Nephrol Dial Transplant (2007) 22: 3478–3486
doi:10.1093/ndt/gfm427
Advance Access publication 4 July 2007
Original Article
Estimating excess glucose, sodium and water deficits in non-ketotic
hyperglycaemia
Ettore Bartoli1, Francesca Guidetti2 and Luca Bergamasco2
1
Chair of Internal Medicine and 2Dipartimento di Medicina Clinica e Sperimentale, Università degli Studi del Piemonte
Orientale ‘A. Avogadro’, Novara, Italy
Abstract
Background. The treatment of solute addition, Na
and water losses in hyperglycaemic hyponatraemia
is guided by clinical judgement rather than by a
quantitative assessment.
Methods. We devised an iteration method to compute
glucose appearance (GA) within the extracellular space,
to obtain the PNa (plasma sodium concentration)
expected by glucose addition only (PNaG). The
difference between this and the actual measurement
(PNa1) was used to compute the attending Na and/or
volume depletion, and the PNa expected during
correction. The equations were validated on computer-built models, where the electrolyte derangements
were simulated, generating true values of plasma
glucose (PG) and Na concentrations, from which
surfeit and deficits were back-calculated with our
formulas.
We also computed GA and PNaG on 43 patients who
were stratified into a group with normal hydration
(PNa1 ¼ PNaG), one with prevalent Na depletion
(PNa1 < PNaG), and one with prevalent volume
depletion (PNa1 > PNaG). The volume conditions
established by our computations were compared by
logistic regression analysis with those assessed from
clinical laboratory data.
Results. The computer simulations demonstrated that
the method gave exact results when only one variable
changed, clinically useful estimates in the presence
of mixed volume and sodium deficits. There was
a strongly significant concordance between the clinical
and the quantitative method (P < 0.001). The latter
predicted the PNa measured after correction of
hyperglycaemia (P < 0.001).
Conclusion. This new method more accurately computes the initial conditions, resulting in a useful
stratification of patients which improves the quantitative evaluation and treatment of hyperosmolar coma.
Keywords: dehydration; extracellular volume;
hyperglycaemia; hyponatraemia; hyperosmolar coma;
NIDDM
Introduction
The main derangement of hyperosmolar coma is
represented by the accumulation of unmetabolized
glucose inside the extracellular space, since the
entrance of glucose into cells is blocked. The rise in
extracellular osmolarity drives a flow of solvent from
cells, diluting the extracellular solutes. Thus, hyponatraemia ensues [1]. It proves difficult in clinical practice
to quantitatively analyse this phenomenon, because
the osmotic diuresis causes a loss of extracellular
fluids that interferes with the quantitative appraisal
of glucose addition and sodium losses. In fact, if the
osmotic diuresis induces a loss of water exceeding
that of solutes, an inappropriately high Na concentration would be measured, unless the patient were to
compensate the preferential water losses with an
adequate intake. A preferential solute loss alone,
and/or coupled with insufficient intake, could instead
worsen hyponatraemia.
The present study was undertaken with the aim of
improving the quantitative appraisal of the glucose
load, and to compute predicted changes in Na
concentration capable of helping to understand
whether solute or water losses are more important in
a single patient. Consequently, the treatment could be
focused on the main abnormality, guided to avoid
abrupt changes in electrolyte concentrations.
Methods
Correspondence and offprint requests to: Prof Ettore Bartoli,
Dipartimento di Medicina Clinica e Sperimentale, Università
degli Studi del Piemonte Orientale ‘A. Avogadro’, Via Solaroli
17, 28100 Novara, Italy. Email: [email protected]
We will deal with a simplified model, similar to that
published by Katz [1], where subfix 0 refers to normal
conditions, 1 to the presence of the derangement, 2 to any
ß The Author [2007]. Published by Oxford University Press on behalf of ERA-EDTA. All rights reserved.
For Permissions, please email: [email protected]
Hyperglycaemic hyponatraemia
3479
Fig. 1. Column A indicates normal conditions with assumed normal values of total body water (TBW0), extra (ECV0) and
intracellular volume (ICV0), Na concentration and solute contents in the two body fluid compartments. P refers to plasma, PNa0 is
the normal plasma Na concentration. Column B: 2000 mM of glucose are added to the extracellular fluid, causing the disequilibrium
condition illustrated. Column C: the accumulation of the osmotically active glucose causes a water shift from cells to interstitium, till the
final equilibrium situation is established. Indicating with subfix 0 the normal conditions, with 1 the deranged state, the true numbers are given
by: Posm1 ¼ (pre-existing osmoles þ added glucose)/TBW ¼ (2PNa0 TBW0 þ added glucose)/TBW0 ¼ (11 200 þ 2000)/40 ¼ 330 mOsm/kg
water; ECV1 ¼ (pre-existing ECV osmoles þ added glucose)/Posm1 ¼ (2PNa0 ECV0 þ added glucose)/Posm1 ¼ (4200 þ 2000)/330 ¼ 18.8 l;
PG1 ¼ added glucose/ECV1 ¼ 2000/18.8 ¼ 106 mM/l; PNa1 ¼ pre-existing Na/ECV1 ¼ PNa0 ECV0/ECV1 ¼ 2100/18.8 ¼ 112 mEq/l.
time during, or at the end of correction. The assumptions are
the following:
system defined exactly. We therefore approach the problem
by computing these values with the following equations:
(i) Posm (plasma osmolarity, mOsm/kg) is accounted
for only by Na and its accompanying anions in
normal conditions, and by these plus glucose during
hyperglycaemia. We did not include baseline normal
glycaemia nor plasma urea [2] in the calculations.
Thus, Posm0 ¼ 2 PNa0; Posm1 ¼ 2 PNa1 þ PG1,
where
PG1 ¼ hyperglycaemic
plasma
glucose
concentration.
(ii) During this acute process, the intracellular content
in osmoles does not change: ICV1 Posm1 ¼
ICV0 Posm0 ¼ ICV0 2 PNa0,
where
ICV ¼
intracellular, ECV ¼ extracellular volume.
(iii) The third assumption is that plasma osmolarity equals
that of cells, and it is given by the total amount of
osmotically active solutes (osmoles) divided by the
total body water (TBW): Posm ¼ (total osmoles
present in body water)/TBW ¼ [2(exchangeable
Na þ K)]/TBW [3]. Consequently: ECV ¼ extracellular
content in osmoles/Posm.
(i) TBW1 ¼ TBW0 {[(2PNa1 þ PG1 2PNa0) TBW0 (2Na)]/(2PNa1 þ PG1)} (1); this formula is exact only
if neither solvent nor sodium contents change:
TBW0 ¼ TBW1 and Na ¼ 0. Under these conditions,
the numerator of the equation is GA. However, this
formula computes a volume change even when not
present, whilst dividing the same numerator by Posm
(given by 2PNa1 – 2PNa0 þ PG1) gives an exact TBW
value. However, V 6¼ 0 is computed as ECV.
(ii) ICV1 ¼ [2PNa0 ICV0/(2PNa1 þ PG1)] (2); this formula is valid independent from changes in solvent
and sodium contents.
(iii) ECV1 ¼ TBW1 ICV1, and ECV ¼ ECV0 ECV1
(3);
(iv) GA ¼ ECV1 PG1 (4); the problem of how to compute
Na requires introducing a new concept, that of
PNaG, which is the plasma Na concentration that
would be measured if there were only GA added to the
system. Clearly, PNa1 PNaG should indicate an
excess Na content when positive, an Na deficit when
negative. Since an Na surfeit is unlikely in hyperosmolar coma, a positive difference indicates a
plausible water deficit. It is important to recognize
that a negative difference should represent an estimate
of Na depletion. PNaG is calculated as if there were
no water nor Na changes: PNaG ¼ pre-existing
Na/ECV1 ¼ PNa0 ECV0/ECV1; it is equal to PNa1
of Figure 1, where only GA was added. Thus:
(v) Na ¼ (PNa1 PNaG) ECV1 (5), where ECV1 is
calculated by (3). Na must be fed into the
equation (1) to compute ECV1 under these conditions.
(vi) When both Na and water contents change, the system
does not recognize any exact mathematical solution,
Hyperosmolar coma develops as portrayed in Figure 1,
which illustrates the water shift from the intra to the
extracellular compartment induced by the hyperosmolarity
caused by the added glucose, attended by dilution of Na with
a fall in PNa. Clinically, the physician tries to back-calculate
the changes in Na and water contents and distribution
from the only available measurements, PNa1 and PG1.
The problem is further complicated by the fact that BW1
(the actual body weight) cannot usually be measured in these
patients neither is the weight known just before the
derangement (BW0). Inspection of Figure 1 shows that the
critical aspect is to correctly compute either GA or ECV1:
once one of these values is known, the second can be easily
calculated (GA ¼ PG1 ECV1; ECV1 ¼ GA/PG1) and the
3480
E. Bartoli et al.
and we must resort to the algorithm of the appendix
to calculate GA and, from this, ECV1 and PNaG. The
formulas (1) to (5) will be referred to, elsewhere in this
text, as the ‘one step method’ or ‘one step procedure’.
Experiments with computer simulations
Figure 1 exemplifies our computer model, as it
illustrates one simulation as an example. The simulations were performed, for a large array of assumed
initial normal conditions, by changing GA in steps,
with or without the imposition of Na and/or TBW,
also modified in discrete steps. For each simulation
the true PNa1 and PG1 are calculated, as shown in
Figure 1, by inserting in the computation formulas the
values of GA, Na and/or TBW imposed upon the
system each time. These values are then considered
as if they were actual measurements during a derangement in a simulated patient, and used to backcalculate, with the formulas detailed above, the deficit
of Na and/or water, and the surfeit of glucose.
To simulate the diagnostic problem, we assume,
for each simulation, that we know exclusively the
initial conditions (those portrayed in column A for the
example shown in Figure 1), and the PG1, PNa1 values
measured during the derangement (generated by the
computer), as if we were in column C.
Experiments on patients with hyperglycaemic
hyponatraemia
We studied 43 patients admitted with hyperglycaemia to the hospital. Inclusion criteria required
PG > 15 mM/l (>270 mg/dl), arterial pH > 7.30, plasma
bicarbonate >15 mEq/l, urine ketones <2þ. Each
patient had a complete history and physical examination taken, and the initial measurements of BW (when
possible), plasma sodium (PNa mEq/l) and creatinine
concentrations (PCr mg/dl), urinalysis, haematocrit
(Hct). These measurements were repeated during or at
the end of treatment. Although the patients had all
routine exams performed, their evaluation for the
purpose of the present study was based only on 21
clinical lab results independently scored: at least 10 had
to be unequivocally present to allow a clinical
evaluation. A patient was considered significantly
dehydrated if >7 out of 10 measurements indicating
low volume were present, rather normally hydrated if
<3 were present. The score attributed to normal values
was zero. PCr was not corrected, since the correction
assumes perfect steady state, probably not present in
these patients. The mixed clinical laboratory symptoms, each followed by its score in brackets were:
blood pressure mmHg: 115/70 (1), <100/60 (2),
diastolic unmeasurable (3); heart rate per min: 84
(1); 92 (2), 100 (3); appearance of skin and
mucosae: dry (3); plasma creatinine: 1.5 (1), 2.0
(2), 2.5 (3); creatinine clearance ml/min: 60 (1), <20
(3); haematocrit %: 40 (1), >54 (3); jugular veins and
bedside estimate of CVP: low (1); extremities: cold (1);
extremities and forehead: sweaty (1); mental state:
clouded (3); peripheral veins: flat (1); urine volume/
24 h: <400 (1), <100 ml (3).
TBW (litres, l) was computed, in normal conditions,
as: TBW0 ¼ 0.6 kg of normal BW when BMI (body
mass index) was 25 or unknown, otherwise it was
calculated as TBW0 ¼ 15 h2, where h is height in
meters, measured in supine position in the hospital
bed. This formula subtends that fat exceeding the
amount expected for a normal BMI of 25 is
anhydrous and contributes to BW, not to TBW.
Thus, TBW0 ¼ 0.6 lean BW ¼ 0.6 normal BMI h2
¼ 0.15 h2, taking 25 as normal BMI. The normal
ICV was: ICV0 ¼ TBW0 0.625; the normal ECV was
ECV0 ¼ TBW0 0.375; normal plasma sodium concentration (PNa0) was 140 mEq/l. The data of the
patients were processed with the same methods used
for computer simulations, computing, both with the
equations (1)–(5) and the iteration method of the
appendix, GA, PNaG, Na and TBW.
An immediate treatment was started by infusing
regular insulin (Humulin R-Lilly, or Actrapid-Novo
Nordisk Farmaceutici) at an initial rate of 20 IU/h,
plus maintenance infusions calculated to replace the
external losses of Na and water. During treatment
which lasted on average, 12 h, the rate of insulin
administration was progressively reduced, according
to plasma glucose measurements, to an average of
8 IU/h. As glucose is metabolized, its corresponding
osmoles disappear from the extracellular fluid, causing
a shift of water into cells: PNa consequently increases.
This phenomenon is the reverse of that portrayed
in Figure 1. The PNa rise can be calculated, at an
unchanged TBW, by the equations reported at the end
of the appendix.
The data were analysed statistically. Means and
standard errors of the means (M SE) were
computed, and their differences tested by paired or
unpaired t-tests. Correlations and regressions between
variables were computed by least square methods,
using a ‘Stat Soft’ software package commercially
available.
Results
The computer simulation experiments establish the
validity of the equations and their range of applicability, being built with assigned values and yielding
true numbers on which to base our calculations.
Table 1 displays the true values of GA and ECV1
generated by the computer, and their paired values
calculated with the iteration method of the appendix
and the formulas (1)–(5) of the methods. The data of
the first horizontal lines refer to simulations where
there was neither Na nor V. The methods give
exact values. The regression equations between true
and calculated data are in unity with the formulas
(1)–(5) and nearly in unity with the appendix method.
When only the TBW changes, the iteration slightly
overestimates the true ECV and GA, while correlation
Hyperglycaemic hyponatraemia
3481
Table 1. The means SEM of the true values yielded by the computer, of the same values computed with the iteration method of the
appendix (identified with A), and with the ‘one-step procedure’ (B), are shown.
Conditions and
variables
Na ¼ 0
TBW ¼ 0 GA 6¼ 0
Na ¼ 0
TBW 6¼ 0 GA 6¼ 0
Na 6¼ 0
TBW ¼ 0 GA 6¼ 0
Na 6¼ 0
TBW 6¼ 0 GA 6¼ 0
True values
Calculated values
Correlations
(True values vs A B)
A. Iteration
method
B. ‘One-step’
method
GA (mM)
937.50 107.32
934.06 106.55
711.98 68.01
ECV1 (L)
16.92 0.20
16.88 0.19
GA (mM)
937.50 107.32
1038.71 120.15
711.98 68.01
ECV1 (L)
15.49 0.23
GA (mM)
937.50 107.32
1088.18 1{2,1}21.94
1132.88 117.50
ECV1 (L)
14.72 0.35
17.15 0.21
Na (mEq)
500.00 56.61
255.86 30.60
GA (mM)
937.50 53.15
ECV1 (L)
13.63 0.17
17.06 0.21
1196.86 67.41
17.32 0.11
13.94 0.12
12.76 0.16
19.04 0.63
317.06 41.32
1132.88 58.19
17.62 0.30
A. Y ¼ 0.99 X þ 3.31
(R2 ¼ 1, P < 0.0001)
B. Y ¼ 0.63 X þ 120.29
(R2 ¼ 0.99, P < 0.0001)
A. Y ¼ 0.96 X þ 0.58
(R2 ¼ 1, P < 0.0001)
B. Y ¼ 0.60 X þ 24.09
(R2 ¼ 1, P < 0.0001)
A. Y ¼ 1.11 X 3.83
(R2 ¼ 0.99, P < 0.0001)
B. Y ¼ 0.63 X þ 120.29
(R2 ¼ 0.99, P < 0.0001)
A. Y ¼ 0.65 X þ 7.03
(R2 ¼ 0.52, P < 0.0001)
B. N.S.
A. Y ¼ 1.11 X þ 41.51
(R2 ¼ 0.97, P < 0.0001)
B. Y ¼ 0.93 X þ 255.40
(R2 ¼ 0.73, P < 0.0001)
A. Y ¼ 0.32 X þ 12.32
(R2 ¼ 0.30, P < 0.001)
B. Y ¼ 1.61 X þ 42.77
(R2 ¼ 0.81, P < 0.0001)
A. Y ¼ 0.51 X þ 3.06
(R2 ¼ 0.92, P < P < 0.0001)
B. Y ¼ 0.72 X þ 43.02
(R2 ¼ 0.97, P < 0.0001)
A. Y ¼ 1.23 X þ 37.78
(R2 ¼ 0.95, P < 0.0001)
B. Y ¼ 0.93 X þ 255.40
(R2 ¼ 0.73, P < 0.0001)
A. Y ¼ 0.30 X þ 13.17
(R2 ¼ 0.22, P < 0.0001)
B. Y ¼ 1.20 X þ 34.10
(R2 ¼ 0.49, P < 0.0001)
At the far right-hand side there are the regression equations and correlation coefficients. Vertically, we report the glucose added, GA mM,
the ECV1 litres, the Na lost, Na mEq. These are shown for computations where only one of these variables changed. The two bottom lines
portray the simultaneous Na and V changes in mixed disorders, when both were being lost. The asterisks on the left side of each mean
refer to the statistical comparisons with the true values, those on the right hand side to the comparisons with the next mean.
¼ P < 0.05; ¼ P < 0.01; ¼ P < 0.001; N.S. ¼ non significant. GA in the first line is underestimated by (4), whilst it is exact
(data not shown), with the part in parenthesis of equation (1).
and regression are significant. Instead, the ‘one-step
method’ strikingly underestimates both, while regression and correlation are not significant.
The data obtained when there was a change in Na
while not in water contents, are reported in the three
lines that follow. The values of Na are shown
together with their paired values calculated with the
formulas. The iteration method calculates more useful
data as the intercept is more reliable than that given
by the ‘one-step procedure’.
The two bottom lines of Table 1 display the true
values generated by the computer, plotted against
the paired values calculated by each method when
both water and Na were lost. The ECV regression is
negative, as it computes large volumes when the true
ones are small, and vice versa, when the ‘one-step
method’ is used. Instead, the data are more reliable
using the iteration method of the appendix, which,
however, underestimates the true value.
Therefore, we applied to patients, who were very
likely suffering from mixed disorders, the iteration
method of the appendix to compute the entity of Na
and water derangements, and the critical value of
PNaG. The data obtained in each patient are reported
in Table 2, together with the clinical score of the
volume conditions. The PNa calculated by GA (PNaG),
and the difference between this value and the actual
PNa1 measurement performed at the same time
(PNa) are included.
Seventeen out of 43 patients with PNa1 – PNaG
within 3 mEq/l, belonged to group 1. They should
have had an ECV compatible with the water shift from
cells, which remained either stable or was partly
depleted iso-osmotically by the osmotic diuresis.
3482
E. Bartoli et al.
Table 2. The table shows the data of the 43 patients studied, aligned vertically. From left to right are shown the total body water (TBW) and
the improved-ECV1 in litres (l), the PNa1 measured at the admission to the ward, the PNaG (calculated as PNaGfe1 mEq/l, see Appendix I),
the measured PG1 mM/l, the measured PNa2 (true value) and its related PNaGfe2 (PNa2 calculated by the fall in plasma glucose during
treatment, PG). The difference between PNa1 and PNaG (PNa) was used to subdivide the patients into groups. The PNa measured are
given without decimal digits, while the calculated values were rounded off at the first decimal digit
Patients
TBW1
(l)
ECV1
(l)
PNa1
(mEq/l)
PNaG
(mEq/l)
PG1
(mM/l)
PNa
(mEq/l)
PG
(mM/l)
True PNa2
(mEq/l)
Calculated
PNa2 (mEq/l)
Dehydratation
score
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
72.0
38.4
45.9
46.8
36.0
42.0
48.0
96.0
50.1
67.2
34.8
54.0
39.0
43.2
40.5
42.6
54.0
55.2
41.4
51.0
51.9
40.8
39.0
27.0
39.6
45.0
49.5
42.0
39.0
36.0
39.0
66.0
55.8
37.8
48.0
30.0
47.4
37.8
48.0
52.0
30.0
30.0
44.4
28.6
15.1
18.1
18.7
14.4
17.1
19.2
18.4
19.8
26.8
14.1
22.8
16.0
17.3
15.9
17.0
21.3
17.3
16.3
20.6
17.5
13.4
15.6
10.7
17.1
17.6
19.2
16.8
16.4
14.1
16.0
15.2
23.4
14.7
18.9
11.8
18.5
14.7
18.6
20.3
12.0
11.7
18.3
129
134
123
126
152
144
126
132
129
131
152
127
133
123
135
136
128
136
131
124
120
135
132
123
130
133
133
137
133
130
131
129
122
127
132
127
128
132
133
130
135
136
116
132.0
133.2
133.0
131.2
131.3
128.9
131.0
135.4
133.1
131.9
129.8
124.3
134.4
131.1
134.1
131.7
132.9
134.9
133.6
129.7
130.2
131.8
131.4
132.0
132.8
134.6
135.1
131.0
135.4
134.4
133.8
132.9
125.1
135.2
133.6
133.9
134.7
135.2
135.4
134.9
131.5
135.2
127.6
26.7
22.6
23.3
29.6
29.3
37.6
30.0
15.0
23.0
27.0
34.3
54.9
18.5
29.8
19.4
27.7
23.5
16.6
21.3
34.7
33.0
27.2
28.8
26.6
24.0
17.6
16.0
30.2
15.0
18.5
20.4
23.7
52.0
15.8
21.2
20.2
17.5
15.8
15.0
16.8
28.3
15.7
42.5
3.0
0.8
10.0
5.2
20.7
15.1
5.0
3.4
4.1
0.9
22.2
2.7
1.4
8.1
0.9
4.3
4.9
1.1
2.6
5.7
10.2
3.2
0.6
9.0
2.8
1.6
2.1
6.0
2.4
4.4
2.8
3.9
3.1
8.2
1.6
6.9
6.7
3.2
2.4
4.9
3.5
0.8
11.6
16.0
13.3
15.6
25.5
18.4
29.0
20.6
5.6
13.8
15.0
28.7
40.9
10.4
23.7
6.5
20.5
12.0
6.7
10.9
28.1
25.6
22.0
19.1
16.8
17.3
9.9
8.0
21.3
5.8
13.8
12.9
14.1
44.2
7.4
15.7
11.0
6.1
11.0
4.8
10.2
17.2
11.8
36.7
134
140
134
138
140
143
132
135
140
134
146
136
137
132
136
140
134
140
137
133
131
134
136
131
140
137
133
142
144
138
136
136
138
137
139
135
120
130
131
129
143
134
139
133.5
138.1
127.2
133.2
158.1
153.2
131.7
133.6
132.9
135.3
161.7
138.3
136.1
129.5
136.9
142.2
131.3
138.0
134.1
131.8
126.8
141.6
137.5
127.5
135.0
135.9
135.3
143.4
134.7
134.0
134.7
133.0
134.2
129.1
136.6
130.1
129.7
135.2
134.4
132.9
140.1
139.6
125.5
8
11
3
2
18
18
0
0
1
1
21
22
1
5
2
2
1
1
3
13
2
3
0
1
13
1
8
12
2
1
5
8
5
1
1
1
12
10
0
2
6
1
11
However, patient 12 was comatose, anuric, with
important dehydration. Patient 2 had mild dehydration, a rise in PCr to 2.4 mg/dl and a heart rate of
92 bpm. Patients 25 and 27 had mental clouding and
dehydration. All other patients of this group had
normal volume and circulatory conditions by clinical
examination, were alert, had normal BP, no renal
failure, indicating an intake of hypotonic fluids capable
of balancing the losses due to the osmotic diuresis. In
this group, the mean PG1 was 22.4 2.3 mM/l, systolic
BP 141 8 and diastolic BP 84 5 mmHg.
Seven out of the remaining 25 patients (numbers 5,
6, 11, 16, 22, 28, 41) had PNa1 – PNaG > 3 mEq/l: they
belonged to group 2, where there was a significant
volume loss. They exhibited clinical signs of
dehydration. The PG1 averaged 30.6 1.5 mM/l, systolic BP 131 16, diastolic BP 78 12 mmHg.
The remaining 19 patients had PNa1 –
PNaG < 3 mEq/l. Therefore, they belonged to
group 3, which should have suffered from a significant
Na depletion caused by the prolonged osmotic diuresis,
compensated by a water intake larger than that of Na.
They did not show clinical signs of dehydration. Their
average PG1 was 25.9 2.2 mM/l, systolic BP 145 4,
diastolic 92 5 mmHg.
Figure 2 shows the plot between the actual
measurements of PNa1 and the values calculated
from GA (PNaG). Regressions and correlations are
significant only for groups 1 and 3. The intercepts
indicate that, extrapolated to a measurement of zero,
Hyperglycaemic hyponatraemia
Fig. 2. The abscissa shows the plasma sodium concentrations (PNa1)
measured on admission to the hospital. These values are plotted
against those calculated by PG1 (predicted values, or PNaG,
given by PNaGfe1 in the appendix). Patients belonging to group
1 are indicated by open circles, to group 2 by triangles, to
group 3 by closed circles. Regressions and correlations are:
PNaG ¼ 0.84 PNa1 þ 21.67, R2 ¼ 0.57, P < 0.001 for group 1;
PNaG ¼ 0.08 PNa1 þ 141.57, R2 ¼ 0.29, n.s. for group 2;
PNaG ¼ 0.52 PNa1 þ 66.15, R2 ¼ 0.63, P < 0.001 for group 3.
Fig. 3. The abscissa shows the plasma sodium concentrations (PNa2)
measured at an intermediate or final point of the correction of the
hyperglycaemia. These values are plotted against those calculated by
the fall in PG1 (expected values). Patients of all groups are included.
Regression and correlation are highly significant (PNa2
expected ¼ 0.88 PNa2 measured þ 16.31, R2 ¼ 0.31, P < 0.0001).
The regression coefficient of 0.88 underscores the fact that the
method underestimates the Na losses and, by consequence, the
predicted PNa2.
there is an overestimate of PNa in group 3 > 1,
suggesting the existence of Na depletion in both,
although significantly (P < 0.001) larger in group 3.
Group 2 did not show any significant correlation.
Figure 3 shows a similar plot between the
actual measurements obtained at the same time
point during correction (PNa2), and the values
computed by the associated changes in PG by
equations (17)–(20). Correlations and regressions are
highly significant.
3483
Fig. 4. The abscissa portrays the differences in plasma glucose
concentrations measured (PG1 – PG2) during correction of the
hyperglycaemia, plotted against the corresponding differences in
the measured plasma sodium concentrations (PNa2 – PNa1). Patients
of group 2 displayed, as expected, no significant correlation, and are
not shown. The equation is: PNa ¼ 0.36 PG 2.60, R2 ¼ 0.38.
The symbols include patients of groups 1 and 3, whose pairs of
values fall about the line which has a slope of 0.36 mEq/l/mM of
(PG1 – PG2), P < 0.0001. This corresponds to 2.2 mEq/l/100 mg% fall
in PG.
Figure 4 portrays the changes in PNa during
correction, plotted against the associated changes in
PG. As expected, PNa rises in all patients, parallel to
the fall in plasma glucose concentration.
The slope indicates that the rate of PNa rise, with
respect to the fall in PG, averaged 2.2 mEq of Na per
litre for 100 mg% drop in glucose concentration.
The logistic regression analysis was executed
between the scores of the mixed clinical laboratory
signs reported in the methods, and the groups stratified
according to the PNa1 – PNaG difference intended as
an independent appraisal of their volume conditions.
There is a significant concordance between the clinical
evaluation and the results of the quantitative analysis
of volume conditions derived from the mathematical
formulas. The overall F-value is 7.33, P < 0.002.
The difference between groups shows the following:
group 1 vs group 2, P < 0.02; group 2 vs group 3,
P < 0.02; group 1 vs group 3, P ¼ n.s.
The prediction of clinical signs in suggesting the
volume status gave the following results: mental state
(F ¼ 112, P < 0.001); appearance of skin and mucosae
(F ¼ 68, P < 0.001); blood pressure (F ¼ 60, P < 0.001);
urine volume (F ¼ 56; P < 0.001); haematocrit (F ¼ 51,
P < 0.001); creatinine clearance (F ¼ 51, P < 0.001);
temperature of extremities and forehead (F ¼ 49,
P < 0.001); filling of peripheral veins (F ¼ 40,
P < 0.001).
Discussion
The present work represents an initial effort in the
direction of a quantitative assessment of hyperosmolar
3484
coma, providing reliable predictions on the desired
modifications of electrolyte concentrations. For this
purpose, we built a simplified mathematical model that
considers only acute changes, affecting exclusively
the content of ECV. It yields exact results when GA
alone changes, whilst it performs less efficiently in the
presence of altered Na or water content and in
mixed derangements. However, the iteration method
described in the appendix circumvents some of these
problems, yielding reliable estimates even with mixed
derangements, at least on computer simulated patients.
Unfortunately, real patients are different from their
computer counterparts, since the rising glucose concentration dictates the onset of an osmotic diuresis
attended by important losses of solvent and Na [4].
Furthermore, the normal BW preceding the onset of
the alteration is seldom known in the emergency room,
where even the actual BW is difficult to measure,
limiting the applicability of more reliable formulas for
estimating the normal TBW [5].
The critical point of the patient evaluation must then
rely on a derived value, rather independent from these
confounding problems, represented by PNaG. It is
calculated, as shown in Figure 1, with the equations of
the methods and of the appendix. It represents the
plasma Na concentration that would be attained if
only GA had been added, without Na and/or water
loss, as could happen, ideally, in anephric patients. The
patient becomes hyponatraemic because of dilution of
his unchanged content of extracellular Na by the water
shifted from cells because of the osmotic effect of
added glucose. Thus, when measured and calculated
PNa are the same (PNa1 ¼ PNaG), the patient should
have no Na and/or water deficit, the ECV expanded,
good clinical and circulatory conditions, and above
all, he would need only insulin plus reintegration
of urinary losses to correct the disorder. The data of
group 1, reported for each patient in Table 2, as well as
those of the logistic analysis, strongly support our
contention. However, four patients appeared dehydrated at physical examination (patients 2, 12, 25 and
27 of Table 2). It is very likely that these subjects had
incurred in a negative Na balance quantitatively
corresponding to that of water, by a combination of
osmotic diuresis and oral intake. Thus, the usefulness
of our calculations cannot be separated from that of
the clinical reasoning, because the PNa1 PNaG
represents, in most instances, the correct clue to a
preserved ECV and Na content, while, in a few cases, it
could be the fortuitous combination of important,
although casually balanced, losses of both solvent and
solutes.
When PNa1 > PNaG, the osmotic diuresis must have
caused a loss of water in excess of that of Na, as
happened in a patient belonging to group 2. The
solvent lost can be calculated as: volume ¼ improvedECV1 (PNa1 – PNaG)/PNa1 (16). We calculated an
average volume loss of 1.1 0.3 l, significantly higher
than that of the other groups (P < 0.001).
When PNa1 < PNaG, there is a clear indication that
the Na deficit prevails over that of water. Since Na is
E. Bartoli et al.
lost through the kidney, this means that the volume
depletion caused by the osmotic diuresis is being
counterbalanced by an appropriate intake of tap
water, restoring the volume while not the salt deficit.
While non-hyperglycaemic hyponatraemia is attended
by ECV contraction, this hyperosmolar form is
accompanied by preserved or even expanded ECV.
The clinical data of group 3 patients and the logistic
regression analysis strongly supports this interpretation. The Na deficit can be calculated as: Na
mEq ¼ improved-ECV1 (PNa1 – PNaG) (15). We calculated an average Na loss of 100 12, significantly
higher than 19 8 mEq of group 1, P < 0.001. Since
(15) underestimates, on average, the salt loss,
the quantity so calculated should be considered a
minimum estimate.
The prediction of the results after treatment
constitutes an additional important aspect of our
view of hyperosmolar coma. In fact, the formulas of
the appendix allow us to compute the predicted PNa2
after correction. Figure 3 shows the paired data,
predicted vs measured, for PNa2. The significance of
the correlation coefficient confirms the validity of the
model on which the predicted values rest.
Our method also allows a prediction on the ratio
between the rise in PNa and the fall of PG during
treatment. This problem has been debated in the
literature [6–11]. Previous estimates are based on the
fact that 100 mg% glucose are equivalent, osmotically,
to 2.8 mEq of Naþ: hence, the rate of rise in PNa was
assumed to be 2.8 mEq/l/100 mg% fall in PG by Welt
[9]. Katz [1], on a single model and with one single
simulation based on TBW of 42 l and a GA of 1000,
calculated a theoretical rate of 1.6 mEq/l/100 mg%
fall in PG. Nguyen and Kurtz [10] assumes the rate
of 1.6 mEq/l/100 mg% fall in PG in his study as a
constitutive element of the equation for calculating
PNa2. Hillier instead [11] calculated empirically a rate
of 2.4 mEq/l/100 mg% fall in PG, which is similar to
ours. This rate, shown in Figure 4, is predicted by our
model, since our calculated PNa2 is strongly correlated
with that measured, as shown by Figure 3, and its
value is included in the ordinate of Figure 4.
The present article was undertaken with the aim of
introducing an important aid to the physician, that is, a
guide to compute the quantitative estimates of the
losses to be reintegrated, allowing the critical appraisal
of clinical data with mathematical calculations. Even
though the model yields exact estimates when certain
conditions are fulfilled, our data support its usefulness
in more extended conditions. The data suggest that the
physician should compute the PNa1 PNaG difference
with a suitable software. From this he will gain an
immediate diagnostic indication, that he can translate
into proper therapeutic options. He could replace the
ongoing urinary losses with the infusion of hemitonic
saline at a rate equal to that of the urine output, an
acceptable approximation to urine Na content during
osmotic diuresis. He will only add the insulin treatment
when the PNa1 PNaG difference is within 3 mEq/l,
provided the patients do not show signs of dehydration
Hyperglycaemic hyponatraemia
and mental clouding. He will infuse more hypotonic
solutions when the difference is positive, isotonic saline
when the difference is negative. He might consider
re-measuring PNa and PG after the calculated deficits
have been replaced: these values could still be below
those predicted, as the method, on average, underestimates the losses. Nevertheless, he would know that
his procedure is correct, and replace the missing
amounts of salt and/or water. Even during treatment
the clinical judgement remains at least as important as
the quantitative calculations, although their combined
use will significantly improve the effectiveness of the
correction.
In conclusion, this study represents the first step into
a way to quantitatively evaluate and treat hyperosmolar coma. The method we propose is exact in a good
number of clinical situations, while it is capable of
furnishing useful estimates of solvent and solute losses,
as well as glucose accumulation, in all other conditions.
Conflict of interest statement. The computer program necessary to
perform the calculations is copyrighted (SIAE registration n8
0604311) but not presently marketed.
References
1. Katz MA. Hyperglycemia-induced hyponatremia–calculation of
expected serum sodium depression. N Engl J Med 1973; 289:
843–844
2. Trinh-Trang-Tan M, Cartron JP, Bankir L. Molecular basis for
the dialysis disequilibrium syndrome: altered aquaporin and
urea transporter expression in the brain. Nephrol Dial Transplant
2005; 20: 1984–1988
3. 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. J Clin Invest 1958; 37:
1236–1256
4. Spira A, Gowrishankar M, Halperin ML. Factors contributing
to the degree of polyuria in a patient with poorly controlled
diabetes mellitus. Am J Kidney Dis 1997; 30: 829–835
5. Watson PE, Watson ID, Batt RD. Total body water volumes for
adult males and females estimated from simple anthropometric
measurements. Am J Clin Nutr 1980; 33: 27–39
6. Roscoe JM, Halperin ML, Rolleston FS, Goldstein MB.
Hyperglycemia-induced hyponatremia: metabolic considerations
in calculation of serum sodium depression. Can Med Assoc J
1975; 112: 452–453
7. Moran SM, Jamison RL. The variable hyponatremic response to
hyperglycemia. West J Med 1985; 142: 49–53
8. Crandall ED. Letter: Serum sodium response to hyperglycemia.
N Engl J Med 1974; 290: 465
9. Welt LG. Concept of osmotic uniformity and the significance of
hypo- and hypernatremia. Med Bull (Ann Arbor) 1958; 24:
486–498
10. Nguyen MK, Kurtz I. Are the total exchangeable sodium, total
exchangeable potassium and total body water the only
determinants of the plasma water sodium concentration?
Nephrol Dial Transplant 2003; 18: 1266–1271
11. Hillier TA, Abbott RD, Barrett EJ. Hyponatremia: evaluating
the correction factor for hyperglycemia. Am J Med 1999; 106:
399–403
Received for publication: 26.10.06
Accepted in revised form: 6.6.07
3485
Appendix
Iteration method to compute GA, PNaG and Na. The
iteration steps are numbered progressively.
(i) Posm1 ¼ (Posm0 TBW0 þ PG1 ECV0)/TBW0
(6); this calculation will underestimate the
true Posm, because PG1 ECV0 underestimates
the glucose appearance, given by GA ¼
PG1 ECV1;
(ii) ECV1 ¼ ECV0(Posm0 þ PG1)/Posm1 (7); this
calculation will give a better estimate of the
ECV, because the underestimate of the numerator (caused by PG1 ECV0), and that of the
denominator (caused by Posm1), compensate
each other;
(iii) PNaG (the Na concentration expected only by
GA) ¼ PNa0 ECV0/ECV1 (8);
(iv) Improved-Posm1 (improved by the initial
estimate of ECV) ¼ [(Posm0 TBW0) þ (PG1 ECV1)]/TBW0 (9);
(v) Improved-ECV1, improved by the second
estimate of Posm1, is given by: ImprovedECV1 ¼ [(ECV0 Posm0) þ (PG1 ECV1)]/
Improved-Posm1 (10);
(vi) Glucose appearance within the ECV, GA (better
estimate) ¼ PG1 Improved-ECV1 (11);
(final
estimate) ¼ PNa0 ECV0/
(vii) PNaGfe1
Improved-ECV1 (12);
predicted
by
glucose
addition
(viii) PosmG
only ¼ PG1 þ 2 PNaGfe1 (13);
(ix) PNa (the difference between the actual PNa1
and PNaGfe1) ¼ PNa1 – PNaGfe1 (14); PNa1 –
PNaGfe1 ¼ 0 indicates no volume and/or Na
depletion, and the above formulas are exact;
when PNa1 < PNaGfe1, the formula: (PNa1 –
PNaGfe1) Improved-ECV1 ¼ Na (15), represents an exact estimate of Na depletion when
V ¼ 0, while it represents a minimum estimate
when V 6¼ 0; when PNa1 > PNaGfe1, the
formula:
(PNa1PNaGfe1) Improved-ECV1/
PNa1 ¼ ECV (16), estimates exactly the
ECV contraction when Na ¼ 0, while it represents a minimum estimate of its change when
Na 6¼ 0.
These calculations were performed in an iterative
mode, whereby each subsequent computation utilized
the value yielded by the preceding equation, following
their progressive numeration.
In the example of Figure 1, the iteration method
yields, at the first run, the following calculations:
GA ¼ 1988.5 while the true value is 2000, a 99.4%
approximation; Posm ¼ 329.7 mOsm/kg, while the true
value is 330; ECV ¼ 18.8 l, equal (within the rounding
off) to the true value; PNa ¼ 112 mEq/l, equal
(within the rounding off) to the true value;
PNa1 PNaGfe1 ¼ 0, which means that the TBW is
unchanged and Na ¼ 0.
If GA were estimated by PG1 ECV0, it would
be computed as 106 15 ¼ 1590 mM, a 79%
3486
approximation (a 21% error). The PNaGfe1 would be
calculated, by consequence, as 116 mEq/l, indicating a
non-existing water loss.
During treatment, as glucose is metabolized and its
osmotic effect fades away, water shifts into cells and
the contraction of ECV is attended by a rise in PNa.
Subfix 1 indicates the values before treatment, while 2
indicates values measured either during or at the end of
treatment:
Glucose disappeared during treatment up to the
time2 of measurement ¼ (PG1 – PG2) Improved-ECV1
E. Bartoli et al.
(17);
Posm2 ¼ (Posm0 TBW0 þ PG2 Improved(18);
ECV2 ¼ [(ECV0 Posm0) þ
ECV1)/TBW0
(PG2 Improved-ECV1)]/Posm2 (19); PNa2¼ PNa1 Improved-ECV1/ECV2 (20). This ‘predicted’ PNa2
can be compared to that actually measured, to check
the accuracy of prediction, which depends upon that
of the equations. An appropriate software is available
to help with these calculations (SIAE registration
8/16/06, number 0604311, informations from
>[email protected]<, quoting reference program
number 0602).