*To the Editor*: In 1990, Xitron introduced a multifrequency impedance device designed for in vivo body composition studies (10). Soon after, Xitron provided software with the device for computing the Cole model of biological tissue (1) and predicting intracellular water (ICW) and extracellular water (ECW) using equations Xitron derived from mixture theory (8, 13). Before introduction of the software, an improvement in the ICW equation was developed and provided internally by Xitron.

The Xitron ECW and first generation ICW volume equations have been published both by Xitron and others (2, 5, 7, 9, 10). On numerous occasions Xitron has indicated that there was a slight difference between Xitron's published ICW equation and that used (2, 4). Because the actual equation has not been published, it is possible that some investigators may have or still are using the first ICW equation (7, 9). In an unpublished analysis, Xitron found that the second ICW equation produces more accurate results then the first. A study published in this journal in 2000 found that Xitron's second ICW equation predicts ICW change more accurately than other published equations (4). Nonetheless, studies are underway to validate the differences between the two equations, and other investigators may wish to do the same. It is also important to make public Xitron's second generation ICW equation so that further progress can be made in the development of theoretically based volume equations (3, 11, 12).

## RATIONALE FOR NEW ICW EQUATION

The first Xitron equations assumed that the relationship between impedance at low frequency and ECW volume was a simple nonlinear mixture effect involving two spaces (conductor and nonconductor). At high frequency we assumed that there were three spaces (ECW, ICW, and nonconductor) and that the relationship between total body water (TBW) resistivity and ECW-ICW ratio was linear.

Internal analysis revealed that the effect of an ECW-ICW ratio change on TBW resistivity is highly nonlinear because the ICW has a 3–4 to 1 greater resistivity than ECW (1, 6). The ECW equation has not changed. The second generation ICW equation presented below assumes two vs. one mixture effect: one at low frequency to account for the relationship between ECW and the reminder of material in the body (considered nonconductor), and one at high frequency to account for the relationship between ECW, ICW, and nonconductor. Using the same analysis as used for the ECW volume equation described in the appendix, with the same assumptions: (1) where Wt is body weight (kg); Ht is height (cm); *K*_{B} is a factor correcting for a whole body measurement between wrist and ankle, relating the relative proportions of the leg, arm, trunk, and height; V_{TBW} is the TBW volume (liters) (ECW + ICW), ρ_{TBW} is the resistivity of the overall fluid (Ω·cm), and R_{INF} is the resistance of the overall fluid (Ω) (i.e., infinite frequency resistance).

Dividing *Eq. 1* by the published ECW equation yields (2) where R_{E} is the value from the model fitting (Ω), where V_{ICW} is the ICW volume (liters).

R_{I} is a model fitting term (Ω), and, from the parallel resistances formed by R_{E} and R_{I}, Thus (3) Simplifying *Eq. 3*, (4) or (5) From the theory of Hanai, for a mixture of two conductive fluids (6) where C_{ICW} = V_{ICW}/V_{TBW}, i.e., the volumetric ratio concentration of ICW in the overall fluid, σ_{TBW}, σ_{ECW}, and σ_{ICW} are the conductivities of the overall fluid, ECW, and ICW, respectively.

Reexpressing this equation using resistivities and the concentration of ECW (rather than ICW) yields (7) where ρ_{ICW} is the resistivity of the ICW (Ω·cm).

Expanding yields (8) Combining yields (9) But, as shown in *Eq. 4*, (10) i.e., (11) Combining *Eqs. 9* and *11* yields (12) which after simplification becomes (13) or (14) Thus, from the measured circuit model resistances and the computed ECW volume, it is possible to calculate the ICW volume using *Eqs. 5* and *14*.

## APPENDIX

As reported, our equation for ECW is as follows: where V_{ECW} is the predicted total ECW volume (liters). ρ_{ECW} is the resistivity of ECW (Ω·cm); and D_{B} is body density (kg/cc).

For a full description of the assumptions and computation of *K*_{B} and values used for *K*_{B}, ρ_{ECW}, ρ_{ICW}, and D_{B}, please refer to the paper by De Lorenzo (2).

- Copyright © 2005 the American Physiological Society