Analytical solution of phosphate kinetics for hemodialysis

Abstract

Chronic kidney diseases imply an ongoing need to remove toxins, with hemodialysis as the preferred treatment modality. We derive analytical expressions for phosphate clearance during dialysis, the single pass (SP) model corresponding to a standard clinical hemodialysis and the multi pass (MP) model, where dialysate is recycled and therefore makes a smaller clinical setting possible such as a transportable dialysis suitcase. For both cases we show that the convective contribution to the dialysate is negligible for the phosphate kinetics and derive simpler expressions. The SP and MP models are calibrated to clinical data of ten patients showing consistency between the models and provide estimates of the kinetic parameters. Immediately after dialysis a rebound effect is observed. We derive a simple formula describing this effect which is valid both posterior to SP or MP dialysis. The analytical formulas provide explanations to observations of previous clinical studies.

Keywords: 34A30; 92-10; 92B99; 92C30.

Fig. 1

Conceptual single pass dialysis

Fig. 2

Multiple pass (MP) HD where the dialysate is recirculated

Fig. 3

Analytical solutions to the SP and MP models of HD with two hours post dialysis rebound. a Four hours SP dialysis, grey curve is numerical integration of Eq. (4), red curve is the formula (6) and blue is the formula when neglecting ultrafiltration, Eq. (8). Black curve is the rebound after dialysis, given by Eq. (44). b Eight hours MP analysis. Curves starting at (0, 2.9) are phosphate concentration in serum, curves starting at (0, 0) are phosphate in dialysate. Grey curves correspond to direct numerical integration of the differential equations. Red curve is the analytical series solution including the first 27 terms. Blue curve is the analytical solution with $Q=0$, Eq. (42). Black curve is the rebound after treatment, Eq. (44). Parameter values in both panels are $K_s=2.6$ L/h, $C_s=2.9$ mmol/L, $K_b=6.9$ L/h, $Q=0.23$ L/h, $V_{b0}=10$ L, $x_0=C_s,y_0=0$ (color figure online)

Fig. 4

SP dialysis. Data points are represented by dots, the full and dashed lines are the SP model, Eq. (45), and post dialysis rebound. The parameter values are listed in Table 3

Fig. 5

MP dialysis. Data points are represented by dots, the full and dashed lines are MP model (Eqs. 46–47), and post dialysis rebound. Parameter values are given in Table 4

Fig. 6

CP dialysis. Data points are represented by dots, the full and dashed lines are the SP and MP models, Eqs. (45–47), with identical parameter values and stipulated curves are post dialysis rebound. Parameter values are given by Table 5