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.
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Notes
The nanoporosity between mineral crystals in the bone matrix is not subject of the present study.
Abbreviations
- 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
- d L :
-
log-Euclidean distance
- E 1 :
-
Young’s modulus in radial direction
- E 2 :
-
Young’s modulus in circumferential direction
- E 3 :
-
Young’s modulus in axial direction
- f i :
-
volume fraction of material constituent i
- G 12 :
-
shear modulus in radial-circumferential plane
- G 13 :
-
shear modulus in radial-axial plane
- G 23 :
-
shear modulus in circumferential-axial plane
- G eff :
-
effective shear modulus
- \(\bar{\mathbf{G}}\) :
-
inverse of the acoustic tensor K
- HU :
-
Hounsfield unit
- HU BM :
-
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 \({\mathbb{P}}_{\rm cyl}\))
- K eff :
-
effective bulk modulus
- n :
-
number of eigenvalues of matrix A
- N c :
-
number of material constituents
- \({\mathbb{P}}_{\rm cyl}\) :
-
Hill’s tensor for a cylindrical inclusion in an infinite matrix
- v i :
-
ith eigenvector of matrix A
- δ ij :
-
Kronecker delta (components of second-order unity tensor)
- \({\varvec{\varepsilon}}\) :
-
(macroscopic) strain tensor
- ɛV :
-
volumetric (macroscopic) strain
- :
-
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
- λ i :
-
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
- ρ i :
-
(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 \({\mathbb{P}}_{\rm cyl}\)
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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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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,isoeff and shear modulus G upp,isoeff ,
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,isoeff and G upp,isoeff to the components of the orthotropic stiffness tensor \({\mathbb{C}}_{\rm eff},\)
Equation (20) implies the following relations23 linking the moduli K low,isoeff and G low,isoeff to the components of the orthotropic compliance tensor \({\mathbb{D}_{\rm eff}}={\mathbb{C}}^{-1}_{\rm eff}\)
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Hellmich, C., Kober, C. & Erdmann, B. Micromechanics-Based Conversion of CT Data into Anisotropic Elasticity Tensors, Applied to FE Simulations of a Mandible. Ann Biomed Eng 36, 108–122 (2008). https://doi.org/10.1007/s10439-007-9393-8
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DOI: https://doi.org/10.1007/s10439-007-9393-8