461535 Regulation of Anemia in Chronic Renal Disease Using Zone Model Predictive Control
Anemia is a state, characterized by a reduced mass of red blood cells (RBCs) or hemoglobin (Hgb), resulting in reduced oxygen carrying capacity and delivery to the body's tissues and organs. Anemia of chronic renal disease patients is primarily due to insufficient endogenous production of erythropoietin, as well as iron deficiency. Those patients with anemia are treated by the administration of recombinant human erythropoietin (rHuEPO) and iron. It has been recognized by the clinical community that while low Hgb levels lead to anemia, too large Hgb levels can increase the risk of mortality for the patient. Hence, effective methods are needed to determine the appropriate dose of rHuEPO and iron to maintain the target Hgb level. Many conventional methods for guiding rHuEPO and iron dosing used by clinicians rely on a set of rules based on past experiences or retrospective studies. Those methods for rHuEPO administration are generally imprecise, and as a result, the patient's Hgb level may move through the target range with large oscillations and overshoots. New effective anemia management methods are needed to avoid the adverse effects associated with increased Hgb levels while minimizing the effects of anemia. In this paper, one proposed solution to managing anemia in renal disease patients is the utilization of Zone Model Predictive Control (ZMPC).
To implement ZMPC, it is necessary to identify a suitable mathematical model to describe the system. Erythropoietin doses and hemoglobin data from 168 different patients over a span of up to 3 years were retrieved from medical records. Erythropoietin doses were typically given once per week, while hemoglobin values were taken from lab work which was done approximately every two weeks. To overcome the variance in the sampling time of Hgb and drug doses, the data was resampled into weekly intervals. A linear interpolation was used between Hgb values and the rHuEPO dose total for the entire trailing week was used. Due to the closed loop nature of the data acquisition, Direct Closed-loop System Identification was used to identify Autoregressive with Exogenous Input (ARX) models for individual patients. The model order and delay used in the model identification were selected based on priori medical knowledge of the life span of red blood cells and the response delay of Erythropoietin doses.
ZMPC is an advanced control system that uses a dynamic mathematical model of a system to optimize control inputs over a prediction horizon to guide the system to a target zone. For anemia management the target Hgb level selected was 9.5 to 11 g/dL which was based on physician recommendations. The ZMPC algorithm from  was adapted to incorporate the use of the identified ARX system models. Using a large target zone can introduce problems in which the output often has a tendency to settle near the boundaries of the target zone. This results in process noise and model mismatch often pushing the system outside of the desired zone. A viable approach to reducing the effect of this problem is to penalize the output in a funnel shape that reduces to a point along the prediction horizon. This is accomplished by introducing exponentially decaying constraints on the slack variable used within the cost function. In this manner, the controller guides the system towards the center of the zone with control actions that are far less aggressive than traditional model predictive control. ZMPC has the advantage of calculating less aggressive control inputs which can improve the robustness of the controller. The disadvantage is the requirement to tune the shape of the funnel constraints.
Computer simulations were used to explore the performance of MPC and ZMPC against the physician's protocol. Assessment of the controller output performance was based on the frequency and magnitude of zone violation. Although not minimized in the cost function, total erythropoietin doses were also compared. To test the performance of the controllers, both process noise and model mismatch were used. The process noise used in the simulations was determined from the residuals of the identified ARX model and the actual data. To represent model mismatch, one patient ARX model was used in the predictive controller while a different patient model was used as a patient simulator. It was found that both model predictive controllers consistently outperformed the physician protocol in simulations including process noise only. In simulations including process noise and model mismatch, both controllers outperformed the physician's protocol in the majority of cases. In some cases including model mismatch, the traditional MPC controller produced control actions that were too aggressive causing oscillations and overshoots in the Hgb levels. This phenomena is shown in Figure 1. The ZMPC controller with funnel constraints produced results consistently better than the physician's protocol in all simulations.
Figure 1. Comparison of traditional MPC, ZMPC using funnel constraints and Physician's Protocol performance in the presence of process noise and model mismatch
 Roubal, Jirka and Vladimir Havlena. "Range Control MPC Approach for Two-dimensional System." (2005).