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 (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