Inverse problems and biomedical imaging

Table of contents: Mathematical and practical background An example: Numerical inversion in optical tomography Some results Future directions

Mathematically, imaging is usually posed in the form of inverse problems. Within the Department of Mathematics, we conduct active research on both linear inverse problems (Professor Peter Kuchment) as well as on nonlinear versions (Professor Wolfgang Bangerth). For linear inverse problems, typical research questions center around the development of formulas that yield a description of the body's interior as a function of the measured data, as well as on theoretical properties of such formulas, for example stability of reconstruction or what parts of a body can be reconstructed if only a limited amount of data is available.

On the other hand, many recent imaging methodologies are nonlinear, i.e. the operation that describes what we measure (e.g. X-ray intensities) is a nonlinear function of the body's properties that we are interested in (e.g. tissue densities or whether a certain location in a body is occupied by a tumor as opposed to healthy tissue), then we have to deal with a nonlinear inverse problem. These cases typically require solution using numerical methods, and the complexity of the problem often results in very large and expensive computations. Therefore, there is a clear need for advanced numerical methods.

An example: Numerical inversion in optical tomography. In collaboration with scientists at Baylor College of Medicine in Houston, Texas, Dr. Wolfgang Bangerth has worked on the development of optical tomography methods for the detection of breast cancer. Traditionally, breast cancer is detected using mammograms. However, using X-rays, this technique might induce cancer itself. Equally importantly, what the X-rays "see" are only changes in the density of material, typically secondary calcification of vessels in the vicinity of a tumor, but not the tumor structure itself. Thus, one would like to a) use non-ionizing radiation, and b) use methods that are specific to tumor tissue types, not just secondary effects of tumors. Optical tomography using light in the visible and near-infrared range, for which human tissue has a high scattering but low absorption coefficient can satify the first condition. The idea for the second requirement is to use fluorescent agents that bind selectively to cancer tissue, excite the agents with light of one wave length and detect the fluorescent light at a different wave length.

The mathematical description of this leads to an inverse problem where one wants to infer the spatial distribution of the fluorescent dye concentration, which is a coefficient in a diffusion-type equation. The accurate resolution of this inherently three-dimensional problem requires the use of adaptive meshes to avoid excessive computing times. In this on-going collaboration, we have shown that the results of adaptive forward simulations closely match actual measurements. More importantly, we were the first to demonstrate that adaptive codes work for optical tomography, and our results show excellent resolution of tumor location and size, much better than the results of competing groups who do not use adaptivity.

Some results. The following pictures show how adaptivity works for detecting a target in a phantom geometry of cubical shape: we illuminate the right face of the geometry with laser light, which propagates into the tissue, and excites fluorescence in the target molecules; the fluorescent light diffuses back to the surface and is detected there. From this detected light, we reconstruct the target molecule concentrations throughout the body. The following pictures show a sequence of meshes used to simulate the light diffusion problem (top row) and to discretize the sought parameter concentrations (bottom row):

The locally refined nature of the meshes is clearly seen. In the bottom row, the red cells show the location of the reconstructed tumor. The following pictures show the accuracy of the reconstruction of location and size of two small tumors located at distances of 10, 6.6, 3.1, and 1.6mm from each other (again, get larger version by clicking on a picture): The red cubes denote the identified targets, while black wireframes show exact location and radius of the targets. Obviously, our method is able to determine these parameters to excellent accuracy except for the last cases where the two identified targets merge. As a sidenote, for practitioners the holy grail is to get to a resolution of about 1mm, so we are pretty close!

A more realistic situation, modeling an experimental setup, is shown here (taken from a SIAM J. Scient. Comput. paper):

Here, we scan a laser widened to illuminate a rectangle over an experimentally obtained geometry. At each position, we take an image of the light intensity in the infrared wavelength range. If we use this information to determine the original fluorescent dye concentration, we get the following image:

One can see the tumor in the center, which we found to be at a location and depth that is compatible to the location found by a surgeon after we had done our experiments.

Future directions. Although the results shown above are already pretty good, there are a number of directions where more research is needed in the future. In particular, the following topics are certainly open questions at present and will be worked on in our group:

• Solvers: Solving the inverse problems outlined above is expensive: our already quite optimized algorithms take 10-20 minutes to compute an image as the one above. That's too much for clinical practice, where the goal is 1-2 minutes. Consequently, we need to investigate new solver techniques, such as multigrid, that can more efficiently solve the problems at hand.
• Regularization: Inverse problems are typically ill-posed, i.e. a little bit of noise in measurement data will produce significant changes in the reconstructed images. To stabilize this process, we add "regularization" to the problem, but it is a difficult and open problem how much regularization to add, and where. We have some ideas in this regard, but will have to investigate their viability in practice.
• Other imaging modalities: The optical tomography application above is only one imaging modality. Others in biomedical research use radiofrequency waves, ultrasound, optoacoustics, and many other ways, and beyond that there are related modalities in neutron and X-ray imaging of cargo containers and suitcases for border security. In collaboration with computer scientists, biomedical engineers, and nuclear engineers, we are working on extending our methods to some of these other modalities.
• Questions beyond imaging: Beyond creating images there are a number of questions that one can ask. For example, how do we optimize our experimental setup so that the reconstructed images are the most accurate or reliable? These are questions in "optimal experimental design" that lead to problems of enormous mathematical and computational complexity. We are working on approaches to tackle these problems, but there is much left to be solved in these areas.