Mathematical methods for motion-aware medical imaging

The proposed project is settled on the interface of basic and application-driven mathematical and medical engineering research and aims at laying foundations for the transfer of novel mathematical methods for motion-aware tomographic imaging into commercial applications. Tomographic devices, in particular magnetic resonance imaging (MRI), constitute essential diagnostic tools of modern medicine. They allow for the non-invasive visualization of organs and tissue inside the body, delivering valuable information, e.g., for early-stage diagnosis as well as treatment planning. The key ingredient that makes tomography possible is the reconstruction of an image of the interior by just measuring indirectly outside this object. This task poses a mathematical problem whose solution is realized in each tomographic device. The research in this project addresses one of the major current challenges for present tomographic devices: The fast and efficient reconstruction of dynamics and patient motion. Currently, physical speed limits in the acquisition of data often prevent such motion-aware imaging. In particular, for MRI, patient movement during acquisition such as breathing causes severe artifacts in the reconstructions, making the repetition of measurements necessary. More importantly, there are many situations where one aims at imaging dynamics for diagnostic purposes, such as pumping and blood-flow motion of the heart. Due to these dynamics being faster than the measurement device, their imaging is presently only possible accepting serious limitations such a long breath- holds, high scan times and patient discomfort. As physical limits are involved, technological progress is meeting its limits and further improvements can only be achieved with more sophisticated image reconstruction techniques. To this aim, mathematical basic research is indispensable, bearing great potential to significantly extend current possibilities in medical imaging. In this respect, with progress in the mathematical theory of optimal transport, new techniques recently emerged that show promise for application in motion-aware medical imaging. It is the goal of this project to carry out the necessary steps for bringing these newly-emerged mathematical techniques to concrete application in motion-aware MRI, building on the previous work and experience of the participating research groups. It centers around a novel framework for jointly estimating motion and image content that bears the potential to overcome many practical limitations of existing methods as well as to open new horizons for diagnostic imaging. In a joint effort of the mathematical and medical engineering project partners, the proposed approach shall be developed up to a level where it can be readily applied in clinical practice and potentially utilized in clinical research studies. Its eventual realization in tomographic devices and clinical routine, however, can not be achieved without cooperation partners from industry. The project shall therefore in particular lay the basis for such corporate partnerships in medical-engineering and tomographic-imaging-related business areas.

The project at the interface of mathematical research and applications in medical imaging dealt with the various challenges associated with the goal of enabling motion-aware image reconstruction in tomography. Tomographic methods, in particular, magnetic resonance imaging (MRI), are tremendously important in modern medicine and indispensable as diagnostic tools. Their ability for non-invasive visualization of organs and tissue in side the body is used in many medical applications such as, for instance, early diagnostics and therapy planning. The reconstruction of images of the inside of the body based on outside measurements is a central mathematical problem in tomography, whose solution is implemented algorithmically in each respective tomographic device. Although medical imaging modalities such as computed tomography ( CT) and MRI are now established for decades, motion during the measuring process still poses a great challenge for image reconstruction. In particular, without motion-aware techniques, breathing or heartbeats cannot be visualized without artifäcts using MRl. The reason for this are physical limits which have the consequence that the required increase of temporal resolution leads to only a fraction of the necessary data to be measurable in each time instance. The mathematical methods developed in the project enable, in spite of these limits, to reconstruct artifact-free images with high temporal and spatial resolution. An essential feature of these developed methods is the exploitation of temporal interdependencies, which was achieved by employing the theory of dynamic optimal transport. In this context, dynamic optimal transport serves as a realistic model for natural motion patterns in the case of missing temporal information. Up to now, this theory has mainly been used for logistic applications. One of the main results of the project is the successful introduction of this mathematical concept for the solution of motion-aware dynamic inverse problems. The latter include basically all reconstruction tasks in medical imaging and MRl and CT in particular. This concept was also analyzed mathematically in detail with respect to the structure of the obtainable reconstructions, in order to guarantee that appropriate numerical methods would provide realistic motion patterns. In this context, it turned out that indeed, dynamics with minimal possible kinetic energy can probably be reconstructed, which yields realistic results in practice. All these results form, on the one hand, the basis for the concrete algorithms developed in the project. On the other hand, they allowed a successful application to motion- aware magnetic resonance imaging, which opened up, in particular, promising perspectives for further improvement, also beyond MRI