CommunicationModeling, identification and nonlinear model predictive control of type I diabetic patient
Introduction
Insulin-dependent diabetes mellitus is a chronic metabolic disorder, which is characterized by an increased blood glucose level. If this level is sustained, several complications appear, like nephropathies, neuropathies or retinopathies. These high levels are caused by decreased or an absence of insulin production by the pancreas. In most of the cases, patients who suffer diabetes type I need a therapy with several insulin injections a day or an insulin pump. None of these approaches reach a physiological condition, but the last one leads to better results because it is closer to the pancreatic function than a few injections. Following these results, an idea is to mimic the pancreas function using a pump controlled by the current glucose level. In order to avoid complications related to the continuous vascular access, the subcutaneous (s.c.) route has been proposed both to sense glucose levels and to inject insulin [1]. This approach is by far superior to the intravenous route due to its management and safety [2], [3].
The system to be controlled presents highly nonlinear dynamics, time delays associated with the insulin absorption process from subcutaneous tissue to blood and with the transfer rate between blood and subcutaneous glucose, and dead times due to the tubing in the monitoring system. Model Predictive Control (MPC) is a control strategy that predicts system outputs based on an appropriate model and calculates future control actions that are necessary to fit a desired output by means of an optimization algorithm [4]. To be reliable, this strategy needs a precise model of the system to be controlled. A linear model cannot deal with the high nonlinearities of the system we are dealing with. In order to overcome this problem it is possible to use neural networks to model it. This strategy, based on the MPC idea, is called Nonlinear Model Predictive Control (NMPC) [4], [5], [6], [7].
In this work, a diabetic patient model is used in order to represent a real patient. An approach for efficiently simulating multiple insulin injections in the subcutaneous tissue is presented. In order to identify the diabetic patient model a Multilayer Perceptron network is used. The above mentioned NMPC control strategy utilises this neural network as a nonlinear model for prediction purposes. The method is tested with a simulated 50 g oral glucose tolerance test (OGTT) corresponding to a charge disturbance, and with a change in the desired glucose level (reference disturbance). In both cases good results were obtained, avoiding the dangerous hypoglycemic condition and reaching physiological glucose concentrations with acceptable delays.
Section snippets
Physiopathological patient model
The patient model is made up of three submodels. The first of these submodels represents subcutaneous insulin dynamics. The next one takes into account general metabolism, considering plasmatic, hepatic and pancreatic concentrations of glucose, insulin and glucagon. Finally, the last submodel corresponds to the interactions between plasmatic and subcutaneous glucose concentrations. Additionally, it includes a dead time owing to the monitoring system. The structure of this model is presented in
Nonlinear identification by neural networks
Predicting the future outputs of the system to be controlled implies that we should obtain an identification of its nonlinear dynamics. There is a considerable overlap between system identification and time series prediction. The NN’s have important properties related to control systems. The usage of NN’s in nonlinear control problems arises from its ability to approximate arbitrary continuous nonlinear functions, in particular multilayer perceptron networks (MLP) [31], [32], [33]. In our
Nonlinear model-based predictive control
There are several difficulties associated with developing a method for glucose control in diabetic patients. We are dealing with a highly nonlinear system where there is a dead time involved in the s.c. glucose sensing, there are delays in the passage from blood glucose to s.c. tissue and in the insulin absorption process, there are constraints in the control action (it is not possible to inject negative quantities nor to exceed a certain injection speed), and essentially the hypoglycemia
Glucose control in a diabetic patient
It can be deduced from the model presented in Section 2 that the process which represents the glucose metabolism in a diabetic individual has three main difficulties for its control: high nonlinearities, dead times and delays, and the access to only one variable of the space state. However, as the sample time around 1 min is a suitable choice, it is possible to make relatively complex calculations. These features suggest the utilisation of a nonlinear MPC strategy. This strategy is analog to
Results
In order to know the relationship between the insulin injected in the subcutaneous tissue and the final glucose concentration, a series of simulations were carried out. In these simulations, initial plasmatic and subcutaneous glucose concentration were 250 mg/dl and 12.5 mmol/l, respectively, and no initial insulin is assumed. As can be seen in Fig. 6, this is a nonlinear relationship. Furthermore, the system is very sensitive to slight insulin injection changes in the zone with greatest
Discussion and conclusion
In this study, modifications to previous diabetic models were presented in order to simulate multiple insulin injections. The insulin absorption from the subcutaneous tissue to the plasma was modeled using an AR model. An implementation of a MLP neural network with only 18 hidden units was used to identify this model, achieving reasonable long time predictions. A NMPC approach was presented in order to control blood glucose concentration.
Maintaining the plasmatic glucose concentration in a
Acknowledgements
The authors gratefully acknowledge the reviewers for their useful comments. This work was partially supported by the Universidad Nacional de Entre Ríos.
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