## A MATHEMATICAL MODEL OF THE DETERMINANTS OF THE PLASMA WATER SODIUM CONCENTRATION: THE TRUTH IS IN THE PROOF

letter to the editor: In his Letter to the Editor, Dorrington (2) questions the validity of our new equation defining the quantitative interrelationship between Gibbs-Donnan equilibrium, osmotic equilibrium, and the plasma water sodium concentration (8). First, Dorrington claims that our mathematical model is “inelegant,” although he states that the derivation is accurate. Specifically, he argues that the derivation of our equation involves “several unwieldy unnecessary repetitions” because “both ‘total exchangeable’ and ‘osmotically inactive’ sodium and potassium are introduced into the equation and then subtracted from each other to obtain surrogates for ‘osmotically active’ concentrations.” It seems that Dorrington is unaware of the fact that the measurement of total body Na^{+} and total body K^{+} requires the use of radioisotope dilution and that this method measures the total exchangeable Na^{+} (Na_{e}) and exchangeable K^{+} (K_{e}) pools (5). Dorrington must also be unaware of the empirical relationship between the plasma water sodium concentration ([Na^{+}]_{pw}), Na_{e}, K_{e}, and total body water (TBW) originally demonstrated by Edelman et al. (5): [Na^{+}]_{pw} = 1.11(Na_{e} + K_{e})/TBW − 25.6 (*Eq. 1*). Indeed, using the technique of radioisotope dilution, Edelman et al. empirically demonstrated that the Na_{e}, K_{e}, and TBW are the major determinants of the [Na^{+}]_{pw}. However, it is also well known that not all Na_{e} and K_{e} are osmotically active (1, 3–5). There is convincing evidence that a portion of Na_{e} is bound in bone and is therefore rendered osmotically inactive (3–5). Likewise, a portion of cellular K^{+} is reduced in its mobility and in its osmotic activity due to its association with anionic groups such as carboxyl groups on proteins or to phosphate groups in creatine phosphate, ATP, proteins, and nucleic acids (1). Since osmotically inactive Na_{e} and K_{e} cannot contribute to the distribution of water between the extracellular and intracellular compartments, osmotically inactive Na_{e} and K_{e} cannot contribute to the modulation of the [Na^{+}]_{pw}. Consequently, the terms (Na_{e} + K_{e})/TBW − (Na_{osm inactive} + K_{osm inactive})/TBW in our equation are used to represent the total osmotically active Na^{+} and K^{+} ions. The terms (Na_{e} + K_{e})/TBW − (Na_{osm inactive} + K_{osm inactive})/TBW are therefore important in demonstrating that not all Na_{e} and K_{e} in *Eq. 1* contribute to the modulation of the [Na^{+}]_{pw}. These two terms are indeed essential in attaining “scientific exactness and precision,” which is the very definition of “elegance” according to the *American Heritage Dictionary*.

Second, Dorrington also argues that our equation cannot account for the modulating effect of Gibbs-Donnan equilibrium on the [Na^{+}]_{pw} because it does not take into consideration the electrical potential across the capillary membrane and the mean number of negative charges on plasma proteins. It is well known that Gibbs-Donnan equilibrium is established when the altered distribution of Na^{+} and Cl^{−} ions results in electrochemical equilibrium (10). At Gibbs-Donnan equilibrium, the chemical gradient is equal in magnitude and opposite in direction to the electrical gradient as depicted by the following equations: (2) (3) where F is Faraday's constant; E_{m} is electrical potential; R is ideal gas constant; T is absolute temperature; [Na^{+}]_{pw} and [Na^{+}]_{ISF} are plasma water and interstitial fluid sodium concentration, respectively; [Cl^{−}]_{pw} and [Cl^{−}]_{ISF} are plasma water and interstitial fluid chloride concentration, respectively.

By combining *Eqs. 2* and *3*, one can demonstrate that the product of the concentrations of Na^{+} and Cl^{−} ions is the same on both sides of the membrane at Gibbs-Donnan equilibrium (9, 10): (4)

Therefore, *Eq. 4* accounts for the modulating effect of Gibbs-Donnan equilibrium on the distribution of Na^{+} and Cl^{−} ions across the capillary membrane. However, based on Dorrington's arguments, *Eq. 4* should also not account for the Gibbs-Donnan effect since *Eq. 4* per se does not account for the electrical potential and the mean number of negative charges on plasma proteins. Although *Eq. 4* does not include terms that directly account for the electrical potential and the mean number of negative charges on plasma proteins, it is widely recognized that the Gibbs-Donnan effect is indeed reflected in *Eq. 4* by the difference in the plasma and interstitial fluid Na^{+} and Cl^{−} concentrations. Specifically, the greater the difference in the plasma and interstitial fluid Na^{+} and Cl^{−} concentrations, the greater is the chemical gradient and therefore the electrical potential imposed by the negatively charged albumin molecules. Similarly, Gibbs-Donnan equilibrium will also lead to the altered distribution of all osmotically active non-Na^{+} ions. The altered distribution of Na^{+} and non-Na^{+} ions will in turn result in a higher plasma osmolality compared with the interstitial fluid osmolality (6, 9). The higher plasma osmolality is a reflection of the fact that there are more diffusible ions in the plasma space to maintain electroneutrality. Specifically, the greater the osmotic inequality between the plasma and interstitial fluid, the greater will be the Gibbs-Donnan effect. Therefore, our equation does account for the Gibbs-Donnan effect by incorporating the osmotic inequality between the plasma and interstitial fluid. Dorrington seems to have missed all these important points.

Dorrington also argues that our formula cannot predict the correct [Na^{+}]_{pw} unless the value of [Na^{+}]_{pw} itself is entered on the right hand side of the equation. Clearly, the [Na^{+}]_{pw} must be a function of the quantity of Na^{+} ions and volume of water in the plasma space. For the sake of simplicity, we calculated the quantity of osmotically active Na^{+} ions by determining the product of [Na^{+}]_{pw} × V_{pw}. However, as clearly stated in the footnote of our article (8), one can easily determine the quantity of plasma osmotically active Na^{+} by calculating the product of V_{pw} and the difference between the plasma osmolality and the sum of the osmolality of all the non-Na^{+} osmoles, a calculation that does not require the value of the [Na^{+}]_{pw}: (5) Using the data in Table 1 (8):

In contrast to what Dorrington asserts, since plasma osmolality can be directly measured, the calculation of the quantity of osmotically active Na^{+} ions based on *Eq. 5* does not require the value of the [Na^{+}]_{pw}.

In a previous article (7), we stated that a strength of our mathematical model is that it will accurately predict the correct [Na^{+}]_{pw} regardless of what estimate of plasma, interstitial fluid, and intracellular fluid volumes one uses to validate the model. Although the relative proportion of the plasma, interstitial fluid, and intracellular fluid volumes is essentially equivalent among individual subjects, the absolute values of the plasma, interstitial fluid, and intracellular fluid volumes may vary significantly among individuals. Therefore, if a mathematical model were to be accurate, it should correctly predict the [Na^{+}]_{pw} in any given individual. Consequently, the fact that our mathematical model accurately predicts the correct [Na^{+}]_{pw} regardless of what estimate of plasma, interstitial fluid, and intracellular fluid volumes one uses is indeed evidence of the validity of our model. Dorrington seems to have missed this important point as well.

Finally, we disagree with Dorrington's inference that our mathematical model is a “spoof.” Ironically, despite acknowledging that the derivation of our formula is correct, Dorrington uses the word “spoof” in characterizing our mathematical model. According to the Roget's *New Millenium Thesaurus*, a partial list of the synonyms of “spoof” includes: “hoax, fake, cheat, deceit, travesty, trickery, phony, prank and sham.” Which of these definitions Dorrington ascribes to is unknown to us. However, all fair-minded individuals who understand and appreciate the language of mathematics would acknowledge that if a quantitative model were to be mathematically accurate and based on sound physiological principles, it cannot be a “spoof”.

Irrespective of what mathematical model one ascribes, to the [Na^{+}]_{pw} must be a function of the quantity of Na^{+} ions and volume of water in the plasma space, i.e. [Na^{+}]_{pw} = quantity of plasma Na^{+}/volume of plasma water (*Eq. 6*). Our mathematical model can account for all the known physiological factors that modulate the numerator and denominator of *Eq. 6*. The model predicts that all osmotically active solutes in the body fluid compartments modulate the [Na^{+}]_{pw}. Since the body fluid compartments are in osmotic equilibrium with each other, all osmotically active solutes determine the distribution of water in the plasma space and modulate the denominator of *Eq. 6*. Moreover, since Gibbs-Donnan equilibrium alters the distribution of Na^{+} and non-Na^{+} ions between the plasma and interstitial fluid, it will modulate the numerator and denominator of *Eq. 6*. The altered distribution of Na^{+} and non-Na^{+} ions between the plasma and interstitial fluid is known to result in a higher plasma osmolality compared with the interstitial fluid osmolality (6, 9). Since Gibbs-Donnan equilibrium modulates the osmolality of the plasma and interstitial fluid, it will determine the distribution of water in the plasma space and therefore the denominator of *Eq. 6*. Moreover, the higher plasma osmolality is also a reflection of the fact that there are more diffusible Na^{+} ions in the plasma space to maintain electroneutrality, as reflected by the numerator in *Eq. 6*. In accounting for the osmotic inequality between the plasma and interstitial fluid, our model therefore takes into consideration the fact that Gibbs-Donnan equilibrium is a determinant of the [Na^{+}]_{pw} by modulating the numerator and denominator of *Eq. 6*. Last, our model predicts that the osmotic coefficient of Na^{+} salts will modulate the [Na^{+}]_{pw} since the osmotic coefficient reflects the osmotic activity of Na^{+} salts. Indeed, the osmotic activity of Na^{+} salts will determine the distribution of water within the plasma space and therefore modulate the denominator of *Eq. 6*.

Importantly, the accurate formulation of the quantitative interrelationship between these various factors resulting from Gibbs-Donnan and osmotic equilibrium indicates that our mathematical model can be an indispensable tool in the analysis of the determinants of the [Na^{+}]_{pw}. In conclusion, the contention that our mathematical model is a “spoof” is completely unfounded, and all of Dorrington's criticisms are unsubstantiated and scientifically inaccurate.

- Copyright © 2008 the American Physiological Society