A generalized mathematical framework for estimating the residue function for arbitrary vascular networks

The microvasculature plays a vital part in the cardiovascular system. Any impairment to its function can lead to significant pathophysiological effects, particularly in organs such as the brain where there is a very tight coupling between structure and function. However, it is extremely difficult to quantify the health of the microvasculature in vivo, other than by assessing perfusion, using techniques such as arterial spin labelling. Recent work has suggested that the flow distribution within a voxel could also be a valuable measure. This can also be measured clinically, but as yet has not been related to the properties of the microvasculature due to the difficulties in modelling and characterizing these strongly inter-connected networks. In this paper, we present a new technique for characterizing an existing physiologically accurate model of the cerebral microvasculature in terms of its residue function. A new analytical mathematical framework for calculation of the residue function, based on the mass transport equation, of any arbitrary network is presented together with results from simulations. We then present a method for characterizing this function, which can be directly related to clinical data, and show how the resulting parameters are affected under conditions of both reduced perfusion and reduced network density. It is found that the residue function parameters are affected in different ways by these two effects, opening up the possibility of using such parameters, when acquired from clinical data, to infer information about both the network properties and the perfusion distribution. These results open up the possibility of obtaining valuable clinical information about the health of the microvasculature in vivo, providing additional tools to clinicians working in cerebrovascular diseases, such as stroke and dementia.