Multi-stability in blood vessel network motifs

The presence of low oxygen, or hypoxia, within tumours can significantly impact patient responses to therapy. Experimental results have shown that transient periods of cycling hypoxia can select for cancers that are difficult to treat and highly invasive. A useful metric for quantifying hypoxia is the haematocrit (ratio of red blood cells to plasma) distribution in the tumour’s micro-circulation.  

 

We model blood flow through a network of blood vessels by assuming ‘Poiseuille’ flow through individual vessels, conserving blood flow and red blood cells, and using a previously proposed phenomenological rules for haematocrit splitting at vessel bifurcations. The blood flow dictates the distribution of haematocrit within the network. We hypothesise that cycling hypoxia can be generated by fluctuations between distinct stable equilibria. We use continuation techniques to find multiple equilibria, and determine their stability as physiologically relevant model parameters are varied. Hence, we showcase our approach by applying it to several vessel network motifs to uncover a link between the existence of equilibria and network geometry that supports our hypothesis.