## 2 Materials and Methods  
The electrical bioimpedance measurements were performed in a system which consists of a cylindrical container with dimensions of \(80\,\mathrm{mm}\) diameter and \(50\,\mathrm{mm}\) height, with a four-electrode measurement channel incorporated. The sample container was designed using computer-aided design (SolidWorks v. 2015; Dassault Systemes, Waltham, MA, USA) and 3D-printed (XYZ Printing Da Vinci 1.0A; Shenkeng District, New Taipei City, Taiwan) with a \(1.75\,\mathrm{mm}\) acrylonitrile butadiene styrene (ABS) plastic filament. The contact electrodes, adapted from M4 \(\times\)\(0.5\,\mathrm{mm}\) stainless steel screws with a \(7\,\mathrm{mm}\) diameter head, were equally spaced, and fitted in the inner surface of the container at a height of \(20\,\mathrm{mm}\), as shown in Figure 1.  
Figure 1: Cylindrical container design specifications. The proposed bioimpedance measurement method, represented in Figure 2, has a sine wave voltage source (GF-320 function signal generator; Instrumentm, Sao Paulo, Brazil) defined by a frequency (\(f\)) and a peak-to-peak amplitude (\(V_{S}\)), along with a low-inductance metal-film resistor (\(R_{sh}\)) of \(120\,\Omega\) and \(1/4\) W. The electrical voltage is applied to the path formed by electrodes 2 and 4, thus establishing the flow of the excitation electric current (\(I_{L}\)) through the milk sample in the container. The solid grey lines represent the idealised electric current flow in the sample under test, while the dashed lines show the expected distribution of the equipotential lines along a plane whose height is defined by the four lateral electrodes. A set of 24 voltage measuring points distributed along such a plane is represented by circular points in Figure 2. All measurements in the experiment were made with a 2-channel (CH1 and CH2) digital oscilloscope (TBS1072B; Tektronix, Beaverton, OR, USA) and passive voltage probes (TPP0101; Tektronix, Beaverton, OR, USA). Several methods for impedance measurement have been proposed in the literature and many of them are based on four-wire sensing or the Kelvin method [31]. Using a pair of wires for excitation separated from another pair of wires for measuring minimises the errors due to cables and electrode impedances. The impedance measurement method proposed in Figure 2 is based on the opposite current injection method, which was first proposed by [PERSON], [PERSON], and [PERSON] for electrical impedance tomography applications [32]. The voltage measurements can be performed at any pair of electrodes in contact with the sample under test. Several voltage-measuring configurations are analysed in [33; 34]. In this work, electrode 2 is shared with the current injection and voltage measurement reference. Therefore, an electrode impedance error can be introduced at this point. However, such impedance can be evaluated and treated as a systematic error. The proposed bioimpedance measurement method can be represented by the electrical equivalent circuit of Figure 3. The generic impedances \(Z_{e}(n)\) represent the effect of the respective electrode (\(n\)) in the circuit. We assume that the electrode impedances \(Z_{e}\) are composed of a parallel resistor-capacitor (RC) network [35; 36]. As we have performed only AC voltage measurements, the electrodes' half-cell potential has been neglected in the model. As the excitation frequency increases, less influence of the electrode impedance  
Figure 2: Proposed bioimpedance measurement method. is observed on the circuit current \(I_{L}\). Furthermore, since the voltage measurements are performed with high input impedance, the electrode impedance \(Z_{e}(1)\) can be neglected. Concerning the voltage measurements, a small influence caused by the electrode impedance \(Z_{e}(2)\) is expected, while the sample impedance \(Z_{L2}\) is much greater than \(Z_{e}(2)\). The sample under test has a total impedance (\(Z_{L1}+Z_{L2}\)) that can be estimated by measuring the voltage between electrodes 4 and 2.