=Paper=
{{Paper
|id=Vol-2675/paper20
|storemode=property
|title=Prediction of Blood Glucose Levels for People with Type 1 Diabetes using Latent-Variable-based Model
|pdfUrl=https://ceur-ws.org/Vol-2675/paper20.pdf
|volume=Vol-2675
|authors=Xiaoyu Sun,Mudassir Rashid,Mert Sevil,Nicole Hobbs,Rachel Brandt,Mohammad Reza Askari,Andrew Shahidehpour,Ali Cinar
|dblpUrl=https://dblp.org/rec/conf/ecai/SunRSHBASC20
}}
==Prediction of Blood Glucose Levels for People with Type 1 Diabetes using Latent-Variable-based Model==
https://ceur-ws.org/Vol-2675/paper20.pdf
Prediction of Blood Glucose Levels for People with Type 1
Diabetes using Latent-Variable-based Model
Xiaoyu Sun and Mudassir Rashid and Mert Sevil and Nicole Hobbs and Rachel Brandt and
Mohammad Reza Askari and Andrew Shahidehpour and Ali Cinar1
Abstract. Regulation of blood glucose concentrations (BGCs) is a insulin, and constraint conditions are taken into account when com-
tough burden for people living with type 1 diabetes mellitus (T1DM). puting the future insulin doses to be infused. The data-driven model-
People with T1DM must administer exogenous insulin to maintain ing techniques have been evaluated in many studies because of their
their BGC within an euglycemic range. Hyperglycemia (high BGC) computational tractability and identification efficiency.
and hypoglycemia (low BGC) can occur because of poor BGC man- The empirical modeling technologies for T1DM include linear
agement. Using recursively identified models to predict the future and nonlinear methods. For nonlinear models, artificial neural net-
BGC values opens novel possibilities for improving the BGC regu- works (ANN) [1], convolutional neural networks (CNN) [26], recur-
lation performance by adjusting the dose of insulin infusion, taking rent neural networks (RNN) [13, 3], and other machine learning and
rescue carbohydrates, or both. The BGC prediction model can also deep learning techniques [17, 11] have been used to model the glu-
benefit the development of artificial pancreas systems. In this paper, cose dynamics. However, the nonlinear models can only be trained
a latent variable (LV) based multivariable statistical modeling ap- with a large data set, and it can be difficult to develop personalized
proach is applied to model BGC dynamics and forecast future BGCs. models or to update the model parameters or structure. For the lin-
The LV-based model is a powerful linear method to build an empir- ear modeling methods, autoregressive model with exogenous inputs
ical model by using the collected data. The model is evaluated with (ARX), and autoregressive moving average model with exogenous
the Ohio T1DM dataset that contains BGCs from continuous glu- inputs (ARMAX) [7, 19, 25] are used to build personalized glucose
cose monitoring (CGM) sensors, basal and bolus insulin information prediction models with recursively updated model parameters where
from insulin pump, and additional information from wristbands or the future BGC values are modeled as a linear combination of his-
reported by the subject. The results indicate that the LV-based model torical measurements from sensors, administered insulin, and other
can predict future BGC values with high accuracy for prediction hori- information such as meal CHOs and exercise. Statistical methods
zons of 30 and 60 minutes. based on latent variables (LVs) are demonstrated to have powerful
abilities for various data analysis tasks, modeling, and process mon-
itoring [20, 21], and has been proven as a good alternative linear
1 Introduction model for type 1 diabetes [24].
In this paper, a novel multivariate statistical method proposed by
People with type 1 diabetes mellitus (T1DM), an immune disorder Nelson and MacGregor [14, 8] where the score vector is estimated
where the pancreas does not produce insulin, must administer ex- using missing data imputation technique is applied to model the glu-
ogenous insulin either by injections several times every day or in- cose dynamics based on LVs derived from principal component anal-
fusion with an insulin pump to maintain their blood glucose con- ysis (PCA). There are three key steps in developing a BGC predic-
centrations (BGCs) within a safe range (70-180 mg/dL) [23]. With- tion model based on the LVs technique. First, a PCA is performed
out effective regulation, people with T1DM may suffer from several on the gathered data to decompose the data into a linear combina-
long-term complications caused by hyperglycemia (high BGC), such tion of scores and loadings. Then, the unobserved variables are esti-
as kidney failure, blindness, and the deterioration of cardiovascular mated as conditional mean values computed from the gathered data
health. They can also have low BGC (hypoglycemia), which may and new measurements. Finally, the score of new observed data is
cause dizziness, diabetes coma, or even death because of the lack of estimated using incomplete observations and the future BGC values
energy for the brain [4]. are predicted. The rest of this paper is organized as follows: Section 2
Artificial pancreas (AP) systems are proposed to provide more re- summarizes the pre-processing of the data set. The LV-based method
liable BGC management by automatically calculating the dose of in- is described and a glucose prediction model is developed in Section
sulin to infuse by using a closed-loop controller incorporating BGCs 3. The Ohio T1DM data set [12] is used to assess the performance
measured by continuous glucose monitoring (CGM) sensors, histori- of the model and the results are given and discussed in Section 4.
cal infused insulin data, and other available information, such as meal Finally, some concluding remarks are provided in Section 5.
carbohydrates (CHOs) and exercise information [15, 9, 16, 18, 2, 10].
The closed-loop controller is the critical component in developing an
AP system. Model-based controllers have become the preferred op- 2 Data Pre-processing
tion in recent in silico and clinical studies where the future BGCs 2.1 Data
predicted by a model, historical CGM measurements, administered
The Ohio T1DM dataset provided by the Blood Glucose Level Pre-
1 Illinois Institute of Technology, USA, email: cinar@iit.edu
diction (BGLP) challenge records eight weeks of data collected from
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
six T1DM patients with subject ID: 540, 544, 552, 567, 584, and 596.
The data set contains BGC values measured by CGM sensors with 300
Raw CGM
a sampling time of 5 minutes, basal and bolus insulin information Filled CGM
from the insulin pump, information collected from wristbands, and
250
events (i.e., meal, exercise, work, illness, etc.) recorded by the sub-
jects themselves (see [12] for details). However, there is no evidence
on the accuracy of the subject-reported information, which may de-
BGL (mg/dL)
200
grade the reliability of glucose prediction model and the wristband
did not work continuously for a long period of time due to its lim-
ited battery power, which causes lots of missed data. Thus, only data 150
collected from CGM sensors and insulin pumps is used in this study.
The insulin infused with an insulin pump includes basal insulin,
100
given continuously since the defined start times, and bolus insulin,
which is usually given to mitigate the effects of the rapid raise of
BGC around meal times. Basal insulin is recorded as “basal” which 50
is the basal insulin infusion rate and “temp basal” which defines the 8000 8050 8100 8150 8200 8250 8300
samples (/5min)
basal insulin infusion rate in specific time intervals to achieve better
regulation of BGCs. There are three types of bolus insulin defined in Figure 1. Illustration of the filled gap in training data
the dataset: normal, normal dual and square dual. For “normal” bolus (Subject ID: 540, 18 CGM measurements are missed)
insulin, a certain dose of insulin is infused immediately to the patient,
while for “normal dual” bolus insulin, a certain percentage of bolus
is given to the subject up front and the remaining insulin amount is where each row can be considered as an observation and each column
given gradually over a longer time duration. In this study, half of is the values of a single variable.
the dose is supposed to be given to the patient immediately and the
remaining half is infused over the following 30 minutes. For “square 3 Methods
dual” bolus insulin, the bolus insulin is administered continuously
in the given time interval. The bolus and basal insulin values are A multivariable statistical technique based on LVs [14, 8] is devel-
converted to U/min and aligned to match the CGM sampling time. oped to predict the future BGC values, where a LV model is devel-
The insulin on board (IOB), which represents the insulin that oped using the PCA algorithm. Then, the conditional mean values are
remains active within the body, is calculated from a physiological estimated from the same distribution, and future BGCs are predicted
model [22] and is found useful in some previous works [1, 6]. The with LVs and incomplete observations.
physiological model is represented as
dC1 (t) 3.1 Latent Variable Model
= u (t) − Kdia C1 (t) (1)
dt Consider a data matrix X (N × M ) that contains N observations
dC2 (t) (rows) and M variables (columns), then a linear combination of an
= Kdia (C1 (t) − C2 (t)) (2)
dt observation x in dataset X can be written as t = p1 x1 +. . .+pM xM ,
IOB (t) = C1 (t) + C2 (t) (3) where t is a new vector in the same space as x. The fundamental
idea behind PCA is to find the loading vector p that maximizes the
where C1 and C2 define two insulin compartments, u (t) is the in-
variation of t, thus the first LV of PCA model can be calculated by
sulin infusion, and Kdia is a constant related to the duration of in-
solving the following problem:
sulin action, set as 0.0182.
The dataset contains missing CGM measurements for in both the arg max tT t = arg max pT X T Xp
(4)
training and testing sets that cause discontinuous CGM curves. The kpk=1 kpk=1
gaps that are not greater than 3 samples in the training dataset are
interpolated using first-order linear model. For the cases where more where p is the vector of regression coefficients. Accordingly, the
than 3 samples are missed, a single variable statistical model similar dataset X can be expressed as X = tpT + E with E denoting the
to the one used to modeling glucose dynamics is used. In the testing residual matrix. We can get more components by solving the follow-
data set, with the intent of on-line implementation, only the historical ing problem:
2
CGM measurements are used to fill the missing gaps until new the arg max X − tpT (5)
kpk=1
CGM observations become available. Fig. 1 shows an example of the
filled gap in the training dataset (Subject ID 540). Traditionally, the successive progress is evaluated by
kXk2 − kEk2
2.2 Batch Generation 100% (6)
kXk2
The pre-processed CGM readings (y) and calculated IOB (IOB)
are arranged as time series and a sliding window of 2 hours length is which is referred to as the percentage of explained variation of t and
chosen to generate batch segments from the training data as it is fixed as 95% in this study. If the first A (A is much smaller than
the rank of X) significant components can summarize sufficiently
y (1) ··· y (24) IOB (1) ··· IOB (24) well the dataset X, then a LV model developed using PCA algorithm
y (2)
Xtrain = ··· y (25) IOB (2) ··· IOB (25) can be expressed as
.. .. .. .. .. ..
. . . . . . X = t1 pT1 + t2 pT2 + ... + tA pTA + E = T1:A P1:A
T
+E (7)
where T1:A = [t1 , . . . , tA ] (N × A) and P1:A = 3.3 Glucose prediction based on LV model
[p1 , . . . , pA ] (M × A) are now matrices containing A score
An accurate glucose prediction model could benefit diabetes man-
vectors and A loading vectors, respectively.
agement and significantly reduce the risk of hypoglycemia. To pre-
dict the future 60 minutes glucose values, the preceding one hour of
3.2 Conditional Mean Replacement Method data is assumed available and marked as z # . Then, the glucose pre-
diction model can be developed and the subsequent future hour of
For a new test observation z that was not used in the model develop- BGC values can be predicted online as follows:
ment but is drawn from the same distribution as observations in X,
the scores τ of the first A significant components of the new obser- 1. The batch data set Xtrain is generated for each subject using the
vation can then be calculated as training dataset.
2. While a new observation z # is available:
T
τ1:A = P1:A z (8) (a) Calculate the similarities between the new object z # and ob-
servations in submatrix X # , select the most similar N obser-
Consider a situation where only part of the object z is observed, it vations from Xtrain to form a new data matrix X.
is nature to assume the first R variables are measured and the remain-
(b) Develop PCA model using data matrix X as stated in (7).
ing (M − R) variables are unobserved, then without loss of gener-
ality, we have (c) Estimate the score vector τ of the new observation z # as stated
in (11).
#
z
z=
z∗ (d) Predict the next 1 hour’s glucose values as stated in (12).
where z # is the observed variables and z ∗ represents the missing The above process was implemented in MATLAB 2019b and
measurements. This induces the following partition in X: the future one hour’s BGC values (12 samples) can be forecasted
by feeding the testing data to the model. The codes are available:
X= X# X∗ https://github.com/xiaoyu1115/BGLP_2020.git.
where X # contains the first R columns of X and X ∗ is made up
4 Results
of the remaining (M − R) columns. Correspondingly, the loading
matrix P can be partitioned as In the clinical study, the training data from Ohio T1DM dataset with
subject ID: 540, 544, 552, 567, 584, and 596 were first divided into
P# two parts: the first 90% of data were used to develop PCA model and
P =
P∗ the model is tested with the remaining 10% of data to determine the
number of observations that should be used in building the LV-based
where P # is the submatrix comprising the first R rows of P and the glucose prediction models. Using this approach, the computational
remaining (M − R) rows are defined as matrix P ∗ . Then, the score burden in the modeling progress can be reduced significantly. Feed-
vector τ1:A of the new observation z can be rewrite as ing both the processed training and testing data into the modeling
process described in Section 3, the future glucose concentrations can
#T # ∗T ∗
τ̂1:A = P1:A z + P1:A z (9) be predicted with prediction horizons of up to 60 minutes. The root
mean square errors (RMSEs) and mean absolute errors (MAEs) of
For a given data matrix X and a new observation z with z # de- the 6 subjects are summarized in Table 1. The prediction results are
noting the measured variables that follow the same distribution as also analysed using Clark Error Grid (CEG) [5] and the percentages
observations in X, the missing variables z ∗ of z can be estimated as of data distributed in Zone A to Zone D are summarized in Table 1
the expected values from the conditional normal distribution: as well. v
u N
∗
∗ #
u1 X
ẑ = E z z ,S (10) RM SE = t (y (i) − ŷ (i))2 (13)
N
i=1
Substituting the expression into the estimator of score vector of
N
the new observation yields: 1 X
M AE = |y (i) − ŷ (i)| (14)
− # N
#T
τ̂1:A = Θ1:A P1:A S ## z (11) i=1
where y is the BGC values measured by CGM sensors, ŷ is the pre-
dicted BGCs, and N is the total data points in the testing dataset for
T S ## S #∗
where S = X X
= is the covariance matrix of each subject.
N −1 S ∗# S ∗∗
X and Θ is the square diagonal matrix consisted of the eigenvalues In Table 1, the RMSE varies from 16.66 to 22.76 mg/dL for the
λ = [λ1 , . . . , λH ] in descending order and H is the rank of X. prediction horizon of 30 minutes with an average value of 19.37
Finally, the unmeasured variables z ∗ in z can then be calculated mg/dL. The RMSE varies from 26.81 to 38.99 mg/dL for the pre-
from the estimated score vector along with the loading matrix: diction horizon of 60 minutes with an average value of 32.59 mg/dL.
It is reasonable to see this increase in RMSE because the unknown
ẑ ∗ = P1:A
∗
τ̂1:A (12) disturbances including meals and exercise that will influence the glu-
cose dynamics significantly. The relationships between the predic-
which yields the future predictions given the past observed glucose tion horizon and RMSE or MAE in Figure 2 also indicate that the
values. prediction accuracy decreases with increasing prediction horizon. An
Table 1. RMSE (mg/dL), MAE (mg/dL) and CEG (%) results of the LV-based model (STD: standard deviation)
PH=30 minutes PH=60 minutes
Subject RMSE MAE Zone A Zone B Zone C Zone D RMSE MAE Zone A Zone B Zone C Zone D
540 20.76 15.23 84.92 12.55 0.03 2.50 38.99 29.75 57.84 35.58 0.24 6.35
544 16.70 11.65 93.23 6.29 0 0.48 26.81 19.77 78.77 19.67 0.07 1.48
552 16.66 12.36 88.05 10.88 0 1.06 29.41 23.08 64.97 31.63 0.09 3.32
567 22.76 15.30 86.03 12.28 0.04 1.60 37.95 28.13 57.93 34.54 0.34 7.15
584 22.22 15.86 85.11 13.87 0 1.02 34.81 26.57 67.66 30.08 0.15 2.11
596 17.12 12.15 90.11 8.31 0 1.57 27.57 20.53 72.06 25.23 0.04 2.67
Mean 19.37 13.76 87.91 10.70 0.01 1.30 32.59 24.64 66.54 29.46 0.15 3.85
STD 2.87 1.89 3.27 2.87 0.02 0.68 5.35 4.12 8.17 6.03 0.12 2.34
average of 87.91% data lie in zone A of CEG which indicates the
high clinical accuracy of the model with prediction horizon of 30
minutes. The percentage of data in zone B increase dramatically as
40 540 the prediction horizon increases to 60 minutes.
544
RMSE(mg/dL)
30 552 The forecasting performance of the model is better for subjects
567
584
544, 552, and 596 than subjects 540, 552, and 567. One of the rea-
20
596 son is there are less noise and sensor failures contained in the dataset
10
provided by subjects 544, 552, and 596. A filter might improve pre-
0 diction performance for this case. And many gaps are observed in
0 2 4 6 8 10 12
Predict Horizon (5min) CGM measurements, which also deteriorate the prediction accuracy
of the model in which the interpolated CGM data were used to pre-
30 540
544
dict the further BGC values.
MAE(mg/dL)
552 The predicted BGC values for subject 544 with prediction horizon
20 567
584 of 30 minutes and 60 minutes are shown in Figure 3. For the pre-
596
10 diction horizon of 30 minutes, the LV-based model can predict the
future BGC values with a comparable high accuracy, and most of the
0
0 2 4 6 8 10 12 hyperglycemia and hypoglycemia events can be forecasted on time.
Predict Horizon (5min) When the prediction horizon is increased to 60 minutes, the predic-
tion accuracy decreases, but is still acceptable. The prediction values
Figure 2. Relationship between prediction horizon and prediction accuracy can still provide an insight on the variation of BGC which can be
used to tune the insulin infusion rate as a reference.
5 Conclusion
In this paper, an online recursively identified LV-based modeling ap-
300
30-min-ahead prediction results proach is developed to predict the future BGC values with predic-
250
Measurements tion horizons of 30 and 60 minutes. With the pre-processed dataset,
Predictions
BGC (mg/dL)
the LV-based model is developed to calculate the LVs using a small
200
submatrix of the training dataset when a new observation is avail-
150
able. The future blood glucose values are predicted as a linear com-
100
bination of estimated scores and loadings. The proposed model is
50
1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
evaluated with the Ohio T1DM dataset and the results demonstrate
the effectiveness of the model. Although the measurement noise is
60-min-ahead prediction results
300 weight-averaged in the LV-based model, it still has a significant in-
Measurements
250 Predictions
fluence on modeling and prediction progress. Online denoising tech-
BGC (mg/dL)
200
niques would be one of the future study directions that might improve
150
the prediction accuracy. Further, integrating other data fields with the
personalized physiological models is a potential approach to improve
100
the prediction performance in the future work.
50
1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
samples (/5min)
ACKNOWLEDGEMENTS
Figure 3. BGC prediction for Subject 544 using LV-based model The work of Xiaoyu Sun was supported by the China Scholarship
Council under grant 201906080136. Funds provided by the Hyosung
S. R. Cho Endowed Chair at Illinois Institute of Technology to Ali
Cinar is gratefully acknowledged.
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