Physiological models for artificial pancreas development

CHAPTER

Physiological models for artificial pancreas development

6

Roberto Visentina,b , Michele Schiavona,b , Rita Basuc , Ananda Basuc , Chiara Dalla Mana , Claudio Cobellia c University

a University of Padova, Department of Information Engineering, Padova (PD), Italy of Virginia School of Medicine, Department of Endocrinology, Charlottesville, VA, USA

6.1 Role of physiological models The glucose system has received considerable attention in the last decades due to diabetes pandemic. Modeling, often in conjunction with glucose tracers, has been a key tool to quantitate the pathophysiological mechanisms and parameters of glucose metabolism in prediabetes and type 2 diabetes (T2D) [1,2]. In contrast, type 1 diabetes (T1D) has received less mechanistic attention, that is, focus was more on advising T1D subjects to customize their lifelong challenging subcutaneous (sc) insulin treatment problem: reach near-normal glycemic control without increasing their risk of hypoglycemia and targeting almost normal-glucose levels to prevent the risk for future multiorgan complications. In the last two decades, diabetes technology has considerably matured, with the introduction of sc glucose sensors and sc insulin pumps and, in the last 10 years, with the sc wearable artificial pancreas (AP), a system consisting in three externalto-the-body devices: an sc glucose sensor, an sc insulin pump, and a control algorithm implemented either on a tablet or directly on the pump. As a result, there has been a resurgence of mechanistic studies in T1D, also using tracers, and modeling has become an integral component of contemporary diabetes technology. In this chapter, we review various classes of models that have helped the AP field. We start with the paradigm-change of the UVA/Padova T1D simulator, which has been accepted by the US FDA in 2008 as a substitute for preclinical (animal) studies for certain insulin treatments including the AP, an unprecedented event in the modeling community. Then, we turn to the oral minimal model method, which has allowed the description of the circadian pattern of insulin sensitivity and glucose absorption in T1D, features which have been subsequently incorporated into the simulator. Models of various insulin molecules, from inhaled to ultrafast sc insulins, and of pramlintide b Equal contribution. The Artificial Pancreas. https://doi.org/10.1016/B978-0-12-815655-1.00015-6 Copyright © 2019 Elsevier Inc. All rights reserved.

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are then discussed. We then move into the modeling of the sc glucose sensor delay and on the successful use of the UVA/Padova T1D simulator to prove safety and efficacy of nonadjunctive use of sc glucose sensors. Finally, we discuss in silico design of robust adaptive algorithms capable to follow intra- and interday T1D subject variability in month-long trials and their role in informing AP clinical trials.

6.2 The University of Virginia/Padova T1D simulator In silico experiments are of enormous value to accelerating diabetes technology development since it is often not possible, appropriate, convenient, or desirable to perform an experiment on human subjects 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 experimenting in silico with the system. Several simulation models have been published since the 1960s, mostly in biomedical engineering journals [3–9], but their impact in the field has been modest. The reason is that all these models were average models, and, as a result, their capabilities were generally limited to predicting a population average that would be observed during a clinical trial. However, given the large observed interindividual variability, an average model approach cannot describe realistically the variety of individual responses to diabetes treatment. Thus, to enable realistic in silico experimentation, it is necessary to have a diabetes simulator equipped with a cohort of in silico subjects that spans sufficiently well the observed interindividual variability of key metabolic parameters in the general population of people with type 1 (T1D) and type 2 (T2D) diabetes.

6.2.1 A serendipitous beginning The story of the FDA-accepted University of Virginia (UVA)/Padova type 1 simulator began largely by chance. In 2006, as part of a NIH program project studying the effects of two-year administration of “youth pills” in elderly men and women, physiological performance, body composition, and bone density were measured in 204 nondiabetic individuals [10]. These subjects underwent a triple-tracer meal protocol, which provided, in addition to plasma glucose and insulin concentrations, model-independent estimates of fundamental fluxes of the glucose system, including the rate of appearance in plasma of ingested carbohydrates, endogenous glucose production, glucose utilization, and insulin secretion [11]. This rich flux and concentration portrait was key to develop a large-scale glucose–insulin model, which was impossible to build from only plasma glucose and insulin concentrations. A model including 18 differential equations with 42 parameters, 33 of which were free, and 9 were derived from steady state constraints, was identified in each individual using a Bayesian forcing function strategy [12,13]. From the model parameter estimates of the 204 subjects participating in this study, the interindividual variability was described in a nondiabetic population. From there, using the joint multivariate probability distribution of the model parameters, any number of virtual subjects could

6.2 The University of Virginia/Padova T1D simulator

be generated by random sampling, thereby producing a virtual “population.” Simultaneously with the events above, and thanks to the advent of minimally invasive subcutaneous (sc) continuous glucose sensors (CGS), increasing academic, industrial, and political effort has been focused on the development of an sc–sc closed-loop control system for diabetes, which is known as the artificial pancreas (AP). Generally, the AP uses a CGS coupled with an sc insulin infusion pump and a control algorithm directing insulin dosing in real time.

6.2.2 Accelerating AP research: the FDA-accepted T1D simulator In September 2006, the Juvenile Diabetes Research Foundation (JDRF) initiated the Artificial Pancreas Project and funded a consortium of university centers in the United States and Europe to carry closed-loop control research. At the time, the regulatory agencies mandated demonstration of the safety and feasibility of AP systems in animals, for example, dogs or pigs, before any testing could begin in humans. This approach is well illustrated by two papers showing the use of the Medtronic AP system first in 8 dogs [14] and then, later, in 10 people with T1D [15]. However, it also became evident that animal studies were slow, not powered for variability and costly, and that a simulator of T1D would allow a cost-effective preclinical testing of AP control strategies by providing direction for subsequent clinical research and ruling out ineffective control scenarios. We argued that a reliable large-scale simulator would account better for intersubject variability than small-size animal trials and would allow for fast and extensive testing of the limits and robustness of AP control algorithms. We therefore set to build a simulation environment based on the data and the expertise accumulated at the University of Padova and the University of Virginia, two groups that were already collaborating on several aspects of diabetes technology. A first necessary modification of the existing model [13] was the substitution of endogenous insulin secretion subsystem with an exogenous sc insulin delivery, that is, an insulin pump. This required describing insulin absorption with a twocompartment model approximating nonmonomeric and monomeric insulin fractions in the sc space. Given the absence in 2006 of tracer studies in T1D similar to those described before for healthy subjects, a more difficult task was the description of interperson variability. To obtain the joint model parameter distributions in T1D, we introduced certain clinically relevant modifications to the models developed in health. The resulting T1D simulation model included 13 differential equations and 35 parameters, 26 of which were free, and 9 were derived from steady-state constraints; see Fig. 6.1. Once the T1D model was built, its validity was tested using number of T1D data sets including adults, adolescents, and children. Now the UVA/Padova simulator is equipped with 300 virtual “subjects”: 100 adults, 100 adolescents, and 100 children, spanning the variability of the T1D population observed in vivo. In addition, the simulator is equipped with models of CGS and insulin pumps. With this technology, any meal and insulin delivery scenario can be tested efficiently in silico, prior to its clinical

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FIGURE 6.1 Scheme of the glucose metabolism model included in the T1D simulator [16,17].

FIGURE 6.2 Three uses of the T1D simulator. Figure taken from [18].

application (Fig. 6.2) [19]. After extensive testing, in January 2008, this simulator was accepted by the US FDA as a substitute to animal trials for the preclinical testing of control strategies in AP studies and has been adopted by the JDRF AP 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 (MPC) algorithm, and in April 2008, an investigational device exemption

6.2 The University of Virginia/Padova T1D simulator

(IDE) was granted by the FDA for a closed-loop control clinical trial. This IDE was issued solely on the basis of in silico testing of the safety and efficacy of AP control algorithm, an event that set a precedent for future clinical studies [20]. In brief, to test the validity of the computer simulation environment independently from the data used for its development, a number of experiments were conducted, aiming to assess the model capability to reflect the variety of clinical situations as closely as possible. These experiments included the following: 1. Reproducing the distribution of insulin correction factors in the T1D population of children and adults, which tests that the variability in the action of insulin administered by control algorithms will reflect the variability in observed insulin action; 2. Reproducing glucose traces in children with T1D observed in clinical trials performed by the Diabetes Research in Children Network (DirecNet) consortium; 3. Reproducing glucose traces of induced moderate hypoglycemia observed in adults in clinical trials at the UVA, which provides comprehensive evaluation of control algorithms during hypoglycemia. 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. This was a paradigm change in the field of T1D research: for the first time, a computer model has been accepted by a regulatory agency as a substitute of animal trials in the testing of insulin treatments. Since its introduction, this simulator enabled an important acceleration of AP studies, with a number of regulatory approvals obtained using in silico testing. A total of 140 candidate control algorithms have been formally evaluated from March 2008 to August 2014: 4 in 2008, 86 in 2009, 32 in 2010, 2 in 2011, 6 in 2012, 3 in 2103, and 7 in 2014. These 140 evaluations represented 16 AP projects, which typically resulted in IDEs being submitted to FDA after final algorithm validation. However, we need to emphasize that good in silico performance of a control algorithm does not guarantee in vivo performance; it only helps to test the stability of the algorithm in extreme situations and to rule out inefficient scenarios. Thus computer simulation is only a prerequisite to, but not a substitute for, clinical trials.

6.2.3 Further developments of the UVA/Padova T1D simulator Since 2012, the AP studies successfully moved to outpatient free-living environment and became longer, with durations of up to several weeks/months [21–24]. These trials are collecting large amounts of data, typically including closed-loop control and an open-loop mode as a comparator. New data became available on hypoglycaemia [16,19] and counterregulation, which allowed an update of the in silico model in 2014 [16]. This new version has been proven to be valid on single-meal scenarios showing that the simulator was capable of well describing glucose variability observed in 24 type 1 diabetes subjects

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FIGURE 6.3 Glucose absorption (left panel ) and insulin sensitivity parameters (right panel ), estimated at breakfast, lunch, and dinner using the Maximum a Posteriori Bayesian approach [vertical bars represent standard error (SE)]. ∗P < 0.05 with respect to breakfast, from Wilcoxon Signed Rank Test. Figure modified from [28].

who received dinner and breakfast in two occasions (open- and closed-loop) for a total of 96 postprandial glucose profiles [25]. The simulator domain of validity was then extended by the introduction of diurnal patterns of insulin sensitivity based on data in 19 T1D subjects who underwent a triple-tracer study [26] (see Section 6.3.2). This has allowed the incorporation of a circadian time-varying insulin sensitivity into the simulator, thus making this technology suitable for running one-day multiple-meal scenarios and enabling a more robust design of AP control algorithms [27]. Finally, another validation of the simulator was done by comparing its predictions to data of 47 T1D subjects from 6 clinical centers, who underwent three randomized 23-h admissions, one open- and two closed-loop. The protocol approximated real life with breakfast, lunch, and dinner and collected 141 daily traces of glucose and insulin concentrations. We used the Maximum a Posteriori Bayesian approach, which exploited both the information provided by the data and the a priori knowledge on model parameters represented by the joint parameter distribution of the simulator. Plasma insulin concentration was used as model-forcing function, that is, assumed to be known without error. The identification of the simulator on a specific person provided an in silico “clone” of this person; thus, the possibility emerged to clone a large number of T1D individuals and to move from single-meal to breakfast/lunch/dinner scenario, thus accounting for intrasubject variation in glucose absorption and insulin sensitivity (Fig. 6.3) [28]. This new feature, together with a model of dawn phenomenon data, has been incorporated in a new version of T1D simulator [17]. This version also includes a more realistic model of sc insulin delivery [29], models of both intradermal and inhaled insulin pharmacokinetics, and new models of error affecting continuous glucose monitoring and self-monitoring of blood glucose devices. This enhanced version of the T1D simulator has been, and still is, extensively used in designing and testing of advanced diabetes technologies including glucose sensors (see Section 6.5), new insulin molecules (see Section 6.4), and new generation of closed-loop control algorithms (see Section 6.7). In particular, this simulator allows us to assess individualization strategies, that is, methods for tuning the control algo-

6.3 The oral glucose minimal model

FIGURE 6.4 Scheme of the oral glucose minimal model (OGMM). Figure modified from [32].

rithm to a specific person [30] and thus making the AP to be adaptive, that is, learning from the behavior in time of a specific person [31].

6.3 The oral glucose minimal model Minimal models are parsimonious descriptions, which allow us to measure crucial processes of glucose metabolism. They are characterized by a relatively small number of parameters, so that these can be estimated from the data, providing insights into variables not easily accessible to measurements. In particular, the so-called Oral Glucose Minimal Model [2,32] describing insulin action on glucose production and disposal will be presented first. This model allows us to estimate insulin sensitivity, an index quantifying the ability of insulin to promote glucose disposal and inhibit endogenous glucose production. The use of the model in the context of AP development is then illustrated. In particular, here we will focus on the results of two studies, one aiming to determine the presence of diurnal pattern in insulin sensitivity in T1D subjects [26,27] and one aiming to assess possible changes in insulin sensitivity due to simple vs. complex carbohydrates ingestion [33].

6.3.1 The model The oral glucose minimal model (OGMM) is an extension of the minimal model proposed in [34] to interpret intravenous glucose tolerance test (IVGTT) data, able to estimate insulin sensitivity during the more physiological oral glucose tolerance test (OGTT) or meal glucose tolerance test (MGTT) [32,35]. It is described by the following equations (Fig. 6.4):  ˙ = −(p1 + X(t))·Q(t) + p1 ·V ·Gb + Ra(t), Q(0) = Gb ·V , Q(t) (6.1) ˙ X(0) = 0, X(t) = −(p2 + X(t)) + p3 ·(I (t) − Ib ),

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where Q is the amount of glucose in the accessible compartment, G is the plasma glucose concentration, I is the plasma insulin concentration, the suffix b denotes basal values, X is th insulin action, V is the distribution volume, and p1 , p2 , and p3 are model parameters. Specifically, p1 is equal to SG, the fractional (i.e., per unit distribution volume) glucose effectiveness (GE) measuring glucose ability per se to promote glucose disposal and inhibit glucose production, p2 is the rate constant describing the dynamics of insulin action, and p3 is the parameter governing the magnitude of insulin action. The insulin sensitivity index is given by SI =

p3 ·V . p2

(6.2)

The parametric description of Ra proposed in [32,35,36] is a piecewise-linear function with known break-point ti and unknown amplitude ki :  Ra(t) =

ki−1 +

ki −ki−1 ti −ti−1 ·(t

− ti−1 )

0

for ti−1 ≤ t ≤ ti , otherwise.

(6.3)

OGMM identification requires a number of assumptions, which were discussed in detail in the original articles. When we have to deal with data of T1D or T2D subjects, estimating SI with precision can be difficult. To improve numerical identifiability of the model, we can use the constraints p1 =

GEZI + SI ·Ib V

(6.4)

to link SI to p1 through the parameter GEZI , glucose effectiveness at zero insulin [37].

6.3.2 Insulin sensitivity: diurnal pattern The above-described model was applied to a dataset of twenty T1DM subjects who, once a day, underwent a triple-tracer mixed meal study during breakfast (B), lunch (L), or dinner (D) in a Latin square design, with identical meal composition. More details on the study protocol can be found in [26]. The results show that there is a trend in this group of subjects exhibiting lower SI at B, but this was not statistically significant due to the large intersubject variability. To highlight the characteristic SI pattern in each subject, each SI value was first normalized to the maximum observed in the same patient. Then the values greater than 60% were labeled as high (h), whereas the values lower than 60% were labeled as low (l). The rationale of this choice is explained in [27]. With this definition, theoretically, there are seven possible classes, each one associated with a particular pattern of SI at B, L, and D. Each subject was univocally associated with one of the above classes (Fig. 6.5). The probability of each class is then calculated as

6.3 The oral glucose minimal model

FIGURE 6.5 Percentage intra-day insulin sensitivity variation at breakfast (B), lunch (L), and dinner (D), clustered among the seven variability classes. Percent values reported on the top of each panel represent the percentage of the population belonging to the respective variability class. Figure modified from [27].

P (Class i) = Ni /Ntot ,

(6.5)

where Ni is the number of subjects belonging to the ith class, and Ntot is the total number of subjects. To implement the intraday variability of SI into the T1D simulator, each in silico subject was randomly assigned to one of the seven classes, according to their estimated probabilities [27]. The fact that a subject belongs to the ith class means that the SI daily pattern of that subject is on average the one associated to the ith class. For instance, if the jth subject, characterized by the insulin sensitivity SIj , belongs to class 5 (l-h-h), its SI daily pattern will be, on average, αSIj -SIj -SIj , respectively, at B-L-D with α < 1. However, deviations from this nominal profile are allowed by modulating the nominal pattern with a multiplicative random noise described by a normal distribution N (μ, σ 2 ) with μ = 1 and σ = 0.2. The parameter σ was chosen in order to explain a random effect deviation up to 40% of the maximum. The actual SI pattern is then transformed into the corresponding time-varying parameter SI (t), that is, an almost stepwise-line signal that varies three times a day (at 4 am, 11 am, and 5 pm) [27].

6.3.3 Insulin sensitivity: simple vs. complex carbohydrates The OGMM was also employed to assess SI during a mixed meal containing simple vs. complex carbohydrates in 16 healthy subjects [33]. Briefly, subjects received, in random order, either simple or mixed meals containing grains. All meals had 50 g glucose and similar macronutrients. Plasma glucose (G) and insulin (I ) were frequently measured after meal ingestion to enable model identification. The results show that SI is significantly lower with simple vs. complex carb (49.0 ± 5.9 vs. 80.6 ± 8.310−5 dL/kg/min per pmol/L, p = 0.004). If, as expected, these observations are confirmed in T1D, then they would have considerable implications on prandial insulin dosing.

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6.4 Models of new molecules 6.4.1 Inhaled insulin The delayed onset of action inherent to the current sc injected insulin analogues makes their optimal administration difficult, particularly, in the presence of reallife perturbations such as meals. Inhaled prandial insulin with its rapid kinetics may overcome some of these delays but also introduces new challenges. Technosphere insulin (TI; MannKind Corporation, Valencia, CA) is a dry powder formulation of recombinant human insulin adsorbed onto Technosphere microparticles [38]. Upon inhalation, these microparticles can reach the deep lung allowing absorption into the systemic circulation with a time to maximum serum insulin concentration of 12–15 min [39]. In a phase III trial in T1DM, TI demonstrated noninferiority to sc prandial insulin Aspart (Novolog) [40]. However, because of the fast onset and short duration of action, the dosing regimen of TI in this study may have been suboptimal. Designing a clinical trial to identify the optimal dosing regimen and the optimal titration rule would be prohibitively expensive because countless combinations would need to be tested. Thus we performed in silico trials translating the pharmacokinetic profile of TI (and insulin Lispro as comparator) into the expected postprandial glucose response following a meal tolerance test [41]. To describe pharmacokinetic/pharmacodynamic (PK/PD) of TI observed in T1D subjects, a PK model of TI insulin has been developed and identified on a T1D population treated with TI [42]. In particular, the PK model is a variation of the single-compartment model described in [43]: I˙TI (t) = −kaTI ·ITI (t) + FTI ·D,

ITI = 0,

(6.6)

where ITI is the amount of insulin in the alveolar space, D (pmol/kg/min) is the TI dose, FTI is the fraction of inhaled insulin, which actually appears in plasma, and kaTI (min−1 ) is the rate constant of pulmonary insulin absorption. This TI PK model has been incorporated into the T1D simulator (Fig. 6.6), and individual PK parameters of TI insulin have been randomly extracted from the joint parameter distribution that has been created using the parameter estimates obtained from model identification on insulin data of T1D subjects [42]. The effect of different dosing regimens of TI on postprandial glucose after a meal test has been explored in 100 virtual patients for premeal and postmeal dosing and for split dosing scenarios. In particular, a meal test with 50 g CHO has been simulated with TI doses ranging from 10 to 80 TI Units, with timing ranging from 0 to 120 min after meal. For all the simulations, the expected risk (e.g., the number of expected hypoglycemic events) and benefit (e.g., the mean plasma glucose of meal test) were evaluated. More than 200 different dosing regimens have been evaluated for each in silico subject and the suitable titration selected (Fig. 6.7). The simulations suggested that postmeal dosing (at 15 or 30 min after start of the meal) and split dosing (with 15 or 30 min split times) results in a flatter postprandial glucose profile than at-meal dosing, as shown in Fig. 6.8 for one illustrative subject. In several virtual

6.4 Models of new molecules

FIGURE 6.6 Scheme of the T1DM simulator incorporating the PK model of TI. Figure modified from [44].

FIGURE 6.7 Scheme of different dosing regimens explored in simulation: premeal dosing (upper panel ), postmeal dosing (middle panel ), split dosing (lower panel ). Figure modified from [44].

patients, the flatter profile allowed for a higher TI dose without increasing the risk for hypoglycemia events. In addition, the simulations revealed that the selection of the titration rule is crucial to achieve optimal treatment benefit. Simulated up-titrations using 20 titration rules identified that the best time to measure postprandial glucose

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FIGURE 6.8 Meal experiments in one illustrative in silico subject. Upper panels : comparison between insulin (left ) and glucose (right ) in response to premeal (red (dark gray in print version) lines ) vs. postmeal (green (light gray in print version) line) administration of three different dosages. Lower panels : comparison between insulin (left ) and glucose (right ) in response to premeal administration (red (dark gray in print version) lines ) vs. split dosing (blue (mid gray in print version) lines ) of three different dosages. Figure modified from [41].

is 150 min after the meal, and the upper threshold for the glucose target should be 150–160 mg/dL. These optimized titration rules can considerably improve the efficacy of TI on postprandial glucose control. Clinical studies are currently planned to validate the results from these in silico meal test simulations.

6.4.2 Subcutaneous UltraFast acting insulin analog The effects of an ultrafast acting insulin analog (Faster Aspart, FiAsp) vs. insulin aspart (Asp) on parameters of postprandial carbohydrate turnover in a cohort of subjects with T1D were assessed using the state-of-the-art triple-tracer mixed-meal method. In a randomized double blind crossover trial, people with T1D received identical sc doses of FiAsp vs. Asp (individualized for each participant) with a standardized mixed meal containing 75 g of carbohydrates. Insulin exposure with FiAsp was 32%

6.4 Models of new molecules

FIGURE 6.9 Endogenous glucose production suppression over the indicated time periods (A) and increase in glucose disappearance and decrease in free fatty acids (B) for FiAsp vs. Asp. Bars are LS means ± SE. Figure modified from [45].

˙ smaller increment in post prandial glucose at 1 hr. This greater leading to a 0.6M, was attributable to 12% greater suppression of endogenous glucose production and 23% higher glucose uptake with FiAsp during the first hour. Interestingly, suppression of plasma free fatty acid concentrations during the first hour was 36% greater with FiAsp vs. Asp likely contributing mechanistically to the differences observed in the parameters of postprandial glucose fluxes (Fig. 6.9) [45].

6.4.3 Modeling of pramlintide: in silico assessment of optimal pramlintide to insulin ratio Pramlintide is a synthetic analog of human amylin, a peptide hormone that is cosecreted with insulin by pancreatic β-cells in response to meal ingestion. It has been demonstrated that amylin plays an important role in regulation of postprandial glucose level since it inhibits glucagon secretion, slows gastric emptying, and reduces food intake by decreasing appetite [46]. However, in individuals with T1D, amylin is deficient [47]. A number of studies proved that pramlintide administration restores the effects of amylin and, in turn, lowers average postprandial blood glucose levels, substantially reduces postprandial blood glucose excursions, and decreases food intake, resulting in a reduction in mealtime insulin requirements [48–54]. Thus, there are several reasons for administering pramlintide together with insulin. However, prandial insulin dose must be reduced in presence of pramlintide because of the slower gastric emptying, and simulated experiments may be of great help in determining the optimal pramlintide-to-insulin (P/I).

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A recent in silico study (Micheletto 2013) with 100 T1D adult virtual subjects with gastro-intestinal absorption parameters modified to account for the dynamic effects of pramlintide, as observed experimentally in humans [55,56], has investigated the optimal P/I ratio. The model of oral glucose absorption adopted in [57] is that presented in [58] and is the same one incorporated in simulator (A.3). Briefly, glucose transit through the stomach and intestine was modeled by two compartments, one for solid and one for liquid phase, whereas a single compartment was used to describe the gut. The model was numerically identified on placebo and pramlintide data of a previous tracer study [56]. The relative variation of gastrointestinal tract parameters due to pramlintide was calculated as p ij =

p ij PRAM − p ij PBO p ij PBO

with i=1,. . . ,15 subjects, j=1,. . . ,5 parameters, (6.7)

where p ij PRAM is the value of parameter j estimated in the presence of pramlintide (PRAM), and p ij PBO is the value of parameter j estimated in the absence of pramlintide (placebo [PBO]) for subject i. The effect of pramlintide on Ra was incorporated in the simulator via modification of the parameters of the virtual subjects in agreement with the empirically determined p ij [56]. The simulation scenario included a meal containing 50 g of carbohydrates given at 8:00 h to n = 100 in silico adults with T1D. Concurrent with the meal, placebo and several different P/I ratios were administered: P/I = 3, 6, 8, 9, 10, and 12 µg of pramlintide/unit (U) of insulin (µg/U). On placebo, the virtual subjects received a premeal insulin bolus based on each individual’s insulin-to-carbohydrate ratio (CR). With pramlintide, two in silico experiments were performed. In the first, the virtual subjects received a premeal insulin bolus identical to the bolus used on placebo, without adjustment for the effects of pramlintide. In the second, the virtual subjects received a premeal insulin bolus lowered to account for the effects of pramlintide. The adjustment of the insulin dose was done iteratively for each subject until postprandial hypoglycemia due to insulin overdose was avoided. Then the same P/I ratios were administered again. Results showed that a P/I ratio of 3 was no more effective than placebo, whereas P/I ratios of 6, 8, 9, 10, and 12 µg/U resulted in significant improvements of postprandial glucose control. It is worth noting that during the second experiment, the CR ratio was increased on average by a mean of 26 ± 2%, which resulted in an average reduction in insulin dosing of 20 ± 2% for P/I ratios of 6–12 (Fig. 6.10). Following the in silico study, a clinical trial was performed in T1D [59]. The aim was to test if the co-administration of insulin and the amylin analogue pramlintide may be superior to separate dosing. Thus, a number of fixed P/I ratios (6, 9, or 12 µg/U insulin) or placebo were administered to the patients. Results show that to

6.5 Modeling subcutaneous glucose sensor delay

FIGURE 6.10 Metrics for the evaluation of efficacy (attenuation of postprandial hyperglycemia) and safety (reduction of hypoglycemia) of different P/I ratios [57]. The red (dark gray in print version) column indicates the optimal P/I ratio. Figure modified from [57].

avoid hypoglycemia, insulin dosage had to be reduced by 30% compared to patient’s usual estimates. All ratios reduced glucose and glucagon increments by > 50% in the 3 hours after the meal without causing hypoglycemia. These results confirmed what was found in silico.

6.5 Modeling subcutaneous glucose sensor delay Understanding interstitial fluid (I SF ) glucose kinetics is fundamental for continuous glucose monitoring (CGM), which is a key component of contemporary diabetes management. I SF is remote from blood and it is well known that I SF glucose is “delayed” with respect to blood glucose (BG). However, less appreciated is that I SF glucose is not simply a shifted-in-time version of BG, but a distorted version of BG [60]. Characterizing quantitatively this distortion is particularly important when using CGM in real-life, given that glucose is sensed in I SF using an sc sensing probe. To do that, a model of BG-I SF glucose kinetics is needed. Several studies have investigated the temporal relationship between BG and I SF glucose in subjects with and without T1D by using different experimental techniques (see [61] for a brief review), showing that a two-compartment model is adequate to describe BG-I SF glucose kinetics [62–65]. However, only recently an innovative multitracer and microdialysis experimental design was performed [66,67] allowing to simultaneously and frequently collect plasma and I SF glucose data in fasting conditions. Thanks to this unique data set, the model of BG-I SF glucose kinetics [61] was accurately identified in humans. The model allows us to quantify the equilibration time τ , a fundamental parameter describing the dynamics of BG-I SF glucose as a combination of model parameters: τ=

1 , k12 + k02

(6.8)

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where τ represents the time constant characterizing the response of the I SF compartment to a unit step glucose infusion in blood (Fig. 6.11). To gain insight into

FIGURE 6.11 The two-compartment model describing the blood-interstitium glucose kinetics [61]. The equilibration time τ is the time required by interstitium glucose to reach the value of 0.63 if a unit step glucose infusion is performed in blood at time 0. Figure modified from [60].

the meaning of the physiological “delay” between BG-I SF glucose, two in silico studies have been conducted by exploiting the BG-I SF glucose model. Two metrics have been introduced to characterize the complexity of the relationship between BG and the distorted (with respect to BG) I SF glucose: the time course of glucose differences (GL, glucose lag) between BG-I SF glucose at each time point and the time course of time differences (T L, time lag) when I SF glucose is equal to BG: GL(ti ) = BG(ti ) − I SF (ti ), T L(ti ) = ti − tk ,

(6.9) (6.10)

where tk (with k ≤ i) is the time where BG assumes the same value of I SF (ti ) glucose. The results show that the relationship between BG-I SF glucose profiles is inherently time-varying, with a complex pattern depending not only on the equilibration time τ but also on the time course of BG profile (Fig. 6.12) [60]. However, at present the only tracer-based quantification of the model kinetics is available in fasting conditions [61], but even in the case of nonsteady-state conditions, the argument still applies. This opens the door to incorporate predictive models or priors into contemporary sc glucose sensors in order to mitigate the delay for taking actions, for example, to predict ahead of time a hypoglycaemic event and take a rescue carb in advance. In a scenario of insulin dosing, for example, in an AP system, this delay calls again for the incorporation of this knowledge in the control algorithm to also compensate for the additional, and more important, delay of sc insulin absorption. Moreover, putting into the problem additional elements related to technological delays will simply make the picture more articulated but will not change the picture.

6.6 Nonadjunctive use of glucose sensors

FIGURE 6.12

Top panel : BG concentration data measured in a healthy subject after a meal and exercise session [68] (black line) and predicted ISF glucose time course (red (mid gray in print version) and blue (dark gray in print version) line for τ = 7.1 min and τ = 20.5 min, respectively). Middle panel : Time course of glucose differences (GL) between BG and ISF glucose at each time point for two equilibration times (red (mid gray in print version) and blue (dark gray in print version) line for τ = 7.1 min and τ = 20.5 min, respectively). Bottom panel : Time course of time differences (TL) for ISF glucose to equilibrate with BG for two equilibration times (red (mid gray in print version) and blue (dark gray in print version) line for τ = 7.1 min and τ = 20.5 min, respectively). Figure modified from [60].

6.6 The UVA/Padova T1D simulator for nonadjunctive use of glucose sensors In the past 10 years, the accuracy of sc glucose sensing has moved from MARD (mean absolute relative difference, a common metric used to compare CGM to reference blood glucose) of 19.7% of the Medtronic RT-Guardian to a 9% of the Dexcom G4 Platinum (with software 505). Does this improved accuracy make sc glucose sensors reliable for insulin treatment decisions in place of self-monitoring blood glucose (SMBG)? A clinical trial addressing this question would be almost impossible since the required number of patients to ensure exploration of the tails of the sensor MARD distribution would be prohibitive. Also, retrospective data are not useful because it is impossible to see what would have happened if insulin dosing was based on CGM rather than SMBG. Determining whether CGM is safe and effective enough to substitute SMBG in diabetes management has therefore become an important topic of investigation for the diabetes community and regulatory agencies. Computer simulation is of critical importance because it allows us to perform in silico clinical trials

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(see also the outcome of a recent FDA panel meeting [69] and commentary [70]. The simulator used in this case includes a patient decision-making model (Fig. 6.13) and defines in silico scenarios that recreate real-life conditions (e.g., 100 adults and 100 pediatric patients, three meals per day with variability in time and amount, and meal bolus behavior). Performance for both CGM and SMBG scenarios were evaluated using standard outcome metrics, like time in severe hypo, time in hypo, time in target, hypo- or hyperglycemic events. Our results (Fig. 6.14) support the nonin-

FIGURE 6.13 Block-scheme of the T1D patient decision-making simulator model. Arrows entering each block are inputs, whereas arrows exiting are causally related outputs. The input of the simulator is the sequence of meals, whereas the output is the BG concentration profile. The simulator includes parameters describing the patient’s physiology and therapy. The picture reports representative time courses for meals in input and BG in output for a simple scenario in which the patient takes 50 g for breakfast at 07:00 am. Figure modified from [71].

FIGURE 6.14 Modeling strategy to realize the 40,000 combinations of physiology, behavior, and hypoawareness on which the glucose sensor nonadjunctive use vs. SMBG has been tested.

6.7 Adaptive AP algorithms

feriority of CGM vs. SMBG [69,70]. Moreover, time below 50 and 70 mg/dL has significantly improved, time between 70 and 180 mg/dL and time above 180 mg/dL have slightly improved, and the number, extent, and duration of hypoglycaemic events have significantly reduced in the CGM vs. SMBG scenario [71].

6.7 Adaptive AP algorithms In the past decade the research has seen unprecedented advances in AP technology, which moved from short-term inpatient studies to short trials at home employing wireless portable wearable AP systems. Several studies were conducted in adults, for example, those using an AP system based on the Modular Model Predictive Control algorithm (MMPC) [72–74] in gradually less structured and less monitored settings: inpatient first [75], 2-day in hotel settings [76,77], and, recently, 2-month evening & night at home [21]. The formerly conducted studies had a limited duration and were restricted to evening and night, thus allowing us to neglect the impact of intraand interday glucose response variability of each subject, for example, to insulin and meals. The latter is a well-known phenomenon and became a major issue with the introduction of longer (week/month) home trials. This large subject-specific variability calls for an adaptive controller. In the following section, we describe in summary an adaptive AP MMPC algorithm based on the Run-to-Run (R2R) approach [78]. The R2R is a well-known learning-type control algorithm [79], which learns information about the control quality from the current run and changes the control variable to apply in the next run. The R2R strategy has already been used for glucose control in T1DM subjects on the basis of a few daily self-monitoring blood glucose (SMBG) measurements [80–84] or using CGM data [31,85,86] to adapt day-by-day basal insulin delivery or the insulin meal bolus. R2R in the AP context was introduced in [87], where the aggressiveness of the controller was adapted by using the maximum and minimum glucose values provided by CGM. Here a much more realistic R2R approach for tuning the MMPC algorithm is described, which adapts the basal insulin delivery during the night and the carbohydrate-to-insulin ratio (CR) during the day, and its in silico test using the new time-varying UVA/Padova T1D simulator.

6.7.1 Run-to-Run strategy for adaptive MPC tuning The MMPC algorithm considered here is the linear model predictive control described in [74]. The principal parameters used for control tuning and individualization are the basal insulin delivery, the carbohydrate-to-insulin ratio (CR), the correction factor (CF), and the body weight (BW). In particular, the MMPC computes an insulin variation with respect to the basal profile, uses CR and CF (taking into account also the insulin on board, that is, the amount of insulin, coming from previous bolus/infusions, that is still active in the body) to compute the insulin reference in the cost function, and BW and CR to tune the control aggressiveness. Thus the adaptive

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MMPC aims to (i) optimize the tuning of the controller parameters and (ii) adapt them to the interday variability. Specifically, the R2R strategy is applied to update both the basal insulin delivery and CR parameter (and thus the meal insulin bolus); the update is applied in the next day (run) on the basis of the performance measured during the previous day (run). The choice of the performance indices is a critical point for the success of the R2R. CGM sensors (usually employed in an AP context) allow including clinically relevant indices into the problem, for example, the percentage of time spent in hypo-, eu-, and hyperglycaemic range, and the average blood glucose (BG). In particular, since a major concern in T1DM therapy is to avoid hypoglycaemia, the updating law is primarily designed to lead to 0 the percentage of time spent in hypoglycaemia (i.e., BG < 70 mg/dL). Once this primary goal is achieved, a secondary updating law is designed to reduce the percentage of time spent above 180 mg/dL and to lead the average BG to the desired target. For each run, the variation of the basal insulin rate is proportional to the applied basal delivery and to the performance indices computed during the previous run. To give priority in avoiding hypoglycemia, a switching condition depending on the percentage of time spent below 70 mg/dL is introduced. A similar updating law is used to optimize the CR values, each of which is assumed to be constant along three daily intervals (postbreakfast/lunch/dinner). The stability of the proposed strategy can be demonstrated by applying the method described in [31], where an R2R approach for adapting a piecewise basal therapy in an open-loop context is proposed. A key assumption is that disjoint intervals are used to update basal insulin or CR: this is an important requirement; otherwise, the problem would move from several scalars to a multivariable framework with a significant increase of complexity both in terms of algorithm tuning and stability analysis.

6.7.2 In silico testing The R2R algorithm described before has been tested on 100 in silico adults of the simulator. A two-month scenario has been simulated, in which three meals per day are administered at 8:00 am, 1:00 pm, and 8:00 pm having 40 g, 80 g, and 60 g of CHO, respectively. Moreover, if the BG falls below 65 mg/dL, the protocol prescribes a rescue CHO dose of 16 g (hypotreatment). Two hypotreatments are separated by at least 30 minutes. The simulations are performed twice, either by using the MMPC strategy described in [74] (CL) or by employing the adaptive MMPC enhanced by the R2R strategy (CLR2R ), in which basal insulin and CR were updated during the night or daytime, respectively. Performance metrics include average BG (M) and standard deviation (SD), percentage of time spent in euglycaemic target range [70–180] mg/dL (Tr ), percentage of time spent above 180 mg/dL (Ta ), and percentage of time spent below 70 mg/dL (Tb ). The M ± SD of simulated BG after one week, four weeks, and eight weeks are shown in Fig. 6.15: the postprandial overshoots detected after lunch and dinner (Fig. 6.15A) are considerably reduced after one month of R2R (week 4, Fig. 6.15B). A further reduction is achieved after 2 months (week 8, Fig. 6.15C), also with a reduced BG variability.

6.8 Conclusions

Numerical comparison of CL vs. CLR2R on the whole experiment duration is reported in Table 6.1, where the improvement shown by CLR2R is evident. Performance indices show that the improvement of CLR2R vs. CL is modest after one week; after one month, the percent time in range, time in tight range, and time above 180 mg/dL are very much improved with respect to week 1. This performance is maintained until the end of the experiment (week 8). The encouraging results achieved in silico in a realistic one-month scenario have opened to an in vivo testing phase [88]. Specifically, 18 T1D patients underwent an outpatient 4-week study, aimed at comparing the performances achieved by CLR2R with respect to the nonadaptive CL (both used for 24/7). The details and results are reported in Chapter 7, Section 7.7.3.

FIGURE 6.15 Comparison of average ± SD glucose time courses in CL (blue (dark gray in print version)) vs. CLR2R (magenta (light gray in print version)) on week 1 (A), week 4 (B), and week 8 (C), respectively. Figure modified from [78].

6.8 Conclusions Modeling, in conjunction with tracers, has become an integral component of contemporary diabetes technology and has helped tremendously the development of the artificial pancreas. In this chapter, we have reviewed the key role of the UVA/Padova

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Table 6.1 Performance Metrics Improvements. Percent improvements obtained using the CLR2R algorithm with respect to the sole CL. M (mg/dL) Tt (%) Ttt (%) Ta (%)

Week 1 Week 4 Week 8 0.86% 4.81% 6.29% 1.69% 8.31% 11.39% 5.60% 28.37% 44.87% 7.94% 33.95% 48.74%

simulator by also giving a development perspective up to its most recent single-day version. In the beginning the simulator has allowed us to move to human studies on the only basis of in silico evidence with an important acceleration of artificial pancreas research. Crucial were also a number of ancillary models, for example, those describing intra- and interday patterns of insulin sensitivity and glucose absorption parameters, and those quantitating the blood glucose-interstitial fluid kinetics: all these models have been subsequently incorporated in the simulator allowing us to move into the new generation of adaptive closed-loop glucose control systems. In the most recent years the simulator has found new territories of application, which are briefly reviewed, notably the testing of new relevant molecules for better glucose control and the generation of in silico evidence of nonadjunctive use of glucose sensors.

Appendix 6.A 6.A.1 UVA/Padova T1D simulator model equation Glucose subsystem ⎧ ˙ p (t) = EGP (t) + Rameal (t) − Uii (t) ⎪ G ⎪ ⎪ ⎪ ⎨ − E(t) − k1 ·Gp (t) + k2 ·Gp t (t), ⎪ ˙ (t) = −U G ⎪ t id (t) + k1 ·Gp (t) − k2 ·Gp t (t), ⎪ ⎪ ⎩G(t) = G (t)/V , p

G

Gp (0) = Gpb , Gt (0) = Gtb , G(0) = Gb .

(A.1)

Insulin subsystem ⎧ ⎪ ⎨I˙p (t) = −(m2 + m4 )·Ip (t) + m1 ·Il (t) + RaI (t), I˙l (t) = −(m1 + m3 )·Il (t) + m2 ·Ip (t), ⎪ ⎩ I (t) = Ip (t)/VI ,

Ip (0) = Ipb , Il (0) = Ilb , I (0) = Ib .

(A.2)

6.A Appendix

Glucose rate of appearance ⎧ Qsto (t) = Qsto1 (t) + Qsto2 (t), ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ ⎪Qsto1 (t) = −kgri ·Qsto1 (t) + D·δ(t), ⎨ ˙ sto2 (t) = −kempt (Qsto )·Qsto2 (t) + kgri ·Qsto1 (t), Q ⎪ ˙ gut (t) = −kabs ·Qgut (t) + kempt (Qsto )·Qsto2 (t), ⎪Q ⎪ ⎪ ⎪ ⎪ ⎪ f ·kabs ·Qgut (t) ⎩Ra , meal (t) = BW

Qsto (0) = 0, Qsto1 (0) = 0, Qsto2 (0) = 0, Qgut (0) = 0,

(A.3)

Rameal (0) = 0,

with kmax − kmin  · tanh[α(Qsto − β·D)] 2  − tanh[β(Qsto − c·D)] + 2 .

kempt (Qsto ) = kmin +

(A.4)

Endogenous glucose production EGP (t) = kp1 − kp2 ·Gp (t) − kp3 ·X L (t) + ξ ·X H (t), X˙ L (t) = −ki · [X L (t) − I  (t)], ˙



I (t) = −ki · [I (t) − I (t)], X˙ H (t) = −kH ·X H (t) + kH · max[(H (t) − Hb ), 0],

EGP = EGPb , (A.5) X L (0) = Ib , 

(A.6)

I (0) = Ib ,

(A.7)

X H (0) = 0.

(A.8)

Glucose utilization Uii (t) = Fcns , [Vm0 + Vmx ·X(t)·(1 + r1 ·risk)]·G(t) , Uid (t) = Km0 + Gt (t) ˙ X(t) = −p2U ·X(t) + p2U ·[I (t) − Ib ], X(0) = 0,

(A.10)

⎧ if G ≥ Gb , ⎪ ⎨0 risk = 10·[f (G)]2 if Gth ≤ G < Gb , ⎪ ⎩ 10·[f (Gth )]2 if G < Gth ,

r

r f (G) = log(G) 2 − log(Gb ) 2 .

(A.12)

(A.9)

(A.11)

with

(A.13)

Renal excretion  E(t) =

ke1 ·[Gp (t) − ke2 ] 0

if Gp (t) > ke2 , if Gp (t) ≤ ke2 .

(A.14)

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External insulin rate of appearance RaI (t) = RaI sc (t) + RaI id (t) + RaI ih (t).

(A.15)

Subcutaneous insulin kinetics 

RaI sc (t) = ka1 ·Isc1 (t) + ka2 ·Isc2 (t), I˙sc1 (t) = −(kd + ka1 )·Isc1 (t) + usc (t − τ ), I˙sc2 (t) = kd ·Isc1 (t) − ka2 ·Isc2 (t),

(A.16)

Isc1 (0) = Isc1ss , Isc2 (0) = Isc2ss .

(A.17)

Intradermal insulin kinetics 

RaI id (t) = idt1 (t) + ka ·Iid2 (t), I˙id1 (t) = −(0.04 + kd )·Iid1 (t) + uid (t), I˙id2 (t) = −ka ·Iid2 (t) + idt2 (t),

(A.18)

Iid1 (0) = Iid1 ss , Iid2 (0) = Iid2 ss ,

(A.19)

where idt1 (t) and idt2 (t) are defined by the transfer functions  T1 (s) =

b1 s + b1

2 =

L {idt1 (t)}   , T2 (s) = L 0.04·Idt1 (t)



b2 s + b2

a2

=

L {idt2 (t)}  . L kd ·Idt2 (t)

Inhaled insulin kinetics RaI ih (t) = kaI ih ·Iih (t),

(A.20)

I˙ih (t) = −kaI ih ·Iih (t) + FI ih ·uih (t),

Iih (0) = 0.

(A.21)

Gsc (0) = Gb .

(A.22)

H (0) = Hb ,

(A.23)

Subcutaneous glucose kinetics ˙ sc (t) = −1/Ts ·Gsc (t) + 1/Ts ·G(t), G

Glucagon kinetics and secretion H˙ (t) = −n·H (t) + SRH (t) + RaH (t), s d (t) + SRH (t), SRH (t) = SRH

⎧ s

b ⎪ (t) − SRH ⎨−ρ· SRH  

˙ sH (t) = SR σ [Gth − G(t)] s b ⎪ + SRH , 0 ⎩−ρ· SRH (t) − max I (t) + 1  dG(t) d ,0 . SRH (t) = δ· max − dt

(A.24) if G(t) ≥ Gb , if G(t) < Gb , (A.25) (A.26)

References

Subcutaneous glucagon kinetics 

H˙ sc1 (t) = −(kh1 + kh2 )·Hsc1 (t), H˙ sc2 (t) = kh1 ·Hsc1 (t) − kh3 ·Hsc2 (t),

Hsc1 (0) = Hsc1b , Hsc2 (0) = Hsc2b ,

RaH (t) = kh3 ·Hsc2 (t).

(A.27) (A.28)

Acknowledgments This work was supported by MIUR, Italian Ministry of Education, Universities and Research (grant FIRB RBFR08CHY6_002); University of Padova (grant CPDA145405/14); EU project ICT FP7-247138 “Bringing the Artificial Pancreas at Home (AP@home)” (Funding Agency: European Union’s Research and Innovation funding programme, FP7 initiative: FP7-ICT2007-2); Juvenile Diabetes Research Foundation (grant no. 17-2011-273); National Institutes of Health (grants DK-R01-085516, DK-DP3-094331, DK-R01-029953, DK-DP3-106785); Novo-Nordisk (research grant NN1218-3922).

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