Blood Glucose Regulation in Type II DiabeticPatientsbyFatemeh EkramM.Sc., Chemical & Petroleum Engineering, Sharif University of Technology,2010B.Sc., Chemical & Petroleum Engineering, Sharif University of Technology,2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemical & Biological Engineering)The University Of British Columbia(Vancouver)February 2016© Fatemeh Ekram, 2016AbstractType II diabetes is the most pervasive diabetic disorder, characterized by insulinresistance, β -cell failure in secreting insulin and impaired regulatory effects ofthe liver on glucose concentration. Although in the initial steps of the disease,it can be controlled by lifestyle management, but most of the patients eventuallyrequire oral diabetic drugs and insulin therapy. The target for the blood glucoseregulation is a certain range rather than a single value and even in this range, it ismore desirable to keep the blood glucose close to the lower bound.Due to ethical issues and physiological restrictions, the number of experimentsthat can be performed on a real subject is limited. Mathematical modeling ofglucose metabolism in the diabetic patient is a safe alternative to provide suffi-cient and reliable information on the medical status of the patient. In this thesis,dynamic model of type II diabetes has been expanded by incorporation of thepharmacokinetic-pharmacodynamic model of different types of insulin and oraldrug to study the impact of several treatment regimens. The most efficient treat-ment has been then selected amongst all possible multiple daily injection regimensaccording to the patient 's individualized response.In this thesis, the feedback control strategy is applied in this thesis to deter-mine the proper insulin dosage continuously infused through insulin pump to reg-ulate the blood glucose level. The logarithm of blood glucose concentration hasbeen used as the controlled variable to reduce the nonlinearity of the glucose-insulin interactions. Also, the proportional-integral controller has been modifiedby scheduling gains calculated by a fuzzy inference system.iiModel predictive control strategy has been proposed in this research for thetime that sufficient measurements of the blood glucose are available. Multiple lin-ear models have been considered to address the nonlinearity of glucose homeosta-sis. On the other hand, the optimization objective function has been adjusted tobetter fulfill the objectives of the blood glucose regulation by considering asym-metric cost function and soft constraints. The optimization problem has beensolved by the application of multi-parametric quadratic programming approachwhich reduces the on-line optimization problem to off-line function evaluation.iiiPrefaceThis thesis entitled ”Blood Glucose Regulation in Type II Diabetic Patients” con-sists of seven chapters. It presents my research during Ph.D. studies, under thesupervision of Professor K. E. Kwok and Professor R. B. Gopaluni at Chemicaland Biological Engineering Department of the University of British Columbia.Preliminary results of Chapter 3 and Chapter 4 of the thesis have been pub-lished in the literature. A version of Chapter 4 considering the blood glucose reg-ulation in severe type II diabetic patients has been prepared for submission. Thematerials presented in Chapter 5 for insulin therapy are combined with the met-formin therapy as presented in Chapter 3 and being prepared in the manuscriptformat for publication. Contributions and collaborations to the published, submit-ted and prepared papers for publication are explained in the following:1. A version of Chapter 3 entitled “Evaluation of Treatment Regimens forBlood Glucose Regulation in Type II Diabetes Using Pharmacokinetic- Phar-macodynamic Modeling” has been published in the proceeding of Chi-nese Control Conference (CCC), 2015. This manuscript has been preparedwith close collaboration of Professor Kwok and Professor Gopaluni. Ms.Barazandegan helped with preparing type II diabetes model for simulation.I incorporated the pharmacokinetic-pharmacodynamic model of differentinsulin types and metformin to evaluate different treatment regimens.2. A version of Chapter 4 has been published:F. Ekram, L. Sun, O. Vahidi, E. Kwok, and R. B. Gopaluni, “A feedback glu-ivcose control strategy for type II diabetes mellitus based on fuzzy logic,“ TheCanadian Journal of Chemical Engineering, vol. 90, no. 6, pp. 14111417,Dec. 2012.This paper has been published with close collaboration of Professor Kwokand Professor Gopaluni. They also have helped in the preparation of the firstdrafts and revision of the final drafts. Dr. Sun and Dr. Vahidi helped withpreparing the first draft of paper using conventional PI controller. I latermodified the draft by introducing the fuzzy inference system to improve theperformance of conventional PI and prepared the final manuscript for thejournal.3. A version of Chapter 4, entitled “Modified Fuzzy Gain Scheduling Con-troller for Maintaining the Blood Glucose Level in Type II Diabetic Pa-tients” has been prepared for submission.This manuscript has been prepared with close collaboration of ProfessorKwok and Professor Gopaluni. Ms. Barazandegan helped with preparingtype II diabetes model for simulation and writing the mathematical modelsection. I reduced the nonlinearity of glucose-insulin interactions in themodel by using the logarithm of the blood glucose measurement to calcu-late the feedback error and then developed and applied the gain schedulingstrategy for blood glucose regulation. I prepared the final manuscript forsubmitting to the journal.4. A version of Chapter 5 combined with PK-PD model of metformin fromChapter 3, entitled “Combination treatment for different stages of type IIdiabetes using metformin therapy and insulin pump”, is being prepared forsubmitting to a journal.This manuscript has been prepared with close collaboration of ProfessorKwok and Professor Gopaluni. Ms. Barazandegan helped with preparingtype II diabetes model for simulation. I incorporated the PK-PD model ofvmetformin into type II diabetes model and developed and applied the predic-tive control strategy for calculating the proper amount of insulin to regulatethe blood glucose concentration. I am preparing the final manuscript for thejournal submission.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Glucose Metabolism . . . . . . . . . . . . . . . . . . . . 21.1.2 Abnormalities of Type II Diabetes Mellitus . . . . . . . . 51.1.3 Treatments for Type II Diabetes Mellitus . . . . . . . . . . 61.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Clinical Control Objectives . . . . . . . . . . . . . . . . . 81.2.2 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . 122 Mathematical Modeling of Type II Diabetes Mellitus . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15vii2.2 The Sorensen Model . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Glucose Sub-model . . . . . . . . . . . . . . . . . . . . . 202.2.2 Insulin Sub-model . . . . . . . . . . . . . . . . . . . . . 242.2.3 Glucagon Sub-model . . . . . . . . . . . . . . . . . . . . 292.3 Type II Diabetes Model . . . . . . . . . . . . . . . . . . . . . . . 303 Evaluation of Treatment Regimens for Blood Glucose Regulation inType II Diabetes Using Pharmacokinetic-Pharmacodynamic Mod-elling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Expansion of Type II Diabetes Mellitus Mathematical Model . . . 373.2.1 Pharmacokinetic-Pharmacodynamic Modeling of Subcu-taneous Injected Insulin . . . . . . . . . . . . . . . . . . . 373.2.2 Pharmacokinetic-Pharmacodynamic Modeling of Metformin 393.2.3 Mathematical Representation of the Glucose AbsorptionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Simulation Results and Discussion . . . . . . . . . . . . . . . . . 443.3.1 Simulation Results for Different Insulin Injection Regimens 453.3.2 Simulation Results for Administration of Metformin . . . 573.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Development of a Controller for Blood Glucose Regulation in TypeII Diabetes Using Proportional-Integral (PI) Control Strategy . . . . 644.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Feedback Control Strategy . . . . . . . . . . . . . . . . . . . . . 674.2.1 Reducing Nonlinearity of Glucose-Insulin Interaction . . . 684.2.2 Gain Scheduling Control Strategy Based on Fuzzy Logic . 714.3 Simulation Results and Discussion . . . . . . . . . . . . . . . . . 754.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84viii5 Development of a Controller for Blood Glucose Regulation in TypeII Diabetes Using Predictive Control Strategy . . . . . . . . . . . . . 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Predictive Control Strategy . . . . . . . . . . . . . . . . . . . . . 905.2.1 Linear Model Predictive Control (LMPC) . . . . . . . . . 905.2.2 Multi-Parametric Quadratic Programming Approach . . . 955.3 Prediction Model for Type II Diabetes . . . . . . . . . . . . . . . 995.4 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Simulation Results and Discussion . . . . . . . . . . . . . . . . . 1025.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Comparison of Controller Performance in response to variations inthe metabolic rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Controller Performance Investigation . . . . . . . . . . . . . . . . 1126.2.1 Variation in Liver Metabolic Rates . . . . . . . . . . . . . 1126.2.2 Variation in Periphery Metabolic Rate . . . . . . . . . . . 1186.2.3 Variation in Pancreatic Insulin Secretion Rate . . . . . . . 1216.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . 1287.1 Research Summary and Conclusion . . . . . . . . . . . . . . . . 1287.2 Research Limitations . . . . . . . . . . . . . . . . . . . . . . . . 1307.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132ixList of TablesTable 1.1 Comparison of type I and II diabetes [1] . . . . . . . . . . . . 2Table 1.2 Oral drugs for diabetes [2] . . . . . . . . . . . . . . . . . . . . 7Table 1.3 Insulin types [3] . . . . . . . . . . . . . . . . . . . . . . . . . 8Table 2.1 The model parameters [4] . . . . . . . . . . . . . . . . . . . . 31Table 2.2 Abnormalities associated with type II diabetes and their corre-sponding equations . . . . . . . . . . . . . . . . . . . . . . . . 32Table 2.3 Parameter estimation results for the glucose sub-model [5] . . . 33Table 3.1 parameter values of insulin PK-PD model [6] . . . . . . . . . . 38Table 3.2 parameter values of metformin PK-PD model [7] . . . . . . . . 40Table 3.3 Parameter values of metformin PK-PD model [8] . . . . . . . . 42Table 3.4 Insulin injection regimen . . . . . . . . . . . . . . . . . . . . . 54Table 3.5 Comparison of different regimens . . . . . . . . . . . . . . . . 55Table 4.1 Comparison of the designed controllers . . . . . . . . . . . . . 86Table 5.1 Prediction models matrices . . . . . . . . . . . . . . . . . . . 100Table 5.2 Parametric solution regions for low blood glucose concentra-tion (< 7mmol/L) . . . . . . . . . . . . . . . . . . . . . . . . 105Table 5.3 Parametric solution regions for normal blood glucose concen-tration (7−11mmol/L) . . . . . . . . . . . . . . . . . . . . . 105xTable 5.4 Parametric solution regions for high blood glucose concentra-tion (> 11mmol/L) . . . . . . . . . . . . . . . . . . . . . . . . 106Table 6.1 Assessment of the blood glucose regulation using the designedcontrollers for the decreased metabolic rates . . . . . . . . . . 124xiList of FiguresFigure 1.1 Glucose homeostasis control mechanism in the body . . . . . 4Figure 1.2 Glycemic control zones . . . . . . . . . . . . . . . . . . . . . 10Figure 2.1 Simplified blood circulatory system[9] . . . . . . . . . . . . . 17Figure 2.2 General representation of a compartment[9] . . . . . . . . . . 18Figure 2.3 Simplified configurations of physiological compartments[9] . 19Figure 2.4 Schematic diagram of glucose sub-model[9] . . . . . . . . . . 21Figure 2.5 Schematic diagram of insulin sub-model[9] . . . . . . . . . . 25Figure 2.6 Schematic diagram of Landahl and Grodskys model[9] . . . . 27Figure 3.1 Schematic diagram of connections among glucose absorptionmodel, metformin and insulin injection PK-PD model and Sorensenmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.2 The simulated patient response to the meal disturbance duringthe day without drug and insulin therapy . . . . . . . . . . . . 45Figure 3.3 plasma insulin profile for regular, NPH, lente and ultralenteinsulin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.4 The simulated patient blood glucose profile after the subcu-taneous insulin injection according to regimen 1(black solidline), regimen 2 (blue dashed line) and regimen 3(red dottedline) to study the effects of changes in regular insulin dose . . 47xiiFigure 3.5 The simulated patient plasma insulin profile after the subcu-taneous insulin injection according to regimen 1(black solidline),regimen 2 (blue dashed line) and regimen 3(red dottedline) to show the insulin resistance in body cells . . . . . . . . 48Figure 3.6 The simulated patient blood glucose profile after the subcu-taneous insulin injection according to regimen 1(black solidline),regimen 4 (blue dashed line) and regimen 5(red dottedline) to study the effects of changes in NPH insulin dose . . . 49Figure 3.7 The simulated patient plasma insulin profile produced by pan-creas without any external insulin injection . . . . . . . . . . 50Figure 3.8 The simulated patient response to the subcutaneous insulin in-jection to study the effects of NPH according to regimen 6(black solid line), lente according to regimen 7 (blue dashedline) and ultralente according to regimen 8 (red dotted line) . . 52Figure 3.9 The simulated patient response to the subcutaneous insulin in-jection to study the effect of changes in the timing of the injec-tion. Glucose profile for regimen 8 (black solid line) appliedat the meal time is compared with the glucose profile of ad-vancing (blue dashed line) and delaying (red dotted line) allthe injections by an hour. . . . . . . . . . . . . . . . . . . . . 53Figure 3.10 The simulated patient response to the subcutaneous insulininjection corresponding to regimen 9 which proposes an ef-ficient blood glucose regulation . . . . . . . . . . . . . . . . . 56Figure 3.11 Glocose loweing effect of metformin in liver, GI tract and pe-riphery (EL, EGI and EP) . . . . . . . . . . . . . . . . . . . . 57Figure 3.12 The simulated patient response to the administration of 500mg metformin (black solid line) and 1000 mg metformin (bluedashed line) two times a day at 8 am and 8 pm . . . . . . . . . 58xiiiFigure 3.13 The simulated patient response to the administration of 500mg (black solid line) and 1000 mg (blue dotted line) met-formin at 8 am and 8 pm along with the insulin injection ac-cording to regimen 9. . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.14 The blood glucose concentration in gut without administra-tion of metformin (black solid line) and with administrationof 1000 mg metformin at 8 am and 8 pm (blue dotted line) . . 60Figure 3.15 The blood glucose concentration in liver without administra-tion of metformin (black solid line) and with administrationof 1000 mg metformin at 8 am and 8 pm (blue dotted line) . . 61Figure 3.16 The blood glucose concentration in periphery without admin-istration of metformin (black solid line) and with administra-tion of 1000 mg metformin at 8 am and 8 pm (blue dottedline) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.1 Variation of the gain for different step changes (the black pointsconnected with the blue line) and the regression line (dashedred line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.2 Variation of the logarithm of the gain for different step changes(the black points connected with the blue line) and the regres-sion line (dashed red line) . . . . . . . . . . . . . . . . . . . . 70Figure 4.3 Block diagram of the feedback control strategy using loga-rithm of the blood glucose as the controlled variable . . . . . . 70Figure 4.4 Block diagram of the feedback control strategy using fuzzyinference system to define weighting factors . . . . . . . . . . 73Figure 4.5 Fuzzy membership functions[10] . . . . . . . . . . . . . . . . 73Figure 4.6 Patient response to 100 g of meal disturbance at 100 min inabsence of insulin infusion . . . . . . . . . . . . . . . . . . . 75Figure 4.7 Response of the conventional PI controller with a 100 g mealdisturbance at 100 min . . . . . . . . . . . . . . . . . . . . . 76xivFigure 4.8 Response of the conventional PI controller with a 100 g mealdisturbance at 100 min with non-linearity reduction in the pa-tient model using log of blood glucose . . . . . . . . . . . . . 77Figure 4.9 Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min . . . . . . . . . . . . . . . . . . . . . 78Figure 4.10 Block diagram of the feedback control strategy using fuzzyinference system to define weighting factors when logarithmof blood glucose is considered as the controlled variable . . . 78Figure 4.11 Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min with nonlinearity reduction in the pa-tient model using log of blood glucose . . . . . . . . . . . . . 79Figure 4.12 Weight factor from the fuzzy inference system . . . . . . . . . 79Figure 4.13 Block diagram of the feedback and feedforward control strategy 80Figure 4.14 Response of the conventional PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100for 3 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.15 Response of the conventional PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100for 3 hours with non-linearity reduction in the patient modelusing log of blood glucose . . . . . . . . . . . . . . . . . . . 82Figure 4.16 Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at100 for 3 hours . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.17 Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100for 3 hours after non-linearity reduction in the patient modelusing log of blood glucose . . . . . . . . . . . . . . . . . . . 84Figure 5.1 Block diagram of model predictive control strategy . . . . . . 103Figure 5.2 The simulated patient response to 100 gr of glucose ingestionas a meal disturbance with no automatic control . . . . . . . . 107xvFigure 5.3 The simulated patient response to 100 gr of glucose ingestionusing model predictive control strategy . . . . . . . . . . . . . 107Figure 5.4 Response of the conventional PI controller with a 100 g mealdisturbance at 100 min . . . . . . . . . . . . . . . . . . . . . 108Figure 5.5 Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min with non-linearity reduction in the pa-tient model using log of blood glucose . . . . . . . . . . . . . 109Figure 6.1 role of the liver in glucose homeostasis . . . . . . . . . . . . . 113Figure 6.2 The simulated patient response to 100 gr of glucose ingestionas a meal disturbance for 50% decrease in hepatic glucose up-take and production rates (black solid line) and before the ratereduction (blue dashed line) . . . . . . . . . . . . . . . . . . 114Figure 6.3 Plasma blood glucose concentration in presence of continu-ous insulin infusion rate of 10 mU/min for the reduced rates(black solid line) and before the reduction (blue dashed line) . 114Figure 6.4 The performance of fuzzy gains-scheduling controller after100 gr of glucose ingestion as a meal disturbance for 50%decrease in hepatic glucose uptake and production rates . . . . 115Figure 6.5 Weight factor from the fuzzy inference system . . . . . . . . . 116Figure 6.6 The performance of model predictive controller after 100 grof glucose ingestion as a meal disturbance for 50% decreasein hepatic glucose uptake and production rates . . . . . . . . . 117Figure 6.7 The simulated patient response to 100 gr of glucose inges-tion as a meal disturbance for 50% decrease in peripheral glu-cose uptake rate (black solid line) and before the rate reduction(blue dashed line) . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 6.8 The performance of fuzzy gains-scheduling controller after100 gr of glucose ingestion as a meal disturbance for 50%decrease in peripheral glucose uptake rate . . . . . . . . . . . 119xviFigure 6.9 The performance of model predictive controller after 100 grof glucose ingestion as a meal disturbance for 50% decreasein peripheral glucose uptake rate . . . . . . . . . . . . . . . . 120Figure 6.10 The simulated patient response to 100 gr of glucose ingestionas a meal disturbance for 50% decrease in pancreatic insulinsecretion rate (black solid line) and before the rate reduction(blue dashed line) . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 6.11 The performance of fuzzy gains-scheduling controller after100 gr of glucose ingestion as a meal disturbance for 50%decrease in pancreatic insulin secretion rate . . . . . . . . . . 122Figure 6.12 The performance of model predictive controller after 100 grof glucose ingestion as a meal disturbance for 50% decreasein pancreatic insulin secretion rate . . . . . . . . . . . . . . . 123Figure 6.13 The performance of model predictive controller after 100 gr ofglucose ingestion as a meal disturbance for 50% decrease inperipheral glucose uptake rate with the administration of 1000mg metformin . . . . . . . . . . . . . . . . . . . . . . . . . . 126xviiNomenclatureThe following nomenclature is adopted throughout the mathematical model de-scription in chapter 2:Model variables in the glucose sub-modelG Glucose concentration (mg/dl)M Multiplier of metabolic rates (dimensionless)Q Vascular blood flow rate (dl/min)r Metabolic production or consumption rate (mg/min)T Transcapillary diffusion time constant (min)t time (min)V Volume (dl)Model variables in the insulin sub-modelI Insulin concentration (mU/l)M Multiplier of metabolic rates (dimensionless)m Labile insulin mass (U)xviiiP Potentiator (dimensionless)Q Vascular blood flow rate (dl/min)R Inhibitor (dimensionless)r Metabolic production or consumption rate (mU/min)S Insulin secretion rate (U/min)T Transcapillary diffusion time constant (min)t time (min)V Volume (dl)X Glucose-enhanced excitation factor (dimensionless)Y Intermediate variable (dimensionless)Model variables in the glucagon sub-modelΓ Normalized glucagon concentration (dimensionless)M Multiplier of metabolic rates (dimensionless)r Metabolic production or consumption rate (dl/min)t time (min)V Volume (dl)First superscriptΓ GlucagonB Basal conditionxixG GlucoseI InsulinSecond superscript∞ Final steady state valueMetabolic rate subscriptsBGU Brain glucose uptakeGGU Gut glucose uptakeHGP Hepatic glucose productionHGU Hepatic glucose uptakeKGE Kidney glucose excretionKIC Kidney insulin clearanceLIC Liver insulin clearanceMΓC Metabolic glucagon clearancePΓC Plasma glucagon clearancePΓR Pancreatic glucagon releasePGU Peripheral glucose uptakePIC Peripheral insulin clearancePIR Pancreatic insulin releaseRBGU Red blood cell glucose uptakexxFirst subscripts∞ Final steady state valueA Hepatic arteryB BrainG GutL LiverP PeripheryS StomachSecond subscripts (if required)C Capillary spaceF Interstitial fluid spacel Liquids SolidxxiAcknowledgmentsThere are many individuals that I am indebted to and who have provided supportand insight to me while undertaking my doctoral degree. First and foremost, Iwould like to express my sincere thanks and appreciation to my doctoral supervi-sors, Professor Ezra Kwok and Professor Bhushan Gopaluni from Department ofChemical and Biological Engineering at UBC. They have been a key influence inmy successes during my Ph.D. appointment. Their extensive knowledge, encour-agement and constructive guidance have been invaluable to me. I have been ableto grow and develop as a researcher due to their outstanding supervision.I would next like to thank my committee members, Professor Jim Lim fromDepartment of Chemical and Biological Engineering and Professor Martin Or-donez from Department of Electrical and Computer Engineering at UBC for theiradvice and guidance.I would like to thank my group members who have been helpful and support-ive friends during my Ph.D. appointment: Omid Vahidi, Melissa Barazandegan,Navid Ghaffari, David Zamar, Hui Tian, Qiugang Lu, Mahdi Yousefi, Lee Rippon,and Aigerim Ongalbayeva.Finally, I would like to express my gratitude to all of my friends near andfar who always made themselves available to listen. I would also like to thank myparents and my sister for their unconditional love and encouragement to undertakemy doctoral degree.xxiiTo my beloved parentsfor their pure love and supportfor understanding the things I saidand the things I didn’t sayTo my adorable sisterfor giving me joy in my heartTo my dear friendsfor being there for methrough the good times and the badxxiiiChapter 1Introduction1.1 BackgroundDiabetes mellitus is a metabolic disorder in which the blood glucose levels are notregulated because of the impaired insulin secretion, action or both. The prevalenceof diabetes in the world is growing at an unprecedented rate and rapidly becominga health concern. The International Diabetes Federation states that ”diabetes cur-rently affects more than 300 million people in the world, representing 6% of theworld's adult population” and ”every ten seconds, two people are diagnosed withdiabetes somewhere in this world” [11].The two main types of diabetes mellitus are type I and type II. Type I dia-betes mellitus is the result of the abrupt pancreatic beta cell destruction due toan auto-immune process while type II diabetes mellitus is a progressive diseasecharacterized by not only the failure of beta cell in secretion of adequate insulinbut also insulin resistance in muscles and adipose tissues [12]. A comparison oftype I and type II diabetes is presented in table 1.1 [1].Diabetes mellitus increases the risk of many serious health problems and canresult in a variety of complications. Diabetes is the leading cause of blindness(retinopathy), kidney failure (nephropathy) and nerve damage (neuropathy). Di-abetic patients are much more likely to have a stroke, heart disease, or a heart1Table 1.1: Comparison of type I and II diabetes [1]feature Type I diabetes Type II diabetesOnset Sudden GradualAge at onset Mostly in children Mostly in adultsBody habitus Thin or normal Often obeseKetoacidosis Common RareAutoantibodies Usually present AbsentEndogenous insulin Low or absent Normal, decreased or increasedprevalence ∼ 10% of diabetic population ∼ 90% of diabetic populationattack[13].1.1.1 Glucose MetabolismThe body cells obtain their required energy from different fuel sources includingcarbohydrates, fats and proteins. Amongst them, the main source of energy forthe body is carbohydrates. Carbohydrates digest in the gastrointestinal tract andproduce a simple sugar called glucose. It is absorbed by the body cells and is usedas the primary energy source. The body keeps the blood glucose concentration in acertain range during the fasting state by producing glucose endogenously throughtwo main pathways:1. Gluconeogenesis: Gluconeogenesis is a metabolic pathway that generatesglucose from non-carbohydrate carbon substrates such as lactate, glycerol,and glucogenic amino acids. It occurs in the liver and kidney.2. Glycogenolysis: Glycogenolysis is a metabolic pathway which results inthe generation of glucose from the breakdown of glycogen. It occurs in theliver and muscles.Liver and kidney release the produced endogenous glucose into the bloodstream while the produced glucose in muscle cells is consumed by themselves.Approximately 85% of endogenous glucose production which is released into the2blood stream is derived from the liver, and the remaining 15% is produced by thekidney [14].A healthy (non-diabetic) body controls the blood glucose level within a certainrange, despite disturbances such as exercise or intake of a meal containing carbo-hydrates, by precise responses of several organs to any changes in circulatingglucose levels [15]. This glucose regulatory control is carried out through feed-back systems reacting mainly on glucose, insulin and glucagon concentrations.Insulin and glucagon are two hormones which play an important role in glucosehomeostasis in the body and are secreted by the beta and alpha cells, respectively,contained in the islets of Langerhans scattered in the pancreas. The effect of thesehormones on glucose metabolism is opposite from one to another (see figure 1.1).Insulin contributes in lowering the blood sugar level by stimulating some bodycells to absorb glucose, suppressing endogenous glucose production and inhibit-ing glucagon secretion.Glucagon, on the other hand, contributes to increase the blood sugar level bystimulating the liver to produce more glucose and inhibiting insulin secretion.When the blood sugar level is high, pancreas secrets more insulin. Secreted in-sulin has negative paracrine action on the alpha cells causing inhibition of glucagonsecretion. Increased concentration of insulin and decreased the concentration ofglucagon lead to higher absorption of the blood glucose by body cells and lowerendogenous glucose production which in turn decrease the level of blood glu-cose concentration. Conversely, when the blood glucose concentration is low, thepancreas secretes more glucagon which inhibits secretion of insulin leading to in-creased endogenous glucose production and lowered absorption of glucose by thebody cells which in turn reduces the blood glucose concentration.Insulin contributes to augment the glucose uptake in the peripheral tissues andthe liver through affecting the activity of different enzymes:1. Insulin enhances the hepatic and peripheral glucose uptake by stimulatingthe activity of hexokinase, an enzyme responsible for glucose phosphoryla-tion which leads to trapping of glucose inside the cell.3Figure 1.1: Glucose homeostasis control mechanism in the body2. Insulin increases the peripheral glucose uptake by regulating the activity ofpyruvate dehydrogenase, a key enzyme in the glycolysis pathway.3. Insulin enhances peripheral and hepatic glucose absorption by stimulat-ing the glycogen synthesis (glycogenesis pathway). Insulin contributes inglycogenesis by activating a group of enzymes directing the glucose throughglycogen synthesis (e.g. glycogen synthase) and by inhibiting enzymes con-tributing the reverse reactions (e.g. glucose-6-phosphatase) [16].41.1.2 Abnormalities of Type II Diabetes MellitusType II diabetes is characterized by multiple abnormalities in some of the bodyorgans such as the liver, the pancreas, muscles and adipose tissues. These abnor-malities are classified as follows:1. Insulin resistance in peripheral tissues: peripheral tissues (i.e. muscle andadipose tissue cells) are dependent on insulin to absorb blood glucose. Stud-ies have shown that in type II diabetic patients, low glucose uptake rates bymuscle cells and adipose tissue cells are caused by peripheral insulin resis-tance and relative insulin deficiency [17–25]. Impairment of several factorsis known to be associated with insulin residence in peripheral tissues. De-fronzo reviewed these factors in his paper [14]. some of the factors are asfollows:• The number of insulin receptors• The affinity of insulin receptors• Insulin intracellular signaling• The number of glucose transporters• Glucose transporter translocation on the cell membrane• Insulin stimulatory effects on glycogenesis• Insulin stimulatory effects on glycolysis2. Reduced hepatic glucose uptake: Some studies have addressed the reducedhepatic glucose uptake as the result of impaired insulin-induced stimulationeffects on hepatic glucose uptake [26–31]. It is believed to be due to theimpairment of insulin stimulation effect on glucose phosphorylation in theliver [31].3. Impaired hepatic glucose production: Many studies have confirmed thathepatic glucose production rate is impaired in type II diabetic patients [20–25, 27, 28, 32–35]. Most of these studies have indicated that insulin-induced5suppression of endogenous glucose production is low in diabetic patients.Basu et al. [34, 35] have demonstrated the impaired effect of insulin sup-pression on both pathways of endogenous glucose production (i.e. gluco-neogenesis and glycogenolysis). however the effects of the glucose sup-pression on hepatic glucose production rate is still normal in type II diabeticpatients [36, 37].4. Impaired pancreatic insulin secretion: Deficiency in the pancreatic insulinproduction has the key role in the development of overt diabetes [14], whichmeans that overt diabetes does not develop unless the pancreas fails to pro-duce insulin properly. Pancreatic insulin secretion in response to a glucosestimulus has a biphasic pattern. Type II diabetic patients exhibit two formsof defective pancreatic insulin secretion. One in the early peak of insulinproduction and the other one is in the overall insulin secretion rate [38–40].5. Glucose resistance: The glucose-induced stimulation of glucose disposalhas been shown to be normal in type II diabetic patients, in an early studydone by Alzaid et al. [33]. However, later studies by Del Prato et al. [36]and Nielsen et al. [37] have indicated that high levels of glucose concentra-tion (particularly above 7 mmol/l) impair the glucose stimulation effect onglucose uptake in type II diabetic patients.1.1.3 Treatments for Type II Diabetes MellitusType II diabetic patients need to get their blood glucose level under control. Bloodglucose concentration in type II diabetic patients can be initially controlled byexercise and healthy dieting. Sooner or later patients need diabetes drugs alongwith diet and lifestyle changes. The short term goal of the treatment is to keep theblood glucose in the normal range and reduce or get rid of the symptoms whilethe long-term goal is to prevent complications.There are six groups of oral medicines and 11 different drugs. Each of the sixgroups of drugs works in a different way to synergize with the disease abnormali-6ties and are summarized in table 2 [2].Some of the severe type II diabetic patients eventually need insulin therapyalong with the oral drugs. The patients under insulin regimen can experiencebetter control of the blood glucose level through the whole day and avoid thesymptoms of high blood glucose level. On the other hand, the pressure on thepancreas to produce insulin will be reduced and benefit longer term health.Table 1.2: Oral drugs for diabetes [2]Metformin This group of drugs, called biguanides,works by keeping the liver from making glucoseand allowing more glucose to enter cells.Glipizide, Glimepiride This group of drugs, called sulfonylureas,& Glyburide works by helping the pancreas make more insulin.Prandin & Starlix This group of drugs, called meglitinides,works by helping the pancreas make more insulin.Precose & Glyset This group of drugs, called alpha-glucosidaseinhibitors, works by keeping the intestinesfrom absorbing glucose as quickly.Januvia This drug, called a dipeptidyl peptidase 4 inhibitor,works by helping the pancreas release insulin.Actos & Avandia This group of drugs, called thiazolidinediones,works by helping the cells use glucose.There are different forms of insulin to treat diabetes. They are classified basedon the following three characteristics:• Onset: the length of time before insulin reaches the bloodstream and beginslowering blood glucose.• Peak time: the time during which insulin is at maximum strength in termsof lowering the blood glucose.• Duration: how long insulin continues to lower blood glucose.Table 1.3 lists the types of injectable insulin with details about onset, peaktime, duration and the role in the blood sugar management [3].7Table 1.3: Insulin types [3]Type of insulin Onset Peak DurationRapid-acting 10-15 min 60-90 min 4-5 hoursFast-acting 30-60 min 2-4 hours 5-8 hoursIntermediate-acting 1-3 hours 5-8 hours Up to 18 hoursLong-acting 3-4 hours 8-15 hours 22-26 hoursRapid-acting insulin covers insulin needs for meals eaten at the same time asthe injection. This type of insulin is used with longer-acting insulin. Short-actinginsulin covers insulin needs for meals eaten within 30-60 minutes. Intermediate-acting insulin covers insulin needs for about half the day or overnight. This type ofinsulin is often combined with rapid- or short-acting insulin. Long-acting insulincovers insulin needs for about one full day. This type of insulin is often combined,when needed, with rapid- or short-acting insulin.Deciding what type of insulin might be best for a patient will depend on manyfactors, such as the body's individualized response to insulin and the lifestylechoices. Multiple daily injection (MDI) therapy and continuous subcutaneousinsulin infusion (CSII) with external insulin pumps are the available techniquesfor insulin delivery to the body. Two studies shows that CSII was as safe andeffective as MDI therapy for the treatment of type 2 diabetic patients [41, 42].1.2 Objectives1.2.1 Clinical Control ObjectivesFrom a clinician's perspective, the goals of insulin treatment need to be individ-ualized for different groups of the population such as the elderly, children, oreven pregnant patients who have diabetes mellitus. It is important to control glu-cose levels within certain targets to avoid long-term complications. Inadequateshort-term control could also cause fatigue, polyuria, polydipsia, blurred visionglucose or infection as well. Unlike in traditional control, optimal control of glu-8cose, however, does not imply minimizing glucose variance around a single targetvalue. Instead, a human body is able to function well as long as the blood glu-cose is within a certain range. In fact, in the general population, a healthy personmay have a fasting blood glucose between 3.5 to 5.7 mmol/L and a random non-fasting blood glucose up to 11.0 mmol/L. A diagnosis of diabetes mellitus canbe made and confirmed by a fasting glucose of more than 7.0 mmol/L or randomglucose of more than 11.0 mmol/L. For most patients, the most desirable targetglucose range is between 4.0 to 6.0 mmol/L during a fasting state, and 5.0 to 10.0mmol/L during the postprandial phase. The rationale for selecting a higher cut-off of 4.0 mmol/L is to allow for a safety margin to avoid severe hypoglycemiabelow 3.0 mmol/L which can cause fatigue, tremor, lightheadedness, sweating,or, in extreme cases, death. When a patient has hyperglycemia, symptoms of fa-tigue, polyuria, polydipsia or infection such as non-healing wounds will becomemore progressively obvious as the blood glucose increases more than 11 mmol/L.Patients having severe hyperglycemia, such as more than 20 mmol/L, could expe-rience extreme fatigue, loss of consciousness or even coma.Figure 1.2 provides a graphical impression of the glucose control objectives.It shows the red zones corresponding to severe hypoglycemia and severe hyper-glycemia which should be avoided completely. A small margin of tolerance be-tween 3 to 4 mmol/L for hypoglycemia and a large margin of tolerance between11 and 14 mmol/L for hyperglycemia are shown in yellow. The middle greenzone is divided into three sub-zones. The postprandial state consists of a 4-hourperiod that immediately commences as a meal is ingested. The postabsorptivestate is defined as a 6-hour period and follows the postprandial state. The fastingstate starts at the end of the postabsorptive state [43]. The graduating shade in thenormoglycemic zone indicates that blood glucose should always trend towardsthe lower range in favor of avoiding low-term complications of diabetes mellitusdue to persistently high blood glucose. Therefore, from a control perspective, thehomeostasis of glucose in a healthy person is a very well regulated non-linear sys-tem which intelligently maintains glucose in a safe and healthy range. For diabetic9Figure 1.2: Glycemic control zonespatients, linear controllers for glucose regulation will not be suitable or sufficientfor achieving such a control objective [44, 45]. Instead, the controller should bedesigned to return the blood glucose from the hypoglycemia and hyperglycemiczones back to the normoglycemic zone quickly but act aggressively to avoid get-ting into the severe hypoglycemia and severe hyperglycemic zones. When theblood glucose is within the normoglycemic zone, the controller should graduallybring the blood glucose to the lower limit of the normoglycemic zone.1.2.2 Thesis ObjectivesTight glycemic control is essential in order to reduce the risks of long-term dia-betic complications for both type I and type II diabetes patients [45]. Inadequateshort-term control could also cause fatigue, polyuria, polydipsia, blurred vision10or infection as well. Type I diabetic patients should be under intensive insulintherapy because of the lack of insulin production in the pancreas. Although dietand exercise have a role to play in type 1 diabetes management, they cannot re-verse the disease or eliminate the need for insulin. However, in initial steps oftype II diabetes mellitus, pancreas is still working so that treatment typically in-cludes lifestyle management such as diet modification and control, regular andappropriate exercise and home blood glucose testing but most of the patients withsevere type II diabetes mellitus eventually require oral anti-diabetic agent, insulintreatment or combination of them to provide adequate glycemic control.The application of dynamic modeling and advanced control techniques hasincreased in every aspect of our lives. The contribution of control engineers insolving the challenging problems in the area of biomedical processes can havesignificant medical impacts [46, 47].Mathematical modeling of glucose metabolism in the diabetic patient is help-ful and safe in providing reliable information without causing serious and irre-versible harm to the subject. Having sufficient amount of data facilitates the devel-opment and evaluation of different treatments and control strategies. Automaticcontrol systems can substitute manual control to provide better regulation of theblood glucose, lessen the care tension on the patients and improve the quality ofthe control.Most of the studies in the field of modeling and control of diabetes have ad-dressed type I. However; type II diabetes is the most pervasive type which affects90% of the diabetes population around the world [48]. Therefore, developing con-trol systems can be very helpful to understand the pathophysiology of the diseaseand find the most efficient therapy to treat properly.Contrary to type I, in which the only problem is the dysfunction of the pan-creas, type II diabetic patients deal with the malfunction of different organs. Thedesired controller for type I diabetic patients should be designed to mimic the be-haviour of real pancreas, while in type II the homeostasis of blood glucose is muchmore complicated than type I patients as discussed in 1.2.1, so that the control ob-11jective is challenging and not as straight forward as for type I because unlike typeI diabetes, multiple abnormalities in different organs as described in section 1.1.2lead to the deterioration of glucose homeostasis in type II diabetic patients [14].The objective of this research is to regulate the blood glucose for type II dia-betes mellitus by:• improving the dynamic model of type II diabetes mellitus• developing efficient control strategies1.2.3 Thesis OutlineChapter 2 describes the mathematical modelling previously developed based onSorensen model [4] for type II diabetes by Vahidi [5, 9]. This model simulated atype II diabetic patient using compartmental modeling approach. In this approach,different organs or parts of the body are represented by a number of compart-ments. The model equations are derived from the mass balance equations overeach compartment. The model comprises three main sub-models which representthe variation of blood glucose, insulin and glucagon concentrations in differentparts of the body. Increasing the number of compartments benefits analysis ofglucose regulation techniques by providing a better representation of the glucoseand insulin concentrations in different organs.Chapter 3 addresses the problem of evaluating different treatment regimens forpatients prescribed with oral agents and/or under multiple daily injections (MDI)therapy. Type II diabetic patients use different types of oral drug and insulin to gettheir blood glucose level under control as listed in 1.1.3. Deciding on the efficientregimen including drug and insulin therapy is a try and error procedure requiringthe performance of several tests on the patients and varies for each individual pa-tient. Since the number of experiments that can be performed on a human body isrestricted, the available physiological model, described in Chapter 2, is expandedby incorporation of pharmacokinetic-pharmacodynamic models for four popular12types of insulin and metformin which is a very common diabetic drug. Such ex-pansion represents the effects of drug and insulin administration on the patient.The model is suitable to assess the efficiency of several treatment regimens forblood glucose regulation including mixtures of short-acting and intermediate orlong-acting insulin and oral administration of metformin.Chapter 4 presents the application of feedback control strategy to determinethe proper insulin dosage delivered with an insulin pump in continuous subcu-taneous insulin infusion (CSII) therapy. Regarding the non-linear homeostasisof blood glucose, conventional PI controller is modified by considering schedul-ing gains generated by a fuzzy inference system. It is shown that the applicationof logarithm of blood glucose concentration as the controlled variable can alsoreduce the nonlinearity of the glucose-insulin interactions and improve the per-formance of the designed controller. In reality, the patients not only rely on anautomatic control device but also act as a feedforward controller and inject extradoses of insulin before each meal. The results of such action are also taken intoconsideration in this chapter.Chapter 5 demonstrates the procedure of designing a controller based on pre-dictive control strategy for blood glucose regulation. Multiple linear models areconsidered to address the nonlinearity of glucose homeostasis and represent theglucose metabolism for different levels of blood glucose. The optimization prob-lem is modified to develop a controller which closely mimics the glucose regula-tory system of the body. The asymmetric objective function is considered whilethe soft constraints allow partial violation of minimum and maximum values of theboundaries. A state estimator is designed to calculate the states associated withthe current measurement of the blood glucose concentration. Multi-parametricquadratic programming approach is applied to solve the optimization problemwhich reduces the on-line optimization problem to off-line function evaluation. Itprovides a look-up table with all optimal solutions which gives a good insight ofpatient's status.Chapter 6 provides the comparison for the performance of the designed con-13trollers in response to variations in the metabolic rates. The metabolic rates arenot only different between patients but may also vary within a patient during thedisease progress. Since the glucose regulatory system mainly deteriorates as theresult of the malfunction of liver, peripheral tissues and pancreas, the performanceof the designed controllers is investigated as the metabolic rates decrease in theseorgans.Finally, Chapter 7 summarizes and concludes the thesis.14Chapter 2Mathematical Modeling of Type IIDiabetes Mellitus2.1 IntroductionThere are several studies on mathematical modeling of glucose regulation in healthyhuman subjects starting with simple linear models by Bolie [49] and Ackerman[50]. Makroglou et al. [51], Mari [52] and Cedersund et al. [53] reviewed thevarious modeling approaches have considered in proposed models. Amongst allthose approaches for modeling the glucose regulation in a human body, the com-partmental modeling approach is one of the most popular approaches. In thisapproach, different organs or parts of the body are represented by a number ofcompartments. The model equations are derived from the mass balance equationsover each compartment.The minimal compartmental model proposed by Bergman et al. [54] was oneof the pioneers of this modeling approach which consists of three compartments.The mass balance equations form the non-linear differential equations represent-ing glucose and insulin concentrations in the body. This model has been usedwidely in many diabetic studies for blood glucose regulations. Increasing thenumber of compartments benefits analysis of glucose regulation techniques by15providing a better representation of the glucose and insulin concentrations in dif-ferent organs while at the same time increases the complexity of the model. Morecomplicated compartmental models were proposed by Cobelli [55], Sorensen [4]and Hovorka [56].Dynamic modeling of glucose metabolism in healthy human body can be mod-ified and adjusted for the patients to be used in studying the physiological be-haviour of type I and type II diabetic patients. Type I diabetic patients have noinsulin production, therefore the model is easily adjusted for type I diabetes mel-litus without changing the structure. The model is ready to be used after settingthe insulin production rate term of the healthy subject's model to zero.A similar approach can be used to develop a model for type II diabetes, how-ever type II diabetes modeling is not as simple as type I diabetes modeling. Type IIdiabetes is associated with multiple abnormalities in different body organs whichleads to the deterioration of glucose homeostasis in type II diabetes. These abnor-malities target the glucose metabolic rates in the related organs and the secretionrates of glucose regulatory hormones such as pancreatic insulin secretion rate.Therefore, the same model structure can be used but with the updated parameters.Some studies developed the model for type II diabetes by this approach [5, 9, 57].In this chapter the equations of Sorensen model is represented along with theparameters updated for type II diabetic patient by Vahidi [4, 9]. The descriptionof the model variables, presented in the equations of this chapter, can be found innomenclature.2.2 The Sorensen ModelSorensen developed his compartmental model of glucose-insulin interactions in ahealthy body by modifying the model previously suggested by Guyton et al. [58].This model considers the regulatory effects of insulin and glucagon hormoneson glucose metabolism. The organs associated in diabetes research including theliver, pancreas, muscles and adipose tissues are represented by individual com-partments which make the model suitable to address the abnormalities of type II16diabetes as well. The Sorensen model comprises three main sub-models repre-senting variations of blood glucose, insulin and glucagon concentrations in thedifferent part of the body. The number of compartments is different in each sub-model and determined by the significance of the organ's job in maintaining therespective solute concentrations.Figure 2.1: Simplified blood circulatory system[9]In Sorensen model, the hormonal effects of epinephrine, cortisol, and growthhormone are assumed to be negligible. Physiology of changes in amino acid andfree fatty acid substrate levels are not considered, and the physiologic parameterssuch as blood flow rates and capillary space volumes are selected to represent atypical 70 Kg adult male. Figure 2.1 shows the simplified blood circulatory system17including the major organs which contribute significantly in glucose productionand consumption. Oxygen-rich blood is pumped from the heart left ventricle andis delivered to all body organs through the arteries. Deoxygenated blood is drainedout of the body organs and delivered to the heart right atrium through the veins.Figure 2.2: General representation of a compartment[9]In compartmental modeling approach, a compartment represents an organ or aspecific part of the human body. Each compartment is generally divided into threewell-mixed spaces (sub-compartments) representing the capillary blood space, theinterstitial fluid space and the intracellular space. A graphical representation of atypical compartment of the Sorensen model is shown in figure 2.2.The arterialblood inflow feeds in the capillary space and the venous blood outflow drains it.The blood components may diffuse through capillary walls into the interstitialfluid and from interstitial fluid to the intracellular space and vice versa.Not all these three zones are considered for modeling different parts of thebody. If the capillary wall is impermeable to a solute and no extravascular ex-change occurs and, therefore, only the capillary blood space is considered, andthe two other spaces are omitted (figure 2.3 a). In the case of high permeabilityof the capillary wall for a solute which leads to a fast equilibrium of the capillaryblood and the interstitial fluid spaces, two spaces are considered as one combined18Figure 2.3: Simplified configurations of physiological compartments[9]sub-compartment with uniform solute concentration (Figure 2.3 b). Likewise, thepermeability of the cell membrane may be high enough for a solute which causes19fast equilibrium of the interstitial fluid and intracellular fluid spaces. Two spacesare combined in this case and considered as one sub-compartment with uniformsolute concentration (Figure 2.3 c). If the permeability of both capillary wall andcell membrane is high enough to a solute which leads to a fast equilibrium of allthree spaces, therefore, all three spaces are combined and considered as one spacewith uniform solute concentration (Figure 2.3 d). Finally, if the rate of solutetransport across the cell membrane is not restricted by the concentration of thesolute in the intracellular fluid space, the intracellular space is omitted (Figure 2.3e). So that at most two of these sub-compartments are physiologically required tomodel a solute transfer from the capillary blood space to the intracellular spacefor each compartment.2.2.1 Glucose Sub-modelA schematic representation of the glucose sub-model is depicted in figure 2.4In this sub-model, the body is divided into six compartments: brain; liver;heart and lungs; periphery (muscles and adipose tissues); gastrointestinal (GI)tract (the stomach and intestinal system); and kidney. The arrows in figure 2.4 rep-resent the blood flow direction. Mass balance equation over each sub-compartmentresults in eight ordinary differential equations constituting the glucose sub-model:V GBCdGBCdt= QGB (GH−GBC)−V GBFT GB(GBC−GBF), (2.1)V GBFdGBFdt=V GBFT GB(GBC−GBF)− rBGU , (2.2)V GHdGHdt= QGB GBC +QGL GL+QGKGK +QGP GPC +QGHGH− rBCU , (2.3)V GGdGGdt= QGG(GH−GG)− rGGU , (2.4)V GLdGLdt= QGA GH +QGGGG−QGL GL+ rHGP− rHGU , (2.5)20Figure 2.4: Schematic diagram of glucose sub-model[9]V GKdGKdt= QGK(GH−GK)− rKGE , (2.6)V GPCdGPCdt= QGP (GH−GPC)−V GPFT GP(GPC−GPF), (2.7)V GPFdGPFdt=V GPFT GP(GPC−GPF)− rPGU , (2.8)where G is the glucose concentration (mg/dl), Q is the vascular blood flow rate(dl/min), V is the volume (dl), T is the transcapillary diffusion time constant (min),21r is the metabolic production or consumption rate (mg/min) and t is time (min).The subscripts of these variables refer to the body organs. Subscript B is the brain,subscript BC is the brain capillary space and subscript BF is the brain interstitialfluid space. Subscript A is the hepatic artery, subscript G is gut, subscript Lisliver and subscript G is GI tract (stomach and intestines). SubscriptP is periphery,subscript PC is the periphery capillary space and subscript PF is the peripheryinterstitial fluid space.The general form of the metabolic production and consumption rates in eachorgan is as follows:r = MI(t)MG(t)MΓ(t)rB, (2.9)where MI , MG and MΓ are the independent multiplicative effect of insulin, glucoseand glucagon on the metabolic rate, respectively. rB is the basal metabolic rate andthe multipliers have the following general form:MC = a+b tanh(cCCB−d), (2.10)where a, b, c and d are the parameters of the model. C is the substance concentra-tion and CB is the basal concentration of the substance.The following equations are used to calculate the glucose metabolic rates:rBGU = 70 (2.11)rRBGU = 10 (2.12)rGGU = 20 (2.13)rPGU = MIPGU MGPGU rBPGU , (2.14)22rBPGU = 35 (2.15)MIPGU = 7.03+6.52tanh(0.338(IPFIBPF−5.82) (2.16)MGPGU =GPFGBPF(2.17)rHGP = MIHGPMGHGPMΓHGPrBHGP, (2.18)rBHGP = 35 (2.19)ddtMIHGP = 0.04(MI∞HGP−MIHGP) (2.20)MI∞HGP = 1.21−1.14tanh[1.66(ILIBL−0.89)] (2.21)MGHGP = 1.42−1.14tanh[0.62(GLGBL−0.497)] (2.22)MΓHGP = 2.7tanh[0.39(ΓΓB]− f (2.23)ddtf = 0.0154[(2.7tanh[0.39( ΓΓB ]−12)− f ] (2.24)rHGU = MIHGU MGHGU rBHGU , (2.25)rBHGU = 20 (2.26)23ddtMIHGU = 0.04(MI∞HGU −MIHGU) (2.27)MI∞HGU = 2.0tanh[0.55(ILIBL] (2.28)MGHGU = 5.66+5.66tanh[2.44(GLGBL−1.48)] (2.29)KGE = 71+71tanh[0.11(GK−460)] 0≤ GK < 460rKGE = 71+71tanh[0.11(GK−460)] GK ≥ 460(2.30)where rBGU is brain glucose uptake rate, rGGU is gut glucose uptake rate, rHGPis hepatic glucose production rate, rHGU is hepatic glucose uptake rate, rKGE iskidney glucose excretion rate, rPGU is peripheral glucose uptake rate and rRBGUis red blood cell glucose uptake rate. G, I and Γ are the concentration of glucose,insulin and glucagon, respectively. Superscript B refers to the basal condition and∞ refer to final steady state value.As equation 2.17 shows, the form of glucose multiplier of peripheral glucoseuptake rate is different from other multipliers. It is a linear function of the periph-eral glucose concentration and has the following general form:MGPGU = a(GPFGBPF)+b (2.31)where a and b are the parameters of glucose multiplier of peripheral glucose up-take rate.2.2.2 Insulin Sub-modelInsulin sub-model comprises of seven compartments: brain; liver; heart andlungs; periphery (muscles and adipose tissues); gastrointestinal (GI) tract (the24stomach and intestinal system); kidney; and pancreas. A schematic representationof the insulin sub-model is depicted in figure 2.5.Figure 2.5: Schematic diagram of insulin sub-model[9]The sub-model equations include mass balance equation over each sub-compartmentexcept for the pancreas compartment. A separate model is considered for the pan-creas to capture the complex mechanism of pancreatic insulin production whichcannot be described by simple mass balance equations. Mass balance equationsover each sub-compartment results in the following equations:The mass balance equation over the compartments in the insulin sub-model25results in following equations:V IBdIBdt= QIB(IH− IB), (2.32)V IHdIHdt= QIBIB+QILIL+QIKIK +QIPIPV −QIHIH , (2.33)V IGdIGdt= QIG(IH− IG), (2.34)V ILdILdt= QIAIH +QIGIG−QILIL+ rPIR− rLIC, (2.35)V IKdIKdt= QIK(IH− IK)− rKIC, (2.36)V IPCdIPCdt= QIP(IH− IPC)−V IPFT IP(IPC− IPF), (2.37)V IPFdIPFdt=V IPFT IP(IPC− IPF)− rPIC, (2.38)where I is the insulin concentration (mU/l), Q is the vascular blood flow rate(dl/min), V is the volume (dl), T is the transcapillary diffusion time constant (min),r is the metabolic production or consumption rate (mg/min) and t is time (min).The subscripts of the variables refer to the body organs. subscript B is the brain,subscript A is the hepatic artery, subscript G is gut, subscript Lis liver and subscriptG is GI tract (stomach and intestines). Subscript P is periphery, subscript PC isthe periphery capillary space and subscript PF is the periphery interstitial fluidspace.The following equations are used to calculate the insulin consumption rates:rLIC = 0.4[QIAIH +QIGIG+ rPIR] (2.39)rKIC = 0.3QIKIK (2.40)26rPIC =IPF[(1−0.150.15QIP)− 20V IPF ](2.41)where R is the inhibitor (dimensionless) and r is the metabolic production or con-sumption rate (mU/min). rKIC is kidney insulin clearance rate, rLIC is liver insulinclearance rate, rPIC is peripheral insulin clearance rate and rPIR is pancreatic in-sulin release rate.As mentioned before, the simulation of the pancreatic insulin release has beendone through a separate model. The changes in blood glucose concentrationmainly stimulates the pancreatic insulin release. A healthy pancreas has a bipha-sic insulin release pattern in response to a glucose concentration step change. Asharp release of insulin for about 5-10 min constitutes the first phase, followed bya gradual increase of insulin release rate, constituting the second phase [59].The pancreatic insulin release model used in the Sorensen model has beenproposed by Landahl and Grodsky [60]. The aim of Landahl and Grodskys modelis to mimic the biphasic behaviour of pancreatic insulin secretion in response to aglucose stimulus. The graphical representation of this model is presented in figure2.6.Figure 2.6: Schematic diagram of Landahl and Grodskys model[9]This model has two compartments. A small labile insulin compartment is as-sumed to exchange insulin with a large storage compartment. Glucose-stimulated27factor, P, regulates the rate at which insulin flows into the labile compartment.The rate of insulin secretion, S, depends on glucose concentration, the amount oflabile insulin, m, and the difference between the instantaneous level of glucose-enhanced excitation factor, X , and its inhibitor, R. This functionality provides amathematical description of the pancreas biphasic response to a glucose stimu-lus. The first phase insulin release is caused by an instantaneous increase in theglucose-enhanced excitation factor (X) followed by a rapid increase in its inhibitor(R). The second phase release results from the direct dependence of the insulinsecretion rate (S) on the glucose stimulus and the gradual increase in the level ofthe labile compartment filling factor (P).The model equations include mass balance equations over compartments andcorrelations between variables. The mass balance equation over each compart-ment results in:dmdt= K′mSKm+ γP−S, (2.42)dmSdt= Km−K′mS− γP, (2.43)It is assumed that the capacity of the storage compartment is large enough andremains at steady state. For a glucose concentration of zero, P is set to zero.Therefore, the steady state mass balance equation around the storage compartmentis:K′mS = Km0, (2.44)where m0 is the labile insulin quantity at a glucose concentration of zero. The restof the equations for the pancreas model are:dPdt= α(P∞−P), (2.45)dRdt= β (X−R), (2.46)28S = [N1Y +N2(X−R)+ξ1ψ]m x> R,S = [N1Y +ξ1ψ]m x≤ R,(2.47)P∞ = Y = X1.11+ξ2ψ, (2.48)X =G3.27H1323.27+5.93G3.02H(2.49)P∞ and Y reflect the glucose-induced stimulation effects on the liable compartmentfilling factor and the insulin secretion rate, respectively.2.2.3 Glucagon Sub-modelThe glucagon sub-model has one mass balance equation over the whole bodyas follows:VΓdΓdt= rPΓR− rPΓC, (2.50)where Γ is the normalized glucagon concentration (dimensionless), V is volume(dl), t is time (min), rPΓC is plasma glucagon clearance rate and rPΓR is pancreaticglucagon release rate.The metabolic rates for the glucagon sub-model are summarized below:rPΓC = 9.1Γ (2.51)rPΓR = MGPΓRMIPΓRrBPΓR (2.52)MGPΓR = 1.31−0.61tanh[1.06(GHGBH−0.47)] (2.53)MIPΓR = 2.93−2.09tanh[4.18(IHIBH−0.62)] (2.54)rBPΓR = 9.1 (2.55)29where M is the multiplier of metabolic rates (dimensionless) and r is the metabolicproduction or consumption rate (dl/min).2.3 Type II Diabetes ModelThe same structure as presented in Sorensen model can be applied to develop amodel for type II diabetes. The parameters of the healthy human body modelshould be modified for type II diabetic patients to incorporate the abnormalitiesassociated with the patients.As mentioned in section 1.1.2, several organ malfunctions cause the deterio-ration of the blood glucose homeostasis and leads to high or low level of bloodglucose in type II diabetic patients. These abnormalities are summarized in thefollowing:• Insulin resistance in peripheral tissues• Impaired insulin mediated effects on hepatic glucose uptake• Impaired insulin suppression effects on endogenous glucose production• Impaired pancreatic insulin secretion both in first phase of release and inoverall secretion rate• Glucose resistance in the liver and peripheral tissuesThe parameters of the Sorensen model are a lot but some of these parameterssuch as capillary space volumes and blood flow rates, are physiological factorsthat are predetermined by the physical characteristics of the body. The modelparameters are listed in table 2.1. The values of these parameters are the same fora healthy person and a diabetic patient and do not need to be updated to describetheir impact on abnormalities of diabetic subjects. The remaining parameters arein equations representing metabolic rates of glucose, insulin and glucagon besidesthe parameters of the pancreas model.30Table 2.1: The model parameters [4]V GBC = 3.5 dl QGB = 5.9 dl/min TGB = 2.1 minV GBF = 4.5 dl QGH = 43.7 dl/min TGP = 5.0 minV GH = 3.5 dl QGA = 2.5 dl/min TIP = 20 minV GL = 25.1 dl QGL = 12.6 dl/min α = 0.0482 min−1V GG = 11.2 dl QGG = 10.1 dl/min β = 0.931 min−1V GK = 6.6 dl QGK = 10.1 dl/min K = 0.00794 min−1V GPC = 10.4 dl QGP = 12.6 dl/min N1 = 0.00747 min−1V GPF = 67.4 dl QIB = 0.45 l/min N2 = 0.0958 min−1V IB = 0.26 l QIH = 3.12 l/min γ = 0.0958 U/minV IH = 0.99 l QIA = 0.18 l/min m0 = 6.33 UV IG = 0.94 l QIK = 0.72 l/minV IL = 1.14 l QIP = 1.05 l/minV IK = 0.51 l QIG = 0.72 l/minV IPF = 6.74 lVΓ = 6.74 lConsidering the abnormalities of type II diabetic patients, parameters whichare proper candidates for parameter estimation are within the insulin secretion rateand glucose metabolic rates. Considering the functional deficiencies of the liver,peripheral tissues and pancreas and their impact on glucose regulatory system,some related parameters of the Sorensen model are chosen for estimation. Theseparameters can be estimated using the available clinical data for type II diabeticpatients through a non-linear optimization problem.Vahidi et al. [5, 9] picked nineteen parameters and estimated them. Table31Table 2.2: Abnormalities associated with type II diabetes and their corre-sponding equationsAbnormalities Corresponding EquationsInsulin resistance in Insulin multiplier inperipheral tissues peripheral glucose uptake rateInsulin-induced stimulation Insulin multiplier inof hepatic glucose uptake hepatic glucose uptake rateInsulin-induced stimulation Insulin multiplier inof hepatic glucose production hepatic glucose production rateGlucose-induced stimulation glucose multiplier inof hepatic glucose uptake hepatic glucose uptake rateGlucose-induced stimulation glucose multiplier inof peripheral glucose uptake peripheral glucose uptake ratePancreatic insulin secretion rate N1 and N2 in the pancreas modelboth in early peak and overall rate2.2 summarizes the abnormalities associated with type II diabetes and their cor-responding model equations whose parameters are selected for estimation. Asshown in equation 2.10, multipliers which represent multiplicative effects of glu-cose, insulin and glucagon on glucose metabolic rates have four parameters. Outof nineteen selected parameters, twelve of them are chosen from the insulin mul-tiplier parameters in peripheral glucose uptake rate, hepatic glucose uptake rateand hepatic glucose production rate; and five others are selected from the glucosemultiplier parameters in hepatic glucose uptake rate and peripheral glucose uptakerate. From the pancreas model, only two parameters N1 and N2 which representthe early peak of pancreatic insulin release and overall pancreatic insulin secre-tion rate, respectively, are sufficient to be chosen for the parameter estimation.The results of the parameter estimation are presented in table 2.3. For the insulinsub-model, the estimation results are N1 = 0.00595 and N2 = 0.0467.Having reliable information from the patient status is helpful to provides in-32Table 2.3: Parameter estimation results for the glucose sub-model [5]a b c dMIPGU 2.551 1.66 0.69 3.454MI∞HGP 1.173 1.073 0.993 1.164Mi∞HGU 0.662 0.731 0.985 0.493MGHGU 1.855 1.85 2.047 1.244MGPGU 0.897 0.103 - -sights for the clinicians to find an effective treatment for each individual. Hu-man experimentation is a way of obtaining useful and reliable information aboutthe medical status of a patient; however, due to ethical issues, physiological re-strictions and high expenses of human experimentation, it is limitedly performedmostly for research studies. Mathematical modeling is a popular alternative inobtaining useful information about the glucose metabolism and understandingthe medical condition of the diabetic patients. The model is also used to eval-uate the different control strategies for administering the proper amount of insulinto the patient body. Furthermore, the type II diabetes model is used to developpharmacokinetic-pharmacodynamic models for various medicines which allowstudying the effectiveness of oral medicines on regulating the blood sugar.33Chapter 3Evaluation of Treatment Regimensfor Blood Glucose Regulation inType II Diabetes UsingPharmacokinetic-PharmacodynamicModelling3.1 IntroductionBlood glucose concentration in type II diabetic patients can be initially controlledby exercise and healthy dieting [61–64]; however, as the disease progresses, med-ication and insulin therapy are needed.The treatment strategies suggested by the clinicians are vary from differenttypes of oral agents to various insulin therapy regimens. The treatment shouldbe individualized for each patient so that the regular measurement of the bloodglucose concentration is needed to get the update information about the subjects´response to each treatment.34Several studies compared the effects of different insulin regimens on variousgroups of type II diabetic patients by conducting experiments and monitoring theblood glucose concentration during a period of time. Wolfsdorf et.al. showed theeffects of insulin regimen with human ultralente before supper and NPH beforebreakfast in children and adolescents [65]. Yki-Jarvinen et.al. reported the resultsof 153 type II diabetic patients treated by five different insulin regimens [66].Wolffenbuttel et.al. also studied 95 elderly patients with type II diabetes who werepoorly controlled, despite diet and maximal doses of oral agents. They comparedthree insulin administration regimens during a 6-month period [67]. However,such studies need regularly performed monitoring of the blood glucose either athome or in the hospital. Also, due to ethical standards and risks posed to thehuman body, the number and variety of experiments that can be performed on apatient is limited.Development of dynamic models, representing a patient in a simulation envi-ronment, is useful to collect sufficient data. In this chapter we modified the modelpresented in Chapter 2 to assess the performance of different control strategiesusing different insulin types, oral agents or the combination of them and choosethe right one for an individual patient.Different types of oral drugs were summarized in table 1.2. Amongst themmetformin is the first-line drug of choice for the treatment of type II diabetes. Itincreases the rate of intestinal glucose consumption, allows more glucose to entercells and keeps the liver from making glucose due to decreased hepatic gluconeo-genesis and increased glycogenesis and lipogenesis [68]. Some studies describedthe pharmacokinetic-pharmacodynamic(PK-PD) model of metformin [7, 69, 70].Pharmacokinetics is defined as the study of the time course of drug absorption,distribution, metabolism, and excretion. Pharmacodynamics refers to the rela-tionship between drug concentration at the site of action and the effects. In thischapter PK-PD model of metformin is incorporated into type II diabetes model toanalyze the glucose-lowering effect of this oral agent on the patients.Oral agents are helpful in initial steps of diabetes, but most of the patients35with type II diabetes finally need insulin therapy to better regulate their bloodglucose level. Insulin can be delivered by multiple daily injections or infusedcontinuously through the insulin pump. Multiple dose injection (MDI) therapy,also known as multiple daily injections, involves several injections a day, gives thediabetic patient more control over their life and their diabetes by using differentcombinations of insulin [71].There are some studies presented the pharmacokinetic-pharmacodynamic (PK-PD) model of different insulin types for simulation purposes. Berger and Rodbarddeveloped a computer program for the simulation of plasma insulin and glucosedynamics after subcutaneous injection of insulin. This program is based on aphysiologic model of minimal complexity, which describes the pharmacokineticsof absorption and clearance of subcutaneous insulin and the dynamics of glucoseutilization as dependent on both prevailing glucose and insulin levels [6, 72]. Wuet.al. reported a two-compartment model that includes glucose and insulin dy-namics and its evaluation using patient data [73]. Therefore, the dynamics ofplasma insulin after the injections of various combinations of short, intermediateand long-acting insulins can be simulated by using PK-PD model. Such model isused, in this chapter to address the problem of finding the most efficient regimenfor multiple daily injections (MDI) therapy during Chapter 4 and 5 deals withdeveloping control strategies for continuous subcutaneous insulin infusion (CSII)therapy using the insulin pump.The following section represents the modification of the mathematical mod-eling for type II diabetes mellitus by incorporating PK-PD models of metforminand four popular types of insulin (regular, NPH, lente and ultralente) to simulatethe patient response to the meal ingestion with different treatment regimens. Theresults are presented and discussed in section 3.3 for different therapies, amongstthem the most efficient one can be applied to the patient.363.2 Expansion of Type II Diabetes MellitusMathematical ModelPharmacokinetic-Pharmacodynamic (PK-PD) modeling of metformin and differ-ent insulin types can be incorporated into the physiological model of type II dia-betes to better understand the underlying kinetic phenomena involved with thesetreatments. The model should also be updated by adding the meal sub-model todescribe the digestion of the carbohydrate content of the meal. The mathematicalmodels are presented in the following subsections.3.2.1 Pharmacokinetic-Pharmacodynamic Modeling ofSubcutaneous Injected InsulinThe PK-PD model of subcutaneously injected insulin consists of two compart-ments, describing the dynamics in subcutaneous injection site and the plasma in-sulin compartment [6, 72].The following two equations define the dependency of insulin absorption oninsulin types and dose.A%(t) = 100− 100.ts(T50%)+ ts(3.1)T50%(D) = a.D+b (3.2)where A% is percent of injected insulin remaining at the absorption site, t is timeafter injection, s characterizes time course of absorption, and T50% is time intervalto permit 50% of injected dose to be absorbed. a and b are parameters to charac-terize dependency of T50% on dose and D is insulin dose (U). a, b and s have beenestimated for four popular types of insulin (regular, NPH, lente and ultralente) andreported in [6] as depicted in table 3.1.37Table 3.1: parameter values of insulin PK-PD model [6]insulin type s a(hr/U) b(hr)regular 2.0 0.05 1.7NPH 2.0 0.18 4.9lente 2.4 0.15 6.2ultralente 2.5 0 13the rate of change of plasma insulin is calculated by adding a first-order plasmaelimination rate to the first derivative of equation 3.1:dAdt=s.ts.(T50%)s.Dt.|(T50%)s+ ts|2 − ke.A (3.3)where A is plasma insulin and ke is first-order elimination constant. Dividingplasma insulin by distribution volume for insulin yields plasma concentration attime t after injection:I(t) =A(t)Vt(3.4)where I is plasma insulin concentration and Vt , is distribution volume for insulinwhich is assumed to be 12 L as reported in [6] for a patient with 70 kg of bodyweight.Type II diabetes model, presented in Chapter 2, is modified by consideringplasma insulin concentration as an input to the heart compartment of insulin sub-model (equation 2.33).383.2.2 Pharmacokinetic-Pharmacodynamic Modeling ofMetforminDue to the multifactorial mechanism of action of metformin, a multi-compartmentPK-PD model of metformin for the treatment of type II diabetes mellitus is used asdescribed in [7, 69, 70]. This model constitutes three compartments including theGI tract (gut), liver and periphery. GI lumen and GI wall are considered as sub-compartments of the gut compartment, because the accumulation of metformin inthe GI wall is not only through the GI lumen, but also via arterial blood supply tothe intestine.The following mass balance equations, with first-order kinetic, are used todescribe the transfer of metformin between different compartments.X˙1 = −X1(kgo+ kgg)+XO (3.5)X˙2 = X1kgg+X4kpg−X2kgl (3.6)X˙3 = X2kgl +X4kpl−X3kl p (3.7)X˙4 = X3kl p−X4(kpl + kpg+ kpo) (3.8)where X1, X2, X3, and X4 are the mass of metformin in the GI lumen, GI wall, liver,and periphery compartments, respectively. XO is the flow rate of metformin as aresult of a single oral ingestion. The rate constants are: kgo, drug elimination viathe fecal route; kgg, drug transfer from the GI lumen to the GI wall compartment;kgl , drug transfer from the GI wall to the liver compartment; kl p and kpl , drugtransfer from the liver to the periphery compartment and vice versa; kpg, drugtransfer from the periphery to the GI wall compartment; and kpo, drug eliminationvia the urination route.In this modelling approach, metformin is distributed to the GI lumen, liver, andperiphery compartment following oral administration. For the oral administration,the pharmacokinetics of metformin from mouth to the GI lumen is described bythe following equation:39XO = Ae−αt +Be−β t (3.9)where α and β are rate constants; A and B represent the contribution of the cor-responding exponentials. These parameters were estimated by optimization usingexperimental data points in [7] and reported in table 3.2.Table 3.2: parameter values of metformin PK-PD model [7]A(mg/min) 2.7×104 kgo(min−1) 1.88×10−3 kpl(min−1) 1.01×10−2B(mg/min) 2.7×104 kgg(min−1) 1.85×10−3 kpg(min−1) 4.13α 0.06 kgl(min−1) 0.46 kpo(min−1) 0.51β 0.1 kl p(min−1) 0.91The transient changes of metformin amounts at different biophases can be ob-tained as indicated in equations 3.5-8. The glucose-lowering effect of metforminmainly involves the stimulation of glucose consumption in the GI tract and pe-riphery (EGI and EP) and the inhibition of glucose production in the liver (EL),of which the metabolic rate of the gut glucose consumption (rPK−PDGGU ) is only at-tributed to the GI wall. These three coefficients EGI , EL, and EP are modified themetabolic rates of the corresponding compartments in the physiological modeldescribed in Chapter 2 to present the behaviour of a type II diabetic patient withthe treatment of metformin.As metformin is known to increase glucose consumption by the gut, the rateof gut glucose consumption (rGGUPK−PD) in the type II diabetic model is modified asshown in equation 3.10,rPK−PDGGU = (1+EGI)rGGU (3.10)40where rGGU is the rate of the gut glucose consumption with no metformin effect(equation 2.13), and EGI is a weight coefficient that represents the increment ofthe rate rGGU following the administration of metformin. Similarly, metformin isknown to lower hepatic glucose production, whose rate (rHGPPK−PD) is modified asshown in equation 3.11,rPK−PDHGP = (1−EL)rHGP (3.11)where rHGP is the rate of the hepatic glucose production without the effect of met-formin for type II diabetic patients (equation 2.18), and EL is a weight coefficientthat indicates the inhibition of glucose production in the liver (L). Also, the rateof the periphery glucose uptake (rPGUPK−PD) is modified to the following equation:rPK−PDPGU = (1+EP)rPGU (3.12)where rPGU is the rate of the periphery glucose uptake without the treatment ofmetformin (equation 2.14), and EP is a weight coefficient that indicates the stim-ulation of glucose consumption in the periphery (P) with the metformin effect.According to the literature published by Stepensky et al. [8], the correspond-ing weight coefficients in three compartments (EGI , EL, and EP) are calculated asfollows:EGI =νGI,max× (X2)nGI(ϕGI,50)nGI +(X2)nGI(3.13)EL =νL,max× (X3)nL(ϕL,50)nL +(X3)nL(3.14)41EP =νP,max× (X4)nP(ϕP,50)nP +(X4)nP(3.15)where ν is the parameter representing the maximum effect of metformin in eachcompartment (νGI,max, νL,max and νP,max); ϕGI,50, ϕL,50 and ϕP,50 are the massof metformin at the biophase that produces 50% of its maximal effect; and nGI ,nL, and nP are the shape factors. The model parameters can be found in [70] assummarized in table 3.3.Table 3.3: Parameter values of metformin PK-PD model [8]νGI,max 0.486 ϕGI,50(µg) 431 nGI 2νL,max 0.378 ϕL,50(µg) 521 nL 5νP,max 0.148 ϕP,50(µg) 1024 nP 53.2.3 Mathematical Representation of the Glucose AbsorptionModelFor clinical evaluation, the oral glucose tolerance test (OGTT) is usually used toassess the patient's insulin reserve and response to a glucose load. The test usesa standard dose of glucose orally administered to determine the body's ability toregulate the blood sugar. A standard amount of oral glucose was administeredto evaluate the efficiencies of the treatments. The mathematical representation ofthe glucose absorption model has been described in [74]. Lehmann and Deutschpresented the rate of gastric emptying of the ingested glucose as a function oftime with a trapezoidal form. The duration of the period (Tmaxge) for which gas-tric emptying is constant and maximal (Vmaxge) is a function of the carbohydratecontent of the meal ingested.Tmaxge = [Ch−12Vmaxge(Tascge +Tdesge)]/Vmaxge (3.16)42where Vmaxge is the maximal rate of gastric emptying and Tascge and Tdesge arethe respective lengths of the ascending and descending branches of the gastricemptying curve. The default values for Tascge and Tdesge are 30 min (0.5 h) andthe maximal rate of gastric emptying is 120 mmol/h. Linear interpolation can beapplied to obtain the rate of gastric emptying for meals containing Ch millimolesof carbohydrate as follows:Gempt = (Vmaxge/Tascge)t; t < TascgeGempt =Vmaxge; Tascge < t ≤ Tascge +TmaxgeGempt =Vmaxge− (Vmaxge/Tdesge)(t−Tascge−Tmaxge);Tascge +Tmaxge ≤ t < Tmaxge +Tascge +TdesgeGempt = 0; elsewhere(3.17)t is the elapsed time from the start of the meal.Figure 3.1 indicates the connection of the glucose absorption model and alsoPK-PD models of insulin injection and metformin with the Sorensen model. ThePK-PD model of metformin is placed into three compartments of glucose sub-model as shown with the arrows. It updates the metabolic rates in gut, liver andperiphery compartments as explained in equations 3.10 to 3.12.As the arrow shows, the glucose absorption model is placed into the gut com-partment of the glucose sub-model and is responsible for the calculation of theglucose appearance rate into the blood stream following an oral glucose intake.The calculated value of the glucose appearance rate is added as a source of glucoseinto the equation 2.4 which represents the mass balance over the gut compartmentof the glucose sub-model.Finally, the PK-PD model of insulin injection is placed into the heart compart-ment of insulin sub-model. The model calculates the plasma insulin concentrationwhich is entered into the blood stream after the injection as the heart circulates theblood. Therefore, the plasma insulin concentration is added to equation 2.33 asthe source of insulin.43Figure 3.1: Schematic diagram of connections among glucose absorptionmodel, metformin and insulin injection PK-PD model and Sorensenmodel.3.3 Simulation Results and DiscussionThe modified model, proposed in the previous section, can be applied to simulatethe response of the patient to the various amount of glucose intake at differenttimes during the day while different combinations of insulin types and dosagesare injected as daily multiple injection regimen. The simulations can also showthe effect of metformin on the glucose concentration for type II diabetic patients.Therefore, the most efficient regimen can be chosen to be applied on the patient.The simulation is considered for the whole day, starting from 6 am. It isassumed that the patient eats 275 g of carbohydrate during the day, 75 g of glucosefor breakfast at 8 AM, 100 g of glucose for lunch at 1 PM and 100 g of glucose fordinner at 8 PM. Without the administration of oral agents and insulin injections,the blood glucose goes higher than 11 mmol/L as shown in figure 3.2.44Figure 3.2: The simulated patient response to the meal disturbance duringthe day without drug and insulin therapy3.3.1 Simulation Results for Different Insulin InjectionRegimensRegular insulin is a short-acting insulin. It has an onset of action 15 to 60 min afterinjection, a peak effect 2 to 4 h after injection, and a duration of action of rangingfrom 5 to 8 h. Neutral protamine Hagedorn insulin (NPH) is an intermediate-acting insulin. NPH has an onset of action 2 h after injection, a peak effect 6 to 10h after injection, and a duration of action ranging from 13 to 20 h. Lente is alsoan intermediate-acting insulin with an onset of action 2.5 h after injection, a peakeffect 6 to 16 h after injection, and a duration of action up to 24 h. Ultralente isvery stable crystalline insulin considered as long-acting insulin that has its peak45activity 8 to 18 h after injection and a duration of action of 30 h. [71].The plasma insulin profile for each of the four types of insulin is shown in fig-ure 3.3 using the pharmacokinetic-pharmacodynamic model presented in section3.2.1. The simulation results are consistent with the known characteristics of theinsulin types regarding the onset of action, peak time and the duration of action. AFigure 3.3: plasma insulin profile for regular, NPH, lente and ultralente in-sulinmixture of intermediate or long-acting insulins in addition to short-acting insulinare used to both keep the basal insulin level of the body in the normal range andalso regulate the blood glucose after food intake. The following results discuss theeffects of changes in insulin dose, type and injection time on the blood glucoseconcentration.46Insulin doseThe blood glucose profile is affected by changing injected insulin dose. To betterillustrate the effects of different insulin amounts on the blood glucose regulation,the body responses after systematically changing the insulin dose has been shownin figures 3.4 and 3.6.Figure 3.4: The simulated patient blood glucose profile after the subcuta-neous insulin injection according to regimen 1(black solid line), reg-imen 2 (blue dashed line) and regimen 3(red dotted line) to study theeffects of changes in regular insulin doseThe black solid line in figure 3.4 demonstrates the result of the injection of5 units of regular insulin at each meal time and 5 units of NPH insulin in themorning and evening (regimen 1) which is considered as the observed profile. Forunderstanding the effects of changes in regular insulin, the NPH insulin dose is47kept the same as regimen 1 while the regular insulin dose is changing. The bluedashed line shows the result when no regular insulin is injected with the meals(regimen 2). The result of an increase in regular insulin from 5 to 10 units isshown by the red dotted line (regimen 3).Figure 3.5: The simulated patient plasma insulin profile after the sub-cutaneous insulin injection according to regimen 1(black solidline),regimen 2 (blue dashed line) and regimen 3(red dotted line) toshow the insulin resistance in body cellsWithout the injection of regular insulin, the blood glucose level goes up tohyperglycemia zone colored in yellow after lunch and dinner. Increasing regularinsulin dose causes severe hypoglycemia which is dangerous for the patient es-pecially during the night. However, an increase in regular insulin does has littleeffect on blood glucose peak after each meal. The reason is that the body cells48become insulin resistant in type II diabetes, so the presence of excess amount ofinsulin does not help the more absorption of glucose by cells and only increasethe plasma insulin level as shown with the red dotted line in figure 3.5.Figure 3.6: The simulated patient blood glucose profile after the sub-cutaneous insulin injection according to regimen 1(black solidline),regimen 4 (blue dashed line) and regimen 5(red dotted line) tostudy the effects of changes in NPH insulin doseThe effect of changing in NPH dose is depicted in figure 3.6. The black solidline presents the result for regimen 1. The blue dashed line and the red dottedline shows the results for a decrease in NPH dose from 5 to zero (regimen 4) andan increase from 5 to 10 (regimen 5), respectively while the regular insulin dosedoes not change. An increase in NPH yields the glucose profile with the severehypoglycemia after lunch and during night. Again the excess amount of NPH can49not help lowering the blood glucose level because of insulin resistance in bodycells.It can be seen that despite of eliminating intermediate or long-acting insulin inregimen 4, the blood glucose profile still is acceptable and kept in normoglycemiazone. This can be understood from the fact that the pancreas still secretes someinsulin in type II diabetes which provides the basal insulin level of the body butis not sufficient for preventing the blood glucose rise after the meals. This canbe demonstrated better by monitoring the plasma insulin concentration while noexternal insulin is injected (see figure 3.7).Figure 3.7: The simulated patient plasma insulin profile produced by pan-creas without any external insulin injection50Insulin typeIn figure 3.8 different types of intermediate- or long-acting insulin are appliedwith regular insulin as short-acting insulin. Combination of regular insulin withNPH, lente and ultralente sufficiently prevent hyperglycemia.Regarding the problem of hypoglycemia, a mixture of regular and ultralentebefore dinner causes a modest reduction in fasting blood glucose level duringthe night (red dotted line). Compared with a mixed dose of regular and ultra-lente, the similar dose of a mixture of regular and NPH (black solid line) or lente(blue dashed line) increases the risk of hypoglycemia while the patient is sleeping.Available clinical data in the literature also shows a modest reduction in fastingblood glucose using ultralente as long-acting insulin which confirms the simula-tion results [65].The only precaution that should be taken into the consideration is that the zincin ultralente retards the onset of action of the regular insulin, and so it should beimmediately injected after withdrawal from the vial.Timing of insulin injectionChoosing the proper time of injection affects the quality of blood glucose regu-lation. Figure 3.9 shows the effect of the time of injection on the blood glucoseprofile.In this figure insulin injection according to regimen 8 is considered as theobserved glucose profile (black solid line) and is used to investigate the effects ofadvancing and delaying all the injections by an hour, applying the identical insulintype and dose to regimen 8, as shown with blue dashed line and red dotted line,respectively.Having the morning injection followed by an hour delay to commence thebreakfast decrease the fasting blood glucose level and causes severe hypoglycemia.This also occurs when the injection happens an hour before the dinner. However,for type II diabetic patient whose pancreas still is capable of secreting insulin anhour delay in insulin injection is not that much a problem and body can main-51Figure 3.8: The simulated patient response to the subcutaneous insulin in-jection to study the effects of NPH according to regimen 6 (black solidline), lente according to regimen 7 (blue dashed line) and ultralenteaccording to regimen 8 (red dotted line)tain the blood glucose in a safe condition for such a short delay in receiving theexternal insulin.The patients under insulin regimen can experience better control of the bloodglucose level through the whole day and avoid the symptoms of high and lowblood glucose level. On the other hand the pressure on the pancreas to produceinsulin will be reduced and benefit longer term health.Table 3.4 summarizes all the discussed regimens. The comparison of the pro-posed regimens helps to propose an efficient regimen for a patient to prevent hypo-glycemia and hyperglycemia while bringing back the blood glucose to the lower52Figure 3.9: The simulated patient response to the subcutaneous insulin in-jection to study the effect of changes in the timing of the injection.Glucose profile for regimen 8 (black solid line) applied at the mealtime is compared with the glucose profile of advancing (blue dashedline) and delaying (red dotted line) all the injections by an hour.bound of normoglycemia zone in a timely manner.Table 3.5 provides better comparison between the regimens by considering thefollowing factors for each glucose profile:• minimum concentration of blood glucose before lunch (BGBmin)• minimum concentration of blood glucose before dinner (BGLmin)• minimum concentration of blood glucose during the night (BGDmin)• maximum concentration of blood glucose after breakfast (BGBmax)53Table 3.4: Insulin injection regimenregimen# insulin breakfast lunch dinnertype at 8 am at 1 pm at 8 pm75 g of glucose 100 g of glucose 100 g of glucose(1) regular 5 units 5 units 5 unitsNPH 5 units - 5 units(2) regular - - -NPH 5 units - 5 units(3) regular 10 units 10 units 10 unitsNPH 5 units - 5 units(4) regular 5 units 5 units 5 unitsNPH - - -(5) regular 5 units 5 units 5 unitsNPH 10 units - 10 units(6) regular 3 units 3 units 3 unitsNPH 7 units - 7 units(7) regular 3 units 3 units 3 unitslente 7 units - 7 units(8) regular 3 units 3 units 3 unitsultralente 7 units - 7 units(9) regular 3 units 5 units 3 unitsultralente 5 units - 5 units• maximum concentration of blood glucose after lunch (BGLmax)• maximum concentration of blood glucose after dinner (BGDmax)• time in minutes needed to bring back the blood glucose level under 7 mmol/Lafter breakfast (tB)• time in minutes needed to bring back the blood glucose level under 7 mmol/Lafter lunch (tL)• time in minutes needed to bring back the blood glucose level under 7 mmol/L54after dinner (tD)Table 3.5: Comparison of different regimensregimen# BGBmin BGLmin BGDmin BGBmax BGLmax BGDmax tB tL tD(1) 4.36 3.88 2.86 8.04 10.87 10.73 269 319 318(2) 5.76 5.3 4.27 10.31 12.36 11.79 297 341 326(3) 4.27 2.09 0.8 8.01 10.8 10.55 269 319 317(4) 4.91 4.83 4.96 8.07 10.94 10.84 271 320 322(5) 4.29 2.79 0.01 8.02 10.85 10.61 268 319 317(6) 4.43 4.11 2.55 8.07 10.86 10.76 270 320 318(7) 4.54 3.75 2.37 8.12 10.87 10.75 271 320 316(8) 5.28 4.35 3.82 8.23 11.05 10.81 277 322 320(8)advanced 4.63 2.74 4 8.4 11 10.76 280 324 320by an hour(8)delayed 5.05 4.03 3.54 8.37 11.13 11 275 321 320by an hour(9) 5.34 4.15 4.29 8.25 10.96 10.82 304 336 337An efficient regimen can keep the minimum concentration of the blood glu-cose in the normal glycemic zone without traversing to hypoglycemia and severehypoglycemia Moreover, the maximum blood glucose concentration should notgo to hyperglycemia and severe hyperglycemia zones. The yellow color in table3.5 shows the occurrence of hypoglycemia or hyperglycemia and the red color isused to demonstrate the severe hypoglycemia state.Table 3.5 shows that regimens 1 and 7 cause hypoglycemia before having din-ner and severe hypoglycemia during the night. Regimens 3 and 5 result in severehypoglycemia both before dinner and during the night. Severe hypoglycemia alsooccurs in regimen 6 during the night and in regimen 8 before dinner when theinjection is advanced by an hour. Hypoglycemia and hyperglycemia are happened55in regimen 8 both when insulin is injected at the meal time and delayed by anhour. Regimen 2 prevents hypoglycemia but causes hyperglycemia after lunchand dinner.Regimen 9 as described in table 3.4 includes the injections of 3 units of regularinsulin in addition to 5 units of ultralente in the morning and evening and 5 unitsof regular insulin with lunch. This regimen assures a reliable regulation of bloodglucose depicted in figure 3.10. The superiority of this regimen over regimen 4is that it is performing better in keeping blood glucose concentration closer to thelower bound of the green zone which is more desirable (see table 3.5).Figure 3.10: The simulated patient response to the subcutaneous insulin in-jection corresponding to regimen 9 which proposes an efficient bloodglucose regulation563.3.2 Simulation Results for Administration of MetforminThe glucose lowering effects of metformin are defined through the weight co-efficients which are used to modify the metabolic rates in liver, GI lumen andperiphery compartments. EL, EGI and EP are calculated according to equations3.13 to 3.15 and depicted in figure 3.11.Figure 3.11: Glocose loweing effect of metformin in liver, GI tract and pe-riphery (EL, EGI and EP)Figure 3.12 shows and compares the effect of administration of 500 and 1000mg of metformin twice during the day, once in the morning with the breakfast at8 am and the second time in the evening with dinner at 8 pm. Metformin itselfcannot prevent hyperglycemia and severe hyperglycemia. The higher amount ofmetformin causes more decrease in blood glucose level after each meal, but the57Figure 3.12: The simulated patient response to the administration of 500 mgmetformin (black solid line) and 1000 mg metformin (blue dashedline) two times a day at 8 am and 8 pmmaximum dosage for metformin should not exceed 2500 mg per day. High dosesof metformin may cause lactic acidosis which is the elevation of lactic acid thatcan cause a serious electrolyte imbalance.Since the mono-therapy with metformin can not assure the satisfactory glycemiccontrol and the patient still experience hyperglycemia, the insulin therapy alone orin combination with metformin can be considered as the treatment. Metformin'smultiple effects not only benefit individuals with glycemic control but also helpwith cardiovascular problems, endocrine problems, retinopathies, nephropathies,cancer or decreased immunity, infections and weight gain. Therefore most pa-58Figure 3.13: The simulated patient response to the administration of 500 mg(black solid line) and 1000 mg (blue dotted line) metformin at 8 amand 8 pm along with the insulin injection according to regimen 9.tients continue the administration of metformin along with insulin therapy.Figure 3.13 presents the blood glucose concentration as the result of the in-sulin injection according to regimen 9 and the administration of different dosagesof metformin in the morning and the evening. The results show the satisfactoryregulation of the blood glucose concentration in the region corresponding to nor-moglycemia without passing through hypoglycemia and hyperglycemia zones ex-cept for a negligible undershoot to hypoglycemia zone for the administration of1000 mg of metformin.Using the diabetic model presented in Chapter 2, the concentration of the glu-cose in other organs can be investigated besides the plasma glucose level. Figures59Figure 3.14: The blood glucose concentration in gut without administrationof metformin (black solid line) and with administration of 1000 mgmetformin at 8 am and 8 pm (blue dotted line)3.14-3.16 illustrates the effect of 1000 mg metformin on the blood glucose con-centration of gut, liver and periphery (blue dotted line) and compares it with theblood glucose profile without administration of metformin (black solid line).The glucose profile in the gut (figure 3.14) assures the effect of metformin indecreasing gastrointestinal absorption of glucose. Figure 3.15 shows a decreasein blood glucose concentration administering metformin which is the result of thereduced livers´ production of glucose. Metformin also improves insulin sensitivityand increase the absorption of glucose in peripheral tissues which causes lowerblood glucose level as shown in figure 3.16.60Figure 3.15: The blood glucose concentration in liver without administrationof metformin (black solid line) and with administration of 1000 mgmetformin at 8 am and 8 pm (blue dotted line)3.4 ConclusionThe incorporation of the pharmacokinetic-pharmacodynamic model for oral agentsand different insulin types into the available physiological model for type II dia-betes mellitus makes it possible to compare various regimens and choose the mostefficient one for each individual patient.The simulation results confirm that mono-therapy with metformin does notassure the satisfactory glycemic regulation in type II diabetic patients and theyeventually need insulin therapy.In multiple daily injection therapy a mixture of short-acting insulin along with61Figure 3.16: The blood glucose concentration in periphery without admin-istration of metformin (black solid line) and with administration of1000 mg metformin at 8 am and 8 pm (blue dotted line)intermediate or long-acting insulin are used to regulate the blood glucose. Theproper selection of the insulin type, dose and injection time affects the quality ofthe blood glucose control. The efficient injection regimen is the one which is ableto prevent hypoglycemia and hyperglycemia while bringing the blood glucoseunder 7 mmol/L in a timely manner during the postprandial state and graduallydecrease it to around 4 mmol/L during the fasting state.Amongst the investigated regimens the one containing mixture of 3:5 unitsregular and ultralente insulin in the morning and evening and 5 units of regularinsulin for the lunch shows the best performance. Combinations of orally admin-istered agents with insulin aid in achieving glycemic goals by taking advantage62of differing mechanisms of action such as the reduced glucose production rate inliver and increase the glucose uptake rate in peripheral tissues and gut.63Chapter 4Development of a Controller forBlood Glucose Regulation in Type IIDiabetes UsingProportional-Integral (PI) ControlStrategy4.1 IntroductionContinuous subcutaneous insulin infusion (CSII) with external pumps is an alter-native to multiple daily injections (MDI) therapy and can be beneficial for type IIdiabetic patients who require intensive insulin therapy.For type II diabetic patients being treated with CSII, a blood glucose auto-matic control system can be very useful and effective. The control objective is notas straight forward as that for type I since type II involves not only the pancre-atic insulin secretion dysfunction but also malfunction of different organs such asinsulin resistance in muscles and adipose tissues and impaired hepatic regulatory64effects [17]. During the past few decades, a variety of glucose control strategiesfor type I diabetic patients has been reported [75].The proportional-integral-derivative (PID) control algorithm is known to beapplicable to a wide variety of dynamic systems, and many past studies focused onconventional or modified linear PID controllers [76]. PID controllers are suitableto control the blood glucose concentration when the detailed knowledge of thepatient's internal metabolic behavior is not available. The output data describingthe patient's response to the specific input is the basis for control design [75].Studies showed that the β cells which are responsible for insulin secretioninclude a 3-phase response very similar to the characteristics of a PID. The threecomponents of the β cell model are proportional to glucose, the rate of the changesin glucose concentration and a slow increment corresponds to an integrator [77].Steil et al. demonstrated a discrete PID control strategy is desirable for glucosecontrol because of its ability to mimic the first- and second-phase responses thatthe pancreatic β cells behave in the secretion of insulin [77, 78].Bellazi et al. derived a proportional-derivative controller with a pole assign-ment strategy and tested in patients [79]. Chee et al. proposed a PID controlsystem based on a sliding-scale approach and successfully tested it in patients inan intensive care unit [80].Gantt et al. developed an asymmetric PI control algorithm to act properly dur-ing hypoglycemia and hyperglycemia states. The glucose measurement has beencompared to the normal value, and the proportional gain of the controller has beenadapted based on the output error. The negative values of the error, correspondingto hypoglycemia state treated more aggressively. The evaluation of the controllerbased on the modified minimal model showed that the asymmetric PI controllerperformed better compared to a normal PI, but still could not completely preventthe hypoglycemia state [81].A switching PID control algorithm with a time-varying setpoint was proposedby Marchetti et al. A decision-making system was designed to determine theproper switching time for the controller which depends on direct blood glucose65measurement. Extended Hovorka model was used as an artificial patient to test theperformance of the controller. The results showed improvements in the controlledblood glucose concentration profile compared with a manual insulin therapy [82].Control scheme based on fuzzy logic control theory have also shown promis-ing results in blood glucose control for type I diabetes and mimic the performanceof the real pancreas [83–85].However, to the best of our knowledge, none of these studies have been appliedto the regulation of blood glucose in type II diabetes mellitus regarding insulintherapy. Although type I and type II diabetic patients share similar features suchas symptoms and long-term complications, the homeostasis of blood glucose intype II diabetic patients is much more complicated than type I patients.The target of blood glucose control is not a simple set point, but rather a rangewith different lower and upper bounds asymmetrically. Regarding the non-linearhomeostasis of blood glucose, linear control strategies like PI controllers can stillbe considered but should be modified to deal with such nonlinearity. In this chap-ter, the nonlinearity between insulin and glucose as the input and output of thecontrol loop is reduced by using the logarithm of blood glucose concentration asthe controlled variable. On the other hand, a gain scheduling (GS) control schemeis developed to address the existing nonlinearity and improve the controller per-formance to achieve a desirable response.Designing a sufficient GS controller depends on two factors. The proper se-lection of scheduling variables that capture the nonlinearities of the process anddetermination of the scheduling function by means of which the gains are changedas a function of the scheduling variable. Fuzzy logic has capabilities which canbe used to implement effective scheduling gains [86].Fuzzy Logic is also a tool for describing and analyzing unconventional con-straints on a process. Fuzzy control systems can be an alternative to simple lin-ear PI control or complicated model-based control, especially when applied tobiomedical systems. Our literature survey found that a two-loop advisory controlscheme for type I diabetic patients was developed by using fuzzy systems [87].66The application of fuzzy logic was proposed to act as an artificial pancreas [88].A fuzzy-based closed-loop control algorithm was also proposed for type I patients[89].In the current chapter, type II diabetic model presented in Chapter 2 is usedas a base for the closed-loop simulations. In the following section, it is shownthat the nonlinearity of the glucose-insulin interactions can be reduced by theapplication of logarithm of blood glucose concentration. Also, a conventional PIcontroller is modified to a gain scheduling (GS) PI controller by penalizing thefeedback error using a fuzzy inference system (FIS) based on clinician knowledgeand recommendations. Finally, the simulation results of a PI controller and thefuzzy-based GS PI controller is presented.4.2 Feedback Control StrategyOnce the response of the body to insulin treatment is known, a clinician can de-termine the dosing for each individual patient considering the caloric intake andexercise level. Many intrinsic and extrinsic factors can affect the blood glucosewhich can vary from day to day. Intrinsic factors may be related to the metabolicrate of the day, stress, anxiety, or even having a simple respiratory tract infection.Exercise and diet are common extrinsic factors which alter the consumption ofglucose and production of insulin. Therefore, monitoring of blood glucose fordiabetic patients should be done more frequently if not continually. The currenttechnological advance in diabetic care has made feedback blood glucose controlpossible. Portable continuous infusion pumps for insulin have been commonlyused by type I patients. Continuous blood glucose sensors are available. How-ever, their usage is less popular because the sensors can only last a few days longand may not be accurate. Therefore, patients usually act as both the feedbackand feedforward controllers rather than relying on an automatic control device.This means that patients always dial-up the infusion rate for a short period beforea meal to avoid hyperglycemia, and make frequent feedback adjustments if theglucose is not within the desired range.67The following sections suggest the possible modifications to develop a feed-back control strategy which sufficiently regulate the blood glucose. The con-troller should be able to address the nonlinearity of the glucose-insulin interac-tions. Therefore, the application of logarithm of blood glucose and schedulinggains for a conventional PI controller are discussed.4.2.1 Reducing Nonlinearity of Glucose-Insulin InteractionThe homeostasis of blood glucose in type II is complicated, and the patients dealwith the malfunction of different organs which eventually leads to the deteriora-tion of glucose homeostasis [14].The model derived for type II diabetes mellitus should be able to representthe interactions between different organs and determine the concentrations of glu-cose and insulin along with the hormonal effects of glucagon. Model equationsinclude mass balance equations over each sub-compartment and results in a setof nonlinear ordinary differential equations.The rate of mass accumulation is thenet additive result of contributions by convection, diffusion, and any metabolicsources or sinks which add or remove mass from the sub-compartments. Sinceinsulin is produced in the pancreas in a complex mechanism which cannot bedescribed by simple mass balance equations, a separate model is considered forinsulin production in the pancreas.The controlled output of this system is the blood glucose concentration, whichis regulated by the insulin infusion rate as the manipulated variable. According tothe nonlinear factors appearing in the model equations, the relationship betweenthe rate of changes in the blood glucose concentration and the infusion rate ofinsulin is highly nonlinear. Fig. 4.1 shows a typical such nonlinear relationship.Each point on this figure represents the gain of the system for specific amount ofstep change in the rate of insulin infusion as shown on the horizontal axis.Cesar Palerm and Lane Desborough in a poster prepared for Medtronic Di-abetes R&D session mentioned that using logarithm of the blood glucose con-centration decrease the nonlinearity and make the data analysis simpler. In this68Figure 4.1: Variation of the gain for different step changes (the black pointsconnected with the blue line) and the regression line (dashed red line)research, the same strategy is considered as a possible method to decrease thenonlinearity of the response for type II diabetes. Figure 4.2 demonstrates the pro-file of the logarithm of the absolute value of the gain for different amounts of stepchange in the insulin infusion rate. The red lines represent the best linear regres-sion model fit to the data in each figure. R2 value for the suggested linear functionwhich relates the gain to the step changes in insulin infusion rate is 0.89, whilethis value for the logarithm of the gains as a linear function of the insulin infusionrate is 0.98. The comparison of these two values confirms that the logarithm ofthe blood glucose can be employed to reduce the system nonlinearity.Figure 4.3 presents the block diagram of the feedback control loop using PIcontroller. The logarithm of the measured blood glucose is used as the feedback69Figure 4.2: Variation of the logarithm of the gain for different step changes(the black points connected with the blue line) and the regression line(dashed red line)Figure 4.3: Block diagram of the feedback control strategy using logarithmof the blood glucose as the controlled variable70signal and compared to the logarithm of the setpoint to calculate the error. Thecontroller calculates the insulin infusion rate based on the amount of error.4.2.2 Gain Scheduling Control Strategy Based on Fuzzy LogicAs described in the previous section, the clinical objective of blood glucose con-trol is not a single target (or a set point in standard control parlance). Due tothe nonlinear homeostasis of blood glucose and asymmetric dynamics of hyper-glycemic and hypoglycemic states a linear controller will not provide satisfactoryperformance. The large number of highly variable intrinsic and extrinsic fac-tors within each patient makes it complicated and unappealing to use adaptivemodel-based predictive control - to both clinicians and patients alike. The gainscheduling (GS) control strategy is proved to be helpful in many areas dealingwith nonlinear dynamics including biological systems [90].The gain scheduling strategy enhances the performance of conventional PIcontrollers by facilitating the variations of controller parameters based on thechanges of the plant operating conditions. An appropriate variable, known asscheduling variable should be chosen to detect the changes in operating condi-tions. Therefore, the gain values can be calculated through the implementationof a scheduling function by means of which the gains are changed as a functionof the scheduling variable. The important consideration in the gain schedulingproblem is the design of the switching logic to obtain a smooth plant response allover the operation range.Fuzzy logic can be useful in the design procedure of scheduling function of GScontroller as shown in previous studies [86]. In 1965, the concept of fuzzy sets wasproposed by Zadeh [91]. A fuzzy set is a set without a clearly defined boundary,characterized by membership function which assigns a degree of membership toeach element. Fuzzy systems can handle linguistic variables whose values arewords. Contrary to variables in mathematics which usually take numerical values,linguistic variables can be defined in fuzzy logic to describe certain facts andexpress the underlying rules of a system. A membership function is assigned71to each linguistic variable. Although linguistic variables are less precise, theymake it easier to incorporate common sense and expert knowledge into the set ofrules to be used in fuzzy inference mechanism and deal with different types ofuncertainties involving in most design problems [92].Fuzzy inference systems consist of three main parts, a fuzzifier, an inferenceengine and a defuzzifier. In the fuzzification step, crisp input data are fuzzifiedusing membership functions which represent linguistic variables. The inferenceengine is built by a set of fuzzy rules based on expert knowledge. By meansof linguistic variables, common sense and knowledge from the clinicians can beeasily incorporated into set of rules and used in fuzzy inference system. Duringthe inferring step, those rules whose antecedents are satisfied will determine thefuzzy output. Finally, a numerical value of the fuzzy output is determined in thedefuzzification step using the pre-defined membership functions of the outputs.Fuzzy logic in combination with gain scheduling control strategy can be a use-ful tool to achieve the clinical control objective. The principle idea is to select themeasured blood glucose concentration as the scheduling variable and apply fuzzylogic to determine the importance of the error and scheduling gains. Figure 4.4 isa simple block diagram to illustrate the control strategy. In this figure, the glucosemeasurement is being used as a feedback signal for the feedback controller and aninput for the fuzzy logic inference system. A weighting factor is generated fromthe inference system and multiplied to the error between the glucose set point andmeasurement.For the regulation of blood glucose in diabetic patients, the division of dif-ferent blood glucose regions does not need to be as precise as the ones shown infigure 1.2. Therefore, in our approach, the linguistic variables are applied to dif-ferent regions for the blood glucose level. A membership function is also assignedto each of these linguistic variables. Figure 4.5 shows the fuzzification design.The Mamdani’s fuzzy inference method is the most commonly seen fuzzymethodology and is used in the current design. The input to this system is bloodglucose level (G), and the output is the weighting factor to penalize the error ac-72Figure 4.4: Block diagram of the feedback control strategy using fuzzy in-ference system to define weighting factorsFigure 4.5: Fuzzy membership functions[10]73cording to the lower panel of figure 4.5. The proper fuzzy sets for inputs andoutputs were defined by the clinical control objectives and experience explainedin section 1.2.1. The set of allowable values of the blood glucose concentrationis partitioned into four fuzzy sets corresponding to hypoglycemia (Hypo), normo-glycemia (NG), postprandial (PP) and hyperglycemia (Hyper) states. When therisk of hypoglycemia or hyperglycemia is high, the penalizing weighting factorshould be increased. On the other hand, in regions near normal glucose levels,only moderate control action may be sufficient. Accordingly, four fuzzy sets werecreated to generate an output as one of the four choices: small (S), normal (N),large (L), and extra large (XL).Based on the definition of the input and output fuzzy sets, four IF-THEN ruleswere defined as follows:• IF input is Hypo then output is XL.• IF input is NG then output is S.• IF input is PP then output is N.• IF input is Hyper then output is L.The output is calculated by the centroid defuzzification method. This methodis also known as the center of gravity or center of area defuzzification and wasdeveloped by Sugeno [93]. It returns the center of the area under the curve.The selection of the tuning parameters of the PI controller can be achieved byan off-line assessment using different performance indices. In this work, the IAEperformance tuning method is used to obtain the controller tuning parameters [94]as follows:IAE =∫ ∞0|e(t)|dt (4.1)744.3 Simulation Results and DiscussionFor a 70 kg diabetic adult patient, the usual daily caloric intake varies dependingon the activity level. Clinically, a healthy person should be able to process 100g of glucose without exceeding 11.1 mmol/L 4 to 5 hours after the intake. Theclosed-loop simulation assumes that the patient's initial blood glucose is around7.0 mmol/L.Figure 4.6: Patient response to 100 g of meal disturbance at 100 min in ab-sence of insulin infusionTo provide a baseline for comparison, figure 4.6 shows the simulated responseof a diabetic patient with no insulin infusion. The meal disturbance of 100 g glu-cose was introduced at time 100 min. Because the model represents a type IIpatient, the body is still able to produce insulin despite the reduced insulin pro-duction, increased insulin resistance, and impaired glucose regulation in the bodytissue. Therefore, the small meal disturbance of 100 g glucose causes the simu-lated blood glucose to increase up to 17 mmol/L before returning to the normo-glycemic range. It takes almost 6 hours after the intake to have the blood glucoseback around 7 mmol/L.As described in the previous section a PI controller is modified by reducingthe nonlinearity between insulin, as the manipulated variable, and glucose, as thecontrolled variable, through using logarithm (log) of blood glucose. Also, a fuzzyinference system is designed as discussed in section 4.2.2 to produce variable75Figure 4.7: Response of the conventional PI controller with a 100 g mealdisturbance at 100 mingains for the PI controller. Figures 4.7 to 4.11 show the simulation results forconventional PI controller and the modified controllers.Figure 4.7 presents the result of the application of conventional PI controller.Based on the IAE performance function, the tuning parameters of the controllerwere determined to be KP = 12.7 and KI = 0.05. The disturbance of 100 g glucoseis introduced at 100 min to the simulated patient using conventional PI controller.PI controller performance is acceptable in preventing hyperglycemia, but it can notavoid hypoglycemia and the blood glucose traversed into the red zone of severehypoglycemia. It demonstrates that a linear PI controller is not suitable for thenon-linear blood glucose system. The bottom part of figure 4.7 shows the rate of76Figure 4.8: Response of the conventional PI controller with a 100 g meal dis-turbance at 100 min with non-linearity reduction in the patient modelusing log of blood glucoseinsulin infusion. The maximum insulin rate that can be delivered by an insulinpump is 100 mU/min and a saturator is applied to limit the controller output to thementioned amount.A conventional PI controller with the same tuning parameters is applied tothe simulated patient using the logarithm of the blood glucose as the controlledvariable. The result is demonstrated in figure 4.8. The conventional PI is capableof preventing hypoglycemia as the result of the reduced nonlinearity as discussedin section 4.2.1. However, the patient is exposed to high blood glucose for twohours as the glucose profile passes through the yellow zone of hyperglycemiabefore returning to normoglycemic zone. It is desired that the controller returns77Figure 4.9: Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 minFigure 4.10: Block diagram of the feedback control strategy using fuzzy in-ference system to define weighting factors when logarithm of bloodglucose is considered as the controlled variable78Figure 4.11: Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min with nonlinearity reduction in the patientmodel using log of blood glucoseFigure 4.12: Weight factor from the fuzzy inference system79the blood glucose to the lower bound of the normoglycemic (dark green) zonewhich is not achieved by this controller design. Comparing to the previous design,less amount of insulin is delivered to the patient (bottom part of figure 4.8).Figure 4.13: Block diagram of the feedback and feedforward control strat-egyIn figure 4.9, the fuzzy-based PI controller can keep the blood glucose withinthe normoglycemic zone most of the time with small undershoots to the hypo-glycemic zone. This clearly shows the nonlinear nature of the fuzzy-logic-basedcontroller which is able to handle the asymmetric control objective of blood glu-cose regulation. The controller can return the blood glucose from the postprandialstate to the fasting state of the normoglycemic zone within 5 hours and maintainit within the safe zone most of the time.The fuzzy gain scheduling PI controller can be considered using the logarithmof the measured blood glucose as the controlled variable to reduce the nonlinearityof the glucose-insulin interactions. The block diagram of such control strategy ispresented in figure 4.10.The performance of the fuzzy-based PI controller with using the logarithmof the blood glucose, presented in figure 4.11 shows even more improvement inblood glucose regulation. The controller returns the blood glucose to the lowerbound of the normoglycemic zone after almost 5 hours without exceeding 11.1mmol/L and prevents hypoglycemia state while keeping the blood glucose levelclose to the lower bound of the normoglycemic zone.80Figure 4.14: Response of the conventional PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100 for 3hoursThe weighting factors which are calculated by fuzzy inference system (FIS)is presented in figure 4.12. According to the importance of avoiding the dangerof hypoglycemia state, the designed FIS assigns higher weights to the low bloodglucose level rather than high blood glucose level as shown in figure 4.12. It helpsthe controller to act more aggressively to prevent the complications caused bylow blood glucose concentration. As long as the blood glucose is in the normo-glycemic zone, the weighting factors are small for the gradual return of the bloodglucose to the lower bound of this zone.In reality, patients always act as the feedforward controller which means thatpatients always provide themselves with some additional insulin before consum-81Figure 4.15: Response of the conventional PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100 for 3hours with non-linearity reduction in the patient model using log ofblood glucoseing food. Figure 4.13 demonstrates the block diagram of combined feedback andfeedforward control strategy.In the next set of simulations, an additional 10 mU/min of insulin is added tothe body for 3 hours when the 100 g of glucose is introduced at 100 min. Figures4.14 and 4.15 respectively present the responses of the conventional PI controllerwithout and with nonlinearity reduction. The performances of the fuzzy-basedPI controller are also depicted in figures 4.16 and 4.17 without and with usinglogarithm of blood glucose, respectively. In all figures, the glucose concentrationshad an initial drop at the time of the additional insulin injection because insulin82Figure 4.16: Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100 for 3hoursenters the body through the interstitial tissue much faster than glucose absorptionthrough the digestive tract. Conventional PI and fuzzy-based PI controllers arealready infused high amounts of insulin to the body and the additional insulininjection has a negligible effect on the quality of control. In the case of using thelogarithm of blood glucose, receiving the additional insulin injection reduces theamount of hyperglycemia using conventional PI controller while the fuzzy-basedPI controller remains acceptable for handling this meal disturbance.83Figure 4.17: Response of the fuzzy-based PI controller with a 100 g mealdisturbance at 100 min and 10 mU/min insulin injection at 100 for 3hours after non-linearity reduction in the patient model using log ofblood glucose4.4 ConclusionsIn this chapter, a conventional PI controller was modified to address the nonlinearhomeostasis of the blood glucose regulation problem. Table 4.1 presents some ofthe characteristics of the designed controllers. The first three rows of table 4.1summarizes the features applied in each of the designed controllers. Some of thecontrollers use the logarithm of the blood glucose to partially reduce the nonlin-earity. Some others modify the feedback error by scheduling gains calculated bya fuzzy inference system. Some of the simulations considered the injection of the84additional insulin.The next four rows show the performance of the designed controllers in pre-venting severe hyperglycemia, hyperglycemia, hypoglycemia and severe hypo-glycemia. It can be seen that all the controllers can handle severe hyperglycemia,but controllers NO.1, NO.3, NO.5 and NO.7 do not avoid severe hypoglycemia(marked in red) and can not be sufficient for the blood glucose regulation. Amongstthe rest, controllers NO.2., NO.4 and NO.6 cause hyperglycemia (marked in yel-low) but the amount of overshoot to the hyperglycemia zone is negligible for con-trollers NO.4 and NO.6. Therefore, these two controllers along with controllerNO.8 can be considered as the potential candidates for the blood glucose regula-tion.All of the controllers are able to bring back the blood glucose concentrationunder 7 mmol/L in about 5 hours. It is desired that the controller gradually bringsback the blood glucose to the lower bound of normoglycemia zone around 4mmol/L. Amongst controllers NO4., NO.6 and NO.8, the blood glucose con-centration at the settling time for controller NO.6 is around 5.5 mmol/L which isstill high for the fasting state. Therefore, controllers NO.4 and NO.8 are bettercandidates since they can keep the blood glucose level in the normal range withthe infusion of less amount of insulin in total which can be seen in the last row oftable 4.1.The comparison of the designed controllers demonstrates the superiority ofa gain-scheduling PI controller based on fuzzy logic using the logarithm of theblood glucose over other controllers for the control of blood glucose for type IIdiabetic patients.85Table 4.1: Comparison of the designed controllersNO.1 NO.2 NO.3 NO.4 NO.5 NO.6 NO.7 NO.8Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.Characteristics 4.7 4.8 4.9 4.11 4.14 4.15 4.16 4.17Is the log of blood glucoseapplied to reduce NO YES NO YES NO YES NO YESthe nonlinearity?Is scheduling gainsconsidered NO NO YES YES NO NO YES YESusing FIS?Is the additional insulininjected with NO NO NO NO YES YES YES YESthe meal disturbance?Does the controllercause NO NO NO NO NO NO NO NOsevere hyperglycemia?Does the controllercause NO YES NO YES NO YES NO NOhyperglycemia?Does the controllercause YES NO YES NO YES NO YES NOhypoglycemia?Does the controllercause YES NO YES NO YES NO YES NOsevere hypoglycemia?How long does it taketo bring the blood glucose 5 5 5 5 5 5 5 5under 7 mmol/L? (hours)what is theblood glucose concentration 2 5.5 3.5 4 2 5.5 3.5 4.5at settling time? (mmol/L)what is thetotal amount of 48.6 3.9 43.1 7.3 48.7 5.1 42.8 7.5infused insulin? (Units)86Chapter 5Development of a Controller forBlood Glucose Regulation in Type IIDiabetes Using Predictive ControlStrategy5.1 IntroductionIt can be argued that after PID controller, model predictive controller (MPC) isthe most influential developed algorithms in control [47]. The concept of predic-tive control can be applied to control a wide variety of processes [95] and hasmade huge inroads into various application areas including biomedical engineer-ing since it is very intuitive and easy to be understood for those with a limitedknowledge of control.The term Model Predictive Control (MPC) designates a range of control strate-gies which make explicit use of the process model to predict the future behaviourof the system. The control signal is obtained by solving an optimization such thatthe future output of the process converges towards the reference trajectory. Theability of MPC to impose constraints on both manipulated and controlled vari-87ables, alongside the possibility of adjusting the optimization objective functionaccording to the problem requirements, introduces it as an alternative to enhancethe performance of blood sugar control systems for diabetic patients. MPC hasthe capability of anticipating the impact of the meal disturbance and adapting theproper amount of insulin injection in time. The clinical acceptable insulin in-jection amount per day is limited; on the other hand insulin pump mechanismrestrains the insulin infusion rate. Both constraints can be considered by definingthe boundaries for the insulin injection rate in the model predictive control de-sign. Furthermore, blood glucose concentration can be bounded with the lowerand upper level of the normoglycemia zone.There are many studies in the literature considering the model predictive con-trol strategy to improve the blood glucose regulation in type I diabetic patientsto pursue the aim of developing artificial pancreas [75]. However, to the bestof our knowledge none of these studies consider the problem of maintaining theblood glucose in normal range for type II diabetic patients. Some of the studiesconsidered the problem for type I are listed below.Parker et al. [96, 97] represented a linear MPC algorithm using a self-developedlinear step-response model as the internal model. The evaluation of the controllerhas been done by the application of the extended version of the Sorensen [4, 74]patient model for type I. The behaviour of the controller was investigated in re-sponse to 50 g oral glucose tolerance tests.Lynch et al. [98, 99] applied the 5-th order linearized minimal model to de-sign a linear MPC. A first order meal disturbance model from either Lehmann etal. [74] or Fisher [100] and a first order model for glucose transport from plasmato interstitium was added to the internal model of the controller. The controllershowed satisfactory results in presence of 50 g glucose meal disturbance and mea-surement and disturbance noise while Sorensen model [4] for type I diabetic pa-tient was used as artificial patient.An advisory mode linear MPC algorithm has been proposed by Gillis et al.[101] including the linearized minimal model modified by sub models of Hovorka88et al. [56] to determine the subcutaneous insulin infusion rate in type I diabeticpatients based on subcutaneous glucose measurements in presence of 50 g glucosemeal ingestion. Lack of the information for individual subject and the modelmismatch problem was compensated by adding an unknown disturbance term.A discrete linear MPC was presented by Magni et al. [102, 103] in which theyavoided the state estimation by using Dalla Man et al. model [57, 104] with input-output representation of insulin rate and blood glucose concentration. Evaluationof the controller performance was done by using the whole glucose insulin modelof Dalla Man et al. [57, 104]. The comparison of the proposed controller with acommon PID control method showed better regulation of the blood glucose, closerto the setpoint, and prevention of hypoglycemia state in presence of the discretelinear MPC.Dua et al. [105] selected Bergman model [54] to develop the model predictivecontroller. They suggested parametric programming approach to reduce the repet-itively solved on-line optimization problem to a simple function calculation whichexplicitly defines the relationship between the current blood glucose concentrationand the optimal insulin injection rate.Non-linear Model predictive controller has been proposed in some studies forthe blood glucose control in type I diabetes mellitus. One of the studies consid-ering the design of a continuous nonlinear MPC was done by Magni et al. [103]including the entire nonlinear glucose-insulin model of Dalla Man et al. [57, 104].Nonlinear MPC in comparison with linear MPC using the same model showedimprovement by reducing post-prandial hyperglycemia.Another nonlinear MPC was proposed by Hovorka et al. [56]. It is a self-adapting nonlinear MPC algorithm based on their 8th order nonlinear glucose-insulin model. Since the insulin sensitivity of glucose consuming cells changesaccording to the day time, physical activity, patient age and health status, some pa-rameters were defined to describe the insulin sensitivity which were re-estimatedat each control step depending on the current blood glucose measurement.However, better performance of nonlinear MPC will achieve in the expense89of the increased computational efforts in comparison with linear MPC. Decisionsshould be made whether such improvement is worthwhile.The mathematical model representing the glucose-insulin interactions is a non-linear model and has a slow dynamic for performing on-line optimization. There-fore, in this chapter model predictive controller is developed using multiple lin-earized models presented in section 5.3. The optimization problem of designingmodel predictive controller was solved using the multi parametric quadratic pro-gramming approach explained in the following section. The formulation of thestate estimator is presented in section 5.4. Section 5.5 represents the simulationresults.5.2 Predictive Control StrategyThe model predictive control (MPC) strategy makes explicit use of the processmodel to predict the future behavior of the system. The various MPC algorithmsuse different types of models to represent the process and use a customized objec-tive function that is to be minimized. The control signal is obtained by solving anoptimization problem such that the future output of the process converges towardsa reference trajectory. It is also known as receding horizon control because ateach instant an open-loop optimal control problem is solved over a finite horizon.Only the first control signal of the sequence is applied to the system. At the nexttime step, the horizon is moved toward the future and a new optimal control prob-lem is solved based on the new measurements over a shifted prediction horizon[106, 107].5.2.1 Linear Model Predictive Control (LMPC)The formulation of the optimization problem for designing the model predictivecontroller was presented in [108]. Consider the general mathematical description90of a discrete-time, linear time invariant state-space system:xt+1 = Axt +Butyt =Cxt(5.1)subject to the following constraints:ymin ≤ yt ≤ ymax,umin ≤ ut ≤ umax,where xt ∈ Rn, ut ∈ Rm, and yt ∈ Rp are the state, input, and output vectors,respectively, subscripts min and max denote lower and upper bounds, respectively,and the matrix pair (A,B) is stabilizable. The linear MPC problem for regulatingsystem (5.1) to the origin is presented as the following quadratic programmingproblem:minUJ(U,xt) = x′t+Ny|t Pxt+Ny|t +Ny−1∑k=0x′t+k|tQxt+k|t +u′t+kRut+k (5.2)subject to:ymin ≤ yt ≤ ymax, k = 1, . . . ,Ncumin ≤ ut ≤ umax, k = 1, . . . ,Ncxt|t = xt ,xt+k+1|t = Axt+k|t +But+k, k ≥ 0yt+k+1|t =Cxt+k|t , k ≥ 0ut+k+1|t = Kxt+k|t , Nu ≤ k ≤ Nywhere U ≡ {ut , . . . ,ut+Nu−1}, Q = Q′ ≥ 0, R = R′ > 0, P ≥ 0 and Nu, Ny, Ncare the input, output, and constraint horizons, respectively, such that Ny ≥ Nu,Nc ≤ Ny−1, and K is a stabilizing state feedback gain. The optimization problem91in 5.2 is solved repetitively at each time t for the current state xt and the vector ofpredicted state variables, xt+1|t , . . . ,xt+k|t at time t +1, . . . , t + k, respectively, andcorresponding optimal control actions are obtained:U∗ = {u∗t , . . . ,u∗t+k+1}, (5.3)The input that is applied to the system is the first control action:ut = u∗t , (5.4)and the procedure is repeated at time t+1, based on the new state xt+1.The state feedback gain, K, and the terminal cost function matrix, P, usuallyare used to guarantee stability for the MPC. K and P can be chosen as the solutionsof the unconstrained, infinite-horizon linear quadratic regulation (LQR) problem,i.e., when: Nc = Nu = Ny = ∞K =−(R+B′PB)−1B′PA,P = (A+BK)′P(A+BK)+K′RK+Q,(5.5)The following relation is derived from equation (5.1),xt+k|t = Akxt +k−1∑j=0A jBut+k−1− j, (5.6)Substituting equation (5.6) in equation (5.2) results in the following quadraticprogramming (QP) problem:J∗(xt) = minU{12U ′HU + x′tFU +12x′tY x(t)}, (5.7)subject to:GU ≤W +Extwhere U ≡ {ut , . . . ,ut+Nu−1} ∈ Rn, s ≡ mNu is the vector of optimization vari-92ables, and H,F,Y,G,W,E are obtained from Q,R and the system matrices. Theterm involving Y in equation (5.7) is usually dropped, since it does not affect theoptimal solution.MPC is applied by repetitively solving the QP problem in (5.7) at each timet ≥ 0 for the current value of the states. Due to this formulation, the solution U∗of the QP is a function U∗(xt) of the state xt . The control action is given by:ut =[I 0 · · · 0]U∗(xt) (5.8)The problem in equation 5.2 obviously describes the constrained linear quadraticregulation problem, while equation 5.7 is the formulation of the MPC as a QP op-timization problem [106, 107, 109–111].The optimization problem can be modified to better address the control ob-jectives for regulating the blood glucose level. It is desired to keep the bloodglucose concentration within the certain range. The fact is that the boundaries ofsuch range can be slightly violated while being still acceptable. On the other handviolation of the lower bound is more crucial because of the higher dangers of hy-poglycemia state for the patient. The following modifications help to improve thecontroller design.Soft constraintsAs discussed the constraints on the output are soft which means that they are pre-ferred but not required to be satisfied and can be partially violated. The constraintscan be modified as:ymin−υl ≤ yt ≤ ymax+υu (5.9)which means that the output is still acceptable to some degree if it violates theminimum and the maximum values less than υl and υu, respectively. υl and υuare chosen by the designer to fulfill the requirements of the patient.The inequality constraints could be expressed in normalized form so they are93fully satisfied if they are negative:yn ={1− yymin for lower boundyymax−1 for upper bound (5.10)where yn is the normalized inequality constraint for the output y. ymin and ymaxare lower and upper bounds of the output, respectively. Therefore, the degree ofacceptance is assigned to the output by using a linear membership function asfollows:µyn =1 yn ≤ 0 fully satisfiedd−ynd 0< yn ≤ d−1 yn ≤ d strongly violated(5.11)where µyn is the membership function for each inequality constraint on the out-put. d is the acceptable tolerance in normalized form. µyn is used to update theconstraint matrices represented in equation (5.7). The constraints can be dividedinto two parts: GU ≤W1 for the input and −Ext ≤W2 for the output, in whichW =W1 +W2. E and W2 are updated by multiplication of µyn to partially allowthe violations in the lower and upper bounds of the output.Asymmetric control objectiveThe membership function, as defined in equation 5.11, is used to update the con-straint on the upper and lower bounds of the output to prevent hyperglycemia andhypoglycemia. Although both hyperglycemia and hypoglycemia states should beavoided to handle diabetes complications, hypoglycemia is much more dangeroussince patients can have a seizure or go into the coma due to low blood glucose.In order to address the asymmetric nature of the control objective, the mem-bership function, µyn is considered to be −10 instead of −1 for the time that thelower bound is strongly violated. This makes the controller act more aggressivelyin preventing hypoglycemia.94Despite the fact that efficient QP solvers are available to solve equation (5.7),computing the input online may require significant computational effort. In thenext section, the parametric optimization approach is discussed to reduce the com-putational effort and also represents a lookup-table to find the proper insulin infu-sion rates as a function of the states.5.2.2 Multi-Parametric Quadratic Programming ApproachAs discussed before, in model predictive control strategy the control signal iscalculated by solving an optimization problem. The optimizer should be run atregular time intervals, which requires large on-line computational efforts. Thus, itis fair to state that an efficient implementation of on-line optimization tools relieson a quick and repetitive on-line computation of optimal control actions.Parametric programming approach can be considered to avoid this repetitivesolution. The optimization problems with a quadratic performance criterion andlinear constraints are formulated as multi-parametric quadratic programs (mp-QP)to design model predictive controller. The input and state variables are treated asoptimization variables and parameters, respectively, and the control variables areobtained as an explicit function of the state variables. Therefore, the on-line op-timization is solved off-line and breaks down into simple function evaluations,at regular time intervals, for the given state of the system to compute the cor-responding control actions. This results in a very small computational effort incomparison with a repetitively solving of an optimization problem.It can be shown that the control law is piece-wise linear and continuous. Thus,the on-line control computation reduces to the simple evaluation of an explicitlydefined piecewise linear function of the states [108, 112–114]. The followinglinear transformation is considered to transform the QP problem 5.7 into a multiparametric programming problem:z≡U +H−1F ′xt (5.12)95The quadratic programming (QP) problem described in (5.7) is then formulatedto the following multi parametric quadratic programming (mp-QP) problem:Vz(xt) = minz12z′Hz (5.13)subject toGz≤W +Sxtwhere S = E +GH−1F ′, z is the vector of optimization variables and xt is thevector of parameters.The main advantage of writing equation (5.2) in the form given in equation(5.13) is that z and U , can be obtained as an affine function of x for the completefeasible space of the states.The following theorem has to be established to proceed with a method to solveequation (5.13). The proofs of this theorem can be found in [112].Theorem 1. Let x0 ∈R be a vector of parameters and (z0,λ0) be a KKT pairfor problem (5.13), where λ0 = λ0(x0) is a vector of non-negative Lagrange mul-tipliers, λ and z0 = z(x0) is feasible in equation 5.13. Also assume that the (i)linear independence constraint satisfaction and (ii) strict complementary slack-ness conditions hold. Then, there exists in the neighbourhood of x0 a unique, oncecontinuously differentiable function [z(x),λ (x)] where z(x) is a unique isolatedminimizer for equation 5.13, and(dz(x0)dxdλ (x0)dx)=−(M0)−1N0 (5.14)whereM0 =H GT1 · · · GTq−λ1G1 −V1 · · ·... . . .−λpGq −Vq96N0 = (Y,λ1S1, ...,λpSp)Twhere Gi denotes the ith row of G, Si denotes the ith row of S, Vi =Giz0−Wi−Six0,Wi denotes the ith row of W, and Y is a null matrix of dimension (s×n).The optimization variable zx can then be obtained as an affine function of thestate xt by exploiting the first-order KarushKuhn Tucker (KKT) conditions forequation 5.13.Theorem 2. Let x be a vector of parameters and assume that assumptions(i) and (ii) of Theorem 1 hold. Then, the optimal z and the associated Lagrangemultipliers λ are affine functions of x.An observation, resulting from Theorems 1 and 2, is given in the next corol-lary.Corollary 1. From Theorem 1 and 2,[z(x)λ (x)]=−(M0)−1N0(x− x0)+[z0λ0](5.15)The results of Theorems 1 and 2 and Corollary 1 are summarized in the fol-lowing theorem:Theorem 3 For the problem in equation (5.13) let x0 be a vector of parametervalues and (z0,λ0) a KKT pair, where λ0 = λ0(x0) is a vector of non-negativeLagrange multipliers, λ , and z0 = z(x0) is feasible in equation (5.13). Also as-sume that (i) the linear independence constraint qualification and (ii) the strictcomplementary slackness conditions hold. Then,[z(x)λ (x)]=−(M0)−1N0(x− x0)+[z0λ0](5.16)whereM0 =H GT1 · · · GTq−λ1G1 −V1 · · ·... . . .−λpGq −Vq97N0 = (Y,λ1S1, ...,λpSp)Twhere Gi denotes the ith row of G, Si denotes the ith row of S, Vi =Giz0−Wi−Six0,Wi denotes the ith row of W, and Y is a null matrix of dimension (s×n).The presented theorem shows that the solution z(x) and λ (x) can be obtainedfor any parameter vector x from equation (5.16), given the solution z0 and λ0 for aspecific vector of parameters x0. Thus, the optimization variable z and eventuallythe control sequence U are linear, affine functions of the state x. Therefore, the se-quence of control actions is obtained as an explicit function of the state. It remainsnow to establish for which values of the parameter (state) x, this solution remainsoptimal. The set of x where the solution of equation (5.16) remains optimal isdefined as the critical region (CR0) and can be obtained as follows.Let the critical region, CRR represent the set of inequalities obtained (i) bysubstituting z(x) into the inactive constraints in equation 5.13, and (ii) from thepositivity of the Lagrange multipliers corresponding to the active constraints, asfollows:CRR = {^Gz(x)≤ ^W +^S x(t),∼λ (x)≥ 0} (5.17)then CR0 is obtained by removing the redundant constraints from CRR as follows:CRR =4{CRR} (5.18)where 4 is an operator which removes the redundant constraints. Since for agiven space of state variables, X , so far we have characterized only a subset of X ,i.e., CR0 ⊆ X , in the next step the rest of the region CRrest , is obtained as follows:CRrest = X−CR0 (5.19)More information on identifying the regions can be found in [115]. The abovesteps, 5.16-5.19, are repeated and a set of z(x), λ (x) and corresponding CR0 is ob-tained. The solution procedure terminates when no more regions can be obtained,i.e., when CRrest =∅. The regions with the same solution can be unified to give a98convex region.The optimal control sequence U∗(x), once z(x) is obtained by equation 5.16,is calculated from equation 5.12.U∗(x) = z(x)−H−1F ′x (5.20)Finally, the feedback control law is given by:ut =[I 0 · · · 0]U∗(xt) (5.21)5.3 Prediction Model for Type II DiabetesThe number of experiments that can be performed on a human body is restricteddue to ethical standards and risks posed to the subject. On the other hand, itis necessary to obtain reliable information from diabetic patients to describe theglucose-insulin interactions. Mathematical modeling can be considered as an al-ternative approach to provide sufficient information about the medical status of thepatient. Dynamic modeling of glucose metabolism is helpful to evaluate controlstrategies developed for diabetic patients to prevent serious and irreversible harmto the subject. The model derived for type II diabetes mellitus should be able torepresent the interactions between different organs and determine the concentra-tions of glucose and insulin along with the hormonal effects of glucagon.The homeostasis of blood glucose in type II is complicated and the patientsdeal with the malfunction of different organs as described in 1.1.2 which eventu-ally leads to the deterioration of glucose homeostasis [14].To develop a control system based on MPC, a linear model of the patient isneeded. Such a model can be built up from a set of experimental data. Becauseof many limitations in carrying out human experimental studies, the available dy-namic model described in Chapter 2 can be used as a simulator of the patient bodyto collect input-output data and develop prediction model for designing MPC.According to the nonlinear factors appearing in the model equations, the re-99lationship between the rate of changes in the blood glucose concentration andthe infusion rate of insulin is highly nonlinear. Multiple linear models have beenconsidered to better represent such a nonlinear behavior of glucose-insulin in-teractions. Each model is a discrete state-space model with two states and wasobtained using the MATLAB system identification toolbox. One of these modelsrepresents the blood glucose concentration corresponding to severe hypoglycemia,hypoglycemia and fasting states (low blood glucose level), the second one is forpostprandial and postabsorptive states of normoglycemia zone (normal blood glu-cose level) and the third one is for the high blood glucose level in hyperglycemiaand severe hyperglycemia states as depicted in figure 1.2. Equation (5.22) showsthe model structure and the matrices for each model are presented in table 5.1xt+1 = Axt +But +Ke(t)yt =Cxt + e(t)(5.22)Table 5.1: Prediction models matricesSevere hypoglycemia, Normoglycemia state Severe hyperglycemiahypoglycemia and fasting states and hyperglycemia statesA[0.792 −0.015−0.042 0.794] [0.849 0.026−0.084 0.791] [0.851 −0.112−0.149 0.682]B[−0.0235 0.0009−0.0816 0.0014] [−0.0017 0.0003−0.018 0.0002] [−0.0061 0.0002−0.0143 0.0002]C[7.234−2.205] [9.85−1.56] [18.13−7.895]K[0.159 0.067] [0.113 0.086] [0.650 0.289]The inputs of these models (u) are insulin infusion rate (mU/min) and mealglucose disturbance (mg/min), and the output (y) is glucose concentration (mmol/L).x is the vector of two states. The sampling time is 5 minutes which is suitable for100the measurement sensor.5.4 State EstimationSince the blood glucose concentration is the only measurement that we have, astate estimator should be designed to obtain the current estimation of the statesat each sampling time. The prediction model is linear and a steady-state Kalmanfilter can be designed.x[t+1] = Ax[t]+B(u[t]+ω[t]y[t] =Cx[t](5.23)Considering equation 5.23 as the model with gaussian noise ω[t] on the inputu[t], the equations of the steady-state Kalman filter are as follows:Measurement update:xˆ[t|t] = xˆ[t|t−1]+M(yυ [t]−Cxˆ[t|t−1]) (5.24)Time update:xˆ[t+1|t] = Axˆ[t|t]+Bu[t] (5.25)yυ is the output measurement, xˆ[t|t− 1] is the estimation of xˆ[t], given past mea-surements up to yυ [t−1] and xˆ[t|t] is the updated estimate based on the last mea-surement, yυ [t].Equation 5.25 predicts the value of the state one step ahead, given the currentestimation (xˆ[t|t]). Equation 5.24 then adjusts this prediction based on the newmeasurement yυ [t+1]. The correction term is a function of the difference betweenthe measured and predicted values of y[t+1]:yυ [t+1]−Cxˆ[t+1|t] (5.26)The gain M in equation 5.24 is chosen to minimize the steady-state covariance101of the estimation error, given the noise covariances as follows:E(ω[t]ω[t]T ) = Q E(υ [t]υ [t]T ) = R E(ω[t]υ [t]T ) = 0 (5.27)Combining equation 5.24 and 5.25 results in the following equation for theKalman filter (equation 5.28) which gives the optimal estimates of the states andthe output.xˆ[t+1|t] = A(I−MC)xˆ[t|t−1]+Bu[t]+AMyυ [t]yˆ[t|t] =C(I−MC)xˆ[t|t−1]+CMyυ [t](5.28)More details on state estimation and Kalman filter can be found in [116, 117]5.5 Simulation Results and DiscussionThe model predictive controller was designed by using multiple linear predictionmodels as discussed in section 5.3. The application of the model predictive con-troller gives the opportunity of imposing constraints on the variables. The firstconstraint confines the blood glucose concentration to be within the desired rangewhich is below 11 mmol/L after a meal ingestion during the postprandial state,and below 7 mmol/L, as close as possible to 4 mmol/L, during the fasting state.The second constraint keeps the insulin infusion rate between 0 to 100 mU/minwhich is suitable for an insulin pump. Soft constraints for the blood glucose con-centration were introduced as shown in equation 5.9 of section 5.2. The allowableviolation degrees, υl and υu, are 1 for both lower and upper bounds of the con-straints on the blood glucose concentration.The block diagram of the control strategy is demonstrated in figure 5.1.For the evaluation of the controller performance, the nonlinear model of thepatient has been used as the simulated patient to predict the current value of theoutput (the blood glucose concentration) for the current input (insulin infusionrate).In order to implement the model predictive controller based on parametric102Figure 5.1: Block diagram of model predictive control strategyprogramming, the states associated with the current measurement of the bloodglucose concentration should be calculated through a state estimator. Therefore,the region corresponding to the states is identified to compute the optimal controlaction.The proper linearized model is selected based on the estimated states from themodels presented in section 5.3 and used to predict the future outputs.The asymmetric objective function and the soft constraints are introducedto the optimizer block. The optimization problem is solved using the multi-parametric quadratic programming approach, which results in partitioning of thespace of the feasible states into polyhedral regions. High prediction horizons in-creased the computational efforts but did not improve the performance that much,so the prediction horizon is set to two. The steps of mp-QP algorithm, explainedin [112], are as follows:• Step 1: for a given space of x solve equation 5.13 by treating x as a freevariable and obtain [x0]103• Step 2: in equation 5.13 fix x = x0 and solve equation 5.13 to obtain [z0,λ0]• Step 3: obtain [z(x),λ (x)] from equation 5.16• Step 4: define CRR as given in equation 5.17• Step 5: from CRR remove redundant inequalities and define the region ofoptimality CR0 as given in equation 5.18• Step 6: define the rest of the region, CRrest, as given in equation 5.19• Step 7: if no more regions to explore, go to the next step, otherwise go tostep 1• Step 8: collect all the solutions and unify a convex combination of the re-gions having the same solution to obtain a compact representationThe control law for each of the partitions of the state-space was obtained byusing the above algorithm. The region boundaries and corresponding control lawwhich is an affine function of the states are reported in tables 5.2, 5.3 and 5.4 forthe model corresponding to low, normal and high blood glucose levels, respec-tively.In order to provide a baseline for the assessment of the designed controller,some standard clinical tests can be performed. Oral glucose tolerance test (OGTT)is a common test in which a standard amount of dissolved glucose in water is givento the patient to see how the body respond to glucose intake in the absence of in-jected insulin. Figure 5.2 presents the blood glucose concentration for introducinga 100 g glucose disturbance at time 100 minutes to the simulated patients whenno control is applied.The result shows an increase in blood glucose concentration which wouldcause the patient to experience hyperglycemia and severe hyperglycemia statesfor three hours before returning to normoglycemia zone. The pancreas of typeII diabetic patients can still secrete some but not the adequate amount of insulin,and the patient needs the exogenous insulin injection. The basal blood glucose104Table 5.2: Parametric solution regions for low blood glucose concentration(< 7mmol/L)Region# Region boundary Insulin Infusion Rate1−11.83 13.18−1 00 −11 00 1x≤7.14001.51.5 [7.23 −2.2]x+4211.83 −13.18−1 00 1x≤−7.140−6.51.5 [−4.59 10.97]x−3.14Table 5.3: Parametric solution regions for normal blood glucose concentra-tion (7−11mmol/L)Region# Region boundary Insulin Infusion Rate1−34.63 38.47−1 00 −11 0x≤−7.25001.5 [9.85 −1.56]x+4234.63 −38.47−1 01 00 1x≤7.2501.51.5 [−22.73 45.17]x−3.27level for the simulated patients is around 7 mmol/L and the available insulin inthe body returns the blood glucose concentration to normoglycemia zone 6 hoursafter the 100 g glucose administration.The model predictive controller is implemented to calculate the proper insulin105Table 5.4: Parametric solution regions for high blood glucose concentration(> 11mmol/L)Region# Region boundary Insulin Infusion Rate1−34.35 −58.08−1 00 −11 0x≤6.55001.5 [18.13 −7.89]x+4234.35 58.08−1 039.2 −17.141 00 1x≤−6.550−41.51.5 [−16.22 50.18]x−2.553−39.2 17.14−1 00 1x≤ 401.5 [−21.07 9.24]xinfusion rate for the insulin pump to regulate the blood glucose concentration.As discussed in section 1.2.1, minimizing glucose variance around a single targetvalue is not the goal if the controller is to mimic the behavior of a healthy bodyin regulating blood glucose. Therefore, the goal is to keep the blood glucoseconcentration within a certain range while gradually attempting to bring it closerto the lower bound of the range.A 100 g glucose bolus was introduced at time 100 minutes as a meal dis-turbance, and the test result is presented in figure 5.3 using the proposed modelpredictive control. The upper part of figure 5.3 represents the blood glucose con-centration (mmol/L) and the lower part shows the insulin infusion rate (mU/min)calculated by the controller. The controller can avoid hyperglycemia state andbring the blood glucose back below 7 mmol/L after 4 hours, during postprandialstate. It gradually decreases the blood glucose near the lower bound of normo-106Figure 5.2: The simulated patient response to 100 gr of glucose ingestion asa meal disturbance with no automatic controlFigure 5.3: The simulated patient response to 100 gr of glucose ingestionusing model predictive control strategy107glycemia zone, without entering the hypoglycemia state.Figure 5.4: Response of the conventional PI controller with a 100 g mealdisturbance at 100 minFor comparison, conventional PI controller and fuzzy-based PI controller us-ing the logarithm of blood glucose, as designed in Chapter 4 are considered andthe results are demonstrated in figure 5.4 and 5.5 for the administration of a 100 gmeal disturbance to the simulated patient at 100 minutes.The proportional gain is considered to be 12.7, and the integral gain is 0.05.The result shows that a conventional PI controller is capable of handling highblood glucose level but is not sufficient for blood glucose regulation since it can-not prevent hypoglycemia. On the other hand, the amount of injected insulin ismuch more than the amount suggested by MPC. The injection rate exceeds the108Figure 5.5: Response of the fuzzy-based PI controller with a 100 g meal dis-turbance at 100 min with non-linearity reduction in the patient modelusing log of blood glucosemaximum infusion rate of the insulin pump, and the controller output is saturated.The excess insulin does not help the glucose absorption since the body cells areinsulin resistant in type II diabetic patients. It would accumulate in the blood andincrease the plasma insulin level.Fuzzy-based PI controller using the logarithm of blood glucose acts properlyas shown in figure 5.5. It can handle the postprandial blood glucose increase witha small overshoot to hyperglycemia zone. Its performance is also acceptable inpreventing hypoglycemia for the patient.In the presence of sufficient amount of data, the model predictive control strat-egy can be considered to design a controller for regulation of blood glucose. How-109ever if collecting data from the patient is impossible, fuzzy-based PI controllerusing the logarithm of blood glucose is a reliable option for automatic control ofthe blood glucose.5.6 ConclusionIn this chapter, model predictive control strategy has been introduced as an alter-native for the blood glucose regulation problem. The optimization problem hasbeen modified by consideration of asymmetric objective function and incorpora-tion of soft constraints to develop a controller which closely mimics the glucoseregulatory system of a healthy individual. The parametric programming approachwhich has been applied to solve the optimization problem, not only reduces thecomputational efforts of solving an on-line optimization problem but also providesa lookup-table with all optimal solutions of the feasible space of states.The multiple linear models for different blood glucose levels have been con-sidered to capture the nonlinear and complex nature of glucose-insulin interac-tions. The designed controller is able to handle the meal disturbance in a timelymanner. The blood glucose concentration is kept in normoglycemia zone sincethe application of asymmetric objective function strictly prevents the risk of thesevere hypoglycemic state. The controller is also capable of avoiding high level ofthe blood glucose during the postprandial state. The blood glucose concentrationis brought back closer to the lower bound during the postabsorptive state and keptas low as desired during the fasting state.110Chapter 6Comparison of ControllerPerformance in response tovariations in the metabolic rates6.1 IntroductionThe parameters of the model described in Chapter 2 were estimated for an indi-vidual patient. These parameters may vary between patients with different ages,genders, weights and activity levels. They may also change within a patient as thedisease is progressing. The malfunctions of liver, peripheral tissues and pancreasresult in the deficiency of the glucose regulatory system in type II diabetic patient.Therefore, the metabolic rates of these organs are subject to more variations fromone patient to another and also within a patient during the course of the disease. Inthis chapter, the performance of the proposed control algorithms are presented andcompared with each other as the result of the variations in the parameters relatedto these organs.The following section investigates the performance of the gain-scheduling PIcontroller and model predictive controller as designed in Chapter 4 and Chapter5, respectively. The body response is plotted for the administration of 100 g oral111glucose as the meal disturbance once without automatic control and next withthe application of controllers. As the disease becomes more severe, the glucosemetabolic rates reduce and worsen the condition of the patient. Therefore, a 50%decrease in glucose uptake and production rates of the liver, peripheral glucoseuptake rate and pancreatic insulin secretion rate is considered. Finally, the resultsare discussed and concluded.6.2 Controller Performance Investigation6.2.1 Variation in Liver Metabolic RatesThe liver plays a central role in maintaining the blood glucose level. It balances theuptake and storage of glucose via glycogenesis during postabsorptive state, afteringestion of glucose-containing meals and the release of glucose via glycogenol-ysis and gluconeogenesis during fasting state, between meals and overnight[118,119]. The need to store or release glucose is primarily signaled by the hormonesinsulin and glucagon. The high levels of insulin and suppressed levels of glucagoncaused by glucose ingestion inhibits endogenous hepatic glucose production andpromote the storage of glucose as glycogen (red path in figure 6.1). When bloodglucose is reduced by small increments in circulating insulin, a rebound increasein glucose output from the liver is the initial or principal mechanism counteractingthe fall in blood glucose concentration (blue path in figure 6.1).As the disease become more severe, liver loses its capability to regulate theblood glucose and hepatic glucose uptake, and production rates decrease. Thesolid black line in figure 6.2 demonstrates the body response to the administra-tion of 100 g glucose at time 100 minutes when both hepatic glucose uptake andproduction rates are decreased by 50%. The blue dashed line provides a baselinefor comparison and shows the blood glucose profile before the rates reduction.The blood glucose goes up to severe hyperglycemia zone. The maximum bloodglucose concentration after the meal ingestion increases from 16.2 to 18 mmol/Las the deficiencies of the liver are worsening.112Figure 6.1: role of the liver in glucose homeostasisAnother test is performed to measure the insulin sensitivity of the body on thesimulated patients. In this test, the plasma insulin concentration is increased andclamped at a high level by a continuous infusion rate of insulin.Figure 6.3 shows the blood glucose concentration while a constant rate ofinsulin at 10 mU/min is being infused. In the case of low blood glucose level,a healthy body converts the glycogen stored in the liver and muscles to glucoseduring glycogenolysis. The hepatic glucose production rate is impaired in type IIdiabetic patients, and the endogenous glucose production is lower than the healthypatient. The liver of the patient who is in the initial steps of the disease (blue113Figure 6.2: The simulated patient response to 100 gr of glucose ingestionas a meal disturbance for 50% decrease in hepatic glucose uptake andproduction rates (black solid line) and before the rate reduction (bluedashed line)Figure 6.3: Plasma blood glucose concentration in presence of continuousinsulin infusion rate of 10 mU/min for the reduced rates (black solidline) and before the reduction (blue dashed line)dashed line) is still capable of releasing glucose to prevent hypoglycemia, but asthe hepatic glucose production rate is reduced, the blood glucose level decreasesto severe hypoglycemia if no glucose ingested (black solid line).This test shows the importance of the better control to prevent hypoglycemiastate since the improper insulin infusion can drastically decrease the blood glucoseconcentration if the hepatic glucose production rate is low.114Figure 6.4: The performance of fuzzy gains-scheduling controller after 100gr of glucose ingestion as a meal disturbance for 50% decrease in hep-atic glucose uptake and production ratesThe designed controllers are expected to handle the situation by determiningthe proper insulin injection dose. An increase in plasma insulin concentrationstimulates the liver to uptake and store more glucose during hyperglycemia state.A decrease in insulin level gives the signal to the liver for taking care of hy-poglycemia state by releasing glucose. Figure 6.4 indicates the performance offuzzy gain-scheduling controller using the logarithm of the blood glucose as thecontrolled variable.The controller keeps the blood glucose in normoglycemia zone for most ofthe period with an overshoot to hyperglycemia zone and a negligible undershootto hypoglycemia zone which is still acceptable since it prevents the severe hyper-115glycemia and hypoglycemia states.As discussed in Chapter 4, two features were considered to enhance conven-tional PI controller for handling the nonlinear dynamic of glucose homeostasis.Using logarithm of the blood glucose reduces the nonlinearity and makes PI con-troller an acceptable, yet simple alternative to the nonlinear control strategies. Onthe other hand, gain scheduling technique helps the controller to be designed forprevention of severe hyperglycemia and hypoglycemia while gradually returningthe blood glucose to the lower bound of normoglycemia zone. The schedulinggains are calculated by a fuzzy inference system which has the advantage of inter-preting the expert knowledge and clinician's insight into the set of rules. Throughsuch rules, the weighting factors are defined to modify the gains based on thecurrent blood glucose level. Figure 6.5 represents the weighting factors for thedesigned controller.Figure 6.5: Weight factor from the fuzzy inference systemThe performance of the model predictive controller is presented in figure 6.6.Model predictive control acts similar to the fuzzy gain-scheduling controller inhandling high blood glucose level. Also, it can completely avoid hypoglycemiastate and bring back the blood glucose to the lower bound of normoglycemia zone,6 hours after the meal disturbance applied.The model predictive controller, as designed in Chapter 5, takes the asym-metric nature of the glucose regulation problem into consideration. Therefore, it116Figure 6.6: The performance of model predictive controller after 100 gr ofglucose ingestion as a meal disturbance for 50% decrease in hepaticglucose uptake and production ratesefficiently avoids hypoglycemia state which causes more complications since itmay put the patient in the coma. The proposed technique for solving the problemreduce the optimization problem into simple function calculations and provide alook-up table which gives an insight to the proper insulin infusion rate at each stateof the patient and makes it easier to be understood by the patients and clinicians.The total amount of insulin infused into the body in the presence of modelpredictive controller through insulin pump is less than the infusion amount asthe fuzzy gain-scheduling controller is applied. The excess amount of insulinincreases the plasma insulin concentration and finally excretes from the body.117Figure 6.7: The simulated patient response to 100 gr of glucose ingestion asa meal disturbance for 50% decrease in peripheral glucose uptake rate(black solid line) and before the rate reduction (blue dashed line)6.2.2 Variation in Periphery Metabolic RateInsulin resistance is a characteristic feature of type II diabetes and plays an im-portant role in the pathogenesis of the disease even before the failure of β -cellsto produce insulin. The causes of insulin resistance can be placed into three cate-gories: abnormal products secreted by β -cells, circulating insulin antagonists andthe reduced response of peripheral tissues to insulin [120].As the disease is progressing, the body cells lose their sensitivity to insulinand as the result, the peripheral glucose uptake rate decreases. Figure 6.7 showsthe body response to 100 g meal disturbance when the peripheral glucose uptakerate is decreased by 50%. The failure of the peripheral tissues to absorb glucosecauses the increase of the blood glucose level to severe hyperglycemia zone up to19 mmol/L, which is 15% higher than the maximum blood glucose level beforethe decrease in the peripheral glucose uptake rate.The insulin acts on cells throughout the body to stimulate uptake and utiliza-tion of glucose. The effects of insulin on glucose metabolism vary depending onthe target tissue, but mainly insulin facilitates the diffusion of glucose into mus-cle, adipose and several other tissues. There are some tissues that do not requireinsulin for efficient uptake of glucose such as brain and the liver.118Figure 6.8: The performance of fuzzy gains-scheduling controller after 100gr of glucose ingestion as a meal disturbance for 50% decrease in pe-ripheral glucose uptake rateIn peripheral tissues, GLUT4 is the major glucose transporter which is usedfor uptake of glucose through the action of insulin. Binding of insulin to thereceptors on cells leads rapidly to the insertion of the glucose transporters onthe plasma membrane, thereby giving the cells the ability to efficiently take upglucose. When blood levels of insulin decrease and insulin receptors are no longeroccupied, the glucose transporters are recycled back into the cell cytoplasm.The study that has been done by Prato et al. shows that hyperinsulinemia,which is the condition of having excess levels of insulin circulating in the bloodthan expected relative to the level of glucose, can normalize total body glucoseuptake [121]. Therefore, the insulin therapy can still benefit type II diabetic pa-119Figure 6.9: The performance of model predictive controller after 100 gr ofglucose ingestion as a meal disturbance for 50% decrease in peripheralglucose uptake ratetients with insulin resistance and increase the glucose uptake in peripheral tissues.Providing more insulin through insulin infusion pump also reduces the burden onthe pancreas to secrete more insulin.Figure 6.8 represent the performance of gain scheduling PI controller base onfuzzy logic. Administration of 100 g glucose 100 minutes after starting the simu-lation, is following by increasing the insulin infusion rate by the controller. Thisincrease in insulin level avoids the blood glucose to go as high as 19 mmol/Lwhich could happen without control (as seen in figure 6.7). However, the pa-tient still experiences severe hyperglycemia for a short period, but the controlleris capable of returning the blood glucose back to normoglycemia zone in 3 hours.120Figure 6.10: The simulated patient response to 100 gr of glucose ingestionas a meal disturbance for 50% decrease in pancreatic insulin secretionrate (black solid line) and before the rate reduction (blue dashed line)The controller performance is also acceptable in bringing the blood glucose tothe lower bound of normoglycemia zone with a negligible undershoot to hypo-glycemia zone and preventing the severe hypoglycemia.The response of the simulated patient with 50% decrease in peripheral glu-cose uptake rate to 100 g meal disturbance at 100 minutes is shown in figure 6.9in the presence of model predictive controller. In comparison with the fuzzy-based PI controller, MPC acts better in preventing severe hyperglycemia and hy-poglycemia. The overshoot to the severe hyperglycemia is negligible, and theblood glucose is brought back to the lower bound of normoglycemia zone withoutany undershoot to the hypoglycemia zone.6.2.3 Variation in Pancreatic Insulin Secretion RateThe insulin secretion rate has been reduced at the time of diagnosis in most of thetype II diabetic patients. The secretion of insulin further diminishes during thecourse of the disease.Insulin secretory capacity depends on both function and mass of β -cells. Somestudies highlighted the pieces of evidence that the reduced insulin secretion resultsfrom two dissociable factors, a decrease in the population of the functional insulin-121Figure 6.11: The performance of fuzzy gains-scheduling controller after 100gr of glucose ingestion as a meal disturbance for 50% decrease inpancreatic insulin secretion rateproducing β -cells and the intrinsic secretory defect. Both of these factors playimportant roles in the development and progression of type II diabetes [122, 123].Figure 6.10 shows the blood glucose concentration profile for the administra-tion of 100 g oral glucose at 100 minutes to the simulated patient. As the resultof the reduced insulin secretion rate the blood glucose level increases and goeshigher than 20 mmol/l. The insulin therapy is helpful since it compensate thelack of the endogenous insulin in the body.The performance of the gain-scheduling PI controller based on fuzzy logicand model predictive controller are presented in figure 6.11 and 6.12, respectively.Both controller responses are satisfactory in preventing the hyperglycemia and hy-122Figure 6.12: The performance of model predictive controller after 100 gr ofglucose ingestion as a meal disturbance for 50% decrease in pancre-atic insulin secretion ratepoglycemia states. The controllers apply the proper insulin infusion rate to reducethe blood glucose concentration. It takes almost 5 hours for both controllers tobring the blood glucose concentration under 7 mmol/L and an extra hour for set-tling down near the lower bound of the normal range which is the desirable levelof the blood glucose in the fasting state.6.3 DiscussionsType II diabetes is a progressive disease and caused by malfunction of differentorgans. The glucose metabolic rates are varying in a patient during the course of123Table 6.1: Assessment of the blood glucose regulation using the designedcontrollers for the decreased metabolic rates50% decrease in hepatic glucose uptake and production ratessevere hypo- normo- hyper- severehypoglycemia glycemia glycemia glycemia hyperglycemiaWithout control(Figure 6.2) 0 0 52 16 32GS-PI controller(Figure 6.4) 0 6 68 26 0MPC(Figure 6.6) 0 0 78 22 050% decrease in peripheral glucose uptake ratesevere hypo- normo- hyper- severehypoglycemia glycemia glycemia glycemia hyperglycemiaWithout control(Figure 6.7) 0 0 44 20 36GS-PI controller(Figure 6.8) 0 10 60 12 18MPC(Figure 6.9) 0 0 72 22 650% decrease in pancreatic insulin secretion ratesevere hypo- normo- hyper- severehypoglycemia glycemia glycemia glycemia hyperglycemiaWithout control(Figure 6.10) 0 0 40 12 48GS-PI controller(Figure 6.11) 0 0 94 6 0MPC(Figure 6.12) 0 0 100 0 0the disease, especially within those organs which play important roles in the reg-ulation of the blood glucose. The variation in the hepatic glucose uptake and pro-124duction rate, peripheral glucose uptake rate, and the pancreatic insulin secretionrate were considered, and the performance of the gain-scheduling PI controllerbased on fuzzy logic and the model predictive controller were presented in sec-tion 6.2 for 50% decrease in each of these metabolic rates. Table 6.1 representsthe percentage of the simulation time spent in each glycemic zone after the inges-tion of 100 g glucose (from 100 minutes to 600 minutes) as the body is trying toregulate the blood glucose level without and with the designed controllers.As the glucose uptake and production rates decrease in the liver, the patientexperiences severe hyperglycemia for 160 minutes (32% of the simulation time) ascan be seen in the first section of table 6.1. The designed controllers are successfulin calculating the proper insulin infusion rate to prevent severe hyperglycemia (seefigures 6.4 and 6.6). In the presence of both controllers, it takes about 2 hoursfor the blood glucose level to return to normal glycemic zone from the yellowsafety margin which is not ideal but still acceptable since the blood glucose isbelow 14 mmol/L. The model predictive controller is capable of preventing bothhypoglycemia and severe hypoglycemia and returning the blood glucose to thelower bound of normoglycemia zone. The performance of the fuzzy-based PIcontroller is also satisfactory for handling hypoglycemia except for a negligibleperiod (6% of the simulation time i.e. 30 minutes) that the blood glucose goes tothe yellow zone of hypoglycemia.In comparison with the decrease of hepatic glucose uptake and productionrate, the reduced peripheral glucose uptake rate causes higher blood glucose con-centration (19 mmol/L) which stays a bit longer in severe hyperglycemia zone(36% of the simulation time) before returning to normoglycemia zone as seen inthe second section of table 6.1. Although the designed controllers are able to re-duce the time spent in the severe hyperglycemia zone from 36% to 18% for GS-PIcontroller and to 6% for MPC, but cannot avoid it completely. Therefore, the oralagents such as metformin can be helpful for the patients with insulin resistance intheir cells to stimulate the peripheral tissues for more absorption of glucose.Figure 6.13 demonstrates the body response to the treatment including both125Figure 6.13: The performance of model predictive controller after 100 grof glucose ingestion as a meal disturbance for 50% decrease in pe-ripheral glucose uptake rate with the administration of 1000 mg met-forminmetformin administration and insulin infusion using the model predictive con-troller for the reduced peripheral glucose uptake rate. Metformin increases theglucose uptake in the peripheral tissues and decreases the blood glucose concen-tration. Therefore the blood glucose level return to normoglycemia zone from hy-perglycemia zone after almost 2 hours without exceeding to severe hyperglycemiazone except for a negligible period.Regarding the problem of low blood glucose level, MPC acts better in avoid-ing both hypoglycemia and severe hypoglycemia, while fuzzy-based PI controllercauses a small undershoot to hypoglycemia zone for 50 minutes (10% of the sim-126ulation time).Finally, the performance of the designed controllers was assessed for 50% de-crease in pancreatic insulin secretion rate as the results have been summarized inthe third section of table 6.1. Insulin therapy can keep the blood glucose in normalrange since it provides the body with the proper amount of insulin to compensatethe lack of pancreatic insulin secretion. It can be seen from table 6.1 that MPCcan ideally keep the blood glucose in normal range all the time. Fuzzy-based PIcontroller also has an acceptable performance with a small overshoot to yellowzone of hyperglycemia for 30 minutes (6% of the simulation time).6.4 ConclusionIt can be concluded that the problem of regulating the blood glucose in type IIdiabetic patients is more complicated than type I. The controller which is designedfor type I diabetic patients should be able to mimic the performance of pancreasin secreting insulin. However, there are other complications involved with type IIdiabetes such as the insulin resistance in the body cells and liver failure to full-fill its role in regulating the blood glucose. Providing insulin through the insulinpump or multiple injections provide intensive care for the patient with type IIdiabetes and prevent further complications. Better control can be achieved withthe treatments combining both oral agents and insulin therapy.127Chapter 7Conclusion and Future Work7.1 Research Summary and ConclusionA person who is diagnosed with type II diabetes mellitus in initial steps can man-age the disease with modification of diet and improvement of the lifestyle by doingappropriate exercise. They should also regularly monitor their blood glucose leveland as the disease becomes serious the patient requires oral anti-diabetic drugs.Metformin is a very common oral agent for diabetic patients. Chapter 3 rep-resents the pharmacokinetic-pharmacodynamic model of metformin which wasincorporated into the available physiological model for type II diabetes mellituswhich was described in Chapter 2. The simulation results were shown for dif-ferent dosages of metformin which demonstrates that in severe type II diabeticpatients mono-therapy with metformin does not assure the satisfactory glycemicregulation and the patients eventually need insulin therapy.Deciding what type of insulin might be best for a patient will depend on manyfactors, such as the body’s individualized response to insulin and the lifestylechoices such as caloric intake and exercise level. Multiple daily injections (MDI)therapy and continuous subcutaneous insulin infusion (CSII) with external insulinpumps are the available techniques for insulin delivery to the body.The pharmacokinetic-pharmacodynamic models for four different insulin types128were presented in Chapter 3 including short, intermediate and long-acting in-sulins. In multiple daily injections therapy, a mixture of short-acting insulin alongwith intermediate or long-acting insulin are used to regulate the blood glucose.The body response to several regimens were depicted in Chapter 3 for compari-son and the most efficient one has been introduced to an individual patient.The monitoring of blood glucose for diabetic patients should be done morefrequently if not continually. The current technological advance in diabetic carehas made feedback blood glucose control possible. Portable continuous infusionpumps for insulin have been commonly used by Type I patients and can be usedsafely by type II patients.Many past studies proposed conventional or modified PID controllers for reg-ulation of blood glucose in type I diabetic patients but the homeostasis of bloodglucose in type II diabetes is an inherently non-linear system. It was shown inChapter 4 that the application of logarithm of the blood glucose reduces the non-linearity of glucose-insulin interactions. A fuzzy inference system has been de-signed as explained in Chapter 4 to generate scheduling gains for PI controllerso that the controller demonstrates an improved performance by having more ag-gressive control for larger deviations from the normal range and less aggressivecontrol for smaller deviations from the normal range. It is also able to handlethe asymmetric control objective in which more tolerance is allowed for hyper-glycemia and less tolerance for hypoglycemia. Simulations results demonstratethe potential benefit of using a gain scheduling PI controller with the logarithm ofblood glucose as the controlled variable over a conventional linear PI controllerfor the control of blood glucose for type II diabetic patients.One of the most common algorithms, relied on dynamic models of the pro-cess, is model predictive control (MPC). During the past few years, MPC hasbeen typically applied in biomedical process control. In case that, enough amountof data is available for a patient, model predictive control strategy was consideredin Chapter 5 to design a controller to regulate the blood glucose. The nonlinearand complex nature of glucose homeostasis in type II diabetes made it difficult to129capture glucose-insulin interactions without using multiple linear models for dif-ferent blood glucose levels. Parametric programming not only reduced the com-putational efforts of solving an on-line optimization problem but also provideda lookup-table with all optimal solutions of the feasible space of states. The in-corporation of soft constraints helped the development of a controller to closelymimic the glucose regulatory system of a healthy individual. The final controllerwas able to handle the meal disturbance in a timely manner without exceeding theupper range of normoglycemia zone. The blood glucose concentration is broughtback closer to the lower bound during the postabsorptive state and kept as lowas desired within a fasting state while the application of asymmetric membershipfunctions prevents the risk of the severe hypoglycemic state.7.2 Research LimitationsDevelopment of models which precisely represent the glucose-insulin dynamicsin the body is the key factor in finding the efficient treatments and control strate-gies for diabetic patients. The parameters of a physiological model should beupdated for each individual based on the available measurements. The access tothe sufficient data for each individual patient is needed for building a meaningfuland successful model of the glucose metabolism. The available data should beof high quality, free of errors, omissions and conflicts, and adequate in quantity.However, the estimation of the parameters is limited as only a few blood glucoseand insulin measurements per day are available in a non-clinical setting.Another limitation of this research is that the expanded mathematical modelof type II diabetes is limited to study the effects of the administration of met-formin on the patients. Metformin is the most common anti-diabetic drug in thebiguanides class which works by keeping the liver from making glucose and al-lowing more glucose to enter cells. There are also other types of drugs which workdifferently to reduce the diabetic abnormalities. Some of them like biguanides andthiazolidinediones help the body cell to absorb and use glucose. Some others suchas sulfonylureas, meglitinides and dipeptidyl peptidase 4 inhibitor help the pan-130creas to produce and release more insulin. Alpha-glucosidase inhibitors is anothergroup of anti-diabetic medication which keeps the intestines from quickly absorb-ing glucose. Biguanides also keep the liver from producing glucose.7.3 Recommendations for Future WorkA potential research area in the study of type II diabetes mellitus is developing theefficient methods of estimating the model parameters of the blood glucose whenthe glucose and insulin concentrations are only available at irregular intervals.Such studies help to overcome the limitations that are caused by the lack of accessto the sufficient data.I have contributed to a study which applies on-line sequential Monte Carlo(SMC) to estimate the states and parameters of the state-space model for typeII diabetic patients under various levels of randomly missing clinical data. Theresults of this study have been published in [124].Another research area which can be taken into more consideration is the studyof different groups of oral drugs which help diabetic patients to deal with variousabnormalities and complications of the disease.The Pharmacokinetic-Pharmacodynamic (PK-PD) model of metformin, whichis the most common oral agent for type II diabetic patients has been incorporatedinto the patient mathematical model. The similar approach can be applied to de-velop the PK-PD model of other anti-diabetic agents to investigate the effects ofeach medicine on the body organs. 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Title | Blood glucose regulation in type II diabetic patients |
Creator |
Ekram, Fatemeh |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Type II diabetes is the most pervasive diabetic disorder, characterized by insulin resistance, β-cell failure in secreting insulin and impaired regulatory effects of the liver on glucose concentration. Although in the initial steps of the disease, it can be controlled by lifestyle management, but most of the patients eventually require oral diabetic drugs and insulin therapy. The target for the blood glucose regulation is a certain range rather than a single value and even in this range, it is more desirable to keep the blood glucose close to the lower bound. Due to ethical issues and physiological restrictions, the number of experiments that can be performed on a real subject is limited. Mathematical modeling of glucose metabolism in the diabetic patient is a safe alternative to provide sufficient and reliable information on the medical status of the patient. In this thesis, dynamic model of type II diabetes has been expanded by incorporation of the pharmacokinetic-pharmacodynamic model of different types of insulin and oral drug to study the impact of several treatment regimens. The most efficient treatment has been then selected amongst all possible multiple daily injection regimens according to the patient's individualized response. In this thesis, the feedback control strategy is applied in this thesis to determine the proper insulin dosage continuously infused through insulin pump to regulate the blood glucose level. The logarithm of blood glucose concentration has been used as the controlled variable to reduce the nonlinearity of the glucose-insulin interactions. Also, the proportional-integral controller has been modified by scheduling gains calculated by a fuzzy inference system. Model predictive control strategy has been proposed in this research for the time that sufficient measurements of the blood glucose are available. Multiple linear models have been considered to address the nonlinearity of glucose homeostasis. On the other hand, the optimization objective function has been adjusted to better fulfill the objectives of the blood glucose regulation by considering asymmetric cost function and soft constraints. The optimization problem has been solved by the application of multi-parametric quadratic programming approach which reduces the on-line optimization problem to off-line function evaluation. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-03-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0225872 |
URI | http://hdl.handle.net/2429/57070 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2016-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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