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 (Na_{e}+ K_{e})/TBW} – 25.6
Where [Na

^{+}] = plasma sodium concentration, Na_{e}=total body exchangeable sodium, K_{e}=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 TBW_{baseline}= [Na^{+}]_{baseline}x TBW_{current}
·
TBW

_{current}= [Na^{+}]_{baseline}x TBW_{baseline}/[Na^{+}]_{current}… (1)
3.
Free water deficit can be expressed as:

·
FWD = TBW

_{baseline}– TBW_{current }… (2)
4.
Then replacing (1) in (2):

·
FWD = TBW

_{baseline}– ([Na^{+}]_{baseline}x TBW_{baseline})/[Na^{+}]_{current}
·
FWD = TBW

_{baseline}x (1 – [Na^{+}]_{baseline}/[Na^{+}]_{current})
5.
If [Na

^{+}]_{baseline}is considered normal at 140 mEq/L then:
·
FWD = TBW

_{baseline}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 TBW_{baseline}= [Na^{+}]_{current}x TBW_{current}
·
TBW

_{baseline}= [Na^{+}]_{current}x TBW_{current}/[Na^{+}]_{baseline}… (1)
3.
Free water deficit can be expressed as:

·
FWD = TBW

_{baseline}– TBW_{current}… (2)
4.
Then replacing (1) in (2):

·
FWD = [Na

^{+}]_{current}x TBW_{current}/[Na^{+}]_{baseline}– TBW_{current}
·
FWD = TBW

_{current}x ([Na^{+}]_{current}/[Na^{+}]_{baseline}- 1)
5.
If [Na

^{+}]_{baseline}is considered normal at 140 mEq/L then:
·
FWD = TBW

_{current}x ([Na^{+}]_{current}/140 - 1)

__Calculating Rate of Infusion of Hypertonic Saline__

**Method # 1: Na deficit formula**

Deriving Na deficit formula

1.
Na deficit = Na

_{goal}– Na_{current}… (1)
2.
Since Na + K = [Na

^{+}] x TBW, then Na = [Na+] x TBW – K … (2)
3.
Replacing (2) in (1)

·
Na deficit = TBW

_{goal}x [Na^{+}]_{goal}– K_{goal}– {TBW_{current}x [Na^{+}]_{current}– K_{current}}
4.
Assuming TBW and K remain constant, so TBW

_{goal}= TBW_{current},_{ }and K_{goal}= K_{current}, then TBW = TBW_{goal}= TBW_{current}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}= (Na_{current}+ K_{current})/TBW_{current}
·
[Na

^{+}]_{current}x TBW_{current}= (Na_{current}+ K_{current}) … (1)
2.
[Na

^{+}]_{goal}will be the new [Na^{+}] that results when we administer 1L of an infusate containing Na_{infusate}and K_{infusate}, then:
·
[Na

^{+}]_{goal}= (Na_{current}+ K_{current}+ Na_{infusate}+ K_{infusate})/(TBW_{current}+ 1) …(2)
3.
Substracting [Na

^{+}]current from both terms of equation (2), then:
·
[Na

^{+}]_{goal}– [Na^{+}]_{current}= (Na_{current}+ K_{current}+ Na_{infusate }+ K_{infusate})/(TBW_{current}+ 1) – [Na^{+}]_{current}_{}

4.
But [Na

^{+}]_{goal}– [Na^{+}]_{current}is the same as change in [Na^{+}], then:
·
Change in [Na

^{+}] = (Na_{current}+ K_{current}+ Na_{infusate}+ K_{infusate})/(TBW_{current}+ 1) – [Na^{+}]_{current}
·
Change in [Na

^{+}] = {(Na_{current}+ K_{current}+ Na_{infusate}+ K_{infusate}) – (TBW_{current}+ 1) x Na_{current}}/(TBW_{current}+ 1)
·
Change in [Na

^{+}] = {Na_{current}+ K_{current}+ Na_{infusate }+ K_{infusate}– ([Na^{+}]_{current}x TBW_{current}–[Na^{+}]_{current})}/(TBW_{current}+ 1) … (3)
5.
Replacing equation (1) in (3), then:

·
Change in [Na

^{+}] = {Na_{current}+ K_{current}+ Na_{infusate}+ K_{infusate}– (Na_{current }+ K_{current}) - [Na^{+}]_{current}}/(TBW + 1)
6.
Cancelling out Na

_{current}+ K_{current}then:
·
Change in [Na

^{+}] = {Na_{infusate}+ K_{infusate}- [Na^{+}]_{current}}/(TBW_{current}+ 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:

Wat's the limitation of using sodium deficit formula?

Method #2: Adrogue-Madias formulawith 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)

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