Grey value images as a basis for finite element models and their mechanical properties

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Grey value images as a basis for finite element models and their mechanical properties K. Szlazak 1, J. Jaroszewicz 1, J. Idaszek 1, A. Dejaco 2, P. Hasslinger 2, V. Vass 2, C. Hellmich 2, W.

1 Grey value images as a basis for finite element models and their mechanical properties K. Szlazak 1, J. Jaroszewicz 1, J. Idaszek 1, A. Dejaco 2, P. Hasslinger 2, V. Vass 2, C. Hellmich 2, W. Swieszkowski 1 1 Faculty of Materials Science and Engineering, Warsaw University of Technology, Warsaw, Poland 2 Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Austria Aims A novel bone composite scaffolds used in tissue engineering are produced using a 3D rapid prototyping method. To this end, ceramics inclusions are combined with an polymeric matrix. Due to the relatively high strength of commonly used ceramics fillers and biodegradability of polymers, composite scaffolds exhibit different mechanical properties in compare to their individual components. On the micro-ct reconstructed images we can generally observe white spots of ceramics with high strength and grey polymer matrix with high elasticity. The aim of this study was to show an application of micro-ct reconstructed images for assigning mechanical properties to each voxel based on grey values to create finite element (FE) model. Method In the first stage of this study the composite scaffolds were prepared and examined by means of micro-ct. We consider scaffold made of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV, Sigma-Aldrich; USA), poly(lactic-co-glycolic) acid (PLGA, Resomer 855S; Boehringer Ingelheim; Germany) and tricalcium phosphate (TCP, Sigma-Aldrich, USA) make up 67.60%, 30.24% and 2.16% of the total volume respectively. Scaffold with customized geometry and controlled internal architecture were fabricated using rapid prototyping method (Bioscaffolder, Syseng, Germany). Cylinder-like sample (fig. 1a) with diameter of 6 mm, height of 2 mm and with an average macroporosity of 55% was obtained and then scanned (with phantom of PHBV) by means of SkyScan 1172 (Bruker) in area which consisted 4 4 fibers and 7 layers (fig. 1b). Pixel size was 6 μm with a source voltage and current of 40 kv and 250 A, respectively. A B Fig. 1. Three dimensional reconstruction of composite scaffold (a) and inner part area which consist 4 4 fibers and 7 layers (b).

2 The method for evaluation of a mechanical properties of each voxel based on grey value images was determined. Each voxel occupied by the solid contains a composite material consisting of PHBV (filling volume fraction f PHBV), PLGA (filling volume fraction f PHBV) and with TCP inclusions (filling volume fraction f TCP). Hence, the volume fractions fulfill f PHBV + f PLGA + f TCP = 1, (1) In our case scaffold peak contained mainly grey value came from PHBV-PLGA matrix (denote as polymer), because of the highest content of polymers in the composition. Based on this GV polymer = GV PHBV = GV PLGA, (2) and f PHBV + f PLGA = f pol ymer, (3) For this PLLA-TCP composite, the average rule for X-ray attenuation coefficients reads as [1-3] μ = f pol ymer μ pol ymer + f TCP μ TCP, (4) where polymer and TCP are the X-ray attenuation coefficients of polymers (mean value) and of TCP, respectively. Since the X-ray attenuation coefficients are linearly related to the grey values GV, it follows from Eq. (4) that GV = f polymer GV polymer + f TCP GV TCP, (5) Eq. (5) can be used to convert voxel-specific grey values into voxel-specific volume fractions of TCP f TCP = GV GV polymer (6), GV TCP GV polymer resulting in volume fractions of TCP ranging from 0 to 100% for grey values from 66 to 255, respectively. Next, this volume fraction enters a micromechanical description of the material found within the individual voxels. We used the Mori-Tanaka scheme [4-6] for determination of the voxelspecific fourth-order (homogenized) stiffness tensor C hom of the polymer-ceramic composite, C hom = 1 f TCP c polymer + f TCP c TCP : [I + P polymer,sph : (c TCP c polymer ) 1 : (7) { 1 f TCP I + f TCP [I + P polymer,sph : (c TCP c polymer ) 1 } 1 where c polymer and c TCP are stiffness tensors of polymer and TCP, repectively. I is unity tensor, P polymer,sph denotes the fourth-order Hill tensor accounting for the spherical shape of the TCP inclusions embedded in the polymer matrix. Finally, use of the elastic properties of polymer matrix and TCP in Eq. (7), while considering Eq. (6), gives access to grey value-specific Young s moduli E hom (GV) and Poisson s ratios ν hom (GV) throughout the scaffold s solid phase; more precisely, the latter are, in a standard fashion, computed from the compliance tensor D hom =(C hom ) -1, according to: E hom hom GV = 1/D 1111 GV, (8) v hom GV = E hom hom GV /D 1122 GV (9)

Results Regular finite element models of the polymer-tcp-based scaffold were generated from the stack of microct images analogously to the procedure described in [7].

All solid finite elements were assigned one pair of Young s modulus and Poisson s ratio. The heterogeneous distribution of elastic properties across the tissue engineering scaffold is depicted in Fig.

3 Results Regular finite element models of the polymer-tcp-based scaffold were generated from the stack of microct images analogously to the procedure described in [7]. Thereby, voxels were combined into one finite element. The grey values GV were converted, according to Eq. (1) - (7) and (8) - (9) into finite element-specific stiffness properties. All solid finite elements were assigned one pair of Young s modulus and Poisson s ratio. The heterogeneous distribution of elastic properties across the tissue engineering scaffold is depicted in Fig. 2. Fig. 2. Cross section of the scaffold - grey value GV image (a) and its finite element representation of Young's modulus E and Poisson s ratio v distribution (b) The FE model was subjected to uniaxial compression conditions in vertical (longitudinal) direction (orthogonal to the rapid prototyped beams, piled on each other) indicated by unit vector z. Based on the Abaqus simulation package (Abaqus , Simulia, USA), a vertical displacement being the height of the scanned scaffold portion measured in z direction, is prescribed at the top surface of the scaffold. The bottom surface is held at fixed displacements in z direction, i.e. in the direction of the applied displacement, while this surface is free to deform in the plane orthogonal to z. The lateral walls of the scaffold are kept stress-free. Abaqus simulation allowed to show minimum principal strain (fig. 3) and stress (fig. 4) distribution in a heterogonous (polymer-ceramics model) and homogenous model (with mean mechanical properties of PHBV and PLGA).

from the heterogeneous (a) and homogenous (b) simulation (middle part of the scaffold was shown) Fig. 4.

4 Fig. 3. Minimum principal strain ε distribution in three dimensional reconstruction through the scaffold, derived from the heterogeneous (a) and homogenous (b) simulation (middle part of the scaffold was shown) Fig. 4. Minimum principal stress σ distribution in three dimensional reconstruction through the scaffold, derived from the heterogeneous (a) and homogenous (b) simulation (middle part of the scaffold was shown)

5 Conclusion Finite element analysis based on micro-ct voxel-specific approach provides not only shape of the bone scaffolds but also an in-depth assessment of them mechanical behaviour. Voxels with assign specific mechanical properties (heterogeneous model) depended on grey values have higher strain and stress under compression in compare to homogenous model. Therefore, it is reasonable to take information about mechanical properties contained in grey values into account. Acknowledgments This work has been supported by the National Centre for Research and Development no. STRATEGMED1/233224/10/NCBR/2014 and by the National Science Centre no. DEC- 2012/07/D/ST8/02606 and partially by a STSM Grant from COST Action NAMABIO MP1005 and NEWGEN MP1301. References: 1. Crawley E.,O., Evans W.,D., Owen G.,M., A theoretical analysis of the accuracy of singleenergy CT bone-mineral measurements. Physics in Medicine and Biology. 1988,33, no. 10, Hellmich C., Kober C., Erdmann B., Micromechanics-based conversion of CT data into anisotropic elasticity tensors, applied to FE simulations of a mandible. Annals of Biomedical Engineering. 2008,36, no.1, Scheiner S., Sinibaldi R., Pichler B., Komlev V., Renghini C., Vitale-Brovarone C., et al., Micromechanics of bone tissue-engineering scaffolds, based on resolution error-cleared computer tomography. Biomaterials ,30(12), Benveniste Y.A., A new approach to the application of mori-tanakas theory in composite materials. Mechanics of Materials. 1987,6, Eshelby J.D., The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,241, no.8, Mori T., Tanaka K., Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica. 1973,21, no.5, Dejaco A., Komlev V.S., Jaroszewicz J., Swieszkowski W., Hellmich C., Micro CT-based multiscale elasticity of double-porous (pre-cracked) hydroxyapatite granules for regenerative medicine. J Biomech /5,45(6),

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