1 Introduction

While there are several methods for measuring total body water (TBW) using bioimpedance analysis (BIA) from the wrist-ankle impedance at 50 kHz [2, 4, 5, 10, 11, 19], the number of BIA methods from measuring extracellular water (ECW) from the same impedance is more limited [7, 17]. ECW and TBW measurements can be useful in several pathologies, such as hemodialysis [8, 13], as the major part of water removed by ultrafiltration comes from ECW, or in cardiac disease, often leading to extracellular edema [12]. In addition, the difference between TBW and ECW yields the intracellular volume (ICW) which gives access to body cell protein mass BCMpro [20] by

$$ {\text{BCM}}_{\text{pro}} = 0.3838\;{\text{ICW}} $$
(1)

At frequencies below 1 kHz, the current will not penetrate cell membranes and only circulates in the ECW. The bioimpedance spectroscopy (BIS) method [3, 18] uses this property to calculate the ECW resistance R e by extrapolating the impedance measured at various frequencies along a circle until the resistance axis, when it becomes a pure resistance at zero frequency (Fig. 1). The human body is then approximated as the sum of 5 cylinders (the limbs and the trunk) by multiplying the resistance-volume relationship for a single cylinder by a dimensionless shape factor K B calculated from the length and perimeters of the limbs and the trunk to give the body resistance R as shown by De Lorenzo et al. [3]

$$ R = \frac{{K_{\text{B}} \rho_{\text{a}} H^{2} }}{{V_{\text{B}} }} $$
(2)

where V B is the body volume, H the height and ρa the apparent tissue resistivity. De Lorenzo et al. [3] obtained a value of 4.3 for the shape coefficient K B from statistical anatomical measurements in adults. The apparent tissue resistivity is given by Hanai’s [6] “mixture” conductivity theory, where c is the volume fraction of non-conducting tissues and ρ the resistivity of fluid inside tissues

Fig. 1
figure 1

Schematic of determination of resistances R e and R ∞ by extrapolation in R–X plane

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$$ \rho_{\text{a}} = \frac{\rho }{{(1 - c)^{3/2} }} $$
(3)

At low frequency, c is equal to 1 − V e/V B, as only ECW is conducting and Eqs. (2) and (3) lead to the following expression for ECW volume in L, where H is the height in cm, and W the body weight in kg

$$ V_{\text{ex}} = k_{\text{e}} \left( {\frac{{H^{2} W^{1/2} }}{{R_{\text{e}} }}} \right)^{2/3} $$
(4)

with

$$ k_{\text{e}} = \left( {\frac{{K_{\text{B}}^{2} \rho _{\text{e}}^{2} }}{{D_{\text{B}} }}} \right)^{1/3} $$
(5)

where D B is the body density and ρe the ECW resistivity. Values of the coefficient k e were determined from measurements of ECW volumes by bromide dilution technique, and found to be 0.306 for men and 0.299 for women, when V e is in liter, the body density D B is 1.05 kg L−1 and ECW resistivity ρe is in Ω cm. When these values of k e are introduced in Eq. (4), ECW resistivities ρe are found equal to 40.3 Ω cm for men and 39.0 Ω cm for women, close to values of saline (40 Ω cm).

In a recent paper [9], we have used Eq. (4) to calculate ECW volumes from the provided by a single frequency foot-to-foot impedancemeter which featured a square signal at 114 kHz, using the low frequency resistance R lf extracted from the top of the square signal. Individual coefficients k ie in this case were determined by substituting R lf to R e in Eq. (4) and equating V e with the ECW volume V ex measured by a Xitron 4200 multifrequency impedancemeter (Xitron Technology, San Diego, Ca, USA) to obtain

$$ k_{\text{ie}} = V_{\text{ex}} \left( {H^{2} W^{1/2} /R_{\text{lf}} } \right)^{ - 2/3} $$
(6)

Mean values of these individual coefficients k ie were used as k e in Eq. (4) to measure ECW with the foot-to-foot impedancemeter. These ECW volumes were found not significantly different from corresponding ones measured by the Xitron.

Because many commercially available impedancemeters operate at a single frequency of 50 kHz, this paper investigates whether the BIS method could be implemented by replacing R e in Eq. (4) by an ECW resistance R e50 deducted only from R 50, the body resistance measured at 50 kHz, rather than by extrapolation of impedances measured at various frequencies, as the Xitron does. The study was first performed using R 50 and R e measured by a Xitron 4200 on two groups of subjects, the second being used for validation and the method was later applied to a third group, using a recently commercialized 50 kHz impedancemeter, the BodyExplorer (Juwell Medical, Gauting, Bavaria, Germany).

2 Material and methods

2.1 Principle of method

Our method consists in estimating the ECW resistance (R e50) from R 50 by assuming that the ratio b = R 50/R e50 remains constant in men and in women, although it may be slightly different between the two sexes. In order to determine this parameter b, we have measured, using a Xitron 4200, R 50 and R e in a first group of 27 men and 28 women (method definition group) from the student body and staff of our university. The parameter b was taken as the mean value of the R 50/R e ratio in men and women, so that the ECW resistance R e50 is given by

$$ R_{{{\text{e}}50}} = R_{50} /b $$
(7)

Then the ECW volume V e50 was calculated from Eq. (8), in which R 50/b was substituted to R e, and V e50 to V ex.

$$ V_{{{\text{e}}50}} = k_{\text{e}} \left( {\frac{{bH^{2} W^{1/2} }}{{R_{50} }}} \right)^{2/3} $$
(8)

with the same values of k e as for the BIS method of Eq. (7), k e = 0.306 for men and 0.299 for women.

For an independent validation of the method, the same impedance measurements were made in a second group (validation group) of 15 men and 16 women and values of V e50 were calculated from Eq. (8) using values of b found for the first group. These values of V e50 were compared with corresponding values of V ex calculated by the Xitron from Eq. (4).

2.2 Subjects and measurement protocol

Whole body impedance data were recorded in the first and second groups of healthy volunteers aged from 16 to 71 years. Their characteristics are summarized in Table 1. Their weight (W) was measured by a Bodymaster Vision scale (Tefal SA, Rumilly, France) and their height (H) by a wall mounted measuring tape. A Xitron 4200 was used in the supine position with four gel electrodes placed on the dorsal surfaces of the right hand and foot. Current electrodes were placed, respectively, proximal to metacarpal and metatarsal phalangeal joints, in accordance with standard tetrapolar electrode placement [2]. Proximal (voltage) electrodes were separated by 5 cm from current ones. This device operates at 50 frequencies between 5 and 1,000 kHz and calculates resistances at zero (R e) and infinite (R ∞) frequencies by extrapolating its data to the real axis in the resistance-reactance plane as shown in Fig. 1, plotted using impedance values given by the Xitron software. The reproducibility of resistances measurements has been checked and found to be within 3–5 Ω or about 1%.

Table 1 Mean values and standard deviations of physical characteristics of the three groups of subjects
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Once the method was validated after comparing values of R e50 with corresponding ones of R e from the Xitron in the second group of subjects, it was applied to the BodyExplorer, a BIA impedancemeter. To this effect, impedance measurements were made on a third group of 21 healthy subjects (11 men and 10 women) using successively the Xitron 4200 and the BodyExplorer supplying a resistance (R 50b), a reactance (X 50b) and phase at 50 kHz according to the protocol described previously. The third group characteristics are also summarized in Table 1. A block diagram showing the purpose and measurements in each group is given in Fig. 2.

Fig. 2
figure 2

Block diagram of impedance measurements in various groups

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2.3 Comparison with other methods for calculating ECW volume at 50 kHz

We have used the equation of Sergi et al. [17]

$$ V_{\text{es}} = - 5.22 + 0.2H^{2} /R_{50} + 0.005H^{2} /X_{50 } + 0.08W + 1.9 + 1.86\;{\text{sex}} $$
(9)

where X 50 is the reactance at 50 kHz and sex is equal to 0 for men and to 1 for women and that of Hannan’s et al. [7]

$$ V_{\text{eh}} = 0.0119H^{2} /X_{50} + 0.123H^{2} /R_{50} + 6.15 $$
(10)

2.4 Statistical analysis

The comparison of ECW resistances and volumes measured by the Xitron at zero frequency and at 50 kHz were presented using linear regressions with squared correlation coefficients (R 2). A Bland–Altman graph [1] was used to present differences V e50 − V ex in the second group, their standard deviations (SD) and limits of agreement (mean ± 2 SD), which include, in principle 95% of these differences. These resistances and ECW volumes by different methods for both groups were compared using paired Student tests (t tests). Results are considered to be significatively different if p values are less than 0.05.

3 Results

3.1 Determination and validation of method using the Xitron 4200

3.1.1 Determination of R e50 from R 50

The mean and SD of R 50, measured by the Xitron at 50 kHz, those of R e, extrapolated by the Xitron at zero frequency and their ratio R 50/R e, are listed in Table 2 for the first and second groups of subjects. This table shows that values of R e are higher than those of R 50, since the resistance decreases with increasing frequency. The values of coefficient b to be used in Eq. (8) are given by the mean values of R 50/R e ratios obtained for the first group which are equal to 0.806 for men and 0.833 for women, as shown in Table 2. The mean values of R e50 calculated by Eq. (7) with these values of b were 622.4 Ω versus 623.4 for R ex, in men, and 742.2 Ω versus 741.6 for Re in women. To avoid any confusion, p values of Student test for comparing R e50 with R e for the first group were deleted as this group was used for determination of R e50.

Table 2 Mean values and SD of ECW resistances R e, resistances R 50 and reactances X 50 measured at 50 kHz by the Xitron in the 3 groups
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Values of R e50 for the second group (Validation group) were calculated using the values of b obtained with the first group, and mean differences with R e were larger, equal to −4.63 Ω for men and 5.99 Ω for women. However, R e50 were not significantly different from corresponding values of R e with p values equal to 0.346 in men and 0.30 in women. Figure 3 shows the comparison of ECW resistances extrapolated from 50 kHz with values of R e given by the Xitron in the first group. Figure 4 displays the same comparison for the second group and shows that values of R e50 were very close to those of R e.

Fig. 3
figure 3

Comparison of ECW resistances extrapolated from 50 kHz (R e50) with R e, extrapolated by the Xitron in the first group

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Fig. 4
figure 4

Comparison of ECW resistances extrapolated from 50 kHz (R e50) with R e, extrapolated by the Xitron in the second group

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3.1.2 Determination of ECW volumes V e50 using the Xitron

Values of V ex, calculated by the Xitron using Eq. (4), and V e50, calculated from Eq. (8) in first and second group, and differences V e50 − V ex, are given in Table 3, together with p values of Student test, deleting again values for the first group. It can be seen that values of V e50 and V ex were close, and not significantly different in the second group, with p values of 0.277 in men and 0.393 in women. Values of V es, ECW volumes calculated from Sergi’s method (Eq. (9)) also shown in Table 3 together with differences V es − V ex, were slightly above those of V ex and V e50 in first group, especially in women where they were significantly different. Values V eh calculated from Hannan’s method (Eq. (10)) are given in the last column of Table 3. Hannan’s method overestimates the ECW as compared to the Xitron by an average of 2 L in men and 3 L in women of first and second groups.

Table 3 Comparison of ECW volumes calculated by our 50 kHz-Bis method V e50, Sergi’s and Hannan’s methods with those given by the Xitron V ex, including values of p tests
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Similar results for the comparison of ECW volumes by different methods were found in the second group. Differences between V e50 and V ex were larger than in the first group and p values were smaller but V e50 were not significantly different from V ex with p values of 0.277 for men and 0.393 for women. Values of V es were not significantly different from those of V ex in men, but they overestimated V ex by 1.0 L in women. As for the first group, values of V eh for the second group overestimated V ex by 1.72 L in men and by 1.85 L in women.

Measurements of V e50 in the third group can also be considered as a validation test for our method, since they use the value of b for the first group. For this third group also, values of V e50 were not significantly different from V ex with p values of 0.545 for men and 0.096 for women.

The detailed comparison of ECW volumes V e50 calculated from R 50 and Eq. (8) with V ex, given in Fig. 5a for men of first group, shows that their values are very close. This is true also for V es calculated by Sergi’s method (Eq. (9)). The same comparison displayed in Fig. 5b for women of first group shows that, if values of V e50 were also very close to those of V ex, this was not the case for V es which overestimated V ex by an average of 1.25 L. Perhaps the coefficient 1.86 of the last term of Eq. (9) was too high. The same comparison for the second group is depicted in Fig. 6. Values of V e50 are very close to the identity line for both men and women, while values of V es for women generally overestimate ECW as compared to the Xitron. A Bland–Altman [1] graph of differences V e50 − V ex is shown in Fig. 7 for the second group. Only one point for men and one for women lie outside the limits of agreement.

Fig. 5
figure 5

a Comparison of V e50 and V es with V ex from Xitron in men of first group. b Comparison of V e50 and V es with V ex from Xitron in women of first group

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Fig. 6
figure 6

Comparison of V e50 and V es with V ex from Xitron in second group

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Fig. 7
figure 7

Bland–Altman graph of comparison between ECW volumes V e50 and V ex in the second group

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3.2 Application to the BodyExplorer

This application was performed on the third group of subjects.

3.2.1 Comparison of resistances measured by BodyExplorer and Xitron

The first step was to verify that resistances R 50b and reactances X 50b at 50 kHz of the BodyExplorer were close to those of the Xitron. Mean values and SD of R 50b and X 50b are given in Table 4 while those of R 50 and X 50 from Xitron in third group are listed in Table 2. Mean values of R 50b− R 50 were 4.16 ± 2.73 Ω or +0.8% in men and 6.79 ± 2.14 in women or +1.07% as can be seen in Table 4.

Table 4 Mean and SD of resistances R 50b and reactances X 50b measured by BodyExplorer in the third group) and of ECW resistances R e50b calculated from R 50b, comparison with corresponding resistances by the Xitron using p values of Student paired tests
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3.2.2 Comparison of ECW volumes in the third group by BodyExplorer and Xitron with different methods

Mean values and SD of these volumes are shown in Table 3. Mean ECW volumes measured by the Xitron (V ex) using R e were 18.61 L in men and 13.89 L in women. Mean ECW volumes V e50 using Eq. (8) and the BodyExplorer resistance R 50b were 18.53 L for men and 13.73 L in women. Both were not significantly different from V ex with respective p values of 0.545 and 0.096. When using Sergi’s equation (9) and R 50b, ECW volumes V es were, respectively, 18.7 L for men (p/V ex = 0.689) and 15.14 L for women, an overestimation of 1.24 L relatively to the Xitron (p = 8 × 10−7). As with the first and second groups, Hannan’s method with R 50b overestimated ECW by 1.8 L in men and by 2.4 L in women. Corresponding graphs of comparison of V e50 with V ex are shown in Fig. 8 for men and in Fig. 9 for women together with linear regressions and correlation coefficients. Figure 8 shows that, in men, our method (V e50) gives results almost identical to those of Xitron while Sergi’s method slightly underestimates ECW as compared to Xitron below 16 L and overestimates it above 22 L by 0.5–1 L. In women, as seen in Fig. 9, values of V e50 are again very close to those of Xitron, with a slight underestimation of 0.08 L, while Sergi’s method (V es) overestimates V ex by 1–1.3 L.

Fig. 8
figure 8

Comparison between V e50 and V es (from BodyExplorer) and V ex (from Xitron) for men of third group

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Fig. 9
figure 9

Comparison between V e50 and V es (from BodyExplorer) and V ex (from Xitron) for women of third group

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4 Discussion and conclusion

Our results show that it seems possible to use the BIS-Hanai method of determination of ECW which, although it is not a gold standard, is often considered as the most accurate impedance method, with a 50 kHz impedancemeter. This was done by substituting to R e, the resistance at 50 kHz divided by 0.806 for men and by 0.833 for women. No significant improvement in accuracy for R e50 was obtained by using the reactance at 50 kHz. Even in the validation group (second group), values of R e50 and V e50 were not significantly different from those given by the Xitron for R e and V ex. While our method gave ECW volumes closer to V ex in second and third group than Sergi’s method in women and than Hannan’s method in men and women, Sergi’s method was as close as ours to V ex for men of second and third group. When our method was applied to the third group, using R e50b from the BodyExplorer and Eq. (8), differences between ECW volumes of the BodyExplorer and Xitron, V e50 − V ex, were very small −0.08 ± 0.44 L for men and −0.16 ± 0.27 L for women, even though our method has been established from a different group of subjects.

The success of our method is due to the fact that, in our population, the R 50/R e ratio varied between narrow limits as its SD represented only 3.1% of its mean value in men and 2.2% in women, and also to the strong correlation which exists between ECW and TBW in healthy individuals as noted in [7, 14]. It would be interesting to verify whether our method would be applicable to a population with abnormal fluid distribution, such as dialysed patients.

Although the Xitron method for ECW cannot be considered as a reference method, it has been compared with bromide dilution data by various authors [3, 18] and more recently by Moissl et al. [15] who found a mean difference of −0.39 ± 1.44 L relatively to dilution data in a population of 120 healthy subjects and 32 renal failure patients.

This work shows that it is possible, in a normal population, to calculate resistances at zero (R e), and at infinite frequency (R ∞) from R 50, as was shown in [16] with nearly the same accuracy as with a multifrequency impedancemeter. However, since our method for estimating TBW from R 50 presented in [16] uses an equation similar to Eq. (8) with different coefficients k and b, a simultaneous use of these two methods would result in uniform ECW/TBW ratios of 0.403 in men and 0.412 in women, which correspond to mean ratios published in the literature [19] but would not be realistic for each individual. However, we feel that our new BIA method for ECW may be useful, as much less BIA correlations are available for ECW than for TBW and the two BIA correlations that we have tested gave results more different from those of the Xitron BIS method even in second and third groups than our method. If our ECW method was combined with a BIA linear correlation of H 2/R 50 and W, such as that of Kushner and Schoeller [11] or Hannan et al. [7], it would give different values of the ECW/TBW ratio in each individual.

Thus, based on these data, our method seems to be an interesting alternative to previous BIA methods, as it combines the rationale of the BIS-Hanai method with the simplicity and low cost of 50 kHz impedancemetry.