The Role of Feedback Control Design in Developing Anemia Management Protocols

Abstract

The optimal use of erythropoiesis stimulating agents to treat anemia of end-stage renal disease remains difficult due to reported associations with adverse events. A patient’s hemoglobin response to these agents cannot be accurately described using population-level models due to many individual factors including chronic inflammation, red blood cell lifespan, and acute blood loss. As a consequence, it is generally understood that current one-size-fits-all anemia management protocols result in suboptimal outcomes. In this paper, we report on our collaboration with the medical community in designing anemia management protocols. In clinical implementation, these new dosing protocols have led to improved outcomes due to their use of control-relevant modelling, model parameter identification, and principles of feedback control. This is an example of medical professionals and control engineers working together to positively affect the performance of anemia management protocols in end-stage renal disease.

Appendix: Erythropoiesis Modelling in ESRD

The dynamics of Hb concentration following the administration of intravenous ESA has been described using a combination of pharmacokinetic (PK) and pharmacodynamics (PD) models.8,23,31 We present the PK (top two relations below) and the production portion of the PD together: for t ≥ 0,

$$ \begin{aligned} \frac{{{\text{d}}E(t)}}{{{\text{d}}t}} = - \frac{{{\text{VE}}(t)}}{{K_{\text{m}} + E(t)}} - \alpha E(t) + {\text{dose}}(t);\;E(0) = 0 \\ E_{p} (t) = E(t) + E_{\text{en}} (t); \\ k_{\text{in}} (t) = \frac{{{\text{SE}}_{p} (t)}}{{C + E_{p} (t)}} \\ \end{aligned} $$

where $V$, $K_{\text{m}}$, C and $S$ are constants, $\alpha$ is the linear clearance constant, $k_{\text{in}}$ is the RBC production-rate, $E_{P}$ denotes EPO in plasma, $E_{\text{en}}$ is endogenous EPO, and $E$ is exogenous ESA. The term ${\text{dose}}(t)$ denotes the ESA dose at time t, which we model as a train of impulses of ESA doses.

The remaining part of the PD model is for the RBC population R(t) at time t ≥ 0, given by8

se

$$ \begin{aligned} \frac{{{\text{d}}R(t)}}{{{\text{d}}t}} = k_{\text{in}} (t - D) - \int\limits_{0}^{\infty } {k_{\text{in}} (t - \lambda - D)} \ell (\lambda ){\text{d}}\lambda ;\;R(0) = R_{0} \\ H(t) = K_{\text{H}} R(t) \\ \end{aligned} $$

where $\ell (\tau )$ is the probability density function (pdf) of the lifespan $\tau$ of an RBC entering the pool at any time t, $D$ is the total time required for EPO-stimulated progenitor cells to progress through their various stages and finally become reticulocytes ready to mature into RBCs, $R_{0}$ denotes the initial RBC population at t = 0, at which time EPO therapy begins, and $K_{\text{H}}$ is the average amount of [Hb] per RBC (mean corpuscular hemoglobin, or MCH, in a complete blood count). We initially assumed that $\ell (\tau )$ is a time-invariant 2nd-order gamma pdf

$$ \ell (\tau ) = \frac{4}{{\mu^{2} }}\tau e^{{{\raise0.7ex\hbox{${ - 2\tau }$} \!\mathord{\left/ {\vphantom {{ - 2\tau } \mu }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\mu $}}}} ,\quad \mu ,\tau > 0 $$

where $\mu$ is the mean RBC lifespan. This assumption was later confirmed in a clinical study.28

We used retrospective data from 44 HD patients collected over a 16-month period20 to establish suitability of this model. The data consisted of administered rHuEPO doses and Hb measurements. Each patient’s model was identified using a training period of ≥ 90 days, which is consistent with the reported range of RBC lifespans of 34–120 days in ESRD patients.31 Each training period was followed by a validation period of similar length. We illustrate estimation results of the parameters in the above nonlinear model corresponding to patient #27 in Fig. 9. This figure shows a training period (days 82–277, shaded in the figure) followed by a validation period (days 278–473) during which the estimated model accurately predicts the actual response, although the match degrades slightly over time. Note that this training/validation sequence followed an initial training period (days 0–82) that produced a model which failed to predict the response beyond that period, and so was re-initiated at day 82. There are several possible explanations for this: (a) the training period was too short relative to the dynamics of the RBC pool (approximated by the RBC lifespan, 92.2 days), and (b) the rapid decrease in Hb evident during days 0–82, occurring during administration of a constant rHuEPO dose, may indicate an event not included in the model, such as gastrointestinal bleeding.

Figure 9

Estimation/Validation results for patient #27. (top) Clinical Hb data (dots), validation-failed estimation results (dashed), parameter estimation results (solid) trained over data from day 82 to 278 (shaded area). (bottom) administered rHuEPO doses.