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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
IEEE Rev Biomed Eng. Author manuscript; available in PMC 2010 October 8.
Published in final edited form as:
PMCID: PMC2951686
NIHMSID: NIHMS235670

Diabetes: Models, Signals, and Control

Abstract

The control of diabetes is an interdisciplinary endeavor, which includes a significant biomedical engineering component, with traditions of success beginning in the early 1960s. It began with modeling of the insulin-glucose system, and progressed to large-scale in silico experiments, and automated closed-loop control (artificial pancreas). Here, we follow these engineering efforts through the last, almost 50 years. We begin with the now classic minimal modeling approach and discuss a number of subsequent models, which have recently resulted in the first in silico simulation model accepted as substitute to animal trials in the quest for optimal diabetes control. We then review metabolic monitoring, with a particular emphasis on the new continuous glucose sensors, on the analyses of their time-series signals, and on the opportunities that they present for automation of diabetes control. Finally, we review control strategies that have been successfully employed in vivo or in silico, presenting a promise for the development of a future artificial pancreas and, in particular, discuss a modular architecture for building closed-loop control systems, including insulin delivery and patient safety supervision layers. We conclude with a brief discussion of the unique interactions between human physiology, behavioral events, engineering modeling and control relevant to diabetes.

Index Terms: Artificial pancreas, automatic control, identification, parameter estimation, physiological systems, sensors, signal processing

I. Introduction

Diabetes is a common metabolic disorder characterized by chronic hyperglycemia that leads to microvascular and macrovascular complications [1]–[5]. These complications include limb loss, blindness, ischemic heart disease, and end-stage renal disease. Diabetes is broadly classified into two categories, type 1 diabetes and type 2 diabetes. Both arise from complex interactions between genes and the environment, however their pathogenesis is distinct. Type 1 diabetes is the result of immune-mediated destruction of the beta-cells in the islets of Langerhans—the site of insulin secretion and production. In general, the disease occurs in childhood and adolescence (although it can occur at all ages) and is characterized by absolute insulin deficiency. Consequently, affected individuals require insulin therapy to control hyperglycemia and sustain life. As a rule, obesity does not play a part in the pathogenesis of type 1 diabetes, although obesity in type 1 diabetes is associated with the development of cardiovascular complications. In contrast, type 2 diabetes occurs because insulin secretion is inadequate and cannot overcome the prevailing defects in insulin action, resulting in hyperglycemia. Excess caloric intake, inactivity, and obesity all play parts in the pathogenesis of type 2 diabetes. In general, it is a disease that occurs with increasing frequency with increasing age and is uncommon before age 40 (although there are important exceptions). In addition, people with type 2 diabetes are more likely to have associated adverse cardiovascular risk factors such as dyslipidemia and hypertension. Prediabetes, i.e., impaired fasting glucose (IFG) and impaired glucose tolerance (IGT), is an intermediate condition in the transition between normality and diabetes. People with IGT or IFG are at high risk of progressing to type 2 diabetes, although this is not inevitable. Both type 2 diabetes and prediabetes are recognized risk factors for overt cardiovascular disease and related metabolic complications and are major components of health care spending [6], [7]. Rapid urbanization and societal affluence of global migrating populations has been suggested as major risk factors for the observed exploding prevalence of prediabetes and type 2 diabetes with consequent rising trends in cardiovascular risks [8]. IFG is a rapidly emerging form of prediabetes with a 20%–30% risk of progression to diabetes over 5–10 years. This risk is even greater if individuals have both IFG and IGT. Furthermore, both IFG and IGT are linked to increased risk for cardiovascular events [6], [7] in the Caucasian population. Ninety percent of the world population with diabetes is type 2 with type 1 diabetes comprising between 5%–10%. It is plausible that the relative frequency of type 1 and type 2 diabetes will change with rising trends in the prevalence of type 2 diabetes, obesity, and prediabetes in the developing world.

Over time, diabetes leads to complications, in particular: diabetic retinopathy, which leads to blindness; diabetic neuropathy, which increases of the risk of foot ulceration and limb loss; and diabetic nephropathy leading to kidney failure. In addition, there is an increased risk of heart disease and stroke with 50% of people with diabetes dying of cardiovascular disease and stroke. Finally, the overall risk of dying among people with diabetes is at least double the risk of their peers without diabetes. The World Health Organization (WHO) estimates than more than 180 million people worldwide have diabetes. This number is likely to more than double by 2030. In 2005, an estimated 1.1 million people died from diabetes. When ranked by cause-specific mortality, diabetes is the fifth cause of death, after communicable diseases, cardiovascular diseases, cancer and injury [8]. Almost 80% of diabetes death occurs in low- and middle-income countries. WHO projects that diabetes deaths will increase by more than 50% in the next ten years without urgent action. Most notably, diabetes is projected to increase by over 80% in upper-middle income countries between 2006 and 2015. Diabetes and its complications impose significant economic consequences on individuals, families, health systems, and countries. WHO estimates that over the next ten years (2005–2015) China will lose $558 billion in national income due to heart disease, stroke, and diabetes alone.

Given the complexity of the disease it is not surprising that diabetes is fought with a battery of tools spanning over several disciplines, from cellular biology to pathophysiology to pharmacology to chemistry, physics, and engineering to transplantation to patient management to health care (Fig. 1). Dynamic system models are an essential ingredient of virtually all of these strategies. However, in this review we have to limit our scope; thus, we will not be able to discuss modeling in the areas of patient management [9], [10] and health care intervention strategies [11]–[14], or in the emerging area of systems biology [15], [16]. We will first focus on two mechanistic, physiologically based classes of models: minimal (coarse) models which describe the key components of the system functionality and are capable of measuring crucial processes of glucose metabolism and insulin control in health and diabetes; maximal (fine-grain) models which include comprehensively all available knowledge about system functionality and are capable to simulate the glucose-insulin system in diabetes, thus making it possible to create simulation scenarios whereby cost effective experiments can be conducted in silico to assess the efficacy of various treatment strategies. Then, we discuss the crucial role of models to enhance the interpretation of glucose and insulin time-series signals. Finally, we discuss recent strategies, in particular Model Predictive Control aiming for closed-loop control of type 1 diabetes (known as artificial pancreas), where models also play an important role.

Fig. 1
Disciplines which contribute to diabetes control.

II. Glucose-Insulin Control System

Glucose concentration is tightly regulated in health by a complex neuro-hormonal control system [1], [2]. Insulin is the primary regulator of glucose homeostasis, i.e., it promotes glucose utilization and inhibits glucose production. A battery of counterregulatory hormones are also at work, i.e., glucagon, epinephrine, cortisol, and growth hormone, which defend, on different time scales, the body from life-threatening hypoglycemia. Both hypoglycemia counterregulation and insulin control are neuro-mediated.

In this review, we focus on the glucose-insulin control system which is not only the most studied in terms of modeling, but also the one where modeling has had major impact in diabetes research and therapy. A high-level scheme of this system is shown in Fig. 2. Glucose is produced (mainly by the liver), distributed, and utilized in both insulin-independent (e.g., central nervous system and red blood cells) and insulin-dependent (muscle and adipose tissues) tissues. Insulin is secreted by pancreatic beta-cells, reaches the system circulation after liver degradation, and is peripherally cleared primarily by the kidneys. The glucose and insulin systems interact by feedback control signals, e.g., if a glucose perturbation occurs (after a meal), beta-cells secrete more insulin in response to increased plasma glucose concentration and in turn insulin signaling promotes glucose utilization and inhibits glucose production so as to bring rapidly and effectively plasma glucose to the preperturbation level. These control interactions are usually referred to as insulin sensitivity and beta-cell responsivity. In pathophysiology, the control is degraded. In type 2 diabetes this degradation is initially presented as prediabetes, characterized by progressive deterioration of both insulin sensitivity and beta-cell responsivity. In type 1 diabetes, beta-cells rapidly become virtually silent and insulin must be provided exogenously by the patient attempting to compensate for hyperglycemia. However, insulin treatment may risk potentially severe hypoglycemia and thus people with type 1 diabetes face a life-long behavior-controlled optimization problem: to maintain strict glycemic control and reduce hyperglycemia, without increasing their risk for hypoglycemia. Blood glucose is both the measurable result of this optimization and the principal feedback signal to the patient for his/her control of diabetes. Several types of models assist the optimization of diabetes control.

Fig. 2
Scheme of the glucose-insulin system.

III. Minimal Models

A. Rationale

Minimal models must be parsimonious and describe the key components of system functionality. Thus, a sound modeling methodology must be used to select a valid model, i.e., a well founded and useful model which fulfills the purpose for which it was formulated [17]. Briefly, it is reasonable to assume that a good minimal model will not be a large-scale one: not every known substrate/hormone needs to be included because the macro-level response of the system would be relatively insensitive to many micro-level relationships. In addition, because it is not possible to estimate the values of all system parameters from in vivo dynamic data, many of the unit processes must be lumped together. Therefore, desirable features of a minimal model include: 1) physiologically based; 2) parameters that can be estimated with reasonable precision from a single dynamic response of the system; 3) parameters that vary within physiologically plausible ranges; and 4) ability to describe the dynamics of the system with the smallest number of identifiable parameters. Given a set of dynamic data, e.g., plasma concentration measurements after a perturbation, one generally proceeds by proposing a series of system models, beginning with the simplest and systematically increasing the complexity by including more known physiological detail. Each of the series of system models is then tested for a priori identifiability and subsequently numerically identified from experimental data in which samples are taken at frequent intervals, and for which measurements are made with utmost care (to reduce the influence of measurement error on the choice of a model). A number of quantitative tools are available to select the most parsimonious from a series of models, including testing the residuals, parameter precision, parsimony criteria, and parameter plausibility. Also of importance is the validation of the model-derived measures against those provided by an independent technique. All these methodological aspects have received systematic attention; books are available which cover in details all aspects of model identification and validation [17], [18]. In Section III-B, we discuss minimal models used to understand/measure glucose metabolism and insulin control; in Section III-C we discuss insulin secretion and glucose control; Section III-D reviews briefly some recent clinical studies where certain models discussed below have been successfully employed.

B. Glucose System

Understanding quantitatively the glucose system requires dynamic data, e.g., glucose or a tracer-glucose perturbation, and a system model, e.g., a compartmental model [18].

1) Insulin Action: Steady State

To study glucose kinetics in steady state, a tracer is needed. Tracer theory shows that linear time-variant compartmental models are accurate [19], thus allowing the use of a sum of exponential models (input–output models) to understand the number of compartments needed to describe the system. Insel et al. [20] were the first showing that a three compartment model is required to describe basal glucose kinetics (Fig. 3, top panel). This was later confirmed by Cobelli et al. [21], albeit a different model structure was postulated (Fig. 3, middle panel). The compartmental structure has been validated in animal experiments [22]. Both models incorporated a priori physiological knowledge to achieve unique identifiability. As noted in [21], the exchange kinetics between compartment 1 and 2 is much faster than between 1 and 3; thus, it is reasonable to use a simpler model (Fig. 3, bottom panel) with compartments 1 and 2 aggregated into a new single compartment 1 (e.g., after the first 2–3 min following tracer injection, only a two-compartment model is resolvable from the data).

Fig. 3
Compartmental model of glucose kinetics in steady state. Upper panel: three compartment model with glucose utilization taking place in plasma and rapidly equilibrating tissues; Middle panel: three compartment model with glucose utilization taking place ...

Steady state allows a safe use of linear modeling strategies, so it is not surprising that the first quantitative studies on the effect of insulin on glucose kinetics were performed by artificially inducing an elevated insulin steady state with glucose clamp at basal level and performing a tracer study. As an example, Fig. 4 shows the model-derived parametric representations of basal and elevated-insulin state obtained in [21] and [23] by identifying the tracer model of Fig. 3 (middle panel). If one compares the parameter values in the two steady states, an insulin effect on the rate constants into and out of compartment 3 is detected, slowly equilibrating tissues, presumably insulin-dependent tissues. In fact, not only parameter k03, which describes irreversible removal, increases, but also the exchange parameters with the accessible pool, k31 and k13, increase and decrease, respectively.

Fig. 4
Model of glucose kinetics in steady state: model-derived parametric representation at basal (upper panel) and elevated insulin (lower panel). EGP denotes endogenous glucose production.

Multiple steady-state tracer data allow not only inference of the effect of insulin on glucose masses, fluxes, and rate parameters, but also on the timing and magnitude of insulin control [20]. A model of insulin kinetics was studied at basal glucose with the glucose clamp technique and it was shown, for the first time, that it is not plasma insulin but insulin in a large slowly equilibrating compartment that mimics the time course of glucose utilization (Fig. 5). The authors performed three separate experiments in each individual: primed continuous infusion of insulin, primed continuous infusion of a glucose tracer, and glucose clamp with a variable infusion, and were able to identify the 12-parameter combined glucose (tracer and trace) and insulin model of Fig. 6. The two steady-state glucose systems were quantified in detail and the model revealed their ability to quantify the action of insulin on glucose utilization (with glucose production suppressed). In order to describe glucose utilization (equal to the glucose infusion rate), this approach followed Sherwin et al. finding [24] where glucose utilization from compartment 2 was related to insulin in the remote compartment 3 by a linear function, k03 = α + β compartment 3, thus deriving a measure of insulin effectiveness, defined as the derivative of glucose utilization with respect to insulin in the remote compartment 3.

Fig. 5
Comparison among insulin concentration in plasma (compartment 1), rapidly (compartment 2) and slowly (compartment 3) equilibrating tissues with glucose utilization measured with glucose clamp technique. Compartment 3 mimics the time course of glucose ...
Fig. 6
Compartmental models of insulin (upper panel), glucose (middle panel), and tracer glucose (lower panel) kinetics.

2) Insulin Action: Nonsteady State

Multiple steady-state tracer studies have been very informative, but also very complex in terms of required protocols and models. Is there a possibility to take advantage of the rich information content of a glucose dynamic perturbation, illustrated in Fig. 7, e.g., an intravenous glucose tolerance test (IVGTT), or a meal or an oral glucose tolerance test (OGTT)? In other words, is it possible to derive a measure of key control points of the glucose regulatory system, such as insulin sensitivity, i.e., the ability of insulin to control glucose production and utilization (Fig. 2), by simply exploiting the information content of the measured plasma glucose and insulin concentrations after the perturbation? The quest for answers inspired the minimal modeling strategy development at the end of the 1970s/early 1980s [25], [26], which has been, and still is, very successful [27], [28]. The simplest model describing the responses of plasma glucose and insulin to an oral or intravenous administration of glucose, was the pioneering study of Bolie in 1961 [29]

Fig. 7
Plasma glucose (upper) and insulin (lower) concentrations measured during IVGTT (right), OGTT (middle), and MTT (left, panel).

{G.(t)=a1G(t)a2I(t)+J(t)I.(t)=a3G(t)a4I(t)
(1)

where G, I denote plasma glucose and insulin, respectively, and J is glucose input which can be either an intravenous injection or the absorption rate of glucose during a meal or an oral glucose tolerance test. The model assumed that glucose disappearance was a linear function of both glucose and insulin, that insulin secretion was proportional to glucose, and that insulin disappeared in proportion to plasma insulin concentration. With various minor modifications this model has been used in conjunction with intravenous injections or infusions [29]–[31] and also during an oral glucose tolerance test [32]–[34] to obtain a four-parameter representation of glucose metabolism in various states of glucose intolerance, including diabetes. This model is a priori and a posteriori (numerically) identifiable, but it is too simple to be an adequate representation of the glucose-insulin system. First, the assumed linear relationship between insulin secretion and glucose has no experimental basis; in fact, the relationship between insulin secretion rate and glucose is dynamically very complex. Second, this model does not explicitly consider the complex interactive control of hepatic glucose production and uptake by glucose and insulin. Thus, this first model can be criticized as representing complex metabolic interactions in too simplistic terms to adequately represent the complex dynamical patterns that characterize the response of the metabolic system to exogenous perturbation.

Dynamic perturbations

The modeling methodology outlined in the Section III-A was first employed to define the so-called glucose minimal model, the goal being the estimation of insulin sensitivity from an IVGTT [25]. To facilitate the model selection process, system decomposition or partition analysis was introduced [35]. In fact, to describe plasma glucose and insulin data measured in an organism there is the need to simultaneously model not only the glucose but also the insulin system and their interactions. This means that, in addition to model insulin action, one also has to model glucose-stimulated insulin secretion. Since models are, by definition, “wrong,” an error in the insulin secretion model would be compensated by an error in the insulin action model, thus introducing a bias in insulin sensitivity. To avoid this interference that is so common in physiological studies, the dynamic contribution of a subsystem can be eliminated. Such a “loop-opening” can be accomplished in a several ways by gross surgical manipulation of the systems, by using an external feedback loop to clamp the level of specific system variables, or by infusing certain substances that inhibit the endogenous elaboration of some feedback signals. All of these techniques are invasive, and most are not applicable to humans, at least on a routine basis. In contrast, model-based system decomposition [25], [35] is an artificial “loop cut”: the system is decomposed in two subsystems which are linked together by measured variables (Fig. 8); the insulin subsystem represents all tissues secreting, distributing and degrading insulin, and the glucose subsystem represents all tissues producing, distributing and metabolizing glucose. When the system is perturbed, e.g., by a glucose injection, and the time courses of plasma glucose, G and insulin I, are measured, then the time courses of G and I can be considered in terms as “input” (assumed known) and “output” (assumed noisy) of the beta-cell and glucose subsystems, respectively. Models are then proposed not for the whole system but for each of the subsystems, independently, thus considerably reducing the difficulties of the modeling exercise. This way, the difficulties of modeling the beta-cell do not interfere with modelling insulin action on glucose-consuming tissues and vice versa (see Section III-C). Seven models of increasing complexity were proposed to explain plasma glucose concentration by using plasma insulin as the known input. The chosen minimal model (Fig. 9, left panel) assumes that glucose kinetics can be described with one compartment (the early portion of glucose data is not considered) and that remote (with respect to plasma) insulin controls both net hepatic glucose balance and peripheral glucose disposal:

{Q.1(t)=NHGB(Q(t),I(t))Rd(Q(t),I(t))+D·δ(t)Q(0)=QbI.(t)=k3·I(t)+k2·[I(t)Ib]I(0)=0G(t)=Q(t)V
(2)

where Q is plasma glucose mass, with Qb denoting its basal value; I is plasma insulin concentration, with Ib denoting its basal value, I′ (t) above basal remote insulin; D is the glucose dose; V is the glucose distribution volume, and k2 and k3 are rate parameters.

Fig. 8
Decomposition of glucose-insulin system into glucose and insulin subsystems.
Fig. 9
Left panel: IVGTT glucose minimal model. Right panel: OGTT/MTT glucose minimal model.

NHGB is the net hepatic glucose balance, which depends upon plasma glucose and remote insulin I′

NHGB(Q(t),I(t))=NHGB0[k5+k6·I(t)]·Q(t)
(3)

and Rd the rate of glucose disappearance form the peripheral tissues, also function of plasma glucose and remote insulin, I

Rd(Q(t),I(t))=Rd0+[k1+k4·I(t)]·Q(t).
(4)

This nonlinear model requires a reparameterization in order to become a priori uniquely identifiable (details in [17])

{Q.(t)=[p1+X(t)]·Q(t)+p1·Qb+D·δ(t)Q(0)=QbX.(t)=p2·X(t)+p3·[I(t)Ib]X(0)=0G(t)=Q(t)V
(5)

with X (t) = (k4 + k6). I′ (t); p1 = k1 + k5; p2 = k3; p3 = k2. (k4 + k6); p4 = NHGB0Rd0 = p1·Qb. X is insulin action, p1 is the fractional (i.e., per unit distribution volume) glucose effectiveness measuring glucose ability per se to promote glucose disposal and inhibit glucose production; p2 is the rate constant of the remote insulin compartment from which insulin action is emanated; p3 is a scale factor governing the amplitude of insulin action. The model allows the estimation of insulin sensitivity as

SIIVGTT=p3p2·V(dl/kg/minperμU/ml).
(6)

A novel feature of the model was that insulin action did not emanate from plasma but from a compartment remote from plasma. This was a model ingredient requested by data and modeling methodology, in agreement with [24]. Only later this remote compartment was experimentally proven in dog studies to be the interstitial fluid [27].

In numerous studies reviewed in [27], insulin sensitivity estimated with the minimal model has been shown to strongly correlate to that measured with the euglycemic-hyperinsulinemic clamp [36], calculated as the steady-state glucose infusion rate divided by basal glucose times the above basal increase in insulin concentration. This technique is usually considered the gold standard to measure insulin sensitivity in humans. However, it is labor-intensive and requires glucose and insulin pumps to be frequently manipulated by a trained physician; moreover, the method is “nonphysiological” since in normal life conditions a constantly elevated insulin concentration not coordinated with a concomitant rise in glucose is never experienced. The minimal model has been widely employed by more than one thousand papers since its introduction in 1979 (counted until mid 2009); in the last five years alone around 55 papers/year have been published.

SIIVGTT is essentially a steady-state measure, i.e., it does not account for how fast or slow insulin action takes place. Recently, a new dynamic insulin sensitivity index, SIDIVGTT, was introduced and shown to provide a more comprehensive picture of insulin action on glucose metabolism than SIIVGTT, especially in diabetic subjects who exhibit both low and slow insulin action [37].

Glucose kinetics requires at least a two compartment description [21]. Undermodeling the system, i.e., using one instead of two compartments during the highly dynamic IVGTT perturbation, can introduce bias in glucose effectiveness and insulin sensitivity, being over- and under-estimated, respectively [38]–[40]. A two compartment glucose minimal model has been proposed [41], [42], but this requires a priori knowledge on the two glucose exchange parameters [21], [23], [43], [44]. By incorporating this a priori knowledge and using a Bayesian Maximum a posteriori estimator, the accuracy of both glucose effectiveness and insulin sensitivity has been shown to improve.

Like the glucose clamp technique, the IVGTT is nonphysiological, albeit to a lesser extent. Because it is important to measure insulin sensitivity in the presence of physiological changes in glucose and insulin concentrations, e.g., during a meal or OGTT, a different approach is needed to describe glucose kinetics in the gastrointestinal tract. The oral glucose minimal model (OMM) builds on the IVGTT minimal model by parametrically describing the rate of appearance of glucose into plasma (Ra) [45]. The model (Fig. 9, right panel) has a new mass balance equation

Q.(t)=[p1+X(t)]·Q(t)+p1·Qb+Ra(t,α)Q(0)=Qb
(7)

with α = [α1, α2, .... αN] the parameter vector describing Ra. Insulin sensitivity, SIORAL, can be derived from model parameters as reported in (6). A piecewise linear description for Ra with eight parameters is reasonably flexible to accommodate meal or OGTT data. The addition of parameters α renders the model more complex and it can be shown that the OMM is not a priori uniquely identifiable because V is nonidentifiable and p1 is nonuniquely identifiable (two solutions). Thus, there is the need to assume V and p1 to be known, usually fixed to population values. To improve numerical identifiability, a Maximum a Posteriori Bayesian estimator is used exploiting some prior on p2 and a constraint on Ra, related to the total amount of glucose appearing in the circulation. SIORAL has been precisely estimated and has been validated by comparison with both multiple tracer [46] and glucose clamp [47] techniques. Given the potential of the method for large scale clinical trials, a reduces protocol, e.g., a 2h-7 samples OGTT, has been designed and show to perform as well as a full protocol, e.g., a 5h-11 samples OGTT [54].

For both OGTT and MTT the new dynamic index SIDORAL has been shown to have the same beneficial effects on assessing insulin action as for IVGTT [37].

Dynamic perturbations with tracer

The IVGTT and oral minimal models are providing a composite “liver plus peripheral tissues” measure of insulin action (Fig. 9). Is it possible to dissect insulin action on the liver and peripheral tissues? Yes, if a glucose tracer is added to the IVGTT or meal/OGTT, thanks to the tracer’s ability to separate glucose utilization from production. The labeled IVGTT, both radio- (e.g., [3 – 3H]-glucose) and stable- (e.g., [6, 6 – 2H2]-glucose) label, was first interpreted with one [48]–[51] and, subsequently, with a two-compartment minimal model [52], [53], [55], [56] which has been shown to be more accurate. The model proposed in [52] and [53] is shown in Fig. 10. It is assumed that insulin-independent glucose utilization takes place in the accessible compartment while insulin-dependent glucose utilization consists of two components, one constant, Rd0, and the other proportional to glycemia. Insulin-dependent glucose utilization is parametrically controlled by insulin in a compartment remote from plasma. Details on model identifiability can be found in [52] and [53]. From the uniquely identifiable parameterization important indices such as glucose effectiveness, glucose clearance, and insulin sensitivity, can be estimated.

Fig. 10
Two compartment model of tracer glucose kinetics. The insulin-independent glucose utilization takes place in the accessible compartment ( Q1) while insulin-dependent glucose utilization consists of two components, one constant, Rd0, and the other ...

More recently, a labeled meal/OGTT was introduced and interpreted with the oral tracer minimal model [57]. A stable isotope glucose tracer (e.g., [1 –13 C]-glucose or [6, 6 – 2H2]-glucose) is normally used and from the oral tracer dose and plasma measurements of glucose tracer one can calculate the exogenous glucose (Gexo), i.e., coming from the meal/OGTT.

The model describing Gexo data is given by [57]

{Q.(t)=[p1+X(t)]·Q(t)+Ra(t,α)Q(0)=0X.(t)=p3·X(t)+p3·[I(t)Ib]X(0)=0Gexo(t)=Q(t)V
(8)

where * denotes tracer variables and parameters.

The disposal insulin sensitivity index, SIORAL, i.e., insulin action on glucose disposal only, is

SIORAL=p3p2·V(dl/kg/minperμU/ml).
(9)

The model shares the same input, Ra(t, α), of the oral minimal model [(7)]. Thus, these two models can be simultaneously identified from the measurements Gexo and G and there is no need for using Bayesian priors for p2 and p2 to improve numerical identifiability. However, knowledge of V* and p1 is still needed.

SIORAL has been validated against the multiple tracer [57] and the glucose clamp [47] techniques.

An unexpected finding commented in [47] was SIORAL to be very close to (sometimes higher than) SIORAL, which implies a negligible (or negative) action of insulin on the liver since hepatic insulin sensitivity SILORAL=SIORALSIORAL. The principal suspect was the mechanistic description of endogenous glucose production (EGP), an ingredient of SIORAL [58]. To address this issue, model-independent EGP profiles [59] were used.

The minimal model EGP description is

EGP(t)=EGPbGEL·[G(t)Gb]XL(t)·G(t)EGP(0)=EGPb
(10)

where EGPb is the basal EGP, GEL liver glucose effectiveness, G is glucose concentration, Gb its basal value, XL is liver insulin action (deviation from basal) which follows the dynamic equation

X.L(t)=p2·XL(t)+p3L·[I(t)Ib]XL(0)=0
(11)

where p2 is a rate constant describing the dynamics of insulin action on glucose production (assumed the same as on glucose utilization) and p3L is the scale factor governing the amplitude of hepatic insulin action.

When tested against EGP data, the model failed, i.e., it was unable to fit the data, and provided imprecise and physiologically implausible parameter estimates. This finding supported our hypothesis that the mechanistic description of EGP included in the minimal model is incorrect, and this may be the cause of the unreliable (negative) estimate of hepatic insulin sensitivity obtained with the minimal model.

Other models were tested, with the goal to find an EGP description that is suitable for incorporation in the minimal model instead of (10); the best one was

EGP(t)=EGPbkG·[G(t)Gb]XL(t)XDer(t)EGP(0)=EGPb
(12)

with XL defined as

{X.1(t)=p2L·X1(t)+p3L·[I(t)Ib]X1(0)=0X.L(t)=p2L·XL(t)+p3L·X1(t)XL(0)=0
(13)

and

XDer(t)={kGR·dG(t)dt,ifdG(t)dt00,ifdG(t)dt<0
(14)

where p2L is a rate constant describing the dynamics of insulin action on glucose production (assumed here to be different from that on glucose utilization), p3L is the scale factor governing the amplitude of hepatic insulin action, and kGR is a parameter governing the magnitude of glucose derivative control.

This model is different from the minimal-model EGP description. First, insulin action it is not multiplied by glucose concentration, which means that glucose and insulin act independently on the liver. In fact, while insulin modulates glucose utilization by moving the glucose transporter GLUT-4 to cell membrane and thus the glucose utilization clearly depends on glucose level, the insulin-stimulated inhibition of production follows different paths. In addition, insulin action on glucose production has a different time course from insulin signalling on glucose disposal. Finally, the model incorporates the accepted notion that a portal insulin signal (here is approximated by glucose derivative) controls the rapid suppression of EGP [60]. Hepatic insulin sensitivity is given by

SILORAL=p3Lp2L·1Gb(dl/kg/minperμU/ml)
(15)

with Gb basal glucose concentration and p2L and p3L defined above.

This model has been recently coupled with oral tracer minimal model (unpublished); thus, hepatic [ SILORAL, (15)] and disposal insulin sensitivity ( SIORAL, (9)] can be simultaneously estimated from a single-tracer experiment using total glucose, exogenous glucose, and insulin concentrations. The whole-body insulin sensitivity is SI=SIORAL+SILORAL, thus, the relative role of the liver versus periphery can be determined in percentage. Our results show that during a meal SILORAL represents 34% of total insulin action.

Both unlabelled and tracer IVGTT and oral minimal models are powerful tools to measure a number of indices characterizing the control that glucose and insulin exert on glucose metabolism. These indices are of utmost importance for the understanding and treatment of pathophysiology, from glucose intolerance to diabetes. Even more importantly, the combined use of the unlabelled and labeled minimal models, thanks to the tracer-to-tracee indistinguishability principle [19], allows us to move from a parametric indices to a flux portrait.

Consider the IVGTT probe. Glucose utilization can be calculated from the uniquely identifiable parameterization of the model of Fig. 10 as

U(t)=k01(t)·Q1(t)+[k02+X(t)]·Q2(t)
(16)

where Q1 and Q2 are glucose masses in compartments 1 and 2, respectively.

Endogenous glucose production can be calculated by using the endogenous glucose concentration, Gend, i.e., the component of total glucose concentration measured in plasma due to glucose production, which can be calculated in a model-independent fashion [61]. Gend is related to the endogenous glucose production, EGP, by the integral equation

Gend(t)=0th(t,τ)·EGP(τ)·dτ+Gb·h(t,0)
(17)

where h (t, τ) is the time-varying impulse response of the glucose system, given by the tracer minimal model of Fig. 10, and Gb is basal glucose. EGP can be estimated by deconvolution, e.g., the stochastic deconvolution method [62]. In [61], EGP estimation has been validated against that obtained by a dual tracer-to-tracee clamp technique, which provided a virtually model-independent estimation (Fig. 11). A similar modeling strategy has also been used in [63].

Fig. 11
Endogenous glucose production estimated with the dual tracer technique (open circles, vertical bars represents standard deviation) and with deconvolution (continuous line).

Recently, Krudys et al. [64] have proposed a structural model of EGP during an IVGTT. EGP depends on the amount of releasable glucose in the liver and has an inhibitory function which is related to remote insulin. The model reconstructed EGP was validated against that obtained via deconvolution and the tracer-to-tracee clamp technique [64]. The model also provided some indices of glucose and insulin control on EGP which have been recently refined [65].

As for the IVGTT, relevant glucose fluxes can also be derived from meal and OGTT data. Glucose rate of appearance, Ra, can be reconstructed from the simultaneous identification of the two models of (7) and (8). Moreover, glucose utilization can be calculated as

U(t)=[p1+X(t)]·Q(t).
(18)

The time course of endogenous glucose production EGP (t) can be reconstructed with the model described in [58] and reported in (12). In this case, in addition to EGP profile, the model provides quantitative indices of hepatic insulin sensitivity [(15)] and glucose effectiveness as well as (unlike the Krudys model) an index quantifying hepatic insulin sensitivity.

Alternatively, EGP(t) can also be estimated by stochastic deconvolution from endogenous glucose concentration data, Gend [66]. Unlike IVGTT, during meal and OGTT the single compartment model can be used to describe glucose kinetics; thus, h (t, τ) is the impulse response of the system of (8). Moreover, if a single glucose tracer is orally administered, h (t, τ) can be determined by fixing V* and p1 to population values.

EGP and Ra time courses reconstructed with both models and deconvolution have been validated against the triple tracer technique [59], [67], [68] which provided virtually model-independent estimates of such glucose fluxes (Fig. 12) [58].

Fig. 12
Comparison between endogenous glucose production (upper panel) and rate of appearance of glucose (lower panel) during a meal reconstructed with models and triple tracer model-independent method.

3) From Whole-Body to Organ/Tissue

While whole-body models can provide important quantitative information on insulin action, it is important but at the same time remarkably difficult, to noninvasively measure the processes of glucose transport and metabolism at the organ level. A crucial target tissue of glucose metabolism is the skeletal muscle. Impaired insulin transport within the muscle is a well-recognized characteristic of a number of metabolic diseases, including type 2 diabetes, obesity, hypertension, and cardiovascular disease [69]. Understanding its causes requires us to segregate and quantify in situ the major individual steps of glucose processing, particularly those of glucose delivery, transport in and out of the cell, and phosphorylation (Fig. 13).

Fig. 13
Major glucose processing: diffusion to/from the intersitium, active transport in and out of the cell, and phosphorylation/metabolism.

There are two major experimental techniques to tackle this problem, both employing tracers with glucose metabolism at steady state. The classical experimental approach is based on the multiple tracer dilution [70]. This model consists of the simultaneous injection, upstream of the organ, of more than one tracer which allows the separate monitoring of the individual steps of glucose metabolism. For example, in the case when the objective of the experiment is the measure of all the elementary processes (convection, diffusion, transport, and metabolism), usually one can simultaneously inject upstream of the organ (artery) and measure downstream (vein) a first tracer that is distributed only in the capillary bed (intravascular tracer), a second that is subject to the bidirectional exchange through the capillary membrane (extracellular tracer), a third that also permeates the cell through the sarcolemma (permeating not metabolizable tracer), and, finally, a fourth that is also metabolized (permeating metabolizable tracer). These tracers must obviously be distinguishable once they reach the organ outflow. The venous outflow curves must then be analyzed by means of a physiological system model. More recently, the positron emission tomography (PET) noninvasive imaging technique was proposed. Using appropriate tracers, PET can provide highly specific and rich biochemical information if applied in dynamic mode, i.e., sequential tissue images acquired following a bolus injection of tracer so that the time course of the tissue metabolism can be monitored. Again, a physiological system model is needed to interpret the data. Unlike the multiple tracer dilution technique, here tracers cannot be injected simultaneously but only sequentially.

Multiple tracer dilution data can be interpreted with both linear distributed parameter (see reviews [18] and [71] and compartmental organ models [72]. The only application to glucose metabolism of distributed parameter models has been in an isolated and perfused heart [70]. In contrast and despite short history, compartmental organ models have been more intensively applied to interpret multiple tracer dilution data in the human skeletal muscle. A compartmental model has been proposed [73] describing the transmembrane transport of glucose. The model has been developed from multiple tracer dilution data obtained in human skeletal muscle in vivo using two tracers, one extracellular (L – [3H]-glucose) and the other permeant, nonmetabolizable ([14C]-3-O-methyl-D-glucose). This allows us to estimate with very good precision the rate constants of glucose transport into and out of the cell. Consequently, this model allowed us to study the enhancing effect of insulin on muscle glucose transport parameters in nondiabetic subjects [73] and identified the presence of a localized defect in insulin control in non-insulin-dependent diabetic (NIDD) patients [74]. This compartmental model has been extended [76] to describe the kinetics of a third tracer, permeant nonmetabolizable ([3H]-glucose). The gain obtained by adding to the experimental protocol a third tracer is immense. This ultimately allowed us to quantify a model of the tracee and therefore study not only the rate constants of transport and phosphorylation but also the bidirectional glucose flux through the cell membrane, the phosphorylation flux, and the intracellular concentration, in nondiabetic, obese and diabetic subjects [75]–[77]. This allowed important physiological results to be obtained. Among these, it was possible to show that the insulin control on both transmembrane transport and phosphorylation flux in subjects affected by NIDDM is much less efficient with respect to nondiabetic subjects [76], [77]. Therefore, the model enabled demonstration of the fact that cellular transport plays a very important role in the insulin resistance associated with NIDDM.

PET data can be analyzed by several modeling strategies with regional compartmental modeling being the most powerful approach to connect organ or tissue concentration data into measures of physiological parameters. In this respect, the brain glucose model by Sokoloff et al. [78] has been a landmark. The selected tracer for studying glucose metabolism in skeletal muscle (but also in the brain and myocardium) is [18F]fluorodeoxyglucose([18F]FDG), a glucose analog. The ideal tracer would be [11C]-glucose but the interpretative model by having to account for all metabolic products along the glycolysis and glycogen synthetic pathways would be unreasonable from the limited information content of PET data. The advantage of [18F]FDG is that a simpler model can be adopted. In fact, [18F]FDG once in the tissue, similarly to glucose, can either be transported back to plasma or can be phosphorilated to [18F]FDG – 6 – phosphate, [18F]FDG – 6 – P. The advantage is that [18F]FDG – 6 – P is trapped in the tissue and released very slowly. In other words, [18F]FDG – 6 – P cannot be metabolized further, while glucose-6-P does so along the glycolysis and glycogenosynthesis pathways. The major disadvantage of [18F]FDG is the necessity to correct for the differences in transport and phosphorylation between the analog [18F]FDG and glucose. A correction factor called lumped constant (LC) can be employed to convert [18F]FDG fractional uptake (but not the [18F]FDG transport rate parameters) to that of glucose. LC values in human skeletal muscle are available [79], [80]. In order to provide rate constants of transport and phosphorylation of [18F]FDG in skeletal muscle, a four-compartment model (plasma, extracellular, tissue [18F]FDG, and [18F]FDG – 6 – P) with five rate constants has been proposed [81]. The model (Fig. 14) is described by

Fig. 14
The 5 k model of [18F]FDG in skeletal muscle: Cp is [18F]FDG plasma arterial concentration, Cc extracellular concentration of [18F]FDG normalized to tissue volume, Ce [18F]FDG tissue concentration, Cm [18F]FDG – 6 – P tissue concentration, ...

C.c(t)=K1Cp(t)(k2+k3)Cc(t)+k4Ce(t)Cc(0)=0C.e(t)=k3Cc(t)(k4+k5)Ce(t)Ce(0)=0C.m(t)=k5Ce(t)Cm(0)=0
(19)

C(t)=(1Vb)(Cc(t)+Ce(t)+Cm(t))+VbCb(t)
(20)

where Cp is [18F]FDG plasma arterial concentration, Cc is extracellular concentration of [18F]FDG normalized to tissue volume, Ce [18F]FDG is tissue concentration, Cm [18F]FDG – 6 – P is tissue concentration, C is total 18F activity concentration in the ROI, K1 [ml/ml/min] and k2 [min−1] are the exchanges between plasma and extracellular space, k3 [min−1], k4 [min−1] is the rate of transport in and out of cell, and k5 [min−1] is the rate of phophorylation. Vb is the fractional blood volume in the region of interest, and Cb is the whole blood tracer concentration. From the model one can calculate the fractional uptake of [18F]FDG, K [ml/ml/min]

K=K1k3k5k2k4+k2k5+k3k5
(21)

and, by using LC value and the glucose basal plasma concentration value, the glucose fractional uptake. [18F]FDG rate constants of transport and phosphorylation are inefficient in obesity and type 2 diabetes, but these defects can be substantially reversed with weight loss [82].

To move to a glucose representation a multi-tracer positron emission tomography (PET) imaging method is needed [83], which allows the simultaneous assessment of blood flow, glucose transport, and phosphorylation in the skeletal muscle. The method employs three different PET tracers (Fig. 15) injected at different times, and allows us to quantify blood flow from [15O]H2O images with one- compartment two-rate constant model; glucose transport from [11C]3-OMG images with a three compartment four-rate constant model; and, finally, glucose phosphorylation by combining [18F]FDG fractional uptake with [11C]3-OMG rate constants. The [11C]3-OMG model is a simpler version of that of (19), (20) since [11C]3-OMG is not phosphorylated. This multi-tracer model-based PET imaging method has shown that glucose transport from plasma into interstitial space is virtually identical to tissue perfusion and not affected by insulin; insulin significantly increases both glucose transport and phosphorylation modulating distribution of control among delivery, transport, and phosphorylation (glucose delivery and transport contribute nearly equally to the control of glucose uptake accounting for 90% of control together); predominately oxidative muscles (soleus) have higher perfusion and higher capacity for glucose phosphorylation than less oxidative muscles (tibialis).

Fig. 15
Employment of PET tracers to study glucose diffusion through capillary membrane, active transport into the cells and metabolism.

C. Insulin System

Here, we describe some models of the insulin system and the control exerted by glucose on insulin secretion.

1) Insulin Kinetics

To study insulin kinetics in the steady state a tracer would be the ideal probe. Tracer studies have been performed, e.g., by using radioactive isotopes of iodine or hydrogen, but ideal tracer prerequisites such as tracer-tracee indistinguishability and nonnegligible perturbation, are not completely met. The majority of insulin kinetic studies have been performed by pulse injection or infusion. However, the administration of a nontrace amount of insulin has two undesired effects: first, it may induce hypoglycemia, which in turn could trigger counterregulatory response that may affect insulin kinetics; second, it inhibits insulin secretion, thus the measured insulin concentration contains a time-varying endogenous component. These confounding effects can be avoided by designing a rather complex experiment, where hypoglycemia is prevented by variable glucose infusion (glucose clamp technique), and endogenous insulin secretion is suppressed by somatostatin infusion.

In the physiological concentration range (up to 100–150 μU/ml) where insulin kinetic is approximately linear, various linear compartmental models have been proposed [84], following the landmark model of Sherwin et al. [24] (already discussed, see Fig. 6, upper panel). Some of these models are shown in Fig. 16. The two-compartment structure is derived by noting that compartments 1 and 2 in the model of Fig. 6 (upper panel) are in rapid equilibrium. SRpost is the posthepatic insulin secretion rate, i.e., the flux of newly secreted insulin that reaches plasma after the first passage through the liver. The two models differ in terms of the site of irreversible loss: plasma (Fig. 17, top panel), or peripheral tissues (Fig. 16, middle panel). Both models are a priori uniquely identifiable. Typical numerical values of their parameters are shown in the figure. In the situation where a two-compartment model cannot be resolved from the data, the single compartment model (Fig. 16, bottom panel) can be used.

Fig. 16
Compartmental models of insulin kinetics. Upper panel: two compartment model with insulin degradation in the accessible compartment; Upper panel: two compartment model with insulin degradation in the remote compartment; Upper panel: one compartment model. ...
Fig. 17
Schematic representation of insulin (upper) and C-peptide (lower) pancreatic secretion and kinetics. Insulin is secreted by the beta-cells in the portal vein and extracted by the liver before it appears in plasma; C-peptide is secreted by the beta-cells, ...

In the supraphysiological range, nonlinear or linear time-varying insulin kinetics model are more appropriate. A relatively simple nonlinear two-compartment model, similar to that of Fig. 16, upper panel, has been proposed by Frost et al. [85], with linear transfer rate k12 and k21 but nonlinear irreversible loss k01 described by the Michaelis-Menten relation. However, the relatively high number of model parameters (six) poses some problems in deriving precise estimates for all of them. A nonlinear five-compartment model has been proposed by Hovorka et al. [86], which also incorporates a description of insulin kinetic at the receptor level. Alternatively, linear time-varying models has also been formulated, e.g., that of Morishima et al. [87], which assumes constant k12 and k21 and time-varying k01 (t).

2) Insulin Secretion

The problem of estimating the secretion profile in vivo during perturbation from plasma concentration measurements is a classic input estimation problem for which deconvolution offers the classic solution. However, it is not possible to infer on pancreatic secretion from plasma insulin concentration data; it is only possible to derive its component appearing in plasma, or the posthepatic insulin secretion, which is approximately equal to 50% of the pancreatic secretion. This problem can be bypassed if C-peptide concentration is measured during the perturbation and used to estimate insulin secretion since C-peptide is secreted equimolarly with insulin, but it is extracted by the liver to a negligible extent (Fig. 17). In other words, plasma C-peptide concentration well reflects, apart from the rapid liver dynamics, C-peptide pancreatic secretion, which coincides with insulin secretion.

Since there is solid evidence that C-peptide kinetics are linear in a wide range of concentration, the relationship between above basal pancreatic secretion (SR, the input), and the above basal C-peptide concentration measurements (C, the output) is the convolution integral

C(t)=0th(tτ)·SR(τ)·dτ
(22)

where h is the impulse response function of the system. SR profile during a perturbation can be reconstructed by solving the inverse problem, which is deriving SR by deconvolution, given C and h. The knowledge of the impulse response h is a prerequisite. This requires an additional experiment on the same subject, consisting of a bolus of C-peptide and a concomitant infusion of somatostatin to inhibit endogenous C-peptide secretion. Usually, C-peptide concentration data are then approximated by a sum of two exponential models that, after normalization to the C-peptide dose, provide the impulse response function [88], [96]. Deconvolution methods have been applied to estimate the secretion profile in various physiological states [89], [90] and during both intravenous and oral glucose tolerance tests [91]–[93].

To eliminate the need for a separate experiment to evaluate the impulse response function, a method has been proposed [94] to derive C-peptide kinetic parameters in an individual based on data about his or her age, weight, height and gender. Secretion reconstructed by deconvolution, whith the impulse function evaluated from either the C-peptide bolus experiment or the population parameters are similar [94], indicating that the population values allow a good prediction of individual secretion profiles.

As for pancreatic insulin secretion, an indirect measurement approach is essential to quantify hepatic insulin extraction in humans since the direct measurement requires invasive protocols, with catheters placed in a artery and hepatic vein [84]. Deconvolution offers a possible solution since by comparing pancreatic secretion rate (SR) obtained from C-peptide data and posthepatic secretion (SRpost) obtained from insulin data—one can estimate hepatic extraction E as

E(t)=SR(t)SRpost(t)SR(t).
(23)

Estimation of SRpost by deconvolution is straightforward if insulin levels are in the physiological range since the kinetic model is linear, while it becomes more problematic if insulin levels are supraphysiological. In this last case, the link between SRpost and insulin concentration (I) is given by

I(t)=0tg(t,τ)·SRpost(τ)·dτ
(24)

with g(t, τ) representing the impulsive response of the linear time-varying insulin kinetic system.

3) Beta-Cell Function

Deconvolution allows us to measure in a virtually model-independent way insulin secretion after a glucose stimulus. However, giving a mechanistic description of pancreatic insulin secretion as a function of plasma glucose concentration has the advantage to provide quantitative indices of beta-cell function.

The minimal modeling methodology is similar to that employed to derive the glucose minimal model: the system is decomposed in two subsystems (Fig. 8.), but this time we look at the insulin subsystem with plasma glucose, G, considered the “input” (assumed known) and C-peptide, C, the “output” (assumed noisy).

Since the secretion model is identified on C-peptide measurements taken in plasma, it must be integrated into a model of whole-body C-peptide kinetics, which has two compartments [88].

During IVGTT, the above-basal insulin secretion [97] is given by (Fig. 18, left)

Fig. 18
Left panel: IVGTT C-peptide minimal model. Right panel: OGTT/MTT C-peptide minimal model.

SR(t)=m·F(t)
(25)

with F the ready releasable insulin described by

F.(t)=m·F(t)+Y(G,t)F(0)=F0
(26)

with F0 the amount of insulin released immediately after the glucose stimulus; Y (G, t) is the provision of new insulin, which depends on glucose level

Y.(G,t)=1T·[Y(G,t)Y(G,)]Y(0)=0
(27)

and

Y(G,)={0ifG(t)<hβ·[G(t)h]ifG(t)h.
(28)

In other words, insulin secretion consists of two components: first and second phase secretion. First phase secretion is portrayed by a rapidly turning-over compartment (2 min) and likely represents exocytosis of previously primed insulin secretory granules (commonly called readily releasable). It exerts derivative control, since it is proportional to the rate of increase of glucose from basal up to the maximum through a parameter [28], Φ1, which defines the first responsivity index

Φ1=F0ΔG
(29)

with ΔG the difference between peak and basal glucose concentration.

Second phase insulin secretion is believed to be derived from the provision and/or docking of new insulin secretory granules that occurs in response to a given (i.e., proportional to) glucose concentration, through a parameter Φ2, which defines the second phase responsivity index, and reaching the releasable pool with a delay time constant, T

Φ2=β.
(30)

The meaning of Φ2 and T can be explained by analyzing model parameters under the above-basal step increase of glucose: provision tends with time constant T towards a steady state, which is linearly related to the glucose step size through parameter Φ2. T presumably represents the time required for new “readily releasable” granules to dock, be primed, then exocytosed. In addition to Φ1 and Φ2, a basal responsivity index can also be calculated, Φb. Finally, a single total index of stimulated beta-cell responsivity, ΦIVGTT, can be derived by combining Φ1 and Φ2. Of note, in [97] it has been shown that population values are also a good predictor of the individual kinetic parameters. However, additional uncertainty is brought in by the population approach [98], which can be taken into account into a Bayesian setting via Markov Chain Monte Carlo [99].

Beta-cell function can also be assessed from an oral test, such as meal or OGTT. An oral glucose test differs from IVGTT in several important aspects including the route of delivery with the associated incretin hormone secretion, the more physiological and smoother changes in glucose, insulin, and C-peptide concentrations, and, during a mixed meal, the presence of non-glucose nutrients stimulation, i.e., amino acids and fat. Various models have been proposed to assess beta-cell function during a meal and OGTT [95], [100]–[102]. All of them share the model of C-peptide kinetics described above, but differ on the assumption on how glucose controls the secretion. The oral C-peptide minimal model proposed by Breda et al. [101] Fig. 18 (right) maintains basically all of the previous model ingredients employed in the IVGTT model, with the exception of the fast turning over insulin releasable pool (F, which is not evident under these conditions) to describe the data. In particular, both a rate of change of glucose component of insulin secretion and a delay between glucose stimulus and beta-cell response have been shown to be necessary to fit the data [103]. Model equations are

SR(t)=Y(G,t)+SRd(G,t)
(31)

with Y(G, t) described by (27) and (28), also called a static component of insulin secretion, and SRd(G, t) the dynamic component of insulin secretion

SRd(G,t)={K·dG(t)dtifdG(t)dt00ifdG(t)dt<0.
(32)

Thus, this model features a dynamic component that senses the rate of change of glucose concentration, and a static component that represents the release of insulin that, after a delay, occurs in proportion to prevailing glucose concentration. Similarly to the IVGTT, where first, Φ1, and second phase, Φ2, beta-cell responsivity indices were defined, from the oral model dynamic, Φd (= K), and static Φs(= β) responsivity indices can be derived. Basal responsivity index, Φb, can be obtained as a ratio between basal secretion and basal glucose. A total responsivity index, ΦORAL, which combines Φd and Φs, can also be derived.

The model has been successfully used by Toffolo et al. [104] during “up&down” intravenous glucose infusion and by Steil et al. [105] for describing hyperglycemic clamp C-peptide data as well as meals, thus providing further independent evidence for its validity. In contrast to the IVGTT model where the derivative component of first phase secretion was operative only during the first few minutes as the plasma glucose concentration increased from a “basal” to “maximal” concentration, the relatively gradual pattern of glucose appearance observed during oral tests necessitated the presence of a secretion component proportional to glucose rate of change that contributed to the model for the first 60–90 minutes. In addition, and similar to IVGTT, a component of insulin secretion proportional to glucose, characterized by a delay time T (presumably reflecting at least in part the time it takes for new granules to reach the releasable pool), that contributed throughout the experimental period, was also necessary. As discussed above, Φd is markedly different from its IVGTT counterparts Φ1. In fact, during IVGTT first phase component only contributed during the first 4–6 minutes and the proportional component for the rest of test. Thus, it is probable that Φd and Φ1 are assessing different aspects of the insulin secretory pathway. In particular, Φd presumably could also reflect multiple distal steps including the rate of granule docking, priming as well as exocytosis.

The model proposed by Hovorka et al. [95] assumes an instantaneous linear control of glucose on insulin secretion; i.e., there is no delay between glucose stimulus and beta-cell response. The model proposed by Cretti et al. [100] describes insulin secretion with the static component of glucose control of the C-peptide minimal model; thus, it is characterized by a delay but does not include any dynamic, i.e., rate of change, glucose control. Interestingly, the same authors have recently included a dynamic control to describe first phase secretion in a subsequent report [106]. The model proposed by Mari et al. [102], similarly to the oral model shown in Fig. 18 right, has both a proportional component and a component responsive to the rate of change of glucose, but there is no delay between glucose signaling and supply of new insulin to the circulation. The authors choose to account for the expected inability of a proportional plus derivative glucose control to account for C-peptide measurements with a time-varying term correcting only the static component of insulin secretion, which has been called the potentiation factor. In simple words, the potentiation factor is a time-varying correction term that mathematically compensates for the proportional plus derivative description deficiency.

4) Disposition Index

It is worth noting that beta-cell function needs to be interpreted in light of the prevailing insulin sensitivity. One possibility is to resort to a normalization of beta-cell function based on the disposition index paradigm, first introduced in 1981 [35], and recently revisited in [28], where beta-cell function is multiplied by insulin sensitivity. This concept, self-evident in Fig. 8, is more clearly illustrated in Fig. 20 (left). While regulation of carbohydrate tolerance is undoubtedly more complex, it is conceived that glucose tolerance of an individual is related to the product of beta-cell function and insulin sensitivity. In essence, different values of tolerance are represented by different hyperbolas, i.e., DI = beta-cell function × insulin sensitivity = const. If an individual’s beta-cells respond to a decrease in insulin sensitivity by adequately increasing insulin secretion (state II) the product of beta-cell function and insulin sensitivity (the disposition index) is unchanged, and normal glucose tolerance is retained. In contrast, if there is an inadequate compensatory increase in beta-cell function to the decreased insulin sensitivity (state 2), the individual develops glucose intolerance. Thanks to its intuitive and reasonable grounds, this measure of beta-cell functionality, which was first introduced for IVGTT, has become the method of choice also with the meal and OGTT test. Thus, disposition indices DI1, DI2, DIIVGTT or DId, DIS, DIORAL can be calculated by multiplying responsivity indices Φ1, Φ2, ΦIVGTT by SIIVGTT (for IVGTT) or by multiplying responsivity indices Φd, Φs, ΦORAL by SIORAL (for meal/OGTT) to determine if the first phase, second phase, global beta-cell function is appropriate in light of the prevailing insulin sensitivity. Another important use of the disposition index paradigm is the monitoring in time of the individual components of tolerance and the assessment of different treatment strategies. It is easily appreciated from Fig. 20 (right) the importance of segregating glucose tolerance into its individual components of beta-cell responsivity and insulin sensitivity: subject x is intolerant due to its poor beta-cell function while subject y has a poor insulin sensitivity and these two individuals need opposite therapy vectors. However, the glucose-insulin feedback system is more complex than the hyperbola paradigm. The relation between beta-cell function and insulin sensitivity is certainly describable by a nonlinear inverse relationship but is in all likelihood more complex than a simple hyperbola, i.e., the relation is more likely DI = beta-cell function×(insulin sensitivity)α = constant. In addition, this simple concept hides several methodological issues which, unless fully appreciated, could lead to errors in interpretation. Some critical questions are: is it true that the hyperbolic relationship holds in a population? How should the disposition indices be used in comparing populations, i.e., should the individual values be averaged, or should the disposition index be estimated directly in the population? How should the population variability accounted for? Also, since the effect of insulin on peripheral tissues is also determined by the amount of insulin to which the tissue is exposed, hepatic insulin extraction may come into play (see the following) and provide yet another dimension to the relationship between insulin secretion and action portrayed in Fig. 20. Some of these aspects have been recently examined in [28] and [107].

Fig. 20
Scheme of the glucose-insulin control system which relates measured plasma concentrations, i.e., glucose and insulin, to glucose fluxes, i.e., rate of appearance, production, utilization, renal extraction, and insulin fluxes, i.e., secretion and degradation. ...

5) Hepatic Insulin Extraction

Minimal models also provide an approach to assess hepatic insulin extraction. In fact, SR can be assessed from the model of C-peptide kinetics and secretion identified from C-pepide and glucose measured during IVGTT or meal or OGTT. By following a similar approach, SRpost can be assessed by insulin and glucose data. In particular, insulin-modified IVGTT experiment, i.e., an IVGTT associated with short insulin infusion given between 20 and 25 minutes after the glucose bolus, offers definitive advantages with respect to the standard IVGTT since the insulin infusion generates an additional disappearance curve that greatly facilitates the simultaneous estimation of insulin kinetic and secretion parameters. Hepatic insulin extraction can then be calculated from SR and SRpost profiles [106]. During meal and OGTT the situation is more complicated, however, a recent study [108] proposes a population model, which, similarly to the widely used Van Cauter formulas [94], allows us to calculate insulin kinetic parameters from subject anthropometric characteristic, such age, gender, body surface area, etc., avoiding the necessity to infuse exogenous insulin. From pre- and post-hepatic insulin secretion rates, hepatic insulin extraction time course can be derived as for the IVGTT. In addition, from pre- and post-hepatic model parameters an index of hepatic extraction can also be calculated [108], [109].

D. Clinical Studies

It is well beyond the scope of this contribution to review model-based phathophysiological studies. However, it may be helpful for the reader to refer to specific instances where an answer to a diabetes-related question has been provided by a systematic use of models. For instance, the battery of oral glucose, C-peptide, and insulin models have been used in studying: the effect of age and gender on glucose metabolism [59]; the effect of anti-aging drugs [110]; the influence of ethnicity [111]; insulin sensitivity and beta-cell function in nondiabetic [112] and obese [113] adolescents; the pathogenesis of prediabetes [114]–[116] and type 2 diabetes [117], [118].

IV Maximal Models

A. Rationale

In contrast to minimal, maximal (fine-grain) models are comprehensive descriptions attempting to fully implement the body of knowledge about metabolic regulation into a generally large, nonlinear model of high order, with a large number of parameters. This class of models cannot, in general, be identified, i.e., without massive experimental investigation on a single individual it is not possible to relate with confidence alterations in the dynamics of blood-borne substances to specific changes in parameters of a comprehensive model. This means that these models are not generally useful for the quantification of specific metabolic relationships—their utility is in the possibility for system simulation.

Simulation is a powerful investigative tool, particularly in the engineering disciplines where the system structure and function is usually “known” and equations can be written based on first principles. In contrast, a physiological system is largely “unknown” in terms of structure and function, i.e., equations can always be written but the problem is model validity. Although strategies for validation of a complex (versus simple) model have been delineated [17], [119], the difficulties of the problem remain.

In metabolism and diabetes, large scale models have been of value as research tool, i.e., to test a theory or incompatibilities of theories. A classic is the insulin secretion model of Grodsky [120] where in order to describe insulin secretion patterns in response to a variety of glucose stimuli, he postulated that insulin granules were not a homogeneous pool. While the threshold hypothesis he introduced (i.e., each granule has a certain glucose threshold above which it releases the content) has gained little support from subsequent experiments, the non-homogeneity of insulin—containing granules pool is today an accepted notion and various beta-cell biology theories have been put forward. Another example is the glucose-insulin simulation model by Sturis et al. [121], which offered an explanation of why wide ultradian oscillations (period of ~120 min in humans) occur in insulin and glucose profiles in various physiological conditions, stating that they can originate from the interaction between the glucose and insulin subsystems without the need to invoke the presence of a pancreatic pacemaker operating at these frequencies.

Another important area for maximal models is their use as test beds for examining the empirical validity of models intended for clinical applications. For instance, in [122] and [123] the consequences of undermodeling the glucose system by using the classic IVGTT single compartment minimal model has been studied by using a richer IVGTT two compartment glucose-insulin simulation model.

Finally, simulation models have been proven useful in a teaching setting as heuristic devices providing easy and quick answers to “what…if” questions. Some good examples on simulators being an efficient library of physiological knowledge are [124]–[126].

A classic simulation is in silico experimentation, which in disciplines like engineering is carried out in everyday life; a recent success is the Boeing 777 jetliner, which has been recognized as the first airplane to be 100% digitally designed and assembled in a computer simulation environment. This important use of simulation has not had the expected impact in metabolism and diabetes. However, there are situations where in silico experiments with complex models could be of enormous value. In fact, it is often not possible, appropriate, convenient, or desirable to perform an experiment on the glucose system, because it cannot be done at all, or it is too difficult, too dangerous, or unethical. In such cases, simulation offers an alternative way of in silico experimenting on the system. Simulation models have been published and used to examine various aspects of diabetes control, e.g., for assessing different control algorithms and different insulin infusion routes [122], [127]–[138]. Although the confidence in their results is certainly higher than that obtained using as simulator the glucose minimal model [25] (extensively misused for this purpose and recently being the core of the Medtronic Virtual Patient [139]), the impact has been very modest. All these models are average population models and as a result their capabilities are generally limited to prediction of population averages that would be observed during clinical trials. An average-model approach is not realistic for in silico experimentation. A different approach is needed in order to provide valuable information about the safety and the limitations of control algorithms, or to guide and focus emphasis of clinical studies, or to rule out ineffective control scenarios in a cost-effective manner prior to human use. It is necessary to have a diabetes simulator equipped with a cohort of in silico subjects that spans sufficiently well the observed interperson variability of key metabolic parameters in the general population of people, say with type 1 or type 2 diabetes.

In the following, we discuss in detail a whole-body glucose-insulin simulator and an organ/cellular-level insulin secretion simulator. The healthy state glucose-insulin simulator is presented first, then prediabetes and type 2 diabetes version are discussed. Finally, we describe the type 1 diabetes simulator, which was recently accepted by FDA as a substitute of preclinical animal trials for certain insulin treatments [140].

B. Healthy State Simulator

The rationale was to identify the glucose-insulin meal simulation model [137], developed on average data, in each of 204 nondiabetic individuals. All these subjects underwent a triple tracer meal protocol which provided virtually model-independent estimates of crucial fluxes of the system, i.e., the rate of appearance in plasma of ingested glucose, glucose production, glucose utilization, and insulin secretion [59]. This flux information was key to developing with confidence alarge scale maximal model, i.e., with only plasma glucose and insulin concentration this exercise is practically impossible since one cannot obtain a good description of the multiple system fluxes (Fig. 20). Fig. 21 shows the database and allows appreciation of the relevance of interindividual variability. Thanks to this rich “concentration and flux” representation, the glucose-insulin system model was identified by resorting to a subsystem forcing function strategy, which minimizes structural uncertainties in modeling the various subsystems. The rationale is shown in Fig. 22. As an example consider the glucose utilization subsystem (see [137] for details). Glucose kinetics is described with a two compartment model. Insulin-independent utilization takes place in the first compartment, is constant, and represents glucose uptake by the brain and erythrocytes (Fcns)

Fig. 21
Mixed meal data base (average of 204 nondiabetic subjects, grey area represents mean± 1SD range). Top panel: glucose (left) and insulin (right) concentrations. Middle panel: endogenous glucose production (left) and glucose rate of appearance (right). ...
Fig. 22
Unit process models and forcing function strategy: endogenous glucose production (top left panel); glucose rate of appearance (top right panel); glucose utilization (bottom left panel); insulin secretion (bottom right panel). Entering arrows represent ...
Uii(t)=Fcns.
(33)

Insulin-dependent utilization takes place in the remote compartment and depends nonlinearly (Michaelis–Menten) from glucose in the tissues

Uid(t)=Vm(X(t))·Gt(t)Km+Gt(t)
(34)

where Gt(t) is glucose mass in the insulin-dependent tissues and Vm(X(t)) is assumed to be linearly dependent on a remote insulin X(t)

Vm(X(t))=Vm0+Vmx·X(t).
(35)

X (pmol/L) is insulin in the interstitial fluid described by

X.(t)=p2U·X(t)+p2U[I(t)Ib]X(0)=0
(36)

where I is plasma insulin, suffix b denotes basal state, and p2U is rate constant of insulin action on the peripheral glucose utilization.

Total glucose utilization U is thus

U(t)=Uii(t)+Uid(t).
(37)

For each of the 204 subjects included in the database, the glucose kinetics model equipped with (33)–(37) was numerically identified using the measured glucose utilization (U) and concentration (G) as output and plasma insulin, endogenous glucose production and glucose rate of appearance as known inputs (Fig. 22, bottom left panel).

The other subsystems of Fig. 22 have been identified following a similar strategy (more details can be found in [137]).

The model consists of 12 differential equations and 35 parameters, 26 of which are free and nine derived from steady-state constraints.

From the 204 subjects model parameters and their joint probability distribution in the healthy population were reconstructed. Since most parameters were approximately log-normally distributed, this probability distribution is uniquely defined by the average vector and the covariance matrix of the log-transformed parameter vector. Given the joint distribution, virtual subjects can be generated, i.e., say 1000 realizations of the log-transformed parameter vector can be sampled randomly from the multivariate normal distribution, thereby producing 1000 virtual in silico “subjects.”

Fig. 23 shows examples of daily glucose patterns in some generated in silico subjects.

Fig. 23
Example of daily glucose concentration in some generated in silico subjects.

C. Prediabetes and Type 2 Diabetes Simulator

A prediabetes and type 2 diabetes simulators would be very useful to assess the efficacy of various drug therapies before performing experiments in humans. A triple tracer protocol meal data base, similar to that available in nondiabetics, was available for 35 prediabetic and 23 type 2 diabetic subjects. The same modeling strategy developed for the Healthy State Simulator was adopted. The model consists of 12 differential equation and 35 parameters (26 of which are free) [137], [138]. The simulator can generate cohorts of virtual subjects, thus enabling various diabetes treatments to be assessed rapidly in cost-effective in silico experiments.

D. Type 1 Diabetes Simulator

A type 1 diabetes version of the simulator would be critical for the preclinical testing of control strategies in artificial pancreas studies. Arguably, large-scale simulations would account better for intersubject variability than small-size animal trials and would allow for more extensive testing of the limits and robustness of control algorithms. A first necessary modification was the substitution of insulin secretion subsystem with an exogenous insulin delivery subsystem, e.g., a subcutaneous (sc) insulin pump an associated model of insulin kinetics and absorption. Subcutaneous insulin transport has been extensively modeled and quantitative models exist [142], [285], [286] which allowed both the estimation of the timing and duration of insulin action from insulin pump data, and their computer simulation in in silico experiments. One approach includes a two-compartment model approximating nonmonomeric and monomeric insulin fractions in the subcutaneous space [285], which can serve as a base for the translation of the insulin signal from the pump to insulin in the circulation.

A much more difficult task was the description of the inter-subject variability, since even single-tracer studies in type 1 diabetes are scarce. In order to obtain parameter joint distributions in type 1 diabetes from those in the healthy state, we assumed that the intersubject variability was the same (same covariance matrix), but certain clinically relevant modifications were introduced in the average vector. The model consists of 13 differential equation and 35 parameters (26 of which are free and nine derived from steady-state constraints). The simulator has been tested by several experiments in adults, adolescents, and children to assess the validity of the cohort of in silico subjects. The simulator (Fig. 24) is equipped with 100 virtual adults, 100 adolescents, and 100 children, spanning the variability of the T1DM population observed in vivo. In addition to virtual “subjects,” the simulator is equipped with mechanisms reproducing CGM errors, e.g., virtual “sensors” that can be placed on the “subjects” for in silico closed-loop control experiments. Subcutaneous insulin delivery is modeled as well, which allows placing virtual “insulin pumps” on the subjects for full-scale open- or closed-loop control experiments with any predefined treatment scenario. In January 2008, the simulator was accepted by the Food and Drug Administration (FDA) as a substitute to animal trials for the preclinical testing of control strategies in artificial pancreas studies and has been adopted by the JDRF Artificial Pancreas Consortium as a primary test bed for new closed-loop control algorithms. The simulator was immediately put to its intended use with the in silico testing of a new Model Predictive Control algorithm [141], and in April 2008, an investigational device exemption (IDE) was granted by the FDA for a closed-loop control clinical trial (see Section VI). This IDE was issued solely on the basis of in silico testing of the safety and effectiveness of the proposed artificial pancreas algorithm, an event that sets a precedent for future preclinical studies. Thus, the following paradigm has emerged: 1) in silico modeling could produce credible preclinical results that could substitute certain animal trials and 2) in silico testing yields these results in a fraction of the time and the cost required for animal trials. However, one needs to emphasize that good in silico performance of a control algorithm does not guarantee in vivo performance; it only helps to test extreme situations and the stability of the algorithm and to rule out inefficient scenarios. Thus, computer simulation is only a prerequisite to, but not a substitute for, clinical trials.

Fig. 24
Employment of the type 1 diabetes simulator for testing closed-loop control algorithm for insulin infusion.

In addition to the simulator described above [137], [247], another simulation environment has been developed with the core glucose-insulin model detailed in [136] which is used by the University of Cambridge in the JDRF Artificial Pancreas Program. The model [136] has been built for being the model ingredient of a closed loop MPC algorithm. The model, coupled to a subcutaneous insulin kinetic model [142], consists of 11 differential equations and 20 parameters. The strategy to describe the population variability differs from that described above. The simulator presently includes 18 type 1 synthetic subjects defined by 20 parameter sets. A subset of six parameters was estimated from experimental data collected in type 1 subjects and the remaining 14 obtained from the literature. An important use has been the assessment of hypoglycemia and hyperglycemia risk during overnight with closed-loop Model Predictive Control versus open-loop insulin delivery [143].

E. Insulin Secretion

Another useful application of simulation models is hypothesis pilot testing. For instance, simulation models have been used to investigate the cellular mechanisms which lead pancreatic beta-cells to secrete insulin. Beta-cells show bursting electrical activity and oscillatory calcium levels and insulin secretion and modeling has contributed significantly to the understanding of the generation of these rhythmic patterns (see reviews in [144] and [145]). However, surprisingly little work has been done on detailed modeling insulin secretion. Already in the 1970s, Grodsky [120] and Cerasi et al. [146], among others, modeled the pancreatic insulin response to various kinds of glucose stimuli, but these models were phenomenological due to the limited knowledge of the beta-cell biology at that time. Only recently has the knowledge of the control of the movement and fusion of insulin granules increased to a level where it is possible to formulate mechanistically based models. Here, we discuss some recent developments on cellular simulation models of insulin secretion serving as research tool at organ level to pilot-test new theories.

Grodsky [120] proposed that insulin was located in “packets,” plausibly the insulin containing granules, but also possibly entire beta-cells. Some of the insulin was stored in a reserve pool, while other insulin packets were located in a labile pool, ready for release in response to glucose. The labile pool was responsible for the first phase of insulin secretion [120], while the reserve pool was responsible for creating a sustained second phase. This basic distinction has been, at least partly, confirmed when the packets are identified with granules [147], [148]. Moreover, Grodsky [120] assumed that the labile pool is heterogeneous in the sense that the packets in the pool have different thresholds with respect to glucose, beyond which they release their content. This assumption was necessary for explaining the so-called staircase experiment, where the glucose concentration was stepped up and each step gave rise to a peak of insulin. There is no supporting evidence for granules having different thresholds [149], but Grodsky [120] mentioned that cells apparently have different thresholds based on electro-physiological measurements. Later, Jonkers and Henquin [150] showed that the number of active cells is a sigmoidal function of the glucose concentration, as assumed by Grodsky [120] or the threshold distribution.

Recently, Pedersen et al. [151] unified the threshold distribution for cells with the pool description for granules, thus providing an updated version of Grodsky’s model, which also takes into account recent knowledge of beta-cell biology. The model scheme is shown in Fig. 25 (top panel). It includes mobilization of secretory granules from a very large reserve pool to the cell periphery, where they attach to the plasma membrane (docking). The granules can mature further (priming) and attach to calcium channels, thus entering the “readily releasable pool” (RRP). Calcium influx provides the signal triggering membrane fusion, and the insulin molecules can then be released into the extracellular space. Also included is the possibility of so-called kiss-and-run exocytosis, where the fusion pore reseals before the granule cargo is released. For the mathematical formulation the mobilized and docked pools have been lumped into a single “intermediate pool” (Fig. 25, lower panel). The glucose-dependent increase in the number of cells showing a calcium signal [150] was included by distinguishing between readily releasable granules in silent and active cells. Therefore, the RRP is heterogeneous in the sense that only granules residing in cells with a threshold for calcium activity below the ambient glucose concentration are allowed to fuse. Hence, the model provides a biologically founded explanation for the heterogeneity assumed by Grodsky [120] and it is able to simulate the characteristic biphasic insulin secretion pattern in response to a step in glucose stimulation, as well as the secretory profile of the staircase stimulation protocol (Fig. 26). The model is a classic compartmental one, except from the description of the RRP. The intermediate pool I develops according to the mass-balance equation

Fig. 25
Upper panel: overview of the in silico model of insulin secretion which includes mobilization of secretory granules from a very large reserve pool to the cell periphery, where they attach to the plasma membrane (docking). The granules can mature further ...
Fig. 26
Simulation results: insulin secretion (SR) in response to the staircase glucose stimulation (G).

dI(t)dt=M(G,t)r·I(t)p+·I(t)+p·0h(g,t)·dg
(38)

where M is the mobilization flux, and r is the rate of reinternalization. The last term describes the flux of granules loosing the capacity of rapid exocytosis. Mobilization has been modeled similarly to Grodsky (1972) by a first-order differential equation.

The RRP is described by a time-varying density function h(g, t) indicating the amount of insulin in the RRP in beta-cells with a threshold between g and g + dg. Granules are primed with rate p+ and are assumed to loose the capacity of rapid exocytosis with rate p Moreover, if the granule is in a triggered beta-cell it will fuse with rate f+. This leads to the equation

dh(g,t)dt=p+·I(t)·j(g)p·h(g,t)f+·h(g,t)·θ(Gg)
(39)

Here, θ(Gg) is the Heaviside step function, which is 1 for G > g and zero otherwise, indicating that fusion only occurs when the threshold is reached. I is total intermediate pool, and the priming flux p+I distributes among cells according to the fraction of cells with threshold g described by the time-constant function j(g). Thus, priming is assumed to occur with the same rate in all cells, but the model takes into account the fraction of cells with the corresponding threshold. The secretion rate is proportional to the size of the fused pool F (Fig. 25, bottom panel).

An interesting property of beta-cells that was included in Grodsky’s model is so-called derivative control, i.e., the fact that the pancreas senses not only the glucose concentration but also its rate of change. Modeling at whole-body level has shown that this property is necessary for explaining data, e.g., both IVGTT and OGTT&MTT [101], [152]. Derivative control arises from the threshold hypothesis as explained by Grodsky [120] and in greater detail by Licko [153]. Due to the threshold on cells, the model by Pedersen [151] also possesses derivative control, which is only active when dG/dt > 0. It is of interest to understand, and is currently under investigation, how the subcellular parameters relate to beta-cell responsivity measured by whole-body minimal models, i.e., Φ1, Φ2 for IVGTT and Φd, Φs for OGTT&MTT, e.g., which steps of glucose stimulated insulin secretion are impaired in diabetics.

V. Signals

A. Rationale

Historically, the use of signal analysis techniques in the study of diabetes physiology started in the late 1970s with the quantification of blood glucose (BG) concentration and other substances, such as insulin, C-peptide, and glucagon during in-hospital monitoring. Some of the principal approaches are discussed in Section V-B.

Routine field observation of BG fluctuation began with the advent of self-monitoring (SMBG), which typically provides 2–5 BG capillary BG samples per day analyzed by portable glucometers. This opened the possibility of studying individuals’ glucose fluctuation (and thus the effectiveness of their therapy) during natural conditions for extended periods of time. The principal methods for analysis of SMBG data include statistical approaches and risk assessment and are presented in Section V-C.

In the last ten years, new continuous glucose monitoring (CGM) systems capable of monitoring glucose concentration frequently (e.g., every 5 minutes) for several days have begun to emerge. Most of CGM systems are minimally invasive and portable, measuring glucose subcutaneously and assessing BG concentration indirectly via interstitial fluid sampling. Even if certain accuracy issues are still unsolved, CGM sensors open new possibilities in diabetes management, showing encouraging treatment results and potential for real-time prevention of hypo- and hyper-glycemia. Methodologies and applications concerning CGM sensor data are discussed in Section V-D.

B. Physiological Signals

1) Glucose-Insulin Oscillations

According to a recent review, for glucose concentration measured in blood four time scales of BG fluctuation have been identified: 5–15 min corresponding to pulsatile secretion of insulin; 60–120 min corresponding to intrinsic oscillatory phenomena; 150–500 min accounting for meals, insulin injection, and other external schedules, and ~700 min corresponding to circadian rhythm [154]. The Nyquist sampling period sufficient to follow the intrinsic blood glucose dynamics in diabetes was estimated at 10 minutes [155]. The temporal relationship between insulin oscillation and plasma glucose excursions has been attributed to a feedback loop between beta cell action and endogenous glucose production [156]. It has been established that the ability of glucose to entrain ultradian insulin secretion patterns is disturbed in diabetic and prediabetic individuals [157]–[159], but the exact role of oscillations in the glucose regulation systems is still under investigation [160].

2) Peak Detection and Spectral/Correlation Analysis

Most of the approaches employed in the 1980s and early 1990s to study glucose and insulin oscillations can be traced back to peak-detection methods and spectral/correlation analysis. Peak-detection methods provide information on pulse amplitude and location [161]–[164], while spectral/correlation analysis seeks to identify cycles in time series [165], [166]. These methods can be easily modified to assess the concordance of peaks and the cross-spectrum/correlation function in order to study the dynamic relationship between paired time series, e.g., glucose and insulin [167]–[170]. However, in situations where pulses are brief, small in amplitude, and irregular, peak detection is difficult. Spectral/correlation analysis is hard to apply to short and noisy time series, or when the observed cycles are inherently irregular. Finally, both peak-detection and spectral/correlation analysis ignore the morphology of the pulses and cannot detect e.g., changes in the pattern of episodic insulin release which is characteristic of some physiological and patho-physiological states. Overcoming these limitations may require the use of different data analysis tools, such as the ApEn described as follows.

3) Hormone Pulsatility and Approximate Entropy

Approximate Entropy (ApEn) is a method developed in the early 1990s [171] to quantify the “regularity” of a time series. ApEn detects changes in underlying episodic behavior not reflected in peak occurrences or amplitudes. To do so, ApEn assigns a nonnegative number to a time series, with larger values corresponding to greater apparent temporal irregularity in the hormone release pattern over time. Two input parameters, m and r, must be specified to compute ApEn, which then measures the logarithmic likelihood that runs in the patterns that are close (within r) for m contiguous observations remain close (within the same tolerance r) on the next incremental comparisons [172]. An extension of this theory has also been proposed to analyze the synchrony of two time series belonging to the same physiologic network through the Cross Approximate Entropy index [173].

To a large extent, ApEn is complementary to peak detection and spectral analysis in that it evaluates both dominant and subordinate patterns in concentration-time series [174]. Given its ability to detect changes in underlying episodic behavior which will not be reflected in pulse occurrences or their characteristics, ApEn acts as a “barometer” of feedback system change in many coupled systems. ApEn has been utilized to probe the regulation of several hormones including growth hormone [175], aldosterone [176], cortisol [177], insulin [178], and glucagon [179]. Specific to diabetes, it has been reported that insulin secretion patterns are more irregular in healthy older and obese individuals and patients with prediabetes or type II diabetes mellitus than in young, nonobese, nondiabetic subjects [156]–[159]. It was hypothesized that this may in part reflect failure of negative feedback by glucose, and that disorderly insulin secretion patterns in aging and prediabetes accompany insulin resistance [160]. Methodologically it has also been discussed how, e.g., in hormone secretion studies, the discrimination power of ApEn from plasma concentration time series can be influenced by their kinetics [180], [181].

C. Self Monitoring of Blood Glucose

1) Blood Glucose Self-Monitoring

Blood glucose self-monitoring (SMBG) is typically comprised of several episodic BG readings per day. When rather large time intervals are monitored in a given patient, several tools can be used to analyze SMBG time series. For instance, in [182] and [183], the expected cyclo-stationary behavior of glucose was investigated by decomposing the SMBG time series into a cyclic component, that expresses the daily pattern, and a trend component, that describes long-term variations, by a Bayesian method implemented by Monte Carlo Markov chains. Another approach, widely accepted to analyze SMBG time series, is risk analysis. A key element to the interpretation of SMBG signals is the use of the BG Risk Space, which was introduced over ten years ago and was based on a transformation of the BG scale that corrects an inherent asymmetry of the hypoglycemic versus hyperglycemic ranges [184]. The steps of the SMBG data risk analysis are as follows.

2) Symmetrization of BG Scale

The BG measurement scale is asymmetric: the hypoglycemic range (below 70 mg/dl) is much narrower numerically than the hyperglycemic range (BG > 180 mg/dl). This asymmetry creates a number of computational problems and challenges. For example, a BG excursion from 180 to 240 mg/dl is much larger numerically than a BG excursion from 70 to 50 mg/dl, yet the second excursion carries much greater risk to the patient. The asymmetry of the BG scale can be corrected by using

f(BG,α,β)=[(ln(BG))αβ]
(40)

where α, β > 0 are parameters determined from the assumptions

A1:f(600,α,β)=f(20,α,β)
(41)

and

A2:f(180,α,β)=f(70,α,β).
(42)

By multiplying by a third parameter γ we fix the minimal and maximal values of the transformed BG range at 10 and 10 respectively. When solved numerically under the restriction α > 0̃, these equations give: α = 1.084, β = 5.381, and γ = 1.509. These parameters are sample independent and were fixed in 1997 [184].

3) BG Risk Space

After fixing the parameters of f(BG), we define the quadratic function r(BG) = 10.f(BG)2, which defines the BG risk space. The function r(BG) ranges from 0 to 100. Its minimum value is 0 and is achieved at BG = 112.5 mg/dl, a safe euglycemic BG reading, while its maximum is reached at the extreme ends of the BG scale 20 mg/dl and 600 mg/dl. Thus, r(BG) can be interpreted as a measure of the risk associated with a certain BG level. The left branch of this parabola identifies the risk of hypoglycemia, while the right branch identifies the risk of hyperglycemia [185]. Fig. 27 visualizes this concept by a 72-hour CGM trace of a patient with T1DM: each CGM reading is processed in two steps: 1) application of the symmetrization formula [184], and 2) application of the quadratic risk function that assigns increasing weights to larger BG deviations towards hypo- or hyper-glycemia [185]. As seen in Fig. 27, a hypoglycemic episode occurring at hour 30 of observation is hardly visible in BG scale, but is well pronounced in risk space. Conversely, a hyperglycemic excursion at hour 54 is numerically reduced in risk space, which corresponds to the notion of relative risk it carries.

Fig. 27
Transforming BG data into risk space equalizing the hypoglycemic and hyperglycemic blood glucose ranges.

The conversion of the BG data into risk values has profound implications not only for the interpretation of the BG signal, but for control as well because similar emphasis is placed on the hypoglycemic and hyperglycemic ranges; the normal BG range (70–180 mg/dl) is given less weight, thus variability contained within normal range carries less risk than excursions outside of this range; excursions into extreme hypo- and hyper-glycemia get progressively increasing risk values.

4) SMBG-Based Risk Metrics

In addition, the conversion of BG data into risk space has resulted in established metrics for the risk for hypoglycemia and glucose variability in general. Let x1, x2, … xn be a series of n BG readings, and let:

  1. rl(BG) = r(BG) if f(BG) < 0 and 0 otherwise;
  2. rh(BG) = r(BG) if f(BG) > 0 and 0 otherwise.

The Low Blood Glucose [Risk] Index (LBGI) and the High BG [Risk] Index (HBGI) are then defined as:

LBGI=1ni=1nrl(xi)
(43)

and

HBGI=1ni=1nrh(xi)
(44)

respectively.

The LBGI, which is based on the left branch of the BG Risk Function r(BG) has been validated as an excellent predictor of future significant hypoglycemia [186], [187]. The LBGI also provides means for classification of the subjects with regard to their long-term risk for hypoglycemia into: minimal, low, moderate and high-risk groups, with LBGI of below 1.1, 1.1–2.5, 2.5–5.0, and above 5.0, respectively [187], and has been used for short term prediction of hypoglycemia as well [188], [189]. By definition, the LBGI is independent from hyperglycemic episodes.

Further, we define the Average Daily Risk Range (ADRR), which is a measure of overall glycemic variability based on r(BG) and computed as follows.

  1. Let x11, x12, … xn1 be a series of n1 SMBG readings taken on Day 1.
  2. Let x12, x22, … xn2 be a series of n2 SMBG readings taken on Day 2.
    ……
  3. Let x1M, x2M, … xnM be a series of nM SMBG readings taken on Day M.

Thus, n1, n2, . . . . , nm > 3 and the number of days of observation M is between 14 and 42 (two to six weeks). Further, let

LRi=max(rl(x1i),rl(x2i),,rl(xni))

and

HRi=max(rh(x1i),rh(x2i),,rh(xni))forday#i;i=1,2,M.

The Average Daily Risk Range is then defined as

ADRR=1Mi=1M[LRi+HRi].
(45)

The ADRR has been shown superior to traditional measures in terms of risk assessment and prediction of extreme glycemic excursions [190]. Specifically, it has been demonstrated that classification of risk for hypoglycemia based on four ADRR categories: low risk: ADRR < 20; low-moderate risk: 20 <= ADRR < 30; moderate-high risk: 30 <= ADRR < 40, and high risk: ADRR > 40, resulted in over six-fold increase in risk for hypoglycemia from the lowest to the highest risk category [190].

D. Continuous Glucose Monitoring Time Series

Subcutaneous continuous glucose monitors (CGM) assist the treatment of diabetes by providing frequent data for the dynamics of BG. Recent studies have documented the benefits of CGM [191]–[193] and charted guidelines for clinical use [194] and its future as a precursor to closed-loop control [195]. However, while CGM has the potential to revolutionize the control of diabetes, it also generates data streams that are both voluminous and complex. The utilization of such data requires an understanding of the physical, biochemical, and mathematical principles and properties involved in this new technology. It is important to know that CGM devices measure glucose concentration in a different compartment—the interstitium. Interstitial glucose (IG) fluctuations are related to BG presumably via diffusion process [196], [197]. This leads to a number of issues, including distortion (which incorporate a time lag) and calibration errors, and necessitates the development of methods for their mitigation. In particular, it is necessary to consider that, since the BG to IG kinetics acts as a low-pass filter, the frequency content of interstitial glucose is different from that of blood glucose [198], [199].

1) CGM Sensor Calibration

CGM sensor calibration accounts for the gradient between BG and IG. Typically, CGM devices are calibrated with capillary glucose, which brings the (generally lower) IG concentration to BG levels. The factors pertinent to successful calibration and the effects of calibration errors have been extensively studied in the past few years [200]–[203]. Alternative calibration methods based on decomposition of the sensor errors and on the use of diffusion models have been studied as well [204], [205]. Various calibration functions are adopted by researchers and industry, but most have a certain linear component using a 2-point linear regression model: y = ax + b, where a and b are calibration parameters which are determined by fitting them against a couple of reference BG(y) and raw CGM current CGM(x) levels collected at the same time. However, this procedure can be suboptimal, because it does not take into account distortions introduced by BG-to-IG kinetics. The DirecNet Study Group [200], analyzed the improvement in CGMS sensor accuracy by retrospectively modifying the number and timing of the calibration points and established that the timing of the calibration points is quite important. In particular, performing calibrations during periods of relative glucose stability, i.e., where the point-to-point difference due to the BG-to-IG kinetics is minimized, significantly improves the accuracy. An approach presented by Kuure-Kinsey et al. [206] uses dual-rate Kalman filter to improve the accuracy of CGM data. The procedure exploits episodic SMBG readings and estimates in real-time the sensor gain. A critical aspect of this algorithm is that it does not embed any BG-to-IG kinetics model. A more comprehensive description of the CGM measurement process was done by Knobbe and Buckingham [201]. In their work, BG-to-IG kinetics model was explicitly taken into account in order to reconstruct BG levels in continuous time from CGM measurements by an Extended Kalman Filter potentially able to deal with a multiplicative calibration error. However, successful calibration would adjust the amplitude of IG fluctuations with respect to BG, but would not eliminate the possible time lag due to BG-to-IG glucose transport and the sensor processing time (instrument delay).

2) CGM versus BG

Because the time lag typically associated with CGM could greatly influence CGM applications, a number of studies were dedicated to its investigation [207]–[211], yielding various results. For example, it was hypothesized that if glucose fall is due to peripheral glucose consumption the physiologic time lag would be negative, i.e., fall in IG would precede fall in BG [196], [212]. In most studies, however, IG lagged behind BG by 4–10 min, regardless of the direction of BG change [196], [207], [208]. The formulation of the push-pull phenomenon offered reconciliation of these results and provided arguments for a more complex BG-IG relationship than a simple constant or directional time lag [211]. In addition, loss of sensitivity and random noise confound CGM data [213]–[215]. Thus, while the accuracy of CGM is increasing, it is still below the accuracy of direct BG measurement [216]–[219] and may be reaching a physiological limit of s.c. glucose monitoring.

Consequently, it is now recognized that the glucose monitor remains the main limiting factor in the development of diabetes control systems [220]–[223]. Less recognized is the fact that the algorithms retrieving CGM data also have a major contribution to the clinical success of these devices. In order to provide the best data possible, CGM signal processing is needed. Of primary importance are methods for: tracking CGM data integrity, including CGM data filtering (denoising); predicting glucose fluctuations which has the potential to mitigate the effects of a time lag; generating predictive alarms that could produce appropriate warning for upcoming extreme glycemic events (e.g., hypoglycemia) and would thereby assist the behavioral self-regulation of diabetes. Available techniques for filtering, prediction, and alert generation have been recently reviewed in [224].

3) Filtering

In general, denoising filters start from the following equation:

y(t)=u(t)+v(t)
(46)

where y(t) is the glucose level measured at time t, u(t) is the true, unknown, glucose level and v(t) is random additive noise. The purpose of filtering is then recovering u(t) from y(t). Given the expected spectral characteristics of signal and noise, e.g., signal is lowpass and noise is white, (causal) low-pass filtering represents the most natural candidate to separate signal from noise in online applications. One major problem in low-pass filtering is that, because signal and noise spectra overlap, it is not possible to remove noise v(t) from the measured signal y(t) without distorting the true signal u(t). In particular, distortion results in a delay affecting the estimate û(t) with respect to the true u(t): the more the filtering, the larger the delay. Thus, a clinically significant filtering issue is reaching a compromise between the regularity of û(t) and its delay with respect to the true u(t).

In the literature, real-time denoising of CGM signals has been addressed, both by sensor manufacturers and university researchers. The information of signal processing in commercial CGM devices is generally proprietary, but some results seem to indicate that moving-average (MA) filters with fixed parameters are often used. Methods which indirectly can address the denoising issue can be found in Chase et al. [214], Knobbe et al. [201] Palerm et al. [226], and Kuure-Kinsey et al. [206]. An explicit dealing with the denoising problem is made in the recent work by Facchinetti et al. [225]. In this paper, a Bayesian estimation approach was implemented by Kalman filtering for the real-time denoising of CGM signals. A key feature of the method is that, thanks to the incorporation of a stochastically based smoothing criterion, it can individualize filter parameters and hence the regularization amount according to the signal-to-noise ratio (SNR) of the specific CGM signal. In particular, the approach is able to cope with different sensors, interindividual and intraindividual SNR variability of CGM data. The performance of this new approach was assessed using Monte Carlo simulations and 24 CGM datasets and compared to a moving average filtering with fixed parameters. Fig. 28 shows two CGM time series (gray profiles) taken from the Monte Carlo simulation of [225] and obtained by adding, to a reference profile, a white Gaussian noise sequence with variance σ2 = 4 mg2/dl2 (top panel) and σ2 = 49 mg2/dl2 (bottom panel). Data from a burn-in interval (shaded box) were used to estimate σ2: notably, the numerical values estimated for σ2 (reported in the shaded box) are very close to the true ones. Thanks to the self-tunable parameters individualization, in both simulations the Kalman filter denoises optimally noisy CGM data (black lines in Fig. 28). On real data, results showed that, for comparable signal denoising, the delay introduced by the Kalman filter is about 35% less than that obtained by MA [225].

Fig. 28
Two simulated CGM profiles obtained with different noise variance. Noisy (gray line) versus Kalman filtered (black) signals. Shaded areas correspond to the burn-in intervals.

4) Prediction

A multitude of methods has been proposed for the near-term (up to 45 minutes) prediction of glucose fluctuations. Most of these methods are based on time-series modeling techniques [227]–[233], but neural networks [234], [235] and other physiological structural models [236] have been used as well. Some of the approaches based on time-series modeling are briefly described as follows.

A typical CGM-based predictor would be based on a local approximation of the CGM time series by a first-order polynomial:

ui=α·ti+β
(47)

or by a first-order autoregressive (AR) model corresponding to the following time-domain difference equation

ui=a·ui1+wi
(48)

where i = 1,2, … n is the order of glucose samples collected until the nth sampling time tn and {wi} is a random white noise process with zero mean and variance σ2. The prediction strategy then works as follows: Let θ be the vector of the parameters of the model employed to describe the glucose time series, at each sampling time tn. A new value of θ is determined by fitting past glucose data un, un−1, un−2, … by weighted linear least squares. Once θ determined, the model is used to calculate the prediction of glucose level Q steps ahead, i.e., ûn+Q. The product Q · Ts, where Ts is the sensor sampling period, gives the so-called prediction horizon (PH). The necessity of having a time-varying θ is obvious in the model of (47). For the AR model of (48), the use of a time-invariant θ e.g., identified in a burn-in interval, would produce inaccurate predictions because of nonstationarity of CGM time series, which, in the case of low-order model, calls for a time-varying modeling strategy [230]. In the estimation of θ the past data participate with different relative weights. A typical choice is to employ exponential weighting, i.e., μk is the weight of the sample taken k instants before the actual sampling time, with the forgetting factor μ within the range (0, 1) termed forgetting factor. In [233], the forgetting factor is modulated in order to account for sudden changes of glucose dynamics.

Reifman et al. [228] employed a different approach, which exploited a high-order AR model (order 10) which was first fitted, in each subject, in a burn-in interval and then used within the prediction algorithm for the rest of the time series. A price to be paid for this approach is an increased model complexity which requires the use of a rather long burn-in interval (about 2000 samples, or ~36 h). Moreover, given the high number of parameters to be estimated, this AR model would be overly sensitive to noise. Indeed, a “regularization constraint” was placed on the AR coefficients in order to decrease their sensitivity to the data. Furthermore, the prediction algorithm of [228] has been assessed on retrospectively smoothed CGM data.

A stochastic nonparametric approach, similar to that employed in [225] for denoising, was proposed for prediction purposes in [237]. The idea of the method is to exploit the available a priori information on the smoothness of CGM signal, formalized through a stochastic model including the multiple integration of a white noise process. After having placed the problem in a state-space setting, a Kalman Filtering methodology is used to predict glucose level within a given PH. The approach was tested on 13 data sets of Minimed CGM data (5 min sampling) during a hypoglycemic clamp (4 hour data). Three different PH were tested, PH = 10, 20, and 30 min. The parameters of the Kalman filter were empirically determined, in a retrospective fashion, in order to “maximize” sensitivity and specificity. The authors reported that the prediction performance, in this well-controlled hypoglycaemic clamp situation, was satisfactory in terms of sensitivity and specificity. No quantitative estimates of the prediction delays were reported.

5) Hypoglycemia and Hyperglycemia Alerts

A special class of predictive methods concerns the generation of alarms forewarning the patient about upcoming extreme glycemic events, e.g., hypo- or hyper-glycemia. These methods have rapidly evolved from a concept [238] to implementation in CGM devices, such as the GuardianRT (Medtronic, Norhtridge, CA) [239] and the Freestyle Navigator (Abbott Diabetes Care, Alameda, CA) [178]. Discussion of the methods for testing of the accuracy and the utility of such alarms has been initiated [178], [240], [241], and the next logical step—prevention of hypoglycemia via shutoff of the insulin pump—has been undertaken [242].

VI. Control

A. Rationale

As already noted in Section II, a patient with type 1 diabetes faces a life-long behavior-controlled optimization problem: the administration of external insulin to control glycemia enters a stochastic scenario where hyperglycemia and hypoglycemia may not be easily prevented by open-loop therapy. The adjustment of therapy, i.e., basal insulin delivery and premeal boluses, on the basis of a few daily fingerstick blood glucose measurements, can be seen as rudimentary way to close the loop. Clearly, the few daily measurements, albeit very important, considerably limit the effectiveness of the feedback action.

Closed-loop glucose control uses in contrast frequent measurements. This subject has been discussed by numerous research papers since the 1960s, and several surveys are now available [223], [243]. The purpose here is to review recent developments by taking into account some guidelines. In particular, attention is focused on the subcutaneous sc-to-sc control route, i.e., on control systems adopting noninvasive sc insulin pumps and sc CGM devices. Therefore, contributions specific to intensive care patients or those dealing with glucagon pumps are not dealt with. Recent years have witnessed the development of more realistic models of glucose metabolism, as well as the first trials testing closed-loop glucose control systems in people. In view of this, another inclusion criterion is the validation platform: we will only consider control algorithms validated in clinical in vivo trials or in advanced in silico experiments, which provide accurate description of dynamic phenomena and/or incorporate interindividual variability [137], [244], [245]. The ideal in silico experiment not only provides a detailed simulation of metabolic processes but is also capable of running simulations on a large virtual population of patients. The importance of these models for testing glucose control algorithms is confirmed by the fact that the FDA has accepted in silico trials conducted with a large-scale in silico model as a substitute of the preclinical animals studies, which are usually needed to authorize clinical trials in humans [140]. In [246] and [247], some guidelines for in silico proof-of-concept testing of artificial pancreas control algorithms are proposed. A critical step of a full-scale in silico testing should involve not only controller software but also hardware elements, including the communication interface between the controller and the glucose sensor and insulin pump [248].

The problem of maintaining glucose levels within a predefined range by acting on insulin delivery is a control problem with a number of more or less specific features. The controlled variable is glucose utilization, the measured output is the sc glucose provided by the CGM, and the clinical criterion for success is plasma glucose. There is one manipulated variable (also called control input), namely the insulin delivered by the sc pump that can be acted upon by either the patient or the control systems to regulate plasma glucose. The system is subjected to disturbances, the most important one being the meals. It is important to note that this disturbance may be announced, approximately known, or even predictable. Such knowledge is routinely exploited in conventional insulin therapy in order to compute premeal boluses. Among other disturbance inputs, one may mention physical exercise that is known to acutely increase glucose utilization and chronically modify insulin sensitivity.

The dynamics of the system linking sc insulin to sc glucose consist of a cascade of three subsystems: the sc insulin having plasma insulin as output, the insulin-glucose metabolism nonlinear model having plasma glucose as output, and the sc glucose subsystem having sc glucose as output. As a result, sc insulin infusion poses major challenges to control algorithms due to the significant time needed for insulin absorption, diffusion, and action. With the advent of new rapid-acting insulin analogues that have been developed to more closely approximate the physiology of meal-related insulin secretion (e.g., lispro, aspart, glulisine) this time is now one hour or less [287], [288], which is still far inferior to the rate of insulin response in health. Such large time delays are relatively inconsequential in a steady (fasting) state, but have a major impact during system disturbances (e.g., meals, exercise). Recently, it has been found that, due to the “smoothing” inherent with sc insulin transport, small 15-min insulin boluses are indistinguishable from continuous basal rate [289], which allows designing control algorithms with up to a 15-min actuation rate, an approach that may be superior to the traditional bolus + continuous basal rate in terms of both computational and insulin pump energy efficiency. Further, the effect of meals on plasma glucose is characterized by an absorption delay whose time constant is in the order of hours. Overall, the sc system dynamics is nonlinear and affected by substantial delays, making the design of effective sc closed-loop control algorithms all but a trivial task. In addition, control must also face the significant inter- and intra-patient variability, meaning that it may be virtually impossible to apply the same controller to different patients and that even the same patient may show large variations at different days. Another issue is the presence of intrinsic input constraints, in that the manipulated input variable, e.g., insulin, is nonnegative. Moreover, there are also output constraints on the controlled variable in that plasma glucose should never go below a hypoglycaemia threshold, e.g., 60 or 70 mg/dl. On the other hand, in order to prevent long-term complications, hyperglycemia should be avoided as well. Finally, it is important to realize that closed-loop control is not without risks. For instance, in presence of hyperglycemia following a meal the regulator is likely to react by delivering more insulin, whose effect will not be immediately apparent due to intrinsic system delays. Then, insulin given in excess and too late may act when meal effect has ceased, so that hypoglycemia becomes unavoidable even if the insulin pump is shut off.

B. Architecture of Glucose Control

The availability of CGM measurements opens the way to different types of closed-loop control strategies, ranging from simple short-term safety-oriented interventions to long-term therapy-optimization schemes. In order to classify and organize the review of the literature, we refer to a recent paper that has proposed a layered architecture for artificial pancreas systems [249] (Fig. 29). The layers are characterized by the time-scale of their operations. At the bottom, the fastest layer is in charge of safety requirements. Possible algorithms include pump shutoff, insulin on board (IOB), and the so-called “brakes.” Immediately above, there is the real-time control layer that decides insulin delivery on the basis of latest CGM data, previous insulin delivery, and meal information. Typical algorithms are either Proportional Integral Derivative (PID) or Model Predictive Control (MPC) regulators. The top layer, called offline control tuning, uses clinical parameters and historical data to tune the real-time control layer. In this case, the methods include individual controller calibration strategies, run-to-run (R2R) control algorithms, and behavioral analysis of patient lifestyle. Each layer processes available information (experimental measurements and patient’s inputs) to take decisions that are passed to a lower layer. Each layer can override commands from an upper layer if this is either useful or necessary: a typical example may be provided by the safety layer zeroing insulin administration decided by the real-time control module.

Fig. 29
Modular layered architecture of the artificial pancreas. The layers work on different time scales: the fastest one deal with safety maintenance, the middle one with real-time closed-loop control, and the top one with tuning and supervision on a daily ...

It is interesting to note that activation of all layers is not strictly necessary. Within a traditional therapy consisting of basal insulin and meal-compensating boluses, use of the CGM information may be limited to safety interventions when the patient is at risk of hypoglycemia. For instance, the real-time control layer may be omitted, in which case the offline layer would just be in charge of adjusting basal insulin and premeal insulin boluses using historical information.

C. Safety Algorithms

The idea behind control algorithms pertaining to the safety layer is to exploit CGM data to improve patient’s safety. Since the main short-term risk is hypoglycemia, the safety algorithms discontinue or reduce basal insulin to prevent dangerous decreases of blood glucose level.

The simplest strategy is pump shutoff when hypoglycemia is predicted. This approach has been shown to reduce the risk of nocturnal hypoglycemia [242], [250]. A possible drawback is that the use of an on–off control law for basal insulin, similar to bang-bang or relay control, may induce undesired oscillations of plasma glucose. In fact, if the basal insulin is higher than that needed to keep the glycemic target, the recovery from hypoglycemia would be followed by application of the basal that will cause a new shut-off occurrence. The cycle of shut-off interventions yields an insulin square wave that induces periodic oscillation of plasma glucose.

An alternative approach relies on insulin-on-board (IOB) computation [251]. The amount of IOB should never exceed a quantity that can cause future hyperglycemia. This assessment of the amount of on-board insulin, though approximate, offers a practical way to prevent hypo phenomena. Moreover, the IOB computation is simple enough to be included within a safety module that should act when upper layers have failed.

The third approach, the so-called “brakes” [252], which assess the risk associated with glucose values and reduce the delivered insulin accordingly. The main advantage, compared to the shut-off approach, is the ability to finely adapt intervention to the estimated risk. A possible improvement may regard risk assessment that could take into account past CGM and insulin history and not only the current CGM value. With this respect, the complex risk prediction algorithms may be incompatible with simplicity and robustness requirements of a safety module.

Recently, it has been proposed that implementation of “semi closed-loop” glucose control, only using the safety module, without the real-time control, could offer a first step towards full-closed loop control [253]. However, the philosophy of pure safety (act only to prevent hypoglycemia) needs to be adopted with caution, particularly in a system characterized by slow dynamics where prevention is much more efficient than correction. Nevertheless, in the presence of a real-time control module, safety algorithms may play an important role, especially if they are based on simple and physically grounded computations and offer a recovery strategy from physical or algorithmic failures of real-time controllers.

D. Real-Time Control

1) Feedback and Feedforward Control

There are two classes of control schemes: open- and closed-loop. Both classes of control schemes aim to keep the controlled variable (e.g., blood glucose) within the admissible or desired range, compensating for disturbances and uncertainties by acting on the manipulated variable (e.g., insulin). The main difference is that open-loop methods do not employ real-time measurements in order to take their decisions, whereas closed-loop control exploits measurements correlated with the variable under control to react to uncertainties and disturbances.

A fully open-loop scheme would correspond to a fixed therapy (for instance consisting of basal insulin administration throughout the day and insulin boluses at meal times), based on patient characteristics, without plasma glucose measurements. The open-loop therapy could, however, exploit knowledge of external disturbances, e.g., adapting the boluses to the predicted meal amount. The use of knowledge on an external disturbance in order to compensate in advance for its effects is a feedforward action. In principle, if the patient dynamics and external disturbances were perfectly known, it would be possible to design an open-loop insulin profile ensuring the desired glycemic control. In real life, however, both the patient dynamics and presence and size of external disturbances are far from being perfectly known. Hence, there is the need of corrections that must be based to the actual patient state. In fact, in the conventional therapy, occasional fingerstick glucose measurements are used to trigger corrective actions in order to react to deviations from the nominal profile that the open-loop control is expected to produce. This gives rise to a kind of partially closed-loop control scheme. However, few daily measurements are insufficient to change the nature of a control strategy which relies heavily on feedforward compensation.

With the commercial availability of CGM, it has become possible to design minimally invasive closed-loop control schemes based on frequent output measurements. In particular, the effect of external disturbances (e.g., meal and exercise, but also changes of insulin sensitivity) can be corrected based on their effect on glucose levels. In presence of an excessive rise of glucose, the controller is alerted by the CGM signal and can increase insulin delivery. Conversely, an undesired decrease of glycemia can trigger the reduction of basal insulin delivery.

A purely closed-loop control scheme would decide instantaneous insulin delivery on the basis of CGM signal alone. This would bring several advantages. First, any undesired glucose level would be accounted for in real time by suitable corrections of insulin rate. Moreover, being based on the measured effects of disturbances, corrections are applied without need for explicit knowledge or modeling of such disturbances. For example, an unexpected meal would be dealt with by reacting to the consequent glucose rise. If all this were possible and effective, improved quality of life would be apparent, as any change of meals and habits would be dealt with similarly to glucose control in health. Unfortunately, the sc delays described above present a major stumbling block on the way to a purely closed-loop control strategy: The action of insulin on plasma glucose is subject to significant delays so that effects of reactions to undesired glucose level may arrive too late to prevent hyper- or hypo-glycemic episodes. This problem is further exacerbated by the inherent delay between plasma glucose and the CGM signal. For a closed-loop controller, the worst case scenario is when a fast acting disturbance has to be counteracted by a manipulated variable whose action is delayed by the system dynamics. In this context, attempts to speed up the responsiveness of the closed-loop system may even result in an unstable behavior. For example, an excessive closed-loop insulin administration following postprandial hyperglycemia is likely to cause a hypoglycemic episode. This kind of intrinsic performance limitation is well known in control: in presence of significant delays in the route from manipulated (s.c. insulin) to the controlled variable (plasma glucose), the designer must settle for a relatively slow response time (i.e., the time needed for desired glycemia to be restored). These considerations suggest that the closed-loop control action should not be abrupt and favour gradual corrections. However, a poorly aggressive control is not likely to provide good postprandial glycaemic attenuation, when a large and sudden disturbance is to be counteracted.

The problem of finding a tradeoff between nocturnal regulation, well suited to mild control actions, and postprandial regulation, calling for prompt and energic correction has been pointed out in [254] with reference to previous closed-loop trials [255]–[257] where good nocturnal regulation was to the detriment of breakfast control quality. This dilemma can be escaped by a control scheme that combines feedforward and feedback actions [254], [258] (Fig. 30). In particular, feedback control actions are applied only as corrections that are summed to a “conventional” insulin administration (feedforward action) made of a basal profile and premeal boluses calibrated on the presumed meal amount. Associated with the feedforward action is a nominal glucose profile which represents the expected consequence of the conventional therapy. The closed-loop regulator bases its actions on the difference between the CGM signal and this nominal profile. If the difference is zero, no closed-loop correction is applied and the patient is subject to the conventional therapy alone. Although in practice the difference will always be nonzero, a well-designed feedforward action will require small-size feedback corrections. The major advantage is the possibility of combining prompt and energic compensation of meals (through the feedforward bolus) with the robustness of closed-loop control capable of adapting to unpredicted events, disturbances, and changes in patient’s dynamics.

Fig. 30
Block diagram of a closed-loop glucose control systems including a feedforward action. Information on meal time and amount is used to generate a feedforward action, typically under the form of a premeal bolus. The insulin control signal is obtained as ...

2) Models for Control

The deployment of a controller, especially a closed-loop one, relies heavily on mathematical models. It is worth noting that the requirements posed to models may vary depending on the different phases: design, tuning, and validation.

Most control design methods make use of compact models, whose main task is capturing system dynamics on the time scale regulation is concerned with. In this respect, linear time-invariant models may be obtained by linearization of either the average insulin-glucose in silico model or the minimal model. In the former case, order reduction methods may be employed to eliminate redundant state variables. Another way to obtain a linear time-invariant model is by black-box identification techniques applied to patient’s data, e.g., to identify ARMAX models. Some nonlinear control strategies such as Nonlinear Model Predictive Control, described as follows, can rely directly on nonlinear models such as the insulin-glucose in silico model or the minimal model (nonlinear black-box models, e.g., Nonlinear ARMAX, can be considered as well).

In the tuning and validation stages, it is convenient to run simulations that mimic the real system dynamics as faithfully as possible, so that large scale simulation models are particularly useful. In control engineering, it is a common practice to design a controller on some simple model (low-order linear time-invariant, for instance) and validate it on a detailed simulator of the system under control. Recalling that the insulin-glucose in silico model characterizes a population of virtual patients it is even possible to tune the controller parameters through in silico trials that compare the performances of different parameters values on the whole in silico population.

In the following review of glucose control strategies, the role and type of associated models will be pinpointed for each control method. In any case, a common feature of models for control design is instrumental to the achievement of satisfactory regulation performances rather than striving for the best possible description of all physiological phenomena.

3) PID Control

The classical PID control scheme is the most widespread in the control of industrial processes due to its simplicity, flexibility, and ease of tuning. In particular, the tuning of the controller parameters can be done without a mathematical model of patient’s metabolism, simply using empirical rules.

It is therefore not surprising that some of the first sc-to-sc control experiments used a PID controller [259], [260], [290], [291]. Among its pros, there is the possibility of relating the PID tuning parameters to the biometric parameters of the patient [259]–[261].

However, one issue concerning PID is the merit of the integral action. In fact, integral action is usually included in industrial regulators because, provided that the closed-loop system is stable, its presence guarantees asymptotic zero-error regulation. In the case of glucose control, this would correspond to the asymptotic convergence of glycemia, in absence disturbances, towards the assigned setpoint. Recalling that meals are to be regarded as disturbances, the assumption is unrealistic, which makes the asymptotic zero-error property much less appealing. There is a further concern, relative to the transient response to disturbances, which is even more critical: Consider a closed-loop stable linear system having in the loop a controller with an integral action and assume that the initial state is at the equilibrium with output equal to the setpoint. If an external transient disturbance is applied, it can be easily demonstrated that the integral over time of the difference between the resulting output and the equilibrium output is equal to zero. The glucose metabolism is nonlinear but nevertheless it may be approximated by a linear system in a neighborhood of the equilibrium point. As observed by [243], this means that after a meal, which makes glycemia exceed the desired value, there will always be an undershoot, i.e., a glucose transient that takes values below the glycemia setpoint and whose area is comparable the area of the previous overshoot above the setpoint. For these reasons, it may be convenient to turn to PD controllers, i.e., without the integral action. In particular, [262] can be regarded as a PD controller, in that is adapts the basal insulin depending of glucose value (proportional action) and its variation in the past 30 min (which replaces the derivative action).

PID control suffers from the same problems already highlighted. Due to the presence of substantial delays in the insulin-to-glucose route, one has to design a scarcely aggressive controller which, however, may react less promptly and effectively to meals than the usual premeal boluses of the conventional therapy. As already mentioned, in order to improve performance, a possibility is to add a feedforward action, so as to recover the promptness of meal compensation. Although MPC strategies appear better suited to incorporate predictions on the future effects of meals, an attempt has been made along this direction by [261], where the authors present a controller switching between PID regulation and bolus administration in proximity of meals. In any case, irrespective of the adopted tuning procedure or the presence of a feedforward action, PID controller will be subject to saturations and therefore an anti-windup implementation is always needed.

4) MPC

In recent years MPC [263] has emerged as the most promising approach to glucose control. The main ingredients of MPC are: the model, the cost function, and the constraints. The model is needed in order to be able to predict the future states and outputs of the system as a function of the current state, future values of the manipulated variables, and future values of measurable or predictable disturbances (Fig. 31). It can be linear or nonlinear, continuous-time or discrete-time, state-space or input-output, black-box, grey-box, or white-box. The cost function measures the quality of closed-loop control. Usually, but not necessarily, it is a quadratic penalty

Fig. 31
MPC prediction scheme: given the model, past inputs and outputs, the future outputs are predicted as a function of future inputs.

J(k)=i=0N(q(yo(k+i)y(k+i))2+(uo(k+i)u(k+i))2)
(49)

on future deviations (whose number is a design parameter called prediction horizon) of the output from the setpoint y0 and may include also a quadratic penalty on future control actions that can be the difference of the input u with respect to a reference u0 or the variation along the time Δu = u(k) − u(k − 1). Finally, there may be constraints on the manipulated variables (insulin delivery rates by the pump is greater than zero and less than some maximal value)

uminuumax
(50)

ΔuminΔuΔumax
(51)

and also on the controlled ones (glycemia in the admissible range)

yminyymax.
(52)

The rationale behind MPC is rather simple: at each time point we compute the sequence of future input moves optimizing the cost function subject to the constraints. Then, only the first control move is applied. At the next step, the procedure is repeated by translating the predictions and control horizons: an optimization is again performed and only the first input move is applied. The principal merit of MPC is that it reduces the control design problem to a sequence of finite-horizon optimization problems, which makes it possible to deal with nonlinear system models, input and output constraints, multiple inputs and outputs, and possible knowledge regarding the dynamics of disturbances. Finally, the tuning of the regulator can follow a rather straightforward trial-and-error procedure: if the control action is sluggish, it suffices to adjust the cost function increasing the weight on the controlled output, i.e., glycemia. In the simplest implementation, after finding reasonable values for the control and prediction horizons (e.g., a unique value covering the duration of typical post-meal transients), the tuning reduces to the calibration of a scalar aggressiveness parameter [270]. As already mentioned, the core of MPC is a mathematical model able to predict the future evolution of the system under control. In fact, different MPC schemes are obtained depending on the nature and complexity of the adopted model.

Nonlinear Model Predictive Control (NMPC) assumes a nonlinear patient model and keeps into account input and output constraints. In this case, explicit solution of the finite-horizon optimization problem does not exist, and the price to pay for an accurate description of the nonlinear dynamics is the need for online iterative optimization within the algorithm. This may preclude the adoption of NMPC within commercial devices, for both engineering and regulatory reasons. Moreover, standard NMPC algorithms assume knowledge of the current state variables, so that it is necessary to include also a state observer or a Kalman filter whose design and tuning may not be straightforward. A final problem is the difficulty of obtaining reliable individual models of insulin-glucose dynamics, as interindividual variability may hinder the use of an average model.

NMPC is however of particular interest as a touchstone for other simpler MPC schemes. A study has indeed demonstrated a distinct improvement over linear MPC [141] on the average in silico patient of the insulin-glucose in silico model. Experiments on real patients using NMPC have also been performed [136], [264], [265]. The need for an individual model has been overcome by online recursive identification of model parameters within a Bayesian setting. Given that experimental data alone may not guarantee parameter identifiability, the Bayesian priors play a key role. Moreover, a formal demonstration of stability and robustness properties of the closed-loop system is obstructed by the complex interplay between on line recursive identification and nonlinear control.

Linear Model Predictive Control (LMPC) uses an approximate linear model of the insulin-glucose dynamics, which produces a substantial algorithmic simplification. The linear model can have different sources. For instance, it may be obtained from the linearization of a more complex nonlinear average patient model around a suitable working point; however such an approach would suffer from the lack of individualization. To overcome this limitation, one possibility is to resort to black-box identification of an individual patient model from data collected on the same individual subject to conventional therapy. In practice, time series of insulin, CGM data, and meal information are used to identify an ARMAX model with two inputs (sc insulin and meals) and one output (sc glucose). A necessary condition to ensure good identifiability properties of ARMAX models is the so-called persistent excitation property of input signals, which should not be collinear between each other and whose spectrum should excite an adequate number of frequencies. Unfortunately, the meals and insulin boluses of the conventional therapy turn out to be collinear and, due to this lack of excitation, the identification algorithm may even fail to correctly estimate the sign of the gains from insulin and meals to sc glucose [266]. As a remedy, it has been proposed to use split and/or delayed insulin boluses to improve the joint excitation properties of the inputs [254], [266] (see also [267] for further optimal design issues in the identification of insulin-glucose models).

A remarkable property of constrained LMPC is the existence of a closed-form solution under the form of a piecewise constant control law that can be computed off line, e.g., in [268] an application to simulated patients is reported. The main drawback of this LMPC scheme, that goes under the name of explicit multiparametric MPC, is the need of finding the appropriate control gain for the current state, a task that involves searching among a potentially very high number of regions in the state space. This can be avoided by computing on line the optimal solution via quadratic programming methods [258], which could be a computationally advantageous alternative to explicit multiparametric MPC. It is worth noting the possibility of including IOB among the constraints of the optimization problem, to embed certain safety constraints already within the real-time control module [258], [269].

It is unlikely that the first commercial realizations of an artificial pancreas will include overly complex computational algorithms. This motivates the interest for the simplest possible LMPC scheme, which is input–output unconstrained LMPC. This particular MPC scheme neglects constraints and uses a linear model in input–output form, e.g., an ARX- or ARMAX-type model. There is no need for online optimization algorithms, because the unconstrained optimization problem admits a closed-form solution that calculates the insulin rate as a simple linear combination of previous insulin rates, previous CGM values, and meal amounts (when known). The use of an input–output model, instead of state-space model, alleviates the need for a state observer, which is a further simplification. A clinical trial on 20 patients has been recently carried out using unconstrained LMPC [255]–[257], showing a five-fold reduction of nocturnal hypoglycemia episodes and an improvement of overnight percent time within the target range of 70–140 mg/dl, with respect to conventional open-loop control [255] (Fig. 32). The LMPC controller was based on an average patient model, but the regulator aggressiveness was personalized keeping into account easy-to-measure individual clinical parameters. To improve breakfast regulation, which was slightly worse than open-loop, an unconstrained LMPC with feedforward action has been proposed [249]. Generalized Predictive Control (GPC), a classic type of input–output unconstrained LMPC, has been experimented with on diabetic swine using both insulin and glucagon as inputs. A first trial used an adaptive GPC (i.e., with model parameters adapted on line) employing insulin or glucagon depending on the sign of the difference between measured glycemia and its target value [271]. In a second trial, GPC was used to compute the insulin rate while, to speed up glucagon administration, the regulation of this second input was decided by a PD controller [272]. The main drawbacks of unconstrained LMPC are the use of a simplified linear model and the neglect of constraints, especially the one on minimal admissible glycemia. However, these drawbacks may be compensated by the advantage of a very simple implementation, especially if this real-time controller is part of a modular architecture including a safety module responsible for hyperglycemia prevention.

Fig. 32
Results of a recent clinical trial that compared conventional open-loop therapy to closed-loop glucose control using Linear Model Predictive Control. Closed-loop control achieved an increase of overnight percent time within the target range and an almost ...

5) Real-Time Detection and Estimation

As already mentioned, controllers based on state feedback require the knowledge of the state of the insulin-glucose system. Even for the simplest models, the available measurements do not give access to the full state vector, so that real time algorithms for state observation are needed (see [143], [273], and [292]). State estimation is important not only for control properties, but can also be easily extended to prediction. In particular, it can be used to predict hypoglycemia [274], [275] and possibly trigger safety actions such as pump shutoff or attenuation. Nevertheless, insofar as a state observer or a Kalman filter rely heavily on a complex model not easily personalized to the specific patient, caution should be used in their use as safety algorithms.

The knowledge of meal time and amount plays an important role in that it can substantially help choosing the correct insulin profile for restoring glycemic control. An ideal artificial pancreas would be completely automatic, dispensing the patient from providing meal information to the device. In this context, a meal detection and estimation module [258], [276], [277] could greatly help the controller reacting properly to the glucose rise following a meal. However, in an s.c.-s.c. glucose control system, the delay introduced by meal detection and estimation would add to the intrinsic delays in the sc-to-sc route, thus degrading the effectiveness of meal compensation compared to the use of an appropriate premeal bolus. In any case, even if the patient were asked to provide meal confirmation to the control device, it would still be necessary to recover from a user’s error by giving the device the capability to automatically detect unannounced meals.

E. Strategies for Control Tuning

1) Measuring Control Performance on Population

Although several classic control metrics exist for assessing the quality of glycemic control [278], some additional issues emerge. In particular, it does not suffice to develop an algorithm that performs satisfactorily on a single subject either real or simulated. As a matter of fact, given the great interindividual variability, it is of paramount importance to guarantee satisfactory performance on the entire population of patients. This motivated the introduction of the so-called Control Variability Grid Analysis (CVGA) [279], which associates to each patient a point in a plane (Fig. 33). The two coordinates correspond, via a nonlinear transformation, to the minimal and maximal glucose value in the considered time interval. The lower left corner is associated with ideal glycemic control while high x-values correspond to hypoglycemic episodes and high y-values to hyperglycemic episodes. The plane is partitioned into nine regions corresponding to different levels of glycemic control quality, from A (best) to E (worst). In this way, the results from a real or simulated trial can be visualized by plotting the patients as a cloud of points onto the CVGA and summarized by counting the percentage of points in the nine regions. Using the CVGA, comparison with either conventional open-loop control or another closed-loop controller is immediate. Of course, a good controller will bring as many patients as possible in the A and B regions. Preliminary to a clinical study, the controller can be applied to an in silico population, representative of the real population, and the performance assessed on the CVGA. If they are unsatisfactory, the controller can be modified and tested again by repeating the in silico trial. The procedure is iterated until glycemic control is acceptable for all the virtual patients.

Fig. 33
An example of CVGA plot. Each patient is represented by a point in the CVGA plane whose coordinates correspond to the minimal and maximal glycemia reached during the monitored time period (note that the axis of the minimal glycemia is reverted). Regulation ...

2) Robustness versus Personalization of Controller Parameters

The ideal artificial pancreas should perform safely and satisfactorily in all patients. To achieve this, the designed controller should be highly robust against uncertainties in the system dynamics, either due to interindividual or intraindividual variability. Experiments conducted in in silico patients have shown that, a fixed controller yields largely disparate performances when applied to different patients; hence, there is need for a personalization of the control algorithm. Given the difficulty of identifying accurate individual models, the direct tuning of controller parameters has been investigated on the basis of few biometric and clinical parameters (body weight, total daily insulin, basal insulin delivery, carb ratio, etc.) characterizing the physiology of each individual. This can be done for both PID regulators [261] and MPC ones [143], [255]–[257]. In particular, for MPC the relation between biometric and physiological patient’s parameters and the optimal scalar aggressiveness parameter of the controller can be determined through a sequence of in silico trials [255]–[257].

3) Run-to-Run Control and Behavioral Analysis

An offline module may be in charge of adapting the control strategy on a daily or weekly basis through the monitoring of the outcomes achieved by the real-time control module. This corresponds to a further closed-loop working on a coarser time scale. This type of problem, called run-to-run (R2R) control, has been extensively studied in the control of chemical and manufacturing processes [293]. The rationale of R2R control is rather simple: the parameter to be adjusted is corrected on the basis of the outcome of the last run. Proportional and proportional-integral control schemes are the most widely used ones. The first applications to glucose control regarded the iterative adjustment of the basal and boluses forming the conventional open-loop therapy [280]–[282]. It goes without saying that if the glucose controller includes a feedforward action, it may still benefit from this kind of R2R control. More recently, R2R control has been applied also to the tuning of the controller parameters [254] where adjustment of the controller aggressiveness is considered. An iterative tuning based on the last 24 hours may also be performed continuously via iterative learning control techniques [283].

Finally, stochastic models of patient’s behavior may be very helpful in order to design and recursively update the parameters (basal and boluses) of the conventional therapy. A formal stochastic model of the process of self-treatment in diabetes (e.g., regular meals, exercise, as well as random treatment deviations) and its potential to generate system challenges can be very useful in that regard, giving a probabilistic interpretation of the observed patterns [294], [295]. Specifically, behavioral self-regulation of diabetes can be approximated by a periodic renewal process that has a significant random component. Such a periodic pattern causes downward and upward BG shifts that, with certain probability, can result in extreme events, such as severe hypoglycemia or major hyperglycemia. The parameters of this process are individual, contingent on behavioral interpretation (a person’s ability to control his/her BG within near-normal limits) and physiology (e.g., a hypoglycemic episode would increase the risk for recurrent hypoglycemia). It has been shown that such an approach is capable of quantifying patterns of diabetes self-management [296] and is particularly useful for understanding the causes of severe hypoglycemia. These studies have demonstrated a quantifiable relationship between stochastic patterns of self-treatment behavior and subsequent occurrence of hypoglycemic episodes [297], [298]. Integrating behavior and physiology into a common framework has also enabled the computer simulation of patterns related to counteregulatory depletion, HAAF, and recurrent hypoglycemia [299].

VII. Conclusion

Approached from a biomedical engineering point of view, the bio-behavioral control of insulin-dependent diabetes (type 1 or insulin-treated type 2) is therefore comprised of: 1) physiologic processes depending on a person’s metabolic parameters such as insulin sensitivity and counterregulation, which could suffer from occasional depletion of counterregulatory reserves occurring with repeated hypoglycemia, and 2) behaviorally triggered processes of glucose fluctuations (e.g., regular postprandial glucose excursions) interrupted by generally random hypoglycemia-triggering behavioral events (e.g., insulin overdose, missed food, or excessive exercise). Fig. 34 presents the dual-layer structure of the engineering understanding of diabetes and places in context the unique combination of mathematical modeling, signal processing, and optimal control reviewed in this contribution. Specifically, Layer 1 represents the puzzle of physiological and behavioral interactions predetermining the specific parameters of each individual with diabetes. Each puzzle piece represents a specific subsystem pertinent to glycemic control: the glucose system, reflecting the appearance and the elimination of exogenous or endogenous carbohydrates, the insulin system that controls the dynamics of insulin-mediated carbohydrate metabolism, the counterregulatory system which is of enormous relevance to diabetes and is presented by hypoglycemia—the primary obstacle to diabetes control, and the patient behavior, which serves as a generator of events (e.g., meals, physical activity, human errors) resulting in metabolic perturbations. Completing the puzzle, the processing of these perturbations depends on individual physiologic parameters of glucose appearance, insulin sensitivity, and counterregulation. The interplay between behavior and biology results in glucose variability, which is the primary observable signal for optimal diabetes control. Layer 2 represents the engineering tools available to contemporary diabetes control. We have discussed several types of mathematical models which over the past 40–50 years have become increasingly elaborate, with recent trends also moving from single individual to population [300] by describing inter-subject variability in a stochastic framework where individual data and anthropometric and metabolic characteristics are explicitly taken into account [301]–[304]. A milestone in this line of research was the introduction of the minimal modeling concept, which established in 1979, the framework for a host of subsequent studies of the human glucose metabolism. A more contemporary milestone has been achieved in 2008, when the first model-based computer simulator of the human metabolic system was accepted as a viable tool for the preclinical testing of control algorithms, essentially alleviating the need for costly and time-consuming animal trials. The progress in modeling was supported by other technological developments, most importantly by the advent of continuous glucose monitoring, which provided detailed signals that could be used as inputs for optimizing glucose control. Finally, a number of control algorithms have been used and are recently under development with the goal to assist, and ultimately automate, the glycemic control in diabetes.

Fig. 34
Dual-layer bio-behavioral structure of diabetes modeling and control: Layer 1 includes the puzzle of physiologic and behavioral characteristics that determine the specifics of each individual. Layer 2 includes the engineering approaches available to support ...

We can therefore conclude that the formal understanding and description of glucose-insulin metabolism in health and diabetes is, arguably, one of the most advanced applications of biomedical engineering to the life sciences. A rich background exists of models, metrics, and algorithms, and this contribution attempts to provide a systematic review of a number of them. We have to admit, however, that many more metabolic models exist that fall out of the scope of this review. This is because our goal is to follow a specific line of research that led from the first comprehensive model of glucose-insulin dynamics, through detailed simulation of the human physiology, to the first attempts for automated closed-loop control. Future reviews will therefore discuss problems not addressed by this manuscript.

Fig. 19
Disposition index paradigm. Left panel: a normal individual could be represent by state I; if beta-cells respond to a decrease in insulin sensitivity by adequately increasing insulin secretion (state II) the product of beta-cell function and insulin sensitivity ...

Acknowledgments

The authors thank Dr. A. Vella and Dr. A. Basu, Mayo Clinic, Rochester, MN; Dr. A. Avogaro, Dr. G. Toffolo, Dr. A. Bertoldo, and Dr. A. Facchinetti, University of Padova, Padova, Italy; Dr. L. Farhi, Dr. M. Breton, and Dr. S. Patek, University of Virginia, Charlottesville, VA, for their contributions to this review.

Biographies

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Claudio Cobelli (M’84–SM’90–F’01) was born in Bressanone (Bolzano), Italy, on February 21, 1946. He received the Ph.D. degree (Laurea) in electrical engineering from the University of Padova, Padova, Italy, in 1970.

From 1970 to 1980, he was a Research Fellow of the Institute of System Science and Biomedical Engineering, National Research Council, Padova. From 1973 to 1975 and 1975 to 1981, he was an Associate Professor of biological systems at the University of Florence and Associate Professor of biomedical engineering at the University of Padova, respectively. In 1981, he became a Full Professor of biomedical engineering at the University of Padova. Since 2000, he has been an Affiliate Professor of bioengineering at the University of Washington, Seattle. Since 2000, he has been Chairman of the Graduate and Ph.D. Program on bioengineering at the University of Padova. His main research activity is in the field of modeling and identification of physiological systems, especially endocrine-metabolic systems. He has published around 280 papers in internationally refereed journals. He is Coeditor of Carbohydrate Metabolism: Quantitative Physiology and Mathematical Modeling (Chichester: Wiley, 1981), Modeling and Control of Biomedical Systems (Oxford: Pergamon, 1989), and Modeling Methodology for Physiology and Medicine (New York: Academic, 2000). He is Coauthor of The Mathematical Modeling of Metabolic and Endocrine Systems (New York: Wiley, 1983); Tracer Kinetics in Biomedical Research: from Data to Model (London: Kluwer Academic/Plenum, 2001) and Introduction to Modeling in Physiology and Medicine (San Diego: Academic, 2008).

Dr. Cobelli is currently an Associate Editor of IEEE Transactions on Biomedical Engineering and Diabetes. He is on the Editorial Board of the American Journal of Physiology: Endocrinology and Metabolism, Diabetes, and Journal of Diabetes Science & Technology. In the past he has been Associate Editor of Mathematical Biosciences and on the Editorial Board of Control Engineering Practice, Diabetes, Nutrition and Metabolism, Diabetologia, and American Journal of Physiology: Modeling in Physiology. He has been Chairman (1999–2004) of the Italian Biomedical Engineering Group and has been Chairman (1990–1993 and 1993–1996) of IFAC TC on Modeling and Control of Biomedical Systems. He is Fellow of BMES.

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Chiara Dalla Man was born in Venice, Italy, on March 2, 1977. She received the Ph.D. degree (Laurea) cum laude in electronics engineering from the University of Padova, Padova, Italy, in 2000. She also received the Ph.D degree in biomedical engineering from the University of Padova, and City University London, U.K., in 2005.

From March to September 2004, she was a Visiting Ph.D. Student at the Centre of Measurements and Information in Medicine, City University London, U.K. From 2005 to 2007, she was a Post-Doctoral Research Fellow with the Department of Information Engineering of Padova University. Since October 2007, she has been an Assistant Professor in the Faculty of Engineering of Padova University. Her research interests include the field of mathematical modeling of metabolic and endocrine systems.

Dr. Dalla Man is on the Editorial Board of Journal of Diabetes Science and Technology.

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Giovanni Sparacino was born in Pordenone, Italy, on November 11, 1967. He received the Doctoral degree in electronics engineering cum laude from the University of Padua, Padua, Italy, in 1992, and the Ph.D. degree in biomedical engineering from the Polytechnic of Milan, Milan, Italy, in 1996.

Since 1997, he has been with the University of Padua: from 1997 to 1998, he was a Research Engineer at the Faculty of Medicine; from 1999 to 2004, he was an Assistant Professor at the Faculty of Engineering; since 2005, he has been an Associate Professor of biomedical engineering at the Faculty of Engineering. His scientific interests include deconvolution and parameter estimation techniques for the study of physiological systems, hormone time-series analysis, continuous glucose monitoring, and measurement and processing of evoked potentials.

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Lalo Magni was born in Bormio, Italy, in 1971. He graduated with full marks and honors (summa cum laude) in computer engineering from the University of Pavia, Pavia, Italy, in 1994. He received the Ph.D. degree in electronic and computer engineering in 1998.

From January 1999 to December 2004, he was an Assistant Professor at the University of Pavia, where he has been an Associate Professor since January 2005. From October 1996 to February 1997 and in March 1998, he was at CESAME, Universitè Catholique de Louvain, Louvain La Neuve, Belgium. From October to November 1997, he was at the University of Twente with the System and Control Group in the Faculty of Applied Mathematics. His current research interests include nonlinear control, predictive control, robust control, process control and glucose concentration control in subjects with diabetes. His research is witnessed by more than 40 papers published in international journals.

Dr. Magni was a Plenary Speaker at the 2nd IFAC Conference “Control Systems Design” (CSD’03), in 2003. In 2005, he was a Keynote Speaker at the NMPC Workshop on Assessment and Future Direction. In 2003, he was a Guest Editor of the Special Issue “Control of Nonlinear Systems with Model Predictive Control” in the International Journal of Robust and Nonlinear Control. He served as an Associate Editor of the IEEE Transactions on Automatic Control. He is an Associate Editor of Automatica. He was subarea Chair for the area “Nonlinear systems optimal and predictive control” at the IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007). He organized the NMPC Workshop on Assessment and Future Direction in September 2008 in Pavia.

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Giuseppe De Nicolao (SM’01) received the degree in electronic engineering from the Polytechnic of Milan, Italy.

From 1987 to 1988, he was with the Biomathematics and Biostatistics Unit of the Institute of Pharmacological Researches “Mario Negri”, Milano. In 1988, he joined the Italian National Research Council (CNR) as a Research Scientist at the Center of System Theory in Milan, Italy. From 1992 to 2000, he was an Associate Professor and, since 2000, he has been a full Professor of model identification in the Department of Computer Science and Systems Engineering of the University of Pavia, Pavia, Italy. In 1991, he held a visiting fellowship at the Department of Systems Engineering of the Australian National University, Canberra. His research interests include Bayesian learning, neural networks, model predictive control, optimal and robust filtering and control, deconvolution techniques, modeling, identification and control of biomedical systems, advanced process control and fault diagnosis for semiconductor manufacturing. On these subjects he has authored or coauthored more than 100 journal papers and is coinventor of two patents.

Dr. De Nicolao was a Keynote Speaker at the IFAC workshop on “Nonlinear model predictive control: Assessment and future directions for research”. From 1999 to 2001, he was an Associate Editor of the IEEE Transactions on Automatic Control and, since 2007, he has been an Associate Editor of Automatica.

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Boris P. Kovatchev received the Ph.D. degree in mathematics (probability and statistics) from Sofia University “St. Kliment Ohridski,” Bulgaria, in 1989.

Currently, he is a Professor in the Department of Psychiatry and Neurobehavioral Sciences and Adjunct Professor of Systems and Information Engineering, University of Virginia, Charlottesville. He is Head of Section Computational Neuroscience and Director of the University of Virginia Diabetes Technology Program. His research expertise is in biomathematics, specifically modeling of biologic and behavioral processes. In the past 15 years he has been involved in various aspects of diabetes technology development, as well as in the development of quantitative strategies for neurobiological problems. Currently, he is the Principal Investigator of two large projects funded by the National Institutes of Health, and the Principal Investigator of the JDRF Artificial Pancreas Project at the University of Virginia. He is also involved in industry-sponsored translational research. He is author of over 100 scientific publications and coauthor of the textbook Invitation to Biomathematics (Academic, 2008). His academic work includes participation in several international boards and NIH study sections. He holds five patents and is author of 15 other inventions that are currently at various stages of the patenting process.

Dr. Kovatchev is an Associate Editor of IEEE Transactions on Biomedical Engineering and member of the Editorial board of the Journal of Diabetes Science and Technology.

Contributor Information

Claudio Cobelli, Department of Information Engineering, University of Padova, Via Gradenigo 6B, 35131 Padova, Italy.

Chiara Dalla Man, Department of Information Engineering, University of Padova, Via Gradenigo 6B, 35131 Padova, Italy.

Giovanni Sparacino, Department of Information Engineering, University of Padova, Via Gradenigo 6B, 35131 Padova, Italy.

Lalo Magni, Department of Computer Engineering and Systems Science, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy.

Giuseppe De Nicolao, Department of Computer Engineering and Systems Science, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy.

Boris P. Kovatchev, Department of Psychiatry and Neurobehavioral Sciences, P.O. Box 40888, University of Virginia, Charlottesville, VA 22903 USA.

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