On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging

Akkari, H.; Bhouri, Imen; Dubois, Patrick; Bedoui, Mohamed Hédi

doi:10.1051/mmnp:2008081

Cited by 13 publications

(12 citation statements)

References 18 publications

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“…Transforming a 2D projected image into a three‐dimensional (3D) structure is a mathematical challenge . However, several kinds of texture analysis methods, such as Fourier conversion, fractal analysis, and run‐length analysis, have been proposed as indirect measurements of 3D trabecular bone microarchitecture . These methods analyze trabecular structures according to different statistical properties of pixels in relation to density, computing a feature strongly related to the 3D parameters of the projected trabecular bone.…”

Section: Estimation Of 3d Indices From a 2d Projected Imagementioning

confidence: 99%

“…(41,42) However, several kinds of texture analysis methods, such as Fourier conversion, fractal analysis, and run-length analysis, have been proposed as indirect measurements of 3D trabecular bone microarchitecture. (41)(42)(43)(44)(45) These methods analyze trabecular structures according to different statistical properties of pixels in relation to density, computing a feature strongly related to the 3D parameters of the projected trabecular bone. These techniques provide a global estimate of bone quality, but they are not direct physical measurements of trabecular parameters.…”

Section: Estimation Of 3d Indices From a 2d Projected Imagementioning

confidence: 99%

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Trabecular Bone Score: A Noninvasive Analytical Method Based Upon the DXA Image

Silva

Leslie

Resch³

et al. 2014

Journal of Bone and Mineral Research

706

487

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The trabecular bone score (TBS) is a gray-level textural metric that can be extracted from the two-dimensional lumbar spine dualenergy X-ray absorptiometry (DXA) image. TBS is related to bone microarchitecture and provides skeletal information that is not captured from the standard bone mineral density (BMD) measurement. Based on experimental variograms of the projected DXA image, TBS has the potential to discern differences between DXA scans that show similar BMD measurements. An elevated TBS value correlates with better skeletal microstructure; a low TBS value correlates with weaker skeletal microstructure. Lumbar spine TBS has been evaluated in cross-sectional and longitudinal studies. The following conclusions are based upon publications reviewed in this article: 1) TBS gives lower values in postmenopausal women and in men with previous fragility fractures than their nonfractured counterparts; 2) TBS is complementary to data available by lumbar spine DXA measurements; 3) TBS results are lower in women who have sustained a fragility fracture but in whom DXA does not indicate osteoporosis or even osteopenia; 4) TBS predicts fracture risk as well as lumbar spine BMD measurements in postmenopausal women; 5) efficacious therapies for osteoporosis differ in the extent to which they influence the TBS; 6) TBS is associated with fracture risk in individuals with conditions related to reduced bone mass or bone quality. Based on these data, lumbar spine TBS holds promise as an emerging technology that could well become a valuable clinical tool in the diagnosis of osteoporosis and in fracture risk assessment.

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Section: Estimation Of 3d Indices From a 2d Projected Imagementioning

confidence: 99%

Section: Estimation Of 3d Indices From a 2d Projected Imagementioning

confidence: 99%

Trabecular Bone Score: A Noninvasive Analytical Method Based Upon the DXA Image

Silva

Leslie

Resch³

et al. 2014

Journal of Bone and Mineral Research

706

487

View full text Add to dashboard Cite

show abstract

“…Although there is a known relationship between d f,3D and d f,slice for the Hausdorff dimension, an alternative approach to describing fractal objects that is computationally challenging to calculate, there is no such established relationship for the Minkowski-Bouligand dimension (61,62). As such, one cannot immediately compare the fractal dimension obtained through 2D confocal imaging to those predicted from percolation theory.…”

Section: Comparison Of Structural Features To Critical Gelation Predictionsmentioning

confidence: 99%

Confocal Rheology Probes the Structure and Mechanics of Collagen through the Sol-Gel Transition

Tran-Ba

Lee

Zhu

et al. 2017

Biophysical Journal

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Fibrillar type I collagen-based hydrogels are commonly used in tissue engineering and as matrices for biophysical studies. Mechanical and structural properties of these gels are known to be governed by the conditions under which fibrillogenesis occurs, exhibiting variation as a function of protein concentration, temperature, pH, and ionic strength. Deeper understanding of how macroscopic structure affects viscoelastic properties of collagen gels over the course of fibrillogenesis provides fundamental insight into biopolymer gel properties and promises enhanced control over the properties of such gels. Here, we investigate type I collagen fibrillogenesis using confocal rheology-simultaneous confocal reflectance microscopy, confocal fluorescence microscopy, and rheology. The multimodal approach allows direct comparison of how viscoelastic properties track the structural evolution of the gel on fiber and network length scales. Quantitative assessment and comparison of each imaging modality and the simultaneously collected rheological measurements show that the presence of a system-spanning structure occurs at a time similar to rheological determinants of gelation. Although this and some rheological measures are consistent with critical gelation through percolation, additional rheological and structural properties of the gel are found to be inconsistent with this theory. This study clarifies how structure sets viscoelasticity during collagen fibrillogenesis and more broadly highlights the utility of multimodal measurements as critical test-beds for theoretical descriptions of complex systems.

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“…Although equation 11predicts that the power-law exponent m of the size dependence of strength is inversely related to the exponent q of the mesh-length distribution, actual experimental verification is difficult since the q involved is for the mesh-length distribution in 3-D but the initial dislocation pattern can only be conveniently imaged in the transmission electron microscope (TEM) as 2-D projections. Even if the TEM tomography technique is used, the sampling is still from thin slices of the 3-D sample, and so is still based on 2-D. Zaiser et al (Zaiser et al, 1999) assumed that the fractal dimension of a 3-D dislocation network is that of its 2-D projection plus one, but more recent work has shown that there is no general relation between the dimensions of a 3-D fractal and its 2-D projections (Akkari et al, 2008). Here, in order to proceed, we take a simplification step by arguing that, instead of equation 7, the ensemble yielding rate ̇ can be estimated by sampling from a 2-D projection of the network.…”

Section: Sampling From 2-d Projections Of 3-d Dislocation Networkmentioning

confidence: 99%

Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength

Ngan

2013

Journal of the Mechanics and Physics of Solids

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It is by now well-known that micron-sized metallic crystals exhibit a smaller-beingstronger size effect: the yield strength σ varies with specimen size D approximately as a power law ~− , and the exponent m has been found to vary within a range of ~0.3 to ~1.0 for different metals. However, little is known about why such a power law comes into play, and what determines the actual value of the exponent m involved. This work shows that if the yield strength is determined by the Taylor interaction mechanism within the initial dislocation network, then for the size dependence of strength to be of the power-law relation observed, it is necessary for the mesh lengths L of the dislocation network to be power-law distributed, i.e. ()~−. In such a case, the exponent m of the size effect is predicted to be inversely proportional to the sum of q the exponent of the mesh-length distribution and n the exponent of the dislocation velocity vs stress law. To verify these predictions, compression experiments on aluminum micro-pillars with different pre-strains from 0 to 15% were carried out. The different pre-strains led to different initial dislocation networks, as well as different exponent m in the size dependence of strength. Box-counting analyses of transmission electron micrographs of the initial dislocation networks showed that the 2-D projected dislocation patterns were approximate fractals. On increasing pre-strain, the exponent m for the size dependence of strength was found to decrease while the fractal dimension of the initial dislocation patterns increased, thus verifying the inverse relationship between the two quantities. These findings show that the commonly observed power-law scaling of strength with size is due to an approximate power-law distribution of the initial dislocation mesh lengths, which also appears to be a robust feature in deformed metals. Furthermore, for a given metal, it is the exponent q of the initial mesh-length distribution which determines the value of the exponent m in the size dependence of strength.

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On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging

Cited by 13 publications

References 18 publications

Trabecular Bone Score: A Noninvasive Analytical Method Based Upon the DXA Image

Trabecular Bone Score: A Noninvasive Analytical Method Based Upon the DXA Image

Confocal Rheology Probes the Structure and Mechanics of Collagen through the Sol-Gel Transition

Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength

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