Mathematical analysis of urea rebound in long-term hemodialysis patients

Quantifying hemodialysis (HD) treatment requires knowledge of the equilibrated concentrations of the post-HD small molecule rebounds. However, measurement of the equilibrated concentrations is only possible after resting in bed after HD for at least 30 min, and this is often impractical. Therefore, we have analyzed mathematically the time course of post-HD urea rebound, and from this, have derived a new formula for predicting its equilibrated concentration. The blood urea nitrogen (BUN) was measured at 10 time points (immediately following HD, and 0.5, 2.5, 5, 7.5, 10, 15, 20, 25, and 30 min post-HD) in 12 anuric HD patients. The absolute change in the urea rebound (DeltaeqBUN) was approximated (DeltaestBUN) using the equation: DeltaestBUN = b -[1-exp x (-c x time (min))] + a x time (min). After the good correlation between DeltaeqBUN and DeltaestBUN, we compared the value of DeltaeqBUN measured at 30 min (DeltaeqBUN(30)) with that calculated (DeltaestBUN(30)) using only four sample points (immediately following HD, and 2.5, 5 and 10 min post-HD). Based on this result, we tried to predict post-HD BUN at 30 min (estBUN(30)). This study was undertaken to determine whether estBUN(30) may be representative of the equilibrated BUN (eqBUN(30)), and to compare with Kt/V using estBUN(30) and eqBUN(30). There was a significant correlation between DeltaeqBUN and DeltaestBUN (0.97 < r < 0.99, P < 0.001). Thus, there was a significant positive linear correlation between eqBUN(30) and estBUN(30) (eqBUN(30): 25.7 +/- 2.25 mg/dL, estBUN(30): 26.3 +/- 2.31 mg/dL; r(2) = 0.99, P < 0.001). A Kt/V measurement was obtained with single pool model using BUN just after HD (Kt/V(sp)), eqBUN(30) (Kt/V(eq)), and estBUN(30) (Kt/V(est)), and with double pool model using Kt/V(sp) (Kt/V(dp)) and was compared with them. Though Kt/V(sp) was significantly higher than Kt/V(eq) (1.26 +/- 0.08 vs. 1.09 +/- 0.07, P < 0.001), there were no differences among Kt/V(eq), Kt/V(est) and Kt/V(dp) (Kt/V(est): 1.06 +/- 0.07, Kt/V(dp): 1.10 +/- 0.07) and all values were clinically acceptable. Furthermore, there was a significant positive linear correlation between Kt/V(eq) and Kt/V(est) (r(2) = 0.98, P < 0.001). In conclusion, we have devised the method to predict equilibrated BUN and calculate double pool Kt/V, which requires samples up to 10 min post-HD.