a multiscale modeling approach to scaffold design and property prediction

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 3 ( 2 0 1 0 ) 5 8 4 – 5 9 3

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Research paper

A multiscale modeling approach to scaffold design andproperty prediction

K.S. Chan∗, W. Liang, W.L. Francis, D.P. Nicolella

Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78238, USA

A R T I C L E I N F O

Article history:

Received 12 July 2008

Received in revised form

12 July 2010

Accepted 13 July 2010

Published online 16 August 2010

Keywords:

Tissue engineering

Scaffold

Multiscale modeling

Structure design

Microstructure

Nanocomposites

A B S T R A C T

The optimum scaffold architecture for bone tissue regeneration is a porous structure

with a narrow range of pore sizes, pore density, and a high degree of interconnectivity

among pores. To achieve such a design, the microstructure of the scaffold material must

be optimized in order to satisfy both biological and mechanical function requirements.

In this paper, we present a multiscale modeling approach for designing a scaffold with

an optimized porosity and mechanical properties made from a two-phase composite of

spherical hydroxyapatite (HAp) particles embedded in a collagen matrix. In particular, first-

principles computation is used to calculate the elastic properties and theoretical strengths

of nanoscaled HAp particles. The constitutive properties of the HAp/collagen composites

are subsequently computed as a function of HAp content via FEM-based micromechanical

modeling. The constitutive relations of the composite are then utilized to optimize the

mechanical properties of a three-dimensional scaffold for either cortical or cancellous bone

by varying the pore size, pore density and volume fractions of HAp in the composite. For the

pore size, pore density, volume fractions of HAp considered, the scaffold can be designed

to match the mechanical properties of cancellous bone, but not those of cortical bone. The

optimized scaffold is one with a pore diameter of 1000 µm, a channel diameter of 100 µm,

27% pore density and at least 20% HAp by volume.c© 2010 Elsevier Ltd. All rights reserved.

d

1. Introduction

Bone–tissue engineering scaffolds address a need to heallarge defects due to trauma, disease, or therapies directedtowards mitigating pathology. To promote bone formation,vasculature-inducing pore geometry is critical in scaffolddesign (Mahmood et al., 2001), since osteogenesis does notproceed in the absence of vascular invasion (Ferguson et al.,1999; Peng et al., 2002). Scaffold architecture, especiallypore size, has been shown to influence microvascularinduction into and around porous biomaterials (Sieminski

∗ Corresponding author. Tel.: +1 210 522 2053; fax: +1 210 522 6965.E-mail address: [email protected] (K.S. Chan).

1751-6161/$ - see front matter c© 2010 Elsevier Ltd. All rights reservedoi:10.1016/j.jmbbm.2010.07.006

and Gooch, 2000; Sanders et al., 2002). Reports suggest thatcertain pore sizes support osteoblastic activities related tovascularization (Jin et al., 2000). Thus, pore dimension andinterconnectivity are key factors in structural design toensure tissue attachment and osteoid formation (Green et al.,2002). The optimum scaffold architecture for bone tissueregeneration is a porous structure with a narrow range of poresizes, pore density, and a high degree of interconnectivityamong pores (Hollister, 2005).

Considerable research has focused on preparing scaffoldsby simple pore generating methods, such as solvent casting

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http://dx.doi.org/10.1016/j.jmbbm.2010.07.006

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 3 ( 2 0 1 0 ) 5 8 4 – 5 9 3 585

and particulate leaching, melt molding, slip casting, gasfoaming, freeze drying, phase separation, and supercriticalfluid processing (Lanza et al., 2000). These methods yieldhighly porous, very tortuous, and structurally weak scaffolds(Shastri et al., 2000). Because of poor nutrient diffusionproperties within the scaffold, these simple scaffolds do notadequately support and sustain the biology of tissue genesis,as evidenced by the occurrence of live cells and tissue atthe scaffold periphery, but lack of tissue ingrowth into thescaffold (Ishaug et al., 1997). Furthermore, many scaffolds arebased on synthetic polymers, such as polylactic-co-glycolicacid (PLGA), which have hydrophobic surfaces, resulting insignificant research to alter their properties through surfacemodification and incorporation of growth factors to improvetheir performance (Lavik et al., 2004).

Aside from biocompatibility, a scaffold must support cellmigration and attachment, provide transport of nutrientsand bi-products, and must provide sufficient mechanicalstrength and stiffness to support both the tissue regenerationprocess and maintain physical form. The essential scaffoldrequirements include porosity, permeability, and the correctpore size for the candidate cell type (Agrawal and Ray,2001). Scaffold architecture, especially pore size and porosity,has been shown to influence microvascular induction intoand around porous biomaterials (Sieminski and Gooch,2000). Reports suggest certain pore sizes support osteoblasticactivities related to vascularization (Jin et al., 2000), andthat pore dimension and interconnectivity are key factors instructural design to ensure tissue attachment and osteoidformation (Green et al., 2002). The ability of bone in-growth increases when the porosity of a ceramic scaffold isincreased, but no noticeable increases have been observedwhen the porosity exceeds 30% (Toth et al., 1995). Optimalpore sizes for bone ingrowth have been reported to be on theorder of 300–400 µm (Tsuruga et al., 1997), while others statethat a pore size of more than 100 µm is not a requisite forbone ingrowth (Itala et al., 2001). Small pores in the range of85 to 150 µm have been found to promote cell adhesion byvirtue of a high specific surface area. This beneficial effect,however, is diminished at pore sizes larger than 300 µmdue to improved cell migration and proliferation (Murphyet al., 2010; Murphy and O’Brien, 2010). These conflictingdata likely result from complex dependence of cell activities,which include adhesion, migration, and proliferation, andwound healing on scaffold architecture such as pore size,porosity level, tortuosity (Jin et al., 2000), and fluid flow(Boardman and Swartz, 2003). A precise pore dimensionalcontrol has been difficult with many scaffold techniques(Borden et al., 2003), but rapid prototyping methods mayprovide an easy mean to control such pore dimensions.The limitations of many scaffold fabrication methods haverequired neovascularization and osteogenesis adapting toscaffold geometry rather than specifically designing scaffoldgeometry to accommodate the biological imperatives ofbone wound healing. Thus, there is a need for a designmethodology that can be used to optimize the mechanicalperformance of a scaffold based on a given set of architecturalconstraints imposed by the biological requirements.

In this paper, we present a multiscale modeling approachintended for designing bone–tissue scaffold with a well-defined and controlled microstructure. This approach, which

(a) (b)

(c)

(d)

Fig. 1 – Schematics illustrate the multiscale approach formodeling a 3D scaffold representative volume element(RVE) using a micro-2D HAp/collagen unit cell at themicrostructural level and a HAp unit cell and the collagenfibril at the molecular level: (a) unite cell of HAp,Ca10(PO4)6(OH)2; (b) schematics of a collagen fibril oftropocollagen molecular chains with cross-linked endsshown in red; (c) FEM mesh of a composite unit celldepicting a HAp particle (1/4 circle) embedded in a collagenmatrix; (d) finite-element model of a bone–tissue scaffoldrepresentative volume element (RVE) with an internal poreof major diameter d1 and minor diameter d2. The internalpores, which are intended as stations for nutrients andgrowth factors, are interconnected via channels of adiameter D. The dimensions of the RVE are l1 × l2 × l3. Allthree length-scales except the collagen fibril at themolecular level are considered in this work.

is shown schematically in Fig. 1, is to represent thegeometry of the scaffold in terms of a three-dimensional(3D) representative volume element (RVE). We have chosenan artificial design with specified shape and dimensionin order to control the pore shape, pore dimensions, andpore density to specified limits that are considered to bedesirable for promoting cell growth and tissue regeneration.This approach differs from other approaches in the literaturewhich generally have little or no control on the pore size,shape, or density. As shown in Fig. 1(d), the RVE containsan internal ellipsoidal pore with two intersecting circularchannels which are linked to form continuous pathways forcells to migrate from one internal pore to another internalpore in a neighboring RVE. These internal pores are designedto serve as stations where nutrients and growth factorscan be stored to stimulate cell migration and growth. Anindividual internal pore of major diameter d1 and minor

586 J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 3 ( 2 0 1 0 ) 5 8 4 – 5 9 3

diameter d2, is situated at the center of a block withdimensions l1 × l2 × l3 and is connected to similar pores inadjacent blocks through cylindrical channels of diameter D.The interconnecting channels are intended as passages tofacilitate inward cell migration. The pore density in the RVEcan be varied by changing the pore diameters (major andminor), the channel diameter, and the dimensions (l1, l2, andl3) of the RVE and is computed as the ratio of voided volume(pore volume plus channel volumes) to the total volume ofthe RVE. The design of the scaffold has been motivated byresults in the literature which indicated that cell growth inscaffolds may be significantly limited by nutrient deliveryas the void space inside the scaffolds is gradually occupiedby cells, thereby restricting nutrient transport to the interiorof the scaffold (Freed et al., 1994). Recent work by Chungand Ho (2010) suggests that internal seeding of cells in theinterior of a scaffold may lead to better cell amounts anduniformities within the scaffold. The smaller diameter ofthe interconnecting channels may be varied to promote celladhesion (Green et al., 2002; Murphy et al., 2010; Murphyand O’Brien, 2010), while the larger pore diameter in theinterior region of the scaffold may be optimized to promotecell migration and proliferation (Murphy et al., 2010; Murphyand O’Brien, 2010).

The scaffold is to be fabricated from a two-phasecomposite material consisting of nano-scaled sphericalhydroxyapatite particles embedded in a collagen matrix.Each FEM element in the RVE is comprised of a two-phasecomposite microstructure with a spherical particle of HApembedded in a collagen matrix with a volume fraction ofthe particle and the matrix being Vα and 1 − Vα, respectively.The value of Vα, and therefore the constitutive properties, areconsidered to vary in each of the FEM elements in the RVE(elastic and plastic). It is envisioned that the pore densityof the RVE can be designed by varying the pore diameters(d1,d2), the channel diameter (D), while the elastic modulusand the yield strength of the RVE that represents the scaffoldcan be designed by varying the HAp content at individualelements of the FEM mesh.

The objective of this investigation is to develop a mul-tiscale approach that allows one to predict the mechanicalperformance, which includes the elastic modulus and yieldstrength, of the 3D scaffold of a given pore geometry on thebasis of properties of constituents at smaller length scalesand higher structural hierarchies. Our approach is illustratedin Fig. 1. First, we utilize a first-principles computationalmethod to compute themechanical properties of nano-scaledHAp particles. Nano-scaled particles have been chosen be-cause it is biocompatible, biodegradable, and has been uti-lized as implants in bone repair in a dog model (Nishikawaet al., 2005) and in a rabbit model (Fukui et al., 2008; Zhouet al., 2006). In the study by Fukui et al. (2008), the nano-HAp/collagen composites were prepared by soaking honey-comb collagen sponge in a nano-HAp suspension to obtaina microstructure of pore sizes in the range of 200–350 µm inthe honeycomb collagen sponge with a HAp content in therange of 2 to 10 wt%. The animal studies (Nishikawa et al.,2005; Fukui et al., 2008; Zhou et al., 2006) indicated that theHAp/collagen composite implants induced new bone growthand was replaced by the newly formed bone tissue. Thus,

nano-HAp/collagen composite is bioresorptive and is attrac-tive as a scaffold material. In addition, the strength of HApincreases with decreasing particle size. A high strength can beattained in HAp when the particle size is in the nano-scaledrange (Ji and Gao, 2004a). The results for HAp and the experi-mental stress–strain data of collagen are utilized as materialinput in an FEM-based micromechanical modeling effort tocompute the constitutive response of HAp/collagen compos-ites at several HAp volume ratios. An atomistic approach wasnot utilized for the collagen matrix because of computationalcomplexities of the tropocollagen molecular chains in a colla-gen fibril (Buehler, 2006, 2008).

2. First-principles modeling at the atomisticscale

We utilized a first-principles computational code, WIEN2K,to calculate the stiffness matrix and theoretical strengthof nano-sized hydroxyapatite (HAp) platelets or particles.Developed by Blaha et al. (1990, 2001), WIEN2K is a first-principles quantum mechanical computational softwarepackage that uses a local density-based full-potentiallinearized augmented plane wave (FPLAW) method forcomputing the electronic structure and total energy of asolid within the density functional theory. To compute thetotal energy of hydroxyapatite, a unit cell model of HAp,Ca10(PO4)6(OH)2, was built using literature data (Kay et al.,1964). The crystal structure of HAp, shown in Fig. 1(a),belongs to the space group P63/m, group no. 176, with latticeparameters a = b = 9.4 A, and c = 6.87A.

The electronic structure calculations were performedusing a fully relativistic core and scalar relativistic valenceelectrons treatment without including magnetic effects. Theexchange–correlation energy term was evaluated within thegeneralized-gradient approximation (Perdew et al., 1996). Themuffin–tin radii of Ca, P, and O were taken to be 2.0, 1.669,and 1.181 au, respectively, where au is the atomic unit. Thenumber of k-points in the irreducible wedge of the Brillouiuzone was 8000 and the computational results converged to apreset charge limit of 0.0001 in a stable manner, yielding atotal energy value that remained constant within the 0.0001to 0.001 Ry range in individual self-consistency cycle (SCF)calculations.

The total energy of the unit cell was computed as afunction of lattice parameters (a,b, and c) and the equilibriumunit cell was established by minimizing the total energywith respect to the lattice parameter a and the aspectratios (c/a and b/a). Subsequently, we minimized the totalenergy by relaxing the atoms by varying an internal structureparameter, ∆, that represents the deviation of oxygen atomsfrom their ideal location in the crystal geometry. Once theequilibrium unit cell was established, the stiffnessmatrix andtheoretical strength of calcium phosphate were computed byapplying a strain and stretching the unit cell in a prescribedcrystallographic direction.

The stiffness matrix and theoretical strength of calciumphosphate were computed by stretching the equilibrium unitcell in a crystallographic direction as a function of an appliedstrain. The computational procedure involves calculation of


a b

Fig. 2 – First-principles computational results for HAp subjected to tensile straining: (a) energy change normalized by thecell volume as a function of the applied strain in the c direction of the HAp unit cell, and (b) tensile stress–strain curve ofHAp derived from computed total energy change as a function of the strain, ε33, in the c direction of the unit cell.

Table 1 – A summary of the elastic contents (Cij),theoretical strengths (σth), surface energy (γs), andcritical thickness (h∗) for hydroxyapatite.

Directionof unitcell

Latticeparameter

(Å)

Cij(GPa)

σth(GPa)

γs(J/m2)

h∗

(nm)

a 9.5303 177.6 21.6 4.06 5.14b 9.5303 177.6 21.6 4.06 5.14c 6.959 192.9 22.0 1.47 1.84

the total unit cell energy as a function of the applied strainsin the selected crystallographic direction. The energy changeswere normalized by the volume, Vo, of the equilibrium unitcell and the results were fitted as a function of the appliedstrain as shown in Fig. 2(a). The first derivative of the totalenergy with respect to strain gives a nonlinear stress–straincurve. The initial slope of this stress–strain curve gives theelastic constant in the selected crystallographic direction,while the maximum stress is the theoretical fracture strengthresulting from the breaking of bonds between atoms. Thetheoretical fracture strengths of HAp were computed in thethree principal lattice (a,b, and c) directions. The computedtensile stress–strain curve in the c direction is presented inFig. 2(b). A summary of the elastic constants and theoreticalstrengths of hydroxyapatite in the three principal directionsis presented in Table 1.

First-principles computational results of the elasticmodulus and theoretical strength of hydroxyapatite wereutilized in conjunction of the Griffith criterion (Griffith,1921) to compute the length scale above which linear-elastic fracture mechanics (LEFM) is applicable for nano-sized platelets of HAp. From the Griffith criterion for brittlefracture, the critical thickness, h∗, of HAp above which LEFMis applicable is given by Griffith (1921) and Ji and Gao (2004a)

h∗ =πEγsσ2th

(1)

where E is the elastic modulus, σth is the theoretical strength,and γs is the surface energy. The computed values of E andσth for HAp in the three principal directions (a,b, and c)of the unit cell are shown in Table 1. The surface energywas computed as one-half of the energy change per unitarea at the peak (theoretical strength) of the theoreticalstress–strain curve computed by the first-principles method.Using these material inputs, the critical length scales for HApare 1.8 nm in the c (platelet thickness) direction, while theyare 5.14 nm for the a and b directions. The computed 2–5 nmrange is in excellent agreement with the experimentallyobserved range of 2–7 nm for HAp platelet thickness (Ji andGao, 2004a). Experimental evidence also indicated that HApplatelet thickness increased from 2 to 3.6 nm with increasingage (15 weeks post conception to 97 years of age) in humanvertebral bodies from Fratzl et al. (2004). The good agreementsuggests that the strength dependence on particle size of HAputilized in the multiscale model is adequate at least for theHAp particles. In applications of the multiscale model, theHAp size is not limited to the nanosized range, but likely tovary in the nanometer to micrometer size range.

The computed elastic moduli are about 180–190 GPa,which are about a factor of two higher than those(100–146 GPa) reported in the literature (Ji and Gao, 2004a,b;Fratzl et al., 2004; Porter, 2004). The computed theoreticalstrengths are 21–22 GPa, which are significantly higher thanthe 100 MPa fracture strength for HAp. The discrepanciesmay be attributed the two sources: (1) the first-principlescomputation was performed for 0 K, and (2) the presenceof defects such as grain boundaries and microcracks in theHAp materials. By incorporating the critical length scale intothe Griffith criterion (Griffith, 1921), the fracture strengthof HAp can be related to the theoretical strength and acharacteristic size or critical thickness, h∗. Such a relationpredicts a decrease in the fracture strength with increasingparticle size according to a power-law with a −1/2 power.


3. Finite-element modeling at the microstruc-tural scale

At the microstructural scale, the stress–strain behavior ofHAp and collagen composites under uniaxial compressionwas simulated using a two-dimensional (2D) compositeunit cell, which is shown in Fig. 1(c), and the finite-element-method (FEM). Although a 3D scaffold is modeled,a 2D micromechanical model was utilized to generatethe stress–strain curves of the nanocomposites. Such anapproach was deemed adequate because the stress–straincurves will be fitted to analytical expressions, which aresubsequently applied to the 3D representative volumeelement of the scaffold. HAp was modeled as a cylindricalparticle embedded in a collagen matrix and the HAp volumepercent was altered by varying the radius of the HAp inthe FEM mesh. The volume percentages of HAp examinedwere 0%, 25%, 50%, and 75%. These volume percentageswere selected in order to provide a sufficient wide range ofstress–strain data so that they could be fitted to analyticalexpressions to cover the full range of nanocomposites from0% to 100% of HAp. The FEM mesh for a 50% HAp and 50%collagen is shown in Fig. 1(c). Two boundary layers of elementsof uniform size were specified along the interface betweenthe matrix and the particles. Opening of the interface wasgoverned by a critical normal tensile stress, but this featurewas not used for compressive loading.

The elastic and fracture properties of the HAp par-ticles were described in terms of the constitutive lawsobtained from first-principles computations, while the con-stitutive behavior of the collagen matrix was described usingthe Ramberg–Osgood relation for elastic–plastic materials ex-hibiting power-law hardening. This Ramberg–Osgood consti-tutive relation is given by

ε =σ

E

[1+ α

(σ

σo

)n−1](2)

where E is Young’s modulus, σo is the yield stress at0.2% plastic strain, and α is a fitting parameter. Eq. (2)was fitted to the stress–strain data of dry, cross-linkedcollagen reported by Thompson and Czermusza (1995). Usingthese constituent properties, the stress–strain response ofthe composite was computed as a function of size andvolume fraction of HAp particles. The computed compressivestress–strain curves for various volume fractions of HAp arepresented in Fig. 3. Also shown is the experimental data(circles) of collagen fitted to the Ramberg–Osgood equation.Fig. 3 shows increasing values of the elastic modulus andyield strength with increasing volume percentages of HAp.Analysis of the composite stress–strain response revealedthat the Ramberg–Osgood constitutive relation, Eq. (2), isstill applicable for the composite, providing that the elasticmodulus, E, the yield strength, σo, and the α parameterare expressed as functions of the volume fraction of theHAp particles. A plot of Young’s modulus as a function ofvolume percentage of HAp is presented in Fig. 4(a), while theyield strength at 0.2% plastic strain is shown as a functionof volume percentage of HAp particles in Fig. 4(b). Thestress–strain response of the HAp/collagen composites atlarger strains (>0.03) was utilized to determine the strength

Fig. 3 – Compressive stress–strain curves of HAp/collagencomposites predicted by FEM on the basis the unit cellgeometry and constituent properties for 0%, 25%, 50%, and70% HAp by volume.

coefficient (k) and the strain-hardening exponent (n) and thecorresponding results are shown in Fig. 4(b). These materialconstants were used as material input for structural analysesof a representative volume element (RVE) of a scaffold, asdescribed in the next section.

4. Scaffold microstructural optimization

Microscale 2D unit cell constitutive data on HA/Collagencomposites was used to determine input values for the LS-DYNA material model, MAT_POWER_LAW_PLASTICITY. Thematerial model is an isotropic plasticity model with rateeffects that uses a power-law hardening rule. The yield strainis a function of plastic strain:

σy = k(εyp + ε̄p)n (3)

where εyp is the elastic strain to yield, ε̄p is the effectiveplastic strain (logarithmic) and n is the hardening exponentand k is the strength coefficient. The elastic strain to yield isdetermined by

εyp =(σyk

)[ 1n

](4)

and the values for modulus, yield strength, k and n were ob-tained from Fig. 4(b). Material models were created for 20%,50% and 80% HAp and incorporated in to a parametric finiteelement model, Fig. 1(d). The new material model was testedby applying a 10% compression strain to a single element andcomparing the FE model results for the 50% HAp materialmodel to the 2D composite unit cell microscale model (Fig. 5).

Using the parametric finite element model shown inFig. 1(d), a study was performed on the effects of geometryvariations on the modulus and yield strength of the 3Dscaffold model. For the parameter study D,d1, and l1 werevaried along with the ratio of d1 to d2 over a total of 16runs (Table 2). The channel diameter D was varied from 100to 400 µm to encompass the optimized pore size range forcell growth (Tsuruga et al., 1997). The internal pore diameterd1 was varied from 500 to 1000 µm in order to optimize


a b

Fig. 4 – Mechanical properties of HAp/collagen composites via FEM modeling: (a) predicted values of Young’s modulus as afunction of volume percents (XHA) of HAp, and (b) predicted yield strength and strength coefficient (k) of HAp/collagencomposites as a function of XHA. The predicted relation between strain hardening exponent, n, and XHA is shown as aninsert.

Fig. 5 – Predicted compressive stress–strain curve for a single solid element model at 50% HAp composition obtained viathe LS-DYNA code and compared to the 50% HAp microscale model.

the overall scaffold porosity and storage volume for nutrientstorage and cell growth. The corresponding values of thepore density are presented in Table 2. For all simulations, themodel was loaded to 10% strain in compression using periodicboundary conditions (Fig. 6). The following equations governthe relationship between the variables in Fig. 1(d):

l2 = l1 l3 =l12

ratio =d1d2. (5)

The results from the parameter study show that the modulusis reduced by an increase in D or d1 and is increased byan increase in l1 for all three HAp levels (Fig. 7). The yieldstrength is increased slightly when D is increased from 100to 200 µm (Fig. 8). Any increases to D beyond 200 µm causesa decrease in the yield strength. The same effect can beseen for the d1 parameter. The yield strength is increasedfrom the mean at 600 µm but then declines. The ratioof d1 to d2, while having little effect on modulus, has asignificant effect on yield strength. A ratio of 2:1 of d1 to d2

increases the yield strength from 11 to 12 MPa (Fig. 8). Forcomparison purposes, the elastic modulus (291 MPa) and theyield strength (5.33 MPa) for cancellous bone (Kopperdahl andKeaveny, 1998; Linde et al., 1991) are also presented in Figs. 7and 8. In all cases considered, the computed properties for the3D scaffold exceed those of the cancellous bone. On the otherhand, the longitudinal compression elastic modulus and theyield strength for cortical bone are 17 GPa and 193 MPa,respectively (Reilly et al., 1974); both are higher than thecomputed properties for the 3D scaffold. Thus, the scaffoldgeometry considered in Table 2 is suitable for cancellous bonetissue, but may not be suitable for cortical bone if adequatestrength is required. Although scaffolds used in the fielddo not generally have similar properties to the tissue theyreplace, adequate strength and matching elastic modulusshould be important considerations for bone–tissue scaffoldsthat are intended for the repair of bone fracture with a criticalsize defect. A matching elastic modulus is essential in orderto avoid stress shielding from the bone tissue. Inadequate


Fig. 6 – Finite element model of the 3D scaffold subjected to uniaxial compression loading to 10% strain. Representativelocal strain profile at 0.2% nominal strain is illustrated.

Fig. 7 – Parameter study results of elastic modulus for various pore diameters (d1 in µm), pore diameter ratio (d1/d2),channel diameters (D in µm), and scaffold length (l1 in µm) at three HAp levels (20%, 50%, and 80% by volume). The nominalelastic modulus for cancellous bone is shown for comparison.

scaffold strength may result in premature fracture of thescaffold and delay proper healing of the bone tissue. It isalso worthy to note that bone tissue is transversely isotropicwith a higher strength in the longitudinal direction thanthose in the radial and circumferential directions. The currentcomposite model based on spherical HAp nanoparticles isisotropic, which gives identical strengths in all directions.To attain transversely isotropic properties, the shape ofthe spherical HAp particle in the composite model needsto be replaced by ellipsoidal, rod-like or platelet-like HApnanoparticles.

5. Discussion

The results of this investigation demonstrated that the mul-tiscale modeling approach provides a viable computational

framework for optimizing the pore dimensions and mechan-ical properties (yield strength and stiffness) of a scaffold forcancellous bone. For the range of pore and channel diame-ters examined, the optimized 3D scaffold is one with a porediameter of 1000 µm, a channel diameter of 100 µm, and apore density of ≈27%. This porosity level is slightly less thanthe 30% porosity reported as the optimum level in the litera-ture (Toth et al., 1995). For the scaffold geometry consideredin this paper, the elastic modulus and yield strength of can-cellous bone can be obtained with a 20 Vol.% HAp in the com-posite. Higher elastic modulus and yield strength values areobtained at 50% and 75% (Fig. 8). Hence, the pore diameter,channel diameter, and pore density can be further increasedwith the desired levels of the yield strength and elastic mod-ulus if the HAp content is allowed to increase in the com-posite. In other words, further improvement in the optimized3D scaffold and pore density geometry may be feasible if one


Fig. 8 – Parameter study results of yield stress for various pore diameters (d1 in µm), pore diameter ratio (d1/d2), channeldiameters (D in µm), and scaffold length (l1 in µm) at three HAp levels (20%, 50%, and 80% by volume). The nominal elasticmodulus for cancellous bone is shown for comparison.

Table 2 – Variable values for the 16 parameter variationruns.

Run D(microns)

d1(microns)

l1(microns)

Ratio Porosity(%)

1−mean 100 500 2500 1 4.222 200 500 2500 1 6.913 300 500 2500 1 11.654 400 500 2500 1 18.445 100 600 2500 1 6.536 100 700 2500 1 9.807 100 800 2500 1 14.148 100 900 2500 1 19.879 100 1000 2500 1 26.99

10 100 500 3000 1 2.6111 100 500 3500 1 1.7512 100 500 4000 1 1.2413 100 500 4500 1 0.9414 100 500 5000 1 0.7115 100 500 2500 0.5 7.3616 100 500 2500 2 13.84

includes the HAp content in the optimization procedure. Itshould also be noted that biological constraints correspond-ing to those for vascularization and angiogenesis have notbeen included in the present optimization analysis, but theyare intended to be included in future work. The results alsoindicate that this particular scaffold design is not as strongas the cortical bone and may not work as a scaffold in a longbone defect if matching elastic modulus and yield strengthare required. In the latter case, a different scaffold designwitha smaller internal pore size, a higher volume fraction of HApparticles, or a stronger collagen matrix with a higher cross-link density might be needed. One of the limitations of thecurrent approach is that the pore and channel diameters arevaried within ranges (5000 µm ≤ d1 < 1000 µm;100 µm ≤ D ≤400 µm) of specified pore size that are known to promote cellgrowth, while the HAp content has been considered for threelevels (25%, 50%, and 75%). Strictly speaking, the HAp con-tent of the composite scaffold may not be optimized because

of the coarse increments in the HAp content. Thus, furtheranalyses with wider ranges of pore size and smaller HApincrements are required to refine and improve the currentmethodology to design scaffolds for cancellous and corticalbone tissues.

A successful scaffold must support cell growth and sustainthe biology of tissue genesis inside the interior regime of thescaffold. The channels of the 3D scaffold are intended to serveas passageways for cell migration into the interior region ofthe 3D scaffold. Once inside the 3D scaffold, sustainmentof cell growth is provided by nutrient station as representedby the central pore inside the 3D scaffold. The channel andpore diameters are motivated from reported data in theliterature that are relevant for cell growth. The optimizationprocedure performed in this work identified the desirablechannel diameter and pore diameter that provide matchingstrength and elastic modulus to the cancellous bone. Sucha scaffold would avoid the stress shielding effect that mayoccur if a scaffold with a higher elastic modulus is used.Furthermore, the smaller diameter (D) of the connectingchannels in the scaffold can be optimized to promote celladhesion (Green et al., 2002; Murphy et al., 2010; Murphy andO’Brien, 2010), while the larger diameter of the central pore,intended as an interior nutrient station, may be optimizedto promote cell migration, growth, and proliferation (Murphyet al., 2010; Murphy and O’Brien, 2010). The use of multiplesize scales to accommodate both cell adhesion and growthmay encourage cell migration into the interior region of thescaffold, thereby eliminating cell aggregation near the outersurfaces or edges of the scaffold.

In the current multiscale model, the constitutive modelfor the collagen matrix is based on the Ramberg–Osgoodrelation whose material constants have been evaluated fromexperimental data. An atomistic computational approachwas not attempted for the collagen matrix because ofthe difficulties in modeling the tropocollagen molecularchains in a collagen fibril. Recently, Buehler (2006, 2008)represented the entire tropocollagen molecule as a collection


of beads of super-atoms whose intermolecular interactionswere described in terms of a Lennard–Jones potential. Anenergy parameter ELJ was used to represent the strength ofthe intermolecular adhesion. In addition, a scalar parameterβ was utilized to represent the increase in the intermolecularadhesion when one cross-link is present at each end of atropocollagen molecule. A collagen fibril is considered to becomprised of a staggered array of tropocollagen molecules.The strength of the collagen fibril then depends on theintermolecular adhesion parameter ELJ in the potential andthe cross-link density parameter β. Such a meso-molecularapproach is quite attractive and may be incorporated intothe current approach in future work, if the computationalrequirement is not too stringent.

We anticipate that a layer-by-layer fabrication process willbe utilized to build the scaffold. Such a fabrication has beendesigned, built, and utilized to fabricate simple scaffolds bylaying micron-sized filaments in a specific pattern (Nicolellaet al., 2008). This computer-aided fabrication process allowsprecise control of local pore size and composition whenmultiple nozzles are used to lay down the layers or filamentsof different materials. This fabrication system is still indevelopment and has not been perfected. To take advantageof this fabrication capability, a model improvement to allowdifferent HAp contents at various locations of the scaffoldis also being considered. In principle, the 3D scaffold ofarbitrary geometry can be fabricated using materials such asHAp, collagen, Poly-L-Lactic Acid (PLLA), or PLGA. The currentmultiscale modeling framework has been developed to allowdesign of scaffolds with controlled gradients in composition,microstructure, and properties. It is envisioned that afteroptimization by computation and simulation, the optimizedscaffold is then fabricated by a layer-by-layer deposition orprinting technique.

6. Conclusions

The conclusions reached as the results of this investigationare as follows:

1. First-principles unit-cell computation provides reasonablepredictions of the elastic moduli, theoretical strengths,and critical dimensions for nano-scaled hydroxyapatiteparticles.

2. A micro-2D unit-cell approach has been shown to providethe constitutive (stress–strain) response of HAp/collagencomposites as a function of HAp content for spherical HApparticles embedded in a collagen matrix.

3. A multiscale modeling approach has been developed andapplied to optimizing the mechanical performance of abone–tissue scaffold by varying the pore size, pore density,and the microstructure of HAp/collagen composites in thebone–tissue scaffold using 3D FEM analyses.

4. For the scaffold pore geometry considered, the elasticmodulus and yield strength of the scaffold can be opti-mized to match or exceed the corresponding properties ofcancellous bone.

Acknowledgements

This research was supported by the Internal ResearchProgram of Southwest Research Institute, through ProjectR9551. The clerical assistance by Ms. L. Mesa in thepreparation of the manuscript is acknowledged.

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a multiscale modeling approach to scaffold design and property prediction

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