Ultrasonic Backscatter Research Articles

Ultrasonic backscattering measurement of hardness gradient distribution in polycrystalline materials

It is crucial to obtain the internal hardness distribution in polycrystalline materials to evaluate the mechanical performance of components and monitor their service life. Current methods, however, fail to meet the non-destructive evaluation needs for materials with hardness gradient distributions. This paper, based on the principle of grain boundary scattering of ultrasound in polycrystalline materials, combined with the Transverse-to-Transverse Singly-Scattered Response (T-T SSR) theory, proposes an ultrasonic SSR model adapted to hardness gradient distributions. The model elucidates the influence of hardness gradient variations and grain dispersion on ultrasonic scattering. Using DREAM.3D, seven different-scale polycrystalline volumes were constructed to assess the relevance of volume-weighted average grain size and spatial correlation of hardness gradient materials. Finally, induction quenching was applied to 40Cr to induce a gradient hardness distribution internally, followed by ultrasonic backscatter experiments. The results indicate that the theoretical model and the spatial variance of measured signals align well over a relatively long time window. For the specimen with minor curvature, the theoretical hardness distribution obtained by the model is accurate, with an average error of 2.55 % compared to destructive testing data. However, the results for the larger curvature reveal limitations in the model.

Inverting Single Crystal Elastic Constants of Polycrystals with Metallographic Measurement and Modified Ultrasonic Backscatter Mode

Inverting Single Crystal Elastic Constants of Polycrystals with Metallographic Measurement and Modified Ultrasonic Backscatter Mode

A numerical simulation method for ultrasonic transmission in cancellous bone based on a four-parameter random growth method

Abstract Quantitative ultrasound (QUS) has been widely used in non-destructive evaluation of bone health in research and clinical practice. To make a more accurate bone evaluation, the transmission characteristics of ultrasound in the bone need to be understood in detail. In the two-dimensional finite element model, cancellous bone is usually simulated by a non-porous structure solid or by approximating bone trabeculae as ellipses, which is different from real bone. However, although the error of the model constructed by bone CT images is small, it needs to be based on real bone samples, and the samples are limited. Therefore, a modeling method of cancellous bone based on four-parameter random growth method was proposed in this paper, and on this basis, numerical simulation of ultrasonic transmission was carried out. Firstly, based on the four-parameter random growth method, the aggregation algorithm is used to concentrate discrete pixels and smooth the edge of pores. Meanwhile, the built-in algorithm ensures the same porosity before and after processing to reduce the discrete pore structure. Secondly, based on COMSOL to establish the simulation model of ultrasonic propagation in cancellous bone, we analyzed the change of acoustic field distribution, discussed the correlation between the porosity of cancellous bone and backscattering coefficient (BSC) based on the ultrasonic backscattering method, and compared the experimental results of CT scan images of bone samples. The experimental results show that the cancellous bone modeling method in this paper has the same conclusion as the method based on CT images, which verifies the feasibility of this method. This method can generate a geometric model of the cancellous bone microstructure with specified porosity and different bone trabecular distribution, which is similar to the real bone structure, and can be directly imported into the finite element software to facilitate the study of bone microstructure related problems. It provides a more convenient method to study the mechanism of ultrasonic propagation in cancellous bone and has important significance in solving the inverse problem of recovering effective bone parameters from the received ultrasonic signals.

Prediction of the Case Depth Based on the Spatial Gradient Feature in Induction-Hardened Steel With a Large Transition Zone by Ultrasonic Backscatter

Prediction of the Case Depth Based on the Spatial Gradient Feature in Induction-Hardened Steel With a Large Transition Zone by Ultrasonic Backscatter

Research on Suspended Particle Size Measurement Based on Ultrasonic Backscattered Amplitude Analysis

Research on Suspended Particle Size Measurement Based on Ultrasonic Backscattered Amplitude Analysis

Clinical performance of ultrasonic backscatter parametric and nonparametric statistics in detecting early hepatic steatosis

Diagnosis of early hepatic steatosis would allow timely intervention. B-mode ultrasound imaging was in question for detecting early steatosis, especially with a variety of concomitant parenchymal disease. This study aimed to use the surgical specimen as a reference standard to elucidate the clinical performance of ultrasonic echogenicity and backscatter parametric and nonparametric statistics in real-world scenarios. Ultrasound radio-frequency (RF) signals of right liver lobe and patient data were collected preoperatively. Surgical specimen was then used to histologically determine staging of steatosis. A backscatter nonparametric statistic (h), a known backscatter parametric statistic, i.e., the Nakagami parameter (m), and a quantitative echo intensity (env) were calculated. Among the 236 patients included in the study, 93 were grade 0 (<5% fat) and 143 were with steatosis. All the env, m and h statistics had shown significant discriminatory power of steatosis grades (AUC = 0.643–0.907 with p-value < 0.001). Mann-Whitney U tests, however, revealed that only the backscatter statistics m and h were significantly different between the groups of grades 0 and 1 steatosis. The two-way ANOVA showed a significant confounding effect of the elevated ALT on env (p-value = 0.028), but no effect on m or h. Additionally, the severe fibrosis was found to be a significant covariate for m and h. Ultrasonic signals acquired from different scanners were found linearly comparable.

Prediction of case depth for deep carburising heat treatments based on the spatial gradient feature of the grain size profile

ABSTRACT Precise evaluation of carburising heat treatment is necessary to avoid failure and degradation of high-performance industrial mechanical components. In this work, a novel non-destructive detection method is proposed for the deep carburised case depth (>2 mm) combining spatial ultrasonic backscatter technique and linear regression model. To create various carburised case depths, three groups of cylindrical 20CrMnMo steel samples were heat-treated in various cycles (Carburising, diffusing, quenching at 850–870°C (holding in range of 2.5–3 h) and tempering). The carburising heat treatment results in a gradient microstructure within the surface region caused by changes in grain size. We employed a transverse-to-transverse scattering model to provide the gradient feature of the grain size profile in the depth direction. The extracted gradient profiles of different carburised case depths exhibit two overlapped growth rates with high- and low-slope factors, respectively. With the increase of the carburised case depth, the positions of the centre points increase and the slope factors decrease. Based on these observations, a modified double sigmoidal function was utilised to parametrise the gradient profile curve. The relationship among the selected gradient feature parameters and various carburised case depths has been established. The predicted results were compared to hardness measurements with good agreement.

Quantitative evaluation of spheroidisation in 15CrMo steel based on deep learning

ABSTRACT 15CrMo steel has good mechanical properties and is widely used in high-temperature pressure components. After long term service, it is easy to cause material spheroidisation. The metallographic and mechanical property tests are used to evaluate the spheroidisation of specimens. Ultrasonic backscattering signal is sensitive to microstructure and contains more microstructure information, which can be used for the evaluation of spheroidisation. However, the ultrasonic backscattering signal is nonlinear, and feature extraction is difficult. In this study, ultrasonic testing is used to scan different spheroidisation specimens, and the ultrasonic backscattering signal is extracted as the input to the deep learning model, which is used to extract features from the backscattering signal. The models are evaluated using classification and regression evaluation metrics. The results show that the proposed CNN-LSTM model has good identification performance for the classification of spheroidisation and the prediction of mechanical properties. The classification accuracy, recall, precision, and F1-score are all 1. Additionally, the maximum predicted RMSE and MAE values are only 2.33 MPa and 1.70 MPa, and the minimum R 2 is only 0.97. The worst prediction is the tensile strength, with an average value of 442.2 MPa and a maximum value of 481.1 MPa.

Backscatter measurement of cancellous bone using the ultrasound transit time spectroscopy.

Recently, ultrasound transit time spectroscopy (UTTS) was proposed as a promising method for bone quantitative ultrasound measurement. Studies have showed that UTTS could estimate the bone volume fraction and other trabecular bone structure in ultrasonic through-transmission measurements. The goal of this study was to explore the feasibility of UTTS to be adapted in ultrasonic backscatter measurement and further evaluate the performance of backscattered ultrasound transit time spectrum (BS-UTTS) in the measurement of cancellous bone density and structure. First, taking ultrasonic attenuation into account, the concept of BS-UTTS was verified on ultrasonic backscatter signals simulated from a set of scatterers with different positions and intensities. Then, in vitro backscatter measurements were performed on 26 bovine cancellous bone specimens. After a logarithmic compression of the BS-UTTS, a linear fitting of the log-compressed BS-UTTS versus ultrasonic propagated distance was performed and the slope and intercept of the fitted line for BS-UTTS were determined. The associations between BS-UTTS parameters and cancellous bone features were analyzed using simple linear regression. The results showed that the BS-UTTS could make an accurate deconvolution of the backscatter signal and predict the position and intensity of the simulated scatterers eliminating phase interference, even the simulated backscatter signal was with a relatively low signal-to-noise ratio. With varied positions and intensities of the scatterers, the slope of the fitted line for the log-compressed BS-UTTS versus ultrasonic propagated distance (i.e., slope of BS-UTTS for short) yield a high agreement (r2 = 99.84%-99.96%) with ultrasonic attenuation in simulated backscatter signal. Compared with the high-density cancellous bone, the low-density specimen showed more abundant backscatter impulse response in the BS-UTTS. The slope of BS-UTTS yield a significant correlation with bone mineral density (r = 0.87; p < 0.001), BV/TV (r = 0.87; p < 0.001), and cancellous bone microstructures (r up to 0.87; p < 0.05). The intercept of BS-UTTS was also significantly correlated with bone densities (r = -0.87; p < 0.001) and trabecular structures (|r|=0.43-0.80; p < 0.05). However, the slope of the BS-UTTS underestimated attenuation when measurements were performed experimentally. In addition, a significant non-linear relationship was observed between the measured attenuation and the attenuation estimated by the slope of the BS-UTTS. This study demonstrated that the UTTS method could be adapted to ultrasonic backscatter measurement of cancellous bone. The derived slope and intercept of BS-UTTS could be used in the measurement of bone density and microstructure. The backscattered ultrasound transit time spectroscopy might have potential in the diagnosis of osteoporosis in the clinic.

Ultrasonic backscattering model for Rayleigh waves in polycrystals with Born and independent scattering approximations

This paper presents theoretical and numerical models for the backscattering of 2D Rayleigh waves in single-phase, untextured polycrystalline materials with statistically equiaxed grains. The theoretical model, based on our prior inclusion-induced Rayleigh wave scattering model and the independent scattering approximation, considers single scattering of Rayleigh-to-Rayleigh (R–R) waves. The numerical finite element model is established to accurately simulate the scattering problem and evaluate the theoretical model. Good quantitative agreement is observed between the theoretical model and the finite element results, especially for weakly scattering materials. The agreement decreases with the increase of the anisotropy index, owing to the reduced applicability of the Born approximation. However, the agreement remains generally good when weak multiple scattering is involved. In addition, the R–R backscattering behaviour of 2D Rayleigh waves is similar to the longitudinal-to-longitudinal and transverse-to-transverse backscattering of bulk waves, with the former exhibiting stronger scattering. These findings establish a foundation for using Rayleigh waves in the quantitative characterisation of polycrystalline materials.

Ultrasonic evaluation of additively manufactured components with embedded structures

In recent years, the use of ultrasonic nondestructive evaluation (NDE) for in situ and ex situ assessment of Additive Manufacturing (AM) parts has garnered interest due to its proficiency in defect detection and material characterization. This study focuses on evaluating the ability to resolve internal structures created by AM and hybrid AM using ultrasonic NDE as well as assessing the impact of microstructure heterogeneity. For this purpose, first the parameter space for laser powder bed fusion (LPBF) of SS316L is mapped in a Lumex Avance 25 system. Based on this parameter mapping, optimal process parameters are identified and sets of process parameters are selected to vary the thermal history of the samples, while minimizing the influence of porosity. Two groups of samples are printed with these parameter sets: a solid group of samples, and a group with internal structures, such that each internal structure sample has an equivalent solid sample. Measurements of ultrasonic velocity, attenuation, and backscatter amplitude are collected and compared against the intended geometry to assess the impact of non-optimal processing on the ability to resolve internal structures with ultrasound. The viability and limitations of using ultrasound to assess internal features created with AM are discussed.

Efficient grain size evaluation based on single direction measurement of ultrasonic backscattering coefficient

&lt;sec&gt;GH4742 nickel-based superalloy exhibits excellent mechanical properties, and grain size is a key factor affecting its performance. A physical model-based ultrasonic backscattering method makes grain size measurement accurate and efficient. Nevertheless, it is constrained by complex models or multiple measurements taken from various beam angles. As a result, a backscattering coefficient method that requires only a single measurement for grain size evaluation is proposed. In contrast to the existing methods, the proposed method solely focuses on the backscattering coefficient component of the backscattering signal. It effectively eliminates the influence of unrelated factors, such as the measurement system and the acoustic field, through the utilization of reference signals.&lt;/sec&gt;&lt;sec&gt;The independent scattering model is employed to derive the backscattering coefficient, which solely pertains to the material itself. The relationship between grain size and backscattering coefficient is described by using a spatial correlation function. To consider the irrelevant factors, an experimental measurement method is developed by using the reference signals. Through numerical calculation and analysis, it has been observed that the backscattering coefficient is closely related to the frequency. When the product of the wavenumber and the grain size is significantly greater than 1 (&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ ka\gg 1 $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), a Stochastic scattering limit is reached. Conversely, when &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ ka\ll 1 $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, a Rayleigh scattering limit is observed. Furthermore, the backscattering coefficient is directly proportional to the grain size. As a general trend, larger grain sizes result in higher backscattering coefficient.&lt;/sec&gt;&lt;sec&gt;Three sets of GH4742 specimens with different grain sizes are prepared for phased array ultrasound experiments. It can be observed that the experimental backscattering coefficients, root mean square (RMS) values, and the amplitude trend of time domain signal are consistent. To perform grain size inversion, the backscattering coefficients in the effective bandwidth range of the probe are selected. By utilizing the least-square method, the theoretical backscattering coefficient is employed to fit the curves of the experimental backscattering coefficients. The evaluation results are compared with those obtained by metallographic analysis. The results show that the grain sizes obtained by the proposed method have a maximum relative error of –22.7% and a minimum relative error of –3.7%.&lt;/sec&gt;

Ultrasonic scattering model and identification experiment of polycrystalline materials

Ultrasonic identification has an important application value for national defense, military affairs, aerospace, nuclear facilities and other high-tech fields. Ultrasonic waves can be used to identify any metal material. At present, the researches focus on algorithms for identifying the difference in ultrasonic signal among materials, but the study on the corresponding identification theory is lacking. In this work, 10 primary models of the microstructure of 2A12 aluminum alloy are established as analogies to the complex microstructures of polycrystalline metallic materials. The grains of these models are different from each other in size, separation distance, shape, arrangement directions and orders. The time-domain ultrasonic echo signals of different microstructures are calculated by making use of the finite element method. The grass-like signals between two echoes are ultrasonic backscattering signals, which are sensitive to any change of microstructure. The backscattering signals between the primary echo and the secondary echo in the ultrasonic echo time domain signals are extracted as ultrasonic fingerprints. The feature difference &lt;i&gt;Q&lt;/i&gt; is defined to quantify the difference in ultrasonic fingerprint of each sample. The results show that the slight variation in microstructure will lead to difference in ultrasonic signal, and the difference caused by the variation in grain size is more distinct. And then, an ultrasonic identification algorithm is proposed, and the identification experiments are conducted on four 2A12 aluminum alloy samples with the same shape. The identification results show that the target sample can be accurately identified by using ultrasonic fingerprints and the ultrasonic fingerprints of the target sample are distinctly different from those of the other samples. The microstructure morphologies of the samples are examined by using scanning electron microscopy (SEM). The SEM results show that there are significant differences in grain size, separation distance and densification between samples although they are the same material. The features of the microstructure in the proposed ultrasonic scattering model in this work are confirmed by the actual y micromorphologies observed in the SEM images. The identification experiments and SEM results demonstrate that the established ultrasonic scattering model is effective. This work can provide a reference for theoretically studying ultrasonic identification and present an idea for developing some new identification algorithms in future.

Collagenase treatment reduces the anisotropy of ultrasonic backscatter in rat myocardium by reducing collagen crosslinks.

Dysregulation of collagen deposition, degradation, and crosslinking in the heart occur in response to increased physiological stress. Collagen content has been associated with ultrasonic backscatter (brightness), and we have shown that the anisotropy of backscatter can be used to measure myofiber alignment, that is, variation in the brightness of a left ventricular short-axis ultrasound. This study investigated collagen's role in anisotropy of ultrasonic backscatter; female Sprague-Dawley rat hearts were treated with a collagenase-containing solution, for either 10 or 30 min, or control solution for 30 min. Serial ultrasound images were acquired at 2.5-min intervals throughout collagenase treatment. Ultrasonic backscatter was assessed from anterior and posterior walls, where collagen fibrils are predominately aligned perpendicular to the angle of insonification, and the lateral and septal walls, where collagen is predominately aligned parallel to the angle of insonification. Collagenase digestion reduced backscatter anisotropy within the myocardium. Collagen remains present in the myocardium throughout collagenase treatment, but crosslinking is altered within 10 min. These data suggest that crosslinking of collagen modulates the anisotropy of ultrasonic backscatter. An Anisotropy Index, derived from differences in backscatter from parallel and perpendicularly aligned fibers, may provide a noninvasive index to monitor the progression and state of myocardial fibrosis.

Effect of transducer position on ultrasonic backscatter measurements of cancellous bone.

Ultrasonic backscatter techniques are being developed to detect changes in bone caused by osteoporosis and other diseases. Backscatter measurements performed at peripheral skeletal sites such as the heel may place the interrogated region of bone tissue in the acoustic near field of the transducer. The purpose of this study is to investigate how measurements in the near field affect backscatter parameters used for ultrasonic bone assessment. Ultrasonic measurements were performed in a water tank using a planar 2.25 MHz transducer. Signals were acquired for five transducer-specimen distances: N/4, N/2, 3 N/4, N, and 5 N/4, where N is the near-field distance, a location that represents the transition from the near field to far field. Five backscatter parameters previously identified as potentially useful for ultrasonic bone assessment purposes were measured: apparent integrated backscatter, frequency slope of apparent backscatter (FSAB), frequency intercept of apparent backscatter, normalized mean of the backscatter difference, and backscatter amplitude decay constant. All five parameters depended on transducer-specimen distance to varying degrees with FSAB exhibiting the greatest dependence on distance. These results suggest that laboratory studies of bone should evaluate the performance of backscatter parameters using transducer-specimen distances that may be encountered clinically including distances where the ultrasonically interrogated region is in the near field of the transducer.

Ultrasonic backscatter measurements of cancellous bone in the near field of a planar transducer

Ultrasonic backscatter techniques may be used to detect changes in bone caused by osteoporosis and other diseases. Typical measurement locations include peripheral skeletal sites such as the heel, which may place the interrogated region of bone tissue in the acoustic near field of the transducer. The purpose of this study is to investigate how measurements in the near field of a planar transducer affect backscatter parameters used for ultrasonic bone assessment. Ultrasonic measurements were performed in a water tank using a planar 2.25 MHz transducer. Signals were acquired for five transducer-specimen distances: N/4, N/2, 3N/4, N, and 5N/4 where N is the near-field distance, a location that represents the transition from the near field to far field. Three backscatter parameters previously identified as potentially useful for ultrasonic bone assessment purposes were investigated: AIB, FSAB, and FIAB. All three parameters depended on transducer-specimen distance to varying degrees with FSAB exhibiting the greatest dependence on distance. These results suggest that laboratory studies of bone should evaluate the performance of backscatter parameters using clinically relevant transducer-specimen distances including distances where the ultrasonically interrogated region is in the near field of the transducer.

The Protocol of Ultrasonic Backscatter Measurements of Musculoskeletal Properties.

This study aims to introduce the protocol for ultrasonic backscatter measurements of musculoskeletal properties based on a novel ultrasonic backscatter bone diagnostic (UBBD) instrument. Dual-energy X-ray absorptiometry (DXA) can be adopted to measure bone mineral density(BMD) in the hip, spine, legs and the whole body. The muscle and fat mass in the legs and the whole body can be also calculated by DXA body composition analysis. Based on the proposed protocol for backscatter measurements by UBBD, ultrasonic backscatter signals can be measured in vivo, deriving three backscatter parameters [apparent integral backscatter (AIB), backscatter signal peak amplitude (BSPA) and the corresponding arrival time (BSPT)]. AIB may provide important diagnostic information about bone properties. BSPA and BSPT may be important indicators of muscle and fat properties. The standardized backscatter measurement protocol of the UBBD instrument may have the potential to evaluate musculoskeletal characteristics, providing help for promoting the application of the backscatter technique in the clinical diagnosis of musculoskeletal disorders (MSDs), such as osteoporosis and muscular atrophy.

A receiver-optimized total focusing method for detectability enhancement of small defects in coarse grained materials

The identification and localization of small defects are among the main objectives of ultrasonic non-destructive testing (NDT). In particular, the inspection of polycrystalline materials remains a challenge due to ultrasonic attenuation and backscatter noise caused by grain scattering. An adaptive ultrasonic array imaging algorithm is proposed in this paper, which is capable of enhancing the signal-to-noise ratio (SNR) of small subwavelength defects. The proposed approach adopts an iterative optimization procedure based on sequential backward selection, and is termed receiver optimized total focusing method (ROTFM). ROTFM is formulated in a baseline subtraction (BS) setting, and it is shown to achieve higher defect SNRs compared to TFM results obtained by directly applying the BS method to measured array data. More importantly, the performance of the proposed ROTFM is robust to time and spatial misalignments, and hence, it alleviates the need for complex compensation schemes when performing imaging computations. Compared to conventional TFM, ROTFM achieved an average SNR improvement of 3.41 dB and 6.05 dB in experiments for four test defects (cracks and holes of sizes 0.5 mm and 1 mm) at the frequencies of 5 MHz and 7 MHz, respectively.

Simultaneous imaging of ultrasonic relative backscatter and attenuation coefficients for quantitative liver steatosis assessment

Prevalence of liver disease is continuously increasing and nonalcoholic fatty liver disease (NAFLD) is the most common etiology. We present an approach to detect the progression of liver steatosis based on quantitative ultrasound (QUS) imaging. This study was performed on a group of 55 rats that were subjected to a control or methionine and choline deficient (MCD) diet known to induce NAFLD. Ultrasound (US) measurements were performed at 2 and 6 weeks. Thereafter, animals were humanely euthanized and livers excised for histological analysis. Relative backscatter and attenuation coefficients were simultaneously estimated from the US data and envelope signal-to-noise ratio was calculated to train a regression model for: (1) fat fraction percentage estimation and (2) performing classification according to Brunt’s criteria in grades (0 <5%; 1, 5–33%; 2, >33–66%; 3, >66%) of liver steatosis. The trained regression model achieved an R^2 of 0.97 (p-value < 0.01) and a RMSE of 3.64. Moreover, the classification task reached an accuracy of 94.55%. Our results suggest that in vivo QUS is a promising noninvasive imaging modality for the early assessment of NAFLD.

Spatial imaging of stratified heterogeneous microstructures: determination of the hardness penetration depth in thermally treated steel parts by laser ultrasound

We introduce an improved non-destructive and non-contact measurement technique for the hardness penetration depth of thermally hardened steel parts based on all-optical laser ultrasound. The hardening of the near-surface region results in a stratified system with layers of different microstructural characteristics caused by changes in grain size. We employ spatially resolved ultrasonic backscattering signals to provide subsurface imaging of the axial and lateral hardness profile. Spatial scans over a range of 30 mm with a lateral step size of 100 μm were obtained within 30 s. We investigated three common steel grades with different microstructural characteristics and hardened layer thicknesses ranging from about 3 to 10 mm. The hardened surface layer and the unhardened core material can be identified. The data evaluation is assisted by a supervised machine learning approach that takes advantage of the spatio-temporal ultrasonic backscattering information. This method also provides an alternative route to define the hardness penetration depth using spatio-temporal scattering characteristics.