Detection of Small Tumors in Microwave Medical Imaging Using Level Sets and Music

We focus on the application of microwaves for the early detection of breast cancer. We investigate the potential of a novel strategy using shapes for modeling the tumor in the breast. An inversion using a shape-based model offers several advantages like well-defined boundaries and the incorporation of an intrinsic regularization that reduces the dimensionality of the inverse problem whereby at the same time stabilizing the reconstruction. We explore novel level-set techniques as a means to detect the tumor without any initialization of its position and size. We present some numerical resonstructions and we compare them with the conventional MUSIC algorithm, in particular with respect to the frequency which is used for the investigation. We show that for different frequencies these two methods show a different qualitative behaviour in the reconstructions. Microwave imaging shows significant promise as a new technique for the early detection of breast cancer (see (5) and references therein). This is so because of the high contrast between the dielectric properties of the healthy breast tissue and the malignant tumors at microwave frequencies. As a consequence, microwave imaging may be used as a clinical complement to the conventional mammography which is based on the attenuation of X-rays that go through the breast. We note that mammographies offer high resolution images but with low contrast. Several image reconstruction algorithms have been investigated during the last years for the detection and location of breast tumors using active microwave imaging. In this application, one is typically not so much interested on the detailed reconstruction of the spatial distribution of the dielectric properties (which would require by far more data than there are usually available), but mainly to answer in a fast, harmless and inex- pensive way the following three questions: (i) whether or not there is a malignant tumor, (ii) its (approximate) location, and (iii) its (approximate) size. Once these questions have been answered reliably, more details can be investigated if necessary by alternative (but then typically more expensive) imaging techniques. In this paper we investigate the use of the level set technique (see (4,7-11) and references given there for details) as a means to detect the presence, location and size of small tumors if their properties are assumed to be known. The main difficulty in this work is the extremely limited view to thedomain of interest due to a very specific source-receiver geometry: all sources and receivers are located at the same side of the domain. Our observation from earlier work (4) has been that in these situations the level set iteration, when initiated with an arbitrary starting guess for the shape, tends to suffer from local minima, which makes it difficult to reliably detect the correct location of the tumor. Therefore, we have investigated an adaptation of our level set approach to this new situation which is able to start without any pre-specified starting guess for the shape. Our algorithm is able to create shapes in any location of the domain. It does so during the early iterations taking into account the data and the sensitivity mapping of the inverse problem. Once a good first approximation for the shape is found, it continues in a completely automatic way with optimizing this shape until the data least squares cost functional is sufficiently reduced. We compare the results of numerical experiments for this new reconstruction algorithm with those of a straightforward (and non-optimized) implementation of the MUSIC algorithm (for a detailed theoretical and numerical investigation of this imaging scheme see for example (1-3,6) and the references given there). Some conclusions of this comparison are given at the end of this paper.