Download Anticipatory Behavior in Blood Glucose Control: Using Meal Prior

Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Anticipatory Behavior in Blood Glucose Control:
Using Meal Prior Probabilities to Prepare for Future Meal Disturbances
Fraser Cameron*, B. Wayne Bequette*
Bruce A. Buckingham**, Darrell M. Wilson**, Hyunjin Lee*, Günter Niemeyer***

*Rensselaer Polytechnic Institute, Troy,NY 12180
USA (Tel: 650-450-0908; e-mail: [email protected]).
**Stanford University, Stanford,CA 94305 USA
*** Willow Garage & Stanford University
Abstract: Automatic regulation of blood glucose in patients with Type 1 Diabetes is challenged by
unknown or unannounced food consumption. Yet we know most food is ingested in discrete meals,
spaced by brief periods of fasting. Treating meals as discrete events, this paper proposes a set of prior
probabilities relating distinct meals independently of daily patterns. Using these prior probabilities,
closed-loop blood glucose control can attain anticipatory behavior, implicitly lowering glucose levels
when patients should be hungry and meals are most likely to occur. This improves overall performance
and individual meal responses. The benefits occur because blood glucose prediction, which anticipates
future food intake, can achieve near zero mean errors even over a long prediction horizon. As a side
effect, such predictors can easily be tuned or tested against unrestricted out-patient data which, by
definition, contains unknown meals. We validate our approach in both prediction and control of blood
glucose levels. We see improved prediction accuracy over 1-4 hour horizons and a significant reduction
in the blood glucose risk index for simulated closed-loop control.
Keywords: Probabilistic models, Adaptive control, Feedforward, Biomedical systems, Biomedical control

1. INTRODUCTION
Patients with Type 1 Diabetes Mellitus must explicitly
regulate their blood glucose (BG) levels to remain healthy.
When done manually, this effort can loosely be separated into
two components: a continuous insulin infusion (basal) to
maintain an appropriate steady-state glucose level and
discrete infusions (bolus) to counteract the effects of meals.
Since meals act roughly as fast as infused insulin, early action
is crucial to preventing large blood glucose excursions. In
fact (Cobry et al. 2010) conclude that boluses should be given
20 minutes before the meal.
In reality, manual BG control is tedious and bolusing is error
prone. (Burdick et al. 2004) show that, on average, teenagers
under manual control forget to compensate for 2.1 meals per
week. New automatic control algorithms aim to supplant
manual attention using insulin pumps and continuous glucose
monitors (CGMs). As with manual control, one of the
primary challenges to achieving tight BG regulation is the
rejection of unknown or unannounced meal disturbances. For
a review of other challenges see (Kumareswaran et al. 2009;
B Wayne Bequette 2005; Claudio Cobelli et al. 2009).
Relying only on feedback control implies a delayed response
to meals and associated significant glucose deviations.
Patients could assist feedback controllers by announcing
meals and/or explicitly administering insulin. However, we
fear the illusion of safety presented by automatic regulation
would only worsen the manual error rate of missed meal
announcements. Automatic control algorithms should be
designed to handle unannounced meals.
Copyright by the
International Federation of Automatic Control (IFAC)
Current controllers handle unannounced meals through
explicit or implicit meal detection (Eyal Dassau et al. 2008;
Lee & B Wayne Bequette 2009). These controllers can use
one of several meal detection algorithms (Cameron &
Niemeyer 2010; Cameron et al. 2009; E. Dassau et al. 2008;
Lee et al. 2009). The meal detection algorithms search for the
signature of a meal in the sensor measurements. By
definition, they can provide detection only after a meal has
occurred.
Fortunately meals are not entirely unpredictable. We know an
individual meal follows a consistent shape (Livesey et al.
1998; A Basu et al. 2003; Vella et al. 2007). We also know
that meal times and sizes follow particular patterns and rules.
This has led (Patek et al. 2009) to predict meal times based
on daily patterns. A possible concern is the detrimental effect
if a patient deviates from a consistent routine.
In this paper we propose an alternate method of anticipation.
We declare meal prior probabilities as a function of time
since previous meals, effectively encoding brief periods of
fasting between meals. While such anticipation cannot match
the certainty of meal announcements, it does allow
preparatory lowering of glucose levels when we expect
patients to be hungry and meals to occur, regardless of
historical patterns.
We show substantial performance improvements, lowering
the average blood glucose risk index (BGRI) (Clarke & B.
Kovatchev 2009; Boris P. Kovatchev et al. 1997) from 6.92
to 2.83 when adding meal priors for a set of ten simulated
patients. Simultaneously, we show that glucose levels remain
reasonable even with two consecutive skipped meals.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
The control benefits result from the meal priors improving
prediction of longer prediction horizons including unknown
future meals. In contrast to other predictors that assume no
meals (Bergman et al. 1979; Hovorka et al. 2004), or stay
limited to short prediction horizons (Gillis et al. 2007), our
prediction can achieve causal zero-mean long-term prediction
errors. As an additional benefit, we can test, tune, and
validate the prediction on existing free-living outpatient data.
meals are only eaten when awake, and c) meals inhibit new
meals while being digested.
After a brief background of relevant prior work, we propose
the novel meal prior probabilities, determine appropriate
parameter selections, and examine the resultant expected
meal behavior. We finally validate our approach in both
prediction and closed-loop control before concluding.
2. BACKGROUND
(Patek et al. 2009) provide a Model Prediction Control
(MPC)-like method that anticipates meals when augmented
with a meal pattern and either meal announcement or a meal
detector. The meal pattern used is generated from historical
data and uses the regime of breakfast in the morning, lunch
around noon, and dinner in the evening. This allows insulin
to be safely administered before a meal, yielding improved
controller performance and safety in the event of a missed
meal. This work is based on historical patterns and so has
reduced benefit for non-standard days.
An alternate implementation by (Cameron & Niemeyer 2010)
avoids historical patterns such as the breakfast, lunch, and
dinner regime. They generate predictions and uncertainties
that can be incorporated into MPC like controllers. (Cameron
2010; Cameron et al. 2011) use the generated predictions and
uncertainties in an MPC-like controller. The probabilistic
framework used for meals is, however, simplistic, and is not
readily based upon actual human behaviour.
(Strubbe & Woods 2004) report on human behaviour from
extensive work characterising the eating behaviour of rats.
They report that food intake occurs as discrete bouts or
meals, and then continue to suggest that circadian activity
pattern, changes in the light cycle, and the size of the
previous meal can have discernable effects. An authoritative
textbook on human physiology (Guyton & Hall 2006)
provides some fundamental biological evidence. They state
that the hypothalamus stimulates the desire for food which is
then quenched by stomach stretch. Further it states that
stomach emptying and thus reinvigoration of hunger is
governed by the small intestine to ensure adequate absorption
of the nutrients in food.
(Winkler et al. 1999; Winkler et al. 1995) actually observed
human behaviour for German men aged between 45-64 over
6293 days. They report a significant variance in the number
of meals eaten each day, with only 19% of patient days
containing 3 meals. 34.4% patient days contained 4 meals,
while 32% had 5 meals.
3. MEAL PRIOR PROBABILITY
The meal prior probabilities should reflect what we know
about how humans consume meals. Consequently, we work
from what we know: a) meals are eaten in discrete bouts, b)
Fig 1. Published Glucose Appearance vs. Time Profiles
3.1 Meals are discrete
We model meals as discrete events that occur entirely in a
single sample period, have a default size of effect ¹m in
mg/dL, and an uncertainty ¾m.
3.2 Patients eat when they are awake
To represent the fact that we do not eat while asleep we begin
by defining a base probability function ¡(k) for each time
step k that is zero at night, and non-zero during the daytime.
Meals are assumed to only start every Ts time steps. Since we
are ignoring meal patterns, ¡(k) is set to a constant
probability ¡d during the day. ¡d represents the probability
of a meal in a given sampling period when there are no meals
in the recent past. The mathematical formulation of ¡(k) is:
(
¡(k) =
¡d Ts if awake and k mod Ts = 0
0
otherwise
(1)
This formulation assumes that we know that when the patient
is awake. In the more general case we would need to
determine the patient’s status through announcement,
accelerometers, a prolonged absence of detected meals, or
assumed using a historical pattern.
3.3 Meals in the recent past inhibit meals now
The speed of meal digestion is dependent on the fat content
of the meal since fat is digested much more slowly than
protein or carbohydrates (Guyton & Hall 2006) and the
stomach empties at a rate that ensures efficient absorption of
the meal. So, carbohydrates in high fat meals flow into the
small intestine slower, and are absorbed slower. Fig. 1 shows
several examples with the lower fat meals finishing faster (R.
Basu et al. 2006; A Basu et al. 2004; Livesey et al. 1998;
Vella et al. 2007).
Continuous glucose monitors provide poor information about
the fat content of a meal. Consequently, we assume the same
digestion pattern m(k ¡ k0 ) for all meals. Specifically, we
assume that all meals follow the glucose appearance vs. time
profile published in (R. Basu et al. 2006). Here, k0 represents
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
the start time of the meal in minutes and m(k ¡ k0 ) is
normalized to sum to 1. The default effect of a meal is equal
to ¹m m(k ¡ k0 ).
day. Assuming 4 meals per day (Winkler et al. 1999)
¹m = 203mg=dL. Diabetics tend to eat fewer carbohydrates
to shrink meal disturbances and ease blood glucose control.
Here we handle the harder case of the average healthy diet.
, but only because they have
Diabetics may have a lower
adapted to facilitate manual blood glucose control. We seek
to not require this adaptation.
Table 1. Published and fitted meal frequencies
Fig. 2: Probability penalty function for a single past meal
The digestion of the previous meal inhibits the eating of
meals now, meaning that the probability of eating a meal now
is lowered by the digestion of all previous meals.
Consequently, the probability of eating a meal now can be the
base probability ¡(k ) multiplied by a penalty for each past
meal:
¶
Q μ
Y
¡(kp ¡k)2
2
¿
P (meal; k) ´ ¡(k)
1¡e
(2)
p=1
where, Q represents the number of past meals (practically
there are 1 to 3 meals in the last 6 hours), kp is the starting
time in minutes for meal p , and ¿ is the time constant that
parameretrizes the digestion pattern for meals. An example
penalty for a single meal is shown in Fig. 2.
Number of
Number of
Reported %
Fitted %
Meals per
Reported
Day
Days
1
7
0.1
1.3
2
71
1.1
6.3
3
1194
19
19
4
2162
34
34
5
2012
32
27
6
847
14
11
7
0
0
1.8
8
0
0
0.1
Mean
4.1
4.2
Ts is the sampling period for potential meal start times. We
arbitrarily set Ts = 5 min. For reference (Patek et al. 2009)
set a similar value to 15 minutes.
¡d , and ¿ representing the base meal probability and the meal
digestion time constant respectively are set to best match the
published frequency of meals per day as published in
(Winkler et al. 1999) and shown in Table 1. The resultant
%
values are ¡d = 3:1 min
and ¿ = 128 min
4. PARAMETER SELECTION
These prior probabilities are defined by several parameters:
¹m, ¾m, m(k ¡ k0 ), Ts, ¡d, and ¿ . Here we describe the
values we used and why.
¹m and ¾mrepresent the mean and standard deviation of the
effect of a meal in mg/dL. This will depend upon the person,
the situation, and the model. For instance, an athlete eating 4
standard meals per day will have a high ¹m, and a low ¾m.
Alternately, if the model lumps the suppression of
endogenous glucose production (removal of glucose) with the
meal effect (addition of glucose) then ¹m will be decreased.
For the control validation we use ¹m = 240 mg/dL, and
¾m = 90 mg/dL based on the known carbohydrate intake.
For the prediction validation in patients aged 3-18, we use
¹m = 210 mg/dL, and ¾m = 75 mg/dL. For reference we
derive ¹m for a generic adult. In general 50% of the caloric
intake should come from carbohydrates (Anderson et al.
2004). With a standard diet of 2,000 Calories per day, and 4
Calories per gram of carbohydrate that corresponds to 250
grams of carbohydrate (g CHO) per day. Using a conversion
between g CHO and mg/dL of 3.25, based on the insulin to
carb ratio and correction factor in (Walsh & Roberts 1994),
the average adult takes in 812 mg/dL from carbohydrates per
Fig. 3: Meal types consumed by time of day
4.1 Reality check
The tuning of the meal prior probabilities only forces them to
fit the expected number of meals per day. So, we predict
three other behaviours to check agreement with reality. a)
does the predicted breakfast behaviour match Winkler et al.
b) How long do we expect between meals and c) what is the
distribution of meals within a day?
(Winkler et al. 1995) show that the first meal of the day,
breakfast, is eaten on 98.5% of days, and is eaten according
to Fig. 3. Fig. 3 shows a wide spread of times for breakfast.
The meal priors expect that the first meal of the day
(breakfast) is eaten within 2 hours of awakening 98.5% of the
time. If we assume that the spread in breakfast times are due
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
5. VALIDATION
in large part to the variation in wake up times for the
participants, then this number can match.
Assuming that a meal has just occurred then the next meal is
penalized. As time passes the probability of the next meal
increases while the penalty drops, and then decreases due to
the chance that it has already occurred. Assuming that k0 = 0
and that the patient is awake, the equation is:
P (next meal; k) = ¡d Ts (1 ¡ e
¡k2
¿2
)
k¡1
Y
(1 ¡ P (next meal; kn ))
(3)
kn
With reference to Fig 4 That shows the time profile of the
probability of the next meal, the most likely spacing between
successive meals is 100 minutes.
We validate the meal prior probabilities in prediction of
blood glucose profiles containing unknown meals (B.
Buckingham et al. 2007) and in closed-loop control.
5.1 Validation in prediction
(Cameron & Niemeyer 2010) previously published a
prediction algorithm without these meal priors. To prevent
skewing from the unknown future meals they were forced to
alter the future insulin when showing results on clinical data.
Now we show results for the same prediction algorithm using
these meal priors to predict and estimate the effects of the
next meal.
To calculate the expected effects of the next meal we simply
convolved the probability of the next meal, generated using
the meal priors and the meal knowledge provided by the
prediction algorithm, with the default meal effect
¹m m(k ¡ k0 ). Note that we can also estimate the uncertainty
of the effect of the next meal.
Fig. 4: The probability of the next meal (Ts =5 min)
Fig. 6: Root mean squared error vs. prediction horizon
Fig. 5: Simulated an published meal density vs. time of day
(Winkler et al. 1999; Winkler et al. 1995) report the average
number of meals per person eaten in a given hour of the day
as shown in blue in Fig 5. To get the simulated version
(shown in green) we simulated using the meal prior
probabilities for 16 hour days with wake up times uniformly
distributed between 6 and 8 AM.
Though roughly similar, these two distributions are not
identical. This is due in large part to the fact that these meal
priors do not include the societal pattern of eating lunch at
noon or 1 PM, and eating dinner between 6 and 8 PM.
However, since we are seeking to use human physiology
instead of patterns, this is acceptable.
Fig. 7: Mean error vs. prediction horizon
Figs. 6 and 7 shows how the error behaves versus the length
of the prediction on clinical data. None of the algorithms are
provided with any information about meals. The first
prediction in black assumes a Kalman filter augmented statespace model that assumes that there are no meals. The error
grows steadily due to known insulin that is not offset by
estimates of the unknown meals in the recent past and in the
future.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
The second algorithm (red) also uses a single state-space
model with a Kalman filter. It differs from the first algorithm
by including a two-state meal model. The Kalman filter
updates the state estimates to fit meals that were occurring.
This algorithm can thus adapt to existing meals, but assumes
that there are no future meals. The effect of adapting to the
existing meals is to lessen the initial growth in the error. It
does not affect the error growth at the longer prediction
horizons.
when anticipating a non-existent meal. However, the
postprandial peaks are uniformly 34 mg/dL higher.
The third algorithm (blue) follows (Cameron & Niemeyer
2010) except using the new meal priors. This algorithm both
adapts to the existing meals, lessening the error growth at
short prediction horizons, and anticipates the effect of future
meals, lessening error growth at longer prediction horizons.
5.2 Validation in control
Fig. 8: Sample case with missed meals with meal priors
Using a control algorithm as described in (Cameron 2010), a
prediction algorithm as described in (Cameron & Niemeyer
2010), and the meal priors as described here we simulated
using the 10 adult patients in the UVa/Padova Metabolic
simulator for Type 1 Diabetes (B.P. Kovatchev et al. 2009).
Each of the 10 patients was simulated for 34.5 hours
containing 6 meals of 50, 70, 90, 25, 50, 70 g CHO at 9:00,
13:00, 5:30, 8:00, 9:00, and 13:00 hours respectively. The
controller starts after 1.5 hours and is told that the patient is
asleep between 22:00 and 6:00.
We show results for this controller with and without the meal
priors, and also for an optimized basal/bolus controller. Since
this controller explicitly minimizes the expected, combined
clinical risk of hypoglycaemia and hyperglycemia there are
no performance related tuning parameters. The basal/bolus
controller has a single basal rate and optimized boluses at the
start of each meal. The boluses were optimized
retrospectively and thus represent a rough lower bound on the
achievable BGRI. The BGRI is defined in (Boris P.
Kovatchev et al. 1997; Clarke & B. Kovatchev 2009). The
BGRI and the time spent in the euglycemic range are shown
in Table 2.
Table 2. Controller validation results
Controller
BGRI
% BG 70-180 mg/dL
Without meal priors
6.92
64
With meal priors
2.83
90
Basal/Bolus
1.64
97
Since this controller anticipates meals, it is possible that it
will provide insulin for an upcoming meal that is dangerous
when the meal is missed. A sample case where dinner and the
evening snack were missed is shown in Fig 8. The top plot
shows the CGM sensor signal (blue) and the true blood
glucose level (green). The middle plot shows the insulin
injected by the controller. The bottom plot shows the ingested
carbohydrates. Recall that the controller anticipates meals
between 3 PM and 10 PM, where there are none in this case.
The lowest blood glucose level in this example is 73 mg/dL,
which is in the euglycemic range.
Fig. 9: Sample case with missed meals without meal priors
6. CONCLUSIONS
This paper develops a mathematical expression for meal prior
probabilities that does not rely on historical eating patterns.
As a result it can adapt to days where the eating schedule is
abnormal.
Predictions using these meal priors achieve significantly
improved long-term prediction accuracy. Specifically, it
removes the prediction bias associated with the effect of
future meals. This enables tuning to patient data where the
future effects of meals are unknown.
Simulations of closed-loop control using these meal priors
significantly reduced the blood glucose risk index. The
benefits come with marginally lower blood glucose values
when patients abstain from meals for long time periods.
We hope such anticipatory behavior will make practical
controllers both more effective and robust, improving the
lives of diabetes patients.
ACKNOWLEDGEMENTS
This work is supported by funding from the Juvenile
Diabetes Research Foundation’s Artificial Pancreas Project.
By comparison the performance for the same patient without
use of the meal priors to anticipate the next meal is shown in
Fig. 9. Here the lowest value is 98, 25 mg/dL higher than
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
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