Micromechanics of bone tissue-engineering scaffolds, based on resolution error-cleared computer tomography
Introduction
Computed tomography (CT), the mathematical basis for which was established in 1917 by Radon [1], and for the first time successfully applied in 1972 [2], is a powerful non-destructive evaluation technique for producing three-dimensional (3D) images, revealing the microstructure of objects [3], [4], [5], [6], [7], [8], [9], [10]. For this purpose, X-rays are sent through the investigated object, and the transmitted intensity is recorded as a 2D image [3]. If this transmission is repeated under different angles in the 3D space, all corresponding images can be transformed into one 3D distribution of X-ray attenuation coefficients, giving access to a 3D image of the object. Numerically, the latter is defined through values related to small volume units, called voxels. Over recent years, CT has become a widely applied tool in science, technology, industry, and medicine, with a particularly strong position in the latter field [6], [11], [12], [13], [14], [15], [16]. There, it supports the clinical practice, but also fundamental research devoted e.g. to development of novel biomaterials or tissue engineering [17], [18], [19], [20], [21], [22], [23], [24] (regenerative medicine). The latter endeavors have motivated the striving for an even finer resolution, in order to reveal, in a 3D fashion, more and more of the microstructures and nanostructures found in biological and biomimetic materials. Evaluation of the obtained data by means of appropriate visualization tools allows for definition of topology and geometry of the investigated objects. This information allows for further data processing and understanding, such as mechanical simulations (e.g. based on Finite Elements) of biostructures [25], [26], [27], [28], [29]. These analyses can serve as basis to quantify mechanical properties of the investigated objects, which helps (i) to replace (or at least to decrease the amount of) additional, often expensive laboratory testing on a large number of specimens, and (ii) to learn about implications of different alternatives to a specific implantation strategy, as well as to support the corresponding decision-making process.
However, although detailed information on the geometry of the investigated objects is accessible, the results of performed mechanical simulations still comprise uncertainties. This is because the mechanical properties of the material constituting the voxels are not directly accessible. In fact, they are mostly guessed on the basis of experience or of some back-analysis; or some additional experimental work would be necessary to have access to them. Aiming at an improvement of this situation, the present paper proposes a new strategy for reliable determination of the voxel-specific material properties based on fundamental knowledge of X-ray physics and continuum micromechanics: Attenuation coefficients from micro-CT scans of a CEL2 glass–ceramic sample with a voxel size of 9 μm [30] (see Section 2) are first cleared from systematic machine-induced resolution errors (see Section 3), and then translated, voxel by voxel, into the voxel-specific CEL2 glass–ceramic composition (see Section 4). Thereby, the skeleton material within a voxel is regarded as a composite built up by dense CEL2 glass–ceramic and by an inclusion-type nanoporosity. Subsequently, the voxel-specific composition (nanoporosity) is translated into voxel-specific elastic properties, in the framework of continuum micromechanics (see Section 5). These voxel-specific elastic properties are fed into Finite Element (FE)-based structural analyses of the macroporous scaffold with (micro-) inhomogeneous skeleton material (see Section 6). To test the reliability of the aforementioned FE analyses, they are used to determine the macroscopic Young's modulus of the CEL2 glass–ceramic scaffold, which is then compared to Young's modulus obtained from ultrasonic experiments on exactly the same sample, as discussed in Section 7.
Section snippets
CEL2 glass–ceramic scaffolds: processing and micro-CT scanning
The investigated scaffold was made of a highly bioactive glass (called CEL2), with the following molar composition: 45% SiO2, 3% P2O5, 26% CaO, 7% MgO, 15% Na2O, 4% K2O, and a 4:1 Na2O/K2O ratio. The raw products (SiO2, Ca3(PO4)2, CaCO3, 4MgCO3Mg(OH)25H2O, Na2CO3, K2CO3) were melted at 1400 °C for 1 h, followed by quenching in cold water, grounding and sieving to a grain size of less than 30 μm [31]. For scaffold production, a polymeric template exhibiting a porous microstructure was impregnated
Clearance of measured CT data from the tomograph-specific instrumental function
While a distribution of values around the right-hand peak in Fig. 1(c) may follow from inhomogeneities in the solid skeleton, inhomogeneities in the air-filled pores do not make physical sense. All pore-related voxels should have the same attenuation coefficient: that of air. In other words, the first peak (Fig. 1(c)) should be a Dirac delta function. This gives us a straightforward access to the tomograph-specific machine error. The deviation of the peak on the left-hand side of Fig. 1(c) from
Voxel-specific composition/nanoporosity
Next, the voxel-specific, machine error-cleared grey values are translated into the voxel-specific composition of the material inside one voxel. Therein, we envision a dense glass–ceramic matrix embedding nanopores (Fig. 5), and the voxel-specific nanoporosity ϕ will be determined from the voxel-specific grey values. Therefore, we recall [39], [40] that the attenuation coefficient of a composite material is the volume average over the attenuation coefficients of the individual components of the
Micromechanics-based voxel-specific elastic properties
In continuum micromechanics [41], a material is understood as a macro-homogeneous, but micro-heterogeneous body filling a representative volume element (RVE) with characteristic length l, l ≫ d, d standing for the characteristic length of inhomogeneities within the RVE, and l ≪ L, L standing for the characteristic lengths of geometry or loading of a structure built up by the material defined on the RVE. In general, the microstructure within one RVE is so complicated that it cannot be described in
Generation of Finite Element mesh
In the SRμCT measurements, the CEL2 glass–ceramic scaffold is resolved by 684 × 732 × 300 voxels. Consideration of all available morphological details (revealed through SRμCT measurements and corrected as shown in Section 3) in the FE-based structural analyses would consequently necessitate the creation of an FE mesh composed of more than 1.5 × 108 elements with eight nodes each, rendering the related numerical simulation, from a computational point of view, as extremely expensive. As a remedy,
Experimental validation of new model approach
The FE-predicted value for the macroscopic Young's modulus of the investigated CEL2 glass–ceramic sample, , is now validated through the ultrasonic measurements as described in Kohlhauser et al. [44]. Velocities of longitudinal ultrasonic waves sent through the CEL2 glass–ceramic samples give experimental access to the 1111-component of the macroscopic stiffness tensor of the herein investigated sample of porous CEL2 glass–ceramic: When considering
Conclusion
3D images of CEL2 glass–ceramic scaffold microstructures, obtained from SRμCT measurements, were utilized for determination of their mechanical deformation behavior and in particular their macroscopic elastic properties. This approach consistently integrates well-accepted fundamentals of X-ray physics, continuum micromechanics, and Finite Element technology, by realizing the following steps:
- (i)
acquisition of the scaffold microstructure by means of SRμCT measurements;
- (ii)
a new strategy to clear the
Acknowledgements
Financial support by ‘KMM-NoE – Network of Excellence on Knowledge-based Multicomponent Materials for Durable and Safe Performance’ (project number NOE 502243-2), sponsored by the European Commission, is gratefully acknowledged. Moreover, the authors are indebted to Christoph Hackspiel, Vienna University of Technology, for providing helpful support with ABAQUS, and to Josef Beiglböck, for assistance with the Supercomputing Center of Vienna University of Technology.
Glossary
- A
- cross-section of the CEL2 glass–ceramic scaffold
- C i
- Finite Element-specific stiffness tensor
- d
- characteristic length of inhomogeneities within the RVE
- dg
- index denoting dense CEL2 glass–ceramic
- dsk
- characteristic length of inhomogeneities within an RVE of solid skeleton material
- macroscopic Young's modulus of the investigated (macroporous) CEL2 glass–ceramic sample, obtained from ultrasonic experiments
- macroscopic Young's modulus of the investigated (macroporous) CEL2 glass–ceramic
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Both authors contributed equally to this work.