1Towards a Multi-scale Computerized Bone Diagnostic System:2D Micro-scale Finite Element Analysis

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Update

Toward a Multiscale Computerized Bone Diagnostic System: 2D Microscale FiniteElement Analysis

Lev Podshivalov, Yaron Holdstein, Anath Fischer, and Pinhas Bar-Yoseph

1Overview

Bone is composed of hierarchical biocomposite materials that are characterized bycomplex multiscale structural geometry and heterogeneous material properties.Bone tissue structure may be classified into five structural levels [1, 2], rangingfrommacro- to nanoscales. Each patient�s bone structure is unique and is influencedby gender, age, lifestyle, and physical condition. Depending on the anatomical site,bone architecture differs significantly at the microstructural level with respect toshape, thickness, directionality, and size. It may also vary at different locations of thesame site, as a result of functionality and locally applied forces characterized bymagnitude and direction. These characteristics are crucial for diagnosis of bonemetabolic diseases.

Metabolic bone diseases are characterized by microarchitectural deterioration ofbone tissue, leading to microfractures; therefore, early diagnosis is a key tointervention. The most widely used technique for assessing bone quality is bonemineral density (BMD) assessment. BMD data are represented as scalar of astatistical value (T-score). Although BMD testing is used regularly, it has a majorlimitation. It does not describe the complex 3D bone microstructure, but ratherprovides an indication using a single scalar value. As a result, the BMD methodoccasionally fails to diagnose risk of osteoporotic fractures [3], leading to unreliablediagnosis and incorrect treatment.

The need for diagnosis of microstructures has led to the development ofemerging 3D microscale scanning methods such as mCT and mMRI, which offerhigh-resolution in vivo and in vitro scanning of bone structures. To date, however,no complete diagnostic method incorporating these technical abilities has beenproposed.

Computer Aided Biomanufacturing, Edited by Roger Narayan and Paul Calvert� 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

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Surface reconstruction and volumetric modeling of the 3D bone structure frommedical images constitute the first stages for analysis. Mesh reconstruction frommCT/mMRI images produced by commercial software often results in a highlydistorted triangulation that is not suitable for finite element (FE) mechanicalanalysis, and therefore requires remeshing and mesh optimization [4]. Theseoperations are time consuming and often require manual user interventions.Moreover, a multiresolution volumetric representation should be created [5–7] inorder to analyze microscale models using the microfinite element method. Themajor limitation of current methods [8] is their inability to handle largebone volumes using commonly available computational resources. Therefore, amultiscale approach is required.

State-of-the-art multiscale FE methods currently utilize parallel computing onlarge, high-resolution models of human bone reconstructed from mCT/mMRIimages [9]. However, these multiscale methods include only two binary scales:macro- and microscales; no intermediate scales can be represented. An adaptiveapproach has recently been proposed [10]. This method is based on modeling anentire model using a macroscale homogeneous model and embedding this modelinto the volume of interest (VOI). Nevertheless, this method does not provide thecomplete multiscale method needed for robust multiscale analysis of bonearchitecture.

The required multiscale analysis approach should take advantage of both thesimplicity and efficiency of the macroscopic models and the accuracy of thedetailed microscopic model, by coupling macro- and microscopic models [11, 12].A new multiscale method must include a multiresolution geometric representa-tion and predefined hierarchical physical criteria capable of computing themicrostructural level locally. The multiscale models should facilitate fast geometricand material transition between the scales, without accumulative errors. Thus, amultiscale approach for mechanical analysis of bone is imperative. Our main goalis to develop an efficient and robust computational technique that will formthe basis of a computerized virtual biopsy system for diagnosing metabolicbone diseases.

2The Multiscale Finite Element Approach

Our approach to multiscale finite element analysis is based on a �digital magnifyingglass� that provides continuous transition between macro- and microscales. In theproposed approach, the computational method for the large-scale high-resolutionproblem is adaptive, so that the computational resourcesmay be used efficiently. Thisis essential because a diagnostic tool for physicians must be able to use commonlyavailable workstations rather than supercomputers or clusters. A detailed descriptionof the algorithm is presented in our full paper [13]. The main stages of the algorithmare as follows: (a) acquisition of mCT/mMRI images; (b) 3D reconstruction ofa multiresolution volumetric representation; (c) multiscale finite element analysis,

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and (d) visualization. The multiscale finite element analysis stage is the key moduleof the proposed approach. It consists of following components:

a) Geometric, domain-based, multiresolution model: The multiresolution modelenables continuous bidirectional transition between the microscale and mac-roscale geometric representations of the bone models. The algorithm forgenerating the progressivemultiresolutionmeshes is based on the edge collapsetechnique, which is applied to the DCEL boundary representation [13, 14].

b) Multiscalematerial properties: Mechanical properties of bone tissue vary at eachintermediate scale due to the discontinuity of the stresses, ranging fromhomogeneousmaterial at the macroscale level to highly heterogeneousmaterialat the microscale level. This variance requires recalculation of the materialcompliance matrix at each scale level, as further discussed in the next section.

c) Multiscale computational method: Handling large-scale high-resolution bonemodel on a single processor is technically difficult due to computationalcomplexity and memory demands. Thus, parallel computation is imperative.Domain decomposition methods are natural in parallel computing [15]. Themodel domain is subdivided into subdomains, and the mechanical problem iscomputed in parallel on all these subdomains. Several domain decompositionmethods are discussed in our detailed paper [13].

In this update, we focus on the multiscale material compliance matrix that is thecore of the mechanical model.

3Domain-Based Multiscale Material Properties

The main challenge when dealing with multiscale computational models is topreserve the prominent geometric features and mechanical properties at all struc-tural levels. Since the multiresolution modeling alters the geometry at every struc-tural level, the mechanical properties also change. Therefore, the local mechanicalproperties should be defined for each level. To this end, we applied the representativevolume element (RVE) homogenization approach to each subdomain of the bonemodel. The main stages in the proposed method are as follows:

a) Calculating the progressive multiresolutionmeshes from themicroscale level tothemacroscale level. Eachmesh represents the complete geometricmodel but ata different level of detail (scales).

b) Calculating the material parameters for each scale using the homogenizationtechnique. In this stage, an anisotropy direction of thematerial is calculated andthe tensor is then transformed in this direction. The local material model isassumed to be an isotropic linearmodel, where the tissuemodel is assumed to bean orthotropic material model.

c) Defining the behavior of material parameters as a function of modelporosity.

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d) Calculating the inverse material model that preserves the global materialproperties for all structural levels. Local material properties for each structurallevel are then computed from this model. At this stage, the local material modelis assumed to be orthotropic to compensate for geometry changes.

e) Verifying the model for all structural levels (scales).

The outcomes of stages (a)–(d) are depicted in Figure 1. In (a)–(c), the first rowdepicts three meshes at the macro-, meso-, and microscales. The corresponding

Figure 1 Calculation of the local material properties for the structural levels. Anisotropy anddirection for all levels (a–c) and local material (d) as a function of porosity (circles: Poisson ratio;squares and rhombuses: Young�s moduli in principal directions; triangles: shear modulus).

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anisotropy directions of these meshes are depicted in the second row. The curve onthe polar coordinates represents a ratio between Young�s moduli for a givenorientation. The principal axes of the material directionality can be calculated fromthis curve. At themicroscale, the tissue is anisotropic due to the cancellous geometry.At the macroscale, the tissue material properties are preserved by replacing anisotropic local material with an orthotropic material, as shown in Figure 1d. At thisscale, the tissue becomes isotropic without correcting the local material properties.Such adjustment compensates for the changes in geometry and preserves tissuematerial properties at all scales.

4Summary and Future Work

This update describes a new approach for a 2Dmultiscale finite element analysis of atrabecular bone structure. The proposed method includes reconstruction of thegeometric multiresolution model, thus facilitating continuous geometric transitionfrommacro- to microscale, and vice versa. In addition, the method includes a bidirec-tional transition from micro- to macroscale of the material models based on the RVEhomogenizationapproach.Ourcurrent researchfocusesonextending this techniquetogeneral 3D bone architecture for a computerized 3D virtual biopsy system.

The proposed method can be used as a stand-alone tool for mechanical analysis ofbone tissue and for diagnostic biopsies. Moreover, it can be used as a tool forsimulating bone regeneration [16] and bone in-growth into bone microscaffolds. Afuture extension may include development of new methods at the cell level forfracture healing [17].

Acknowledgments

This study has been partially supported by the Samuel and Anne Tolkowsky chair atthe Technion, by the Irwin and Joan Jacobs Fellowship 2008, and by the Phyllis andJoseph Gurwin Fund for Scientific Advancement at the Technion 2009.

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