Digital Image Analysis Techniques Research Articles

Experimental study on the effect of circulating liquid velocity on bubble size distribution and turbulence characteristic in an external loop airlift reactor

AbstractThe influence of circulating liquid velocity on bubble size distribution (BSD) and turbulence characteristics of an external loop airlift reactor was studied in this paper. The instantaneous and time‐averaged velocity in the riser were studied using particle image velocimetry (PIV), and BSDs were measured by digital image analysis technique based on machine learning. Then turbulence kinetic energy (TKE) and energy dissipation rate (EDR) were calculated through the velocity field. The results indicate that as the circulating liquid velocity increased, the peak value of BSD rapidly decreased from nearly 6 mm to approximately 2 mm. The radial velocity of the liquid gradually decreased and changed direction, eventually increasing again. TKE first decreased and then increased. Compared with bubble flow with a BSD peak of 2–6 mm, bubble flow with a BSD peak of 2 mm had larger TKE. The radial movement of bubbles had great influence on the turbulence characteristics. This study demonstrates that selecting an appropriate circulating liquid velocity can reduce the diameter of bubbles while obtaining greater TKE, thereby improving the mass transfer and reaction efficiency in external loop airlift reactors (EL‐ALR).

Exploring water-absorbing capacity: a digital image analysis of seeds from 120 wheat varieties

Wheat is a staple food crop that provides a significant portion of the world's daily caloric intake, serving as a vital source of carbohydrates and dietary fiber for billions of people. Seed shape studies of wheat typically involve the use of digital image analysis software to quantify various seed shape parameters such as length, width, area, aspect ratio, roundness, and symmetry. This study presents a comprehensive investigation into the water-absorbing capacity of seeds from 120 distinct wheat lines, leveraging digital image analysis techniques facilitated by SmartGrain software. Water absorption is a pivotal process in the early stages of seed germination, directly influencing plant growth and crop yield. SmartGrain, a powerful image analysis tool, was employed to extract precise quantitative data from digital images of wheat seeds, enabling the assessment of various seed traits in relation to their water-absorbing capacity. The analysis revealed significant transformations in seed characteristics as they absorbed water, including changes in size, weight, shape, and more. Through statistical analysis and correlation assessments, we identified robust relationships between these seed traits, both before and after water treatment. Principal Component Analysis (PCA) and Agglomerative Hierarchical Clustering (AHC) were employed to categorize genotypes with similar trait patterns, providing insights valuable for crop breeding and genetic research. Multiple linear regression analysis further elucidated the influence of specific seed traits, such as weight, width, and distance, on water-absorbing capacity. Our study contributes to a deeper understanding of seed development, imbibition, and the crucial role of water absorption in wheat. These insights have practical implications in agriculture, offering opportunities to optimize breeding programs for improved water absorption in wheat genotypes. The integration of SmartGrain software with advanced statistical methods enhances the reliability and significance of our findings, paving the way for more efficient and resilient wheat crop production. Significant changes in wheat seed shape parameters were observed after imbibition, with notable increases in area, perimeter, length, width, and weight. The length-to-width ratio (LWR) and circularity displayed opposite trends, with higher values before imbibition and lower values after imbibition.

Mixing phenomena in circular and rectangular cross-sectional T-mixers: Experimental and numerical assessment

In the present work, the mixing performance of square and circular cross-sectional millimetre-sized T-mixers was experimentally and numerically investigated. The pure dilution of non-reacting species (blue and yellow food dyes) and the chemical reaction of two-coloured reactants (i.e., strong acid and strong base with a pH indicator) were captured experimentally by adopting a low-cost colorimetric digital image analysis technique. Turbulence was modelled by k − ε model coupled with the β-PDF approach to account for turbulence-chemistry interactions. The circular cross-sectional T-mixer showed better mixing efficiency than the square one. Moreover, an even larger mixing performances were obtained by setting unbalanced fluid flow rates. To exploit the low computational demanding k − ε model, the Cε2 parameter was tuned, showing good reliability in describing the phenomena in the mixers. Mixing time estimations were based on both time averages and instantaneous quantities, and some possible limits of the approaches commonly adopted in the literature were identified.

Hydrodynamics of a cold circulating fluidized bed for methanol-to-olefins (MTO) process: experimental and computational investigation

The present study aims to investigate the hydrodynamic behavior of gas–solid flow in a pseudo-two-dimensional cold circulation fluidized bed specifically designed to mimic a plane cut of an industrial methanol-to-olefins (MTO) bed. The solid velocity pattern was experimentally examined using the Particle Image Velocimetry with Digital Image Analysis (PIV-DIA) technique. The hydrodynamics of this pseudo-2D bed were studied using the twoPhaseEulerFoam solver in OpenFOAM, and the adjusted solver was validated through solid particle velocity and solid volume fraction comparisons, showing good agreement with experimental data. Additionally, the study explores different geometries, including symmetric and asymmetric baffles, to understand their impact on the mixing characteristics of the circulating fluidized bed for MTO. The findings highlight the significant role of asymmetric baffles in enhancing mixing behavior. Furthermore, the DIA technique was employed to study the effects of bubbles on gas–solid mixing, revealing that the low mixing region of the bed limited the diameter of bubbles.

Mitigation of reverse faulting in foundation soils using geosynthetic-encased granular columns

In this study, a series of reduced model tests was conducted on soil foundations reinforced by Geosynthetic-Encased Granular Columns (GECs) placed across a reverse fault. These tests aimed at evaluating the effectiveness, reinforcing mechanism and optimal GEC horizontal spacing to mitigate the ground surface deformation associated with reverse faulting. For comparison, reduced model tests were also performed on unreinforced and Geosynthetic-Reinforced Soil (GRS) foundations. The reduced model tests were conducted to simulate a prototype 3-m-thick foundation layer subjected to a reverse fault displacement of 0.9 m. Digital Image Analysis (DIA) techniques were adopted to determine the surface displacement profile, angular distortion and shear rupture propagation considering various reverse fault offsets. Test results revealed that the GEC foundation can considerably reduce the fault-induced angular distortion at the ground surface. A reduction of 23.3% on the maximum angular distortion at the ground surface was observed as the fault displacement reached 30% of the foundation height (S/H = 30%), indicating the GEC foundation can mitigate the risk of surface fault hazards associated with large reverse fault movement. Two mechanisms, shear rupture diffusion and diversion effects, were identified for the GEC foundation, depending on the magnitude of the fault displacement and the system stiffness of the GEC foundation. Because of complex mechanisms between diversion and diffusion of the shear rupture in the GEC foundations, an optimal GEC spacing of Sh/dc = 3.3 was observed, which exhibited the most significant reduction in βmax at large fault offsets.

Optimized detection and segmentation of nuclei in gastric cancer images using stain normalization and blurred artifact removal

Histological analysis with microscopy is the gold standard to diagnose and stage cancer, where slides or whole slide images are analyzed for cell morphological and spatial features by pathologists. The nuclei of cancerous cells are characterized by nonuniform chromatin distribution, irregular shapes, and varying size. As nucleus area and shape alone carry prognostic value, detection and segmentation of nuclei are among the most important steps in disease grading. However, evaluation of nuclei is a laborious, time-consuming, and subjective process with large variation among pathologists. Recent advances in digital pathology have allowed significant applications in nuclei detection, segmentation, and classification, but automated image analysis is greatly affected by staining factors, scanner variability, and imaging artifacts, requiring robust image preprocessing, normalization, and segmentation methods for clinically satisfactory results. In this paper, we aimed to evaluate and compare the digital image analysis techniques used in clinical pathology and research in the setting of gastric cancer. A literature review was conducted to evaluate potential methods of improving nuclei detection. Digitized images of 35 patients from a retrospective cohort of gastric adenocarcinoma at Oulu University Hospital in 1987–2016 were annotated for nuclei (n = 9085) by expert pathologists and 14 images of different cancer types from public TCGA dataset with annotated nuclei (n = 7000) were used as a comparison to evaluate applicability in other cancer types. The detection and segmentation accuracy with the selected color normalization and stain separation techniques were compared between the methods. The extracted information can be supplemented by patient’s medical data and fed to the existing statistical clinical tools or subjected to subsequent AI-assisted classification and prediction models. The performance of each method is evaluated by several metrics against the annotations done by expert pathologists. The F1-measure of 0.854 ± 0.068 is achieved with color normalization for the gastric cancer dataset, and 0.907 ± 0.044 with color deconvolution for the public dataset, showing comparable results to the earlier state-of-the-art works. The developed techniques serve as a basis for further research on application and interpretability of AI-assisted tools for gastric cancer diagnosis.

Bubble dynamics properties of B-particles in a quasi-2D gas-solid fluidized bed: Computational particle fluid dynamics numerical simulation and post-processed by digital image analysis technique

Bubble dynamics properties play a crucial and significant role in the design and optimization of gas-solid fluidized beds. In this study, the bubble dynamics properties of four B-particles were investigated in a quasi-two-dimensional (quasi-2D) fluidized bed, including bubble equivalent diameter, bubble size distribution, average bubble density, bubble aspect ratio, bubble hold-up, bed expansion ratio, bubble radial position, and bubble velocity. The studies were performed by computational particle fluid dynamics (CPFD) numerical simulation and post-processed with digital image analysis (DIA) technique, at superficial gas velocities ranging from 2umf to 7umf. The simulated results shown that the CPFD simulation combining with DIA technique post-processing could be used as a reliable method for simulating bubble dynamics properties in quasi-2D gas-solid fluidized beds. However, it seemed not desirable for the simulation of bubble motion near the air distributor at higher superficial gas velocity from the simulated average bubble density distribution. The superficial gas velocity significantly affected the bubble equivalent diameter and evolution, while it had little influence on bubble size distribution and bubble aspect ratio distribution for the same particles. Both time-averaged bubble hold-up and bed expansion ratio increased with the increase of superficial gas velocity. Two core-annular flow structures could be found in the fluidized bed for all cases. The average bubble rising velocity increased with the increasing bubble equivalent diameter. For bubble lateral movement, the smaller bubbles might be more susceptible, and superficial gas velocity had a little influence on the absolute lateral velocity of bubbles. The simulated results presented a valuable and novel approach for studying bubble dynamics properties. The comprehensive understanding of bubble dynamics behaviors in quasi-2D gas-solid fluidized beds would provide support in the design, operation, and optimization of gas-solid fluidized bed reactors.

Experimental investigation on the effect of superficial gas velocity on bubble dynamics properties of B particles in gas–solid fluidized bed reactor using digital image analysis technique

The bubble dynamics properties in gas–solid particle system play a crucial role in the design and optimization of gas–solid fluidized beds, and are significantly influenced by superficial gas velocity. In present study, the effect of superficial gas velocity on bubble dynamics properties in a classical B particle system were investigated at gas velocity ranging from (0.41 to 0.59) m·s−1 with umf = 0.067 m·s−1 in a quasi-two dimensional (quasi-2D) fluidized bed domain, including bubble equivalent diameter, bubble size distribution, bubble density, aspect ratio, bubble hold-up, bed expansion ratio, bubble rising and lateral velocity. Those were derived from the taken image by digital image analysis (DIA) technique. The results show that superficial gas velocity has a significant effect on the bubble equivalent diameter, while it has little effect on bubble size distribution (BSD), average bubble density distribution and bubble aspect ratio distribution for the same particle system. Both time-averaged bed expansion ratio and bubble hold-up increase with the increase of superficial gas velocity under the range of gas velocity in this work. Two uniform core-annular flows form in the bed for all cases. The bubble rising velocity coefficients of developed correlation range from 0.69 to 1.00. For bubble lateral movement, smaller bubbles may be more susceptible, and superficial gas velocity has a little influence on the absolute lateral velocity of bubbles. Overall understanding the effect of gas velocity on bubbles dynamics properties is good for the design and optimization of fluidized bed reactors, leading to the better process outcomes and performance.

Performance of Manufactured and Recycled Steel Fibres in Restraining Concrete Plastic Shrinkage Cracks.

Early-age plastic shrinkage cracks can reduce the durability of concrete slabs by creating direct paths for the ingress of aggressive agents and thus accelerating degradation due to environmental attack, in particular, in hot and windy environments. The elimination of such cracks is essential for durable and sustainable concrete structures. This paper parametrically investigates the effect of manufactured steel fibres (MSF) and recycled tyre steel fibres (RTSF) on restraining plastic shrinkage and micro cracks at different dosages (10, 20, and 30 kg/m3). The plastic shrinkage tests were carried out in a specially designed chamber, according to ASTM C1579. Various environmental conditions are examined, and their impact on compressive strength and crack potential is assessed. A digital image analysis technique is used to measure length, width, and the area of the crack on the exposed surface to gain additional insights into crack behaviour. The results show a slight early-age (one-day) increase in compressive strength for the concrete exposed to the various environmental conditions, mostly as a result of higher temperatures. Through the use of the crack reduction ratio (CRR), both RTSF and MSF are shown to be successful in controlling plastic shrinkage and micro cracks, with the RTSF being superior due to the fact that they are better distributed in the concrete volume. The addition of 30 kg/m3 of RTSF was effective in preventing crack development in most environments or restraining cracks in extremely harsh environments. The adoption of these results will lead to more sustainable concrete slabs in the harsher environmental conditions created by climate change.

Analiza nivoa greške (ANG)

ELA (Error Level Analysis) is a digital image analysis technique that has gained popularity in recent years in the field of modern forensics. It is a powerful method that can help experts determine if an image has been altered or manipulated in any way. ELA works by analyzing the levels of compression within an image and identifying areas that have been edited or manipulated at a different level than the rest of the image. This information can be crucial in criminal investigations, where the authenticity of an image can be key to solving a case. In this article, we will explore the use of ELA in modern forensics, its benefits, and limitations, as well as its potential impact on the future of forensic investigation.

Computational fluid dynamic study of polycaprolactone nanoparticles precipitation in a co‐flow capillary micro‐tube

AbstractPolycaprolactone nanoparticles (NPs) were produced in the co‐flow glass capillary device with tip dimension. NPs size was 990 nm for continuous phase velocity, and and 426 nm for . The droplet formation process in the co‐flow microchannel was also simulated using computational fluid dynamic (CFD). Employing a digital image analysis technique, a cut‐off value of 0.81 for the dispersed phase volume fraction was proposed for the determination of droplet size. A scenario was expressed for NP formation from micro‐droplets. Moreover, after conducting 27 CFD simulations, a dimensionless exponential form correlation was proposed for droplet size estimation. In this generalized map, there were three distinct regions based on the relative capillary numbers of continuous to dispersed phases. It was revealed that at a constant dispersed phase velocity, by increasing the continuous phase velocity, more small NPs are formed. The results show that the ratio of micro‐droplets to NPs size was between 735 and 755.

Investigation of damage characteristics of coral reef limestone under uniaxial compression based on pore structure

The variation of the biological source and diagenetic environment results in significant differences in pore structure and mechanical properties of coral reef limestone (CRL) with various structures. This study analyzed uniaxial compression behavior and pore structural characteristics of four types of CRL using the uniaxial compression test and computerized tomography (CT) scanning test. The influence mechanism of CRL's physical properties on its uniaxial compression characteristics was revealed from a microscopic perspective. The results suggested that the uniaxial compressive curves of different structural types of CRLs exhibited “single-peak” or “multi-peak” characteristics, in which Coral Framework Limestone (CFL) and Coral Boulder Limestone (CBL) mainly experienced brittle failure while Coral Gravel Limestone (CGL) and Coral Calcarenite Limestone (CCL) underwent ductile failure with the increase of porosity. Through a CT scanning test, the influence of pore structure on the damage mechanism of CRL was explored. According to the digital image analysis technique, the influence of CRL diagenesis on the pore fractal dimension was investigated from the perspective of geological genesis, and an empirical formula for estimating uniaxial compressive strength (UCS) of CRL was established based on the fractal dimension. Furthermore, a quality evaluation method for CRL suitable for practical engineering was proposed. The research results can provide a reference for the estimation of UCS and quality classification evaluation of CRL in the construction of island reefs.

Fracture resistance of polymeric wastes modified asphalt using R-curve and digital image correlation

Characterising the fracture properties of asphalt mixtures is a critical step toward a better pavement design. This study aims to investigate the mechanical response of asphalt mixtures modified with crumb rubber and plastic waste. These modified mixtures have been introduced in road construction based on the basic understanding of their mechanical properties. The R-curve based cumulative fracture energy and digital image correlation (DIC) technique were used to assess the fracture behaviour of the modified asphalt. The asphalt mixtures were examined under a monotonic semi-circular bending test for unconditioned, aged- and moisture-conditioned samples. The digital image analysis technique was used to construct the resistance curve based on the cumulative fracture energy. The DIC technique was used to capture the sample’s strain field for the control and modified mixtures. Results showed that the rubberised asphalt promoted crack propagation, whilst plastic waste resisted cracking by prolonging the crack initiation phase compared to the conventional asphalt. Strain fields derived by DIC demonstrated different mechanical contributions for both modifiers under the fracture test.

Digital Histological Study of Neocortical Grey and White Matter Tau Burden Across Tauopathies.

3R/4R-tau species are found in Alzheimer disease (AD) and ∼50% of Lewy body dementias at autopsy (LBD+tau); 4R-tau accumulations are found in progressive supranuclear palsy (PSP) and corticobasal degeneration (CBD). Digital image analysis techniques can elucidate patterns of tau pathology more precisely than traditional methods but repeatability across centers is unclear. We calculated regional percentage areas occupied by tau pathological inclusions from the middle frontal cortex (MFC), superior temporal cortex (STC), and angular gyrus (ANG) from cases from the University of Pennsylvania and the University of California San Diego with AD, LBD+tau, PSP, or CBD (n = 150) using QuPath. In both cohorts, AD and LBD+tau had the highest grey and white matter tau burden in the STC (p ≤ 0.04). White matter tau burden was relatively higher in 4R-tauopathies than 3R/4R-tauopathies (p < 0.003). Grey and white matter tau were correlated in all diseases (R2=0.43-0.79, p < 0.04) with the greatest increase of white matter per unit grey matter tau observed in PSP (p < 0.02 both cohorts). Grey matter tau negatively correlated with MMSE in AD and LBD+tau (r = -4.4 to -5.4, p ≤ 0.02). These data demonstrate the feasibility of cross-institutional digital histology studies that generate finely grained measurements of pathology which can be used to support biomarker development and models of disease progression.

Bubble behaviors of geldart B particle in a pseudo two-dimensional pressurized fluidized bed

Pressurized fluidized beds have been developed in quite a few industrial applications because of intensified heat and mass transfer and chemical reaction. The bubble behaviors under elevated pressure, strongly influencing the fluidization and reaction conversion of the whole system, are of great research significance. In this work, the bubble behaviors of Geldart B particle in a pseudo two-dimensional (2D) pressurized fluidized bed were experimentally studied based on digital image analysis technique. The effects of pressure and fluidization gas velocity on the general bubble behaviors (i.e., size, shape and spatial distribution) and the dynamic characteristics, such as the time-evolution of voidage distribution and local flow regimes, were comprehensively investigated. Results show that increasing pressure reduces the stability of bubbles and facilitates gas passing through the emulsion phase, resulting in the “smoother” fluidization state with smaller bubbles and declined bubble fraction and standard deviation. The equivalent bubble diameter and bubble aspect ratio increase with the increasing gas velocity while decrease as pressure rises. The elevated pressure reduces bubbles extension in the vertical direction, prohibits the “short pass” of fluidization gas in large oblong bubbles/slugs and benefits the gas–solid interaction. The flow regimes variation with gas velocity is affected by the elevated pressure, and demonstrates different features in different local positions of the bed.

Centrifuge model and numerical studies of strip footing on reinforced transparent soils

This paper presents the results of centrifuge model tests to investigate the deformation behaviour of unreinforced and reinforced transparent soil foundations under strip loading. A digital image analysis technique was employed to obtain the soil displacement field and strain distribution of reinforcements. Two-dimensional (2D) numerical models were developed and verified using the test results. The soil was modelled as a linearly-elastic perfectly-plastic material with Mohr–Coulomb failure criterion. The reinforcement was characterised using a linearly-elastic model considering rupture behaviour. Moreover, a parametric study was conducted to investigate the load–settlement response of foundations, distribution of reinforcement tension and failure sequence of reinforcements. The experimental and numerical studies show that the results obtained from the numerical simulations are in good agreement with the results of the centrifuge model tests. The 2D finite difference model developed using the user-defined functions coded into the FLAC programme can better simulate the progressive failure of the reinforcement layers in the tests. The failure sequence of reinforcement layers is not affected by the modulus and internal friction angle of soil and the reinforcement length, but is closely related to the combined effect of spacing and number of reinforcement layers and the combined effect of reinforcement stiffness and strength.

Overall particles flow pattern in a two-zone gas-solid fluidized bed with a secondary-gas stream

Fluidized bed reactors with multiple input streams have been developed over the last decades to satisfy industries in the scope of multiphase catalytic reactions, and two-zone fluidized beds have been developed to perform reaction and catalyst regeneration simultaneously. However, the presence of the secondary gas nozzle in the bed significantly impacts the hydrodynamics and efficiency. Particle image velocimetry (PIV) and digital image analysis (DIA) techniques have been used to study different particle flow patterns in a two-zone pseudo-2D fluidized bed with lateral gas injection. The secondary flow nozzles' location has been changed, and different particle flow patterns have been observed in the fluidized bed. It was found that at low excess gas flow rates, when the lateral nozzle is located at the middle part of the bed, particles move downwardly at the bed center and upwardly near the walls; while when the lateral nozzle is located at the top part of the bed, particles have a reverse movement trend. At higher gas velocities for both mentioned nozzle locations, particles move upwardly at the bed center and downwardly near the walls. The formation of large bubbles was less probable when the nozzle was located at the middle of the bed.

Study of curling mechanism by precision kinematic measurements of curling stone’s motion

Why do curling stones curl? That is a question physicists are often asked, yet no answer has been established. Stones rotating clockwise curl right, contrary to our naive expectations. After a century of debate between contradicting hypotheses, this paper provides a possible answer based on experimental evidence. A digital image analysis technique was used to perform precision kinematic measurements of a curling stone’s motion to identify the curling mechanism. We observed a significant left–right asymmetric friction due to velocity dependence on the friction coefficient. Combined with the discrete point-like nature of the friction between ice and stone, swinging around slow-side friction points has been concluded as the dominant origin of the curling. Many new angular momentum transfer phenomena have been found, supporting this conclusion.

A Digital Image Analysis Technique for Improved Strain Measurement in Geosynthetic Tensile Testing

Abstract Tensile testing of geosynthetics for measurement of strength and stiffness plays a critical role in the selection and specification of these materials for civil engineering applications. Measurement of the tensile properties of geotextiles is commonly performed via the wide-width strip method, where wide-width specimens are gradually strained to failure while being held in tight-gripping jaws or wrapped roller grips. During testing, material strains are inferred from the relative displacement of the grips. This approach does not properly account for slippage of the material in the gripping jaws or around the roller grip windings. Moreover, it inherently assumes a uniform distribution of strain across the tested specimen, which is often not the case at failure, as can readily be observed from visual examination of tested specimens. To address these issues, this paper utilizes a simple digital image analysis technique, implemented using open source software tools, to improve strain measurements in geosynthetic tensile testing. The presented technique is simple to use, low-cost, and can be deployed easily using readily available technology, making this technique useful for geotechnical engineering practitioners and laboratory testing managers in a production environment. Results from wide-width geotextile tensile tests on a polypropylene woven fabric are presented and analyzed using this technique; tests conducted using both tight-gripping jaws and wrapped roller grips show significant slippage, stress concentrations, and nonuniform strains. The technique that was utilized allowed for improved strain characterization across the entirety of a given tested specimen, which in turn yielded enhanced interpretation of geotextile test results.

Deep learning and morphometric approach for Sex determination of the lumbar vertebrae in a Thai population.

Sex determination is a fundamental step in biological profile estimation from skeletal remains in forensic anthropology. This study proposes deep learning and morphometric technique to perform sex determination from lumbar vertebrae in a Thai population. A total of 1100 lumbar vertebrae (L1-L5) from 220 Thai individuals (110 males and 110 females) were obtained from the Forensic Osteology Research Center, Faculty of Medicine, Chiang Mai University, Thailand. In addition, two linear measurements of superior and inferior endplates from the digital caliper and image analysis were carried out for morphometric technique. Deep learning applied image classification to the superior and inferior endplates of the lumbar vertebral body. All lumbar vertebrae images are included in the dataset to increase the number of images per class. The accuracy determined the performance of each technique. The results showed the accuracies of 82.7%, 90.0%, and 92.5% for digital caliper, image analysis, and deep learning techniques, respectively. The lumbar vertebrae L1-L5 exhibit sexual dimorphism and can be used in sex estimation. Deep learning is more accurate in determining sex than the morphometric method. In addition, the subjectivity and errors in the measurement are decreased. Finally, this study presented an alternative approach to determining sex from lumbar vertebrae when the more traditionally used skeletal elements are incomplete or absent.