Abstract
Knowledge of patient fluid distribution would be useful clinically. Both singlefrequency (SF) and impedance modeling approaches are proposed. The high intercorrelation between body water compartments makes determining the best approach difficult. This study was conducted to evaluate the merits of an SF approach. Mathematical simulation was performed to determine the effect of tissue change on resistance and reactance. Dilution results were reanalyzed, and resistance and parallel reactance were used to predict the intracellular water for two groups. Results indicated that the amount of intracellular and extracellular water conduction at any SF can vary with tissue change, and reactance at any SF is affected by all tissue parameters. Modeling provided a good prediction of dilution intracellular and extracellular water, but an SF method did not. Intracellular, extracellular, and total body water were equally predicted at all frequencies by SF resistance and parallel reactance. Extracellular and intracellular water are best measured through modeling, because only at the zero and infinite frequencies are the results sensitive only to extracellular and intracellular water. At all other frequencies there are other effects.
 bioimpedance spectroscopy
 body cell mass
 extracellular water
 intracellular water
 total body water
knowledge of patient fluid distribution would be very useful for guiding drug and renal replacement therapy as well as nutritional support. Drug and renal replacement therapy to alleviate excess fluid, as well as drug dosage, is presently prescribed using only gross estimates of fluid distribution. Inaccurate information about fluid distribution can change drug disposition and requirement and cause serious acute and longterm complications for a dialysis patient. Nutritional status is generally assessed with often unreliable chemical and anthropometric methods. Dilution methods are tedious, and magnetic resonance imaging, computer axial tomography, and whole body counting are expensive and not suited for routine use. Some investigators are attempting to use an Xraydetermined fatfree mass to assess nutritional status, but this method cannot distinguish the contribution of water to the estimate of fatfree mass. A loss in body cell mass (BCM) with a concurrent increase in extracellular water (ECW) (25, 36) would result in no detectable change in fatfree mass. A simple, inexpensive, accurate, and reliable noninvasive method of determining fluid distribution is needed. Impedance methods are increasingly being used to fulfill this need.
Bioimpedance spectroscopy (BIS), which means fitting measured impedance spectral data to a physical model, is a wellknown analytic technique (19). All the underlying principles used today in the impedancebody composition field came from the use of this technique in biophysics (8). In review, there is little conduction through skeletal muscle tissue at low frequency (1–5 kHz), and the impedance is principally a function of the ECW. As frequency increases, conduction through the intracellular water (ICW) increases. At high frequency (10–100 MHz) the ICW becomes fully conductive, and the impedance is a function of both ECW and ICW (Fig.1). This phenomenon is caused by cell membrane capacitance (C _{m}) and named βdispersion (4, 31). Thus at very low and very high frequencies the overall impedance is essentially independent of theC _{m}, whereas at the mid or characteristic frequency (f _{c}) the dependence on the value ofC _{m} is at a maximum. The f _{c}can also be defined as the frequency of maximum reactance. Two different phenomena discovered at very low frequency (<1 kHz) and very high frequency (>100 MHz) are named α and γdispersions, respectively (31). Impedance spectral data measured on biological tissue produce a semicircle with a suppressed center when resistance and reactance are plotted (Fig. 2). The physical model most widely used since 1940 (23) to interpret this phenomenon is the Cole model (3). The Cole model consists of the terms resistance ECW (R_{E}), resistance ICW (R_{I}),C _{m}, and the exponent α. It should be noted that the relationship between R_{E} and R_{I} and their respective volumes is not simple, because of the nonlinear effects of the concentration of nonconductor on the impedance (4, 41). Thus R_{E} and R_{I} are only model terms, but for simplicity R_{E} and R_{I} are expressed as resistance ECW and ICW, respectively. The Cole model is computed by using nonlinear curve fitting to extrapolate the data to the low and highfrequency limits (19) (Figs. 1 and 2). These limits are known as resistance at the zero frequency (R_{0}), which is the same as R_{E}, and resistance at the infinite frequency (R_{∞}). R_{I} is computed as 1/R_{I} = 1/R_{∞} − 1/R_{0}. Although the Cole model is a mathematical model and cannot be correctly represented as an equivalent electronic circuit, an analog representation is shown in Fig.3. Because biological tissue consists of multiple parameters, modeling is considered essential, because it is the only means of independently analyzing the different parameters (19).
Thomasset was the first to attempt to apply this biophysics knowledge clinically in 1963 (34). He proposed use of dualfrequency impedance (i.e., 1 and 100 kHz) as a measure of ECW and total body water (TBW), respectively (12, 34). An overwhelming number of different impedance methods and multiple regression equations have since been introduced. The field is now divided along three major lines of reasoning. The first is the 50kHz singlefrequency impedance method that was originally proposed as a measure of TBW (12) and fatfree mass (18,26). With little scientific explanation, this method evolved into a measure of both ECW and TBW (16) and now into a measure of TBW and BCM (2, 14, 15, 24, 33). The second is prediction of ECW and TBW by low (e.g., 1–5 kHz) and high (e.g., 100–500 kHz)frequency impedance (5, 32). The third is use of a BIS or Cole modeling approach (4, 6, 11, 13, 3840).
Because ECW and ICW are tightly regulated biologically and are parts of the TBW, impedance measured at virtually any frequency in the 1kHz to 1MHz range equally predicts ECW, ICW, and TBW (40). This high intercorrelation between the body water compartments has made determining the best approach difficult. A plethora of singlefrequency impedance equations have been published, but cross validation and detection of small changes in volume have generally been difficult. As defined by the Cole model (3), biological tissue consists of four parameters. A singlefrequency impedance consists of two data (i.e., resistance and reactance). It may be that singlefrequency impedance equations do not cross validate well or detect small changes because they rely on the tissue elements having relative uniformity between individuals and when their tissue changes. There is also good reason to believe that at any frequency other than zero and infinity the impedance measurement would be affected, because a different amount of volume would be measured when a change inf _{c} occurred. To test these and other research questions, we conducted an evaluation. Impedance spectral data measured before and after infusion during cardiac surgery and before and after hemodialysis were compared with the results of mathematical simulation. This determined the sensitivity of resistance and reactance to changes in the parameters of the Cole model. Previous deuterium and sodium bromide results reported on cardiac surgery patients (28) were reevaluated after use of different apparent ECW and ICW resistivity constants in the volume equation we developed from mixture theory (4). This analysis was conducted because only the differences between methods had been reported (28), and it has been discovered that the scaling of the apparent resistivity depends on dilution method and protocol (4). To ascertain the validity of the recently proposed parallel reactance (X_{P}) model (14, 15), we computed X_{P} at multiple frequencies and predicted the deuteriumsodium bromide space (i.e., ICW) measured on the cardiac surgery patients. To test for sample dependency, total body potassium (TBK) measured on a second sample was compared with TBK predicted by X_{P} at multiple frequencies.
MATERIALS AND METHODS
The cardiac surgery study was conducted at Henry Ford Hospital and approved by the Henry Ford Human Research Committee; written informed consent was obtained from each subject. Eight men, after elective coronary artery bypass graft surgery, elected to participate. On the patients’ arrival at the intensive care unit, body weight was measured to the nearest 1 kg with a standard balance, with the patient dressed in a hospital gown. Body height was measured to the nearest 1 cm with a stadiometer. Within 2 h of the patients’ arrival, two blood samples were taken to establish baseline deuterium and sodium bromide concentrations. Patients then were measured preoperatively (time 1) with a singlefrequency device (1 frequency at 50 kHz; model BIA 101, RJL Systems, Detroit, MI). Patients were then immediately measured with a multiplefrequency impedance device (44 frequencies logarithmically spaced between 1 kHz and 1.348 MHz; model 4000B, Xitron Technologies, San Diego, CA). Patients were placed in a supine position on their beds with limbs slightly abducted. Skin current electrodes (model IS4000, Xitron Technologies) were placed on the right dorsal surface of the hand and foot at the metacarpals and metatarsals. Voltagedetector skin electrodes were placed at the right pisiform prominence of the wrist and between the medial and lateral malleoli at the ankle. Electrode placement was marked for subsequent measurements, and, when possible, electrodes were left in place for the duration of the study.
Within 2 h after surgery, the patients underwent bioimpedance testing again (time 2). Patients also received 10 g of intravenous solution of deuterium (9 ml of 99.9%^{2}H_{2}O) and 30 mmol of sodium bromide (10 ml of a 3 mmol/ml solution of sodium bromide) over 2 min through a central venous catheter. After 4 h of equilibration, two additional blood samples were taken to determine postequilibration concentrations (17, 29, 35). To predict ECW and TBW volumes, bioimpedance was measured again immediately after blood sampling (time 3). Blood samples were collected in serum separator tubes, and the clotted samples were centrifuged for 30 min. Mass spectroscopy (Metabolic Solutions, Boston, MA) was used for deuterium analysis, and TBW was calculated using the method of Schoeller et al. (30). Sodium bromide concentrations were measured using an inductively coupled plasma mass spectrometer (PlasmaQuad2+, VG Elemental, Winsford, Cheshire, UK). Corrected sodium bromide space was then estimated using the method of Price et al. (29), whereby a correction is made for sodium bromide uptake into the red blood cells and for Donnan equilibration. Dilution ICW was computed by deuteriumsodium bromide.
The effects of isotonic saline infusion during surgery and after recovery from surgery on the Cole model parameters were analyzed. Impedance measurements were taken on day 2, 24 h postdilution steadystate equilibration (time 4), and onday 3, 48 h postdilution steadystate equilibration (time 5). Patients received an average of 2.1 liters of infused fluid during surgery.
For the singlefrequency method, ECW and TBW volumes were predicted by applying the measured resistance and reactance measured attime 3 to the same statistical equations used by Patel et al. (28; see also Refs. 24 and 37). For the BIS method a nonlinear curvefitting program described previously (4) was used to fit impedance and phase spectral data to the Cole model. Cole model terms R_{E} and R_{I} were used in a volume equation we developed from mixture theory to predict ECW and ICW volumes, respectively (4). TBW was computed as ECW + ICW. Recorded fluid inputs and outputs were used to calculate estimated body weight at the time of dilution steady state from baseline. As discussed previously (4), the mixturevolume equation utilizes apparent ECW and ICW resistivity constants. The constants used by Patel et al. (28) were previously established from deuterium and sodium bromide data (38) and cross validated (27). The male ECW and ICW constants used to predict ECW and ICW were 214 and 824 Ω ⋅ cm, respectively (28). Also, these terms are only apparent resistivities, because they are affected by geometry when a wristtoankle measurement is made. (For a full discussion see Ref. 4.) For the statistical analysis the Microsoft Excel program was used. In addition to the descriptive statistics, Pearson’s product moment correlation and standard error of estimate (SEE) statistics were computed for the relationship among dilution, singlefrequency, and BISdetermined ECW, TBW, and ICW volumes.
Resistivity constants computed from deuterium and sodium bromide data collected by De Lorenzo et al. (4) were then used to repredict ECW and ICW volume. The male values used for ECW and ICW were 174 and 1,177 Ω ⋅ cm, respectively. The correlation, SEE, mean, and mean differences were recomputed. New resistivity constants for the ECW and ICW were then computed by regressing the samples computed by Cole model terms R_{E} and R_{I} against their dilutiondetermined ECW and ICW volumes, respectively. The equation and methods we used to compute resistivity have been described previously (4, 11). The ECW and ICW resistivity constants became 229 and 1,054 Ω ⋅ cm, respectively.
Reactance at 5, 10, 49, 100, 204, 424, and 876 kHz measured attime 3 was transformed into X_{P} by using the recently proposed equation: X_{P} = reactance + resistance^{2}/reactance (14, 15). X_{P} was then used in the male multiple regression equations published by Kotler et al. (14) to predict the dilution ICW: 254 * Ht^{2}/X_{P}+ 1,493, where Ht is height (simple linear equation), 59.06 * Ht^{1.6}/
To test for sampledependent variation, we conducted the same X_{P} analysis on a second group of 48 healthy Italian men 26–57 yr of age. This study, which was performed at University of Rome “Tor Vergata,” was approved by their Medical Ethical Committee. The subjects volunteered to participate in the study (4), and written informed consent was obtained from all participants. On the subjects’ arrival in the morning in an overnightfasted state, body weight was measured to the nearest 0.05 kg with a standard balance, with the subjects dressed in swimming clothes. Body height was measured to the nearest 1 mm with a stadiometer. After the measurement of weight and height and with the subjects still in the fasting state, ^{40}K was measured with a whole body counter. The whole body counter was formed by a cell 2.5 m wide and 3 m high of 10cmthick lead bricks, the door of which was formed by a 22cmthick iron slab. The room was continuously ventilated. A single 20.3 × 10.2cm thalliumactivated sodium iodine crystal was positioned above the subject, who was measured in a sitting position and dressed only in paper pajamas. TBK was calculated as ^{40}K * 8,474.6 (7). The coefficient of variation for TBK was 2–3%. ICW volume was computed by assuming that TBK is present only in the ICW and that potassium concentration in the ICW is 150 mM (7). After the measurement of weight, height, and TBK, wristtoankle (i.e., whole body) impedance spectra, consisting of 21 frequencies ranging from 1 kHz to 1.348 MHz, were measured with the same model multiplefrequency impedance device described above. The measurements were taken within the first several minutes after the subjects assumed a supine position. The measurements were taken on the left side of the body with use of disposable electrocardiograph electrodes (5 cm^{2}, 3M, Minneapolis, MN) and in accordance with the standard wristtoankle protocol discussed above (38). The impedance and phase spectral data were fit to the Cole model. The equation described above (14, 15) was used to transform reactance at 5, 10, 50, 100, 200, 500, and 748 kHz into X_{P}. Transformed X_{P} was used in the same equations to predict the TBKdetermined ICW of 48 healthy Italian men. Predicting ICW with a TBK equation has validity, and any error in assuming the potassium concentration in the ICW would only result in a scaling difference (7). The frequencies differed from those used for the cardiac patients, because different software, which requested a different set of frequencies from the device, was used.
To investigate the effects of fluid and solute removal through hemodialysis, impedance spectral data measured on 16 patients before and after hemodialysis were fit to the Cole model and analyzed. This study, which was conducted at the Karolinska Institute, was approved by the Huddinge Hospital Medical Ethical Committee. The patients (10 men and 6 women), ranging in age from 35 to 84 yr, volunteered to participate in the study. Informed consent was obtained from all participants. The patients were randomly selected from 28 hemodialysis patients. The amount of fluid removed by ultrafiltration ranged from 0.08 to 3.96 liters. Treatment lasted for 4–4.5 h. Blood flow was 250–300 ml/min, dialysate flow was 500 ml/min, dialysate sodium concentration was 141 mM, and bicarbonate concentration was 34 mM. Dialysis machines (model AK 100, Gambro) and several types of dialyzers (models AC 130 and 170, Baxter, and models GEF 15 and GEF 18, Gambro) were used. Temperature did not change during the procedure. On the patients’ arrival in the dialysis unit and after hemodialysis, body weight was measured to the nearest 0.1 kg with a standard balance. Body height was measured to the nearest 1 cm with a stadiometer. After the patients were in a supine position on a bed for 20 min, wristtoankle (i.e., whole body) impedance spectra, consisting of 50 frequencies ranging from 5 to 500 kHz, were measured with the multiplefrequency device described above. The measurements were taken on the side opposite the side with the arteriovenous fistula by use of skin electrodes (model IS4000, Xitron Technologies) and in accordance with the standard wristtoankle protocol discussed above. Electrodes were left in place throughout the study, and patients remained in a supine position throughout the treatment. After termination of treatment, impedance spectra were measured again, and the data were fit to the Cole model. Descriptive statistics were computed, and the spectral data were graphically formatted. The difference in impedance before and after dialysis at the low and highfrequency limits (i.e., R_{0} and R_{∞}) and at 5 and 500 kHz was compared.
To simulate the sensitivity of resistance and reactance to changes in the parameters of the Cole model, we computed resistance and reactance at multiple frequencies in the 100Hz to 10MHz frequency range (i.e., βdispersion) (3, 31) for nominal data and with the modeled parameters changed. Nominal data were Cole modeling results obtained previously on one healthy adult woman (27) and previously reported dispersion data outside the βdispersion range on skeletal muscle tissue (3, 31). At each frequency, resistance and reactance were computed with the Cole model term α set at 0.63 and 1, after 5 liters were added to the ICW and ECW, and forf _{c} changed from 40 to 100 kHz. The resistance and reactance data were then plotted vs. log frequency with use of a Microsoft Excel Spreadsheet program.
To properly evaluate the singlefrequency X_{P} proposal, we performed a mathematical modeling evaluation (MATLAB, Math Works, Natick, MA). This was performed to ascertain the interaction between X_{P} and the biophysical parameters expressed in the Cole model. Use of the Cole model had validity, because it is the model most widely used to interpret impedance measurements of biological tissue (23), and measured data in vivo fit it with high precision (4). Impedance spectral data of one healthy Asian man were fit to the Cole model. Nominal values were 594.9 Ω for R_{E}, 935.6 Ω for R_{I}, 2.85 nF forC
_{m}, and 0.7 for α. For frequencies of 5, 50, and 200 kHz and 1 MHz, we individually varied R_{E}, R_{I},C
_{m}, and α ±20% in 1% increments. The following equations were used to compute X_{P} at each 1% increment for each frequency ( f). For the Cole model, we have the impedance (Z) equation
The results were then divided by the nominal value to express the result uniformly as a ratio. Then the X_{P} ratio change was plotted vs. the change in ratio for R_{E}, R_{I},C _{m}, and α.
RESULTS
The physical characteristics of the cardiac surgery patients, as well as their dilutiondetermined ECW, ICW, and TBW volumes, are shown in Table 1. Table2 displays their Cole modeling results. The impedance data measured on the eight male cardiac surgery patients corresponded extremely well to the Cole model. The results of Cole modeling computed at five different time points before and after cardiac surgery are shown in Table 3. R_{E} and R_{I} are not simply related to volume (4). However, these data suggest that the infusion of fluid affected both the ECW and ICW, since both R_{E} and R_{I} changed. The infused fluid resulted in a decrease in R_{E} and R_{I} and a concurrent increase inf _{c}.
As shown in Table 4, for the cardiac surgery patients, singlefrequency impedance predicted TBW well, with little mean difference compared with dilution. The correlation between singlefrequency impedance and dilution ECW was reasonable, but the SEE was quite high, and there was a mean difference of 1.7 liters (Table 4). For singlefrequency ICW, which was determined by subtracting the predicted ECW from the predicted TBW, both the correlation and SEE values were poor, and the mean difference was large (Table 4).
Except for mean difference, the BIS prediction of the cardiac surgery patients’ dilution ECW, ICW, and TBW volumes was good for all three sets of resistivity constants (Table 4). For ECW the correlation and SEE values were the same for each constant (Table 4). For ICW, correlation and SEE varied slightly. This establishes that ECW resistivity, as used in our mixturevolume equation, is purely a scalar term, having no effect on correlation or SEE. ICW resistivity is effectively a scalar, since the nonlinearity was slight (Table 4). Most noticeable about Table 4 is that the use of different resistivity constants dramatically changed the mean difference between the BIS and dilutiondetermined volumes. The variation in scaling caused by the dilution method equally affects the singlefrequency method. These results are considered further in thediscussion.
For the cardiac surgery patients, the strongest prediction of dilution ICW with X_{P} was achieved by using Kotler’s base exponential equation at all frequencies (14) (Table5). Similar predictions of dilution ICW were discovered at all the measured frequencies. Most interesting was that the best prediction produced by X_{P} was not at 49 kHz but at 204 kHz. As shown in Table 6, the prediction of dilution ICW with use of resistance alone, which is in the X_{P} equation, was strong. X_{P} also predicted dilution ECW and TBW with high correlation and reasonable SEE values at all frequencies measured (Table 7). The descriptive characteristics of the 48 healthy Italian men used to further analyze the X_{P} model are shown in Table8; their Cole modeling results are shown in Table 9. As found with the cardiac surgery patients, similar predictions of TBK were produced by X_{P} at all frequencies (Table10). X_{P} computed at 50 kHz again did not provide the best prediction of TBK. The best prediction was provided by X_{P} at 10 kHz, which is different by a factor of 20 from that which predicted best for the cardiac surgery patients (Table 5). The TBK predicted by resistance alone was almost as good as that predicted by the inclusion of reactance (Table 11).
The descriptive characteristics of the hemodialysis patients are shown in Table 12; their Cole modeling results before and after hemodialysis are shown in Table13. As found with the cardiac surgery patients and healthy Italian men, the measured impedance data fit the Cole model well. Opposite to the results after infusion of fluid (Table3), R_{E} increased rather than decreased with ultrafiltration (Table 13). R_{I} remained relatively stable, suggesting that most of the fluid was removed from the ECW.C _{m} changed considerably, and α remained significantly lower than that found for healthy subjects (Table 9). The effect Cole model term α is believed to represent a distribution of time constants caused by different cell sizes and shapes (3). If this is the case, a decrease in α would suggest a widening in cell size and shape. Thef _{c}, rather than increasing, as when fluid was infused, decreased when fluid was removed (Table 13).
The results of the mathematical modeling revealed virtually no change in X_{P} at 50 kHz with a ±20% change in R_{E} and a change of only ∼2% at the extremes of frequency (Fig.4). A ±20% change in R_{I} caused an ∼15% change in X_{P} at 50 kHz and a very large change (e.g., ∼30%) at 1 MHz (Fig. 5). A ±20% change inC _{m} caused an ∼4% change in X_{P} at 50 kHz, and the effect was slightly nonlinear. At 1 MHz the effect of a ±20% change in C _{m}resulted in an ∼12% change in X_{P} (Fig.6). A ±20% change in α (from 0.56 to 0.84, ±20% from the nominal value of 0.7; Table 9) caused an ∼26% change in X_{P} at 50 kHz, and the result was quite nonlinear. The second largest effect on X_{P} was caused by a change in α at 1 MHz (Fig. 7).
The frequencyresistance plot (Fig. 8) suggests that at frequencies low enough to ensure that the measurement is solely dependent on ECW (below a few hundred hertz), the effects of a different phenomenon (αdispersion) have become significant (4,31). The resistance measured at the common 50kHz frequency (2, 18, 33) was only partially conducting through the ICW; thus it is clearly dependent on both ECW andf _{c}. This also confirms the criticism that 50kHz measurements of TBW are dependent on high ECWTBW intercorrelation (40). It is well known that ECW and ICW are not fully measured until >10 MHz (4, 31). This is supported by in vivo data (Fig. 1). Even if it were technically feasible to measure to such high frequencies, the influence of another unwanted phenomenon (γdispersion) would still need to be removed. To predict TBW with any single frequency requires the use of a TBW resistivity term and the assumption that it is fixed. The resistivity of ECW is different from the resistivity of ICW by a factor of 3–4 (4, 9); thus the sensitivity of a single highfrequency measurement to changes in ECW and ICW is different. A simple change in the ECWtoICW ratio will significantly alter TBW resistivity and cause error, even when ion concentration has not changed. This error can be removed by solving for Cole model terms R_{E} and R_{I} and determining independent resistivities for the ECW and ICW, respectively. From both theoretical (Fig. 8) and measured data (Fig. 1), it can be readily demonstrated that at any single frequency the amount of ECW and ICW conduction varies not only between subjects but whenf _{c} changes. A change in R_{E}, R_{I}, orC _{m} can cause a change in f _{c}. When f _{c} changes, the amount of conduction through the ECW and ICW at any fixed frequency changes. Becausef _{c} decreases when fluid is removed through ultrafiltration, it was not surprising that, between pre and postdialysis, impedance at the lowfrequency limit (i.e., R_{0}) changed 18.9% more than at 5 kHz (Table 14). Nor was it surprising that impedance changed 9.2% less at the highfrequency limit (i.e., R_{∞}) than at 500 kHz (Table 14). Also not unexpected, impedance was greater at R_{0} than at 5 kHz and less at R_{∞} than at 500 kHz. These findings are considered more thoroughly in thediscussion.
The frequencyreactance plot (Fig. 9) suggests that reactance is sensitive to all body composition parameters in the frequency range aroundf _{c} and is sensitive to the other dispersions (i.e., α and γ) for frequencies significantly different fromf _{c}. This indicates a tenuous relationship between reactance at any single frequency and any single body composition parameter, with any apparent relationship being accentuated by the high correlation between parameters (40). Furthermore, the sensitivity of reactance is extremely dependent on the relationship between the frequency of measurement andf _{c} and is symmetrical aboutf _{c}. Phase is a function of the ratio of resistance to reactance [arctan(reactance/resistance)]; thus it is sensitive to all the problems associated with both singlefrequency resistance and reactance.
DISCUSSION
In review of Figs. 1, 8, and 9, the debate concerning the best single or dualfrequency impedance to use for measuring ECW and ICW appears irrelevant (5, 15). Apart from the error caused by not accounting for the different ECW and ICW resistivities, the proportion of current conducting through the ICW at any single frequency varies. This is supported by the hemodialysis results (Table 14). The change in impedance before and after hemodialysis at R_{0} was 18.9% greater than that at 5 kHz. With a decreasedf _{c} after dialysis (Table 13), conduction through the ICW at 5 kHz would be greater after than before dialysis. More ICW conduction would alter the impedance, because an increased volume (i.e., ICW) is being measured, making the measurement less sensitive to changes in ECW. Similarly, because ultrafiltration principally affects the ECW, one would expect the impedance to change less at R_{∞} than that at R_{0}, because both ECW and ICW are being measured. As also shown in Table 14, between pre and postdialysis the impedance changed 9.2% more at 500 kHz than at R_{∞}. Inasmuch asf _{c} decreases after dialysis (Table 13), there is more conduction through the ICW at 500 kHz than before dialysis. By use of curve fitting to determine the impedance at R_{0} and R_{∞}, the measurement becomes independent off _{c}, and error caused by the change inf _{c} is effectively removed (Fig. 2). Although R_{I} can be computed by the impedance measured at other frequencies, the error in calculating R_{I} can be as high as 200% by not using R_{∞} (13).
The previous conclusion that singlefrequency impedance and BIS methods provide similar ECW information appears tenuous. A 50kHz resistance had been used to predict TBW and fatfree mass, then the predicted fatfree mass and reactance were used to predict ECW (28). There is little relationship between reactance and ECW (32); thus, one datum (i.e., resistance at 50 kHz) should not be used to predict two variables. This is not intended as criticism, for such statistical reasoning is widespread, but rather an example of how a statistical equation that has no scientific basis can produce misleading results (33). The lack of scientific reasoning and reliance on statistical methods, despite high intercorrelation between variables (40), has caused a great deal of confusion in the field.
The finding that the ECW and ICW resistivities are scalar or multiplicative terms was replicated in this study (4, 22, 39) (Table4). Different dilution methods (e. g., sodium bromide vs. sulfate) produce significantly differentsized ECW and TBW spaces (7). It is now believed that the size of the space measured by dilution varies for even the same methods when different protocols and analysis techniques are used (4, 39). This is further supported by this study, since both deuterium and sodium bromide were used (Table 4). Had Patel et al. (28) previously used the resistivity constants computed by De Lorenzo (4) rather than those by Van Loan et al. (38), the mean TBW difference would have been very small, but then the mean difference for ECW would have been large (Table 4). Because each of these studies used deuterium and sodium bromide, it is difficult to determine which scaling is physiologically correct.
Several findings emerge from this reassessment of the dilution results on the cardiac surgery patients. The measured impedance spectral data corresponded well to the Cole model (Table 2), and the impedance spectroscopy approach provided a better prediction of ECW and ICW than did singlefrequency impedance (Table 4). The singlefrequency TBW result may appear adequate, but resistance at 50 kHz is a tenuous prediction of TBW, because, as shown in Figs. 1 and 8, only a portion of the ICW is measured at this frequency. Thus any 50kHz prediction of TBW would rely on high intercorrelation between ECW and TBW. The further a subject’s ECW and ICW volume deviated from the sample used to regress the equation, the less precise the prediction would become. This must be so, because the ECW and ICW resistivities are different by a factor of 3–4 (3, 9), andf _{c} can change when any tissue variable changes (Fig. 9). ECW can also be predicted by a 50kHz impedance by forcing a fit to an equation. That such an approach is simply forcing an equation to fit the data, rather than a measurement, is evidenced by the poor prediction of ICW (i.e., predicted TBW − predicted ECW) shown in Table 4. On the other hand, similar predictions were obtained for ECW, ICW, and TBW when the BIS approach was used (Table 4).
We previously stated that reactance can only be associated withC _{m}, because ICW is a resistive, not a reactive, medium (4). This is true, but the results of mathematical modeling revealed that X_{P} is strongly influenced by R_{I} (Fig. 5). The Cole model predicts that it will be. The problem is that X_{P} is also highly sensitive to changes in C _{m} and α (Figs. 6 and 7). AlthoughC _{m} and α are considerably decreased in the cardiac surgery patients (∼16 and 14%, respectively; Table 2) compared with healthy men (Table 9),C _{m} and α change dramatically in theory (Fig. 9) and in practice (Tables 3 and 13). Bestoso and Mehta (1) observed a mean 50% increase inC _{m} after fluid and solute removal by hemodialysis. We have discoveredC _{m} to be considerably decreased in clinical populations (Tables 3 and 13) compared with healthy subjects (Table 9) and on the individual level as much as 75% (Table 15). The exponent α was observed to be extremely decreased (i.e., 0.45) in severely depleted patients compared with healthy subjects (i.e., 0.7; unpublished observations). This study also discovered that α is lower in clinical populations (Tables 3 and 13) and is considerably lower on the individual level (Table 15).
Variation in C _{m}and α was less in patients who had fluid infused (Tables 3 and16) than in those who had fluid removed through ultrafiltration (Tables 13 and 15). We suspect this is due to chemical changes that occur during hemodialysis. ThatC _{m} increases with fluid removal and decreases with overhydration is very interesting. Theoretically, C _{m}changes only when there is a change in the thickness of the cell membrane (10). What is interesting about Figs.10 and 11is that on recovery (time 5) the frequency response approaches that of patients before cardiac surgery (time 1). Most noticeable about Figs. 12 and13 is the higherf _{c} for this dialysis patient than for healthy subjects (Table 9) and its change from 124 kHz before to 71 kHz after dialysis (Table 15). It can be seen in Fig. 10 that the data follow the common S curve that is easily fit. However, in Fig. 12 the curvature is less discernible. We previously reported (4) that modeling with both impedance and phase is essential, because phase has a much broader range of sensitivity to change than impedance (Figs. 11 and 13). Deurenberg et al. (5) stated that fitting the model with impedance alone is adequate. However, with the data shown in Fig. 12, the results of modeling by use of impedance alone would be far less precise, because there would be no discernible curve to follow without phase (19).
Similar to a singlefrequency resistancepredicted ECW or TBW, an X_{P}predicted ICW is dependent on the elements in the tissue having relative uniformity. The mathematical modeling revealed that X_{P} as presented previously (14, 15) is merely a simplification of the Cole model. Van Marken Lichtenbelt et al. (40) and others discovered that resistance at any frequency predicts ECW, ICW, and TBW with virtually equal precision. We have found that X_{P} also predicts ICW, ECW, and TBW with equal precision at any frequency measured (Tables 5, 7, and 10). The best prediction of ICW with X_{P}was not even produced by the proposed 49–50 kHz (14, 15) and was sample dependent. Furthermore, R_{S}alone provided similar predictions (Tables 6 and 11). From the name of this new theory, “parallel reactance,” it would seem that reactance should be providing most if not all the prediction, but the opposite is the case. The correlation and percent SEE values reported by Kotler et al. (14) for the X_{P}predicted BCM were only 0.04 and 1.1% better, respectively, than those using R_{S} alone. The same resistance was also used to predict TBW (14). Lukaski (15), who is promoting an X_{P} BCM prediction, previously promoted reactance as a measure of ECW (16), but confusingly it was reported to be invalid by Kotler’s laboratory, because reactance was contributing virtually nothing to the prediction (32).
No theoretical basis for predicting BCM with X_{P} has been reported (14), and the statement was made that “a major uncertainty in the theory underlying BIA [bioelectrical impedance analysis] whether the body’s ionic circuit is arranged as a series or parallel circuit” (14). This statement should have been accompanied by a reference. It was suggested that the improved correlation between X_{P} and BCM was proof that X_{P} was superior to X_{S} for predicting BCM (14). It is true that impedance measured at any single frequency can be interpreted as a series or parallel circuit, with both resulting in two final elements (resistance and reactance). The problem is that biological tissue consists of more than two elements. No reference was given, because there is no rigorous biophysical research to support this claim or a singlefrequency prediction of cell volume. Single biological cells have been interpreted in biophysics since 1925 as a threeelement model, with an R_{E} in parallel with a series C _{m} and R_{I} (8). Cole (3) added an important fourth term to the model (α) to represent the effects of an infinite number of seriesC _{m} and R_{I} values (cells) in a cell suspension (tissue) (Fig. 3). The “major” uncertainty proclaimed applies only to the singlefrequency approach. The improved correlation discovered when BCM was predicted with a parallel, rather than a series, 50kHz reactance is meaningless, because X_{S} is not used for anything, and there is no scientific basis for either (14).
It had also been suggested that the improved correlation between X_{P} and BCM was proof that a “logarithmic transformation of X_{P} or impedance may be more representative of the body’s complex shape” (14). This is interesting, because an improvement in correlation for men from 0.83 to 0.84 for BCM but not from 0.77 to 0.78 for TBW was considered to be significant (14). It was also assumed that if the regression coefficients, SEEs, means, and standard deviations for the model and validation groups were not different statistically, the derived model was valid (14). On this assumption, final results were reported using equations based on the entire group. Although correlation and SEE values for the model and validation groups were reported, the regression coefficients and mean offsets were not (14). Thus a crossvalidation study was purported to have been performed, but interestingly the information needed to evaluate it was not provided. If the parameters of an equation vary randomly and are sample dependent, the scientific basis of the equation can be challenged, and it will likely be poor at detecting change. In this regard, it is tenuous that the scaling coefficients (i.e., statistically derived ICW resistivity) reported for the entire group were 59.06 for men and 1.30 for women (14). Such a difference in ICW resistivity is beyond unrealistic and suggests that something was severely wrong. It is also tenuous that the predictions across the board were significantly improved when the sexes were combined. The most logical explanation is that the increased sample size reduced the effects of significant random error. This is supported by Fig. 1 in Ref. 14, in that there were many outliers, some by as much as 50%. Possibly this explains why plots of differences between methods were not reported.
In this recent study the prediction of TBK by X_{P} was also discussed, but BCM was interchanged with TBK throughout the article (14). The authors also did not report any absolute values for TBK or BCM predicted by X_{P}, only correlation and percent SEE values. It is important to note that BCM is only a concept, and any relationship between TBK and BCM derives from the relationship between ICW and BCM. ICW can be determined by dilution (39, 40) or by dividing TBK by an assumed ICW potassium concentration (7, 25). Classically, the ICW TBK is expressed as 150 mM (7). BCM is then computed by dividing ICW by an assumed cell hydration constant [e.g., ICW/0.70 (25)]. Variation in the ICW TBK and BCM constants will cause a scaling error in the predicted ICW and BCM. However, for a strong correlation to emerge between TBK and BCM, the TBK ICW and ICW BCM relationships must be very similar between subjects. Without a description of exactly how BCM was determined, we could not fully replicate this previous study (14). To calculate ICW from the X_{P}predicted TBK, we assumed an ICW TBK concentration of 150 mM (Tables 5 and 10). It is curious why absolute values for predicted ICW, BCM, TBW, and ECW were not reported in this previous study (14). Both ICW and TBW had been measured by criterion methods, so ECW could have been predicted. As discussed and shown in Table 4, if two of the three volume spaces are correctly determined, the third should also be reasonably predicted (4, 39). Nevertheless, we found a considerable mean difference between ICW determined by dilution (Table 1) and TBK (Table 8) and ICW predicted by Kotler’s X_{P} equation (Tables 5and 10). It also varied considerably with frequency. With such a poor theoretical basis, frequency dependency, sensitivity to multiple tissue variables, and a high intercorrelation between variables, it is difficult to discern what exactly X_{P} is measuring. This is why we conducted mathematical modeling. What X_{P} is measuring is very important, because if it does not physically relate to TBK, its ability to measure change will be tenuous, because the relationship of X_{P} to ICW is unknown. All these factors may explain in part why the correlation between X_{P} and TBK change reported previously was poor (r = 0.58) (14). We have reported since 1992, against rhetorical argument (5), that the relationship between impedance and ECW and ICW volume is far more complicated than the simple Ht^{2}/resistance index used (4). Such an approach assumes that only one material is being measured. It is well known that the proportion of different conductors (ECW and ICW) (9) and nonconductors (bone and fat) has a nonlinear effect on impedance (4, 41). Albeit randomly discovered, what is important about the findings of Kotler et al. (14) is that the exponents they discovered statistically for height were very close to the 1.5 exponent we reported in 1992 using Hanai mixture theory (4). We also found that Ht^{1.5} provided a simple form of mixture theory, but it relied on differences in body composition being purely due to differences in fluid rather than differences in both conductor and nonconductor. By modifying the equation to incorporate total object volume, we found that it was possible to account for both conductor and nonconductor effects (4).
In conclusion, the results of this study suggest that the Cole model may be applicable to multiple clinical populations. This study has demonstrated that the common 50kHz frequency used to predict TBW is only partially conducting through the ICW. Thus any prediction of TBW provided by impedance measured at 50 kHz is achieved only because of high ECWTBW intercorrelation. The problem is that the proportion of ECW and ICW measured is not fixed. Whenf _{c} changes, the proportion of ECW and ICW measured changes, and this is true for all singlefrequency measurements. Only by fitting to the zero and infinite frequencies through modeling can this error be removed. An additional problem of using a singlefrequency measurement to predict TBW is that the sensitivity of a single highfrequency measurement to changes in ECW and ICW is different. This is because the ECW and ICW have significantly different resistivities. Thus a simple change in ECWtoICW ratio will alter TBW resistivity and cause error. Through modeling, this error can be removed by independently determining ECW and ICW resistivities and volumes and computing TBW as ECW + ICW. This study has established that resistivity is a scalar term dependent on the dilution method used. Thus, before future studies place blame for the cause of the difference, a more extensive evaluation should be conducted. For the cardiac surgery patients the BIS prediction of ICW was good but the singlefrequency prediction was not. This indicated that the BIS method is providing more of a measurement, whereas a singlefrequency approach relies on the tissue parameters having relative uniformity between individuals. This study found that, at any single frequency, both resistance and X_{P} provide similar predictions of ICWBCM. Furthermore, the best prediction was not even produced by X_{P} computed at the proposed 50kHz frequency, and the best predicting frequency was sample dependent. This suggests that X_{P} relies on consistency in the tissue parameters and that these parameters remain unchanged. This is supported by the finding that although X_{P} is sensitive to ICW, it is also highly sensitive to Cole model termsC _{m} and α. Thus X_{P} is sensitive to parameters other than that which it is purported to measure. ECW and ICW are best predicted through modeling, because only at the zero and infinite frequencies are the results sensitive to only ECW and ICW. At all other frequencies there are other effects.
Acknowledgments
The work conducted by K. Katzarski was supported by Hospital International, Lyon, France.
Footnotes

Address for reprint requests: J. Matthie, Medical Dept., Xitron Technologies, Inc., 6295D Ferris Sq., San Diego, CA 92121 (Email:jmatthie{at}xitrontech.com).
 Copyright © 1998 the American Physiological Society