Edelman is the Root of Almost All Good in Nephrology

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Almost all the formulas we use in the management of the disorders of water homeostasis are derived from the Edelman equation. I am presenting where these formulas come from for the math aficionados.
 
Edelman equation
·
Original Edelman equation (J Clin Invest. 1958;37:1236-56):
[Na+]
= {1.1 x (Nae + Ke)/TBW} – 25.6
Where [Na+] = plasma sodium
concentration, Nae=total body exchangeable sodium, Ke=total
body exchangeable potassium, TBW = total body water.
·
Simplified Edelman equation: [Na+] = (Na
+ K)/TBW
·         [Na+] x TBW = Na + K
·         Na + K = [Na+] x TBW
Calculating Free
Water Deficit (FWD)
 
Method #1 (Using
baseline weight, certainty about what % of body weight is water)
 
1.
Assuming only pure water has been lost, the total
body sodium and potassium remain constant so the total body sodium and
potassium at baseline (Na + K)baseline and the total body sodium and
potassium after water loss (Na + K)current are equal:
·
(Na + K)baseline = (Na + K)current
2.
Total body sodium and potassium can be expressed
as sodium concentration ([Na+]) multiplied by total body water (TBW):
·
[Na+]baseline x TBWbaseline
= [Na+]baseline x TBWcurrent
·
TBWcurrent = [Na+]baseline
x TBWbaseline/[Na+]current … (1)
3.
Free water deficit can be expressed as:
·
FWD = TBWbaseline – TBWcurrent …
(2)
4.
Then replacing (1) in (2):
·
FWD = TBWbaseline – ([Na+]baseline
x TBWbaseline)/[Na+]current
·
FWD = TBWbaseline x (1 – [Na+]baseline/[Na+]current)
5.
If [Na+]baseline is
considered normal at 140 mEq/L then:
·
FWD = TBWbaseline x (1 – 140/[Na+]current)
Method #2 (Using
current weight, uncertainty about what % of body weight is water)
 
1.
Assuming only pure water has been lost, the total
body sodium and potassium remain constant so the total body sodium and
potassium at baseline (Na + K)baseline and the total body sodium and
potassium after water loss (Na + K)current are equal:
·
(Na + K)baseline = (Na + K)current
2.
Sodium and potassium masses can be expressed as
sodium concentration ([Na+]) multiplied by total body water (TBW):
·
[Na+]baseline x TBWbaseline
= [Na+]current x TBWcurrent
·
TBWbaseline = [Na+]current
x TBWcurrent/[Na+]baseline … (1)
3.
Free water deficit can be expressed as:
·
FWD = TBWbaseline – TBWcurrent
… (2)
4.
Then replacing (1) in (2):
·
FWD = [Na+]current x TBWcurrent/[Na+]baseline
– TBWcurrent
·
FWD = TBWcurrent x ([Na+]current/[Na+]baseline
– 1)
5.
If [Na+]baseline is
considered normal at 140 mEq/L then:
·
FWD = TBWcurrent x ([Na+]current/140
– 1)
Calculating Rate
of Infusion of Hypertonic Saline
 
Method # 1: Na
deficit formula
 
Deriving Na deficit formula
1.
Na deficit = Nagoal – Nacurrent
… (1)
2.
Since Na + K = [Na+] x TBW, then Na =
[Na+] x TBW – K … (2)
3.
Replacing (2) in (1)
·
Na deficit = TBWgoal x [Na+]goal
– Kgoal – {TBWcurrent  x [Na+]current – Kcurrent}
4.
Assuming TBW and K remain constant, so TBWgoal
= TBWcurrent, and Kgoal = Kcurrent,
then TBW = TBWgoal = TBWcurrent and K is cancelled out
from equation:
·
Na deficit = TBW x [Na+]goal
– TBW x [Na+]current
·
Na deficit = TBW x ([Na+]goal
– [Na+]current)
5.
Since now we aim for an increase in [Na+]
of 6 mEq/L, so [Na+]goal – [Na+]current =
6 mEq/L then:
·
Na deficit = TBW x 6 mEq/L
Calculating volume of infusate
·
Volume of infusate = Na deficit x (1000 mL/513
mEq)
Calculating rate of infusion
·
Rate of infusion = volume of infusate/24h
Method #2: Adrogue-Madias
formula
 
Deriving Adrogue-Madias formula
1.
[Na+] = (Na + K)/TBW … (Edelman
equation)
·
[Na+]current = (Nacurrent
+ Kcurrent)/TBWcurrent
·
[Na+]current x TBWcurrent
= (Nacurrent + Kcurrent) … (1)
2.
[Na+]goal will be the new
[Na+] that results when we administer 1L of an infusate containing
Nainfusate and Kinfusate, then:
·
[Na+]goal = (Nacurrent
+ Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent
+ 1) …(2)
3.
Substracting [Na+]current from both
terms of equation (2), then:
·
[Na+]goal – [Na+]current
= (Nacurrent + Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent
+ 1) – [Na+]current
4.
But [Na+]goal – [Na+]current
is the same as change in [Na+], then:
·
Change in [Na+] = (Nacurrent
+ Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent
+ 1) – [Na+]current
·
Change in [Na+] = {(Nacurrent
+ Kcurrent + Nainfusate + Kinfusate) – (TBWcurrent
+ 1) x Nacurrent}/(TBWcurrent + 1)
·
Change in [Na+] = {Nacurrent
+ Kcurrent + Nainfusate + Kinfusate – ([Na+]current
x TBWcurrent –[Na+]current)}/(TBWcurrent
+ 1) … (3)
5.
Replacing equation (1) in (3), then:
·
Change in [Na+] = {Nacurrent
+ Kcurrent + Nainfusate + Kinfusate – (Nacurrent
+ Kcurrent) – [Na+]current}/(TBW + 1)
6.
Cancelling out Nacurrent + Kcurrent
then:
·
Change in [Na+] = {Nainfusate
+ Kinfusate – [Na+]current}/(TBWcurrent
+ 1)
Calculating volume of infusate
·
Volume of infusate = {1000 mL x (Change in [Na+])goal}/(Change
in [Na+])
·
Volume of infusate = {1000 mL x 6 mEq/L}/(Change
in [Na+])
Calculating rate of infusion
·
Rate of infusion = volume of infusate/24h

2 comments

  1. Method #2: Adrogue-Madias formula with some corrections ^_^

    Deriving Adrogue-Madias formula

    (A) [Na+] = (Na + K)/TBW (Edelman equation)

    [Na+]current = (Nacurrent + Kcurrent)/TBWcurrent

    [Na+]current x TBWcurrent = (Nacurrent + Kcurrent) (1)

    (B) [Na+]goal will be the new [Na+] that results when we administer 1 L of an infusate containing Nainfusate and Kinfusate, then:

    [Na+]goal = (Nacurrent + Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent + 1 L) (2)

    (C) Substracting [Na+]current from both terms of equation (2), then:

    [Na+]goal – [Na+]current = (Nacurrent + Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent + 1 L) – [Na+]current

    (D) But [Na+]goal – [Na+]current is the same as change in [Na+], then:

    Change in [Na+] = (Nacurrent + Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent + 1 L) – [Na+]current

    Change in [Na+] = {(Nacurrent + Kcurrent + Nainfusate + Kinfusate) +
    – (TBWcurrent + 1 L) x [Na+]current}/(TBWcurrent + 1 L)

    Change in [Na+] = {Nacurrent + Kcurrent + Nainfusate + Kinfusate +
    – ([Na+]current x TBWcurrent + [Na+]current x 1 L)}/(TBWcurrent + 1 L) (3)

    (E) Replacing equation (1) in (3), then:

    Change in [Na+] = {Nacurrent + Kcurrent + Nainfusate + Kinfusate +
    – (Nacurrent + Kcurrent) – [Na+]current x 1 L}/(TBW + 1 L)

    (F) Cancelling out Nacurrent + Kcurrent then:

    Change in [Na+] = {Nainfusate + Kinfusate – [Na+]current x 1 L}/(TBWcurrent + 1 L)

  2. Wat's the limitation of using sodium deficit formula?

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