Urban, Matthew W.

We present an inverse problem methodology based on the Error in Constitutive Equations (ECE) approach for the identification of material properties in the context of frequency-domain elastodynamics. In the ECE approach, we define a cost functional based on an energy norm that connects a set of kinematically admissible displacements and a set of dynamically admissible stresses. The set of kinematically admissible displacements is composed of fields that satisfy essential boundary conditions and possess sufficient regularity (i.e. smoothness). The set of dynamically admissible stresses is composed of fields that satisfy conservation of linear momentum and natural (i.e. traction) boundary conditions. The inverse problem is solved by finding material properties along with admissible displacement and stress fields such that the ECE functional is minimized. Experimental data is introduced in the formulation as a quadratic penalty term added to the ECE functional. The talk will focus on the reconstruction of elastic and viscoelastic properties in heterogeneous materials in the context of frequency-domain dynamics. Our findings indicate that ECE methods provide faster and more accurate results than conventional least-squares minimization. We will show numerical and experimental results that demonstrate the salient features of the method.

The cardiovascular diseases such as atherosclerosis, coronary artery disease, hypertension and diastolic dysfunction have been associated with arterial stiffening and decreased ventricular compliance. Noninvasive techniques capable of quantifying elasticity and viscosity of cardiovascular tissues could facilitate early diagnosis and improve treatment. Here, we present a technique that uses focused ultrasound radiation force to excite mechanical waves in the tissue of interest and pulse echo to track the tissue deformation. Analysis of tissue deformation as a function of time using Fourier transforms allows calculation of the phase wave velocity dispersion (change of velocity as a function of frequency) of various modes of deformation. Continuum mechanics equations governing the motion of a viscoelastic plate and a tube are used to model the myocardial wall and arteries, respectively. Dispersion equations are derived for the two geometries and fit to the measured velocity dispersion to estimate tissue elasticity and viscosity. ECG-gated in vivo measurements of porcine myocardial and arterial elasticity and viscosity through a heart cycle are reported. The results show that both elasticity and viscosity increase during systole and decrease during diastole in the myocardium and arteries, consistent with underlying physiology.

New vibration pulses are proposed to increase the power of shear waves induced by ultrasound radiation force in a tissue region with a preferred spectral distribution. The new pulses are sparsely sampled from an orthogonal frequency wave composed of several sinusoidal signals. Those sinusoidal signals have different frequencies and are orthogonal to each other. The phase and amplitude of each sinusoidal signal are adjusted to control the shape of the orthogonal frequency wave. Amplitude of the sinusoidal signal is increased as its frequency increases to compensate for higher loss at higher frequency in the tissue region. The new vibration pulses and detection pulses can be interleaved for array transducer applications. The experimental results show that the new vibration pulses significantly increases induced tissue vibration with the same peak ultrasound intensity, compared with the binary vibration pulses.

In the past several decades, the fields of ultrasound and magnetic resonance elastography have shown promising results in noninvasive estimates of mechanical properties of soft tissues. These techniques often rely on measuring shear wave velocity due to an external or internal source of force and relating the velocity to viscoelasticity of the tissue. The mathematical relationship between the measured velocity and material properties of the myocardial wall, arteries, and other organs with non-negligible boundary conditions is often complicated and computationally expensive. A simple relationship between the Lamb–Rayleigh dispersion and the shear wave dispersion is derived for both the velocity and attenuation. The relationship shows that the shear wave velocity is around 20% higher than the Lamb–Rayleigh velocity and that the shear wave attenuation is about 20% lower than the Lamb–Rayleigh attenuation. Results of numerical simulations in the frequency range 0–500 Hz are presented.

Diastolic dysfunction is characterized by the stiffening of the left-ventricular myocardium. An imaging technique capable of quantifying viscoelasticity of the left-ventricle would aid clinicians in evaluating diastolic function. Lamb wave dispersion ultrasound vibrometry (LDUV) is an ultrasound-based method for measuring viscoelasticity of plate-like materials by exciting Lamb waves and measuring change of wave velocity as a function of frequency (dispersion). Lamb wave dispersion equation is fit to the dispersion data to estimate mechanical properties. We report in vivo transthoracic studies aimed at quantifying elasticity and viscosity of a beating porcine left-ventricle using LDUV. A single ultrasound probe operating at 5 MHz was used for impulse excitation in the myocardium and to track the wave motion. A Fourier space-time analysis of the motion was used to obtain Lamb wave velocity dispersion. ECG-gated transthoracic in vivo measurements of group velocity, elasticity, and viscosity throughout a single heart cycle were obtained. Group velocity, elasticity, and viscosity in the frequency range from 50 to 500 Hz increased by at least two-fold from diastole to systole, consistent with contraction and relaxation of the myocardium.

Tissue mechanical properties can be obtained by algebraic direct inversion (ADI) of the shear-wave motion differential equation, which is insensitive to reflection and geometry of the pushing beam. Shear waves normally generated by a focused ultrasound pushing beam have limited spatial extent in depth and are transient in time, leading to noisy and unstable ADI results. A comb-shape acoustic radiation force distribution can generate shear waves with longer spatial extent and time duration, facilitating more robust ADI. For 400 μs, a linear array transducer simultaneously transmitted four unfocused pushing beams (12 on elements for each pushing beam, 8 off elements between beams) into a tissue-mimicking phantom (∼1.5 kPa) with a cylindrical inclusion (∼4 kPa). The ultrasound system (Verasonics Inc.) then immediately switched to flash imaging mode (frame rate = 2 kHz, spatial resolution = 0.31 mm) to measure shear-wave displacements in a 38 mm by 30 mm domain that was used for ADI. The reconstructed 2D shear elasticity map provides accurate shear elasticity estimates (background = 1.5 kPa with 7% variance; inclusion = 3.9 kPa with 15% variance) and excellent contrast between the background and inclusion.

In the field of tissue engineering, monitoring the elasticity of developing tissue cultures is important for evaluating their growth and maturation. In this work, we used wave propagation methods to measure the elasticity of thin tissue-mimicking layers to test the feasibility of measuring the elasticity of developing tissue cultures. We developed finite element and analytical models for investigating this problem and found good agreement between the results from both models. We used ultrasound radiation force to induce propagating waves, and the waves are measured using high frame rate ultrasound imaging at 10 kHz. The wave modes were identified and compared with an analytic model for Rayleigh wave propagation in a thin elastic layer bound to a solid substrate. The same gelatin mixture was used to make phantoms 1, 4, and 50 mm thick, where the phantom with 50 mm thickness was used as control and evaluated with shear wave imaging methods. The analytic model was used to fit data from experiments in the large block and thin layers of 1 and 4 mm thick, and the measured shear moduli were 22.7, 21.8, and 22.5 kPa, respectively. These results provide a validation for measurement of elasticity in thin elastic layers.

This paper describes the principles and initial experimental results of a new Doppler method called vibro-acoustic Doppler (VAD). VAD uses acoustic response of a moving object to a highly localized dynamic radiation force of ultrasound to measure the velocity of the object. The low frequency radiation force is exerted by two co-focused ultrasound beams with slightly different frequencies. The acoustic response of the object is detected by a hydrophone. A formula that describes the relation between the Doppler frequency shift of the emitted acoustic field and the velocity of the moving object is reported. To verify the theory, experiments are conducted on a moving string and a fluid flow phantom. Results show that the error in velocity measurement is less than 9.1% for either phantom. An advantage of this method over the traditional ultrasound Doppler is that velocity measurement with VAD is almost independent of the angle between the ultrasound beam and motion direction. It is shows that in the worst case, the error is 10.3% for a ±30 deg angle variation. Potential biomedical applications of VAD will be discussed.

Propagating shear waves in tissue can be measured with high frame rate Doppler or correlation methods. The measured characteristics of the shear waves, such as speed versus frequency, can be used to deduce material properties such as complex viscoelastic modulus using physics models appropriate to the geometry and properties of the tissue. This inverse problem is characterized by calculating the storage and loss modulus as a function of frequency and requires appropriate tissue motion, which in turn requires optimized dynamic radiation force distributions. We will discuss novel radiation force distributions that provide enhanced tissue motions appropriate to dealing with the inverse problem of determining tissue material properties from ultrasonically measured tissue motion.

The response of an embedded sphere in a viscoelastic medium excited by acoustic radiation force has been studied in both the time- and frequency-domains. This model is important because it can be used to characterize the viscoelastic properties of the medium by fitting the response to the theoretical model. The Kelvin–Voigt model has been used exclusively in these models. An extension to the previously reported models is described so that any viscoelastic rheological model can be used. This theoretical development describes the generalized embedded sphere response both in the time and frequency domains. Comparing the results from derivations in both domains showed very good agreement with a median absolute error (MAE) ranging from 0.0044 to 0.0072. Good agreement is demonstrated with finite element model simulations and the theory with a MAE of 0.006. Lastly, results for characterization of gelatin and rubber materials with the new theory are shown where the MAE values were used to determine which rheological model best describes the measured responses.