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The data published here is subject to copyright of
Institute of Fundamental Technological Research.
You may use numerical data downloaded from this page for
your non-commercial research purposes provided that, whenever you publish
your results related to this material, you make reference
to [1] (cubic or prismatic cells)
or [2] (qcs cell)
and to this site. |
Files containing tabularized results of analysis for equivalent trabecular microstructures can be downloaded (see notes below).
| te=0.6 | te=0.8 | te=1.0 | te=1.2 | te=1.4 | |
| cubic cell | elastic constants | elastic constants | elastic constants (cubic symmetry) |
elastic constants | elastic constants |
| prismatic cell | elastic constants | elastic constants | elastic constants (transv. isotropy) |
elastic constants | elastic constants |
| qcs cell |
elastic constants fabric data |
elastic constants fabric data |
elastic constants fabric data (transv. isotropy) |
elastic constants fabric data |
elastic constants fabric data |
The files are in CSV format, with comma ',' as the column separator
and with dot '.' as the decimal point.
Each record (line) in a file corresponds to one microstructural instance
(a shape parameter set). First data in the record are values of the
shape parameters. Subsequent columns contain values of apparent density,
surface density, and
particular material constants. The qcs data additionally contain values of
principal fabric measures (MIL, VO, and square-rooted SLD)
gathered in separate files.
Only intercepts contained within trabeculae are considered in MIL calculations.
Column descriptions are given in the first record of each file.
Note 1 (on different sets of tabularized data). The data downloaded through links in the table above have been recomputed in 2020 and thus they are newer than those described in the papers [1],[2]. They were obtained for somewhat enhanced cell geometry and mesh generator and they may thus slightly differ numerically from the older ones. Only the fabric data were not updated and are fully compatible with those described in [2]. The readers interested in downloading precisely the same numerical data as those reported in [1] should rather use the following links to the historical data for cubic and prismatic cell models (both obtained for te=1.0). To get the data fully compatible with [2] for qcs cell please use the following links for te=0.6, te=0.8, te=1.0, te=1.2 and te=1.4.
Note 2 (on smoothness of results). Even though changes of tc, th, tv and te are continuous, the corresponding changes in f.e. meshes may not be so - to the contrary, mesh topology may change from instance to instance. This inevitably leads to some "roughness" of result graphs. Even the monotonicity of results, expected for most of the measured material constants, is locally impaired at isolated points. This may cause numerical problems if the data is used "as is" in computations in which gradients of the data are important. Thus, it is advised to perform some smoothening procedure before employing the data in computations. Smoothening must be done with caution, for instance users are strongly discouraged from high-order polynomial approximations (as they do not preserve expected monotonicities and yield spurious local extrema).
Note 3 (on dimensionality and scaling of results).
The equivalent microstructure cells are dimensionless.
So are the computed and tabularized data.
Some of them are dimensionless by the definition and can be used
with no scaling;
let us mention here: the apparent density,
the Poisson ratios (vij), and the
principal values of volume orientation (VOi).
Values of elastic moduli
(Cij, Ei, Gij)
need only to be scaled by the actual value of the tissue
elastic modulus E
(unit value of the modulus was assumed in the computations).
Principal values of the mean intercept length (MILi) and
the star length distribution (SLDi)
have the dimension of length unit and need to be multiplied by the quotient
of the real cell size to the model unit dimension.
The surface density has the dimension of inverse length unit, thus its scaling
consists in dividing the tabularized result values by the mentioned quotient.
Since, for a given real trabecular microstructure,
it is not possible to define and measure a quantity
that would correspond to the unit cell dimension,
determination of the quotient may be problematic.
It is, though, possible to measure the mean trabecular spacing,
understood here as a mean distance between central axes of
neighboring trabeculae (some authors prefer to report the `trabecular number',
Tb.N, which is exactly inverse of the trabecular spacing defined above).
The mean spacing is also easy to measure for the equivalent cell models
and is displayed in the table below for each cell type.
Thus, for instance, if trabecular spacing measured for a real microstructure
is 0.75mm (0.00075m) and we try to model this real microstructure with
the tranversely isotropic `qcs' cell (te=1 and the model spacing is 1 unit)
for which we have computed e.g. the dimensionless principal value of MIL1=0.22,
then the corresponding scaled MIL1 value will be obtained by multiplying
0.22 by the real spacing 0.75mm which gives 0.165mm.
If, however, te is different than 1 then we have to divide
the result by the non-unit model spacing taken from the table below.
| cubic cell | prismatic cell | qcs cell | |
| volume at te=1 (in unit3) | 1 | 0.75*√3 ≈ 1.299 | 0.5*√2 ≈ 0.707107 |
| volume (in unit3) | te | te*0.75*√3 ≈ 1.299*te | te*0.5*√2 ≈ 0.707107*te |
| trabecular spacing at te=1 (in units) | 1 | 1 | 1 |
| mean trabecular spacing (in units) | (2+te)/3 | [3 + te + √(3+te2)] / 6 | [ te + 0.5*√(3+te2) + 2*te / √(1+3*te2) ] / 3 |
[1]
P. Kowalczyk. Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells.
Journal of Biomechanics, 36:961-972, 2003.
[2]
P. Kowalczyk. Orthotropic properties of cancellous bone modelled as parameterized cellular material.
Computer Methods in Biomechanics and Biomedical Engineering, 9:135-147, 2006.