New analytical solution for Pyle-Popovich's peritoneal dialysis model

Continuous Ambulatory Peritoneal Dialysis (CAPD) is one of the standard treatments for kidney disease patients. A washing solution, called dialysate, is put into the peritoneal cavity to remove waste products and excess amounts of water in CAPD. The dialysate is exchanged four to five times a day by the patient. However, it is not easy to prescribe CAPD therapy, which may have precluded popularization of CAPD therapy. Popovich et al. constructed a mathematical model (P-P model) that applies to the prescription of the treatment schedule. It requires, however, a number of iterative calculations to obtain an exact numerical solution because the model is a set of nonlinear simultaneous ordinary differential equations. In this paper, the authors derived a new approximated analytical solution by employing a time-discrete technique, assuming all the parameters to be constant within each piecewise period of time for the P-P model. We have also described an algorithm of a numerical calculation with the new solution for determining a set of unknown parameters in the P-P model. We compare the validation of the new solution for clinical use with another analytical solution (Vonesh's solution). The new analytical solution consists of a forward solution (FW solution), that is the solution for the plasma and dialysate concentrations from t_i to t_i+1 (t₁ < t_i+1), and a backward solution (BW solution) from t₁ to t_i-1 (t_i-1 < t_i). The unknown parameters were determined by employing the Newton-Raphson method, a trial-andor method and the modified Powell method in combination with FW and BW solutions. The new analytical solution show an excellent agreement with the exact numerical solution for entire dwelling time. Moreover, optimized parameters with the new analytical solution show much smaller discrepancy than those with Vonesh's solution. Although the proposed method requires a slightly longer calculation time than Vonesh's, it can simulate concentrations in plasma and dialysate for an entire single exchange in CAPD using the clinical data measured at arbitrary time. The proposed method may be useful for determining unknown parameters as well as for prescribing CAPD treatment.