Adaptive constrained constructive optimisation for complex vascularisation processes

Abstract

Mimicking angiogenetic processes in vascular territories acquires importance in the analysis of the multi-scale circulatory cascade and the coupling between blood flow and cell function. The present work extends, in several aspects, the Constrained Constructive Optimisation (CCO) algorithm to tackle complex automatic vascularisation tasks. The main extensions are based on the integration of adaptive optimisation criteria and multi-staged space-filling strategies which enhance the modelling capabilities of CCO for specific vascular architectures. Moreover, this vascular outgrowth can be performed either from scratch or from an existing network of vessels. Hence, the vascular territory is defined as a partition of vascular, avascular and carriage domains (the last one contains vessels but not terminals) allowing one to model complex vascular domains. In turn, the multi-staged space-filling approach allows one to delineate a sequence of biologically-inspired stages during the vascularisation process by exploiting different constraints, optimisation strategies and domain partitions stage by stage, improving the consistency with the architectural hierarchy observed in anatomical structures. With these features, the aDaptive CCO (DCCO) algorithm proposed here aims at improving the modelled network anatomy. The capabilities of the DCCO algorithm are assessed with a number of anatomically realistic scenarios.

Parallel generation of extensive vascular networks with application to an archetypal human kidney model

“…Nonetheless, we can change the optimality criterion by modifying F . For instance, to account for the energetic cost associated with angiogenesis [ 30 ], F is defined as follows: where V ref and l ref are characteristic volume and length of the perfusion domain, r ref is a characteristic radius for the arterial tree, c v , c p and c d are, respectively, the volumetric, proteolytic and diffusion non-negative coefficients such that c v + c p + c d = 1. Parameter γ characterizes the branching power law which governs the relation between parent and daughter vessels.…”

“…For each vessel , we take as possible bifurcation points, , n bif points inside the triangle and check if , the modified parent vessel v p , , the sibling vessel v con , and , the new terminal vessel v n , are valid segments, i.e. the segments satisfy geometrical and hemodynamic constraints as detailed in [ 30 ]. If they are valid, we remove the original segment , add the new ones to a provisional tree and evaluate .…”

“…However, it is important to highlight that lack of convergence can be obtained for unrealistically small vessels (smaller than capillaries), for which the viscosity model is also questionable. For a detailed explanation of the DCCO algorithm, the reader is referred to [ 30 ].…”

“…In light of the previous context, the main novelty of the present work is to develop a parallel strategy to efficiently generate extensive vascular networks using a CCO-based approach. Given a spatial domain to be vascularized, the first stage consists of creating a baseline network using a recently proposed aDaptive Constrained Constructive Optimization method (DCCO) [ 30 ]. At a second stage, we split the domain into non-overlapping subdomains, each of which is independently vascularized using the baseline network as the underlying vascular substrate.…”

“…, x bif x d p , the sibling vessel v con , and x bif x d p , the new terminal vessel v n , are valid segments, i.e. the segments satisfy geometrical and hemodynamic constraints as detailed in[30]. If they are valid, we remove the original segment x…”