Tensor Shape Research Articles

Dendrimer building toolkit: Model building and characterization of various dendrimer architectures

We have developed a graphical user interface based dendrimer builder toolkit (DBT) which can be used to generate the dendrimer configuration of desired generation for various dendrimer architectures. The validation of structures generated by this tool was carried out by studying the structural properties of two well known classes of dendrimers: ethylenediamine cored poly(amidoamine) (PAMAM) dendrimer, diaminobutyl cored poly(propylene imine) (PPI) dendrimer. Using full atomistic molecular dynamics (MD) simulation we have calculated the radius of gyration, shape tensor and monomer density distribution for PAMAM and PPI dendrimer at neutral and high pH. A good agreement between the available simulation and experimental (small angle X-ray and neutron scattering; SAXS, SANS) results and calculated radius of gyration was observed. With this validation we have used DBT to build another new class of nitrogen cored poly(propyl ether imine) dendrimer and study it's structural features using all atomistic MD simulation. DBT is a versatile tool and can be easily used to generate other dendrimer structures with different chemistry and topology. The use of general amber force field to describe the intra-molecular interactions allows us to integrate this tool easily with the widely used molecular dynamics software AMBER. This makes our tool a very useful utility which can help to facilitate the study of dendrimer interaction with nucleic acids, protein and lipid bilayer for various biological applications.

Long-term declarative memory deficits in diffuse TBI: Correlations with cortical thickness, white matter integrity and hippocampal volume

We investigated structural brain damage in subjects who had suffered severe and diffuse traumatic brain injury (TBI), and examined its relationship with declarative memory impairment. Cortical thickness, diffusion tensor imaging (DTI), and volumetric and shape data of the hippocampus were assessed in a group of 26 adults with severe TBI in the chronic stage and 22 healthy matched controls. Declarative memory was evaluated by Rey’s Auditory Verbal Learning Test (RAVLT). TBI patients performed significantly worse than controls on all RAVLT measures. The group comparison for cortical thickness and DTI revealed a pattern of widespread atrophy in TBI patients. In the TBI group DTI measures correlated with cortical thickness in the prefrontal and parietal regions, including the precuneus. Declarative memory correlated with both cortical thickness and DTI measures. However, although hippocampal volume was significantly decreased in TBI patients, no correlations were found. Multiple regression analysis of all the structural measures revealed that decreases in Fractional anisotropy (FA) and thinning of the left parietal region were the best predictors of memory impairment. In conclusion, cortical thickness reductions in the left hemisphere and a lack of white matter integrity are the main contributors to long-term impairment in declarative memory among patients suffering from severe and diffuse TBI. In this study the hippocampus did not make a significant contribution to memory dysfunctions, suggesting that damage to this structure is compensated for by other regions, with the definitive sequelae being mainly explained by alterations in cortico-subcortical connectivity.

Full tensor registration of diffusion tensor magnetic resonance imaging for assessment of cardiac pathologies

Background Cardiac fiber architecture can be characterized through DTI, resulting in a better understanding of the electrical and mechanical behavior of the heart. Structural remodeling of the myocardium may also be observed. Most cardiac DTI studies are performed ex-vivo and only recently have in-vivo measurements become possible. However, in-vivo imaging remains challenging due to signal attenuation from heart motion. Previous research to create a cardiac DTI atlas used registration of scalar images such as fractional anisotropy maps and anatomical MRIs. This study aims to register the DT images using an algorithm that relies on the full tensor information and tensor orientation to drive the registration in order to compare a population of normal hearts with a population of hearts with dyssynchronous heart failure. Methods Data Acquisition: 7 normal and 4 failing canine hearts were imaged using DTI with the long axis aligned with the scanner’s z-axis. Diffusion was measured on a 1.5T GE scanner in a minimum of 16 different directions, bvalue = 1500s/mm2 and FOV varied from 8 to 10cm. Data was provided by the Center for Cardiovascular Bioinformatics and Modeling. DTI Registration: Our registration algorithm matches the tensor orientation to find the registration transformation. The source was warped into alignment with the target by applying a transformation that matches tensor size, shape and orientation most closely. Images were aligned using affine transformations and the Preservation of Principal Direction (PPD) reorientation strategy. Registration was performed with global optimization using Powell’s local optimization combined with Simulated Annealing global optimization. Results

ON DETERMINING THE SHAPE OF MATTER DISTRIBUTIONS

A basic property of objects, like galaxies and halos that form in cosmological structure formation simulations, is their shape. Here, we critically investigate shape determination methods that are commonly used in the literature. It is found that using an enclosed integration volume and weight factors r^{-2} and r_{ell}^{-2} (elliptical radius) for the contribution of each particle or volume element in the shape tensor leads to biased axis ratios and smoothing of details when calculating the local shape as a function of distance from the center. To determine the local shape of matter distributions as a function of distance for well resolved objects (typically more than O(10^4) particles), we advocate a method that (1) uses an ellipsoidal shell (homoeoid) as an integration volume without any weight factors in the shape tensor and (2) removes subhalos.

Determination of Grade and Subtype of Meningiomas by Using Histogram Analysis of Diffusion-Tensor Imaging Metrics

To determine whether histogram analysis of diffusion-tensor (DT) magnetic resonance (MR) imaging metrics, including tensor shape measurements, can help determine the grades and subtypes of meningiomas. The institutional review board approved this HIPAA-compliant study. Nine atypical, three anaplastic, and 39 typical meningiomas were retrospectively studied. The 39 typical meningiomas included one secretory meningioma and 11 fibroblastic, 11 transitional, 14 meningothelial, and two angiomatous meningiomas. DT imaging metrics, including fractional anisotropy, mean diffusivity, linear anisotropy coefficient, planar anisotropy coefficient (CP), spherical anisotropy coefficient (CS), and eigenvalue skewness (SK), as well as normalized signal intensity from contrast-enhanced T1- and T2-weighted images, were measured from the enhancing region of the tumor. Mean, variance, skewness, and kurtosis were extracted from the histograms. A two-level decision tree was designed, and a multivariate logistic regression analysis was used at each level to determine the best model for classification. Histogram skewness of SK and kurtosis of SK were significantly higher in atypical and anaplastic meningiomas than in typical meningiomas (P<.01). Among typical meningiomas, significant differences in histogram measures of CP and CS between fibroblastic meningiomas and other subtypes were observed (P<.01). The best model for differentiating atypical and anaplastic meningiomas from typical meningiomas consisted of mean and skewness of SK and kurtosis of T1 signal intensity, with an area under the receiver operating characteristic curve (AUC) of 0.946. The best model for differentiating fibroblastic meningiomas from other subtypes consisted of skewness of T2 signal intensity and kurtosis of CP (AUC, 0.970). Histogram analysis of DT imaging metrics can help determine the grades and subtypes of meningiomas, which can better assist in surgical planning.

Advanced fit of the diffusion kurtosis tensor by directional weighting and regularization

The diffusional kurtosis is an indicator for diffusion restrictions in biological tissue. It is observed experimentally that the kurtosis is largest for directions perpendicular to the fiber direction in white matter. The directional dependence of the kurtosis can be described by the diffusion kurtosis tensor. Since the intention of diffusion kurtosis imaging is to detect diffusion restrictions, the fit of the kurtosis tensor should be dominated by directions perpendicular to the fibers. In this work, it is shown that the basic approach, which is solving the occurring linear system by a pseudoinverse matrix, may completely fail in this regard if the diffusion is highly anisotropic. This problem is solved by adapting the weights of the fit--and thus emphasizing directions of restricted water motion--using a direct fit of the kurtosis tensor to the measured kurtosis values. Moreover, due to its large number of degrees of freedom, the kurtosis tensor can assume complicated shapes resulting in a fit which is sensitive to noise. This article demonstrates that the quality of the kurtosis tensor calculation can be further improved if the fit is regularized by suppressing too large and too small kurtosis tensor values and thus restricting the possible tensor shapes.

Differentiation of Brain Abscesses from Necrotic Glioblastomas and Cystic Metastatic Brain Tumors with Diffusion Tensor Imaging

The differentiation of abscesses from glioblastomas and metastases may not always be possible on the basis of DWI. Our hypothesis was that differences in diffusion properties as detected by DTI allow differentiation of abscess from glioblastomas and metastasis. Furthermore, diagnostic performance of tensor metrics quantifying anisotropy or tensor shapes is better than that of ADC in measuring mean diffusivity for this purpose. DTI was performed in 15 abscesses, 15 necrotic glioblastomas, and 26 cystic metastases. In each lesion, manually segmented into 4 regions of interest (ie, cystic cavity, enhancing rim, and immediate [edema most adjacent to the enhancing rim] and distant zones of edema), FA, ADC, C(l), C(p), and C(s) values were measured and statistically compared among groups and evaluated with ROC curve analysis. The presence of a hyperintense FA rim (a rim of edematous tissue that was hyperintense on the FA map) was assessed visually. Abscess was significantly different from glioblastoma for all tensor metrics measured in the cystic cavity and immediate zone of edema and for all except C(l) in the enhancing rim. Abscess was significantly different from metastasis for all tensor metrics measured in the cystic cavity and enhancing rim and for FA, ADC, and C(l) in immediate zone of edema. The incidence of a hyperintense FA rim was significantly higher in glioblastoma and metastasis compared with abscess. The 3 tensor metrics with the highest performance in differentiating abscess from glioblastoma and metastasis were FA, C(l), and C(s) of the cystic cavity. DTI is able to differentiate abscess from glioblastoma and metastasis. FA, C(l), and C(s) outperformed ADC in diagnostic performance comparisons.

Two‐step anomalous diffusion tensor imaging

We extend the formalism of anomalous diffusion imaging to include directional anisotropy of fitted parameters. The resulting technique is termed anomalous diffusion tensor imaging (aDTI), and allows the directional properties of the distributed diffusion coefficient (α) and the anomalous diffusion exponent, (γ) to be analysed using the same analytical techniques as regular diffusion tensor imaging (DTI). Together, these parameters quantify the rate of diffusion (α) and the complexity of the diffusion environment (γ). We generated tensor images for the anomalous exponent tensor (Γ) and distributed diffusivity tensor (A) from in vivo human brain data and present images of eigenvalues, eigenvectors, Trace/3 (Tr), fractional anisotropy (FA) and tensor shape measures. In white matter, A is found to have a median Tr = 0.56 × 10(- 3) mm(2) s(- 1), FA = 0.58 and Γ Tr = 0.69, FA = 0.13. We observed that white matter shows a similar anisotropic geometry for the distributed diffusion tensor as for the regular diffusion tensor, whereas the anomalous exponent tensor exhibits a different shape characteristic which may be informative of tissue microstructure.

Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures

Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.

Dynamics of droplet with viscoelastic interface

A theoretical model for droplet dynamics and rheology of blends with viscoelastic interface and interfacial slip is suggested in this work. The interfacial constitutive equation, so called as the Boussinesq–Scriven equation, is integrated in the pertubation analysis on the flow field inside and outside the droplet, while the interfacial slip is introduced via Navier slip boundary conditions. The results of small deformation analysis is extended to larger deformation using the ellipsoidal shape tensor. The effect of Boussinesq number, the interfacial viscosity ratio and the interfacial thickness on the characteristic relaxation time of droplet, the steady deformation of droplet and the shear viscosity of blends are discussed in detail. The model is validated by certain experimental data using interfacial viscosities that are reasonably measured in experiments.

DTI measures in crossing-fibre areas: Increased diffusion anisotropy reveals early white matter alteration in MCI and mild Alzheimer's disease

Though mild cognitive impairment is an intermediate clinical state between healthy aging and Alzheimer's disease (AD), there are very few whole-brain voxel-wise diffusion MRI studies directly comparing changes in healthy control, mild cognitive impairment (MCI) and AD subjects. Here we report whole-brain findings from a comprehensive study of diffusion tensor indices and probabilistic tractography obtained in a very large population of healthy controls, MCI and probable AD subjects. As expected from the literature, all diffusion indices converged to show that the cingulum bundle, the uncinate fasciculus, the entire corpus callosum and the superior longitudinal fasciculus are the most affected white matter tracts in AD. Significant differences between MCI and AD were essentially confined to the corpus callosum. More importantly, we introduce for the first time in a degenerative disorder an application of a recently developed tensor index, the “mode” of anisotropy, as well as probabilistic crossing-fibre tractography. The mode of anisotropy specifies the type of anisotropy as a continuous measure reflecting differences in shape of the diffusion tensor ranging from planar (e.g., in regions of crossing fibres from two fibre populations of similar density or regions of “kissing” fibres) to linear (e.g., in regions where one fibre population orientation predominates), while probabilistic crossing-fibre tractography allows to accurately trace pathways from a crossing-fibre region. Remarkably, when looking for whole-brain diffusion differences between MCI patients and healthy subjects, the only region with significant abnormalities was a region of crossing fibres in the centrum semiovale, showing an increased mode of anisotropy. The only white matter region demonstrating a significant difference in correlations between neuropsychological scores and a diffusion measure (mode of anisotropy) across the three groups was the same region of crossing fibres. Further examination using probabilistic tractography established explicitly and quantitatively that this previously unreported increase of mode and co-localised increase of fractional anisotropy was explained by a relative preservation of motor-related projection fibres (at this early stage of the disease) crossing the association fibres of the superior longitudinal fasciculus. These findings emphasise the benefit of looking at the more complex regions in which spared and affected pathways are crossing to detect very early alterations of the white matter that could not be detected in regions consisting of one fibre population only. Finally, the methods used in this study may have general applicability for other degenerative disorders and, beyond the clinical sphere, they could contribute to a better quantification and understanding of subtle effects generated by normal processes such as visuospatial attention or motor learning.

DYNAMIC SHAPE STYLE ANALYSIS: BILINEAR AND MULTILINEAR HUMAN IDENTIFICATION WITH TEMPORAL NORMALIZATION

Modeling and analyzing the dynamic shape of human motion is a challenging task owing to temporal variations in the shape and multiple sources of observed shape variations such as viewpoint, motion speed, clothing, etc. We present a new framework for dynamic shape analysis based on temporal normalization and factorized shape style analysis. Using a nonlinear generative model with motion manifold embedding in a low-dimensional space, we detect cycles of periodic motion like gait in different views and synthesize temporally-aligned shape sequences from the same type of motion at different speeds. The bilinear analysis of temporally-aligned shape sequences decomposes dynamic motion into time-invariant shape style factors and time-dependent motion factors. We extend the bilinear model into a tensor shape model, a multilinear decomposition of dynamic shape sequences for view-invariant shape style representations. The shape style is a view-invariant, time-invariant, and speed-invariant shape signature and is used as a feature vector for human identification. The shape style can be adapted to new environmental conditions by iterative estimation of style and content factors to reflect new observation conditions. We present the experimental results of gait recognition using the CMU Mobo gait database and the USF gait challenging database.

Superquadric Glyphs for Symmetric Second-Order Tensors

Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.

Problemas asociados al equilibrio en estructuras de membrana con bordes rígidos

This paper presents the equilibrium analysis of a membrane with rigid boundary. The idea of using membranes in applications such as footbridges, a new technology being developed in Spain, implies a more accurate analysis procedure. Due to the tension stresses, membrane is identifi ed to a negative gaussian curvature surface. Equilibrium is directly expressed by means of partial differential equations, in terms of the membrane shape and stress tensor. Starting from these equations, two dual approaches can be defi ned, namely direct problem and dual problem. Both problems are analyzed, studying their possible solutions in order to obtain practical results. In particular, the main analytical aspects of the direct problem are discussed, a numerical resolution procedure is proposed and, fi nally, analytical and numerical solution examples are presented.

Differentiation between glioblastomas and solitary brain metastases using diffusion tensor imaging

The purpose of this study is to determine whether diffusion tensor imaging (DTI) metrics including tensor shape measures such as linear and planar anisotropy coefficients (CL and CP) can help differentiate glioblastomas from solitary brain metastases. Sixty-three patients with histopathologic diagnosis of glioblastomas (22 men, 16 women, mean age 58.4 years) and brain metastases (13 men, 12 women, mean age 56.3 years) were included in this study. Contrast-enhanced T1-weighted, fluid-attenuated inversion recovery (FLAIR) images, fractional anisotropy (FA), apparent diffusion coefficient (ADC), CL and CP maps were co-registered and each lesion was semi-automatically subdivided into four regions: central, enhancing, immediate peritumoral and distant peritumoral. DTI metrics as well as the normalized signal intensity from the contrast-enhanced T1-weighted images were measured from each region. Univariate and multivariate logistic regression analyses were employed to determine the best model for classification. The results demonstrated that FA, CL and CP from glioblastomas were significantly higher than those of brain metastases from all segmented regions (p<0.05), and the differences from the enhancing regions were most significant (p<0.001). FA and CL from the enhancing region had the highest prediction accuracy when used alone with an area under the curve of 0.90. The best logistic regression model included three parameters (ADC, FA and CP) from the enhancing part, resulting in 92% sensitivity, 100% specificity and area under the curve of 0.98. We conclude that DTI metrics, used individually or combined, have a potential as a non-invasive measure to differentiate glioblastomas from metastases.

Three‐dimensional mapping of lingual myoarchitecture by diffusion tensor MRI

This study was performed to assess the feasibility of investigating the complex lingual myoarchitecture through segmentation of muscles from diffusion tensor imaging (DTI) data. The primary eigenvectors were found to be adequate for delineating the superior and inferior longitudinalis, genioglossus, and hyoglossus. The tertiary eigenvector orientations effectively revealed the homogeneous and systematic change of muscle orientation in the tongue core. In the longitudinalis near the tongue tip, the secondary eigenvectors were oriented in the radial direction. Lingual muscles were segmented using two methods: modified directional correlation (DC) and tensor coherence (TC) methods. The DC method, based on one eigenvector, was found to be inadequate for lingual muscle segmentation, whereas the TC method, based on the tensor shape and orientation, was used successfully to segment most lingual muscles. The segmentation result was used to report the diffusion tensor properties of individual lingual muscles. Also found was a continuous change in skewness of the intrinsic tongue core from negative in the anterior region to positive in the posterior region. DTI and the proposed segmentation method provide an adequate means of imaging and visualizing the complex, compartmentalized musculature of the tongue. The potential for in vivo research and clinical applications is demonstrated.

Can we differentiate true white matter fibers from pseudofibers inside a brain abscess cavity using geometrical diffusion tensor imaging metrics?

High fractional anisotropy (FA) usually reflects the orientation and integrity of white matter (WM) fibers. Other regions of increased FA have been described, such as brain abscesses, developing cortex, and areas of hemorrhage. It may not be possible to differentiate true fibers from the pseudofibers found inside an abscess cavity on the basis of FA and mean diffusivity (MD). The aim of this study was to differentiate true WM fibers from pseudo WM tracts inside the abscess cavity using geometrical diffusion tensor imaging metrics [linear anisotropy (CL), planar anisotropy (CP), and spherical anisotropy (CS)]. Diffusion tensor imaging was performed in 42 patients with brain abscess and 10 age/sex-matched controls. Automated segmentation using Java-based software divided the abscess cavity into two sub-regions with FA < 0.20 and FA > or = 0.20. Quantitation was carried out on the sub-regions of the abscess cavity with FA > or = 0.20. In healthy controls, regions of interest were placed on the corpus callosum, posterior limb of the internal capsule, and periventricular and subcortical WM. Significantly increased CP values were observed inside the abscess cavity compared with various normal WM regions. Significantly increased FA and CL values were observed in the abscess cavity compared with subcortical WM only. However decreased FA and CL values were observed in the cavity compared with the corpus callosum, posterior limb of the internal capsule, and periventricular WM. The 95% confidence intervals of means for the abscess cavity were well separated from those for WM in the case of CL and CP; however, they overlapped in the case of FA, MD, and CS. High CP with low CL inside the abscess cavity suggests that the shape of the diffusion tensor is predominantly planar, whereas it is linear in WM tracts. These geometrical indices may have advantages over FA for differentiating true from pseudo WM tracts inside the abscess cavity.

Diffusion Tensor Analysis With Invariant Gradients and Rotation Tangents

Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.

Diffusion tensor MRI in rat models of invasive and well‐demarcated brain tumors

Diffusion tensor imaging (DTI) and its metrics, such as mean diffusivity (MD) and fractional anisotropy (FA), have been used to detect the extent of brain tumors and understand tumor growth and its influence on the surrounding tissue. However, there are conflicting reports on how DTI metrics can be used for tumor diagnosis. The physiological interpretation of these metrics in terms of tumor growth is also not clear. The objective of this study was to investigate the DTI parameters in two rat brain tumor models (9L and F98) with different patterns of aggressiveness by longitudinal monitoring of tumor growth and comparing the DTI parameters of these two tumor models. In addition to the standard DTI metrics, MD and FA, we measured other metrics representing diffusion tensor shape, such as linear and planar anisotropy coefficients (CL and CP), and orientational coherence measured by lattice index (LI), to characterize the two tumor models. The 9L tumor had higher FA, CL, and LI than the F98 tumor. F98 had a larger difference in anisotropies between tumor and peritumor regions than 9L. From the eigenvalues, it was found that the increase in CL and trace of the 9L tumor was due to an increase in the primary eigenvalue, whereas the increase in CP in the peritumor region was due to an increase in both primary and secondary eigenvalues and a decrease in tertiary eigenvalue. Our results indicate that shape-oriented anisotropy measures, such as CL and CP, and orientational coherence measures, such as LI, can provide useful information in differentiating these two tumor models and also differentiating tumor from peritumoral regions.

Classification of spacelike surfaces in spacetime

A classification of two-dimensional surfaces embedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels and provides, among other things, a refinement of the concepts of trapped, umbilical and extremal surfaces, which split into several different classes. The classification raises new important questions and opens many possible new lines of research. These, together with some applications and examples, are briefly considered.