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Annals of Biomedical Engineering

, Volume 45, Issue 6, pp 1555–1559 | Cite as

Erratum to: Anisotropic Permeability of Trabecular Bone and Its Relationship to Fabric and Architecture: A Computational Study

  • T. C. Kreipke
  • G. L. Niebur
Erratum
  • 585 Downloads

Erratum to: Annals of Biomedical Engineering (2017) DOI: 10.1007/s10439-017-1805-9

This erratum is to amend the data presented in this paper. The authors have found an error in an algorithm used to post-process the finite element results. This error affected the data in Tables 2, 3, and 4, as well as Figures 2, 3, 5, and 6. The coefficients of the regression models changed by 1 to 2%, and the coefficients of determination were also altered. These changes did not affect the conclusions of the paper. In addition to the data in the tables, the error in the alignment of fabric tensor relative to the permeability reported in the text changed from 29.8 to 30.5 ± 19.6° for the primary eigen vector, from 53.8 to 52.6 ± 26.7° for the secondary eigen vector, and from 49.1 to 48.2 ± 25.0° for the tertiary eigen vector. There were no changes in the statistical analyses of the eigen vector directions.
Figure 2

Permeability approached the Darcy flow regime as the ratio of viscosity to pressure gradient increased, and was within 1% of its steady state permeability at a magnitude of 3 × 10−6 Pa s/Pa m−1, indicated by the dashed line. For a pressure gradient of 20 Pa/mm, the working fluid should have a viscosity greater than 0.06 Pa s.

Figure 3

The permeability magnitude (a) and direction (b), the fabric magnitude (c) and direction (d), and trabecular spacing (f) converged to a steady state value for regions of interest greater than 3 mm (about 10 trabecular spacings). The SMI continued to decrease as the region of interest increased. SMI (e) did not converge, suggesting that it may have more spatial variability than other parameters.

Figure 5

Permeability and fabric were more closely aligned along the direction of the primary eigenvalue compared to the transverse directions. The alignment was not sensitive to the selected boundary conditions. (*,† p < 0.05 compared to other directions within boundary condition, two factor ANOVA).

Figure 6

Eigenvalues of the calculated permeability tensors were fit to a Kozeny-Carman like model with and without fabric (a), a power-law relationship using stepwise regression (b), and a power-law model based on only porosity and fabric (c). A one-for-one line is shown. N = (3 directions x 30 samples)=90 for each relationship. The resulting regression equations and coefficients of determination are given in Tables 3 and 4.

Table 2

Power law coefficients for the principal values of permeability as a function of porosity for each site separated by the respective permeability eigenvalues. Coefficients sharing the same superscript letter within each site are not significantly different from one another (p > 0.05).

Species

Site

\( k_{ii} = A\phi^{b} \left( {{\text{m}}^{2} } \right) \)

R 2

A (x 10−9)

b

Primary

    

 Porcine

Proximal

Neck

6.16a

3.32a

0.63

Central

Neck

8.31b

3.27a

0.34

Distal

Neck

11.7a

2.49a

0.10

 Human

Greater Trochanter

74.1a

9.37b

0.74

Femoral Neck

63.1c

5.10a

0.92

Secondary

    

 Porcine

Proximal

Neck

2.82d

2.98d

0.94

Central

Neck

8.13d

5.37e

0.67

Distal

Neck

13.5d

5.84d

0.77

 Human

Greater Trochanter

67.6d

12.1e

0.95

Femoral Neck

64.6e

9.42e

0.97

Tertiary

    

 Porcine

Proximal

Neck

2.69g

3.92g

0.79

Central

Neck

4.47g

5.18g

0.77

Distal

Neck

5.13g

5.23g

0.84

 Human

Greater Trochanter

25.1g

11.1h

0.71

Femoral Neck

38.9h

8.74h

0.93

Table 3

Kozeny-Carman relationships for permeability based on the constrained flow simulations. S v has units of mm−1 and k has units of m2. All other parameters are dimensionless. Coefficients of determination are for the log-log fit.

Formulation

Model Fit

Adj. R 2

Isotropic

\( {\mathbf{k}} = (10^{ - 6} )\frac{{\phi^{2.88} }}{{9.333S_{v}^{2} }} \)

0.79

Anisotropic

\( {\mathbf{k}} = \left( {10^{ - 6} } \right)\frac{{\phi^{2.85} }}{{1.122S_{v}^{2} }}{\mathbf{H}}^{1.86} \)

0.97

Table 4

Permeability relationships based on stepwise regression or on only porosity and fabric. The regressions were not different between the two boundary conditions studied. k is in units of m 2 and Tb.Sp. is in units of mm 2. All other parameters are dimensionless. Coefficients of determination are for the log-log fit.

Boundary Condition

Formulation

Model Fit

Adj. R2

Unconstrained Flow

Stepwise

k=0.380(10−6)H 1.86 Tb.Sp. 2.00(3-SMI)−1.35

0.91

Porosity/Fabric

k=0.254(10−6) ϕ 7.37 H 1.86

0.81

Constrained Flow

Stepwise

k=0.327(10−6)H 1.85 Tb.Sp. 2.06(3-SMI)−1.32

0.97

Porosity/Fabric

k=0.219(10−6)ϕ 7.33 H 1.84

0.85

Finally, we note that in the methods section, the calculated fluid flux was normalized to the total volume of interest, consistent with experimental measurements.

Copyright information

© Biomedical Engineering Society 2017

Authors and Affiliations

  1. 1.Tissue Mechanics Laboratory, Bioengineering Graduate Program and Department of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre DameUSA
  2. 2.Notre DameUSA

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