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Geometry and Computational Anatomy

Periodic Reporting for period 1 - GEOCA (Geometry and Computational Anatomy)

Reporting period: 2015-03-01 to 2017-02-28

Computing techniques in medical imaging have a wide range of applications in medicine and bioscience. The objective of computational anatomy (CA), which is a novel field within medical imaging, is modeling and analysis of biological variability of the human anatomy. Suggested applications cover the simulation of average anatomies and normal variations, the discovery of structural differences between healthy and diseased populations, and the detection and classification of pathologies from structural anomalies. A specific example, addressed in this project together with medical experts, is the problem of tracking organs undergoing deformations as a result of breathing in the thorax and imaged via computed tomography (CT). To this end, we have developed a mathematical framework for robust computer algorithms for this and similar types of problems. Our algorithms, implemented in open-source software, could have significant impact for example on radiation therapy of the lung, where accurate estimation of organ deformations during treatment impacts dose calculation and treatment decisions.
The research in this project proceeded in three steps:

1. First, development of new state-of-the-art numerical methods for computational anatomy. A limiting factor in previous state-of-the-art algorithms is that they require large computational resources, i.e. the computational complexity is high. We have developed new algorithms with lower computational complexity, yet equal or better results in quality (especially for CT lung images). Implementations of these algorithms have been released as open-source software.

2. Second, mathematical analysis of robustness of the algorithm, so called “convergence analysis”. This is important to assert that the developed algorithms produce reliable results. Although work remains, we have so far established convergence analysis for one of the developed algorithms.

3. Third, as mathematicians, it is important to collaborate with medical experts, to assert that our results are disseminated and, ultimately, come to societal use. In this project we have worked together with experts on thoracic imaging (lung and heart region). We have demonstrated that our algorithm outperforms previous state-of-the-art in both computational complexity and quality. In the attached Figure 1 the Jacobian determinant, measuring compression and expansion, is illustrated for our new method (left) and previous state-of-the-art methods (middle and right). Notice that the left image gives a better results, since expansion (blue color) only occurs in regions of low density (essentially where there is no tissue).
The best outcome of this project so far is the new mathematical framework, where density matching problems (for example, registration of CT images of lungs) are solved using nonlinear gradient flows. Potentially, our developed methods could be used in commercial medical software, used at medical institutions for improved practices. The software can also be used by medical researchers, as a mean towards discovering structural differences between healthy and diseased populations. The long-term outcome of such research could be improved tools and protocols for the diagnosis of, e.g. Alzheimer's disease. The most promising societal application so far is as part of a tool for planning radiation therapy of the lung, where accurate estimation of organ deformations are utterly important in order to produce more optimized dose calculations and treatment decisions.
Figure 1: Compression ratio for new method (left) and previous state-of-the-art (middle and right)