Review and Analysis of Blood Glucose (BG) Models for Type 1 Diabetic Patients

Published: September 12, 2011

r 2011 American Chemical Society 12041 dx.doi.org/10.1021/ie2004779 | Ind. Eng. Chem. Res. 2011, 50, 12041–12066

ARTICLE

pubs.acs.org/IECR

Review and Analysis of Blood Glucose (BG) Models for Type 1 DiabeticPatientsNaviyn Prabhu Balakrishnan, Gade Pandu Rangaiah,* and Lakshminarayanan Samavedham

Department of Chemical and Biomolecular Engineering, National University of Singapore, Kent Ridge Campus, 4 Engineering Drive 4,Singapore 117576

bS Supporting Information

ABSTRACT: Blood glucose (BG) regulation in type 1 diabetic patients has been investigated by researchers for a long time. Manymathematical models mimicking the physiological behavior of diabetic patients have been developed to predict BG variations.Models characterizingmeal absorption and physical activities have also been developed in the literature, as they play a significant rolein altering BG levels. Hence, existing glucose-insulin dynamic models dating back from early 1960s are reviewed along with anoverview of meal absorption and exercise effect models. The available knowledge-driven BG models have been classified intodifferent families based on their origin for development. Also, five knowledge-driven BG models (with at least one model from afamily) have been analyzed by either varying basal insulin or meal ingestion. The available meal absorption models have also beensimulated to compare and analyze them for different meal sizes. The major objective of the analysis is to study the BG dynamics ofdifferent models at their nominal parameter values, under varying basal insulin doses and meal ingestion. Similar analysis has beenperformed on 10 adult patient models in a recent benchmark simulator for comparison. These results will be useful forunderstanding the responses of different BG models at their nominal parameter values and for preliminary selection of a suitabletreatment model(s) for a patient.

1. INTRODUCTION

According to International Diabetes Federation, in the year2010, about 4 million deaths within the 20�79 age groups acrossthe globe may have been due to diabetic complications.28 WorldHealth Organization (WHO) has defined diabetes as a chronicdisease which results in increased blood glucose (BG) concen-tration due to the inability of pancreas to secrete insulin or due toineffective usage of produced insulin.31 Generally, the normal BGlevels reported in literature vary in a narrow range of 70�110mg/dl34 after 2�3 h of a meal or following an overnight fast. Inhealthy individuals, α- and β-cells of pancreas play an active rolein maintaining the BG levels within the specified range (seeFigure.1 for the normal glucose-insulin regulatory mechanism).When there is an increased BG concentration, β-cells of pancreassecrete insulin hormone which stimulates: (i) hepatic glucoseuptake, and (ii) extra-hepatic glucose uptake from blood. Con-versely, when low BG level exists, α-cells of pancreas secreteglucagon hormone which acts on liver cells to breakdown thestored glycogen into glucose and release it into blood therebyachieving the normoglycemia levels. When BG levels are out ofthe narrow range for a prolonged period of time, it will leadto hyperglycemic (BG > 120 mg/dl) or hypoglycemic (BG < 60mg/dl) conditions.38

Diabetes occurs when there is lengthened hyperglycemiccondition due to some abnormalities related to the pancreaticβ-cells. One such abnormality is complete destruction of theinsulin secreting β-cells due to autoimmune disorder, whichis characterized as type 1 diabetes mellitus (T1DM). Patientswith T1DM are completely dependent on exogenous insulin.39

Another defect may be due to insulin insensitivity or pancreaticinability to secrete sufficient insulin or combination of both,

which is defined as type 2 diabetes mellitus (T2DM). Thediabetic state provoked by prolonged hyperglycemia may leadto complications like retinopathy, neuropathy, nephropathy,coronary heart disease, and peripheral vascular disease. On theother hand, hypoglycemia induced by low BG levels lead tocomplications starting from nervousness, sweating, intense hun-ger progressing toward drowsiness, coma, and sometimes evendeath. Eventually, the serious complications of both hyperglyce-mia and hypoglycemia necessitate the importance of ensuring theBG levels within the therapeutic levels for sustaining a healthylife. Hence, proper amounts of calculated insulin should bedelivered at right times based on the meal intake such that theBG concentration is within the minimum therapeutic window(60�120 mg/dl) However, calculation of optimal insulin doserequirement by real-time experimentation on patients will not bea safe and efficient treatment method. Instead, testing on virtualpatients that mimic the real physiological behavior is easy, safeand can help improve the quality of treatment. Virtual diabeticpatients can be created by the development of mathematicalmodels that mimic the physiological behaviors related to glucose-insulin regulatory mechanism. Mathematically, biological sys-tems are viewed as distributed parameter, stochastic, nonlineardynamical systems that exhibit multicompartmental interactions.43

Blood glucose regulation is one such biological system withinteractions existing between the organs/cells related to glucose-insulin dynamics. There are two main classes of mathematical

Received: March 10, 2011Accepted: September 12, 2011Revised: September 8, 2011

12042 dx.doi.org/10.1021/ie2004779 |Ind. Eng. Chem. Res. 2011, 50, 12041–12066

Industrial & Engineering Chemistry Research ARTICLE

models developed for predicting the BG values: (i) data-drivenmodels, and (ii) knowledge-driven models. The former modelsare developed based on the input�output data without con-sidering any physiology.44�64 Usually, the output of thesemodels is the BG concentration and the input data can be avariety of information like previously measured BG values, diet,exercise, insulin therapy, body temperature, etc. On the otherhand, a priori knowledge on physiology behind glucose-insulinregulatory system is very much essential for development ofknowledge-driven models. Some models of this class containminimum number of equations that focus on the basic physio-logical aspects only1,2,5,6,8,9,11,12,15,21,22,27 while other modelswith larger number of equations cover the detailed physiologybehind glucose-insulin dynamics.3,18�20 Furthermore, the inputs/disturbances like insulin dose, meals and exercise influence thevariations in BG concentration of diabetic subjects. Mathe-matical models characterizing the meal absorption rate inplasma,1,5,13,27,35 insulin kinetics7,65 and exercise effects41,66,67 havealso been developed in the literature. The development ofmathematical models has facilitated the virtual evaluation ofBG control algorithms. Many control studies have employedmathematical models for their testing.1,9,23,24,29,32,35,68�80

The reviews published so far mainly focus on the available knowl-edge-driven BG models,34,81 model-based control studies38,82�86

or on both.87�89 However, most of them do not provide anysimulation results. Even the simulations presented in Makroglouet al.34 and Cobelli et al.88 have not included the BG models fromdifferent research families. A review showing the simulation ofmodels focusing on subcutaneous insulin kinetics has beenpublished.65 However, there are no reviews which analyze theBG models obtained from different families of knowledge-drivenBG models. Further, available reviews mainly focus on the knowl-edge-driven models but not on the empirical models.

The main objectives of this article are to review the availableBG models with a summary on the model based glycaemiccontrol literature, and to analyze the relative BG dynamicscaptured by different knowledge-driven models. The reviewpart of this article covers both the data- and knowledge-drivenglucose-insulin dynamic models along with meal absorptionand exercise effect models. Through this review, the availableknowledge-driven BG models are classified into differentfamilies based on their origin of development and modifications.

The analysis part focuses on study of the BG dynamics capturedby five important knowledge-driven glucose-insulin dynamicmodels (at least one model from each family) at their nominalparameter values, either by varying the manipulated variable (i.e.,basal insulin) or the disturbance variable (i.e., meal ingestion).Similarly, BG dynamics of ten adult patients obtained from arecent simulator (known as UVa T1DM simulator, which wasaccepted by Food and Drug Administration (FDA) for an insilico study25) has been studied. The 10 patients are mimicked bythe five models in terms of their basal glucose levels and basalinsulin requirements, whereas nominal values are used for theother parameters of thesemodels. The resulting BG dynamics arecompared and analyzed in order to understand BG dynamicspredicted by nominal parameter models for adult benchmarkpatients. All the five models are developed as specific Simulinklibraries in MATLAB.

The article is organized in the following way. Section 2 focuseson the review of glucose-insulin dynamic models (Subsection2.1), meal absorptionmodels (Subsection 2.2), and exercise effectmodels (Subsection 2.3), and finally control studies (Subsection2.4). Section 3 of the article focuses on the analysis of BG modelswith varying step changes in insulin inputs (Subsection 3.1), mealabsorption dynamics (Subsection 3.2), and BG models withdifferent meal scenarios at basal insulin doses (Subsection 3.3).Section 4 presents the conclusions of this article.

2. REVIEW ON T1DM MODELS

2.1.Models onGlucose-InsulinDynamics.The available BGmodels for glucose-insulin dynamics can be broadly classified as:(i) data-driven (or empirical) models; and (ii) knowledge-drivenmodels. Subsections 2.1.1 and 2.1.2 summarize the importantdata- and knowledge-driven BG prediction models reported inthe open literature. The BG concentration mentioned elsewherein this paper refers to the plasma glucose concentration. Biolo-gically, whole BG concentration is not equal to plasma concen-tration as plasma is cell free liquid after the whole blood iscentrifuged. However, literature tends to use these two termsinterchangeably, and there is still confusion and debate regardingthis. The differences between plasma, serum and blood glucosehave been outlined in Schrot et al.90

Figure 1. Normal glucose-insulin regulatorymechanism. (a)High blood glucose (BG) levels signal pancreas to release insulin (b) Pancreaticβ-cells secreteinsulin (c1) Insulin stimulates “extra hepatic glucose uptake” in the form of glycogen from (c2) blood Insulin stimulates “hepatic glucose uptake”(d) Low BG levels signal pancreas to release glucagon (e) Pancreatic α-cells secrete glucagon (f) Glucagon stimulates breakdown of glycogen in the liver.



2.1.1. Data-Driven Models. Empirical (or data-driven) modelsare the black box models which relate the input(s) given to thepatients with the output, which is usually BG concentration. Inthesemodels, future BG values of a patient will be predicted usingrecently recorded BG values and other inputs that can influencethe BG levels. The most attractive aspect is that simple modelspredicting BG concentrations can be developed in a short periodof time. Implementation of controllers on these models will beeasier as the model structure is easily identifiable.61,91�93 How-ever, these models cannot provide any insights on the levels ofglucose and insulin in various tissue and organs, as they do notconsider any physiology. Also, these models often require themost recent data from patients to predict the BG changes in thenear future which may not be practicable due to factors likeadditional measurement cost, patient’s compliance to frequentBG measurements etc.Kotanko et al.54 employed an empirical modeling technique

called universal process modeling algorithm (UPMA) to predictthe BG levels with the data collected from two insulin dependentdiabetic patients for a period of 700 days. This data containedinformation like daily therapy, BG measurements, physicalactivities, and diet of the patients. The UPMA used the data ofpreceding month as a reference library to model the BG changesof following month. Predictions of the UPMA were comparedwith the predictions of two experienced physicians. This studyconcluded that BG values predicted by UPMA possessed sig-nificant correlation with the observed BG values (correlationcoefficients ranging from 0.45 to 0.73) when compared to that ofthe predictions made by the two physicians (correlation coeffi-cients ranging from 0.05 to 0.1).Artificial neural networks (ANN) have been used to forecast

the BG levels in T1DM patients. Sandham et al.59 investigatedthe feasibility of predicting BG concentration in two T1DMpatients by using recurrent ANN (RANN). The BG level of pasttime was used to predict the present/future BG level. The inputvector used to predict BG levels in T1DM patients includes:insulin (type, time and site of injection), diet (total carbohydrates(CHO) and meal time), exercise (duration, mobility, strength,endurance, and time), BG level and a vector comprising of otherparameters (stress, illness, pregnancy). The prediction results byANNwere found to be very close to themeasured values of BG inboth the patients.Ghevondian et al.50 used a back-propagation trained ANN

algorithm to estimate the BG levels in insulin dependent diabeticpatients by using the input data containing noninvasively mea-sured physiological parameters such as skin impedance and heartrate. Tresp and his colleagues62 have shown that the RANN incombination with linear error model (developed to consider theuncertainty in the system and to handle missing BG observations)gave good results in predicting BG levels of a T1DMmale patient.Mougiakakou et al.56 presented two models based on the combi-nation of compartmental models and ANN to simulate theglucose-insulin metabolism in four T1DM children. The insulinkinetics and glucose absorption from the gut were characterized inthe form of compartmental models while the glucose dynamicswas given byANN.Onemodel used feed forwardANN(FFANN)and the other used RANN trainedwith real time recurrent learningalgorithm (RTRL). The outputs from compartment models weretreated as the inputs of ANN models. The results showed asuperior performance by RANN in all the four children.Time seriesmodels are yet another category of empiricalmodels

which can be used to develop linear and nonlinear dynamicmodels

by relating the input and output data. Bremer et al.46 assessedthe ability of empirically developed linear autoregressive (AR)models (identified by using the auto correlation function (ACF)estimates) to predict the future BGvalues. Results showed that thelinear models were able to predict BG values in a short horizon (ofup to 30 min) from the recent past data in both diabetic (T1DM)and nondiabetic patients. Subsequently, AR models for predictingBG levels in different prediction horizons were developed bySparacino et al.60 and Reifman et al.58 from continuous glucosedata of 28 ambulatory T1DM and 9 T1DM subjects, respectively.Van Herpe et al.64 developed an input-output time series modelusing the data obtained from 41 patients in an intensive care unit(ICU). A second order autoregressive exogenous input (ARX)model was found to be the optimal model for this data set. Inanother work, Van Herpe et al.63 used the data of 15 ICU patientsto develop an adaptive input-output time series model by employ-ing the significant input variables identified in their previousstudy.64 An initial model was developed based on the data of eightrandomly selected patients by using ordinary least-squares (OLS).Two different validation approaches (adaptive and nonadaptiveapproach) were followedwith the remaining data set. The adaptiveapproach applied weighted least-squares such that the modelcoefficients of the initial model were updated based on the recentdata from a specific validation patient. Themodel coefficients werekept constant for the nonadaptive approach. The adaptive modelpredictions were superior to the initial model predictions. More-over, the new model was able to make four hours ahead BGpredictions. Both themodels of VanHerpe et al.63,64 were found toexhibit better predicting ability and clinical interpretation ofestimated coefficients. However, the former work64 did not followany adaptive approach to address the issue of developing patient-specific model.Finan and his co-workers48 developed AR, ARX and autore-

gressive moving average exogenous input (ARMAX) modelsusing ambulatory data of two T1DM patients. They also devel-oped these models for the data obtained by simulating a T1DMnonlinear physiological model1,2 with exogenous insulin andmeal disturbances as inputs. The results of this study48 recom-mended AR or ARX models for ambulatory data and ARMAXmodel for the simulated data. Kazama et al.53 calculated an oralglucose insulin sensitivity index based on an AR model, devel-oped by using oral glucose tolerance test (OGTT) data sets of115 different patients who were grouped as normal, impairedglucose tolerance and T2DM subjects. The new sensitivityindex was found to be comparable with the standard insulinsensitivity index obtained from euglycemic hyperinsulinemicclamp method. Cescon and Johansson47 showed that state-space(SS) model predictions performed better when compared toARX and ARMAX model predictions, for the data obtainedfrom one T1DM patient. However, robustness of this approachacross a large group of patient population was not examined.Apart from these regular time series models, Eren-Oruklu andcolleagues have developed subject-specific recursive algorithmsfor the development of time series models predicting the futureBG concentrations.94,95 Recently, Eren-Oruklu et al.94 developedrecursive autoregressive moving average55 model and evaluatedthe recursive model to further predict the hypoglycemia toprovide hypoglycemic alarms.A study examining the possibility of developing universal

data-driven time series models for predicting subcutaneous BGconcentrations in both T1DM and T2DM patients has beenpublished recently.49 This study developed AR models of order



30 which could make 30 min ahead short-term BG predictions.Recently, application of mobile phones (called as Few Touchapplication) for self-management of T1DM by assembling a dataset with BG concentration, exercise, diet and insulin medicationhas been proposed.96 Aside from the above-mentioned lineardata-driven models, some nonlinear models have also beendeveloped. Mitsis et al.55 developed Volterra-type nonlineardynamic empirical models with the data obtained from simula-tion of minimal model.21 A nonlinear ARX (NARX) model wasdeveloped from Intravenous Glucose Tolerance Test (IVGTT)data to derive an index for insulin sensitivity.51 The calculatedinsulin sensitivity index was found to coincide with the sensitivitycalculated from minimal model for normal patients. In case ofpatients with slow insulin dynamics, the new index was foundto be more accurate than minimal model index as NARXmodel possessed better prediction capability than its counterpart.Rollins and colleagues97,98 made significant contributions indeveloping nonlinear block-oriented Weiner models for T2DMpatients. Recently, this group has employed Weiner-block mod-eling methodology to predict the BG levels inT2DM patientsusing a large set of strongly correlated noninvasive inputs (nearly35 sets of input variables related to stress, activity and food).97

In this work,97 it has also been mentioned that this sort ofmodeling methodology has been tried for one T1DM patientwith known insulin amounts, which showed correlation coeffi-cient (rfit) value of 0.96 in the validation.ANNmodels discussed above have used the data sets obtained

from 1 or 2 patients to predict the changes in BG concentrations.Though these studies showed successful results in BG predictions,the application of ANN over a large set of data obtained fromdifferent patients has not been reported. On the other hand, practicalapplicability of ANN for real time BG predictions is challenging dueto the requirement of expertise in estimation of network connectionweights for the design of efficient ANN. Time seriesmodels can helpin this regard as inmost cases BG concentrations in the near futureare predicted from the past BG data, without any demand oftechnical skills from patients or clinicians. However, only very fewstudies on time series models have shown the ability to developpatient-specific models by using adaptive modeling techniques.63 Ashort summary of various data-driven models developed for pre-dicting BG dynamics is presented in the Appendix (Table 6).2.1.2. Knowledge-DrivenModels.The knowledge-drivenmod-

els consider the physiological interactions related to glucose-insulin metabolism. The effort required to develop these modelsis much larger when compared to that of data-driven models. Theknowledge-driven models can further be categorized as lumped orsemiempirical models, and comprehensive models. The formercategory models possess limited number of dynamic equations,in which various organs or tissues in the body are lumped intoone or more compartments (like extracellular and intracellularcompartments). The parameters of lumped models are estimatedfrom the data collected via clinical tests (like Intravenous GlucoseTolerance Test, IVGTT;Oral glucose tolerance test, OGTT; etc),and hence they are also termed as semiempirical models. In thedevelopment of comprehensive models, detailed physiology isconsidered. The glucose and insulin distributions in various organsare modeled separately with the available experimental data in theliterature. The interaction between these organs is also captured bythese models. The comprehensive models are highly complex andrequire more development time than the lumped models.2.1.2 (a). Lumped or Semiempirical Models. Bolie was a

pioneer in developingmathematicalmodels basedon the underlying

physiological features in glucose-insulin regulatory system. In1961, he proposed a linear model with two ordinary differentialequations (ODEs) representing glucose and insulin concentra-tions, based on his condensed theory.8 This theory assumed thatthe liver, pancreas and peripheral tissues are in communicationwith each other by means of a single compartment throughoutwhich the glucose and insulin concentration changes are dis-tributed rapidly and uniformly. However, this model did notconsider the action of the kidneys and the intravascular-extra-vascular differences in concentrations of insulin and glucose. Asimilar model with two ODEs for denoting glucose and insulinkinetics was developed in 1965.17

Nevertheless, an actual turnaround in diabetes modelingoccurred with the arrival of the Bergman’s minimal model inthe late seventies and early eighties of the last century.21,22 Thepopularity of this model in diabetic research has been quantita-tively reported in the reviews.34,81 In 2002, approximately 500studies in the literature were related to the minimal model.34,81

Theminimal model proposed by Bergman and his colleagues wasdeveloped after elaborate statistical and physiological analysis ofglucose disappearance and associated plasma insulin appearanceduring IVGTT in canines.21 The original model was also testedon human beings to determine important physiological param-eters like glucose effectiveness, insulin sensitivity and pancreaticresponsiveness.22,99 Theminimal model mimics a healthy subjectby using three ODEs for plasma glucose, plasma insulin andremote insulin concentration without considering any externaldisturbances. The major advantages of this model are its struc-tural simplicity and ability to estimate the significant physiologi-cal parameters in a minimally invasive manner using the BG andplasma insulin data obtained from clinical tests.The minimal model has seen many modifications sub-

sequently.29,32,35,37,40 Mostly, these modifications introduceddiabetic condition to the original minimal model which wasdeveloped based on IVGTT data obtained from normal subjects.In 1985, Furler et al.29 proposed a modified minimal model forcontrolling BG level in diabetic patients by introducing exogen-ous insulin supply. Ollerton32 and Fisher35 modified the minimalmodel to mimic T1DM patients by (i) adding exogenous glucoseinfusion term to glucose dynamics equation, and (ii) omittingpancreatic insulin secretion term and adding exogenous insulininfusion term in the plasma insulin component. In 2002, theFisher model was further modified by Lynch and Bequette37 byincluding a new equation which related subcutaneous glucoseconcentration to the BG concentration. In 2006, Van Herpe40

and his colleagues proposed a modified minimal model whichcan be valid for a longer time horizon to control BG levels inintensive care unit (ICU) patients. Roy et al.42 developed anextended minimal model by considering a physiological aspectthat free fatty acids (FFA) play an important role in alteringglucose-insulin regulatory mechanism. This extended minimalmodel incorporated a new ODE representing plasma FFA dy-namics. The interaction of plasma FFA with glucose and insulinwas also explained in the model. Also, Roy et al.41 have developedan extended minimal model incorporating various exerciseeffects on glucose and insulin dynamics.Despite its popularity, the drawbacks of Bergman’s mini-

mal model have also been discussed in the literature.34,100�102

Studies have shown that minimal model overestimates glucoseeffectiveness101 and underestimates insulin sensitivity.102 Caumoet al.100 have provided a more detailed explanation of this byshowing monocompartmental approximation of glucose system



in the original minimal model as a reason for over- and under-estimation. Further, they have proposed that a two compart-mental model for glucose system (accessible and inaccessiblecompartments) can give precise values for glucose effectivenessand insulin sensitivity. Also, Cobelli et al.103 have stated thatfailure of the minimal model to separate glucose production fromutilizationmay result in a biased parameter description. They alsoproposed a revised minimal model (which can provide the pa-rametric measure of glucose utilization) to overcome the limita-tion of the minimal model in estimating insulin sensitivity andglucose effectiveness. This revised model was developed basedon IVGTT performed in six canines with cold and radio-labeled104

glucose infusions. However, additional cost and technologyinvolved in employing radio-labeled glucose infusion makes thepractical applicability of this revised model in human populationdoubtful. Taking this into consideration, Cobelli et al.10 formu-lated a new two-compartmental minimal model for glucosekinetics by using a Bayesian approach. This model was developedbased on standard IVGTT in normal human subjects. Thedeveloped model10 was found to exhibit an improved glucoseeffectiveness and insulin sensitivity when compared to that of theoriginal minimal model.21,22

In 1989, Berger and Rodbard9 contributed a computer pro-gram based on a pharmacokinetic-pharmacodynamic modelfor simulating the insulin and glucose dynamics after subcuta-neous insulin injection. The pharmacokinetic model representinginsulin action can be used to calculate the time courses of plasmainsulin for various combinations of insulin formulations whilethe pharmacodynamic glucose model can be used to predictexpected time course of plasma glucose in response to a changein carbohydrate intake, insulin dose, timing, or regimen. In 1992,Lehmann and Deutsch27 developed a new semiempirical modelof glucose-insulin interaction in T1DM. This model employedBerger and Rodbard’s pharmacokinetic model of insulin action9

with a new pharmacodynamic glucose model developed fromthe experimental data of Guyton et al.16 This pharmacodynamicmodel27 assumed a single compartment for extracellular glucosewith systemic appearance of glucose via gut absorption and nethepatic glucose balance as inputs, and peripheral and insulinindependent glucose utilization and renal glucose excretion asoutputs. The most significant part of Lehmann and Deutsch’smodel is the description of the amount of glucose in the gutfollowing meal ingestion. This meal model is discussed moreelaborately in Section 3.1.Various clinical (in vivo) and in vitro studies have demon-

strated that pancreatic insulin secretion follows rapid smallamplitude and slower large amplitude oscillatory behavior de-pending on the glucose infusion.105�112 Therefore, Sturis et al.11

developed a physiological model consisting of six ODEs and fivenonlinear equations in order to study the possible mechanismsbehind the slow oscillations of insulin and glucose. The firstthree differential equations of this model simulate concentrationsof plasma insulin, intercellular insulin and plasma glucose,respectively. The last three differential equations simulate thephysiological delay of plasma insulin to inhibit hepatic glucoseproduction. This model was developed based on the followingphysiological events which occur in sequence: (i) glucose stim-ulates pancreatic insulin secretion, (ii) insulin stimulates glucoseuptake, (iii) insulin inhibits hepatic glucose production and(iv) glucose enhances its own uptake. The five nonlinearequations of this model11 characterize the pancreatic insulinsecretion, glucose utilization (by brain, fat, and muscle cells) and

physiological delay. These nonlinear functions were developedbased on experimental results.109,113�115

In 2000, Tolic et al.26 extended the model of Sturis et al. bydeveloping two detailed models of insulin receptor dynamics andconnecting them with the existing model.11 This extension wasdone to examine the possible effects of receptor dynamics on thebehavior of original model. However, a proper clinical inter-pretation of the three variables corresponding to the physiolo-gical delay of plasma insulin was not established. Hence, Bennettet al.30 proposed a model by incorporating an explicit time delayinto the glucose dynamic equation of the original model toexclude the three delay variables which had no clinical signifi-cance. This resulted in reduction of original six ODEs model11 tothree ODEs model30 with an explicit time delay. Following this,Li et al.33 proposed a newmodel by introducing two explicit timedelays into the ODEs to simulate insulin and glucose dynamics.However, these models11,26,33 were not validated with the clinicaldata. Very recently, Chen et al.36 has slightly modified Li et al.’s33

model and verified it on a clinical data set obtained from 56subjects. The results have shown that the model could practicallyreflect the dynamic and oscillation behavior on diabetic patients.In 2002, Hovorka and his co-workers2 contributed one of the

most recent semiempirical BG models to the diabetes literature.This model was developed based on the clinical data collectedduring IVGTT performed in six human subjects using dualtracers and native glucose as inputs. Dual tracer technique wasemployed to characterize the insulin action on glucose transport/distribution, glucose disposal and endogenous glucose produc-tion (EGP). Structurally, this model used two compartmentsfor glucose kinetics and single compartment for insulin action(i.e., each action is represented by one ODE). The two ODEsrepresenting the accessible and inaccessible compartments rep-resent the glucose absorption, distribution and disposal.Model equations representing insulin subsystem can be foundin Hovorka et al.1 The insulin subsystem was modeled as a two-compartment chain for denoting the absorption of subcuta-neously infused insulin formulation and a single compartmentfor describing the distribution and disposal of subcutaneouslyinfused insulin in plasma. A complete structure of Hovorka’ssemiempirical model can be seen in the open literature.1 Also,Wilinska et al.7 investigated eleven postulated insulin kineticmodels on the continuous subcutaneous insulin (Lispro) infu-sion (CSII) data obtained from seven T1DMpatients. This studyresulted in identification of the best insulin kineticmodel (amongthe eleven models) involving combined slow and fast absorptionwith local degradation of insulin at the injected site. Recently, theoptimal parameters of the combined Hovorka2 (glucose andinsulin action subsystem)�Wilinska7 (insulin subsystem) model(HWM) have been identified through the model based design ofclinical tests.116

More recently, Fabietti et al.5 developed amathematical modelof insulin-glucose kinetics in T1DM with an aim to evaluate theeffectiveness of the new model in designing control algorithmsfor artificial pancreas. This model consists of two main subsys-tems for explaining insulin and glucose dynamics. The glucosesubsystem was modeled as two compartments representing BGand interstitial glucose concentrations. The interstitial compart-ment was modeled by following Regitting et al.117 The glucoseblock in this model incorporates terms which differentiate theinsulin-dependent and insulin-independent glucose utiliza-tion. Also, the model has provisions for glucose contributionsfrom ingested food, hepatic release and intravenous glucose



administration. The meal model was developed based on theresults of Arleth et al.,118 which will be discussed in Section 3.On the other hand, insulin subsystem was modeled as threecompartments representing insulin concentrations in subcuta-neous, plasma and remote space. In another work, Fabietti et al.6

validated this model with the data obtained from two differentclinical conditions: intravenous glucose measurement and sub-cutaneous insulin infusion, and intravenous glucose measure-ment and intravenous insulin infusion.In 2007, Dalla Man and colleagues12 presented a new model

describing the physiological events that occur in the postprandialstate. Glucose subsystem was modeled as a two compartmentalstructure by using two ODEs, one explaining the glucose mass inplasma and rapidly equilibrating tissues and another denoting theglucose mass in slow equilibrating tissues. The insulin subsystemwas also represented by a two compartmental structure with twoODEs for insulin masses in plasma and liver. The model involvedfour unit processes, namely, endogenous glucose production(EGP), glucose absorption via gut wall, glucose utilization, andendogenous insulin secretion. The mathematical representationof EGP used in this model12 was based on an another work ofDalla Man et al.,15 which quantified the effect of insulin andglucose on liver to suppress the EGP. The physiological model ofglucose absorption rate used in this model was from reference 13which will be discussed in Section 3.1. The insulin-dependent(tissues) and independent (brain and erythrocytes) glucoseutilizations during meals were modeled based on the resultsof Cobelli et al.3,4,119 Finally, the unit process for pancreatic in-sulin secretion was adopted from the literature120,121 whichmodeled the pancreatic insulin secretion responses with respectto increasing and decreasing glucose concentration.Dalla Man et al.12 fitted the parameters of the resulting

model to a large database of normal subjects and also to a smallerdatabase of T2DM patients. Based on this model,12 a newsimulation software of glucose insulin model (GIM), whichcould be used to simulate the life of normal, T1DM andT2DM subjects, was developed.14 This model12 was modifiedfor T1DM patients14 by excluding pancreatic insulin secretionterm and by including a new model of subcutaneous insulinkinetics based on exogenous insulin infusion from Nucciet al.65 In 2009, Kovatchev et al.25 developed another simulator(available as “UVa simulator”) for performing in silico preclinicaltrials in T1DM subjects based on the glucose-insulin model ofDalla Man et al.12,14 This simulator was proven to capture theimportant BG fluctuations during meal challenges in T1DMpatients.25 Also, it has been reported25 that the Food and DrugAdministration (FDA) has accepted this simulator as an alter-native to animal studies in the preclinical testing of a closed-loopcontrol strategy. However, it should be noted that FDA allowedthis simulator as an investigator device exemption application(IDE) for a specific instance alone but not as a universalalternative to all the animal studies related to T1DM.2.1.2 (b). Comprehensive Models. Tiran et al. developed

detailed physiological models for glucose19 and insulin18 dy-namics based on physiological parameters incorporating bloodflow rates and circulatory paths of specific tissues and organsrelated to insulin-glucose dynamics. In the year 1975, Tiranet al.19 developed the circulatory model for glucose dynamics bymathematically representing major organs and tissues (like gut,kidneys, brain, heart, lungs, and periphery) as several compart-ments. After four years, Tiran et al.18 developed a circulationand organs model for insulin dynamics (COMID) based on the

glucose dynamics model.19 During the development of insulindynamic model,18 insulin disappearance in gut, heart, lungs,brain, and hepatic artery was ignored due to the fact that theseorgans make only small contribution to insulin disappearance.However, the validity of these models was tested only on diabeticcanines but not on human subjects.In 1978, Guyton et al.16 synthesized a comprehensive math-

ematical model of glucose metabolism in a normal subject basedon the physiological knowledge. Thismodel divided human bodyinto six compartments to study the glucose-insulin homeostasisin liver, brain, pancreas, kidney, peripheral tissues and centralvascular organs. The model structure was identified from morethan 100 IVGTTs and literature available at that time. In 1985,Sorensen20 updated this model by providing a more elaboratemathematical explanation. Parker et al.23,24 updated Sorenson’smodel by including generalized meal disturbances and param-eters for uncertainty analysis. The updated model is a 19thorder physiological model with 11, 7, and 1 differential equation-(s) characterizing glucose, insulin and glucagon subsystems,respectively. The glucose subsystem of this model consists ofsix compartments namely, brain, heart/lungs, gut, liver, kidney,and periphery (representing both skeletal muscle and adiposetissue). In compartments like heart/lungs, kidney, liver, and gut,the glucose concentrations were assumed to be homogeneousas they have a single well-mixed space. On the other hand,brain and periphery compartments have two well-mixed spaces(with uniform substrate concentration in each of them). This isbecause the capillary wall permeability in these compartments isadequately low which leads to slow equilibration of substratesbetween the capillary and interstitial spaces.122 The insulinsubsystem of this model also has a similar structure as that ofglucose with a difference that the brain has only one well-mixedspace as the capillary wall of brain is impermeable to insulin.Each compartment has arterial blood as input and venous bloodas output.In 1992, Puckett et al.123 proposed a multicompartment

PK model structure to explain glucose-insulin interactions atvarious tissue and organ levels, which is very similar to Sorenson’supdated model. However, Puckett’s model did not consider theglucagon effect, considered in Sorenson’s model. Also, theformer model encompasses submodel and functions for repre-senting the glucose and insulin independent uptake, glucosedependent uptake, insulin and glucose dependent uptake, glu-cose input forcing functions and liver glucose production. In1995, Puckett and Lightfoot124 proposed a three pool compart-ment submodel to describe the subcutaneous insulin absorption.This model along with various submodels is made available as aneducational simulator “GLUCOSIM”.125

In the early eighties, Cobelli et al.3,4 developed anotherdetailed physiological model to characterize the glucose, insulinand glucagon kinetics. The glucose dynamics were modeledby a single compartment which considered net hepatic balance,renal excretion, and insulin-dependent and independent glu-cose utilizations. Insulin kinetics were represented by a five-compartment model which explains the quantity of pancreaticstored insulin, promptly releasable insulin by pancreas, insulinin plasma, liver, and in interstitial fluids. The glucagon sub-system was described by a single compartment which explainedthe quantity of glucagon in plasma and interstitial fluids. Theresulting model has seven nonlinear functions which modelthe hepatic glucose production and uptake, renal excretion, andperipheral insulin-dependent and independent glucose utilizations.



The model used two test inputs for exogenous glucose andinsulin infusions. Cobelli and Mari126 used this detailed modelto analyze the causes of inability of exogenous insulin dosingsystems in diabetic subjects to recover the normal metaboliclevels. Finally, based on the origin of development and modifica-tions seen by each model, a family tree has been drawn for thefirst principle models as shown in Figure 2.Importance of Structural Identifiability in Knowledge�

Driven models. The drawback in using lumped knowledge-driven models is that these models, with minimal parameters inthem, may not mimic the dynamics of the glucose regulatorysystem at various organ and/or tissue levels. Conversely, detailedknowledge-driven models possess more number of parameterswhich characterize the dynamics of glucose regulatory system attissue/organ level. In both these types of knowledge-drivenmodels, it is very much essential to identify the parameters whichare physiologically meaningful and correct for the model to bepractically useful. Unique parameter identification relies on anumber of factors such as the structure of the model, the type ofdata collected (BG data and/or plasma insulin data), samplinginstants at which the data are collected and the level of inputperturbations (like pattern of oral/intravenous glucose ingestionand/or pattern of intravenous/subcutaneous insulin dose) dur-ing the clinical tests. Hence, structural127 or a priori128,129 globalidentifiability analysis has often been seen as a prerequisite toexperimental design127 and parameter estimation.127�129

Particularly, diabetes models have been identified with thedata such as BG data and/or plasma insulin data from the

standard clinical tests like IVGTT, OGTT, postprandial glucosetest (PGT), euglycemic hyperinsulinemic clamp test (ECT), andso on. However, these clinical tests should be designed in anoptimal way such that the information obtained from the tests issufficient enough to estimate the parameters precisely. As quotedin Galvanin et al.,116 the parameters of the well-known Bergman’sminimal model22 were found to be estimable with a modificationin the input perturbations of the standard IVGTT. Further,literature studies have shown that insulin infusion after glucoseinjection in the standard IVGTT can improve the parameterestimation.130,131 In 2009, Galvanin et al.116 identified theparameters of Hovorka-Wilinska model by modifying and opti-mizing the test protocols of PGT and OGTT with the aid of atechnique called model based design of experiments.In 2010, Chin et al.132 have performed structural identifiability

analysis on the original minimal model21 and extended minimalmodel.22 It has been reported that only extended minimal modelis globally structurally identifiable if both glucose and insulinlevels are measured. Also, Chin et al.132 have performed thestructural identifiability analysis on the euglycemic hyperinsuli-nemic clamp model133,134 for glucose-insulin dynamics. Theyconcluded that the euglycemic hyperinsulinemic clamp model isstructurally globally identifiable if glucose infusion is the onlyinput to the model. Very recently, Watson et al.127 have proposeda general glucose homeostatic model which includes a novelplasma insulin compartment that mimics the three phases ofinsulin secretion which is similar to that of a proportional-integral-derivative (PID) controller. Structural identifiabilityanalysis on this model has revealed two unidentifiable parameters(glucose production rate and proportional control coefficient)from IVGTT and hyperglycemic clamps; of these, the formerparameter could be measured from a tracer experiment andthe latter could be estimated by other means as reported inWatson et al.127

2.2. Models on Meal-Absorption Dynamics. Available mealabsorption models from the literature are discussed with theirmathematical equations in this section. The structure of availablemeal-absorption models can be seen in Figure 32.2.1. Fisher Meal Model (FIMM). Fisher35 developed a meal

model to explain the rate at which glucose enters bloodstreamafter intestinal absorption frommeal (Ug(t)) with an assumptionthat oral glucose ingestion commences at time (t) = 0 min.Recently, Farmer et al.78 presented the Fisher model as follows:

UgðtÞ ¼ 0; t < tmealB� expð � kabstÞ; t g tmeal

8<: ð1Þ

Fisher35 used the parameter values: B = 0.5 mg/min and k =0.05 min�1. However, Farmer et al.78 explained that the param-eter B of this model as the total size of meal given by:

B ¼ Dmealkabs ð2Þwhere Dmeal represents the CHO content of the given meal

and kabs represents the absorption rate. Equation 2 resemblesthe solution of a first order ODE, which indicates that Fishermodeled the intestinal glucose absorption rate from meal as asingle compartment (shown in Figure 3(A)).2.2.2. Lehmann and Deutsch Meal Model (LDMM). Lehmann

and Deutsch27 proposed a meal model to mimic the glucosedynamics starting from oral meal ingestion to glucose absorptioninto blood via the gut wall as shown in Figure 3(B). Theymodeled

Figure 2. Family tree of knowledge-driven (first principle) BG modelsavailable in the literature.



the rate of glucose appearance in plasma compartment via the gutwall Ug(t) as follows:

UgðtÞ ¼ kabsGgut ð3Þ

dGgut

dt¼ Gempt � kabsGgut ð4Þ

where Ggut represents the amount of glucose in the gut afterthe uptake of meals containing CHOs and kabs denotes the rate

constant of glucose absorption from gut. The model representsthe duration of period for which gastric emptying (Gempt) isconstant and maximal as a function of CHO content in theingested meal. Gempt curves result in a trapezoidal or triangularfunction for meals containing large or small quantity of CHO,respectively. The triangular function occurs for Gempt of smallCHO meals as there will not be sufficient time for the curve toplateau out. Lehmann and Deutsch27 defined the critical level ofCHO (Dcrit) which differentiates the meal size (as large or small)

Figure 3. Schematic representation of meal absorption models.



as follows:

Dcrit ¼½ðTascge þ TdesgeÞVmaxge�

2ð5Þ

For large meal sizes Gempt was modeled as:

Gempt ¼

ðVmaxge =TascgeÞt; t < TascgeVmaxge ;Tascge < t e Tascge þ Tmaxge

Vmaxge � ðVmaxge =TdesgeÞðt � Tascge � TmaxgeÞ;Tascge þ Tmaxge < t e Tascge þ Tmaxge þ Tdesge

0; otherwise

8>>>>>>>><>>>>>>>>:

ð6ÞIn eq 7, Tmaxge is the duration of the period for which Gempt is

constant andmaximal (Vmaxge) which is the characteristic of largemeal. Tascge and Tdesge are the lengths of ascending and descend-ing branches of Gempt curve. For large meals (Dmeal > Dcrit),Tmaxge was mathematically characterized by

Tmaxge ¼ ½Dmeal � VmaxgeðTascge þ TdesgeÞ�Vmaxge

ð7Þ

For small meal sizes (Dmeal < Dcrit),

Tascge ¼ Tdesge ¼ 2Dmeal

Vmaxgeð8Þ

Lehmann and Deutsch27 calculated the parameter values (kabs andVmaxge) of the meal model from the experimental results ofGuyton et al.16 Thus, when compared to the meal model ofFisher,35 this model gives a better physiological interpretationwithits parameters being estimated from the experimental data.2.2.3. Hovorka Meal Model (HMM). Hovorka et al.1 formu-

lated a meal model for their glucose-insulin dynamic model. Thegut absorption rate was modeled as a two compartmental chainwith identical transfer rates 1/tmax,G as shown in Figure 3(C). Thefirst compartment of this model explains the total amount ofglucose present initially (i.e., before intestinal absorption). Thesecond compartment receives input from its predecessor, and itdescribes the intestinal absorption of glucose into blood (UG(t)) as

UGðtÞ ¼ Dmeal f t eð� t=tmax,GÞ

t2max ,Gð9Þ

where tmax,G is the time-to-maximum appearance rate of glucose inthe accessible glucose compartment and f is a dimensionlessquantity which represents the fraction of CHO absorbed. Hovorkaet al.1 defined tmax,G and f as constants with values 40 min and 0.8,respectively. These values were obtained from Livesey et al.135

2.2.4. Fabietti Meal Model (FMM). Fabietti et al.5,6 used a mealmodel based on Arleth et al.118 to describe the glucose input viameals by using a time dependent input which denotes the foodintake rate. Gut absorption was modeled as the sum of glucoseabsorption rates of three classes of carbohydrates (sugar (Ag),fast absorption starch (As), and slow absorption starch (Am)):

UgðtÞ ¼ Ag þ As þ Am ð10ÞAg, As, and Am were obtained mathematically in the formof transfer functions based on the experimental results of

Arleth et al.118 as follows:

AgðsÞ ¼ ð1� FsÞ 16:6ðs þ 1:44Þðs þ 135Þ RiðsÞ ð11Þ

AsðsÞ ¼ Fsð1� FmÞ 467ðs þ 1:61Þðs þ 7:20Þðs þ 7:18Þ RiðsÞ

ð12Þ

AmðsÞ ¼ FsFm75:1

ðs þ 0:466Þðs þ 5:54Þðs þ 5:86Þðs þ 6:43Þ RiðsÞ

ð13ÞwhereRi is the rate of carbohydrate ingestion duringmeal, Fs is theoverall fraction of starch, and Fm is the fraction of mixed meal intotal amount of starch. The term (1�Fs) in eq 11 represents thefraction of sugar. Unlike other meal models, this model gives thegut absorption rates for each class of CHOs in the ingested meal.2.2.5. Dalla Man Meal Model (DMM). Dalla Man et al.13

developed a model for explaining the rate of appearance ofglucose in plasma after meal ingestion based on the experimentaldata obtained from labeled oral glucose tolerance test of 41 subjectsand labeled mixed meal ingestion of 20 subjects. Glucose intestinalabsorption was modeled by assuming a two compartmentalstructure for glucose passage in stomach, one for the solid phase(Gsto1) and another for liquid phase (Gsto2), and a single compart-ment for glucose mass in the upper small intestine (Ggut).

Gsto ¼ Gsto1 þ Gsto2 ð14Þ

dGsto1

dt¼ � kgriGsto1 þ DmealdðtÞ ð15Þ

dGsto2

dt¼ � GemptðGstoÞ:Gsto2 þ kgriGsto1 ð16Þ

dGgut

dt¼ � kabsGgut þ GemptðGstoÞ:Gsto2 ð17Þ

UgðtÞ ¼ f 3 kabs 3Ggut ð18Þwhere Gsto is the mass of total glucose in the stomach. kabs and kgridenote the rate constants of intestinal absorption and grinding,respectively. Gempt(Gsto) in eq 16 denotes the rate constant ofgastric emptyingwhichwasmodeled as a nonlinear function ofGsto

and kabs in Dalla Man et al.13 as follows:

GemptðGstoÞ ¼ Gmin þ Gmax � Gmin

2ftanh½αðGsto � bDÞ�

� tanh½βðGsto � cDÞ� þ 2gð19Þ

Gmax corresponds to maximum gastric emptying when sto-mach has the ingested glucose and Gmin is the minimum gastricemptying when stomach has very less glucose. α is the rate atwhich Gmax decreases to a minimum Gmin and β is the rate atwhich Gmin recovers back to Gmax. Parameters b and c are thepercentage doses at which Gempt decreases at (Gmax � Gmin)/2and comes back to (Gmax � Gmin)/2, respectively. The param-eter values corresponding to this glucose absorption model canbe found in ref 12. As mentioned previously, this meal model13 in



combination with glucose-insulin dynamic model12 was used inthe development of simulators like GIM14 and UVa.25

2.3. Models on Exercise Effects. Exercise plays a significantrole in introducing changes to the body like increased heart rateand oxygen consumption,136 elevated glucose uptake by theexercising muscle,137 increased hepatic glucose production138

and decreased plasma insulin concentration.139 However, therewill be an increased hepatic glucose release only during the epi-sodes of mild to moderate exercise. During prolonged exerciseperiods, the hepatic glucose release rate decreases due to thelimited storage of glycogen in liver.140 Ultimately, glycogenolysis(process involving break down of glycogen into glucose) ratedecreases which directs to net decrease in hepatic glucose pro-duction during extended exercise periods.141 Such intense ex-ercise episodes may lead to hypoglycemia due to the imbalancebetween glucose uptake and hepatic glucose production.140,142

Thus, it is evident that exercise effects alter the glucose-insulinregulatory system. Exercise effects are also considered as distur-bances to glucose-insulin dynamicmodels likemeals. An overviewof models considering exercise effects will be given in this section.In 2002, Lenart and Parker67 incorporated the effects of exercise

into the updated physiological glucose-insulin-glucagon model ofSorenson.20,24 Their exercise model quantified the exercise level of aperson by using percentage of a person’s maximum oxygen con-sumption rate. The influence of exercise on blood flow distribution,peripheral glucose uptake, hepatic glucose production, and insulinuptake has beenmodeled in the literature.67The experimentalworksof Ahlborg et al.137,142 were used to validate the exercise model. Thismodel was mainly developed for introducing mild and moderateintensity exercise effects into the physiological model.In early 2007, Kim et al.66 developed a comprehensive model

to predict the fuel homeostasis during exercise. This compre-hensive model divided the entire body into seven compartmentsincluding the brain, heart, liver, gastrointestinal tract, skeletalmuscle, adipose tissue, and other tissues. The hormonal changesduring exercise were predicted by introducing the glucagon-insulin controller to the model. Themodel was validated with thedata obtained from normal subjects who experienced moderate

intensity exercise for a period of 1 h. This model predicted thedynamics changes of hard to measure metabolic pathways likehepatic glycogenolysis, and gluconeogenesis. Another fascinatingthing about the model of Kim et al.66 is its ability to predict themetabolic responses in each tissue during exercise.However, the complex nature of the above two exercise

models66,67 induce difficulties in tailoring them for a specificpatient in order tomake thesemodels clinically acceptable. Inmid2007, Roy and Parker41 incorporated exercise effects into Berg-man’s minimal model.22 The extended minimal model41 consistsof three additional variables representing rates of glucose uptake,hepatic glucose production, and insulin removal from the circu-latory system due to the effects of exercise. Also, the interactionsof these new variables with the plasma insulin and glucosedynamics were mathematically represented in the extendedmodel. Experimental data from the existing literature140,142�144

were used to estimate the parameters of the newly proposedminimal model with exercise effects.41 The simplistic nature ofthis new model can help in designing patient specific models forclinical use. Nevertheless, this simple model cannot predict thedynamics of the metabolic pathways measured by Kim et al.66

2.4. Control Studies on BG Regulation. As stated earlier inthe introductory part, there will be no pancreatic insulin secretionin case of T1DM subjects. Hence, the treatment goal under thesecircumstances should be maintaining BG values within thetherapeutic limits via controlled delivery of exogenous insulindoses at proper time intervals depending on the meals and otherdisturbances. This has been accomplished with the aid of multi-ple insulin injections, insulin pumps, insulin jets, and insulin penswhere a predetermined insulin dose is administered. However,the open loop nature of insulin delivery in the above-mentionedtreatment options poses serious risk of significant and frequentvariations in glucose concentrations, which may lead to hypo- orhyperglycemia in the treated patients. A safer and efficienttreatment protocol should encompass an external insulin deliv-ery mode that can control the glucose regulatory mechanism asthat of the pancreas in healthy individuals. A closed loop BGcontrol system (as shown in Figure 4) can serve as an artificial

Figure 4. Closed loop BG control system (also termed as “artificial pancreas”) for T1DM patients.



pancreas inmonitoring the BG levels. In a closed loop system, therecent BG concentration measured by the sensor(s) will bepassed on to the controller which is expected to possess a detailedroadmap of all possible scenarios. The controller calculates theoptimal amount of insulin dose to be delivered at a particulartime instant and commands the insulin infusion pump to deliverthe optimal dose into the patient. Various control algorithmshave been proposed in the literature for optimally regulating theexternal insulin infusion based on the recent sensormeasured BGconcentrations and the desired BG target. Reviews38,82�85,87,145

have already elaborately discussed most of the BG controlalgorithms. Hence, only a brief overview of the various BGcontrol algorithms will be presented in this article.The existing control strategies can be broadly classified as

open, closed and semiclosed loop strategies. Open loop strategiesallows injection of predetermined insulin doses based on eventslike meal, exercise, etc. and is mostly followed by physicians inpractice. Recently, a model based insulin therapy schedulingbased on mixed integer nonlinear optimization approach hasbeen tested on clinical data sets for evaluating optimal injectiontime, type and insulin dosage.146 A closed loop strategy deliversinsulin doses continuously based on continuous monitoring ofBG levels.1,23,24,70�77,79,80,93,147�180 The first approach to closedloop control was initiated by Kadish155 in the year 1964, whoused an on�off system comprising of intravenous insulin andglucose (or glucagon) infusion based on continuous BG mon-itoring in an actual diabetic subject. The semiclosed loopprograms possess continuous insulin delivery based on inter-mittent measurement of BG levels.29,32,35,69

Chee et al.87 categorized the existing control algorithms as (i)model-less or empirical control approach and (ii) model basedcontrol approach. In the former approach, a control rule isformulated by mere employment of the clinical data as the basis(dose�response relationship) without any consideration of un-derlying theory. Control based on curve fitting,147�150 lookuptable,69,181 and rule based control182�184 belong to this approach.Albisser et al.147�149 contributed one of the earliest controlequations based on curve fitting. This algorithm was latermodified as reported in the literature87,145 and formed the basisfor the algorithm used in Biostator, the first commercial deviceintroduced by Clemens et al.185 Being the first device, Biostatorencountered some challenging limitations.145 The lookup tablescontain various BG ranges with an assigned external insulin dosefor each range. Even today, BG levels in most of the ambulatorypatients are controlled by administering insulin using the lookuptable oriented approach. In fact, the control by this approach isattained in an open loop manner, where blood is sampledsporadically (literature has shown 1 h181 and 3 h69 BG sampling)and insulin infusion is manually adjusted using the lookup tables.Control based on expert rules are yet another kind of model-lessapproach that can be found in clinical practice and also inliterature.182,184 Basically, insulin infusion rates are increased/decreased every 2 h based on BGmeasurement183,184 or a simplenomogram.182 This method is similar to the lookup table withonly difference being insulin infusion rate, which is increased ordecreased from the previous rate in rule based control methodand fixed for each BG range in lookup table method.On the other hand, the model based control approach

seeks the aid of a physiological model to control the BGlevels within safer limits. In other words, the diabetic patientshown in Figure 4 is replaced by an equivalent mathematicalmodel for testing the proposed control algorithm. Various

mathematical models discussed in the earlier part of thisarticle have been employed for implementing classical con-trol algorithms like proportional-integral-derivative (PID, PD,PI),68,73�76,78,152,153,159,161,166,167,170,171,173�176,186 feed forwardcontrol78,161,167 to compensate for meal disturbances, and ad-vanced control algorithms like MPC,1,23,37,70,72,154,172 run-to-run,162�165 H∞,

24,77,156,169 fuzzy and fuzzy PID control,151,160,177,179

neural network based controllers,71,159 etc. Some of these controlalgorithms have also been tested on animals166,173,186 and even ondiabetic patients.75,76,152,153,155,163,171,175 Recently, a novel combina-tion of model predictive control and iterative learning control(MPILC) has been proposed byWang et al.178 This combinationhas been proven to be robust for random variations in meal timingsand amounts.More recently, the concept of incorporating supervisorylevel control to monitor the controller performance and detect thefaults in patient’s glucose insulin dynamics has been investigated byusing multivariate statistical techniques like principal componentanalysis (PCA).187 Finan et al.187 applied PCA on simulated andclinical data to distinguish between normal and abnormal days ofT1DM subjects. The results of Finan et al.187 show the possibility ofemploying PCA as a monitoring tool in the supervisory controlconcept to detect faults in patient’s dynamics. A brief summary of thevarious model based control approaches with details like modelemployed, glucose measurement and insulin infusion routes, anddisturbances employed is presented in the Appendix (Table 7).

3. ANALYSIS AND SIMULATION OF T1DM MODELS

As mentioned earlier, data-driven models are advantageous overknowledge-driven models in terms of structural simplicity and lesstime required for model development. However, empirical modelssuffer from several drawbacks compared to knowledge-drivenmodels. The main reason for these is that the former models donot consider any physiological basis. Often empirical models canonly predict BG values with the recent BG recordings and otherinputs, while knowledge-driven models can be helpful in predictingthe glucose, insulin and even glucagon concentrations in variousorgans and tissue. Hence, some of the knowledge-driven modelsdiscussed above (minimal model,21,22,35,37 Hovorka,1,2 Fabietti,5,6

Sturis,11 and Parker20,23,24,122) will be analyzed in this section, basedon the simulations performed on them in MATLAB’s Simulinkenvironment. These models are selected so as to cover at least onemodel from each family of Figure 2. All the fivemodels are simulatedusing their nominal parameter values. The benchmark simulatorused in this work is from Cobelli family. The diagrammaticrepresentation of the selected models is shown in Figure 5.

Mathematical equations of the modified minimal model(MODMM)35 alone is given in this article and the correspondingnominal parameter values can be found in Table 1. The equations ofthe other models can be seen in the literature.1,5,6,11,20,24,26,122 Thenominal parameter values of the other models used in this study aregiven in the Supporting Information. In this work, all the fivemodelsare simulated using their nominal parameter values listed in Table 1(for MODMM) and the Supporting Information (for Hovorkamodel (HM), Fabietti model (FM), Sturis model (SM), and Parkermodel (PM)). The MODMM as per Fisher35 is as follows:

dGdt

¼ � p1G� XðG þ GbÞ þ UGðtÞ ð20Þ

dXdt

¼ � p2X þ p3I ð21Þ



dIdt

¼ � nðI þ IbÞ þ UIðtÞ=VI ð22Þ

where G, X, and I refer to difference in plasma glucose, remoteinsulin, and plasma insulin concentrations from their basalvalues, respectively. UG (t) and UI(t) denote the meal distur-bance function and exogenous insulin infusion, respectively.

The physiological meaning and the values of the parameters inthe above equations are summarized in Table 1.

Originally, all the five models were developed and validatedusing different patient cohorts. They were developed usingdifferent adult patient cohorts with average ages of 33 ( 3,31.17 ( 9.4, and 26.9 ( 1.2 years for HM, FM, and SM,respectively (ages for MODMM and PM were not revealed in

Figure 5. Diagrammatic representation of the five models used for analysis.



their original works), and average BMI of 22.7( 0.6 kg/m2, 26(2.7 kg/m2, and 22.9 ( 1.2 kg/m2 for HM, FM, and SMrespectively. The nominal parameter values in the five modelsare the average parameter values of a particular cohort. The BGdynamics of five models are studied in this work either by varyingthe manipulated variable (i.e., the basal insulin dose) or byvarying the disturbance variable, (i.e., the meal ingestion). As isthe common practice in control studies, all other parameters ofthese models are kept at their nominal values throughout theanalysis in this work. The UVa simulator has three differentpatient cohorts (children, adolescents and adults) based on age.Since all the five models were developed for adult patients, onlyadult patient cohort was considered in the UVa simulator also, tomaintain consistency.3.1. Open Loop Simulations of BGModels and Benchmark

Patients. The five important models and ten adult benchmarkpatients are simulated at various basal glucose levels (GB). Thebasal insulin dose (UI, B) required to attain a particular GB is theonly source of insulin input in all the simulations. From the basalvalues, the manipulated variable UI, B is perturbed for differentstep changes and the corresponding BG dynamics are obtained.The parameters of all the models are at their nominal values, asthese values are mostly used in the control studies. Subsection3.1.1 focuses on the open loop simulations performed on the fiveBG models at GB levels that are frequently used in the literature.Subsection 3.1.2 presents the open loop simulations performedon the benchmark patients of UVa simulator and on the five BGmodels. The objective, simulation and the results involved inthese two analyses are discussed separately. The main motive of

the simulations shown in this section is to study and highlight theBG dynamics of different models (at their nominal parametervalues) for various basal insulin inputs. Structural identifiabilityanalysis followed by parameter estimation should be performedusing the data from the same patient cohorts, for a complete andconsistent comparison of the models.3.1.1. Open Loop Simulations of BGModels for Step Changes

in UI, B at GB = 81 and 108 mg/dl. Objective. The objective ofthe simulations in this section is to study the BG dynamics of thefive models at their nominal parameter values. The BG dynamicsof each model is analyzed for various step changes inUI, B (whichis the manipulated variable), without any meal ingestion.Simulation. The basalUI, B required for maintainingGB in the

absence of meals is calculated for all the five models and thecorresponding values are listed in Table 2. Actually, researchersstudied these models for different basal glucose values like81 mg/dl23,24,35,37,116 and 108�110 mg/dl.1,11,26,79 Hence,UI, B values required to maintain the GB at two different levels(81 and 108 mg/dl) are calculated for each model. Amongthe 5 models, FM and HM require higher and lower externalbasal insulin dose to maintain GB, respectively. In the case ofMODMM, UI, B values remain the same for both GB values asfound in literature.35,37,79 This is because UI, B values inMODMM vary based on the IB value (as UI, B = nVIIB) whichis 15 mU/min for both GB values as in the literature.35,37,79

Unlike MODMM, UI, B of other models depend on GB valuesonly. Hence, UI, B values for the remaining 4 models vary ina way that a higher basal dose is required for maintaining GB at81 mg/dl when compared to the dose required for 108 mg/dl.

Table 1. Physiological Significance and Nominal Values of Parameters in the MODMM

parameter significance value

p1 (min�1) rate of glucose removal from plasma into liver or periphery 0.028735

p2 (min�1) rate of insulin disappearance in remote compartment 0.028344

p3(min�1) rate of insulin appearance in remote compartment 5.035 � 10�5

VI insulin distribution volume 12

n (min�1) fractional disappearance rate constant for endogenous insulin 5/54

IB (mU/l) subject’s basal insulin level 15

GB (mg/dl) subject’s basal glucose level 81

Table 2. Basal Glucose and Insulin, and Percentage Change in BG Due to Step Changes in the External Basal Insulin Dose

diabetic model GB (mg/dl) UI,B(mU/min)

percentage decrease in BGb percentage increase in BGb

hypoglycemic risk

(Yes- if BG < 60 mg/dl)

positive step change in UI,B negative step change in UI,B

5% 15% 25% 5% 15% 25%

MODMM 81 16.67 4.4 12.2 18.9 4.8 16.2 30.1 no

108 16.67 4.4 12.2 18.8 4.8 16.1 30.2 no

Hovorka model (HM)a 81 7.99 3.5 10.3 16.9 28.8 101.5 133.8 no

108 7.53 20.8 29.4 34.2 25.0 62.7 86.3 no

Fabietti model (FM) 81 24.43 13.1 33.8 49.6 15.3 38.5 54.8 yes

108 21.70 9.1 29.4 44.7 4.6 15.5 29.2 yes

Sturis model (SM) 81 10.96 11.2 34 55.1 11.1 33 55.1 yes

108 9.29 6.9 20.8 35.1 6.9 21.3 36.3 no

Parker model (PM) 81 22.3 4.3 12.8 15.7 4.8 14.6 24.2 no

108 14.57 2.3 6.9 11.7 2.3 7 11.9 noa UI,Bwas calculated for an 82 kg person b Percent decrease/increase in BG = (New steady state BG value � GB) � 100/GB.



In order to study the glucose dynamics of each model underboth GB conditions (i.e., 81 and 108 mg/dl), simulations areperformed by introducing(5%,(15%, and(25% step changesin the external basal insulin doses without any meal intake.Results and Discussion. The resulting BG variation and

percentage decrease/increase in BG level from the basal statefor the specified step variations in the basal insulin input can befound in Figure 6 and Table 2, respectively. The BG profiles ofSM (in Figures 6(D) and (I)) and PM (in Figure 4(E) and (J))and the respective percentage BG variations from the basal BGlevels (in Table 2) reveal proportionate increase or decreaseof BG for(5%,(15%, and(25% step changes in the insulininput. The symmetric BG profiles of SM and PM indicate thatthese systems possess high open-loop linearity for theirnominal parameter values. Also, the BG profile of MODMM(Figures 6(A) and (F)) shows almost symmetric BG trends for(5% and (15% insulin step changes. Hence, linear controlalgorithms should be sufficient for controlling BG levels inMODMM, PM, and SM. This is evident from the modelbased control works23,79 related to PM and MODMM, whichhave identified linear model predictive controller (MPC) asan efficient BG control algorithm. In contrast, HM exhibits ahigh degree of nonlinearity as its BG profiles are more sensitiveto negative insulin step changes compared to those to positiveinsulin step changes as shown in Figures 6(B) and (G). Hence,the use of nonlinear control algorithms like nonlinear MPC(NMPC) for HM has been studied in the literature.1 Interest-ingly, BG profiles of FM show symmetric trends when the basalglucose level is at 81 mg/dl (Figure 6(C)) and asymmetric

behavior for GB at 108 mg/dl (Figure 6(H)), whereas othermodels (MODMM, HM, SM, and PM) show either symmetricor asymmetric trends for the simulations at both GB values.Hypoglycemic episodes are more risky when compared to

hyperglycemic episodes, as the former may lead to serious com-plications like coma and death. Even BG level up to 300 mg/dl isdefined as permissible limit in the literature38 which is not thecase for BG < 60 mg/dl. Hence, the possibility of hypoglycemicrisk when these models are simulated for various insulin stepchanges is analyzed. Mainly, FM and SM show the risk ofhypoglycemia for 15% and 25% insulin step change (bold valuesin Table 2) atGB of 81 mg/dl. Also, FM produces hypoglycemicrisk for 25% insulin step change atGB of 108 mg/dl. There is norisk of hypoglycemia observed in the other models (MODMM,HM, and PM) until the positive step change of 25% in theexogenously infused insulin. It is also important to note thatnone of these models have shown serious BG elevations (>300mg/dl) for negative step changes in the insulin input.3.1.2. Open Loop Simulations of the Benchmark Patients

and BG Models at Actual GB Values of Benchmark Patients.Objective. The main objective is to study the BG dynamics ofeach adult patient in UVa simulator and to analyze which of thefive nominal parameter models are closer to each adult patient interms of BG dynamics. The interpatient differences on the fivemodels are incorporated only in terms ofGB andUI,B values. Otherparameters of all themodels aremaintained at their nominal valuesas most of the control studies employ these nominal values.Simulation. In this study, the UVa simulator is used as the

benchmark to compare the BG profiles of other selected models.

Figure 6. Variations in BG profiles for step changes inUI, B ((5%,( 15% and(25%) atGB = 81 mg/dl (top row) and 108 mg/dl (bottom row). Notethat different scale is used in the y-axis in some plots for clarity and conciseness. The star symbols in the subplots (B), (G), and (H) indicate nonlinearitydue to asymmetric BG trends; smiley faces in subplots (A), (B), (E), (F), (G), (I), and (J) represent no hypoglycemic risk and frowning faces in thesubplots (C), (D), and (H) represent existence of hypoglycemic risk.



The reason for this is that FDA has permitted the simulator as anIDE application once.25 However, by doing so, we do not claim thatthe UVa simulator can represent the response data from the realpatients. Instead of having no benchmark to compare the modelprofiles with, we opted to use UVa simulator because it has receivedsome recognition from FDA. The actual GB values of 10 T1DMadult patients used by the UVa simulator vary from 135.3 to 152.8mg/dl. The UI,B doses required to maintain the actual GB values inthe 10 adult patients of UVa simulator are calculated (filled circles inFigure 7). As already mentioned interpatient differences on the fivemodels are introduced only terms ofGB andUI,B values. Hence,UI,B

doses required to maintain the actual GB values (corresponding toten adult patients) are calculated for each model at their nominalparameter values. TheUI,B dose of each patient to maintain his/herGB is found to be different for each of the five models, as shown inFigure 7. This difference in UI,B for the same patient may be due tothe structural differences of each model and it also depends on theactual basal glucose value (GB) of a patient. In most of the patients,UI,B doses by MODMM and FM are closer to those of the UVasimulator. The UI,B values calculated by HM, SM, and PM are veryclose to each other but much lower than the benchmark values. Insome cases (like Patients 4 and 9), the MODMM calculated dosesalmost coincide with the values calculated by the simulator. How-ever, the model calculatedUI,B doses of the remaining eight patientsare much less than the benchmark requirement. All the ten adultUVa patients and the similar ten patients generated by the fivemodels (in terms of GB and UI,B values) are simulated for (5%,(15%, and (25% step changes in UI, B.Results and Discussion. The actual GB values of the 10

patients in the UVa simulator are much higher than thefrequently used GB values (i.e., 81 or 108 mg/dl) in theliterature.1,11,23,24,26,35,37,79,116 For higher step changes in UI, B,there exists a risk of hypoglycemia for low GB values (like 81 or108 mg/dl), as has been confirmed in Subsection 3.1.1. Simula-tion of the 10 benchmark patients at their actual GB values haveshown that none of the patients encounter hypoglycemic riskfor (5%, (15%, and (25% step changes in UI, B doses (resultsnot shown here for brevity). However, severe hyperglycemicepisodes (BG > 300 mg/dl) are reported in UVa patients 6 and 7

for �25% external insulin step changes. Also, these simulationsindicate that patients 3, 4, 6, and 7 exhibit nonlinear BG dynamicsfor varying step changes in UI,B.The BG dynamics of the 10 patients due to (5%, ( 15%,

and(25% step changes in the UI,B values are obtained for all thefive models and are compared with the benchmark. For brevity,only the BG profiles of Patient 9 are shown here (Figure 8).The closer models to each benchmark patient have beenidentified in terms of the final steady state BG values for eachstep change in UI,B. The degree of closeness (Δ(n %) of a modelfor a particular step change in UI,B is quantified as: (% change ofBG)UVa � (% change of BG)nearest model. The closer models andΔ(n % values for each patient are summarized in Table 3. Thelowest and highest values of Δ(n% for nearer models are 0%(for +25% step change in patient 9 and for �5% change inpatients 8 and 10) and 114.5% (for�25% step change in patient 7),respectively. Formost of the step changes inUI,B,Table 3 shows thatMODMMis closer to patients 1, 2, 5, and 8,HM is closer to patients3, 4, 6, and 7, SM and FM are closer to patients 9 and 10. Also, forUVa patients (no: 3, 4, 6, 7) who exhibited nonlinear BG dynamicsfor the step changes in UI,B, a nonlinear model like HM is found tobe the closest model for majority of the step changes. Hence,nonlinear BG models can be better for treatment for patients withnonlinear BGdynamics.Nevertheless,morework should be done toidentify a particular model that is suitabe to devise the treatmentstrategy. Overall, this analysis has given a choice of a few BGmodelsfor eachUVa patient.Out of these two or three competingmodels, asingle representative treatment model can be selected for eachpatient via parameter identification and model-based experimentaldesign. Such identification is beyond the scope of this article, andwill be considered in our future work.3.2. Simulation of Meal Absorption Models. Objective.

To analyze the meal absorption dynamics of the available mealmodels at their nominal parameter values. Meal absorptiondynamics for two different sizes of meal ingestion is studied.Simulation. The meal models described in Subsection 2.2

are implemented in Simulink. These models are simulated for500 min to observe the rate of glucose appearance in plasma aftermeal ingestion for two differentmeal sizes of 50 and 10 g (Figure 9).

Figure 7. UI,B required to maintain the actual GB values of 10 patients, given by the models and the benchmark.



The meal model parameter values used for the simulations aresummarized in the Supporting Information. In all the models,meal disturbance is given at 120 min. Hence, zero glucoseabsorption rate prevails for the first two hours in all the models(as shown in Figure 9), after which the glucose absorption

profile of each model attains a peak and settles back to theinitial level.Results and Discussion. The peak absorption rate and settling

time are different for eachmeal model. Figure 9 and Table 4 showthat there exists variation between models in terms of peak

Figure 8. Variations in BG profiles of patient 9 for step changes inUI, B: (A), (C), and (E) for�5%,�15%, and�25% step changes; and (B), (D), and(F) for 5%, 15%, and 25% step changes in UI,B.

Table 3. Closer Model and Its Degree of Closeness to the Benchmark Patients for (n% Step Changes in UI, Ba

patient

closer model and its degree of closeness (Δ(n%) to the benchmark patient

for +n% step change in UI,B for �n% step change in UI,B

no. 1 FM (Δ5% = �1.3%, Δ15% = �0.3%) SM (Δ25% =0.4%) MODMM (Δ�5% = 1.5%, Δ�15% = 5.8%, Δ�25% = 2.9%)

no. 2 PM (Δ5% = 1.1%) FM (Δ15% = �2.6%, Δ25% = 0.5%) MODMM (Δ�5% = 4.2%, Δ�15% = 0.5%, Δ�25% = �0.7%)

no. 3 FM (Δ5% = �0.1%) MODMM (Δ15% = �0.4%, Δ25% = �1%) HM (Δ�5% = �0.9%, and Δ�25% = �1.3%, Δ�15% = 1.7%)

no. 4 FM (Δ5% = 0.7%) SM (Δ15% = 10.7%) HM (Δ25% = �13.6%) HM (Δ�5% = �1.9%, Δ�15% = 16.8%, Δ�25% = 47.5%)

no. 5 MODMM (Δ5% = �0.8%, Δ15% = �1.3%, and Δ25% = �3.4%) PM (Δ�5% = 0.4%) FM (Δ�15% = 0.5%, Δ�25% = �0.7%)

no. 6 HM (Δ5% = �5.7%, and Δ15% = �3.5, Δ25% = 4.8%) HM (Δ�5% = �4.5%, Δ�15% = 31.4%, Δ�25% = 62.5%)

no. 7 HM (Δ5% = �1%, Δ15% = �1.4%, Δ25% = 4.1%) HM (Δ�5% = 2.4%, Δ�15% = 59.1%, Δ�25% = 114.5%)

no. 8 MODMM (Δ5% = 1.4%, Δ15% = 2.8%) SM (Δ25% = 7.6%) SM (Δ�5% = 0%) MODMM (Δ�15% = �1.3%, Δ�25% = 2.6%)

no. 9 SM (Δ5% = 0.2%) FM (Δ15% = 0.1%, Δ25% = 0%) MODMM (Δ�5% = 0.2%) SM (Δ�15% = �0.4%, Δ�25% = �2.3%)

no. 10 FM (Δ5% = �0.4%, Δ15% = 0.8, Δ25% = �0.6%) MODMM (Δ�5% = 0%) SM (Δ�15% = �1.3%, Δ�25% = �2.9%)aΔ(n% = (% decrease/increase of BG)UVa � (%decrease/increase of BG)nearest model; In other words, Δ(n% can be taken as a measure of closeness(in terms of the newly attained steady state BG value, after ( step change in the UI,B) of the diabetic model to the UVa patient.



absorption rate and settling time. These differences may be dueto the basis on which a model was developed. For instance,FIMM attains the peak value immediately after meal ingestion(i.e., at 120 min itself peak is reached) and the absorption profiletends to settle within 80 min of meal ingestion. Also, whencompared to other models, the peak value of glucose absorptionrate is very high for both large (2500 mg/min for 50 g meal) andsmall meal (500 mg/min for 10 g meal) sizes. This may be due tothe structural difference of FIMM from the other models.Practically, meal digestion does not occur immediately after foodintake as shown by FIMM profile in Figure 9. When the foodreaches stomach through esophagus, grinding, and gastric emp-tying occurs after which it enters small intestine. In the upper halfof the small intestine, glucose absorption into blood circulationstarts slowly and reaches a peak value after which the absorptionrate restores back to the initial level gradually. The one compart-mental model structure of FIMM (shown in Figure 3(A)) is notsufficient to explain the glucose transit in the gastrointestinaltract before intestinal absorption.Other meal models describe the glucose absorption rate from

the beginning of oral meal intake. The two compartment modelof Hovorka et al.1 allowed a time of 40min to reach themaximumabsorption rate, which is observed well in the simulated resultsof Figure 9. Though HMM explained the meal absorptionwithout showing the transit of glucose in stomach and gastricemptying, the proposed structure mimics the absorption profilefrom the commencement of meals. This means at least a twocompartment structure is required to capture the absorption

profile from the start of meal intake. The same is the case withFMMwhich models the meal absorption through three differenttransfer functions for different types of meal as shown inFigure 3(D). In fact, the time taken to reach the peak absorptionis almost the same in HMM and FMM for both the meal sizes(Table 4). The LDMM and DMM explained the glucoseabsorption from meal by considering the gastric emptying rateas shown in Figure 3(C) and (E). The glucose absorption profilesof LDMM showed different peak times for 50 and 10 g meals(Figure 9).However, other meal models show same peak time for both

meal sizes as shown in Table 4 and Figure 9. LDMM differ-entiated the gastric emptying rate for large and small mealsizes by using trapezoidal and triangular functions, respectively.Hence, large meal size requires a long time of 139 min(trapezoidal function) and small meal size needs a shorter timeof 48 min (triangular function) to reach the peak absorption rate.The gastric emptying function used by DMM does not showsuch differentiation and hence the peak time of absorption is thesame (21 min) for both large and small size meals, like someother models. However, Dalla Man et al.13 showed that LDMM’sabsorption profile was not able to fit real experimental dataobtained from 41 subjects after OGTT and mixed meal intake.Also, DMM was developed based on these experiments. Hence,it seems that the meal size differentiation of LDMM does notmatch with the reality, and DMMmodel is more consistent withthe actual physiological behavior.

Figure 9. Meal absorption profiles of (A) 50 g CHO meal; and (B) 10 g CHO meal.

Table 4. Peak Glucose Absorption Rate and Time Taken to Reach the Peak Absorption, Predicted by Different Meal AbsorptionModels for Large and Small Meal Sizes

meal model

larger meal size (50 g CHO) smaller meal size (10 g CHO)

peak glucose absorption rate in

plasma (mg/min)

time taken to reach maximum

absorption rateapeak glucose absorption rate in

plasma (mg/min)

time taken to reach maximum

absorption ratea

FIMM 2500 0 500 0

LDMM 314 139 114 48

HMM 368 40 74 40

FMM 439 44 88 44

DMM 358 21 72 21aTime taken to reach maximum absorption rate = Peak glucose absorption time - Meal ingestion time.



3.3. Simulation of BG Dynamic Models for Different MealScenarios. Objective. To study the impact of meal absorp-tion dynamics on the BG dynamics in different models (attheir nominal parameter values) for single and multiple mealscenarios.Simulation. The MM, SM, and PM use LDMM as their

meal model, as literature studies have used some of thesecombinations.23,24,37 HM, and FM use their own respective mealmodel (i.e., HMM and FMM). The results of these models arecompared with the bench mark (UVa simulator), which uses ameal model similar to DMM. Different meal models are used forthe five BG models, in order to study the variations of thesemodels in predicting BG levels from the real patient behavior. Assaid earlier, this work persumes the benchmark patients as realpatients for comparison. For single meal scenario, meal contain-ing 50 g of CHO is introduced at second hour. Single mealscenario is tested for all the 10 patients in order to find whichmodel is more sensitive to meal disturbances. In case of multiplemeal condition, meals containing 35 g, 50 g, 10 g, and 60 g CHOare introduced at second, seventh, 12th, and 15th hour, respec-tively, to mimic typical meals in a day. Both these scenarios areimplemented on the five models and the resulting BG profiles arecompared with that of benchmark for all the 10 patients. All thefive BG models are simulated with their nominal parametervalues listed in the Supporting Information at theGB values of the

10 benchmark patients. UI,B calculated using the five models,and the benchmark (shown in Figure 7) is the only source ofexogenous insulin for all the 10 patients. The actual intentionof the simulations shown in this section is to study andhighlight the BG dynamics of different models (at theirnominal parameter values) for various meal scenarios. Struc-tural identifiability analysis followed by parameter estimationshould be performed for a complete and consistent compar-ison of the models.Results and Discussion. The BG profiles after single meal

uptake indicate that SM is themost sensitive tomeal disturbancesfor all the 10 patients (only the profiles of two patients are shownin Figures 10 and 11 for brevity). HM takes longer time to returnto the basal BG values for all the patients, which has beenreported as a structural limitation of HM.116 However, somebenchmark patients (patient 6) take a longer time to return to thebasal values. HM. FM and MM are closer to patient 2 in termsof the maximum BG value attained after every meal uptake(Figure 10). On the other hand, patient 6 is closer to the peaks ofPM for the first twomeals and to the peaks of SM andHM for thelast two meal disturbances (Figure 11B).The maximum and minimum BG values attained by each

patient during a day with four meals and basal insulin dosecan be found in Table 5. Magni et al.188 categorized T1DMpatients into different zones based on the maximum and

Figure 10. BG changes of patient 2 (in the UVa simulator) and models studied, due to meal disturbances: (A) Single meal scenario; and (B) Multiplemeals scenario.

Figure 11. BG changes of patient 6 (in the UVa simulator) and models studied, due to meal disturbances: (A) Single meal scenario and (B) Multiplemeals scenarios.



minimum BG values recorded in a day. This study shows thatpatients with minimum BG > 110 mg/dl and maximum BG of180�300 mg/dl belong to gentle hyperglycemic zone and patientswith maximum BG > 300 mg/dl are categorized into severehyperglycemic zone. The maximum and minimum BG values inTable 5 show that the 10 patients fall into either of these twozones. The extreme BG values predicted by MODMM, FM, andPM vary between 130 and 300 mg/dl. These three models fall ingentle hyperglycemic zone due to multiple meals during a day.However, in the case of HM and SM, all the patients are at severehyperglycemic zone (i.e., maximum BG > 300 mg/dl). The max-imum BG values predicted by HM for 50% of patients vary from300 mg/dl to 400 mg/dl and for the remaining 50% patients thesevalues are even higher (>400 mg/dl). The SM predicts that all thepatients encounter maximum BG values greater than 400 mg/dl.Hence, mere UI,B dose alone is not sufficient to avoid severehyperglycemic risks (i.e., when BG > 300 mg/dl) in all thepatients of HM and SM. The results of benchmark simulatorshow that 40% of the patients show gentle hyperglycemicdeviation (i.e., extreme BG < 300 mg/dl) while the remainingpatients encounter severe hyperglycemic risk. However, noneof the five models have shown such a distribution of patientsbetween the gentle and severe hyperglycemic zones. Thesemodels place all the patients either in the gentle hyperglycemic(MODMM, FM, and PM) or in the severe hyperglycemicranges (HM, and SM).

4. CONCLUSIONS

Progressive development of knowledge- and data- driven BGdynamic models starting from early 1960s until today has beenreviewed in this article. An overview of the available mealabsorption and exercise effect models as well as a brief summaryof BG control studies has been provided. Open loop simulationstudies have been performed to study the BG dynamics ofdifferent BG models and ten adult patients of a benchmarksimulator. The analysis is on studying the BG dynamics forvarying basal insulin inputs, the meal absorption dynamics forvariousmeal sizes, and the impact ofmeal absorption dynamics onBG dynamics for different meal scenarios. The results highlightthe different dynamics and features of available BG and mealmodels. For example, the analysis has shown that most of theBG models at their nominal parameter values follow linear BGdynamics while a few exhibit nonlinear BG dynamics whereas 3 ofthe 10 adult patients in the simulator show nonlinear dynamics.The reviews and analyses of this work are useful for further studieson BG regulation and for the preliminary selection of BG andmeal models for the treatment of a particular adult patient.

’APPENDIX

The appendix comprises the summary of various data drivenmodels andmodel based control studies available in the literaturein Tables 6 and 7.

Table 5. Maximum and Minimum BG Values Attained in a Day (24 h) with Four Meals and Basal Insulin Dose

patient

maximum BG attained in a day (mg/dl) minimum BG attained in a day (mg/dl)

MODMM HM FM PM SM UVa MODMM HM FM PM SM UVa

no. 001 229 348 253 273 434 250 139 207 151 141 182 174

no. 002 226 332 251 268 429 242 137 201 149 139 178 153

no. 003 237 403 262 292 452 302 148 228 158 151 196 182

no. 004 241 480 267 299 428 355 151 247 163 155 184 212

no. 005 233 368 256 282 442 255 143 215 153 146 189 169

no. 006 226 455 250 266 427 370 136 237 149 138 177 256

no. 007 225 370 250 265 425 305 136 212 148 137 176 207

no. 008 233 350 256 283 444 216 144 210 154 147 190 164

no. 009 235 426 259 288 446 309 146 233 156 148 193 194

no. 010 243 433 270 304 465 315 153 243 165 158 206 187

Table 6. Data-Driven BG Prediction Models Reported in the Literature; See the Text for More Details

year (author) model type data source, size model inputs

1992 (Kotanko et al.54) UPMA 2 T1DM subjects diet, physical activity, insulin therapy, preceding BG values

ANN

1998 (Sandham et al.59) RANN 2 T1DM subjects insulin, diet, exercise, past BG levels with a vector of stress,

illness, and pregnancy

1999 (Tresp et al.62) RANN + Linear error

model

1 T1DM subject insulin therapy, meal intake, exercise, and BG

1999 (Ghevondian et al.50) ANN 2 T1DM subjects skin impedance, and heart rate

2006 (Mougiakakou et al.56) FFANN and RANN 2 T1DM children insulin doses, CHO intake and measured BG (past values)

Time Series Models

1999 (Bremer et al.46) AR 22 ambulatory T1DM,

nondiabetic and T1DM

subjects.

BG data

2006 (Van Herpe et al.64) ARX 41 ICU patients insulin, total carbohydrate (CHO) calories, body

temperature and dopamine



Table 6. Continuedyear (author) model type data source, size model inputs

2006 (Van Herpe et al.63) OLS, and WLS models 15 ICU patients insulin, total carbohydrate (CHO) calories, body

temperature and dopamine

2007 (Sparacino et al.60) Polynomial, and AR

models

28 ambulatory T1DM patients continuous glucose monitoring (CGM) from GlucoDay

2007 (Reifman et al.58) AR 15 T1DM subjects continuous subcutaneous BG measurements from iSense

CGM

2008 (Finan et al.48) AR, ARX, and ARMAX 2 ambulatory T1DM subjects

and 1 simulated T1DM

insulin therapy, CGM, andmeals (for ambulatory patients)

with stress and exercise (simulated data)

2008 (Kazama et al.53) AR 40 normal, 34 impaired glucose

tolerance, and 41 T2DM

OGTT data

2009 (Ghosh et al.51) NARX 18 subjects IVGTT data

2009 (Cescon et al.47) AR, ARMAX, and SS 1 T1DM age, body mass index, weight, insulin therapy, meal intakes,

and measured BG

2010 (Gani et al.49) AR 27 T1DM, and 7 T2DM CGM data from iSense, Guardian RT, and DexCom

2010 (Skrovseth et al.96) AR 1 T1DM, and 12 T2DM measured BG data

Table 7. Model/Subject Based Control Algorithms along with Insulin Infusion-Glucose Measurement Routes and Disturbancesa

year (author) control strategy employed model(s)/ subject(s)

insulin infusion�glucose

measurement routes disturbances introduced

1964 (Kadish155) on�off subject i.v.�i.v. glucose or glucagon

1984 (Chisolm et al.69),

and 1985 (Furler et al.29)

semiclosed loop MODMM29 i.v.�i.v. meal

1989 (Ollerton32) optimal control theory MODMM22,32 i.v.�i.v.

1991 (Fisher35) optimal control theory,

and semiclosed loop

MODMM22,35 i.v.�i.v. single meal

1999 (Parker et al.23) MPC Sorenson20�Parker23 i.v.�arterial single meal

2000 (Parker et al.24) H∞ control Sorenson20 � Parker23 i.v.�rterial single meal

2002 (Lynch et al.37) MPC MODMM37 i.v.�s.c. single meal

2002 (Steil et al.173,174) PID diabetic canines s.c.�s.c. two meals/day

2002 (Renard et al.75,76) PD T1DM subjects i.p.�i.v. three meals/day

2003 (Chee et al.152) PI ICU patients i.v.�s.c

2003 (Chee et al.153) PID ICU patients i.v.�s.c. oral glucose

2003 (Matsuo et al.186) PD beagle canines + PK model i.p.�s.c. two meals/day

2004 (Hovorka et al.1) NMPC HM 1 + T1DM patients for

validation

s.c.�i.v. meal

2004 (Ramprasad et al.74) robust PID Sorenson20�Parker23 i.v.�arterial single and multiple meals

2004 (Ruiz-Velazquez et al.77) H∞ control Sorenson20�Parker23 i.v.�periphery two meals/day

2004 (Dua et al.70) MPC MODMM22,35 i.v.�i.v. meal

2005 (El-Jabali et al.71) neural controller ANN model meals, exercise

2006 (Schaller et al.172) MPC subjects+ HM 1 s.c.�i.v./s.c. meal

2006 (Renard et al.171) PID subjects i.p.�i.v. multiple meals

2006 (Steil et al.175) PID subjects s.c.�s.c. five meals containing 15 g

CHO in juice

2006(Owens et al.162) run-to-run MODMM9,21,22 i.v.�i.v. three meals

2006 (Marchetti et al.73) switching PID HM 1-Wilinska7 s.c.�i.v. single meal

2006 (Canonico et al.68) PDD2 FM5,6 s.c.+ i.v.� i.v. three meals/day

2006 (Panteleon et al.166) PID diabetic canines s.c.�i.v. single meal

2006 (Campos-Delgado

et al.151)

fuzzy control Sorenson20�Parker;23 Berger et al.9 s.c.�arterial three meals/day

2007(Li et al.159) PID based on ANN MODMM22,35 i.v.�i.v. meal

2007(Palerm et al.164) run-to-run HM1-Wilinska7 s.c.�i.v. three meals/day

2007 (Palerm et al.163) run-to-run subjects s.c.�s.c. two meals/day

2007 (Magni et al.72) MPC Dalla Man et al.12,13,15 s.c.�s.c. three meals/day



’ASSOCIATED CONTENT

bS Supporting Information. Nominal parameter values ofthe five BG models and meal absorption models that have beenanalyzed in this article. This material is available free of charge viathe Internet at http://pubs.acs.org.

’AUTHOR INFORMATION

Corresponding Author*Fax: (+65) 6779 1936; e-mail: [email protected].

’ACKNOWLEDGMENT

We thank National University of Singapore (NUS) for thefinancial support.

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Table 7. Continued

year (author) control strategy employed model(s)/ subject(s)

insulin infusion�glucose

measurement routes disturbances introduced

2008 (Marchetti et al.161) switching PID, FF, FB-FF HM1-Wilinska7 s.c.�i.v. single meal

2008 (Palerm et al.165) run-to-run HM1-Wilinska7 with a model

for circadian change

s.c.�i.v. three meals/day

2008 (Susanto-Lee et al.177) fuzzy PID MODMM29 i.v.�i.v.

2008 (Yasini et al.179) fuzzy logic MODMM22,35 i.v.�i.v. two glucose meals/day

2009 (Li et al.160) fuzzy PID MODMM22,35 i.v.�i.v. single meal

2009 (Percival et al.167) PID and FFP control HM 1-Wilinska7 s.c.�i.v. multiple meals

2009 (Dua et al.154) MPC MODMM22,35 i.v.�i.v. meal

2009 (Farmer et al.78) PID, FF, FF/FB, P MODMM;22,35 HM 1;

Sorenson20�Parker;23i.v.�i.v. meal; and exercise

2009 (Eren-Oruklu et al.80) adaptive model based control Glucosim,125 and HM 1-Wilinska7 s.c.�s.c. four meals/day

2009 (Reddy et al.170) PI MODMM22,35 i.v.�i.v. meal

2010 (Quiroz et al.169) H∞ control Sorenson20�Parker23 i.v.�peripheral meals, exercise

2010 (Kamnath et al.156) H∞ control Sorenson20�Parker23 i.v.�arterial two meals/day

2010 (Wang et al.178) MPILC Dalla Man et al.12,13,15,25 s.c.�s.c. multiple mealsa P: proportional; I: integral; D: derivative; FF: feed forward; FB: feed backward; i.v.: intravenous; s.c.: subcutaneous; i.p.: intra-peritoneal.



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Documents

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