A mathematical model of umbilical venous pulsation
Pulsations in the fetal heart propagate through the precordial vein and the ductus venosus but are normally not transmitted into the umbilical vein. Pulsations in the umbilical vein do occur, however, in early pregnancy and in pathological conditions. Such transmission into the umbilical vein is poorly understood. In this paper we hypothesize that the mechanical properties and the dimensions of the vessels do influence the umbilical venous pulsations, in addition to the magnitude of the pressure and flow waves generated in the fetal atria. To support this hypothesis we established a mathematical model of the umbilical vein/ductus venosus bifurcation. The umbilical vein was modeled as a compliant reservoir and the umbilical vein pressure was assumed to be equal to the stagnation pressure at the ductus venosus inlet. We calculated the index of pulsation of the umbilical vein pressure ((max-min)/mean), the reflection and transmission factors at the ductus venosus inlet, numerically and with estimates. Typical dimensions in the physiological range for the human fetus were used, while stiffness parameters were taken from fetal sheep. We found that wave transmission and reflection in the umbilical vein ductus venosus bifurcation depend on the impedance ratio between the umbilical vein and the ductus venosus, as well as the ratio of the mean velocity and the pulse wave velocity in the ductus venosus. Accordingly, the pulsations initiated by the fetal heart are transmitted upstream and may arrive in the umbilical vein with amplitudes depending on the impedance ratio and the ratio between the mean velocity and the pulse wave velocity in the ductus venosus.