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LETTER TO THE EDITOR

Second generation mixture theory equation for estimating intracellular water using bioimpedance spectroscopy

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 In an unpublished analysis, Xitron found that the second ICW equation produces more accurate results then the first.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), _TBW is the resistivity of the overall fluid (·cm), and R_INF is the resistance of the overall fluid () (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 (),

where V_ICW is the ICW volume (liters).

R_I is a model fitting term (), 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, _TBW, _ECW, and _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 _ICW is the resistivity of the ICW (·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).

_ECW is the resistivity of ECW (·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, _ECW, _ICW, and D_B, please refer to the paper by De Lorenzo (2).

REFERENCES

1. Cole KS. Permeability and impermeability of cell membranes for ions. Cold Spring Harb Symp Quant Biol 8: 110–122, 1940.[Abstract/Free Full Text]
2. De Lorenzo A, Andreoli A, Matthie J, and Withers P. Predicting body cell mass with bioimpedance by using theoretical methods: a technological review [published erratum appears in J Appl Physiol 83(6): following table of contents, 1997]. J Appl Physiol 82: 1542–1558, 1997.[Abstract/Free Full Text]
3. De Vries PMJM, Meijer JH, Vlaanderen K, Visser V, Oe PL, Donker AJM, and Schneider H. Measurement of transcellular fluid shift during haemodialysis. Med Biol Eng Comput 27: 152–158, 1989.[CrossRef][Web of Science][Medline]
4. Earthman CP, Matthie JR, Reid PM, Harper IT, Ravussin E, and Howell WH. A comparison of bioimpedance methods for detection of body cell mass change in HIV infection. J Appl Physiol 88: 944–956, 2000.[Abstract/Free Full Text]
5. Ellis KJ, Shypailo RJ, and Wong WW. Measurement of body water by multifrequency bioelectrical impedance spectroscopy in a multiethnic pediatric population. Am J Clin Nutr 70: 847–853, 1999.[Abstract/Free Full Text]
6. Geddes LA and Baker LE. The specific resistance of biological material: a compendium of data for the biomedical engineer and physiologist. Med Biol Eng 5: 271–293, 1967.[CrossRef][Web of Science][Medline]
7. Gudivaka R, Schoeller DA, Kushner RF, and Bolt MJG. Single- and multifrequency models for bioelectrical impedance analysis of body water compartments. J Appl Physiol 87: 1087–1096, 1999.[Abstract/Free Full Text]
8. Hanai T. Electrical properties of emulsions. In: Emulsion Science, edited by Sherman PH. London: Academic, 1968, p. 354–477.
9. Ho LT, Kushner RF, Schoeller DA, Gudivaka R, and Spiegel DM. Bioimpedance analysis of total body water in hemodialysis patients. Kidney Int 46: 1438–1442, 1994.[Web of Science][Medline]
10. Matthie JR, Withers PO, Van Loan MD, and Mayclin PL. Development of a commercial complex bio-impedance spectroscopic (CBIS) system for determining intracellular water (ICW) and extracellular water (ECW) volumes. In: Proceedings of the 8th International Conference on Electrical Bio-impedance. Kuopio, Finland: University of Kuopio, Finland, 1992, p. 203–205.
11. Van Kreel BK, Cox-Reyven N, and Soeters P. Determination of total body water by multifrequency bio-electric impedance: development of several models. Med Biol Eng Comput 36: 337–345, 1998.[Medline]
12. Ward LC, Elia M, and Cornish BH. Potential errors in the application of mixture theory to multifrequency bioelectrical impedance analysis. Physiol Meas 19: 53–60, 1998.[Medline]
13. Withers P. Multi-frequency impedance measurements of extracellular fluid volume. Physiol Meas 16: 71–76, 1995.[Web of Science][Medline]

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