Rigorous mathematical optimization of synthetic hepatic vascular trees

TL;DR

This paper presents a rigorous mathematical optimization framework for generating synthetic hepatic vascular trees, improving structural accuracy and flexibility over existing methods by integrating nonlinear optimization and topology search.

Contribution

It introduces a novel NLP-based formulation for optimal tree geometry and combines it with topology optimization, validated against real human liver data.

Findings

Synthetic trees match corrosion cast data better than standard methods

The framework accommodates trifurcations and boundary condition changes

Generated trees are more physiologically accurate

Abstract

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure to the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.