Umbilical flow distribution to the liver and the ductus venosus in human fetuses during gestation: an anatomy-based mathematical modeling

The partitioning of umbilical vein blood flow between fetal liver and ductus venosus may be an indicator of the fetal well-being, because the goal of the ductus venosus is to supply oxygen and nutrients to heart and brain. Both distribution and blood flow rate of the umbilical vein are functions of the local vascular impedances that, in turn, depend on the anatomical features of the related vessels. In order to investigate the venous blood flows in human fetuses during a normal gestation, a simple lumped parameter mathematical model was developed on the basis of some information achievable by ultrasonographic techniques. Particularly, the diameter and length of umbilical vein and ductus venosus and the volume of the liver were used to derive the vascular impedances. Three different impedance models were adopted for the umbilical vein, the ductus venosus and the hepatic circulation. A linear model described viscous hydraulic dissipations through the umbilical vein, while a quadratic pressure-flow relationship was used for the ductus venosus due to the irregular local hemodynamics at its inlet. Finally, the equivalent impedance of the whole hepatic network was related to the hepatic volume assuming a tree-like, symmetric and self-similar fractal geometry. The hepatic vascular resistances predicted according to the fractal analysis were quite consistent with some experimental measurements in fetal lambs. In agreement with clinical observations, the model predicted blood flows through the ductus venosus and umbilical vein increasing (from about 25 to 75 ml/min and from about 45 to 370 ml/min, respectively) throughout the gestation (20–40 weeks), while the flow fraction shunted via the ductus venosus diminishes (from about 50 to 20%).