In this article an inherently parallel algorithm to approximate the operator exponential is presented. The construction is based on the integral representation of the operator exponential and allows arbitrarily large time steps constituting a major advantage compared to classical schemes. The algorithm rests on the efficient solution of several elliptic problems depending on a complex parameter. We prove Besov regularity of the solutions to these elliptic problems. This result implies the efficiency of adaptive methods applied to the elliptic problems and leads to a complexity estimate for the complete algorithm. In the numerical experiments the efficiency of the new scheme is demonstrated by comparison to a single step method of second order.

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During the diagnosis of ischemic strokes, the Circle of Willis and its surrounding vessels are the arteries of interest. Their visualization in case of an acute stroke is often enabled by Computed Tomography Angiography (CTA). Still, the identification and analysis of the cerebral arteries remain time consuming in such scans due to a large number of peripheral vessels which may disturb the visual impression. In previous work we proposed VirtualDSA++, an algorithm designed to segment and label the cerebrovascular tree on CTA scans. Especially with stroke patients, labeling is a delicate procedure, as in the worst case whole hemispheres may not be present due to impeded perfusion. Hence, we extended the labeling mechanism for the cerebral arteries to identify occluded vessels. In the work at hand, we place the algorithm in a clinical context by evaluating the labeling and occlusion detection on stroke patients, where we have achieved labeling sensitivities comparable to other works between 92 % and 95 %. To the best of our knowledge, ours is the first work to address labeling and occlusion detection at once, whereby a sensitivity of 67 % and a specificity of 81 % were obtained for the latter. VirtualDSA++ also automatically segments and models the intracranial system, which we further used in a deep learning driven follow up work. We present the generic concept of iterative systematic search for pathways on all nodes of said model, which enables new interactive features. Exemplary, we derive in detail, firstly, the interactive planning of vascular interventions like the mechanical thrombectomy and secondly, the interactive suppression of vessel structures that are not of interest in diagnosing strokes (like veins). We discuss both features as well as further possibilities emerging from the proposed concept.