The spectrum of the backscattered response from ultrasonic scans of weakly scattering media such as tissue can provide additional contrast compared to ultrasound B-mode images and, in the case of medical diagnosis, increased sensitivity and specificity to disease conditions. A modeling approach that accounts for transducer beam diffraction effects, predicts the response from a three-dimensional spatial map of acoustic properties that is several wavelengths in size containing features that are on the order of the size of a cell, and relates the structure of the medium to the spectrum of the response can be used to predict and interpret this contrast. Currently, no modeling approach has been proposed to accomplish this task which meets these criteria.This dissertation presents a systematic approach to modeling the backscattered spectrum of the response from weakly scattering random media consisting of discrete scatterers located in the beam of a transducer. The approach builds on the concept of the transducer spatial impulse response and extends the model to generate the frequency-domain response from collections of discrete scatterers with a specified size, shape, and orientation. Single spherical and cylindrical scatterers were studied first, and the backscattered response throughout the focal plane was related to the plane-wave response or intensity form factor. The model predicted a response at the focus which closely agreed over the bandwidth (1.5 to 3 MHz) of a weakly focused transducer with the intensity form factor for scatterers with a diameter equal to half of a beamwidth. A bias was predicted for the same scatterers with respect to the form factor for a more highly focused transducer, where the scatterers were a full beamwidth in diameter. The bias was quantified in terms of root mean squared error (RMSE) between predicted and plane-wave responses, and the response for the more focused transducer was found to correspond closely (RMSE < 3 dB) to the response for a scatterer that was 2% smaller that the true scatter size for both cylindrical and spherical scatterers. Experimental measurements were conducted to validate model predictions. Single fish eggs and water cylinders in agar were scanned throughout the focal plane of a single-element transducer, and the response was compared to the model response for a weakly scattering sphere or cylinder. A comparison of measurement scans of the same spherical or cylindrical scatterer with two transducers having the same diameter and center frequency but different focal lengths revealed a shift in the response due to diffraction effects consistent with model predictions. The measured response for the more strongly focused transducer corresponded closely (RMSE = 2.61 dB, 1.5 to 3 MHz) to the form factor for a scatter that was 1.1% and 1.4% smaller (sphere and cylinder, respectively) than the response for the more weakly focused transducer. The model response likewise predicted 1.6% and 2.0% differences for spheres and cylinders, respectively. Plane-wave models were evaluated using estimated acoustic properties for the fish egg and water cylinder. The best-fit form factor for the predicted plane-wave response differed from the actual scatter size by 2.4% for fish eggs and 0.4% for fluid cylinders. Next, the model response for collections of identical scatterers was studied.Scatterer positions were independent, and low number densities (<10%) were considered, for which the average plane-wave response is known to be equal to the plane-wave response for a single scatterer. The plane-wave response was accurately estimated (RMSE < 3 dB) over the transducer bandwidth (1.5 to 3 MHz) from averages of independent realizations of collections of spheres and cylinders with radii (500 µm) close to a wavelength (650 µm) in size. This response included a normalization function to account for the transducer focal plane directivity. Containers of insufficient size (axial size < 5λ) were found to produce a deterministic bias within the bandwidth (1.5 to 3 MHz) of the simulation transducer for the collections considered, which resulted in deviations from the expected response of 15 dB. Sphere and cylinder modeling parameters were used to evaluate a new scattering model based on randomly oriented spheroids. The scattering model was considered as an acoustic model for tissue and predicted a smooth response compared to collections of spheres or cylinders, whose response contains nulls. An analytical expression was derived as the form factor for the model, and the expression was compared to model simulations of randomly oriented spheroids. The results agreed closely (RMSE < 1 dB, 1.5 to 3MHz) for three different spheroid size configurations ranging from nearly spherical (B/A = 1.18, where B and A are major and minor axis radii) to strongly non-spherical (B/A = 5.2).The modeling approach was found to consistently represent transducer scans of weakly scattering single scatterers, and predicted important extensions for the class of media for classification with spectral quantitative ultrasound as well as quantified the bias that results for physically large scatterers compared to a transducer beamwidth. The model was used to validate a new form factor model for scattering in tissues and predict the response from a tissue sample based on its histology.
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