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Physical principles underlying the application of the techniques of computed tomography to quantitative imaging of the ultrasonic attenuation within soft tissue specimens are explored. From a phenomenological model of the propagation of ultrasound through inhomogeneous media, appropriate methods are reviewed that provide attenuation measurements consistent with the imaging equations of computed tomography. Specifically, the use of a phase-insensitive acoustoelectric receiving transducer is demonstrated to eliminate phase cancellation errors. Attenuation reconstructions based on the slope of the attenuation coefficient as a function of frequency are shown to reduce image artifacts arising from reflection and refraction. A new technique is described to compensate for the frequency dependence of the transmitter directivity pattern in the computation of the slope. Reconstructed images of excised hearts are used to illustrate these techniques. The use of ultrasonic computed tomography to identify tissue pathology in vitro is demonstrated using myocardial infarction in the dog as a model. Results of attenuation and time-of-flight reconstructions are compared, with both methods demonstrated to be capable of differentiating normal myocardium from infarct. The consequences of anisotropy, i.e., the dependence of the attenuation on the direction of ultrasonic propagation, are discussed in the context of computed tomographic imaging.