IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, . 59, . 11, NOVEMBER 2012 2390 0885–3010/$25.00 © 2012 IEEE The Variance of Quantitative Estimates in Shear Wave Imaging: Theory and Experiments Thomas Deffieux, Jean-Luc Gennisson, Benoit Larrat, Mathias Fink, and Mickael Tanter Abstract—In this paper, we investigate the relationship be- tween the estimated shear modulus produced in shear wave imaging and the acquisition parameters. Using the framework of estimation theory and the Cramer–Rao lower bound applied both to the estimation of the velocity field variance and to the estimation of the shear wave travel time, we can derive the analytical formulation of the shear modulus variance σ μ 2 using relevant physical parameters such as the shear wave frequency, bandwidth, and ultrasonic parameters. This variance corre- sponds to the reproducibility of shear modulus reconstruction for a deterministic, quasi-homogeneous, and purely elastic me- dium. We thus consider the shear wave propagation as a deter- ministic process which is then corrupted during its observation by electronic noise and speckle decorrelation caused by shear- ing. A good correlation was found between analytical, numeri- cal, and experimental results, which indicates that this formu- lation is well suited to understand the parameters’ influence in those cases. The analytical formula stresses the importance of high-frequency and wideband shear waves for good estimation. Stiffer media are more difficult to assess reliably with identical acquisition signal-to-noise ratios, and a tradeoff between the reconstruction resolution of the shear modulus maps and the shear modulus variance is demonstrated. We then propose to use this formulation as a physical ground for a pixel-based quality measure that could be helpful for improving the recon- struction of real-time shear modulus maps for clinical applica- tions. I. I T  recent development of quantitative elastography methods is regarded as a highly promising step toward improved diagnosis. Since its introduction, many studies have been published by several groups demonstrating sev- eral clinical applications such as breast cancer detection [1]–[4], liver fibrosis staging [5]–[9], kidney monitoring [10], thyroid gland [11] and prostate cancer detection [12], and musculoskeletal monitoring [13]–[15], ophthalmologic [16], cardiac [17], or vascular applications [18]. Some clini- cal applications, such as liver fibrosis staging [19], require a very accurate and quantitative estimation of the shear modulus to make a diagnosis because the estimated value is directly compared with a pre-established quantitative scale. A quantitative approach is also mandatory for the longitudinal monitoring of drug treatment efficacy (such as chemotherapy, antiangiogenic, or antivascular treat- ments). To allow a quantitative estimation of the shear mod- ulus of tissue, dynamic elastography techniques—or in other words, shear-wave-based imaging techniques—use mechanical shear waves whose propagation velocity is di- rectly linked to the shear modulus of the medium, i.e., its stiffness. After generation of shear waves in the medium, their propagation is recorded either in real time or using a stroboscopic acquisition. By solving the wave equation, it is then possible to retrieve a quantitative value of the shear modulus. There are generally two groups of dynamic elastography techniques depending on the regime of the shear waves used. In steady-state elastography, such as in magnetic resonance (MR) elastography, a monochro- matic shear wave is generated and imaged with a strobo- scopic acquisition, but in 3-D, using an MR imaging sys- tem. These techniques were first developed by Muthupillai and Manduca [20], [21] and are used today for clinical research. Although they allow the full 3-D reconstruction of the shear wave field, the overall complex patterns of the steady-state shear waves require the complete solving of the shear wave equation, which leads to difficulties with regard to robustness and bias because the estimation of second-order derivatives is required [22], [23]. Sonoelastic- ity [24] is another example of a steady-state elastography technique, in which shear waves are acquired with an ul- trasound system. Conversely, in transient elastography, such as in 1-D transient elastography (TE) [25], shear wave elasticity im- aging (SWEI) [26], acoustic radiation force imaging with shear wave speed estimation (ARFI SWS) [27], supersonic shear imaging (SSI) [28], shear wave dispersion ultrasound vibrometrey (SDUV) [29], spatially modulated ultrasound radiation force (SMURF) [30], dynamic micro elastog- raphy (DME) [31] or transient MR elastography [32] (t- MRE), a transient—and consequently wideband—shear wave is generated and tracked during its propagation ei- ther in real time or using a gated acquisition. The key dif- ference is that the shear wave is imaged before it can reach the boundaries and is generally considered as a transient wave front. Under this assumption, the shear modulus can then be estimated by solving the eikonal equation for the wave front instead of the full wave equation with much simpler algorithms, such as a time-of-flight algorithm [2] which locally estimates the shear group velocity along the propagation direction. This approach is more robust to Manuscript received November 18, 2011; accepted July 6, 2012. This work was partly supported by the French national agency for research on AIDS and Viral Hepatitis (ANRS). The authors are with the Institut Langevin, Ondes et Images, Ecole Supérieure de Physique et de Chemie Industrielles (ESPCI, ParisTech), Centre National de la Recherche Scientifique (CNRS) UMR 7587, INSERM U979, France (e-mail: tdeffieux@gmail.com). DOI http://dx.doi.org/10.1109/TUFFC.2012.2472