Nonlinear Elasticity Imaging Research Articles

Nonlinear ultrasonic imaging of damage in composite materials using a higher harmonic and modulated multi-path reciprocal method

Structural health monitoring has become an important factor in the assessment of defects/damage in material components. Ultrasonic methods generally incorporate a sparse array of sensors/transducers as they provide a low number of piezoelectric sensors per area, thus providing savings with regard to system costs and weight. Many structural health monitoring techniques rely on linear ultrasonic effects such as reflections, amplitude changes, time of arrival and wave scattering effects which rely on precise baseline measurements. In this work, a nonlinear ultrasonic method based on a sparse array of surface-bonded ultrasonic transducers was used to evaluate the second harmonic and modulated elastic responses from a damaged medium. A complex composite stiffened panel with barely visible impact damage was evaluated. The points closest to damage are found on the paths between transmitter–receiver pairs through a reciprocal relationship of nonlinear elastic parameters and a statistical approach was used to select a cloud of points so that a two-dimensional image of the damaged region is created. Experimental results revealed that the second-order nonlinear parameter provided accurate damage localisation and imaging and the use of modulation bands further improved imaging accuracy and damage localisation. The maximum error between the calculated and the real damage area centres was only 1.3 mm. The proposed nonlinear elastic multi-path imaging technique based on higher harmonic generation and modulation coupled with a statistical approach provides damage detection without a priori knowledge of the material characteristics, which is in direct contrast to conventional linear ultrasonic methods which rely on precise measurement of elastic wave effects both before and after the initiation of damage.

Nonlinear elastic imaging with amplitude and frequency modulated low frequency sources

The use of low frequency waves is the most practical means to investigate nonlinear elastic properties of hysteretic media, such as the strain dependence of wave velocity. Indeed, the rapid increase in damping as a function of frequency makes high frequency waves unable to provide sufficient strain energy to successfully excite nonlinearity. The drawback is that low frequency waves are not suitable for imaging, due to their long wavelength and resulting intrinsic averaging of nonlinear effects over large spatial scales. To address this, we propose here an amplitude modulation of swept sources that allows us to correlate the frequency at which harmonics are detected with the position of the source of nonlinearity, taking advantage of the different strain distributions in the sample at different frequencies.

Shear Induced Non-Linear Elasticity Imaging: Elastography for Compound Deformations.

The goal of non-linear ultrasound elastography is to characterize tissue mechanical properties under finite deformations. Existing methods produce high contrast non-linear elastograms under conditions of pure uni-axial compression, but exhibit bias errors of 10-50% when the applied deformation deviates from the uni-axial condition. Since freehand transducer motion generally does not produce pure uniaxial compression, a motion-agnostic non-linearity estimator is desirable for clinical translation. Here we derive an expression for measurement of the Non-Linear Shear Modulus (NLSM) of tissue subject to combined shear and axial deformations. This method gives consistent nonlinear elasticity estimates irrespective of the type of applied deformation, with a reduced bias in NLSM values to 6-13%. The method combines quasi-static strain imaging with Single-Track Location-Shear Wave Elastography (STL-SWEI) to generate local estimates of axial strain, shear strain, and Shear Wave Speed (SWS). These local values were registered and non-linear elastograms reconstructed with a novel nonlinear shear modulus estimation scheme for general deformations. Results on tissue mimicking phantoms were validated with mechanical measurements and multiphysics simulations for all deformation types with an error in NLSM of 6-13%. Quantitative performance metrics with the new compound-motion tracking strategy reveal a 10-15 dB improvement in Signal-to-Noise Ratio (SNR) for simple shear versus pure compressive deformation for NLSM elastograms of homogeneous phantoms. Similarly, the Contrast-to-Noise Ratio (CNR) of NLSM elastograms of inclusion phantoms improved by 25-30% for simple shear over pure uni-axial compression. Our results show that high fidelity NLSM estimates may be obtained at ~30% lower strain under conditions of shear deformation as opposed axial compression. The reduction in strain required could reduce sonographer effort and improve scan safety.

Advancing elastic wave imaging using thermal susceptibility of acoustic nonlinearity

Despite proven effectiveness in characterizing material degradation and embryonic defects, the use of acoustic nonlinearity is restricted by its intrinsic vulnerability to measurement contamination and to fluctuations in ambient temperature in particular. Analytically, we shed light on the susceptibility of acoustic nonlinearity embodied in elastic waves to ambient temperature. Rather than eliminating or compensating for such thermal susceptibility, we subtly exploit it to advance nonlinear elastic wave imaging. Experimental validation corroborates theoretical prediction, spotlighting the capacity of the approach to improve the precision of material characterization using nonlinear elastic waves and therefore to enhance the accuracy of anomaly imaging when other nonlinearity sources interfere with the extraction of nonlinear attributes of elastic waves.

Nonlinear bulk elasticity imaging using dual frequency ultrasound.

The nonlinear acoustic bulk properties of tissue, e.g., the coefficient of nonlinearity, βn, or the nonlinear bulk elasticity, βp=βnκ0, have been shown to be promising parameters for tissue characterization due to their sensitivity to tissue structure. Previously developed methods for imaging these parameters using single frequency ultrasound have shown success in a laboratory setting using the transmission mode. In the pulse-echo mode, however, unknown absorption, diffraction, and speckle produce unreliable estimates and instability, causing these methods to have achieved no clinical relevance. In this paper, a pulse-echo method for measurement of the nonlinear bulk elasticity is presented using a dual frequency approach. The method is less sensitive to diffraction and absorption due to a separate low frequency manipulation wave. The technique is tested in both simulations and in vitro in a heterogeneous phantom with two regions of different nonlinear properties. Both in simulations and in vitro, a spatial βp map is produced where the two regions are clearly distinguished. In addition, the quantitative estimates of βp obtained are close to the expected values, making the method a promising first step toward in vivo imaging of nonlinear bulk properties.

Nonlinear Shear Modulus Estimation With Bi-Axial Motion Registered Local Strain.

Nonlinear elasticity imaging provides additional information about tissue behavior that is potentially diagnostic and avoids errors inherent in applying a linear elastic model to tissue under large strains. Nonlinear elasticity imaging is challenging to perform due to the large deformations required to obtain sufficient tissue strain to elicit nonlinear behavior. This work uses a method of axial and lateral displacement tracking to estimate local axial strain with simultaneous measurement of shear modulus at multiple compression levels. By following the change in apparent shear modulus and the stress deduced from the strain maps, we are able to accurately quantify nonlinear shear modulus (NLSM). We have validated our technique with a mechanical NLSM measurement system. Our results demonstrate that 2-D tracking provides more consistent NLSM estimates than those obtained by 1-D (axial) tracking alone, especially where lateral motion is significant. The elastographic contrast-to-noise ratio in heterogeneous phantoms was 12.5%-60% higher using our method than that of 1-D tracking. Our method is less susceptible to mechanical variations, with deviations in mean elastic values of 2%-4% versus 5%-37% for 1-D tracking.

Nonlinear elastic imaging of barely visible impact damage in composite structures using a constructive nonlinear array sweep technique

Linear and nonlinear ultrasound imaging methods highlight different damage features: the linear method detects large stiffness changes, while the nonlinear technique identifies small impedance mismatches, such as microcracks or closed delaminations. Typically, nonlinear ultrasound techniques detect damage/defects in materials by measuring higher order harmonics. These harmonics can be difficult to measure due to low magnitude and signal to noise ratios (SNR): hence large excitation amplitudes are needed, which can further complicate the reliability of these methods as equipment nonlinearities can be generated. To overcome these issues, exciting at specific frequencies, known as local defect resonances (LDR), produce a much larger displacements at the damaged regions. However, estimation of LDR is time-consuming, complex and not an easily automated process.A coupled baseline-free linear and nonlinear ultrasonic imaging approach is proposed, using a Constructive Nonlinear Array Sweep excitation and an image subtraction method for identifying damage in layered materials. The signal sweep method uses a narrow band frequency excitation to increase the probability of detection of a LDR frequency. The novel imaging approach was employed using laser vibrometry measurements in various complex composite structures to assess barely visible impact damage, critical for the aircraft industry. The results showed better estimation of impact damage when compared to classical linear or nonlinear ultrasonic methods leading to improved reliability of aircraft inspections.

Nonlinear coda wave interferometry: Characterizing damage in complex solids

In this talk, we report results on the nonlinear interactions of ultrasonic coda waves, in reverberating or multiple scattering mesoscopic solid media. Using the method of coda wave interferometry (CWI), we analyze the effect of mixing a coda wave with an additional lower frequency pump wave. The extracted CWI parameters, known to be highly sensitive to small geometric or elastic modifications of the tested medium, are shown to be pump-amplitude dependent and to capture finely the results of the nonlinear interactions. Although nonlinear self-action effects with coda waves have been reported in unconsolidated granular media, they are difficult to implement in cracked solids or concrete. Instead, the reported nonlinear CWI class of methods (NCWI) shows robustness, a high sensitivity, and has been applied successfully to various complex media and structures. We show through several examples on « model » media (cracked glass plates) and on concrete structures, that NCWI can be useful for the nondestructive evaluation of complex solids that are strongly scattering at usual probing frequencies. Preliminary results and prospects in nonlinear elastic properties imaging and quantitative evaluation with NCWI are discussed.

Nonlinear elasticity imaging with dual frequency ultrasound

Elasticity imaging has shown to enhance diagnostic capabilities of disease as tissue elasticity is often associated with pathological condition. Methods such as Acoustic Radiation Force Impulse imaging and Supersonic Shear Wave imaging have shown good results at shallow depths. Second-order ultrasound field (SURF) imaging is a dual band imaging technique that utilizes a low frequency (LF) pulse to manipulate the nonlinear elasticity of the medium observed by a co-propagating high frequency pulse (HF). The manipulation of the material parameters causes the HF to experience a change in propagation velocity which depends on the manipulation pressure and nonlinear elasticity of the medium. The change in propagation velocity causes an accumulative delay or advancement compared to a single HF pulse, called the nonlinear propagation delay (NPD). By transmitting multiple pulse complexes with different LF polarities the technique has proven capabilities for suppression of multiple scattering noise. By observing local variations in the development of NPD, it is possible to estimate the nonlinear elasticity variation of the medium. Simulations with k-Wave toolbox have shown the methods feasibility, also at greater depths. In-vitro and in-vivo experiments have also yielded promising results.

Nonlinear Coda Wave Interferometry: Detecting, quantifying, and locating damage in complex solids

In this talk, we report results on the nonlinear interactions of ultrasonic coda waves with lower pump waves, in reverberating or multiple scattering mesoscopic solid media. Using the method of coda wave interferometry (CWI), we analyze the effect of mixing a coda wave with an additional lower frequency pump wave. The extracted CWI parameters, known to be highly sensitive to small geometric or elastic modifications of the tested medium, are shown to be pump-amplitude dependent and to capture finely the results of the nonlinear interactions. Although nonlinear self-action effects with coda waves have been reported in unconsolidated granular media, they are difficult to implement in cracked solids or concrete. Instead, the reported nonlinear CWI class of methods (NCWI) shows robustness, a high sensitivity, and has been applied successfully to various complex media and structures. We show through several examples on « model » media (cracked glass plates) and on concrete structures, that NCWI can be useful for the nondestructive evaluation of complex solids that are strongly scattering at usual probing frequencies. Preliminary results and prospects in nonlinear elastic properties imaging and quantitative evaluation with NCWI are discussed.

Beyond conventional migration: non-linear elastic subsalt imaging with transmissions and two-sided illumination

SUMMARY Source–receiver interferometric imaging can be used to synthesize a subsurface acoustic or elastic image, consisting of a zero-time, zero-offset response (or Green’s function) between a colocated pseudo-source and pseudo-receiver placed at each point in the subsurface image. However, if the imaging process does not properly account for multiple reflections, and enclosing boundaries of sources and receivers are not available, the image shows artefacts, poorly illuminated areas and distorted image amplitudes. Here we demonstrate with numerical examples that two-sided non-linear imaging provides the best elastic pure-mode (PP and SS) and converted-mode (PS) images, having higher resolution and more uniform illumination than those obtained from both one-sided linear imaging and from other intermediate steps of imaging (e.g. non-linear one-sided, linear two-sided). We also propose practical approaches to construct the additional fields required by two-sided non-linear imaging without the need for a detailed velocity model and receivers (and/or sources) in the subsurface. Moreover, when conversions are used for imaging, ‘true-amplitude’ images (here true-amplitude means properlyretrievingamplitudesthatrepresentthezero-time,zero-offsetelasticresponse)should theoretically vanish because neither P-to-S or S-to-P conversions arise at zero-time and zerooffset. Applying a correction procedure that accounts for the polarity reversal in PS (or SP) single-shot images helps with their structural interpretation but results in an unphysical estimate of the subsurface response and uninterpretable amplitudes. This suggests that there are advantages in exploiting pure-mode SS reflections/transmissions, in addition to converted waves only, because they require no polarity correction and the resulting image contains meaningful amplitudes that are proportional to the local shear-wave properties of the medium.

Progress in inferring desmoplastic stromal tissue microstructure noninvasively with ultrasound nonlinear elasticity imaging

Desmoplasia is the abnormal growth of fibrous tissue that is often associated with malignancy in breast and other solid tumors. The collagen comprising the extracellular tissue matrix tends to be more dense than in normal breast tissue, and to exhibit altered microstructure. The distinctive mechanical properties of solid tumors are often attributed to the abnormal density and microstructure of their collagen extracellular matrices. For example, a higher concentration of collagen implies a larger linear elastic modulus, while a decrease in fiber tortuosity would mean a smaller “toe” region in the stress-strain curve. This talk describes progress of inferring local tissue microstructural properties from noninvasive, in vivo measurement of nonlinear elastic properties of breast tissue. An ultrasound elasticity-imaging based approach for determining the average local micro-structural properties of tissue was developed and implemented. It utilizes multiscale homogenization methods to determine effective macroscopic properties from assumed microstructural arrangements of collagen fibers. The resulting nonlinear constitutive equation is then used in a model-based elastic inverse problem solver to obtain local microstructural tissue properties from macroscopic observations of tissue deformation. Its potential in diagnosing malignant breast tumors in a small set of patients was also examined. [NSF Grant 50201109; NIH NCI-R01CA140271.]

A nonlinear elasticity phantom containing spherical inclusions

The strain image contrast of some in vivo breast lesions changes with increasing applied load. This change is attributed to differences in the nonlinear elastic properties of the constituent tissues suggesting some potential to help classify breast diseases by their nonlinear elastic properties. A phantom with inclusions and long-term stability is desired to serve as a test bed for nonlinear elasticity imaging method development, testing, etc. This study reports a phantom designed to investigate nonlinear elastic properties with ultrasound elastographic techniques. The phantom contains four spherical inclusions and was manufactured from a mixture of gelatin, agar and oil. The phantom background and each of the inclusions have distinct Young's modulus and nonlinear mechanical behavior. This phantom was subjected to large deformations (up to 20%) while scanning with ultrasound, and changes in strain image contrast and contrast-to-noise ratio between inclusion and background, as a function of applied deformation, were investigated. The changes in contrast over a large deformation range predicted by the finite element analysis (FEA) were consistent with those experimentally observed. Therefore, the paper reports a procedure for making phantoms with predictable nonlinear behavior, based on independent measurements of the constituent materials, and shows that the resulting strain images (e.g., strain contrast) agree with that predicted with nonlinear FEA.

Nonlinear elastic imaging using reciprocal time reversal and third order symmetry analysis

This paper presents a nonlinear imaging method for the detection of the nonlinear signature due to impact damage in complex anisotropic solids with diffuse field conditions. The proposed technique, based on a combination of an inverse filtering approach with phase symmetry analysis and frequency modulated excitation signals, is applied to a number of waveforms containing the nonlinear impulse responses of the medium. Phase symmetry analysis was used to characterize the third order nonlinearity of the structure by exploiting its invariant properties with the phase angle of the input waveforms. Then, a "virtual" reciprocal time reversal imaging process, using only one broadcasting transducer and one receiving transducer, was used to insonify the defect taking advantage of multiple linear scattering as mode conversion and boundary reflections. The robustness of this technique was experimentally demonstrated on a damaged sandwich panel, and the nonlinear source, induced by low-velocity impact loading, was retrieved with a high level of accuracy. Its minimal processing requirements make this method a valid alternative to the traditional nonlinear elastic wave spectroscopy techniques for materials showing either classical or non-classical nonlinear behavior.

On the use of a chaotic cavity transducer in nonlinear elastic imaging

We discuss the advantages of “chaotic cavity transducer focusing” to enhance the localization of microdamage in conjunction with nonlinear elastic wave spectroscopy methods. Chaotic cavity transducer focusing is defined as the hardware-software combination of a piezoelectric ceramic glued on a cavity of chaotic shape on the one hand with the reciprocal time reversal (or inverse filter) technique on the other hand. Additional optimization through the use of sweeps and inverse filtering techniques are discussed as well. The technique is applied to image a crack in a steel sample.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.

Solution of the nonlinear elasticity imaging inverse problem: the compressible case

We discuss and solve an inverse problem in nonlinear elasticity imaging in which we recover spatial distributions of hyperelastic material properties from measured displacement fields. This problem has applications to elasticity imaging of soft tissue because the strain dependence of the apparent stiffness may potentially be used to differentiate between malignant and normal tissues. We account for the geometric and material nonlinearity of the tissues by assuming a known hyperelastic model for the soft tissue. We formulate the problem as the minimization of a cost function representing the difference between the measured and predicted displacement fields. We minimize the cost function with respect to the spatial distribution of material properties using a gradient-based (quasi-Newton) optimization approach. We calculate the gradient efficiently using the adjoint method and a continuation strategy in the material properties. We present numerical examples that demonstrate the feasibility of the approach.

Combination of digital mammography with semi-automated 3D breast ultrasound.

This paper describes work aimed at combining 3D ultrasound with full-field digital mammography via a semi-automatic prototype ultrasound scanning mechanism attached to the digital mammography system gantry. Initial efforts to obtain high x-ray and ultrasound image quality through a compression paddle are proving successful. Registration between the x-ray mammogram and ultrasound image volumes is quite promising when the breast is stably compressed. This prototype system takes advantage of many synergies between the co-registered digital mammography and pulse-echo ultrasound image data used for breast cancer detection and diagnosis. In addition, innovative combinations of advanced US and X-ray applications are being implemented and tested along with the basic modes. The basic and advanced applications are those that should provide relatively independent information about the breast tissues. Advanced applications include x-ray tomosynthesis, for 3D delineation of mammographic structures, and non-linear elasticity and 3D color flow imaging by ultrasound, for mechanical and physiological information unavailable from conventional, non-contrast x-ray and ultrasound imaging.

Nonlinear elasticity imaging:theory and phantom study

In tissue the Young's modulus cannot be assumed constant over a wide deformation range. For example, direct mechanical measurements on human prostate show up to a threefold increase in Young's modulus over a 10% deformation. In conventional elasticity imaging, these effects produce strain-dependent elastic contrast. Ignoring these effects generally leads to suboptimal contrast (stiffer tissues at lower strain are contrasted against softer tissues at higher strain), but measuring the nonlinear behavior results in enhanced tissue differentiation. To demonstrate the methods extracting nonlinear elastic properties, both simulations and measurements were performed on an agar-gelatin phantom. Multiple frames of phase-sensitive ultrasound data are acquired as the phantom is deformed by 12%. All interframe displacement data are brought back to the geometry of the first frame to form a three-dimensional (3-D) data set (depth, lateral, and preload dimensions). Data are fit to a 3-D second order polynomial model for each pixel that adjusts for deformation irregularities. For the phantom geometry and elastic properties considered in this paper, reconstructed frame-to-frame strain images using this model result in improved contrast to noise ratios (CNR) at all preload levels, without any sacrifice in spatial resolution. From the same model, strain hardening at all preload levels can be extracted. This is an independent contrast mechanism. Its maximum CNR occurs at 5.13% preload, and it is a 54% improvement over the best case (preload 10.6%) CNR for frame-to-frame strain reconstruction. Actual phantom measurements confirm the essential features of the simulation. Results show that modeling of the nonlinear elastic behavior has the potential to both increase detectability in elasticity imaging and provide a new independent mechanism for tissue differentiation.

Nonlinear Elasticity Imaging: Theory and Phantom Study

In tissue the Young's modulus cannot be assumed constant over a wide deformation range. For example, direct mechanical measurements on human prostate show up to a threefold increase in Young's modulus over a 10% deformation. In conventional elasticity imaging, these effects produce strain-dependent elastic contrast. Ignoring these effects generally leads to suboptimal contrast (stiffer tissues at lower strain are contrasted against softer tissues at higher strain), but measuring the nonlinear behavior results in enhanced tissue differentiation. To demonstrate the methods extracting nonlinear elastic properties, both simulations and measurements were performed on an agar-gelatin phantom. Multiple frames of phase-sensitive ultrasound data are acquired as the phantom is deformed by 12%. All interframe displacement data are brought back to the geometry of the first frame to form a three-dimensional (3-D) data set (depth, lateral, and preload dimensions). Data are fit to a 3-D second order polynomial model for each pixel that adjusts for deformation irregularities. For the phantom geometry and elastic properties considered in this paper, reconstructed frame-to-frame strain images using this model result in improved contrast to noise ratios (CNR) at all preload levels, without any sacrifice in spatial resolution. From the same model, strain hardening at all preload levels can be extracted. This is an independent contrast mechanism. Its maximum CNR occurs at 5.13% preload, and it is a 54% improvement over the best case (preload 10.6%) CNR for frame-to-frame strain reconstruction. Actual phantom measurements confirm the essential features of the simulation. Results show that modeling of the nonlinear elastic behavior has the potential to both increase detectability in elasticity imaging and provide a new independent mechanism for tissue differentiation.