Osteoporosis is a frequent skeletal disease characterised both by loss of bone mineral mass and deterioration of
cancellous bone micro-architecture. It can be caused by mechanical disuse, estrogen deficiency or natural age-related
resorption process. Numerical analysis of high-resolution images of the trabecular network is recognised as a powerful
tool for assessment of structural characteristics. Using μCT images of 73 thoracic and 78 lumbar human vertebral
specimens in vitro with isotropic resolution of 26μm we simulate bone atrophy as random resorption of bone surface
voxels. Global morphological and topological characteristics provided by four Minkowski Functionals (MF) are
calculated for two numerical resorption models with and without conservation of global topological connectivity of the
trabecular network, which simulates different types of bone loss in osteoporosis, as it has been described in males and
females. Diagnostic performance of morphological and topological characteristics as a function of relative bone loss is
evaluated by a correlation analysis with respect to experimentally measured Maximum Compressive Strength (MCS). In
both resorption models the second MF, which coincides with bone surface fraction BS/TV, demonstrates almost constant
value of Pearson's correlation coefficient with respect to the relative bone loss ▵BV/TV. This morphological
characteristic does not vary considerably under age-related random resorption and can be used for predicting bone
strength in the elderly. The third and fourth MF demonstrate an increasing correlation coefficients with MCS after
applying random bone surface thinning without preserving topological connectivity, what can be used for improvement
of evaluation of the current state of the structure.
KEYWORDS: Bone, Neck, 3D image processing, Distance measurement, In vitro testing, Image segmentation, Control systems, Anisotropy, Adaptive optics, Medical research
According to Wolff's law bone remodels in response to the mechanical stresses it experiences so as to produce a
minimal-weight structure that is adapted to its applied stresses. Consequently, the inner bone structure should show signs
of adaptation to external forces acting on the bone. To test this paradigm, we investigate the relations between bone
volume and structure for the trabecular bone using 3D μCT images taken from two different sites in the femur in vitro,
namely from the femoral neck (88 specimens) and femoral trochanter (126 specimens). We determine the local structure
of the trabecular network as well as its alignment with the direction of the external force acting on the bone by
calculating isotropic (α) and anisotropic scaling indices (αz). Comparing global structure measures derived from the
scaling indices (mean, variance) with the bone mass (BV/TV) we find that all correlations obey very accurately power
laws with scaling exponents of 0.48 and 0.45 (<α>), -1.45 and -1.59 (var(μz)), 0.50 and 0.44 (<α>) and -1,47 and -1.32
(var(μz)) (neck and trochanter respectively). Thus, the relations for the isotropic scaling indices turn out to be siteindependent,
albeit the mechanical stress to which the femoral neck is exposed is much larger than that for the
trochanter. We find, however, differences in the degree of alignment of the trabeculae as reflected by the moments of the
distribution of the anisotropic scaling indices. In summary, the mass-structure scaling relations of the bone probes taken
from the two different sites of the femur show surprisingly small variations. Thus, a naïve interpretation of Wolff's law
may not universally valid.
We analyze μ-CT tomographic images of human trabecular bone in vitro. We consider a sample consisting
of 201 bone specimens harvested from six different skeletal sites with bone fraction in the range BV/TV ε
[0.04, 0.075]. Using the local characterization of the bone trabecular network given by isotropic and anisotropic
scaling indices, we apply classification algorithms in order to reveal structural similarities in the sample. The
classification procedures based on isotropic and anisotropic scaling indices lead to different clustering solutions.
This comparison helps revealing interesting site specific structural features connected to the intrinsic anisotropy
of the trabecular network.
We investigate the utility of amplitude remapping of magnetic resonance (MR)-images for making the analysis
of such images more independent of the MR-device, the selected sequence, and its parameters. To this end,
we analyze the morphological structure of trabecular bones using weighted scaling indices and Minkowski
functionals in the context of osteoporosis. After remapping the amplitude distribution of MR-images onto
a normal distribution with zero mean and unit variance, we study how the diagnostic performance of the
structure measures is affected by this remapping. The diagnostic performance of the scaling index method
is stable under the remapping for both spin echo (SE) and gradient echo (GE) sequences: The area under
curve (AUC) value from the ROC analysis changes only slightly from 0.76 (original image) to 0.74 (remapped
image) for the SE sequence and from 0.78 to 0.77 for the GE sequence. For the Minkowski functionals, the
diagnostic performance suffers significantly for the SE sequence, whereas it is much more robust for the GE
sequence. Therefore, the scaling index method should be the method of choice when analyzing MR-images
after amplitude remapping. We also find that in the scaling index analysis, the remapping makes the results
much more consistent between the SE and the GE sequence by bringing the histograms of the scaling indices
closer together. Thus, the amplitude remapping can be used as a first step to standardize the scaling index
analysis between different sequences of an MRI device.
We analyse μ-CT tomographic images of human trabecular bone in vitro. We consider a sample consisting of
201 bone specimens harvested from six different skeletal sites within a narrow range of bone fraction values.
Using the characterization of the trabecular bone network given by local Minkowski Functionals, we apply
classification algorithms in order to reveal structural similarities in the sample. Clusters show some interesting
specific structural features, like compact, porous, and fragmented structures. The contribution of the different
skeletal sites to these clusters indicate some variability due to intrinsic structural differences of the specific
skeletal site.
The quantitative characterization of images showing tissue probes being visualized by e.g. CT or MR is of great interest
in many fields of medical image analysis. A proper quantification of the information content in such images can be
realized by calculating well-suited texture measures, which are able to capture the main characteristics of the image
structures under study. Using test images showing the complex trabecular structure of the inner bone of a healthy and
osteoporotic patient we propose and apply a novel statistical framework, with which one can systematically assess the
sensitivity of texture measures to scale-dependent higher order correlations (HOCs). To this end, so-called surrogate
images are generated, in which the linear properties are exactly preserved, while parts of the higher order correlations (if
present) are wiped out in a scale dependent manner. This is achieved by dedicated Fourier phase shuffling techniques.
We compare three commonly used classes of texture measures, namely spherical Mexican hat wavelets (SMHW),
Minkowski functionals (MF) and scaling indices (SIM). While the SMHW were sensitive to HOCs on small scales
(Significance S=19-23), the MF and SIM could detect the HOCs very well for the larger scales (S = 39 (MF) and S = 29
(SIM)). Thus the three classes of texture measures are complimentary with respect to their ability to detect scaledependent
HOCs. The MF and SIM are, however, slightly preferable, because they are more sensitive to HOCs on length
scales, which the important structural elements, i.e. the trabeculae, are considered to have.
The assessment of trabecular bone microarchitecture by numerical analysis of high resolution magnetic resonance
(HRMR) images provides global and local structural characteristics, which improve the understanding of the
progression of osteoporosis and its diagnosis. In the present work we apply the finite elements method (FEM), which
models the biomechanical behaviour of the bone, the scaling index method (SIM), which describes the topology of the
structure on a local level, and Minkowski Functionals (MF), which are global topological characteristics, for analysing
3D HRMR images of 48 distal radius specimens in vitro. Diagnostic performance of texture measures derived from the
numerical methods is compared with regard to the prevalence of vertebral fractures. Both topological methods show
significantly better results than those obtained using bone mineral density (BMD) measurement and the failure load
estimated by FEM. The receiver operating characteristic analysis for differentiating subjects with and without fractures
reveals area under the curve of 0.63 for BMD, 0.66 for maximum compressive strength as determined in a
biomechanical test, 0.72 for critical load estimated by FEM, 0.79 for MF4 and 0.86 for SIM, i.e. local topological
characteristics derived by SIM suit best for diagnosing osteoporosis. The combination of FEM and SIM on tissue level
shows that in both weak and strong bones the plate-like substructure of the trabecular network are the main load bearing
part of the inner bone and that the relative amount of plates to rods is the most important characteristic for the
prediction of bone strength.
According to Wolff's law bone remodels in response to the mechanical stresses it experiences so as to produce a
minimal-weight structure that is adapted to its applied stresses. Here, we investigate the relations between bone volume
and structure for the trabecular bone using 3D μCT images taken from different skeletal sites in vitro, namely from the
distal radii (96 specimens), thoracic (73 specimens) and lumbar vertebrae (78 specimens). We determine the local
structure of the trabecular network by calculating isotropic and anisotropic scaling indices (α, αz). These measures have
been proven to be able to discriminate rod- from sheet-like structures and to quantify the alignment of structures with
respect to a preferential direction as given by the direction of the external force. Comparing global structure measures
derived from the scaling indices (mean, standard deviation) with the bone mass (BV/TV) we find that all correlations
obey very accurately power laws with scaling exponents of 0.14, 0.12, 0.15 (<α⪆), -0.2, -017, -0.17 (σ(αz)), 0.09, 0.05,
0.07 (⪅αz⪆) and -0.20, -0.11 ,-0.13 (σ(αz)); distal radius, thoracic vertebra and lumbar vertebra respectively. Thus, these
relations turn out to be site-independent, albeit the mechanical stresses to which the bones of the forearm and the spine
are exposed, are quite different. The similar alignment might not be in agreement with a universal validity of Wolff's
law. On the other hand, such universal power law relations may allow to develop additional diagnostic means to better
assess healthy and osteoporotic bone.
The quantitative characterization of tissue probes as visualized by CT or MR is of great interest in many fields of
medical image analysis. A proper quantification of the information content in such images can be realized by calculating
well-suited texture measures, which are able to capture the main characteristics of the image structures under study.
Using test images showing the complex trabecular structure of the inner bone of a healthy and osteoporotic patient we
propose and apply a novel statistical framework, with which one can systematically assess the sensitivity of the chosen
texture measures to higher order correlations (HOCs), i.e. correlations not being captured by linear methods like the
power spectrum. To this end, so-called surrogate images are generated, in which the linear properties are preserved,
while parts or all higher order correlations are wiped out. This is achieved by dedicated Fourier phase shuffling
techniques. We compare three commonly used classes of texture measures, namely spherical Mexican hat wavelets
(SMHW), Minkowski functionals (MF) and scaling indices (SIM). While the SMHW yield only very poor sensitivity to
HOCs in both cases, the MF and SIM could detect the HOCs very well with significance up to S = 320σ (MF) and S =
150σ (SIM). The relative performance of the MF and SIM differed significantly for the healthy and osteoporotic bone.
Thus, MF and SIM are preferable for a proper quantification of the bone structure. They depict complementary aspects
of it and thus should both be used for characterising the trabecular bone.
Spine fractures are the most frequent complication of osteoporosis, a disease characterized by low bone mass and
structural deterioration of bone tissue. In case of the spine, the trabecular network plays the main role in load carrying
and distribution. A correct description of mechanical properties of this bone structure helps to differentiate between
strong and weak bones and can be useful for fracture prediction and treatment monitoring. By means of the finite
element method (FEM), applied to μCT images, we modelled biomechanical processes in probes during loading and
correlated the estimated failure load with the maximum compressive strength (MCS), obtained in real biomechanical
tests. We studied a sample of 151 specimens taken from the trabecular part of human vertebrae in vitro, visualised using
μCT imaging at an isotropic resolution of 26μm and tested by uniaxial compression. Besides the standard way of
estimating failure load, which takes into account only strong micro-fractures, we also included small micro-fractures,
what improved the correlation with MCS (Pearson's correlation coefficient r=0.78 vs. r=0.58). This correlation
coefficient was larger than that for both the standard morphometric parameters (r=0.73 for bone volume fraction) and for
texture measures defined by the local (an-) isotropic scaling indices method (r=0.55) and Minkowski Functionals
(r=0.61). However, the performance of the FEM was different for subsamples selected according to the MCS value. The
correlation increased for strong specimens (r=0.88), slightly decreased for weak specimens (r=0.68) and markedly
dropped for specimens with medium MCS, e.g. between 60<MCS<120, r=0.26.
Osteoporosis is bone disease which leads to low bone mass and the deterioration of the bone micro-architecture.
Rarefied bone structures are more susceptible to fractures which are the worst complications of osteoporosis. Bone
mineral density is considered to be the standard technique for predicting the bone strength and the effects of drug
therapy. However, other properties of the bone like the trabecular structure and connectivity may also contribute.
Here, we analyze μ-CT tomographic images for a sample of 151 specimens taken from human vertebrae in vitro.
Using the local structural characterization of the bone trabecular network given by isotropic and anisotropic
scaling indices, we generate structural decompositions of the μ-CT image and quantify the resulting patterns
applying topological measures, namely the Minkowski Functionals (MF). The values of the MF are then used to
assess the biomechanical properties of trabecular bone via a correlation analysis. Biomechanical properties were
quantified by the maximum compressive strength calculated in an uniaxial compression test. We compare our
results with those obtained using standard global histomorphometric parameters and the bone fraction BV/TV .
Results obtained using structural decompositions obtained from anisotropic scaling indices were superior to those
given by isotropic scaling indices. The highest correlation coefficient (r = 0.72) was better than those obtained
for the standard global histomorphometric parameters and only comparable with the one given by BV/TV. Our
results suggest that plate-like and dense column-like structures aligned along the direction of the external force
play a relevant role for the prediction of bone strength.
KEYWORDS: Signal to noise ratio, Bone, Magnetic resonance imaging, 3D metrology, Spatial resolution, Image resolution, 3D image processing, Image analysis, Visualization, Image quality
3.0 Tesla MRI devices are becoming popular in clinical applications since they render images with a higher signal-tonoise ratio than the former 1.5 Tesla MRI devices. Here, we investigate if higher signal-to-noise ratio can be beneficial for a quantitative image analysis in the context of bone research. We performed a detailed analysis of the effect of noise on the performance of 2D morphometric linear measures and a 3D nonlinear measure with respect to their correlation with biomechanical properties of the bone expressed by the maximum compressive strength. The performance of both 2D and 3D texture measures was relatively insensitive to superimposed artificial noise. This finding suggests that MR sequences for visualizing bone structures at 3T should rather be optimized to spatial resolution (or scanning time) than to signal-to-noise ratio.
High resolution magnetic resonance (HRMR) imaging can reveal major characteristics of trabecular bone. The quantification of this trabecular micro architecture can be useful for better understanding the progression of osteoporosis and improve its diagnosis. In the present work we applied the scaling index method (SIM) and Minkowski Functionals (MF) for analysing tomographic images of distal radius specimens in vitro. For both methods, the correlation with the maximum compressive strength (MCS) as determined in a biomechanical test and the diagnostic performance with regard to the spine fracture status were calculated. Both local SIM and global MF methods showed significantly better results compared to bone mineral density measured by quantitative computed tomography. The receiver operating characteristic analysis for differentiating fractured and non-fractured subjects revealed area under the curve (AUC) values of 0.716 for BMD, 0.897 for SIM and 0.911 for MF. The correlation coefficients with MCS were 0.6771 for BMD, 0.843 for SIM and 0.772 for MF. We simulated the effect of perturbations, namely noise effects and intensity variations. Overall, MF method was more sensitive to noise than SIM. A combination of SIM and MF methods could, however, increase AUC values from 0.85 to 0.89 and correlation coefficients from 0.71 to 0.82. In conclusion, local SIM and global MF techniques can successfully be applied for analysing HRMR image data. Since these methods are complementary, their combination offers a new possibility of describing MR images of the trabecular bone, especially noisy ones.
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