Eur J Appl Physiol (2007) 101:133–142 DOI 10.1007/s00421-007-0488-6
ORIGINAL ARTICLE
A new method to distinguish the hyponatremia of electrolyte loss from that due to pure solvent changes E. Bartoli Æ L. Castello Æ L. Bergamasco Æ P. P. Sainaghi
Accepted: 30 April 2007 / Published online: 5 June 2007 Springer-Verlag 2007
Abstract Estimates of solute and solvent changes during electrolyte abnormalities are valid only when either total body water (TBW) or solute content do not change, while it cannot be established which one of these is altered. The present paper provides a method capable of distinguishing these two different conditions. When only solvent changes, the respective concentration ratios of plasma (P) solutes PCl/ PNa, POAN/PNa, PCl/POAN (POAN = anions other than Cl) remain unchanged. Moreover, PNa1/PNa0 (the ratio of PNa during the derangement over the normal value, indicated by subfix 1 and 0, respectively) = PCl1/PCl0 = POAN1/POAN0. When these constraints are met, the abnormality is due only to a TBW change, which is easily calculated and corrected. When they are not met, the exact change in Na content is correctly calculated assuming no variation in TBW. These calculations could still be useful even in the presence of TBW modifications, where they represent minimum estimates of electrolyte losses. The formulas were validated by computer simulations generating true electrolyte concentrations, which were then used to back calculate the changes in their contents and extra/intra-cellular volumes. Since the predicted results were significantly correlated with the true data, the method was transferred to 24 patients with electrolyte disturbances, who met the above constraints. The calculated volume changes were significantly correlated
E. Bartoli (&) L. Castello L. Bergamasco P. P. Sainaghi Chair of Internal Medicine, Dipartimento di Medicina Clinica e Sperimentale, Universita` degli Studi del Piemonte Orientale ‘‘A. Avogadro’’, Via Solaroli 17, 28100 Novara, Italy e-mail:
[email protected] P. P. Sainaghi e-mail:
[email protected]
with those obtained by body weight measurements (regression coefficient = 0.94, P < 0.0001), while the quantitative estimates of Na deficit predicted the PNa values measured after corrective treatment (P < 0.0001). This new method may prove valuable in diagnosing and treating electrolyte derangements. Keywords Hyponatremia Body fluids volumes Total body water Extracellular volume Sodium homeostasis
Introduction The quantitative evaluation of water and Na derangements was defined in pioneer studies (Edelman et al. 1958). The formulas proposed to estimate Na and water surfeit and deficit, though still valid, rely upon the assumption that either total body water (TBW) or body solutes are unchanged. In the former circumstance the variations in solute, in the latter the changes in solvent content can be exactly established. It is mathematically impossible to distinguish between these two different events which could generate identically deranged values of plasma Na concentration (PNa), such that only the history and the physical examination can determine the clinical judgment as to which alteration, or combination thereof, is present (Castello et al. 2005a). Even the assumption that either solutes or water are unchanged with respect to normal conditions is seldom true (Castello et al. 2005b; Bartoli et al. 2002). Therefore, any new method that could help distinguishing between Na and solvent losses or gains, and quantitate them, could represent a significant advance in diagnosis and treatment of electrolyte imbalance. It might turn particularly useful in the dangerous hyponatremia of
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marathon runners, where a prompt treatment is frequently necessary (Almond et al. 2005). The present paper describes such a new method.
Methods Model and calculations In agreement with the initial (Edelman et al. 1958) and more recent (Nguyen and Kurtz 2003) literature, it is assumed that all solutes contributing to osmotic pressure are represented by the exchangeable Na, K, and their accompanying anions. Therefore, the plasma osmolality in milliosmoles per kg of solvent (Posm mOsm/kg) is equal to 2PNa. This, in turn, is equal to that of cells, even though due to a K concentration in cellular water equal to that of Na in the extracellular water. It is also assumed that acute Na and water derangements are not attended by intracellular K content. The working model is portrayed in Figs. 1 and 2. Figure 1a, in the middle, portrays a simplified schematic representation of TBW and its compartments, the extra (ECV) and intracellular (ICV) volumes, with their normal electrolyte content and concentration. Figure 1b, on the right, depicts the changes due to a net water gain, while Fig. 1c, on the left, illustrates those occurring after a net solvent loss. It can be appreciated that changes in volume alone do not alter the respective ratios between the indi-
a
vidual species (Na, sodium, Cl, chloride, OAN, anions other than chloride) present in solution, such that they remain unmodified with respect to baseline after water subtraction or addition. Thus, there is indeed a way to establish whether TBW alone has changed, a condition that formerly could have been only assumed: it consists in computing the PCl/PNa, POAN/PNa, PCl/POAN ratios and to compare them with normal values: if not different, they establish the presence of volume change alone. Moreover, the ratios between concentrations at time 1 (during a derangement) over those measured at time 0 (normal conditions) are identical for each solutes species. Under these circumstances we can write: PNa0 TBW0 ¼ PNa1 TBW1 ;
ð1Þ
TBW1 ¼ TBW0 PNa0 =PNa1 ;
ð2Þ
where P indicates the plasma concentration. Similar equations, omitted for brevity, can be written for Cl and OAN. Consequently: PNa1 =PNa0 ¼ PCl1 =PCl0 ¼ POAN1 = POAN0 :
ð3Þ
The variation in solute content cannot be calculated unless TBW is held constant. Thus, up to the present time, there was no exact way to establish whether solvent or solute changed. However, the changes in solute concentrations lend themselves to the development of an empirical method,
TBW = 40 L
PNa0 = 140 PCl0 = 105 POAN0 = 35
PCl1/PCl0 = 1.14 PNa1/PNa0 = 1.14 POAN1/POAN0 = 1.14
c
PCl1/PCl0 = 0.89 PNa1/PNa0 = 0.89 POAN1/POAN0 = 0.89 E C V = 15 L
ICV = 25 L
TBW = 35.2 L PCl0/PNa0 = 0.75 POAN0/PCl0 = 0.33 POAN0/PNa0 = 0.25
PNa1 = 159 PCl1 = 119.3 POAN1 = 39.7
- 4.8 L E C V = 13 . 2 L
H2O
ICV = 22 L
PCl1/PNa1 = 0.75 POAN1/PCl1 = 0.33 POAN1/PNa1 = 0.25
Fig. 1 a In the centre depicts a normal condition of TBW (the large rectangle) subdivided into ECV, the minor rectangle, and ICV. The normal Na and Cl concentrations in mEq/l are shown, together with the respective solute ratios reported below. b On the right shows the effects of a gain of 4.8 l of water on the volume of the compartments as well as the respective solute ratios, and the ratios for each solute
123
TBW = 44,8 L
b
PNa1 = 125 PCl1 = 93.75 POAN1 = 31.25
+ 4 .8 L E C V = 1 6. 8 L
ICV = 28 L
PCl1/PNa1 = 0.75 POAN1/PCl1 = 0.33 POAN1/PNa1 = 0.25
between the deranged (subfix 1) and the normal (subfix 0) values. c On the left shows the effects of subtracting 4.8 l of solvent. The respective solute ratios remain equal to those of the normal conditions depicted in a, while the individual ratios reach exactly the same value for each solute in the conditions depicted in b and c
Eur J Appl Physiol (2007) 101:133–142
135
a
TBW = 40 L
PNa0 = 140 PCl0 = 105 POAN0 = 35
PCl1/PCl0 = 1.15 PNa1/PNa0 = 1.13 POAN1/POAN0 = 1.10
PCl1/PCl0 = 0.85 PNa1/PNa0 = 0.89 POAN1/POAN0 = 1.01 E C V = 15 L
c
ICV = 25 L
TBW = 40 L
TBW = 40 L PCl0/PNa0 = 0.75 POAN0/PCl0 = 0.33 POAN0/PNa0 = 0.25
PNa1 = 159 PCl1 = 120.8 POAN1 = 38.1
b
PNa1 = 125 PCl1 = 89.6 POAN1 = 35.4
Na + 760 mEq
- 600 mEq Cl
E CV = 1 8
ICV = 22
PCl1/PNa1 = 0.76 POAN1/PCl1 = 0.31 POAN1/PNa1 = 0.24
+ 600 mEq
- 500 mEq
OA N + 160 mEq
- 100 mEq
E C V = 12 L
ICV = 28 L
PCl1/PNa1 = 0.72 POAN1/PCl1 = 0.39 POAN1/PNa1 = 0.28
Fig. 2 ‘‘Side a’’ in the centre is identical to that of Fig. 1. In ‘‘side b’’, 600 mEq of Na are withdrawn, yielding the same sodium concentration as that shown in Fig. 1b, while the ECV contracts as compared to its expansion in Fig. 1b. The ICV is instead expanded to the same extent in Figs. 1b and 2b. Na is lost with 500 mEq Cl and 100 mEq OAN. The respective solute ratios change with respect to Fig. 1b, and the final to initial solute ratios are different for the
different solutes, except for Na, where it is the same with volume addition and equivalent solute subtraction. c Shows the addition of 760 mEq of NaCl, causing a rise in Na concentration identical to that observed in Fig. 1c. Na is added with 600 mEq Cl and 160 mEq OAN. Again the respective solute ratios are different from normal, while the final to initial solute ratios are different from those in Fig. 1c, except for Na
which allows rather accurate estimates of solute losses and gain under the constraint that TBW remains unchanged. In fact, under these circumstances the PCl/PNa, POAN/PNa, PCl/POAN ratios measured will differ from the same ratios obtained under normal conditions, and PNa1/PNa0 will be different from PCl1/PCl0 and from POAN1/POAN0, since solute losses matching exactly solvent losses are an extremely unlikely event. This is shown by Fig. 2b, which illustrates the changes occurring after a loss of Na without water capable of generating an identical Na concentration as that due to solvent gain in Fig. 1b. Figure 2c shows the changes that occur with a solute gain capable of causing a plasma Na concentration as that generated by solvent loss in Fig. 1c. The change in Na content is:
If the anions are lost unevenly, as almost inescapably happens, although their sum must be equal to the Na content, and their cumulative change must equal the Na change, their respective concentration ratios will be different from those measured at time zero. In turn, Na can be re-concentrated by any attending volume loss to become even equal or higher than its initial concentration. For any given value of PNa1, the following equations are valid:
DNa ¼ ðPNa0 PNa1 Þ TBW0 ;
to be used when TBW0 is not known;
ð4Þ
when TBW is unchanged. The change in anions must equal that of Na. Thus:
DTBW ¼ TBW0 ðPNa0 PNa1 Þ=PNa1 ;
ð6Þ
equivalent (as math derivation and results) to: DTBW ¼ TBW1 ðPNa0 PNa1 Þ=PNa0 ;
DNa ¼ ðPNa0 PNa1 Þ TBW0 ;
ð6bisÞ
ð4Þ
hence, DNa ¼ DCl þ DOAN ¼ ½ðPCl0 PCl1 Þ þ ðPOAN0 POAN1 Þ TBW0 :
ð5Þ
The rather chronic effect of anions bound to glycosaminoglycans (Tietze et al. 2003, Schafflhuber et al. 2007) does not interfere with the presently considered acute effects, as Eq. 5 deals only with fast exchangeable species.
PNa1 ¼ DNa=DTBW;
ð7Þ
this demonstrates that there is pure solute loss without water when the reciprocal solute ratios differ from normal and Eq. 7 does not yield the measured value of PCll and POAN1, when substituting DNa for the calculated DCl or DOAN.
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When TBW is unchanged, a loss of solutes will cause a change in the intracellular volume, assuming that its solutes will remain the same, computed as: ICV1 ¼ PNa0 ICV0 =PNa1 ;
ð8Þ
while ECV1 will be TBW0 ICV1 ;
ð9Þ
we shall call this ECV1 as ECVSH (due to water shift, SH, from cells to interstitium). For the same PNa1 the ICV1 will be the same during a change in TBW alone, as during Na loss. When the above conditions are not completely satisfied, the formula to calculate Na depletion (Eq. 4) will still give a minimum estimate, which could result clinically useful. This is based on the additional assumption that the intracellular solute contents do not change during acute derangement in Na and water homeostasis. Therefore, our method computes firstly the reciprocal ratios between the concentration of solutes, then the ratios between actual and normal solute concentrations: when the reciprocal ratios are unchanged with respect to normal, and actual to normal ratios are identical for all solutes, the water excess or deficit is calculated by Eqs. 6 or 6bis; instead, when the requirements for an exclusive NaCl loss are met, the change in Na content is given exactly by Eq. 4. Finally, sodium concentration is given by the extracellular normal content minus its loss, divided by the ECVSH, and so will the concentrations of chloride and other anions: PNa1 ¼ ðPNa0 ECV0 DNaÞ=ECVSH ;
ð10Þ
PCl1 and POAN1 will be computed with similar formulas. Any other condition not meeting the constraints discussed above will not yield exact values and will not be solved correctly with the mathematical equations herein discussed and presented. Materials
Patient studies We examined the data of 45 patients admitted to a General Internal Medicine ward with acute electrolyte derangements. To be included the patients must have had a change in either Na, or Cl or OAN plasma concentrations >10 mEq/l and the initial measurement of body weight (BW1), that must have been available both on admission to the ward and at the end of treatment. We applied to each of these patients the above formulas to compute their body water volumes and amounts of electrolytes residual and lost. The calculations were based exclusively on PNa and BW measured, and on the assumed normal values shown in Fig. 1a (Castello et al. 2005a). The validation of the calculation with the new method on patients can be accomplished only by checking the accuracy of predictions during the correction of the disorders. This was done by re-measuring PNa at a time, during correction, when Na and water balance were known. These concentration are indicated by the subfix 2. Since the correction was performed by adding or subtracting the losses or gains calculated with the formulas of the new method, the discrepancy between measured and predicted values was considered an estimate of the reliability of the methods. The subfix 0 indicates then the normal situation, where values are assumed based upon normal electrolyte concentrations and content. Subfix 1 refers to the time of admission to hospital, when the altered concentrations and BW were measured, subfix 2 to any time point during correction, when plasma concentrations and body weight were re-measured. The Na and water balance between times 1 and 2 was determined by measuring the urinary outputs and i.v. infusates. During treatment up to the time 2 of measurement: Posm2 ¼ ðPosm0 TBW0 2 DNa 2 Na-lost þ 2 the added amounts of NaÞ= ðTBW1 þ H2 O-added H2 O-lostÞ;
ð11Þ
this is equivalent to: Posm2 ¼ ðPosm1 TBW1 þ 2 Na balanceÞ=
The study was performed on computer simulations and on patients. Computer simulations We devised a large series of simulated clinical conditions like those shown by Figs. 1 and 2, with different TBWs associated to variable ECV/ICV ratios, DTBWs and to a variety of losses of Na and its accompanying anions. A flow chart describing representative simulations based on Figs. 1 and 2 is reported in Appendix I.
123
ðTBW1 þ water balanceÞ;
ð11bisÞ
used when TBW0 is not known; ECV2 ¼ ðECV0 Posm0 2 DNa 2 Na-lost þ 2 the added amounts of NaÞ=Posm2 ;
ð12Þ
where Posm2 is computed from (11); in turn, this is equivalent to: ECV2 ¼ ðECV1 Posm1 þ 2 Na balanceÞ=Posm2 ; ð12bisÞ
Eur J Appl Physiol (2007) 101:133–142
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Results
used if ECV0 is unknown; PNa2 ¼ ðPNa1 ECV1 þ Na balanceÞ=ECV2 ;
ð13Þ
To compute the change in TBW of the patients, we need a different estimate of the Na change from that given by Eq. 4, as the correction does not necessarily reach the desired normal PNa0 of 140 mEq/l. Hence: Estimated DNa ¼ ðPNa2 ECV2 Þ ðPNa1 ECV1 Þ; ð14Þ at this point we compute the TBW present at time
2
Estimated TBW2 ¼ ðPNa1 TBW1
as: ð15Þ
þ Estimated DNaÞ=PNa2 ; finally, the calculated volume change is: Calculated DTBW ¼ TBW1 Estimated TBW2 :
ð16Þ
These calculations for patients are outlined in Appendix II. The data were analyzed statistically. Correlations and regressions between variables were computed by least square methods. All these computations were performed by a ‘‘Stat soft’’ software package commercially available.
The simulations were performed for a large variety of absolute TBW values, each one associated to step changes in DNa and/or DTBW. Appendix I exemplifies the calculations of four different simulations from the data of Figs. 1 and 2. Preliminary data showed that the PCl/PNa ratio must lie within ±0.005 of the normal value, and the final Na content/final Cl content must be £ 1.25. Outside this range the calculations of solute changes are less reliable. However, up to a PCl/PNa ratio within ±0.035 of the normal range, the error in computing the true values seems still compatible with a suitable clinical use. The results of these computer simulations, when the respective plasma solute ratios were unchanged, are shown in Fig. 3. The true values built in the models are plotted in abscissa against those calculated, shown in the vertical axis. True and calculated data are identical for ECV1 (Fig. 3a) and solute changes (not shown), to the point that in the figure they identify the identity line, with a slope of 1.00 and an intercept of 0.00, when the ratios remain normal. The individual data are not shown, as they lie inside the line of identity, equal to that of regression. The results of computer-simulated experiments where the PCl/ PNa ratios were within ±0.005 of normal, are also reported
a
C al c ula te d E C V 1 (Liters)
22
20
b
C a l c ul a t ed E C V 1 (Liters)
20
18
18
16
16 14
14
12
12
10 8
10 10
12
14
16
18
20
True ECV1 (Liters)
Fig. 3 ‘‘Side a’’ reports in abscissa the true values of the extracellular volume (ECV) generated in computer simulated models with known TBW , ECV0, and with assigned values of solvent added to or subtracted from the body fluids. They are plotted against their paired values computed by PNa1, PCl1 according to the formulas reported in the methods (4). The regression line traced is superimposed to, and indistinguishable from, the line of identity, and originates from the conditions of normality of the respective solute ratios, yielding: ECV1 calculated ¼ 0:00 þ 1:00 true ECV1 : Correlation coefficient (1.00) and regression are highly significant (P < 0.0001). The individual data are not reported as they are distributed inside the identity line. The symbols shown (open circles) refer instead to data obtained when the ratios were within ±0.005 from the normal value of 0.75. The regressions equation fitting the
8
10
12
14
16
18
20
22
True ECV1 (Liters)
open circles (not shown) is: ECV1 calculated ¼ 0:00 þ 0:999 true ECV1 ðP\0:0001Þ: ‘‘Side b’’ is identical to the previous one, while it reports simulations performed with a degree of tolerance in ratio derangement within ±0.035 from the normal value of 0.75. The calculated regression equations are shown by the hatched lines: that above the continuous identity line (identical to that of ‘‘side a’’), fits the open circles, and originates from ratios differing negatively from the normal value of 0.75 for PCl/PNa, down to 0.715. The regression e q u a t i o n i s : ECV1 calculated ¼ 2:00 þ 1:03 true ECV1 ; R2 ¼ 0:67; P\0:005: The hatched line below that of identity fits the closed circles, which individuate data characterized by positive differences in the respective ratios from 0.75 up to 0.785. The r e g r e s s i o n e q u a t i o n i s : ECV1 calculated ¼ 2:00 þ 0:98 true ECV1 ; R2 ¼ 0:77; P\0:005
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138
in Fig. 3. These data, instead, are identified by the individual symbols, lying outside the continuous line of identity, which is determined by PCl1/PNa1 = PCl0/PNa0. The regression (not shown in Fig. 3a) is: ECV calculated = 0.0 + 0.999True ECV, P < 0.0001. The maximum error in estimating the true ECV value is £ 5% when PCl1/PNa1 lies between 0.755 and 0.745. The data, not shown in the figure, become less predictable when the ratios increase, and poorly predictable when the ratios exceed ±0.035 of the normal value. However, the difference between the actual and the normal ratio is known, it is either positive or negative, and this permits to reduce the error. As shown in Fig. 3b by the full circles, that individuate results with ratios differing by a positive value up to 0.035 from normal, the method underestimates the true ECV by the value of the intercept, –2 l, while the slope (lower hatched line) is close to unity (0.98, P < 0.005). The results are exactly symmetrical (open circles) when the ratios differ from normal by identical negative values (Calculated ECV = 2 + 1.03True ECV, P < 0.005). Thus, when the PCl/PNa ratio <1% of normal, the calculations can be performed with a maximum error of 2%, whilst when the derangement of the normal PCl/PNa ratio (=0.75) ranges between 0.785 and 0.715, the calculations can be performed with a similar error by using the empirical equations shown in Fig. 3b, according to the sign of the difference of the ratio from normal. Beyond these limits the calculations can be misleading and their use cannot be recommended. We therefore extended these calculations to patients resorting to the same formulas used for, and to the empirical equations yielded by the computer simulations. The changes calculated and the residual amounts, as well as the ECV when the derangement was present can be validated by predicting the changes in electrolyte concentration during a quantitative treatment that introduces known amounts of salts and water while recording their simultaneous losses. This implies using Eqs. 11–13. Balance studies were executed on 24 out of 45 patients studied during correction of their electrolyte abnormalities. The correction was carried out by infusing the Na and/or water deficit (or withdrawing the surfeits) while simultaneously replacing the external losses by appropriate i.v. infusions. The individual data are reported in Table 1. Figure 4 reports the results of these calculations as PNa2 predicted versus that actually measured. The Figure plots the pairs of values and demonstrates the presence of a significant correlation and regression. Figure 5 demonstrates a significant correlation and regression between the changes in TBW volume calculated and those measured by the changes in BW. A flow chart describing the calculations performed on the data of Table 1 to build Figs. 4 and 5 is reported in Appendix II.
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Eur J Appl Physiol (2007) 101:133–142
Discussion The quantitative assessment of electrolyte derangements is important to understand the generation of the events leading to the imbalance, and to guide a treatment capable of yielding predictable results. The methods presently used to compute surfeit and deficit of solvent and solutes rests on the assumption that either TBW or solute contents are unchanged during the derangements. When the assumption postulating a stable body water holds true, the difference in PNa multiplied by TBW correctly measures the change in Na content (Edelman et al. 1958). However, an identical change in Na concentration can be generated by a change in volume, as demonstrated by Figs. 1 and 2. There is no way, mathematically, to select the right clinical derangement. The present paper demonstrates that it is possible to establish mathematically whether TBW or solute contents are unchanged, by examining the respective ratios between different solutes, and between normal and deranged states for each solute. Figures 1 and 2 illustrate these constraints, while Figs. 3 and 4 show that the results of the computations are exact when the constraints are completely satisfied, as demonstrated by the superimpositions between the calculated regression lines and the lines of identity between true and calculated data. Thus, a number of clinical conditions, present in 53% of our patients, can be quantitatively examined and solved, and their correction guided on an unimpeachable basis, by resorting to a very simple calculation of solute ratios. Even when the assumption of constancy in TBW cannot be established with certainty, still the formulas will yield a minimum estimate of solute losses and a maximal estimate of ECV. It should be understood that even the exact calculation is no better than the assumption it rests upon, such that solute changes are very unlikely to cause losses in the same exact proportions as their respective ratios in plasma. Were such an unlikely circumstance to occur, a true volume deficit will be computed as water excess, as the respective solute ratios will be unchanged with respect to normal, and the ratios of measured over initial normal values for each solute will be identical. Therefore, the clinical judgement on the volume conditions of the patients retains its full exclusive reliability, such that physicians and physiologists must match the results of their calculations with the conclusions of physical examination and history, to avoid a treatment that might worsen the ominous brain oedema (Arieff and Guisado 1976). The clinical conditions on which this mathematical method can be applied seem limited, and their usefulness apparently minor. However, considering that presently the guesses of expert physicians can be mistaken by a large margin, we could extend the usefulness of the formulas to
Eur J Appl Physiol (2007) 101:133–142
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Table 1 The 24 patients out of the 45 studied in whom body weight (BW), Na and water balance could be measured are shown vertically Before correction
Correction Estimated DNa (mEq)
Estimated DH2O (L)
Patients PNa1 (mEq/l)
BW1 (Kg)
1
127
73.0
+4.1
0
–3.0
136.3
134
70.0
2
111
61.5
+7.6
+420
–1.0
125.8
126
60.5
3
125
48.0
–432
+280
0.0
134.7
134
48.0
4
125
60.0
–540
+228
0.0
131.3
130
60.0
5
120
96.5
–1,158
+778
0.0
133.4
132
96.5
6
121
60.0
–311
–5.0
130.5
132
55.0
7 8
113 121
66.5 69.0
+625 +296
0.0 0.0
128.7 128.2
128 129
66.5 69.0
9
117
43.0
+4.2
+270
–0.5
130.0
132
42.5
10
127
59.0
+3.3
–200
–4.0
136.8
134
55.0
11
109
52.0
+616
0.0
128.7
131
52.0
12
109
72.0
+353
–5.0
132.5
129
67.0
13
108
80.0
–1,536
+1,100
–2.0
136.6
135
78.0
14
128
38.0
–274
+117
0.0
133.1
134
38.0
15
118
68.0
–898
+285
0.0
125.0
128
68.0
16
117
44.0
–607
+384
+1.0
126.7
124
45.0
17
126
44.0
–370
+128
0.0
130.8
131
44.0
18
113
44.0
–713
+512
+1.0
127.6
125
45.0
19
129
72.0
–475
+256
0.0
134.9
134
72.0
20
108
57.0
–1,094
+1,200
+1.5
137.1
137
58.5
21
112
52.0
–874
+861
+2.0
131.2
131
54.0
22 23
126 128
54.0 68.0
+3.2 +3.5
+140 +255
0.0 –1.0
130.3 137.6
132 135
54.0 67.0
24
118
41.0
+3.9
+77
–0.5
123.6
124
40.5
+4.9 –1,077 –787
–967 +9.6
Na balance (mEq)
After correction Water balance (DKg)
Calculated PNa2 (mEq/L)
Measured PNa2 (mEq/L)
BW2 (Kg)
From left to right the Table displays PNa1 (the plasma Na concentration measured during the derangement, in mEq/l), the BW1 measured during the derangement, the calculated changes (D) in Na and/or water, the balance of Na and/or water during correction, the predicted (the plasma Na concentration calculated) and measured PNa2 following correction, and its corresponding BW2· DH2O = BW1 – BW2, rounded off to 0 when <0.3 kg (the error of the scale). The DNa was calculated with (4) when PCl/PNa was > 0.785 < 0.715. Within this range, DNa was assumed to be nil, while only a change in volume was assumed to be present. The predicted PNa2 was calculated with Eq. 13. BW2 is not = BW0, having been measured at the end of the balance period, that might have ended before the normal PNa had been reached. TBW0 = BW10.6PNa1/PNa0 when PCl1/PNa1 > 0.715 < 0.785, = BW10.6 when the ratios were outside this range. A flow chart reporting these computations is shown in Appendix II
include a number of derangements within the range of what we could consider a clinically acceptable error. The results of the simulations indicate that accepting as unchanged ratios a deviation within ±5% from the value established by the assumptions, the maximum error in calculations could be less than 9% of the true value, applying the empirical formulas of Fig. 3. These would help to establish a satisfactory treatment, with significant advantage for the patient. Therefore, our data suggest that the confidence limits of the ratios should be set within ±5%, and this higher tolerance will still be clinically adjuvant with respect to conventional ways of treatment. During treatment, when solutes and solvent were added to the residual amounts calculated with Eqs. 4 and 6, we predicted the changes in Na concentration by Eq. 13, and those in volume by Eq. 16, as shown by Figs. 4 and 5.
These results validate, within the confidence limits of the statistical variability, the method of calculation. This may be important in long distance runners, whose hyponatremia is mainly due to skin water and solute losses, attended by oral replacement of the solvent only (Almond et al. 2005). An exact calculation of their Na deficit would help a quantitative correction, avoiding under or over-treatment and the risk of cerebral oedema. In conclusion, traditional ways to calculate water or solute losses are based on the assumption that solutes are unchanged when solvent derangements are calculated, and vice versa. Only the clinical judgement can establish whether to consider a single patient volume depleted or expanded, as compared to solute loaded or depleted. The present paper describes a way to check this assumption and to verify whether and how to apply formulas to calculate
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Appendix I
PNa expected mEq/L
138 136
Flow chart describing the steps of computer simulation and of calculations of Na and water surfeits or deficits, omitting OAN for the sake of simplicity. Example I: Looking at Fig. 1a, we have: A: Simulation
134 132 130 128 126
1.
The total content in Osmoles is 2 PNa0 TBW0 = 280 40 = 11,200 mOsm;
2.
Adding 4.8 L of water (Fig. 1b) changes the osmolality to: Posm1 = total Osmoles/TBW = 11,200/44.8 = 250 mOsm/kg;
3.
ECV1 = 2 PNa0 ECV0/Posm1 = 16.8 l;
124 122 122 124 126 128 130 132 134 136 138 140
PNa measured mEq/L Fig. 4 The plasma sodium concentrations (PNa2) are in the abscissa and are measured at a time point of correction where Na and water losses were known. These are plotted against the corresponding values accounted for by applying Eq. 13. There is a satisfactory correlation (R2 = 0.81, P < 0.0001) and regression (PNa calculated = 0.00 + 1.00 · PNa measured, P < 0.0001) between the paired values. We included eleven cases already published (Castello et al. 2005b), recalculated with the present formulas. Sentence changed as shown
4.
PNa1 = PNa0 ECV0/ECV1 = 140 15/ 16.8 = 125 mEq/l;
5.
PCl1 = PCl0 ECV0/ECV1 = 105 15/16.8 = 93.75;
6.
PCl1/PNa1 = 93.75/125 = 0.75 = PCl0/PNa0; PNa0 = 125/140 = 0.89 = PCl1/PCl0;
PNa1/
B: Calculation using only the assumed normal values of PNa0 and TBW0, and the measured PNa1. Since the plasma solute ratios are unchanged, we calculate the pure water change:
Calculated 5.0 ∆TBW 4.0 (Liters) 3.0
1.
2.0 1.0
Example II: Looking at Fig. 1c, we have: A: Simulation
0.0 -3.0
-2.0
-1.0 0.0 -1.0
1.0
2.0
3.0
4.0
5.0
True ∆TBW (Liters)
2.
Subtracting 4.8 l of water from the normal content of Fig. 1a changes the osmolality to: Posm1 = total Osmoles/TBW = 11,200/35.2 = 318.2 mOsm/kg;
3.
ECV1 = 2 PNa0 ECV0/Posm1 = 13.2 l;
4.
PNa 1 = PNa 0 ECV 0 /ECV 1 = 140 15/13.2 = 159 mEq/l;
5.
PCl1 = PCl0 ECV0/ECV1 = 105 15/13.2 = 119.3;
6.
PCl1/PNa1 = 119.3/159 = 0.75 = PCl0/PNa0; PNa0 = 159/140 = 1.136 = PCl1/PCl0;
-2.0 -3.0
Fig. 5 The true change in TBW measured by the difference in body weight (BW1 – BW2) is plotted in abscissa, against the paired values computed by Eq. 16. When the PCl/PNa ratio was between 0.715 and 0.785, the data are indicated by the full circles (patients deemed to have suffered only volume changes), by open circles outside this range (patients deemed to have suffered only solute changes). The regression equation is: Calculated DTBW ¼ 0:00 þ 0:96 True DTBW; P \ 0:0001; R2 ¼ 0:97 for all symbols; ¼ 0:00 þ 0:93 True DTBW; P\0:0001; R2 ¼ 0:97; for the ‘‘pure’’ water derangements. Details of these calculations are reported in Appendix II
solute or solvent changes. This new way was validated by computer simulations as well as on real patients, and its application could be extended from the limited cases, where the assumptions are fully met, to physiological (marathon runners) and clinical conditions where the error is not large enough to offset the usefulness of quantitative estimates and treatments.
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DH2O = (PNa0 – PNa1) TBW0/PNa1 = (140 – 125) 40/125 = + 4.8 l;
PNa1/
B: Calculation using only the assumed normal values of PNa0 and TBW0, and the measured PNa1. Since the plasma solute ratios are unchanged, we calculate the pure water change: 1.
DH2O = (PNa0 – PNa1) TBW0/PNa1 = (140 – 159) 40/159 = –4.8 l; Example III: Looking at Fig. 2a, we have:
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A: Simulation 1.
The total content in Osmoles (Fig. 2a) 2 PNa0 TBW0 = 280 40 = 11,200 mOsm;
is
2.
Subtracting 600 mEq of Na (Fig. 1b) changes the osmolality to:
Posm1 ¼ total Osmoles=TBW ¼ ð11200 2 600Þ=40
B: Calculation using only the assumed normal values of PNa0 and TBW0, and the measured PNa1. As the plasma solute ratios are importantly modified, we calculate the pure solute and ECV change: 1.
DNa = (PNa1 – PNa0) TBW0 = (128.3 – 140) 40 = – 468 mEq;
2.
ECV1 = (2 PNa0 ECV0– 2 DNa)/ Posm1 = (4,200 – 2 468)/256.4 = 12.7 l; in this example, both the Na deficit and the fall in ECV are underestimated by 132 mEq and 1 l, respectively.
¼ 250 mOsm=kg; 3.
ECV1 = (2 PNa0 ECV0 – 2 DNa)/Posm1 = (2 140 15 – 2 600)/250 = 12 L;
4.
PNa1 = (PNa0 ECV0 – DNa)/ECV1 = (140 15 – 600)/ 12 = 125 mEq/L;
5.
PCl1 = (PCl0 ECV0 – DCl)/ECV1 = (105 15 – 500)/ 12 = 89.6;
6.
PCl1/PNa1 = 89.6/125 = 0.717 „ PCl0/PNa0; PNa1/ PNa0 = 125/140 = 0.89 „ PCl1/PCl0 = 0.85;
B: Calculation using only the assumed normal values of PNa0 and TBW0, and the measured PNa1. Since the plasma solute ratios are importantly modified, we calculate the pure solute change: 1.
DNa = (PNa1 – PNa0) TBW0 = (125 – 140) 40 = – 600 mEq;
Omitting for simplicity the example from Figure 2c, we can proceed with: Example IV: We withdraw 600 mEq of Na, 600 mEq of Cl, and 1 l of water from Figure 1a. A: Simulation 1. 2.
The total content in Osmoles is 2 PNa0 TBW0 = 280 40 = 11,200 mOsm; Subtracting 600 mEq of Na (Figure 1b) changes the osmolality to:
Posm1 ¼ total Osmoles=TBW ¼ ð11200 2 600Þ=39 ¼ 256:4 mOsm=kg;
Appendix II Flow chart for computing data of patients reported in Table 1. Example I. A: Calculation of water surfeit in patient number 6, whose PCl/PNa ratio indicated pure volume excess. Since we do not know TBW0, instead of Eq. 6 we apply the equivalent equation 6bis: 1. 2.
DH2O = (PNa0 – PNa1) BW1 0.6/PNa0 = (140 – 121) 60 0.6/140 = + 4.9 L; True DH2O = BW1 – BW2 = 60.0 – 55.0 = 5 L;
B: Calculation of predicted PNa2 at the end of correction of the derangement. The balance study executed during correction registered a net Na balance of –311 mEq. As TBW0 is unknown: 1. Posm2 ¼ ½ðPosm1 TBW1 þ 2 Na balanceÞ= ðTBW1 þ water balanceÞ; ¼ ½ð2 121 60 0:6 2 311Þ=ð36 4:9Þ ¼ 260 mOsm=kg; ð11bisÞ 2. ECV2 = (ECV1 Posm1 + 2 Na balance)/Posm2 ; (12bis) since ECV0 is unknown, ECV1 must be calculated firstly:
3.
ECV1 = (2 PNa0 ECV0 – 2 DNa)/ Posm1 = (4,200 – 2 600)/256.4 = 11.7 l;
4.
PNa1 = (PNa0 ECV0 – DNa)/ECV1 = (140 15 – 600)/ 11.7 = 128.2 mEq/l;
5.
PCl1 = (PCl0 ECV0 – DCl)/ECV1 = (105 15 – 600)/ 11.7 = 83.3;
6.
PCl1/PNa1 = 83.3/128.2 = 0.65 „ PCl0/PNa0; PNa1/ PNa0 = 128.2/140 = 0.916 „ PCl1/PCl0 = 0.793;
Calculated TBW0 = BW1 0.6 – DH2O = 60 0.6 – 4.9 = 31.1 l; (b) Calculated ECV0 = calculated TBW0 0.375 = 31.1 0.375 = 11.7 l; since PNa0 ECV0 = PNa1 ECV1 in pure volume loss: (c) ECV1 = PNa0 calculated ECV0/PNa1 = 140 11.7/ 121 = 13.5 L; = BW1 0.6 0.375 = 60 0.6 0.375 = 13.5 l (direct calculation); (d) ECV2 = (13.5 2 121 – 2 311)/260 = 10.1; (a)
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3.
(d) Estimated DNa ¼ ðPNa2 ECV2 Þ ðPNa1 ECV1 Þ; ¼ ð 135 22:3Þ ð108 18Þ
PNa2 ¼38; ðPNa1 ECV1 DNaÞ=ECV2 ; ¼38; ð121 13:5 311Þ=10:1 ¼ 131 mEq=l;
¼ 1066 mEq; 4.
Since the correction was interrupted before reaching the normal PNa, the volume change must be estimated as follows:
ð14Þ
(e) Estimated TBW2 ¼½ðPNa1 TBW1 ÞþEstimated DNa=PNa2 ;
(a)
¼½ð108 48Þ þ 1066=135 ¼ 46:3l;
Estimated DNa ¼ ðPNa2 ECV2 Þ ðPNa1 ECV1 Þ;
ð15Þ
¼ ð132 10:1Þ ð121 13:5Þ ¼ 300 mEq; ð14Þ
(f) Calculated DTBW ¼38; TBW1 Estimated TBW2 ; ¼38; 48 46:3 ¼ 1:7 l:
(b) EstimatedTBW2 ¼½ðPNa1 TBW1 ÞþEstimatedDNa=PNa2 ; ¼½ð121 36Þ300=131¼31:0l; ð15Þ (c) Calculated DTBW ¼ TBW1 Estimated TBW2 ;
ð16Þ An appropriate software is available to ease these calculations (SIAE registration 8/16/06, number 0604311, informations from ‘‘
[email protected]’’, quoting reference program number 0601).
ð16Þ
¼ 36 31 ¼ 5:0 l:
References Example II: Calculation of pure Na loss in patient 13, whose PCl/PNa indicated pure Na deficit. We compute: 1.
2.
DNa = (PNa1 – PNa0) BW1 0.6 = (108 – 140) 80 0.6 = – 1,536 mEq; even though the preceding calculation assumes a ‘‘pure’’ Na loss, our formulas allow us to compute an associated volume change; The volume change computation requires using (11bis)–(16)
(a) Posm2 ¼ ½ðPosm1 TBW1 þ 2 Na balanceÞ= ðTBW1 þ water balanceÞ; ¼ ½ð2 108 80 0:6 þ 2 1100Þ= ð 80 0:6 2Þ ¼ 273 mOsm=kg; ð11bisÞ (b) ECV2 ¼ ðECV1 Posm1 þ 2 Na balanceÞ=Posm2 ; ¼ ð80 0:6 0:375 216 þ 2200Þ=273 ¼ 22:3 l;
ð12bisÞ
(c) PNa2 ¼ ðPNa1 ECV1 þ Na balanceÞ=ECV2 ; ¼ ð108 80 0:6 0:375 þ 1100Þ=22:3 ¼ 136:6;
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ð13Þ
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