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Plasma Insulin Cognizant Predictive Control for Artificial Pancreas
Mudassir Rashid, Iman Hajizadeh and Ali Cinar
Abstract— In the present work, an adaptive model predictive
control (MPC) algorithm is designed to effectively compute the
optimal exogenous insulin delivery for artificial pancreas sys-
tems. The proposed MPC is designed using adaptive models that
are recursively identified through subspace-based techniques to
characterize the transient dynamics of glycemic measurements
without requiring any information on the time and amount
of carbohydrate consumption. A dynamic safety constraint
derived from the estimation of plasma insulin concentration
(PIC) is incorporated in the proposed MPC algorithm for the
efficacy and reliability of the artificial pancreas system. The
MPC algorithm, cognizant of the PIC, computes the optimal
control solution to regulate blood glucose concentration while
mitigating aggressive control actions (excessive insulin doses)
when sufficient insulin is present in the bloodstream, thereby
minimizing the risk of hypoglycemia. The efficiency of the
proposed MPC algorithm is demonstrated using simulation
studies.
I. Introduction
Type 1 diabetes mellitus (T1DM) is a chronic disease
characterized by the autoimmune destruction of insulin-
producing pancreatic beta cells, resulting in the inability
of the pancreas to produce sufficient insulin to maintain
euglycemia. As a result, people with T1DM depend on
exogenous insulin that is administered either by multiple daily
insulin injections or by a continuous subcutaneous insulin
infusion pump to control their blood glucose concentration
(BGC). Recent advances in T1DM therapy involve the
development of fully automated insulin delivery systems,
called the artificial pancreas (AP), that use appropriate control
algorithms to compute the insulin dose based on recurrent
measurements from continuous glucose monitoring (CGM)
sensors. Computing the exogenous insulin infusion rates
exclusively on feedback from lagged measurements of glucose
concentration in the interstitial fluid without considering
the previously administered insulin administration may lead
to hypoglycemia (
BGC <
70
mg/dL
) as a result of over-
correction. Therefore, a critical element of a safe and
effective fully automated AP system is a feedback control
law that is cognizant of the previously administered insulin.
A number of AP systems are proposed based on a variety of
control algorithms, such as proportional-integral-derivative
(PID) control [1]–[3], fuzzy logic (FL) control [4], adaptive
neural networks, and model predictive control (MPC) [5]–
[14]. Among these control techniques, MPC is particularly
Department of Chemical and Biological Engineering, Illinois In-
stitute of Technology, Chicago, IL 60616.
mrashid3@iit.edu
,
ihajizad@hawk.iit.edu and cinar@iit.edu
Financial support from the National Institutes of Health (NIH) under the
grants 1DP3DK101075-01 and 1DP3DK101077-01 is gratefully acknowl-
edged.
attractive because it can explicitly consider constraints in
the computation of the optimal control actions. Moreover,
in contrast to PID and FL control [15], the formulation
of MPC algorithms is not restricted by the type of model,
objective function, or constraints. This inherent flexibility
of MPC is exploited to design controllers that predict the
future dynamic glycemic evolution over a finite-time horizon
to determine the optimal insulin infusion rate with respect
to a specified performance index. Despite the theoretical
advantages of MPC techniques, the closed-loop performance
of predictive control algorithms is predicated on definitive
models of the glucose–insulin dynamics. High fidelity
glycemic predictive models are not readily available, however,
due to the significant inter- and intra-subject variability of
human physiology. The model accuracy notwithstanding, the
performance of feedback control algorithms is also inhibited
by the CGM measurements that are affected by varying sensor
errors and distortions as well as uncertain time-delays in
CGM response to carbohydrate consumption. Consequently,
the risk of hypoglycemia is a concern in closed-loop insulin
therapy.
Effective feedback control is necessary to maintain eug-
lycemia, and the greater time spent in a safe glycemic range
can effectively delay the onset and slow the progression of
serious diabetes related complications. Although advances in
diabetes therapeutics have improved glucose regulation, peo-
ple with T1DM still suffer from long-term ailments as a result
of prolonged hyperglycemia (
BGC >
180
mg/dL
), including
cardiovascular complications, nephropathy, neuropathy, and
retinopathy. To alleviate such adversity, it is desirable to
maintain BGC levels closer to the lower values within a
safe glycemic range, though suppressing the blood glucose
levels towards the lower end of the allowable range increases
the probability of hypoglycemic episodes. The risk of hypo-
glycemic episodes, however, can be attenuated by formulating
controllers cognizant of the previously administered unreacted
insulin still present in the body [16].
The injected insulin (basal or bolus) gradually accumulates
in the bloodstream and is eventually utilized by the body. One
factor that prolongs the utilization of administered insulin
is the significant time delays involved in the diffusion and
absorption of the subcutaneously injected insulin analogues.
The amount of previously administered insulin that is present
in the blood or the subcutaneous space is referred to as the
insulin on board (IOB) [17], [18]. The IOB is typically
determined in insulin pumps through static approximations of
the insulin action curves. The significant time-varying delays
induced by the unsteady rates of insulin diffusion, absorption
and utilization and the diurnal variations in the metabolic state
2018 Annual American Control Conference (ACC)
June 27–29, 2018. Wisconsin Center, Milwaukee, USA
978-1-5386-5427-9/$31.00 ©2018 AACC 3589
of individuals have significant effects on the dynamics of the
glucose–insulin system. Therefore, the insulin decay profiles
and action curves used in the calculation of the IOB are
not accurate enough over the diverse conditions encountered
throughout the day to be reliably used in an AP control system
[19].
In contrast to the conventional IOB calculations based
on approximated insulin decay curves, accurate estimates
of the concentration of insulin in the bloodstream, termed
plasma insulin concentration (PIC), can be obtained by using
CGM measurements with adaptive observers designed for
simultaneous state and parameter estimation [19]. Such PIC
estimation approaches typically incorporate reliable glucose–
insulin dynamic models with nonlinear filtering algorithms.
The estimated PIC can be subsequently used to design a
predictive control algorithm that is dynamically constrained
by the estimated PIC and thus considers the insulin concentra-
tion in the bloodstream as part of the optimal control solution.
Incorporating PIC constraints in the optimal control problems
can prevent insulin stacking that may lead to hypoglycemia.
Avoiding extreme glycemic excursions can thus yield a safe
and reliable AP system even in the presence of significant
uncertainty in the system.
Motivated by the above considerations, an MPC algorithm
that is cognizant of the estimated PIC is proposed in this work
for use in AP systems. A recursive subspace-based system
identification approach is used to identify a linear, time-
varying state-space model to characterize the glycemic dy-
namics without requiring onerous and obscure information on
the time and amount of carbohydrate consumption [20]. The
identified adaptive model with an insulin compartment model
that translates insulin to PIC for use in the predictive model is
employed to design the MPC algorithm. The large degree of
design flexibility afforded by MPC algorithms is leveraged to
develop a glycemic controller that manipulates the exogenous
insulin delivery while satisfying a dynamic safety constraint
that limits the insulin infusion rate when PIC levels are high.
The control algorithm also manipulates the objective penalty
weights in response to hypo- and hyperglycemic excursions to
improve control performance. The efficacy of the proposed
PIC cognizant MPC is demonstrated using the University of
Virginia/Padova (UVa/Padova) metabolic simulator [21].
II. Plasma Insulin Concentration Cognizant
Model Predictive Control
In this section, we briefly describe the state and pa-
rameter estimation approach for determining the values of
time-varying parameters and the meal effect for consumed
carbohydrates [19]. Then, the structure of the model deter-
mined through the recursive system identification approach
is outlined. Finally, a detailed description of the risk indexes
for negotiating the penalty weights in the objective function
of the optimal control problem is provided, followed by the
formulation of the MPC algorithm.
A. Plasma Insulin Concentration Estimation
The PIC and the uncertain model parameters, including
the effects of carbohydrates consumed, are simultaneously
estimated using the infused insulin inputs and CGM output
measurements. To this end, Hovorka’s glucose–insulin dy-
namic model is utilized to design the joint state and parameter
estimator [22]. The model is of the general form
xk+1=f(xk,uk, θk)+wk,w∼ N (0,Rw)
yk=g(xk,uk, θk)+vk,v∼ N (0,Rv)(1)
where
x∈Rn
,
u∈Rm
and
y∈Rp
denote the vectors of state,
input, and output variables, respectively, with
f:Rn×m→Rn
and
g:Rn×m→Rp
obtained from Hovorka’s model [22]. In
this work, the outputs are the CGM measurements and the
input variable is the infused insulin. Further,
w∈Rn
and
v∈Rp
denote the vectors of process and measurement noises,
respectively, and
θ
denotes the uncertain model parameters.
In the simultaneous state and parameter estimation approach,
the unscented Kalman filter algorithm is used to recursively
compute both the state and parameter estimates
ˆxi,ˆ
θi
at the
i
th sampling instance. To achieve the simultaneous estimation,
the original problem is transformed by treating the parameters
to be estimated as additional states as follows:
x0
k+1=f0x0
k,uk
yk=g0x0
k,uk(2)
where x0
kBhxT
kθT
kiT
is the augmented state vector and
f0x0
k,ukBf(xk,uk, θk)
θk
g0x0
k,ukBg(xk,uk, θk)
Furthermore, the augmented process and measurement noise
covariances are defined as
R0
wBdiag {Rw,Rθ}
and
R0
vBR
,
respectively, as well as the augmented state estimation error
covariance
P0
kBdiag nPx
k,Pθ
ko
. Bounds on the states and pa-
rameters can be similarly augmented as
x0
L≤x0
k≤x0
U
, where
x0
LBxT
LθT
LT
and
x0
UBhxT
UθT
UiT
. The augmented
quantities can be used with the standard UKF state estimator
framework to perform simultaneous state and parameter
estimation. Employing the state and parameter estimation
approach with Hovorka’s model allows for simultaneously
estimating the PIC, which is a state variable in the model,
and the meal effect, along with other time-varying parameters.
B. Recursive System Identification Algorithm
In this work, a recursive subspace-based empirical model-
ing algorithm based on the predictor-based subspace identifi-
cation (PBSID) method is used to determine linear dynamic
models for designing the predictive controller [20], [23]. The
proposed system identification method is able to provide a
stable, time-varying, and individualized state-space model
for predicting the future CGM measurement outputs with
the insulin infusion rate and biometric variables as inputs.
3590
Adaptive identification allows the model to be valid over
various daily life conditions without requiring onerous and
obscure information on carbohydrate consumption. The
identified model is of the form
˜xk+1=Ak˜xk+Bkuk−d−3+˜wk,˜w∼ N 0,˜
Rw
yk=Ck˜xk+˜vk,˜v∼ N 0,˜
Rv(3)
where
˜x∈R˜n
denotes the vector of state variables for the
identified system, and the delayed input is
uk−d−3
. Note that
the abrupt and discrete insulin variations are challenging
for empirical modeling, thus complicating the direct use
of the administered insulin in the subspace identification
approach. Therefore, to improve the prediction ability of the
identified model, the administered insulin is filtered through
a compartment model extracted from Hovorka’s model with
time-varying model parameters estimated using the UKF
algorithm. This results in the past administered insulin
uk−d−4
being translated to
PICk−d−1
as a variable of the state
vector
˜xk
, resulting in the predicted CGM
ˆyk
. Accordingly,
given
uk−1
and thus
PICk+3
, the prediction
ˆyk+d+3
can be
determined off-line. Therefore, the off-line calculated CGM
prediction of
ˆyk+d+3
is used in this work to determine the
appropriate values for the PIC bounds and the risk indexes.
C. Glycemic and Plasma Insulin Risk Indexes
1) Glycemic Risk Index: A glycemic risk index (GRI) is
used to determine the weighting matrix, denoted
Qˆyk+d+3
,
for penalizing the deviations of the outputs from their nominal
set-point [24]. To this end, the time-varying positive semi-
definite weighting matrix
Qk
is defined as
QkBQˆyk+d+3
.
The glycemic risk index asymmetrically increases the set-
point tracking weight when the off-line predicted CGM
ˆyk+d+3
diverges from the target range. Since hypoglycemic
events have serious short-term implications, the set-point
penalty increases rapidly in response to hypoglycemic ex-
cursions and more gradually in hyperglycemic excursions. A
plot of the glycemic risk index is given in Fig. 1.
2) Plasma Insulin Risk Index: A plasma insulin risk
index (PIRI), denoted
γk+d+3
, is defined to manipulate the
weighting matrix for penalizing the amount of input actuation
(aggressiveness of insulin dosing) depending on the estimated
PIC, thus suppressing the infusion rate if sufficient insulin
is present in the bloodstream. To this end, the time-varying
positive definite weighting matrix
Rk
is developed from the
0 50 80 140 180 220 300 400
CGM (mg/dL)
0
0.2
0.4
0.6
0.8
1
Glycemic penalty index (GPI)
Fig. 1. Plot of the glycemic risk index
PIRI as RkBRγk+d+3, with γk+d+3defined as
γk+d+3BPICk+d+3
PICbasal,k+d+32
(4)
where
PICbasal,k+d+3BIdb,k+d+3
VI·ke,k
(5)
and
Idb
is the patient specific (possibly time-varying) basal
insulin rate that is known in practice, and
VI
and
ke
are
parameters of Hovorka’s model. Furthermore, the parameter
ke
is estimated on-line using the UKF and the CGM output
measurements. A plot of the plasma insulin risk index is
given in Fig. 2. Note that as the penalty weight on the input
action increases, and dosing becomes less aggressive, if the
estimated PIC in high.
D. Plasma Insulin Concentration Bounds
In the proposed MPC, the estimated future PIC is dynam-
ically bounded with updated constraints at each sampling
time depending on the value of the CGM measurements.
For instance, if the CGM measurement values are elevated,
the bounds on the PIC are increased to ensure sufficient
insulin is administered to regulate the glucose concentration.
Furthermore, the PIC bounds also constrain the search
space in the optimization problem, thus reducing the search
space and convergence of the optimization in the proposed
MPC. The PIC bounds are determined based on the CGM
measurements as
hd
PICL,kd
PICU,kiBΓˆyk+d+3(6)
where the function
Γˆyk+d+3
defines the lower and upper
bounds on the normalized PIC values, denoted
d
PICL
and
d
PICU
(respectively), through the estimated CGM
ˆyk+d+3
. The
nominal PIC bounds can be determined from the normalized
PIC bounds as
PICL,kPICU,kBhd
PICL,kd
PICU,ki·PICbasal,k+d+3(7)
Therefore, appropriate PIC bounds can be determined based
on each subject’s basal rate and the CGM measurement. A
plot of the PIC bounds is given in Fig. 3.
E. Model Predictive Control Formulation
In this subsection, we propose a novel adaptive MPC algo-
rithm cognizant of the PIC for computing the optimal insulin
infusion rate. The proposed MPC formulation employs the
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled PIC (PICk/PIC b asal )
0
5
10
15
PIC penalty index
Fig. 2. Plot of the plasma insulin risk index
3591
Fig. 3. Plot of the plasma insulin concentration bounds
glycemic and PIC risk indexes that manipulate the penalty
weighting matrices in the cost function. To this end, the MPC
computes the optimal insulin infusion over a finite horizon
using the identified time-varying subspace-based models by
solving at each
i
th sampling instance the following quadratic
programming problem
u∗
knP−d−4
k=0Barg min
u∈U
J
nP,iˆyi+d+3, γk+d+3,{uk}nP−d−4
k=0
s.t.
ˆ
˜xk+1=Akˆ
˜xk+Bkuk−d−3,∀k∈ZnP−1
0
ˆyk=Ckˆ
˜xk,∀k∈ZnP
0
ˆ
˜x0=ˆ
˜xix∈ X
(8)
with the objective function
J
nP,i(·)B
nP
Õ
k=0
eT
kQiek+
nP−d−4
Õ
k=0
uT
kRiuk
where
ˆ
˜x
and
ˆy
denote the predicted states and outputs,
respectively, for the prediction/control horizon
nP
,
u∈Rm
denotes the vector of constrained input variables, taking
values in a nonempty convex set
U ⊆ Rm
with
UB
{u∈Rm:umin ≤u≤umax}
,
umin ∈Rm
and
umax ∈Rm
denote the lower and upper bounds on the manipulated input,
respectively, and
ekBˆyk−ysp
. The index
ZnP
0
represents
all integers in a set as
ZnP
0B{0, . . . , nP}
. The nonempty
convex set
X ⊆ R˜n
with
XBx∈R˜n:xmin ≤x≤xmax
,
xmin ∈R˜n
and
xmax ∈R˜n
denote the lower and upper bounds
on the state variables, respectively, with one of the states
as the estimated PIC that is constrained through the PIC
bounds. Furthermore,
ˆ
˜xi
provides an initialization of the
state vector,
Q≥
0,
QkBQˆyk+d+3
is a positive semi-
definite symmetric matrix used to penalize the deviations
of the outputs from their nominal set-point, and
R>
0,
RkBRγk+d+3
is a strictly positive definite symmetric
matrix to penalize the manipulated input variables.
III. Results
The efficacy of the proposed PIC cognizant MPC is
demonstrated using the UVa/Padova metabolic simulator [21].
The subjects are simulated for three days with varying times
and quantities of meals consumed on each day, as detailed
in Table I. The meal information is not utilized in the MPC
algorithm as the controller is designed to regulate BGC in
the presence of significant disturbances such as unannounced
meals. The controller set-point is specified to be 110
mg/dL
.
In practice, such low glycemic set-points are avoided due to
fear of hypoglycemia, though a controller that is aware of
the previously administered insulin will moderate aggressive
inputs (decrease insulin dosing) when sufficient insulin has
been delivered. The model is designed with a delay of order
d=1with regards to the effect of PIC on CGM.
The quantitative evaluation of the closed-loop results based
on the proposed MPC algorithm are presented in Table II,
which gives the percentage of samples in defined glycemic
ranges and selected statistics for the glucose measurements. It
is readily observed that no hypoglycemia occurs as the BGC is
never below 70
mg/dL
. Furthermore, the average percentage
of time spent in the target range (
BGC ∈[70,180]mg/dL
)
and the higher tier of
BGC ∈[180,250]mg/dL
are 71
.
14%
and 27
.
79%, respectively. The minimum and maximum
observed BGC values across all experiments are 72 and
267
ml/dL
, respectively. Therefore, for the majority of time
BGC is tightly controlled to be within the safe range. Overall,
the results demonstrate that the proposed PIC cognizant
MPC is able to regulate BGC effectively without requiring
meal announcement, as shown by the significant disturbances
caused by the diverse timing and amounts of meals, while
mitigating severe hypo- and hyperglycemic excursions.
A representative glycemic trajectory and corresponding
insulin dosing decisions made by the proposed PIC cognizant
MPC are shown in Fig. 4. It is evident that the glucose
values stay within or close to the target range for most of the
duration of the experiment. Notice that the basal insulin, a
constant low dose of insulin continuously infused to regulate
blood glucose levels at a consistent level during periods of
fasting, is sometimes reduced or even completely shutoff.
This is done automatically by the controller when the PIC
is higher than the upper limit and thus no additional insulin
infusion is necessary, while the continuation of the basal
rate may result in hypoglycemia as a result of overcorrection.
Furthermore, the insulin boluses, typically a single large
dose of insulin administered before meal consumption to
counteract the postprandial rise blood glucose levels, occur
in in close proximity to the unannounced meals. Therefore,
the proposed PIC cognizant MPC is aware of the insulin
concentration in the bloodstream to avoid overcorrection yet
capable of effectively regulating the glucose levels.
IV. Discussion
The PIRI is used to manipulate the penalty weights of the
objective function and is specified using the CGM value of
TABLE I
Meal scenario for three days closed-loop experiment
using the UVa/Padova metabolic simulator
Meal First day Second day Third day
Time Amount Time Amount Time Amount
Breakfast 09:45 53 g 09:10 61 g 09:00 83 g
Lunch 13:30 63 g 13:45 77 g 14:00 44 g
Dinner 17:45 83 g 18:00 72 g 18:20 75 g
Snack 21:30 34 g 22:00 22 g 22:30 28 g
3592
TABLE II
Results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the
UVA/Padova metabolic simulator
Subject Percent of time in range Statistics
<55 [55,70) [70,180) [180,250)>250 Mean SD Min Max
Adult 1 0.0 0.0 81.2 18.8 0.0 143.0 35.9 96.0 210.0
Adult 2 0.0 0.0 64.0 34.2 1.7 156.5 48.6 91.0 263.0
Adult 3 0.0 0.0 75.8 24.2 0.0 157.2 33.4 91.0 242.0
Adult 4 0.0 0.0 61.8 36.6 1.5 160.7 48.5 76.0 258.0
Adult 5 0.0 0.0 65.7 30.8 3.6 157.9 48.5 72.0 267.0
Adult 6 0.0 0.0 69.4 29.9 0.7 151.7 46.5 87.0 253.0
Adult 7 0.0 0.0 65.1 31.7 3.2 160.1 47.9 100.0 262.0
Adult 8 0.0 0.0 71.7 28.3 0.0 148.8 45.6 88.0 242.0
Adult 9 0.0 0.0 75.3 24.7 0.0 146.5 40.0 92.0 223.0
Adult 10 0.0 0.0 81.4 18.6 0.0 141.6 37.0 90.0 214.0
Average 0.0 0.0 71.14 27.79 1.07 152.4 43.19 88.3 243.4
Fig. 4. Closed-loop results of PIC cognizant MPC for a selected subject (Adult 1) of the UVa/Padova metabolic simulator
ˆyk+d+3
. The daily basal insulin rate for each patient should
be specified appropriately as it affects the weightings of the
MPC objective function through the PIRI. Furthermore, the
daily basal insulin rate is also used to define the bound
constraints for the estimated PIC. The PIC bounds, along with
the risk indexes, govern the aggressiveness of the controller.
A minimum bound for the PIC is considered in the MPC
formulation to enforce the controller to suggest a safe amount
of insulin to derive the CGM towards the specified set-point
target value in a reasonable amount of time. A maximum
bound is considered to avoid giving large doses of insulin
that may cause hypoglycemia as a result of overcorrection.
By bounding the PIC estimates, the insulin doses are also
constrained.
The insulin concentration in the bloodsteam should be
maintained within a safe range. If the PIC decreases
to extreme low values (less than the
PI Cbasal
value that
characterizes the PIC without disturbances and only steady
basal insulin infusion), then the BGC may rise rapidly in
response to meal consumption. The low PIC value may cause
hyperglycemia, and consequently a large bolus to derive the
high BGC towards the set-point. However, the significant
delays in the glucose–insulin dynamics may result in an
overcorrection of the high glucose values, which may ad-
versely lead to hypoglycemia. Such abrupt and counteracting
behavior should be avoided for effective glucose regulation.
One approach to ensure such unfavorable dynamics are
avoided is to effectively negotiate the trade-offs between
the opposing criteria of the cost function. To this end,
the glycemic and plasma insulin risk indexes are defined to
maintain PIC close to the basal value under normal conditions
and thus increase the effectiveness of the AP therapy.
Since bound constraints are considered for the PIC, and
as PIC is one of states in the model used in designing the
predictive controller, the MPC prediction horizon should be
specified sufficiently large to capture the prolonged effects of
insulin on the glucose measurements. Specifically, the pre-
diction horizon should be large enough that the peak effect of
3593
the administered insulin is evident in the predicted future PIC
values. The
tmax,I
parameter in Hovorka’s model characterizes
the time duration of PIC reaching its peak value in response
to administered insulin. Therefore, the prediction horizon of
the predictive controller should be defined according to the
time duration of the insulin subsystem. The adaptive and per-
sonalized PIC estimator is able to provide accurate estimates
of the insulin present in the bloodstream for direct use in the
control algorithm. The presented results are based on an MPC
controller without incorporating any additional AP modules
like the meal detection module that automatically recognizes
carbohydrate consumption and suggests appropriate boluses.
Such modules have the potential to further improve the closed-
loop performance of the proposed PIC cognizant MPC for
use in safe and reliable AP systems.
V. Conclusions
In this work, an adaptive MPC algorithm is designed to
effectively compute the optimal exogenous insulin delivery for
AP systems. The proposed MPC is designed using adaptive
models that are recursively identified through subspace-based
techniques to characterize the transient dynamics of glycemic
measurements without requiring any information on the time
and amount of carbohydrate consumption. A dynamic safety
constraint derived from the estimation of PIC is incorporated
in the proposed MPC algorithm for the efficacy and reliability
of the AP system. The MPC algorithm, cognizant of the PIC,
computes the optimal control solution to regulate BGC while
mitigating aggressive control actions (excessive insulin doses)
when sufficient insulin is present in the bloodstream, thereby
minimizing the risk of hypoglycemia.
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