Mathematical model for interpretation of doppler velocity waveform indices

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Abstract
Various empirical indices such as the pulsatility index (PI) are widely used for quantitative analysis of Doppler ultrasound velocity waveforms. The physical interpretation of these indices was studied using a mathematical model. Although the method has more general applicability, this particular study was concerned with the umbilical-placental circulation. A lumped element electrical circuit equivalent was used, with each arterial branch represented by a resistor and a capacitor. The placental villous bed was modelled by a two-step parallel branching structure. Placental vascular disease was modelled either as obliteration of a fraction of the terminal branches, or as a fractional decrease in the radius of the vessels. The main features of both normal and abnormal umbilical artery waveforms can be reproduced by this simple model. Theoretical relationships between the velocity waveform indices and the lumped resistances and capacitance of the system were obtained for different input pressure functions. Over a wide range of physically reasonable conditions, the umbilical artery PI is approximately proportional to the ratio of the placental resistance to the umbilial artery resistance. The PI also depends on the pulsatility of the input pressure waveform. The Fourier pulsatility index was evaluated for an arbitrary pressure function, and shown to behave like (PI)2 for the umbilical artery waveform.

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