A mathematical characterization of anatomically consistent blood capillary networks

Blood microcirculation is the site of control of tissue perfusion, blood-tissue exchange, and tissue blood volume. Despite the many irregularities, almost ubiquitously, one can recognize in microcirculation vessels a hierarchy of arterioles and venules, organized in tree-like structures, and capillary plexi, organized in net-like structures. Whilst for arterioles and venules it may be envisageable to obtain geometries needed for numerical simulations from imaging techniques, the size and numerosity of capillaries makes this task much more cumbersome. For this reason, it is interesting to study approaches to generate in silico -derived artifacts of capillary networks, even in view of machine-learning based approaches which require a large amount of samples for training. Artificial networks must correctly reflect proper metrics and topology, which in turn, will ensure with proper boundary conditions a physiological blood flux in the net and a sufficient nutrient distribution in the surrounding tissues. In this paper, we introduce the sequence of curves whose limit is the space filling Hilbert curve and we discuss its inherent properties and we obtain the backbone of the artificial capillary network from a suitable element of this sequence. The backbone represents a significant synthesis of basic metric features of the network and, in this context, its properties can be studied analytically. In this framework, the Hilbert curve is a malleable entity which allows to shape the backbone according to the physical indicators. In particular, two significant factors are shown to control the network topology and scaling: the iteration step of the Hilbert curve generation and the characteristic length of the REV, respectively. Based on the points we generate for a certain iteration step, we then obtain via spline interpolation a smoothed version of the curve, which fine–tunes the tortuosity. A volumetric construction is obtained building a tubular neighborhood of the backbone, whose metrics can be computed and tuned as well. Numerical simulations of the blood flow in the obtained geometry show the physical fields occurring in the artificial network.