The Method of Small-Volume Expansions for Medical Imaging

Inverse problems in medical imaging are in their most general form ill-posed. They literally have no solution. If, however, in advance we have additional structural information or supply missing information, then we may be able to determine specific features about what we wish to image with a satisfactory resolution and accuracy. One such type of information can be that the imaging problem is to find unknown small anomalies with significantly different parameters from those of the surrounding medium. These anomalies may represent potential tumors at early stage. Over the last few years, the method of small-volume expansions has been developed for the imaging of such anomalies. The aim of this chapter is to provide a synthetic exposition of the method, a technique that has proven useful in dealing with many medical imaging problems. The method relies on deriving asymptotics. Such asymptotics have been investigated in the case of the conduction equation, the elasticity equation, the Helmholtz equation, the Maxwell system, the wave equation, the heat equation, and the Stokes system. A remarkable feature of this method is that it allows a stable and accurate reconstruction of the location and of some geometric features of the anomalies, even with moderately noisy data. In this chapter we first provide asymptotic expansions for internal and boundary perturbations due to the presence of small anomalies. We then apply the asymptotic formulas for the purpose of identifying the location and certain properties of the shape of the anomalies. We shall restrict ourselves to conductivity and elasticity imaging and single out simple fundamental algorithms. We should emphasize that, since biological tissues are nearly incompressible, the model problem in elasticity imaging we shall deal with is the Stokes system rather than the Laḿe system. The method of small-volume expansions also applies to the optical tomography and microwave imaging. However, these techniques are not discussed here. We refer the interested reader to, for instance, [2]. © 2009 Springer-Verlag Berlin Heidelberg.