J Appl Physiol 100: 1293–1300, 2006.
First published December 15, 2005; doi:10.1152/japplphysiol.01274.2005.
TRANSLATIONAL PHYSIOLOGY
Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality
of body fluid compartments, and plasma water sodium concentration
Minhtri K. Nguyen and Ira Kurtz
Division of Nephrology, David Geffen School of Medicine at UCLA, Los Angeles, California
Submitted 4 October 2005; accepted in final form 12 December 2005
plasma water sodium concentration; hyponatremia
the plasma water Na⫹ and Cl⫺
concentrations and interstitial fluid (ISF) Na⫹ and Cl⫺ concentrations are different despite the high permeability of Na⫹
and Cl⫺ ions across the capillary membrane, which separates
these two fluid compartments (23). This difference in ionic
concentrations between the plasma and the ISF is attributed to
the much higher concentration of proteins in the plasma compared with the ISF. Proteins are large-molecular-weight substances and therefore do not cross the capillary membrane
easily. The low protein permeability across capillary membranes is responsible for causing ionic concentration differences between the plasma and ISF and is known as the
Gibbs-Donnan effect or Gibbs-Donnan equilibrium (23).
Negatively charged, nonpermeant proteins present predominantly in the plasma space will attract positively charged ions
and repel negatively charged ions (23). The passive distribution of cations and anions is altered to preserve electroneutralIT IS WELL RECOGNIZED THAT
Address for reprint requests and other correspondence: M. K. Nguyen,
Division of Nephrology, David Geffen School of Medicine at UCLA, 10833
Le Conte Ave., Rm. 7-155 Factor Bldg., Los Angeles, CA 90095 (e-mail:
[email protected]).
http://www. jap.org
ity in the plasma and ISF. As a result, the diffusible cation
concentration is higher in the compartment containing nondiffusible, anionic proteins, whereas diffusible anion concentration is lower in the protein-containing compartment. GibbsDonnan equilibrium is established when the altered distribution
of cations and anions results in electrochemical equilibrium. It
is also well recognized that another consequence of the GibbsDonnan effect is that there are more osmotically active particles in the plasma space than in the ISF at equilibrium (13, 23).
Consequently, the plasma osmolality is slightly greater than the
osmolality of the ISF and intracellular fluid (ICF). Indeed, the
plasma osmolality is typically 1 mosmol/l H2O greater than
that of the ISF and ICF (13). In addition to the modulating
effect of Gibbs-Donnan equilibrium on the [Na⫹]pw and
plasma osmolality, alterations in the osmolality of the ISF and
ICF will also lead to changes in the [Na⫹]pw and plasma
osmolality due to intercompartmental H2O shift since the body
fluid compartments are in osmotic equilibrium. Presently, there
are no formulas in the literature that have determined the
mathematical relationships between Gibbs-Donnan equilibrium, osmolality of body fluids (plasma, ISF, and ICF), and
[Na⫹]pw. In this article, based on the principles of GibbsDonnan and osmotic equilibrium, we derive for the first time a
new equation that quantitatively predicts the effect of changes
in negatively charged plasma proteins on the osmolality of all
body fluid compartments and the [Na⫹]pw.
MATHEMATICAL DERIVATION
Quantification of the Effect of Gibbs-Donnan Equilibrium on
the [Na⫹]pw by Modulating the Osmolality of the Body
Fluid Compartments
It is well known that the plasma osmolality is slightly greater
than the osmolality of the ISF and ICF owing to the GibbsDonnan equilibrium (13, 23). The plasma osmolality is typically 1 mosmol/l H2O greater than that of the ISF and intracellular compartment (13).
Therefore: plasma osmolality ⬎ total body osmolality
To equate plasma osmolality to the total body osmolality,
one has to introduce a correction factor for the incremental
effect of Gibbs-Donnan equilibrium on the plasma osmolality.
This correction factor, which will be termed g, is equal to the
ratio of plasma osmolality/total body osmolality and is therefore unitless.
The costs of publication of this article were defrayed in part by the payment
of page charges. The article must therefore be hereby marked “advertisement”
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
8750-7587/06 $8.00 Copyright © 2006 the American Physiological Society
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Nguyen, Minhtri K., and Ira Kurtz. Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid
compartments, and plasma water sodium concentration. J Appl
Physiol 100: 1293–1300, 2006. First published December 15, 2005;
doi:10.1152/japplphysiol.01274.2005.—The presence of negatively
charged, impermeant proteins in the plasma space alters the distribution of diffusible ions in the plasma and interstitial fluid (ISF)
compartments to preserve electroneutrality. We have derived a new
mathematical model to define the quantitative interrelationship between the Gibbs-Donnan equilibrium, the osmolality of body fluid
compartments, and the plasma water Na⫹ concentration ([Na⫹]pw)
and validated the model using empirical data from the literature. The
new model can account for the alterations in all ionic concentrations
(Na⫹ and non-Na⫹ ions) between the plasma and ISF due to GibbsDonnan equilibrium. In addition to the effect of Gibbs-Donnan equilibrium on Na⫹ distribution between plasma and ISF, our model
predicts that the altered distribution of osmotically active non-Na⫹
ions will also have a modulating effect on the [Na⫹]pw by affecting
the distribution of H2O between the plasma and ISF. The new
physiological insights provided by this model can for the first time
provide a basis for understanding quantitatively how changes in the
plasma protein concentration modulate the [Na⫹]pw. Moreover, this
model defines all known physiological factors that may modulate the
[Na⫹]pw and is especially helpful in conceptually understanding the
pathophysiological basis of the dysnatremias.
1294
GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
Therefore:
Plasma osmolality ⫽ g ⫻ total body osmolality
(1)
Total number of osmoles
Total body water
(2)
Total body osmolality
␪pw ⫻ Vpw ⫹ ␪ISF ⫻ VISF ⫹ ␪ICF ⫻ VICF
(3)
⫽
TBW
␪ pw ⫻ TBW
␪pw ⫻ Vpw ⫹ ␪ISF ⫻ VISF ⫹ ␪ICF ⫻ VICF
(4)
We now define the components of plasma osmolality and
total body osmolality:
Plasma osmolality ⫽
Napw ⫹ Kpw ⫹ osmolpw
Vpw
(5)
It is well known that not all exchangeable Na⫹ (Nae) and
exchangeable K⫹ (Ke) are osmotically active because there is
abundant evidence for the existence of osmotically inactive
Na⫹ and K⫹ storage in bone and skin (3, 5, 8 –10, 14, 25,
27–29). Hence, only osmotically active exchangeable Na⫹ and
K⫹ contribute to the distribution of water between the extracellular and intracellular spaces.
Rearranging:
关Na⫹兴pw
Na osm active ⫹ Kosm active ⫹ osmolECF ⫹ osmolICF
⫽
(6)
TBW
where Napw is plasma water Na⫹, Kpw is plasma water K⫹,
osmolpw is osmotically active plasma water non-Na⫹ non-K⫹
osmoles, Naosm active is osmotically active Na⫹, Kosm active is
osmotically active K⫹, osmolECF is osmotically active extracellular non-Na⫹ and non-K⫹ osmoles, and osmolICF is osmotically active intracellular non-Na⫹ and non-K⫹ osmoles.
Since:
(1)
冋
⫽g
册
Naosm active ⫹ Kosm active osmolECF ⫹ osmolICF
⫹
TBW
TBW
⫺ 关K⫹兴pw ⫺
冋
关Na ⫹兴pw ⫽ g
Nae ⫹ Ke ⫺ 共Naosm inactive ⫹ Kosm inactive兲
TBW
共Naosm active ⫹ Kosm active ⫹ osmolECF ⫹ osmolICF兲
⫻
TBW
Let [Na⫹]pw ⫽ plasma water Na⫹ concentration and
[K⫹]pw ⫽ plasma water K⫹ concentration.
Since: Na pw/Vpw ⫽ 关Na⫹兴pw and Kpw/Vpw ⫽ 关K⫹兴pw
J Appl Physiol • VOL
册
osmolECF ⫹ osmolICF
osmolpw
⫹
⫺ 关K⫹兴pw ⫺
TBW
Vpw
冋
关Na ⫹兴pw ⫽ g
Nae ⫹ Ke 共Naosm inactive ⫹ Kosm inactive兲
⫺
TBW
TBW
册
osmolpw
osmolECF ⫹ osmolICF
⫺ 关K⫹兴pw ⫺
⫹
TBW
Vpw
(10)
(11)
where [Na⫹]pw is expressed in milliosmoles per liter H2O.
To convert [Na⫹]pw in Eq. 11 from milliosmoles per liter
H2O to millimoles per liter H2O, one needs to divide both sides
of Eq. 11 by the osmotic coefficient Ø (13, 16):
冋
关Na ⫹兴*pw ⫽ g/Ø
Nae ⫹ Ke 共Naosm inactive ⫹ Kosm inactive兲
⫺
TBW
TBW
册 冋
册
(12)
osmolpw
osmolECF ⫹ osmolICF
⫺ 1/Ø 关K⫹兴pw ⫹
⫹
TBW
Vpw
where [Na⫹]*pw is expressed in millimoles per liter H2O, Ø is
osmotic coefficient of Na⫹ salts, and ␪ is osmolality;
g⫽
(7)
osmolpw
Vpw
Since total exchangeable Na⫹ (Nae) and exchangeable K⫹
(Ke) consists of osmotically active and osmotically inactive
exchangeable Na⫹ and K⫹:
Let Nae ⫽ total exchangeable Na⫹; Ke ⫽ total exchangeable
K⫹; Naosm inactive ⫽ osmotically inactive Na⫹; and Kosm inactive ⫽
osmotically inactive K⫹.
Hence: Nae ⫽ Naosm active ⫹ Naosm inactive and Ke ⫽
Kosm active ⫹ Kosm inactive
Therefore:
Na pw ⫹ Kpw ⫹ osmolpw
⫽g
Vpw
(9)
␪ pw ⫻ TBW
␪pw ⫻ Vpw ⫹ ␪ISF ⫻ VISF ⫹ ␪ICF ⫻ VICF
where: g ⫽
Total body osmolality
Plasma osmolality ⫽ g ⫻ total body osmolality
册
(8)
␪ pw ⫻ TBW
␪pw ⫻ Vpw ⫹ ␪ISF ⫻ VISF ⫹ ␪ICF ⫻ VICF
This equation quantitatively predicts the effect of changes in
negatively charged plasma proteins on the osmolality of all
body fluid compartments and the [Na⫹]pw.
Clinical Validity of Equation 11
Using the data from Table 1, one can demonstrate the
clinical validity of Eq. 11. Because the solutes in Table 1 are
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where ␪pw is plasma osmolality, Vpw is plasma water volume,
␪ISF is ISF osmolality, VISF is ISF volume, ␪ICF is ICF
osmolality, VICF is ICF volume, and TBW is total body water.
The unit of osmolality is expressed throughout the derivation in
milliosmoles per liter H2O rather than milliosmoles per kilogram H2O assuming the density of water is 1 kg/l.
Therefore:
g⫽
冋
osmolpw
Vpw
Naosm active ⫹ Kosm active osmolECF ⫹ osmolICF
⫹
⫽g
TBW
TBW
where g is plasma osmolality/total body osmolality.
Since:
Total body osmolality ⫽
关Na ⫹兴pw ⫹ 关K⫹兴pw ⫹
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GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
Na osm active ⫽ 共关Na⫹兴pw ⫻ Vpw兲a ⫹ 关Na⫹兴ISF ⫻ VISF ⫹ 关Na⫹兴ICF
Table 1. Osmolar substances in the extracellular and
intracellular compartments
Plasma,
mosmol/l H2O
⫻ VICF ⫽ 2,722 mosmol
Intracellular,
mosmol/l H2O
142
4.2
1.3
0.8
108
24
2
0.5
139
4.0
1.2
0.7
108
28.3
2
0.5
2
0.2
1.2
2
0.2
1.2
5.6
1.2
4
4.8
301.8
5.6
0.2
4
3.9
300.8
4
4
10
301.2
282.0
281.0
281.0
5,443
5,423
⫹
K osm active ⫽ 关K 兴pw ⫻ Vpw ⫹ 关K 兴ISF ⫻ VISF ⫹ 关K⫹兴ICF ⫻ VICF
⫽ 3,568.6 mosmol
14
140
0
20
4
10
11
1
45
14
8
9
1.5
5
3.7
Na osm active ⫹ Kosm active ⫽ 2,722 mosmol ⫹ 3,568.6 mosmol
⫽ 6,290.6 mosmol
⫹
⫹
osmol pw ⫽ 共␪pw ⫺ 关Na ⫹ K 兴pw兲 ⫻ Vpw ⫽ 共301.8 mosmol/l
b
⫺ 146.2 mosmol/l兲 ⫻ 3 liters ⫽ 466.8 mosmol
osmol ISF ⫽ 共␪ISF ⫺ 关Na⫹ ⫹ K⫹兴ISF兲 ⫻ VISF ⫽ 共300.8 mosmol/l
⫺ 143 mosmol/l兲 ⫻ 14 liters ⫽ 2,209.2 mosmol
osmol ICF ⫽ 共␪ICF ⫺ 关Na⫹ ⫹ K⫹兴ICF兲 ⫻ VICF ⫽ 共301.2 mosmol/l
⫺154 mosmol/l兲 ⫻ 25 liters ⫽ 3,680 mosmol
osmol pw
⫽ ␪pw ⫺ 关Na⫹ ⫹ K⫹兴pw ⫽ 301.8 mosmol/l
Vpw
⫺146.2 mosmol/l⫽155.6 mosmol/l
g⫽
5,423
Data are adapted from Ref. 13. *Corrected osmolar activity accounts for the
reduced osmotic activity of ionic particles in solution.
expressed in milliosmoles per liter H2O, Eq. 11 will be utilized
in this example. It is also important to realize that the measured
Na⫹ and K⫹ ions in the plasma, ISF, and ICF in Table 1
include only the osmotically active Na⫹ and K⫹ ions, which is
reflected by the terms (Nae ⫹ Ke)/TBW ⫺ (Naosm inactive ⫹
Kosm inactive)/TBW in Eq. 11. Table 1 includes only the osmotically active Na⫹ and K⫹ ions because the distribution of H2O
depends solely on osmotically active solute particles, as reflected by the fact that the osmolality of the plasma, ISF, and
ICF are essentially equal (with the exception that the plasma
osmolality is slightly greater than the ISF osmolality owing to
the Gibbs-Donnan equilibrium). In other words, if the measured Na⫹ and K⫹ ions in the plasma, ISF, and ICF in Table
1 were to include both osmotically active and inactive Na⫹ and
K⫹ ions, the calculated osmolality of the plasma, ISF, and ICF
cannot be equal to one another. Moreover, the determination of
the [Na⫹] and [K⫹] in the plasma, ISF, and ICF by conventional laboratory techniques measures only the osmotically
active Na⫹ and K⫹ ions. Exchangeable but osmotically inactive bound Na⫹ and K⫹ ions (e.g., exchangeable bound,
osmotically inactive Na⫹ in bone) are not measured by these
techniques. The measurement of total exchangeable Na⫹ and
total exchangeable K⫹ would require isotope dilution methodology.
In this example, we will calculate the [Na⫹]pw in an average
70-kg man in whom the TBW is ⬃42 liters, of which 25 liters
are in the ICF, 14 liters are in the ISF, and 3 liters are in the
plasma space.
Since: Nae ⫽ Naosm active ⫹ Naosm inactive and Ke ⫽
Kosm active ⫹ Kosm inactive
Therefore: Nae ⫹ Ke ⫺ (Naosm inactive ⫹ Kosm inactive) ⫽
Naosm active ⫹ Kosm active
J Appl Physiol • VOL
⫹
␪ pw
共␪pw ⫻ Vpw ⫹ ␪ISF ⫻ VISF ⫹ ␪ICF ⫻ VICF兲/TBW
⫽ 301.8/301.1
Na
⫹
K
共Na
⫹
K
e
e
osm inactive
osm inactive兲
关Na ⫹兴pw ⫽ g
⫺
TBW
TBW
(11)
osmolECF ⫹ osmolICF
osmol
pw
⫹
⫺ 关K⫹兴pw ⫺
TBW
Vpw
冋
册
Since the terms (Nae ⫹ Ke)/TBW ⫺ (Naosm inactive ⫹
Kosm inactive/TBW) represent the osmotically active Na⫹ and
K⫹ ions:
冋
关Na ⫹兴pw ⫽ g
册
共Naosm active ⫹ Kosm active兲 osmolECF ⫹ osmolICF
⫹
TBW
TBW
⫺ 关K⫹兴pw ⫺
osmolpw
Vpw
Since osmolECF ⫽ osmolpw ⫹ osmolISF:
冋
关Na ⫹兴pw ⫽ g
共Naosm active ⫹ Kosm active兲
TBW
册
osmolpw
osmolpw ⫹ osmolISF ⫹ osmolICF
⫺ 关K⫹兴pw ⫺
TBW
Vpw
关Na ⫹兴pw ⫽ 共301.8/301.1兲关共6,290.6 ⫹ 466.8 ⫹ 2,209.2
⫹
⫹ 3,680兲/42兴 ⫺ 4.2 ⫺ 155.6
关Na ⫹兴pw ⫽ 142 mOsm/l
a
The osmolal quantity of plasma osmotically active Na⫹ can also be
determined by calculating the product of Vpw and the difference between the
plasma osmolality and the sum of the osmolality of all the non-Na⫹ osmoles,
i.e. (plasma osmolality ⫺ non-Na⫹ osmolality) ⫻ Vpw. This calculation does
not require the value of the [Na⫹]pw .
b
The plasma non-Na⫹ and non-K⫹ osmoles can also be calculated by
simply adding the quantity of all the individual plasma non-Na⫹ and non-K⫹
osmoles, a calculation that does not require the value of the [Na⫹]pw .
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Na⫹
K⫹
Ca2⫹
Mg2⫹
Cl⫺
HCO⫺
3
⫺
HPO2⫺
4 , H2PO4
2⫺
SO4
Phosphocreatine
Carnosine
Amino acids
Creatine
Lactate
Adenosine triphosphate
Hexose monophosphate
Glucose
Protein
Urea
Others
Total mosmol/l
Corrected osmolar activity,
mosmol/l*
Total osmotic pressure at
37°C, mmHg
Interstitial,
mosmol/l H2O
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GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
DISCUSSION
Modulation of [Na⫹]pw by Alterations in the Osmolality of
Body Fluids
In this article, we examine quantitatively the interrelationship between the Gibbs-Donnan equilibrium, osmolality of the
body fluids (plasma, ISF, and ICF), and [Na⫹]pw. By altering
the distribution of diffusible ions between the plasma and ISF,
Gibbs-Donnan equilibrium has a modulating effect on the
osmolality of the plasma and ISF. Because the body fluid
compartments are in osmotic equilibrium (13), changes in the
osmolality in any one compartment will alter the distribution of
H2O in the other two compartments. Consequently, the GibbsDonnan equilibrium will also have a modulating effect on the
osmolality of the ICF as well.
Because the [Na⫹]pw is determined by the quantity of Na⫹
ions in the plasma space and the plasma water volume, changes
in the osmolality in any body fluid compartment will have a
modulating effect on the [Na⫹]pw by altering the distribution of
H2O in the plasma space. It is therefore not surprising that a
determinant of the [Na⫹]pw is the osmolality of the body fluid
compartments. Indeed, as depicted in Eq. 12, the osmolality of
the plasma, ISF, and ICF compartments has a modulating
effect on the [Na⫹]pw as reflected by the term g. Moreover,
although the plasma, ISF, and ICF osmolality can have a direct
effect on the [Na⫹]pw completely independent of g, the osmolality of the plasma, ISF, and ICF also has a separate quantitative effect on the [Na⫹]pw mediated through the GibbsDonnan equilibrium. Indeed, the plasma, ISF, and ICF osmolality can affect the [Na⫹]pw completely independent of g
because the body fluid compartments are in osmotic equilibrium. However, if osmotic equilibrium is the only mechanism
modulating the [Na⫹]pw, then the [Na⫹]pw will be equal to the
[Na⫹]ISF and the plasma and ISF osmolality will be exactly
J Appl Physiol • VOL
equal. In reality, the [Na⫹]pw is greater than the [Na⫹]ISF and
the plasma osmolality is slightly greater than the ISF osmolality owing to the effect of Gibbs-Donnan equilibrium in modulating the distribution of Na⫹ and non-Na⫹ ions between the
plasma and ISF. This fact is accounted for by the GibbsDonnan correction factor g in Eq. 12.
Osmotic Equilibrium of Body Fluid Compartments:
Osmotically Active and Inactive Exchangeable Na⫹ and K⫹
Because osmotic equilibration depends only on osmotically
active solutes, the presence of osmotically inactive Nae and Ke
has to be accounted for in Eq. 12. Indeed, there is convincing
evidence for the existence of an osmotically inactive Na⫹ and
K⫹ reservoir (3, 5, 8 –10, 14, 25, 27–29). Recently, Titze et al.
(28) reported Na⫹ accumulation in an osmotically inactive
form in human subjects in a terrestrial space station simulation
study and suggested the existence of an osmotically inactive
Na⫹ reservoir that exchanges Na⫹ with the extracellular space.
Titze et al. (27) also showed that salt-sensitive Dahl rats (which
developed hypertension if fed a high-sodium diet) were characterized by a reduced osmotically inactive sodium storage
capacity compared with Sprague-Dawley rats, thereby resulting in fluid accumulation and high blood pressure. In addition,
Heer et al. (14) demonstrated positive Na⫹ balance in healthy
subjects on a metabolic ward without increases in body weight,
expansion of the extracellular space, or plasma sodium concentration. These authors, therefore, suggested that there is
osmotic inactivation of exchangeable Na⫹. Moreover, there is
also evidence that a portion of intracellular K⫹ is bound and is
therefore osmotically inactive (5). Together, these findings
provide convincing evidence for the existence of an osmotically inactive Na⫹ and K⫹ reservoir. Because osmotically
inactive Nae and Ke cannot contribute to the distribution of
water between the extracellular and intracellular spaces, osmotically inactive exchangeable Na⫹ and K⫹ cannot contribute to the modulation of the [Na⫹]pw and therefore are accounted for quantitatively by the (Naosm inactive ⫹ Kosm inactive)/
TBW term in Eq. 12.
It is also evident that any osmotically active non-Na⫹
osmoles are involved in the distribution of H2O between the
body fluid compartments. The terms Ke, [K⫹]pw, osmolECF,
osmolICF, and osmolpw account for the fact that non-Na⫹
osmoles also alter the distribution of H2O between the body
fluid compartments. Moreover, because the osmotic activity of
most ionic particles is slightly less than one owing to electrical
interactions between the ions (16), the osmotic coefficient Ø in
Eq. 12 accounts for the effectiveness of Na⫹ ions as independent osmotically active particles under physiological conditions.
Role of Gibbs-Donnan Equilibrium in Modulating the
Osmolality of the Body Fluid Compartments
It is well recognized that Gibbs-Donnan equilibrium modulates the plasma osmolality by altering the distribution of Na⫹
and Cl⫺ ions in the plasma and ISF to preserve electroneutrality (13, 23). Because of the presence of negatively charged,
impermeant proteins in the plasma space, diffusible cation
concentration is higher and diffusible anion concentration is
lower in the plasma compartment. As a result of this GibbsDonnan effect, the total osmolar concentration is slightly
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As demonstrated in this example, Eq. 11 precisely predicts
that the [Na⫹]pw is 142 mosmol/l H2O.
Because Gibbs-Donnan equilibrium alters the distribution of
Na⫹ and non-Na⫹ ions between the plasma and ISF, Eqs. 11
and 12 account for the alterations in all ionic concentrations
(Na⫹ and non-Na⫹ ions) between the plasma and ISF (as
reflected by the changes in the osmolality of plasma and ISF).
Equations 11 and 12 therefore take into consideration the fact
that the altered distribution of osmotically active non-Na⫹ ions
due to the Gibbs-Donnan equilibrium will also have a modulating effect on the [Na⫹]pw by affecting the distribution of
H2O between the plasma and ISF. Moreover, in contrast to our
previous analysis (18), Eqs. 11 and 12 account for the fact that
alterations in the plasma protein concentration will result in
changes in the distribution of Na⫹ and non-Na⫹ ions between
the plasma and ISF due to Gibbs-Donnan equilibrium. The
alterations in the concentrations of Na⫹ and non-Na⫹ ions
between the plasma and ISF due to changes in the plasma
protein concentration will be reflected by the changes in the
osmolality of the plasma and ISF, which are accounted for by
the terms ␪pw and ␪ISF in the Gibbs-Donnan correction factor g
(i.e., ␪pw ⫽ [Na⫹]pw ⫹ [K⫹]pw ⫹ [Ca2⫹]pw ⫹ [Mg2⫹]pw ⫹
⫺
2⫺
[Cl⫺]pw ⫹ [HCO⫺
3 ]pw ⫹ [H2PO4 ]pw ⫹ [SO4 ]pw ⫹ [pro⫹
⫹
tein]pw ⫹ etc.; and ␪ISF ⫽ [Na ]ISF ⫹ [K ]ISF ⫹ [Ca2⫹]ISF ⫹
⫺
[Mg2⫹]ISF ⫹ [Cl⫺]ISF ⫹ [HCO⫺
3 ]ISF ⫹ [H2PO4 ]ISF ⫹
2⫺
[SO4 ]ISF ⫹ etc.).
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GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
Analysis of the Pathogenesis of the Dysnatremias
Historically, the pathogenesis of the dysnatremias has focused on the effect of changes in the mass balance of Nae, Ke,
and TBW on the [Na⫹]pw. This pathophysiological approach is
based on the empirical relationship between the [Na⫹]pw and
Nae, Ke, and TBW originally demonstrated by Edelman et al.
(10) where [Na⫹]pw ⫽ 1.11(Nae ⫹ Ke)/TBW ⫺ 25.6. However, this approach is inadequate in that it fails to explain
quantitatively how several known physiological factors that do
not alter the value of the (Nae ⫹ Ke)/TBW term are capable of
modulating the [Na⫹]pw. These physiological parameters that
modulate the [Na⫹]pw are components of the slope and yintercept of the Edelman equation. It is important to appreciate
that Eq. 12 is in essence a mathematical analysis of all the
physiological parameters modulating the [Na⫹]pw as implicated in the Edelman equation. Equation 12 can be rearranged
as follows:
J Appl Physiol • VOL
关Na ⫹兴*pw ⫽ g/Ø
⫺
冋
共Nae ⫹ Ke兲
共Naosm inactive ⫹ Kosm inactive兲
⫺ g/Ø
TBW
TBW
册 冋
册
osmolpw
共osmolECF ⫹ osmolICF兲
⫺ 1/Ø 关K⫹兴pw ⫹
TBW
Vpw
Therefore, the slope of 1.11 in Edelman’s equation is represented by the term g/Ø in Eq. 12, whereas the y-intercept of
⫺25.6 is represented by the terms:
⫺ g/Ø
冋
册
共Naosm inactive ⫹ Kosm inactive兲 (osmolECF ⫹ osmolICF)
⫺
TBW
TBW
冋
⫺ 1/Ø 关K⫹兴pw ⫹
册
osmolpw
Vpw
Therefore, Eq. 12 is especially helpful in explaining quantitatively how physiological factors that do not alter the value
of the (Nae ⫹ Ke)/TBW term are capable of modulating the
[Na⫹]pw by altering the value of the slope and y-intercept in the
Edelman equation.
Association of Hypoalbuminemia and Hyponatremia
Alterations in the distribution of Na⫹ and non-Na⫹ ions
between the plasma and ISF resulting from changes in the
plasma protein concentration will be reflected by the changes
in the plasma and ISF osmolality. Equation 12 accounts for the
effect of changes in the plasma protein concentration on the
Gibbs-Donnan equilibrium since the Gibbs-Donnan correction
factor g includes the terms ␪pw and ␪ISF. Therefore, clinical
conditions characterized by hypoalbuminemia would be expected to change the [Na⫹]pw by altering the slope of the
Edelman equation, which will be reflected by changes in the
term g in Eq. 12.
Indeed, Ferreira da Cunha et al. (11) demonstrated a
significant association between hypoalbuminemia and hyponatremia and that the plasma sodium concentration ([Na⫹]p)
varies proportionally with incremental changes in the albumin concentration. Similarly, Dandona et al. (6) also reported a strong association between low [Na⫹]p and albumin
concentration. These authors also reported several case
reports of “hypoalbuminemic hyponatremia” in patients in
whom correction of the hypoalbuminemia with albumin
infusion resulted in significant improvement in their hyponatremia. The authors attributed the improvement in the
hyponatremia with albumin infusions to the nonosmotic
suppression of antidiuretic hormone secretion. However, the
marked improvement in the hyponatremia in these patients
cannot be due to a significant increase in the urinary free
water excretion because the urinary osmolality remained
inappropriately elevated after albumin infusion in these
hyponatremic patients (urine osmolality ranged from 350 to
430 mosmol/l H2O after albumin infusion). Therefore, the
improvement in the hyponatremia with the correction of the
hypoalbuminemia in these patients likely reflected the alterations in the distribution of Na⫹ ions between the plasma
and ISF resulting from the Gibbs-Donnan effect. Specifically, an increase in the negatively charged albumin concentration will attract positively charged Na⫹ ions into the
plasma space, thereby leading to an increase in the [Na⫹]p.
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greater in the plasma compartment; this extra osmotic force
from diffusible ions is added to the osmotic forces exerted by
the anionic proteins. The end result of the Gibbs-Donnan effect
is that more water moves into the plasma compartment than
would be predicted on the basis of the protein concentration
alone. A steady state is achieved in which the plasma osmolality is maintained at a slightly greater osmolality than the ISF
because the capillary hydrostatic pressure opposes the osmotic
movement of water into the plasma space.
Interestingly, cells contain a significant concentration of
large-molecular-weight, impermeant anionic proteins that also
lead to the Gibbs-Donnan equilibrium across the cell membrane. As a result of the Gibbs-Donnan equilibrium, electroneutrality would be preserved but there will be more particles
intracellularly. The higher intracellular osmolality would lead
to osmotic water movement down its concentration gradient,
and the cell would tend to swell. This osmotic water movement
would upset the Gibbs-Donnan effect, and the ions would
diffuse to reestablish the Gibbs-Donnan equilibrium. There
would again be an osmotic gradient across the cell membrane,
resulting in further water movement down its concentration
gradient. Consequently, this is an unstable situation that, if
unopposed, would lead to significant cell swelling. In preventing cell swelling, cells pump Na⫹ ions out of the cell actively
to compensate for the Gibbs-Donnan effect due to impermeant
intracellular proteins. The Na⫹-K⫹ ATPase renders the cell
membrane effectively impermeant to Na⫹ ions, thereby creating a double-Donnan effect (21). Therefore, in maintaining cell
volume, the Gibbs-Donnan effect due to ISF Na⫹ balances the
Gibbs-Donnan effect due to impermeant intracellular proteins
(21). As a result of this double-Donnan effect, there is no net
osmotic pressure across the cell membrane, and the system is
stable (21).
The net result of the Gibbs-Donnan effect is that the plasma
osmolality is typically 1 mosmol/l H2O greater than that of the
ISF and intracellular compartment, which have the same osmolality (13). Therefore, a correction factor for the GibbsDonnan effect termed g is incorporated in Eq. 12 to account for
this small osmotic inequality. Not surprisingly, the determinants of g are the osmolality of the plasma, ISF, and ICF.
1298
GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
Non-Na⫹ and Non-K⫹ Osmoles-Induced Hyponatremia
Because 0.93 ⫻ [Na⫹]pw ⫽ [Na⫹]p (1, 7):
关Na ⫹兴pw ⫽ 1.11
共Nae ⫹ Ke兲
⫺ 25.6
TBW
Multiplying both sides of the equation by 0.93 to convert
[Na⫹]pw to [Na⫹]p (1, 7):
0.93 ⫻ 关Na ⫹兴pw ⫽ 1.03
共Nae ⫹ Ke兲
⫺ 23.8
TBW
J Appl Physiol • VOL
关Na ⫹兴p ⫽ 1.03
共Nae ⫹ Ke兲
⫺ 23.8
TBW
Because there is an expected decrease of 0.59 mM/l in the
[Na⫹]p for each millimolar increment in the plasma maltose
concentration (22):
关Na ⫹兴p ⫽ 1.03
共Nae ⫹ Ke兲
⫺ 23.8 ⫺ 共0.59兲共⌬关maltose兴p兲
TBW
where ⌬[maltose]p is change in plasma maltose concentration
(mM/l).
Similarly, the y-intercept is not constant in hyperglycemia-induced dilutional hyponatremia resulting from the
translocation of water and will vary directly with the plasma
glucose concentration. Previously, it has been shown that
there is an expected decrease of 1.6 mM/l in the [Na⫹]p for
each 100 mg/dl increment in the plasma glucose concentration due to the translocation of water from the ICF to the
ECF (15). Hence, to predict the dilutional effect of hyperglycemia on the [Na⫹]p attributable to the osmotic shift of
water, the [Na⫹]p can be predicted from the following
equation (17):
关Na ⫹兴p ⫽ 1.03共Nae ⫹ Ke兲/TBW ⫺ 23.8
⫺ 共1.6/100兲共关glucose兴p ⫺ 120兲
where [glucose]p is plasma glucose concentration (mg/dl).
Determinants of Osmotic Activity
The osmotic activity of a solute depends on its ability to
move randomly in solution. Therefore, any factors that
reduce the random movement of a solute will reduce its
osmotic activity. It is well known that not all Nae and Ke are
osmotically active (3, 5, 8 –10, 14, 25, 27–29). Indeed, there
is abundant evidence that a portion of Nae is bound in bone
and skin and is therefore rendered osmotically inactive (3,
8 –10, 14, 25, 27–29). Likewise, a portion of cellular K⫹ is
reduced in its mobility and in its osmotic activity owing to
its association with anionic groups such as carboxyl groups
on proteins or to phosphate groups in creatine phosphate,
ATP, proteins, and nucleic acids (5). Because osmotically
inactive Nae and Ke cannot contribute to the distribution of
water between the extracellular and intracellular compartments, osmotically inactive Nae and Ke cannot contribute to
the modulation of the [Na⫹]pw and therefore are accounted
for quantitatively by the (Naosm inactive ⫹ Kosm inactive)/TBW
term in Eq. 12.
It is also well known that the osmotic activity of most ions
is slightly less than one owing to the electrostatic interactions between ions (16). Because ionic interactions in
plasma reduce the random movement of Na⫹ ions, each
mole of Na⫹ ions does not exert exactly one osmole of
osmotic activity (16). The osmotic coefficient Ø in Eq. 12
therefore accounts for the effectiveness of Na⫹ salts as
independent osmotically active particles under physiological conditions and is reflected in the slope of the Edelman
equation. Finally, the osmotic activity of a solute will
depend on its impermeability to cross cell membranes. For
instance, urea is considered to be an ineffective osmole
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Non-Na⫹ and non-K⫹ osmoles are capable of modulating
the [Na⫹]pw by altering the value of the y-intercept in the
Edelman equation. For example, mannitol administration in
patients with impaired renal function has been shown to
result in the development of hyponatremia (2, 4). Mannitol
is confined to the extracellular fluid after intravenous administration. The resultant increased osmolality of the extracellular fluid will in turn promote the osmotic shift of
water from the intracellular compartment to the extracellular
compartment. Therefore, hyponatremia may result when
mannitol is given to patients with impaired renal function
because the mannitol is poorly cleared by the kidney.
Similarly, contrast-induced translocational hyponatremia
has been described in patients with advanced kidney disease
undergoing cardiac catheterization (26). In addition, maltose
and sucrose used in commercial intravenous immunoglobulin preparations as well as hyperglycemia can promote the
translocation of water from the intracellular compartment to
the extracellular compartment, thereby resulting in dilutional hyponatremia (15, 20, 22).
Because the osmotic shift of water from the ICF to the ECF
does not alter the value of the (Nae ⫹ Ke)/TBW term in the
Edelman equation, non-Na⫹ and non-K⫹ osmoles therefore
induce changes in the [Na⫹]pw by altering the magnitude of the
y-intercept in the Edelman equation. Specifically, non-Na⫹ and
non-K⫹ osmoles will lead to an increase in the ratio (osmolECF ⫹
osmolICF)/TBW. During the non-Na⫹ and non-K⫹ osmolesinduced osmotic shift of water from the intracellular compartment to the extracellular space, the TBW remains constant
because the change in intracellular volume is equal to the
change in extracellular volume. The (osmolECF ⫹ osmolICF)/
TBW term in Eq. 12 increases because non-Na⫹ and non-K⫹
osmoles increase the osmolECF term whereas the TBW remains
unchanged. Secondly, the [K⫹]pw will also be affected by the
non-Na⫹ and non-K⫹ osmoles-induced osmotic shift of water
and subsequent cellular K⫹ efflux induced by the increase in
intracellular K⫹ concentration and solvent drag induced by
hyperosmolality. Finally, non-Na⫹ and non-K⫹ osmoles will
increase the term osmolpw/Vpw, thereby lowering the [Na⫹]pw
by promoting the osmotic movement of water into the plasma
space.
Non-Na⫹ and non-K⫹ osmoles therefore induce changes in
the [Na⫹]pw by altering the magnitude of the y-intercept in the
Edelman equation. In maltose-induced hyponatremia, it has
been shown that there is a decrease in the [Na⫹]p of 0.59 mM/l
for each millimolar increase in the plasma maltose concentration (22). As a result, to predict the dilutional effect of maltose
on the [Na⫹]p attributable to the osmotic shift of water, the
correction factor of 0.59 can be incorporated into the Edelman
equation:
GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION
because its concentration is equal throughout the body fluid
compartments owing to its high permeability across cell
membranes (12). This fact can be quantitatively demonstrated by calculating the [Na⫹]pw assuming that the plasma
urea concentration increases from a value of 4 to 14
mosmol/l H2O. By using Table 1, the [Na⫹]pw is calculated
as follows:
Na osm active ⫽ 2,722 mosmol
K osm active ⫽ 3,568.6 mosmol
Na osm active ⫹ Kosm active ⫽ 2,722 mosmol ⫹ 3,568.6 mosmol
osmol pw ⫽ 496.8 mosmol
osmol ISF ⫽ 2,349.2 mosmol
osmol ICF ⫽ 3,930 mosmol
⫹
g ⫽ 311.8/311.1
共Naosm active ⫹ Kosm active兲
TBW
册
osmolpw ⫹ osmolISF ⫹ osmolICF
osmolpw
⫺ 关K⫹兴pw ⫺
TBW
Vpw
关Na ⫹兴pw ⫽ 共311.8/311.1兲关共6,290.6 ⫹ 496.8 ⫹ 2,349.2
⫹ 3,930兲/42兴 ⫺ 4.2 ⫺ 165.6
关Na ⫹兴pw ⫽ 142 mosmol/l
Changes in the plasma urea concentration therefore do not
alter the [Na⫹]pw as demonstrated quantitatively by the application of Eq. 11.
Summary
Changes in the osmolality of the body fluid compartments
have long been known to result in intercompartmental H2O
shifts, thereby leading to alterations in the [Na⫹]pw. In this
article, we derive an equation that depicts the quantitative
interrelationship between the Gibbs-Donnan equilibrium, osmolality of body fluids (plasma, ISF, and ICF), and [Na⫹]pw.
This equation also defines all the known factors that determine
the magnitude of the [Na⫹]pw. Moreover, Eq. 12 accounts for
the alterations in all ionic concentrations (Na⫹ and non-Na⫹
ions) between the plasma and ISF. It is important to take into
consideration the fact that the altered distribution of osmotically active non-Na⫹ ions due to the Gibbs-Donnan equilibrium will also have a modulating effect on the [Na⫹]pw by
affecting the distribution of H2O between the plasma and ISF.
Furthermore, Eq. 12 can account for alterations in the distribution of Na⫹ and non-Na⫹ ions between the plasma and ISF
resulting from changes in the Gibbs-Donnan equilibrium induced by changes in the plasma protein concentration. Importantly, this new equation can be an indispensable tool in the
analysis of the pathophysiology of the dysnatremias. In particular, Eq. 12 is especially helpful in explaining quantitatively
J Appl Physiol • VOL
GRANTS
This work was supported by grants to I. Kurtz from the Max Factor Family
Foundation, Richard and Hinda Rosenthal Foundation, Fredricka Taubitz fund,
and National Institute of Diabetes and Digestive and Kidney Diseases Grants
DK-63125, DK-58563, and DK-07789.
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osmol pw
⫽ 165.6 mosm/l
Vpw
冋
how physiological factors that do not alter the value of the
(Nae ⫹ Ke)/TBW term are capable of modulating the [Na⫹]pw
by altering the value of the slope and y-intercept in the
Edelman equation.
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