3D structural analysis: sensitivity of Minkowski functionals.

The Minkowski functionals, a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space, are the complete set of additive morphological measures and can be simply calculated from local contributions. As such, they have found a wide range of applications. We consider the effects of distortions (drift, noise and blurring) on the morphological properties of complex random models, representative of a wide range of structure. Such distortions arise experimentally in imaging techniques due to diffraction, absorption and sample drift. The question is asked, how critically these distortions effect image quality as measured by the Minkowski functionals. Defining a length scale based on the two-point correlation function, we consider how distortion at different scales can lead to quantitative errors in morphological measures.

Boolean reconstructions of complex materials: Integral geometric approach.

We show that for the Boolean model of random composite media one can, from a single image of a system at any particle fraction, define a set of parameters which allows one to accurately reconstruct the medium for all other phase fractions. The morphological characterization is based on a family of measures known in integral geometry which provides powerful formulas for the Boolean model. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from the single three-dimensional image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples.

Investigation of microstructural features in regenerating bone using micro computed tomography.

We illustrate some of the uses of micro-computed tomography (micro-CT) to study tissue-engineered bone using a micro-CT facility for imaging and visualizing biomaterials in three dimensions (3-D). The micro-CT is capable of acquiring 3D X-ray CT images made up of 2000(3) voxels on specimens up to 5 cm in extent with resolutions down to 2 microm. This allows the 3-D structure of tissue-engineered materials to be imaged across orders of magnitude in resolution. This capability is used to examine an explanted, tissue-engineered bone material based on a polycaprolactone scaffold and autologous bone marrow cells. Imaging of the tissue-engineered bone at a scale of 1 cm and resolutions of 10 microm allows one to visualize the complex ingrowth of bone into the polymer scaffold. From a theoretical viewpoint the voxel data may also be used to calculate expected mechanical properties of the tissue-engineered implant. These observations illustrate the benefits of tomography over traditional techniques for the characterization of bone morphology and interconnectivity. As the method is nondestructive it can perform a complimentary role to current histomorphometric techniques.

Reconstructing complex materials via effective grain shapes.

We introduce a powerful method based on integral geometry and the Kac theorem for the spectrum of the Laplace operator to define the effective shape of an inclusion in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using the effective inclusion shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Use of the equivalent shape in effective medium formulations leads to good predictions. The method is verified for a sedimentary rock sample.

Euler-Poincaré characteristics of classes of disordered media.

We consider a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space that are sensitive to the morphology of disordered structures. These measures embody information from every order of the correlation function but can be calculated simply by summing over local contributions. We compute the evolution of the measures with density for a range of disordered microstructural models; particle-based models, amorphous microstructures, and cellular and foamlike structures. Analytic results for the particle-based models are given and the computational algorithm verified. Computational results for the different microstructures exhibit a range of qualitative behavior. A length scale is derived based on two-point autocorrelation functions to allow qualitative comparison between the different structures. We compute the morphological parameters for the experimental microstructure of a sandstone sample and compare them to three common stochastic model systems for porous media. None of the statistical models are able to accurately reproduce the morphology of the sandstone.