Technical advances in ultrasound supported by mathematics include computed tomography (inverse scattering), scatterer number density calculations (statistics), wave elastic tissue interaction (viscoelasticity), ferroelectric transducer development (ceramic physics), and wave equation modeling of ultrasound in viscoelastic materials such as tissue. The mathematical model behind ultrasound computed tomography is presented in section 14.1.3.
Mathematics and physics have greatly influenced the development of ultrasonic imaging, and many challenging problems from physics and the mathematical sciences remain to be solved. Beam forming in nonhomogeneous and usually nonisotropic materials such as biologic tissue is not at all well developed theoretically. Adaptive beam forming, which corrects for variations in refractive index within the imaged field, is a problem that is not solved. Inverse scattering is not a solved problem for geometries in which the sound is either traversing (forward scattering) or reflecting (backscattering) from the object. Acoustic models for the behavior of transducers with the complicated geometries of today's scanners are not well developed.
Following are some viewpoints on the various aspects of ultrasonic imaging.
The instrumentation of ultrasonic imaging consists of the transducer, a beam former, and a signal analysis and display component.
There are three important areas of development pertaining to transducers: (1) field distributions, (2) acoustics and vibration, and (3) electromechanical properties of piezoelectric and ferroelectric materials.
The seminal references in this area are, of course, related to the classic diffraction theory of Huygens and various analyses by Rayleigh, Kirchoff, and Sommerfeld. This theory is particularly applicable to monochromatic radiation and enables accurate calculations of the field distribution from ultrasound radiators under certain conditions. Because it is limited to monochromatic ultrasonic energy, this approach is not well suited to wideband ultrasound imaging.
A method called the impulse response model was developed first by Oberhettinger in 1960 and subsequently refined by Stepanishen in the early 1970s