ELECTRONIC SUPPLEMENTAL MATERIAL
Abnormalities of serum potassium concentration in dialysis-associated
hyperglycemia and their correction with insulin: A unique
clinical/physiologic exercise in internal potassium balance
Antonios H. Tzamaloukas, Todd S. Ing, Moses S. Elisaf, Dominic S.C. Raj, Kostas C. Siamopoulos,
Mark Rohrscheib, Glen H. Murata
APPENDIX
Abbreviations and Definitions
[K+]e = extracellular potassium concentration, [K+]i = intracellular potassium concentration
[Na+]e = extracellular sodium concentration, [Na+]i = intracellular sodium concentration
[Cl-]e = extracellular chloride concentration, [Cl-]i = intracellular chloride concentration
EΚ = transmembrane electrical potential gradient calculated by the Nernst equation
EM = transmembrane electrical potential gradient calculated by the Goldman-Hodgkin-Katz equation
The Nernst equation:
EΚ = (RT/F)×ln([K+]e/[ K+]i)
where R is the gas constant (8.314 joules per Kelvin), T is the absolute temperature (Kelvin), and F is
Faraday’s constant (9.65×104 coulomb/mol). For example: At room temperature, RT/F = 25 and if [K+]e
is 4 mmol/L and [K+]i is 145 mmol/L, the value of the Nernst equation will be EΚ = 25×ln(4/145) = - 90
mVolt [22].
The Goldman-Hodgkin-Katz equation:
Em = (RT/F)×ln{(PK×[ K+]e + PNa×[Na+]e + PCl×[Cl-]i)/(PK×[K+]i + PNa×[Na+]i + PCl×[Cl-]e)}
where the prefix P represents the membrane permeability for potassium (K), sodium (Na) and chloride
(Cl) respectively.
Computation of insulin effects on internal potassium balance
For the calculations presented, it was assumed that either at euglycemia or in the fasting state extracellular
volume (Ve) = 14 L, intracellular volume (Vi) = 28 L, [K+]e = 4 mmol/L, [K+]i = 145 mmol/L, and
therefore [K+]e/[K+]i = 0.0276 and initial EΚ = -90 mVolt (Nernst equation).
Total body water = Ve + Vi, or 42 L {1}
Total body potassium = Ve×[K+]e + Vi×[K+]i, or 4116 mmol {2}
A. Development of dialysis-associated hyperglycemia. Total body potassium constant
The following calculations were made for changes in internal potassium balance occurring during
development of dialysis-associated hyperglycemia (DH). The changes that occur during correction of DH
with insulin have exactly the reverse direction and magnitude.
1. Effects of hypertonicity on the extracellular and intracellular volumes
This stage describes the effects of hypertonicity on body water distribution between the intracellular and
extracellular compartments during development of DH, while EK remains at -90 mVolt. The following
equations compute the fractional changes in extracellular volume (ΔV/Ve) and intracellular volume
(ΔV/Vi) during development of DH [31]:
ΔV/Ve = (Vi/Ve)×Δ[Glu]/{(Vi/Ve + 1)×(2×[Na+] + [Glu]) + Δ[Glu]}
ΔV/Vi = -Δ[Glu]/{(Vi/Ve + 1)×(2×[Na+] + [Glu]) + Δ[Glu]}
{3}
{4}
where [Glu] in mmol/L, and [Na+] are respectively serum glucose and serum sodium concentrations at
euglycemia, and Δ[Glu] the increase in serum glucose concentration. Example: [Na+] = 140 mmol/L,
[Glu] = 5.6 mmol/L (100 mg/dL), and (Δ[Glu]) = 61.1 mmol/L (1100 mg/dL). From equation 3, ΔV/Ve
= 0.133 L/L, and hyperglycemic extracellular volume = 14×(1 + 0.133) = 15.867 L. From equation 4,
ΔV/Vi = -0.067 L/L, and hyperglycemic intracellular volume = 28×(1 – 0.067) = 26.134 L. If [K+]e1 and
[K+]i1 are respectively the extracellular and intracellular potassium concentrations at hyperglycemia, and
because total body potassium content will not change during development of DH, from equation 2:
15.867×[K+]e1 + 26.134×[K+]i1 = 4116 mmol
{5}
Because also the transmembrane electrical potential gradient was also assumed to stay the same at -90
mvolt during development of DH:
[K+]e1 /[K+]i1 = 0.0276 {6}
From equations 5 and 6, [K+]e1 = 4.3 mmol/L and [K+]i1 = 154.9 mmol/L.
2. Effects of insulin absence through depolarization of the cell membrane
In this second stage, we assumed that insulin absence led to a decrease (absolutely) in EK from -90 to 80 mVolt. At an EK of -80 mVolt, the relation between the intracellular and extracellular potassium
concentrations, [K+]e2 and [K+]i2 respectively, will be from the Nernst equation:
[K+]e2 /[K+]i2 = 0.0408 {7}
From equations 5 and 7, [K+]e2 = 6.3 mmol/L and /[K+]i2 = 153.7 mmol/L.
By varying the euglycemic value of [K+]e it can be shown that the magnitude of the hyperkalemia
during development of both the hypertonicity and the cell membrane depolarization in DH depends on the
euglycemic [K+]e value.
B. Food intake: Changes in total body potassium
Assume that a subject consumed a large meal containing 80 mmol of potassium. According to known
physiologic principles, roughly 50% of the ingested potassium, or 40 mmol, will appear in the urine
within the first few hours after the meal. If the other 40 mmol of potassium had remained in the
extracellular compartment, total extracellular potassium would be 4×14 + 40 = 96 mmol, [K+]e would rise
to 96/14 = 6.9 mmol/L, the ratio [K+]e/[K+]i would be 6.9/145 = 0.0476, and EΚ would decrease to -76.1
mVolt (Nernst equation), increasing the arrythmogenic potential. If the secreted insulin returns the EΚ to
– 90 mVolt, the ratio [K+]e/[K+]i will be again 0.0276. In this case: Total body potassium = 14×6.9 +
28×145, or, after simplification:
[K+]e + 2×[ K+]i = 6.9 + 2×145 {8}, and
[K+]e/[ K+]i = 0.0276
{9}
By solving equations 8 and 9, final [K+]e = 4.04 mmol/L and final [K+]i will be 146.4 mmol/L.