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

Article

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.

Keywords

Computer tomography Finite Element analyses X-ray attenuation coefficients Hounsfield units Bone Anisotropy Inhomogeneity Continuum micromechanics 

Nomenclature

A

6 × 6 matrix representing fourth-order tensor

\({\mathbb{c}}\)

fourth-order stiffness tensor

\(\hat{\mathbf{c}}\)

compressed matrix notation of fourth-order tensor \({{\mathbb{c}}}\) (Kelvin notation)

\({\mathbb{c}}_{\rm {H_{2}O}}\)

stiffness tensor of water

\({{\mathbb{c}}}_{\rm BM}\)

stiffness tensor of (extravascular) solid bone matrix

\({\mathbb{C}}_{\rm eff}\)

effective stiffness tensor of the macroscopic (porous) bone material (bone microstructure)

\({\mathbb{C}}^{\rm low}_{\rm eff}\)

lower bound for effective stiffness tensor

\({\mathbb{C}}^{\rm upp}_{\rm eff}\)

upper bound for effective stiffness tensor

\({\mathbb{C}}_{\rm eff}^{\rm hex}\)

hexagonal average of effective stiffness tensor

\({\mathbb{C}}_{\rm eff}^{\rm TI}\)

transversely isotropic average of effective stiffness tensor

\({\mathbb{C}}_{\rm eff}^{\rm iso}\)

isotropic stiffness closest to effective orthotropic stiffness tensor

\(\hat{\mathbf C}_{\rm eff}\)

compressed matrix notation of orthotropic effective stiffness tensor

\(\hat{\mathbf C}_{\rm eff}^{\rm iso}\)

compressed matrix notation of isotropic stiffness closest to effective orthotropic stiffness tensor

\({{\mathbb{d}}}\)

fourth-order compliance tensor

dL

log-Euclidean distance

E1

Young’s modulus in radial direction

E2

Young’s modulus in circumferential direction

E3

Young’s modulus in axial direction

fi

volume fraction of material constituent i

G12

shear modulus in radial-circumferential plane

G13

shear modulus in radial-axial plane

G23

shear modulus in circumferential-axial plane

Geff

effective shear modulus

\(\bar{\mathbf{G}}\)

inverse of the acoustic tensor K

HU

Hounsfield unit

HUBM

Hounsfield unit of (extravascular) solid bone matrix

\({\mathbb{I}}\)

fourth-order unity tensor

\({\mathbb{J}}\)

volumetric part of \({\mathbb{I}}\)

\({\mathbb{K}}\)

deviatoric part of \({\mathbb{I}}\)

K

second-order acoustic tensor (entering the expression for \({\mathbb{P}}_{\rm cyl}\))

Keff

effective bulk modulus

n

number of eigenvalues of matrix A

Nc

number of material constituents

\({\mathbb{P}}_{\rm cyl}\)

Hill’s tensor for a cylindrical inclusion in an infinite matrix

vi

ith eigenvector of matrix A

δij

Kronecker delta (components of second-order unity tensor)

\({\varvec{\varepsilon}}\)

(macroscopic) strain tensor

ɛV

volumetric (macroscopic) strain

Open image in new window

fourth-order tensor entering the expression for \({\mathbb{P}}_{\rm cyl}\)

\(\Upphi\)

Euler angle in Laws’ integral expression for Hill’s tensor \({\mathbb{P}}_{\rm cyl}\)

ϕ

vascular porosity, related to Haversian canals and intertrabecular space

φ

polar coordinate, used for rotation and averaging of orthotropic material properties

λi

ith eigenvalue of matrix A

μ

X-ray intensity attenuation coefficient of composite material (bone)

\(\mu_{\rm {H_{2}O}}\)

attenuation coefficient of water

μBM

attenuation coefficient of (extravascular) solid bone matrix

ν12

Poisson’s ratio in radial-circumferential plane

ν13

Poisson’s ratio in radial-axial plane

ν23

Poisson’s ratio in circumferential-axial plane

ρi

(real) mass density of material constituent i

θ

spherical coordinate, used for rotation and averaging of orthotropic material properties

\(\Uptheta\)

Euler angle in Laws’ integral expression for Hill’s tensor \({\mathbb{P}}_{\rm cyl}\)

\({\varvec{\xi}}\)

unit vector in Laws’ integral expression for Hill’s tensor \({\mathbb{P}}_{\rm cyl}\)

Notes

Acknowledgments

The reported Finite Element simulations are part of a research project on the human mandible, which is under the medical courtesy of Robert Sader, Frankfurt University, Germany, and Hans-Florian Zeilhofer, Basle University Hospital, Switzerland. The authors are grateful for the implantology-related medical advice of Stefan Stuebinger, Frankfurt University, Germany, and for the support of Sherin Torabia in the initial phase of our study on relations between CT data and elastic properties of bony organs, in the course of her Master’s thesis completed at Vienna University of Technology, under the supervision of the first author.84,85

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Copyright information

© Biomedical Engineering Society 2007

Authors and Affiliations

  • Christian Hellmich
    • 1
  • Cornelia Kober
    • 2
  • Bodo Erdmann
    • 3
  1. 1.Vienna University of Technology (TU Wien)ViennaAustria
  2. 2.Osnabrück University of Applied SciencesOsnabrueckGermany
  3. 3.Zuse InstituteBerlin-DahlemGermany

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