(390b) A Robust Hybrid Model Predictive Control Framework for Hill Curve Model-Based Systems

The Hill curve, introduced by A.V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of haemoglobin [1], is an empirical relation exhibiting an S shape characteristic used widely as part of diverse mathematical models in different fields. Examples include: (i) biosciences as a descriptive-deterministic model [2], (ii) in physiology, to describe nonlinear and saturable mechanisms as the renal uptake of aminoglycosides by proximal tubular cells [3], the tubuloglomerular feedback of the glomerular filtration in the kidney [3, 4], and the effect of ligand binding on the conductance of voltage-dependent ion channels [5], (iii) in pharmacology, on the rational basis of the receptor occupancy theory [6] as a model for describing the drug concentrationâ??effect relationship [7], for the treatment of many diseases.

In this work we present a general framework for modelling and advanced control of systems involving the Hill curve as part of their mathematical model. The developed framework features (i) a piece wise affine modelling representation of the Hill curve [8], (ii) a hybrid mixed integer optimisation formulation leading to a hybrid model predictive control (hMPC) formulation [9], (iii) a state-of-the art multiparametric mixed integer quadratic programming (mp-MIQP) solution step for the derivation of explicit controllers [10] and (iv) advanced robust control and estimation strategies addressing issues related to variability and uncertainties. The framework will be demonstrated with its application to two representative biomedical systems (i) the intravenous anaesthesia process [11] where robust advanced control strategies are presented for the administration of Propofol to control the level of the depth of anaestheisa for a set of 12 real patients data in the induction and maintenance phases; and (ii) the acute myeloid leukemia process [12] where robust personalised treatment schedules for the delivery of chemotherapy are derived and performed for 6 real patient data.

References

1. Hill, A.V., The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Physiol, 1910. 40.

2. Michaelis, L. and M.L. Menten, Die kinetic der Invertinwirkung. Biochem. Z., 1913. 61: p. 530-546.

3. Rougier, F., et al., Aminoglycoside nephrotoxicity: Modeling, simulation, and control. Antimicrobial Agents and Chemotherapy, 2003. 47(3): p. 1010-1016.

4. Schnermann, J., et al., Tubuloglomerular feedback control of renal vascular resistance, in Handbook of renal physiology, O.U. Press, Editor. 1992: Oxford, UK. p. 1675â??1705.

5. Haynes, L.W., A.R. Kay, and K.W. Yau, Single cyclic GMP-activated channel activity in excised patches of rod outer segment membrane. Nature, 1986. 321(6065): p. 66-70.

6. Wagner, J.G., Kinetics of pharmacologic response I. Proposed relationships between response and drug concentration in the intact animal and man. Journal of Theoretical Biology, 1968. 20(2): p. 173-201.

7. Csajka, C. and D. Verotta, Pharmacokinetic-pharmacodynamic modelling: History and perspectives. Journal of Pharmacokinetics and Pharmacodynamics, 2006. 33(3): p. 227-279.

8. Nascu, I., R. Oberdieck, and E. Pistikopoulos, An explicit Hybrid Model Predictive Control Strategy for Intravenous Anaesthesia. 9th IFAC Symposium on Biological and Medical Systems, 2015. in press.

9. Bemporad, A. and M. Morari, Control of systems integrating logic, dynamics, and constraints. Automatica, 1999. 35(3): p. 407-427.

10. Oberdieck, R. and E. Pistikopoulos, Explicit hybrid model-predictive control: The exact solution. Automatica, 2014. submitted.

11. NaÅ?cu, I., et al., Advanced model-based control studies for the induction and maintenance of intravenous anaesthesia. IEEE Transactions on Biomedical Engineering, 2015. 62(3): p. 832-841.

12. Pefani, E., et al., Chemotherapy Drug Scheduling for the Induction Treatment of patients with Acute Myeloid Leukemia. IEEE Trans Biomed Eng, 2014.