Optimizing Non-Intersecting Synthetic Vascular Trees in Nonconvex Organs

Objective: The understanding of the mechanisms driving vascular development is still limited. Techniques to generate vascular trees synthetically have been developed to tackle this problem. However, most algorithms are limited to single trees inside convex perfusion volumes. We introduce a new framework for generating multiple trees inside general nonconvex perfusion volumes. Methods: Our framework combines topology optimization and global geometry optimization into a single algorithmic approach. Our first contribution is defining a baseline problem based on Murray's original formulation, which accommodates efficient solution algorithms. The problem of finding the global minimum is cast into a nonlinear optimization problem (NLP) with merely super-linear solution effort. Our second contribution extends the NLP to constrain multiple vascular trees inside any nonconvex boundary while avoiding intersections. We test our framework against a benchmark of an anatomic region of brain tissue and a vasculature of the human liver. Results: In all cases, the total tree energy is improved significantly compared to local approaches. Conclusion: By avoiding intersections globally, we can reproduce key physiological features such as parallel running inflow vessels and tortuous vessels. Significance: The ability to generate non-intersecting vascular trees inside nonconvex organs can improve the functional assessment of organs.