# Edelman is the Root of Almost All Good in Nephrology

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
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

1. (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?