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Phenomenological analysis of renal regulation of sodium and potassium balance
Kidney International, Vol. 27 (1985), pp. 837—841
EDITORIAL REVIEW
Phenomenological analysis of renal regulation of sodium and potassium balance MACKENZIE WALSER Department of Pharmacology and Experimental Therapeutics and Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
Extracellular fluid volume (ECF) is maintained within normal limits principally by the renal regulation of sodium excretion. Despite a wealth of studies on specific mechanisms, such as the renin-angiotensin-aldosterone system, there is little information as to exactly how ECF varies in response to variations in salt intake. For example, when intake and output of sodium are equal, what is the relationship between ECF and output (or intake)? When salt intake changes in a step fashion, what is the
Strauss et al [2] apparently recognized. However, the slope of their linear diagram is 1.0, while the exponential decay constant they observed was 0.7 15 0.028 day'. As shown below, both constants should be the same. Renal regulation of potassium excretion, from the phenom-
enological point of view, has received even less attention, despite many studies of the roles of aldosterone, distal sodium delivery, and other factors. It apparently has not been deter-
time course of the resulting changes in ECF and sodium mined how much (if any) total body potassium varies with excretion?
Ludwig [1] was the first to recognize that ECF varied in response to salt intake. Strauss et al [21 suggested that "the amount of sodium excreted in the urine on any one day is a function of total body sodium on that day and particularly of that moiety which is in excess of the body's needs." Although their diagram indicates that this function should be a linear one, they did not offer any evidence for linearity. Hollenberg [3, 4] extended this concept by proposing that sodium balance in a normal subject is regulated about a set point defined as "the amount of sodium chloride in his body when he is in balance on no salt intake." He also proposed that ECF is a linear function of the logarithm of sodium intake (or output) as shown by the following statement: ". . . one would anticipate that the impact on the system of moving from a l0-mEq to a 30-mEq daily intake would be equivalent to that in moving from 30 mEq to 100 mEq and even more dramatically, on moving from 100 to 300 mEq daily" [3]. Bonventre and Leaf [5, 6] disputed the idea of a fixed set point, suggesting that each sodium intake eventually leads to a new steady-state level of body sodium. However, they evidently accepted the view that the relationship between total body sodium and sodium intake or output at steady-state should be logarithmic, as indicated by their Figure 1. None of these authors has derived mathematical formulations of these relationships or presented data to support their hypotheses, except Strauss et al [2], who showed that sodium excretion decays exponentially in response to a very low salt intake, an observation that has been confirmed repeatedly [7—9]. This latter observation suggests that sodium excretion is a linear function, not a logarithmic function, of body sodium, as
potassium intake or how fast adjustments occur. The purpose of this report is to examine what the consequences would be, in these terms, of a linear dependence of sodium and potassium
excretion on total body sodium and potassium, respectively, and to compare these predictions with recalculations of re-
ported studies in which normal subjects have been given varying intakes of sodium or potassium. The model
It is well known that rates of renal excretion of sodium and potassium can fall to a minute fraction of the usual rates during dietary deprivation or frank deficiency. Therefore, a value for bodily content of each ion, A0, must exist below which renal excretion effectively ceases (excretion by other routes such as the gut being ignored). The hypothesis to be tested is that the excretory rate is proportional to the excess of bodily content,
A1, at any time t, over A0, which will be termed the "zero point." The term "set point" will not be used for this quantity to avoid the implication that the system is designed to regulate body content at this value. The letter "I" designates absorbed intake1. Thus,
dA/dt = I
—
k(A
—
A0) where k is a proportionality constant. Integration yields A1 = (I
+ kA0) (k' + Ce_kt)
(1)
(2)
where C is the constant of integration.
'If absorbed intake were to exceed renal excretory capacity, A, would Received for publication March 19, 1984, and in revised form October 4, 1984
© 1985 by the International Society of Nephrology
expand without limit. However, there is no evidence that absorbed intake of sodium or potassium can exceed the enormous renal excretory
capacity for these ions that is demonstrable, for example, during intravenous loading.
837
838
Walser
Table 1. Slope, kNa, of urinary sodium excretion as a function of sodium balance in normal subjects, derived from various reports
Reference
No. of subjects
Weinberger et al [12] Sealey et al [13] Crabbé, Ross, and Thorn [141 Braunwald, Plauth, and Morrow [15] Leaf and Couter [16] Leaf, Couter, and Newburgh [17] Casson et al [18] Kirkendall et al [19] Brown, Brown, and Krishan [20] Ross and Winternitz [71 Epstein and Hollenberg [81 Carey [9] Strauss et al [2]
14 3
5 2 3 3
5 8 8 8 89
8 15
Average
Method bNa
bN, bN. bNa bNa bNa
kNa, day' 0.72 0.77 0.77 0.70 0.73
0.74
Wt
1.13
YD inulin
0.60
ENa
1.12
t112 t112
t1,2 t1,2
r 0,91
0.98 0.83 0.95 0.77 0.74 0.98
0.66 0.94 0.64 0.72
0.04 0.03 0.03
0.996
0,79
0.05
0.90
Abbreviations: bNa, change in cumulative sodium balance; V inulin, change in volume of distribution of inulin; Wt, change in weight; t112, exponential decay of urinary sodium on a low sodium diet; r, correlation coefficient between cumulative sodium balance and urinary sodium excretion.
Thus, when intake is changed at time to from one constant elapse between intravenous administration of sodium and the value, I, to another, 12, A1 will approach its new limit according physiological alterations necessary to bring about the relationto an exponential approach to a limit with time constant k as ship postulated in Eq. (1), this should occasion no surprise. The follows: renal response occurred within a few hours in the experiments cited above. In consequence, only 24-hr excretion rates will be A1 — A0 =+_— '2—kt (3) used to test the model proposed.
k k Results were compiled from experiments in which normal Steady-state body content, A, on constant intake, I, of any subjects were given two or more levels of sodium intake, and magnitude will be a linear function of intake, because the 24-hr sodium excretion was reported. Cumulative sodium bal-
ance during the course of the experiments could be calculated from published graphs or tables in six studies of varying sodium A = A0 + I/k (4) intake. Changes in cumulative sodium balance during varying Urinary excretion, U, is the difference between intake and intake also were evaluated using three other types of data: (1) rate of change of body content. Thus, estimating dA/dt by changes in body weight, multiplied by 140 mEq/kg to yield a differentiation of Eq. (3) Ut is expressed as shown: rough approximation of changes in body sodium (one study); (2) changes in the product, plasma sodium concentration times 12 — dA1/dt = 12 + (I — 12) e_kt (5) extracellular fluid volume as estimated by inulin space (one At steady-state, urinary excretion, U, is equal to intake and study); (3) changes in exchangeable sodium, determined by isotopic measurement (one study). The linear regression of dA/dt equals 0. Hence, from Eq. (1), urinary sodium excretion on changes in total body sodium, U = k (A — A0) (6) estimated by one of these methods, was calculated for all four The following points should be noted: (1) Neither negative types of data. The calculated slopes and correlation coefficients exponential term will disappear as t approaches infinity:
values of intake nor negative values of A, A0, or k are for each of these nine studies are shown in Table 1. Four considered; (2) Eq. (4) defines a series of set points that depend on the value of I.
Sodium homeostasis: compilation of data from literature to test the proposed hypothesis
The renal response to sodium loading and deprivation has been examined repeatedly. A brief literature review makes clear that the present hypothesis is not applicable to short-term experiments. Papper, Belsky, and Bleifer [101, for example, showed that the rate of excretion of a sodium load subsequent to an intravenous infusion of sodium chloride is much more rapid in subjects on a high sodium intake than in those on a low
sodium intake. This finding, also reported by others [111, demonstrates that when total body sodium is changed abruptly, Eq. (1) cannot apply instantaneously. Since some interval must
additional studies identified report the exponential slope of falling sodium excretion in response to a very low sodium diet which is another way to estimate kNa, as noted above. In one of these studies [7] variability was not reported. The table reveals that there is some variability in the average kNa among these 13 studies. These differences may be attributable to variations in experimental design; however, the average correlation in the balance studies is quite high, 0.90, and the SE of the slope in three of the four t112 studies averages only 4% of the slope. Figures 1 and 2 display a few examples of cumulative balance studies. The linear model clearly provides a good description of the 24-hr excretion data in all of these studies. kNa in the 13 reports averages 0.79 0.05 day'. In the nine cumulative balance studies, the average value (0.81 day') is very close to the average in the four t112 studies (0.74 day). As Eq. (3) shows, the overall average of 0.79 day means that a
839
Analysis of sodium and potassium homeostasis
Table 2. Slope, kK, of urinary potassium excretion as a function of potassium balance in normal subjects, calculated from various reports
.
300
No. of Reference
200
>,, 100
z
100
200
400
300
500
600
Cumulative sodium balance
subjects
0.98 0.75
0.99
16
2
0.71
0.90 0.98 0.89 0.88 0.96
8
Costill, Cote, and Fink [21] Horwitz, Margolius, and Keiser [22] Gann et al [23] Brunner et al [24] Cannon, Ames, and Laragh [25] Malhotra et al [26] Bartter [27] Voors et al [28]
0.39 0.42 0.37 0.55 0.65
1
4 1 1
7
0.60
Average
0.91
0.91
0.93
0.08
Fig. 1. Three representative studies of sodium loading in normal subjects from literature, recalculated to display 24-hr urinary sodium excretion as a function of cumulative sodium balance. The lines were calculated by least squares linear regression. Sources used are: crosses, Braunwald, Plauth, and Morrow [15]; solid circles, Sealey et al [13]; open circles, Kirkendall et al [19].
r
kK, day
Abbreviation: r, correlation coefficient between cumulative potassium balance and urinary potassium excretion.
200
1200
x
.
1000
a
0.
0
Lu
800
E
Q)
0. 0.
Lu
50
600
E
>-
D 400
—150 —100
—50 Cumulative
200
.
________________________________________
200
400
600
800
1000
1200
1400
0
+50
+100
+150 +200
potassium balance, mEq
Fig. 3. Urinary potassium excretion in normal subjects fed high or low potassium diets, recalculated as a function of cumulative potassium balance from three representative studies. The linear regression lines for each study are shown. Sources used are: crosses, Brunner et a! [24]; solid circles, Costill, Cote, and Fink [21]; open circles, Bartter [27].
Cumulative sodium balance, mEq
Fig. 2. Urinary sodium excretion as a function of cumulative sodium
on cumulative balance in each study and the correlation coefficient, which averages 0.93, are shown. A few examples are illustrated in Figure 3. The model appears consistent with reported data. kK for the nine studies averages 0.60 0.08 day. At step change in sodium intake will be followed by a change in steady-state the change in total body potassium is, on the sodium excretion which will approach its new steady-state average, 1.7 times the change in intake. Thus, the half-time of value with a half-time of 0.693/0.79 days or 21 hr on the average. approach of urinary potassium excretion to a new steady-state As Eq. (4) shows, the change in total body sodium at steady- value following a change in potassium intake should be, on the balance in 14 normal subjects, recalculated from data reported by Weinberger et al [12]. The regression line is shown. Different results were seen in subjects in whom potassium losses were replaced.
state will be, on the average, 1.3 sodium intake.
0.1 times the change in daily
Potassium homeostasis: compilation of data from literature to test the proposed hypothesis
The results of nine published studies in normal subjects in which urinary potassium excretion is reported on varying intakes of potassium are summarized in Table 2. Cumulative potassium balance has been recalculated in each case from daily measured potassium balance, or from an estimate of potassium
balance as intake minus urinary excretion minus 10 mEq/day (for stool potassium). The linear regression slope of excretion
average, 0.693/0.60 days or 28 hr.
Sodium excretion is another factor determining potassium excretion [29]. Subjects on a high salt intake not only exhibit
increased potassium excretion on constant intake, thereby lowering the zero point for potassium excretion, but also exhibit
an accelerated adaptation of urinary potassium excretion to a change in intake; subjects on sodium-restricted diets show the opposite trends. Both the slope and intercept of potassium homeostasis correlate positively with respect to sodium intake. These observations would need to be considered in a more complete description of potassium homeostasis. Furthermore, other factors may modify the relationship between total body potassium and renal potassium excretion, including adaptation
840
Walser
to altered dietary intake, acid-base balance, distal tubular flow rate, and mineralocorticoid activity [291. Some of these factors may account for the differences between the results of the nine studies summarized in Table 2.
the zero point, change in the slope, or both. Re-examination of pathological states in this light may prove useful. MACKENZIE WALSER
Baltimore, Maryland, USA
Appendix When sodium or potassium intake is changed, 2 or 3 days are The purpose of this analysis has been to clarify the relation- required for a new steady-state to be established, as shown by ship between intake, bodily content, and excretion of sodium the studies cited. In compiling the data for Tables 1 and 2, all and potassium. Despite numerous studies of regulatory mecha- observations were used, including those on the first few days of Discussion
nisms involved in sodium and potassium homeostasis, it appears there have been few attempts to describe these fundamental relationships as observed in normal human subjects. Indeed, the term "homeostasis" is often used with the implication that regulatory mechanisms prevent any variation in the homeostatic variable in response to external influences. This was not Cannon's view [30] and is also not characteristic of servomechanisms in general. The published studies of sodium and potassium balance in response to variations in sodium or potassium intake, summarized herein, make clear that the bodily content of both ions is not constant, but depends on intake. Because both sodium and potassium are excreted predominantly by the kidney, the inference that excretion of each ion is a function of body content is almost inescapable. There appears to be no evidence for an effect of intake on excretion, independent of body content, for either sodium or potassium. Adapta-
tion to low or high potassium intake can occur, but this is presumably adaptation to low or high body content, not intake per se. The function proposed in this study to describe this relationship is the simplest one conceivable, in view of the knowledge that excretion of either ion by the kidney can fall to zero. As such, it is almost certain to be an oversimplification. Nevertheless, the relationship between published data and
the predictions of this model is quite close—too close, it
a change in intake, The average number of days on a given intake is 8 days for Table 1 studies and 5 days for Table 2 studies. Inclusion of data before the steady-state should tend to underestimate potassium somewhat. Ideally, Eq. (3) instead of Eq. (6) should be fitted to these early data. However, exclusion of data obtained during the first 2 days had no discernible effect on the overall means. Acknowledgments I am indebted to Dr. K. L. Zierler for helpful comments. Reprint requests to Dr. M. Walser, Department of Pharmacology, Johns Hopkins University School of Medicine, 725 N. Wolfe Street, Baltimore, Maryland 21205, USA
References 1. LUDWIG C: Manuscript notes of lectures 1869—1870, in A Textbook
of Pharmacology,
Therapeutics and Materia Medica, edited by BRUNTON TL, Philadelphia, Lea Brothers & Co., 1885, p 503 2. STRAuss MB, LAMDIN E, SMITH WP, BLEIFER DJ: Surfeit and deficit of sodium: a kinetic concept of sodium excretion. Arch mt
Med 102:527—536, 1958 3. H0LLENBERG NK: Set point for sodium homeostasis: surfeit, deficit, and their implications. Kidney mt 17:423—429, 1980 4. HOLLENBERG NK: Surfeit, deficit, and the set point for sodium homeostasis. Kidney mt 21:883—884, 1982 5. BONVENTRE
JV, LEAF A: Sodium homeostasis: steady states
appears, to refute the model at least regarding 24-hr excretion data. On the other hand, sodium excretion when observed over
without a set point. Kidney mt 21:881—883, 1982 6. BONVENTRE JV, LEAF A: Response of Drs. Bonventre and Leaf. Kidney mt 21:885, 1982
shorter intervals clearly does not follow this model. This
7. Ross EJ, WINTERNITz WW: The effect of an aldosterone antagonist on the renal response to sodium restriction, Clin Sci 20:143—148,
observation does not mean that the model is invalid, but rather that it does not describe short-term variations in sodium excre-
tion in response to intravenous loads (nor does it describe diurnal variations). As noted above, some finite interval of time
must elapse when ECF is changed before those mechanisms that are responsible for simple proportionality between ECF and excretion can be activated. The fact that observations made during the few hours of intravenous loading deviate from the model suggest that these mechanisms require more than a few hours to achieve their effect. Additional data clearly are needed to test this model. In
particular, it has not been shown that the time constant of approach to a new steady-state value for ECF is the same when salt intake is increased as it is when it is decreased (although it is usually assumed that this is the case.) No data appear to be available to assess the rate of decline or increase of potassium excretion as potassium intake is changed.
1960
8. EPSTEIN M, HOLLENBERG NK: Age as a determinant of renal sodium conservation in normal man. J Lab Clin Med 87:411-417, 1976
9. CAREY RM: Evidence for a splanchnic sodium input monitor regulating renal sodium excretion in man: lack of dependence upon aldosterone. Circ Res 43:19—23, 1978 10. PAPPER 5, BELSKY JL, BLEIFER KB: The response to the administration of an isotonic sodium chloride-lactate solution in patients with essential hypertension. J Clin Invest 39:876—884, 1960 11. BALDWIN DS, BIGGS AW, GOLDRING W, HULET WH, CHA5I5 H:
Exaggerated natriuresis in essential hypertension. Am J Med 24:893—902, 1958 12. WEINBERGER MB, LUFT FC, BLOCH R, HENRY DP, PRATT JH, WEYMAN AE, RANKIN LI, Muiut,x RH, WILLIS LR, GRIM CE:
The blood pressure-raising effects of high dietary sodium intake:
racial differences and the role of potassium. J Am CoIl Nutr 1:139—148, 1982
13. SEALEY JE, BUHLER FR, LARAGH JH, MANNING EL, BRUNNER
BR: Aldosterone excretion. Physiological variations in man mea-
sured by radioimmunoassay or double-isotope dilution. Circ Res The applicability of this model to abnormalities of sodium 3 1:367—378, 1972 homeostasis is of interest. If the proportionality between so- 14. CRABBE J, Ross EJ, THORN GW: The significance of the secretion dium excretion and ECF is maintained in pathological states, it of aldosterone during dietary sodium deprivation in normal subis clear that this relationship can be altered through change in jects. J Clin Endocrinol Metab 18:1159—1177, 1958
Analysis of sodium and potassium homeostasis
841
15. BRAUNWALD E, PLAUTH WH JR, MORROW AG: A method for the
Control of aldosterone secretion by change of body potassium in
detection and quantification of impaired sodium excretion: results of an oral sodium tolerance test in normal subjects and in patients
normal man. Am J Physiol 207:104—108, 1964 24. BRUNNER HR, BAER L, SEALEY JE, LEDINGHAM JGG, LARAGH
with heart disease. Circulation 32:223—23 1, 1965
16. LEAF A, COUTER WT: Evidence that renal sodium excretion by normal human subjects is regulated by adrenal cortical activity. J Clin Invest 28:1067—1081, 1949 17. LEAF A, COUTER WT, NEWBURGH LH: Some effects of variation
in sodium intake and of different sodium salts in normal subjects. J
JH: The influence of potassium administration and of potassium deprivation on plasma renin in normal and hypertensive subjects. J Clin Invest 49:2128—2138, 1970 25. CANNON PJ, AMES RP, LARAGH JH: Relation between potassium
balance and aldosterone secretion in normal subjects and in patients with hypertensive or renal tubular disease. J Clin Invest
Clin Invest 28:1082—1090, 1949 18. CASSON IF, LEE MR, BROWNJOHN AM, PARSONS FM, DAvIsoN AM, WiLL EJ, CLAYDEN AD: Failure of renal dopamine response to salt loading in chronic renal disease. Br MedJ 286:503—506, 1983
45:865—879, 1966 26. MALHOTRA MS, SRIDHARAN K, VENKATASWAMY Y, RAI RM, PICHAN G, RADHAKRISHNAN U, GROVER SK: Effect of restricted
19. KIRKENDALL WM, CONNOR WE, ABBOUD F, RASTOGI SP, ANDERSON TA, FRY M: The effect of dietary sodium chloride on blood pressure, body fluids, electrolytes, renal function, and serum lipids of normotensive man. J Lab Clin Med 87:418—434, 1976
during heat stress. Eur J App! Physiol 47:169—179, 1981 27. BARTTER FC: The role of aldosterone in the regulation of body fluid
20. BROWN Wi JR, BROWN FR, KRISHAN I: Exchangeable sodium and
blood volume in normotensive and hypertensive humans on high and low sodium intake. Circulation 43:508—519, 1971
21. COSTILL DL, COTE R, FINK WJ: Dietary potassium and heavy exercise: effects on muscle water and electrolytes. Am J Clin Nutr 36:266—275, 1982
22. HoRwITz D, MARGOLIUS HS, KEISER HR: Effects of dietary potassium and race on urinary excretion of kallikrein and aldosterone in man. J Clin Endocrinol Metab 47:296—299, 1978 23. GANN DS, DELEA CS, GILL JR JR, THOMAS JP, BARTTER FC:
potassium intake on its excretion and on physiological responses
volume and composition. Scand J Clin Lab Invest l0(Suppl)31:50—61, 1958
28. VooRs AW, DALFERES ER JR, FRANK GC, ARISTIMuN0 GG, BERENSON GS: Relation between ingested potassium and sodium balance in young blacks and whites. Am J Clin Nutr 37:583—594, 1983
29. GIEBLSCH G: K transport along the nephron, in Physiology and Pathology of Electrolyte Metabolism, edited by SELDIN GW, GIEBISCH G, New York, Raven Press, in press 30. CANNON WB: Organization for physiological homeostasis. Physiol Rev 9:399—431, 1929
EDITORIAL REVIEW
Phenomenological analysis of renal regulation of sodium and potassium balance MACKENZIE WALSER Department of Pharmacology and Experimental Therapeutics and Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
Extracellular fluid volume (ECF) is maintained within normal limits principally by the renal regulation of sodium excretion. Despite a wealth of studies on specific mechanisms, such as the renin-angiotensin-aldosterone system, there is little information as to exactly how ECF varies in response to variations in salt intake. For example, when intake and output of sodium are equal, what is the relationship between ECF and output (or intake)? When salt intake changes in a step fashion, what is the
Strauss et al [2] apparently recognized. However, the slope of their linear diagram is 1.0, while the exponential decay constant they observed was 0.7 15 0.028 day'. As shown below, both constants should be the same. Renal regulation of potassium excretion, from the phenom-
enological point of view, has received even less attention, despite many studies of the roles of aldosterone, distal sodium delivery, and other factors. It apparently has not been deter-
time course of the resulting changes in ECF and sodium mined how much (if any) total body potassium varies with excretion?
Ludwig [1] was the first to recognize that ECF varied in response to salt intake. Strauss et al [21 suggested that "the amount of sodium excreted in the urine on any one day is a function of total body sodium on that day and particularly of that moiety which is in excess of the body's needs." Although their diagram indicates that this function should be a linear one, they did not offer any evidence for linearity. Hollenberg [3, 4] extended this concept by proposing that sodium balance in a normal subject is regulated about a set point defined as "the amount of sodium chloride in his body when he is in balance on no salt intake." He also proposed that ECF is a linear function of the logarithm of sodium intake (or output) as shown by the following statement: ". . . one would anticipate that the impact on the system of moving from a l0-mEq to a 30-mEq daily intake would be equivalent to that in moving from 30 mEq to 100 mEq and even more dramatically, on moving from 100 to 300 mEq daily" [3]. Bonventre and Leaf [5, 6] disputed the idea of a fixed set point, suggesting that each sodium intake eventually leads to a new steady-state level of body sodium. However, they evidently accepted the view that the relationship between total body sodium and sodium intake or output at steady-state should be logarithmic, as indicated by their Figure 1. None of these authors has derived mathematical formulations of these relationships or presented data to support their hypotheses, except Strauss et al [2], who showed that sodium excretion decays exponentially in response to a very low salt intake, an observation that has been confirmed repeatedly [7—9]. This latter observation suggests that sodium excretion is a linear function, not a logarithmic function, of body sodium, as
potassium intake or how fast adjustments occur. The purpose of this report is to examine what the consequences would be, in these terms, of a linear dependence of sodium and potassium
excretion on total body sodium and potassium, respectively, and to compare these predictions with recalculations of re-
ported studies in which normal subjects have been given varying intakes of sodium or potassium. The model
It is well known that rates of renal excretion of sodium and potassium can fall to a minute fraction of the usual rates during dietary deprivation or frank deficiency. Therefore, a value for bodily content of each ion, A0, must exist below which renal excretion effectively ceases (excretion by other routes such as the gut being ignored). The hypothesis to be tested is that the excretory rate is proportional to the excess of bodily content,
A1, at any time t, over A0, which will be termed the "zero point." The term "set point" will not be used for this quantity to avoid the implication that the system is designed to regulate body content at this value. The letter "I" designates absorbed intake1. Thus,
dA/dt = I
—
k(A
—
A0) where k is a proportionality constant. Integration yields A1 = (I
+ kA0) (k' + Ce_kt)
(1)
(2)
where C is the constant of integration.
'If absorbed intake were to exceed renal excretory capacity, A, would Received for publication March 19, 1984, and in revised form October 4, 1984
© 1985 by the International Society of Nephrology
expand without limit. However, there is no evidence that absorbed intake of sodium or potassium can exceed the enormous renal excretory
capacity for these ions that is demonstrable, for example, during intravenous loading.
837
838
Walser
Table 1. Slope, kNa, of urinary sodium excretion as a function of sodium balance in normal subjects, derived from various reports
Reference
No. of subjects
Weinberger et al [12] Sealey et al [13] Crabbé, Ross, and Thorn [141 Braunwald, Plauth, and Morrow [15] Leaf and Couter [16] Leaf, Couter, and Newburgh [17] Casson et al [18] Kirkendall et al [19] Brown, Brown, and Krishan [20] Ross and Winternitz [71 Epstein and Hollenberg [81 Carey [9] Strauss et al [2]
14 3
5 2 3 3
5 8 8 8 89
8 15
Average
Method bNa
bN, bN. bNa bNa bNa
kNa, day' 0.72 0.77 0.77 0.70 0.73
0.74
Wt
1.13
YD inulin
0.60
ENa
1.12
t112 t112
t1,2 t1,2
r 0,91
0.98 0.83 0.95 0.77 0.74 0.98
0.66 0.94 0.64 0.72
0.04 0.03 0.03
0.996
0,79
0.05
0.90
Abbreviations: bNa, change in cumulative sodium balance; V inulin, change in volume of distribution of inulin; Wt, change in weight; t112, exponential decay of urinary sodium on a low sodium diet; r, correlation coefficient between cumulative sodium balance and urinary sodium excretion.
Thus, when intake is changed at time to from one constant elapse between intravenous administration of sodium and the value, I, to another, 12, A1 will approach its new limit according physiological alterations necessary to bring about the relationto an exponential approach to a limit with time constant k as ship postulated in Eq. (1), this should occasion no surprise. The follows: renal response occurred within a few hours in the experiments cited above. In consequence, only 24-hr excretion rates will be A1 — A0 =+_— '2—kt (3) used to test the model proposed.
k k Results were compiled from experiments in which normal Steady-state body content, A, on constant intake, I, of any subjects were given two or more levels of sodium intake, and magnitude will be a linear function of intake, because the 24-hr sodium excretion was reported. Cumulative sodium bal-
ance during the course of the experiments could be calculated from published graphs or tables in six studies of varying sodium A = A0 + I/k (4) intake. Changes in cumulative sodium balance during varying Urinary excretion, U, is the difference between intake and intake also were evaluated using three other types of data: (1) rate of change of body content. Thus, estimating dA/dt by changes in body weight, multiplied by 140 mEq/kg to yield a differentiation of Eq. (3) Ut is expressed as shown: rough approximation of changes in body sodium (one study); (2) changes in the product, plasma sodium concentration times 12 — dA1/dt = 12 + (I — 12) e_kt (5) extracellular fluid volume as estimated by inulin space (one At steady-state, urinary excretion, U, is equal to intake and study); (3) changes in exchangeable sodium, determined by isotopic measurement (one study). The linear regression of dA/dt equals 0. Hence, from Eq. (1), urinary sodium excretion on changes in total body sodium, U = k (A — A0) (6) estimated by one of these methods, was calculated for all four The following points should be noted: (1) Neither negative types of data. The calculated slopes and correlation coefficients exponential term will disappear as t approaches infinity:
values of intake nor negative values of A, A0, or k are for each of these nine studies are shown in Table 1. Four considered; (2) Eq. (4) defines a series of set points that depend on the value of I.
Sodium homeostasis: compilation of data from literature to test the proposed hypothesis
The renal response to sodium loading and deprivation has been examined repeatedly. A brief literature review makes clear that the present hypothesis is not applicable to short-term experiments. Papper, Belsky, and Bleifer [101, for example, showed that the rate of excretion of a sodium load subsequent to an intravenous infusion of sodium chloride is much more rapid in subjects on a high sodium intake than in those on a low
sodium intake. This finding, also reported by others [111, demonstrates that when total body sodium is changed abruptly, Eq. (1) cannot apply instantaneously. Since some interval must
additional studies identified report the exponential slope of falling sodium excretion in response to a very low sodium diet which is another way to estimate kNa, as noted above. In one of these studies [7] variability was not reported. The table reveals that there is some variability in the average kNa among these 13 studies. These differences may be attributable to variations in experimental design; however, the average correlation in the balance studies is quite high, 0.90, and the SE of the slope in three of the four t112 studies averages only 4% of the slope. Figures 1 and 2 display a few examples of cumulative balance studies. The linear model clearly provides a good description of the 24-hr excretion data in all of these studies. kNa in the 13 reports averages 0.79 0.05 day'. In the nine cumulative balance studies, the average value (0.81 day') is very close to the average in the four t112 studies (0.74 day). As Eq. (3) shows, the overall average of 0.79 day means that a
839
Analysis of sodium and potassium homeostasis
Table 2. Slope, kK, of urinary potassium excretion as a function of potassium balance in normal subjects, calculated from various reports
.
300
No. of Reference
200
>,, 100
z
100
200
400
300
500
600
Cumulative sodium balance
subjects
0.98 0.75
0.99
16
2
0.71
0.90 0.98 0.89 0.88 0.96
8
Costill, Cote, and Fink [21] Horwitz, Margolius, and Keiser [22] Gann et al [23] Brunner et al [24] Cannon, Ames, and Laragh [25] Malhotra et al [26] Bartter [27] Voors et al [28]
0.39 0.42 0.37 0.55 0.65
1
4 1 1
7
0.60
Average
0.91
0.91
0.93
0.08
Fig. 1. Three representative studies of sodium loading in normal subjects from literature, recalculated to display 24-hr urinary sodium excretion as a function of cumulative sodium balance. The lines were calculated by least squares linear regression. Sources used are: crosses, Braunwald, Plauth, and Morrow [15]; solid circles, Sealey et al [13]; open circles, Kirkendall et al [19].
r
kK, day
Abbreviation: r, correlation coefficient between cumulative potassium balance and urinary potassium excretion.
200
1200
x
.
1000
a
0.
0
Lu
800
E
Q)
0. 0.
Lu
50
600
E
>-
D 400
—150 —100
—50 Cumulative
200
.
________________________________________
200
400
600
800
1000
1200
1400
0
+50
+100
+150 +200
potassium balance, mEq
Fig. 3. Urinary potassium excretion in normal subjects fed high or low potassium diets, recalculated as a function of cumulative potassium balance from three representative studies. The linear regression lines for each study are shown. Sources used are: crosses, Brunner et a! [24]; solid circles, Costill, Cote, and Fink [21]; open circles, Bartter [27].
Cumulative sodium balance, mEq
Fig. 2. Urinary sodium excretion as a function of cumulative sodium
on cumulative balance in each study and the correlation coefficient, which averages 0.93, are shown. A few examples are illustrated in Figure 3. The model appears consistent with reported data. kK for the nine studies averages 0.60 0.08 day. At step change in sodium intake will be followed by a change in steady-state the change in total body potassium is, on the sodium excretion which will approach its new steady-state average, 1.7 times the change in intake. Thus, the half-time of value with a half-time of 0.693/0.79 days or 21 hr on the average. approach of urinary potassium excretion to a new steady-state As Eq. (4) shows, the change in total body sodium at steady- value following a change in potassium intake should be, on the balance in 14 normal subjects, recalculated from data reported by Weinberger et al [12]. The regression line is shown. Different results were seen in subjects in whom potassium losses were replaced.
state will be, on the average, 1.3 sodium intake.
0.1 times the change in daily
Potassium homeostasis: compilation of data from literature to test the proposed hypothesis
The results of nine published studies in normal subjects in which urinary potassium excretion is reported on varying intakes of potassium are summarized in Table 2. Cumulative potassium balance has been recalculated in each case from daily measured potassium balance, or from an estimate of potassium
balance as intake minus urinary excretion minus 10 mEq/day (for stool potassium). The linear regression slope of excretion
average, 0.693/0.60 days or 28 hr.
Sodium excretion is another factor determining potassium excretion [29]. Subjects on a high salt intake not only exhibit
increased potassium excretion on constant intake, thereby lowering the zero point for potassium excretion, but also exhibit
an accelerated adaptation of urinary potassium excretion to a change in intake; subjects on sodium-restricted diets show the opposite trends. Both the slope and intercept of potassium homeostasis correlate positively with respect to sodium intake. These observations would need to be considered in a more complete description of potassium homeostasis. Furthermore, other factors may modify the relationship between total body potassium and renal potassium excretion, including adaptation
840
Walser
to altered dietary intake, acid-base balance, distal tubular flow rate, and mineralocorticoid activity [291. Some of these factors may account for the differences between the results of the nine studies summarized in Table 2.
the zero point, change in the slope, or both. Re-examination of pathological states in this light may prove useful. MACKENZIE WALSER
Baltimore, Maryland, USA
Appendix When sodium or potassium intake is changed, 2 or 3 days are The purpose of this analysis has been to clarify the relation- required for a new steady-state to be established, as shown by ship between intake, bodily content, and excretion of sodium the studies cited. In compiling the data for Tables 1 and 2, all and potassium. Despite numerous studies of regulatory mecha- observations were used, including those on the first few days of Discussion
nisms involved in sodium and potassium homeostasis, it appears there have been few attempts to describe these fundamental relationships as observed in normal human subjects. Indeed, the term "homeostasis" is often used with the implication that regulatory mechanisms prevent any variation in the homeostatic variable in response to external influences. This was not Cannon's view [30] and is also not characteristic of servomechanisms in general. The published studies of sodium and potassium balance in response to variations in sodium or potassium intake, summarized herein, make clear that the bodily content of both ions is not constant, but depends on intake. Because both sodium and potassium are excreted predominantly by the kidney, the inference that excretion of each ion is a function of body content is almost inescapable. There appears to be no evidence for an effect of intake on excretion, independent of body content, for either sodium or potassium. Adapta-
tion to low or high potassium intake can occur, but this is presumably adaptation to low or high body content, not intake per se. The function proposed in this study to describe this relationship is the simplest one conceivable, in view of the knowledge that excretion of either ion by the kidney can fall to zero. As such, it is almost certain to be an oversimplification. Nevertheless, the relationship between published data and
the predictions of this model is quite close—too close, it
a change in intake, The average number of days on a given intake is 8 days for Table 1 studies and 5 days for Table 2 studies. Inclusion of data before the steady-state should tend to underestimate potassium somewhat. Ideally, Eq. (3) instead of Eq. (6) should be fitted to these early data. However, exclusion of data obtained during the first 2 days had no discernible effect on the overall means. Acknowledgments I am indebted to Dr. K. L. Zierler for helpful comments. Reprint requests to Dr. M. Walser, Department of Pharmacology, Johns Hopkins University School of Medicine, 725 N. Wolfe Street, Baltimore, Maryland 21205, USA
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