- Email: [email protected]
Pharmacodynamic Model Kernel
8th IFAC Symposium on Biological and Medical Systems The International Federation of Automatic Control August 29-31, 2012. Budapest, Hungary
A new Perspective on Closed-Loop Glucose Control using a Physiology-Based Pharmacokinetic / Pharmacodynamic Model Kernel Stephan Schaller*/**, Stefan Willmann*, Lukas Schaupp***, Thomas Pieber***, Andreas Schuppert*/**, Joerg Lippert*, Thomas Eissing* * Computational Systems Biology, Bayer Technology Services GmbH, Leverkusen, Germany, (e-mail: stephan.schaller@ bayer.com) ** Aachen Institute for Advanced Study in Computational Engineering Sciences, RWTH Aachen, Aachen, Germany (e-mail: [email protected]) *** Department of Internal Medicine, Medical University of Graz, Graz, Austria, (e-mail: [email protected]) Abstract: After decades of research, Automated Glucose Control (AGC) is still out of reach for everyday control of blood glucose. The inter- and intra-individual variability of glucose dynamics largely arising from variability in insulin absorption, distribution, and action, and related physiological lag-times remain a core problem in the development of suitable control algorithms. Here, we demonstrate the potential of approaches based on detailed models to improve controller performance by comparing two control algorithms. The performance of fading memory proportional derivative control (FMPD) with varying degrees of model-based information on the effects of insulin on board (EIOB) is compared to an ideal nonlinear model predictive control (MPC) scheme. We conclude, that largescale in silico models of the glucose metabolism can provide a framework to improve diabetes research, the development of automatic control strategies for diabetes and ultimately every day diabetes management. Keywords: automatic glucose control, decision support, diabetes, insulin on board, physiology-based, pharmacokinetic/pharmacodynamic model.
of-the-art in glucose modelling used for model-based glucose control.
1. INTRODUCTION Blood glucose levels within the human body are controlled by a complex system involving neuronal, hormonal and metabolic signalling networks. However, the body’s autonomous control of blood glucose is strongly impaired in patients with Diabetes Mellitus and glucose levels start to spiral out of control exposing patients with diabetes to risks associated with hyper- and hypoglycaemia. Effective control of blood glucose levels may improve the risk profile of most patients with diabetes. Improved management of diabetes offers exceptionally good prospects, both in clinical and economic terms (Kovatchev 2011). Towards this goal, automated, controlled insulin delivery concepts are an active area of research aiming for an artificial, technical substitution of the natural regulatory system missing in diabetes patients. Mathematical models of the relevant processes form the basis for such automatic control approaches. Structures of early mathematical models of the glucoseinsulin metabolism (GIM) only vaguely reflect the underlying physiological properties. However, model-based glucose control requires models with increased predictive power. Thus, the idea to move towards more physiologically-based models became prominent. The UVa/Padova simulator, based on a model by Dalla Man et al. (Dalla Man et al. 2007), accepted by the Federal Drug Administration (FDA) to replace animal testing of glucose controllers (Kovatchev et al. 2009), and the Cambridge Model, developed by Hovorka et al. (Hovorka et al. 2008) for closed loop glucose control contain physiological aspects and represent the current state978-3-902823-10-6/12/$20.00 © 2012 IFAC
However, these models can still not be considered as full physiological models because they miss representations of blood flows and explicit organ volumes. Indeed, they appear like a step backwards from a model developed by Sorensen (Sorensen 1985) more than 25 years ago, which represents a real physiological model and can be considered unique in this respect. Sorensen used mass-balance equations to describe blood-flow and distribution-, metabolisation- and excretion dynamics of glucose, insulin, and even glucagon, which was neglected in most other models. A major drawback of Sorensen’s approach is the lack of a framework for the individualization of the model, such that inter-individual variability cannot be addressed. Complex models thus either require an extensive database or additional assumptions which can introduce additional error. This may be the reason why on the one hand this model was used only for in silico studies in closed loop control (Parker et al. 1999), and has never been used as a kernel for automated closed loop control in a clinical trial and why on the other hand current state-ofthe-art models rely on a more parsimonious but easy to individualize model structure. However, the parsimonious mechanistic representations of the GIM in current model descriptions can partly be made accountable for one or more of the following remaining issues of current model-based glucose control approaches: 1) The selected model design results in a reduced correspondence between important physiology and its
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8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
mathematical representation; 2) The resulting limited predictive quality has to be compensated for by a more robust controller (and time-variant parameter-sets) on the cost of reduced controller aggressiveness; 3) The models lack a direct quantification of metabolic processes; 4) The models are generally not applicable to a wide range of inputs and physiologic input sites and last but not least 5) No framework for the a-priory individualization based on anthropologic data is provided.
neglected for small molecules like glucose. For insulin and glucagon, in addition to lymph flow the exchange across capillary walls is described assuming convection and diffusion through two types of endothelial pores (Rippe and Haraldsson 1994) based on the hydrodynamic radius and the molecular weight of the compound.
To address these challenges, a detailed physiology-based pharmacokinetic/pharmacodynamic (PBPK/PD) model was developed and combined with state-of-the-art control algorithms for automatic glucose control in a post-hoc insilico study. The focus of this paper, however, is not on the model details but on the contributions that this modelling approach can make to the automatic glucose control community. 2. INDIVIDUALIZED PHYSIOLOGY-BASED MODEL It is not within the scope of this paper to present the developed PBPK/PD model of the GIM in full detail. Thus we will give a short overview on the modelling concept and the relevant aspects of glucose metabolism within the model. 2.1 Model Development PBPK models describe the mechanisms underlying the absorption, distribution, metabolism and excretion (ADME) of a substance within the body at an in-depth level of detail. PBPK models are based to a large extend on prior information, either taken from collections of anatomical and physiological data or calculated from drug-dependent properties, regarding an organism’s anatomy and physiology and its changes associated with age, weight, height, gender and race (e.g. age dependence of organ weights, blood flows, etc.) as well as an understanding of how active processes (e.g. metabolic rates, transport/clearance rates), scale with relation to these properties. Based on such basic physicochemical parameters of a substance, generic PK prediction models are automatically parameterized. These models can then be used to simulate drug concentration profiles in various organs and tissues (Eissing et al. 2011). Once this adult PBPK model has been established, it can be used to extrapolate to individuals, be it adults, children or elderly outside the selected cohort (Strougo et al. 2012). The PBPK models for glucose, insulin and glucagon have been established in PK-Sim® 4.2 and coupled in MoBi® 2.3, commercial software packages for PBPK and molecular biology modelling (Willmann et al. 2003, Eissing et al. 2011). Model identification, model parameterization, and the development of control algorithms have been conducted using the MoBi® Toolbox for MATLAB® 2.2. PBPK models include all relevant organs which are connected by arterial and venous blood flow. Each organ is generally divided into four sub-compartments as shown in Figure 1. For larger molecules like insulin and glucagon, distribution by lymph flow becomes relevant which can be
Figure 1: Organ representation in PK-Sim® with four subcompartments (intracellular, interstitial, plasma and blood-cells) and the corresponding flow rates and transports. C: concentrations, Q: flow rates, P × SA permeability surface area products, K partition coefficient, Vmax, KM: Michaelis-Menten constants. From Willmann et al. (Willmann et al. 2003). The PK of glucose considers the following ADME-related mechanisms characterized by oral absorption of glucose and meals from the GI-Tract based on a recent in-house development of a detailed compartmental absorption model (Thelen et al. 2011) reflecting detailed knowledge of human gastrointestinal (GI) physiology such as anatomical dimensions and mucosal blood flow. The compartmental GItract was complemented by known mechanistics like facilitating and active sodium dependent glucose transporters GLUT2 and SGLT1; Glucose distribution is described by the default transport processes included in PK-Sim®. Additional processes for cellular glucose uptake, metabolisation, and excretion were implemented to reflect known glucose distribution physiology via tissue specific facilitating transporters, GLUT-2 in the liver and intestinal mucosa, GLUT-3 in the brain, and insulin sensitive GLUT-4 in fat tissue and skeletal muscle. The PK of insulin is modelled using the 2-pore model of PK-Sim® and is characterized by absorption after subcutaneous administration for which the standard PBPK model was complemented by a subcutaneous depot which incorporated the model by (Tarin et al. 2005) for subcutaneous insulin absorption. Distribution is modelled with standard mechanisms of PK-Sim®. Additionally a simple model was developed for receptor-mediated transcytosis (Barrett et al. 2011) introducing a transendothelial transfer-state. Uptake, metabolisation, and excretion (or degradation) is a feature of all insulin sensitive tissues like muscle, fat, the kidney and the liver and mediated primarily by the insulin receptor. To take these processes into account the receptor model from (Sedaghat et al. 2002) was adapted and integrated.
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The PK model of glucagon is also described by the 2-pore model formalism of PK-Sim®, similar to insulin. The glucagon model was mostly kept at its default parameterization, which is reflecting general knowledge for the PK of proteins as the biological processes effecting distribution of glucagon are less well understood as for glucose and insulin. The model was only extended by processes for subcutaneous absorption, similar to insulin. For glucagon production and secretion the simple glucagon secretion model from (Sorensen 1985) was used. Like insulin, glucagon is cleared by receptor-mediated endocytosis. However, due to the limited knowledge about the mechanistics of the receptor binding processes of glucagon, a simple single state representation of receptor dynamics was chosen. Through the implementation as a specific process, all rates of transport and metabolisation of the GIM model are scaled dependent organ surface areas and volumes in relation to the organ volumes of a mean adult male individual, i.e. corresponding to an allometric scaling implemented for organ volumes in PK-Sim®. The pharmacodynamics of the PBPK/PD GIM model for T1DM are loosely based on the Sorensen model equations (Sorensen 1985). In addition to the more detailed compartmental structure of models developed within PKSim®, the pharmacodynamic interactions from Sorensen were also refined and implemented within MoBi® . The generic modulating functions M used for the description of several of the pharmacodynamic effects of glucose, insulin and glucagon were inspired by the threshold functions in (Sorensen 1985) and are defined as: n⋅S ( Crel ⋅ S ) = V0 + Vmax (K m )n + (Crel ⋅ S )n⋅S
glucose effectiveness was reduced to dampen the fast decrease from high glucose levels (Figure 3) at basal insulin and consequently the effect of insulin was increased for high insulin levels (Figure 4).
Figure 2: Representation of the effect of glucagon (as Crel ) on liver glucose production (MNHGP, as modelled by the generalized Hill-function in (1)) as a family of curves with respect to changes in the sensitivity (S, left plot) and responsiveness (SR, right plot) of the PD function. Corresponding data is shown as squares (Sorensen 1985).
Figure 3: Corrected PD-function of MGHGU. Green: function used by Sorensen. Blue: refitted generalized Hill-equation used here. Corresponding data (Sorensen 1985) is shown as squares.
R
M
R
(1)
with effector responsiveness SR, effector sensitivity S, the relative effector (e.g. insulin) concentration Crel the reaction rate at zero concentration V0, the maximal change in the rate of reaction Vmax the concentration of half maximal change in rate of reaction Km and the cooperativity exponent n. The effect of changes in SR, S and Crel are shown exemplary for glucagon dependent hepatic glucose production in Figure 2. The reason why generalized Hill-functions were used instead of hyperbolic tangent functions for the description of the threshold function is the intuitive parameterization for changes in sensitivity and responsiveness. Within this paper, only deviations from the threshold functions from Sorensen will be briefly described. Liver glucose homeostasis is mainly defined as a balance of two processes: hepatic glucose uptake (HGU) and hepatic glucose production (HGP) controlled by glucose (MGHGU/P), insulin (MIHGU/P), and glucagon (MNHGP). With the implementation of the insulin receptor model the effector of MGHGP/U, was changed to total amount of phosphorylated insulin receptor IRp. Earlier publications and own simulations showed that glucose effectiveness of the original Sorensen model seemed to be excessive (Steil et al. 2005). Thus
Figure 4: Corrected PD-function of MIHGP (bold lines) and MIHGU (slim lines). Green: functions used by Sorensen. Blue: refitted generalized Hill-equation used here. Corresponding data (Sorensen 1985) is shown as squares. It is important to note that we did not adjust basal hepatic glucose production for subjects with type 1 diabetes as it is generally done in state-of-the-art models of glucose metabolism (Magni et al. 2007). Instead we considered the circumstance that insulin in healthy subjects is endogenous. This means that the liver is saturated with insulin, whereas in subjects with T1DM, where insulin is exogenous, the liver is exposed to a lower level of insulin and insulin exerts only a fraction of its glucose lowering effect on the liver. This natural distribution of insulin is accounted for in the physiology-based PK/PD model due to its detailed description of the human physiology and distributive fluid flows.
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Peripheral glucose uptake, as mentioned before, is mainly managed by insulin sensitive glucose transporter GLUT4 stimulated by a downstream signalling cascade of the insulin receptor model (Sedaghat et al. 2002). Glucose dependent GLUT4 transport rate is modelled as in (1). And Vmax is a function of membrane GLUT4 concentrations translocated upon stimulation by insulin. 2.2 Patient Cohort and Individualized Models Individualized models were generated using data collected from 12 patients during a 2-phase randomized crossover trial (2PRCT) with patients with T1DM for the comparison of continuous subcutaneous insulin infusion (CSII) and a model predictive control algorithm (MPC) (Hovorka et al. 2004).
of insulin, an insulin-to-carbohydrate ratio value, meal CHO content, and the individuals’ body weight (BW) were used. 3.2 Model Predictive Control In model predictive control the current control action is obtained by solving at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the system as the initial state. The optimization yields an optimal control sequence from which the first control in the sequence is applied to the system as the implicit predictive control law (Rawlings and Mayne 2009). We used the full model in a nonlinear MPC strategy. To save computational time and to increase the convergence of the optimization problem, also only the first control in the input sequence which was applied was optimized. This approach is motivated by the large dead-time of s.c. infusion of insulin. The results of our study demonstrate that this approach is sufficient for s.c. glucose control. 3.3 Insulin on Board Time delays associated with physiological lag-times do complicate glycaemic control. Without the information on remaining circulating insulin levels a controller may, as glucose levels rise during a meal, overdose on insulin. For that reason, recent control algorithms use simple decay functions describing the pharmacokinetics of “insulin-onboard” (IOB) (Ellingsen et al. 2009) to constrain insulin infusion to avoid accumulation of insulin within the body.
Figure 5: Simulated and measured peripheral venous blood glucose (top) and insulin (bottom) from Subject 2 controlled by the CSII protocol. Insulin was administered in piecewise constant rate intervals of 15 min in combination with meal boluses. Participating subjects received three meals per day. Glucose measurements were taken intravenously, although for the algorithm evaluation a simulated s.c.-s.c. route was mimicked by delaying glucose measurements by 30 min. Insulin was administered subcutaneously. An exemplary result of the model individualization is shown in Figure 5. 3. AUTOMATIC GLUCOSE CONTROL 3.1 Fading Memory Proportional Derivative Control The development of a FMPD controller is based on basic concepts of feedback control. The FMPD mimics the normal physiology of the pancreatic beta-cell (Gopakumaran et al. 2005) which consists of a first phase (derivative component) response to the rise in blood glucose, and a second phase (proportional component) response to the hyperglycaemic state. A generalized formulation of FMPD control can be found in (Gopakumaran et al. 2005). We additionally used small-gain integral control for basal insulin infusion to adapt to inter-individual variability of insulin sensitivity. To calculate feed-forward insulin-boluses for informed in-time (not ahead of meal time) meal control, total daily dose (TDD)
Figure 6: Surface plot of the pharmacodynamic effect of insulin at the liver (MIHGU) and the mesh plot of the constraint function based on the saturation of the pharmacodynamic effect of insulin at the liver (EIOBliv). Values for MIHGU are relative (as depicted in Figure 4 for SI = 1). Values for EIOBliv are scaled for illustration and depend on the weighting factors for the constraint. Here, we advance this method by now using the effect of insulin on board (EIOB) and the effect of missing insulin on board (EMIOB), as expressed by the insulin dependent pharmacodynamic functions, i.e. MIHGU, MIHGP and rate of insulin mediated peripheral translocation of glucose transporter 4 (GLUT4trans) of the PBPK/PD model to guide
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the rate of insulin infusion. The function for EIOB was calculated as depicted in (1), based on the corresponding modulating function using an increased cooperativity exponent n for a more prominent switch-like behaviour and a shifted Km value. The concept of the EIOB constraint is depicted in Figure 6. Whereas MIHGU and GLUT4trans, were used to constrain additional insulin application, MIHGP was used, as an indicator for the effect of missing insulin, to initiate supplementary insulin infusion once insulin levels dropped too low to forestall excessive hepatic glucose production. The EIOB function was used in FMPD as a scaling factor of the controller gains and as an additional penalty in the cost function of the MPC. 4. RESULTS The in silico trials for the evaluation of the integrated algorithms followed the same protocol as the 2PRCT. Control inputs are defined as piecewise constant segments of 15 min. The control input is calculated every 15 min. During the trial, patients (N = 12) received 4 meals. After the start of the trial, dinner was the first meal at t = 240 min. On the next day, subjects received their meals at t = 1020 min (breakfast), t = 1320 min (lunch), and t = 1680 min (dinner). Meal information was transferred to the controller at the time of meal onset, so no insulin infusions prior to the meal were injected for better meal rejection.
Figure 7: Comparison of blood glucose control using the FMPD algorithm without model-based safety constraint EIOB (mean +- std, red dashed line and shaded red area) versus control by MPC (mean +- std, blue solid line and shaded blue area). Curves in the upper area represent glucose levels; curves in the lower area represent insulin infusion rates. The thick green line represents the target value for MPC, the dashed green line for PMPD. The black dashed lines are the chosen threshold values to characterize either hypo- (lower line, 3.3 mmol/l) or hyperglycaemia (upper line, 10 mmol/l). For MPC a prediction horizon of two hours was chosen. The model-free FMPD was evaluated without constraint, with EIOB only and with both, EIOB and EMIOB. For MPC, EIOB is explicitly accounted for in the cost function. The following results reflect the mean results obtained with the 12 in silico patients.
Direct comparison of model-free FMPD and MPC shows that control by FMPD without model-based constraints (EIOB) is inferior to control by MPC (Figure 7). Although the adapting integral part of FMPD improves control after some time, variability of glucose levels is still immense. Except for the postprandial rise in glucose levels, MPC glucose levels remained exactly on target value with almost no notable variation within the 12 subjects. Also, no episodes of postprandial hypoglycaemia could be observed. Although no robustness properties were analysed here, this strongly indicates that the full PBPK/PD model can be used as an internal model within an MPC scheme. Combining the standard FMPD controller with the modelbased EIOB constraint markedly improved quality of control (Figure 8). Due to the constraining properties of the EIOB function, the aggressiveness of the controller gains could be increased significantly, allowing faster and more effective counter-regulation of meal disturbances without the risk of postprandial hypoglycaemia, although, glucose levels did slightly undershoot. Including the function of EMIOB to the FMPD control to supplement missing insulin further improved quality of control.
Figure 8: Comparison of blood glucose control using the FMPD algorithm model-based safety constraint EIOB (mean +- std, red dashed line and shaded red area) versus FMPD with EIOB and EMIOB (mean +- std, blue solid line and shaded blue area). The results show, that EIOB in general improves the performance of FMPD and that EMIOB additionally reduce nocturnal glucose oscillations. Curves in the upper area represent glucose levels; curves in the lower area represent insulin infusion rates. The thick green line represents the target value for glucose levels. The black dashed lines are the chosen threshold values to characterize either hypo- (lower line, 3.3 mmol/l) or hyperglycaemia (upper line, 10 mmol/l). The effect of EMIOB is particularly prominent during nocturnal control (at t=400-1000 min). EMIOB prevents the suspension of the insulin pump at low insulin levels during the postprandial undershoot of glucose. The smoother insulin infusion profile leads to more stable basal insulin levels, reducing hepatic counter-regulation and thereby reducing delayed nocturnal glucose surges. 5. CONCLUSIONS The results show that model-based predictive concepts such as the MPC are superior to model-free reactive concepts like
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the FMPD controller when it comes to systems with significant time-delays. However, if model-free reactive control concepts are combined with safety systems based on model predictions, i.e. supplemented with observer-based state feedback, their efficiency and safety can be improved dramatically. When comparing the two control concepts with respect to the ease of use, the biggest advantage of MPC is the intuitive way in which the cost function is designed. Although the model-free FMPD controller does not require an individualized model the controller gains need to be tuned for each patient individually. For the MPC, the same costfunction can be used for all patients as long as a fitted individualized model is available. In its improved version combined with the EIOB constraint, the performance of FMPD also depends on a fitted individualized model but individual tuning of the FMPD controller gains is no longer necessary. Thus, the benefit from detailed PBPK/PD models is twofold. First, reliable models of the glucose-insulin metabolism that can be combined with available control concepts improve glycaemic control and second, these more detailed models provide a powerful tool for in silico development and evaluation of control concepts. The developed PKPK/PD model will be the first of its kind evaluated as a kernel for automated closed loop control in a clinical trial as will be conducted in the near future. 6. ACKNOWLEDGEMENTS This work was performed in the framework of FP7 Integrated Project REACTION partially funded by the European Commission under Grant Agreement 248590. REFERENCES Barrett, E. J., Wang, H., Upchurch, C. T. and Liu, Z. (2011) 'Insulin regulates its own delivery to skeletal muscle by feed-forward actions on the vasculature', Am J Physiol Endocrinol Metab, 301(2), E252-63. Dalla Man, C., Rizza, R. A. and Cobelli, C. (2007) 'Meal simulation model of the glucose-insulin system', IEEE Trans Biomed Eng, 54(10), 1740-9. Eissing, T., Kuepfer, L., Becker, C., Block, M., Coboeken, K., Gaub, T., Goerlitz, L., Jaeger, J., Loosen, R., Ludewig, B., Meyer, M., Niederalt, C., Sevestre, M., Siegmund, H. U., Solodenko, J., Thelen, K., Telle, U., Weiss, W., Wendl, T., Willmann, S. and Lippert, J. (2011) 'A computational systems biology software platform for multiscale modeling and simulation: integrating whole-body physiology, disease biology, and molecular reaction networks', Front Physiol, 2, 4. Ellingsen, C., Dassau, E., Zisser, H., Grosman, B., Percival, M. W., Jovanovic, L. and Doyle, F. J., 3rd (2009) 'Safety constraints in an artificial pancreatic beta cell: an implementation of model predictive control with insulin on board', J Diabetes Sci Technol, 3(3), 536-44. Gopakumaran, B., Duman, H. M., Overholser, D. P., Federiuk, I. F., Quinn, M. J., Wood, M. D. and Ward, W.
K. (2005) 'A novel insulin delivery algorithm in rats with type 1 diabetes: the fading memory proportionalderivative method', Artif Organs, 29(8), 599-607. Hovorka, R., Chassin, L. J., Ellmerer, M., Plank, J. and Wilinska, M. E. (2008) 'A simulation model of glucose regulation in the critically ill', Physiol Meas, 29(8), 95978. Hovorka, R., Chassin, L. J., Wilinska, M. E., Canonico, V., Akwi, J. A., Federici, M. O., Massi-Benedetti, M., Hutzli, I., Zaugg, C., Kaufmann, H., Both, M., Vering, T., Schaller, H. C., Schaupp, L., Bodenlenz, M. and Pieber, T. R. (2004) 'Closing the loop: the adicol experience', Diabetes Technol Ther, 6(3), 307-18. Kovatchev, B. (2011) 'Closed loop control for type 1 diabetes', BMJ, 342, d1911. Kovatchev, B. P., Breton, M., Man, C. D. and Cobelli, C. (2009) 'In Silico Preclinical Trials: A Proof of Concept in Closed-Loop Control of Type 1 Diabetes', J Diabetes Sci Technol, 3(1), 44-55. Magni, L., Raimondo, D. M., Bossi, L., Man, C. D., De Nicolao, G., Kovatchev, B. and Cobelli, C. (2007) 'Model predictive control of type 1 diabetes: an in silico trial', J Diabetes Sci Technol, 1(6), 804-12. Parker, R. S., Doyle, F. J., 3rd and Peppas, N. A. (1999) 'A model-based algorithm for blood glucose control in type I diabetic patients', IEEE Trans Biomed Eng, 46(2), 148-57. Rawlings, J. B. and Mayne, D. Q. (2009) Model Predictive Control Theory and Design, Nob Hill Pub. Rippe, B. and Haraldsson, B. (1994) 'Transport of macromolecules across microvascular walls: the two-pore theory', Physiol Rev, 74(1), 163-219. Sedaghat, A. R., Sherman, A. and Quon, M. J. (2002) 'A mathematical model of metabolic insulin signaling pathways', Am J Physiol Endocrinol Metab, 283(5), E1084-101. Sorensen, J. T. (1985) A Physiologic Model of Glucose Metabolism in Man and its Use to Design and Assess Improved Insulin Therapies for Diabetes, unpublished thesis (PhD Thesis), MIT. Steil, G. M., Clark, B., Kanderian, S. and Rebrin, K. (2005) 'Modeling insulin action for development of a closed-loop artificial pancreas', Diabetes Technol Ther, 7(1), 94-108. Strougo, A., Eissing, T., Yassen, A., Willmann, S., Danhof, M. and Freijer, J. (2012) 'First dose in children: physiological insights into pharmacokinetic scaling approaches and their implications in paediatric drug development', J Pharmacokinet Pharmacodyn. Tarin, C., Teufel, E., Pico, J., Bondia, J. and Pfleiderer, H. J. (2005) 'Comprehensive pharmacokinetic model of insulin Glargine and other insulin formulations', IEEE Trans Biomed Eng, 52(12), 1994-2005. Thelen, K., Coboeken, K., Willmann, S., Burghaus, R., Dressman, J. B. and Lippert, J. (2011) 'Evolution of a detailed physiological model to simulate the gastrointestinal transit and absorption process in humans, part 1: oral solutions', J Pharm Sci, 100(12), 5324-45. Willmann, S., Lippert, J. and al., E. (2003) 'PK-Sim®: a physiologically based pharmacokinetic ‘whole-body’ model', Biosilico, 1, 121-124.
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A new Perspective on Closed-Loop Glucose Control using a Physiology-Based Pharmacokinetic / Pharmacodynamic Model Kernel Stephan Schaller*/**, Stefan Willmann*, Lukas Schaupp***, Thomas Pieber***, Andreas Schuppert*/**, Joerg Lippert*, Thomas Eissing* * Computational Systems Biology, Bayer Technology Services GmbH, Leverkusen, Germany, (e-mail: stephan.schaller@ bayer.com) ** Aachen Institute for Advanced Study in Computational Engineering Sciences, RWTH Aachen, Aachen, Germany (e-mail: [email protected]) *** Department of Internal Medicine, Medical University of Graz, Graz, Austria, (e-mail: [email protected]) Abstract: After decades of research, Automated Glucose Control (AGC) is still out of reach for everyday control of blood glucose. The inter- and intra-individual variability of glucose dynamics largely arising from variability in insulin absorption, distribution, and action, and related physiological lag-times remain a core problem in the development of suitable control algorithms. Here, we demonstrate the potential of approaches based on detailed models to improve controller performance by comparing two control algorithms. The performance of fading memory proportional derivative control (FMPD) with varying degrees of model-based information on the effects of insulin on board (EIOB) is compared to an ideal nonlinear model predictive control (MPC) scheme. We conclude, that largescale in silico models of the glucose metabolism can provide a framework to improve diabetes research, the development of automatic control strategies for diabetes and ultimately every day diabetes management. Keywords: automatic glucose control, decision support, diabetes, insulin on board, physiology-based, pharmacokinetic/pharmacodynamic model.
of-the-art in glucose modelling used for model-based glucose control.
1. INTRODUCTION Blood glucose levels within the human body are controlled by a complex system involving neuronal, hormonal and metabolic signalling networks. However, the body’s autonomous control of blood glucose is strongly impaired in patients with Diabetes Mellitus and glucose levels start to spiral out of control exposing patients with diabetes to risks associated with hyper- and hypoglycaemia. Effective control of blood glucose levels may improve the risk profile of most patients with diabetes. Improved management of diabetes offers exceptionally good prospects, both in clinical and economic terms (Kovatchev 2011). Towards this goal, automated, controlled insulin delivery concepts are an active area of research aiming for an artificial, technical substitution of the natural regulatory system missing in diabetes patients. Mathematical models of the relevant processes form the basis for such automatic control approaches. Structures of early mathematical models of the glucoseinsulin metabolism (GIM) only vaguely reflect the underlying physiological properties. However, model-based glucose control requires models with increased predictive power. Thus, the idea to move towards more physiologically-based models became prominent. The UVa/Padova simulator, based on a model by Dalla Man et al. (Dalla Man et al. 2007), accepted by the Federal Drug Administration (FDA) to replace animal testing of glucose controllers (Kovatchev et al. 2009), and the Cambridge Model, developed by Hovorka et al. (Hovorka et al. 2008) for closed loop glucose control contain physiological aspects and represent the current state978-3-902823-10-6/12/$20.00 © 2012 IFAC
However, these models can still not be considered as full physiological models because they miss representations of blood flows and explicit organ volumes. Indeed, they appear like a step backwards from a model developed by Sorensen (Sorensen 1985) more than 25 years ago, which represents a real physiological model and can be considered unique in this respect. Sorensen used mass-balance equations to describe blood-flow and distribution-, metabolisation- and excretion dynamics of glucose, insulin, and even glucagon, which was neglected in most other models. A major drawback of Sorensen’s approach is the lack of a framework for the individualization of the model, such that inter-individual variability cannot be addressed. Complex models thus either require an extensive database or additional assumptions which can introduce additional error. This may be the reason why on the one hand this model was used only for in silico studies in closed loop control (Parker et al. 1999), and has never been used as a kernel for automated closed loop control in a clinical trial and why on the other hand current state-ofthe-art models rely on a more parsimonious but easy to individualize model structure. However, the parsimonious mechanistic representations of the GIM in current model descriptions can partly be made accountable for one or more of the following remaining issues of current model-based glucose control approaches: 1) The selected model design results in a reduced correspondence between important physiology and its
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mathematical representation; 2) The resulting limited predictive quality has to be compensated for by a more robust controller (and time-variant parameter-sets) on the cost of reduced controller aggressiveness; 3) The models lack a direct quantification of metabolic processes; 4) The models are generally not applicable to a wide range of inputs and physiologic input sites and last but not least 5) No framework for the a-priory individualization based on anthropologic data is provided.
neglected for small molecules like glucose. For insulin and glucagon, in addition to lymph flow the exchange across capillary walls is described assuming convection and diffusion through two types of endothelial pores (Rippe and Haraldsson 1994) based on the hydrodynamic radius and the molecular weight of the compound.
To address these challenges, a detailed physiology-based pharmacokinetic/pharmacodynamic (PBPK/PD) model was developed and combined with state-of-the-art control algorithms for automatic glucose control in a post-hoc insilico study. The focus of this paper, however, is not on the model details but on the contributions that this modelling approach can make to the automatic glucose control community. 2. INDIVIDUALIZED PHYSIOLOGY-BASED MODEL It is not within the scope of this paper to present the developed PBPK/PD model of the GIM in full detail. Thus we will give a short overview on the modelling concept and the relevant aspects of glucose metabolism within the model. 2.1 Model Development PBPK models describe the mechanisms underlying the absorption, distribution, metabolism and excretion (ADME) of a substance within the body at an in-depth level of detail. PBPK models are based to a large extend on prior information, either taken from collections of anatomical and physiological data or calculated from drug-dependent properties, regarding an organism’s anatomy and physiology and its changes associated with age, weight, height, gender and race (e.g. age dependence of organ weights, blood flows, etc.) as well as an understanding of how active processes (e.g. metabolic rates, transport/clearance rates), scale with relation to these properties. Based on such basic physicochemical parameters of a substance, generic PK prediction models are automatically parameterized. These models can then be used to simulate drug concentration profiles in various organs and tissues (Eissing et al. 2011). Once this adult PBPK model has been established, it can be used to extrapolate to individuals, be it adults, children or elderly outside the selected cohort (Strougo et al. 2012). The PBPK models for glucose, insulin and glucagon have been established in PK-Sim® 4.2 and coupled in MoBi® 2.3, commercial software packages for PBPK and molecular biology modelling (Willmann et al. 2003, Eissing et al. 2011). Model identification, model parameterization, and the development of control algorithms have been conducted using the MoBi® Toolbox for MATLAB® 2.2. PBPK models include all relevant organs which are connected by arterial and venous blood flow. Each organ is generally divided into four sub-compartments as shown in Figure 1. For larger molecules like insulin and glucagon, distribution by lymph flow becomes relevant which can be
Figure 1: Organ representation in PK-Sim® with four subcompartments (intracellular, interstitial, plasma and blood-cells) and the corresponding flow rates and transports. C: concentrations, Q: flow rates, P × SA permeability surface area products, K partition coefficient, Vmax, KM: Michaelis-Menten constants. From Willmann et al. (Willmann et al. 2003). The PK of glucose considers the following ADME-related mechanisms characterized by oral absorption of glucose and meals from the GI-Tract based on a recent in-house development of a detailed compartmental absorption model (Thelen et al. 2011) reflecting detailed knowledge of human gastrointestinal (GI) physiology such as anatomical dimensions and mucosal blood flow. The compartmental GItract was complemented by known mechanistics like facilitating and active sodium dependent glucose transporters GLUT2 and SGLT1; Glucose distribution is described by the default transport processes included in PK-Sim®. Additional processes for cellular glucose uptake, metabolisation, and excretion were implemented to reflect known glucose distribution physiology via tissue specific facilitating transporters, GLUT-2 in the liver and intestinal mucosa, GLUT-3 in the brain, and insulin sensitive GLUT-4 in fat tissue and skeletal muscle. The PK of insulin is modelled using the 2-pore model of PK-Sim® and is characterized by absorption after subcutaneous administration for which the standard PBPK model was complemented by a subcutaneous depot which incorporated the model by (Tarin et al. 2005) for subcutaneous insulin absorption. Distribution is modelled with standard mechanisms of PK-Sim®. Additionally a simple model was developed for receptor-mediated transcytosis (Barrett et al. 2011) introducing a transendothelial transfer-state. Uptake, metabolisation, and excretion (or degradation) is a feature of all insulin sensitive tissues like muscle, fat, the kidney and the liver and mediated primarily by the insulin receptor. To take these processes into account the receptor model from (Sedaghat et al. 2002) was adapted and integrated.
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The PK model of glucagon is also described by the 2-pore model formalism of PK-Sim®, similar to insulin. The glucagon model was mostly kept at its default parameterization, which is reflecting general knowledge for the PK of proteins as the biological processes effecting distribution of glucagon are less well understood as for glucose and insulin. The model was only extended by processes for subcutaneous absorption, similar to insulin. For glucagon production and secretion the simple glucagon secretion model from (Sorensen 1985) was used. Like insulin, glucagon is cleared by receptor-mediated endocytosis. However, due to the limited knowledge about the mechanistics of the receptor binding processes of glucagon, a simple single state representation of receptor dynamics was chosen. Through the implementation as a specific process, all rates of transport and metabolisation of the GIM model are scaled dependent organ surface areas and volumes in relation to the organ volumes of a mean adult male individual, i.e. corresponding to an allometric scaling implemented for organ volumes in PK-Sim®. The pharmacodynamics of the PBPK/PD GIM model for T1DM are loosely based on the Sorensen model equations (Sorensen 1985). In addition to the more detailed compartmental structure of models developed within PKSim®, the pharmacodynamic interactions from Sorensen were also refined and implemented within MoBi® . The generic modulating functions M used for the description of several of the pharmacodynamic effects of glucose, insulin and glucagon were inspired by the threshold functions in (Sorensen 1985) and are defined as: n⋅S ( Crel ⋅ S ) = V0 + Vmax (K m )n + (Crel ⋅ S )n⋅S
glucose effectiveness was reduced to dampen the fast decrease from high glucose levels (Figure 3) at basal insulin and consequently the effect of insulin was increased for high insulin levels (Figure 4).
Figure 2: Representation of the effect of glucagon (as Crel ) on liver glucose production (MNHGP, as modelled by the generalized Hill-function in (1)) as a family of curves with respect to changes in the sensitivity (S, left plot) and responsiveness (SR, right plot) of the PD function. Corresponding data is shown as squares (Sorensen 1985).
Figure 3: Corrected PD-function of MGHGU. Green: function used by Sorensen. Blue: refitted generalized Hill-equation used here. Corresponding data (Sorensen 1985) is shown as squares.
R
M
R
(1)
with effector responsiveness SR, effector sensitivity S, the relative effector (e.g. insulin) concentration Crel the reaction rate at zero concentration V0, the maximal change in the rate of reaction Vmax the concentration of half maximal change in rate of reaction Km and the cooperativity exponent n. The effect of changes in SR, S and Crel are shown exemplary for glucagon dependent hepatic glucose production in Figure 2. The reason why generalized Hill-functions were used instead of hyperbolic tangent functions for the description of the threshold function is the intuitive parameterization for changes in sensitivity and responsiveness. Within this paper, only deviations from the threshold functions from Sorensen will be briefly described. Liver glucose homeostasis is mainly defined as a balance of two processes: hepatic glucose uptake (HGU) and hepatic glucose production (HGP) controlled by glucose (MGHGU/P), insulin (MIHGU/P), and glucagon (MNHGP). With the implementation of the insulin receptor model the effector of MGHGP/U, was changed to total amount of phosphorylated insulin receptor IRp. Earlier publications and own simulations showed that glucose effectiveness of the original Sorensen model seemed to be excessive (Steil et al. 2005). Thus
Figure 4: Corrected PD-function of MIHGP (bold lines) and MIHGU (slim lines). Green: functions used by Sorensen. Blue: refitted generalized Hill-equation used here. Corresponding data (Sorensen 1985) is shown as squares. It is important to note that we did not adjust basal hepatic glucose production for subjects with type 1 diabetes as it is generally done in state-of-the-art models of glucose metabolism (Magni et al. 2007). Instead we considered the circumstance that insulin in healthy subjects is endogenous. This means that the liver is saturated with insulin, whereas in subjects with T1DM, where insulin is exogenous, the liver is exposed to a lower level of insulin and insulin exerts only a fraction of its glucose lowering effect on the liver. This natural distribution of insulin is accounted for in the physiology-based PK/PD model due to its detailed description of the human physiology and distributive fluid flows.
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Peripheral glucose uptake, as mentioned before, is mainly managed by insulin sensitive glucose transporter GLUT4 stimulated by a downstream signalling cascade of the insulin receptor model (Sedaghat et al. 2002). Glucose dependent GLUT4 transport rate is modelled as in (1). And Vmax is a function of membrane GLUT4 concentrations translocated upon stimulation by insulin. 2.2 Patient Cohort and Individualized Models Individualized models were generated using data collected from 12 patients during a 2-phase randomized crossover trial (2PRCT) with patients with T1DM for the comparison of continuous subcutaneous insulin infusion (CSII) and a model predictive control algorithm (MPC) (Hovorka et al. 2004).
of insulin, an insulin-to-carbohydrate ratio value, meal CHO content, and the individuals’ body weight (BW) were used. 3.2 Model Predictive Control In model predictive control the current control action is obtained by solving at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the system as the initial state. The optimization yields an optimal control sequence from which the first control in the sequence is applied to the system as the implicit predictive control law (Rawlings and Mayne 2009). We used the full model in a nonlinear MPC strategy. To save computational time and to increase the convergence of the optimization problem, also only the first control in the input sequence which was applied was optimized. This approach is motivated by the large dead-time of s.c. infusion of insulin. The results of our study demonstrate that this approach is sufficient for s.c. glucose control. 3.3 Insulin on Board Time delays associated with physiological lag-times do complicate glycaemic control. Without the information on remaining circulating insulin levels a controller may, as glucose levels rise during a meal, overdose on insulin. For that reason, recent control algorithms use simple decay functions describing the pharmacokinetics of “insulin-onboard” (IOB) (Ellingsen et al. 2009) to constrain insulin infusion to avoid accumulation of insulin within the body.
Figure 5: Simulated and measured peripheral venous blood glucose (top) and insulin (bottom) from Subject 2 controlled by the CSII protocol. Insulin was administered in piecewise constant rate intervals of 15 min in combination with meal boluses. Participating subjects received three meals per day. Glucose measurements were taken intravenously, although for the algorithm evaluation a simulated s.c.-s.c. route was mimicked by delaying glucose measurements by 30 min. Insulin was administered subcutaneously. An exemplary result of the model individualization is shown in Figure 5. 3. AUTOMATIC GLUCOSE CONTROL 3.1 Fading Memory Proportional Derivative Control The development of a FMPD controller is based on basic concepts of feedback control. The FMPD mimics the normal physiology of the pancreatic beta-cell (Gopakumaran et al. 2005) which consists of a first phase (derivative component) response to the rise in blood glucose, and a second phase (proportional component) response to the hyperglycaemic state. A generalized formulation of FMPD control can be found in (Gopakumaran et al. 2005). We additionally used small-gain integral control for basal insulin infusion to adapt to inter-individual variability of insulin sensitivity. To calculate feed-forward insulin-boluses for informed in-time (not ahead of meal time) meal control, total daily dose (TDD)
Figure 6: Surface plot of the pharmacodynamic effect of insulin at the liver (MIHGU) and the mesh plot of the constraint function based on the saturation of the pharmacodynamic effect of insulin at the liver (EIOBliv). Values for MIHGU are relative (as depicted in Figure 4 for SI = 1). Values for EIOBliv are scaled for illustration and depend on the weighting factors for the constraint. Here, we advance this method by now using the effect of insulin on board (EIOB) and the effect of missing insulin on board (EMIOB), as expressed by the insulin dependent pharmacodynamic functions, i.e. MIHGU, MIHGP and rate of insulin mediated peripheral translocation of glucose transporter 4 (GLUT4trans) of the PBPK/PD model to guide
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the rate of insulin infusion. The function for EIOB was calculated as depicted in (1), based on the corresponding modulating function using an increased cooperativity exponent n for a more prominent switch-like behaviour and a shifted Km value. The concept of the EIOB constraint is depicted in Figure 6. Whereas MIHGU and GLUT4trans, were used to constrain additional insulin application, MIHGP was used, as an indicator for the effect of missing insulin, to initiate supplementary insulin infusion once insulin levels dropped too low to forestall excessive hepatic glucose production. The EIOB function was used in FMPD as a scaling factor of the controller gains and as an additional penalty in the cost function of the MPC. 4. RESULTS The in silico trials for the evaluation of the integrated algorithms followed the same protocol as the 2PRCT. Control inputs are defined as piecewise constant segments of 15 min. The control input is calculated every 15 min. During the trial, patients (N = 12) received 4 meals. After the start of the trial, dinner was the first meal at t = 240 min. On the next day, subjects received their meals at t = 1020 min (breakfast), t = 1320 min (lunch), and t = 1680 min (dinner). Meal information was transferred to the controller at the time of meal onset, so no insulin infusions prior to the meal were injected for better meal rejection.
Figure 7: Comparison of blood glucose control using the FMPD algorithm without model-based safety constraint EIOB (mean +- std, red dashed line and shaded red area) versus control by MPC (mean +- std, blue solid line and shaded blue area). Curves in the upper area represent glucose levels; curves in the lower area represent insulin infusion rates. The thick green line represents the target value for MPC, the dashed green line for PMPD. The black dashed lines are the chosen threshold values to characterize either hypo- (lower line, 3.3 mmol/l) or hyperglycaemia (upper line, 10 mmol/l). For MPC a prediction horizon of two hours was chosen. The model-free FMPD was evaluated without constraint, with EIOB only and with both, EIOB and EMIOB. For MPC, EIOB is explicitly accounted for in the cost function. The following results reflect the mean results obtained with the 12 in silico patients.
Direct comparison of model-free FMPD and MPC shows that control by FMPD without model-based constraints (EIOB) is inferior to control by MPC (Figure 7). Although the adapting integral part of FMPD improves control after some time, variability of glucose levels is still immense. Except for the postprandial rise in glucose levels, MPC glucose levels remained exactly on target value with almost no notable variation within the 12 subjects. Also, no episodes of postprandial hypoglycaemia could be observed. Although no robustness properties were analysed here, this strongly indicates that the full PBPK/PD model can be used as an internal model within an MPC scheme. Combining the standard FMPD controller with the modelbased EIOB constraint markedly improved quality of control (Figure 8). Due to the constraining properties of the EIOB function, the aggressiveness of the controller gains could be increased significantly, allowing faster and more effective counter-regulation of meal disturbances without the risk of postprandial hypoglycaemia, although, glucose levels did slightly undershoot. Including the function of EMIOB to the FMPD control to supplement missing insulin further improved quality of control.
Figure 8: Comparison of blood glucose control using the FMPD algorithm model-based safety constraint EIOB (mean +- std, red dashed line and shaded red area) versus FMPD with EIOB and EMIOB (mean +- std, blue solid line and shaded blue area). The results show, that EIOB in general improves the performance of FMPD and that EMIOB additionally reduce nocturnal glucose oscillations. Curves in the upper area represent glucose levels; curves in the lower area represent insulin infusion rates. The thick green line represents the target value for glucose levels. The black dashed lines are the chosen threshold values to characterize either hypo- (lower line, 3.3 mmol/l) or hyperglycaemia (upper line, 10 mmol/l). The effect of EMIOB is particularly prominent during nocturnal control (at t=400-1000 min). EMIOB prevents the suspension of the insulin pump at low insulin levels during the postprandial undershoot of glucose. The smoother insulin infusion profile leads to more stable basal insulin levels, reducing hepatic counter-regulation and thereby reducing delayed nocturnal glucose surges. 5. CONCLUSIONS The results show that model-based predictive concepts such as the MPC are superior to model-free reactive concepts like
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the FMPD controller when it comes to systems with significant time-delays. However, if model-free reactive control concepts are combined with safety systems based on model predictions, i.e. supplemented with observer-based state feedback, their efficiency and safety can be improved dramatically. When comparing the two control concepts with respect to the ease of use, the biggest advantage of MPC is the intuitive way in which the cost function is designed. Although the model-free FMPD controller does not require an individualized model the controller gains need to be tuned for each patient individually. For the MPC, the same costfunction can be used for all patients as long as a fitted individualized model is available. In its improved version combined with the EIOB constraint, the performance of FMPD also depends on a fitted individualized model but individual tuning of the FMPD controller gains is no longer necessary. Thus, the benefit from detailed PBPK/PD models is twofold. First, reliable models of the glucose-insulin metabolism that can be combined with available control concepts improve glycaemic control and second, these more detailed models provide a powerful tool for in silico development and evaluation of control concepts. The developed PKPK/PD model will be the first of its kind evaluated as a kernel for automated closed loop control in a clinical trial as will be conducted in the near future. 6. ACKNOWLEDGEMENTS This work was performed in the framework of FP7 Integrated Project REACTION partially funded by the European Commission under Grant Agreement 248590. REFERENCES Barrett, E. J., Wang, H., Upchurch, C. T. and Liu, Z. (2011) 'Insulin regulates its own delivery to skeletal muscle by feed-forward actions on the vasculature', Am J Physiol Endocrinol Metab, 301(2), E252-63. Dalla Man, C., Rizza, R. A. and Cobelli, C. (2007) 'Meal simulation model of the glucose-insulin system', IEEE Trans Biomed Eng, 54(10), 1740-9. Eissing, T., Kuepfer, L., Becker, C., Block, M., Coboeken, K., Gaub, T., Goerlitz, L., Jaeger, J., Loosen, R., Ludewig, B., Meyer, M., Niederalt, C., Sevestre, M., Siegmund, H. U., Solodenko, J., Thelen, K., Telle, U., Weiss, W., Wendl, T., Willmann, S. and Lippert, J. (2011) 'A computational systems biology software platform for multiscale modeling and simulation: integrating whole-body physiology, disease biology, and molecular reaction networks', Front Physiol, 2, 4. Ellingsen, C., Dassau, E., Zisser, H., Grosman, B., Percival, M. W., Jovanovic, L. and Doyle, F. J., 3rd (2009) 'Safety constraints in an artificial pancreatic beta cell: an implementation of model predictive control with insulin on board', J Diabetes Sci Technol, 3(3), 536-44. Gopakumaran, B., Duman, H. M., Overholser, D. P., Federiuk, I. F., Quinn, M. J., Wood, M. D. and Ward, W.
K. (2005) 'A novel insulin delivery algorithm in rats with type 1 diabetes: the fading memory proportionalderivative method', Artif Organs, 29(8), 599-607. Hovorka, R., Chassin, L. J., Ellmerer, M., Plank, J. and Wilinska, M. E. (2008) 'A simulation model of glucose regulation in the critically ill', Physiol Meas, 29(8), 95978. Hovorka, R., Chassin, L. J., Wilinska, M. E., Canonico, V., Akwi, J. A., Federici, M. O., Massi-Benedetti, M., Hutzli, I., Zaugg, C., Kaufmann, H., Both, M., Vering, T., Schaller, H. C., Schaupp, L., Bodenlenz, M. and Pieber, T. R. (2004) 'Closing the loop: the adicol experience', Diabetes Technol Ther, 6(3), 307-18. Kovatchev, B. (2011) 'Closed loop control for type 1 diabetes', BMJ, 342, d1911. Kovatchev, B. P., Breton, M., Man, C. D. and Cobelli, C. (2009) 'In Silico Preclinical Trials: A Proof of Concept in Closed-Loop Control of Type 1 Diabetes', J Diabetes Sci Technol, 3(1), 44-55. Magni, L., Raimondo, D. M., Bossi, L., Man, C. D., De Nicolao, G., Kovatchev, B. and Cobelli, C. (2007) 'Model predictive control of type 1 diabetes: an in silico trial', J Diabetes Sci Technol, 1(6), 804-12. Parker, R. S., Doyle, F. J., 3rd and Peppas, N. A. (1999) 'A model-based algorithm for blood glucose control in type I diabetic patients', IEEE Trans Biomed Eng, 46(2), 148-57. Rawlings, J. B. and Mayne, D. Q. (2009) Model Predictive Control Theory and Design, Nob Hill Pub. Rippe, B. and Haraldsson, B. (1994) 'Transport of macromolecules across microvascular walls: the two-pore theory', Physiol Rev, 74(1), 163-219. Sedaghat, A. R., Sherman, A. and Quon, M. J. (2002) 'A mathematical model of metabolic insulin signaling pathways', Am J Physiol Endocrinol Metab, 283(5), E1084-101. Sorensen, J. T. (1985) A Physiologic Model of Glucose Metabolism in Man and its Use to Design and Assess Improved Insulin Therapies for Diabetes, unpublished thesis (PhD Thesis), MIT. Steil, G. M., Clark, B., Kanderian, S. and Rebrin, K. (2005) 'Modeling insulin action for development of a closed-loop artificial pancreas', Diabetes Technol Ther, 7(1), 94-108. Strougo, A., Eissing, T., Yassen, A., Willmann, S., Danhof, M. and Freijer, J. (2012) 'First dose in children: physiological insights into pharmacokinetic scaling approaches and their implications in paediatric drug development', J Pharmacokinet Pharmacodyn. Tarin, C., Teufel, E., Pico, J., Bondia, J. and Pfleiderer, H. J. (2005) 'Comprehensive pharmacokinetic model of insulin Glargine and other insulin formulations', IEEE Trans Biomed Eng, 52(12), 1994-2005. Thelen, K., Coboeken, K., Willmann, S., Burghaus, R., Dressman, J. B. and Lippert, J. (2011) 'Evolution of a detailed physiological model to simulate the gastrointestinal transit and absorption process in humans, part 1: oral solutions', J Pharm Sci, 100(12), 5324-45. Willmann, S., Lippert, J. and al., E. (2003) 'PK-Sim®: a physiologically based pharmacokinetic ‘whole-body’ model', Biosilico, 1, 121-124.
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