaffolds, can be used as bone replacement when other procedures, such as autologous bone grafting, are not feasible or involve high risk for the patient. Engineering scaffolds have to provide an optimum environment for cells to develop and culture into tissues meeting biological, physical, and mechanical properties of the replaced tissues. Thus, at the design stage, different parameters have to be considered to accomplish the requirements. The performance of a scaffold usually depends on an optimal combination between the dimensional and geometrical characteristics of the pores and the resultant mechanical properties of the structure. Generally, large and interconnected pores are preferred for bone ingrowth leading to structures with high porosity; on the other hand, the mechanical properties have to be similar to those of the bone to avoid mismatches with the surrounding tissues which can affect the operation or longevity of the implant as a result of the effect known as stress shielding [2]. The choice of a proper material is another signiﬁcant issue in the design of the scaffolds, since, besides the intrinsic mechanical characteristics, the biocompatibility is a fundamental aspect to be

Symmetry 2018, 10, 361; doi:10.3390/sym10090361 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 361 2 of 13 taken into account. Biodegradable polymeric materials have been considered [3] to facilitate cell ingrowth due to material resorption over time along with the adequate mechanical environment provided by the structure of the scaffold. For load bearing applications investigated in this study, biocompatible metallic scaffolds are expected to fulfill the required mechanical strength. In particular, the use of titanium and its alloys, such as Ti-6Al-4V, is well reported in literature [4–6] since they have an excellent strength-to-weight ratio, toughness, and most importantly, the biocompatibility and corrosion resistance of their naturally forming surface oxide [7], making them particularly suitable for these applications. The advent of new manufacturing technologies, such as additive manufacturing (AM), has expanded the potentiality in the design of these structures since with these techniques, tailored design specification implants and highly complex and custom-fitting medical devices can be provided [8]. The modelling of these structures can be conducted with different approaches. The simpler approach uses parametric Computer-Aided Design (CAD) modelling in which a unit cell is generated from solid features that are combined together. A three-dimensional (3D) periodic array of the unit cell can be carried out along three mutually perpendicular directions to obtain the final architecture. Usually, lattice-based geometries [5,6,9] such as cubic, diamond lattice, rhombic dodecahedron, or similar, are generated with this method since more complex geometries are difficult to model. Another approach, which was used in this paper, is the method of implicit surface modelling (ISM). ISM is a highly flexible approach for the generation of cellular structures providing a compact representation of potentially complex surfaces. It allows the description of scaffold architectures using a single mathematical equation, with the capability to introduce different pore shapes and architectural features, including pore size gradients [10]. This method was used to model scaffolds based on the class of surfaces known as triply periodic minimal surfaces (TPMS). These surfaces, frequently occurring in nature, are considered very attractive for these kind of applications [11,12]. They are, mathematically, defined as surfaces with zero mean curvature at every point and periodicity in three directions mutually perpendicular [13]. Structures based on these surfaces, have shown to provide high values of permeability [14], parameter related to geometrical characteristics such as pore shape and size, high surface to volume ratio, and topological complexity for cells ingrowth. Furthermore, it has been shown [15] that their curved configuration can promote cell proliferation more than scaffold architectures based on lattice cells. Lattice based architectures are well reported in literature and they have been studied both experimentally and numerically. The scaffolds based on TPMS surfaces can be potentially advantageous compared to those for the properties abovementioned. Moreover, the lattice-based architectures are characterised by the presence of sharp corners that should lead to stress concentrations; the curved surfaces of the scaffolds based on TPMS surfaces should contribute to attenuate this effect. Among the various TPMS surfaces that have been described, the level surface known as Schwarz’s Primitive (P) minimal surface has been considered in this study. Examples of applications of this surface to scaffold design is more limited in literature with respect to other surfaces belonging to this class such as Gyroid [16] or Diamond [17]. Starting from the basic equation of the surface and by considering various parameters that are varied within suitable ranges, solid models of the cells can be obtained. Scaffolds with uniform geometrical characteristics can be obtained with the procedure previously described for lattice-based architectures, that is, by replicating a unit cell along three orthogonal directions. Most of the structures reported in literature are arranged in this way. In addition, scaffolds whose characteristics locally vary, for example, with a linear gradient of porosity along one or more directions, can be obtained with the combination of cells with different geometrical configurations resulting in structures with a more complex architecture with respect to those with a uniform arrangement. Scaffolds in which either porosity or pore architecture vary throughout the structure are of particular interest, since they can facilitate the optimisation of different, and sometimes conflicting, requirements. In fact, generally, high porosity supports more cell growth and penetration but it limits Symmetry 2018, 10, 361 3 of 13 the load bearing capability of the scaffold. Furthermore, this arrangement is more similar to that of real bone which is intrinsically stochastic, although the configuration and the porosity distribution is different in the various zones of the human body. Different approaches can be used to model structures with variable architectures. Scaffolds with uneven structural configurations can be obtained by assembling unit cells arranged in a library by using programmed algorithms [18] or by using mathematical expressions that are manipulated operating on the terms related to the porosity domain [19,20]. In Reference [21], geometrical and topological information on the bone tissue are obtained from medical images; the complex geometry is then partitioned into functionally uniform regions to generate structures with gradients of porosity. In this paper a method based on ISM has been implemented and scaffolds with variable pore architecture were modelled. These structures are based on the TPMS Primitive (P) cell whose geometrical characteristics were locally varied so as to obtain the different architectures. The characteristics of a scaffold in terms of distribution of the porosity should be chosen according to the specific function the implant has to carry out. From this perspective, scaffolds with a linear gradient of porosity in a direction and structures with a radial gradient, or similar, are of interest. These types of scaffolds were considered in this study and structurally analysed in relation to a scaffold obtained with a P-unit cell replicated in the 3D space. The numerical evaluation of the modelled scaffolds was done by means of finite element analysis (FEA) considering the structures made of a biocompatible titanium alloy and were subjected to compression load. In this way, the geometrical configuration of the structures considered was related to the mechanical response in terms of strength and stiffness.

2. Methodology The methodology proposed here for scaffold modelling was based on surfaces suitably derived from the Schwarz Primitive, which belongs to the class of the minimal surfaces. Several approaches exist for modelling the minimal surfaces. These approaches are: parametric, implicit, or based on boundary methods. In the parametric approach, the minimal surface is described by a parametric equation. However, only a few minimal surfaces can be described with simple analytical functions in Cartesian space, such as, for example, the catenoid or the helicoid [22]. Minimal surfaces can be generated parametrically using the Enneper–Weierstrass representation. However, since the Weierstrass function is known only for a few minimal surfaces, the parametric method is generally not used for describing this class of surfaces [15]. Boundary methods include a number of approaches aimed at the modelling of discrete TPMS. These methods are based on the progressive refinement of a first polygonal model defined by its boundary, subjected to a variety of boundary and inter-polygonal constraints [15]. In the Plateau method (1873), a polygon in R3 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. In this paper, as stated in the introduction, the implicit method is used for the modelling of the scaffolds derived from the Schwarz Primitive. Using this approach, the surface of the scaffold is described by means of an implicit f that is a continuous scalar-valued function over the domain R3. 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:

f(x, y, z, s, k) = cos (x/s) + cos (y/s) + cos (z/s) − k. (1) Symmetry 2018, 10, 361 4 of 13 Symmetry 2018, 10, x FOR PEER REVIEW 4 of 13 Symmetry 2018, 10, x FOR PEER REVIEW 4 of 13 TheThe two two mathematical mathematical parameters, parameters, ss andand kk inin EquationEquation (1), (1), can can be be used used for for modifying modifying the the architecturearchitectureThe oftwo of the the mathematical P-surface. P-surface. Under parameters,Under the the boundary boundarys and k conditions in conditionsEquation x = (1),[x−π, = can π], [− π bey ,= π used[],−π,y π]= for and [ − modifyingπ ,z π= ][− andπ, π], thez = [−πthearchitecture, πP],-surface theP-surface described of the describedP-surface by Equation. byUnder Equation (1) the delimits boundary (1) delimitsa single conditions cell a single with x cubic= cell [−π, with symmetryπ], cubicy = [−π, symmetryof π] side and 2π z (= ofP [-−cell sideπ, inπ], 2 π (P-cellthethe in following).P-surface the following). described When When in by Equation Equation in Equation (1)(1) delimitsk=0 (1), thek = aP 0,single-surface the P-surfacecell is with a minim cubic is a minimalalsymmetry surface surface of satisfying side satisfying2π ( allP-cell the inall theproperties propertiesthe following). of of this this Whenclass class of in of surfaces. surfaces.Equation When When(1) kk= 0kis, is non the non-zero, -Pzero,-surface the the Pis- P-surfacesurface a minim is isalcharacteri characterised surfaces ed satisfying by by a constant a constant all the valuevalueproperties of of the the mean ofmean this curvature curvatureclass of surfaces. at at each each point, Whenpoint, andandk is itnonit partitionspartitions-zero, the the the P- surface3D 3D space space is intocharacteri into two two phasess phases,ed by, noa noconstant longer longer havinghavingvalue the ofthe samethe same mean volume. volume curvature If. If the the at two twoeachphases phases point, respectivelyrespectivelyand it partitions identify identify the 3D a a solid solidspace and andinto a atwovoid void phases region, region,, nothis thislonger last last observationobservationhaving the implies impliessame thatvolume that the the. P-cellIf Pthe-cell istwo characterisedis characteriphases respectivelysed by by a porositya porosity identify different different a solidfrom and from a 0.5. void0.5. Figure Figure region,1 shows1 thisshows last the Schwarztheobservation Schwarz minimal minimal implies surface thatsurface which the whichP is-cell obtained is obtainedcharacteri setting settingsedk by= 0 ka= and 0porosityands =s= 1 indifferent Equation from (1). (1). 0.5. Figure 1 shows the Schwarz minimal surface which is obtained setting k=0and s=1 in Equation (1).

FigureFigure 1. 1.The The Schwarz Schwarz Primitive Primitive minimalminimal surface ( (kk == 0 0 and and s s= =1). 1). Figure 1. The Schwarz Primitive minimal surface (k = 0 and s = 1). In Figure 2, different models of P-cell are represented for s=1 and at the varying of the k In Figure2, different models of P-cell are represented for s = 1 and at the varying of the k parameter.In Figure In this 2, different figure, the models surface of in P - redcell is are the represented P-surface shown for s= 1 in and Figure at the 1 and varying the k ofvalues the k parameter.increaseparameter. going In this In inward. ﬁgure,this figure the, surface the surface in red in is red the is P-surface the P-surface shown shown in Figure in 1 Figure and the 1 andk values the k increasevalues goingincrease inward. going inward.

Figure 2. Different models of the P-cell at the varying of the k parameter and s =1 (The Schwarz Figure 2. Different models of the P-cell at the varying of the k parameter and s = 1 (The Schwarz PrimitiveFigure 2. is Different indicated modelsin red) . of the P-cell at the varying of the k parameter and s =1 (The Schwarz Primitive is indicated in red). Primitive is indicated in red). The variation of the parameter s in Equation (1) has the same effect of a scaling transformation (uniformTheThe variation if variation the s-value of theof theis parameter the paramete same alongsr ins in Equation with Equation the directions (1) (1) has has the the x same, samey and effect effectz) with ofof respect aa scalingscaling to transformation transformationits barycenter (uniformfollowed(uniform if theby if thethes-value intersections-value is theis the same of same the along scaled along with surfacewith the the with directions directions the cubicx x,,y ycell.and and zz)) withwith respect to to its its barycenter barycenter followedfollowedIn byFigure theby the intersection3a, intersectionb, different of models of the the scaled scaled of the surface surfaceP-cell with with k the =the0 are cubic cubic shown cell. cell. by varying the s parameter (the SchwarzIn FigureIn PrimitiveFigure3a,b, 3a differentis,b ,always different models in modelsred). of Figure theof the P-cell 3a P -scellhows with with thek =k =P 00- arecellsare shownshown obtained by varying for s ≥ 1. the theFigure ss parameterparameter 3b reports (the (the SchwarztheSchwarz P- Primitivecells Primitive for s≤is 1; alwaysis in always this incase, in red). red to Figure)point. Figure out3a 3a showsthe shows geomet the the P-cellsrical P-cells differences obtained obtained foramong for ss ≥≥ the1.1. Figure Figurevarious 3b3b cells, reports a thesectionalthe P-cells P-cells forview fors ≤of s≤ 1;the 1; in insurfaces this thiscase, case, is shown toto pointpoint. out the geomet geometricalrical differences differences among among the the various various cells, cells, a a sectionalsectional view view of of the the surfaces surfacesis is shown.shown.

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(a) (b)

Figure 3.Different models of P-cells at the varying of the s parameter and k=0(the Schwarz Primitive is Figure 3. Different models of P-cells at the varying of the s parameter and k = 0 (the Schwarz Primitive indicated in red): (a) P-cells for s ≥ 1; (b) Half-P-cells for s ≤ 1. is indicated in red): (a) P-cells for s ≥ 1; (b) Half-P-cells for s ≤ 1.

When k or s are varied(a) in Equation (1), some important geometric properties(b) of the cell also vary, Whensuch ask orporositys are, variedpore size, in area Equation-volume ratio (1),, someand so important on. These properties geometric are properties directly related of the to cell the also vary, suchtargetFigure as requirementsporosity 3.Different, pore models for size the of, design area-volumeP-cells atof the customi varying ratios,ed andof bonethe so s parameter scaffolds. on. These and A propertiessolid k=0(the model Schwarz are of Primitivethe directly P-cell is relatedcan be to indicated in red): (a) P-cells for s ≥ 1; (b) Half-P-cells for s ≤ 1. the targetobtained requirements from the tessellated for the design surface of by customised an operation bone of offsetting scaffolds. that A makes solid the model P-isosurface of the P-cell solid. can The geometry of the solid P-cell is influenced not only by the parameters k and s of the mathematical be obtainedWhen from k or s the are tessellatedvaried in Equation surface (1), by some an operationimportant geometric of offsetting properties that makesof the cell the also P-isosurface vary, formulation of its most external surface, but also by the thickness t added during the offsetting solid.such The as geometryporosity, pore of size the, area solid-volume P-cell ratio is, inﬂuencedand so on. These not onlyproperties by the are parameters directly relatedk and to thes of the operation. target requirements for the design of customised bone scaffolds. A solid model of the P-tcell can be mathematicalThe formulationscaffolds model ofled its in most this study external are described surface, butby an also implicit bythe function thickness similar toadded Equation during obtained from the tessellated surface by an operation of offsetting that makes the P-isosurface solid. the offsetting(1), where operation. k is a continuous scalar-valued function over the domain R3. For a cubic scaffold with a The geometry of the solid P-cell is influenced not only by the parameters k and s of the mathematical Thenumber scaffolds of cells modelled along one in this side study equal are to N, described the domain by an of implicitthe function function is given similar by the to boundaryEquation (1), formulation of its most external surface, but also by the thickness t added during the offsetting where conditions:k is a continuous x = [−Nπ, scalar-valued Nπ], y = [−Nπ, function Nπ], and over z = the[−Nπ, domain Nπ]. R3. For a cubic scaffold with a number operation. of cells alongThe onefunctions side equalfor k that to N, have the been domain considered of the in function this paper is givenare shown by the in the boundary right column conditions: of The scaffolds modelled in this study are described by an implicit function similar to Equation x = [−NTableπ,N 1π (where], y = [ −C Nis πan,N arbitraryπ], and constant)z = [−N πand,N identifyπ]. different types of scaffolds, as reported in the (1), where k is a continuous scalar-valued function over the domain R3. For a cubic scaffold with a left column of the table. numberThe functions of cells alongfor k that one side have equal been to considered N, the domain in this of the paper function are shownis given in by the the right boundary column of Table1 (where C is an arbitrary constant) and identify different types of scaffolds, as reported in the conditions: x = [−Nπ, Nπ],Table y = 1.[−RelationshipNπ, Nπ], and between z = [− theNπ, type Nπ]. of scaffold and k(x, y, z). left columnThe offunctions the table. for k that have been considered in this paper are shown in the right column of Table 1 (where C is an arbitrary constant)Type of and Scaffold identify differentk(x,y types,z) of scaffolds, as reported in the left column of the table.Table 1. RelationshipLinear between the type of scaffoldC∙z and k(x, y, z). Radial C∙(x2 + y2) Table 1.RelationshipTypeQuadratic of between Scaffold the type ofC∙ scaffoldk(x(x2−,y,2z) a )nd k(x, y, z).

Type Linearof Scaffold k(x,y,zC) ·z In Figure 4, P-surface with s=1, RadialN = 7 and a quadraticC ·function(x2 + y2) for k (with C = 0.002) is shown. Linear C∙z Quadratic C·(x2 − y2) Radial C∙(x2 + y2) Quadratic C∙(x2−y2) In Figure4, P-surface with s = 1, N = 7 and a quadratic function for k (with C = 0.002) is shown. In Figure 4, P-surface with s=1, N = 7 and a quadratic function for k (with C = 0.002) is shown.

Figure 4.Quadratic P-surface, with s = 1 and N = 7.

FigureFigure 4. 4.QuadraticQuadratic PP-surface,-surface, with with s =s 1= and 1 and N =N 7. = 7.

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InIn particular, particular, this this surface surface represents represents the most the most external external face of face the scaffold of the designated scaffold designated as ‘quadratic as’ in‘quadratic Table1.’ in Table 1. TheThe value value of ofthe the constant constantC has C been has established been established to balance to the balance two following the two opposite following requirements: opposite requirements: - the gradient of k function along x and y must be as large as possible; - 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 - the values of k function must be such as to preserve the integrity of the P-cell and that of its openings. This requirement is essential to allow the cells to gradually and progressively populate lateral openings. This requirement is essential to allow the cells to gradually and progressively the ducts of the scaffold, regenerating the bone tissues. The satisfaction of this aspect requires populate the ducts of the scaffold, regenerating the bone tissues. The satisfaction of this aspect that ranges of variability, suitable for the two parameters k and s, are identiﬁed. Since in this requires that ranges of variability, suitable for the two parameters k and s, are identified. Since paper s has been assumed equal to 1, k must vary between −1 and 1 into the function domain in this paper s has been assumed equal to 1, k must vary between −1 and 1 into the function identiﬁed by the boundary conditions x = [−7π, 7π] and y = [−7π, 7π]. domain identified by the boundary conditions x = [−7π, 7π] and y = [−7π, 7π]. An algorithm, based on MATLAB functions for implicit surfaces representation, carried out the An algorithm, based on MATLAB functions for implicit surfaces representation, carried out the P-surface modelling, described in the form of tessellated model (STL model). Along the diagonals of P-surface modelling, described in the form of tessellated model (STL model). Along the diagonals of the top view of the surface, the P-cells are very close to the Schwarz Primitive of Figure1. Furthermore, the top view of the surface, the P-cells are very close to the Schwarz Primitive of Figure 1. Furthermore, the porosity and the pore size of the cell increase quadratically along the x-axis and decrease, with the the porosity and the pore size of the cell increase quadratically along the x-axis and decrease, with same law, along the y-axis. the same law, along the y-axis. Starting from the P-surface of Figure4, a solid scaffold was modelled according to a procedure Starting from the P-surface of Figure 4, a solid scaffold was modelled according to a procedure whose main steps are reported in Figure5. whose main steps are reported in Figure 5.

(b) (c)

(a)

Figure 5. Main steps of the methodology used for the modelling of a scaffold with variable architecture: Figure 5.Main steps of the methodology used for the modelling of a scaffold with variable (a) Scaffold surface modelling and scaling to a cubic structure of given side; (b) Offset with an assigned architecture: (a) Scaffold surface modelling and scaling to a cubic structure of given side; (b) Offset thickness; (c) Planar cuts and scaffold surface closing. with an assigned thickness; (c) Planar cuts and scaffold surface closing.

TheThe P-surfaceP-surface was wa ﬁrstlys firstly scaled scaled to obtainto obtain a cubic a cubic cell of cell assigned of assigned side (Figure side (Figure5a). Then, 5a the). Then, internal the faceinternal of the face scaffold of the was scaffold generated was generated starting from starting the external from the one external by an offsettingone by an operation offsetting with operation offset distancewith offset equal distance to the thicknessequal to thet (Figure thickness5b). Int (Figure order to 5b obtain). In order a scaffold to obtain with aa given scaffold number with ofa given cells, severalnumber planar of cells, cuts several were planar performed cuts were along performed the three along directions the threex, y, directions and z. In Figurex, y, and5, forz. In example, Figure 5, thesefor example, planar cutsthese resized planar the cuts quadratic resized the scaffold, quadratic bringing scaffold, it from bringing N = 7 it to from N = 5.N Finally,= 7 to N the= 5. scaffold Finally, wasthe convertedscaffold wa intos converted a solid structure into a solid (Figure structure5c). These (Figure steps, 5 aimingc). These at achievingsteps, aiming the ﬁnal at achieving structure the of thefinal scaffold, structure were of the performed scaffold, in we ar CADe performed environment in a CAD by CATIA environment software. by CATIA software. AA similarsimilar procedureprocedure waswas usedused toto obtainobtain thethe otherother structures.structures.

3.3. ResultsResults andand DiscussionDiscussion TheThe methodology methodology described described in thein the previous previous paragraph paragraph was used was to used obtain to representative obtain representative models ofmodels scaffolds of scaffolds with different with diffe geometricalrent geometrical characteristics. characteristics.

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Symmetry 2018, 10, x FOR PEER REVIEW 7 of 13 A preliminary analysis was conducted on unit P-cells starting from Equation (1). By considering the parametersA preliminaryk and analysiss previously was conducted discussed, on solid unit cells P-cells with starting different from geometrical Equation configurations (1). By considering were thefinally parameters obtained. k Their and characterisations previously discussed, in terms solid of geometrical cells with properties different andgeometrical performance configurations under load wereprovided finally useful obtained. information, Their mainly characteri in relationsation in with terms porosity of geometrical and stiffness properties of the different and performance geometries, underfor the loaddevelopment provided of useful the scaffolds. information, mainly in relation with porosity and stiffness of the differentThe geometries, 3D models werefor the made development of 5 × 5 ×of 5the cells. scaffolds. Previous studies [23,24] have shown that many propertiesThe 3D of models a porous were scaffold made and, of 5 in × particular,5 × 5 cells the. Previous elastic modulus,studies [23,24 can be] have numerically shown that predicted many propertiesby considering of a porous a lower scaffold amount and, of unit in particular, cells, rather the than elastic the wholemodulus, structure. can be Itnumerically has been shown predicted that bythere considering is no benefit a lower to model amount structures of unit greater cells, rather than 5 than3 cells; the in whole fact, the structure. difference It has between been theshown value that of 3 thereconvergence is no benefit of the to simulation model structures and the calculatedgreater than value 5 cells; is, in in percentage, fact, the difference negligible. between The models the value made of aconvergence limited number of the of simulation cells provide and the the additional calculated benefit value of is, reduced in percentage, computational negligible. cost The in the models Finite madeElements of a FE limited simulations number compared of cells provide with larger the models.additional benefit of reduced computational cost in the FiniteThe valueElements of the FE thickness simulations assumed compared for the with models larger was models. 0.240 mm. The selection of this parameter is relatedThe value to the rangeof the of thickness values of assumed high porosity for the suitable models for thewas scaffolds.0.240 mm The. The choice selection of modelling of this parameterstructures byis related introducing to the a range constant of values thickness of high into porosity the surface suitable of the for P-cells, the scaffolds instead. of The considering choice of modellingsolid models structures made with by introducing P-surfaces entirelya constant filled thickness in the innerinto the portion, surface as of reported the P-cells, elsewhere instead [20 of], consideringhas the advantage solid models to increase made the with interconnectivity P-surfaces entirely of the filledcells which in the is inner desired portion, for tissue as reported growth. elsewheFurthermore,re [20], in has all thethe modelledadvantage structures, to increase a limitedthe interconnectivity gradient of porosity of thewas cells introduced which is desired in order for to tissueavoid abrupt growth. changes Furthermore, between in adjacent all the cells.modelled structures, a limited gradient of porosity was introducedThe first in configurationorder to avoid consideredabrupt changes was relativebetween to adjacent structures cells. with a linear gradient of porosity alongThe the first vertical configuration direction. considered This is a property was relative that should to structures be desired with for a linear some applications,gradient of porosity such as alonglinkage the replacement, vertical direction. to mimic This the is incipient a property tissue. that should be desired for some applications, such as linkageFigure replacement,6 shows a to linear mimic model the incipient with a gradient tissue. along the vertical ( z) direction equal to 0.05. Figure 6 shows a linear model with a gradient along the vertical (z) direction equal to 0.05.

Figure 6. ScaffoldScaffold with with a a linear gradient in the vertical ( z) direction.

In this model,model,cells cells with with lower lower porosity porosity are are located located at theat the bottom bottom portion portion of the of model the model while while those thosewith the with largest the valueslargest arevalues obtained are obtained at the top. at The the porosity top. The of porosity this structure of this varies structure between varies 0.88 andbetween 0.91. 0.88 andTwo 0.91. other representative models made of 5 × 5 × 5 cells are reported in Figure7. Two other representative models made of 5 × 5 × 5 cells are reported in Figure 7.

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(a) (b)

Figure 7. Scaffolds with variable architecture: (a) radial, (b) quadratic. Figure 7. Scaffolds with variable architecture: (a) radial, (b) quadratic.

The scaffold in Figure 7a is relative to a structure corresponding to the radial configuration as definedThe in scaffold Table 1. in This Figure structure7a is relativeis characteri to as structureed by the corresponding presence of the to cells the with radial larger configuration values of porosityas defined at inthe Table outer1. Thispart structureleading to is a characterised configuration by that the should presence be ofinteresting the cells withto attribute larger valuesto the peripheralof porosity areas at the bone outer ingrowth part leading while to a the configuration central areas that should should accomplish be interesting the loadto attribute bearing, to forthe example,peripheral when areas the bone scaffold ingrowth is designed while the to central fill in areas cylindrical should accomplishvoids. The structure the load bearing, reported for in example, Figure 7bwhen is relative the scaffold to a scaffold is designed with to a fill quadratic in cylindrical architecture voids. The, as structuremathematically reported defined in Figure in the7b isprevious relative paragraph.to a scaffold with a quadratic architecture, as mathematically defined in the previous paragraph. TheThe models models of of the the scaffolds were analyanalysedsed with FE numerical simulations conducted conducted on on the the commercialcommercial software software Abaqus. Abaqus. The The material material chosen chosen was was a titanium a titanium alloy alloy (Ti-6Al (Ti-6Al-4V),-4V), a material a material with biocompatiblewith biocompatible properties, properties, frequently frequently reported reported in literature in literature for for bone bone implants. implants. The The mechanical ν propropertiesperties are assumed as: E E = 110 110 GPa, GPa, ν=0.3. 0.3. BrickBrick linear elementselements werewere usedused for for the the discretisation discretisation of of each each structure. structure. It hasIt has to beto observedbe observed that thatthe models the models of the of scaffolds the scaffolds based based on the on P-cells the P are-cells characterised are characteri bys theed presence by the presence of curved of surfaces curved surfaceswhose discretisation, whose discreti tosation, obtain to affordable obtain affordable results, requiresresults, arequires mesh with a mesh a large with number a large ofnumber elements. of elements.To limit the To computational limit the computational effort, a quarter effort, ofa quarter each model of each was model considered was considered and symmetry and symmetry boundary boundaryconditions conditio were appliedns were to simulate applied the to whole simulate structure. the whole A sensitivity structure. analysis A sensitiv wasity performed analysis for was an performedoptimal compromise for an optimal between compromise accuracy between and computational accuracy and time computational which conducted time towhich meshes conducted of about to2,500,000 meshes elements.of about 2,500,000 elements. TheThe models models were were subjected subjected to a to uniaxial a uniaxial compression compression load. In load particular,. In particular, the tests were the tests performed were performedby applying by a uniformapplying displacement, a uniform displacement, within the material within elasticthe material limit, to elastic the top limit, surface to the of thetop structure surface ofcorresponding the structure tocorresponding 0.1% of the compressive to 0.1% of the strain, compressive while the lowerstrain, surfacewhile the was lower fully su constrained.rface was fully constrained.The first model analyzed was the structure reported in Figure6, which is characterized by a linear gradientThe offirst porosity model alonganalyzed the zwas direction. the structure The uniaxial reported compression in Figure load 6, which was applied is characterized in the direction by a linearof the gradient (z).of porosity Figure8 showsalong the an iso-colourz direction. representation The uniaxial ofcompr the compressiveession load stress,was applied expressed in the in directionMPa, obtained of the forgradient this model. (z). Figure 8 shows an iso-colour representation of the compressive stress, expressed in MPa, obtained for this model.

Symmetry 20182018,, 1010,, 361x FOR PEER REVIEW 9 of 13 Symmetry 2018, 10, x FOR PEER REVIEW 9 of 13

Figure 8. Iso-colour representation of the compressive stress in a linear scaffold. Figure 8.8. IsoIso-colour-colour representation of the compressive stressstress inin aa linearlinear scaffold.scaffold. In the figure, both the 3D representation of the compressive stress distribution and a frontal In the figure, both the 3D representation of the compressive stress distribution and a frontal viewIn are the depicted. figure, both The the figure 3D representationshows, qualitatively, of the compressive a non-uniform stress distribution distribution of and the a compressive frontal view view are depicted. The figure shows, qualitatively, a non-uniform distribution of the compressive arestress depicted. along the The structure, figure shows, which qualitatively, can be explained a non-uniform in relation distribution to the distribution of the compressive of the porosity. stress stress along the structure, which can be explained in relation to the distribution of the porosity. alongWider the areas structure, of compression which can stress, be explained with respect in relation to the to areas the distribution of tensile stress, of the porosity.are identified Wider in areas the Wider areas of compression stress, with respect to the areas of tensile stress, are identified in the ofupper compression layers where stress, the with cells respect with to lar theger areas values of tensile of porosity stress, are are identifiedlocated. On in the upperother layershand, wherein the upper layers where the cells with larger values of porosity are located. On the other hand, in the thelowest cells layer with of larger cells, valueswhere the of porosity porosity are takes located. the lowest On the value, other these hand, areas in theare lowestmore comparable layer of cells, in lowest layer of cells, where the porosity takes the lowest value, these areas are more comparable in whereterms of the extension. porosity takes the lowest value, these areas are more comparable in terms of extension. terms of extension. Figure9 9 reports reports an an iso-colour iso-colour representation representation of of the the compressive compressive stress stress distribution distribution relative, relative, Figure 9 reports an iso-colour representation of the compressive stress distribution relative, respectively,respectively, toto thethe radialradial scaffoldscaffold (Figure(Figure9 a)9a) and and the the quadratic quadratic scaffold scaffold (Figure (Figure9b). 9b). respectively, to the radial scaffold (Figure 9a) and the quadratic scaffold (Figure 9b).

(a) (b) (a) (b) Figure 9. Iso-colour representation of the compressive stress: (a) radial scaffold; (b) quadratic scaffold. Figure 9. Iso-colour representation of the compressive stress: (a) radial scaffold; (b) quadratic Figure 9. Iso-colour representation of the compressive stress: (a) radial scaffold; (b) quadratic scaffold. Thescaffold. stress distribution in the radial scaffold appears less homogeneous with respect to the quadraticThe stress scaffold, distributio thus, showingn in the a trendradial similar scaffold to appears that reported less homogeneous for the linear structure. with respect As for to the The stress distribution in the radial scaffold appears less homogeneous with respect to the quadratic scaffold,scaffold, Figure thus, 9showingb reports a the trend stress similar obtained to that by applyingreported thefor loadthe linear in the structure.y direction As similarly for the quadratic scaffold, thus, showing a trend similar to that reported for the linear structure. As for the toquadratic the radial scaffold, scaffold. Figure However, 9b reports considering the stress the obtained resultant by different applying arrangement the load in of the the cellsy direction in the quadratic scaffold, Figure 9b reports the stress obtained by applying the load in the y direction orthogonalsimilarly to directions,the radial scaffold. the model However, was also consider analyseding the under resultant uniaxial different compression arrangement by applying of the cells the similarly to the radial scaffold. However, considering the resultant different arrangement of the cells loadin the on orthogonal each of the directions, other orthogonal the model directions. was also Thisanaly analysissed under showed uniaxial a trend compression in stress distribution,by applying in the orthogonal directions, the model was also analysed under uniaxial compression by applying qualitatively,the load on similar each of to the that other obtained orthogonal by applying directions the compression. This analysis load in show the yeddirection. a trend in stress the load on each of the other orthogonal directions. This analysis showed a trend in stress distribution,These results qualitatively, together with similar the analysisto that obtained of the geometrical by applying conﬁguration the compression of the quadratic load in model, the y distribution, qualitatively, similar to that obtained by applying the compression load in the y brieﬂydirection. described in the previous paragraph, led to verify a potential similarity of behaviour with direction.

Symmetry 2018, 10, x FOR PEER REVIEW 10 of 13 Symmetry 2018, 10, 361 10 of 13 Symmetry 2018, 10, x FOR PEER REVIEW 10 of 13 These results together with the analysis of the geometrical configuration of the quadratic model, briefly described in the previous paragraph, led to verify a potential similarity of behaviour with a These results together with the analysis of the geometrical configuration of the quadratic model, scaffolda scaffold of ofconstant constant porosity porosity obtained obtained by replicating by replicating a unit a unitP-cell P-cell in three in three orthogonal orthogonal directions. directions. The briefly described in the previous paragraph, led to verify a potential similarity of behaviour with a analogyThe analogy between between the twothe two models models was was confirmed conﬁrmed by by the the results results obtained obtained with with the the FE FE analysis analysis scaffold of constant porosity obtained by replicating a unit P-cell in three orthogonal directions. The conductedconducted on on this this model model,, similarly similarly to to the the structures structures previous previouslyly considered. considered. Figure Figure 1010 showsshows the the analogy between the two models was confirmed by the results obtained with the FE analysis isoiso-colour-colour representation representation of of the the compressive compressive stress stress distribution distribution for for the the model model with with constant constant porosity. porosity. conducted on this model, similarly to the structures previously considered. Figure 10 shows the iso-colour representation of the compressive stress distribution for the model with constant porosity.

FigureFigure 10. 10. IsoIso-colour-colour representation representation of of the the compressive compressive stress stress in in a a uniform uniform scaffold. scaffold.

Figure 10. Iso-colour representation of the compressive stress in a uniform scaffold. ThisThis uniform scaffoldscaffold hashas thethe same same thickness thickness as as the the variable variable scaffolds scaffolds previously previously considered considered and and a porosity value of 0.89. a porosityThis valueuniform of 0.89.scaffold has the same thickness as the variable scaffolds previously considered To further investigate the scaffolds analysed, the effective elastic modulus (Eeff.), given by the and Toa porosity further investigatevalue of 0.89. the scaffolds analysed, the effective elastic modulus (Eeff.), given by the ratio ratiobetween between the homogenised the homogeni stresssed and stress the and applied the strain, applied was strain, evaluated was evaluatedfor each variable for each architecture variable To further investigate the scaffolds analysed, the effective elastic modulus (Eeff.), given by the architecture considered in comparison with the scaffold obtained by replicating a unit P-cell. The consideredratio between in comparison the homogeni withsed the stress scaffold and obtained the applied by replicating strain, was a evaluatedunit P-cell. for The each mechanical variable mechanical properties of a porous structure can be determined via FEA [25] based on Hooke’slaw, propertiesarchitecture of aconsidered porous structure in comparison can be determined with the viascaffold FEA [obtained25] based by on replicating Hooke’slaw, a andunit the P- effectivecell. The and the effective elastic modulus of a porous structure can be expressed by means of the relationship: elasticmechanical modulus properties of a porous of a structureporous structure can be expressed can be determined by means ofvia the FEA relationship: [25] based on Hooke’slaw, and the effective elastic modulus of a porousE effstructure. = (FR/A)/ caneA be expressed by means of the relationship:(2) Eeff. = (FR/A)/eA (2) where FR is the reaction force calculated by theE effFE. = solver, (FR/A)/ eAA is the total cross-sectional area, and eA is(2) thewhere applied FR is strain. the reaction Since the force applied calculated strain by is theknown, FE solver, the parameter A is the total can cross-sectionalbe estimated. This area, parameter and eA is where FR is the reaction force calculated by the FE solver, A is the total cross-sectional area, and eA is ofthe stiffness applied represents strain. Since an theaverage applied value strain for isthe known, structures the parameter with variable can bearchitectures; estimated. Thishowever parameter, it is the applied strain. Since the applied strain is known, the parameter can be estimated. This parameter usefulof stiffness to globally represents evaluate an average them. Figure value 11 for shows the structures a diagram with which variable summari architectures;ses the results however, obtained. it is 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. useful to globally evaluate them. Figure 11 shows a diagram which summarises the results obtained.

Figure 11.Effective elastic modulus for scaffolds with different architecture.

FigureFigure 11.11.Effective elastic modulus for scaffolds withwith differentdifferent architecture.architecture. The model indicated in the diagram as ‘linear’, corresponding to the architecture shown in Figure 6, and that indicated as ‘radial’ (Figure 7a) have a slightly lower stiffness with respect to the The model indicated in the diagram as ‘linear’, corresponding to the architecture shown in Figure 6, and that indicated as ‘radial’ (Figure 7a) have a slightly lower stiffness with respect to the

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The model indicated in the diagram as ‘linear’, corresponding to the architecture shown in Figure6, and that indicated as ‘radial’ (Figure7a) have a slightly lower stiffness with respect to the scaffold ‘quadratic’ (Figure7b). With regard to this last structure, the diagram reports both the valuesof the effective elastic modulus obtained by the application of a compression load, respectively, in the x and in the y direction. The value obtained by the application of the load in the z direction was found to have an intermediate value between the two reported. From the diagram it is possible to observe that the stiffness of this complex structure can take a lower or a larger value with respect to that of the scaffold with a uniform architecture, according to the direction of the loading, maintaining a relatively uniform distribution of the compression stress, as was observed in the previous analysis relative to strength. Consequently, this kind of structure could be useful in bone repairing when different stiffness is required in orthogonal directions, since this parameter can be adjusted in preferred directions. Finally, it can be observed that the calculated values of the stiffness for the scaffolds analysed fall within the range of that of the human cortical bone which is estimated around 15 GPa with a large deviation depending on several factors; thus, these structures can potentially satisfy mechanical requirements for the applications proposed in this study.

4. Conclusions The procedure for the modelling of porous structures based on the P-surface implemented, allowed the design of representative models made of cells whose geometrical characteristics vary in 3D space according to different mathematical distributions, potentially useful to accomplish the different biological and mechanical requirements for bone scaffold applications. An original MATLAB implementation has been developed for modelling parametrically a generic surface represented by an implicit function in the form of a tessellated model. In particular, different P surfaces have been generated, which differ from each other for the various analytical expressions used to describe the k parameter of Equation (1). Each kind of structure modelled and analysed has peculiar characteristics that could be adapted to the design of customised implants. The ﬁnite element simulations allowed the analysis of the different structures in terms of compressive stress and stiffness, highlighting the effect of the local variation of their geometry, also in relation to a scaffold obtained by the replication of a constant unit cell. A decrease of the global stiffness was observed for the linear and the radial scaffold with respect to the uniform structure, while the behaviour of the quadratic scaffold was more comparable to that. In continuing this research, a widening of the types of models with variable architecture will be considered, also including structures based on different types of TPMS surfaces, and a comparative numerical FE analysis is foreseen to be used for the selection of the most appropriate—in terms of performance—architectures. These will be manufactured with the AM technology and experimentally tested in order to assess them, and also compared with the results obtained from the FE analysis. Finally, in vitro tests will be performed.

Author Contributions: The authors contributed equally to this work. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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