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):

Original Edelman equation (J Clin Invest. 1958;37:1236-56):

**[Na+]**

= {1.1 x (Nae + Ke)/TBW} – 25.6

= {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.

concentration, Nae=total body exchangeable sodium, Ke=total

body exchangeable potassium, TBW = total body water.

·

Simplified Edelman equation: [Na+] = (Na

+ K)/TBW

Simplified Edelman equation: [Na+] = (Na

+ K)/TBW

· [Na+] x TBW = Na + K

· Na + K = [Na+] x TBW

__Calculating Free__

Water Deficit (FWD)Water Deficit (FWD)

**Method #1 (Using**

baseline weight, certainty about what % of body weight is water)

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:

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

(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):

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

[Na+]baseline x TBWbaseline

= [Na+]baseline x TBWcurrent

·

TBWcurrent = [Na+]baseline

x TBWbaseline/[Na+]current … (1)

TBWcurrent = [Na+]baseline

x TBWbaseline/[Na+]current … (1)

3.

Free water deficit can be expressed as:

Free water deficit can be expressed as:

·

FWD = TBWbaseline – TBWcurrent …

(2)

FWD = TBWbaseline – TBWcurrent …

(2)

4.

Then replacing (1) in (2):

Then replacing (1) in (2):

·

FWD = TBWbaseline – ([Na+]baseline

x TBWbaseline)/[Na+]current

FWD = TBWbaseline – ([Na+]baseline

x TBWbaseline)/[Na+]current

·

FWD = TBWbaseline x (1 – [Na+]baseline/[Na+]current)

FWD = TBWbaseline x (1 – [Na+]baseline/[Na+]current)

5.

If [Na+]baseline is

considered normal at 140 mEq/L then:

If [Na+]baseline is

considered normal at 140 mEq/L then:

·

FWD = TBWbaseline x (1 – 140/[Na+]current)

FWD = TBWbaseline x (1 – 140/[Na+]current)

**Method #2 (Using**

current weight, uncertainty about what % of body weight is water)

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:

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

(Na + K)baseline = (Na + K)current

2.

Sodium and potassium masses can be expressed as

sodium concentration ([Na+]) multiplied by total body water (TBW):

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

[Na+]baseline x TBWbaseline

= [Na+]current x TBWcurrent

·

TBWbaseline = [Na+]current

x TBWcurrent/[Na+]baseline … (1)

TBWbaseline = [Na+]current

x TBWcurrent/[Na+]baseline … (1)

3.

Free water deficit can be expressed as:

Free water deficit can be expressed as:

·

FWD = TBWbaseline – TBWcurrent

… (2)

FWD = TBWbaseline – TBWcurrent

… (2)

4.

Then replacing (1) in (2):

Then replacing (1) in (2):

·

FWD = [Na+]current x TBWcurrent/[Na+]baseline

– TBWcurrent

FWD = [Na+]current x TBWcurrent/[Na+]baseline

– TBWcurrent

·

FWD = TBWcurrent x ([Na+]current/[Na+]baseline

– 1)

FWD = TBWcurrent x ([Na+]current/[Na+]baseline

– 1)

5.

If [Na+]baseline is

considered normal at 140 mEq/L then:

If [Na+]baseline is

considered normal at 140 mEq/L then:

·

FWD = TBWcurrent x ([Na+]current/140

– 1)

FWD = TBWcurrent x ([Na+]current/140

– 1)

__Calculating Rate__

of Infusion of Hypertonic Salineof Infusion of Hypertonic Saline

**Method # 1: Na**

deficit formula

deficit formula

Deriving Na deficit formula

1.

Na deficit = Nagoal – Nacurrent

… (1)

Na deficit = Nagoal – Nacurrent

… (1)

2.

Since Na + K = [Na+] x TBW, then Na =

[Na+] x TBW – K … (2)

Since Na + K = [Na+] x TBW, then Na =

[Na+] x TBW – K … (2)

3.

Replacing (2) in (1)

Replacing (2) in (1)

·

Na deficit = TBWgoal x [Na+]goal

– Kgoal – {TBWcurrent x [Na+]current – Kcurrent}

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:

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

– TBW x [Na+]current

·

Na deficit = TBW x ([Na+]goal

– [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:

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

Na deficit = TBW x 6 mEq/L

Calculating volume of infusate

·

Volume of infusate = Na deficit x (1000 mL/513

mEq)

Volume of infusate = Na deficit x (1000 mL/513

mEq)

Calculating rate of infusion

·

Rate of infusion = volume of infusate/24h

Rate of infusion = volume of infusate/24h

**Method #2: Adrogue-Madias**

formula

formula

Deriving Adrogue-Madias formula

1.

[Na+] = (Na + K)/TBW … (Edelman

equation)

[Na+] = (Na + K)/TBW … (Edelman

equation)

·

[Na+]current = (Nacurrent

+ Kcurrent)/TBWcurrent

[Na+]current = (Nacurrent

+ Kcurrent)/TBWcurrent

·

[Na+]current x TBWcurrent

= (Nacurrent + Kcurrent) … (1)

[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 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)

[Na+]goal = (Nacurrent

+ Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent

+ 1) …(2)

3.

Substracting [Na+]current from both

terms of equation (2), then:

Substracting [Na+]current from both

terms of equation (2), then:

·

[Na+]goal – [Na+]current

= (Nacurrent + Kcurrent + Nainfusate + Kinfusate)/(TBWcurrent

+ 1) – [Na+]current

[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:

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) – [Na+]current

·

Change in [Na+] = {(Nacurrent

+ Kcurrent + Nainfusate + Kinfusate) – (TBWcurrent

+ 1) x Nacurrent}/(TBWcurrent + 1)

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)

Change in [Na+] = {Nacurrent

+ Kcurrent + Nainfusate + Kinfusate – ([Na+]current

x TBWcurrent –[Na+]current)}/(TBWcurrent

+ 1) … (3)

5.

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

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

·

Change in [Na+] = {Nacurrent

+ Kcurrent + Nainfusate + Kinfusate – (Nacurrent

+ Kcurrent) – [Na+]current}/(TBW + 1)

Change in [Na+] = {Nacurrent

+ Kcurrent + Nainfusate + Kinfusate – (Nacurrent

+ Kcurrent) – [Na+]current}/(TBW + 1)

6.

Cancelling out Nacurrent + Kcurrent

then:

Cancelling out Nacurrent + Kcurrent

then:

·

Change in [Na+] = {Nainfusate

+ Kinfusate – [Na+]current}/(TBWcurrent

+ 1)

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 (Change in [Na+])goal}/(Change

in [Na+])

·

Volume of infusate = {1000 mL x 6 mEq/L}/(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

Rate of infusion = volume of infusate/24h

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)

Wat's the limitation of using sodium deficit formula?