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Design of hierarchical vascular networks
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Biomanufacturing of organ-specific tissues with high cellular density and embedded vascular channels

Procedure

When templating a complex hierarchical vascular network (Fig. 3, C and D), the channel diameters were computed to ensure an equivalent shear stress within each segment of the network, independent of their diameter. Assuming Poiseuille flow, we determined the channel radius R and the flow Q, along with the pressure P at each node by solving the following set of equations (in MATLAB). On the basis of mass conservation, the sum of flow coming to a node is zero$∑iQi=0$

The flow Q through each segment is proportional to the fourth power of its radius R4 and the difference of pressure across ΔP, as given by$Q=πR48ηl∆P$where η is the viscosity of the culture medium and l is the length of a given segment. The shear stress exerted on a vessel wall, $τ=4ηQπR3$, should be constant throughout the network, hence$QiRi3=Q1R13$

For the vascular network depicted in Fig. 3 (C and D), the ratio between the largest vessels (inlet and outlet) and the smallest one was found to be 1.3. To print these channels at a constant deposition rate, the nozzle velocity was modulated proportionally to the inverse of the square of the diameter of each channel (or to its cross-sectional area).

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