Micromechanics-Based Conversion of CT Data into Anisotropic Elasticity Tensors, Applied to FE Simulations of a Mandible

Abstract

Computer Tomographic (CT) image data have become a standard basis for structural analyses of bony organs. In this context, regression functions between stiffness components and Hounsfields units (HU) from CT, related to X-ray attenuation coefficients, are widely used for the definition of the (actually inhomogeneous and anisotropic) material behavior inside the organ. Herein, we suggest to derive the functional dependence of the fully orthotropic stiffness tensors on the Hounsfield units from the physical information contained in the X-ray attenuation coefficients: (i) Based on voxel average rules for the X-ray attenuation coefficients, we assign to each voxel the volume fraction occupied by water (marrow) and that occupied by solid bone matrix. (ii) By means of a continuum micromechanics representation for bone, which is based on voxel-invariant (species and whole bone-specific) stiffness properties of solid bone matrix and of water, we convert the aforementioned volume fractions into voxel-specific orthotropic stiffness tensor components. The micromechanics model, in combination with the average rule for X-ray attenuation coefficients, predicts a quasi-linear relationship between axial Young’s modulus and HU, and highly nonlinear relationships for both circumferential and radial Young’s moduli as well as for the shear moduli in all principal material directions. Corresponding whole-organ Finite Element (FE) analyses of a partially edentulous human mandible characterized by atrophy of the alveolar ridge show that volumetric strain concentrations/peaks within the organ are decreased when considering material anisotropy, and increased when considering material inhomogeneity.

Appendices

Appendix I: Time and Voxel-Independent Mineral Content of Solid Bone Matrix

Computerized quantitative contact microradiography14 revealed that the mineral content of the solid bone matrix (also called degree of mineralization of bone DMB) averaged over whole human iliac bones does not change with age, and (see Fig. 3 of the aforementioned reference) that the mineral content of the solid bone matrix averaged over millimeter-sized domains does not significantly vary in space. Quantitative backscattered electron imaging76 revealed that the mineral content of the solid bone matrix in human trabecular iliac and vertebral samples, averaged over an area of some square millimeters, does not vary with age. Raman microscopy1 revealed that the mineral content of the solid bone matrix averaged over human femoral cortices does not significantly change with age. Synchrotron Micro Computer Tomography studies16 revealed that the mineral content of the solid bone matrix (also called tissue mineralization MIN) averaged over an entire human radius is equal to that averaged over a 1-mm-thick external layer.

Appendix II: Bulk and Shear Moduli Related to Isotropic Voigt and Reuss Averages

The isotropic Voigt bound ${\mathbb{C}}^{\rm upp,iso}_{\rm eff},$ Eq. (19), can be given in terms of bulk modulus K ^upp,iso_eff and shear modulus G ^upp,iso_eff ,

$$ {\mathbb{C}}^{\rm upp,iso}_{\rm eff} = 3 {\mathbb{J}} K_{\rm eff}^{\rm upp,iso} + 2 {\mathbb{K}} G_{\rm eff}^{\rm upp,iso} $$

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where ${\mathbb{K}}$ is the deviatoric part of the fourth-order unity tensor ${\mathbb{I}},\ {\mathbb{K}}={\mathbb{I}}- {\mathbb{J}}.$ Equation (19) implies the following relations23 linking the moduli K ^upp,iso_eff and G ^upp,iso_eff to the components of the orthotropic stiffness tensor ${\mathbb{C}}_{\rm eff},$

$$ \begin{aligned} K_{\rm eff}^{\rm upp,iso}=& \frac{1}{9} (C_{\rm eff,1111}+C_{\rm eff,2222}+C_{\rm eff,3333}) \\ &+\frac{2}{9} (C_{\rm eff,1122}+C_{\rm eff,2233}+C_{\rm eff,3311})\\ \end{aligned} $$

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$$ \begin{aligned} G_{\rm eff}^{\rm upp,iso}=& \frac{1}{15} (C_{\rm eff,1111}+C_{\rm eff,2222}+C_{\rm eff,3333} - C_{\rm eff,1122}-C_{\rm eff,2233}-C_{\rm eff,3311})\\ & +\frac{1}{5} (C_{\rm eff,2323}+C_{\rm eff,1313}+C_{\rm eff,1212}) \end{aligned} $$

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Equation (20) implies the following relations23 linking the moduli K ^low,iso_eff and G ^low,iso_eff to the components of the orthotropic compliance tensor ${\mathbb{D}_{\rm eff}}={\mathbb{C}}^{-1}_{\rm eff}$

$$ \begin{aligned} K_{\rm eff}^{\rm low,iso}=& \left\{(D_{\rm eff,1111}+D_{\rm eff,2222}+D_{\rm eff,3333}) \right. \\ & \left. +2 (D_{\rm eff,1122}+D_{\rm eff,2233}+D_{\rm eff,3311}) \right\}^{-1} \end{aligned} $$

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$$ \begin{aligned} G_{\rm eff}^{\rm low,iso}=& \left\{ (4/15) (D_{\rm eff,1111}+D_{\rm eff,2222}+D_{\rm eff,3333} - \right.\\ & D_{\rm eff,1122}-D_{\rm eff,2233}-D_{\rm eff,3311})\\ & \left. +(4/5) (C_{\rm eff,2323}+C_{\rm eff,1313}+C_{\rm eff,1212}) \right\}^{-1} \end{aligned} $$

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