Advancing our understanding of the glucose system via modeling: a perspective.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 5, MAY 2014

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Advancing Our Understanding of the Glucose System via Modeling: A Perspective Claudio Cobelli∗ , Fellow, IEEE, Chiara Dalla Man, Morten Gram Pedersen, Alessandra Bertoldo, and Gianna Toffolo

Abstract—The glucose story begins with Claude Bernard’s discovery of glycogen and milieu interieur, continued with Banting’s and Best’s discovery of insulin and with Rudolf Schoenheimer’s paradigm of dynamic body constituents. Tracers and compartmental models allowed moving to the first quantitative pictures of the system and stimulated important developments in terms of modeling methodology. Three classes of multiscale models, models to measure, models to simulate, and models to control the glucose system, are reviewed in their historical development with an eye to the future. Index Terms—Artificial pancreas, diabetes, glucose control, insulin action, insulin secretion, minimal models, simulation models. Fig. 1.

Scheme of the glucose–insulin system.

I. INTRODUCTION HE glucose system has received considerable attention in the last decades due to the diabetes pandemia [1]. Given the complexity of the disease it is not surprising that it is fought with a battery of tools spanning over several disciplines, from biology to pathophysiology to pharmacology to chemistry, physics and engineering, to transplantation to health care. Biomedical engineering has allowed important advancements in the field, particularly in four areas: technology, mathematical modeling, signal processing, and control. Here, we review the role of dynamic system modeling in the quantitative understanding of the glucose system and its progressive pathophysiological derangement from prediabetes to type 2 or type 1 diabetes. Models to measure and to simulate, both at whole-body and organ/tissue levels, will be reviewed. Given the nature of this issue of IEEE BME Transactions and the vastness of the glucose system literature, we will provide here a personal story on major advancements of our understanding of the glucose system via modeling in a past–present–future perspective. The story is supported by numerous references but we have to refer the reader to the relevant literature for additional models, techniques, and results on model-based pathophysiological studies.

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Manuscript received September 26, 2013; revised December 9, 2013 and January 7, 2014; accepted January 8, 2014. Date of publication March 11, 2014; date of current version April 17, 2014. This work was supported in part by the FP7 EU Project AP@home, Italian Ministero dell’Univerisità e della Ricerca project FIRB 2008, Juvenile Diabetes Research Foundation Simulation Core Facility Grant. Asterisk indicates corresponding author. ∗ C. Cobelli is with the Department of Information Engineering, University of Padova, I-35131 Padova, Italy (e-mail: [email protected]). C. Dalla Man, M. G. Pedersen, A. Bertoldo, and G. Toffolo are with the Department of Information Engineering, University of Padova, I-35131 Padova, Italy (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2014.2310514

II. GLUCOSE SYSTEM Before diving into some history, it may help to refresh some fundamentals on the glucose system. Glucose concentration is tightly regulated in health by a complex neurohormonal control system, which involves several hormones like insulin, glucagon, epinephrine, cortisol, and growth hormone. Here focus is on insulin control where modeling has had major impact. A highlevel scheme is shown in Fig. 1. Glucose is produced (mainly by the liver), distributed, and utilized in both insulin-independent (e.g., central nervous system and red blood cells) and insulindependent (muscle and adipose tissues). Insulin is secreted by pancreatic beta-cells, reaches the system circulation after partial liver degradation, and is peripherally cleared primarily by the kidneys. The glucose and insulin systems interact by feedback control signals, e.g., if plasma glucose concentration increases (after a meal), beta-cells secrete more insulin 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. In pathophysiology, the control is degraded, in prediabetes and type 2 diabetes due to a progressive deterioration of both insulin sensitivity and beta-cell responsivity, in type 1 diabetes due to beta-cell destruction, so that insulin must be provided exogenously to compensate for hyperglycemia. People with type 1 diabetes face a life-long behaviorcontrolled optimization problem: to maintain strict glycemic control and reduce hyperglycemia, without increasing their risk for hypoglycemia. III. MILIEU INTERIEUR, THE DISCOVERY OF INSULIN AND THE DYNAMIC STATE OF BODY CONSTITUENTS The story starts with the founding father of modern physiology, Claude Bernard for whom 2013 has marked the 200th birthday. When Claude Bernard started exploring metabolism and diabetes, various hypotheses were circulating in the medical community. One of them was that glucose was transported by

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the lymphatic system into the blood and burned there. Some assumed that the lung was the place where this “burning” occurred. He summarized his findings in [2] where he states: “Normally there is always sugar in the blood of the heart and the liver. The sugar is formed by the liver; this is independent of the nutrition with sugar or carbohydrates.” Now another question had to be answered in what form is the sugar stored in the liver? After many experiments, in February 1855, Bernard isolated glycogen. Another key contribution was [3] where he first describes the “milieu intérieur.” He writes “I think I was the first to express the idea that for animals there are in fact two environments, one milieu which is outside the body and an inner milieu, in which the components of living tissues are embedded. The real existence of the animal does not take place in the external world but inside the liquid medium of circulating organic fluid. This fluid is the expression of all local nutrition and the source and mouth of elementary exchange” [4]. The discovery of insulin by Banting and Best in 1922 [5], which allowed to save the life of people with type 1 diabetes, was the next breakthrough in the understanding the glucose system; however, it took 40 years to be able to measure insulin concentration with radioimmunoassay methods in the circulation [6]. Similarly glucagon and epinephrine, the counterregulatory hormones, were discovered in 1923 and 1924, respectively, but it became possible to measure their concentrations in plasma only in 1959 and 1968, respectively. Another conceptual move forward in the characterization of the milieu intèrieur was allowed by the introduction of isotopes to trace the movement of substances (tracee) in the body: R. Schoenheimer in 1942, after some revolutionary studies involving the use of stable and radioactive tracers on intermediatebrk metabolism, formulated in a famous book [7] the concept of “the dynamic state of body constituents” by which at any time the concentration of a substance in the circulation, e.g., of a substrate or a hormone, is the result of production/secretion, distribution, exchange with other body pools, and utilization/degradation. IV. TRACERS AND COMPARTMENTAL MODELS The dynamic state of body constituents was a qualitative paradigm and the quantitative conversion of tracer and tracee levels in the circulation (accessible to measurement) into tracee fluxes of production, distribution, and metabolism (not accessible) was a difficult problem, especially in vivo. There was the need to develop dynamic differential equation models of the system, able to interpret the plasma measurements, and thus to tackle problems like model structure determination, model identification, and validation. Studies employing radioactive glucose tracers increased in the 1940s, and especially after World War 2 when radioactive isotopes became commercially available (it took another 30 years to see the first glucose stable isotope tracer study in children [8]). The increased number of animal and human tracer studies stimulated the development and theoretical formalization of modeling methodologies and approaches. In 1948, Sheppard introduced for the first time the term compartment, albeit it was implicitly there in some earlier studies on tracer [9] and drug [10], [11] kinetics, and provided the first

Fig. 2. Generic n-compartment model. Compartments 1, i, j, and n are evidenced with r S i denoting the rth species in which substance S is present in compartment i, r ai the corresponding specific activity, i.e., the ratio of tracer to tracee, and ρi j the flux of the substance from i to j [12].

multicompartment model of tracer kinetics in a steady-state tracee system described by a system of linear time-invariant differential equations [12] (see Fig. 2). V. FROM ANALOG TO DIGITAL COMPUTERS Handling linear differential equation models in the 1950s was computationally challenging. The analytic solution (sum of exponentials) was feasible for the two- and some threecompartment model structures but pen and pencil modeling was soon realized not to be a viable approach—sometimes even transcending the ability or the patience of the investigator— given the complexity of the problem. Popular alternatives in the 1950s were analog models, e.g., hydrodynamic and electrical models [13], [14] (see Fig. 3). A significant step forward was made possible in the 1960s by the introduction of analog computers, ranging from simple to complex configurations, with, respectively, tens to hundreds operational amplifiers. Simulation was relatively easy with the analog computers, while parameter estimation was hard to perform. The first digital computers were becoming available around this time. In the 1962 book by Sheppard [15] on compartmental models—the first published—there was, in addition to Chapter 6 on analog simulation of compartmental models, Chapter 7 (pp. 124–151) devoted to their numerical solution. In the Preface Sheppard noted “In the mathematical field of digital computation methods, we must continue to be prepared for rapid obsolescence. As one example, during the preparation of Chapter 7, a revised FOR TRANSIT system appeared, and the programming instructions had to be altered accordingly. The IBM 650 may soon cease to be the machine of choice, but it would seem that, for more modern IBM machines at least, some of the basic principles will carry into the new systems.” New momentum in

COBELLI et al.: ADVANCING OUR UNDERSTANDING OF THE GLUCOSE SYSTEM VIA MODELING: A PERSPECTIVE

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Fig. 4. Compartmental model describing a physiological control system, e.g., a substrate–hormone system. In addition to material fluxes in both the substrate and hormone subsystem (continuous lines), control signals (dashed lines) from substrate to hormone and hormone to substrate subsystems are described.

Fig. 3.

Hydraulic [13] (upper) and electric [14] (lower panel) models.

the use of digital computers for modeling the glucose, and in general, metabolic systems was brought by Mones Berman at NIH, Bethesda, MD, USA, [16]. The debate between numerical versus analogic was vivid on one hand, Berman wrote [16], “The disadvantages of a conventional analog computer are that 1) it does not provide a measure of a “good” fit, 2) it does not provide a systematic way for adjusting parameter values to improve a fit, and 3) it has no provision for indicating uniqueness of fit or uncertainties in the values of the parameters”; on the other side, Ackerman et al., in a review paper [17] argued that the only advantage of digital computing consists in the possibility of automatically comparing the simulated curve with the experimental data. The digital path was set and in a footnote of a paper, Berman [18] announced the software SAAM (Simulation, Analyses, and Modeling)—approximately 150 subroutines and 10 000 lines of FORTRAN for computers with 32 K memory—entered the public domain in 1967 [18]. VI. MODELING METHODOLOGY The 1950/1960s saw some important methodological contributions. Berman and Schoenfeld [19] addressed for the first time the a priori identifiability problem for linear compartmental models—they did not use the term explicitly and discussed the problem of information content of data versus model structure. An important theoretical contribution was that of Hearon [20] who defined the structural stability properties of linear compartmental models. Tracer theory was extended to study tracee systems also in nonsteady state, i.e., after a perturbation like a meal,

physical activity or a drug: tracer kinetics is still described by linear differential equations but parameters become time-variant (as a result of nonlinearity) [21]. Compartmental models moved out of the tracer context and nonlinear compartmental models were formalized to describe physiological control systems, e.g., production, distribution, utilization of glucose, secretion, distribution, degradation of insulin, and the feedback glucose and insulin signals (see Fig. 4). Later, in the 1970/1980s, the methodological problems posed by linear, but also nonlinear, compartmental models saw a new cultural wave: the physicist approach to problems was substituted by a bioengineering and system and control approach. The identifiably problem was first posed by Bellman and Astrom [22] and subsequently attacked by various investigators (see the review by Cobelli and Di Stefano [23]). The stability properties of nonlinear compartmental models were investigated [24], [25]. The numerical identification of models was posed in the correct theoretical setting with tools including test of residuals, parameter precision, and parsimony criteria (see the review [26]), the optimal design of an identification experiment was tackled ([27]), and a model validation methodology established [28], in particular testing the model-derived measures against those provided by independent techniques. Books were published offering a consolidated methodology for modeling endocrine and metabolic systems [21], [29]. The methodological advancements largely merged also into new software, e.g., the conversational versions of SAAM, CONSAM [30], and SAAMII [31], and ADAPT [32], which had a more general breadth by including, in addition to the classical least squares parameter estimation, maximum likelihood, and Bayesian methods. In the following years, the identifiability problem was tackled also for nonlinear models with new methods, based on concepts of differential algebra [33], [34], and specific software was made available [35]. Also new books were published [36], [37]. Of note has been the increasing use of Bayesian methods given the increased a priori knowledge that has become available. Finally, small size clinical studies moved from the research center to the field and this required the adoption of new parameter estimation methods. In fact the methods discussed above are applicable to an individual in a data rich (sampling) context (compared to model complexity) and/or in the presence of independent a priori knowledge on model parameters. Population approaches are alternative

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methods developed for estimating model parameters in a data poor (sampling) context but with the number of subjects comparatively large. Among them, the nonlinear mixed-effects modeling approach [38] has become increasingly important, since it is able to compensate the lack of information by borrowing the knowledge spread on the population and include it during the estimation step. This modeling approach has a long-standing tradition in pharmacological studies and it has only recently been applied to the glucose system with definitive benefits [39], [40]. VII. MODELS TO MEASURE—MINIMAL Minimal (coarse-grained) models—as opposed to maximal (fine-grained) discussed later—are parsimonious descriptions of the key components of the system functionality and are capable of measuring crucial processes of glucose metabolism, thus also improving our understanding of the system. A. Glucose and Insulin Fluxes Glucose. Glucose production and utilization vary as an effect of a perturbation, e.g., a meal, due to endocrine and nervous control mechanisms. To circumvent the need of explicitly describing these controls, Steele [41] proposed to use a glucose tracer and interpret the data with a single compartment model with a time-varying parameter, surrogating the system nonlinearities [41], [42]. The model allowed calculating the rate of appearance Ra, and disappearance Rd, of glucose from the mass balance equation. An important contribution was provided by Norwich [42] and Radziuk [43], they proposed an ingenious tracer clamp infusion protocol, which renders the estimation of Ra less model-dependent, i.e., with a perfect clamp, the Ra can be calculated from the tracer infusion rate with an algebraic equation. They also proposed moving to a two compartment time-varying model to make the approach more robust. This new approach was validated in dogs [43], and later put on more solid theoretical grounds [44]. Then, new experimental guidelines for accurate estimation of Ra and Rd were provided [45]. The increased availability and use of stable glucose isotopes has stimulated the generalization to the tracer-to-tracee clamp technique [46]. Today the clamp technique has become a standard to measure glucose fluxes. Depending on the question being asked, both dual or triple tracer protocols are implemented, the rule being that if one is interested in n fluxes it is necessary to use n + 1 tracers [47], e.g., to estimate glucose production and Ra after a meal one has to use three tracers [48], [49]. Insulin. The first attempt to quantitatively assess the insulin secretion profile after a glucose stimulus was posed as a classical input estimation problem for which deconvolution offers the classic solution [50]: from the knowledge of the insulin impulse response (a single compartment is adequate) and plasma insulin concentrations, the rate of insulin delivery can be reconstructed by a rather empirical deconvolution method. A step forward was the method employed in [51], which improved not only the impulse response characterization by using an insulin tracer but also the deconvolution method by employing the regularization techniques published in the mid-1960s by Philipps [52] and Twomey [53].

Fig. 5.

Analog computer implementation of the Bolie model [58].

However, it is not possible to reconstruct pancreatic secretion from plasma insulin concentration since the insulin secreted in the portal vein is degraded by the liver before appearing in the circulation. This problem was bypassed when the hormone C-peptide was discovered: since C-peptide is secreted equimolarly with insulin, but it is extracted by the liver to a negligible extent, deconvolution of plasma C-peptide data provides the time course of pancreatic insulin secretion. Eaton [54] and Polonsky [55] were the first in attacking the problem, although the deconvolution method was less refined than that used by Pilo et al. [51]. The knowledge of the C-peptide impulse response (a two compartment linear model) requires ideally an additional experiment on the subject, but a method was proposed in [56] that allows C-peptide kinetic parameters to be derived in an individual based on subject anthropometric characteristics (age, weight, height, and gender). Nowadays, the state-of-the-art method to estimate pancreatic insulin secretion is to apply the deconvolution technique proposed by [57] to C-peptide data and use the Van Cauter method [54] to individualize C-peptide kinetics. B. Insulin and Glucose Control Insulin. The pioneer was V. Bolie [58] who proposed a linear model to describe the responses of plasma glucose and insulin to an intravenous glucose tolerance test (IVGTT) with glucose disappearance a linear function of both glucose and insulin, insulin secretion proportional to glucose, and hepatic glucose production constant. He also proposed the analog computer implementation of Fig. 5. The model was subsequently extended to an oral glucose tolerance test (OGTT) in [59] to obtain a four-parameter representation of glucose metabolism in various states of glucose intolerance, including diabetes. This linear model was simplistic, but we must consider that at that time insulin was still difficult to measure and the authors fitted the model on plasma glucose data only.


An elegant tracer study with the glucose system at steady state by Insel et al. [60]—at basal glucose and basal insulin, and at basal glucose and elevated insulin, this latter realized by exogenously infusing insulin and glucose—advanced the field by assessing timing and magnitude of insulin action. Linear three-compartment models were used to describe glucose and insulin kinetics, and in order to describe glucose utilization in insulin-dependent tissues, it was necessary to relate the compartmental disappearance rate constant to insulin in a large, slowly equilibrating compartment, thus confirming the finding of a year before by Sherwin et al. [61], who showed that it is not plasma insulin, but insulin in a remote compartment, that controls glucose utilization. In 1979, Bergman and Cobelli [62] introduced the minimal model method to describe an IVGTT, thus arriving at an index of insulin action, called insulin sensitivity, without the use of tracers. The idea was that minimal models must be parsimonious, i.e., they only describe the key components of the system. Desirable features of a minimal model include 1) physiology based; 2) parameters estimated with reasonable precision; 3) parameter values within physiologically plausible ranges; and 4) system dynamics described with the smallest number of identifiable parameters. One generally proceeds by proposing a series of system models, beginning with the simplest and systematically increasing the complexity by including more known physiological details. Each model is first tested for a priori identifiability, subsequently numerically identified from the data, and finally the most parsimonious model is selected by using the identification/validation criteria described before. To facilitate the model selection process, system partition was introduced (see Fig. 6, upper panel). In fact, to describe plasma glucose and insulin data it is necessary to simultaneously model not only the glucose, but also the insulin system and their interactions. This means that, in addition to modeling insulin action, one has also to model glucose-stimulated insulin secretion. Since models are, by definition, wrong, an error in the insulin model would be compensated by an error in the glucose model, thus introducing a bias in insulin sensitivity. To avoid this interference, the dynamic contribution of a subsystem should be eliminated. The authors developed a conceptual “loop cut”: the system is partitioned in two subsystems which are linked together by measured variables, the insulin and the glucose subsystems. When the system is perturbed, e.g., by a glucose injection, and the time courses of plasma glucose and insulin are measured, then their time course can be considered as “input” (assumed known) and “output” (to be fitted) of the insulin and glucose subsystems, respectively. Models are then proposed not for the whole system but for each of the two subsystems, independently, thus considerably reducing the difficulties of modeling. Seven models of increasing complexity were proposed to explain plasma glucose concentration by using plasma insulin as the known input. The chosen minimal model was the nonlinear model shown in Fig. 6 (middle panel, left): it assumes that glucose kinetics can be described with one compartment and that remote (with respect to plasma) insulin controls both net hepatic glucose balance and peripheral glucose disposal. The remote insulin finding [61] was thus

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Fig. 6. Upper panel: Partition of the glucose–insulin system into glucose and insulin subsystems. Middle panels: The glucose (left) and C-peptide (right) IVGTT minimal models. Lower panels: The disposition index paradigm. Left: a normal individual is represented by state I; if beta-cells respond to a decrease in insulin sensitivity by adequately increasing insulin secretion (state II) 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. Right: 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 poor insulin sensitivity; these two individuals need opposite therapy vectors.

confirmed on an independent data set—only later this remote compartment was experimentally proven to be the interstitial fluid [63]. The model provides an index of insulin sensitivity, which has been validated in numerous studies against the independent glucose clamp technique and has been widely employed in more than 1000 papers. This index is essentially a steady-state measure, i.e., it provides the magnitude of insulin sensitivity but does not account for how fast or slow insulin action takes place. A new index, called dynamic insulin sensitivity, has been introduced to incorporate also the timing of insulin action [64]. As shown in [65], glucose kinetics requires at least a twocompartment model. Undermodeling the system during a highly dynamic perturbation like the IVGTT, introduces an underestimation in insulin sensitivity. An improved two compartment glucose minimal model has been proposed [66]: since the added complexity renders the model nonidentifiable, a Bayesian maximum a posteriori (MAP) estimator exploiting a priori knowledge on the two glucose exchange parameters has been used for its identification. The IVGTT establishes glucose and insulin concentrations that are not seen in the normal life, while it would be desirable to measure insulin sensitivity in the presence of physiological conditions, e.g., during a meal (MTT) or OGTT. To this purpose, the oral glucose minimal model has been developed. It

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has a similar structure to the IVGTT model apart from the input: the known injected glucose dose is substituted by a parametric function, e.g., a piecewise linear function, describing the rate of glucose appearance in plasma through the gastrointestinal tract [67]. This added complexity renders the model nonidentifiable, i.e., there is the need to assume some parameter values and to use a MAP estimator. The model provides an index of insulin sensitivity, which has been validated against independent techniques [68], [69]. Also for MTT/OGTT, the dynamic insulin sensitivity index can be calculated [64]. Both the IVGTT and MTT/OGTT minimal models provide a composite measure of insulin action, i.e., the net effect of insulin to inhibit glucose production and stimulate glucose utilization. It is possible to dissect insulin action into its two individual components by adding a glucose tracer to the IVGTT or MTT/OGTT, thanks to the tracer’s ability to separate glucose utilization from production. The labeled IVGTT singlecompartment model came first [70], [71] later improved by a two-compartment version [72]. More recently, a stable labeled MTT/OGTT model was proposed in [73] and subsequently refined in [74]. The indices of disposal and liver insulin sensitivity have been validated against the independent euglycemichyperinsulinemic clamp technique, e.g., for the MTT/OGTT in [69], [75]. Of note is that the combined use of the tracer and tracee models can also provide glucose fluxes, i.e., one can arrive at the flux portrait by using a different experimental/modeling strategy than that described in the Glucose Fluxes section. For instance during MTT, they provide the rate of appearance of glucose, its rate of disappearance and hepatic glucose production, a flux portrait [74] which has been validated against that provided by the tracer clamp technique [47]–[49]. While whole-body models can provide an overall measure of insulin action, it would be important to measure insulin action at the organ/tissue level, e.g., the skeletal muscle, by quantitating the effect of insulin on the individual steps of glucose metabolism, i.e., transport from plasma to interstitium, transport from interstitium to cell, and phosphorylation. Understanding which metabolic step is impaired, e.g., in prediabetes or type 2 diabetes can guide a targeted therapeutic strategy. Direct measurement in vivo of these individual steps is not possible, and two model-based approaches are available both employing tracers with glucose at steady state: the classical multiple tracer dilution technique and the more recent technique based on positron emission tomography (PET). The multitracer dilution technique consists of the simultaneous injection, upstream of the organ, of more than one tracer, which allows separate monitoring of the individual steps of glucose metabolism. Multiple tracer data have been interpreted with both linear distributed and lumped (compartmental) parameter models. The only application of distributed parameter models to glucose metabolism has been in an isolated and perfused heart [76]. In contrast, compartmental models have been intensively applied to interpret multiple tracer dilution data in the human forearm skeletal muscle. First a two-tracer compartmental model was developed to measure transmembrane glucose transport [77], subsequently extended to a three-tracer model


to also measure glucose phosphorylation [78]. These models allowed important pathophysiological findings in diabetes, e.g., they enabled demonstrating that cellular transport plays a very important role in the defective insulin action in diabetes [79]. PET is an imaging technique that allows deriving highly specific and rich biochemical information if applied in dynamic mode, i.e., sequential tissue images acquired following a bolus injection of radiotracer so that the time course of the tissue behavior is monitored. Quantitative PET information can be extracted at whole-organ level (i.e., comparable with the triple tracer technique) as well as at region of interest (i.e., a specific area/volume of the organ) or voxel level. The glucose model of the brain by Sokoloff et al. [80] has been a landmark for quantitative PET metabolic studies. The model, originally proposed for 2-deoxy-D-[14 C]glucose ([14 C]DG), a glucose analog, and quantified in the rat from autoradiography data, was immediately extended to the PET tracer [18 F]fluorodeoxyglucose ([18 F]FDG), another glucose analog. The advantage of using an analog, instead of the ideal [11 C]glucose tracer, is that the end-product of phosphorylation is trapped in the tissue, thus reducing significantly the model complexity; the disadvantage is the necessity to correct for the differences in transport and phosphorylation between the analog and glucose with a correction factor, called lumped constant (LC). The model assumes that the interstitial and intracellular spaces are in equilibrium and has three rate constants: one describing the transport from blood into tissue, a second transport back to blood, and a third intracellular phosphorylation. From the three rate constants one can calculate the fractional uptake of [18 F]FDG in the brain, and, knowing LC and the plasma glucose concentration, the fractional uptake of glucose. This three-rate-constant model (see Fig. 7) has been extensively used not only in brain but also in myocardium [18 F]FDG dynamic PET studies. A more complex five-rate-constant model (see Fig. 7) is needed for studying glucose metabolism in the skeletal muscle [81]. In fact, an additional compartment is needed to account for the difference between arterial and interstitial concentrations, thus introducing the two new rate constants of [18 F]FDG exchange between plasma and extracellular space. Also with this model, by using the skeletal muscle LC, one can derive the glucose fractional uptake. The model has revealed inefficient transport and phosphorylation [18 F]FDG rate constants in obesity and type 2 diabetes, but also the plasticity of the system, i.e., defects can be substantially reversed with weight loss [82]. The LC allows moving from [18 F]FDG to glucose fractional uptake but not to the glucose transport and phosphorylation rate constants. To this end, a multiple tracer approach is needed with three different PET tracers injected sequentially [83]. This multitracer PET imaging method allows quantification of blood flow from [15 O]H2 O images with a one-compartment two-rateconstant model; glucose transport from [11 C]3-OMG images with a three-compartment four-rate-constant model, and, finally, glucose phosphorylation by combining [18 F]FDG fractional uptake with [11 C]3-OMG rate constants. This method has shown that glucose transport from plasma into interstitial space is not affected by insulin while insulin increases both glucose transport and phosphorylation. In addition, the study has


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Fig. 7. Two-tissue compartment three-rate constant model developed by Sokoloff et al. to quantify brain [1 8 F]FDG dynamic images (upper panel) and the three-tissue compartment five-rate constant to quantify skeletal muscle [1 8 F]FDG dynamic images (lower panel).

elucidated that predominately oxidative muscles (soleus) have higher perfusion and higher capacity for glucose phosphorylation than less oxidative muscles (tibialis). A promising imaging approach alternative to PET is an MRI approach, based on the use of the chemical exchange saturation transfer (CEST) which has been shown to enable sensitive brain imaging of the dynamics of 2-deoxy-D-glucose uptake and metabolism without the necessity of any isotopically labeled molecules [84]. Insulin. Deconvolution allows measuring insulin secretion after a glucose stimulus. However, a mechanistic description of pancreatic insulin secretion as a function of plasma glucose concentration has the advantage of providing quantitative indices of beta-cell function. The first models describing insulin during an IVGTT were those of Licko [85] and Hagander [86]. In particular, starting from the cellular model of biphasic insulin secretion by Grodsky discussed later [87], Licko developed a whole-body IVGTT model by making a number of assumptions and proposed for the first time beta-cell function indices, i.e., first- and second-phase responsivities. A few years later, the minimal modeling methodology similar to that employed to derive the IVGTT glucose minimal model was applied to the insulin subsystem, having, with reference to Fig. 6, plasma glucose as the “input” (assumed known), and insulin as the “output” [88]. While models based on insulin data allowed post hepatic insulin delivery to be quantified, an improved beta-cell function parametric portrait was later obtained by models based on C-peptide data. The IVGTT model [89] shown in Fig. 6, (middle panel, right) integrates the secretion model into the twocompartment model of C-peptide kinetics, since it is identified on C-peptide plasma measurements. Insulin secretion is modeled with two components: first-phase secretion, likely representing exocytosis of previously primed insulin secretory granules, is portrayed as the release of insulin from a rapidly turningover compartment (2 min). Glucose exerts a derivative control, since first-phase secretion is assumed to be proportional to the increase of glucose from basal up to the maximum, through a pa-

rameter that defines the first-phase responsivity. Second-phase insulin secretion is believed to be derived from the provision and/or docking of new insulin secretory granules, and is assumed to be proportional to glucose concentration through a parameter that defines the second-phase responsivity. The seconnd-phase secretion term includes a delay, presumably representing the time required for new granules to dock, be primed and then exocytosed. In the following years, beta-cell function has also been assessed from a more physiological oral test, i.e., MTT/OGTT [90], by properly adapting the IVGTT minimal model to the more gradual changes in glucose, insulin, and C-peptide concentrations. From the oral model, two responsivity indices can be derived as well, related to the dynamic (i.e., proportional to the rate of change) and the static (i.e., proportional to) glucose control. Future developments in the modeling of insulin secretion include the incorporation of the action of gut hormones, e.g., the glucagon-like-peptide 1 (GLP-1) [91]. Since the glucose–insulin system is a negative feedback control system, 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 (DI) paradigm, first introduced in [28], [92], and then revisited first in [93] and recently in [38], where betacell function is multiplied by insulin sensitivity. This concept is clearly illustrated in Fig. 6 (lower left panel): while regulation of carbohydrate tolerance is undoubtedly more complex, it is conceivable that beta-cells’ ability to respond to a decrease in insulin sensitivity by adequately increasing insulin secretion can be assessed by measuring the product of beta-cell function and insulin sensitivity. Thanks to its intuitive and reasonable grounds, first- and second-phase disposition indices, first introduced for IVGTT, have become the method of choice also for MTT/OGTT. They allow us to assess if the two phases of betacell function are appropriate in light of the prevailing insulin sensitivity, to monitor their variations in time, and to quantify

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the effect of different treatment strategies, as illustrated in Fig. 6 (lower right panel). However, the glucose–insulin feedback system is more complex than the hyperbola paradigm. The relation between betacell function and insulin sensitivity is in all likelihood more complex than a rectangular hyperbola, i.e., a power function DI = beta-cell function x (insulin sensitivity)α = constant. This issue, as well as several methodological issues which, unless fully appreciated, could lead to errors in interpretation, have been thoroughly addressed in [40]. Since DI is a conceptual surrogate of the efficiency of glucose homeostasis in a given individual, it provides key phenotypic information, to be considered in addition to other classical parameters such as fasting plasma glucose concentrations or HbA1c. In addition, the novel paradigm of Genome Wide Association Studies has identified more than 30 genes associated with type 2 diabetes and, more in general, to glucose related traits. One of the most challenging goals of current research is now to link such phenotypic and genotypic information, to explore the hereditary component of complex multifactorial diseases like diabetes. Probabilistic relations between genetic and phenotypic variables encoded by Bayesian Networks have been successfully used in some diseases [94]–[96], but not yet diabetes. However, polymorphisms identified so far account for only a part of the phenotype across the population, thus indicating the need to extend the integration approach to “Big Data” of heterogeneous nature, including genetic variations, gene expression, protein expression, and posttranslational modifications, epigenetic regulators, metabolite abundance. Methods/models/algorithms/tools already available should evolve toward approaches able to handle, explore, integrate, and interpret this huge amount of data. VIII. MODELS TO SIMULATE—MAXIMAL Minimal models can be used for simulation but in this context large-scale, maximal (fine-grain) models usually play a more relevant role. These models are comprehensive descriptions attempting to fully implement the body of knowledge about a system into a generally large, nonlinear model of high order, with several parameters, thus improving our understanding of the system. This class of models cannot, in general, be identified from the data and, thus, they are not usable for the quantification of specific metabolic relationships. Their utility is in the possibility for system simulation. Simulation is a powerful investigative tool, widely used in engineering disciplines. Given that technological system structure and function are usually “known” and equations can be written based on first principles, simulation can be used for in silico experimentation (a landmark is the Boeing 777 jetliner, the first airplane to be 100% digitally designed and assembled in a computer simulation environment). In contrast, a physiological system, like the glucose–insulin control system is largely “unknown” in terms of structure and function. Below, we discuss two classical uses of large-scale models and simulation, the developing and testing of theories of insulin signaling and secretion, and the substitution of animal trial with in silico experiments in type 1 diabetes.

Fig. 8.

Insulin signaling model [97].

A. Theories of Insulin Signaling The insulin signaling pathway is a complex cascade of signals triggered by binding of insulin to its receptor and culminating in several biological responses by means of the activation of two major signaling subpathways: PI3K-AKT/PKB pathway, which is responsible for most of the metabolic actions of insulin, and Ras-MAPK pathway, which regulates expression of some transcription factors and cooperates with the PI3K pathway in controlling cell growth and differentiation. In 2002, Sedaghat et al. [97] proposed a model of insulin receptors and the PI3K-AKT/PKB signaling pathway, able to represent the available knowledge and to gain new insight regarding the molecular mechanisms underlying the insulin signal transduction pathway. The model integrated previously validated models of insulin receptor binding kinetics, receptor recycling, and GLUT4 translocation in a comprehensive model with 23 state variables (see Fig. 8). The model exploited kinetic parameters and experimental data mostly taken from the literature and referring to 3T3-L1 adypocytes, and was used to run predictions of the system under different conditions. Despite being dated 2002, the model is still a landmark and used by many groups since it contains several of the most accepted mechanisms and intermediates in the downstream signaling controlling glucose uptake, such as activation of phosphoinositide 3-kinase (PI3K), phosphorylation of protein kinase B (PKB), and translocation of GLUT4 to the plasma membrane. In recent years, other aspects of insulin signaling pathways were modeled, e.g., related to AMPK-mTOR subnetwork [98], [99] and to the crosstalk of insulin receptor with two other prosurvival signaling pathways such as epidermal growth factor receptor (EGFR) and insulin-like growth factor-1 receptor (IGF1R) [100]. B. Theories of Insulin Secretion The dynamics and control of insulin secretion were investigated in vitro in the 1960s and 1970s in the perfused pancreas mainly from the rat. In response to a step in glucose concentration, insulin is secreted in a characteristic biphasic pattern consisting of a first-phase lasting ∼5 min followed by a sustained second phase [101]. First-phase secretion is blunted and the second phase is reduced in early phases of diabetes [102]. Hence, this dynamical response is considered a signature of healthy beta-cell function, and its disturbance has been suggested to be a predictor of diabetes [102].


Fig. 9. Left panel: Grodsky’s [87] model with a large reserve pool and a small labile pool of insulin packets with different thresholds. Right panel: The model [114] with a pool of docked insulin granules (D), a readily releasable pool (RRP), and a pool of fused granules (F) releasing insulin. The model assumes that beta-cells have different activation threshold with respect to the glucose concentration (G) by distinguishing between RRP granules in active cells (denoted H(G), filled circles) and in silent cells (open circles).

The biphasic secretion pattern is inherent to the beta-cells, and not a result of systemic regulation, since it is consistently found in isolated pancreases [87], [101], groups of pancreatic islets [103], [104], and even single islets [105], [106]. In the 1970s, Grodsky et al. [85], [87], [107], and Cerasi et al. [108] developed models of the pancreatic insulin response to various patterns of glucose stimuli. Because of the limited knowledge of the beta-cell biology at that time, these early models were phenomenological. Only recently has our knowledge of the events leading to exocytosis of insulin granules reached a level that allows us to formulate mechanistically based models. Cerasi et al. [108] suggested that the dynamics of insulin secretion was due to time-dependent inhibitory and potentiating signals, still unidentified [109], which act on insulin secretion at different times. For biphasic insulin secretion, inhibition is responsible for creating the nadir after the first-phase peak, while potentiation acts later to produce the second phase. As an alternative to Cerasi’s signal hypothesis, Grodsky et al. [107] proposed that insulin was located in “packets,” plausibly the insulin containing granules, but possibly entire beta cells. In this model, part of the insulin is stored in a reserve pool, while other insulin packets belong to a labile and releasable pool. The rapid release of the labile pool results in the first phase of insulin secretion [87], [107], while the reserve pool is responsible for the sustained second phase. The distinction between reserve and “readily releasable” insulin has been at least partly confirmed when the packets are identified with granules [110], [111]. However, a modification of this conceptually simple model is needed to explain the so-called staircase experiment where the glucose concentration is increased in consecutive steps, each step giving rise to a peak of insulin. Grodsky [87] (see Fig. 9, left panel) assumed that the packets in the labile pool have different thresholds with respect to glucose beyond which they release their content. The resulting mathematical model can reproduce the staircase and many other experiments. Both the signal limited and pool models were unable to describe all the data obtained with different glucose stimulation patterns [112]. Combining the two ideas by adding timevarying (phenomenological) signals to the two-pool model without threshold, Landahl and Grodsky [113] satisfactorily reproduced all of the investigated patterns.

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Recently, an update of Grodsky’s model has been provided based on data of cell-to-cell heterogeneity with respect to their activation threshold [114]. In agreement with earlier electrophysiological findings [115], recent Ca2+ imaging experiments [116], [117] showed that the beta-cells have different glucose thresholds for triggering Ca2+ influx. Above this threshold, the cytosolic Ca2+ concentration changes little with glucose [116], [117]. The subcellular model [114] describes the dynamics of granule pools in the entire pancreatic population of beta-cells (see Fig. 9, right panel). Granules mobilize from a reserve pool to a pool of “docked” granules at the plasma membrane. The granules can mature further (priming) to gain release competence and enter the “readily releasable pool” (RRP). Calcium influx then triggers exocytosis and insulin release from the RRP. The glucose-dependent increase in the number of cells showing a calcium signal 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. Because of the heterogeneity of the RRP, the model can simulate the staircase protocol [114]. Each peak of insulin secretion in response to the steps of the staircase is due to the activation of cells with a threshold between the two glucose concentrations defining the step. The recruitment of more cells into their active state leads to the release of their RRP, i.e., each step of glucose recruits an additional part of the total pancreatic RRP to release insulin. The model has shortcoming similar to the ones of Grodsky’s model. Current work focuses on extending the model by including the fact that Ca2+ shows phasic dynamics, which likely contribute to the various insulin secretion patterns [104], [118]. Such a combined model will include both granule pools and time-varying signals, in the form of Ca2+ , much in the spirit of the combined model by Landahl and Grodsky [113]. Using multiscale modeling the relations between the secretion minimal models and the subcellular events described in the mechanistic model have been investigated [119], [120]. Both the oral and the IVGTT minimal secretion models can be derived from the cellular model, underlining that the two minimal models reflect the same underlying biology. The analysis revealed that the first-phase (IVGTT) secretion and the dynamic (oral) secretion both reflect the amount of readily releasable insulin, but also that the dynamic secretion is shaped by the threshold distribution for cell activation as well as the dynamics of mobilization and docking. Second phase and static secretion reflect a combination of mobilization, docking, priming, and recruitment of new cells. Future work will address the multiscale aspects of a model including Ca2+ dynamics, and look into the mechanistic effects of incretins. A first attempt for a better understanding of the minimal model including GLP-1 [91] was performed in [121]. C. In Silico Experiments in Type 1 Diabetes There are situations where in silico experiments with largescale models could be of enormous value. In fact, it is often not

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Fig. 11. Type 1 diabetes simulator equipped with glucose sensor and insulin pump models allows testing of closed-loop control algorithm for insulin infusion [130].

Fig. 10.

Glucose metabolism simulation model [128].

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 form of in silico experimentation on the system. A number of simulation models have been published in the last four decades and used in particular to examine the performance of control algorithms and insulin infusion routes for the therapy of type 1 diabetes [122]–[128], but the impact has been very modest. An example of such models, which was popular in the 1970s, was that of Foster et al. [128] shown in Fig. 10. However, all these models are “average,” meaning that they are only able to simulate average population dynamics, but not the interindividual variability. The average-model approach is not sufficient for realistic in silico experimentation with control scenarios, where facing with intersubject variability is particularly challenging. A good simulator should be equipped with a cohort of in silico subjects that spans sufficiently well the observed interperson variability of key metabolic parameters, thus providing better information about controller safety and limitations than small-size animal trials. Building on the large-scale model developed in the healthy state from a rich triple tracer meal data set [129], a type 1 diabetes simulator has been developed, able to realistically describe intersubject variability. This was a paradigm change in the field of type 1 diabetes: for the first time a computer model has been accepted by a regulatory agency as a substitute of animal trials for certain insulin treatments [130]. In this simulator, a virtual “human” is described as a combination of several glucose and insulin subsystems (see Fig. 11). In summary, the model consists of 13 differential equations and 35 parameters for each subject. The simulator is equipped with 100 virtual adults, 100 adolescents, and 100 children, spanning the variability of type 1 diabetes population observed in vivo.

Each virtual subject is represented by a model parameter vector, which is randomly extracted from an appropriate joint parameter distribution. With this technology, any meal and insulin delivery scenario can be pilot-tested very efficiently in silico, prior to its clinical application. This simulator has been adopted by the JDRF Artificial Pancreas Consortium and has allowed an important acceleration of closed-loop studies with a number of regulatory approvals obtained based on simulation only. The simulator has been used by 32 research groups in academia, by five companies active in the field of diabetes and has led to 63 publications in peer reviewed journals. Recently new data and models have become available, in particular on hypoglycemia and counterregulation, and a new version of the simulator has been accepted by FDA [131]. In the future, the capabilities of the simulator will be expanded by incorporating intraday variability of key signals, e.g., insulin sensitivity.

IX. MODELS TO CONTROL A patient with type 1 diabetes faces a lifelong behaviorcontrolled optimization problem: the administration of external insulin to control glycemia enters a stochastic scenario where hyperglycemia and hypoglycemia may not be easily prevented by standard open-loop therapy. The adjustment of insulin therapy on the basis of a few daily fingerstick blood glucose measurements is a rudimentary way to technologically restore the missing loop, but the few daily measurements considerably limit the efficacy of the feedback action. To close the loop a system combining a glucose sensor, a control algorithm, and an insulin infusion device is needed, the so-called artificial pancreas (AP). AP developments can be traced back 50 years when the possibility for external BG regulation was established by studies in people with type 1 diabetes using intravenous glucose measurement and infusion of insulin and glucose. After the pioneering work by Kadish in 1964 [132] expectations for effectively closing the loop were inspired by the nearly simultaneous work of five teams reporting closed-loop control results between 1974 and 1978: Albisser et al. [133], Pfeiffer et al. [134], Mirouze et al. [135] Kraegen et al. [136], and Shichiri et al. [137]. In 1977, one of these realizations [138] resulted in the first commercial device—the Biostator ([138], Fig. 12, upper panel).


Fig. 12. Upper panel: The Biostator (courtesy of William Clarke, University of Virginia) [149]. Lower Panel: The outpatient AP system [153], [154].

Although the intravenous route of glucose sensing and insulin infusion is unsuitable for outpatient use, these devices proved the feasibility of external glucose control and stimulated further technology development. In 1979, landmark studies by Pickup et al. [139] and Tamborlane et al. [140] showed that the subcutaneous (SC) route was feasible for continuous insulin delivery. Three years later Shichiri et al. tested a prototype of a wearable AP [141]. In the late 1980s, an implantable system was introduced using intravenous glucose sensing and intraperitoneal insulin infusion [142]. This technology was tested in clinical trials for long-term use [143], [144], but its use remained limited due to the extensive surgical procedures needed for sensor and pump implantation. In all early intravenous and intraperitoneal AP systems, the control algorithms were proportional-derivative (PD) or proportional-integral-derivative (PID) using BG values, BG area under the curve, and BG rate of change to calculate the insulin dose. However, these algorithms have inherent limitations that hinder their use in SC systems due to unavoidable time lags in SC glucose sensing and insulin action. Newer controllers known as model-predictive control (MPC) avoid these limitations by using in their calculations a model of the glucose system of the person being controlled. The new wave—SC artificial pancreas—developed after minimally invasive SC glucose sensing was commercially intro-

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duced in 1999 by the MiniMed continuous glucose monitoring (CGM) system. It provided evidence for the feasibility of the SC–SC route for fully automated BG PD control [145]. Since then, there has been an increase in AP research facilitated by important support by several bodies, including JDRF, NIH, and EU. In particular, JDRF has supported the development of the type 1 diabetes simulator discussed in the previous section, which has greatly accelerated human studies. Despite important developments in sensor and pump technology, the AP must cope with the delays and inaccuracies in both glucose sensing and insulin delivery described in the previous sections. This is particularly difficult when a system disturbance, e.g., a meal, occurs and triggers rapid glucose rise that is substantially faster than the time needed for insulin absorption and action. The problem with inherent delays is that any attempt to speed up the responsiveness of the closed loop may result in unstable system behavior and system oscillation. Thus, a sound controller design must consider a relatively slow response. However, a slow response cannot provide good attenuation of postprandial glucose peaks. Hence, the principal AP control dilemma: find a tradeoff between slow-pace regulation well suited to mild control actions applicable to quasi-steady state (e.g., overnight), and postprandial regulation calling for prompt and energetic corrections [1], [146]. Historically, this problem was clearly demonstrated by the first closed-loop experiments that used PID algorithms. Because PID is purely reactive, responding to changes in glucose concentration after they have occurred, it suffers most from the problems described above. To improve PID performance, one possibility is to add a feed-forward action—a regular premeal bolus—which helps with meal compensation as demonstrated in clinical studies [147]. To mitigate hypoglycemic events, an insulin negative feedback on insulin delivery rate has also been introduced [148]. The PID as well as other control strategies, e.g., fuzzy logic, are continuously theoretically evolving and have been used in several studies (see [1], [144] for reviews). Another control strategy which in these last years has become central to AP design by several groups [(see [1], [144] for reviews] is MPC. The new wave of MPC control design is based on prediction of glucose dynamics using a model of the patient and, as a result, appears better suited for mitigation of time delays due to SC glucose sensing and insulin infusion. In addition, MPC is a better platform for incorporation of predictions of the effects of meals, and for introduction of constraints on insulin delivery rate and glucose values that safeguard against insulin overdose or extreme BG fluctuations. In some sense, an MPC algorithm works as a chess strategy (see Fig. 13). Based on past game (glucose) history, a several-moves-ahead strategy (insulin infusion rate) is planned, but only the first move, e.g., the next 15-min insulin infusion is implemented; after the response of the opponent, the strategy is reassessed, but only the second move (the 30-min insulin infusion rate) is implemented, and so on. In reality, glucose prediction may be different from the actual glucose measurement or an unexpected event may happen; with this strategy these events are taken into account in the next plan.

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Fig. 13. Upper: The concept of MPC. At each step, future glucose levels are predicted and insulin delivery strategy is mapped several steps ahead. Then, the first insulin delivery step is implemented, and the situation is reassessed with new glucose data. The process is very similar to a chess game in which several moves are planned ahead, and after the implementation of the first move the position is reassessed given the response of the opponent. Lower: The critical stage of the famous chess game between Leonid Stein (white) and Lajos Portisch (black), Stockholm, 1962 (courtesy of Leon Farhi,, University of Virginia) [149].

Additionally, an MPC system must cope with inter- and intrapatient variabilities. Some strategies are discussed in [1] and [149] and include: identification of an individualized model; use of a customizable controller individually tuned through a “control aggressiveness” parameter calculated from a few routine biometric and clinical data of each individual; “learning” capabilities by using run-to-run control algorithms of the specifics of patients’ daily routine (e.g., timing of meals) and then optimize the response to a subsequent meal using this information; account for circadian fluctuation in insulin sensitivity. Finally, the MPC system should also have certain feed-forward capabilities, i.e., the ability to use a combination of feed-forward (e.g., patient-initiated) and feedback (controller-initiated) insulin delivery that can partially solve the tradeoff between slow-pace regulation in quasi-steady state and prompt correction of meals. Several successful clinical trials using MPC with single (insulin) and dual (insulin and glucagon) hormone delivery were published between 2008 and 2011 (see the review [149]). Automated communication between CGM devices, insulin pumps, and control algorithms was made possible in 2008 when a new research platform—the Artificial Pancreas Software (APS)— was introduced [150]. Another important design element was the concept of modular approach to AP design [149], [151], which has the advantage to allow sequential development, clinical testing, and ambulatory acceptance of elements (modules) of the closed-loop system. The various modules have different responsibilities, e.g., prevention of hypoglycemia, postprandial


insulin correction boluses, basal rate control, and administration of premeal boluses, and act on different time scales. At the bottom, the fastest layer is in charge of safety requirements. Immediately above, there is the real-time control layer deciding insulin delivery based on latest CGM data, previous insulin delivery, and meal information. The top layer (offline control tuning) is in charge of tuning the real-time control layer using clinical parameters and historical data. The structure is hierarchical, i.e., each layer processes available information (sensor and patient inputs) in order to take decisions that are passed to a lower layer. If necessary, commands from an upper layer can be overridden: a typical example is the safety layer canceling insulin delivery suggested by the real-time control module. Based on modular architecture, various increasingly complex configurations of an AP system become possible. For example, a relatively simple control system responsible only for night-time basal rate regulation has been successfully tested as a first step to AP. More recently, a new modular control-to-range algorithm has been introduced, which resulted in reduced average glucose without increasing patients’ risk for hypoglycemia [152]. However, none of these previous systems had an AP system suitable for outpatient use. The critical missing features were portability and a user interface designed to be operated by the patient. The AP transition to portability and ambulatory use began in 2011 with the introduction of the diabetes assistant (DiAs)—the first portable outpatient artificial pancreas platform, which uses an Android smartphone as a computational platform (see Fig. 12, lower panel). In October 2011, DiAs was used in two pilot trials of portable outpatient AP done simultaneously at the Universities of Padova, Italy, and Montpellier, France [153]. These 2-day trials enabled a feasibility study of ambulatory closed-loop control conducted at Universities of Virginia (VA), Padova, and Montpellier, and at Sansum Diabetes Research Institute, Santa Barbara, CA, USA, [154]. At present multisite randomized cross-over trial comparing the safety of closed-loop control to state-of-the art sensor-augmented insulin pump therapy in an outpatient setting are being conducted. X. CONCLUSION Since the early history of mathematical modeling in physiology and medicine, the glucose system has stimulated the development of modeling methodologies and approaches, and due to its inherent complexity it represented an important test bed for assessing their validity. The progress made along the years in the glucose area is representative of the evolution of modeling in life sciences and of its multiple purposes, from mathematically describing the system in order to measure nonaccessible parameters and variables, from simulating mechanisms and processes to designing control algorithms, spanning from molecular, cellular, and organ level, to whole-body and population studies. The present challenge is now multiscale modeling, aiming to effectively capture biological interdependencies and interactions across multiple scales. This has only very recently commenced in the field of diabetes modeling, but given the dramatically increasing prevalence of diabetes with its large socioeconomical and personal consequences, every effort should be done in


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[154] B. P. Kovatchev, E. Renard, C. Cobelli, H. C. Zisser, P. Keith-Hynes, S. M. Anderson, S. A. Brown, D. R. Chernavvsky, M. D. Breton, A. Farret, M. J. Pelletier, J. Place, D. Bruttomesso, S. Del Favero, R. Visentin, A. Filipe, R. Scotton, A. Avogaro, and F. J. Doyle III, “Feasibility of outpatient fully integrated closed-loop control: First studies of wearable artificial pancreas,” Diabetes Care, vol. 36, pp. 1851–1858, Jul. 2013. Claudio Cobelli (F’03) received the M.Sc. degree in electrical engineering from the University of Padova, Padova, Italy, in 1970. He is currently a Full Professor of biomedical engineering at Padova University, Italy, since 1981. His research activity regards mainly modeling and control of metabolic systems, with focus on glucose metabolism, and is mainly supported by NIH, JDRF, and the EU. He is a fellow of the BME, and AIBME. In 2010, he received the Diabetes Technology Artificial Pancreas Award (www.dei.unipd.it/∼cobelli). Chiara Dalla Man received the M.Sc. degree in electrical engineering and the Ph.D degree in bioengineering, both from the University of Padova, Padova, Italy, in 2000 and 2005, respectively. She is currently an Assistant Professor of biomedical at Padova University, Italy, since 2007. Her research interests include mathematical modeling of metabolic and endocrine systems (www.dei.unipd.it/∼dallaman).

Morten Gram Pedersen received the B.Sc. degree in mathematics and physics in 1992, the M.Sc. degree in mathematics in 2002, both from the University of Copenhagen, Denmark, and the Ph.D degree in applied mathematics from the Technical University of Denmark, Denmark, in 2006. He is currently an Assistant Professor of biomedical engineering at Padova University, Italy, since 2011. His main research interests include theoretical studies of beta-cells, insulin secretion, and exocytosis (www.dei.unipd.it/∼pedersen). Alessandra Bertoldo received the M.Sc. degree in electrical engineering from the University of Padova, Padova, Italy, in 1994, and the Ph.D. degree in bioengineering from the Politecnico of Milano, Italy in 1999. She is currently an Assistant Professor of biomedical engineering at the University of Padova, Padova, Italy, since 2002. Her research interests include the development of mathematical models for analysis and control of biological systems and to the quantification of functional positron emission tomography and magnetic resonance images (www.dei.unipd.it/∼bertoldo). Gianna Toffolo received the M.Sc. degree in electrical engineering from the University of Padova, Padova, Italy, in 1978. She is currently a Full Professor of biological signal processing at the Padova University, Italy, since 2001. Her research activity mainly regards modeling of biological and physiological systems, and includes methodological aspects as well as specific applications in biology, physiology, and medicine, with particular focus on endocrine-metabolic systems (www.dei.unipd.it/∼toffolo).

Theme Introduction: Advancing Our Understanding of Hallucinations.

Advancing our quantitative understanding of radiotherapy normal tissue morbidity.

Stones in 2014: Advancing our understanding--aetiology, prevention and treatment.

Advancing our understanding of infant bronchiolitis through phenotyping and endotyping: clinical and molecular approaches.

Barking up the right tree: advancing our understanding and treatment of lymphoma with a spontaneous canine model.

Understanding the mTOR signaling pathway via mathematical modeling.

Advancing drug discovery: a pharmaceutics perspective.

Advancing Our Understanding of the Psychological Impact of PICU Hospitalization: Is It Time to Pause and Reflect?

Using a Virtual Environment to Examine How Children Cross Streets: Advancing Our Understanding of How Injury Risk Arises.

Advancing the accessibility of psychotherapy: learning from our international colleagues.

Advancing research transdisciplinarity within our discipline.

Improving our fundamental understanding of the role of aerosol-cloud interactions in the climate system.

Improving our understanding of gastroparesis.

Understanding Acupuncture Based on ZHENG Classification from System Perspective.

Dextran sodium sulfate colitis murine model: An indispensable tool for advancing our understanding of inflammatory bowel diseases pathogenesis.

Advancing the understanding of autism disease mechanisms through genetics.

Recent advances in our understanding of the vitamin D endocrine system.

Motor-pattern-generating networks in invertebrates: modeling our way toward understanding.

Cancer Metabolism: A Modeling Perspective.

Making Sense of Dynamic Systems: How Our Understanding of Stocks and Flows Depends on a Global Perspective.

[Research in Radiology: our perspective].

Dynamic modeling of the hydrogel molecular filter in a metamaterial biosensing system for glucose concentration estimation.

Enhancing our understanding of teen-driver crashes.

Shaping our understanding of endothermic thermoregulation.