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Reconsidering the Edelman equation: impact of plasma sodium concentration, edema and body weight

  • Jetta J. Oppelaar
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Microcirculation, Amsterdam, The Netherlands
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  • Mart D. Vuurboom
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Microcirculation, Amsterdam, The Netherlands
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  • Eliane F.E. Wenstedt
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Microcirculation, Amsterdam, The Netherlands
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  • Frans J. van Ittersum
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam UMC location Vrije Universiteit Amsterdam, Department of Internal Medicine, Section of Nephrology, Boelelaan 1117, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Atherosclerosis & Ischemic Syndromes, Amsterdam, The Netherlands
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  • L. Vogt
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Microcirculation, Amsterdam, The Netherlands
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  • Rik H.G. Olde Engberink
    Correspondence
    Corresponding author at: Room D3-206, Meibergdreef 9, 1105 AZ.
    Affiliations
    Amsterdam UMC location University of Amsterdam, Department of Internal Medicine, Section of Nephrology, Meibergdreef 9, Amsterdam, The Netherlands

    Amsterdam Cardiovascular Sciences, Microcirculation, Amsterdam, The Netherlands
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Open AccessPublished:April 04, 2022DOI:https://doi.org/10.1016/j.ejim.2022.03.027

      Highlights

      • The Edelman equation defines how cations (Na++K+) in body water predict plasma Na+
      • A two compartment view of sodium homeostasis is too limited to determine plasma Na+
      • Weight and edema affect the relation between plasma Na+ and exchangeable cations
      • The Edelman equation is different for subjects with hyponatremia
      • Using Edelman-derived formulas one should consider plasma Na+, edema and weight

      Abstract

      Background

      Guidelines recommend treatment of dysnatremias to be guided by formulas based on the Edelman equation. This equation describes the relation between plasma sodium concentration and exchangeable cations. However, this formula does not take into account clinical parameters that have recently been associated with local tissue sodium accumulation, which occurs without concurrent water retention. We investigated to what extent such clinical factors affect the Edelman equation and dysnatremia treatment.

      Methods

      We performed a post-hoc analysis with original data of the Edelman study. Linear regression was used to examine the effect of age, sex, weight, edema, total body water (TBW) and heart and kidney failure on the Edelman equation. With attenuated correction, we corrected for measurement errors of both variables. Using piecewise regression, we analyzed whether the Edelman association differs for different plasma sodium concentrations.

      Results

      Data was available for 82 patients; 57 males and 25 females with a mean (SD) age of 57 (15) years. The slope of the Edelman equation was significantly affected by weight (p=0.01) and edema (p=0.03). Also, below and above plasma sodium levels of 133 mmol/L the slope of the Edelman equation was significantly different (1.25 x0025vs 0.58x0025, p<0.01).

      Conclusion

      Edelman's equation's coefficients are significantly affected by weight, edema and plasma sodium, possibly reflecting differences in tissue sodium accumulation capacity. The performance of Edelman-based formulas in clinical settings may be improved by taking these clinical characteristics into account.

      Keywords

      Abbreviations:

      [Na+]p (plasma sodium concentration), [Na+]pw (plasma water sodium concentration), Ke (Exchangeable potassium), Nae (Exchangeable sodium), TBW (Total body water)

      1. Introduction

      Hypernatremia and hyponatremia, are commonly encountered electrolyte disturbances in clinical practice [
      • Upadhyay A
      • Jaber BL
      • Madias NE.
      Incidence and Prevalence of Hyponatremia.
      ,
      • Palevsky PM
      • Bhagrath R
      • Greenberg A.
      Hypernatremia in hospitalized patients.
      ]. The pathophysiology underlying these disturbances is complex and clinicians often face diagnostic and therapeutic difficulties [
      • Hoorn EJ
      • Zietse R.
      Hyponatremia revisited: translating physiology to practice.
      ,
      • Sam R
      • Feizi I.
      Understanding Hypernatremia.
      ]. Adequate treatment is critical, since these electrolyte disorders are associated with increased morbidity and mortality, yet overly rapid correction can lead to severe brain injury [
      Sterns RH. Disorders of Plasma Sodium — Causes, Consequences, and Correction.
      ]. Treatment is often guided by formulas predicting changes in plasma sodium concentration following hypertonic saline infusion or correction of water deficit.
      In 1958, Edelman et al. studied the relation between plasma water sodium concentration [Na+]pw and exchangeable sodium (Nae) plus exchangeable potassium (Ke) per total body water (TBW) by using measurements based on Na and K isotopes [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ]. The analysis showed a very strong association between [Na+]pw and total exchangeable cation concentration per TBW, which could be described as: [Na+]pw = 1.11 * (Nae + Ke)/TBW – 25.6. This equation, which links plasma sodium concentration to intracellular (Ke/TBW) and extracellular solute concentration (Nae/TBW), implies a two-compartment model of body water. In this traditional concept, osmotic equilibration is present between the intra- and extracellular space, and changes in plasma sodium concentration will induce water shifts between the intracellular and extracellular compartment. This two-compartment view has influenced our understanding of water balance and osmoregulation since the second half of the 20th century. Furthermore, this equation has provided the basis for the development of formulas such as the Adrogué-Madias, Nguyen-Kurtz and Barsoum-Levine formula, which are recommended by guidelines to guide infusion strategies for dysnatremias [
      • Adrogué HJ
      • Madias NE.
      Aiding fluid prescription for the dysnatremias.
      ,
      • Nguyen MK
      • Kurtz I.
      A new quantitative approach to the treatment of the dysnatremias.
      ,
      • Barsoum NR
      • Levine BS.
      Current prescriptions for the correction of hyponatraemia and hypernatraemia: are they too simple?.
      ,
      • Spasovski G
      • Vanholder R
      • Allolio B
      • Annane D
      • Ball S
      • Bichet D
      • et al.
      Clinical practice guideline on diagnosis and treatment of hyponatraemia.
      ,
      • Hoorn EJ
      • Tuut MK
      • Hoorntje SJ
      • van Saase JL
      • Zietse R
      • Geers AB.
      Dutch guideline for the management of electrolyte disorders–2012 revision.
      ]. Multiple clinical studies in patients with dysnatremia have, however, demonstrated that these formulas often fail to accurately predict the course of serum sodium concentration during treatment [
      • Lindner G
      • Schwarz C
      • Kneidinger N
      • Kramer L
      • Oberbauer R
      • Druml W.
      Can we really predict the change in serum sodium levels? An analysis of currently proposed formulae in hypernatraemic patients.
      ,
      • Hanna RM
      • Yang W-T
      • Lopez EA
      • Riad JN
      • Wilson J.
      The utility and accuracy of four equations in predicting sodium levels in dysnatremic patients.
      ,
      • Liamis G
      • Kalogirou M
      • Saugos V
      • Elisaf M.
      Therapeutic approach in patients with dysnatraemias.
      ]. Also, controlled balance studies in healthy participants have shown that these formulas are not able to estimate changes in plasma sodium concentration accurately [
      • Olde Engberink RH
      • Rorije NM
      • van den Born BH
      • Vogt L
      Quantification of nonosmotic sodium storage capacity following acute hypertonic saline infusion in healthy individuals.
      ,
      • Wouda RD
      • Dekker SEI
      • Reijm J
      • Olde Engberink RHG
      • Vogt L
      Effects of Water Loading on Observed and Predicted Plasma Sodium, and Fluid and Urine Cation Excretion in Healthy Individuals.
      ].
      In the last decade, experimental and clinical studies demonstrated that sodium can accumulate in supraphysiological concentrations in the skin and other tissues without concurrent water retention, which is facilitated by negatively charged glycosaminoglycans (GAGs) [
      • Olde Engberink RHG
      • Selvarajah V
      Clinical impact of tissue sodium storage.
      ,
      • Titze J
      • Shakibaei M
      • Schafflhuber M
      • Schulze-Tanzil G
      • Porst M
      • Schwind KH
      • et al.
      Glycosaminoglycan polymerization may enable osmotically inactive Na+ storage in the skin.
      ,
      • Fischereder M
      • Michalke B
      • Schmockel E
      • Habicht A
      • Kunisch R
      • Pavelic I
      • et al.
      Sodium storage in human tissues is mediated by glycosaminoglycan expression.
      ,
      • Oppelaar JJ
      • Rorije NMG
      • Olde Engberink RHG
      • Chahid Y
      • van Vlies N
      • Verberne HJ
      • et al.
      Perturbed body fluid distribution and osmoregulation in response to high salt intake in patients with hereditary multiple exostoses.
      ]. Kopp et al. showed that tissue sodium accumulation is involved in the pathophysiology of plasma sodium disorders, since skin and muscle sodium content significantly increase during hypernatremia and normalize after correction of the plasma sodium concentration [
      • Kopp C
      • Linz P
      • Hammon M
      • Schöfl C
      • Grauer M
      • Eckardt K-U
      • et al.
      Seeing the sodium in a patient with hypernatremia.
      ]. Although the 95% confidence interval of Edelman's y-intercept is not documented, it has been hypothesized, by Edelman himself and colleagues in later publications, that the above mentioned y-intercept of -25.6 reflects the fixed fraction of isotopically measured Nae+Ke that is not dissolved in water, possibly representing tissue sodium storage [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ,
      • Sterns RH.
      Formulas for fixing serum sodium: curb your enthusiasm.
      ,
      • Nguyen MK
      • Kurtz I.
      Determinants of plasma water sodium concentration as reflected in the Edelman equation: role of osmotic and Gibbs-Donnan equilibrium.
      ].
      In this post-hoc analysis of data from the Edelman study, we reassessed the association between plasma sodium and total body cation concentration per TBW by taking into account clinical characteristics that are involved in tissue sodium storage capacity.

      2. Methods

      2.1 Data extraction and preparation

      Data were extracted from the original publication by Edelman et al. [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ] and verified by two authors (MV and ROE). For this post-hoc analysis, all subjects with complete data on plasma water sodium concentration ([Na+]pw) and exchangeable cations per total body water ((Nae + Ke)/TBW) were selected. Serial measurements were identified in three subjects, but we only included the last measurement of each subject in the analysis. In the original paper, Edelman and colleagues used [Na+]pw expressed as mmol/L of serum water instead of the clinically used plasma sodium concentration [Na+]p to correct for potential alterations in lipid or lipoprotein contents, which may affect plasma sodium concentrations. To facilitate clinical interpretation of our results, we calculated [Na+]p by multiplying [Na+]pw * 0.93, whereas only [Na+]pw was used for our analysis. The amount of exchangeable cations and TBW values were obtained by Edelman et al. by using isotopes for sodium (Na24), potassium (K42) and water (D2O). For every subject, the obtained values were published in the original paper. We recalculated (Nae + Ke)/TBW using the provided data for Nae, Ke and TBW. We used ordinary least squares regression, as method for our linear regression, to explore the association between [Na+]pw and (Nae + Ke)/TBW in the complete group.

      2.2 Statistical analysis in subgroups

      The data was split in subgroups of patient characteristics that were previously shown to affect tissue sodium storage: age, sex, weight, TBW, the presence of edema, heart failure or kidney failure [
      • Kopp C
      • Linz P
      • Hammon M
      • Schöfl C
      • Grauer M
      • Eckardt K-U
      • et al.
      Seeing the sodium in a patient with hypernatremia.
      ,
      • Hammon M
      • Grossmann S
      • Linz P
      • Seuss H
      • Hammon R
      • Rosenhauer D
      • et al.
      3 Tesla (23)Na Magnetic Resonance Imaging During Acute Kidney Injury.
      ,
      • Kopp C
      • Linz P
      • Dahlmann A
      • Hammon M
      • Jantsch J
      • Muller DN
      • et al.
      23Na magnetic resonance imaging-determined tissue sodium in healthy subjects and hypertensive patients.
      ,
      • Wang P
      • Deger MS
      • Kang H
      • Ikizler TA
      • Titze J
      • Gore JC.
      Sex differences in sodium deposition in human muscle and skin.
      ,
      • Roth S
      • Markó L
      • Birukov A
      • Hennemuth A
      • Kühnen P
      • Jones A
      • et al.
      Tissue Sodium Content and Arterial Hypertension in Obese Adolescents.
      ,
      • Hammon M
      • Grossmann S
      • Linz P
      • Kopp C
      • Dahlmann A
      • Garlichs C
      • et al.
      23Na Magnetic Resonance Imaging of the Lower Leg of Acute Heart Failure Patients during Diuretic Treatment.
      ,
      • Schneider MP
      • Raff U
      • Kopp C
      • Scheppach JB
      • Toncar S
      • Wanner C
      • et al.
      Skin Sodium Concentration Correlates with Left Ventricular Hypertrophy in CKD.
      ]. For continuous variables, we used a median split to create subgroups. First, we explored the relationship between [Na+]pw and (Nae + Ke)/TBW in these subgroups. Differences in slope were analyzed by exploring the significance of the interaction term between (Nae + Ke)/TBW and subgroup category variables. For age and weight, continuous variables were used in the interaction analysis. Since edema and sex are important contributors to weight of the subjects, we also tested if differences in slopes and intercept for weight and edema were present in subgroups of males or females. If the interaction between (Nae + Ke)/TBW and subgroup covariate was not significant, we tested for differences in y-intercept values among regression lines by fitting a linear model without the subgroup interaction term, but including the subgroup as independent variable. Using piecewise regression (also known as segmented regression), we tested for differences in slope and y-intercept for the regression between [Na+]pw and increasing values of (Nae + Ke)/TBW, without adjustment for subgroup characteristics. Second, following up on the original Edelman analysis, we performed a sensitivity analysis in which we repeated the linear regression analysis in all subgroups by using attenuated correction of the correlation coefficient (r’). In the original paper, Edelman et al. used attenuated correction as a method to correct for potential measurement errors in x- and y- variables. The formulas for this correction method are provided in the original paper. This method uses standard deviations of reproducibility, which Edelman and colleagues obtained from observations on pooled specimens or by serial estimations in given subjects. Identical to Edelman's analysis, we used the standard deviations of reproducibility provided in the paper for our post-hoc analysis: [Na+]pw = 1.05 mEq, Nae = 65 mEq, Ke = 57 mEq, TBW = 0.71 liter. With r’ and standard deviations (SD) of [Na+]pw and (Nae + Ke)/TBW, we calculated slope [r’ * SD[Na+)pwSD(Nae+Ke)TBW] and y-intercept [mean [Na+]pw – (slope * mean Nae+KeTBW)]. The derived slopes and y-intercepts for attenuated regression analysis in different subgroups were compared by using an independent samples t-test.

      2.3 Improvement of the Edelman equation

      We selected patient characteristics that affected the Edelman slope or y-intercept in our subgroup analyses and used the Chi-square test to analyze whether inclusion of these patient characteristics in the original Edelman equation significantly improved the association between [Na+]pw and exchangeable cations per TBW. To test the influence of a variable on the y-intercept, we added the patient characteristic as independent variable to the formula. To test the influence of a variable on the slope of the relationship, we added the characteristic as interaction term to the formula. For this analysis we used continuous data for continuous independent variables.

      2.4 Data presentation

      Continuous data are shown as mean and SD for data following a normal distribution and median with interquartile range (IQR) for non-parametric variables. Slopes and y-intercept are reported with 95% confidence intervals (CI). All statistical analyses were performed using RStudio version 3.6.1 (R Foundation for Statistical Computing, Vienna, Austria). Figures were also acquired using RStudio. A p-value <0.05 was considered to indicate statistical significance.

      3. Results

      3.1 Study population

      Edelman reported data for 98 observations in 95 patients, which are characterized by large heterogeneity in clinical and metabolic status [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ]. Complete data on [Na+]pw, total body cation content and TBW was available for 85 measurements in 82 patients (Fig. 1).
      Fig. 1
      Fig. 1Flow chart of inclusion. The original Edelman study reported data for 98 individual measurements
      [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ]
      . After exclusion of measurements with missing data for (Nae + Ke)/TBW and identification of serial measurements of which the first measurement was excluded, 82 observations remained for our analysis.
      Clinical characteristics of all included subjects are depicted in Table 1. Two third of the subjects were male (n=57). [Na+]pw ranged from 111.2 to 161.5 mmol/L plasma water, with a mean of 140.2 (9.9) mmol/L plasma water. This corresponds to [Na+]p levels ranging from 103.4 to 150.2 mmol/L, with a mean [Na+]p concentration of 130.4 (9.2) mmol/L. Of all subjects, 67% had any degree of edema (transudate >0) and 37% of participants suffered from heart disease. A precise overview of all medical diagnoses of included subjects is displayed in supplemental table 1. Median weight of males was 62.4 (53.5 - 77.2) kg and the median weight of females was 51.8 (40.0 - 60.9) kg.
      Table 1Clinical characteristics of study subjects.
      Characteristicn=82
      Age (years)57 (15)
      Male n (%)57 (69.5)
      Weight (kg)59.8 (51.1-70.7)
      Disease n (%)
      Heart disease30 (36.6)
      Liver disease24 (29.3)
      Neurological disease7 (8.5)
      Kidney disease4 (4.9)
      Lung disease4 (4.9)
      Gastrointestinal disease3 (3.7)
      Miscellaneous10 (12.2)
      Transudate n (%)
      027 (32.9)
      +111 (13.4)
      +217 (20.7)
      +311 (13.4)
      +416 (19.5)
      Plasma water sodium (mmol/l water)140.2 (9.9)
      Plasma sodium (mmol/l)130.4 (9.2)
      Serum potassium (mmol/l)4.3 (0.8)
      Serum chloride (mmol/l)97.0 (10.2)
      Serum bicarbonate (mmol/l)25.9 (5.1)
      Serum osmolarity (mOsm/kg)254.4 (17.6)
      Arterial pH7.4 (0.1)
      Data are depicted as mean (SD) or median (IQR) depending on data distribution. Data were obtained from Edelman et al[6]. Edelman's corrected serum osmolarity is displayed for serum osmolarity, for which Edelman corrected measured osmolarity for the osmotic contributions of glucose and urea by osmolmeasured[glucose]plasma18[nonproteinnitrogen]plasma2.8. Weight and serum osmolarity were available for 81 subjects. Serum chloride was available for 52 subjects. Serum bicarbonate was available for 67 subjects. Arterial pH was available for 66 subjects. Transudate is an ordinal variable, with 0 for no transudate.
      To evaluate the influence of body composition, we calculated body water percentage by dividing TBW (measured by isotopes) by body weight. In males, the mean body water percentage was 61% of body weight and 36 males (63%) had a body water percentage within the normal range of 55-70% [
      • Akram M
      • Hamid A.
      A comprehensive review on water balance.
      ]. In females, the mean water percentage was 54.3% of body weight and 60% of all included females had a water percentage within the normal range of 45-60% [
      • Akram M
      • Hamid A.
      A comprehensive review on water balance.
      ]. Since 60-75% of total body potassium is found in muscles, the isotopically measured exchangeable potassium concentration can be considered a proxy for muscle mass. We observed that in the subjects, Ke was significantly correlated with body weight in the complete group and males and females separately (r=0.82, r=0.81 and r=0.81 respectively). The mean potassium content per total body weight was 34.9 mmol/kg in men, which is considerably lower than the normal range for healthy men (50-55 mmol/kg)[31]. In females, the mean potassium content of 31.8 mmol/kg was also lower than the normal range for healthy females of 40-45 mmol/kg [
      • Navarro MP
      • Vaquero MP.
      POTASSIUM | Physiology.
      ].

      3.2 Recalculation of the Edelman equation results in different slope and y-intercept

      Recalculation of (Nae+Ke)/TBW resulted in slight differences with the reported values (median difference 0.04 mmol/L), although in four subjects differences were notably larger (0.83, 0.84, 1.98 and 2.39 mmol/L). With recalculated (Nae+Ke)/TBW and all observations with complete data for [Na+]pw, exchangeable cations and TBW (n=85), we performed ordinary least squares linear regression which reproduced previous recalculations of the slope (0.93 [95% CI, 0.80 to 1.07]) and y-intercept (1.4 [95% CI, -18.8 to 21.6]) by Nguyen et al. [
      • Nguyen MK
      • Nguyen D-S
      • Nguyen M-K.
      Osmotically inactive sodium and potassium storage: lessons learned from the Edelman and Boling data.
      ]. When serial measurements were excluded, our regression resulted in a slightly different slope (0.94 [95% CI, 0.79 to 1.08]) and y-intercept (0.8 [95% CI, -20.6 to 22.2]). Attenuated correction resulted in a replication of Edelman's corrected regression coefficient (r’) of 0.92. Yet, recalculation of slope and y-intercept resulted in a different slope (1.04 [95% CI, 0.89 to 1.19]) and y-intercept of (-15.0 [95% CI, -36.7 to 6.6). For an unknown reason, observation 32 and 80a were not included in the original regression figure (Figure 7) published by Edelman et al. [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ]. Exclusion of these two observations had only minor effect on the attenuated slope (1.05 [95% CI, 0.90 to 1.19]) and no effect on the y-intercept. The differences between reported and recalculated (Nae+ Ke)/TBW also had a minor effect on the attenuated corrected slope (1.05 [95% CI, 0.90 to 1.19]) but not on the y-intercept. Exclusion of the second serial measurement, instead of the first, in three subjects with duplicate measurements, resulted in an attenuated corrected slope of 1.01 (95% CI, 0.85 to 1.16) and y-intercept of -9.3 (95% CI, -32.9 to 14.4). However this also does not explain the differences with the reported slope and y-intercept by Edelman et al. [
      • Edelman IS
      • Leibman J
      • O'Meara MP
      • Birkenfeld LW.
      Interrelations between serum sodium concentration, serum osmolarity and total exchangeable sodium, total exchangeable potassium and total body water.
      ]

      3.3 Coefficients are significantly affected by weight, edema and plasma sodium concentration

      To determine if individual patient characteristics could alter the relation between [Na+]pw and total body cation concentration per TBW, we calculated the slope and y-intercept for various subgroups which are depicted in Table 2.
      Table 2Recalculated regression coefficients for subgroups without attenuated correction.
      Subgroup (IQR)nR2Slope (95% CI)p for slopey-intercept (95% CI)p for y-intercept
      SexMale570.650.89 (0.72-1.07)0.526.4 (-19.7-32.4)0.06
      Female250.731.00 (0.75-1.25)-6.5 (-44.5-31.4)
      Age (years)≤ 58.5

      (40.-51.0)
      410.651.01 (0.77-1.25)0.87-9.6 (-45.1-25.9)0.84
      > 58.5

      (64.0-74.0)
      410.700.88 (0.70-1.07)8.6 (-18.8-36.0)
      Weight (kg)≤ 59.8

      (43.3 – 55.2)
      420.771.10 (0.91-1.29)0.01-23.8 (-51.7-4.1)NA
      > 59.8

      (64.9 – 83.6)
      390.550.72 (0.51-0.93)*34.3 (2.5-66.1)*
      EdemaNo270.510.69 (0.43-0.96)0.0338.1 (-3.1-79.2)NA
      Yes550.711.04 (0.86-1.23)-14.9 (-41.7-11.8)
      TBW (L)≤ 35.1410.690.99 (0.78-1.21)0.08-7.1 (-38.9-24.8)0.50
      > 35.1410.650.89 (0.68-1.09)8.0 (-22.6-38.6)
      Heart diseaseNo520.721.00 (0.83-1.17)0.36-9.0 (-34.8-16.9)0.28
      Yes300.620.87 (0.61-1.12)12.4 (-26.0-50.8)
      Kidney diseaseNo780.640.91 (0.76-1.06)0.514.8 (-18.1-27.7)0.31
      Yes40.921.08 (0.28-1.87)-22.2 (-136.6-92.2)
      Regression coefficients for the relation between [Na+]pw and (Nae+Ke)/TBW. Median split was used to create subgroups for continuous variables, but continuous data was used in the analysis. Edema was defined as transudate ≥1. R2 is the adjusted r squared from the ordinary least square regression. P for slope term reflects significance of the variable interaction term. P for y-intercept reflects significance for the variable as covariate in regression, if slope was not significant. *p<0.05 versus the coefficient for the complete group regression analysis. IQR, interquartile range. NA, x003Dnot applicable.
      In our main (non-attenuated corrected) analysis, we found that the slope of the regression line was significantly affected by weight (pinteraction 0.01) and the presence of edema (pinteraction 0.03). Sex did not significantly affect the slope. The modifying effect of weight was most pronounced in males, with a significant steeper slope for males with a weight below the median of 62.4 kg. For females no differences in weight categories was observed (pinteraction=0.33) (Fig. 2).
      Fig. 2
      Fig. 2Different slopes between subgroups are observed. A. Non-attenuated corrected linear regression graph for the complete group. B. Non-attenuated corrected regression graph for subgroups based on edema. C. Non-attenuated corrected linear regression graph for males in different weight categories based on median split. We observed a steeper slope for males with weight <62.4 kg when compared to males with weight >62.4 kg (pinteraction = <0.01) together with a more negative y-intercept. D. Non-attenuated corrected linear regression graph for females in different weight categories based on median split. No effect interaction between weight and (Nae + Ke)/TBW could be observed for females (pinteraction = 0.33). The shaded gray areas represent the confidence intervals of the regression lines. Analysis are performed with [Na]pw as dependent variable, [Na]p on the y-axis is for clinical interpretation only.
      Additional analyses for differences in y-intercepts (with subgroup added as independent variable) did not reveal the presence of parallel regression lines with significant differences in y-intercept. In our sensitivity analysis, attenuated correction of the regression coefficient resulted in similar data: we found significantly different slopes and y-intercepts for subgroups of weight and borderline significance for the presence of edema (Supplemental table 2).
      Piecewise regression analysis showed a significant decrease in slope in the Edelman regression line above 149.0 mmol/L (Nae+Ke)/TBW, corresponding with a plasma water sodium level of 142.6 mmol/L plasma water (i.e. 132.7 mmol/L [Na+]p) (Fig. 3). A significant difference in slopes remains present when weight and edema are included as covariates in our piecewise regression model (breakpoint at 148.9 mmol/L (Nae+Ke)/TBW for a subject of 60 kg without edema this corresponds to 143.3 mmol/L [Na+]pw and 133.3 mmol/L plasma sodium, p= <0.01).
      Fig. 3
      Fig. 3Piecewise regression shows a significant decrease in slope. Regression graph for [Na+]pw and (Nae + Ke)/TBW. Piecewise regression shows a significant decrease in slope above 149.0 mmol/L (Nae + Ke)/TBW [95% CI, 143.4 - 154.7] (slope of 1.25 x0025vs 0.58x0025, p<0.01). The analysis is performed with [Na]pw as dependent variable, [Na]p on the y-axis is for clinical interpretation only.
      In a sensitivity analysis based on attenuated correction analysis, the outcome of the piecewise regression analysis was similar (Supplemental Table 2). In another sensitivity analysis the change in slope remained significant after exclusion of two apparent outliers.

      3.4 Inclusion of weight significantly improves the estimated relationship in the Edelman equation

      We then tested whether the patient characteristics that were identified to affect the association between [Na+]pw and (Nae+Ke/TBW) can potentially improve the Edelman equation. Chi-square analysis showed that adding a continuous interaction term for weight significantly improved this association (Table 3). Adding an interaction term for edema did not improve the relationship (p=0.09). When the interaction term of edema was added to the equation including the interaction of weight, the adjusted R2 improved significantly (p=0.02). The addition of an interaction term for sex on top of the interaction terms for weight and edema further improved the adjusted R2 (p=0.04). In the subgroup analysis that includes solely males, the original Edelman equation could only be improved by addition of the weight interaction term. In a sensitivity analysis based on attenuated correction analysis we found similar results.
      Table 3Analysis of Variance Table to compare the original Edelman equation with models including interaction terms for subgroups.
      GroupModel AModel BSum of squaresDfFp-value
      ModelR2ModelR2
      Complete group (n=82)Original Edelman
      The original Edelman for the complete group (with complete data for [Na+]pw, exchangeable cations and weight) was expressed by [Na+]pw = 1.04 + 0.94*(Nae+Ke)/TBW.
      0.68A + interaction term for weight0.70228.3123.89760.02
      Complete group (n=82)Original Edelman0.68A + interaction term for edema0.69151.5722.50240.09
      Complete group (n=82)Original Edelman + interaction term for weight0.70A + interaction term for edema0.73238.3324.43140.02
      Complete group (n=82)Original Edelman + interaction term for weight AND Edema0.73A + interaction term for sex0.74176.5623.50180.04
      Males only (n=57)Original Edelman
      The original Edelman equation for the males only group (with complete data for [Na+]pw, exchangeable cations and weight) is expressed by [Na+]pw = 6.50 + 0.89*(Nae+Ke)/TBW, alternative models are depicted in supplementary appendix A. R2, adjusted r-squared from linear regression. Df, degrees of freedom. F, F-statistic.
      0.66A + Interaction term for weight0.72402.8527.5529<0.01
      Males only (n=57)Original Edelman + interaction term for weight0.72A + Interaction term for edema0.7393.68221.81120.17
      Analysis of Variance Table showing the results of the comparison between the original Edelman equation (without attenuated correction) and adding continuous interaction terms for subgroups.
      low asterisk The original Edelman for the complete group (with complete data for [Na+]pw, exchangeable cations and weight) was expressed by [Na+]pw = 1.04 + 0.94*(Nae+Ke)/TBW.
      § The original Edelman equation for the males only group (with complete data for [Na+]pw, exchangeable cations and weight) is expressed by [Na+]pw = 6.50 + 0.89*(Nae+Ke)/TBW, alternative models are depicted in supplementary appendix A. R2, adjusted r-squared from linear regression. Df, degrees of freedom. F, F-statistic.
      Taking into account the (Nae+Ke/TBW) of 149 mmol/L (i.e. 132.7 mmol/L [Na+]p) breakpoint in regression resulted in a significantly improved adjusted R2 of 0.70, whereas adjusted R2 was 0.67 without taking into account this breakpoint (p=0.01).

      4. Discussion

      In this post-hoc analysis of the original data of the Edelman study, we show that the association between plasma water sodium concentration and exchangeable cation concentration per TBW is affected by body weight, the presence of edema, and is different for subjects with profound hyponatremia and subjects with normal or high plasma sodium concentration. These patient characteristics have been previously related to tissue sodium storage.
      In recent years, 23Na-MRI studies have identified clinical factors associated with increased skin sodium content. Normotensive obese adolescents were found to have a 23% higher skin sodium content when compared to their age-matched normotensive controls [
      • Roth S
      • Markó L
      • Birukov A
      • Hennemuth A
      • Kühnen P
      • Jones A
      • et al.
      Tissue Sodium Content and Arterial Hypertension in Obese Adolescents.
      ]. Also, skin sodium content in males was, depending on age, 13-25% higher than in females [
      • Kopp C
      • Linz P
      • Dahlmann A
      • Hammon M
      • Jantsch J
      • Muller DN
      • et al.
      23Na magnetic resonance imaging-determined tissue sodium in healthy subjects and hypertensive patients.
      ]. Furthermore, heart failure patients with peripheral edema had significant increased skin sodium content, which was mobilized directly after loop diuretic therapy [
      • Hammon M
      • Grossmann S
      • Linz P
      • Kopp C
      • Dahlmann A
      • Garlichs C
      • et al.
      23Na Magnetic Resonance Imaging of the Lower Leg of Acute Heart Failure Patients during Diuretic Treatment.
      ].
      In our post-hoc analysis, we show that weight and edema significantly influence the association between [Na+]pw and (Nae+Ke/TBW). When evaluating the effect of weight in the Edelman population, however, it is important to realize that no data on body composition were published and that overall mean body weight was rather low. It remains therefore unclear whether the study participants with higher weight represent persons with increased length or higher fat mass, or vice versa whether lower weight is the result of short stature or muscle wasting. Since fat, muscle and interstitial tissue have different sodium accumulating capabilities, more information about body weight composition is important for drawing final conclusions. Considering the reported low mean body weight and normal ratio of body weight-to-total body water, the Edelman population is unlikely to be overweight. This is supported by a mean estimated BMI of 21-23 kg/m2 assuming that patients had an average length that is reported for persons born around 1900 [

      Average height of men by year of birth. Online at OurWorldInData.org. Retrieved from: https://ourworldindata.org/grapher/average-height-of-men?tab=chart&country=∼x223CUSAJanuary 06, 2022.

      ]. The low levels of total body potassium content per kg body weight are likely to reflect low muscle mass. The increase in slope that was seen in the low weight and edema subgroups may thus represent a relative increase in extracellular volume and exchangeable cations due to edema or a decrease in intracellular volume as a result of muscle wasting. This hypothesis is in line with previous 23Na-MRI data demonstrating that muscle wasting and edema is associated with higher tissue sodium content [
      • Hammon M
      • Grossmann S
      • Linz P
      • Kopp C
      • Dahlmann A
      • Garlichs C
      • et al.
      23Na Magnetic Resonance Imaging of the Lower Leg of Acute Heart Failure Patients during Diuretic Treatment.
      ,
      • Gerlach DA
      • Schopen K
      • Linz P
      • Johannes B
      • Titze J
      • Zange J
      • et al.
      Atrophy of calf muscles by unloading results in an increase of tissue sodium concentration and fat fraction decrease: a 23Na MRI physiology study.
      ].
      Plasma sodium concentrations itself may also affect the sodium storage compartment. Kopp et al. demonstrated significant muscle sodium accumulation in a hypernatremic patient (not paralleled by commensurate water retention), which decreased after normalization of plasma sodium concentration [
      • Kopp C
      • Linz P
      • Hammon M
      • Schöfl C
      • Grauer M
      • Eckardt K-U
      • et al.
      Seeing the sodium in a patient with hypernatremia.
      ]. In line with this finding, we demonstrated that the Edelman association is different for low and high plasma sodium concentration. We observed that above 142.6 mmol/L plasma water (i.e. 132.7 mmol/L [Na+]p), plasma sodium concentration increased less for a similar increase in total body cation content per liter body water, which is in line with the concept of tissue sodium storage. On the contrary, the steeper slope at lower [Na+]pw concentration levels might represent sodium release from the dynamic sodium compartment. Such a dynamic protection mechanism against osmotic stress is also observed in aquatic species that modulate the sulfation grade of glycosaminoglycans in response to changes in environmental sodium concentrations to facilitate osmotic inactivation of sodium [
      • Nader HB
      • Medeiros MGL
      • Paiva J
      • Paiva VMP
      • Jerônimo SMB
      • Ferreira TMPC
      • et al.
      A correlation between the sulfated glycosaminoglycan concentration and degree of salinity of the “habitat” in fifteen species of the classes Crustacea, Pelecypoda and Gastropoda.
      ]. Likewise, increased tissue sodium content has been described in rats and humans after high salt intake, as well as tissue sodium release after salt restriction [
      • Braconnier P
      • Milani B
      • Loncle N
      • Lourenco JM
      • Brito W
      • Delacoste J
      • et al.
      Short-term changes in dietary sodium intake influence sweat sodium concentration and muscle sodium content in healthy individuals.
      ,
      • Schafflhuber M
      • Volpi N
      • Dahlmann A
      • Hilgers KF
      • Maccari F
      • Dietsch P
      • et al.
      Mobilization of osmotically inactive Na+ by growth and by dietary salt restriction in rats.
      ,
      • Titze J
      • Lang R
      • Ilies C
      • Schwind KH
      • Kirsch KA
      • Dietsch P
      • et al.
      Osmotically inactive skin Na+ storage in rats.
      ]. The dynamics of tissue sodium accumulation is further supported by the observation that healthy individuals can mobilize tissue sodium within 30 minutes after an hypotonic fluid load [
      • Wouda RD
      • Dekker SEI
      • Reijm J
      • Olde Engberink RHG
      • Vogt L
      Effects of Water Loading on Observed and Predicted Plasma Sodium, and Fluid and Urine Cation Excretion in Healthy Individuals.
      ].
      The influence of tissue sodium accumulation on the Edelman equation is subject of debate. Mathematical studies have extensively described the theoretical background of the slope and y-intercept of Edelman's equation and concluded that despite dynamic tissue sodium accumulation, Edelman's equation adequately predicts [Na+]pw [
      • Nguyen MK
      • Nguyen D-S
      • Nguyen M-K.
      Osmotically inactive sodium and potassium storage: lessons learned from the Edelman and Boling data.
      ,
      • Nguyen MK
      • Kurtz I.
      Determinants of plasma water sodium concentration as reflected in the Edelman equation: role of osmotic and Gibbs-Donnan equilibrium.
      ]. This conclusion is in contrast with experimental and clinical studies into human sodium balance demonstrating large discrepancies between formula-based expected versus clinically measured plasma sodium concentrations [
      • Lindner G
      • Schwarz C
      • Kneidinger N
      • Kramer L
      • Oberbauer R
      • Druml W.
      Can we really predict the change in serum sodium levels? An analysis of currently proposed formulae in hypernatraemic patients.
      ,
      • Hanna RM
      • Yang W-T
      • Lopez EA
      • Riad JN
      • Wilson J.
      The utility and accuracy of four equations in predicting sodium levels in dysnatremic patients.
      ,
      • Liamis G
      • Kalogirou M
      • Saugos V
      • Elisaf M.
      Therapeutic approach in patients with dysnatraemias.
      ,
      • Carroll HJ
      • Gotterer R
      • Altshuler B.
      Exchangeable Sodium, Body Potassium, and Body Water in Previously Edematous Cardiac Patients.
      ]. A retrospective cohort study in hypernatremic intensive care patients demonstrated that none of the available formulas could predict the course of serum sodium in the individual patient [
      • Lindner G
      • Schwarz C
      • Kneidinger N
      • Kramer L
      • Oberbauer R
      • Druml W.
      Can we really predict the change in serum sodium levels? An analysis of currently proposed formulae in hypernatraemic patients.
      ]. In a volume depleted subgroup of 15 hypo- and hypernatremic patients the observed plasma sodium concentrations after 24 hours of treatment were on average 5.6 mmol/L higher than expected according to the Adrogue-Madias formula [
      • Liamis G
      • Kalogirou M
      • Saugos V
      • Elisaf M.
      Therapeutic approach in patients with dysnatraemias.
      ]. Such large inconsistencies are clinically relevant and could impact treatment and outcome.
      We explored whether our findings could help to understand the clinical limitations of the Edelman equation. We demonstrated that inclusion of weight, edema and the piecewise regression breakpoint significantly improved the R2 of the Edelman equation. If we use the data from the male participants of the Edelman study, we can calculate expected [Na+]pw with a formula with interaction terms for weight and edema and with the original formula without these clinical parameters. When interaction terms of weight and edema are included, mean expected [Na+]pw is 8.0 mmol/L lower than when calculations would be based on the original formula without taking these clinical parameters into account. Also, if a hyponatremic patient with a plasma sodium concentration of 120 mmol/L and a body weight of 70kg (corresponding to approximately 42L TBW) is treated with hypertonic saline (3% NaCl, 513 mmol/L) the calculated expected change in serum sodium is 25% higher when the adjusted slope of 1.25 is used in the Adrogue-Madias formula (slope * Infusate(Na+)serum(Na+)Totalbodywater+1) instead of the assumed slope of 1.x0025 Future studies are needed to investigate whether inclusion of these variables have the potential to improve the treatment of dysnatremias.
      In our study we were unable to reproduce the exact slope and y-intercept reported by Edelman, although the corrected regression coefficient (r’=0.92) could be replicated. Selection of our study group did not influence recalculation of slope and y-intercept. The use of more modern statistical analysis programs and rounding of coefficients might play a role, however, since multiple analysis steps are included this is difficult to establish. With Passing-Bablok regression analysis, a modern regression technique which - like attenuated correction which was used in the original Edelman paper - take errors in x and y variables into account, Edelman's slope and y-intercept could also not be exactly replicated [
      • Nguyen MK
      • Nguyen D-S
      • Nguyen M-K.
      Osmotically inactive sodium and potassium storage: lessons learned from the Edelman and Boling data.
      ]. Considering the minor discrepancies and the robustness of our data in multiple sensitivity analyses, these inconsistencies are unlikely to influence our conclusions.
      Limitations of our post-hoc analysis are closely related to the limitations of the original Edelman study [
      • Ring T.
      Quantitative analysis of the dysnatremias.
      ]. The main limitation is the lack of serial measurement in the Edelman population. Our data is thus limited to analyses among subjects and not within subjects, which would better approximate the effect of infusion strategies in clinical practice. Also, included subjects were highly heterogeneous with a wide spectrum of significant comorbidities reported. Noticeably, only one subject with severe hypernatremia (plasma sodium >150mmol/L) was included and the majority of included subjects had hyponatremia. Our results may therefore limitedly translate to hypernatremic patients. Furthermore, the original Edelman paper did not include information about medical treatment of included patients. Hence, we are not able to investigate the influence of drugs on the Edelman association. Since our post-hoc analysis is lacking information about tissue sodium content, we cannot prove the role of tissue sodium content. Further studies into developing formulas for treatment of dysnatremias should include serial measurements of multiple patients to investigate the effect of tissue sodium on the course of serum sodium concentrations.

      5. Conclusion

      We demonstrate that body weight, edema and plasma sodium concentration are clinical parameters that significantly affect the Edelman equation coefficients. The absence of these parameters in Edelman-derived formulas is likely to contribute to the inaccurateness of these formulas. Future studies with serial measurements are needed to further understand sodium physiology and assess whether these and other patient characteristics could help to predict the course of dysnatremias, optimize treatment.

      Funding

      This work was supported by the Dutch Kidney Foundation (grant number 19OP016 to R.H.G. Olde Engberink).

      Declaration of interest

      None.

      Appendix. Supplementary materials

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