Macromolecular Mapping and Inverse Problems in MRI

Macromolecular mapping plays an essential role in biomedical and clinical research. A prime example of this is the determination of myelination patterns in the central nervous system (CNS). Myelin is a protein- and lipid-rich substance that potentiates electrical impulse transmission along axons; disorders of myelin are central to certain pathologies, including multiple sclerosis, and implicated in many others, including Alzheimer’s disease. However, conventional MRI provides maps only of water, with adjustable contrast provided by relaxation or diffusion characteristics of the water within an imaging pixel. This can provide indirect information about macromolecular content, but these measurements are notoriously non-specific. A more direct measure of macromolecular content can be achieved by a multi-component mathematical analysis of the MRI signal to distinguish between relatively unbound water and water that is to a greater extent motion-constrained. This is a much more complicated mathematical problem. Conventional MRI is a Fourier technique, with image reconstruction having the attractive property of being mathematically well-conditioned; noise in the data is transmitted to the image, but not magnified. In contrast, extraction of the macromolecular signal in MR relaxometry and related techniques is performed most often via the inverse Laplace transform, a form of the classically ill-posed inverse problem of solving the Fredholm equation of the first kind. This results in parameter estimates that can be extremely sensitive to noise. As a result, specialized methods must be undertaken to produce useful results. The inverse problems perspective has proven to be enormously fruitful in this setting. We will discuss this framework, and our applications to myelin mapping in the CNS and proteoglycan mapping in cartilage. The goals of our work are twofold: to improve the capacity of MR to evaluate tissue pathology, and to develop methods for application to inverse problems more generally.