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Acta Biomaterialia 4 (2008) 1715–1724

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Finite element modeling as a tool for predicting the fracturebehavior of robocast scaffolds

Pedro Miranda *, Antonia Pajares, Fernando Guiberteau

Departamento de Ingenierıa Mecanica, Energetica y de los Materiales, Universidad de Extremadura, Avda de Elvas s/n, 06071 Badajoz, Spain

Received 10 January 2008; received in revised form 13 May 2008; accepted 22 May 2008Available online 5 June 2008

Abstract

The use of finite element modeling to calculate the stress fields in complex scaffold structures and thus predict their mechanical behav-ior during service (e.g., as load-bearing bone implants) is evaluated. The method is applied to identifying the fracture modes and esti-mating the strength of robocast hydroxyapatite and b-tricalcium phosphate scaffolds, consisting of a three-dimensional lattice ofinterpenetrating rods. The calculations are performed for three testing configurations: compression, tension and shear. Different testingorientations relative to the calcium phosphate rods are considered for each configuration. The predictions for the compressive configu-rations are compared to experimental data from uniaxial compression tests.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Finite element analysis; Robocasting; Scaffolds; Strength; Fracture modes

1. Introduction

Tissue engineering for bone regeneration requires theuse of a porous scaffold to serve as a template for cell inter-actions and the formation of the extracellular matrix, aswell as to provide structural support for the newly formedtissue [1]. These scaffolds should be able to withstand somedegree of loading during their use in vivo, besides provid-ing the required biological response. However, since por-ous scaffolds are inherently weak (especially those basedon calcium phosphates) [2], it is necessary to optimize theirmechanical response so that they can provide the appropri-ate support while bone regenerates and they are slowlyresorbed [3]. Characterizing and predicting the mechanicalbehavior of these complex three-dimensional (3D) struc-tures is therefore an essential task.

Finite element modeling (FEM) has been shown to becapable of predicting the behavior of complex structures,such as multilayer systems [4–10], provided the mechanical

1742-7061/$ - see front matter � 2008 Acta Materialia Inc. Published by Else

doi:10.1016/j.actbio.2008.05.020

* Corresponding author. Tel.: +34 924 28 9600; fax: +34 924 28 9601.E-mail address: pmiranda@unex.es (P. Miranda).

properties of the materials comprising the structure areknown. Consequently, the objective of the present workwas to analyze the capabilities of finite element simulationas a technique for calculating the stress fields in complexscaffolds and thereby estimating their mechanical behaviorunder different loading configurations. In particular, FEMis applied to identify the fracture modes and estimate thestrength of hydroxyapatite (HA) and b-tricalcium phos-phate (b-TCP) scaffolds fabricated by robocasting.

Robocasting, or direct-write assembly, is a solid free-form fabrication (SFF) technique that consists of therobotic deposition of highly concentrated colloidal suspen-sions (inks) capable of fully supporting their own weightduring assembly [11–13]. Robocasting allows ceramic scaf-folds to be built using water-based inks with minimalorganic content (<1 wt.%) and without the need for a sac-rificial support material or mold. Using this technique anetwork of semisolid ink rods is printed directly, hencethe term direct-write assembly, by extrusion through adeposition nozzle to get the desired 3D structure. Fig. 1shows a schematic diagram of the process.

The prototype structure considered for this studyconsisted of a 3D tetragonal lattice of interpenetrating rods

vier Ltd. All rights reserved.

Fig. 1. Scheme of the robocasting fabrication process. The ceramicscaffold is built layer by layer from a computer design. A three-axis roboticarm moves the injection syringe while pressing the ceramic ink through theconical deposition nozzle, immersed in an oil bath, to create a self-supporting 3D network of ceramic rods.

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– fabricated from either HA or b-TCP inks. The mechani-cal behavior of these scaffolds was simulated by FEMunder three testing configurations: uniaxial compression,tension and shear. Different testing orientations relativeto the rods comprising the structure were considered foreach configuration. The most probable fracture modewas determined from the location of the maximum tensilestresses in the structure. This approach had been foundto give excellent results in an earlier study of the compres-sive fracture modes of HA scaffolds [14]. The predictedcompressive, tensile and shear strengths of the scaffoldsfor each testing orientation were estimated from theFEM results assuming a critical stress criterion. Finally,the predictions for the compressive configuration werecompared to experimental data from uniaxial compressiontests.

2. Experimental procedure

Details of the fabrication of the HA and b-TCP inks andthe corresponding robocast scaffolds have been presentedelsewhere [2,15] and, as they are used here only for the exper-imental verification of the FEM results obtained in thiswork, only the relevant data will be included in this section.

The 3D scaffolds, consisting of a lattice of ceramic rods,were constructed layer by layer via extrusion of the corre-

sponding inks through conical deposition nozzles, using acomputer-controlled robotic deposition device (3D Inks,Stillwater, OK), as shown schematically in Fig. 1. Thedeposition was done in a non-wetting oil bath to preventnon-uniform drying during assembly. The samples wereremoved from the bath and dried in air at room tempera-ture for 24 h, followed by a burn-out at 400 �C (ramp at1 �C min�1 and a 1 h hold) to evaporate organics. Theywere finally sintered at 1300 �C (ramp at 3 �C min�1 anda 2 h hold). The resulting scaffolds were imaged (Fig. 2)using scanning electron microscopy (SEM) (S-4300SE/N,Hitachi, USA). The average rod diameter was measuredto be d = 220 ± 10 lm and the average center-to-centerspacing between rods in the printing plane s = 300 ±10 lm, with a rod overlap between adjacent layers of about50 ± 10 lm, yielding a vertical spacing between like layersof around h = 340 ± 10 lm (Fig. 3).

The intrinsic mechanical properties of the individual cal-cium phosphate rods were evaluated to provide essentialinput parameters for the numerical simulations. Instru-mented indentation (Nanotest, Micro Materials Ltd.,Wrexham, UK) was used to determine the elastic moduliof the HA and b-TCP rods. Berkovich indentation testswere performed on polished sections (to 1 lm finish) per-pendicular to the rod axis. Single indentations of about5.5 lm depth and 40 lm side were placed at the center ofthe rods. This indent size is large enough compared tothe grain sizes in both materials (3.2 ± 0.5 and7.4 ± 0.7 lm for HA and b-TCP, respectively [2]) to pro-vide meaningful information about the mechanical proper-ties of the rods (and not of individual grains) but smallenough to avoid a significant influence of the free surfaceof the rods – ASTM E 384 recommends a center-to-sidespacing between Vickers indents of 2.5 times the diagonal.

The inert fracture strength (i.e., the strength in theabsence of chemical fatigue or slow crack growth) of therods was estimated from high-load-resolution three-pointbending tests (Electroforce 3200, Bose Corp., Eden Prairie,MN) performed on individual rods printed and sintered forthe purpose. The tests were performed in air at a constantcrosshead speed of 30 mm min–1. Since the fracture loadfor rods of 220 lm diameter was close to the sensitivitylimit of the load cell, thicker rods of 360 lm diameter wereused instead. Although the geometry and size of these iso-lated rods deviate from the real situation, these tests are thebest possible means available to measure this materialproperty (much more accurate than testing bulk samplesof like materials, for example). Besides, in the opinion ofthe authors, the actual values of the rod strength shouldlay within the errors of these imperfect measurements.

Finite element simulations were carried out using ABA-QUS/Standard� software (Simulia, Providence, RI). Thealgorithm models an approximately cubic scaffold(2 � 2 � 2.09 mm) consisting of 12 alternating orthogonallayers of parallel HA or b-TCP rods of 220 lm diameterspaced 300 lm from center to center, with each layer over-lapping the adjacent ones by 50 lm (Fig. 3). The FEM grid

Fig. 2. SEM micrographs showing the morphology of HA (a, c) and b-TCP (b, d) scaffolds after sintering at 1300 �C for 2 h: (a, b) printing plane view and(c, d) cross-section view.

Fig. 3. Finite element grid used to simulate the scaffolds, consisting of 12layers of rods of 220 lm diameter, d, spaced s = 300 lm from center-to-center, and with a layer overlap of 50 lm that yields a like-layer spacing ofh = 340 lm along the printing direction (direction 3). The dimensions ofthe elements at the rod surfaces are around 25 lm, but become largertowards the interior.

P. Miranda et al. / Acta Biomaterialia 4 (2008) 1715–1724 1717

consists of more than 1.25 � 106 linear tetrahedral elementsof about 25 lm in the vicinity of the external surfaces of therods, but progressively increasing in size towards their inte-rior. Isotropic elastic behavior is assumed for the whole

system, using the elastic moduli obtained from the indenta-tion tests and a Poisson’s ratio of 0.28 [16] for bothmaterials.

The scaffold is placed between two parallel rigid plates(Fig. 4), already in contact with the structure from thebeginning. The contact is considered frictionless for thecompressive test simulations and infinitely rough andsticky for the tensile and shear test simulations. One ofthe rigid plates is fixed while the other moves under theaction of a linearly increasing applied force (up to500 N). For both the uniaxial tensile and compressive testsimulations, two different orientations were considered,one with the load applied perpendicular to the printingplane (Fig. 4a and c) and the other with the load appliedparallel to the rods (Fig. 4b and d), i.e., along the equiva-lent axis 1 or 2 in Fig. 3. For the shear test simulationsthe three different load orientations depicted in Fig. 4e–gwere considered.

The evolution of stresses with the applied stress – esti-mated by normalizing the applied load by the correspond-ing initial cross-sectional area of the simulated sample –was recorded during the simulation. The predictedstrengths (compressive, tensile, or shear) of the scaffolds,i.e., the applied stress acting on the structure at failure,for each testing configuration were calculated assuming acritical stress criterion. First the greatest tensile stress act-ing at each point in the structure, rt, is determined fromthe FEM calculated stress tensor. Failure is then presumedto occur when the maximum of rt equals the inert fracturestrength, rF, of the rods (as determined from the bending

Fig. 4. Scheme of the different testing configurations simulated in this study: compression along (a) the printing direction or (b) a rod axis; tension along(c) the printing direction or (d) a rod axis; shear (e) on the printing plane along a rod axis, or on a plane orthogonal to a rod axis along (f) the printingdirection or (g) a rod axis. The rigid plates used in the FEM simulations to transfer the load to the structure are indicated. The bottom plate is held fixedwhile a load is applied to the top plate in the direction indicated by the arrows.

Table 1Intrinsic mechanical properties of HA and b-TCP rods

E* (GPa) ma E (GPa) rF (MPa)

HA 83 ± 4 0.28 82 ± 4 68 ± 12TCP 38 ± 8 0.28 36 ± 7 27 ± 9

a From Ref. [16].

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tests) – this implicitly assumes an opening fracture mode(mode I), which is that most commonly activated in brittlematerials [17]. For the calculation of the maximum value ofrt, the near-contact regions were ignored as they are notrepresentative of the macroscopic behavior of the structureunder each load configuration (i.e., uniaxial tension, com-pression or shear), but rather of the boundary conditionsselected for the simulations. Furthermore, contact stresseswill only produce localized damage [14] and would not beresponsible for the ultimate failure of the structure exceptunder high-loading-rate, e.g., impact, conditions.

Experimental uniaxial compression tests were per-formed along the aforementioned two directions (Fig. 4aand b) on cubic blocks of about 2 mm side cut from the sin-tered HA and b-TCP scaffolds. The tests were carried outon a universal testing machine (AG-IS10kN, ShimadzuCorp., Kyoto, Japan) in air at a constant crosshead speedof 0.6 mm min�1 (the rate of 30 mm min�1 used in bendingtests was impractical for the compression tests because itcaused premature failure from contact damage and thusa slower rate was selected). The compressive strength of

the structure was estimated as the maximum applied loaddivided by the square external cross-section of the sample,and the results were compared with the FEM predictions.More than 10 samples were tested in each case in orderto get statistically reliable values.

3. Results and discussion

Table 1 summarizes the results of the characterization ofthe individual rods comprising the HA and b-TCP scaf-folds. The elastic modulus, E, of each material was calcu-lated from the effective modulus, E* = E/(1 � m2),obtained from the indentation tests, assuming a value of

P. Miranda et al. / Acta Biomaterialia 4 (2008) 1715–1724 1719

0.28 [16] for the Poisson’s ratio. As can be clearly appreci-ated, the HA rods exhibited more than twice the stiffnessand strength of the b-TCP rods.

These elastic properties were used as input for the FEMsimulations of the different testing configurations depictedin Fig. 4. The FEM results confirmed that the stressesdeveloped in the scaffolds for each configuration are obvi-ously independent of the elastic modulus of the materialconsidered, as shown for the case of uniaxial compressionin Fig. 5, where the maximum of rt is represented vs. theapplied stress for both HA and b-TCP. Consequently, theanalysis that follows of the stress fields generated and theassociated damage modes that are to be expected in eachcase is applicable to any scaffold with a similar geometry.Evidently, the strains do vary from one material toanother, and the elastic modulus of the material compris-ing the individual rods will determine the effective modulusof the scaffold.

Fig. 6 shows the contours of the greatest tensile stress,rt, developed in the scaffold during uniaxial compressionat 250 N load (which is around the critical load for fracturein the experimental tests) for the two orientations consid-ered (Fig. 4a and b). When compression is applied perpen-dicularly to the printing plane (Fig. 6a), i.e., along direction3, rt presents a maximum at the centers of the unsupportedsegments – specifically at the top and bottom surfaces ofthose segments – of all the rods (although not so clearly vis-ible in Fig. 6a for those aligned along direction 2). At thosepoints, rt is directed along the corresponding rod axis (i.e.,rt = r11 or rt = r22) and therefore, assuming an openingfracture mode (mode I), which, as mentioned above, isthe most commonly activated in brittle materials [17], will

Fig. 5. Plot of maximum tensile stress in the structure (maximum of rt) vs.the applied stress (applied load normalized by initial cross-sectional area)for both HA and b-TCP scaffolds in the two compression directionsconsidered: perpendicular to the printing plane (direction 3, Fig. 4a) andalong the rod axes (direction 1 or 2, Fig. 4b). Note that the results areindependent of the material.

induce cracking perpendicular to the rod axes over theentire structure at those points. This type of cracking hasindeed been observed in experimental compressive testswith this same orientation [14]. Of course, crack initiationdoes not necessarily occur at the exact location of the max-imum tensile stresses because it also depends on the loca-tion of the largest flaws [18].

When the compressive load is applied along one of therod axes (directions 1 or 2), the maximum tensile stressesare again located on the top and bottom surfaces of thehorizontal rods (Fig. 6b), but now close to the joints withthe vertical rods (see the inset). The orientation of thesemaximum tensile stresses is such that it will induce cracking(again, assuming mode I fracture) of the horizontal rodsalong those joints, effectively detaching them from the ver-tical rods that remain largely in compression. This predic-tion was also consistent with the curvature of the cracksurfaces observed in the experimental tests [14]. The rela-tive magnitude of the maximum stresses in the two orienta-tions will be discussed in detail below.

For the case of uniaxial tension tests, Fig. 7 shows the rt

contours developed in the scaffold at 250 N load (for com-parison with Fig. 6) for the two orientations considered(Fig. 4c and d). Note that, since stresses increase linearlywith load (Fig. 5), contours at any other load can be calcu-lated just by multiplying the stresses in the legend by theappropriate factor. When the loading is along direction 3(Fig. 7a), and ignoring the values near the contacts, rt pre-sents its maximum values at the joints between adjacentrod layers, as clearly shown in Fig. 7a for the rods alignedalong direction 1 but also present in those aligned alongdirection 2. At a certain load, these maximum tensile stres-ses will induce fracture along those joints, separating eachlayer from its neighbors. When the tension is applied alongone of the rod axes (directions 1 or 2), the tensile stressesare largely located on the vertical rods (Fig. 7b), with theirmaxima in the free segments of those rods close to (thoughnot exactly at) the joints with the horizontal rods (seeinset). The orientation of these maximum tensile stressesis largely along the rod axes, so that cracks originated bythem will tend to break the vertical rods transversally.

Fig. 8 shows analogous contour plots of rt at 250 N loadfor the three different shear configurations selected (Fig.4e–g). For the sake of clarity, the direction of the appliedload is included in the figures (recall that the bottom sur-face is kept fixed). Although the three configurations aresignificantly different, the stress contours generated exhibitmany similarities. For all three orientations, the tensilestresses organize into bands of alternating intensity ori-ented at 45� with respect to the load axis. The maxima ofthese stresses, disregarding the near-contact field, arelocated at the joints between the different rod layers, theedges at the intersections acting as stress concentrators.Therefore, for the three configurations, cracking will occurat these locations and the different rod layers will tend todetach from each other. However, the actual cracking pro-cess will be slightly different in the case of the configuration

Fig. 6. FEM-generated stress contours corresponding to the highest tensile stress, rt, generated when a compressive load of 250 N is applied (a)orthogonal to the rods (direction 3) or (b) along a rod axis (direction 1, equivalent to direction 2).

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in Fig. 8c, where adjacent layers tend to rotate relative toeach other instead of simply one displacing the others, asin the cases of Fig. 8a and b.

Concerning the relative intensities of the maximumstresses developed in each of the configurations analyzed,Fig. 9 shows the absolute maximum value of rt anywherein the scaffold vs. the applied stress (i.e., applied load nor-malized by the corresponding scaffold’s external cross-sec-tion) for each testing configuration. This representationand the conclusions that derive from it are independentof the material considered and therefore have a universalvalidity for any scaffold with the same morphology. Ascan be clearly appreciated, there are very large differencesin the maximum tensile stress levels that develop in thescaffolds depending on the type of load applied. Asexpected, compressive loads are the mildest, with stressesabout three times lower than those that develop in the ten-sion tests and more than an order of magnitude lower thanthose of the shear configurations. Of the two compression

configurations considered, the stresses are higher whenthe load is applied perpendicular to the printing plane(direction 3), but this situation is reversed under tensileloads, which yield higher stresses when applied along therod axes (direction 1 or 2). The most deleterious configura-tion analyzed is that corresponding to shear load applied inplanes perpendicular to the printing plane, along the direc-tion of the rod axes (Fig. 4g). The torsion-like relative dis-placement of the rod layers generates a stress field aroundthe joint that is significantly greater than in the two othershear configurations, which nonetheless both give stresslevels more than twice as large as those of the tension con-figurations. An immediate, though anticipated, conclusionfrom these results is that brittle porous scaffolds fabricatedby robocasting, regardless of the material they are made of,will exhibit far superior strength properties under compres-sive loads; tensile and shear stresses acting on them shouldbe minimized, if not altogether suppressed, to improvetheir mechanical performance.

Fig. 7. FEM-generated stress contours corresponding to the greatest tensile stress, rt, generated when a tensile load of 250 N is applied (a) orthogonal tothe rods (direction 3) or (b) along a rod axis (direction 1, equivalent to direction 2).

P. Miranda et al. / Acta Biomaterialia 4 (2008) 1715–1724 1721

From the plots in Fig. 9 it is simple to predict themechanical strength of a scaffold for each testing configura-tion by using a critical stress criterion, provided the intrin-sic strength of the constituent material is known. Indeed,these plots allow one to determine the applied stress atwhich the maximum tensile stress in the scaffold will equalthe strength of the material, i.e., the structure strengthunder that type of load. Fig. 10 shows the strength valuescalculated following this procedure for the two materialsconsidered in this study, HA and b-TCP, using the valuesof the inert strength, rF, from Table 1. The error bars rep-resent standard deviations of the predictions due to uncer-tainties in the measurement of rF from the corresponding

three-point bending tests. As expected, the compressivestrength values exhibited by both materials are significantlyhigher than their corresponding tensile and, especially,shear strengths. That the HA values are more than twicethose corresponding to b-TCP is a simple reflection of theirrespective inert strengths.

Finally, Fig. 11 shows the comparison between the pre-dicted compressive strength and the experimental resultsobtained in the uniaxial compressive tests performed onthe actual scaffolds for the two testing orientations consid-ered (Fig. 4a and b). Again, the error bars represent thestandard deviation of the data. The FEM predictions areseen to agree with the experimental compressive strengths

Fig. 8. FEM-generated stress contours corresponding to the greatest tensile stress, rt, generated when a shear load of 250 N is applied on the printingplane along a rod axis (direction 1, equivalent to direction 2) (a), or on a plane orthogonal to a rod axis (direction 1, equivalent to direction 2) along theprinting direction (direction 3) (b) or along the perpendicular rod axis (direction 2, equivalent to direction 1) (c).

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within uncertainties for both the HA and the b-TCP scaf-folds. The apparently systematic slight (within the errors)overestimate of the compressive strength, especially notice-

able for b-TCP, could be attributed to the use of an inertstrength value, while the actual experiments were per-formed at a low speed, where slow crack growth could be

Fig. 9. Plot of maximum tensile stress in the scaffold (maximum of rt) vs.the applied stress (applied load normalized by initial cross-sectional area)for all the testing configurations analyzed: compression (two solid lines),tension (two dashed lines) and shear (three lines with dots). Legendnotation is the same as used in Fig. 4.

Fig. 10. Bar graph showing HA and b-TCP scaffold strength valuespredicted by FEM for the different testing configurations, using the inertstrength, rF, values of Table 1. Error bars represent standard deviations ofpredictions due to uncertainties in the measurement of rF. Label notationon the x-axis is the same as used in Fig. 4.

Fig. 11. Bar graph showing comparison between experimental measure-ments and FEM predictions of the compressive strength of HA and b-TCPscaffolds for the two compression orientations considered. Error barsrepresent standard deviations of the data.

P. Miranda et al. / Acta Biomaterialia 4 (2008) 1715–1724 1723

reducing the fracture strength of the scaffold. The TCPstrength values used to calculate FEM estimates are lessreliable than the corresponding HA values because, evenafter increasing the rod diameters, the critical loads forcracking lay close to the sensitivity threshold of the loadcell used in the three-point bending test, which couldexplain the more significant deviation of FEM predictionfrom experimental values for this material. In sum, there-

fore, these results confirm the ability of FEM to predictthe fracture behavior of robocast scaffolds both qualita-tively (i.e., to predict their fracture modes [14]) and quanti-tatively (i.e., to calculate their strength), which was themain objective of the present work.

4. Conclusions and implications

Computer-aided fabrication methods, such as robocast-ing, are ideal for fabricating scaffolds with a designed porestructure that enables their biological and mechanicalresponses to be optimized to levels unattainable with othertechniques. In particular, optimizing the mechanical per-formance of porous brittle scaffolds is critical since that isprecisely what prevents their widespread application forregeneration of load-bearing bone defects. To performsuch optimization, it is necessary to have a predictive toolas an aid in the design of the most favorable scaffold archi-tecture. As the present results show, FEM could be thattool, as it enables the analysis and prediction of failuremodes and the quantitative evaluation of importantmechanical parameters, such as strength.

In particular, the present study has successfully analyzedthe stress fields and the expected fracture modes underseven different loading conditions (Fig. 4) of a robocastscaffold consisting of a tetragonal lattice of ceramic rods.From these results (Fig. 9), a key guideline for the designof brittle robocast scaffolds was derived: implant shapeand location should be designed to minimize the tensileand, above all, shear stresses acting on the structure. Thislast type of load is extremely prejudicial for the mechanicalperformance of the scaffold as it generates intense tensilestresses that can lead to premature failure of the system.

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Fortunately, in most bone implant applications the loadswill be largely compressive and therefore scaffold perfor-mance will not be so seriously jeopardized. For this partic-ular type of load, another recommendation would be topreferentially orient the scaffolds so that the compressionis parallel to one of the rod axes in order to minimize thetensile stresses generated in the structure.

The FEM results were also used to predict the fracturestrength of HA and b-TCP scaffolds fabricated by thistechnique under each type of load by applying a criticaltensile stress criterion using their respective inert strengths,as measured in three-point bending tests, as the criticalstress values. The compressive strengths predicted by theFEM simulations (for both materials in the two orienta-tions considered) were successfully validated by compari-son with experimental uniaxial compression test data,justifying the suitability of the present methodologicalapproach for the optimization task at hand.

Indeed, predictions obtained by FEM could allow oneto determine how the different geometrical variables (rodthickness, spacing, interpenetration, relative angles, etc.)affect the mechanical behavior of the structure. Work inthis direction is currently in progress. The results of thosestudies will make it possible to create an intelligent designof the scaffold’s initial computer-aided design model andthereby optimize its mechanical performance. Althoughwe have focused here on the estimation of fracture modesand strength, nothing prevents this method from beingapplied to the optimization of other mechanical parame-ters, such as the elastic modulus of the structure. In sum,FEM is a powerful tool for predicting the mechanical per-formance of scaffolds fabricated by robocasting or otherSFF techniques, and thus optimizing their mechanical per-formance through the intelligent design of their geometry,which is an unavoidable prerequisite for their use in load-bearing bone tissue engineering applications.

Acknowledgements

This work was supported by the Ministerio de Educa-cion y Ciencia (Spanish Government) and the Fondo So-cial Europeo (MAT2006-08720).

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