Monte Carlo study of noise-induced phase angle errors in low-frequency tetrapolar bioimpedance measurements

Phase angles measured at low frequencies with the tetrapolar technique are sometimes scattered or even positive. We used Monte Carlo simulations of lock-in detection to quantify how large phase errors that measurement noise alone can cause. For a signal-to-noise ratio after demodulation (SNR_out) of about 3 or higher, the standard deviation of the phase error was close to 57° divided by SNR_out, and 95% of the errors were within about twice this value. At SNR_out = 10 the error standard deviation was 6°; at SNR_out = 1 it was 64° and 16% of the measurements were wrong by more than 90°. As SNR_out decreased towards zero, the phase gradually approached a completely random value. In an illustrative low-frequency tetrapolar measurement, the absolute phase errors were small (standard deviation 0.007° at 1 Hz and 0.04° at 0.1 Hz), but the true tissue phase was even smaller. The error was therefore 1.2 times the true phase at 1 Hz and 37 times at 0.1 Hz, and the measured phase had the wrong sign in 20% and 49% of the measurements, respectively. The sign of a small phase angle is only reliable (wrong in less than 1% of measurements) when the true phase is larger than 2.3 times the error standard deviation.