Dynamical control of the dialysis process. Part II: An improved algorithm for the solution of a tracking problem.

An efficient algorithm for the optimization of process parameters during dialysis has been developed. By solving a tracking-problem for prescribed time courses of distinguished variables, it is possible to compute optimal concentrations of electrolytes in dialysate as well as an optimal rate of ultrafiltration. These variables are indirectly influencing the status of the patient and can be directly modelled. They are describing the important exchange processes between blood and dialysate as well as between the different distribution spaces within the patient during dialysis. Their time courses are determined by an individually identifiable patient model. The tracking problem was treated as a dynamic optimization problem, and a continuous descent procedure which is usually employed for solving unconstrained static optimization problems has been adapted in such a manner that it is applicable for the solution of this problem. The used method is characterized by its simple mode of application, short solution time and moderate storage need. Especially in cases of contradictional requirements for desired time courses of model outputs the used optimization method performs well.

Dynamical control of the dialysis process. Part I: Structural considerations and first mathematical approach.

Individual optimization of the dialysis process requires the (open-loop or closed-loop) control of many different variables, e.g. plasma ion concentrations, acid base state, volemic state and hemodynamic quantities. For this purpose a general concept for multiple-input-multiple-output (MIMO) control of the dialysis process is presented. The controlled variables have been differentiated into variables which can be modeled mechanistically (primary controlled variables, PCVs) and (hemodynamic) variables for which no mechanistic model has been developed up to now (secondary controlled variables, SCVs). Accordingly the controller is decomposed into two stages. Stage 1 contains an expert system which links the PCVs to the SCVs and provides the generation of optimal profiles for the PCVs with respect to maximum hemodynamic stability of the patient. Stage 2 is a tracking controller for the PCVs. An algorithm for the multidimensional tracking problem at stage 2 has been developed. It can be used for open-loop and future closed-loop control. The algorithm has been tested for 4 controlled (plasma Na+, plasma K+, plasma volume and ratio between intra- and extracellular volume) and 3 control variables (dialysate Na+, dialysate K+, ultrafiltration rate) up to now. It renders possible the exact tracking of the prescribed trajectories as long as all points are reachable under consideration of all physical and physiological boundary conditions. If they are not, appropriate weighting of the conflicting optimization goals must be applied. An extension towards more than 4 controlled variables is possible on principle. Main advantages of the method are its mathematical simplicity and the applicability of standard optimization subroutines.