Ultrasound imaging is a ubiquitous part of the modern medical diagnostics toolbox. It has widespread applications to many areas of medicine, including angiology, cardiology, nephrology, urology, and obstetrics. It is often preferred over other imaging modalities, such as x-ray computed tomography (CAT) and magnetic resonance imaging (MRI) because it is non-invasive, non-ionizing, inexpensive, and has excellent penetration depth in the body. The design, optimization, and manufacturing of ultrasound transducers used in ultrasound imaging is a challenging engineering problem. Faced with a variety of different imaging environments, ultrasound transducers must often be optimized for performance in very specific applications. This is especially true for catheter-based solutions, such as intracardiac and intravascular ultrasound, where imaging performance is strongly dependent on the strength of backscatter from tissue due to significant limitations in device size, electronics, and signal-to-noise ratio. Currently, there is a need for the accurate and fast simulation of the imaging process used in ultrasound imaging, including the ability to capture the effects of backscatter from a variety of different tissues. This thesis discusses the development of simulation tools for the quantitative simulation of tissue backscatter and blood motion from acoustic fields coupled to spatial array transducers, based on an application of the Rayleigh speckle model to the linear systems model for acoustic diffraction from spatial array transducers. These simulation tools have potential applications in the field of medical ultrasonics, with particular attention to the areas of transducer design and optimization, beamforming and array processing, and image reconstruction. We demonstrate how the simulation tools developed here can be used to characterize array imaging performance and to investigate reconstruction performance of common flow algorithms for Doppler ultrasound imaging.