et. al [10] tout it as the operative mechanism for coagulative necrosis in their apparatus. Our current level of understanding therefore invites conflicting interpretations, hence, uncertainties in treatment protocols that effectively limit the technique. To further develop and qualify tissue ablation by focused ultrasound it is essential that we obtain a better understanding of the tissue response phenomena. They are fundamentally nonlinear since the initial focused beam rapidly changes tissue

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... absorption and wave speed, which change beam focus and promotes scattering, which change tissue properties, and so on. This complexity has precluded comprehensive model studies of thermal ablation. Linear models are used [5, 11] effectively, but cannot reproduce treatmentlimiting behavior, while nonlinear models [12], e.g., based on the KZK equation [13] , are currently too idealized for application to scattering, cavitating focal tissue. The solution we propose is a discrete numerical model that is not limited by common mathematical and material idealizations. Furthermore, it must be versatile and easily applied by researchers and manufacturers who need to design safer, faster therapy protocols and transducers. Our basis is PZFlex [14], a 2D/3D, time-domain finite element code for electromechanical analysis that is widely used by the U.S. ultrasound system industry for linear, transient transducer modeling. PZFlex was originally developed for largescale propagation, structural dynamics, and nonlinear modeling, hence, provides a near-ideal foundation for the present work. This paper examines a number of modeling issues, including focused transducer design, the relevance of shocking in water and tissue, increased focal heating from nonlinearity, and evolution of "bubbly" regions emanating from the focal zone. Our work is in preparation for quantitative validations against laboratory experiments. It is expected that the following model results and general analysis techniques will help us understand anomalous focused ultrasound phenomena, highlight critical tissue characterization needs, augment experiment planning and interpretation, and help optimize treatments. Last but not least, it may reduce the number of in vivo experiments and sacrificed animals currently necessary to validate different focused ultrasound treatments. FOCUSED TRANSDUCER The canonical focused ultrasound transducer is an unbacked spherical cap mounted in a fixture. The element may be solid piezoceramic, annular, or composite. Flat phased arrays are also used for their dynamic focus and scan capability. In the simple case of axial symmetry, the transducer and complete propagation path can often be modeled with a discrete method, depending on fnumber and frequency. It is otherwise necessary to divide the model into coupled zones, e.g., the transduction zone, the linear propagation zone, and the nonlinear focal zone. For example, from ABSTRACT We describe experimental finite element modeling of tissue ablation by focused ultrasound. Emphasis is on nonlinear coupling of high intensity sound, temperature, and tissue properties. The numerical basis for modeling nonlinearity is an incrementally linear, timedomain, finite element algorithm solving the electromechanical and bioheat equations in 2D/3D inhomogeneous elastic and acoustic media. Nonstandard modeling issues examined include harmonic generation/absorption and focal "bubble" evolution with consistent sound and thermal redistribution. The nonlinear pressure-density relation generates harmonics that increase absorption and heating, particularly in the focal zone. In the tissues modeled, harmonic heating is negligible for peak focal intensities of a few kW/cm 2 . As the focal hot spot ablates tissue it may also generate "bubbles." Prefocal growth of a bubbly region is modeled using a simple boiling threshold and strong coupling between the scattered ultrasound and temperature redistribution as the region spreads. Generally, these experiments are intended to develop a more comprehensive modeling basis for quantifying tissue ablation phenomenology.