Automated Insulin Delivery for Type 1 Diabetes Mellitus Patients using Gaussian Process-based Model Predictive Control
Lukas Ortmann, Dawei Shi, Eyal Dassau, Francis J. Doyle III, Berno J.E. Misgeld, Steffen Leonhardt
AAutomated Insulin Delivery for Type 1 Diabetes Mellitus Patients usingGaussian Process-based Model Predictive Control
Lukas Ortmann , Dawei Shi , Eyal Dassau , Francis J. Doyle III , Berno J.E. Misgeld , Steffen Leonhardt Abstract — The human insulin-glucose metabolism is a time-varying process, which is partly caused by the changinginsulin sensitivity of the body. This insulin sensitivity followsa circadian rhythm and its effects should be anticipated byany automated insulin delivery system. This paper presents anextension of our previous work on automated insulin deliveryby developing a controller suitable for humans with Type 1Diabetes Mellitus. Furthermore, we enhance the controller witha new kernel function for the Gaussian Process and deal withnoisy measurements, as well as, the noisy training data forthe Gaussian Process, arising therefrom. This enables us tomove the proposed control algorithm, a combination of ModelPredictive Controller and a Gaussian Process, closer towardsclinical application. Simulation results on the University ofVirginia/Padova FDA-accepted metabolic simulator are presen-ted for a meal schedule with random carbohydrate sizes andrandom times of carbohydrate uptake to show the performanceof the proposed control scheme.
Index Terms — Artificial Pancreas, Insulin Sensitivity, ModelPredictive Control, Gaussian Process.
I. I
NTRODUCTION
Approximately 415 million people had diabetes mellitus in2015 and this number is assumed to increase to 642 millionby the year 2040 [1]. Of these patients, around 10% have type1 diabetes mellitus and they are therefore not able to controltheir blood glucose (BG) level without exogenous insulininjections. On the one side, the goal of these exogenousinjections is to prevent hyperglycemia (BG > 180 mg/dl)and secondary complications arising therefrom. On the otherside, the induced insulin can lead to insulin-induced hypo-glycemia (BG < 70 mg/dl), which can be life-threatening.An automated insulin delivery device can help the patientsachieve both these goals and patients do not need to controlthere BG level manually anymore. Recent developments ininsulin pumps and glucose sensors enable us to solve theproblem of insulin delivery in closed-loop, which leads toimproved BG regulation and which is the goal of the artificialpancreas project [2], [3].A variety of factors in the insulin-glucose metabolismchange over time and call for adaptive control techniques.These factors are e.g. the irregular pattern of exercises [4],
This work was partially funded by the German Academic NationalFoundation and by the United States National Institutes of Health undergrants DP3DK104057 and UC4DK108483. Philips Chair for Medical Information Technology, Helmholtz-Institute, RWTH Aachen University, 52074 Aachen, Germany. [email protected] Harvard John A. Paulson School of Engineering & Applied Sciences,Harvard University, Cambridge, MA 02138, USA. the quasi-periodic appearance of meals [5] and the diurnalchanges of the insulin sensitivity [6]. There are many pro-posed control algorithms in the literature that focus on therequirement of adaptation. Based on the patient’s reaction toboluses given during the last days, the carbohydrate (CHO)to insulin ratio is adapted in [7], to enhance the rejectionof meal intakes. Run-to-run control was also used in [8] toimprove the tracking performance and to control the bloodglucose level to a tighter zone, by updating the basal infusionrate of patients based on past measurements. In comparisonto these studies, where only a few BG measurements wereavailable during the day, the following studies used conti-nuous glucose measurements. The periodic appearance ofmeals was used in [5] to improve the control performancewith a Model Predictive Iterative Learning Controller. In [9],run-to-run adaptation of the basal rate was used during nighttime, while the carbohydrate-to-insulin ratio, which is usedto calculate meal boluses, was adapted during the day. Thecontrol penalty in the cost function of a zone MPC wasadapted in [10] to reduce the mean glucose level, whilenot increasing the risk of hypoglycemia. Both insulin andthe counterregulatory hormone glucagon were used in aclinical study on pigs, where the authors used a GeneralizedPredictive Control approach [11]. In [12], the controller wasadapted by Gain Scheduling based on the blood glucose con-centration [12], whereas the controller was switched basedon an estimate of the insulin sensitivity in [13]. Anotherpublication where the insulin sensitivity is included into thecontroller to enhance the control performance is [6], wherethe insulin sensitivity was included into the input constraintof a Model Predictive Controller (MPC). In [14] the changinginsulin sensitivity was included into a run-to-run controller,which adjusts the carbohydrate-to-insulin ratio and the basalinsulin delivery rate. The authors in [15] provide a reviewof adaptive controllers for BG regulation, including Self-Tuning-Regulators, Minimum Variance Control, GeneralizedPredictive Control and Linear Quadratic Regulators. Our ideaof facilitating Gaussian Processes in the field of glucosecontrol is also used in [16] to determine personalized linearpatient models.In this paper, we present an extension of our work onincorporating information about the changing insulin sensi-tivity into a controller for the insulin-glucose metabolism[17]. In contrast to previously published results on inclu-ding the insulin sensitivity, we determine the effect of thechanging insulin sensitivity during closed-loop. Due to the
Published on
American Control Conference (ACC),
July 2019. https://doi.org/10.23919/ACC.2019.8815258 a r X i v : . [ ee ss . S Y ] J a n oodInsulin Glucose levelStatePredictionsReference Fig. 1. Block diagram of the proposed controller, consisting of anUnscented Kalman Filter, the Gaussian Process and a Model PredictiveController. periodicity of the insulin sensitivity, we can anticipate theupcoming effect of the changing insulin sensitivity by usingthe collected data. We extract the effect of the changinginsulin sensitivity from the state estimate which is providedby an Unscented Kalman Filter [18]. The collected data ofthe effect of the changing insulin sensitivity is given to aGaussian Process which predicts the future values of theeffect. These predictions are then given to a MPC whichincorporates the information into the optimization problem,when calculating the insulin injections. A block diagram ofthe proposed control structure is provided in Fig. 1.The newly developed content of this paper is that wepresent a controller parametrized for humans, in comparisonto the original controller, which was using a model forGöttingen Minipigs. We also provide a new kernel functionfor the Gaussian Process that is fading out old data and iscapable of dealing with the measurement noise we are facingin applications and our simulation model. By postprocessingthe collected training data for the Gaussian Process, thecontroller becomes insensitive to unannounced meals andmeal sizes and times can be arbitrary. To show the advantagesof including a learning part into the glucose controller weshow simulation results on the FDA-accepted University ofVirginia/Padova (UVA/Padova) metabolic simulator [19].The paper is structured as follows: In Section II we give areview of our previous work. We explain how we adapt themetabolic simulator to show a changing insulin sensitivityand present the model used inside the MPC in III. Thepostprocessing of the training data, the new kernel functionfor the Gaussian Process and its predictions are shown inSection IV. Afterwards, the results are shown in Section Vand we give a conclusion in Section VI.II. R
EVIEW OF P REVIOUS W ORK
Here, we give an overview of our previous work. Weexplain how the training data is extracted from the state esti-mates and the insulin injections. Furthermore, we describethe Model Predictive Controller that is provided with thepredictions of the Gaussian Process.
A. Training Data Calculation
The training data for the Gaussian Process is calculatedduring closed-loop control from the state estimate of the UKFand the insulin input. Every new glucose measurement gives a new state estimate, which is then used to derive a newtraining data point. To do so, we start with the time-varyinglinear model given in (12), that we split up into a modeldescribing the insulin glucose metabolism ( ˆ Ax ( t ) ) and a partthat describes the effect of the changing insulin sensitivity( A k IS ( t ) x ( t ) ): ˙ x ( t ) = A ( k IS ( t )) x ( t ) + B u ( t )= ˆ Ax ( t ) + A k IS ( t ) x ( t ) + B u ( t ) . (1)The second part is then interpreted as a disturbance u k IS ( t ) induced by the changing insulin sensitivity, which enters thesystem through B k IS . This leads to: ˙ x ( t ) = ˆ Ax ( t ) + B u ( t ) + B k IS u k IS ( t ) . (2)This system is now discretized and the notation [ · ] i isintroduced to refer to the i th row of a vector/matrix variable.Furthermore, we denote [ x ] i ∗ as the row in the A-matrix of(12) which is time-varying. With the discretized system, x k +1 = ˆ A d x k + B d u k + B k IS d u k IS k , (3)we calculated the disturbance as follows: u k IS k − = (cid:16) [ x k ] i ∗ − (cid:104) ˆ A d x k − (cid:105) i ∗ − [ B d u k − ] i ∗ (cid:17) / (cid:104) B k IS d (cid:105) i ∗ . (4)The training data for the Gaussian Process consists of thesedata points and there corresponding time stamps. For moredetails, see [17]. B. Controller
The controller that we are using is a MPC that is suppliedwith the information about the predicted effect of the chan-ging insulin sensitivity u k IS k . We will denote this combinationof Gaussian Process and MPC as GP-MPC. The GaussianProcess is predicting the effect for the complete predictionhorizon of the MPC formulation. If we are referring to MPC,we mean the same controller but without the predictions ofthe Gaussian Process. The GP-MPC is defined as follows: J ∗ → N − ( x ) = min U → N − J → N − ( x , U → N − ) s.t. x k +1 = ˆ A d x k + B d u k + B k IS d u k IS k , x = x ( t ) , y k = C d x k , y N = 0 , ≤ u k + u basal ≤ u max , (5)where the cost function is J → N − ( · , · ) = N − (cid:88) k =0 y Tk Qy k + ( u k − u ssk ) T R ( u k − u ssk ) . (6)The matrices can be obtained by discretizing the system in(12) for the nominal insulin sensitivity ( k IS ( t ) = 1 ) and theydescribe the linearization of the model around the steadystate x basal for the basal input u basal . Therefore, when theMPC steers the state x k to zero, this corresponds to the basalconditions of the patient. The vector x k is the derivationof the state from the linearization point x basal , u k is thederivation of the insulin injection form its basal rate u basal in mU/min, u k IS k is the predicted influence of the changingnsulin sensitivity provided by the Gaussian Process, y N isthe blood glucose level in mg/dl, N is the prediction horizonof the MPC and Q and R are the weights of the MPC. Toenable reference tracking in the presence of the disturbance u k IS , we introduce u ssk , that is needed to reject the disturbance u k IS k at steady state and is calculated by solving the linearequations (cid:20) ˆ A d − I B d C d (cid:21) (cid:20) x ssk u ssk (cid:21) = (cid:20) − B k IS d u k IS k (cid:21) , (7) ∀ k ∈ [0 , N − . The constraint finite time optimal control(CFTOC) problem in (5) is solved with the Yalmip tool-box [20] and the first element of the solution is given as acommand to the insulin pump and the optimization problemis solved again once a new measurement is available. Theparameters of the MPC are chosen as follows. The predictionhorizon is N = 30 and the sample time of the model inthe MPC is 5 minutes. The prediction horizon is therefore2.5 hours, which covers the most important dynamics of theinsulin-glucose metabolism. The weights of the MPC arechosen as Q = 1 and R = 10 and the input is constraint toa maximum of u max = 0 . U/min.III. S
YSTEM M ODELING
In this section, we describe how we adapt the simulationmodel which we use to evaluate the control performance, toinclude a changing insulin sensitivity. We also describe thechanges made to the model within the MPC, such that itdescribes the human insulin-glucose metabolism.
A. Simulation Model
To evaluate the performance of the proposed controlalgorithm we use the UVA/Padova simulator. The constantinsulin sensitivity of the simulator is adapted such that thepatient shows a diurnal change in insulin sensitivity. Todo so, we follow the approach in [21] and [14], wherethe parameters V mx and k p are changed depending on theinsulin sensitivity: V mx ( t ) = V nominal mx · k IS ( t ) k p ( t ) = k nominal p · k IS ( t ) . (8)We use the insulin sensitivity curve in Fig. 2, which is basedon data from [6]. B. Controller Model
The model used inside the Model Predictive Controller isbased on the Lunze model [22]. This model is a simplificati-on of the Sorensen model [23] and was originally developedfor Göttingen Minipigs. The advantage of this model, forour purposes, is that the insulin sensitivity is specificallymodeled by its own parameter and only effects one stateof the model. This makes it easier to extract the effect ofthe changing insulin sensitivity from the state estimate. Areduced order nonlinear version of the Lunze model [24]is used in the Unscented Kalman Filter and a linearizedversion is used inside the CFTOC problem of the MPC andto obtain the training data for the Gaussian Process. The Lunze model was completely reparametrized to adapt it tothe insulin-glucose metabolism of humans by using input-output data of the Dalla Man model and publicly availabledata from [25]. We choose to adapt the Lunze model to theDalla Man model, because it is accepted and has been used tosupport FDA regulated clinical studies. The gastro-intestinaltract and the subcutaneous insulin route were reparametrizedusing the data from [25], while the remaining parameterswere obtained using input-output data of patient 1 of thesimulator. IV. G
AUSSIAN P ROCESS
This section presents the newly developed postprocessingof the training data and the new combination of kernelfunctions. We also show the collected training data andthe predictions made by the newly parametrized GaussianProcess.
A. Postprocessing the Training Data
Food intakes disturb the training data calculation andshould therefore be announced to the controller. Due tomodel mismatch in the gastro-intestinal tract, the error dueto the food intake cannot be completely calculated andwe therefore discard training data that are collected afteran announced food intake. Using the model of the gastro-intestinal tract, we disable the training data collection, whilethe rate of glucose entering the blood stream is larger than150 mg/min.Furthermore, we discard training data points which are outof range, which we define by | u k IS k | > mg/dl. This secondtrigger for discarding training data is important to copewith forgotten meal announcements, so called unannouncedmeals. Otherwise, the unannounced rise in the glucose levelwill be interpreted as a drop in insulin sensitivity (high u k IS k values) and therefore corrupts the training data.It is also possible to discard data points when the patientannounces an exercise or unusual physical activity, becausethese drain glucose from the system and would alter thetraining data. Also, more advanced techniques could be usedto determine if the collected training data points should bediscarded and it is also possible to let the patient interactwith the controller to benefit form the patients knowledgeand experience. For example, if there is a large peak in thetraining data values, the controller could ask the patient ifhe/she forgot to announce a meal and if so what the meal sizewas. With the information about the meal size, the corruptedtraining data can then be recalculated. B. Kernel Function and Predictions
Once the training data is collected, it is used by theGaussian Process to predict future values of the effect ofthe changing insulin sensitivity. These values can then beincluded in the CFTOC problem of the MPC. We use thefollowing two kernel functions to define the correlationbetween our data points. Namely, a periodic kernel functionand an exponential kernel function: k P ( t, t (cid:48) ; l P , λ ) = exp (cid:16) ( − (cid:16) πλ ( t − t (cid:48) ) (cid:17) ) /l P (cid:17) . (9) ABLE IH
YPERPARAMETERS OF THE G AUSSIAN P ROCESS .Parameter θ [1] λ [min] l P [1]Value .
071 1440 0 . Parameter l E [min] σ n [1]Value . · . k E ( t, t (cid:48) ; l E ) = exp (( −| t − t (cid:48) | ) /l E ) (10)The periodic kernel functions enables us to have a highcorrelation between data points that were collected at thesame time of day. The exponential kernel function gives usthe opportunity to fade out old data points over time byreducing their correlation to the prediction. Finally, we adda noise term to the kernel function, because the calculatedtraining data is affected by the measurement noise. This leadsto: k C ( t, t (cid:48) ; η ) = θ · k E ( t, t (cid:48) ; l E ) · k P ( t, t (cid:48) ; l P , λ ) + σ n . (11)The hyperparameters of the kernel function are lumped into η := [ θ , l E , l P , λ, σ n ] and their values can be seen in Table I.The periodic length, λ , is chosen to be equal to the length ofthe diurnal insulin sensitivity rhythm, which is 24 hours. Thelength scale of the exponential kernel function, l E , is chosensuch that the correlation of data points that are 3 days oldis reduced to 90%. The length scale of the periodic kernelfunction, l P , and the variance, θ , are determined throughhyperparameter optimization and then fixed at these values.Finally, the noise standard deviation, σ n is adapted to thenoise level.A set of training data points is presented in Fig. 2. In thefigure, it can be seen that training data points are discarded(either because a meal input is present or because theirabsolute value is out of range). Furthermore, we can seethe prediction of the Gaussian Process using this data andthe variance of the prediction. Please note that only thepredictions for the next 2.5 hours are going to be used inthe CFTOC problem and that the peaks in the training dataare not present in the prediction, because these peaks areaveraged out. V. R ESULTS
To evaluate the performance of the proposed controlscheme, we use the UVA/Padova simulator and comparethe developed GP-MPC with the MPC (same controller butwithout the predictions of the Gaussian Process). We simu-late meal intakes with randomized timing and carbohydratesizes for patient 1 of the simulator. This patient has a fastingglucose level of 122 mg/dl, which is the control reference,and a basal insulin injection of 20.4 mU/min. To keep themeal times and sizes reasonable, breakfast is eaten between7am and 9am and has sizes between 40 g and 60 g. Lunchis eaten between 12pm and 2pm and dinner is eaten between6pm and 8pm, both with a size between 60 g and 90 g. Theexact meal sizes and meal times for the 7 day simulation canbe found in Table III. We first present the performance for
Time [d] -2-1.5-1-0.500.511.52 D i s t u r ban c e [ m g / d l ] I n s u li n s en s i t i v i t y [ m g / m i n m U ] Training Data and Predictions
Training Data95% Confidence IntervalPredictionInsulin Sensitivity
Fig. 2. Training data for the Gaussian Process and its prediction andvariance, as well as, the insulin sensitivity rhythm. The training data isdisjoint, because of the postprocessing explained in Section IV-A.TABLE IIR
ESULTS WITH ANNOUNCED MEALS .Day and night OvernightMetric MPC GP-MPC MPC GP-MPC% time<54 mg/dl 0.0 0.0 0.0 0.0<60 mg /dl 0.3 0.0 1.1 0.0<70 mg/dl 2.4 0.3 7.8 1.070-140 mg/dl 45.6 47.4 79.0 76.170-180 mg/dl 72.1 75.2 91.5 98.8>180 mg/dl 24.8 24.0 0.0 0.0>250 mg/dl 0.2 1.0 0.0 0.0>300 mg/dl 0.0 0.0 0.0 0.0Mean glucose [mg/dl] 146.4 149.2 109.5 120.6Median glucose [mg/dl] 141.0 141.0 109.0 123.0SD glucose [mg/dl] 44.9 40.9 26.3 22.3Coefficient of glucose variation 0.3 0.3 0.2 0.2Mean glucose at 7:00am [mg/dl] 134.1 125.3 — — announced meals and then show how the closed-loop systemis behaving when a patient forgets to announce a meal.The performance of the GP-MPC and MPC for announcedmeals can be seen in Fig. 3 and Fig. 4. The upper panelof Fig. 3 shows that with the GP-MPC there are no eventsof hypoglycemia after the second night and the drops thatcan be seen for the MPC during the night are prevented.Furthermore, the glucose level around breakfast is lower andcloser to the reference, which leads to lower glucose peaksafter the patient eats breakfast. In the closeup in Fig. 4, theperformance improvement, coming from the predictions ofthe Gaussian Process, can be seen more clearly. The glucoselevel is closer to the reference value for most of the day.Only after dinner the glucose level rises higher than theone with the MPC, but this is due to the low glucose levelwith the MPC before dinner, which is 35 mg/dl below thefasting glucose level. The metrics in Table II underline theperformance advantage of the GP-MPC. There are less eventsof hypoglycemia and the standard deviation of the glucoselevel can be reduced. The time in range for 70-180 mg/dl can ( k IS ( t )) = − .
70 0 .
32 0 .
38 0 0 0 0 0 0 0 0 .
024 00 . − .
56 0 0 − .
010 5 . − .
63 5 .
62 0 − .
030 0 01 .
45 0 − .
75 1 .
30 0 0 0 0 0 0 0 00 0 0 . − . − . · k IS ( t ) 0 0 0 0 0 0 00 0 0 0 − .
091 0 0 0 0 0 .
075 0 0 − . − . − .
08 0 0 0 − . . − .
015 0 0 0 0 00 0 0 0 − . − .
04 0 − . − .
025 0 0 00 0 0 0 0 0 0 0 0 . − .
011 0 00 0 0 0 0 0 0 0 0 0 − .
078 0 . − . ,B = (cid:2) . . (cid:3) T , C = (cid:2) (cid:3) (12) TABLE IIIM
EAL SIZES IN g CARBOHYDRATES AND MEAL TIMES .Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7Breakfast 50 g 40 g 60 g 55 g 50 g 40 g 60 g8am 7:30am 9am 8:30am 9am 7am 7:30amLunch 90 g 70 g 80 g 75 g 80 g 70 g 85 g1pm 12pm 12:30pm 1pm 12pm 1:30pm 12:30pmDinner 75 g 60 g 85 g 90 g 85 g 90 g 70 g7pm 7:30pm 6:30pm 6pm 7:30pm 6:30pm 8pm be increased during day and night through incorporating theGaussian Process. For the range 70-140 mg/dl, the GP-MPCoutperforms the MPC during the day. During the night, thevalue for the GP-MPC is slightly worse than for the MPC,which originates from the glucose level being closer to thefasting glucose level before dinner. The glucose level with theGP-MPC is higher before dinner, which leads to glucose levelabove 180 mg/dl during the night. In general, the set point forthe GP-MPC could be lowered in comparison to the MPC,which would reduce the events of hyperglycemia, whilenot introducing events of hypoglycemia. This is anotheradvantage of adopting insulin sensitivity anticipation.In Fig. 5 and Fig. 6 we show how the controller isreacting to an unannounced meal of 60 g of CHO, whichis indicated by the orange triangle. In the lower panels, onecan see that the insulin bolus is missing, because the mealis not announced and in the upper panels one can see thesubsequent large peak in the glucose level. This unannouncedfood intake leads to training data that is out of range andgets discarded, as explained in Section IV-A. Therefore, thetraining data does not become corrupted and the controllerwill not introduce a peak in the insulin injections on the nextday. In the closeup in Fig. 6, the behavior of the controllerduring the following day can be seen and it shows that thecontrol performance is not altered by the unannounced meal.VI. C
ONCLUSION
This paper presents an enhancement to our previous workby developing a controller suitable for humans that is ableto deal with measurement noise and unannounced meals.Now, the Gaussian Process fades out old training dataand is insensitive to noise. Furthermore, the performance C G M l e v e l [ m g / d l ] GP-MPC vs MPC
MPC GP-MPC
Time [d] I n s u li n i n j e c t i on [ U / m i n ] Fig. 3. Performance of GP-MPC and MPC for announced meals. advantage of anticipating the changing insulin sensitivity isshown on a FDA-accepted simulation model, which leadsthe developed controller towards clinical application. Theproposed integration of a controller with a machine learningtechnique has proven to be effective and could enhanceother glucose control strategies. Finally, through learningthe patient’s specific insulin sensitivity circadian rhythmwith fading memories, the proposed controller enables andimproves personalized health care.R
EFERENCES[1] K. Ogurtsova, J. da Rocha Fernandes, Y. Huang, U. Linnenkamp,L. Guariguata, N. Cho, D. Cavan, J. Shaw, and L. Makaroff, “Idfdiabetes atlas: Global estimates for the prevalence of diabetes for 2015and 2040,”
Diabetes Res. Clin. Pract. , vol. 128, pp. 40–50, 2017.[2] H. Thabit and R. Hovorka, “Coming of age: The artificial pancreas fortype 1 diabetes,”
Diabetologia , vol. 59, no. 9, pp. 1795–1805, 2016.[3] F. J. Doyle III, L. M. Huyett, J. B. Lee, H. C. Zisser, and E. Dassau,“Closed-loop artificial pancreas systems: Engineering the algorithms,”
Diabetes Care , vol. 37, no. 5, pp. 1191–1197, 2014.[4] B. Zinman, N. Ruderman, B. Campaigne, J. Devlin, and S. Schneider,“Physical activity/exercise and diabetes mellitus,”
Diabetes Care ,vol. 26, no. suppl 1, pp. s73–s77, 2003.[5] Y. Wang, E. Dassau, and F. J. Doyle III, “Closed-loop control ofartificial pancreatic beta -cell in type 1 diabetes mellitus using modelpredictive iterative learning control,”
IEEE Trans. Biomed. Eng. ,vol. 57, no. 2, pp. 211–219, 2010. C G M l e v e l [ m g / d l ] GP-MPC vs MPC
MPC GP-MPC
Time [d] I n s u li n i n j e c t i on [ U / m i n ] Fig. 4. Closeup of day 5 for GP-MPC and MPC for announced meals. C G M l e v e l [ m g / d l ] GP-MPC vs MPC
MPC GP-MPC
Time [d] I n s u li n i n j e c t i on [ U / m i n ] Fig. 5. Performance of GP-MPC and MPC for an unannounced break-fast (60 g) on day 3, which is indicated by the orange triangle. C G M l e v e l [ m g / d l ] GP-MPC vs MPC
MPC GP-MPC
Time [d] I n s u li n i n j e c t i on [ U / m i n ] Fig. 6. Closeup for GP-MPC and MPC of the period after the unannouncedmeal, which is indicated by the orange triangle. [6] C. Toffanin, H. Zisser, F. J. Doyle III, and E. Dassau, “Dynamic insulinon board: Incorporation of circadian insulin sensitivity variation,”
J. Diabetes Sci. Technol. , vol. 7, no. 4, pp. 928–940, 2013.[7] C. C. Palerm, H. Zisser, L. Jovanoviˇc, and F. J. Doyle III, “A run-to-runframework for prandial insulin dosing: Handling real-life uncertainty,”
Int. J. Robust. Nonlin. , vol. 17, no. 13, pp. 1194–1213, 2007.[8] C. Palerm, H. Zisser, L. Jovanoviˇc, and F. J. Doyle III, “A run-to-run control strategy to adjust basal insulin infusion rates in type 1diabetes,”
J. Process Control , vol. 18, no. 3, pp. 258–265, 2008.[9] C. Toffanin, R. Visentin, M. Messori, F. Di Palma, L. Magni, andC. Cobelli, “Towards a run-to-run adaptive artificial pancreas: In silicoresults,”
IEEE Trans. Biomed. Eng. , 2017.[10] D. Shi, E. Dassau, and F. J. Doyle III, “Adaptive zone model predictivecontrol of artificial pancreas based on glucose-and velocity-dependentcontrol penalties,”
IEEE Trans. on Biomed. Eng. , 2018.[11] F. H. El-Khatib, J. Jiang, and E. R. Damiano, “Adaptive closed-loopcontrol provides blood-glucose regulation using dual subcutaneousinsulin and glucagon infusion in diabetic swine,”
J. Diabetes Sci.Technol. , vol. 1, no. 2, pp. 181–192, 2007.[12] P. H. Colmegna, R. S. Sánchez-Peña, R. Gondhalekar, E. Dassau, andF. J. Doyle III, “Switched LPV glucose control in type 1 diabetes,”
IEEE Trans. Biomed. Eng. , vol. 63, no. 6, pp. 1192–1200, 2016.[13] B. J. Misgeld, P. G. Tenbrock, K. Lunze, and S. Leonhardt, “Discreteblood glucose control in diabetic göttingen minipigs,”
Processes ,vol. 4, no. 3, p. 22, 2016.[14] C. Toffanin, R. Visentin, M. Messori, F. Di Palma, L. Magni, andC. Cobelli, “Toward a run-to-run adaptive artificial pancreas: In silicoresults,”
IEEE Trans. Biomed. Eng. , vol. 65, no. 3, pp. 479–488, 2018.[15] K. Turksoy and A. Cinar, “Adaptive control of artificial pancreassystems-a review,”
J. Healthc. Eng. , vol. 5, no. 1, pp. 1–22, 2014.[16] M. Messori, C. Toffanin, S. Del Favero, G. De Nicolao, C. Cobelli, andL. Magni, “Model individualization for artificial pancreas,”
Computermethods and programs in biomedicine , 2016.[17] L. Ortmann, D. Shi, E. Dassau, F. J. Doyle, S. Leonhardt, andB. J. Misgeld, “Gaussian process-based model predictive control ofblood glucose for patients with type 1 diabetes mellitus,” in
ControlConference (ASCC), 2017 11th Asian . IEEE, 2017, pp. 1092–1097.[18] B. J. Misgeld, P. G. Tenbrock, and S. Leonhardt, “Reduced-orderfiltering for insulin sensitivity estimation under external disturbances,”in
Amer. Control Conf.
IEEE, 2017, pp. 1444–1449.[19] C. D. Man, F. Micheletto, D. Lv, M. Breton, B. Kovatchev, andC. Cobelli, “The uva/padova type 1 diabetes simulator: New features,”
J. Diabetes Sci. Technol. , vol. 8, no. 1, pp. 26–34, 2014.[20] J. Löfberg, “Yalmip: A toolbox for modeling and optimization inmatlab,” in
Proc. of the CACSD Conf. , Taipei, Taiwan, 2004.[21] R. Visentin, C. Dalla Man, Y. C. Kudva, A. Basu, and C. Cobelli,“Circadian variability of insulin sensitivity: Physiological input forin silico artificial pancreas,”
Diabetes Technology & Therapeutics ,vol. 17, no. 1, pp. 1–7, 2015.[22] K. Lunze, A. Woitok, M. Walter, M. D. Brendel, M. Afify, R. Tolba,and S. Leonhardt, “Analysis and modelling of glucose metabolism indiabetic göttingen minipigs,”
Biomed. Signal Process Control , vol. 13,pp. 132 – 141, 2014.[23] J. T. Sorensen, “A physiologic model of glucose metabolism inman and its use to design and assess improved insulin therapies fordiabetes,” Ph.D. dissertation, MIT, 1985.[24] B. J. Misgeld, P. G. Tenbrock, K. Lunze, J. W. Dietrich, and S. Leon-hardt, “Estimation of insulin sensitivity in diabetic göttingen minipigs,”