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Assessment of type II diabetes mellitus Barazandegan, Melissa 2016

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Assessment of Type II DiabetesMellitusbyMelissa BarazandeganB.Sc., Sharif University of Technology, 2007M.Sc., Sharif University of Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Chemical and Biological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2016c© Melissa Barazandegan 2016AbstractSeveral methods have been proposed to evaluate a person’s insulin sensitivity (ISI). How-ever, all are neither easy nor inexpensive to implement. Therefore, the purpose of thisresearch is to develop a new ISI that can be easily and accurately obtained by patientsthemselves without costly, time consuming and inconvenient testing methods. In this the-sis, the proposed testing method has been simulated on the computerized model of the typeII diabetic-patients to estimate the ISI. The proposed new ISI correlates well with the ISIcalled M-value obtained from the gold standard but elaborate euglycemic hyperinsulinemicclamp (r = 0.927, p = 0.0045).In this research, using a stochastic nonlinear state-space model, the insulin-glucose dy-namics in type II diabetes mellitus is modeled. If only a few blood glucose and insulinmeasurements per day are available in a non-clinical setting, estimating the parametersof such a model is difficult. Therefore, when the glucose and insulin concentrations areonly available at irregular intervals, developing a predictive model of the blood glucose ofa person with type II diabetes mellitus is important. To overcome these difficulties, undervarious levels of randomly missing clinical data, we resort to online Sequential Monte Carloestimation of states and parameters of the state-space model for type II diabetic patients.This method is efficient in monitoring and estimating the dynamics of the peripheral glu-cose, insulin and incretins concentration when 10%, 25% and 50% of the simulated clinicaldata were randomly removed.iiAbstractVariabilities such as insulin sensitivity, carbohydrates intake, exercise, and more makecontrolling blood glucose level a complex problem. In patients with advanced TIIDM,the control of blood glucose level may fail even under insulin pump therapy. Therefore,building a reliable model-based fault detection (FD) system to detect failures in control-ling blood glucose level is critical. In this thesis, we propose utilizing a validated robustmodel-based FD technique for detecting faults in the insulin infusion system and detectingpatients organ dysfunction. Our results show that the proposed technique is capable ofdetecting disconnection in insulin infusion systems and detecting peripheral and hepaticinsulin resistance.iiiPrefaceThis thesis entitled “Assessment of type II diabetes mellitus” presents my research duringmy PhD studies at Chemical and Biological Engineering Department of the University ofBritish Columbia. I led and performed my PhD research under the supervision of ProfessorK. E. Kwok and Professor R. B. Gopaluni. This thesis includes six Chapters. Contributionsand collaborations to the published papers or submitted papers for publication are conciselyexplained in the following:• A version of chapter 3 has been presented at the 37th Canadian Medical and Bio-logical Engineering Society (CMBES 37) and won the 3rd place prize presentationaward. A version of this chapter has been published first online. Melissa Barazan-degan, Fatemeh Ekram, Ezra Kwok, Bhushan Gopaluni, “Simple Self-AdministeredMethod for Assessing Insulin Sensitivity in Diabetic Patients”, Journal of Medicaland Biological Engineering, pp 1-9, First online: 08 April 2016 [1]. This paper hasbeen prepared with close collaboration of Professor Kwok and Professor Gopaluni.They also have helped in revision of the final draft. Dr. Ekram helped in preparationof the initial drafts of the paper.• A version of chapter 4 has been published. Melissa Barazandegan, Fatemeh Ekram,Ezra Kwok, Bhushan Gopaluni, Aditya Tulsyan, “Assessment of type II diabetes mel-litus using irregularly sampled measurements with missing data”, Bioprocess BiosystEng, Volume 38, Issue 4, pp 615-629, 2015 [2]. This paper has been published withivPrefaceclose collaboration of Professor Kwok and Professor Gopaluni. They also have helpedin revision of the final drafts. Dr. Ekram helped in preparation of the first drafts ofthe paper and Mr. Tulsyan helped in preparing SIR particle filtering model undermissing data.• A version of chapter 5, Melissa Barazandegan, Ezra Kwok, Bhushan Gopaluni, “Model-Based Detection of Organ Dysfunction and Faults in Insulin Infusion Devices forType 2 Diabetic Patients” has been presented and published as a proceeding paperin American Control Conference 2016 (ACC2016) [? ]. This paper has been preparedwith close collaboration of Professor Kwok and Professor Gopaluni. They also havehelped in revision of the final drafts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Glucose homeostasis . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Type II diabetes mellitus and the related metabolic abnormalities . 41.1.3 Evaluation of the health status of diabetic patients . . . . . . . . . 61.1.4 Qualitative and quantitative evaluation of abnormal metabolism be-haviour in diabetic patients . . . . . . . . . . . . . . . . . . . . . . 7viTable of Contents1.2 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Mathematical modeling of type II diabetes mellitus . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The Sorensen model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Glucose sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Incretins sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 Insulin sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.4 Glucagon sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Type II diabetes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Selection of model parameters for estimation . . . . . . . . . . . . . 312.3.2 Nonlinear optimization problem . . . . . . . . . . . . . . . . . . . . 333 A novel and simple self-administered method for assessing insulin sensi-tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1 Hyperinsulinemic euglycemic insulin clamp technique . . . . . . . . 363.1.2 Modified minimal model (MINMOD) analysis in conjunction withthe frequently sampled intravenous glucose tolerance test (FSIVGTT)363.1.3 Homeostasis model assessment (HOMA) of insulin resistance (HOMA-IR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.4 Insulin sensitivity indices investigated from oral glucose tolerancetest (OGTT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Clinical data used for model development . . . . . . . . . . . . . . . . . . . 39viiTable of Contents3.3 Proposed self-assessment method for estimation of insulin sensitivity . . . . 423.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 Parameters estimation results . . . . . . . . . . . . . . . . . . . . . 443.4.2 Quantitative estimation of insulin sensitivity . . . . . . . . . . . . . 613.4.3 Comparison of various insulin sensitivity indices obtained from OGTT704 Assessment of type II diabetes mellitus using irregularly sampled mea-surements with missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Mathematical model preparation . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Response models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3.1 States and parameters estimation of the response model . . . . . . 794.3.2 Recursive bayesian estimation . . . . . . . . . . . . . . . . . . . . . 824.3.3 Sequential monte carlo (SMC) . . . . . . . . . . . . . . . . . . . . . 834.4 Online state and parameter estimation in nonlinear state-space models . . 844.4.1 Complete clinical data . . . . . . . . . . . . . . . . . . . . . . . . . 854.4.2 Irregular clinical data . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5.1 Clinical data used for model development . . . . . . . . . . . . . . . 884.5.2 On-line states and parameters estimation results . . . . . . . . . . . 904.5.3 Application of SIR particle filtering in detection of organ dysfunctionin diabetic patients under irregular clinical data . . . . . . . . . . . 1024.5.4 strengths and limitations of the SIR particle filtering in clinical prac-tice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104viiiTable of Contents5 Model-based detection of organ dysfunction and faults in insulin infusiondevices for type II diabetic patients . . . . . . . . . . . . . . . . . . . . . . 1065.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.1.1 Theoretical background on fault detection approaches . . . . . . . . 1095.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3 Fault detection of glucose-insulin system . . . . . . . . . . . . . . . . . . . 1135.3.1 Detection of insulin pump disconnection (Case 1) . . . . . . . . . . 1155.3.2 Detection of kinked insulin pump tubing (Case 2) . . . . . . . . . . 1165.3.3 Detection of organ dysfunction . . . . . . . . . . . . . . . . . . . . . 1186 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . 1226.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.1.1 A simple self-administered method for assessing insulin sensitivity intype II diabetic patients . . . . . . . . . . . . . . . . . . . . . . . . 1226.1.2 Assessment of type II diabetes mellitus using irregularly sampledmeasurements with missing data . . . . . . . . . . . . . . . . . . . . 1236.1.3 Model-based detection of organ dysfunction and faults in insulin in-fusion devices for type 2 diabetic patients . . . . . . . . . . . . . . . 1236.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . 124Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142ixList of Tables1.1 Comparison of type I and II diabetes [8] . . . . . . . . . . . . . . . . . . . . 22.1 The model parameters [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Abnormalities associated with type II diabetes and their corresponding equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Diagnosis of diabetes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Mean plasma glucose and insulin levels during OGTT . . . . . . . . . . . . 413.3 Parameter estimation results for glucose sub-model (subject 1). . . . . . . . 453.4 Parameter estimation results for insulin sub-model (subject 1). . . . . . . . 453.5 Pearson correlations with M-Value and results of correlation comparisons [1] 724.1 Parameter estimation results for insulin sub-model after 600 sampling timeduring IIVGIT test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Variation of the parameters cGHGP , cGHGU , dI∞HGP and K12 in glucose sub-model after 2400 sampling time during OGTT test . . . . . . . . . . . . . . 96A.1 Gender and body weight of diabetic subjects [7] . . . . . . . . . . . . . . . . 142A.2 Normalized GLP-1 concentration data set (pmol/l) of diabetic subjects forOGTT [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142xList of TablesA.3 Normalized GIP concentration data set (pmol/l) of diabetic subjects forOGTT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.4 Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for OGTT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.5 Normalized peripheral insulin concentration data set (pmol/l) of diabeticsubjects for OGTT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.6 Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for IIVGIT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 146A.7 Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for IIVGIT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 147A.8 Normalized peripheral insulin concentration data set (pmol/l) of diabeticsubjects for IIVGIT test [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.9 : Intravenous glucose infusion amount (g) to diabetic subjects during IIVGITtest [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149xiList of Figures1.1 Glucose homeostasis control mechanism in the body [7] . . . . . . . . . . . 32.1 Simplified blood circulatory system [7] . . . . . . . . . . . . . . . . . . . . . 152.2 Shematic diagram of insulin submodel [51] . . . . . . . . . . . . . . . . . . . 162.3 Shematic diagram of glucose submodel [7] . . . . . . . . . . . . . . . . . . . 172.4 General representation of a compartment[7] . . . . . . . . . . . . . . . . . . 182.5 Simplified configurations of physiological compartments[7] . . . . . . . . . . 192.6 Biphasic response of a healthy pancreas to a glucose concentration stepchange [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Schematic diagram of Landahl and Grodskys model [7] . . . . . . . . . . . . 273.1 Plasma glucose and insulin concentration profile in subject #1, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Plasma glucose and insulin concentration profile in subject #2, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Plasma glucose and insulin concentration profile in subject #3, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Plasma glucose and insulin concentration profile in subject #4, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 49xiiList of Figures3.5 Plasma glucose and insulin concentration profile in subject #5, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Plasma glucose and insulin concentration profile in subject #6, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 513.7 Plasma glucose and insulin concentration profile in subject #7, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 523.8 Plasma glucose and insulin concentration profile in subject #8, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 533.9 Plasma glucose and insulin concentration profile in subject #9, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 543.10 Plasma glucose and insulin concentration profile in subject #10, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 553.11 Plasma glucose and insulin concentration profile in subject #11, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 563.12 Plasma glucose and insulin concentration profile in subject #12, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 573.13 Plasma glucose and insulin concentration profile in subject #13, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 583.14 Plasma glucose and insulin concentration profile in subject #14, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 593.15 Plasma glucose and insulin concentration profile in subject #15, the clinicaldata (•), the model results (−) . . . . . . . . . . . . . . . . . . . . . . . . . 603.16 Effect of insulin injection in subject #1, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 61xiiiList of Figures3.17 Effect of insulin injection in subject #2, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 623.18 Effect of insulin injection in subject #3, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 623.19 Effect of insulin injection in subject #4, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 633.20 Effect of insulin injection in subject #5, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 633.21 Effect of insulin injection in subject #6, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 643.22 Effect of insulin injection in subject #7, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 643.23 Effect of insulin injection in subject #8, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 653.24 Effect of insulin injection in subject #9, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . . . 653.25 Effect of insulin injection in subject #10, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 663.26 Effect of insulin injection in subject #11, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 663.27 Effect of insulin injection in subject #12, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 673.28 Effect of insulin injection in subject #13, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 67xivList of Figures3.29 Effect of insulin injection in subject #14, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 683.30 Effect of insulin injection in subject #15, two 10 mU/kg insulin injectionsat 20 and 50 min respectively (-), and no injection (–) . . . . . . . . . . . . 683.31 Correlation between the ISI and the M-value for the fifteen subjects; r =0.927, p = 0.0045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1 Variations of the parameters N1, N2, ξ1 and ξ2 in the Pancreatic insulinrelease model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Variations of the parameters α, γ, K in the Pancreatic insulin release model 934.3 Peripheral insulin concentration for type II diabetic subjects during theIIVGIT test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4 Variations of the parameters cIPGU , cI∞HGU , dIPGU and dI∞HGU in glucose sub-model after 2400 sampling time during OGTT test . . . . . . . . . . . . . . 974.5 Parameter estimation results for glucose sub-model after 2400 sampling timeduring OGTT test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.6 Peripheral glucose concentration for type II diabetic subjects during theOGTT test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.7 Peripheral insulin concentration for type II diabetic subjects during theOGTT test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.8 Incretins concentration for type II diabetic subjects during the OGTT test 1014.9 The probability density function of cIPGU from Bayesian identification atsampling time number 1, 600, 607 and 620 (area under each curve is unitary).The y-axis quantity is unit-less. . . . . . . . . . . . . . . . . . . . . . . . . . 1024.10 Variation of different glucose metabolic rates . . . . . . . . . . . . . . . . . 104xvList of Figures5.1 Model-based fault diagnosis Scheme [97, 113, 114] . . . . . . . . . . . . . . . 1105.2 Response of the PI controller with a 75 gr meal disturbance at time = 500min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3 Detection of insulin pump disconnection. The blue solid line curve representsno fault and red dashed curve represents fault in Case 1 . . . . . . . . . . . 1155.4 Detection of kinked insulin pump tubing. The blue solid line curve representsno fault and red dashed curve represents fault in Case 2 . . . . . . . . . . . 1175.5 Detection of peripheral insulin resistance. The blue solid line curve repre-sents no fault and red dashed curve represents fault in Case 3 . . . . . . . . 1195.6 Detection of hepatic insulin resistance. The blue solid line curve representsno fault and red dashed curve represents fault in Case 4 . . . . . . . . . . . 120xviNomenclatureModel variables in the glucose sub-modelD Oral glucose amount (mg)G Glucose concentration (mg/dl)M Multiplier of metabolic rates (dimensionless)Q Vascular blood flow rate (dl/min)q Glucose amount in GI tract (mg)r Metabolic production or consumption rate (mg/min)Ra Rate of glucose appearance in the blood stream(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)xviiNomenclatureM Multiplier of metabolic rates (dimensionless)m Labile insulin mass (U)P 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)xviiiNomenclatureModel variables in the incretins sub-modelΨ Incretins concentration (pmol/l)r Metabolic production or consumption rate (pmol/min)t time (min)V Volume (dl)First superscriptΓ GlucagonB Basal conditionG GlucoseI InsulinM ncretinsSecond superscript∞ Final steady state valueMetabolic rate subscriptsBGU Brain glucose uptakeGGU Gut glucose uptakeHGP Hepatic glucose productionxixNomenclatureHGU Hepatic glucose uptakeIΨR Intestinal incretins releaseKGE Kidney glucose excretionKIC Kidney insulin clearanceLIC Liver insulin clearanceMΓC Metabolic glucagon clearancePΓC Plasma glucagon clearancePΓR Pancreatic glucagon releasePΨC Plasma incretins clearancePGU Peripheral glucose uptakePIC Peripheral insulin clearancePIR Pancreatic insulin releaseRBCU Red blood cell glucose uptakeFirst superscripts∞ Final steady state valueA Hepatic arteryB BrainG GutxxNomenclatureH Heart and lungsL LiverP PeripheryS StomachSecond subscripts (if required)C Capillary spaceF Interstitial fluid spacel Liquids SolidxxiAcknowledgementsAll praise is due to God Almighty, the most gracious and the most merciful, who pro-tects, preserves and supports us during day and night. Thank you, God, for your insight,guidance, companionship, integrity, and for all the sacrifices you made.I would like to express my sincere thanks and appreciation to my doctoral supervisors,Professor Ezra Kwok and Professor Bhushan Gopaluni from Chemical and Biological En-gineering Department at the University of British Columbia. You have been a tremendousmentor for me. I would like to thank you for encouraging my research and for allowing meto grow as a research scientist. Your advice on both research as well as on my career havebeen priceless.I would like to thank my control group members who have been helpful and supportivefriends during my Ph.D. appointment: Omid Vahidi, Fatemeh Ekram, and Aditya Tulsyan.I would like to thank my parents, Mehdi and Mojgan, and especially my sister, Sogol,for their unconditional love, encouragement and support during my research.Finally I would like to express my gratitude to my husband, Mohammadali, and myson, Aubtin, for being so supportive throughout my graduate program.xxiiDedicationThis thesis is dedicated to the loveliest creature on earth, my mother, who sacrificed herselffor me, to my father for all his support, to my sweet sister, to my lovely husband and tomy beloved son.xxiiiChapter 1Introduction1.1 BackgroundDiabetes Mellitus is one of the leading diseases in the developed world. According to theInternational Diabetes Federation, the prevalence of diabetes is growing rapidly in theworld. Diabetes mellitus occurs when the blood glucose levels are not regulated due to theimpaired insulin secretion, action or both. Insulin is a key hormone secreted from β-cellsin the pancreas that regulates glucose homeostasis [3–6]. Diabetes mellitus is generallycategorized into three groups [7]:• Type I diabetes or insulin dependent diabetes mellitus (IDDM), in which the bodyis unable to produce insulin due to the autoimmune destruction of the beta cells inthe pancreas. Therefore, the body becomes insulin dependent and daily insulin dosesmust be supplied to the type I diabetic patients to survive. Most often, it occurs inchildhood, but the disease can also develop in adults in their late 30s and early 40s.• Type II diabetes or non-insulin dependent diabetes mellitus (NIDDM), in which thepancreas does not produce enough insulin or the human body cells become resistanceagainst insulin [3]. Type II diabetic patients require gradual treatment and are notin emergency need of medical attention.• Gestational diabetes, which develops during pregnancy (gestation). It occurs in preg-11.1. Backgroundnant women who have never had diabetes before. Gestational diabetes causes highblood glucose level that can affect woman’s pregnancy and the baby’s health.A comparison of type I and type II diabetes is presented in table 1.1 [8].Table 1.1: Comparison of type I and II diabetes [8]Feature Type I diabetes Type II diabetesOnset Sudden GradualAge at onset Any age (Mostly in children) Mostly in adultsBody habitus Thin or normal Often obeseKetoacidosis Common RareAutoantibodies Usually present AbsentEndogenous insulin Low or absent Normal, decreased or increasedConcordance in identical twins %50 %90Prevalence ∼ 10% of diabetic population ∼ 90% of diabetic population1.1.1 Glucose homeostasisWhen the glucose, a simple sugar, is produced from digestion of carbohydrates in thegastrointestinal tract, it is absorbed by the body cells to provide the primary energy source.The blood glucose concentration is maintained at a constant level during fast by producingendogenous glucose through two main metabolic pathways [7]:• Gluconeogenesis, in which glucose is generated from non-carbohydrate carbon sub-strates such as lactate, glycerol, and glucogenic amino acids. In this metabolic path-way, the endogenous glucose is produced by the liver and kidney and is released intothe blood stream.• Glycogenolysis, in which glucose is generated from breakdown of glycogen. In thismetabolic pathway, the endogenous glucose is produced by the liver and muscles.The endogenous glucose produced by the liver is released into the blood stream whilethe produced glucose in muscle cells is consumed by themselves.21.1. BackgroundFigure 1.1: Glucose homeostasis control mechanism in the body [7]Approximately 15% of endogenous glucose production released into the blood streamis derived from the kidney, and the remaining 85% is produced by the liver. [9].In normal (non-diabetic) subjects, the blood glucose level is controlled within an ap-proximate range of 60-150 mg/dl, despite disturbances such as exercise or intake of a mealcontaining carbohydrates [10]. The blood glucose level is regulated through feedback sys-tems reacting mainly on glucose, insulin and glucagon concentrations. Insulin and glucagonare two hormones in the body secreted from the β and α cells of the pancreas, respectively.These hormones play an important role in glucose homeostasis in the body, however, theeffects of glucagon are opposite to those of insulin (see Figure 1.1). Insulin contributes inlowering the blood sugar level by stimulating some body cells to absorb glucose, suppressingendogenous glucose production and inhibiting glucagon secretion. When the blood sugar31.1. Backgroundlevel is high, insulin is secreted from β − cells of pancreas to:• stimulate the body cells to absorb glucose• suppress endogenous glucose production• inhibit glucagon secretion from α− cells of pancreasConversely, when the blood glucose concentration is low, glucagon is secreted from α−cellsof pancreas to:• stimulate the liver to produce more glucose• inhibit insulin secretion from β − cells of pancreas1.1.2 Type II diabetes mellitus and the related metabolic abnormalitiesType II diabetes occurs when the pancreas does not produce enough insulin or the humanbody cells become resistance against insulin [3]. Type II diabetes is characterized bymultiple abnormalities in a number of body organs such as the liver, the pancreas, musclesand adipose tissues. These abnormalities are classified as follows:• Insulin resistance in peripheral tissues: Peripheral tissues (i.e. muscle and adi-pose tissue cells) absorb blood glucose by sensing insulin hormone. Insulin resistancehappens when the sensitivity of peripheral cells to the metabolic action of insulin isdecreased due to genetic factors, environmental factors, obesity, hypertension, dys-lipidemias, and/or coronary artery diseases [11]. Impairment of the following factorsis known to be associated with insulin residence in peripheral tissues [9]:– The number of insulin receptors– The affinity of insulin receptors41.1. Background– Insulin intracellular signalling– The number of glucose transporters– Glucose transporter translocation on the cell membrane– Insulin stimulatory effects on glycogenesis– Insulin stimulatory effects on glycolysis• Reduced hepatic glucose uptake: It is believed that reduce in hepatic glucoseuptake rate is due to the impairment of insulin stimulation effect on glucose phos-phorylation in the liver [12].• Impaired hepatic glucose production: Many studies have confirmed that typeII diabetic patients have impaired hepatic glucose production rate and low insulin-induced suppression of endogenous glucose production [13–17]. The impaired effectof insulin suppression on both pathways of endogenous glucose production (i.e. glu-coneogenesis and glycogenolysis) have been demonstrated by Basu et al. [16, 17].• Impaired pancreatic insulin secretion: Deficiency and failure in the pancreaticinsulin production shows the development of overt diabetes [9]. Pancreatic insulinsecretion in response to a glucose stimulus has a biphasic pattern. Early peak ofinsulin production and the overall insulin secretion rate are the two forms of defectivepancreatic insulin secretion in type II diabetic patients[18–20].• Glucose resistance: When the levels of glucose concentration is less than 130mg/dl, glucose-induced stimulation of glucose disposal is normal in type II diabeticpatients [21]. However, high levels of glucose concentration (particularly above 130mg/dl) impair the glucose stimulation effect on glucose uptake rate in type II diabeticpatients [22, 23].51.1. Background1.1.3 Evaluation of the health status of diabetic patientsThere are different clinical tests used for assessing the glucose metabolism in different bodyorgans to evaluate the health status of diabetic patients. A brief explanation of some ofthese clinical tests is as follows [7]:• Oral glucose tolerance test (OGTT): This test is usually used to diagnose dia-betes, insulin resistance, impaired beta cell function, and sometimes reactive hypo-glycemia and acromegaly, or rarer disorders of carbohydrate metabolism. First, thefasting plasma glucose is tested. Then, to determine how body is able to clear glu-cose from the blood, a glass of dissolved glucose in water is given to the patient andblood samples are taken afterwards up to four times to measure the blood glucose.Depending on different standards, the dose of glucose may vary from 50 gr to 100 gr.• Euglycemic hyperinsulinemic clamp (EHIC): In this test, the plasma glucoseconcentration is held constant at basal levels by intravenous glucose infusion. Mean-while, the plasma insulin concentration is raised and maintained at 100 µU/ml by acontinuous infusion of insulin. When the steady-state is achieved, the glucose infusionrate equals glucose uptake by all the tissues in the body. Therefore, for more sensitiveinsulin tissues, more glucose infusion is needed. The hyperinsulinemic clamps is ameasure of insulin resistance.• Hyperglycemic clamp (HGC): In this test, the plasma glucose concentration israised to 125 mg/dl above basal levels by intravenous glucose infusion and no insulininjection. Since the plasma glucose concentration is held constant, at steady states,the rate of glucose infusion is an index of insulin secretion and glucose metabolism.The hyperglycemic clamps are often used to assess insulin secretion capacity.61.1. Background• Intravenous glucose tolerance test (IVGTT): This test is similar to the OGTTtest. However, instead of oral glucose consumption, glucose is infused intravenouslyinto the patient’s body. Then, variation of glucose concentration is measured frompatient’s blood samples. Measurements of glucose concentration show how the bodyclears glucose from the body.• Insulin suppression test: Similar to the EHIC test, this test is used to measureinsulin sensitivity. In this test, somatostatin is injected to suppress endogenous se-cretion of glucose and insulin while a constant rate of glucose and insulin is infusedintravenously. Blood samples are taken from the subject in specific times during thetest. At steady-states, the plasma insulin concentration is the same in all subjects,and the value of the plasma glucose concentration provides a direct estimate of in-sulin resistance. Body with high level of insulin resistance, has the higher value ofthe plasma glucose concentration at steady-state.1.1.4 Qualitative and quantitative evaluation of abnormal metabolismbehaviour in diabetic patientsAs described in previous section, there are different clinical test that their results used toevaluate abnormal behaviour of body organs in diabetic patients. To quantify the medicalcondition of healthy and diabetic patients, many studies proposed different indices in theliterature such as [7]:• Insulin Sensitivity Index (ISI), which quantifies the ability of insulin to stimulatebody glucose disposal.• Glucose Effectiveness Index (GEI), which measures the ability of glucose per se tomediate its rate of disappearance and to inhibit hepatic glucose production.71.2. Thesis objectivesMany different definitions for these two indices are reported in the literature. Directmeasurements of insulin sensitivity are proposed via the following test in the literature:• Euglycemic hyperinsulinemic clamp test [24, 25]• Insulin Sensitivity Tolerance (IST) test [26–28]In addition, indirect measurement of insulin sensitivity is proposed in the literature fromfrequently sampled intravenous glucose tolerance test (FSIVGTT) [29–35]. In this method,from the results of FSIVGTT test the parameters of the minimal model are determined[36]. Later, the obtained parameters are used to define insulin sensitivity and glucoseeffectiveness indices. Also, different surrogate indices for insulin sensitivity have been alsodefined in the literature using fasting insulin and glucose measurements as follows:• Homeostasis model assessment (HOMA) [37–39],• Quantitative insulin sensitivity check index (QUICKI) [25, 40–43]• Oral Glucose Tolerance test (OGTT): Matsuda index [44], Stumvoll index [45], Avi-gnon index [46], oral glucose insulin sensitivity index [47], Gutt index [48], andBelfiore index [49] are the insulin sensitivity indices obtained from this test by usingdifferent sampling protocols during OGTT test.1.2 Thesis objectivesThe application of dynamic mathematical modelling has increased in every aspect of ourlives. Mathematical modelling of glucose metabolism in diabetic patient is helpful in pro-viding reliable information without causing serious and irreversible harm to the subject.Most of the studies in the field of modelling of diabetes have addressed type I diabetesmellitus. However, type II diabetes is the most pervasive type which affects 90% of the81.2. Thesis objectivesdiabetes population around the world [50]. Type I diabetes mellitus is characterized bypancreas dysfunction, however, type II diabetic patients deal with multiple abnormalitiesin a number of body organs such as the liver, the pancreas, muscles and adipose tissues.Therefore, studying, modelling and simulating physiological behaviour of type II diabeticpatients is much more complicated than type I diabetic patients.In the light of aforementioned above, the goal of my Ph.D. research is to benefit oursociety in managing diabetes mellitus, which is one of the most prevalent diseases affectingat least 285 million people worldwide. The objective of my Ph.D. research mainly isfocused on employing a clinically-relevant physiological model of type II diabetes mellitusto improve the management of blood glucose level and fault detection features suitable formonitoring and control. This objective can be achieved in the following three steps:1. Type II diabetes is characterized by multiple abnormalities in a number of body or-gans. Insulin resistance is one of the abnormalities happening when the sensitivityof peripheral cells to the metabolic action of insulin is decreased. The ability ofinsulin to stimulate body glucose disposal can be characterized by an insulin sen-sitivity index (ISI). Several methods have been proposed for evaluating a person’sinsulin sensitivity from an oral glucose tolerance test (OGTT) and the euglycemicinsulin clamp technique. However, none are easy or inexpensive to implement sincethe plasma insulin concentration, as a key variable for assessing the insulin sensi-tivity index (ISI), is required to be clinically measured at specific times. Therefore,my first thesis objective is using clinically-relevant physiological model of type II di-abetes mellitus to develop a simple self-administered testing method for estimatingthe insulin sensitivity index that can be easily and accurately obtained by patientsthemselves without costly, time-consuming, and inconvenient testing methods.2. Mathematical modelling of glucose metabolism in diabetic patient provides useful in-91.2. Thesis objectivesformation to diabetic patients of dangerous metabolic conditions, enables physiciansto review past therapy, estimates future blood glucose levels, and provides therapyrecommendations.The insulin-glucose dynamics in type II diabetes mellitus can bemodelled by using a stochastic nonlinear state-space model. Estimating the parame-ters of such a model is difficult as only a few blood glucose and insulin measurementsper day are available in a non-clinical setting. Therefore, my second thesis objectiveis to develop a predictive model of the blood glucose of a person with type II diabetesmellitus when the glucose and insulin concentrations are only available at irregularintervals. The results of this study can be used to inform type II diabetic patientsof their medical conditions, enable physicians to review past therapy, estimate fu-ture blood glucose levels, provide therapeutic recommendations and even design astabilizing control system for blood glucose regulation.3. Controlling blood glucose level for patient with type II diabetes mellitus (TIIDM)has been influenced by many variables with significant levels of variability, such asinsulin sensitivity, carbohydrates intake, exercise, and more. These variabilities makecontrolling blood glucose level a complex problem. In patients with advanced typeII diabetes mellitus, when the body fails to regulate blood glucose level, an externalloop including an insulin pump and a glucose measurement device can be used inmaintaining glucose regulation. However, the control of blood glucose level may faileven in patient with insulin pump therapy. Therefore, my third thesis objective is tobuild a reliable model-based fault detection system to detect failures in controllingblood glucose level. The results of this study is helpful to detect faults in the insulininfusion system and detect patient’s organ dysfunction.101.3. Thesis outline1.3 Thesis outlineMy thesis is organized as follows:• In chapter 2, the mathematical modelling developed by Vahidi et al. [7, 51] for typeII diabetes is described. The Vahidi model results from initial work by Guyton et al.[52], which was updated by Sorensen [53]. This model is a much more detailed modelcompared to the compartmental minimal modelling (MINMOD) approach proposedby Bergman [36]. The MINMOD includes three nonlinear differential equations repre-senting variations of plasma insulin and glucose concentrations. However, the Vahidimodel consists of more compartments for better representation of the glucose andinsulin concentrations in different parts of a human body. Their application of ad-ditional compartments allows for a more accurate simulation of the physiologicaldynamics and individual abnormalities for type II diabetic patients.• In chapter 3, the feasibility of using the mathematical compartment model proposedby Vahidi et al. [7, 51] to estimate insulin sensitivity has been described. A simplemethod for conveniently estimating insulin sensitivity by patients themselves hasbeen developed and evaluated.• In chapter 4, the nonlinear states and the parameters of Vahidi model in the presenceof 10%, 25% and 50% of randomly missing clinical observations have been estimatedby implementing a Bayesian filtering method.• In chapter 5, faults in insulin infusion system and organs dysfunction are detected intype II diabetic patients using the model-based fault detection technique based on aSequential Monte Carlo (SMC) filtering method.• Finally, chapter 6 summarizes the thesis and provides recommendations on future111.3. Thesis outlineworks.12Chapter 2Mathematical modeling of type IIdiabetes mellitus2.1 IntroductionGlucose-insulin interactions in a healthy human body have been mathematically modelledin many studies. Initially, Bolie [54] and Ackerman et al. [55] proposed a simple linearmodel, and later, more complicated nonlinear models have been proposed. Among thoseapproaches, the compartmental modelling approach is the most popular one. In this ap-proach, different organs or parts of the body are represented by compartments, and themodel equations are derived from the mass balance equations over each compartment. Thecompartmental minimal model (MINMOD) of Bergman et al. [36] has been widely used inmany studies. The MINMOD includes three nonlinear differential equations representingvariations of plasma insulin and glucose concentrations. Later, more complicated compart-mental models have been proposed including more compartments for better understandingof the organ’s behaviour [53, 56, 57].The physiological behaviour of type I and type II diabetes can be developed by adjustingthe structures of healthy human models. For, instance, since the pancreas in type I diabeticpatients does not produce insulin hormone, type I diabetes mellitus model can be simplyadjusted from healthy human models by setting the insulin production rate term to zero132.2. The Sorensen model[7].However, using the similar approach for type II diabetes modelling is not as simple astype I diabetes modelling since type II diabetes is associated with multiple abnormalities indifferent body organs. In type II diabetic patients, all the body organs are still functioningand the organ’s abnormalities affects the glucose metabolic rates, the glucose regulatorysecretion rates, and the pancreatic insulin secretion rates. Therefore, type II diabetesmellitus model can be developed from the same structure of the healthy human model butwith the modified parameters. This approach has been used by Dalla Man et al. [58] andVahidi et al.. [7]. Vahidi model is based on a healthy human body model proposed bySorensen [53], which is adjusted and validated for type II diabetes using available clinicaldata of diabetic patients.In the following sections, the equations of Sorensen model are presented along with theparameters updated for type II diabetic patient by Vahidi [7, 53]. The description of themodel variables can be found in Nomenclature.2.2 The Sorensen modelSorensen modified the compartmental model of glucose-insulin interactions in a healthybody developed by Guyton et al. [52]. In this, model, the regulatory effects of insulin andglucagon hormones on glucose metabolism are considered. However, the hormonal effectsof epinephrine, cortisol, and growth hormone are assumed to be negligible. Also, thephysiology of changes in amino acid and free fatty acid substrate levels are not considered.In this model, the physiologic parameters such as blood flow rates and capillary spacevolumes are selected to represent a typical 70 kg adult male.The simplified blood circulatory system contributing significantly in glucose productionand consumption is shown in Figure 2.1. The heart left ventricle pumps the Oxygen-rich142.2. The Sorensen modelblood and deliver it to all body organs through the arteries. the body organs drain outdeoxygenated blood and deliver it to the heart right atrium through the veins.Figure 2.1: Simplified blood circulatory system [7]The Sorensen model contains three main sub-models representing blood glucose, insulinand glucagon concentrations in the body and their interactions.Each sub-model is divided into individual numbers of compartments representing spe-cific parts or organs of a human body. The number of compartments is different in eachsub-model. As can be seen in Figure. 2.2, the insulin sub-model has seven compartments:brain, liver, heart and lungs, periphery, gut, kidney, and pancreas. The blocks representdifferent compartments and the arrows indicate the blood flow directions.152.2. The Sorensen modelFigure 2.2: Shematic diagram of insulin submodel [51]The glucose sub-model is similar to the insulin sub-model except that the pancreascompartment is excluded. The glucose sub-model can be seen in Figure. 2.3.162.2. The Sorensen modelFigure 2.3: Shematic diagram of glucose submodel [7]Since the glucagon concentration is considered to be identical in all parts of the body,only one compartment is used in the glucagon sub-model [59]. Each compartment is gen-erally divided into the following three well-mixed spaces, which are shown in Figure 2.4:• Capillary blood space• Interstitial fluid space• Intracellular space172.2. The Sorensen modelFigure 2.4: General representation of a compartment[7]As can be seen from Figure 2.4, the capillary space is fed in by arterial blood inflowand drained by venous blood outflow. The blood components may diffuse through capillarywalls to the interstitial fluid and from interstitial fluid to the intracellular space and viceversa. Due to the following reasons, maximum two of these sub-compartments are phys-iologically required to be considered in modelling of solute transport from the capillaryblood space to the intracellular space (Figure 2.5):• The capillary wall may not allowing fluid to pass through a solute and no extravascu-lar exchange occurs. Therefore, both the interstitial fluid and the intracellular fluidspaces are omitted and only the capillary blood space is considered (Figure 2.5 a).• The capillary wall may be very permeable to a solute leading to a fast equilibrium ofthe capillary blood and the interstitial fluid spaces. In this case, both of the capillaryblood and the interstitial fluid spaces are considered as a combined sub-compartmentwith uniform solute concentration (Figure 2.5 b).• The cell membrane may be very permeable to a solute leading to a fast equilib-rium of the interstitial fluid and intracellular fluid spaces. Therefore, both of the182.2. The Sorensen modelinterstitial fluid and intracellular fluid spaces are combined and considered as onesub-compartment with uniform solute concentration (Figure 2.5 c).• The capillary wall and cell membrane are both very permeableto a solute leading toa fast equilibrium of all three spaces. Therefore, all three spaces are combined andconsidered as one space with uniform solute concentration (Figure 2.5 d).• The concentration of the solute in the intracellular fluid space restricts the rateof solute transport across the cell membrane. Therefore, the intracellular space isomitted (Figure 2.5 e).Figure 2.5: Simplified configurations of physiological compartments[7]In the following sections, the mass balance equations over each sub-compartment arepresented.192.2. The Sorensen model2.2.1 Glucose sub-modelAs explained in section 2.2, the glucose sub-model 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. Mass balance equations over each sub-compartment results the following eight ordinary differential equations:V GBCdGBCdt= QGB(GH −GBC)−V GBFTGB(GBC −GBF ), (2.1)V GBFdGBFdt=V GBFTGB(GBC −GBF )− rBGU , (2.2)V GHdGHdt= QGBGBC +QGLGL +QGKGK +QGPGPC +QGHGH − rBCU , (2.3)V GGdGGdt= QGG(GH −GG)− rGGU , (2.4)V GLdGLdt= QGAGH +QGGGG −QGLGL + rHGP − rHGU , (2.5)V GKdGKdt= QGK(GH −GK)− rKGE , (2.6)V GPCdGPCdt= QGP (GH −GPC)−V GPFTGP(GPC −GPF ), (2.7)V GPFdGPFdt=V GPFTGP(GPC −GPF )− rPGU , (2.8)where G is the glucose concentration (mg/dl), Q is the vascular blood flow rate (dl/min), Vis the volume (dl), T is the transcapillary diffusion time constant (min), r is the metabolicproduction or consumption rate (mg/min) and t is time (min). The subscripts of thesevariables refer to the body organs. Subscript B is the brain, subscript BC is the braincapillary space and subscript BF is the brain interstitial fluid space. Subscript A is thehepatic artery, subscript G is gut, subscript L is liver and subscript G is GI tract (stomach202.2. The Sorensen modeland intestines). Subscript P is periphery, subscript PC is the periphery capillary spaceand subscript PF is the periphery interstitial fluid space.The general form of the metabolic production and consumption rates in each organ isas follows [53]:r = M I(t)MG(t)MΓ(t)rB, (2.9)where M I , MG and MΓ are the independent multiplicative effect of insulin, glucose andglucagon on the metabolic rate, respectively. rB is the basal metabolic rate and the mul-tipliers have the following general form:MC = a+ b tanh[c(CCB− d)], (2.10)where a, b, c and d are the parameters of the model. C is the substance concentrationand CB is the basal concentration of the substance. The following equations are used tocalculate the glucose metabolic rates [53]:rBGU = 70 (2.11)rRBGU = 10 (2.12)rGGU = 20 (2.13)rPGU = MIPGUMGPGUrBPGU , (2.14)rBPGU = 35 (2.15)M IPGU = 7.03 + 6.52 tanh(0.338(IPFIBPF− 5.82) (2.16)MGPGU =GPFGBPF(2.17)212.2. The Sorensen modelrHGP = MIHGPMGHGPMΓHGP rBHGP , (2.18)rBHGP = 35 (2.19)ddtM IHGP = 0.04(MI∞HGP −M IHGP ) (2.20)M I∞HGP = 1.21− 1.14 tanh[1.66(ILIBL− 0.89)] (2.21)MGHGP = 1.42− 1.14 tanh[0.62(GLGBL− 0.497)] (2.22)MΓHGP = 2.7 tanh[0.39(ΓΓB]− f (2.23)ddtf = 0.0154[(2.7 tanh[0.39( ΓΓB]− 12)− f ] (2.24)rHGU = MIHGUMGHGUrBHGU , (2.25)rBHGU = 20 (2.26)ddtM IHGU = 0.04(MI∞HGU −M IHGU ) (2.27)M I∞HGU = 2.0 tanh[0.55(ILIBL] (2.28)MGHGU = 5.66 + 5.66 tanh[2.44(GLGBL− 1.48)] (2.29)KGE = 71 + 71 tanh[0.11(GK − 460)] 0 ≤ GK < 460rKGE = 71 + 71 tanh[0.11(GK − 460)] GK ≥ 460(2.30)where rBGU is brain glucose uptake rate, rRBGU is red blood cell glucose uptake rate,rGGU is gut glucose uptake rate, rHGP is hepatic glucose production rate, rHGU is hepaticglucose uptake rate, rKGE is kidney glucose excretion rate, and rPGU is peripheral glucoseuptake rate. G, I and Γ are the concentration of glucose, insulin and glucagon, respectively.222.2. The Sorensen modelSuperscript B refers to the basal condition and ∞ refer to final steady state value.In equation 2.17, the glucose multiplier of peripheral glucose uptake rate is differentfrom other multipliers. It is a linear function of the peripheral glucose concentration andhas the following general form:MGPGU = a(GPFGBPF) + b (2.31)where a and b are the parameters of glucose multiplier of peripheral glucose uptake rate.The glucose absorption model that calculates the glucose appearance rate into the bloodstream following an oral glucose intake is considered in the gut compartment of the glucosesub-model as follows:dqSsdt= −k12qSs +Dδ(t), (2.32)dqSIdt= −kemptqSs + k12qSI , (2.33)dqintdt= −kabsqint + kemptqSI , (2.34)kempt = kmin +kmax − kmin2{tanh[ϕ1(qSs + qSI − x1D)]− tanh[ϕ2(qSs + qSI − x2D)] + 2},(2.35)ϕ1 =52D(1− x1) , (2.36)ϕ2 =52Dx2, (2.37)Ra = fkabsqint, (2.38)where δ(t) is the impulse function. x1, x2 and f are constant and their values are 0.9.0.82, and 0.00236 respectively [58]. D is the amount of oral glucose intake (mg). The232.2. The Sorensen modelparameters that are unknown and need to be estimated are k12, kmin, kmax, and kabs.2.2.2 Incretins sub-modelThe incretins production is calculated from the following differential equation:dψdt= ς kempt qS2 − rIΨP , (2.39)where ψ is the amount of produced incretins, kempt qS2 is the rate of glucose entrance tothe small intestine, rIΨP is the rate of incretins absorption into the blood stream, and ς isa constant. rIΨP is calculated from the following equation:rIΨP =ΨτΨ, (2.40)where τΨ is the time constant of the incretins absorption process into the blood stream.The mass balance equation over the incretins compartment results in:V Ψdψdt= rIΨP − rPΨC , (2.41)where V Ψ=11.31 (l) is the incretins distribution volume, Ψ is the blood incretins concen-tration and rPΨC the rate of plasma incretins clearance, which depends on the incretinsconcentration. The clearance rate is calculated from the following equation:rPΨC = rMΨC ψ, (2.42)The parameters that are unknown and need to be estimated are ς, τΨ, and rMΨC .242.2. The Sorensen model2.2.3 Insulin sub-modelAs explained in section 2.2, the insulin sub-model is divided into seven compartments:brain; liver; heart and lungs; periphery (muscles and adipose tissues); gastrointestinal (GI)tract (the stomach and intestinal system); kidney; and pancreas. Since pancreatic insulinproduction is a complex mechanism that cannot be described by simple mass balance equa-tions, the insulin sub-model comprises mass balance equations over each sub-compartmentexcept for the pancreas compartment. Therefore, a separate model is considered for thepancreas.Mass balance equations over each sub-compartment results in the following equations:V IBdIBdt= QIB(IH − IB), (2.43)V IHdIHdt= QIBIB +QILIL +QIKIK +QIP IPV −QIHIH , (2.44)V IGdIGdt= QIG(IH − IG), (2.45)V ILdILdt= QIAIH +QIGIG −QILIL + rPIR − rLIC , (2.46)V IKdIKdt= QIK(IH − IK)− rKIC , (2.47)V IPCdIPCdt= QIP (IH − IPC)−V IPFT IP(IPC − IPF ), (2.48)V IPFdIPFdt=V IPFT IP(IPC − IPF )− rPIC , (2.49)where I is the insulin concentration (mU/l), Q is the vascular blood flow rate (dl/min), Vis the volume (dl), T is the transcapillary diffusion time constant (min), r is the metabolicproduction or consumption rate (mg/min) and t is time (min). The subscripts of thevariables refer to the body organs. subscript B is the brain, subscript A is the hepatic252.2. The Sorensen modelartery, subscript G is gut, subscript L is liver and subscript G is GI tract (stomach andintestines). Subscript P is periphery, subscript PC is the periphery capillary space andsubscript PF is the periphery interstitial fluid space.The following equations are used to calculate the insulin consumption rates:rLIC = 0.4[QIAIH +QIGIG + rP IR] (2.50)rKIC = 0.3QIKIK (2.51)rPIC =IPF[( 1−0.150.15QIP)− 20V IPF](2.52)where R is the inhibitor (dimensionless) and r is the metabolic production or consumptionrate (mU/min). rKIC is kidney insulin clearance rate, rLIC is liver insulin clearance rate,rPIC is peripheral insulin clearance rate and rPIR is pancreatic insulin release rate.Pancreatic insulin releasePancreatic insulin release is mainly stimulated by blood glucose concentration changes.Insulin release pattern in response to a glucose concentration step change has a biphasicin a healthy pancreas (see Figure 2.6). As can be seen from Figure 2.6, there is a sharprelease of insulin about 5-10 min in the first phase [7].262.2. The Sorensen modelFigure 2.6: Biphasic response of a healthy pancreas to a glucose concentration step change[7]To mimic the biphasic behaviour of pancreatic insulin secretion in response to a glucosestimulus, Landahl and Grodsky [60] proposed the pancreatic insulin release model presentedin Figure 2.7.Figure 2.7: Schematic diagram of Landahl and Grodskys model [7]In the pancreas model, insulin is exchanged between a small labile insulin unit and alarge stored insulin unit. Glucose-stimulated factor, P , regulates the rate at which insulinflows into the labile compartment. The rate of insulin secretion from the labile insulincompartment is a function of the glucose concentration, the amount of labile insulin m, and272.2. The Sorensen modelthe instantaneous level of glucose-enhanced excitation factor X and its inhibitor R. Thefirst phase insulin release is caused by an instantaneous increase in the glucose-enhancedexcitation factor (X) followed by a rapid increase in its inhibitor (R). The second phaserelease results from the direct dependence of the insulin secretion rate (S) on the glucosestimulus and the gradual increase in the level of the labile compartment filling factor (P ).The pancreas model equations including mass balance equations over its compartmentsand correlations between variables results in:dmdt= K ′mSKm+ γP − S, (2.53)dmSdt= Km−K ′mS − γP, (2.54)It is assumed that the capacity of the storage compartment is large enough and remainsat steady state. For a glucose concentration of zero, P is set to zero. Therefore, the steadystate mass balance equation around the storage compartment is:K ′mS = Km0, (2.55)where m0 is the labile insulin quantity at a glucose concentration of zero. The rest of theequations for the pancreas model are:dPdt= α(P∞ − P ), (2.56)dRdt= β(X −R), (2.57)S = [N1Y +N2(X −R) + ξ1ψ]m x > R,S = [N1Y + ξ1ψ]m x ≤ R,(2.58)282.2. The Sorensen modelP∞ = Y = X1.11 + ξ2ψ, (2.59)X =G3.27H1323.27 + 5.93G3.02H(2.60)P∞ and Y reflect the glucose-induced stimulation effects on the liable compartment fillingfactor and the insulin secretion rate, respectively. The parameters that are unknown andneed to be estimated are α, β, K, N1, N2, γ, ξ1 and ξ2.2.2.4 Glucagon sub-modelThe glucagon sub-model has one mass balance equation over the whole body as follows:V ΓdΓdt= rPΓR − rPΓC , (2.61)The metabolic rates for the glucagon sub-model are summarized below:rPΓC = 9.1Γ (2.62)rPΓR = MGPΓRMIPΓRrBPΓR (2.63)MGPΓR = 1.31− 0.61 tanh[1.06(GHGBH− 0.47)] (2.64)M IPΓR = 2.93− 2.09 tanh[4.18(IHIBH− 0.62)] (2.65)rBPΓR = 9.1 (2.66)The Sorensen model parameters are summarized in Table 2.1.292.3. Type II diabetes modelTable 2.1: The model parameters [53]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 m0 = 6.33 UV GG = 11.2 dl QGG = 10.1 dl/minV GK = 6.6 dl QGK = 10.1 dl/minV GPC = 10.4 dl QGP = 12.6 dl/minV GPF = 67.4 dl QIB = 0.45 l/minV IB = 0.26 l QIH = 3.12 l/minV IH = 0.99 l QIA = 0.18 l/minV 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 lV Γ = 6.74 l2.3 Type II diabetes modelTo develop a model for type II diabetes, the same structure of Sorensen model can be used.However, the parameters of the healthy human body model should be modified based onthe blood glucose and insulin concentrations sampled from type II diabetic patients duringstandard clinical test.As mentioned in section 1.1.2, type II diabetes mellitus is characterized by several organmalfunctions. These abnormalities are summarized as follows:302.3. Type II diabetes model• 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 in overallsecretion rate• Glucose resistance in the liver and peripheral tissues2.3.1 Selection of model parameters for estimationThe parameters showed in Table 2.1 in section 2.2.4 for a healthy person, represent thephysical characteristics of the body, which are the same for diabetic patients. These pa-rameters do not need to be updated in type II diabetes model.However, from the abnormalities of type II diabetic patients, the parameters within theinsulin secretion rate and glucose metabolic rates should be modified. These parametersshould be estimated using the available clinical data for type II diabetic patients througha non-linear optimization problem.Table 2.2 summarizes the abnormalities associated with type II diabetes and theircorresponding model equations.312.3. Type II diabetes modelTable 2.2: Abnormalities associated with type II diabetes and their corresponding equationsAbnormalities Corresponding EquationsInsulin resistance in Insulin multiplier inperipheral tissues peripheral glucose uptake rate (equation 2.16)Insulin-induced stimulation Insulin multiplier inof hepatic glucose uptake hepatic glucose uptake rate (equation 2.28)Insulin-induced stimulation Insulin multiplier inof hepatic glucose production hepatic glucose production rate (equation 2.21)Glucose-induced stimulation glucose multiplier inof hepatic glucose uptake hepatic glucose uptake rate (equation 2.29)Glucose-induced stimulation glucose multiplier inof peripheral glucose uptake peripheral glucose uptake rate (equation 2.31)Pancreatic insulin secretion rate N1 and N2 in the pancreas model (equation 2.58)both in early peak and overall rateVahidi et al. [7, 51] estimated the following parameters in their type II diabetes mellitusmodel:• From the glucose sub-model, parameters of the glucose metabolic rates and someparameters of the glucose absorption model have been considered for the parameterestimation. the glucose metabolic rates in the glucose sub-model has the generalform of equation 2.9 and the multipliers have the general form of equation 2.10.Considering equation 2.10, a, b, c and d are the parameters of the glucose metabolicrates. To reduce the number of parameters for estimation c and d are selected forthe parameter estimation and a and b are considered to be unchanged.• The glucose absorption model equations are represented by equations 2.32 to 2.38.The model parameters that have been chosen for parameter estimation are k12, kmin,kmax, and kabs.322.3. Type II diabetes model• From the insulin sub-model, some parameters from the pancreas model have beenchosen for parameter estimation. The pancreas model is represented by equations2.53 to 2.60 from which N1, N2, K, γ, α and β are selected for parameter estimation.• The hormonal effects of incretins on the pancreatic insulin production are included inequation 2.58. The parameters representing the hormonal effects of incretins on thepancreatic insulin secretion rate are ξ1 and ξ2, which are considered for parameterestimation.• The incretins sub-model is represented by equations 2.39 to 2.42. It has three pa-rameters (i.e. ς, τΨ and rMΨC), which all of them are selected for the parameterestimation.2.3.2 Nonlinear optimization problemVahidi et al. [7, 51] has used a set of available clinical data to estimate the parameters of themodel by solving a nonlinear optimization problem. The model parameters are estimatedthrough an iterative optimization algorithm using a sequential quadratic programming(SQP) method. In each iteration, the new values of the estimated parameters are used tosolve the model equations.Estimation of the modified model parameters were carried out by minimizing the devi-ation of model predictions from the available measurements of peripheral glucose, insulinconcentrations. The deviation of model predictions from the measured clinical data isminimized through the following objective function:minΘn∑j=1[(Gj − Gˆj)2 + (Ij − Iˆj)2 + (Ψj − Ψˆj)2]. (2.67)where Gj and Ij , and Ψj are peripheral glucose, insulin, and incretins concentrations at332.3. Type II diabetes modeltime j obtained from the model respectively; Gˆj , Iˆj , Ψˆj are the corresponding clinicalmeasurements; n is the number of samples in the clinical data set; and Θ is the vector ofparameters that should be estimated [51].Different parameters of metabolic rates will be obtained after the optimization pro-cedure for each type II diabetic patient since there are different peripheral glucose, andinsulin concentrations profile for different patients.This optimization problem contains totally four constraints; three constraints for theinsulin multipliers in peripheral glucose uptake rate, hepatic glucose uptake rate, andhepatic glucose production rate; and one constraint for the glucose multiplier in hepaticglucose uptake rate.These constraints express that the value of the multiplier must be set to 1 at basalconditions. The general form of the constraints based on equation 2.10 is:a+ b tanh[c(1− d)] = 1, (2.68)34Chapter 3A novel and simpleself-administered method forassessing insulin sensitivity3.1 IntroductionInsulin is a key hormone secreted from β-cells in the pancreas that regulates glucosehomeostasis. Type II diabetes is characterized by both insulin resistance and decreas-ing βcell mass [61]. Insulin resistance happens when the sensitivity of peripheral cells tothe metabolic action of insulin is decreased due to genetic or environmental factors, obe-sity, hypertension, dyslipidemias, and/or coronary artery diseases. The ability of insulin tostimulate body glucose disposal can be characterized by an insulin sensitivity index (ISI)[11, 25, 29, 62].Various methods have been developed for determining the presence and degree of insulinresistance. In Section 1.1.4, some of the common methods usually employed in diabetesresearch are briefly described. These methods are summarized in the next following sec-tions.353.1. Introduction3.1.1 Hyperinsulinemic euglycemic insulin clamp techniqueThe hyperinsulinemic euglycemic insulin clamp technique has been widely used as a goldstandard for understanding insulin resistance in vivo [24]. Defronzo et al. [24] in 1979developed this test. In this test, by a continuous infusion of insulin, the plasma insulinconcentration is raised and clamped at around 100 µU/l. At the same time, by glucoseinjection via a negative feedback principle, the plasma glucose concentration is kept con-stant at basal levels. Endogenous glucose production rate is suppressed by high insulinconcentration to almost zero. At steady state conditions, the rate of glucose infusion rateis equal to the glucose uptake rate by all body tissues and is therefore a measure of thebody insulin sensitivity. This is the only information that can be obtained from this test.This method is labor-intensive, expensive, and limiting for large-scale clinical studies [63].3.1.2 Modified minimal model (MINMOD) analysis in conjunctionwith the frequently sampled intravenous glucose tolerance test(FSIVGTT)More accurate and less labor-intensive than the insulin clamp technique is a modified min-imal model (MINMOD) analysis in conjunction with the frequently sampled intravenousglucose tolerance test (FSIVGTT) [32] for the estimation of insulin sensitivity. This methodproposed by Yang et al. [32] in 1987. In this method, the results of FSIVGTT test is usedto determine the parameters of the minimal model [29] and then, obtained parametersare used to define insulin sensitivity and glucose effectiveness indices. They attempted toimprove the precision of the estimation of insulin sensitivity (SI) from the minimal modeltechnique by modifying insulin dynamics during a frequently sampled intravenous glucosetolerance test (FSIGT). However, the FSIVGTT is still restrictive for large studies [25, 63].363.1. Introduction3.1.3 Homeostasis model assessment (HOMA) of insulin resistance(HOMA-IR)Homeostasis model assessment (HOMA) of insulin resistance (HOMA-IR), fasting plasmainsulin [64], and the fasting-glucose-to-insulin ratio [65] are simple indices of insulin resis-tance compared with the insulin clamp test. HOMA proposed by Matthews et al. [37] in1985 is a structural computer model of the glucose-insulin feedback system in the homeo-static (overnight-fasted) state. A number of nonlinear empirical equations describing thefunctions of organs and tissues involved in glucose regulation are included in their model.By solving these equations numerically, glucose, insulin, and C-peptide concentrations arepredicted in the fasting steady state for any combination of pancreatic β-cell function andinsulin sensitivity (or resistance). From these predictions, the deduction of β-cell functionand insulin sensitivity from pairs of fasting glucose and insulin (or C-peptide) measure-ments can be taken.Matthews et al. [37] demonstrated that in only a few patients with type II diabetes,the homeostasis model assessment of insulin resistance (HOMA-IR) is closely correlatedwith the insulin sensitivity index assessed by euglycemic clamp. Also it was reported in[66] and [67] that in a relatively greater number of diabetic subjects, HOMA-IR provided agood correlation in the clamp studies. However, some investigators recognized that whenthe insulin secretion decreases in patients with advanced type II diabetes, the HOMA-IRshows relatively low value since the HOMA-IR is a product of fasting glucose and insulinlevels.373.1. Introduction3.1.4 Insulin sensitivity indices investigated from oral glucose tolerancetest (OGTT)Recently, several methods have been investigated from oral glucose tolerance test (OGTT).Cederholm and Wibell [68] proposed a formula for ISI that uses the OGTT based onfour timed samples of insulin and glucose (at 0, 30, 60, and 120 min). It has fairlygood agreement with more complicated procedures, such as the clamp test and the insulinsuppression test.Stumvoll et al. [45] disclosed that the insulin sensitivity in non-diabetic subjects canbe assessed from OGTT. Simple ISI for type II diabetic patients were derived based onthe OGTT by Matsuda and DeFronzo [44]. Their index was correlated to clamp-derivedinsulin sensitivity.Gutt et al. [48] devised a formula for an insulin sensitivity index, ISI0,120 that uses thefasting (0 min) and 120 min post-oral glucose (OGTT) insulin and glucose concentrations.Their data showed that ISI0,120 correlated well, when applied prospectively in comparativestudies, with the insulin sensitivity index obtained from the euglycemic hyperinsulinemicclamp. Their correlation was demonstrably superior to other indices of insulin sensitivitysuch as the HOMA formula presented by Matthews et al. [37], and performed comparablyto the computerized HOMA index.Although the above methods are relatively easy to conduct, accurate, and adaptableto both population studies and clinical settings, they are not inexpensive, self-monitoring,and convenient since the plasma insulin level must be measured at a specific time as a keyvariable for calculating these indices in medical labs.In this chapter, we propose a new ISI estimated from capillary blood glucose measure-ments. Our approach is to evaluate the feasibility of using the mathematical compartmentmodel proposed by Vahidi et al. [7, 51] to estimate insulin sensitivity.383.2. Clinical data used for model developmentIn the next section, the model of 15 diabetic patients have been developed using theavailable clinical data from OGTT. In section 3.3, 15 simulated patients’ models were usedfor the development and evaluation of a self-assessment method for obtaining the ISI.3.2 Clinical data used for model developmentThe aim of this study is to develop a simple measure of insulin sensitivity by using a self-assessment test without laboratory requirements. From a literature review of OGTT, itwas found that the pattern of glucose response to insulin varies from patient to patient.To ensure that the proposed test for estimating the ISI is valid for all available patterns ofglucose and insulin concentrations, different sets of blood glucose and insulin measurementsmust be used for the estimation of the Vahidi model parameters. Different sets of clinicaldata for type II diabetic patients have been published in the literature from the 2-h 75-g OGTT. Based on the Canadian Diabetes Association 2013 criteria [69], the diagnosticcriteria for diabetes are summarized in Table 3.1.Table 3.1: Diagnosis of diabetesType FPG (mg/dl) 2-h PG (mg/dl)Normal <110 <140Impaired Glucose Tolerance (IGT) <110 140-199Impaired Fasting Glucose (IFG) 110-125 <140Combined IFG and IGT 110-125 140-199Type II diabetes ≥126 ≥200From our literature survey, it was found that the insulin concentration profile duringan OGTT can be grouped in to a few patterns. Hayashi et al. [70] derived four possiblepatterns of insulin profile from a study involving 400 non-diabetic Japanese Americans.They concluded that the insulin concentration pattern during an OGTT strongly predicts393.2. Clinical data used for model developmentthe development of type II diabetes and is correlated with measures of insulin sensitivity.Bakari and Onyemelukwe [71] studied the plasma insulin pattern both in the fasting stateand in response to a standard OGTT in 42 type II diabetic Nigerians and 36 healthycontrol subjects. They found that the type II diabetic patients demonstrated both fastingand post-OGTT hypoinsulinaemia. Therefore, for our model development, 15 availablepatterns of glucose and insulin concentrations during the 2-h 75-g OGTT for diabetic andnon-diabetic subjects were included and presented in details in Table 3.2.403.2.ClinicaldatausedformodeldevelopmentTable 3.2: Mean plasma glucose and insulin levels during OGTTSubjectPlasma glucose during OGTT (mg/dl) Plasma insulin during OGTT (µU/ml)Reference0 min 30 min 60 min 90 min 120 min 0 min 30 min 60 min 90 min 120 min1 175.86 249.84 315.00 338.40 323.64 4.20 5.50 6.01 6.98 9.92 [71]2 71.10 135.90 124.92 116.10 101.34 5.72 15.58 13.67 10.48 8.03 [71]3 75.29 125.71 129.13 108.50 84.67 8.18 30.00 33.05 33.47 16.77 [72]4 80.00 120.40 110.40 92.10 76.50 7.00 38.40 31.10 21.90 9.30 [72]5 71.30 130.20 145.00 122.40 91.60 9.20 23.10 34.70 41.90 21.90 [72]6 74.00 121.00 177.00 180.00 154.00 9.00 13.00 35.00 46.00 41.00 [72]7 71.00 125.00 134.00 103.00 80.00 7.00 62.00 58.00 36.00 20.00 [72]8 72.00 118.00 115.00 92.00 62.00 10.00 12.00 35.00 20.00 14.00 [72]9 89.90 160.2 134.20 - 109.00 11.30 98.90 68.40 - 43.70 [70]10 90.90 154.80 124.70 - 130.80 11.60 109.80 53.90 - 71 [70]11 93.30 166.20 171.40 - 122.10 11.70 66.80 103.90 - 58.30 [70]12 95.50 171.30 193.30 - 159.10 12.70 59.60 86.70 - 118.90 [70]13 91.30 158.10 148.50 - 144.80 14.90 96.40 74.80 - 130.20 [70]14 153.40 238.40 292.58 278.68 239.89 6.47 18.88 22.00 20.64 14.57 [73, 74]15 97.75 164.68 154.54 110.50 87.61 5.52 37.75 42.63 19.58 7.89 [73, 74]413.3. Proposed self-assessment method for estimation of insulin sensitivityAfter the Vahidi model had been developed, 15 simulated patients’ models were usedfor the development and evaluation of a self-assessment method for obtaining the ISI. Thenext section describes the development of the proposed method for obtaining the ISI.3.3 Proposed self-assessment method for estimation ofinsulin sensitivitySeveral authors proposed various indices for measuring insulin sensitivity by using fastingstate or OGTT data and correlated the indices with the data obtained from the hyperinsu-linemic euglycemic clamp test. Formulas proposed for calculating the ISI are based on theintercorrelations between the concentrations of glucose and insulin and other parameters.However, they all require the measurements of plasma insulin levels sampled at specifictimes by laboratory equipment, which is expensive and inconvenient. Therefore, a morepractical method for obtaining the ISI is the focus of this research.A practical test for obtaining the ISI should not require plasma insulin measurementsand only need capillary blood glucose measurements. Capillary blood glucose refers to theblood glucose concentration measured from capillary blood vessels. This is most commonlydone by a finger prick test by a diabetic patient. The plasma insulin measurement refersto the actual insulin concentration in a persons blood sampled and measured by a labtechnician.For type II diabetic patients, the body is suffering from some insulin resistance andit requires larger amounts of insulin either from the pancreas or from injections to lowertheir plasma glucose level compared to that of an insulin-sensitive body. For those withsevere insulin resistance, the normal physiological response to a given amount of insulin isblunted. As a result, higher levels of insulin are needed to achieve a proper effect.423.3. Proposed self-assessment method for estimation of insulin sensitivityIn light of this, we propose a simple testing approach, in which the simulated patientstake a dose of oral glucose ingestion followed by multiple insulin injections at differenttimes. The proposed test is considered clinically acceptable and safe as the insulin dosagecan be selected with a large safety margin. We have conducted extensive simulation withdifferent combinations of testing protocols on the fifteen simulated patients using the Vahidimodel. After the extensive simulations, we have found that the ISI can be estimatedby patients completing a simple testing protocol, which includes two procedures on twoseparate occasions.In the first procedure, the fifteen simulated subjects were given a single dose of 75-gglucose. The plasma glucose concentrations of the fifteen subjects were sampled in orderto check how their bodies suppress the plasma glucose level with no insulin injection.In the second procedure, a single dose of 75-g glucose was given to the fifteen simulatedsubjects. Then, 10 mU/kg insulin was injected twice subcutaneously into the body of thesimulated subjects 20 and 50 min after glucose consumption since the major response toa moderate load occurs within 15 min of glucose ingestion [75, 76]. The plasma glucoseconcentrations of the fifteen subjects were sampled in order to check how their bodiesregulate the plasma glucose level with two insulin injections.After statistical evaluation, it was found that the differences in the plasma glucoseconcentration profile of each subject from the first and second procedures can be used todefine a formula for the ISI. The formula adopted for the estimation of the ISI is describedin Section 3.4.2. This section also shows how well the proposed index correlates with theISI (called M-value) obtained from the euglycemic insulin clamp technique.433.4. Results and discussion3.4 Results and discussionThe Vahidi model includes a set of nonlinear ordinary differential equations and algebraicequations. The model parameters are estimated through an iterative optimization algo-rithm using an SQP method, as described in Section 2.67. The estimated parametersare then used to solve the Vahidi model equations. The optimization was carried out inMATLAB.3.4.1 Parameters estimation resultsSince different patterns of glucose and insulin concentrations result in different sets ofparameters in the Vahidi model, for each subject in Table 3.2, a set of parameters wasestimated using the nonlinear optimization algorithm described in Section 2.67. As anexample, using the blood glucose and insulin concentration data of subject 1 in Table3.2, the parameters of the glucose metabolic rates have been considered for the parameterestimation. As the model equations in Section 2.2.1 shows, the glucose metabolic rates inthe glucose sub-model has the general form of equation 2.9 and the multipliers have thegeneral form of equation 2.10.Considering equation 2.10, parameters a, b, c and d in the hepatic glucose production(HGP) rate, the hepatic glucose uptake (HGU) rate, and the peripheral glucose uptakerate (PGU) are selected to be estimated. Also, considering equation 2.58 in Section 2.2.3,the parameters N1 and N2 in the pancreatic insulin release model are estimated.As an example, the estimated model parameters for the glucose and insulin sub-modelspresented above are shown in Table 3.3 and Table 3.4, respectively for subject 1.443.4. Results and discussionTable 3.3: Parameter estimation results for glucose sub-model (subject 1).Multiplier in equation (2.10) a b c dM IPGU 7.035 6.516 0.15 4.000M I∞HGP 1.425 1.406 0.607 0.241M I∞HGU 0.001 2.000 1.500 0.001MGHGU 5.664 5.658 2.013 1.678Table 3.4: Parameter estimation results for insulin sub-model (subject 1).Parameter in equation (2.58) ValueN1 1.096N2 0.654The model estimation results of the fifteen subjects presented in Table 3.2, are shownin Figures. 3.1 and 3.15.453.4. Results and discussion0 20 40 60 80 100 120150200250300350Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120456789Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #1Figure 3.1: Plasma glucose and insulin concentration profile in subject #1, the clinicaldata (•), the model results (−)463.4. Results and discussion0 20 40 60 80 100 1206080100120140Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 1205101520Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #2Figure 3.2: Plasma glucose and insulin concentration profile in subject #2, the clinicaldata (•), the model results (−)473.4. Results and discussion0 20 40 60 80 100 1206080100120140160Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 12001020304050Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #3Figure 3.3: Plasma glucose and insulin concentration profile in subject #3, the clinicaldata (•), the model results (−)483.4. Results and discussion0 20 40 60 80 100 1206080100120140Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 12001020304050Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject # 4Figure 3.4: Plasma glucose and insulin concentration profile in subject #4, the clinicaldata (•), the model results (−)493.4. Results and discussion0 20 40 60 80 100 12050100150200250Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 1200102030405060Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #5Figure 3.5: Plasma glucose and insulin concentration profile in subject #5, the clinicaldata (•), the model results (−)503.4. Results and discussion0 20 40 60 80 100 12050100150200250300Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120020406080Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #6Figure 3.6: Plasma glucose and insulin concentration profile in subject #6, the clinicaldata (•), the model results (−)513.4. Results and discussion0 20 40 60 80 100 12050100150200250Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120020406080100Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #7Figure 3.7: Plasma glucose and insulin concentration profile in subject #7, the clinicaldata (•), the model results (−)523.4. Results and discussion0 20 40 60 80 100 1206080100120140160Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 12001020304050Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #8Figure 3.8: Plasma glucose and insulin concentration profile in subject #8, the clinicaldata (•), the model results (−)533.4. Results and discussion0 20 40 60 80 100 12080100120140160180Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120050100150Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #9Figure 3.9: Plasma glucose and insulin concentration profile in subject #9, the clinicaldata (•), the model results (−)543.4. Results and discussion0 20 40 60 80 100 12080100120140160Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120050100150Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #10Figure 3.10: Plasma glucose and insulin concentration profile in subject #10, the clinicaldata (•), the model results (−)553.4. Results and discussion0 20 40 60 80 100 12080100120140160180Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120050100150Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #11Figure 3.11: Plasma glucose and insulin concentration profile in subject #11, the clinicaldata (•), the model results (−)563.4. Results and discussion0 20 40 60 80 100 120050100150200250300Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120050100150Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #12Figure 3.12: Plasma glucose and insulin concentration profile in subject #12, the clinicaldata (•), the model results (−)573.4. Results and discussion0 20 40 60 80 100 12080100120140160Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120050100150Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #13Figure 3.13: Plasma glucose and insulin concentration profile in subject #13, the clinicaldata (•), the model results (−)583.4. Results and discussion0 20 40 60 80 100 120150200250300350400Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 12051015202530Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #14Figure 3.14: Plasma glucose and insulin concentration profile in subject #14, the clinicaldata (•), the model results (−)593.4. Results and discussion0 20 40 60 80 100 12050100150200250Time (min)Plasma Glucose concentration (mg/dl)  Clinical dataThe model result0 20 40 60 80 100 120020406080Time (min)Plasma Insulin concentration (µU/ml)  Clinical dataThe model resultSubject #15Figure 3.15: Plasma glucose and insulin concentration profile in subject #15, the clinicaldata (•), the model results (−)The goodness of fit between the model estimation and the available clinical data setcan be calculated using different cost functions in MATLAB. In this study, the goodnessof fit is calculated using the mean square error (MSE) as a cost function:MSE =|x− xref |Ns − 1 (3.1)where x is the glucose or insulin concentration matrix estimated by the model, xref is theavailable glucose or insulin concentration from Table 3.2 as the reference, and Ns is thenumber of actual measured clinical data. From equation (3.1), the overall average goodnessof fit for all fifteen subjects is 92%. The simulated trends are reasonably consistent with603.4. Results and discussionthe actual clinical data from both a visual inspection and the average goodness of fit.3.4.2 Quantitative estimation of insulin sensitivityIn order to validate the proposed protocol for estimating the ISI, the M-values from theeuglycemic insulin clamp test were obtained for the fifteen subjects from the simulatedmodels. To perform the euglycemic insulin clamp test on the simulated bodies of thefifteen subjects with the Vahidi model, the plasma insulin concentration was acutely raisedand maintained at 100 µU/ml by a continuous infusion of insulin. Meanwhile, the plasmaglucose concentration was held constant at basal levels by a variable glucose infusion inMATLAB. Then, proposed testing protocols described in Section 3.3 were applied to thefifteen simulated subjects.The plasma glucose concentration profiles of each subject from the first and secondprocedures are plotted in Figures. 3.16-3.30.0 50 100 150 200 250 300 350140160180200220240260280300320340Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 1Figure 3.16: Effect of insulin injection in subject #1, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)613.4. Results and discussion0 50 100 150 200 250 300 35060708090100110120130140Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 2Figure 3.17: Effect of insulin injection in subject #2, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 35060708090100110120130140150160Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 3Figure 3.18: Effect of insulin injection in subject #3, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)623.4. Results and discussion0 50 100 150 200 250 300 35060708090100110120130140Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 4Figure 3.19: Effect of insulin injection in subject #4, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 3506080100120140160180200220240Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 5Figure 3.20: Effect of insulin injection in subject #5, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)633.4. Results and discussion0 50 100 150 200 250 300 35050100150200250300Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 6Figure 3.21: Effect of insulin injection in subject #6, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 350406080100120140160180200220Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 7Figure 3.22: Effect of insulin injection in subject #7, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)643.4. Results and discussion0 50 100 150 200 250 300 350708090100110120130140150160Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 8Figure 3.23: Effect of insulin injection in subject #8, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 350406080100120140160180Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 9Figure 3.24: Effect of insulin injection in subject #9, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)653.4. Results and discussion0 50 100 150 200 250 300 35060708090100110120130140150160Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 10Figure 3.25: Effect of insulin injection in subject #10, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 3508090100110120130140150160170180Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 11Figure 3.26: Effect of insulin injection in subject #11, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)663.4. Results and discussion0 50 100 150 200 250 300 35050100150200Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 12Figure 3.27: Effect of insulin injection in subject #12, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 35060708090100110120130140150160Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 13Figure 3.28: Effect of insulin injection in subject #13, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)673.4. Results and discussion0 50 100 150 200 250 300 350100150200250300350400Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 14Figure 3.29: Effect of insulin injection in subject #14, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)0 50 100 150 200 250 300 3506080100120140160180200220240Time (min)Peripheral Glucose Concentration (mg/dl)Subject # 15Figure 3.30: Effect of insulin injection in subject #15, two 10 mU/kg insulin injections at20 and 50 min respectively (-), and no injection (–)From Figures. 3.16-3.30, the plasma glucose level for insulin-sensitive subjects 2, 4, 5,683.4. Results and discussion8, 9, and 10 were suppressed significantly after the two insulin injections. However, theperipheral glucose concentration profile did not change or were suppressed slightly afterthe two insulin injections for insulin-resistant subjects 1, 3, 6, 7, 11, 12, 13, 14, and 15.In the same figure, the maximum differences between plasma glucose levels in theinsulin-sensitive subjects occur almost at 60 min and 80 min after glucose consumptionbecause of the two insulin injections. In statistics, multiple linear regression is an approachfor modelling the relationship between two or more explanatory variables denoted X anda response variable y by fitting an equation to observed data. To find a new ISI, step-wisemultiple regression analysis was performed with the M-value as the dependent variable (y)and the glucose concentrations at fasting (0 min), 60 min, and 80 min after ingestionof 75-g glucose as the three independent variables (X) in MATLAB. The obtained ISIequation from the multiple regression analysis is:ISI = 44.071− 0.1534× FPG− 0.1855×G60min + 0.182×G80min− ( 1.95FPG+6.81G60min− 5.88G80min)× 103(3.2)where FPG, G60min , and G80min are the peripheral glucose concentrations in mg/dl atfasting (0 min), 60 min, and 80 min after ingestion of a 75-g glucose, respectively.The means and standard deviations were computed in MATLAB for the defined insulinsensitivity and M-values. Pearsons r coefficient was used for the calculation of correlationsbetween these two measures. The scatter plot of the relationship between the M-value andthe ISI from equation ((3.2)) for each subject is shown in Figure. 3.31. Both the Pearsonscoefficient (r = 0.927) and the p-value (p = 0.0045) indicate a strong correlation betweenthe new ISI and the M-value from the euglycemic clamp test.693.4. Results and discussionFigure 3.31: Correlation between the ISI and the M-value for the fifteen subjects; r =0.927, p = 0.0045Previous ISIs derived from the OGTT data require the measurements of plasma insulinlevels at specific times by laboratory equipment, which is inconvenient, time-consuming,and expensive. The proposed ISI can be estimated from data collected by diabetic patientswho need to frequently monitor their status without the need for expensive laboratoryfacilities.3.4.3 Comparison of various insulin sensitivity indices obtained fromOGTTThe derivations of other indices obtained during the OGTT are briefly presented here. Theindex of whole-body insulin sensitivity derived by Matsuda and DeFronzo [44] calculatesinsulin sensitivity from plasma glucose (mg/dl) and insulin (mU/l) concentrations in thefasting state and during the OGTT. Stumvoll et al. [45] proposed several ISI equations,703.4. Results and discussionwhich were obtained from multiple linear regression analysis. The equations calculatethe insulin sensitivity from plasma glucose (mmol/l) and insulin (pmol/l) concentrationsduring the OGTT. The Gutt index (ISI0,120) [48] was adopted from the ISI proposed byCederholm and Wibell [68]. The calculation of ISI0,120 (mgl2mmol−1mlU−1min−1) onlyuses the fasting (0 min) and 120 min concentrations of glucose and insulin during theOGTT.These three ISIs calculated from the OGTT data are shown in Table 3.5 to comparethe correlation of each index with the M-value. Table 3.5 shows the Pearsons correlation ofeach measurement of insulin sensitivity with the M-value computed in MATLAB. As canbe seen from Table 3.5, the correlation of the proposed ISI with the M-values is significantlystronger than those of the other indices, (r = 0.927, p = 0.0045). Although Table 3.5 showsa very promising and convenient ISI estimation, a proper comparison should be done byapplying the proposed ISI protocol to real subjects. This can be a part of future studiesresearch.713.4.ResultsanddiscussionTable 3.5: Pearson correlations with M-Value and results of correlation comparisons [1]Measure Formula Correlation with M-value ReferenceISIMatsuda10000√FPG×FPI×Gmean×Imean r = −0.43 p = 0.1 [44]ISIStumvoll 0.156− 0.0000459× I120 min − 0.0000321× FPI − 0.0054×G120 min r = 0.47 p = 0.0794 [45]ISIGutt75000+(FPG−G120 min)×0.19×BWGmean×log(Imean) r = −0.2965 p = 0.28 [48]Proposed ISI equation (3.2) r = 0.927 p = 0.0045 -72Chapter 4Assessment of type II diabetesmellitus using irregularly sampledmeasurements with missing data4.1 IntroductionIn diabetic patients, the glucose metabolic rates represent the health status of the liver,muscles and adipose tissues. To measure the glucose metabolic rates in the type II dia-betic patients, the measurement of the glucose and insulin concentrations in different partsof the body are needed. However, clinical measurements of all necessary concentrationsdeep inside different organs or tissues are just not practical or realistic. Therefore, physi-cians mostly rely on a few measurements from patients’ blood and/or capillary glucosemeasurements at regular or irregular intervals for clinical decisions [77].Previous studies have shown that important clinical data may be missing owing todifferent reasons such as inability to record clinical results, infrequent sampling by patients,and illegible hand writing. Lack of complete knowledge about the health status of thediabetic patients poses more problems to physicians in managing type II diabetes whilethey need time oriented clinical data of past and present status of diabetic patients [78–80].Since only a few blood glucose measurements per day are available in a non-clinical734.1. Introductionsetting, developing a predictive model of the blood glucose of a person with type II diabetesmellitus is important. Such a model may provide useful information to diabetic patients ofdangerous metabolic conditions, enable physicians to review past therapy, estimate futureblood glucose levels, and provide therapy recommendations. It can also be used in thedesign of a stabilizing control system for blood glucose regulations [81, 82].Many studies proposed on-line identification of type I diabetes mellitus using neuralnetwork modelling approaches [82–85]. Tresp et al. [82] developed a predictive model ofthe blood glucose of a person with type I diabetes mellitus with partially missing clinicaldata by using a combination of a nonlinear recurrent neural network and a linear errormodel. However, developing a nonlinear state-space model for type II diabetes mellitusthat can easily deal with missing data has received limited attention.The goal of this work is to develop a blood glucose predictive model for a type IIdiabetic patient and the model can be estimated by using patient data collected undernormal everyday conditions rather than a well-controlled environment typically done in aclinical facility. Such a model should be able to detect dangerous metabolic states of apatient, and optimize the patient’s therapy.In this study, we use online Bayesian estimation framework to estimate a stochasticnonlinear model for type II diabetes mellitus using clinical data with missing data atrandom intervals. We adopt the detailed nonlinear model developed by Vahidi et al. [7, 51]for type II diabetes since the Vahidi model is a much more detailed model comparingwith the MINMOD approach. The Vahidi model is able to effectively model individualabnormalities by characterizing distinct compartments as the faulty organs. To artificiallycreate clinical data sets with missing data at random intervals, we then randomly remove10%, 25% and 50% of the original available data obtained from the Vahidi model. At theend, glucose, insulin, and incretins concentrations, as well as the parameters of different744.2. Mathematical model preparationcompartments, are estimated from clinical data with missing data at random intervals.These estimates can then be used to measure the glucose metabolic rates in differentorgans of the type II diabetic patients.There is an extensive discussion on estimating the states and the parameters of thenonlinear state-space models from partially missing data using mathematical approachessuch as Bayesian filters, Particle filters (PFs), Expectation-Maximization (EM) algorithms,Sequential Importance Resampling (SIR) particle filters [86–89]. In many studies, Baysianestimation has been used in metabolism and physiological modelling [90–92]. Among thesemethods, we use the SIR based PF proposed by Tulsyan et al. [93] for online Bayesianestimation of the states and the parameters of the Vahidi model since it needs less compu-tational cost when a large number of unknown states and parameters must be estimatedsimultaneously. To do this, a clinical data set, as well as a prior information on the un-known states and parameters of the Vahidi model, are needed. This kind of informationcan be gathered from physical considerations and population studies.In this study, the estimation of the Vahidi model parameters are carried out by the SIRparticle filtering method for the data sets containing randomly deleted simulated data.4.2 Mathematical model preparationThe continuous mathematical model of type II diabetes developed by Vahidi et al. [7, 51]has been described previously in chapter 2. In order to estimate the unknown parametersof the Vahidi model using particle filtering method, the model must be discretized. Todiscretize the model, any discretization method is possible to be used. In this study,simply the fixed-step backward difference approximation has been used since it was accurateenough to discretize the model. The following equation represents general discretization754.2. Mathematical model preparationusing fixed-step backward difference approximation:dydx=yi − yi−1∆x, (4.1)The above equation is applied to all ordinary differential equations explained in chapter2. The model can be then rewritten in state space general form as follows:xt+1 = f(yt, θt, ut) + vt, (4.2a)yt = g(xt, θt, ut) + wt, (4.2b)where:• f is the state function representing equations 2.1 to 2.8, 2.20, 2.24, 2.27, 2.43 to 2.49,2.53, 2.56, 2.57, and 2.61.• g is the measurement dynamic function representing equations 2.7 and 2.48.• t is the sampling time index.• xt is the vector of states.• ut is the vector of inputs.• yt is the vector of measurements.• θ is the vector of model parameters, which are constant values.• vt and wt are state and measurement noise sequences with known probability den-sity functions with zero mean. Since all real systems normally incorporate differentenvironmental noises affecting the measurements and also mathematical models nor-mally have some uncertainties, these noise sequences are added to the model statesand outputs to address usual measurement noises and also model uncertainties.764.3. Response modelsIn type II diabetes model g represents the measurement dynamic function variables andŷt is the vector of concentration of either insulin, glucose, or glucagon at sampling time tmonitored and recorded by several sensors and measuring devices. These devices recordpatient’s critical variables yt (output) in response to the test action ut (input) implementedat some point in time indexed by t. For example, in the intravenous glucose infusion test,insulin concentration measurements as output yt are recorded at regular intervals againstthe infused glucose concentration as input ut. A summary description of this on-lineestimation method is provided in the next following sections.4.3 Response modelsIn clinical trials, several sensors and measuring devices were used for monitoring the re-sponse of a patient to a clinical test. Let us assume that we have a sequence of time-taggedclinical measurements y1:t = {y1, y2, . . . , yt} corresponding to the input action u1:t = {u1, u2, . . . , ut},and that we are interested in predicting the response yt+1 for some known input action ut+1.Such predictions are valuable to the physician assessing the health of the patient duringclinical trials. To solve this problem, we can assume yt+1 is independent of {u1:t, y1:t}, inwhich case, the prediction of yt+1 is impossible. Alternatively, we can assume yt+1 dependson the trend recorded in the past data {u1:t, y1:t}. For the latter assumption– which is truefor any causal system– a response model1 is useful in predicting the response of a patientto a clinical test.A reliable response model should not only accurately model a patient’s physical andbiological response to a clinical test, it should also account for the various uncertaintiessuch as modelling and measurement errors. For example, random measurement errors can1A response model is a mathematical model describing the dynamics of the key internal states of apatient in response to a clinical test.774.3. Response modelsbe modelled by viewing yt as a random realization of a stochastic process. In this work, weuse stochastic state-space models (SSMs) to represent a response model. Mathematically,a SSM can be represented as:xt+1 = f(yt, θt, ut) + vt, (4.3a)yt = g(xt, θt, ut) + wt, (4.3b)where xt describes evolution of the internal states of the patient. Physically, xt models thecomplete response of a patient subject to a clinical test. Given the states xt, inputs ut andmodel parameters θt at time t, the internal states evolve to xt+1. vt in equation(4.3a) isthe state noise, which accounts for the unknown and unmeasured variations in the statesnot captured by the response model. Due to the non-zero random state noise vt, the statesare not precisely known. Equation (4.3b) describes how sensor readings yt relate to thestates xt and parameters θt. wt in equation (4.3b) is the noise term, which accounts forthe random sensor noise.In clinical trials, measurements of only a few critical states are available at our disposal.This is because the high cost or lack of appropriate sensing technology or devices precludesmeasurement of all but key internal states. The state-space modelling framework is general,and can be used to represent a wide class of response models, including the type II diabetesmellitus response model given in section 2.1.In this study, we use equation (4.3) for real-time estimation of the critical responsevariables, such as blood glucose, insulin and incretins concentrations during clinical trial ofpatients with type II diabetes mellitus. Monitoring these variables is critical as it enablesthe physicians to review past therapy, estimate future blood glucose levels and providetherapy recommendations. To predict the critical variables using equation (4.3), the modelstates and parameters, which are typically unknown for a patient need to be estimated first.784.3. Response modelsGiven the state and parameter estimates, the model predictions at t can be computed as:ŷt = g(x̂t, θ̂t, ut), (4.4)where ŷt is the response predictions and x̂t and θ̂t are the parameter and state estimates,respectively. Ideally, given an accurate estimate of the states and parameters, the modelpredictions should match the clinical measurements as closely as possible. Any standardestimation approach involves fitting the model using available clinical measurements; how-ever, data fitting is not straightforward for SSMs because of the following reasons: 1) thestates are stochastic, which makes estimation of both states and parameters challenging and2) the clinical measurements are assumed to be irregularly sampled. In the next section,we explain how the unknown states and parameters of the response model are estimated.Remark. There is a much larger appeal to use state-space modelling framework torepresent response models. From equation 4.4, it is evident that computing the responsepredictions using a SSM also requires estimation of all the internal states of the patient.Thus, any method designed to compute the response predictions gives away estimation ofall the internal states as a side product. This is of immense value to a physician, consideringonly a handful of the internal states are actually measured.4.3.1 States and parameters estimation of the response modelIn most real systems, all the physical states are not measurable due to the various inherentrestrictions. Therefore, by applying a series of mathematical calculations called “statesobserver” or “states filtering”, unknown model states are estimated by input and outputmeasurements from the real system [7]. For instance, in the type II diabetes model, obtain-ing measurements of blood glucose and insulin concentrations from different body organsare extremely difficult, dangerous for the subjects and in most of the times clinically im-794.3. Response modelspossible. However, having this type of information is necessary in evaluating the behaviourof body organs. An alternative is to estimate these concentrations using available mea-surements from peripheral tissues along with a mathematical model and a states estimatoralgorithm [7].If the models is deterministic (the output of the model is fully determined by theparameter values and the initial conditions), the unknown states and parameters are oftengenerated using an observer such as Luenberger observer. Otherwise, for stochastic models(the output of the model is unpredictable due to the influence of a random variable), afilter is used for estimation [94].In the stochastic state-space model, when the model parameter θ is known, on-lineinference about the state process xt given the observations yt is a so-called optimal filtering[94]. Kalman filtering methods rely on the model being linear and noise being Gaussian[95]. Extensions to nonlinear systems have also been considered in the literature such asextended Kalman filters (EKF), and unscented Kalman filters (UKF). These suboptimalfilters either approximate the nonlinear system through linearization and/or assume thatthe noise is Gaussian. The approximations are often not satisfactory.However, Sequential Monte Carlo (SMC) methods, also known as particle methodsdo not require linearization of the stochastic state-spaces model or Gaussianity of themeasurement noises [96]. Particle filtering algorithm are a class of sequential simulation-based algorithms to approximate the posterior distributions of interest. This algorithmis a powerful state estimation method whose accuracy is independent on the degree ofmodel nonlinearity and is able to be improved by increasing the number of particles unlikeKalman filter [94].In the stochastic state-space model, when the parameter θ is unknown and needs to beestimated from the data either in an on-line or off-line manner, the following methods are804.3. Response modelsused [94]:• Bayesian or Maximum Likelihood (ML)• Off-line (batch) or on-line (recursive)In the past 15 years, several algorithms have been proposed to solve the simultaneousstate-parameter estimation problem in real-time using likelihood and Bayesian derivedmethods. The recent review paper by Kantas et al. [94] provided a detailed exposition ofon-line and off-line methods for parameter estimation using Bayesian and likelihood basedmethods. Simultaneous on-line Bayesian estimators is performed by filtering an extendedvector of states and parameters using an adaptive sequential-importance-resampling (SIR)filter with a kernel density estimation method [93].The existing literature for on-line state-parameter estimation using Bayesian and like-lihood based methods assumes that measurement will be available at all sampling time;however, in practice, measurements may not be available at all sampling time instants.Tulsyan et al. [93] proposed on-line Bayesian state and parameter estimation in non-linearstate- space models (SSMs) with non-Gaussian noise under missing measurements. Theyused a particle based SIR filtering approach due to the inherent limitations of the EKF andUKF based simultaneous state-parameter estimators. They selected the SIR filter sinceit is relatively less sensitive to large process noise and is computationally less expensive.Furthermore, the importance weights are easily evaluated and the importance functionscan be easily sampled [93].In the light of aforementioned above, in this study, we apply a particle based SIRfiltering approach proposed by Tulsyan et al. [93] to estimate the real-time states andparameters of the stochastic nonlinear state-space type II diabetes model under irregularlysampled clinical measurements.814.3. Response modelsA brief description of the standard particle filter algorithm is presented below to providethe necessary background for the algorithm developed in the following section.4.3.2 Recursive bayesian estimationThe Sequential Monte Carlo (SMC) approach is a recursive Bayesian estimation methodfor nonlinear and non-Gaussian filtering problems. In this approach, given a sequenceof measurements, the probability density function (PDF) of the current state xt is es-timated. Let us assume that we have a sequence of time-tagged clinical measurementsy1:t = {y1, y2, . . . , yt} corresponding to the input action u1:t = {u1, u2, . . . , ut}, and we areinterested in predicting the density of the state, i.e. p(xt|y1:t), for every iteration. Usingthe Bayes’ theorem and Total law of probability, p(xt|y1:t) can be recursively computed intwo steps, which are the update and prediction steps as shown below:Update Step:p(xt|y1:t) ∝ py(yt|xt)p(xt|y1:t−1). (4.5)Prediction Step:p(xt|y1:t−1) =∫px(xt|xt−1)p(xt−1|y1:t−1)dxt. (4.6)It is assumed that the PDF of the initial time step, p(x0|y0), is known. Equations (4.5)and (4.6) do not have analytical solutions for nonlinear processes with Gaussian noise. TheSequential Monte Carlo algorithms make these complex integrals tractable through the useof efficient sampling strategies.824.3. Response models4.3.3 Sequential monte carlo (SMC)The basic idea of SMC is the recursive computation of any given target PDF pi(x) (e.g.,p(xt|y1:t)) by generating a set of random particles (samples) and associated weights fromthe target PDF pi(x). Furthermore, due to its generality and robustness, it has becomean important alternative to the extended Kalman filter (EKF) and unscented Kalmanfilter (UKF). Unlike Kalman filter method, in the particle filtering method, the exploitedapproximation does not involve linearization around current estimates [97].Consider the state space model given by equations (4.3a) and (4.3b). Rather thandirect solving the integrals in equations (4.5) and (4.6), the Bayesian recursive estimationis implemented via Monte Carlo sampling. For estimating the PDF, the two pieces ofinformation required at each time step t: the samples xit and their associated weightsωit. From a known density called importance density function, q(xt|y1:t), samples xit areassumed to be generated. The corresponding weights of the samples are defined as:ωit =p(xit|y1:t)q(xit|y1:t)(4.7)and after normalization the weights become:ωit =ωit∑Ni=1 ωit(4.8)where N is the number of samples used. The samples at time step t, xit ∼ q(xit|y1:t), arecomputed by multiplying the existing samples, xit−1 ∼ q(xit−1|y1:t−1), and the new state,xit ∼ q(xit|xit−1, y1:t) if the importance density function is chosen to be factorized such that:q(xt|y1:t) = q(xt|xt−1, yt)q(xt−1|y1:t−1) (4.9)834.4. Online state and parameter estimation in nonlinear state-space modelsand the updated weight ωit associated with xit can be obtained according to:ωit ∝ ωit−1p(yt|xit)p(xit|xit−1)q(xit|xit−1, y1:t)(4.10)It is useful to assume that q(xit|xit−1, y1:t) = q(xit|xit−1, yt) when only a filtered estimateof p(xt|y1:t) is required. Then, the importance density only depends on xt−1 and yt. Underthis assumption, equation 4.11 can be rewritten as:ωit ∝ ωit−1p(yt|xit)p(xit|xit−1)q(xit|xit−1, yt)(4.11)and the filtered density p(xt|y1:t) is approximated as follows:p(xt|y1:t) ≈N∑i=1ωitδ(xt − xit) (4.12)where δ(.) is a multi-dimensional Dirac function. xit is the ith sample that approximatesthe distribution, and the coefficient ωit is the corresponding weight. As N →∞, the abovedensity approximation approaches the true filtered density p(xt|y1:t). In the next section,a description of real-time Bayesian estimation is provided.4.4 Online state and parameter estimation in nonlinearstate-space modelsOur objective is to estimate zt in real-time using clinical data {u1:t; y1:t}. Let zt = {xt, θt}denote an extended vector of unknown states and parameters. It is further assumed thatthe clinical measurements are recorded at irregular times, such that only a subset of y1:tis available for estimation at t. For notational convenience, we dispense with the input ut844.4. Online state and parameter estimation in nonlinear state-space modelsin the succeeding discussions; however, the method presented in this study holds with theinputs included.In the Bayesian framework, the variables to be estimated are assumed to be randomvariables. The states are inherently random due to the noise in equation (4.3a); and theparameters, which are unknown but non-random are assumed to be random, such thatzt = {xt, θt} is a vector of random variables. To set up the Bayesian estimation, weassume z0 to be distributed with a prior density p(z0|y1:0). Also, we assume the stateand measurement noise are independent and identically distributed (i.i.d) zero mean finitevariance Gaussian sequences with the probability density functions (PDF) px(.) and py(.)known a priori.4.4.1 Complete clinical dataFirst we consider the estimation problem using the complete clinical data set. Assumingy1:t to be available, the real-time Bayesian estimation of zt at t involves computing theposterior density p(zt|y1:t). Here, p(zt|y1:t) is a probabilistic representation of the statisticalinformation available on zt conditioned on the clinical measurements y1:t. Based on theBayes’ theorem and Total law of probability presented in equations (4.5) and (4.6), p(zt|y1:t)can be recursively computed as shown below:Update Step:p(zt|y1:t) ∝ py(yt|zt)p(zt|y1:t−1). (4.13)Prediction Step:p(zt|y1:t−1) =∫pz(zt|zt−1)p(zt−1|y1:t−1)dzt. (4.14)854.4. Online state and parameter estimation in nonlinear state-space modelsIn equation (4.13), py(yt|zt) is the measurement noise distribution or the likelihoodfunction indicating how likely it is for zt to have generated the clinical measurement yt.p(zt|y1:t−1) is a one-step-ahead prior density representing statistical information on zt priorto the recorded clinical measurement yt. The prior density p(zt|y1:t−1) is computed usingequation (4.14), where pz(zt|zt−1) is the joint state and parameter noise distribution andp(zt−1|y1:t−1) is the posterior distribution at t− 1.Starting with p(z0|y0:1), in principle, the recurrence relation between equations (4.13)and (4.14) provides a complete Bayesian solution to the state and parameter estimationproblem under complete clinical data. Finally using p(zt|y1:t), the estimate of ẑt at t can becomputed as the mean of the posterior density, such that the estimate step can be definedas:ẑt =∫ztp(zt|y1:t)dzt, (4.15)where ẑt = {x̂t, θ̂t} is the state and parameter estimation at t. Note that other values, suchas the mode or median of p(zt|y1:t) can also be selected as the point estimate.4.4.2 Irregular clinical dataFrom section 4.4.1, it is evident that if yt is not measured at t, the posterior densityp(zt|y1:t) cannot be computed using equation (4.13). In such situations, the estimates att, in presence of irregular data can be computed by replacing p(zt|y1:t) in equation (4.15)using the one-step ahead prior density p(zt|y1:t−1).Now assuming yt+1 to be available at t + 1, the posterior density for zt+1, given clin-ical measurements {y1:t−1, yt}, i.e., p(zt+1|y1:t−1, yt+1) can be computed using the Bayes’theorem and the law of total probability, such that the update step shows that:864.4. Online state and parameter estimation in nonlinear state-space modelsp(zt+1|y1:t−1, yt+1) ∝ p(yt+1|zt+1)p(zt+1|y1:t−1). (4.16)where p(zt+1|y1:t−1, yt+1) is the posterior density for zt+1 and p(zt+1|y1:t−1) is a two-stepahead prior density computed using the law of total probability, i.e.:p(zt+1|y1:t−1) =∫pz(zt+1|zt)p(zt|y1:t−1)dzt. (4.17)Substituting equation (4.14) into equation (4.17) yields the prediction step:p(zt+1|y1:t−1) =∫∫pz(zt+1|zt)pz(zt|zt−1)p(zt−1|y1:t−1)dzt−1:t. (4.18)Similar to section 4.4.1, having computed p(zt+1|y1:t−1, yt+1), the estimate of zt+1 canbe computed by replacing the density by p(zt+1|y1:t−1, yt+1) in equation (4.15). Note thatthe method proposed in this section is general and can naturally be extended to handleconsecutively missing measurements as well.The Bayesian approach developed in section 4.4 provides an excellent framework forreal-time state and parameter estimation under complete and irregular clinical measure-ments. Computing the Bayesian solution requires evaluation of the multiple integrals in theprediction and estimation steps. Unfortunately, except for linear systems with Gaussianstate and measurement noise, or when the states and parameters take on only finite values,the Bayesian solution cannot be solved exactly with finite computing capabilities.This study uses a sequential Monte-Carlo (SMC)-based adaptive sequential-importance-resampling (SIR) filter proposed by Tulsyan et al. [93] to numerically approximate theBayesian solution. In the next section, using SMC method, the estimation results of statesand parameters of the state-space model for type II diabetic patients under various levelsof randomly missing clinical data are presented.874.5. Results and discussion4.5 Results and discussionIn this section, the efficiency of the SIR filtering method in handling missing measure-ments for estimation of the nonlinear stochastic model for type II diabetes mellitus isdemonstrated. All the simulations were conducted on a 2.90 GHz CPU with 8 GB RAMMac using MATLAB 2012b. On-line estimation of states and all the parameters cited in thereference [7] by SIR filtering leads to large memory requirements and computational com-plexity. To reduce the computation load, only the parameters of type II diabetic subjectsthat have considerable effects on peripheral glucose, insulin and incretins concentrationswere chosen for estimation while keeping all other non-essential model parameters constant.4.5.1 Clinical data used for model developmentThe states and the parameters of the Vahidi model were estimated using two different clin-ical tests, oral glucose tolerance test (OGTT) and isoglycemic intravenous glucose infusiontest (IIVGIT) performed by Knop et al. [73, 74]. Ten type II diabetic patients (eight menand two women) have been selected for the tests.A 50 g glucose tolerance test (OGTT test) is performed in the first test and 17 bloodsamples are taken from the subjects during the test to determine how quickly glucagonsuppression occurred. Blood was sampled 15, 10 and 0 minutes before, and after theingestion of glucose at 5, 10, 15, 20, 30, 40, 45, 50, 60, 70, 90, 120, 150, 180 and 240minutes.In the second test, isoglycemic intravenous glucose infusion test (IIVGIT test) is carriedout to mimic the plasma glucose profile obtained from the OGTT test. Therefore, the sameamount of glucose was injected intravenously to the diabetic subjects in the IIVGIT test.Blood was sampled every 5 minutes [73, 74]. 20 blood samples are taken from the subjectsduring the second test. Details about the experiments and the subjects’ characteristics are884.5. Results and discussionavailable in [73, 74].Since the Sorensen model is proposed for a typical 70 kg subject and the clinical datasets, which we used are from subjects with different body weights, all clinical data is scaledto a 70 kg body weight using the following equation:C = (CC − CB)W70+ CB (4.19)where C is the substance concentration, W is the subjects body weight (kg), CC refers tothe concentration from original clinical data, and CB refers to the concentration at basalcondition. The normalized values of the clinical data sets for a 70 kg are provided inAppendix A.The information included in data sets are peripheral glucose, insulin and incretins(GLP-1 plus GIP) concentrations. The data from both tests are used for estimating theparameters of the model using SIR particle filtering method as follows:• From the IIVGIT test:– Incretins concentrations are not used since no secretion of incretins occurs duringthe IIVGIT test.– Insulin concentrations are used to estimate the parameters of the pancreasmodel.• From the OGTT test:– The rest of the integrated model parameters including the parameters of theglucose sub-model which also comprises the parameters of the glucose absorptionmodel and the parameters representing the hormonal effects of incretins on thepancreatic insulin production are all estimated using OGTT test data set.894.5. Results and discussion4.5.2 On-line states and parameters estimation resultsTo apply the SIR particle filtering method, a prior information on the unknown parametersof the Vahidi model from both OGTT and IIVGIT is needed. This information can beobtained from equation (2.67) by minimizing the deviation of model predictions from theavailable clinical measurements of peripheral glucose, insulin and incretins concentrationsdescribed in section 4.5.1.After the prior information on the unknown parameters obtained, the estimation ofthe states and parameters of the Vahidi model were estimated using SIR particle filteringmethod. Firstly, the parameters of the pancreas model were estimated from the isoglycemicintravenous glucose infusion test (IIVGIT) test since no secretion of incretins occurs duringthe IIVGIT test. In the model parameter estimation, the peripheral insulin concentrationin the Vahidi model [7] was considered as a measurement yk. For implementing the SIRfiltering based on the discrepancies between the Vahidi model and the Knop’s experimentaldata, the following parameters were selected:• The number of particles N = 5000• The sampling time used for discretizing the Vahidi model ∆k = 0.4 min• The maximum states noises vk ∼ N (0, 0.001)• The measurement noise wk ∼ N (0, 0.001)A priori information on {x0; θ} includes the lower (LB) and the upper bound (UB)based on the physiological considerations. Four simulation experiments were carried out toevaluate the effectiveness of our proposed method to identify patient models with incom-plete data. In the four experiments, 0%, 10%, 25% and 50% of available data simulatedfrom the Vahidi model were removed randomly. For example, when 10% of data were904.5. Results and discussionconsidered missing, a peripheral insulin concentration at each sampling time was removedfrom the original data set if a uniformly distributed random variable q in the interval (0,1)is less than 0.1. Similar experiments were done with 25% and 50% missing data [88].The parameter values after 600 samples from each of these experiments are shown inTable 4.1. The detailed information about these parameters were presented previouslyin section 2.2.3 from equation 2.53 to 2.58 in detail. For all the experiments, all theparameters except θ6 and θ7 converged to the neighbourhood of the original values after acertain number of iterations. θ6 and θ7 are not estimated precisely since the sensitivity ofthe KLD in kernel smoothing algorithm to changes in θ6 and θ7 is smaller than its variance.Table 4.1: Parameter estimation results for insulin sub-model after 600 sampling timeduring IIVGIT testParameters [7] OriginalValues Percentage of Missing insulin measurements0% 10% 25% 50%θ1 : α(min)−1 0.6152 0.6168 0.5728 0.5803 0.5159θ2 : γ(U/min) 2.3665 2.3422 2.1860 2.3667 2.5200θ3 : K(min)−1 0.0572 0.0565 0.0560 0.0561 0.0544θ4 : N1(min)−1 0.0499 0.0496 0.0519 0.0481 0.0474θ5 : N2(min)−1 0.0001490 0.0001489 0.0001482 0.0001500 0.0001506θ6 : ξ1(min)−1 0.000124 0.000125 0.000126 0.000125 0.000141θ7 : ξ2(min)−1 0.00270 0.00271 0.00280 0.00152 0.00122Variations of the parameters N1, N2, ξ1 and ξ2 are shown in Figure. 4.1 and thevariations of the parameters α, γ, and K are shown in Figure. 4.2.914.5. Results and discussionFigure 4.1: Variations of the parameters N1, N2, ξ1 and ξ2 in the Pancreatic insulin releasemodel924.5. Results and discussionFigure 4.2: Variations of the parameters α, γ, K in the Pancreatic insulin release model934.5. Results and discussionVariations of the peripheral insulin concentrations during the IIVGIT test are shownin Fig. 4.3a in which, r shows the percentage of missing observations. From the Fig. 4.3a,the dynamics of peripheral insulin concentration can be estimated reasonably well withphysiological responses in all the experiments even when 50% of the simulated clinical datawere absent. Fig. 4.3b presents the goodness of fit between the estimated output andthe measured output performed with MATLAB System Identification Toolbox by usingnormalized root mean square error (NRMSE) as a cost function. Based on the NRMSEmeasure, the goodness of fit between the simulated peripheral insulin concentration andthe available measurements are more than 80%.Figure 4.3: Peripheral insulin concentration for type II diabetic subjects during the IIVGITtestSecondly, from the oral glucose tolerance test (OGTT), the peripheral insulin concen-tration, peripheral glucose concentration and incretins concentrations were considered as944.5. Results and discussionmeasurements yk for estimation of the rest of the model parameters. These parametersconsist of the parameters of the glucose sub-model including the parameters of the glucoseabsorption model and the parameters describing the hormonal effects of incretins on thepancreatic insulin production as described previously in section 2.2.1.From the glucose sub-model, parameters of the glucose metabolic rates and some param-eters of the glucose absorption model have been considered for the parameter estimation.As the model equations in section 2.2.1 shows, the glucose metabolic rates in the glucosesub-model has the general form of equation 2.9 and the multipliers have the general formof equation 2.10. Considering equation 2.10, a, b, c and d are the parameters of the glucosemetabolic rates. To reduce the number of parameters for estimation, c and d are selectedfor the parameter estimation and a and b are considered to be unchanged.Therefore, parameters c and d in the hepatic glucose production (HGP) rate, the hepaticglucose uptake (HGU) rate, and the peripheral glucose uptake rate (PGU) are selected tobe estimated. For implementing the SIR filtering based on the discrepancies between theVahidi model and the Knop’s experimental data, the following parameters were selected:• The number of particles N = 20000• The sampling time used for discretizing the Vahidi model ∆k = 0.1 min• The maximum states noises vk ∼ N (0, 0.7)• The measurement noise wk ∼ N (0, 0.7)The parameter values after 2400 samples from each of these experiments are shown inTable 4.2. In all the experiments, the estimated parameters, except θ1 and θ4 convergedto the neighbourhood of the original values after a certain number of iterations. θ1 and θ4are not estimated precisely since the sensitivity of the KLD in kernel smoothing algorithmto changes in θ1 and θ4 is smaller than its variance.954.5. Results and discussionTable 4.2: Variation of the parameters cGHGP , cGHGU , dI∞HGP and K12 in glucose sub-modelafter 2400 sampling time during OGTT testParameters [7] OriginalValues Percentage of Missing insulin measurements0% 10% 25% 50%θ1 : cGHGP 1.0385 1.0352 1.0885 1.9743 2.0649θ2 : cGHGU 2.03 1.97 1.84 2.23 1.53θ3 : dI∞HGP 0.3648 0.3676 0.3610 0.3625 0.3667θ4 : K12(min)−1 0.0783 0.0796 0.0798 0.0842 0.0616θ5 : cIPGU 0.0970 0.0965 0.0965 0.0902 0.1300θ6 : cI∞HGU 3.2606 3.3125 2.9543 2.9625 2.8057θ7 : dIPGU 2.752 2.747 2.724 2.897 2.909θ8 : dI∞HGU 0.0031 0.0030 0.0028 0.0030 0.0038Variations of the parameters cGHGP , cGHGU , dI∞HGP and K12 are shown in Figure. 4.4 andthe variations of the parameters cIPGU , cI∞HGU , dIPGU and dI∞HGU are shown in Figure. 4.5.964.5. Results and discussionFigure 4.4: Variations of the parameters cIPGU , cI∞HGU , dIPGU and dI∞HGU in glucose sub-modelafter 2400 sampling time during OGTT test974.5. Results and discussionFigure 4.5: Parameter estimation results for glucose sub-model after 2400 sampling timeduring OGTT testVariations of the peripheral glucose, insulin, and incretins concentrations during theOGTT test after 2400 iterations are shown in Figs. 5.2a-5.4a. r shows the percentageof missing observations. From the Figs. 5.2a-5.4a, the dynamics of peripheral glucose,insulin, and incretins concentration can be estimated reasonably well with physiologicalresponses in all the experiments even when 50% of the clinical data is missing. Figs. 5.2b-5.4b present the goodness of fit between the estimated output and the measured outputperformed with MATLAB System Identification Toolbox by using the normalized rootmean square error (NRMSE) as a cost function. From Figs. 5.2b-5.4b, the goodness of984.5. Results and discussionfit between the simulated peripheral glucose, insulin and incretins concentration and theiravailable measurements were almost 80% in all the experiments except in Fig. 5.4b when25% and 50% of peripheral insulin measurements removed randomly. Comparing to Fig.4.3b in the IIVGIT test , the peripheral insulin concentration was not estimated preciselyin the OGTT test since only the parameters of the incretins sub-model and the parametersof the glucose sub-model were estimated in order to reduce the computational complexity.Figure 4.6: Peripheral glucose concentration for type II diabetic subjects during the OGTTtest994.5. Results and discussionFigure 4.7: Peripheral insulin concentration for type II diabetic subjects during the OGTTtest1004.5. Results and discussionFigure 4.8: Incretins concentration for type II diabetic subjects during the OGTT testThe probability density function of the parameter CIPGU , one of the parameters of theperipheral glucose uptake rate in the insulin sub-model in [98], is reported in Fig. 5.5. Assomewhat expected, the posterior provided by SIR filtering method is concentrated aroundits original value after about 620 sampling times.1014.5. Results and discussion050005000100020000.095 0.096 0.097 0.098 0.099 0.1 0.101010002000Probability density functionCIPGUIteration #607Iteration #620Iteration #600Iteration #1d)c)b)a)Figure 4.9: The probability density function of cIPGU from Bayesian identification at sam-pling time number 1, 600, 607 and 620 (area under each curve is unitary). The y-axisquantity is unit-less.4.5.3 Application of SIR particle filtering in detection of organdysfunction in diabetic patients under irregular clinical dataIn this section, the application of the adaptive sequential-importance-resampling (SIR)filter in the estimation of the glucose, insulin, and incretins concentrations in differentparts of the body under irregularly sampled clinical data is presented. These estimates areused for calculating the glucose metabolic rates in different organs of the type II diabeticpatients using irregularly sampled data. Then, by comparing the glucose metabolic rate ofeach organ in the diabetic patients with the glucose metabolic rate of the same organ in a1024.5. Results and discussionnormal subject, the abnormal functioning of certain organs is detected and identified.Using the states and parameters of the Vahidi model estimated in the previous section,the glucose metabolic rates in the peripheral tissues and the liver are calculated fromequations (2.9) and (2.10). Figure 5.6 shows the glucose metabolic rates in peripheraltissues and the liver compared with the healthy subjects. According to Figs. 5.6a and5.6b, the peripheral glucose uptake rate and hepatic glucose uptake rate in type II diabeticpatients for all experiments are less than the corresponding values in the healthy subjectsdue to insulin sensitivity in peripheral tissues and dysfunction in the liver of the diabeticpatients. Decreased rate of glucose infusion shows that the overall insulin sensitivity of thebody is decreased about 54% in diabetic patients. Even under the presence of 50% missingdata, the abnormalities of the liver and adipose tissues are detectable, which provides morephysiological information to physicians.1034.5. Results and discussion0 50 100 150 200 2500100200300400500Time (min)Peripheral glucose uptake rate (mg/min)  0 50 100 150 200 2500100200300400500Time (min)Hepatic glucose uptake rate (mg/min)  r=0%r=10%r=25%r=50%Normal subjectr=0%r=10%r=25%r=50%Normal subjecta)b)Figure 4.10: Variation of different glucose metabolic rates4.5.4 strengths and limitations of the SIR particle filtering in clinicalpracticeThe primary practical advantage of the SIR particle filtering method comparing to thetraditional statistical methods is its independence from the degree of nonlinearity of themodel unlike extended Kalman filtering [77]. An additional advantage in a clinical practiceis that the SIR filtering approach is readily adaptable to sequential updating of informa-tion obtained from owner history, clinical examination of diabetic patients, and results ofdifferent diagnostic tests [99]. It exhibits good performance even for systems with largeprocess or measurement noise.Furthermore, the accuracy of the particle filtering method can be improved by increas-1044.5. Results and discussioning the number of particles used in the estimation algorithm. However, particles size over1000 can be computationally intensive and time consuming [88, 99]. The SIR based PFused in this study, needs less computational cost when a large number of unknown statesand parameters must be estimated simultaneously since the marginal probability distribu-tion of each parameter and state can be obtained from a posteriori probability distributionof the model parameters and states [92].105Chapter 5Model-based detection of organdysfunction and faults in insulininfusion devices for type IIdiabetic patients5.1 IntroductionDiabetes is a disease characterized by abnormal glycemic values due to the inability of thepancreas to produce insulin (Type I diabetes) or to the inefficiency of insulin secretion andaction (type II diabetes). Patients affected by diabetes need to monitor their glycemic levelduring all day and control it inside the normal range of 70-180 mg/dl as much as possible.Glycemic level can be controlled by being in diet, doing exercise and taking oral med-ication. However, in patients with severe type II diabetes mellitus, insulin treatment isneeded like type I diabetes. The insulin treatment can be either multiple daily injectionregimens (MDIR) or a continuous subcutaneous insulin infusion (CSII) pump [59]. Re-cently, new technologies have been developed in order to improve and facilitate diabetestherapy such as [100]:1065.1. Introduction• sensors for continuous glucose monitoring (CGM), minimally invasive devices, whichmeasure real-time glucose levels and returns the value in every 1 to 5 minutes for upto 7 days.• pumps for continuous subcutaneous insulin infusion (CSII), which allow a more ef-fective and physiological delivery of insulin. Moreover, the sensor-augmented pump,which are the simple combination of pumps in a single device, makes a further re-duction of time spent in hypoglycemia and hyperglycemia.The availability of CGM sensors and CSII pumps gave the idea of developing an artificialpancreas. These system is based on a closed-loop control algorithm in which CGM measuresthe glycemic value as a receiving input and the optimal insulin dosage as an output of thecontroller is infused by CSII pump to keep glycemia in the normal range [100].The usage of CGM sensors and CSII pumps have made more progress, efficacy andsafety in improving the quality of life of people with diabetes compared to the usual multipledaily injection therapy [59, 101]. In such a system, detection of possible failures in eitherthe CGM sensor or CSII pump is crucial for safety.During day-time while the patient is awake, failures are less critical since they can befixed by patients. However, in the night-time while the patient is asleep, failures are moredangerous. The possible failures in insulin pump therapy are listed as follows [101]:• Occlusions in the infusion set; An occlusion is any blockage that prevents the pumpfrom delivering insulin properly. Occlusions may occur for any of the following rea-sons:– If pressure is being applied to the tubing or the infusion site– If the cannula has been bent during insertion– Kinked insulin pump tubing1075.1. Introduction– Crystals forming in the insulin and causing blockages at the cannula• Disconnection or leakage in the infusion set• Presence of the bubbles in insulin pumpsIf any of the aforementioned issues happen during the insulin pump therapy, the bodywill not get the intended full insulin dose, which can lead to higher than normal bloodglucose level. In addition, even under safe insulin pump therapy, the control of the bloodglucose level may fail due to the organ dysfunction progression. Multiple abnormalities indifferent body organs are listed as follows [77, 102]:• Resistance of muscles and adipose tissues against the secreted insulin• Impaired insulin-induced suppression of hepatic glucose production• Abnormal hepatic glucose uptake rate• Deficiency in pancreatic insulin production rateThe goal of using fault detection (FD) of glucose-insulin system is to detect the glucosecontrol failure. FD technique has been used previously in many literatures to detect failuresin insulin pump therapy for type I diabetes mellitus.A multivariate statistical technique proposed by Finan et al. [103] detects insulin pumpleakages and glucose sensor bias. Fecchinetti et al. [100] proposed a model-based approachusing a Kalman estimator for detecting failures in both continuous subcutaneous insulin in-fusion (CSII) and continuous glucose monitoring (CGM) to improve safety during overnightglycemic control. Herrero et al. [101] proposed utilizing a validated robust model-basedfault detection technique based on the interval analysis for detecting disconnections of theinsulin infusion set.1085.1. IntroductionAll the aforementioned studies used the Bergman minimal model (MINMOD) includingthree nonlinear differential equations representing variations of plasma insulin and glucoseconcentrations for type I diabetic patients. However, in this study, we used the detailednonlinear compartmental type II diabetes model developed by Vahidi et al. [51].In the previous chapter, we estimated the states and the parameters of the Vahidimodel using a sequential monte carlo (SMC) filtering method called particle filters. In thischapter, we propose for the first time to our knowledge, the application of the model-basedFD algorithm based on a sequential monte carlo (SMC) method to detect either the faultsin insulin pump therapy or the organ abnormalities in type II diabetic patients. In thenext section, we present definitions of FD and a theoretical background.5.1.1 Theoretical background on fault detection approachesThere is a large volume of literature on fault detection. In the last four decades, a varietyof techniques to solve a number of process monitoring and fault detection problems havebeen developed. Many of these techniques are described in the survey papers by Basseville[104], Frank [105], Isermann[106], and Willsky [107] and in the books by Willsky [108],Basseville and Nikiforov[109], Pouliezos and Stavrakakis [110] and the references therein.The primary objectives of all FD methods are to detect any deviation from the normalbehaviour of the process by providing an alarm tool [111].There are two approaches for detection of failures [96]:• Model-based approaches, which are based on a physical model of the process.• History-based approaches, which rely on large historical data sets.A model-based approaches often tend to be more powerful and provide a better per-formance if the process is well modelled [112]. The model-based approaches typically areconsist of two procedures [97, 113, 114]:1095.1. Introduction1. extracting fault symptoms from the process, and residual evaluation2. decision making based on the residual evaluationResidual generation in model based approaches is the most important step, whichis non-trivial in processes with unmeasured state variables [97, 113, 114]. A schematicrepresentation of model-based fault diagnosis is shown in Figure. 5.1.Figure 5.1: Model-based fault diagnosis Scheme [97, 113, 114]If the models is deterministic, the residuals are often generated using an observer.Otherwise, for stochastic models, a filter has been used. Typically, for residuals generation,the model-based methods rely on the model being linear and noise being Gaussian [95].The model-based methods have been extended to nonlinear systems in the literature.However, these extensions are based on suboptimal state estimators such as extendedKalman filters (EKF), and unscented Kalman filters (UKF). In these suboptimal filters,the nonlinear system has been approximated through linearization and/or the noise hasbeen assumed Gaussian. The approximations are often not satisfactory and lead to a high1105.2. Problem statementrate of false alarms.In the light of aforementioned above, we use the model-based FD algorithm based ona sequential monte carlo (SMC) method called particle filter to detect either the faultsin insulin pump therapy or the organ abnormalities in type II diabetic patients. Theproposed approach does not require linearization of the Vahidi model or Gaussianity of themeasurement noises [96].Since the SMC methods are computationally intensive, their implementation needs thehigh performance computers. There is some existing literature on the use of SMC forfault detection [96, 115–117]. In these studies, the SMC algorithms proposed are basedon the log-likelihood test of observed data under a null and an alternate hypothesis. Inthese approaches the likelihood function is estimated under both hypotheses and the like-lihood function is driven through an approximation that is not applicable to all types ofnonlinearities.In this study, we propose the application of the model-based FD algorithm based on asequential monte carlo (SMC) method to detect either the faults in insulin pump therapyor the organ abnormalities in type II diabetic patients.This chapter is organized as follows. In the following section, the model-based faultdetection based on the SMC filtering method is discussed. In section 5.3, the proposedtechnique for detecting disconnection in insulin infusion systems and detecting organs de-ficiency are explained.5.2 Problem statementIn order to use the model-based fault detection technique, the discrete format of the Vahidimodel described in section 4.2 in equation 4.3 has been used. The measurement and statenoises presented in equations (4.3a) and (4.3b) are assumed to enter the process in a1115.2. Problem statementlinear fashion in linear processes and, to some extent, in nonlinear processes. Therefore,based on this fundamental assumption, faults can be detected simply by generating andmonitoring the prediction errors (or residuals) between the process measurements andmodel predictions. The one step-ahead predictions from equation (4.3b) can be written as:yˆt = g(xt|t−1, ut, θ) (5.1)where xt|t−1 is the one step-ahead prediction of the state, yˆt is the one-step ahead predictionof the output. Then the prediction error or the residual can be simply written as:rˆt = yt − yˆt (5.2)When there are no changes in the glucose-insulin system, the density function of theresiduals, rˆt, must closely follow the measurement noise, wt. Any deviation of the residualsfrom this density function implies a fault in the glucose-insulin system.Measurements of glucose and insulin concentrations in different parts of the body re-quire complex clinical facilities and in some cases may risk the life of the patient. Therefore,clinical measurements of all required concentrations are not possible. The commonly avail-able clinical data include peripheral insulin and glucose concentrations only. Therefore,a straightforward residual analysis, as presented in equations (5.1) and (5.2) is difficult.To overcome this problem, we use a sequential monte carlo (SMC) filtering method calledparticle filters on a nonlinear model of a group of diabetic patients to estimate the glucoseand insulin concentrations in different parts of the body. SMC filtering method previouslydescribed in details in section 4.3.3.1125.3. Fault detection of glucose-insulin system5.3 Fault detection of glucose-insulin systemIn this section, the efficiency of the sequential monte carlo (SMC) filtering method forfailure detection in a glucose-insulin system is demonstrated. To build a model-basedfault detection system, we need to have reliable type II diabetes model. In the previouschapter, we used the data provided by Knop et al. [73, 74] to estimate the parameters ofthe Vahidi model. In this chapter, we use the same parameters to simulate the body of thetype II diabetic patient, which is under closed-loop insulin pump therapy. The closed-loopsimulation assumes that the patient’s initial blood glucose is at 115.63 mg/dl. The mealdisturbance of 75 gr glucose was introduced at time 500 min. A PI controller with tuningparameters KC = 0.22 and KI = 0.44 is simulated to control the patient’s blood glucoselevel at 90 mg/dl. Fig. 5.2 presents the responses of the PI controller. At time 500 minglucose level increases due to 75 gr meal disturbance. However, the PI controller is ableto control the blood glucose level at 90 mg/dl.1135.3. Fault detection of glucose-insulin system0 200 400 600 800 1000 1200 1400 160050100150Glucose Conc. (mg/dl)Time (min)0 200 400 600 800 1000 1200 1400 16000204060Insulin Infusion Rate (mU/min)Time (min)Figure 5.2: Response of the PI controller with a 75 gr meal disturbance at time = 500min.In the following sections, four different fault cases are simulated and the SMC filteringmethod is used to detect the faults. For implementing the SMC filtering, based on thediscrepancies between the Vahidi model and the Knop’s experimental data, the followingparameters were selected:• The number of particles N = 25• The sampling time used for discretizing the Vahidi model ∆t = 0.2 min• The maximum states noises vt ∼ N (0, 0.001)• The measurement noise wt ∼ N (0, 0.001)1145.3. Fault detection of glucose-insulin systemAll the simulations were conducted on a 2.90 GHz CPU with 8 GB RAM Mac usingMATLAB 2012b.5.3.1 Detection of insulin pump disconnection (Case 1)In case 1, we assume that the insulin pump is disconnected 800 min after the diabeticpatient consumes the 75 gr meal at 500 min. Figure. 5.3a shows the response of the PIcontroller when there is no fault and one when the insulin pump is disconnected at time1300 min.0 200 400 600 800 1000 1200 1400 160050100150Glucose      Conc. (mg/dl)0 200 400 600 800 1000 1200 1400 1600−50050Residual0 200 400 600 800 1000 1200 1400 160000.51Alarm Signal0 200 400 600 800 1000 1200 1400 1600050Time (min)Insulin Infusion Rate (mU/min)  b)a)c)d)Figure 5.3: Detection of insulin pump disconnection. The blue solid line curve representsno fault and red dashed curve represents fault in Case 1The residual or prediction error between the measured and the predicted blood glucose1155.3. Fault detection of glucose-insulin systemlevel is shown in Fig. 5.3b. The residual closely follows the measurement noise, wt beforeinsulin pump gets disconnected. However, it deviates from this density function after time1300 min.Figure. 5.3c shows the alarm signal, which stays at zero while the prediction errorbetween predicted and measured blood glucose level is less than 10 mg/dl. Otherwise, itstays at one. Figure. 5.3d shows that the insulin pump is disconnected at time 1300 minand the insulin infusion rate is zero after time 1300 min.5.3.2 Detection of kinked insulin pump tubing (Case 2)In case 2, we assume that the insulin pump is kinked 800 min after the diabetic patientconsumes the 75 gr meal at 500 min. Figure. 5.4a shows the response of the PI controllerwhen there is no fault and one when the insulin pump is kinked after time 1300 min. Tosimulate the kinked insulin pump tubing, we assume that the PI controller output (insulininfusion rate) is fluctuated between 0 to 6 mU/min after 1300 min.1165.3. Fault detection of glucose-insulin system0 200 400 600 800 1000 1200 1400 160050100150Glucose        Conc. (mg/dl)0 200 400 600 800 1000 1200 1400 1600−20020Residual0 200 400 600 800 1000 1200 1400 160000.51Alarm Signal0 200 400 600 800 1000 1200 1400 1600−50050Time (min)Insulin InfusionRate (mU/min)   a)b)c)d)Figure 5.4: Detection of kinked insulin pump tubing. The blue solid line curve representsno fault and red dashed curve represents fault in Case 2The residual or prediction error between the measured and the predicted blood glucoselevel is shown in Fig. 5.4b. The residual closely follows the measurement noise, wt beforeinsulin pump gets kinked. However, it deviates from this density function after time 1300min.Figure. 5.4c shows the alarm signal, which stays at zero while the prediction errorbetween predicted and measured blood glucose level is less than 10 mg/dl. Otherwise, itstays at one. Figure. 5.4d shows that the insulin pump is kinked at time 1300 min andthe insulin infusion rate is fluctuated.1175.3. Fault detection of glucose-insulin system5.3.3 Detection of organ dysfunctionThe glucose metabolic rates in the liver, muscles and adipose tissues represent the behaviourof those organs. From section 2.1, these rates are calculated by measuring the glucose andinsulin concentrations in different parts of the body using complex clinical facilities. How-ever, only peripheral insulin and glucose concentrations are available in common clinicaldata.In the following sections the glucose concentration in peripheral tissues and the liverare estimated using the SMC filtering method. Then, using the estimated glucose concen-trations in peripheral tissue and liver, peripheral glucose uptake rate and hepatic glucoseuptake rate are calculated. Decrease in the glucose uptake rate and hepatic uptake ratewill provide insight into the insulin resistance of peripheral tissues and liver.Detection of peripheral insulin resistance (Case 3)In case 3, we assume that the peripheral glucose uptake rate, rPGU , is decreased by 80%from the previous rate in the simulated patient under insulin pump therapy. The meal dis-turbance of 75 gr glucose was introduced at time 500 min. Figure. 5.5a shows the responseof the PI controller when the insulin sensitivity of the peripheral tissues is decreased by80%.1185.3. Fault detection of glucose-insulin system0 200 400 600 800 1000 1200 1400 16000200400Glucose      Conc. (mg/dl)0 200 400 600 800 1000 1200 1400 1600−2000200Residual0 200 400 600 800 1000 1200 1400 160000.51Alarm Signal0 200 400 600 800 1000 1200 1400 16000100200Time (min)r PGU(mg/min)a)b)c)d)Figure 5.5: Detection of peripheral insulin resistance. The blue solid line curve representsno fault and red dashed curve represents fault in Case 3The residual or prediction error between the measured and the predicted blood glucoselevel is shown in Fig. 5.5b. The residual deviates significantly during the meal ingestion.Figure. 5.5c shows the alarm signal, which stays at zero while the prediction error betweenpredicted and measured blood glucose level is less than 10 mg/dl. Otherwise, it stays atone.In Fig. 5.5d, the profile of the peripheral glucose uptake rate is shown. The increasein blood glucose level during the meal ingestion at time 500 min is due to %80 increase ininsulin resistance of the peripheral tissues.1195.3. Fault detection of glucose-insulin systemDetection of hepatic insulin resistance (Case 4)In case 4, we assume that the hepatic glucose uptake rate, rHGU , is decreased by 80%from the previous rate. The meal disturbance of 75 gr glucose was consumed at time 500min. Figure. 5.6a shows the response of the PI controller when there is no hepatic insulinresistance and one when the hepatic insulin resistance decreased by 80%.0 200 400 600 800 1000 1200 1400 16000100200Glucose      Conc. (mg/dl)0 200 400 600 800 1000 1200 1400 1600−50050Residual0 200 400 600 800 1000 1200 1400 160000.51Alarm Signal0 200 400 600 800 1000 1200 1400 16000100200Time (min)r HGU(mg/min)a)b)c)d)Figure 5.6: Detection of hepatic insulin resistance. The blue solid line curve represents nofault and red dashed curve represents fault in Case 4The residual or prediction error between the measured and the predicted blood glucoselevel is shown in Fig. 5.6b. The residual deviates significantly during the meal ingestion.Figure. 5.6c shows the alarm signal, which stays at zero while the prediction error betweenpredicted and measured blood glucose level is less than 10 mg/dl. Otherwise, it stays at1205.3. Fault detection of glucose-insulin systemone.In Fig. 5.6d, the profile of the hepatic glucose uptake rate is shown. The increase inblood glucose level during the meal ingestion at time 500 min is due to %80 increase inhepatic insulin resistance.121Chapter 6Conclusions and recommendations6.1 SummaryThis thesis focuses on assessment of type II diabetes mellitus using compartmental mathe-matical modelling. The outcomes of this research are summarized in the following sections.6.1.1 A simple self-administered method for assessing insulinsensitivity in type II diabetic patientsChapter 3 presented a simple self-administered method for assessing insulin sensitivityin type II diabetic patients. In this chapter, the feasibility of using the mathematicalcompartment model proposed by Vahidi et al. [7, 51] to estimate insulin sensitivity has beenevaluated. Fifteen sets of OGTT data from diabetic patients published in the literaturehave been used to estimate the Vahidi model parameters. From the estimated modelparameters, a simple method for conveniently estimating insulin sensitivity by patientsthemselves has been developed and evaluated. It is shown that the proposed methodyields an ISI measure, which is strongly correlated with the M-value obtained from theeuglycemic clamp test (r =0.927, p= 0.0045).1226.1. Summary6.1.2 Assessment of type II diabetes mellitus using irregularly sampledmeasurements with missing dataIn chapter 4, we have identified the nonlinear states and the parameters of glucose, insulinand incretins sub-model developed by Vahidi et.al [7] for type II diabetes mellitus in thepresence of 10%, 25% and 50% of randomly missing clinical observations by employing theSequential Importance Resampling (SIR) filtering method. The motivation for this studyoriginates from the lack of complete knowledge about the health status of the diabeticpatients. In addition, only a few blood glucose measurements per day are available in anon-clinical setting due to different reasons like unreadable hand writing, inability to recordclinical results, and infrequent sampling by patients.It is shown that by implementing an on-line SIR particle filtering method to the Vahidimodel developed for type II diabetes mellitus, we are able to estimate the dynamics of theplasma glucose, insulin and incretins concentration under the presence of maximum 50%available clinical data. In addition, the goodness of fit between the simulated peripheralglucose, insulin and incretins concentration and their available measurements were almost80% in the most of the experiments. The results of this study can be used to inform typeII diabetic patients of their medical conditions, enable physicians to review past therapy,estimate future blood glucose levels, provide therapeutic recommendations and even designa stabilizing control system for blood glucose regulation.6.1.3 Model-based detection of organ dysfunction and faults in insulininfusion devices for type 2 diabetic patientsIn chapter 5, we have used the type II diabetes model developed by Vahidi et.al [51].We employed model-based fault detection technique based on a Sequential Monte Carlo(SMC) filtering method for detecting faults in insulin infusion system and detecting organ1236.2. Recommendations for future workdysfunction. The proposed approach has been demonstrated (in silico) to be an effectivetool for detecting disconnection faults in insulin infusion systems and detecting organdeficiencies. In the future work, we intend to extend this work to develop an algorithmcapable of detecting and isolating faults simultaneously in a glucose-insulin system.6.2 Recommendations for future workVarious medications are available for type II diabetic subjects such as insulin, sulfonylureas,meglitinides, biguanides and thiazolidinediones that decrease the blood glucose level in typeII diabetic patients. Specific type of medication is prescribed to the diabetic patients basedon the type and severity of abnormalities. Therefore, the information obtained from thedeveloped type II diabetes model would be helpful in prescribing a suitable medication forthe diabetic patients. In addition, the chance of prescribing an efficient medication for thepatients in a safe and cost effective way has been increased.In the light of aforementioned above, the research area, which can be taken into moreconsideration is the study of development of pharmacokinetic-pharmacodynamic (PK-PD)models for different medicines. The term “pharmacokinetics” refers to a branch of phar-macology studying the fate of an external substance administered to a live organism. Theterm “pharmacodynamic” refers to another branch of pharmacology examining the bio-chemical and physiological effects of a medicine on a live organism. The PK-PD modelsstudy the effect of different medicines on the patients safely without any administrationthat may be harmful for the patients.A pharmacokinetic model can be attached to the type II diabetes model representinghow the medication is distributed into the body organs and consumed by them. For thepharmacodynamic part, structural modification should be attached into the type II diabetesmodel to represent the effects of the medication on body organs. I have contributed in1246.2. Recommendations for future workdeveloping a pharmacokinetic pharmacodynamic model for metformin and four types ofinsulin (regular, NPH, lente and ultralente) whose preliminary results have been publishedin [76]. Similarly, PK-PD models for other medications can be developed for studying theeffects of them on lowering the blood sugar level.125Bibliography[1] Melissa Barazandegan, Fatemeh Ekram, Ezra Kwok, and Bhushan Gopaluni. SimpleSelf-Administered Method for Assessing Insulin Sensitivity in Diabetic Patients. J.Med. Biol. Eng., 36(2):197–205, apr 2016.[2] Melissa Barazandegan, Fatemeh Ekram, Ezra Kwok, Bhushan Gopaluni, and AdityaTulsyan. Assessment of type II diabetes mellitus using irregularly sampled measure-ments with missing data. Bioprocess Biosyst. 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Eng., 27(3):313 – 326, 2003.141Appendix AThe numerical values of the clinical data sets used in chapter 4 are provided here in detailed.These values are normalized by the body weight of the subjects shown in Table A.1.Table A.1: Gender and body weight of diabetic subjects [7]Subject1 2 3 4 5 6 7 8 9 10Body weight Kg 73.0 81.0 106.0 74.0 53.0 65.5 77.5 69.0 69.0 69.5Gender M M M M F F M M M MTable A.2: Normalized GLP-1 concentration data set (pmol/l) of diabetic subjects forOGTT [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 17.7 19.7 18.0 18.3 15.7 11.7 22.7 8.0 9.7 9.3 15.115 41.0 28.2 30.1 15.9 28.8 14.8 25.3 9.0 9.0 16.0 21.830 26.4 42.0 36.2 32.8 28.8 48.5 33.0 29.7 13.9 19.9 31.145 29.5 36.3 36.2 18.0 36.4 65.3 27.5 35.6 16.9 30.8 33.260 36.8 28.2 39.2 22.2 29.5 45.7 25.3 22.8 17.9 29.9 29.790 22.2 16.6 30.1 18.0 20.5 20.4 23.0 18.8 22.8 21.9 21.4120 12.8 10.8 33.1 11.6 13.6 15.7 24.1 12.9 13.9 16.0 16.5150 9.7 10.8 39.2 14.8 21.2 18.5 23.0 13.9 10.5 14.0 17.6180 14.9 13.1 28.6 21.2 16.7 14.8 14.2 16.9 7.0 15.0 16.2240 9.7 14.3 19.5 16.9 11.4 12.0 9.8 10.0 10.0 16.0 12.9142Appendix A.Table A.3: Normalized GIP concentration data set (pmol/l) of diabetic subjects for OGTTtest [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 9.0 6.7 9.0 36.0 16.3 12.3 30.3 8.7 11.7 42.3 18.215 89.3 80.0 127.1 77.2 61.5 47.6 125.2 26.7 19.9 100.6 75.530 59.1 118.1 133.2 144.9 67.6 78.5 107.5 32.7 45.5 110.5 89.745 83.0 95.0 115.0 142.8 72.9 55.1 108.6 28.7 57.3 111.5 87.060 69.5 95.0 90.8 113.2 61.5 60.7 105.3 35.6 62.3 118.5 81.290 72.6 39.5 68.1 66.7 59.2 40.1 78.7 26.7 51.4 93.6 59.7120 33.0 22.1 31.7 40.2 24.4 30.7 41.0 15.9 24.8 32.1 29.6150 15.3 11.7 9.0 18.0 13.1 17.6 20.0 8.0 17.4 21.2 15.1180 10.0 11.7 10.5 19.1 10.0 14.8 8.9 10.0 10.0 24.1 12.9240 6.9 10.5 12.0 10.6 12.3 9.2 17.8 9.0 7.1 12.2 10.8143Appendix A.Table A.4: Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for OGTT test [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 8.4 9.1 7.5 10.4 7.6 7.3 8.3 6.2 12.0 8.4 8.55 8.1 8.8 6.7 9.9 7.2 6.7 8.0 6.1 11.6 8.3 8.110 8.8 9.9 8.3 10.8 8.5 7.5 8.1 6.6 12.5 8.7 9.015 10.0 11.3 9.6 11.5 9.3 8.3 9.3 8.0 12.8 9.7 10.020 11.1 13.2 10.7 12.5 10.1 9.1 10.7 7.9 14.1 10.8 11.030 12.3 15.8 13.6 16.4 11.0 11.4 14.4 10.0 15.1 12.5 13.240 13.2 18.0 14.6 17.6 12.0 13.0 15.3 11.2 16.8 14.5 14.650 14.7 18.6 16.4 18.6 12.7 12.7 16.4 11.7 17.5 15.8 15.560 16.3 19.4 17.3 19.6 13.1 12.7 17.0 11.5 18.5 16.9 16.370 16.8 19.6 16.3 19.6 13.4 11.3 16.8 11.6 19.1 17.7 16.290 18.4 16.7 15.5 19.3 12.7 9.6 13.5 11.3 20.2 17.5 15.5120 17.5 14.0 13.3 17.4 10.9 7.8 10.1 9.7 19.2 13.6 13.3150 16.4 12.0 10.2 15.4 8.6 5.8 8.3 7.6 17.5 11.0 11.3180 14.3 9.6 7.7 14.0 7.5 5.2 7.1 6.5 16.2 9.0 9.7240 12.2 7.2 5.7 12.4 6.4 5.3 5.4 5.5 14.7 6.4 8.1144Appendix A.Table A.5: Normalized peripheral insulin concentration data set (pmol/l) of diabetic sub-jects for OGTT test [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 43 50 126 48 52 32 27 17 22 33 4510 33 71 102 42 68 34 34 14 29 39 4620 75 122 216 29 122 89 68 40 37 62 8630 109 153 379 67 114 197 99 77 34 83 13140 66 202 399 55 152 202 99 91 57 59 13850 68 180 408 81 201 155 133 59 53 102 14460 81 184 446 75 199 196 152 48 47 102 15370 73 85 393 50 161 198 179 57 35 165 14090 82 107 414 65 175 173 129 59 36 194 143120 49 111 378 52 107 102 74 38 24 78 101150 63 87 250 63 60 38 40 28 24 64 72180 50 69 184 42 66 26 30 14 33 65 58240 38 44 81 46 42 30 14 5 22 28 35145Appendix A.Table A.6: Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for IIVGIT test [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 9.3 10.8 7.5 10.4 6.7 7.4 7.2 5.5 12.4 8.8 8.65 8.7 11.0 6.4 9.7 7.5 6.3 6.3 4.8 11.9 8.3 8.110 9.1 11.1 8.9 10.7 8.3 7.7 8.2 6.6 12.3 8.6 9.115 9.7 11.1 10.7 11.9 9.2 9.2 9.7 8.0 13.5 10.5 10.320 10.4 13.3 12.0 12.1 9.9 9.5 10.3 8.7 14.0 11.4 11.225 10.8 14.3 13.1 12.8 10.3 9.7 10.6 9.3 14.0 11.8 11.730 11.4 15.4 15.2 14.0 11.3 10.5 12.3 10.5 15.3 13.1 12.935 12.3 16.0 16.1 13.5 12.3 10.8 13.1 10.5 15.7 13.8 13.440 13.2 17.0 18.7 15.9 13.3 11.5 15.4 11.3 16.6 14.3 14.745 14.0 17.7 18.7 17.8 14.3 11.9 16.7 10.1 16.8 14.6 15.350 14.6 18.3 18.6 18.6 14.4 13.8 17.4 12.2 17.3 15.7 16.160 16.3 19.0 17.9 19.4 14.1 13.9 16.9 12.1 18.0 17.4 16.570 16.9 19.8 17.2 19.9 12.8 12.4 16.9 12.3 19.1 17.9 16.590 17.4 18.0 14.9 20.2 11.5 9.6 15.2 12.9 19.7 17.8 15.7120 17.9 15.1 12.3 18.1 10.3 7.4 11.9 11.8 19.9 14.9 13.9150 16.1 12.9 9.8 16.2 8.8 6.0 8.2 9.9 18.2 12.5 11.9180 15.0 11.0 8.3 13.8 7.3 5.3 7.1 8.3 17.1 10.4 10.4240 12.6 8.8 5.8 12.3 6.1 5.2 5.2 6.3 15.3 7.3 8.5146Appendix A.Table A.7: Normalized peripheral glucose concentration data set (mmol/l) of diabeticsubjects for IIVGIT test [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 9.3 10.8 7.5 10.4 6.7 7.4 7.2 5.5 12.4 8.8 8.65 8.7 11.0 6.4 9.7 7.5 6.3 6.3 4.8 11.9 8.3 8.110 9.1 11.1 8.9 10.7 8.3 7.7 8.2 6.6 12.3 8.6 9.115 9.7 11.1 10.7 11.9 9.2 9.2 9.7 8.0 13.5 10.5 10.320 10.4 13.3 12.0 12.1 9.9 9.5 10.3 8.7 14.0 11.4 11.225 10.8 14.3 13.1 12.8 10.3 9.7 10.6 9.3 14.0 11.8 11.730 11.4 15.4 15.2 14.0 11.3 10.5 12.3 10.5 15.3 13.1 12.935 12.3 16.0 16.1 13.5 12.3 10.8 13.1 10.5 15.7 13.8 13.440 13.2 17.0 18.7 15.9 13.3 11.5 15.4 11.3 16.6 14.3 14.745 14.0 17.7 18.7 17.8 14.3 11.9 16.7 10.1 16.8 14.6 15.350 14.6 18.3 18.6 18.6 14.4 13.8 17.4 12.2 17.3 15.7 16.160 16.3 19.0 17.9 19.4 14.1 13.9 16.9 12.1 18.0 17.4 16.570 16.9 19.8 17.2 19.9 12.8 12.4 16.9 12.3 19.1 17.9 16.590 17.4 18.0 14.9 20.2 11.5 9.6 15.2 12.9 19.7 17.8 15.7120 17.9 15.1 12.3 18.1 10.3 7.4 11.9 11.8 19.9 14.9 13.9150 16.1 12.9 9.8 16.2 8.8 6.0 8.2 9.9 18.2 12.5 11.9180 15.0 11.0 8.3 13.8 7.3 5.3 7.1 8.3 17.1 10.4 10.4240 12.6 8.8 5.8 12.3 6.1 5.2 5.2 6.3 15.3 7.3 8.5147Appendix A.Table A.8: Normalized peripheral insulin concentration data set (pmol/l) of diabetic sub-jects for IIVGIT test [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100 27 74 111 46 45 33 17 7 17 32 4110 21 73 90 46 47 44 21 26 18 30 4220 22 84 152 36 47 51 28 28 11 30 4930 20 80 209 43 63 55 31 26 12 36 5740 18 98 244 45 62 78 39 31 16 47 6850 22 106 297 57 67 99 44 30 19 49 7960 25 101 334 75 79 126 73 25 15 66 9270 27 105 259 91 64 117 71 23 17 47 8290 37 101 349 75 70 108 62 24 20 42 89120 36 95 294 64 84 68 72 19 23 62 82150 41 100 197 47 56 53 55 21 21 16 61180 50 110 161 53 40 21 29 17 24 16 52240 30 50 91 35 31 19 17 10 20 27 33148Appendix A.Table A.9: : Intravenous glucose infusion amount (g) to diabetic subjects during IIVGITtest [7]Time (min)SubjectMean1 2 3 4 5 6 7 8 9 100-15 1.53 0.2 13.8 2.4 8 3.2 6 6.4 2.8 2.6 4.715-30 4.41 11 10.6 6.4 8.8 7.6 11.2 10 7.8 8.2 8.6230-45 9.21 8.6 17.4 13.2 12.4 11.6 12.8 3.8 8.4 8.4 10.6245-60 7.48 8.6 4.8 12.2 3.8 7.8 8 7.8 8.2 11.2 8.0260-75 5.95 6.6 2.8 7.6 1.4 0.8 4.8 5.8 9.2 8.6 5.3875-90 4.99 1.4 0.8 3.2 2 0.4 2 4.2 7.2 6.2 3.2690-105 6.33 0 0.2 2 2.8 0.2 0.2 3 7 0.4 2.24105-120 0.96 0 0 2 3 0.4 0 1.4 0.4 0 0.82120-135 0.58 0 0 0.6 3.2 0.2 0 0 0.2 0 0.48135-150 0.38 0 0 0.4 0 0.2 0 0 0 0 0.1150-165 0.58 0 0 0 0 0 0 0 0 0 0.06165-180 0 0 0 0 0 0 0 0 0 0 0180-195 0.19 0 0 0 0 0 0 0 0 0 0.02195-240 0 0 0 0 0 0 0 0 0 0 0Total 42.6 36.4 50.4 50 45.4 32.4 45 42.4 51.2 45.6 44.32149

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