AVM modelling by multi-branching tube flow: large flow rates and dual solutions

Cerebral arteriovenous malformations (AVMs) present a common yet complex clinical challenge, through 'steal' phenomena, haemorrhage risks and epilepsy effects, aspects which are little understood even for individual lesions. The main difficulty lies in understanding the detailed haemodynamics of AVMs and especially the enhanced through-flow associated with steal. Mathematically, as a basic step, the paper investigates a nonlinear inviscid model for the planar incompressible flow of fluid through a branched geometry consisting of a single feeding mother tube which splits into two or more non-aligned daughter tubes. Recurrence relations between the unknown flow profiles in the daughter tubes and the incoming rotational flow profile in the mother tube are derived, analysed, and solved in detail in order to find the total flow rate. The results show greatly enhanced through-flow arising, for a fixed value of the total downstream flow area, either from non-unique solutions to the problem or more particularly from an increase in the number of daughter tubes, or from both, depending on the distribution of pressure differences applied across the branching region and the total downstream flow area. Extensions of the basic flow model are noted, along with comparisons with recent direct numerical simulations and discussion of possible repercussions in the context of treatment and clinical observations of enhanced through-flows in AVMs.