Improving on the Adrogué-Madias Formula

Abstract

The Adrogué-Madias (A-M) formula is correct as written, but technically it only works when adding one liter of an intravenous (IV) fluid. For all other volumes, the A-M algorithm gives an approximate answer, one that diverges further from the truth as the IV volume is increased. If one liter of an IV fluid is calculated to change the serum sodium by some amount, then it was long assumed that giving a fraction of the liter would change the serum sodium by a proportional amount. We challenged that assumption and now prove that the A-M change in [sodium] is not scalable in a linear way. Rather, the delta [Na] needs to be scaled in a way that accounts for the actual volume of IV fluid being given. This is accomplished by our improved version of the A-M formula in a mathematically rigorous way. Our equation accepts any IV fluid volume, eliminates the illogical infinities, and, most importantly, incorporates the scaling step so that it cannot be forgotten. However, the non-linear scaling makes it harder to obtain a desired delta [Na]. Therefore, we reversed the equation so that clinicians can enter the desired delta [Na], keeping the rate of sodium correction safe, and then get an answer in terms of the volume of IV fluid to infuse. The improved equation can also unify the A-M formula with the corollary A-M loss equation wherein one liter of urine is lost. The method is to treat loss as a negative volume. Since the new equation is just as straightforward as the original formula, we believe that the improved form of A-M is ready for immediate use, alongside frequent [sodium] monitoring.