Abstract
De Lorenzo, A., A. Andreoli, J. Matthie, and P. Withers.Predicting body cell mass with bioimpedance by using theoretical methods: a technological review. J. Appl. Physiol. 82(5): 1542–1558, 1997.—The body cell mass (BCM), defined as intracellular water (ICW), was estimated in 73 healthy men and women by total body potassium (TBK) and by bioimpedance spectroscopy (BIS). In 14 other subjects, extracellular water (ECW) and total body water (TBW) were measured by bromide dilution and deuterium oxide dilution, respectively. For all subjects, impedance spectral data were fit to the Cole model, and ECW and ICW volumes were predicted by using model electrical resistance terms R_{E} and R_{I} in an equation derived from Hanai mixture theory, respectively. The BIS ECW prediction bromide dilution wasr = 0.91, standard error of the estimate (SEE) 0.90 liter. The BIS TBW prediction of deuterium space was r = 0.95, SEE 1.33 liters. The BIS ICW prediction of the dilutiondetermined ICW wasr = 0.87, SEE 1.69 liters. The BIS ICW prediction of the TBKdetermined ICW for the 73 subjects wasr = 0.85, SEE = 2.22 liters. These results add further support to the validity of the Hanai theory, the equation used, and the conclusion that ECW and ICW volume can be predicted by an approach based solely on fundamental principles.
 bioimpedance spectroscopy
 extracellular water
 total body water
the first successful validation of extracellular water (ECW), TBW (total body water), and intracellular water (ICW) by using bioimpedance spectroscopy (BIS) methods was reported in 1992 (30). BIS, which implies fitting complex impedance (Z) data measured at multiple frequencies to a biophysical model, has been used extensively in biophysics and is the technique from which derive all the underlying theories for body impedance analysis (BIA) (48). Although a number of studies have been reported that used the methods employed in this study, the BIS principles and the rationale for the methods employed remain poorly understood. Partly because of the complex nature of BIS, there has been very little crosstransfer of BIS information to the field of human body composition; furthermore, these methods have not been fully reported by the investigators responsible for their development (30). As a result, these methods have been only partially reported with insufficient underlying discussion.
Several singlefrequency Z studies have reported a prediction of body cell mass (BCM; Refs. 3, 42), but the scientific bases of the approaches used were not well supported, and the relationships reported may have simply been a result of high intercorrelation between variables (48). Results should be associated with ICW or total body potassium (TBK) rather than BCM, because BCM is a concept that can only be defined by ICW or TBK (33). Any relationship between Z and BCM can only emerge from the relationship between Z and cell volume. What becomes important then is the best approach of predicting cell volume with Z.
This study reports the relationship between a BISpredicted ICW and a TBKpredicted ICW, and the relationship between a BISpredicted volume of ECW, ICW, and TBW compared with dilutiondetermined volumes. This manuscript also provides a description of the principles of BIS and a discussion of the simplified single and dualfrequency methods relative to wellknown BIS principles.
Physical Principles
Variations in heterogeneous tissues cause interfaces, separating regions of different properties, to trap or release electrical charge as a stimulus potential is changed. The time lag between the stimulus potential and the change in charge in these interfaces creates a frequency (f)dependent Z (i.e., dispersion). The dispersion found in the lowfrequency (LF) radio range (1 kHz to 100 MHz), which is of interest for predicting ECW and ICW volume, is known as βdispersion and is caused by cell membrane capacitance (C _{m}) (40) (Fig.1). With direct current (DC), there is no conduction through a capacitor. Thus, in the LF range of βdispersion, there is minimal conduction through the cells because of the high Z of theC _{m}, and conductivity is governed primarily by the properties of the ECW. Asf increases into alternating current (AC), the Z of theC _{m} decreases, allowing more current to flow into the ICW compartment. Because of the change in polarity that occurs with AC current, the cell membrane charges and discharges the current at the rate of thef. The Z decreases withf, because the amount of conducting volume is increasing. At higher frequencies (HF), the rate of charge and discharge becomes such that the effect of theC _{m} diminishes to insignificant proportions, and the current flows through both the ECW and ICW compartments in proportions dependent on their relative conductivity and volumes (5) (Fig. 2). Thus, at both very low and very high frequencies, the overall Z is essentially independent of theC _{m}, whereas at the mid or characteristic frequency (f _{c}), the dependence on the value of theC _{m} is at a maximum.
If the complex Z [resistance (R) and reactance (X)] of skeletal muscle tissue is measured, and thef varies from low to high, a series of values is derived that can be represented by complex points. The curve formed by these points is called an impedance locus, and its shape is a result of the electrical and structural characteristics of the tissue (5). The mathematical model that is used most often to describe both theoretical and experimental data on skeletal muscle tissue is known as the Cole model. It produces a semicircular relationship between R and X, with a depressed center when plotted (5) (Fig. 3). Modeling is considered essential, because it is the only means of independently analyzing the individual components of a heterogeneous material system (5, 25). The Cole model can be viewed as the equivalent electrical circuit shown in Fig. 4.
To accurately predict volume, the mixture effects need to be accounted for, because the relationship between R and body water volume is nonlinear (9, 17). These mixture effects are greater at LF because the conductor (i.e., ECW) represents only 25% of the total body volume compared with a concentration of nonconductor of 75%. At HF, the concentration of nonconductor is much less (e.g., 40%). Good examples of the mixture effects are the change in resistivity (ρ) that occurs with a change in hematocrit (14) and that plasma (i.e., ECW) is four to sixfold more conductive than is skeletal muscle tissue measured at 1 kHz (14). At 1 kHz, there is little conduction through the cells; thus the ρ should be similar to that of plasma. It is dramatically increased because the cells are nonconductive at LF and restrict the flow of current. Hanai (17) developed a theoretical equation that describes the effect on the apparent conductivity of a conducting material having a restricting concentration of nonconductive material in suspension. Hanai postulated that the theory could be applied to tissues with nonconductive concentrations ranging from 10 to 90%. To employ his theory, we have constructed an equation that considers the ECW to be such a medium at LF, where the ECW is the conductive material, and all remaining items (including ICW because it is surrounded by cell membrane) in the body are the restricting nonconductive material. At HF, the combination of both ECW and ICW forms the conductive medium, and all remaining items in the body form the restrictive material. Previous results have supported that this theory can be used in vivo to predict ECW, TBW, and ICW volume (35, 47).
SUBJECTS AND METHODS
A group of 87 healthy Italian men (n = 77) and women (n = 10), ages 21–57 yr, volunteered to participate in this study. Written informed consent was obtained from all participants. The study protocol was approved by the Medical Ethical Committee of the University of “Tor Vergata” in Rome.
On arriving in the morning in an overnightfasted state, subjects were weighed in swimming clothes, and body weight (Wt) was measured with a standard balance to the nearest 0.05 kg. Body height (Ht) was measured with a stadiometer to the nearest 1 mm. After the measurements of Wt and Ht were made, and still in the fasting state, all 87 subjects had their ^{40}K measured by a whole body counter, formed by a cell 2.5 m wide and 3 m high of 10cmthick lead bricks, the door to which was formed by a 22cmthick iron slab. The room was continuously ventilated. A single 20.3 × 10.2 thalliumactivated sodium iodine crystal was positioned above the subject, who was measured in a sitting position and dressed in only paper pajamas. TBK was calculated as^{40}K ∗ 8,474.6 (11). The correlation of variation (CV) for TBK was found to be between 2 and 3%. ICW was computed from TBK, assuming that potassium is only present in the intracellular fluid and assuming a potassium concentration in the intracellular fluid of 150 mmol/l (11).
For 14 of the men, out of the total sample of 87 men and women, TBW and ECW were measured by dilution methods. Deuterium oxide (D_{2}O) was used for the determination of TBW, and sodium bromide (NaBr) was used for the determination of ECW. While still in a fasted state and after the Z measurements were taken, the subjects drank 50 g of a solution containing 10 g of D_{2}O (99.8%; Carlo Erba) and 1.3 g of NaBr (Carlo Erba) in 38.7 g of tap water. After a 3h equilibration time, urine was collected for the determination of D_{2}O, and a venous blood sample was taken for determining plasma NaBr. The subjects remained in a fasted state throughout the equilibration period and refrained from voiding. No baseline measurement samples of either urine or plasma were taken. Enrichment of D_{2}O in the urine sample was measured after sublimation with infrared spectrophotometry, as described by Lukaski and Johnson (24). A correction of 5% for nonaqueous dilution was used (11). The venous blood sample was centrifuged (3,000 revolutions/min for 15 min) to separate the plasma. The plasma was then stored in sealed plastic tubes at −20°C until analysis. Bromide enrichment was measured in the plasma by highpressure liquid chromatography (32). The accuracy of the NaBr measurements by this technique is considered to be within 1% (50). ECW was calculated by using a 10% correction for nonextracellular distribution and 5% for the Donnan equilibration (11,50). ICW was calculated as the difference between TBW and ECW.
After the measurement of Wt, Ht, and TBK; before ingestion of the D_{2}O and NaBr solution; and with subjects still in a fastedstate; single wristtoankle (i.e., whole body) complex Z measurements were taken in all 87 subjects by using a BIS analyzer (model 4000B, Xitron Technologies, San Diego, CA). R and X were measured, and the corresponding Z, phase (θ), was computed from R and X at 21 f ranging from 1 kHz to 1.248 MHz. The measurements were taken within the first several minutes after the subjects assumed a supine position. No correction was made for orthostatic fluid shifts. The measurements were taken on the left side of the body, with the use of disposable electrocardiogram electrodes (5 cm^{2}; 3M, Minneapolis, MN) and in accordance with the standard wristtoankle protocol (47). Data were transmitted directly from the analyzer into an ASCII format file via an RS232 interface to a personal computer and controlled by the software program supplied with the device.
The Z and θ spectra data were fit to an enhanced Cole model (5) to account for any time delay (T_{d}) effects (see appendix ), using the nonlinear curvefitting software developed for the device. The ECW and ICW volumes were predicted by using modeled Re and Ri values in equations formulated previously (47) from Hanai mixture theory (17) (appendix ). The BIS TBW was calculated as ECW + ICW. The constants used fork _{ECW} (i.e., men = 0.306, women = 0.316) andk _{ρ} (men = 3.82, women = 3.40) had been scaled to D_{2}O and NaBr data collected in a previous study (47). Although only the constants for men were used in this study, the constants for women were included for discussion purposes because a previous study discovered a gender difference (47). When reference ECW and TBW data are available for only one gender, the software automatically and arbitrarily scales the other gender’s constant by the same percentage difference discovered previously. Further research needs to be done to determine whether there truly is a gender difference in these terms. Expressing the abovek _{ECW} andk _{ρ} terms as apparent ECW and ICW resistivity (ρ_{ECW} and ρ_{ICW}, respectively), they become 214 and 206, and 824 and 797 for ρ_{ECW} and ρ_{ICW} in men and women, respectively. These terms will be expressed as apparent ρ in the remainder of the document. The male constants from a previous study (47) listed above were used to predict the dilution ECW, TBW, and ICW of the 14 men. The TBKdetermined ICW of these 14 men was also predicted by both BIS and dilutionICW volumes. Then, new constants for men were computed from the dilution volumes measured on the sample of 14 men, as described in appendix . The new ρ_{ICW} constant was then used to predict the BISICW volume for the other 73 men and women and crossvalidated against their TBK determined ICW.
The Excel program was used for the statistical analysis. In addition to the descriptive statistics, the Pearson’s product moment correlation (r) and standard error of estimate (SEE) statistics were computed. BlandAltman plots were also constructed to display the individual subject differences between the BISpredicted water volumes and those determined by TBK and dilution methods.
RESULTS
Table 1 displays the physical characteristics of the two subject groups. Table2 provides the Cole modeling results for the two subject groups. Of the 87 subjects, according to the criteria rating the fit to the Cole model (appendix ), 21 subjects were rated as 0, 64 were rated as 1, and the remaining 2 were rated as 2. The mean correlation of fit to the Cole model, using scalar Z, was 0.998. With the use of the constants for men from a previous study (47), the dilution ECW was predicted asr = 0.91, SEE = 0.90 liters, with a mean of 21.03 liters and a mean difference of 2.69 liters. Dilution TBW was predicted as r = 0.95, SEE = 1.33 liters, with a mean of 41.02 liters and a mean difference of −4.46 liters. Dilution ICW (as TBW − ECW) was predicted asr = 0.87, SEE = 1.69 liters, with a mean of 19.99 liters and a mean difference of −7.14 liters (Table3).
The new ρ_{ECW} and ρ_{ICW} constants computed from the dilution sample of 14 men were 174.32 and 1,177.94, respectively. For discussion purposes, the computed contants for women became 167.8 and 1,139.34 for ρ_{ECW} and ρ_{ICW}, respectively. Using the new ρ_{ICW} constant for men, the prediction of the TBK ICW for the 73 subjects with BIS ICW wasr = 0.85, SEE = 2.22 liters, with virtually no mean difference (i.e., 0.08 liter). When the TBK ICW was predicted by gender (men = 63, women = 10) the correlations and SEE were similar to that of the total group, but there was a slight mean difference (i.e., 0.15 and −0.38). For the 14 men, the correlation and SEE values for the BIS and dilutionpredicted ICW and the TBKpredicted ICW were r = 0.56 and 0.57, and SEE = 2.68 and 2.32 liters, respectively.
To determine the effect that scaling of ρ_{ECW} and ρ_{ICW} had on the correlation and SEE, the new constants for men were used to repredict the dilution ECW, TBW, and ICW on the 14 male subjects. The correlation and SEE remained identical for ECW (i.e., r = 0.91, SEE = 0.90 liter) with no mean difference. For TBW, the correlation decreased slightly (i.e., r = 0.95–0.94) and the SEE increased slightly (i.e., 1.33–1.41 liters) with no mean difference. For ICW, the correlation decreased slightly (i.e., r = 0.87–0.80) and the SEE decreased very slightly (i.e., 0.01 liter) with no mean difference. Thus, ρ_{ECW} is purely a scalar and has no affect on correlation or SEE for ECW. Similarly, ρ_{ICW} is effectively a scalar, since changing it only slightly alters the prediction of ICW because the nonlinearity is slight. Figures 57 display the plotted differences between the BISpredicted ECW, ICW, and TBW volumes and the dilutionpredicted volumes. Figure 8displays the plotted differences between the BIS and the TBKpredicted ICW. The ECW prediction was achieved by using the exponent 1.5 predicted by Hanai theory.
DISCUSSION
Effects of mixture, scaling, and reference methods.
We constructed an equation from Hanai theory (17) because we wanted to account for as many error sources as possible, and we believed the relationship between R and volume should be explained scientifically rather than randomly through multipleregression analysis. ρ is dependent on the concentration of nonconductor present in a mixture, giving rise to an empirical exponent ranging from 1.43 for very small spheres to 1.53 for packed cylinders (7, 17). Hanai theory predicts an exponent, 1.5. The exponent 1.5 was recently confirmed in vitro in human blood (9). A linear equation computed from multipleregression analysis is not well suited to nonlinear effects. Thus it did not seem prudent to solve for a fivedimensional nonlinear biophysical model (i.e., Cole) and then use an overly simplistic volume theory (i.e., Ht^{2}/R) that assumes only one material is being measured. Predicting volume with an equation formed by scientific principles would enhance its utility and address the error sources directly rather than accounting for them statistically, which offers no scientific explanation.
Although the successful prediction of ECW and ICW volume with the equation used in this study has been reported (35, 47), it had not been reported that when we regressed an exponent against NaBr space (47), the highest correlation was achieved by using the exponent 1.5 predicted by Hanai theory. This finding strongly suggests the presence of mixture effects, and the strong predictions by using the exponent 1.5 (10, 35, 47) support the validity of Hanai’s theory. As a spherical theory developed in the emulsion sciences, the reasons why Hanai’s theory should not work are many, but they do not explain the strength of prediction this theory provides or the emergence of the exact theoretical exponent. In the absence of a more applicable theory, we use the developed equation, because we believe the errors of not accounting for mixture effects are greater than the inadequacy of the theory.
Because the strength of the BIS prediction (r and SEE) is independent of scaling, and TBW is determined by ECW + ICW, a good prediction of a dilution TBW would suggest a strong ECW and ICW prediction. This would only not be the case if there were exactly offsetting errors in the ECW and ICW prediction. Thus it is probable that the BIS prediction of ICW was better than that of dilution because ther and SEE were better for the BIS prediction of TBW than that of ICW. The inaccuracies of determining ICW by dilution have not been adequately discussed, nor is it clear whether the errors are additive or propagated. For the 14 subjects, both the dilution and BISpredicted ICW were poorly correlated to the TBKICW. That the TBKICW prediction improved substantially for the larger sample of 73 subjects suggests that the poor correlations were caused by outlying data in a small sample. That both BIS and dilution ICW were highly correlated with each other, as well as both poorly correlated to the TBK ICW in the 14 subjects, suggests that the lower correlation between BIS ICW and TBK ICW in the entire sample was due to the error in TBK rather than BIS. Nevertheless, the strength of the discovered relationships among the BIS ICW and dilution and TBK ICW suggests that BCM can be determined by BIS.
The only other equation derived from Hanai theory, or any mixture theory for that matter, was never validated, and its sensitivity to change was poor (9). We believe the results achieved in this and other studies can be attributed to viewing the body as having three compartments (i.e., ECW, ICW, and the remainder) rather than only the two used previously (i.e., ECW, ICW) (9). We also believe this to be attributed to the fact that the Hanai equation describes the effect on conductivity of the material, not the overall conductance. Thus, its use is volume dependent. To apply mixture theory, total volume must be known and is provided by body Wt/body density (D_{b}). D_{b} varies between individuals, but the range is generally within 1–1.07 kg/l (20). The effect on ρ_{ECW} in this range is ±1%, because it is only dependent on the cube root of D_{b}. We do not use body Wt as a fudge factor to improve the correlation (8) but rather as a theoretically required term to measure of total body volume. Like D_{b}, body Wt is expressed in cube root form; thus its contribution to the prediction of body water is reduced by 2/3.
The fact that changing ρ_{ECW} had no effect on correlation or SEE for ECW, and only slightly for ICW when ρ_{ICW} was changed demonstrates the scaling nature of these constants and that they have no effect on the scientific relationship between the BIS and dilution volumes. Large offsets were observed when the constants derived from a previous study were used (47), but these constants, which were derived from D_{2}O and NaBr (47), have been crossvalidated (10, 35). As such, it is troubling that ρ_{ECW} would decrease by 19% and ρ_{ICW} increase by 30%. This difference could have been caused by error in the BIS method. However, the high correlations and low SEE values observed, and the fact that the sample studied was similar to those samples used to calibrate (47) and crossvalidate (35) these terms, suggest otherwise. Because subjects were healthy, it is unlikely that there were any major differences in ion concentration. Even so, a 5 mmol change in ion has been found to affect ECW only 1–2% and ICW only 4–5% (38).
Different dilution methods produce differently sized ECW and TBW spaces, e.g., sulfate (^{35}SO_{4}) space being typically 20% smaller than NaBr space (11). Thus, a ρ_{ECW} calibrated to NaBr would predict an ECW space scaled 20% larger than^{35}SO_{4}space. We have noted (28) that Van Marken Lichtenbelt et al. obtained slightly higherr ^{2} and lower SEE values by using the methods described in this study, but D_{2}O space was underpredicted by −6.3 liters, and the NaBr space overpredicted by 3.0 liters. The NaBrtoD_{2}O space ratio was in the expected range of 0.40 and 0.42 in this study (Table 1) and the study reported by Van Marken Lichtenbelt (48), respectively. In contrast, the BIS ECWTBW ratio predicted in this study by the previous constants was 0.51 (Table 3). This supports that the ρ_{ECW} constant computed previously (47) may be scaling ECW too large, but this would equally occur if ICW were underestimated. That ICW, and thus TBW, may have been scaled too low was supported by the finding that the percent TBW of body Wt was 61% by dilution and 55% by BIS (Tables 1and 3). Further evidence that the new ρ_{ICW} constant may have validity was that TBKICW, which was independent from dilution, was predicted with very little mean difference. It is of concern that the previously determined constants are predicting ECW and TBW with little mean difference at some laboratories but not others. There is nothing apparent in the dilution methods used in this study, by Van Marken Lichtenbelt et al. (48) or Van Loan et al. (47). For D_{2}O, each used accepted protocols (e.g., fasted state, dosage, and equilibration time). Both we and Van Loan et al. (47) analyzed D_{2}O enrichment with an accepted infrared spectrophotometry method and Van Marken Lichtenbelt et al. (48) with an accepted isotoperatio mass spectrometry approach. All three laboratories corrected for isotope fractionation. The only variable identified was that we did not make a baseline measure of D_{2}O, which potentially could lead to an underprediction of TBW space. However, this is unlikely to explain such a large scaling difference. We also did not account for D_{2}O lost in the urine, but the subjects were measured in a fasted state and refrained from drinking or eating; thus, this is unlikely to explain such offset. Similarly, for NaBr, all three laboratories used accepted administration and analytical protocols, including 10% corrections for nonextracellular distribution and 5% Donnan equilibration. As did Van Marken Lichtenbelt et al. (48), we measured NaBr concentrations with the accepted anionexchange chromatographic method (32, 50), and Van Loan et al. (47) used a fluorescentexcitation technique. The only variables not accounted for were basal NaBr concentrations and NaBr lost in the urine during the equilibration. However, over such a short period of time and with the subjects being in a fasted state, the loss in NaBr in the urine would be small, as would be the error caused by not subtracting baseline NaBr from the plasma after administration (32). There are other variations (such as hydration status and metabolic rates) (11), but these also would not explain such large offsets. As such, there was little difference in the methods used. The offset could be attributed to the small sample size (n = 24) originally used to compute ρ_{ECW} and ρ_{ICW} (47), but if correctly determined there should not be such large deviations between individuals or samples. The validity of these terms is supported by their correspondence to biophysics results and crossvalidation.
If BIS or even the same dilution methods have such variation, it will be difficult to completely standardize the BIS prediction of ECW and ICW. To establish whether this variability is BIS or dilution based, ρ_{ECW} and ρ_{ICW} should be computed from D_{2}O and NaBr collected from a large, wellstandardized, multiplelaboratory study. Later studies could then use the same methods to judge how well these terms hold up. However, if constants can be derived that allow ECW and ICW to be predicted close to an accepted reality, the change in volume may become the most relevant clinically. The change in BISpredicted volume has been reported to be quite good (10, 18). Despite the small gender difference discovered in ρ_{ECW}and ρ_{ICW} (35, 47), this may be only a samplespecific phenomenon. Isolating ρ_{ECW} and ρ_{ICW} will allow investigation of the specific effects of temperature and ion concentration on ECW and ICW, rather than using a gross singlefrequency tissue measurement.
Effects of geometry on ρ.
A wristankle measurement would be inappropriate for patients with ascites, but the vast majority of subjects do not have such conditions (29). The good predictions of body water reported by this and many other studies using a wristankle measurement supports that body water is evenly distributed in healthy subjects. If not, good predictions of body water would not be possible. To evaluate the error caused by making a wristankle measurement and determine the validity of ρ_{ECW} and ρ_{ICW} computed previously (47), we compared the values of these terms to results reported in biophysics for plasma and ICW. First, we used standard anthropometric values for the ratios of arm, leg, and trunk lengths and girths (45) in the equation listed in appendix to compute a geometry constantK _{B} and remove the geometry effects on ρ. Albeit a rough approximation, because the arms, legs, and trunk are not perfect cylinders and the fraction of ECW and ICW is not constant between segments, an approximation should be possible. The value forK _{B} was computed to be 4.3. The longitudinal ρ of human skeletal muscle tissue measured at 1 kHz has generally been reported to be 200–300 Ω ⋅ cm. (14). These reported measures for apparent ρ were obtained from direct measurements on skeletal muscle tissue (14) and thus were corrected for the mixture effects caused by the ICW contained in that muscle. Assuming the fluid distribution found in a previous study (47) to be representative of healthy adults (45% ECW in TBW and 73% TBW in FFM), the concentration of nonconductive material in the skeletal muscle tissue was estimated to be 67.2%. By usingEq. C4, this yielded a ρ for ECW of nominally 250 ⋅ (1 − 0.672)^{1.5} or ρ_{ECW} of 47 Ω ⋅ cm. Our results indicated thatk _{ECW} was nominally 0.311; by using a nominal D_{b} of 1.05 kg/l, this relates to a ρ_{ECW} of 41 Ω ⋅ cm, which is in reasonable agreement with both the ρ calculated from skeletal muscle tissue measurements and to the ρ for pure ECW reported (50–60 Ω ⋅ cm)(14). Thus the distribution of ECW throughout the body was very consistent, and there was no significant error caused by making a wristankle measurement. Similarly, the values found fork _{ρ} (i.e., 3.6) and α (i.e., 0.7; Table 2) were in reasonable agreement with the values of 0.3 and 0.6 previously reported in biophysics (5, 14), respectively. These findings have been replicated in a pediatric sample where the computed ρ_{ECW} was within 5% (49) of the value discovered in healthy adults (47). It is uncertain how the mixture effects will be adequately accounted for with a segmental measurement.
Principles of fitting data to a biophysical model.
Plots are used to construct an applicable physical or mathematical model. Once a model has been constructed, computing the components of the model becomes the focus (5, 25, 40). The Cole model can be computed graphically or mathematically by drawing or fitting the best fitting curve through the data and extrapolating each end of the curve to where it intercepts the resistance axis (known as R_{0} and R_{∞}) (Figure 3). R_{0} equals R_{E}; thus, once R_{0} an R_{∞} are known, R_{I} can be determined by 1/R_{∞} − 1/R_{0} = 1/R_{I}. Fitting is generally performed mathematically because it is far more precise (25). Modeling can also be performed either manually or mathematically by simply fitting a circle through the measured R and X data (6, 43) but this approach does not include f and thus is two dimensional, using onethird less data to determine and crosscheck the best fit. This method also provides no estimate ofC _{m} but most importantly does not allow for the effects of T_{d} to be removed. Network analysis is a wellknown analytical technique, and the common method used for fitting data to a network model is to simultaneously fitf and weighted Z and θ data, using nonlinear least squares curve fitting (25). The important data are not at LF and HF but in the middle surroundingf _{c} because these data have greater certainty. To accurately fit and extrapolate a curve requires adequate data on either side off _{c}. Properly weighting the raw data is essential for obtaining the most accurate fit to the model (25) because measurements may have greater uncertainties at LF and HF. By weighting, the data with more certainty (i.e., middle) make a greater contribution to the overall fit to the model (25). Determining what weights to use is not easily decided (25), and simultaneously fitting a multidimensional nonlinear equation is an art, with the raw data having a very complicated relationship to the final fitted parameters. Evaluating the accuracy of fit should be determined by comparing the offset to fit to the expected measurement uncertainty at each f (25). Because of weighting and the Cole model being multidimensional, evaluating the fit with a twodimensional statistical analysis [e.g., root mean square error (RMSE) of R and X] would be meaningless (8, 27, 43). Weighting can be determined by measuring the range of error at eachf, according to the expected error in the measured quantities (e.g., accuracy specifications of the device), or as we use by weighting the error rather than the data by comparing the expected error with the actual error (25). Limitations must be enforced to prevent the software from forcing a fit. As many frequencies as possible should be used because solving for five unknowns requires at least five data, and all data are potentially contaminated with error (e.g., interference) and thus are uncertain (25). The ability to delete data that are significantly decreasing the overall accuracy of fit is a highly desirable capability (25). The accuracy of resolving a model is a function of the square root of the number of extra data pairs (i.e., Z and θ) over the number of variables in the model. A 16:1 increase in data provides a 4:1 improvement. However, processing time is effectively the square of the number of data pairs. Thus, a 16:1 increase in data takes 256 times longer to compute. We presently measure at 50 frequencies logarithmically spaced from 5 kHz to 1 MHz to balance between accuracy and processing time. It is important to space the frequencies logarithmically to ensure a proper density of data. We fit with both Z and θ (rather than Z alone, or R and X) to ensure the best possible accuracy of fit. If only Z is modeled (8), the overall accuracy is reduced (25). Using Z and θ provides twice the amount of data, and θ is an extremely important discriminating variable because it has a much broader range of sensitivity to change than Z. However, using θ requires that the time delay (T_{d}) effects must be accounted for (37). We fit with Z and θ vs. R and X because the weighting introduced by X would enhance the importance of frequencies furthest away from f _{c}and thus emphasize the opposite to what is needed. Furthermore, fitting against R and X is more complex and considerably slower because X is nonlinear.
Effects of f invariant T_{d}.
As published (30) and provided in Xitron’s product literature since 1992, we recommend investigators extend the HF range of the measurement by removing the effects of T_{d} by multiplying the Cole equation by the factore ^{−jwTd}, wheree is natural number, j is √−1, and w is f in radians/s. Despite the confusion this parameter has caused (8, 43), all conductors exhibit a T_{d} that causes a linear θ shift with f. Conductor length would be an obvious cause for T_{d} (copper wire having a T_{d} of ∼1.2 ns/ft), whereby an 8ft conductor length (e.g., wrist to ankle) would produce 10 ns of delay (37). However, a longer T_{d} of 32.4 ns was observed in the female subjects (Table 2). This is because T_{d} can also be caused by interaction between contact R, stray capacitance and transmission line effects, with the latter including conductor length (wrist to ankle) and the conductor (i.e., body) position relative to ground (floor, bed, table, and so on) (15). Only conductor length is a true T_{d} effect, but in the 1 kHz1 MHzf range the other effects also give rise to a linear θ shift with f and thus can be approximated as a T_{d}. For simplicity, all effects will be labeled as T_{d} throughout the discussion. The various causes of T_{d} were simulated by using the widely available SPICE circuitsimulation software program (Intusoft, San Pedro, CA) and using the SPICE file shown in Table 4. T_{d} were modeled by a highZ transmission line with conductor length set by T_{d}, and the body Z was modeled by a simple threeelement model (no α) using an R_{E} = 680, R_{I} = 900, andC _{m} = 2.8 nF. Figure 9 shows a plot of θ vs.f in the 1 kHz1 MHzf range for body T_{d} of 0, 15, and 30 ns of T_{d} and a characteristic transmission line impedance (Z_{c}) of 300 Ω. Because the body is usually suspended some distance from a ground plane, the body behaves like a transmission line (15). Figure10 shows θ vs.f by using the same threeelement values for R_{E}, R_{I} andC _{m}, a fixed conductor length T_{d} of 15 ns, and a Z_{c} of 150, 300, 450, and 600 Ω, respectively.
For convention, θ in these plots is expressed as having positive polarity. As shown in Figs. 9 and 10, there is a significant linear θ shift with f caused by both conductor length (wrist to ankle) and conductor (body) relative to ground. Adjustment of the various parameters (e.g., distance from ground plane) in the simulation circuit changed the magnitude and frequency at which T_{d} effects emerged. As shown in Figs. 11 and12, there can be considerable variations in T_{d} between subjects. Different devices, subjects, and environments will cause different responses to the variables causing T_{d} (15). The interaction between variables resulted in an effect larger than their sum and begin to affect Z slightly over 1 MHz. As shown in Fig. 10, the effects caused by conductor relative to ground can cause a negative T_{d} and opposite effect on θ, thus explaining the negative T_{d}shown in Table 2. Although less accurate than modeling, the T_{d} effects can be removed manually by the methods described in appendix .
It is disappointing that several investigators would not use or even mention our suggested methods and information on T_{d}, then cause undue confusion by reporting an intermediate result and incorrectly attributing the deviation in the raw X data from the Cole model to measurement error (8, 43), particularly when these methods had been disclosed and used to successfully predict ECW and ICW volumes for the first time using Cole model terms R_{E} and R_{I} (30, 47). The effects of T_{d} are linear on θ not on X, and in the f range of interest T_{d} only significantly affects HF θ shifts not Z (37). If T_{d} were not a valid term, the correlation of fit of Z (which is the √R^{2} + X^{2}) would also be seriously reduced, whereas it is not (i.e., 0.998; Table 2). Stroud et al. (43) should have questioned their conclusions, because if the measurement was poor, the data would not have corresponded so well to an electronic circuit. Similarly, Deurenberg et al. (8) should have questioned their conclusions when there was no deviation of Z from the model at HF (8), and as stated, the final fit to the model included data up to 500 kHz. To review appendix , it can readily be seen that any f significantly affecting the overall fit will be deleted.
The practical reasons for modeling for T_{d} are simple. There are T_{d} effects in all raw data to varying degrees (15); thus as much of the T_{d} effects as possible should be removed from the analysis. Fortunately, in thef range of interest, all the effects of T_{d} cause a linear θ shift with f. Because the Cole model (i.e.,C _{m}) causes a nonlinear θ shift with f, the effects of T_{d} can be effectively modeled and significantly removed. Our modeling program and information on T_{d} were widely distributed in 1992. As such, it would not be difficult to adjust the various parameters to push out to higher frequencies where the effects of T_{d} become dominant. However, as discovered by Stroud et al. (43), without removing the effects of T_{d}, only useful data up to 500–600 kHz will be obtained (43). Table5 is an output file (.MDL file) generated from our fitting software on one of the subjects of this study. As shown, few data were deleted from the final fit, and frequencies up to 1 MHz were included. Without removing the effects of T_{d}, inclusion of such HF data would not be possible. The problem with not modeling for T_{d} is that there are variations in the environment in which measurements will be performed, and it is not uncommon, particularly in the clinical setting, to observe f _{c}s >500 kHz. With usable data only to 500–600 kHz, it would not be possible to accurately fit for Ri. Although the physics underlying T_{d} is important, what is important to the prediction of ECW and ICW is to remove the effects of T_{d} so the highest possiblef range can be included. Even with lower f _{c}, the closer the actual data are to R_{∞}, the better the calculation of R_{I} will become.
Dual and singlefrequency measurements.
Biophysicists have been fitting Z data to models since the early 1920s. However, before 1963 when a dual LFHF Z approach was first introduced as a measure of ECW and TBW (44), measuring Z was much more difficult. Additionally, solving a fivedimensional nonlinear equation by hand without a microprocessor would have been extremely tedious. Although pragmatic at the time, it can readily demonstrated (Figs. 1 and 13) that for some subjects where the current is fully conducting through the ICW does not occur until >10 MHz. A f of 10 MHz is 100fold away from 100 kHz. Thomasset (44) reported in 1963 that 100 kHz was too low af. Even if higher frequencies could be measured, the effects of gamma dispersion must still be avoided (Fig.1). Similarly, the effects of αdispersion become dominant near 1 kHz and must be avoided (Fig. 1). Most importantly, the proportion of current conducting through the cells at “any” singlef is not fixed but varies withf _{c}, andf _{c} varies between individuals, as well as in the same individual when R_{E}, R_{I}, Cm, or α is altered (5, 22, 26, 40; Figs. 1, 13, and 14). By fitting the data to the R_{0} and R_{∞}, the above error sources are removed. Jaffrin (19) reported that the overestimation of R_{I} can be as high as 200% by not using R_{∞} (19).
As shown in Figs. 1 and 13, a 50kHz measurement is neither a measure of ECW or TBW but rather some of both. No single HF measurement is a measure of TBW, including R_{∞} and Z at f _{c} as promoted by Cornish et al. (6, 27), but rather a measure of two significantly different fluids. The ρ_{ECW} has been reported to be 50–60 Ω ⋅ cm (14) and the ρ_{ICW} to be 200–300 Ω ⋅ cm (5). It has been previously assumed that the ρ of TBW is constant. Obviously this assumption is invalid because a simple change in the ECWICW ratio would dramatically change it. This error can be reduced, as we have done, by using the measured R_{E} and R_{I} with the previously established constants ρ_{ECW} and ρ_{ICW} to determine the actual relative proportions of ECW and ICW. From this, one can establish the ρ of TBW. The equation shown uses a linear mixture effect; however, in practice a nonlinear ECWICW mixture effect was used. The difference is insignificant in healthy subjects or when small changes are not of concern.
The prediction of ECW is inherently better than ICW and TBW for both technical and theoretical reasons. The prediction of ECW is achieved directly from model term R_{E}, whereas ICW is predicted effectively by the difference between two large numbers (R_{∞} and R_{0}). Thus, a 0.1% error in R_{∞} is ≈0.5% error in the predicted ICW. Although there is a call for a return to a HF and LF approach because R_{∞} is more variable than a fixed HF (8), the above error sources can never be resolved with a fixedf nonmodeling approach; thus, it is fraught with error. On the other hand, the repeatability and accuracy of solving for R_{∞} and thus ICW is technical rather than theoretical in nature. Improvements in the measurement should reduce the variability in predicting ICW.
Parallel reactance, phase angle, and cell membrane capacitance.
Series reactance at 50 kHz (X_{s}) had been proposed as a measure of ECW (41) and as a measure of the extracellular mass/BCM ratio (42). A 50kHz parallel X model (X_{p}) has now been proposed as a measure of BCM (3, 23). To support this proposal, Lukaski (23) performed a progressive potato study to demonstrate thef dependence of biological tissue (23) and drew on the statement by Foster et al. (12) that Z can be interpreted as either a parallel or series circuit and both resulting in two final elements (real and imaginary). The later is absolutely true for any single f measurement, but biological tissue consists of more than two elements. Z at any singlef can be interpreted as a parallel or series circuit, but the field is concerned with how to interpret the Z of biological tissue. According to Fricke (13), Schwan (40), and Cole (5), single biological cells can be represented as a seriesparallel network having three elements: R_{E} in parallel with a seriesC _{m} and R_{I}. Cole (5) added an exponent (α) to the model to represent the distribution effects observed on biological cell suspensions and tissues. The Cole model is used most often to interpret Z measured on biological tissue and consists of four elements (31). Based on the belief that how biophysicists interpret Z measurements has merit, we use the Cole model. To do this, we fit all real and imaginary data (i.e., corresponding Z and θ) to the Cole model to discriminate the component parts of the tissue.
If previous work in biophysics does have validity, the use of R and X at any single f to predict ECW or ICW would be an oversimplification and is dependent on the elements in the tissue having relative uniformity between individuals. Any relationship between BCM and X is likely a function of the relationship between X and C _{m} because ICW is a resistive not capacitive medium. It has been suggested that as the cell swells, the membrane becomes thinner andC _{m} increases, and the opposite occurs when the cell shrinks (16). However, X at any single f is not merelyC _{m}, andC _{m} can only be computed by modeling for all the elements in the Cole model (5, 25,40). Which variable is affecting X at any singlef cannot be determined. However, since R_{E}, R_{I}, and α tend to be tightly regulated and vary within narrow limits, X would tenuously reflectC _{m} and give rise to a correlation to cell volume.
To investigate the strength of the relationship betweenC _{m} and the variables related toC _{m} and BCM as defined by TBK, we investigated their correlation to TBK. As shown in Table 6, weight was strongly correlated to TBK, but variables other than BCM can cause a change in weight.C _{m} alone was strongly correlated to TBK and improved when expressed as Ht^{2} ×C _{m}. There was a poor direct relationship between X_{s} and X_{p} and TBK andC _{m}, respectively. X_{s} and X_{p} were moderately correlated to TBK when expressed as Ht^{2}/X_{s}or Ht^{2}/X_{p}, respectively, but Ht alone was highly correlated.f _{C} was also correlated to TBK, but mathematicallyf _{c} is dominated by C _{m} and R_{E}, and since this healthy population would have a narrow range in R_{E} in relation to the other model parameters, the relationship betweenf _{c} and TBK was most likely dominated byC _{m}. As shown in Table 6, there was a strong relationship betweenf _{c} andC _{m}. Use off _{c} cannot be supported, because it is affected by all the variables in the model, but as expected,f _{c} predicted TBK better than X at 50 kHz. The meanf _{c} for this healthy sample was ≈60 kHz (Table 2); thus, X would approximatef _{c}. However,f _{c} was more highly correlated than X simply becausef _{c} more closely reflects C _{m}, whereas the strength of the relationship between X andC _{m} varied withf _{c}, which ranged in this sample from 43 to 110 kHz. The strong relationship betweenC _{m} and TBK was expected becauseC _{m} is a function of cell surface area. The moderate relationship betweenC _{m} and weight reflects this dependence. However,C _{m} is also affected by the aspect ratio (length to crosssectional area) of the body’s conductor. With an identical total cell volume, a greater conductor length would cause lessC _{m}, whereas a greater conductor crosssectional area would cause higherC _{m}. Although error caused by aspect ratio is removed by length^{2} ×C _{m}, this is only true for a uniform cylinder. Further refinements inC _{m} would need to be made by accounting forK _{B}(appendix ). As discussed above,C _{m} is also affected by the thickness of the cell membrane. With the errors caused by aspect ratio and cell membrane thickness, it is unclear why a surface measurement (i.e.,C _{m}) would be used to reflect what is inside the cells when it can be determined more directly by a resistivepredicted ICW (ICW_{R}).
A single 50kHzfrequency measure of X and θ, and an RX graph have been proposed as measures of fluid distribution and discriminating indexes of health and disease (2, 36). Recently, θangle spectrum analysis (that is, θ vs. f) has been proposed as descriptive of body water and body composition (4). θ angle is a function of the ratio of R and X; thus, both θ and an RX graph would be sensitive to the same errors and uncertainties as any single frequency of R and X. The sensitivity of θ is extremely dependent on X, which in turn is extremely dependent on the relationship between the frequency of measurement andf _{c}, and is symmetrical aboutf _{c}. X and θ at 50 kHz change dramatically whenf _{c} changes, simply because 50 kHz is a fixed point on the changing curve (22, 26) (Fig. 14). X simply changes more than R because X is only 5% of the total Z; thus, a slight change inf _{c} causes a greater percentage change in X. In 1974 (Fig. 14; see Ref. 34), it was observed that dialysis patients had a lower X and θ measured at 50 kHz predialysis and that it returned to that observed in healthy subjects postdialysis. Lofgren (22) attributed the cause of this to the change in f _{c} 23 yr previously. The change in X and θ with a change in fluid distribution has given the incorrect impression that X and θ are somehow directly related to fluid distribution. A change in fluid distribution does changef _{c}, which in turn changes X and θ, but this is principally caused by a change in ECW (and C _{m}, as discussed above). Again, the problem with using X, θ, or f _{c} to reflect any body composition parameter is that these variables are affected by all the elements in the tissue (appendix ). One can question the utility or need for X, θ,f _{c}, or an RX graph when which variable is causing their change cannot be determined and they have no theoretical basis. On the other hand, R_{E} and R_{I} at least relate in theory to a physical object (i.e., ECW and ICW).
On the same individual,C _{m} is determined by total cell volume and membrane thickness and porosity (5). Thus, any Ht^{2} ×C _{m} relationship would be a function of these three parameters. An ICW_{R} can be used to predict total cell volume, and then if removed from theK _{B} corrected Ht^{2} ×C _{m} relationship by using Ht^{2} ×C _{m}/ICW_{R}, the remaining index would reflect cell membrane thickness and porosity. Such a measure might have several applications. Scheltinga et al. (39) observed that as the severity of sepsis increased,C _{m} decreased to the point where there was virtually no β dispersion. This corresponds to what Lukaski (23) observed on a cooked potato. It is well known that with cell death or cell destruction, the cell membrane loses its high resistive properties. During dialysis, ECW changes are on the order of 20–30%, R_{I} varies little, but both F_{c} andC _{m} can change by as much as 2:1 (1, 19) (Fig. 14). As shown in Table 2, the mean C _{m} for the male subjects of this study was 2.32 nF. Bestoso et al. (1) discovered a mean increase ofC _{m} in men from pre to postdialysis of 64% (1.64–2.47 nF), with the postdialysisC _{m} being quite close to that measured in the men in this study. Thus, if Scharfetter’s (38) estimates are correct that a 5mmol change in ion affects the ICW 4%, the error caused by ion on ICW_{R}, as well as the error in predicting ICW, would be insignificant compared with the percentage change in C _{m}. Use of C _{m} for studying cell membrane health is an exciting area of research that awaits further investigation.
In conclusion, there has been very poor crosstransfer of information from the fields of physics and engineering to the field of human body composition. It does not help that the principles of BIS and mixture theory are rather complicated for many investigators. However, Z is an engineering and physicsbased technique, and the principles and merits of modeling and mixture theory have been known for a very long time. Multiplefrequency devices safe for human studies and modeling programs have now been available for >4 yr; thus, the lack of appropriate equipment is no longer a valid reason for not using modeling. Due to the high intercorrelation (48) among ECW, ICW, and TBW, one can always correlate a limited single or dualfrequency measurement to body water, but this leads to populationspecific equations and a reduced sensitivity to change, which is why Z measurement has not yet reached its full potential. Until investigators begin using the proven and accepted fundamental techniques used in other fields of science (i.e., modeling) that use Z measurements (e.g., biophysics, advanced materials research, and chemical engineering), the use of Z in clinical medicine will remain what it is today—a technique that generates many papers but has no real clinical application.
Footnotes

Address for reprint requests: J. Matthie, Medical Dept., Xitron Technologies, Inc., 6295 Ferris Sq., Suite D, San Diego, CA 92121 or A. De Lorenzo, Dept. of Human Physiol., Univ. of Rome “Tor Vergata,” via O. Raimondo 1, I00173, Rome, Italy.
 Copyright © 1997 the American Physiological Society
Appendix
Modeling
The Z and θ spectra data were fit to the ColeCole model (5),Eq. EA1 , using iterative nonlinear curvefitting software. The modeling program evaluated the weighted least square error of both Z and θ, where the weighting is established by the published accuracy specifications of the instrument, and removed any f that would significantly decrease the total weighted least square error. In addition to the correlation of fit using scalar Z, the program established the accuracy of fit to the model as follows:
1) Mean offset to fit <½ the instrument measurement specifications.
2) Mean offset to fit less than the instrument measurement specifications.
3) Mean offset to fit <2 × the instrument measurement specifications.
4) Mean offset to fit <5 × the instrument measurement specifications.
5) Mean offset to fit >5 × the instrument measurement specifications.
To prevent the program from deleting frequencies solely to “force” a fit to the model, the following limitations were enforced in the software
1) A maximum of 25% of the frequencies (f) may be deleted.
2) Within any 3:1 range off, at least onef must remain.
3) Onlyf whose Z and θ lay more than the instrument specification from the curve may be deleted.
4) Only onef is deleted per iteration of fitting.
5) Af is only deleted if it results in the maximum improvement in resultant fit; this is not necessarily thef whose Z and θ lay farthest from the fit.
The Cole model was extended to allow for thef invariant time delay (T_{d}) caused by the speed at which electrical information is transferred through a conductor (15,37). The error introduced by this fixed T_{d} was modeled as a θ error that increases linearly with f. This linear θ error was mathematically modeled by multiplyingEq. EA1
by the factore
^{−jwTd}. Thus the overall modeled equation was
fC was computed after the model components (R_{E}, R_{I}, CM, T_{d}, and α) had been determined by solving the equation
Appendix
Theoretical Volume Equations
The ECW and ICW volumes were predicted from the modeled R_{E}and R_{I} by using equations formulated from Hanai’s theory, which describes the effect that a concentration of nonconductive material has on the apparent resistivity (ρ) of the surrounding conductive fluid, and is
From Eq. B1, with the following assumptions, we derived a set of equations as follows
The following assumptions were made:
1) The volumetric concentration of nonconductive elements in the body at low frequencies (LF) is given by
2) The volumetric concentration of nonconductive elements in the body at high frequencies (HF) is given by
3) V_{Tot} is body Wt/D_{b}.
4) The total volume of a body fluid can be described by
5) The factors D_{b},K _{B}, and ρ_{F} can be considered largely constant.
6) The Hanai equation is applicable at HF and LF to mixtures found in the human body.
By using Eqs. B2 andB4, predicted V_{ECW} and V_{ICW} were computed, from which predicted TBW was computed by using the following equations
Computing the Constants
k
_{ECW} is established as the mean value of
Appendix
Derivation of K_{B}
It should be noted that the derivation forK _{B} shown here is only an approximation for the purposes of confirming whether its use results in a ρ_{ECW} value that is within the range measured by other investigators.
The resistance (R) of a cylinder, measured longitudinally, is given by
Restating Eq. C1 in terms of the cylinder length and circumference
If we consider the body to be formed by five cylinders (the legs, the arms, and the trunk), then the volume of the body is given by
When we measure the Z between the wrist and the ankle, the measured value will be
Combining Eqs. C4,C5, andC6 yields
Appendix
Removing the effects of T_{d}
T_{d} can be removed through modeling, but it is only applicable when the actual measured data are used as input to fitting a program against a model. In this case, an additional multiplicative term must be added to the model (Eq. EA1
), which yields no change in amplitude but instead yields a linear θ shift with increasingf. As shown, the user will have one more “constant” term (K in this example) in the model
The modeling approach to removing T_{d} performs the best, allowing the user to extend the usable f range up to ∼1 MHz, thus allowing for higher accuracy modeling. However, this technique is the most complex and may take excessive computing knowledge or time. The user can also approximate the observed θ error manually and recalculate the set of measurement data including a correction for this effect. It should be noted that, without employing the full model as outlined above, it is not possible to separately model this effect alone because the biological effects will “skew” the results. The manual method does offer the advantage of being very fast to implement, and it may be performed either by computer or by hand. The technique does not offer the accuracy of this first method, however, because the correction is only optimized over relatively few data points. These data do, however, increase the usefulf range of the measured data up to several hundred kiloherz.
Because there is little biological θ shift caused byC
_{m} at HF, choose a HF and assume that all the observed θ shift is caused by T_{d}. If 1MHz data (as an example) are selected, note the measured θ shift at 1 MHz as φ1_{MHz}, then correct the measured θ shifts by subtracting φ1_{MHz}× f/1,000 from all measured θ shifts. The measured Z data need no correction. The final, corrected, resistance (R) and reactance (X) data may be computed as follows